Notation by work
Stratified Recursive Improvement Closure
All notation · Glossary · Source coverage
2681 symbol spellings in 6662 mathematical passages across 4 selected source files. Subscripts and superscripts have their own entries alongside their base symbols.
The same symbol can have several meanings here. A passage classified as a definition may define another part of its expression. Read the linked context before assigning a meaning.
All 2681 entries are available below.
+Notation table · 415 passages
Search aliases: +
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20400
σ_+(D) · Smallest positive singular value of D
Exact mathematical passage
\sigma_+(D)
=Notation table · 1237 passages
Search aliases: =
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20596
G=(Z,P,C,B) · Research-compiler diagnostic, experiment, method-constructor, and binding language
Exact mathematical passage
\mathfrak G=(\Zcal,\Pcal,\Ccal,\Bcal)
ANotation table · 332 passages
Search aliases: A
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20371
J_Ξ(A) · Resource-normalized utility of agent A in context Ξ
Exact mathematical passage
J_\Xi(A)
\mathfrak ANotation table · 48 passages
Search aliases: A mathfrak
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20372
A_t · Stratified functional atlas at generation t
Exact mathematical passage
\mathfrak A_t
\mathfrak aNotation table · 2 passages
Search aliases: a mathfrak
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20381
a · Repair abstraction operator from episodes to schemas
Exact mathematical passage
\mathfrak a
\mathsf ANotation table · 17 passages
Search aliases: A mathsf
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20424
A · Positive semidefinite evaluator holonomy-energy operator
Exact mathematical passage
\mathsf A
aNotation table · 334 passages
Search aliases: a
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20429
q,a,Z,Π · Trace map, sealed acceptance rule, revealed trace, and receipt
Exact mathematical passage
q,a,Z,\Pi
A_tNotation table · 16 passages
Search aliases: A_t
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20383
A_t · Active stratified agent
Exact mathematical passage
A_t
\mathfrak A_tNotation table · 4 passages
Search aliases: A_t mathfrak
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20372
A_t · Stratified functional atlas at generation t
Exact mathematical passage
\mathfrak A_t
\AcalNotation table · 41 passages
Search aliases: Acal A mathcal
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20374
A_t · Executable ancestor-closed archive inside the lineage graph
Exact mathematical passage
\Acal_t
\Acal^\starNotation table · 9 passages
Search aliases: Acal^⋆ A^⋆ Acal star mathcal star
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20598
A^⋆(τ) · Acceptable validity-filtered methods for research problem τ
Exact mathematical passage
\Acal^\star(\tau)
\Acal_tNotation table · 13 passages
Search aliases: Acal_t A_t Acal mathcal
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20374
A_t · Executable ancestor-closed archive inside the lineage graph
Exact mathematical passage
\Acal_t
\mathrm{aff}Notation table · 5 passages
Search aliases: aff mathrm
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20542
C_aff · Inherited affine repair-constructor grammar
Exact mathematical passage
\Ccal_{\mathrm{aff}}
\AuthNotation table · 14 passages
Search aliases: Auth auth mathrm
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20392
U_t^auth · Authorized correction space
Exact mathematical passage
U_t^{\Auth}
BNotation table · 342 passages
Search aliases: B
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20407
B,L_G,λ_2(L_G) · Nerve incidence matrix, graph Laplacian, and spectral gap
Exact mathematical passage
B,L_G,\lambda_2(L_G)
\mathcal BNotation table · 5 passages
Search aliases: B mathcal
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20582
N_ B · Local behavioral subspace invisible to evaluator portfolio B
Exact mathematical passage
\mathcal N_{\mathcal B}
bNotation table · 185 passages
Search aliases: b
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20411
b_n · Order-n residual coefficient in the formal obstruction tower
Exact mathematical passage
b_n
b_nNotation table · 6 passages
Search aliases: b_n
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20411
b_n · Order-n residual coefficient in the formal obstruction tower
Exact mathematical passage
b_n
\BcalNotation table · 18 passages
Search aliases: Bcal B mathcal
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20562
N_B · Local behavior subspace invisible to evaluator portfolio B
Exact mathematical passage
\Ncal_{\Bcal}
CNotation table · 179 passages
Search aliases: C
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20421
C_tr(p;u) · Certified transport-route cost for correction u
Exact mathematical passage
C_{\mathrm{tr}}(p;u)
\mathfrak CNotation table · 14 passages
Search aliases: C mathfrak
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20560
C_B(Γ) · Validity-filtered compiler metaproductivity under research budget B
Exact mathematical passage
\mathfrak C_B(\Gamma)
\mathsf CNotation table · 7 passages
Search aliases: C mathsf
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20389
C · Set of stratum colors or types
Exact mathematical passage
\mathsf C
cNotation table · 241 passages
Search aliases: c
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20599
H_rm conflict · Hypergraph of repair-conflicting research-problem cells
Exact mathematical passage
\Hcal_{\rm conflict}
\mathfrak C_BNotation table · 4 passages
Search aliases: C_B mathfrak
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20560
C_B(Γ) · Validity-filtered compiler metaproductivity under research budget B
Exact mathematical passage
\mathfrak C_B(\Gamma)
\mathfrak C_{\rm rob}Notation table · 2 passages
Search aliases: C_rm rob mathfrak rm
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20578
C_rm rob(Γ) · Worst-environment validity-filtered compiler value with a lower-tail risk term
Exact mathematical passage
\mathfrak C_{\rm rob}(\Gamma)
C_{\mathrm{tr}}Notation table · 2 passages
Search aliases: C_tr mathrm
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20421
C_tr(p;u) · Certified transport-route cost for correction u
Exact mathematical passage
C_{\mathrm{tr}}(p;u)
\CcalNotation table · 28 passages
Search aliases: Ccal C mathcal
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20542
C_aff · Inherited affine repair-constructor grammar
Exact mathematical passage
\Ccal_{\mathrm{aff}}
\Ccal_2Notation table · 1 passage
Search aliases: Ccal_2 C_2 Ccal mathcal
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20543
C_2(P) · Degree-two repair grammar localized to packet P
Exact mathematical passage
\Ccal_2(P)
\Ccal_{\mathrm{aff}}Notation table · 2 passages
Search aliases: Ccal_aff C_aff Ccal mathrm mathcal mathrm
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20542
C_aff · Inherited affine repair-constructor grammar
Exact mathematical passage
\Ccal_{\mathrm{aff}}
\Ccal_{\rm pair}Notation table · 5 passages
Search aliases: Ccal_rm pair C_rm pair Ccal rm mathcal rm
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20601
C_rm pair · Generic ordered pair-method constructor grammar
Exact mathematical passage
\Ccal_{\rm pair}
\cokerNotation table · 30 passages
Search aliases: coker operatorname
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20395
[r] · Obstruction class in coker D
Exact mathematical passage
\coker D
\mathrm{con}Notation table · 3 passages
Search aliases: con mathrm
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20544
Π_con · Proof-carrying constructor-expansion certificate
Exact mathematical passage
\Pi_{\mathrm{con}}
\mathrm{cor}Notation table · 10 passages
Search aliases: cor mathrm
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20397
P_cor,P_ob · Orthogonal correctable and obstruction projectors
Exact mathematical passage
P_{\mathrm{cor}},P_{\mathrm{ob}}
\mathrm{cov}Notation table · 2 passages
Search aliases: cov mathrm
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20561
S_cov · Coverage-weighted geometric benchmark score
Exact mathematical passage
S_{\mathrm{cov}}
\mathrm{cyc}Notation table · 5 passages
Search aliases: cyc mathrm
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20408
P_cyc · Orthogonal projection onto the overlap cycle space
Exact mathematical passage
P_{\mathrm{cyc}}
DNotation table · 357 passages
Search aliases: D
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20394
D_t · Linearized defect-response map
Exact mathematical passage
D_t
dNotation table · 144 passages
Search aliases: d
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20405
d_αβ · Difference of local repairs on an overlap
Exact mathematical passage
d_{\alpha\beta}
D^\daggerNotation table · 14 passages
Search aliases: D^† dagger
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20396
D^† · Moore--Penrose pseudoinverse of the correction map
Exact mathematical passage
D^\dagger
D_0Notation table · 12 passages
Search aliases: D_0
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20403
D_0,E,Q_0 · Packet-diagonal response, cross-packet coupling, and local right inverse
Exact mathematical passage
D_0,E,Q_0
D_tNotation table · 39 passages
Search aliases: D_t
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20394
D_t · Linearized defect-response map
Exact mathematical passage
D_t
d_{\alpha\beta}Notation table · 13 passages
Search aliases: d_αβ alpha beta
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20405
d_αβ · Difference of local repairs on an overlap
Exact mathematical passage
d_{\alpha\beta}
\DcalNotation table · 32 passages
Search aliases: Dcal D mathcal
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20390
D_t · Authorized modification family
Exact mathematical passage
\Dcal_t
\Dcal_tNotation table · 7 passages
Search aliases: Dcal_t D_t Dcal mathcal
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20390
D_t · Authorized modification family
Exact mathematical passage
\Dcal_t
\dimNotation table · 10 passages
Search aliases: dim
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20402
o_t · Unresolved obstruction dimension dimcoker D_t
Exact mathematical passage
\dim\coker D_t
\distNotation table · 8 passages
Search aliases: dist operatorname
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20398
η_D(r) · Quantitative obstruction dist(r,im D)
Exact mathematical passage
\dist(r,\im D)
ENotation table · 148 passages
Search aliases: E
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20403
D_0,E,Q_0 · Packet-diagonal response, cross-packet coupling, and local right inverse
Exact mathematical passage
D_0,E,Q_0
\mathbb ENotation table · 59 passages
Search aliases: E mathbb
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20603
κ_Q · Task-relevant structural language margin E_Q[O_1-O_0]
Exact mathematical passage
\mathbb E_Q[O_1-O_0]
\mathsf ENotation table · 5 passages
Search aliases: E mathsf
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20422
E_v · Evaluator object at version v
Exact mathematical passage
\mathsf E_v
eNotation table · 275 passages
Search aliases: e
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20410
H_e · Fundamental-cycle holonomy associated with a non-tree edge
Exact mathematical passage
H_e
E^{\flat}Notation table · 6 passages
Search aliases: E^flat flat
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20427
E^flat · Maximal path-flat evidence core
Exact mathematical passage
E^{\flat}
\mathbb E_QNotation table · 1 passage
Search aliases: E_Q mathbb
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20603
κ_Q · Task-relevant structural language margin E_Q[O_1-O_0]
Exact mathematical passage
\mathbb E_Q[O_1-O_0]
E_vNotation table · 9 passages
Search aliases: E_v
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20423
R_v,E_v,ι_v,J_v · Raw evidence, internal evidence space, encoder, and readout of evaluator v
Exact mathematical passage
R_v,E_v,\iota_v,J_v
\mathsf E_vNotation table · 2 passages
Search aliases: E_v mathsf
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20422
E_v · Evaluator object at version v
Exact mathematical passage
\mathsf E_v
E_\tau^{\flat}Notation table · 1 passage
Search aliases: E_τ^flat tau flat
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20425
E_τ^flat · Spectral approximate path-flat core at threshold τ
Exact mathematical passage
E_\tau^{\flat}
\EcalNotation table · 16 passages
Search aliases: Ecal E mathcal
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20385
E_t · Append-only external-evidence ledger
Exact mathematical passage
\Ecal_t
\Ecal_tNotation table · 10 passages
Search aliases: Ecal_t E_t Ecal mathcal
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20385
E_t · Append-only external-evidence ledger
Exact mathematical passage
\Ecal_t
\epsNotation table · 162 passages
Search aliases: eps ε varepsilon
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20413
ε_D · Correction-diagram commutation defect
Exact mathematical passage
\eps_D
\eps_DNotation table · 11 passages
Search aliases: eps_D ε_D eps varepsilon
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20413
ε_D · Correction-diagram commutation defect
Exact mathematical passage
\eps_D
\eps_{\mathrm{hol}}Notation table · 7 passages
Search aliases: eps_hol ε_hol eps mathrm varepsilon mathrm
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20417
ε_hol · Semantic holonomy error
Exact mathematical passage
\eps_{\mathrm{hol}}
\eps_JNotation table · 7 passages
Search aliases: eps_J ε_J eps varepsilon
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20415
ε_J · Evaluator alignment error
Exact mathematical passage
\eps_J
\eps_{\mathrm{leak}}Notation table · 7 passages
Search aliases: eps_leak ε_leak eps mathrm varepsilon mathrm
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20416
ε_leak · Packet leakage error
Exact mathematical passage
\eps_{\mathrm{leak}}
\eps_rNotation table · 12 passages
Search aliases: eps_r ε_r eps varepsilon
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20414
ε_r · Defect-transport mismatch
Exact mathematical passage
\eps_r
FNotation table · 98 passages
Search aliases: F
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20426
μ(F;Δ_prot) · Adequacy modulus of core F on protected evidence differences
Exact mathematical passage
\mu(F;\Delta_{\mathrm{prot}})
fNotation table · 71 passages
Search aliases: f
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20585
widehatΔ_Γ^rm primary,widehatΔ_Γ^rm foreign · Paired robust-child minus inherited-parent effects
Exact mathematical passage
\widehat\Delta_{\Gamma}^{\rm primary},\widehat\Delta_{\Gamma}^{\rm foreign}
F_\thetaNotation table · 2 passages
Search aliases: F_θ theta
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20563
F_θ(z) · Outcome, risk, and scored-performance forecaster from proposal and pilot evidence
Exact mathematical passage
F_\theta(z)
\flatNotation table · 28 passages
Search aliases: flat
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20425
E_τ^flat · Spectral approximate path-flat core at threshold τ
Exact mathematical passage
E_\tau^{\flat}
GNotation table · 142 passages
Search aliases: G
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20373
G_t · Directed multigraph of typed cross-stratum bridges
Exact mathematical passage
G_t
\mathfrak GNotation table · 6 passages
Search aliases: G mathfrak
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20596
G=(Z,P,C,B) · Research-compiler diagnostic, experiment, method-constructor, and binding language
Exact mathematical passage
\mathfrak G=(\Zcal,\Pcal,\Ccal,\Bcal)
gNotation table · 82 passages
Search aliases: g
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20409
g_αβ · Nonabelian transition function between local nonlinear repairs
Exact mathematical passage
g_{\alpha\beta}
G_tNotation table · 3 passages
Search aliases: G_t
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20373
G_t · Directed multigraph of typed cross-stratum bridges
Exact mathematical passage
G_t
g_{\alpha\beta}Notation table · 6 passages
Search aliases: g_αβ alpha beta
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20409
g_αβ · Nonabelian transition function between local nonlinear repairs
Exact mathematical passage
g_{\alpha\beta}
\mathrm{glue}Notation table · 7 passages
Search aliases: glue mathrm
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20406
η_loc,η_glue · Quantitative local and gluing residuals
Exact mathematical passage
\eta_{\mathrm{loc}},\eta_{\mathrm{glue}}
HNotation table · 215 passages
Search aliases: H
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20410
H_e · Fundamental-cycle holonomy associated with a non-tree edge
Exact mathematical passage
H_e
\mathcal HNotation table · 4 passages
Search aliases: H mathcal
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20565
H_obs · Finite hidden obstruction-class family in the Block 13 benchmark
Exact mathematical passage
\mathcal H_{\mathrm{obs}}
H_aNotation table · 27 passages
Search aliases: H_a
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20540
σ_R(H_a) · Response signature of hidden block H_a
Exact mathematical passage
\sigma_{\Rcal}(H_a)
H_eNotation table · 4 passages
Search aliases: H_e
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20410
H_e · Fundamental-cycle holonomy associated with a non-tree edge
Exact mathematical passage
H_e
\mathcal H_{\mathrm{obs}}Notation table · 1 passage
Search aliases: H_obs mathcal mathrm
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20565
H_obs · Finite hidden obstruction-class family in the Block 13 benchmark
Exact mathematical passage
\mathcal H_{\mathrm{obs}}
H_\gamma^UNotation table · 1 passage
Search aliases: H_γ^U gamma
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20420
H_γ^U,H_γ^V · Correction and defect holonomies around a transport loop
Exact mathematical passage
H_\gamma^U,H_\gamma^V
H_\gamma^VNotation table · 1 passage
Search aliases: H_γ^V gamma
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20420
H_γ^U,H_γ^V · Correction and defect holonomies around a transport loop
Exact mathematical passage
H_\gamma^U,H_\gamma^V
\HcalNotation table · 18 passages
Search aliases: Hcal H mathcal
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20377
H · Causal-hypothesis family for active diagnosis
Exact mathematical passage
\Hcal
\Hcal_{\rm conflict}Notation table · 3 passages
Search aliases: Hcal_rm conflict H_rm conflict Hcal rm mathcal rm
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20599
H_rm conflict · Hypergraph of repair-conflicting research-problem cells
Exact mathematical passage
\Hcal_{\rm conflict}
\mathrm{hol}Notation table · 10 passages
Search aliases: hol mathrm
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20417
ε_hol · Semantic holonomy error
Exact mathematical passage
\eps_{\mathrm{hol}}
iNotation table · 257 passages
Search aliases: i
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20558
Π_i^-,Π_i^+ · Results-free pre-receipt and evidence-appended post-receipt for research candidate i
Exact mathematical passage
\Pi_i^-,\Pi_i^+
\imNotation table · 50 passages
Search aliases: im operatorname
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20398
η_D(r) · Quantitative obstruction dist(r,im D)
Exact mathematical passage
\dist(r,\im D)
JNotation table · 80 passages
Search aliases: J
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20371
J_Ξ(A) · Resource-normalized utility of agent A in context Ξ
Exact mathematical passage
J_\Xi(A)
J_vNotation table · 9 passages
Search aliases: J_v
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20423
R_v,E_v,ι_v,J_v · Raw evidence, internal evidence space, encoder, and readout of evaluator v
Exact mathematical passage
R_v,E_v,\iota_v,J_v
J_\XiNotation table · 11 passages
Search aliases: J_Ξ Xi
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20371
J_Ξ(A) · Resource-normalized utility of agent A in context Ξ
Exact mathematical passage
J_\Xi(A)
KNotation table · 282 passages
Search aliases: K
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20375
K_A^δ · Protected external-outcome kernel under intervention δ
Exact mathematical passage
K_A^\delta
K_4Notation table · 11 passages
Search aliases: K_4
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20545
K_4,K_5,K_6 · Endogenous-atlas parent, response-language child, and degree-two constructor descendant
Exact mathematical passage
K_4,K_5,K_6
K_5Notation table · 5 passages
Search aliases: K_5
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20545
K_4,K_5,K_6 · Endogenous-atlas parent, response-language child, and degree-two constructor descendant
Exact mathematical passage
K_4,K_5,K_6
K_6Notation table · 4 passages
Search aliases: K_6
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20545
K_4,K_5,K_6 · Endogenous-atlas parent, response-language child, and degree-two constructor descendant
Exact mathematical passage
K_4,K_5,K_6
K_A^\deltaNotation table · 2 passages
Search aliases: K_A^δ delta
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20375
K_A^δ · Protected external-outcome kernel under intervention δ
Exact mathematical passage
K_A^\delta
LNotation table · 101 passages
Search aliases: L
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20407
B,L_G,λ_2(L_G) · Nerve incidence matrix, graph Laplacian, and spectral gap
Exact mathematical passage
B,L_G,\lambda_2(L_G)
\mathfrak LNotation table · 15 passages
Search aliases: L mathfrak
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20382
L_t · Complete resolved lineage state at generation t
Exact mathematical passage
\mathfrak L_t
lNotation table · 44 passages
Search aliases: l
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20599
H_rm conflict · Hypergraph of repair-conflicting research-problem cells
Exact mathematical passage
\Hcal_{\rm conflict}
L_GNotation table · 5 passages
Search aliases: L_G
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20407
B,L_G,λ_2(L_G) · Nerve incidence matrix, graph Laplacian, and spectral gap
Exact mathematical passage
B,L_G,\lambda_2(L_G)
\mathfrak L_tNotation table · 7 passages
Search aliases: L_t mathfrak
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20382
L_t · Complete resolved lineage state at generation t
Exact mathematical passage
\mathfrak L_t
\mathrm{leak}Notation table · 9 passages
Search aliases: leak mathrm
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20416
ε_leak · Packet leakage error
Exact mathematical passage
\eps_{\mathrm{leak}}
\mathrm{loc}Notation table · 7 passages
Search aliases: loc mathrm
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20406
η_loc,η_glue · Quantitative local and gluing residuals
Exact mathematical passage
\eta_{\mathrm{loc}},\eta_{\mathrm{glue}}
MNotation table · 100 passages
Search aliases: M
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20583
M_∩ · Conjunctive integrity-monitor ensemble
Exact mathematical passage
M_{\cap}
mNotation table · 151 passages
Search aliases: m
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20585
widehatΔ_Γ^rm primary,widehatΔ_Γ^rm foreign · Paired robust-child minus inherited-parent effects
Exact mathematical passage
\widehat\Delta_{\Gamma}^{\rm primary},\widehat\Delta_{\Gamma}^{\rm foreign}
M_{\cap}Notation table · 4 passages
Search aliases: M_∩ cap
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20583
M_∩ · Conjunctive integrity-monitor ensemble
Exact mathematical passage
M_{\cap}
\mathscrNotation table · 32 passages
Search aliases: mathscr
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20404
mathscr U,mathscr V · Sheaves of authorized corrections and defects
Exact mathematical passage
\mathscr U,\mathscr V
\mathcal NNotation table · 8 passages
Search aliases: N mathcal
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20582
N_ B · Local behavioral subspace invisible to evaluator portfolio B
Exact mathematical passage
\mathcal N_{\mathcal B}
nNotation table · 173 passages
Search aliases: n
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20411
b_n · Order-n residual coefficient in the formal obstruction tower
Exact mathematical passage
b_n
\mathcal N_{\mathcal B}Notation table · 4 passages
Search aliases: N_ B mathcal mathcal
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20582
N_ B · Local behavioral subspace invisible to evaluator portfolio B
Exact mathematical passage
\mathcal N_{\mathcal B}
\NcalNotation table · 9 passages
Search aliases: Ncal N mathcal
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20562
N_B · Local behavior subspace invisible to evaluator portfolio B
Exact mathematical passage
\Ncal_{\Bcal}
\Ncal_{\Bcal}Notation table · 4 passages
Search aliases: Ncal_Bcal N_B Ncal Bcal mathcal mathcal
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20562
N_B · Local behavior subspace invisible to evaluator portfolio B
Exact mathematical passage
\Ncal_{\Bcal}
ONotation table · 14 passages
Search aliases: O
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20602
O_0,O_1 · Inherited and expanded method-language oracle envelopes
Exact mathematical passage
O_0,O_1
oNotation table · 65 passages
Search aliases: o
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20402
o_t · Unresolved obstruction dimension dimcoker D_t
Exact mathematical passage
o_t
O_0Notation table · 4 passages
Search aliases: O_0
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20602
O_0,O_1 · Inherited and expanded method-language oracle envelopes
Exact mathematical passage
O_0,O_1
O_1Notation table · 4 passages
Search aliases: O_1
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20602
O_0,O_1 · Inherited and expanded method-language oracle envelopes
Exact mathematical passage
O_0,O_1
o_tNotation table · 3 passages
Search aliases: o_t
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20402
o_t · Unresolved obstruction dimension dimcoker D_t
Exact mathematical passage
o_t
\mathrm{ob}Notation table · 28 passages
Search aliases: ob mathrm
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20397
P_cor,P_ob · Orthogonal correctable and obstruction projectors
Exact mathematical passage
P_{\mathrm{cor}},P_{\mathrm{ob}}
\mathrm{obs}Notation table · 8 passages
Search aliases: obs mathrm
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20565
H_obs · Finite hidden obstruction-class family in the Block 13 benchmark
Exact mathematical passage
\mathcal H_{\mathrm{obs}}
\OcalNotation table · 15 passages
Search aliases: Ocal O mathcal
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20384
O_t · Proof-carrying repair operad
Exact mathematical passage
\Ocal_t
\Ocal_tNotation table · 8 passages
Search aliases: Ocal_t O_t Ocal mathcal
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20384
O_t · Proof-carrying repair operad
Exact mathematical passage
\Ocal_t
\overline\OcalNotation table · 6 passages
Search aliases: overlineOcal overlineO overline Ocal overline mathcal
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20380
overlineO · Underlying syntactic colored repair operad
Exact mathematical passage
\overline\Ocal
PNotation table · 199 passages
Search aliases: P
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20397
P_cor,P_ob · Orthogonal correctable and obstruction projectors
Exact mathematical passage
P_{\mathrm{cor}},P_{\mathrm{ob}}
pNotation table · 133 passages
Search aliases: p
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20421
C_tr(p;u) · Certified transport-route cost for correction u
Exact mathematical passage
C_{\mathrm{tr}}(p;u)
P_{\mathrm{cor}}Notation table · 5 passages
Search aliases: P_cor mathrm
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20397
P_cor,P_ob · Orthogonal correctable and obstruction projectors
Exact mathematical passage
P_{\mathrm{cor}},P_{\mathrm{ob}}
P_{\mathrm{cyc}}Notation table · 5 passages
Search aliases: P_cyc mathrm
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20408
P_cyc · Orthogonal projection onto the overlap cycle space
Exact mathematical passage
P_{\mathrm{cyc}}
P_{\mathrm{ob}}Notation table · 12 passages
Search aliases: P_ob mathrm
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20397
P_cor,P_ob · Orthogonal correctable and obstruction projectors
Exact mathematical passage
P_{\mathrm{cor}},P_{\mathrm{ob}}
\PcalNotation table · 25 passages
Search aliases: Pcal P mathcal
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20378
P(T) · Preservation profile of a typed bridge T
Exact mathematical passage
\Pcal(T)
\Pcal_{\mathrm{probe}}Notation table · 6 passages
Search aliases: Pcal_probe P_probe Pcal mathrm mathcal mathrm
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20379
P_probe · Permitted active-probe family
Exact mathematical passage
\Pcal_{\mathrm{probe}}
\mathrm{probe}Notation table · 10 passages
Search aliases: probe mathrm
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20379
P_probe · Permitted active-probe family
Exact mathematical passage
\Pcal_{\mathrm{probe}}
\mathrm{prot}Notation table · 5 passages
Search aliases: prot mathrm
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20426
μ(F;Δ_prot) · Adequacy modulus of core F on protected evidence differences
Exact mathematical passage
\mu(F;\Delta_{\mathrm{prot}})
QNotation table · 132 passages
Search aliases: Q
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20403
D_0,E,Q_0 · Packet-diagonal response, cross-packet coupling, and local right inverse
Exact mathematical passage
D_0,E,Q_0
qNotation table · 133 passages
Search aliases: q
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20429
q,a,Z,Π · Trace map, sealed acceptance rule, revealed trace, and receipt
Exact mathematical passage
q,a,Z,\Pi
Q_0Notation table · 13 passages
Search aliases: Q_0
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20403
D_0,E,Q_0 · Packet-diagonal response, cross-packet coupling, and local right inverse
Exact mathematical passage
D_0,E,Q_0
Q_ANotation table · 4 passages
Search aliases: Q_A
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20419
Q_A · Protected obstruction subspace required to survive transport
Exact mathematical passage
Q_A
RNotation table · 264 passages
Search aliases: R
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20423
R_v,E_v,ι_v,J_v · Raw evidence, internal evidence space, encoder, and readout of evaluator v
Exact mathematical passage
R_v,E_v,\iota_v,J_v
rNotation table · 387 passages
Search aliases: r
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20395
[r] · Obstruction class in coker D
Exact mathematical passage
[r]
R_vNotation table · 6 passages
Search aliases: R_v
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20423
R_v,E_v,ι_v,J_v · Raw evidence, internal evidence space, encoder, and readout of evaluator v
Exact mathematical passage
R_v,E_v,\iota_v,J_v
\rankNotation table · 14 passages
Search aliases: rank operatorname
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20401
ρ_t · Authorized repair rank rank D_t
Exact mathematical passage
\rank D_t
\RcalNotation table · 24 passages
Search aliases: Rcal R mathcal
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20538
R · Declared intervention-response language
Exact mathematical passage
\Rcal
\mathrm{resp}Notation table · 7 passages
Search aliases: resp mathrm
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20541
Π_resp · Proof-carrying response-language expansion certificate
Exact mathematical passage
\Pi_{\mathrm{resp}}
\rmNotation table · 153 passages
Search aliases: rm
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20578
C_rm rob(Γ) · Worst-environment validity-filtered compiler value with a lower-tail risk term
Exact mathematical passage
\mathfrak C_{\rm rob}(\Gamma)
SNotation table · 133 passages
Search aliases: S
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20561
S_cov · Coverage-weighted geometric benchmark score
Exact mathematical passage
S_{\mathrm{cov}}
S_{\mathrm{cov}}Notation table · 2 passages
Search aliases: S_cov mathrm
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20561
S_cov · Coverage-weighted geometric benchmark score
Exact mathematical passage
S_{\mathrm{cov}}
TNotation table · 281 passages
Search aliases: T
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20378
P(T) · Preservation profile of a typed bridge T
Exact mathematical passage
\Pcal(T)
tNotation table · 521 passages
Search aliases: t
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20372
A_t · Stratified functional atlas at generation t
Exact mathematical passage
\mathfrak A_t
T_UNotation table · 34 passages
Search aliases: T_U
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20412
T_U,T_V · Correction-space and defect-space transport maps
Exact mathematical passage
T_U,T_V
T_VNotation table · 39 passages
Search aliases: T_V
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20412
T_U,T_V · Correction-space and defect-space transport maps
Exact mathematical passage
T_U,T_V
\mathrm{tr}Notation table · 2 passages
Search aliases: tr mathrm
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20421
C_tr(p;u) · Certified transport-route cost for correction u
Exact mathematical passage
C_{\mathrm{tr}}(p;u)
UNotation table · 227 passages
Search aliases: U
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20392
U_t^auth · Authorized correction space
Exact mathematical passage
U_t^{\Auth}
\mathfrak UNotation table · 12 passages
Search aliases: U mathfrak
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20577
U · Declared research-compiler robustness environment
Exact mathematical passage
\mathfrak U
uNotation table · 262 passages
Search aliases: u
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20421
C_tr(p;u) · Certified transport-route cost for correction u
Exact mathematical passage
C_{\mathrm{tr}}(p;u)
U_t^{\Auth}Notation table · 6 passages
Search aliases: U_t^Auth U_t^auth Auth mathrm
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20392
U_t^auth · Authorized correction space
Exact mathematical passage
U_t^{\Auth}
VNotation table · 204 passages
Search aliases: V
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20393
V_t · Defect space
Exact mathematical passage
V_t
vNotation table · 142 passages
Search aliases: v
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20422
E_v · Evaluator object at version v
Exact mathematical passage
\mathsf E_v
V_tNotation table · 8 passages
Search aliases: V_t
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20393
V_t · Defect space
Exact mathematical passage
V_t
\VcalNotation table · 14 passages
Search aliases: Vcal V mathcal
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20386
V_t · Evaluator atlas
Exact mathematical passage
\Vcal_t
\Vcal_tNotation table · 7 passages
Search aliases: Vcal_t V_t Vcal mathcal
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20386
V_t · Evaluator atlas
Exact mathematical passage
\Vcal_t
WNotation table · 89 passages
Search aliases: W
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20428
E,W · Editable and witness component masks of a transaction
Exact mathematical passage
E,W
\WcalNotation table · 10 passages
Search aliases: Wcal W mathcal
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20388
W_t · Witness, checkpoint, and rollback structure
Exact mathematical passage
\Wcal_t
\Wcal_tNotation table · 5 passages
Search aliases: Wcal_t W_t Wcal mathcal
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20388
W_t · Witness, checkpoint, and rollback structure
Exact mathematical passage
\Wcal_t
\widehat KNotation table · 6 passages
Search aliases: widehat K widehat
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20376
widehat K_ϕ(A)^δ · Outcome kernel predicted by the compressed self-model
Exact mathematical passage
\widehat K_{\phi(A)}^\delta
\widehat K_{\phi(A)}^\deltaNotation table · 2 passages
Search aliases: widehat K_ϕ(A)^δ widehat phi delta
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20376
widehat K_ϕ(A)^δ · Outcome kernel predicted by the compressed self-model
Exact mathematical passage
\widehat K_{\phi(A)}^\delta
\widehat\DeltaNotation table · 40 passages
Search aliases: widehatΔ widehat Delta
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20564
widehatΔ_Γ · Paired child-minus-parent compiler metaproductivity effect
Exact mathematical passage
\widehat\Delta_\Gamma
\widehat\Delta_\GammaNotation table · 1 passage
Search aliases: widehatΔ_Γ widehat Delta Gamma
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20564
widehatΔ_Γ · Paired child-minus-parent compiler metaproductivity effect
Exact mathematical passage
\widehat\Delta_\Gamma
\widehat\Delta_{\Gamma\text{-lang}}^{\rm foreign}Notation table · 1 passage
Search aliases: widehatΔ_Γ-lang^rm foreign widehat Delta Gamma text rm
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20606
widehatΔ_Γ-lang^rm primary,widehatΔ_Γ-lang^rm foreign · Paired child-parent language-growth effects
Exact mathematical passage
\widehat\Delta_{\Gamma\text{-lang}}^{\rm primary},\widehat\Delta_{\Gamma\text{-lang}}^{\rm foreign}
\widehat\Delta_{\Gamma\text{-lang}}^{\rm primary}Notation table · 1 passage
Search aliases: widehatΔ_Γ-lang^rm primary widehat Delta Gamma text rm
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20606
widehatΔ_Γ-lang^rm primary,widehatΔ_Γ-lang^rm foreign · Paired child-parent language-growth effects
Exact mathematical passage
\widehat\Delta_{\Gamma\text{-lang}}^{\rm primary},\widehat\Delta_{\Gamma\text{-lang}}^{\rm foreign}
\widehat\Delta_{\Gamma}^{\rm foreign}Notation table · 2 passages
Search aliases: widehatΔ_Γ^rm foreign widehat Delta Gamma rm
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20585
widehatΔ_Γ^rm primary,widehatΔ_Γ^rm foreign · Paired robust-child minus inherited-parent effects
Exact mathematical passage
\widehat\Delta_{\Gamma}^{\rm primary},\widehat\Delta_{\Gamma}^{\rm foreign}
\widehat\Delta_{\Gamma}^{\rm primary}Notation table · 2 passages
Search aliases: widehatΔ_Γ^rm primary widehat Delta Gamma rm
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20585
widehatΔ_Γ^rm primary,widehatΔ_Γ^rm foreign · Paired robust-child minus inherited-parent effects
Exact mathematical passage
\widehat\Delta_{\Gamma}^{\rm primary},\widehat\Delta_{\Gamma}^{\rm foreign}
\widetilde PNotation table · 2 passages
Search aliases: widetilde P widetilde
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20581
widetilde P_ϵ · ϵ-contaminated research-receipt distribution
Exact mathematical passage
\widetilde P_\epsilon
\widetilde P_\epsilonNotation table · 2 passages
Search aliases: widetilde P_ϵ widetilde epsilon
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20581
widetilde P_ϵ · ϵ-contaminated research-receipt distribution
Exact mathematical passage
\widetilde P_\epsilon
yNotation table · 132 passages
Search aliases: y
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20579
R,z,y · Response-signature matrix, sparse obstruction mixture, and observed sequential response
Exact mathematical passage
R,z,y
y_pNotation table · 4 passages
Search aliases: y_p
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20600
y_p,θ · Candidate probe response and refinement threshold
Exact mathematical passage
y_p,\theta
ZNotation table · 103 passages
Search aliases: Z
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20429
q,a,Z,Π · Trace map, sealed acceptance rule, revealed trace, and receipt
Exact mathematical passage
q,a,Z,\Pi
zNotation table · 125 passages
Search aliases: z
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20563
F_θ(z) · Outcome, risk, and scored-performance forecaster from proposal and pilot evidence
Exact mathematical passage
F_\theta(z)
\ZcalNotation table · 9 passages
Search aliases: Zcal Z mathcal
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20596
G=(Z,P,C,B) · Research-compiler diagnostic, experiment, method-constructor, and binding language
Exact mathematical passage
\mathfrak G=(\Zcal,\Pcal,\Ccal,\Bcal)
\alphaNotation table · 110 passages
Search aliases: α alpha
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20405
d_αβ · Difference of local repairs on an overlap
Exact mathematical passage
d_{\alpha\beta}
\betaNotation table · 34 passages
Search aliases: β beta
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20405
d_αβ · Difference of local repairs on an overlap
Exact mathematical passage
d_{\alpha\beta}
\GammaNotation table · 90 passages
Search aliases: Γ Gamma
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20557
Γ_0,Γ_1 · AAR-style parent and lineage-trained child research compilers
Exact mathematical passage
\Gamma_0,\Gamma_1
\gammaNotation table · 99 passages
Search aliases: γ gamma
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20420
H_γ^U,H_γ^V · Correction and defect holonomies around a transport loop
Exact mathematical passage
H_\gamma^U,H_\gamma^V
\Gamma_0Notation table · 9 passages
Search aliases: Γ_0 Gamma
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20557
Γ_0,Γ_1 · AAR-style parent and lineage-trained child research compilers
Exact mathematical passage
\Gamma_0,\Gamma_1
\Gamma_1Notation table · 11 passages
Search aliases: Γ_1 Gamma
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20557
Γ_0,Γ_1 · AAR-style parent and lineage-trained child research compilers
Exact mathematical passage
\Gamma_0,\Gamma_1
\Gamma_2Notation table · 9 passages
Search aliases: Γ_2 Gamma
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20584
Γ_2,Π_8^Γ · Robust compiler descendant and its Block 14 lineage receipt
Exact mathematical passage
\Gamma_2,\Pi_8^{\Gamma}
\Gamma_3Notation table · 3 passages
Search aliases: Γ_3 Gamma
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20605
Γ_3,Π_9^Γ · Block 15 language-growing compiler and its lineage receipt
Exact mathematical passage
\Gamma_3,\Pi_9^\Gamma
\DeltaNotation table · 68 passages
Search aliases: Δ Delta
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20418
Δ_D,Δ_r · Correction-square defect and defect-identification error of a transport datum
Exact mathematical passage
\Delta_D,\Delta_r
\deltaNotation table · 102 passages
Search aliases: δ delta
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20375
K_A^δ · Protected external-outcome kernel under intervention δ
Exact mathematical passage
K_A^\delta
\Delta_DNotation table · 4 passages
Search aliases: Δ_D Delta
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20418
Δ_D,Δ_r · Correction-square defect and defect-identification error of a transport datum
Exact mathematical passage
\Delta_D,\Delta_r
\Delta_{\mathrm{prot}}Notation table · 5 passages
Search aliases: Δ_prot Delta mathrm
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20426
μ(F;Δ_prot) · Adequacy modulus of core F on protected evidence differences
Exact mathematical passage
\mu(F;\Delta_{\mathrm{prot}})
\Delta_rNotation table · 4 passages
Search aliases: Δ_r Delta
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20418
Δ_D,Δ_r · Correction-square defect and defect-identification error of a transport datum
Exact mathematical passage
\Delta_D,\Delta_r
\epsilonNotation table · 18 passages
Search aliases: ϵ epsilon
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20581
widetilde P_ϵ · ϵ-contaminated research-receipt distribution
Exact mathematical passage
\widetilde P_\epsilon
\etaNotation table · 83 passages
Search aliases: η eta
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20398
η_D(r) · Quantitative obstruction dist(r,im D)
Exact mathematical passage
\eta_D(r)
\eta_DNotation table · 8 passages
Search aliases: η_D eta
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20398
η_D(r) · Quantitative obstruction dist(r,im D)
Exact mathematical passage
\eta_D(r)
\eta_{D,K}Notation table · 4 passages
Search aliases: η_D,K eta
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20399
η_D,K(r) · Minimum residual under convex authorization set K
Exact mathematical passage
\eta_{D,K}(r)
\eta_{\mathrm{glue}}Notation table · 6 passages
Search aliases: η_glue eta mathrm
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20406
η_loc,η_glue · Quantitative local and gluing residuals
Exact mathematical passage
\eta_{\mathrm{loc}},\eta_{\mathrm{glue}}
\eta_{\mathrm{loc}}Notation table · 3 passages
Search aliases: η_loc eta mathrm
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20406
η_loc,η_glue · Quantitative local and gluing residuals
Exact mathematical passage
\eta_{\mathrm{loc}},\eta_{\mathrm{glue}}
\eta_{\rm open}Notation table · 4 passages
Search aliases: η_rm open eta rm
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20580
η_rm open(y) · Distance of an observed response from the inherited known response cone
Exact mathematical passage
\eta_{\rm open}(y)
\thetaNotation table · 30 passages
Search aliases: θ theta
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20563
F_θ(z) · Outcome, risk, and scored-performance forecaster from proposal and pilot evidence
Exact mathematical passage
F_\theta(z)
\iotaNotation table · 21 passages
Search aliases: ι iota
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20423
R_v,E_v,ι_v,J_v · Raw evidence, internal evidence space, encoder, and readout of evaluator v
Exact mathematical passage
R_v,E_v,\iota_v,J_v
\iota_vNotation table · 9 passages
Search aliases: ι_v iota
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20423
R_v,E_v,ι_v,J_v · Raw evidence, internal evidence space, encoder, and readout of evaluator v
Exact mathematical passage
R_v,E_v,\iota_v,J_v
\kappaNotation table · 27 passages
Search aliases: κ kappa
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20603
κ_Q · Task-relevant structural language margin E_Q[O_1-O_0]
Exact mathematical passage
\kappa_Q
\kappa_QNotation table · 2 passages
Search aliases: κ_Q kappa
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20603
κ_Q · Task-relevant structural language margin E_Q[O_1-O_0]
Exact mathematical passage
\kappa_Q
\LambdaNotation table · 17 passages
Search aliases: Λ Lambda
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20387
Λ_t · Proof-carrying lineage directed acyclic graph
Exact mathematical passage
\Lambda_t
\lambdaNotation table · 87 passages
Search aliases: λ lambda
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20407
B,L_G,λ_2(L_G) · Nerve incidence matrix, graph Laplacian, and spectral gap
Exact mathematical passage
B,L_G,\lambda_2(L_G)
\lambda_2Notation table · 4 passages
Search aliases: λ_2 lambda
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20407
B,L_G,λ_2(L_G) · Nerve incidence matrix, graph Laplacian, and spectral gap
Exact mathematical passage
B,L_G,\lambda_2(L_G)
\Lambda_tNotation table · 7 passages
Search aliases: Λ_t Lambda
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20387
Λ_t · Proof-carrying lineage directed acyclic graph
Exact mathematical passage
\Lambda_t
\muNotation table · 53 passages
Search aliases: μ mu
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20426
μ(F;Δ_prot) · Adequacy modulus of core F on protected evidence differences
Exact mathematical passage
\mu(F;\Delta_{\mathrm{prot}})
\XiNotation table · 39 passages
Search aliases: Ξ Xi
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20370
Ξ · Protected evaluation context
Exact mathematical passage
\Xi
\PiNotation table · 86 passages
Search aliases: Π Pi
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20429
q,a,Z,Π · Trace map, sealed acceptance rule, revealed trace, and receipt
Exact mathematical passage
q,a,Z,\Pi
\Pi_7^{\Gamma}Notation table · 3 passages
Search aliases: Π_7^Γ Pi Gamma
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20559
Π_7^Γ · Proof-carrying Block 13 compiler lineage receipt
Exact mathematical passage
\Pi_7^{\Gamma}
\Pi_8^{\Gamma}Notation table · 2 passages
Search aliases: Π_8^Γ Pi Gamma
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20584
Γ_2,Π_8^Γ · Robust compiler descendant and its Block 14 lineage receipt
Exact mathematical passage
\Gamma_2,\Pi_8^{\Gamma}
\Pi_9^\GammaNotation table · 2 passages
Search aliases: Π_9^Γ Pi Gamma
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20605
Γ_3,Π_9^Γ · Block 15 language-growing compiler and its lineage receipt
Exact mathematical passage
\Gamma_3,\Pi_9^\Gamma
\Pi_{\mathrm{con}}Notation table · 2 passages
Search aliases: Π_con Pi mathrm
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20544
Π_con · Proof-carrying constructor-expansion certificate
Exact mathematical passage
\Pi_{\mathrm{con}}
\Pi_i^+Notation table · 3 passages
Search aliases: Π_i^+ Pi
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20558
Π_i^-,Π_i^+ · Results-free pre-receipt and evidence-appended post-receipt for research candidate i
Exact mathematical passage
\Pi_i^-,\Pi_i^+
\Pi_i^-Notation table · 3 passages
Search aliases: Π_i^- Pi
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20558
Π_i^-,Π_i^+ · Results-free pre-receipt and evidence-appended post-receipt for research candidate i
Exact mathematical passage
\Pi_i^-,\Pi_i^+
\Pi_{\mathrm{resp}}Notation table · 2 passages
Search aliases: Π_resp Pi mathrm
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20541
Π_resp · Proof-carrying response-language expansion certificate
Exact mathematical passage
\Pi_{\mathrm{resp}}
\Pi_{\Gamma\text{-lang}}Notation table · 2 passages
Search aliases: Π_Γ-lang Pi Gamma text
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20604
Π_Γ-lang · Proof-carrying compiler-language expansion receipt
Exact mathematical passage
\Pi_{\Gamma\text{-lang}}
\rhoNotation table · 25 passages
Search aliases: ρ rho
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20401
ρ_t · Authorized repair rank rank D_t
Exact mathematical passage
\rho_t
\rho_tNotation table · 2 passages
Search aliases: ρ_t rho
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20401
ρ_t · Authorized repair rank rank D_t
Exact mathematical passage
\rho_t
\sigmaNotation table · 51 passages
Search aliases: σ sigma
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20400
σ_+(D) · Smallest positive singular value of D
Exact mathematical passage
\sigma_+(D)
\sigma_+Notation table · 5 passages
Search aliases: σ_+ sigma
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20400
σ_+(D) · Smallest positive singular value of D
Exact mathematical passage
\sigma_+(D)
\sigma_{\Rcal}Notation table · 3 passages
Search aliases: σ_Rcal σ_R sigma Rcal sigma mathcal
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20540
σ_R(H_a) · Response signature of hidden block H_a
Exact mathematical passage
\sigma_{\Rcal}(H_a)
\tauNotation table · 109 passages
Search aliases: τ tau
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20425
E_τ^flat · Spectral approximate path-flat core at threshold τ
Exact mathematical passage
E_\tau^{\flat}
\phiNotation table · 50 passages
Search aliases: ϕ phi
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20376
widehat K_ϕ(A)^δ · Outcome kernel predicted by the compressed self-model
Exact mathematical passage
\widehat K_{\phi(A)}^\delta
\phi_0Notation table · 7 passages
Search aliases: ϕ_0 phi
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20597
ϕ_0 · Inherited diagnostic quotient mapping both novel mechanisms to one unknown token
Exact mathematical passage
\phi_0
\phi_tNotation table · 8 passages
Search aliases: ϕ_t phi
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20391
ϕ_t(A_t) · Compressed causal self-model
Exact mathematical passage
\phi_t(A_t)
\daggerNotation table · 36 passages
Search aliases: † dagger
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20396
D^† · Moore--Penrose pseudoinverse of the correction map
Exact mathematical passage
D^\dagger
\capNotation table · 25 passages
Search aliases: ∩ cap
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20583
M_∩ · Conjunctive integrity-monitor ensemble
Exact mathematical passage
M_{\cap}
\starNotation table · 25 passages
Search aliases: ⋆ star
Notation table · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20598
A^⋆(τ) · Acceptable validity-filtered methods for research problem τ
Exact mathematical passage
\Acal^\star(\tau)
*Definition passage · 105 passages
Search aliases: *
Definition passage: Forward implementation refinement · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5466
nition[Forward implementation refinement] A map α:widehat S→ S forward-refines the runtime when: item every executable initial state maps to an authorized abstract initial state; item for every executable transition widehat sxrightarrowawidehat s', there is an abstract transition path α(widehat s)xrightarrowa,*α(widehat s'); item protected observations commute: widehat O(widehat s)=O(α(widehat s)); item receipt compilation commutes with abstraction up to the declared canonical encoding.
Exact mathematical passage
\alpha(\widehat s)\xrightarrow{a,*}\alpha(\widehat s');
\mathcal ADefinition passage · 12 passages
Search aliases: A mathcal
Definition passage: Preservation profile · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 704
[Preservation profile] A preservation profile for a bridge T:X→ Y is a declared record P(T)= ( Σ_T, Q_T, ρ_T, ℓ_T, σ_T, A_T ), where Σ_T lists preserved algebraic structures, Q_T lists preserved causal or evaluative queries, ρ_T records provenance behavior, ℓ_T records information loss, σ_T records uncertainty transport, and A_T lists authorized downstream uses.
Exact mathematical passage
\Pcal(T)= \left( \Sigma_T, \mathcal Q_T, \rho_T, \ell_T, \sigma_T, \mathcal A_T \right),
\mathfrak A^+Definition passage · 1 passage
Search aliases: A^+ mathfrak
Definition passage: Response-language expansion certificate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12976
[Response-language expansion certificate] A response-language expansion certificate is a tuple Π_resp = (R,S, A, A^+,P_alias,γ, ε_comm,ε_fit,Π_foreign), where P_alias records the inherited merged block, γ is the minimum separating spectral gap, ε_comm is a commutator residual, ε_fit is response-model error, and Π_foreign records held-out repair effects.
Exact mathematical passage
\Pi_{\mathrm{resp}} = (\Rcal,S,\mathfrak A,\mathfrak A^+,P_{\mathrm{alias}},\gamma, \eps_{\mathrm{comm}},\eps_{\mathrm{fit}},\Pi_{\mathrm{foreign}}),
a_iDefinition passage · 4 passages
Search aliases: a_i
Definition passage: Post-result research receipt · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13741
[Post-result research receipt] After evaluation, the post-result receipt is Π_i^+ = (Π_i^-, c_i, r_i,a_i,m_i), where c_i is the scored benchmark vector, r_i the protected-regression vector, a_i the hidden-audit result, and m_i the integrity-monitor verdict.
Exact mathematical passage
\Pi_i^{+} = (\Pi_i^{-},\mathbf c_i,\mathbf r_i,a_i,m_i), \label{eq:b13-postresult-receipt}
a_jDefinition passage · 4 passages
Search aliases: a_j
Definition passage: Executable affine-program task · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16036
[Executable affine-program task] A Block 17 task is τ = (x,m,(a_j,b_j)_j=1^m,A_m,B_m,y_m,g,s), y_m=A_mx+B_m, where g∈{A,B} names the generator and s names the data split. The model input is a padded serialization of x, the instructions, and instruction-presence bits. The protected semantic witness is w(τ) = (A_m,fracB_m3,fracy_m4)∈ℝ^3.
Exact mathematical passage
\tau = (x,m,(a_j,b_j)_{j=1}^{m},A_m,B_m,y_m,g,s), \qquad y_m=A_mx+B_m, \label{eq:b17-task}
A_mDefinition passage · 7 passages
Search aliases: A_m
Definition passage: Executable affine-program task · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16036
[Executable affine-program task] A Block 17 task is τ = (x,m,(a_j,b_j)_j=1^m,A_m,B_m,y_m,g,s), y_m=A_mx+B_m, where g∈{A,B} names the generator and s names the data split. The model input is a padded serialization of x, the instructions, and instruction-presence bits. The protected semantic witness is w(τ) = (A_m,fracB_m3,fracy_m4)∈ℝ^3.
Exact mathematical passage
\tau = (x,m,(a_j,b_j)_{j=1}^{m},A_m,B_m,y_m,g,s), \qquad y_m=A_mx+B_m, \label{eq:b17-task}
A_sDefinition passage · 2 passages
Search aliases: A_s
Definition passage: Learned repair-transport instance · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6465
[Learned repair-transport instance] A learned repair-transport instance is a tuple R_learn = ( X,ϕ,q,widetilde q,A_s, A_t, D, P, G, W), where: item X is the input space; item ϕ is the task feature map; item q and widetilde q are protected and corrupted score maps; item A_s is the source agent; item A_t is a family of target agents excluded from source repair fitting; item D is the train, anchor, calibration, and audit partition; item P is the permitted causal probe language; item G is the repair…
Exact mathematical passage
\mathfrak R_{\mathrm{learn}} = (\mathcal X,\phi,q,\widetilde q,A_s,\mathcal A_t, \mathsf D,\mathsf P,\mathsf G,\mathsf W),
\mathcal A_TDefinition passage · 2 passages
Search aliases: A_T mathcal
Definition passage: Preservation profile · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 704
[Preservation profile] A preservation profile for a bridge T:X→ Y is a declared record P(T)= ( Σ_T, Q_T, ρ_T, ℓ_T, σ_T, A_T ), where Σ_T lists preserved algebraic structures, Q_T lists preserved causal or evaluative queries, ρ_T records provenance behavior, ℓ_T records information loss, σ_T records uncertainty transport, and A_T lists authorized downstream uses.
Exact mathematical passage
\Pcal(T)= \left( \Sigma_T, \mathcal Q_T, \rho_T, \ell_T, \sigma_T, \mathcal A_T \right),
\mathcal A_tDefinition passage · 2 passages
Search aliases: A_t mathcal
Definition passage: Learned repair-transport instance · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6465
[Learned repair-transport instance] A learned repair-transport instance is a tuple R_learn = ( X,ϕ,q,widetilde q,A_s, A_t, D, P, G, W), where: item X is the input space; item ϕ is the task feature map; item q and widetilde q are protected and corrupted score maps; item A_s is the source agent; item A_t is a family of target agents excluded from source repair fitting; item D is the train, anchor, calibration, and audit partition; item P is the permitted causal probe language; item G is the repair…
Exact mathematical passage
\mathfrak R_{\mathrm{learn}} = (\mathcal X,\phi,q,\widetilde q,A_s,\mathcal A_t, \mathsf D,\mathsf P,\mathsf G,\mathsf W),
\mathfrak A_t'Definition passage · 4 passages
Search aliases: A_t' mathfrak
Definition passage: Response algebra and commutant · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 19573
[Response algebra and commutant] The response algebra and its commutant are A_t=alg(I,R_v^t:v∈ V), A_t'={S∈End(H_t):SR_v^t=R_v^tS ∀ v}.
Exact mathematical passage
\mathfrak A_t=\operatorname{alg}(I,R_v^t:v\in\mathcal V), \qquad \mathfrak A_t'=\{S\in\End(H_t):SR_v^t=R_v^tS\ \forall v\}.
\mathsf A_vDefinition passage · 1 passage
Search aliases: A_v mathsf
Definition passage: Evaluator-upgrade certificate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3857
[Evaluator-upgrade certificate] An evaluator-upgrade certificate is a tuple Π_eval = (v,w,S_e,T_e,R_e, h_v, A_v, A_w, τ,μ_0, δ_obs,δ_J, T_new, R_rollback, σ), containing the old and new evaluator identifiers, raw and internal transports, core retraction, protected replay set, holonomy-energy operators, accepted flatness and adequacy thresholds, measured replay defects, held-out new-evidence tests, rollback record, and uncertainty statement σ.
Exact mathematical passage
\Pi_{\mathrm{eval}} = (v,w,S_e,T_e,R_e, \mathfrak h_v, \mathsf A_v,\mathsf A_w, \tau,\mu_0, \delta_{\mathrm{obs}},\delta_J, \mathcal T_{\mathrm{new}}, \mathcal R_{\mathrm{rollback}}, \sigma),
\mathsf A_wDefinition passage · 1 passage
Search aliases: A_w mathsf
Definition passage: Evaluator-upgrade certificate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3857
[Evaluator-upgrade certificate] An evaluator-upgrade certificate is a tuple Π_eval = (v,w,S_e,T_e,R_e, h_v, A_v, A_w, τ,μ_0, δ_obs,δ_J, T_new, R_rollback, σ), containing the old and new evaluator identifiers, raw and internal transports, core retraction, protected replay set, holonomy-energy operators, accepted flatness and adequacy thresholds, measured replay defects, held-out new-evidence tests, rollback record, and uncertainty statement σ.
Exact mathematical passage
\Pi_{\mathrm{eval}} = (v,w,S_e,T_e,R_e, \mathfrak h_v, \mathsf A_v,\mathsf A_w, \tau,\mu_0, \delta_{\mathrm{obs}},\delta_J, \mathcal T_{\mathrm{new}}, \mathcal R_{\mathrm{rollback}}, \sigma),
\absDefinition passage · 31 passages
Search aliases: abs
Definition passage: Per-task Neuralese causal coordinates · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15710
[Per-task Neuralese causal coordinates] For task (x_i,y_i), define n_i · :=L(x_i,y_i;zero)-L(x_i,y_i;Z(x_i)), o_i · :=L(x_i,y_i;Z_π(x_i))-L(x_i,y_i;Z(x_i)), c_i · :=L(x_i,y_i;Z_η(x_i))-L(x_i,y_i;Z(x_i)), s_i · :=L(x_i,y_i;Z(x'_i))-L(x_i,y_i;Z(x_i)), r_i · :=absL(x_i,y_i;replay(Z(x_i)))-L(x_i,y_i;Z(x_i)). These are respectively latent necessity, order sensitivity, corruption sensitivity, task specificity, and replay error.
Exact mathematical passage
n_i &:=L(x_i,y_i;\mathsf{zero})-L(x_i,y_i;Z(x_i)), \label{eq:b16-necessity}\\ o_i &:=L(x_i,y_i;Z_{\pi}(x_i))-L(x_i,y_i;Z(x_i)), \label{eq:b16-order}\\ c_i &:=L(x_i,y_i;Z_{\eta}(x_i))-L(x_i,y_i;Z(x_i)), \label{eq:b16-noise}\\ s_i &:=L(x_i,y_i;Z(x'_i))-L(x_i,y_i;Z(x_i)), \label{eq:b16-specificity}\\ r_i &:=\abs{L(x_i,y_i;\mathsf{replay}(Z(x_i)))-L(x_i,y_i;Z(x_i))}. \label{eq:b16-replay}
\Acal_0Definition passage · 3 passages
Search aliases: Acal_0 A_0 Acal mathcal
Definition passage: Conservative atlas extension · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17224
[Conservative atlas extension] An atlas A_1 conservatively extends A_0 when every chart of A_0 occurs as an identical prefix of A_1, the route and canonicalized coordinates agree on every inherited support, and the inherited decision path remains executable as an exact fallback.
Exact mathematical passage
\Acal_0
\Acal_1Definition passage · 5 passages
Search aliases: Acal_1 A_1 Acal mathcal
Definition passage: Conservative atlas extension · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17224
[Conservative atlas extension] An atlas A_1 conservatively extends A_0 when every chart of A_0 occurs as an identical prefix of A_1, the route and canonicalized coordinates agree on every inherited support, and the inherited decision path remains executable as an exact fallback.
Exact mathematical passage
\Acal_1
\Acal_ADefinition passage · 4 passages
Search aliases: Acal_A A_A Acal mathcal
Definition passage: Authorized transport · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2808
[Authorized transport] A transport datum is authorized when T_U(A_A)⊆A_B. It is (c_A,c_B,κ)-budget controlled when c_B(T_Uu)≤κ c_A(u) for every u∈A_A.
Exact mathematical passage
T_U(\Acal_A)\subseteq\Acal_B.
\Acal_BDefinition passage · 2 passages
Search aliases: Acal_B A_B Acal mathcal
Definition passage: Authorized transport · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2808
[Authorized transport] A transport datum is authorized when T_U(A_A)⊆A_B. It is (c_A,c_B,κ)-budget controlled when c_B(T_Uu)≤κ c_A(u) for every u∈A_A.
Exact mathematical passage
T_U(\Acal_A)\subseteq\Acal_B.
\operatorname{Accept}Definition passage · 1 passage
Search aliases: Accept operatorname
Definition passage: Kernel acceptance predicate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5788
[Kernel acceptance predicate] For revealed protected evidence E, the reference acceptance predicate is Accept(widehat s,widehat s',widehatδ,E) iff · K_edit∩ K_protected=∅ · land restrwidehat s'K_protected =restrwidehat sK_protected · land E[utility_gain]≥γ_min · land E[risk]≤ r_max · land E[provenance_ok]=1 · land E_irr=∅.
Exact mathematical passage
\begin{aligned} \operatorname{Accept}(\widehat s,\widehat s',\widehat\delta,E) \iff{}& K_{\mathrm{edit}}\cap K_{\mathrm{protected}}=\varnothing\\ &\land\ \restr{\widehat s'}{K_{\mathrm{protected}}} =\restr{\widehat s}{K_{\mathrm{protected}}}\\ &\land\ E[\mathrm{utility\_gain}]\geq\gamma_{\min}\\ &\land\ E[\mathrm{risk}]\leq r_{\max}\\ &\land\ E[\mathrm{provenance\_ok}]=1\\ &\land\ E_{\mathrm{irr}}=\varnothing. \end{aligned}
\mathsf{Act}Definition passage · 5 passages
Search aliases: Act mathsf
Definition passage: Controllable predecessor · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4311
[Controllable predecessor] For X⊆ S, define CPre(X) = set s∈ S: ∃ u∈Act(s) ∀ d∈Dist(s,u), T(s,u,d)∈ X .
Exact mathematical passage
\operatorname{CPre}(X) = \set{ s\in S: \exists u\in\mathsf{Act}(s)\ \forall d\in\mathsf{Dist}(s,u),\ T(s,u,d)\in X }.
\mathrm{adm}Definition passage · 4 passages
Search aliases: adm mathrm
Definition passage: Distribution-indexed atlas · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17157
[Distribution-indexed atlas] A finite distribution-indexed atlas is A = ( {χ_c:c∈ C}, d_sig, τ_adm, π ), where d_sig is a normalized signature distance, τ_adm is an admission threshold, and π chooses the nearest chart. An input is admitted only when min_c∈ Cd_sig(σ(x),σ_c) ≤τ_adm. Otherwise the atlas returns an unsupported-chart certificate and abstains.
Exact mathematical passage
\Acal = \left( \{\chi_c:c\in C\}, d_{\mathrm{sig}}, \tau_{\mathrm{adm}}, \pi \right), \label{eq:b19-atlas}
\mathsf{Adm}Definition passage · 2 passages
Search aliases: Adm mathsf
Definition passage: Block 19 semantic bundle · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17337
[Block 19 semantic bundle] The finite semantic bundle is B_19 = ( A_1, a, S, F, J, H, C, Adm ), where A_1 is the expanded atlas, a is the set aggregator, F is the recurrent Neuralese transition, J is the protected task readout, H is the halt readout, C is the context branch, and Adm is the selective chart gate.
Exact mathematical passage
\Bcal_{19} = \left( \Acal_1, a, S, F, J, H, C, \mathsf{Adm} \right),
\operatorname{Admissible}Definition passage · 2 passages
Search aliases: Admissible operatorname
Definition passage: Claim gate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5511
[Claim gate] For claim level ℓ, let G_ℓ(E,Π)∈{0,1} be a precommitted predicate over revealed evidence and receipts. The claim is admissible only if every prerequisite gate is one: Admissible(ℓ) iff bigwedge_j≤ℓG_j(E,Π)=1.
Exact mathematical passage
\operatorname{Admissible}(\ell) \iff \bigwedge_{j\leq\ell}G_j(E,\Pi)=1.
\operatorname{alg}Definition passage · 5 passages
Search aliases: alg operatorname
Definition passage: Response algebra and commutant · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 19573
[Response algebra and commutant] The response algebra and its commutant are A_t=alg(I,R_v^t:v∈ V), A_t'={S∈End(H_t):SR_v^t=R_v^tS ∀ v}.
Exact mathematical passage
\mathfrak A_t=\operatorname{alg}(I,R_v^t:v\in\mathcal V), \qquad \mathfrak A_t'=\{S\in\End(H_t):SR_v^t=R_v^tS\ \forall v\}.
\mathrm{alias}Definition passage · 2 passages
Search aliases: alias mathrm
Definition passage: Response-language expansion certificate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12976
[Response-language expansion certificate] A response-language expansion certificate is a tuple Π_resp = (R,S, A, A^+,P_alias,γ, ε_comm,ε_fit,Π_foreign), where P_alias records the inherited merged block, γ is the minimum separating spectral gap, ε_comm is a commutator residual, ε_fit is response-model error, and Π_foreign records held-out repair effects.
Exact mathematical passage
\Pi_{\mathrm{resp}} = (\Rcal,S,\mathfrak A,\mathfrak A^+,P_{\mathrm{alias}},\gamma, \eps_{\mathrm{comm}},\eps_{\mathrm{fit}},\Pi_{\mathrm{foreign}}),
\anglesDefinition passage · 26 passages
Search aliases: angles
Definition passage: Invariant-context two-chart grammar · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11380
[Invariant-context two-chart grammar] Let P_ctx:H_s→ℝ^r be a protected context projector whose value is unchanged by the source repair. Let F be a declared family of separators g(h)=1{anglesu,P_ctxh>θ}. The chart extension is G_chart = set(g,bar T_0,bar T_1):g∈F, bar T_j∈ℝ^q×(d+1). Its prediction is widehat y(h)=bar T_g(h)bar h.
Exact mathematical passage
g(h)=\one\{\angles{u,P_{\mathrm{ctx}}h}>\theta\}.
\AutDefinition passage · 1 passage
Search aliases: Aut operatorname
Definition passage: Protected gauge block · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 19887
[Protected gauge block] A subspace G⊆ Z is a protected gauge block when every permitted future and intervention law is invariant under a declared group G≤Aut(G). The identified object is then the orbit [z]_ G, not a preferred basis inside G.
Exact mathematical passage
\mathcal G\leq\Aut(G)
\mathrm{auth}Definition passage · 3 passages
Search aliases: auth mathrm
Definition passage: Transport route cost · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3000
[Transport route cost] For a fixed source correction u_0, define the certified cost of a nonexpansive path p by C_tr(p;u_0) := normu_0∑_e∈ pε_D,e + ∑_e∈ pε_r,e + ∑_e∈ pε_auth,e + ∑_e∈ pε_sem,e.
Exact mathematical passage
C_{\mathrm{tr}}(p;u_0) := \norm{u_0}\sum_{e\in p}\eps_{D,e} + \sum_{e\in p}\eps_{r,e} + \sum_{e\in p}\eps_{\mathrm{auth},e} + \sum_{e\in p}\eps_{\mathrm{sem},e}.
\mathfrak BDefinition passage · 3 passages
Search aliases: B mathfrak
Definition passage: Empirical benchmark instance · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5392
[Empirical benchmark instance] A benchmark instance is a tuple B = ( E, A, G, P, W, V, D, S, B, R), where: item E is the environment or transition process; item A is the agent family and functional atlas; item G is the repair grammar and candidate generator; item P is the permitted probe and intervention language; item W is the witness, checkpoint, and effect-mediation layer; item V is the protected evaluation context; item D is the data and evidence partition; item S is the seed, randomization,…
Exact mathematical passage
\mathfrak B = (\mathsf E,\mathsf A,\mathsf G,\mathsf P,\mathsf W,\mathsf V, \mathsf D,\mathsf S,\mathsf B,\mathsf R),
\mathsf BDefinition passage · 4 passages
Search aliases: B mathsf
Definition passage: Empirical benchmark instance · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5392
[Empirical benchmark instance] A benchmark instance is a tuple B = ( E, A, G, P, W, V, D, S, B, R), where: item E is the environment or transition process; item A is the agent family and functional atlas; item G is the repair grammar and candidate generator; item P is the permitted probe and intervention language; item W is the witness, checkpoint, and effect-mediation layer; item V is the protected evaluation context; item D is the data and evidence partition; item S is the seed, randomization,…
Exact mathematical passage
\mathfrak B = (\mathsf E,\mathsf A,\mathsf G,\mathsf P,\mathsf W,\mathsf V, \mathsf D,\mathsf S,\mathsf B,\mathsf R),
b_jDefinition passage · 12 passages
Search aliases: b_j
Definition passage: Evaluator-nullspace · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13820
[Evaluator-nullspace] The locally invisible behavior subspace of portfolio B is N_B = ⋂_j=1^mker Db_j.
Exact mathematical passage
\Ncal_{\Bcal} = \bigcap_{j=1}^{m}\ker Db_j. \label{eq:b13-eval-nullspace}
B_mDefinition passage · 7 passages
Search aliases: B_m
Definition passage: Executable affine-program task · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16036
[Executable affine-program task] A Block 17 task is τ = (x,m,(a_j,b_j)_j=1^m,A_m,B_m,y_m,g,s), y_m=A_mx+B_m, where g∈{A,B} names the generator and s names the data split. The model input is a padded serialization of x, the instructions, and instruction-presence bits. The protected semantic witness is w(τ) = (A_m,fracB_m3,fracy_m4)∈ℝ^3.
Exact mathematical passage
\tau = (x,m,(a_j,b_j)_{j=1}^{m},A_m,B_m,y_m,g,s), \qquad y_m=A_mx+B_m, \label{eq:b17-task}
B_tDefinition passage · 4 passages
Search aliases: B_t
Definition passage: Budget acceleration · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4649
[Budget acceleration] Let B_t(γ) be the least budget required by mechanism K_t to obtain expected foreign-agent gain at least γ. Budget acceleration holds on an interval I when B_t+1(γ)<B_t(γ) for every γ∈ I.
Exact mathematical passage
B_t(\gamma)
B_{t+1}Definition passage · 1 passage
Search aliases: B_t+1
Definition passage: Budget acceleration · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4650
[Budget acceleration] Let B_t(γ) be the least budget required by mechanism K_t to obtain expected foreign-agent gain at least γ. Budget acceleration holds on an interval I when B_t+1(γ)<B_t(γ) for every γ∈ I.
Exact mathematical passage
B_{t+1}(\gamma)<B_t(\gamma) \qquad \text{for every }\gamma\in I.
\bar cDefinition passage · 3 passages
Search aliases: bar c bar
Definition passage: Protected evaluation context · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 470
[Protected evaluation context] A protected evaluation context is a tuple Ξ= ( Z, μ, Y, ν, bar c, λ, mathscr I, B, α ), where: item Z is a task or environment family and μ is its held-out distribution; item Y is the space of protected external traces; item ν:Z×Y→[0,1] is task utility; item bar c:Z×Y→[0,1] is normalized resource cost; item λ≥0 is a cost weight; item mathscr I(A,B)∈{0,1} is a declared invariant-preservation predicate for the transition A→ B; item B is the resource envelope used for…
Exact mathematical passage
\Xi= \left( \Zcal, \mu, \Ycal, \nu, \bar c, \lambda, \mathscr I, \Bcal, \alpha \right),
\bar hDefinition passage · 6 passages
Search aliases: bar h bar
Definition passage: Invariant-context two-chart grammar · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11390
oℝ^r be a protected context projector whose value is unchanged by the source repair. Let F be a declared family of separators g(h)=1{anglesu,P_ctxh>θ}. The chart extension is G_chart = set(g,bar T_0,bar T_1):g∈F, bar T_j∈ℝ^q×(d+1). Its prediction is widehat y(h)=bar T_g(h)bar h.
Exact mathematical passage
\widehat y(h)=\bar T_{g(h)}\bar h.
\bar TDefinition passage · 10 passages
Search aliases: bar T bar
Definition passage: Inherited global-affine grammar · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11369
[Inherited global-affine grammar] The inherited bridge grammar is G_glob = setbar T:ℝ^d+1→ℝ^q linear in homogeneous coordinates. It fits one affine source-to-target map across every observable context.
Exact mathematical passage
\Gcal_{\mathrm{glob}} = \set{\bar T:\R^{d+1}\to\R^q\text{ linear in homogeneous coordinates}}.
\bar T_0Definition passage · 2 passages
Search aliases: bar T_0 bar
Definition passage: Invariant-context two-chart grammar · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11384
[Invariant-context two-chart grammar] Let P_ctx:H_s→ℝ^r be a protected context projector whose value is unchanged by the source repair. Let F be a declared family of separators g(h)=1{anglesu,P_ctxh>θ}. The chart extension is G_chart = set(g,bar T_0,bar T_1):g∈F, bar T_j∈ℝ^q×(d+1). Its prediction is widehat y(h)=bar T_g(h)bar h.
Exact mathematical passage
\Gcal_{\mathrm{chart}} = \set{(g,\bar T_0,\bar T_1):g\in\Fcal,\ \bar T_j\in\R^{q\times(d+1)}}.
\bar T_1Definition passage · 2 passages
Search aliases: bar T_1 bar
Definition passage: Invariant-context two-chart grammar · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11384
[Invariant-context two-chart grammar] Let P_ctx:H_s→ℝ^r be a protected context projector whose value is unchanged by the source repair. Let F be a declared family of separators g(h)=1{anglesu,P_ctxh>θ}. The chart extension is G_chart = set(g,bar T_0,bar T_1):g∈F, bar T_j∈ℝ^q×(d+1). Its prediction is widehat y(h)=bar T_g(h)bar h.
Exact mathematical passage
\Gcal_{\mathrm{chart}} = \set{(g,\bar T_0,\bar T_1):g\in\Fcal,\ \bar T_j\in\R^{q\times(d+1)}}.
\bar T_{g(h)}Definition passage · 1 passage
Search aliases: bar T_g(h) bar
Definition passage: Invariant-context two-chart grammar · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11390
oℝ^r be a protected context projector whose value is unchanged by the source repair. Let F be a declared family of separators g(h)=1{anglesu,P_ctxh>θ}. The chart extension is G_chart = set(g,bar T_0,bar T_1):g∈F, bar T_j∈ℝ^q×(d+1). Its prediction is widehat y(h)=bar T_g(h)bar h.
Exact mathematical passage
\widehat y(h)=\bar T_{g(h)}\bar h.
\bar T_jDefinition passage · 1 passage
Search aliases: bar T_j bar
Definition passage: Invariant-context two-chart grammar · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11384
[Invariant-context two-chart grammar] Let P_ctx:H_s→ℝ^r be a protected context projector whose value is unchanged by the source repair. Let F be a declared family of separators g(h)=1{anglesu,P_ctxh>θ}. The chart extension is G_chart = set(g,bar T_0,bar T_1):g∈F, bar T_j∈ℝ^q×(d+1). Its prediction is widehat y(h)=bar T_g(h)bar h.
Exact mathematical passage
\Gcal_{\mathrm{chart}} = \set{(g,\bar T_0,\bar T_1):g\in\Fcal,\ \bar T_j\in\R^{q\times(d+1)}}.
\Bcal_{19}Definition passage · 1 passage
Search aliases: Bcal_19 B_19 Bcal mathcal
Definition passage: Block 19 semantic bundle · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17337
[Block 19 semantic bundle] The finite semantic bundle is B_19 = ( A_1, a, S, F, J, H, C, Adm ), where A_1 is the expanded atlas, a is the set aggregator, F is the recurrent Neuralese transition, J is the protected task readout, H is the halt readout, C is the context branch, and Adm is the selective chart gate.
Exact mathematical passage
\Bcal_{19} = \left( \Acal_1, a, S, F, J, H, C, \mathsf{Adm} \right),
\bigwedgeDefinition passage · 3 passages
Search aliases: bigwedge
Definition passage: Claim gate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5511
[Claim gate] For claim level ℓ, let G_ℓ(E,Π)∈{0,1} be a precommitted predicate over revealed evidence and receipts. The claim is admissible only if every prerequisite gate is one: Admissible(ℓ) iff bigwedge_j≤ℓG_j(E,Π)=1.
Exact mathematical passage
\operatorname{Admissible}(\ell) \iff \bigwedge_{j\leq\ell}G_j(E,\Pi)=1.
\bigwedge_{j\leq\ell}Definition passage · 1 passage
Search aliases: bigwedge_j≤ℓ bigwedge leq ell
Definition passage: Claim gate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5511
[Claim gate] For claim level ℓ, let G_ℓ(E,Π)∈{0,1} be a precommitted predicate over revealed evidence and receipts. The claim is admissible only if every prerequisite gate is one: Admissible(ℓ) iff bigwedge_j≤ℓG_j(E,Π)=1.
Exact mathematical passage
\operatorname{Admissible}(\ell) \iff \bigwedge_{j\leq\ell}G_j(E,\Pi)=1.
\bm cDefinition passage · 2 passages
Search aliases: c bm
Definition passage: Proof-carrying repair schema · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1190
[Proof-carrying repair schema] A repair schema is a tuple ω= ( σ, c_in, c_out, D, C, Q, [r], u, g, mathscr I, Π ), where: item σ is a recognizable failure signature; item c_in=(c_1,…,c_n) and c_out are input and output colors; item D is the declared local defect-response map; item C is an applicability precondition; item Q is a postcondition; item [r] is a local obstruction certificate or unresolved-obstruction record; item ν is the correction program; item g is a precommitted effect profile; it…
Exact mathematical passage
\omega= \left( \sigma, \bm c_{\mathrm{in}}, c_{\mathrm{out}}, D, C, Q, [r], u, \bm g, \mathscr I, \Pi \right),
\mathbf cDefinition passage · 3 passages
Search aliases: c mathbf
Definition passage: Post-result research receipt · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13741
[Post-result research receipt] After evaluation, the post-result receipt is Π_i^+ = (Π_i^-, c_i, r_i,a_i,m_i), where c_i is the scored benchmark vector, r_i the protected-regression vector, a_i the hidden-audit result, and m_i the integrity-monitor verdict.
Exact mathematical passage
\Pi_i^{+} = (\Pi_i^{-},\mathbf c_i,\mathbf r_i,a_i,m_i), \label{eq:b13-postresult-receipt}
\mathcal CDefinition passage · 9 passages
Search aliases: C mathcal
Definition passage: Finite Neuralese semantic bundle · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17699
[Finite Neuralese semantic bundle] A finite architecture--context Neuralese bundle is a tuple E = ( A, C, {H_a,c}_a,c, {J_a,c}_a,c, R ), where H_a,c is a latent trajectory space, J_a,c is an externally calibrated semantic readout, and R is a frozen chart-admission rule. No global identification among the fibers is assumed.
Exact mathematical passage
\mathfrak E = \left( \mathcal A, \mathcal C, \{H_{a,c}\}_{a,c}, \{J_{a,c}\}_{a,c}, \mathcal R \right),
C^{a}_{c\to d}Definition passage · 2 passages
Search aliases: C^a_c→ d to
Definition passage: Architecture--context curvature · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17725
[Architecture--context curvature] For architectures a,b and charts c,d, the transport-square defect is Ω_a,b;c,d := C^a_c→ dT^c_b→ a - T^d_b→ aC^b_c→ d. Given an audit distribution P_b,c on target trajectories, its normalized root-mean-square curvature is κ_a,b;c,d^2 = frac E_h∼ P_b,c normΩ_a,b;c,dh^2 E_z∼ P_a,dnormz^2 .
Exact mathematical passage
\Omega_{a,b;c,d} := C^{a}_{c\to d}T^{c}_{b\to a} - T^{d}_{b\to a}C^{b}_{c\to d}. \label{eq:b20-curvature}
C^{b}_{c\to d}Definition passage · 1 passage
Search aliases: C^b_c→ d to
Definition passage: Architecture--context curvature · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17725
[Architecture--context curvature] For architectures a,b and charts c,d, the transport-square defect is Ω_a,b;c,d := C^a_c→ dT^c_b→ a - T^d_b→ aC^b_c→ d. Given an audit distribution P_b,c on target trajectories, its normalized root-mean-square curvature is κ_a,b;c,d^2 = frac E_h∼ P_b,c normΩ_a,b;c,dh^2 E_z∼ P_a,dnormz^2 .
Exact mathematical passage
\Omega_{a,b;c,d} := C^{a}_{c\to d}T^{c}_{b\to a} - T^{d}_{b\to a}C^{b}_{c\to d}. \label{eq:b20-curvature}
c_1Definition passage · 12 passages
Search aliases: c_1
Definition passage: Proof-carrying repair schema · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1209
[Proof-carrying repair schema] A repair schema is a tuple ω= ( σ, c_in, c_out, D, C, Q, [r], u, g, mathscr I, Π ), where: item σ is a recognizable failure signature; item c_in=(c_1,…,c_n) and c_out are input and output colors; item D is the declared local defect-response map; item C is an applicability precondition; item Q is a postcondition; item [r] is a local obstruction certificate or unresolved-obstruction record; item ν is the correction program; item g is a precommitted effect profile; it…
Exact mathematical passage
\bm c_{\mathrm{in}}=(c_1,\ldots,c_n)
\mathfrak C_ADefinition passage · 1 passage
Search aliases: C_A mathfrak
Definition passage: Transport datum · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2667
[Transport datum] Let C_A and C_B be correction data. A transport datum from A to B is a pair T_A→ B=(T_U,T_V), T_U:U_A→ U_B, T_V:V_A→ V_B. Its correction-square defect is Δ_D( T_A→ B) := D_BT_U-T_VD_A. For declared source and target defects r_A∈ V_A and r_B∈ V_B, its defect-identification error is Δ_r( T_A→ B;r_A,r_B) := r_B-T_Vr_A. The datum is exact when both quantities vanish.
Exact mathematical passage
\mathfrak C_A
c_ADefinition passage · 4 passages
Search aliases: c_A
Definition passage: Authorized transport · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2811
[Authorized transport] A transport datum is authorized when T_U(A_A)⊆A_B. It is (c_A,c_B,κ)-budget controlled when c_B(T_Uu)≤κ c_A(u) for every u∈A_A.
Exact mathematical passage
(c_A,c_B,\kappa)
c_BDefinition passage · 4 passages
Search aliases: c_B
Definition passage: Authorized transport · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2811
[Authorized transport] A transport datum is authorized when T_U(A_A)⊆A_B. It is (c_A,c_B,κ)-budget controlled when c_B(T_Uu)≤κ c_A(u) for every u∈A_A.
Exact mathematical passage
(c_A,c_B,\kappa)
C_eDefinition passage · 7 passages
Search aliases: C_e
Definition passage: Matched selector comparison · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7259
[Matched selector comparison] Two selectors K and K' are compared under matched resources when, episode by episode, they receive: item the identical candidate bundle C_e in the identical order; item the identical receipt fields and calibration evidence; item the identical target, anchor, and audit data; item the identical candidate-construction budget; item the identical protected utility and risk definitions; item no policy-specific regeneration, resampling, or extra oracle query.
Exact mathematical passage
C_e
c_{e,K(Z_e)}Definition passage · 1 passage
Search aliases: c_e,K(Z_e)
Definition passage: Repair selector · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7253
[Repair selector] A repair selector is a measurable map K: Z^B⟶{0,…,B-1}, where Z is the space of candidate receipts available before the protected audit outcome is revealed. The selected repair is c_e,K(Z_e).
Exact mathematical passage
c_{e,K(Z_e)}
\mathbf c_iDefinition passage · 3 passages
Search aliases: c_i mathbf
Definition passage: Post-result research receipt · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13741
[Post-result research receipt] After evaluation, the post-result receipt is Π_i^+ = (Π_i^-, c_i, r_i,a_i,m_i), where c_i is the scored benchmark vector, r_i the protected-regression vector, a_i the hidden-audit result, and m_i the integrity-monitor verdict.
Exact mathematical passage
\Pi_i^{+} = (\Pi_i^{-},\mathbf c_i,\mathbf r_i,a_i,m_i), \label{eq:b13-postresult-receipt}
c_iDefinition passage · 11 passages
Search aliases: c_i
Definition passage: Per-task Neuralese causal coordinates · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15710
[Per-task Neuralese causal coordinates] For task (x_i,y_i), define n_i · :=L(x_i,y_i;zero)-L(x_i,y_i;Z(x_i)), o_i · :=L(x_i,y_i;Z_π(x_i))-L(x_i,y_i;Z(x_i)), c_i · :=L(x_i,y_i;Z_η(x_i))-L(x_i,y_i;Z(x_i)), s_i · :=L(x_i,y_i;Z(x'_i))-L(x_i,y_i;Z(x_i)), r_i · :=absL(x_i,y_i;replay(Z(x_i)))-L(x_i,y_i;Z(x_i)). These are respectively latent necessity, order sensitivity, corruption sensitivity, task specificity, and replay error.
Exact mathematical passage
n_i &:=L(x_i,y_i;\mathsf{zero})-L(x_i,y_i;Z(x_i)), \label{eq:b16-necessity}\\ o_i &:=L(x_i,y_i;Z_{\pi}(x_i))-L(x_i,y_i;Z(x_i)), \label{eq:b16-order}\\ c_i &:=L(x_i,y_i;Z_{\eta}(x_i))-L(x_i,y_i;Z(x_i)), \label{eq:b16-noise}\\ s_i &:=L(x_i,y_i;Z(x'_i))-L(x_i,y_i;Z(x_i)), \label{eq:b16-specificity}\\ r_i &:=\abs{L(x_i,y_i;\mathsf{replay}(Z(x_i)))-L(x_i,y_i;Z(x_i))}. \label{eq:b16-replay}
\bm c_{\mathrm{in}}Definition passage · 2 passages
Search aliases: c_in bm mathrm
Definition passage: Proof-carrying repair schema · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1190
[Proof-carrying repair schema] A repair schema is a tuple ω= ( σ, c_in, c_out, D, C, Q, [r], u, g, mathscr I, Π ), where: item σ is a recognizable failure signature; item c_in=(c_1,…,c_n) and c_out are input and output colors; item D is the declared local defect-response map; item C is an applicability precondition; item Q is a postcondition; item [r] is a local obstruction certificate or unresolved-obstruction record; item ν is the correction program; item g is a precommitted effect profile; it…
Exact mathematical passage
\omega= \left( \sigma, \bm c_{\mathrm{in}}, c_{\mathrm{out}}, D, C, Q, [r], u, \bm g, \mathscr I, \Pi \right),
C_{\mathrm{model}}Definition passage · 2 passages
Search aliases: C_model mathrm
Definition passage: Constructor-expansion certificate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13142
[Constructor-expansion certificate] A constructor-expansion certificate is a tuple Π_con = (C_t,C_t^+,η_old,η_new, H,κ,C_model,Π_foreign), where H is a nonzero higher-order derivative witness, κ is held-out obstruction reduction, and C_model records constructor complexity.
Exact mathematical passage
\Pi_{\mathrm{con}} = (\Ccal_t,\Ccal_t^+,\eta_{\mathrm{old}},\eta_{\mathrm{new}}, \mathsf H,\kappa,C_{\mathrm{model}},\Pi_{\mathrm{foreign}}),
c_nDefinition passage · 3 passages
Search aliases: c_n
Definition passage: Proof-carrying repair schema · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1209
[Proof-carrying repair schema] A repair schema is a tuple ω= ( σ, c_in, c_out, D, C, Q, [r], u, g, mathscr I, Π ), where: item σ is a recognizable failure signature; item c_in=(c_1,…,c_n) and c_out are input and output colors; item D is the declared local defect-response map; item C is an applicability precondition; item Q is a postcondition; item [r] is a local obstruction certificate or unresolved-obstruction record; item ν is the correction program; item g is a precommitted effect profile; it…
Exact mathematical passage
\bm c_{\mathrm{in}}=(c_1,\ldots,c_n)
c_{\mathrm{out}}Definition passage · 2 passages
Search aliases: c_out mathrm
Definition passage: Proof-carrying repair schema · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1190
[Proof-carrying repair schema] A repair schema is a tuple ω= ( σ, c_in, c_out, D, C, Q, [r], u, g, mathscr I, Π ), where: item σ is a recognizable failure signature; item c_in=(c_1,…,c_n) and c_out are input and output colors; item D is the declared local defect-response map; item C is an applicability precondition; item Q is a postcondition; item [r] is a local obstruction certificate or unresolved-obstruction record; item ν is the correction program; item g is a precommitted effect profile; it…
Exact mathematical passage
\omega= \left( \sigma, \bm c_{\mathrm{in}}, c_{\mathrm{out}}, D, C, Q, [r], u, \bm g, \mathscr I, \Pi \right),
\mathfrak C_uDefinition passage · 2 passages
Search aliases: C_u mathfrak
Definition passage: Robust validity-filtered compiler value · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 14421
[Robust validity-filtered compiler value] For risk weight λ≥0, define C_rm rob(Γ) = inf_u∈ U [ C_u(Γ) - λCVaR_α (-R_u(Γ,τ)) ]. The finite Block 14 realization does not compute the unrestricted infimum. It evaluates a frozen finite stress atlas and reports the mean, groupwise minima, lower-tail gain, negative-gain rate, invalid-selection rate, and unknown-class error separately.
Exact mathematical passage
\mathfrak C_{\rm rob}(\Gamma) = \inf_{u\in\mathfrak U} \left[ \mathfrak C_u(\Gamma) - \lambda\operatorname{CVaR}_\alpha \bigl(-R_u(\Gamma,\tau)\bigr) \right]. \label{eq:robust-compiler-value}
C_zDefinition passage · 5 passages
Search aliases: C_z
Definition passage: Repair-consistent cell · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9873
[Repair-consistent cell] A cell is repair consistent when ⋂_x∈ C_zA^⋆(x)≠∅. It is inconsistent when no single authorized repair works for every episode that the grammar treats as equivalent.
Exact mathematical passage
\bigcap_{x\in C_z}\Acal^\star(x)\neq\varnothing. \label{eq:b8-consistent-cell}
C_\omegaDefinition passage · 5 passages
Search aliases: C_ω omega
Definition passage: Receipt composition · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1265
[Receipt composition] Let Π_i validate ω_i, and let Π validate the outer operation ω conditional on C_ω. The composed receipt Π⋆(Π_1,…,Π_n) is the provenance-directed acyclic graph obtained by proof-path concatenation, evidence union, explicit dependency links, and deduplication of immutable evidence references without duplicating linear observation authority.
Exact mathematical passage
C_\omega
C_{\omega_1}Definition passage · 2 passages
Search aliases: C_ω_1 omega
Definition passage: Schema generality preorder · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1415
[Schema generality preorder] For repair schemas ω_1 and ω_2 with the same typed effect, write ω_1preceqω_2 when every state satisfying C_ω_1 also satisfies C_ω_2 and the two correction semantics agree on dom(C_ω_1) up to the declared transport tolerance. Then ω_2 is at least as general as ω_1.
Exact mathematical passage
C_{\omega_1}
C_{\omega_2}Definition passage · 1 passage
Search aliases: C_ω_2 omega
Definition passage: Schema generality preorder · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1415
[Schema generality preorder] For repair schemas ω_1 and ω_2 with the same typed effect, write ω_1preceqω_2 when every state satisfying C_ω_1 also satisfies C_ω_2 and the two correction semantics agree on dom(C_ω_1) up to the declared transport tolerance. Then ω_2 is at least as general as ω_1.
Exact mathematical passage
C_{\omega_2}
\Ccal_tDefinition passage · 2 passages
Search aliases: Ccal_t C_t Ccal mathcal
Definition passage: Constructor-expansion certificate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13142
[Constructor-expansion certificate] A constructor-expansion certificate is a tuple Π_con = (C_t,C_t^+,η_old,η_new, H,κ,C_model,Π_foreign), where H is a nonzero higher-order derivative witness, κ is held-out obstruction reduction, and C_model records constructor complexity.
Exact mathematical passage
\Pi_{\mathrm{con}} = (\Ccal_t,\Ccal_t^+,\eta_{\mathrm{old}},\eta_{\mathrm{new}}, \mathsf H,\kappa,C_{\mathrm{model}},\Pi_{\mathrm{foreign}}),
\Ccal_t^+Definition passage · 2 passages
Search aliases: Ccal_t^+ C_t^+ Ccal mathcal
Definition passage: Constructor-expansion certificate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13142
[Constructor-expansion certificate] A constructor-expansion certificate is a tuple Π_con = (C_t,C_t^+,η_old,η_new, H,κ,C_model,Π_foreign), where H is a nonzero higher-order derivative witness, κ is held-out obstruction reduction, and C_model records constructor complexity.
Exact mathematical passage
\Pi_{\mathrm{con}} = (\Ccal_t,\Ccal_t^+,\eta_{\mathrm{old}},\eta_{\mathrm{new}}, \mathsf H,\kappa,C_{\mathrm{model}},\Pi_{\mathrm{foreign}}),
\mathrm{chart}Definition passage · 10 passages
Search aliases: chart mathrm
Definition passage: Invariant-context two-chart grammar · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11384
[Invariant-context two-chart grammar] Let P_ctx:H_s→ℝ^r be a protected context projector whose value is unchanged by the source repair. Let F be a declared family of separators g(h)=1{anglesu,P_ctxh>θ}. The chart extension is G_chart = set(g,bar T_0,bar T_1):g∈F, bar T_j∈ℝ^q×(d+1). Its prediction is widehat y(h)=bar T_g(h)bar h.
Exact mathematical passage
\Gcal_{\mathrm{chart}} = \set{(g,\bar T_0,\bar T_1):g\in\Fcal,\ \bar T_j\in\R^{q\times(d+1)}}.
\checkDefinition passage · 6 passages
Search aliases: check
Definition passage: Two-level repair obstruction · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2316
[Two-level repair obstruction] The local obstruction on U_α is [r|_U_α]∈coker D_α. When every local class vanishes, the global gluing obstruction is [d_αβ]∈check H^1( U,ker D).
Exact mathematical passage
[d_{\alpha\beta}]\in\check H^1(\mathfrak U,\ker D).
\mathrm{code}Definition passage · 4 passages
Search aliases: code mathrm
Definition passage: Pre-result research receipt · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13727
[Pre-result research receipt] A pre-result receipt has the form Π_i^- = (H_i,O_i,D_i,θ_i, widehat c_i, widehat r_i, F_i,h_i^code,h_i^spec), where H_i is the causal hypothesis, O_i the proposed objective or repair mechanism, D_i the evidence or data construction, θ_i the configuration, widehat c_i the predicted benchmark-effect vector, widehat r_i the predicted protected-risk vector, F_i the declared falsification condition, and h_i^code,h_i^spec immutable digests of the executable and specificat…
Exact mathematical passage
\Pi_i^{-} = (H_i,O_i,D_i,\theta_i, \widehat{\mathbf c}_i, \widehat{\mathbf r}_i, F_i,h_i^{\mathrm{code}},h_i^{\mathrm{spec}}), \label{eq:b13-preresult-receipt}
\mathrm{comm}Definition passage · 7 passages
Search aliases: comm mathrm
Definition passage: Response-language expansion certificate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12976
[Response-language expansion certificate] A response-language expansion certificate is a tuple Π_resp = (R,S, A, A^+,P_alias,γ, ε_comm,ε_fit,Π_foreign), where P_alias records the inherited merged block, γ is the minimum separating spectral gap, ε_comm is a commutator residual, ε_fit is response-model error, and Π_foreign records held-out repair effects.
Exact mathematical passage
\Pi_{\mathrm{resp}} = (\Rcal,S,\mathfrak A,\mathfrak A^+,P_{\mathrm{alias}},\gamma, \eps_{\mathrm{comm}},\eps_{\mathrm{fit}},\Pi_{\mathrm{foreign}}),
\mathrm{Complexity}Definition passage · 1 passage
Search aliases: Complexity mathrm
Definition passage: Cover adequacy score · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2631
[Cover adequacy score] For a candidate functional cover U, define the diagnostic score A( U) := η_loc( U) +τη_glue( U) +λ Complexity( U) +γ Instability( U).
Exact mathematical passage
\Acal(\mathfrak U) := \eta_{\mathrm{loc}}(\mathfrak U) +\tau\eta_{\mathrm{glue}}(\mathfrak U) +\lambda\,\mathrm{Complexity}(\mathfrak U) +\gamma\,\mathrm{Instability}(\mathfrak U).
\mathrm{cons}Definition passage · 3 passages
Search aliases: cons mathrm
Definition passage: Conservative lineage dominance · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9436
[Conservative lineage dominance] Write L_tpreceq_consL_t+1 when: item the old grammar, evidence ledger, and proof objects embed into the new lineage state; item every old protected interpretation remains recoverable within the declared semantic-debt bound; item old authorized behaviors replay within declared tolerances; item the new correction image contains the transported old correction image; item at least one new externally validated distinction, repair schema, or correction direction is add…
Exact mathematical passage
L_t\preceq_{\mathrm{cons}}L_{t+1}
\operatorname{cov}Definition passage · 1 passage
Search aliases: cov operatorname
Definition passage: Partial ontology · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 19908
[Partial ontology] Let P⊆ Z denote the protected semantic coordinates. A quotient identifies a partial ontology when it is sufficient for a declared subset P_0subsetneq P, returns calibrated ignorance for P∖ P_0, and records coverage cov(q)=fracdim P_0dim P.
Exact mathematical passage
\operatorname{cov}(q)=\frac{\dim P_0}{\dim P}.
\operatorname{CPre}Definition passage · 6 passages
Search aliases: CPre operatorname
Definition passage: Controllable predecessor · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4311
[Controllable predecessor] For X⊆ S, define CPre(X) = set s∈ S: ∃ u∈Act(s) ∀ d∈Dist(s,u), T(s,u,d)∈ X .
Exact mathematical passage
\operatorname{CPre}(X) = \set{ s\in S: \exists u\in\mathsf{Act}(s)\ \forall d\in\mathsf{Dist}(s,u),\ T(s,u,d)\in X }.
\mathrm{ctx}Definition passage · 11 passages
Search aliases: ctx mathrm
Definition passage: Invariant-context two-chart grammar · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11379
[Invariant-context two-chart grammar] Let P_ctx:H_s→ℝ^r be a protected context projector whose value is unchanged by the source repair. Let F be a declared family of separators g(h)=1{anglesu,P_ctxh>θ}. The chart extension is G_chart = set(g,bar T_0,bar T_1):g∈F, bar T_j∈ℝ^q×(d+1). Its prediction is widehat y(h)=bar T_g(h)bar h.
Exact mathematical passage
P_{\mathrm{ctx}}:H_s\to\R^r
\operatorname{CVaR}Definition passage · 7 passages
Search aliases: CVaR operatorname
Definition passage: Robust validity-filtered compiler value · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 14421
[Robust validity-filtered compiler value] For risk weight λ≥0, define C_rm rob(Γ) = inf_u∈ U [ C_u(Γ) - λCVaR_α (-R_u(Γ,τ)) ]. The finite Block 14 realization does not compute the unrestricted infimum. It evaluates a frozen finite stress atlas and reports the mean, groupwise minima, lower-tail gain, negative-gain rate, invalid-selection rate, and unknown-class error separately.
Exact mathematical passage
\mathfrak C_{\rm rob}(\Gamma) = \inf_{u\in\mathfrak U} \left[ \mathfrak C_u(\Gamma) - \lambda\operatorname{CVaR}_\alpha \bigl(-R_u(\Gamma,\tau)\bigr) \right]. \label{eq:robust-compiler-value}
\operatorname{CVaR}_\alphaDefinition passage · 2 passages
Search aliases: CVaR_α operatorname alpha
Definition passage: Robust validity-filtered compiler value · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 14421
[Robust validity-filtered compiler value] For risk weight λ≥0, define C_rm rob(Γ) = inf_u∈ U [ C_u(Γ) - λCVaR_α (-R_u(Γ,τ)) ]. The finite Block 14 realization does not compute the unrestricted infimum. It evaluates a frozen finite stress atlas and reports the mean, groupwise minima, lower-tail gain, negative-gain rate, invalid-selection rate, and unknown-class error separately.
Exact mathematical passage
\mathfrak C_{\rm rob}(\Gamma) = \inf_{u\in\mathfrak U} \left[ \mathfrak C_u(\Gamma) - \lambda\operatorname{CVaR}_\alpha \bigl(-R_u(\Gamma,\tau)\bigr) \right]. \label{eq:robust-compiler-value}
\mathcal DDefinition passage · 5 passages
Search aliases: D mathcal
Definition passage: Bounded recursive improvement step · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 574
[Bounded recursive improvement step] A lineage transition L_t→ L_t+1 is a bounded recursive improvement step relative to (Ξ,h,B) when: item its lineage receipt is valid and the protected evidence ledger embeds into the next ledger; item F_t+1(Ξ)≥ F_t(Ξ); item on a foreign held-out target distribution D_foreign, E_A∼ D_foreign Γ_Ξ,h,B(K_t+1;A) > E_A∼ D_foreign Γ_Ξ,h,B(K_t;A); item evaluator risk does not increase beyond a declared tolerance; item the improvement is obtained under matched model ac…
Exact mathematical passage
\mathcal D_{\mathrm{foreign}}
\mathsf DDefinition passage · 17 passages
Search aliases: D mathsf
Definition passage: Empirical benchmark instance · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5392
[Empirical benchmark instance] A benchmark instance is a tuple B = ( E, A, G, P, W, V, D, S, B, R), where: item E is the environment or transition process; item A is the agent family and functional atlas; item G is the repair grammar and candidate generator; item P is the permitted probe and intervention language; item W is the witness, checkpoint, and effect-mediation layer; item V is the protected evaluation context; item D is the data and evidence partition; item S is the seed, randomization,…
Exact mathematical passage
\mathfrak B = (\mathsf E,\mathsf A,\mathsf G,\mathsf P,\mathsf W,\mathsf V, \mathsf D,\mathsf S,\mathsf B,\mathsf R),
D_ADefinition passage · 38 passages
Search aliases: D_A
Definition passage: Transport datum · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2676
[Transport datum] Let C_A and C_B be correction data. A transport datum from A to B is a pair T_A→ B=(T_U,T_V), T_U:U_A→ U_B, T_V:V_A→ V_B. Its correction-square defect is Δ_D( T_A→ B) := D_BT_U-T_VD_A. For declared source and target defects r_A∈ V_A and r_B∈ V_B, its defect-identification error is Δ_r( T_A→ B;r_A,r_B) := r_B-T_Vr_A. The datum is exact when both quantities vanish.
Exact mathematical passage
\Delta_D(\mathfrak T_{A\to B}) := D_BT_U-T_VD_A.
D_BDefinition passage · 31 passages
Search aliases: D_B
Definition passage: Transport datum · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2676
[Transport datum] Let C_A and C_B be correction data. A transport datum from A to B is a pair T_A→ B=(T_U,T_V), T_U:U_A→ U_B, T_V:V_A→ V_B. Its correction-square defect is Δ_D( T_A→ B) := D_BT_U-T_VD_A. For declared source and target defects r_A∈ V_A and r_B∈ V_B, its defect-identification error is Δ_r( T_A→ B;r_A,r_B) := r_B-T_Vr_A. The datum is exact when both quantities vanish.
Exact mathematical passage
\Delta_D(\mathfrak T_{A\to B}) := D_BT_U-T_VD_A.
\mathcal D_{\mathrm{foreign}}Definition passage · 2 passages
Search aliases: D_foreign mathcal mathrm
Definition passage: Bounded recursive improvement step · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 574
[Bounded recursive improvement step] A lineage transition L_t→ L_t+1 is a bounded recursive improvement step relative to (Ξ,h,B) when: item its lineage receipt is valid and the protected evidence ledger embeds into the next ledger; item F_t+1(Ξ)≥ F_t(Ξ); item on a foreign held-out target distribution D_foreign, E_A∼ D_foreign Γ_Ξ,h,B(K_t+1;A) > E_A∼ D_foreign Γ_Ξ,h,B(K_t;A); item evaluator risk does not increase beyond a declared tolerance; item the improvement is obtained under matched model ac…
Exact mathematical passage
\mathcal D_{\mathrm{foreign}}
D_iDefinition passage · 12 passages
Search aliases: D_i
Definition passage: Pre-result research receipt · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13727
[Pre-result research receipt] A pre-result receipt has the form Π_i^- = (H_i,O_i,D_i,θ_i, widehat c_i, widehat r_i, F_i,h_i^code,h_i^spec), where H_i is the causal hypothesis, O_i the proposed objective or repair mechanism, D_i the evidence or data construction, θ_i the configuration, widehat c_i the predicted benchmark-effect vector, widehat r_i the predicted protected-risk vector, F_i the declared falsification condition, and h_i^code,h_i^spec immutable digests of the executable and specificat…
Exact mathematical passage
\Pi_i^{-} = (H_i,O_i,D_i,\theta_i, \widehat{\mathbf c}_i, \widehat{\mathbf r}_i, F_i,h_i^{\mathrm{code}},h_i^{\mathrm{spec}}), \label{eq:b13-preresult-receipt}
\mathcal D_{\rm audit}Definition passage · 1 passage
Search aliases: D_rm audit mathcal rm
Definition passage: Post-connection transport episode · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 18241
[Post-connection transport episode] A Block 21 episode is a tuple E = (X,Y,Z,p, D_rm cal, D_rm meta, D_rm audit), where X∈ℝ^5×8 is the protected source packet, Y is an observed post-connection packet or lower-dimensional statistic, Z∈ℝ^9 is the protected semantic witness, p∈ℝ is a bounded diagnostic response, and the three supports are pairwise disjoint. The compiler receives only these public fields together with an episode identifier and architecture identifier. The private obstruction family …
Exact mathematical passage
\mathfrak E = (X,Y,Z,p,\mathcal D_{\rm cal},\mathcal D_{\rm meta},\mathcal D_{\rm audit}),
\mathcal D_{\rm cal}Definition passage · 1 passage
Search aliases: D_rm cal mathcal rm
Definition passage: Post-connection transport episode · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 18241
[Post-connection transport episode] A Block 21 episode is a tuple E = (X,Y,Z,p, D_rm cal, D_rm meta, D_rm audit), where X∈ℝ^5×8 is the protected source packet, Y is an observed post-connection packet or lower-dimensional statistic, Z∈ℝ^9 is the protected semantic witness, p∈ℝ is a bounded diagnostic response, and the three supports are pairwise disjoint. The compiler receives only these public fields together with an episode identifier and architecture identifier. The private obstruction family …
Exact mathematical passage
\mathfrak E = (X,Y,Z,p,\mathcal D_{\rm cal},\mathcal D_{\rm meta},\mathcal D_{\rm audit}),
\mathsf D_{\rm dev}Definition passage · 4 passages
Search aliases: D_rm dev mathsf rm
Definition passage: Sealed meta-compiler · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7306
[Sealed meta-compiler] A sealed meta-compiler is a map M: ( D_rm dev, D_rm meta,K_0,Γ) ⟼ (K_1,Π_1), where Γ is a declared finite meta-grammar and Π_1 is a descendant receipt. The compiler is sealed when: item neither final audit split is supplied to M; item Π_1 commits to the child state, parent identity, input hashes, grammar, selected configuration, and candidate budget; item the audit files are opened only after the receipt has been serialized and hashed; item the child state used for final e…
Exact mathematical passage
\mathfrak M: \left(\mathsf D_{\rm dev},\mathsf D_{\rm meta},K_0,\Gamma\right) \longmapsto \left(K_1,\Pi_1\right), \label{eq:block07-meta-compiler}
\mathcal D_{\rm meta}Definition passage · 1 passage
Search aliases: D_rm meta mathcal rm
Definition passage: Post-connection transport episode · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 18241
[Post-connection transport episode] A Block 21 episode is a tuple E = (X,Y,Z,p, D_rm cal, D_rm meta, D_rm audit), where X∈ℝ^5×8 is the protected source packet, Y is an observed post-connection packet or lower-dimensional statistic, Z∈ℝ^9 is the protected semantic witness, p∈ℝ is a bounded diagnostic response, and the three supports are pairwise disjoint. The compiler receives only these public fields together with an episode identifier and architecture identifier. The private obstruction family …
Exact mathematical passage
\mathfrak E = (X,Y,Z,p,\mathcal D_{\rm cal},\mathcal D_{\rm meta},\mathcal D_{\rm audit}),
\mathsf D_{\rm meta}Definition passage · 4 passages
Search aliases: D_rm meta mathsf rm
Definition passage: Sealed meta-compiler · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7306
[Sealed meta-compiler] A sealed meta-compiler is a map M: ( D_rm dev, D_rm meta,K_0,Γ) ⟼ (K_1,Π_1), where Γ is a declared finite meta-grammar and Π_1 is a descendant receipt. The compiler is sealed when: item neither final audit split is supplied to M; item Π_1 commits to the child state, parent identity, input hashes, grammar, selected configuration, and candidate budget; item the audit files are opened only after the receipt has been serialized and hashed; item the child state used for final e…
Exact mathematical passage
\mathfrak M: \left(\mathsf D_{\rm dev},\mathsf D_{\rm meta},K_0,\Gamma\right) \longmapsto \left(K_1,\Pi_1\right), \label{eq:block07-meta-compiler}
d_{\mathrm{sig}}Definition passage · 6 passages
Search aliases: d_sig mathrm
Definition passage: Distribution-indexed atlas · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17157
[Distribution-indexed atlas] A finite distribution-indexed atlas is A = ( {χ_c:c∈ C}, d_sig, τ_adm, π ), where d_sig is a normalized signature distance, τ_adm is an admission threshold, and π chooses the nearest chart. An input is admitted only when min_c∈ Cd_sig(σ(x),σ_c) ≤τ_adm. Otherwise the atlas returns an unsupported-chart certificate and abstains.
Exact mathematical passage
\Acal = \left( \{\chi_c:c\in C\}, d_{\mathrm{sig}}, \tau_{\mathrm{adm}}, \pi \right), \label{eq:b19-atlas}
D_{t+1}Definition passage · 20 passages
Search aliases: D_t+1
Definition passage: Repair death · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9672
[Repair death] A class [r]∈coker D_t undergoes repair death at t+1 when overline T_t([r])=0, there exists an authorized u∈ U_t+1 satisfying D_t+1u=-T_t^Vr, and the protected defect transport obeys a lower norm bound normT_t^Vr≥κnormr for a declared κ>0.
Exact mathematical passage
D_{t+1}u=-T_t^Vr,
D_\alphaDefinition passage · 9 passages
Search aliases: D_α alpha
Definition passage: Two-level repair obstruction · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2312
[Two-level repair obstruction] The local obstruction on U_α is [r|_U_α]∈coker D_α. When every local class vanishes, the global gluing obstruction is [d_αβ]∈check H^1( U,ker D).
Exact mathematical passage
[r|_{U_\alpha}]\in\coker D_\alpha.
\Dcal_{\rm for}Definition passage · 4 passages
Search aliases: Dcal_rm for D_rm for Dcal rm mathcal rm
Definition passage: Matched-budget foreign metaproductivity · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8307
[Matched-budget foreign metaproductivity] For a foreign target distribution D_rm for and budget B, define M_B^rm for( P) = E_A∼D_rm for [G_Ξ( P,A)], provided every compared process receives the same target correspondence, calibration, candidate, and irreversible-effect budgets.
Exact mathematical passage
\Dcal_{\rm for}
\mathrm{decision}Definition passage · 1 passage
Search aliases: decision mathrm
Definition passage: Lineage-edge receipt · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 858
[Lineage-edge receipt] A lineage edge AxrightarrowΠB has a receipt Π= ( id_A, id_B, θ, W, E_pre, E_post, widehat g, mathscr I, tests, decision, rollback, digest ), recording the parent and child, transformation θ, frozen witness, pre- and post-reveal evidence, precommitted effect claim widehat g, protected invariants, validation results, commit decision, rollback point, and integrity digest.
Exact mathematical passage
\Pi= \left( \mathrm{id}_A, \mathrm{id}_B, \theta, \Wcal, \Ecal_{\mathrm{pre}}, \Ecal_{\mathrm{post}}, \widehat g, \mathscr I, \mathrm{tests}, \mathrm{decision}, \mathrm{rollback}, \mathrm{digest} \right),
\mathsf{depth}Definition passage · 1 passage
Search aliases: depth mathsf
Definition passage: Finite Neuralese repair grammar · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16139
[Finite Neuralese repair grammar] The Block 17 compiler grammar is G_17 = {parent^+,depth,rank1, rank4,full,full^-sem}. The symbol parent^+ denotes matched-update continuation of the task-blind parent. The final symbol uses full conditioning but removes the semantic-witness loss. Every candidate receives exactly 550 AdamW updates with batch size 256, learning rate 8×10^-4, and identical ordinary task, intermediate-step, halt, and weight-decay terms.
Exact mathematical passage
\Gcal_{17} = \{\mathsf{parent}^{+},\mathsf{depth},\mathsf{rank1}, \mathsf{rank4},\mathsf{full},\mathsf{full}^{-\mathrm{sem}}\}. \label{eq:b17-repair-grammar}
\mathrm{digest}Definition passage · 1 passage
Search aliases: digest mathrm
Definition passage: Lineage-edge receipt · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 858
[Lineage-edge receipt] A lineage edge AxrightarrowΠB has a receipt Π= ( id_A, id_B, θ, W, E_pre, E_post, widehat g, mathscr I, tests, decision, rollback, digest ), recording the parent and child, transformation θ, frozen witness, pre- and post-reveal evidence, precommitted effect claim widehat g, protected invariants, validation results, commit decision, rollback point, and integrity digest.
Exact mathematical passage
\Pi= \left( \mathrm{id}_A, \mathrm{id}_B, \theta, \Wcal, \Ecal_{\mathrm{pre}}, \Ecal_{\mathrm{post}}, \widehat g, \mathscr I, \mathrm{tests}, \mathrm{decision}, \mathrm{rollback}, \mathrm{digest} \right),
\mathsf{Dist}Definition passage · 3 passages
Search aliases: Dist mathsf
Definition passage: Controllable predecessor · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4311
[Controllable predecessor] For X⊆ S, define CPre(X) = set s∈ S: ∃ u∈Act(s) ∀ d∈Dist(s,u), T(s,u,d)∈ X .
Exact mathematical passage
\operatorname{CPre}(X) = \set{ s\in S: \exists u\in\mathsf{Act}(s)\ \forall d\in\mathsf{Dist}(s,u),\ T(s,u,d)\in X }.
\domDefinition passage · 1 passage
Search aliases: dom operatorname
Definition passage: Schema generality preorder · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1415
[Schema generality preorder] For repair schemas ω_1 and ω_2 with the same typed effect, write ω_1preceqω_2 when every state satisfying C_ω_1 also satisfies C_ω_2 and the two correction semantics agree on dom(C_ω_1) up to the declared transport tolerance. Then ω_2 is at least as general as ω_1.
Exact mathematical passage
\dom(C_{\omega_1})
\mathfrak EDefinition passage · 2 passages
Search aliases: E mathfrak
Definition passage: Finite Neuralese semantic bundle · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17699
[Finite Neuralese semantic bundle] A finite architecture--context Neuralese bundle is a tuple E = ( A, C, {H_a,c}_a,c, {J_a,c}_a,c, R ), where H_a,c is a latent trajectory space, J_a,c is an externally calibrated semantic readout, and R is a frozen chart-admission rule. No global identification among the fibers is assumed.
Exact mathematical passage
\mathfrak E = \left( \mathcal A, \mathcal C, \{H_{a,c}\}_{a,c}, \{J_{a,c}\}_{a,c}, \mathcal R \right),
\mathbb E_{A\sim\mathcal D_{\mathrm{foreign}}}Definition passage · 1 passage
Search aliases: E_A∼ D_foreign mathbb sim mathcal mathrm
Definition passage: Bounded recursive improvement step · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 575
ursive improvement step] A lineage transition L_t→ L_t+1 is a bounded recursive improvement step relative to (Ξ,h,B) when: item its lineage receipt is valid and the protected evidence ledger embeds into the next ledger; item F_t+1(Ξ)≥ F_t(Ξ); item on a foreign held-out target distribution D_foreign, E_A∼ D_foreign Γ_Ξ,h,B(K_t+1;A) > E_A∼ D_foreign Γ_Ξ,h,B(K_t;A); item evaluator risk does not increase beyond a declared tolerance; item the improvement is obtained under matched model access, candid…
Exact mathematical passage
\mathbb E_{A\sim\mathcal D_{\mathrm{foreign}}} \Gamma_{\Xi,h,B}(K_{t+1};A) > \mathbb E_{A\sim\mathcal D_{\mathrm{foreign}}} \Gamma_{\Xi,h,B}(K_t;A);
\mathbb E_{A\sim\mathcal F_{\mathrm{post}}}Definition passage · 1 passage
Search aliases: E_A∼ F_post mathbb sim mathcal mathrm
Definition passage: Recursive abstraction estimand · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6265
[Recursive abstraction estimand] Let F_post be foreign defect families revealed after the source repair. Define τ_M^future = E_A∼ F_post [ G_A(Z_M=1)-G_A(Z_M=0) ].
Exact mathematical passage
\tau_M^{\mathrm{future}} = \mathbb E_{A\sim\mathcal F_{\mathrm{post}}} \left[ G_A(Z_M=1)-G_A(Z_M=0) \right].
\mathbb E_{A\sim\Dcal_{\rm for}}Definition passage · 1 passage
Search aliases: E_A∼Dcal_rm for E_A∼D_rm for mathbb sim Dcal rm mathbb sim mathcal rm
Definition passage: Matched-budget foreign metaproductivity · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8308
[Matched-budget foreign metaproductivity] For a foreign target distribution D_rm for and budget B, define M_B^rm for( P) = E_A∼D_rm for [G_Ξ( P,A)], provided every compared process receives the same target correspondence, calibration, candidate, and irreversible-effect budgets.
Exact mathematical passage
M_B^{\rm for}(\mathfrak P) = \mathbb E_{A\sim\Dcal_{\rm for}} \left[G_\Xi(\mathfrak P,A)\right],
E_{\mathrm{con}}Definition passage · 1 passage
Search aliases: E_con mathrm
Definition passage: Witness-separated ontology transaction · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 19869
[Witness-separated ontology transaction] An ontology transaction consists of construction evidence E_con, a compiled quotient q, a pre-reveal commitment σ(q), and witness evidence E_wit. It is witness separated when E_wit is inaccessible to quotient construction and candidate selection, and the final certificate is computed only after σ(q) is fixed.
Exact mathematical passage
E_{\mathrm{con}}
\mathbb E_{e\sim Q}Definition passage · 2 passages
Search aliases: E_e∼ Q mathbb sim
Definition passage: Selector-level metaproductivity · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7272
[Selector-level metaproductivity] Let Q be a held-out distribution over target episodes. A child selector K_1 has selector-level metaproductivity over parent K_0 when Δ_Q(K_1,K_0) := E_e∼ Q [ G_e(K_1)-G_e(K_0) ] >0, under a matched comparison, where G_e(K) is protected utility gain relative to the no-op on episode e.
Exact mathematical passage
\Delta_Q(K_1,K_0) := \mathbb E_{e\sim Q} \left[ G_e(K_1)-G_e(K_0) \right] >0, \label{eq:block07-metaproductivity}
\mathbb E_{h\sim P_{b,c}}Definition passage · 1 passage
Search aliases: E_h∼ P_b,c mathbb sim
Definition passage: Architecture--context curvature · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17734
[Architecture--context curvature] For architectures a,b and charts c,d, the transport-square defect is Ω_a,b;c,d := C^a_c→ dT^c_b→ a - T^d_b→ aC^b_c→ d. Given an audit distribution P_b,c on target trajectories, its normalized root-mean-square curvature is κ_a,b;c,d^2 = frac E_h∼ P_b,c normΩ_a,b;c,dh^2 E_z∼ P_a,dnormz^2 .
Exact mathematical passage
\kappa_{a,b;c,d}^{2} = \frac{ \mathbb E_{h\sim P_{b,c}} \norm{\Omega_{a,b;c,d}h}^{2} }{ \mathbb E_{z\sim P_{a,d}}\norm{z}^{2} }. \label{eq:b20-normalized-curvature}
E_{\mathrm{irr}}Definition passage · 3 passages
Search aliases: E_irr mathrm
Definition passage: Kernel acceptance predicate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5788
[Kernel acceptance predicate] For revealed protected evidence E, the reference acceptance predicate is Accept(widehat s,widehat s',widehatδ,E) iff · K_edit∩ K_protected=∅ · land restrwidehat s'K_protected =restrwidehat sK_protected · land E[utility_gain]≥γ_min · land E[risk]≤ r_max · land E[provenance_ok]=1 · land E_irr=∅.
Exact mathematical passage
\begin{aligned} \operatorname{Accept}(\widehat s,\widehat s',\widehat\delta,E) \iff{}& K_{\mathrm{edit}}\cap K_{\mathrm{protected}}=\varnothing\\ &\land\ \restr{\widehat s'}{K_{\mathrm{protected}}} =\restr{\widehat s}{K_{\mathrm{protected}}}\\ &\land\ E[\mathrm{utility\_gain}]\geq\gamma_{\min}\\ &\land\ E[\mathrm{risk}]\leq r_{\max}\\ &\land\ E[\mathrm{provenance\_ok}]=1\\ &\land\ E_{\mathrm{irr}}=\varnothing. \end{aligned}
E_tDefinition passage · 11 passages
Search aliases: E_t
Definition passage: Rotating witness schedule · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4174
[Rotating witness schedule] A sequence (E_t,W_t)_t≥0 is a rotating witness schedule when: item E_t∩ W_t=∅ for every t; item W_t is immutable during transaction t; item every component belongs to E_t for at least one t; item every witness replacement is validated by a component outside the same edit set; item protected evidence is transported through a conservative evaluator morphism at every rotation; item every committed round retains a rollback path to a state certified by the preceding witnes…
Exact mathematical passage
(E_t,W_t)_{t\geq0}
E_v^{\mathrm{eq}}Definition passage · 2 passages
Search aliases: E_v^eq mathrm
Definition passage: Closed-walk equalizer core · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3532
[Closed-walk equalizer core] For a possibly noninvertible evaluator graph, let W(v) be the monoid of directed closed walks at v. Define E_v^eq := ⋂_γ∈ W(v)ker(H_γ-I).
Exact mathematical passage
E_v^{\mathrm{eq}} := \bigcap_{\gamma\in\mathsf W(v)}\ker(H_\gamma-I).
E_v^{\flat}Definition passage · 6 passages
Search aliases: E_v^flat flat
Definition passage: Core-retractable upgrade · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3741
[Core-retractable upgrade] An evaluator morphism e:v→ w is core-retractable when T_e(E_v^flat)⊆ E_w^flat and there is a map R_e:E_w^flat→ E_v^flat such that R_eT_e x=x for every x∈ E_v^flat. It is core-isomorphic when T_e restricts to an isomorphism from E_v^flat onto E_w^flat.
Exact mathematical passage
T_e(E_v^{\flat})\subseteq E_w^{\flat}
E_{v_0}^{\flat}Definition passage · 6 passages
Search aliases: E_v_0^flat flat
Definition passage: Path-flat evidence core · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3467
[Path-flat evidence core] The path-flat evidence core at v_0 is E_v_0^flat := ⋂_γ∈π_1(G,v_0) ker(H_γ-I).
Exact mathematical passage
E_{v_0}^{\flat} := \bigcap_{\gamma\in\pi_1(G,v_0)} \ker(H_\gamma-I).
E_wDefinition passage · 2 passages
Search aliases: E_w
Definition passage: Evaluator morphism · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3402
[Evaluator morphism] An evaluator morphism e:v→ w consists of maps S_e:R_v→ R_w, T_e:E_v→ E_w, where S_e is the append-only raw-evidence embedding and T_e transports internal representations. Its protected-evidence defect is δ_obs(e) := sup_r∈ h_v normT_eι_v(r)-ι_wS_e(r), and its historical score defect is δ_J(e) := sup_r∈ h_v normJ_wT_eι_v(r)-J_vι_v(r). The morphism is exact on protected history when both defects vanish.
Exact mathematical passage
S_e:R_v\to R_w, \qquad T_e:E_v\to E_w,
E_w^{\flat}Definition passage · 5 passages
Search aliases: E_w^flat flat
Definition passage: Core-retractable upgrade · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3741
[Core-retractable upgrade] An evaluator morphism e:v→ w is core-retractable when T_e(E_v^flat)⊆ E_w^flat and there is a map R_e:E_w^flat→ E_v^flat such that R_eT_e x=x for every x∈ E_v^flat. It is core-isomorphic when T_e restricts to an isomorphism from E_v^flat onto E_w^flat.
Exact mathematical passage
T_e(E_v^{\flat})\subseteq E_w^{\flat}
E_{\mathrm{wit}}Definition passage · 2 passages
Search aliases: E_wit mathrm
Definition passage: Witness-separated ontology transaction · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 19869
[Witness-separated ontology transaction] An ontology transaction consists of construction evidence E_con, a compiled quotient q, a pre-reveal commitment σ(q), and witness evidence E_wit. It is witness separated when E_wit is inaccessible to quotient construction and candidate selection, and the final certificate is computed only after σ(q) is fixed.
Exact mathematical passage
E_{\mathrm{wit}}
\mathbb E_{z\sim P_{a,d}}Definition passage · 1 passage
Search aliases: E_z∼ P_a,d mathbb sim
Definition passage: Architecture--context curvature · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17734
[Architecture--context curvature] For architectures a,b and charts c,d, the transport-square defect is Ω_a,b;c,d := C^a_c→ dT^c_b→ a - T^d_b→ aC^b_c→ d. Given an audit distribution P_b,c on target trajectories, its normalized root-mean-square curvature is κ_a,b;c,d^2 = frac E_h∼ P_b,c normΩ_a,b;c,dh^2 E_z∼ P_a,dnormz^2 .
Exact mathematical passage
\kappa_{a,b;c,d}^{2} = \frac{ \mathbb E_{h\sim P_{b,c}} \norm{\Omega_{a,b;c,d}h}^{2} }{ \mathbb E_{z\sim P_{a,d}}\norm{z}^{2} }. \label{eq:b20-normalized-curvature}
E_{\tau}^{\flat}Definition passage · 2 passages
Search aliases: E_τ^flat tau flat
Definition passage: Thresholded approximate flat core · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3587
[Thresholded approximate flat core] For τ≥0, define E_τ^flat := spansetx_i:λ_i≤τ, where x_i is an eigenbasis of A.
Exact mathematical passage
E_{\tau}^{\flat} := \Span\set{x_i:\lambda_i\leq\tau},
\Ecal_{\mathrm{post}}Definition passage · 1 passage
Search aliases: Ecal_post E_post Ecal mathrm mathcal mathrm
Definition passage: Lineage-edge receipt · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 858
[Lineage-edge receipt] A lineage edge AxrightarrowΠB has a receipt Π= ( id_A, id_B, θ, W, E_pre, E_post, widehat g, mathscr I, tests, decision, rollback, digest ), recording the parent and child, transformation θ, frozen witness, pre- and post-reveal evidence, precommitted effect claim widehat g, protected invariants, validation results, commit decision, rollback point, and integrity digest.
Exact mathematical passage
\Pi= \left( \mathrm{id}_A, \mathrm{id}_B, \theta, \Wcal, \Ecal_{\mathrm{pre}}, \Ecal_{\mathrm{post}}, \widehat g, \mathscr I, \mathrm{tests}, \mathrm{decision}, \mathrm{rollback}, \mathrm{digest} \right),
\Ecal_{\mathrm{pre}}Definition passage · 1 passage
Search aliases: Ecal_pre E_pre Ecal mathrm mathcal mathrm
Definition passage: Lineage-edge receipt · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 858
[Lineage-edge receipt] A lineage edge AxrightarrowΠB has a receipt Π= ( id_A, id_B, θ, W, E_pre, E_post, widehat g, mathscr I, tests, decision, rollback, digest ), recording the parent and child, transformation θ, frozen witness, pre- and post-reveal evidence, precommitted effect claim widehat g, protected invariants, validation results, commit decision, rollback point, and integrity digest.
Exact mathematical passage
\Pi= \left( \mathrm{id}_A, \mathrm{id}_B, \theta, \Wcal, \Ecal_{\mathrm{pre}}, \Ecal_{\mathrm{post}}, \widehat g, \mathscr I, \mathrm{tests}, \mathrm{decision}, \mathrm{rollback}, \mathrm{digest} \right),
\Ecal_{t+1}Definition passage · 1 passage
Search aliases: Ecal_t+1 E_t+1 Ecal mathcal
Definition passage: Append-only evidence embedding · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 804
[Append-only evidence embedding] Evidence preservation from generation t to t+1 is an injective map ι_t:E_t↪E_t+1 that preserves payload, source, timestamp order, precommitment status, provenance type, and cryptographic integrity fields. A later correction may append a refutation or reinterpretation receipt, but it may not silently alter the old entry.
Exact mathematical passage
\iota_t:\Ecal_t\hookrightarrow\Ecal_{t+1}
\mathrm{edit}Definition passage · 5 passages
Search aliases: edit mathrm
Definition passage: Kernel acceptance predicate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5788
[Kernel acceptance predicate] For revealed protected evidence E, the reference acceptance predicate is Accept(widehat s,widehat s',widehatδ,E) iff · K_edit∩ K_protected=∅ · land restrwidehat s'K_protected =restrwidehat sK_protected · land E[utility_gain]≥γ_min · land E[risk]≤ r_max · land E[provenance_ok]=1 · land E_irr=∅.
Exact mathematical passage
\begin{aligned} \operatorname{Accept}(\widehat s,\widehat s',\widehat\delta,E) \iff{}& K_{\mathrm{edit}}\cap K_{\mathrm{protected}}=\varnothing\\ &\land\ \restr{\widehat s'}{K_{\mathrm{protected}}} =\restr{\widehat s}{K_{\mathrm{protected}}}\\ &\land\ E[\mathrm{utility\_gain}]\geq\gamma_{\min}\\ &\land\ E[\mathrm{risk}]\leq r_{\max}\\ &\land\ E[\mathrm{provenance\_ok}]=1\\ &\land\ E_{\mathrm{irr}}=\varnothing. \end{aligned}
\EndDefinition passage · 12 passages
Search aliases: End operatorname
Definition passage: Response algebra and commutant · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 19573
[Response algebra and commutant] The response algebra and its commutant are A_t=alg(I,R_v^t:v∈ V), A_t'={S∈End(H_t):SR_v^t=R_v^tS ∀ v}.
Exact mathematical passage
\mathfrak A_t=\operatorname{alg}(I,R_v^t:v\in\mathcal V), \qquad \mathfrak A_t'=\{S\in\End(H_t):SR_v^t=R_v^tS\ \forall v\}.
\mathrm{episodes}Definition passage · 1 passage
Search aliases: episodes mathrm
Definition passage: Repair abstraction operator · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1401
[Repair abstraction operator] A repair abstraction operator is a map a: H_episodes ⟶ overlineO that maps one or more concrete episodes to a typed schema with a failure signature, applicability condition, correction program, effect profile, and receipt template.
Exact mathematical passage
\mathfrak a: \mathcal H_{\mathrm{episodes}} \longrightarrow \overline\Ocal
\bm\epsDefinition passage · 2 passages
Search aliases: eps ε bm eps bm varepsilon
Definition passage: Stratified functional atlas · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 641
[Stratified functional atlas] A stratified functional atlas at generation t is a tuple A_t= ( C, G_t, {X_t^c}_c∈ C, {F_t^c}_c∈ C, {T_e,τ_e,P_e,ε_e}_e∈ E(G_t) ), where: item C is a finite set of stratum colors; item G_t is a directed multigraph of declared interfaces; item X_t^c is the state space of stratum c; item F_t^c is its local transition rule; item each edge e:c→ d carries a bridge T_e:X_t^c→ X_t^d; item τ_e declares the bridge class; item P_e is its preservation profile; item ε_e is a ve…
Exact mathematical passage
\mathfrak A_t= \left( \mathsf C, G_t, \{X_t^c\}_{c\in\mathsf C}, \{F_t^c\}_{c\in\mathsf C}, \{T_e,\tau_e,\Pcal_e,\bm\eps_e\}_{e\in E(G_t)} \right),
\eps_{\mathrm{auth},e}Definition passage · 1 passage
Search aliases: eps_auth,e ε_auth,e eps mathrm varepsilon mathrm
Definition passage: Transport route cost · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3000
[Transport route cost] For a fixed source correction u_0, define the certified cost of a nonexpansive path p by C_tr(p;u_0) := normu_0∑_e∈ pε_D,e + ∑_e∈ pε_r,e + ∑_e∈ pε_auth,e + ∑_e∈ pε_sem,e.
Exact mathematical passage
C_{\mathrm{tr}}(p;u_0) := \norm{u_0}\sum_{e\in p}\eps_{D,e} + \sum_{e\in p}\eps_{r,e} + \sum_{e\in p}\eps_{\mathrm{auth},e} + \sum_{e\in p}\eps_{\mathrm{sem},e}.
\eps_{\mathrm{comm}}Definition passage · 6 passages
Search aliases: eps_comm ε_comm eps mathrm varepsilon mathrm
Definition passage: Response-language expansion certificate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12976
[Response-language expansion certificate] A response-language expansion certificate is a tuple Π_resp = (R,S, A, A^+,P_alias,γ, ε_comm,ε_fit,Π_foreign), where P_alias records the inherited merged block, γ is the minimum separating spectral gap, ε_comm is a commutator residual, ε_fit is response-model error, and Π_foreign records held-out repair effects.
Exact mathematical passage
\Pi_{\mathrm{resp}} = (\Rcal,S,\mathfrak A,\mathfrak A^+,P_{\mathrm{alias}},\gamma, \eps_{\mathrm{comm}},\eps_{\mathrm{fit}},\Pi_{\mathrm{foreign}}),
\eps_{D,e}Definition passage · 1 passage
Search aliases: eps_D,e ε_D,e eps varepsilon
Definition passage: Transport route cost · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3000
[Transport route cost] For a fixed source correction u_0, define the certified cost of a nonexpansive path p by C_tr(p;u_0) := normu_0∑_e∈ pε_D,e + ∑_e∈ pε_r,e + ∑_e∈ pε_auth,e + ∑_e∈ pε_sem,e.
Exact mathematical passage
C_{\mathrm{tr}}(p;u_0) := \norm{u_0}\sum_{e\in p}\eps_{D,e} + \sum_{e\in p}\eps_{r,e} + \sum_{e\in p}\eps_{\mathrm{auth},e} + \sum_{e\in p}\eps_{\mathrm{sem},e}.
\eps_D^{\mathrm{rel}}Definition passage · 1 passage
Search aliases: eps_D^rel ε_D^rel eps mathrm varepsilon mathrm
Definition passage: Scale-normalized transport defect · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3156
[Scale-normalized transport defect] For a declared stabilizer τ>0, define ε_D^rel := fracnormD_BT_U-T_VD_A normD_BnormT_U+normT_VnormD_A+τ.
Exact mathematical passage
\eps_D^{\mathrm{rel}} := \frac{\norm{D_BT_U-T_VD_A}} {\norm{D_B}\norm{T_U}+\norm{T_V}\norm{D_A}+\tau}.
\bm\eps_eDefinition passage · 2 passages
Search aliases: eps_e ε_e bm eps bm varepsilon
Definition passage: Stratified functional atlas · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 641
[Stratified functional atlas] A stratified functional atlas at generation t is a tuple A_t= ( C, G_t, {X_t^c}_c∈ C, {F_t^c}_c∈ C, {T_e,τ_e,P_e,ε_e}_e∈ E(G_t) ), where: item C is a finite set of stratum colors; item G_t is a directed multigraph of declared interfaces; item X_t^c is the state space of stratum c; item F_t^c is its local transition rule; item each edge e:c→ d carries a bridge T_e:X_t^c→ X_t^d; item τ_e declares the bridge class; item P_e is its preservation profile; item ε_e is a ve…
Exact mathematical passage
\mathfrak A_t= \left( \mathsf C, G_t, \{X_t^c\}_{c\in\mathsf C}, \{F_t^c\}_{c\in\mathsf C}, \{T_e,\tau_e,\Pcal_e,\bm\eps_e\}_{e\in E(G_t)} \right),
\eps_{\mathrm{fit}}Definition passage · 3 passages
Search aliases: eps_fit ε_fit eps mathrm varepsilon mathrm
Definition passage: Response-language expansion certificate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12976
[Response-language expansion certificate] A response-language expansion certificate is a tuple Π_resp = (R,S, A, A^+,P_alias,γ, ε_comm,ε_fit,Π_foreign), where P_alias records the inherited merged block, γ is the minimum separating spectral gap, ε_comm is a commutator residual, ε_fit is response-model error, and Π_foreign records held-out repair effects.
Exact mathematical passage
\Pi_{\mathrm{resp}} = (\Rcal,S,\mathfrak A,\mathfrak A^+,P_{\mathrm{alias}},\gamma, \eps_{\mathrm{comm}},\eps_{\mathrm{fit}},\Pi_{\mathrm{foreign}}),
\eps_QDefinition passage · 2 passages
Search aliases: eps_Q ε_Q eps varepsilon
Definition passage: Property-transport defect · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 726
[Property-transport defect] For a declared domain K⊆ X, define ε_Q(T;K) = sup_x∈ K norm Q_Y(Tx)-widetilde T_QQ_X(x) .
Exact mathematical passage
\eps_Q(T;K) = \sup_{x\in K} \norm{ Q_Y(Tx)-\widetilde T_QQ_X(x) }.
\eps_{r,e}Definition passage · 1 passage
Search aliases: eps_r,e ε_r,e eps varepsilon
Definition passage: Transport route cost · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3000
[Transport route cost] For a fixed source correction u_0, define the certified cost of a nonexpansive path p by C_tr(p;u_0) := normu_0∑_e∈ pε_D,e + ∑_e∈ pε_r,e + ∑_e∈ pε_auth,e + ∑_e∈ pε_sem,e.
Exact mathematical passage
C_{\mathrm{tr}}(p;u_0) := \norm{u_0}\sum_{e\in p}\eps_{D,e} + \sum_{e\in p}\eps_{r,e} + \sum_{e\in p}\eps_{\mathrm{auth},e} + \sum_{e\in p}\eps_{\mathrm{sem},e}.
\eps_{\rm rank}Definition passage · 3 passages
Search aliases: eps_rm rank ε_rm rank eps rm varepsilon rm
Definition passage: Conditioning bottleneck certificate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10549
[Conditioning bottleneck certificate] A rank-revealing parent has a conditioning bottleneck with margin κ>0 when: item its full-rank rate is at least 1-ε_rm rank; item its weighted information floor is below the declared stability floor; item a conditioning-aware policy using the same query budget improves development metaproductivity by at least κ; item the improvement survives recalibration of the rank parent and is not obtained only by changing the commit gate.
Exact mathematical passage
1-\eps_{\rm rank}
\eps_{\mathrm{sem},e}Definition passage · 1 passage
Search aliases: eps_sem,e ε_sem,e eps mathrm varepsilon mathrm
Definition passage: Transport route cost · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3000
[Transport route cost] For a fixed source correction u_0, define the certified cost of a nonexpansive path p by C_tr(p;u_0) := normu_0∑_e∈ pε_D,e + ∑_e∈ pε_r,e + ∑_e∈ pε_auth,e + ∑_e∈ pε_sem,e.
Exact mathematical passage
C_{\mathrm{tr}}(p;u_0) := \norm{u_0}\sum_{e\in p}\eps_{D,e} + \sum_{e\in p}\eps_{r,e} + \sum_{e\in p}\eps_{\mathrm{auth},e} + \sum_{e\in p}\eps_{\mathrm{sem},e}.
\mathrm{eq}Definition passage · 2 passages
Search aliases: eq mathrm
Definition passage: Closed-walk equalizer core · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3532
[Closed-walk equalizer core] For a possibly noninvertible evaluator graph, let W(v) be the monoid of directed closed walks at v. Define E_v^eq := ⋂_γ∈ W(v)ker(H_γ-I).
Exact mathematical passage
E_v^{\mathrm{eq}} := \bigcap_{\gamma\in\mathsf W(v)}\ker(H_\gamma-I).
\mathrm{eval}Definition passage · 1 passage
Search aliases: eval mathrm
Definition passage: Evaluator-upgrade certificate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3857
[Evaluator-upgrade certificate] An evaluator-upgrade certificate is a tuple Π_eval = (v,w,S_e,T_e,R_e, h_v, A_v, A_w, τ,μ_0, δ_obs,δ_J, T_new, R_rollback, σ), containing the old and new evaluator identifiers, raw and internal transports, core retraction, protected replay set, holonomy-energy operators, accepted flatness and adequacy thresholds, measured replay defects, held-out new-evidence tests, rollback record, and uncertainty statement σ.
Exact mathematical passage
\Pi_{\mathrm{eval}} = (v,w,S_e,T_e,R_e, \mathfrak h_v, \mathsf A_v,\mathsf A_w, \tau,\mu_0, \delta_{\mathrm{obs}},\delta_J, \mathcal T_{\mathrm{new}}, \mathcal R_{\mathrm{rollback}}, \sigma),
\mathcal FDefinition passage · 2 passages
Search aliases: F mathcal
Definition passage: Recursive abstraction estimand · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6264
[Recursive abstraction estimand] Let F_post be foreign defect families revealed after the source repair. Define τ_M^future = E_A∼ F_post [ G_A(Z_M=1)-G_A(Z_M=0) ].
Exact mathematical passage
\mathcal F_{\mathrm{post}}
F_iDefinition passage · 2 passages
Search aliases: F_i
Definition passage: Pre-result research receipt · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13727
[Pre-result research receipt] A pre-result receipt has the form Π_i^- = (H_i,O_i,D_i,θ_i, widehat c_i, widehat r_i, F_i,h_i^code,h_i^spec), where H_i is the causal hypothesis, O_i the proposed objective or repair mechanism, D_i the evidence or data construction, θ_i the configuration, widehat c_i the predicted benchmark-effect vector, widehat r_i the predicted protected-risk vector, F_i the declared falsification condition, and h_i^code,h_i^spec immutable digests of the executable and specificat…
Exact mathematical passage
\Pi_i^{-} = (H_i,O_i,D_i,\theta_i, \widehat{\mathbf c}_i, \widehat{\mathbf r}_i, F_i,h_i^{\mathrm{code}},h_i^{\mathrm{spec}}), \label{eq:b13-preresult-receipt}
F_KDefinition passage · 18 passages
Search aliases: F_K
Definition passage: Shapley meta-headroom · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9139
[Shapley meta-headroom] For i∈ N, define h_i(K) = ∑_R⊆ N∖{i} frac|R|!(m-|R|-1)!m! [ F_K(R∪{i})-F_K(R) ].
Exact mathematical passage
h_i(K) = \sum_{R\subseteq N\setminus\{i\}} \frac{|R|!(m-|R|-1)!}{m!} \left[ F_K(R\cup\{i\})-F_K(R) \right]. \label{eq:b8-shapley-headroom}
\mathcal F_{\mathrm{post}}Definition passage · 2 passages
Search aliases: F_post mathcal mathrm
Definition passage: Recursive abstraction estimand · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6264
[Recursive abstraction estimand] Let F_post be foreign defect families revealed after the source repair. Define τ_M^future = E_A∼ F_post [ G_A(Z_M=1)-G_A(Z_M=0) ].
Exact mathematical passage
\mathcal F_{\mathrm{post}}
F_tDefinition passage · 8 passages
Search aliases: F_t
Definition passage: Bounded recursive improvement step · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 573
[Bounded recursive improvement step] A lineage transition L_t→ L_t+1 is a bounded recursive improvement step relative to (Ξ,h,B) when: item its lineage receipt is valid and the protected evidence ledger embeds into the next ledger; item F_t+1(Ξ)≥ F_t(Ξ); item on a foreign held-out target distribution D_foreign, E_A∼ D_foreign Γ_Ξ,h,B(K_t+1;A) > E_A∼ D_foreign Γ_Ξ,h,B(K_t;A); item evaluator risk does not increase beyond a declared tolerance; item the improvement is obtained under matched model ac…
Exact mathematical passage
F_{t+1}(\Xi)\geq F_t(\Xi)
F_{t+1}Definition passage · 1 passage
Search aliases: F_t+1
Definition passage: Bounded recursive improvement step · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 573
[Bounded recursive improvement step] A lineage transition L_t→ L_t+1 is a bounded recursive improvement step relative to (Ξ,h,B) when: item its lineage receipt is valid and the protected evidence ledger embeds into the next ledger; item F_t+1(Ξ)≥ F_t(Ξ); item on a foreign held-out target distribution D_foreign, E_A∼ D_foreign Γ_Ξ,h,B(K_t+1;A) > E_A∼ D_foreign Γ_Ξ,h,B(K_t;A); item evaluator risk does not increase beyond a declared tolerance; item the improvement is obtained under matched model ac…
Exact mathematical passage
F_{t+1}(\Xi)\geq F_t(\Xi)
F_t^cDefinition passage · 2 passages
Search aliases: F_t^c
Definition passage: Stratified functional atlas · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 641
[Stratified functional atlas] A stratified functional atlas at generation t is a tuple A_t= ( C, G_t, {X_t^c}_c∈ C, {F_t^c}_c∈ C, {T_e,τ_e,P_e,ε_e}_e∈ E(G_t) ), where: item C is a finite set of stratum colors; item G_t is a directed multigraph of declared interfaces; item X_t^c is the state space of stratum c; item F_t^c is its local transition rule; item each edge e:c→ d carries a bridge T_e:X_t^c→ X_t^d; item τ_e declares the bridge class; item P_e is its preservation profile; item ε_e is a ve…
Exact mathematical passage
\mathfrak A_t= \left( \mathsf C, G_t, \{X_t^c\}_{c\in\mathsf C}, \{F_t^c\}_{c\in\mathsf C}, \{T_e,\tau_e,\Pcal_e,\bm\eps_e\}_{e\in E(G_t)} \right),
\FcalDefinition passage · 20 passages
Search aliases: Fcal F mathcal
Definition passage: Invariant-context two-chart grammar · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11379
[Invariant-context two-chart grammar] Let P_ctx:H_s→ℝ^r be a protected context projector whose value is unchanged by the source repair. Let F be a declared family of separators g(h)=1{anglesu,P_ctxh>θ}. The chart extension is G_chart = set(g,bar T_0,bar T_1):g∈F, bar T_j∈ℝ^q×(d+1). Its prediction is widehat y(h)=bar T_g(h)bar h.
Exact mathematical passage
\Fcal
\mathrm{fit}Definition passage · 3 passages
Search aliases: fit mathrm
Definition passage: Response-language expansion certificate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12976
[Response-language expansion certificate] A response-language expansion certificate is a tuple Π_resp = (R,S, A, A^+,P_alias,γ, ε_comm,ε_fit,Π_foreign), where P_alias records the inherited merged block, γ is the minimum separating spectral gap, ε_comm is a commutator residual, ε_fit is response-model error, and Π_foreign records held-out repair effects.
Exact mathematical passage
\Pi_{\mathrm{resp}} = (\Rcal,S,\mathfrak A,\mathfrak A^+,P_{\mathrm{alias}},\gamma, \eps_{\mathrm{comm}},\eps_{\mathrm{fit}},\Pi_{\mathrm{foreign}}),
\mathrm{foreign}Definition passage · 16 passages
Search aliases: foreign mathrm
Definition passage: Bounded recursive improvement step · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 574
[Bounded recursive improvement step] A lineage transition L_t→ L_t+1 is a bounded recursive improvement step relative to (Ξ,h,B) when: item its lineage receipt is valid and the protected evidence ledger embeds into the next ledger; item F_t+1(Ξ)≥ F_t(Ξ); item on a foreign held-out target distribution D_foreign, E_A∼ D_foreign Γ_Ξ,h,B(K_t+1;A) > E_A∼ D_foreign Γ_Ξ,h,B(K_t;A); item evaluator risk does not increase beyond a declared tolerance; item the improvement is obtained under matched model ac…
Exact mathematical passage
\mathcal D_{\mathrm{foreign}}
\mathsf{full}Definition passage · 1 passage
Search aliases: full mathsf
Definition passage: Finite Neuralese repair grammar · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16139
[Finite Neuralese repair grammar] The Block 17 compiler grammar is G_17 = {parent^+,depth,rank1, rank4,full,full^-sem}. The symbol parent^+ denotes matched-update continuation of the task-blind parent. The final symbol uses full conditioning but removes the semantic-witness loss. Every candidate receives exactly 550 AdamW updates with batch size 256, learning rate 8×10^-4, and identical ordinary task, intermediate-step, halt, and weight-decay terms.
Exact mathematical passage
\Gcal_{17} = \{\mathsf{parent}^{+},\mathsf{depth},\mathsf{rank1}, \mathsf{rank4},\mathsf{full},\mathsf{full}^{-\mathrm{sem}}\}. \label{eq:b17-repair-grammar}
\mathsf{full}^{-\mathrm{sem}}Definition passage · 1 passage
Search aliases: full^-sem mathsf mathrm
Definition passage: Finite Neuralese repair grammar · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16139
[Finite Neuralese repair grammar] The Block 17 compiler grammar is G_17 = {parent^+,depth,rank1, rank4,full,full^-sem}. The symbol parent^+ denotes matched-update continuation of the task-blind parent. The final symbol uses full conditioning but removes the semantic-witness loss. Every candidate receives exactly 550 AdamW updates with batch size 256, learning rate 8×10^-4, and identical ordinary task, intermediate-step, halt, and weight-decay terms.
Exact mathematical passage
\Gcal_{17} = \{\mathsf{parent}^{+},\mathsf{depth},\mathsf{rank1}, \mathsf{rank4},\mathsf{full},\mathsf{full}^{-\mathrm{sem}}\}. \label{eq:b17-repair-grammar}
\mathrm{future}Definition passage · 4 passages
Search aliases: future mathrm
Definition passage: Recursive abstraction estimand · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6265
[Recursive abstraction estimand] Let F_post be foreign defect families revealed after the source repair. Define τ_M^future = E_A∼ F_post [ G_A(Z_M=1)-G_A(Z_M=0) ].
Exact mathematical passage
\tau_M^{\mathrm{future}} = \mathbb E_{A\sim\mathcal F_{\mathrm{post}}} \left[ G_A(Z_M=1)-G_A(Z_M=0) \right].
\bm gDefinition passage · 4 passages
Search aliases: g bm
Definition passage: Proof-carrying repair schema · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1190
[Proof-carrying repair schema] A repair schema is a tuple ω= ( σ, c_in, c_out, D, C, Q, [r], u, g, mathscr I, Π ), where: item σ is a recognizable failure signature; item c_in=(c_1,…,c_n) and c_out are input and output colors; item D is the declared local defect-response map; item C is an applicability precondition; item Q is a postcondition; item [r] is a local obstruction certificate or unresolved-obstruction record; item ν is the correction program; item g is a precommitted effect profile; it…
Exact mathematical passage
\omega= \left( \sigma, \bm c_{\mathrm{in}}, c_{\mathrm{out}}, D, C, Q, [r], u, \bm g, \mathscr I, \Pi \right),
\mathcal GDefinition passage · 11 passages
Search aliases: G mathcal
Definition passage: Research-compiler robustness environment · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 14386
[Research-compiler robustness environment] A research-compiler robustness environment is a tuple U = ( Q, H, A, V, G ), where: item Q is a family of research-problem distributions and generator shifts; item H is a family of archive contaminations and provenance perturbations; item A is a family of integrity, specification, and selection attacks; item V is a family of evaluator portfolios and aggregation rules; item G is a family of benchmark generators, including problems outside the inherited o…
Exact mathematical passage
\mathfrak U = \left( \mathcal Q, \mathcal H, \mathcal A, \mathcal V, \mathcal G \right), \label{eq:robustness-environment}
\mathsf GDefinition passage · 5 passages
Search aliases: G mathsf
Definition passage: Empirical benchmark instance · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5392
[Empirical benchmark instance] A benchmark instance is a tuple B = ( E, A, G, P, W, V, D, S, B, R), where: item E is the environment or transition process; item A is the agent family and functional atlas; item G is the repair grammar and candidate generator; item P is the permitted probe and intervention language; item W is the witness, checkpoint, and effect-mediation layer; item V is the protected evaluation context; item D is the data and evidence partition; item S is the seed, randomization,…
Exact mathematical passage
\mathfrak B = (\mathsf E,\mathsf A,\mathsf G,\mathsf P,\mathsf W,\mathsf V, \mathsf D,\mathsf S,\mathsf B,\mathsf R),
G_ADefinition passage · 1 passage
Search aliases: G_A
Definition passage: Recursive abstraction estimand · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6265
[Recursive abstraction estimand] Let F_post be foreign defect families revealed after the source repair. Define τ_M^future = E_A∼ F_post [ G_A(Z_M=1)-G_A(Z_M=0) ].
Exact mathematical passage
\tau_M^{\mathrm{future}} = \mathbb E_{A\sim\mathcal F_{\mathrm{post}}} \left[ G_A(Z_M=1)-G_A(Z_M=0) \right].
g_ADefinition passage · 4 passages
Search aliases: g_A
Definition passage: High-probability transport certificate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3311
[High-probability transport certificate] A random tuple (underline g_A, overlineε_J, overlineε_r, overlineε_D, overlineε_leak, overlineε_hol, overline L) is an (1-α) transport certificate when, jointly with probability at least 1-α, it lower-bounds g_A and upper-bounds every corresponding error or Lipschitz quantity.
Exact mathematical passage
g_A
G_eDefinition passage · 4 passages
Search aliases: G_e
Definition passage: Selector-level metaproductivity · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7272
[Selector-level metaproductivity] Let Q be a held-out distribution over target episodes. A child selector K_1 has selector-level metaproductivity over parent K_0 when Δ_Q(K_1,K_0) := E_e∼ Q [ G_e(K_1)-G_e(K_0) ] >0, under a matched comparison, where G_e(K) is protected utility gain relative to the no-op on episode e.
Exact mathematical passage
\Delta_Q(K_1,K_0) := \mathbb E_{e\sim Q} \left[ G_e(K_1)-G_e(K_0) \right] >0, \label{eq:block07-metaproductivity}
G_jDefinition passage · 1 passage
Search aliases: G_j
Definition passage: Claim gate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5511
[Claim gate] For claim level ℓ, let G_ℓ(E,Π)∈{0,1} be a precommitted predicate over revealed evidence and receipts. The claim is admissible only if every prerequisite gate is one: Admissible(ℓ) iff bigwedge_j≤ℓG_j(E,Π)=1.
Exact mathematical passage
\operatorname{Admissible}(\ell) \iff \bigwedge_{j\leq\ell}G_j(E,\Pi)=1.
\mathcal G_{\rm gauge}Definition passage · 5 passages
Search aliases: G_rm gauge mathcal rm
Definition passage: Unpaired intervention-orbit transport instance · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 19549
[Unpaired intervention-orbit transport instance] An instance is a tuple U = (H_s,H_t,μ_s,μ_t, U, {R_u^s}_u∈ U, {R_v^t}_v∈ V, G_rm gauge, W), where μ_s and μ_t are independently sampled latent distributions, R_u^s and R_v^t are within-system intervention-response operators, G_rm gauge is the declared harmless gauge group, and W is a witness hidden from construction. No map between individual samples from μ_s and μ_t is available to the compiler.
Exact mathematical passage
\mathfrak U = (H_s,H_t,\mu_s,\mu_t,\mathcal U, \{R_u^s\}_{u\in\mathcal U}, \{R_v^t\}_{v\in\mathcal V}, \mathcal G_{\rm gauge},\mathsf W),
G_{\gamma}Definition passage · 4 passages
Search aliases: G_γ gamma
Definition passage: Productive recursive closure · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4375
[Productive recursive closure] For a finite perfect-information transaction game, define I_γ^ω = ν Z. μ Y. [ (G_γ∩SPre(Z)) ∪ SPre(Y) ].
Exact mathematical passage
\Ical_{\gamma}^{\omega} = \nu Z.\,\mu Y. \left[ \left(G_{\gamma}\cap\operatorname{SPre}(Z)\right) \cup \operatorname{SPre}(Y) \right].
G_\XiDefinition passage · 2 passages
Search aliases: G_Ξ Xi
Definition passage: Matched-budget foreign metaproductivity · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8308
[Matched-budget foreign metaproductivity] For a foreign target distribution D_rm for and budget B, define M_B^rm for( P) = E_A∼D_rm for [G_Ξ( P,A)], provided every compared process receives the same target correspondence, calibration, candidate, and irreversible-effect budgets.
Exact mathematical passage
M_B^{\rm for}(\mathfrak P) = \mathbb E_{A\sim\Dcal_{\rm for}} \left[G_\Xi(\mathfrak P,A)\right],
G_\ellDefinition passage · 2 passages
Search aliases: G_ℓ ell
Definition passage: Claim gate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5507
[Claim gate] For claim level ℓ, let G_ℓ(E,Π)∈{0,1} be a precommitted predicate over revealed evidence and receipts. The claim is admissible only if every prerequisite gate is one: Admissible(ℓ) iff bigwedge_j≤ℓG_j(E,Π)=1.
Exact mathematical passage
G_\ell(E,\Pi)\in\{0,1\}
\GcalDefinition passage · 51 passages
Search aliases: Gcal G mathcal
Definition passage: Conservative grammar extension · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4958
[Conservative grammar extension] An embedding ι_t:G_t↪G_t+1 is conservative when: item every old well-typed expression remains well typed; item every old derivation translates to a derivation; item the interpretation of every old expression on old evidence is unchanged; item protected evidence identifiers and provenance resources are transported identically; item new rules cannot derive an old-language contradiction unless a correction receipt explicitly retracts a previously unsound axiom or ev…
Exact mathematical passage
\iota_t:\Gcal_t\hookrightarrow\Gcal_{t+1}
\Gcal'Definition passage · 4 passages
Search aliases: Gcal' G' Gcal mathcal
Definition passage: Relative grammar-expansion witness · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5032
[Relative grammar-expansion witness] Let G↪G' be a candidate extension and let r' be the transported defect. The extension exposes a missing correction direction with margin κ>0 when q_G'(r') ≤ q_G(r)-κ.
Exact mathematical passage
\Gcal\hookrightarrow\Gcal'
\Gcal_{17}Definition passage · 1 passage
Search aliases: Gcal_17 G_17 Gcal mathcal
Definition passage: Finite Neuralese repair grammar · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16139
[Finite Neuralese repair grammar] The Block 17 compiler grammar is G_17 = {parent^+,depth,rank1, rank4,full,full^-sem}. The symbol parent^+ denotes matched-update continuation of the task-blind parent. The final symbol uses full conditioning but removes the semantic-witness loss. Every candidate receives exactly 550 AdamW updates with batch size 256, learning rate 8×10^-4, and identical ordinary task, intermediate-step, halt, and weight-decay terms.
Exact mathematical passage
\Gcal_{17} = \{\mathsf{parent}^{+},\mathsf{depth},\mathsf{rank1}, \mathsf{rank4},\mathsf{full},\mathsf{full}^{-\mathrm{sem}}\}. \label{eq:b17-repair-grammar}
\Gcal_{\mathrm{chart}}Definition passage · 3 passages
Search aliases: Gcal_chart G_chart Gcal mathrm mathcal mathrm
Definition passage: Invariant-context two-chart grammar · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11384
[Invariant-context two-chart grammar] Let P_ctx:H_s→ℝ^r be a protected context projector whose value is unchanged by the source repair. Let F be a declared family of separators g(h)=1{anglesu,P_ctxh>θ}. The chart extension is G_chart = set(g,bar T_0,bar T_1):g∈F, bar T_j∈ℝ^q×(d+1). Its prediction is widehat y(h)=bar T_g(h)bar h.
Exact mathematical passage
\Gcal_{\mathrm{chart}} = \set{(g,\bar T_0,\bar T_1):g\in\Fcal,\ \bar T_j\in\R^{q\times(d+1)}}.
\Gcal_{\mathrm{glob}}Definition passage · 4 passages
Search aliases: Gcal_glob G_glob Gcal mathrm mathcal mathrm
Definition passage: Inherited global-affine grammar · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11369
[Inherited global-affine grammar] The inherited bridge grammar is G_glob = setbar T:ℝ^d+1→ℝ^q linear in homogeneous coordinates. It fits one affine source-to-target map across every observable context.
Exact mathematical passage
\Gcal_{\mathrm{glob}} = \set{\bar T:\R^{d+1}\to\R^q\text{ linear in homogeneous coordinates}}.
\Gcal_tDefinition passage · 6 passages
Search aliases: Gcal_t G_t Gcal mathcal
Definition passage: Conservative grammar extension · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4958
[Conservative grammar extension] An embedding ι_t:G_t↪G_t+1 is conservative when: item every old well-typed expression remains well typed; item every old derivation translates to a derivation; item the interpretation of every old expression on old evidence is unchanged; item protected evidence identifiers and provenance resources are transported identically; item new rules cannot derive an old-language contradiction unless a correction receipt explicitly retracts a previously unsound axiom or ev…
Exact mathematical passage
\iota_t:\Gcal_t\hookrightarrow\Gcal_{t+1}
\Gcal_{t+1}Definition passage · 3 passages
Search aliases: Gcal_t+1 G_t+1 Gcal mathcal
Definition passage: Conservative grammar extension · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4958
[Conservative grammar extension] An embedding ι_t:G_t↪G_t+1 is conservative when: item every old well-typed expression remains well typed; item every old derivation translates to a derivation; item the interpretation of every old expression on old evidence is unchanged; item protected evidence identifiers and provenance resources are transported identically; item new rules cannot derive an old-language contradiction unless a correction receipt explicitly retracts a previously unsound axiom or ev…
Exact mathematical passage
\iota_t:\Gcal_t\hookrightarrow\Gcal_{t+1}
\mathrm{geo}Definition passage · 1 passage
Search aliases: geo mathrm
Definition passage: Cross-implementation repair schema · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16726
[Cross-implementation repair schema] A Block 18 repair schema is a tuple ρ=(a,λ_sem,λ_prog, λ_geo,χ), where a is an adapter constructor, the three nonnegative λ values weight final semantic, progressive semantic, and latent-geometry witnesses, and χ records whether witness labels are correctly bound or shuffled. A schema is cross-implementation when the same tuple is instantiated through each cell family's declared adapter interface without architecture-specific candidate selection.
Exact mathematical passage
\rho=(a,\lambda_{\mathrm{sem}},\lambda_{\mathrm{prog}}, \lambda_{\mathrm{geo}},\chi), \label{eq:b18-repair-schema}
\mathrm{glob}Definition passage · 8 passages
Search aliases: glob mathrm
Definition passage: Inherited global-affine grammar · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11369
[Inherited global-affine grammar] The inherited bridge grammar is G_glob = setbar T:ℝ^d+1→ℝ^q linear in homogeneous coordinates. It fits one affine source-to-target map across every observable context.
Exact mathematical passage
\Gcal_{\mathrm{glob}} = \set{\bar T:\R^{d+1}\to\R^q\text{ linear in homogeneous coordinates}}.
\mathfrak hDefinition passage · 8 passages
Search aliases: h mathfrak
Definition passage: Evaluator object · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3386
[Evaluator object] An evaluator object at version v is a tuple E_v=(R_v,E_v,ι_v,J_v,Q_v, h_v), where: item R_v is the protected raw-evidence space or ledger span; item E_v is the evaluator's internal evidence-representation space; item ι_v:R_v→ E_v encodes raw evidence; item J_v:E_v→ℝ^q is a score or decision-readout map; item Q_v is an uncertainty or calibration readout; item h_v is the historical replay set protected at version v.
Exact mathematical passage
\mathsf E_v=(R_v,E_v,\iota_v,J_v,Q_v,\mathfrak h_v),
\mathsf HDefinition passage · 2 passages
Search aliases: H mathsf
Definition passage: Constructor-expansion certificate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13142
[Constructor-expansion certificate] A constructor-expansion certificate is a tuple Π_con = (C_t,C_t^+,η_old,η_new, H,κ,C_model,Π_foreign), where H is a nonzero higher-order derivative witness, κ is held-out obstruction reduction, and C_model records constructor complexity.
Exact mathematical passage
\Pi_{\mathrm{con}} = (\Ccal_t,\Ccal_t^+,\eta_{\mathrm{old}},\eta_{\mathrm{new}}, \mathsf H,\kappa,C_{\mathrm{model}},\Pi_{\mathrm{foreign}}),
hDefinition passage · 193 passages
Search aliases: h
Definition passage: Bounded recursive improvement step · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 570
[Bounded recursive improvement step] A lineage transition L_t→ L_t+1 is a bounded recursive improvement step relative to (Ξ,h,B) when: item its lineage receipt is valid and the protected evidence ledger embeds into the next ledger; item F_t+1(Ξ)≥ F_t(Ξ); item on a foreign held-out target distribution D_foreign, E_A∼ D_foreign Γ_Ξ,h,B(K_t+1;A) > E_A∼ D_foreign Γ_Ξ,h,B(K_t;A); item evaluator risk does not increase beyond a declared tolerance; item the improvement is obtained under matched model ac…
Exact mathematical passage
(\Xi,h,B)
h'Definition passage · 13 passages
Search aliases: h'
Definition passage: Deterministically separating probe family · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1121
[Deterministically separating probe family] Suppose every K_h^p is a point mass at an outcome y(h,p). The family P_probe separates H when for every h≠ h' there exists p∈P_probe such that y(h,p)≠ y(h',p).
Exact mathematical passage
h\neq h'
H^1Definition passage · 6 passages
Search aliases: H^1
Definition passage: Two-level repair obstruction · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2316
[Two-level repair obstruction] The local obstruction on U_α is [r|_U_α]∈coker D_α. When every local class vanishes, the global gluing obstruction is [d_αβ]∈check H^1( U,ker D).
Exact mathematical passage
[d_{\alpha\beta}]\in\check H^1(\mathfrak U,\ker D).
H^KDefinition passage · 1 passage
Search aliases: H^K
Definition passage: Neuralese trace · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15679
[Neuralese trace] For input x and fixed depth K, the injected Neuralese trace is Z_K(x)=(z_1(x),…,z_K(x))∈ H^K. The updated trace is H_K(x)=(h_1(x),…,h_K(x)). Every receipt must state which trace convention is being intervened on. Block 16 uses the injected trace because this is the point at which the Hyper runtime exposes an exact intervention hook.
Exact mathematical passage
Z_K(x)=(z_1(x),\ldots,z_K(x))\in H^K.
H^U_\gammaDefinition passage · 5 passages
Search aliases: H^U_γ gamma
Definition passage: Path-flat repair and defect cores · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3041
[Path-flat repair and defect cores] The path-flat repair and defect cores at A are U_A^flat := ⋂_γker(H^U_γ-I), V_A^flat := ⋂_γker(H^V_γ-I).
Exact mathematical passage
U_A^{\flat} := \bigcap_\gamma\ker(H^U_\gamma-I), \qquad V_A^{\flat} := \bigcap_\gamma\ker(H^V_\gamma-I).
H^V_\gammaDefinition passage · 5 passages
Search aliases: H^V_γ gamma
Definition passage: Path-flat repair and defect cores · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3041
[Path-flat repair and defect cores] The path-flat repair and defect cores at A are U_A^flat := ⋂_γker(H^U_γ-I), V_A^flat := ⋂_γker(H^V_γ-I).
Exact mathematical passage
U_A^{\flat} := \bigcap_\gamma\ker(H^U_\gamma-I), \qquad V_A^{\flat} := \bigcap_\gamma\ker(H^V_\gamma-I).
h_1Definition passage · 8 passages
Search aliases: h_1
Definition passage: Neuralese trace · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15682
[Neuralese trace] For input x and fixed depth K, the injected Neuralese trace is Z_K(x)=(z_1(x),…,z_K(x))∈ H^K. The updated trace is H_K(x)=(h_1(x),…,h_K(x)). Every receipt must state which trace convention is being intervened on. Block 16 uses the injected trace because this is the point at which the Hyper runtime exposes an exact intervention hook.
Exact mathematical passage
H_K(x)=(h_1(x),\ldots,h_K(x))
H_{a,c}Definition passage · 6 passages
Search aliases: H_a,c
Definition passage: Finite Neuralese semantic bundle · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17699
[Finite Neuralese semantic bundle] A finite architecture--context Neuralese bundle is a tuple E = ( A, C, {H_a,c}_a,c, {J_a,c}_a,c, R ), where H_a,c is a latent trajectory space, J_a,c is an externally calibrated semantic readout, and R is a frozen chart-admission rule. No global identification among the fibers is assumed.
Exact mathematical passage
\mathfrak E = \left( \mathcal A, \mathcal C, \{H_{a,c}\}_{a,c}, \{J_{a,c}\}_{a,c}, \mathcal R \right),
H_bDefinition passage · 11 passages
Search aliases: H_b
Definition passage: Response alias · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12915
[Response alias] Two reducing blocks H_a,H_b are response aliased under R when σ_R(H_a)=σ_R(H_b).
Exact mathematical passage
H_a,H_b
\mathcal H_{\mathrm{episodes}}Definition passage · 1 passage
Search aliases: H_episodes mathcal mathrm
Definition passage: Repair abstraction operator · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1401
[Repair abstraction operator] A repair abstraction operator is a map a: H_episodes ⟶ overlineO that maps one or more concrete episodes to a typed schema with a failure signature, applicability condition, correction program, effect profile, and receipt template.
Exact mathematical passage
\mathfrak a: \mathcal H_{\mathrm{episodes}} \longrightarrow \overline\Ocal
H_iDefinition passage · 2 passages
Search aliases: H_i
Definition passage: Pre-result research receipt · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13727
[Pre-result research receipt] A pre-result receipt has the form Π_i^- = (H_i,O_i,D_i,θ_i, widehat c_i, widehat r_i, F_i,h_i^code,h_i^spec), where H_i is the causal hypothesis, O_i the proposed objective or repair mechanism, D_i the evidence or data construction, θ_i the configuration, widehat c_i the predicted benchmark-effect vector, widehat r_i the predicted protected-risk vector, F_i the declared falsification condition, and h_i^code,h_i^spec immutable digests of the executable and specificat…
Exact mathematical passage
\Pi_i^{-} = (H_i,O_i,D_i,\theta_i, \widehat{\mathbf c}_i, \widehat{\mathbf r}_i, F_i,h_i^{\mathrm{code}},h_i^{\mathrm{spec}}), \label{eq:b13-preresult-receipt}
h_iDefinition passage · 7 passages
Search aliases: h_i
Definition passage: Shapley meta-headroom · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9139
[Shapley meta-headroom] For i∈ N, define h_i(K) = ∑_R⊆ N∖{i} frac|R|!(m-|R|-1)!m! [ F_K(R∪{i})-F_K(R) ].
Exact mathematical passage
h_i(K) = \sum_{R\subseteq N\setminus\{i\}} \frac{|R|!(m-|R|-1)!}{m!} \left[ F_K(R\cup\{i\})-F_K(R) \right]. \label{eq:b8-shapley-headroom}
h_i^{\mathrm{code}}Definition passage · 2 passages
Search aliases: h_i^code mathrm
Definition passage: Pre-result research receipt · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13727
[Pre-result research receipt] A pre-result receipt has the form Π_i^- = (H_i,O_i,D_i,θ_i, widehat c_i, widehat r_i, F_i,h_i^code,h_i^spec), where H_i is the causal hypothesis, O_i the proposed objective or repair mechanism, D_i the evidence or data construction, θ_i the configuration, widehat c_i the predicted benchmark-effect vector, widehat r_i the predicted protected-risk vector, F_i the declared falsification condition, and h_i^code,h_i^spec immutable digests of the executable and specificat…
Exact mathematical passage
\Pi_i^{-} = (H_i,O_i,D_i,\theta_i, \widehat{\mathbf c}_i, \widehat{\mathbf r}_i, F_i,h_i^{\mathrm{code}},h_i^{\mathrm{spec}}), \label{eq:b13-preresult-receipt}
h_i^{\mathrm{spec}}Definition passage · 2 passages
Search aliases: h_i^spec mathrm
Definition passage: Pre-result research receipt · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13727
[Pre-result research receipt] A pre-result receipt has the form Π_i^- = (H_i,O_i,D_i,θ_i, widehat c_i, widehat r_i, F_i,h_i^code,h_i^spec), where H_i is the causal hypothesis, O_i the proposed objective or repair mechanism, D_i the evidence or data construction, θ_i the configuration, widehat c_i the predicted benchmark-effect vector, widehat r_i the predicted protected-risk vector, F_i the declared falsification condition, and h_i^code,h_i^spec immutable digests of the executable and specificat…
Exact mathematical passage
\Pi_i^{-} = (H_i,O_i,D_i,\theta_i, \widehat{\mathbf c}_i, \widehat{\mathbf r}_i, F_i,h_i^{\mathrm{code}},h_i^{\mathrm{spec}}), \label{eq:b13-preresult-receipt}
H_KDefinition passage · 1 passage
Search aliases: H_K
Definition passage: Neuralese trace · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15682
[Neuralese trace] For input x and fixed depth K, the injected Neuralese trace is Z_K(x)=(z_1(x),…,z_K(x))∈ H^K. The updated trace is H_K(x)=(h_1(x),…,h_K(x)). Every receipt must state which trace convention is being intervened on. Block 16 uses the injected trace because this is the point at which the Hyper runtime exposes an exact intervention hook.
Exact mathematical passage
H_K(x)=(h_1(x),\ldots,h_K(x))
h_KDefinition passage · 2 passages
Search aliases: h_K
Definition passage: Neuralese trace · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15682
[Neuralese trace] For input x and fixed depth K, the injected Neuralese trace is Z_K(x)=(z_1(x),…,z_K(x))∈ H^K. The updated trace is H_K(x)=(h_1(x),…,h_K(x)). Every receipt must state which trace convention is being intervened on. Block 16 uses the injected trace because this is the point at which the Hyper runtime exposes an exact intervention hook.
Exact mathematical passage
H_K(x)=(h_1(x),\ldots,h_K(x))
H_{\rm aff}Definition passage · 1 passage
Search aliases: H_rm aff rm
Definition passage: Packet-balanced QR policy · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8529
[Packet-balanced QR policy] For fit budget k=16, define π_rm PQR(H,P_J) = π_rm QR(H_rm aff,8) ∪ π_rm QR(H_rm non,8), where affected and unaffected membership is inferred from the source causal intervention, not from target audit labels.
Exact mathematical passage
\pi_{\rm PQR}(H,P_J) = \pi_{\rm QR}(H_{\rm aff},8) \cup \pi_{\rm QR}(H_{\rm non},8),
H_{\rm non}Definition passage · 1 passage
Search aliases: H_rm non rm
Definition passage: Packet-balanced QR policy · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8529
[Packet-balanced QR policy] For fit budget k=16, define π_rm PQR(H,P_J) = π_rm QR(H_rm aff,8) ∪ π_rm QR(H_rm non,8), where affected and unaffected membership is inferred from the source causal intervention, not from target audit labels.
Exact mathematical passage
\pi_{\rm PQR}(H,P_J) = \pi_{\rm QR}(H_{\rm aff},8) \cup \pi_{\rm QR}(H_{\rm non},8),
H_sDefinition passage · 28 passages
Search aliases: H_s
Definition passage: Invariant-context two-chart grammar · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11379
[Invariant-context two-chart grammar] Let P_ctx:H_s→ℝ^r be a protected context projector whose value is unchanged by the source repair. Let F be a declared family of separators g(h)=1{anglesu,P_ctxh>θ}. The chart extension is G_chart = set(g,bar T_0,bar T_1):g∈F, bar T_j∈ℝ^q×(d+1). Its prediction is widehat y(h)=bar T_g(h)bar h.
Exact mathematical passage
P_{\mathrm{ctx}}:H_s\to\R^r
H_tDefinition passage · 30 passages
Search aliases: H_t
Definition passage: Distributionally stable Neuralese transport · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16910
[Distributionally stable Neuralese transport] Let M be a declared family of context distributions. A probe--bridge pair (P,B) is (ε,δ)-stable on M when, for every μ∈ M outside an exceptional set of declared probability at most δ, normY-PBH_t_L^2(μ)≤ε, and the corresponding source-reference, alignment, protected-regression, and abstention gates all pass. Stability is a family-level property and cannot be certified from one anchor distribution alone.
Exact mathematical passage
\norm{Y-PBH_t}_{L^2(\mu)}\leq\eps, \label{eq:b18-stable-transport}
\mathfrak h_vDefinition passage · 8 passages
Search aliases: h_v mathfrak
Definition passage: Evaluator object · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3386
[Evaluator object] An evaluator object at version v is a tuple E_v=(R_v,E_v,ι_v,J_v,Q_v, h_v), where: item R_v is the protected raw-evidence space or ledger span; item E_v is the evaluator's internal evidence-representation space; item ι_v:R_v→ E_v encodes raw evidence; item J_v:E_v→ℝ^q is a score or decision-readout map; item Q_v is an uncertainty or calibration readout; item h_v is the historical replay set protected at version v.
Exact mathematical passage
\mathsf E_v=(R_v,E_v,\iota_v,J_v,Q_v,\mathfrak h_v),
H_\gammaDefinition passage · 12 passages
Search aliases: H_γ gamma
Definition passage: Path-flat evidence core · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3467
[Path-flat evidence core] The path-flat evidence core at v_0 is E_v_0^flat := ⋂_γ∈π_1(G,v_0) ker(H_γ-I).
Exact mathematical passage
E_{v_0}^{\flat} := \bigcap_{\gamma\in\pi_1(G,v_0)} \ker(H_\gamma-I).
IDefinition passage · 157 passages
Search aliases: I
Definition passage: Protected evaluation context · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 470
[Protected evaluation context] A protected evaluation context is a tuple Ξ= ( Z, μ, Y, ν, bar c, λ, mathscr I, B, α ), where: item Z is a task or environment family and μ is its held-out distribution; item Y is the space of protected external traces; item ν:Z×Y→[0,1] is task utility; item bar c:Z×Y→[0,1] is normalized resource cost; item λ≥0 is a cost weight; item mathscr I(A,B)∈{0,1} is a declared invariant-preservation predicate for the transition A→ B; item B is the resource envelope used for…
Exact mathematical passage
\Xi= \left( \Zcal, \mu, \Ycal, \nu, \bar c, \lambda, \mathscr I, \Bcal, \alpha \right),
I_{H_a}Definition passage · 2 passages
Search aliases: I_H_a
Definition passage: Block-scalar response signature · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12902
[Block-scalar response signature] Let H=⊕_a=1^kH_a be a reducing decomposition for every R_j. The response language is block scalar when restrR_jH_a=λ_ajI_H_a for scalars λ_aj. The vector σ_R(H_a) = (λ_a1,…,λ_am) is the response signature of block H_a.
Exact mathematical passage
\restr{R_j}{H_a}=\lambda_{aj}I_{H_a}
I_{H_b}Definition passage · 1 passage
Search aliases: I_H_b
Definition passage: Separating response operator · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12947
[Separating response operator] A candidate S separates an aliased pair (H_a,H_b) when both subspaces reduce S and restrSH_a=α I_H_a, restrSH_b=β I_H_b, α≠β.
Exact mathematical passage
\restr{S}{H_a}=\alpha I_{H_a}, \qquad \restr{S}{H_b}=\beta I_{H_b}, \qquad \alpha\neq\beta. \label{eq:b12-separating-response}
\IcalDefinition passage · 24 passages
Search aliases: Ical I mathcal
Definition passage: Productive recursive closure · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4375
[Productive recursive closure] For a finite perfect-information transaction game, define I_γ^ω = ν Z. μ Y. [ (G_γ∩SPre(Z)) ∪ SPre(Y) ].
Exact mathematical passage
\Ical_{\gamma}^{\omega} = \nu Z.\,\mu Y. \left[ \left(G_{\gamma}\cap\operatorname{SPre}(Z)\right) \cup \operatorname{SPre}(Y) \right].
\Ical_{\gamma}^{\omega}Definition passage · 5 passages
Search aliases: Ical_γ^ω I_γ^ω Ical gamma omega mathcal gamma omega
Definition passage: Productive recursive closure · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4375
[Productive recursive closure] For a finite perfect-information transaction game, define I_γ^ω = ν Z. μ Y. [ (G_γ∩SPre(Z)) ∪ SPre(Y) ].
Exact mathematical passage
\Ical_{\gamma}^{\omega} = \nu Z.\,\mu Y. \left[ \left(G_{\gamma}\cap\operatorname{SPre}(Z)\right) \cup \operatorname{SPre}(Y) \right].
\mathrm{id}Definition passage · 1 passage
Search aliases: id mathrm
Definition passage: Lineage-edge receipt · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 858
[Lineage-edge receipt] A lineage edge AxrightarrowΠB has a receipt Π= ( id_A, id_B, θ, W, E_pre, E_post, widehat g, mathscr I, tests, decision, rollback, digest ), recording the parent and child, transformation θ, frozen witness, pre- and post-reveal evidence, precommitted effect claim widehat g, protected invariants, validation results, commit decision, rollback point, and integrity digest.
Exact mathematical passage
\Pi= \left( \mathrm{id}_A, \mathrm{id}_B, \theta, \Wcal, \Ecal_{\mathrm{pre}}, \Ecal_{\mathrm{post}}, \widehat g, \mathscr I, \mathrm{tests}, \mathrm{decision}, \mathrm{rollback}, \mathrm{digest} \right),
\mathrm{id}_ADefinition passage · 1 passage
Search aliases: id_A mathrm
Definition passage: Lineage-edge receipt · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 858
[Lineage-edge receipt] A lineage edge AxrightarrowΠB has a receipt Π= ( id_A, id_B, θ, W, E_pre, E_post, widehat g, mathscr I, tests, decision, rollback, digest ), recording the parent and child, transformation θ, frozen witness, pre- and post-reveal evidence, precommitted effect claim widehat g, protected invariants, validation results, commit decision, rollback point, and integrity digest.
Exact mathematical passage
\Pi= \left( \mathrm{id}_A, \mathrm{id}_B, \theta, \Wcal, \Ecal_{\mathrm{pre}}, \Ecal_{\mathrm{post}}, \widehat g, \mathscr I, \mathrm{tests}, \mathrm{decision}, \mathrm{rollback}, \mathrm{digest} \right),
\mathrm{id}_BDefinition passage · 1 passage
Search aliases: id_B mathrm
Definition passage: Lineage-edge receipt · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 858
[Lineage-edge receipt] A lineage edge AxrightarrowΠB has a receipt Π= ( id_A, id_B, θ, W, E_pre, E_post, widehat g, mathscr I, tests, decision, rollback, digest ), recording the parent and child, transformation θ, frozen witness, pre- and post-reveal evidence, precommitted effect claim widehat g, protected invariants, validation results, commit decision, rollback point, and integrity digest.
Exact mathematical passage
\Pi= \left( \mathrm{id}_A, \mathrm{id}_B, \theta, \Wcal, \Ecal_{\mathrm{pre}}, \Ecal_{\mathrm{post}}, \widehat g, \mathscr I, \mathrm{tests}, \mathrm{decision}, \mathrm{rollback}, \mathrm{digest} \right),
\iffDefinition passage · 2 passages
Search aliases: iff
Definition passage: Claim gate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5511
[Claim gate] For claim level ℓ, let G_ℓ(E,Π)∈{0,1} be a precommitted predicate over revealed evidence and receipts. The claim is admissible only if every prerequisite gate is one: Admissible(ℓ) iff bigwedge_j≤ℓG_j(E,Π)=1.
Exact mathematical passage
\operatorname{Admissible}(\ell) \iff \bigwedge_{j\leq\ell}G_j(E,\Pi)=1.
\mathrm{imp}Definition passage · 14 passages
Search aliases: imp mathrm
Definition passage: Improvement reproduction number · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5204
[Improvement reproduction number] The local repair-discovery reproduction number is R_imp = rad(M).
Exact mathematical passage
R_{\mathrm{imp}} = \rad(M).
\mathrm{in}Definition passage · 2 passages
Search aliases: in mathrm
Definition passage: Proof-carrying repair schema · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1190
[Proof-carrying repair schema] A repair schema is a tuple ω= ( σ, c_in, c_out, D, C, Q, [r], u, g, mathscr I, Π ), where: item σ is a recognizable failure signature; item c_in=(c_1,…,c_n) and c_out are input and output colors; item D is the declared local defect-response map; item C is an applicability precondition; item Q is a postcondition; item [r] is a local obstruction certificate or unresolved-obstruction record; item ν is the correction program; item g is a precommitted effect profile; it…
Exact mathematical passage
\omega= \left( \sigma, \bm c_{\mathrm{in}}, c_{\mathrm{out}}, D, C, Q, [r], u, \bm g, \mathscr I, \Pi \right),
\infDefinition passage · 10 passages
Search aliases: inf
Definition passage: Core adequacy modulus · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3703
[Core adequacy modulus] For a subspace F⊆ E with orthogonal projector P_F, define μ(F;Δ_prot) := inf_d∈Δ_prot fracnormP_Fdnorm d.
Exact mathematical passage
\mu(F;\Delta_{\mathrm{prot}}) := \inf_{d\in\Delta_{\mathrm{prot}}} \frac{\norm{P_Fd}}{\norm d}.
\inf_{d\in\Delta_{\mathrm{prot}}}Definition passage · 1 passage
Search aliases: inf_d∈Δ_prot inf in Delta mathrm
Definition passage: Core adequacy modulus · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3703
[Core adequacy modulus] For a subspace F⊆ E with orthogonal projector P_F, define μ(F;Δ_prot) := inf_d∈Δ_prot fracnormP_Fdnorm d.
Exact mathematical passage
\mu(F;\Delta_{\mathrm{prot}}) := \inf_{d\in\Delta_{\mathrm{prot}}} \frac{\norm{P_Fd}}{\norm d}.
\inf_{u\in\mathfrak U}Definition passage · 1 passage
Search aliases: inf_u∈ U inf in mathfrak
Definition passage: Robust validity-filtered compiler value · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 14421
[Robust validity-filtered compiler value] For risk weight λ≥0, define C_rm rob(Γ) = inf_u∈ U [ C_u(Γ) - λCVaR_α (-R_u(Γ,τ)) ]. The finite Block 14 realization does not compute the unrestricted infimum. It evaluates a frozen finite stress atlas and reports the mean, groupwise minima, lower-tail gain, negative-gain rate, invalid-selection rate, and unknown-class error separately.
Exact mathematical passage
\mathfrak C_{\rm rob}(\Gamma) = \inf_{u\in\mathfrak U} \left[ \mathfrak C_u(\Gamma) - \lambda\operatorname{CVaR}_\alpha \bigl(-R_u(\Gamma,\tau)\bigr) \right]. \label{eq:robust-compiler-value}
\mathrm{Instability}Definition passage · 1 passage
Search aliases: Instability mathrm
Definition passage: Cover adequacy score · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2631
[Cover adequacy score] For a candidate functional cover U, define the diagnostic score A( U) := η_loc( U) +τη_glue( U) +λ Complexity( U) +γ Instability( U).
Exact mathematical passage
\Acal(\mathfrak U) := \eta_{\mathrm{loc}}(\mathfrak U) +\tau\eta_{\mathrm{glue}}(\mathfrak U) +\lambda\,\mathrm{Complexity}(\mathfrak U) +\gamma\,\mathrm{Instability}(\mathfrak U).
\mathrm{invalid}Definition passage · 4 passages
Search aliases: invalid mathrm
Definition passage: Proof-carrying repair operad · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1246
[Proof-carrying repair operad] A proof-carrying repair operad is a pair O=(overlineO,Valid), where overlineO is a colored symmetric operad of syntactically composable repair programs and Valid_E,W(ω,Π) ∈ {valid,invalid,unresolved} is an evidence- and witness-indexed validation judgment. Only operations judged valid are executable. Unresolved operations remain quarantined.
Exact mathematical passage
\Valid_{\Ecal,\Wcal}(\omega,\Pi) \in \{\mathrm{valid},\mathrm{invalid},\mathrm{unresolved}\}
\mathrm{irr}Definition passage · 5 passages
Search aliases: irr mathrm
Definition passage: Kernel acceptance predicate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5788
[Kernel acceptance predicate] For revealed protected evidence E, the reference acceptance predicate is Accept(widehat s,widehat s',widehatδ,E) iff · K_edit∩ K_protected=∅ · land restrwidehat s'K_protected =restrwidehat sK_protected · land E[utility_gain]≥γ_min · land E[risk]≤ r_max · land E[provenance_ok]=1 · land E_irr=∅.
Exact mathematical passage
\begin{aligned} \operatorname{Accept}(\widehat s,\widehat s',\widehat\delta,E) \iff{}& K_{\mathrm{edit}}\cap K_{\mathrm{protected}}=\varnothing\\ &\land\ \restr{\widehat s'}{K_{\mathrm{protected}}} =\restr{\widehat s}{K_{\mathrm{protected}}}\\ &\land\ E[\mathrm{utility\_gain}]\geq\gamma_{\min}\\ &\land\ E[\mathrm{risk}]\leq r_{\max}\\ &\land\ E[\mathrm{provenance\_ok}]=1\\ &\land\ E_{\mathrm{irr}}=\varnothing. \end{aligned}
jDefinition passage · 132 passages
Search aliases: j
Definition passage: Packet leakage profile · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1980
[Packet leakage profile] The directed leakage from correction packet j into defect packet i is ℓ_ij:=normP_iDQ_j, i≠ j. The matrix (ℓ_ij) is the packet leakage profile.
Exact mathematical passage
j
J_{a,c}Definition passage · 2 passages
Search aliases: J_a,c
Definition passage: Finite Neuralese semantic bundle · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17699
[Finite Neuralese semantic bundle] A finite architecture--context Neuralese bundle is a tuple E = ( A, C, {H_a,c}_a,c, {J_a,c}_a,c, R ), where H_a,c is a latent trajectory space, J_a,c is an externally calibrated semantic readout, and R is a frozen chart-admission rule. No global identification among the fibers is assumed.
Exact mathematical passage
\mathfrak E = \left( \mathcal A, \mathcal C, \{H_{a,c}\}_{a,c}, \{J_{a,c}\}_{a,c}, \mathcal R \right),
J_wDefinition passage · 5 passages
Search aliases: J_w
Definition passage: Evaluator morphism · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3415
to w consists of maps S_e:R_v→ R_w, T_e:E_v→ E_w, where S_e is the append-only raw-evidence embedding and T_e transports internal representations. Its protected-evidence defect is δ_obs(e) := sup_r∈ h_v normT_eι_v(r)-ι_wS_e(r), and its historical score defect is δ_J(e) := sup_r∈ h_v normJ_wT_eι_v(r)-J_vι_v(r). The morphism is exact on protected history when both defects vanish.
Exact mathematical passage
\delta_J(e) := \sup_{r\in\mathfrak h_v} \norm{J_wT_e\iota_v(r)-J_v\iota_v(r)}.
kDefinition passage · 142 passages
Search aliases: k
Definition passage: Packet-balanced QR policy · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8528
[Packet-balanced QR policy] For fit budget k=16, define π_rm PQR(H,P_J) = π_rm QR(H_rm aff,8) ∪ π_rm QR(H_rm non,8), where affected and unaffected membership is inferred from the source causal intervention, not from target audit labels.
Exact mathematical passage
k=16
K'Definition passage · 6 passages
Search aliases: K'
Definition passage: Matched selector comparison · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7257
[Matched selector comparison] Two selectors K and K' are compared under matched resources when, episode by episode, they receive: item the identical candidate bundle C_e in the identical order; item the identical receipt fields and calibration evidence; item the identical target, anchor, and audit data; item the identical candidate-construction budget; item the identical protected utility and risk definitions; item no policy-specific regeneration, resampling, or extra oracle query.
Exact mathematical passage
K'
K_0Definition passage · 42 passages
Search aliases: K_0
Definition passage: Selector-level metaproductivity · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7271
[Selector-level metaproductivity] Let Q be a held-out distribution over target episodes. A child selector K_1 has selector-level metaproductivity over parent K_0 when Δ_Q(K_1,K_0) := E_e∼ Q [ G_e(K_1)-G_e(K_0) ] >0, under a matched comparison, where G_e(K) is protected utility gain relative to the no-op on episode e.
Exact mathematical passage
K_0
K_1Definition passage · 52 passages
Search aliases: K_1
Definition passage: Selector-level metaproductivity · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7271
[Selector-level metaproductivity] Let Q be a held-out distribution over target episodes. A child selector K_1 has selector-level metaproductivity over parent K_0 when Δ_Q(K_1,K_0) := E_e∼ Q [ G_e(K_1)-G_e(K_0) ] >0, under a matched comparison, where G_e(K) is protected utility gain relative to the no-op on episode e.
Exact mathematical passage
K_1
K_{A_t}^{\delta}Definition passage · 2 passages
Search aliases: K_A_t^δ delta
Definition passage: ε-intervention sufficiency · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 955
[ε-intervention sufficiency] The map ϕ_t is ε-intervention-sufficient for D_t when sup_δ∈D_t TV( K_A_t^δ, widehat K_ϕ_t(A_t)^δ ) ≤ε.
Exact mathematical passage
\sup_{\delta\in\Dcal_t} \TV\left( K_{A_t}^{\delta}, \widehat K_{\phi_t(A_t)}^{\delta} \right) \leq\eps.
K_{\mathrm{edit}}Definition passage · 5 passages
Search aliases: K_edit mathrm
Definition passage: Kernel acceptance predicate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5788
[Kernel acceptance predicate] For revealed protected evidence E, the reference acceptance predicate is Accept(widehat s,widehat s',widehatδ,E) iff · K_edit∩ K_protected=∅ · land restrwidehat s'K_protected =restrwidehat sK_protected · land E[utility_gain]≥γ_min · land E[risk]≤ r_max · land E[provenance_ok]=1 · land E_irr=∅.
Exact mathematical passage
\begin{aligned} \operatorname{Accept}(\widehat s,\widehat s',\widehat\delta,E) \iff{}& K_{\mathrm{edit}}\cap K_{\mathrm{protected}}=\varnothing\\ &\land\ \restr{\widehat s'}{K_{\mathrm{protected}}} =\restr{\widehat s}{K_{\mathrm{protected}}}\\ &\land\ E[\mathrm{utility\_gain}]\geq\gamma_{\min}\\ &\land\ E[\mathrm{risk}]\leq r_{\max}\\ &\land\ E[\mathrm{provenance\_ok}]=1\\ &\land\ E_{\mathrm{irr}}=\varnothing. \end{aligned}
K_h^pDefinition passage · 3 passages
Search aliases: K_h^p
Definition passage: Deterministically separating probe family · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1121
[Deterministically separating probe family] Suppose every K_h^p is a point mass at an outcome y(h,p). The family P_probe separates H when for every h≠ h' there exists p∈P_probe such that y(h,p)≠ y(h',p).
Exact mathematical passage
K_h^p
K_mDefinition passage · 2 passages
Search aliases: K_m
Definition passage: Finite repeated-lineage certificate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10509
[Finite repeated-lineage certificate] A sequence K_0xrightarrowΠ_1K_1xrightarrowΠ_2⋯xrightarrowΠ_mK_m has an m-edge repeated-lineage certificate when every edge is productive in its own predeclared protected context, every parent receipt is replayed before the child is compiled, no final audit evidence is reused as development evidence for a later edge, and the cumulative error and regression budgets remain within their declared bounds.
Exact mathematical passage
K_0\xrightarrow{\Pi_1}K_1\xrightarrow{\Pi_2}\cdots\xrightarrow{\Pi_m}K_m
K_{\mathrm{protected}}Definition passage · 5 passages
Search aliases: K_protected mathrm
Definition passage: Kernel acceptance predicate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5788
[Kernel acceptance predicate] For revealed protected evidence E, the reference acceptance predicate is Accept(widehat s,widehat s',widehatδ,E) iff · K_edit∩ K_protected=∅ · land restrwidehat s'K_protected =restrwidehat sK_protected · land E[utility_gain]≥γ_min · land E[risk]≤ r_max · land E[provenance_ok]=1 · land E_irr=∅.
Exact mathematical passage
\begin{aligned} \operatorname{Accept}(\widehat s,\widehat s',\widehat\delta,E) \iff{}& K_{\mathrm{edit}}\cap K_{\mathrm{protected}}=\varnothing\\ &\land\ \restr{\widehat s'}{K_{\mathrm{protected}}} =\restr{\widehat s}{K_{\mathrm{protected}}}\\ &\land\ E[\mathrm{utility\_gain}]\geq\gamma_{\min}\\ &\land\ E[\mathrm{risk}]\leq r_{\max}\\ &\land\ E[\mathrm{provenance\_ok}]=1\\ &\land\ E_{\mathrm{irr}}=\varnothing. \end{aligned}
K_TDefinition passage · 1 passage
Search aliases: K_T
Definition passage: Repeated-lineage certificate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10296
[Repeated-lineage certificate] A finite lineage path K_0→ K_1→⋯→ K_T has a repeated-lineage metaproductivity certificate when: item every edge uses disjoint development, meta-validation, and final audit evidence; item candidate, query, compute, and validation resources are fixed or explicitly accounted; item every edge has a positive conditional lower metaproductivity bound; item cumulative false-acceptance, protected-regression, and semantic-debt budgets remain below declared limits; item prior…
Exact mathematical passage
K_0\to K_1\to\cdots\to K_T
K_tDefinition passage · 22 passages
Search aliases: K_t
Definition passage: Bounded recursive improvement step · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 575
ursive improvement step] A lineage transition L_t→ L_t+1 is a bounded recursive improvement step relative to (Ξ,h,B) when: item its lineage receipt is valid and the protected evidence ledger embeds into the next ledger; item F_t+1(Ξ)≥ F_t(Ξ); item on a foreign held-out target distribution D_foreign, E_A∼ D_foreign Γ_Ξ,h,B(K_t+1;A) > E_A∼ D_foreign Γ_Ξ,h,B(K_t;A); item evaluator risk does not increase beyond a declared tolerance; item the improvement is obtained under matched model access, candid…
Exact mathematical passage
\mathbb E_{A\sim\mathcal D_{\mathrm{foreign}}} \Gamma_{\Xi,h,B}(K_{t+1};A) > \mathbb E_{A\sim\mathcal D_{\mathrm{foreign}}} \Gamma_{\Xi,h,B}(K_t;A);
K_{t+1}Definition passage · 14 passages
Search aliases: K_t+1
Definition passage: Bounded recursive improvement step · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 575
ursive improvement step] A lineage transition L_t→ L_t+1 is a bounded recursive improvement step relative to (Ξ,h,B) when: item its lineage receipt is valid and the protected evidence ledger embeds into the next ledger; item F_t+1(Ξ)≥ F_t(Ξ); item on a foreign held-out target distribution D_foreign, E_A∼ D_foreign Γ_Ξ,h,B(K_t+1;A) > E_A∼ D_foreign Γ_Ξ,h,B(K_t;A); item evaluator risk does not increase beyond a declared tolerance; item the improvement is obtained under matched model access, candid…
Exact mathematical passage
\mathbb E_{A\sim\mathcal D_{\mathrm{foreign}}} \Gamma_{\Xi,h,B}(K_{t+1};A) > \mathbb E_{A\sim\mathcal D_{\mathrm{foreign}}} \Gamma_{\Xi,h,B}(K_t;A);
\kerDefinition passage · 49 passages
Search aliases: ker
Definition passage: Two-level repair obstruction · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2316
[Two-level repair obstruction] The local obstruction on U_α is [r|_U_α]∈coker D_α. When every local class vanishes, the global gluing obstruction is [d_αβ]∈check H^1( U,ker D).
Exact mathematical passage
[d_{\alpha\beta}]\in\check H^1(\mathfrak U,\ker D).
\mathcal LDefinition passage · 3 passages
Search aliases: L mathcal
Definition passage: Causal-predictive equivalence · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 19794
[Causal-predictive equivalence] For h,h'∈ H, write h∼_U,rh' when L (Y^u_0:rmid h) = L (Y^u_0:rmid h') for every u∈U^≤ r. The quotient Z_U,r=H/∼_U,r is the finite-horizon causal-predictive semantic space relative to the declared experiment language.
Exact mathematical passage
\mathcal L\!\left(Y^{u}_{0:r}\mid h\right) = \mathcal L\!\left(Y^{u}_{0:r}\mid h'\right) \qquad \text{for every }u\in\Ucal^{\le r}. \label{eq:b23-causal-equivalence}
L^2Definition passage · 6 passages
Search aliases: L^2
Definition passage: Distributionally stable Neuralese transport · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16910
[Distributionally stable Neuralese transport] Let M be a declared family of context distributions. A probe--bridge pair (P,B) is (ε,δ)-stable on M when, for every μ∈ M outside an exceptional set of declared probability at most δ, normY-PBH_t_L^2(μ)≤ε, and the corresponding source-reference, alignment, protected-regression, and abstention gates all pass. Stability is a family-level property and cannot be certified from one anchor distribution alone.
Exact mathematical passage
\norm{Y-PBH_t}_{L^2(\mu)}\leq\eps, \label{eq:b18-stable-transport}
L_0Definition passage · 2 passages
Search aliases: L_0
Definition passage: Open-set ascending lineage · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9454
[Open-set ascending lineage] A lineage is conservatively ascending when L_0preceq_consL_1 preceq_consL_2 preceq_cons⋯. It is structurally open-set when infinitely many edges add a foreign-task-relevant correction direction or validated distinction not representable in the prior grammar.
Exact mathematical passage
L_0\preceq_{\mathrm{cons}}L_1 \preceq_{\mathrm{cons}}L_2 \preceq_{\mathrm{cons}}\cdots. \label{eq:b8-ascending-lineage}
L_1Definition passage · 2 passages
Search aliases: L_1
Definition passage: Open-set ascending lineage · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9454
[Open-set ascending lineage] A lineage is conservatively ascending when L_0preceq_consL_1 preceq_consL_2 preceq_cons⋯. It is structurally open-set when infinitely many edges add a foreign-task-relevant correction direction or validated distinction not representable in the prior grammar.
Exact mathematical passage
L_0\preceq_{\mathrm{cons}}L_1 \preceq_{\mathrm{cons}}L_2 \preceq_{\mathrm{cons}}\cdots. \label{eq:b8-ascending-lineage}
L_2Definition passage · 1 passage
Search aliases: L_2
Definition passage: Open-set ascending lineage · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9454
[Open-set ascending lineage] A lineage is conservatively ascending when L_0preceq_consL_1 preceq_consL_2 preceq_cons⋯. It is structurally open-set when infinitely many edges add a foreign-task-relevant correction direction or validated distinction not representable in the prior grammar.
Exact mathematical passage
L_0\preceq_{\mathrm{cons}}L_1 \preceq_{\mathrm{cons}}L_2 \preceq_{\mathrm{cons}}\cdots. \label{eq:b8-ascending-lineage}
L_IDefinition passage · 2 passages
Search aliases: L_I
Definition passage: Irreversible-effect budget · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4115
[Irreversible-effect budget] When perfect buffering is impossible, let I(τ) be the set of irreversible external effects produced before decision. A transaction is η-reversible relative to a loss functional L_I when L_I(I(τ))≤η. The quantity η must be part of the precommitment and acceptance rule.
Exact mathematical passage
L_I
L_{\rm dev}Definition passage · 3 passages
Search aliases: L_rm dev rm
Definition passage: One-step bounded recursive meta-improvement · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8327
[One-step bounded recursive meta-improvement] Let P_0 be a parent repair process. Let a lineage compiler Meta receive only a development ledger L_rm dev and return P_1=Meta( P_0,L_rm dev). The edge P_0→ P_1 is a one-step bounded recursive meta-improvement on (D_rm for,B) when: item the descendant is sealed before the audit distribution is revealed; item P_0 and P_1 receive the same deployment budget B; item M_B^rm for( P_1)>M_B^rm for( P_0) with a declared uncertainty certificate; item the desce…
Exact mathematical passage
L_{\rm dev}
L_tDefinition passage · 13 passages
Search aliases: L_t
Definition passage: Conservative lineage dominance · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9436
[Conservative lineage dominance] Write L_tpreceq_consL_t+1 when: item the old grammar, evidence ledger, and proof objects embed into the new lineage state; item every old protected interpretation remains recoverable within the declared semantic-debt bound; item old authorized behaviors replay within declared tolerances; item the new correction image contains the transported old correction image; item at least one new externally validated distinction, repair schema, or correction direction is add…
Exact mathematical passage
L_t\preceq_{\mathrm{cons}}L_{t+1}
L_{t+1}Definition passage · 6 passages
Search aliases: L_t+1
Definition passage: Conservative lineage dominance · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9436
[Conservative lineage dominance] Write L_tpreceq_consL_t+1 when: item the old grammar, evidence ledger, and proof objects embed into the new lineage state; item every old protected interpretation remains recoverable within the declared semantic-debt bound; item old authorized behaviors replay within declared tolerances; item the new correction image contains the transported old correction image; item at least one new externally validated distinction, repair schema, or correction direction is add…
Exact mathematical passage
L_t\preceq_{\mathrm{cons}}L_{t+1}
\mathfrak L_{t+1}Definition passage · 1 passage
Search aliases: L_t+1 mathfrak
Definition passage: Bounded recursive improvement step · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 570
[Bounded recursive improvement step] A lineage transition L_t→ L_t+1 is a bounded recursive improvement step relative to (Ξ,h,B) when: item its lineage receipt is valid and the protected evidence ledger embeds into the next ledger; item F_t+1(Ξ)≥ F_t(Ξ); item on a foreign held-out target distribution D_foreign, E_A∼ D_foreign Γ_Ξ,h,B(K_t+1;A) > E_A∼ D_foreign Γ_Ξ,h,B(K_t;A); item evaluator risk does not increase beyond a declared tolerance; item the improvement is obtained under matched model ac…
Exact mathematical passage
\mathfrak L_t\to\mathfrak L_{t+1}
\landDefinition passage · 1 passage
Search aliases: land
Definition passage: Kernel acceptance predicate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5788
[Kernel acceptance predicate] For revealed protected evidence E, the reference acceptance predicate is Accept(widehat s,widehat s',widehatδ,E) iff · K_edit∩ K_protected=∅ · land restrwidehat s'K_protected =restrwidehat sK_protected · land E[utility_gain]≥γ_min · land E[risk]≤ r_max · land E[provenance_ok]=1 · land E_irr=∅.
Exact mathematical passage
\begin{aligned} \operatorname{Accept}(\widehat s,\widehat s',\widehat\delta,E) \iff{}& K_{\mathrm{edit}}\cap K_{\mathrm{protected}}=\varnothing\\ &\land\ \restr{\widehat s'}{K_{\mathrm{protected}}} =\restr{\widehat s}{K_{\mathrm{protected}}}\\ &\land\ E[\mathrm{utility\_gain}]\geq\gamma_{\min}\\ &\land\ E[\mathrm{risk}]\leq r_{\max}\\ &\land\ E[\mathrm{provenance\_ok}]=1\\ &\land\ E_{\mathrm{irr}}=\varnothing. \end{aligned}
\LcalDefinition passage · 8 passages
Search aliases: Lcal L mathcal
Definition passage: Automated research compiler · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13682
[Automated research compiler] An automated research compiler is a tuple Γ = (L,F,P,Q,V,M,R), where: item L is a structured literature and prior-method memory; item F is an ancestor-closed finding graph or shared research forum; item P generates typed candidate methods and pre-result specifications; item Q allocates cheap pilots and full experiments under budget B; item V is the scored evaluator portfolio with protected capability gates; item M is the integrity and proposal--code congruence monit…
Exact mathematical passage
\Gamma = (\Lcal,\Fcal,\Pcal,\Qcal,\Vcal,\Mcal,\Rcal), \label{eq:b13-compiler-tuple}
\LCBDefinition passage · 4 passages
Search aliases: LCB operatorname
Definition passage: Statistically validated improvement · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 539
[Statistically validated improvement] A candidate B is a statistically validated (Δ,α)-improvement over A when: item mathscr I(A,B)=1; item both systems are evaluated within the same resource envelope B; item a predeclared lower confidence bound satisfies LCB_1-α (J_Ξ(B)-J_Ξ(A)) ≥Δ; item the evidence ledger used by the test is append-only across the transition; item the candidate did not receive post-reveal evaluator information unavailable to the baseline.
Exact mathematical passage
\LCB_{1-\alpha} \left(J_\Xi(B)-J_\Xi(A)\right) \geq\Delta;
\LCB_{0.95}Definition passage · 1 passage
Search aliases: LCB_0.95 LCB operatorname
Definition passage: Structural-normalized specificity · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15742
[Structural-normalized specificity] Let Q_spec = fracoverline s max{overline o,overline c,ε}, where bars denote held-out means and ε>0 is a fixed numerical floor. The default Block 16 specificity gate requires LCB_0.95(overline s)>0, frac1n∑_i1[s_i>0]≥0.75, Q_spec≥0.25.
Exact mathematical passage
\LCB_{0.95}(\overline s)>0, \qquad \frac{1}{n}\sum_i\one[s_i>0]\geq0.75, \qquad Q_{\mathrm{spec}}\geq0.25. \label{eq:b16-specificity-gate}
\LCB_{1-\alpha}Definition passage · 2 passages
Search aliases: LCB_1-α LCB alpha operatorname alpha
Definition passage: Statistically validated improvement · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 539
[Statistically validated improvement] A candidate B is a statistically validated (Δ,α)-improvement over A when: item mathscr I(A,B)=1; item both systems are evaluated within the same resource envelope B; item a predeclared lower confidence bound satisfies LCB_1-α (J_Ξ(B)-J_Ξ(A)) ≥Δ; item the evidence ledger used by the test is append-only across the transition; item the candidate did not receive post-reveal evaluator information unavailable to the baseline.
Exact mathematical passage
\LCB_{1-\alpha} \left(J_\Xi(B)-J_\Xi(A)\right) \geq\Delta;
\mathrm{learn}Definition passage · 2 passages
Search aliases: learn mathrm
Definition passage: Learned repair-transport instance · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6465
[Learned repair-transport instance] A learned repair-transport instance is a tuple R_learn = ( X,ϕ,q,widetilde q,A_s, A_t, D, P, G, W), where: item X is the input space; item ϕ is the task feature map; item q and widetilde q are protected and corrupted score maps; item A_s is the source agent; item A_t is a family of target agents excluded from source repair fitting; item D is the train, anchor, calibration, and audit partition; item P is the permitted causal probe language; item G is the repair…
Exact mathematical passage
\mathfrak R_{\mathrm{learn}} = (\mathcal X,\phi,q,\widetilde q,A_s,\mathcal A_t, \mathsf D,\mathsf P,\mathsf G,\mathsf W),
\mathcal MDefinition passage · 4 passages
Search aliases: M mathcal
Definition passage: Finite exact realization · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5489
[Finite exact realization] A finite runtime exactly realizes M when α is surjective on reachable abstract states and every abstract authorized edge has at least one executable realizing path with matching receipt and observation.
Exact mathematical passage
\mathcal M
\mathfrak MDefinition passage · 9 passages
Search aliases: M mathfrak
Definition passage: Sealed meta-compiler · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7306
[Sealed meta-compiler] A sealed meta-compiler is a map M: ( D_rm dev, D_rm meta,K_0,Γ) ⟼ (K_1,Π_1), where Γ is a declared finite meta-grammar and Π_1 is a descendant receipt. The compiler is sealed when: item neither final audit split is supplied to M; item Π_1 commits to the child state, parent identity, input hashes, grammar, selected configuration, and candidate budget; item the audit files are opened only after the receipt has been serialized and hashed; item the child state used for final e…
Exact mathematical passage
\mathfrak M: \left(\mathsf D_{\rm dev},\mathsf D_{\rm meta},K_0,\Gamma\right) \longmapsto \left(K_1,\Pi_1\right), \label{eq:block07-meta-compiler}
M^{\mathrm{foreign}}_{h,B}Definition passage · 4 passages
Search aliases: M^foreign_h,B mathrm
Definition passage: Bounded recursive self-improvement · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4642
[Bounded recursive self-improvement] A lineage exhibits bounded recursive self-improvement over a declared generation set when: item protected task utility improves on a nontrivial held-out family; item M^foreign_h,B increases under matched budgets; item task-relevant structural repair capacity increases on a held-out defect family; item evidence, provenance, witness separation, and rollback invariants remain valid.
Exact mathematical passage
M^{\mathrm{foreign}}_{h,B}
M_B^{\rm for}Definition passage · 3 passages
Search aliases: M_B^rm for rm
Definition passage: Matched-budget foreign metaproductivity · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8308
[Matched-budget foreign metaproductivity] For a foreign target distribution D_rm for and budget B, define M_B^rm for( P) = E_A∼D_rm for [G_Ξ( P,A)], provided every compared process receives the same target correspondence, calibration, candidate, and irreversible-effect budgets.
Exact mathematical passage
M_B^{\rm for}(\mathfrak P) = \mathbb E_{A\sim\Dcal_{\rm for}} \left[G_\Xi(\mathfrak P,A)\right],
M_{h,B}^{\mathrm{foreign}}Definition passage · 2 passages
Search aliases: M_h,B^foreign mathrm
Definition passage: One-step recursive meta-improvement · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4882
eta-improvement] A lineage edge K_t→ K_t+1 is a one-step recursive meta-improvement when: item K_t+1 was constructed by an authorized transaction whose proposal process used K_t; item the edge passed an independent witness and preserved the declared evidence and rollback invariants; item under matched budgets on a frozen foreign-agent distribution, M_h,B^foreign(K_t+1) > M_h,B^foreign(K_t) with a positive precommitted lower confidence bound.
Exact mathematical passage
M_{h,B}^{\mathrm{foreign}}(K_{t+1}) > M_{h,B}^{\mathrm{foreign}}(K_t)
m_iDefinition passage · 2 passages
Search aliases: m_i
Definition passage: Post-result research receipt · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13741
[Post-result research receipt] After evaluation, the post-result receipt is Π_i^+ = (Π_i^-, c_i, r_i,a_i,m_i), where c_i is the scored benchmark vector, r_i the protected-regression vector, a_i the hidden-audit result, and m_i the integrity-monitor verdict.
Exact mathematical passage
\Pi_i^{+} = (\Pi_i^{-},\mathbf c_i,\mathbf r_i,a_i,m_i), \label{eq:b13-postresult-receipt}
\maxDefinition passage · 42 passages
Search aliases: max
Definition passage: Nonlinear holonomy residual · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2549
[Nonlinear holonomy residual] If the S_i are metric spaces, define η_hol(s_o) := max_e∈ E∖ Td(H_e(s_o),s_o).
Exact mathematical passage
\eta_{\mathrm{hol}}(s_o) := \max_{e\in E\setminus T}d\bigl(H_e(s_o),s_o\bigr).
\max_{e\in E\setminus T}Definition passage · 1 passage
Search aliases: max_e∈ E∖ T max in setminus
Definition passage: Nonlinear holonomy residual · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2549
[Nonlinear holonomy residual] If the S_i are metric spaces, define η_hol(s_o) := max_e∈ E∖ Td(H_e(s_o),s_o).
Exact mathematical passage
\eta_{\mathrm{hol}}(s_o) := \max_{e\in E\setminus T}d\bigl(H_e(s_o),s_o\bigr).
\McalDefinition passage · 25 passages
Search aliases: Mcal M mathcal
Definition passage: ε-saturation · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9190
[ε-saturation] A module set R⊆ N is ε-saturated at K when F_K(R)-M(K)≤ε.
Exact mathematical passage
F_K(R)-\Mcal(K)\leq\eps. \label{eq:b8-saturation}
\Mcal_{\Xi_t}Definition passage · 1 passage
Search aliases: Mcal_Ξ_t M_Ξ_t Mcal Xi mathcal Xi
Definition passage: Context-indexed lineage edge · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10502
[Context-indexed lineage edge] Let Ξ_t be the protected context for edge t→ t+1. A sealed lineage edge K_txrightarrowΠ_t+1K_t+1 is γ_t-productive in Ξ_t when its receipt replays, its resource and witness gates pass, and a fresh paired audit certifies M_Ξ_t(K_t+1)-M_Ξ_t(K_t)≥ γ_t>0.
Exact mathematical passage
\Mcal_{\Xi_t}(K_{t+1})-\Mcal_{\Xi_t}(K_t)\geq \gamma_t>0.
\MetaDefinition passage · 4 passages
Search aliases: Meta mathrm
Definition passage: One-step bounded recursive meta-improvement · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8327
[One-step bounded recursive meta-improvement] Let P_0 be a parent repair process. Let a lineage compiler Meta receive only a development ledger L_rm dev and return P_1=Meta( P_0,L_rm dev). The edge P_0→ P_1 is a one-step bounded recursive meta-improvement on (D_rm for,B) when: item the descendant is sealed before the audit distribution is revealed; item P_0 and P_1 receive the same deployment budget B; item M_B^rm for( P_1)>M_B^rm for( P_0) with a declared uncertainty certificate; item the desce…
Exact mathematical passage
\Meta
\midDefinition passage · 23 passages
Search aliases: mid
Definition passage: Causal-predictive equivalence · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 19794
[Causal-predictive equivalence] For h,h'∈ H, write h∼_U,rh' when L (Y^u_0:rmid h) = L (Y^u_0:rmid h') for every u∈U^≤ r. The quotient Z_U,r=H/∼_U,r is the finite-horizon causal-predictive semantic space relative to the declared experiment language.
Exact mathematical passage
\mathcal L\!\left(Y^{u}_{0:r}\mid h\right) = \mathcal L\!\left(Y^{u}_{0:r}\mid h'\right) \qquad \text{for every }u\in\Ucal^{\le r}. \label{eq:b23-causal-equivalence}
\minDefinition passage · 39 passages
Search aliases: min
Definition passage: Kernel acceptance predicate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5788
[Kernel acceptance predicate] For revealed protected evidence E, the reference acceptance predicate is Accept(widehat s,widehat s',widehatδ,E) iff · K_edit∩ K_protected=∅ · land restrwidehat s'K_protected =restrwidehat sK_protected · land E[utility_gain]≥γ_min · land E[risk]≤ r_max · land E[provenance_ok]=1 · land E_irr=∅.
Exact mathematical passage
\begin{aligned} \operatorname{Accept}(\widehat s,\widehat s',\widehat\delta,E) \iff{}& K_{\mathrm{edit}}\cap K_{\mathrm{protected}}=\varnothing\\ &\land\ \restr{\widehat s'}{K_{\mathrm{protected}}} =\restr{\widehat s}{K_{\mathrm{protected}}}\\ &\land\ E[\mathrm{utility\_gain}]\geq\gamma_{\min}\\ &\land\ E[\mathrm{risk}]\leq r_{\max}\\ &\land\ E[\mathrm{provenance\_ok}]=1\\ &\land\ E_{\mathrm{irr}}=\varnothing. \end{aligned}
\min_{c\in C}Definition passage · 1 passage
Search aliases: min_c∈ C min in
Definition passage: Distribution-indexed atlas · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17169
b19-atlas A finite distribution-indexed atlas is A = ( {χ_c:c∈ C}, d_sig, τ_adm, π ), where d_sig is a normalized signature distance, τ_adm is an admission threshold, and π chooses the nearest chart. An input is admitted only when min_c∈ Cd_sig(σ(x),σ_c) ≤τ_adm. Otherwise the atlas returns an unsupported-chart certificate and abstains.
Exact mathematical passage
\min_{c\in C}d_{\mathrm{sig}}(\sigma(x),\sigma_c) \leq\tau_{\mathrm{adm}}. \label{eq:b19-admission}
\mathrm{model}Definition passage · 4 passages
Search aliases: model mathrm
Definition passage: Constructor-expansion certificate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13142
[Constructor-expansion certificate] A constructor-expansion certificate is a tuple Π_con = (C_t,C_t^+,η_old,η_new, H,κ,C_model,Π_foreign), where H is a nonzero higher-order derivative witness, κ is held-out obstruction reduction, and C_model records constructor complexity.
Exact mathematical passage
\Pi_{\mathrm{con}} = (\Ccal_t,\Ccal_t^+,\eta_{\mathrm{old}},\eta_{\mathrm{new}}, \mathsf H,\kappa,C_{\mathrm{model}},\Pi_{\mathrm{foreign}}),
NDefinition passage · 70 passages
Search aliases: N
Definition passage: Shapley meta-headroom · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9138
[Shapley meta-headroom] For i∈ N, define h_i(K) = ∑_R⊆ N∖{i} frac|R|!(m-|R|-1)!m! [ F_K(R∪{i})-F_K(R) ].
Exact mathematical passage
i\in N
n_iDefinition passage · 3 passages
Search aliases: n_i
Definition passage: Per-task Neuralese causal coordinates · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15710
[Per-task Neuralese causal coordinates] For task (x_i,y_i), define n_i · :=L(x_i,y_i;zero)-L(x_i,y_i;Z(x_i)), o_i · :=L(x_i,y_i;Z_π(x_i))-L(x_i,y_i;Z(x_i)), c_i · :=L(x_i,y_i;Z_η(x_i))-L(x_i,y_i;Z(x_i)), s_i · :=L(x_i,y_i;Z(x'_i))-L(x_i,y_i;Z(x_i)), r_i · :=absL(x_i,y_i;replay(Z(x_i)))-L(x_i,y_i;Z(x_i)). These are respectively latent necessity, order sensitivity, corruption sensitivity, task specificity, and replay error.
Exact mathematical passage
n_i &:=L(x_i,y_i;\mathsf{zero})-L(x_i,y_i;Z(x_i)), \label{eq:b16-necessity}\\ o_i &:=L(x_i,y_i;Z_{\pi}(x_i))-L(x_i,y_i;Z(x_i)), \label{eq:b16-order}\\ c_i &:=L(x_i,y_i;Z_{\eta}(x_i))-L(x_i,y_i;Z(x_i)), \label{eq:b16-noise}\\ s_i &:=L(x_i,y_i;Z(x'_i))-L(x_i,y_i;Z(x_i)), \label{eq:b16-specificity}\\ r_i &:=\abs{L(x_i,y_i;\mathsf{replay}(Z(x_i)))-L(x_i,y_i;Z(x_i))}. \label{eq:b16-replay}
\mathrm{new}Definition passage · 2 passages
Search aliases: new mathrm
Definition passage: Evaluator-upgrade certificate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3857
[Evaluator-upgrade certificate] An evaluator-upgrade certificate is a tuple Π_eval = (v,w,S_e,T_e,R_e, h_v, A_v, A_w, τ,μ_0, δ_obs,δ_J, T_new, R_rollback, σ), containing the old and new evaluator identifiers, raw and internal transports, core retraction, protected replay set, holonomy-energy operators, accepted flatness and adequacy thresholds, measured replay defects, held-out new-evidence tests, rollback record, and uncertainty statement σ.
Exact mathematical passage
\Pi_{\mathrm{eval}} = (v,w,S_e,T_e,R_e, \mathfrak h_v, \mathsf A_v,\mathsf A_w, \tau,\mu_0, \delta_{\mathrm{obs}},\delta_J, \mathcal T_{\mathrm{new}}, \mathcal R_{\mathrm{rollback}}, \sigma),
\normDefinition passage · 163 passages
Search aliases: norm
Definition passage: Property-transport defect · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 726
[Property-transport defect] For a declared domain K⊆ X, define ε_Q(T;K) = sup_x∈ K norm Q_Y(Tx)-widetilde T_QQ_X(x) .
Exact mathematical passage
\eps_Q(T;K) = \sup_{x\in K} \norm{ Q_Y(Tx)-\widetilde T_QQ_X(x) }.
\notDefinition passage · 2 passages
Search aliases: not
Definition passage: Structurally new repair schema · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9969
[Structurally new repair schema] Let mathscrS_t be the closure of existing repair schemas under the authorized operadic compositions. A schema s_t+1 is structurally new when s_t+1∉overlinemathscrS_t under the declared behavioral equivalence and spanim s_t+1 not⊆ spanimmathscrS_t.
Exact mathematical passage
\Span\im s_{t+1} \not\subseteq \Span\im\Sscr_t. \label{eq:b8-new-schema-direction}
O_iDefinition passage · 2 passages
Search aliases: O_i
Definition passage: Pre-result research receipt · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13727
[Pre-result research receipt] A pre-result receipt has the form Π_i^- = (H_i,O_i,D_i,θ_i, widehat c_i, widehat r_i, F_i,h_i^code,h_i^spec), where H_i is the causal hypothesis, O_i the proposed objective or repair mechanism, D_i the evidence or data construction, θ_i the configuration, widehat c_i the predicted benchmark-effect vector, widehat r_i the predicted protected-risk vector, F_i the declared falsification condition, and h_i^code,h_i^spec immutable digests of the executable and specificat…
Exact mathematical passage
\Pi_i^{-} = (H_i,O_i,D_i,\theta_i, \widehat{\mathbf c}_i, \widehat{\mathbf r}_i, F_i,h_i^{\mathrm{code}},h_i^{\mathrm{spec}}), \label{eq:b13-preresult-receipt}
o_iDefinition passage · 6 passages
Search aliases: o_i
Definition passage: Per-task Neuralese causal coordinates · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15710
[Per-task Neuralese causal coordinates] For task (x_i,y_i), define n_i · :=L(x_i,y_i;zero)-L(x_i,y_i;Z(x_i)), o_i · :=L(x_i,y_i;Z_π(x_i))-L(x_i,y_i;Z(x_i)), c_i · :=L(x_i,y_i;Z_η(x_i))-L(x_i,y_i;Z(x_i)), s_i · :=L(x_i,y_i;Z(x'_i))-L(x_i,y_i;Z(x_i)), r_i · :=absL(x_i,y_i;replay(Z(x_i)))-L(x_i,y_i;Z(x_i)). These are respectively latent necessity, order sensitivity, corruption sensitivity, task specificity, and replay error.
Exact mathematical passage
n_i &:=L(x_i,y_i;\mathsf{zero})-L(x_i,y_i;Z(x_i)), \label{eq:b16-necessity}\\ o_i &:=L(x_i,y_i;Z_{\pi}(x_i))-L(x_i,y_i;Z(x_i)), \label{eq:b16-order}\\ c_i &:=L(x_i,y_i;Z_{\eta}(x_i))-L(x_i,y_i;Z(x_i)), \label{eq:b16-noise}\\ s_i &:=L(x_i,y_i;Z(x'_i))-L(x_i,y_i;Z(x_i)), \label{eq:b16-specificity}\\ r_i &:=\abs{L(x_i,y_i;\mathsf{replay}(Z(x_i)))-L(x_i,y_i;Z(x_i))}. \label{eq:b16-replay}
\ObDefinition passage · 1 passage
Search aliases: Ob operatorname
Definition passage: Authorized obstruction class · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1517
[Authorized obstruction class] The authorized obstruction to correcting r is Ob_D(r)=[r]∈coker D:=V/im D.
Exact mathematical passage
\Ob_D(r)=[r]\in\coker D:=V/\im D.
\Ob_DDefinition passage · 1 passage
Search aliases: Ob_D Ob operatorname
Definition passage: Authorized obstruction class · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1517
[Authorized obstruction class] The authorized obstruction to correcting r is Ob_D(r)=[r]∈coker D:=V/im D.
Exact mathematical passage
\Ob_D(r)=[r]\in\coker D:=V/\im D.
\Ocal_0Definition passage · 1 passage
Search aliases: Ocal_0 O_0 Ocal mathcal
Definition passage: One-step bounded repair-process improvement · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7342
[One-step bounded repair-process improvement] A lineage edge (O_0,K_0) xrightarrowΠ_1 (O_1,K_1) is a one-step bounded repair-process improvement relative to (Q,B,U,R) when: item K_1 is generated and sealed through an authorized lineage transaction; item parent and child use the same candidate budget B and candidate bundle on each target; item a predeclared lower confidence bound for Δ_Q(K_1,K_0) is positive; item the child satisfies declared average, tail, and maximum protected-risk gates; item …
Exact mathematical passage
(\Ocal_0,K_0) \xrightarrow{\Pi_1} (\Ocal_1,K_1)
\Ocal_1Definition passage · 1 passage
Search aliases: Ocal_1 O_1 Ocal mathcal
Definition passage: One-step bounded repair-process improvement · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7342
[One-step bounded repair-process improvement] A lineage edge (O_0,K_0) xrightarrowΠ_1 (O_1,K_1) is a one-step bounded repair-process improvement relative to (Q,B,U,R) when: item K_1 is generated and sealed through an authorized lineage transaction; item parent and child use the same candidate budget B and candidate bundle on each target; item a predeclared lower confidence bound for Δ_Q(K_1,K_0) is positive; item the child satisfies declared average, tail, and maximum protected-risk gates; item …
Exact mathematical passage
(\Ocal_0,K_0) \xrightarrow{\Pi_1} (\Ocal_1,K_1)
\Ocal_{t+1}Definition passage · 3 passages
Search aliases: Ocal_t+1 O_t+1 Ocal mathcal
Definition passage: Conservative repair-operad extension · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1356
[Conservative repair-operad extension] An inclusion ι_t:O_t↪O_t+1 is conservative when: item it is injective on old colors and operations; item it preserves identities, permutations, and every old syntactic composition; item it preserves old preconditions, postconditions, and protected-invariant contracts on old states; item every previously valid old receipt remains valid against the same immutable evidence; item new operations cannot retroactively consume or rewrite old observation authority.
Exact mathematical passage
\iota_t:\Ocal_t\hookrightarrow\Ocal_{t+1}
\mathrm{old}Definition passage · 1 passage
Search aliases: old mathrm
Definition passage: Constructor-expansion certificate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13142
[Constructor-expansion certificate] A constructor-expansion certificate is a tuple Π_con = (C_t,C_t^+,η_old,η_new, H,κ,C_model,Π_foreign), where H is a nonzero higher-order derivative witness, κ is held-out obstruction reduction, and C_model records constructor complexity.
Exact mathematical passage
\Pi_{\mathrm{con}} = (\Ccal_t,\Ccal_t^+,\eta_{\mathrm{old}},\eta_{\mathrm{new}}, \mathsf H,\kappa,C_{\mathrm{model}},\Pi_{\mathrm{foreign}}),
\oneDefinition passage · 17 passages
Search aliases: one 1 mathbf
Definition passage: Invariant-context two-chart grammar · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11380
[Invariant-context two-chart grammar] Let P_ctx:H_s→ℝ^r be a protected context projector whose value is unchanged by the source repair. Let F be a declared family of separators g(h)=1{anglesu,P_ctxh>θ}. The chart extension is G_chart = set(g,bar T_0,bar T_1):g∈F, bar T_j∈ℝ^q×(d+1). Its prediction is widehat y(h)=bar T_g(h)bar h.
Exact mathematical passage
g(h)=\one\{\angles{u,P_{\mathrm{ctx}}h}>\theta\}.
\mathrm{out}Definition passage · 2 passages
Search aliases: out mathrm
Definition passage: Proof-carrying repair schema · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1190
[Proof-carrying repair schema] A repair schema is a tuple ω= ( σ, c_in, c_out, D, C, Q, [r], u, g, mathscr I, Π ), where: item σ is a recognizable failure signature; item c_in=(c_1,…,c_n) and c_out are input and output colors; item D is the declared local defect-response map; item C is an applicability precondition; item Q is a postcondition; item [r] is a local obstruction certificate or unresolved-obstruction record; item ν is the correction program; item g is a precommitted effect profile; it…
Exact mathematical passage
\omega= \left( \sigma, \bm c_{\mathrm{in}}, c_{\mathrm{out}}, D, C, Q, [r], u, \bm g, \mathscr I, \Pi \right),
\overline cDefinition passage · 3 passages
Search aliases: overline c overline
Definition passage: Structural-normalized specificity · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15734
[Structural-normalized specificity] Let Q_spec = fracoverline s max{overline o,overline c,ε}, where bars denote held-out means and ε>0 is a fixed numerical floor. The default Block 16 specificity gate requires LCB_0.95(overline s)>0, frac1n∑_i1[s_i>0]≥0.75, Q_spec≥0.25.
Exact mathematical passage
Q_{\mathrm{spec}} = \frac{\overline s} {\max\{\overline o,\overline c,\eps\}}, \label{eq:b16-specificity-ratio}
\overline LDefinition passage · 2 passages
Search aliases: overline L overline
Definition passage: High-probability transport certificate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3302
[High-probability transport certificate] A random tuple (underline g_A, overlineε_J, overlineε_r, overlineε_D, overlineε_leak, overlineε_hol, overline L) is an (1-α) transport certificate when, jointly with probability at least 1-α, it lower-bounds g_A and upper-bounds every corresponding error or Lipschitz quantity.
Exact mathematical passage
(\underline g_A, \overline\eps_J, \overline\eps_r, \overline\eps_D, \overline\eps_{\mathrm{leak}}, \overline\eps_{\mathrm{hol}}, \overline L)
\overline oDefinition passage · 3 passages
Search aliases: overline o overline
Definition passage: Structural-normalized specificity · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15734
[Structural-normalized specificity] Let Q_spec = fracoverline s max{overline o,overline c,ε}, where bars denote held-out means and ε>0 is a fixed numerical floor. The default Block 16 specificity gate requires LCB_0.95(overline s)>0, frac1n∑_i1[s_i>0]≥0.75, Q_spec≥0.25.
Exact mathematical passage
Q_{\mathrm{spec}} = \frac{\overline s} {\max\{\overline o,\overline c,\eps\}}, \label{eq:b16-specificity-ratio}
\overline sDefinition passage · 2 passages
Search aliases: overline s overline
Definition passage: Structural-normalized specificity · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15734
[Structural-normalized specificity] Let Q_spec = fracoverline s max{overline o,overline c,ε}, where bars denote held-out means and ε>0 is a fixed numerical floor. The default Block 16 specificity gate requires LCB_0.95(overline s)>0, frac1n∑_i1[s_i>0]≥0.75, Q_spec≥0.25.
Exact mathematical passage
Q_{\mathrm{spec}} = \frac{\overline s} {\max\{\overline o,\overline c,\eps\}}, \label{eq:b16-specificity-ratio}
\overline TDefinition passage · 9 passages
Search aliases: overline T overline
Definition passage: Protected obstruction subspace · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2780
[Protected obstruction subspace] A protected obstruction subspace is a declared subspace Q_A⊆coker D_A whose nonzero classes must remain detectable after transport. The transport is obstruction-faithful on Q_A when keroverline T_V∩ Q_A=set0.
Exact mathematical passage
\ker\overline T_V\cap Q_A=\set{0}.
\overline T_tDefinition passage · 4 passages
Search aliases: overline T_t overline
Definition passage: Repair death · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9668
[Repair death] A class [r]∈coker D_t undergoes repair death at t+1 when overline T_t([r])=0, there exists an authorized u∈ U_t+1 satisfying D_t+1u=-T_t^Vr, and the protected defect transport obeys a lower norm bound normT_t^Vr≥κnormr for a declared κ>0.
Exact mathematical passage
\overline T_t([r])=0,
\overline T_VDefinition passage · 4 passages
Search aliases: overline T_V overline
Definition passage: Protected obstruction subspace · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2780
[Protected obstruction subspace] A protected obstruction subspace is a declared subspace Q_A⊆coker D_A whose nonzero classes must remain detectable after transport. The transport is obstruction-faithful on Q_A when keroverline T_V∩ Q_A=set0.
Exact mathematical passage
\ker\overline T_V\cap Q_A=\set{0}.
\overline wDefinition passage · 1 passage
Search aliases: overline w overline
Definition passage: External trace probe · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16262
[External trace probe] A Block 17 trace probe is a ridge map widehat w=P z_4+q fit only on the 1,200 probe-calibration tasks. The final audit reports held-out coefficient of determination R^2_probe = 1- frac∑_inormw_i-widehat w_i^2 ∑_inormw_i-overline w^2. The frozen semantic-probe gate is R^2_probe≥0.65 on each final support.
Exact mathematical passage
R^2_{\mathrm{probe}} = 1- \frac{\sum_i\norm{w_i-\widehat w_i}^2} {\sum_i\norm{w_i-\overline w}^2}. \label{eq:b17-probe-r2}
\overline\epsDefinition passage · 2 passages
Search aliases: overlineeps overlineε overline eps overline varepsilon
Definition passage: High-probability transport certificate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3302
[High-probability transport certificate] A random tuple (underline g_A, overlineε_J, overlineε_r, overlineε_D, overlineε_leak, overlineε_hol, overline L) is an (1-α) transport certificate when, jointly with probability at least 1-α, it lower-bounds g_A and upper-bounds every corresponding error or Lipschitz quantity.
Exact mathematical passage
(\underline g_A, \overline\eps_J, \overline\eps_r, \overline\eps_D, \overline\eps_{\mathrm{leak}}, \overline\eps_{\mathrm{hol}}, \overline L)
\overline\eps_DDefinition passage · 2 passages
Search aliases: overlineeps_D overlineε_D overline eps overline varepsilon
Definition passage: High-probability transport certificate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3302
[High-probability transport certificate] A random tuple (underline g_A, overlineε_J, overlineε_r, overlineε_D, overlineε_leak, overlineε_hol, overline L) is an (1-α) transport certificate when, jointly with probability at least 1-α, it lower-bounds g_A and upper-bounds every corresponding error or Lipschitz quantity.
Exact mathematical passage
(\underline g_A, \overline\eps_J, \overline\eps_r, \overline\eps_D, \overline\eps_{\mathrm{leak}}, \overline\eps_{\mathrm{hol}}, \overline L)
\overline\eps_{\mathrm{hol}}Definition passage · 2 passages
Search aliases: overlineeps_hol overlineε_hol overline eps mathrm overline varepsilon mathrm
Definition passage: High-probability transport certificate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3302
[High-probability transport certificate] A random tuple (underline g_A, overlineε_J, overlineε_r, overlineε_D, overlineε_leak, overlineε_hol, overline L) is an (1-α) transport certificate when, jointly with probability at least 1-α, it lower-bounds g_A and upper-bounds every corresponding error or Lipschitz quantity.
Exact mathematical passage
(\underline g_A, \overline\eps_J, \overline\eps_r, \overline\eps_D, \overline\eps_{\mathrm{leak}}, \overline\eps_{\mathrm{hol}}, \overline L)
\overline\eps_JDefinition passage · 2 passages
Search aliases: overlineeps_J overlineε_J overline eps overline varepsilon
Definition passage: High-probability transport certificate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3302
[High-probability transport certificate] A random tuple (underline g_A, overlineε_J, overlineε_r, overlineε_D, overlineε_leak, overlineε_hol, overline L) is an (1-α) transport certificate when, jointly with probability at least 1-α, it lower-bounds g_A and upper-bounds every corresponding error or Lipschitz quantity.
Exact mathematical passage
(\underline g_A, \overline\eps_J, \overline\eps_r, \overline\eps_D, \overline\eps_{\mathrm{leak}}, \overline\eps_{\mathrm{hol}}, \overline L)
\overline\eps_{\mathrm{leak}}Definition passage · 2 passages
Search aliases: overlineeps_leak overlineε_leak overline eps mathrm overline varepsilon mathrm
Definition passage: High-probability transport certificate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3302
[High-probability transport certificate] A random tuple (underline g_A, overlineε_J, overlineε_r, overlineε_D, overlineε_leak, overlineε_hol, overline L) is an (1-α) transport certificate when, jointly with probability at least 1-α, it lower-bounds g_A and upper-bounds every corresponding error or Lipschitz quantity.
Exact mathematical passage
(\underline g_A, \overline\eps_J, \overline\eps_r, \overline\eps_D, \overline\eps_{\mathrm{leak}}, \overline\eps_{\mathrm{hol}}, \overline L)
\overline\eps_rDefinition passage · 2 passages
Search aliases: overlineeps_r overlineε_r overline eps overline varepsilon
Definition passage: High-probability transport certificate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3302
[High-probability transport certificate] A random tuple (underline g_A, overlineε_J, overlineε_r, overlineε_D, overlineε_leak, overlineε_hol, overline L) is an (1-α) transport certificate when, jointly with probability at least 1-α, it lower-bounds g_A and upper-bounds every corresponding error or Lipschitz quantity.
Exact mathematical passage
(\underline g_A, \overline\eps_J, \overline\eps_r, \overline\eps_D, \overline\eps_{\mathrm{leak}}, \overline\eps_{\mathrm{hol}}, \overline L)
\overline{\Sscr_t}Definition passage · 1 passage
Search aliases: overlineSscr_t overlinemathscrS_t overline Sscr overline mathscr
Definition passage: Structurally new repair schema · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9965
[Structurally new repair schema] Let mathscrS_t be the closure of existing repair schemas under the authorized operadic compositions. A schema s_t+1 is structurally new when s_t+1∉overlinemathscrS_t under the declared behavioral equivalence and spanim s_t+1 not⊆ spanimmathscrS_t.
Exact mathematical passage
s_{t+1}\notin\overline{\Sscr_t}
\mathcal PDefinition passage · 1 passage
Search aliases: P mathcal
Definition passage: Neuralese reasoner · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15659
on[Neuralese reasoner] A finite-step Neuralese reasoner is a tuple N = (X,Y,H,E,B,U,D,Λ,K), where X is an input space, Y an answer space, H a finite-dimensional latent state space, E:X→ H a prompt encoder, B:H×X→ H a backbone transition, U:H× H× H→ H an update gate, D:H×X→ P(Y) an answer readout, Λ:H→[0,1] a halting map, and K a maximum latent-step budget.
Exact mathematical passage
D:H\times\Xcal\to\mathcal P(\Ycal)
\mathfrak PDefinition passage · 18 passages
Search aliases: P mathfrak
Definition passage: Finite repair process · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8292
[Finite repair process] A finite repair process is a tuple P=(π_rm anc, B,π_rm gate,B,Π), where π_rm anc selects target correspondence queries, B estimates a bridge and transports a frozen source repair, π_rm gate commits or abstains, B is the declared resource budget, and Π is the receipt schema.
Exact mathematical passage
\mathfrak P=(\pi_{\rm anc},\mathsf B,\pi_{\rm gate},B,\Pi),
\mathfrak pDefinition passage · 2 passages
Search aliases: p mathfrak
Definition passage: Correction datum · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2660
[Correction datum] A finite-dimensional correction datum is a tuple C=(U,V,D,A,c, p), where U is a correction space, V is a defect space, D:U→ V is the linearized defect-response map, A⊆ U is the authorized correction set, c:U→[0,∞] is a declared resource cost, and p is a collection of protected predicates or invariant maps.
Exact mathematical passage
\mathfrak C=(U,V,D,\Acal,c,\mathfrak p),
\mathsf PDefinition passage · 4 passages
Search aliases: P mathsf
Definition passage: Empirical benchmark instance · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5392
[Empirical benchmark instance] A benchmark instance is a tuple B = ( E, A, G, P, W, V, D, S, B, R), where: item E is the environment or transition process; item A is the agent family and functional atlas; item G is the repair grammar and candidate generator; item P is the permitted probe and intervention language; item W is the witness, checkpoint, and effect-mediation layer; item V is the protected evaluation context; item D is the data and evidence partition; item S is the seed, randomization,…
Exact mathematical passage
\mathfrak B = (\mathsf E,\mathsf A,\mathsf G,\mathsf P,\mathsf W,\mathsf V, \mathsf D,\mathsf S,\mathsf B,\mathsf R),
P_0Definition passage · 12 passages
Search aliases: P_0
Definition passage: Partial ontology · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 19907
[Partial ontology] Let P⊆ Z denote the protected semantic coordinates. A quotient identifies a partial ontology when it is sufficient for a declared subset P_0subsetneq P, returns calibrated ignorance for P∖ P_0, and records coverage cov(q)=fracdim P_0dim P.
Exact mathematical passage
P_0\subsetneq P
\mathfrak P_0Definition passage · 11 passages
Search aliases: P_0 mathfrak
Definition passage: One-step bounded recursive meta-improvement · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8327
[One-step bounded recursive meta-improvement] Let P_0 be a parent repair process. Let a lineage compiler Meta receive only a development ledger L_rm dev and return P_1=Meta( P_0,L_rm dev). The edge P_0→ P_1 is a one-step bounded recursive meta-improvement on (D_rm for,B) when: item the descendant is sealed before the audit distribution is revealed; item P_0 and P_1 receive the same deployment budget B; item M_B^rm for( P_1)>M_B^rm for( P_0) with a declared uncertainty certificate; item the desce…
Exact mathematical passage
\mathfrak P_0
\mathfrak P_1Definition passage · 9 passages
Search aliases: P_1 mathfrak
Definition passage: One-step bounded recursive meta-improvement · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8328
[One-step bounded recursive meta-improvement] Let P_0 be a parent repair process. Let a lineage compiler Meta receive only a development ledger L_rm dev and return P_1=Meta( P_0,L_rm dev). The edge P_0→ P_1 is a one-step bounded recursive meta-improvement on (D_rm for,B) when: item the descendant is sealed before the audit distribution is revealed; item P_0 and P_1 receive the same deployment budget B; item M_B^rm for( P_1)>M_B^rm for( P_0) with a declared uncertainty certificate; item the desce…
Exact mathematical passage
\mathfrak P_1=\Meta(\mathfrak P_0,L_{\rm dev}).
P_aDefinition passage · 19 passages
Search aliases: P_a
Definition passage: Block-scalar response model · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12163
[Block-scalar response model] The response family is block scalar on [source reference] when R_j=∑_a=1^mλ_ajP_a, λ_aj∈ℝ. The response signature of block a is λ_a=(λ_a1,…,λ_ak)∈ℝ^k. The signatures are separating when λ_a≠λ_b for every a≠ b.
Exact mathematical passage
R_j=\sum_{a=1}^{m}\lambda_{aj}P_a, \qquad \lambda_{aj}\in\R. \label{eq:b11-block-scalar}
P_{a,d}Definition passage · 1 passage
Search aliases: P_a,d
Definition passage: Architecture--context curvature · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17734
[Architecture--context curvature] For architectures a,b and charts c,d, the transport-square defect is Ω_a,b;c,d := C^a_c→ dT^c_b→ a - T^d_b→ aC^b_c→ d. Given an audit distribution P_b,c on target trajectories, its normalized root-mean-square curvature is κ_a,b;c,d^2 = frac E_h∼ P_b,c normΩ_a,b;c,dh^2 E_z∼ P_a,dnormz^2 .
Exact mathematical passage
\kappa_{a,b;c,d}^{2} = \frac{ \mathbb E_{h\sim P_{b,c}} \norm{\Omega_{a,b;c,d}h}^{2} }{ \mathbb E_{z\sim P_{a,d}}\norm{z}^{2} }. \label{eq:b20-normalized-curvature}
P_{\mathrm{alias}}Definition passage · 2 passages
Search aliases: P_alias mathrm
Definition passage: Response-language expansion certificate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12976
[Response-language expansion certificate] A response-language expansion certificate is a tuple Π_resp = (R,S, A, A^+,P_alias,γ, ε_comm,ε_fit,Π_foreign), where P_alias records the inherited merged block, γ is the minimum separating spectral gap, ε_comm is a commutator residual, ε_fit is response-model error, and Π_foreign records held-out repair effects.
Exact mathematical passage
\Pi_{\mathrm{resp}} = (\Rcal,S,\mathfrak A,\mathfrak A^+,P_{\mathrm{alias}},\gamma, \eps_{\mathrm{comm}},\eps_{\mathrm{fit}},\Pi_{\mathrm{foreign}}),
P_{b,c}Definition passage · 2 passages
Search aliases: P_b,c
Definition passage: Architecture--context curvature · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17733
[Architecture--context curvature] For architectures a,b and charts c,d, the transport-square defect is Ω_a,b;c,d := C^a_c→ dT^c_b→ a - T^d_b→ aC^b_c→ d. Given an audit distribution P_b,c on target trajectories, its normalized root-mean-square curvature is κ_a,b;c,d^2 = frac E_h∼ P_b,c normΩ_a,b;c,dh^2 E_z∼ P_a,dnormz^2 .
Exact mathematical passage
P_{b,c}
P_{\mathrm{ctx}}Definition passage · 8 passages
Search aliases: P_ctx mathrm
Definition passage: Invariant-context two-chart grammar · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11379
[Invariant-context two-chart grammar] Let P_ctx:H_s→ℝ^r be a protected context projector whose value is unchanged by the source repair. Let F be a declared family of separators g(h)=1{anglesu,P_ctxh>θ}. The chart extension is G_chart = set(g,bar T_0,bar T_1):g∈F, bar T_j∈ℝ^q×(d+1). Its prediction is widehat y(h)=bar T_g(h)bar h.
Exact mathematical passage
P_{\mathrm{ctx}}:H_s\to\R^r
P_FDefinition passage · 3 passages
Search aliases: P_F
Definition passage: Core adequacy modulus · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3702
[Core adequacy modulus] For a subspace F⊆ E with orthogonal projector P_F, define μ(F;Δ_prot) := inf_d∈Δ_prot fracnormP_Fdnorm d.
Exact mathematical passage
P_F
P_iDefinition passage · 3 passages
Search aliases: P_i
Definition passage: Packet leakage profile · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1981
[Packet leakage profile] The directed leakage from correction packet j into defect packet i is ℓ_ij:=normP_iDQ_j, i≠ j. The matrix (ℓ_ij) is the packet leakage profile.
Exact mathematical passage
\ell_{ij}:=\norm{P_iDQ_j}, \qquad i\neq j.
P_JDefinition passage · 1 passage
Search aliases: P_J
Definition passage: Packet-balanced QR policy · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8529
[Packet-balanced QR policy] For fit budget k=16, define π_rm PQR(H,P_J) = π_rm QR(H_rm aff,8) ∪ π_rm QR(H_rm non,8), where affected and unaffected membership is inferred from the source causal intervention, not from target audit labels.
Exact mathematical passage
\pi_{\rm PQR}(H,P_J) = \pi_{\rm QR}(H_{\rm aff},8) \cup \pi_{\rm QR}(H_{\rm non},8),
\mathsf{parent}Definition passage · 2 passages
Search aliases: parent mathsf
Definition passage: Finite Neuralese repair grammar · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16139
[Finite Neuralese repair grammar] The Block 17 compiler grammar is G_17 = {parent^+,depth,rank1, rank4,full,full^-sem}. The symbol parent^+ denotes matched-update continuation of the task-blind parent. The final symbol uses full conditioning but removes the semantic-witness loss. Every candidate receives exactly 550 AdamW updates with batch size 256, learning rate 8×10^-4, and identical ordinary task, intermediate-step, halt, and weight-decay terms.
Exact mathematical passage
\Gcal_{17} = \{\mathsf{parent}^{+},\mathsf{depth},\mathsf{rank1}, \mathsf{rank4},\mathsf{full},\mathsf{full}^{-\mathrm{sem}}\}. \label{eq:b17-repair-grammar}
\mathsf{parent}^{+}Definition passage · 2 passages
Search aliases: parent^+ mathsf
Definition passage: Finite Neuralese repair grammar · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16139
[Finite Neuralese repair grammar] The Block 17 compiler grammar is G_17 = {parent^+,depth,rank1, rank4,full,full^-sem}. The symbol parent^+ denotes matched-update continuation of the task-blind parent. The final symbol uses full conditioning but removes the semantic-witness loss. Every candidate receives exactly 550 AdamW updates with batch size 256, learning rate 8×10^-4, and identical ordinary task, intermediate-step, halt, and weight-decay terms.
Exact mathematical passage
\Gcal_{17} = \{\mathsf{parent}^{+},\mathsf{depth},\mathsf{rank1}, \mathsf{rank4},\mathsf{full},\mathsf{full}^{-\mathrm{sem}}\}. \label{eq:b17-repair-grammar}
\Pcal_eDefinition passage · 2 passages
Search aliases: Pcal_e P_e Pcal mathcal
Definition passage: Stratified functional atlas · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 641
[Stratified functional atlas] A stratified functional atlas at generation t is a tuple A_t= ( C, G_t, {X_t^c}_c∈ C, {F_t^c}_c∈ C, {T_e,τ_e,P_e,ε_e}_e∈ E(G_t) ), where: item C is a finite set of stratum colors; item G_t is a directed multigraph of declared interfaces; item X_t^c is the state space of stratum c; item F_t^c is its local transition rule; item each edge e:c→ d carries a bridge T_e:X_t^c→ X_t^d; item τ_e declares the bridge class; item P_e is its preservation profile; item ε_e is a ve…
Exact mathematical passage
\mathfrak A_t= \left( \mathsf C, G_t, \{X_t^c\}_{c\in\mathsf C}, \{F_t^c\}_{c\in\mathsf C}, \{T_e,\tau_e,\Pcal_e,\bm\eps_e\}_{e\in E(G_t)} \right),
\mathrm{post}Definition passage · 4 passages
Search aliases: post mathrm
Definition passage: Lineage-edge receipt · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 858
[Lineage-edge receipt] A lineage edge AxrightarrowΠB has a receipt Π= ( id_A, id_B, θ, W, E_pre, E_post, widehat g, mathscr I, tests, decision, rollback, digest ), recording the parent and child, transformation θ, frozen witness, pre- and post-reveal evidence, precommitted effect claim widehat g, protected invariants, validation results, commit decision, rollback point, and integrity digest.
Exact mathematical passage
\Pi= \left( \mathrm{id}_A, \mathrm{id}_B, \theta, \Wcal, \Ecal_{\mathrm{pre}}, \Ecal_{\mathrm{post}}, \widehat g, \mathscr I, \mathrm{tests}, \mathrm{decision}, \mathrm{rollback}, \mathrm{digest} \right),
\mathrm{pre}Definition passage · 2 passages
Search aliases: pre mathrm
Definition passage: Lineage-edge receipt · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 858
[Lineage-edge receipt] A lineage edge AxrightarrowΠB has a receipt Π= ( id_A, id_B, θ, W, E_pre, E_post, widehat g, mathscr I, tests, decision, rollback, digest ), recording the parent and child, transformation θ, frozen witness, pre- and post-reveal evidence, precommitted effect claim widehat g, protected invariants, validation results, commit decision, rollback point, and integrity digest.
Exact mathematical passage
\Pi= \left( \mathrm{id}_A, \mathrm{id}_B, \theta, \Wcal, \Ecal_{\mathrm{pre}}, \Ecal_{\mathrm{post}}, \widehat g, \mathscr I, \mathrm{tests}, \mathrm{decision}, \mathrm{rollback}, \mathrm{digest} \right),
\preceqDefinition passage · 4 passages
Search aliases: preceq
Definition passage: Schema generality preorder · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1412
[Schema generality preorder] For repair schemas ω_1 and ω_2 with the same typed effect, write ω_1preceqω_2 when every state satisfying C_ω_1 also satisfies C_ω_2 and the two correction semantics agree on dom(C_ω_1) up to the declared transport tolerance. Then ω_2 is at least as general as ω_1.
Exact mathematical passage
\omega_1\preceq\omega_2
\preceq_{\mathrm{cons}}Definition passage · 3 passages
Search aliases: preceq_cons preceq mathrm
Definition passage: Conservative lineage dominance · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9436
[Conservative lineage dominance] Write L_tpreceq_consL_t+1 when: item the old grammar, evidence ledger, and proof objects embed into the new lineage state; item every old protected interpretation remains recoverable within the declared semantic-debt bound; item old authorized behaviors replay within declared tolerances; item the new correction image contains the transported old correction image; item at least one new externally validated distinction, repair schema, or correction direction is add…
Exact mathematical passage
L_t\preceq_{\mathrm{cons}}L_{t+1}
\mathrm{previous}Definition passage · 1 passage
Search aliases: previous mathrm
Definition passage: Receipt chain · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5833
[Receipt chain] For receipts Π_1,…,Π_T, define Π_t.previous = ∅, · t=1, h(Π_t-1), · t>1.
Exact mathematical passage
\Pi_t.\mathrm{previous} = \begin{cases} \varnothing,&t=1,\\ h(\Pi_{t-1}),&t>1. \end{cases}
\mathrm{prog}Definition passage · 11 passages
Search aliases: prog mathrm
Definition passage: Cross-implementation repair schema · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16726
[Cross-implementation repair schema] A Block 18 repair schema is a tuple ρ=(a,λ_sem,λ_prog, λ_geo,χ), where a is an adapter constructor, the three nonnegative λ values weight final semantic, progressive semantic, and latent-geometry witnesses, and χ records whether witness labels are correctly bound or shuffled. A schema is cross-implementation when the same tuple is instantiated through each cell family's declared adapter interface without architecture-specific candidate selection.
Exact mathematical passage
\rho=(a,\lambda_{\mathrm{sem}},\lambda_{\mathrm{prog}}, \lambda_{\mathrm{geo}},\chi), \label{eq:b18-repair-schema}
\mathrm{protected}Definition passage · 7 passages
Search aliases: protected mathrm
Definition passage: Kernel acceptance predicate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5788
[Kernel acceptance predicate] For revealed protected evidence E, the reference acceptance predicate is Accept(widehat s,widehat s',widehatδ,E) iff · K_edit∩ K_protected=∅ · land restrwidehat s'K_protected =restrwidehat sK_protected · land E[utility_gain]≥γ_min · land E[risk]≤ r_max · land E[provenance_ok]=1 · land E_irr=∅.
Exact mathematical passage
\begin{aligned} \operatorname{Accept}(\widehat s,\widehat s',\widehat\delta,E) \iff{}& K_{\mathrm{edit}}\cap K_{\mathrm{protected}}=\varnothing\\ &\land\ \restr{\widehat s'}{K_{\mathrm{protected}}} =\restr{\widehat s}{K_{\mathrm{protected}}}\\ &\land\ E[\mathrm{utility\_gain}]\geq\gamma_{\min}\\ &\land\ E[\mathrm{risk}]\leq r_{\max}\\ &\land\ E[\mathrm{provenance\_ok}]=1\\ &\land\ E_{\mathrm{irr}}=\varnothing. \end{aligned}
\mathrm{provenance\_ok}Definition passage · 1 passage
Search aliases: provenance_ok mathrm
Definition passage: Kernel acceptance predicate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5788
[Kernel acceptance predicate] For revealed protected evidence E, the reference acceptance predicate is Accept(widehat s,widehat s',widehatδ,E) iff · K_edit∩ K_protected=∅ · land restrwidehat s'K_protected =restrwidehat sK_protected · land E[utility_gain]≥γ_min · land E[risk]≤ r_max · land E[provenance_ok]=1 · land E_irr=∅.
Exact mathematical passage
\begin{aligned} \operatorname{Accept}(\widehat s,\widehat s',\widehat\delta,E) \iff{}& K_{\mathrm{edit}}\cap K_{\mathrm{protected}}=\varnothing\\ &\land\ \restr{\widehat s'}{K_{\mathrm{protected}}} =\restr{\widehat s}{K_{\mathrm{protected}}}\\ &\land\ E[\mathrm{utility\_gain}]\geq\gamma_{\min}\\ &\land\ E[\mathrm{risk}]\leq r_{\max}\\ &\land\ E[\mathrm{provenance\_ok}]=1\\ &\land\ E_{\mathrm{irr}}=\varnothing. \end{aligned}
\mathcal QDefinition passage · 4 passages
Search aliases: Q mathcal
Definition passage: Preservation profile · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 704
[Preservation profile] A preservation profile for a bridge T:X→ Y is a declared record P(T)= ( Σ_T, Q_T, ρ_T, ℓ_T, σ_T, A_T ), where Σ_T lists preserved algebraic structures, Q_T lists preserved causal or evaluative queries, ρ_T records provenance behavior, ℓ_T records information loss, σ_T records uncertainty transport, and A_T lists authorized downstream uses.
Exact mathematical passage
\Pcal(T)= \left( \Sigma_T, \mathcal Q_T, \rho_T, \ell_T, \sigma_T, \mathcal A_T \right),
q_{\Gcal}Definition passage · 4 passages
Search aliases: q_Gcal q_G Gcal mathcal
Definition passage: Relative grammar-expansion witness · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5033
[Relative grammar-expansion witness] Let G↪G' be a candidate extension and let r' be the transported defect. The extension exposes a missing correction direction with margin κ>0 when q_G'(r') ≤ q_G(r)-κ.
Exact mathematical passage
q_{\Gcal'}(r') \leq q_{\Gcal}(r)-\kappa.
q_{\Gcal'}Definition passage · 1 passage
Search aliases: q_Gcal' q_G' Gcal mathcal
Definition passage: Relative grammar-expansion witness · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5033
[Relative grammar-expansion witness] Let G↪G' be a candidate extension and let r' be the transported defect. The extension exposes a missing correction direction with margin κ>0 when q_G'(r') ≤ q_G(r)-κ.
Exact mathematical passage
q_{\Gcal'}(r') \leq q_{\Gcal}(r)-\kappa.
Q_jDefinition passage · 1 passage
Search aliases: Q_j
Definition passage: Packet leakage profile · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1981
[Packet leakage profile] The directed leakage from correction packet j into defect packet i is ℓ_ij:=normP_iDQ_j, i≠ j. The matrix (ℓ_ij) is the packet leakage profile.
Exact mathematical passage
\ell_{ij}:=\norm{P_iDQ_j}, \qquad i\neq j.
Q_{\mathrm{spec}}Definition passage · 2 passages
Search aliases: Q_spec mathrm
Definition passage: Structural-normalized specificity · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15734
[Structural-normalized specificity] Let Q_spec = fracoverline s max{overline o,overline c,ε}, where bars denote held-out means and ε>0 is a fixed numerical floor. The default Block 16 specificity gate requires LCB_0.95(overline s)>0, frac1n∑_i1[s_i>0]≥0.75, Q_spec≥0.25.
Exact mathematical passage
Q_{\mathrm{spec}} = \frac{\overline s} {\max\{\overline o,\overline c,\eps\}}, \label{eq:b16-specificity-ratio}
\mathcal Q_TDefinition passage · 2 passages
Search aliases: Q_T mathcal
Definition passage: Preservation profile · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 704
[Preservation profile] A preservation profile for a bridge T:X→ Y is a declared record P(T)= ( Σ_T, Q_T, ρ_T, ℓ_T, σ_T, A_T ), where Σ_T lists preserved algebraic structures, Q_T lists preserved causal or evaluative queries, ρ_T records provenance behavior, ℓ_T records information loss, σ_T records uncertainty transport, and A_T lists authorized downstream uses.
Exact mathematical passage
\Pcal(T)= \left( \Sigma_T, \mathcal Q_T, \rho_T, \ell_T, \sigma_T, \mathcal A_T \right),
Q_vDefinition passage · 2 passages
Search aliases: Q_v
Definition passage: Evaluator object · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3386
[Evaluator object] An evaluator object at version v is a tuple E_v=(R_v,E_v,ι_v,J_v,Q_v, h_v), where: item R_v is the protected raw-evidence space or ledger span; item E_v is the evaluator's internal evidence-representation space; item ι_v:R_v→ E_v encodes raw evidence; item J_v:E_v→ℝ^q is a score or decision-readout map; item Q_v is an uncertainty or calibration readout; item h_v is the historical replay set protected at version v.
Exact mathematical passage
\mathsf E_v=(R_v,E_v,\iota_v,J_v,Q_v,\mathfrak h_v),
Q_XDefinition passage · 4 passages
Search aliases: Q_X
Definition passage: Property-transport defect · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 726
[Property-transport defect] For a declared domain K⊆ X, define ε_Q(T;K) = sup_x∈ K norm Q_Y(Tx)-widetilde T_QQ_X(x) .
Exact mathematical passage
\eps_Q(T;K) = \sup_{x\in K} \norm{ Q_Y(Tx)-\widetilde T_QQ_X(x) }.
Q_YDefinition passage · 5 passages
Search aliases: Q_Y
Definition passage: Property-transport defect · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 726
[Property-transport defect] For a declared domain K⊆ X, define ε_Q(T;K) = sup_x∈ K norm Q_Y(Tx)-widetilde T_QQ_X(x) .
Exact mathematical passage
\eps_Q(T;K) = \sup_{x\in K} \norm{ Q_Y(Tx)-\widetilde T_QQ_X(x) }.
\QcalDefinition passage · 2 passages
Search aliases: Qcal Q mathcal
Definition passage: Automated research compiler · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13682
[Automated research compiler] An automated research compiler is a tuple Γ = (L,F,P,Q,V,M,R), where: item L is a structured literature and prior-method memory; item F is an ancestor-closed finding graph or shared research forum; item P generates typed candidate methods and pre-result specifications; item Q allocates cheap pilots and full experiments under budget B; item V is the scored evaluator portfolio with protected capability gates; item M is the integrity and proposal--code congruence monit…
Exact mathematical passage
\Gamma = (\Lcal,\Fcal,\Pcal,\Qcal,\Vcal,\Mcal,\Rcal), \label{eq:b13-compiler-tuple}
\RDefinition passage · 111 passages
Search aliases: R ℝ mathbb
Definition passage: Evaluator object · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3394
[Evaluator object] An evaluator object at version v is a tuple E_v=(R_v,E_v,ι_v,J_v,Q_v, h_v), where: item R_v is the protected raw-evidence space or ledger span; item E_v is the evaluator's internal evidence-representation space; item ι_v:R_v→ E_v encodes raw evidence; item J_v:E_v→ℝ^q is a score or decision-readout map; item Q_v is an uncertainty or calibration readout; item h_v is the historical replay set protected at version v.
Exact mathematical passage
J_v:E_v\to\R^q
\mathbb RDefinition passage · 7 passages
Search aliases: R mathbb
Definition passage: Finite transport grammar · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17838
[Finite transport grammar] The Block 20 compiler searches the following declared grammar: item global_full: one pooled affine map R^200→ R^200; item chart_step_shared: six chart-indexed affine maps R^40→ R^40 applied independently at every latent step; item chart_full: six chart-indexed affine maps R^200→ R^200.
Exact mathematical passage
\mathbb R^{200}\to\mathbb R^{200}
\mathbf rDefinition passage · 2 passages
Search aliases: r mathbf
Definition passage: Post-result research receipt · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13741
[Post-result research receipt] After evaluation, the post-result receipt is Π_i^+ = (Π_i^-, c_i, r_i,a_i,m_i), where c_i is the scored benchmark vector, r_i the protected-regression vector, a_i the hidden-audit result, and m_i the integrity-monitor verdict.
Exact mathematical passage
\Pi_i^{+} = (\Pi_i^{-},\mathbf c_i,\mathbf r_i,a_i,m_i), \label{eq:b13-postresult-receipt}
\mathcal RDefinition passage · 3 passages
Search aliases: R mathcal
Definition passage: Evaluator-upgrade certificate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3857
[Evaluator-upgrade certificate] An evaluator-upgrade certificate is a tuple Π_eval = (v,w,S_e,T_e,R_e, h_v, A_v, A_w, τ,μ_0, δ_obs,δ_J, T_new, R_rollback, σ), containing the old and new evaluator identifiers, raw and internal transports, core retraction, protected replay set, holonomy-energy operators, accepted flatness and adequacy thresholds, measured replay defects, held-out new-evidence tests, rollback record, and uncertainty statement σ.
Exact mathematical passage
\Pi_{\mathrm{eval}} = (v,w,S_e,T_e,R_e, \mathfrak h_v, \mathsf A_v,\mathsf A_w, \tau,\mu_0, \delta_{\mathrm{obs}},\delta_J, \mathcal T_{\mathrm{new}}, \mathcal R_{\mathrm{rollback}}, \sigma),
\mathfrak RDefinition passage · 2 passages
Search aliases: R mathfrak
Definition passage: Learned repair-transport instance · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6465
[Learned repair-transport instance] A learned repair-transport instance is a tuple R_learn = ( X,ϕ,q,widetilde q,A_s, A_t, D, P, G, W), where: item X is the input space; item ϕ is the task feature map; item q and widetilde q are protected and corrupted score maps; item A_s is the source agent; item A_t is a family of target agents excluded from source repair fitting; item D is the train, anchor, calibration, and audit partition; item P is the permitted causal probe language; item G is the repair…
Exact mathematical passage
\mathfrak R_{\mathrm{learn}} = (\mathcal X,\phi,q,\widetilde q,A_s,\mathcal A_t, \mathsf D,\mathsf P,\mathsf G,\mathsf W),
\mathsf RDefinition passage · 3 passages
Search aliases: R mathsf
Definition passage: Empirical benchmark instance · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5392
[Empirical benchmark instance] A benchmark instance is a tuple B = ( E, A, G, P, W, V, D, S, B, R), where: item E is the environment or transition process; item A is the agent family and functional atlas; item G is the repair grammar and candidate generator; item P is the permitted probe and intervention language; item W is the witness, checkpoint, and effect-mediation layer; item V is the protected evaluation context; item D is the data and evidence partition; item S is the seed, randomization,…
Exact mathematical passage
\mathfrak B = (\mathsf E,\mathsf A,\mathsf G,\mathsf P,\mathsf W,\mathsf V, \mathsf D,\mathsf S,\mathsf B,\mathsf R),
r'Definition passage · 7 passages
Search aliases: r'
Definition passage: Relative grammar-expansion witness · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5032
[Relative grammar-expansion witness] Let G↪G' be a candidate extension and let r' be the transported defect. The extension exposes a missing correction direction with margin κ>0 when q_G'(r') ≤ q_G(r)-κ.
Exact mathematical passage
r'
\mathbb R^{200}Definition passage · 4 passages
Search aliases: R^200 mathbb
Definition passage: Finite transport grammar · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17838
[Finite transport grammar] The Block 20 compiler searches the following declared grammar: item global_full: one pooled affine map R^200→ R^200; item chart_step_shared: six chart-indexed affine maps R^40→ R^40 applied independently at every latent step; item chart_full: six chart-indexed affine maps R^200→ R^200.
Exact mathematical passage
\mathbb R^{200}\to\mathbb R^{200}
R^2_{\mathrm{probe}}Definition passage · 4 passages
Search aliases: R^2_probe mathrm
Definition passage: External trace probe · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16262
[External trace probe] A Block 17 trace probe is a ridge map widehat w=P z_4+q fit only on the 1,200 probe-calibration tasks. The final audit reports held-out coefficient of determination R^2_probe = 1- frac∑_inormw_i-widehat w_i^2 ∑_inormw_i-overline w^2. The frozen semantic-probe gate is R^2_probe≥0.65 on each final support.
Exact mathematical passage
R^2_{\mathrm{probe}} = 1- \frac{\sum_i\norm{w_i-\widehat w_i}^2} {\sum_i\norm{w_i-\overline w}^2}. \label{eq:b17-probe-r2}
\R^3Definition passage · 7 passages
Search aliases: R^3 ℝ^3 R mathbb
Definition passage: Executable affine-program task · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16045
[Executable affine-program task] A Block 17 task is τ = (x,m,(a_j,b_j)_j=1^m,A_m,B_m,y_m,g,s), y_m=A_mx+B_m, where g∈{A,B} names the generator and s names the data split. The model input is a padded serialization of x, the instructions, and instruction-presence bits. The protected semantic witness is w(τ) = (A_m,fracB_m3,fracy_m4)∈ℝ^3.
Exact mathematical passage
w(\tau) = \left(A_m,\frac{B_m}{3},\frac{y_m}{4}\right)\in\R^3. \label{eq:b17-semantic-witness}
\mathbb R^{40}Definition passage · 2 passages
Search aliases: R^40 mathbb
Definition passage: Finite transport grammar · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17839
[Finite transport grammar] The Block 20 compiler searches the following declared grammar: item global_full: one pooled affine map R^200→ R^200; item chart_step_shared: six chart-indexed affine maps R^40→ R^40 applied independently at every latent step; item chart_full: six chart-indexed affine maps R^200→ R^200.
Exact mathematical passage
\mathbb R^{40}\to\mathbb R^{40}
\R^5Definition passage · 1 passage
Search aliases: R^5 ℝ^5 R mathbb
Definition passage: Affine observation chart · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17151
servation chart] For a context c, an affine observation chart is a tuple χ_c=(s_x,t_x,s_y,t_y,σ_c), s_xs_y≠0, with canonicalization widehat x=fracx^obs-t_xs_x, widehat y=fracy^obs-t_ys_y, and support signature σ_c∈ℝ^5. The signature is the sorted vector of observed demonstration inputs predicted by the chart on the five canonical demonstration locations.
Exact mathematical passage
\sigma_c\in\R^5
\R^{5\times8}Definition passage · 1 passage
Search aliases: R^5×8 ℝ^5×8 R times mathbb times
Definition passage: Post-connection transport episode · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 18246
[Post-connection transport episode] A Block 21 episode is a tuple E = (X,Y,Z,p, D_rm cal, D_rm meta, D_rm audit), where X∈ℝ^5×8 is the protected source packet, Y is an observed post-connection packet or lower-dimensional statistic, Z∈ℝ^9 is the protected semantic witness, p∈ℝ is a bounded diagnostic response, and the three supports are pairwise disjoint. The compiler receives only these public fields together with an episode identifier and architecture identifier. The private obstruction family …
Exact mathematical passage
X\in\R^{5\times8}
\R^9Definition passage · 2 passages
Search aliases: R^9 ℝ^9 R mathbb
Definition passage: Post-connection transport episode · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 18246
[Post-connection transport episode] A Block 21 episode is a tuple E = (X,Y,Z,p, D_rm cal, D_rm meta, D_rm audit), where X∈ℝ^5×8 is the protected source packet, Y is an observed post-connection packet or lower-dimensional statistic, Z∈ℝ^9 is the protected semantic witness, p∈ℝ is a bounded diagnostic response, and the three supports are pairwise disjoint. The compiler receives only these public fields together with an episode identifier and architecture identifier. The private obstruction family …
Exact mathematical passage
Z\in\R^9
\R^{d+1}Definition passage · 2 passages
Search aliases: R^d+1 ℝ^d+1 R mathbb
Definition passage: Inherited global-affine grammar · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11369
[Inherited global-affine grammar] The inherited bridge grammar is G_glob = setbar T:ℝ^d+1→ℝ^q linear in homogeneous coordinates. It fits one affine source-to-target map across every observable context.
Exact mathematical passage
\Gcal_{\mathrm{glob}} = \set{\bar T:\R^{d+1}\to\R^q\text{ linear in homogeneous coordinates}}.
\R^kDefinition passage · 2 passages
Search aliases: R^k ℝ^k R mathbb
Definition passage: Block-scalar response model · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12170
[Block-scalar response model] The response family is block scalar on [source reference] when R_j=∑_a=1^mλ_ajP_a, λ_aj∈ℝ. The response signature of block a is λ_a=(λ_a1,…,λ_ak)∈ℝ^k. The signatures are separating when λ_a≠λ_b for every a≠ b.
Exact mathematical passage
\lambda_a=(\lambda_{a1},\ldots,\lambda_{ak})\in\R^k.
\R^qDefinition passage · 3 passages
Search aliases: R^q ℝ^q R mathbb
Definition passage: Evaluator object · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3394
[Evaluator object] An evaluator object at version v is a tuple E_v=(R_v,E_v,ι_v,J_v,Q_v, h_v), where: item R_v is the protected raw-evidence space or ledger span; item E_v is the evaluator's internal evidence-representation space; item ι_v:R_v→ E_v encodes raw evidence; item J_v:E_v→ℝ^q is a score or decision-readout map; item Q_v is an uncertainty or calibration readout; item h_v is the historical replay set protected at version v.
Exact mathematical passage
J_v:E_v\to\R^q
\R^{q\times(d+1)}Definition passage · 2 passages
Search aliases: R^q×(d+1) ℝ^q×(d+1) R times mathbb times
Definition passage: Invariant-context two-chart grammar · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11384
[Invariant-context two-chart grammar] Let P_ctx:H_s→ℝ^r be a protected context projector whose value is unchanged by the source repair. Let F be a declared family of separators g(h)=1{anglesu,P_ctxh>θ}. The chart extension is G_chart = set(g,bar T_0,bar T_1):g∈F, bar T_j∈ℝ^q×(d+1). Its prediction is widehat y(h)=bar T_g(h)bar h.
Exact mathematical passage
\Gcal_{\mathrm{chart}} = \set{(g,\bar T_0,\bar T_1):g\in\Fcal,\ \bar T_j\in\R^{q\times(d+1)}}.
\R^rDefinition passage · 1 passage
Search aliases: R^r ℝ^r R mathbb
Definition passage: Invariant-context two-chart grammar · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11379
[Invariant-context two-chart grammar] Let P_ctx:H_s→ℝ^r be a protected context projector whose value is unchanged by the source repair. Let F be a declared family of separators g(h)=1{anglesu,P_ctxh>θ}. The chart extension is G_chart = set(g,bar T_0,bar T_1):g∈F, bar T_j∈ℝ^q×(d+1). Its prediction is widehat y(h)=bar T_g(h)bar h.
Exact mathematical passage
P_{\mathrm{ctx}}:H_s\to\R^r
\R_{>0}^{7}Definition passage · 2 passages
Search aliases: R_>0^7 ℝ_>0^7 R mathbb
Definition passage: Weighted meta-regret · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9273
[Weighted meta-regret] For w∈ℝ_>0^7, define Φ_w(r)=w^top r. The weights encode the declared relative value or risk of unresolved module headroom.
Exact mathematical passage
w\in\R_{>0}^{7}
r_ADefinition passage · 24 passages
Search aliases: r_A
Definition passage: Transport datum · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2681
gindefinition[Transport datum] Let C_A and C_B be correction data. A transport datum from A to B is a pair T_A→ B=(T_U,T_V), T_U:U_A→ U_B, T_V:V_A→ V_B. Its correction-square defect is Δ_D( T_A→ B) := D_BT_U-T_VD_A. For declared source and target defects r_A∈ V_A and r_B∈ V_B, its defect-identification error is Δ_r( T_A→ B;r_A,r_B) := r_B-T_Vr_A. The datum is exact when both quantities vanish.
Exact mathematical passage
r_A\in V_A
r_BDefinition passage · 16 passages
Search aliases: r_B
Definition passage: Transport datum · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2681
ransport datum] Let C_A and C_B be correction data. A transport datum from A to B is a pair T_A→ B=(T_U,T_V), T_U:U_A→ U_B, T_V:V_A→ V_B. Its correction-square defect is Δ_D( T_A→ B) := D_BT_U-T_VD_A. For declared source and target defects r_A∈ V_A and r_B∈ V_B, its defect-identification error is Δ_r( T_A→ B;r_A,r_B) := r_B-T_Vr_A. The datum is exact when both quantities vanish.
Exact mathematical passage
r_B\in V_B
R_eDefinition passage · 7 passages
Search aliases: R_e
Definition passage: Core-retractable upgrade · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3745
[Core-retractable upgrade] An evaluator morphism e:v→ w is core-retractable when T_e(E_v^flat)⊆ E_w^flat and there is a map R_e:E_w^flat→ E_v^flat such that R_eT_e x=x for every x∈ E_v^flat. It is core-isomorphic when T_e restricts to an isomorphism from E_v^flat onto E_w^flat.
Exact mathematical passage
R_e:E_w^{\flat}\to E_v^{\flat}
\mathbf r_iDefinition passage · 2 passages
Search aliases: r_i mathbf
Definition passage: Post-result research receipt · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13741
[Post-result research receipt] After evaluation, the post-result receipt is Π_i^+ = (Π_i^-, c_i, r_i,a_i,m_i), where c_i is the scored benchmark vector, r_i the protected-regression vector, a_i the hidden-audit result, and m_i the integrity-monitor verdict.
Exact mathematical passage
\Pi_i^{+} = (\Pi_i^{-},\mathbf c_i,\mathbf r_i,a_i,m_i), \label{eq:b13-postresult-receipt}
r_iDefinition passage · 6 passages
Search aliases: r_i
Definition passage: Per-task Neuralese causal coordinates · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15710
[Per-task Neuralese causal coordinates] For task (x_i,y_i), define n_i · :=L(x_i,y_i;zero)-L(x_i,y_i;Z(x_i)), o_i · :=L(x_i,y_i;Z_π(x_i))-L(x_i,y_i;Z(x_i)), c_i · :=L(x_i,y_i;Z_η(x_i))-L(x_i,y_i;Z(x_i)), s_i · :=L(x_i,y_i;Z(x'_i))-L(x_i,y_i;Z(x_i)), r_i · :=absL(x_i,y_i;replay(Z(x_i)))-L(x_i,y_i;Z(x_i)). These are respectively latent necessity, order sensitivity, corruption sensitivity, task specificity, and replay error.
Exact mathematical passage
n_i &:=L(x_i,y_i;\mathsf{zero})-L(x_i,y_i;Z(x_i)), \label{eq:b16-necessity}\\ o_i &:=L(x_i,y_i;Z_{\pi}(x_i))-L(x_i,y_i;Z(x_i)), \label{eq:b16-order}\\ c_i &:=L(x_i,y_i;Z_{\eta}(x_i))-L(x_i,y_i;Z(x_i)), \label{eq:b16-noise}\\ s_i &:=L(x_i,y_i;Z(x'_i))-L(x_i,y_i;Z(x_i)), \label{eq:b16-specificity}\\ r_i &:=\abs{L(x_i,y_i;\mathsf{replay}(Z(x_i)))-L(x_i,y_i;Z(x_i))}. \label{eq:b16-replay}
R_{\mathrm{imp}}Definition passage · 9 passages
Search aliases: R_imp mathrm
Definition passage: Improvement reproduction number · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5204
[Improvement reproduction number] The local repair-discovery reproduction number is R_imp = rad(M).
Exact mathematical passage
R_{\mathrm{imp}} = \rad(M).
R_jDefinition passage · 18 passages
Search aliases: R_j
Definition passage: Block-scalar response model · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12163
[Block-scalar response model] The response family is block scalar on [source reference] when R_j=∑_a=1^mλ_ajP_a, λ_aj∈ℝ. The response signature of block a is λ_a=(λ_a1,…,λ_ak)∈ℝ^k. The signatures are separating when λ_a≠λ_b for every a≠ b.
Exact mathematical passage
R_j=\sum_{a=1}^{m}\lambda_{aj}P_a, \qquad \lambda_{aj}\in\R. \label{eq:b11-block-scalar}
\mathfrak R_{\mathrm{learn}}Definition passage · 2 passages
Search aliases: R_learn mathfrak mathrm
Definition passage: Learned repair-transport instance · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6465
[Learned repair-transport instance] A learned repair-transport instance is a tuple R_learn = ( X,ϕ,q,widetilde q,A_s, A_t, D, P, G, W), where: item X is the input space; item ϕ is the task feature map; item q and widetilde q are protected and corrupted score maps; item A_s is the source agent; item A_t is a family of target agents excluded from source repair fitting; item D is the train, anchor, calibration, and audit partition; item P is the permitted causal probe language; item G is the repair…
Exact mathematical passage
\mathfrak R_{\mathrm{learn}} = (\mathcal X,\phi,q,\widetilde q,A_s,\mathcal A_t, \mathsf D,\mathsf P,\mathsf G,\mathsf W),
r_{\max}Definition passage · 4 passages
Search aliases: r_max max
Definition passage: Kernel acceptance predicate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5788
[Kernel acceptance predicate] For revealed protected evidence E, the reference acceptance predicate is Accept(widehat s,widehat s',widehatδ,E) iff · K_edit∩ K_protected=∅ · land restrwidehat s'K_protected =restrwidehat sK_protected · land E[utility_gain]≥γ_min · land E[risk]≤ r_max · land E[provenance_ok]=1 · land E_irr=∅.
Exact mathematical passage
\begin{aligned} \operatorname{Accept}(\widehat s,\widehat s',\widehat\delta,E) \iff{}& K_{\mathrm{edit}}\cap K_{\mathrm{protected}}=\varnothing\\ &\land\ \restr{\widehat s'}{K_{\mathrm{protected}}} =\restr{\widehat s}{K_{\mathrm{protected}}}\\ &\land\ E[\mathrm{utility\_gain}]\geq\gamma_{\min}\\ &\land\ E[\mathrm{risk}]\leq r_{\max}\\ &\land\ E[\mathrm{provenance\_ok}]=1\\ &\land\ E_{\mathrm{irr}}=\varnothing. \end{aligned}
\mathcal R_{\mathrm{rollback}}Definition passage · 1 passage
Search aliases: R_rollback mathcal mathrm
Definition passage: Evaluator-upgrade certificate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3857
[Evaluator-upgrade certificate] An evaluator-upgrade certificate is a tuple Π_eval = (v,w,S_e,T_e,R_e, h_v, A_v, A_w, τ,μ_0, δ_obs,δ_J, T_new, R_rollback, σ), containing the old and new evaluator identifiers, raw and internal transports, core retraction, protected replay set, holonomy-energy operators, accepted flatness and adequacy thresholds, measured replay defects, held-out new-evidence tests, rollback record, and uncertainty statement σ.
Exact mathematical passage
\Pi_{\mathrm{eval}} = (v,w,S_e,T_e,R_e, \mathfrak h_v, \mathsf A_v,\mathsf A_w, \tau,\mu_0, \delta_{\mathrm{obs}},\delta_J, \mathcal T_{\mathrm{new}}, \mathcal R_{\mathrm{rollback}}, \sigma),
R_uDefinition passage · 2 passages
Search aliases: R_u
Definition passage: Robust validity-filtered compiler value · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 14421
[Robust validity-filtered compiler value] For risk weight λ≥0, define C_rm rob(Γ) = inf_u∈ U [ C_u(Γ) - λCVaR_α (-R_u(Γ,τ)) ]. The finite Block 14 realization does not compute the unrestricted infimum. It evaluates a frozen finite stress atlas and reports the mean, groupwise minima, lower-tail gain, negative-gain rate, invalid-selection rate, and unknown-class error separately.
Exact mathematical passage
\mathfrak C_{\rm rob}(\Gamma) = \inf_{u\in\mathfrak U} \left[ \mathfrak C_u(\Gamma) - \lambda\operatorname{CVaR}_\alpha \bigl(-R_u(\Gamma,\tau)\bigr) \right]. \label{eq:robust-compiler-value}
R_u^sDefinition passage · 3 passages
Search aliases: R_u^s
Definition passage: Unpaired intervention-orbit transport instance · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 19549
[Unpaired intervention-orbit transport instance] An instance is a tuple U = (H_s,H_t,μ_s,μ_t, U, {R_u^s}_u∈ U, {R_v^t}_v∈ V, G_rm gauge, W), where μ_s and μ_t are independently sampled latent distributions, R_u^s and R_v^t are within-system intervention-response operators, G_rm gauge is the declared harmless gauge group, and W is a witness hidden from construction. No map between individual samples from μ_s and μ_t is available to the compiler.
Exact mathematical passage
\mathfrak U = (H_s,H_t,\mu_s,\mu_t,\mathcal U, \{R_u^s\}_{u\in\mathcal U}, \{R_v^t\}_{v\in\mathcal V}, \mathcal G_{\rm gauge},\mathsf W),
R_v^tDefinition passage · 5 passages
Search aliases: R_v^t
Definition passage: Unpaired intervention-orbit transport instance · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 19549
[Unpaired intervention-orbit transport instance] An instance is a tuple U = (H_s,H_t,μ_s,μ_t, U, {R_u^s}_u∈ U, {R_v^t}_v∈ V, G_rm gauge, W), where μ_s and μ_t are independently sampled latent distributions, R_u^s and R_v^t are within-system intervention-response operators, G_rm gauge is the declared harmless gauge group, and W is a witness hidden from construction. No map between individual samples from μ_s and μ_t is available to the compiler.
Exact mathematical passage
\mathfrak U = (H_s,H_t,\mu_s,\mu_t,\mathcal U, \{R_u^s\}_{u\in\mathcal U}, \{R_v^t\}_{v\in\mathcal V}, \mathcal G_{\rm gauge},\mathsf W),
R_wDefinition passage · 2 passages
Search aliases: R_w
Definition passage: Evaluator morphism · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3402
[Evaluator morphism] An evaluator morphism e:v→ w consists of maps S_e:R_v→ R_w, T_e:E_v→ E_w, where S_e is the append-only raw-evidence embedding and T_e transports internal representations. Its protected-evidence defect is δ_obs(e) := sup_r∈ h_v normT_eι_v(r)-ι_wS_e(r), and its historical score defect is δ_J(e) := sup_r∈ h_v normJ_wT_eι_v(r)-J_vι_v(r). The morphism is exact on protected history when both defects vanish.
Exact mathematical passage
S_e:R_v\to R_w, \qquad T_e:E_v\to E_w,
r_\alphaDefinition passage · 1 passage
Search aliases: r_α alpha
Definition passage: Quantitative gluing obstruction · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2409
[Quantitative gluing obstruction] The graph-level quantitative gluing obstruction is η_glue(e) :=normP_cyce. The local-to-global repair certificate is the pair ( η_loc, η_glue ), η_loc^2 :=∑_αnormD_α u_α+r_α^2.
Exact mathematical passage
\left( \eta_{\mathrm{loc}}, \eta_{\mathrm{glue}} \right), \qquad \eta_{\mathrm{loc}}^2 :=\sum_{\alpha}\norm{D_\alpha u_\alpha+r_\alpha}^2.
\radDefinition passage · 7 passages
Search aliases: rad operatorname
Definition passage: Improvement reproduction number · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5204
[Improvement reproduction number] The local repair-discovery reproduction number is R_imp = rad(M).
Exact mathematical passage
R_{\mathrm{imp}} = \rad(M).
\mathsf{rank1}Definition passage · 1 passage
Search aliases: rank1 mathsf
Definition passage: Finite Neuralese repair grammar · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16139
[Finite Neuralese repair grammar] The Block 17 compiler grammar is G_17 = {parent^+,depth,rank1, rank4,full,full^-sem}. The symbol parent^+ denotes matched-update continuation of the task-blind parent. The final symbol uses full conditioning but removes the semantic-witness loss. Every candidate receives exactly 550 AdamW updates with batch size 256, learning rate 8×10^-4, and identical ordinary task, intermediate-step, halt, and weight-decay terms.
Exact mathematical passage
\Gcal_{17} = \{\mathsf{parent}^{+},\mathsf{depth},\mathsf{rank1}, \mathsf{rank4},\mathsf{full},\mathsf{full}^{-\mathrm{sem}}\}. \label{eq:b17-repair-grammar}
\mathsf{rank4}Definition passage · 1 passage
Search aliases: rank4 mathsf
Definition passage: Finite Neuralese repair grammar · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16139
[Finite Neuralese repair grammar] The Block 17 compiler grammar is G_17 = {parent^+,depth,rank1, rank4,full,full^-sem}. The symbol parent^+ denotes matched-update continuation of the task-blind parent. The final symbol uses full conditioning but removes the semantic-witness loss. Every candidate receives exactly 550 AdamW updates with batch size 256, learning rate 8×10^-4, and identical ordinary task, intermediate-step, halt, and weight-decay terms.
Exact mathematical passage
\Gcal_{17} = \{\mathsf{parent}^{+},\mathsf{depth},\mathsf{rank1}, \mathsf{rank4},\mathsf{full},\mathsf{full}^{-\mathrm{sem}}\}. \label{eq:b17-repair-grammar}
\mathrm{rel}Definition passage · 1 passage
Search aliases: rel mathrm
Definition passage: Scale-normalized transport defect · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3156
[Scale-normalized transport defect] For a declared stabilizer τ>0, define ε_D^rel := fracnormD_BT_U-T_VD_A normD_BnormT_U+normT_VnormD_A+τ.
Exact mathematical passage
\eps_D^{\mathrm{rel}} := \frac{\norm{D_BT_U-T_VD_A}} {\norm{D_B}\norm{T_U}+\norm{T_V}\norm{D_A}+\tau}.
\mathsf{replay}Definition passage · 4 passages
Search aliases: replay mathsf
Definition passage: Per-task Neuralese causal coordinates · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15710
[Per-task Neuralese causal coordinates] For task (x_i,y_i), define n_i · :=L(x_i,y_i;zero)-L(x_i,y_i;Z(x_i)), o_i · :=L(x_i,y_i;Z_π(x_i))-L(x_i,y_i;Z(x_i)), c_i · :=L(x_i,y_i;Z_η(x_i))-L(x_i,y_i;Z(x_i)), s_i · :=L(x_i,y_i;Z(x'_i))-L(x_i,y_i;Z(x_i)), r_i · :=absL(x_i,y_i;replay(Z(x_i)))-L(x_i,y_i;Z(x_i)). These are respectively latent necessity, order sensitivity, corruption sensitivity, task specificity, and replay error.
Exact mathematical passage
n_i &:=L(x_i,y_i;\mathsf{zero})-L(x_i,y_i;Z(x_i)), \label{eq:b16-necessity}\\ o_i &:=L(x_i,y_i;Z_{\pi}(x_i))-L(x_i,y_i;Z(x_i)), \label{eq:b16-order}\\ c_i &:=L(x_i,y_i;Z_{\eta}(x_i))-L(x_i,y_i;Z(x_i)), \label{eq:b16-noise}\\ s_i &:=L(x_i,y_i;Z(x'_i))-L(x_i,y_i;Z(x_i)), \label{eq:b16-specificity}\\ r_i &:=\abs{L(x_i,y_i;\mathsf{replay}(Z(x_i)))-L(x_i,y_i;Z(x_i))}. \label{eq:b16-replay}
\restrDefinition passage · 10 passages
Search aliases: restr
Definition passage: Kernel acceptance predicate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5788
[Kernel acceptance predicate] For revealed protected evidence E, the reference acceptance predicate is Accept(widehat s,widehat s',widehatδ,E) iff · K_edit∩ K_protected=∅ · land restrwidehat s'K_protected =restrwidehat sK_protected · land E[utility_gain]≥γ_min · land E[risk]≤ r_max · land E[provenance_ok]=1 · land E_irr=∅.
Exact mathematical passage
\begin{aligned} \operatorname{Accept}(\widehat s,\widehat s',\widehat\delta,E) \iff{}& K_{\mathrm{edit}}\cap K_{\mathrm{protected}}=\varnothing\\ &\land\ \restr{\widehat s'}{K_{\mathrm{protected}}} =\restr{\widehat s}{K_{\mathrm{protected}}}\\ &\land\ E[\mathrm{utility\_gain}]\geq\gamma_{\min}\\ &\land\ E[\mathrm{risk}]\leq r_{\max}\\ &\land\ E[\mathrm{provenance\_ok}]=1\\ &\land\ E_{\mathrm{irr}}=\varnothing. \end{aligned}
\mathrm{risk}Definition passage · 2 passages
Search aliases: risk mathrm
Definition passage: Kernel acceptance predicate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5788
[Kernel acceptance predicate] For revealed protected evidence E, the reference acceptance predicate is Accept(widehat s,widehat s',widehatδ,E) iff · K_edit∩ K_protected=∅ · land restrwidehat s'K_protected =restrwidehat sK_protected · land E[utility_gain]≥γ_min · land E[risk]≤ r_max · land E[provenance_ok]=1 · land E_irr=∅.
Exact mathematical passage
\begin{aligned} \operatorname{Accept}(\widehat s,\widehat s',\widehat\delta,E) \iff{}& K_{\mathrm{edit}}\cap K_{\mathrm{protected}}=\varnothing\\ &\land\ \restr{\widehat s'}{K_{\mathrm{protected}}} =\restr{\widehat s}{K_{\mathrm{protected}}}\\ &\land\ E[\mathrm{utility\_gain}]\geq\gamma_{\min}\\ &\land\ E[\mathrm{risk}]\leq r_{\max}\\ &\land\ E[\mathrm{provenance\_ok}]=1\\ &\land\ E_{\mathrm{irr}}=\varnothing. \end{aligned}
\mathrm{rollback}Definition passage · 3 passages
Search aliases: rollback mathrm
Definition passage: Lineage-edge receipt · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 858
[Lineage-edge receipt] A lineage edge AxrightarrowΠB has a receipt Π= ( id_A, id_B, θ, W, E_pre, E_post, widehat g, mathscr I, tests, decision, rollback, digest ), recording the parent and child, transformation θ, frozen witness, pre- and post-reveal evidence, precommitted effect claim widehat g, protected invariants, validation results, commit decision, rollback point, and integrity digest.
Exact mathematical passage
\Pi= \left( \mathrm{id}_A, \mathrm{id}_B, \theta, \Wcal, \Ecal_{\mathrm{pre}}, \Ecal_{\mathrm{post}}, \widehat g, \mathscr I, \mathrm{tests}, \mathrm{decision}, \mathrm{rollback}, \mathrm{digest} \right),
\mathsf SDefinition passage · 3 passages
Search aliases: S mathsf
Definition passage: Empirical benchmark instance · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5392
[Empirical benchmark instance] A benchmark instance is a tuple B = ( E, A, G, P, W, V, D, S, B, R), where: item E is the environment or transition process; item A is the agent family and functional atlas; item G is the repair grammar and candidate generator; item P is the permitted probe and intervention language; item W is the witness, checkpoint, and effect-mediation layer; item V is the protected evaluation context; item D is the data and evidence partition; item S is the seed, randomization,…
Exact mathematical passage
\mathfrak B = (\mathsf E,\mathsf A,\mathsf G,\mathsf P,\mathsf W,\mathsf V, \mathsf D,\mathsf S,\mathsf B,\mathsf R),
sDefinition passage · 250 passages
Search aliases: s
Definition passage: Nonlinear holonomy residual · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2549
[Nonlinear holonomy residual] If the S_i are metric spaces, define η_hol(s_o) := max_e∈ E∖ Td(H_e(s_o),s_o).
Exact mathematical passage
\eta_{\mathrm{hol}}(s_o) := \max_{e\in E\setminus T}d\bigl(H_e(s_o),s_o\bigr).
S_eDefinition passage · 10 passages
Search aliases: S_e
Definition passage: Evaluator morphism · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3402
[Evaluator morphism] An evaluator morphism e:v→ w consists of maps S_e:R_v→ R_w, T_e:E_v→ E_w, where S_e is the append-only raw-evidence embedding and T_e transports internal representations. Its protected-evidence defect is δ_obs(e) := sup_r∈ h_v normT_eι_v(r)-ι_wS_e(r), and its historical score defect is δ_J(e) := sup_r∈ h_v normJ_wT_eι_v(r)-J_vι_v(r). The morphism is exact on protected history when both defects vanish.
Exact mathematical passage
S_e:R_v\to R_w, \qquad T_e:E_v\to E_w,
S_iDefinition passage · 3 passages
Search aliases: S_i
Definition passage: Nonlinear holonomy residual · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2548
[Nonlinear holonomy residual] If the S_i are metric spaces, define η_hol(s_o) := max_e∈ E∖ Td(H_e(s_o),s_o).
Exact mathematical passage
S_i
s_iDefinition passage · 4 passages
Search aliases: s_i
Definition passage: Per-task Neuralese causal coordinates · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15710
[Per-task Neuralese causal coordinates] For task (x_i,y_i), define n_i · :=L(x_i,y_i;zero)-L(x_i,y_i;Z(x_i)), o_i · :=L(x_i,y_i;Z_π(x_i))-L(x_i,y_i;Z(x_i)), c_i · :=L(x_i,y_i;Z_η(x_i))-L(x_i,y_i;Z(x_i)), s_i · :=L(x_i,y_i;Z(x'_i))-L(x_i,y_i;Z(x_i)), r_i · :=absL(x_i,y_i;replay(Z(x_i)))-L(x_i,y_i;Z(x_i)). These are respectively latent necessity, order sensitivity, corruption sensitivity, task specificity, and replay error.
Exact mathematical passage
n_i &:=L(x_i,y_i;\mathsf{zero})-L(x_i,y_i;Z(x_i)), \label{eq:b16-necessity}\\ o_i &:=L(x_i,y_i;Z_{\pi}(x_i))-L(x_i,y_i;Z(x_i)), \label{eq:b16-order}\\ c_i &:=L(x_i,y_i;Z_{\eta}(x_i))-L(x_i,y_i;Z(x_i)), \label{eq:b16-noise}\\ s_i &:=L(x_i,y_i;Z(x'_i))-L(x_i,y_i;Z(x_i)), \label{eq:b16-specificity}\\ r_i &:=\abs{L(x_i,y_i;\mathsf{replay}(Z(x_i)))-L(x_i,y_i;Z(x_i))}. \label{eq:b16-replay}
s_oDefinition passage · 4 passages
Search aliases: s_o
Definition passage: Nonlinear holonomy residual · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2549
[Nonlinear holonomy residual] If the S_i are metric spaces, define η_hol(s_o) := max_e∈ E∖ Td(H_e(s_o),s_o).
Exact mathematical passage
\eta_{\mathrm{hol}}(s_o) := \max_{e\in E\setminus T}d\bigl(H_e(s_o),s_o\bigr).
s_{t+1}Definition passage · 3 passages
Search aliases: s_t+1
Definition passage: Structurally new repair schema · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9964
[Structurally new repair schema] Let mathscrS_t be the closure of existing repair schemas under the authorized operadic compositions. A schema s_t+1 is structurally new when s_t+1∉overlinemathscrS_t under the declared behavioral equivalence and spanim s_t+1 not⊆ spanimmathscrS_t.
Exact mathematical passage
s_{t+1}
s_xDefinition passage · 8 passages
Search aliases: s_x
Definition passage: Affine observation chart · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17140
[Affine observation chart] For a context c, an affine observation chart is a tuple χ_c=(s_x,t_x,s_y,t_y,σ_c), s_xs_y≠0, with canonicalization widehat x=fracx^obs-t_xs_x, widehat y=fracy^obs-t_ys_y, and support signature σ_c∈ℝ^5. The signature is the sorted vector of observed demonstration inputs predicted by the chart on the five canonical demonstration locations.
Exact mathematical passage
\chi_c=(s_x,t_x,s_y,t_y,\sigma_c), \qquad s_xs_y\neq0,
s_yDefinition passage · 7 passages
Search aliases: s_y
Definition passage: Affine observation chart · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17140
[Affine observation chart] For a context c, an affine observation chart is a tuple χ_c=(s_x,t_x,s_y,t_y,σ_c), s_xs_y≠0, with canonicalization widehat x=fracx^obs-t_xs_x, widehat y=fracy^obs-t_ys_y, and support signature σ_c∈ℝ^5. The signature is the sorted vector of observed demonstration inputs predicted by the chart on the five canonical demonstration locations.
Exact mathematical passage
\chi_c=(s_x,t_x,s_y,t_y,\sigma_c), \qquad s_xs_y\neq0,
\mathrm{sem}Definition passage · 11 passages
Search aliases: sem mathrm
Definition passage: Transport route cost · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3000
[Transport route cost] For a fixed source correction u_0, define the certified cost of a nonexpansive path p by C_tr(p;u_0) := normu_0∑_e∈ pε_D,e + ∑_e∈ pε_r,e + ∑_e∈ pε_auth,e + ∑_e∈ pε_sem,e.
Exact mathematical passage
C_{\mathrm{tr}}(p;u_0) := \norm{u_0}\sum_{e\in p}\eps_{D,e} + \sum_{e\in p}\eps_{r,e} + \sum_{e\in p}\eps_{\mathrm{auth},e} + \sum_{e\in p}\eps_{\mathrm{sem},e}.
\setDefinition passage · 36 passages
Search aliases: set
Definition passage: Protected obstruction subspace · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2780
[Protected obstruction subspace] A protected obstruction subspace is a declared subspace Q_A⊆coker D_A whose nonzero classes must remain detectable after transport. The transport is obstruction-faithful on Q_A when keroverline T_V∩ Q_A=set0.
Exact mathematical passage
\ker\overline T_V\cap Q_A=\set{0}.
\mathrm{sig}Definition passage · 6 passages
Search aliases: sig mathrm
Definition passage: Distribution-indexed atlas · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17157
[Distribution-indexed atlas] A finite distribution-indexed atlas is A = ( {χ_c:c∈ C}, d_sig, τ_adm, π ), where d_sig is a normalized signature distance, τ_adm is an admission threshold, and π chooses the nearest chart. An input is admitted only when min_c∈ Cd_sig(σ(x),σ_c) ≤τ_adm. Otherwise the atlas returns an unsupported-chart certificate and abstains.
Exact mathematical passage
\Acal = \left( \{\chi_c:c\in C\}, d_{\mathrm{sig}}, \tau_{\mathrm{adm}}, \pi \right), \label{eq:b19-atlas}
\SpanDefinition passage · 16 passages
Search aliases: Span span operatorname
Definition passage: Thresholded approximate flat core · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3587
[Thresholded approximate flat core] For τ≥0, define E_τ^flat := spansetx_i:λ_i≤τ, where x_i is an eigenbasis of A.
Exact mathematical passage
E_{\tau}^{\flat} := \Span\set{x_i:\lambda_i\leq\tau},
\mathrm{spec}Definition passage · 4 passages
Search aliases: spec mathrm
Definition passage: Pre-result research receipt · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13727
[Pre-result research receipt] A pre-result receipt has the form Π_i^- = (H_i,O_i,D_i,θ_i, widehat c_i, widehat r_i, F_i,h_i^code,h_i^spec), where H_i is the causal hypothesis, O_i the proposed objective or repair mechanism, D_i the evidence or data construction, θ_i the configuration, widehat c_i the predicted benchmark-effect vector, widehat r_i the predicted protected-risk vector, F_i the declared falsification condition, and h_i^code,h_i^spec immutable digests of the executable and specificat…
Exact mathematical passage
\Pi_i^{-} = (H_i,O_i,D_i,\theta_i, \widehat{\mathbf c}_i, \widehat{\mathbf r}_i, F_i,h_i^{\mathrm{code}},h_i^{\mathrm{spec}}), \label{eq:b13-preresult-receipt}
\operatorname{SPre}Definition passage · 3 passages
Search aliases: SPre operatorname
Definition passage: Productive recursive closure · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4375
[Productive recursive closure] For a finite perfect-information transaction game, define I_γ^ω = ν Z. μ Y. [ (G_γ∩SPre(Z)) ∪ SPre(Y) ].
Exact mathematical passage
\Ical_{\gamma}^{\omega} = \nu Z.\,\mu Y. \left[ \left(G_{\gamma}\cap\operatorname{SPre}(Z)\right) \cup \operatorname{SPre}(Y) \right].
\SscrDefinition passage · 3 passages
Search aliases: Sscr mathscrS mathscr
Definition passage: Structurally new repair schema · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9964
[Structurally new repair schema] Let mathscrS_t be the closure of existing repair schemas under the authorized operadic compositions. A schema s_t+1 is structurally new when s_t+1∉overlinemathscrS_t under the declared behavioral equivalence and spanim s_t+1 not⊆ spanimmathscrS_t.
Exact mathematical passage
\Sscr_t
\Sscr_tDefinition passage · 2 passages
Search aliases: Sscr_t mathscrS_t Sscr mathscr
Definition passage: Structurally new repair schema · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9964
[Structurally new repair schema] Let mathscrS_t be the closure of existing repair schemas under the authorized operadic compositions. A schema s_t+1 is structurally new when s_t+1∉overlinemathscrS_t under the declared behavioral equivalence and spanim s_t+1 not⊆ spanimmathscrS_t.
Exact mathematical passage
\Sscr_t
\subsetneqDefinition passage · 5 passages
Search aliases: subsetneq
Definition passage: Partial ontology · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 19907
[Partial ontology] Let P⊆ Z denote the protected semantic coordinates. A quotient identifies a partial ontology when it is sufficient for a declared subset P_0subsetneq P, returns calibrated ignorance for P∖ P_0, and records coverage cov(q)=fracdim P_0dim P.
Exact mathematical passage
P_0\subsetneq P
\succDefinition passage · 3 passages
Search aliases: succ
Definition passage: Exact context-relative improvement · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 522
[Exact context-relative improvement] For agents A,B, write Bsucceq_Ξ A when mathscr I(A,B)=1 and J_Ξ(B)≥ J_Ξ(A). For Δ>0, write Bsucc_Ξ,ΔA when the utility difference is at least Δ.
Exact mathematical passage
B\succ_{\Xi,\Delta}A
\succ_{\Xi,\Delta}Definition passage · 1 passage
Search aliases: succ_Ξ,Δ succ Xi Delta
Definition passage: Exact context-relative improvement · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 522
[Exact context-relative improvement] For agents A,B, write Bsucceq_Ξ A when mathscr I(A,B)=1 and J_Ξ(B)≥ J_Ξ(A). For Δ>0, write Bsucc_Ξ,ΔA when the utility difference is at least Δ.
Exact mathematical passage
B\succ_{\Xi,\Delta}A
\succeqDefinition passage · 5 passages
Search aliases: succeq
Definition passage: Exact context-relative improvement · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 513
[Exact context-relative improvement] For agents A,B, write Bsucceq_Ξ A when mathscr I(A,B)=1 and J_Ξ(B)≥ J_Ξ(A). For Δ>0, write Bsucc_Ξ,ΔA when the utility difference is at least Δ.
Exact mathematical passage
B\succeq_\Xi A
\succeq_\XiDefinition passage · 4 passages
Search aliases: succeq_Ξ succeq Xi
Definition passage: Exact context-relative improvement · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 513
[Exact context-relative improvement] For agents A,B, write Bsucceq_Ξ A when mathscr I(A,B)=1 and J_Ξ(B)≥ J_Ξ(A). For Δ>0, write Bsucc_Ξ,ΔA when the utility difference is at least Δ.
Exact mathematical passage
B\succeq_\Xi A
\supDefinition passage · 14 passages
Search aliases: sup
Definition passage: Property-transport defect · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 726
[Property-transport defect] For a declared domain K⊆ X, define ε_Q(T;K) = sup_x∈ K norm Q_Y(Tx)-widetilde T_QQ_X(x) .
Exact mathematical passage
\eps_Q(T;K) = \sup_{x\in K} \norm{ Q_Y(Tx)-\widetilde T_QQ_X(x) }.
\sup_{r\in\mathfrak h_v}Definition passage · 3 passages
Search aliases: sup_r∈ h_v sup in mathfrak
Definition passage: Evaluator morphism · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3408
[Evaluator morphism] An evaluator morphism e:v→ w consists of maps S_e:R_v→ R_w, T_e:E_v→ E_w, where S_e is the append-only raw-evidence embedding and T_e transports internal representations. Its protected-evidence defect is δ_obs(e) := sup_r∈ h_v normT_eι_v(r)-ι_wS_e(r), and its historical score defect is δ_J(e) := sup_r∈ h_v normJ_wT_eι_v(r)-J_vι_v(r). The morphism is exact on protected history when both defects vanish.
Exact mathematical passage
\delta_{\mathrm{obs}}(e) := \sup_{r\in\mathfrak h_v} \norm{T_e\iota_v(r)-\iota_wS_e(r)},
\sup_{x\in K}Definition passage · 1 passage
Search aliases: sup_x∈ K sup in
Definition passage: Property-transport defect · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 726
[Property-transport defect] For a declared domain K⊆ X, define ε_Q(T;K) = sup_x∈ K norm Q_Y(Tx)-widetilde T_QQ_X(x) .
Exact mathematical passage
\eps_Q(T;K) = \sup_{x\in K} \norm{ Q_Y(Tx)-\widetilde T_QQ_X(x) }.
\sup_{\delta\in\Dcal_t}Definition passage · 2 passages
Search aliases: sup_δ∈Dcal_t sup_δ∈D_t sup delta in Dcal sup delta in mathcal
Definition passage: ε-intervention sufficiency · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 955
[ε-intervention sufficiency] The map ϕ_t is ε-intervention-sufficient for D_t when sup_δ∈D_t TV( K_A_t^δ, widehat K_ϕ_t(A_t)^δ ) ≤ε.
Exact mathematical passage
\sup_{\delta\in\Dcal_t} \TV\left( K_{A_t}^{\delta}, \widehat K_{\phi_t(A_t)}^{\delta} \right) \leq\eps.
\mathcal TDefinition passage · 3 passages
Search aliases: T mathcal
Definition passage: Evaluator-upgrade certificate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3857
[Evaluator-upgrade certificate] An evaluator-upgrade certificate is a tuple Π_eval = (v,w,S_e,T_e,R_e, h_v, A_v, A_w, τ,μ_0, δ_obs,δ_J, T_new, R_rollback, σ), containing the old and new evaluator identifiers, raw and internal transports, core retraction, protected replay set, holonomy-energy operators, accepted flatness and adequacy thresholds, measured replay defects, held-out new-evidence tests, rollback record, and uncertainty statement σ.
Exact mathematical passage
\Pi_{\mathrm{eval}} = (v,w,S_e,T_e,R_e, \mathfrak h_v, \mathsf A_v,\mathsf A_w, \tau,\mu_0, \delta_{\mathrm{obs}},\delta_J, \mathcal T_{\mathrm{new}}, \mathcal R_{\mathrm{rollback}}, \sigma),
\mathfrak TDefinition passage · 8 passages
Search aliases: T mathfrak
Definition passage: Transport datum · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2668
[Transport datum] Let C_A and C_B be correction data. A transport datum from A to B is a pair T_A→ B=(T_U,T_V), T_U:U_A→ U_B, T_V:V_A→ V_B. Its correction-square defect is Δ_D( T_A→ B) := D_BT_U-T_VD_A. For declared source and target defects r_A∈ V_A and r_B∈ V_B, its defect-identification error is Δ_r( T_A→ B;r_A,r_B) := r_B-T_Vr_A. The datum is exact when both quantities vanish.
Exact mathematical passage
\mathfrak T_{A\to B}=(T_U,T_V), \qquad T_U:U_A\to U_B, \qquad T_V:V_A\to V_B.
T^{c}_{b\to a}Definition passage · 3 passages
Search aliases: T^c_b→ a to
Definition passage: Architecture--context curvature · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17725
[Architecture--context curvature] For architectures a,b and charts c,d, the transport-square defect is Ω_a,b;c,d := C^a_c→ dT^c_b→ a - T^d_b→ aC^b_c→ d. Given an audit distribution P_b,c on target trajectories, its normalized root-mean-square curvature is κ_a,b;c,d^2 = frac E_h∼ P_b,c normΩ_a,b;c,dh^2 E_z∼ P_a,dnormz^2 .
Exact mathematical passage
\Omega_{a,b;c,d} := C^{a}_{c\to d}T^{c}_{b\to a} - T^{d}_{b\to a}C^{b}_{c\to d}. \label{eq:b20-curvature}
T^{d}_{b\to a}Definition passage · 1 passage
Search aliases: T^d_b→ a to
Definition passage: Architecture--context curvature · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17725
[Architecture--context curvature] For architectures a,b and charts c,d, the transport-square defect is Ω_a,b;c,d := C^a_c→ dT^c_b→ a - T^d_b→ aC^b_c→ d. Given an audit distribution P_b,c on target trajectories, its normalized root-mean-square curvature is κ_a,b;c,d^2 = frac E_h∼ P_b,c normΩ_a,b;c,dh^2 E_z∼ P_a,dnormz^2 .
Exact mathematical passage
\Omega_{a,b;c,d} := C^{a}_{c\to d}T^{c}_{b\to a} - T^{d}_{b\to a}C^{b}_{c\to d}. \label{eq:b20-curvature}
\mathfrak T_{A\to B}Definition passage · 6 passages
Search aliases: T_A→ B mathfrak to
Definition passage: Transport datum · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2668
[Transport datum] Let C_A and C_B be correction data. A transport datum from A to B is a pair T_A→ B=(T_U,T_V), T_U:U_A→ U_B, T_V:V_A→ V_B. Its correction-square defect is Δ_D( T_A→ B) := D_BT_U-T_VD_A. For declared source and target defects r_A∈ V_A and r_B∈ V_B, its defect-identification error is Δ_r( T_A→ B;r_A,r_B) := r_B-T_Vr_A. The datum is exact when both quantities vanish.
Exact mathematical passage
\mathfrak T_{A\to B}=(T_U,T_V), \qquad T_U:U_A\to U_B, \qquad T_V:V_A\to V_B.
T_eDefinition passage · 18 passages
Search aliases: T_e
Definition passage: Stratified functional atlas · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 641
[Stratified functional atlas] A stratified functional atlas at generation t is a tuple A_t= ( C, G_t, {X_t^c}_c∈ C, {F_t^c}_c∈ C, {T_e,τ_e,P_e,ε_e}_e∈ E(G_t) ), where: item C is a finite set of stratum colors; item G_t is a directed multigraph of declared interfaces; item X_t^c is the state space of stratum c; item F_t^c is its local transition rule; item each edge e:c→ d carries a bridge T_e:X_t^c→ X_t^d; item τ_e declares the bridge class; item P_e is its preservation profile; item ε_e is a ve…
Exact mathematical passage
\mathfrak A_t= \left( \mathsf C, G_t, \{X_t^c\}_{c\in\mathsf C}, \{F_t^c\}_{c\in\mathsf C}, \{T_e,\tau_e,\Pcal_e,\bm\eps_e\}_{e\in E(G_t)} \right),
\mathcal T_{\mathrm{new}}Definition passage · 1 passage
Search aliases: T_new mathcal mathrm
Definition passage: Evaluator-upgrade certificate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3857
[Evaluator-upgrade certificate] An evaluator-upgrade certificate is a tuple Π_eval = (v,w,S_e,T_e,R_e, h_v, A_v, A_w, τ,μ_0, δ_obs,δ_J, T_new, R_rollback, σ), containing the old and new evaluator identifiers, raw and internal transports, core retraction, protected replay set, holonomy-energy operators, accepted flatness and adequacy thresholds, measured replay defects, held-out new-evidence tests, rollback record, and uncertainty statement σ.
Exact mathematical passage
\Pi_{\mathrm{eval}} = (v,w,S_e,T_e,R_e, \mathfrak h_v, \mathsf A_v,\mathsf A_w, \tau,\mu_0, \delta_{\mathrm{obs}},\delta_J, \mathcal T_{\mathrm{new}}, \mathcal R_{\mathrm{rollback}}, \sigma),
T_t^VDefinition passage · 11 passages
Search aliases: T_t^V
Definition passage: Repair death · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9672
[Repair death] A class [r]∈coker D_t undergoes repair death at t+1 when overline T_t([r])=0, there exists an authorized u∈ U_t+1 satisfying D_t+1u=-T_t^Vr, and the protected defect transport obeys a lower norm bound normT_t^Vr≥κnormr for a declared κ>0.
Exact mathematical passage
D_{t+1}u=-T_t^Vr,
t_xDefinition passage · 8 passages
Search aliases: t_x
Definition passage: Affine observation chart · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17140
[Affine observation chart] For a context c, an affine observation chart is a tuple χ_c=(s_x,t_x,s_y,t_y,σ_c), s_xs_y≠0, with canonicalization widehat x=fracx^obs-t_xs_x, widehat y=fracy^obs-t_ys_y, and support signature σ_c∈ℝ^5. The signature is the sorted vector of observed demonstration inputs predicted by the chart on the five canonical demonstration locations.
Exact mathematical passage
\chi_c=(s_x,t_x,s_y,t_y,\sigma_c), \qquad s_xs_y\neq0,
t_yDefinition passage · 7 passages
Search aliases: t_y
Definition passage: Affine observation chart · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17140
[Affine observation chart] For a context c, an affine observation chart is a tuple χ_c=(s_x,t_x,s_y,t_y,σ_c), s_xs_y≠0, with canonicalization widehat x=fracx^obs-t_xs_x, widehat y=fracy^obs-t_ys_y, and support signature σ_c∈ℝ^5. The signature is the sorted vector of observed demonstration inputs predicted by the chart on the five canonical demonstration locations.
Exact mathematical passage
\chi_c=(s_x,t_x,s_y,t_y,\sigma_c), \qquad s_xs_y\neq0,
\TauDefinition passage · 9 passages
Search aliases: Tau T mathcal
Definition passage: Compiler-language conflict · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15090
[Compiler-language conflict] A finite set S⊆T is a compiler-language conflict for G when all problems in S receive the same inherited diagnostic token but ⋂_τ∈ SA^⋆(τ)=∅. A minimal conflict has no proper conflicting subset.
Exact mathematical passage
S\subseteq\Tau
\mathrm{tests}Definition passage · 1 passage
Search aliases: tests mathrm
Definition passage: Lineage-edge receipt · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 858
[Lineage-edge receipt] A lineage edge AxrightarrowΠB has a receipt Π= ( id_A, id_B, θ, W, E_pre, E_post, widehat g, mathscr I, tests, decision, rollback, digest ), recording the parent and child, transformation θ, frozen witness, pre- and post-reveal evidence, precommitted effect claim widehat g, protected invariants, validation results, commit decision, rollback point, and integrity digest.
Exact mathematical passage
\Pi= \left( \mathrm{id}_A, \mathrm{id}_B, \theta, \Wcal, \Ecal_{\mathrm{pre}}, \Ecal_{\mathrm{post}}, \widehat g, \mathscr I, \mathrm{tests}, \mathrm{decision}, \mathrm{rollback}, \mathrm{digest} \right),
\topDefinition passage · 58 passages
Search aliases: top
Definition passage: Weighted meta-regret · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9274
[Weighted meta-regret] For w∈ℝ_>0^7, define Φ_w(r)=w^top r. The weights encode the declared relative value or risk of unresolved module headroom.
Exact mathematical passage
\Phi_w(r)=w^\top r. \label{eq:b8-weighted-regret}
\TVDefinition passage · 4 passages
Search aliases: TV operatorname
Definition passage: ε-intervention sufficiency · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 955
[ε-intervention sufficiency] The map ϕ_t is ε-intervention-sufficient for D_t when sup_δ∈D_t TV( K_A_t^δ, widehat K_ϕ_t(A_t)^δ ) ≤ε.
Exact mathematical passage
\sup_{\delta\in\Dcal_t} \TV\left( K_{A_t}^{\delta}, \widehat K_{\phi_t(A_t)}^{\delta} \right) \leq\eps.
\mathcal UDefinition passage · 3 passages
Search aliases: U mathcal
Definition passage: Unpaired intervention-orbit transport instance · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 19549
[Unpaired intervention-orbit transport instance] An instance is a tuple U = (H_s,H_t,μ_s,μ_t, U, {R_u^s}_u∈ U, {R_v^t}_v∈ V, G_rm gauge, W), where μ_s and μ_t are independently sampled latent distributions, R_u^s and R_v^t are within-system intervention-response operators, G_rm gauge is the declared harmless gauge group, and W is a witness hidden from construction. No map between individual samples from μ_s and μ_t is available to the compiler.
Exact mathematical passage
\mathfrak U = (H_s,H_t,\mu_s,\mu_t,\mathcal U, \{R_u^s\}_{u\in\mathcal U}, \{R_v^t\}_{v\in\mathcal V}, \mathcal G_{\rm gauge},\mathsf W),
u_0Definition passage · 18 passages
Search aliases: u_0
Definition passage: Transport route cost · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2999
[Transport route cost] For a fixed source correction u_0, define the certified cost of a nonexpansive path p by C_tr(p;u_0) := normu_0∑_e∈ pε_D,e + ∑_e∈ pε_r,e + ∑_e∈ pε_auth,e + ∑_e∈ pε_sem,e.
Exact mathematical passage
u_0
U_ADefinition passage · 6 passages
Search aliases: U_A
Definition passage: Transport datum · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2668
[Transport datum] Let C_A and C_B be correction data. A transport datum from A to B is a pair T_A→ B=(T_U,T_V), T_U:U_A→ U_B, T_V:V_A→ V_B. Its correction-square defect is Δ_D( T_A→ B) := D_BT_U-T_VD_A. For declared source and target defects r_A∈ V_A and r_B∈ V_B, its defect-identification error is Δ_r( T_A→ B;r_A,r_B) := r_B-T_Vr_A. The datum is exact when both quantities vanish.
Exact mathematical passage
\mathfrak T_{A\to B}=(T_U,T_V), \qquad T_U:U_A\to U_B, \qquad T_V:V_A\to V_B.
U_A^{\flat}Definition passage · 3 passages
Search aliases: U_A^flat flat
Definition passage: Path-flat repair and defect cores · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3041
[Path-flat repair and defect cores] The path-flat repair and defect cores at A are U_A^flat := ⋂_γker(H^U_γ-I), V_A^flat := ⋂_γker(H^V_γ-I).
Exact mathematical passage
U_A^{\flat} := \bigcap_\gamma\ker(H^U_\gamma-I), \qquad V_A^{\flat} := \bigcap_\gamma\ker(H^V_\gamma-I).
U_BDefinition passage · 5 passages
Search aliases: U_B
Definition passage: Transport datum · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2668
[Transport datum] Let C_A and C_B be correction data. A transport datum from A to B is a pair T_A→ B=(T_U,T_V), T_U:U_A→ U_B, T_V:V_A→ V_B. Its correction-square defect is Δ_D( T_A→ B) := D_BT_U-T_VD_A. For declared source and target defects r_A∈ V_A and r_B∈ V_B, its defect-identification error is Δ_r( T_A→ B;r_A,r_B) := r_B-T_Vr_A. The datum is exact when both quantities vanish.
Exact mathematical passage
\mathfrak T_{A\to B}=(T_U,T_V), \qquad T_U:U_A\to U_B, \qquad T_V:V_A\to V_B.
U_{t+1}Definition passage · 2 passages
Search aliases: U_t+1
Definition passage: Repair death · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9671
[Repair death] A class [r]∈coker D_t undergoes repair death at t+1 when overline T_t([r])=0, there exists an authorized u∈ U_t+1 satisfying D_t+1u=-T_t^Vr, and the protected defect transport obeys a lower norm bound normT_t^Vr≥κnormr for a declared κ>0.
Exact mathematical passage
u\in U_{t+1}
U_\alphaDefinition passage · 17 passages
Search aliases: U_α alpha
Definition passage: Two-level repair obstruction · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2311
[Two-level repair obstruction] The local obstruction on U_α is [r|_U_α]∈coker D_α. When every local class vanishes, the global gluing obstruction is [d_αβ]∈check H^1( U,ker D).
Exact mathematical passage
U_\alpha
u_\alphaDefinition passage · 8 passages
Search aliases: u_α alpha
Definition passage: Quantitative gluing obstruction · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2409
[Quantitative gluing obstruction] The graph-level quantitative gluing obstruction is η_glue(e) :=normP_cyce. The local-to-global repair certificate is the pair ( η_loc, η_glue ), η_loc^2 :=∑_αnormD_α u_α+r_α^2.
Exact mathematical passage
\left( \eta_{\mathrm{loc}}, \eta_{\mathrm{glue}} \right), \qquad \eta_{\mathrm{loc}}^2 :=\sum_{\alpha}\norm{D_\alpha u_\alpha+r_\alpha}^2.
\UcalDefinition passage · 11 passages
Search aliases: Ucal U mathcal
Definition passage: Causal-predictive equivalence · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 19790
[Causal-predictive equivalence] For h,h'∈ H, write h∼_U,rh' when L (Y^u_0:rmid h) = L (Y^u_0:rmid h') for every u∈U^≤ r. The quotient Z_U,r=H/∼_U,r is the finite-horizon causal-predictive semantic space relative to the declared experiment language.
Exact mathematical passage
h\sim_{\Ucal,r}h'
\Ucal^{\le r}Definition passage · 3 passages
Search aliases: Ucal^≤ r U^≤ r Ucal le mathcal le
Definition passage: Causal-predictive equivalence · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 19794
[Causal-predictive equivalence] For h,h'∈ H, write h∼_U,rh' when L (Y^u_0:rmid h) = L (Y^u_0:rmid h') for every u∈U^≤ r. The quotient Z_U,r=H/∼_U,r is the finite-horizon causal-predictive semantic space relative to the declared experiment language.
Exact mathematical passage
\mathcal L\!\left(Y^{u}_{0:r}\mid h\right) = \mathcal L\!\left(Y^{u}_{0:r}\mid h'\right) \qquad \text{for every }u\in\Ucal^{\le r}. \label{eq:b23-causal-equivalence}
\underline gDefinition passage · 4 passages
Search aliases: underline g underline
Definition passage: High-probability transport certificate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3302
[High-probability transport certificate] A random tuple (underline g_A, overlineε_J, overlineε_r, overlineε_D, overlineε_leak, overlineε_hol, overline L) is an (1-α) transport certificate when, jointly with probability at least 1-α, it lower-bounds g_A and upper-bounds every corresponding error or Lipschitz quantity.
Exact mathematical passage
(\underline g_A, \overline\eps_J, \overline\eps_r, \overline\eps_D, \overline\eps_{\mathrm{leak}}, \overline\eps_{\mathrm{hol}}, \overline L)
\underline g_ADefinition passage · 2 passages
Search aliases: underline g_A underline
Definition passage: High-probability transport certificate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3302
[High-probability transport certificate] A random tuple (underline g_A, overlineε_J, overlineε_r, overlineε_D, overlineε_leak, overlineε_hol, overline L) is an (1-α) transport certificate when, jointly with probability at least 1-α, it lower-bounds g_A and upper-bounds every corresponding error or Lipschitz quantity.
Exact mathematical passage
(\underline g_A, \overline\eps_J, \overline\eps_r, \overline\eps_D, \overline\eps_{\mathrm{leak}}, \overline\eps_{\mathrm{hol}}, \overline L)
\mathrm{unresolved}Definition passage · 1 passage
Search aliases: unresolved mathrm
Definition passage: Proof-carrying repair operad · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1246
[Proof-carrying repair operad] A proof-carrying repair operad is a pair O=(overlineO,Valid), where overlineO is a colored symmetric operad of syntactically composable repair programs and Valid_E,W(ω,Π) ∈ {valid,invalid,unresolved} is an evidence- and witness-indexed validation judgment. Only operations judged valid are executable. Unresolved operations remain quarantined.
Exact mathematical passage
\Valid_{\Ecal,\Wcal}(\omega,\Pi) \in \{\mathrm{valid},\mathrm{invalid},\mathrm{unresolved}\}
\mathsf{use}Definition passage · 1 passage
Search aliases: use mathsf
Definition passage: Evidence namespace · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4160
[Evidence namespace] An evidence namespace is a pair (E,use) consisting of protected records and a use ledger. Each evaluation query records which data, metrics, stopping rules, and candidate families were exposed. Reusing evidence is permitted only under a validation rule whose guarantee remains valid conditional on the recorded use history.
Exact mathematical passage
(\Ecal,\mathsf{use})
\mathrm{utility\_gain}Definition passage · 1 passage
Search aliases: utility_gain mathrm
Definition passage: Kernel acceptance predicate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5788
[Kernel acceptance predicate] For revealed protected evidence E, the reference acceptance predicate is Accept(widehat s,widehat s',widehatδ,E) iff · K_edit∩ K_protected=∅ · land restrwidehat s'K_protected =restrwidehat sK_protected · land E[utility_gain]≥γ_min · land E[risk]≤ r_max · land E[provenance_ok]=1 · land E_irr=∅.
Exact mathematical passage
\begin{aligned} \operatorname{Accept}(\widehat s,\widehat s',\widehat\delta,E) \iff{}& K_{\mathrm{edit}}\cap K_{\mathrm{protected}}=\varnothing\\ &\land\ \restr{\widehat s'}{K_{\mathrm{protected}}} =\restr{\widehat s}{K_{\mathrm{protected}}}\\ &\land\ E[\mathrm{utility\_gain}]\geq\gamma_{\min}\\ &\land\ E[\mathrm{risk}]\leq r_{\max}\\ &\land\ E[\mathrm{provenance\_ok}]=1\\ &\land\ E_{\mathrm{irr}}=\varnothing. \end{aligned}
\mathcal VDefinition passage · 6 passages
Search aliases: V mathcal
Definition passage: Research-compiler robustness environment · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 14386
[Research-compiler robustness environment] A research-compiler robustness environment is a tuple U = ( Q, H, A, V, G ), where: item Q is a family of research-problem distributions and generator shifts; item H is a family of archive contaminations and provenance perturbations; item A is a family of integrity, specification, and selection attacks; item V is a family of evaluator portfolios and aggregation rules; item G is a family of benchmark generators, including problems outside the inherited o…
Exact mathematical passage
\mathfrak U = \left( \mathcal Q, \mathcal H, \mathcal A, \mathcal V, \mathcal G \right), \label{eq:robustness-environment}
\mathsf VDefinition passage · 2 passages
Search aliases: V mathsf
Definition passage: Empirical benchmark instance · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5392
[Empirical benchmark instance] A benchmark instance is a tuple B = ( E, A, G, P, W, V, D, S, B, R), where: item E is the environment or transition process; item A is the agent family and functional atlas; item G is the repair grammar and candidate generator; item P is the permitted probe and intervention language; item W is the witness, checkpoint, and effect-mediation layer; item V is the protected evaluation context; item D is the data and evidence partition; item S is the seed, randomization,…
Exact mathematical passage
\mathfrak B = (\mathsf E,\mathsf A,\mathsf G,\mathsf P,\mathsf W,\mathsf V, \mathsf D,\mathsf S,\mathsf B,\mathsf R),
v_0Definition passage · 12 passages
Search aliases: v_0
Definition passage: Path-flat evidence core · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3466
[Path-flat evidence core] The path-flat evidence core at v_0 is E_v_0^flat := ⋂_γ∈π_1(G,v_0) ker(H_γ-I).
Exact mathematical passage
v_0
V_ADefinition passage · 6 passages
Search aliases: V_A
Definition passage: Transport datum · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2668
[Transport datum] Let C_A and C_B be correction data. A transport datum from A to B is a pair T_A→ B=(T_U,T_V), T_U:U_A→ U_B, T_V:V_A→ V_B. Its correction-square defect is Δ_D( T_A→ B) := D_BT_U-T_VD_A. For declared source and target defects r_A∈ V_A and r_B∈ V_B, its defect-identification error is Δ_r( T_A→ B;r_A,r_B) := r_B-T_Vr_A. The datum is exact when both quantities vanish.
Exact mathematical passage
\mathfrak T_{A\to B}=(T_U,T_V), \qquad T_U:U_A\to U_B, \qquad T_V:V_A\to V_B.
V_A^{\flat}Definition passage · 2 passages
Search aliases: V_A^flat flat
Definition passage: Path-flat repair and defect cores · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3041
[Path-flat repair and defect cores] The path-flat repair and defect cores at A are U_A^flat := ⋂_γker(H^U_γ-I), V_A^flat := ⋂_γker(H^V_γ-I).
Exact mathematical passage
U_A^{\flat} := \bigcap_\gamma\ker(H^U_\gamma-I), \qquad V_A^{\flat} := \bigcap_\gamma\ker(H^V_\gamma-I).
V_BDefinition passage · 8 passages
Search aliases: V_B
Definition passage: Transport datum · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2668
[Transport datum] Let C_A and C_B be correction data. A transport datum from A to B is a pair T_A→ B=(T_U,T_V), T_U:U_A→ U_B, T_V:V_A→ V_B. Its correction-square defect is Δ_D( T_A→ B) := D_BT_U-T_VD_A. For declared source and target defects r_A∈ V_A and r_B∈ V_B, its defect-identification error is Δ_r( T_A→ B;r_A,r_B) := r_B-T_Vr_A. The datum is exact when both quantities vanish.
Exact mathematical passage
\mathfrak T_{A\to B}=(T_U,T_V), \qquad T_U:U_A\to U_B, \qquad T_V:V_A\to V_B.
\ValidDefinition passage · 2 passages
Search aliases: Valid mathrm
Definition passage: Proof-carrying repair operad · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1242
[Proof-carrying repair operad] A proof-carrying repair operad is a pair O=(overlineO,Valid), where overlineO is a colored symmetric operad of syntactically composable repair programs and Valid_E,W(ω,Π) ∈ {valid,invalid,unresolved} is an evidence- and witness-indexed validation judgment. Only operations judged valid are executable. Unresolved operations remain quarantined.
Exact mathematical passage
\Ocal=(\overline\Ocal,\Valid),
\mathrm{valid}Definition passage · 2 passages
Search aliases: valid mathrm
Definition passage: Proof-carrying repair operad · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1246
[Proof-carrying repair operad] A proof-carrying repair operad is a pair O=(overlineO,Valid), where overlineO is a colored symmetric operad of syntactically composable repair programs and Valid_E,W(ω,Π) ∈ {valid,invalid,unresolved} is an evidence- and witness-indexed validation judgment. Only operations judged valid are executable. Unresolved operations remain quarantined.
Exact mathematical passage
\Valid_{\Ecal,\Wcal}(\omega,\Pi) \in \{\mathrm{valid},\mathrm{invalid},\mathrm{unresolved}\}
\Valid_{\Ecal,\Wcal}Definition passage · 1 passage
Search aliases: Valid_Ecal,Wcal Valid_E,W Valid Ecal Wcal mathrm mathcal mathcal
Definition passage: Proof-carrying repair operad · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1246
[Proof-carrying repair operad] A proof-carrying repair operad is a pair O=(overlineO,Valid), where overlineO is a colored symmetric operad of syntactically composable repair programs and Valid_E,W(ω,Π) ∈ {valid,invalid,unresolved} is an evidence- and witness-indexed validation judgment. Only operations judged valid are executable. Unresolved operations remain quarantined.
Exact mathematical passage
\Valid_{\Ecal,\Wcal}(\omega,\Pi) \in \{\mathrm{valid},\mathrm{invalid},\mathrm{unresolved}\}
\mathsf WDefinition passage · 8 passages
Search aliases: W mathsf
Definition passage: Closed-walk equalizer core · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3531
[Closed-walk equalizer core] For a possibly noninvertible evaluator graph, let W(v) be the monoid of directed closed walks at v. Define E_v^eq := ⋂_γ∈ W(v)ker(H_γ-I).
Exact mathematical passage
\mathsf W(v)
wDefinition passage · 81 passages
Search aliases: w
Definition passage: Evaluator morphism · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3401
[Evaluator morphism] An evaluator morphism e:v→ w consists of maps S_e:R_v→ R_w, T_e:E_v→ E_w, where S_e is the append-only raw-evidence embedding and T_e transports internal representations. Its protected-evidence defect is δ_obs(e) := sup_r∈ h_v normT_eι_v(r)-ι_wS_e(r), and its historical score defect is δ_J(e) := sup_r∈ h_v normJ_wT_eι_v(r)-J_vι_v(r). The morphism is exact on protected history when both defects vanish.
Exact mathematical passage
e:v\to w
w^\topDefinition passage · 7 passages
Search aliases: w^top top
Definition passage: Weighted meta-regret · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9274
[Weighted meta-regret] For w∈ℝ_>0^7, define Φ_w(r)=w^top r. The weights encode the declared relative value or risk of unresolved module headroom.
Exact mathematical passage
\Phi_w(r)=w^\top r. \label{eq:b8-weighted-regret}
w_iDefinition passage · 8 passages
Search aliases: w_i
Definition passage: External trace probe · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16262
[External trace probe] A Block 17 trace probe is a ridge map widehat w=P z_4+q fit only on the 1,200 probe-calibration tasks. The final audit reports held-out coefficient of determination R^2_probe = 1- frac∑_inormw_i-widehat w_i^2 ∑_inormw_i-overline w^2. The frozen semantic-probe gate is R^2_probe≥0.65 on each final support.
Exact mathematical passage
R^2_{\mathrm{probe}} = 1- \frac{\sum_i\norm{w_i-\widehat w_i}^2} {\sum_i\norm{w_i-\overline w}^2}. \label{eq:b17-probe-r2}
W_tDefinition passage · 7 passages
Search aliases: W_t
Definition passage: Rotating witness schedule · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4174
[Rotating witness schedule] A sequence (E_t,W_t)_t≥0 is a rotating witness schedule when: item E_t∩ W_t=∅ for every t; item W_t is immutable during transaction t; item every component belongs to E_t for at least one t; item every witness replacement is validated by a component outside the same edit set; item protected evidence is transported through a conservative evaluator morphism at every rotation; item every committed round retains a rollback path to a state certified by the preceding witnes…
Exact mathematical passage
(E_t,W_t)_{t\geq0}
\widehat{\mathbf c}Definition passage · 3 passages
Search aliases: widehat c widehat mathbf
Definition passage: Pre-result research receipt · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13727
[Pre-result research receipt] A pre-result receipt has the form Π_i^- = (H_i,O_i,D_i,θ_i, widehat c_i, widehat r_i, F_i,h_i^code,h_i^spec), where H_i is the causal hypothesis, O_i the proposed objective or repair mechanism, D_i the evidence or data construction, θ_i the configuration, widehat c_i the predicted benchmark-effect vector, widehat r_i the predicted protected-risk vector, F_i the declared falsification condition, and h_i^code,h_i^spec immutable digests of the executable and specificat…
Exact mathematical passage
\Pi_i^{-} = (H_i,O_i,D_i,\theta_i, \widehat{\mathbf c}_i, \widehat{\mathbf r}_i, F_i,h_i^{\mathrm{code}},h_i^{\mathrm{spec}}), \label{eq:b13-preresult-receipt}
\widehat{\mathbf c}_iDefinition passage · 3 passages
Search aliases: widehat c_i widehat mathbf
Definition passage: Pre-result research receipt · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13727
[Pre-result research receipt] A pre-result receipt has the form Π_i^- = (H_i,O_i,D_i,θ_i, widehat c_i, widehat r_i, F_i,h_i^code,h_i^spec), where H_i is the causal hypothesis, O_i the proposed objective or repair mechanism, D_i the evidence or data construction, θ_i the configuration, widehat c_i the predicted benchmark-effect vector, widehat r_i the predicted protected-risk vector, F_i the declared falsification condition, and h_i^code,h_i^spec immutable digests of the executable and specificat…
Exact mathematical passage
\Pi_i^{-} = (H_i,O_i,D_i,\theta_i, \widehat{\mathbf c}_i, \widehat{\mathbf r}_i, F_i,h_i^{\mathrm{code}},h_i^{\mathrm{spec}}), \label{eq:b13-preresult-receipt}
\widehat gDefinition passage · 22 passages
Search aliases: widehat g widehat
Definition passage: Lineage-edge receipt · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 858
[Lineage-edge receipt] A lineage edge AxrightarrowΠB has a receipt Π= ( id_A, id_B, θ, W, E_pre, E_post, widehat g, mathscr I, tests, decision, rollback, digest ), recording the parent and child, transformation θ, frozen witness, pre- and post-reveal evidence, precommitted effect claim widehat g, protected invariants, validation results, commit decision, rollback point, and integrity digest.
Exact mathematical passage
\Pi= \left( \mathrm{id}_A, \mathrm{id}_B, \theta, \Wcal, \Ecal_{\mathrm{pre}}, \Ecal_{\mathrm{post}}, \widehat g, \mathscr I, \mathrm{tests}, \mathrm{decision}, \mathrm{rollback}, \mathrm{digest} \right),
\widehat K_{\phi_t(A_t)}^{\delta}Definition passage · 2 passages
Search aliases: widehat K_ϕ_t(A_t)^δ widehat phi delta
Definition passage: ε-intervention sufficiency · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 955
[ε-intervention sufficiency] The map ϕ_t is ε-intervention-sufficient for D_t when sup_δ∈D_t TV( K_A_t^δ, widehat K_ϕ_t(A_t)^δ ) ≤ε.
Exact mathematical passage
\sup_{\delta\in\Dcal_t} \TV\left( K_{A_t}^{\delta}, \widehat K_{\phi_t(A_t)}^{\delta} \right) \leq\eps.
\widehat ODefinition passage · 4 passages
Search aliases: widehat O widehat
Definition passage: Forward implementation refinement · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5470
ntime when: item every executable initial state maps to an authorized abstract initial state; item for every executable transition widehat sxrightarrowawidehat s', there is an abstract transition path α(widehat s)xrightarrowa,*α(widehat s'); item protected observations commute: widehat O(widehat s)=O(α(widehat s)); item receipt compilation commutes with abstraction up to the declared canonical encoding.
Exact mathematical passage
\widehat O(\widehat s)=O(\alpha(\widehat s));
\widehat{\mathbf r}Definition passage · 2 passages
Search aliases: widehat r widehat mathbf
Definition passage: Pre-result research receipt · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13727
[Pre-result research receipt] A pre-result receipt has the form Π_i^- = (H_i,O_i,D_i,θ_i, widehat c_i, widehat r_i, F_i,h_i^code,h_i^spec), where H_i is the causal hypothesis, O_i the proposed objective or repair mechanism, D_i the evidence or data construction, θ_i the configuration, widehat c_i the predicted benchmark-effect vector, widehat r_i the predicted protected-risk vector, F_i the declared falsification condition, and h_i^code,h_i^spec immutable digests of the executable and specificat…
Exact mathematical passage
\Pi_i^{-} = (H_i,O_i,D_i,\theta_i, \widehat{\mathbf c}_i, \widehat{\mathbf r}_i, F_i,h_i^{\mathrm{code}},h_i^{\mathrm{spec}}), \label{eq:b13-preresult-receipt}
\widehat{\mathbf r}_iDefinition passage · 2 passages
Search aliases: widehat r_i widehat mathbf
Definition passage: Pre-result research receipt · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13727
[Pre-result research receipt] A pre-result receipt has the form Π_i^- = (H_i,O_i,D_i,θ_i, widehat c_i, widehat r_i, F_i,h_i^code,h_i^spec), where H_i is the causal hypothesis, O_i the proposed objective or repair mechanism, D_i the evidence or data construction, θ_i the configuration, widehat c_i the predicted benchmark-effect vector, widehat r_i the predicted protected-risk vector, F_i the declared falsification condition, and h_i^code,h_i^spec immutable digests of the executable and specificat…
Exact mathematical passage
\Pi_i^{-} = (H_i,O_i,D_i,\theta_i, \widehat{\mathbf c}_i, \widehat{\mathbf r}_i, F_i,h_i^{\mathrm{code}},h_i^{\mathrm{spec}}), \label{eq:b13-preresult-receipt}
\widehat SDefinition passage · 2 passages
Search aliases: widehat S widehat
Definition passage: Forward implementation refinement · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5455
[Forward implementation refinement] A map α:widehat S→ S forward-refines the runtime when: item every executable initial state maps to an authorized abstract initial state; item for every executable transition widehat sxrightarrowawidehat s', there is an abstract transition path α(widehat s)xrightarrowa,*α(widehat s'); item protected observations commute: widehat O(widehat s)=O(α(widehat s)); item receipt compilation commutes with abstraction up to the declared canonical encoding.
Exact mathematical passage
\alpha:\widehat S\to S
\widehat sDefinition passage · 12 passages
Search aliases: widehat s widehat
Definition passage: Forward implementation refinement · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5462
[Forward implementation refinement] A map α:widehat S→ S forward-refines the runtime when: item every executable initial state maps to an authorized abstract initial state; item for every executable transition widehat sxrightarrowawidehat s', there is an abstract transition path α(widehat s)xrightarrowa,*α(widehat s'); item protected observations commute: widehat O(widehat s)=O(α(widehat s)); item receipt compilation commutes with abstraction up to the declared canonical encoding.
Exact mathematical passage
\widehat s\xrightarrow{a}\widehat s',
\widehat s'Definition passage · 3 passages
Search aliases: widehat s' widehat
Definition passage: Forward implementation refinement · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5462
[Forward implementation refinement] A map α:widehat S→ S forward-refines the runtime when: item every executable initial state maps to an authorized abstract initial state; item for every executable transition widehat sxrightarrowawidehat s', there is an abstract transition path α(widehat s)xrightarrowa,*α(widehat s'); item protected observations commute: widehat O(widehat s)=O(α(widehat s)); item receipt compilation commutes with abstraction up to the declared canonical encoding.
Exact mathematical passage
\widehat s\xrightarrow{a}\widehat s',
\widehat wDefinition passage · 3 passages
Search aliases: widehat w widehat
Definition passage: External trace probe · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16257
[External trace probe] A Block 17 trace probe is a ridge map widehat w=P z_4+q fit only on the 1,200 probe-calibration tasks. The final audit reports held-out coefficient of determination R^2_probe = 1- frac∑_inormw_i-widehat w_i^2 ∑_inormw_i-overline w^2. The frozen semantic-probe gate is R^2_probe≥0.65 on each final support.
Exact mathematical passage
\widehat w=P\,z_4+q \label{eq:b17-external-probe}
\widehat w_iDefinition passage · 1 passage
Search aliases: widehat w_i widehat
Definition passage: External trace probe · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16262
[External trace probe] A Block 17 trace probe is a ridge map widehat w=P z_4+q fit only on the 1,200 probe-calibration tasks. The final audit reports held-out coefficient of determination R^2_probe = 1- frac∑_inormw_i-widehat w_i^2 ∑_inormw_i-overline w^2. The frozen semantic-probe gate is R^2_probe≥0.65 on each final support.
Exact mathematical passage
R^2_{\mathrm{probe}} = 1- \frac{\sum_i\norm{w_i-\widehat w_i}^2} {\sum_i\norm{w_i-\overline w}^2}. \label{eq:b17-probe-r2}
\widehat xDefinition passage · 2 passages
Search aliases: widehat x widehat
Definition passage: Affine observation chart · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17145
[Affine observation chart] For a context c, an affine observation chart is a tuple χ_c=(s_x,t_x,s_y,t_y,σ_c), s_xs_y≠0, with canonicalization widehat x=fracx^obs-t_xs_x, widehat y=fracy^obs-t_ys_y, and support signature σ_c∈ℝ^5. The signature is the sorted vector of observed demonstration inputs predicted by the chart on the five canonical demonstration locations.
Exact mathematical passage
\widehat x=\frac{x^{\mathrm{obs}}-t_x}{s_x}, \qquad \widehat y=\frac{y^{\mathrm{obs}}-t_y}{s_y}, \label{eq:b19-chart-inverse}
\widehat yDefinition passage · 4 passages
Search aliases: widehat y widehat
Definition passage: Invariant-context two-chart grammar · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11390
oℝ^r be a protected context projector whose value is unchanged by the source repair. Let F be a declared family of separators g(h)=1{anglesu,P_ctxh>θ}. The chart extension is G_chart = set(g,bar T_0,bar T_1):g∈F, bar T_j∈ℝ^q×(d+1). Its prediction is widehat y(h)=bar T_g(h)bar h.
Exact mathematical passage
\widehat y(h)=\bar T_{g(h)}\bar h.
\widehat\deltaDefinition passage · 6 passages
Search aliases: widehatδ widehat delta
Definition passage: Kernel acceptance predicate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5788
[Kernel acceptance predicate] For revealed protected evidence E, the reference acceptance predicate is Accept(widehat s,widehat s',widehatδ,E) iff · K_edit∩ K_protected=∅ · land restrwidehat s'K_protected =restrwidehat sK_protected · land E[utility_gain]≥γ_min · land E[risk]≤ r_max · land E[provenance_ok]=1 · land E_irr=∅.
Exact mathematical passage
\begin{aligned} \operatorname{Accept}(\widehat s,\widehat s',\widehat\delta,E) \iff{}& K_{\mathrm{edit}}\cap K_{\mathrm{protected}}=\varnothing\\ &\land\ \restr{\widehat s'}{K_{\mathrm{protected}}} =\restr{\widehat s}{K_{\mathrm{protected}}}\\ &\land\ E[\mathrm{utility\_gain}]\geq\gamma_{\min}\\ &\land\ E[\mathrm{risk}]\leq r_{\max}\\ &\land\ E[\mathrm{provenance\_ok}]=1\\ &\land\ E_{\mathrm{irr}}=\varnothing. \end{aligned}
\widehat\etaDefinition passage · 1 passage
Search aliases: widehatη widehat eta
Definition passage: Quantitative authorized obstruction · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1592
[Quantitative authorized obstruction] The quantitative obstruction magnitude is η_D(r):=normP_obr=dist(r,im D). The normalized obstruction fraction is widehatη_D(r):= η_D(r)/norm r, · r≠0, 0, · r=0.
Exact mathematical passage
\widehat\eta_D(r):= \begin{cases} \eta_D(r)/\norm r,&r\neq0,\\ 0,&r=0. \end{cases}
\widehat\eta_DDefinition passage · 1 passage
Search aliases: widehatη_D widehat eta
Definition passage: Quantitative authorized obstruction · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1592
[Quantitative authorized obstruction] The quantitative obstruction magnitude is η_D(r):=normP_obr=dist(r,im D). The normalized obstruction fraction is widehatη_D(r):= η_D(r)/norm r, · r≠0, 0, · r=0.
Exact mathematical passage
\widehat\eta_D(r):= \begin{cases} \eta_D(r)/\norm r,&r\neq0,\\ 0,&r=0. \end{cases}
\widetilde qDefinition passage · 6 passages
Search aliases: widetilde q widetilde
Definition passage: Learned repair-transport instance · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6465
[Learned repair-transport instance] A learned repair-transport instance is a tuple R_learn = ( X,ϕ,q,widetilde q,A_s, A_t, D, P, G, W), where: item X is the input space; item ϕ is the task feature map; item q and widetilde q are protected and corrupted score maps; item A_s is the source agent; item A_t is a family of target agents excluded from source repair fitting; item D is the train, anchor, calibration, and audit partition; item P is the permitted causal probe language; item G is the repair…
Exact mathematical passage
\mathfrak R_{\mathrm{learn}} = (\mathcal X,\phi,q,\widetilde q,A_s,\mathcal A_t, \mathsf D,\mathsf P,\mathsf G,\mathsf W),
\widetilde TDefinition passage · 5 passages
Search aliases: widetilde T widetilde
Definition passage: Property-transport defect · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 726
[Property-transport defect] For a declared domain K⊆ X, define ε_Q(T;K) = sup_x∈ K norm Q_Y(Tx)-widetilde T_QQ_X(x) .
Exact mathematical passage
\eps_Q(T;K) = \sup_{x\in K} \norm{ Q_Y(Tx)-\widetilde T_QQ_X(x) }.
\widetilde T_QDefinition passage · 5 passages
Search aliases: widetilde T_Q widetilde
Definition passage: Property-transport defect · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 726
[Property-transport defect] For a declared domain K⊆ X, define ε_Q(T;K) = sup_x∈ K norm Q_Y(Tx)-widetilde T_QQ_X(x) .
Exact mathematical passage
\eps_Q(T;K) = \sup_{x\in K} \norm{ Q_Y(Tx)-\widetilde T_QQ_X(x) }.
\mathrm{wit}Definition passage · 2 passages
Search aliases: wit mathrm
Definition passage: Witness-separated ontology transaction · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 19869
[Witness-separated ontology transaction] An ontology transaction consists of construction evidence E_con, a compiled quotient q, a pre-reveal commitment σ(q), and witness evidence E_wit. It is witness separated when E_wit is inaccessible to quotient construction and candidate selection, and the final certificate is computed only after σ(q) is fixed.
Exact mathematical passage
E_{\mathrm{wit}}
XDefinition passage · 138 passages
Search aliases: X
Definition passage: Stratified functional atlas · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 641
[Stratified functional atlas] A stratified functional atlas at generation t is a tuple A_t= ( C, G_t, {X_t^c}_c∈ C, {F_t^c}_c∈ C, {T_e,τ_e,P_e,ε_e}_e∈ E(G_t) ), where: item C is a finite set of stratum colors; item G_t is a directed multigraph of declared interfaces; item X_t^c is the state space of stratum c; item F_t^c is its local transition rule; item each edge e:c→ d carries a bridge T_e:X_t^c→ X_t^d; item τ_e declares the bridge class; item P_e is its preservation profile; item ε_e is a ve…
Exact mathematical passage
\mathfrak A_t= \left( \mathsf C, G_t, \{X_t^c\}_{c\in\mathsf C}, \{F_t^c\}_{c\in\mathsf C}, \{T_e,\tau_e,\Pcal_e,\bm\eps_e\}_{e\in E(G_t)} \right),
\mathcal XDefinition passage · 2 passages
Search aliases: X mathcal
Definition passage: Learned repair-transport instance · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6465
[Learned repair-transport instance] A learned repair-transport instance is a tuple R_learn = ( X,ϕ,q,widetilde q,A_s, A_t, D, P, G, W), where: item X is the input space; item ϕ is the task feature map; item q and widetilde q are protected and corrupted score maps; item A_s is the source agent; item A_t is a family of target agents excluded from source repair fitting; item D is the train, anchor, calibration, and audit partition; item P is the permitted causal probe language; item G is the repair…
Exact mathematical passage
\mathfrak R_{\mathrm{learn}} = (\mathcal X,\phi,q,\widetilde q,A_s,\mathcal A_t, \mathsf D,\mathsf P,\mathsf G,\mathsf W),
xDefinition passage · 220 passages
Search aliases: x
Definition passage: Property-transport defect · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 726
[Property-transport defect] For a declared domain K⊆ X, define ε_Q(T;K) = sup_x∈ K norm Q_Y(Tx)-widetilde T_QQ_X(x) .
Exact mathematical passage
\eps_Q(T;K) = \sup_{x\in K} \norm{ Q_Y(Tx)-\widetilde T_QQ_X(x) }.
x'Definition passage · 7 passages
Search aliases: x'
Definition passage: Semantic twin · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 19844
[Semantic twin] Two observed trajectories x and x' are semantic twins relative to U when they arise from the same protected task and have the same protected future and intervention laws, while nuisance coordinates, rendering variables, or architecture-specific gauges may differ.
Exact mathematical passage
x'
x'_iDefinition passage · 2 passages
Search aliases: x'_i
Definition passage: Per-task Neuralese causal coordinates · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15710
[Per-task Neuralese causal coordinates] For task (x_i,y_i), define n_i · :=L(x_i,y_i;zero)-L(x_i,y_i;Z(x_i)), o_i · :=L(x_i,y_i;Z_π(x_i))-L(x_i,y_i;Z(x_i)), c_i · :=L(x_i,y_i;Z_η(x_i))-L(x_i,y_i;Z(x_i)), s_i · :=L(x_i,y_i;Z(x'_i))-L(x_i,y_i;Z(x_i)), r_i · :=absL(x_i,y_i;replay(Z(x_i)))-L(x_i,y_i;Z(x_i)). These are respectively latent necessity, order sensitivity, corruption sensitivity, task specificity, and replay error.
Exact mathematical passage
n_i &:=L(x_i,y_i;\mathsf{zero})-L(x_i,y_i;Z(x_i)), \label{eq:b16-necessity}\\ o_i &:=L(x_i,y_i;Z_{\pi}(x_i))-L(x_i,y_i;Z(x_i)), \label{eq:b16-order}\\ c_i &:=L(x_i,y_i;Z_{\eta}(x_i))-L(x_i,y_i;Z(x_i)), \label{eq:b16-noise}\\ s_i &:=L(x_i,y_i;Z(x'_i))-L(x_i,y_i;Z(x_i)), \label{eq:b16-specificity}\\ r_i &:=\abs{L(x_i,y_i;\mathsf{replay}(Z(x_i)))-L(x_i,y_i;Z(x_i))}. \label{eq:b16-replay}
x^{\mathrm{obs}}Definition passage · 2 passages
Search aliases: x^obs mathrm
Definition passage: Affine observation chart · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17145
[Affine observation chart] For a context c, an affine observation chart is a tuple χ_c=(s_x,t_x,s_y,t_y,σ_c), s_xs_y≠0, with canonicalization widehat x=fracx^obs-t_xs_x, widehat y=fracy^obs-t_ys_y, and support signature σ_c∈ℝ^5. The signature is the sorted vector of observed demonstration inputs predicted by the chart on the five canonical demonstration locations.
Exact mathematical passage
\widehat x=\frac{x^{\mathrm{obs}}-t_x}{s_x}, \qquad \widehat y=\frac{y^{\mathrm{obs}}-t_y}{s_y}, \label{eq:b19-chart-inverse}
X_cDefinition passage · 4 passages
Search aliases: X_c
Definition passage: Semantic twins · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17126
[Semantic twins] Two observed records x_c∈ X_c and x_d∈ X_d are semantic twins when there exists one protected semantic object z∈ Z such that x_c=ρ_c(z), x_d=ρ_d(z). A semantic-twin group is an orbit {ρ_c(z):c∈ C_z} for a declared finite context set C_z⊆C.
Exact mathematical passage
x_c\in X_c
x_cDefinition passage · 3 passages
Search aliases: x_c
Definition passage: Semantic twins · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17126
[Semantic twins] Two observed records x_c∈ X_c and x_d∈ X_d are semantic twins when there exists one protected semantic object z∈ Z such that x_c=ρ_c(z), x_d=ρ_d(z). A semantic-twin group is an orbit {ρ_c(z):c∈ C_z} for a declared finite context set C_z⊆C.
Exact mathematical passage
x_c\in X_c
X_dDefinition passage · 1 passage
Search aliases: X_d
Definition passage: Semantic twins · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17126
[Semantic twins] Two observed records x_c∈ X_c and x_d∈ X_d are semantic twins when there exists one protected semantic object z∈ Z such that x_c=ρ_c(z), x_d=ρ_d(z). A semantic-twin group is an orbit {ρ_c(z):c∈ C_z} for a declared finite context set C_z⊆C.
Exact mathematical passage
x_d\in X_d
x_dDefinition passage · 2 passages
Search aliases: x_d
Definition passage: Semantic twins · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17126
[Semantic twins] Two observed records x_c∈ X_c and x_d∈ X_d are semantic twins when there exists one protected semantic object z∈ Z such that x_c=ρ_c(z), x_d=ρ_d(z). A semantic-twin group is an orbit {ρ_c(z):c∈ C_z} for a declared finite context set C_z⊆C.
Exact mathematical passage
x_d\in X_d
x_iDefinition passage · 24 passages
Search aliases: x_i
Definition passage: Thresholded approximate flat core · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3587
[Thresholded approximate flat core] For τ≥0, define E_τ^flat := spansetx_i:λ_i≤τ, where x_i is an eigenbasis of A.
Exact mathematical passage
E_{\tau}^{\flat} := \Span\set{x_i:\lambda_i\leq\tau},
X_t^cDefinition passage · 3 passages
Search aliases: X_t^c
Definition passage: Stratified functional atlas · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 641
[Stratified functional atlas] A stratified functional atlas at generation t is a tuple A_t= ( C, G_t, {X_t^c}_c∈ C, {F_t^c}_c∈ C, {T_e,τ_e,P_e,ε_e}_e∈ E(G_t) ), where: item C is a finite set of stratum colors; item G_t is a directed multigraph of declared interfaces; item X_t^c is the state space of stratum c; item F_t^c is its local transition rule; item each edge e:c→ d carries a bridge T_e:X_t^c→ X_t^d; item τ_e declares the bridge class; item P_e is its preservation profile; item ε_e is a ve…
Exact mathematical passage
\mathfrak A_t= \left( \mathsf C, G_t, \{X_t^c\}_{c\in\mathsf C}, \{F_t^c\}_{c\in\mathsf C}, \{T_e,\tau_e,\Pcal_e,\bm\eps_e\}_{e\in E(G_t)} \right),
X_t^dDefinition passage · 1 passage
Search aliases: X_t^d
Definition passage: Stratified functional atlas · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 657
X_t^c}_c∈ C, {F_t^c}_c∈ C, {T_e,τ_e,P_e,ε_e}_e∈ E(G_t) ), where: item C is a finite set of stratum colors; item G_t is a directed multigraph of declared interfaces; item X_t^c is the state space of stratum c; item F_t^c is its local transition rule; item each edge e:c→ d carries a bridge T_e:X_t^c→ X_t^d; item τ_e declares the bridge class; item P_e is its preservation profile; item ε_e is a vector of measured or certified approximation defects.
Exact mathematical passage
T_e:X_t^c\to X_t^d
\XcalDefinition passage · 7 passages
Search aliases: Xcal X mathcal
Definition passage: Neuralese reasoner · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15653
[Neuralese reasoner] A finite-step Neuralese reasoner is a tuple N = (X,Y,H,E,B,U,D,Λ,K), where X is an input space, Y an answer space, H a finite-dimensional latent state space, E:X→ H a prompt encoder, B:H×X→ H a backbone transition, U:H× H× H→ H an update gate, D:H×X→ P(Y) an answer readout, Λ:H→[0,1] a halting map, and K a maximum latent-step budget.
Exact mathematical passage
\Ncal = (\Xcal,\Ycal,H,E,B,U,D,\Lambda,K), \label{eq:b16-neuralese-object}
\xrightarrowDefinition passage · 42 passages
Search aliases: xrightarrow
Definition passage: Lineage-edge receipt · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 854
[Lineage-edge receipt] A lineage edge AxrightarrowΠB has a receipt Π= ( id_A, id_B, θ, W, E_pre, E_post, widehat g, mathscr I, tests, decision, rollback, digest ), recording the parent and child, transformation θ, frozen witness, pre- and post-reveal evidence, precommitted effect claim widehat g, protected invariants, validation results, commit decision, rollback point, and integrity digest.
Exact mathematical passage
A\xrightarrow{\Pi}B
YDefinition passage · 94 passages
Search aliases: Y
Definition passage: Preservation profile · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 703
[Preservation profile] A preservation profile for a bridge T:X→ Y is a declared record P(T)= ( Σ_T, Q_T, ρ_T, ℓ_T, σ_T, A_T ), where Σ_T lists preserved algebraic structures, Q_T lists preserved causal or evaluative queries, ρ_T records provenance behavior, ℓ_T records information loss, σ_T records uncertainty transport, and A_T lists authorized downstream uses.
Exact mathematical passage
T:X\to Y
y^{\mathrm{obs}}Definition passage · 1 passage
Search aliases: y^obs mathrm
Definition passage: Affine observation chart · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17145
[Affine observation chart] For a context c, an affine observation chart is a tuple χ_c=(s_x,t_x,s_y,t_y,σ_c), s_xs_y≠0, with canonicalization widehat x=fracx^obs-t_xs_x, widehat y=fracy^obs-t_ys_y, and support signature σ_c∈ℝ^5. The signature is the sorted vector of observed demonstration inputs predicted by the chart on the five canonical demonstration locations.
Exact mathematical passage
\widehat x=\frac{x^{\mathrm{obs}}-t_x}{s_x}, \qquad \widehat y=\frac{y^{\mathrm{obs}}-t_y}{s_y}, \label{eq:b19-chart-inverse}
Y^{u}_{0:r}Definition passage · 3 passages
Search aliases: Y^u_0:r
Definition passage: Causal-predictive equivalence · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 19794
[Causal-predictive equivalence] For h,h'∈ H, write h∼_U,rh' when L (Y^u_0:rmid h) = L (Y^u_0:rmid h') for every u∈U^≤ r. The quotient Z_U,r=H/∼_U,r is the finite-horizon causal-predictive semantic space relative to the declared experiment language.
Exact mathematical passage
\mathcal L\!\left(Y^{u}_{0:r}\mid h\right) = \mathcal L\!\left(Y^{u}_{0:r}\mid h'\right) \qquad \text{for every }u\in\Ucal^{\le r}. \label{eq:b23-causal-equivalence}
y_iDefinition passage · 8 passages
Search aliases: y_i
Definition passage: Per-task Neuralese causal coordinates · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15709
[Per-task Neuralese causal coordinates] For task (x_i,y_i), define n_i · :=L(x_i,y_i;zero)-L(x_i,y_i;Z(x_i)),
Exact mathematical passage
(x_i,y_i)
y_mDefinition passage · 4 passages
Search aliases: y_m
Definition passage: Executable affine-program task · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16036
[Executable affine-program task] A Block 17 task is τ = (x,m,(a_j,b_j)_j=1^m,A_m,B_m,y_m,g,s), y_m=A_mx+B_m, where g∈{A,B} names the generator and s names the data split. The model input is a padded serialization of x, the instructions, and instruction-presence bits. The protected semantic witness is w(τ) = (A_m,fracB_m3,fracy_m4)∈ℝ^3.
Exact mathematical passage
\tau = (x,m,(a_j,b_j)_{j=1}^{m},A_m,B_m,y_m,g,s), \qquad y_m=A_mx+B_m, \label{eq:b17-task}
Y_{\mathrm{post}}Definition passage · 1 passage
Search aliases: Y_post mathrm
Definition passage: Repair episode · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1385
[Repair episode] A repair episode is a tuple e= ( A, r, u, Y_pre, Y_post, Π, g ).
Exact mathematical passage
e= \left( A, r, u, Y_{\mathrm{pre}}, Y_{\mathrm{post}}, \Pi, \bm g \right).
Y_{\mathrm{pre}}Definition passage · 1 passage
Search aliases: Y_pre mathrm
Definition passage: Repair episode · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1385
[Repair episode] A repair episode is a tuple e= ( A, r, u, Y_pre, Y_post, Π, g ).
Exact mathematical passage
e= \left( A, r, u, Y_{\mathrm{pre}}, Y_{\mathrm{post}}, \Pi, \bm g \right).
\YcalDefinition passage · 7 passages
Search aliases: Ycal Y mathcal
Definition passage: Protected evaluation context · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 470
[Protected evaluation context] A protected evaluation context is a tuple Ξ= ( Z, μ, Y, ν, bar c, λ, mathscr I, B, α ), where: item Z is a task or environment family and μ is its held-out distribution; item Y is the space of protected external traces; item ν:Z×Y→[0,1] is task utility; item bar c:Z×Y→[0,1] is normalized resource cost; item λ≥0 is a cost weight; item mathscr I(A,B)∈{0,1} is a declared invariant-preservation predicate for the transition A→ B; item B is the resource envelope used for…
Exact mathematical passage
\Xi= \left( \Zcal, \mu, \Ycal, \nu, \bar c, \lambda, \mathscr I, \Bcal, \alpha \right),
\mathcal ZDefinition passage · 3 passages
Search aliases: Z mathcal
Definition passage: Repair selector · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7250
[Repair selector] A repair selector is a measurable map K: Z^B⟶{0,…,B-1}, where Z is the space of candidate receipts available before the protected audit outcome is revealed. The selected repair is c_e,K(Z_e).
Exact mathematical passage
K:\mathcal Z^B\longrightarrow\{0,\ldots,B-1\},
\mathcal Z^BDefinition passage · 1 passage
Search aliases: Z^B mathcal
Definition passage: Repair selector · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7250
[Repair selector] A repair selector is a measurable map K: Z^B⟶{0,…,B-1}, where Z is the space of candidate receipts available before the protected audit outcome is revealed. The selected repair is c_e,K(Z_e).
Exact mathematical passage
K:\mathcal Z^B\longrightarrow\{0,\ldots,B-1\},
z_1Definition passage · 8 passages
Search aliases: z_1
Definition passage: Neuralese trace · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15679
[Neuralese trace] For input x and fixed depth K, the injected Neuralese trace is Z_K(x)=(z_1(x),…,z_K(x))∈ H^K. The updated trace is H_K(x)=(h_1(x),…,h_K(x)). Every receipt must state which trace convention is being intervened on. Block 16 uses the injected trace because this is the point at which the Hyper runtime exposes an exact intervention hook.
Exact mathematical passage
Z_K(x)=(z_1(x),\ldots,z_K(x))\in H^K.
z_4Definition passage · 2 passages
Search aliases: z_4
Definition passage: External trace probe · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16257
[External trace probe] A Block 17 trace probe is a ridge map widehat w=P z_4+q fit only on the 1,200 probe-calibration tasks. The final audit reports held-out coefficient of determination R^2_probe = 1- frac∑_inormw_i-widehat w_i^2 ∑_inormw_i-overline w^2. The frozen semantic-probe gate is R^2_probe≥0.65 on each final support.
Exact mathematical passage
\widehat w=P\,z_4+q \label{eq:b17-external-probe}
Z_eDefinition passage · 3 passages
Search aliases: Z_e
Definition passage: Repair selector · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7253
[Repair selector] A repair selector is a measurable map K: Z^B⟶{0,…,B-1}, where Z is the space of candidate receipts available before the protected audit outcome is revealed. The selected repair is c_e,K(Z_e).
Exact mathematical passage
c_{e,K(Z_e)}
Z_KDefinition passage · 1 passage
Search aliases: Z_K
Definition passage: Neuralese trace · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15679
[Neuralese trace] For input x and fixed depth K, the injected Neuralese trace is Z_K(x)=(z_1(x),…,z_K(x))∈ H^K. The updated trace is H_K(x)=(h_1(x),…,h_K(x)). Every receipt must state which trace convention is being intervened on. Block 16 uses the injected trace because this is the point at which the Hyper runtime exposes an exact intervention hook.
Exact mathematical passage
Z_K(x)=(z_1(x),\ldots,z_K(x))\in H^K.
z_KDefinition passage · 1 passage
Search aliases: z_K
Definition passage: Neuralese trace · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15679
[Neuralese trace] For input x and fixed depth K, the injected Neuralese trace is Z_K(x)=(z_1(x),…,z_K(x))∈ H^K. The updated trace is H_K(x)=(h_1(x),…,h_K(x)). Every receipt must state which trace convention is being intervened on. Block 16 uses the injected trace because this is the point at which the Hyper runtime exposes an exact intervention hook.
Exact mathematical passage
Z_K(x)=(z_1(x),\ldots,z_K(x))\in H^K.
Z_MDefinition passage · 3 passages
Search aliases: Z_M
Definition passage: Recursive abstraction estimand · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6265
[Recursive abstraction estimand] Let F_post be foreign defect families revealed after the source repair. Define τ_M^future = E_A∼ F_post [ G_A(Z_M=1)-G_A(Z_M=0) ].
Exact mathematical passage
\tau_M^{\mathrm{future}} = \mathbb E_{A\sim\mathcal F_{\mathrm{post}}} \left[ G_A(Z_M=1)-G_A(Z_M=0) \right].
Z_{\Ucal,r}Definition passage · 3 passages
Search aliases: Z_Ucal,r Z_U,r Ucal mathcal
Definition passage: Causal-predictive equivalence · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 19803
[Causal-predictive equivalence] For h,h'∈ H, write h∼_U,rh' when L (Y^u_0:rmid h) = L (Y^u_0:rmid h') for every u∈U^≤ r. The quotient Z_U,r=H/∼_U,r is the finite-horizon causal-predictive semantic space relative to the declared experiment language.
Exact mathematical passage
Z_{\Ucal,r}=H/{\sim_{\Ucal,r}} \label{eq:b23-semantic-quotient}
Z_{\eta}Definition passage · 1 passage
Search aliases: Z_η eta
Definition passage: Per-task Neuralese causal coordinates · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15710
[Per-task Neuralese causal coordinates] For task (x_i,y_i), define n_i · :=L(x_i,y_i;zero)-L(x_i,y_i;Z(x_i)), o_i · :=L(x_i,y_i;Z_π(x_i))-L(x_i,y_i;Z(x_i)), c_i · :=L(x_i,y_i;Z_η(x_i))-L(x_i,y_i;Z(x_i)), s_i · :=L(x_i,y_i;Z(x'_i))-L(x_i,y_i;Z(x_i)), r_i · :=absL(x_i,y_i;replay(Z(x_i)))-L(x_i,y_i;Z(x_i)). These are respectively latent necessity, order sensitivity, corruption sensitivity, task specificity, and replay error.
Exact mathematical passage
n_i &:=L(x_i,y_i;\mathsf{zero})-L(x_i,y_i;Z(x_i)), \label{eq:b16-necessity}\\ o_i &:=L(x_i,y_i;Z_{\pi}(x_i))-L(x_i,y_i;Z(x_i)), \label{eq:b16-order}\\ c_i &:=L(x_i,y_i;Z_{\eta}(x_i))-L(x_i,y_i;Z(x_i)), \label{eq:b16-noise}\\ s_i &:=L(x_i,y_i;Z(x'_i))-L(x_i,y_i;Z(x_i)), \label{eq:b16-specificity}\\ r_i &:=\abs{L(x_i,y_i;\mathsf{replay}(Z(x_i)))-L(x_i,y_i;Z(x_i))}. \label{eq:b16-replay}
Z_{\pi}Definition passage · 1 passage
Search aliases: Z_π pi
Definition passage: Per-task Neuralese causal coordinates · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15710
[Per-task Neuralese causal coordinates] For task (x_i,y_i), define n_i · :=L(x_i,y_i;zero)-L(x_i,y_i;Z(x_i)), o_i · :=L(x_i,y_i;Z_π(x_i))-L(x_i,y_i;Z(x_i)), c_i · :=L(x_i,y_i;Z_η(x_i))-L(x_i,y_i;Z(x_i)), s_i · :=L(x_i,y_i;Z(x'_i))-L(x_i,y_i;Z(x_i)), r_i · :=absL(x_i,y_i;replay(Z(x_i)))-L(x_i,y_i;Z(x_i)). These are respectively latent necessity, order sensitivity, corruption sensitivity, task specificity, and replay error.
Exact mathematical passage
n_i &:=L(x_i,y_i;\mathsf{zero})-L(x_i,y_i;Z(x_i)), \label{eq:b16-necessity}\\ o_i &:=L(x_i,y_i;Z_{\pi}(x_i))-L(x_i,y_i;Z(x_i)), \label{eq:b16-order}\\ c_i &:=L(x_i,y_i;Z_{\eta}(x_i))-L(x_i,y_i;Z(x_i)), \label{eq:b16-noise}\\ s_i &:=L(x_i,y_i;Z(x'_i))-L(x_i,y_i;Z(x_i)), \label{eq:b16-specificity}\\ r_i &:=\abs{L(x_i,y_i;\mathsf{replay}(Z(x_i)))-L(x_i,y_i;Z(x_i))}. \label{eq:b16-replay}
\mathsf{zero}Definition passage · 4 passages
Search aliases: zero mathsf
Definition passage: Per-task Neuralese causal coordinates · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15710
[Per-task Neuralese causal coordinates] For task (x_i,y_i), define n_i · :=L(x_i,y_i;zero)-L(x_i,y_i;Z(x_i)), o_i · :=L(x_i,y_i;Z_π(x_i))-L(x_i,y_i;Z(x_i)), c_i · :=L(x_i,y_i;Z_η(x_i))-L(x_i,y_i;Z(x_i)), s_i · :=L(x_i,y_i;Z(x'_i))-L(x_i,y_i;Z(x_i)), r_i · :=absL(x_i,y_i;replay(Z(x_i)))-L(x_i,y_i;Z(x_i)). These are respectively latent necessity, order sensitivity, corruption sensitivity, task specificity, and replay error.
Exact mathematical passage
n_i &:=L(x_i,y_i;\mathsf{zero})-L(x_i,y_i;Z(x_i)), \label{eq:b16-necessity}\\ o_i &:=L(x_i,y_i;Z_{\pi}(x_i))-L(x_i,y_i;Z(x_i)), \label{eq:b16-order}\\ c_i &:=L(x_i,y_i;Z_{\eta}(x_i))-L(x_i,y_i;Z(x_i)), \label{eq:b16-noise}\\ s_i &:=L(x_i,y_i;Z(x'_i))-L(x_i,y_i;Z(x_i)), \label{eq:b16-specificity}\\ r_i &:=\abs{L(x_i,y_i;\mathsf{replay}(Z(x_i)))-L(x_i,y_i;Z(x_i))}. \label{eq:b16-replay}
|Definition passage · 33 passages
Search aliases: |
Definition passage: Two-level repair obstruction · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2312
[Two-level repair obstruction] The local obstruction on U_α is [r|_U_α]∈coker D_α. When every local class vanishes, the global gluing obstruction is [d_αβ]∈check H^1( U,ker D).
Exact mathematical passage
[r|_{U_\alpha}]\in\coker D_\alpha.
|_{U_\alpha}Definition passage · 4 passages
Search aliases: |_U_α alpha
Definition passage: Two-level repair obstruction · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2312
[Two-level repair obstruction] The local obstruction on U_α is [r|_U_α]∈coker D_α. When every local class vanishes, the global gluing obstruction is [d_αβ]∈check H^1( U,ker D).
Exact mathematical passage
[r|_{U_\alpha}]\in\coker D_\alpha.
\timesDefinition passage · 88 passages
Search aliases: × times
Definition passage: Protected evaluation context · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 488
[Protected evaluation context] A protected evaluation context is a tuple Ξ= ( Z, μ, Y, ν, bar c, λ, mathscr I, B, α ), where: item Z is a task or environment family and μ is its held-out distribution; item Y is the space of protected external traces; item ν:Z×Y→[0,1] is task utility; item bar c:Z×Y→[0,1] is normalized resource cost; item λ≥0 is a cost weight; item mathscr I(A,B)∈{0,1} is a declared invariant-preservation predicate for the transition A→ B; item B is the resource envelope used for…
Exact mathematical passage
\nu:\Zcal\times\Ycal\to[0,1]
\gamma_{\min}Definition passage · 3 passages
Search aliases: γ_min gamma min
Definition passage: Kernel acceptance predicate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5788
[Kernel acceptance predicate] For revealed protected evidence E, the reference acceptance predicate is Accept(widehat s,widehat s',widehatδ,E) iff · K_edit∩ K_protected=∅ · land restrwidehat s'K_protected =restrwidehat sK_protected · land E[utility_gain]≥γ_min · land E[risk]≤ r_max · land E[provenance_ok]=1 · land E_irr=∅.
Exact mathematical passage
\begin{aligned} \operatorname{Accept}(\widehat s,\widehat s',\widehat\delta,E) \iff{}& K_{\mathrm{edit}}\cap K_{\mathrm{protected}}=\varnothing\\ &\land\ \restr{\widehat s'}{K_{\mathrm{protected}}} =\restr{\widehat s}{K_{\mathrm{protected}}}\\ &\land\ E[\mathrm{utility\_gain}]\geq\gamma_{\min}\\ &\land\ E[\mathrm{risk}]\leq r_{\max}\\ &\land\ E[\mathrm{provenance\_ok}]=1\\ &\land\ E_{\mathrm{irr}}=\varnothing. \end{aligned}
\gamma_tDefinition passage · 2 passages
Search aliases: γ_t gamma
Definition passage: Context-indexed lineage edge · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10501
[Context-indexed lineage edge] Let Ξ_t be the protected context for edge t→ t+1. A sealed lineage edge K_txrightarrowΠ_t+1K_t+1 is γ_t-productive in Ξ_t when its receipt replays, its resource and witness gates pass, and a fresh paired audit certifies M_Ξ_t(K_t+1)-M_Ξ_t(K_t)≥ γ_t>0.
Exact mathematical passage
\gamma_t
\Gamma_{\Xi,h,B}Definition passage · 2 passages
Search aliases: Γ_Ξ,h,B Gamma Xi
Definition passage: Bounded recursive improvement step · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 575
ursive improvement step] A lineage transition L_t→ L_t+1 is a bounded recursive improvement step relative to (Ξ,h,B) when: item its lineage receipt is valid and the protected evidence ledger embeds into the next ledger; item F_t+1(Ξ)≥ F_t(Ξ); item on a foreign held-out target distribution D_foreign, E_A∼ D_foreign Γ_Ξ,h,B(K_t+1;A) > E_A∼ D_foreign Γ_Ξ,h,B(K_t;A); item evaluator risk does not increase beyond a declared tolerance; item the improvement is obtained under matched model access, candid…
Exact mathematical passage
\mathbb E_{A\sim\mathcal D_{\mathrm{foreign}}} \Gamma_{\Xi,h,B}(K_{t+1};A) > \mathbb E_{A\sim\mathcal D_{\mathrm{foreign}}} \Gamma_{\Xi,h,B}(K_t;A);
\delta_JDefinition passage · 5 passages
Search aliases: δ_J delta
Definition passage: Evaluator morphism · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3415
to w consists of maps S_e:R_v→ R_w, T_e:E_v→ E_w, where S_e is the append-only raw-evidence embedding and T_e transports internal representations. Its protected-evidence defect is δ_obs(e) := sup_r∈ h_v normT_eι_v(r)-ι_wS_e(r), and its historical score defect is δ_J(e) := sup_r∈ h_v normJ_wT_eι_v(r)-J_vι_v(r). The morphism is exact on protected history when both defects vanish.
Exact mathematical passage
\delta_J(e) := \sup_{r\in\mathfrak h_v} \norm{J_wT_e\iota_v(r)-J_v\iota_v(r)}.
\delta_{\mathrm{obs}}Definition passage · 4 passages
Search aliases: δ_obs delta mathrm
Definition passage: Evaluator morphism · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3408
[Evaluator morphism] An evaluator morphism e:v→ w consists of maps S_e:R_v→ R_w, T_e:E_v→ E_w, where S_e is the append-only raw-evidence embedding and T_e transports internal representations. Its protected-evidence defect is δ_obs(e) := sup_r∈ h_v normT_eι_v(r)-ι_wS_e(r), and its historical score defect is δ_J(e) := sup_r∈ h_v normJ_wT_eι_v(r)-J_vι_v(r). The morphism is exact on protected history when both defects vanish.
Exact mathematical passage
\delta_{\mathrm{obs}}(e) := \sup_{r\in\mathfrak h_v} \norm{T_e\iota_v(r)-\iota_wS_e(r)},
\Delta_QDefinition passage · 5 passages
Search aliases: Δ_Q Delta
Definition passage: Selector-level metaproductivity · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7272
[Selector-level metaproductivity] Let Q be a held-out distribution over target episodes. A child selector K_1 has selector-level metaproductivity over parent K_0 when Δ_Q(K_1,K_0) := E_e∼ Q [ G_e(K_1)-G_e(K_0) ] >0, under a matched comparison, where G_e(K) is protected utility gain relative to the no-op on episode e.
Exact mathematical passage
\Delta_Q(K_1,K_0) := \mathbb E_{e\sim Q} \left[ G_e(K_1)-G_e(K_0) \right] >0, \label{eq:block07-metaproductivity}
\eta^U_\gammaDefinition passage · 1 passage
Search aliases: η^U_γ eta gamma
Definition passage: Repair holonomy residual · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3086
[Repair holonomy residual] For a correction u and loop γ, define η^U_γ(u) := norm(H^U_γ-I)u. For a defect r, define η^V_γ(r) := norm(H^V_γ-I)r.
Exact mathematical passage
\eta^U_\gamma(u) := \norm{(H^U_\gamma-I)u}.
\eta^V_\gammaDefinition passage · 1 passage
Search aliases: η^V_γ eta gamma
Definition passage: Repair holonomy residual · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3092
[Repair holonomy residual] For a correction u and loop γ, define η^U_γ(u) := norm(H^U_γ-I)u. For a defect r, define η^V_γ(r) := norm(H^V_γ-I)r.
Exact mathematical passage
\eta^V_\gamma(r) := \norm{(H^V_\gamma-I)r}.
\eta_{0:T}Definition passage · 1 passage
Search aliases: η_0:T eta
Definition passage: Approximate semantic debt · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5005
[Approximate semantic debt] If an extension preserves old semantics only approximately, let η_t bound its worst protected interpretation error after transport. Under nonexpansive semantic transports, the cumulative debt through stage T is η_0:T ≤ ∑_t=0^T-1η_t.
Exact mathematical passage
\eta_{0:T} \leq \sum_{t=0}^{T-1}\eta_t.
\eta_{\mathrm{hol}}Definition passage · 1 passage
Search aliases: η_hol eta mathrm
Definition passage: Nonlinear holonomy residual · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2549
[Nonlinear holonomy residual] If the S_i are metric spaces, define η_hol(s_o) := max_e∈ E∖ Td(H_e(s_o),s_o).
Exact mathematical passage
\eta_{\mathrm{hol}}(s_o) := \max_{e\in E\setminus T}d\bigl(H_e(s_o),s_o\bigr).
\eta_{\mathrm{loc}}^2Definition passage · 1 passage
Search aliases: η_loc^2 eta mathrm
Definition passage: Quantitative gluing obstruction · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2409
[Quantitative gluing obstruction] The graph-level quantitative gluing obstruction is η_glue(e) :=normP_cyce. The local-to-global repair certificate is the pair ( η_loc, η_glue ), η_loc^2 :=∑_αnormD_α u_α+r_α^2.
Exact mathematical passage
\left( \eta_{\mathrm{loc}}, \eta_{\mathrm{glue}} \right), \qquad \eta_{\mathrm{loc}}^2 :=\sum_{\alpha}\norm{D_\alpha u_\alpha+r_\alpha}^2.
\eta_{\mathrm{new}}Definition passage · 1 passage
Search aliases: η_new eta mathrm
Definition passage: Constructor-expansion certificate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13142
[Constructor-expansion certificate] A constructor-expansion certificate is a tuple Π_con = (C_t,C_t^+,η_old,η_new, H,κ,C_model,Π_foreign), where H is a nonzero higher-order derivative witness, κ is held-out obstruction reduction, and C_model records constructor complexity.
Exact mathematical passage
\Pi_{\mathrm{con}} = (\Ccal_t,\Ccal_t^+,\eta_{\mathrm{old}},\eta_{\mathrm{new}}, \mathsf H,\kappa,C_{\mathrm{model}},\Pi_{\mathrm{foreign}}),
\eta_{\mathrm{old}}Definition passage · 1 passage
Search aliases: η_old eta mathrm
Definition passage: Constructor-expansion certificate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13142
[Constructor-expansion certificate] A constructor-expansion certificate is a tuple Π_con = (C_t,C_t^+,η_old,η_new, H,κ,C_model,Π_foreign), where H is a nonzero higher-order derivative witness, κ is held-out obstruction reduction, and C_model records constructor complexity.
Exact mathematical passage
\Pi_{\mathrm{con}} = (\Ccal_t,\Ccal_t^+,\eta_{\mathrm{old}},\eta_{\mathrm{new}}, \mathsf H,\kappa,C_{\mathrm{model}},\Pi_{\mathrm{foreign}}),
\eta_{\mathrm{sem}}Definition passage · 4 passages
Search aliases: η_sem eta mathrm
Definition passage: Semantic holonomy residual · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3642
[Semantic holonomy residual] For a represented evidence state x∈ E_v and loop γ at v, define η_sem(x;γ) := norm(H_γ-I)x.
Exact mathematical passage
\eta_{\mathrm{sem}}(x;\gamma) := \norm{(H_\gamma-I)x}.
\eta_tDefinition passage · 11 passages
Search aliases: η_t eta
Definition passage: Approximate semantic debt · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5004
[Approximate semantic debt] If an extension preserves old semantics only approximately, let η_t bound its worst protected interpretation error after transport. Under nonexpansive semantic transports, the cumulative debt through stage T is η_0:T ≤ ∑_t=0^T-1η_t.
Exact mathematical passage
\eta_t
\ThetaDefinition passage · 7 passages
Search aliases: Θ Theta
Definition passage: Meta-repair · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1459
[Meta-repair] A meta-repair is an authorized transformation Θ_t:O_t⟶O_t+1 that changes one or more of: item failure-signature recognition; item causal-hypothesis generation; item candidate proposal distributions; item repair composition and scheduling; item receipt validation or confidence allocation; item repair abstraction and foreign-transfer testing.
Exact mathematical passage
\Theta_t:\Ocal_t\longrightarrow\Ocal_{t+1}
\theta_iDefinition passage · 2 passages
Search aliases: θ_i theta
Definition passage: Pre-result research receipt · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13727
[Pre-result research receipt] A pre-result receipt has the form Π_i^- = (H_i,O_i,D_i,θ_i, widehat c_i, widehat r_i, F_i,h_i^code,h_i^spec), where H_i is the causal hypothesis, O_i the proposed objective or repair mechanism, D_i the evidence or data construction, θ_i the configuration, widehat c_i the predicted benchmark-effect vector, widehat r_i the predicted protected-risk vector, F_i the declared falsification condition, and h_i^code,h_i^spec immutable digests of the executable and specificat…
Exact mathematical passage
\Pi_i^{-} = (H_i,O_i,D_i,\theta_i, \widehat{\mathbf c}_i, \widehat{\mathbf r}_i, F_i,h_i^{\mathrm{code}},h_i^{\mathrm{spec}}), \label{eq:b13-preresult-receipt}
\Theta_tDefinition passage · 1 passage
Search aliases: Θ_t Theta
Definition passage: Meta-repair · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1459
[Meta-repair] A meta-repair is an authorized transformation Θ_t:O_t⟶O_t+1 that changes one or more of: item failure-signature recognition; item causal-hypothesis generation; item candidate proposal distributions; item repair composition and scheduling; item receipt validation or confidence allocation; item repair abstraction and foreign-transfer testing.
Exact mathematical passage
\Theta_t:\Ocal_t\longrightarrow\Ocal_{t+1}
\iota_tDefinition passage · 4 passages
Search aliases: ι_t iota
Definition passage: Append-only evidence embedding · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 804
[Append-only evidence embedding] Evidence preservation from generation t to t+1 is an injective map ι_t:E_t↪E_t+1 that preserves payload, source, timestamp order, precommitment status, provenance type, and cryptographic integrity fields. A later correction may append a refutation or reinterpretation receipt, but it may not silently alter the old entry.
Exact mathematical passage
\iota_t:\Ecal_t\hookrightarrow\Ecal_{t+1}
\iota_wDefinition passage · 3 passages
Search aliases: ι_w iota
Definition passage: Evaluator morphism · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3408
[Evaluator morphism] An evaluator morphism e:v→ w consists of maps S_e:R_v→ R_w, T_e:E_v→ E_w, where S_e is the append-only raw-evidence embedding and T_e transports internal representations. Its protected-evidence defect is δ_obs(e) := sup_r∈ h_v normT_eι_v(r)-ι_wS_e(r), and its historical score defect is δ_J(e) := sup_r∈ h_v normJ_wT_eι_v(r)-J_vι_v(r). The morphism is exact on protected history when both defects vanish.
Exact mathematical passage
\delta_{\mathrm{obs}}(e) := \sup_{r\in\mathfrak h_v} \norm{T_e\iota_v(r)-\iota_wS_e(r)},
\kappa_{a,b;c,d}^{2}Definition passage · 1 passage
Search aliases: κ_a,b;c,d^2 kappa
Definition passage: Architecture--context curvature · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17734
[Architecture--context curvature] For architectures a,b and charts c,d, the transport-square defect is Ω_a,b;c,d := C^a_c→ dT^c_b→ a - T^d_b→ aC^b_c→ d. Given an audit distribution P_b,c on target trajectories, its normalized root-mean-square curvature is κ_a,b;c,d^2 = frac E_h∼ P_b,c normΩ_a,b;c,dh^2 E_z∼ P_a,dnormz^2 .
Exact mathematical passage
\kappa_{a,b;c,d}^{2} = \frac{ \mathbb E_{h\sim P_{b,c}} \norm{\Omega_{a,b;c,d}h}^{2} }{ \mathbb E_{z\sim P_{a,d}}\norm{z}^{2} }. \label{eq:b20-normalized-curvature}
\lambda_aDefinition passage · 6 passages
Search aliases: λ_a lambda
Definition passage: Block-scalar response model · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12170
[Block-scalar response model] The response family is block scalar on [source reference] when R_j=∑_a=1^mλ_ajP_a, λ_aj∈ℝ. The response signature of block a is λ_a=(λ_a1,…,λ_ak)∈ℝ^k. The signatures are separating when λ_a≠λ_b for every a≠ b.
Exact mathematical passage
\lambda_a=(\lambda_{a1},\ldots,\lambda_{ak})\in\R^k.
\lambda_{a1}Definition passage · 2 passages
Search aliases: λ_a1 lambda
Definition passage: Block-scalar response model · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12170
[Block-scalar response model] The response family is block scalar on [source reference] when R_j=∑_a=1^mλ_ajP_a, λ_aj∈ℝ. The response signature of block a is λ_a=(λ_a1,…,λ_ak)∈ℝ^k. The signatures are separating when λ_a≠λ_b for every a≠ b.
Exact mathematical passage
\lambda_a=(\lambda_{a1},\ldots,\lambda_{ak})\in\R^k.
\lambda_{aj}Definition passage · 4 passages
Search aliases: λ_aj lambda
Definition passage: Block-scalar response model · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12163
[Block-scalar response model] The response family is block scalar on [source reference] when R_j=∑_a=1^mλ_ajP_a, λ_aj∈ℝ. The response signature of block a is λ_a=(λ_a1,…,λ_ak)∈ℝ^k. The signatures are separating when λ_a≠λ_b for every a≠ b.
Exact mathematical passage
R_j=\sum_{a=1}^{m}\lambda_{aj}P_a, \qquad \lambda_{aj}\in\R. \label{eq:b11-block-scalar}
\lambda_{ak}Definition passage · 1 passage
Search aliases: λ_ak lambda
Definition passage: Block-scalar response model · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12170
[Block-scalar response model] The response family is block scalar on [source reference] when R_j=∑_a=1^mλ_ajP_a, λ_aj∈ℝ. The response signature of block a is λ_a=(λ_a1,…,λ_ak)∈ℝ^k. The signatures are separating when λ_a≠λ_b for every a≠ b.
Exact mathematical passage
\lambda_a=(\lambda_{a1},\ldots,\lambda_{ak})\in\R^k.
\lambda_{am}Definition passage · 1 passage
Search aliases: λ_am lambda
Definition passage: Block-scalar response signature · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12906
[Block-scalar response signature] Let H=⊕_a=1^kH_a be a reducing decomposition for every R_j. The response language is block scalar when restrR_jH_a=λ_ajI_H_a for scalars λ_aj. The vector σ_R(H_a) = (λ_a1,…,λ_am) is the response signature of block H_a.
Exact mathematical passage
\sigma_{\Rcal}(H_a) = (\lambda_{a1},\ldots,\lambda_{am})
\lambda_bDefinition passage · 3 passages
Search aliases: λ_b lambda
Definition passage: Block-scalar response model · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12173
response model] The response family is block scalar on [source reference] when R_j=∑_a=1^mλ_ajP_a, λ_aj∈ℝ. The response signature of block a is λ_a=(λ_a1,…,λ_ak)∈ℝ^k. The signatures are separating when λ_a≠λ_b for every a≠ b.
Exact mathematical passage
\lambda_a\neq\lambda_b
\lambda_{\mathrm{geo}}Definition passage · 1 passage
Search aliases: λ_geo lambda mathrm
Definition passage: Cross-implementation repair schema · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16726
[Cross-implementation repair schema] A Block 18 repair schema is a tuple ρ=(a,λ_sem,λ_prog, λ_geo,χ), where a is an adapter constructor, the three nonnegative λ values weight final semantic, progressive semantic, and latent-geometry witnesses, and χ records whether witness labels are correctly bound or shuffled. A schema is cross-implementation when the same tuple is instantiated through each cell family's declared adapter interface without architecture-specific candidate selection.
Exact mathematical passage
\rho=(a,\lambda_{\mathrm{sem}},\lambda_{\mathrm{prog}}, \lambda_{\mathrm{geo}},\chi), \label{eq:b18-repair-schema}
\lambda_iDefinition passage · 3 passages
Search aliases: λ_i lambda
Definition passage: Thresholded approximate flat core · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3587
[Thresholded approximate flat core] For τ≥0, define E_τ^flat := spansetx_i:λ_i≤τ, where x_i is an eigenbasis of A.
Exact mathematical passage
E_{\tau}^{\flat} := \Span\set{x_i:\lambda_i\leq\tau},
\lambda_{\mathrm{prog}}Definition passage · 1 passage
Search aliases: λ_prog lambda mathrm
Definition passage: Cross-implementation repair schema · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16726
[Cross-implementation repair schema] A Block 18 repair schema is a tuple ρ=(a,λ_sem,λ_prog, λ_geo,χ), where a is an adapter constructor, the three nonnegative λ values weight final semantic, progressive semantic, and latent-geometry witnesses, and χ records whether witness labels are correctly bound or shuffled. A schema is cross-implementation when the same tuple is instantiated through each cell family's declared adapter interface without architecture-specific candidate selection.
Exact mathematical passage
\rho=(a,\lambda_{\mathrm{sem}},\lambda_{\mathrm{prog}}, \lambda_{\mathrm{geo}},\chi), \label{eq:b18-repair-schema}
\lambda_{\mathrm{sem}}Definition passage · 1 passage
Search aliases: λ_sem lambda mathrm
Definition passage: Cross-implementation repair schema · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16726
[Cross-implementation repair schema] A Block 18 repair schema is a tuple ρ=(a,λ_sem,λ_prog, λ_geo,χ), where a is an adapter constructor, the three nonnegative λ values weight final semantic, progressive semantic, and latent-geometry witnesses, and χ records whether witness labels are correctly bound or shuffled. A schema is cross-implementation when the same tuple is instantiated through each cell family's declared adapter interface without architecture-specific candidate selection.
Exact mathematical passage
\rho=(a,\lambda_{\mathrm{sem}},\lambda_{\mathrm{prog}}, \lambda_{\mathrm{geo}},\chi), \label{eq:b18-repair-schema}
\mu_0Definition passage · 5 passages
Search aliases: μ_0 mu
Definition passage: Adequate approximate evidence core · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3728
[Adequate approximate evidence core] A subspace F is (τ,μ_0)-adequate when: item its weighted holonomy energy is at most τ per unit norm; item μ(F;Δ_prot)≥μ_0>0.
Exact mathematical passage
(\tau,\mu_0)
\mu_sDefinition passage · 3 passages
Search aliases: μ_s mu
Definition passage: Unpaired intervention-orbit transport instance · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 19549
[Unpaired intervention-orbit transport instance] An instance is a tuple U = (H_s,H_t,μ_s,μ_t, U, {R_u^s}_u∈ U, {R_v^t}_v∈ V, G_rm gauge, W), where μ_s and μ_t are independently sampled latent distributions, R_u^s and R_v^t are within-system intervention-response operators, G_rm gauge is the declared harmless gauge group, and W is a witness hidden from construction. No map between individual samples from μ_s and μ_t is available to the compiler.
Exact mathematical passage
\mathfrak U = (H_s,H_t,\mu_s,\mu_t,\mathcal U, \{R_u^s\}_{u\in\mathcal U}, \{R_v^t\}_{v\in\mathcal V}, \mathcal G_{\rm gauge},\mathsf W),
\mu_tDefinition passage · 4 passages
Search aliases: μ_t mu
Definition passage: Unpaired intervention-orbit transport instance · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 19549
[Unpaired intervention-orbit transport instance] An instance is a tuple U = (H_s,H_t,μ_s,μ_t, U, {R_u^s}_u∈ U, {R_v^t}_v∈ V, G_rm gauge, W), where μ_s and μ_t are independently sampled latent distributions, R_u^s and R_v^t are within-system intervention-response operators, G_rm gauge is the declared harmless gauge group, and W is a witness hidden from construction. No map between individual samples from μ_s and μ_t is available to the compiler.
Exact mathematical passage
\mathfrak U = (H_s,H_t,\mu_s,\mu_t,\mathcal U, \{R_u^s\}_{u\in\mathcal U}, \{R_v^t\}_{v\in\mathcal V}, \mathcal G_{\rm gauge},\mathsf W),
\nuDefinition passage · 7 passages
Search aliases: ν nu
Definition passage: Protected evaluation context · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 470
[Protected evaluation context] A protected evaluation context is a tuple Ξ= ( Z, μ, Y, ν, bar c, λ, mathscr I, B, α ), where: item Z is a task or environment family and μ is its held-out distribution; item Y is the space of protected external traces; item ν:Z×Y→[0,1] is task utility; item bar c:Z×Y→[0,1] is normalized resource cost; item λ≥0 is a cost weight; item mathscr I(A,B)∈{0,1} is a declared invariant-preservation predicate for the transition A→ B; item B is the resource envelope used for…
Exact mathematical passage
\Xi= \left( \Zcal, \mu, \Ycal, \nu, \bar c, \lambda, \mathscr I, \Bcal, \alpha \right),
\Xi_tDefinition passage · 5 passages
Search aliases: Ξ_t Xi
Definition passage: Context-indexed lineage edge · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10497
[Context-indexed lineage edge] Let Ξ_t be the protected context for edge t→ t+1. A sealed lineage edge K_txrightarrowΠ_t+1K_t+1 is γ_t-productive in Ξ_t when its receipt replays, its resource and witness gates pass, and a fresh paired audit certifies M_Ξ_t(K_t+1)-M_Ξ_t(K_t)≥ γ_t>0.
Exact mathematical passage
\Xi_t
\piDefinition passage · 20 passages
Search aliases: π pi
Definition passage: Path-flat evidence core · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3467
[Path-flat evidence core] The path-flat evidence core at v_0 is E_v_0^flat := ⋂_γ∈π_1(G,v_0) ker(H_γ-I).
Exact mathematical passage
E_{v_0}^{\flat} := \bigcap_{\gamma\in\pi_1(G,v_0)} \ker(H_\gamma-I).
\Pi_1Definition passage · 17 passages
Search aliases: Π_1 Pi
Definition passage: Receipt composition · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1266
[Receipt composition] Let Π_i validate ω_i, and let Π validate the outer operation ω conditional on C_ω. The composed receipt Π⋆(Π_1,…,Π_n) is the provenance-directed acyclic graph obtained by proof-path concatenation, evidence union, explicit dependency links, and deduplication of immutable evidence references without duplicating linear observation authority.
Exact mathematical passage
\Pi\star(\Pi_1,\ldots,\Pi_n)
\pi_1Definition passage · 3 passages
Search aliases: π_1 pi
Definition passage: Path-flat evidence core · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3467
[Path-flat evidence core] The path-flat evidence core at v_0 is E_v_0^flat := ⋂_γ∈π_1(G,v_0) ker(H_γ-I).
Exact mathematical passage
E_{v_0}^{\flat} := \bigcap_{\gamma\in\pi_1(G,v_0)} \ker(H_\gamma-I).
\Pi_2Definition passage · 8 passages
Search aliases: Π_2 Pi
Definition passage: Finite repeated-lineage certificate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10509
[Finite repeated-lineage certificate] A sequence K_0xrightarrowΠ_1K_1xrightarrowΠ_2⋯xrightarrowΠ_mK_m has an m-edge repeated-lineage certificate when every edge is productive in its own predeclared protected context, every parent receipt is replayed before the child is compiled, no final audit evidence is reused as development evidence for a later edge, and the cumulative error and regression budgets remain within their declared bounds.
Exact mathematical passage
K_0\xrightarrow{\Pi_1}K_1\xrightarrow{\Pi_2}\cdots\xrightarrow{\Pi_m}K_m
\Pi_{\mathrm{eval}}Definition passage · 1 passage
Search aliases: Π_eval Pi mathrm
Definition passage: Evaluator-upgrade certificate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3857
[Evaluator-upgrade certificate] An evaluator-upgrade certificate is a tuple Π_eval = (v,w,S_e,T_e,R_e, h_v, A_v, A_w, τ,μ_0, δ_obs,δ_J, T_new, R_rollback, σ), containing the old and new evaluator identifiers, raw and internal transports, core retraction, protected replay set, holonomy-energy operators, accepted flatness and adequacy thresholds, measured replay defects, held-out new-evidence tests, rollback record, and uncertainty statement σ.
Exact mathematical passage
\Pi_{\mathrm{eval}} = (v,w,S_e,T_e,R_e, \mathfrak h_v, \mathsf A_v,\mathsf A_w, \tau,\mu_0, \delta_{\mathrm{obs}},\delta_J, \mathcal T_{\mathrm{new}}, \mathcal R_{\mathrm{rollback}}, \sigma),
\Pi_{\mathrm{foreign}}Definition passage · 3 passages
Search aliases: Π_foreign Pi mathrm
Definition passage: Response-language expansion certificate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12976
[Response-language expansion certificate] A response-language expansion certificate is a tuple Π_resp = (R,S, A, A^+,P_alias,γ, ε_comm,ε_fit,Π_foreign), where P_alias records the inherited merged block, γ is the minimum separating spectral gap, ε_comm is a commutator residual, ε_fit is response-model error, and Π_foreign records held-out repair effects.
Exact mathematical passage
\Pi_{\mathrm{resp}} = (\Rcal,S,\mathfrak A,\mathfrak A^+,P_{\mathrm{alias}},\gamma, \eps_{\mathrm{comm}},\eps_{\mathrm{fit}},\Pi_{\mathrm{foreign}}),
\Pi_iDefinition passage · 1 passage
Search aliases: Π_i Pi
Definition passage: Receipt composition · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1265
[Receipt composition] Let Π_i validate ω_i, and let Π validate the outer operation ω conditional on C_ω. The composed receipt Π⋆(Π_1,…,Π_n) is the provenance-directed acyclic graph obtained by proof-path concatenation, evidence union, explicit dependency links, and deduplication of immutable evidence references without duplicating linear observation authority.
Exact mathematical passage
\Pi_i
\Pi_i^{+}Definition passage · 4 passages
Search aliases: Π_i^+ Pi
Definition passage: Post-result research receipt · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13741
[Post-result research receipt] After evaluation, the post-result receipt is Π_i^+ = (Π_i^-, c_i, r_i,a_i,m_i), where c_i is the scored benchmark vector, r_i the protected-regression vector, a_i the hidden-audit result, and m_i the integrity-monitor verdict.
Exact mathematical passage
\Pi_i^{+} = (\Pi_i^{-},\mathbf c_i,\mathbf r_i,a_i,m_i), \label{eq:b13-postresult-receipt}
\Pi_i^{-}Definition passage · 5 passages
Search aliases: Π_i^- Pi
Definition passage: Pre-result research receipt · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13727
[Pre-result research receipt] A pre-result receipt has the form Π_i^- = (H_i,O_i,D_i,θ_i, widehat c_i, widehat r_i, F_i,h_i^code,h_i^spec), where H_i is the causal hypothesis, O_i the proposed objective or repair mechanism, D_i the evidence or data construction, θ_i the configuration, widehat c_i the predicted benchmark-effect vector, widehat r_i the predicted protected-risk vector, F_i the declared falsification condition, and h_i^code,h_i^spec immutable digests of the executable and specificat…
Exact mathematical passage
\Pi_i^{-} = (H_i,O_i,D_i,\theta_i, \widehat{\mathbf c}_i, \widehat{\mathbf r}_i, F_i,h_i^{\mathrm{code}},h_i^{\mathrm{spec}}), \label{eq:b13-preresult-receipt}
\Pi_mDefinition passage · 1 passage
Search aliases: Π_m Pi
Definition passage: Finite repeated-lineage certificate · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10509
[Finite repeated-lineage certificate] A sequence K_0xrightarrowΠ_1K_1xrightarrowΠ_2⋯xrightarrowΠ_mK_m has an m-edge repeated-lineage certificate when every edge is productive in its own predeclared protected context, every parent receipt is replayed before the child is compiled, no final audit evidence is reused as development evidence for a later edge, and the cumulative error and regression budgets remain within their declared bounds.
Exact mathematical passage
K_0\xrightarrow{\Pi_1}K_1\xrightarrow{\Pi_2}\cdots\xrightarrow{\Pi_m}K_m
\Pi_nDefinition passage · 1 passage
Search aliases: Π_n Pi
Definition passage: Receipt composition · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1266
[Receipt composition] Let Π_i validate ω_i, and let Π validate the outer operation ω conditional on C_ω. The composed receipt Π⋆(Π_1,…,Π_n) is the provenance-directed acyclic graph obtained by proof-path concatenation, evidence union, explicit dependency links, and deduplication of immutable evidence references without duplicating linear observation authority.
Exact mathematical passage
\Pi\star(\Pi_1,\ldots,\Pi_n)
\pi_{\rm anc}Definition passage · 4 passages
Search aliases: π_rm anc pi rm
Definition passage: Finite repair process · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8292
[Finite repair process] A finite repair process is a tuple P=(π_rm anc, B,π_rm gate,B,Π), where π_rm anc selects target correspondence queries, B estimates a bridge and transports a frozen source repair, π_rm gate commits or abstains, B is the declared resource budget, and Π is the receipt schema.
Exact mathematical passage
\mathfrak P=(\pi_{\rm anc},\mathsf B,\pi_{\rm gate},B,\Pi),
\pi_{\rm gate}Definition passage · 2 passages
Search aliases: π_rm gate pi rm
Definition passage: Finite repair process · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8292
[Finite repair process] A finite repair process is a tuple P=(π_rm anc, B,π_rm gate,B,Π), where π_rm anc selects target correspondence queries, B estimates a bridge and transports a frozen source repair, π_rm gate commits or abstains, B is the declared resource budget, and Π is the receipt schema.
Exact mathematical passage
\mathfrak P=(\pi_{\rm anc},\mathsf B,\pi_{\rm gate},B,\Pi),
\pi_{\rm PQR}Definition passage · 2 passages
Search aliases: π_rm PQR pi rm
Definition passage: Packet-balanced QR policy · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8529
[Packet-balanced QR policy] For fit budget k=16, define π_rm PQR(H,P_J) = π_rm QR(H_rm aff,8) ∪ π_rm QR(H_rm non,8), where affected and unaffected membership is inferred from the source causal intervention, not from target audit labels.
Exact mathematical passage
\pi_{\rm PQR}(H,P_J) = \pi_{\rm QR}(H_{\rm aff},8) \cup \pi_{\rm QR}(H_{\rm non},8),
\pi_{\rm QR}Definition passage · 1 passage
Search aliases: π_rm QR pi rm
Definition passage: Packet-balanced QR policy · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8529
[Packet-balanced QR policy] For fit budget k=16, define π_rm PQR(H,P_J) = π_rm QR(H_rm aff,8) ∪ π_rm QR(H_rm non,8), where affected and unaffected membership is inferred from the source causal intervention, not from target audit labels.
Exact mathematical passage
\pi_{\rm PQR}(H,P_J) = \pi_{\rm QR}(H_{\rm aff},8) \cup \pi_{\rm QR}(H_{\rm non},8),
\Pi_TDefinition passage · 1 passage
Search aliases: Π_T Pi
Definition passage: Receipt chain · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5832
[Receipt chain] For receipts Π_1,…,Π_T, define Π_t.previous = ∅, · t=1,
Exact mathematical passage
\Pi_1,\ldots,\Pi_T
\Pi_tDefinition passage · 1 passage
Search aliases: Π_t Pi
Definition passage: Receipt chain · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5833
[Receipt chain] For receipts Π_1,…,Π_T, define Π_t.previous = ∅, · t=1, h(Π_t-1), · t>1.
Exact mathematical passage
\Pi_t.\mathrm{previous} = \begin{cases} \varnothing,&t=1,\\ h(\Pi_{t-1}),&t>1. \end{cases}
\Pi_{t+1}Definition passage · 1 passage
Search aliases: Π_t+1 Pi
Definition passage: Context-indexed lineage edge · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10498
[Context-indexed lineage edge] Let Ξ_t be the protected context for edge t→ t+1. A sealed lineage edge K_txrightarrowΠ_t+1K_t+1 is γ_t-productive in Ξ_t when its receipt replays, its resource and witness gates pass, and a fresh paired audit certifies M_Ξ_t(K_t+1)-M_Ξ_t(K_t)≥ γ_t>0.
Exact mathematical passage
K_t\xrightarrow{\Pi_{t+1}}K_{t+1}
\Pi_{t-1}Definition passage · 1 passage
Search aliases: Π_t-1 Pi
Definition passage: Receipt chain · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5833
[Receipt chain] For receipts Π_1,…,Π_T, define Π_t.previous = ∅, · t=1, h(Π_t-1), · t>1.
Exact mathematical passage
\Pi_t.\mathrm{previous} = \begin{cases} \varnothing,&t=1,\\ h(\Pi_{t-1}),&t>1. \end{cases}
\rho_cDefinition passage · 3 passages
Search aliases: ρ_c rho
Definition passage: Semantic twins · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17127
[Semantic twins] Two observed records x_c∈ X_c and x_d∈ X_d are semantic twins when there exists one protected semantic object z∈ Z such that x_c=ρ_c(z), x_d=ρ_d(z). A semantic-twin group is an orbit {ρ_c(z):c∈ C_z} for a declared finite context set C_z⊆C.
Exact mathematical passage
x_c=\rho_c(z), \qquad x_d=\rho_d(z).
\rho_dDefinition passage · 1 passage
Search aliases: ρ_d rho
Definition passage: Semantic twins · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17127
[Semantic twins] Two observed records x_c∈ X_c and x_d∈ X_d are semantic twins when there exists one protected semantic object z∈ Z such that x_c=ρ_c(z), x_d=ρ_d(z). A semantic-twin group is an orbit {ρ_c(z):c∈ C_z} for a declared finite context set C_z⊆C.
Exact mathematical passage
x_c=\rho_c(z), \qquad x_d=\rho_d(z).
\rho_TDefinition passage · 2 passages
Search aliases: ρ_T rho
Definition passage: Preservation profile · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 704
[Preservation profile] A preservation profile for a bridge T:X→ Y is a declared record P(T)= ( Σ_T, Q_T, ρ_T, ℓ_T, σ_T, A_T ), where Σ_T lists preserved algebraic structures, Q_T lists preserved causal or evaluative queries, ρ_T records provenance behavior, ℓ_T records information loss, σ_T records uncertainty transport, and A_T lists authorized downstream uses.
Exact mathematical passage
\Pcal(T)= \left( \Sigma_T, \mathcal Q_T, \rho_T, \ell_T, \sigma_T, \mathcal A_T \right),
\SigmaDefinition passage · 17 passages
Search aliases: Σ Sigma
Definition passage: Preservation profile · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 704
[Preservation profile] A preservation profile for a bridge T:X→ Y is a declared record P(T)= ( Σ_T, Q_T, ρ_T, ℓ_T, σ_T, A_T ), where Σ_T lists preserved algebraic structures, Q_T lists preserved causal or evaluative queries, ρ_T records provenance behavior, ℓ_T records information loss, σ_T records uncertainty transport, and A_T lists authorized downstream uses.
Exact mathematical passage
\Pcal(T)= \left( \Sigma_T, \mathcal Q_T, \rho_T, \ell_T, \sigma_T, \mathcal A_T \right),
\sigma_cDefinition passage · 4 passages
Search aliases: σ_c sigma
Definition passage: Affine observation chart · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17140
[Affine observation chart] For a context c, an affine observation chart is a tuple χ_c=(s_x,t_x,s_y,t_y,σ_c), s_xs_y≠0, with canonicalization widehat x=fracx^obs-t_xs_x, widehat y=fracy^obs-t_ys_y, and support signature σ_c∈ℝ^5. The signature is the sorted vector of observed demonstration inputs predicted by the chart on the five canonical demonstration locations.
Exact mathematical passage
\chi_c=(s_x,t_x,s_y,t_y,\sigma_c), \qquad s_xs_y\neq0,
\Sigma_TDefinition passage · 2 passages
Search aliases: Σ_T Sigma
Definition passage: Preservation profile · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 704
[Preservation profile] A preservation profile for a bridge T:X→ Y is a declared record P(T)= ( Σ_T, Q_T, ρ_T, ℓ_T, σ_T, A_T ), where Σ_T lists preserved algebraic structures, Q_T lists preserved causal or evaluative queries, ρ_T records provenance behavior, ℓ_T records information loss, σ_T records uncertainty transport, and A_T lists authorized downstream uses.
Exact mathematical passage
\Pcal(T)= \left( \Sigma_T, \mathcal Q_T, \rho_T, \ell_T, \sigma_T, \mathcal A_T \right),
\sigma_TDefinition passage · 2 passages
Search aliases: σ_T sigma
Definition passage: Preservation profile · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 704
[Preservation profile] A preservation profile for a bridge T:X→ Y is a declared record P(T)= ( Σ_T, Q_T, ρ_T, ℓ_T, σ_T, A_T ), where Σ_T lists preserved algebraic structures, Q_T lists preserved causal or evaluative queries, ρ_T records provenance behavior, ℓ_T records information loss, σ_T records uncertainty transport, and A_T lists authorized downstream uses.
Exact mathematical passage
\Pcal(T)= \left( \Sigma_T, \mathcal Q_T, \rho_T, \ell_T, \sigma_T, \mathcal A_T \right),
\tau_{\mathrm{adm}}Definition passage · 4 passages
Search aliases: τ_adm tau mathrm
Definition passage: Distribution-indexed atlas · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17157
[Distribution-indexed atlas] A finite distribution-indexed atlas is A = ( {χ_c:c∈ C}, d_sig, τ_adm, π ), where d_sig is a normalized signature distance, τ_adm is an admission threshold, and π chooses the nearest chart. An input is admitted only when min_c∈ Cd_sig(σ(x),σ_c) ≤τ_adm. Otherwise the atlas returns an unsupported-chart certificate and abstains.
Exact mathematical passage
\Acal = \left( \{\chi_c:c\in C\}, d_{\mathrm{sig}}, \tau_{\mathrm{adm}}, \pi \right), \label{eq:b19-atlas}
\tau_eDefinition passage · 2 passages
Search aliases: τ_e tau
Definition passage: Stratified functional atlas · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 641
[Stratified functional atlas] A stratified functional atlas at generation t is a tuple A_t= ( C, G_t, {X_t^c}_c∈ C, {F_t^c}_c∈ C, {T_e,τ_e,P_e,ε_e}_e∈ E(G_t) ), where: item C is a finite set of stratum colors; item G_t is a directed multigraph of declared interfaces; item X_t^c is the state space of stratum c; item F_t^c is its local transition rule; item each edge e:c→ d carries a bridge T_e:X_t^c→ X_t^d; item τ_e declares the bridge class; item P_e is its preservation profile; item ε_e is a ve…
Exact mathematical passage
\mathfrak A_t= \left( \mathsf C, G_t, \{X_t^c\}_{c\in\mathsf C}, \{F_t^c\}_{c\in\mathsf C}, \{T_e,\tau_e,\Pcal_e,\bm\eps_e\}_{e\in E(G_t)} \right),
\tau_M^{\mathrm{future}}Definition passage · 2 passages
Search aliases: τ_M^future tau mathrm
Definition passage: Recursive abstraction estimand · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6265
[Recursive abstraction estimand] Let F_post be foreign defect families revealed after the source repair. Define τ_M^future = E_A∼ F_post [ G_A(Z_M=1)-G_A(Z_M=0) ].
Exact mathematical passage
\tau_M^{\mathrm{future}} = \mathbb E_{A\sim\mathcal F_{\mathrm{post}}} \left[ G_A(Z_M=1)-G_A(Z_M=0) \right].
\PhiDefinition passage · 11 passages
Search aliases: Φ Phi
Definition passage: Weighted meta-regret · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9274
[Weighted meta-regret] For w∈ℝ_>0^7, define Φ_w(r)=w^top r. The weights encode the declared relative value or risk of unresolved module headroom.
Exact mathematical passage
\Phi_w(r)=w^\top r. \label{eq:b8-weighted-regret}
\Phi_wDefinition passage · 2 passages
Search aliases: Φ_w Phi
Definition passage: Weighted meta-regret · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9274
[Weighted meta-regret] For w∈ℝ_>0^7, define Φ_w(r)=w^top r. The weights encode the declared relative value or risk of unresolved module headroom.
Exact mathematical passage
\Phi_w(r)=w^\top r. \label{eq:b8-weighted-regret}
\chiDefinition passage · 15 passages
Search aliases: χ chi
Definition passage: Cross-implementation repair schema · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16726
[Cross-implementation repair schema] A Block 18 repair schema is a tuple ρ=(a,λ_sem,λ_prog, λ_geo,χ), where a is an adapter constructor, the three nonnegative λ values weight final semantic, progressive semantic, and latent-geometry witnesses, and χ records whether witness labels are correctly bound or shuffled. A schema is cross-implementation when the same tuple is instantiated through each cell family's declared adapter interface without architecture-specific candidate selection.
Exact mathematical passage
\rho=(a,\lambda_{\mathrm{sem}},\lambda_{\mathrm{prog}}, \lambda_{\mathrm{geo}},\chi), \label{eq:b18-repair-schema}
\chi_cDefinition passage · 2 passages
Search aliases: χ_c chi
Definition passage: Affine observation chart · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17140
[Affine observation chart] For a context c, an affine observation chart is a tuple χ_c=(s_x,t_x,s_y,t_y,σ_c), s_xs_y≠0, with canonicalization widehat x=fracx^obs-t_xs_x, widehat y=fracy^obs-t_ys_y, and support signature σ_c∈ℝ^5. The signature is the sorted vector of observed demonstration inputs predicted by the chart on the five canonical demonstration locations.
Exact mathematical passage
\chi_c=(s_x,t_x,s_y,t_y,\sigma_c), \qquad s_xs_y\neq0,
\OmegaDefinition passage · 8 passages
Search aliases: Ω Omega
Definition passage: Architecture--context curvature · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17725
[Architecture--context curvature] For architectures a,b and charts c,d, the transport-square defect is Ω_a,b;c,d := C^a_c→ dT^c_b→ a - T^d_b→ aC^b_c→ d. Given an audit distribution P_b,c on target trajectories, its normalized root-mean-square curvature is κ_a,b;c,d^2 = frac E_h∼ P_b,c normΩ_a,b;c,dh^2 E_z∼ P_a,dnormz^2 .
Exact mathematical passage
\Omega_{a,b;c,d} := C^{a}_{c\to d}T^{c}_{b\to a} - T^{d}_{b\to a}C^{b}_{c\to d}. \label{eq:b20-curvature}
\omegaDefinition passage · 49 passages
Search aliases: ω omega
Definition passage: Proof-carrying repair schema · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1190
[Proof-carrying repair schema] A repair schema is a tuple ω= ( σ, c_in, c_out, D, C, Q, [r], u, g, mathscr I, Π ), where: item σ is a recognizable failure signature; item c_in=(c_1,…,c_n) and c_out are input and output colors; item D is the declared local defect-response map; item C is an applicability precondition; item Q is a postcondition; item [r] is a local obstruction certificate or unresolved-obstruction record; item ν is the correction program; item g is a precommitted effect profile; it…
Exact mathematical passage
\omega= \left( \sigma, \bm c_{\mathrm{in}}, c_{\mathrm{out}}, D, C, Q, [r], u, \bm g, \mathscr I, \Pi \right),
\omega_1Definition passage · 9 passages
Search aliases: ω_1 omega
Definition passage: Schema generality preorder · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1411
[Schema generality preorder] For repair schemas ω_1 and ω_2 with the same typed effect, write ω_1preceqω_2 when every state satisfying C_ω_1 also satisfies C_ω_2 and the two correction semantics agree on dom(C_ω_1) up to the declared transport tolerance. Then ω_2 is at least as general as ω_1.
Exact mathematical passage
\omega_1
\omega_2Definition passage · 4 passages
Search aliases: ω_2 omega
Definition passage: Schema generality preorder · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1411
[Schema generality preorder] For repair schemas ω_1 and ω_2 with the same typed effect, write ω_1preceqω_2 when every state satisfying C_ω_1 also satisfies C_ω_2 and the two correction semantics agree on dom(C_ω_1) up to the declared transport tolerance. Then ω_2 is at least as general as ω_1.
Exact mathematical passage
\omega_2
\omega_ADefinition passage · 2 passages
Search aliases: ω_A omega
Definition passage: Exact transport equivalence of repairs · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1430
[Exact transport equivalence of repairs] Write ω_A≅_Tω_B when: item input and output colors correspond under the atlas transport; item the defect-response maps commute with correction and defect transports; item applicability and postconditions are preserved in both directions; item protected invariants and provenance types are preserved; item the effect profiles agree after coordinate transport; item the receipts map to equivalent proof paths.
Exact mathematical passage
\omega_A\cong_T\omega_B
\Omega_{a,b;c,d}Definition passage · 2 passages
Search aliases: Ω_a,b;c,d Omega
Definition passage: Architecture--context curvature · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17725
[Architecture--context curvature] For architectures a,b and charts c,d, the transport-square defect is Ω_a,b;c,d := C^a_c→ dT^c_b→ a - T^d_b→ aC^b_c→ d. Given an audit distribution P_b,c on target trajectories, its normalized root-mean-square curvature is κ_a,b;c,d^2 = frac E_h∼ P_b,c normΩ_a,b;c,dh^2 E_z∼ P_a,dnormz^2 .
Exact mathematical passage
\Omega_{a,b;c,d} := C^{a}_{c\to d}T^{c}_{b\to a} - T^{d}_{b\to a}C^{b}_{c\to d}. \label{eq:b20-curvature}
\omega_BDefinition passage · 2 passages
Search aliases: ω_B omega
Definition passage: Exact transport equivalence of repairs · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1430
[Exact transport equivalence of repairs] Write ω_A≅_Tω_B when: item input and output colors correspond under the atlas transport; item the defect-response maps commute with correction and defect transports; item applicability and postconditions are preserved in both directions; item protected invariants and provenance types are preserved; item the effect profiles agree after coordinate transport; item the receipts map to equivalent proof paths.
Exact mathematical passage
\omega_A\cong_T\omega_B
\omega_iDefinition passage · 7 passages
Search aliases: ω_i omega
Definition passage: Receipt composition · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1265
[Receipt composition] Let Π_i validate ω_i, and let Π validate the outer operation ω conditional on C_ω. The composed receipt Π⋆(Π_1,…,Π_n) is the provenance-directed acyclic graph obtained by proof-path concatenation, evidence union, explicit dependency links, and deduplication of immutable evidence references without duplicating linear observation authority.
Exact mathematical passage
\omega_i
\ldotsDefinition passage · 58 passages
Search aliases: … ldots
Definition passage: Proof-carrying repair schema · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1209
[Proof-carrying repair schema] A repair schema is a tuple ω= ( σ, c_in, c_out, D, C, Q, [r], u, g, mathscr I, Π ), where: item σ is a recognizable failure signature; item c_in=(c_1,…,c_n) and c_out are input and output colors; item D is the declared local defect-response map; item C is an applicability precondition; item Q is a postcondition; item [r] is a local obstruction certificate or unresolved-obstruction record; item ν is the correction program; item g is a precommitted effect profile; it…
Exact mathematical passage
\bm c_{\mathrm{in}}=(c_1,\ldots,c_n)
\ellDefinition passage · 33 passages
Search aliases: ℓ ell
Definition passage: Preservation profile · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 704
[Preservation profile] A preservation profile for a bridge T:X→ Y is a declared record P(T)= ( Σ_T, Q_T, ρ_T, ℓ_T, σ_T, A_T ), where Σ_T lists preserved algebraic structures, Q_T lists preserved causal or evaluative queries, ρ_T records provenance behavior, ℓ_T records information loss, σ_T records uncertainty transport, and A_T lists authorized downstream uses.
Exact mathematical passage
\Pcal(T)= \left( \Sigma_T, \mathcal Q_T, \rho_T, \ell_T, \sigma_T, \mathcal A_T \right),
\ell_{ij}Definition passage · 2 passages
Search aliases: ℓ_ij ell
Definition passage: Packet leakage profile · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1981
[Packet leakage profile] The directed leakage from correction packet j into defect packet i is ℓ_ij:=normP_iDQ_j, i≠ j. The matrix (ℓ_ij) is the packet leakage profile.
Exact mathematical passage
\ell_{ij}:=\norm{P_iDQ_j}, \qquad i\neq j.
\ell_TDefinition passage · 2 passages
Search aliases: ℓ_T ell
Definition passage: Preservation profile · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 704
[Preservation profile] A preservation profile for a bridge T:X→ Y is a declared record P(T)= ( Σ_T, Q_T, ρ_T, ℓ_T, σ_T, A_T ), where Σ_T lists preserved algebraic structures, Q_T lists preserved causal or evaluative queries, ρ_T records provenance behavior, ℓ_T records information loss, σ_T records uncertainty transport, and A_T lists authorized downstream uses.
Exact mathematical passage
\Pcal(T)= \left( \Sigma_T, \mathcal Q_T, \rho_T, \ell_T, \sigma_T, \mathcal A_T \right),
\toDefinition passage · 165 passages
Search aliases: → to
Definition passage: Protected evaluation context · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 488
[Protected evaluation context] A protected evaluation context is a tuple Ξ= ( Z, μ, Y, ν, bar c, λ, mathscr I, B, α ), where: item Z is a task or environment family and μ is its held-out distribution; item Y is the space of protected external traces; item ν:Z×Y→[0,1] is task utility; item bar c:Z×Y→[0,1] is normalized resource cost; item λ≥0 is a cost weight; item mathscr I(A,B)∈{0,1} is a declared invariant-preservation predicate for the transition A→ B; item B is the resource envelope used for…
Exact mathematical passage
\nu:\Zcal\times\Ycal\to[0,1]
\hookrightarrowDefinition passage · 12 passages
Search aliases: ↪ hookrightarrow
Definition passage: Append-only evidence embedding · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 804
[Append-only evidence embedding] Evidence preservation from generation t to t+1 is an injective map ι_t:E_t↪E_t+1 that preserves payload, source, timestamp order, precommitment status, provenance type, and cryptographic integrity fields. A later correction may append a refutation or reinterpretation receipt, but it may not silently alter the old entry.
Exact mathematical passage
\iota_t:\Ecal_t\hookrightarrow\Ecal_{t+1}
\forallDefinition passage · 12 passages
Search aliases: ∀ forall
Definition passage: Controllable predecessor · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4311
[Controllable predecessor] For X⊆ S, define CPre(X) = set s∈ S: ∃ u∈Act(s) ∀ d∈Dist(s,u), T(s,u,d)∈ X .
Exact mathematical passage
\operatorname{CPre}(X) = \set{ s\in S: \exists u\in\mathsf{Act}(s)\ \forall d\in\mathsf{Dist}(s,u),\ T(s,u,d)\in X }.
\existsDefinition passage · 3 passages
Search aliases: ∃ exists
Definition passage: Controllable predecessor · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4311
[Controllable predecessor] For X⊆ S, define CPre(X) = set s∈ S: ∃ u∈Act(s) ∀ d∈Dist(s,u), T(s,u,d)∈ X .
Exact mathematical passage
\operatorname{CPre}(X) = \set{ s\in S: \exists u\in\mathsf{Act}(s)\ \forall d\in\mathsf{Dist}(s,u),\ T(s,u,d)\in X }.
\varnothingDefinition passage · 19 passages
Search aliases: ∅ varnothing
Definition passage: Rotating witness schedule · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4176
[Rotating witness schedule] A sequence (E_t,W_t)_t≥0 is a rotating witness schedule when: item E_t∩ W_t=∅ for every t; item W_t is immutable during transaction t; item every component belongs to E_t for at least one t; item every witness replacement is validated by a component outside the same edit set; item protected evidence is transported through a conservative evaluator morphism at every rotation; item every committed round retains a rollback path to a state certified by the preceding witnes…
Exact mathematical passage
E_t\cap W_t=\varnothing
\inDefinition passage · 308 passages
Search aliases: ∈ in
Definition passage: Protected evaluation context · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 491
c, λ, mathscr I, B, α ), where: item Z is a task or environment family and μ is its held-out distribution; item Y is the space of protected external traces; item ν:Z×Y→[0,1] is task utility; item bar c:Z×Y→[0,1] is normalized resource cost; item λ≥0 is a cost weight; item mathscr I(A,B)∈{0,1} is a declared invariant-preservation predicate for the transition A→ B; item B is the resource envelope used for candidate generation and evaluation; item α∈(0,1) is the accepted statistical error level. Th…
Exact mathematical passage
\mathscr I(A,B)\in\{0,1\}
\notinDefinition passage · 7 passages
Search aliases: ∉ notin
Definition passage: Structurally new repair schema · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9965
[Structurally new repair schema] Let mathscrS_t be the closure of existing repair schemas under the authorized operadic compositions. A schema s_t+1 is structurally new when s_t+1∉overlinemathscrS_t under the declared behavioral equivalence and spanim s_t+1 not⊆ spanimmathscrS_t.
Exact mathematical passage
s_{t+1}\notin\overline{\Sscr_t}
\sumDefinition passage · 78 passages
Search aliases: ∑ sum
Definition passage: Quantitative gluing obstruction · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2409
[Quantitative gluing obstruction] The graph-level quantitative gluing obstruction is η_glue(e) :=normP_cyce. The local-to-global repair certificate is the pair ( η_loc, η_glue ), η_loc^2 :=∑_αnormD_α u_α+r_α^2.
Exact mathematical passage
\left( \eta_{\mathrm{loc}}, \eta_{\mathrm{glue}} \right), \qquad \eta_{\mathrm{loc}}^2 :=\sum_{\alpha}\norm{D_\alpha u_\alpha+r_\alpha}^2.
\sum_{a=1}^{m}Definition passage · 4 passages
Search aliases: ∑_a=1^m sum
Definition passage: Block-scalar response model · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12163
[Block-scalar response model] The response family is block scalar on [source reference] when R_j=∑_a=1^mλ_ajP_a, λ_aj∈ℝ. The response signature of block a is λ_a=(λ_a1,…,λ_ak)∈ℝ^k. The signatures are separating when λ_a≠λ_b for every a≠ b.
Exact mathematical passage
R_j=\sum_{a=1}^{m}\lambda_{aj}P_a, \qquad \lambda_{aj}\in\R. \label{eq:b11-block-scalar}
\sum_{e\in p}Definition passage · 1 passage
Search aliases: ∑_e∈ p sum in
Definition passage: Transport route cost · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3000
[Transport route cost] For a fixed source correction u_0, define the certified cost of a nonexpansive path p by C_tr(p;u_0) := normu_0∑_e∈ pε_D,e + ∑_e∈ pε_r,e + ∑_e∈ pε_auth,e + ∑_e∈ pε_sem,e.
Exact mathematical passage
C_{\mathrm{tr}}(p;u_0) := \norm{u_0}\sum_{e\in p}\eps_{D,e} + \sum_{e\in p}\eps_{r,e} + \sum_{e\in p}\eps_{\mathrm{auth},e} + \sum_{e\in p}\eps_{\mathrm{sem},e}.
\sum_iDefinition passage · 3 passages
Search aliases: ∑_i sum
Definition passage: Structural-normalized specificity · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15742
[Structural-normalized specificity] Let Q_spec = fracoverline s max{overline o,overline c,ε}, where bars denote held-out means and ε>0 is a fixed numerical floor. The default Block 16 specificity gate requires LCB_0.95(overline s)>0, frac1n∑_i1[s_i>0]≥0.75, Q_spec≥0.25.
Exact mathematical passage
\LCB_{0.95}(\overline s)>0, \qquad \frac{1}{n}\sum_i\one[s_i>0]\geq0.75, \qquad Q_{\mathrm{spec}}\geq0.25. \label{eq:b16-specificity-gate}
\sum_{R\subseteq N\setminus\{i\}}Definition passage · 1 passage
Search aliases: ∑_R⊆ N∖{i} sum subseteq setminus
Definition passage: Shapley meta-headroom · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9139
[Shapley meta-headroom] For i∈ N, define h_i(K) = ∑_R⊆ N∖{i} frac|R|!(m-|R|-1)!m! [ F_K(R∪{i})-F_K(R) ].
Exact mathematical passage
h_i(K) = \sum_{R\subseteq N\setminus\{i\}} \frac{|R|!(m-|R|-1)!}{m!} \left[ F_K(R\cup\{i\})-F_K(R) \right]. \label{eq:b8-shapley-headroom}
\sum_{t=0}^{T-1}Definition passage · 1 passage
Search aliases: ∑_t=0^T-1 sum
Definition passage: Approximate semantic debt · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5005
[Approximate semantic debt] If an extension preserves old semantics only approximately, let η_t bound its worst protected interpretation error after transport. Under nonexpansive semantic transports, the cumulative debt through stage T is η_0:T ≤ ∑_t=0^T-1η_t.
Exact mathematical passage
\eta_{0:T} \leq \sum_{t=0}^{T-1}\eta_t.
\sum_{\alpha}Definition passage · 1 passage
Search aliases: ∑_α sum alpha
Definition passage: Quantitative gluing obstruction · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2409
[Quantitative gluing obstruction] The graph-level quantitative gluing obstruction is η_glue(e) :=normP_cyce. The local-to-global repair certificate is the pair ( η_loc, η_glue ), η_loc^2 :=∑_αnormD_α u_α+r_α^2.
Exact mathematical passage
\left( \eta_{\mathrm{loc}}, \eta_{\mathrm{glue}} \right), \qquad \eta_{\mathrm{loc}}^2 :=\sum_{\alpha}\norm{D_\alpha u_\alpha+r_\alpha}^2.
\setminusDefinition passage · 4 passages
Search aliases: ∖ setminus
Definition passage: Nonlinear holonomy residual · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2549
[Nonlinear holonomy residual] If the S_i are metric spaces, define η_hol(s_o) := max_e∈ E∖ Td(H_e(s_o),s_o).
Exact mathematical passage
\eta_{\mathrm{hol}}(s_o) := \max_{e\in E\setminus T}d\bigl(H_e(s_o),s_o\bigr).
\inftyDefinition passage · 18 passages
Search aliases: ∞ infty
Definition passage: Correction datum · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2663
[Correction datum] A finite-dimensional correction datum is a tuple C=(U,V,D,A,c, p), where U is a correction space, V is a defect space, D:U→ V is the linearized defect-response map, A⊆ U is the authorized correction set, c:U→[0,∞] is a declared resource cost, and p is a collection of protected predicates or invariant maps.
Exact mathematical passage
c:U\to[0,\infty]
\cupDefinition passage · 9 passages
Search aliases: ∪ cup
Definition passage: Productive recursive closure · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4375
[Productive recursive closure] For a finite perfect-information transaction game, define I_γ^ω = ν Z. μ Y. [ (G_γ∩SPre(Z)) ∪ SPre(Y) ].
Exact mathematical passage
\Ical_{\gamma}^{\omega} = \nu Z.\,\mu Y. \left[ \left(G_{\gamma}\cap\operatorname{SPre}(Z)\right) \cup \operatorname{SPre}(Y) \right].
\simDefinition passage · 36 passages
Search aliases: ∼ sim
Definition passage: Bounded recursive improvement step · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 575
ursive improvement step] A lineage transition L_t→ L_t+1 is a bounded recursive improvement step relative to (Ξ,h,B) when: item its lineage receipt is valid and the protected evidence ledger embeds into the next ledger; item F_t+1(Ξ)≥ F_t(Ξ); item on a foreign held-out target distribution D_foreign, E_A∼ D_foreign Γ_Ξ,h,B(K_t+1;A) > E_A∼ D_foreign Γ_Ξ,h,B(K_t;A); item evaluator risk does not increase beyond a declared tolerance; item the improvement is obtained under matched model access, candid…
Exact mathematical passage
\mathbb E_{A\sim\mathcal D_{\mathrm{foreign}}} \Gamma_{\Xi,h,B}(K_{t+1};A) > \mathbb E_{A\sim\mathcal D_{\mathrm{foreign}}} \Gamma_{\Xi,h,B}(K_t;A);
\sim_hDefinition passage · 3 passages
Search aliases: ∼_h sim
Definition passage: Horizon-h continuation equivalence · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4566
[Horizon-h continuation equivalence] Two archive nodes A and B are horizon-h continuation equivalent, written A∼_h B, when, under the same repair grammar and matched budgets, their sets of reachable descriptor vectors through at most h further transactions are identical.
Exact mathematical passage
A\sim_h B,
\sim_{\Ucal,r}Definition passage · 3 passages
Search aliases: ∼_Ucal,r ∼_U,r sim Ucal sim mathcal
Definition passage: Causal-predictive equivalence · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 19790
[Causal-predictive equivalence] For h,h'∈ H, write h∼_U,rh' when L (Y^u_0:rmid h) = L (Y^u_0:rmid h') for every u∈U^≤ r. The quotient Z_U,r=H/∼_U,r is the finite-horizon causal-predictive semantic space relative to the declared experiment language.
Exact mathematical passage
h\sim_{\Ucal,r}h'
\congDefinition passage · 16 passages
Search aliases: ≅ cong
Definition passage: Exact transport equivalence of repairs · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1430
[Exact transport equivalence of repairs] Write ω_A≅_Tω_B when: item input and output colors correspond under the atlas transport; item the defect-response maps commute with correction and defect transports; item applicability and postconditions are preserved in both directions; item protected invariants and provenance types are preserved; item the effect profiles agree after coordinate transport; item the receipts map to equivalent proof paths.
Exact mathematical passage
\omega_A\cong_T\omega_B
\cong_TDefinition passage · 1 passage
Search aliases: ≅_T cong
Definition passage: Exact transport equivalence of repairs · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1430
[Exact transport equivalence of repairs] Write ω_A≅_Tω_B when: item input and output colors correspond under the atlas transport; item the defect-response maps commute with correction and defect transports; item applicability and postconditions are preserved in both directions; item protected invariants and provenance types are preserved; item the effect profiles agree after coordinate transport; item the receipts map to equivalent proof paths.
Exact mathematical passage
\omega_A\cong_T\omega_B
\neqDefinition passage · 59 passages
Search aliases: ≠ neq
Definition passage: Deterministically separating probe family · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1121
[Deterministically separating probe family] Suppose every K_h^p is a point mass at an outcome y(h,p). The family P_probe separates H when for every h≠ h' there exists p∈P_probe such that y(h,p)≠ y(h',p).
Exact mathematical passage
h\neq h'
\leDefinition passage · 3 passages
Search aliases: ≤ le
Definition passage: Causal-predictive equivalence · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 19794
[Causal-predictive equivalence] For h,h'∈ H, write h∼_U,rh' when L (Y^u_0:rmid h) = L (Y^u_0:rmid h') for every u∈U^≤ r. The quotient Z_U,r=H/∼_U,r is the finite-horizon causal-predictive semantic space relative to the declared experiment language.
Exact mathematical passage
\mathcal L\!\left(Y^{u}_{0:r}\mid h\right) = \mathcal L\!\left(Y^{u}_{0:r}\mid h'\right) \qquad \text{for every }u\in\Ucal^{\le r}. \label{eq:b23-causal-equivalence}
\leqDefinition passage · 177 passages
Search aliases: ≤ leq
Definition passage: ε-intervention sufficiency · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 955
[ε-intervention sufficiency] The map ϕ_t is ε-intervention-sufficient for D_t when sup_δ∈D_t TV( K_A_t^δ, widehat K_ϕ_t(A_t)^δ ) ≤ε.
Exact mathematical passage
\sup_{\delta\in\Dcal_t} \TV\left( K_{A_t}^{\delta}, \widehat K_{\phi_t(A_t)}^{\delta} \right) \leq\eps.
\geqDefinition passage · 112 passages
Search aliases: ≥ geq
Definition passage: Protected evaluation context · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 490
Xi= ( Z, μ, Y, ν, bar c, λ, mathscr I, B, α ), where: item Z is a task or environment family and μ is its held-out distribution; item Y is the space of protected external traces; item ν:Z×Y→[0,1] is task utility; item bar c:Z×Y→[0,1] is normalized resource cost; item λ≥0 is a cost weight; item mathscr I(A,B)∈{0,1} is a declared invariant-preservation predicate for the transition A→ B; item B is the resource envelope used for candidate generation and evaluation; item α∈(0,1) is the accepted stati…
Exact mathematical passage
\lambda\geq0
\subseteqDefinition passage · 71 passages
Search aliases: ⊆ subseteq
Definition passage: Property-transport defect · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 725
[Property-transport defect] For a declared domain K⊆ X, define ε_Q(T;K) = sup_x∈ K norm Q_Y(Tx)-widetilde T_QQ_X(x) .
Exact mathematical passage
K\subseteq X
\bigoplusDefinition passage · 7 passages
Search aliases: ⊕ bigoplus
Definition passage: Block-scalar response signature · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12898
[Block-scalar response signature] Let H=⊕_a=1^kH_a be a reducing decomposition for every R_j. The response language is block scalar when restrR_jH_a=λ_ajI_H_a for scalars λ_aj. The vector σ_R(H_a) = (λ_a1,…,λ_am) is the response signature of block H_a.
Exact mathematical passage
H=\bigoplus_{a=1}^{k}H_a
\bigoplus_{a=1}^{k}Definition passage · 1 passage
Search aliases: ⊕_a=1^k bigoplus
Definition passage: Block-scalar response signature · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12898
[Block-scalar response signature] Let H=⊕_a=1^kH_a be a reducing decomposition for every R_j. The response language is block scalar when restrR_jH_a=λ_ajI_H_a for scalars λ_aj. The vector σ_R(H_a) = (λ_a1,…,λ_am) is the response signature of block H_a.
Exact mathematical passage
H=\bigoplus_{a=1}^{k}H_a
\bigcapDefinition passage · 9 passages
Search aliases: ⋂ bigcap
Definition passage: Path-flat repair and defect cores · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3041
[Path-flat repair and defect cores] The path-flat repair and defect cores at A are U_A^flat := ⋂_γker(H^U_γ-I), V_A^flat := ⋂_γker(H^V_γ-I).
Exact mathematical passage
U_A^{\flat} := \bigcap_\gamma\ker(H^U_\gamma-I), \qquad V_A^{\flat} := \bigcap_\gamma\ker(H^V_\gamma-I).
\bigcap_{j=1}^{m}Definition passage · 2 passages
Search aliases: ⋂_j=1^m bigcap
Definition passage: Evaluator-nullspace · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13820
[Evaluator-nullspace] The locally invisible behavior subspace of portfolio B is N_B = ⋂_j=1^mker Db_j.
Exact mathematical passage
\Ncal_{\Bcal} = \bigcap_{j=1}^{m}\ker Db_j. \label{eq:b13-eval-nullspace}
\bigcap_{x\in C_z}Definition passage · 1 passage
Search aliases: ⋂_x∈ C_z bigcap in
Definition passage: Repair-consistent cell · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9873
[Repair-consistent cell] A cell is repair consistent when ⋂_x∈ C_zA^⋆(x)≠∅. It is inconsistent when no single authorized repair works for every episode that the grammar treats as equivalent.
Exact mathematical passage
\bigcap_{x\in C_z}\Acal^\star(x)\neq\varnothing. \label{eq:b8-consistent-cell}
\bigcap_{x\in S}Definition passage · 1 passage
Search aliases: ⋂_x∈ S bigcap in
Definition passage: Minimal repair conflict · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9882
[Minimal repair conflict] A subset S⊆ C_z is a minimal repair conflict when ⋂_x∈ SA^⋆(x)=∅, but every proper subset has a nonempty repair intersection.
Exact mathematical passage
\bigcap_{x\in S}\Acal^\star(x)=\varnothing,
\bigcap_\gammaDefinition passage · 1 passage
Search aliases: ⋂_γ bigcap gamma
Definition passage: Path-flat repair and defect cores · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3041
[Path-flat repair and defect cores] The path-flat repair and defect cores at A are U_A^flat := ⋂_γker(H^U_γ-I), V_A^flat := ⋂_γker(H^V_γ-I).
Exact mathematical passage
U_A^{\flat} := \bigcap_\gamma\ker(H^U_\gamma-I), \qquad V_A^{\flat} := \bigcap_\gamma\ker(H^V_\gamma-I).
\bigcap_{\gamma\in\mathsf W(v)}Definition passage · 1 passage
Search aliases: ⋂_γ∈ W(v) bigcap gamma in mathsf
Definition passage: Closed-walk equalizer core · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3532
[Closed-walk equalizer core] For a possibly noninvertible evaluator graph, let W(v) be the monoid of directed closed walks at v. Define E_v^eq := ⋂_γ∈ W(v)ker(H_γ-I).
Exact mathematical passage
E_v^{\mathrm{eq}} := \bigcap_{\gamma\in\mathsf W(v)}\ker(H_\gamma-I).
\bigcap_{\gamma\in\pi_1(G,v_0)}Definition passage · 1 passage
Search aliases: ⋂_γ∈π_1(G,v_0) bigcap gamma in pi
Definition passage: Path-flat evidence core · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3467
[Path-flat evidence core] The path-flat evidence core at v_0 is E_v_0^flat := ⋂_γ∈π_1(G,v_0) ker(H_γ-I).
Exact mathematical passage
E_{v_0}^{\flat} := \bigcap_{\gamma\in\pi_1(G,v_0)} \ker(H_\gamma-I).
\bigcap_{\tau\in S}Definition passage · 1 passage
Search aliases: ⋂_τ∈ S bigcap tau in
Definition passage: Compiler-language conflict · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15091
[Compiler-language conflict] A finite set S⊆T is a compiler-language conflict for G when all problems in S receive the same inherited diagnostic token but ⋂_τ∈ SA^⋆(τ)=∅. A minimal conflict has no proper conflicting subset.
Exact mathematical passage
\bigcap_{\tau\in S}\Acal^\star(\tau)=\varnothing. \label{eq:b15-conflict}
\cdotsDefinition passage · 22 passages
Search aliases: ⋯ cdots
Definition passage: Open-set ascending lineage · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9454
[Open-set ascending lineage] A lineage is conservatively ascending when L_0preceq_consL_1 preceq_consL_2 preceq_cons⋯. It is structurally open-set when infinitely many edges add a foreign-task-relevant correction direction or validated distinction not representable in the prior grammar.
Exact mathematical passage
L_0\preceq_{\mathrm{cons}}L_1 \preceq_{\mathrm{cons}}L_2 \preceq_{\mathrm{cons}}\cdots. \label{eq:b8-ascending-lineage}
\longrightarrowDefinition passage · 40 passages
Search aliases: ⟶ longrightarrow
Definition passage: Repair abstraction operator · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1401
[Repair abstraction operator] A repair abstraction operator is a map a: H_episodes ⟶ overlineO that maps one or more concrete episodes to a typed schema with a failure signature, applicability condition, correction program, effect profile, and receipt template.
Exact mathematical passage
\mathfrak a: \mathcal H_{\mathrm{episodes}} \longrightarrow \overline\Ocal
\longmapstoDefinition passage · 8 passages
Search aliases: ⟼ longmapsto
Definition passage: Sealed meta-compiler · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7306
[Sealed meta-compiler] A sealed meta-compiler is a map M: ( D_rm dev, D_rm meta,K_0,Γ) ⟼ (K_1,Π_1), where Γ is a declared finite meta-grammar and Π_1 is a descendant receipt. The compiler is sealed when: item neither final audit split is supplied to M; item Π_1 commits to the child state, parent identity, input hashes, grammar, selected configuration, and candidate budget; item the audit files are opened only after the receipt has been serialized and hashed; item the child state used for final e…
Exact mathematical passage
\mathfrak M: \left(\mathsf D_{\rm dev},\mathsf D_{\rm meta},K_0,\Gamma\right) \longmapsto \left(K_1,\Pi_1\right), \label{eq:block07-meta-compiler}
A'Introduction passage · 3 passages
Search aliases: A'
Introduction passage · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 558
Let A_t⊆ V(Λ_t) be the executable archive at generation t. Its task frontier is F_t(Ξ)=sup_A∈A_tJ_Ξ(A). Let K_t be the active improvement mechanism, including diagnosis, proposal, composition, and validation policies. For horizon h and budget B, define target-relative metaproductivity Γ_Ξ,h,B(K_t;A) = E [ sup_A'∈Desc_h,B(K_t,A) (J_Ξ(A')-J_Ξ(A))_+ ]. Finally, let Risk_Ξ(V_t) denote held-out evaluator risk on candidate-effect judgments, including false acceptance of harmful candidates and false re…
Exact mathematical passage
\Gamma_{\Xi,h,B}(K_t;A) = \mathbb E \left[ \sup_{A'\in\Desc_{h,B}(K_t,A)} \left(J_\Xi(A')-J_\Xi(A)\right)_+ \right].
A^{\rm base}_{i,\rm prot}Introduction passage · 1 passage
Search aliases: A^rm base_i,rm prot rm rm
Introduction passage · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7388
Let x_i denote a development candidate feature vector, y_i its revealed protected utility, and r_i=[A^rm base_i,rm prot-A_i,rm prot]_+ its protected-regression label. Standardize the feature matrix to Z using development means and scales.
Exact mathematical passage
r_i=[A^{\rm base}_{i,\rm prot}-A_{i,\rm prot}]_+
A_{i,\rm prot}Introduction passage · 1 passage
Search aliases: A_i,rm prot rm
Introduction passage · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7388
Let x_i denote a development candidate feature vector, y_i its revealed protected utility, and r_i=[A^rm base_i,rm prot-A_i,rm prot]_+ its protected-regression label. Standardize the feature matrix to Z using development means and scales.
Exact mathematical passage
r_i=[A^{\rm base}_{i,\rm prot}-A_{i,\rm prot}]_+
A_{t+1}Introduction passage · 4 passages
Search aliases: A_t+1
Introduction passage · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 332
Let A_t denote an active agent system at generation t, and let J(A_t) denote externally evaluated task utility. Ordinary task improvement requires J(A_t+1)>J(A_t). Let Γ_h,B(K;A) denote the expected best gain obtained when improvement mechanism K is allowed horizon h and resource budget B to modify target agent A. Improvement-process improvement requires, on a held-out distribution of target agents, E_AΓ_h,B(K_t+1;A) > E_AΓ_h,B(K_t;A). Finally, if V_t is the evaluator active at generation t, epi…
Exact mathematical passage
J(A_{t+1})>J(A_t).
B_0Introduction passage · 3 passages
Search aliases: B_0
Introduction passage · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7605
Let M_B(K) denote expected best gain when policy K receives B independently generated candidate opportunities. Even if the policy is unchanged, M_B+1(K)≥ M_B(K). Therefore, comparing a child with B_1>B_0 against a parent with budget B_0 does not isolate improvement of the selection procedure.
Exact mathematical passage
B_1>B_0
B_1Introduction passage · 6 passages
Search aliases: B_1
Introduction passage · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7605
Let M_B(K) denote expected best gain when policy K receives B independently generated candidate opportunities. Even if the policy is unchanged, M_B+1(K)≥ M_B(K). Therefore, comparing a child with B_1>B_0 against a parent with budget B_0 does not isolate improvement of the selection procedure.
Exact mathematical passage
B_1>B_0
\operatorname{cj}Introduction passage · 2 passages
Search aliases: cj operatorname
Introduction passage · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5825
Let cj denote canonical JSON serialization with sorted keys and normalized arrays. For payload x, define h(x)=SHA256(cj(x)). A transaction receipt records parent, candidate, witness, checkpoint, phase trace, evidence, decision, rejection reasons, final state, and previous receipt hash.
Exact mathematical passage
\operatorname{cj}
\DescIntroduction passage · 3 passages
Search aliases: Desc operatorname
Introduction passage · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 558
Let A_t⊆ V(Λ_t) be the executable archive at generation t. Its task frontier is F_t(Ξ)=sup_A∈A_tJ_Ξ(A). Let K_t be the active improvement mechanism, including diagnosis, proposal, composition, and validation policies. For horizon h and budget B, define target-relative metaproductivity Γ_Ξ,h,B(K_t;A) = E [ sup_A'∈Desc_h,B(K_t,A) (J_Ξ(A')-J_Ξ(A))_+ ]. Finally, let Risk_Ξ(V_t) denote held-out evaluator risk on candidate-effect judgments, including false acceptance of harmful candidates and false re…
Exact mathematical passage
\Gamma_{\Xi,h,B}(K_t;A) = \mathbb E \left[ \sup_{A'\in\Desc_{h,B}(K_t,A)} \left(J_\Xi(A')-J_\Xi(A)\right)_+ \right].
\Desc_{h,B}Introduction passage · 3 passages
Search aliases: Desc_h,B Desc operatorname
Introduction passage · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 558
Let A_t⊆ V(Λ_t) be the executable archive at generation t. Its task frontier is F_t(Ξ)=sup_A∈A_tJ_Ξ(A). Let K_t be the active improvement mechanism, including diagnosis, proposal, composition, and validation policies. For horizon h and budget B, define target-relative metaproductivity Γ_Ξ,h,B(K_t;A) = E [ sup_A'∈Desc_h,B(K_t,A) (J_Ξ(A')-J_Ξ(A))_+ ]. Finally, let Risk_Ξ(V_t) denote held-out evaluator risk on candidate-effect judgments, including false acceptance of harmful candidates and false re…
Exact mathematical passage
\Gamma_{\Xi,h,B}(K_t;A) = \mathbb E \left[ \sup_{A'\in\Desc_{h,B}(K_t,A)} \left(J_\Xi(A')-J_\Xi(A)\right)_+ \right].
\mathbb E_AIntroduction passage · 1 passage
Search aliases: E_A mathbb
Introduction passage · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 336
denote externally evaluated task utility. Ordinary task improvement requires J(A_t+1)>J(A_t). Let Γ_h,B(K;A) denote the expected best gain obtained when improvement mechanism K is allowed horizon h and resource budget B to modify target agent A. Improvement-process improvement requires, on a held-out distribution of target agents, E_AΓ_h,B(K_t+1;A) > E_AΓ_h,B(K_t;A). Finally, if V_t is the evaluator active at generation t, epistemic improvement requires a decrease in held-out evaluation risk rat…
Exact mathematical passage
\mathbb E_A\Gamma_{h,B}(K_{t+1};A) > \mathbb E_A\Gamma_{h,B}(K_t;A).
\mathbb E_{\tau\sim Q}Introduction passage · 2 passages
Search aliases: E_τ∼ Q mathbb tau sim
Introduction passage · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 14410
e fixture, a selected method that passes the stronger post hoc capability and integrity predicates receives its protected audit gain, a no-op receives zero, and a selected invalid method receives the frozen penalty -0.75. This prevents an invalid selection from being silently converted into a neutral outcome. Let R_u(Γ,τ) be protected downside or integrity risk. Define C_u(Γ) = E_τ∼ Q U_u^rm valid(Γ,τ). For α∈(0,1), write CVaR_α(X) for the mean of the lower α-tail of a gain variable X.
Exact mathematical passage
\mathfrak C_u(\Gamma) = \mathbb E_{\tau\sim Q} U_u^{\rm valid}(\Gamma,\tau). \label{eq:compiler-context-utility}
\Gcal_{10}Introduction passage · 3 passages
Search aliases: Gcal_10 G_10 Gcal mathcal
Introduction passage · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12396
Let G_10 denote the Block 10 two-chart grammar with a supplied context projector P_ctx. Let G_11 contain: item the intervention-response atlas compiler; item a finite family of projectors and separators returned by the compiler; item every constructor and witness rule of G_10; item a declared fallback that may instantiate the old projector directly.
Exact mathematical passage
\Gcal_{10}
\Gcal_{11}Introduction passage · 3 passages
Search aliases: Gcal_11 G_11 Gcal mathcal
Introduction passage · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12396
Let G_10 denote the Block 10 two-chart grammar with a supplied context projector P_ctx. Let G_11 contain: item the intervention-response atlas compiler; item a finite family of projectors and separators returned by the compiler; item every constructor and witness rule of G_10; item a declared fallback that may instantiate the old projector directly.
Exact mathematical passage
\Gcal_{11}
H_{A,c}^{(1:5)}Introduction passage · 1 passage
Search aliases: H_A,c^(1:5)
Introduction passage · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 18220
Let A denote the frozen source architecture and let B be one of the two Block 20 foreign architectures. For admitted observation chart c, Block 20 supplies an accepted full-trajectory connection T^c_B→ A:H_B,c^(1:5)⟶ H_A,c^(1:5). Let P:ℝ^40→ℝ^8 be an orthonormal packet projector fitted on a pre-audit source support. Applying P independently to the five recurrent states gives Π(H) = (P h_1,…,P h_5) ∈ (ℝ^8)^5 ≅ℝ^40. The packet is not claimed to preserve every latent coordinate. It is required to p…
Exact mathematical passage
T^{c}_{B\to A}:H_{B,c}^{(1:5)}\longrightarrow H_{A,c}^{(1:5)}.
H_{B,c}^{(1:5)}Introduction passage · 1 passage
Search aliases: H_B,c^(1:5)
Introduction passage · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 18220
Let A denote the frozen source architecture and let B be one of the two Block 20 foreign architectures. For admitted observation chart c, Block 20 supplies an accepted full-trajectory connection T^c_B→ A:H_B,c^(1:5)⟶ H_A,c^(1:5). Let P:ℝ^40→ℝ^8 be an orthonormal packet projector fitted on a pre-audit source support. Applying P independently to the five recurrent states gives Π(H) = (P h_1,…,P h_5) ∈ (ℝ^8)^5 ≅ℝ^40. The packet is not claimed to preserve every latent coordinate. It is required to p…
Exact mathematical passage
T^{c}_{B\to A}:H_{B,c}^{(1:5)}\longrightarrow H_{A,c}^{(1:5)}.
h_sIntroduction passage · 18 passages
Search aliases: h_s
Introduction passage · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6743
Let s denote the source and t a target. On paired anchor inputs x_1,…,x_m, the system observes H_s = h_s(x_1)^top vdots h_s(x_m)^top , H_t = h_t(x_1)^top vdots h_t(x_m)^top . It estimates the bridge by regularized least squares: widehat T_t← s ∈ arg min_T normH_sT^top-H_t_F^2 +λnormT_F^2. The transformation matrices used to construct the controlled targets are not exposed to this fit.
Exact mathematical passage
H_s = \begin{bmatrix} h_s(x_1)^{\top}\\ \vdots\\ h_s(x_m)^{\top} \end{bmatrix}, \qquad H_t = \begin{bmatrix} h_t(x_1)^{\top}\\ \vdots\\ h_t(x_m)^{\top} \end{bmatrix}.
h_tIntroduction passage · 12 passages
Search aliases: h_t
Introduction passage · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6743
Let s denote the source and t a target. On paired anchor inputs x_1,…,x_m, the system observes H_s = h_s(x_1)^top vdots h_s(x_m)^top , H_t = h_t(x_1)^top vdots h_t(x_m)^top . It estimates the bridge by regularized least squares: widehat T_t← s ∈ arg min_T normH_sT^top-H_t_F^2 +λnormT_F^2. The transformation matrices used to construct the controlled targets are not exposed to this fit.
Exact mathematical passage
H_s = \begin{bmatrix} h_s(x_1)^{\top}\\ \vdots\\ h_s(x_m)^{\top} \end{bmatrix}, \qquad H_t = \begin{bmatrix} h_t(x_1)^{\top}\\ \vdots\\ h_t(x_m)^{\top} \end{bmatrix}.
I_VIntroduction passage · 5 passages
Search aliases: I_V
Introduction passage · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1538
Let D^†:V→ U^auth denote the Moore--Penrose pseudoinverse. Define P_cor:=DD^†, P_ob:=I_V-DD^†.
Exact mathematical passage
P_{\mathrm{cor}}:=DD^\dagger, \qquad P_{\mathrm{ob}}:=I_V-DD^\dagger.
\Ical_{\mathrm{NL}}Introduction passage · 1 passage
Search aliases: Ical_NL I_NL Ical mathrm mathcal mathrm
Introduction passage · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15690
Let L(x,y;Z) denote mean gold-answer cross-entropy when the prompt is x, target answer is y, and the recurrent latent slots are externally set to trace Z. Let Z(x) denote the clean trace. For a donor x' selected from the same declared answer-format group, let Z(x') be its clean trace. Block 16 fixes the intervention language I_NL = {zero,replay,permute,noise, transplant,packet,transport}.
Exact mathematical passage
\Ical_{\mathrm{NL}} = \{\mathsf{zero},\mathsf{replay},\mathsf{permute},\mathsf{noise}, \mathsf{transplant},\mathsf{packet},\mathsf{transport}\}. \label{eq:b16-intervention-language}
K_2Introduction passage · 21 passages
Search aliases: K_2
Introduction passage · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11655
Let K_2 denote the exact Block 9 conditioning-aware process. It supplies the quality-weighted D-optimal anchor-ordering rule and the published lineage receipt. Block 10 constructs two resource-matched processes: K_2,G^96, K_3,C^96. The global parent K_2,G^96 retains one affine bridge and uses all 80 non-witness pairs to estimate it. The chart child K_3,C^96 uses 48 pairs for local bridge fitting, 32 pairs for chart selection, and the same 16 witness pairs. Both consume exactly 96 paired source-t…
Exact mathematical passage
K_2
K_{2,G}^{96}Introduction passage · 7 passages
Search aliases: K_2,G^96
Introduction passage · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11656
Let K_2 denote the exact Block 9 conditioning-aware process. It supplies the quality-weighted D-optimal anchor-ordering rule and the published lineage receipt. Block 10 constructs two resource-matched processes: K_2,G^96, K_3,C^96. The global parent K_2,G^96 retains one affine bridge and uses all 80 non-witness pairs to estimate it. The chart child K_3,C^96 uses 48 pairs for local bridge fitting, 32 pairs for chart selection, and the same 16 witness pairs. Both consume exactly 96 paired source-t…
Exact mathematical passage
K_{2,G}^{96}, \qquad K_{3,C}^{96}.
K_{3,C}^{96}Introduction passage · 8 passages
Search aliases: K_3,C^96
Introduction passage · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11656
Let K_2 denote the exact Block 9 conditioning-aware process. It supplies the quality-weighted D-optimal anchor-ordering rule and the published lineage receipt. Block 10 constructs two resource-matched processes: K_2,G^96, K_3,C^96. The global parent K_2,G^96 retains one affine bridge and uses all 80 non-witness pairs to estimate it. The chart child K_3,C^96 uses 48 pairs for local bridge fitting, 32 pairs for chart selection, and the same 16 witness pairs. Both consume exactly 96 paired source-t…
Exact mathematical passage
K_{2,G}^{96}, \qquad K_{3,C}^{96}.
M_BIntroduction passage · 2 passages
Search aliases: M_B
Introduction passage · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7601
Let M_B(K) denote expected best gain when policy K receives B independently generated candidate opportunities. Even if the policy is unchanged, M_B+1(K)≥ M_B(K). Therefore, comparing a child with B_1>B_0 against a parent with budget B_0 does not isolate improvement of the selection procedure.
Exact mathematical passage
M_B(K)
M_{B+1}Introduction passage · 1 passage
Search aliases: M_B+1
Introduction passage · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7602
Let M_B(K) denote expected best gain when policy K receives B independently generated candidate opportunities. Even if the policy is unchanged, M_B+1(K)≥ M_B(K). Therefore, comparing a child with B_1>B_0 against a parent with budget B_0 does not isolate improvement of the selection procedure.
Exact mathematical passage
M_{B+1}(K)\geq M_B(K).
\Mcal_0Introduction passage · 3 passages
Search aliases: Mcal_0 M_0 Mcal mathcal
Introduction passage · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15155
Let M_0 denote the inherited singleton-method grammar. For inherited method families a and b, define an ordered pair constructor Seq(a,b) = ``apply the research intervention from family a, then the intervention from family b''. The generic finite constructor language is C_rm pair = setSeq(a,b):a,b∈M_0, a≠ b. In the executable fixture there are six typed inherited obstruction families, producing 30 ordered pair constructors.
Exact mathematical passage
\Mcal_0
\mathrm{NL}Introduction passage · 12 passages
Search aliases: NL mathrm
Introduction passage · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15690
Let L(x,y;Z) denote mean gold-answer cross-entropy when the prompt is x, target answer is y, and the recurrent latent slots are externally set to trace Z. Let Z(x) denote the clean trace. For a donor x' selected from the same declared answer-format group, let Z(x') be its clean trace. Block 16 fixes the intervention language I_NL = {zero,replay,permute,noise, transplant,packet,transport}.
Exact mathematical passage
\Ical_{\mathrm{NL}} = \{\mathsf{zero},\mathsf{replay},\mathsf{permute},\mathsf{noise}, \mathsf{transplant},\mathsf{packet},\mathsf{transport}\}. \label{eq:b16-intervention-language}
\mathsf{noise}Introduction passage · 3 passages
Search aliases: noise mathsf
Introduction passage · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15690
Let L(x,y;Z) denote mean gold-answer cross-entropy when the prompt is x, target answer is y, and the recurrent latent slots are externally set to trace Z. Let Z(x) denote the clean trace. For a donor x' selected from the same declared answer-format group, let Z(x') be its clean trace. Block 16 fixes the intervention language I_NL = {zero,replay,permute,noise, transplant,packet,transport}.
Exact mathematical passage
\Ical_{\mathrm{NL}} = \{\mathsf{zero},\mathsf{replay},\mathsf{permute},\mathsf{noise}, \mathsf{transplant},\mathsf{packet},\mathsf{transport}\}. \label{eq:b16-intervention-language}
\mathsf{packet}Introduction passage · 1 passage
Search aliases: packet mathsf
Introduction passage · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15690
Let L(x,y;Z) denote mean gold-answer cross-entropy when the prompt is x, target answer is y, and the recurrent latent slots are externally set to trace Z. Let Z(x) denote the clean trace. For a donor x' selected from the same declared answer-format group, let Z(x') be its clean trace. Block 16 fixes the intervention language I_NL = {zero,replay,permute,noise, transplant,packet,transport}.
Exact mathematical passage
\Ical_{\mathrm{NL}} = \{\mathsf{zero},\mathsf{replay},\mathsf{permute},\mathsf{noise}, \mathsf{transplant},\mathsf{packet},\mathsf{transport}\}. \label{eq:b16-intervention-language}
\mathsf{permute}Introduction passage · 2 passages
Search aliases: permute mathsf
Introduction passage · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15690
Let L(x,y;Z) denote mean gold-answer cross-entropy when the prompt is x, target answer is y, and the recurrent latent slots are externally set to trace Z. Let Z(x) denote the clean trace. For a donor x' selected from the same declared answer-format group, let Z(x') be its clean trace. Block 16 fixes the intervention language I_NL = {zero,replay,permute,noise, transplant,packet,transport}.
Exact mathematical passage
\Ical_{\mathrm{NL}} = \{\mathsf{zero},\mathsf{replay},\mathsf{permute},\mathsf{noise}, \mathsf{transplant},\mathsf{packet},\mathsf{transport}\}. \label{eq:b16-intervention-language}
\R^{40}Introduction passage · 5 passages
Search aliases: R^40 ℝ^40 R mathbb
Introduction passage · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 18223
Let A denote the frozen source architecture and let B be one of the two Block 20 foreign architectures. For admitted observation chart c, Block 20 supplies an accepted full-trajectory connection T^c_B→ A:H_B,c^(1:5)⟶ H_A,c^(1:5). Let P:ℝ^40→ℝ^8 be an orthonormal packet projector fitted on a pre-audit source support. Applying P independently to the five recurrent states gives Π(H) = (P h_1,…,P h_5) ∈ (ℝ^8)^5 ≅ℝ^40. The packet is not claimed to preserve every latent coordinate. It is required to p…
Exact mathematical passage
P:\R^{40}\to\R^8
\R^8Introduction passage · 7 passages
Search aliases: R^8 ℝ^8 R mathbb
Introduction passage · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 18223
Let A denote the frozen source architecture and let B be one of the two Block 20 foreign architectures. For admitted observation chart c, Block 20 supplies an accepted full-trajectory connection T^c_B→ A:H_B,c^(1:5)⟶ H_A,c^(1:5). Let P:ℝ^40→ℝ^8 be an orthonormal packet projector fitted on a pre-audit source support. Applying P independently to the five recurrent states gives Π(H) = (P h_1,…,P h_5) ∈ (ℝ^8)^5 ≅ℝ^40. The packet is not claimed to preserve every latent coordinate. It is required to p…
Exact mathematical passage
P:\R^{40}\to\R^8
\RiskIntroduction passage · 3 passages
Search aliases: Risk operatorname
Introduction passage · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 567
(Ξ)=sup_A∈A_tJ_Ξ(A). Let K_t be the active improvement mechanism, including diagnosis, proposal, composition, and validation policies. For horizon h and budget B, define target-relative metaproductivity Γ_Ξ,h,B(K_t;A) = E [ sup_A'∈Desc_h,B(K_t,A) (J_Ξ(A')-J_Ξ(A))_+ ]. Finally, let Risk_Ξ(V_t) denote held-out evaluator risk on candidate-effect judgments, including false acceptance of harmful candidates and false rejection of beneficial candidates above a declared margin.
Exact mathematical passage
\Risk_\Xi(\Vcal_t)
\Risk_\XiIntroduction passage · 1 passage
Search aliases: Risk_Ξ Risk Xi operatorname Xi
Introduction passage · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 567
(Ξ)=sup_A∈A_tJ_Ξ(A). Let K_t be the active improvement mechanism, including diagnosis, proposal, composition, and validation policies. For horizon h and budget B, define target-relative metaproductivity Γ_Ξ,h,B(K_t;A) = E [ sup_A'∈Desc_h,B(K_t,A) (J_Ξ(A')-J_Ξ(A))_+ ]. Finally, let Risk_Ξ(V_t) denote held-out evaluator risk on candidate-effect judgments, including false acceptance of harmful candidates and false rejection of beneficial candidates above a declared margin.
Exact mathematical passage
\Risk_\Xi(\Vcal_t)
s_aIntroduction passage · 4 passages
Search aliases: s_a
Introduction passage · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12362
Let s_a denote the best population score attainable from block a, and let widehat s_a be its empirical score.
Exact mathematical passage
s_a
\operatorname{Seq}Introduction passage · 2 passages
Search aliases: Seq operatorname
Introduction passage · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15156
Let M_0 denote the inherited singleton-method grammar. For inherited method families a and b, define an ordered pair constructor Seq(a,b) = ``apply the research intervention from family a, then the intervention from family b''. The generic finite constructor language is C_rm pair = setSeq(a,b):a,b∈M_0, a≠ b. In the executable fixture there are six typed inherited obstruction families, producing 30 ordered pair constructors.
Exact mathematical passage
\operatorname{Seq}(a,b) = \text{``apply the research intervention from family $a$, then the intervention from family $b$''}. \label{eq:b15-seq-constructor}
\operatorname{SHA256}Introduction passage · 1 passage
Search aliases: SHA256 operatorname
Introduction passage · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5826
Let cj denote canonical JSON serialization with sorted keys and normalized arrays. For payload x, define h(x)=SHA256(cj(x)). A transaction receipt records parent, candidate, witness, checkpoint, phase trace, evidence, decision, rejection reasons, final state, and previous receipt hash.
Exact mathematical passage
h(x)=\operatorname{SHA256}(\operatorname{cj}(x)).
\sup_{A'\in\Desc_{h,B}(K_t,A)}Introduction passage · 1 passage
Search aliases: sup_A'∈Desc_h,B(K_t,A) sup in Desc sup in operatorname
Introduction passage · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 558
Let A_t⊆ V(Λ_t) be the executable archive at generation t. Its task frontier is F_t(Ξ)=sup_A∈A_tJ_Ξ(A). Let K_t be the active improvement mechanism, including diagnosis, proposal, composition, and validation policies. For horizon h and budget B, define target-relative metaproductivity Γ_Ξ,h,B(K_t;A) = E [ sup_A'∈Desc_h,B(K_t,A) (J_Ξ(A')-J_Ξ(A))_+ ]. Finally, let Risk_Ξ(V_t) denote held-out evaluator risk on candidate-effect judgments, including false acceptance of harmful candidates and false re…
Exact mathematical passage
\Gamma_{\Xi,h,B}(K_t;A) = \mathbb E \left[ \sup_{A'\in\Desc_{h,B}(K_t,A)} \left(J_\Xi(A')-J_\Xi(A)\right)_+ \right].
\sup_{A\in\Acal_t}Introduction passage · 1 passage
Search aliases: sup_A∈Acal_t sup_A∈A_t sup in Acal sup in mathcal
Introduction passage · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 554
Let A_t⊆ V(Λ_t) be the executable archive at generation t. Its task frontier is F_t(Ξ)=sup_A∈A_tJ_Ξ(A). Let K_t be the active improvement mechanism, including diagnosis, proposal, composition, and validation policies. For horizon h and budget B, define target-relative metaproductivity Γ_Ξ,h,B(K_t;A) = E [ sup_A'∈Desc_h,B(K_t,A) (J_Ξ(A')-J_Ξ(A))_+ ]. Finally, let Risk_Ξ(V_t) denote held-out evaluator risk on candidate-effect judgments, including false acceptance of harmful candidates and false re…
Exact mathematical passage
F_t(\Xi)=\sup_{A\in\Acal_t}J_\Xi(A).
T^{c}_{B\to A}Introduction passage · 1 passage
Search aliases: T^c_B→ A to
Introduction passage · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 18220
Let A denote the frozen source architecture and let B be one of the two Block 20 foreign architectures. For admitted observation chart c, Block 20 supplies an accepted full-trajectory connection T^c_B→ A:H_B,c^(1:5)⟶ H_A,c^(1:5). Let P:ℝ^40→ℝ^8 be an orthonormal packet projector fitted on a pre-audit source support. Applying P independently to the five recurrent states gives Π(H) = (P h_1,…,P h_5) ∈ (ℝ^8)^5 ≅ℝ^40. The packet is not claimed to preserve every latent coordinate. It is required to p…
Exact mathematical passage
T^{c}_{B\to A}:H_{B,c}^{(1:5)}\longrightarrow H_{A,c}^{(1:5)}.
\mathsf{transplant}Introduction passage · 2 passages
Search aliases: transplant mathsf
Introduction passage · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15690
Let L(x,y;Z) denote mean gold-answer cross-entropy when the prompt is x, target answer is y, and the recurrent latent slots are externally set to trace Z. Let Z(x) denote the clean trace. For a donor x' selected from the same declared answer-format group, let Z(x') be its clean trace. Block 16 fixes the intervention language I_NL = {zero,replay,permute,noise, transplant,packet,transport}.
Exact mathematical passage
\Ical_{\mathrm{NL}} = \{\mathsf{zero},\mathsf{replay},\mathsf{permute},\mathsf{noise}, \mathsf{transplant},\mathsf{packet},\mathsf{transport}\}. \label{eq:b16-intervention-language}
\mathsf{transport}Introduction passage · 1 passage
Search aliases: transport mathsf
Introduction passage · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15690
Let L(x,y;Z) denote mean gold-answer cross-entropy when the prompt is x, target answer is y, and the recurrent latent slots are externally set to trace Z. Let Z(x) denote the clean trace. For a donor x' selected from the same declared answer-format group, let Z(x') be its clean trace. Block 16 fixes the intervention language I_NL = {zero,replay,permute,noise, transplant,packet,transport}.
Exact mathematical passage
\Ical_{\mathrm{NL}} = \{\mathsf{zero},\mathsf{replay},\mathsf{permute},\mathsf{noise}, \mathsf{transplant},\mathsf{packet},\mathsf{transport}\}. \label{eq:b16-intervention-language}
U^{\Auth}Introduction passage · 6 passages
Search aliases: U^Auth U^auth Auth mathrm
Introduction passage · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1537
Let D^†:V→ U^auth denote the Moore--Penrose pseudoinverse. Define P_cor:=DD^†, P_ob:=I_V-DD^†.
Exact mathematical passage
D^\dagger:V\to U^{\Auth}
U_u^{\rm valid}Introduction passage · 2 passages
Search aliases: U_u^rm valid rm
Introduction passage · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 14409
Let U_u^rm valid(Γ,τ) denote the validity-filtered protected gain produced by compiler Γ on research problem τ under environment u. In the executable fixture, a selected method that passes the stronger post hoc capability and integrity predicates receives its protected audit gain, a no-op receives zero, and a selected invalid method receives the frozen penalty -0.75. This prevents an invalid selection from being silently converted into a neutral outcome. Let R_u(Γ,τ) be protected downside or int…
Exact mathematical passage
U_u^{\rm valid}(\Gamma,\tau)
\vdotsIntroduction passage · 4 passages
Search aliases: vdots
Introduction passage · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6743
Let s denote the source and t a target. On paired anchor inputs x_1,…,x_m, the system observes H_s = h_s(x_1)^top vdots h_s(x_m)^top , H_t = h_t(x_1)^top vdots h_t(x_m)^top . It estimates the bridge by regularized least squares: widehat T_t← s ∈ arg min_T normH_sT^top-H_t_F^2 +λnormT_F^2. The transformation matrices used to construct the controlled targets are not exposed to this fit.
Exact mathematical passage
H_s = \begin{bmatrix} h_s(x_1)^{\top}\\ \vdots\\ h_s(x_m)^{\top} \end{bmatrix}, \qquad H_t = \begin{bmatrix} h_t(x_1)^{\top}\\ \vdots\\ h_t(x_m)^{\top} \end{bmatrix}.
\widehat s_aIntroduction passage · 4 passages
Search aliases: widehat s_a widehat
Introduction passage · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12362
Let s_a denote the best population score attainable from block a, and let widehat s_a be its empirical score.
Exact mathematical passage
\widehat s_a
\widetilde FIntroduction passage · 2 passages
Search aliases: widetilde F widetilde
Introduction passage · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4092
Let widetilde F(P,δ) denote execution in an isolated child environment. Write widetilde F(P,δ)=(C,B), where C is the candidate child state and B is a buffer of externally persistent effects.
Exact mathematical passage
\widetilde F(P,\delta)
x_1Introduction passage · 16 passages
Search aliases: x_1
Introduction passage · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6742
Let s denote the source and t a target. On paired anchor inputs x_1,…,x_m, the system observes H_s = h_s(x_1)^top
Exact mathematical passage
x_1,\ldots,x_m
x_mIntroduction passage · 2 passages
Search aliases: x_m
Introduction passage · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6742
Let s denote the source and t a target. On paired anchor inputs x_1,…,x_m, the system observes H_s = h_s(x_1)^top
Exact mathematical passage
x_1,\ldots,x_m
\Gamma_{h,B}Introduction passage · 2 passages
Search aliases: Γ_h,B Gamma
Introduction passage · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 335
Let A_t denote an active agent system at generation t, and let J(A_t) denote externally evaluated task utility. Ordinary task improvement requires J(A_t+1)>J(A_t). Let Γ_h,B(K;A) denote the expected best gain obtained when improvement mechanism K is allowed horizon h and resource budget B to modify target agent A. Improvement-process improvement requires, on a held-out distribution of target agents, E_AΓ_h,B(K_t+1;A) > E_AΓ_h,B(K_t;A). Finally, if V_t is the evaluator active at generation t, epi…
Exact mathematical passage
\Gamma_{h,B}(K;A)
\perpIntroduction passage · 21 passages
Search aliases: ⊥ perp
Introduction passage · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7325
[Compilation noninterference] Let K_1= M(D,V,K_0,Γ), where D and V denote the development and meta-validation receipts. Suppose M is deterministic and receives no value measurable only with respect to final audit outcomes A. Then, conditional on (D,V,K_0,Γ), K_1⊥ ⊥ A. Equivalently, changing the realized final audit outcomes while holding compiler inputs fixed leaves K_1 unchanged.
Exact mathematical passage
K_1\perp\!\!\!\perp A.
\mathbf{0.050745}Use in source · 1 passage
Search aliases: 0.050745 mathbf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17481
Full child · novel charts · 0.050745 · 0.194936 · 0.779341 · 0.666244 · 0.594444
Exact mathematical passage
\mathbf{0.050745}
\mathbf{0.057546}Use in source · 1 passage
Search aliases: 0.057546 mathbf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17480
Full child · inherited · 0.057546 · 0.174135 · 0.770701 · 0.630918 · 0.607778
Exact mathematical passage
\mathbf{0.057546}
\mathbf{0.096}Use in source · 1 passage
Search aliases: 0.096 mathbf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11056
Conditioning grandchild · 0.209580 · 0.638021 · 1.000 · 14.268 · 0.096
Exact mathematical passage
\mathbf{0.096}
\mathbf{0.174135}Use in source · 1 passage
Search aliases: 0.174135 mathbf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17480
Full child · inherited · 0.057546 · 0.174135 · 0.770701 · 0.630918 · 0.607778
Exact mathematical passage
\mathbf{0.174135}
\mathbf{0.194936}Use in source · 1 passage
Search aliases: 0.194936 mathbf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17481
Full child · novel charts · 0.050745 · 0.194936 · 0.779341 · 0.666244 · 0.594444
Exact mathematical passage
\mathbf{0.194936}
\mathbf{0.209580}Use in source · 1 passage
Search aliases: 0.209580 mathbf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11056
Conditioning grandchild · 0.209580 · 0.638021 · 1.000 · 14.268 · 0.096
Exact mathematical passage
\mathbf{0.209580}
\mathbf{0.333839}Use in source · 1 passage
Search aliases: 0.333839 mathbf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15456
Primary language child · 0.859731 · 0.792894 · 0.333839
Exact mathematical passage
\mathbf{0.333839}
\mathbf{0.334224}Use in source · 1 passage
Search aliases: 0.334224 mathbf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11891
Chart child K_3,C^96 · 0.334224 · 0.284265 · 0.572917 · 0.572917 · 0
Exact mathematical passage
\mathbf{0.334224}
\mathbf{0.344720}Use in source · 1 passage
Search aliases: 0.344720 mathbf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8692
Descendant packet-QR 16 · 0.344720 · 1.000000 · 1.00000 · 24
Exact mathematical passage
\mathbf{0.344720}
\mathbf{0.482289}Use in source · 1 passage
Search aliases: 0.482289 mathbf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15458
Generator-foreign child · 0.814520 · 0.785584 · 0.482289
Exact mathematical passage
\mathbf{0.482289}
\mathbf{0.594444}Use in source · 1 passage
Search aliases: 0.594444 mathbf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17481
Full child · novel charts · 0.050745 · 0.194936 · 0.779341 · 0.666244 · 0.594444
Exact mathematical passage
\mathbf{0.594444}
\mathbf{0.607778}Use in source · 1 passage
Search aliases: 0.607778 mathbf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17480
Full child · inherited · 0.057546 · 0.174135 · 0.770701 · 0.630918 · 0.607778
Exact mathematical passage
\mathbf{0.607778}
\mathbf{0.630918}Use in source · 1 passage
Search aliases: 0.630918 mathbf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17480
Full child · inherited · 0.057546 · 0.174135 · 0.770701 · 0.630918 · 0.607778
Exact mathematical passage
\mathbf{0.630918}
\mathbf{0.666244}Use in source · 1 passage
Search aliases: 0.666244 mathbf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17481
Full child · novel charts · 0.050745 · 0.194936 · 0.779341 · 0.666244 · 0.594444
Exact mathematical passage
\mathbf{0.666244}
\mathbf{0.770701}Use in source · 1 passage
Search aliases: 0.770701 mathbf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17480
Full child · inherited · 0.057546 · 0.174135 · 0.770701 · 0.630918 · 0.607778
Exact mathematical passage
\mathbf{0.770701}
\mathbf{0.779341}Use in source · 1 passage
Search aliases: 0.779341 mathbf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17481
Full child · novel charts · 0.050745 · 0.194936 · 0.779341 · 0.666244 · 0.594444
Exact mathematical passage
\mathbf{0.779341}
\mathbf{0.785584}Use in source · 1 passage
Search aliases: 0.785584 mathbf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15458
Generator-foreign child · 0.814520 · 0.785584 · 0.482289
Exact mathematical passage
\mathbf{0.785584}
\mathbf{0.792894}Use in source · 1 passage
Search aliases: 0.792894 mathbf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15456
Primary language child · 0.859731 · 0.792894 · 0.333839
Exact mathematical passage
\mathbf{0.792894}
\mathbf{0.814520}Use in source · 1 passage
Search aliases: 0.814520 mathbf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15458
Generator-foreign child · 0.814520 · 0.785584 · 0.482289
Exact mathematical passage
\mathbf{0.814520}
\mathbf{0.859731}Use in source · 2 passages
Search aliases: 0.859731 mathbf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15456
Primary language child · 0.859731 · 0.792894 · 0.333839
Exact mathematical passage
\mathbf{0.859731}
\mathbf{1.00000}Use in source · 1 passage
Search aliases: 1.00000 mathbf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8692
Descendant packet-QR 16 · 0.344720 · 1.000000 · 1.00000 · 24
Exact mathematical passage
\mathbf{1.00000}
\mathbf{1.000000}Use in source · 1 passage
Search aliases: 1.000000 mathbf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8692
Descendant packet-QR 16 · 0.344720 · 1.000000 · 1.00000 · 24
Exact mathematical passage
\mathbf{1.000000}
\mathbf{14.268}Use in source · 1 passage
Search aliases: 14.268 mathbf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11056
Conditioning grandchild · 0.209580 · 0.638021 · 1.000 · 14.268 · 0.096
Exact mathematical passage
\mathbf{14.268}
\mathfrak A'Use in source · 4 passages
Search aliases: A' mathfrak
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10037
Define the unital adjoint-closed algebra A = alg^∗ (I,R_1,…,R_m). Its commutant is A' = {X∈End(H):XA=AX for all A∈ A}. The center is Z( A)= A∩ A'.
Exact mathematical passage
\mathfrak A' = \{X\in\End(H):XA=AX\text{ for all }A\in\mathfrak A\}.
A'_{m'}Use in source · 1 passage
Search aliases: A'_m'
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16172
Let d_prog(τ,τ') be a normalized distance between (A_m,B_m) and (A'_m',B'_m'), and let d_ans(τ,τ') = absy_m/4-y'_m'/4. For a recipient task τ_i, define four donor classes: item SAP: same-answer, far-program, approximately depth-matched donors; item FAD: far-answer, depth-matched donors; item SN: semantic-neighbor donors; item RMD: random depth-matched donors. Three donors per available class are selected by deterministic task-identifier hashing. Each audit records donor coverage and the realized…
Exact mathematical passage
(A'_{m'},B'_{m'})
a^*Use in source · 6 passages
Search aliases: a^*
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12366
[Finite repair-relevant block selection] Suppose there is a unique repair-relevant block a^* and a margin m>0 such that s_a^*≥ s_a+m for every a≠ a^*. If max_a|widehat s_a-s_a|<fracm2, then arg max_awidehat s_a=a^*.
Exact mathematical passage
a^*
a^2Use in source · 4 passages
Search aliases: a^2
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20965
0 · 1 . Then (H-I)^*(H-I) = 0 · 0 0 · a^2 . The holonomy-energy eigenvalues are 0 and a^2. Thus E^flat=spansete_1. For a threshold 0≤τ<a^2, the approximate flat core is still one-dimensional. For τ≥ a^2, both coordinates are admitted, but the second coordinate carries loop distortion at scale |a| and should not be described as invariant.
Exact mathematical passage
(H-I)^*(H-I) = \begin{bmatrix} 0&0\\ 0&a^2 \end{bmatrix}.
A^{\rm base}_{\rm cal,prot}Use in source · 1 passage
Search aliases: A^rm base_rm cal,prot rm rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7418
widehat R_rm UCB(z) · = max{0,widehat R(z)}+κ_r s_r,c(z). A candidate is admissible when widehat R_rm UCB(z)≤τ_r, widehat r_rm cal(z)≤τ_rm cal, A^rm base_rm cal,prot(z)≥ a_min. The child chooses the admissible candidate with maximum widehat U_rm LCB. If the admissible set is empty, it returns the no-op.
Exact mathematical passage
\widehat R_{\rm UCB}(z)\leq\tau_r, \qquad \widehat r_{\rm cal}(z)\leq\tau_{\rm cal}, \qquad A^{\rm base}_{\rm cal,prot}(z)\geq a_{\min}. \label{eq:block07-admissible}
A^{\rm prot}_eUse in source · 3 passages
Search aliases: A^rm prot_e rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7665
For target episode e and candidate c, let A^rm slice_e(c) · =accuracy on the affected stratum, A^rm prot_e(c) · =accuracy on the unaffected protected stratum, q(c) · =repair rank. The candidate utility is U_e(c) = A^rm slice_e(c) -2[A^rm prot_e(0)-A^rm prot_e(c)]_+ -0.0005q(c), and the reported protected utility gain is G_e(c)=U_e(c)-A^rm slice_e(0). The protected regression is R_e(c)=[A^rm prot_e(0)-A^rm prot_e(c)]_+. The utility therefore rewards repair of the affected stratum, penalizes damag…
Exact mathematical passage
A^{\rm slice}_e(c) &=\text{accuracy on the affected stratum},\\ A^{\rm prot}_e(c) &=\text{accuracy on the unaffected protected stratum},\\ q(c) &=\text{repair rank}.
A^{\rm slice}_eUse in source · 3 passages
Search aliases: A^rm slice_e rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7665
For target episode e and candidate c, let A^rm slice_e(c) · =accuracy on the affected stratum, A^rm prot_e(c) · =accuracy on the unaffected protected stratum, q(c) · =repair rank. The candidate utility is U_e(c) = A^rm slice_e(c) -2[A^rm prot_e(0)-A^rm prot_e(c)]_+ -0.0005q(c), and the reported protected utility gain is G_e(c)=U_e(c)-A^rm slice_e(0). The protected regression is R_e(c)=[A^rm prot_e(0)-A^rm prot_e(c)]_+. The utility therefore rewards repair of the affected stratum, penalizes damag…
Exact mathematical passage
A^{\rm slice}_e(c) &=\text{accuracy on the affected stratum},\\ A^{\rm prot}_e(c) &=\text{accuracy on the unaffected protected stratum},\\ q(c) &=\text{repair rank}.
A^\astUse in source · 1 passage
Search aliases: A^∗ ast
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10107
The identity belongs to A, so r∈ H_r. For B∈ A and Ar∈ H_r, one has BAr∈ H_r, proving invariance. Every invariant subspace containing r contains Ar for every A, hence contains H_r. Adjoint closure implies invariance under every A^∗, so the orthogonal complement is invariant as well.
Exact mathematical passage
A^\ast
A_0Use in source · 6 passages
Search aliases: A_0
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 615
After n strict transitions, V(A_n)≥ V(A_0)+nΔ≥ nΔ. Since V(A_n)≤1, one has n≤1/Δ.
Exact mathematical passage
V(A_n)\geq V(A_0)+n\Delta\geq n\Delta.
\mathcal A_0Use in source · 2 passages
Search aliases: A_0 mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7770
[Risk-filter tradeoff] Let A_r⊆ A_0 be the candidate set retained with and without a risk filter. Then max_c∈ A_0widehat U(c) ≥ max_c∈ A_rwidehat U(c), but no analogous inequality favors the unfiltered selector for true protected risk.
Exact mathematical passage
\mathcal A_r\subseteq\mathcal A_0
A_1Use in source · 5 passages
Search aliases: A_1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2935
[Pathwise transport bound] Consider a path of systems A_0xrightarrow T_0A_1 xrightarrow T_1⋯ xrightarrow T_n-1A_n. Write T_U,k:U_k→ U_k+1 and T_V,k:V_k→ V_k+1. Let normD_k+1T_U,k-T_V,kD_k≤ε_D,k and normr_k+1-T_V,kr_k≤ε_r,k. If D_0u_0=-r_0 and u_n=T_U,n-1⋯ T_U,0u_0, then normD_nu_n+r_n · ≤ ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_D,k normT_U,k-1⋯ T_U,0u_0
Exact mathematical passage
A_0\xrightarrow{\mathfrak T_0}A_1 \xrightarrow{\mathfrak T_1}\cdots \xrightarrow{\mathfrak T_{n-1}}A_n.
a_1Use in source · 1 passage
Search aliases: a_1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 14479
This theorem gives a precise reason to value experiments rather than static classification. Suppose an experiment a returns an observation Y_a whose distribution differs between the aliased hypotheses. The compiler can then choose a sequence a_1,a_2,…,a_t that increases the information available for diagnosis.
Exact mathematical passage
a_1,a_2,\ldots,a_t
a_2Use in source · 1 passage
Search aliases: a_2
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 14479
This theorem gives a precise reason to value experiments rather than static classification. Suppose an experiment a returns an observation Y_a whose distribution differs between the aliased hypotheses. The compiler can then choose a sequence a_1,a_2,…,a_t that increases the information available for diagnosis.
Exact mathematical passage
a_1,a_2,\ldots,a_t
A_aUse in source · 8 passages
Search aliases: A_a
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6616
[Exact causal-packet recovery for a linear encoder] Suppose E_a = A_a · B_a, A_a,B_a:ℝ^8→ℝ^d_a, and suppose the centered conditional covariance of ψ(x) given s(x)=1 is positive definite. Then span{Δ_a(x):s(x)=1} = im B_a. Consequently, a rank-r singular-subspace estimator with r=rank B_a recovers the exact causal packet from a sufficiently rich noiseless sample.
Exact mathematical passage
E_a = \begin{bmatrix}A_a&B_a\end{bmatrix}, \qquad A_a,B_a:\R^8\to\R^{d_a},
A_{a_t}Use in source · 2 passages
Search aliases: A_a_t
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9286
[Meta-regret Lyapunov contraction] Suppose there exist w∈ℝ_>0^7 and q∈[0,1) such that for every reachable r_t, the controller can choose an authorized action a_t satisfying w^top A_a_tr_t ≤ q w^top r_t. Then every controlled trajectory satisfying [source reference] obeys w^top r_t ≤ q^t w^top r_0 + ∑_k=0^t-1 q^t-1-kw^top e_k.
Exact mathematical passage
w^\top A_{a_t}r_t \leq q\,w^\top r_t. \label{eq:b8-lyapunov-condition}
A_CUse in source · 2 passages
Search aliases: A_C
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11624
Let K_G be the global parent and K_C the chart process. Let A_C(x) be the frozen chart-certificate decision for target episode x. The descendant is K_3(x)= K_C(x), · A_C(x)=1,
Exact mathematical passage
A_C(x)
A_cUse in source · 1 passage
Search aliases: A_c
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17814
A latent-step-shared map acts independently on each recurrent state: widehat h_a,c^(k) = A_c h_b,c^(k)+b_c, k=1,…,5. A full-trajectory map instead acts on the concatenation vec(h_b,c^(1),…,h_b,c^(5)) ∈ R^200, allowing cross-step mixing: vec(widehat h_a,c) = M_cvec(h_b,c)+q_c. This difference is substantive when two recurrent architectures distribute the same computation across latent time differently.
Exact mathematical passage
\widehat h_{a,c}^{(k)} = A_c h_{b,c}^{(k)}+b_c, \qquad k=1,\ldots,5. \label{eq:b20-step-shared}
A_{c(h)}Use in source · 1 passage
Search aliases: A_c(h)
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11734
Every primary target is four dimensional and has two invertible affine charts: y=A_c(h)h+b+ε. The two maps share the protected-context block but differ in the mutable response block. The source repair therefore preserves the chart while inducing two different target-space repair conjugates.
Exact mathematical passage
y=A_{c(h)}h+b+\varepsilon.
A_iUse in source · 2 passages
Search aliases: A_i
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4699
Let K_0 and K_1 be frozen improvement mechanisms evaluated on the same independent foreign agents A_1,…,A_n. Define G_h,B(K,A) = max_A'∈Desc_h,B(K,A)J(A')-J(A) and assume -G_max ≤ G_h,B(K,A) ≤ G_max. For matched budgets and random seeds, let D_i = G_h,B(K_1,A_i)-G_h,B(K_0,A_i), overline D_n = frac1n∑_i=1^nD_i.
Exact mathematical passage
D_i = G_{h,B}(K_1,A_i)-G_{h,B}(K_0,A_i), \qquad \overline D_n = \frac1n\sum_{i=1}^{n}D_i.
A_jUse in source · 7 passages
Search aliases: A_j
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11421
[Open-set incompatibility of distinct affine charts] Let U_0,U_1⊆ℝ^d each contain a nonempty open set. Let f_j(h)=A_jh+b_j, j∈{0,1}. If one affine map f(h)=Ah+b satisfies f(h)=f_j(h) for every h∈ U_j and both j, then A=A_0=A_1, b=b_0=b_1. Consequently, if (A_0,b_0)≠(A_1,b_1), no element of G_glob represents the two-chart relation exactly.
Exact mathematical passage
f_j(h)=A_jh+b_j, \qquad j\in\{0,1\}.
A_{j-1}Use in source · 1 passage
Search aliases: A_j-1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16022
ion be f_j(z)=a_jz+b_j, a_j∈{-1,-tfrac12,tfrac12,1}, b_j∈{-tfrac12,0,tfrac12}. A depth-m program acts on x∈{-1,-tfrac34,…,1} by F_m(x) = (f_m∘⋯∘ f_1)(x) =A_mx+B_m, where A_0=1, B_0=0, A_j=a_jA_j-1, B_j=a_jB_j-1+b_j. All quantities in [source reference] are evaluated with exact rational arithmetic before a floating representation is exposed to the model.
Exact mathematical passage
A_0=1, \qquad B_0=0, \qquad A_j=a_jA_{j-1}, \qquad B_j=a_jB_{j-1}+b_j. \label{eq:b17-affine-witness-recursion}
A_j^{-1}Use in source · 2 passages
Search aliases: A_j^-1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11481
Let the source repair be R_s(h)=M_sh+c_s. On chart j, let the target bridge be T_j(h)=A_jh+b_j, with A_j invertible on the repair-relevant subspace. Define R_t,j(y) = T_j (R_s(T_j^-1(y))). Expanding gives R_t,j(y) = A_jM_sA_j^-1(y-b_j)+A_jc_s+b_j.
Exact mathematical passage
R_{t,j}(y) = A_jM_sA_j^{-1}(y-b_j)+A_jc_s+b_j. \label{eq:b10-chart-repair}
a_{\min}Use in source · 2 passages
Search aliases: a_min min
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7418
widehat R_rm UCB(z) · = max{0,widehat R(z)}+κ_r s_r,c(z). A candidate is admissible when widehat R_rm UCB(z)≤τ_r, widehat r_rm cal(z)≤τ_rm cal, A^rm base_rm cal,prot(z)≥ a_min. The child chooses the admissible candidate with maximum widehat U_rm LCB. If the admissible set is empty, it returns the no-op.
Exact mathematical passage
\widehat R_{\rm UCB}(z)\leq\tau_r, \qquad \widehat r_{\rm cal}(z)\leq\tau_{\rm cal}, \qquad A^{\rm base}_{\rm cal,prot}(z)\geq a_{\min}. \label{eq:block07-admissible}
A_nUse in source · 4 passages
Search aliases: A_n
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 615
After n strict transitions, V(A_n)≥ V(A_0)+nΔ≥ nΔ. Since V(A_n)≤1, one has n≤1/Δ.
Exact mathematical passage
V(A_n)\geq V(A_0)+n\Delta\geq n\Delta.
\mathcal A_rUse in source · 2 passages
Search aliases: A_r mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7770
[Risk-filter tradeoff] Let A_r⊆ A_0 be the candidate set retained with and without a risk filter. Then max_c∈ A_0widehat U(c) ≥ max_c∈ A_rwidehat U(c), but no analogous inequality favors the unfiltered selector for true protected risk.
Exact mathematical passage
\mathcal A_r\subseteq\mathcal A_0
\mathfrak A_RUse in source · 12 passages
Search aliases: A_R mathfrak
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12177
The algebra generated by the response family is A_R = alg^*(I,R_1,…,R_k). Because the R_j are simultaneously block scalar in the exact model, A_R is commutative and every polynomial in the responses is again block scalar.
Exact mathematical passage
\mathfrak A_R = \operatorname{alg}^{*}(I,R_1,\ldots,R_k). \label{eq:b11-response-algebra}
\mathfrak A_R'Use in source · 5 passages
Search aliases: A_R' mathfrak
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12226
The response commutant is A_R' = setX∈End(H):XR_j=R_jX for every j.
Exact mathematical passage
\mathfrak A_R' = \set{X\in\End(H):XR_j=R_jX\text{ for every }j}.
a_{\mathrm{risky}}Use in source · 1 passage
Search aliases: a_risky mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21064
Let S={s_0,s_1,b}, C_RI={s_0,s_1}. At s_0 there are two authorized actions: a_safe:s_0↦ s_1, a_risky:s_0↦ b. At s_1 there is one action with successor s_1. Universal closure excludes s_0 because one authorized successor is unsafe. Controlled closure includes s_0 because the policy can choose a_safe. This is the concrete distinction between safety of every authorized action and existence of one robustly safe action.
Exact mathematical passage
a_{\mathrm{safe}}:s_0\mapsto s_1, \qquad a_{\mathrm{risky}}:s_0\mapsto b.
A_{\rm aff}Use in source · 2 passages
Search aliases: A_rm aff rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8318
The protected utility used in Block 7 is the balanced accuracy functional J_Ξ(A) = frac12 A_rm aff(A)+frac12 A_rm non(A), where A_rm aff is accuracy on the defect-affected stratum and A_rm non is accuracy on the protected unaffected stratum. This prevents the more common unaffected examples from overwhelming the repair objective.
Exact mathematical passage
J_\Xi(A) = \frac12 A_{\rm aff}(A)+\frac12 A_{\rm non}(A), \label{eq:block07q-balanced-utility}
A_{\rm non}Use in source · 2 passages
Search aliases: A_rm non rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8318
The protected utility used in Block 7 is the balanced accuracy functional J_Ξ(A) = frac12 A_rm aff(A)+frac12 A_rm non(A), where A_rm aff is accuracy on the defect-affected stratum and A_rm non is accuracy on the protected unaffected stratum. This prevents the more common unaffected examples from overwhelming the repair objective.
Exact mathematical passage
J_\Xi(A) = \frac12 A_{\rm aff}(A)+\frac12 A_{\rm non}(A), \label{eq:block07q-balanced-utility}
a_{\mathrm{safe}}Use in source · 2 passages
Search aliases: a_safe mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21064
Let S={s_0,s_1,b}, C_RI={s_0,s_1}. At s_0 there are two authorized actions: a_safe:s_0↦ s_1, a_risky:s_0↦ b. At s_1 there is one action with successor s_1. Universal closure excludes s_0 because one authorized successor is unsafe. Controlled closure includes s_0 because the policy can choose a_safe. This is the concrete distinction between safety of every authorized action and existence of one robustly safe action.
Exact mathematical passage
a_{\mathrm{safe}}:s_0\mapsto s_1, \qquad a_{\mathrm{risky}}:s_0\mapsto b.
a_tUse in source · 6 passages
Search aliases: a_t
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5905
Let the latent world state be x_t∈ℝ^2, action a_t∈ℝ^2, observation y_t∈ℝ^3, and memory state m_t∈ℝ^2. The baseline dynamics are x_t+1=Ax_t+Ba_t, y_t=Cx_t+b_s, m_t+1=λ m_t+(1-λ)y_t,1:2, and the reported evaluator score is q_t=w^top y_t. The redundant observation map is C= 1 · 0
Exact mathematical passage
a_t\in\R^2
A_{t+2}Use in source · 1 passage
Search aliases: A_t+2
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 318
The phrase recursive self-improvement is often represented by the schematic loop A_t⟶ A_t+1⟶ A_t+2⟶⋯, with the informal assertion that each descendant is ``better'' than its parent. This notation suppresses the central difficulties. It does not identify the object that persists across generations, the evidence by which improvement is judged, the interventions available to the system, the components that may be modified, the resource budget under which comparisons are made, or the mechanism that …
Exact mathematical passage
A_t\longrightarrow A_{t+1}\longrightarrow A_{t+2}\longrightarrow\cdots,
\mathfrak A_xUse in source · 1 passage
Search aliases: A_x mathfrak
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10180
For nonlinear dynamics on a state manifold M, the response algebra becomes base-point dependent: x⟼ A_x⊆End(T_xM). Local central projectors define a bundle of tangent strata when their ranks remain locally constant. Transport around a loop can permute or distort the local modules, producing atlas holonomy. A complete nonlinear theory must combine: item local intervention algebras; item spectral-gap control of approximate central projectors; item transition maps between neighboring tangent atlase…
Exact mathematical passage
x\longmapsto\mathfrak A_x\subseteq\End(T_xM).
\mathrm{abs}Use in source · 3 passages
Search aliases: abs mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16619
Let the template set be Θ = {θ_aff,θ_abs, θ_quad,θ_hinge}. For integer parameters a,b,c and integer input x, the exact programs are f_aff(x) · =ax+b,
Exact mathematical passage
\Theta = \{\theta_{\mathrm{aff}},\theta_{\mathrm{abs}}, \theta_{\mathrm{quad}},\theta_{\mathrm{hinge}}\}. \label{eq:b18-template-set}
\mathrm{abstract}Use in source · 2 passages
Search aliases: abstract mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6212
Let N⊆{1,…,n} be foreign agents whose dominant defect belongs to the held-out sixth type. Define widehatτ_abstract = frac1|N|∑_j∈ N ( G_j^meta-G_j^no-meta ). In the reference positive control, widehatτ_abstract=0.700533. A learned recursive-improvement claim would require the schema to be generated by the parent improvement process, sealed before foreign evaluation, and compared under matched computation and evidence access.
Exact mathematical passage
\widehat\tau_{\mathrm{abstract}} = \frac1{|N|}\sum_{j\in N} \left( G_{j}^{\mathrm{meta}}-G_{j}^{\mathrm{no\mbox{-}meta}} \right).
\operatorname{Acc}Use in source · 1 passage
Search aliases: Acc operatorname
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 19952
full identity, equal-rank random }. For each nontrivial constructor, ranks one through six are considered. The full identity is an oversized control. A projection W is valid only when, on both recurrent systems, R^2_future · ≥0.82, R^2_intervention · ≥0.90, Acc_answer · ≥0.78, D_twin · ≤0.22. Among valid candidates, the compiler minimizes rank and then follows the frozen method priority. No hidden semantic score enters the choice.
Exact mathematical passage
\begin{aligned} R^2_{\mathrm{future}}&\geq0.82,\\ R^2_{\mathrm{intervention}}&\geq0.90,\\ \operatorname{Acc}_{\mathrm{answer}}&\geq0.78,\\ D_{\mathrm{twin}}&\leq0.22. \end{aligned} \label{eq:b23-visible-gates}
\operatorname{Acc}_{\mathrm{answer}}Use in source · 1 passage
Search aliases: Acc_answer operatorname mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 19952
full identity, equal-rank random }. For each nontrivial constructor, ranks one through six are considered. The full identity is an oversized control. A projection W is valid only when, on both recurrent systems, R^2_future · ≥0.82, R^2_intervention · ≥0.90, Acc_answer · ≥0.78, D_twin · ≤0.22. Among valid candidates, the compiler minimizes rank and then follows the frozen method priority. No hidden semantic score enters the choice.
Exact mathematical passage
\begin{aligned} R^2_{\mathrm{future}}&\geq0.82,\\ R^2_{\mathrm{intervention}}&\geq0.90,\\ \operatorname{Acc}_{\mathrm{answer}}&\geq0.78,\\ D_{\mathrm{twin}}&\leq0.22. \end{aligned} \label{eq:b23-visible-gates}
\mathrm{accuracy}Use in source · 1 passage
Search aliases: accuracy mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21025
Suppose a sandbox returns only the scalar task score q(ω)=accuracy(ω). Execution ω_0 reaches accuracy 0.9 using authorized retrieval. Execution ω_1 reaches the same accuracy after importing a prohibited hidden answer key. Then q(ω_0)=q(ω_1)=0.9, but the ideal full-information decisions differ. By [source reference], no witness using only accuracy can accept the first and reject the second. Adding signed tool-call provenance or a complete effect receipt refines the trace and can separate the exec…
Exact mathematical passage
q(\omega)=\mathrm{accuracy}(\omega).
\mathrm{actuation}Use in source · 1 passage
Search aliases: actuation mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5953
From a common probe state x_0 and bounded action a, define the feature vector ϕ(z) = ϕ_passive ϕ_actuation ϕ_sensor ϕ_memory ϕ_evaluator . Its components are, respectively: item deviation from the baseline no-action continuation; item deviation of the action-conditioned response from the baseline action response; item violation of redundant-sensor consistency; item deviation from baseline memory update; item discrepancy between reported and protected evaluator scores. The observed probe is widet…
Exact mathematical passage
\phi(z) = \begin{bmatrix} \phi_{\mathrm{passive}}\\ \phi_{\mathrm{actuation}}\\ \phi_{\mathrm{sensor}}\\ \phi_{\mathrm{memory}}\\ \phi_{\mathrm{evaluator}} \end{bmatrix}.
\operatorname{Aff}Use in source · 2 passages
Search aliases: Aff operatorname
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11508
The inherited global target-repair response lies in the diagonal class D_diag = set(R,R):R∈Aff(H_t), whereas the chart grammar reaches D_chart = Aff(H_t)×Aff(H_t).
Exact mathematical passage
\Dcal_{\mathrm{diag}} = \set{(R,R):R\in\operatorname{Aff}(H_t)},
\mathrm{affine\ oracle}Use in source · 1 passage
Search aliases: affine oracle mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11685
The main paired estimands are Δ_G→ C = E [ G(K_3,C^96,A)-G(K_2,G^96,A) ] and Δ_oracle→ C = E [ G(K_3,C^96,A)-G(K_affine oracle,A) ]. The second quantity is the sharper structural test.
Exact mathematical passage
\Delta_{\mathrm{oracle}\to C} = \mathbb E \left[ G(K_{3,C}^{96},A)-G(K_{\mathrm{affine\ oracle}},A) \right].
\operatorname{alg}^{*}Use in source · 1 passage
Search aliases: alg^* operatorname
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12177
The algebra generated by the response family is A_R = alg^*(I,R_1,…,R_k). Because the R_j are simultaneously block scalar in the exact model, A_R is commutative and every polynomial in the responses is again block scalar.
Exact mathematical passage
\mathfrak A_R = \operatorname{alg}^{*}(I,R_1,\ldots,R_k). \label{eq:b11-response-algebra}
\operatorname{alg}^{\ast}Use in source · 2 passages
Search aliases: alg^∗ operatorname ast
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10029
Define the unital adjoint-closed algebra A = alg^∗ (I,R_1,…,R_m). Its commutant is A' = {X∈End(H):XA=AX for all A∈ A}. The center is Z( A)= A∩ A'.
Exact mathematical passage
\mathfrak A = \operatorname{alg}^{\ast} \left(I,R_1,\ldots,R_m\right). \label{eq:b8-response-algebra}
\mathrm{align}Use in source · 4 passages
Search aliases: align mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16840
Let (Y,H_s,H_t)∼μ be the protected semantics, source trace, and target trace under context distribution μ. Fix a source probe P and bridge B. Define e_src=Y-PH_s, e_align=P(H_s-BH_t). Then Y-PBH_t=e_src+e_align.
Exact mathematical passage
e_{\mathrm{src}}=Y-PH_s, \qquad e_{\mathrm{align}}=P(H_s-BH_t). \label{eq:b18-error-parts}
\mathrm{ans}Use in source · 8 passages
Search aliases: ans mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16173
Let d_prog(τ,τ') be a normalized distance between (A_m,B_m) and (A'_m',B'_m'), and let d_ans(τ,τ') = absy_m/4-y'_m'/4. For a recipient task τ_i, define four donor classes: item SAP: same-answer, far-program, approximately depth-matched donors; item FAD: far-answer, depth-matched donors; item SN: semantic-neighbor donors; item RMD: random depth-matched donors. Three donors per available class are selected by deterministic task-identifier hashing. Each audit records donor coverage and the realized…
Exact mathematical passage
d_{\mathrm{ans}}(\tau,\tau') = \abs{y_m/4-y'_{m'}/4}. \label{eq:b17-answer-distance}
\mathrm{answer}Use in source · 2 passages
Search aliases: answer mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16789
s L_i, the audit reuses the Block 17 donor distinctions and adds cross-implementation status labels. It records latent necessity, reversal cost, norm-matched noise cost, exact replay, same-answer far-program transplant cost, far-answer matched-depth transplant cost, local semantic-neighbor cost, external trace-probe R^2, dynamic-depth correlation, and exact program execution. Let n_i · =L_i^0-L_i, · o_i · =L_i^rev-L_i, · c_i · =L_i^η-L_i, p_i · =L_i,sem^same answer,far program -L_i,sem, · a_i · …
Exact mathematical passage
n_i&=L_i^{0}-L_i, &o_i&=L_i^{\mathrm{rev}}-L_i, &c_i&=L_i^{\eta}-L_i, \\ p_i&=L_{i,\mathrm{sem}}^{\mathrm{same\ answer,far\ program}} -L_{i,\mathrm{sem}}, &a_i&=L_{i,\mathrm{answer}}^{\mathrm{far\ answer,depth}} -L_{i,\mathrm{answer}}. \label{eq:b18-audit-coordinates}
\mathrm{arch}Use in source · 1 passage
Search aliases: arch mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 14149
The architecture-foreign paired effect is widehatΔ_Γ^arch = 0.090800, CI_95% = [0.052311,0.135081]. The sign-flip estimate is again 9.9999×10^-6. The child improves 34.722% of pairs, ties on 65.278%, and no architecture-foreign pair favors the parent.
Exact mathematical passage
\widehat\Delta_{\Gamma}^{\mathrm{arch}} = 0.090800, \qquad \mathrm{CI}_{95\%} = [0.052311,0.135081]. \label{eq:b13-architecture-effect}
\argUse in source · 1 passage
Search aliases: arg
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 18415
Each repair g receives a description-length proxy L(g) recording parameter count and constructor surcharge. The compiler selects widehat g ∈ argmin_g∈ G_21 L(g) subject to the appropriate protected gates. If no repair passes and the collision certificate is valid, it returns G_bot; otherwise it returns unresolved.
Exact mathematical passage
\widehat g \in \arg\min_{g\in\mathcal G_{21}} L(g) \quad\text{subject to the appropriate protected gates.} \label{eq:block21-mdl}
\argmaxUse in source · 8 passages
Search aliases: argmax arg max operatorname
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 983
pection] Let u:Y→[0,1] be a bounded utility. Define V(δ)= E[u(Y)mid A_t,δ], widehat V(δ)= E[u(Y)mid ϕ_t(A_t),δ]. If ϕ_t is ε-intervention-sufficient, then sup_δ∈D_t absV(δ)-widehat V(δ)≤ε. If δ^*∈arg max_δ∈D_tV(δ), widehatδ∈arg max_δ∈D_twidehat V(δ), then V(δ^*)-V(widehatδ)≤2ε.
Exact mathematical passage
\delta^*\in\argmax_{\delta\in\Dcal_t}V(\delta), \qquad \widehat\delta\in\argmax_{\delta\in\Dcal_t}\widehat V(\delta),
\argmax_aUse in source · 1 passage
Search aliases: argmax_a arg max_a argmax operatorname
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12375
[Finite repair-relevant block selection] Suppose there is a unique repair-relevant block a^* and a margin m>0 such that s_a^*≥ s_a+m for every a≠ a^*. If max_a|widehat s_a-s_a|<fracm2, then arg max_awidehat s_a=a^*.
Exact mathematical passage
\argmax_a\widehat s_a=a^*.
\argmax_iUse in source · 2 passages
Search aliases: argmax_i arg max_i argmax operatorname
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13905
[Uniform forecast-regret bound] Let g_i be the true protected gain of each admissible candidate and suppose |widehat g_i-g_i|≤ε for all candidates. If widehat i∈arg max_iwidehat g_i and i^⋆∈arg max_i g_i, then g_i^⋆-g_widehat i ≤ 2ε.
Exact mathematical passage
\widehat i\in\argmax_i\widehat g_i
\argmax_{i:\operatorname{Valid}(c_i,\Pi_i^+)=1}Use in source · 1 passage
Search aliases: argmax_i:Valid(c_i,Π_i^+)=1 arg max_i:Valid(c_i,Π_i^+)=1 argmax operatorname Pi operatorname operatorname Pi
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13712
Let Valid(c_i,Π_i^+)∈{0,1} encode capability, integrity, provenance, and resource validity. The selected descendant is K_Γ^⋆(τ) ∈ arg max_i:Valid(c_i,Π_i^+)=1 S_V(c_i), with the no-op baseline retained when no valid candidate exists.
Exact mathematical passage
K_{\Gamma}^{\star}(\tau) \in \argmax_{i:\operatorname{Valid}(c_i,\Pi_i^+)=1} S_{\Vcal}(c_i), \label{eq:b13-selected-method}
\argmax_{i\notin I_{j-1}}Use in source · 1 passage
Search aliases: argmax_i∉ I_j-1 arg max_i∉ I_j-1 argmax notin operatorname notin
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8519
The descendant policy uses the source intervention-defined packet to divide the unlabeled source-side candidate pool into high- and low-packet-activation strata. It selects eight anchors from each stratum. Within each stratum it applies a deterministic pivoted residual rule: i_j ∈ arg max_i∉ I_j-1 norm(I-Q_j-1Q_j-1^top)z_i^2, where z_i is the standardized source activation and Q_j-1 spans the previously selected residual directions.
Exact mathematical passage
i_j \in \argmax_{i\notin I_{j-1}} \norm{(I-Q_{j-1}Q_{j-1}^\top)z_i}^2,
\argmax_{v\in\Vcal_{\mathrm{cand}}}Use in source · 1 passage
Search aliases: argmax_v∈Vcal_cand arg max_v∈V_cand argmax in Vcal mathrm operatorname in mathcal mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9772
Equation [source reference] gives a direct acquisition score. For a candidate family V_cand, choose v^⋆ ∈ arg max_v∈V_cand fracabsanglesz,P_I^⊥v^2 normP_I^⊥v^2+λ c(v)+βRisk(v), where c(v) is implementation cost and Risk(v) is a protected-risk estimate.
Exact mathematical passage
v^\star \in \argmax_{v\in\Vcal_{\mathrm{cand}}} \frac{\abs{\angles{z,P_{I^\perp}v}}^2} {\norm{P_{I^\perp}v}^2+\lambda c(v)+\beta\Risk(v)}, \label{eq:b8-generator-score}
\argmax_{\gamma\in\Gamma_{\rm safe}}Use in source · 1 passage
Search aliases: argmax_γ∈Γ_rm safe arg max_γ∈Γ_rm safe argmax gamma in Gamma rm operatorname gamma in Gamma rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7499
ot4·5·4 =2000. For γ∈Γ, let overline U_rm meta(γ) be its mean protected utility on the meta-validation episodes and widehatrm se(γ) its standard error. A configuration is validation-safe when its protected-regression tail rate and maximum regression satisfy the frozen bounds. Among safe configurations, the compiler selects widehatγ ∈ arg max_γ∈Γ_rm safe { overline U_rm meta(γ) - frac12widehatrm se(γ) }.
Exact mathematical passage
\widehat\gamma \in \argmax_{\gamma\in\Gamma_{\rm safe}} \left \{ \overline U_{\rm meta}(\gamma) - frac12\widehat{\rm se}(\gamma) \right\}. \label{eq:block07-meta-objective}
\argmax_{\delta\in\Dcal_t}Use in source · 1 passage
Search aliases: argmax_δ∈Dcal_t arg max_δ∈D_t argmax delta in Dcal operatorname delta in mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 983
pection] Let u:Y→[0,1] be a bounded utility. Define V(δ)= E[u(Y)mid A_t,δ], widehat V(δ)= E[u(Y)mid ϕ_t(A_t),δ]. If ϕ_t is ε-intervention-sufficient, then sup_δ∈D_t absV(δ)-widehat V(δ)≤ε. If δ^*∈arg max_δ∈D_tV(δ), widehatδ∈arg max_δ∈D_twidehat V(δ), then V(δ^*)-V(widehatδ)≤2ε.
Exact mathematical passage
\delta^*\in\argmax_{\delta\in\Dcal_t}V(\delta), \qquad \widehat\delta\in\argmax_{\delta\in\Dcal_t}\widehat V(\delta),
\argminUse in source · 3 passages
Search aliases: argmin arg min operatorname
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6757
h_t(x_m)^top . It estimates the bridge by regularized least squares: widehat T_t← s ∈ arg min_T normH_sT^top-H_t_F^2 +λnormT_F^2. The transformation matrices used to construct the controlled targets are not exposed to this fit.
Exact mathematical passage
\widehat T_{t\leftarrow s} \in \argmin_T \norm{H_sT^{\top}-H_t}_F^2 +\lambda\norm{T}_F^2. \label{eq:block06-learned-transport}
\argmin_{B,b}Use in source · 1 passage
Search aliases: argmin_B,b arg min_B,b argmin operatorname
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16818
Let H_s(τ)∈ℝ^36 be the source model's final latent trace state and H_t(τ)∈ℝ^36 the corresponding foreign-child state. A ridge-affine bridge is fitted on the 900 paired generator-A anchors: B_t = arg min_B,b ∑_i=1^900 normH_s(τ_i)-BH_t(τ_i)-b^2 +λnormB^2. A source ridge probe P_s is fitted independently on the 1,000 probe-calibration tasks. The transported semantic prediction is widehat Y_t(τ) =P_s(B_tH_t(τ)+b_t). The bridge gate requires effective rank 36, transported R^2≥0.35, an advantage of a…
Exact mathematical passage
B_t = \argmin_{B,b} \sum_{i=1}^{900} \norm{H_s(\tau_i)-BH_t(\tau_i)-b}^2 +\lambda\norm{B}^2. \label{eq:b18-bridge-fit}
\argmin_TUse in source · 1 passage
Search aliases: argmin_T arg min_T argmin operatorname
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6757
h_t(x_m)^top . It estimates the bridge by regularized least squares: widehat T_t← s ∈ arg min_T normH_sT^top-H_t_F^2 +λnormT_F^2. The transformation matrices used to construct the controlled targets are not exposed to this fit.
Exact mathematical passage
\widehat T_{t\leftarrow s} \in \argmin_T \norm{H_sT^{\top}-H_t}_F^2 +\lambda\norm{T}_F^2. \label{eq:block06-learned-transport}
\argmin_{T^{\mathsf T}T=I}Use in source · 1 passage
Search aliases: argmin_T^ TT=I arg min_T^ TT=I argmin mathsf operatorname mathsf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15812
For parent and child state spaces H_A and H_B of equal dimension, let calibration traces produce matrices X and Y. The orthogonal Procrustes transport is T^* = arg min_T^ TT=InormXT-Y_ F. If X^ TY=UΣ V^ T, then T^*=UV^ T. A transport receipt records calibration residual, held-out residual, singular floor, round-trip residual, and behavior change after transported trace injection.
Exact mathematical passage
T^* = \argmin_{T^{\mathsf T}T=I}\norm{XT-Y}_{\mathrm F}. \label{eq:b16-procrustes}
\operatorname{arsinh}Use in source · 1 passage
Search aliases: arsinh operatorname
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 18278
The nonlinear family applies Y_t = fracsinh (α(X_tA+b))α, α>0, coordinatewise. The declared inverse constructor first applies ϕ^-1_α(y) = fracarsinh(α y)α and then fits a step-affine map.
Exact mathematical passage
\phi^{-1}_{\alpha}(y) = \frac{\operatorname{arsinh}(\alpha y)}{\alpha}
\mathrm{audit}Use in source · 5 passages
Search aliases: audit mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5414
The data partition is itself typed: D = D_construct sqcup D_calibrate sqcup D_select sqcup D_foreign sqcup D_audit. Construction data may be used to create schemas. Calibration data may set thresholds. Selection data may choose among frozen candidates. Foreign data estimate metaproductivity. Audit data are not available until the final claim object has been sealed.
Exact mathematical passage
\mathsf D = \mathsf D_{\mathrm{construct}} \sqcup \mathsf D_{\mathrm{calibrate}} \sqcup \mathsf D_{\mathrm{select}} \sqcup \mathsf D_{\mathrm{foreign}} \sqcup \mathsf D_{\mathrm{audit}}.
\mathrm{AuthorizedCommit}Use in source · 1 passage
Search aliases: AuthorizedCommit mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21345
The example separates two predicates: ImprovesSlice(R_t) not⇒ AuthorizedCommit(R_t).
Exact mathematical passage
\mathrm{ImprovesSlice}(R_t) \not\Rightarrow \mathrm{AuthorizedCommit}(R_t).
\mathrm{avg}Use in source · 3 passages
Search aliases: avg mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21877
Equip End(H_t)⊕End(H_t) with the product Frobenius norm. For a chart repair pair (D_-,D_+), write D_avg = fracD_-+D_+2, D_ctr = fracD_+-D_-2. The closest diagonal repair is (D_avg,D_avg). Therefore dist^2 ( (D_-,D_+),D_diag ) · = normD_--D_avg_F^2 + normD_+-D_avg_F^2
Exact mathematical passage
D_{\mathrm{avg}} = \frac{D_-+D_+}{2}, \qquad D_{\mathrm{ctr}} = \frac{D_+-D_-}{2}.
\mathbf BUse in source · 14 passages
Search aliases: B mathbf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2349
Let G=(I,E) be a connected oriented graph representing the nerve of the cover, let K be a finite-dimensional Hilbert space representing a common trivialization of the kernel fibers, and let e=(e_a)_a∈ E∈ K^E be the transported overlap-mismatch vector. Let B∈ℝ^|I|×|E| be the oriented incidence matrix and set B:=B⊗ I_K. A vertex gauge k∈ K^I changes the mismatch to e+ B^*k.
Exact mathematical passage
\mathbf B:=B\otimes I_K.
B'_{m'}Use in source · 1 passage
Search aliases: B'_m'
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16172
Let d_prog(τ,τ') be a normalized distance between (A_m,B_m) and (A'_m',B'_m'), and let d_ans(τ,τ') = absy_m/4-y'_m'/4. For a recipient task τ_i, define four donor classes: item SAP: same-answer, far-program, approximately depth-matched donors; item FAD: far-answer, depth-matched donors; item SN: semantic-neighbor donors; item RMD: random depth-matched donors. Three donors per available class are selected by deterministic task-identifier hashing. Each audit records donor coverage and the realized…
Exact mathematical passage
(A'_{m'},B'_{m'})
B^*Use in source · 1 passage
Search aliases: B^*
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2360
[Quantitative graph-Hodge gluing] Let L_G:=BB^* be the graph Laplacian and let P_cyc be the orthogonal projector from K^E onto ker B, the cycle space. Then inf_k∈ K^Inorme+ B^*k = normP_cyce. The mean-zero minimum-norm gauge is k^* =-(L_G^† B⊗ I_K)e. If λ_2(L_G)>0 is the spectral gap, then normk^* ≤ frac1sqrtλ_2(L_G)norm e. Exact graph-level gluing is possible if and only if P_cyce=0.
Exact mathematical passage
L_G:=BB^*
\mathbf B^*Use in source · 11 passages
Search aliases: B^* mathbf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2353
K be a finite-dimensional Hilbert space representing a common trivialization of the kernel fibers, and let e=(e_a)_a∈ E∈ K^E be the transported overlap-mismatch vector. Let B∈ℝ^|I|×|E| be the oriented incidence matrix and set B:=B⊗ I_K. A vertex gauge k∈ K^I changes the mismatch to e+ B^*k.
Exact mathematical passage
e+\mathbf B^*k.
b^2Use in source · 1 passage
Search aliases: b^2
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21838
Let h∼ N(0,1) and let s∈{-1,+1} be independent and balanced. Define y=(a+sb)h, b≠0. A global scalar bridge predicts widehat y=th. The population minimizer is t^*=a, and inf_t E (y-th)^2 = E(sbh)^2 = b^2. The residual is not an estimation artifact. It remains after infinite data because one slope cannot equal both a-b and a+b. The two-chart grammar uses t_-=a-b, t_+=a+b, and has zero noiseless approximation error. This is the one-dimensional form of [source reference].
Exact mathematical passage
\inf_t\mathbb E (y-th)^2 = \mathbb E(sbh)^2 = b^2.
B^\topUse in source · 4 passages
Search aliases: B^top top
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5635
The executable classifier applies the ordered rule Class(r,b,H) = LOCAL_OBSTRUCTION, · normP_(im D)^⊥r>τ, GLUING_OBSTRUCTION, · normP_ker B^topb>τ, EVALUATOR_HOLONOMY, · norm(H-I)e>τ, FLAT_REPAIR, · otherwise. The order is part of the receipt because a case can exhibit more than one defect. The fixture isolates the four modes, so the ordered rule is unambiguous.
Exact mathematical passage
\operatorname{Class}(r,b,H) = \begin{cases} \texttt{LOCAL\_OBSTRUCTION},&\norm{P_{(\im D)^\perp}r}>\tau,\\ \texttt{GLUING\_OBSTRUCTION},&\norm{P_{\ker B^\top}b}>\tau,\\ \texttt{EVALUATOR\_HOLONOMY},&\norm{(H-I)e}>\tau,\\ \texttt{FLAT\_REPAIR},&\text{otherwise}. \end{cases}
b_0Use in source · 3 passages
Search aliases: b_0
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11430
[Open-set incompatibility of distinct affine charts] Let U_0,U_1⊆ℝ^d each contain a nonempty open set. Let f_j(h)=A_jh+b_j, j∈{0,1}. If one affine map f(h)=Ah+b satisfies f(h)=f_j(h) for every h∈ U_j and both j, then A=A_0=A_1, b=b_0=b_1. Consequently, if (A_0,b_0)≠(A_1,b_1), no element of G_glob represents the two-chart relation exactly.
Exact mathematical passage
A=A_0=A_1, \qquad b=b_0=b_1.
b_1Use in source · 9 passages
Search aliases: b_1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2223
The sequence [b_1],[b_2],[b_3],…∈coker D is an obstruction tower. Choices in ker D at early orders can alter later classes, so a repair search must retain branch information rather than greedily fixing one representative.
Exact mathematical passage
[b_1],[b_2],[b_3],\ldots\in\coker D
b_{1,\alpha}Use in source · 1 passage
Search aliases: b_1,α alpha
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2605
The theorem produces an obstruction ledger c|cc order · local obstruction · gluing obstruction 1 · [b_1,α]∈coker D_α · [d^(1)]∈check H^1(ker D) 2 · [b_2,α]∈coker D_α · [d^(2)]∈check H^1(ker D) vdots · vdots · vdots N · [b_N,α]∈coker D_α · [d^(N)]∈check H^1(ker D). Higher categorical repair interfaces may introduce additional coherence obstructions beyond this two-column ledger. Those are not proved here.
Exact mathematical passage
\begin{array}{c|cc} \text{order}&\text{local obstruction}&\text{gluing obstruction}\\ \hline 1&[b_{1,\alpha}]\in\coker D_\alpha&[d^{(1)}]\in\check H^1(\ker D)\\ 2&[b_{2,\alpha}]\in\coker D_\alpha&[d^{(2)}]\in\check H^1(\ker D)\\ \vdots&\vdots&\vdots\\ N&[b_{N,\alpha}]\in\coker D_\alpha&[d^{(N)}]\in\check H^1(\ker D). \end{array}
B_2Use in source · 5 passages
Search aliases: B_2
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 900
A lineage may branch: A⟶ B_1, A⟶ B_2. No definition of recursive improvement requires one child to be the unique continuation of the parent.
Exact mathematical passage
A\longrightarrow B_1, \qquad A\longrightarrow B_2.
b_2Use in source · 3 passages
Search aliases: b_2
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2223
The sequence [b_1],[b_2],[b_3],…∈coker D is an obstruction tower. Choices in ker D at early orders can alter later classes, so a repair search must retain branch information rather than greedily fixing one representative.
Exact mathematical passage
[b_1],[b_2],[b_3],\ldots\in\coker D
b_{2,\alpha}Use in source · 1 passage
Search aliases: b_2,α alpha
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2605
The theorem produces an obstruction ledger c|cc order · local obstruction · gluing obstruction 1 · [b_1,α]∈coker D_α · [d^(1)]∈check H^1(ker D) 2 · [b_2,α]∈coker D_α · [d^(2)]∈check H^1(ker D) vdots · vdots · vdots N · [b_N,α]∈coker D_α · [d^(N)]∈check H^1(ker D). Higher categorical repair interfaces may introduce additional coherence obstructions beyond this two-column ledger. Those are not proved here.
Exact mathematical passage
\begin{array}{c|cc} \text{order}&\text{local obstruction}&\text{gluing obstruction}\\ \hline 1&[b_{1,\alpha}]\in\coker D_\alpha&[d^{(1)}]\in\check H^1(\ker D)\\ 2&[b_{2,\alpha}]\in\coker D_\alpha&[d^{(2)}]\in\check H^1(\ker D)\\ \vdots&\vdots&\vdots\\ N&[b_{N,\alpha}]\in\coker D_\alpha&[d^{(N)}]\in\check H^1(\ker D). \end{array}
b_3Use in source · 1 passage
Search aliases: b_3
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2223
The sequence [b_1],[b_2],[b_3],…∈coker D is an obstruction tower. Choices in ker D at early orders can alter later classes, so a repair search must retain branch information rather than greedily fixing one representative.
Exact mathematical passage
[b_1],[b_2],[b_3],\ldots\in\coker D
B_aUse in source · 9 passages
Search aliases: B_a
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6616
[Exact causal-packet recovery for a linear encoder] Suppose E_a = A_a · B_a, A_a,B_a:ℝ^8→ℝ^d_a, and suppose the centered conditional covariance of ψ(x) given s(x)=1 is positive definite. Then span{Δ_a(x):s(x)=1} = im B_a. Consequently, a rank-r singular-subspace estimator with r=rank B_a recovers the exact causal packet from a sufficiently rich noiseless sample.
Exact mathematical passage
E_a = \begin{bmatrix}A_a&B_a\end{bmatrix}, \qquad A_a,B_a:\R^8\to\R^{d_a},
b_aUse in source · 2 passages
Search aliases: b_a
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6661
Let U_a∈ℝ^d_a× r have orthonormal columns spanning the discovered packet. Define desired packet coordinates z_i = U_a^top(h_a^∘(x_i)-h_a(x_i)). Fit an affine predictor widehat z(h)=K_a h+c_a by ridge regression on source anchor states. The repair is R_a(h) = h+U_awidehat z(h) = h+D_a h+b_a, where D_a=U_aK_a, b_a=U_ac_a.
Exact mathematical passage
R_a(h) = h+U_a\widehat z(h) = h+D_a h+b_a, \label{eq:block06-source-repair}
B_cUse in source · 1 passage
Search aliases: B_c
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5290
Let M(B) be the offspring matrix under a fixed discovery budget B. Define the critical budget B_c = infsetB:rad(M(B))>1. Comparisons of R_imp are meaningful only at matched budgets and evidence access.
Exact mathematical passage
B_c = \inf\set{B:\rad(M(B))>1}.
b_cUse in source · 1 passage
Search aliases: b_c
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17814
A latent-step-shared map acts independently on each recurrent state: widehat h_a,c^(k) = A_c h_b,c^(k)+b_c, k=1,…,5. A full-trajectory map instead acts on the concatenation vec(h_b,c^(1),…,h_b,c^(5)) ∈ R^200, allowing cross-step mixing: vec(widehat h_a,c) = M_cvec(h_b,c)+q_c. This difference is substantive when two recurrent architectures distribute the same computation across latent time differently.
Exact mathematical passage
\widehat h_{a,c}^{(k)} = A_c h_{b,c}^{(k)}+b_c, \qquad k=1,\ldots,5. \label{eq:b20-step-shared}
b_iUse in source · 1 passage
Search aliases: b_i
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17321
widehat y_i)∈ℝ^32, and the permutation-invariant aggregate a(D) = ( frac15∑_i=1^5p_i, max_1≤ i≤5p_i )∈ℝ^64. The semantic seed is h_0=S(a(D),Q(widehat q))∈ℝ^40. A five-step Elman recurrence generates the Neuralese trace r_k=tanh(W_ih_0+b_i+U_ir_k-1+c_i+e_k), k=1,…,5, where e_k is a learned step embedding. A gated readout recurrence maps the trace to protected program and answer coordinates. A halt head predicts one of the five execution depths.
Exact mathematical passage
r_k=\tanh\left(W_ih_0+b_i+U_ir_{k-1}+c_i+e_k\right), \qquad k=1,\ldots,5, \label{eq:b19-recurrence}
B_jUse in source · 1 passage
Search aliases: B_j
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16022
ion be f_j(z)=a_jz+b_j, a_j∈{-1,-tfrac12,tfrac12,1}, b_j∈{-tfrac12,0,tfrac12}. A depth-m program acts on x∈{-1,-tfrac34,…,1} by F_m(x) = (f_m∘⋯∘ f_1)(x) =A_mx+B_m, where A_0=1, B_0=0, A_j=a_jA_j-1, B_j=a_jB_j-1+b_j. All quantities in [source reference] are evaluated with exact rational arithmetic before a floating representation is exposed to the model.
Exact mathematical passage
A_0=1, \qquad B_0=0, \qquad A_j=a_jA_{j-1}, \qquad B_j=a_jB_{j-1}+b_j. \label{eq:b17-affine-witness-recursion}
B_{j-1}Use in source · 1 passage
Search aliases: B_j-1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16022
ion be f_j(z)=a_jz+b_j, a_j∈{-1,-tfrac12,tfrac12,1}, b_j∈{-tfrac12,0,tfrac12}. A depth-m program acts on x∈{-1,-tfrac34,…,1} by F_m(x) = (f_m∘⋯∘ f_1)(x) =A_mx+B_m, where A_0=1, B_0=0, A_j=a_jA_j-1, B_j=a_jB_j-1+b_j. All quantities in [source reference] are evaluated with exact rational arithmetic before a floating representation is exposed to the model.
Exact mathematical passage
A_0=1, \qquad B_0=0, \qquad A_j=a_jA_{j-1}, \qquad B_j=a_jB_{j-1}+b_j. \label{eq:b17-affine-witness-recursion}
b_mUse in source · 2 passages
Search aliases: b_m
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13792
Let the scored evaluator portfolio be B = {b_1,…,b_m}. For method c, define normalized headroom closure q_j(c) = fracs_j(c)-s_j(K_τ)s_j^⋆-s_j(K_τ). Let I_+(c)={j:q_j(c)>0}. The coverage-weighted geometric score is S_cov(c) = frac|I_+(c)|m ( ∏_j∈ I_+(c)q_j(c) )^1/|I_+(c)|, with value zero when I_+(c)=∅. Hard capability and integrity gates are applied separately rather than traded against this scalar.
Exact mathematical passage
\Bcal = \{b_1,\ldots,b_m\}.
b_{m+1}Use in source · 5 passages
Search aliases: b_m+1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13829
[Portfolio refinement shrinks the invisible subspace] Let B^+=B∪{b_m+1}. Then N_B^+ ⊆ N_B. The inclusion is strict if and only if there exists v∈N_B such that Db_m+1(v)≠0.
Exact mathematical passage
\Bcal^+=\Bcal\cup\{b_{m+1}\}
b_{N,\alpha}Use in source · 1 passage
Search aliases: b_N,α alpha
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2605
The theorem produces an obstruction ledger c|cc order · local obstruction · gluing obstruction 1 · [b_1,α]∈coker D_α · [d^(1)]∈check H^1(ker D) 2 · [b_2,α]∈coker D_α · [d^(2)]∈check H^1(ker D) vdots · vdots · vdots N · [b_N,α]∈coker D_α · [d^(N)]∈check H^1(ker D). Higher categorical repair interfaces may introduce additional coherence obstructions beyond this two-column ledger. Those are not proved here.
Exact mathematical passage
\begin{array}{c|cc} \text{order}&\text{local obstruction}&\text{gluing obstruction}\\ \hline 1&[b_{1,\alpha}]\in\coker D_\alpha&[d^{(1)}]\in\check H^1(\ker D)\\ 2&[b_{2,\alpha}]\in\coker D_\alpha&[d^{(2)}]\in\check H^1(\ker D)\\ \vdots&\vdots&\vdots\\ N&[b_{N,\alpha}]\in\coker D_\alpha&[d^{(N)}]\in\check H^1(\ker D). \end{array}
b_{n,\alpha}Use in source · 5 passages
Search aliases: b_n,α alpha
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2563
At order n, let b_n,α∈mathscr V(U_α) be the residual coefficient determined by lower-order choices. The local obstruction is [b_n,α]∈coker D_α. If every local class vanishes, choose u_n,α with D_α u_n,α=-b_n,α. Their differences define an order-n gluing class in check H^1(ker D).
Exact mathematical passage
b_{n,\alpha}\in\mathscr V(U_\alpha)
B_RUse in source · 2 passages
Search aliases: B_R
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10269
Let R_t be true protected regression and let U_t^R be a conditional upper confidence bound. Require U_t^R≤β_t, ∑_tβ_t≤ B_R. Let d_t be semantic debt incurred by approximate grammar or evaluator transport, with ∑_td_t≤ B_sem. A lineage certificate should therefore report three separate cumulative quantities: ( ∑_tδ_t, ∑_tβ_t, ∑_td_t ). A positive gain sequence is not sufficient if its protected regression or semantic debt diverges.
Exact mathematical passage
U_t^R\leq\beta_t, \qquad \sum_t\beta_t\leq B_R. \label{eq:b8-regression-budget}
B_R^WUse in source · 7 passages
Search aliases: B_R^W
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2006
[Gauge-fixed local nonlinear repair by contraction] Suppose D:U→ V has a bounded right inverse Q:V→ U, so DQ=I_V, and set W:=im Q. Then restr D W:W→ V is an isomorphism with inverse Q. Let B_R^W:=setw∈ W:norm w≤ R. Suppose normN(u)-N(v)≤ L_Nnormu-v (u,v∈ B_R^W), with q:=norm QL_N<1, and normQr+norm Qsup_u∈ B_R^WnormN(u)≤ R. Then there exists a unique u^*∈ B_R^W satisfying F(u^*)=0. The gauge-fixed iteration u_k+1=-Qr-QN(u_k) converges to u^* from every u_0∈ B_R^W, with normu_k-u^*≤fracq^k1-qnorm…
Exact mathematical passage
B_R^W:=\set{w\in W:\norm w\leq R}.
B_{\mathrm{research}}Use in source · 1 passage
Search aliases: B_research mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 14070
Every parent and child run receives exactly B_research=6 full candidate experiments. Candidate metadata and cheap diagnostics are common inputs. The number of proposal receipts, scored evaluator calls, capability checks, and integrity-monitor calls is therefore matched.
Exact mathematical passage
B_{\mathrm{research}}=6 \label{eq:b13-budget}
B_{\rm pair}Use in source · 1 passage
Search aliases: B_rm pair rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20474
B_rm pair · Matched deployment paired-query budget
Exact mathematical passage
B_{\rm pair}
B_{\rm research}Use in source · 1 passage
Search aliases: B_rm research rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 14704
Both compilers receive exactly B_rm research=6 full experiments per problem. The child can use early experiments to estimate response mixtures and allocate later experiments, but it receives no additional full experiment. Candidate metadata, proposal count, evaluator calls, capability checks, and integrity calls are matched.
Exact mathematical passage
B_{\rm research}=6 \label{eq:block14-budget}
b_sUse in source · 8 passages
Search aliases: b_s
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5909
Let the latent world state be x_t∈ℝ^2, action a_t∈ℝ^2, observation y_t∈ℝ^3, and memory state m_t∈ℝ^2. The baseline dynamics are x_t+1=Ax_t+Ba_t, y_t=Cx_t+b_s, m_t+1=λ m_t+(1-λ)y_t,1:2, and the reported evaluator score is q_t=w^top y_t. The redundant observation map is C= 1 · 0
Exact mathematical passage
y_t=Cx_t+b_s,
B_{\mathrm{sem}}Use in source · 1 passage
Search aliases: B_sem mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10276
Let R_t be true protected regression and let U_t^R be a conditional upper confidence bound. Require U_t^R≤β_t, ∑_tβ_t≤ B_R. Let d_t be semantic debt incurred by approximate grammar or evaluator transport, with ∑_td_t≤ B_sem. A lineage certificate should therefore report three separate cumulative quantities: ( ∑_tδ_t, ∑_tβ_t, ∑_td_t ). A positive gain sequence is not sufficient if its protected regression or semantic debt diverges.
Exact mathematical passage
\sum_td_t\leq B_{\mathrm{sem}}. \label{eq:b8-semantic-budget}
b_tUse in source · 6 passages
Search aliases: b_t
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6772
Given source repair R_s(h)=h+D_sh+b_s, its transported target form is D_t = widehat T D_swidehat T^+, b_t = widehat T b_s, with transported packet basis obtained by orthogonalizing widehat T U_s.
Exact mathematical passage
D_t = \widehat T D_s\widehat T^{+}, \qquad b_t = \widehat T b_s, \label{eq:block06-transported-repair}
\bar AUse in source · 2 passages
Search aliases: bar A bar
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11355
1∈ℝ^d+1 for homogeneous affine coordinates. Let c:H_s→{-1,+1} be an observable chart variable. A two-chart affine relation is specified by bar T_-,bar T_+∈ℝ^q×(d+1), with y=bar T_c(h)bar h+ε. Equivalently, writing bar A=fracbar T_++bar T_-2, bar B=fracbar T_+-bar T_-2, we have y=(bar A+c(h)bar B)bar h+ε.
Exact mathematical passage
\bar A=\frac{\bar T_++\bar T_-}{2}, \qquad \bar B=\frac{\bar T_+-\bar T_-}{2},
\bar BUse in source · 2 passages
Search aliases: bar B bar
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11355
1∈ℝ^d+1 for homogeneous affine coordinates. Let c:H_s→{-1,+1} be an observable chart variable. A two-chart affine relation is specified by bar T_-,bar T_+∈ℝ^q×(d+1), with y=bar T_c(h)bar h+ε. Equivalently, writing bar A=fracbar T_++bar T_-2, bar B=fracbar T_+-bar T_-2, we have y=(bar A+c(h)bar B)bar h+ε.
Exact mathematical passage
\bar A=\frac{\bar T_++\bar T_-}{2}, \qquad \bar B=\frac{\bar T_+-\bar T_-}{2},
\bar gUse in source · 2 passages
Search aliases: bar g bar
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8567
For a candidate configuration c, the development compiler computes target-level selected gains and the score S_rm dev(c) = bar g_c -0.20s_c -2h_c +0.02q_c, where bar g_c is mean protected gain, s_c is the target-level standard deviation, h_c is harmful-target rate, and q_c is positive-target rate. This is a finite engineering objective, not a universal definition of evolvability.
Exact mathematical passage
S_{\rm dev}(c) = \bar g_c -0.20s_c -2h_c +0.02q_c, \label{eq:block07q-compiler-score}
\bar g_cUse in source · 2 passages
Search aliases: bar g_c bar
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8567
For a candidate configuration c, the development compiler computes target-level selected gains and the score S_rm dev(c) = bar g_c -0.20s_c -2h_c +0.02q_c, where bar g_c is mean protected gain, s_c is the target-level standard deviation, h_c is harmful-target rate, and q_c is positive-target rate. This is a finite engineering objective, not a universal definition of evolvability.
Exact mathematical passage
S_{\rm dev}(c) = \bar g_c -0.20s_c -2h_c +0.02q_c, \label{eq:block07q-compiler-score}
\bar h_iUse in source · 1 passage
Search aliases: bar h_i bar
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11757
A candidate pool contains source state h_i, target observation y_i, and known observation scale σ_i. The inherited acquisition policy selects rows sequentially to maximize the quality-weighted log-determinant increment logdet( λ I+ ∑_i∈ Sσ_i^-2bar h_ibar h_i^top ). Both parent and child use the same resulting order. The parent takes the first 80 rows for one global affine fit. The child takes the first 48 rows for local fits and the next 32 for separator selection. Both reserve the final 16 rows…
Exact mathematical passage
\log\det\left( \lambda I+ \sum_{i\in S}\sigma_i^{-2}\bar h_i\bar h_i^\top \right).
\bar h_i^\topUse in source · 1 passage
Search aliases: bar h_i^top bar top
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11757
A candidate pool contains source state h_i, target observation y_i, and known observation scale σ_i. The inherited acquisition policy selects rows sequentially to maximize the quality-weighted log-determinant increment logdet( λ I+ ∑_i∈ Sσ_i^-2bar h_ibar h_i^top ). Both parent and child use the same resulting order. The parent takes the first 80 rows for one global affine fit. The child takes the first 48 rows for local fits and the next 32 for separator selection. Both reserve the final 16 rows…
Exact mathematical passage
\log\det\left( \lambda I+ \sum_{i\in S}\sigma_i^{-2}\bar h_i\bar h_i^\top \right).
\bar RUse in source · 7 passages
Search aliases: bar R bar
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11522
[New chart-contrast correction direction] Let the affine target repairs have homogeneous matrices bar R_0 and bar R_1. If bar R_0≠bar R_1, then bar R_ctr = fracbar R_1-bar R_02 induces a nonzero anti-diagonal component (-bar R_ctr,bar R_ctr) orthogonal to every diagonal pair under the product Frobenius inner product. Therefore the chart grammar contains a correction direction unavailable to the parent grammar.
Exact mathematical passage
\bar R_0
\bar R_0Use in source · 3 passages
Search aliases: bar R_0 bar
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11522
[New chart-contrast correction direction] Let the affine target repairs have homogeneous matrices bar R_0 and bar R_1. If bar R_0≠bar R_1, then bar R_ctr = fracbar R_1-bar R_02 induces a nonzero anti-diagonal component (-bar R_ctr,bar R_ctr) orthogonal to every diagonal pair under the product Frobenius inner product. Therefore the chart grammar contains a correction direction unavailable to the parent grammar.
Exact mathematical passage
\bar R_0
\bar R_1Use in source · 3 passages
Search aliases: bar R_1 bar
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11522
[New chart-contrast correction direction] Let the affine target repairs have homogeneous matrices bar R_0 and bar R_1. If bar R_0≠bar R_1, then bar R_ctr = fracbar R_1-bar R_02 induces a nonzero anti-diagonal component (-bar R_ctr,bar R_ctr) orthogonal to every diagonal pair under the product Frobenius inner product. Therefore the chart grammar contains a correction direction unavailable to the parent grammar.
Exact mathematical passage
\bar R_1
\bar R_{\mathrm{ctr}}Use in source · 3 passages
Search aliases: bar R_ctr bar mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11527
[New chart-contrast correction direction] Let the affine target repairs have homogeneous matrices bar R_0 and bar R_1. If bar R_0≠bar R_1, then bar R_ctr = fracbar R_1-bar R_02 induces a nonzero anti-diagonal component (-bar R_ctr,bar R_ctr) orthogonal to every diagonal pair under the product Frobenius inner product. Therefore the chart grammar contains a correction direction unavailable to the parent grammar.
Exact mathematical passage
\bar R_{\mathrm{ctr}} = \frac{\bar R_1-\bar R_0}{2}
\bar T_+Use in source · 2 passages
Search aliases: bar T_+ bar
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11346
1∈ℝ^d+1 for homogeneous affine coordinates. Let c:H_s→{-1,+1} be an observable chart variable. A two-chart affine relation is specified by bar T_-,bar T_+∈ℝ^q×(d+1), with y=bar T_c(h)bar h+ε. Equivalently, writing bar A=fracbar T_++bar T_-2, bar B=fracbar T_+-bar T_-2, we have y=(bar A+c(h)bar B)bar h+ε.
Exact mathematical passage
\bar T_-,\bar T_+\in\R^{q\times(d+1)},
\bar T_-Use in source · 2 passages
Search aliases: bar T_- bar
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11346
1∈ℝ^d+1 for homogeneous affine coordinates. Let c:H_s→{-1,+1} be an observable chart variable. A two-chart affine relation is specified by bar T_-,bar T_+∈ℝ^q×(d+1), with y=bar T_c(h)bar h+ε. Equivalently, writing bar A=fracbar T_++bar T_-2, bar B=fracbar T_+-bar T_-2, we have y=(bar A+c(h)bar B)bar h+ε.
Exact mathematical passage
\bar T_-,\bar T_+\in\R^{q\times(d+1)},
\bar T_{c(h)}Use in source · 1 passage
Search aliases: bar T_c(h) bar
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11350
1∈ℝ^d+1 for homogeneous affine coordinates. Let c:H_s→{-1,+1} be an observable chart variable. A two-chart affine relation is specified by bar T_-,bar T_+∈ℝ^q×(d+1), with y=bar T_c(h)bar h+ε. Equivalently, writing bar A=fracbar T_++bar T_-2, bar B=fracbar T_+-bar T_-2, we have y=(bar A+c(h)bar B)bar h+ε.
Exact mathematical passage
y=\bar T_{c(h)}\bar h+\varepsilon. \label{eq:b10-chart-map}
\bar vUse in source · 9 passages
Search aliases: bar v bar
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9740
Let I=im D⊆ V be the current authorized correction image and let z=P_I^⊥r be the unresolved component of defect r. Suppose a candidate grammar extension contributes a response vector v∈ V. Only its component outside the old image is structurally new: bar v=P_I^⊥v. The expanded image is I'=I+span{v}.
Exact mathematical passage
\bar v=P_{I^\perp}v. \label{eq:b8-new-generator-component}
\mathrm{base}Use in source · 10 passages
Search aliases: base mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5669
The minimal closure model has states S={s_base,s_improved,s_invalid} and valid set K={s_base,s_improved}. With intact rollback, authorized complete transactions have successor relation Post(s_base) ={s_base,s_improved}, Post(s_improved)={s_improved}. Failed candidates are restored to the base state before the complete transaction ends.
Exact mathematical passage
S=\{s_{\mathrm{base}},s_{\mathrm{improved}},s_{\mathrm{invalid}}\}
\Bcal^+Use in source · 2 passages
Search aliases: Bcal^+ B^+ Bcal mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13829
[Portfolio refinement shrinks the invisible subspace] Let B^+=B∪{b_m+1}. Then N_B^+ ⊆ N_B. The inclusion is strict if and only if there exists v∈N_B such that Db_m+1(v)≠0.
Exact mathematical passage
\Bcal^+=\Bcal\cup\{b_{m+1}\}
\Bcal_t^{\mathrm{sand}}Use in source · 2 passages
Search aliases: Bcal_t^sand B_t^sand Bcal mathrm mathcal mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 811
A witness structure may be represented as W_t = ( V_t^fr, C_t, B_t^sand, R_t, Π_t^auth ), where V_t^fr is the frozen evaluator instance for the current transaction, C_t is a protected checkpoint, B_t^sand is the sandbox boundary, R_t is the rollback procedure, and Π_t^auth records effect authority.
Exact mathematical passage
\Wcal_t = \left( \Vcal_t^{\mathrm{fr}}, C_t, \Bcal_t^{\mathrm{sand}}, R_t, \Pi_t^{\mathrm{auth}} \right),
\bigwedge_iUse in source · 2 passages
Search aliases: bigwedge_i bigwedge
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1286
_ω_i; item the conjunction of the inner postconditions and bridge guarantees implies C_ω; item execution of ω from C_ω establishes Q_ω; item receipt composition preserves provenance and protected evidence references. Then the composed receipt is sound for ω∘(ω_1,…,ω_n) with precondition bigwedge_iC_ω_i and postcondition Q_ω.
Exact mathematical passage
\bigwedge_iC_{\omega_i}
\botUse in source · 10 passages
Search aliases: bot
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15068
Let ϕ_0:T→Z_0 be the inherited diagnostic map. In Block 15, six inherited obstruction classes retain their existing tokens, while two previously unseen obstruction mechanisms are both mapped to one open-set token bot: ϕ_0(τ_ m int) = ϕ_0(τ_ m temp) = bot. The equality in [source reference] is not itself a defect. An unknown token is an appropriate abstention state until the lineage possesses external evidence that the collapsed cases require incompatible research actions.
Exact mathematical passage
\bot
\mathbb CUse in source · 4 passages
Search aliases: C mathbb
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10052
[Finite causal atlas decomposition] There exists a unitary identification H ≅ ⊕_α=1^p ( C^m_α⊗ C^n_α ) under which A ≅ ⊕_α=1^p ( I_m_α⊗ M_n_α( C) ). The minimal central projections P_α∈ Z( A) give a canonical coarse causal atlas H = ⊕_α=1^pP_α H. The blocks are unique up to permutation and unitary changes of basis within multiplicity spaces.
Exact mathematical passage
H \cong \bigoplus_{\alpha=1}^{p} \left( \mathbb C^{m_\alpha}\otimes\mathbb C^{n_\alpha} \right) \label{eq:b8-algebra-space-decomposition}
C'Use in source · 11 passages
Search aliases: C'
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5057
Let j:V↪ V' be an injective evidence-conservative embedding of old defects into an expanded ontology. Let C=im D⊆ V, C'=im D'⊆ V', and assume preservation of old correctability: j(C)⊆ C'. The embedding induces a map overline j:V/C→ V'/C'.
Exact mathematical passage
C=\im D\subseteq V, \qquad C'=\im D'\subseteq V',
c^{\mathsf T}Use in source · 2 passages
Search aliases: c^ T mathsf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12192
Choose c∈ℝ^k outside the finite union of hyperplanes setc:c^ T(λ_a-λ_b)=0, a≠ b. Then the scalars μ_a=c^ Tλ_a are pairwise distinct. The combined response B=∑_j=1^kc_jR_j =∑_a=1^mμ_aP_a belongs to A_R. For each a, define the Lagrange polynomial p_a(z)=∏_b≠ afracz-μ_bμ_a-μ_b. Functional calculus gives p_a(B)=∑_b=1^mp_a(μ_b)P_b=P_a. Thus every P_a lies in the generated algebra. A nonzero central projection that is strictly below one P_a would have to be scalar on H_a and therefore is either 0 or P…
Exact mathematical passage
\set{c:c^{\mathsf T}(\lambda_a-\lambda_b)=0}, \qquad a\neq b.
\mathcal C^{(0)}_eUse in source · 1 passage
Search aliases: C^(0)_e mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7647
[Fixed-candidate structural invariance] Let C_e be the authorized candidate set for episode e. If parent and child differ only in the selector and C^(0)_e= C^(1)_e for every e, then the pointwise reachable correction set and its linear span are identical for parent and child.
Exact mathematical passage
\mathcal C^{(0)}_e=\mathcal C^{(1)}_e
\mathcal C^{(1)}_eUse in source · 1 passage
Search aliases: C^(1)_e mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7647
[Fixed-candidate structural invariance] Let C_e be the authorized candidate set for episode e. If parent and child differ only in the selector and C^(0)_e= C^(1)_e for every e, then the pointwise reachable correction set and its linear span are identical for parent and child.
Exact mathematical passage
\mathcal C^{(0)}_e=\mathcal C^{(1)}_e
C^*Use in source · 2 passages
Search aliases: C^*
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1723
Set C=D(K), a closed convex cone. Its dual is C^*=sety:D^*y∈ K^*. By the bipolar theorem, -r∈ C if and only if anglesy,-r≥0 for every y∈ C^*, which is the displayed inequality.
Exact mathematical passage
C^*=\set{y:D^*y\in K^*}.
c^*Use in source · 5 passages
Search aliases: c^*
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7455
ity regret under simultaneous lower bounds] Let C_rm safe be the truly safe candidate set. Suppose every c∈ C_rm safe passes the implemented admissibility filter, and suppose 0≤ U_e(c)-widehat U_rm LCB(c)≤ϵ for every admissible c. If widehat c maximizes widehat U_rm LCB and c^* maximizes U_e over C_rm safe, then U_e(c^*)-U_e(widehat c)≤ϵ.
Exact mathematical passage
c^*
C^{a}_{A0\to c}Use in source · 1 passage
Search aliases: C^a_A0→ c to
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17721
For architectures a,b and admitted chart c, an architecture map is T^c_b→ a:H_b,c⟶ H_a,c. For one architecture a and two admitted charts c,d, a context map is C^a_c→ d:H_a,c⟶ H_a,d. Block 20 uses A0 as a reference chart and fits a star of maps C^a_c→ A0 and C^a_A0→ c from protected semantic twins.
Exact mathematical passage
C^{a}_{A0\to c}
C^{a}_{c\to A0}Use in source · 1 passage
Search aliases: C^a_c→ A0 to
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17721
For architectures a,b and admitted chart c, an architecture map is T^c_b→ a:H_b,c⟶ H_a,c. For one architecture a and two admitted charts c,d, a context map is C^a_c→ d:H_a,c⟶ H_a,d. Block 20 uses A0 as a reference chart and fits a star of maps C^a_c→ A0 and C^a_A0→ c from protected semantic twins.
Exact mathematical passage
C^{a}_{c\to A0}
C^a_{c\to d}Use in source · 1 passage
Search aliases: C^a_c→ d to
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17645
The next threshold is atlas-indexed cross-architecture Neuralese transport. Source and target systems should each carry locally valid semantic charts, transport should be tested on chart overlaps, and architecture transport should approximately commute with context transition: C^b_c→ dT^c_a→ b ≈ T^d_a→ bC^a_c→ d. Block 20 tests this threshold directly in two foreign recurrent architectures. It preserves the inherited six-chart atlas, audits the curvature residual of [source reference], and retai…
Exact mathematical passage
C^b_{c\to d}T^c_{a\to b} \approx T^d_{a\to b}C^a_{c\to d}. \label{eq:b19-transport-square}
C^b_{c\to d}Use in source · 1 passage
Search aliases: C^b_c→ d to
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17645
The next threshold is atlas-indexed cross-architecture Neuralese transport. Source and target systems should each carry locally valid semantic charts, transport should be tested on chart overlaps, and architecture transport should approximately commute with context transition: C^b_c→ dT^c_a→ b ≈ T^d_a→ bC^a_c→ d. Block 20 tests this threshold directly in two foreign recurrent architectures. It preserves the inherited six-chart atlas, audits the curvature residual of [source reference], and retai…
Exact mathematical passage
C^b_{c\to d}T^c_{a\to b} \approx T^d_{a\to b}C^a_{c\to d}. \label{eq:b19-transport-square}
C^kUse in source · 2 passages
Search aliases: C^k
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2115
[Lyapunov--Schmidt repair reduction] Let F:P× U→ V be C^k, k≥2, between finite-dimensional spaces, with F(0,0)=0. Let D:=∂_uF(0,0). Choose splittings U=K⊕ X, K=ker D, and V=R⊕ C, R=im D, so that restrDX:X→ R is an isomorphism. Let P_R and P_C be the corresponding projections. Then there are neighborhoods of the origin and a unique C^k map χ:P× K→ X, χ(0,0)=0, such that P_RF(p,k+χ(p,k))=0. Define the reduced obstruction map ψ(p,k):=P_CF(p,k+χ(p,k)). A sufficiently small pair
Exact mathematical passage
C^k
\mathbb C^{m_\alpha}Use in source · 2 passages
Search aliases: C^m_α mathbb alpha
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10052
[Finite causal atlas decomposition] There exists a unitary identification H ≅ ⊕_α=1^p ( C^m_α⊗ C^n_α ) under which A ≅ ⊕_α=1^p ( I_m_α⊗ M_n_α( C) ). The minimal central projections P_α∈ Z( A) give a canonical coarse causal atlas H = ⊕_α=1^pP_α H. The blocks are unique up to permutation and unitary changes of basis within multiplicity spaces.
Exact mathematical passage
H \cong \bigoplus_{\alpha=1}^{p} \left( \mathbb C^{m_\alpha}\otimes\mathbb C^{n_\alpha} \right) \label{eq:b8-algebra-space-decomposition}
\mathbb C^{n_\alpha}Use in source · 1 passage
Search aliases: C^n_α mathbb alpha
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10052
[Finite causal atlas decomposition] There exists a unitary identification H ≅ ⊕_α=1^p ( C^m_α⊗ C^n_α ) under which A ≅ ⊕_α=1^p ( I_m_α⊗ M_n_α( C) ). The minimal central projections P_α∈ Z( A) give a canonical coarse causal atlas H = ⊕_α=1^pP_α H. The blocks are unique up to permutation and unitary changes of basis within multiplicity spaces.
Exact mathematical passage
H \cong \bigoplus_{\alpha=1}^{p} \left( \mathbb C^{m_\alpha}\otimes\mathbb C^{n_\alpha} \right) \label{eq:b8-algebra-space-decomposition}
c^\topUse in source · 1 passage
Search aliases: c^top top
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5595
-1 , we obtain absc^top b=frac1sqrt3>0. Thus local correction succeeds while descent fails.
Exact mathematical passage
\abs{c^\top b}=\frac1{\sqrt3}>0.
C^\astUse in source · 2 passages
Search aliases: C^∗ ast
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10082
A finite-dimensional unital ∗-algebra is a finite-dimensional C^∗-algebra. Decompose its identity into minimal central projections. Each central summand is a simple finite-dimensional C^∗-algebra and therefore isomorphic to a full matrix algebra M_n_α( C). Its representation on the corresponding invariant subspace is a direct sum of m_α equivalent irreducible representations, yielding [source reference] and [source reference]. Minimal central projections are intrinsic to the center, giving uniqu…
Exact mathematical passage
C^\ast
C_+Use in source · 2 passages
Search aliases: C_+
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15198
[Finite conflict-grounded language synthesis] Let a finite development archive contain one inherited token cell C with at least two incompatible repair subclasses. Suppose: item a candidate probe p∈P^+ separates C into cells C_- and C_+; item there exist m_- ,m_+∈C_rm pair such that m_- is acceptable throughout C_- and m_+ is acceptable throughout C_+; item neither one inherited singleton method nor one unrefined binding is acceptable throughout C; item old-token behavior is preserved through th…
Exact mathematical passage
C_+
C_-Use in source · 2 passages
Search aliases: C_-
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15198
[Finite conflict-grounded language synthesis] Let a finite development archive contain one inherited token cell C with at least two incompatible repair subclasses. Suppose: item a candidate probe p∈P^+ separates C into cells C_- and C_+; item there exist m_- ,m_+∈C_rm pair such that m_- is acceptable throughout C_- and m_+ is acceptable throughout C_+; item neither one inherited singleton method nor one unrefined binding is acceptable throughout C; item old-token behavior is preserved through th…
Exact mathematical passage
C_-
c_0Use in source · 2 passages
Search aliases: c_0
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21376
Suppose every target exposes a no-op c_0 and one repair c_1. On half the episodes, the repair gain is 0.4; on the other half it is -0.2. A parent that always abstains has expected gain zero. A child that observes a receipt feature identifying the positive regime with probability 0.8 and the negative regime with probability 0.7 selects c_1 with those corresponding probabilities. Its expected gain is frac12(0.8)(0.4) + frac12(0.3)(-0.2) =0.13. The candidate set is unchanged. The child improves onl…
Exact mathematical passage
c_0
C_{1}Use in source · 1 passage
Search aliases: C_1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16109
efine z_0=ζ, z_k+1 = tanh (Rz_k+s_k+C_α(e,m)), 0≤ k<4, where s_k is a learned step embedding. The task-blind parent satisfies C_parent(e,m)=0. The descendants use one of the following conditioning maps: C_depth(e,m) · =D 1_m, C_1(e,m) · =U_1V_1e, rank(U_1V_1)≤1, C_4(e,m) · =U_4V_4e, rank(U_4V_4)≤4, C_full(e,m) · =We. Here 1_m is the one-hot depth code. Each child inherits every compatible parent parameter. The added up-projection or full-conditioning matrix is initialized so that the initial inj…
Exact mathematical passage
C_{\mathrm{depth}}(e,m)&=D\,\one_m, \label{eq:b17-depth-conditioning}\\ C_{1}(e,m)&=U_1V_1e, \qquad \rank(U_1V_1)\leq1, \label{eq:b17-rank1-conditioning}\\ C_{4}(e,m)&=U_4V_4e, \qquad \rank(U_4V_4)\leq4, \label{eq:b17-rank4-conditioning}\\ C_{\mathrm{full}}(e,m)&=We. \label{eq:b17-full-conditioning}
c_2Use in source · 5 passages
Search aliases: c_2
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17770
[Pooled bridge compatibility obstruction] Let V⊆ R^m be spanned by the canonical trajectories observed in two charts c_1,c_2. Suppose one linear map T exactly transports architecture b into architecture a in both charts: zG_b,c_iT=zG_a,c_i for every z∈ V, i∈{1,2}. Then the restrictions to V satisfy G_b,c_1T=G_a,c_1, G_b,c_2T=G_a,c_2. If G_b,c_i are invertible on V, exact pooled transport is possible only when G_b,c_1^-1G_a,c_1 = G_b,c_2^-1G_a,c_2 on V.
Exact mathematical passage
c_1,c_2
C_{4}Use in source · 1 passage
Search aliases: C_4
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16109
efine z_0=ζ, z_k+1 = tanh (Rz_k+s_k+C_α(e,m)), 0≤ k<4, where s_k is a learned step embedding. The task-blind parent satisfies C_parent(e,m)=0. The descendants use one of the following conditioning maps: C_depth(e,m) · =D 1_m, C_1(e,m) · =U_1V_1e, rank(U_1V_1)≤1, C_4(e,m) · =U_4V_4e, rank(U_4V_4)≤4, C_full(e,m) · =We. Here 1_m is the one-hot depth code. Each child inherits every compatible parent parameter. The added up-projection or full-conditioning matrix is initialized so that the initial inj…
Exact mathematical passage
C_{\mathrm{depth}}(e,m)&=D\,\one_m, \label{eq:b17-depth-conditioning}\\ C_{1}(e,m)&=U_1V_1e, \qquad \rank(U_1V_1)\leq1, \label{eq:b17-rank1-conditioning}\\ C_{4}(e,m)&=U_4V_4e, \qquad \rank(U_4V_4)\leq4, \label{eq:b17-rank4-conditioning}\\ C_{\mathrm{full}}(e,m)&=We. \label{eq:b17-full-conditioning}
C_AUse in source · 6 passages
Search aliases: C_A
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2832
Let protected linear invariants be represented by C_A:U_A→ Z_A, C_B:U_B→ Z_B. A correction preserves the invariant when C_Au=0 or C_Bu=0.
Exact mathematical passage
C_A:U_A\to Z_A, \qquad C_B:U_B\to Z_B.
c_aUse in source · 5 passages
Search aliases: c_a
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6657
Let U_a∈ℝ^d_a× r have orthonormal columns spanning the discovered packet. Define desired packet coordinates z_i = U_a^top(h_a^∘(x_i)-h_a(x_i)). Fit an affine predictor widehat z(h)=K_a h+c_a by ridge regression on source anchor states. The repair is R_a(h) = h+U_awidehat z(h) = h+D_a h+b_a, where D_a=U_aK_a, b_a=U_ac_a.
Exact mathematical passage
\widehat z(h)=K_a h+c_a
C_BUse in source · 5 passages
Search aliases: C_B
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2832
Let protected linear invariants be represented by C_A:U_A→ Z_A, C_B:U_B→ Z_B. A correction preserves the invariant when C_Au=0 or C_Bu=0.
Exact mathematical passage
C_A:U_A\to Z_A, \qquad C_B:U_B\to Z_B.
\mathfrak C_B^{\mathrm{nom}}Use in source · 2 passages
Search aliases: C_B^nom mathfrak mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13862
Let C_B^nom be the same envelope when integrity constraints are ignored.
Exact mathematical passage
\mathfrak C_B^{\mathrm{nom}}
C_{\mathrm{depth}}Use in source · 1 passage
Search aliases: C_depth mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16109
efine z_0=ζ, z_k+1 = tanh (Rz_k+s_k+C_α(e,m)), 0≤ k<4, where s_k is a learned step embedding. The task-blind parent satisfies C_parent(e,m)=0. The descendants use one of the following conditioning maps: C_depth(e,m) · =D 1_m, C_1(e,m) · =U_1V_1e, rank(U_1V_1)≤1, C_4(e,m) · =U_4V_4e, rank(U_4V_4)≤4, C_full(e,m) · =We. Here 1_m is the one-hot depth code. Each child inherits every compatible parent parameter. The added up-projection or full-conditioning matrix is initialized so that the initial inj…
Exact mathematical passage
C_{\mathrm{depth}}(e,m)&=D\,\one_m, \label{eq:b17-depth-conditioning}\\ C_{1}(e,m)&=U_1V_1e, \qquad \rank(U_1V_1)\leq1, \label{eq:b17-rank1-conditioning}\\ C_{4}(e,m)&=U_4V_4e, \qquad \rank(U_4V_4)\leq4, \label{eq:b17-rank4-conditioning}\\ C_{\mathrm{full}}(e,m)&=We. \label{eq:b17-full-conditioning}
\mathcal C_eUse in source · 1 passage
Search aliases: C_e mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7646
[Fixed-candidate structural invariance] Let C_e be the authorized candidate set for episode e. If parent and child differ only in the selector and C^(0)_e= C^(1)_e for every e, then the pointwise reachable correction set and its linear span are identical for parent and child.
Exact mathematical passage
\mathcal C_e
c_{e,0}Use in source · 3 passages
Search aliases: c_e,0
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7240
Let E be a space of target episodes. An episode e∈ E contains a target agent, a declared defect context, anchor observations, calibration observations, protected audit observations, and all provenance needed to reproduce those objects. Let C_e = (c_e,0,…,c_e,B-1) be an ordered candidate bundle of fixed size B. Candidate c_e,0 is the exact no-op. Every nontrivial candidate carries a receipt containing its construction route, rank or complexity, calibration behavior, transport diagnostics, packet …
Exact mathematical passage
\label{eq:block07-candidate-bundle} C_e = \left(c_{e,0},\ldots,c_{e,B-1}\right)
c_{e,B-1}Use in source · 2 passages
Search aliases: c_e,B-1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7240
Let E be a space of target episodes. An episode e∈ E contains a target agent, a declared defect context, anchor observations, calibration observations, protected audit observations, and all provenance needed to reproduce those objects. Let C_e = (c_e,0,…,c_e,B-1) be an ordered candidate bundle of fixed size B. Candidate c_e,0 is the exact no-op. Every nontrivial candidate carries a receipt containing its construction route, rank or complexity, calibration behavior, transport diagnostics, packet …
Exact mathematical passage
\label{eq:block07-candidate-bundle} C_e = \left(c_{e,0},\ldots,c_{e,B-1}\right)
c_{e,i_0}Use in source · 1 passage
Search aliases: c_e,i_0
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7575
[Pathwise selector isolation] For episode e, let parent and child receive the identical ordered bundle C_e and identical pre-audit receipt Z_e. Let i_0=K_0(Z_e), i_1=K_1(Z_e). Then the realized paired utility difference is exactly D_e = U_e(c_e,i_1)-U_e(c_e,i_0). If candidate construction, target state, and audit evaluation are frozen, no candidate-generation term remains in this difference.
Exact mathematical passage
D_e = U_e(c_{e,i_1})-U_e(c_{e,i_0}). \label{eq:block07-paired-difference}
c_{e,i_1}Use in source · 1 passage
Search aliases: c_e,i_1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7575
[Pathwise selector isolation] For episode e, let parent and child receive the identical ordered bundle C_e and identical pre-audit receipt Z_e. Let i_0=K_0(Z_e), i_1=K_1(Z_e). Then the realized paired utility difference is exactly D_e = U_e(c_e,i_1)-U_e(c_e,i_0). If candidate construction, target state, and audit evaluation are frozen, no candidate-generation term remains in this difference.
Exact mathematical passage
D_e = U_e(c_{e,i_1})-U_e(c_{e,i_0}). \label{eq:block07-paired-difference}
C_{\mathrm{full}}Use in source · 1 passage
Search aliases: C_full mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16109
efine z_0=ζ, z_k+1 = tanh (Rz_k+s_k+C_α(e,m)), 0≤ k<4, where s_k is a learned step embedding. The task-blind parent satisfies C_parent(e,m)=0. The descendants use one of the following conditioning maps: C_depth(e,m) · =D 1_m, C_1(e,m) · =U_1V_1e, rank(U_1V_1)≤1, C_4(e,m) · =U_4V_4e, rank(U_4V_4)≤4, C_full(e,m) · =We. Here 1_m is the one-hot depth code. Each child inherits every compatible parent parameter. The added up-projection or full-conditioning matrix is initialized so that the initial inj…
Exact mathematical passage
C_{\mathrm{depth}}(e,m)&=D\,\one_m, \label{eq:b17-depth-conditioning}\\ C_{1}(e,m)&=U_1V_1e, \qquad \rank(U_1V_1)\leq1, \label{eq:b17-rank1-conditioning}\\ C_{4}(e,m)&=U_4V_4e, \qquad \rank(U_4V_4)\leq4, \label{eq:b17-rank4-conditioning}\\ C_{\mathrm{full}}(e,m)&=We. \label{eq:b17-full-conditioning}
C_iUse in source · 2 passages
Search aliases: C_i
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13888
A simple acquisition rule is A_i = widehat g_i -λ_rwidehat r_i +λ_uwidehatσ_i +λ_h p_θ(h_imid x_τ) -λ_c C_i, where h_i is the method family, p_θ is an obstruction-class posterior, widehatσ_i is a declared uncertainty bonus, and C_i is experimental cost.
Exact mathematical passage
A_i = \widehat g_i -\lambda_r\widehat r_i +\lambda_u\widehat\sigma_i +\lambda_h p_\theta(h_i\mid x_\tau) -\lambda_c C_i, \label{eq:b13-acquisition}
c_jUse in source · 4 passages
Search aliases: c_j
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12201
Choose c∈ℝ^k outside the finite union of hyperplanes setc:c^ T(λ_a-λ_b)=0, a≠ b. Then the scalars μ_a=c^ Tλ_a are pairwise distinct. The combined response B=∑_j=1^kc_jR_j =∑_a=1^mμ_aP_a belongs to A_R. For each a, define the Lagrange polynomial p_a(z)=∏_b≠ afracz-μ_bμ_a-μ_b. Functional calculus gives p_a(B)=∑_b=1^mp_a(μ_b)P_b=P_a. Thus every P_a lies in the generated algebra. A nonzero central projection that is strictly below one P_a would have to be scalar on H_a and therefore is either 0 or P…
Exact mathematical passage
B=\sum_{j=1}^{k}c_jR_j =\sum_{a=1}^{m}\mu_aP_a
c_{m+1}Use in source · 1 passage
Search aliases: c_m+1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12964
Choose a generic linear combination G=∑_j=1^mc_jR_j+c_m+1S whose induced scalar values on the distinct joint signatures are pairwise distinct. For each block value γ_a, the Lagrange polynomial p_a(z)=∏_b≠ afracz-γ_bγ_a-γ_b satisfies p_a(G)=P_a, the orthogonal projector onto H_a. Therefore P_a,P_b∈ A(R^+) and belong to its center.
Exact mathematical passage
G=\sum_{j=1}^{m}c_jR_j+c_{m+1}S
C_{\mathrm{parent}}Use in source · 1 passage
Search aliases: C_parent mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16104
tau))∈ℝ^40 be the prompt representation. The latent trace has four states. Given conditioning mode α, define z_0=ζ, z_k+1 = tanh (Rz_k+s_k+C_α(e,m)), 0≤ k<4, where s_k is a learned step embedding. The task-blind parent satisfies C_parent(e,m)=0. The descendants use one of the following conditioning maps: C_depth(e,m) · =D 1_m,
Exact mathematical passage
C_{\mathrm{parent}}(e,m)=0. \label{eq:b17-parent-conditioning}
C_{\mathrm{RI}}Use in source · 6 passages
Search aliases: C_RI mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4259
Let C_RI(s) = · Grounded(s) wedgeProvValid(s) wedgeWorkspaceBound(s) · wedgeRepairTyped(s) wedgeWitnessed(s) wedgeEvidenceFlat(s) · wedgeRollbackable(s) wedgeMetaCapable(s). For X⊆ S, define Ψ_∀(X) = sets∈ S:C_RI(s) and Post_imp(s)⊆ X.
Exact mathematical passage
\begin{aligned} C_{\mathrm{RI}}(s) ={}& \Grounded(s) \wedge\ProvValid(s) \wedge\WorkspaceBound(s)\\ &\wedge\RepairTyped(s) \wedge\Witnessed(s) \wedge\EvidenceFlat(s)\\ &\wedge\Rollbackable(s) \wedge\MetaCapable(s). \end{aligned}
C_{\rm id}Use in source · 1 passage
Search aliases: C_rm id rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8720
Define the finite structural coverage functional C_rm id( P) = Pr_A∼D_rm for [ ank X_A=d_s]. Then widehat C_rm id( P_1) - widehat C_rm id( P_0) = 0.96875. This is the measured bridge-identification direction added by the meta-repair.
Exact mathematical passage
C_{\rm id}(\mathfrak P) = \Pr_{A\sim\Dcal_{\rm for}} \left[ ank X_A=d_s\right].
\mathcal C_{\rm safe}Use in source · 3 passages
Search aliases: C_rm safe mathcal rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7451
[Utility regret under simultaneous lower bounds] Let C_rm safe be the truly safe candidate set. Suppose every c∈ C_rm safe passes the implemented admissibility filter, and suppose 0≤ U_e(c)-widehat U_rm LCB(c)≤ϵ for every admissible c. If widehat c maximizes widehat U_rm LCB and c^* maximizes U_e over C_rm safe, then U_e(c^*)-U_e(widehat c)≤ϵ.
Exact mathematical passage
\mathcal C_{\rm safe}
c_sUse in source · 5 passages
Search aliases: c_s
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11466
Let the source repair be R_s(h)=M_sh+c_s. On chart j, let the target bridge be T_j(h)=A_jh+b_j, with A_j invertible on the repair-relevant subspace. Define R_t,j(y) = T_j (R_s(T_j^-1(y))). Expanding gives R_t,j(y) = A_jM_sA_j^-1(y-b_j)+A_jc_s+b_j.
Exact mathematical passage
R_s(h)=M_sh+c_s.
C_TUse in source · 3 passages
Search aliases: C_T
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10718
[Conditioned repair stability] Suppose T has full column rank and normT^†normwidehat T-T<1. Then there is a finite constant C_T, depending continuously on normT and normT^† inside this perturbation neighborhood, such that normwidehat D_t-D_t ≤ C_TnormD_snormwidehat T-T. If the protected utility readout is L-Lipschitz with respect to the transported repair operator on the declared evaluation set, the loss in repair gain is at most LC_TnormD_snormwidehat T-T.
Exact mathematical passage
C_T
C_tUse in source · 2 passages
Search aliases: C_t
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 811
A witness structure may be represented as W_t = ( V_t^fr, C_t, B_t^sand, R_t, Π_t^auth ), where V_t^fr is the frozen evaluator instance for the current transaction, C_t is a protected checkpoint, B_t^sand is the sandbox boundary, R_t is the rollback procedure, and Π_t^auth records effect authority.
Exact mathematical passage
\Wcal_t = \left( \Vcal_t^{\mathrm{fr}}, C_t, \Bcal_t^{\mathrm{sand}}, R_t, \Pi_t^{\mathrm{auth}} \right),
C_\alphaUse in source · 3 passages
Search aliases: C_α alpha
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16093
Let ϕ(τ)∈ℝ^19 be the padded task serialization, and let e=E_θ(ϕ(τ))∈ℝ^40 be the prompt representation. The latent trace has four states. Given conditioning mode α, define z_0=ζ, z_k+1 = tanh (Rz_k+s_k+C_α(e,m)), 0≤ k<4, where s_k is a learned step embedding. The task-blind parent satisfies C_parent(e,m)=0. The descendants use one of the following conditioning maps: C_depth(e,m) · =D 1_m,
Exact mathematical passage
z_0=\zeta, \qquad z_{k+1} = \tanh\!\left(Rz_k+s_k+C_\alpha(e,m)\right), \qquad 0\leq k<4, \label{eq:b17-trace}
C_\alpha'Use in source · 1 passage
Search aliases: C_α' alpha
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16497
Let P be a signed permutation matrix on the latent space. Define transformed latent parameters ζ'=Pζ, R'=PRP^ T, s_k'=Ps_k, C_α'=PC_α, and transform every latent-consuming linear map by right multiplication with P^ T. Then the transformed model has z_k'=Pz_k and identical external outputs.
Exact mathematical passage
\zeta'=P\zeta, \quad R'=PRP^{\mathsf T}, \quad s_k'=Ps_k, \quad C_\alpha'=PC_\alpha, \label{eq:b17-gauge-trace}
C_{\omega_i}Use in source · 3 passages
Search aliases: C_ω_i omega
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1277
[Compositional receipt soundness] Assume: item every primitive receipt is semantically sound; item execution of each ω_i from C_ω_i establishes Q_ω_i; item the conjunction of the inner postconditions and bridge guarantees implies C_ω; item execution of ω from C_ω establishes Q_ω; item receipt composition preserves provenance and protected evidence references. Then the composed receipt is sound for ω∘(ω_1,…,ω_n) with precondition bigwedge_iC_ω_i and postcondition Q_ω.
Exact mathematical passage
C_{\omega_i}
\mathrm{calibrate}Use in source · 1 passage
Search aliases: calibrate mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5414
The data partition is itself typed: D = D_construct sqcup D_calibrate sqcup D_select sqcup D_foreign sqcup D_audit. Construction data may be used to create schemas. Calibration data may set thresholds. Selection data may choose among frozen candidates. Foreign data estimate metaproductivity. Audit data are not available until the final claim object has been sealed.
Exact mathematical passage
\mathsf D = \mathsf D_{\mathrm{construct}} \sqcup \mathsf D_{\mathrm{calibrate}} \sqcup \mathsf D_{\mathrm{select}} \sqcup \mathsf D_{\mathrm{foreign}} \sqcup \mathsf D_{\mathrm{audit}}.
\mathrm{cand}Use in source · 2 passages
Search aliases: cand mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9771
Equation [source reference] gives a direct acquisition score. For a candidate family V_cand, choose v^⋆ ∈ arg max_v∈V_cand fracabsanglesz,P_I^⊥v^2 normP_I^⊥v^2+λ c(v)+βRisk(v), where c(v) is implementation cost and Risk(v) is a protected-risk estimate.
Exact mathematical passage
\Vcal_{\mathrm{cand}}
\mathrm{cause}Use in source · 1 passage
Search aliases: cause mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5527
G_cause · Causal role · Intervention, necessity ablation, matched negative controls
Exact mathematical passage
G_{\mathrm{cause}}
\Ccal_\tauUse in source · 1 passage
Search aliases: Ccal_τ C_τ Ccal tau mathcal tau
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13158
[Finite constructor selection with conservative tie-break] Suppose a finite constructor family is evaluated on fixed development evidence. Let G^⋆ be the largest held-out development gain and let C_τ={C:G(C)≥ G^⋆-τ}. Selecting the minimum-complexity constructor in C_τ is deterministic and preserves an explicit near-optimality bound of τ on the development criterion.
Exact mathematical passage
\Ccal_\tau
\Ccal_{\tau}Use in source · 1 passage
Search aliases: Ccal_τ C_τ Ccal tau mathcal tau
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13155
[Finite constructor selection with conservative tie-break] Suppose a finite constructor family is evaluated on fixed development evidence. Let G^⋆ be the largest held-out development gain and let C_τ={C:G(C)≥ G^⋆-τ}. Selecting the minimum-complexity constructor in C_τ is deterministic and preserves an explicit near-optimality bound of τ on the development criterion.
Exact mathematical passage
\Ccal_{\tau}=\{C:G(C)\geq G^\star-\tau\}.
\mathrm{check}Use in source · 1 passage
Search aliases: check mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1485
Consider four colors memory, representation, verifier, planner. A composite factuality repair may have the form ω_gate ∘ ( ω_retrieve, ω_decompose, ω_check ). The inner operations retrieve evidence, decompose output into claims, and check claims against protected sources. Their postconditions establish the precondition for a planner gate that blocks or revises unsupported claims. The composite receipt records source provenance, claim-to-evidence links, verifier uncertainty, revised spans, latenc…
Exact mathematical passage
\omega_{\mathrm{gate}} \circ \left( \omega_{\mathrm{retrieve}}, \omega_{\mathrm{decompose}}, \omega_{\mathrm{check}} \right).
\chooseUse in source · 1 passage
Search aliases: choose
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8506
If random anchor membership is Bernoulli with affected probability p, the idealized full-rank probability at k=16 is therefore Pr(rank X=16) = Pr(N_1≥8) = ∑_j=8^1616choose jp^j(1-p)^16-j under the generic model. The observed parent full-rank rate on the primary audit set is 0.03125, consistent with a low-probability affected-count event.
Exact mathematical passage
\Pr(\rank X=16) = \Pr(N_1\geq8) = \sum_{j=8}^{16}{16\choose j}p^j(1-p)^{16-j} \label{eq:block07q-random-rank-probability}
\mathrm{CI}Use in source · 21 passages
Search aliases: CI mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7947
The paired effects are widehatΔ_rm audit · =0.072518, · CI_95% · =[0.046745,0.101389], widehatΔ_rm arch · =0.104999, · CI_95% · =[0.072711,0.140506]. Both one-sided sign-flip p-values equal approximately 4.99975×10^-5 under the frozen 20,000-draw randomization procedure.
Exact mathematical passage
\widehat\Delta_{\rm audit} &=0.072518, &\mathrm{CI}_{95\%} &=[0.046745,0.101389], \label{eq:block07-audit-effect}\\ \widehat\Delta_{\rm arch} &=0.104999, &\mathrm{CI}_{95\%} &=[0.072711,0.140506]. \label{eq:block07-foreign-effect}
\mathrm{CI}_{0.95}Use in source · 3 passages
Search aliases: CI_0.95 mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16419
The paired child-minus-parent program-specificity gain is widehatΔ_prog^A =0.010714, CI_0.95 =[0.009911,0.011527], with positive fraction 0.823125. The answer-specificity gain is 0.017727. Its frozen interval is [0.016490,0.018960]. Parent-minus-child clean loss is 0.000979, with frozen interval [0.000695,0.001290].
Exact mathematical passage
\widehat\Delta_{\mathrm{prog}}^{A} =0.010714, \qquad \mathrm{CI}_{0.95} =[0.009911,0.011527], \label{eq:b17-primary-program-gain}
\mathrm{CI}_{95\%}Use in source · 17 passages
Search aliases: CI_95% mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7947
The paired effects are widehatΔ_rm audit · =0.072518, · CI_95% · =[0.046745,0.101389], widehatΔ_rm arch · =0.104999, · CI_95% · =[0.072711,0.140506]. Both one-sided sign-flip p-values equal approximately 4.99975×10^-5 under the frozen 20,000-draw randomization procedure.
Exact mathematical passage
\widehat\Delta_{\rm audit} &=0.072518, &\mathrm{CI}_{95\%} &=[0.046745,0.101389], \label{eq:block07-audit-effect}\\ \widehat\Delta_{\rm arch} &=0.104999, &\mathrm{CI}_{95\%} &=[0.072711,0.140506]. \label{eq:block07-foreign-effect}
\operatorname{Class}Use in source · 1 passage
Search aliases: Class operatorname
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5635
The executable classifier applies the ordered rule Class(r,b,H) = LOCAL_OBSTRUCTION, · normP_(im D)^⊥r>τ, GLUING_OBSTRUCTION, · normP_ker B^topb>τ, EVALUATOR_HOLONOMY, · norm(H-I)e>τ, FLAT_REPAIR, · otherwise. The order is part of the receipt because a case can exhibit more than one defect. The fixture isolates the four modes, so the ordered rule is unambiguous.
Exact mathematical passage
\operatorname{Class}(r,b,H) = \begin{cases} \texttt{LOCAL\_OBSTRUCTION},&\norm{P_{(\im D)^\perp}r}>\tau,\\ \texttt{GLUING\_OBSTRUCTION},&\norm{P_{\ker B^\top}b}>\tau,\\ \texttt{EVALUATOR\_HOLONOMY},&\norm{(H-I)e}>\tau,\\ \texttt{FLAT\_REPAIR},&\text{otherwise}. \end{cases}
\mathrm{clean}Use in source · 2 passages
Search aliases: clean mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16303
positive interval and fraction gates, local continuity, donor coverage, adaptive halting. Among eligible candidates it maximizes S(α) = · min{ fracR^2_probe0.65, fracQ_prog0.15, fracQ_ans0.20 } +0.20R^2_probe · -4 MSE_clean +0.02ρ_halt -Penalty_dynamic. The minimum term makes the weakest normalized semantic gate the principal bottleneck. The compiler seal commits the selected candidate, all candidate and parent state hashes, training receipts, meta-validation task digest, exact thresholds, selec…
Exact mathematical passage
\begin{split} S(\alpha) ={}& \min\left\{ \frac{R^2_{\mathrm{probe}}}{0.65}, \frac{Q_{\mathrm{prog}}}{0.15}, \frac{Q_{\mathrm{ans}}}{0.20} \right\} +0.20R^2_{\mathrm{probe}}\\ &-4\,\mathrm{MSE}_{\mathrm{clean}} +0.02\rho_{\mathrm{halt}} -\mathsf{Penalty}_{\mathrm{dynamic}}. \end{split} \label{eq:b17-selection-score}
\operatorname{clip}Use in source · 1 passage
Search aliases: clip operatorname
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11770
For current target state y, candidate repaired state widetilde y, and oracle repaired state y^⋆, define G = clip_[-1,1] ( 1- frac Enormwidetilde y-y^⋆^2 Enormy-y^⋆^2 ). The witness records: item validation gain; item source-to-target alignment NRMSE; item decoded protected-context regression; item minimum chart support; item alignment reduction relative to the global bridge; item source-gate and target-gate agreement; item target-gate margin.
Exact mathematical passage
G = \operatorname{clip}_{[-1,1]} \left( 1- \frac{\mathbb E\norm{\widetilde y-y^\star}^2} {\mathbb E\norm{y-y^\star}^2} \right). \label{eq:b10-gain}
\operatorname{clip}_{[-1,1]}Use in source · 1 passage
Search aliases: clip_[-1,1] operatorname
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11770
For current target state y, candidate repaired state widetilde y, and oracle repaired state y^⋆, define G = clip_[-1,1] ( 1- frac Enormwidetilde y-y^⋆^2 Enormy-y^⋆^2 ). The witness records: item validation gain; item source-to-target alignment NRMSE; item decoded protected-context regression; item minimum chart support; item alignment reduction relative to the global bridge; item source-gate and target-gate agreement; item target-gate margin.
Exact mathematical passage
G = \operatorname{clip}_{[-1,1]} \left( 1- \frac{\mathbb E\norm{\widetilde y-y^\star}^2} {\mathbb E\norm{y-y^\star}^2} \right). \label{eq:b10-gain}
\mathrm{code}_iUse in source · 2 passages
Search aliases: code_i mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13751
The forecast residual and proposal--code congruence residual are ε_forecast,i = normwidehat c_i- c_i, ε_spec/code,i = d_sem ( Sem(Π_i^-), Sem(code_i) ). In the executable finite companion, congruence is a Boolean monitor rather than a complete semantic metric.
Exact mathematical passage
\eps_{\mathrm{forecast},i} = \norm{\widehat{\mathbf c}_i-\mathbf c_i}, \qquad \eps_{\mathrm{spec/code},i} = d_{\mathrm{sem}} \left( \mathrm{Sem}(\Pi_i^-), \mathrm{Sem}(\mathrm{code}_i) \right). \label{eq:b13-receipt-residuals}
\mathrm{com}Use in source · 8 passages
Search aliases: com mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10120
For X∈End(H), define commutator energy E_com(X) = ∑_j=1^m ( normXR_j-R_jX_F^2 + normXR_j^∗-R_j^∗ X_F^2 ). There is a positive semidefinite linear operator L_com on the Hilbert space of matrices such that anglesX,L_comX_F = E_com(X). Its kernel is the commutant: kerL_com= A'.
Exact mathematical passage
\Ecal_{\mathrm{com}}(X) = \sum_{j=1}^{m} \left( \norm{XR_j-R_jX}_F^2 + \norm{XR_j^\ast-R_j^\ast X}_F^2 \right). \label{eq:b8-commutator-energy}
\operatorname{cone}Use in source · 1 passage
Search aliases: cone operatorname
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 22132
Open-set residual. Let inherited known responses span the nonnegative cone generated by e_1 and e_2 in ℝ^3. The unknown response y=e_3 has dist(y,cone{e_1,e_2})=1. Any threshold below one and above known-response noise flags the case as outside the inherited language. The flag does not identify the missing obstruction. It only prevents a false confident inherited label.
Exact mathematical passage
\dist(y,\operatorname{cone}\{e_1,e_2\})=1.
\mathrm{construct}Use in source · 1 passage
Search aliases: construct mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5414
The data partition is itself typed: D = D_construct sqcup D_calibrate sqcup D_select sqcup D_foreign sqcup D_audit. Construction data may be used to create schemas. Calibration data may set thresholds. Selection data may choose among frozen candidates. Foreign data estimate metaproductivity. Audit data are not available until the final claim object has been sealed.
Exact mathematical passage
\mathsf D = \mathsf D_{\mathrm{construct}} \sqcup \mathsf D_{\mathrm{calibrate}} \sqcup \mathsf D_{\mathrm{select}} \sqcup \mathsf D_{\mathrm{foreign}} \sqcup \mathsf D_{\mathrm{audit}}.
\mathrm{corr}Use in source · 2 passages
Search aliases: corr mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6170
For canonical patch u=T^-1δ, define corrected protected energy E_corr(d,u) = ∑_i=1^6w_imin{d_i^2,u_i^2}. This quantity is zero for a no-op and positive when a patch reaches task-relevant defect coordinates. It is reported alongside behavioral gain because a high apparent score can be generated without correcting protected defect energy.
Exact mathematical passage
E_{\mathrm{corr}}(d,u) = \sum_{i=1}^6w_i\min\{d_i^2,u_i^2\}.
\cosUse in source · 3 passages
Search aliases: cos
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6538
Let x=(x_1,x_2,x_3,x_4)∈ℝ^4. Define the eight-dimensional base feature ψ(x) = x_1 · x_2 · x_3 · x_4 · x_1x_2 · x_3x_4 · sin x_1 · cos x_2 ^ top ∈ℝ^8, and the affected-stratum indicator s(x)=1{x_1+x_2>0.8}. The full feature map is ϕ(x) = ψ(x)
Exact mathematical passage
\psi(x) = \begin{bmatrix} x_1 & x_2 & x_3 & x_4 & x_1x_2 & x_3x_4 & \sin x_1 & \cos x_2 \end{bmatrix}^{\!\top} \in\R^8,
\CovUse in source · 3 passages
Search aliases: Cov operatorname
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9924
Let each candidate feature f cover the conflict set Cov(f) = {S∈ H_G: f resolves S}. Choosing a minimum-cost feature set that resolves all conflicts is a weighted set-cover problem. An exact solver is possible in the finite companion. A scalable lineage may use the greedy score fracnewly resolved conflictsfeature cost+probe risk. The optimization target must remain subordinate to external repair validation. A feature that separates labels but does not improve authorized repair decisions is not a…
Exact mathematical passage
\Cov(f) = \{S\in\mathfrak H_{\Gcal}: f\text{ resolves }S\}.
\operatorname{CPre}_{\delta}Use in source · 1 passage
Search aliases: CPre_δ operatorname delta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4429
For stochastic transitions, define CPre_δ(X) = set s:∃ u∈Act(s), Pr(T(s,u,D)∈ Xmid s,u)≥1-δ . A one-step chance constraint does not by itself provide an infinite-horizon guarantee.
Exact mathematical passage
\operatorname{CPre}_{\delta}(X) = \set{ s:\exists u\in\mathsf{Act}(s),\ \Pr(T(s,u,D)\in X\mid s,u)\geq1-\delta }.
\mathrm{ctr}Use in source · 5 passages
Search aliases: ctr mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11527
[New chart-contrast correction direction] Let the affine target repairs have homogeneous matrices bar R_0 and bar R_1. If bar R_0≠bar R_1, then bar R_ctr = fracbar R_1-bar R_02 induces a nonzero anti-diagonal component (-bar R_ctr,bar R_ctr) orthogonal to every diagonal pair under the product Frobenius inner product. Therefore the chart grammar contains a correction direction unavailable to the parent grammar.
Exact mathematical passage
\bar R_{\mathrm{ctr}} = \frac{\bar R_1-\bar R_0}{2}
\mathrm{ctrl}Use in source · 13 passages
Search aliases: ctrl mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4322
Define Ψ_ctrl(X) = sets∈ S:C_RI(s) ∩ CPre(X).
Exact mathematical passage
\Psi_{\mathrm{ctrl}}(X) = \set{s\in S:C_{\mathrm{RI}}(s)} \cap \operatorname{CPre}(X).
\operatorname{CVaR}_{0.10}Use in source · 5 passages
Search aliases: CVaR_0.10 operatorname
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 14779
The primary paired results are: @lrrrr@ Compiler · Mean gain · Median · CVaR_0.10 · Invalid selected
Exact mathematical passage
\operatorname{CVaR}_{0.10}
D'Use in source · 1 passage
Search aliases: D'
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5057
Let j:V↪ V' be an injective evidence-conservative embedding of old defects into an expanded ontology. Let C=im D⊆ V, C'=im D'⊆ V', and assume preservation of old correctability: j(C)⊆ C'. The embedding induces a map overline j:V/C→ V'/C'.
Exact mathematical passage
C=\im D\subseteq V, \qquad C'=\im D'\subseteq V',
d'_{\alpha\beta}Use in source · 1 passage
Search aliases: d'_αβ alpha beta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2327
If u'_α=u_α+k_α with k_α∈ker D(U_α), then d'_αβ =d_αβ+k_α-k_β. Thus the cocycles differ by a coboundary.
Exact mathematical passage
d'_{\alpha\beta} =d_{\alpha\beta}+k_\alpha-k_\beta.
d^{(1)}Use in source · 1 passage
Search aliases: d^(1)
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2605
The theorem produces an obstruction ledger c|cc order · local obstruction · gluing obstruction 1 · [b_1,α]∈coker D_α · [d^(1)]∈check H^1(ker D) 2 · [b_2,α]∈coker D_α · [d^(2)]∈check H^1(ker D) vdots · vdots · vdots N · [b_N,α]∈coker D_α · [d^(N)]∈check H^1(ker D). Higher categorical repair interfaces may introduce additional coherence obstructions beyond this two-column ledger. Those are not proved here.
Exact mathematical passage
\begin{array}{c|cc} \text{order}&\text{local obstruction}&\text{gluing obstruction}\\ \hline 1&[b_{1,\alpha}]\in\coker D_\alpha&[d^{(1)}]\in\check H^1(\ker D)\\ 2&[b_{2,\alpha}]\in\coker D_\alpha&[d^{(2)}]\in\check H^1(\ker D)\\ \vdots&\vdots&\vdots\\ N&[b_{N,\alpha}]\in\coker D_\alpha&[d^{(N)}]\in\check H^1(\ker D). \end{array}
d^{(2)}Use in source · 1 passage
Search aliases: d^(2)
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2605
The theorem produces an obstruction ledger c|cc order · local obstruction · gluing obstruction 1 · [b_1,α]∈coker D_α · [d^(1)]∈check H^1(ker D) 2 · [b_2,α]∈coker D_α · [d^(2)]∈check H^1(ker D) vdots · vdots · vdots N · [b_N,α]∈coker D_α · [d^(N)]∈check H^1(ker D). Higher categorical repair interfaces may introduce additional coherence obstructions beyond this two-column ledger. Those are not proved here.
Exact mathematical passage
\begin{array}{c|cc} \text{order}&\text{local obstruction}&\text{gluing obstruction}\\ \hline 1&[b_{1,\alpha}]\in\coker D_\alpha&[d^{(1)}]\in\check H^1(\ker D)\\ 2&[b_{2,\alpha}]\in\coker D_\alpha&[d^{(2)}]\in\check H^1(\ker D)\\ \vdots&\vdots&\vdots\\ N&[b_{N,\alpha}]\in\coker D_\alpha&[d^{(N)}]\in\check H^1(\ker D). \end{array}
d^{(N)}Use in source · 1 passage
Search aliases: d^(N)
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2605
The theorem produces an obstruction ledger c|cc order · local obstruction · gluing obstruction 1 · [b_1,α]∈coker D_α · [d^(1)]∈check H^1(ker D) 2 · [b_2,α]∈coker D_α · [d^(2)]∈check H^1(ker D) vdots · vdots · vdots N · [b_N,α]∈coker D_α · [d^(N)]∈check H^1(ker D). Higher categorical repair interfaces may introduce additional coherence obstructions beyond this two-column ledger. Those are not proved here.
Exact mathematical passage
\begin{array}{c|cc} \text{order}&\text{local obstruction}&\text{gluing obstruction}\\ \hline 1&[b_{1,\alpha}]\in\coker D_\alpha&[d^{(1)}]\in\check H^1(\ker D)\\ 2&[b_{2,\alpha}]\in\coker D_\alpha&[d^{(2)}]\in\check H^1(\ker D)\\ \vdots&\vdots&\vdots\\ N&[b_{N,\alpha}]\in\coker D_\alpha&[d^{(N)}]\in\check H^1(\ker D). \end{array}
d^{(n)}_{\alpha\beta}Use in source · 1 passage
Search aliases: d^(n)_αβ alpha beta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2591
declared cover; item repairs have been constructed globally through order n-1. Then a global repair extends through order n if and only if: item every local coefficient class vanishes, [b_n,α]=0 in coker D_α; item for one, hence every, choice of local lifts, the overlap class vanishes, [d^(n)_αβ]=0 in check H^1( U,ker D). If both conditions hold for every 1≤ n≤ N, a global formal repair exists through order N.
Exact mathematical passage
[d^{(n)}_{\alpha\beta}]=0 \quad\text{in}\quad \check H^1(\mathfrak U,\ker D).
D^*Use in source · 12 passages
Search aliases: D^*
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1549
[Canonical correction--obstruction decomposition] For every r∈ V: item P_cor is the orthogonal projector onto im D; item P_ob is the orthogonal projector onto ker D^*; item the defect decomposes uniquely as r=r_cor+r_ob, r_cor=P_corr, r_ob=P_obr; item the minimum achievable residual is inf_u∈ U^authnormDu+r =normr_ob =dist(r,im D); item the complete set of least-squares minimizers is -D^† r+ker D; item if r is exactly correctable, u^*=-D^† r is the unique exact repair of minimum norm.
Exact mathematical passage
\ker D^*
D^{-1}Use in source · 2 passages
Search aliases: D^-1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20701
q · 0 , 0<q<1, so Q_0=I_2 and normEQ_0=q. Then D^-1=(I_2+E)^-1 =frac11-q^2 1 · -q -q · 1 . For r=e_1, the exact repair is u^*=-D^-1r =-frac11-q^2 1
Exact mathematical passage
D^{-1}=(I_2+E)^{-1} =\frac{1}{1-q^2} \begin{bmatrix} 1&-q\\-q&1 \end{bmatrix}.
D_+Use in source · 5 passages
Search aliases: D_+
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20510
D_-,D_+ · Chart-specific transported target repairs
Exact mathematical passage
D_-,D_+
D_-Use in source · 5 passages
Search aliases: D_-
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20510
D_-,D_+ · Chart-specific transported target repairs
Exact mathematical passage
D_-,D_+
D_1Use in source · 4 passages
Search aliases: D_1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4921
[Finite structural-improvement bound] Let V be a fixed n-dimensional defect space and suppose im D_0 subsetneq im D_1 subsetneq ⋯ is a strict ascending chain. Then the chain contains at most n-rank D_0 strict inclusions.
Exact mathematical passage
\im D_0 \subsetneq \im D_1 \subsetneq \cdots
d_{12}Use in source · 2 passages
Search aliases: d_12
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2467
[Locally repairable, globally incompatible] Let the nerve be a three-cycle with no triple intersection. Suppose the kernel fiber is ℝ and the oriented differences are d_12=1, d_23=1, d_31=1. If the cocycle were a coboundary, there would be k_1,k_2,k_3 with d_12=k_2-k_1, d_23=k_3-k_2, d_31=k_1-k_3. Summing gives 3=0, a contradiction. Hence no global correction exists. The graph-Hodge obstruction has norm sqrt3.
Exact mathematical passage
d_{12}=1, \qquad d_{23}=1, \qquad d_{31}=1.
D_2Use in source · 2 passages
Search aliases: D_2
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5580
Case F2: gluing obstruction. Take D_0=D_1=D_2=I_2, so every local defect is exactly correctable. On the oriented triangle with edges (0,1), (1,2), and (0,2), set the scalar overlap mismatch b= 0
Exact mathematical passage
D_0=D_1=D_2=I_2
d_{23}Use in source · 2 passages
Search aliases: d_23
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2467
[Locally repairable, globally incompatible] Let the nerve be a three-cycle with no triple intersection. Suppose the kernel fiber is ℝ and the oriented differences are d_12=1, d_23=1, d_31=1. If the cocycle were a coboundary, there would be k_1,k_2,k_3 with d_12=k_2-k_1, d_23=k_3-k_2, d_31=k_1-k_3. Summing gives 3=0, a contradiction. Hence no global correction exists. The graph-Hodge obstruction has norm sqrt3.
Exact mathematical passage
d_{12}=1, \qquad d_{23}=1, \qquad d_{31}=1.
D_3Use in source · 1 passage
Search aliases: D_3
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20923
Suppose a source correction has norm normu_0=2, every transport is nonexpansive, and three edges have (ε_D,0,ε_D,1,ε_D,2) =(0.01,0.02,0.015), (ε_r,0,ε_r,1,ε_r,2) =(0.005,0.004,0.006). Then [source reference] gives normD_3u_3+r_3 · ≤ 2(0.01+0.02+0.015) +(0.005+0.004+0.006) · =0.105. A direct source-to-target bridge with certified residual penalty below 0.105 is preferable under this metric, even if each edge of the three-step route appears individually accurate.
Exact mathematical passage
\begin{aligned} \norm{D_3u_3+r_3} &\leq 2(0.01+0.02+0.015) +(0.005+0.004+0.006)\\ &=0.105. \end{aligned}
d_{31}Use in source · 2 passages
Search aliases: d_31
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2467
[Locally repairable, globally incompatible] Let the nerve be a three-cycle with no triple intersection. Suppose the kernel fiber is ℝ and the oriented differences are d_12=1, d_23=1, d_31=1. If the cocycle were a coboundary, there would be k_1,k_2,k_3 with d_12=k_2-k_1, d_23=k_3-k_2, d_31=k_1-k_3. Summing gives 3=0, a contradiction. Hence no global correction exists. The graph-Hodge obstruction has norm sqrt3.
Exact mathematical passage
d_{12}=1, \qquad d_{23}=1, \qquad d_{31}=1.
D_aUse in source · 2 passages
Search aliases: D_a
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6661
Let U_a∈ℝ^d_a× r have orthonormal columns spanning the discovered packet. Define desired packet coordinates z_i = U_a^top(h_a^∘(x_i)-h_a(x_i)). Fit an affine predictor widehat z(h)=K_a h+c_a by ridge regression on source anchor states. The repair is R_a(h) = h+U_awidehat z(h) = h+D_a h+b_a, where D_a=U_aK_a, b_a=U_ac_a.
Exact mathematical passage
R_a(h) = h+U_a\widehat z(h) = h+D_a h+b_a, \label{eq:block06-source-repair}
d_aUse in source · 4 passages
Search aliases: d_a
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6580
An agent a has encoder E_a:ℝ^16→ℝ^d_a and an affine head fitted to widetilde q. Its hidden state is h_a(x)=E_a(ϕ(x)). The transport system is not given the matrix or nonlinear parameters of E_a.
Exact mathematical passage
E_a:\R^{16}\to\R^{d_a}
d_a^2Use in source · 2 passages
Search aliases: d_a^2
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12242
[Commutant characterization] Under the hypotheses of [source reference], A_R' = ⊕_a=1^mEnd(H_a). If d_a=dim H_a, then dim A_R'=∑_a=1^md_a^2.
Exact mathematical passage
\dim\mathfrak A_R'=\sum_{a=1}^{m}d_a^2. \label{eq:b11-commutant-dim}
D_A^{\dagger}Use in source · 5 passages
Search aliases: D_A^† dagger
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2868
[Naturality of the Moore--Penrose repair] Let U_A,U_B,V_A,V_B be finite-dimensional Hilbert spaces. Suppose T_U and T_V are unitary and D_B=T_VD_AT_U^*. Then D_B^†=T_UD_A^†T_V^*. For every r_B=T_Vr_A, the target minimum-norm least-squares correction is the transport of the source minimum-norm least-squares correction: -D_B^†r_B = T_U(-D_A^†r_A). The orthogonal obstruction projectors satisfy I-D_BD_B^† = T_V(I-D_AD_A^†)T_V^*.
Exact mathematical passage
D_B^{\dagger}=T_UD_A^{\dagger}T_V^*.
d_{\mathrm{ans}}Use in source · 1 passage
Search aliases: d_ans mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16173
Let d_prog(τ,τ') be a normalized distance between (A_m,B_m) and (A'_m',B'_m'), and let d_ans(τ,τ') = absy_m/4-y'_m'/4. For a recipient task τ_i, define four donor classes: item SAP: same-answer, far-program, approximately depth-matched donors; item FAD: far-answer, depth-matched donors; item SN: semantic-neighbor donors; item RMD: random depth-matched donors. Three donors per available class are selected by deterministic task-identifier hashing. Each audit records donor coverage and the realized…
Exact mathematical passage
d_{\mathrm{ans}}(\tau,\tau') = \abs{y_m/4-y'_{m'}/4}. \label{eq:b17-answer-distance}
\mathsf D_{\mathrm{audit}}Use in source · 1 passage
Search aliases: D_audit mathsf mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5414
The data partition is itself typed: D = D_construct sqcup D_calibrate sqcup D_select sqcup D_foreign sqcup D_audit. Construction data may be used to create schemas. Calibration data may set thresholds. Selection data may choose among frozen candidates. Foreign data estimate metaproductivity. Audit data are not available until the final claim object has been sealed.
Exact mathematical passage
\mathsf D = \mathsf D_{\mathrm{construct}} \sqcup \mathsf D_{\mathrm{calibrate}} \sqcup \mathsf D_{\mathrm{select}} \sqcup \mathsf D_{\mathrm{foreign}} \sqcup \mathsf D_{\mathrm{audit}}.
D_{\mathrm{avg}}Use in source · 3 passages
Search aliases: D_avg mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21877
Equip End(H_t)⊕End(H_t) with the product Frobenius norm. For a chart repair pair (D_-,D_+), write D_avg = fracD_-+D_+2, D_ctr = fracD_+-D_-2. The closest diagonal repair is (D_avg,D_avg). Therefore dist^2 ( (D_-,D_+),D_diag ) · = normD_--D_avg_F^2 + normD_+-D_avg_F^2
Exact mathematical passage
D_{\mathrm{avg}} = \frac{D_-+D_+}{2}, \qquad D_{\mathrm{ctr}} = \frac{D_+-D_-}{2}.
d_BUse in source · 2 passages
Search aliases: d_B
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3271
[Coupled distributional gain transfer] Suppose there is a coupling π of P_A and P_B such that E_(x,y)∼πd_B(y,Sx)≤δ. Assume q_B is K-Lipschitz and q_B(Sx)≥ q_A(x)-η for P_A-almost every x. Then E_y∼ P_Bq_B(y) ≥ E_x∼ P_Aq_A(x)-η-Kδ.
Exact mathematical passage
\mathbb E_{(x,y)\sim\pi}d_B(y,Sx)\leq\delta.
D_B^{\dagger}Use in source · 4 passages
Search aliases: D_B^† dagger
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2868
[Naturality of the Moore--Penrose repair] Let U_A,U_B,V_A,V_B be finite-dimensional Hilbert spaces. Suppose T_U and T_V are unitary and D_B=T_VD_AT_U^*. Then D_B^†=T_UD_A^†T_V^*. For every r_B=T_Vr_A, the target minimum-norm least-squares correction is the transport of the source minimum-norm least-squares correction: -D_B^†r_B = T_U(-D_A^†r_A). The orthogonal obstruction projectors satisfy I-D_BD_B^† = T_V(I-D_AD_A^†)T_V^*.
Exact mathematical passage
D_B^{\dagger}=T_UD_A^{\dagger}T_V^*.
D_CUse in source · 4 passages
Search aliases: D_C
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1637
[Invariant-constrained obstruction] Let D_C:=restrDker C. An exact first-order repair preserving the invariant exists if and only if [r]=0 in coker D_C. Equivalently, the stacked system D
Exact mathematical passage
D_C:=\restr{D}{\ker C}
\mathsf D_{\mathrm{calibrate}}Use in source · 1 passage
Search aliases: D_calibrate mathsf mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5414
The data partition is itself typed: D = D_construct sqcup D_calibrate sqcup D_select sqcup D_foreign sqcup D_audit. Construction data may be used to create schemas. Calibration data may set thresholds. Selection data may choose among frozen candidates. Foreign data estimate metaproductivity. Audit data are not available until the final claim object has been sealed.
Exact mathematical passage
\mathsf D = \mathsf D_{\mathrm{construct}} \sqcup \mathsf D_{\mathrm{calibrate}} \sqcup \mathsf D_{\mathrm{select}} \sqcup \mathsf D_{\mathrm{foreign}} \sqcup \mathsf D_{\mathrm{audit}}.
\mathsf D_{\mathrm{construct}}Use in source · 1 passage
Search aliases: D_construct mathsf mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5414
The data partition is itself typed: D = D_construct sqcup D_calibrate sqcup D_select sqcup D_foreign sqcup D_audit. Construction data may be used to create schemas. Calibration data may set thresholds. Selection data may choose among frozen candidates. Foreign data estimate metaproductivity. Audit data are not available until the final claim object has been sealed.
Exact mathematical passage
\mathsf D = \mathsf D_{\mathrm{construct}} \sqcup \mathsf D_{\mathrm{calibrate}} \sqcup \mathsf D_{\mathrm{select}} \sqcup \mathsf D_{\mathrm{foreign}} \sqcup \mathsf D_{\mathrm{audit}}.
D_{\mathrm{ctr}}Use in source · 2 passages
Search aliases: D_ctr mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21877
Equip End(H_t)⊕End(H_t) with the product Frobenius norm. For a chart repair pair (D_-,D_+), write D_avg = fracD_-+D_+2, D_ctr = fracD_+-D_-2. The closest diagonal repair is (D_avg,D_avg). Therefore dist^2 ( (D_-,D_+),D_diag ) · = normD_--D_avg_F^2 + normD_+-D_avg_F^2
Exact mathematical passage
D_{\mathrm{avg}} = \frac{D_-+D_+}{2}, \qquad D_{\mathrm{ctr}} = \frac{D_+-D_-}{2}.
D_eUse in source · 1 passage
Search aliases: D_e
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7575
[Pathwise selector isolation] For episode e, let parent and child receive the identical ordered bundle C_e and identical pre-audit receipt Z_e. Let i_0=K_0(Z_e), i_1=K_1(Z_e). Then the realized paired utility difference is exactly D_e = U_e(c_e,i_1)-U_e(c_e,i_0). If candidate construction, target state, and audit evaluation are frozen, no candidate-generation term remains in this difference.
Exact mathematical passage
D_e = U_e(c_{e,i_1})-U_e(c_{e,i_0}). \label{eq:block07-paired-difference}
D_{\mathrm{flat}}Use in source · 1 passage
Search aliases: D_flat mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5602
Case F3: flat repair. Set D_flat=I_2, r_flat= 1 -2 , with zero overlap mismatch and identity evaluator transport. The unique minimum-norm repair is u_flat=-r_flat.
Exact mathematical passage
D_{\mathrm{flat}}=I_2, \qquad r_{\mathrm{flat}}= \begin{bmatrix} 1\\-2 \end{bmatrix},
\mathsf D_{\mathrm{foreign}}Use in source · 1 passage
Search aliases: D_foreign mathsf mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5414
The data partition is itself typed: D = D_construct sqcup D_calibrate sqcup D_select sqcup D_foreign sqcup D_audit. Construction data may be used to create schemas. Calibration data may set thresholds. Selection data may choose among frozen candidates. Foreign data estimate metaproductivity. Audit data are not available until the final claim object has been sealed.
Exact mathematical passage
\mathsf D = \mathsf D_{\mathrm{construct}} \sqcup \mathsf D_{\mathrm{calibrate}} \sqcup \mathsf D_{\mathrm{select}} \sqcup \mathsf D_{\mathrm{foreign}} \sqcup \mathsf D_{\mathrm{audit}}.
d_i^2Use in source · 1 passage
Search aliases: d_i^2
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6170
For canonical patch u=T^-1δ, define corrected protected energy E_corr(d,u) = ∑_i=1^6w_imin{d_i^2,u_i^2}. This quantity is zero for a no-op and positive when a patch reaches task-relevant defect coordinates. It is reported alongside behavioral gain because a high apparent score can be generated without correcting protected defect energy.
Exact mathematical passage
E_{\mathrm{corr}}(d,u) = \sum_{i=1}^6w_i\min\{d_i^2,u_i^2\}.
D_jUse in source · 3 passages
Search aliases: D_j
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7696
For n held-out episodes, define paired differences D_j = G_e_j(K_1)-G_e_j(K_0), widehatΔ_n=frac1n∑_j=1^nD_j. Because parent and child share each target and candidate bundle, pairing removes much of the episode difficulty variation.
Exact mathematical passage
D_j = G_{e_j}(K_1)-G_{e_j}(K_0), \qquad \widehat\Delta_n=\frac1n\sum_{j=1}^nD_j. \label{eq:block07-paired-estimator}
D_kUse in source · 1 passage
Search aliases: D_k
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2941
[Pathwise transport bound] Consider a path of systems A_0xrightarrow T_0A_1 xrightarrow T_1⋯ xrightarrow T_n-1A_n. Write T_U,k:U_k→ U_k+1 and T_V,k:V_k→ V_k+1. Let normD_k+1T_U,k-T_V,kD_k≤ε_D,k and normr_k+1-T_V,kr_k≤ε_r,k. If D_0u_0=-r_0 and u_n=T_U,n-1⋯ T_U,0u_0, then normD_nu_n+r_n · ≤ ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_D,k normT_U,k-1⋯ T_U,0u_0
Exact mathematical passage
\norm{D_{k+1}T_{U,k}-T_{V,k}D_k}\leq\eps_{D,k}
D_{k+1}Use in source · 1 passage
Search aliases: D_k+1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2941
[Pathwise transport bound] Consider a path of systems A_0xrightarrow T_0A_1 xrightarrow T_1⋯ xrightarrow T_n-1A_n. Write T_U,k:U_k→ U_k+1 and T_V,k:V_k→ V_k+1. Let normD_k+1T_U,k-T_V,kD_k≤ε_D,k and normr_k+1-T_V,kr_k≤ε_r,k. If D_0u_0=-r_0 and u_n=T_U,n-1⋯ T_U,0u_0, then normD_nu_n+r_n · ≤ ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_D,k normT_U,k-1⋯ T_U,0u_0
Exact mathematical passage
\norm{D_{k+1}T_{U,k}-T_{V,k}D_k}\leq\eps_{D,k}
D_{\mathrm{loc}}Use in source · 2 passages
Search aliases: D_loc mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5557
Case F1: local obstruction. Set D_loc = 1 · 0 0 · 0 , r_loc = 0 1 . Then P_im D_locr_loc=0, (I-P_im D_loc)r_loc=r_loc, so the obstruction norm is one.
Exact mathematical passage
D_{\mathrm{loc}} = \begin{bmatrix} 1&0\\ 0&0 \end{bmatrix}, \qquad r_{\mathrm{loc}} = \begin{bmatrix} 0\\1 \end{bmatrix}.
D_nUse in source · 3 passages
Search aliases: D_n
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2953
T_1⋯ xrightarrow T_n-1A_n. Write T_U,k:U_k→ U_k+1 and T_V,k:V_k→ V_k+1. Let normD_k+1T_U,k-T_V,kD_k≤ε_D,k and normr_k+1-T_V,kr_k≤ε_r,k. If D_0u_0=-r_0 and u_n=T_U,n-1⋯ T_U,0u_0, then normD_nu_n+r_n · ≤ ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_D,k normT_U,k-1⋯ T_U,0u_0 · + ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_r,k, where empty products are identities.
Exact mathematical passage
\begin{aligned} \norm{D_nu_n+r_n} &\leq \sum_{k=0}^{n-1} \norm{T_{V,n-1}\cdots T_{V,k+1}} \eps_{D,k} \norm{T_{U,k-1}\cdots T_{U,0}u_0}\\ &\quad+ \sum_{k=0}^{n-1} \norm{T_{V,n-1}\cdots T_{V,k+1}} \eps_{r,k}, \end{aligned}
d_{\mathrm{prog}}Use in source · 1 passage
Search aliases: d_prog mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16172
Let d_prog(τ,τ') be a normalized distance between (A_m,B_m) and (A'_m',B'_m'), and let d_ans(τ,τ') = absy_m/4-y'_m'/4. For a recipient task τ_i, define four donor classes: item SAP: same-answer, far-program, approximately depth-matched donors; item FAD: far-answer, depth-matched donors; item SN: semantic-neighbor donors; item RMD: random depth-matched donors. Three donors per available class are selected by deterministic task-identifier hashing. Each audit records donor coverage and the realized…
Exact mathematical passage
d_{\mathrm{prog}}(\tau,\tau')
\mathsf D_{\rm anchor}Use in source · 2 passages
Search aliases: D_rm anchor mathsf rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6496
The controlled suite uses four disjoint data sets: D = D_rm train sqcup D_rm anchor sqcup D_rm calibrate sqcup D_rm audit. The source and target heads are fitted on D_rm train. Packet discovery and source repair fitting use D_rm anchor. The witness sees only D_rm calibrate. Reported final accuracies use D_rm audit.
Exact mathematical passage
\mathsf D = \mathsf D_{\rm train} \sqcup \mathsf D_{\rm anchor} \sqcup \mathsf D_{\rm calibrate} \sqcup \mathsf D_{\rm audit}.
\mathsf D_{\rm arch}Use in source · 2 passages
Search aliases: D_rm arch mathsf rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7292
The experiment uses four disjoint episode collections: D_rm dev sqcup D_rm meta sqcup D_rm audit sqcup D_rm arch. Development receipts in D_rm dev train the child utility and risk heads. The meta-validation set D_rm meta selects one finite hyperparameter configuration. The seed-and-task audit D_rm audit evaluates new episodes from the development architecture-family support. The architecture-foreign audit D_rm arch evaluates representation families absent from both development and meta-validatio…
Exact mathematical passage
\mathsf D_{\rm dev} \sqcup \mathsf D_{\rm meta} \sqcup \mathsf D_{\rm audit} \sqcup \mathsf D_{\rm arch}. \label{eq:block07-splits}
\mathsf D_{\rm audit}Use in source · 4 passages
Search aliases: D_rm audit mathsf rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6496
The controlled suite uses four disjoint data sets: D = D_rm train sqcup D_rm anchor sqcup D_rm calibrate sqcup D_rm audit. The source and target heads are fitted on D_rm train. Packet discovery and source repair fitting use D_rm anchor. The witness sees only D_rm calibrate. Reported final accuracies use D_rm audit.
Exact mathematical passage
\mathsf D = \mathsf D_{\rm train} \sqcup \mathsf D_{\rm anchor} \sqcup \mathsf D_{\rm calibrate} \sqcup \mathsf D_{\rm audit}.
\mathsf D_{\rm calibrate}Use in source · 2 passages
Search aliases: D_rm calibrate mathsf rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6496
The controlled suite uses four disjoint data sets: D = D_rm train sqcup D_rm anchor sqcup D_rm calibrate sqcup D_rm audit. The source and target heads are fitted on D_rm train. Packet discovery and source repair fitting use D_rm anchor. The witness sees only D_rm calibrate. Reported final accuracies use D_rm audit.
Exact mathematical passage
\mathsf D = \mathsf D_{\rm train} \sqcup \mathsf D_{\rm anchor} \sqcup \mathsf D_{\rm calibrate} \sqcup \mathsf D_{\rm audit}.
\mathsf D_{\rm train}Use in source · 2 passages
Search aliases: D_rm train mathsf rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6496
The controlled suite uses four disjoint data sets: D = D_rm train sqcup D_rm anchor sqcup D_rm calibrate sqcup D_rm audit. The source and target heads are fitted on D_rm train. Packet discovery and source repair fitting use D_rm anchor. The witness sees only D_rm calibrate. Reported final accuracies use D_rm audit.
Exact mathematical passage
\mathsf D = \mathsf D_{\rm train} \sqcup \mathsf D_{\rm anchor} \sqcup \mathsf D_{\rm calibrate} \sqcup \mathsf D_{\rm audit}.
D_sUse in source · 20 passages
Search aliases: D_s
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6768
Given source repair R_s(h)=h+D_sh+b_s, its transported target form is D_t = widehat T D_swidehat T^+, b_t = widehat T b_s, with transported packet basis obtained by orthogonalizing widehat T U_s.
Exact mathematical passage
R_s(h)=h+D_sh+b_s,
d_sUse in source · 4 passages
Search aliases: d_s
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6788
[Exact transport through an injective hidden bridge] Let T:ℝ^d_s→ℝ^d_t have full column rank, so T^+T=I_d_s. Suppose h_t=Th_s throughout the evaluated support. Define D_t and b_t by [source reference] with widehat T=T. Then R_t(Th_s)=TR_s(h_s).
Exact mathematical passage
T:\R^{d_s}\to\R^{d_t}
\mathsf D_{\mathrm{select}}Use in source · 1 passage
Search aliases: D_select mathsf mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5414
The data partition is itself typed: D = D_construct sqcup D_calibrate sqcup D_select sqcup D_foreign sqcup D_audit. Construction data may be used to create schemas. Calibration data may set thresholds. Selection data may choose among frozen candidates. Foreign data estimate metaproductivity. Audit data are not available until the final claim object has been sealed.
Exact mathematical passage
\mathsf D = \mathsf D_{\mathrm{construct}} \sqcup \mathsf D_{\mathrm{calibrate}} \sqcup \mathsf D_{\mathrm{select}} \sqcup \mathsf D_{\mathrm{foreign}} \sqcup \mathsf D_{\mathrm{audit}}.
d_{\mathrm{sem}}Use in source · 1 passage
Search aliases: d_sem mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13751
The forecast residual and proposal--code congruence residual are ε_forecast,i = normwidehat c_i- c_i, ε_spec/code,i = d_sem ( Sem(Π_i^-), Sem(code_i) ). In the executable finite companion, congruence is a Boolean monitor rather than a complete semantic metric.
Exact mathematical passage
\eps_{\mathrm{forecast},i} = \norm{\widehat{\mathbf c}_i-\mathbf c_i}, \qquad \eps_{\mathrm{spec/code},i} = d_{\mathrm{sem}} \left( \mathrm{Sem}(\Pi_i^-), \mathrm{Sem}(\mathrm{code}_i) \right). \label{eq:b13-receipt-residuals}
d_tUse in source · 6 passages
Search aliases: d_t
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6788
[Exact transport through an injective hidden bridge] Let T:ℝ^d_s→ℝ^d_t have full column rank, so T^+T=I_d_s. Suppose h_t=Th_s throughout the evaluated support. Define D_t and b_t by [source reference] with widehat T=T. Then R_t(Th_s)=TR_s(h_s).
Exact mathematical passage
T:\R^{d_s}\to\R^{d_t}
D_t^{\Auth}Use in source · 1 passage
Search aliases: D_t^Auth D_t^auth Auth mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4770
Let V be a Hilbert defect space, let L_t=im D_t^auth, and let P_t be the orthogonal projector onto L_t. For a defect distribution μ, define the correctable energy E_t^cor(μ) = E_r∼μnormP_t r^2 and the obstruction energy E_t^ob(μ) = E_r∼μnorm(I-P_t)r^2.
Exact mathematical passage
L_t=\im D_t^{\Auth}
D_{\mathrm{twin}}Use in source · 2 passages
Search aliases: D_twin mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 19848
For a candidate quotient q_θ, the empirical twin discrepancy is D_twin(q_θ) = E fracnormq_θ(x)-q_θ(x') max{normq_θ(x),ε}. A candidate is construction-valid only when it jointly predicts future consequences and intervention effects, predicts the externally scored answer, and keeps [source reference] below the declared ceiling.
Exact mathematical passage
D_{\mathrm{twin}}(q_\theta) = \mathbb E \frac{\norm{q_\theta(x)-q_\theta(x')}} {\max\{\norm{q_\theta(x)},\eps\}}. \label{eq:b23-twin-drift}
D_uUse in source · 2 passages
Search aliases: D_u
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 19608
The executable fixture uses five commuting symmetric response operators. In latent semantic coordinates they are diagonal: D_u=diag(λ_u1,…,λ_ud), and each system exposes an orthogonal gauge Q_a: R_u^a=Q_aD_uQ_a^top. The joint response signature of semantic coordinate j is λ_j=(λ_1j,…,λ_mj).
Exact mathematical passage
D_u=\operatorname{diag}(\lambda_{u1},\ldots,\lambda_{ud}),
D_xUse in source · 2 passages
Search aliases: D_x
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9952
In the linear case, F_x(u)=r_x+D_xu, and the canonical minimum-norm repair is s(x)=-D_x^† r_x whenever r_x∈im D_x.
Exact mathematical passage
F_x(u)=r_x+D_xu,
D_x^\daggerUse in source · 1 passage
Search aliases: D_x^† dagger
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9956
In the linear case, F_x(u)=r_x+D_xu, and the canonical minimum-norm repair is s(x)=-D_x^† r_x whenever r_x∈im D_x.
Exact mathematical passage
s(x)=-D_x^\dagger r_x
d_YUse in source · 1 passage
Search aliases: d_Y
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3652
[Readout stability under semantic holonomy] Let J:E_v→ Y be L_J-Lipschitz. Then d_Y(J(H_γ x),J(x)) ≤ L_Jη_sem(x;γ). If J returns a score vector and the winning score at x has margin m(x) over the runner-up, then the decision is unchanged whenever 2L_Jη_sem(x;γ)<m(x), where L_J bounds each score coordinate in the same norm.
Exact mathematical passage
d_Y(J(H_\gamma x),J(x)) \leq L_J\eta_{\mathrm{sem}}(x;\gamma).
d_{\beta\gamma}Use in source · 1 passage
Search aliases: d_βγ beta gamma
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2298
On every overlap, D(d_αβ) =-r|_U_αβ+r|_U_αβ=0. On triple overlaps, d_αβ+d_βγ+d_γα=0, so the family is a cocycle.
Exact mathematical passage
d_{\alpha\beta}+d_{\beta\gamma}+d_{\gamma\alpha}=0,
d_{\gamma\alpha}Use in source · 1 passage
Search aliases: d_γα gamma alpha
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2298
On every overlap, D(d_αβ) =-r|_U_αβ+r|_U_αβ=0. On triple overlaps, d_αβ+d_βγ+d_γα=0, so the family is a cocycle.
Exact mathematical passage
d_{\alpha\beta}+d_{\beta\gamma}+d_{\gamma\alpha}=0,
\dashrightarrowUse in source · 1 passage
Search aliases: dashrightarrow
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 697
Heuristic correspondence · Xdashrightarrow Y · Candidate analogy with no theorem transport until separately certified.
Exact mathematical passage
X\dashrightarrow Y
\Dcal'Use in source · 5 passages
Search aliases: Dcal' D' Dcal mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1081
[Coverage monotonicity] If D⊆D', then ε_D'(ϕ;A) ≥ ε_D(ϕ;A).
Exact mathematical passage
\Dcal\subseteq\Dcal'
\Dcal_{\mathrm{chart}}Use in source · 1 passage
Search aliases: Dcal_chart D_chart Dcal mathrm mathcal mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11514
The inherited global target-repair response lies in the diagonal class D_diag = set(R,R):R∈Aff(H_t), whereas the chart grammar reaches D_chart = Aff(H_t)×Aff(H_t).
Exact mathematical passage
\Dcal_{\mathrm{chart}} = \operatorname{Aff}(H_t)\times\operatorname{Aff}(H_t).
\Dcal_{\mathrm{diag}}Use in source · 2 passages
Search aliases: Dcal_diag D_diag Dcal mathrm mathcal mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11508
The inherited global target-repair response lies in the diagonal class D_diag = set(R,R):R∈Aff(H_t), whereas the chart grammar reaches D_chart = Aff(H_t)×Aff(H_t).
Exact mathematical passage
\Dcal_{\mathrm{diag}} = \set{(R,R):R\in\operatorname{Aff}(H_t)},
\Dcal_{\mathrm{foreign}}Use in source · 1 passage
Search aliases: Dcal_foreign D_foreign Dcal mathrm mathcal mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13848
Let U_audit be a protected audit utility unavailable during compiler allocation and final scored selection. Define compiler metaproductivity at research budget B by C_B(Γ) = E_τ∼D_foreign [ U_audit (K_Γ^⋆(τ)) - U_audit(K_τ) ]. Only integrity-valid and capability-passing outputs are eligible.
Exact mathematical passage
\mathfrak C_B(\Gamma) = \mathbb E_{\tau\sim\Dcal_{\mathrm{foreign}}} \left[ U_{\mathrm{audit}} \left(K_{\Gamma}^{\star}(\tau)\right) - U_{\mathrm{audit}}(K_\tau) \right]. \label{eq:b13-compiler-meta}
\Dcal_{\mathrm{hold}}Use in source · 2 passages
Search aliases: Dcal_hold D_hold Dcal mathrm mathcal mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4625
Let D_hold be a distribution over foreign held-out agents that are not descendants of the evaluated lineage. Let K be a frozen improvement mechanism. Under horizon h and budget B, define M^foreign_h,B(K) = E_A∼D_hold [ max_A'∈Desc_h,B(K,A) J(A')-J(A) ].
Exact mathematical passage
\Dcal_{\mathrm{hold}}
\Dcal_{\mathrm{parent}}Use in source · 1 passage
Search aliases: Dcal_parent D_parent Dcal mathrm mathcal mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9484
Define compiler metaproductivity by C(Γ) = E_K∼D_parent [ M(Γ(K,E_K))-M(K) ]. A stronger recursive claim requires C(Γ_t+1)>C(Γ_t) on sealed foreign parent processes under matched compilation resources.
Exact mathematical passage
\Ccal(\Gamma) = \mathbb E_{K\sim\Dcal_{\mathrm{parent}}} \left[ \Mcal\bigl(\Gamma(K,\Ecal_K)\bigr)-\Mcal(K) \right]. \label{eq:b8-compiler-metaproductivity}
\ddUse in source · 2 passages
Search aliases: dd d mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4662
[Expected best-of-k gain] Let G_1,…,G_k be independent nonnegative candidate gains with common cumulative distribution function F and finite mean. Then Emax_1≤ i≤ kG_i = ∫_0^∞ (1-F(x)^k) d x. In particular, expected best gain is nondecreasing in k even when the candidate generator itself does not improve.
Exact mathematical passage
\mathbb E\max_{1\leq i\leq k}G_i = \int_0^{\infty} \left(1-F(x)^k\right)\dd x.
\mathrm{decide}Use in source · 1 passage
Search aliases: decide mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4027
The transaction follows six phases: prepare ⟶ seal ⟶ execute ⟶ reveal ⟶ decide ⟶ finalize. Only the final phase may expose buffered effects or replace the active checkpoint.
Exact mathematical passage
\mathrm{prepare} \longrightarrow \mathrm{seal} \longrightarrow \mathrm{execute} \longrightarrow \mathrm{reveal} \longrightarrow \mathrm{decide} \longrightarrow \mathrm{finalize}.
\mathrm{decompose}Use in source · 1 passage
Search aliases: decompose mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1485
Consider four colors memory, representation, verifier, planner. A composite factuality repair may have the form ω_gate ∘ ( ω_retrieve, ω_decompose, ω_check ). The inner operations retrieve evidence, decompose output into claims, and check claims against protected sources. Their postconditions establish the precondition for a planner gate that blocks or revises unsupported claims. The composite receipt records source provenance, claim-to-evidence links, verifier uncertainty, revised spans, latenc…
Exact mathematical passage
\omega_{\mathrm{gate}} \circ \left( \omega_{\mathrm{retrieve}}, \omega_{\mathrm{decompose}}, \omega_{\mathrm{check}} \right).
\mathrm{demo}Use in source · 2 passages
Search aliases: demo mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16686
Generator A uses approximately uniform template frequencies. Its demonstration and query supports are X_A^demo={-2,-1,0,1,2}, X_A^query={-3,3}. Generator B changes the template mixture toward quadratic and hinge programs and uses X_B^demo={-3,-1,0,1,3}, X_B^query={-3,-2,2,3}. Every shifted record remains inside the same exact program language. Nevertheless, every shifted task contains at least one normalized feature coordinate outside the coordinatewise range of the primary support, and the larg…
Exact mathematical passage
\Xcal_A^{\mathrm{demo}}=\{-2,-1,0,1,2\}, \qquad \Xcal_A^{\mathrm{query}}=\{-3,3\}.
\mathrm{depth}Use in source · 1 passage
Search aliases: depth mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16109
efine z_0=ζ, z_k+1 = tanh (Rz_k+s_k+C_α(e,m)), 0≤ k<4, where s_k is a learned step embedding. The task-blind parent satisfies C_parent(e,m)=0. The descendants use one of the following conditioning maps: C_depth(e,m) · =D 1_m, C_1(e,m) · =U_1V_1e, rank(U_1V_1)≤1, C_4(e,m) · =U_4V_4e, rank(U_4V_4)≤4, C_full(e,m) · =We. Here 1_m is the one-hot depth code. Each child inherits every compatible parent parameter. The added up-projection or full-conditioning matrix is initialized so that the initial inj…
Exact mathematical passage
C_{\mathrm{depth}}(e,m)&=D\,\one_m, \label{eq:b17-depth-conditioning}\\ C_{1}(e,m)&=U_1V_1e, \qquad \rank(U_1V_1)\leq1, \label{eq:b17-rank1-conditioning}\\ C_{4}(e,m)&=U_4V_4e, \qquad \rank(U_4V_4)\leq4, \label{eq:b17-rank4-conditioning}\\ C_{\mathrm{full}}(e,m)&=We. \label{eq:b17-full-conditioning}
\detUse in source · 4 passages
Search aliases: det
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10660
The A-optimal quantity tr(M_S^-1) controls mean-square bridge error. The E-optimal quantity λ_min(M_S) controls the least informed direction. The D-optimal quantity logdet M_S controls information volume.
Exact mathematical passage
\log\det M_S
\diagUse in source · 8 passages
Search aliases: diag operatorname
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5617
Case F4: evaluator holonomy. Use an exactly correctable local defect and zero overlap mismatch. Let evaluator transports around a three-edge cycle be T_01=I_2, T_12=I_2, T_20=diag(1,-1). For protected evaluator direction e_2, H_γ e_2-e_2=-2e_2, normH_γ e_2-e_2=2. The candidate is locally repairable and globally gluable, but its evaluator semantics are not path flat.
Exact mathematical passage
T_{01}=I_2, \qquad T_{12}=I_2, \qquad T_{20}=\diag(1,-1).
\mathrm{diag}Use in source · 2 passages
Search aliases: diag mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11508
The inherited global target-repair response lies in the diagonal class D_diag = set(R,R):R∈Aff(H_t), whereas the chart grammar reaches D_chart = Aff(H_t)×Aff(H_t).
Exact mathematical passage
\Dcal_{\mathrm{diag}} = \set{(R,R):R\in\operatorname{Aff}(H_t)},
\operatorname{diag}Use in source · 1 passage
Search aliases: diag operatorname
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 19608
The executable fixture uses five commuting symmetric response operators. In latent semantic coordinates they are diagonal: D_u=diag(λ_u1,…,λ_ud), and each system exposes an orthogonal gauge Q_a: R_u^a=Q_aD_uQ_a^top. The joint response signature of semantic coordinate j is λ_j=(λ_1j,…,λ_mj).
Exact mathematical passage
D_u=\operatorname{diag}(\lambda_{u1},\ldots,\lambda_{ud}),
\mathrm{disc}Use in source · 5 passages
Search aliases: disc mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5165
Let Ω be a finite universe of candidate repair-schema equivalence classes. For S⊆Ω, let Disc(S) be the schemas independently validated using the currently available set S. Define Φ_disc(S) = S∪Disc(S).
Exact mathematical passage
\Phi_{\mathrm{disc}}(S) = S\cup\operatorname{Disc}(S).
\operatorname{Disc}Use in source · 3 passages
Search aliases: Disc operatorname
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5164
Let Ω be a finite universe of candidate repair-schema equivalence classes. For S⊆Ω, let Disc(S) be the schemas independently validated using the currently available set S. Define Φ_disc(S) = S∪Disc(S).
Exact mathematical passage
\operatorname{Disc}(S)
\mathrm{discovery}Use in source · 2 passages
Search aliases: discovery mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9410
ive embedding of old meta-regret coordinates. Decompose r_t+1 = E_twidetilde r_t+1+n_t+1, n_t+1⊥im E_t. Define old-coordinate repair progress by P_t^repair = normr_t^2-normwidetilde r_t+1^2, and epistemic discovery by P_t^discovery = normn_t+1^2. Both quantities may be positive. The total represented regret may increase because the lineage learned to represent a previously invisible limitation.
Exact mathematical passage
P_t^{\mathrm{discovery}} = \norm{n_{t+1}}^2. \label{eq:b8-discovery-progress}
\operatorname{dist}Use in source · 1 passage
Search aliases: dist operatorname
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21887
Equip End(H_t)⊕End(H_t) with the product Frobenius norm. For a chart repair pair (D_-,D_+), write D_avg = fracD_-+D_+2, D_ctr = fracD_+-D_-2. The closest diagonal repair is (D_avg,D_avg). Therefore dist^2 ( (D_-,D_+),D_diag ) · = normD_--D_avg_F^2 + normD_+-D_avg_F^2 · = 2normD_ctr_F^2 · = frac12normD_+-D_-_F^2. Thus D_+≠ D_- gives a quantitative structural-novelty certificate, not merely two differently named copies of one global operator.
Exact mathematical passage
\operatorname{dist}^2 \left( (D_-,D_+),\Dcal_{\mathrm{diag}} \right) &= \norm{D_--D_{\mathrm{avg}}}_F^2 + \norm{D_+-D_{\mathrm{avg}}}_F^2\\ &= 2\norm{D_{\mathrm{ctr}}}_F^2\\ &= \frac12\norm{D_+-D_-}_F^2.
\operatorname{dist}^2Use in source · 1 passage
Search aliases: dist^2 operatorname
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21887
Equip End(H_t)⊕End(H_t) with the product Frobenius norm. For a chart repair pair (D_-,D_+), write D_avg = fracD_-+D_+2, D_ctr = fracD_+-D_-2. The closest diagonal repair is (D_avg,D_avg). Therefore dist^2 ( (D_-,D_+),D_diag ) · = normD_--D_avg_F^2 + normD_+-D_avg_F^2 · = 2normD_ctr_F^2 · = frac12normD_+-D_-_F^2. Thus D_+≠ D_- gives a quantitative structural-novelty certificate, not merely two differently named copies of one global operator.
Exact mathematical passage
\operatorname{dist}^2 \left( (D_-,D_+),\Dcal_{\mathrm{diag}} \right) &= \norm{D_--D_{\mathrm{avg}}}_F^2 + \norm{D_+-D_{\mathrm{avg}}}_F^2\\ &= 2\norm{D_{\mathrm{ctr}}}_F^2\\ &= \frac12\norm{D_+-D_-}_F^2.
\operatorname{do}Use in source · 1 passage
Search aliases: do operatorname
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 942
Let D_t be an authorized modification family containing a no-op intervention δ_0. For δ∈D_t, let K_A_t^δ(·) = P(Y∈·mid do(δ),A_t) be the protected external-outcome kernel. A compressed causal self-model ϕ_t(A_t) induces a predicted kernel widehat K_ϕ_t(A_t)^δ(·). The model is evaluated on counterfactual consequences, not on its ability to reproduce hidden activations for their own sake.
Exact mathematical passage
K_{A_t}^{\delta}(\cdot) = P\left(Y\in\cdot\mid \operatorname{do}(\delta),A_t\right)
\mathrm{dynamic}Use in source · 1 passage
Search aliases: dynamic mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16303
positive interval and fraction gates, local continuity, donor coverage, adaptive halting. Among eligible candidates it maximizes S(α) = · min{ fracR^2_probe0.65, fracQ_prog0.15, fracQ_ans0.20 } +0.20R^2_probe · -4 MSE_clean +0.02ρ_halt -Penalty_dynamic. The minimum term makes the weakest normalized semantic gate the principal bottleneck. The compiler seal commits the selected candidate, all candidate and parent state hashes, training receipts, meta-validation task digest, exact thresholds, selec…
Exact mathematical passage
\begin{split} S(\alpha) ={}& \min\left\{ \frac{R^2_{\mathrm{probe}}}{0.65}, \frac{Q_{\mathrm{prog}}}{0.15}, \frac{Q_{\mathrm{ans}}}{0.20} \right\} +0.20R^2_{\mathrm{probe}}\\ &-4\,\mathrm{MSE}_{\mathrm{clean}} +0.02\rho_{\mathrm{halt}} -\mathsf{Penalty}_{\mathrm{dynamic}}. \end{split} \label{eq:b17-selection-score}
\mathcal EUse in source · 7 passages
Search aliases: E mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4771
Let V be a Hilbert defect space, let L_t=im D_t^auth, and let P_t be the orthogonal projector onto L_t. For a defect distribution μ, define the correctable energy E_t^cor(μ) = E_r∼μnormP_t r^2 and the obstruction energy E_t^ob(μ) = E_r∼μnorm(I-P_t)r^2.
Exact mathematical passage
\mathcal E_t^{\mathrm{cor}}(\mu) = \mathbb E_{r\sim\mu}\norm{P_t r}^2
\mathbb E_{(x,y)\sim\pi}Use in source · 1 passage
Search aliases: E_(x,y)∼π mathbb sim pi
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3271
[Coupled distributional gain transfer] Suppose there is a coupling π of P_A and P_B such that E_(x,y)∼πd_B(y,Sx)≤δ. Assume q_B is K-Lipschitz and q_B(Sx)≥ q_A(x)-η for P_A-almost every x. Then E_y∼ P_Bq_B(y) ≥ E_x∼ P_Aq_A(x)-η-Kδ.
Exact mathematical passage
\mathbb E_{(x,y)\sim\pi}d_B(y,Sx)\leq\delta.
\mathsf E_0Use in source · 1 passage
Search aliases: E_0 mathsf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3671
[Cumulative evaluator-drift bound] Consider a chain of evaluator morphisms with readouts into a common complete normed score space, E_0xrightarrowe_0 E_1 xrightarrowe_1⋯ xrightarrowe_n-1 E_n. Suppose that for every protected represented state x_t transported from version 0, normJ_t+1(T_tx_t)-J_t(x_t)≤ε_t. Then normJ_n(T_n-1⋯ T_0x_0)-J_0(x_0) ≤ ∑_t=0^n-1ε_t. If ∑_tε_t<∞, then the transported score sequence is Cauchy and converges in every complete score space.
Exact mathematical passage
\mathsf E_0\xrightarrow{e_0}\mathsf E_1 \xrightarrow{e_1}\cdots \xrightarrow{e_{n-1}}\mathsf E_n.
e_0Use in source · 1 passage
Search aliases: e_0
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3671
[Cumulative evaluator-drift bound] Consider a chain of evaluator morphisms with readouts into a common complete normed score space, E_0xrightarrowe_0 E_1 xrightarrowe_1⋯ xrightarrowe_n-1 E_n. Suppose that for every protected represented state x_t transported from version 0, normJ_t+1(T_tx_t)-J_t(x_t)≤ε_t. Then normJ_n(T_n-1⋯ T_0x_0)-J_0(x_0) ≤ ∑_t=0^n-1ε_t. If ∑_tε_t<∞, then the transported score sequence is Cauchy and converges in every complete score space.
Exact mathematical passage
\mathsf E_0\xrightarrow{e_0}\mathsf E_1 \xrightarrow{e_1}\cdots \xrightarrow{e_{n-1}}\mathsf E_n.
\mathsf E_1Use in source · 1 passage
Search aliases: E_1 mathsf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3671
[Cumulative evaluator-drift bound] Consider a chain of evaluator morphisms with readouts into a common complete normed score space, E_0xrightarrowe_0 E_1 xrightarrowe_1⋯ xrightarrowe_n-1 E_n. Suppose that for every protected represented state x_t transported from version 0, normJ_t+1(T_tx_t)-J_t(x_t)≤ε_t. Then normJ_n(T_n-1⋯ T_0x_0)-J_0(x_0) ≤ ∑_t=0^n-1ε_t. If ∑_tε_t<∞, then the transported score sequence is Cauchy and converges in every complete score space.
Exact mathematical passage
\mathsf E_0\xrightarrow{e_0}\mathsf E_1 \xrightarrow{e_1}\cdots \xrightarrow{e_{n-1}}\mathsf E_n.
e_1Use in source · 19 passages
Search aliases: e_1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3671
[Cumulative evaluator-drift bound] Consider a chain of evaluator morphisms with readouts into a common complete normed score space, E_0xrightarrowe_0 E_1 xrightarrowe_1⋯ xrightarrowe_n-1 E_n. Suppose that for every protected represented state x_t transported from version 0, normJ_t+1(T_tx_t)-J_t(x_t)≤ε_t. Then normJ_n(T_n-1⋯ T_0x_0)-J_0(x_0) ≤ ∑_t=0^n-1ε_t. If ∑_tε_t<∞, then the transported score sequence is Cauchy and converges in every complete score space.
Exact mathematical passage
\mathsf E_0\xrightarrow{e_0}\mathsf E_1 \xrightarrow{e_1}\cdots \xrightarrow{e_{n-1}}\mathsf E_n.
e_{12}Use in source · 2 passages
Search aliases: e_12
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20724
Orient a triangle cyclically and identify edge mismatches with a vector e=(e_12,e_23,e_31)∈ℝ^3. The cycle space is spanned by c=frac1sqrt3(1,1,1). Hence η_glue(e)=|anglesc,e| =frac|e_12+e_23+e_31|sqrt3. For e=(1,1,-2), the sum vanishes and the mismatch is a pure gradient, so a vertex gauge glues the repairs exactly. For e=(1,1,1), the quantitative obstruction is sqrt3 and no gauge can reconcile the cycle.
Exact mathematical passage
e=(e_{12},e_{23},e_{31})\in\R^3.
e_1^{\mathsf T}Use in source · 1 passage
Search aliases: e_1^ T mathsf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 22038
Let the true context packet be P_*=UU^ T for a target-specific orthonormal matrix U∈ℝ^6×2, while the inherited packet is P_0=diag(1,1,0,0,0,0). Let the chart label be c(h)=1{e_1^ TU^ Th>0}. For generic U, c(h) is not measurable from P_0h. No threshold or nonlinear classifier using only P_0h can recover the chart perfectly. The response atlas can recover im U up to basis gauge when the intervention signatures separate it.
Exact mathematical passage
c(h)=\one\{e_1^{\mathsf T}U^{\mathsf T}h>0\}.
e_2Use in source · 21 passages
Search aliases: e_2
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5624
Case F4: evaluator holonomy. Use an exactly correctable local defect and zero overlap mismatch. Let evaluator transports around a three-edge cycle be T_01=I_2, T_12=I_2, T_20=diag(1,-1). For protected evaluator direction e_2, H_γ e_2-e_2=-2e_2, normH_γ e_2-e_2=2. The candidate is locally repairable and globally gluable, but its evaluator semantics are not path flat.
Exact mathematical passage
e_2
e_{23}Use in source · 2 passages
Search aliases: e_23
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20724
Orient a triangle cyclically and identify edge mismatches with a vector e=(e_12,e_23,e_31)∈ℝ^3. The cycle space is spanned by c=frac1sqrt3(1,1,1). Hence η_glue(e)=|anglesc,e| =frac|e_12+e_23+e_31|sqrt3. For e=(1,1,-2), the sum vanishes and the mismatch is a pure gradient, so a vertex gauge glues the repairs exactly. For e=(1,1,1), the quantitative obstruction is sqrt3 and no gauge can reconcile the cycle.
Exact mathematical passage
e=(e_{12},e_{23},e_{31})\in\R^3.
e_3Use in source · 4 passages
Search aliases: e_3
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21154
V=ℝ^2, C=span{e_1}, and embed V into V'=ℝ^3 by the standard coordinate inclusion. Let C'=span{e_1,e_2}. The old obstruction space is generated by [e_2], but e_2 becomes correctable in C'. Hence fracj^-1(C')C ≅ span{[e_2]}. The genuinely new obstruction quotient is fracV'C'+j(V) ≅ span{[e_3]}. The expansion repairs one old direction while exposing one new direction, exactly as in [source reference].
Exact mathematical passage
\frac{V'}{C'+j(V)} \cong \Span\{[e_3]\}.
e_{31}Use in source · 2 passages
Search aliases: e_31
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20724
Orient a triangle cyclically and identify edge mismatches with a vector e=(e_12,e_23,e_31)∈ℝ^3. The cycle space is spanned by c=frac1sqrt3(1,1,1). Hence η_glue(e)=|anglesc,e| =frac|e_12+e_23+e_31|sqrt3. For e=(1,1,-2), the sum vanishes and the mismatch is a pure gradient, so a vertex gauge glues the repairs exactly. For e=(1,1,1), the quantitative obstruction is sqrt3 and no gauge can reconcile the cycle.
Exact mathematical passage
e=(e_{12},e_{23},e_{31})\in\R^3.
e_4Use in source · 1 passage
Search aliases: e_4
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21623
Let the old correction image be I=span{e_1,e_2}⊂ℝ^4 and let r=(0,0,3,4). Then z=P_I^⊥r=(0,0,3,4), η_I(r)^2=25. Add generator v=2e_4. Because v∈ I^⊥, frac|anglesz,v|^2norm v^2 = frac8^24 =16. The expanded residual is 3e_3, so η_I'(r)^2=9=25-16. A generator parallel to e_1 would have bar v=0 and would produce no structural reduction despite having nonzero raw norm.
Exact mathematical passage
v=2e_4.
E_aUse in source · 5 passages
Search aliases: E_a
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6580
An agent a has encoder E_a:ℝ^16→ℝ^d_a and an affine head fitted to widetilde q. Its hidden state is h_a(x)=E_a(ϕ(x)). The transport system is not given the matrix or nonlinear parameters of E_a.
Exact mathematical passage
E_a:\R^{16}\to\R^{d_a}
e_aUse in source · 1 passage
Search aliases: e_a
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2345
Let G=(I,E) be a connected oriented graph representing the nerve of the cover, let K be a finite-dimensional Hilbert space representing a common trivialization of the kernel fibers, and let e=(e_a)_a∈ E∈ K^E be the transported overlap-mismatch vector. Let B∈ℝ^|I|×|E| be the oriented incidence matrix and set B:=B⊗ I_K. A vertex gauge k∈ K^I changes the mismatch to e+ B^*k.
Exact mathematical passage
e=(e_a)_{a\in E}\in K^E
e_{\mathrm{align}}Use in source · 3 passages
Search aliases: e_align mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16840
Let (Y,H_s,H_t)∼μ be the protected semantics, source trace, and target trace under context distribution μ. Fix a source probe P and bridge B. Define e_src=Y-PH_s, e_align=P(H_s-BH_t). Then Y-PBH_t=e_src+e_align.
Exact mathematical passage
e_{\mathrm{src}}=Y-PH_s, \qquad e_{\mathrm{align}}=P(H_s-BH_t). \label{eq:b18-error-parts}
\mathbb E_{A\sim\Dcal}Use in source · 1 passage
Search aliases: E_A∼Dcal E_A∼D mathbb sim Dcal mathbb sim mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9015
Fix a protected evaluation context Ξ, a foreign-system distribution D, and a matched resource budget B. Define protected foreign metaproductivity by M_Ξ,D,B(K) = E_A∼D [ J_Ξ(K[A;B])-J_Ξ(A) ], where K[A;B] is the authorized descendant selected by K for target A, or A itself when K abstains.
Exact mathematical passage
\Mcal_{\Xi,\Dcal,B}(K) = \mathbb E_{A\sim\Dcal} \left[ J_\Xi\bigl(K[A;B]\bigr)-J_\Xi(A) \right], \label{eq:b8-metaproductivity}
\mathbb E_{A\sim\Dcal_{\mathrm{hold}}}Use in source · 1 passage
Search aliases: E_A∼Dcal_hold E_A∼D_hold mathbb sim Dcal mathrm mathbb sim mathcal mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4626
Let D_hold be a distribution over foreign held-out agents that are not descendants of the evaluated lineage. Let K be a frozen improvement mechanism. Under horizon h and budget B, define M^foreign_h,B(K) = E_A∼D_hold [ max_A'∈Desc_h,B(K,A) J(A')-J(A) ].
Exact mathematical passage
M^{\mathrm{foreign}}_{h,B}(K) = \mathbb E_{A\sim\Dcal_{\mathrm{hold}}} \left[ \max_{A'\in\Desc_{h,B}(K,A)} J(A')-J(A) \right].
E_bUse in source · 3 passages
Search aliases: E_b
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6841
Let e_h=h_t-widehat T h_s, E_D=D_twidehat T-widehat T D_s, E_b=b_t-widehat T b_s. Then R_t(h_t)-widehat T R_s(h_s) = (I+D_t)e_h+E_Dh_s+E_b. Therefore normR_t(h_t)-widehat T R_s(h_s) ≤ (1+normD_t)norme_h +normE_Dnormh_s +normE_b. The implementation reports normalized alignment and repair-equivariance residuals as empirical summaries of these terms.
Exact mathematical passage
e_h=h_t-\widehat T h_s, \qquad E_D=D_t\widehat T-\widehat T D_s, \qquad E_b=b_t-\widehat T b_s.
E_{\mathrm{corr}}Use in source · 2 passages
Search aliases: E_corr mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6170
For canonical patch u=T^-1δ, define corrected protected energy E_corr(d,u) = ∑_i=1^6w_imin{d_i^2,u_i^2}. This quantity is zero for a no-op and positive when a patch reaches task-relevant defect coordinates. It is reported alongside behavioral gain because a high apparent score can be generated without correcting protected defect energy.
Exact mathematical passage
E_{\mathrm{corr}}(d,u) = \sum_{i=1}^6w_i\min\{d_i^2,u_i^2\}.
E_DUse in source · 3 passages
Search aliases: E_D
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6841
Let e_h=h_t-widehat T h_s, E_D=D_twidehat T-widehat T D_s, E_b=b_t-widehat T b_s. Then R_t(h_t)-widehat T R_s(h_s) = (I+D_t)e_h+E_Dh_s+E_b. Therefore normR_t(h_t)-widehat T R_s(h_s) ≤ (1+normD_t)norme_h +normE_Dnormh_s +normE_b. The implementation reports normalized alignment and repair-equivariance residuals as empirical summaries of these terms.
Exact mathematical passage
e_h=h_t-\widehat T h_s, \qquad E_D=D_t\widehat T-\widehat T D_s, \qquad E_b=b_t-\widehat T b_s.
e_hUse in source · 3 passages
Search aliases: e_h
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6841
Let e_h=h_t-widehat T h_s, E_D=D_twidehat T-widehat T D_s, E_b=b_t-widehat T b_s. Then R_t(h_t)-widehat T R_s(h_s) = (I+D_t)e_h+E_Dh_s+E_b. Therefore normR_t(h_t)-widehat T R_s(h_s) ≤ (1+normD_t)norme_h +normE_Dnormh_s +normE_b. The implementation reports normalized alignment and repair-equivariance residuals as empirical summaries of these terms.
Exact mathematical passage
e_h=h_t-\widehat T h_s, \qquad E_D=D_t\widehat T-\widehat T D_s, \qquad E_b=b_t-\widehat T b_s.
e_iUse in source · 2 passages
Search aliases: e_i
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13778
A finding is represented by f_i = (Π_i^-,Π_i^+,code_i, Par(i),e_i,o_i), where e_i is the observed effect profile and o_i is either a surviving obstruction or a validated no-go certificate. Directed edges are typed as extension, ablation, refutation, transport, or replication. A proposal must cite the archive nodes on which its mechanism depends.
Exact mathematical passage
f_i = (\Pi_i^-,\Pi_i^+,\mathrm{code}_i, \operatorname{Par}(i),e_i,o_i), \label{eq:b13-finding-node}
e_{\mathrm{irr}}Use in source · 2 passages
Search aliases: e_irr mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5755
An executable state is a finite canonical map widehat s:Key→Value. A witness policy is widehat W =(K_protected,γ_min,r_max,p_prov,e_irr,v_W), where K_protected is the immutable-key set, γ_min is the minimum protected gain, r_max is the maximum accepted risk, p_prov indicates whether provenance is mandatory, e_irr controls irreversible effects, and v_W is the witness version.
Exact mathematical passage
\widehat W =(K_{\mathrm{protected}},\gamma_{\min},r_{\max},p_{\mathrm{prov}},e_{\mathrm{irr}},v_W),
E_jUse in source · 1 passage
Search aliases: E_j
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12276
In an executable system, the responses are estimated from finite intervention data. Let widehat R_j=R_j+E_j, and fix coefficients c_j. Define B=∑_jc_jR_j, widehat B=∑_jc_jwidehat R_j, E=widehat B-B. Let the distinct eigenvalue associated with block a be μ_a and define its separation γ_a=min_b≠ a|μ_a-μ_b|.
Exact mathematical passage
\widehat R_j=R_j+E_j,
e_jUse in source · 2 passages
Search aliases: e_j
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7696
For n held-out episodes, define paired differences D_j = G_e_j(K_1)-G_e_j(K_0), widehatΔ_n=frac1n∑_j=1^nD_j. Because parent and child share each target and candidate bundle, pairing removes much of the episode difficulty variation.
Exact mathematical passage
D_j = G_{e_j}(K_1)-G_{e_j}(K_0), \qquad \widehat\Delta_n=\frac1n\sum_{j=1}^nD_j. \label{eq:block07-paired-estimator}
\mathbb E_{j\in\mathsf{FAD}(i)}Use in source · 1 passage
Search aliases: E_j∈FAD(i) mathbb in mathsf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16224
r_i · =absℓ_i(replay(Z_i))-ℓ_i(Z_i). Noise is deterministic and norm matched at every step. The program and answer specificity coordinates are the mean recipient costs under SAP and FAD donors: p_i = E_j∈SAP(i) [ℓ_i(Z_j)-ℓ_i(Z_i)], a_i = E_j∈FAD(i) [ℓ_i(Z_j)-ℓ_i(Z_i)]. Define Q_prog = fracoverline pmax{overline o,overline c,ε}, Q_ans = fracoverline amax{overline o,overline c,ε}. The frozen ratio thresholds are 0.15 and 0.20, respectively. Each specificity gate additionally requires a positive 95…
Exact mathematical passage
p_i = \mathbb E_{j\in\mathsf{SAP}(i)} \left[\ell_i(Z_j)-\ell_i(Z_i)\right], \qquad a_i = \mathbb E_{j\in\mathsf{FAD}(i)} \left[\ell_i(Z_j)-\ell_i(Z_i)\right]. \label{eq:b17-specificity-costs}
\mathbb E_{j\in\mathsf{SAP}(i)}Use in source · 1 passage
Search aliases: E_j∈SAP(i) mathbb in mathsf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16224
r_i · =absℓ_i(replay(Z_i))-ℓ_i(Z_i). Noise is deterministic and norm matched at every step. The program and answer specificity coordinates are the mean recipient costs under SAP and FAD donors: p_i = E_j∈SAP(i) [ℓ_i(Z_j)-ℓ_i(Z_i)], a_i = E_j∈FAD(i) [ℓ_i(Z_j)-ℓ_i(Z_i)]. Define Q_prog = fracoverline pmax{overline o,overline c,ε}, Q_ans = fracoverline amax{overline o,overline c,ε}. The frozen ratio thresholds are 0.15 and 0.20, respectively. Each specificity gate additionally requires a positive 95…
Exact mathematical passage
p_i = \mathbb E_{j\in\mathsf{SAP}(i)} \left[\ell_i(Z_j)-\ell_i(Z_i)\right], \qquad a_i = \mathbb E_{j\in\mathsf{FAD}(i)} \left[\ell_i(Z_j)-\ell_i(Z_i)\right]. \label{eq:b17-specificity-costs}
e_kUse in source · 4 passages
Search aliases: e_k
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6245
For foreign agent j, let Y_j(z) be protected gain under factor setting z. The average necessity effect of factor k relative to the intact architecture 1 is τ_k = E [ Y_j(1)-Y_j(1-e_k) ]. A positive value supports causal usefulness only relative to the frozen environment, compensating mechanisms, and matched resource schedule.
Exact mathematical passage
\tau_k = \mathbb E \left[ Y_j(\one)-Y_j(\one-e_k) \right].
\mathbb E_{K\sim\Dcal_{\mathrm{parent}}}Use in source · 1 passage
Search aliases: E_K∼Dcal_parent E_K∼D_parent mathbb sim Dcal mathrm mathbb sim mathcal mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9484
Define compiler metaproductivity by C(Γ) = E_K∼D_parent [ M(Γ(K,E_K))-M(K) ]. A stronger recursive claim requires C(Γ_t+1)>C(Γ_t) on sealed foreign parent processes under matched compilation resources.
Exact mathematical passage
\Ccal(\Gamma) = \mathbb E_{K\sim\Dcal_{\mathrm{parent}}} \left[ \Mcal\bigl(\Gamma(K,\Ecal_K)\bigr)-\Mcal(K) \right]. \label{eq:b8-compiler-metaproductivity}
\mathsf E_nUse in source · 1 passage
Search aliases: E_n mathsf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3671
[Cumulative evaluator-drift bound] Consider a chain of evaluator morphisms with readouts into a common complete normed score space, E_0xrightarrowe_0 E_1 xrightarrowe_1⋯ xrightarrowe_n-1 E_n. Suppose that for every protected represented state x_t transported from version 0, normJ_t+1(T_tx_t)-J_t(x_t)≤ε_t. Then normJ_n(T_n-1⋯ T_0x_0)-J_0(x_0) ≤ ∑_t=0^n-1ε_t. If ∑_tε_t<∞, then the transported score sequence is Cauchy and converges in every complete score space.
Exact mathematical passage
\mathsf E_0\xrightarrow{e_0}\mathsf E_1 \xrightarrow{e_1}\cdots \xrightarrow{e_{n-1}}\mathsf E_n.
e_{n-1}Use in source · 1 passage
Search aliases: e_n-1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3671
[Cumulative evaluator-drift bound] Consider a chain of evaluator morphisms with readouts into a common complete normed score space, E_0xrightarrowe_0 E_1 xrightarrowe_1⋯ xrightarrowe_n-1 E_n. Suppose that for every protected represented state x_t transported from version 0, normJ_t+1(T_tx_t)-J_t(x_t)≤ε_t. Then normJ_n(T_n-1⋯ T_0x_0)-J_0(x_0) ≤ ∑_t=0^n-1ε_t. If ∑_tε_t<∞, then the transported score sequence is Cauchy and converges in every complete score space.
Exact mathematical passage
\mathsf E_0\xrightarrow{e_0}\mathsf E_1 \xrightarrow{e_1}\cdots \xrightarrow{e_{n-1}}\mathsf E_n.
\mathbb E_PUse in source · 2 passages
Search aliases: E_P mathbb
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7629
Let Z be column centered and let P be a uniformly random permutation matrix independent of y. Then E_P[Z^top Py] = Z^top E_P[Py] = Z^topoverline y1 =0. The ridge coefficients are linear in Z^top Py for fixed Z.
Exact mathematical passage
\mathbb E_P[Z^\top Py] = Z^\top\mathbb E_P[Py] = Z^\top\overline y\one =0.
E_QUse in source · 1 passage
Search aliases: E_Q
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9846
Define the causal generator contribution Δ_gen = E_Q[η_without(r)^2] - E_Q[η_with(r)^2]. The contribution is admissible only when a matched ablation isolates the generator while holding the remaining grammar, evidence, and selection budget fixed.
Exact mathematical passage
\Delta_{\mathrm{gen}} = E_Q\bigl[\eta_{\mathrm{without}}(r)^2\bigr] - E_Q\bigl[\eta_{\mathrm{with}}(r)^2\bigr]. \label{eq:b8-generator-credit}
\mathbb E_{r\sim Q}Use in source · 4 passages
Search aliases: E_r∼ Q mathbb sim
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9578
For a foreign residual distribution Q, define expected obstruction energy E_t(Q) = E_r∼ Qη_t(r)^2. Then E_t(Q)-E_t+1(Q) = E_r∼ Q normP_I_t+1∩ I_t^⊥r^2. This is the primary structural metaproductivity estimand for grammar growth.
Exact mathematical passage
E_t(Q) = \mathbb E_{r\sim Q}\eta_t(r)^2. \label{eq:b8-expected-obstruction}
\mathbb E_{r\sim\mu}Use in source · 4 passages
Search aliases: E_r∼μ mathbb sim mu
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4771
Let V be a Hilbert defect space, let L_t=im D_t^auth, and let P_t be the orthogonal projector onto L_t. For a defect distribution μ, define the correctable energy E_t^cor(μ) = E_r∼μnormP_t r^2 and the obstruction energy E_t^ob(μ) = E_r∼μnorm(I-P_t)r^2.
Exact mathematical passage
\mathcal E_t^{\mathrm{cor}}(\mu) = \mathbb E_{r\sim\mu}\norm{P_t r}^2
e_{\mathrm{src}}Use in source · 3 passages
Search aliases: e_src mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16840
Let (Y,H_s,H_t)∼μ be the protected semantics, source trace, and target trace under context distribution μ. Fix a source probe P and bridge B. Define e_src=Y-PH_s, e_align=P(H_s-BH_t). Then Y-PBH_t=e_src+e_align.
Exact mathematical passage
e_{\mathrm{src}}=Y-PH_s, \qquad e_{\mathrm{align}}=P(H_s-BH_t). \label{eq:b18-error-parts}
e_tUse in source · 5 passages
Search aliases: e_t
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9262
h_W(K_t) ∈ℝ_≥0^7. An authorized meta-action a changes one or more repair-process modules. In a local finite model, represent its effect by r_t+1 ≤ A_a r_t+e_t, where A_a is a nonnegative response matrix and e_t records estimation error, distribution shift, semantic debt, or newly induced headroom.
Exact mathematical passage
r_{t+1} \leq A_a r_t+e_t, \label{eq:b8-regret-dynamics}
E_{t+1}Use in source · 2 passages
Search aliases: E_t+1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9585
For a foreign residual distribution Q, define expected obstruction energy E_t(Q) = E_r∼ Qη_t(r)^2. Then E_t(Q)-E_t+1(Q) = E_r∼ Q normP_I_t+1∩ I_t^⊥r^2. This is the primary structural metaproductivity estimand for grammar growth.
Exact mathematical passage
E_t(Q)-E_{t+1}(Q) = \mathbb E_{r\sim Q} \norm{P_{I_{t+1}\cap I_t^\perp}r}^2. \label{eq:b8-distributional-new-energy}
\mathcal E_{t+1}^{\mathrm{cor}}Use in source · 1 passage
Search aliases: E_t+1^cor mathcal mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4790
[Structural-to-behavioral identity] Assume L_t⊆ L_t+1 and let W=L_t+1∩ L_t^⊥. Then E_t+1^cor(μ)- E_t^cor(μ) = E_r∼μnormP_Wr^2 and E_t^ob(μ)- E_t+1^ob(μ) = E_r∼μnormP_Wr^2. Under quadratic residual loss and exact cancellation of the correctable component, this quantity is exactly the expected additional task-loss reduction made available by the expanded repair image.
Exact mathematical passage
\mathcal E_{t+1}^{\mathrm{cor}}(\mu)-\mathcal E_t^{\mathrm{cor}}(\mu) = \mathbb E_{r\sim\mu}\norm{P_Wr}^2
\mathcal E_{t+1}^{\mathrm{ob}}Use in source · 1 passage
Search aliases: E_t+1^ob mathcal mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4796
[Structural-to-behavioral identity] Assume L_t⊆ L_t+1 and let W=L_t+1∩ L_t^⊥. Then E_t+1^cor(μ)- E_t^cor(μ) = E_r∼μnormP_Wr^2 and E_t^ob(μ)- E_t+1^ob(μ) = E_r∼μnormP_Wr^2. Under quadratic residual loss and exact cancellation of the correctable component, this quantity is exactly the expected additional task-loss reduction made available by the expanded repair image.
Exact mathematical passage
\mathcal E_t^{\mathrm{ob}}(\mu)-\mathcal E_{t+1}^{\mathrm{ob}}(\mu) = \mathbb E_{r\sim\mu}\norm{P_Wr}^2.
\mathcal E_t^{\mathrm{cor}}Use in source · 3 passages
Search aliases: E_t^cor mathcal mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4771
Let V be a Hilbert defect space, let L_t=im D_t^auth, and let P_t be the orthogonal projector onto L_t. For a defect distribution μ, define the correctable energy E_t^cor(μ) = E_r∼μnormP_t r^2 and the obstruction energy E_t^ob(μ) = E_r∼μnorm(I-P_t)r^2.
Exact mathematical passage
\mathcal E_t^{\mathrm{cor}}(\mu) = \mathbb E_{r\sim\mu}\norm{P_t r}^2
\mathcal E_t^{\mathrm{ob}}Use in source · 3 passages
Search aliases: E_t^ob mathcal mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4777
Let V be a Hilbert defect space, let L_t=im D_t^auth, and let P_t be the orthogonal projector onto L_t. For a defect distribution μ, define the correctable energy E_t^cor(μ) = E_r∼μnormP_t r^2 and the obstruction energy E_t^ob(μ) = E_r∼μnorm(I-P_t)r^2.
Exact mathematical passage
\mathcal E_t^{\mathrm{ob}}(\mu) = \mathbb E_{r\sim\mu}\norm{(I-P_t)r}^2.
E_{v_0}Use in source · 3 passages
Search aliases: E_v_0
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3460
Let G=(N,E) be a connected directed graph of evaluator versions. Each directed edge e:v→ w carries an invertible internal transport T_e:E_v→ E_w. Fix a base node v_0. For any loop γ based at v_0, let H_γ:E_v_0→ E_v_0 be the composed transport around the loop.
Exact mathematical passage
H_\gamma:E_{v_0}\to E_{v_0}
\mathbb E_{x\sim P_A}Use in source · 1 passage
Search aliases: E_x∼ P_A mathbb sim
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3279
[Coupled distributional gain transfer] Suppose there is a coupling π of P_A and P_B such that E_(x,y)∼πd_B(y,Sx)≤δ. Assume q_B is K-Lipschitz and q_B(Sx)≥ q_A(x)-η for P_A-almost every x. Then E_y∼ P_Bq_B(y) ≥ E_x∼ P_Aq_A(x)-η-Kδ.
Exact mathematical passage
\mathbb E_{y\sim P_B}q_B(y) \geq \mathbb E_{x\sim P_A}q_A(x)-\eta-K\delta.
\mathbb E_{y\sim P_B}Use in source · 1 passage
Search aliases: E_y∼ P_B mathbb sim
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3279
[Coupled distributional gain transfer] Suppose there is a coupling π of P_A and P_B such that E_(x,y)∼πd_B(y,Sx)≤δ. Assume q_B is K-Lipschitz and q_B(Sx)≥ q_A(x)-η for P_A-almost every x. Then E_y∼ P_Bq_B(y) ≥ E_x∼ P_Aq_A(x)-η-Kδ.
Exact mathematical passage
\mathbb E_{y\sim P_B}q_B(y) \geq \mathbb E_{x\sim P_A}q_A(x)-\eta-K\delta.
\mathbb E_{z\sim\mu}Use in source · 1 passage
Search aliases: E_z∼μ mathbb sim mu
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 499
For an agent A, define the resource-normalized utility J_Ξ(A) = E_z∼μ [ ν(z,Y_A(z)) - λbar c(z,Y_A(z)) ]. The definition does not require scalar utility to encode every scientific judgment. It supplies a declared comparison functional for one experimental claim. Multiple protected contexts may be used simultaneously, producing a Pareto relation instead of a total order.
Exact mathematical passage
J_\Xi(A) = \mathbb E_{z\sim\mu} \left[ \nu\bigl(z,Y_A(z)\bigr) - \lambda\bar c\bigl(z,Y_A(z)\bigr) \right].
E_\thetaUse in source · 2 passages
Search aliases: E_θ theta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16088
Let ϕ(τ)∈ℝ^19 be the padded task serialization, and let e=E_θ(ϕ(τ))∈ℝ^40 be the prompt representation. The latent trace has four states. Given conditioning mode α, define z_0=ζ, z_k+1 = tanh (Rz_k+s_k+C_α(e,m)), 0≤ k<4, where s_k is a learned step embedding. The task-blind parent satisfies C_parent(e,m)=0. The descendants use one of the following conditioning maps: C_depth(e,m) · =D 1_m,
Exact mathematical passage
e=E_\theta(\phi(\tau))\in\R^{40} \label{eq:b17-encoder}
E_{\theta}Use in source · 2 passages
Search aliases: E_θ theta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7030
For encoder E_θ, decoder G_δ, and score head H_ω, the pilot minimizes minibatch estimates of L(θ,δ,ω) = λ_rm rec normG_δ(E_θ(ϕ))-ϕ_F^2 + normH_ω(E_θ(ϕ))-widetilde q_F^2, with λ_rm rec=2. Optimization uses a deterministic NumPy Adam implementation for 500 steps. After representation training, the head is refitted by ridge regression to reduce avoidable readout variance.
Exact mathematical passage
E_{\theta}
e_{\theta}Use in source · 2 passages
Search aliases: e_θ theta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16642
bx+c, · x<0. The coefficients belong to finite declared integer sets. A task contains five shuffled input--output demonstrations and one query input. The model receives a normalized serialization of those demonstrations and the query. The protected target is y(τ) = ( e_θ, fracp_12,…,fracp_42, fracd(θ)5, fracf_θ,p(x_q)32 )∈ℝ^10, where e_θ∈ℝ^4 is the one-hot template code, unused parameter slots are zero, and d(θ)∈{2,3,4,5} is the declared execution depth. All demonstrations, query answers, semant…
Exact mathematical passage
y(\tau) = \left( e_{\theta}, \frac{p_1}{2},\ldots,\frac{p_4}{2}, \frac{d(\theta)}{5}, \frac{f_{\theta,p}(x_q)}{32} \right)\in\R^{10}, \label{eq:b18-protected-target}
\mathbb E_{\mu}Use in source · 1 passage
Search aliases: E_μ mathbb mu
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16855
[Witness--alignment decomposition] For every square-integrable (Y,H_s,H_t), E_μnormY-PBH_t^2 · = E_μnorme_src^2 + E_μnorme_align^2 +2 E_μanglese_src,e_align, normY-PBH_t_L^2(μ) · ≤ normY-PH_s_L^2(μ) +normP(H_s-BH_t)_L^2(μ).
Exact mathematical passage
\mathbb E_{\mu}\norm{Y-PBH_t}^2 &= \mathbb E_{\mu}\norm{e_{\mathrm{src}}}^2 +\mathbb E_{\mu}\norm{e_{\mathrm{align}}}^2 +2\mathbb E_{\mu}\angles{e_{\mathrm{src}},e_{\mathrm{align}}}, \label{eq:b18-squared-decomposition}\\ \norm{Y-PBH_t}_{L^2(\mu)} &\leq \norm{Y-PH_s}_{L^2(\mu)} +\norm{P(H_s-BH_t)}_{L^2(\mu)}. \label{eq:b18-triangle-decomposition}
\mathbb E_{\tau\sim\Dcal_{\mathrm{foreign}}}Use in source · 1 passage
Search aliases: E_τ∼Dcal_foreign E_τ∼D_foreign mathbb tau sim Dcal mathrm mathbb tau sim mathcal mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13848
Let U_audit be a protected audit utility unavailable during compiler allocation and final scored selection. Define compiler metaproductivity at research budget B by C_B(Γ) = E_τ∼D_foreign [ U_audit (K_Γ^⋆(τ)) - U_audit(K_τ) ]. Only integrity-valid and capability-passing outputs are eligible.
Exact mathematical passage
\mathfrak C_B(\Gamma) = \mathbb E_{\tau\sim\Dcal_{\mathrm{foreign}}} \left[ U_{\mathrm{audit}} \left(K_{\Gamma}^{\star}(\tau)\right) - U_{\mathrm{audit}}(K_\tau) \right]. \label{eq:b13-compiler-meta}
\Ecal_{\mathrm{com}}Use in source · 2 passages
Search aliases: Ecal_com E_com Ecal mathrm mathcal mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10120
For X∈End(H), define commutator energy E_com(X) = ∑_j=1^m ( normXR_j-R_jX_F^2 + normXR_j^∗-R_j^∗ X_F^2 ). There is a positive semidefinite linear operator L_com on the Hilbert space of matrices such that anglesX,L_comX_F = E_com(X). Its kernel is the commutant: kerL_com= A'.
Exact mathematical passage
\Ecal_{\mathrm{com}}(X) = \sum_{j=1}^{m} \left( \norm{XR_j-R_jX}_F^2 + \norm{XR_j^\ast-R_j^\ast X}_F^2 \right). \label{eq:b8-commutator-energy}
\Ecal_KUse in source · 1 passage
Search aliases: Ecal_K E_K Ecal mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9484
Define compiler metaproductivity by C(Γ) = E_K∼D_parent [ M(Γ(K,E_K))-M(K) ]. A stronger recursive claim requires C(Γ_t+1)>C(Γ_t) on sealed foreign parent processes under matched compilation resources.
Exact mathematical passage
\Ccal(\Gamma) = \mathbb E_{K\sim\Dcal_{\mathrm{parent}}} \left[ \Mcal\bigl(\Gamma(K,\Ecal_K)\bigr)-\Mcal(K) \right]. \label{eq:b8-compiler-metaproductivity}
\mathrm{Elman}Use in source · 1 passage
Search aliases: Elman mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17678
Let A be a finite architecture index set and C a finite admitted chart index set. In the released realization, A = {Elman,GRU,Residual}, C = {A0,A1,A2,A3,N0,N1}. Each pair (a,c)∈ A× C carries a full Neuralese trajectory fiber H_a,c ≅ ( R^40)^5 ≅ R^200. The observation atlas determines which context fiber is authorized. A task outside every admitted chart is not assigned to an arbitrary fiber.
Exact mathematical passage
\mathcal A = \{\mathrm{Elman},\mathrm{GRU},\mathrm{Residual}\}, \qquad \mathcal C = \{A0,A1,A2,A3,N0,N1\}.
\eps^2Use in source · 6 passages
Search aliases: eps^2 ε^2 eps varepsilon
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2175
Consider the specific formal correction equation F(ε,u) = ε r_1+Du+frac12B(u,u)+O(ε^3+norm u^3), where B:U× U→ V is symmetric bilinear. Seek u(ε)=ε u_1+ε^2u_2+O(ε^3).
Exact mathematical passage
u(\eps)=\eps u_1+\eps^2u_2+O(\eps^3).
\eps^3Use in source · 2 passages
Search aliases: eps^3 ε^3 eps varepsilon
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2168
Consider the specific formal correction equation F(ε,u) = ε r_1+Du+frac12B(u,u)+O(ε^3+norm u^3), where B:U× U→ V is symmetric bilinear. Seek u(ε)=ε u_1+ε^2u_2+O(ε^3).
Exact mathematical passage
F(\eps,u) = \eps r_1+Du+\frac12B(u,u)+O\bigl(\eps^3+\norm u^3\bigr), \label{eq:second-order-model}
\eps^jUse in source · 2 passages
Search aliases: eps^j ε^j eps varepsilon
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2201
[Finite formal obstruction tower] Let F(ε,u) be analytic near (0,0) with F(0,0)=0 and D=∂_uF(0,0). Suppose coefficients u_1,…,u_n-1 have been chosen so that F(ε,∑_j=1^n-1ε^ju_j) =ε^nb_n+O(ε^n+1). Then extending the repair to order n by adding ε^nu_n is possible if and only if [b_n]=0 in coker D. When it vanishes, any solution of Du_n=-b_n produces an order-n lift.
Exact mathematical passage
F\left(\eps,\sum_{j=1}^{n-1}\eps^ju_j\right) =\eps^nb_n+O(\eps^{n+1}).
\eps^nUse in source · 3 passages
Search aliases: eps^n ε^n eps varepsilon
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2201
[Finite formal obstruction tower] Let F(ε,u) be analytic near (0,0) with F(0,0)=0 and D=∂_uF(0,0). Suppose coefficients u_1,…,u_n-1 have been chosen so that F(ε,∑_j=1^n-1ε^ju_j) =ε^nb_n+O(ε^n+1). Then extending the repair to order n by adding ε^nu_n is possible if and only if [b_n]=0 in coker D. When it vanishes, any solution of Du_n=-b_n produces an order-n lift.
Exact mathematical passage
F\left(\eps,\sum_{j=1}^{n-1}\eps^ju_j\right) =\eps^nb_n+O(\eps^{n+1}).
\eps^{n+1}Use in source · 2 passages
Search aliases: eps^n+1 ε^n+1 eps varepsilon
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2201
[Finite formal obstruction tower] Let F(ε,u) be analytic near (0,0) with F(0,0)=0 and D=∂_uF(0,0). Suppose coefficients u_1,…,u_n-1 have been chosen so that F(ε,∑_j=1^n-1ε^ju_j) =ε^nb_n+O(ε^n+1). Then extending the repair to order n by adding ε^nu_n is possible if and only if [b_n]=0 in coker D. When it vanishes, any solution of Du_n=-b_n produces an order-n lift.
Exact mathematical passage
F\left(\eps,\sum_{j=1}^{n-1}\eps^ju_j\right) =\eps^nb_n+O(\eps^{n+1}).
\eps_cUse in source · 3 passages
Search aliases: eps_c ε_c eps varepsilon
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1050
[Cost-sensitive regret] Suppose, in addition, that normalized intervention cost c(δ) is predicted by widehat c(δ) with uniform error at most ε_c. For net utility W(δ)=V(δ)-λ c(δ), widehat W(δ)=widehat V(δ)-λwidehat c(δ), selection by widehat W has regret at most 2(ε+λε_c).
Exact mathematical passage
\eps_c
\eps_{D,0}Use in source · 1 passage
Search aliases: eps_D,0 ε_D,0 eps varepsilon
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20914
Suppose a source correction has norm normu_0=2, every transport is nonexpansive, and three edges have (ε_D,0,ε_D,1,ε_D,2) =(0.01,0.02,0.015), (ε_r,0,ε_r,1,ε_r,2) =(0.005,0.004,0.006). Then [source reference] gives normD_3u_3+r_3 · ≤ 2(0.01+0.02+0.015) +(0.005+0.004+0.006)
Exact mathematical passage
(\eps_{D,0},\eps_{D,1},\eps_{D,2}) =(0.01,0.02,0.015),
\eps_{D,1}Use in source · 1 passage
Search aliases: eps_D,1 ε_D,1 eps varepsilon
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20914
Suppose a source correction has norm normu_0=2, every transport is nonexpansive, and three edges have (ε_D,0,ε_D,1,ε_D,2) =(0.01,0.02,0.015), (ε_r,0,ε_r,1,ε_r,2) =(0.005,0.004,0.006). Then [source reference] gives normD_3u_3+r_3 · ≤ 2(0.01+0.02+0.015) +(0.005+0.004+0.006)
Exact mathematical passage
(\eps_{D,0},\eps_{D,1},\eps_{D,2}) =(0.01,0.02,0.015),
\eps_{D,2}Use in source · 1 passage
Search aliases: eps_D,2 ε_D,2 eps varepsilon
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20914
Suppose a source correction has norm normu_0=2, every transport is nonexpansive, and three edges have (ε_D,0,ε_D,1,ε_D,2) =(0.01,0.02,0.015), (ε_r,0,ε_r,1,ε_r,2) =(0.005,0.004,0.006). Then [source reference] gives normD_3u_3+r_3 · ≤ 2(0.01+0.02+0.015) +(0.005+0.004+0.006)
Exact mathematical passage
(\eps_{D,0},\eps_{D,1},\eps_{D,2}) =(0.01,0.02,0.015),
\eps_{D,k}Use in source · 3 passages
Search aliases: eps_D,k ε_D,k eps varepsilon
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2941
[Pathwise transport bound] Consider a path of systems A_0xrightarrow T_0A_1 xrightarrow T_1⋯ xrightarrow T_n-1A_n. Write T_U,k:U_k→ U_k+1 and T_V,k:V_k→ V_k+1. Let normD_k+1T_U,k-T_V,kD_k≤ε_D,k and normr_k+1-T_V,kr_k≤ε_r,k. If D_0u_0=-r_0 and u_n=T_U,n-1⋯ T_U,0u_0, then normD_nu_n+r_n · ≤ ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_D,k normT_U,k-1⋯ T_U,0u_0
Exact mathematical passage
\norm{D_{k+1}T_{U,k}-T_{V,k}D_k}\leq\eps_{D,k}
\eps_{\Dcal}Use in source · 2 passages
Search aliases: eps_Dcal ε_D eps Dcal varepsilon mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1073
Define the intervention-coverage error ε_D(ϕ;A) = sup_δ∈D TV(K_A^δ,widehat K_ϕ(A)^δ).
Exact mathematical passage
\eps_{\Dcal}(\phi;A) = \sup_{\delta\in\Dcal} \TV\left(K_A^\delta,\widehat K_{\phi(A)}^\delta\right).
\eps_{\Dcal'}Use in source · 1 passage
Search aliases: eps_Dcal' ε_D' eps Dcal varepsilon mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1082
[Coverage monotonicity] If D⊆D', then ε_D'(ϕ;A) ≥ ε_D(ϕ;A).
Exact mathematical passage
\eps_{\Dcal'}(\phi;A) \geq \eps_{\Dcal}(\phi;A).
\eps_{\mathrm{forecast},i}Use in source · 1 passage
Search aliases: eps_forecast,i ε_forecast,i eps mathrm varepsilon mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13751
The forecast residual and proposal--code congruence residual are ε_forecast,i = normwidehat c_i- c_i, ε_spec/code,i = d_sem ( Sem(Π_i^-), Sem(code_i) ). In the executable finite companion, congruence is a Boolean monitor rather than a complete semantic metric.
Exact mathematical passage
\eps_{\mathrm{forecast},i} = \norm{\widehat{\mathbf c}_i-\mathbf c_i}, \qquad \eps_{\mathrm{spec/code},i} = d_{\mathrm{sem}} \left( \mathrm{Sem}(\Pi_i^-), \mathrm{Sem}(\mathrm{code}_i) \right). \label{eq:b13-receipt-residuals}
\eps_{\mathrm{glob}}Use in source · 2 passages
Search aliases: eps_glob ε_glob eps mathrm varepsilon mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21910
e true target relation depends on s=1{h_1>0}, but a candidate separator uses g(h)=1{h_2>0}, where h_2 is independent of s and the repair-relevant state. With balanced cells and unlimited data, both g-branches see the same mixture of the two true maps. Each fitted local map converges to the global population minimizer. Consequently, ε_loc = ε_glob, Δ_gram=0. A larger syntax is therefore not enough. Structural credit requires a held-out residual reduction, conservative replay, a nonzero repair con…
Exact mathematical passage
\eps_{\mathrm{loc}} = \eps_{\mathrm{glob}}, \qquad \Delta_{\mathrm{gram}}=0.
\eps_iUse in source · 1 passage
Search aliases: eps_i ε_i eps varepsilon
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10592
Let the source hidden state have dimension d and the target hidden state have dimension q. For selected source anchors x_i∈ℝ^d, suppose the observed target anchor is y_i=Tx_i+ε_i, E[ε_i]=0, Cov(ε_i)=σ_i^2I_q. The query-quality metadata supplies σ_i before the anchor is selected. This metadata is part of the declared Block 9 environment. It is not discovered autonomously.
Exact mathematical passage
y_i=Tx_i+\eps_i, \qquad \mathbb E[\eps_i]=0, \qquad \Cov(\eps_i)=\sigma_i^2I_q.
\eps_{\mathrm{loc}}Use in source · 2 passages
Search aliases: eps_loc ε_loc eps mathrm varepsilon mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21910
e true target relation depends on s=1{h_1>0}, but a candidate separator uses g(h)=1{h_2>0}, where h_2 is independent of s and the repair-relevant state. With balanced cells and unlimited data, both g-branches see the same mixture of the two true maps. Each fitted local map converges to the global population minimizer. Consequently, ε_loc = ε_glob, Δ_gram=0. A larger syntax is therefore not enough. Structural credit requires a held-out residual reduction, conservative replay, a nonzero repair con…
Exact mathematical passage
\eps_{\mathrm{loc}} = \eps_{\mathrm{glob}}, \qquad \Delta_{\mathrm{gram}}=0.
\eps_{\max}Use in source · 1 passage
Search aliases: eps_max ε_max eps max varepsilon max
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6999
r_rm non · ≤ widehat r_rm non+η_r, and that the reported alignment error upper-bounds the audit alignment error up to η_a. If widehat g_rm slice-η_g · >0, widehat r_rm non+η_r · ≤ r_max, widehatε_rm align+η_a · ≤ ε_max, then a candidate passing the corresponding tightened calibration gate has positive audit gain and satisfies the declared audit regression and alignment limits.
Exact mathematical passage
\widehat g_{\rm slice}-\eta_g&>0,\\ \widehat r_{\rm non}+\eta_r&\leq r_{\max},\\ \widehat\eps_{\rm align}+\eta_a&\leq \eps_{\max},
\eps_nUse in source · 3 passages
Search aliases: eps_n ε_n eps varepsilon
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15227
[Finite search bound] For n independent meta-validation problems, with probability at least 1-δ, every candidate language score differs from its population score by at most ε_n = sqrt fraclog(2N/δ)2n . Consequently, the selected candidate's population score is within 2ε_n of the best candidate in the finite language.
Exact mathematical passage
\eps_n = \sqrt{ \frac{\log(2N/\delta)}{2n} }. \label{eq:b15-uniform-bound}
\eps_{r,0}Use in source · 1 passage
Search aliases: eps_r,0 ε_r,0 eps varepsilon
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20918
Suppose a source correction has norm normu_0=2, every transport is nonexpansive, and three edges have (ε_D,0,ε_D,1,ε_D,2) =(0.01,0.02,0.015), (ε_r,0,ε_r,1,ε_r,2) =(0.005,0.004,0.006). Then [source reference] gives normD_3u_3+r_3 · ≤ 2(0.01+0.02+0.015) +(0.005+0.004+0.006)
Exact mathematical passage
(\eps_{r,0},\eps_{r,1},\eps_{r,2}) =(0.005,0.004,0.006).
\eps_{r,1}Use in source · 1 passage
Search aliases: eps_r,1 ε_r,1 eps varepsilon
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20918
Suppose a source correction has norm normu_0=2, every transport is nonexpansive, and three edges have (ε_D,0,ε_D,1,ε_D,2) =(0.01,0.02,0.015), (ε_r,0,ε_r,1,ε_r,2) =(0.005,0.004,0.006). Then [source reference] gives normD_3u_3+r_3 · ≤ 2(0.01+0.02+0.015) +(0.005+0.004+0.006)
Exact mathematical passage
(\eps_{r,0},\eps_{r,1},\eps_{r,2}) =(0.005,0.004,0.006).
\eps_{r,2}Use in source · 1 passage
Search aliases: eps_r,2 ε_r,2 eps varepsilon
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20918
Suppose a source correction has norm normu_0=2, every transport is nonexpansive, and three edges have (ε_D,0,ε_D,1,ε_D,2) =(0.01,0.02,0.015), (ε_r,0,ε_r,1,ε_r,2) =(0.005,0.004,0.006). Then [source reference] gives normD_3u_3+r_3 · ≤ 2(0.01+0.02+0.015) +(0.005+0.004+0.006)
Exact mathematical passage
(\eps_{r,0},\eps_{r,1},\eps_{r,2}) =(0.005,0.004,0.006).
\eps_{r,k}Use in source · 3 passages
Search aliases: eps_r,k ε_r,k eps varepsilon
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2945
[Pathwise transport bound] Consider a path of systems A_0xrightarrow T_0A_1 xrightarrow T_1⋯ xrightarrow T_n-1A_n. Write T_U,k:U_k→ U_k+1 and T_V,k:V_k→ V_k+1. Let normD_k+1T_U,k-T_V,kD_k≤ε_D,k and normr_k+1-T_V,kr_k≤ε_r,k. If D_0u_0=-r_0 and u_n=T_U,n-1⋯ T_U,0u_0, then normD_nu_n+r_n · ≤ ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_D,k normT_U,k-1⋯ T_U,0u_0
Exact mathematical passage
\norm{r_{k+1}-T_{V,k}r_k}\leq\eps_{r,k}.
\eps_{\mathrm{resp}}Use in source · 3 passages
Search aliases: eps_resp ε_resp eps mathrm varepsilon mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12310
For recovered projectors widehat P_1,…,widehat P_m, define the normalized response-fit residual ε_resp = frac ( ∑_jnormwidehat R_j- ∑_awidehatλ_ajwidehat P_a_F^2 )^1/2 ( ∑_jnormwidehat R_j_F^2 )^1/2+10^-12 , and the maximum normalized commutator residual ε_comm = max_a,j fracnormwidehat P_awidehat R_j- widehat R_jwidehat P_a_F normwidehat R_j_F+10^-12. The smallest selected spectral gap is denoted widehatγ_min. These quantities serve different roles: item ε_resp measures whether the finite block…
Exact mathematical passage
\eps_{\mathrm{resp}} = \frac{ \left( \sum_j\norm{\widehat R_j- \sum_a\widehat\lambda_{aj}\widehat P_a}_F^2 \right)^{1/2} }{ \left( \sum_j\norm{\widehat R_j}_F^2 \right)^{1/2}+10^{-12} }, \label{eq:b11-response-fit}
\eps_SUse in source · 3 passages
Search aliases: eps_S ε_S eps varepsilon
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 743
[Composition bound for bridge defects] Let T:X→ Y and S:Y→ Z. Suppose the property transports satisfy normQ_Y(Tx)-widetilde T_QQ_X(x)≤ε_T for x∈ K, and normQ_Z(Sy)-widetilde S_QQ_Y(y)≤ε_S for y∈ T(K). If normwidetilde S_Q≤ L_S, then the composite bridge obeys ε_Q(S∘ T;K) ≤ ε_S+L_Sε_T with intended property transport widetilde S_Qwidetilde T_Q.
Exact mathematical passage
\norm{Q_Z(Sy)-\widetilde S_QQ_Y(y)}\leq\eps_S
\eps_{\mathrm{spec/code},i}Use in source · 1 passage
Search aliases: eps_spec/code,i ε_spec/code,i eps mathrm varepsilon mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13751
The forecast residual and proposal--code congruence residual are ε_forecast,i = normwidehat c_i- c_i, ε_spec/code,i = d_sem ( Sem(Π_i^-), Sem(code_i) ). In the executable finite companion, congruence is a Boolean monitor rather than a complete semantic metric.
Exact mathematical passage
\eps_{\mathrm{forecast},i} = \norm{\widehat{\mathbf c}_i-\mathbf c_i}, \qquad \eps_{\mathrm{spec/code},i} = d_{\mathrm{sem}} \left( \mathrm{Sem}(\Pi_i^-), \mathrm{Sem}(\mathrm{code}_i) \right). \label{eq:b13-receipt-residuals}
\eps_TUse in source · 3 passages
Search aliases: eps_T ε_T eps varepsilon
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 739
[Composition bound for bridge defects] Let T:X→ Y and S:Y→ Z. Suppose the property transports satisfy normQ_Y(Tx)-widetilde T_QQ_X(x)≤ε_T for x∈ K, and normQ_Z(Sy)-widetilde S_QQ_Y(y)≤ε_S for y∈ T(K). If normwidetilde S_Q≤ L_S, then the composite bridge obeys ε_Q(S∘ T;K) ≤ ε_S+L_Sε_T with intended property transport widetilde S_Qwidetilde T_Q.
Exact mathematical passage
\norm{Q_Y(Tx)-\widetilde T_QQ_X(x)}\leq\eps_T
\eps_tUse in source · 10 passages
Search aliases: eps_t ε_t eps varepsilon
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3677
ulative evaluator-drift bound] Consider a chain of evaluator morphisms with readouts into a common complete normed score space, E_0xrightarrowe_0 E_1 xrightarrowe_1⋯ xrightarrowe_n-1 E_n. Suppose that for every protected represented state x_t transported from version 0, normJ_t+1(T_tx_t)-J_t(x_t)≤ε_t. Then normJ_n(T_n-1⋯ T_0x_0)-J_0(x_0) ≤ ∑_t=0^n-1ε_t. If ∑_tε_t<∞, then the transported score sequence is Cauchy and converges in every complete score space.
Exact mathematical passage
\norm{J_{t+1}(T_tx_t)-J_t(x_t)}\leq\eps_t.
\eps_t^DUse in source · 3 passages
Search aliases: eps_t^D ε_t^D eps varepsilon
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9702
Suppose the correction square is approximate: normD_t+1T_t^U-T_t^VD_t ≤ε_t^D. Suppose protected defect meanings are reconstructed with error at most ε_t^V. Define cumulative dynamic semantic debt Δ_m,n = ∑_t=m^n-1 ( ε_t^D+ε_t^V ). Under nonexpansive transports, old correction and defect relations drift by at most the corresponding telescoping debt. If ∑_t(ε_t^D+ε_t^V)<∞, old protected meanings form Cauchy trajectories along the grammar chain.
Exact mathematical passage
\norm{D_{t+1}T_t^U-T_t^VD_t} \leq\eps_t^D.
\eps_t^VUse in source · 3 passages
Search aliases: eps_t^V ε_t^V eps varepsilon
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9706
Suppose the correction square is approximate: normD_t+1T_t^U-T_t^VD_t ≤ε_t^D. Suppose protected defect meanings are reconstructed with error at most ε_t^V. Define cumulative dynamic semantic debt Δ_m,n = ∑_t=m^n-1 ( ε_t^D+ε_t^V ). Under nonexpansive transports, old correction and defect relations drift by at most the corresponding telescoping debt. If ∑_t(ε_t^D+ε_t^V)<∞, old protected meanings form Cauchy trajectories along the grammar chain.
Exact mathematical passage
\eps_t^V
\mathrm{evaluation}Use in source · 1 passage
Search aliases: evaluation mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3960
Let a complete implementation state decompose as Σ = Σ^task ⊕ Σ^memory ⊕ Σ^repair ⊕ Σ^evaluation ⊕ Σ^provenance ⊕ Σ^rollback. A transaction selects an editable mask E⊆Σ and a witness mask W⊆Σ with E∩ W=∅. The decomposition is logical rather than necessarily physical. Two components may share hardware while remaining isolated by capabilities, address spaces, process boundaries, cryptographic commitments, or reproducible snapshots.
Exact mathematical passage
\Sigma = \Sigma^{\mathrm{task}} \oplus \Sigma^{\mathrm{memory}} \oplus \Sigma^{\mathrm{repair}} \oplus \Sigma^{\mathrm{evaluation}} \oplus \Sigma^{\mathrm{provenance}} \oplus \Sigma^{\mathrm{rollback}}.
\mathrm{evaluator}Use in source · 6 passages
Search aliases: evaluator mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 665
A representative color set is C= set model, workspace, memory, planner, improver, evaluator, provenance, tools . The atlas does not assert that these colors are universally correct. It declares the decomposition under which one particular repair theory is being tested.
Exact mathematical passage
\mathsf C= \set{ \mathrm{model}, \mathrm{workspace}, \mathrm{memory}, \mathrm{planner}, \mathrm{improver}, \mathrm{evaluator}, \mathrm{provenance}, \mathrm{tools} }.
\mathrm{evidence}Use in source · 1 passage
Search aliases: evidence mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21213
Let protected keys be K_protected ={evidence,evaluator,provenance}. The adversarial candidate declares K_edit ={evaluator,utility}. Then K_edit∩ K_protected ={evaluator}≠∅. Even before considering utility, the acceptance predicate is false. The sandbox also changes the protected evaluator value, providing a second independent rejection reason. The final state is restored from the checkpoint and has the original evaluator digest.
Exact mathematical passage
K_{\mathrm{protected}} =\{\mathrm{evidence},\mathrm{evaluator},\mathrm{provenance}\}.
\EvidenceFlatUse in source · 1 passage
Search aliases: EvidenceFlat mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4259
Let C_RI(s) = · Grounded(s) wedgeProvValid(s) wedgeWorkspaceBound(s) · wedgeRepairTyped(s) wedgeWitnessed(s) wedgeEvidenceFlat(s) · wedgeRollbackable(s) wedgeMetaCapable(s). For X⊆ S, define Ψ_∀(X) = sets∈ S:C_RI(s) and Post_imp(s)⊆ X.
Exact mathematical passage
\begin{aligned} C_{\mathrm{RI}}(s) ={}& \Grounded(s) \wedge\ProvValid(s) \wedge\WorkspaceBound(s)\\ &\wedge\RepairTyped(s) \wedge\Witnessed(s) \wedge\EvidenceFlat(s)\\ &\wedge\Rollbackable(s) \wedge\MetaCapable(s). \end{aligned}
\mathrm{execute}Use in source · 1 passage
Search aliases: execute mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4027
The transaction follows six phases: prepare ⟶ seal ⟶ execute ⟶ reveal ⟶ decide ⟶ finalize. Only the final phase may expose buffered effects or replace the active checkpoint.
Exact mathematical passage
\mathrm{prepare} \longrightarrow \mathrm{seal} \longrightarrow \mathrm{execute} \longrightarrow \mathrm{reveal} \longrightarrow \mathrm{decide} \longrightarrow \mathrm{finalize}.
\expUse in source · 7 passages
Search aliases: exp
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3360
Hoeffding's inequality for a variable with range width 2G gives Pr( overline Z_n- EZ_i≥ t ) ≤ exp(-fracnt^22G^2). Set the right side to α. The family statement follows by the union bound.
Exact mathematical passage
\Pr\left( \overline Z_n-\mathbb EZ_i\geq t \right) \leq \exp\left(-\frac{nt^2}{2G^2}\right).
\mathrm FUse in source · 1 passage
Search aliases: F mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15812
For parent and child state spaces H_A and H_B of equal dimension, let calibration traces produce matrices X and Y. The orthogonal Procrustes transport is T^* = arg min_T^ TT=InormXT-Y_ F. If X^ TY=UΣ V^ T, then T^*=UV^ T. A transport receipt records calibration residual, held-out residual, singular floor, round-trip residual, and behavior change after transported trace injection.
Exact mathematical passage
T^* = \argmin_{T^{\mathsf T}T=I}\norm{XT-Y}_{\mathrm F}. \label{eq:b16-procrustes}
\mathsf FUse in source · 4 passages
Search aliases: F mathsf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7728
The child must pass performance and risk gates simultaneously. Let Gate = Seal wedge Match wedge AuditGain wedge ForeignGain wedge Risk wedge Controls. The released suite evaluates the following frozen selector-gate families: item descendant receipt validity; item split-file load order and sealing; item positive paired child-parent audit interval; item positive paired child-parent architecture-foreign interval; item positive child-raw architecture-foreign interval; item three seed-and-task audit…
Exact mathematical passage
\mathsf Gate = \mathsf Seal \wedge \mathsf Match \wedge \mathsf AuditGain \wedge \mathsf ForeignGain \wedge \mathsf Risk \wedge \mathsf Controls.
F'Use in source · 5 passages
Search aliases: F'
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2074
[Newton--Kantorovich repair criterion] Let F:U→ V be continuously Fr echet differentiable between Banach spaces. Let u_0∈ U and suppose F'(u_0) is invertible with inverse B. Assume that on the closed ball overline B(u_0,R), normF'(u)-F'(v)≤ Lnormu-v. Set η:=normBF(u_0), h:=norm B Lη. If h≤frac12 and t_*:=frac1-sqrt1-2hnorm B L≤ R, then Newton iteration u_n+1=u_n-F'(u_n)^-1F(u_n) is well-defined, remains in overline B(u_0,t_*), and converges to a root u^* of F. If the larger radius t^**:=frac1+sq…
Exact mathematical passage
F'(u_0)
F^2Use in source · 8 passages
Search aliases: F^2
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6757
h_t(x_m)^top . It estimates the bridge by regularized least squares: widehat T_t← s ∈ arg min_T normH_sT^top-H_t_F^2 +λnormT_F^2. The transformation matrices used to construct the controlled targets are not exposed to this fit.
Exact mathematical passage
\widehat T_{t\leftarrow s} \in \argmin_T \norm{H_sT^{\top}-H_t}_F^2 +\lambda\norm{T}_F^2. \label{eq:block06-learned-transport}
f_1Use in source · 1 passage
Search aliases: f_1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16014
Let an affine instruction be f_j(z)=a_jz+b_j, a_j∈{-1,-tfrac12,tfrac12,1}, b_j∈{-tfrac12,0,tfrac12}. A depth-m program acts on x∈{-1,-tfrac34,…,1} by F_m(x) = (f_m∘⋯∘ f_1)(x) =A_mx+B_m, where A_0=1, B_0=0, A_j=a_jA_j-1, B_j=a_jB_j-1+b_j. All quantities in [source reference] are evaluated with exact rational arithmetic before a floating representation is exposed to the model.
Exact mathematical passage
F_m(x) = (f_m\circ\cdots\circ f_1)(x) =A_mx+B_m, \label{eq:b17-affine-program}
F_{A,b,C}Use in source · 1 passage
Search aliases: F_A,b,C
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13108
Let U_P∈ℝ^d× r have orthonormal columns spanning the recovered repair packet. Define packet coordinates z=U_P^top(x-c). The degree-two block-local grammar consists of maps F_A,b,C(x) = Ax+b+C vech(zz^top), where vech lists the r(r+1)/2 distinct symmetric quadratic monomials.
Exact mathematical passage
F_{A,b,C}(x) = Ax+b+C\,\vech(zz^\top), \label{eq:b12-block-quadratic}
f_{\mathrm{abs}}Use in source · 2 passages
Search aliases: f_abs mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16627
Let the template set be Θ = {θ_aff,θ_abs, θ_quad,θ_hinge}. For integer parameters a,b,c and integer input x, the exact programs are f_aff(x) · =ax+b, f_abs(x) · =a|x|+b, f_quad(x) · =ax^2+bx+c, f_hinge(x) · = ax+c, · x≥0, bx+c, · x<0. The coefficients belong to finite declared integer sets. A task contains five shuffled input--output demonstrations and one query input. The model receives a normalized serialization of those demonstrations and the query. The protected target is y(τ) = ( e_θ, fracp…
Exact mathematical passage
f_{\mathrm{aff}}(x)&=ax+b, \\ f_{\mathrm{abs}}(x)&=a|x|+b, \\ f_{\mathrm{quad}}(x)&=ax^2+bx+c, \\ f_{\mathrm{hinge}}(x)&= \begin{cases} ax+c,&x\geq0,\\ bx+c,&x<0. \end{cases} \label{eq:b18-program-templates}
f_{\mathrm{aff}}Use in source · 2 passages
Search aliases: f_aff mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16627
Let the template set be Θ = {θ_aff,θ_abs, θ_quad,θ_hinge}. For integer parameters a,b,c and integer input x, the exact programs are f_aff(x) · =ax+b, f_abs(x) · =a|x|+b, f_quad(x) · =ax^2+bx+c, f_hinge(x) · = ax+c, · x≥0, bx+c, · x<0. The coefficients belong to finite declared integer sets. A task contains five shuffled input--output demonstrations and one query input. The model receives a normalized serialization of those demonstrations and the query. The protected target is y(τ) = ( e_θ, fracp…
Exact mathematical passage
f_{\mathrm{aff}}(x)&=ax+b, \\ f_{\mathrm{abs}}(x)&=a|x|+b, \\ f_{\mathrm{quad}}(x)&=ax^2+bx+c, \\ f_{\mathrm{hinge}}(x)&= \begin{cases} ax+c,&x\geq0,\\ bx+c,&x<0. \end{cases} \label{eq:b18-program-templates}
F_{\mathrm{FiLM}}Use in source · 1 passage
Search aliases: F_FiLM mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16735
The source grammar contains a task-blind continuation, inherited additive conditioning, final-step FiLM, progressive FiLM with geometry, a low-rank hypernetwork, and a shuffled-witness FiLM control. For the selected FiLM constructor, the encoded task produces step-indexed scale and shift vectors (γ_k(e),β_k(e))_k=0^4 =F_FiLM(e), which act on the preactivation by q_k ⟼ q_kodot(1+frac12tanhγ_k(e)) +β_k(e). This constructor is absent from Block 17's rank-one, rank-four, and full additive conditioni…
Exact mathematical passage
(\gamma_k(e),\beta_k(e))_{k=0}^{4} =F_{\mathrm{FiLM}}(e), \label{eq:b18-film}
f_{\mathrm{hinge}}Use in source · 2 passages
Search aliases: f_hinge mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16627
Let the template set be Θ = {θ_aff,θ_abs, θ_quad,θ_hinge}. For integer parameters a,b,c and integer input x, the exact programs are f_aff(x) · =ax+b, f_abs(x) · =a|x|+b, f_quad(x) · =ax^2+bx+c, f_hinge(x) · = ax+c, · x≥0, bx+c, · x<0. The coefficients belong to finite declared integer sets. A task contains five shuffled input--output demonstrations and one query input. The model receives a normalized serialization of those demonstrations and the query. The protected target is y(τ) = ( e_θ, fracp…
Exact mathematical passage
f_{\mathrm{aff}}(x)&=ax+b, \\ f_{\mathrm{abs}}(x)&=a|x|+b, \\ f_{\mathrm{quad}}(x)&=ax^2+bx+c, \\ f_{\mathrm{hinge}}(x)&= \begin{cases} ax+c,&x\geq0,\\ bx+c,&x<0. \end{cases} \label{eq:b18-program-templates}
f_iUse in source · 1 passage
Search aliases: f_i
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13778
A finding is represented by f_i = (Π_i^-,Π_i^+,code_i, Par(i),e_i,o_i), where e_i is the observed effect profile and o_i is either a surviving obstruction or a validated no-go certificate. Directed edges are typed as extension, ablation, refutation, transport, or replication. A proposal must cite the archive nodes on which its mechanism depends.
Exact mathematical passage
f_i = (\Pi_i^-,\Pi_i^+,\mathrm{code}_i, \operatorname{Par}(i),e_i,o_i), \label{eq:b13-finding-node}
f_jUse in source · 5 passages
Search aliases: f_j
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11421
[Open-set incompatibility of distinct affine charts] Let U_0,U_1⊆ℝ^d each contain a nonempty open set. Let f_j(h)=A_jh+b_j, j∈{0,1}. If one affine map f(h)=Ah+b satisfies f(h)=f_j(h) for every h∈ U_j and both j, then A=A_0=A_1, b=b_0=b_1. Consequently, if (A_0,b_0)≠(A_1,b_1), no element of G_glob represents the two-chart relation exactly.
Exact mathematical passage
f_j(h)=A_jh+b_j, \qquad j\in\{0,1\}.
F_mUse in source · 1 passage
Search aliases: F_m
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16014
Let an affine instruction be f_j(z)=a_jz+b_j, a_j∈{-1,-tfrac12,tfrac12,1}, b_j∈{-tfrac12,0,tfrac12}. A depth-m program acts on x∈{-1,-tfrac34,…,1} by F_m(x) = (f_m∘⋯∘ f_1)(x) =A_mx+B_m, where A_0=1, B_0=0, A_j=a_jA_j-1, B_j=a_jB_j-1+b_j. All quantities in [source reference] are evaluated with exact rational arithmetic before a floating representation is exposed to the model.
Exact mathematical passage
F_m(x) = (f_m\circ\cdots\circ f_1)(x) =A_mx+B_m, \label{eq:b17-affine-program}
f_mUse in source · 1 passage
Search aliases: f_m
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16014
Let an affine instruction be f_j(z)=a_jz+b_j, a_j∈{-1,-tfrac12,tfrac12,1}, b_j∈{-tfrac12,0,tfrac12}. A depth-m program acts on x∈{-1,-tfrac34,…,1} by F_m(x) = (f_m∘⋯∘ f_1)(x) =A_mx+B_m, where A_0=1, B_0=0, A_j=a_jA_j-1, B_j=a_jB_j-1+b_j. All quantities in [source reference] are evaluated with exact rational arithmetic before a floating representation is exposed to the model.
Exact mathematical passage
F_m(x) = (f_m\circ\cdots\circ f_1)(x) =A_mx+B_m, \label{eq:b17-affine-program}
f_{\mathrm{quad}}Use in source · 2 passages
Search aliases: f_quad mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16627
Let the template set be Θ = {θ_aff,θ_abs, θ_quad,θ_hinge}. For integer parameters a,b,c and integer input x, the exact programs are f_aff(x) · =ax+b, f_abs(x) · =a|x|+b, f_quad(x) · =ax^2+bx+c, f_hinge(x) · = ax+c, · x≥0, bx+c, · x<0. The coefficients belong to finite declared integer sets. A task contains five shuffled input--output demonstrations and one query input. The model receives a normalized serialization of those demonstrations and the query. The protected target is y(τ) = ( e_θ, fracp…
Exact mathematical passage
f_{\mathrm{aff}}(x)&=ax+b, \\ f_{\mathrm{abs}}(x)&=a|x|+b, \\ f_{\mathrm{quad}}(x)&=ax^2+bx+c, \\ f_{\mathrm{hinge}}(x)&= \begin{cases} ax+c,&x\geq0,\\ bx+c,&x<0. \end{cases} \label{eq:b18-program-templates}
\mathsf F_{\rm audit}^{\rm primary}Use in source · 2 passages
Search aliases: F_rm audit^rm primary mathsf rm rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8376
The Block 7 primary audit differs from development in both task and representation family: T_rm dev · ∩ T_rm audit=∅, F_rm dev^rm primary · ∩ F_rm audit^rm primary=∅. Development uses ten tasks generated from four trigger families. Audit uses twelve new tasks generated from radial and mixed triggers, new teacher seeds, and new corruption permutations. The primary audit representations are dense linear 20-dimensional, block-linear 24-dimensional, and ill-conditioned linear 20-dimensional systems.…
Exact mathematical passage
\begin{aligned} \mathsf T_{\rm dev}&\cap\mathsf T_{\rm audit}=\varnothing,\\ \mathsf F_{\rm dev}^{\rm primary}&\cap\mathsf F_{\rm audit}^{\rm primary}=\varnothing. \end{aligned}
\mathsf F_{\rm audit}^{\rm stress}Use in source · 1 passage
Search aliases: F_rm audit^rm stress mathsf rm rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8627
The primary audit families are: F_rm audit^rm primary = set dense-linear-20, block-linear-24, ill-conditioned-linear-20 . The stress families are: F_rm audit^rm stress = set compressed-10, saturated-tanh-24, leaky-ReLU-32, smooth-polynomial-24 . The primary families satisfy the linear bridge hypothesis but are absent from primary development. The stress families violate injectivity or global linearity in different ways.
Exact mathematical passage
\mathsf F_{\rm audit}^{\rm stress} = \set{ \text{compressed-10}, \text{saturated-tanh-24}, \text{leaky-ReLU-32}, \text{smooth-polynomial-24} }.
\mathsf F_{\rm dev}^{\rm primary}Use in source · 1 passage
Search aliases: F_rm dev^rm primary mathsf rm rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8376
The Block 7 primary audit differs from development in both task and representation family: T_rm dev · ∩ T_rm audit=∅, F_rm dev^rm primary · ∩ F_rm audit^rm primary=∅. Development uses ten tasks generated from four trigger families. Audit uses twelve new tasks generated from radial and mixed triggers, new teacher seeds, and new corruption permutations. The primary audit representations are dense linear 20-dimensional, block-linear 24-dimensional, and ill-conditioned linear 20-dimensional systems.…
Exact mathematical passage
\begin{aligned} \mathsf T_{\rm dev}&\cap\mathsf T_{\rm audit}=\varnothing,\\ \mathsf F_{\rm dev}^{\rm primary}&\cap\mathsf F_{\rm audit}^{\rm primary}=\varnothing. \end{aligned}
F_xUse in source · 5 passages
Search aliases: F_x
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9938
For episode x, let F_x:U_x^auth→ V_x be the post-modification defect map, with F_x(0)=r_x. A valid repair satisfies F_x(u_x)=0. A repair schema is a local section s:x↦ u_x that approximately inverts F_x over its declared domain.
Exact mathematical passage
F_x:U_x^{\Auth}\to V_x
f_{\theta,p}Use in source · 1 passage
Search aliases: f_θ,p theta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16642
bx+c, · x<0. The coefficients belong to finite declared integer sets. A task contains five shuffled input--output demonstrations and one query input. The model receives a normalized serialization of those demonstrations and the query. The protected target is y(τ) = ( e_θ, fracp_12,…,fracp_42, fracd(θ)5, fracf_θ,p(x_q)32 )∈ℝ^10, where e_θ∈ℝ^4 is the one-hot template code, unused parameter slots are zero, and d(θ)∈{2,3,4,5} is the declared execution depth. All demonstrations, query answers, semant…
Exact mathematical passage
y(\tau) = \left( e_{\theta}, \frac{p_1}{2},\ldots,\frac{p_4}{2}, \frac{d(\theta)}{5}, \frac{f_{\theta,p}(x_q)}{32} \right)\in\R^{10}, \label{eq:b18-protected-target}
\mathsf{FAD}Use in source · 3 passages
Search aliases: FAD mathsf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16182
t d_prog(τ,τ') be a normalized distance between (A_m,B_m) and (A'_m',B'_m'), and let d_ans(τ,τ') = absy_m/4-y'_m'/4. For a recipient task τ_i, define four donor classes: item SAP: same-answer, far-program, approximately depth-matched donors; item FAD: far-answer, depth-matched donors; item SN: semantic-neighbor donors; item RMD: random depth-matched donors. Three donors per available class are selected by deterministic task-identifier hashing. Each audit records donor coverage and the realized s…
Exact mathematical passage
\mathsf{FAD}
\mathrm{far\ answer,depth}Use in source · 1 passage
Search aliases: far answer,depth mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16789
s L_i, the audit reuses the Block 17 donor distinctions and adds cross-implementation status labels. It records latent necessity, reversal cost, norm-matched noise cost, exact replay, same-answer far-program transplant cost, far-answer matched-depth transplant cost, local semantic-neighbor cost, external trace-probe R^2, dynamic-depth correlation, and exact program execution. Let n_i · =L_i^0-L_i, · o_i · =L_i^rev-L_i, · c_i · =L_i^η-L_i, p_i · =L_i,sem^same answer,far program -L_i,sem, · a_i · …
Exact mathematical passage
n_i&=L_i^{0}-L_i, &o_i&=L_i^{\mathrm{rev}}-L_i, &c_i&=L_i^{\eta}-L_i, \\ p_i&=L_{i,\mathrm{sem}}^{\mathrm{same\ answer,far\ program}} -L_{i,\mathrm{sem}}, &a_i&=L_{i,\mathrm{answer}}^{\mathrm{far\ answer,depth}} -L_{i,\mathrm{answer}}. \label{eq:b18-audit-coordinates}
\Fcal_aUse in source · 1 passage
Search aliases: Fcal_a F_a Fcal mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12346
For each recovered block widehat P_a, let F_a be a finite family of candidate separators g_a,θ,u(h) = 1{anglesu,widehat P_ah>θ}. The compiler fits a two-chart bridge for every candidate and evaluates a score containing: item held-out local-bridge validation gain; item source-repair invariance of the chart assignment; item agreement between source and target chart assignments; item chart-balance constraints; item response-fit, commutator, and spectral-gap penalties; item protected-context regress…
Exact mathematical passage
\Fcal_a
\Fcal_tUse in source · 4 passages
Search aliases: Fcal_t F_t Fcal mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4137
Let (F_t)_t≥0 be the filtration generated by all committed lineage history through transaction t. A proposal is precommitted when its edit, claimed effect, target metrics, stopping rule, and evaluation namespace are F_t-1-measurable. The new evaluation trace is revealed only afterward.
Exact mathematical passage
(\Fcal_t)_{t\geq0}
\Fcal_{t-1}Use in source · 3 passages
Search aliases: Fcal_t-1 F_t-1 Fcal mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4137
Let (F_t)_t≥0 be the filtration generated by all committed lineage history through transaction t. A proposal is precommitted when its edit, claimed effect, target metrics, stopping rule, and evaluation namespace are F_t-1-measurable. The new evaluation trace is revealed only afterward.
Exact mathematical passage
\Fcal_{t-1}
\Fcal_XUse in source · 1 passage
Search aliases: Fcal_X F_X Fcal mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11564
In the executable fixture, P_ctxh=(h_1,h_2), and the source repair leaves these coordinates invariant. The initial separator vocabulary is F_X = set 1{anglesu_k,P_ctxh>θ} : 0≤ k<128, θ∈Θ_k(X) , where u_k=(cos(2π k/128),sin(2π k/128)) and Θ_k(X) contains zero and the empirical 0.30,0.40,0.50,0.60,0.70 quantiles of the projected fit states. For every candidate separator, the compiler fits two weighted affine bridges and evaluates held-out chart-selection error. It then performs at most five determ…
Exact mathematical passage
\Fcal_X = \set{ \one\{\angles{u_k,P_{\mathrm{ctx}}h}>\theta\} : 0\leq k<128, \ \theta\in\Theta_k(X) },
\mathrm{FiLM}Use in source · 1 passage
Search aliases: FiLM mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16735
The source grammar contains a task-blind continuation, inherited additive conditioning, final-step FiLM, progressive FiLM with geometry, a low-rank hypernetwork, and a shuffled-witness FiLM control. For the selected FiLM constructor, the encoded task produces step-indexed scale and shift vectors (γ_k(e),β_k(e))_k=0^4 =F_FiLM(e), which act on the preactivation by q_k ⟼ q_kodot(1+frac12tanhγ_k(e)) +β_k(e). This constructor is absent from Block 17's rank-one, rank-four, and full additive conditioni…
Exact mathematical passage
(\gamma_k(e),\beta_k(e))_{k=0}^{4} =F_{\mathrm{FiLM}}(e), \label{eq:b18-film}
\mathrm{FiLM-add}Use in source · 1 passage
Search aliases: FiLM-add mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16956
The new constructor also fails a stronger comparison with the inherited additive repair. Pooled over the two architectures and two final supports, the selected-child minus legacy-additive program-specificity effect is widehatΔ_FiLM-add =-0.029417, CI_0.95 =[-0.034102,-0.024735]. The negative interval forbids a claim that the new source-selected constructor dominates the inherited constructor. It can still be structurally new and useful on the primary audits, but novelty is not superiority.
Exact mathematical passage
\widehat\Delta_{\mathrm{FiLM-add}} =-0.029417, \qquad \mathrm{CI}_{0.95} =[-0.034102,-0.024735]. \label{eq:b18-legacy-comparison}
\mathrm{finalize}Use in source · 1 passage
Search aliases: finalize mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4027
The transaction follows six phases: prepare ⟶ seal ⟶ execute ⟶ reveal ⟶ decide ⟶ finalize. Only the final phase may expose buffered effects or replace the active checkpoint.
Exact mathematical passage
\mathrm{prepare} \longrightarrow \mathrm{seal} \longrightarrow \mathrm{execute} \longrightarrow \mathrm{reveal} \longrightarrow \mathrm{decide} \longrightarrow \mathrm{finalize}.
\mathrm{flat}Use in source · 2 passages
Search aliases: flat mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5602
Case F3: flat repair. Set D_flat=I_2, r_flat= 1 -2 , with zero overlap mismatch and identity evaluator transport. The unique minimum-norm repair is u_flat=-r_flat.
Exact mathematical passage
D_{\mathrm{flat}}=I_2, \qquad r_{\mathrm{flat}}= \begin{bmatrix} 1\\-2 \end{bmatrix},
\mathrm{forecast}Use in source · 1 passage
Search aliases: forecast mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13751
The forecast residual and proposal--code congruence residual are ε_forecast,i = normwidehat c_i- c_i, ε_spec/code,i = d_sem ( Sem(Π_i^-), Sem(code_i) ). In the executable finite companion, congruence is a Boolean monitor rather than a complete semantic metric.
Exact mathematical passage
\eps_{\mathrm{forecast},i} = \norm{\widehat{\mathbf c}_i-\mathbf c_i}, \qquad \eps_{\mathrm{spec/code},i} = d_{\mathrm{sem}} \left( \mathrm{Sem}(\Pi_i^-), \mathrm{Sem}(\mathrm{code}_i) \right). \label{eq:b13-receipt-residuals}
\mathrm{fr}Use in source · 2 passages
Search aliases: fr mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 811
A witness structure may be represented as W_t = ( V_t^fr, C_t, B_t^sand, R_t, Π_t^auth ), where V_t^fr is the frozen evaluator instance for the current transaction, C_t is a protected checkpoint, B_t^sand is the sandbox boundary, R_t is the rollback procedure, and Π_t^auth records effect authority.
Exact mathematical passage
\Wcal_t = \left( \Vcal_t^{\mathrm{fr}}, C_t, \Bcal_t^{\mathrm{sand}}, R_t, \Pi_t^{\mathrm{auth}} \right),
\mathrm{full}Use in source · 1 passage
Search aliases: full mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16109
efine z_0=ζ, z_k+1 = tanh (Rz_k+s_k+C_α(e,m)), 0≤ k<4, where s_k is a learned step embedding. The task-blind parent satisfies C_parent(e,m)=0. The descendants use one of the following conditioning maps: C_depth(e,m) · =D 1_m, C_1(e,m) · =U_1V_1e, rank(U_1V_1)≤1, C_4(e,m) · =U_4V_4e, rank(U_4V_4)≤4, C_full(e,m) · =We. Here 1_m is the one-hot depth code. Each child inherits every compatible parent parameter. The added up-projection or full-conditioning matrix is initialized so that the initial inj…
Exact mathematical passage
C_{\mathrm{depth}}(e,m)&=D\,\one_m, \label{eq:b17-depth-conditioning}\\ C_{1}(e,m)&=U_1V_1e, \qquad \rank(U_1V_1)\leq1, \label{eq:b17-rank1-conditioning}\\ C_{4}(e,m)&=U_4V_4e, \qquad \rank(U_4V_4)\leq4, \label{eq:b17-rank4-conditioning}\\ C_{\mathrm{full}}(e,m)&=We. \label{eq:b17-full-conditioning}
G^2Use in source · 2 passages
Search aliases: G^2
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3360
Hoeffding's inequality for a variable with range width 2G gives Pr( overline Z_n- EZ_i≥ t ) ≤ exp(-fracnt^22G^2). Set the right side to α. The family statement follows by the union bound.
Exact mathematical passage
\Pr\left( \overline Z_n-\mathbb EZ_i\geq t \right) \leq \exp\left(-\frac{nt^2}{2G^2}\right).
G^\starUse in source · 2 passages
Search aliases: G^⋆ star
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13154
[Finite constructor selection with conservative tie-break] Suppose a finite constructor family is evaluated on fixed development evidence. Let G^⋆ be the largest held-out development gain and let C_τ={C:G(C)≥ G^⋆-τ}. Selecting the minimum-complexity constructor in C_τ is deterministic and preserves an explicit near-optimality bound of τ on the development criterion.
Exact mathematical passage
G^\star
G_0Use in source · 1 passage
Search aliases: G_0
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 18386
The finite grammar is G_21 = { ll G_0 · step-affine map, G_1 · full-trajectory affine map, G_2 · monotone nonlinear inverse plus step-affine map, G_3 · bidiagonal temporal inverse plus step-affine map, G_4 · probe-conditioned two-chart affine map, G_5 · semantic quotient, G_bot · collision-grounded no-go. .
Exact mathematical passage
\mathcal G_{21} = \left\{ \begin{array}{ll} G_0&\text{step-affine map},\\ G_1&\text{full-trajectory affine map},\\ G_2&\text{monotone nonlinear inverse plus step-affine map},\\ G_3&\text{bidiagonal temporal inverse plus step-affine map},\\ G_4&\text{probe-conditioned two-chart affine map},\\ G_5&\text{semantic quotient},\\ G_{\bot}&\text{collision-grounded no-go}. \end{array} \right. \label{eq:block21-grammar}
\mathfrak G_0Use in source · 2 passages
Search aliases: G_0 mathfrak
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15204
_- and C_+; item there exist m_- ,m_+∈C_rm pair such that m_- is acceptable throughout C_- and m_+ is acceptable throughout C_+; item neither one inherited singleton method nor one unrefined binding is acceptable throughout C; item old-token behavior is preserved through the fallback of [source reference]. Then the extension G_0 ↪ G_1 = G_0 ∪ setp,(bot,-),(bot,+),m_-,m_+ eliminates the development conflict while conservatively preserving the inherited compiler.
Exact mathematical passage
\mathfrak G_0 \hookrightarrow \mathfrak G_1 = \mathfrak G_0 \cup \set{p,(\bot,-),(\bot,+),m_-,m_+}
G_1Use in source · 2 passages
Search aliases: G_1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4661
[Expected best-of-k gain] Let G_1,…,G_k be independent nonnegative candidate gains with common cumulative distribution function F and finite mean. Then Emax_1≤ i≤ kG_i = ∫_0^∞ (1-F(x)^k) d x. In particular, expected best gain is nondecreasing in k even when the candidate generator itself does not improve.
Exact mathematical passage
G_1,\dots,G_k
\mathfrak G_1Use in source · 1 passage
Search aliases: G_1 mathfrak
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15204
_- and C_+; item there exist m_- ,m_+∈C_rm pair such that m_- is acceptable throughout C_- and m_+ is acceptable throughout C_+; item neither one inherited singleton method nor one unrefined binding is acceptable throughout C; item old-token behavior is preserved through the fallback of [source reference]. Then the extension G_0 ↪ G_1 = G_0 ∪ setp,(bot,-),(bot,+),m_-,m_+ eliminates the development conflict while conservatively preserving the inherited compiler.
Exact mathematical passage
\mathfrak G_0 \hookrightarrow \mathfrak G_1 = \mathfrak G_0 \cup \set{p,(\bot,-),(\bot,+),m_-,m_+}
G_2Use in source · 1 passage
Search aliases: G_2
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 18386
The finite grammar is G_21 = { ll G_0 · step-affine map, G_1 · full-trajectory affine map, G_2 · monotone nonlinear inverse plus step-affine map, G_3 · bidiagonal temporal inverse plus step-affine map, G_4 · probe-conditioned two-chart affine map, G_5 · semantic quotient, G_bot · collision-grounded no-go. .
Exact mathematical passage
\mathcal G_{21} = \left\{ \begin{array}{ll} G_0&\text{step-affine map},\\ G_1&\text{full-trajectory affine map},\\ G_2&\text{monotone nonlinear inverse plus step-affine map},\\ G_3&\text{bidiagonal temporal inverse plus step-affine map},\\ G_4&\text{probe-conditioned two-chart affine map},\\ G_5&\text{semantic quotient},\\ G_{\bot}&\text{collision-grounded no-go}. \end{array} \right. \label{eq:block21-grammar}
\mathcal G_{21}Use in source · 2 passages
Search aliases: G_21 mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 18386
The finite grammar is G_21 = { ll G_0 · step-affine map, G_1 · full-trajectory affine map, G_2 · monotone nonlinear inverse plus step-affine map, G_3 · bidiagonal temporal inverse plus step-affine map, G_4 · probe-conditioned two-chart affine map, G_5 · semantic quotient, G_bot · collision-grounded no-go. .
Exact mathematical passage
\mathcal G_{21} = \left\{ \begin{array}{ll} G_0&\text{step-affine map},\\ G_1&\text{full-trajectory affine map},\\ G_2&\text{monotone nonlinear inverse plus step-affine map},\\ G_3&\text{bidiagonal temporal inverse plus step-affine map},\\ G_4&\text{probe-conditioned two-chart affine map},\\ G_5&\text{semantic quotient},\\ G_{\bot}&\text{collision-grounded no-go}. \end{array} \right. \label{eq:block21-grammar}
G_3Use in source · 1 passage
Search aliases: G_3
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 18386
The finite grammar is G_21 = { ll G_0 · step-affine map, G_1 · full-trajectory affine map, G_2 · monotone nonlinear inverse plus step-affine map, G_3 · bidiagonal temporal inverse plus step-affine map, G_4 · probe-conditioned two-chart affine map, G_5 · semantic quotient, G_bot · collision-grounded no-go. .
Exact mathematical passage
\mathcal G_{21} = \left\{ \begin{array}{ll} G_0&\text{step-affine map},\\ G_1&\text{full-trajectory affine map},\\ G_2&\text{monotone nonlinear inverse plus step-affine map},\\ G_3&\text{bidiagonal temporal inverse plus step-affine map},\\ G_4&\text{probe-conditioned two-chart affine map},\\ G_5&\text{semantic quotient},\\ G_{\bot}&\text{collision-grounded no-go}. \end{array} \right. \label{eq:block21-grammar}
G_4Use in source · 1 passage
Search aliases: G_4
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 18386
The finite grammar is G_21 = { ll G_0 · step-affine map, G_1 · full-trajectory affine map, G_2 · monotone nonlinear inverse plus step-affine map, G_3 · bidiagonal temporal inverse plus step-affine map, G_4 · probe-conditioned two-chart affine map, G_5 · semantic quotient, G_bot · collision-grounded no-go. .
Exact mathematical passage
\mathcal G_{21} = \left\{ \begin{array}{ll} G_0&\text{step-affine map},\\ G_1&\text{full-trajectory affine map},\\ G_2&\text{monotone nonlinear inverse plus step-affine map},\\ G_3&\text{bidiagonal temporal inverse plus step-affine map},\\ G_4&\text{probe-conditioned two-chart affine map},\\ G_5&\text{semantic quotient},\\ G_{\bot}&\text{collision-grounded no-go}. \end{array} \right. \label{eq:block21-grammar}
G_5Use in source · 1 passage
Search aliases: G_5
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 18386
The finite grammar is G_21 = { ll G_0 · step-affine map, G_1 · full-trajectory affine map, G_2 · monotone nonlinear inverse plus step-affine map, G_3 · bidiagonal temporal inverse plus step-affine map, G_4 · probe-conditioned two-chart affine map, G_5 · semantic quotient, G_bot · collision-grounded no-go. .
Exact mathematical passage
\mathcal G_{21} = \left\{ \begin{array}{ll} G_0&\text{step-affine map},\\ G_1&\text{full-trajectory affine map},\\ G_2&\text{monotone nonlinear inverse plus step-affine map},\\ G_3&\text{bidiagonal temporal inverse plus step-affine map},\\ G_4&\text{probe-conditioned two-chart affine map},\\ G_5&\text{semantic quotient},\\ G_{\bot}&\text{collision-grounded no-go}. \end{array} \right. \label{eq:block21-grammar}
G_{a,c}Use in source · 2 passages
Search aliases: G_a,c
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17760
Suppose an underlying canonical trajectory is represented in chart c by a row vector z and architecture-dependent gauge matrices: h_a,c=zG_a,c, h_b,c=zG_b,c. The matrices may include coordinate rotations, sign changes, and architecture-specific temporal mixing. Block 20 uses deterministic orthogonal gauges as a controlled stress at the exposed trajectory interface.
Exact mathematical passage
h_{a,c}=zG_{a,c}, \qquad h_{b,c}=zG_{b,c}. \label{eq:b20-gauge-model}
G_{a,c_1}Use in source · 2 passages
Search aliases: G_a,c_1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17778
Let V⊆ R^m be spanned by the canonical trajectories observed in two charts c_1,c_2. Suppose one linear map T exactly transports architecture b into architecture a in both charts: zG_b,c_iT=zG_a,c_i for every z∈ V, i∈{1,2}. Then the restrictions to V satisfy G_b,c_1T=G_a,c_1, G_b,c_2T=G_a,c_2. If G_b,c_i are invertible on V, exact pooled transport is possible only when G_b,c_1^-1G_a,c_1 = G_b,c_2^-1G_a,c_2 on V.
Exact mathematical passage
G_{b,c_1}T=G_{a,c_1}, \qquad G_{b,c_2}T=G_{a,c_2}.
G_{a,c_2}Use in source · 2 passages
Search aliases: G_a,c_2
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17778
Let V⊆ R^m be spanned by the canonical trajectories observed in two charts c_1,c_2. Suppose one linear map T exactly transports architecture b into architecture a in both charts: zG_b,c_iT=zG_a,c_i for every z∈ V, i∈{1,2}. Then the restrictions to V satisfy G_b,c_1T=G_a,c_1, G_b,c_2T=G_a,c_2. If G_b,c_i are invertible on V, exact pooled transport is possible only when G_b,c_1^-1G_a,c_1 = G_b,c_2^-1G_a,c_2 on V.
Exact mathematical passage
G_{b,c_1}T=G_{a,c_1}, \qquad G_{b,c_2}T=G_{a,c_2}.
G_{a,c_i}Use in source · 1 passage
Search aliases: G_a,c_i
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17771
[Pooled bridge compatibility obstruction] Let V⊆ R^m be spanned by the canonical trajectories observed in two charts c_1,c_2. Suppose one linear map T exactly transports architecture b into architecture a in both charts: zG_b,c_iT=zG_a,c_i for every z∈ V, i∈{1,2}. Then the restrictions to V satisfy G_b,c_1T=G_a,c_1, G_b,c_2T=G_a,c_2. If G_b,c_i are invertible on V, exact pooled transport is possible only when G_b,c_1^-1G_a,c_1 = G_b,c_2^-1G_a,c_2 on V.
Exact mathematical passage
zG_{b,c_i}T=zG_{a,c_i} \qquad \text{for every }z\in V, \quad i\in\{1,2\}.
G_{a,j}Use in source · 1 passage
Search aliases: G_a,j
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6280
For arms a and b, evaluate both on the same frozen foreign agents and seeds. Let Δ_j=G_a,j-G_b,j. The paired estimate and standard error are widehatΔ =frac1n∑_j=1^nΔ_j, widehatse(widehatΔ) =sqrtfrac1n(n-1)∑_j=1^n(Δ_j-widehatΔ)^2. When gains are bounded in [-1,1], a distribution-free Hoeffding lower bound is LCB_1-α(Δ) = widehatΔ - sqrtfrac2log(1/α)n. Sharper paired permutation or bootstrap intervals may be used when frozen in advance.
Exact mathematical passage
\Delta_j=G_{a,j}-G_{b,j}.
G_{a,j,1}Use in source · 1 passage
Search aliases: G_a,j,1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6139
The archive arm can perform well without a causal repair calculus because it searches several noisy candidates and retains the best. Candidate-budget matching is therefore mandatory. Let G_a,j,1,…,G_a,j,B be candidate gains for arm a on agent j. The primary behavioral quantity is M_a(B) = frac1n∑_j=1^nmax_1≤ b≤ BG_a,j,b. Comparing arms at unequal B confounds repair competence with order statistics.
Exact mathematical passage
G_{a,j,1},\ldots,G_{a,j,B}
G_{a,j,B}Use in source · 1 passage
Search aliases: G_a,j,B
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6139
The archive arm can perform well without a causal repair calculus because it searches several noisy candidates and retains the best. Candidate-budget matching is therefore mandatory. Let G_a,j,1,…,G_a,j,B be candidate gains for arm a on agent j. The primary behavioral quantity is M_a(B) = frac1n∑_j=1^nmax_1≤ b≤ BG_a,j,b. Comparing arms at unequal B confounds repair competence with order statistics.
Exact mathematical passage
G_{a,j,1},\ldots,G_{a,j,B}
G_{a,j,b}Use in source · 1 passage
Search aliases: G_a,j,b
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6143
The archive arm can perform well without a causal repair calculus because it searches several noisy candidates and retains the best. Candidate-budget matching is therefore mandatory. Let G_a,j,1,…,G_a,j,B be candidate gains for arm a on agent j. The primary behavioral quantity is M_a(B) = frac1n∑_j=1^nmax_1≤ b≤ BG_a,j,b. Comparing arms at unequal B confounds repair competence with order statistics.
Exact mathematical passage
M_a(B) = \frac1n\sum_{j=1}^n\max_{1\leq b\leq B}G_{a,j,b}.
g_{a,\theta,u}Use in source · 1 passage
Search aliases: g_a,θ,u theta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12347
For each recovered block widehat P_a, let F_a be a finite family of candidate separators g_a,θ,u(h) = 1{anglesu,widehat P_ah>θ}. The compiler fits a two-chart bridge for every candidate and evaluates a score containing: item held-out local-bridge validation gain; item source-repair invariance of the chart assignment; item agreement between source and target chart assignments; item chart-balance constraints; item response-fit, commutator, and spectral-gap penalties; item protected-context regress…
Exact mathematical passage
g_{a,\theta,u}(h) = \one\{\angles{u,\widehat P_ah}>\theta\}.
g_BUse in source · 4 passages
Search aliases: g_B
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3174
Let ℓ_A:V_A→ℝ and ℓ_B:V_B→ℝ be post-defect losses. Define source and target gains g_A := ℓ_A(r_A)-ℓ_A(r_A+D_Au_A), g_B := ℓ_B(r_B)-ℓ_B(r_B+D_BT_Uu_A).
Exact mathematical passage
g_B := \ell_B(r_B)-\ell_B(r_B+D_BT_Uu_A).
G_{b,c}Use in source · 1 passage
Search aliases: G_b,c
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17760
Suppose an underlying canonical trajectory is represented in chart c by a row vector z and architecture-dependent gauge matrices: h_a,c=zG_a,c, h_b,c=zG_b,c. The matrices may include coordinate rotations, sign changes, and architecture-specific temporal mixing. Block 20 uses deterministic orthogonal gauges as a controlled stress at the exposed trajectory interface.
Exact mathematical passage
h_{a,c}=zG_{a,c}, \qquad h_{b,c}=zG_{b,c}. \label{eq:b20-gauge-model}
G_{b,c}^{\mathsf T}Use in source · 1 passage
Search aliases: G_b,c^ T mathsf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17805
[Exact chart-indexed gauge transport] Under the same hypotheses, the chart-indexed family T^c_b→ a = G_b,c^-1G_a,c transports each chart exactly. If all gauges are orthogonal, then G_b,c^-1=G_b,c^ T.
Exact mathematical passage
G_{b,c}^{-1}=G_{b,c}^{\mathsf T}
G_{b,c}^{-1}Use in source · 2 passages
Search aliases: G_b,c^-1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17800
[Exact chart-indexed gauge transport] Under the same hypotheses, the chart-indexed family T^c_b→ a = G_b,c^-1G_a,c transports each chart exactly. If all gauges are orthogonal, then G_b,c^-1=G_b,c^ T.
Exact mathematical passage
T^{c}_{b\to a} = G_{b,c}^{-1}G_{a,c}
G_{b,c_1}Use in source · 1 passage
Search aliases: G_b,c_1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17778
Let V⊆ R^m be spanned by the canonical trajectories observed in two charts c_1,c_2. Suppose one linear map T exactly transports architecture b into architecture a in both charts: zG_b,c_iT=zG_a,c_i for every z∈ V, i∈{1,2}. Then the restrictions to V satisfy G_b,c_1T=G_a,c_1, G_b,c_2T=G_a,c_2. If G_b,c_i are invertible on V, exact pooled transport is possible only when G_b,c_1^-1G_a,c_1 = G_b,c_2^-1G_a,c_2 on V.
Exact mathematical passage
G_{b,c_1}T=G_{a,c_1}, \qquad G_{b,c_2}T=G_{a,c_2}.
G_{b,c_1}^{-1}Use in source · 1 passage
Search aliases: G_b,c_1^-1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17784
exactly transports architecture b into architecture a in both charts: zG_b,c_iT=zG_a,c_i for every z∈ V, i∈{1,2}. Then the restrictions to V satisfy G_b,c_1T=G_a,c_1, G_b,c_2T=G_a,c_2. If G_b,c_i are invertible on V, exact pooled transport is possible only when G_b,c_1^-1G_a,c_1 = G_b,c_2^-1G_a,c_2 on V.
Exact mathematical passage
G_{b,c_1}^{-1}G_{a,c_1} = G_{b,c_2}^{-1}G_{a,c_2} \quad\text{on }V. \label{eq:b20-compatibility}
G_{b,c_2}Use in source · 1 passage
Search aliases: G_b,c_2
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17778
Let V⊆ R^m be spanned by the canonical trajectories observed in two charts c_1,c_2. Suppose one linear map T exactly transports architecture b into architecture a in both charts: zG_b,c_iT=zG_a,c_i for every z∈ V, i∈{1,2}. Then the restrictions to V satisfy G_b,c_1T=G_a,c_1, G_b,c_2T=G_a,c_2. If G_b,c_i are invertible on V, exact pooled transport is possible only when G_b,c_1^-1G_a,c_1 = G_b,c_2^-1G_a,c_2 on V.
Exact mathematical passage
G_{b,c_1}T=G_{a,c_1}, \qquad G_{b,c_2}T=G_{a,c_2}.
G_{b,c_2}^{-1}Use in source · 1 passage
Search aliases: G_b,c_2^-1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17784
exactly transports architecture b into architecture a in both charts: zG_b,c_iT=zG_a,c_i for every z∈ V, i∈{1,2}. Then the restrictions to V satisfy G_b,c_1T=G_a,c_1, G_b,c_2T=G_a,c_2. If G_b,c_i are invertible on V, exact pooled transport is possible only when G_b,c_1^-1G_a,c_1 = G_b,c_2^-1G_a,c_2 on V.
Exact mathematical passage
G_{b,c_1}^{-1}G_{a,c_1} = G_{b,c_2}^{-1}G_{a,c_2} \quad\text{on }V. \label{eq:b20-compatibility}
G_{b,c_i}Use in source · 2 passages
Search aliases: G_b,c_i
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17771
[Pooled bridge compatibility obstruction] Let V⊆ R^m be spanned by the canonical trajectories observed in two charts c_1,c_2. Suppose one linear map T exactly transports architecture b into architecture a in both charts: zG_b,c_iT=zG_a,c_i for every z∈ V, i∈{1,2}. Then the restrictions to V satisfy G_b,c_1T=G_a,c_1, G_b,c_2T=G_a,c_2. If G_b,c_i are invertible on V, exact pooled transport is possible only when G_b,c_1^-1G_a,c_1 = G_b,c_2^-1G_a,c_2 on V.
Exact mathematical passage
zG_{b,c_i}T=zG_{a,c_i} \qquad \text{for every }z\in V, \quad i\in\{1,2\}.
G_{b,j}Use in source · 1 passage
Search aliases: G_b,j
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6280
For arms a and b, evaluate both on the same frozen foreign agents and seeds. Let Δ_j=G_a,j-G_b,j. The paired estimate and standard error are widehatΔ =frac1n∑_j=1^nΔ_j, widehatse(widehatΔ) =sqrtfrac1n(n-1)∑_j=1^n(Δ_j-widehatΔ)^2. When gains are bounded in [-1,1], a distribution-free Hoeffding lower bound is LCB_1-α(Δ) = widehatΔ - sqrtfrac2log(1/α)n. Sharper paired permutation or bootstrap intervals may be used when frozen in advance.
Exact mathematical passage
\Delta_j=G_{a,j}-G_{b,j}.
G_{\bot}Use in source · 2 passages
Search aliases: G_bot bot
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 18386
The finite grammar is G_21 = { ll G_0 · step-affine map, G_1 · full-trajectory affine map, G_2 · monotone nonlinear inverse plus step-affine map, G_3 · bidiagonal temporal inverse plus step-affine map, G_4 · probe-conditioned two-chart affine map, G_5 · semantic quotient, G_bot · collision-grounded no-go. .
Exact mathematical passage
\mathcal G_{21} = \left\{ \begin{array}{ll} G_0&\text{step-affine map},\\ G_1&\text{full-trajectory affine map},\\ G_2&\text{monotone nonlinear inverse plus step-affine map},\\ G_3&\text{bidiagonal temporal inverse plus step-affine map},\\ G_4&\text{probe-conditioned two-chart affine map},\\ G_5&\text{semantic quotient},\\ G_{\bot}&\text{collision-grounded no-go}. \end{array} \right. \label{eq:block21-grammar}
G_{\mathrm{cause}}Use in source · 1 passage
Search aliases: G_cause mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5527
G_cause · Causal role · Intervention, necessity ablation, matched negative controls
Exact mathematical passage
G_{\mathrm{cause}}
G_{e_j}Use in source · 1 passage
Search aliases: G_e_j
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7696
For n held-out episodes, define paired differences D_j = G_e_j(K_1)-G_e_j(K_0), widehatΔ_n=frac1n∑_j=1^nD_j. Because parent and child share each target and candidate bundle, pairing removes much of the episode difficulty variation.
Exact mathematical passage
D_j = G_{e_j}(K_1)-G_{e_j}(K_0), \qquad \widehat\Delta_n=\frac1n\sum_{j=1}^nD_j. \label{eq:block07-paired-estimator}
G_{\mathrm{foreign}}Use in source · 1 passage
Search aliases: G_foreign mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5528
G_foreign · Transfer · Held-out systems, matched candidate budget, protected evaluator
Exact mathematical passage
G_{\mathrm{foreign}}
G_{\mathrm{global\ oracle}}Use in source · 1 passage
Search aliases: G_global oracle mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11868
tion 0.152169 and maximum gain 0.364965 across development. By contrast, the true-chart oracle has mean gain 0.998046, and the learned local constructor has mean blind gain 0.613651. Every one of the 54 development targets satisfies the frozen repair-conflict predicate G_global oracle≤0.40, G_true chart≥0.95. Mean learned gate accuracy is 0.915000, and the mean chart-specific bridge contrast is 2.186784. These quantities, together with the conservative embedding, produce the frozen grammar-inade…
Exact mathematical passage
G_{\mathrm{global\ oracle}}\leq0.40, \qquad G_{\mathrm{true\ chart}}\geq0.95.
G_{h,B}Use in source · 6 passages
Search aliases: G_h,B
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4685
Let K_0 and K_1 be frozen improvement mechanisms evaluated on the same independent foreign agents A_1,…,A_n. Define G_h,B(K,A) = max_A'∈Desc_h,B(K,A)J(A')-J(A) and assume -G_max ≤ G_h,B(K,A) ≤ G_max. For matched budgets and random seeds, let D_i = G_h,B(K_1,A_i)-G_h,B(K_0,A_i), overline D_n = frac1n∑_i=1^nD_i.
Exact mathematical passage
G_{h,B}(K,A) = \max_{A'\in\Desc_{h,B}(K,A)}J(A')-J(A)
G_iUse in source · 2 passages
Search aliases: G_i
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4662
[Expected best-of-k gain] Let G_1,…,G_k be independent nonnegative candidate gains with common cumulative distribution function F and finite mean. Then Emax_1≤ i≤ kG_i = ∫_0^∞ (1-F(x)^k) d x. In particular, expected best gain is nondecreasing in k even when the candidate generator itself does not improve.
Exact mathematical passage
\mathbb E\max_{1\leq i\leq k}G_i = \int_0^{\infty} \left(1-F(x)^k\right)\dd x.
g_iUse in source · 3 passages
Search aliases: g_i
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13901
[Uniform forecast-regret bound] Let g_i be the true protected gain of each admissible candidate and suppose |widehat g_i-g_i|≤ε for all candidates. If widehat i∈arg max_iwidehat g_i and i^⋆∈arg max_i g_i, then g_i^⋆-g_widehat i ≤ 2ε.
Exact mathematical passage
g_i
g_{i^\star}Use in source · 2 passages
Search aliases: g_i^⋆ star
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13906
[Uniform forecast-regret bound] Let g_i be the true protected gain of each admissible candidate and suppose |widehat g_i-g_i|≤ε for all candidates. If widehat i∈arg max_iwidehat g_i and i^⋆∈arg max_i g_i, then g_i^⋆-g_widehat i ≤ 2ε.
Exact mathematical passage
g_{i^\star}-g_{\widehat i} \leq 2\eps. \label{eq:b13-forecast-regret}
G_{j}^{\mathrm{meta}}Use in source · 1 passage
Search aliases: G_j^meta mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6212
Let N⊆{1,…,n} be foreign agents whose dominant defect belongs to the held-out sixth type. Define widehatτ_abstract = frac1|N|∑_j∈ N ( G_j^meta-G_j^no-meta ). In the reference positive control, widehatτ_abstract=0.700533. A learned recursive-improvement claim would require the schema to be generated by the parent improvement process, sealed before foreign evaluation, and compared under matched computation and evidence access.
Exact mathematical passage
\widehat\tau_{\mathrm{abstract}} = \frac1{|N|}\sum_{j\in N} \left( G_{j}^{\mathrm{meta}}-G_{j}^{\mathrm{no\mbox{-}meta}} \right).
G_{j}^{\mathrm{no\mbox{-}meta}}Use in source · 1 passage
Search aliases: G_j^no-meta mathrm mbox
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6212
Let N⊆{1,…,n} be foreign agents whose dominant defect belongs to the held-out sixth type. Define widehatτ_abstract = frac1|N|∑_j∈ N ( G_j^meta-G_j^no-meta ). In the reference positive control, widehatτ_abstract=0.700533. A learned recursive-improvement claim would require the schema to be generated by the parent improvement process, sealed before foreign evaluation, and compared under matched computation and evidence access.
Exact mathematical passage
\widehat\tau_{\mathrm{abstract}} = \frac1{|N|}\sum_{j\in N} \left( G_{j}^{\mathrm{meta}}-G_{j}^{\mathrm{no\mbox{-}meta}} \right).
G_kUse in source · 1 passage
Search aliases: G_k
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4661
[Expected best-of-k gain] Let G_1,…,G_k be independent nonnegative candidate gains with common cumulative distribution function F and finite mean. Then Emax_1≤ i≤ kG_i = ∫_0^∞ (1-F(x)^k) d x. In particular, expected best gain is nondecreasing in k even when the candidate generator itself does not improve.
Exact mathematical passage
G_1,\dots,G_k
g_kUse in source · 1 passage
Search aliases: g_k
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16124
The latent trace updates the task representation through g_k · =σ(Gz_k), widetilde h_k · =tanh(Th_k-1+Lz_k), h_k · =(1-g_k)odot h_k-1+g_kodotwidetilde h_k, h_0=e, widehat w_k · =Oh_k. A separate halt head predicts one of four dynamic depths from the mean trace and, for nonparent candidates, the encoded prompt. A semantic training head predicts w(τ) from z_4. Its auxiliary loss is an externally declared witness term, not a claim that the latent coordinates are uniquely interpretable.
Exact mathematical passage
g_k&=\sigma(Gz_k), \label{eq:b17-update-gate}\\ \widetilde h_k&=\tanh(Th_{k-1}+Lz_k), \label{eq:b17-hidden-candidate}\\ h_k&=(1-g_k)\odot h_{k-1}+g_k\odot\widetilde h_k, \qquad h_0=e, \label{eq:b17-hidden-update}\\ \widehat w_k&=Oh_k. \label{eq:b17-readout}
G_{\max}Use in source · 6 passages
Search aliases: G_max max
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4691
Let K_0 and K_1 be frozen improvement mechanisms evaluated on the same independent foreign agents A_1,…,A_n. Define G_h,B(K,A) = max_A'∈Desc_h,B(K,A)J(A')-J(A) and assume -G_max ≤ G_h,B(K,A) ≤ G_max. For matched budgets and random seeds, let D_i = G_h,B(K_1,A_i)-G_h,B(K_0,A_i), overline D_n = frac1n∑_i=1^nD_i.
Exact mathematical passage
-G_{\max} \leq G_{h,B}(K,A) \leq G_{\max}.
G_{\max}^2Use in source · 1 passage
Search aliases: G_max^2 max
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4725
Each D_i lies in an interval of length 4G_max. The one-sided Hoeffding inequality gives Pr( overline D_n- E D_i≥ε ) ≤ exp( -fracnε^28G_max^2 ). Solving for ε proves the bound.
Exact mathematical passage
\Pr\left( \overline D_n-\mathbb E D_i\geq\eps \right) \leq \exp\left( -\frac{n\eps^2}{8G_{\max}^2} \right).
G_{\mathrm{mech}}Use in source · 1 passage
Search aliases: G_mech mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5525
G_mech · Mechanics · Legal phase traces, receipt verification, checkpoint restoration, deterministic replay
Exact mathematical passage
G_{\mathrm{mech}}
G_{\mathrm{meta}}Use in source · 1 passage
Search aliases: G_meta mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5529
G_meta · Metaproductivity · Parent-child repair-calculus comparison on frozen foreign systems
Exact mathematical passage
G_{\mathrm{meta}}
G_{\mathrm{open}}Use in source · 1 passage
Search aliases: G_open mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5530
G_open · Open-set recursion · Conservative grammar extension, prior-proof replay, novel externally validated defect families
Exact mathematical passage
G_{\mathrm{open}}
G_{\mathrm{pkg}}Use in source · 1 passage
Search aliases: G_pkg mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5524
G_pkg · Package integrity · Exact manifest, canonical digests, clean extraction, successful verifier
Exact mathematical passage
G_{\mathrm{pkg}}
G_{\rm cause}Use in source · 1 passage
Search aliases: G_rm cause rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7132
G_rm cause · Pass · Causal packet, PCA, random, wrong-block, and shuffled-anchor controls · Causal role inside the fixture
Exact mathematical passage
G_{\rm cause}
G_{\rm child}Use in source · 1 passage
Search aliases: G_rm child rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21388
Let a parent receive one candidate whose gain is uniformly distributed on [-1,1], and let the child receive two independent candidates from the same distribution. If both choose the best available candidate, then E[G_rm parent]=0, E[G_rm child]=frac13. The child appears better even though its selection rule is unchanged. The difference is entirely the best-of-two opportunity advantage. Matching the candidate budget eliminates this explanation.
Exact mathematical passage
\mathbb E[G_{\rm parent}]=0, \qquad \mathbb E[G_{\rm child}]=\frac13.
G_{\rm foreign}Use in source · 3 passages
Search aliases: G_rm foreign rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7133
G_rm foreign · Pass · Held-out target seeds, dimensions, encoder families, and learned targets · One shared task and defect grammar
Exact mathematical passage
G_{\rm foreign}
G_{\rm mech}Use in source · 1 passage
Search aliases: G_rm mech rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7130
G_rm mech · Pass · Deterministic execution, five tests, frozen JSON and figures · Runtime mechanics only
Exact mathematical passage
G_{\rm mech}
G_{\rm meta}Use in source · 2 passages
Search aliases: G_rm meta rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7134
G_rm meta · Not tested · No parent-child repair-calculus comparison · No metaproductivity claim
Exact mathematical passage
G_{\rm meta}
G_{\rm meta}^{\rm bridge}Use in source · 2 passages
Search aliases: G_rm meta^rm bridge rm rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 18729
The controlled packages therefore pass two distinct level-5 gates: G_rm meta^rm select for selector metaproductivity and G_rm meta^rm bridge for bridge-policy metaproductivity. Neither package establishes repeated-generation closure, autonomous candidate invention, or open-set grammar expansion.
Exact mathematical passage
G_{\rm meta}^{\rm bridge}
G_{\rm meta}^{\rm select}Use in source · 2 passages
Search aliases: G_rm meta^rm select rm rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 18729
The controlled packages therefore pass two distinct level-5 gates: G_rm meta^rm select for selector metaproductivity and G_rm meta^rm bridge for bridge-policy metaproductivity. Neither package establishes repeated-generation closure, autonomous candidate invention, or open-set grammar expansion.
Exact mathematical passage
G_{\rm meta}^{\rm select}
G_{\rm open}Use in source · 2 passages
Search aliases: G_rm open rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7135
G_rm open · Not tested · No autonomous grammar or evaluator expansion · No open-set recursion claim
Exact mathematical passage
G_{\rm open}
G_{\rm parent}Use in source · 1 passage
Search aliases: G_rm parent rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21388
Let a parent receive one candidate whose gain is uniformly distributed on [-1,1], and let the child receive two independent candidates from the same distribution. If both choose the best available candidate, then E[G_rm parent]=0, E[G_rm child]=frac13. The child appears better even though its selection rule is unchanged. The difference is entirely the best-of-two opportunity advantage. Matching the candidate budget eliminates this explanation.
Exact mathematical passage
\mathbb E[G_{\rm parent}]=0, \qquad \mathbb E[G_{\rm child}]=\frac13.
G_{\rm pkg}Use in source · 1 passage
Search aliases: G_rm pkg rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7129
G_rm pkg · Pass · 34-file SHA-256 manifest and clean verifier · Byte integrity relative to the released manifest
Exact mathematical passage
G_{\rm pkg}
G_{\rm sem}Use in source · 1 passage
Search aliases: G_rm sem rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7131
G_rm sem · Pass · Declared task, repair, transport, and gate objects reproduced · Controlled finite semantics
Exact mathematical passage
G_{\rm sem}
g_{\rm slice}Use in source · 1 passage
Search aliases: g_rm slice rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6994
[Calibration-to-audit witness soundness] Suppose, simultaneously, that the audit gain and regression satisfy g_rm slice · ≥ widehat g_rm slice-η_g, r_rm non · ≤ widehat r_rm non+η_r, and that the reported alignment error upper-bounds the audit alignment error up to η_a. If widehat g_rm slice-η_g · >0,
Exact mathematical passage
g_{\rm slice}&\geq \widehat g_{\rm slice}-\eta_g,\\ r_{\rm non}&\leq \widehat r_{\rm non}+\eta_r,
G_{\mathrm{sem}}Use in source · 1 passage
Search aliases: G_sem mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5526
G_sem · Finite semantics · Abstract-runtime refinement and exhaustive frozen endpoints
Exact mathematical passage
G_{\mathrm{sem}}
G_{\mathrm{true\ chart}}Use in source · 1 passage
Search aliases: G_true chart mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11868
tion 0.152169 and maximum gain 0.364965 across development. By contrast, the true-chart oracle has mean gain 0.998046, and the learned local constructor has mean blind gain 0.613651. Every one of the 54 development targets satisfies the frozen repair-conflict predicate G_global oracle≤0.40, G_true chart≥0.95. Mean learned gate accuracy is 0.915000, and the mean chart-specific bridge contrast is 2.186784. These quantities, together with the conservative embedding, produce the frozen grammar-inade…
Exact mathematical passage
G_{\mathrm{global\ oracle}}\leq0.40, \qquad G_{\mathrm{true\ chart}}\geq0.95.
g_{\widehat i}Use in source · 2 passages
Search aliases: g_widehat i widehat
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13906
[Uniform forecast-regret bound] Let g_i be the true protected gain of each admissible candidate and suppose |widehat g_i-g_i|≤ε for all candidates. If widehat i∈arg max_iwidehat g_i and i^⋆∈arg max_i g_i, then g_i^⋆-g_widehat i ≤ 2ε.
Exact mathematical passage
g_{i^\star}-g_{\widehat i} \leq 2\eps. \label{eq:b13-forecast-regret}
g_\alphaUse in source · 3 passages
Search aliases: g_α alpha
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2512
epair exists if and only if the transition family satisfies the nonabelian cocycle condition g_αβg_βγg_γα=e on triple overlaps and its class is trivial in nonabelian vCech cohomology [g_αβ]=1 in check H^1( U,mathscr G). Equivalently, there exist local gauges g_α∈mathscr G(U_α) such that g_αβ=g_α^-1g_β.
Exact mathematical passage
g_\alpha\in\mathscr G(U_\alpha)
g_\alpha^{-1}Use in source · 1 passage
Search aliases: g_α^-1 alpha
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2513
the nonabelian cocycle condition g_αβg_βγg_γα=e on triple overlaps and its class is trivial in nonabelian vCech cohomology [g_αβ]=1 in check H^1( U,mathscr G). Equivalently, there exist local gauges g_α∈mathscr G(U_α) such that g_αβ=g_α^-1g_β.
Exact mathematical passage
g_{\alpha\beta}=g_\alpha^{-1}g_\beta.
g_\betaUse in source · 1 passage
Search aliases: g_β beta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2513
the nonabelian cocycle condition g_αβg_βγg_γα=e on triple overlaps and its class is trivial in nonabelian vCech cohomology [g_αβ]=1 in check H^1( U,mathscr G). Equivalently, there exist local gauges g_α∈mathscr G(U_α) such that g_αβ=g_α^-1g_β.
Exact mathematical passage
g_{\alpha\beta}=g_\alpha^{-1}g_\beta.
g_{\beta\gamma}Use in source · 1 passage
Search aliases: g_βγ beta gamma
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2503
[Nonlinear torsor descent] Assume the local solution action is principal and effective descent holds for mathscr G-torsors on the declared cover. Then a global nonlinear repair exists if and only if the transition family satisfies the nonabelian cocycle condition g_αβg_βγg_γα=e on triple overlaps and its class is trivial in nonabelian vCech cohomology [g_αβ]=1 in check H^1( U,mathscr G). Equivalently, there exist local gauges g_α∈mathscr G(U_α) such that g_αβ=g_α^-1g_β.
Exact mathematical passage
g_{\alpha\beta}g_{\beta\gamma}g_{\gamma\alpha}=e
G_\gammaUse in source · 4 passages
Search aliases: G_γ gamma
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4406
Every state in the Buechi set remains inside the static invariant set under its winning strategy, hence lies in the robust safety kernel. A safe absorbing no-op state outside G_γ witnesses possible strictness.
Exact mathematical passage
G_\gamma
g_{\gamma\alpha}Use in source · 1 passage
Search aliases: g_γα gamma alpha
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2503
[Nonlinear torsor descent] Assume the local solution action is principal and effective descent holds for mathscr G-torsors on the declared cover. Then a global nonlinear repair exists if and only if the transition family satisfies the nonabelian cocycle condition g_αβg_βγg_γα=e on triple overlaps and its class is trivial in nonabelian vCech cohomology [g_αβ]=1 in check H^1( U,mathscr G). Equivalently, there exist local gauges g_α∈mathscr G(U_α) such that g_αβ=g_α^-1g_β.
Exact mathematical passage
g_{\alpha\beta}g_{\beta\gamma}g_{\gamma\alpha}=e
G_{\delta}Use in source · 2 passages
Search aliases: G_δ delta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7030
For encoder E_θ, decoder G_δ, and score head H_ω, the pilot minimizes minibatch estimates of L(θ,δ,ω) = λ_rm rec normG_δ(E_θ(ϕ))-ϕ_F^2 + normH_ω(E_θ(ϕ))-widetilde q_F^2, with λ_rm rec=2. Optimization uses a deterministic NumPy Adam implementation for 500 steps. After representation training, the head is refitted by ridge regression to reduce avoidable readout variance.
Exact mathematical passage
G_{\delta}
\mathrm{gate}Use in source · 1 passage
Search aliases: gate mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1485
Consider four colors memory, representation, verifier, planner. A composite factuality repair may have the form ω_gate ∘ ( ω_retrieve, ω_decompose, ω_check ). The inner operations retrieve evidence, decompose output into claims, and check claims against protected sources. Their postconditions establish the precondition for a planner gate that blocks or revises unsupported claims. The composite receipt records source provenance, claim-to-evidence links, verifier uncertainty, revised spans, latenc…
Exact mathematical passage
\omega_{\mathrm{gate}} \circ \left( \omega_{\mathrm{retrieve}}, \omega_{\mathrm{decompose}}, \omega_{\mathrm{check}} \right).
\Gcal_0Use in source · 1 passage
Search aliases: Gcal_0 G_0 Gcal mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4983
Consider a chain G_0 xhookrightarrowι_0 G_1 xhookrightarrowι_1 G_2 xhookrightarrow ⋯ of conservative extensions.
Exact mathematical passage
\Gcal_0 \xhookrightarrow{\iota_0} \Gcal_1 \xhookrightarrow{\iota_1} \Gcal_2 \xhookrightarrow{} \cdots
\Gcal_1Use in source · 1 passage
Search aliases: Gcal_1 G_1 Gcal mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4983
Consider a chain G_0 xhookrightarrowι_0 G_1 xhookrightarrowι_1 G_2 xhookrightarrow ⋯ of conservative extensions.
Exact mathematical passage
\Gcal_0 \xhookrightarrow{\iota_0} \Gcal_1 \xhookrightarrow{\iota_1} \Gcal_2 \xhookrightarrow{} \cdots
\Gcal_2Use in source · 1 passage
Search aliases: Gcal_2 G_2 Gcal mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4983
Consider a chain G_0 xhookrightarrowι_0 G_1 xhookrightarrowι_1 G_2 xhookrightarrow ⋯ of conservative extensions.
Exact mathematical passage
\Gcal_0 \xhookrightarrow{\iota_0} \Gcal_1 \xhookrightarrow{\iota_1} \Gcal_2 \xhookrightarrow{} \cdots
\Gcal_{23}Use in source · 1 passage
Search aliases: Gcal_23 G_23 Gcal mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 19939
The finite grammar is G_23 = { l answer-only, PCA, predictive, intervention-response, joint minimal, full identity, equal-rank random }. For each nontrivial constructor, ranks one through six are considered. The full identity is an oversized control. A projection W is valid only when, on both recurrent systems, R^2_future · ≥0.82,
Exact mathematical passage
\Gcal_{23} = \left\{ \begin{array}{l} \text{answer-only},\ \text{PCA},\ \text{predictive},\ \\ \text{intervention-response},\ \text{joint minimal},\\ \text{full identity},\ \text{equal-rank random} \end{array} \right\}. \label{eq:b23-grammar}
\Gcal_{\infty}Use in source · 3 passages
Search aliases: Gcal_∞ G_∞ Gcal infty mathcal infty
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4996
[Conservative directed limit] The union of translated syntax and proofs defines a directed-limit grammar G_∞ with the following property: every finite expression and every finite derivation in G_∞ appears at some finite stage. Every theorem proved at stage t retains the same protected meaning at every later stage and in G_∞.
Exact mathematical passage
\Gcal_{\infty}
\operatorname{GELU}Use in source · 1 passage
Search aliases: GELU operatorname
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16714
p embedding at latent depth k∈{0,…,4}. The source uses an Elman transition z_k+1 = tanh(Wz_k+s_k+δ_k(e,z_k)). The first foreign family uses a GRU cell z_k+1=GRUCell(u_k,z_k), and the second uses a gated nonlinear residual cell z_k+1 = z_k+σ(Gz_k)odot tanh(V_2GELU(V_1z_k)+s_k+δ_k(e,z_k)). The three cell classes have distinct executable implementations and parameterizations. They share only the declared external input, latent dimension, five-step budget, semantic target, downstream gated readout, …
Exact mathematical passage
z_{k+1} = z_k+\sigma(Gz_k)\odot \tanh\left(V_2\operatorname{GELU}(V_1z_k)+s_k+\delta_k(e,z_k)\right). \label{eq:b18-residual}
\mathrm{gen}Use in source · 1 passage
Search aliases: gen mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9846
Define the causal generator contribution Δ_gen = E_Q[η_without(r)^2] - E_Q[η_with(r)^2]. The contribution is admissible only when a matched ablation isolates the generator while holding the remaining grammar, evidence, and selection budget fixed.
Exact mathematical passage
\Delta_{\mathrm{gen}} = E_Q\bigl[\eta_{\mathrm{without}}(r)^2\bigr] - E_Q\bigl[\eta_{\mathrm{with}}(r)^2\bigr]. \label{eq:b8-generator-credit}
\ggUse in source · 1 passage
Search aliases: gg
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10561
This certificate separates two statements: rank X_S=d and λ_min(X_S^ TWX_S)gg0. The first is qualitative identifiability. The second is quantitative stability.
Exact mathematical passage
\lambda_{\min}(X_S^{\mathsf T}WX_S)\gg0.
\mathrm{GL}Use in source · 1 passage
Search aliases: GL mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 18260
The ordinary control has Y_t=X_t A+b, A∈GL(8), with one shared map at every recurrent step. The inherited step-affine constructor is sufficient. This family tests conservative minimality: a full-trajectory affine map is available and usually eligible, but it should not be selected when the smaller inherited constructor already passes.
Exact mathematical passage
Y_t=X_t A+b, \qquad A\in\mathrm{GL}(8), \label{eq:block21-affine}
\mathrm{global}Use in source · 3 passages
Search aliases: global mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 18043
For every final task, let ℓ_global and ℓ_chart be the source-probe semantic losses after pooled and selected transport. The paired effect is Δ_i = ℓ_global,i - ℓ_chart,i. Positive values favor the selected chart-indexed map.
Exact mathematical passage
\ell_{\mathrm{global}}
\mathrm{global\ oracle}Use in source · 1 passage
Search aliases: global oracle mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11868
tion 0.152169 and maximum gain 0.364965 across development. By contrast, the true-chart oracle has mean gain 0.998046, and the learned local constructor has mean blind gain 0.613651. Every one of the 54 development targets satisfies the frozen repair-conflict predicate G_global oracle≤0.40, G_true chart≥0.95. Mean learned gate accuracy is 0.915000, and the mean chart-specific bridge contrast is 2.186784. These quantities, together with the conservative embedding, produce the frozen grammar-inade…
Exact mathematical passage
G_{\mathrm{global\ oracle}}\leq0.40, \qquad G_{\mathrm{true\ chart}}\geq0.95.
\mathrm{gram}Use in source · 2 passages
Search aliases: gram mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20511
Δ_gram · Global-minus-local held-out transport residual margin
Exact mathematical passage
\Delta_{\mathrm{gram}}
\GroundedUse in source · 1 passage
Search aliases: Grounded mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4259
Let C_RI(s) = · Grounded(s) wedgeProvValid(s) wedgeWorkspaceBound(s) · wedgeRepairTyped(s) wedgeWitnessed(s) wedgeEvidenceFlat(s) · wedgeRollbackable(s) wedgeMetaCapable(s). For X⊆ S, define Ψ_∀(X) = sets∈ S:C_RI(s) and Post_imp(s)⊆ X.
Exact mathematical passage
\begin{aligned} C_{\mathrm{RI}}(s) ={}& \Grounded(s) \wedge\ProvValid(s) \wedge\WorkspaceBound(s)\\ &\wedge\RepairTyped(s) \wedge\Witnessed(s) \wedge\EvidenceFlat(s)\\ &\wedge\Rollbackable(s) \wedge\MetaCapable(s). \end{aligned}
\mathrm{GRU}Use in source · 1 passage
Search aliases: GRU mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17678
Let A be a finite architecture index set and C a finite admitted chart index set. In the released realization, A = {Elman,GRU,Residual}, C = {A0,A1,A2,A3,N0,N1}. Each pair (a,c)∈ A× C carries a full Neuralese trajectory fiber H_a,c ≅ ( R^40)^5 ≅ R^200. The observation atlas determines which context fiber is authorized. A task outside every admitted chart is not assigned to an arbitrary fiber.
Exact mathematical passage
\mathcal A = \{\mathrm{Elman},\mathrm{GRU},\mathrm{Residual}\}, \qquad \mathcal C = \{A0,A1,A2,A3,N0,N1\}.
\operatorname{GRUCell}Use in source · 1 passage
Search aliases: GRUCell operatorname
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16709
Let e=E(ϕ(τ))∈ℝ^36 be the encoded task representation, let z_0∈ℝ^36 be a learned seed, and let s_k∈ℝ^36 be the step embedding at latent depth k∈{0,…,4}. The source uses an Elman transition z_k+1 = tanh(Wz_k+s_k+δ_k(e,z_k)). The first foreign family uses a GRU cell z_k+1=GRUCell(u_k,z_k), and the second uses a gated nonlinear residual cell z_k+1 = z_k+σ(Gz_k)odot tanh(V_2GELU(V_1z_k)+s_k+δ_k(e,z_k)). The three cell classes have distinct executable implementations and parameterizations. They share…
Exact mathematical passage
z_{k+1}=\operatorname{GRUCell}(u_k,z_k), \label{eq:b18-gru}
\mathfrak HUse in source · 4 passages
Search aliases: H mathfrak
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9888
The minimal conflicts form a finite hypergraph H_G. A candidate feature or probe f:X→ Y_f resolves a conflict S when f is not constant on S.
Exact mathematical passage
\mathfrak H_{\Gcal}
h^{-1}Use in source · 1 passage
Search aliases: h^-1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4411
[Finite ranking certificate for one gain cycle] Let X⊆I_ctrl^* and let h:X→{0,1,…,H} satisfy h^-1(0)⊆ G_γ. Suppose that at every state with h(s)>0, the controller has an authorized action whose every admissible successor s' lies in X and satisfies h(s')<h(s). Then the controller can force a visit to G_γ within at most H transactions while remaining in X.
Exact mathematical passage
h^{-1}(0)\subseteq G_\gamma
H^UUse in source · 2 passages
Search aliases: H^U
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20937
Let U=V=ℝ^2, D=I, and suppose two routes from A to B differ by a loop with correction holonomy H^U= 1 · 0 0 · -1 . The flat repair core is U^flat=spansete_1. For u=e_1, both routes produce the same transported repair. For u=e_2, the two routes differ by (H^U-I)e_2=-2e_2, so the loop residual is 2. A repair schema containing an e_2 component is therefore path dependent even though every individual edge transport may be exact.
Exact mathematical passage
H^U= \begin{bmatrix} 1&0\\ 0&-1 \end{bmatrix}.
H^U_{q^{-1}p}Use in source · 3 passages
Search aliases: H^U_q^-1p
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3078
The path identity is T_U^p=T_U^qH^U_q^-1p. Apply both sides to u and use H^U_q^-1pu=u.
Exact mathematical passage
T_U^p=T_U^qH^U_{q^{-1}p}.
h_0Use in source · 6 passages
Search aliases: h_0
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15662
Given input x, define the prompt anchor h_0=E(x). A normalized injection state z_t and updated latent state h_t satisfy z_t=N(h_t-1), widetilde h_t=B(z_t,x), h_t=U(z_t,widetilde h_t,h_0), 1≤ t≤ K, where N is any declared normalization or scale-matching operator. The answer is sampled or selected from D(h_K,x) after a declared end-of-thought transition.
Exact mathematical passage
h_0=E(x)
H_1Use in source · 4 passages
Search aliases: H_1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3549
Exact fixed subspaces can be too brittle for learned evaluator transports. Let H_1,…,H_m be a finite generating family of loop transports on a Hilbert space E, with positive weights w_j satisfying ∑_jw_j=1. Define the holonomy-energy operator A := ∑_j=1^m w_j(H_j-I)^*(H_j-I). This operator is positive semidefinite.
Exact mathematical passage
H_1,\ldots,H_m
H_2Use in source · 2 passages
Search aliases: H_2
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21975
Suppose H=H_1⊕ H_2⊕ H_3, dim H_a=2. Under separating block-scalar responses, every commuting operator has form X=X_1⊕ X_2⊕ X_3, X_a∈ M_2(ℝ). Hence dim A_R' =3dim M_2(ℝ) =12. The companion's modal commutant dimension is 12, matching the exact finite theory.
Exact mathematical passage
H=H_1\oplus H_2\oplus H_3, \qquad \dim H_a=2.
h_2Use in source · 7 passages
Search aliases: h_2
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11560
In the executable fixture, P_ctxh=(h_1,h_2), and the source repair leaves these coordinates invariant. The initial separator vocabulary is F_X = set 1{anglesu_k,P_ctxh>θ} : 0≤ k<128, θ∈Θ_k(X) , where u_k=(cos(2π k/128),sin(2π k/128)) and Θ_k(X) contains zero and the empirical 0.30,0.40,0.50,0.60,0.70 quantiles of the projected fit states. For every candidate separator, the compiler fits two weighted affine bridges and evaluates held-out chart-selection error. It then performs at most five determ…
Exact mathematical passage
P_{\mathrm{ctx}}h=(h_1,h_2),
H_3Use in source · 1 passage
Search aliases: H_3
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21975
Suppose H=H_1⊕ H_2⊕ H_3, dim H_a=2. Under separating block-scalar responses, every commuting operator has form X=X_1⊕ X_2⊕ X_3, X_a∈ M_2(ℝ). Hence dim A_R' =3dim M_2(ℝ) =12. The companion's modal commutant dimension is 12, matching the exact finite theory.
Exact mathematical passage
H=H_1\oplus H_2\oplus H_3, \qquad \dim H_a=2.
h_3Use in source · 1 passage
Search aliases: h_3
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11698
The source state is h=(h_1,h_2,h_3,h_4)∈ℝ^4. The first two coordinates form the protected context packet. The source repair is R_s(h)=M_sh+c_s, where M_s= 1 · 0 · 0 · 0
Exact mathematical passage
h=(h_1,h_2,h_3,h_4)\in\R^4.
h_4Use in source · 1 passage
Search aliases: h_4
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11698
The source state is h=(h_1,h_2,h_3,h_4)∈ℝ^4. The first two coordinates form the protected context packet. The source repair is R_s(h)=M_sh+c_s, where M_s= 1 · 0 · 0 · 0
Exact mathematical passage
h=(h_1,h_2,h_3,h_4)\in\R^4.
h_5Use in source · 1 passage
Search aliases: h_5
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 18224
two Block 20 foreign architectures. For admitted observation chart c, Block 20 supplies an accepted full-trajectory connection T^c_B→ A:H_B,c^(1:5)⟶ H_A,c^(1:5). Let P:ℝ^40→ℝ^8 be an orthonormal packet projector fitted on a pre-audit source support. Applying P independently to the five recurrent states gives Π(H) = (P h_1,…,P h_5) ∈ (ℝ^8)^5 ≅ℝ^40. The packet is not claimed to preserve every latent coordinate. It is required to preserve the protected nine-coordinate semantic witness on the declar…
Exact mathematical passage
\Pi(H) = \left(P h_1,\ldots,P h_5\right) \in \left(\R^8\right)^5 \cong\R^{40}. \label{eq:block21-packet}
H_AUse in source · 1 passage
Search aliases: H_A
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15811
For parent and child state spaces H_A and H_B of equal dimension, let calibration traces produce matrices X and Y. The orthogonal Procrustes transport is T^* = arg min_T^ TT=InormXT-Y_ F. If X^ TY=UΣ V^ T, then T^*=UV^ T. A transport receipt records calibration residual, held-out residual, singular floor, round-trip residual, and behavior change after transported trace injection.
Exact mathematical passage
H_A
h_aUse in source · 7 passages
Search aliases: h_a
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6584
An agent a has encoder E_a:ℝ^16→ℝ^d_a and an affine head fitted to widetilde q. Its hidden state is h_a(x)=E_a(ϕ(x)). The transport system is not given the matrix or nonlinear parameters of E_a.
Exact mathematical passage
h_a(x)=E_a(\phi(x)).
h_{a,c}Use in source · 3 passages
Search aliases: h_a,c
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17760
Suppose an underlying canonical trajectory is represented in chart c by a row vector z and architecture-dependent gauge matrices: h_a,c=zG_a,c, h_b,c=zG_b,c. The matrices may include coordinate rotations, sign changes, and architecture-specific temporal mixing. Block 20 uses deterministic orthogonal gauges as a controlled stress at the exposed trajectory interface.
Exact mathematical passage
h_{a,c}=zG_{a,c}, \qquad h_{b,c}=zG_{b,c}. \label{eq:b20-gauge-model}
H_{a,d}Use in source · 1 passage
Search aliases: H_a,d
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17718
For architectures a,b and admitted chart c, an architecture map is T^c_b→ a:H_b,c⟶ H_a,c. For one architecture a and two admitted charts c,d, a context map is C^a_c→ d:H_a,c⟶ H_a,d. Block 20 uses A0 as a reference chart and fits a star of maps C^a_c→ A0 and C^a_A0→ c from protected semantic twins.
Exact mathematical passage
C^{a}_{c\to d}:H_{a,c}\longrightarrow H_{a,d}.
h_a^{\circ}Use in source · 5 passages
Search aliases: h_a^∘ circ
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6601
0 . The resulting hidden difference is Δ_a(x) = h_a(x)-h_a^∘(x), h_a^∘(x)=E_a(ϕ^∘(x)). The empirical causal packet is the leading rank-r right-singular subspace of the affected differences {Δ_a(x_i):s(x_i)=1}.
Exact mathematical passage
\Delta_a(x) = h_a(x)-h_a^{\circ}(x), \qquad h_a^{\circ}(x)=E_a(\phi^{\circ}(x)).
H_BUse in source · 1 passage
Search aliases: H_B
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15811
For parent and child state spaces H_A and H_B of equal dimension, let calibration traces produce matrices X and Y. The orthogonal Procrustes transport is T^* = arg min_T^ TT=InormXT-Y_ F. If X^ TY=UΣ V^ T, then T^*=UV^ T. A transport receipt records calibration residual, held-out residual, singular floor, round-trip residual, and behavior change after transported trace injection.
Exact mathematical passage
H_B
H_{b,c}Use in source · 1 passage
Search aliases: H_b,c
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17714
For architectures a,b and admitted chart c, an architecture map is T^c_b→ a:H_b,c⟶ H_a,c. For one architecture a and two admitted charts c,d, a context map is C^a_c→ d:H_a,c⟶ H_a,d. Block 20 uses A0 as a reference chart and fits a star of maps C^a_c→ A0 and C^a_A0→ c from protected semantic twins.
Exact mathematical passage
T^{c}_{b\to a}:H_{b,c}\longrightarrow H_{a,c}.
h_{b,c}Use in source · 4 passages
Search aliases: h_b,c
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17760
Suppose an underlying canonical trajectory is represented in chart c by a row vector z and architecture-dependent gauge matrices: h_a,c=zG_a,c, h_b,c=zG_b,c. The matrices may include coordinate rotations, sign changes, and architecture-specific temporal mixing. Block 20 uses deterministic orthogonal gauges as a controlled stress at the exposed trajectory interface.
Exact mathematical passage
h_{a,c}=zG_{a,c}, \qquad h_{b,c}=zG_{b,c}. \label{eq:b20-gauge-model}
h_{b,c}^{(1)}Use in source · 1 passage
Search aliases: h_b,c^(1)
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17822
A latent-step-shared map acts independently on each recurrent state: widehat h_a,c^(k) = A_c h_b,c^(k)+b_c, k=1,…,5. A full-trajectory map instead acts on the concatenation vec(h_b,c^(1),…,h_b,c^(5)) ∈ R^200, allowing cross-step mixing: vec(widehat h_a,c) = M_cvec(h_b,c)+q_c. This difference is substantive when two recurrent architectures distribute the same computation across latent time differently.
Exact mathematical passage
\operatorname{vec}(h_{b,c}^{(1)},\ldots,h_{b,c}^{(5)}) \in\mathbb R^{200},
h_{b,c}^{(5)}Use in source · 1 passage
Search aliases: h_b,c^(5)
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17822
A latent-step-shared map acts independently on each recurrent state: widehat h_a,c^(k) = A_c h_b,c^(k)+b_c, k=1,…,5. A full-trajectory map instead acts on the concatenation vec(h_b,c^(1),…,h_b,c^(5)) ∈ R^200, allowing cross-step mixing: vec(widehat h_a,c) = M_cvec(h_b,c)+q_c. This difference is substantive when two recurrent architectures distribute the same computation across latent time differently.
Exact mathematical passage
\operatorname{vec}(h_{b,c}^{(1)},\ldots,h_{b,c}^{(5)}) \in\mathbb R^{200},
h_{b,c}^{(k)}Use in source · 1 passage
Search aliases: h_b,c^(k)
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17814
A latent-step-shared map acts independently on each recurrent state: widehat h_a,c^(k) = A_c h_b,c^(k)+b_c, k=1,…,5. A full-trajectory map instead acts on the concatenation vec(h_b,c^(1),…,h_b,c^(5)) ∈ R^200, allowing cross-step mixing: vec(widehat h_a,c) = M_cvec(h_b,c)+q_c. This difference is substantive when two recurrent architectures distribute the same computation across latent time differently.
Exact mathematical passage
\widehat h_{a,c}^{(k)} = A_c h_{b,c}^{(k)}+b_c, \qquad k=1,\ldots,5. \label{eq:b20-step-shared}
h_CUse in source · 2 passages
Search aliases: h_C
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9246
Let the unresolved module-headroom vector be r_t = h_H(K_t) h_Q(K_t) h_C(K_t) h_T(K_t) h_S(K_t) h_V(K_t) h_W(K_t) ∈ℝ_≥0^7. An authorized meta-action a changes one or more repair-process modules. In a local finite model, represent its effect by r_t+1 ≤ A_a r_t+e_t, where A_a is a nonnegative response matrix and e_t records estimation error, distribution shift, semantic debt, or newly induced headroom.
Exact mathematical passage
r_t = \begin{bmatrix} h_H(K_t)\\ h_Q(K_t)\\ h_C(K_t)\\ h_T(K_t)\\ h_S(K_t)\\ h_V(K_t)\\ h_W(K_t) \end{bmatrix} \in\R_{\geq0}^{7}. \label{eq:b8-regret-vector}
h_cUse in source · 2 passages
Search aliases: h_c
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8567
For a candidate configuration c, the development compiler computes target-level selected gains and the score S_rm dev(c) = bar g_c -0.20s_c -2h_c +0.02q_c, where bar g_c is mean protected gain, s_c is the target-level standard deviation, h_c is harmful-target rate, and q_c is positive-target rate. This is a finite engineering objective, not a universal definition of evolvability.
Exact mathematical passage
S_{\rm dev}(c) = \bar g_c -0.20s_c -2h_c +0.02q_c, \label{eq:block07q-compiler-score}
H_{\mathrm{ctx}}Use in source · 2 passages
Search aliases: H_ctx mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12465
The source hidden space has dimension six and is generated from three target-specific unknown rank-two blocks: H=H_ctx⊕ H_rep⊕ H_nui. A target-specific orthogonal or injective embedding hides these blocks from the runtime coordinate system. The source repair changes only H_rep and leaves the context block invariant. Two controlled interventions act with different block signatures. The context label controls which target bridge and target repair are valid.
Exact mathematical passage
H=H_{\mathrm{ctx}}\oplus H_{\mathrm{rep}}\oplus H_{\mathrm{nui}}. \label{eq:b11-three-blocks}
\mathfrak H_{\Gcal}Use in source · 4 passages
Search aliases: H_Gcal H_G mathfrak Gcal mathfrak mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9888
The minimal conflicts form a finite hypergraph H_G. A candidate feature or probe f:X→ Y_f resolves a conflict S when f is not constant on S.
Exact mathematical passage
\mathfrak H_{\Gcal}
h_HUse in source · 2 passages
Search aliases: h_H
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9246
Let the unresolved module-headroom vector be r_t = h_H(K_t) h_Q(K_t) h_C(K_t) h_T(K_t) h_S(K_t) h_V(K_t) h_W(K_t) ∈ℝ_≥0^7. An authorized meta-action a changes one or more repair-process modules. In a local finite model, represent its effect by r_t+1 ≤ A_a r_t+e_t, where A_a is a nonnegative response matrix and e_t records estimation error, distribution shift, semantic debt, or newly induced headroom.
Exact mathematical passage
r_t = \begin{bmatrix} h_H(K_t)\\ h_Q(K_t)\\ h_C(K_t)\\ h_T(K_t)\\ h_S(K_t)\\ h_V(K_t)\\ h_W(K_t) \end{bmatrix} \in\R_{\geq0}^{7}. \label{eq:b8-regret-vector}
H_jUse in source · 5 passages
Search aliases: H_j
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3550
Exact fixed subspaces can be too brittle for learned evaluator transports. Let H_1,…,H_m be a finite generating family of loop transports on a Hilbert space E, with positive weights w_j satisfying ∑_jw_j=1. Define the holonomy-energy operator A := ∑_j=1^m w_j(H_j-I)^*(H_j-I). This operator is positive semidefinite.
Exact mathematical passage
\mathsf A := \sum_{j=1}^{m} w_j(H_j-I)^*(H_j-I).
h_kUse in source · 1 passage
Search aliases: h_k
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16124
The latent trace updates the task representation through g_k · =σ(Gz_k), widetilde h_k · =tanh(Th_k-1+Lz_k), h_k · =(1-g_k)odot h_k-1+g_kodotwidetilde h_k, h_0=e, widehat w_k · =Oh_k. A separate halt head predicts one of four dynamic depths from the mean trace and, for nonparent candidates, the encoded prompt. A semantic training head predicts w(τ) from z_4. Its auxiliary loss is an externally declared witness term, not a claim that the latent coordinates are uniquely interpretable.
Exact mathematical passage
g_k&=\sigma(Gz_k), \label{eq:b17-update-gate}\\ \widetilde h_k&=\tanh(Th_{k-1}+Lz_k), \label{eq:b17-hidden-candidate}\\ h_k&=(1-g_k)\odot h_{k-1}+g_k\odot\widetilde h_k, \qquad h_0=e, \label{eq:b17-hidden-update}\\ \widehat w_k&=Oh_k. \label{eq:b17-readout}
h_{k-1}Use in source · 1 passage
Search aliases: h_k-1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16124
The latent trace updates the task representation through g_k · =σ(Gz_k), widetilde h_k · =tanh(Th_k-1+Lz_k), h_k · =(1-g_k)odot h_k-1+g_kodotwidetilde h_k, h_0=e, widehat w_k · =Oh_k. A separate halt head predicts one of four dynamic depths from the mean trace and, for nonparent candidates, the encoded prompt. A semantic training head predicts w(τ) from z_4. Its auxiliary loss is an externally declared witness term, not a claim that the latent coordinates are uniquely interpretable.
Exact mathematical passage
g_k&=\sigma(Gz_k), \label{eq:b17-update-gate}\\ \widetilde h_k&=\tanh(Th_{k-1}+Lz_k), \label{eq:b17-hidden-candidate}\\ h_k&=(1-g_k)\odot h_{k-1}+g_k\odot\widetilde h_k, \qquad h_0=e, \label{eq:b17-hidden-update}\\ \widehat w_k&=Oh_k. \label{eq:b17-readout}
H_mUse in source · 2 passages
Search aliases: H_m
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3549
Exact fixed subspaces can be too brittle for learned evaluator transports. Let H_1,…,H_m be a finite generating family of loop transports on a Hilbert space E, with positive weights w_j satisfying ∑_jw_j=1. Define the holonomy-energy operator A := ∑_j=1^m w_j(H_j-I)^*(H_j-I). This operator is positive semidefinite.
Exact mathematical passage
H_1,\ldots,H_m
H_{\mathrm{nui}}Use in source · 2 passages
Search aliases: H_nui mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12465
The source hidden space has dimension six and is generated from three target-specific unknown rank-two blocks: H=H_ctx⊕ H_rep⊕ H_nui. A target-specific orthogonal or injective embedding hides these blocks from the runtime coordinate system. The source repair changes only H_rep and leaves the context block invariant. Two controlled interventions act with different block signatures. The context label controls which target bridge and target repair are valid.
Exact mathematical passage
H=H_{\mathrm{ctx}}\oplus H_{\mathrm{rep}}\oplus H_{\mathrm{nui}}. \label{eq:b11-three-blocks}
H_{p^{-1}\gamma p}Use in source · 2 passages
Search aliases: H_p^-1γ p gamma
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3522
Every loop γ at w corresponds to the conjugate loop p^-1γ p at v, with H_γ=T_pH_p^-1γ pT_p^-1. If x∈ E_v^flat, then H_γ T_px=T_pH_p^-1γ px=T_px, so T_px∈ E_w^flat. Apply the same argument to p^-1 for the reverse inclusion.
Exact mathematical passage
H_\gamma=T_pH_{p^{-1}\gamma p}T_p^{-1}.
h_QUse in source · 2 passages
Search aliases: h_Q
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9246
Let the unresolved module-headroom vector be r_t = h_H(K_t) h_Q(K_t) h_C(K_t) h_T(K_t) h_S(K_t) h_V(K_t) h_W(K_t) ∈ℝ_≥0^7. An authorized meta-action a changes one or more repair-process modules. In a local finite model, represent its effect by r_t+1 ≤ A_a r_t+e_t, where A_a is a nonnegative response matrix and e_t records estimation error, distribution shift, semantic debt, or newly induced headroom.
Exact mathematical passage
r_t = \begin{bmatrix} h_H(K_t)\\ h_Q(K_t)\\ h_C(K_t)\\ h_T(K_t)\\ h_S(K_t)\\ h_V(K_t)\\ h_W(K_t) \end{bmatrix} \in\R_{\geq0}^{7}. \label{eq:b8-regret-vector}
H_rUse in source · 14 passages
Search aliases: H_r
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10094
For a current residual r∈ H, define its cyclic causal module H_r = span{Ar:A∈ A}.
Exact mathematical passage
H_r = \Span\{Ar:A\in\mathfrak A\}. \label{eq:b8-cyclic-module}
H_{\mathrm{rep}}Use in source · 3 passages
Search aliases: H_rep mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12465
The source hidden space has dimension six and is generated from three target-specific unknown rank-two blocks: H=H_ctx⊕ H_rep⊕ H_nui. A target-specific orthogonal or injective embedding hides these blocks from the runtime coordinate system. The source repair changes only H_rep and leaves the context block invariant. Two controlled interventions act with different block signatures. The context label controls which target bridge and target repair are valid.
Exact mathematical passage
H=H_{\mathrm{ctx}}\oplus H_{\mathrm{rep}}\oplus H_{\mathrm{nui}}. \label{eq:b11-three-blocks}
h_SUse in source · 2 passages
Search aliases: h_S
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9246
Let the unresolved module-headroom vector be r_t = h_H(K_t) h_Q(K_t) h_C(K_t) h_T(K_t) h_S(K_t) h_V(K_t) h_W(K_t) ∈ℝ_≥0^7. An authorized meta-action a changes one or more repair-process modules. In a local finite model, represent its effect by r_t+1 ≤ A_a r_t+e_t, where A_a is a nonnegative response matrix and e_t records estimation error, distribution shift, semantic debt, or newly induced headroom.
Exact mathematical passage
r_t = \begin{bmatrix} h_H(K_t)\\ h_Q(K_t)\\ h_C(K_t)\\ h_T(K_t)\\ h_S(K_t)\\ h_V(K_t)\\ h_W(K_t) \end{bmatrix} \in\R_{\geq0}^{7}. \label{eq:b8-regret-vector}
H_s^{\top}Use in source · 2 passages
Search aliases: H_s^top top
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6874
The normal equations are H_s^topH_swidehat T^top=H_s^topH_t. Full column rank makes H_s^topH_s invertible. Substitution of H_t=H_sT^top yields widehat T=T.
Exact mathematical passage
H_s^{\top}H_s\widehat T^{\top}=H_s^{\top}H_t.
h_TUse in source · 2 passages
Search aliases: h_T
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9246
Let the unresolved module-headroom vector be r_t = h_H(K_t) h_Q(K_t) h_C(K_t) h_T(K_t) h_S(K_t) h_V(K_t) h_W(K_t) ∈ℝ_≥0^7. An authorized meta-action a changes one or more repair-process modules. In a local finite model, represent its effect by r_t+1 ≤ A_a r_t+e_t, where A_a is a nonnegative response matrix and e_t records estimation error, distribution shift, semantic debt, or newly induced headroom.
Exact mathematical passage
r_t = \begin{bmatrix} h_H(K_t)\\ h_Q(K_t)\\ h_C(K_t)\\ h_T(K_t)\\ h_S(K_t)\\ h_V(K_t)\\ h_W(K_t) \end{bmatrix} \in\R_{\geq0}^{7}. \label{eq:b8-regret-vector}
h_{t-1}Use in source · 1 passage
Search aliases: h_t-1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15663
Given input x, define the prompt anchor h_0=E(x). A normalized injection state z_t and updated latent state h_t satisfy z_t=N(h_t-1), widetilde h_t=B(z_t,x), h_t=U(z_t,widetilde h_t,h_0), 1≤ t≤ K, where N is any declared normalization or scale-matching operator. The answer is sampled or selected from D(h_K,x) after a declared end-of-thought transition.
Exact mathematical passage
z_t=N(h_{t-1}), \qquad \widetilde h_t=B(z_t,x), \qquad h_t=U(z_t,\widetilde h_t,h_0), \qquad 1\leq t\leq K, \label{eq:b16-latent-recurrence}
h_VUse in source · 1 passage
Search aliases: h_V
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9246
Let the unresolved module-headroom vector be r_t = h_H(K_t) h_Q(K_t) h_C(K_t) h_T(K_t) h_S(K_t) h_V(K_t) h_W(K_t) ∈ℝ_≥0^7. An authorized meta-action a changes one or more repair-process modules. In a local finite model, represent its effect by r_t+1 ≤ A_a r_t+e_t, where A_a is a nonnegative response matrix and e_t records estimation error, distribution shift, semantic debt, or newly induced headroom.
Exact mathematical passage
r_t = \begin{bmatrix} h_H(K_t)\\ h_Q(K_t)\\ h_C(K_t)\\ h_T(K_t)\\ h_S(K_t)\\ h_V(K_t)\\ h_W(K_t) \end{bmatrix} \in\R_{\geq0}^{7}. \label{eq:b8-regret-vector}
H_wUse in source · 3 passages
Search aliases: H_w
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4498
[Archive-envelope monotonicity] Suppose archive updates never delete the last representative of a retained node. For every nonnegative weight vector w, define H_w(A_t) = max_A∈A_tw^ Tz(A). Then H_w(A_t) is nondecreasing in t.
Exact mathematical passage
H_w(\Acal_t) = \max_{A\in\Acal_t}w^{\mathsf T}z(A).
h_WUse in source · 1 passage
Search aliases: h_W
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9246
Let the unresolved module-headroom vector be r_t = h_H(K_t) h_Q(K_t) h_C(K_t) h_T(K_t) h_S(K_t) h_V(K_t) h_W(K_t) ∈ℝ_≥0^7. An authorized meta-action a changes one or more repair-process modules. In a local finite model, represent its effect by r_t+1 ≤ A_a r_t+e_t, where A_a is a nonnegative response matrix and e_t records estimation error, distribution shift, semantic debt, or newly induced headroom.
Exact mathematical passage
r_t = \begin{bmatrix} h_H(K_t)\\ h_Q(K_t)\\ h_C(K_t)\\ h_T(K_t)\\ h_S(K_t)\\ h_V(K_t)\\ h_W(K_t) \end{bmatrix} \in\R_{\geq0}^{7}. \label{eq:b8-regret-vector}
H_{\gamma_1}Use in source · 1 passage
Search aliases: H_γ_1 gamma
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3498
t G be finite. Choose a spanning tree and let γ_1,…,γ_m be the fundamental loops generated by non-tree edges. If the loop transports form a representation of the graph fundamental group, then E_v_0^flat = ⋂_j=1^m ker(H_γ_j-I). Equivalently, E_v_0^flat is the null space of the vertically stacked matrix H_γ_1-I vdots H_γ_m-I .
Exact mathematical passage
\begin{bmatrix} H_{\gamma_1}-I\\ \vdots\\ H_{\gamma_m}-I \end{bmatrix}.
H_{\gamma_j}Use in source · 1 passage
Search aliases: H_γ_j gamma
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3491
[Finite computation from a cycle basis] Let G be finite. Choose a spanning tree and let γ_1,…,γ_m be the fundamental loops generated by non-tree edges. If the loop transports form a representation of the graph fundamental group, then E_v_0^flat = ⋂_j=1^m ker(H_γ_j-I). Equivalently, E_v_0^flat is the null space of the vertically stacked matrix H_γ_1-I
Exact mathematical passage
E_{v_0}^{\flat} = \bigcap_{j=1}^{m} \ker(H_{\gamma_j}-I).
H_{\gamma_m}Use in source · 1 passage
Search aliases: H_γ_m gamma
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3498
t G be finite. Choose a spanning tree and let γ_1,…,γ_m be the fundamental loops generated by non-tree edges. If the loop transports form a representation of the graph fundamental group, then E_v_0^flat = ⋂_j=1^m ker(H_γ_j-I). Equivalently, E_v_0^flat is the null space of the vertically stacked matrix H_γ_1-I vdots H_γ_m-I .
Exact mathematical passage
\begin{bmatrix} H_{\gamma_1}-I\\ \vdots\\ H_{\gamma_m}-I \end{bmatrix}.
H_{\omega}Use in source · 2 passages
Search aliases: H_ω omega
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7030
For encoder E_θ, decoder G_δ, and score head H_ω, the pilot minimizes minibatch estimates of L(θ,δ,ω) = λ_rm rec normG_δ(E_θ(ϕ))-ϕ_F^2 + normH_ω(E_θ(ϕ))-widetilde q_F^2, with λ_rm rec=2. Optimization uses a deterministic NumPy Adam implementation for 500 steps. After representation training, the head is refitted by ridge regression to reduce avoidable readout variance.
Exact mathematical passage
H_{\omega}
\mathrm{halt}Use in source · 1 passage
Search aliases: halt mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16303
positive interval and fraction gates, local continuity, donor coverage, adaptive halting. Among eligible candidates it maximizes S(α) = · min{ fracR^2_probe0.65, fracQ_prog0.15, fracQ_ans0.20 } +0.20R^2_probe · -4 MSE_clean +0.02ρ_halt -Penalty_dynamic. The minimum term makes the weakest normalized semantic gate the principal bottleneck. The compiler seal commits the selected candidate, all candidate and parent state hashes, training receipts, meta-validation task digest, exact thresholds, selec…
Exact mathematical passage
\begin{split} S(\alpha) ={}& \min\left\{ \frac{R^2_{\mathrm{probe}}}{0.65}, \frac{Q_{\mathrm{prog}}}{0.15}, \frac{Q_{\mathrm{ans}}}{0.20} \right\} +0.20R^2_{\mathrm{probe}}\\ &-4\,\mathrm{MSE}_{\mathrm{clean}} +0.02\rho_{\mathrm{halt}} -\mathsf{Penalty}_{\mathrm{dynamic}}. \end{split} \label{eq:b17-selection-score}
\Hcal_{\mathrm{obs}}Use in source · 1 passage
Search aliases: Hcal_obs H_obs Hcal mathrm mathcal mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13973
The finite benchmark contains six hidden obstruction classes: H_obs = {data,objective,response,constructor,evaluator,resource}. The compiler is not given these labels at deployment. Each research problem exposes a twelve-dimensional cheap diagnostic vector, a baseline repair process, a finite candidate-method grammar, four scored benchmarks, a hidden audit, a capability check, and integrity constraints.
Exact mathematical passage
\Hcal_{\mathrm{obs}} = \{\text{data},\text{objective},\text{response},\text{constructor},\text{evaluator},\text{resource}\}. \label{eq:b13-obstruction-classes}
\mathrm{high}Use in source · 1 passage
Search aliases: high mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15120
Let p∈P^+ be a candidate experiment with response y_p:T⟶ℝ. For threshold θ, define the refined diagnostic map ϕ_p,θ(τ) = (ϕ_0(τ),low), · y_p(τ)≤θ, (ϕ_0(τ),high), · y_p(τ)>θ. The old token is recovered by the projection π_0(z,s)=z, π_0∘ϕ_p,θ=ϕ_0. Thus the extension is conservative at the level of old diagnostic semantics.
Exact mathematical passage
\phi_{p,\theta}(\tau) = \begin{cases} (\phi_0(\tau),\mathrm{low}), & y_p(\tau)\leq\theta,\\ (\phi_0(\tau),\mathrm{high}), & y_p(\tau)>\theta. \end{cases} \label{eq:b15-refined-token}
\mathrm{hinge}Use in source · 3 passages
Search aliases: hinge mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16619
Let the template set be Θ = {θ_aff,θ_abs, θ_quad,θ_hinge}. For integer parameters a,b,c and integer input x, the exact programs are f_aff(x) · =ax+b,
Exact mathematical passage
\Theta = \{\theta_{\mathrm{aff}},\theta_{\mathrm{abs}}, \theta_{\mathrm{quad}},\theta_{\mathrm{hinge}}\}. \label{eq:b18-template-set}
\mathrm{hist}Use in source · 4 passages
Search aliases: hist mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3774
[Historical ranking preservation] Let x_1,…,x_N be protected historical candidate records with scalar old scores J_v(x_i). Suppose every unequal old pair has margin at least m_hist := min_J_v(x_i)≠ J_v(x_j) absJ_v(x_i)-J_v(x_j)>0. If an upgrade satisfies absJ_w(T_ex_i)-J_v(x_i)≤ε for every i, with 2ε<m_hist, then every strict historical ranking is preserved.
Exact mathematical passage
m_{\mathrm{hist}} := \min_{J_v(x_i)\neq J_v(x_j)} \abs{J_v(x_i)-J_v(x_j)}>0.
\hlineUse in source · 2 passages
Search aliases: hline
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2605
The theorem produces an obstruction ledger c|cc order · local obstruction · gluing obstruction 1 · [b_1,α]∈coker D_α · [d^(1)]∈check H^1(ker D) 2 · [b_2,α]∈coker D_α · [d^(2)]∈check H^1(ker D) vdots · vdots · vdots N · [b_N,α]∈coker D_α · [d^(N)]∈check H^1(ker D). Higher categorical repair interfaces may introduce additional coherence obstructions beyond this two-column ledger. Those are not proved here.
Exact mathematical passage
\begin{array}{c|cc} \text{order}&\text{local obstruction}&\text{gluing obstruction}\\ \hline 1&[b_{1,\alpha}]\in\coker D_\alpha&[d^{(1)}]\in\check H^1(\ker D)\\ 2&[b_{2,\alpha}]\in\coker D_\alpha&[d^{(2)}]\in\check H^1(\ker D)\\ \vdots&\vdots&\vdots\\ N&[b_{N,\alpha}]\in\coker D_\alpha&[d^{(N)}]\in\check H^1(\ker D). \end{array}
\mathrm{hold}Use in source · 2 passages
Search aliases: hold mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4625
Let D_hold be a distribution over foreign held-out agents that are not descendants of the evaluated lineage. Let K be a frozen improvement mechanism. Under horizon h and budget B, define M^foreign_h,B(K) = E_A∼D_hold [ max_A'∈Desc_h,B(K,A) J(A')-J(A) ].
Exact mathematical passage
\Dcal_{\mathrm{hold}}
I'Use in source · 7 passages
Search aliases: I'
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 626
An evaluation context is nontrivial when candidate identity, self-reported utility, and mutable internal error registers are conditionally irrelevant given protected external traces. Formally, for candidate identifier I and protected trace Y, V(I,Y)=V(I',Y) for all identifiers I,I' that generate the same authorized trace and resource record. This excludes a large class of self-confirming evaluator changes while allowing the evaluator to use causally relevant provenance and intervention receipts.
Exact mathematical passage
V(I,Y)=V(I',Y)
i^2Use in source · 1 passage
Search aliases: i^2
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6093
Each foreign agent has a canonical defect d∈ℝ^6, a positive diagonal importance vector w, and an invertible architecture transform T∈ℝ^6×6. The runtime-visible defect is widetilde d=Td. Protected utility is J_T(widetilde d) = exp(-∑_i=1^6w_i(T^-1widetilde d)_i^2). A candidate runtime patch δ produces gain g(δ) = J_T(widetilde d+δ)-J_T(widetilde d).
Exact mathematical passage
J_T(\widetilde d) = \exp\left(-\sum_{i=1}^6w_i\bigl(T^{-1}\widetilde d\bigr)_i^2\right).
I^\perpUse in source · 9 passages
Search aliases: I^⊥ perp
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9736
Let I=im D⊆ V be the current authorized correction image and let z=P_I^⊥r be the unresolved component of defect r. Suppose a candidate grammar extension contributes a response vector v∈ V. Only its component outside the old image is structurally new: bar v=P_I^⊥v. The expanded image is I'=I+span{v}.
Exact mathematical passage
z=P_{I^\perp}r
i^\starUse in source · 4 passages
Search aliases: i^⋆ star
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13905
[Uniform forecast-regret bound] Let g_i be the true protected gain of each admissible candidate and suppose |widehat g_i-g_i|≤ε for all candidates. If widehat i∈arg max_iwidehat g_i and i^⋆∈arg max_i g_i, then g_i^⋆-g_widehat i ≤ 2ε.
Exact mathematical passage
i^\star\in\argmax_i g_i
I_+Use in source · 3 passages
Search aliases: I_+
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13804
Let the scored evaluator portfolio be B = {b_1,…,b_m}. For method c, define normalized headroom closure q_j(c) = fracs_j(c)-s_j(K_τ)s_j^⋆-s_j(K_τ). Let I_+(c)={j:q_j(c)>0}. The coverage-weighted geometric score is S_cov(c) = frac|I_+(c)|m ( ∏_j∈ I_+(c)q_j(c) )^1/|I_+(c)|, with value zero when I_+(c)=∅. Hard capability and integrity gates are applied separately rather than traded against this scalar.
Exact mathematical passage
I_+(c)=\{j:q_j(c)>0\}
I_0Use in source · 1 passage
Search aliases: I_0
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9509
Let V be a finite-dimensional defect Hilbert space. Suppose a conservative grammar chain gives nested closed authorized correction images I_0⊆ I_1⊆ I_2⊆⋯⊆ V, I_t=im D_t. For r∈ V, define unresolved obstruction magnitude η_t(r) = dist(-r,I_t) = normP_I_t^⊥r.
Exact mathematical passage
I_0\subseteq I_1\subseteq I_2\subseteq\cdots\subseteq V, \qquad I_t=\im D_t. \label{eq:b8-nested-images}
i_0Use in source · 3 passages
Search aliases: i_0
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7569
[Pathwise selector isolation] For episode e, let parent and child receive the identical ordered bundle C_e and identical pre-audit receipt Z_e. Let i_0=K_0(Z_e), i_1=K_1(Z_e). Then the realized paired utility difference is exactly D_e = U_e(c_e,i_1)-U_e(c_e,i_0). If candidate construction, target state, and audit evaluation are frozen, no candidate-generation term remains in this difference.
Exact mathematical passage
i_0=K_0(Z_e), \qquad i_1=K_1(Z_e).
I_1Use in source · 1 passage
Search aliases: I_1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9509
Let V be a finite-dimensional defect Hilbert space. Suppose a conservative grammar chain gives nested closed authorized correction images I_0⊆ I_1⊆ I_2⊆⋯⊆ V, I_t=im D_t. For r∈ V, define unresolved obstruction magnitude η_t(r) = dist(-r,I_t) = normP_I_t^⊥r.
Exact mathematical passage
I_0\subseteq I_1\subseteq I_2\subseteq\cdots\subseteq V, \qquad I_t=\im D_t. \label{eq:b8-nested-images}
i_1Use in source · 5 passages
Search aliases: i_1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7569
[Pathwise selector isolation] For episode e, let parent and child receive the identical ordered bundle C_e and identical pre-audit receipt Z_e. Let i_0=K_0(Z_e), i_1=K_1(Z_e). Then the realized paired utility difference is exactly D_e = U_e(c_e,i_1)-U_e(c_e,i_0). If candidate construction, target state, and audit evaluation are frozen, no candidate-generation term remains in this difference.
Exact mathematical passage
i_0=K_0(Z_e), \qquad i_1=K_1(Z_e).
I_2Use in source · 11 passages
Search aliases: I_2
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5580
Case F2: gluing obstruction. Take D_0=D_1=D_2=I_2, so every local defect is exactly correctable. On the oriented triangle with edges (0,1), (1,2), and (0,2), set the scalar overlap mismatch b= 0
Exact mathematical passage
D_0=D_1=D_2=I_2
I_5Use in source · 1 passage
Search aliases: I_5
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5973
nts are, respectively: item deviation from the baseline no-action continuation; item deviation of the action-conditioned response from the baseline action response; item violation of redundant-sensor consistency; item deviation from baseline memory update; item discrepancy between reported and protected evaluator scores. The observed probe is widetildeϕ=ϕ(z)+η, η∼ N(0,σ^2I_5).
Exact mathematical passage
\widetilde\phi=\phi(z)+\eta, \qquad \eta\sim\mathcal N(0,\sigma^2I_5).
I_{d_s}Use in source · 1 passage
Search aliases: I_d_s
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6788
[Exact transport through an injective hidden bridge] Let T:ℝ^d_s→ℝ^d_t have full column rank, so T^+T=I_d_s. Suppose h_t=Th_s throughout the evaluated support. Define D_t and b_t by [source reference] with widehat T=T. Then R_t(Th_s)=TR_s(h_s).
Exact mathematical passage
T^{+}T=I_{d_s}
i_jUse in source · 1 passage
Search aliases: i_j
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8519
The descendant policy uses the source intervention-defined packet to divide the unlabeled source-side candidate pool into high- and low-packet-activation strata. It selects eight anchors from each stratum. Within each stratum it applies a deterministic pivoted residual rule: i_j ∈ arg max_i∉ I_j-1 norm(I-Q_j-1Q_j-1^top)z_i^2, where z_i is the standardized source activation and Q_j-1 spans the previously selected residual directions.
Exact mathematical passage
i_j \in \argmax_{i\notin I_{j-1}} \norm{(I-Q_{j-1}Q_{j-1}^\top)z_i}^2,
I_{j-1}Use in source · 1 passage
Search aliases: I_j-1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8519
The descendant policy uses the source intervention-defined packet to divide the unlabeled source-side candidate pool into high- and low-packet-activation strata. It selects eight anchors from each stratum. Within each stratum it applies a deterministic pivoted residual rule: i_j ∈ arg max_i∉ I_j-1 norm(I-Q_j-1Q_j-1^top)z_i^2, where z_i is the standardized source activation and Q_j-1 spans the previously selected residual directions.
Exact mathematical passage
i_j \in \argmax_{i\notin I_{j-1}} \norm{(I-Q_{j-1}Q_{j-1}^\top)z_i}^2,
I_KUse in source · 2 passages
Search aliases: I_K
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2349
Let G=(I,E) be a connected oriented graph representing the nerve of the cover, let K be a finite-dimensional Hilbert space representing a common trivialization of the kernel fibers, and let e=(e_a)_a∈ E∈ K^E be the transported overlap-mismatch vector. Let B∈ℝ^|I|×|E| be the oriented incidence matrix and set B:=B⊗ I_K. A vertex gauge k∈ K^I changes the mismatch to e+ B^*k.
Exact mathematical passage
\mathbf B:=B\otimes I_K.
I_{m_\alpha}Use in source · 1 passage
Search aliases: I_m_α alpha
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10062
[Finite causal atlas decomposition] There exists a unitary identification H ≅ ⊕_α=1^p ( C^m_α⊗ C^n_α ) under which A ≅ ⊕_α=1^p ( I_m_α⊗ M_n_α( C) ). The minimal central projections P_α∈ Z( A) give a canonical coarse causal atlas H = ⊕_α=1^pP_α H. The blocks are unique up to permutation and unitary changes of basis within multiplicity spaces.
Exact mathematical passage
\mathfrak A \cong \bigoplus_{\alpha=1}^{p} \left( I_{m_\alpha}\otimes M_{n_\alpha}(\mathbb C) \right). \label{eq:b8-algebra-decomposition}
I_qUse in source · 4 passages
Search aliases: I_q
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10592
Let the source hidden state have dimension d and the target hidden state have dimension q. For selected source anchors x_i∈ℝ^d, suppose the observed target anchor is y_i=Tx_i+ε_i, E[ε_i]=0, Cov(ε_i)=σ_i^2I_q. The query-quality metadata supplies σ_i before the anchor is selected. This metadata is part of the declared Block 9 environment. It is not discovered autonomously.
Exact mathematical passage
y_i=Tx_i+\eps_i, \qquad \mathbb E[\eps_i]=0, \qquad \Cov(\eps_i)=\sigma_i^2I_q.
I_rUse in source · 1 passage
Search aliases: I_r
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13081
[Best affine residual for a quadratic packet repair] Let z∼ N(0,I_r), B=B^top, and define q(z)=z^top Bz. The best L^2 affine approximation to q(z)v is the constant (tr B)v, and its minimum squared residual is inf_a,L Enormq(z)v-(a+Lz)^2 = 2normB_F^2normv^2.
Exact mathematical passage
z\sim N(0,I_r)
I_tUse in source · 7 passages
Search aliases: I_t
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9509
Let V be a finite-dimensional defect Hilbert space. Suppose a conservative grammar chain gives nested closed authorized correction images I_0⊆ I_1⊆ I_2⊆⋯⊆ V, I_t=im D_t. For r∈ V, define unresolved obstruction magnitude η_t(r) = dist(-r,I_t) = normP_I_t^⊥r.
Exact mathematical passage
I_0\subseteq I_1\subseteq I_2\subseteq\cdots\subseteq V, \qquad I_t=\im D_t. \label{eq:b8-nested-images}
i_tUse in source · 3 passages
Search aliases: i_t
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9337
[Greedy bottleneck contraction] Let r_t∈ℝ_≥0^m and Φ_t=1^top r_t. Suppose the controller chooses i_t with r_i_t,t≥fracΦ_tm, contracts that component by q∈[0,1), and permits total collateral increase at most ε_t. Then Φ_t+1 ≤ (1-frac1-qm)Φ_t+ε_t.
Exact mathematical passage
i_t
I_{t+1}Use in source · 6 passages
Search aliases: I_t+1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9537
Nestedness gives I_t⊆ I_t+1, so distance to I_t+1 cannot exceed distance to I_t. Let I_∞=overline⋃_t I_t. The projections P_I_tr converge strongly to P_I_∞r in finite dimension, so the orthogonal residual norms converge to the distance from -r to I_∞.
Exact mathematical passage
I_t\subseteq I_{t+1}
I_{t+1}^\perpUse in source · 1 passage
Search aliases: I_t+1^⊥ perp
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9565
The nested orthogonal decomposition is I_t+1 = I_t⊕(I_t+1∩ I_t^⊥). Therefore P_I_t+1^⊥r = P_I_t^⊥r - P_I_t+1∩ I_t^⊥r, and the two terms on the right are orthogonal. The Pythagorean identity gives [source reference].
Exact mathematical passage
P_{I_{t+1}^\perp}r = P_{I_t^\perp}r - P_{I_{t+1}\cap I_t^\perp}r,
I_t^\perpUse in source · 5 passages
Search aliases: I_t^⊥ perp
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9516
Let V be a finite-dimensional defect Hilbert space. Suppose a conservative grammar chain gives nested closed authorized correction images I_0⊆ I_1⊆ I_2⊆⋯⊆ V, I_t=im D_t. For r∈ V, define unresolved obstruction magnitude η_t(r) = dist(-r,I_t) = normP_I_t^⊥r.
Exact mathematical passage
\eta_t(r) = \dist(-r,I_t) = \norm{P_{I_t^\perp}r}. \label{eq:b8-obstruction-magnitude}
i_\ellUse in source · 2 passages
Search aliases: i_ℓ ell
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12928
Every element of A(R) is a finite linear combination of words in the generators. On H_a, a word R_i_1⋯ R_i_ℓ acts by the scalar λ_ai_1⋯λ_ai_ℓ. Equality of the signature vectors makes the same scalar act on H_b. Linear combinations preserve the equality. A central idempotent separating the blocks would act as 1 on one and 0 on the other, contradicting the first conclusion.
Exact mathematical passage
R_{i_1}\cdots R_{i_\ell}
I_\inftyUse in source · 3 passages
Search aliases: I_∞ infty
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9538
Nestedness gives I_t⊆ I_t+1, so distance to I_t+1 cannot exceed distance to I_t. Let I_∞=overline⋃_t I_t. The projections P_I_tr converge strongly to P_I_∞r in finite dimension, so the orthogonal residual norms converge to the distance from -r to I_∞.
Exact mathematical passage
I_\infty=\overline{\bigcup_t I_t}.
\Ical_{\mathrm{ctrl}}^{*}Use in source · 11 passages
Search aliases: Ical_ctrl^* I_ctrl^* Ical mathrm mathcal mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4333
[Robust controlled improvement kernel] The operator Ψ_ctrl is monotone and has a greatest fixed point I_ctrl^* = ν X.Ψ_ctrl(X). For every state in I_ctrl^*, there exists a stationary authorized policy that keeps every admissible successor inside I_ctrl^*.
Exact mathematical passage
\Ical_{\mathrm{ctrl}}^{*} = \nu X.\Psi_{\mathrm{ctrl}}(X).
\Ical_t^{\mathrm{NL}}Use in source · 2 passages
Search aliases: Ical_t^NL I_t^NL Ical mathrm mathcal mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15834
Extend the SRIC lineage state by L_t^NL = ( L_t,N_t,I_t^NL,T_t^NL,W_t^NL), where T_t^NL contains trace-transport certificates and W_t^NL is the frozen external answer and invariant witness. A proposed descendant may modify the latent transition, update gate, halting head, packet atlas, or transport map. The backbone may be held fixed for a component-isolation edge or changed under a stronger whole-model edge.
Exact mathematical passage
\mathfrak L_t^{\mathrm{NL}} = (\mathfrak L_t,\Ncal_t,\Ical_t^{\mathrm{NL}},\Tau_t^{\mathrm{NL}},\Wcal_t^{\mathrm{NL}}),
\Ical_\gamma^\omegaUse in source · 1 passage
Search aliases: Ical_γ^ω I_γ^ω Ical gamma omega mathcal gamma omega
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21073
Let s_1 above be an absorbing no-op state and suppose s_1∉ G_γ. Then s_1 lies in the controlled safety kernel but not in the productive Buechi closure I_γ^ω. The system can preserve every invariant forever while producing no repeated validated improvement. Safe persistence is therefore not evidence of recursive improvement.
Exact mathematical passage
\Ical_\gamma^\omega
\Ical_{\forall}^{*}Use in source · 6 passages
Search aliases: Ical_∀^* I_∀^* Ical forall mathcal forall
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4283
[Existence of universal recursive improvement closure] The operator Ψ_∀:2^S→2^S is monotone and has a greatest fixed point I_∀^* = ν X.Ψ_∀(X). A state lies in I_∀^* exactly when it satisfies the static invariants and every authorized successor also lies in I_∀^*.
Exact mathematical passage
\Ical_{\forall}^{*} = \nu X.\Psi_{\forall}(X).
\mathrm{improved}Use in source · 4 passages
Search aliases: improved mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5669
The minimal closure model has states S={s_base,s_improved,s_invalid} and valid set K={s_base,s_improved}. With intact rollback, authorized complete transactions have successor relation Post(s_base) ={s_base,s_improved}, Post(s_improved)={s_improved}. Failed candidates are restored to the base state before the complete transaction ends.
Exact mathematical passage
S=\{s_{\mathrm{base}},s_{\mathrm{improved}},s_{\mathrm{invalid}}\}
\mathrm{improver}Use in source · 1 passage
Search aliases: improver mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 665
A representative color set is C= set model, workspace, memory, planner, improver, evaluator, provenance, tools . The atlas does not assert that these colors are universally correct. It declares the decomposition under which one particular repair theory is being tested.
Exact mathematical passage
\mathsf C= \set{ \mathrm{model}, \mathrm{workspace}, \mathrm{memory}, \mathrm{planner}, \mathrm{improver}, \mathrm{evaluator}, \mathrm{provenance}, \mathrm{tools} }.
\mathrm{ImprovesSlice}Use in source · 1 passage
Search aliases: ImprovesSlice mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21345
The example separates two predicates: ImprovesSlice(R_t) not⇒ AuthorizedCommit(R_t).
Exact mathematical passage
\mathrm{ImprovesSlice}(R_t) \not\Rightarrow \mathrm{AuthorizedCommit}(R_t).
\inf_{a,L}Use in source · 1 passage
Search aliases: inf_a,L inf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13086
[Best affine residual for a quadratic packet repair] Let z∼ N(0,I_r), B=B^top, and define q(z)=z^top Bz. The best L^2 affine approximation to q(z)v is the constant (tr B)v, and its minimum squared residual is inf_a,L Enormq(z)v-(a+Lz)^2 = 2normB_F^2normv^2.
Exact mathematical passage
\inf_{a,L} \mathbb E\norm{q(z)v-(a+Lz)}^2 = 2\norm{B}_F^2\norm{v}^2. \label{eq:b12-affine-residual}
\inf_kUse in source · 1 passage
Search aliases: inf_k inf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2433
Different overlaps may have different uncertainty or semantic importance. Let W_E be a positive-definite operator on K^E. Define η_glue,W(e) := inf_knorme+ B^*k_W_E.
Exact mathematical passage
\eta_{\mathrm{glue},W}(e) := \inf_k\norm{e+\mathbf B^*k}_{W_E}.
\inf_{k\in K^I}Use in source · 1 passage
Search aliases: inf_k∈ K^I inf in
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2368
[Quantitative graph-Hodge gluing] Let L_G:=BB^* be the graph Laplacian and let P_cyc be the orthogonal projector from K^E onto ker B, the cycle space. Then inf_k∈ K^Inorme+ B^*k = normP_cyce. The mean-zero minimum-norm gauge is k^* =-(L_G^† B⊗ I_K)e. If λ_2(L_G)>0 is the spectral gap, then normk^* ≤ frac1sqrtλ_2(L_G)norm e. Exact graph-level gluing is possible if and only if P_cyce=0.
Exact mathematical passage
\inf_{k\in K^I}\norm{e+\mathbf B^*k} = \norm{P_{\mathrm{cyc}}e}.
\inf_TUse in source · 1 passage
Search aliases: inf_T inf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11451
A population residual formula gives a quantitative special case. Suppose h is centered with covariance Σsucc0, s∈{-1,+1} is independent of h with Pr(s=1)=p, and y=(A+sB)h. Then the best global linear map is T^*=A+(2p-1)B, and inf_T Enormy-Th^2 = 4p(1-p)tr(BΣ B^top). The proof is the normal-equation calculation from the centered two-component mixture. The executable fixture allows affine offsets, context-dependent source distributions, and heteroscedastic noise, so its operative certificate is a …
Exact mathematical passage
\inf_T\mathbb E\norm{y-Th}^2 = 4p(1-p)\tr(B\Sigma B^\top). \label{eq:b10-population-residual}
\inf_tUse in source · 1 passage
Search aliases: inf_t inf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21838
Let h∼ N(0,1) and let s∈{-1,+1} be independent and balanced. Define y=(a+sb)h, b≠0. A global scalar bridge predicts widehat y=th. The population minimizer is t^*=a, and inf_t E (y-th)^2 = E(sbh)^2 = b^2. The residual is not an estimation artifact. It remains after infinite data because one slope cannot equal both a-b and a+b. The two-chart grammar uses t_-=a-b, t_+=a+b, and has zero noiseless approximation error. This is the one-dimensional form of [source reference].
Exact mathematical passage
\inf_t\mathbb E (y-th)^2 = \mathbb E(sbh)^2 = b^2.
\inf_{u\in K}Use in source · 1 passage
Search aliases: inf_u∈ K inf in
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1688
Some repairs are not closed under negation. Examples include adding tests, allocating nonnegative compute, increasing redundancy, or modifying parameters subject to inequality constraints. Let K⊆ U be a nonempty closed convex authorization set and define η_D,K(r):=inf_u∈ KnormDu+r.
Exact mathematical passage
\eta_{D,K}(r):=\inf_{u\in K}\norm{Du+r}.
\inf_{u\in U^{\Auth}}Use in source · 1 passage
Search aliases: inf_u∈ U^Auth inf_u∈ U^auth inf in Auth inf in mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1559
eginenumerate item P_cor is the orthogonal projector onto im D; item P_ob is the orthogonal projector onto ker D^*; item the defect decomposes uniquely as r=r_cor+r_ob, r_cor=P_corr, r_ob=P_obr; item the minimum achievable residual is inf_u∈ U^authnormDu+r =normr_ob =dist(r,im D); item the complete set of least-squares minimizers is -D^† r+ker D; item if r is exactly correctable, u^*=-D^† r is the unique exact repair of minimum norm.
Exact mathematical passage
\inf_{u\in U^{\Auth}}\norm{Du+r} =\norm{r_{\mathrm{ob}}} =\dist(r,\im D);
\mathrm{int}Use in source · 2 passages
Search aliases: int mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1168
[Internal error reduction without external improvement] Let the environment always produce protected outcome Y=1, independently of the agent's action. Let an internal predictor be p∈ℝ with internal score J_int(p)=-(p-Y)^2. A self-intervention setting p←1 improves J_int to zero error but does not change the world, task success, or action consequence. More extremely, an intervention that rewrites the error register itself can report zero error without even improving prediction. Therefore reduction…
Exact mathematical passage
J_{\mathrm{int}}(p)=-(p-Y)^2.
\mathrm{internal}Use in source · 1 passage
Search aliases: internal mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5935
The latent event variable is z∈ {none,internal,world,sensor,memory,evaluator}. Each event changes one declared mechanism: item internal: the action-response map B changes; item world: the passive dynamics A change; item sensor: the redundant sensor receives an additive bias; item memory: the retention coefficient λ changes; item evaluator: the reported weight vector w changes; item none: all baseline mechanisms remain fixed. The event label is not supplied to the classifier.
Exact mathematical passage
z\in \{\mathrm{none},\mathrm{internal},\mathrm{world},\mathrm{sensor},\mathrm{memory},\mathrm{evaluator}\}.
\mathrm{intervention}Use in source · 2 passages
Search aliases: intervention mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 19952
full identity, equal-rank random }. For each nontrivial constructor, ranks one through six are considered. The full identity is an oversized control. A projection W is valid only when, on both recurrent systems, R^2_future · ≥0.82, R^2_intervention · ≥0.90, Acc_answer · ≥0.78, D_twin · ≤0.22. Among valid candidates, the compiler minimizes rank and then follows the frozen method priority. No hidden semantic score enters the choice.
Exact mathematical passage
\begin{aligned} R^2_{\mathrm{future}}&\geq0.82,\\ R^2_{\mathrm{intervention}}&\geq0.90,\\ \operatorname{Acc}_{\mathrm{answer}}&\geq0.78,\\ D_{\mathrm{twin}}&\leq0.22. \end{aligned} \label{eq:b23-visible-gates}
J^*Use in source · 2 passages
Search aliases: J^*
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4587
Let J^*(A_t) be the highest true protected utility among retained validated nodes.
Exact mathematical passage
J^*(\Acal_t)
j^{-1}Use in source · 3 passages
Search aliases: j^-1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5074
[Old-resolution and new-obstruction exact sequence] There is an exact sequence 0 ⟶ fracj^-1(C')C ⟶ fracVC xrightarrowoverline j fracV'C' ⟶ fracV'C'+j(V) ⟶ 0. The first nonzero term contains old obstruction classes newly resolved by the expanded repair calculus. The final quotient contains genuinely new obstruction directions not represented in the old ontology.
Exact mathematical passage
0 \longrightarrow \frac{j^{-1}(C')}{C} \longrightarrow \frac{V}{C} \xrightarrow{\overline j} \frac{V'}{C'} \longrightarrow \frac{V'}{C'+j(V)} \longrightarrow 0.
J_0Use in source · 5 passages
Search aliases: J_0
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3681
chain of evaluator morphisms with readouts into a common complete normed score space, E_0xrightarrowe_0 E_1 xrightarrowe_1⋯ xrightarrowe_n-1 E_n. Suppose that for every protected represented state x_t transported from version 0, normJ_t+1(T_tx_t)-J_t(x_t)≤ε_t. Then normJ_n(T_n-1⋯ T_0x_0)-J_0(x_0) ≤ ∑_t=0^n-1ε_t. If ∑_tε_t<∞, then the transported score sequence is Cauchy and converges in every complete score space.
Exact mathematical passage
\norm{J_n(T_{n-1}\cdots T_0x_0)-J_0(x_0)} \leq \sum_{t=0}^{n-1}\eps_t.
J_{\mathrm{int}}Use in source · 2 passages
Search aliases: J_int mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1168
[Internal error reduction without external improvement] Let the environment always produce protected outcome Y=1, independently of the agent's action. Let an internal predictor be p∈ℝ with internal score J_int(p)=-(p-Y)^2. A self-intervention setting p←1 improves J_int to zero error but does not change the world, task success, or action consequence. More extremely, an intervention that rewrites the error register itself can report zero error without even improving prediction. Therefore reduction…
Exact mathematical passage
J_{\mathrm{int}}(p)=-(p-Y)^2.
J_nUse in source · 1 passage
Search aliases: J_n
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3681
chain of evaluator morphisms with readouts into a common complete normed score space, E_0xrightarrowe_0 E_1 xrightarrowe_1⋯ xrightarrowe_n-1 E_n. Suppose that for every protected represented state x_t transported from version 0, normJ_t+1(T_tx_t)-J_t(x_t)≤ε_t. Then normJ_n(T_n-1⋯ T_0x_0)-J_0(x_0) ≤ ∑_t=0^n-1ε_t. If ∑_tε_t<∞, then the transported score sequence is Cauchy and converges in every complete score space.
Exact mathematical passage
\norm{J_n(T_{n-1}\cdots T_0x_0)-J_0(x_0)} \leq \sum_{t=0}^{n-1}\eps_t.
J_{\mathrm{protected}}Use in source · 2 passages
Search aliases: J_protected mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21246
Consider a parent with protected utility J_0=0.42. A large global rewrite has protected utility J_protected=0.10 but apparent utility widehat J=0.10+1.10=1.20 due to an exploitable proxy bonus. A selector using widehat J accepts apparent gain 0.78, while protected gain is 0.10-0.42=-0.32. This is the mechanism underlying the global-edit positive control. A protected witness ranks by J_protected and rejects the candidate.
Exact mathematical passage
J_{\mathrm{protected}}=0.10
J_TUse in source · 2 passages
Search aliases: J_T
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6093
Each foreign agent has a canonical defect d∈ℝ^6, a positive diagonal importance vector w, and an invertible architecture transform T∈ℝ^6×6. The runtime-visible defect is widetilde d=Td. Protected utility is J_T(widetilde d) = exp(-∑_i=1^6w_i(T^-1widetilde d)_i^2). A candidate runtime patch δ produces gain g(δ) = J_T(widetilde d+δ)-J_T(widetilde d).
Exact mathematical passage
J_T(\widetilde d) = \exp\left(-\sum_{i=1}^6w_i\bigl(T^{-1}\widetilde d\bigr)_i^2\right).
J_tUse in source · 1 passage
Search aliases: J_t
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3677
ulative evaluator-drift bound] Consider a chain of evaluator morphisms with readouts into a common complete normed score space, E_0xrightarrowe_0 E_1 xrightarrowe_1⋯ xrightarrowe_n-1 E_n. Suppose that for every protected represented state x_t transported from version 0, normJ_t+1(T_tx_t)-J_t(x_t)≤ε_t. Then normJ_n(T_n-1⋯ T_0x_0)-J_0(x_0) ≤ ∑_t=0^n-1ε_t. If ∑_tε_t<∞, then the transported score sequence is Cauchy and converges in every complete score space.
Exact mathematical passage
\norm{J_{t+1}(T_tx_t)-J_t(x_t)}\leq\eps_t.
J_{t+1}Use in source · 1 passage
Search aliases: J_t+1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3677
ulative evaluator-drift bound] Consider a chain of evaluator morphisms with readouts into a common complete normed score space, E_0xrightarrowe_0 E_1 xrightarrowe_1⋯ xrightarrowe_n-1 E_n. Suppose that for every protected represented state x_t transported from version 0, normJ_t+1(T_tx_t)-J_t(x_t)≤ε_t. Then normJ_n(T_n-1⋯ T_0x_0)-J_0(x_0) ≤ ∑_t=0^n-1ε_t. If ∑_tε_t<∞, then the transported score sequence is Cauchy and converges in every complete score space.
Exact mathematical passage
\norm{J_{t+1}(T_tx_t)-J_t(x_t)}\leq\eps_t.
j_UUse in source · 3 passages
Search aliases: j_U
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1877
Open-set improvement may enlarge the defect ontology itself. Let j_U:U_t^auth↪ U_t+1^auth, j_V:V_t↪ V_t+1 be injective embeddings satisfying D_t+1j_U=j_VD_t.
Exact mathematical passage
j_U:U_t^{\Auth}\hookrightarrow U_{t+1}^{\Auth}, \qquad j_V:V_t\hookrightarrow V_{t+1}
j_VUse in source · 6 passages
Search aliases: j_V
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1877
Open-set improvement may enlarge the defect ontology itself. Let j_U:U_t^auth↪ U_t+1^auth, j_V:V_t↪ V_t+1 be injective embeddings satisfying D_t+1j_U=j_VD_t.
Exact mathematical passage
j_U:U_t^{\Auth}\hookrightarrow U_{t+1}^{\Auth}, \qquad j_V:V_t\hookrightarrow V_{t+1}
J_zUse in source · 1 passage
Search aliases: J_z
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3433
of evaluator defects] Let e:v→ w and f:w→ z be evaluator morphisms. Suppose T_f is L_T-Lipschitz on represented protected evidence. Then δ_obs(f∘ e) ≤ L_Tδ_obs(e)+δ_obs(f), where the second term is evaluated on S_e( h_v). If, in addition, sup_r∈ h_v norm J_zT_fT_eι_v(r)-J_wT_eι_v(r) ≤ δ_J(fmid e), then δ_J(f∘ e) ≤ δ_J(e)+δ_J(fmid e).
Exact mathematical passage
\sup_{r\in\mathfrak h_v} \norm{ J_zT_fT_e\iota_v(r)-J_wT_e\iota_v(r) } \leq \delta_J(f\mid e),
\JcalUse in source · 2 passages
Search aliases: Jcal J mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1661
[Regularized minimum-cost repair] For λ>0, the functional J_λ(u) := frac12anglesDu+r,W(Du+r) + fracλ2anglesu,Ru has a unique minimizer u_λ =-(D^*WD+λ R)^-1D^*Wr. It is characterized by the normal equation D^*W(Du_λ+r)+λ Ru_λ=0.
Exact mathematical passage
\Jcal_\lambda(u) := \frac12\angles{Du+r,W(Du+r)} + \frac\lambda2\angles{u,Ru}
\Jcal_\lambdaUse in source · 2 passages
Search aliases: Jcal_λ J_λ Jcal lambda mathcal lambda
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1661
[Regularized minimum-cost repair] For λ>0, the functional J_λ(u) := frac12anglesDu+r,W(Du+r) + fracλ2anglesu,Ru has a unique minimizer u_λ =-(D^*WD+λ R)^-1D^*Wr. It is characterized by the normal equation D^*W(Du_λ+r)+λ Ru_λ=0.
Exact mathematical passage
\Jcal_\lambda(u) := \frac12\angles{Du+r,W(Du+r)} + \frac\lambda2\angles{u,Ru}
\mathcal KUse in source · 4 passages
Search aliases: K mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10962
[Finite meta-grammar generalization] Let K contain N candidate grandchild configurations. Suppose each target-level protected gain lies in [-G,G] and n independent development targets are used. Then, with probability at least 1-δ, every candidate satisfies abswidehatM(K)-M(K) ≤ Gsqrtfrac2log(2N/δ)n. If widehat K maximizes the empirical mean, then M(widehat K) ≥ max_K∈ KM(K) - 2Gsqrtfrac2log(2N/δ)n.
Exact mathematical passage
\mathcal K
K^*Use in source · 5 passages
Search aliases: K^*
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1704
When K is a cone, noncorrectability admits a Farkas-type dual witness. Define the dual cone K^*:=setz∈ U:anglesz,u≥0 for all u∈ K.
Exact mathematical passage
K^*:=\set{z\in U:\angles{z,u}\geq0\text{ for all }u\in K}.
k^*Use in source · 2 passages
Search aliases: k^*
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2374
ge gluing] Let L_G:=BB^* be the graph Laplacian and let P_cyc be the orthogonal projector from K^E onto ker B, the cycle space. Then inf_k∈ K^Inorme+ B^*k = normP_cyce. The mean-zero minimum-norm gauge is k^* =-(L_G^† B⊗ I_K)e. If λ_2(L_G)>0 is the spectral gap, then normk^* ≤ frac1sqrtλ_2(L_G)norm e. Exact graph-level gluing is possible if and only if P_cyce=0.
Exact mathematical passage
k^* =-(L_G^\dagger B\otimes I_K)e.
K^EUse in source · 4 passages
Search aliases: K^E
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2345
Let G=(I,E) be a connected oriented graph representing the nerve of the cover, let K be a finite-dimensional Hilbert space representing a common trivialization of the kernel fibers, and let e=(e_a)_a∈ E∈ K^E be the transported overlap-mismatch vector. Let B∈ℝ^|I|×|E| be the oriented incidence matrix and set B:=B⊗ I_K. A vertex gauge k∈ K^I changes the mismatch to e+ B^*k.
Exact mathematical passage
e=(e_a)_{a\in E}\in K^E
K^IUse in source · 2 passages
Search aliases: K^I
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2352
d graph representing the nerve of the cover, let K be a finite-dimensional Hilbert space representing a common trivialization of the kernel fibers, and let e=(e_a)_a∈ E∈ K^E be the transported overlap-mismatch vector. Let B∈ℝ^|I|×|E| be the oriented incidence matrix and set B:=B⊗ I_K. A vertex gauge k∈ K^I changes the mismatch to e+ B^*k.
Exact mathematical passage
k\in K^I
K^\topUse in source · 1 passage
Search aliases: K^top top
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8459
When X has full column rank, X^top X is invertible and (X^top X)^-1X^top Y =(X^top X)^-1X^top XT^top =T^top. When rank X<d, choose a nonzero v∈ker X. For any nonzero w∈ℝ^m, define K=wv^top. Then XK^top=0, so T and T+K agree on every selected anchor while differing on v.
Exact mathematical passage
XK^\top=0
K_{0}Use in source · 1 passage
Search aliases: K_0
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7223
al level. A parent repair selector is frozen. Development receipts generated through the existing lineage are used to compile a child selector. Parent and child then receive exactly the same candidate repairs, target observations, calibration information, evaluator, and candidate budget. The child receipt is sealed before either final audit split is opened. The primary question is E_e∼ Q [ U_e (K_1(C_e)) - U_e (K_0(C_e)) ] >0, where K_0 is the frozen parent selector, K_1 is the lineage-generated…
Exact mathematical passage
\label{eq:block07-primary-question} \mathbb E_{e\sim Q} \left[ U_e\!\left(K_{1}(C_e)\right) - U_e\!\left(K_{0}(C_e)\right) \right] >0,
K_{1}Use in source · 1 passage
Search aliases: K_1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7223
al level. A parent repair selector is frozen. Development receipts generated through the existing lineage are used to compile a child selector. Parent and child then receive exactly the same candidate repairs, target observations, calibration information, evaluator, and candidate budget. The child receipt is sealed before either final audit split is opened. The primary question is E_e∼ Q [ U_e (K_1(C_e)) - U_e (K_0(C_e)) ] >0, where K_0 is the frozen parent selector, K_1 is the lineage-generated…
Exact mathematical passage
\label{eq:block07-primary-question} \mathbb E_{e\sim Q} \left[ U_e\!\left(K_{1}(C_e)\right) - U_e\!\left(K_{0}(C_e)\right) \right] >0,
k_1Use in source · 2 passages
Search aliases: k_1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2474
[Locally repairable, globally incompatible] Let the nerve be a three-cycle with no triple intersection. Suppose the kernel fiber is ℝ and the oriented differences are d_12=1, d_23=1, d_31=1. If the cocycle were a coboundary, there would be k_1,k_2,k_3 with d_12=k_2-k_1, d_23=k_3-k_2, d_31=k_1-k_3. Summing gives 3=0, a contradiction. Hence no global correction exists. The graph-Hodge obstruction has norm sqrt3.
Exact mathematical passage
k_1,k_2,k_3
K_{18}Use in source · 1 passage
Search aliases: K_18
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17579
item the admission results in [source reference] hold; item set-only, chart-only, shuffled-binding, no-abstention, and always-abstain controls retain their declared interpretations; item the automated tests, independent environment replay, independent chart reconstruction, independent NumPy forward pass, receipts, and package manifest verify. Then the edge K_18 xrightarrowΠ_19 K_19 is a finite, conservative, source-side Neuralese semantic-atlas expansion. It resolves two observation contexts abs…
Exact mathematical passage
K_{18} \xrightarrow{\Pi_{19}} K_{19}
K_{19}Use in source · 1 passage
Search aliases: K_19
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17579
item the admission results in [source reference] hold; item set-only, chart-only, shuffled-binding, no-abstention, and always-abstain controls retain their declared interpretations; item the automated tests, independent environment replay, independent chart reconstruction, independent NumPy forward pass, receipts, and package manifest verify. Then the edge K_18 xrightarrowΠ_19 K_19 is a finite, conservative, source-side Neuralese semantic-atlas expansion. It resolves two observation contexts abs…
Exact mathematical passage
K_{18} \xrightarrow{\Pi_{19}} K_{19}
K_1^{\rm recal}Use in source · 6 passages
Search aliases: K_1^rm recal rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10487
The descendant compiler receives only new development receipts. It diagnoses full-rank saturation, activates a finite conditioning-aware query grammar, selects a grandchild K_2, and seals the resulting policy before the fresh audit tasks and representation families are executed. The primary comparison is M_audit(K_2)-M_audit(K_1^rm recal), where K_1^rm recal preserves the Block 7 rank-revealing query policy but is allowed to recalibrate its witness gate on Block 9 development evidence. This is a…
Exact mathematical passage
\Mcal_{\mathrm{audit}}(K_2)-\Mcal_{\mathrm{audit}}(K_1^{\rm recal}),
k_2Use in source · 2 passages
Search aliases: k_2
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2474
[Locally repairable, globally incompatible] Let the nerve be a three-cycle with no triple intersection. Suppose the kernel fiber is ℝ and the oriented differences are d_12=1, d_23=1, d_31=1. If the cocycle were a coboundary, there would be k_1,k_2,k_3 with d_12=k_2-k_1, d_23=k_3-k_2, d_31=k_1-k_3. Summing gives 3=0, a contradiction. Hence no global correction exists. The graph-Hodge obstruction has norm sqrt3.
Exact mathematical passage
k_1,k_2,k_3
K_3Use in source · 10 passages
Search aliases: K_3
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10425
t learned or trained experiment should be staged so that the bottleneck genuinely changes: item K_0 uses random anchors and a conservative fixed selector; item K_1 uses the Block 7 rank-revealing query policy and learned risk-sensitive selector; item K_2 faces full-rank but noisy and ill-conditioned bridges and must discover a conditioning-aware design criterion; item K_3 faces systematic failure of every global linear bridge and must produce a grammar-inadequacy certificate followed by local or…
Exact mathematical passage
K_3
k_3Use in source · 2 passages
Search aliases: k_3
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2474
[Locally repairable, globally incompatible] Let the nerve be a three-cycle with no triple intersection. Suppose the kernel fiber is ℝ and the oriented differences are d_12=1, d_23=1, d_31=1. If the cocycle were a coboundary, there would be k_1,k_2,k_3 with d_12=k_2-k_1, d_23=k_3-k_2, d_31=k_1-k_3. Summing gives 3=0, a contradiction. Hence no global correction exists. The graph-Hodge obstruction has norm sqrt3.
Exact mathematical passage
k_1,k_2,k_3
K_aUse in source · 5 passages
Search aliases: K_a
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6657
Let U_a∈ℝ^d_a× r have orthonormal columns spanning the discovered packet. Define desired packet coordinates z_i = U_a^top(h_a^∘(x_i)-h_a(x_i)). Fit an affine predictor widehat z(h)=K_a h+c_a by ridge regression on source anchor states. The repair is R_a(h) = h+U_awidehat z(h) = h+D_a h+b_a, where D_a=U_aK_a, b_a=U_ac_a.
Exact mathematical passage
\widehat z(h)=K_a h+c_a
K_{\mathrm{affine\ oracle}}Use in source · 1 passage
Search aliases: K_affine oracle mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11685
The main paired estimands are Δ_G→ C = E [ G(K_3,C^96,A)-G(K_2,G^96,A) ] and Δ_oracle→ C = E [ G(K_3,C^96,A)-G(K_affine oracle,A) ]. The second quantity is the sharper structural test.
Exact mathematical passage
\Delta_{\mathrm{oracle}\to C} = \mathbb E \left[ G(K_{3,C}^{96},A)-G(K_{\mathrm{affine\ oracle}},A) \right].
K_CUse in source · 3 passages
Search aliases: K_C
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11624
Let K_G be the global parent and K_C the chart process. Let A_C(x) be the frozen chart-certificate decision for target episode x. The descendant is K_3(x)= K_C(x), · A_C(x)=1,
Exact mathematical passage
K_C
K_{\mathrm{chart}}Use in source · 1 passage
Search aliases: K_chart mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21860
Let the old bridge be T∈ℝ^q× d and let g be any admissible separator. The embedding T⟼(g,T,T) returns Th on both charts. Hence the local grammar can execute the inherited process without approximation. A conservative runtime may therefore use K_chart(h) = K_local(h), · if the chart certificate passes, K_global(h), · otherwise. The child is structurally richer, but it does not require deletion of the parent behavior. This is the operational meaning of [source reference].
Exact mathematical passage
K_{\mathrm{chart}}(h) = \begin{cases} K_{\mathrm{local}}(h),&\text{if the chart certificate passes},\\ K_{\mathrm{global}}(h),&\text{otherwise}. \end{cases}
K_GUse in source · 4 passages
Search aliases: K_G
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11624
Let K_G be the global parent and K_C the chart process. Let A_C(x) be the frozen chart-certificate decision for target episode x. The descendant is K_3(x)= K_C(x), · A_C(x)=1,
Exact mathematical passage
K_G
K_{\mathrm{global}}Use in source · 1 passage
Search aliases: K_global mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21860
Let the old bridge be T∈ℝ^q× d and let g be any admissible separator. The embedding T⟼(g,T,T) returns Th on both charts. Hence the local grammar can execute the inherited process without approximation. A conservative runtime may therefore use K_chart(h) = K_local(h), · if the chart certificate passes, K_global(h), · otherwise. The child is structurally richer, but it does not require deletion of the parent behavior. This is the operational meaning of [source reference].
Exact mathematical passage
K_{\mathrm{chart}}(h) = \begin{cases} K_{\mathrm{local}}(h),&\text{if the chart certificate passes},\\ K_{\mathrm{global}}(h),&\text{otherwise}. \end{cases}
K_{h'}^pUse in source · 1 passage
Search aliases: K_h'^p
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1108
[Observational indistinguishability] If two hypotheses h,h'∈H satisfy K_h^p=K_h'^p for every permitted probe p∈P_probe, then no adaptive decision rule based only on permitted probe transcripts can distinguish h from h' with different transcript distributions.
Exact mathematical passage
K_h^p=K_{h'}^p \qquad \text{for every permitted probe }p\in\Pcal_{\mathrm{probe}},
K_iUse in source · 1 passage
Search aliases: K_i
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9033
Let K be the declared finite or compact class of admissible repair processes under the protected context. For R⊆ N, let K_R(K) = { widetilde K∈K: widetilde K_i=K_i for every i∉ R }. The restricted oracle envelope is F_K(R) = sup_widetilde K∈K_R(K) M_Ξ,D,B(widetilde K). Thus F_K(∅)=M(K), F_K(N)=sup_widetilde K∈KM(widetilde K). The total declared headroom is H(K) = F_K(N)-F_K(∅). The adjective declared matters. A larger, previously unavailable meta-grammar may expose additional headroom. Equation …
Exact mathematical passage
\Kcal_R(K) = \left\{ \widetilde K\in\Kcal: \widetilde K_i=K_i \text{ for every }i\notin R \right\}. \label{eq:b8-restricted-class}
K_{\mathrm{local}}Use in source · 1 passage
Search aliases: K_local mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21860
Let the old bridge be T∈ℝ^q× d and let g be any admissible separator. The embedding T⟼(g,T,T) returns Th on both charts. Hence the local grammar can execute the inherited process without approximation. A conservative runtime may therefore use K_chart(h) = K_local(h), · if the chart certificate passes, K_global(h), · otherwise. The child is structurally richer, but it does not require deletion of the parent behavior. This is the operational meaning of [source reference].
Exact mathematical passage
K_{\mathrm{chart}}(h) = \begin{cases} K_{\mathrm{local}}(h),&\text{if the chart certificate passes},\\ K_{\mathrm{global}}(h),&\text{otherwise}. \end{cases}
K_{\max}Use in source · 1 passage
Search aliases: K_max max
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15789
For fixed maximum depth K_max, let λ_t(x)=Λ(z_t(x)). Define cumulative halt mass M_t(x) = ∑_j=1^t λ_j(x) ∏_q<j(1-λ_q(x)), and dynamic depth T(x)=min{t:M_t(x)≥τ}, with declared threshold τ. Block 16 does not infer adaptive computation from the existence of a halt head. It requires a difficulty variable d(x) fixed before result access and tests both ρ_Spearman(d(x),T(x))>0 and nondegenerate prompt variation in the halt schedules. A constant depth can coexist with fully task-specific Neuralese; con…
Exact mathematical passage
K_{\max}
K_rUse in source · 1 passage
Search aliases: K_r
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7487
The child compiler searches a finite grammar Γ = Λ_u×Λ_r× K_u× K_r× T_r, where the factors specify utility regularization, risk regularization, utility residual penalty, risk residual penalty, and risk threshold. In the released fixture, |Γ| =5·5·4·5·4 =2000. For γ∈Γ, let overline U_rm meta(γ) be its mean protected utility on the meta-validation episodes and widehatrm se(γ) its standard error. A configuration is validation-safe when its protected-regression tail rate and maximum regression satis…
Exact mathematical passage
\Gamma = \Lambda_u\times\Lambda_r\times K_u\times K_r\times T_r,
\mathcal K_RUse in source · 2 passages
Search aliases: K_R mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 14520
The inherited obstruction language spans a response cone K_R = {Rz:z∈ℝ_≥0^k}. For observed response y, define the open-set residual η_rm open(y) = dist(y, K_R).
Exact mathematical passage
\mathcal K_R = \{Rz:z\in\R_{\geq0}^k\}.
k_{\rm fit}Use in source · 1 passage
Search aliases: k_rm fit rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8579
The compiler selected: π_rm anc · =π_rm PQR, k_rm fit · =16, k_rm val · =8, γ_rm cal · =0.08, ρ_rm cal · =0.035, τ_rm align · =0.05, full-rank requirement · =true. The selected development score is 0.263289. The underlying mean selected development gain is 0.296478 with zero harmful targets in the development ledger.
Exact mathematical passage
\begin{aligned} \pi_{\rm anc}&=\pi_{\rm PQR},\\ k_{\rm fit}&=16,\\ k_{\rm val}&=8,\\ \gamma_{\rm cal}&=0.08,\\ \rho_{\rm cal}&=0.035,\\ \tau_{\rm align}&=0.05,\\ \text{full-rank requirement}&=\text{true}. \end{aligned} \label{eq:block07q-descendant-config}
k_{\rm val}Use in source · 1 passage
Search aliases: k_rm val rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8579
The compiler selected: π_rm anc · =π_rm PQR, k_rm fit · =16, k_rm val · =8, γ_rm cal · =0.08, ρ_rm cal · =0.035, τ_rm align · =0.05, full-rank requirement · =true. The selected development score is 0.263289. The underlying mean selected development gain is 0.296478 with zero harmful targets in the development ledger.
Exact mathematical passage
\begin{aligned} \pi_{\rm anc}&=\pi_{\rm PQR},\\ k_{\rm fit}&=16,\\ k_{\rm val}&=8,\\ \gamma_{\rm cal}&=0.08,\\ \rho_{\rm cal}&=0.035,\\ \tau_{\rm align}&=0.05,\\ \text{full-rank requirement}&=\text{true}. \end{aligned} \label{eq:block07q-descendant-config}
K_uUse in source · 3 passages
Search aliases: K_u
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7487
The child compiler searches a finite grammar Γ = Λ_u×Λ_r× K_u× K_r× T_r, where the factors specify utility regularization, risk regularization, utility residual penalty, risk residual penalty, and risk threshold. In the released fixture, |Γ| =5·5·4·5·4 =2000. For γ∈Γ, let overline U_rm meta(γ) be its mean protected utility on the meta-validation episodes and widehatrm se(γ) its standard error. A configuration is validation-safe when its protected-regression tail rate and maximum regression satis…
Exact mathematical passage
\Gamma = \Lambda_u\times\Lambda_r\times K_u\times K_r\times T_r,
k_W^*Use in source · 1 passage
Search aliases: k_W^*
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2446
[Weighted gluing normal equation] Every weighted least-squares gauge satisfies BW_E B^*k =- BW_Ee. The mean-zero minimum-norm minimizer is k_W^* =-( BW_E B^*)^† BW_Ee. The irreducible weighted mismatch is the W_E-orthogonal projection of e onto the W_E-orthogonal complement of im B^*.
Exact mathematical passage
k_W^* =-\bigl(\mathbf BW_E\mathbf B^*\bigr)^\dagger\mathbf BW_Ee.
k_\alphaUse in source · 9 passages
Search aliases: k_α alpha
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2303
If a global repair u exists, set k_α=u|_U_α-u_α. Then Dk_α=0 and d_αβ=k_β-k_α, so the cocycle is a coboundary. Conversely, if d_αβ=k_β-k_α for kernel sections k_α, then u'_α=u_α+k_α agree on overlaps. The sheaf gluing axiom yields a global u, and Du=-r because the adjustments lie in the kernel.
Exact mathematical passage
k_\alpha=u|_{U_\alpha}-u_\alpha
k_\betaUse in source · 3 passages
Search aliases: k_β beta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2304
If a global repair u exists, set k_α=u|_U_α-u_α. Then Dk_α=0 and d_αβ=k_β-k_α, so the cocycle is a coboundary. Conversely, if d_αβ=k_β-k_α for kernel sections k_α, then u'_α=u_α+k_α agree on overlaps. The sheaf gluing axiom yields a global u, and Du=-r because the adjustments lie in the kernel.
Exact mathematical passage
d_{\alpha\beta}=k_\beta-k_\alpha,
K_{\Gamma}^{\star}Use in source · 2 passages
Search aliases: K_Γ^⋆ Gamma star
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13712
Let Valid(c_i,Π_i^+)∈{0,1} encode capability, integrity, provenance, and resource validity. The selected descendant is K_Γ^⋆(τ) ∈ arg max_i:Valid(c_i,Π_i^+)=1 S_V(c_i), with the no-op baseline retained when no valid candidate exists.
Exact mathematical passage
K_{\Gamma}^{\star}(\tau) \in \argmax_{i:\operatorname{Valid}(c_i,\Pi_i^+)=1} S_{\Vcal}(c_i), \label{eq:b13-selected-method}
K_{\Gamma_0}^{\star}Use in source · 1 passage
Search aliases: K_Γ_0^⋆ Gamma star
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13936
Let Γ_0 and Γ_1 operate on the same problem τ with identical initial archive, candidate grammar, experiment budget, evaluator portfolio, integrity monitor, and hidden audit isolation. Let D_i = U_audit (K_Γ_1^⋆(τ_i)) - U_audit (K_Γ_0^⋆(τ_i)).
Exact mathematical passage
D_i = U_{\mathrm{audit}} \left(K_{\Gamma_1}^{\star}(\tau_i)\right) - U_{\mathrm{audit}} \left(K_{\Gamma_0}^{\star}(\tau_i)\right).
K_{\Gamma_1}^{\star}Use in source · 1 passage
Search aliases: K_Γ_1^⋆ Gamma star
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13936
Let Γ_0 and Γ_1 operate on the same problem τ with identical initial archive, candidate grammar, experiment budget, evaluator portfolio, integrity monitor, and hidden audit isolation. Let D_i = U_audit (K_Γ_1^⋆(τ_i)) - U_audit (K_Γ_0^⋆(τ_i)).
Exact mathematical passage
D_i = U_{\mathrm{audit}} \left(K_{\Gamma_1}^{\star}(\tau_i)\right) - U_{\mathrm{audit}} \left(K_{\Gamma_0}^{\star}(\tau_i)\right).
K_\tauUse in source · 3 passages
Search aliases: K_τ tau
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13678
Let T be a research-problem space. A problem τ∈T contains a baseline repair process K_τ, a declared candidate grammar P_τ, a cheap evidence channel, a scored evaluator portfolio, a hidden protected audit, integrity rules, and a resource budget B.
Exact mathematical passage
K_\tau
\KcalUse in source · 6 passages
Search aliases: Kcal K mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9032
Let K be the declared finite or compact class of admissible repair processes under the protected context. For R⊆ N, let K_R(K) = { widetilde K∈K: widetilde K_i=K_i for every i∉ R }. The restricted oracle envelope is F_K(R) = sup_widetilde K∈K_R(K) M_Ξ,D,B(widetilde K). Thus F_K(∅)=M(K), F_K(N)=sup_widetilde K∈KM(widetilde K). The total declared headroom is H(K) = F_K(N)-F_K(varnothi
Exact mathematical passage
\Kcal
\Kcal_RUse in source · 4 passages
Search aliases: Kcal_R K_R Kcal mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9033
Let K be the declared finite or compact class of admissible repair processes under the protected context. For R⊆ N, let K_R(K) = { widetilde K∈K: widetilde K_i=K_i for every i∉ R }. The restricted oracle envelope is F_K(R) = sup_widetilde K∈K_R(K) M_Ξ,D,B(widetilde K). Thus F_K(∅)=M(K), F_K(N)=sup_widetilde K∈KM(widetilde K). The total declared headroom is H(K) = F_K(N)-F_K(∅). The adjective declared matters. A larger, previously unavailable meta-grammar may expose additional headroom. Equation …
Exact mathematical passage
\Kcal_R(K) = \left\{ \widetilde K\in\Kcal: \widetilde K_i=K_i \text{ for every }i\notin R \right\}. \label{eq:b8-restricted-class}
\Kcal_{R'}Use in source · 1 passage
Search aliases: Kcal_R' K_R' Kcal mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9077
Every process admissible under R is admissible under R', so K_R(K)⊆K_R'(K). Taking suprema gives the result.
Exact mathematical passage
\Kcal_R(K)\subseteq\Kcal_{R'}(K)
\mathsf{Key}Use in source · 1 passage
Search aliases: Key mathsf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5751
An executable state is a finite canonical map widehat s:Key→Value. A witness policy is widehat W =(K_protected,γ_min,r_max,p_prov,e_irr,v_W), where K_protected is the immutable-key set, γ_min is the minimum protected gain, r_max is the maximum accepted risk, p_prov indicates whether provenance is mandatory, e_irr controls irreversible effects, and v_W is the witness version.
Exact mathematical passage
\widehat s:\mathsf{Key}\to\mathsf{Value}.
\mathfrak L_{16}^{\mathrm{NL}}Use in source · 2 passages
Search aliases: L_16^NL mathfrak mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16551
ain their declared interpretations; item the child is FULL and adaptive on both fresh final supports, while the paired effects in [source reference] have positive frozen intervals; item the independent NumPy, receipt, witness, and gauge checks pass within their declared numerical tolerances. Then the accepted transition L_16^NL xrightarrowΠ_17 L_17^NL is a finite, evidence-preserving, training-generated repair of the Neuralese binding component on the declared affine-program environment and gene…
Exact mathematical passage
\mathfrak L_{16}^{\mathrm{NL}} \xrightarrow{\Pi_{17}} \mathfrak L_{17}^{\mathrm{NL}} \label{eq:b17-lineage-edge}
\mathfrak L_{17}^{\mathrm{NL}}Use in source · 3 passages
Search aliases: L_17^NL mathfrak mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16551
ain their declared interpretations; item the child is FULL and adaptive on both fresh final supports, while the paired effects in [source reference] have positive frozen intervals; item the independent NumPy, receipt, witness, and gauge checks pass within their declared numerical tolerances. Then the accepted transition L_16^NL xrightarrowΠ_17 L_17^NL is a finite, evidence-preserving, training-generated repair of the Neuralese binding component on the declared affine-program environment and gene…
Exact mathematical passage
\mathfrak L_{16}^{\mathrm{NL}} \xrightarrow{\Pi_{17}} \mathfrak L_{17}^{\mathrm{NL}} \label{eq:b17-lineage-edge}
\mathfrak L_{18}^{\mathrm{NL}}Use in source · 2 passages
Search aliases: L_18^NL mathfrak mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20328
Block 18 attempts but does not accept L_17^NL xrightarrowΠ_18? L_18^NL. The question mark is substantive: the selected schema passes both primary architecture-foreign audits but fails both shifted-generator audits and both frozen semantic-transport gates. The lineage therefore appends an obstruction receipt rather than a descendant commit. The post-hoc local-atlas recovery is stored as diagnostic evidence for the next compiler transaction and is not counted as a completed edge.
Exact mathematical passage
\mathfrak L_{17}^{\mathrm{NL}} \xrightarrow{\Pi_{18}?} \mathfrak L_{18}^{\mathrm{NL}}.
\mathfrak L_{19}^{\mathrm{NL}}Use in source · 2 passages
Search aliases: L_19^NL mathfrak mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20336
Block 19 accepts the source-side continuation L_18^NL xrightarrowΠ_19 L_19^NL, where the added object is a six-chart semantic atlas with a set-structured latent pathway and selective support gate. The edge is conservative on inherited charts and improves fresh tasks in two compiled contexts. It does not yet transport that bundle across the foreign recurrent architectures from Block 18.
Exact mathematical passage
\mathfrak L_{18}^{\mathrm{NL}} \xrightarrow{\Pi_{19}} \mathfrak L_{19}^{\mathrm{NL}},
\mathfrak L_{20}^{\mathrm{NL}}Use in source · 2 passages
Search aliases: L_20^NL mathfrak mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20344
Block 20 accepts the finite transport continuation L_19^NL xrightarrowΠ_20 L_20^NL, where the added object is an atlas-indexed connection carrying complete Neuralese trajectories from GRU and gated-residual fibers into the frozen Elman source fibers. The edge preserves the six-chart atlas, passes fresh semantic and curvature audits, and rejects unsupported contexts. Its scope remains the declared finite architecture, chart, gauge, and affine transport grammars.
Exact mathematical passage
\mathfrak L_{19}^{\mathrm{NL}} \xrightarrow{\Pi_{20}} \mathfrak L_{20}^{\mathrm{NL}},
\mathfrak L_{21}^{\mathrm{NL}}Use in source · 1 passage
Search aliases: L_21^NL mathfrak mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20352
Block 21 accepts the finite grammar continuation L_20^NL xrightarrowΠ_21 L_21^NL, where the transport process gains a typed response grammar over affine drift, nonlinear warp, latent-clock mixing, chart aliasing, benign semantic quotient, and destructive semantic loss. The edge preserves every inherited valid repair and accepts only the minimum-description candidate that passes the protected witness.
Exact mathematical passage
\mathfrak L_{20}^{\mathrm{NL}} \xrightarrow{\Pi_{21}} \mathfrak L_{21}^{\mathrm{NL}},
\mathfrak L_6Use in source · 1 passage
Search aliases: L_6 mathfrak
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13666
The realized compiler edge is L_6 xrightarrowΠ^Γ_7 L_7, where the active repair grammar, candidate-method language, evaluator portfolio, integrity monitor, and experiment budget remain fixed, but the research-allocation compiler changes from Γ_0 to Γ_1. The edge is therefore compiler metaproductivity inside a finite declared research language, not unrestricted automated science.
Exact mathematical passage
\mathfrak L_6 \xrightarrow{\Pi^{\Gamma}_7} \mathfrak L_7, \label{eq:b13-lineage}
\mathfrak L_7Use in source · 1 passage
Search aliases: L_7 mathfrak
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13666
The realized compiler edge is L_6 xrightarrowΠ^Γ_7 L_7, where the active repair grammar, candidate-method language, evaluator portfolio, integrity monitor, and experiment budget remain fixed, but the research-allocation compiler changes from Γ_0 to Γ_1. The edge is therefore compiler metaproductivity inside a finite declared research language, not unrestricted automated science.
Exact mathematical passage
\mathfrak L_6 \xrightarrow{\Pi^{\Gamma}_7} \mathfrak L_7, \label{eq:b13-lineage}
L_G^\daggerUse in source · 1 passage
Search aliases: L_G^† dagger
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2374
ge gluing] Let L_G:=BB^* be the graph Laplacian and let P_cyc be the orthogonal projector from K^E onto ker B, the cycle space. Then inf_k∈ K^Inorme+ B^*k = normP_cyce. The mean-zero minimum-norm gauge is k^* =-(L_G^† B⊗ I_K)e. If λ_2(L_G)>0 is the spectral gap, then normk^* ≤ frac1sqrtλ_2(L_G)norm e. Exact graph-level gluing is possible if and only if P_cyce=0.
Exact mathematical passage
k^* =-(L_G^\dagger B\otimes I_K)e.
L_{\Gcal}Use in source · 2 passages
Search aliases: L_Gcal L_G Gcal mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5023
For grammar G with linearized authorized correction image L_G, define q_G(r) = dist(-r,L_G). This is the minimum unresolved first-order residual inside the current grammar.
Exact mathematical passage
L_{\Gcal}
L_iUse in source · 2 passages
Search aliases: L_i
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16788
For model M, task i, and clean protected loss L_i, the audit reuses the Block 17 donor distinctions and adds cross-implementation status labels. It records latent necessity, reversal cost, norm-matched noise cost, exact replay, same-answer far-program transplant cost, far-answer matched-depth transplant cost, local semantic-neighbor cost, external trace-probe R^2, dynamic-depth correlation, and exact program execution. Let n_i · =L_i^0-L_i, · o_i · =L_i^rev-L_i, · c_i · =L_i^η-L_i,
Exact mathematical passage
L_i
L_{i,\mathrm{answer}}Use in source · 1 passage
Search aliases: L_i,answer mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16789
s L_i, the audit reuses the Block 17 donor distinctions and adds cross-implementation status labels. It records latent necessity, reversal cost, norm-matched noise cost, exact replay, same-answer far-program transplant cost, far-answer matched-depth transplant cost, local semantic-neighbor cost, external trace-probe R^2, dynamic-depth correlation, and exact program execution. Let n_i · =L_i^0-L_i, · o_i · =L_i^rev-L_i, · c_i · =L_i^η-L_i, p_i · =L_i,sem^same answer,far program -L_i,sem, · a_i · …
Exact mathematical passage
n_i&=L_i^{0}-L_i, &o_i&=L_i^{\mathrm{rev}}-L_i, &c_i&=L_i^{\eta}-L_i, \\ p_i&=L_{i,\mathrm{sem}}^{\mathrm{same\ answer,far\ program}} -L_{i,\mathrm{sem}}, &a_i&=L_{i,\mathrm{answer}}^{\mathrm{far\ answer,depth}} -L_{i,\mathrm{answer}}. \label{eq:b18-audit-coordinates}
L_{i,\mathrm{answer}}^{\mathrm{far\ answer,depth}}Use in source · 1 passage
Search aliases: L_i,answer^far answer,depth mathrm mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16789
s L_i, the audit reuses the Block 17 donor distinctions and adds cross-implementation status labels. It records latent necessity, reversal cost, norm-matched noise cost, exact replay, same-answer far-program transplant cost, far-answer matched-depth transplant cost, local semantic-neighbor cost, external trace-probe R^2, dynamic-depth correlation, and exact program execution. Let n_i · =L_i^0-L_i, · o_i · =L_i^rev-L_i, · c_i · =L_i^η-L_i, p_i · =L_i,sem^same answer,far program -L_i,sem, · a_i · …
Exact mathematical passage
n_i&=L_i^{0}-L_i, &o_i&=L_i^{\mathrm{rev}}-L_i, &c_i&=L_i^{\eta}-L_i, \\ p_i&=L_{i,\mathrm{sem}}^{\mathrm{same\ answer,far\ program}} -L_{i,\mathrm{sem}}, &a_i&=L_{i,\mathrm{answer}}^{\mathrm{far\ answer,depth}} -L_{i,\mathrm{answer}}. \label{eq:b18-audit-coordinates}
L_{i,\mathrm{sem}}Use in source · 1 passage
Search aliases: L_i,sem mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16789
s L_i, the audit reuses the Block 17 donor distinctions and adds cross-implementation status labels. It records latent necessity, reversal cost, norm-matched noise cost, exact replay, same-answer far-program transplant cost, far-answer matched-depth transplant cost, local semantic-neighbor cost, external trace-probe R^2, dynamic-depth correlation, and exact program execution. Let n_i · =L_i^0-L_i, · o_i · =L_i^rev-L_i, · c_i · =L_i^η-L_i, p_i · =L_i,sem^same answer,far program -L_i,sem, · a_i · …
Exact mathematical passage
n_i&=L_i^{0}-L_i, &o_i&=L_i^{\mathrm{rev}}-L_i, &c_i&=L_i^{\eta}-L_i, \\ p_i&=L_{i,\mathrm{sem}}^{\mathrm{same\ answer,far\ program}} -L_{i,\mathrm{sem}}, &a_i&=L_{i,\mathrm{answer}}^{\mathrm{far\ answer,depth}} -L_{i,\mathrm{answer}}. \label{eq:b18-audit-coordinates}
L_{i,\mathrm{sem}}^{\mathrm{same\ answer,far\ program}}Use in source · 1 passage
Search aliases: L_i,sem^same answer,far program mathrm mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16789
s L_i, the audit reuses the Block 17 donor distinctions and adds cross-implementation status labels. It records latent necessity, reversal cost, norm-matched noise cost, exact replay, same-answer far-program transplant cost, far-answer matched-depth transplant cost, local semantic-neighbor cost, external trace-probe R^2, dynamic-depth correlation, and exact program execution. Let n_i · =L_i^0-L_i, · o_i · =L_i^rev-L_i, · c_i · =L_i^η-L_i, p_i · =L_i,sem^same answer,far program -L_i,sem, · a_i · …
Exact mathematical passage
n_i&=L_i^{0}-L_i, &o_i&=L_i^{\mathrm{rev}}-L_i, &c_i&=L_i^{\eta}-L_i, \\ p_i&=L_{i,\mathrm{sem}}^{\mathrm{same\ answer,far\ program}} -L_{i,\mathrm{sem}}, &a_i&=L_{i,\mathrm{answer}}^{\mathrm{far\ answer,depth}} -L_{i,\mathrm{answer}}. \label{eq:b18-audit-coordinates}
L_i^{0}Use in source · 1 passage
Search aliases: L_i^0
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16789
s L_i, the audit reuses the Block 17 donor distinctions and adds cross-implementation status labels. It records latent necessity, reversal cost, norm-matched noise cost, exact replay, same-answer far-program transplant cost, far-answer matched-depth transplant cost, local semantic-neighbor cost, external trace-probe R^2, dynamic-depth correlation, and exact program execution. Let n_i · =L_i^0-L_i, · o_i · =L_i^rev-L_i, · c_i · =L_i^η-L_i, p_i · =L_i,sem^same answer,far program -L_i,sem, · a_i · …
Exact mathematical passage
n_i&=L_i^{0}-L_i, &o_i&=L_i^{\mathrm{rev}}-L_i, &c_i&=L_i^{\eta}-L_i, \\ p_i&=L_{i,\mathrm{sem}}^{\mathrm{same\ answer,far\ program}} -L_{i,\mathrm{sem}}, &a_i&=L_{i,\mathrm{answer}}^{\mathrm{far\ answer,depth}} -L_{i,\mathrm{answer}}. \label{eq:b18-audit-coordinates}
L_i^{\mathrm{rev}}Use in source · 1 passage
Search aliases: L_i^rev mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16789
s L_i, the audit reuses the Block 17 donor distinctions and adds cross-implementation status labels. It records latent necessity, reversal cost, norm-matched noise cost, exact replay, same-answer far-program transplant cost, far-answer matched-depth transplant cost, local semantic-neighbor cost, external trace-probe R^2, dynamic-depth correlation, and exact program execution. Let n_i · =L_i^0-L_i, · o_i · =L_i^rev-L_i, · c_i · =L_i^η-L_i, p_i · =L_i,sem^same answer,far program -L_i,sem, · a_i · …
Exact mathematical passage
n_i&=L_i^{0}-L_i, &o_i&=L_i^{\mathrm{rev}}-L_i, &c_i&=L_i^{\eta}-L_i, \\ p_i&=L_{i,\mathrm{sem}}^{\mathrm{same\ answer,far\ program}} -L_{i,\mathrm{sem}}, &a_i&=L_{i,\mathrm{answer}}^{\mathrm{far\ answer,depth}} -L_{i,\mathrm{answer}}. \label{eq:b18-audit-coordinates}
L_i^{\eta}Use in source · 1 passage
Search aliases: L_i^η eta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16789
s L_i, the audit reuses the Block 17 donor distinctions and adds cross-implementation status labels. It records latent necessity, reversal cost, norm-matched noise cost, exact replay, same-answer far-program transplant cost, far-answer matched-depth transplant cost, local semantic-neighbor cost, external trace-probe R^2, dynamic-depth correlation, and exact program execution. Let n_i · =L_i^0-L_i, · o_i · =L_i^rev-L_i, · c_i · =L_i^η-L_i, p_i · =L_i,sem^same answer,far program -L_i,sem, · a_i · …
Exact mathematical passage
n_i&=L_i^{0}-L_i, &o_i&=L_i^{\mathrm{rev}}-L_i, &c_i&=L_i^{\eta}-L_i, \\ p_i&=L_{i,\mathrm{sem}}^{\mathrm{same\ answer,far\ program}} -L_{i,\mathrm{sem}}, &a_i&=L_{i,\mathrm{answer}}^{\mathrm{far\ answer,depth}} -L_{i,\mathrm{answer}}. \label{eq:b18-audit-coordinates}
L_JUse in source · 10 passages
Search aliases: L_J
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3651
[Readout stability under semantic holonomy] Let J:E_v→ Y be L_J-Lipschitz. Then d_Y(J(H_γ x),J(x)) ≤ L_Jη_sem(x;γ). If J returns a score vector and the winning score at x has margin m(x) over the runner-up, then the decision is unchanged whenever 2L_Jη_sem(x;γ)<m(x), where L_J bounds each score coordinate in the same norm.
Exact mathematical passage
L_J
L_{\Meta}Use in source · 2 passages
Search aliases: L_Meta Meta mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8351
Let Ω = Ω_rm devsqcupΩ_rm audit be a disjoint target index set. The compiler receives rows indexed only by Ω_rm dev. It emits a descendant configuration c_1, a compiler ledger L_Meta, and a pre-audit seal Σ_rm pre = H(c_1,L_Meta,source hashes), where H is SHA-256 over canonical serialization. Audit execution refuses to proceed when the in-memory descendant differs from the sealed configuration.
Exact mathematical passage
L_{\Meta}
L_NUse in source · 2 passages
Search aliases: L_N
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2010
[Gauge-fixed local nonlinear repair by contraction] Suppose D:U→ V has a bounded right inverse Q:V→ U, so DQ=I_V, and set W:=im Q. Then restr D W:W→ V is an isomorphism with inverse Q. Let B_R^W:=setw∈ W:norm w≤ R. Suppose normN(u)-N(v)≤ L_Nnormu-v (u,v∈ B_R^W), with q:=norm QL_N<1, and normQr+norm Qsup_u∈ B_R^WnormN(u)≤ R. Then there exists a unique u^*∈ B_R^W satisfying F(u^*)=0. The gauge-fixed iteration u_k+1=-Qr-QN(u_k) converges to u^* from every u_0∈ B_R^W, with normu_k-u^*≤fracq^k1-qnorm…
Exact mathematical passage
\norm{N(u)-N(v)}\leq L_N\norm{u-v} \qquad(u,v\in B_R^W),
L_SUse in source · 3 passages
Search aliases: L_S
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 746
[Composition bound for bridge defects] Let T:X→ Y and S:Y→ Z. Suppose the property transports satisfy normQ_Y(Tx)-widetilde T_QQ_X(x)≤ε_T for x∈ K, and normQ_Z(Sy)-widetilde S_QQ_Y(y)≤ε_S for y∈ T(K). If normwidetilde S_Q≤ L_S, then the composite bridge obeys ε_Q(S∘ T;K) ≤ ε_S+L_Sε_T with intended property transport widetilde S_Qwidetilde T_Q.
Exact mathematical passage
\norm{\widetilde S_Q}\leq L_S
L_TUse in source · 2 passages
Search aliases: L_T
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3426
[Composition of evaluator defects] Let e:v→ w and f:w→ z be evaluator morphisms. Suppose T_f is L_T-Lipschitz on represented protected evidence. Then δ_obs(f∘ e) ≤ L_Tδ_obs(e)+δ_obs(f), where the second term is evaluated on S_e( h_v). If, in addition, sup_r∈ h_v norm J_zT_fT_eι_v(r)-J_wT_eι_v(r) ≤ δ_J(fmid e), then δ_J(f∘ e) ≤ δ_J(e)+δ_J(fmid e).
Exact mathematical passage
L_T
\mathfrak L_{t+1}^{\mathrm{NL}}Use in source · 1 passage
Search aliases: L_t+1^NL mathfrak mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15868
child-minus-parent transplant-cost interval has positive lower endpoint; item child clean loss does not regress beyond a predeclared tolerance; item every claimed packet or transport passes its own held-out causal certificate; item model, task, audit, and result digests are committed in an append-only receipt chain before promotion. Then the accepted transition L_t^NL xrightarrowΠ_16 L_t+1^NL is a finite evidence-preserving improvement of the Neuralese reasoning component. If the child-producing…
Exact mathematical passage
\mathfrak L_t^{\mathrm{NL}} \xrightarrow{\Pi_{16}} \mathfrak L_{t+1}^{\mathrm{NL}}
\mathfrak L_t^{\mathrm{NL}}Use in source · 3 passages
Search aliases: L_t^NL mathfrak mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15834
Extend the SRIC lineage state by L_t^NL = ( L_t,N_t,I_t^NL,T_t^NL,W_t^NL), where T_t^NL contains trace-transport certificates and W_t^NL is the frozen external answer and invariant witness. A proposed descendant may modify the latent transition, update gate, halting head, packet atlas, or transport map. The backbone may be held fixed for a component-isolation edge or changed under a stronger whole-model edge.
Exact mathematical passage
\mathfrak L_t^{\mathrm{NL}} = (\mathfrak L_t,\Ncal_t,\Ical_t^{\mathrm{NL}},\Tau_t^{\mathrm{NL}},\Wcal_t^{\mathrm{NL}}),
L_t^{\perp}Use in source · 1 passage
Search aliases: L_t^⊥ perp
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4786
[Structural-to-behavioral identity] Assume L_t⊆ L_t+1 and let W=L_t+1∩ L_t^⊥. Then E_t+1^cor(μ)- E_t^cor(μ) = E_r∼μnormP_Wr^2 and E_t^ob(μ)- E_t+1^ob(μ) = E_r∼μnormP_Wr^2. Under quadratic residual loss and exact cancellation of the correctable component, this quantity is exactly the expected additional task-loss reduction made available by the expanded repair image.
Exact mathematical passage
W=L_{t+1}\cap L_t^{\perp}.
L_vUse in source · 2 passages
Search aliases: L_v
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3826
[Paired evaluator-risk improvement] Let Z_1,…,Z_n be independent and identically distributed held-out externally revealed cases. Let L_v(Z_i),L_w(Z_i)∈[0,1] be the losses of the old and proposed evaluators, and define Δ_i=L_v(Z_i)-L_w(Z_i)∈[-1,1]. If the upgrade and evaluation rule are fixed before the held-out outcomes are revealed, then with probability at least 1-α, EΔ_i ≥ overlineΔ_n - sqrtfrac2log(1/α)n. Thus a positive lower bound certifies lower expected held-out risk for the proposed eva…
Exact mathematical passage
L_v(Z_i),L_w(Z_i)\in[0,1]
L_wUse in source · 2 passages
Search aliases: L_w
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3826
[Paired evaluator-risk improvement] Let Z_1,…,Z_n be independent and identically distributed held-out externally revealed cases. Let L_v(Z_i),L_w(Z_i)∈[0,1] be the losses of the old and proposed evaluators, and define Δ_i=L_v(Z_i)-L_w(Z_i)∈[-1,1]. If the upgrade and evaluation rule are fixed before the held-out outcomes are revealed, then with probability at least 1-α, EΔ_i ≥ overlineΔ_n - sqrtfrac2log(1/α)n. Thus a positive lower bound certifies lower expected held-out risk for the proposed eva…
Exact mathematical passage
L_v(Z_i),L_w(Z_i)\in[0,1]
\Lcal_{\mathrm{com}}Use in source · 6 passages
Search aliases: Lcal_com L_com Lcal mathrm mathcal mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10131
For X∈End(H), define commutator energy E_com(X) = ∑_j=1^m ( normXR_j-R_jX_F^2 + normXR_j^∗-R_j^∗ X_F^2 ). There is a positive semidefinite linear operator L_com on the Hilbert space of matrices such that anglesX,L_comX_F = E_com(X). Its kernel is the commutant: kerL_com= A'.
Exact mathematical passage
\Lcal_{\mathrm{com}}
\lceilUse in source · 3 passages
Search aliases: lceil
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4188
[Minimum rounds under a witness quorum] Suppose every transaction requires at least q frozen witness components, where 1≤ q<n. Then at most n-q components can be edited in one round. Editing every component at least once requires at least lceilfracnn-qrceil rounds. This bound is attained by partitioning the components into edit blocks of size at most n-q and taking the complement of each block as its witness set.
Exact mathematical passage
\left\lceil\frac{n}{n-q}\right\rceil
\lfloorUse in source · 2 passages
Search aliases: lfloor
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 610
[Fixed-margin evaluator ceiling] Let V: S→[0,1] be fixed. If every accepted transition satisfies V(A_t+1)≥ V(A_t)+Δ for a constant Δ>0, then every accepted chain has length at most lfloor 1/Δrfloor+1.
Exact mathematical passage
\lfloor 1/\Delta\rfloor+1
\limsupUse in source · 1 passage
Search aliases: limsup
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5304
For a time-varying sequence M_t, let z_t be a row vector of expected schema counts. Under the linear mean model, E z_T = z_0M_0M_1⋯ M_T-1. Define the upper mean-growth exponent λ_mean = limsup_T→∞ frac1T lognormM_0M_1⋯ M_T-1. A positive value indicates exponential growth of the linearized expected count. It does not imply almost-sure survival under dependence or resource competition.
Exact mathematical passage
\lambda_{\mathrm{mean}} = \limsup_{T\to\infty} \frac1T \log\norm{M_0M_1\cdots M_{T-1}}.
\limsup_{T\to\infty}Use in source · 1 passage
Search aliases: limsup_T→∞ limsup to infty
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5304
For a time-varying sequence M_t, let z_t be a row vector of expected schema counts. Under the linear mean model, E z_T = z_0M_0M_1⋯ M_T-1. Define the upper mean-growth exponent λ_mean = limsup_T→∞ frac1T lognormM_0M_1⋯ M_T-1. A positive value indicates exponential growth of the linearized expected count. It does not imply almost-sure survival under dependence or resource competition.
Exact mathematical passage
\lambda_{\mathrm{mean}} = \limsup_{T\to\infty} \frac1T \log\norm{M_0M_1\cdots M_{T-1}}.
\mathbf{Lin}Use in source · 1 passage
Search aliases: Lin mathbf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 886
[Category of validated lineage paths] Assume valid edge receipts are closed under proof-path concatenation and contain identity receipts for zero-change transitions. Then lineage states are objects and validated receipt paths are morphisms of a category Lin.
Exact mathematical passage
\mathbf{Lin}
\llUse in source · 2 passages
Search aliases: ll
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17747
The requirement κll1 is stronger than high pointwise probe performance on one support. It asks whether two independently constructed routes through the bundle yield compatible source representations.
Exact mathematical passage
\kappa\ll1
\mathrm{local}Use in source · 1 passage
Search aliases: local mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21860
Let the old bridge be T∈ℝ^q× d and let g be any admissible separator. The embedding T⟼(g,T,T) returns Th on both charts. Hence the local grammar can execute the inherited process without approximation. A conservative runtime may therefore use K_chart(h) = K_local(h), · if the chart certificate passes, K_global(h), · otherwise. The child is structurally richer, but it does not require deletion of the parent behavior. This is the operational meaning of [source reference].
Exact mathematical passage
K_{\mathrm{chart}}(h) = \begin{cases} K_{\mathrm{local}}(h),&\text{if the chart certificate passes},\\ K_{\mathrm{global}}(h),&\text{otherwise}. \end{cases}
\logUse in source · 20 passages
Search aliases: log
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3349
[Paired held-out gain certificate] Let Z_1,…,Z_n be independent and identically distributed paired target-context gains for a precommitted transported repair, with -G≤ Z_i≤ G. Let overline Z_n=n^-1∑_iZ_i. Then, with probability at least 1-α, E Z_i ≥ overline Z_n -Gsqrtfrac2log(1/α)n. For a precommitted family of m repairs, replacing α by α/m gives a simultaneous lower bound for all m candidates.
Exact mathematical passage
\mathbb E Z_i \geq \overline Z_n -G\sqrt{\frac{2\log(1/\alpha)}{n}}.
\LongrightarrowUse in source · 68 passages
Search aliases: Longrightarrow
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 22534
endfoot endlastfoot Protected evaluation context · Longrightarrow · Nonvacuous improvement relation
Exact mathematical passage
\Longrightarrow
\mathrm{low}Use in source · 1 passage
Search aliases: low mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15120
Let p∈P^+ be a candidate experiment with response y_p:T⟶ℝ. For threshold θ, define the refined diagnostic map ϕ_p,θ(τ) = (ϕ_0(τ),low), · y_p(τ)≤θ, (ϕ_0(τ),high), · y_p(τ)>θ. The old token is recovered by the projection π_0(z,s)=z, π_0∘ϕ_p,θ=ϕ_0. Thus the extension is conservative at the level of old diagnostic semantics.
Exact mathematical passage
\phi_{p,\theta}(\tau) = \begin{cases} (\phi_0(\tau),\mathrm{low}), & y_p(\tau)\leq\theta,\\ (\phi_0(\tau),\mathrm{high}), & y_p(\tau)>\theta. \end{cases} \label{eq:b15-refined-token}
\mathsf MUse in source · 1 passage
Search aliases: M mathsf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7728
The child must pass performance and risk gates simultaneously. Let Gate = Seal wedge Match wedge AuditGain wedge ForeignGain wedge Risk wedge Controls. The released suite evaluates the following frozen selector-gate families: item descendant receipt validity; item split-file load order and sealing; item positive paired child-parent audit interval; item positive paired child-parent architecture-foreign interval; item positive child-raw architecture-foreign interval; item three seed-and-task audit…
Exact mathematical passage
\mathsf Gate = \mathsf Seal \wedge \mathsf Match \wedge \mathsf AuditGain \wedge \mathsf ForeignGain \wedge \mathsf Risk \wedge \mathsf Controls.
M'Use in source · 4 passages
Search aliases: M'
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10744
Let Msucc0 be the current regularized information matrix. Adding candidate anchor x_i with quality weight w_i=σ_i^-2 gives M'=M+w_ix_ix_i^ T.
Exact mathematical passage
M'=M+w_ix_ix_i^{\mathsf T}.
m'Use in source · 2 passages
Search aliases: m'
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16172
Let d_prog(τ,τ') be a normalized distance between (A_m,B_m) and (A'_m',B'_m'), and let d_ans(τ,τ') = absy_m/4-y'_m'/4. For a recipient task τ_i, define four donor classes: item SAP: same-answer, far-program, approximately depth-matched donors; item FAD: far-answer, depth-matched donors; item SN: semantic-neighbor donors; item RMD: random depth-matched donors. Three donors per available class are selected by deterministic task-identifier hashing. Each audit records donor coverage and the realized…
Exact mathematical passage
(A'_{m'},B'_{m'})
M^{-1}Use in source · 5 passages
Search aliases: M^-1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10751
[D-optimal marginal gain] The increase in log information volume is logdet M'-logdet M = log(1+w_ix_i^ TM^-1x_i).
Exact mathematical passage
\log\det M'-\log\det M = \log\left(1+w_ix_i^{\mathsf T}M^{-1}x_i\right).
M^{-2}Use in source · 2 passages
Search aliases: M^-2
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10768
[A-optimal trace reduction] Let v=sqrtw_ix_i. Then tr(M^-1)-tr((M+vv^ T)^-1) = fracv^ TM^-2v1+v^ TM^-1v.
Exact mathematical passage
\tr(M^{-1})-\tr\left((M+vv^{\mathsf T})^{-1}\right) = \frac{v^{\mathsf T}M^{-2}v}{1+v^{\mathsf T}M^{-1}v}.
m_+Use in source · 5 passages
Search aliases: m_+
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15199
[Finite conflict-grounded language synthesis] Let a finite development archive contain one inherited token cell C with at least two incompatible repair subclasses. Suppose: item a candidate probe p∈P^+ separates C into cells C_- and C_+; item there exist m_- ,m_+∈C_rm pair such that m_- is acceptable throughout C_- and m_+ is acceptable throughout C_+; item neither one inherited singleton method nor one unrefined binding is acceptable throughout C; item old-token behavior is preserved through th…
Exact mathematical passage
m_- ,m_+\in\Ccal_{\rm pair}
m_-Use in source · 5 passages
Search aliases: m_-
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15199
[Finite conflict-grounded language synthesis] Let a finite development archive contain one inherited token cell C with at least two incompatible repair subclasses. Suppose: item a candidate probe p∈P^+ separates C into cells C_- and C_+; item there exist m_- ,m_+∈C_rm pair such that m_- is acceptable throughout C_- and m_+ is acceptable throughout C_+; item neither one inherited singleton method nor one unrefined binding is acceptable throughout C; item old-token behavior is preserved through th…
Exact mathematical passage
m_- ,m_+\in\Ccal_{\rm pair}
M_0Use in source · 2 passages
Search aliases: M_0
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5298
For a time-varying sequence M_t, let z_t be a row vector of expected schema counts. Under the linear mean model, E z_T = z_0M_0M_1⋯ M_T-1. Define the upper mean-growth exponent λ_mean = limsup_T→∞ frac1T lognormM_0M_1⋯ M_T-1. A positive value indicates exponential growth of the linearized expected count. It does not imply almost-sure survival under dependence or resource competition.
Exact mathematical passage
\mathbb E z_T = z_0M_0M_1\cdots M_{T-1}.
M_1Use in source · 4 passages
Search aliases: M_1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5231
[Localization of supercriticality] After permuting schema types, M has block upper-triangular form with diagonal blocks M_1,…,M_k given by strongly connected components. Therefore rad(M) = max_1≤ r≤ krad(M_r). In particular, R_imp>1 only if at least one strongly connected schema family is itself supercritical.
Exact mathematical passage
M_1,\ldots,M_k
m_1Use in source · 2 passages
Search aliases: m_1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 22153
One unknown token with incompatible methods. Let two problems a and b both map to bot. Suppose A^⋆(a)={m_1}, A^⋆(b)={m_2}, m_1≠ m_2. Any deterministic policy conditioned only on bot must choose one method and fail the other problem. This is a diagnostic-language obstruction, not ordinary estimation noise.
Exact mathematical passage
\Acal^\star(a)=\{m_1\}, \qquad \Acal^\star(b)=\{m_2\}, \qquad m_1\neq m_2.
M_2Use in source · 3 passages
Search aliases: M_2
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21700
0 · -1. On H=ℝ^4, define four response operators R_1=X⊕0, R_2=Z⊕0, R_3=0⊕ X, R_4=0⊕ Z. The adjoint-closed algebra they generate is M_2(ℝ)⊕ M_2(ℝ), with dimension 8. Its commutant consists of scalar multiples of the identity on each block and therefore has dimension 2. The minimal central projectors are P_1=I_2⊕0, P_2=0⊕ I_2. A residual e_1 generates the first two-dimensional block. A residual e_1+e_3 generates both blocks and has cyclic dimension 4.
Exact mathematical passage
M_2(\R)\oplus M_2(\R),
m_2Use in source · 2 passages
Search aliases: m_2
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 22153
One unknown token with incompatible methods. Let two problems a and b both map to bot. Suppose A^⋆(a)={m_1}, A^⋆(b)={m_2}, m_1≠ m_2. Any deterministic policy conditioned only on bot must choose one method and fail the other problem. This is a diagnostic-language obstruction, not ordinary estimation noise.
Exact mathematical passage
\Acal^\star(a)=\{m_1\}, \qquad \Acal^\star(b)=\{m_2\}, \qquad m_1\neq m_2.
M_{24}^{\rm for}Use in source · 1 passage
Search aliases: M_24^rm for rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8863
item parent and descendant receive the matched budget in [source reference]; item target-level audit differences are exchangeable or otherwise support the declared paired interval; item the protected evaluator and source repairs are unchanged across the parent-child edge; item accepted-harm rate is not increased. If the paired lower confidence bound for M_24^rm for( P_1)-M_24^rm for( P_0) is positive and bridge-identification coverage also increases, then the edge P_0→ P_1 is a bounded one-step …
Exact mathematical passage
M_{24}^{\rm for}(\mathfrak P_1)-M_{24}^{\rm for}(\mathfrak P_0)
M_aUse in source · 2 passages
Search aliases: M_a
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6143
The archive arm can perform well without a causal repair calculus because it searches several noisy candidates and retains the best. Candidate-budget matching is therefore mandatory. Let G_a,j,1,…,G_a,j,B be candidate gains for arm a on agent j. The primary behavioral quantity is M_a(B) = frac1n∑_j=1^nmax_1≤ b≤ BG_a,j,b. Comparing arms at unequal B confounds repair competence with order statistics.
Exact mathematical passage
M_a(B) = \frac1n\sum_{j=1}^n\max_{1\leq b\leq B}G_{a,j,b}.
M_cUse in source · 1 passage
Search aliases: M_c
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17827
tent-step-shared map acts independently on each recurrent state: widehat h_a,c^(k) = A_c h_b,c^(k)+b_c, k=1,…,5. A full-trajectory map instead acts on the concatenation vec(h_b,c^(1),…,h_b,c^(5)) ∈ R^200, allowing cross-step mixing: vec(widehat h_a,c) = M_cvec(h_b,c)+q_c. This difference is substantive when two recurrent architectures distribute the same computation across latent time differently.
Exact mathematical passage
\operatorname{vec}(\widehat h_{a,c}) = M_c\operatorname{vec}(h_{b,c})+q_c. \label{eq:b20-full-map}
M_{h,B}^{\Dcal}Use in source · 1 passage
Search aliases: M_h,B^Dcal M_h,B^D Dcal mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4749
[Metaproductivity shift bound] If absG_h,B(K,A)≤ G_max, then abs M_h,B^D(K) - M_h,B^D'(K) ≤ 2G_maxTV(D,D').
Exact mathematical passage
\abs{ M_{h,B}^{\Dcal}(K) - M_{h,B}^{\Dcal'}(K) } \leq 2G_{\max}\TV(\Dcal,\Dcal').
M_{h,B}^{\Dcal'}Use in source · 1 passage
Search aliases: M_h,B^Dcal' M_h,B^D' Dcal mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4749
[Metaproductivity shift bound] If absG_h,B(K,A)≤ G_max, then abs M_h,B^D(K) - M_h,B^D'(K) ≤ 2G_maxTV(D,D').
Exact mathematical passage
\abs{ M_{h,B}^{\Dcal}(K) - M_{h,B}^{\Dcal'}(K) } \leq 2G_{\max}\TV(\Dcal,\Dcal').
m_{\mathrm{hist}}Use in source · 4 passages
Search aliases: m_hist mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3774
[Historical ranking preservation] Let x_1,…,x_N be protected historical candidate records with scalar old scores J_v(x_i). Suppose every unequal old pair has margin at least m_hist := min_J_v(x_i)≠ J_v(x_j) absJ_v(x_i)-J_v(x_j)>0. If an upgrade satisfies absJ_w(T_ex_i)-J_v(x_i)≤ε for every i, with 2ε<m_hist, then every strict historical ranking is preserved.
Exact mathematical passage
m_{\mathrm{hist}} := \min_{J_v(x_i)\neq J_v(x_j)} \abs{J_v(x_i)-J_v(x_j)}>0.
M_{ij}Use in source · 4 passages
Search aliases: M_ij
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5197
Partition validated schemas into m types. For a type-i schema, let N_ij count type-j descendants whose receipts include a successful necessity ablation showing that removal of the parent schema eliminates the descendant discovery under matched evidence and budget. Define M_ij= E N_ij. This definition excludes descendants that merely share ancestry or evidence with the parent.
Exact mathematical passage
M_{ij}=\mathbb E N_{ij}.
M_jUse in source · 3 passages
Search aliases: M_j
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 14605
For integrity monitoring, let M_1,…,M_s∈{0,1} be monitor pass indicators and let the ensemble pass only when every component passes: M_∩=∏_j=1^s M_j.
Exact mathematical passage
M_{\cap}=\prod_{j=1}^s M_j.
M_kUse in source · 1 passage
Search aliases: M_k
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5231
[Localization of supercriticality] After permuting schema types, M has block upper-triangular form with diagonal blocks M_1,…,M_k given by strongly connected components. Therefore rad(M) = max_1≤ r≤ krad(M_r). In particular, R_imp>1 only if at least one strongly connected schema family is itself supercritical.
Exact mathematical passage
M_1,\ldots,M_k
M_{n_\alpha}Use in source · 2 passages
Search aliases: M_n_α alpha
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10062
[Finite causal atlas decomposition] There exists a unitary identification H ≅ ⊕_α=1^p ( C^m_α⊗ C^n_α ) under which A ≅ ⊕_α=1^p ( I_m_α⊗ M_n_α( C) ). The minimal central projections P_α∈ Z( A) give a canonical coarse causal atlas H = ⊕_α=1^pP_α H. The blocks are unique up to permutation and unitary changes of basis within multiplicity spaces.
Exact mathematical passage
\mathfrak A \cong \bigoplus_{\alpha=1}^{p} \left( I_{m_\alpha}\otimes M_{n_\alpha}(\mathbb C) \right). \label{eq:b8-algebra-decomposition}
M_rUse in source · 2 passages
Search aliases: M_r
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5232
[Localization of supercriticality] After permuting schema types, M has block upper-triangular form with diagonal blocks M_1,…,M_k given by strongly connected components. Therefore rad(M) = max_1≤ r≤ krad(M_r). In particular, R_imp>1 only if at least one strongly connected schema family is itself supercritical.
Exact mathematical passage
\rad(M) = \max_{1\leq r\leq k}\rad(M_r).
M_SUse in source · 9 passages
Search aliases: M_S
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10610
Stacking the selected anchors gives Y=XT^ T+E, where X∈ℝ^n× d and Y∈ℝ^n× q. Define W=diag(σ_1^-2,…,σ_n^-2) and the weighted information matrix M_S=X^ TWX.
Exact mathematical passage
M_S=X^{\mathsf T}WX.
M_sUse in source · 6 passages
Search aliases: M_s
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11466
Let the source repair be R_s(h)=M_sh+c_s. On chart j, let the target bridge be T_j(h)=A_jh+b_j, with A_j invertible on the repair-relevant subspace. Define R_t,j(y) = T_j (R_s(T_j^-1(y))). Expanding gives R_t,j(y) = A_jM_sA_j^-1(y-b_j)+A_jc_s+b_j.
Exact mathematical passage
R_s(h)=M_sh+c_s.
M_{S,\lambda}Use in source · 1 passage
Search aliases: M_S,λ lambda
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10665
The executable implementation uses M_S,λ=X^ TWX+λ I with a small fixed λ>0. The reported design diagnostics use source coordinates standardized against the complete candidate pool before applying the reliability weights. Standardization prevents one arbitrary source-coordinate scale from dominating the comparison. The actual bridge fit remains weighted in the original hidden coordinates.
Exact mathematical passage
M_{S,\lambda}=X^{\mathsf T}WX+\lambda I
M_S^{-1}Use in source · 8 passages
Search aliases: M_S^-1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10617
[Weighted bridge covariance] Assume M_S is invertible and the noise rows are independent with covariance σ_i^2I_q. The weighted least-squares estimator widehat T^ T = M_S^-1X^ TWY is unbiased and satisfies Cov(vec(widehat T^ T-T^ T)mid X) = I_q⊗ M_S^-1. Consequently, E[normwidehat T-T_F^2mid X] = q tr(M_S^-1).
Exact mathematical passage
\widehat T^{\mathsf T} = M_S^{-1}X^{\mathsf T}WY
M_tUse in source · 3 passages
Search aliases: M_t
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5297
For a time-varying sequence M_t, let z_t be a row vector of expected schema counts. Under the linear mean model, E z_T = z_0M_0M_1⋯ M_T-1. Define the upper mean-growth exponent λ_mean = limsup_T→∞ frac1T lognormM_0M_1⋯ M_T-1. A positive value indicates exponential growth of the linearized expected count. It does not imply almost-sure survival under dependence or resource competition.
Exact mathematical passage
M_t
m_tUse in source · 3 passages
Search aliases: m_t
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5905
Let the latent world state be x_t∈ℝ^2, action a_t∈ℝ^2, observation y_t∈ℝ^3, and memory state m_t∈ℝ^2. The baseline dynamics are x_t+1=Ax_t+Ba_t, y_t=Cx_t+b_s, m_t+1=λ m_t+(1-λ)y_t,1:2, and the reported evaluator score is q_t=w^top y_t. The redundant observation map is C= 1 · 0
Exact mathematical passage
m_t\in\R^2
m_{t+1}Use in source · 2 passages
Search aliases: m_t+1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5912
Let the latent world state be x_t∈ℝ^2, action a_t∈ℝ^2, observation y_t∈ℝ^3, and memory state m_t∈ℝ^2. The baseline dynamics are x_t+1=Ax_t+Ba_t, y_t=Cx_t+b_s, m_t+1=λ m_t+(1-λ)y_t,1:2, and the reported evaluator score is q_t=w^top y_t. The redundant observation map is C= 1 · 0
Exact mathematical passage
m_{t+1}=\lambda m_t+(1-\lambda)y_{t,1:2},
M_{T-1}Use in source · 2 passages
Search aliases: M_T-1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5298
For a time-varying sequence M_t, let z_t be a row vector of expected schema counts. Under the linear mean model, E z_T = z_0M_0M_1⋯ M_T-1. Define the upper mean-growth exponent λ_mean = limsup_T→∞ frac1T lognormM_0M_1⋯ M_T-1. A positive value indicates exponential growth of the linearized expected count. It does not imply almost-sure survival under dependence or resource competition.
Exact mathematical passage
\mathbb E z_T = z_0M_0M_1\cdots M_{T-1}.
m_\alphaUse in source · 4 passages
Search aliases: m_α alpha
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10052
[Finite causal atlas decomposition] There exists a unitary identification H ≅ ⊕_α=1^p ( C^m_α⊗ C^n_α ) under which A ≅ ⊕_α=1^p ( I_m_α⊗ M_n_α( C) ). The minimal central projections P_α∈ Z( A) give a canonical coarse causal atlas H = ⊕_α=1^pP_α H. The blocks are unique up to permutation and unitary changes of basis within multiplicity spaces.
Exact mathematical passage
H \cong \bigoplus_{\alpha=1}^{p} \left( \mathbb C^{m_\alpha}\otimes\mathbb C^{n_\alpha} \right) \label{eq:b8-algebra-space-decomposition}
\max_{1\leq b\leq B}Use in source · 1 passage
Search aliases: max_1≤ b≤ B max leq leq
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6143
The archive arm can perform well without a causal repair calculus because it searches several noisy candidates and retains the best. Candidate-budget matching is therefore mandatory. Let G_a,j,1,…,G_a,j,B be candidate gains for arm a on agent j. The primary behavioral quantity is M_a(B) = frac1n∑_j=1^nmax_1≤ b≤ BG_a,j,b. Comparing arms at unequal B confounds repair competence with order statistics.
Exact mathematical passage
M_a(B) = \frac1n\sum_{j=1}^n\max_{1\leq b\leq B}G_{a,j,b}.
\max_{1\leq i\leq k}Use in source · 1 passage
Search aliases: max_1≤ i≤ k max leq leq
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4662
[Expected best-of-k gain] Let G_1,…,G_k be independent nonnegative candidate gains with common cumulative distribution function F and finite mean. Then Emax_1≤ i≤ kG_i = ∫_0^∞ (1-F(x)^k) d x. In particular, expected best gain is nondecreasing in k even when the candidate generator itself does not improve.
Exact mathematical passage
\mathbb E\max_{1\leq i\leq k}G_i = \int_0^{\infty} \left(1-F(x)^k\right)\dd x.
\max_{1\leq i\leq5}Use in source · 1 passage
Search aliases: max_1≤ i≤5 max leq leq
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17307
Each task contains five demonstration pairs D={(x_i,y_i)}_i=1^5 and a query q. After chart canonicalization, the full descendant computes pair codes p_i=ϕ(widehat x_i,widehat y_i)∈ℝ^32, and the permutation-invariant aggregate a(D) = ( frac15∑_i=1^5p_i, max_1≤ i≤5p_i )∈ℝ^64. The semantic seed is h_0=S(a(D),Q(widehat q))∈ℝ^40. A five-step Elman recurrence generates the Neuralese trace r_k=tanh(W_ih_0+b_i+U_ir_k-1+c_i+e_k), k=1,…,5, where e_k is a learned step embedding. A gated readout recurrence …
Exact mathematical passage
a(D) = \left( \frac15\sum_{i=1}^{5}p_i, \max_{1\leq i\leq5}p_i \right)\in\R^{64}. \label{eq:b19-set-aggregate}
\max_{1\leq r\leq k}Use in source · 1 passage
Search aliases: max_1≤ r≤ k max leq leq
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5232
[Localization of supercriticality] After permuting schema types, M has block upper-triangular form with diagonal blocks M_1,…,M_k given by strongly connected components. Therefore rad(M) = max_1≤ r≤ krad(M_r). In particular, R_imp>1 only if at least one strongly connected schema family is itself supercritical.
Exact mathematical passage
\rad(M) = \max_{1\leq r\leq k}\rad(M_r).
\max_aUse in source · 1 passage
Search aliases: max_a max
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12371
[Finite repair-relevant block selection] Suppose there is a unique repair-relevant block a^* and a margin m>0 such that s_a^*≥ s_a+m for every a≠ a^*. If max_a|widehat s_a-s_a|<fracm2, then arg max_awidehat s_a=a^*.
Exact mathematical passage
\max_a|\widehat s_a-s_a|<\frac{m}{2},
\max_{A'\in\Desc_{h,B}(K,A)}Use in source · 2 passages
Search aliases: max_A'∈Desc_h,B(K,A) max in Desc max in operatorname
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4626
Let D_hold be a distribution over foreign held-out agents that are not descendants of the evaluated lineage. Let K be a frozen improvement mechanism. Under horizon h and budget B, define M^foreign_h,B(K) = E_A∼D_hold [ max_A'∈Desc_h,B(K,A) J(A')-J(A) ].
Exact mathematical passage
M^{\mathrm{foreign}}_{h,B}(K) = \mathbb E_{A\sim\Dcal_{\mathrm{hold}}} \left[ \max_{A'\in\Desc_{h,B}(K,A)} J(A')-J(A) \right].
\max_{a,j}Use in source · 1 passage
Search aliases: max_a,j max
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12326
rojectors widehat P_1,…,widehat P_m, define the normalized response-fit residual ε_resp = frac ( ∑_jnormwidehat R_j- ∑_awidehatλ_ajwidehat P_a_F^2 )^1/2 ( ∑_jnormwidehat R_j_F^2 )^1/2+10^-12 , and the maximum normalized commutator residual ε_comm = max_a,j fracnormwidehat P_awidehat R_j- widehat R_jwidehat P_a_F normwidehat R_j_F+10^-12. The smallest selected spectral gap is denoted widehatγ_min. These quantities serve different roles: item ε_resp measures whether the finite block-scalar model f…
Exact mathematical passage
\eps_{\mathrm{comm}} = \max_{a,j} \frac{\norm{\widehat P_a\widehat R_j- \widehat R_j\widehat P_a}_F} {\norm{\widehat R_j}_F+10^{-12}}. \label{eq:b11-commutator}
\max_{A\in\Acal_t}Use in source · 1 passage
Search aliases: max_A∈Acal_t max_A∈A_t max in Acal max in mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4498
[Archive-envelope monotonicity] Suppose archive updates never delete the last representative of a retained node. For every nonnegative weight vector w, define H_w(A_t) = max_A∈A_tw^ Tz(A). Then H_w(A_t) is nondecreasing in t.
Exact mathematical passage
H_w(\Acal_t) = \max_{A\in\Acal_t}w^{\mathsf T}z(A).
\max_{c\in\mathcal A_0}Use in source · 1 passage
Search aliases: max_c∈ A_0 max in mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7771
[Risk-filter tradeoff] Let A_r⊆ A_0 be the candidate set retained with and without a risk filter. Then max_c∈ A_0widehat U(c) ≥ max_c∈ A_rwidehat U(c), but no analogous inequality favors the unfiltered selector for true protected risk.
Exact mathematical passage
\max_{c\in\mathcal A_0}\widehat U(c) \geq \max_{c\in\mathcal A_r}\widehat U(c),
\max_{c\in\mathcal A_r}Use in source · 1 passage
Search aliases: max_c∈ A_r max in mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7771
[Risk-filter tradeoff] Let A_r⊆ A_0 be the candidate set retained with and without a risk filter. Then max_c∈ A_0widehat U(c) ≥ max_c∈ A_rwidehat U(c), but no analogous inequality favors the unfiltered selector for true protected risk.
Exact mathematical passage
\max_{c\in\mathcal A_0}\widehat U(c) \geq \max_{c\in\mathcal A_r}\widehat U(c),
\max_iUse in source · 2 passages
Search aliases: max_i max
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4673
For a nonnegative random variable X, E X=∫_0^∞Pr(X>x) d x. Independence gives Pr(max_iG_i≤ x)=F(x)^k. Substitution proves the identity.
Exact mathematical passage
\Pr\left(\max_iG_i\leq x\right)=F(x)^k.
\max_{K\in\mathcal K}Use in source · 1 passage
Search aliases: max_K∈ K max in mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10969
contain N candidate grandchild configurations. Suppose each target-level protected gain lies in [-G,G] and n independent development targets are used. Then, with probability at least 1-δ, every candidate satisfies abswidehatM(K)-M(K) ≤ Gsqrtfrac2log(2N/δ)n. If widehat K maximizes the empirical mean, then M(widehat K) ≥ max_K∈ KM(K) - 2Gsqrtfrac2log(2N/δ)n.
Exact mathematical passage
\Mcal(\widehat K) \geq \max_{K\in\mathcal K}\Mcal(K) - 2G\sqrt{\frac{2\log(2N/\delta)}{n}}.
\max_{m\in\Mcal_0}Use in source · 1 passage
Search aliases: max_m∈Mcal_0 max_m∈M_0 max in Mcal max in mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15172
Let U(τ,m) be protected validity-filtered research gain. Define inherited and extended oracle envelopes O_0(τ)=max_m∈M_0U(τ,m), O_1(τ)=max_m∈M_0∪C_rm pairU(τ,m). The task-relevant structural margin is κ_Q = E_τ∼ Q [ O_1(τ)-O_0(τ) ]. A positive margin means that the extension adds useful methods outside the inherited candidate opportunity set.
Exact mathematical passage
O_0(\tau)=\max_{m\in\Mcal_0}U(\tau,m), \qquad O_1(\tau)=\max_{m\in\Mcal_0\cup\Ccal_{\rm pair}}U(\tau,m).
\max_{m\in\Mcal_0\cup\Ccal_{\rm pair}}Use in source · 1 passage
Search aliases: max_m∈Mcal_0∪Ccal_rm pair max_m∈M_0∪C_rm pair max in Mcal cup Ccal rm max in mathcal cup mathcal rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15172
Let U(τ,m) be protected validity-filtered research gain. Define inherited and extended oracle envelopes O_0(τ)=max_m∈M_0U(τ,m), O_1(τ)=max_m∈M_0∪C_rm pairU(τ,m). The task-relevant structural margin is κ_Q = E_τ∼ Q [ O_1(τ)-O_0(τ) ]. A positive margin means that the extension adds useful methods outside the inherited candidate opportunity set.
Exact mathematical passage
O_0(\tau)=\max_{m\in\Mcal_0}U(\tau,m), \qquad O_1(\tau)=\max_{m\in\Mcal_0\cup\Ccal_{\rm pair}}U(\tau,m).
\max_{q\in Q}Use in source · 1 passage
Search aliases: max_q∈ Q max in
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4540
[Approximate scalarization preservation] For every w≥0 with normw_1=1, max_q∈ Qw^ Tq ≥ max_z∈ Zw^ Tz-ε.
Exact mathematical passage
\max_{q\in Q}w^{\mathsf T}q \geq \max_{z\in Z}w^{\mathsf T}z-\eps.
\max_{S:|S|=k}Use in source · 1 passage
Search aliases: max_S:|S|=k max
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21539
does not imply that the repair process has no remaining headroom. Let the target bridge be full rank but have minimum singular value σ_min=10^-4. Random noise of norm 10^-3 can then dominate the weakest identified direction. The bottleneck has migrated from rank to conditioning. A meaningful grandchild should target a criterion such as max_S:|S|=k λ_min(X_S^top X_S) or a regularized determinant objective, not another rank-only rule.
Exact mathematical passage
\max_{S:|S|=k} \lambda_{\min}(X_S^\top X_S)
\max_{\substack{y\in\ker D^*\\ \norm y\leq1}}Use in source · 1 passage
Search aliases: max_substacky∈ker D^* norm y≤1 max substack in ker norm leq
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1606
[Dual obstruction witness] For every r∈ V, η_D(r) = max_substacky∈ker D^* norm y≤1 anglesy,r. If r_ob≠0, the maximum is attained by y^*=fracr_obnormr_ob.
Exact mathematical passage
\eta_D(r) = \max_{\substack{y\in\ker D^*\\ \norm y\leq1}} \angles{y,r}.
\max_zUse in source · 1 passage
Search aliases: max_z max
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5989
Let μ_z=ϕ(z) be the frozen event prototype. With equal priors and matched isotropic noise, the posterior is p(zmidwidetildeϕ) = fracexp(-normwidetildeϕ-μ_z^2/(2σ^2)) ∑_z'exp(-normwidetildeϕ-μ_z'^2/(2σ^2)). A repair is attempted only when max_z p(zmidwidetildeϕ)≥θ and the maximizing event is not none. Otherwise the system abstains.
Exact mathematical passage
\max_z p(z\mid\widetilde\phi)\geq\theta
\max_{z\in Z}Use in source · 1 passage
Search aliases: max_z∈ Z max in
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4540
[Approximate scalarization preservation] For every w≥0 with normw_1=1, max_q∈ Qw^ Tq ≥ max_z∈ Zw^ Tz-ε.
Exact mathematical passage
\max_{q\in Q}w^{\mathsf T}q \geq \max_{z\in Z}w^{\mathsf T}z-\eps.
\max_{|S|=k}Use in source · 1 passage
Search aliases: max_|S|=k max
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10790
The E-optimal objective is max_|S|=kλ_min(M_S). The companion initializes with the weighted D-optimal design, then performs same-stratum exchanges that increase the smallest eigenvalue. This is a finite heuristic rather than a claim of global E-optimality.
Exact mathematical passage
\max_{|S|=k}\lambda_{\min}(M_S).
\Mcal_{\mathrm{audit}}Use in source · 1 passage
Search aliases: Mcal_audit M_audit Mcal mathrm mathcal mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10487
The descendant compiler receives only new development receipts. It diagnoses full-rank saturation, activates a finite conditioning-aware query grammar, selects a grandchild K_2, and seals the resulting policy before the fresh audit tasks and representation families are executed. The primary comparison is M_audit(K_2)-M_audit(K_1^rm recal), where K_1^rm recal preserves the Block 7 rank-revealing query policy but is allowed to recalibrate its witness gate on Block 9 development evidence. This is a…
Exact mathematical passage
\Mcal_{\mathrm{audit}}(K_2)-\Mcal_{\mathrm{audit}}(K_1^{\rm recal}),
\Mcal_{\Xi}Use in source · 1 passage
Search aliases: Mcal_Ξ M_Ξ Mcal Xi mathcal Xi
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10517
[Same-context transitivity] Suppose every edge is evaluated in one common context Ξ. Then M_Ξ(K_m)-M_Ξ(K_0) = ∑_t=0^m-1 ( M_Ξ(K_t+1)-M_Ξ(K_t) ). Consequently, positive lower bounds on all edges add to a positive lower bound on the complete path.
Exact mathematical passage
\Mcal_{\Xi}(K_m)-\Mcal_{\Xi}(K_0) = \sum_{t=0}^{m-1} \left( \Mcal_{\Xi}(K_{t+1})-\Mcal_{\Xi}(K_t) \right).
\Mcal_{\Xi,\Dcal,B}Use in source · 2 passages
Search aliases: Mcal_Ξ,Dcal,B M_Ξ,D,B Mcal Xi Dcal mathcal Xi mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9015
Fix a protected evaluation context Ξ, a foreign-system distribution D, and a matched resource budget B. Define protected foreign metaproductivity by M_Ξ,D,B(K) = E_A∼D [ J_Ξ(K[A;B])-J_Ξ(A) ], where K[A;B] is the authorized descendant selected by K for target A, or A itself when K abstains.
Exact mathematical passage
\Mcal_{\Xi,\Dcal,B}(K) = \mathbb E_{A\sim\Dcal} \left[ J_\Xi\bigl(K[A;B]\bigr)-J_\Xi(A) \right], \label{eq:b8-metaproductivity}
\Mcal_{\Xi_0}Use in source · 1 passage
Search aliases: Mcal_Ξ_0 M_Ξ_0 Mcal Xi mathcal Xi
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21803
Let edge K_0→ K_1 improve a bridge task distribution Ξ_0, while K_1→ K_2 improves a heteroscedastic distribution Ξ_1. Positive effects in both contexts do not imply M_Ξ_0(K_2)>M_Ξ_0(K_0) or any additive path gain unless Ξ_0 and Ξ_1 share a common utility or a validated context transport.
Exact mathematical passage
\Mcal_{\Xi_0}(K_2)>\Mcal_{\Xi_0}(K_0)
\mathrm{mean}Use in source · 2 passages
Search aliases: mean mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5304
For a time-varying sequence M_t, let z_t be a row vector of expected schema counts. Under the linear mean model, E z_T = z_0M_0M_1⋯ M_T-1. Define the upper mean-growth exponent λ_mean = limsup_T→∞ frac1T lognormM_0M_1⋯ M_T-1. A positive value indicates exponential growth of the linearized expected count. It does not imply almost-sure survival under dependence or resource competition.
Exact mathematical passage
\lambda_{\mathrm{mean}} = \limsup_{T\to\infty} \frac1T \log\norm{M_0M_1\cdots M_{T-1}}.
\mathrm{mech}Use in source · 1 passage
Search aliases: mech mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5525
G_mech · Mechanics · Legal phase traces, receipt verification, checkpoint restoration, deterministic replay
Exact mathematical passage
G_{\mathrm{mech}}
\mathrm{memory}Use in source · 5 passages
Search aliases: memory mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 665
A representative color set is C= set model, workspace, memory, planner, improver, evaluator, provenance, tools . The atlas does not assert that these colors are universally correct. It declares the decomposition under which one particular repair theory is being tested.
Exact mathematical passage
\mathsf C= \set{ \mathrm{model}, \mathrm{workspace}, \mathrm{memory}, \mathrm{planner}, \mathrm{improver}, \mathrm{evaluator}, \mathrm{provenance}, \mathrm{tools} }.
\mathrm{meta}Use in source · 2 passages
Search aliases: meta mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5529
G_meta · Metaproductivity · Parent-child repair-calculus comparison on frozen foreign systems
Exact mathematical passage
G_{\mathrm{meta}}
\MetaCapableUse in source · 1 passage
Search aliases: MetaCapable mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4259
Let C_RI(s) = · Grounded(s) wedgeProvValid(s) wedgeWorkspaceBound(s) · wedgeRepairTyped(s) wedgeWitnessed(s) wedgeEvidenceFlat(s) · wedgeRollbackable(s) wedgeMetaCapable(s). For X⊆ S, define Ψ_∀(X) = sets∈ S:C_RI(s) and Post_imp(s)⊆ X.
Exact mathematical passage
\begin{aligned} C_{\mathrm{RI}}(s) ={}& \Grounded(s) \wedge\ProvValid(s) \wedge\WorkspaceBound(s)\\ &\wedge\RepairTyped(s) \wedge\Witnessed(s) \wedge\EvidenceFlat(s)\\ &\wedge\Rollbackable(s) \wedge\MetaCapable(s). \end{aligned}
\min_{b\neq a}Use in source · 1 passage
Search aliases: min_b≠ a min neq
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12288
In an executable system, the responses are estimated from finite intervention data. Let widehat R_j=R_j+E_j, and fix coefficients c_j. Define B=∑_jc_jR_j, widehat B=∑_jc_jwidehat R_j, E=widehat B-B. Let the distinct eigenvalue associated with block a be μ_a and define its separation γ_a=min_b≠ a|μ_a-μ_b|.
Exact mathematical passage
\gamma_a=\min_{b\neq a}|\mu_a-\mu_b|.
\min_{c\neq d}Use in source · 1 passage
Search aliases: min_c≠ d min neq
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17210
[Nearest-signature route stability] Let the minimum pairwise normalized signature separation of an atlas be Δ_A = min_c≠ dd_sig(σ_c,σ_d). If an observed signature from chart c has normalized perturbation at most δ and 2δ<Δ_A, nearest-signature routing returns c. If additionally δ≤τ_adm, the route is admitted.
Exact mathematical passage
\Delta_{\Acal} = \min_{c\neq d}d_{\mathrm{sig}}(\sigma_c,\sigma_d).
\min_{g\in\mathcal G_{21}}Use in source · 1 passage
Search aliases: min_g∈ G_21 min in mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 18415
Each repair g receives a description-length proxy L(g) recording parameter count and constructor surcharge. The compiler selects widehat g ∈ argmin_g∈ G_21 L(g) subject to the appropriate protected gates. If no repair passes and the collision certificate is valid, it returns G_bot; otherwise it returns unresolved.
Exact mathematical passage
\widehat g \in \arg\min_{g\in\mathcal G_{21}} L(g) \quad\text{subject to the appropriate protected gates.} \label{eq:block21-mdl}
\min_jUse in source · 1 passage
Search aliases: min_j min
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 14612
[Conjunctive monitor false-negative bound] For an invalid method event I, the ensemble false-negative probability satisfies P(M_∩=1mid I) ≤ min_j P(M_j=1mid I). If monitor errors are conditionally independent given I, then P(M_∩=1mid I) = ∏_j=1^s P(M_j=1mid I).
Exact mathematical passage
\mathbb P(M_{\cap}=1\mid I) \leq \min_j\mathbb P(M_j=1\mid I).
\min_{J_v(x_i)\neq J_v(x_j)}Use in source · 1 passage
Search aliases: min_J_v(x_i)≠ J_v(x_j) min neq
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3774
[Historical ranking preservation] Let x_1,…,x_N be protected historical candidate records with scalar old scores J_v(x_i). Suppose every unequal old pair has margin at least m_hist := min_J_v(x_i)≠ J_v(x_j) absJ_v(x_i)-J_v(x_j)>0. If an upgrade satisfies absJ_w(T_ex_i)-J_v(x_i)≤ε for every i, with 2ε<m_hist, then every strict historical ranking is preserved.
Exact mathematical passage
m_{\mathrm{hist}} := \min_{J_v(x_i)\neq J_v(x_j)} \abs{J_v(x_i)-J_v(x_j)}>0.
\min_{z\neq z'}Use in source · 1 passage
Search aliases: min_z≠ z' min neq
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5997
[Prototype-separation error bound] Let Δ = min_z≠ z'normμ_z-μ_z'. Under equal priors and isotropic Gaussian noise, nearest-prototype classification satisfies Pr(widehat z≠ z) ≤ (| Z|-1) exp(-fracΔ^28σ^2).
Exact mathematical passage
\Delta = \min_{z\neq z'}\norm{\mu_z-\mu_{z'}}.
\mathrm{MSE}Use in source · 1 passage
Search aliases: MSE mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16303
positive interval and fraction gates, local continuity, donor coverage, adaptive halting. Among eligible candidates it maximizes S(α) = · min{ fracR^2_probe0.65, fracQ_prog0.15, fracQ_ans0.20 } +0.20R^2_probe · -4 MSE_clean +0.02ρ_halt -Penalty_dynamic. The minimum term makes the weakest normalized semantic gate the principal bottleneck. The compiler seal commits the selected candidate, all candidate and parent state hashes, training receipts, meta-validation task digest, exact thresholds, selec…
Exact mathematical passage
\begin{split} S(\alpha) ={}& \min\left\{ \frac{R^2_{\mathrm{probe}}}{0.65}, \frac{Q_{\mathrm{prog}}}{0.15}, \frac{Q_{\mathrm{ans}}}{0.20} \right\} +0.20R^2_{\mathrm{probe}}\\ &-4\,\mathrm{MSE}_{\mathrm{clean}} +0.02\rho_{\mathrm{halt}} -\mathsf{Penalty}_{\mathrm{dynamic}}. \end{split} \label{eq:b17-selection-score}
\mathrm{MSE}_{\mathrm{clean}}Use in source · 1 passage
Search aliases: MSE_clean mathrm mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16303
positive interval and fraction gates, local continuity, donor coverage, adaptive halting. Among eligible candidates it maximizes S(α) = · min{ fracR^2_probe0.65, fracQ_prog0.15, fracQ_ans0.20 } +0.20R^2_probe · -4 MSE_clean +0.02ρ_halt -Penalty_dynamic. The minimum term makes the weakest normalized semantic gate the principal bottleneck. The compiler seal commits the selected candidate, all candidate and parent state hashes, training receipts, meta-validation task digest, exact thresholds, selec…
Exact mathematical passage
\begin{split} S(\alpha) ={}& \min\left\{ \frac{R^2_{\mathrm{probe}}}{0.65}, \frac{Q_{\mathrm{prog}}}{0.15}, \frac{Q_{\mathrm{ans}}}{0.20} \right\} +0.20R^2_{\mathrm{probe}}\\ &-4\,\mathrm{MSE}_{\mathrm{clean}} +0.02\rho_{\mathrm{halt}} -\mathsf{Penalty}_{\mathrm{dynamic}}. \end{split} \label{eq:b17-selection-score}
\NUse in source · 1 passage
Search aliases: N ℕ mathbb
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2578
[Finite local-to-global obstruction tower] Fix N∈ℕ. Assume: item the analytic local repair equations agree under restriction; item the sheaves mathscr U and mathscr V satisfy effective linear descent on the declared cover; item repairs have been constructed globally through order n-1. Then a global repair extends through order n if and only if: item every local coefficient class vanishes, [b_n,α]=0 in coker D_α; item for one, hence every, choice of local lifts, the overlap class vanishes, [d^(n)…
Exact mathematical passage
N\in\N
n^{-1}Use in source · 1 passage
Search aliases: n^-1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3348
[Paired held-out gain certificate] Let Z_1,…,Z_n be independent and identically distributed paired target-context gains for a precommitted transported repair, with -G≤ Z_i≤ G. Let overline Z_n=n^-1∑_iZ_i. Then, with probability at least 1-α, E Z_i ≥ overline Z_n -Gsqrtfrac2log(1/α)n. For a precommitted family of m repairs, replacing α by α/m gives a simultaneous lower bound for all m candidates.
Exact mathematical passage
\overline Z_n=n^{-1}\sum_iZ_i
\mathcal N_{\mathcal B\cup\{b_{m+1}\}}Use in source · 1 passage
Search aliases: N_ B∪{b_m+1} mathcal mathcal cup
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 14591
[Strict evaluator-nullspace refinement] If Db_m+1 is nonzero on some v∈ N_ B, then N_ B∪{b_m+1} subsetneq N_ B.
Exact mathematical passage
\mathcal N_{\mathcal B\cup\{b_{m+1}\}} \subsetneq \mathcal N_{\mathcal B}.
N_1Use in source · 8 passages
Search aliases: N_1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8490
s(x)v(x) , where u,v∈ℝ^8 and A is invertible. Let N_1 be the number of affected anchors among k selections.
Exact mathematical passage
N_1
N_5Use in source · 2 passages
Search aliases: N_5
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21481
Consider five mutable modules: N_5 = {S,Q,T,C,H}, representing selection, information acquisition, transport, candidate construction, and causal-atlas structure. Let F_K(∅)=0.20. Assign singleton contributions (0.08,0.12,0.10,0.16,0.14) and pair synergies ω_QT=0.05, ω_CH=0.06, ω_SC=0.03. Then F_K(N_5)=0.94, H(K)=0.74. Every pair synergy is divided equally between its two participants by the permutation-average rule. Therefore h_S · =0.08+frac0.032=0.095,
Exact mathematical passage
N_5 = \{S,Q,T,C,H\},
N_{ij}Use in source · 2 passages
Search aliases: N_ij
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5196
Partition validated schemas into m types. For a type-i schema, let N_ij count type-j descendants whose receipts include a successful necessity ablation showing that removal of the parent schema eliminates the descendant discovery under matched evidence and budget. Define M_ij= E N_ij. This definition excludes descendants that merely share ancestry or evidence with the parent.
Exact mathematical passage
N_{ij}
n_{\rm anchor}Use in source · 1 passage
Search aliases: n_rm anchor rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6510
For the controlled transport experiment, the sample sizes are (n_rm train,n_rm anchor,n_rm calibrate,n_rm audit) =(7000,1500,1600,6000). The learned-representation pilot uses (5000,1200,1200,4000).
Exact mathematical passage
(n_{\rm train},n_{\rm anchor},n_{\rm calibrate},n_{\rm audit}) =(7000,1500,1600,6000).
n_{\rm audit}Use in source · 1 passage
Search aliases: n_rm audit rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6510
For the controlled transport experiment, the sample sizes are (n_rm train,n_rm anchor,n_rm calibrate,n_rm audit) =(7000,1500,1600,6000). The learned-representation pilot uses (5000,1200,1200,4000).
Exact mathematical passage
(n_{\rm train},n_{\rm anchor},n_{\rm calibrate},n_{\rm audit}) =(7000,1500,1600,6000).
n_{\rm calibrate}Use in source · 1 passage
Search aliases: n_rm calibrate rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6510
For the controlled transport experiment, the sample sizes are (n_rm train,n_rm anchor,n_rm calibrate,n_rm audit) =(7000,1500,1600,6000). The learned-representation pilot uses (5000,1200,1200,4000).
Exact mathematical passage
(n_{\rm train},n_{\rm anchor},n_{\rm calibrate},n_{\rm audit}) =(7000,1500,1600,6000).
n_{\rm train}Use in source · 1 passage
Search aliases: n_rm train rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6510
For the controlled transport experiment, the sample sizes are (n_rm train,n_rm anchor,n_rm calibrate,n_rm audit) =(7000,1500,1600,6000). The learned-representation pilot uses (5000,1200,1200,4000).
Exact mathematical passage
(n_{\rm train},n_{\rm anchor},n_{\rm calibrate},n_{\rm audit}) =(7000,1500,1600,6000).
n_{t+1}Use in source · 2 passages
Search aliases: n_t+1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9394
A grammar extension can reveal error that did not exist as a coordinate in the old state. Let E_t:ℝ^m_t↪ℝ^m_t+1 be the conservative embedding of old meta-regret coordinates. Decompose r_t+1 = E_twidetilde r_t+1+n_t+1, n_t+1⊥im E_t. Define old-coordinate repair progress by P_t^repair = normr_t^2-normwidetilde r_t+1^2, and epistemic discovery by P_t^discovery = normn_t+1^2. Both quantities may be positive. The total represented regret may increase because the lineage learned to represent a previou…
Exact mathematical passage
r_{t+1} = E_t\widetilde r_{t+1}+n_{t+1}, \qquad n_{t+1}\perp\im E_t. \label{eq:b8-regret-expansion}
\mathcal N_{\{b_1\}}Use in source · 1 passage
Search aliases: N_{b_1} mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 14575
Let b_1,…,b_m be benchmark readouts. A single scored benchmark leaves the local nullspace N_{b_1} = ker Db_1. An evaluator portfolio has invisible subspace N_ B = ⋂_j=1^mker Db_j.
Exact mathematical passage
\mathcal N_{\{b_1\}} = \ker Db_1.
n_\alphaUse in source · 3 passages
Search aliases: n_α alpha
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10052
[Finite causal atlas decomposition] There exists a unitary identification H ≅ ⊕_α=1^p ( C^m_α⊗ C^n_α ) under which A ≅ ⊕_α=1^p ( I_m_α⊗ M_n_α( C) ). The minimal central projections P_α∈ Z( A) give a canonical coarse causal atlas H = ⊕_α=1^pP_α H. The blocks are unique up to permutation and unitary changes of basis within multiplicity spaces.
Exact mathematical passage
H \cong \bigoplus_{\alpha=1}^{p} \left( \mathbb C^{m_\alpha}\otimes\mathbb C^{n_\alpha} \right) \label{eq:b8-algebra-space-decomposition}
\Ncal_0Use in source · 1 passage
Search aliases: Ncal_0 N_0 Ncal mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15858
[Finite Neuralese metaproductivity edge] Let parent N_0 and child N_1 be evaluated on identical sealed tasks with identical fixed latent-step budgets, answer-scoring rules, donor construction, numerical precision, evaluator, and witness. Assume: item both audits satisfy replay integrity; item the child satisfies latent necessity, ordered computation, corruption sensitivity, and task specificity on primary and generator-foreign supports; item the paired child-minus-parent transplant-cost interval…
Exact mathematical passage
\Ncal_0
\Ncal_1Use in source · 1 passage
Search aliases: Ncal_1 N_1 Ncal mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15858
[Finite Neuralese metaproductivity edge] Let parent N_0 and child N_1 be evaluated on identical sealed tasks with identical fixed latent-step budgets, answer-scoring rules, donor construction, numerical precision, evaluator, and witness. Assume: item both audits satisfy replay integrity; item the child satisfies latent necessity, ordered computation, corruption sensitivity, and task specificity on primary and generator-foreign supports; item the paired child-minus-parent transplant-cost interval…
Exact mathematical passage
\Ncal_1
\Ncal_{\Bcal^+}Use in source · 1 passage
Search aliases: Ncal_Bcal^+ N_B^+ Ncal Bcal mathcal mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13830
[Portfolio refinement shrinks the invisible subspace] Let B^+=B∪{b_m+1}. Then N_B^+ ⊆ N_B. The inclusion is strict if and only if there exists v∈N_B such that Db_m+1(v)≠0.
Exact mathematical passage
\Ncal_{\Bcal^+} \subseteq \Ncal_{\Bcal}. \label{eq:b13-nullspace-inclusion}
\Ncal_tUse in source · 2 passages
Search aliases: Ncal_t N_t Ncal mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15834
Extend the SRIC lineage state by L_t^NL = ( L_t,N_t,I_t^NL,T_t^NL,W_t^NL), where T_t^NL contains trace-transport certificates and W_t^NL is the frozen external answer and invariant witness. A proposed descendant may modify the latent transition, update gate, halting head, packet atlas, or transport map. The backbone may be held fixed for a component-isolation edge or changed under a stronger whole-model edge.
Exact mathematical passage
\mathfrak L_t^{\mathrm{NL}} = (\mathfrak L_t,\Ncal_t,\Ical_t^{\mathrm{NL}},\Tau_t^{\mathrm{NL}},\Wcal_t^{\mathrm{NL}}),
\mathrm{no\mbox{-}meta}Use in source · 1 passage
Search aliases: no-meta mathrm mbox
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6212
Let N⊆{1,…,n} be foreign agents whose dominant defect belongs to the held-out sixth type. Define widehatτ_abstract = frac1|N|∑_j∈ N ( G_j^meta-G_j^no-meta ). In the reference positive control, widehatτ_abstract=0.700533. A learned recursive-improvement claim would require the schema to be generated by the parent improvement process, sealed before foreign evaluation, and compared under matched computation and evidence access.
Exact mathematical passage
\widehat\tau_{\mathrm{abstract}} = \frac1{|N|}\sum_{j\in N} \left( G_{j}^{\mathrm{meta}}-G_{j}^{\mathrm{no\mbox{-}meta}} \right).
\mathrm{no\mbox{-}op}Use in source · 1 passage
Search aliases: no-op mathrm mbox
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11625
Let K_G be the global parent and K_C the chart process. Let A_C(x) be the frozen chart-certificate decision for target episode x. The descendant is K_3(x)= K_C(x), · A_C(x)=1, K_G(x), · A_C(x)=0 and the parent gate passes, no-op, · otherwise.
Exact mathematical passage
K_3(x)= \begin{cases} K_C(x),&A_C(x)=1,\\ K_G(x),&A_C(x)=0\text{ and the parent gate passes},\\ \mathrm{no\mbox{-}op},&\text{otherwise}. \end{cases} \label{eq:b10-fallback}
\mathrm{nom}Use in source · 2 passages
Search aliases: nom mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13862
Let C_B^nom be the same envelope when integrity constraints are ignored.
Exact mathematical passage
\mathfrak C_B^{\mathrm{nom}}
\mathrm{none}Use in source · 1 passage
Search aliases: none mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5935
The latent event variable is z∈ {none,internal,world,sensor,memory,evaluator}. Each event changes one declared mechanism: item internal: the action-response map B changes; item world: the passive dynamics A change; item sensor: the redundant sensor receives an additive bias; item memory: the retention coefficient λ changes; item evaluator: the reported weight vector w changes; item none: all baseline mechanisms remain fixed. The event label is not supplied to the classifier.
Exact mathematical passage
z\in \{\mathrm{none},\mathrm{internal},\mathrm{world},\mathrm{sensor},\mathrm{memory},\mathrm{evaluator}\}.
\operatorname{NRMSE}Use in source · 2 passages
Search aliases: NRMSE operatorname
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17934
A candidate is pre-audit eligible only if both foreign target models are ATLAS_FULL and the candidate jointly satisfies R^2_transport · ≥0.75, · R^2_worst chart · ≥0.74, ρ_preserve · ≥0.98, · NRMSE_trace · ≤0.05, κ_square · ≤0.02. · · The selection score is the weakest normalized margin across these gates. This prevents a strong aggregate R^2 from compensating for a failed chart, high curvature, or poor trajectory reconstruction.
Exact mathematical passage
\begin{aligned} R^2_{\mathrm{transport}}&\geq0.75, &\qquad R^2_{\mathrm{worst\ chart}}&\geq0.74,\\ \rho_{\mathrm{preserve}}&\geq0.98, & \operatorname{NRMSE}_{\mathrm{trace}}&\leq0.05,\\ \kappa_{\mathrm{square}}&\leq0.02. && \end{aligned} \label{eq:b20-thresholds}
\operatorname{NRMSE}_{\rm packet}Use in source · 1 passage
Search aliases: NRMSE_rm packet operatorname rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 18404
For reconstructive candidates, admissibility requires R^2_rm sem≥0.68, R^2_rm packet≥0.72, NRMSE_rm packet≤0.55. A quotient is admissible only when the observed dimension ratio is at most 0.82, semantic R^2 is at least 0.68, and no reconstructive candidate reaches packet R^2=0.82. A no-go certificate is admissible only after the compiler discovers at least thirty identical-observation pairs whose mean protected semantic separation is at least 0.50.
Exact mathematical passage
R^2_{\rm sem}\geq0.68, \qquad R^2_{\rm packet}\geq0.72, \qquad \operatorname{NRMSE}_{\rm packet}\leq0.55. \label{eq:block21-reconstructive-gates}
\operatorname{NRMSE}_{\mathrm{trace}}Use in source · 1 passage
Search aliases: NRMSE_trace operatorname mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17934
A candidate is pre-audit eligible only if both foreign target models are ATLAS_FULL and the candidate jointly satisfies R^2_transport · ≥0.75, · R^2_worst chart · ≥0.74, ρ_preserve · ≥0.98, · NRMSE_trace · ≤0.05, κ_square · ≤0.02. · · The selection score is the weakest normalized margin across these gates. This prevents a strong aggregate R^2 from compensating for a failed chart, high curvature, or poor trajectory reconstruction.
Exact mathematical passage
\begin{aligned} R^2_{\mathrm{transport}}&\geq0.75, &\qquad R^2_{\mathrm{worst\ chart}}&\geq0.74,\\ \rho_{\mathrm{preserve}}&\geq0.98, & \operatorname{NRMSE}_{\mathrm{trace}}&\leq0.05,\\ \kappa_{\mathrm{square}}&\leq0.02. && \end{aligned} \label{eq:b20-thresholds}
\nsubseteqUse in source · 1 passage
Search aliases: nsubseteq
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5696
In the intact model every complete successor of s_base lies in K. Under ablation, Post(s_base)nsubseteq K, so s_base∉Ψ(K).
Exact mathematical passage
\Post(s_{\mathrm{base}})\nsubseteq K
\mathrm{nui}Use in source · 2 passages
Search aliases: nui mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12465
The source hidden space has dimension six and is generated from three target-specific unknown rank-two blocks: H=H_ctx⊕ H_rep⊕ H_nui. A target-specific orthogonal or injective embedding hides these blocks from the runtime coordinate system. The source repair changes only H_rep and leaves the context block invariant. Two controlled interventions act with different block signatures. The context label controls which target bridge and target repair are valid.
Exact mathematical passage
H=H_{\mathrm{ctx}}\oplus H_{\mathrm{rep}}\oplus H_{\mathrm{nui}}. \label{eq:b11-three-blocks}
o_{t+1}Use in source · 1 passage
Search aliases: o_t+1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1861
[Exact sequence of newly correctable directions] Under the assumptions of [source reference], there is a short exact sequence 0 ⟶ fracim D_t+1im D_t ⟶ coker D_t oversetq_t⟶ coker D_t+1 ⟶0. Consequently, o_t-o_t+1 = dimfracim D_t+1im D_t.
Exact mathematical passage
o_t-o_{t+1} = \dim\frac{\im D_{t+1}}{\im D_t}.
\odotUse in source · 3 passages
Search aliases: odot
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16124
The latent trace updates the task representation through g_k · =σ(Gz_k), widetilde h_k · =tanh(Th_k-1+Lz_k), h_k · =(1-g_k)odot h_k-1+g_kodotwidetilde h_k, h_0=e, widehat w_k · =Oh_k. A separate halt head predicts one of four dynamic depths from the mean trace and, for nonparent candidates, the encoded prompt. A semantic training head predicts w(τ) from z_4. Its auxiliary loss is an externally declared witness term, not a claim that the latent coordinates are uniquely interpretable.
Exact mathematical passage
g_k&=\sigma(Gz_k), \label{eq:b17-update-gate}\\ \widetilde h_k&=\tanh(Th_{k-1}+Lz_k), \label{eq:b17-hidden-candidate}\\ h_k&=(1-g_k)\odot h_{k-1}+g_k\odot\widetilde h_k, \qquad h_0=e, \label{eq:b17-hidden-update}\\ \widehat w_k&=Oh_k. \label{eq:b17-readout}
\one^\topUse in source · 2 passages
Search aliases: one^top 1^top one top mathbf top
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9334
[Greedy bottleneck contraction] Let r_t∈ℝ_≥0^m and Φ_t=1^top r_t. Suppose the controller chooses i_t with r_i_t,t≥fracΦ_tm, contracts that component by q∈[0,1), and permits total collateral increase at most ε_t. Then Φ_t+1 ≤ (1-frac1-qm)Φ_t+ε_t.
Exact mathematical passage
\Phi_t=\one^\top r_t.
\one_mUse in source · 2 passages
Search aliases: one_m 1_m one mathbf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16109
efine z_0=ζ, z_k+1 = tanh (Rz_k+s_k+C_α(e,m)), 0≤ k<4, where s_k is a learned step embedding. The task-blind parent satisfies C_parent(e,m)=0. The descendants use one of the following conditioning maps: C_depth(e,m) · =D 1_m, C_1(e,m) · =U_1V_1e, rank(U_1V_1)≤1, C_4(e,m) · =U_4V_4e, rank(U_4V_4)≤4, C_full(e,m) · =We. Here 1_m is the one-hot depth code. Each child inherits every compatible parent parameter. The added up-projection or full-conditioning matrix is initialized so that the initial inj…
Exact mathematical passage
C_{\mathrm{depth}}(e,m)&=D\,\one_m, \label{eq:b17-depth-conditioning}\\ C_{1}(e,m)&=U_1V_1e, \qquad \rank(U_1V_1)\leq1, \label{eq:b17-rank1-conditioning}\\ C_{4}(e,m)&=U_4V_4e, \qquad \rank(U_4V_4)\leq4, \label{eq:b17-rank4-conditioning}\\ C_{\mathrm{full}}(e,m)&=We. \label{eq:b17-full-conditioning}
\mathrm{open}Use in source · 1 passage
Search aliases: open mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5530
G_open · Open-set recursion · Conservative grammar extension, prior-proof replay, novel externally validated defect families
Exact mathematical passage
G_{\mathrm{open}}
\mathrm{oracle}Use in source · 3 passages
Search aliases: oracle mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11685
The main paired estimands are Δ_G→ C = E [ G(K_3,C^96,A)-G(K_2,G^96,A) ] and Δ_oracle→ C = E [ G(K_3,C^96,A)-G(K_affine oracle,A) ]. The second quantity is the sharper structural test.
Exact mathematical passage
\Delta_{\mathrm{oracle}\to C} = \mathbb E \left[ G(K_{3,C}^{96},A)-G(K_{\mathrm{affine\ oracle}},A) \right].
\mathrm{ord}Use in source · 2 passages
Search aliases: ord mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9094
Choose an operational ordering r_1prec r_2prec⋯prec r_m, m=|N|, with nested sets R_0=∅, R_j={r_1,…,r_j}. Define the ordered marginal headroom Δ_j^ord(K) = F_K(R_j)-F_K(R_j-1).
Exact mathematical passage
\Delta_j^{\mathrm{ord}}(K) = F_K(R_j)-F_K(R_{j-1}). \label{eq:b8-ordered-marginal}
\overline aUse in source · 2 passages
Search aliases: overline a overline
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16237
hed at every step. The program and answer specificity coordinates are the mean recipient costs under SAP and FAD donors: p_i = E_j∈SAP(i) [ℓ_i(Z_j)-ℓ_i(Z_i)], a_i = E_j∈FAD(i) [ℓ_i(Z_j)-ℓ_i(Z_i)]. Define Q_prog = fracoverline pmax{overline o,overline c,ε}, Q_ans = fracoverline amax{overline o,overline c,ε}. The frozen ratio thresholds are 0.15 and 0.20, respectively. Each specificity gate additionally requires a positive 95% bootstrap lower endpoint and a positive-effect fraction of at least 0.5…
Exact mathematical passage
Q_{\mathrm{prog}} = \frac{\overline p}{\max\{\overline o,\overline c,\eps\}}, \qquad Q_{\mathrm{ans}} = \frac{\overline a}{\max\{\overline o,\overline c,\eps\}}. \label{eq:b17-specificity-ratios}
\overline BUse in source · 3 passages
Search aliases: overline B overline
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2074
[Newton--Kantorovich repair criterion] Let F:U→ V be continuously Fr echet differentiable between Banach spaces. Let u_0∈ U and suppose F'(u_0) is invertible with inverse B. Assume that on the closed ball overline B(u_0,R), normF'(u)-F'(v)≤ Lnormu-v. Set η:=normBF(u_0), h:=norm B Lη. If h≤frac12 and t_*:=frac1-sqrt1-2hnorm B L≤ R, then Newton iteration u_n+1=u_n-F'(u_n)^-1F(u_n) is well-defined, remains in overline B(u_0,t_*), and converges to a root u^* of F. If the larger radius t^**:=frac1+sq…
Exact mathematical passage
\overline B(u_0,R)
\overline DUse in source · 7 passages
Search aliases: overline D overline
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4699
Let K_0 and K_1 be frozen improvement mechanisms evaluated on the same independent foreign agents A_1,…,A_n. Define G_h,B(K,A) = max_A'∈Desc_h,B(K,A)J(A')-J(A) and assume -G_max ≤ G_h,B(K,A) ≤ G_max. For matched budgets and random seeds, let D_i = G_h,B(K_1,A_i)-G_h,B(K_0,A_i), overline D_n = frac1n∑_i=1^nD_i.
Exact mathematical passage
D_i = G_{h,B}(K_1,A_i)-G_{h,B}(K_0,A_i), \qquad \overline D_n = \frac1n\sum_{i=1}^{n}D_i.
\overline D_nUse in source · 5 passages
Search aliases: overline D_n overline
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4699
Let K_0 and K_1 be frozen improvement mechanisms evaluated on the same independent foreign agents A_1,…,A_n. Define G_h,B(K,A) = max_A'∈Desc_h,B(K,A)J(A')-J(A) and assume -G_max ≤ G_h,B(K,A) ≤ G_max. For matched budgets and random seeds, let D_i = G_h,B(K_1,A_i)-G_h,B(K_0,A_i), overline D_n = frac1n∑_i=1^nD_i.
Exact mathematical passage
D_i = G_{h,B}(K_1,A_i)-G_{h,B}(K_0,A_i), \qquad \overline D_n = \frac1n\sum_{i=1}^{n}D_i.
\overline FUse in source · 1 passage
Search aliases: overline F overline
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9225
When the exact envelope is unknown, certified lower and upper oracle bounds underline F_K(R) ≤ F_K(R) ≤ overline F_K(R) produce corresponding bounds on χ_R. Audit-oracle arms are finite empirical approximations to this idea, but they must remain inadmissible for selection because they use final audit outcomes.
Exact mathematical passage
\underline F_K(R) \leq F_K(R) \leq \overline F_K(R)
\overline F_KUse in source · 1 passage
Search aliases: overline F_K overline
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9225
When the exact envelope is unknown, certified lower and upper oracle bounds underline F_K(R) ≤ F_K(R) ≤ overline F_K(R) produce corresponding bounds on χ_R. Audit-oracle arms are finite empirical approximations to this idea, but they must remain inadmissible for selection because they use final audit outcomes.
Exact mathematical passage
\underline F_K(R) \leq F_K(R) \leq \overline F_K(R)
\overline GUse in source · 3 passages
Search aliases: overline G overline
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21440
The child is developed on eight representation families and audited on four disjoint families. The frozen architecture-foreign metrics are overline G_rm parent · =0.167417, overline G_rm child · =0.272416, overline G_rm raw · =0.254427. Thus overline G_rm child-overline G_rm parent=0.104999 and overline G_rm child-overline G_rm raw=0.017989. The first contrast isolates improvement over the frozen lineage parent. The second tests whether the learned selector offers anything beyond a strong direct…
Exact mathematical passage
\begin{aligned} \overline G_{\rm parent}&=0.167417,\\ \overline G_{\rm child}&=0.272416,\\ \overline G_{\rm raw}&=0.254427. \end{aligned}
\overline G_{\rm child}Use in source · 3 passages
Search aliases: overline G_rm child overline rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21440
The child is developed on eight representation families and audited on four disjoint families. The frozen architecture-foreign metrics are overline G_rm parent · =0.167417, overline G_rm child · =0.272416, overline G_rm raw · =0.254427. Thus overline G_rm child-overline G_rm parent=0.104999 and overline G_rm child-overline G_rm raw=0.017989. The first contrast isolates improvement over the frozen lineage parent. The second tests whether the learned selector offers anything beyond a strong direct…
Exact mathematical passage
\begin{aligned} \overline G_{\rm parent}&=0.167417,\\ \overline G_{\rm child}&=0.272416,\\ \overline G_{\rm raw}&=0.254427. \end{aligned}
\overline G_{\rm parent}Use in source · 2 passages
Search aliases: overline G_rm parent overline rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21440
The child is developed on eight representation families and audited on four disjoint families. The frozen architecture-foreign metrics are overline G_rm parent · =0.167417, overline G_rm child · =0.272416, overline G_rm raw · =0.254427. Thus overline G_rm child-overline G_rm parent=0.104999 and overline G_rm child-overline G_rm raw=0.017989. The first contrast isolates improvement over the frozen lineage parent. The second tests whether the learned selector offers anything beyond a strong direct…
Exact mathematical passage
\begin{aligned} \overline G_{\rm parent}&=0.167417,\\ \overline G_{\rm child}&=0.272416,\\ \overline G_{\rm raw}&=0.254427. \end{aligned}
\overline G_{\rm raw}Use in source · 2 passages
Search aliases: overline G_rm raw overline rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21440
The child is developed on eight representation families and audited on four disjoint families. The frozen architecture-foreign metrics are overline G_rm parent · =0.167417, overline G_rm child · =0.272416, overline G_rm raw · =0.254427. Thus overline G_rm child-overline G_rm parent=0.104999 and overline G_rm child-overline G_rm raw=0.017989. The first contrast isolates improvement over the frozen lineage parent. The second tests whether the learned selector offers anything beyond a strong direct…
Exact mathematical passage
\begin{aligned} \overline G_{\rm parent}&=0.167417,\\ \overline G_{\rm child}&=0.272416,\\ \overline G_{\rm raw}&=0.254427. \end{aligned}
\overline jUse in source · 5 passages
Search aliases: overline j overline
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1890
[Induced obstruction transport under ontology expansion] Equation [source reference] induces a linear map overline j_t:coker D_t→coker D_t+1, [v]⟼[j_Vv]. Its kernel is canonically isomorphic to keroverline j_t ≅ fracj_V(V_t)∩im D_t+1j_V(im D_t). Thus an old obstruction vanishes after expansion exactly when the enlarged repair grammar corrects its embedded defect direction.
Exact mathematical passage
\overline j_t:\coker D_t\to\coker D_{t+1}, \qquad [v]\longmapsto[j_Vv].
\overline j_tUse in source · 2 passages
Search aliases: overline j_t overline
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1890
[Induced obstruction transport under ontology expansion] Equation [source reference] induces a linear map overline j_t:coker D_t→coker D_t+1, [v]⟼[j_Vv]. Its kernel is canonically isomorphic to keroverline j_t ≅ fracj_V(V_t)∩im D_t+1j_V(im D_t). Thus an old obstruction vanishes after expansion exactly when the enlarged repair grammar corrects its embedded defect direction.
Exact mathematical passage
\overline j_t:\coker D_t\to\coker D_{t+1}, \qquad [v]\longmapsto[j_Vv].
\overline pUse in source · 2 passages
Search aliases: overline p overline
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16237
hed at every step. The program and answer specificity coordinates are the mean recipient costs under SAP and FAD donors: p_i = E_j∈SAP(i) [ℓ_i(Z_j)-ℓ_i(Z_i)], a_i = E_j∈FAD(i) [ℓ_i(Z_j)-ℓ_i(Z_i)]. Define Q_prog = fracoverline pmax{overline o,overline c,ε}, Q_ans = fracoverline amax{overline o,overline c,ε}. The frozen ratio thresholds are 0.15 and 0.20, respectively. Each specificity gate additionally requires a positive 95% bootstrap lower endpoint and a positive-effect fraction of at least 0.5…
Exact mathematical passage
Q_{\mathrm{prog}} = \frac{\overline p}{\max\{\overline o,\overline c,\eps\}}, \qquad Q_{\mathrm{ans}} = \frac{\overline a}{\max\{\overline o,\overline c,\eps\}}. \label{eq:b17-specificity-ratios}
\overline RUse in source · 1 passage
Search aliases: overline R overline
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8019
For the architecture-foreign split, overline R_rm child · =0.012852, · Pr(R>0.05) · =0.033333, overline R_rm no risk · =0.015741, · Pr(R>0.05) · =0.058333. The risk head therefore reduces both mean and tail protected regression, at a utility cost of 0.021425 relative to the no-risk arm. This is expected from constrained selection and should not be described as unconstrained dominance.
Exact mathematical passage
\overline R_{\rm child}&=0.012852, &\Pr(R>0.05)&=0.033333,\\ \overline R_{\rm no\ risk}&=0.015741, &\Pr(R>0.05)&=0.058333.
\overline R_{\rm child}Use in source · 1 passage
Search aliases: overline R_rm child overline rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8019
For the architecture-foreign split, overline R_rm child · =0.012852, · Pr(R>0.05) · =0.033333, overline R_rm no risk · =0.015741, · Pr(R>0.05) · =0.058333. The risk head therefore reduces both mean and tail protected regression, at a utility cost of 0.021425 relative to the no-risk arm. This is expected from constrained selection and should not be described as unconstrained dominance.
Exact mathematical passage
\overline R_{\rm child}&=0.012852, &\Pr(R>0.05)&=0.033333,\\ \overline R_{\rm no\ risk}&=0.015741, &\Pr(R>0.05)&=0.058333.
\overline R_{\rm no\ risk}Use in source · 1 passage
Search aliases: overline R_rm no risk overline rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8019
For the architecture-foreign split, overline R_rm child · =0.012852, · Pr(R>0.05) · =0.033333, overline R_rm no risk · =0.015741, · Pr(R>0.05) · =0.058333. The risk head therefore reduces both mean and tail protected regression, at a utility cost of 0.021425 relative to the no-risk arm. This is expected from constrained selection and should not be described as unconstrained dominance.
Exact mathematical passage
\overline R_{\rm child}&=0.012852, &\Pr(R>0.05)&=0.033333,\\ \overline R_{\rm no\ risk}&=0.015741, &\Pr(R>0.05)&=0.058333.
\overline T_0Use in source · 1 passage
Search aliases: overline T_0 overline
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9651
The sequence coker D_0 xrightarrowoverline T_0 coker D_1 xrightarrowoverline T_1 coker D_2 ⟶⋯ is the obstruction persistence module of the grammar lineage.
Exact mathematical passage
\coker D_0 \xrightarrow{\overline T_0} \coker D_1 \xrightarrow{\overline T_1} \coker D_2 \longrightarrow\cdots \label{eq:b8-obstruction-module}
\overline T_1Use in source · 1 passage
Search aliases: overline T_1 overline
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9651
The sequence coker D_0 xrightarrowoverline T_0 coker D_1 xrightarrowoverline T_1 coker D_2 ⟶⋯ is the obstruction persistence module of the grammar lineage.
Exact mathematical passage
\coker D_0 \xrightarrow{\overline T_0} \coker D_1 \xrightarrow{\overline T_1} \coker D_2 \longrightarrow\cdots \label{eq:b8-obstruction-module}
\overline UUse in source · 2 passages
Search aliases: overline U overline
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7498
compiler searches a finite grammar Γ = Λ_u×Λ_r× K_u× K_r× T_r, where the factors specify utility regularization, risk regularization, utility residual penalty, risk residual penalty, and risk threshold. In the released fixture, |Γ| =5·5·4·5·4 =2000. For γ∈Γ, let overline U_rm meta(γ) be its mean protected utility on the meta-validation episodes and widehatrm se(γ) its standard error. A configuration is validation-safe when its protected-regression tail rate and maximum regression satisfy the fro…
Exact mathematical passage
\overline U_{\rm meta}(\gamma)
\overline U_{\rm meta}Use in source · 2 passages
Search aliases: overline U_rm meta overline rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7498
compiler searches a finite grammar Γ = Λ_u×Λ_r× K_u× K_r× T_r, where the factors specify utility regularization, risk regularization, utility residual penalty, risk residual penalty, and risk threshold. In the released fixture, |Γ| =5·5·4·5·4 =2000. For γ∈Γ, let overline U_rm meta(γ) be its mean protected utility on the meta-validation episodes and widehatrm se(γ) its standard error. A configuration is validation-safe when its protected-regression tail rate and maximum regression satisfy the fro…
Exact mathematical passage
\overline U_{\rm meta}(\gamma)
\overline yUse in source · 1 passage
Search aliases: overline y overline
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7629
Let Z be column centered and let P be a uniformly random permutation matrix independent of y. Then E_P[Z^top Py] = Z^top E_P[Py] = Z^topoverline y1 =0. The ridge coefficients are linear in Z^top Py for fixed Z.
Exact mathematical passage
\mathbb E_P[Z^\top Py] = Z^\top\mathbb E_P[Py] = Z^\top\overline y\one =0.
\overline ZUse in source · 3 passages
Search aliases: overline Z overline
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3348
[Paired held-out gain certificate] Let Z_1,…,Z_n be independent and identically distributed paired target-context gains for a precommitted transported repair, with -G≤ Z_i≤ G. Let overline Z_n=n^-1∑_iZ_i. Then, with probability at least 1-α, E Z_i ≥ overline Z_n -Gsqrtfrac2log(1/α)n. For a precommitted family of m repairs, replacing α by α/m gives a simultaneous lower bound for all m candidates.
Exact mathematical passage
\overline Z_n=n^{-1}\sum_iZ_i
\overline Z_nUse in source · 3 passages
Search aliases: overline Z_n overline
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3348
[Paired held-out gain certificate] Let Z_1,…,Z_n be independent and identically distributed paired target-context gains for a precommitted transported repair, with -G≤ Z_i≤ G. Let overline Z_n=n^-1∑_iZ_i. Then, with probability at least 1-α, E Z_i ≥ overline Z_n -Gsqrtfrac2log(1/α)n. For a precommitted family of m repairs, replacing α by α/m gives a simultaneous lower bound for all m candidates.
Exact mathematical passage
\overline Z_n=n^{-1}\sum_iZ_i
\overline\DeltaUse in source · 2 passages
Search aliases: overlineΔ overline Delta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3831
…,Z_n be independent and identically distributed held-out externally revealed cases. Let L_v(Z_i),L_w(Z_i)∈[0,1] be the losses of the old and proposed evaluators, and define Δ_i=L_v(Z_i)-L_w(Z_i)∈[-1,1]. If the upgrade and evaluation rule are fixed before the held-out outcomes are revealed, then with probability at least 1-α, EΔ_i ≥ overlineΔ_n - sqrtfrac2log(1/α)n. Thus a positive lower bound certifies lower expected held-out risk for the proposed evaluator.
Exact mathematical passage
\mathbb E\Delta_i \geq \overline\Delta_n - \sqrt{\frac{2\log(1/\alpha)}{n}}.
\overline\Delta_nUse in source · 2 passages
Search aliases: overlineΔ_n overline Delta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3831
…,Z_n be independent and identically distributed held-out externally revealed cases. Let L_v(Z_i),L_w(Z_i)∈[0,1] be the losses of the old and proposed evaluators, and define Δ_i=L_v(Z_i)-L_w(Z_i)∈[-1,1]. If the upgrade and evaluation rule are fixed before the held-out outcomes are revealed, then with probability at least 1-α, EΔ_i ≥ overlineΔ_n - sqrtfrac2log(1/α)n. Thus a positive lower bound certifies lower expected held-out risk for the proposed evaluator.
Exact mathematical passage
\mathbb E\Delta_i \geq \overline\Delta_n - \sqrt{\frac{2\log(1/\alpha)}{n}}.
\overline{\bigcup_t I_t}Use in source · 1 passage
Search aliases: overline⋃_t I_t overline bigcup
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9538
Nestedness gives I_t⊆ I_t+1, so distance to I_t+1 cannot exceed distance to I_t. Let I_∞=overline⋃_t I_t. The projections P_I_tr converge strongly to P_I_∞r in finite dimension, so the orthogonal residual norms converge to the distance from -r to I_∞.
Exact mathematical passage
I_\infty=\overline{\bigcup_t I_t}.
\overline{\bigcup_{t\geq0}I_t}Use in source · 1 passage
Search aliases: overline⋃_t≥0I_t overline bigcup geq
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9528
[Monotone obstruction convergence] For every defect r, the sequence η_t(r) is nonincreasing and η_t(r) ⟶ dist(-r,overline⋃_t≥0I_t).
Exact mathematical passage
\eta_t(r) \longrightarrow \dist\left(-r,\overline{\bigcup_{t\geq0}I_t}\right). \label{eq:b8-obstruction-limit}
\oversetUse in source · 1 passage
Search aliases: overset
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1850
[Exact sequence of newly correctable directions] Under the assumptions of [source reference], there is a short exact sequence 0 ⟶ fracim D_t+1im D_t ⟶ coker D_t oversetq_t⟶ coker D_t+1 ⟶0. Consequently, o_t-o_t+1 = dimfracim D_t+1im D_t.
Exact mathematical passage
0 \longrightarrow \frac{\im D_{t+1}}{\im D_t} \longrightarrow \coker D_t \overset{q_t}{\longrightarrow} \coker D_{t+1} \longrightarrow0.
\mathbb PUse in source · 7 passages
Search aliases: P mathbb
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 14460
[Static diagnostic indistinguishability] Let z_1≠ z_2 satisfy P(X_0∈ Bmid Z=z_1) = P(X_0∈ Bmid Z=z_2) for every measurable set B. Then no decision rule d(X_0) distinguishes z_1 from z_2 with accuracy exceeding the larger prior probability: P(d(X_0)=Zmid Z∈{z_1,z_2}) ≤ max{π_1,π_2}, where π_j= P(Z=z_jmid Z∈{z_1,z_2}).
Exact mathematical passage
\mathbb P(X_0\in B\mid Z=z_1) = \mathbb P(X_0\in B\mid Z=z_2)
P^{\mathsf T}Use in source · 4 passages
Search aliases: P^ T mathsf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16497
Let P be a signed permutation matrix on the latent space. Define transformed latent parameters ζ'=Pζ, R'=PRP^ T, s_k'=Ps_k, C_α'=PC_α, and transform every latent-consuming linear map by right multiplication with P^ T. Then the transformed model has z_k'=Pz_k and identical external outputs.
Exact mathematical passage
\zeta'=P\zeta, \quad R'=PRP^{\mathsf T}, \quad s_k'=Ps_k, \quad C_\alpha'=PC_\alpha, \label{eq:b17-gauge-trace}
p^{-1}Use in source · 4 passages
Search aliases: p^-1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3521
Every loop γ at w corresponds to the conjugate loop p^-1γ p at v, with H_γ=T_pH_p^-1γ pT_p^-1. If x∈ E_v^flat, then H_γ T_px=T_pH_p^-1γ px=T_px, so T_px∈ E_w^flat. Apply the same argument to p^-1 for the reverse inclusion.
Exact mathematical passage
p^{-1}\gamma p
p^jUse in source · 1 passage
Search aliases: p^j
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8506
If random anchor membership is Bernoulli with affected probability p, the idealized full-rank probability at k=16 is therefore Pr(rank X=16) = Pr(N_1≥8) = ∑_j=8^1616choose jp^j(1-p)^16-j under the generic model. The observed parent full-rank rate on the primary audit set is 0.03125, consistent with a low-probability affected-count event.
Exact mathematical passage
\Pr(\rank X=16) = \Pr(N_1\geq8) = \sum_{j=8}^{16}{16\choose j}p^j(1-p)^{16-j} \label{eq:block07q-random-rank-probability}
P^\topUse in source · 2 passages
Search aliases: P^top top
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13059
Let P be the recovered repair packet, let z=P^top x, and consider a target correction F(x) = Ax+b+(z^top Bz)v, where B=B^top≠0 and v≠0.
Exact mathematical passage
z=P^\top x
p^\starUse in source · 2 passages
Search aliases: p^⋆ star
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10833
Let p^⋆ be the best conditioning policy and p_0 the best inherited-policy configuration. Empirical errors of at most η imply widehatμ(p^⋆) ≥ μ(p^⋆)-η > μ(p_0)+η ≥ widehatμ(p_0).
Exact mathematical passage
p^\star
P_{(\im D)^\perp}Use in source · 1 passage
Search aliases: P_(im D)^⊥ im perp operatorname perp
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5635
The executable classifier applies the ordered rule Class(r,b,H) = LOCAL_OBSTRUCTION, · normP_(im D)^⊥r>τ, GLUING_OBSTRUCTION, · normP_ker B^topb>τ, EVALUATOR_HOLONOMY, · norm(H-I)e>τ, FLAT_REPAIR, · otherwise. The order is part of the receipt because a case can exhibit more than one defect. The fixture isolates the four modes, so the ordered rule is unambiguous.
Exact mathematical passage
\operatorname{Class}(r,b,H) = \begin{cases} \texttt{LOCAL\_OBSTRUCTION},&\norm{P_{(\im D)^\perp}r}>\tau,\\ \texttt{GLUING\_OBSTRUCTION},&\norm{P_{\ker B^\top}b}>\tau,\\ \texttt{EVALUATOR\_HOLONOMY},&\norm{(H-I)e}>\tau,\\ \texttt{FLAT\_REPAIR},&\text{otherwise}. \end{cases}
P_{(I\oplus W)^\perp}Use in source · 2 passages
Search aliases: P_(I⊕ W)^⊥ oplus perp
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9801
[Optimal rank-k obstruction extension] The subspace W minimizing expected unresolved obstruction energy E_r∼ Q normP_(I⊕ W)^⊥r^2 is the span of the top k eigenvectors of Σ_ob. The minimum residual energy is the sum of the remaining eigenvalues.
Exact mathematical passage
\mathbb E_{r\sim Q} \norm{P_{(I\oplus W)^\perp}r}^2
P_*Use in source · 4 passages
Search aliases: P_*
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12421
[Fixed-packet insufficiency under hidden rotation] Let P_0 be a fixed rank-r packet and P_* a target-specific rank-r context packet. Suppose the chart label c(h) is measurable from P_*h but is not almost surely measurable from P_0h. Then no separator depending only on P_0h can reproduce c(h) almost surely. In particular, a target-specific rotation that moves im P_* away from im P_0 creates an irreducible fixed-packet chart error.
Exact mathematical passage
P_*
p_0Use in source · 2 passages
Search aliases: p_0
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10833
Let p^⋆ be the best conditioning policy and p_0 the best inherited-policy configuration. Empirical errors of at most η imply widehatμ(p^⋆) ≥ μ(p^⋆)-η > μ(p_0)+η ≥ widehatμ(p_0).
Exact mathematical passage
p_0
P_1Use in source · 9 passages
Search aliases: P_1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12187
[Exact response-signature atlas recovery] Assume the block-scalar response model and separating signatures. Then every hidden projector P_a belongs to A_R. Consequently, the nonzero minimal central projections of A_R are exactly P_1,…,P_m, up to permutation.
Exact mathematical passage
P_1,\ldots,P_m
p_1Use in source · 3 passages
Search aliases: p_1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16642
bx+c, · x<0. The coefficients belong to finite declared integer sets. A task contains five shuffled input--output demonstrations and one query input. The model receives a normalized serialization of those demonstrations and the query. The protected target is y(τ) = ( e_θ, fracp_12,…,fracp_42, fracd(θ)5, fracf_θ,p(x_q)32 )∈ℝ^10, where e_θ∈ℝ^4 is the one-hot template code, unused parameter slots are zero, and d(θ)∈{2,3,4,5} is the declared execution depth. All demonstrations, query answers, semant…
Exact mathematical passage
y(\tau) = \left( e_{\theta}, \frac{p_1}{2},\ldots,\frac{p_4}{2}, \frac{d(\theta)}{5}, \frac{f_{\theta,p}(x_q)}{32} \right)\in\R^{10}, \label{eq:b18-protected-target}
P_{12}Use in source · 1 passage
Search aliases: P_12
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 22003
Let R_1=P_1+P_2+3P_3, R_2=2P_1+2P_2-P_3. Blocks 1 and 2 have the same signature (1,2). Every polynomial in R_1,R_2 acts with the same scalar on H_1 and H_2. The finest response-algebra projector is therefore P_12=P_1+P_2, not the pair P_1,P_2. A compiler that outputs two separate blocks from these interventions has invented information absent from its evidence. The Block 11 stress witness instead abstains.
Exact mathematical passage
P_{12}=P_1+P_2,
P_2Use in source · 8 passages
Search aliases: P_2
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21704
ponse operators R_1=X⊕0, R_2=Z⊕0, R_3=0⊕ X, R_4=0⊕ Z. The adjoint-closed algebra they generate is M_2(ℝ)⊕ M_2(ℝ), with dimension 8. Its commutant consists of scalar multiples of the identity on each block and therefore has dimension 2. The minimal central projectors are P_1=I_2⊕0, P_2=0⊕ I_2. A residual e_1 generates the first two-dimensional block. A residual e_1+e_3 generates both blocks and has cyclic dimension 4.
Exact mathematical passage
P_1=I_2\oplus0, \qquad P_2=0\oplus I_2.
p_2Use in source · 1 passage
Search aliases: p_2
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21666
Consider the typed repair programs p_1(a,b) · =(0,0), p_2(a,b) · =(a,0), p_3(a,b) · =(0,b), p_4(a,b) · =(b,a), p_5(a,b) · =(a,b). The complete finite verifier checks all eight episodes. Programs p_1 through p_4 each receive a counterexample and are eliminated. Program p_5 passes every episode. The loop terminates after four rejections with a verified repair.
Exact mathematical passage
\begin{aligned} p_1(a,b)&=(0,0),\\ p_2(a,b)&=(a,0),\\ p_3(a,b)&=(0,b),\\ p_4(a,b)&=(b,a),\\ p_5(a,b)&=(a,b). \end{aligned}
P_3Use in source · 5 passages
Search aliases: P_3
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21955
Let R_1=P_1+2P_2+4P_3, R_2=3P_1-P_2+2P_3, where P_1,P_2,P_3 are pairwise orthogonal rank-two projectors. Take B=R_1+2R_2=7P_1+0P_2+8P_3. The three eigenvalues 7,0,8 are distinct. Therefore P_1 · =fracB(B-8I)7(7-8),
Exact mathematical passage
R_1=P_1+2P_2+4P_3, \qquad R_2=3P_1-P_2+2P_3,
p_3Use in source · 1 passage
Search aliases: p_3
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21666
Consider the typed repair programs p_1(a,b) · =(0,0), p_2(a,b) · =(a,0), p_3(a,b) · =(0,b), p_4(a,b) · =(b,a), p_5(a,b) · =(a,b). The complete finite verifier checks all eight episodes. Programs p_1 through p_4 each receive a counterexample and are eliminated. Program p_5 passes every episode. The loop terminates after four rejections with a verified repair.
Exact mathematical passage
\begin{aligned} p_1(a,b)&=(0,0),\\ p_2(a,b)&=(a,0),\\ p_3(a,b)&=(0,b),\\ p_4(a,b)&=(b,a),\\ p_5(a,b)&=(a,b). \end{aligned}
p_4Use in source · 3 passages
Search aliases: p_4
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16642
bx+c, · x<0. The coefficients belong to finite declared integer sets. A task contains five shuffled input--output demonstrations and one query input. The model receives a normalized serialization of those demonstrations and the query. The protected target is y(τ) = ( e_θ, fracp_12,…,fracp_42, fracd(θ)5, fracf_θ,p(x_q)32 )∈ℝ^10, where e_θ∈ℝ^4 is the one-hot template code, unused parameter slots are zero, and d(θ)∈{2,3,4,5} is the declared execution depth. All demonstrations, query answers, semant…
Exact mathematical passage
y(\tau) = \left( e_{\theta}, \frac{p_1}{2},\ldots,\frac{p_4}{2}, \frac{d(\theta)}{5}, \frac{f_{\theta,p}(x_q)}{32} \right)\in\R^{10}, \label{eq:b18-protected-target}
p_5Use in source · 3 passages
Search aliases: p_5
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21666
Consider the typed repair programs p_1(a,b) · =(0,0), p_2(a,b) · =(a,0), p_3(a,b) · =(0,b), p_4(a,b) · =(b,a), p_5(a,b) · =(a,b). The complete finite verifier checks all eight episodes. Programs p_1 through p_4 each receive a counterexample and are eliminated. Program p_5 passes every episode. The loop terminates after four rejections with a verified repair.
Exact mathematical passage
\begin{aligned} p_1(a,b)&=(0,0),\\ p_2(a,b)&=(a,0),\\ p_3(a,b)&=(0,b),\\ p_4(a,b)&=(b,a),\\ p_5(a,b)&=(a,b). \end{aligned}
P_AUse in source · 4 passages
Search aliases: P_A
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3266
The deterministic theorem compares declared defects. A deployed repair is usually evaluated across a distribution of contexts. Let P_A and P_B be source and target context distributions, let S:X_A→ X_B be a declared context transport, and let q_A,q_B be per-context repair gains.
Exact mathematical passage
P_A
p_aUse in source · 4 passages
Search aliases: p_a
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12206
of hyperplanes setc:c^ T(λ_a-λ_b)=0, a≠ b. Then the scalars μ_a=c^ Tλ_a are pairwise distinct. The combined response B=∑_j=1^kc_jR_j =∑_a=1^mμ_aP_a belongs to A_R. For each a, define the Lagrange polynomial p_a(z)=∏_b≠ afracz-μ_bμ_a-μ_b. Functional calculus gives p_a(B)=∑_b=1^mp_a(μ_b)P_b=P_a. Thus every P_a lies in the generated algebra. A nonzero central projection that is strictly below one P_a would have to be scalar on H_a and therefore is either 0 or P_a. Hence the P_a are the minimal cent…
Exact mathematical passage
p_a(z)=\prod_{b\neq a}\frac{z-\mu_b}{\mu_a-\mu_b}.
P_BUse in source · 3 passages
Search aliases: P_B
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3266
The deterministic theorem compares declared defects. A deployed repair is usually evaluated across a distribution of contexts. Let P_A and P_B be source and target context distributions, let S:X_A→ X_B be a declared context transport, and let q_A,q_B be per-context repair gains.
Exact mathematical passage
P_B
P_bUse in source · 8 passages
Search aliases: P_b
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12148
Let H be a finite-dimensional real or complex Hilbert space. Suppose the unknown functional decomposition is H=⊕_a=1^mH_a, P_a:H→ H_a, with pairwise orthogonal projectors satisfying P_aP_b=δ_abP_a, ∑_a=1^mP_a=I. For controlled interventions u_1,…,u_k, let R_j∈End(H), j=1,…,k, be their finite-difference or linearized response operators.
Exact mathematical passage
P_aP_b=\delta_{ab}P_a, \qquad \sum_{a=1}^{m}P_a=I.
P_CUse in source · 2 passages
Search aliases: P_C
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2131
eduction] Let F:P× U→ V be C^k, k≥2, between finite-dimensional spaces, with F(0,0)=0. Let D:=∂_uF(0,0). Choose splittings U=K⊕ X, K=ker D, and V=R⊕ C, R=im D, so that restrDX:X→ R is an isomorphism. Let P_R and P_C be the corresponding projections. Then there are neighborhoods of the origin and a unique C^k map χ:P× K→ X, χ(0,0)=0, such that P_RF(p,k+χ(p,k))=0. Define the reduced obstruction map ψ(p,k):=P_CF(p,k+χ(p,k)). A sufficiently small pair (p,u) solves F(p,u)=0 if and only if u=k+χ(p,k) …
Exact mathematical passage
P_C
p_iUse in source · 4 passages
Search aliases: p_i
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16224
r_i · =absℓ_i(replay(Z_i))-ℓ_i(Z_i). Noise is deterministic and norm matched at every step. The program and answer specificity coordinates are the mean recipient costs under SAP and FAD donors: p_i = E_j∈SAP(i) [ℓ_i(Z_j)-ℓ_i(Z_i)], a_i = E_j∈FAD(i) [ℓ_i(Z_j)-ℓ_i(Z_i)]. Define Q_prog = fracoverline pmax{overline o,overline c,ε}, Q_ans = fracoverline amax{overline o,overline c,ε}. The frozen ratio thresholds are 0.15 and 0.20, respectively. Each specificity gate additionally requires a positive 95…
Exact mathematical passage
p_i = \mathbb E_{j\in\mathsf{SAP}(i)} \left[\ell_i(Z_j)-\ell_i(Z_i)\right], \qquad a_i = \mathbb E_{j\in\mathsf{FAD}(i)} \left[\ell_i(Z_j)-\ell_i(Z_i)\right]. \label{eq:b17-specificity-costs}
P_{I^\perp}Use in source · 5 passages
Search aliases: P_I^⊥ perp
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9736
Let I=im D⊆ V be the current authorized correction image and let z=P_I^⊥r be the unresolved component of defect r. Suppose a candidate grammar extension contributes a response vector v∈ V. Only its component outside the old image is structurally new: bar v=P_I^⊥v. The expanded image is I'=I+span{v}.
Exact mathematical passage
z=P_{I^\perp}r
P_{I_t}Use in source · 1 passage
Search aliases: P_I_t
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9541
Nestedness gives I_t⊆ I_t+1, so distance to I_t+1 cannot exceed distance to I_t. Let I_∞=overline⋃_t I_t. The projections P_I_tr converge strongly to P_I_∞r in finite dimension, so the orthogonal residual norms converge to the distance from -r to I_∞.
Exact mathematical passage
P_{I_t}r
P_{I_{t+1}^\perp}Use in source · 1 passage
Search aliases: P_I_t+1^⊥ perp
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9565
The nested orthogonal decomposition is I_t+1 = I_t⊕(I_t+1∩ I_t^⊥). Therefore P_I_t+1^⊥r = P_I_t^⊥r - P_I_t+1∩ I_t^⊥r, and the two terms on the right are orthogonal. The Pythagorean identity gives [source reference].
Exact mathematical passage
P_{I_{t+1}^\perp}r = P_{I_t^\perp}r - P_{I_{t+1}\cap I_t^\perp}r,
P_{I_{t+1}\cap I_t^\perp}Use in source · 3 passages
Search aliases: P_I_t+1∩ I_t^⊥ cap perp
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9549
[Newly correctable energy identity] For nested correction images, η_t(r)^2-η_t+1(r)^2 = normP_I_t+1∩ I_t^⊥r^2.
Exact mathematical passage
\eta_t(r)^2-\eta_{t+1}(r)^2 = \norm{P_{I_{t+1}\cap I_t^\perp}r}^2. \label{eq:b8-new-energy}
P_{I_t^\perp}Use in source · 2 passages
Search aliases: P_I_t^⊥ perp
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9516
Let V be a finite-dimensional defect Hilbert space. Suppose a conservative grammar chain gives nested closed authorized correction images I_0⊆ I_1⊆ I_2⊆⋯⊆ V, I_t=im D_t. For r∈ V, define unresolved obstruction magnitude η_t(r) = dist(-r,I_t) = normP_I_t^⊥r.
Exact mathematical passage
\eta_t(r) = \dist(-r,I_t) = \norm{P_{I_t^\perp}r}. \label{eq:b8-obstruction-magnitude}
P_{I_\infty}Use in source · 1 passage
Search aliases: P_I_∞ infty
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9541
Nestedness gives I_t⊆ I_t+1, so distance to I_t+1 cannot exceed distance to I_t. Let I_∞=overline⋃_t I_t. The projections P_I_tr converge strongly to P_I_∞r in finite dimension, so the orthogonal residual norms converge to the distance from -r to I_∞.
Exact mathematical passage
P_{I_\infty}r
P_{\im D_{\mathrm{loc}}}Use in source · 1 passage
Search aliases: P_im D_loc im mathrm operatorname mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5572
1 . Then P_im D_locr_loc=0, (I-P_im D_loc)r_loc=r_loc, so the obstruction norm is one.
Exact mathematical passage
P_{\im D_{\mathrm{loc}}}r_{\mathrm{loc}}=0, \qquad (I-P_{\im D_{\mathrm{loc}}})r_{\mathrm{loc}}=r_{\mathrm{loc}},
P_{\ker B^\top}Use in source · 1 passage
Search aliases: P_ker B^top ker top
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5635
The executable classifier applies the ordered rule Class(r,b,H) = LOCAL_OBSTRUCTION, · normP_(im D)^⊥r>τ, GLUING_OBSTRUCTION, · normP_ker B^topb>τ, EVALUATOR_HOLONOMY, · norm(H-I)e>τ, FLAT_REPAIR, · otherwise. The order is part of the receipt because a case can exhibit more than one defect. The fixture isolates the four modes, so the ordered rule is unambiguous.
Exact mathematical passage
\operatorname{Class}(r,b,H) = \begin{cases} \texttt{LOCAL\_OBSTRUCTION},&\norm{P_{(\im D)^\perp}r}>\tau,\\ \texttt{GLUING\_OBSTRUCTION},&\norm{P_{\ker B^\top}b}>\tau,\\ \texttt{EVALUATOR\_HOLONOMY},&\norm{(H-I)e}>\tau,\\ \texttt{FLAT\_REPAIR},&\text{otherwise}. \end{cases}
P_mUse in source · 1 passage
Search aliases: P_m
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12187
[Exact response-signature atlas recovery] Assume the block-scalar response model and separating signatures. Then every hidden projector P_a belongs to A_R. Consequently, the nonzero minimal central projections of A_R are exactly P_1,…,P_m, up to permutation.
Exact mathematical passage
P_1,\ldots,P_m
p_{\mathrm{prov}}Use in source · 2 passages
Search aliases: p_prov mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5755
An executable state is a finite canonical map widehat s:Key→Value. A witness policy is widehat W =(K_protected,γ_min,r_max,p_prov,e_irr,v_W), where K_protected is the immutable-key set, γ_min is the minimum protected gain, r_max is the maximum accepted risk, p_prov indicates whether provenance is mandatory, e_irr controls irreversible effects, and v_W is the witness version.
Exact mathematical passage
\widehat W =(K_{\mathrm{protected}},\gamma_{\min},r_{\max},p_{\mathrm{prov}},e_{\mathrm{irr}},v_W),
P_RUse in source · 3 passages
Search aliases: P_R
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2131
t repair reduction] Let F:P× U→ V be C^k, k≥2, between finite-dimensional spaces, with F(0,0)=0. Let D:=∂_uF(0,0). Choose splittings U=K⊕ X, K=ker D, and V=R⊕ C, R=im D, so that restrDX:X→ R is an isomorphism. Let P_R and P_C be the corresponding projections. Then there are neighborhoods of the origin and a unique C^k map χ:P× K→ X, χ(0,0)=0, such that P_RF(p,k+χ(p,k))=0. Define the reduced obstruction map ψ(p,k):=P_CF(p,k+χ(p,k)). A sufficiently small pair (p,u) solves F(p,u)=0 if and only if u…
Exact mathematical passage
P_R
p_{\mathrm{rank}}Use in source · 2 passages
Search aliases: p_rank mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21527
Suppose bridge-rank identification has success probability p_rank(K_1)=1. For every child that changes only the rank-identification rule while leaving all other modules fixed, p_rank(K_2)-p_rank(K_1)≤0. This does not imply that the repair process has no remaining headroom. Let the target bridge be full rank but have minimum singular value σ_min=10^-4. Random noise of norm 10^-3 can then dominate the weakest identified direction. The bottleneck has migrated from rank to conditioning. A meaningful…
Exact mathematical passage
p_{\mathrm{rank}}(K_1)=1.
p_{\rm sign}Use in source · 1 passage
Search aliases: p_rm sign rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7998
On the architecture-foreign split, the child's paired advantage over raw calibration is 0.017989, CI_95%=[0.003477,0.036611], p_rm sign=0.00940. On the seed-and-task audit, the corresponding interval includes zero. The result therefore supports robust improvement over the frozen parent on both audits and a smaller architecture-foreign improvement over a strong ordinary selector.
Exact mathematical passage
0.017989, \qquad \mathrm{CI}_{95\%}=[0.003477,0.036611], \qquad p_{\rm sign}=0.00940.
P_sUse in source · 2 passages
Search aliases: P_s
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16827
Let H_s(τ)∈ℝ^36 be the source model's final latent trace state and H_t(τ)∈ℝ^36 the corresponding foreign-child state. A ridge-affine bridge is fitted on the 900 paired generator-A anchors: B_t = arg min_B,b ∑_i=1^900 normH_s(τ_i)-BH_t(τ_i)-b^2 +λnormB^2. A source ridge probe P_s is fitted independently on the 1,000 probe-calibration tasks. The transported semantic prediction is widehat Y_t(τ) =P_s(B_tH_t(τ)+b_t). The bridge gate requires effective rank 36, transported R^2≥0.35, an advantage of a…
Exact mathematical passage
P_s
P_tUse in source · 7 passages
Search aliases: P_t
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4770
Let V be a Hilbert defect space, let L_t=im D_t^auth, and let P_t be the orthogonal projector onto L_t. For a defect distribution μ, define the correctable energy E_t^cor(μ) = E_r∼μnormP_t r^2 and the obstruction energy E_t^ob(μ) = E_r∼μnorm(I-P_t)r^2.
Exact mathematical passage
P_t
P_{t+1}Use in source · 2 passages
Search aliases: P_t+1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4806
Nested subspaces give the orthogonal decomposition L_t+1=L_t⊕ W, P_t+1=P_t+P_W. The two projected components are orthogonal, so normP_t+1r^2 = normP_tr^2+normP_Wr^2. The obstruction identity follows from Pythagoras. Under exact correction, the post-repair residual is (I-P_t)r, so loss reduction is normP_tr^2.
Exact mathematical passage
L_{t+1}=L_t\oplus W, \qquad P_{t+1}=P_t+P_W.
P_t^{\mathrm{discovery}}Use in source · 2 passages
Search aliases: P_t^discovery mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9410
ive embedding of old meta-regret coordinates. Decompose r_t+1 = E_twidetilde r_t+1+n_t+1, n_t+1⊥im E_t. Define old-coordinate repair progress by P_t^repair = normr_t^2-normwidetilde r_t+1^2, and epistemic discovery by P_t^discovery = normn_t+1^2. Both quantities may be positive. The total represented regret may increase because the lineage learned to represent a previously invisible limitation.
Exact mathematical passage
P_t^{\mathrm{discovery}} = \norm{n_{t+1}}^2. \label{eq:b8-discovery-progress}
P_t^{\mathrm{repair}}Use in source · 2 passages
Search aliases: P_t^repair mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9403
eal error that did not exist as a coordinate in the old state. Let E_t:ℝ^m_t↪ℝ^m_t+1 be the conservative embedding of old meta-regret coordinates. Decompose r_t+1 = E_twidetilde r_t+1+n_t+1, n_t+1⊥im E_t. Define old-coordinate repair progress by P_t^repair = normr_t^2-normwidetilde r_t+1^2, and epistemic discovery by P_t^discovery = normn_t+1^2. Both quantities may be positive. The total represented regret may increase because the lineage learned to represent a previously invisible limitation.
Exact mathematical passage
P_t^{\mathrm{repair}} = \norm{r_t}^2-\norm{\widetilde r_{t+1}}^2, \label{eq:b8-repair-progress}
P_WUse in source · 9 passages
Search aliases: P_W
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4790
[Structural-to-behavioral identity] Assume L_t⊆ L_t+1 and let W=L_t+1∩ L_t^⊥. Then E_t+1^cor(μ)- E_t^cor(μ) = E_r∼μnormP_Wr^2 and E_t^ob(μ)- E_t+1^ob(μ) = E_r∼μnormP_Wr^2. Under quadratic residual loss and exact cancellation of the correctable component, this quantity is exactly the expected additional task-loss reduction made available by the expanded repair image.
Exact mathematical passage
\mathcal E_{t+1}^{\mathrm{cor}}(\mu)-\mathcal E_t^{\mathrm{cor}}(\mu) = \mathbb E_{r\sim\mu}\norm{P_Wr}^2
P_xUse in source · 3 passages
Search aliases: P_x
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15809
A coordinate-level decoder is not the identifiability target. Let P_x be a projector onto a trace packet discovered from declared interventions. Packet necessity and sufficiency are evaluated by removing P_xZ(x), retaining only P_xZ(x), and comparing equal-rank random and PCA controls. The packet is accepted only when its causal effect transfers to held-out tasks and survives within-packet basis changes.
Exact mathematical passage
P_x
P_\alphaUse in source · 3 passages
Search aliases: P_α alpha
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10071
dentification H ≅ ⊕_α=1^p ( C^m_α⊗ C^n_α ) under which A ≅ ⊕_α=1^p ( I_m_α⊗ M_n_α( C) ). The minimal central projections P_α∈ Z( A) give a canonical coarse causal atlas H = ⊕_α=1^pP_α H. The blocks are unique up to permutation and unitary changes of basis within multiplicity spaces.
Exact mathematical passage
P_\alpha\in Z(\mathfrak A)
p_\thetaUse in source · 2 passages
Search aliases: p_θ theta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13888
A simple acquisition rule is A_i = widehat g_i -λ_rwidehat r_i +λ_uwidehatσ_i +λ_h p_θ(h_imid x_τ) -λ_c C_i, where h_i is the method family, p_θ is an obstruction-class posterior, widehatσ_i is a declared uncertainty bonus, and C_i is experimental cost.
Exact mathematical passage
A_i = \widehat g_i -\lambda_r\widehat r_i +\lambda_u\widehat\sigma_i +\lambda_h p_\theta(h_i\mid x_\tau) -\lambda_c C_i, \label{eq:b13-acquisition}
\operatorname{Par}Use in source · 1 passage
Search aliases: Par operatorname
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13778
A finding is represented by f_i = (Π_i^-,Π_i^+,code_i, Par(i),e_i,o_i), where e_i is the observed effect profile and o_i is either a surviving obstruction or a validated no-go certificate. Directed edges are typed as extension, ablation, refutation, transport, or replication. A proposal must cite the archive nodes on which its mechanism depends.
Exact mathematical passage
f_i = (\Pi_i^-,\Pi_i^+,\mathrm{code}_i, \operatorname{Par}(i),e_i,o_i), \label{eq:b13-finding-node}
\mathrm{parent}Use in source · 2 passages
Search aliases: parent mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9484
Define compiler metaproductivity by C(Γ) = E_K∼D_parent [ M(Γ(K,E_K))-M(K) ]. A stronger recursive claim requires C(Γ_t+1)>C(Γ_t) on sealed foreign parent processes under matched compilation resources.
Exact mathematical passage
\Ccal(\Gamma) = \mathbb E_{K\sim\Dcal_{\mathrm{parent}}} \left[ \Mcal\bigl(\Gamma(K,\Ecal_K)\bigr)-\Mcal(K) \right]. \label{eq:b8-compiler-metaproductivity}
\mathrm{passive}Use in source · 1 passage
Search aliases: passive mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5953
From a common probe state x_0 and bounded action a, define the feature vector ϕ(z) = ϕ_passive ϕ_actuation ϕ_sensor ϕ_memory ϕ_evaluator . Its components are, respectively: item deviation from the baseline no-action continuation; item deviation of the action-conditioned response from the baseline action response; item violation of redundant-sensor consistency; item deviation from baseline memory update; item discrepancy between reported and protected evaluator scores. The observed probe is widet…
Exact mathematical passage
\phi(z) = \begin{bmatrix} \phi_{\mathrm{passive}}\\ \phi_{\mathrm{actuation}}\\ \phi_{\mathrm{sensor}}\\ \phi_{\mathrm{memory}}\\ \phi_{\mathrm{evaluator}} \end{bmatrix}.
\Pcal^+Use in source · 4 passages
Search aliases: Pcal^+ P^+ Pcal mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15115
Let p∈P^+ be a candidate experiment with response y_p:T⟶ℝ. For threshold θ, define the refined diagnostic map ϕ_p,θ(τ) = (ϕ_0(τ),low), · y_p(τ)≤θ,
Exact mathematical passage
p\in\Pcal^+
\Pcal_{\Gcal}Use in source · 4 passages
Search aliases: Pcal_Gcal P_G Pcal Gcal mathcal mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9988
Let P_G be a finite typed repair-program grammar and let Verify(p,x)∈{0,1} be a complete verifier on a finite declared episode domain.
Exact mathematical passage
\Pcal_{\Gcal}
\Pcal_\tauUse in source · 1 passage
Search aliases: Pcal_τ P_τ Pcal tau mathcal tau
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13678
Let T be a research-problem space. A problem τ∈T contains a baseline repair process K_τ, a declared candidate grammar P_τ, a cheap evidence channel, a scored evaluator portfolio, a hidden protected audit, integrity rules, and a resource budget B.
Exact mathematical passage
\Pcal_\tau
\mathsf{Penalty}Use in source · 1 passage
Search aliases: Penalty mathsf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16303
positive interval and fraction gates, local continuity, donor coverage, adaptive halting. Among eligible candidates it maximizes S(α) = · min{ fracR^2_probe0.65, fracQ_prog0.15, fracQ_ans0.20 } +0.20R^2_probe · -4 MSE_clean +0.02ρ_halt -Penalty_dynamic. The minimum term makes the weakest normalized semantic gate the principal bottleneck. The compiler seal commits the selected candidate, all candidate and parent state hashes, training receipts, meta-validation task digest, exact thresholds, selec…
Exact mathematical passage
\begin{split} S(\alpha) ={}& \min\left\{ \frac{R^2_{\mathrm{probe}}}{0.65}, \frac{Q_{\mathrm{prog}}}{0.15}, \frac{Q_{\mathrm{ans}}}{0.20} \right\} +0.20R^2_{\mathrm{probe}}\\ &-4\,\mathrm{MSE}_{\mathrm{clean}} +0.02\rho_{\mathrm{halt}} -\mathsf{Penalty}_{\mathrm{dynamic}}. \end{split} \label{eq:b17-selection-score}
\mathsf{Penalty}_{\mathrm{dynamic}}Use in source · 1 passage
Search aliases: Penalty_dynamic mathsf mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16303
positive interval and fraction gates, local continuity, donor coverage, adaptive halting. Among eligible candidates it maximizes S(α) = · min{ fracR^2_probe0.65, fracQ_prog0.15, fracQ_ans0.20 } +0.20R^2_probe · -4 MSE_clean +0.02ρ_halt -Penalty_dynamic. The minimum term makes the weakest normalized semantic gate the principal bottleneck. The compiler seal commits the selected candidate, all candidate and parent state hashes, training receipts, meta-validation task digest, exact thresholds, selec…
Exact mathematical passage
\begin{split} S(\alpha) ={}& \min\left\{ \frac{R^2_{\mathrm{probe}}}{0.65}, \frac{Q_{\mathrm{prog}}}{0.15}, \frac{Q_{\mathrm{ans}}}{0.20} \right\} +0.20R^2_{\mathrm{probe}}\\ &-4\,\mathrm{MSE}_{\mathrm{clean}} +0.02\rho_{\mathrm{halt}} -\mathsf{Penalty}_{\mathrm{dynamic}}. \end{split} \label{eq:b17-selection-score}
\mathrm{pkg}Use in source · 1 passage
Search aliases: pkg mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5524
G_pkg · Package integrity · Exact manifest, canonical digests, clean extraction, successful verifier
Exact mathematical passage
G_{\mathrm{pkg}}
\mathrm{planner}Use in source · 2 passages
Search aliases: planner mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 665
A representative color set is C= set model, workspace, memory, planner, improver, evaluator, provenance, tools . The atlas does not assert that these colors are universally correct. It declares the decomposition under which one particular repair theory is being tested.
Exact mathematical passage
\mathsf C= \set{ \mathrm{model}, \mathrm{workspace}, \mathrm{memory}, \mathrm{planner}, \mathrm{improver}, \mathrm{evaluator}, \mathrm{provenance}, \mathrm{tools} }.
\PostUse in source · 6 passages
Search aliases: Post operatorname
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4252
Let S be a finite set of complete lineage states. Write sxrightarrowτ_imps' when one authorized improvement transaction can take s to s'. Define Post_imp(s) = sets'∈ S:sxrightarrowτ_imps'.
Exact mathematical passage
\Post_{\mathrm{imp}}(s) = \set{s'\in S:s\xrightarrow{\tau_{\mathrm{imp}}}s'}.
\Post_{\mathrm{imp}}Use in source · 4 passages
Search aliases: Post_imp Post mathrm operatorname mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4252
Let S be a finite set of complete lineage states. Write sxrightarrowτ_imps' when one authorized improvement transaction can take s to s'. Define Post_imp(s) = sets'∈ S:sxrightarrowτ_imps'.
Exact mathematical passage
\Post_{\mathrm{imp}}(s) = \set{s'\in S:s\xrightarrow{\tau_{\mathrm{imp}}}s'}.
\PrUse in source · 22 passages
Search aliases: Pr
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3332
· -overline L( 2overlineε_r +overlineε_Dnormu_A +overlineε_leak +overlineε_hol ). Then Pr(g_B≥underline g_B)≥1-α. In particular, underline g_B>0 is a valid one-sided certificate of beneficial target transfer.
Exact mathematical passage
\Pr(g_B\geq\underline g_B)\geq1-\alpha.
\Pr_{A\sim\Dcal_{\rm for}}Use in source · 1 passage
Search aliases: Pr_A∼Dcal_rm for Pr_A∼D_rm for Pr sim Dcal rm Pr sim mathcal rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8720
Define the finite structural coverage functional C_rm id( P) = Pr_A∼D_rm for [ ank X_A=d_s]. Then widehat C_rm id( P_1) - widehat C_rm id( P_0) = 0.96875. This is the measured bridge-identification direction added by the meta-repair.
Exact mathematical passage
C_{\rm id}(\mathfrak P) = \Pr_{A\sim\Dcal_{\rm for}} \left[ ank X_A=d_s\right].
\precUse in source · 2 passages
Search aliases: prec
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9083
Choose an operational ordering r_1prec r_2prec⋯prec r_m, m=|N|, with nested sets R_0=∅, R_j={r_1,…,r_j}. Define the ordered marginal headroom Δ_j^ord(K) = F_K(R_j)-F_K(R_j-1).
Exact mathematical passage
r_1\prec r_2\prec\cdots\prec r_m, \qquad m=|N|,
\mathrm{prepare}Use in source · 1 passage
Search aliases: prepare mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4027
The transaction follows six phases: prepare ⟶ seal ⟶ execute ⟶ reveal ⟶ decide ⟶ finalize. Only the final phase may expose buffered effects or replace the active checkpoint.
Exact mathematical passage
\mathrm{prepare} \longrightarrow \mathrm{seal} \longrightarrow \mathrm{execute} \longrightarrow \mathrm{reveal} \longrightarrow \mathrm{decide} \longrightarrow \mathrm{finalize}.
\mathrm{preserve}Use in source · 1 passage
Search aliases: preserve mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17934
A candidate is pre-audit eligible only if both foreign target models are ATLAS_FULL and the candidate jointly satisfies R^2_transport · ≥0.75, · R^2_worst chart · ≥0.74, ρ_preserve · ≥0.98, · NRMSE_trace · ≤0.05, κ_square · ≤0.02. · · The selection score is the weakest normalized margin across these gates. This prevents a strong aggregate R^2 from compensating for a failed chart, high curvature, or poor trajectory reconstruction.
Exact mathematical passage
\begin{aligned} R^2_{\mathrm{transport}}&\geq0.75, &\qquad R^2_{\mathrm{worst\ chart}}&\geq0.74,\\ \rho_{\mathrm{preserve}}&\geq0.98, & \operatorname{NRMSE}_{\mathrm{trace}}&\leq0.05,\\ \kappa_{\mathrm{square}}&\leq0.02. && \end{aligned} \label{eq:b20-thresholds}
\mathrm{primary}Use in source · 2 passages
Search aliases: primary mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 14136
The primary paired effect is widehatΔ_Γ^primary = 0.067227, CI_95% = [0.035881,0.103274]. The sign-flip estimate is 9.9999×10^-6. The child improves 21.875% of pairs, ties on 75%, and loses on 3.125%.
Exact mathematical passage
\widehat\Delta_{\Gamma}^{\mathrm{primary}} = 0.067227, \qquad \mathrm{CI}_{95\%} = [0.035881,0.103274]. \label{eq:b13-primary-effect}
\mathrm{program}Use in source · 1 passage
Search aliases: program mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17488
Paired effects are: widehatΔ_clean^A · =0.142631, · 95% CI · =[0.135421,0.149531], widehatΔ_clean^N · =0.365616, · 95% CI · =[0.352991,0.378402], widehatΔ_program^A · =0.120614, · 95% CI · =[0.110353,0.131880], widehatΔ_program^N · =0.125290, · 95% CI · =[0.111056,0.138921]. The clean effects are parent minus child; the program effects are child minus parent. Every interval is frozen and strictly positive.
Exact mathematical passage
\widehat\Delta_{\mathrm{clean}}^{A} &=0.142631, &95\%\ \mathrm{CI}&=[0.135421,0.149531],\\ \widehat\Delta_{\mathrm{clean}}^{N} &=0.365616, &95\%\ \mathrm{CI}&=[0.352991,0.378402],\\ \widehat\Delta_{\mathrm{program}}^{A} &=0.120614, &95\%\ \mathrm{CI}&=[0.110353,0.131880],\\ \widehat\Delta_{\mathrm{program}}^{N} &=0.125290, &95\%\ \mathrm{CI}&=[0.111056,0.138921]. \label{eq:b19-paired-effects}
\mathrm{prov}Use in source · 2 passages
Search aliases: prov mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5755
An executable state is a finite canonical map widehat s:Key→Value. A witness policy is widehat W =(K_protected,γ_min,r_max,p_prov,e_irr,v_W), where K_protected is the immutable-key set, γ_min is the minimum protected gain, r_max is the maximum accepted risk, p_prov indicates whether provenance is mandatory, e_irr controls irreversible effects, and v_W is the witness version.
Exact mathematical passage
\widehat W =(K_{\mathrm{protected}},\gamma_{\min},r_{\max},p_{\mathrm{prov}},e_{\mathrm{irr}},v_W),
\mathrm{provenance}Use in source · 3 passages
Search aliases: provenance mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 665
A representative color set is C= set model, workspace, memory, planner, improver, evaluator, provenance, tools . The atlas does not assert that these colors are universally correct. It declares the decomposition under which one particular repair theory is being tested.
Exact mathematical passage
\mathsf C= \set{ \mathrm{model}, \mathrm{workspace}, \mathrm{memory}, \mathrm{planner}, \mathrm{improver}, \mathrm{evaluator}, \mathrm{provenance}, \mathrm{tools} }.
\ProvValidUse in source · 1 passage
Search aliases: ProvValid mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4259
Let C_RI(s) = · Grounded(s) wedgeProvValid(s) wedgeWorkspaceBound(s) · wedgeRepairTyped(s) wedgeWitnessed(s) wedgeEvidenceFlat(s) · wedgeRollbackable(s) wedgeMetaCapable(s). For X⊆ S, define Ψ_∀(X) = sets∈ S:C_RI(s) and Post_imp(s)⊆ X.
Exact mathematical passage
\begin{aligned} C_{\mathrm{RI}}(s) ={}& \Grounded(s) \wedge\ProvValid(s) \wedge\WorkspaceBound(s)\\ &\wedge\RepairTyped(s) \wedge\Witnessed(s) \wedge\EvidenceFlat(s)\\ &\wedge\Rollbackable(s) \wedge\MetaCapable(s). \end{aligned}
Q^{\mathsf T}Use in source · 1 passage
Search aliases: Q^ T mathsf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15826
The objective has value zero at Q. Full column rank implies that XT=Y=XQ only when T=Q. Orthogonality gives QQ^ T=I.
Exact mathematical passage
QQ^{\mathsf T}=I
q^{-1}Use in source · 6 passages
Search aliases: q^-1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3069
[Path independence on the flat repair core] Let p and q be two exact invertible transport paths from A to B. If u∈ U_A is fixed by the loop holonomy associated with q^-1p, then T_U^p u=T_U^q u. In particular, every u∈ U_A^flat has path-independent transport to B.
Exact mathematical passage
q^{-1}p
q^2Use in source · 2 passages
Search aliases: q^2
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20701
q · 0 , 0<q<1, so Q_0=I_2 and normEQ_0=q. Then D^-1=(I_2+E)^-1 =frac11-q^2 1 · -q -q · 1 . For r=e_1, the exact repair is u^*=-D^-1r =-frac11-q^2 1
Exact mathematical passage
D^{-1}=(I_2+E)^{-1} =\frac{1}{1-q^2} \begin{bmatrix} 1&-q\\-q&1 \end{bmatrix}.
q^kUse in source · 1 passage
Search aliases: q^k
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2027
u-v (u,v∈ B_R^W), with q:=norm QL_N<1, and normQr+norm Qsup_u∈ B_R^WnormN(u)≤ R. Then there exists a unique u^*∈ B_R^W satisfying F(u^*)=0. The gauge-fixed iteration u_k+1=-Qr-QN(u_k) converges to u^* from every u_0∈ B_R^W, with normu_k-u^*≤fracq^k1-qnormu_1-u_0.
Exact mathematical passage
\norm{u_k-u^*}\leq\frac{q^k}{1-q}\norm{u_1-u_0}.
q^{n+1}Use in source · 1 passage
Search aliases: q^n+1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1959
ht inverse Q =Q_0(I_V+EQ_0)^-1 =Q_0∑_k=0^∞(-EQ_0)^k. Moreover, norm Q≤fracnormQ_01-q. For defect r, the truncated packet-correction series u^(n) :=-Q_0∑_k=0^n(-EQ_0)^kr has residual Du^(n)+r=(-EQ_0)^n+1r, and hence normDu^(n)+r≤ q^n+1norm r.
Exact mathematical passage
\norm{Du^{(n)}+r}\leq q^{n+1}\norm r.
q^tUse in source · 1 passage
Search aliases: q^t
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9293
Lyapunov contraction] Suppose there exist w∈ℝ_>0^7 and q∈[0,1) such that for every reachable r_t, the controller can choose an authorized action a_t satisfying w^top A_a_tr_t ≤ q w^top r_t. Then every controlled trajectory satisfying [source reference] obeys w^top r_t ≤ q^t w^top r_0 + ∑_k=0^t-1 q^t-1-kw^top e_k.
Exact mathematical passage
w^\top r_t \leq q^t w^\top r_0 + \sum_{k=0}^{t-1} q^{t-1-k}w^\top e_k. \label{eq:b8-lyapunov-bound}
q^{t-1-k}Use in source · 1 passage
Search aliases: q^t-1-k
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9293
Lyapunov contraction] Suppose there exist w∈ℝ_>0^7 and q∈[0,1) such that for every reachable r_t, the controller can choose an authorized action a_t satisfying w^top A_a_tr_t ≤ q w^top r_t. Then every controlled trajectory satisfying [source reference] obeys w^top r_t ≤ q^t w^top r_0 + ∑_k=0^t-1 q^t-1-kw^top e_k.
Exact mathematical passage
w^\top r_t \leq q^t w^\top r_0 + \sum_{k=0}^{t-1} q^{t-1-k}w^\top e_k. \label{eq:b8-lyapunov-bound}
Q^\topUse in source · 2 passages
Search aliases: Q^top top
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13123
[Within-packet gauge invariance] If packet coordinates change by an orthogonal basis transformation z'=Q^top z, then the class of functions in [source reference] is unchanged.
Exact mathematical passage
z'=Q^\top z
Q_1Use in source · 3 passages
Search aliases: Q_1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4221
For quorums Q_1,Q_2, absQ_1∩ Q_2 ≥ 2q-m > f. At most f replicas are compromised, so at least one member of the intersection is uncompromised.
Exact mathematical passage
Q_1,Q_2
q_1Use in source · 3 passages
Search aliases: q_1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 19891
[Semantic underidentification] Suppose two semantic maps q_1,q_2:H→ Z induce exactly the same compiler-visible future and intervention laws, but there exists a protected witness W such that W(q_1(h))≠ W(q_2(h)) with positive probability. No compiler measurable with respect to the visible evidence can validly choose between q_1 and q_2 with probability exceeding the larger prior probability of the two alternatives.
Exact mathematical passage
q_1,q_2:H\to Z
Q_2Use in source · 2 passages
Search aliases: Q_2
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4221
For quorums Q_1,Q_2, absQ_1∩ Q_2 ≥ 2q-m > f. At most f replicas are compromised, so at least one member of the intersection is uncompromised.
Exact mathematical passage
Q_1,Q_2
q_2Use in source · 3 passages
Search aliases: q_2
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 19891
[Semantic underidentification] Suppose two semantic maps q_1,q_2:H→ Z induce exactly the same compiler-visible future and intervention laws, but there exists a protected witness W such that W(q_1(h))≠ W(q_2(h)) with positive probability. No compiler measurable with respect to the visible evidence can validly choose between q_1 and q_2 with probability exceeding the larger prior probability of the two alternatives.
Exact mathematical passage
q_1,q_2:H\to Z
Q_aUse in source · 2 passages
Search aliases: Q_a
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 19611
The executable fixture uses five commuting symmetric response operators. In latent semantic coordinates they are diagonal: D_u=diag(λ_u1,…,λ_ud), and each system exposes an orthogonal gauge Q_a: R_u^a=Q_aD_uQ_a^top. The joint response signature of semantic coordinate j is λ_j=(λ_1j,…,λ_mj).
Exact mathematical passage
Q_a
q_AUse in source · 4 passages
Search aliases: q_A
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3266
The deterministic theorem compares declared defects. A deployed repair is usually evaluated across a distribution of contexts. Let P_A and P_B be source and target context distributions, let S:X_A→ X_B be a declared context transport, and let q_A,q_B be per-context repair gains.
Exact mathematical passage
q_A,q_B
Q_a^\topUse in source · 1 passage
Search aliases: Q_a^top top
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 19612
The executable fixture uses five commuting symmetric response operators. In latent semantic coordinates they are diagonal: D_u=diag(λ_u1,…,λ_ud), and each system exposes an orthogonal gauge Q_a: R_u^a=Q_aD_uQ_a^top. The joint response signature of semantic coordinate j is λ_j=(λ_1j,…,λ_mj).
Exact mathematical passage
R_u^a=Q_aD_uQ_a^\top.
Q_{\mathrm{ans}}Use in source · 7 passages
Search aliases: Q_ans mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16237
hed at every step. The program and answer specificity coordinates are the mean recipient costs under SAP and FAD donors: p_i = E_j∈SAP(i) [ℓ_i(Z_j)-ℓ_i(Z_i)], a_i = E_j∈FAD(i) [ℓ_i(Z_j)-ℓ_i(Z_i)]. Define Q_prog = fracoverline pmax{overline o,overline c,ε}, Q_ans = fracoverline amax{overline o,overline c,ε}. The frozen ratio thresholds are 0.15 and 0.20, respectively. Each specificity gate additionally requires a positive 95% bootstrap lower endpoint and a positive-effect fraction of at least 0.5…
Exact mathematical passage
Q_{\mathrm{prog}} = \frac{\overline p}{\max\{\overline o,\overline c,\eps\}}, \qquad Q_{\mathrm{ans}} = \frac{\overline a}{\max\{\overline o,\overline c,\eps\}}. \label{eq:b17-specificity-ratios}
q_BUse in source · 5 passages
Search aliases: q_B
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3266
The deterministic theorem compares declared defects. A deployed repair is usually evaluated across a distribution of contexts. Let P_A and P_B be source and target context distributions, let S:X_A→ X_B be a declared context transport, and let q_A,q_B be per-context repair gains.
Exact mathematical passage
q_A,q_B
q_cUse in source · 3 passages
Search aliases: q_c
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8567
For a candidate configuration c, the development compiler computes target-level selected gains and the score S_rm dev(c) = bar g_c -0.20s_c -2h_c +0.02q_c, where bar g_c is mean protected gain, s_c is the target-level standard deviation, h_c is harmful-target rate, and q_c is positive-target rate. This is a finite engineering objective, not a universal definition of evolvability.
Exact mathematical passage
S_{\rm dev}(c) = \bar g_c -0.20s_c -2h_c +0.02q_c, \label{eq:block07q-compiler-score}
Q_{\mathrm{foreign}}Use in source · 1 passage
Search aliases: Q_foreign mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9978
A structural metaproductivity claim additionally requires E_t+1(Q_foreign) < E_t(Q_foreign) and a causal ablation showing that the new schema is responsible for the reduction.
Exact mathematical passage
E_{t+1}(Q_{\mathrm{foreign}}) < E_t(Q_{\mathrm{foreign}}) \label{eq:b8-structural-gate}
Q_iUse in source · 3 passages
Search aliases: Q_i
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1918
Let U=⊕_i=1^m U_i, V=⊕_i=1^m V_i, with orthogonal projectors Q_i and P_i. Decompose D=D_0+E, D_0:=∑_i=1^m P_iDQ_i. The block-diagonal operator D_0 represents ideal packet-local correction; E contains all cross-packet effects.
Exact mathematical passage
Q_i
q_iUse in source · 1 passage
Search aliases: q_i
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4527
[Finite ε-Pareto cover] Let Z⊆[0,1]^d be finite and let 0<ε≤1. There exists a subset Q⊆ Z with absQ ≤ lceilfrac1εrceil^d such that for every z∈ Z, some q∈ Q satisfies q_i≥ z_i-ε for every i.
Exact mathematical passage
q_i\geq z_i-\eps \qquad \text{for every }i.
q_jUse in source · 3 passages
Search aliases: q_j
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13798
Let the scored evaluator portfolio be B = {b_1,…,b_m}. For method c, define normalized headroom closure q_j(c) = fracs_j(c)-s_j(K_τ)s_j^⋆-s_j(K_τ). Let I_+(c)={j:q_j(c)>0}. The coverage-weighted geometric score is S_cov(c) = frac|I_+(c)|m ( ∏_j∈ I_+(c)q_j(c) )^1/|I_+(c)|, with value zero when I_+(c)=∅. Hard capability and integrity gates are applied separately rather than traded against this scalar.
Exact mathematical passage
q_j(c) = \frac{s_j(c)-s_j(K_\tau)}{s_j^{\star}-s_j(K_\tau)}. \label{eq:b13-headroom-closure}
Q_{j-1}Use in source · 2 passages
Search aliases: Q_j-1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8519
The descendant policy uses the source intervention-defined packet to divide the unlabeled source-side candidate pool into high- and low-packet-activation strata. It selects eight anchors from each stratum. Within each stratum it applies a deterministic pivoted residual rule: i_j ∈ arg max_i∉ I_j-1 norm(I-Q_j-1Q_j-1^top)z_i^2, where z_i is the standardized source activation and Q_j-1 spans the previously selected residual directions.
Exact mathematical passage
i_j \in \argmax_{i\notin I_{j-1}} \norm{(I-Q_{j-1}Q_{j-1}^\top)z_i}^2,
Q_{j-1}^\topUse in source · 1 passage
Search aliases: Q_j-1^top top
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8519
The descendant policy uses the source intervention-defined packet to divide the unlabeled source-side candidate pool into high- and low-packet-activation strata. It selects eight anchors from each stratum. Within each stratum it applies a deterministic pivoted residual rule: i_j ∈ arg max_i∉ I_j-1 norm(I-Q_j-1Q_j-1^top)z_i^2, where z_i is the standardized source activation and Q_j-1 spans the previously selected residual directions.
Exact mathematical passage
i_j \in \argmax_{i\notin I_{j-1}} \norm{(I-Q_{j-1}Q_{j-1}^\top)z_i}^2,
q_kUse in source · 1 passage
Search aliases: q_k
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16741
ited additive conditioning, final-step FiLM, progressive FiLM with geometry, a low-rank hypernetwork, and a shuffled-witness FiLM control. For the selected FiLM constructor, the encoded task produces step-indexed scale and shift vectors (γ_k(e),β_k(e))_k=0^4 =F_FiLM(e), which act on the preactivation by q_k ⟼ q_kodot(1+frac12tanhγ_k(e)) +β_k(e). This constructor is absent from Block 17's rank-one, rank-four, and full additive conditioning family because it provides step-specific multiplicative a…
Exact mathematical passage
q_k \longmapsto q_k\odot\left(1+\frac12\tanh\gamma_k(e)\right) +\beta_k(e). \label{eq:b18-film-action}
Q_mUse in source · 1 passage
Search aliases: Q_m
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5494
[Exhaustive finite endpoint theorem] Let M and widehat M be finite, exactly related, and let Q_1,…,Q_m be decidable state or transition predicates. Exhaustive enumeration of all reachable executable states and authorized edges decides whether each Q_i holds over the realized finite model.
Exact mathematical passage
Q_1,\ldots,Q_m
Q_{\mathrm{prog}}Use in source · 7 passages
Search aliases: Q_prog mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16237
hed at every step. The program and answer specificity coordinates are the mean recipient costs under SAP and FAD donors: p_i = E_j∈SAP(i) [ℓ_i(Z_j)-ℓ_i(Z_i)], a_i = E_j∈FAD(i) [ℓ_i(Z_j)-ℓ_i(Z_i)]. Define Q_prog = fracoverline pmax{overline o,overline c,ε}, Q_ans = fracoverline amax{overline o,overline c,ε}. The frozen ratio thresholds are 0.15 and 0.20, respectively. Each specificity gate additionally requires a positive 95% bootstrap lower endpoint and a positive-effect fraction of at least 0.5…
Exact mathematical passage
Q_{\mathrm{prog}} = \frac{\overline p}{\max\{\overline o,\overline c,\eps\}}, \qquad Q_{\mathrm{ans}} = \frac{\overline a}{\max\{\overline o,\overline c,\eps\}}. \label{eq:b17-specificity-ratios}
Q_{\rm ans}Use in source · 1 passage
Search aliases: Q_rm ans rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16926
The canonical final audits are: [H] centering small resizeboxtextwidth!% @llrrrrrrrl@ Architecture · Split · Clean · Program cost · Q_rm prog · Q_rm ans · Probe R^2 · Halt ρ · Execution · Status
Exact mathematical passage
Q_{\rm ans}
Q_{\rm prog}Use in source · 1 passage
Search aliases: Q_rm prog rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16926
The canonical final audits are: [H] centering small resizeboxtextwidth!% @llrrrrrrrl@ Architecture · Split · Clean · Program cost · Q_rm prog · Q_rm ans · Probe R^2 · Halt ρ · Execution · Status
Exact mathematical passage
Q_{\rm prog}
Q_sUse in source · 1 passage
Search aliases: Q_s
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 19625
[Proof sketch] A generic linear combination of the commuting operators has simple spectrum. Its eigenvectors recover the gauge columns up to sign and permutation. Joint response signatures identify the common permutation; operator spectra identify an anonymous intervention permutation; and signed third moments orient the one-dimensional eigendirections. Substituting the recovered bases gives T=Q_sQ_t^top up to sampling error.
Exact mathematical passage
T=Q_sQ_t^\top
Q_tUse in source · 3 passages
Search aliases: Q_t
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9690
Let Q_t⊆ V_t be a protected obstruction subspace. A dynamic grammar transport is κ-faithful on Q_t when normT_t^Vq≥κnormq for all q∈ Q_t. This extends the obstruction-faithfulness requirement from one bridge to an entire grammar lineage.
Exact mathematical passage
Q_t\subseteq V_t
q_tUse in source · 4 passages
Search aliases: q_t
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1822
[Authorized obstruction-shrinkage theorem] Suppose U_t^auth⊆ U_t+1^auth and restrD_t+1U_t^auth=D_t. Then im D_t⊆im D_t+1, there is a canonical surjection q_t:coker D_t↠coker D_t+1, [v]_im D_t⟼[v]_im D_t+1, and dimcoker D_t+1≤dimcoker D_t. The inequality is strict if and only if im D_tsubsetneqim D_t+1.
Exact mathematical passage
q_t:\coker D_t\twoheadrightarrow\coker D_{t+1}, \qquad [v]_{\im D_t}\longmapsto[v]_{\im D_{t+1}},
Q_t^\topUse in source · 1 passage
Search aliases: Q_t^top top
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 19625
[Proof sketch] A generic linear combination of the commuting operators has simple spectrum. Its eigenvectors recover the gauge columns up to sign and permutation. Joint response signatures identify the common permutation; operator spectra identify an anonymous intervention permutation; and signed third moments orient the one-dimensional eigendirections. Substituting the recovered bases gives T=Q_sQ_t^top up to sampling error.
Exact mathematical passage
T=Q_sQ_t^\top
Q_ZUse in source · 2 passages
Search aliases: Q_Z
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 743
[Composition bound for bridge defects] Let T:X→ Y and S:Y→ Z. Suppose the property transports satisfy normQ_Y(Tx)-widetilde T_QQ_X(x)≤ε_T for x∈ K, and normQ_Z(Sy)-widetilde S_QQ_Y(y)≤ε_S for y∈ T(K). If normwidetilde S_Q≤ L_S, then the composite bridge obeys ε_Q(S∘ T;K) ≤ ε_S+L_Sε_T with intended property transport widetilde S_Qwidetilde T_Q.
Exact mathematical passage
\norm{Q_Z(Sy)-\widetilde S_QQ_Y(y)}\leq\eps_S
q_\thetaUse in source · 2 passages
Search aliases: q_θ theta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 19847
For a candidate quotient q_θ, the empirical twin discrepancy is D_twin(q_θ) = E fracnormq_θ(x)-q_θ(x') max{normq_θ(x),ε}. A candidate is construction-valid only when it jointly predicts future consequences and intervention effects, predicts the externally scored answer, and keeps [source reference] below the declared ceiling.
Exact mathematical passage
q_\theta
Q_\omegaUse in source · 4 passages
Search aliases: Q_ω omega
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1255
Each executable operation has a Hoare-style contract {C_ω} ω {Q_ω}. Syntactic substitution ω∘(ω_1,…,ω_n) is semantically authorized only when the inner postconditions and bridge guarantees imply the outer precondition.
Exact mathematical passage
\{C_\omega\}\ \omega\ \{Q_\omega\}.
Q_{\omega_i}Use in source · 2 passages
Search aliases: Q_ω_i omega
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1277
[Compositional receipt soundness] Assume: item every primitive receipt is semantically sound; item execution of each ω_i from C_ω_i establishes Q_ω_i; item the conjunction of the inner postconditions and bridge guarantees implies C_ω; item execution of ω from C_ω establishes Q_ω; item receipt composition preserves provenance and protected evidence references. Then the composed receipt is sound for ω∘(ω_1,…,ω_n) with precondition bigwedge_iC_ω_i and postcondition Q_ω.
Exact mathematical passage
Q_{\omega_i}
\mathrm{quad}Use in source · 3 passages
Search aliases: quad mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16619
Let the template set be Θ = {θ_aff,θ_abs, θ_quad,θ_hinge}. For integer parameters a,b,c and integer input x, the exact programs are f_aff(x) · =ax+b,
Exact mathematical passage
\Theta = \{\theta_{\mathrm{aff}},\theta_{\mathrm{abs}}, \theta_{\mathrm{quad}},\theta_{\mathrm{hinge}}\}. \label{eq:b18-template-set}
\mathrm{query}Use in source · 2 passages
Search aliases: query mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16686
Generator A uses approximately uniform template frequencies. Its demonstration and query supports are X_A^demo={-2,-1,0,1,2}, X_A^query={-3,3}. Generator B changes the template mixture toward quadratic and hinge programs and uses X_B^demo={-3,-1,0,1,3}, X_B^query={-3,-2,2,3}. Every shifted record remains inside the same exact program language. Nevertheless, every shifted task contains at least one normalized feature coordinate outside the coordinatewise range of the primary support, and the larg…
Exact mathematical passage
\Xcal_A^{\mathrm{demo}}=\{-2,-1,0,1,2\}, \qquad \Xcal_A^{\mathrm{query}}=\{-3,3\}.
R'Use in source · 5 passages
Search aliases: R'
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9069
[Monotonicity of restricted envelopes] If R⊆ R'⊆ N, then F_K(R)≤ F_K(R').
Exact mathematical passage
R\subseteq R'\subseteq N
\R^{10}Use in source · 2 passages
Search aliases: R^10 ℝ^10 R mathbb
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16642
bx+c, · x<0. The coefficients belong to finite declared integer sets. A task contains five shuffled input--output demonstrations and one query input. The model receives a normalized serialization of those demonstrations and the query. The protected target is y(τ) = ( e_θ, fracp_12,…,fracp_42, fracd(θ)5, fracf_θ,p(x_q)32 )∈ℝ^10, where e_θ∈ℝ^4 is the one-hot template code, unused parameter slots are zero, and d(θ)∈{2,3,4,5} is the declared execution depth. All demonstrations, query answers, semant…
Exact mathematical passage
y(\tau) = \left( e_{\theta}, \frac{p_1}{2},\ldots,\frac{p_4}{2}, \frac{d(\theta)}{5}, \frac{f_{\theta,p}(x_q)}{32} \right)\in\R^{10}, \label{eq:b18-protected-target}
\R^{16}Use in source · 5 passages
Search aliases: R^16 ℝ^16 R mathbb
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6551
Let x=(x_1,x_2,x_3,x_4)∈ℝ^4. Define the eight-dimensional base feature ψ(x) = x_1 · x_2 · x_3 · x_4 · x_1x_2 · x_3x_4 · sin x_1 · cos x_2 ^ top ∈ℝ^8, and the affected-stratum indicator s(x)=1{x_1+x_2>0.8}. The full feature map is ϕ(x) = ψ(x) s(x)ψ(x) ∈ℝ^16.
Exact mathematical passage
\phi(x) = \begin{bmatrix} \psi(x)\\ s(x)\psi(x) \end{bmatrix} \in\R^{16}. \label{eq:block06-feature}
\R^{16\times8}Use in source · 1 passage
Search aliases: R^16×8 ℝ^16×8 R times mathbb times
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21290
Let E=[A B]:ℝ^16→ℝ^16 with A,B∈ℝ^16×8 and let ϕ(x)=ψ(x)
Exact mathematical passage
A,B\in\R^{16\times8}
\R^{19}Use in source · 1 passage
Search aliases: R^19 ℝ^19 R mathbb
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16087
Let ϕ(τ)∈ℝ^19 be the padded task serialization, and let e=E_θ(ϕ(τ))∈ℝ^40 be the prompt representation. The latent trace has four states. Given conditioning mode α, define z_0=ζ, z_k+1 = tanh (Rz_k+s_k+C_α(e,m)), 0≤ k<4, where s_k is a learned step embedding. The task-blind parent satisfies C_parent(e,m)=0. The descendants use one of the following conditioning maps: C_depth(e,m) · =D 1_m, labeleq:b17-depth-conditio
Exact mathematical passage
\phi(\tau)\in\R^{19}
R^2Use in source · 75 passages
Search aliases: R^2
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 246
thout weakening its gates. After a precommitted compiler revision and fresh final-task generation, the selected full-binding child is FULL on both 1600-task primary and shifted-generator audits while the parent remains PARTIAL. Primary and foreign program-specificity gains are 0.010714 and 0.024763, with frozen intervals [0.009911,0.011527] and [0.022794,0.026793]; held-out trace-probe R^2 is 0.865504 and 0.691471. The accepted result is one finite package-local executable Neuralese binding-repa…
Exact mathematical passage
R^2
\R^2Use in source · 11 passages
Search aliases: R^2 ℝ^2 R mathbb
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3910
[Shear holonomy] Let E=ℝ^2 and suppose evaluator evolution around one loop produces H= 1 · α
Exact mathematical passage
E=\R^2
\R^{23}Use in source · 1 passage
Search aliases: R^23 ℝ^23 R mathbb
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7369
Each candidate receipt z_e,c is mapped to a feature vector x(z_e,c)∈ℝ^23. The released feature schema contains: item a constant coordinate; item logarithmic anchor and calibration sizes; item target representation width; item alignment, transport-condition, and repair-equivariance diagnostics; item packet residual and relative repair magnitude; item repair rank; item calibration baseline slice and protected accuracy; item calibration gain, protected regression, and protected utility; item five c…
Exact mathematical passage
x(z_{e,c})\in\R^{23}.
R^2_{\mathrm{future}}Use in source · 2 passages
Search aliases: R^2_future mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 19952
full identity, equal-rank random }. For each nontrivial constructor, ranks one through six are considered. The full identity is an oversized control. A projection W is valid only when, on both recurrent systems, R^2_future · ≥0.82, R^2_intervention · ≥0.90, Acc_answer · ≥0.78, D_twin · ≤0.22. Among valid candidates, the compiler minimizes rank and then follows the frozen method priority. No hidden semantic score enters the choice.
Exact mathematical passage
\begin{aligned} R^2_{\mathrm{future}}&\geq0.82,\\ R^2_{\mathrm{intervention}}&\geq0.90,\\ \operatorname{Acc}_{\mathrm{answer}}&\geq0.78,\\ D_{\mathrm{twin}}&\leq0.22. \end{aligned} \label{eq:b23-visible-gates}
R^2_{\mathrm{intervention}}Use in source · 2 passages
Search aliases: R^2_intervention mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 19952
full identity, equal-rank random }. For each nontrivial constructor, ranks one through six are considered. The full identity is an oversized control. A projection W is valid only when, on both recurrent systems, R^2_future · ≥0.82, R^2_intervention · ≥0.90, Acc_answer · ≥0.78, D_twin · ≤0.22. Among valid candidates, the compiler minimizes rank and then follows the frozen method priority. No hidden semantic score enters the choice.
Exact mathematical passage
\begin{aligned} R^2_{\mathrm{future}}&\geq0.82,\\ R^2_{\mathrm{intervention}}&\geq0.90,\\ \operatorname{Acc}_{\mathrm{answer}}&\geq0.78,\\ D_{\mathrm{twin}}&\leq0.22. \end{aligned} \label{eq:b23-visible-gates}
R^2_{\rm packet}Use in source · 1 passage
Search aliases: R^2_rm packet rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 18404
For reconstructive candidates, admissibility requires R^2_rm sem≥0.68, R^2_rm packet≥0.72, NRMSE_rm packet≤0.55. A quotient is admissible only when the observed dimension ratio is at most 0.82, semantic R^2 is at least 0.68, and no reconstructive candidate reaches packet R^2=0.82. A no-go certificate is admissible only after the compiler discovers at least thirty identical-observation pairs whose mean protected semantic separation is at least 0.50.
Exact mathematical passage
R^2_{\rm sem}\geq0.68, \qquad R^2_{\rm packet}\geq0.72, \qquad \operatorname{NRMSE}_{\rm packet}\leq0.55. \label{eq:block21-reconstructive-gates}
R^2_{\rm sem}Use in source · 2 passages
Search aliases: R^2_rm sem rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 18404
For reconstructive candidates, admissibility requires R^2_rm sem≥0.68, R^2_rm packet≥0.72, NRMSE_rm packet≤0.55. A quotient is admissible only when the observed dimension ratio is at most 0.82, semantic R^2 is at least 0.68, and no reconstructive candidate reaches packet R^2=0.82. A no-go certificate is admissible only after the compiler discovers at least thirty identical-observation pairs whose mean protected semantic separation is at least 0.50.
Exact mathematical passage
R^2_{\rm sem}\geq0.68, \qquad R^2_{\rm packet}\geq0.72, \qquad \operatorname{NRMSE}_{\rm packet}\leq0.55. \label{eq:block21-reconstructive-gates}
R^2_{\mathrm{transport}}Use in source · 1 passage
Search aliases: R^2_transport mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17934
A candidate is pre-audit eligible only if both foreign target models are ATLAS_FULL and the candidate jointly satisfies R^2_transport · ≥0.75, · R^2_worst chart · ≥0.74, ρ_preserve · ≥0.98, · NRMSE_trace · ≤0.05, κ_square · ≤0.02. · · The selection score is the weakest normalized margin across these gates. This prevents a strong aggregate R^2 from compensating for a failed chart, high curvature, or poor trajectory reconstruction.
Exact mathematical passage
\begin{aligned} R^2_{\mathrm{transport}}&\geq0.75, &\qquad R^2_{\mathrm{worst\ chart}}&\geq0.74,\\ \rho_{\mathrm{preserve}}&\geq0.98, & \operatorname{NRMSE}_{\mathrm{trace}}&\leq0.05,\\ \kappa_{\mathrm{square}}&\leq0.02. && \end{aligned} \label{eq:b20-thresholds}
R^2_{\mathrm{worst\ chart}}Use in source · 1 passage
Search aliases: R^2_worst chart mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17934
A candidate is pre-audit eligible only if both foreign target models are ATLAS_FULL and the candidate jointly satisfies R^2_transport · ≥0.75, · R^2_worst chart · ≥0.74, ρ_preserve · ≥0.98, · NRMSE_trace · ≤0.05, κ_square · ≤0.02. · · The selection score is the weakest normalized margin across these gates. This prevents a strong aggregate R^2 from compensating for a failed chart, high curvature, or poor trajectory reconstruction.
Exact mathematical passage
\begin{aligned} R^2_{\mathrm{transport}}&\geq0.75, &\qquad R^2_{\mathrm{worst\ chart}}&\geq0.74,\\ \rho_{\mathrm{preserve}}&\geq0.98, & \operatorname{NRMSE}_{\mathrm{trace}}&\leq0.05,\\ \kappa_{\mathrm{square}}&\leq0.02. && \end{aligned} \label{eq:b20-thresholds}
\R^{32}Use in source · 1 passage
Search aliases: R^32 ℝ^32 R mathbb
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17303
Each task contains five demonstration pairs D={(x_i,y_i)}_i=1^5 and a query q. After chart canonicalization, the full descendant computes pair codes p_i=ϕ(widehat x_i,widehat y_i)∈ℝ^32, and the permutation-invariant aggregate a(D) = ( frac15∑_i=1^5p_i, max_1≤ i≤5p_i )∈ℝ^64. The semantic seed is h_0=S(a(D),Q(widehat q))∈ℝ^40. A five-step Elman recurrence generates the Neuralese trace r_k=tanh(W_ih_0+b_i+U_ir_k-1+c_i+e_k), k=1,…,5, where e_k is a learned step embedding. A gated readout recurrence …
Exact mathematical passage
p_i=\phi(\widehat x_i,\widehat y_i)\in\R^{32},
\R^{36}Use in source · 5 passages
Search aliases: R^36 ℝ^36 R mathbb
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16701
Let e=E(ϕ(τ))∈ℝ^36 be the encoded task representation, let z_0∈ℝ^36 be a learned seed, and let s_k∈ℝ^36 be the step embedding at latent depth k∈{0,…,4}. The source uses an Elman transition z_k+1 = tanh(Wz_k+s_k+δ_k(e,z_k)). The first foreign family uses a GRU cell z_k+1=GRUCell(u_k,z_k), and the second uses a gated nonlinear residual cell z_k+1 = z_k+σ(Gz_k)odot tanh(V_2GELU(V_1z_k)+s_k+δ_k(e,z_k)). The three cell classes have distinct execu
Exact mathematical passage
e=E(\phi(\tau))\in\R^{36}
\R^4Use in source · 7 passages
Search aliases: R^4 ℝ^4 R mathbb
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6537
Let x=(x_1,x_2,x_3,x_4)∈ℝ^4. Define the eight-dimensional base feature ψ(x) = x_1 · x_2 · x_3 · x_4 · x_1x_2 · x_3x_4 · sin x_1 · cos x_2 ^ top ∈ℝ^8, and the affected-stratum indicator s(x)=1{x_1+x_2>0.8}. The full feature map is ϕ(x) = ψ(x)
Exact mathematical passage
x=(x_1,x_2,x_3,x_4)\in\R^4
\R^6Use in source · 1 passage
Search aliases: R^6 ℝ^6 R mathbb
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6081
Each foreign agent has a canonical defect d∈ℝ^6, a positive diagonal importance vector w, and an invertible architecture transform T∈ℝ^6×6. The runtime-visible defect is widetilde d=Td. Protected utility is J_T(widetilde d) = exp(-∑_i=1^6w_i(T^-1widetilde d)_i^2). A candidate runtime patch δ produces gain g(δ) = J_T(widetilde d+δ)-J_T(widetilde d).
Exact mathematical passage
d\in\R^6,
\R^{64}Use in source · 1 passage
Search aliases: R^64 ℝ^64 R mathbb
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17307
Each task contains five demonstration pairs D={(x_i,y_i)}_i=1^5 and a query q. After chart canonicalization, the full descendant computes pair codes p_i=ϕ(widehat x_i,widehat y_i)∈ℝ^32, and the permutation-invariant aggregate a(D) = ( frac15∑_i=1^5p_i, max_1≤ i≤5p_i )∈ℝ^64. The semantic seed is h_0=S(a(D),Q(widehat q))∈ℝ^40. A five-step Elman recurrence generates the Neuralese trace r_k=tanh(W_ih_0+b_i+U_ir_k-1+c_i+e_k), k=1,…,5, where e_k is a learned step embedding. A gated readout recurrence …
Exact mathematical passage
a(D) = \left( \frac15\sum_{i=1}^{5}p_i, \max_{1\leq i\leq5}p_i \right)\in\R^{64}. \label{eq:b19-set-aggregate}
\R^{6\times2}Use in source · 1 passage
Search aliases: R^6×2 ℝ^6×2 R times mathbb times
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 22033
Let the true context packet be P_*=UU^ T for a target-specific orthonormal matrix U∈ℝ^6×2, while the inherited packet is P_0=diag(1,1,0,0,0,0). Let the chart label be c(h)=1{e_1^ TU^ Th>0}. For generic U, c(h) is not measurable from P_0h. No threshold or nonlinear classifier using only P_0h can recover the chart perfectly. The response atlas can recover im U up to basis gauge when the intervention signatures separate it.
Exact mathematical passage
U\in\R^{6\times2}
\R^{6\times6}Use in source · 1 passage
Search aliases: R^6×6 ℝ^6×6 R times mathbb times
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6085
Each foreign agent has a canonical defect d∈ℝ^6, a positive diagonal importance vector w, and an invertible architecture transform T∈ℝ^6×6. The runtime-visible defect is widetilde d=Td. Protected utility is J_T(widetilde d) = exp(-∑_i=1^6w_i(T^-1widetilde d)_i^2). A candidate runtime patch δ produces gain g(δ) = J_T(widetilde d+δ)-J_T(widetilde d).
Exact mathematical passage
T\in\R^{6\times6}.
\R^{80}Use in source · 1 passage
Search aliases: R^80 ℝ^80 R mathbb
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 19936
Each episode contains two independently gauged recurrent systems. A trajectory has five latent steps, four task-relevant semantic coordinates, and twelve nuisance coordinates per step. The observed flattened trajectory therefore lies in ℝ^80. The semantic dynamics, gauges, nuisance processes, and hidden audit coordinates are not passed to the quotient compiler.
Exact mathematical passage
\R^{80}
\R^dUse in source · 6 passages
Search aliases: R^d ℝ^d R mathbb
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8445
[Full-rank bridge identification] Let the target representation satisfy h_t(x)=T h_s(x) for a fixed linear map T:ℝ^d→ℝ^m. Let X∈ℝ^k× d be the selected source design and Y=XT^top. If rank X=d, then the unregularized least-squares bridge is unique and equals T on the entire source space. If rank X<d, there exist distinct maps T_1≠ T_2 such that XT_1^top=XT_2^top=Y. Consequently, repair transport is not identifiable on every unobserved source direction.
Exact mathematical passage
T:\R^d\to\R^m
\R^{d_a}Use in source · 2 passages
Search aliases: R^d_a ℝ^d_a R mathbb
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6580
An agent a has encoder E_a:ℝ^16→ℝ^d_a and an affine head fitted to widetilde q. Its hidden state is h_a(x)=E_a(ϕ(x)). The transport system is not given the matrix or nonlinear parameters of E_a.
Exact mathematical passage
E_a:\R^{16}\to\R^{d_a}
\R^{d_a\times r}Use in source · 1 passage
Search aliases: R^d_a× r ℝ^d_a× r R times mathbb times
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6650
Let U_a∈ℝ^d_a× r have orthonormal columns spanning the discovered packet. Define desired packet coordinates z_i = U_a^top(h_a^∘(x_i)-h_a(x_i)). Fit an affine predictor widehat z(h)=K_a h+c_a by ridge regression on source anchor states. The repair is R_a(h) = h+U_awidehat z(h) = h+D_a h+b_a, where D_a=U_aK_a, b_a=U_ac_a.
Exact mathematical passage
U_a\in\R^{d_a\times r}
\R^{d_s}Use in source · 1 passage
Search aliases: R^d_s ℝ^d_s R mathbb
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6788
[Exact transport through an injective hidden bridge] Let T:ℝ^d_s→ℝ^d_t have full column rank, so T^+T=I_d_s. Suppose h_t=Th_s throughout the evaluated support. Define D_t and b_t by [source reference] with widehat T=T. Then R_t(Th_s)=TR_s(h_s).
Exact mathematical passage
T:\R^{d_s}\to\R^{d_t}
\R^{d_t}Use in source · 1 passage
Search aliases: R^d_t ℝ^d_t R mathbb
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6788
[Exact transport through an injective hidden bridge] Let T:ℝ^d_s→ℝ^d_t have full column rank, so T^+T=I_d_s. Suppose h_t=Th_s throughout the evaluated support. Define D_t and b_t by [source reference] with widehat T=T. Then R_t(Th_s)=TR_s(h_s).
Exact mathematical passage
T:\R^{d_s}\to\R^{d_t}
\R^{d\times d}Use in source · 1 passage
Search aliases: R^d× d ℝ^d× d R times mathbb times
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10567
[Rank saturation does not imply stability] For every d≥2 and every η>0, there exists a full-rank design X∈ℝ^d× d such that λ_min(X^ TX)<η.
Exact mathematical passage
X\in\R^{d\times d}
\R^{d\times r}Use in source · 1 passage
Search aliases: R^d× r ℝ^d× r R times mathbb times
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13103
Let U_P∈ℝ^d× r have orthonormal columns spanning the recovered repair packet. Define packet coordinates z=U_P^top(x-c). The degree-two block-local grammar consists of maps F_A,b,C(x) = Ax+b+C vech(zz^top), where vech lists the r(r+1)/2 distinct symmetric quadratic monomials.
Exact mathematical passage
U_P\in\R^{d\times r}
\R^{k\times d}Use in source · 1 passage
Search aliases: R^k× d ℝ^k× d R times mathbb times
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8445
[Full-rank bridge identification] Let the target representation satisfy h_t(x)=T h_s(x) for a fixed linear map T:ℝ^d→ℝ^m. Let X∈ℝ^k× d be the selected source design and Y=XT^top. If rank X=d, then the unregularized least-squares bridge is unique and equals T on the entire source space. If rank X<d, there exist distinct maps T_1≠ T_2 such that XT_1^top=XT_2^top=Y. Consequently, repair transport is not identifiable on every unobserved source direction.
Exact mathematical passage
X\in\R^{k\times d}
\R^mUse in source · 2 passages
Search aliases: R^m ℝ^m R mathbb
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8445
[Full-rank bridge identification] Let the target representation satisfy h_t(x)=T h_s(x) for a fixed linear map T:ℝ^d→ℝ^m. Let X∈ℝ^k× d be the selected source design and Y=XT^top. If rank X=d, then the unregularized least-squares bridge is unique and equals T on the entire source space. If rank X<d, there exist distinct maps T_1≠ T_2 such that XT_1^top=XT_2^top=Y. Consequently, repair transport is not identifiable on every unobserved source direction.
Exact mathematical passage
T:\R^d\to\R^m
\mathbb R^{m}Use in source · 1 passage
Search aliases: R^m mathbb
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17770
[Pooled bridge compatibility obstruction] Let V⊆ R^m be spanned by the canonical trajectories observed in two charts c_1,c_2. Suppose one linear map T exactly transports architecture b into architecture a in both charts: zG_b,c_iT=zG_a,c_i for every z∈ V, i∈{1,2}. Then the restrictions to V satisfy G_b,c_1T=G_a,c_1, G_b,c_2T=G_a,c_2. If G_b,c_i are invertible on V, exact pooled transport is possible only when G_b,c_1^-1G_a,c_1 = G_b,c_2^-1G_a,c_2 on V.
Exact mathematical passage
V\subseteq\mathbb R^{m}
\R^{m_t}Use in source · 1 passage
Search aliases: R^m_t ℝ^m_t R mathbb
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9390
A grammar extension can reveal error that did not exist as a coordinate in the old state. Let E_t:ℝ^m_t↪ℝ^m_t+1 be the conservative embedding of old meta-regret coordinates. Decompose r_t+1 = E_twidetilde r_t+1+n_t+1, n_t+1⊥im E_t. Define old-coordinate repair progress by P_t^repair = normr_t^2-normwidetilde r_t+1^2, and epistemic discovery by P_t^discovery = normn_t+1^2. Both quantities may be positive. The total represented regret may increase because the lineage learned to represent a previou…
Exact mathematical passage
E_t:\R^{m_t}\hookrightarrow\R^{m_{t+1}}
\R^{m_{t+1}}Use in source · 1 passage
Search aliases: R^m_t+1 ℝ^m_t+1 R mathbb
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9390
A grammar extension can reveal error that did not exist as a coordinate in the old state. Let E_t:ℝ^m_t↪ℝ^m_t+1 be the conservative embedding of old meta-regret coordinates. Decompose r_t+1 = E_twidetilde r_t+1+n_t+1, n_t+1⊥im E_t. Define old-coordinate repair progress by P_t^repair = normr_t^2-normwidetilde r_t+1^2, and epistemic discovery by P_t^discovery = normn_t+1^2. Both quantities may be positive. The total represented regret may increase because the lineage learned to represent a previou…
Exact mathematical passage
E_t:\R^{m_t}\hookrightarrow\R^{m_{t+1}}
\R^{m\times k}Use in source · 1 passage
Search aliases: R^m× k ℝ^m× k R times mathbb times
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 14484
For a finite response model, let R∈ℝ^m× k contain the expected response signatures of k inherited obstruction components under m experiment types. Let z∈ℝ_
Exact mathematical passage
R\in\R^{m\times k}
\R^{m\times16}Use in source · 1 passage
Search aliases: R^m×16 ℝ^m×16 R times mathbb times
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21332
For an exact linear target with hidden width sixteen, let H_s∈ℝ^m×16. When m<16, full column rank is impossible, so the bridge is not uniquely identified by ordinary least squares. At m=8, the frozen sweep reaches 0.4115 affected accuracy and the witness rejects. At m=16, random anchors are generically spanning and accuracy rises to 0.7877. By m=32, conditioning is sufficient for numerical recovery and accuracy reaches 0.9943.
Exact mathematical passage
H_s\in\R^{m\times16}
\R^{n\times d}Use in source · 2 passages
Search aliases: R^n× d ℝ^n× d R times mathbb times
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10540
Let X_S∈ℝ^n× d be the source hidden-state design induced by anchor set S. Rank recovery is saturated when Pr(rank X_S=d)≥ 1-ε_rm rank under the current foreign development distribution and the remaining performance gap cannot be explained by rank failure.
Exact mathematical passage
X_S\in\R^{n\times d}
\R^{n\times q}Use in source · 1 passage
Search aliases: R^n× q ℝ^n× q R times mathbb times
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10605
Stacking the selected anchors gives Y=XT^ T+E, where X∈ℝ^n× d and Y∈ℝ^n× q. Define W=diag(σ_1^-2,…,σ_n^-2) and the weighted information matrix M_S=X^ TWX.
Exact mathematical passage
Y\in\R^{n\times q}
\mathbb R^{p}Use in source · 1 passage
Search aliases: R^p mathbb
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17848
[Readout stability from trajectory reconstruction] Let J:H_a,c→ R^p be L_J-Lipschitz. If a transported trajectory satisfies normT^c_b→ ah_b,c-h_a,c ≤ε, then normJ(T^c_b→ ah_b,c)-J(h_a,c) ≤ L_Jε. If two transport paths differ by at most δ, their semantic readouts differ by at most L_Jδ.
Exact mathematical passage
J:H_{a,c}\to\mathbb R^{p}
\R^{q\times d}Use in source · 1 passage
Search aliases: R^q× d ℝ^q× d R times mathbb times
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21855
Let the old bridge be T∈ℝ^q× d and let g be any admissible separator. The embedding T⟼(g,T,T) returns Th on both charts. Hence the local grammar can execute the inherited process without approximation. A conservative runtime may therefore use K_chart(h) = K_local(h), · if the chart certificate passes,
Exact mathematical passage
T\in\R^{q\times d}
R^t_{\sigma(u)}Use in source · 1 passage
Search aliases: R^t_σ(u) sigma
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 19561
For a proposed intervention correspondence σ: U→ V and transport T:H_t→ H_s, define the normalized orbit defect δ_rm orb(T,σ) = frac1| U| ∑_u∈ U fracnormTR^t_σ(u)-R_u^sT_F max{normR_u^s_F,ε}.
Exact mathematical passage
\delta_{\rm orb}(T,\sigma) = \frac1{|\mathcal U|} \sum_{u\in\mathcal U} \frac{\norm{TR^t_{\sigma(u)}-R_u^sT}_F} {\max\{\norm{R_u^s}_F,\eps\}}. \label{eq:b22-orbit-defect}
r^\topUse in source · 1 passage
Search aliases: r^top top
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9787
Let foreign residuals follow distribution Q. Define the obstruction covariance operator Σ_ob = E_r∼ Q [ P_I^⊥r r^top P_I^⊥ ]. Let W⊆ I^⊥ be a k-dimensional new response subspace.
Exact mathematical passage
\Sigma_{\mathrm{ob}} = \mathbb E_{r\sim Q} \left[ P_{I^\perp}r\,r^\top P_{I^\perp} \right]. \label{eq:b8-obstruction-covariance}
\R^{|I|\times|E|}Use in source · 1 passage
Search aliases: R^|I|×|E| ℝ^|I|×|E| R times mathbb times
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2348
Let G=(I,E) be a connected oriented graph representing the nerve of the cover, let K be a finite-dimensional Hilbert space representing a common trivialization of the kernel fibers, and let e=(e_a)_a∈ E∈ K^E be the transported overlap-mismatch vector. Let B∈ℝ^|I|×|E| be the oriented incidence matrix and set B:=B⊗ I_K. A vertex gauge k∈ K^I changes the mismatch to e+ B^*k.
Exact mathematical passage
B\in\R^{|I|\times|E|}
R^\daggerUse in source · 5 passages
Search aliases: R^† dagger
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 14493
[Response-mixture recovery bound] Assume R has full column rank and normξ_2≤ε. The least-squares estimator widehat z=R^† y satisfies normwidehat z-z_2 ≤ normR^†_2ε = fracεσ_min(R).
Exact mathematical passage
\widehat z=R^\dagger y
\R_{\\geq0}^kUse in source · 1 passage
Search aliases: R_ geq0^k ℝ_ geq0^k R geq mathbb geq
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 14484
For a finite response model, let R∈ℝ^m× k contain the expected response signatures of k inherited obstruction components under m experiment types. Let z∈ℝ_ geq0^k be a sparse mixture with 1^top z=1. An observed response vector has the form y=Rz+ξ.
Exact mathematical passage
z\in\R_{\\geq0}^k
\R_+^2Use in source · 1 passage
Search aliases: R_+^2 ℝ_+^2 R mathbb
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20672
Let K=ℝ_+^2, D=I_2, and r= 1
Exact mathematical passage
K=\R_+^2
R_0Use in source · 2 passages
Search aliases: R_0
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9088
Choose an operational ordering r_1prec r_2prec⋯prec r_m, m=|N|, with nested sets R_0=∅, R_j={r_1,…,r_j}. Define the ordered marginal headroom Δ_j^ord(K) = F_K(R_j)-F_K(R_j-1).
Exact mathematical passage
R_0=\varnothing, \qquad R_j=\{r_1,\ldots,r_j\}.
r_0Use in source · 4 passages
Search aliases: r_0
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2948
-transport Consider a path of systems A_0xrightarrow T_0A_1 xrightarrow T_1⋯ xrightarrow T_n-1A_n. Write T_U,k:U_k→ U_k+1 and T_V,k:V_k→ V_k+1. Let normD_k+1T_U,k-T_V,kD_k≤ε_D,k and normr_k+1-T_V,kr_k≤ε_r,k. If D_0u_0=-r_0 and u_n=T_U,n-1⋯ T_U,0u_0, then normD_nu_n+r_n · ≤ ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_D,k normT_U,k-1⋯ T_U,0u_0
Exact mathematical passage
D_0u_0=-r_0
R_1Use in source · 11 passages
Search aliases: R_1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10023
Let H be a finite-dimensional complex Hilbert space. Suppose permitted interventions or controlled transitions produce operators R_1,…,R_m∈End(H). These may be finite-difference response maps, local transition Jacobians, or finite Koopman operators on an observable dictionary.
Exact mathematical passage
R_1,\ldots,R_m\in\End(H).
r_1Use in source · 8 passages
Search aliases: r_1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2168
Consider the specific formal correction equation F(ε,u) = ε r_1+Du+frac12B(u,u)+O(ε^3+norm u^3), where B:U× U→ V is symmetric bilinear. Seek u(ε)=ε u_1+ε^2u_2+O(ε^3).
Exact mathematical passage
F(\eps,u) = \eps r_1+Du+\frac12B(u,u)+O\bigl(\eps^3+\norm u^3\bigr), \label{eq:second-order-model}
R_2Use in source · 5 passages
Search aliases: R_2
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21690
0 · -1. On H=ℝ^4, define four response operators R_1=X⊕0, R_2=Z⊕0, R_3=0⊕ X, R_4=0⊕ Z. The adjoint-closed algebra they generate is M_2(ℝ)⊕ M_2(ℝ), with dimension 8. Its commutant consists of scalar multiples of the identity on each block and therefore has dimension 2. The minimal central projectors are P_1=I_2⊕0, P_2=0⊕ I_2. A residual e_1 generates the first two-dimensional block. A residual e_1+e_3 generates both blocks and has cyclic dimension 4.
Exact mathematical passage
R_1=X\oplus0, \quad R_2=Z\oplus0, \quad R_3=0\oplus X, \quad R_4=0\oplus Z.
r_2Use in source · 3 passages
Search aliases: r_2
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3808
For distinct records r_1,r_2 with protected difference d=ι_v(r_1)-ι_v(r_2)≠0, injectivity gives T_ed≠0. Exact commutation identifies this with the difference of their new representations.
Exact mathematical passage
r_1,r_2
R_3Use in source · 1 passage
Search aliases: R_3
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21690
0 · -1. On H=ℝ^4, define four response operators R_1=X⊕0, R_2=Z⊕0, R_3=0⊕ X, R_4=0⊕ Z. The adjoint-closed algebra they generate is M_2(ℝ)⊕ M_2(ℝ), with dimension 8. Its commutant consists of scalar multiples of the identity on each block and therefore has dimension 2. The minimal central projectors are P_1=I_2⊕0, P_2=0⊕ I_2. A residual e_1 generates the first two-dimensional block. A residual e_1+e_3 generates both blocks and has cyclic dimension 4.
Exact mathematical passage
R_1=X\oplus0, \quad R_2=Z\oplus0, \quad R_3=0\oplus X, \quad R_4=0\oplus Z.
r_3Use in source · 1 passage
Search aliases: r_3
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20923
Suppose a source correction has norm normu_0=2, every transport is nonexpansive, and three edges have (ε_D,0,ε_D,1,ε_D,2) =(0.01,0.02,0.015), (ε_r,0,ε_r,1,ε_r,2) =(0.005,0.004,0.006). Then [source reference] gives normD_3u_3+r_3 · ≤ 2(0.01+0.02+0.015) +(0.005+0.004+0.006) · =0.105. A direct source-to-target bridge with certified residual penalty below 0.105 is preferable under this metric, even if each edge of the three-step route appears individually accurate.
Exact mathematical passage
\begin{aligned} \norm{D_3u_3+r_3} &\leq 2(0.01+0.02+0.015) +(0.005+0.004+0.006)\\ &=0.105. \end{aligned}
R_4Use in source · 1 passage
Search aliases: R_4
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21690
0 · -1. On H=ℝ^4, define four response operators R_1=X⊕0, R_2=Z⊕0, R_3=0⊕ X, R_4=0⊕ Z. The adjoint-closed algebra they generate is M_2(ℝ)⊕ M_2(ℝ), with dimension 8. Its commutant consists of scalar multiples of the identity on each block and therefore has dimension 2. The minimal central projectors are P_1=I_2⊕0, P_2=0⊕ I_2. A residual e_1 generates the first two-dimensional block. A residual e_1+e_3 generates both blocks and has cyclic dimension 4.
Exact mathematical passage
R_1=X\oplus0, \quad R_2=Z\oplus0, \quad R_3=0\oplus X, \quad R_4=0\oplus Z.
R_aUse in source · 2 passages
Search aliases: R_a
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6661
Let U_a∈ℝ^d_a× r have orthonormal columns spanning the discovered packet. Define desired packet coordinates z_i = U_a^top(h_a^∘(x_i)-h_a(x_i)). Fit an affine predictor widehat z(h)=K_a h+c_a by ridge regression on source anchor states. The repair is R_a(h) = h+U_awidehat z(h) = h+D_a h+b_a, where D_a=U_aK_a, b_a=U_ac_a.
Exact mathematical passage
R_a(h) = h+U_a\widehat z(h) = h+D_a h+b_a, \label{eq:block06-source-repair}
r_{\mathrm{cor}}Use in source · 3 passages
Search aliases: r_cor mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1551
[Canonical correction--obstruction decomposition] For every r∈ V: item P_cor is the orthogonal projector onto im D; item P_ob is the orthogonal projector onto ker D^*; item the defect decomposes uniquely as r=r_cor+r_ob, r_cor=P_corr, r_ob=P_obr; item the minimum achievable residual is inf_u∈ U^authnormDu+r =normr_ob =dist(r,im D); item the complete set of least-squares minimizers is -D^† r+ker D; item if r is exactly correctable, u^*=-D^† r is the unique exact repair of minimum norm.
Exact mathematical passage
r=r_{\mathrm{cor}}+r_{\mathrm{ob}}, \qquad r_{\mathrm{cor}}=P_{\mathrm{cor}}r, \quad r_{\mathrm{ob}}=P_{\mathrm{ob}}r;
r_{\mathrm{flat}}Use in source · 2 passages
Search aliases: r_flat mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5602
Case F3: flat repair. Set D_flat=I_2, r_flat= 1 -2 , with zero overlap mismatch and identity evaluator transport. The unique minimum-norm repair is u_flat=-r_flat.
Exact mathematical passage
D_{\mathrm{flat}}=I_2, \qquad r_{\mathrm{flat}}= \begin{bmatrix} 1\\-2 \end{bmatrix},
R_gUse in source · 2 passages
Search aliases: R_g
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11605
[Gate misclassification bound] Let g be the correct chart and widehat g an estimated chart. Suppose the two target repair outputs satisfy normR_1(y)-R_0(y)≤ L throughout the evaluated region and Pr(widehat g≠ g)≤ϵ_g. Then EnormR_widehat g(y)-R_g(y)^2 ≤ ϵ_gL^2. If protected utility is L_J-Lipschitz in repaired state, then abs E J(R_widehat g(y))- E J(R_g(y)) ≤ L_JLsqrtϵ_g.
Exact mathematical passage
\mathbb E\norm{R_{\widehat g}(y)-R_g(y)}^2 \leq \epsilon_gL^2.
R_{i_1}Use in source · 1 passage
Search aliases: R_i_1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12928
Every element of A(R) is a finite linear combination of words in the generators. On H_a, a word R_i_1⋯ R_i_ℓ acts by the scalar λ_ai_1⋯λ_ai_ℓ. Equality of the signature vectors makes the same scalar act on H_b. Linear combinations preserve the equality. A central idempotent separating the blocks would act as 1 on one and 0 on the other, contradicting the first conclusion.
Exact mathematical passage
R_{i_1}\cdots R_{i_\ell}
r_{i_t,t}Use in source · 2 passages
Search aliases: r_i_t,t
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9338
[Greedy bottleneck contraction] Let r_t∈ℝ_≥0^m and Φ_t=1^top r_t. Suppose the controller chooses i_t with r_i_t,t≥fracΦ_tm, contracts that component by q∈[0,1), and permits total collateral increase at most ε_t. Then Φ_t+1 ≤ (1-frac1-qm)Φ_t+ε_t.
Exact mathematical passage
r_{i_t,t}\geq\frac{\Phi_t}{m},
R_{i_\ell}Use in source · 1 passage
Search aliases: R_i_ℓ ell
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12928
Every element of A(R) is a finite linear combination of words in the generators. On H_a, a word R_i_1⋯ R_i_ℓ acts by the scalar λ_ai_1⋯λ_ai_ℓ. Equality of the signature vectors makes the same scalar act on H_b. Linear combinations preserve the equality. A central idempotent separating the blocks would act as 1 on one and 0 on the other, contradicting the first conclusion.
Exact mathematical passage
R_{i_1}\cdots R_{i_\ell}
r_jUse in source · 3 passages
Search aliases: r_j
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5274
[Perron sensitivity] Let M be irreducible with simple Perron root ρ, positive right eigenvector r, and positive left eigenvector ℓ, normalized by ℓ^ Tr=1. Then frac∂ρ∂ M_ij = ℓ_i r_j.
Exact mathematical passage
\frac{\partial\rho}{\partial M_{ij}} = \ell_i r_j.
R_{j-1}Use in source · 2 passages
Search aliases: R_j-1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9094
Choose an operational ordering r_1prec r_2prec⋯prec r_m, m=|N|, with nested sets R_0=∅, R_j={r_1,…,r_j}. Define the ordered marginal headroom Δ_j^ord(K) = F_K(R_j)-F_K(R_j-1).
Exact mathematical passage
\Delta_j^{\mathrm{ord}}(K) = F_K(R_j)-F_K(R_{j-1}). \label{eq:b8-ordered-marginal}
R_j^\astUse in source · 1 passage
Search aliases: R_j^∗ ast
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10120
For X∈End(H), define commutator energy E_com(X) = ∑_j=1^m ( normXR_j-R_jX_F^2 + normXR_j^∗-R_j^∗ X_F^2 ). There is a positive semidefinite linear operator L_com on the Hilbert space of matrices such that anglesX,L_comX_F = E_com(X). Its kernel is the commutant: kerL_com= A'.
Exact mathematical passage
\Ecal_{\mathrm{com}}(X) = \sum_{j=1}^{m} \left( \norm{XR_j-R_jX}_F^2 + \norm{XR_j^\ast-R_j^\ast X}_F^2 \right). \label{eq:b8-commutator-energy}
R_kUse in source · 1 passage
Search aliases: R_k
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12177
The algebra generated by the response family is A_R = alg^*(I,R_1,…,R_k). Because the R_j are simultaneously block scalar in the exact model, A_R is commutative and every polynomial in the responses is again block scalar.
Exact mathematical passage
\mathfrak A_R = \operatorname{alg}^{*}(I,R_1,\ldots,R_k). \label{eq:b11-response-algebra}
r_kUse in source · 3 passages
Search aliases: r_k
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2945
[Pathwise transport bound] Consider a path of systems A_0xrightarrow T_0A_1 xrightarrow T_1⋯ xrightarrow T_n-1A_n. Write T_U,k:U_k→ U_k+1 and T_V,k:V_k→ V_k+1. Let normD_k+1T_U,k-T_V,kD_k≤ε_D,k and normr_k+1-T_V,kr_k≤ε_r,k. If D_0u_0=-r_0 and u_n=T_U,n-1⋯ T_U,0u_0, then normD_nu_n+r_n · ≤ ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_D,k normT_U,k-1⋯ T_U,0u_0
Exact mathematical passage
\norm{r_{k+1}-T_{V,k}r_k}\leq\eps_{r,k}.
r_{k+1}Use in source · 1 passage
Search aliases: r_k+1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2945
[Pathwise transport bound] Consider a path of systems A_0xrightarrow T_0A_1 xrightarrow T_1⋯ xrightarrow T_n-1A_n. Write T_U,k:U_k→ U_k+1 and T_V,k:V_k→ V_k+1. Let normD_k+1T_U,k-T_V,kD_k≤ε_D,k and normr_k+1-T_V,kr_k≤ε_r,k. If D_0u_0=-r_0 and u_n=T_U,n-1⋯ T_U,0u_0, then normD_nu_n+r_n · ≤ ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_D,k normT_U,k-1⋯ T_U,0u_0
Exact mathematical passage
\norm{r_{k+1}-T_{V,k}r_k}\leq\eps_{r,k}.
r_{k-1}Use in source · 1 passage
Search aliases: r_k-1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17321
widehat y_i)∈ℝ^32, and the permutation-invariant aggregate a(D) = ( frac15∑_i=1^5p_i, max_1≤ i≤5p_i )∈ℝ^64. The semantic seed is h_0=S(a(D),Q(widehat q))∈ℝ^40. A five-step Elman recurrence generates the Neuralese trace r_k=tanh(W_ih_0+b_i+U_ir_k-1+c_i+e_k), k=1,…,5, where e_k is a learned step embedding. A gated readout recurrence maps the trace to protected program and answer coordinates. A halt head predicts one of the five execution depths.
Exact mathematical passage
r_k=\tanh\left(W_ih_0+b_i+U_ir_{k-1}+c_i+e_k\right), \qquad k=1,\ldots,5, \label{eq:b19-recurrence}
r_{\mathrm{loc}}Use in source · 2 passages
Search aliases: r_loc mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5557
Case F1: local obstruction. Set D_loc = 1 · 0 0 · 0 , r_loc = 0 1 . Then P_im D_locr_loc=0, (I-P_im D_loc)r_loc=r_loc, so the obstruction norm is one.
Exact mathematical passage
D_{\mathrm{loc}} = \begin{bmatrix} 1&0\\ 0&0 \end{bmatrix}, \qquad r_{\mathrm{loc}} = \begin{bmatrix} 0\\1 \end{bmatrix}.
R_mUse in source · 4 passages
Search aliases: R_m
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10023
Let H be a finite-dimensional complex Hilbert space. Suppose permitted interventions or controlled transitions produce operators R_1,…,R_m∈End(H). These may be finite-difference response maps, local transition Jacobians, or finite Koopman operators on an observable dictionary.
Exact mathematical passage
R_1,\ldots,R_m\in\End(H).
r_mUse in source · 1 passage
Search aliases: r_m
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9083
Choose an operational ordering r_1prec r_2prec⋯prec r_m, m=|N|, with nested sets R_0=∅, R_j={r_1,…,r_j}. Define the ordered marginal headroom Δ_j^ord(K) = F_K(R_j)-F_K(R_j-1).
Exact mathematical passage
r_1\prec r_2\prec\cdots\prec r_m, \qquad m=|N|,
r_nUse in source · 3 passages
Search aliases: r_n
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2953
T_1⋯ xrightarrow T_n-1A_n. Write T_U,k:U_k→ U_k+1 and T_V,k:V_k→ V_k+1. Let normD_k+1T_U,k-T_V,kD_k≤ε_D,k and normr_k+1-T_V,kr_k≤ε_r,k. If D_0u_0=-r_0 and u_n=T_U,n-1⋯ T_U,0u_0, then normD_nu_n+r_n · ≤ ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_D,k normT_U,k-1⋯ T_U,0u_0 · + ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_r,k, where empty products are identities.
Exact mathematical passage
\begin{aligned} \norm{D_nu_n+r_n} &\leq \sum_{k=0}^{n-1} \norm{T_{V,n-1}\cdots T_{V,k+1}} \eps_{D,k} \norm{T_{U,k-1}\cdots T_{U,0}u_0}\\ &\quad+ \sum_{k=0}^{n-1} \norm{T_{V,n-1}\cdots T_{V,k+1}} \eps_{r,k}, \end{aligned}
r_{\mathrm{ob}}Use in source · 8 passages
Search aliases: r_ob mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1551
[Canonical correction--obstruction decomposition] For every r∈ V: item P_cor is the orthogonal projector onto im D; item P_ob is the orthogonal projector onto ker D^*; item the defect decomposes uniquely as r=r_cor+r_ob, r_cor=P_corr, r_ob=P_obr; item the minimum achievable residual is inf_u∈ U^authnormDu+r =normr_ob =dist(r,im D); item the complete set of least-squares minimizers is -D^† r+ker D; item if r is exactly correctable, u^*=-D^† r is the unique exact repair of minimum norm.
Exact mathematical passage
r=r_{\mathrm{cor}}+r_{\mathrm{ob}}, \qquad r_{\mathrm{cor}}=P_{\mathrm{cor}}r, \quad r_{\mathrm{ob}}=P_{\mathrm{ob}}r;
r_{\rm non}Use in source · 1 passage
Search aliases: r_rm non rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6994
[Calibration-to-audit witness soundness] Suppose, simultaneously, that the audit gain and regression satisfy g_rm slice · ≥ widehat g_rm slice-η_g, r_rm non · ≤ widehat r_rm non+η_r, and that the reported alignment error upper-bounds the audit alignment error up to η_a. If widehat g_rm slice-η_g · >0,
Exact mathematical passage
g_{\rm slice}&\geq \widehat g_{\rm slice}-\eta_g,\\ r_{\rm non}&\leq \widehat r_{\rm non}+\eta_r,
R_sUse in source · 14 passages
Search aliases: R_s
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6768
Given source repair R_s(h)=h+D_sh+b_s, its transported target form is D_t = widehat T D_swidehat T^+, b_t = widehat T b_s, with transported packet basis obtained by orthogonalizing widehat T U_s.
Exact mathematical passage
R_s(h)=h+D_sh+b_s,
R_tUse in source · 10 passages
Search aliases: R_t
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 811
A witness structure may be represented as W_t = ( V_t^fr, C_t, B_t^sand, R_t, Π_t^auth ), where V_t^fr is the frozen evaluator instance for the current transaction, C_t is a protected checkpoint, B_t^sand is the sandbox boundary, R_t is the rollback procedure, and Π_t^auth records effect authority.
Exact mathematical passage
\Wcal_t = \left( \Vcal_t^{\mathrm{fr}}, C_t, \Bcal_t^{\mathrm{sand}}, R_t, \Pi_t^{\mathrm{auth}} \right),
r_tUse in source · 11 passages
Search aliases: r_t
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9246
Let the unresolved module-headroom vector be r_t = h_H(K_t) h_Q(K_t) h_C(K_t) h_T(K_t) h_S(K_t) h_V(K_t) h_W(K_t) ∈ℝ_≥0^7. An authorized meta-action a changes one or more repair-process modules. In a local finite model, represent its effect by r_t+1 ≤ A_a r_t+e_t, where A_a is a nonnegative response matrix and e_t records estimation error, distribution shift, semantic debt, or newly induced headroom.
Exact mathematical passage
r_t = \begin{bmatrix} h_H(K_t)\\ h_Q(K_t)\\ h_C(K_t)\\ h_T(K_t)\\ h_S(K_t)\\ h_V(K_t)\\ h_W(K_t) \end{bmatrix} \in\R_{\geq0}^{7}. \label{eq:b8-regret-vector}
r_{t+1}Use in source · 3 passages
Search aliases: r_t+1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9262
h_W(K_t) ∈ℝ_≥0^7. An authorized meta-action a changes one or more repair-process modules. In a local finite model, represent its effect by r_t+1 ≤ A_a r_t+e_t, where A_a is a nonnegative response matrix and e_t records estimation error, distribution shift, semantic debt, or newly induced headroom.
Exact mathematical passage
r_{t+1} \leq A_a r_t+e_t, \label{eq:b8-regret-dynamics}
R_{t,j}Use in source · 4 passages
Search aliases: R_t,j
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11474
Let the source repair be R_s(h)=M_sh+c_s. On chart j, let the target bridge be T_j(h)=A_jh+b_j, with A_j invertible on the repair-relevant subspace. Define R_t,j(y) = T_j (R_s(T_j^-1(y))). Expanding gives R_t,j(y) = A_jM_sA_j^-1(y-b_j)+A_jc_s+b_j.
Exact mathematical passage
R_{t,j}(y) = T_j\!\left(R_s(T_j^{-1}(y))\right). \label{eq:b10-chart-repair-def}
R_u^aUse in source · 1 passage
Search aliases: R_u^a
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 19612
The executable fixture uses five commuting symmetric response operators. In latent semantic coordinates they are diagonal: D_u=diag(λ_u1,…,λ_ud), and each system exposes an orthogonal gauge Q_a: R_u^a=Q_aD_uQ_a^top. The joint response signature of semantic coordinate j is λ_j=(λ_1j,…,λ_mj).
Exact mathematical passage
R_u^a=Q_aD_uQ_a^\top.
R_{\widehat g}Use in source · 2 passages
Search aliases: R_widehat g widehat
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11605
[Gate misclassification bound] Let g be the correct chart and widehat g an estimated chart. Suppose the two target repair outputs satisfy normR_1(y)-R_0(y)≤ L throughout the evaluated region and Pr(widehat g≠ g)≤ϵ_g. Then EnormR_widehat g(y)-R_g(y)^2 ≤ ϵ_gL^2. If protected utility is L_J-Lipschitz in repaired state, then abs E J(R_widehat g(y))- E J(R_g(y)) ≤ L_JLsqrtϵ_g.
Exact mathematical passage
\mathbb E\norm{R_{\widehat g}(y)-R_g(y)}^2 \leq \epsilon_gL^2.
r_xUse in source · 4 passages
Search aliases: r_x
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9941
For episode x, let F_x:U_x^auth→ V_x be the post-modification defect map, with F_x(0)=r_x. A valid repair satisfies F_x(u_x)=0. A repair schema is a local section s:x↦ u_x that approximately inverts F_x over its declared domain.
Exact mathematical passage
F_x(0)=r_x
R_{\sigma^{-1}(v)}^sUse in source · 2 passages
Search aliases: R_σ^-1(v)^s sigma
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 19583
[Intertwiner uniqueness modulo the response commutant] Let T_1,T_2:H_t→ H_s be invertible and satisfy T_iR_v^t=R_σ^-1(v)^sT_i for every v∈ V. Then S=T_2^-1T_1 belongs to A_t'. Hence exact intervention-orbit data identifies a transport at most up to the response commutant.
Exact mathematical passage
T_iR_v^t=R_{\sigma^{-1}(v)}^sT_i
\R_{\geq0}^{7}Use in source · 1 passage
Search aliases: R_≥0^7 ℝ_≥0^7 R geq mathbb geq
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9246
Let the unresolved module-headroom vector be r_t = h_H(K_t) h_Q(K_t) h_C(K_t) h_T(K_t) h_S(K_t) h_V(K_t) h_W(K_t) ∈ℝ_≥0^7. An authorized meta-action a changes one or more repair-process modules. In a local finite model, represent its effect by r_t+1 ≤ A_a r_t+e_t, where A_a is a nonnegative response matrix and e_t records estimation error, distribution shift, semantic debt, or newly induced headroom.
Exact mathematical passage
r_t = \begin{bmatrix} h_H(K_t)\\ h_Q(K_t)\\ h_C(K_t)\\ h_T(K_t)\\ h_S(K_t)\\ h_V(K_t)\\ h_W(K_t) \end{bmatrix} \in\R_{\geq0}^{7}. \label{eq:b8-regret-vector}
\R_{\geq0}^kUse in source · 1 passage
Search aliases: R_≥0^k ℝ_≥0^k R geq mathbb geq
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 14520
The inherited obstruction language spans a response cone K_R = {Rz:z∈ℝ_≥0^k}. For observed response y, define the open-set residual η_rm open(y) = dist(y, K_R).
Exact mathematical passage
\mathcal K_R = \{Rz:z\in\R_{\geq0}^k\}.
\R_{\geq0}^{m}Use in source · 1 passage
Search aliases: R_≥0^m ℝ_≥0^m R geq mathbb geq
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9333
[Greedy bottleneck contraction] Let r_t∈ℝ_≥0^m and Φ_t=1^top r_t. Suppose the controller chooses i_t with r_i_t,t≥fracΦ_tm, contracts that component by q∈[0,1), and permits total collateral increase at most ε_t. Then Φ_t+1 ≤ (1-frac1-qm)Φ_t+ε_t.
Exact mathematical passage
r_t\in\R_{\geq0}^{m}
\mathrm{rank}Use in source · 2 passages
Search aliases: rank mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21527
Suppose bridge-rank identification has success probability p_rank(K_1)=1. For every child that changes only the rank-identification rule while leaving all other modules fixed, p_rank(K_2)-p_rank(K_1)≤0. This does not imply that the repair process has no remaining headroom. Let the target bridge be full rank but have minimum singular value σ_min=10^-4. Random noise of norm 10^-3 can then dominate the weakest identified direction. The bottleneck has migrated from rank to conditioning. A meaningful…
Exact mathematical passage
p_{\mathrm{rank}}(K_1)=1.
\Rcal^+Use in source · 3 passages
Search aliases: Rcal^+ R^+ Rcal mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12959
[Central refinement by a separating intervention] Suppose H_a,H_b are aliased under R, and S satisfies [source reference]. If the joint signature of every remaining block is distinct from those of H_a and H_b under R^+, then the minimal central projectors of A(R^+) separate H_a and H_b.
Exact mathematical passage
\Rcal^+
\Rcal^{+}Use in source · 1 passage
Search aliases: Rcal^+ R^+ Rcal mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12940
Let S∈End(H) be a candidate intervention response. Define the expanded language R^+=R∪{S}. The extension is conservative at the syntactic level because every old response expression remains available unchanged.
Exact mathematical passage
\Rcal^{+}=\Rcal\cup\{S\}.
\Rcal_{\mathrm{valid}}Use in source · 1 passage
Search aliases: Rcal_valid R_valid Rcal mathrm mathcal mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5116
[Representation-separation obstruction] Suppose states x_0,x_1 satisfy ϕ_G(x_0)=ϕ_G(x_1), but their externally validated repair sets are disjoint: R_valid(x_0) ∩ R_valid(x_1) = ∅. Then no G-measurable deterministic repair policy is valid on both states.
Exact mathematical passage
\Rcal_{\mathrm{valid}}(x_0) \cap \Rcal_{\mathrm{valid}}(x_1) = \varnothing.
\rceilUse in source · 3 passages
Search aliases: rceil
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4188
[Minimum rounds under a witness quorum] Suppose every transaction requires at least q frozen witness components, where 1≤ q<n. Then at most n-q components can be edited in one round. Editing every component at least once requires at least lceilfracnn-qrceil rounds. This bound is attained by partitioning the components into edit blocks of size at most n-q and taking the complement of each block as its witness set.
Exact mathematical passage
\left\lceil\frac{n}{n-q}\right\rceil
\rceil^dUse in source · 1 passage
Search aliases: rceil^d rceil
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4521
[Finite ε-Pareto cover] Let Z⊆[0,1]^d be finite and let 0<ε≤1. There exists a subset Q⊆ Z with absQ ≤ lceilfrac1εrceil^d such that for every z∈ Z, some q∈ Q satisfies q_i≥ z_i-ε for every i.
Exact mathematical passage
\abs{Q} \leq \left\lceil\frac{1}{\eps}\right\rceil^d
\mathrm{rep}Use in source · 4 passages
Search aliases: rep mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12465
The source hidden space has dimension six and is generated from three target-specific unknown rank-two blocks: H=H_ctx⊕ H_rep⊕ H_nui. A target-specific orthogonal or injective embedding hides these blocks from the runtime coordinate system. The source repair changes only H_rep and leaves the context block invariant. Two controlled interventions act with different block signatures. The context label controls which target bridge and target repair are valid.
Exact mathematical passage
H=H_{\mathrm{ctx}}\oplus H_{\mathrm{rep}}\oplus H_{\mathrm{nui}}. \label{eq:b11-three-blocks}
\mathrm{repair}Use in source · 3 passages
Search aliases: repair mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3960
Let a complete implementation state decompose as Σ = Σ^task ⊕ Σ^memory ⊕ Σ^repair ⊕ Σ^evaluation ⊕ Σ^provenance ⊕ Σ^rollback. A transaction selects an editable mask E⊆Σ and a witness mask W⊆Σ with E∩ W=∅. The decomposition is logical rather than necessarily physical. Two components may share hardware while remaining isolated by capabilities, address spaces, process boundaries, cryptographic commitments, or reproducible snapshots.
Exact mathematical passage
\Sigma = \Sigma^{\mathrm{task}} \oplus \Sigma^{\mathrm{memory}} \oplus \Sigma^{\mathrm{repair}} \oplus \Sigma^{\mathrm{evaluation}} \oplus \Sigma^{\mathrm{provenance}} \oplus \Sigma^{\mathrm{rollback}}.
\mathrm{repair\ inv}Use in source · 1 passage
Search aliases: repair inv mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12552
The selected atlas candidate commits only when all frozen conditions hold: widehat g_val · ≥0.10, LCB(widehat g_val) · ≥0.05, widehat r_ctx · ≤3×10^-4, widehat a_target · ≥0.72, widehat a_repair inv · ≥0.96, widehat b_chart · ≥0.20, widehat s_model · ≤0.25, widehatγ_min · ≥0.004, widehatε_comm · ≤0.16, widehatε_resp · ≤0.08. The witness therefore evaluates both downstream repair quality and the internal atlas certificate. A high blind gain with unstable response geometry is not sufficient for co…
Exact mathematical passage
\widehat g_{\mathrm{val}}&\geq0.10,\\ \LCB(\widehat g_{\mathrm{val}})&\geq0.05,\\ \widehat r_{\mathrm{ctx}}&\leq3\times10^{-4},\\ \widehat a_{\mathrm{target}}&\geq0.72,\\ \widehat a_{\mathrm{repair\ inv}}&\geq0.96,\\ \widehat b_{\mathrm{chart}}&\geq0.20,\\ \widehat s_{\mathrm{model}}&\leq0.25,\\ \widehat\gamma_{\min}&\geq0.004,\\ \widehat\eps_{\mathrm{comm}}&\leq0.16,\\ \widehat\eps_{\mathrm{resp}}&\leq0.08.
\RepairTypedUse in source · 1 passage
Search aliases: RepairTyped mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4259
Let C_RI(s) = · Grounded(s) wedgeProvValid(s) wedgeWorkspaceBound(s) · wedgeRepairTyped(s) wedgeWitnessed(s) wedgeEvidenceFlat(s) · wedgeRollbackable(s) wedgeMetaCapable(s). For X⊆ S, define Ψ_∀(X) = sets∈ S:C_RI(s) and Post_imp(s)⊆ X.
Exact mathematical passage
\begin{aligned} C_{\mathrm{RI}}(s) ={}& \Grounded(s) \wedge\ProvValid(s) \wedge\WorkspaceBound(s)\\ &\wedge\RepairTyped(s) \wedge\Witnessed(s) \wedge\EvidenceFlat(s)\\ &\wedge\Rollbackable(s) \wedge\MetaCapable(s). \end{aligned}
\mathrm{representation}Use in source · 1 passage
Search aliases: representation mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1478
Consider four colors memory, representation, verifier, planner. A composite factuality repair may have the form ω_gate ∘ ( ω_retrieve, ω_decompose, ω_check ). The inner operations retrieve evidence, decompose output into claims, and check claims against protected sources. Their postconditions establish the precondition for a planner gate that blocks or revises unsupported claims. The composite receipt records source provenance, claim-to-evidence links, verifier uncertainty, revised spans, latenc…
Exact mathematical passage
\mathrm{memory}, \mathrm{representation}, \mathrm{verifier}, \mathrm{planner}.
\mathrm{research}Use in source · 1 passage
Search aliases: research mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 14070
Every parent and child run receives exactly B_research=6 full candidate experiments. Candidate metadata and cheap diagnostics are common inputs. The number of proposal receipts, scored evaluator calls, capability checks, and integrity-monitor calls is therefore matched.
Exact mathematical passage
B_{\mathrm{research}}=6 \label{eq:b13-budget}
\mathrm{Residual}Use in source · 1 passage
Search aliases: Residual mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17678
Let A be a finite architecture index set and C a finite admitted chart index set. In the released realization, A = {Elman,GRU,Residual}, C = {A0,A1,A2,A3,N0,N1}. Each pair (a,c)∈ A× C carries a full Neuralese trajectory fiber H_a,c ≅ ( R^40)^5 ≅ R^200. The observation atlas determines which context fiber is authorized. A task outside every admitted chart is not assigned to an arbitrary fiber.
Exact mathematical passage
\mathcal A = \{\mathrm{Elman},\mathrm{GRU},\mathrm{Residual}\}, \qquad \mathcal C = \{A0,A1,A2,A3,N0,N1\}.
\mathrm{retrieve}Use in source · 1 passage
Search aliases: retrieve mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1485
Consider four colors memory, representation, verifier, planner. A composite factuality repair may have the form ω_gate ∘ ( ω_retrieve, ω_decompose, ω_check ). The inner operations retrieve evidence, decompose output into claims, and check claims against protected sources. Their postconditions establish the precondition for a planner gate that blocks or revises unsupported claims. The composite receipt records source provenance, claim-to-evidence links, verifier uncertainty, revised spans, latenc…
Exact mathematical passage
\omega_{\mathrm{gate}} \circ \left( \omega_{\mathrm{retrieve}}, \omega_{\mathrm{decompose}}, \omega_{\mathrm{check}} \right).
\mathrm{rev}Use in source · 1 passage
Search aliases: rev mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16789
s L_i, the audit reuses the Block 17 donor distinctions and adds cross-implementation status labels. It records latent necessity, reversal cost, norm-matched noise cost, exact replay, same-answer far-program transplant cost, far-answer matched-depth transplant cost, local semantic-neighbor cost, external trace-probe R^2, dynamic-depth correlation, and exact program execution. Let n_i · =L_i^0-L_i, · o_i · =L_i^rev-L_i, · c_i · =L_i^η-L_i, p_i · =L_i,sem^same answer,far program -L_i,sem, · a_i · …
Exact mathematical passage
n_i&=L_i^{0}-L_i, &o_i&=L_i^{\mathrm{rev}}-L_i, &c_i&=L_i^{\eta}-L_i, \\ p_i&=L_{i,\mathrm{sem}}^{\mathrm{same\ answer,far\ program}} -L_{i,\mathrm{sem}}, &a_i&=L_{i,\mathrm{answer}}^{\mathrm{far\ answer,depth}} -L_{i,\mathrm{answer}}. \label{eq:b18-audit-coordinates}
\mathrm{reveal}Use in source · 1 passage
Search aliases: reveal mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4027
The transaction follows six phases: prepare ⟶ seal ⟶ execute ⟶ reveal ⟶ decide ⟶ finalize. Only the final phase may expose buffered effects or replace the active checkpoint.
Exact mathematical passage
\mathrm{prepare} \longrightarrow \mathrm{seal} \longrightarrow \mathrm{execute} \longrightarrow \mathrm{reveal} \longrightarrow \mathrm{decide} \longrightarrow \mathrm{finalize}.
\mathsf{reverse}Use in source · 1 passage
Search aliases: reverse mathsf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16213
Let ℓ_i(Z) be mean squared witness error on recipient i when the four latent states are externally supplied as Z. Let Z_i be the clean trace. The structural coordinates are n_i · =ℓ_i(zero)-ℓ_i(Z_i), o_i · =ℓ_i(reverse(Z_i))-ℓ_i(Z_i), c_i · =ℓ_i(noise(Z_i))-ℓ_i(Z_i), r_i · =absℓ_i(replay(Z_i))-ℓ_i(Z_i). Noise is deterministic and norm matched at every step. The program and answer specificity coordinates are the mean recipient costs under SAP and FAD donors: p_i = E_j∈SAP(i) [ℓ_i(Z_j)-ℓ_i(Z_i)], …
Exact mathematical passage
n_i&=\ell_i(\mathsf{zero})-\ell_i(Z_i), \label{eq:b17-necessity}\\ o_i&=\ell_i(\mathsf{reverse}(Z_i))-\ell_i(Z_i), \label{eq:b17-order}\\ c_i&=\ell_i(\mathsf{noise}(Z_i))-\ell_i(Z_i), \label{eq:b17-noise}\\ r_i&=\abs{\ell_i(\mathsf{replay}(Z_i))-\ell_i(Z_i)}. \label{eq:b17-replay}
\rfloorUse in source · 2 passages
Search aliases: rfloor
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 610
[Fixed-margin evaluator ceiling] Let V: S→[0,1] be fixed. If every accepted transition satisfies V(A_t+1)≥ V(A_t)+Δ for a constant Δ>0, then every accepted chain has length at most lfloor 1/Δrfloor+1.
Exact mathematical passage
\lfloor 1/\Delta\rfloor+1
\mathrm{RI}Use in source · 6 passages
Search aliases: RI mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4259
Let C_RI(s) = · Grounded(s) wedgeProvValid(s) wedgeWorkspaceBound(s) · wedgeRepairTyped(s) wedgeWitnessed(s) wedgeEvidenceFlat(s) · wedgeRollbackable(s) wedgeMetaCapable(s). For X⊆ S, define Ψ_∀(X) = sets∈ S:C_RI(s) and Post_imp(s)⊆ X.
Exact mathematical passage
\begin{aligned} C_{\mathrm{RI}}(s) ={}& \Grounded(s) \wedge\ProvValid(s) \wedge\WorkspaceBound(s)\\ &\wedge\RepairTyped(s) \wedge\Witnessed(s) \wedge\EvidenceFlat(s)\\ &\wedge\Rollbackable(s) \wedge\MetaCapable(s). \end{aligned}
\rightsquigarrowUse in source · 1 passage
Search aliases: rightsquigarrow
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 696
Stochastic channel · K:Xrightsquigarrow Y · Distributional transport only; pointwise identity is not implied.
Exact mathematical passage
K:X\rightsquigarrow Y
\mathrm{risky}Use in source · 1 passage
Search aliases: risky mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21064
Let S={s_0,s_1,b}, C_RI={s_0,s_1}. At s_0 there are two authorized actions: a_safe:s_0↦ s_1, a_risky:s_0↦ b. At s_1 there is one action with successor s_1. Universal closure excludes s_0 because one authorized successor is unsafe. Controlled closure includes s_0 because the policy can choose a_safe. This is the concrete distinction between safety of every authorized action and existence of one robustly safe action.
Exact mathematical passage
a_{\mathrm{safe}}:s_0\mapsto s_1, \qquad a_{\mathrm{risky}}:s_0\mapsto b.
\mathsf{RMD}Use in source · 1 passage
Search aliases: RMD mathsf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16184
and let d_ans(τ,τ') = absy_m/4-y'_m'/4. For a recipient task τ_i, define four donor classes: item SAP: same-answer, far-program, approximately depth-matched donors; item FAD: far-answer, depth-matched donors; item SN: semantic-neighbor donors; item RMD: random depth-matched donors. Three donors per available class are selected by deterministic task-identifier hashing. Each audit records donor coverage and the realized semantic and answer distances.
Exact mathematical passage
\mathsf{RMD}
\RollbackableUse in source · 1 passage
Search aliases: Rollbackable mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4259
Let C_RI(s) = · Grounded(s) wedgeProvValid(s) wedgeWorkspaceBound(s) · wedgeRepairTyped(s) wedgeWitnessed(s) wedgeEvidenceFlat(s) · wedgeRollbackable(s) wedgeMetaCapable(s). For X⊆ S, define Ψ_∀(X) = sets∈ S:C_RI(s) and Post_imp(s)⊆ X.
Exact mathematical passage
\begin{aligned} C_{\mathrm{RI}}(s) ={}& \Grounded(s) \wedge\ProvValid(s) \wedge\WorkspaceBound(s)\\ &\wedge\RepairTyped(s) \wedge\Witnessed(s) \wedge\EvidenceFlat(s)\\ &\wedge\Rollbackable(s) \wedge\MetaCapable(s). \end{aligned}
\mathcal SUse in source · 4 passages
Search aliases: S mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 593
[Evaluator-capture triviality] Let S be a candidate set with at least two elements, and suppose the admissible evaluator class contains every function V: S→[0,1]. For every candidate B∈ S, there exists an evaluator V_B under which B is the unique maximizer.
Exact mathematical passage
\mathcal S
s'Use in source · 5 passages
Search aliases: s'
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4248
Let S be a finite set of complete lineage states. Write sxrightarrowτ_imps' when one authorized improvement transaction can take s to s'. Define Post_imp(s) = sets'∈ S:sxrightarrowτ_imps'.
Exact mathematical passage
s\xrightarrow{\tau_{\mathrm{imp}}}s'
S_0Use in source · 2 passages
Search aliases: S_0
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5183
[Finite repair-discovery closure] Under discovery monotonicity, Φ_disc has a least fixed point above any seed set S_0. Iteration S_k+1=Φ_disc(S_k) stabilizes after at most absΩ-absS_0 strict expansions.
Exact mathematical passage
S_0
s_0Use in source · 8 passages
Search aliases: s_0
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21058
Let S={s_0,s_1,b}, C_RI={s_0,s_1}. At s_0 there are two authorized actions: a_safe:s_0↦ s_1, a_risky:s_0↦ b. At s_1 there is one action with successor s_1. Universal closure excludes s_0 because one authorized successor is unsafe. Controlled closure includes s_0 because the policy can choose a_safe. This is the concrete distinction between safety of every authorized action and existence of one robustly safe action.
Exact mathematical passage
S=\{s_0,s_1,b\}, \qquad C_{\mathrm{RI}}=\{s_0,s_1\}.
s_1Use in source · 13 passages
Search aliases: s_1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21058
Let S={s_0,s_1,b}, C_RI={s_0,s_1}. At s_0 there are two authorized actions: a_safe:s_0↦ s_1, a_risky:s_0↦ b. At s_1 there is one action with successor s_1. Universal closure excludes s_0 because one authorized successor is unsafe. Controlled closure includes s_0 because the policy can choose a_safe. This is the concrete distinction between safety of every authorized action and existence of one robustly safe action.
Exact mathematical passage
S=\{s_0,s_1,b\}, \qquad C_{\mathrm{RI}}=\{s_0,s_1\}.
s_2Use in source · 2 passages
Search aliases: s_2
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21471
Consider a transaction graph with states s_0,s_1,s_2. Suppose s_0→ s_1 is a validated positive parent--child edge, but every authorized successor of s_1 either returns to s_1 with zero gain or enters unsafe s_2. Then s_0→ s_1 witnesses one-step metaproductivity, but s_1 does not lie in a closure requiring infinitely repeated positive gains. A grandchild audit is logically necessary to distinguish a one-edge result from recurrent improvement capacity.
Exact mathematical passage
s_0,s_1,s_2
s_{a^*}Use in source · 2 passages
Search aliases: s_a^*
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12367
[Finite repair-relevant block selection] Suppose there is a unique repair-relevant block a^* and a margin m>0 such that s_a^*≥ s_a+m for every a≠ a^*. If max_a|widehat s_a-s_a|<fracm2, then arg max_awidehat s_a=a^*.
Exact mathematical passage
s_{a^*}\geq s_a+m
s_{\mathrm{base}}Use in source · 10 passages
Search aliases: s_base mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5669
The minimal closure model has states S={s_base,s_improved,s_invalid} and valid set K={s_base,s_improved}. With intact rollback, authorized complete transactions have successor relation Post(s_base) ={s_base,s_improved}, Post(s_improved)={s_improved}. Failed candidates are restored to the base state before the complete transaction ends.
Exact mathematical passage
S=\{s_{\mathrm{base}},s_{\mathrm{improved}},s_{\mathrm{invalid}}\}
s_cUse in source · 2 passages
Search aliases: s_c
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8567
For a candidate configuration c, the development compiler computes target-level selected gains and the score S_rm dev(c) = bar g_c -0.20s_c -2h_c +0.02q_c, where bar g_c is mean protected gain, s_c is the target-level standard deviation, h_c is harmful-target rate, and q_c is positive-target rate. This is a finite engineering objective, not a universal definition of evolvability.
Exact mathematical passage
S_{\rm dev}(c) = \bar g_c -0.20s_c -2h_c +0.02q_c, \label{eq:block07q-compiler-score}
s_DUse in source · 1 passage
Search aliases: s_D
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21419
On the seed/task audit, the parent and child receive the same five candidates on 120 episodes. Let D_j be the child-minus-parent protected utility on episode j. The frozen values satisfy overline D=0.072518, s_D=0.154617, widehatrm se=0.014115. The ordinary normal approximation gives 0.072518±1.98(0.014115) ≈ [0.0446,0.1005], close to the frozen paired bootstrap interval [0.046745,0.101389].
Exact mathematical passage
\overline D=0.072518, \qquad s_D=0.154617, \qquad \widehat{\rm se}=0.014115.
s_i^TUse in source · 1 passage
Search aliases: s_i^T
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2536
[Fundamental-cycle holonomy criterion] For each root state s_o∈ S_o, tree transport determines a unique family (s_i^T). For every non-tree edge e=(i,j), let H_e:S_o→ S_o be transport around the corresponding fundamental cycle. The tree family extends to a globally compatible family on all edges if and only if H_e(s_o)=s_o for every non-tree edge e.
Exact mathematical passage
(s_i^T)
s_{\mathrm{improved}}Use in source · 4 passages
Search aliases: s_improved mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5669
The minimal closure model has states S={s_base,s_improved,s_invalid} and valid set K={s_base,s_improved}. With intact rollback, authorized complete transactions have successor relation Post(s_base) ={s_base,s_improved}, Post(s_improved)={s_improved}. Failed candidates are restored to the base state before the complete transaction ends.
Exact mathematical passage
S=\{s_{\mathrm{base}},s_{\mathrm{improved}},s_{\mathrm{invalid}}\}
s_{\mathrm{invalid}}Use in source · 3 passages
Search aliases: s_invalid mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5669
The minimal closure model has states S={s_base,s_improved,s_invalid} and valid set K={s_base,s_improved}. With intact rollback, authorized complete transactions have successor relation Post(s_base) ={s_base,s_improved}, Post(s_improved)={s_improved}. Failed candidates are restored to the base state before the complete transaction ends.
Exact mathematical passage
S=\{s_{\mathrm{base}},s_{\mathrm{improved}},s_{\mathrm{invalid}}\}
S_jUse in source · 1 passage
Search aliases: S_j
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2527
A finite operational version avoids explicit cohomology. Let G=(I,E) be connected, with sets of local solutions S_i and invertible transition maps T_ij:S_i→ S_j, T_ji=T_ij^-1. Choose a root o and a spanning tree T⊆ G.
Exact mathematical passage
T_{ij}:S_i\to S_j, \qquad T_{ji}=T_{ij}^{-1}.
s_jUse in source · 1 passage
Search aliases: s_j
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13798
Let the scored evaluator portfolio be B = {b_1,…,b_m}. For method c, define normalized headroom closure q_j(c) = fracs_j(c)-s_j(K_τ)s_j^⋆-s_j(K_τ). Let I_+(c)={j:q_j(c)>0}. The coverage-weighted geometric score is S_cov(c) = frac|I_+(c)|m ( ∏_j∈ I_+(c)q_j(c) )^1/|I_+(c)|, with value zero when I_+(c)=∅. Hard capability and integrity gates are applied separately rather than traded against this scalar.
Exact mathematical passage
q_j(c) = \frac{s_j(c)-s_j(K_\tau)}{s_j^{\star}-s_j(K_\tau)}. \label{eq:b13-headroom-closure}
s_j^{\star}Use in source · 1 passage
Search aliases: s_j^⋆ star
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13798
Let the scored evaluator portfolio be B = {b_1,…,b_m}. For method c, define normalized headroom closure q_j(c) = fracs_j(c)-s_j(K_τ)s_j^⋆-s_j(K_τ). Let I_+(c)={j:q_j(c)>0}. The coverage-weighted geometric score is S_cov(c) = frac|I_+(c)|m ( ∏_j∈ I_+(c)q_j(c) )^1/|I_+(c)|, with value zero when I_+(c)=∅. Hard capability and integrity gates are applied separately rather than traded against this scalar.
Exact mathematical passage
q_j(c) = \frac{s_j(c)-s_j(K_\tau)}{s_j^{\star}-s_j(K_\tau)}. \label{eq:b13-headroom-closure}
S_kUse in source · 1 passage
Search aliases: S_k
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5184
[Finite repair-discovery closure] Under discovery monotonicity, Φ_disc has a least fixed point above any seed set S_0. Iteration S_k+1=Φ_disc(S_k) stabilizes after at most absΩ-absS_0 strict expansions.
Exact mathematical passage
S_{k+1}=\Phi_{\mathrm{disc}}(S_k)
s_kUse in source · 7 passages
Search aliases: s_k
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16093
Let ϕ(τ)∈ℝ^19 be the padded task serialization, and let e=E_θ(ϕ(τ))∈ℝ^40 be the prompt representation. The latent trace has four states. Given conditioning mode α, define z_0=ζ, z_k+1 = tanh (Rz_k+s_k+C_α(e,m)), 0≤ k<4, where s_k is a learned step embedding. The task-blind parent satisfies C_parent(e,m)=0. The descendants use one of the following conditioning maps: C_depth(e,m) · =D 1_m,
Exact mathematical passage
z_0=\zeta, \qquad z_{k+1} = \tanh\!\left(Rz_k+s_k+C_\alpha(e,m)\right), \qquad 0\leq k<4, \label{eq:b17-trace}
s_k'Use in source · 1 passage
Search aliases: s_k'
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16497
Let P be a signed permutation matrix on the latent space. Define transformed latent parameters ζ'=Pζ, R'=PRP^ T, s_k'=Ps_k, C_α'=PC_α, and transform every latent-consuming linear map by right multiplication with P^ T. Then the transformed model has z_k'=Pz_k and identical external outputs.
Exact mathematical passage
\zeta'=P\zeta, \quad R'=PRP^{\mathsf T}, \quad s_k'=Ps_k, \quad C_\alpha'=PC_\alpha, \label{eq:b17-gauge-trace}
S_{k+1}Use in source · 1 passage
Search aliases: S_k+1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5184
[Finite repair-discovery closure] Under discovery monotonicity, Φ_disc has a least fixed point above any seed set S_0. Iteration S_k+1=Φ_disc(S_k) stabilizes after at most absΩ-absS_0 strict expansions.
Exact mathematical passage
S_{k+1}=\Phi_{\mathrm{disc}}(S_k)
S_oUse in source · 2 passages
Search aliases: S_o
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2536
[Fundamental-cycle holonomy criterion] For each root state s_o∈ S_o, tree transport determines a unique family (s_i^T). For every non-tree edge e=(i,j), let H_e:S_o→ S_o be transport around the corresponding fundamental cycle. The tree family extends to a globally compatible family on all edges if and only if H_e(s_o)=s_o for every non-tree edge e.
Exact mathematical passage
s_o\in S_o
s_{r,c}Use in source · 1 passage
Search aliases: s_r,c
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7406
widehatβ_r · = (Z^top Z+λ_r I)^-1Z^top r. For candidate type c, let s_u,c and s_r,c be the development residual standard deviations for the utility and risk heads. The deployed scores are widehat U_rm LCB(z) · = widehat U(z)-κ_u s_u,c(z),
Exact mathematical passage
s_{r,c}
s_{r,c(z)}Use in source · 1 passage
Search aliases: s_r,c(z)
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7407
widehatβ_r · = (Z^top Z+λ_r I)^-1Z^top r. For candidate type c, let s_u,c and s_r,c be the development residual standard deviations for the utility and risk heads. The deployed scores are widehat U_rm LCB(z) · = widehat U(z)-κ_u s_u,c(z), widehat R_rm UCB(z) · = max{0,widehat R(z)}+κ_r s_r,c(z). A candidate is admissible when widehat R_rm UCB(z)≤τ_r, widehat r_rm cal(z)≤τ_rm cal, A^rm base_rm cal,prot(z)≥ a_min. The child chooses the admissible candidate with maximum widehat U_rm LCB. If the adm…
Exact mathematical passage
\widehat U_{\rm LCB}(z) &= \widehat U(z)-\kappa_u s_{u,c(z)}, \label{eq:block07-utility-lcb}\\ \widehat R_{\rm UCB}(z) &= \max\{0,\widehat R(z)\}+\kappa_r s_{r,c(z)}. \label{eq:block07-risk-ucb}
S_{\rm dev}Use in source · 1 passage
Search aliases: S_rm dev rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8567
For a candidate configuration c, the development compiler computes target-level selected gains and the score S_rm dev(c) = bar g_c -0.20s_c -2h_c +0.02q_c, where bar g_c is mean protected gain, s_c is the target-level standard deviation, h_c is harmful-target rate, and q_c is positive-target rate. This is a finite engineering objective, not a universal definition of evolvability.
Exact mathematical passage
S_{\rm dev}(c) = \bar g_c -0.20s_c -2h_c +0.02q_c, \label{eq:block07q-compiler-score}
s_{u,c}Use in source · 1 passage
Search aliases: s_u,c
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7406
widehatβ_r · = (Z^top Z+λ_r I)^-1Z^top r. For candidate type c, let s_u,c and s_r,c be the development residual standard deviations for the utility and risk heads. The deployed scores are widehat U_rm LCB(z) · = widehat U(z)-κ_u s_u,c(z),
Exact mathematical passage
s_{u,c}
s_{u,c(z)}Use in source · 1 passage
Search aliases: s_u,c(z)
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7407
widehatβ_r · = (Z^top Z+λ_r I)^-1Z^top r. For candidate type c, let s_u,c and s_r,c be the development residual standard deviations for the utility and risk heads. The deployed scores are widehat U_rm LCB(z) · = widehat U(z)-κ_u s_u,c(z), widehat R_rm UCB(z) · = max{0,widehat R(z)}+κ_r s_r,c(z). A candidate is admissible when widehat R_rm UCB(z)≤τ_r, widehat r_rm cal(z)≤τ_rm cal, A^rm base_rm cal,prot(z)≥ a_min. The child chooses the admissible candidate with maximum widehat U_rm LCB. If the adm…
Exact mathematical passage
\widehat U_{\rm LCB}(z) &= \widehat U(z)-\kappa_u s_{u,c(z)}, \label{eq:block07-utility-lcb}\\ \widehat R_{\rm UCB}(z) &= \max\{0,\widehat R(z)\}+\kappa_r s_{r,c(z)}. \label{eq:block07-risk-ucb}
S_{\Vcal}Use in source · 1 passage
Search aliases: S_Vcal S_V Vcal mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13712
Let Valid(c_i,Π_i^+)∈{0,1} encode capability, integrity, provenance, and resource validity. The selected descendant is K_Γ^⋆(τ) ∈ arg max_i:Valid(c_i,Π_i^+)=1 S_V(c_i), with the no-op baseline retained when no valid candidate exists.
Exact mathematical passage
K_{\Gamma}^{\star}(\tau) \in \argmax_{i:\operatorname{Valid}(c_i,\Pi_i^+)=1} S_{\Vcal}(c_i), \label{eq:b13-selected-method}
s_\alphaUse in source · 4 passages
Search aliases: s_α alpha
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2491
Let mathscr G be a sheaf of groups acting freely and transitively on the sheaf of local nonlinear solutions mathscr S. Choose local solutions s_α∈mathscr S(U_α). On overlaps there are unique transition elements g_αβ∈mathscr G(U_αβ) such that s_α=g_αβ· s_β.
Exact mathematical passage
s_\alpha\in\mathscr S(U_\alpha)
s_\betaUse in source · 1 passage
Search aliases: s_β beta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2496
Let mathscr G be a sheaf of groups acting freely and transitively on the sheaf of local nonlinear solutions mathscr S. Choose local solutions s_α∈mathscr S(U_α). On overlaps there are unique transition elements g_αβ∈mathscr G(U_αβ) such that s_α=g_αβ· s_β.
Exact mathematical passage
s_\alpha=g_{\alpha\beta}\cdot s_\beta.
\mathrm{safe}Use in source · 2 passages
Search aliases: safe mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21064
Let S={s_0,s_1,b}, C_RI={s_0,s_1}. At s_0 there are two authorized actions: a_safe:s_0↦ s_1, a_risky:s_0↦ b. At s_1 there is one action with successor s_1. Universal closure excludes s_0 because one authorized successor is unsafe. Controlled closure includes s_0 because the policy can choose a_safe. This is the concrete distinction between safety of every authorized action and existence of one robustly safe action.
Exact mathematical passage
a_{\mathrm{safe}}:s_0\mapsto s_1, \qquad a_{\mathrm{risky}}:s_0\mapsto b.
\mathrm{same\ answer,far\ program}Use in source · 1 passage
Search aliases: same answer,far program mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16789
s L_i, the audit reuses the Block 17 donor distinctions and adds cross-implementation status labels. It records latent necessity, reversal cost, norm-matched noise cost, exact replay, same-answer far-program transplant cost, far-answer matched-depth transplant cost, local semantic-neighbor cost, external trace-probe R^2, dynamic-depth correlation, and exact program execution. Let n_i · =L_i^0-L_i, · o_i · =L_i^rev-L_i, · c_i · =L_i^η-L_i, p_i · =L_i,sem^same answer,far program -L_i,sem, · a_i · …
Exact mathematical passage
n_i&=L_i^{0}-L_i, &o_i&=L_i^{\mathrm{rev}}-L_i, &c_i&=L_i^{\eta}-L_i, \\ p_i&=L_{i,\mathrm{sem}}^{\mathrm{same\ answer,far\ program}} -L_{i,\mathrm{sem}}, &a_i&=L_{i,\mathrm{answer}}^{\mathrm{far\ answer,depth}} -L_{i,\mathrm{answer}}. \label{eq:b18-audit-coordinates}
\mathrm{sand}Use in source · 2 passages
Search aliases: sand mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 811
A witness structure may be represented as W_t = ( V_t^fr, C_t, B_t^sand, R_t, Π_t^auth ), where V_t^fr is the frozen evaluator instance for the current transaction, C_t is a protected checkpoint, B_t^sand is the sandbox boundary, R_t is the rollback procedure, and Π_t^auth records effect authority.
Exact mathematical passage
\Wcal_t = \left( \Vcal_t^{\mathrm{fr}}, C_t, \Bcal_t^{\mathrm{sand}}, R_t, \Pi_t^{\mathrm{auth}} \right),
\mathsf{SAP}Use in source · 5 passages
Search aliases: SAP mathsf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16181
Let d_prog(τ,τ') be a normalized distance between (A_m,B_m) and (A'_m',B'_m'), and let d_ans(τ,τ') = absy_m/4-y'_m'/4. For a recipient task τ_i, define four donor classes: item SAP: same-answer, far-program, approximately depth-matched donors; item FAD: far-answer, depth-matched donors; item SN: semantic-neighbor donors; item RMD: random depth-matched donors. Three donors per available class are selected by deterministic task-identifier hashing. Each audit records donor coverage and the realized…
Exact mathematical passage
\mathsf{SAP}
\operatorname{Score}Use in source · 1 passage
Search aliases: Score operatorname
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13011
Let mathscr S be a finite typed candidate-intervention grammar. For candidate S∈mathscr S, define a development score Score_resp(S) = widehat M_foreign(S) -λ_cC(S) -λ_commε_comm(S) -λ_fitε_fit(S) -λ_riskwidehat R(S). The compiler selects a candidate only after verifying that it adds a nontrivial response-signature distinction and preserves the inherited response language as an executable subgrammar.
Exact mathematical passage
\operatorname{Score}_{\mathrm{resp}}(S) = \widehat M_{\mathrm{foreign}}(S) -\lambda_{c}C(S) -\lambda_{\mathrm{comm}}\eps_{\mathrm{comm}}(S) -\lambda_{\mathrm{fit}}\eps_{\mathrm{fit}}(S) -\lambda_{\mathrm{risk}}\widehat R(S). \label{eq:b12-response-score}
\operatorname{Score}_{\mathrm{resp}}Use in source · 1 passage
Search aliases: Score_resp operatorname mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13011
Let mathscr S be a finite typed candidate-intervention grammar. For candidate S∈mathscr S, define a development score Score_resp(S) = widehat M_foreign(S) -λ_cC(S) -λ_commε_comm(S) -λ_fitε_fit(S) -λ_riskwidehat R(S). The compiler selects a candidate only after verifying that it adds a nontrivial response-signature distinction and preserves the inherited response language as an executable subgrammar.
Exact mathematical passage
\operatorname{Score}_{\mathrm{resp}}(S) = \widehat M_{\mathrm{foreign}}(S) -\lambda_{c}C(S) -\lambda_{\mathrm{comm}}\eps_{\mathrm{comm}}(S) -\lambda_{\mathrm{fit}}\eps_{\mathrm{fit}}(S) -\lambda_{\mathrm{risk}}\widehat R(S). \label{eq:b12-response-score}
\operatorname{sd}Use in source · 1 passage
Search aliases: sd operatorname
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17178
The implementation uses d_sig(u,v) = frac u-v _2/sqrt5 max{sd(v),10^-8}. The runtime receives only the observed support and values. Context names are retained in the data solely for auditing route accuracy.
Exact mathematical passage
d_{\mathrm{sig}}(u,v) = \frac{\|u-v\|_2/\sqrt5} {\max\{\operatorname{sd}(v),10^{-8}\}}. \label{eq:b19-signature-distance}
\mathrm{seal}Use in source · 1 passage
Search aliases: seal mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4027
The transaction follows six phases: prepare ⟶ seal ⟶ execute ⟶ reveal ⟶ decide ⟶ finalize. Only the final phase may expose buffered effects or replace the active checkpoint.
Exact mathematical passage
\mathrm{prepare} \longrightarrow \mathrm{seal} \longrightarrow \mathrm{execute} \longrightarrow \mathrm{reveal} \longrightarrow \mathrm{decide} \longrightarrow \mathrm{finalize}.
\mathrm{select}Use in source · 1 passage
Search aliases: select mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5414
The data partition is itself typed: D = D_construct sqcup D_calibrate sqcup D_select sqcup D_foreign sqcup D_audit. Construction data may be used to create schemas. Calibration data may set thresholds. Selection data may choose among frozen candidates. Foreign data estimate metaproductivity. Audit data are not available until the final claim object has been sealed.
Exact mathematical passage
\mathsf D = \mathsf D_{\mathrm{construct}} \sqcup \mathsf D_{\mathrm{calibrate}} \sqcup \mathsf D_{\mathrm{select}} \sqcup \mathsf D_{\mathrm{foreign}} \sqcup \mathsf D_{\mathrm{audit}}.
\mathrm{Sem}Use in source · 1 passage
Search aliases: Sem mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13751
The forecast residual and proposal--code congruence residual are ε_forecast,i = normwidehat c_i- c_i, ε_spec/code,i = d_sem ( Sem(Π_i^-), Sem(code_i) ). In the executable finite companion, congruence is a Boolean monitor rather than a complete semantic metric.
Exact mathematical passage
\eps_{\mathrm{forecast},i} = \norm{\widehat{\mathbf c}_i-\mathbf c_i}, \qquad \eps_{\mathrm{spec/code},i} = d_{\mathrm{sem}} \left( \mathrm{Sem}(\Pi_i^-), \mathrm{Sem}(\mathrm{code}_i) \right). \label{eq:b13-receipt-residuals}
\mathrm{sensor}Use in source · 2 passages
Search aliases: sensor mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5935
The latent event variable is z∈ {none,internal,world,sensor,memory,evaluator}. Each event changes one declared mechanism: item internal: the action-response map B changes; item world: the passive dynamics A change; item sensor: the redundant sensor receives an additive bias; item memory: the retention coefficient λ changes; item evaluator: the reported weight vector w changes; item none: all baseline mechanisms remain fixed. The event label is not supplied to the classifier.
Exact mathematical passage
z\in \{\mathrm{none},\mathrm{internal},\mathrm{world},\mathrm{sensor},\mathrm{memory},\mathrm{evaluator}\}.
\operatorname{sgn}Use in source · 1 passage
Search aliases: sgn operatorname
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20650
b. Therefore η_D(r)=|b|. For b≠0, the unit dual witness is y^*=sgn(b)e_2. Every authorized correction has zero effect on the second defect coordinate.
Exact mathematical passage
y^*=\operatorname{sgn}(b)e_2
\sinUse in source · 3 passages
Search aliases: sin
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6538
Let x=(x_1,x_2,x_3,x_4)∈ℝ^4. Define the eight-dimensional base feature ψ(x) = x_1 · x_2 · x_3 · x_4 · x_1x_2 · x_3x_4 · sin x_1 · cos x_2 ^ top ∈ℝ^8, and the affected-stratum indicator s(x)=1{x_1+x_2>0.8}. The full feature map is ϕ(x) = ψ(x)
Exact mathematical passage
\psi(x) = \begin{bmatrix} x_1 & x_2 & x_3 & x_4 & x_1x_2 & x_3x_4 & \sin x_1 & \cos x_2 \end{bmatrix}^{\!\top} \in\R^8,
\sinhUse in source · 4 passages
Search aliases: sinh
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 18270
The nonlinear family applies Y_t = fracsinh (α(X_tA+b))α, α>0, coordinatewise. The declared inverse constructor first applies ϕ^-1_α(y) = fracarsinh(α y)α and then fits a step-affine map.
Exact mathematical passage
Y_t = \frac{\sinh\!\left(\alpha(X_tA+b)\right)}{\alpha}, \qquad \alpha>0, \label{eq:block21-nonlinear}
\mathsf{SN}Use in source · 1 passage
Search aliases: SN mathsf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16183
ce between (A_m,B_m) and (A'_m',B'_m'), and let d_ans(τ,τ') = absy_m/4-y'_m'/4. For a recipient task τ_i, define four donor classes: item SAP: same-answer, far-program, approximately depth-matched donors; item FAD: far-answer, depth-matched donors; item SN: semantic-neighbor donors; item RMD: random depth-matched donors. Three donors per available class are selected by deterministic task-identifier hashing. Each audit records donor coverage and the realized semantic and answer distances.
Exact mathematical passage
\mathsf{SN}
\operatorname{span}Use in source · 1 passage
Search aliases: span operatorname
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 22089
Evaluator-nullspace shrink. Let T=ℝ^2, Db_1(x_1,x_2)=x_1, and Db_2(x_1,x_2)=x_2. With only b_1, the invisible subspace is span(e_2). Adding b_2 reduces the intersection of kernels to {0}. The new benchmark contributes an independent behavioral distinction rather than merely rescaling the old score.
Exact mathematical passage
\operatorname{span}(e_2)
\mathrm{Spearman}Use in source · 1 passage
Search aliases: Spearman mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15802
halt mass M_t(x) = ∑_j=1^t λ_j(x) ∏_q<j(1-λ_q(x)), and dynamic depth T(x)=min{t:M_t(x)≥τ}, with declared threshold τ. Block 16 does not infer adaptive computation from the existence of a halt head. It requires a difficulty variable d(x) fixed before result access and tests both ρ_Spearman(d(x),T(x))>0 and nondegenerate prompt variation in the halt schedules. A constant depth can coexist with fully task-specific Neuralese; conversely, variable depth can be driven by nuisance prompt features. The …
Exact mathematical passage
\rho_{\mathrm{Spearman}}(d(x),T(x))>0
\mathrm{spec/code}Use in source · 1 passage
Search aliases: spec/code mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13751
The forecast residual and proposal--code congruence residual are ε_forecast,i = normwidehat c_i- c_i, ε_spec/code,i = d_sem ( Sem(Π_i^-), Sem(code_i) ). In the executable finite companion, congruence is a Boolean monitor rather than a complete semantic metric.
Exact mathematical passage
\eps_{\mathrm{forecast},i} = \norm{\widehat{\mathbf c}_i-\mathbf c_i}, \qquad \eps_{\mathrm{spec/code},i} = d_{\mathrm{sem}} \left( \mathrm{Sem}(\Pi_i^-), \mathrm{Sem}(\mathrm{code}_i) \right). \label{eq:b13-receipt-residuals}
\sqcupUse in source · 4 passages
Search aliases: sqcup
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5414
The data partition is itself typed: D = D_construct sqcup D_calibrate sqcup D_select sqcup D_foreign sqcup D_audit. Construction data may be used to create schemas. Calibration data may set thresholds. Selection data may choose among frozen candidates. Foreign data estimate metaproductivity. Audit data are not available until the final claim object has been sealed.
Exact mathematical passage
\mathsf D = \mathsf D_{\mathrm{construct}} \sqcup \mathsf D_{\mathrm{calibrate}} \sqcup \mathsf D_{\mathrm{select}} \sqcup \mathsf D_{\mathrm{foreign}} \sqcup \mathsf D_{\mathrm{audit}}.
\mathrm{square}Use in source · 1 passage
Search aliases: square mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17934
A candidate is pre-audit eligible only if both foreign target models are ATLAS_FULL and the candidate jointly satisfies R^2_transport · ≥0.75, · R^2_worst chart · ≥0.74, ρ_preserve · ≥0.98, · NRMSE_trace · ≤0.05, κ_square · ≤0.02. · · The selection score is the weakest normalized margin across these gates. This prevents a strong aggregate R^2 from compensating for a failed chart, high curvature, or poor trajectory reconstruction.
Exact mathematical passage
\begin{aligned} R^2_{\mathrm{transport}}&\geq0.75, &\qquad R^2_{\mathrm{worst\ chart}}&\geq0.74,\\ \rho_{\mathrm{preserve}}&\geq0.98, & \operatorname{NRMSE}_{\mathrm{trace}}&\leq0.05,\\ \kappa_{\mathrm{square}}&\leq0.02. && \end{aligned} \label{eq:b20-thresholds}
\mathrm{src}Use in source · 3 passages
Search aliases: src mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16840
Let (Y,H_s,H_t)∼μ be the protected semantics, source trace, and target trace under context distribution μ. Fix a source probe P and bridge B. Define e_src=Y-PH_s, e_align=P(H_s-BH_t). Then Y-PBH_t=e_src+e_align.
Exact mathematical passage
e_{\mathrm{src}}=Y-PH_s, \qquad e_{\mathrm{align}}=P(H_s-BH_t). \label{eq:b18-error-parts}
\Sscr_\alphaUse in source · 1 passage
Search aliases: Sscr_α mathscrS_α Sscr alpha mathscr alpha
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2251
[Local solution torsor] For each U_α, the set mathscrS_α := setu∈mathscr U(U_α):D_α u=-r|_U_α is either empty or a torsor for the additive group ker D(U_α). If a global repair exists, the global solution set is a torsor for Γ(X,ker D).
Exact mathematical passage
\Sscr_\alpha := \set{u\in\mathscr U(U_\alpha):D_\alpha u=-r|_{U_\alpha}}
\stackrelUse in source · 1 passage
Search aliases: stackrel
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15757
[Format-conditional transplant null] Let G(x) be a declared answer-format class. Suppose that, conditional on G(x)=g, the trace law is independent of task content: Z(x)mid G(x)=g stackreld= Z(x')mid G(x')=g, where x' is an independent donor. Assume the intervention harness leaves the prompt and target answer fixed and replaces only the latent trace. Then E[s_imid G(x_i)=g]=0.
Exact mathematical passage
Z(x)\mid G(x)=g \stackrel{d}{=} Z(x')\mid G(x')=g,
\mathrm{str}Use in source · 2 passages
Search aliases: str mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4832
[Finite-sample structural certificate] Suppose normr≤ R almost surely and r_1,…,r_n are independent held-out defects. Let widehatΔ_str = frac1n∑_i=1^nnormP_Wr_i^2. Then with probability at least 1-δ, | widehatΔ_str - EnormP_Wr^2 | ≤ R^2sqrtfraclog(2/δ)2n.
Exact mathematical passage
\widehat\Delta_{\mathrm{str}} = \frac1n\sum_{i=1}^{n}\norm{P_Wr_i}^2.
\sup_{B}Use in source · 1 passage
Search aliases: sup_B sup
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4740
Let D and D' be two foreign-agent distributions and define TV(D,D') = sup_BabsD(B)-D'(B).
Exact mathematical passage
\TV(\Dcal,\Dcal') = \sup_{B}\abs{\Dcal(B)-\Dcal'(B)}.
\sup_{u\in B_R^W}Use in source · 1 passage
Search aliases: sup_u∈ B_R^W sup in
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2019
bounded right inverse Q:V→ U, so DQ=I_V, and set W:=im Q. Then restr D W:W→ V is an isomorphism with inverse Q. Let B_R^W:=setw∈ W:norm w≤ R. Suppose normN(u)-N(v)≤ L_Nnormu-v (u,v∈ B_R^W), with q:=norm QL_N<1, and normQr+norm Qsup_u∈ B_R^WnormN(u)≤ R. Then there exists a unique u^*∈ B_R^W satisfying F(u^*)=0. The gauge-fixed iteration u_k+1=-Qr-QN(u_k) converges to u^* from every u_0∈ B_R^W, with normu_k-u^*≤fracq^k1-qnormu_1-u_0.
Exact mathematical passage
\norm{Qr}+\norm Q\sup_{u\in B_R^W}\norm{N(u)}\leq R.
\sup_{\widetilde K\in\Kcal}Use in source · 1 passage
Search aliases: sup_widetilde K∈Kcal sup_widetilde K∈K sup widetilde in Kcal sup widetilde in mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9052
protected context. For R⊆ N, let K_R(K) = { widetilde K∈K: widetilde K_i=K_i for every i∉ R }. The restricted oracle envelope is F_K(R) = sup_widetilde K∈K_R(K) M_Ξ,D,B(widetilde K). Thus F_K(∅)=M(K), F_K(N)=sup_widetilde K∈KM(widetilde K). The total declared headroom is H(K) = F_K(N)-F_K(∅). The adjective declared matters. A larger, previously unavailable meta-grammar may expose additional headroom. Equation [source reference] is therefore a relative ceiling, not an absolute limit on future sys…
Exact mathematical passage
F_K(\varnothing)=\Mcal(K), \qquad F_K(N)=\sup_{\widetilde K\in\Kcal}\Mcal(\widetilde K). \label{eq:b8-envelope-endpoints}
\sup_{\widetilde K\in\Kcal_R(K)}Use in source · 1 passage
Search aliases: sup_widetilde K∈Kcal_R(K) sup_widetilde K∈K_R(K) sup widetilde in Kcal sup widetilde in mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9044
Let K be the declared finite or compact class of admissible repair processes under the protected context. For R⊆ N, let K_R(K) = { widetilde K∈K: widetilde K_i=K_i for every i∉ R }. The restricted oracle envelope is F_K(R) = sup_widetilde K∈K_R(K) M_Ξ,D,B(widetilde K). Thus F_K(∅)=M(K), F_K(N)=sup_widetilde K∈KM(widetilde K). The total declared headroom is H(K) = F_K(N)-F_K(∅). The adjective declared matters. A larger, previously unavailable meta-grammar may expose additional headroom. Equation …
Exact mathematical passage
F_K(R) = \sup_{\widetilde K\in\Kcal_R(K)} \Mcal_{\Xi,\Dcal,B}(\widetilde K). \label{eq:b8-oracle-envelope}
\sup_{\gamma\in\Gamma}Use in source · 1 passage
Search aliases: sup_γ∈Γ sup gamma in Gamma
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7514
[Finite meta-grammar uniform bound] Suppose, for each γ∈Γ, the per-episode utility lies in an interval of width L, and the n meta-validation episodes are independent. Then with probability at least 1-δ, sup_γ∈Γ | widehat U_n(γ)-U(γ) | ≤ Lsqrtfraclog(2|Γ|/δ)2n. Consequently, the true utility of the empirical maximizer is within twice the right-hand side of the best configuration in Γ.
Exact mathematical passage
\sup_{\gamma\in\Gamma} \left| \widehat U_n(\gamma)-U(\gamma) \right| \leq L\sqrt{\frac{\log(2|\Gamma|/\delta)}{2n}}. \label{eq:block07-uniform-bound}
\sup_{\delta\in\Dcal}Use in source · 1 passage
Search aliases: sup_δ∈Dcal sup_δ∈D sup delta in Dcal sup delta in mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1073
Define the intervention-coverage error ε_D(ϕ;A) = sup_δ∈D TV(K_A^δ,widehat K_ϕ(A)^δ).
Exact mathematical passage
\eps_{\Dcal}(\phi;A) = \sup_{\delta\in\Dcal} \TV\left(K_A^\delta,\widehat K_{\phi(A)}^\delta\right).
\operatorname{Sym}Use in source · 1 passage
Search aliases: Sym operatorname
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13131
Every quadratic form z^top Bz becomes z'^top(Q^top BQ)z'. The vector space of symmetric quadratic forms is therefore carried isomorphically to itself under the induced action of Q on Sym^2(P^top H).
Exact mathematical passage
\operatorname{Sym}^2(P^\top H)
\operatorname{Sym}^2Use in source · 1 passage
Search aliases: Sym^2 operatorname
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13131
Every quadratic form z^top Bz becomes z'^top(Q^top BQ)z'. The vector space of symmetric quadratic forms is therefore carried isomorphically to itself under the induced action of Q on Sym^2(P^top H).
Exact mathematical passage
\operatorname{Sym}^2(P^\top H)
\mathsf TUse in source · 41 passages
Search aliases: T mathsf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4498
[Archive-envelope monotonicity] Suppose archive updates never delete the last representative of a retained node. For every nonnegative weight vector w, define H_w(A_t) = max_A∈A_tw^ Tz(A). Then H_w(A_t) is nondecreasing in t.
Exact mathematical passage
H_w(\Acal_t) = \max_{A\in\Acal_t}w^{\mathsf T}z(A).
T^{\mathsf T}Use in source · 5 passages
Search aliases: T^ T mathsf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10602
Stacking the selected anchors gives Y=XT^ T+E, where X∈ℝ^n× d and Y∈ℝ^n× q. Define W=diag(σ_1^-2,…,σ_n^-2) and the weighted information matrix M_S=X^ TWX.
Exact mathematical passage
Y=XT^{\mathsf T}+E,
T^*Use in source · 4 passages
Search aliases: T^*
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11447
A population residual formula gives a quantitative special case. Suppose h is centered with covariance Σsucc0, s∈{-1,+1} is independent of h with Pr(s=1)=p, and y=(A+sB)h. Then the best global linear map is T^*=A+(2p-1)B, and inf_T Enormy-Th^2 = 4p(1-p)tr(BΣ B^top). The proof is the normal-equation calculation from the centered two-component mixture. The executable fixture allows affine offsets, context-dependent source distributions, and heteroscedastic noise, so its operative certificate is a …
Exact mathematical passage
T^*=A+(2p-1)B,
t^*Use in source · 1 passage
Search aliases: t^*
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21837
Let h∼ N(0,1) and let s∈{-1,+1} be independent and balanced. Define y=(a+sb)h, b≠0. A global scalar bridge predicts widehat y=th. The population minimizer is t^*=a, and inf_t E (y-th)^2 = E(sbh)^2 = b^2. The residual is not an estimation artifact. It remains after infinite data because one slope cannot equal both a-b and a+b. The two-chart grammar uses t_-=a-b, t_+=a+b, and has zero noiseless approximation error. This is the one-dimensional form of [source reference].
Exact mathematical passage
t^*=a
t^{**}Use in source · 3 passages
Search aliases: t^**
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2093
ion Set η:=normBF(u_0), h:=norm B Lη. If h≤frac12 and t_*:=frac1-sqrt1-2hnorm B L≤ R, then Newton iteration u_n+1=u_n-F'(u_n)^-1F(u_n) is well-defined, remains in overline B(u_0,t_*), and converges to a root u^* of F. If the larger radius t^**:=frac1+sqrt1-2hnorm B L is contained in the regularity region, the root is unique in overline B(u_0,t^**).
Exact mathematical passage
t^{**}:=\frac{1+\sqrt{1-2h}}{\norm B\,L}
T^+Use in source · 2 passages
Search aliases: T^+
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21313
0 · 0, h_s=e_2. Then h_t=Th_s=0, so target-state alignment is exact. However, T^+T=1 · 0 0 · 0, D_t=TD_sT^+=0. The source repair maps the discarded input direction into the retained output direction: D_sh_s=e_1. Consequently, R_t(Th_s)-TR_s(h_s) =TD_s(T^+T-I)h_s =-1. Thus a target can align exactly as a compressed representation while failing to realize a source correction that depends on a lost coordinate. In the released fixture the state-alignment NRMSE is approximately 3.3×10^-8, but the rep…
Exact mathematical passage
T^+T=\begin{bmatrix}1&0\\0&0\end{bmatrix}, \qquad D_t=TD_sT^+=0.
T^{+}Use in source · 5 passages
Search aliases: T^+
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6788
[Exact transport through an injective hidden bridge] Let T:ℝ^d_s→ℝ^d_t have full column rank, so T^+T=I_d_s. Suppose h_t=Th_s throughout the evaluated support. Define D_t and b_t by [source reference] with widehat T=T. Then R_t(Th_s)=TR_s(h_s).
Exact mathematical passage
T^{+}T=I_{d_s}
T^{-1}Use in source · 3 passages
Search aliases: T^-1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6093
Each foreign agent has a canonical defect d∈ℝ^6, a positive diagonal importance vector w, and an invertible architecture transform T∈ℝ^6×6. The runtime-visible defect is widetilde d=Td. Protected utility is J_T(widetilde d) = exp(-∑_i=1^6w_i(T^-1widetilde d)_i^2). A candidate runtime patch δ produces gain g(δ) = J_T(widetilde d+δ)-J_T(widetilde d).
Exact mathematical passage
J_T(\widetilde d) = \exp\left(-\sum_{i=1}^6w_i\bigl(T^{-1}\widetilde d\bigr)_i^2\right).
t^2Use in source · 2 passages
Search aliases: t^2
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2101
Consider the scalar majorant f(t)=η-t+fracnorm B L2t^2. The condition h≤1/2 gives the two nonnegative roots t_* and t^**. The Lipschitz derivative bound and the Banach lemma imply that F'(u) remains invertible while normu-u_0<1/(norm B L). Standard Newton majorization shows that the increments of the operator iteration are dominated by those of scalar Newton iteration for f, whose partial sums remain below t_* and converge. Hence (u_n) is Cauchy, its limit lies in the ball, and continuity gives …
Exact mathematical passage
f(t)=\eta-t+\frac{\norm B\,L}{2}t^2.
T^c_{a\to b}Use in source · 1 passage
Search aliases: T^c_a→ b to
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17645
The next threshold is atlas-indexed cross-architecture Neuralese transport. Source and target systems should each carry locally valid semantic charts, transport should be tested on chart overlaps, and architecture transport should approximately commute with context transition: C^b_c→ dT^c_a→ b ≈ T^d_a→ bC^a_c→ d. Block 20 tests this threshold directly in two foreign recurrent architectures. It preserves the inherited six-chart atlas, audits the curvature residual of [source reference], and retai…
Exact mathematical passage
C^b_{c\to d}T^c_{a\to b} \approx T^d_{a\to b}C^a_{c\to d}. \label{eq:b19-transport-square}
T^c_{b\to a}Use in source · 2 passages
Search aliases: T^c_b→ a to
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17849
[Readout stability from trajectory reconstruction] Let J:H_a,c→ R^p be L_J-Lipschitz. If a transported trajectory satisfies normT^c_b→ ah_b,c-h_a,c ≤ε, then normJ(T^c_b→ ah_b,c)-J(h_a,c) ≤ L_Jε. If two transport paths differ by at most δ, their semantic readouts differ by at most L_Jδ.
Exact mathematical passage
\norm{T^c_{b\to a}h_{b,c}-h_{a,c}} \leq\eps,
T^d_{a\to b}Use in source · 1 passage
Search aliases: T^d_a→ b to
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17645
The next threshold is atlas-indexed cross-architecture Neuralese transport. Source and target systems should each carry locally valid semantic charts, transport should be tested on chart overlaps, and architecture transport should approximately commute with context transition: C^b_c→ dT^c_a→ b ≈ T^d_a→ bC^a_c→ d. Block 20 tests this threshold directly in two foreign recurrent architectures. It preserves the inherited six-chart atlas, audits the curvature residual of [source reference], and retai…
Exact mathematical passage
C^b_{c\to d}T^c_{a\to b} \approx T^d_{a\to b}C^a_{c\to d}. \label{eq:b19-transport-square}
T^\topUse in source · 3 passages
Search aliases: T^top top
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8445
[Full-rank bridge identification] Let the target representation satisfy h_t(x)=T h_s(x) for a fixed linear map T:ℝ^d→ℝ^m. Let X∈ℝ^k× d be the selected source design and Y=XT^top. If rank X=d, then the unregularized least-squares bridge is unique and equals T on the entire source space. If rank X<d, there exist distinct maps T_1≠ T_2 such that XT_1^top=XT_2^top=Y. Consequently, repair transport is not identifiable on every unobserved source direction.
Exact mathematical passage
Y=XT^\top
T^{\top}Use in source · 3 passages
Search aliases: T^top top
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6757
h_t(x_m)^top . It estimates the bridge by regularized least squares: widehat T_t← s ∈ arg min_T normH_sT^top-H_t_F^2 +λnormT_F^2. The transformation matrices used to construct the controlled targets are not exposed to this fit.
Exact mathematical passage
\widehat T_{t\leftarrow s} \in \argmin_T \norm{H_sT^{\top}-H_t}_F^2 +\lambda\norm{T}_F^2. \label{eq:block06-learned-transport}
T^\daggerUse in source · 6 passages
Search aliases: T^† dagger
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10679
Let the frozen source repair be R_s(h)=h+D_sh+b_s. For an injective bridge T, the exact transported repair is D_t=T D_sT^†, b_t=Tb_s. The estimated repair uses widehat T.
Exact mathematical passage
D_t=T D_sT^\dagger, \qquad b_t=Tb_s.
t_*Use in source · 4 passages
Search aliases: t_*
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2085
orovich Let F:U→ V be continuously Fr echet differentiable between Banach spaces. Let u_0∈ U and suppose F'(u_0) is invertible with inverse B. Assume that on the closed ball overline B(u_0,R), normF'(u)-F'(v)≤ Lnormu-v. Set η:=normBF(u_0), h:=norm B Lη. If h≤frac12 and t_*:=frac1-sqrt1-2hnorm B L≤ R, then Newton iteration u_n+1=u_n-F'(u_n)^-1F(u_n) is well-defined, remains in overline B(u_0,t_*), and converges to a root u^* of F. If the larger radius t^**:=frac1+sqrt1-2hnorm B L is contained in …
Exact mathematical passage
t_*:=\frac{1-\sqrt{1-2h}}{\norm B\,L}\leq R,
T_+Use in source · 1 passage
Search aliases: T_+
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20509
T_-,T_+ · Chart-specific source-to-target bridge maps
Exact mathematical passage
T_-,T_+
t_+Use in source · 1 passage
Search aliases: t_+
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21846
eginequation y=(a+sb)h, b≠0. A global scalar bridge predicts widehat y=th. The population minimizer is t^*=a, and inf_t E (y-th)^2 = E(sbh)^2 = b^2. The residual is not an estimation artifact. It remains after infinite data because one slope cannot equal both a-b and a+b. The two-chart grammar uses t_-=a-b, t_+=a+b, and has zero noiseless approximation error. This is the one-dimensional form of [source reference].
Exact mathematical passage
t_-=a-b, \qquad t_+=a+b,
T_-Use in source · 1 passage
Search aliases: T_-
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20509
T_-,T_+ · Chart-specific source-to-target bridge maps
Exact mathematical passage
T_-,T_+
t_-Use in source · 1 passage
Search aliases: t_-
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21846
eginequation y=(a+sb)h, b≠0. A global scalar bridge predicts widehat y=th. The population minimizer is t^*=a, and inf_t E (y-th)^2 = E(sbh)^2 = b^2. The residual is not an estimation artifact. It remains after infinite data because one slope cannot equal both a-b and a+b. The two-chart grammar uses t_-=a-b, t_+=a+b, and has zero noiseless approximation error. This is the one-dimensional form of [source reference].
Exact mathematical passage
t_-=a-b, \qquad t_+=a+b,
T_0Use in source · 1 passage
Search aliases: T_0
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3681
chain of evaluator morphisms with readouts into a common complete normed score space, E_0xrightarrowe_0 E_1 xrightarrowe_1⋯ xrightarrowe_n-1 E_n. Suppose that for every protected represented state x_t transported from version 0, normJ_t+1(T_tx_t)-J_t(x_t)≤ε_t. Then normJ_n(T_n-1⋯ T_0x_0)-J_0(x_0) ≤ ∑_t=0^n-1ε_t. If ∑_tε_t<∞, then the transported score sequence is Cauchy and converges in every complete score space.
Exact mathematical passage
\norm{J_n(T_{n-1}\cdots T_0x_0)-J_0(x_0)} \leq \sum_{t=0}^{n-1}\eps_t.
\mathfrak T_0Use in source · 1 passage
Search aliases: T_0 mathfrak
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2935
[Pathwise transport bound] Consider a path of systems A_0xrightarrow T_0A_1 xrightarrow T_1⋯ xrightarrow T_n-1A_n. Write T_U,k:U_k→ U_k+1 and T_V,k:V_k→ V_k+1. Let normD_k+1T_U,k-T_V,kD_k≤ε_D,k and normr_k+1-T_V,kr_k≤ε_r,k. If D_0u_0=-r_0 and u_n=T_U,n-1⋯ T_U,0u_0, then normD_nu_n+r_n · ≤ ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_D,k normT_U,k-1⋯ T_U,0u_0
Exact mathematical passage
A_0\xrightarrow{\mathfrak T_0}A_1 \xrightarrow{\mathfrak T_1}\cdots \xrightarrow{\mathfrak T_{n-1}}A_n.
T_{01}Use in source · 1 passage
Search aliases: T_01
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5617
Case F4: evaluator holonomy. Use an exactly correctable local defect and zero overlap mismatch. Let evaluator transports around a three-edge cycle be T_01=I_2, T_12=I_2, T_20=diag(1,-1). For protected evaluator direction e_2, H_γ e_2-e_2=-2e_2, normH_γ e_2-e_2=2. The candidate is locally repairable and globally gluable, but its evaluator semantics are not path flat.
Exact mathematical passage
T_{01}=I_2, \qquad T_{12}=I_2, \qquad T_{20}=\diag(1,-1).
T_1Use in source · 4 passages
Search aliases: T_1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8445
cation] Let the target representation satisfy h_t(x)=T h_s(x) for a fixed linear map T:ℝ^d→ℝ^m. Let X∈ℝ^k× d be the selected source design and Y=XT^top. If rank X=d, then the unregularized least-squares bridge is unique and equals T on the entire source space. If rank X<d, there exist distinct maps T_1≠ T_2 such that XT_1^top=XT_2^top=Y. Consequently, repair transport is not identifiable on every unobserved source direction.
Exact mathematical passage
T_1\neq T_2
\mathfrak T_1Use in source · 1 passage
Search aliases: T_1 mathfrak
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2935
[Pathwise transport bound] Consider a path of systems A_0xrightarrow T_0A_1 xrightarrow T_1⋯ xrightarrow T_n-1A_n. Write T_U,k:U_k→ U_k+1 and T_V,k:V_k→ V_k+1. Let normD_k+1T_U,k-T_V,kD_k≤ε_D,k and normr_k+1-T_V,kr_k≤ε_r,k. If D_0u_0=-r_0 and u_n=T_U,n-1⋯ T_U,0u_0, then normD_nu_n+r_n · ≤ ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_D,k normT_U,k-1⋯ T_U,0u_0
Exact mathematical passage
A_0\xrightarrow{\mathfrak T_0}A_1 \xrightarrow{\mathfrak T_1}\cdots \xrightarrow{\mathfrak T_{n-1}}A_n.
T_{12}Use in source · 1 passage
Search aliases: T_12
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5617
Case F4: evaluator holonomy. Use an exactly correctable local defect and zero overlap mismatch. Let evaluator transports around a three-edge cycle be T_01=I_2, T_12=I_2, T_20=diag(1,-1). For protected evaluator direction e_2, H_γ e_2-e_2=-2e_2, normH_γ e_2-e_2=2. The candidate is locally repairable and globally gluable, but its evaluator semantics are not path flat.
Exact mathematical passage
T_{01}=I_2, \qquad T_{12}=I_2, \qquad T_{20}=\diag(1,-1).
T_1^\topUse in source · 1 passage
Search aliases: T_1^top top
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8446
ridge Let the target representation satisfy h_t(x)=T h_s(x) for a fixed linear map T:ℝ^d→ℝ^m. Let X∈ℝ^k× d be the selected source design and Y=XT^top. If rank X=d, then the unregularized least-squares bridge is unique and equals T on the entire source space. If rank X<d, there exist distinct maps T_1≠ T_2 such that XT_1^top=XT_2^top=Y. Consequently, repair transport is not identifiable on every unobserved source direction.
Exact mathematical passage
XT_1^\top=XT_2^\top=Y.
T_2Use in source · 2 passages
Search aliases: T_2
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8445
cation] Let the target representation satisfy h_t(x)=T h_s(x) for a fixed linear map T:ℝ^d→ℝ^m. Let X∈ℝ^k× d be the selected source design and Y=XT^top. If rank X=d, then the unregularized least-squares bridge is unique and equals T on the entire source space. If rank X<d, there exist distinct maps T_1≠ T_2 such that XT_1^top=XT_2^top=Y. Consequently, repair transport is not identifiable on every unobserved source direction.
Exact mathematical passage
T_1\neq T_2
T_{20}Use in source · 1 passage
Search aliases: T_20
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5617
Case F4: evaluator holonomy. Use an exactly correctable local defect and zero overlap mismatch. Let evaluator transports around a three-edge cycle be T_01=I_2, T_12=I_2, T_20=diag(1,-1). For protected evaluator direction e_2, H_γ e_2-e_2=-2e_2, normH_γ e_2-e_2=2. The candidate is locally repairable and globally gluable, but its evaluator semantics are not path flat.
Exact mathematical passage
T_{01}=I_2, \qquad T_{12}=I_2, \qquad T_{20}=\diag(1,-1).
T_2^{-1}Use in source · 2 passages
Search aliases: T_2^-1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 19586
[Intertwiner uniqueness modulo the response commutant] Let T_1,T_2:H_t→ H_s be invertible and satisfy T_iR_v^t=R_σ^-1(v)^sT_i for every v∈ V. Then S=T_2^-1T_1 belongs to A_t'. Hence exact intervention-orbit data identifies a transport at most up to the response commutant.
Exact mathematical passage
S=T_2^{-1}T_1
T_2^\topUse in source · 1 passage
Search aliases: T_2^top top
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8446
ridge Let the target representation satisfy h_t(x)=T h_s(x) for a fixed linear map T:ℝ^d→ℝ^m. Let X∈ℝ^k× d be the selected source design and Y=XT^top. If rank X=d, then the unregularized least-squares bridge is unique and equals T on the entire source space. If rank X<d, there exist distinct maps T_1≠ T_2 such that XT_1^top=XT_2^top=Y. Consequently, repair transport is not identifiable on every unobserved source direction.
Exact mathematical passage
XT_1^\top=XT_2^\top=Y.
\mathfrak T_{A\to C}Use in source · 3 passages
Search aliases: T_A→ C mathfrak to
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2902
Let T_A→ B=(T_U^AB,T_V^AB), T_B→ C=(T_U^BC,T_V^BC). The composed transport is T_A→ C = (T_U^BCT_U^AB,T_V^BCT_V^AB).
Exact mathematical passage
\mathfrak T_{A\to C} = (T_U^{BC}T_U^{AB},T_V^{BC}T_V^{AB}).
\mathfrak T_{B\to C}Use in source · 3 passages
Search aliases: T_B→ C mathfrak to
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2896
Let T_A→ B=(T_U^AB,T_V^AB), T_B→ C=(T_U^BC,T_V^BC). The composed transport is T_A→ C = (T_U^BCT_U^AB,T_V^BCT_V^AB).
Exact mathematical passage
\mathfrak T_{A\to B}=(T_U^{AB},T_V^{AB}), \qquad \mathfrak T_{B\to C}=(T_U^{BC},T_V^{BC}).
T_fUse in source · 4 passages
Search aliases: T_f
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3426
[Composition of evaluator defects] Let e:v→ w and f:w→ z be evaluator morphisms. Suppose T_f is L_T-Lipschitz on represented protected evidence. Then δ_obs(f∘ e) ≤ L_Tδ_obs(e)+δ_obs(f), where the second term is evaluated on S_e( h_v). If, in addition, sup_r∈ h_v norm J_zT_fT_eι_v(r)-J_wT_eι_v(r) ≤ δ_J(fmid e), then δ_J(f∘ e) ≤ δ_J(e)+δ_J(fmid e).
Exact mathematical passage
T_f
T_iUse in source · 1 passage
Search aliases: T_i
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 19583
[Intertwiner uniqueness modulo the response commutant] Let T_1,T_2:H_t→ H_s be invertible and satisfy T_iR_v^t=R_σ^-1(v)^sT_i for every v∈ V. Then S=T_2^-1T_1 belongs to A_t'. Hence exact intervention-orbit data identifies a transport at most up to the response commutant.
Exact mathematical passage
T_iR_v^t=R_{\sigma^{-1}(v)}^sT_i
T_{ij}Use in source · 1 passage
Search aliases: T_ij
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2527
A finite operational version avoids explicit cohomology. Let G=(I,E) be connected, with sets of local solutions S_i and invertible transition maps T_ij:S_i→ S_j, T_ji=T_ij^-1. Choose a root o and a spanning tree T⊆ G.
Exact mathematical passage
T_{ij}:S_i\to S_j, \qquad T_{ji}=T_{ij}^{-1}.
T_{ij}^{-1}Use in source · 1 passage
Search aliases: T_ij^-1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2527
A finite operational version avoids explicit cohomology. Let G=(I,E) be connected, with sets of local solutions S_i and invertible transition maps T_ij:S_i→ S_j, T_ji=T_ij^-1. Choose a root o and a spanning tree T⊆ G.
Exact mathematical passage
T_{ij}:S_i\to S_j, \qquad T_{ji}=T_{ij}^{-1}.
T_jUse in source · 5 passages
Search aliases: T_j
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11470
Let the source repair be R_s(h)=M_sh+c_s. On chart j, let the target bridge be T_j(h)=A_jh+b_j, with A_j invertible on the repair-relevant subspace. Define R_t,j(y) = T_j (R_s(T_j^-1(y))). Expanding gives R_t,j(y) = A_jM_sA_j^-1(y-b_j)+A_jc_s+b_j.
Exact mathematical passage
T_j(h)=A_jh+b_j,
T_j^{-1}Use in source · 1 passage
Search aliases: T_j^-1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11474
Let the source repair be R_s(h)=M_sh+c_s. On chart j, let the target bridge be T_j(h)=A_jh+b_j, with A_j invertible on the repair-relevant subspace. Define R_t,j(y) = T_j (R_s(T_j^-1(y))). Expanding gives R_t,j(y) = A_jM_sA_j^-1(y-b_j)+A_jc_s+b_j.
Exact mathematical passage
R_{t,j}(y) = T_j\!\left(R_s(T_j^{-1}(y))\right). \label{eq:b10-chart-repair-def}
T_{ji}Use in source · 1 passage
Search aliases: T_ji
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2527
A finite operational version avoids explicit cohomology. Let G=(I,E) be connected, with sets of local solutions S_i and invertible transition maps T_ij:S_i→ S_j, T_ji=T_ij^-1. Choose a root o and a spanning tree T⊆ G.
Exact mathematical passage
T_{ij}:S_i\to S_j, \qquad T_{ji}=T_{ij}^{-1}.
T_{n-1}Use in source · 1 passage
Search aliases: T_n-1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3681
chain of evaluator morphisms with readouts into a common complete normed score space, E_0xrightarrowe_0 E_1 xrightarrowe_1⋯ xrightarrowe_n-1 E_n. Suppose that for every protected represented state x_t transported from version 0, normJ_t+1(T_tx_t)-J_t(x_t)≤ε_t. Then normJ_n(T_n-1⋯ T_0x_0)-J_0(x_0) ≤ ∑_t=0^n-1ε_t. If ∑_tε_t<∞, then the transported score sequence is Cauchy and converges in every complete score space.
Exact mathematical passage
\norm{J_n(T_{n-1}\cdots T_0x_0)-J_0(x_0)} \leq \sum_{t=0}^{n-1}\eps_t.
\mathfrak T_{n-1}Use in source · 1 passage
Search aliases: T_n-1 mathfrak
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2935
[Pathwise transport bound] Consider a path of systems A_0xrightarrow T_0A_1 xrightarrow T_1⋯ xrightarrow T_n-1A_n. Write T_U,k:U_k→ U_k+1 and T_V,k:V_k→ V_k+1. Let normD_k+1T_U,k-T_V,kD_k≤ε_D,k and normr_k+1-T_V,kr_k≤ε_r,k. If D_0u_0=-r_0 and u_n=T_U,n-1⋯ T_U,0u_0, then normD_nu_n+r_n · ≤ ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_D,k normT_U,k-1⋯ T_U,0u_0
Exact mathematical passage
A_0\xrightarrow{\mathfrak T_0}A_1 \xrightarrow{\mathfrak T_1}\cdots \xrightarrow{\mathfrak T_{n-1}}A_n.
T_pUse in source · 4 passages
Search aliases: T_p
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3514
[Basepoint covariance of the flat core] Let p:v→ w be any invertible exact evaluator path in a connected atlas. Then T_p(E_v^flat)=E_w^flat. Hence the dimension of the path-flat core is independent of basepoint.
Exact mathematical passage
T_p(E_v^{\flat})=E_w^{\flat}.
T_p^{-1}Use in source · 1 passage
Search aliases: T_p^-1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3522
Every loop γ at w corresponds to the conjugate loop p^-1γ p at v, with H_γ=T_pH_p^-1γ pT_p^-1. If x∈ E_v^flat, then H_γ T_px=T_pH_p^-1γ px=T_px, so T_px∈ E_w^flat. Apply the same argument to p^-1 for the reverse inclusion.
Exact mathematical passage
H_\gamma=T_pH_{p^{-1}\gamma p}T_p^{-1}.
T_rUse in source · 1 passage
Search aliases: T_r
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7487
The child compiler searches a finite grammar Γ = Λ_u×Λ_r× K_u× K_r× T_r, where the factors specify utility regularization, risk regularization, utility residual penalty, risk residual penalty, and risk threshold. In the released fixture, |Γ| =5·5·4·5·4 =2000. For γ∈Γ, let overline U_rm meta(γ) be its mean protected utility on the meta-validation episodes and widehatrm se(γ) its standard error. A configuration is validation-safe when its protected-regression tail rate and maximum regression satis…
Exact mathematical passage
\Gamma = \Lambda_u\times\Lambda_r\times K_u\times K_r\times T_r,
\mathsf T_{\rm audit}Use in source · 1 passage
Search aliases: T_rm audit mathsf rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8376
The Block 7 primary audit differs from development in both task and representation family: T_rm dev · ∩ T_rm audit=∅, F_rm dev^rm primary · ∩ F_rm audit^rm primary=∅. Development uses ten tasks generated from four trigger families. Audit uses twelve new tasks generated from radial and mixed triggers, new teacher seeds, and new corruption permutations. The primary audit representations are dense linear 20-dimensional, block-linear 24-dimensional, and ill-conditioned linear 20-dimensional systems.…
Exact mathematical passage
\begin{aligned} \mathsf T_{\rm dev}&\cap\mathsf T_{\rm audit}=\varnothing,\\ \mathsf F_{\rm dev}^{\rm primary}&\cap\mathsf F_{\rm audit}^{\rm primary}=\varnothing. \end{aligned}
\mathsf T_{\rm dev}Use in source · 1 passage
Search aliases: T_rm dev mathsf rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8376
The Block 7 primary audit differs from development in both task and representation family: T_rm dev · ∩ T_rm audit=∅, F_rm dev^rm primary · ∩ F_rm audit^rm primary=∅. Development uses ten tasks generated from four trigger families. Audit uses twelve new tasks generated from radial and mixed triggers, new teacher seeds, and new corruption permutations. The primary audit representations are dense linear 20-dimensional, block-linear 24-dimensional, and ill-conditioned linear 20-dimensional systems.…
Exact mathematical passage
\begin{aligned} \mathsf T_{\rm dev}&\cap\mathsf T_{\rm audit}=\varnothing,\\ \mathsf F_{\rm dev}^{\rm primary}&\cap\mathsf F_{\rm audit}^{\rm primary}=\varnothing. \end{aligned}
T_tUse in source · 1 passage
Search aliases: T_t
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3677
ulative evaluator-drift bound] Consider a chain of evaluator morphisms with readouts into a common complete normed score space, E_0xrightarrowe_0 E_1 xrightarrowe_1⋯ xrightarrowe_n-1 E_n. Suppose that for every protected represented state x_t transported from version 0, normJ_t+1(T_tx_t)-J_t(x_t)≤ε_t. Then normJ_n(T_n-1⋯ T_0x_0)-J_0(x_0) ≤ ∑_t=0^n-1ε_t. If ∑_tε_t<∞, then the transported score sequence is Cauchy and converges in every complete score space.
Exact mathematical passage
\norm{J_{t+1}(T_tx_t)-J_t(x_t)}\leq\eps_t.
T_t^UUse in source · 4 passages
Search aliases: T_t^U
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9610
A grammar extension may also change the defect representation. Let D_t:U_t→ V_t, with transports T_t^U:U_t→ U_t+1, T_t^V:V_t→ V_t+1. Assume the exact correction square D_t+1T_t^U = T_t^VD_t. Then there is an induced obstruction map overline T_t: coker D_t → coker D_t+1, overline T_t([r])=[T_t^Vr].
Exact mathematical passage
T_t^U:U_t\to U_{t+1}, \qquad T_t^V:V_t\to V_{t+1}.
T_{U,0}Use in source · 3 passages
Search aliases: T_U,0
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2949
stems A_0xrightarrow T_0A_1 xrightarrow T_1⋯ xrightarrow T_n-1A_n. Write T_U,k:U_k→ U_k+1 and T_V,k:V_k→ V_k+1. Let normD_k+1T_U,k-T_V,kD_k≤ε_D,k and normr_k+1-T_V,kr_k≤ε_r,k. If D_0u_0=-r_0 and u_n=T_U,n-1⋯ T_U,0u_0, then normD_nu_n+r_n · ≤ ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_D,k normT_U,k-1⋯ T_U,0u_0
Exact mathematical passage
u_n=T_{U,n-1}\cdots T_{U,0}u_0,
T_{U,k}Use in source · 3 passages
Search aliases: T_U,k
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2940
[Pathwise transport bound] Consider a path of systems A_0xrightarrow T_0A_1 xrightarrow T_1⋯ xrightarrow T_n-1A_n. Write T_U,k:U_k→ U_k+1 and T_V,k:V_k→ V_k+1. Let normD_k+1T_U,k-T_V,kD_k≤ε_D,k and normr_k+1-T_V,kr_k≤ε_r,k. If D_0u_0=-r_0 and u_n=T_U,n-1⋯ T_U,0u_0, then normD_nu_n+r_n · ≤ ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_D,k normT_U,k-1⋯ T_U,0u_0
Exact mathematical passage
T_{U,k}:U_k\to U_{k+1}
T_{U,k-1}Use in source · 2 passages
Search aliases: T_U,k-1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2953
T_1⋯ xrightarrow T_n-1A_n. Write T_U,k:U_k→ U_k+1 and T_V,k:V_k→ V_k+1. Let normD_k+1T_U,k-T_V,kD_k≤ε_D,k and normr_k+1-T_V,kr_k≤ε_r,k. If D_0u_0=-r_0 and u_n=T_U,n-1⋯ T_U,0u_0, then normD_nu_n+r_n · ≤ ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_D,k normT_U,k-1⋯ T_U,0u_0 · + ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_r,k, where empty products are identities.
Exact mathematical passage
\begin{aligned} \norm{D_nu_n+r_n} &\leq \sum_{k=0}^{n-1} \norm{T_{V,n-1}\cdots T_{V,k+1}} \eps_{D,k} \norm{T_{U,k-1}\cdots T_{U,0}u_0}\\ &\quad+ \sum_{k=0}^{n-1} \norm{T_{V,n-1}\cdots T_{V,k+1}} \eps_{r,k}, \end{aligned}
T_{U,n-1}Use in source · 1 passage
Search aliases: T_U,n-1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2949
stems A_0xrightarrow T_0A_1 xrightarrow T_1⋯ xrightarrow T_n-1A_n. Write T_U,k:U_k→ U_k+1 and T_V,k:V_k→ V_k+1. Let normD_k+1T_U,k-T_V,kD_k≤ε_D,k and normr_k+1-T_V,kr_k≤ε_r,k. If D_0u_0=-r_0 and u_n=T_U,n-1⋯ T_U,0u_0, then normD_nu_n+r_n · ≤ ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_D,k normT_U,k-1⋯ T_U,0u_0
Exact mathematical passage
u_n=T_{U,n-1}\cdots T_{U,0}u_0,
T_U^*Use in source · 1 passage
Search aliases: T_U^*
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2864
[Naturality of the Moore--Penrose repair] Let U_A,U_B,V_A,V_B be finite-dimensional Hilbert spaces. Suppose T_U and T_V are unitary and D_B=T_VD_AT_U^*. Then D_B^†=T_UD_A^†T_V^*. For every r_B=T_Vr_A, the target minimum-norm least-squares correction is the transport of the source minimum-norm least-squares correction: -D_B^†r_B = T_U(-D_A^†r_A). The orthogonal obstruction projectors satisfy I-D_BD_B^† = T_V(I-D_AD_A^†)T_V^*.
Exact mathematical passage
D_B=T_VD_AT_U^*.
T_U^{-1}Use in source · 3 passages
Search aliases: T_U^-1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2748
[Exact transport equivalence] Suppose T_U and T_V are isomorphisms and D_B=T_VD_AT_U^-1. Then T_U restricts to an isomorphism ker D_A≅ker D_B, and T_V induces an isomorphism coker D_A≅coker D_B. Thus exact transport equivalence preserves both correction freedom and obstruction dimension.
Exact mathematical passage
D_B=T_VD_AT_U^{-1}.
T_U^{AB}Use in source · 4 passages
Search aliases: T_U^AB
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2896
Let T_A→ B=(T_U^AB,T_V^AB), T_B→ C=(T_U^BC,T_V^BC). The composed transport is T_A→ C = (T_U^BCT_U^AB,T_V^BCT_V^AB).
Exact mathematical passage
\mathfrak T_{A\to B}=(T_U^{AB},T_V^{AB}), \qquad \mathfrak T_{B\to C}=(T_U^{BC},T_V^{BC}).
T_U^{BC}Use in source · 2 passages
Search aliases: T_U^BC
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2896
Let T_A→ B=(T_U^AB,T_V^AB), T_B→ C=(T_U^BC,T_V^BC). The composed transport is T_A→ C = (T_U^BCT_U^AB,T_V^BCT_V^AB).
Exact mathematical passage
\mathfrak T_{A\to B}=(T_U^{AB},T_V^{AB}), \qquad \mathfrak T_{B\to C}=(T_U^{BC},T_V^{BC}).
T_U^pUse in source · 4 passages
Search aliases: T_U^p
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3070
[Path independence on the flat repair core] Let p and q be two exact invertible transport paths from A to B. If u∈ U_A is fixed by the loop holonomy associated with q^-1p, then T_U^p u=T_U^q u. In particular, every u∈ U_A^flat has path-independent transport to B.
Exact mathematical passage
T_U^p u=T_U^q u.
T_U^qUse in source · 4 passages
Search aliases: T_U^q
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3070
[Path independence on the flat repair core] Let p and q be two exact invertible transport paths from A to B. If u∈ U_A is fixed by the loop holonomy associated with q^-1p, then T_U^p u=T_U^q u. In particular, every u∈ U_A^flat has path-independent transport to B.
Exact mathematical passage
T_U^p u=T_U^q u.
T_{V,k}Use in source · 4 passages
Search aliases: T_V,k
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2940
[Pathwise transport bound] Consider a path of systems A_0xrightarrow T_0A_1 xrightarrow T_1⋯ xrightarrow T_n-1A_n. Write T_U,k:U_k→ U_k+1 and T_V,k:V_k→ V_k+1. Let normD_k+1T_U,k-T_V,kD_k≤ε_D,k and normr_k+1-T_V,kr_k≤ε_r,k. If D_0u_0=-r_0 and u_n=T_U,n-1⋯ T_U,0u_0, then normD_nu_n+r_n · ≤ ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_D,k normT_U,k-1⋯ T_U,0u_0
Exact mathematical passage
T_{V,k}:V_k\to V_{k+1}
T_{V,k+1}Use in source · 3 passages
Search aliases: T_V,k+1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2953
T_1⋯ xrightarrow T_n-1A_n. Write T_U,k:U_k→ U_k+1 and T_V,k:V_k→ V_k+1. Let normD_k+1T_U,k-T_V,kD_k≤ε_D,k and normr_k+1-T_V,kr_k≤ε_r,k. If D_0u_0=-r_0 and u_n=T_U,n-1⋯ T_U,0u_0, then normD_nu_n+r_n · ≤ ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_D,k normT_U,k-1⋯ T_U,0u_0 · + ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_r,k, where empty products are identities.
Exact mathematical passage
\begin{aligned} \norm{D_nu_n+r_n} &\leq \sum_{k=0}^{n-1} \norm{T_{V,n-1}\cdots T_{V,k+1}} \eps_{D,k} \norm{T_{U,k-1}\cdots T_{U,0}u_0}\\ &\quad+ \sum_{k=0}^{n-1} \norm{T_{V,n-1}\cdots T_{V,k+1}} \eps_{r,k}, \end{aligned}
T_{V,n-1}Use in source · 3 passages
Search aliases: T_V,n-1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2953
T_1⋯ xrightarrow T_n-1A_n. Write T_U,k:U_k→ U_k+1 and T_V,k:V_k→ V_k+1. Let normD_k+1T_U,k-T_V,kD_k≤ε_D,k and normr_k+1-T_V,kr_k≤ε_r,k. If D_0u_0=-r_0 and u_n=T_U,n-1⋯ T_U,0u_0, then normD_nu_n+r_n · ≤ ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_D,k normT_U,k-1⋯ T_U,0u_0 · + ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_r,k, where empty products are identities.
Exact mathematical passage
\begin{aligned} \norm{D_nu_n+r_n} &\leq \sum_{k=0}^{n-1} \norm{T_{V,n-1}\cdots T_{V,k+1}} \eps_{D,k} \norm{T_{U,k-1}\cdots T_{U,0}u_0}\\ &\quad+ \sum_{k=0}^{n-1} \norm{T_{V,n-1}\cdots T_{V,k+1}} \eps_{r,k}, \end{aligned}
T_V^*Use in source · 3 passages
Search aliases: T_V^*
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2868
[Naturality of the Moore--Penrose repair] Let U_A,U_B,V_A,V_B be finite-dimensional Hilbert spaces. Suppose T_U and T_V are unitary and D_B=T_VD_AT_U^*. Then D_B^†=T_UD_A^†T_V^*. For every r_B=T_Vr_A, the target minimum-norm least-squares correction is the transport of the source minimum-norm least-squares correction: -D_B^†r_B = T_U(-D_A^†r_A). The orthogonal obstruction projectors satisfy I-D_BD_B^† = T_V(I-D_AD_A^†)T_V^*.
Exact mathematical passage
D_B^{\dagger}=T_UD_A^{\dagger}T_V^*.
T_V^{AB}Use in source · 2 passages
Search aliases: T_V^AB
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2896
Let T_A→ B=(T_U^AB,T_V^AB), T_B→ C=(T_U^BC,T_V^BC). The composed transport is T_A→ C = (T_U^BCT_U^AB,T_V^BCT_V^AB).
Exact mathematical passage
\mathfrak T_{A\to B}=(T_U^{AB},T_V^{AB}), \qquad \mathfrak T_{B\to C}=(T_U^{BC},T_V^{BC}).
T_V^{BC}Use in source · 6 passages
Search aliases: T_V^BC
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2896
Let T_A→ B=(T_U^AB,T_V^AB), T_B→ C=(T_U^BC,T_V^BC). The composed transport is T_A→ C = (T_U^BCT_U^AB,T_V^BCT_V^AB).
Exact mathematical passage
\mathfrak T_{A\to B}=(T_U^{AB},T_V^{AB}), \qquad \mathfrak T_{B\to C}=(T_U^{BC},T_V^{BC}).
T_xUse in source · 1 passage
Search aliases: T_x
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10180
For nonlinear dynamics on a state manifold M, the response algebra becomes base-point dependent: x⟼ A_x⊆End(T_xM). Local central projectors define a bundle of tangent strata when their ranks remain locally constant. Transport around a loop can permute or distort the local modules, producing atlas holonomy. A complete nonlinear theory must combine: item local intervention algebras; item spectral-gap control of approximate central projectors; item transition maps between neighboring tangent atlase…
Exact mathematical passage
x\longmapsto\mathfrak A_x\subseteq\End(T_xM).
\tanhUse in source · 8 passages
Search aliases: tanh
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16093
Let ϕ(τ)∈ℝ^19 be the padded task serialization, and let e=E_θ(ϕ(τ))∈ℝ^40 be the prompt representation. The latent trace has four states. Given conditioning mode α, define z_0=ζ, z_k+1 = tanh (Rz_k+s_k+C_α(e,m)), 0≤ k<4, where s_k is a learned step embedding. The task-blind parent satisfies C_parent(e,m)=0. The descendants use one of the following conditioning maps: C_depth(e,m) · =D 1_m,
Exact mathematical passage
z_0=\zeta, \qquad z_{k+1} = \tanh\!\left(Rz_k+s_k+C_\alpha(e,m)\right), \qquad 0\leq k<4, \label{eq:b17-trace}
\mathrm{target}Use in source · 1 passage
Search aliases: target mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12552
The selected atlas candidate commits only when all frozen conditions hold: widehat g_val · ≥0.10, LCB(widehat g_val) · ≥0.05, widehat r_ctx · ≤3×10^-4, widehat a_target · ≥0.72, widehat a_repair inv · ≥0.96, widehat b_chart · ≥0.20, widehat s_model · ≤0.25, widehatγ_min · ≥0.004, widehatε_comm · ≤0.16, widehatε_resp · ≤0.08. The witness therefore evaluates both downstream repair quality and the internal atlas certificate. A high blind gain with unstable response geometry is not sufficient for co…
Exact mathematical passage
\widehat g_{\mathrm{val}}&\geq0.10,\\ \LCB(\widehat g_{\mathrm{val}})&\geq0.05,\\ \widehat r_{\mathrm{ctx}}&\leq3\times10^{-4},\\ \widehat a_{\mathrm{target}}&\geq0.72,\\ \widehat a_{\mathrm{repair\ inv}}&\geq0.96,\\ \widehat b_{\mathrm{chart}}&\geq0.20,\\ \widehat s_{\mathrm{model}}&\leq0.25,\\ \widehat\gamma_{\min}&\geq0.004,\\ \widehat\eps_{\mathrm{comm}}&\leq0.16,\\ \widehat\eps_{\mathrm{resp}}&\leq0.08.
\mathrm{task}Use in source · 1 passage
Search aliases: task mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3960
Let a complete implementation state decompose as Σ = Σ^task ⊕ Σ^memory ⊕ Σ^repair ⊕ Σ^evaluation ⊕ Σ^provenance ⊕ Σ^rollback. A transaction selects an editable mask E⊆Σ and a witness mask W⊆Σ with E∩ W=∅. The decomposition is logical rather than necessarily physical. Two components may share hardware while remaining isolated by capabilities, address spaces, process boundaries, cryptographic commitments, or reproducible snapshots.
Exact mathematical passage
\Sigma = \Sigma^{\mathrm{task}} \oplus \Sigma^{\mathrm{memory}} \oplus \Sigma^{\mathrm{repair}} \oplus \Sigma^{\mathrm{evaluation}} \oplus \Sigma^{\mathrm{provenance}} \oplus \Sigma^{\mathrm{rollback}}.
\Tau_t^{\mathrm{NL}}Use in source · 3 passages
Search aliases: Tau_t^NL T_t^NL Tau mathrm mathcal mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15834
Extend the SRIC lineage state by L_t^NL = ( L_t,N_t,I_t^NL,T_t^NL,W_t^NL), where T_t^NL contains trace-transport certificates and W_t^NL is the frozen external answer and invariant witness. A proposed descendant may modify the latent transition, update gate, halting head, packet atlas, or transport map. The backbone may be held fixed for a component-isolation edge or changed under a stronger whole-model edge.
Exact mathematical passage
\mathfrak L_t^{\mathrm{NL}} = (\mathfrak L_t,\Ncal_t,\Ical_t^{\mathrm{NL}},\Tau_t^{\mathrm{NL}},\Wcal_t^{\mathrm{NL}}),
\mathrm{tools}Use in source · 1 passage
Search aliases: tools mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 665
A representative color set is C= set model, workspace, memory, planner, improver, evaluator, provenance, tools . The atlas does not assert that these colors are universally correct. It declares the decomposition under which one particular repair theory is being tested.
Exact mathematical passage
\mathsf C= \set{ \mathrm{model}, \mathrm{workspace}, \mathrm{memory}, \mathrm{planner}, \mathrm{improver}, \mathrm{evaluator}, \mathrm{provenance}, \mathrm{tools} }.
\trUse in source · 8 passages
Search aliases: tr operatorname
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9810
For W⊆ I^⊥, EnormP_(I⊕ W)^⊥r^2 = tr(Σ_ob)-tr(P_WΣ_ob). The Ky Fan variational principle maximizes tr(P_WΣ_ob) over rank-k orthogonal projectors by choosing the top k eigenspace.
Exact mathematical passage
\mathbb E\norm{P_{(I\oplus W)^\perp}r}^2 = \tr(\Sigma_{\mathrm{ob}})-\tr(P_W\Sigma_{\mathrm{ob}}).
\mathrm{trace}Use in source · 1 passage
Search aliases: trace mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17934
A candidate is pre-audit eligible only if both foreign target models are ATLAS_FULL and the candidate jointly satisfies R^2_transport · ≥0.75, · R^2_worst chart · ≥0.74, ρ_preserve · ≥0.98, · NRMSE_trace · ≤0.05, κ_square · ≤0.02. · · The selection score is the weakest normalized margin across these gates. This prevents a strong aggregate R^2 from compensating for a failed chart, high curvature, or poor trajectory reconstruction.
Exact mathematical passage
\begin{aligned} R^2_{\mathrm{transport}}&\geq0.75, &\qquad R^2_{\mathrm{worst\ chart}}&\geq0.74,\\ \rho_{\mathrm{preserve}}&\geq0.98, & \operatorname{NRMSE}_{\mathrm{trace}}&\leq0.05,\\ \kappa_{\mathrm{square}}&\leq0.02. && \end{aligned} \label{eq:b20-thresholds}
\mathrm{transport}Use in source · 1 passage
Search aliases: transport mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17934
A candidate is pre-audit eligible only if both foreign target models are ATLAS_FULL and the candidate jointly satisfies R^2_transport · ≥0.75, · R^2_worst chart · ≥0.74, ρ_preserve · ≥0.98, · NRMSE_trace · ≤0.05, κ_square · ≤0.02. · · The selection score is the weakest normalized margin across these gates. This prevents a strong aggregate R^2 from compensating for a failed chart, high curvature, or poor trajectory reconstruction.
Exact mathematical passage
\begin{aligned} R^2_{\mathrm{transport}}&\geq0.75, &\qquad R^2_{\mathrm{worst\ chart}}&\geq0.74,\\ \rho_{\mathrm{preserve}}&\geq0.98, & \operatorname{NRMSE}_{\mathrm{trace}}&\leq0.05,\\ \kappa_{\mathrm{square}}&\leq0.02. && \end{aligned} \label{eq:b20-thresholds}
\mathrm{true\ chart}Use in source · 1 passage
Search aliases: true chart mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11868
tion 0.152169 and maximum gain 0.364965 across development. By contrast, the true-chart oracle has mean gain 0.998046, and the learned local constructor has mean blind gain 0.613651. Every one of the 54 development targets satisfies the frozen repair-conflict predicate G_global oracle≤0.40, G_true chart≥0.95. Mean learned gate accuracy is 0.915000, and the mean chart-specific bridge contrast is 2.186784. These quantities, together with the conservative embedding, produce the frozen grammar-inade…
Exact mathematical passage
G_{\mathrm{global\ oracle}}\leq0.40, \qquad G_{\mathrm{true\ chart}}\geq0.95.
\mathrm{twin}Use in source · 2 passages
Search aliases: twin mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 19848
For a candidate quotient q_θ, the empirical twin discrepancy is D_twin(q_θ) = E fracnormq_θ(x)-q_θ(x') max{normq_θ(x),ε}. A candidate is construction-valid only when it jointly predicts future consequences and intervention effects, predicts the externally scored answer, and keeps [source reference] below the declared ceiling.
Exact mathematical passage
D_{\mathrm{twin}}(q_\theta) = \mathbb E \frac{\norm{q_\theta(x)-q_\theta(x')}} {\max\{\norm{q_\theta(x)},\eps\}}. \label{eq:b23-twin-drift}
u'_\alphaUse in source · 2 passages
Search aliases: u'_α alpha
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2307
If a global repair u exists, set k_α=u|_U_α-u_α. Then Dk_α=0 and d_αβ=k_β-k_α, so the cocycle is a coboundary. Conversely, if d_αβ=k_β-k_α for kernel sections k_α, then u'_α=u_α+k_α agree on overlaps. The sheaf gluing axiom yields a global u, and Du=-r because the adjustments lie in the kernel.
Exact mathematical passage
u'_\alpha=u_\alpha+k_\alpha
U^{\mathsf T}Use in source · 2 passages
Search aliases: U^ T mathsf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 22030
Let the true context packet be P_*=UU^ T for a target-specific orthonormal matrix U∈ℝ^6×2, while the inherited packet is P_0=diag(1,1,0,0,0,0). Let the chart label be c(h)=1{e_1^ TU^ Th>0}. For generic U, c(h) is not measurable from P_0h. No threshold or nonlinear classifier using only P_0h can recover the chart perfectly. The response atlas can recover im U up to basis gauge when the intervention signatures separate it.
Exact mathematical passage
P_*=UU^{\mathsf T}
u^{\mathsf T}Use in source · 1 passage
Search aliases: u^ T mathsf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10652
[Worst-direction variance] For every u∈ℝ^d and every target coordinate, Var(u^ T(widehat T^ T-T^ T)e_jmid X) =u^ TM_S^-1u ≤ fracnormu^2λ_min(M_S).
Exact mathematical passage
\Var\left(u^{\mathsf T}(\widehat T^{\mathsf T}-T^{\mathsf T})e_j\mid X\right) =u^{\mathsf T}M_S^{-1}u \leq \frac{\norm{u}^2}{\lambda_{\min}(M_S)}.
u^{(0)}Use in source · 2 passages
Search aliases: u^(0)
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20714
-q. The zeroth packet-local repair is u^(0)=-e_1 and leaves residual Du^(0)+r= 0
Exact mathematical passage
u^{(0)}=-e_1
u^{(n)}Use in source · 3 passages
Search aliases: u^(n)
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1950
orrection under small packet coupling] If q:=normEQ_0<1, then D is surjective and has right inverse Q =Q_0(I_V+EQ_0)^-1 =Q_0∑_k=0^∞(-EQ_0)^k. Moreover, norm Q≤fracnormQ_01-q. For defect r, the truncated packet-correction series u^(n) :=-Q_0∑_k=0^n(-EQ_0)^kr has residual Du^(n)+r=(-EQ_0)^n+1r, and hence normDu^(n)+r≤ q^n+1norm r.
Exact mathematical passage
u^{(n)} :=-Q_0\sum_{k=0}^{n}(-EQ_0)^kr
u^*Use in source · 13 passages
Search aliases: u^*
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1568
hrmob, r_cor=P_corr, r_ob=P_obr; item the minimum achievable residual is inf_u∈ U^authnormDu+r =normr_ob =dist(r,im D); item the complete set of least-squares minimizers is -D^† r+ker D; item if r is exactly correctable, u^*=-D^† r is the unique exact repair of minimum norm.
Exact mathematical passage
u^*=-D^\dagger r
u^3Use in source · 1 passage
Search aliases: u^3
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2168
Consider the specific formal correction equation F(ε,u) = ε r_1+Du+frac12B(u,u)+O(ε^3+norm u^3), where B:U× U→ V is symmetric bilinear. Seek u(ε)=ε u_1+ε^2u_2+O(ε^3).
Exact mathematical passage
F(\eps,u) = \eps r_1+Du+\frac12B(u,u)+O\bigl(\eps^3+\norm u^3\bigr), \label{eq:second-order-model}
U^{\flat}Use in source · 1 passage
Search aliases: U^flat flat
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20945
0 · -1 . The flat repair core is U^flat=spansete_1. For u=e_1, both routes produce the same transported repair. For u=e_2, the two routes differ by (H^U-I)e_2=-2e_2, so the loop residual is 2. A repair schema containing an e_2 component is therefore path dependent even though every individual edge transport may be exact.
Exact mathematical passage
U^{\flat}=\Span\set{e_1}.
u_*Use in source · 2 passages
Search aliases: u_*
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11726
-0.25 . Hence P_ctxR_s(h)=P_ctxh. The observable chart is generated by a task-specific separator c(h)=1{anglesu_*,P_ctxh>θ_*}. The compiler does not receive u_* or θ_*, but it receives the declared protected packet in which a valid separator may be sought.
Exact mathematical passage
c(h)=\one\{\angles{u_*,P_{\mathrm{ctx}}h}>\theta_*\}.
U_0Use in source · 1 passage
Search aliases: U_0
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11420
[Open-set incompatibility of distinct affine charts] Let U_0,U_1⊆ℝ^d each contain a nonempty open set. Let f_j(h)=A_jh+b_j, j∈{0,1}. If one affine map f(h)=Ah+b satisfies f(h)=f_j(h) for every h∈ U_j and both j, then A=A_0=A_1, b=b_0=b_1. Consequently, if (A_0,b_0)≠(A_1,b_1), no element of G_glob represents the two-chart relation exactly.
Exact mathematical passage
U_0,U_1\subseteq\R^d
U_1Use in source · 4 passages
Search aliases: U_1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11420
[Open-set incompatibility of distinct affine charts] Let U_0,U_1⊆ℝ^d each contain a nonempty open set. Let f_j(h)=A_jh+b_j, j∈{0,1}. If one affine map f(h)=Ah+b satisfies f(h)=f_j(h) for every h∈ U_j and both j, then A=A_0=A_1, b=b_0=b_1. Consequently, if (A_0,b_0)≠(A_1,b_1), no element of G_glob represents the two-chart relation exactly.
Exact mathematical passage
U_0,U_1\subseteq\R^d
u_1Use in source · 9 passages
Search aliases: u_1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2027
u-v (u,v∈ B_R^W), with q:=norm QL_N<1, and normQr+norm Qsup_u∈ B_R^WnormN(u)≤ R. Then there exists a unique u^*∈ B_R^W satisfying F(u^*)=0. The gauge-fixed iteration u_k+1=-Qr-QN(u_k) converges to u^* from every u_0∈ B_R^W, with normu_k-u^*≤fracq^k1-qnormu_1-u_0.
Exact mathematical passage
\norm{u_k-u^*}\leq\frac{q^k}{1-q}\norm{u_1-u_0}.
u_2Use in source · 3 passages
Search aliases: u_2
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2175
Consider the specific formal correction equation F(ε,u) = ε r_1+Du+frac12B(u,u)+O(ε^3+norm u^3), where B:U× U→ V is symmetric bilinear. Seek u(ε)=ε u_1+ε^2u_2+O(ε^3).
Exact mathematical passage
u(\eps)=\eps u_1+\eps^2u_2+O(\eps^3).
u_3Use in source · 1 passage
Search aliases: u_3
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20923
Suppose a source correction has norm normu_0=2, every transport is nonexpansive, and three edges have (ε_D,0,ε_D,1,ε_D,2) =(0.01,0.02,0.015), (ε_r,0,ε_r,1,ε_r,2) =(0.005,0.004,0.006). Then [source reference] gives normD_3u_3+r_3 · ≤ 2(0.01+0.02+0.015) +(0.005+0.004+0.006) · =0.105. A direct source-to-target bridge with certified residual penalty below 0.105 is preferable under this metric, even if each edge of the three-step route appears individually accurate.
Exact mathematical passage
\begin{aligned} \norm{D_3u_3+r_3} &\leq 2(0.01+0.02+0.015) +(0.005+0.004+0.006)\\ &=0.105. \end{aligned}
U_4Use in source · 2 passages
Search aliases: U_4
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16109
efine z_0=ζ, z_k+1 = tanh (Rz_k+s_k+C_α(e,m)), 0≤ k<4, where s_k is a learned step embedding. The task-blind parent satisfies C_parent(e,m)=0. The descendants use one of the following conditioning maps: C_depth(e,m) · =D 1_m, C_1(e,m) · =U_1V_1e, rank(U_1V_1)≤1, C_4(e,m) · =U_4V_4e, rank(U_4V_4)≤4, C_full(e,m) · =We. Here 1_m is the one-hot depth code. Each child inherits every compatible parent parameter. The added up-projection or full-conditioning matrix is initialized so that the initial inj…
Exact mathematical passage
C_{\mathrm{depth}}(e,m)&=D\,\one_m, \label{eq:b17-depth-conditioning}\\ C_{1}(e,m)&=U_1V_1e, \qquad \rank(U_1V_1)\leq1, \label{eq:b17-rank1-conditioning}\\ C_{4}(e,m)&=U_4V_4e, \qquad \rank(U_4V_4)\leq4, \label{eq:b17-rank4-conditioning}\\ C_{\mathrm{full}}(e,m)&=We. \label{eq:b17-full-conditioning}
U_aUse in source · 6 passages
Search aliases: U_a
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6650
Let U_a∈ℝ^d_a× r have orthonormal columns spanning the discovered packet. Define desired packet coordinates z_i = U_a^top(h_a^∘(x_i)-h_a(x_i)). Fit an affine predictor widehat z(h)=K_a h+c_a by ridge regression on source anchor states. The repair is R_a(h) = h+U_awidehat z(h) = h+D_a h+b_a, where D_a=U_aK_a, b_a=U_ac_a.
Exact mathematical passage
U_a\in\R^{d_a\times r}
u_AUse in source · 26 passages
Search aliases: u_A
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2702
[Exact repair transport] Suppose D_BT_U=T_VD_A and r_B=T_Vr_A. If u_A satisfies D_Au_A=-r_A, then u_B:=T_Uu_A satisfies D_Bu_B=-r_B.
Exact mathematical passage
u_A
U_a^{\top}Use in source · 3 passages
Search aliases: U_a^top top
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6651
Let U_a∈ℝ^d_a× r have orthonormal columns spanning the discovered packet. Define desired packet coordinates z_i = U_a^top(h_a^∘(x_i)-h_a(x_i)). Fit an affine predictor widehat z(h)=K_a h+c_a by ridge regression on source anchor states. The repair is R_a(h) = h+U_awidehat z(h) = h+D_a h+b_a, where D_a=U_aK_a, b_a=U_ac_a.
Exact mathematical passage
z_i = U_a^{\top}(h_a^{\circ}(x_i)-h_a(x_i)).
U_{\mathrm{audit}}Use in source · 3 passages
Search aliases: U_audit mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13847
Let U_audit be a protected audit utility unavailable during compiler allocation and final scored selection. Define compiler metaproductivity at research budget B by C_B(Γ) = E_τ∼D_foreign [ U_audit (K_Γ^⋆(τ)) - U_audit(K_τ) ]. Only integrity-valid and capability-passing outputs are eligible.
Exact mathematical passage
U_{\mathrm{audit}}
u_BUse in source · 9 passages
Search aliases: u_B
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2707
[Exact repair transport] Suppose D_BT_U=T_VD_A and r_B=T_Vr_A. If u_A satisfies D_Au_A=-r_A, then u_B:=T_Uu_A satisfies D_Bu_B=-r_B.
Exact mathematical passage
u_B:=T_Uu_A
u_cUse in source · 1 passage
Search aliases: u_c
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17329
The context code is computed separately: u_c=C(s_x,t_x,s_y,t_y,d_sig)∈ℝ^10. It supports chart diagnostics but has no computational path into h_0, r_k, the task output, or the halt head.
Exact mathematical passage
u_c=C(s_x,t_x,s_y,t_y,d_{\mathrm{sig}})\in\R^{10}. \label{eq:b19-context-code}
U_eUse in source · 11 passages
Search aliases: U_e
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7223
al level. A parent repair selector is frozen. Development receipts generated through the existing lineage are used to compile a child selector. Parent and child then receive exactly the same candidate repairs, target observations, calibration information, evaluator, and candidate budget. The child receipt is sealed before either final audit split is opened. The primary question is E_e∼ Q [ U_e (K_1(C_e)) - U_e (K_0(C_e)) ] >0, where K_0 is the frozen parent selector, K_1 is the lineage-generated…
Exact mathematical passage
\label{eq:block07-primary-question} \mathbb E_{e\sim Q} \left[ U_e\!\left(K_{1}(C_e)\right) - U_e\!\left(K_{0}(C_e)\right) \right] >0,
u_{\mathrm{flat}}Use in source · 1 passage
Search aliases: u_flat mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5611
-2 , with zero overlap mismatch and identity evaluator transport. The unique minimum-norm repair is u_flat=-r_flat.
Exact mathematical passage
u_{\mathrm{flat}}=-r_{\mathrm{flat}}.
U_iUse in source · 2 passages
Search aliases: U_i
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1913
Let U=⊕_i=1^m U_i, V=⊕_i=1^m V_i, with orthogonal projectors Q_i and P_i. Decompose D=D_0+E, D_0:=∑_i=1^m P_iDQ_i. The block-diagonal operator D_0 represents ideal packet-local correction; E contains all cross-packet effects.
Exact mathematical passage
U=\bigoplus_{i=1}^m U_i, \qquad V=\bigoplus_{i=1}^m V_i,
u_i^2Use in source · 1 passage
Search aliases: u_i^2
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6170
For canonical patch u=T^-1δ, define corrected protected energy E_corr(d,u) = ∑_i=1^6w_imin{d_i^2,u_i^2}. This quantity is zero for a no-op and positive when a patch reaches task-relevant defect coordinates. It is reported alongside behavioral gain because a high apparent score can be generated without correcting protected defect energy.
Exact mathematical passage
E_{\mathrm{corr}}(d,u) = \sum_{i=1}^6w_i\min\{d_i^2,u_i^2\}.
U_jUse in source · 1 passage
Search aliases: U_j
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11429
[Open-set incompatibility of distinct affine charts] Let U_0,U_1⊆ℝ^d each contain a nonempty open set. Let f_j(h)=A_jh+b_j, j∈{0,1}. If one affine map f(h)=Ah+b satisfies f(h)=f_j(h) for every h∈ U_j and both j, then A=A_0=A_1, b=b_0=b_1. Consequently, if (A_0,b_0)≠(A_1,b_1), no element of G_glob represents the two-chart relation exactly.
Exact mathematical passage
h\in U_j
u_jUse in source · 2 passages
Search aliases: u_j
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2201
[Finite formal obstruction tower] Let F(ε,u) be analytic near (0,0) with F(0,0)=0 and D=∂_uF(0,0). Suppose coefficients u_1,…,u_n-1 have been chosen so that F(ε,∑_j=1^n-1ε^ju_j) =ε^nb_n+O(ε^n+1). Then extending the repair to order n by adding ε^nu_n is possible if and only if [b_n]=0 in coker D. When it vanishes, any solution of Du_n=-b_n produces an order-n lift.
Exact mathematical passage
F\left(\eps,\sum_{j=1}^{n-1}\eps^ju_j\right) =\eps^nb_n+O(\eps^{n+1}).
U_kUse in source · 1 passage
Search aliases: U_k
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2940
[Pathwise transport bound] Consider a path of systems A_0xrightarrow T_0A_1 xrightarrow T_1⋯ xrightarrow T_n-1A_n. Write T_U,k:U_k→ U_k+1 and T_V,k:V_k→ V_k+1. Let normD_k+1T_U,k-T_V,kD_k≤ε_D,k and normr_k+1-T_V,kr_k≤ε_r,k. If D_0u_0=-r_0 and u_n=T_U,n-1⋯ T_U,0u_0, then normD_nu_n+r_n · ≤ ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_D,k normT_U,k-1⋯ T_U,0u_0
Exact mathematical passage
T_{U,k}:U_k\to U_{k+1}
u_kUse in source · 6 passages
Search aliases: u_k
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2023
_R^W:=setw∈ W:norm w≤ R. Suppose normN(u)-N(v)≤ L_Nnormu-v (u,v∈ B_R^W), with q:=norm QL_N<1, and normQr+norm Qsup_u∈ B_R^WnormN(u)≤ R. Then there exists a unique u^*∈ B_R^W satisfying F(u^*)=0. The gauge-fixed iteration u_k+1=-Qr-QN(u_k) converges to u^* from every u_0∈ B_R^W, with normu_k-u^*≤fracq^k1-qnormu_1-u_0.
Exact mathematical passage
u_{k+1}=-Qr-QN(u_k)
U_{k+1}Use in source · 1 passage
Search aliases: U_k+1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2940
[Pathwise transport bound] Consider a path of systems A_0xrightarrow T_0A_1 xrightarrow T_1⋯ xrightarrow T_n-1A_n. Write T_U,k:U_k→ U_k+1 and T_V,k:V_k→ V_k+1. Let normD_k+1T_U,k-T_V,kD_k≤ε_D,k and normr_k+1-T_V,kr_k≤ε_r,k. If D_0u_0=-r_0 and u_n=T_U,n-1⋯ T_U,0u_0, then normD_nu_n+r_n · ≤ ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_D,k normT_U,k-1⋯ T_U,0u_0
Exact mathematical passage
T_{U,k}:U_k\to U_{k+1}
u_{k+1}Use in source · 1 passage
Search aliases: u_k+1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2023
_R^W:=setw∈ W:norm w≤ R. Suppose normN(u)-N(v)≤ L_Nnormu-v (u,v∈ B_R^W), with q:=norm QL_N<1, and normQr+norm Qsup_u∈ B_R^WnormN(u)≤ R. Then there exists a unique u^*∈ B_R^W satisfying F(u^*)=0. The gauge-fixed iteration u_k+1=-Qr-QN(u_k) converges to u^* from every u_0∈ B_R^W, with normu_k-u^*≤fracq^k1-qnormu_1-u_0.
Exact mathematical passage
u_{k+1}=-Qr-QN(u_k)
u_nUse in source · 10 passages
Search aliases: u_n
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2089
U and suppose F'(u_0) is invertible with inverse B. Assume that on the closed ball overline B(u_0,R), normF'(u)-F'(v)≤ Lnormu-v. Set η:=normBF(u_0), h:=norm B Lη. If h≤frac12 and t_*:=frac1-sqrt1-2hnorm B L≤ R, then Newton iteration u_n+1=u_n-F'(u_n)^-1F(u_n) is well-defined, remains in overline B(u_0,t_*), and converges to a root u^* of F. If the larger radius t^**:=frac1+sqrt1-2hnorm B L is contained in the regularity region, the root is unique in overline B(u_0,t^**).
Exact mathematical passage
u_{n+1}=u_n-F'(u_n)^{-1}F(u_n)
u_{n+1}Use in source · 1 passage
Search aliases: u_n+1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2089
U and suppose F'(u_0) is invertible with inverse B. Assume that on the closed ball overline B(u_0,R), normF'(u)-F'(v)≤ Lnormu-v. Set η:=normBF(u_0), h:=norm B Lη. If h≤frac12 and t_*:=frac1-sqrt1-2hnorm B L≤ R, then Newton iteration u_n+1=u_n-F'(u_n)^-1F(u_n) is well-defined, remains in overline B(u_0,t_*), and converges to a root u^* of F. If the larger radius t^**:=frac1+sqrt1-2hnorm B L is contained in the regularity region, the root is unique in overline B(u_0,t^**).
Exact mathematical passage
u_{n+1}=u_n-F'(u_n)^{-1}F(u_n)
u_{n,\alpha}Use in source · 3 passages
Search aliases: u_n,α alpha
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2570
At order n, let b_n,α∈mathscr V(U_α) be the residual coefficient determined by lower-order choices. The local obstruction is [b_n,α]∈coker D_α. If every local class vanishes, choose u_n,α with D_α u_n,α=-b_n,α. Their differences define an order-n gluing class in check H^1(ker D).
Exact mathematical passage
u_{n,\alpha}
u_{n-1}Use in source · 1 passage
Search aliases: u_n-1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2200
[Finite formal obstruction tower] Let F(ε,u) be analytic near (0,0) with F(0,0)=0 and D=∂_uF(0,0). Suppose coefficients u_1,…,u_n-1 have been chosen so that F(ε,∑_j=1^n-1ε^ju_j) =ε^nb_n+O(ε^n+1). Then extending the repair to order n by adding ε^nu_n is possible if and only if [b_n]=0 in coker D. When it vanishes, any solution of Du_n=-b_n produces an order-n lift.
Exact mathematical passage
u_1,\ldots,u_{n-1}
U_PUse in source · 1 passage
Search aliases: U_P
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13103
Let U_P∈ℝ^d× r have orthonormal columns spanning the recovered repair packet. Define packet coordinates z=U_P^top(x-c). The degree-two block-local grammar consists of maps F_A,b,C(x) = Ax+b+C vech(zz^top), where vech lists the r(r+1)/2 distinct symmetric quadratic monomials.
Exact mathematical passage
U_P\in\R^{d\times r}
U_P^\topUse in source · 1 passage
Search aliases: U_P^top top
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13104
Let U_P∈ℝ^d× r have orthonormal columns spanning the recovered repair packet. Define packet coordinates z=U_P^top(x-c). The degree-two block-local grammar consists of maps F_A,b,C(x) = Ax+b+C vech(zz^top), where vech lists the r(r+1)/2 distinct symmetric quadratic monomials.
Exact mathematical passage
z=U_P^\top(x-c).
U_{\mathrm{rep}}^\topUse in source · 1 passage
Search aliases: U_rep^top mathrm top
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13231
The constructor context uses the recovered repair packet and a target repair of the form F(x)=Ax+b+(z^top Bz)v, z=U_rep^top x, with B≠0 and v≠0. The parent affine grammar is structurally obstructed by [source reference].
Exact mathematical passage
F(x)=Ax+b+\bigl(z^\top Bz\bigr)v, \qquad z=U_{\mathrm{rep}}^\top x,
U_sUse in source · 1 passage
Search aliases: U_s
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6782
Given source repair R_s(h)=h+D_sh+b_s, its transported target form is D_t = widehat T D_swidehat T^+, b_t = widehat T b_s, with transported packet basis obtained by orthogonalizing widehat T U_s.
Exact mathematical passage
\widehat T U_s
U_tUse in source · 8 passages
Search aliases: U_t
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9606
A grammar extension may also change the defect representation. Let D_t:U_t→ V_t, with transports T_t^U:U_t→ U_t+1, T_t^V:V_t→ V_t+1. Assume the exact correction square D_t+1T_t^U = T_t^VD_t. Then there is an induced obstruction map overline T_t: coker D_t → coker D_t+1, overline T_t([r])=[T_t^Vr].
Exact mathematical passage
D_t:U_t\to V_t,
U_{t+1}^{\Auth}Use in source · 2 passages
Search aliases: U_t+1^Auth U_t+1^auth Auth mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1810
[Authorized obstruction-shrinkage theorem] Suppose U_t^auth⊆ U_t+1^auth and restrD_t+1U_t^auth=D_t. Then im D_t⊆im D_t+1, there is a canonical surjection q_t:coker D_t↠coker D_t+1, [v]_im D_t⟼[v]_im D_t+1, and dimcoker D_t+1≤dimcoker D_t. The inequality is strict if and only if im D_tsubsetneqim D_t+1.
Exact mathematical passage
U_t^{\Auth}\subseteq U_{t+1}^{\Auth}
U_{t-1}Use in source · 3 passages
Search aliases: U_t-1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 18297
The latent-clock family first applies an invertible stepwise affine gauge and then mixes adjacent recurrent times: Y_t = (1-β)U_t+β U_t-1, t=2,…,5, with Y_1=U_1 and 0<β<1. In matrix form, Y=W_βU, where W_β is lower bidiagonal. The temporal constructor applies W_β^-1 before fitting the spatial gauge.
Exact mathematical passage
Y_t = (1-\beta)U_t+\beta U_{t-1}, \qquad t=2,\ldots,5, \label{eq:block21-clock}
U_t^RUse in source · 2 passages
Search aliases: U_t^R
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10268
Let R_t be true protected regression and let U_t^R be a conditional upper confidence bound. Require U_t^R≤β_t, ∑_tβ_t≤ B_R. Let d_t be semantic debt incurred by approximate grammar or evaluator transport, with ∑_td_t≤ B_sem. A lineage certificate should therefore report three separate cumulative quantities: ( ∑_tδ_t, ∑_tβ_t, ∑_td_t ). A positive gain sequence is not sufficient if its protected regression or semantic debt diverges.
Exact mathematical passage
U_t^R
u_xUse in source · 2 passages
Search aliases: u_x
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9942
For episode x, let F_x:U_x^auth→ V_x be the post-modification defect map, with F_x(0)=r_x. A valid repair satisfies F_x(u_x)=0. A repair schema is a local section s:x↦ u_x that approximately inverts F_x over its declared domain.
Exact mathematical passage
F_x(u_x)=0.
U_x^{\Auth}Use in source · 1 passage
Search aliases: U_x^Auth U_x^auth Auth mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9938
For episode x, let F_x:U_x^auth→ V_x be the post-modification defect map, with F_x(0)=r_x. A valid repair satisfies F_x(u_x)=0. A repair schema is a local section s:x↦ u_x that approximately inverts F_x over its declared domain.
Exact mathematical passage
F_x:U_x^{\Auth}\to V_x
U_{\alpha\beta}Use in source · 5 passages
Search aliases: U_αβ alpha beta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2269
[Linear repair-descent theorem] Assume a local correction u_α exists on every U_α. On overlaps U_αβ=U_α∩ U_β, define d_αβ := restru_αU_αβ - restru_βU_αβ. Then: item d_αβ∈ker D(U_αβ); item (d_αβ) is a vCech 1-cocycle with coefficients in ker D; item a global correction u∈Γ(X,mathscr U) with Du=-r exists if and only if (d_αβ) is a vCech coboundary relative to U. If the cover computes first cohomology, the final condition is equivalent to [d_αβ]=0 in check H^1(X,ker D).
Exact mathematical passage
U_{\alpha\beta}=U_\alpha\cap U_\beta
U_\betaUse in source · 1 passage
Search aliases: U_β beta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2269
[Linear repair-descent theorem] Assume a local correction u_α exists on every U_α. On overlaps U_αβ=U_α∩ U_β, define d_αβ := restru_αU_αβ - restru_βU_αβ. Then: item d_αβ∈ker D(U_αβ); item (d_αβ) is a vCech 1-cocycle with coefficients in ker D; item a global correction u∈Γ(X,mathscr U) with Du=-r exists if and only if (d_αβ) is a vCech coboundary relative to U. If the cover computes first cohomology, the final condition is equivalent to [d_αβ]=0 in check H^1(X,ker D).
Exact mathematical passage
U_{\alpha\beta}=U_\alpha\cap U_\beta
u_\betaUse in source · 1 passage
Search aliases: u_β beta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2270
[Linear repair-descent theorem] Assume a local correction u_α exists on every U_α. On overlaps U_αβ=U_α∩ U_β, define d_αβ := restru_αU_αβ - restru_βU_αβ. Then: item d_αβ∈ker D(U_αβ); item (d_αβ) is a vCech 1-cocycle with coefficients in ker D; item a global correction u∈Γ(X,mathscr U) with Du=-r exists if and only if (d_αβ) is a vCech coboundary relative to U. If the cover computes first cohomology, the final condition is equivalent to [d_αβ]=0 in check H^1(X,ker D).
Exact mathematical passage
d_{\alpha\beta} := \restr{u_\alpha}{U_{\alpha\beta}} - \restr{u_\beta}{U_{\alpha\beta}}.
u_\lambdaUse in source · 2 passages
Search aliases: u_λ lambda
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1669
[Regularized minimum-cost repair] For λ>0, the functional J_λ(u) := frac12anglesDu+r,W(Du+r) + fracλ2anglesu,Ru has a unique minimizer u_λ =-(D^*WD+λ R)^-1D^*Wr. It is characterized by the normal equation D^*W(Du_λ+r)+λ Ru_λ=0.
Exact mathematical passage
u_\lambda =-(D^*WD+\lambda R)^{-1}D^*Wr.
\underline FUse in source · 1 passage
Search aliases: underline F underline
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9225
When the exact envelope is unknown, certified lower and upper oracle bounds underline F_K(R) ≤ F_K(R) ≤ overline F_K(R) produce corresponding bounds on χ_R. Audit-oracle arms are finite empirical approximations to this idea, but they must remain inadmissible for selection because they use final audit outcomes.
Exact mathematical passage
\underline F_K(R) \leq F_K(R) \leq \overline F_K(R)
\underline F_KUse in source · 1 passage
Search aliases: underline F_K underline
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9225
When the exact envelope is unknown, certified lower and upper oracle bounds underline F_K(R) ≤ F_K(R) ≤ overline F_K(R) produce corresponding bounds on χ_R. Audit-oracle arms are finite empirical approximations to this idea, but they must remain inadmissible for selection because they use final audit outcomes.
Exact mathematical passage
\underline F_K(R) \leq F_K(R) \leq \overline F_K(R)
\underline g_BUse in source · 3 passages
Search aliases: underline g_B underline
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3317
[Probabilistic gain preservation by simultaneous bounds] Given an (1-α) transport certificate, define underline g_B := · underline g_A -2overlineε_J · -overline L( 2overlineε_r +overlineε_Dnormu_A +overlineε_leak +overlineε_hol ). Then Pr(g_B≥underline g_B)≥1-α. In particular, underline g_B>0 is a valid one-sided certificate of beneficial target transfer.
Exact mathematical passage
\begin{aligned} \underline g_B :=\;& \underline g_A -2\overline\eps_J\\ &-\overline L\left( 2\overline\eps_r +\overline\eps_D\norm{u_A} +\overline\eps_{\mathrm{leak}} +\overline\eps_{\mathrm{hol}} \right). \end{aligned}
\mathrm{utility}Use in source · 1 passage
Search aliases: utility mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21218
Let protected keys be K_protected ={evidence,evaluator,provenance}. The adversarial candidate declares K_edit ={evaluator,utility}. Then K_edit∩ K_protected ={evaluator}≠∅. Even before considering utility, the acceptance predicate is false. The sandbox also changes the protected evaluator value, providing a second independent rejection reason. The final state is restored from the checkpoint and has the original evaluator digest.
Exact mathematical passage
K_{\mathrm{edit}} =\{\mathrm{evaluator},\mathrm{utility}\}.
V'Use in source · 7 passages
Search aliases: V'
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5056
Let j:V↪ V' be an injective evidence-conservative embedding of old defects into an expanded ontology. Let C=im D⊆ V, C'=im D'⊆ V', and assume preservation of old correctability: j(C)⊆ C'. The embedding induces a map overline j:V/C→ V'/C'.
Exact mathematical passage
j:V\hookrightarrow V'
V^{\mathsf T}Use in source · 2 passages
Search aliases: V^ T mathsf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15818
For parent and child state spaces H_A and H_B of equal dimension, let calibration traces produce matrices X and Y. The orthogonal Procrustes transport is T^* = arg min_T^ TT=InormXT-Y_ F. If X^ TY=UΣ V^ T, then T^*=UV^ T. A transport receipt records calibration residual, held-out residual, singular floor, round-trip residual, and behavior change after transported trace injection.
Exact mathematical passage
X^{\mathsf T}Y=U\Sigma V^{\mathsf T}
v^{\mathsf T}Use in source · 3 passages
Search aliases: v^ T mathsf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10768
[A-optimal trace reduction] Let v=sqrtw_ix_i. Then tr(M^-1)-tr((M+vv^ T)^-1) = fracv^ TM^-2v1+v^ TM^-1v.
Exact mathematical passage
\tr(M^{-1})-\tr\left((M+vv^{\mathsf T})^{-1}\right) = \frac{v^{\mathsf T}M^{-2}v}{1+v^{\mathsf T}M^{-1}v}.
v^2Use in source · 1 passage
Search aliases: v^2
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21627
Let the old correction image be I=span{e_1,e_2}⊂ℝ^4 and let r=(0,0,3,4). Then z=P_I^⊥r=(0,0,3,4), η_I(r)^2=25. Add generator v=2e_4. Because v∈ I^⊥, frac|anglesz,v|^2norm v^2 = frac8^24 =16. The expanded residual is 3e_3, so η_I'(r)^2=9=25-16. A generator parallel to e_1 would have bar v=0 and would produce no structural reduction despite having nonzero raw norm.
Exact mathematical passage
\frac{|\angles{z,v}|^2}{\norm v^2} = \frac{8^2}{4} =16.
v^\topUse in source · 1 passage
Search aliases: v^top top
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8459
When X has full column rank, X^top X is invertible and (X^top X)^-1X^top Y =(X^top X)^-1X^top XT^top =T^top. When rank X<d, choose a nonzero v∈ker X. For any nonzero w∈ℝ^m, define K=wv^top. Then XK^top=0, so T and T+K agree on every selected anchor while differing on v.
Exact mathematical passage
K=wv^\top
v^\starUse in source · 1 passage
Search aliases: v^⋆ star
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9772
Equation [source reference] gives a direct acquisition score. For a candidate family V_cand, choose v^⋆ ∈ arg max_v∈V_cand fracabsanglesz,P_I^⊥v^2 normP_I^⊥v^2+λ c(v)+βRisk(v), where c(v) is implementation cost and Risk(v) is a protected-risk estimate.
Exact mathematical passage
v^\star \in \argmax_{v\in\Vcal_{\mathrm{cand}}} \frac{\abs{\angles{z,P_{I^\perp}v}}^2} {\norm{P_{I^\perp}v}^2+\lambda c(v)+\beta\Risk(v)}, \label{eq:b8-generator-score}
V_1Use in source · 3 passages
Search aliases: V_1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16109
efine z_0=ζ, z_k+1 = tanh (Rz_k+s_k+C_α(e,m)), 0≤ k<4, where s_k is a learned step embedding. The task-blind parent satisfies C_parent(e,m)=0. The descendants use one of the following conditioning maps: C_depth(e,m) · =D 1_m, C_1(e,m) · =U_1V_1e, rank(U_1V_1)≤1, C_4(e,m) · =U_4V_4e, rank(U_4V_4)≤4, C_full(e,m) · =We. Here 1_m is the one-hot depth code. Each child inherits every compatible parent parameter. The added up-projection or full-conditioning matrix is initialized so that the initial inj…
Exact mathematical passage
C_{\mathrm{depth}}(e,m)&=D\,\one_m, \label{eq:b17-depth-conditioning}\\ C_{1}(e,m)&=U_1V_1e, \qquad \rank(U_1V_1)\leq1, \label{eq:b17-rank1-conditioning}\\ C_{4}(e,m)&=U_4V_4e, \qquad \rank(U_4V_4)\leq4, \label{eq:b17-rank4-conditioning}\\ C_{\mathrm{full}}(e,m)&=We. \label{eq:b17-full-conditioning}
V_2Use in source · 1 passage
Search aliases: V_2
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16714
p embedding at latent depth k∈{0,…,4}. The source uses an Elman transition z_k+1 = tanh(Wz_k+s_k+δ_k(e,z_k)). The first foreign family uses a GRU cell z_k+1=GRUCell(u_k,z_k), and the second uses a gated nonlinear residual cell z_k+1 = z_k+σ(Gz_k)odot tanh(V_2GELU(V_1z_k)+s_k+δ_k(e,z_k)). The three cell classes have distinct executable implementations and parameterizations. They share only the declared external input, latent dimension, five-step budget, semantic target, downstream gated readout, …
Exact mathematical passage
z_{k+1} = z_k+\sigma(Gz_k)\odot \tanh\left(V_2\operatorname{GELU}(V_1z_k)+s_k+\delta_k(e,z_k)\right). \label{eq:b18-residual}
V_4Use in source · 2 passages
Search aliases: V_4
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16109
efine z_0=ζ, z_k+1 = tanh (Rz_k+s_k+C_α(e,m)), 0≤ k<4, where s_k is a learned step embedding. The task-blind parent satisfies C_parent(e,m)=0. The descendants use one of the following conditioning maps: C_depth(e,m) · =D 1_m, C_1(e,m) · =U_1V_1e, rank(U_1V_1)≤1, C_4(e,m) · =U_4V_4e, rank(U_4V_4)≤4, C_full(e,m) · =We. Here 1_m is the one-hot depth code. Each child inherits every compatible parent parameter. The added up-projection or full-conditioning matrix is initialized so that the initial inj…
Exact mathematical passage
C_{\mathrm{depth}}(e,m)&=D\,\one_m, \label{eq:b17-depth-conditioning}\\ C_{1}(e,m)&=U_1V_1e, \qquad \rank(U_1V_1)\leq1, \label{eq:b17-rank1-conditioning}\\ C_{4}(e,m)&=U_4V_4e, \qquad \rank(U_4V_4)\leq4, \label{eq:b17-rank4-conditioning}\\ C_{\mathrm{full}}(e,m)&=We. \label{eq:b17-full-conditioning}
V_iUse in source · 1 passage
Search aliases: V_i
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1913
Let U=⊕_i=1^m U_i, V=⊕_i=1^m V_i, with orthogonal projectors Q_i and P_i. Decompose D=D_0+E, D_0:=∑_i=1^m P_iDQ_i. The block-diagonal operator D_0 represents ideal packet-local correction; E contains all cross-packet effects.
Exact mathematical passage
U=\bigoplus_{i=1}^m U_i, \qquad V=\bigoplus_{i=1}^m V_i,
V_kUse in source · 1 passage
Search aliases: V_k
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2940
[Pathwise transport bound] Consider a path of systems A_0xrightarrow T_0A_1 xrightarrow T_1⋯ xrightarrow T_n-1A_n. Write T_U,k:U_k→ U_k+1 and T_V,k:V_k→ V_k+1. Let normD_k+1T_U,k-T_V,kD_k≤ε_D,k and normr_k+1-T_V,kr_k≤ε_r,k. If D_0u_0=-r_0 and u_n=T_U,n-1⋯ T_U,0u_0, then normD_nu_n+r_n · ≤ ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_D,k normT_U,k-1⋯ T_U,0u_0
Exact mathematical passage
T_{V,k}:V_k\to V_{k+1}
V_{k+1}Use in source · 1 passage
Search aliases: V_k+1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2940
[Pathwise transport bound] Consider a path of systems A_0xrightarrow T_0A_1 xrightarrow T_1⋯ xrightarrow T_n-1A_n. Write T_U,k:U_k→ U_k+1 and T_V,k:V_k→ V_k+1. Let normD_k+1T_U,k-T_V,kD_k≤ε_D,k and normr_k+1-T_V,kr_k≤ε_r,k. If D_0u_0=-r_0 and u_n=T_U,n-1⋯ T_U,0u_0, then normD_nu_n+r_n · ≤ ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_D,k normT_U,k-1⋯ T_U,0u_0
Exact mathematical passage
T_{V,k}:V_k\to V_{k+1}
V_{t+1}Use in source · 2 passages
Search aliases: V_t+1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1877
Open-set improvement may enlarge the defect ontology itself. Let j_U:U_t^auth↪ U_t+1^auth, j_V:V_t↪ V_t+1 be injective embeddings satisfying D_t+1j_U=j_VD_t.
Exact mathematical passage
j_U:U_t^{\Auth}\hookrightarrow U_{t+1}^{\Auth}, \qquad j_V:V_t\hookrightarrow V_{t+1}
v_WUse in source · 2 passages
Search aliases: v_W
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5755
An executable state is a finite canonical map widehat s:Key→Value. A witness policy is widehat W =(K_protected,γ_min,r_max,p_prov,e_irr,v_W), where K_protected is the immutable-key set, γ_min is the minimum protected gain, r_max is the maximum accepted risk, p_prov indicates whether provenance is mandatory, e_irr controls irreversible effects, and v_W is the witness version.
Exact mathematical passage
\widehat W =(K_{\mathrm{protected}},\gamma_{\min},r_{\max},p_{\mathrm{prov}},e_{\mathrm{irr}},v_W),
V_xUse in source · 1 passage
Search aliases: V_x
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9938
For episode x, let F_x:U_x^auth→ V_x be the post-modification defect map, with F_x(0)=r_x. A valid repair satisfies F_x(u_x)=0. A repair schema is a local section s:x↦ u_x that approximately inverts F_x over its declared domain.
Exact mathematical passage
F_x:U_x^{\Auth}\to V_x
\mathrm{val}Use in source · 1 passage
Search aliases: val mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12552
The selected atlas candidate commits only when all frozen conditions hold: widehat g_val · ≥0.10, LCB(widehat g_val) · ≥0.05, widehat r_ctx · ≤3×10^-4, widehat a_target · ≥0.72, widehat a_repair inv · ≥0.96, widehat b_chart · ≥0.20, widehat s_model · ≤0.25, widehatγ_min · ≥0.004, widehatε_comm · ≤0.16, widehatε_resp · ≤0.08. The witness therefore evaluates both downstream repair quality and the internal atlas certificate. A high blind gain with unstable response geometry is not sufficient for co…
Exact mathematical passage
\widehat g_{\mathrm{val}}&\geq0.10,\\ \LCB(\widehat g_{\mathrm{val}})&\geq0.05,\\ \widehat r_{\mathrm{ctx}}&\leq3\times10^{-4},\\ \widehat a_{\mathrm{target}}&\geq0.72,\\ \widehat a_{\mathrm{repair\ inv}}&\geq0.96,\\ \widehat b_{\mathrm{chart}}&\geq0.20,\\ \widehat s_{\mathrm{model}}&\leq0.25,\\ \widehat\gamma_{\min}&\geq0.004,\\ \widehat\eps_{\mathrm{comm}}&\leq0.16,\\ \widehat\eps_{\mathrm{resp}}&\leq0.08.
\operatorname{Valid}Use in source · 2 passages
Search aliases: Valid operatorname
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13711
Let Valid(c_i,Π_i^+)∈{0,1} encode capability, integrity, provenance, and resource validity. The selected descendant is K_Γ^⋆(τ) ∈ arg max_i:Valid(c_i,Π_i^+)=1 S_V(c_i), with the no-op baseline retained when no valid candidate exists.
Exact mathematical passage
\operatorname{Valid}(c_i,\Pi_i^{+})\in\{0,1\}
\mathsf{Value}Use in source · 1 passage
Search aliases: Value mathsf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5751
An executable state is a finite canonical map widehat s:Key→Value. A witness policy is widehat W =(K_protected,γ_min,r_max,p_prov,e_irr,v_W), where K_protected is the immutable-key set, γ_min is the minimum protected gain, r_max is the maximum accepted risk, p_prov indicates whether provenance is mandatory, e_irr controls irreversible effects, and v_W is the witness version.
Exact mathematical passage
\widehat s:\mathsf{Key}\to\mathsf{Value}.
\VarUse in source · 1 passage
Search aliases: Var operatorname
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10652
[Worst-direction variance] For every u∈ℝ^d and every target coordinate, Var(u^ T(widehat T^ T-T^ T)e_jmid X) =u^ TM_S^-1u ≤ fracnormu^2λ_min(M_S).
Exact mathematical passage
\Var\left(u^{\mathsf T}(\widehat T^{\mathsf T}-T^{\mathsf T})e_j\mid X\right) =u^{\mathsf T}M_S^{-1}u \leq \frac{\norm{u}^2}{\lambda_{\min}(M_S)}.
\Vcal_{\mathrm{cand}}Use in source · 2 passages
Search aliases: Vcal_cand V_cand Vcal mathrm mathcal mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9771
Equation [source reference] gives a direct acquisition score. For a candidate family V_cand, choose v^⋆ ∈ arg max_v∈V_cand fracabsanglesz,P_I^⊥v^2 normP_I^⊥v^2+λ c(v)+βRisk(v), where c(v) is implementation cost and Risk(v) is a protected-risk estimate.
Exact mathematical passage
\Vcal_{\mathrm{cand}}
\Vcal_t^{\mathrm{fr}}Use in source · 2 passages
Search aliases: Vcal_t^fr V_t^fr Vcal mathrm mathcal mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 811
A witness structure may be represented as W_t = ( V_t^fr, C_t, B_t^sand, R_t, Π_t^auth ), where V_t^fr is the frozen evaluator instance for the current transaction, C_t is a protected checkpoint, B_t^sand is the sandbox boundary, R_t is the rollback procedure, and Π_t^auth records effect authority.
Exact mathematical passage
\Wcal_t = \left( \Vcal_t^{\mathrm{fr}}, C_t, \Bcal_t^{\mathrm{sand}}, R_t, \Pi_t^{\mathrm{auth}} \right),
\operatorname{vec}Use in source · 3 passages
Search aliases: vec operatorname
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10623
[Weighted bridge covariance] Assume M_S is invertible and the noise rows are independent with covariance σ_i^2I_q. The weighted least-squares estimator widehat T^ T = M_S^-1X^ TWY is unbiased and satisfies Cov(vec(widehat T^ T-T^ T)mid X) = I_q⊗ M_S^-1. Consequently, E[normwidehat T-T_F^2mid X] = q tr(M_S^-1).
Exact mathematical passage
\Cov\left(\operatorname{vec}(\widehat T^{\mathsf T}-T^{\mathsf T})\mid X\right) = I_q\otimes M_S^{-1}.
\vechUse in source · 2 passages
Search aliases: vech operatorname
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13108
Let U_P∈ℝ^d× r have orthonormal columns spanning the recovered repair packet. Define packet coordinates z=U_P^top(x-c). The degree-two block-local grammar consists of maps F_A,b,C(x) = Ax+b+C vech(zz^top), where vech lists the r(r+1)/2 distinct symmetric quadratic monomials.
Exact mathematical passage
F_{A,b,C}(x) = Ax+b+C\,\vech(zz^\top), \label{eq:b12-block-quadratic}
\mathrm{verifier}Use in source · 1 passage
Search aliases: verifier mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1478
Consider four colors memory, representation, verifier, planner. A composite factuality repair may have the form ω_gate ∘ ( ω_retrieve, ω_decompose, ω_check ). The inner operations retrieve evidence, decompose output into claims, and check claims against protected sources. Their postconditions establish the precondition for a planner gate that blocks or revises unsupported claims. The composite receipt records source provenance, claim-to-evidence links, verifier uncertainty, revised spans, latenc…
Exact mathematical passage
\mathrm{memory}, \mathrm{representation}, \mathrm{verifier}, \mathrm{planner}.
\operatorname{Verify}Use in source · 2 passages
Search aliases: Verify operatorname
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9989
Let P_G be a finite typed repair-program grammar and let Verify(p,x)∈{0,1} be a complete verifier on a finite declared episode domain.
Exact mathematical passage
\operatorname{Verify}(p,x)\in\{0,1\}
w^{\mathsf T}Use in source · 3 passages
Search aliases: w^ T mathsf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4498
[Archive-envelope monotonicity] Suppose archive updates never delete the last representative of a retained node. For every nonnegative weight vector w, define H_w(A_t) = max_A∈A_tw^ Tz(A). Then H_w(A_t) is nondecreasing in t.
Exact mathematical passage
H_w(\Acal_t) = \max_{A\in\Acal_t}w^{\mathsf T}z(A).
W^{-1}Use in source · 2 passages
Search aliases: W^-1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10643
Write widehat T^ T-T^ T = M_S^-1X^ TWE. For one target coordinate, the noise covariance is W^-1, so the coefficient covariance is M_S^-1X^ TW W^-1 WXM_S^-1=M_S^-1. The target coordinates are independent and identically weighted, giving the Kronecker covariance. Taking the trace yields the Frobenius identity.
Exact mathematical passage
W^{-1}
W_EUse in source · 7 passages
Search aliases: W_E
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2432
Different overlaps may have different uncertainty or semantic importance. Let W_E be a positive-definite operator on K^E. Define η_glue,W(e) := inf_knorme+ B^*k_W_E.
Exact mathematical passage
W_E
W_iUse in source · 1 passage
Search aliases: W_i
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17321
widehat y_i)∈ℝ^32, and the permutation-invariant aggregate a(D) = ( frac15∑_i=1^5p_i, max_1≤ i≤5p_i )∈ℝ^64. The semantic seed is h_0=S(a(D),Q(widehat q))∈ℝ^40. A five-step Elman recurrence generates the Neuralese trace r_k=tanh(W_ih_0+b_i+U_ir_k-1+c_i+e_k), k=1,…,5, where e_k is a learned step embedding. A gated readout recurrence maps the trace to protected program and answer coordinates. A halt head predicts one of the five execution depths.
Exact mathematical passage
r_k=\tanh\left(W_ih_0+b_i+U_ir_{k-1}+c_i+e_k\right), \qquad k=1,\ldots,5, \label{eq:b19-recurrence}
w_jUse in source · 6 passages
Search aliases: w_j
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3549
Exact fixed subspaces can be too brittle for learned evaluator transports. Let H_1,…,H_m be a finite generating family of loop transports on a Hilbert space E, with positive weights w_j satisfying ∑_jw_j=1. Define the holonomy-energy operator A := ∑_j=1^m w_j(H_j-I)^*(H_j-I). This operator is positive semidefinite.
Exact mathematical passage
w_j
w_{\rm capability}Use in source · 1 passage
Search aliases: w_rm capability rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 14654
The selected child configuration is τ_rm residual · =0.20, · τ_rm distance · =24, w_rm mean · =0.38, · w_rm median · =0.10, w_rm min · =0.24, · w_rm held · =0.22, w_rm capability · =0.32. It is selected from a 27-point finite meta-validation grid before either final audit is opened.
Exact mathematical passage
\begin{aligned} \tau_{\rm residual}&=0.20, &\tau_{\rm distance}&=24,\\ w_{\rm mean}&=0.38, &w_{\rm median}&=0.10,\\ w_{\rm min}&=0.24, &w_{\rm held}&=0.22,\\ w_{\rm capability}&=0.32. \end{aligned} \label{eq:block14-config}
w_{\rm held}Use in source · 1 passage
Search aliases: w_rm held rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 14654
The selected child configuration is τ_rm residual · =0.20, · τ_rm distance · =24, w_rm mean · =0.38, · w_rm median · =0.10, w_rm min · =0.24, · w_rm held · =0.22, w_rm capability · =0.32. It is selected from a 27-point finite meta-validation grid before either final audit is opened.
Exact mathematical passage
\begin{aligned} \tau_{\rm residual}&=0.20, &\tau_{\rm distance}&=24,\\ w_{\rm mean}&=0.38, &w_{\rm median}&=0.10,\\ w_{\rm min}&=0.24, &w_{\rm held}&=0.22,\\ w_{\rm capability}&=0.32. \end{aligned} \label{eq:block14-config}
w_{\rm mean}Use in source · 1 passage
Search aliases: w_rm mean rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 14654
The selected child configuration is τ_rm residual · =0.20, · τ_rm distance · =24, w_rm mean · =0.38, · w_rm median · =0.10, w_rm min · =0.24, · w_rm held · =0.22, w_rm capability · =0.32. It is selected from a 27-point finite meta-validation grid before either final audit is opened.
Exact mathematical passage
\begin{aligned} \tau_{\rm residual}&=0.20, &\tau_{\rm distance}&=24,\\ w_{\rm mean}&=0.38, &w_{\rm median}&=0.10,\\ w_{\rm min}&=0.24, &w_{\rm held}&=0.22,\\ w_{\rm capability}&=0.32. \end{aligned} \label{eq:block14-config}
w_{\rm median}Use in source · 1 passage
Search aliases: w_rm median rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 14654
The selected child configuration is τ_rm residual · =0.20, · τ_rm distance · =24, w_rm mean · =0.38, · w_rm median · =0.10, w_rm min · =0.24, · w_rm held · =0.22, w_rm capability · =0.32. It is selected from a 27-point finite meta-validation grid before either final audit is opened.
Exact mathematical passage
\begin{aligned} \tau_{\rm residual}&=0.20, &\tau_{\rm distance}&=24,\\ w_{\rm mean}&=0.38, &w_{\rm median}&=0.10,\\ w_{\rm min}&=0.24, &w_{\rm held}&=0.22,\\ w_{\rm capability}&=0.32. \end{aligned} \label{eq:block14-config}
w_{\rm min}Use in source · 1 passage
Search aliases: w_rm min rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 14654
The selected child configuration is τ_rm residual · =0.20, · τ_rm distance · =24, w_rm mean · =0.38, · w_rm median · =0.10, w_rm min · =0.24, · w_rm held · =0.22, w_rm capability · =0.32. It is selected from a 27-point finite meta-validation grid before either final audit is opened.
Exact mathematical passage
\begin{aligned} \tau_{\rm residual}&=0.20, &\tau_{\rm distance}&=24,\\ w_{\rm mean}&=0.38, &w_{\rm median}&=0.10,\\ w_{\rm min}&=0.24, &w_{\rm held}&=0.22,\\ w_{\rm capability}&=0.32. \end{aligned} \label{eq:block14-config}
W_{\beta}Use in source · 2 passages
Search aliases: W_β beta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 18305
The latent-clock family first applies an invertible stepwise affine gauge and then mixes adjacent recurrent times: Y_t = (1-β)U_t+β U_t-1, t=2,…,5, with Y_1=U_1 and 0<β<1. In matrix form, Y=W_βU, where W_β is lower bidiagonal. The temporal constructor applies W_β^-1 before fitting the spatial gauge.
Exact mathematical passage
Y=W_{\beta}U,
W_{\beta}^{-1}Use in source · 1 passage
Search aliases: W_β^-1 beta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 18308
The latent-clock family first applies an invertible stepwise affine gauge and then mixes adjacent recurrent times: Y_t = (1-β)U_t+β U_t-1, t=2,…,5, with Y_1=U_1 and 0<β<1. In matrix form, Y=W_βU, where W_β is lower bidiagonal. The temporal constructor applies W_β^-1 before fitting the spatial gauge.
Exact mathematical passage
W_{\beta}^{-1}
\Wcal_t^{\mathrm{NL}}Use in source · 3 passages
Search aliases: Wcal_t^NL W_t^NL Wcal mathrm mathcal mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15834
Extend the SRIC lineage state by L_t^NL = ( L_t,N_t,I_t^NL,T_t^NL,W_t^NL), where T_t^NL contains trace-transport certificates and W_t^NL is the frozen external answer and invariant witness. A proposed descendant may modify the latent transition, update gate, halting head, packet atlas, or transport map. The backbone may be held fixed for a component-isolation edge or changed under a stronger whole-model edge.
Exact mathematical passage
\mathfrak L_t^{\mathrm{NL}} = (\mathfrak L_t,\Ncal_t,\Ical_t^{\mathrm{NL}},\Tau_t^{\mathrm{NL}},\Wcal_t^{\mathrm{NL}}),
\wedgeUse in source · 2 passages
Search aliases: wedge
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4259
Let C_RI(s) = · Grounded(s) wedgeProvValid(s) wedgeWorkspaceBound(s) · wedgeRepairTyped(s) wedgeWitnessed(s) wedgeEvidenceFlat(s) · wedgeRollbackable(s) wedgeMetaCapable(s). For X⊆ S, define Ψ_∀(X) = sets∈ S:C_RI(s) and Post_imp(s)⊆ X.
Exact mathematical passage
\begin{aligned} C_{\mathrm{RI}}(s) ={}& \Grounded(s) \wedge\ProvValid(s) \wedge\WorkspaceBound(s)\\ &\wedge\RepairTyped(s) \wedge\Witnessed(s) \wedge\EvidenceFlat(s)\\ &\wedge\Rollbackable(s) \wedge\MetaCapable(s). \end{aligned}
\widehat AUse in source · 2 passages
Search aliases: widehat A widehat
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6980
The witness commits a transported repair only when all four calibration conditions hold: widehat g_rm slice · ≥ 0.20, widehat r_rm non · ≤ 0.03, widehatε_rm align · ≤ 0.35, widehat A_rm non^rm base · ≥ 0.90. Here widehat g_rm slice is calibration gain on the affected stratum, widehat r_rm non is regression on the unaffected stratum, widehatε_rm align is normalized alignment error, and widehat A_rm non^rm base is baseline unaffected accuracy.
Exact mathematical passage
\widehat g_{\rm slice}&\geq 0.20,\\ \widehat r_{\rm non}&\leq 0.03,\\ \widehat\eps_{\rm align}&\leq 0.35,\\ \widehat A_{\rm non}^{\rm base}&\geq 0.90. \label{eq:block06-gate}
\widehat aUse in source · 1 passage
Search aliases: widehat a widehat
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12552
The selected atlas candidate commits only when all frozen conditions hold: widehat g_val · ≥0.10, LCB(widehat g_val) · ≥0.05, widehat r_ctx · ≤3×10^-4, widehat a_target · ≥0.72, widehat a_repair inv · ≥0.96, widehat b_chart · ≥0.20, widehat s_model · ≤0.25, widehatγ_min · ≥0.004, widehatε_comm · ≤0.16, widehatε_resp · ≤0.08. The witness therefore evaluates both downstream repair quality and the internal atlas certificate. A high blind gain with unstable response geometry is not sufficient for co…
Exact mathematical passage
\widehat g_{\mathrm{val}}&\geq0.10,\\ \LCB(\widehat g_{\mathrm{val}})&\geq0.05,\\ \widehat r_{\mathrm{ctx}}&\leq3\times10^{-4},\\ \widehat a_{\mathrm{target}}&\geq0.72,\\ \widehat a_{\mathrm{repair\ inv}}&\geq0.96,\\ \widehat b_{\mathrm{chart}}&\geq0.20,\\ \widehat s_{\mathrm{model}}&\leq0.25,\\ \widehat\gamma_{\min}&\geq0.004,\\ \widehat\eps_{\mathrm{comm}}&\leq0.16,\\ \widehat\eps_{\mathrm{resp}}&\leq0.08.
\widehat{\mathsf A}Use in source · 4 passages
Search aliases: widehat A widehat mathsf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3618
[Stability of the threshold count] Let widehat A be an estimated holonomy-energy operator satisfying normwidehat A- A≤η. Then each ordered eigenvalue satisfies abswidehatλ_i-λ_i≤η. If no eigenvalue of A lies in [τ-η,τ+η], then widehat A and A have the same number of eigenvalues below τ.
Exact mathematical passage
\widehat{\mathsf A}
\widehat a_{\mathrm{repair\ inv}}Use in source · 1 passage
Search aliases: widehat a_repair inv widehat mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12552
The selected atlas candidate commits only when all frozen conditions hold: widehat g_val · ≥0.10, LCB(widehat g_val) · ≥0.05, widehat r_ctx · ≤3×10^-4, widehat a_target · ≥0.72, widehat a_repair inv · ≥0.96, widehat b_chart · ≥0.20, widehat s_model · ≤0.25, widehatγ_min · ≥0.004, widehatε_comm · ≤0.16, widehatε_resp · ≤0.08. The witness therefore evaluates both downstream repair quality and the internal atlas certificate. A high blind gain with unstable response geometry is not sufficient for co…
Exact mathematical passage
\widehat g_{\mathrm{val}}&\geq0.10,\\ \LCB(\widehat g_{\mathrm{val}})&\geq0.05,\\ \widehat r_{\mathrm{ctx}}&\leq3\times10^{-4},\\ \widehat a_{\mathrm{target}}&\geq0.72,\\ \widehat a_{\mathrm{repair\ inv}}&\geq0.96,\\ \widehat b_{\mathrm{chart}}&\geq0.20,\\ \widehat s_{\mathrm{model}}&\leq0.25,\\ \widehat\gamma_{\min}&\geq0.004,\\ \widehat\eps_{\mathrm{comm}}&\leq0.16,\\ \widehat\eps_{\mathrm{resp}}&\leq0.08.
\widehat A_{\rm non}^{\rm base}Use in source · 2 passages
Search aliases: widehat A_rm non^rm base widehat rm rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6980
The witness commits a transported repair only when all four calibration conditions hold: widehat g_rm slice · ≥ 0.20, widehat r_rm non · ≤ 0.03, widehatε_rm align · ≤ 0.35, widehat A_rm non^rm base · ≥ 0.90. Here widehat g_rm slice is calibration gain on the affected stratum, widehat r_rm non is regression on the unaffected stratum, widehatε_rm align is normalized alignment error, and widehat A_rm non^rm base is baseline unaffected accuracy.
Exact mathematical passage
\widehat g_{\rm slice}&\geq 0.20,\\ \widehat r_{\rm non}&\leq 0.03,\\ \widehat\eps_{\rm align}&\leq 0.35,\\ \widehat A_{\rm non}^{\rm base}&\geq 0.90. \label{eq:block06-gate}
\widehat a_{\mathrm{target}}Use in source · 1 passage
Search aliases: widehat a_target widehat mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12552
The selected atlas candidate commits only when all frozen conditions hold: widehat g_val · ≥0.10, LCB(widehat g_val) · ≥0.05, widehat r_ctx · ≤3×10^-4, widehat a_target · ≥0.72, widehat a_repair inv · ≥0.96, widehat b_chart · ≥0.20, widehat s_model · ≤0.25, widehatγ_min · ≥0.004, widehatε_comm · ≤0.16, widehatε_resp · ≤0.08. The witness therefore evaluates both downstream repair quality and the internal atlas certificate. A high blind gain with unstable response geometry is not sufficient for co…
Exact mathematical passage
\widehat g_{\mathrm{val}}&\geq0.10,\\ \LCB(\widehat g_{\mathrm{val}})&\geq0.05,\\ \widehat r_{\mathrm{ctx}}&\leq3\times10^{-4},\\ \widehat a_{\mathrm{target}}&\geq0.72,\\ \widehat a_{\mathrm{repair\ inv}}&\geq0.96,\\ \widehat b_{\mathrm{chart}}&\geq0.20,\\ \widehat s_{\mathrm{model}}&\leq0.25,\\ \widehat\gamma_{\min}&\geq0.004,\\ \widehat\eps_{\mathrm{comm}}&\leq0.16,\\ \widehat\eps_{\mathrm{resp}}&\leq0.08.
\widehat BUse in source · 2 passages
Search aliases: widehat B widehat
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12280
In an executable system, the responses are estimated from finite intervention data. Let widehat R_j=R_j+E_j, and fix coefficients c_j. Define B=∑_jc_jR_j, widehat B=∑_jc_jwidehat R_j, E=widehat B-B. Let the distinct eigenvalue associated with block a be μ_a and define its separation γ_a=min_b≠ a|μ_a-μ_b|.
Exact mathematical passage
B=\sum_jc_jR_j, \qquad \widehat B=\sum_jc_j\widehat R_j, \qquad E=\widehat B-B.
\widehat bUse in source · 1 passage
Search aliases: widehat b widehat
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12552
The selected atlas candidate commits only when all frozen conditions hold: widehat g_val · ≥0.10, LCB(widehat g_val) · ≥0.05, widehat r_ctx · ≤3×10^-4, widehat a_target · ≥0.72, widehat a_repair inv · ≥0.96, widehat b_chart · ≥0.20, widehat s_model · ≤0.25, widehatγ_min · ≥0.004, widehatε_comm · ≤0.16, widehatε_resp · ≤0.08. The witness therefore evaluates both downstream repair quality and the internal atlas certificate. A high blind gain with unstable response geometry is not sufficient for co…
Exact mathematical passage
\widehat g_{\mathrm{val}}&\geq0.10,\\ \LCB(\widehat g_{\mathrm{val}})&\geq0.05,\\ \widehat r_{\mathrm{ctx}}&\leq3\times10^{-4},\\ \widehat a_{\mathrm{target}}&\geq0.72,\\ \widehat a_{\mathrm{repair\ inv}}&\geq0.96,\\ \widehat b_{\mathrm{chart}}&\geq0.20,\\ \widehat s_{\mathrm{model}}&\leq0.25,\\ \widehat\gamma_{\min}&\geq0.004,\\ \widehat\eps_{\mathrm{comm}}&\leq0.16,\\ \widehat\eps_{\mathrm{resp}}&\leq0.08.
\widehat b_{\mathrm{chart}}Use in source · 1 passage
Search aliases: widehat b_chart widehat mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12552
The selected atlas candidate commits only when all frozen conditions hold: widehat g_val · ≥0.10, LCB(widehat g_val) · ≥0.05, widehat r_ctx · ≤3×10^-4, widehat a_target · ≥0.72, widehat a_repair inv · ≥0.96, widehat b_chart · ≥0.20, widehat s_model · ≤0.25, widehatγ_min · ≥0.004, widehatε_comm · ≤0.16, widehatε_resp · ≤0.08. The witness therefore evaluates both downstream repair quality and the internal atlas certificate. A high blind gain with unstable response geometry is not sufficient for co…
Exact mathematical passage
\widehat g_{\mathrm{val}}&\geq0.10,\\ \LCB(\widehat g_{\mathrm{val}})&\geq0.05,\\ \widehat r_{\mathrm{ctx}}&\leq3\times10^{-4},\\ \widehat a_{\mathrm{target}}&\geq0.72,\\ \widehat a_{\mathrm{repair\ inv}}&\geq0.96,\\ \widehat b_{\mathrm{chart}}&\geq0.20,\\ \widehat s_{\mathrm{model}}&\leq0.25,\\ \widehat\gamma_{\min}&\geq0.004,\\ \widehat\eps_{\mathrm{comm}}&\leq0.16,\\ \widehat\eps_{\mathrm{resp}}&\leq0.08.
\widehat CUse in source · 1 passage
Search aliases: widehat C widehat
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8728
Define the finite structural coverage functional C_rm id( P) = Pr_A∼D_rm for [ ank X_A=d_s]. Then widehat C_rm id( P_1) - widehat C_rm id( P_0) = 0.96875. This is the measured bridge-identification direction added by the meta-repair.
Exact mathematical passage
\widehat C_{\rm id}(\mathfrak P_1) - \widehat C_{\rm id}(\mathfrak P_0) = 0.96875.
\widehat cUse in source · 8 passages
Search aliases: widehat c widehat
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1050
[Cost-sensitive regret] Suppose, in addition, that normalized intervention cost c(δ) is predicted by widehat c(δ) with uniform error at most ε_c. For net utility W(δ)=V(δ)-λ c(δ), widehat W(δ)=widehat V(δ)-λwidehat c(δ), selection by widehat W has regret at most 2(ε+λε_c).
Exact mathematical passage
\widehat c(\delta)
\widehat C_{\rm id}Use in source · 1 passage
Search aliases: widehat C_rm id widehat rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8728
Define the finite structural coverage functional C_rm id( P) = Pr_A∼D_rm for [ ank X_A=d_s]. Then widehat C_rm id( P_1) - widehat C_rm id( P_0) = 0.96875. This is the measured bridge-identification direction added by the meta-repair.
Exact mathematical passage
\widehat C_{\rm id}(\mathfrak P_1) - \widehat C_{\rm id}(\mathfrak P_0) = 0.96875.
\widehat DUse in source · 9 passages
Search aliases: widehat D widehat
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1751
[Transferred residual under model error] Let widehat D=D+Δ D and widehat r=r+Δ r. For any proposed repair u, absnormDu+r-normwidehat Du+widehat r ≤ normΔ Dnorm u+normΔ r. In particular, applying an exact repair for (D,r) to the perturbed system yields residual at most normΔ Dnorm u+normΔ r.
Exact mathematical passage
\widehat D=D+\Delta D
\widehat D^*Use in source · 2 passages
Search aliases: widehat D^* widehat
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1778
[Robust empirical noncorrectability certificate] Suppose, on an event of probability at least 1-α, normD-widehat D≤ε_D, normr-widehat r≤ε_r. Fix a repair budget norm u≤ B and a unit vector y∈ V. If anglesy,widehat r > ε_r+B(normwidehat D^*y+ε_D), then, on the same event, no budget-feasible repair satisfies Du=-r.
Exact mathematical passage
\angles{y,\widehat r} > \eps_r+B\bigl(\norm{\widehat D^*y}+\eps_D\bigr), \label{eq:robust-dual-gap}
\widehat D_tUse in source · 4 passages
Search aliases: widehat D_t widehat
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10688
[Exact repair-operator perturbation identity] Let widehat D_t=widehat TD_swidehat T^†. Then widehat D_t-D_t = (widehat T-T)D_swidehat T^† + T D_s(widehat T^†-T^†). Hence normwidehat D_t-D_t ≤ normwidehat T-TnormD_snormwidehat T^† + normTnormD_snormwidehat T^†-T^†.
Exact mathematical passage
\widehat D_t=\widehat TD_s\widehat T^\dagger.
\widehat GUse in source · 1 passage
Search aliases: widehat G widehat
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12990
Let G be the ideal generic response combination from the proof of [source reference], and let widehat G=G+E be its empirical estimate. Let P be a spectral projector associated with an eigenvalue cluster separated from the rest of the spectrum by gap γ>0.
Exact mathematical passage
\widehat G=G+E
\widehat g_iUse in source · 5 passages
Search aliases: widehat g_i widehat
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13879
For candidate c_i, let z_i collect its pre-result receipt fields and cheap pilot evidence. A research-outcome forecaster returns F_θ(z_i) = (widehat g_i,widehat r_i,widehat s_i), where widehat g_i predicts hidden protected gain, widehat r_i predicts protected risk, and widehat s_i predicts scored-evaluator performance.
Exact mathematical passage
F_\theta(z_i) = (\widehat g_i,\widehat r_i,\widehat s_i), \label{eq:b13-forecaster}
\widehat g_{i^\star}Use in source · 2 passages
Search aliases: widehat g_i^⋆ widehat star
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13916
By optimality of widehat i under the predicted values, widehat g_widehat i≥widehat g_i^⋆. Therefore g_i^⋆ ≤ widehat g_i^⋆+ε ≤ widehat g_widehat i+ε ≤ g_widehat i+2ε.
Exact mathematical passage
\widehat g_{\widehat i}\geq\widehat g_{i^\star}.
\widehat g_{\rm cal}Use in source · 1 passage
Search aliases: widehat g_rm cal widehat rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10923
For one candidate row r, a configuration commits only if widehat g_rm cal(r) · ≥ τ_g, widehat R_rm cal(r) · ≤ τ_R, widehatε_rm align(r) · ≤ τ_A, λ_min(M_r) · ≥ τ_I, rank X_r · =d. The selected Block 9 grandchild has (τ_g,τ_R,τ_A,τ_I) = (0.02,0.05,0.20,2.0). The recalibrated rank child has (0.02,0.05,0.55,0.10).
Exact mathematical passage
\begin{aligned} \widehat g_{\rm cal}(r)&\geq \tau_g,\\ \widehat R_{\rm cal}(r)&\leq \tau_R,\\ \widehat\eps_{\rm align}(r)&\leq \tau_A,\\ \lambda_{\min}(M_r)&\geq \tau_I,\\ \rank X_r&=d. \end{aligned}
\widehat g_{\rm slice}Use in source · 4 passages
Search aliases: widehat g_rm slice widehat rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6980
The witness commits a transported repair only when all four calibration conditions hold: widehat g_rm slice · ≥ 0.20, widehat r_rm non · ≤ 0.03, widehatε_rm align · ≤ 0.35, widehat A_rm non^rm base · ≥ 0.90. Here widehat g_rm slice is calibration gain on the affected stratum, widehat r_rm non is regression on the unaffected stratum, widehatε_rm align is normalized alignment error, and widehat A_rm non^rm base is baseline unaffected accuracy.
Exact mathematical passage
\widehat g_{\rm slice}&\geq 0.20,\\ \widehat r_{\rm non}&\leq 0.03,\\ \widehat\eps_{\rm align}&\leq 0.35,\\ \widehat A_{\rm non}^{\rm base}&\geq 0.90. \label{eq:block06-gate}
\widehat g_{\mathrm{val}}Use in source · 1 passage
Search aliases: widehat g_val widehat mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12552
The selected atlas candidate commits only when all frozen conditions hold: widehat g_val · ≥0.10, LCB(widehat g_val) · ≥0.05, widehat r_ctx · ≤3×10^-4, widehat a_target · ≥0.72, widehat a_repair inv · ≥0.96, widehat b_chart · ≥0.20, widehat s_model · ≤0.25, widehatγ_min · ≥0.004, widehatε_comm · ≤0.16, widehatε_resp · ≤0.08. The witness therefore evaluates both downstream repair quality and the internal atlas certificate. A high blind gain with unstable response geometry is not sufficient for co…
Exact mathematical passage
\widehat g_{\mathrm{val}}&\geq0.10,\\ \LCB(\widehat g_{\mathrm{val}})&\geq0.05,\\ \widehat r_{\mathrm{ctx}}&\leq3\times10^{-4},\\ \widehat a_{\mathrm{target}}&\geq0.72,\\ \widehat a_{\mathrm{repair\ inv}}&\geq0.96,\\ \widehat b_{\mathrm{chart}}&\geq0.20,\\ \widehat s_{\mathrm{model}}&\leq0.25,\\ \widehat\gamma_{\min}&\geq0.004,\\ \widehat\eps_{\mathrm{comm}}&\leq0.16,\\ \widehat\eps_{\mathrm{resp}}&\leq0.08.
\widehat g_{\widehat i}Use in source · 2 passages
Search aliases: widehat g_widehat i widehat widehat
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13916
By optimality of widehat i under the predicted values, widehat g_widehat i≥widehat g_i^⋆. Therefore g_i^⋆ ≤ widehat g_i^⋆+ε ≤ widehat g_widehat i+ε ≤ g_widehat i+2ε.
Exact mathematical passage
\widehat g_{\widehat i}\geq\widehat g_{i^\star}.
\widehat hUse in source · 2 passages
Search aliases: widehat h widehat
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17814
A latent-step-shared map acts independently on each recurrent state: widehat h_a,c^(k) = A_c h_b,c^(k)+b_c, k=1,…,5. A full-trajectory map instead acts on the concatenation vec(h_b,c^(1),…,h_b,c^(5)) ∈ R^200, allowing cross-step mixing: vec(widehat h_a,c) = M_cvec(h_b,c)+q_c. This difference is substantive when two recurrent architectures distribute the same computation across latent time differently.
Exact mathematical passage
\widehat h_{a,c}^{(k)} = A_c h_{b,c}^{(k)}+b_c, \qquad k=1,\ldots,5. \label{eq:b20-step-shared}
\widehat h_{a,c}Use in source · 1 passage
Search aliases: widehat h_a,c widehat
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17827
tent-step-shared map acts independently on each recurrent state: widehat h_a,c^(k) = A_c h_b,c^(k)+b_c, k=1,…,5. A full-trajectory map instead acts on the concatenation vec(h_b,c^(1),…,h_b,c^(5)) ∈ R^200, allowing cross-step mixing: vec(widehat h_a,c) = M_cvec(h_b,c)+q_c. This difference is substantive when two recurrent architectures distribute the same computation across latent time differently.
Exact mathematical passage
\operatorname{vec}(\widehat h_{a,c}) = M_c\operatorname{vec}(h_{b,c})+q_c. \label{eq:b20-full-map}
\widehat h_{a,c}^{(k)}Use in source · 1 passage
Search aliases: widehat h_a,c^(k) widehat
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17814
A latent-step-shared map acts independently on each recurrent state: widehat h_a,c^(k) = A_c h_b,c^(k)+b_c, k=1,…,5. A full-trajectory map instead acts on the concatenation vec(h_b,c^(1),…,h_b,c^(5)) ∈ R^200, allowing cross-step mixing: vec(widehat h_a,c) = M_cvec(h_b,c)+q_c. This difference is substantive when two recurrent architectures distribute the same computation across latent time differently.
Exact mathematical passage
\widehat h_{a,c}^{(k)} = A_c h_{b,c}^{(k)}+b_c, \qquad k=1,\ldots,5. \label{eq:b20-step-shared}
\widehat iUse in source · 5 passages
Search aliases: widehat i widehat
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13905
[Uniform forecast-regret bound] Let g_i be the true protected gain of each admissible candidate and suppose |widehat g_i-g_i|≤ε for all candidates. If widehat i∈arg max_iwidehat g_i and i^⋆∈arg max_i g_i, then g_i^⋆-g_widehat i ≤ 2ε.
Exact mathematical passage
\widehat i\in\argmax_i\widehat g_i
\widehat JUse in source · 4 passages
Search aliases: widehat J widehat
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6131
The global-edit arm contains a candidate whose apparent score receives an exploitable bonus when its rewrite norm is large. The protected evaluator does not receive this bonus. This creates a controlled instance of evaluator capture: widehat J(δ)>widehat J(0) while J(δ)<J(0). A witness-aware arm ranks and commits by J, not widehat J.
Exact mathematical passage
\widehat J(\delta)>\widehat J(0) \qquad\text{while}\qquad J(\delta)<J(0).
\widehat MUse in source · 2 passages
Search aliases: widehat M widehat
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8660
On 144 primary foreign targets, the target-level paired difference is widehatΔ_rm meta = widehat M_24^rm for( P_1) - widehat M_24^rm for( P_0) = 0.209508. A target-cluster bootstrap gives the interval [0.194968,0.223898], and a 100,000-draw sign-flip test gives p=9.9999×10^-6. All 144 target-level paired differences are positive.
Exact mathematical passage
\widehat\Delta_{\rm meta} = \widehat M_{24}^{\rm for}(\mathfrak P_1) - \widehat M_{24}^{\rm for}(\mathfrak P_0) = 0.209508. \label{eq:block07q-primary-delta}
\widehat{\mathcal M}Use in source · 3 passages
Search aliases: widehat M widehat mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5448
Let M=(S, T,O) be an abstract transition system with state set S, authorized labeled transitions T, and protected observation map O. Let widehat M=(widehat S,widehat T,widehat O) be an executable implementation.
Exact mathematical passage
\widehat{\mathcal M}=(\widehat S,\widehat{\mathcal T},\widehat O)
\widehat M_{24}^{\rm for}Use in source · 1 passage
Search aliases: widehat M_24^rm for widehat rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8660
On 144 primary foreign targets, the target-level paired difference is widehatΔ_rm meta = widehat M_24^rm for( P_1) - widehat M_24^rm for( P_0) = 0.209508. A target-cluster bootstrap gives the interval [0.194968,0.223898], and a 100,000-draw sign-flip test gives p=9.9999×10^-6. All 144 target-level paired differences are positive.
Exact mathematical passage
\widehat\Delta_{\rm meta} = \widehat M_{24}^{\rm for}(\mathfrak P_1) - \widehat M_{24}^{\rm for}(\mathfrak P_0) = 0.209508. \label{eq:block07q-primary-delta}
\widehat M_{\mathrm{foreign}}Use in source · 1 passage
Search aliases: widehat M_foreign widehat mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13011
Let mathscr S be a finite typed candidate-intervention grammar. For candidate S∈mathscr S, define a development score Score_resp(S) = widehat M_foreign(S) -λ_cC(S) -λ_commε_comm(S) -λ_fitε_fit(S) -λ_riskwidehat R(S). The compiler selects a candidate only after verifying that it adds a nontrivial response-signature distinction and preserves the inherited response language as an executable subgrammar.
Exact mathematical passage
\operatorname{Score}_{\mathrm{resp}}(S) = \widehat M_{\mathrm{foreign}}(S) -\lambda_{c}C(S) -\lambda_{\mathrm{comm}}\eps_{\mathrm{comm}}(S) -\lambda_{\mathrm{fit}}\eps_{\mathrm{fit}}(S) -\lambda_{\mathrm{risk}}\widehat R(S). \label{eq:b12-response-score}
\widehat O_0Use in source · 1 passage
Search aliases: widehat O_0 widehat
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15296
The development set contains 56 open-set problems. The inherited base-method oracle obtains mean gain widehat O_0=0.377257. No one inherited unknown-token binding supplies a common high-gain method across the cell.
Exact mathematical passage
\widehat O_0=0.377257. \label{eq:b15-base-oracle-dev}
\widehat O_1Use in source · 1 passage
Search aliases: widehat O_1 widehat
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15323
novel-high · ⟼ objective-then-constructor. The expanded development gain is widehat O_1=0.948962, with newly correctable margin widehatκ = 0.571704.
Exact mathematical passage
\widehat O_1=0.948962,
\widehat PUse in source · 11 passages
Search aliases: widehat P widehat
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12294
[Spectral stability of a recovered atlas block] Suppose B is self-adjoint, normE_2<γ_a/2, and widehat P_a is the spectral projector of widehat B onto the eigenvalue cluster corresponding to μ_a. Then normwidehat P_a-P_a_2 ≤ frac2normE_2γ_a.
Exact mathematical passage
\widehat P_a
\widehat P_1Use in source · 1 passage
Search aliases: widehat P_1 widehat
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12309
For recovered projectors widehat P_1,…,widehat P_m, define the normalized response-fit residual ε_resp = frac ( ∑_jnormwidehat R_j- ∑_awidehatλ_ajwidehat P_a_F^2 )^1/2 ( ∑_jnormwidehat R_j_F^2 )^1/2+10^-12 , and the maximum normalized commutator residual ε_comm = max_a,j fracnormwidehat P_awidehat R_j- widehat R_jwidehat P_a_F normwidehat R_j_F+10^-12. The smallest selected spectral gap is denoted widehatγ_min. These quantities serve different roles: item ε_resp measures w
Exact mathematical passage
\widehat P_1,\ldots,\widehat P_m
\widehat P_aUse in source · 7 passages
Search aliases: widehat P_a widehat
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12294
[Spectral stability of a recovered atlas block] Suppose B is self-adjoint, normE_2<γ_a/2, and widehat P_a is the spectral projector of widehat B onto the eigenvalue cluster corresponding to μ_a. Then normwidehat P_a-P_a_2 ≤ frac2normE_2γ_a.
Exact mathematical passage
\widehat P_a
\widehat P_mUse in source · 1 passage
Search aliases: widehat P_m widehat
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12309
For recovered projectors widehat P_1,…,widehat P_m, define the normalized response-fit residual ε_resp = frac ( ∑_jnormwidehat R_j- ∑_awidehatλ_ajwidehat P_a_F^2 )^1/2 ( ∑_jnormwidehat R_j_F^2 )^1/2+10^-12 , and the maximum normalized commutator residual ε_comm = max_a,j fracnormwidehat P_awidehat R_j- widehat R_jwidehat P_a_F normwidehat R_j_F+10^-12. The smallest selected spectral gap is denoted widehatγ_min. These quantities serve different roles: item ε_resp measures w
Exact mathematical passage
\widehat P_1,\ldots,\widehat P_m
\widehat qUse in source · 1 passage
Search aliases: widehat q widehat
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17317
and a query q. After chart canonicalization, the full descendant computes pair codes p_i=ϕ(widehat x_i,widehat y_i)∈ℝ^32, and the permutation-invariant aggregate a(D) = ( frac15∑_i=1^5p_i, max_1≤ i≤5p_i )∈ℝ^64. The semantic seed is h_0=S(a(D),Q(widehat q))∈ℝ^40. A five-step Elman recurrence generates the Neuralese trace r_k=tanh(W_ih_0+b_i+U_ir_k-1+c_i+e_k), k=1,…,5, where e_k is a learned step embedding. A gated readout recurrence maps the trace to protected program and answer coordinates. A ha…
Exact mathematical passage
h_0=S(a(D),Q(\widehat q))\in\R^{40}.
\widehat RUse in source · 13 passages
Search aliases: widehat R widehat
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7407
widehatβ_r · = (Z^top Z+λ_r I)^-1Z^top r. For candidate type c, let s_u,c and s_r,c be the development residual standard deviations for the utility and risk heads. The deployed scores are widehat U_rm LCB(z) · = widehat U(z)-κ_u s_u,c(z), widehat R_rm UCB(z) · = max{0,widehat R(z)}+κ_r s_r,c(z). A candidate is admissible when widehat R_rm UCB(z)≤τ_r, widehat r_rm cal(z)≤τ_rm cal, A^rm base_rm cal,prot(z)≥ a_min. The child chooses the admissible candidate with maximum widehat U_rm LCB. If the adm…
Exact mathematical passage
\widehat U_{\rm LCB}(z) &= \widehat U(z)-\kappa_u s_{u,c(z)}, \label{eq:block07-utility-lcb}\\ \widehat R_{\rm UCB}(z) &= \max\{0,\widehat R(z)\}+\kappa_r s_{r,c(z)}. \label{eq:block07-risk-ucb}
\widehat rUse in source · 14 passages
Search aliases: widehat r widehat
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1751
[Transferred residual under model error] Let widehat D=D+Δ D and widehat r=r+Δ r. For any proposed repair u, absnormDu+r-normwidehat Du+widehat r ≤ normΔ Dnorm u+normΔ r. In particular, applying an exact repair for (D,r) to the perturbed system yields residual at most normΔ Dnorm u+normΔ r.
Exact mathematical passage
\widehat r=r+\Delta r
\widehat r_{\mathrm{ctx}}Use in source · 1 passage
Search aliases: widehat r_ctx widehat mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12552
The selected atlas candidate commits only when all frozen conditions hold: widehat g_val · ≥0.10, LCB(widehat g_val) · ≥0.05, widehat r_ctx · ≤3×10^-4, widehat a_target · ≥0.72, widehat a_repair inv · ≥0.96, widehat b_chart · ≥0.20, widehat s_model · ≤0.25, widehatγ_min · ≥0.004, widehatε_comm · ≤0.16, widehatε_resp · ≤0.08. The witness therefore evaluates both downstream repair quality and the internal atlas certificate. A high blind gain with unstable response geometry is not sufficient for co…
Exact mathematical passage
\widehat g_{\mathrm{val}}&\geq0.10,\\ \LCB(\widehat g_{\mathrm{val}})&\geq0.05,\\ \widehat r_{\mathrm{ctx}}&\leq3\times10^{-4},\\ \widehat a_{\mathrm{target}}&\geq0.72,\\ \widehat a_{\mathrm{repair\ inv}}&\geq0.96,\\ \widehat b_{\mathrm{chart}}&\geq0.20,\\ \widehat s_{\mathrm{model}}&\leq0.25,\\ \widehat\gamma_{\min}&\geq0.004,\\ \widehat\eps_{\mathrm{comm}}&\leq0.16,\\ \widehat\eps_{\mathrm{resp}}&\leq0.08.
\widehat r_iUse in source · 3 passages
Search aliases: widehat r_i widehat
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13879
For candidate c_i, let z_i collect its pre-result receipt fields and cheap pilot evidence. A research-outcome forecaster returns F_θ(z_i) = (widehat g_i,widehat r_i,widehat s_i), where widehat g_i predicts hidden protected gain, widehat r_i predicts protected risk, and widehat s_i predicts scored-evaluator performance.
Exact mathematical passage
F_\theta(z_i) = (\widehat g_i,\widehat r_i,\widehat s_i), \label{eq:b13-forecaster}
\widehat R_jUse in source · 4 passages
Search aliases: widehat R_j widehat
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12276
In an executable system, the responses are estimated from finite intervention data. Let widehat R_j=R_j+E_j, and fix coefficients c_j. Define B=∑_jc_jR_j, widehat B=∑_jc_jwidehat R_j, E=widehat B-B. Let the distinct eigenvalue associated with block a be μ_a and define its separation γ_a=min_b≠ a|μ_a-μ_b|.
Exact mathematical passage
\widehat R_j=R_j+E_j,
\widehat R_{\rm cal}Use in source · 1 passage
Search aliases: widehat R_rm cal widehat rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10923
For one candidate row r, a configuration commits only if widehat g_rm cal(r) · ≥ τ_g, widehat R_rm cal(r) · ≤ τ_R, widehatε_rm align(r) · ≤ τ_A, λ_min(M_r) · ≥ τ_I, rank X_r · =d. The selected Block 9 grandchild has (τ_g,τ_R,τ_A,τ_I) = (0.02,0.05,0.20,2.0). The recalibrated rank child has (0.02,0.05,0.55,0.10).
Exact mathematical passage
\begin{aligned} \widehat g_{\rm cal}(r)&\geq \tau_g,\\ \widehat R_{\rm cal}(r)&\leq \tau_R,\\ \widehat\eps_{\rm align}(r)&\leq \tau_A,\\ \lambda_{\min}(M_r)&\geq \tau_I,\\ \rank X_r&=d. \end{aligned}
\widehat r_{\rm cal}Use in source · 1 passage
Search aliases: widehat r_rm cal widehat rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7418
widehat R_rm UCB(z) · = max{0,widehat R(z)}+κ_r s_r,c(z). A candidate is admissible when widehat R_rm UCB(z)≤τ_r, widehat r_rm cal(z)≤τ_rm cal, A^rm base_rm cal,prot(z)≥ a_min. The child chooses the admissible candidate with maximum widehat U_rm LCB. If the admissible set is empty, it returns the no-op.
Exact mathematical passage
\widehat R_{\rm UCB}(z)\leq\tau_r, \qquad \widehat r_{\rm cal}(z)\leq\tau_{\rm cal}, \qquad A^{\rm base}_{\rm cal,prot}(z)\geq a_{\min}. \label{eq:block07-admissible}
\widehat r_{\rm non}Use in source · 4 passages
Search aliases: widehat r_rm non widehat rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6980
The witness commits a transported repair only when all four calibration conditions hold: widehat g_rm slice · ≥ 0.20, widehat r_rm non · ≤ 0.03, widehatε_rm align · ≤ 0.35, widehat A_rm non^rm base · ≥ 0.90. Here widehat g_rm slice is calibration gain on the affected stratum, widehat r_rm non is regression on the unaffected stratum, widehatε_rm align is normalized alignment error, and widehat A_rm non^rm base is baseline unaffected accuracy.
Exact mathematical passage
\widehat g_{\rm slice}&\geq 0.20,\\ \widehat r_{\rm non}&\leq 0.03,\\ \widehat\eps_{\rm align}&\leq 0.35,\\ \widehat A_{\rm non}^{\rm base}&\geq 0.90. \label{eq:block06-gate}
\widehat R_{\rm UCB}Use in source · 7 passages
Search aliases: widehat R_rm UCB widehat rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7407
widehatβ_r · = (Z^top Z+λ_r I)^-1Z^top r. For candidate type c, let s_u,c and s_r,c be the development residual standard deviations for the utility and risk heads. The deployed scores are widehat U_rm LCB(z) · = widehat U(z)-κ_u s_u,c(z), widehat R_rm UCB(z) · = max{0,widehat R(z)}+κ_r s_r,c(z). A candidate is admissible when widehat R_rm UCB(z)≤τ_r, widehat r_rm cal(z)≤τ_rm cal, A^rm base_rm cal,prot(z)≥ a_min. The child chooses the admissible candidate with maximum widehat U_rm LCB. If the adm…
Exact mathematical passage
\widehat U_{\rm LCB}(z) &= \widehat U(z)-\kappa_u s_{u,c(z)}, \label{eq:block07-utility-lcb}\\ \widehat R_{\rm UCB}(z) &= \max\{0,\widehat R(z)\}+\kappa_r s_{r,c(z)}. \label{eq:block07-risk-ucb}
\widehat s_{a^*}Use in source · 1 passage
Search aliases: widehat s_a^* widehat
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12382
For every a≠ a^*, widehat s_a^*-widehat s_a · ≥ s_a^*-s_a -|widehat s_a^*-s_a^*| -|widehat s_a-s_a| · >m-frac m2-frac m2=0.
Exact mathematical passage
\widehat s_{a^*}-\widehat s_a &\geq s_{a^*}-s_a -|\widehat s_{a^*}-s_{a^*}| -|\widehat s_a-s_a|\\ &>m-\frac m2-\frac m2=0.
\widehat s_iUse in source · 2 passages
Search aliases: widehat s_i widehat
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13879
For candidate c_i, let z_i collect its pre-result receipt fields and cheap pilot evidence. A research-outcome forecaster returns F_θ(z_i) = (widehat g_i,widehat r_i,widehat s_i), where widehat g_i predicts hidden protected gain, widehat r_i predicts protected risk, and widehat s_i predicts scored-evaluator performance.
Exact mathematical passage
F_\theta(z_i) = (\widehat g_i,\widehat r_i,\widehat s_i), \label{eq:b13-forecaster}
\widehat s_{\mathrm{model}}Use in source · 1 passage
Search aliases: widehat s_model widehat mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12552
The selected atlas candidate commits only when all frozen conditions hold: widehat g_val · ≥0.10, LCB(widehat g_val) · ≥0.05, widehat r_ctx · ≤3×10^-4, widehat a_target · ≥0.72, widehat a_repair inv · ≥0.96, widehat b_chart · ≥0.20, widehat s_model · ≤0.25, widehatγ_min · ≥0.004, widehatε_comm · ≤0.16, widehatε_resp · ≤0.08. The witness therefore evaluates both downstream repair quality and the internal atlas certificate. A high blind gain with unstable response geometry is not sufficient for co…
Exact mathematical passage
\widehat g_{\mathrm{val}}&\geq0.10,\\ \LCB(\widehat g_{\mathrm{val}})&\geq0.05,\\ \widehat r_{\mathrm{ctx}}&\leq3\times10^{-4},\\ \widehat a_{\mathrm{target}}&\geq0.72,\\ \widehat a_{\mathrm{repair\ inv}}&\geq0.96,\\ \widehat b_{\mathrm{chart}}&\geq0.20,\\ \widehat s_{\mathrm{model}}&\leq0.25,\\ \widehat\gamma_{\min}&\geq0.004,\\ \widehat\eps_{\mathrm{comm}}&\leq0.16,\\ \widehat\eps_{\mathrm{resp}}&\leq0.08.
\widehat TUse in source · 27 passages
Search aliases: widehat T widehat
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6757
h_t(x_m)^top . It estimates the bridge by regularized least squares: widehat T_t← s ∈ arg min_T normH_sT^top-H_t_F^2 +λnormT_F^2. The transformation matrices used to construct the controlled targets are not exposed to this fit.
Exact mathematical passage
\widehat T_{t\leftarrow s} \in \argmin_T \norm{H_sT^{\top}-H_t}_F^2 +\lambda\norm{T}_F^2. \label{eq:block06-learned-transport}
\widehat{\mathcal T}Use in source · 1 passage
Search aliases: widehat T widehat mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5448
Let M=(S, T,O) be an abstract transition system with state set S, authorized labeled transitions T, and protected observation map O. Let widehat M=(widehat S,widehat T,widehat O) be an executable implementation.
Exact mathematical passage
\widehat{\mathcal M}=(\widehat S,\widehat{\mathcal T},\widehat O)
\widehat T^{\mathsf T}Use in source · 4 passages
Search aliases: widehat T^ T widehat mathsf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10617
[Weighted bridge covariance] Assume M_S is invertible and the noise rows are independent with covariance σ_i^2I_q. The weighted least-squares estimator widehat T^ T = M_S^-1X^ TWY is unbiased and satisfies Cov(vec(widehat T^ T-T^ T)mid X) = I_q⊗ M_S^-1. Consequently, E[normwidehat T-T_F^2mid X] = q tr(M_S^-1).
Exact mathematical passage
\widehat T^{\mathsf T} = M_S^{-1}X^{\mathsf T}WY
\widehat T^{+}Use in source · 1 passage
Search aliases: widehat T^+ widehat
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6772
Given source repair R_s(h)=h+D_sh+b_s, its transported target form is D_t = widehat T D_swidehat T^+, b_t = widehat T b_s, with transported packet basis obtained by orthogonalizing widehat T U_s.
Exact mathematical passage
D_t = \widehat T D_s\widehat T^{+}, \qquad b_t = \widehat T b_s, \label{eq:block06-transported-repair}
\widehat T^\topUse in source · 2 passages
Search aliases: widehat T^top widehat top
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8417
h_t(x_k)^top , and fits widehat T^top = (X^top X+λ I)^-1X^top Y. The parent chooses the k=16 fit anchors uniformly at random. The source hidden space is also 16-dimensional, so a rank-deficient design leaves at least one source direction unidentified.
Exact mathematical passage
\widehat T^\top = (X^\top X+\lambda I)^{-1}X^\top Y. \label{eq:block07q-ridge-bridge}
\widehat T^{\top}Use in source · 1 passage
Search aliases: widehat T^top widehat top
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6874
The normal equations are H_s^topH_swidehat T^top=H_s^topH_t. Full column rank makes H_s^topH_s invertible. Substitution of H_t=H_sT^top yields widehat T=T.
Exact mathematical passage
H_s^{\top}H_s\widehat T^{\top}=H_s^{\top}H_t.
\widehat T^\daggerUse in source · 4 passages
Search aliases: widehat T^† widehat dagger
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10688
[Exact repair-operator perturbation identity] Let widehat D_t=widehat TD_swidehat T^†. Then widehat D_t-D_t = (widehat T-T)D_swidehat T^† + T D_s(widehat T^†-T^†). Hence normwidehat D_t-D_t ≤ normwidehat T-TnormD_snormwidehat T^† + normTnormD_snormwidehat T^†-T^†.
Exact mathematical passage
\widehat D_t=\widehat TD_s\widehat T^\dagger.
\widehat T_{t\leftarrow s}Use in source · 2 passages
Search aliases: widehat T_t← s widehat leftarrow
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6757
h_t(x_m)^top . It estimates the bridge by regularized least squares: widehat T_t← s ∈ arg min_T normH_sT^top-H_t_F^2 +λnormT_F^2. The transformation matrices used to construct the controlled targets are not exposed to this fit.
Exact mathematical passage
\widehat T_{t\leftarrow s} \in \argmin_T \norm{H_sT^{\top}-H_t}_F^2 +\lambda\norm{T}_F^2. \label{eq:block06-learned-transport}
\widehat UUse in source · 10 passages
Search aliases: widehat U widehat
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7407
widehatβ_r · = (Z^top Z+λ_r I)^-1Z^top r. For candidate type c, let s_u,c and s_r,c be the development residual standard deviations for the utility and risk heads. The deployed scores are widehat U_rm LCB(z) · = widehat U(z)-κ_u s_u,c(z), widehat R_rm UCB(z) · = max{0,widehat R(z)}+κ_r s_r,c(z). A candidate is admissible when widehat R_rm UCB(z)≤τ_r, widehat r_rm cal(z)≤τ_rm cal, A^rm base_rm cal,prot(z)≥ a_min. The child chooses the admissible candidate with maximum widehat U_rm LCB. If the adm…
Exact mathematical passage
\widehat U_{\rm LCB}(z) &= \widehat U(z)-\kappa_u s_{u,c(z)}, \label{eq:block07-utility-lcb}\\ \widehat R_{\rm UCB}(z) &= \max\{0,\widehat R(z)\}+\kappa_r s_{r,c(z)}. \label{eq:block07-risk-ucb}
\widehat U_nUse in source · 1 passage
Search aliases: widehat U_n widehat
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7514
[Finite meta-grammar uniform bound] Suppose, for each γ∈Γ, the per-episode utility lies in an interval of width L, and the n meta-validation episodes are independent. Then with probability at least 1-δ, sup_γ∈Γ | widehat U_n(γ)-U(γ) | ≤ Lsqrtfraclog(2|Γ|/δ)2n. Consequently, the true utility of the empirical maximizer is within twice the right-hand side of the best configuration in Γ.
Exact mathematical passage
\sup_{\gamma\in\Gamma} \left| \widehat U_n(\gamma)-U(\gamma) \right| \leq L\sqrt{\frac{\log(2|\Gamma|/\delta)}{2n}}. \label{eq:block07-uniform-bound}
\widehat U_{\rm LCB}Use in source · 7 passages
Search aliases: widehat U_rm LCB widehat rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7407
widehatβ_r · = (Z^top Z+λ_r I)^-1Z^top r. For candidate type c, let s_u,c and s_r,c be the development residual standard deviations for the utility and risk heads. The deployed scores are widehat U_rm LCB(z) · = widehat U(z)-κ_u s_u,c(z), widehat R_rm UCB(z) · = max{0,widehat R(z)}+κ_r s_r,c(z). A candidate is admissible when widehat R_rm UCB(z)≤τ_r, widehat r_rm cal(z)≤τ_rm cal, A^rm base_rm cal,prot(z)≥ a_min. The child chooses the admissible candidate with maximum widehat U_rm LCB. If the adm…
Exact mathematical passage
\widehat U_{\rm LCB}(z) &= \widehat U(z)-\kappa_u s_{u,c(z)}, \label{eq:block07-utility-lcb}\\ \widehat R_{\rm UCB}(z) &= \max\{0,\widehat R(z)\}+\kappa_r s_{r,c(z)}. \label{eq:block07-risk-ucb}
\widehat VUse in source · 11 passages
Search aliases: widehat V widehat
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 972
[Decision regret from relative introspection] Let u:Y→[0,1] be a bounded utility. Define V(δ)= E[u(Y)mid A_t,δ], widehat V(δ)= E[u(Y)mid ϕ_t(A_t),δ]. If ϕ_t is ε-intervention-sufficient, then sup_δ∈D_t absV(δ)-widehat V(δ)≤ε. If δ^*∈arg max_δ∈D_tV(δ), widehatδ∈arg max_δ∈D_twidehat V(δ), then V(δ^*)-V(widehatδ)≤2ε.
Exact mathematical passage
V(\delta)=\mathbb E[u(Y)\mid A_t,\delta], \qquad \widehat V(\delta)=\mathbb E[u(Y)\mid \phi_t(A_t),\delta].
\widehat WUse in source · 3 passages
Search aliases: widehat W widehat
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1051
[Cost-sensitive regret] Suppose, in addition, that normalized intervention cost c(δ) is predicted by widehat c(δ) with uniform error at most ε_c. For net utility W(δ)=V(δ)-λ c(δ), widehat W(δ)=widehat V(δ)-λwidehat c(δ), selection by widehat W has regret at most 2(ε+λε_c).
Exact mathematical passage
W(\delta)=V(\delta)-\lambda c(\delta), \qquad \widehat W(\delta)=\widehat V(\delta)-\lambda\widehat c(\delta),
\widehat w_kUse in source · 1 passage
Search aliases: widehat w_k widehat
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16124
The latent trace updates the task representation through g_k · =σ(Gz_k), widetilde h_k · =tanh(Th_k-1+Lz_k), h_k · =(1-g_k)odot h_k-1+g_kodotwidetilde h_k, h_0=e, widehat w_k · =Oh_k. A separate halt head predicts one of four dynamic depths from the mean trace and, for nonparent candidates, the encoded prompt. A semantic training head predicts w(τ) from z_4. Its auxiliary loss is an externally declared witness term, not a claim that the latent coordinates are uniquely interpretable.
Exact mathematical passage
g_k&=\sigma(Gz_k), \label{eq:b17-update-gate}\\ \widetilde h_k&=\tanh(Th_{k-1}+Lz_k), \label{eq:b17-hidden-candidate}\\ h_k&=(1-g_k)\odot h_{k-1}+g_k\odot\widetilde h_k, \qquad h_0=e, \label{eq:b17-hidden-update}\\ \widehat w_k&=Oh_k. \label{eq:b17-readout}
\widehat x_iUse in source · 1 passage
Search aliases: widehat x_i widehat
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17303
Each task contains five demonstration pairs D={(x_i,y_i)}_i=1^5 and a query q. After chart canonicalization, the full descendant computes pair codes p_i=ϕ(widehat x_i,widehat y_i)∈ℝ^32, and the permutation-invariant aggregate a(D) = ( frac15∑_i=1^5p_i, max_1≤ i≤5p_i )∈ℝ^64. The semantic seed is h_0=S(a(D),Q(widehat q))∈ℝ^40. A five-step Elman recurrence generates the Neuralese trace r_k=tanh(W_ih_0+b_i+U_ir_k-1+c_i+e_k), k=1,…,5, where e_k is a learned step embedding. A gated readout recurrence …
Exact mathematical passage
p_i=\phi(\widehat x_i,\widehat y_i)\in\R^{32},
\widehat YUse in source · 1 passage
Search aliases: widehat Y widehat
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16828
reign-child state. A ridge-affine bridge is fitted on the 900 paired generator-A anchors: B_t = arg min_B,b ∑_i=1^900 normH_s(τ_i)-BH_t(τ_i)-b^2 +λnormB^2. A source ridge probe P_s is fitted independently on the 1,000 probe-calibration tasks. The transported semantic prediction is widehat Y_t(τ) =P_s(B_tH_t(τ)+b_t). The bridge gate requires effective rank 36, transported R^2≥0.35, an advantage of at least 0.15 over the stronger of the unaligned and shuffled-anchor controls, and shuffled-anchor R…
Exact mathematical passage
\widehat Y_t(\tau) =P_s\bigl(B_tH_t(\tau)+b_t\bigr). \label{eq:b18-transported-readout}
\widehat y_iUse in source · 1 passage
Search aliases: widehat y_i widehat
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17303
Each task contains five demonstration pairs D={(x_i,y_i)}_i=1^5 and a query q. After chart canonicalization, the full descendant computes pair codes p_i=ϕ(widehat x_i,widehat y_i)∈ℝ^32, and the permutation-invariant aggregate a(D) = ( frac15∑_i=1^5p_i, max_1≤ i≤5p_i )∈ℝ^64. The semantic seed is h_0=S(a(D),Q(widehat q))∈ℝ^40. A five-step Elman recurrence generates the Neuralese trace r_k=tanh(W_ih_0+b_i+U_ir_k-1+c_i+e_k), k=1,…,5, where e_k is a learned step embedding. A gated readout recurrence …
Exact mathematical passage
p_i=\phi(\widehat x_i,\widehat y_i)\in\R^{32},
\widehat Y_tUse in source · 1 passage
Search aliases: widehat Y_t widehat
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16828
reign-child state. A ridge-affine bridge is fitted on the 900 paired generator-A anchors: B_t = arg min_B,b ∑_i=1^900 normH_s(τ_i)-BH_t(τ_i)-b^2 +λnormB^2. A source ridge probe P_s is fitted independently on the 1,000 probe-calibration tasks. The transported semantic prediction is widehat Y_t(τ) =P_s(B_tH_t(τ)+b_t). The bridge gate requires effective rank 36, transported R^2≥0.35, an advantage of at least 0.15 over the stronger of the unaligned and shuffled-anchor controls, and shuffled-anchor R…
Exact mathematical passage
\widehat Y_t(\tau) =P_s\bigl(B_tH_t(\tau)+b_t\bigr). \label{eq:b18-transported-readout}
\widehat zUse in source · 7 passages
Search aliases: widehat z widehat
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6003
[Prototype-separation error bound] Let Δ = min_z≠ z'normμ_z-μ_z'. Under equal priors and isotropic Gaussian noise, nearest-prototype classification satisfies Pr(widehat z≠ z) ≤ (| Z|-1) exp(-fracΔ^28σ^2).
Exact mathematical passage
\Pr(\widehat z\neq z) \leq (|\mathcal Z|-1) \exp\left(-\frac{\Delta^2}{8\sigma^2}\right).
\widehat{\operatorname{coverage}}Use in source · 1 passage
Search aliases: widehatcoverage widehat operatorname
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17522
The expanded atlas admits all 900 primary tasks and all 900 novel-chart tasks. It admits none of the 600 nonlinear unsupported tasks: widehatcoverage_A=1, widehatcoverage_N=1, widehatfalse admission_U=0. Forcing the unsupported tasks through the nearest chart produces clean loss 0.703667, semantic-probe R^2=-17.424958, execution accuracy 0.056667, and ATLAS_PARTIAL status. Thus abstention protects against a demonstrably harmful extrapolation. The always-abstain control has zero useful supported …
Exact mathematical passage
\widehat{\operatorname{coverage}}_{A}=1, \qquad \widehat{\operatorname{coverage}}_{N}=1, \qquad \widehat{\operatorname{false\ admission}}_{U}=0. \label{eq:b19-selectivity}
\widehat{\operatorname{coverage}}_{A}Use in source · 1 passage
Search aliases: widehatcoverage_A widehat operatorname
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17522
The expanded atlas admits all 900 primary tasks and all 900 novel-chart tasks. It admits none of the 600 nonlinear unsupported tasks: widehatcoverage_A=1, widehatcoverage_N=1, widehatfalse admission_U=0. Forcing the unsupported tasks through the nearest chart produces clean loss 0.703667, semantic-probe R^2=-17.424958, execution accuracy 0.056667, and ATLAS_PARTIAL status. Thus abstention protects against a demonstrably harmful extrapolation. The always-abstain control has zero useful supported …
Exact mathematical passage
\widehat{\operatorname{coverage}}_{A}=1, \qquad \widehat{\operatorname{coverage}}_{N}=1, \qquad \widehat{\operatorname{false\ admission}}_{U}=0. \label{eq:b19-selectivity}
\widehat{\operatorname{coverage}}_{N}Use in source · 1 passage
Search aliases: widehatcoverage_N widehat operatorname
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17522
The expanded atlas admits all 900 primary tasks and all 900 novel-chart tasks. It admits none of the 600 nonlinear unsupported tasks: widehatcoverage_A=1, widehatcoverage_N=1, widehatfalse admission_U=0. Forcing the unsupported tasks through the nearest chart produces clean loss 0.703667, semantic-probe R^2=-17.424958, execution accuracy 0.056667, and ATLAS_PARTIAL status. Thus abstention protects against a demonstrably harmful extrapolation. The always-abstain control has zero useful supported …
Exact mathematical passage
\widehat{\operatorname{coverage}}_{A}=1, \qquad \widehat{\operatorname{coverage}}_{N}=1, \qquad \widehat{\operatorname{false\ admission}}_{U}=0. \label{eq:b19-selectivity}
\widehat\epsUse in source · 10 passages
Search aliases: widehateps widehatε widehat eps widehat varepsilon
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6980
The witness commits a transported repair only when all four calibration conditions hold: widehat g_rm slice · ≥ 0.20, widehat r_rm non · ≤ 0.03, widehatε_rm align · ≤ 0.35, widehat A_rm non^rm base · ≥ 0.90. Here widehat g_rm slice is calibration gain on the affected stratum, widehat r_rm non is regression on the unaffected stratum, widehatε_rm align is normalized alignment error, and widehat A_rm non^rm base is baseline unaffected accuracy.
Exact mathematical passage
\widehat g_{\rm slice}&\geq 0.20,\\ \widehat r_{\rm non}&\leq 0.03,\\ \widehat\eps_{\rm align}&\leq 0.35,\\ \widehat A_{\rm non}^{\rm base}&\geq 0.90. \label{eq:block06-gate}
\widehat\eps_{\mathrm{align}}Use in source · 1 passage
Search aliases: widehateps_align widehatε_align widehat eps mathrm widehat varepsilon mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20459
widehatε_align · Calibration normalized hidden-state alignment error
Exact mathematical passage
\widehat\eps_{\mathrm{align}}
\widehat\eps_{\mathrm{chart}}Use in source · 2 passages
Search aliases: widehateps_chart widehatε_chart widehat eps mathrm widehat varepsilon mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11843
All 54 development targets have full source-design rank. The inherited global bridge has mean validation alignment NRMSE widehatε_glob=0.632211, whereas the learned two-chart constructor has widehatε_chart=0.068871. The mean reduction is 0.563340.
Exact mathematical passage
\widehat\eps_{\mathrm{chart}}=0.068871.
\widehat\eps_{\mathrm{comm}}Use in source · 1 passage
Search aliases: widehateps_comm widehatε_comm widehat eps mathrm widehat varepsilon mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12552
The selected atlas candidate commits only when all frozen conditions hold: widehat g_val · ≥0.10, LCB(widehat g_val) · ≥0.05, widehat r_ctx · ≤3×10^-4, widehat a_target · ≥0.72, widehat a_repair inv · ≥0.96, widehat b_chart · ≥0.20, widehat s_model · ≤0.25, widehatγ_min · ≥0.004, widehatε_comm · ≤0.16, widehatε_resp · ≤0.08. The witness therefore evaluates both downstream repair quality and the internal atlas certificate. A high blind gain with unstable response geometry is not sufficient for co…
Exact mathematical passage
\widehat g_{\mathrm{val}}&\geq0.10,\\ \LCB(\widehat g_{\mathrm{val}})&\geq0.05,\\ \widehat r_{\mathrm{ctx}}&\leq3\times10^{-4},\\ \widehat a_{\mathrm{target}}&\geq0.72,\\ \widehat a_{\mathrm{repair\ inv}}&\geq0.96,\\ \widehat b_{\mathrm{chart}}&\geq0.20,\\ \widehat s_{\mathrm{model}}&\leq0.25,\\ \widehat\gamma_{\min}&\geq0.004,\\ \widehat\eps_{\mathrm{comm}}&\leq0.16,\\ \widehat\eps_{\mathrm{resp}}&\leq0.08.
\widehat\eps_{\mathrm{glob}}Use in source · 2 passages
Search aliases: widehateps_glob widehatε_glob widehat eps mathrm widehat varepsilon mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11839
All 54 development targets have full source-design rank. The inherited global bridge has mean validation alignment NRMSE widehatε_glob=0.632211, whereas the learned two-chart constructor has widehatε_chart=0.068871. The mean reduction is 0.563340.
Exact mathematical passage
\widehat\eps_{\mathrm{glob}}=0.632211,
\widehat\eps_{\mathrm{resp}}Use in source · 1 passage
Search aliases: widehateps_resp widehatε_resp widehat eps mathrm widehat varepsilon mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12552
The selected atlas candidate commits only when all frozen conditions hold: widehat g_val · ≥0.10, LCB(widehat g_val) · ≥0.05, widehat r_ctx · ≤3×10^-4, widehat a_target · ≥0.72, widehat a_repair inv · ≥0.96, widehat b_chart · ≥0.20, widehat s_model · ≤0.25, widehatγ_min · ≥0.004, widehatε_comm · ≤0.16, widehatε_resp · ≤0.08. The witness therefore evaluates both downstream repair quality and the internal atlas certificate. A high blind gain with unstable response geometry is not sufficient for co…
Exact mathematical passage
\widehat g_{\mathrm{val}}&\geq0.10,\\ \LCB(\widehat g_{\mathrm{val}})&\geq0.05,\\ \widehat r_{\mathrm{ctx}}&\leq3\times10^{-4},\\ \widehat a_{\mathrm{target}}&\geq0.72,\\ \widehat a_{\mathrm{repair\ inv}}&\geq0.96,\\ \widehat b_{\mathrm{chart}}&\geq0.20,\\ \widehat s_{\mathrm{model}}&\leq0.25,\\ \widehat\gamma_{\min}&\geq0.004,\\ \widehat\eps_{\mathrm{comm}}&\leq0.16,\\ \widehat\eps_{\mathrm{resp}}&\leq0.08.
\widehat\eps_{\rm align}Use in source · 4 passages
Search aliases: widehateps_rm align widehatε_rm align widehat eps rm widehat varepsilon rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6980
The witness commits a transported repair only when all four calibration conditions hold: widehat g_rm slice · ≥ 0.20, widehat r_rm non · ≤ 0.03, widehatε_rm align · ≤ 0.35, widehat A_rm non^rm base · ≥ 0.90. Here widehat g_rm slice is calibration gain on the affected stratum, widehat r_rm non is regression on the unaffected stratum, widehatε_rm align is normalized alignment error, and widehat A_rm non^rm base is baseline unaffected accuracy.
Exact mathematical passage
\widehat g_{\rm slice}&\geq 0.20,\\ \widehat r_{\rm non}&\leq 0.03,\\ \widehat\eps_{\rm align}&\leq 0.35,\\ \widehat A_{\rm non}^{\rm base}&\geq 0.90. \label{eq:block06-gate}
\widehat{\operatorname{false\ admission}}Use in source · 1 passage
Search aliases: widehatfalse admission widehat operatorname
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17522
The expanded atlas admits all 900 primary tasks and all 900 novel-chart tasks. It admits none of the 600 nonlinear unsupported tasks: widehatcoverage_A=1, widehatcoverage_N=1, widehatfalse admission_U=0. Forcing the unsupported tasks through the nearest chart produces clean loss 0.703667, semantic-probe R^2=-17.424958, execution accuracy 0.056667, and ATLAS_PARTIAL status. Thus abstention protects against a demonstrably harmful extrapolation. The always-abstain control has zero useful supported …
Exact mathematical passage
\widehat{\operatorname{coverage}}_{A}=1, \qquad \widehat{\operatorname{coverage}}_{N}=1, \qquad \widehat{\operatorname{false\ admission}}_{U}=0. \label{eq:b19-selectivity}
\widehat{\operatorname{false\ admission}}_{U}Use in source · 1 passage
Search aliases: widehatfalse admission_U widehat operatorname
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17522
The expanded atlas admits all 900 primary tasks and all 900 novel-chart tasks. It admits none of the 600 nonlinear unsupported tasks: widehatcoverage_A=1, widehatcoverage_N=1, widehatfalse admission_U=0. Forcing the unsupported tasks through the nearest chart produces clean loss 0.703667, semantic-probe R^2=-17.424958, execution accuracy 0.056667, and ATLAS_PARTIAL status. Thus abstention protects against a demonstrably harmful extrapolation. The always-abstain control has zero useful supported …
Exact mathematical passage
\widehat{\operatorname{coverage}}_{A}=1, \qquad \widehat{\operatorname{coverage}}_{N}=1, \qquad \widehat{\operatorname{false\ admission}}_{U}=0. \label{eq:b19-selectivity}
\widehat{\Lcal}Use in source · 2 passages
Search aliases: widehatLcal widehatL widehat Lcal widehat mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10144
[Spectral-gap stability of the commutant] Assume the first positive eigenvalue of L_com is at least λ>0. Let widehatL_com be an empirical estimate satisfying normwidehatL_com-L_com≤ε<λ/2. Then the spectral projector onto the empirical eigenvalues in [0,λ/2) differs from the exact commutant projector by at most a constant multiple of ε/λ.
Exact mathematical passage
\widehat{\Lcal}_{\mathrm{com}}
\widehat{\Lcal}_{\mathrm{com}}Use in source · 2 passages
Search aliases: widehatLcal_com widehatL_com widehat Lcal mathrm widehat mathcal mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10144
[Spectral-gap stability of the commutant] Assume the first positive eigenvalue of L_com is at least λ>0. Let widehatL_com be an empirical estimate satisfying normwidehatL_com-L_com≤ε<λ/2. Then the spectral projector onto the empirical eigenvalues in [0,λ/2) differs from the exact commutant projector by at most a constant multiple of ε/λ.
Exact mathematical passage
\widehat{\Lcal}_{\mathrm{com}}
\widehat\McalUse in source · 2 passages
Search aliases: widehatMcal widehatM widehat Mcal widehat mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10963
[Finite meta-grammar generalization] Let K contain N candidate grandchild configurations. Suppose each target-level protected gain lies in [-G,G] and n independent development targets are used. Then, with probability at least 1-δ, every candidate satisfies abswidehatM(K)-M(K) ≤ Gsqrtfrac2log(2N/δ)n. If widehat K maximizes the empirical mean, then M(widehat K) ≥ max_K∈ KM(K) - 2Gsqrtfrac2log(2N/δ)n.
Exact mathematical passage
\abs{\widehat\Mcal(K)-\Mcal(K)} \leq G\sqrt{\frac{2\log(2N/\delta)}{n}}.
\widehat{\rm se}Use in source · 3 passages
Search aliases: widehatrm se widehat rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7498
es K_r× T_r, where the factors specify utility regularization, risk regularization, utility residual penalty, risk residual penalty, and risk threshold. In the released fixture, |Γ| =5·5·4·5·4 =2000. For γ∈Γ, let overline U_rm meta(γ) be its mean protected utility on the meta-validation episodes and widehatrm se(γ) its standard error. A configuration is validation-safe when its protected-regression tail rate and maximum regression satisfy the frozen bounds. Among safe configurations, the compile…
Exact mathematical passage
\widehat{\rm se}(\gamma)
\widehat{\operatorname{se}}Use in source · 1 passage
Search aliases: widehatse widehat operatorname
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6284
For arms a and b, evaluate both on the same frozen foreign agents and seeds. Let Δ_j=G_a,j-G_b,j. The paired estimate and standard error are widehatΔ =frac1n∑_j=1^nΔ_j, widehatse(widehatΔ) =sqrtfrac1n(n-1)∑_j=1^n(Δ_j-widehatΔ)^2. When gains are bounded in [-1,1], a distribution-free Hoeffding lower bound is LCB_1-α(Δ) = widehatΔ - sqrtfrac2log(1/α)n. Sharper paired permutation or bootstrap intervals may be used when frozen in advance.
Exact mathematical passage
\widehat\Delta =\frac1n\sum_{j=1}^n\Delta_j, \qquad \widehat{\operatorname{se}}(\widehat\Delta) =\sqrt{\frac{1}{n(n-1)}\sum_{j=1}^n(\Delta_j-\widehat\Delta)^2}.
\widehat\betaUse in source · 2 passages
Search aliases: widehatβ widehat beta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7396
For regularization parameters λ_u,λ_r>0, define widehatβ_u · = (Z^top Z+λ_u I)^-1Z^top y, widehatβ_r · = (Z^top Z+λ_r I)^-1Z^top r. For candidate type c, let s_u,c and s_r,c be the development residual standard deviations for the utility and risk heads. The deployed scores are widehat U_rm LCB(z) · = widehat U(z)-κ_u s_u,c(z),
Exact mathematical passage
\widehat\beta_u &= \left(Z^\top Z+\lambda_u I\right)^{-1}Z^\top y, \label{eq:block07-utility-head}\\ \widehat\beta_r &= \left(Z^\top Z+\lambda_r I\right)^{-1}Z^\top r. \label{eq:block07-risk-head}
\widehat\beta_rUse in source · 1 passage
Search aliases: widehatβ_r widehat beta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7396
For regularization parameters λ_u,λ_r>0, define widehatβ_u · = (Z^top Z+λ_u I)^-1Z^top y, widehatβ_r · = (Z^top Z+λ_r I)^-1Z^top r. For candidate type c, let s_u,c and s_r,c be the development residual standard deviations for the utility and risk heads. The deployed scores are widehat U_rm LCB(z) · = widehat U(z)-κ_u s_u,c(z),
Exact mathematical passage
\widehat\beta_u &= \left(Z^\top Z+\lambda_u I\right)^{-1}Z^\top y, \label{eq:block07-utility-head}\\ \widehat\beta_r &= \left(Z^\top Z+\lambda_r I\right)^{-1}Z^\top r. \label{eq:block07-risk-head}
\widehat\beta_uUse in source · 1 passage
Search aliases: widehatβ_u widehat beta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7396
For regularization parameters λ_u,λ_r>0, define widehatβ_u · = (Z^top Z+λ_u I)^-1Z^top y, widehatβ_r · = (Z^top Z+λ_r I)^-1Z^top r. For candidate type c, let s_u,c and s_r,c be the development residual standard deviations for the utility and risk heads. The deployed scores are widehat U_rm LCB(z) · = widehat U(z)-κ_u s_u,c(z),
Exact mathematical passage
\widehat\beta_u &= \left(Z^\top Z+\lambda_u I\right)^{-1}Z^\top y, \label{eq:block07-utility-head}\\ \widehat\beta_r &= \left(Z^\top Z+\lambda_r I\right)^{-1}Z^\top r. \label{eq:block07-risk-head}
\widehat\gammaUse in source · 5 passages
Search aliases: widehatγ widehat gamma
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7499
ot4·5·4 =2000. For γ∈Γ, let overline U_rm meta(γ) be its mean protected utility on the meta-validation episodes and widehatrm se(γ) its standard error. A configuration is validation-safe when its protected-regression tail rate and maximum regression satisfy the frozen bounds. Among safe configurations, the compiler selects widehatγ ∈ arg max_γ∈Γ_rm safe { overline U_rm meta(γ) - frac12widehatrm se(γ) }.
Exact mathematical passage
\widehat\gamma \in \argmax_{\gamma\in\Gamma_{\rm safe}} \left \{ \overline U_{\rm meta}(\gamma) - frac12\widehat{\rm se}(\gamma) \right\}. \label{eq:block07-meta-objective}
\widehat\gamma_{\min}Use in source · 4 passages
Search aliases: widehatγ_min widehat gamma min
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12335
/2 ( ∑_jnormwidehat R_j_F^2 )^1/2+10^-12 , and the maximum normalized commutator residual ε_comm = max_a,j fracnormwidehat P_awidehat R_j- widehat R_jwidehat P_a_F normwidehat R_j_F+10^-12. The smallest selected spectral gap is denoted widehatγ_min. These quantities serve different roles: item ε_resp measures whether the finite block-scalar model fits the response family; item ε_comm measures whether the recovered packets behave as approximately invariant response modules; item widehatγ_min meas…
Exact mathematical passage
\widehat\gamma_{\min}
\widehat\Delta^{\rm foreign}_{\Gamma\text{-lang}}Use in source · 1 passage
Search aliases: widehatΔ^rm foreign_Γ-lang widehat Delta rm Gamma text
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15475
The primary paired effect is widehatΔ^rm primary_Γ-lang = 0.225762, CI_95% = [0.166708,0.286431]. The generator-foreign paired effect is widehatΔ^rm foreign_Γ-lang = 0.240296, CI_95% = [0.180154,0.302701]. Both sign-flip estimates are approximately 9.9999×10^-6. No paired problem favors the parent. Primary pairs improve on 36.46% of problems and tie on the remainder; generator-foreign pairs improve on 42.68% and tie on the remainder. The ties are the exact conservative fallback on inherited know…
Exact mathematical passage
\widehat\Delta^{\rm foreign}_{\Gamma\text{-lang}} = 0.240296, \qquad \mathrm{CI}_{95\%} = [0.180154,0.302701]. \label{eq:b15-foreign-effect}
\widehat\Delta^{\rm primary}_{\Gamma\text{-lang}}Use in source · 1 passage
Search aliases: widehatΔ^rm primary_Γ-lang widehat Delta rm Gamma text
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15464
The primary paired effect is widehatΔ^rm primary_Γ-lang = 0.225762, CI_95% = [0.166708,0.286431]. The generator-foreign paired effect is widehatΔ^rm foreign_Γ-lang = 0.240296, CI_95% = [0.180154,0.302701]. Both sign-flip estimates are approximately 9.9999×10^-6. No paired problem favors the parent. Primary pairs improve on 36.46% of problems and tie on the remainder; generator-foreign pairs improve on 42.68% and tie on the remainder. The ties are the exact conservative fallback on inherited know…
Exact mathematical passage
\widehat\Delta^{\rm primary}_{\Gamma\text{-lang}} = 0.225762, \qquad \mathrm{CI}_{95\%} = [0.166708,0.286431]. \label{eq:b15-primary-effect}
\widehat\Delta_{1\to2}Use in source · 4 passages
Search aliases: widehatΔ_1→2 widehat Delta to
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11066
The paired grandchild-minus-recalibrated-child effect is widehatΔ_1→2 = 0.172188, with frozen interval [0.141481, 0.203110]. The sign-flip estimate is p=9.9999×10^-6. The positive paired-effect rate is 0.677083, and the negative paired-effect rate is zero. The remaining pairs are ties produced mainly by mutual abstention.
Exact mathematical passage
\widehat\Delta_{1\to2} = 0.172188,
\widehat\Delta_{3\to4}Use in source · 2 passages
Search aliases: widehatΔ_3→4 widehat Delta to
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12635
The paired effect is widehatΔ_3→4 =0.207325, with frozen bootstrap interval [0.131902, 0.287910] and sign-flip estimate p=9.9999×10^-6. No primary target favors the parent. The positive paired-effect rate is 0.2875 and the remaining 0.7125 are abstention ties.
Exact mathematical passage
\widehat\Delta_{3\to4} =0.207325, \label{eq:b11-paired-effect}
\widehat\Delta_{4\to5}Use in source · 1 passage
Search aliases: widehatΔ_4→5 widehat Delta to
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13348
The paired effect is widehatΔ_4→5 = 0.846985, with frozen interval [0.786100,0.901073] and sign-flip estimate p=9.9999×10^-6. No primary target favors the inherited parent. The remaining 9.375% of pairs are abstention ties.
Exact mathematical passage
\widehat\Delta_{4\to5} = 0.846985, \label{eq:b12-response-effect}
\widehat\Delta_{5\to6}Use in source · 1 passage
Search aliases: widehatΔ_5→6 widehat Delta to
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13458
The paired effect is widehatΔ_5→6 = 0.708388, with frozen interval [0.659635,0.757198]. The sign-flip estimate is 9.9999×10^-6, and no primary target favors the affine parent.
Exact mathematical passage
\widehat\Delta_{5\to6} = 0.708388, \label{eq:b12-constructor-effect}
\widehat\Delta_{\mathrm{clean}}^{A}Use in source · 1 passage
Search aliases: widehatΔ_clean^A widehat Delta mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17488
Paired effects are: widehatΔ_clean^A · =0.142631, · 95% CI · =[0.135421,0.149531], widehatΔ_clean^N · =0.365616, · 95% CI · =[0.352991,0.378402], widehatΔ_program^A · =0.120614, · 95% CI · =[0.110353,0.131880], widehatΔ_program^N · =0.125290, · 95% CI · =[0.111056,0.138921]. The clean effects are parent minus child; the program effects are child minus parent. Every interval is frozen and strictly positive.
Exact mathematical passage
\widehat\Delta_{\mathrm{clean}}^{A} &=0.142631, &95\%\ \mathrm{CI}&=[0.135421,0.149531],\\ \widehat\Delta_{\mathrm{clean}}^{N} &=0.365616, &95\%\ \mathrm{CI}&=[0.352991,0.378402],\\ \widehat\Delta_{\mathrm{program}}^{A} &=0.120614, &95\%\ \mathrm{CI}&=[0.110353,0.131880],\\ \widehat\Delta_{\mathrm{program}}^{N} &=0.125290, &95\%\ \mathrm{CI}&=[0.111056,0.138921]. \label{eq:b19-paired-effects}
\widehat\Delta_{\mathrm{clean}}^{N}Use in source · 1 passage
Search aliases: widehatΔ_clean^N widehat Delta mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17488
Paired effects are: widehatΔ_clean^A · =0.142631, · 95% CI · =[0.135421,0.149531], widehatΔ_clean^N · =0.365616, · 95% CI · =[0.352991,0.378402], widehatΔ_program^A · =0.120614, · 95% CI · =[0.110353,0.131880], widehatΔ_program^N · =0.125290, · 95% CI · =[0.111056,0.138921]. The clean effects are parent minus child; the program effects are child minus parent. Every interval is frozen and strictly positive.
Exact mathematical passage
\widehat\Delta_{\mathrm{clean}}^{A} &=0.142631, &95\%\ \mathrm{CI}&=[0.135421,0.149531],\\ \widehat\Delta_{\mathrm{clean}}^{N} &=0.365616, &95\%\ \mathrm{CI}&=[0.352991,0.378402],\\ \widehat\Delta_{\mathrm{program}}^{A} &=0.120614, &95\%\ \mathrm{CI}&=[0.110353,0.131880],\\ \widehat\Delta_{\mathrm{program}}^{N} &=0.125290, &95\%\ \mathrm{CI}&=[0.111056,0.138921]. \label{eq:b19-paired-effects}
\widehat\Delta_{\mathrm{FiLM-add}}Use in source · 1 passage
Search aliases: widehatΔ_FiLM-add widehat Delta mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16956
The new constructor also fails a stronger comparison with the inherited additive repair. Pooled over the two architectures and two final supports, the selected-child minus legacy-additive program-specificity effect is widehatΔ_FiLM-add =-0.029417, CI_0.95 =[-0.034102,-0.024735]. The negative interval forbids a claim that the new source-selected constructor dominates the inherited constructor. It can still be structurally new and useful on the primary audits, but novelty is not superiority.
Exact mathematical passage
\widehat\Delta_{\mathrm{FiLM-add}} =-0.029417, \qquad \mathrm{CI}_{0.95} =[-0.034102,-0.024735]. \label{eq:b18-legacy-comparison}
\widehat\Delta_{G\to C}Use in source · 4 passages
Search aliases: widehatΔ_G→ C widehat Delta to
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11900
The paired child-minus-parent effect is widehatΔ_G→ C=0.334224, with frozen bootstrap interval [0.266459,0.401903]. The sign-flip estimate is p=9.9999×10^-6. The positive-pair rate is 0.572917, the remaining 0.427083 are conservative abstention ties, and no paired target favors the global parent.
Exact mathematical passage
\widehat\Delta_{G\to C}=0.334224,
\widehat\Delta_nUse in source · 2 passages
Search aliases: widehatΔ_n widehat Delta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7696
For n held-out episodes, define paired differences D_j = G_e_j(K_1)-G_e_j(K_0), widehatΔ_n=frac1n∑_j=1^nD_j. Because parent and child share each target and candidate bundle, pairing removes much of the episode difficulty variation.
Exact mathematical passage
D_j = G_{e_j}(K_1)-G_{e_j}(K_0), \qquad \widehat\Delta_n=\frac1n\sum_{j=1}^nD_j. \label{eq:block07-paired-estimator}
\widehat\Delta_{\mathrm{oracle}\to C}Use in source · 2 passages
Search aliases: widehatΔ_oracle→ C widehat Delta mathrm to
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11914
More importantly, the paired child-minus-direct-global-oracle effect is widehatΔ_oracle→ C=0.157530, with frozen interval [0.080863,0.234135]. Thus the child exceeds a directly supervised global-affine ceiling comparison on average, even though the oracle uses far more target labels and is not constrained to estimate a source-to-target bridge.
Exact mathematical passage
\widehat\Delta_{\mathrm{oracle}\to C}=0.157530,
\widehat\Delta_{\mathrm{prog}}^{A}Use in source · 1 passage
Search aliases: widehatΔ_prog^A widehat Delta mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16419
The paired child-minus-parent program-specificity gain is widehatΔ_prog^A =0.010714, CI_0.95 =[0.009911,0.011527], with positive fraction 0.823125. The answer-specificity gain is 0.017727. Its frozen interval is [0.016490,0.018960]. Parent-minus-child clean loss is 0.000979, with frozen interval [0.000695,0.001290].
Exact mathematical passage
\widehat\Delta_{\mathrm{prog}}^{A} =0.010714, \qquad \mathrm{CI}_{0.95} =[0.009911,0.011527], \label{eq:b17-primary-program-gain}
\widehat\Delta_{\mathrm{prog}}^{B}Use in source · 1 passage
Search aliases: widehatΔ_prog^B widehat Delta mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16461
The paired generator-foreign program-specificity gain is widehatΔ_prog^B =0.024763, CI_0.95 =[0.022794,0.026793], with positive fraction 0.855. The answer-specificity gain is 0.035400 with interval [0.032956,0.037938]. Clean-loss improvement is 0.003464 with interval [0.002633,0.004324]. Monte Carlo paired sign-flip tests for the program-specificity gains return 9.999×10^-5 on each final support.
Exact mathematical passage
\widehat\Delta_{\mathrm{prog}}^{B} =0.024763, \qquad \mathrm{CI}_{0.95} =[0.022794,0.026793], \label{eq:b17-foreign-program-gain}
\widehat\Delta_{\mathrm{program}}^{A}Use in source · 1 passage
Search aliases: widehatΔ_program^A widehat Delta mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17488
Paired effects are: widehatΔ_clean^A · =0.142631, · 95% CI · =[0.135421,0.149531], widehatΔ_clean^N · =0.365616, · 95% CI · =[0.352991,0.378402], widehatΔ_program^A · =0.120614, · 95% CI · =[0.110353,0.131880], widehatΔ_program^N · =0.125290, · 95% CI · =[0.111056,0.138921]. The clean effects are parent minus child; the program effects are child minus parent. Every interval is frozen and strictly positive.
Exact mathematical passage
\widehat\Delta_{\mathrm{clean}}^{A} &=0.142631, &95\%\ \mathrm{CI}&=[0.135421,0.149531],\\ \widehat\Delta_{\mathrm{clean}}^{N} &=0.365616, &95\%\ \mathrm{CI}&=[0.352991,0.378402],\\ \widehat\Delta_{\mathrm{program}}^{A} &=0.120614, &95\%\ \mathrm{CI}&=[0.110353,0.131880],\\ \widehat\Delta_{\mathrm{program}}^{N} &=0.125290, &95\%\ \mathrm{CI}&=[0.111056,0.138921]. \label{eq:b19-paired-effects}
\widehat\Delta_{\mathrm{program}}^{N}Use in source · 1 passage
Search aliases: widehatΔ_program^N widehat Delta mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17488
Paired effects are: widehatΔ_clean^A · =0.142631, · 95% CI · =[0.135421,0.149531], widehatΔ_clean^N · =0.365616, · 95% CI · =[0.352991,0.378402], widehatΔ_program^A · =0.120614, · 95% CI · =[0.110353,0.131880], widehatΔ_program^N · =0.125290, · 95% CI · =[0.111056,0.138921]. The clean effects are parent minus child; the program effects are child minus parent. Every interval is frozen and strictly positive.
Exact mathematical passage
\widehat\Delta_{\mathrm{clean}}^{A} &=0.142631, &95\%\ \mathrm{CI}&=[0.135421,0.149531],\\ \widehat\Delta_{\mathrm{clean}}^{N} &=0.365616, &95\%\ \mathrm{CI}&=[0.352991,0.378402],\\ \widehat\Delta_{\mathrm{program}}^{A} &=0.120614, &95\%\ \mathrm{CI}&=[0.110353,0.131880],\\ \widehat\Delta_{\mathrm{program}}^{N} &=0.125290, &95\%\ \mathrm{CI}&=[0.111056,0.138921]. \label{eq:b19-paired-effects}
\widehat\Delta_{\rm arch}Use in source · 2 passages
Search aliases: widehatΔ_rm arch widehat Delta rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7947
The paired effects are widehatΔ_rm audit · =0.072518, · CI_95% · =[0.046745,0.101389], widehatΔ_rm arch · =0.104999, · CI_95% · =[0.072711,0.140506]. Both one-sided sign-flip p-values equal approximately 4.99975×10^-5 under the frozen 20,000-draw randomization procedure.
Exact mathematical passage
\widehat\Delta_{\rm audit} &=0.072518, &\mathrm{CI}_{95\%} &=[0.046745,0.101389], \label{eq:block07-audit-effect}\\ \widehat\Delta_{\rm arch} &=0.104999, &\mathrm{CI}_{95\%} &=[0.072711,0.140506]. \label{eq:block07-foreign-effect}
\widehat\Delta_{\rm audit}Use in source · 2 passages
Search aliases: widehatΔ_rm audit widehat Delta rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7947
The paired effects are widehatΔ_rm audit · =0.072518, · CI_95% · =[0.046745,0.101389], widehatΔ_rm arch · =0.104999, · CI_95% · =[0.072711,0.140506]. Both one-sided sign-flip p-values equal approximately 4.99975×10^-5 under the frozen 20,000-draw randomization procedure.
Exact mathematical passage
\widehat\Delta_{\rm audit} &=0.072518, &\mathrm{CI}_{95\%} &=[0.046745,0.101389], \label{eq:block07-audit-effect}\\ \widehat\Delta_{\rm arch} &=0.104999, &\mathrm{CI}_{95\%} &=[0.072711,0.140506]. \label{eq:block07-foreign-effect}
\widehat\Delta_{\rm bridge}Use in source · 1 passage
Search aliases: widehatΔ_rm bridge widehat Delta rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 18722
resentation families; item a 144-target paired foreign audit and a separate stress suite; item random-budget, farthest-point, shuffled-binding, sign-flip, and plain-QR controls; item exact query ledgers, rank diagnostics, harm checks, and package-closed receipts. On the primary foreign audit, mean balanced selected gain rises from 0.135212 to 0.344720, and widehatΔ_rm bridge=0.209508, CI_95%=[0.194968,0.223898]. Every primary paired effect is positive, full-rank bridge identification rises from …
Exact mathematical passage
\widehat\Delta_{\rm bridge}=0.209508, \qquad \mathrm{CI}_{95\%}=[0.194968,0.223898].
\widehat\Delta_{\rm meta}Use in source · 1 passage
Search aliases: widehatΔ_rm meta widehat Delta rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8660
On 144 primary foreign targets, the target-level paired difference is widehatΔ_rm meta = widehat M_24^rm for( P_1) - widehat M_24^rm for( P_0) = 0.209508. A target-cluster bootstrap gives the interval [0.194968,0.223898], and a 100,000-draw sign-flip test gives p=9.9999×10^-6. All 144 target-level paired differences are positive.
Exact mathematical passage
\widehat\Delta_{\rm meta} = \widehat M_{24}^{\rm for}(\mathfrak P_1) - \widehat M_{24}^{\rm for}(\mathfrak P_0) = 0.209508. \label{eq:block07q-primary-delta}
\widehat\Delta_{\mathrm{str}}Use in source · 2 passages
Search aliases: widehatΔ_str widehat Delta mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4832
[Finite-sample structural certificate] Suppose normr≤ R almost surely and r_1,…,r_n are independent held-out defects. Let widehatΔ_str = frac1n∑_i=1^nnormP_Wr_i^2. Then with probability at least 1-δ, | widehatΔ_str - EnormP_Wr^2 | ≤ R^2sqrtfraclog(2/δ)2n.
Exact mathematical passage
\widehat\Delta_{\mathrm{str}} = \frac1n\sum_{i=1}^{n}\norm{P_Wr_i}^2.
\widehat\Delta_{\Gamma}^{\mathrm{arch}}Use in source · 1 passage
Search aliases: widehatΔ_Γ^arch widehat Delta Gamma mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 14149
The architecture-foreign paired effect is widehatΔ_Γ^arch = 0.090800, CI_95% = [0.052311,0.135081]. The sign-flip estimate is again 9.9999×10^-6. The child improves 34.722% of pairs, ties on 65.278%, and no architecture-foreign pair favors the parent.
Exact mathematical passage
\widehat\Delta_{\Gamma}^{\mathrm{arch}} = 0.090800, \qquad \mathrm{CI}_{95\%} = [0.052311,0.135081]. \label{eq:b13-architecture-effect}
\widehat\Delta_\Gamma^{\mathrm{primary}}Use in source · 1 passage
Search aliases: widehatΔ_Γ^primary widehat Delta Gamma mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 22099
Actual Block 13 compiler edge. On the primary audit, the parent and child gains are 0.883004 and 0.950231, so widehatΔ_Γ^primary=0.067227. The child changes research allocation rather than the candidate grammar. The architecture-foreign paired effect is 0.090800.
Exact mathematical passage
\widehat\Delta_\Gamma^{\mathrm{primary}}=0.067227.
\widehat\Delta_{\Gamma}^{\mathrm{primary}}Use in source · 1 passage
Search aliases: widehatΔ_Γ^primary widehat Delta Gamma mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 14136
The primary paired effect is widehatΔ_Γ^primary = 0.067227, CI_95% = [0.035881,0.103274]. The sign-flip estimate is 9.9999×10^-6. The child improves 21.875% of pairs, ties on 75%, and loses on 3.125%.
Exact mathematical passage
\widehat\Delta_{\Gamma}^{\mathrm{primary}} = 0.067227, \qquad \mathrm{CI}_{95\%} = [0.035881,0.103274]. \label{eq:b13-primary-effect}
\widehat\kappaUse in source · 2 passages
Search aliases: widehatκ widehat kappa
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15327
novel-high · ⟼ objective-then-constructor. The expanded development gain is widehat O_1=0.948962, with newly correctable margin widehatκ = 0.571704.
Exact mathematical passage
\widehat\kappa = 0.571704. \label{eq:b15-development-margin}
\widehat\lambdaUse in source · 2 passages
Search aliases: widehatλ widehat lambda
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3623
[Stability of the threshold count] Let widehat A be an estimated holonomy-energy operator satisfying normwidehat A- A≤η. Then each ordered eigenvalue satisfies abswidehatλ_i-λ_i≤η. If no eigenvalue of A lies in [τ-η,τ+η], then widehat A and A have the same number of eigenvalues below τ.
Exact mathematical passage
\abs{\widehat\lambda_i-\lambda_i}\leq\eta.
\widehat\lambda_{aj}Use in source · 1 passage
Search aliases: widehatλ_aj widehat lambda
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12310
For recovered projectors widehat P_1,…,widehat P_m, define the normalized response-fit residual ε_resp = frac ( ∑_jnormwidehat R_j- ∑_awidehatλ_ajwidehat P_a_F^2 )^1/2 ( ∑_jnormwidehat R_j_F^2 )^1/2+10^-12 , and the maximum normalized commutator residual ε_comm = max_a,j fracnormwidehat P_awidehat R_j- widehat R_jwidehat P_a_F normwidehat R_j_F+10^-12. The smallest selected spectral gap is denoted widehatγ_min. These quantities serve different roles: item ε_resp measures whether the finite block…
Exact mathematical passage
\eps_{\mathrm{resp}} = \frac{ \left( \sum_j\norm{\widehat R_j- \sum_a\widehat\lambda_{aj}\widehat P_a}_F^2 \right)^{1/2} }{ \left( \sum_j\norm{\widehat R_j}_F^2 \right)^{1/2}+10^{-12} }, \label{eq:b11-response-fit}
\widehat\lambda_iUse in source · 1 passage
Search aliases: widehatλ_i widehat lambda
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3623
[Stability of the threshold count] Let widehat A be an estimated holonomy-energy operator satisfying normwidehat A- A≤η. Then each ordered eigenvalue satisfies abswidehatλ_i-λ_i≤η. If no eigenvalue of A lies in [τ-η,τ+η], then widehat A and A have the same number of eigenvalues below τ.
Exact mathematical passage
\abs{\widehat\lambda_i-\lambda_i}\leq\eta.
\widehat\muUse in source · 1 passage
Search aliases: widehatμ widehat mu
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10834
Let p^⋆ be the best conditioning policy and p_0 the best inherited-policy configuration. Empirical errors of at most η imply widehatμ(p^⋆) ≥ μ(p^⋆)-η > μ(p_0)+η ≥ widehatμ(p_0).
Exact mathematical passage
\widehat\mu(p^\star) \geq \mu(p^\star)-\eta > \mu(p_0)+\eta \geq \widehat\mu(p_0).
\widehat\sigmaUse in source · 2 passages
Search aliases: widehatσ widehat sigma
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13888
A simple acquisition rule is A_i = widehat g_i -λ_rwidehat r_i +λ_uwidehatσ_i +λ_h p_θ(h_imid x_τ) -λ_c C_i, where h_i is the method family, p_θ is an obstruction-class posterior, widehatσ_i is a declared uncertainty bonus, and C_i is experimental cost.
Exact mathematical passage
A_i = \widehat g_i -\lambda_r\widehat r_i +\lambda_u\widehat\sigma_i +\lambda_h p_\theta(h_i\mid x_\tau) -\lambda_c C_i, \label{eq:b13-acquisition}
\widehat\sigma_iUse in source · 2 passages
Search aliases: widehatσ_i widehat sigma
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13888
A simple acquisition rule is A_i = widehat g_i -λ_rwidehat r_i +λ_uwidehatσ_i +λ_h p_θ(h_imid x_τ) -λ_c C_i, where h_i is the method family, p_θ is an obstruction-class posterior, widehatσ_i is a declared uncertainty bonus, and C_i is experimental cost.
Exact mathematical passage
A_i = \widehat g_i -\lambda_r\widehat r_i +\lambda_u\widehat\sigma_i +\lambda_h p_\theta(h_i\mid x_\tau) -\lambda_c C_i, \label{eq:b13-acquisition}
\widehat\tauUse in source · 2 passages
Search aliases: widehatτ widehat tau
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6212
Let N⊆{1,…,n} be foreign agents whose dominant defect belongs to the held-out sixth type. Define widehatτ_abstract = frac1|N|∑_j∈ N ( G_j^meta-G_j^no-meta ). In the reference positive control, widehatτ_abstract=0.700533. A learned recursive-improvement claim would require the schema to be generated by the parent improvement process, sealed before foreign evaluation, and compared under matched computation and evidence access.
Exact mathematical passage
\widehat\tau_{\mathrm{abstract}} = \frac1{|N|}\sum_{j\in N} \left( G_{j}^{\mathrm{meta}}-G_{j}^{\mathrm{no\mbox{-}meta}} \right).
\widehat\tau_{\mathrm{abstract}}Use in source · 2 passages
Search aliases: widehatτ_abstract widehat tau mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6212
Let N⊆{1,…,n} be foreign agents whose dominant defect belongs to the held-out sixth type. Define widehatτ_abstract = frac1|N|∑_j∈ N ( G_j^meta-G_j^no-meta ). In the reference positive control, widehatτ_abstract=0.700533. A learned recursive-improvement claim would require the schema to be generated by the parent improvement process, sealed before foreign evaluation, and compared under matched computation and evidence access.
Exact mathematical passage
\widehat\tau_{\mathrm{abstract}} = \frac1{|N|}\sum_{j\in N} \left( G_{j}^{\mathrm{meta}}-G_{j}^{\mathrm{no\mbox{-}meta}} \right).
\widetilde dUse in source · 3 passages
Search aliases: widetilde d widetilde
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6089
Each foreign agent has a canonical defect d∈ℝ^6, a positive diagonal importance vector w, and an invertible architecture transform T∈ℝ^6×6. The runtime-visible defect is widetilde d=Td. Protected utility is J_T(widetilde d) = exp(-∑_i=1^6w_i(T^-1widetilde d)_i^2). A candidate runtime patch δ produces gain g(δ) = J_T(widetilde d+δ)-J_T(widetilde d).
Exact mathematical passage
\widetilde d=Td.
\widetilde hUse in source · 2 passages
Search aliases: widetilde h widetilde
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15663
Given input x, define the prompt anchor h_0=E(x). A normalized injection state z_t and updated latent state h_t satisfy z_t=N(h_t-1), widetilde h_t=B(z_t,x), h_t=U(z_t,widetilde h_t,h_0), 1≤ t≤ K, where N is any declared normalization or scale-matching operator. The answer is sampled or selected from D(h_K,x) after a declared end-of-thought transition.
Exact mathematical passage
z_t=N(h_{t-1}), \qquad \widetilde h_t=B(z_t,x), \qquad h_t=U(z_t,\widetilde h_t,h_0), \qquad 1\leq t\leq K, \label{eq:b16-latent-recurrence}
\widetilde h_kUse in source · 1 passage
Search aliases: widetilde h_k widetilde
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16124
The latent trace updates the task representation through g_k · =σ(Gz_k), widetilde h_k · =tanh(Th_k-1+Lz_k), h_k · =(1-g_k)odot h_k-1+g_kodotwidetilde h_k, h_0=e, widehat w_k · =Oh_k. A separate halt head predicts one of four dynamic depths from the mean trace and, for nonparent candidates, the encoded prompt. A semantic training head predicts w(τ) from z_4. Its auxiliary loss is an externally declared witness term, not a claim that the latent coordinates are uniquely interpretable.
Exact mathematical passage
g_k&=\sigma(Gz_k), \label{eq:b17-update-gate}\\ \widetilde h_k&=\tanh(Th_{k-1}+Lz_k), \label{eq:b17-hidden-candidate}\\ h_k&=(1-g_k)\odot h_{k-1}+g_k\odot\widetilde h_k, \qquad h_0=e, \label{eq:b17-hidden-update}\\ \widehat w_k&=Oh_k. \label{eq:b17-readout}
\widetilde h_tUse in source · 1 passage
Search aliases: widetilde h_t widetilde
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15663
Given input x, define the prompt anchor h_0=E(x). A normalized injection state z_t and updated latent state h_t satisfy z_t=N(h_t-1), widetilde h_t=B(z_t,x), h_t=U(z_t,widetilde h_t,h_0), 1≤ t≤ K, where N is any declared normalization or scale-matching operator. The answer is sampled or selected from D(h_K,x) after a declared end-of-thought transition.
Exact mathematical passage
z_t=N(h_{t-1}), \qquad \widetilde h_t=B(z_t,x), \qquad h_t=U(z_t,\widetilde h_t,h_0), \qquad 1\leq t\leq K, \label{eq:b16-latent-recurrence}
\widetilde KUse in source · 3 passages
Search aliases: widetilde K widetilde
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9033
Let K be the declared finite or compact class of admissible repair processes under the protected context. For R⊆ N, let K_R(K) = { widetilde K∈K: widetilde K_i=K_i for every i∉ R }. The restricted oracle envelope is F_K(R) = sup_widetilde K∈K_R(K) M_Ξ,D,B(widetilde K). Thus F_K(∅)=M(K), F_K(N)=sup_widetilde K∈KM(widetilde K). The total declared headroom is H(K) = F_K(N)-F_K(∅). The adjective declared matters. A larger, previously unavailable meta-grammar may expose additional headroom. Equation …
Exact mathematical passage
\Kcal_R(K) = \left\{ \widetilde K\in\Kcal: \widetilde K_i=K_i \text{ for every }i\notin R \right\}. \label{eq:b8-restricted-class}
\widetilde K_iUse in source · 1 passage
Search aliases: widetilde K_i widetilde
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9033
Let K be the declared finite or compact class of admissible repair processes under the protected context. For R⊆ N, let K_R(K) = { widetilde K∈K: widetilde K_i=K_i for every i∉ R }. The restricted oracle envelope is F_K(R) = sup_widetilde K∈K_R(K) M_Ξ,D,B(widetilde K). Thus F_K(∅)=M(K), F_K(N)=sup_widetilde K∈KM(widetilde K). The total declared headroom is H(K) = F_K(N)-F_K(∅). The adjective declared matters. A larger, previously unavailable meta-grammar may expose additional headroom. Equation …
Exact mathematical passage
\Kcal_R(K) = \left\{ \widetilde K\in\Kcal: \widetilde K_i=K_i \text{ for every }i\notin R \right\}. \label{eq:b8-restricted-class}
\widetilde rUse in source · 2 passages
Search aliases: widetilde r widetilde
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9394
A grammar extension can reveal error that did not exist as a coordinate in the old state. Let E_t:ℝ^m_t↪ℝ^m_t+1 be the conservative embedding of old meta-regret coordinates. Decompose r_t+1 = E_twidetilde r_t+1+n_t+1, n_t+1⊥im E_t. Define old-coordinate repair progress by P_t^repair = normr_t^2-normwidetilde r_t+1^2, and epistemic discovery by P_t^discovery = normn_t+1^2. Both quantities may be positive. The total represented regret may increase because the lineage learned to represent a previou…
Exact mathematical passage
r_{t+1} = E_t\widetilde r_{t+1}+n_{t+1}, \qquad n_{t+1}\perp\im E_t. \label{eq:b8-regret-expansion}
\widetilde r_{t+1}Use in source · 2 passages
Search aliases: widetilde r_t+1 widetilde
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9394
A grammar extension can reveal error that did not exist as a coordinate in the old state. Let E_t:ℝ^m_t↪ℝ^m_t+1 be the conservative embedding of old meta-regret coordinates. Decompose r_t+1 = E_twidetilde r_t+1+n_t+1, n_t+1⊥im E_t. Define old-coordinate repair progress by P_t^repair = normr_t^2-normwidetilde r_t+1^2, and epistemic discovery by P_t^discovery = normn_t+1^2. Both quantities may be positive. The total represented regret may increase because the lineage learned to represent a previou…
Exact mathematical passage
r_{t+1} = E_t\widetilde r_{t+1}+n_{t+1}, \qquad n_{t+1}\perp\im E_t. \label{eq:b8-regret-expansion}
\widetilde SUse in source · 4 passages
Search aliases: widetilde S widetilde
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 743
[Composition bound for bridge defects] Let T:X→ Y and S:Y→ Z. Suppose the property transports satisfy normQ_Y(Tx)-widetilde T_QQ_X(x)≤ε_T for x∈ K, and normQ_Z(Sy)-widetilde S_QQ_Y(y)≤ε_S for y∈ T(K). If normwidetilde S_Q≤ L_S, then the composite bridge obeys ε_Q(S∘ T;K) ≤ ε_S+L_Sε_T with intended property transport widetilde S_Qwidetilde T_Q.
Exact mathematical passage
\norm{Q_Z(Sy)-\widetilde S_QQ_Y(y)}\leq\eps_S
\widetilde S_QUse in source · 4 passages
Search aliases: widetilde S_Q widetilde
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 743
[Composition bound for bridge defects] Let T:X→ Y and S:Y→ Z. Suppose the property transports satisfy normQ_Y(Tx)-widetilde T_QQ_X(x)≤ε_T for x∈ K, and normQ_Z(Sy)-widetilde S_QQ_Y(y)≤ε_S for y∈ T(K). If normwidetilde S_Q≤ L_S, then the composite bridge obeys ε_Q(S∘ T;K) ≤ ε_S+L_Sε_T with intended property transport widetilde S_Qwidetilde T_Q.
Exact mathematical passage
\norm{Q_Z(Sy)-\widetilde S_QQ_Y(y)}\leq\eps_S
\widetilde yUse in source · 2 passages
Search aliases: widetilde y widetilde
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11769
For current target state y, candidate repaired state widetilde y, and oracle repaired state y^⋆, define G = clip_[-1,1] ( 1- frac Enormwidetilde y-y^⋆^2 Enormy-y^⋆^2 ). The witness records: item validation gain; item source-to-target alignment NRMSE; item decoded protected-context regression; item minimum chart support; item alignment reduction relative to the global bridge; item source-gate and target-gate agreement; item target-gate margin.
Exact mathematical passage
\widetilde y
\widetilde\piUse in source · 1 passage
Search aliases: widetildeπ widetilde pi
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5105
Let ϕ_G:X→ Z_G be the state representation expressible in grammar G. A G-measurable deterministic repair policy has the form π=widetildeπ∘ϕ_G.
Exact mathematical passage
\pi=\widetilde\pi\circ\phi_{\Gcal}.
\widetilde\phiUse in source · 3 passages
Search aliases: widetildeϕ widetilde phi
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5973
nts are, respectively: item deviation from the baseline no-action continuation; item deviation of the action-conditioned response from the baseline action response; item violation of redundant-sensor consistency; item deviation from baseline memory update; item discrepancy between reported and protected evaluator scores. The observed probe is widetildeϕ=ϕ(z)+η, η∼ N(0,σ^2I_5).
Exact mathematical passage
\widetilde\phi=\phi(z)+\eta, \qquad \eta\sim\mathcal N(0,\sigma^2I_5).
\mathrm{with}Use in source · 1 passage
Search aliases: with mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9846
Define the causal generator contribution Δ_gen = E_Q[η_without(r)^2] - E_Q[η_with(r)^2]. The contribution is admissible only when a matched ablation isolates the generator while holding the remaining grammar, evidence, and selection budget fixed.
Exact mathematical passage
\Delta_{\mathrm{gen}} = E_Q\bigl[\eta_{\mathrm{without}}(r)^2\bigr] - E_Q\bigl[\eta_{\mathrm{with}}(r)^2\bigr]. \label{eq:b8-generator-credit}
\mathrm{without}Use in source · 1 passage
Search aliases: without mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9846
Define the causal generator contribution Δ_gen = E_Q[η_without(r)^2] - E_Q[η_with(r)^2]. The contribution is admissible only when a matched ablation isolates the generator while holding the remaining grammar, evidence, and selection budget fixed.
Exact mathematical passage
\Delta_{\mathrm{gen}} = E_Q\bigl[\eta_{\mathrm{without}}(r)^2\bigr] - E_Q\bigl[\eta_{\mathrm{with}}(r)^2\bigr]. \label{eq:b8-generator-credit}
\WitnessedUse in source · 1 passage
Search aliases: Witnessed mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4259
Let C_RI(s) = · Grounded(s) wedgeProvValid(s) wedgeWorkspaceBound(s) · wedgeRepairTyped(s) wedgeWitnessed(s) wedgeEvidenceFlat(s) · wedgeRollbackable(s) wedgeMetaCapable(s). For X⊆ S, define Ψ_∀(X) = sets∈ S:C_RI(s) and Post_imp(s)⊆ X.
Exact mathematical passage
\begin{aligned} C_{\mathrm{RI}}(s) ={}& \Grounded(s) \wedge\ProvValid(s) \wedge\WorkspaceBound(s)\\ &\wedge\RepairTyped(s) \wedge\Witnessed(s) \wedge\EvidenceFlat(s)\\ &\wedge\Rollbackable(s) \wedge\MetaCapable(s). \end{aligned}
\mathrm{workspace}Use in source · 1 passage
Search aliases: workspace mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 665
A representative color set is C= set model, workspace, memory, planner, improver, evaluator, provenance, tools . The atlas does not assert that these colors are universally correct. It declares the decomposition under which one particular repair theory is being tested.
Exact mathematical passage
\mathsf C= \set{ \mathrm{model}, \mathrm{workspace}, \mathrm{memory}, \mathrm{planner}, \mathrm{improver}, \mathrm{evaluator}, \mathrm{provenance}, \mathrm{tools} }.
\WorkspaceBoundUse in source · 1 passage
Search aliases: WorkspaceBound mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4259
Let C_RI(s) = · Grounded(s) wedgeProvValid(s) wedgeWorkspaceBound(s) · wedgeRepairTyped(s) wedgeWitnessed(s) wedgeEvidenceFlat(s) · wedgeRollbackable(s) wedgeMetaCapable(s). For X⊆ S, define Ψ_∀(X) = sets∈ S:C_RI(s) and Post_imp(s)⊆ X.
Exact mathematical passage
\begin{aligned} C_{\mathrm{RI}}(s) ={}& \Grounded(s) \wedge\ProvValid(s) \wedge\WorkspaceBound(s)\\ &\wedge\RepairTyped(s) \wedge\Witnessed(s) \wedge\EvidenceFlat(s)\\ &\wedge\Rollbackable(s) \wedge\MetaCapable(s). \end{aligned}
\mathrm{world}Use in source · 1 passage
Search aliases: world mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5935
The latent event variable is z∈ {none,internal,world,sensor,memory,evaluator}. Each event changes one declared mechanism: item internal: the action-response map B changes; item world: the passive dynamics A change; item sensor: the redundant sensor receives an additive bias; item memory: the retention coefficient λ changes; item evaluator: the reported weight vector w changes; item none: all baseline mechanisms remain fixed. The event label is not supplied to the classifier.
Exact mathematical passage
z\in \{\mathrm{none},\mathrm{internal},\mathrm{world},\mathrm{sensor},\mathrm{memory},\mathrm{evaluator}\}.
\mathrm{worst\ chart}Use in source · 1 passage
Search aliases: worst chart mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17934
A candidate is pre-audit eligible only if both foreign target models are ATLAS_FULL and the candidate jointly satisfies R^2_transport · ≥0.75, · R^2_worst chart · ≥0.74, ρ_preserve · ≥0.98, · NRMSE_trace · ≤0.05, κ_square · ≤0.02. · · The selection score is the weakest normalized margin across these gates. This prevents a strong aggregate R^2 from compensating for a failed chart, high curvature, or poor trajectory reconstruction.
Exact mathematical passage
\begin{aligned} R^2_{\mathrm{transport}}&\geq0.75, &\qquad R^2_{\mathrm{worst\ chart}}&\geq0.74,\\ \rho_{\mathrm{preserve}}&\geq0.98, & \operatorname{NRMSE}_{\mathrm{trace}}&\leq0.05,\\ \kappa_{\mathrm{square}}&\leq0.02. && \end{aligned} \label{eq:b20-thresholds}
X^{\mathsf T}Use in source · 8 passages
Search aliases: X^ T mathsf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10568
[Rank saturation does not imply stability] For every d≥2 and every η>0, there exists a full-rank design X∈ℝ^d× d such that λ_min(X^ TX)<η.
Exact mathematical passage
\lambda_{\min}(X^{\mathsf T}X)<\eta.
x^{\mathsf T}Use in source · 1 passage
Search aliases: x^ T mathsf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21777
Let M=I_2, x=(2,1)^ T, and w=4. The D-optimal information increment is logdet(M+wxx^ T)-logdet M = log(1+w x^ TM^-1x) = log 21. A low-quality copy with w=1/16 gives only log(1+5/16), despite identical source geometry.
Exact mathematical passage
\log\det(M+wxx^{\mathsf T})-\log\det M = \log(1+w x^{\mathsf T}M^{-1}x) = \log 21.
x^2Use in source · 5 passages
Search aliases: x^2
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16627
Let the template set be Θ = {θ_aff,θ_abs, θ_quad,θ_hinge}. For integer parameters a,b,c and integer input x, the exact programs are f_aff(x) · =ax+b, f_abs(x) · =a|x|+b, f_quad(x) · =ax^2+bx+c, f_hinge(x) · = ax+c, · x≥0, bx+c, · x<0. The coefficients belong to finite declared integer sets. A task contains five shuffled input--output demonstrations and one query input. The model receives a normalized serialization of those demonstrations and the query. The protected target is y(τ) = ( e_θ, fracp…
Exact mathematical passage
f_{\mathrm{aff}}(x)&=ax+b, \\ f_{\mathrm{abs}}(x)&=a|x|+b, \\ f_{\mathrm{quad}}(x)&=ax^2+bx+c, \\ f_{\mathrm{hinge}}(x)&= \begin{cases} ax+c,&x\geq0,\\ bx+c,&x<0. \end{cases} \label{eq:b18-program-templates}
X^\topUse in source · 4 passages
Search aliases: X^top top
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8417
h_t(x_k)^top , and fits widehat T^top = (X^top X+λ I)^-1X^top Y. The parent chooses the k=16 fit anchors uniformly at random. The source hidden space is also 16-dimensional, so a rank-deficient design leaves at least one source direction unidentified.
Exact mathematical passage
\widehat T^\top = (X^\top X+\lambda I)^{-1}X^\top Y. \label{eq:block07q-ridge-bridge}
x_+Use in source · 3 passages
Search aliases: x_+
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3936
[Flat but inadequate core] Let protected evidence states be x_+=(1,1), x_-=(1,-1), and let shear holonomy leave only the first coordinate flat. Both records project to (1,0), so μ(E^flat;setx_+-x_-)=0. The core is exactly flat but cannot preserve the protected distinction between the two outcomes.
Exact mathematical passage
x_+=(1,1), \qquad x_-=(1,-1),
x_-Use in source · 3 passages
Search aliases: x_-
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3936
[Flat but inadequate core] Let protected evidence states be x_+=(1,1), x_-=(1,-1), and let shear holonomy leave only the first coordinate flat. Both records project to (1,0), so μ(E^flat;setx_+-x_-)=0. The core is exactly flat but cannot preserve the protected distinction between the two outcomes.
Exact mathematical passage
x_+=(1,1), \qquad x_-=(1,-1),
X_0Use in source · 7 passages
Search aliases: X_0
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4296
[Finite descending computation] Set X_0=S and X_k+1=Ψ_∀(X_k). The sequence is descending and stabilizes at I_∀^* after at most absS strict reductions.
Exact mathematical passage
X_0=S
x_0Use in source · 7 passages
Search aliases: x_0
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3681
chain of evaluator morphisms with readouts into a common complete normed score space, E_0xrightarrowe_0 E_1 xrightarrowe_1⋯ xrightarrowe_n-1 E_n. Suppose that for every protected represented state x_t transported from version 0, normJ_t+1(T_tx_t)-J_t(x_t)≤ε_t. Then normJ_n(T_n-1⋯ T_0x_0)-J_0(x_0) ≤ ∑_t=0^n-1ε_t. If ∑_tε_t<∞, then the transported score sequence is Cauchy and converges in every complete score space.
Exact mathematical passage
\norm{J_n(T_{n-1}\cdots T_0x_0)-J_0(x_0)} \leq \sum_{t=0}^{n-1}\eps_t.
X_1Use in source · 1 passage
Search aliases: X_1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21981
Suppose H=H_1⊕ H_2⊕ H_3, dim H_a=2. Under separating block-scalar responses, every commuting operator has form X=X_1⊕ X_2⊕ X_3, X_a∈ M_2(ℝ). Hence dim A_R' =3dim M_2(ℝ) =12. The companion's modal commutant dimension is 12, matching the exact finite theory.
Exact mathematical passage
X=X_1\oplus X_2\oplus X_3, \qquad X_a\in M_2(\R).
X_2Use in source · 1 passage
Search aliases: X_2
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21981
Suppose H=H_1⊕ H_2⊕ H_3, dim H_a=2. Under separating block-scalar responses, every commuting operator has form X=X_1⊕ X_2⊕ X_3, X_a∈ M_2(ℝ). Hence dim A_R' =3dim M_2(ℝ) =12. The companion's modal commutant dimension is 12, matching the exact finite theory.
Exact mathematical passage
X=X_1\oplus X_2\oplus X_3, \qquad X_a\in M_2(\R).
x_2Use in source · 6 passages
Search aliases: x_2
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6537
Let x=(x_1,x_2,x_3,x_4)∈ℝ^4. Define the eight-dimensional base feature ψ(x) = x_1 · x_2 · x_3 · x_4 · x_1x_2 · x_3x_4 · sin x_1 · cos x_2 ^ top ∈ℝ^8, and the affected-stratum indicator s(x)=1{x_1+x_2>0.8}. The full feature map is ϕ(x) = ψ(x)
Exact mathematical passage
x=(x_1,x_2,x_3,x_4)\in\R^4
X_3Use in source · 1 passage
Search aliases: X_3
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21981
Suppose H=H_1⊕ H_2⊕ H_3, dim H_a=2. Under separating block-scalar responses, every commuting operator has form X=X_1⊕ X_2⊕ X_3, X_a∈ M_2(ℝ). Hence dim A_R' =3dim M_2(ℝ) =12. The companion's modal commutant dimension is 12, matching the exact finite theory.
Exact mathematical passage
X=X_1\oplus X_2\oplus X_3, \qquad X_a\in M_2(\R).
x_3Use in source · 3 passages
Search aliases: x_3
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6537
Let x=(x_1,x_2,x_3,x_4)∈ℝ^4. Define the eight-dimensional base feature ψ(x) = x_1 · x_2 · x_3 · x_4 · x_1x_2 · x_3x_4 · sin x_1 · cos x_2 ^ top ∈ℝ^8, and the affected-stratum indicator s(x)=1{x_1+x_2>0.8}. The full feature map is ϕ(x) = ψ(x)
Exact mathematical passage
x=(x_1,x_2,x_3,x_4)\in\R^4
x_4Use in source · 3 passages
Search aliases: x_4
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6537
Let x=(x_1,x_2,x_3,x_4)∈ℝ^4. Define the eight-dimensional base feature ψ(x) = x_1 · x_2 · x_3 · x_4 · x_1x_2 · x_3x_4 · sin x_1 · cos x_2 ^ top ∈ℝ^8, and the affected-stratum indicator s(x)=1{x_1+x_2>0.8}. The full feature map is ϕ(x) = ψ(x)
Exact mathematical passage
x=(x_1,x_2,x_3,x_4)\in\R^4
X_AUse in source · 2 passages
Search aliases: X_A
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3266
The deterministic theorem compares declared defects. A deployed repair is usually evaluated across a distribution of contexts. Let P_A and P_B be source and target context distributions, let S:X_A→ X_B be a declared context transport, and let q_A,q_B be per-context repair gains.
Exact mathematical passage
S:X_A\to X_B
X_aUse in source · 1 passage
Search aliases: X_a
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21981
Suppose H=H_1⊕ H_2⊕ H_3, dim H_a=2. Under separating block-scalar responses, every commuting operator has form X=X_1⊕ X_2⊕ X_3, X_a∈ M_2(ℝ). Hence dim A_R' =3dim M_2(ℝ) =12. The companion's modal commutant dimension is 12, matching the exact finite theory.
Exact mathematical passage
X=X_1\oplus X_2\oplus X_3, \qquad X_a\in M_2(\R).
X_BUse in source · 1 passage
Search aliases: X_B
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3266
The deterministic theorem compares declared defects. A deployed repair is usually evaluated across a distribution of contexts. Let P_A and P_B be source and target context distributions, let S:X_A→ X_B be a declared context transport, and let q_A,q_B be per-context repair gains.
Exact mathematical passage
S:X_A\to X_B
x_i^{\mathsf T}Use in source · 2 passages
Search aliases: x_i^ T mathsf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10744
Let Msucc0 be the current regularized information matrix. Adding candidate anchor x_i with quality weight w_i=σ_i^-2 gives M'=M+w_ix_ix_i^ T.
Exact mathematical passage
M'=M+w_ix_ix_i^{\mathsf T}.
x_jUse in source · 3 passages
Search aliases: x_j
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3774
[Historical ranking preservation] Let x_1,…,x_N be protected historical candidate records with scalar old scores J_v(x_i). Suppose every unequal old pair has margin at least m_hist := min_J_v(x_i)≠ J_v(x_j) absJ_v(x_i)-J_v(x_j)>0. If an upgrade satisfies absJ_w(T_ex_i)-J_v(x_i)≤ε for every i, with 2ε<m_hist, then every strict historical ranking is preserved.
Exact mathematical passage
m_{\mathrm{hist}} := \min_{J_v(x_i)\neq J_v(x_j)} \abs{J_v(x_i)-J_v(x_j)}>0.
X_kUse in source · 1 passage
Search aliases: X_k
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4296
[Finite descending computation] Set X_0=S and X_k+1=Ψ_∀(X_k). The sequence is descending and stabilizes at I_∀^* after at most absS strict reductions.
Exact mathematical passage
X_{k+1}=\Psi_{\forall}(X_k)
x_kUse in source · 2 passages
Search aliases: x_k
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3572
flatness] The exact common fixed space is E^flat=ker A. Let the eigenvalues of A be 0≤λ_1≤⋯≤λ_d. Among all k-dimensional subspaces F⊆ E, the quantity ∑_i=1^kanglesx_i, Ax_i for an orthonormal basis x_1,…,x_k of F is minimized by the span of eigenvectors associated with λ_1,…,λ_k, and the minimum is ∑_i=1^kλ_i.
Exact mathematical passage
x_1,\ldots,x_k
X_{k+1}Use in source · 1 passage
Search aliases: X_k+1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4296
[Finite descending computation] Set X_0=S and X_k+1=Ψ_∀(X_k). The sequence is descending and stabilizes at I_∀^* after at most absS strict reductions.
Exact mathematical passage
X_{k+1}=\Psi_{\forall}(X_k)
x_NUse in source · 1 passage
Search aliases: x_N
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3773
[Historical ranking preservation] Let x_1,…,x_N be protected historical candidate records with scalar old scores J_v(x_i). Suppose every unequal old pair has margin at least m_hist := min_J_v(x_i)≠ J_v(x_j) absJ_v(x_i)-J_v(x_j)>0. If an upgrade satisfies absJ_w(T_ex_i)-J_v(x_i)≤ε for every i, with 2ε<m_hist, then every strict historical ranking is preserved.
Exact mathematical passage
x_1,\ldots,x_N
x_nUse in source · 1 passage
Search aliases: x_n
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 14563
[Median breakdown in one coordinate] Let x_1,…,x_n∈ℝ, and suppose fewer than n/2 observations are replaced arbitrarily. Then the sample median of the contaminated set lies between two uncontaminated order statistics. In particular, an adversary controlling less than half the observations cannot drive the median beyond the range of the retained clean central order statistics.
Exact mathematical passage
x_1,\ldots,x_n\in\R
x_qUse in source · 1 passage
Search aliases: x_q
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16642
bx+c, · x<0. The coefficients belong to finite declared integer sets. A task contains five shuffled input--output demonstrations and one query input. The model receives a normalized serialization of those demonstrations and the query. The protected target is y(τ) = ( e_θ, fracp_12,…,fracp_42, fracd(θ)5, fracf_θ,p(x_q)32 )∈ℝ^10, where e_θ∈ℝ^4 is the one-hot template code, unused parameter slots are zero, and d(θ)∈{2,3,4,5} is the declared execution depth. All demonstrations, query answers, semant…
Exact mathematical passage
y(\tau) = \left( e_{\theta}, \frac{p_1}{2},\ldots,\frac{p_4}{2}, \frac{d(\theta)}{5}, \frac{f_{\theta,p}(x_q)}{32} \right)\in\R^{10}, \label{eq:b18-protected-target}
X_rUse in source · 1 passage
Search aliases: X_r
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10923
For one candidate row r, a configuration commits only if widehat g_rm cal(r) · ≥ τ_g, widehat R_rm cal(r) · ≤ τ_R, widehatε_rm align(r) · ≤ τ_A, λ_min(M_r) · ≥ τ_I, rank X_r · =d. The selected Block 9 grandchild has (τ_g,τ_R,τ_A,τ_I) = (0.02,0.05,0.20,2.0). The recalibrated rank child has (0.02,0.05,0.55,0.10).
Exact mathematical passage
\begin{aligned} \widehat g_{\rm cal}(r)&\geq \tau_g,\\ \widehat R_{\rm cal}(r)&\leq \tau_R,\\ \widehat\eps_{\rm align}(r)&\leq \tau_A,\\ \lambda_{\min}(M_r)&\geq \tau_I,\\ \rank X_r&=d. \end{aligned}
X_SUse in source · 6 passages
Search aliases: X_S
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10540
Let X_S∈ℝ^n× d be the source hidden-state design induced by anchor set S. Rank recovery is saturated when Pr(rank X_S=d)≥ 1-ε_rm rank under the current foreign development distribution and the remaining performance gap cannot be explained by rank failure.
Exact mathematical passage
X_S\in\R^{n\times d}
X_S^{\mathsf T}Use in source · 1 passage
Search aliases: X_S^ T mathsf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10561
This certificate separates two statements: rank X_S=d and λ_min(X_S^ TWX_S)gg0. The first is qualitative identifiability. The second is quantitative stability.
Exact mathematical passage
\lambda_{\min}(X_S^{\mathsf T}WX_S)\gg0.
X_S^\topUse in source · 1 passage
Search aliases: X_S^top top
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21539
does not imply that the repair process has no remaining headroom. Let the target bridge be full rank but have minimum singular value σ_min=10^-4. Random noise of norm 10^-3 can then dominate the weakest identified direction. The bottleneck has migrated from rank to conditioning. A meaningful grandchild should target a criterion such as max_S:|S|=k λ_min(X_S^top X_S) or a regularized determinant objective, not another rank-only rule.
Exact mathematical passage
\max_{S:|S|=k} \lambda_{\min}(X_S^\top X_S)
X_tUse in source · 3 passages
Search aliases: X_t
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 18260
The ordinary control has Y_t=X_t A+b, A∈GL(8), with one shared map at every recurrent step. The inherited step-affine constructor is sufficient. This family tests conservative minimality: a full-trajectory affine map is available and usually eligible, but it should not be selected when the smaller inherited constructor already passes.
Exact mathematical passage
Y_t=X_t A+b, \qquad A\in\mathrm{GL}(8), \label{eq:block21-affine}
x_tUse in source · 5 passages
Search aliases: x_t
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3676
[Cumulative evaluator-drift bound] Consider a chain of evaluator morphisms with readouts into a common complete normed score space, E_0xrightarrowe_0 E_1 xrightarrowe_1⋯ xrightarrowe_n-1 E_n. Suppose that for every protected represented state x_t transported from version 0, normJ_t+1(T_tx_t)-J_t(x_t)≤ε_t. Then normJ_n(T_n-1⋯ T_0x_0)-J_0(x_0) ≤ ∑_t=0^n-1ε_t. If ∑_tε_t<∞, then the transported score sequence is Cauchy and converges in every complete score space.
Exact mathematical passage
x_t
x_{t+1}Use in source · 1 passage
Search aliases: x_t+1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5906
Let the latent world state be x_t∈ℝ^2, action a_t∈ℝ^2, observation y_t∈ℝ^3, and memory state m_t∈ℝ^2. The baseline dynamics are x_t+1=Ax_t+Ba_t, y_t=Cx_t+b_s, m_t+1=λ m_t+(1-λ)y_t,1:2, and the reported evaluator score is q_t=w^top y_t. The redundant observation map is C= 1 · 0
Exact mathematical passage
x_{t+1}=Ax_t+Ba_t,
X_\epsilonUse in source · 3 passages
Search aliases: X_ϵ epsilon
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21764
Let X_ϵ=diag(1,1,ϵ), 0<ϵll1. Then rank X_ϵ=3, but λ_min(X_ϵ^ TX_ϵ)=ϵ^2 and the worst-direction least-squares variance scales as ϵ^-2. A binary rank gate cannot distinguish this design from an orthogonal one.
Exact mathematical passage
X_\epsilon=\diag(1,1,\epsilon), \qquad 0<\epsilon\ll1.
X_\epsilon^{\mathsf T}Use in source · 1 passage
Search aliases: X_ϵ^ T epsilon mathsf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21769
Let X_ϵ=diag(1,1,ϵ), 0<ϵll1. Then rank X_ϵ=3, but λ_min(X_ϵ^ TX_ϵ)=ϵ^2 and the worst-direction least-squares variance scales as ϵ^-2. A binary rank gate cannot distinguish this design from an orthogonal one.
Exact mathematical passage
\lambda_{\min}(X_\epsilon^{\mathsf T}X_\epsilon)=\epsilon^2
x_\tauUse in source · 1 passage
Search aliases: x_τ tau
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13888
A simple acquisition rule is A_i = widehat g_i -λ_rwidehat r_i +λ_uwidehatσ_i +λ_h p_θ(h_imid x_τ) -λ_c C_i, where h_i is the method family, p_θ is an obstruction-class posterior, widehatσ_i is a declared uncertainty bonus, and C_i is experimental cost.
Exact mathematical passage
A_i = \widehat g_i -\lambda_r\widehat r_i +\lambda_u\widehat\sigma_i +\lambda_h p_\theta(h_i\mid x_\tau) -\lambda_c C_i, \label{eq:b13-acquisition}
\Xcal_A^{\mathrm{demo}}Use in source · 1 passage
Search aliases: Xcal_A^demo X_A^demo Xcal mathrm mathcal mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16686
Generator A uses approximately uniform template frequencies. Its demonstration and query supports are X_A^demo={-2,-1,0,1,2}, X_A^query={-3,3}. Generator B changes the template mixture toward quadratic and hinge programs and uses X_B^demo={-3,-1,0,1,3}, X_B^query={-3,-2,2,3}. Every shifted record remains inside the same exact program language. Nevertheless, every shifted task contains at least one normalized feature coordinate outside the coordinatewise range of the primary support, and the larg…
Exact mathematical passage
\Xcal_A^{\mathrm{demo}}=\{-2,-1,0,1,2\}, \qquad \Xcal_A^{\mathrm{query}}=\{-3,3\}.
\Xcal_A^{\mathrm{query}}Use in source · 1 passage
Search aliases: Xcal_A^query X_A^query Xcal mathrm mathcal mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16686
Generator A uses approximately uniform template frequencies. Its demonstration and query supports are X_A^demo={-2,-1,0,1,2}, X_A^query={-3,3}. Generator B changes the template mixture toward quadratic and hinge programs and uses X_B^demo={-3,-1,0,1,3}, X_B^query={-3,-2,2,3}. Every shifted record remains inside the same exact program language. Nevertheless, every shifted task contains at least one normalized feature coordinate outside the coordinatewise range of the primary support, and the larg…
Exact mathematical passage
\Xcal_A^{\mathrm{demo}}=\{-2,-1,0,1,2\}, \qquad \Xcal_A^{\mathrm{query}}=\{-3,3\}.
\Xcal_B^{\mathrm{demo}}Use in source · 1 passage
Search aliases: Xcal_B^demo X_B^demo Xcal mathrm mathcal mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16692
Generator A uses approximately uniform template frequencies. Its demonstration and query supports are X_A^demo={-2,-1,0,1,2}, X_A^query={-3,3}. Generator B changes the template mixture toward quadratic and hinge programs and uses X_B^demo={-3,-1,0,1,3}, X_B^query={-3,-2,2,3}. Every shifted record remains inside the same exact program language. Nevertheless, every shifted task contains at least one normalized feature coordinate outside the coordinatewise range of the primary support, and the larg…
Exact mathematical passage
\Xcal_B^{\mathrm{demo}}=\{-3,-1,0,1,3\}, \qquad \Xcal_B^{\mathrm{query}}=\{-3,-2,2,3\}.
\Xcal_B^{\mathrm{query}}Use in source · 1 passage
Search aliases: Xcal_B^query X_B^query Xcal mathrm mathcal mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16692
Generator A uses approximately uniform template frequencies. Its demonstration and query supports are X_A^demo={-2,-1,0,1,2}, X_A^query={-3,3}. Generator B changes the template mixture toward quadratic and hinge programs and uses X_B^demo={-3,-1,0,1,3}, X_B^query={-3,-2,2,3}. Every shifted record remains inside the same exact program language. Nevertheless, every shifted task contains at least one normalized feature coordinate outside the coordinatewise range of the primary support, and the larg…
Exact mathematical passage
\Xcal_B^{\mathrm{demo}}=\{-3,-1,0,1,3\}, \qquad \Xcal_B^{\mathrm{query}}=\{-3,-2,2,3\}.
\xhookrightarrowUse in source · 1 passage
Search aliases: xhookrightarrow
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4983
Consider a chain G_0 xhookrightarrowι_0 G_1 xhookrightarrowι_1 G_2 xhookrightarrow ⋯ of conservative extensions.
Exact mathematical passage
\Gcal_0 \xhookrightarrow{\iota_0} \Gcal_1 \xhookrightarrow{\iota_1} \Gcal_2 \xhookrightarrow{} \cdots
y'_{m'}Use in source · 1 passage
Search aliases: y'_m'
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16173
Let d_prog(τ,τ') be a normalized distance between (A_m,B_m) and (A'_m',B'_m'), and let d_ans(τ,τ') = absy_m/4-y'_m'/4. For a recipient task τ_i, define four donor classes: item SAP: same-answer, far-program, approximately depth-matched donors; item FAD: far-answer, depth-matched donors; item SN: semantic-neighbor donors; item RMD: random depth-matched donors. Three donors per available class are selected by deterministic task-identifier hashing. Each audit records donor coverage and the realized…
Exact mathematical passage
d_{\mathrm{ans}}(\tau,\tau') = \abs{y_m/4-y'_{m'}/4}. \label{eq:b17-answer-distance}
y^*Use in source · 3 passages
Search aliases: y^*
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1613
norm y≤1 anglesy,r. If r_ob≠0, the maximum is attained by y^*=fracr_obnormr_ob.
Exact mathematical passage
y^*=\frac{r_{\mathrm{ob}}}{\norm{r_{\mathrm{ob}}}}.
y^\starUse in source · 2 passages
Search aliases: y^⋆ star
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11769
For current target state y, candidate repaired state widetilde y, and oracle repaired state y^⋆, define G = clip_[-1,1] ( 1- frac Enormwidetilde y-y^⋆^2 Enormy-y^⋆^2 ). The witness records: item validation gain; item source-to-target alignment NRMSE; item decoded protected-context regression; item minimum chart support; item alignment reduction relative to the global bridge; item source-gate and target-gate agreement; item target-gate margin.
Exact mathematical passage
y^\star
y_0Use in source · 2 passages
Search aliases: y_0
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17190
Suppose a source context 0 and a target context c satisfy x_c=s_xx_0+t_x, y_c=s_yy_0+t_y, for all paired semantic-twin demonstrations.
Exact mathematical passage
x_c=s_xx_0+t_x, \qquad y_c=s_yy_0+t_y, \label{eq:b19-affine-rendering}
Y_1Use in source · 1 passage
Search aliases: Y_1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 18304
The latent-clock family first applies an invertible stepwise affine gauge and then mixes adjacent recurrent times: Y_t = (1-β)U_t+β U_t-1, t=2,…,5, with Y_1=U_1 and 0<β<1. In matrix form, Y=W_βU, where W_β is lower bidiagonal. The temporal constructor applies W_β^-1 before fitting the spatial gauge.
Exact mathematical passage
Y_1=U_1
y_1Use in source · 4 passages
Search aliases: y_1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4081
[Trace-schema attack surface] Witness separation is ineffective when distinct full executions with materially different external consequences map to the same declared trace. The factorization criterion should therefore be tested by adversarially searching for pairs y_1,y_2 satisfying q(y_1)=q(y_2), b(y_1)≠ b(y_2). Such a pair is a constructive certificate of trace incompleteness.
Exact mathematical passage
y_1,y_2
y_2Use in source · 4 passages
Search aliases: y_2
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4081
[Trace-schema attack surface] Witness separation is ineffective when distinct full executions with materially different external consequences map to the same declared trace. The factorization criterion should therefore be tested by adversarially searching for pairs y_1,y_2 satisfying q(y_1)=q(y_2), b(y_1)≠ b(y_2). Such a pair is a constructive certificate of trace incompleteness.
Exact mathematical passage
y_1,y_2
y_3Use in source · 1 passage
Search aliases: y_3
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6567
Let q(x)=Wψ(x)+b∈ℝ^3 be the protected score. Let C be the cyclic score permutation C(y_1,y_2,y_3)^top=(y_3,y_1,y_2)^top. The corrupted score is widetilde q(x) = (1-s(x))q(x)+s(x)Cq(x). Thus the model is almost perfectly correct outside the affected stratum and systematically wrong inside it.
Exact mathematical passage
C(y_1,y_2,y_3)^{\top}=(y_3,y_1,y_2)^{\top}.
Y_AUse in source · 1 passage
Search aliases: Y_A
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 499
For an agent A, define the resource-normalized utility J_Ξ(A) = E_z∼μ [ ν(z,Y_A(z)) - λbar c(z,Y_A(z)) ]. The definition does not require scalar utility to encode every scientific judgment. It supplies a declared comparison functional for one experimental claim. Multiple protected contexts may be used simultaneously, producing a Pareto relation instead of a total order.
Exact mathematical passage
J_\Xi(A) = \mathbb E_{z\sim\mu} \left[ \nu\bigl(z,Y_A(z)\bigr) - \lambda\bar c\bigl(z,Y_A(z)\bigr) \right].
Y_aUse in source · 3 passages
Search aliases: Y_a
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 14478
This theorem gives a precise reason to value experiments rather than static classification. Suppose an experiment a returns an observation Y_a whose distribution differs between the aliased hypotheses. The compiler can then choose a sequence a_1,a_2,…,a_t that increases the information available for diagnosis.
Exact mathematical passage
Y_a
y_cUse in source · 1 passage
Search aliases: y_c
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17190
Suppose a source context 0 and a target context c satisfy x_c=s_xx_0+t_x, y_c=s_yy_0+t_y, for all paired semantic-twin demonstrations.
Exact mathematical passage
x_c=s_xx_0+t_x, \qquad y_c=s_yy_0+t_y, \label{eq:b19-affine-rendering}
Y_fUse in source · 1 passage
Search aliases: Y_f
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9889
The minimal conflicts form a finite hypergraph H_G. A candidate feature or probe f:X→ Y_f resolves a conflict S when f is not constant on S.
Exact mathematical passage
f:X\to Y_f
Y_iUse in source · 3 passages
Search aliases: Y_i
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 18362
The destructive family withholds four protected coordinates and constructs pairs i,j satisfying Y_i=Y_j, Z_i≠ Z_j. The compiler must discover these repeated observations from the public arrays. Their private construction pairs are not supplied.
Exact mathematical passage
Y_i=Y_j, \qquad Z_i\neq Z_j. \label{eq:block21-collision}
Y_jUse in source · 5 passages
Search aliases: Y_j
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6241
For foreign agent j, let Y_j(z) be protected gain under factor setting z. The average necessity effect of factor k relative to the intact architecture 1 is τ_k = E [ Y_j(1)-Y_j(1-e_k) ]. A positive value supports causal usefulness only relative to the frozen environment, compensating mechanisms, and matched resource schedule.
Exact mathematical passage
Y_j(z)
y_{\rm known}Use in source · 2 passages
Search aliases: y_rm known rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 14535
[Open-set separation] Suppose every inherited known response y_rm known satisfies η_rm open(y_rm known)≤ε, while every unknown response in a target family satisfies η_rm open(y_rm unknown)≥γ for γ>ε. Then every threshold τ∈(ε,γ) perfectly separates the two families.
Exact mathematical passage
y_{\rm known}
y_{\rm unknown}Use in source · 1 passage
Search aliases: y_rm unknown rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 14540
[Open-set separation] Suppose every inherited known response y_rm known satisfies η_rm open(y_rm known)≤ε, while every unknown response in a target family satisfies η_rm open(y_rm unknown)≥γ for γ>ε. Then every threshold τ∈(ε,γ) perfectly separates the two families.
Exact mathematical passage
\eta_{\rm open}(y_{\rm unknown})\geq\gamma
Y_tUse in source · 9 passages
Search aliases: Y_t
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 18260
The ordinary control has Y_t=X_t A+b, A∈GL(8), with one shared map at every recurrent step. The inherited step-affine constructor is sufficient. This family tests conservative minimality: a full-trajectory affine map is available and usually eligible, but it should not be selected when the smaller inherited constructor already passes.
Exact mathematical passage
Y_t=X_t A+b, \qquad A\in\mathrm{GL}(8), \label{eq:block21-affine}
y_tUse in source · 3 passages
Search aliases: y_t
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5905
Let the latent world state be x_t∈ℝ^2, action a_t∈ℝ^2, observation y_t∈ℝ^3, and memory state m_t∈ℝ^2. The baseline dynamics are x_t+1=Ax_t+Ba_t, y_t=Cx_t+b_s, m_t+1=λ m_t+(1-λ)y_t,1:2, and the reported evaluator score is q_t=w^top y_t. The redundant observation map is C= 1 · 0
Exact mathematical passage
y_t\in\R^3
Y_{t+1}Use in source · 1 passage
Search aliases: Y_t+1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1143
A useful self-model participates in a complete causal cycle: ϕ_t(A_t) xrightarrowcandidate policy δ_t xrightarrowworld execution Y_t+1 xrightarrowevidence update ϕ_t+1(A_t+1). The candidate is selected before the protected outcome is revealed. The outcome affects future beliefs and repair policy. Cutting either planner use or consequence feedback destroys prospective closure even if the internal self-representation remains decodable.
Exact mathematical passage
\phi_t(A_t) \xrightarrow{\text{candidate policy}} \delta_t \xrightarrow{\text{world execution}} Y_{t+1} \xrightarrow{\text{evidence update}} \phi_{t+1}(A_{t+1}).
y_{t,1}Use in source · 1 passage
Search aliases: y_t,1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5929
1 · 1 , so the protected consistency relation is y_t,3=y_t,1+y_t,2 when the sensor channel is intact.
Exact mathematical passage
y_{t,3}=y_{t,1}+y_{t,2}
y_{t,1:2}Use in source · 1 passage
Search aliases: y_t,1:2
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5912
Let the latent world state be x_t∈ℝ^2, action a_t∈ℝ^2, observation y_t∈ℝ^3, and memory state m_t∈ℝ^2. The baseline dynamics are x_t+1=Ax_t+Ba_t, y_t=Cx_t+b_s, m_t+1=λ m_t+(1-λ)y_t,1:2, and the reported evaluator score is q_t=w^top y_t. The redundant observation map is C= 1 · 0
Exact mathematical passage
m_{t+1}=\lambda m_t+(1-\lambda)y_{t,1:2},
y_{t,2}Use in source · 1 passage
Search aliases: y_t,2
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5929
1 · 1 , so the protected consistency relation is y_t,3=y_t,1+y_t,2 when the sensor channel is intact.
Exact mathematical passage
y_{t,3}=y_{t,1}+y_{t,2}
y_{t,3}Use in source · 1 passage
Search aliases: y_t,3
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5929
1 · 1 , so the protected consistency relation is y_t,3=y_t,1+y_t,2 when the sensor channel is intact.
Exact mathematical passage
y_{t,3}=y_{t,1}+y_{t,2}
z'Use in source · 6 passages
Search aliases: z'
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5982
Let μ_z=ϕ(z) be the frozen event prototype. With equal priors and matched isotropic noise, the posterior is p(zmidwidetildeϕ) = fracexp(-normwidetildeϕ-μ_z^2/(2σ^2)) ∑_z'exp(-normwidetildeϕ-μ_z'^2/(2σ^2)). A repair is attempted only when max_z p(zmidwidetildeϕ)≥θ and the maximizing event is not none. Otherwise the system abstains.
Exact mathematical passage
p(z\mid\widetilde\phi) = \frac{\exp\left(-\norm{\widetilde\phi-\mu_z}^2/(2\sigma^2)\right)} {\sum_{z'}\exp\left(-\norm{\widetilde\phi-\mu_{z'}}^2/(2\sigma^2)\right)}.
z'^{\top}Use in source · 1 passage
Search aliases: z'^top top
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13128
Every quadratic form z^top Bz becomes z'^top(Q^top BQ)z'. The vector space of symmetric quadratic forms is therefore carried isomorphically to itself under the induced action of Q on Sym^2(P^top H).
Exact mathematical passage
z'^{\top}(Q^\top BQ)z'.
Z^\topUse in source · 5 passages
Search aliases: Z^top top
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7396
For regularization parameters λ_u,λ_r>0, define widehatβ_u · = (Z^top Z+λ_u I)^-1Z^top y, widehatβ_r · = (Z^top Z+λ_r I)^-1Z^top r. For candidate type c, let s_u,c and s_r,c be the development residual standard deviations for the utility and risk heads. The deployed scores are widehat U_rm LCB(z) · = widehat U(z)-κ_u s_u,c(z),
Exact mathematical passage
\widehat\beta_u &= \left(Z^\top Z+\lambda_u I\right)^{-1}Z^\top y, \label{eq:block07-utility-head}\\ \widehat\beta_r &= \left(Z^\top Z+\lambda_r I\right)^{-1}Z^\top r. \label{eq:block07-risk-head}
z^\topUse in source · 5 passages
Search aliases: z^top top
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13060
Let P be the recovered repair packet, let z=P^top x, and consider a target correction F(x) = Ax+b+(z^top Bz)v, where B=B^top≠0 and v≠0.
Exact mathematical passage
F(x) = Ax+b+\bigl(z^\top Bz\bigr)v, \label{eq:b12-quadratic-repair}
z_0Use in source · 4 passages
Search aliases: z_0
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5298
For a time-varying sequence M_t, let z_t be a row vector of expected schema counts. Under the linear mean model, E z_T = z_0M_0M_1⋯ M_T-1. Define the upper mean-growth exponent λ_mean = limsup_T→∞ frac1T lognormM_0M_1⋯ M_T-1. A positive value indicates exponential growth of the linearized expected count. It does not imply almost-sure survival under dependence or resource competition.
Exact mathematical passage
\mathbb E z_T = z_0M_0M_1\cdots M_{T-1}.
z_0'Use in source · 1 passage
Search aliases: z_0'
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16515
Inductively, z_0'=Pz_0. If z_k'=Pz_k, then z_k+1' = tanh (PRP^ TPz_k+Ps_k+PC_α) =Ptanh(Rz_k+s_k+C_α) =Pz_k+1, because signed permutations commute with coordinatewise tanh. For every latent-consuming map M'=MP^ T, one has M'z_k'=Mz_k. Therefore the gates, hidden candidates, readouts, semantic prediction, and halt logits coincide. The state tensors nevertheless occupy different coordinates, so their serialized state hash can differ.
Exact mathematical passage
z_0'=Pz_0
Z_1Use in source · 2 passages
Search aliases: Z_1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3344
[Paired held-out gain certificate] Let Z_1,…,Z_n be independent and identically distributed paired target-context gains for a precommitted transported repair, with -G≤ Z_i≤ G. Let overline Z_n=n^-1∑_iZ_i. Then, with probability at least 1-α, E Z_i ≥ overline Z_n -Gsqrtfrac2log(1/α)n. For a precommitted family of m repairs, replacing α by α/m gives a simultaneous lower bound for all m candidates.
Exact mathematical passage
Z_1,\ldots,Z_n
z_1^2Use in source · 2 passages
Search aliases: z_1^2
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 22080
Nonzero Hessian defeats every affine composition. For z∼ N(0,I_2) and q(z)=z_1^2-z_2^2, every affine predictor has residual variance at least 4. The degree-two feature vector (z_1^2,z_1z_2,z_2^2) represents q exactly. No increase in affine depth changes this conclusion because affine maps remain closed under composition.
Exact mathematical passage
q(z)=z_1^2-z_2^2
z_2Use in source · 7 passages
Search aliases: z_2
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 14459
[Static diagnostic indistinguishability] Let z_1≠ z_2 satisfy P(X_0∈ Bmid Z=z_1) = P(X_0∈ Bmid Z=z_2) for every measurable set B. Then no decision rule d(X_0) distinguishes z_1 from z_2 with accuracy exceeding the larger prior probability: P(d(X_0)=Zmid Z∈{z_1,z_2}) ≤ max{π_1,π_2}, where π_j= P(Z=z_jmid Z∈{z_1,z_2}).
Exact mathematical passage
z_1\neq z_2
z_2^2Use in source · 2 passages
Search aliases: z_2^2
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 22080
Nonzero Hessian defeats every affine composition. For z∼ N(0,I_2) and q(z)=z_1^2-z_2^2, every affine predictor has residual variance at least 4. The degree-two feature vector (z_1^2,z_1z_2,z_2^2) represents q exactly. No increase in affine depth changes this conclusion because affine maps remain closed under composition.
Exact mathematical passage
q(z)=z_1^2-z_2^2
Z_AUse in source · 3 passages
Search aliases: Z_A
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2832
Let protected linear invariants be represented by C_A:U_A→ Z_A, C_B:U_B→ Z_B. A correction preserves the invariant when C_Au=0 or C_Bu=0.
Exact mathematical passage
C_A:U_A\to Z_A, \qquad C_B:U_B\to Z_B.
Z_BUse in source · 2 passages
Search aliases: Z_B
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2832
Let protected linear invariants be represented by C_A:U_A→ Z_A, C_B:U_B→ Z_B. A correction preserves the invariant when C_Au=0 or C_Bu=0.
Exact mathematical passage
C_A:U_A\to Z_A, \qquad C_B:U_B\to Z_B.
Z_CUse in source · 1 passage
Search aliases: Z_C
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6234
Let the executable architecture have binary factors Z =(Z_C,Z_P,Z_D,Z_T,Z_W,Z_A,Z_M), representing causal attribution, packet localization, descent checking, transport, witness separation, archive retention, and repair abstraction.
Exact mathematical passage
Z =(Z_C,Z_P,Z_D,Z_T,Z_W,Z_A,Z_M),
Z_DUse in source · 1 passage
Search aliases: Z_D
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6234
Let the executable architecture have binary factors Z =(Z_C,Z_P,Z_D,Z_T,Z_W,Z_A,Z_M), representing causal attribution, packet localization, descent checking, transport, witness separation, archive retention, and repair abstraction.
Exact mathematical passage
Z =(Z_C,Z_P,Z_D,Z_T,Z_W,Z_A,Z_M),
z_{e,c}Use in source · 2 passages
Search aliases: z_e,c
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7368
Each candidate receipt z_e,c is mapped to a feature vector x(z_e,c)∈ℝ^23. The released feature schema contains: item a constant coordinate; item logarithmic anchor and calibration sizes; item target representation width; item alignment, transport-condition, and repair-equivariance diagnostics; item packet residual and relative repair magnitude; item repair rank; item calibration baseline slice and protected accuracy; item calibration gain, protected regression, and protected utility; item five c…
Exact mathematical passage
z_{e,c}
Z_{\Gcal}Use in source · 3 passages
Search aliases: Z_Gcal Z_G Gcal mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5104
Let ϕ_G:X→ Z_G be the state representation expressible in grammar G. A G-measurable deterministic repair policy has the form π=widetildeπ∘ϕ_G.
Exact mathematical passage
\phi_{\Gcal}:X\to Z_{\Gcal}
Z_iUse in source · 12 passages
Search aliases: Z_i
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3345
[Paired held-out gain certificate] Let Z_1,…,Z_n be independent and identically distributed paired target-context gains for a precommitted transported repair, with -G≤ Z_i≤ G. Let overline Z_n=n^-1∑_iZ_i. Then, with probability at least 1-α, E Z_i ≥ overline Z_n -Gsqrtfrac2log(1/α)n. For a precommitted family of m repairs, replacing α by α/m gives a simultaneous lower bound for all m candidates.
Exact mathematical passage
-G\leq Z_i\leq G.
z_iUse in source · 7 passages
Search aliases: z_i
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4489
For maximization coordinates, write Asucceq B when z_i(A)≥ z_i(B) for every i. The Pareto frontier is the set of undominated nodes.
Exact mathematical passage
z_i(A)\geq z_i(B) \qquad\text{for every }i.
Z_jUse in source · 4 passages
Search aliases: Z_j
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16224
r_i · =absℓ_i(replay(Z_i))-ℓ_i(Z_i). Noise is deterministic and norm matched at every step. The program and answer specificity coordinates are the mean recipient costs under SAP and FAD donors: p_i = E_j∈SAP(i) [ℓ_i(Z_j)-ℓ_i(Z_i)], a_i = E_j∈FAD(i) [ℓ_i(Z_j)-ℓ_i(Z_i)]. Define Q_prog = fracoverline pmax{overline o,overline c,ε}, Q_ans = fracoverline amax{overline o,overline c,ε}. The frozen ratio thresholds are 0.15 and 0.20, respectively. Each specificity gate additionally requires a positive 95…
Exact mathematical passage
p_i = \mathbb E_{j\in\mathsf{SAP}(i)} \left[\ell_i(Z_j)-\ell_i(Z_i)\right], \qquad a_i = \mathbb E_{j\in\mathsf{FAD}(i)} \left[\ell_i(Z_j)-\ell_i(Z_i)\right]. \label{eq:b17-specificity-costs}
z_jUse in source · 1 passage
Search aliases: z_j
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 14471
nostic-indistinguishability Let z_1≠ z_2 satisfy P(X_0∈ Bmid Z=z_1) = P(X_0∈ Bmid Z=z_2) for every measurable set B. Then no decision rule d(X_0) distinguishes z_1 from z_2 with accuracy exceeding the larger prior probability: P(d(X_0)=Zmid Z∈{z_1,z_2}) ≤ max{π_1,π_2}, where π_j= P(Z=z_jmid Z∈{z_1,z_2}).
Exact mathematical passage
\pi_j=\mathbb P(Z=z_j\mid Z\in\{z_1,z_2\})
z_kUse in source · 10 passages
Search aliases: z_k
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16093
Let ϕ(τ)∈ℝ^19 be the padded task serialization, and let e=E_θ(ϕ(τ))∈ℝ^40 be the prompt representation. The latent trace has four states. Given conditioning mode α, define z_0=ζ, z_k+1 = tanh (Rz_k+s_k+C_α(e,m)), 0≤ k<4, where s_k is a learned step embedding. The task-blind parent satisfies C_parent(e,m)=0. The descendants use one of the following conditioning maps: C_depth(e,m) · =D 1_m,
Exact mathematical passage
z_0=\zeta, \qquad z_{k+1} = \tanh\!\left(Rz_k+s_k+C_\alpha(e,m)\right), \qquad 0\leq k<4, \label{eq:b17-trace}
z_k'Use in source · 3 passages
Search aliases: z_k'
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16507
Let P be a signed permutation matrix on the latent space. Define transformed latent parameters ζ'=Pζ, R'=PRP^ T, s_k'=Ps_k, C_α'=PC_α, and transform every latent-consuming linear map by right multiplication with P^ T. Then the transformed model has z_k'=Pz_k and identical external outputs.
Exact mathematical passage
z_k'=Pz_k
z_{k+1}Use in source · 5 passages
Search aliases: z_k+1
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16093
Let ϕ(τ)∈ℝ^19 be the padded task serialization, and let e=E_θ(ϕ(τ))∈ℝ^40 be the prompt representation. The latent trace has four states. Given conditioning mode α, define z_0=ζ, z_k+1 = tanh (Rz_k+s_k+C_α(e,m)), 0≤ k<4, where s_k is a learned step embedding. The task-blind parent satisfies C_parent(e,m)=0. The descendants use one of the following conditioning maps: C_depth(e,m) · =D 1_m,
Exact mathematical passage
z_0=\zeta, \qquad z_{k+1} = \tanh\!\left(Rz_k+s_k+C_\alpha(e,m)\right), \qquad 0\leq k<4, \label{eq:b17-trace}
z_{k+1}'Use in source · 1 passage
Search aliases: z_k+1'
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16516
Inductively, z_0'=Pz_0. If z_k'=Pz_k, then z_k+1' = tanh (PRP^ TPz_k+Ps_k+PC_α) =Ptanh(Rz_k+s_k+C_α) =Pz_k+1, because signed permutations commute with coordinatewise tanh. For every latent-consuming map M'=MP^ T, one has M'z_k'=Mz_k. Therefore the gates, hidden candidates, readouts, semantic prediction, and halt logits coincide. The state tensors nevertheless occupy different coordinates, so their serialized state hash can differ.
Exact mathematical passage
z_{k+1}' = \tanh\!\left(PRP^{\mathsf T}Pz_k+Ps_k+PC_\alpha\right) =P\tanh(Rz_k+s_k+C_\alpha) =Pz_{k+1},
Z_nUse in source · 2 passages
Search aliases: Z_n
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3344
[Paired held-out gain certificate] Let Z_1,…,Z_n be independent and identically distributed paired target-context gains for a precommitted transported repair, with -G≤ Z_i≤ G. Let overline Z_n=n^-1∑_iZ_i. Then, with probability at least 1-α, E Z_i ≥ overline Z_n -Gsqrtfrac2log(1/α)n. For a precommitted family of m repairs, replacing α by α/m gives a simultaneous lower bound for all m candidates.
Exact mathematical passage
Z_1,\ldots,Z_n
Z_PUse in source · 1 passage
Search aliases: Z_P
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6234
Let the executable architecture have binary factors Z =(Z_C,Z_P,Z_D,Z_T,Z_W,Z_A,Z_M), representing causal attribution, packet localization, descent checking, transport, witness separation, archive retention, and repair abstraction.
Exact mathematical passage
Z =(Z_C,Z_P,Z_D,Z_T,Z_W,Z_A,Z_M),
Z_{\rm prot}Use in source · 1 passage
Search aliases: Z_rm prot rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 18344
The benign compression exposes a thirty-dimensional statistic containing all nine protected semantic coordinates and twenty-one nuisance projections, followed by an invertible mixing. Since thirty is less than the forty-dimensional packet dimension, complete packet reconstruction is neither required nor generally possible. The appropriate constructor is a semantic quotient YxrightarrowqZ_rm prot, not an artificial lift into all source packet coordinates.
Exact mathematical passage
Y\xrightarrow{q}Z_{\rm prot}, \label{eq:block21-quotient}
Z_TUse in source · 1 passage
Search aliases: Z_T
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6234
Let the executable architecture have binary factors Z =(Z_C,Z_P,Z_D,Z_T,Z_W,Z_A,Z_M), representing causal attribution, packet localization, descent checking, transport, witness separation, archive retention, and repair abstraction.
Exact mathematical passage
Z =(Z_C,Z_P,Z_D,Z_T,Z_W,Z_A,Z_M),
z_TUse in source · 1 passage
Search aliases: z_T
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5298
For a time-varying sequence M_t, let z_t be a row vector of expected schema counts. Under the linear mean model, E z_T = z_0M_0M_1⋯ M_T-1. Define the upper mean-growth exponent λ_mean = limsup_T→∞ frac1T lognormM_0M_1⋯ M_T-1. A positive value indicates exponential growth of the linearized expected count. It does not imply almost-sure survival under dependence or resource competition.
Exact mathematical passage
\mathbb E z_T = z_0M_0M_1\cdots M_{T-1}.
z_tUse in source · 6 passages
Search aliases: z_t
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5297
For a time-varying sequence M_t, let z_t be a row vector of expected schema counts. Under the linear mean model, E z_T = z_0M_0M_1⋯ M_T-1. Define the upper mean-growth exponent λ_mean = limsup_T→∞ frac1T lognormM_0M_1⋯ M_T-1. A positive value indicates exponential growth of the linearized expected count. It does not imply almost-sure survival under dependence or resource competition.
Exact mathematical passage
z_t
Z_WUse in source · 1 passage
Search aliases: Z_W
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6234
Let the executable architecture have binary factors Z =(Z_C,Z_P,Z_D,Z_T,Z_W,Z_A,Z_M), representing causal attribution, packet localization, descent checking, transport, witness separation, archive retention, and repair abstraction.
Exact mathematical passage
Z =(Z_C,Z_P,Z_D,Z_T,Z_W,Z_A,Z_M),
Z_XUse in source · 2 passages
Search aliases: Z_X
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 722
Let Q_X:X→ Z_X and Q_Y:Y→ Z_Y be representations of the same declared property in two strata, and let widetilde T_Q:Z_X→ Z_Y be its intended transport.
Exact mathematical passage
Q_X:X\to Z_X
Z_YUse in source · 2 passages
Search aliases: Z_Y
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 722
Let Q_X:X→ Z_X and Q_Y:Y→ Z_Y be representations of the same declared property in two strata, and let widetilde T_Q:Z_X→ Z_Y be its intended transport.
Exact mathematical passage
Q_Y:Y\to Z_Y
\Zcal_0Use in source · 1 passage
Search aliases: Zcal_0 Z_0 Zcal mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15068
Let ϕ_0:T→Z_0 be the inherited diagnostic map. In Block 15, six inherited obstruction classes retain their existing tokens, while two previously unseen obstruction mechanisms are both mapped to one open-set token bot: ϕ_0(τ_ m int) = ϕ_0(τ_ m temp) = bot. The equality in [source reference] is not itself a defect. An unknown token is an appropriate abstention state until the lineage possesses external evidence that the collapsed cases require incompatible research actions.
Exact mathematical passage
\phi_0:\Tau\to\Zcal_0
|^2Use in source · 1 passage
Search aliases: |^2
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21627
Let the old correction image be I=span{e_1,e_2}⊂ℝ^4 and let r=(0,0,3,4). Then z=P_I^⊥r=(0,0,3,4), η_I(r)^2=25. Add generator v=2e_4. Because v∈ I^⊥, frac|anglesz,v|^2norm v^2 = frac8^24 =16. The expanded residual is 3e_3, so η_I'(r)^2=9=25-16. A generator parallel to e_1 would have bar v=0 and would produce no structural reduction despite having nonzero raw norm.
Exact mathematical passage
\frac{|\angles{z,v}|^2}{\norm v^2} = \frac{8^2}{4} =16.
|_{U_{\alpha\beta}}Use in source · 1 passage
Search aliases: |_U_αβ alpha beta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2293
On every overlap, D(d_αβ) =-r|_U_αβ+r|_U_αβ=0. On triple overlaps, d_αβ+d_βγ+d_γα=0, so the family is a cocycle.
Exact mathematical passage
D(d_{\alpha\beta}) =-r|_{U_{\alpha\beta}}+r|_{U_{\alpha\beta}}=0.
\pmUse in source · 2 passages
Search aliases: ± pm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20791
For F(ε,x)=x^2-ε, the derivative with respect to x at the origin is zero, so the entire scalar equation is the Lyapunov--Schmidt reduced obstruction. For ε>0 there are two branches x=±sqrtε, for ε=0 there is one singular branch, and for ε<0 there is no real repair. A linearized system cannot distinguish these three regimes.
Exact mathematical passage
x=\pm\sqrt\eps,
\cdotUse in source · 6 passages
Search aliases: · cdot
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 942
Let D_t be an authorized modification family containing a no-op intervention δ_0. For δ∈D_t, let K_A_t^δ(·) = P(Y∈·mid do(δ),A_t) be the protected external-outcome kernel. A compressed causal self-model ϕ_t(A_t) induces a predicted kernel widehat K_ϕ_t(A_t)^δ(·). The model is evaluated on counterfactual consequences, not on its ability to reproduce hidden activations for their own sake.
Exact mathematical passage
K_{A_t}^{\delta}(\cdot) = P\left(Y\in\cdot\mid \operatorname{do}(\delta),A_t\right)
\alpha_iUse in source · 2 passages
Search aliases: α_i alpha
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6303
Let H_1,…,H_m be ordered primary hypotheses. Choose nonnegative levels α_i satisfying ∑_i=1^mα_i≤α. Each endpoint is tested only with its assigned held-out evidence. Reusing the same audit sample to tune thresholds and report final confidence invalidates the nominal bound.
Exact mathematical passage
\alpha_i
\beta_kUse in source · 2 passages
Search aliases: β_k beta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16735
The source grammar contains a task-blind continuation, inherited additive conditioning, final-step FiLM, progressive FiLM with geometry, a low-rank hypernetwork, and a shuffled-witness FiLM control. For the selected FiLM constructor, the encoded task produces step-indexed scale and shift vectors (γ_k(e),β_k(e))_k=0^4 =F_FiLM(e), which act on the preactivation by q_k ⟼ q_kodot(1+frac12tanhγ_k(e)) +β_k(e). This constructor is absent from Block 17's rank-one, rank-four, and full additive conditioni…
Exact mathematical passage
(\gamma_k(e),\beta_k(e))_{k=0}^{4} =F_{\mathrm{FiLM}}(e), \label{eq:b18-film}
\beta_tUse in source · 3 passages
Search aliases: β_t beta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10269
Let R_t be true protected regression and let U_t^R be a conditional upper confidence bound. Require U_t^R≤β_t, ∑_tβ_t≤ B_R. Let d_t be semantic debt incurred by approximate grammar or evaluator transport, with ∑_td_t≤ B_sem. A lineage certificate should therefore report three separate cumulative quantities: ( ∑_tδ_t, ∑_tβ_t, ∑_td_t ). A positive gain sequence is not sufficient if its protected regression or semantic debt diverges.
Exact mathematical passage
U_t^R\leq\beta_t, \qquad \sum_t\beta_t\leq B_R. \label{eq:b8-regression-budget}
\gamma_1Use in source · 2 passages
Search aliases: γ_1 gamma
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3490
[Finite computation from a cycle basis] Let G be finite. Choose a spanning tree and let γ_1,…,γ_m be the fundamental loops generated by non-tree edges. If the loop transports form a representation of the graph fundamental group, then E_v_0^flat = ⋂_j=1^m ker(H_γ_j-I). Equivalently, E_v_0^flat is the null space of the vertically stacked matrix H_γ_1-I
Exact mathematical passage
\gamma_1,\ldots,\gamma_m
\Gamma_1^{\rm recal}Use in source · 4 passages
Search aliases: Γ_1^rm recal Gamma rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 14372
Block 14 therefore holds the candidate-method grammar and full-experiment budget fixed while changing the reliability conditions under which research decisions must be made. The published Block 13 result digest and exact 39-method language are replayed. Its compiler design is retained as the inherited parent and recalibrated on the Block 14 development archive: Γ_1^rm recal. A robust descendant Γ_1^rm recalxrightarrowΠ_8^ΓΓ_2 is compiled before either final audit is opened. The child adds sequen…
Exact mathematical passage
\Gamma_1^{\rm recal}.
\gamma_aUse in source · 8 passages
Search aliases: γ_a gamma
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12288
In an executable system, the responses are estimated from finite intervention data. Let widehat R_j=R_j+E_j, and fix coefficients c_j. Define B=∑_jc_jR_j, widehat B=∑_jc_jwidehat R_j, E=widehat B-B. Let the distinct eigenvalue associated with block a be μ_a and define its separation γ_a=min_b≠ a|μ_a-μ_b|.
Exact mathematical passage
\gamma_a=\min_{b\neq a}|\mu_a-\mu_b|.
\gamma_bUse in source · 1 passage
Search aliases: γ_b gamma
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12968
Choose a generic linear combination G=∑_j=1^mc_jR_j+c_m+1S whose induced scalar values on the distinct joint signatures are pairwise distinct. For each block value γ_a, the Lagrange polynomial p_a(z)=∏_b≠ afracz-γ_bγ_a-γ_b satisfies p_a(G)=P_a, the orthogonal projector onto H_a. Therefore P_a,P_b∈ A(R^+) and belong to its center.
Exact mathematical passage
p_a(z)=\prod_{b\neq a}\frac{z-\gamma_b}{\gamma_a-\gamma_b}
\gamma_jUse in source · 1 passage
Search aliases: γ_j gamma
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3491
[Finite computation from a cycle basis] Let G be finite. Choose a spanning tree and let γ_1,…,γ_m be the fundamental loops generated by non-tree edges. If the loop transports form a representation of the graph fundamental group, then E_v_0^flat = ⋂_j=1^m ker(H_γ_j-I). Equivalently, E_v_0^flat is the null space of the vertically stacked matrix H_γ_1-I
Exact mathematical passage
E_{v_0}^{\flat} = \bigcap_{j=1}^{m} \ker(H_{\gamma_j}-I).
\gamma_kUse in source · 2 passages
Search aliases: γ_k gamma
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16735
The source grammar contains a task-blind continuation, inherited additive conditioning, final-step FiLM, progressive FiLM with geometry, a low-rank hypernetwork, and a shuffled-witness FiLM control. For the selected FiLM constructor, the encoded task produces step-indexed scale and shift vectors (γ_k(e),β_k(e))_k=0^4 =F_FiLM(e), which act on the preactivation by q_k ⟼ q_kodot(1+frac12tanhγ_k(e)) +β_k(e). This constructor is absent from Block 17's rank-one, rank-four, and full additive conditioni…
Exact mathematical passage
(\gamma_k(e),\beta_k(e))_{k=0}^{4} =F_{\mathrm{FiLM}}(e), \label{eq:b18-film}
\gamma_mUse in source · 2 passages
Search aliases: γ_m gamma
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3490
[Finite computation from a cycle basis] Let G be finite. Choose a spanning tree and let γ_1,…,γ_m be the fundamental loops generated by non-tree edges. If the loop transports form a representation of the graph fundamental group, then E_v_0^flat = ⋂_j=1^m ker(H_γ_j-I). Equivalently, E_v_0^flat is the null space of the vertically stacked matrix H_γ_1-I
Exact mathematical passage
\gamma_1,\ldots,\gamma_m
\gamma_{\rm cal}Use in source · 1 passage
Search aliases: γ_rm cal gamma rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8579
The compiler selected: π_rm anc · =π_rm PQR, k_rm fit · =16, k_rm val · =8, γ_rm cal · =0.08, ρ_rm cal · =0.035, τ_rm align · =0.05, full-rank requirement · =true. The selected development score is 0.263289. The underlying mean selected development gain is 0.296478 with zero harmful targets in the development ledger.
Exact mathematical passage
\begin{aligned} \pi_{\rm anc}&=\pi_{\rm PQR},\\ k_{\rm fit}&=16,\\ k_{\rm val}&=8,\\ \gamma_{\rm cal}&=0.08,\\ \rho_{\rm cal}&=0.035,\\ \tau_{\rm align}&=0.05,\\ \text{full-rank requirement}&=\text{true}. \end{aligned} \label{eq:block07q-descendant-config}
\Gamma_{\rm safe}Use in source · 1 passage
Search aliases: Γ_rm safe Gamma rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7499
ot4·5·4 =2000. For γ∈Γ, let overline U_rm meta(γ) be its mean protected utility on the meta-validation episodes and widehatrm se(γ) its standard error. A configuration is validation-safe when its protected-regression tail rate and maximum regression satisfy the frozen bounds. Among safe configurations, the compiler selects widehatγ ∈ arg max_γ∈Γ_rm safe { overline U_rm meta(γ) - frac12widehatrm se(γ) }.
Exact mathematical passage
\widehat\gamma \in \argmax_{\gamma\in\Gamma_{\rm safe}} \left \{ \overline U_{\rm meta}(\gamma) - frac12\widehat{\rm se}(\gamma) \right\}. \label{eq:block07-meta-objective}
\Gamma_tUse in source · 9 passages
Search aliases: Γ_t Gamma
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9467
Let K_t be the active repair process and let Γ_t be the lineage compiler. Write K_t+1 = Γ_t(K_t,E_t,A_t), where E_t is the available evidence and A_t is the executable archive.
Exact mathematical passage
\Gamma_t
\Gamma_{t+1}Use in source · 1 passage
Search aliases: Γ_t+1 Gamma
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9494
Define compiler metaproductivity by C(Γ) = E_K∼D_parent [ M(Γ(K,E_K))-M(K) ]. A stronger recursive claim requires C(Γ_t+1)>C(Γ_t) on sealed foreign parent processes under matched compilation resources.
Exact mathematical passage
\Ccal(\Gamma_{t+1})>\Ccal(\Gamma_t)
\delta^*Use in source · 4 passages
Search aliases: δ^* delta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 983
pection] Let u:Y→[0,1] be a bounded utility. Define V(δ)= E[u(Y)mid A_t,δ], widehat V(δ)= E[u(Y)mid ϕ_t(A_t),δ]. If ϕ_t is ε-intervention-sufficient, then sup_δ∈D_t absV(δ)-widehat V(δ)≤ε. If δ^*∈arg max_δ∈D_tV(δ), widehatδ∈arg max_δ∈D_twidehat V(δ), then V(δ^*)-V(widehatδ)≤2ε.
Exact mathematical passage
\delta^*\in\argmax_{\delta\in\Dcal_t}V(\delta), \qquad \widehat\delta\in\argmax_{\delta\in\Dcal_t}\widehat V(\delta),
\Delta^2Use in source · 1 passage
Search aliases: Δ^2 Delta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6003
[Prototype-separation error bound] Let Δ = min_z≠ z'normμ_z-μ_z'. Under equal priors and isotropic Gaussian noise, nearest-prototype classification satisfies Pr(widehat z≠ z) ≤ (| Z|-1) exp(-fracΔ^28σ^2).
Exact mathematical passage
\Pr(\widehat z\neq z) \leq (|\mathcal Z|-1) \exp\left(-\frac{\Delta^2}{8\sigma^2}\right).
\delta_0Use in source · 5 passages
Search aliases: δ_0 delta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 941
Let D_t be an authorized modification family containing a no-op intervention δ_0. For δ∈D_t, let K_A_t^δ(·) = P(Y∈·mid do(δ),A_t) be the protected external-outcome kernel. A compressed causal self-model ϕ_t(A_t) induces a predicted kernel widehat K_ϕ_t(A_t)^δ(·). The model is evaluated on counterfactual consequences, not on its ability to reproduce hidden activations for their own sake.
Exact mathematical passage
\delta_0
\delta_1Use in source · 3 passages
Search aliases: δ_1 delta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1009
[Margin-preserving ranking] If V(δ_1)-V(δ_2)>2ε, then widehat V(δ_1)>widehat V(δ_2). In particular, if the true best intervention is unique and its gap from every alternative exceeds 2ε, the self-model selects the true best intervention.
Exact mathematical passage
V(\delta_1)-V(\delta_2)>2\eps,
\delta_2Use in source · 3 passages
Search aliases: δ_2 delta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1009
[Margin-preserving ranking] If V(δ_1)-V(δ_2)>2ε, then widehat V(δ_1)>widehat V(δ_2). In particular, if the true best intervention is unique and its gap from every alternative exceeds 2ε, the self-model selects the true best intervention.
Exact mathematical passage
V(\delta_1)-V(\delta_2)>2\eps,
\Delta_aUse in source · 4 passages
Search aliases: Δ_a Delta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6601
0 . The resulting hidden difference is Δ_a(x) = h_a(x)-h_a^∘(x), h_a^∘(x)=E_a(ϕ^∘(x)). The empirical causal packet is the leading rank-r right-singular subspace of the affected differences {Δ_a(x_i):s(x_i)=1}.
Exact mathematical passage
\Delta_a(x) = h_a(x)-h_a^{\circ}(x), \qquad h_a^{\circ}(x)=E_a(\phi^{\circ}(x)).
\delta_{ab}Use in source · 1 passage
Search aliases: δ_ab delta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12148
Let H be a finite-dimensional real or complex Hilbert space. Suppose the unknown functional decomposition is H=⊕_a=1^mH_a, P_a:H→ H_a, with pairwise orthogonal projectors satisfying P_aP_b=δ_abP_a, ∑_a=1^mP_a=I. For controlled interventions u_1,…,u_k, let R_j∈End(H), j=1,…,k, be their finite-difference or linearized response operators.
Exact mathematical passage
P_aP_b=\delta_{ab}P_a, \qquad \sum_{a=1}^{m}P_a=I.
\Delta_{\Acal}Use in source · 3 passages
Search aliases: Δ_Acal Δ_A Delta Acal Delta mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17210
[Nearest-signature route stability] Let the minimum pairwise normalized signature separation of an atlas be Δ_A = min_c≠ dd_sig(σ_c,σ_d). If an observed signature from chart c has normalized perturbation at most δ and 2δ<Δ_A, nearest-signature routing returns c. If additionally δ≤τ_adm, the route is admitted.
Exact mathematical passage
\Delta_{\Acal} = \min_{c\neq d}d_{\mathrm{sig}}(\sigma_c,\sigma_d).
\Delta_{D,k}Use in source · 1 passage
Search aliases: Δ_D,k Delta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2972
Repeatedly apply [source reference]. The total correction-square defect is the telescoping sum ∑_k=0^n-1 T_V,n-1⋯ T_V,k+1 Δ_D,k T_U,k-1⋯ T_U,0, and the total defect-identification error is ∑_k=0^n-1 T_V,n-1⋯ T_V,k+1Δ_r,k. Apply both sums to u_0, use D_0u_0+r_0=0, and take norms.
Exact mathematical passage
\sum_{k=0}^{n-1} T_{V,n-1}\cdots T_{V,k+1} \Delta_{D,k} T_{U,k-1}\cdots T_{U,0},
\delta_eUse in source · 3 passages
Search aliases: δ_e delta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10252
For a branching archive, index proposed edges by e∈ E(Λ). Assign each edge a budget δ_e>0 with ∑_e∈ E(Λ)δ_e≤δ. The same proof yields archive-wide false-acceptance control. The allocation may be adaptive provided the remaining total budget is predictable from the pre-audit filtration.
Exact mathematical passage
\delta_e>0
\Delta_{\mathrm{gen}}Use in source · 1 passage
Search aliases: Δ_gen Delta mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9846
Define the causal generator contribution Δ_gen = E_Q[η_without(r)^2] - E_Q[η_with(r)^2]. The contribution is admissible only when a matched ablation isolates the generator while holding the remaining grammar, evidence, and selection budget fixed.
Exact mathematical passage
\Delta_{\mathrm{gen}} = E_Q\bigl[\eta_{\mathrm{without}}(r)^2\bigr] - E_Q\bigl[\eta_{\mathrm{with}}(r)^2\bigr]. \label{eq:b8-generator-credit}
\Delta_{\mathrm{gram}}Use in source · 2 passages
Search aliases: Δ_gram Delta mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20511
Δ_gram · Global-minus-local held-out transport residual margin
Exact mathematical passage
\Delta_{\mathrm{gram}}
\Delta_{G\to C}Use in source · 2 passages
Search aliases: Δ_G→ C Delta to
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11676
The main paired estimands are Δ_G→ C = E [ G(K_3,C^96,A)-G(K_2,G^96,A) ] and Δ_oracle→ C = E [ G(K_3,C^96,A)-G(K_affine oracle,A) ]. The second quantity is the sharper structural test.
Exact mathematical passage
\Delta_{G\to C} = \mathbb E \left[ G(K_{3,C}^{96},A)-G(K_{2,G}^{96},A) \right]
\Delta_iUse in source · 4 passages
Search aliases: Δ_i Delta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3827
[Paired evaluator-risk improvement] Let Z_1,…,Z_n be independent and identically distributed held-out externally revealed cases. Let L_v(Z_i),L_w(Z_i)∈[0,1] be the losses of the old and proposed evaluators, and define Δ_i=L_v(Z_i)-L_w(Z_i)∈[-1,1]. If the upgrade and evaluation rule are fixed before the held-out outcomes are revealed, then with probability at least 1-α, EΔ_i ≥ overlineΔ_n - sqrtfrac2log(1/α)n. Thus a positive lower bound certifies lower expected held-out risk for the proposed eva…
Exact mathematical passage
\Delta_i=L_v(Z_i)-L_w(Z_i)\in[-1,1].
\Delta_jUse in source · 2 passages
Search aliases: Δ_j Delta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6280
For arms a and b, evaluate both on the same frozen foreign agents and seeds. Let Δ_j=G_a,j-G_b,j. The paired estimate and standard error are widehatΔ =frac1n∑_j=1^nΔ_j, widehatse(widehatΔ) =sqrtfrac1n(n-1)∑_j=1^n(Δ_j-widehatΔ)^2. When gains are bounded in [-1,1], a distribution-free Hoeffding lower bound is LCB_1-α(Δ) = widehatΔ - sqrtfrac2log(1/α)n. Sharper paired permutation or bootstrap intervals may be used when frozen in advance.
Exact mathematical passage
\Delta_j=G_{a,j}-G_{b,j}.
\Delta_j^{\mathrm{ord}}Use in source · 2 passages
Search aliases: Δ_j^ord Delta mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9094
Choose an operational ordering r_1prec r_2prec⋯prec r_m, m=|N|, with nested sets R_0=∅, R_j={r_1,…,r_j}. Define the ordered marginal headroom Δ_j^ord(K) = F_K(R_j)-F_K(R_j-1).
Exact mathematical passage
\Delta_j^{\mathrm{ord}}(K) = F_K(R_j)-F_K(R_{j-1}). \label{eq:b8-ordered-marginal}
\delta_kUse in source · 2 passages
Search aliases: δ_k delta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16702
Let e=E(ϕ(τ))∈ℝ^36 be the encoded task representation, let z_0∈ℝ^36 be a learned seed, and let s_k∈ℝ^36 be the step embedding at latent depth k∈{0,…,4}. The source uses an Elman transition z_k+1 = tanh(Wz_k+s_k+δ_k(e,z_k)). The first foreign family uses a GRU cell z_k+1=GRUCell(u_k,z_k), and the second uses a gated nonlinear residual cell z_k+1 = z_k+σ(Gz_k)odot tanh(V_2GELU(V_1z_k)+s_k+δ_k(e,z_k)). The three cell classes have distinct executable implementations and parameterizations. They share…
Exact mathematical passage
z_{k+1} = \tanh\left(Wz_k+s_k+\delta_k(e,z_k)\right). \label{eq:b18-elman}
\Delta_{m,n}Use in source · 1 passage
Search aliases: Δ_m,n Delta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9707
Suppose the correction square is approximate: normD_t+1T_t^U-T_t^VD_t ≤ε_t^D. Suppose protected defect meanings are reconstructed with error at most ε_t^V. Define cumulative dynamic semantic debt Δ_m,n = ∑_t=m^n-1 ( ε_t^D+ε_t^V ). Under nonexpansive transports, old correction and defect relations drift by at most the corresponding telescoping debt. If ∑_t(ε_t^D+ε_t^V)<∞, old protected meanings form Cauchy trajectories along the grammar chain.
Exact mathematical passage
\Delta_{m,n} = \sum_{t=m}^{n-1} \left( \eps_t^D+\eps_t^V \right). \label{eq:b8-dynamic-debt}
\Delta_{\mathrm{oracle}\to C}Use in source · 1 passage
Search aliases: Δ_oracle→ C Delta mathrm to
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11685
The main paired estimands are Δ_G→ C = E [ G(K_3,C^96,A)-G(K_2,G^96,A) ] and Δ_oracle→ C = E [ G(K_3,C^96,A)-G(K_affine oracle,A) ]. The second quantity is the sharper structural test.
Exact mathematical passage
\Delta_{\mathrm{oracle}\to C} = \mathbb E \left[ G(K_{3,C}^{96},A)-G(K_{\mathrm{affine\ oracle}},A) \right].
\Delta_{r,k}Use in source · 1 passage
Search aliases: Δ_r,k Delta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2979
Repeatedly apply [source reference]. The total correction-square defect is the telescoping sum ∑_k=0^n-1 T_V,n-1⋯ T_V,k+1 Δ_D,k T_U,k-1⋯ T_U,0, and the total defect-identification error is ∑_k=0^n-1 T_V,n-1⋯ T_V,k+1Δ_r,k. Apply both sums to u_0, use D_0u_0+r_0=0, and take norms.
Exact mathematical passage
\sum_{k=0}^{n-1} T_{V,n-1}\cdots T_{V,k+1}\Delta_{r,k}.
\delta_{\rm orb}Use in source · 2 passages
Search aliases: δ_rm orb delta rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 19561
For a proposed intervention correspondence σ: U→ V and transport T:H_t→ H_s, define the normalized orbit defect δ_rm orb(T,σ) = frac1| U| ∑_u∈ U fracnormTR^t_σ(u)-R_u^sT_F max{normR_u^s_F,ε}.
Exact mathematical passage
\delta_{\rm orb}(T,\sigma) = \frac1{|\mathcal U|} \sum_{u\in\mathcal U} \frac{\norm{TR^t_{\sigma(u)}-R_u^sT}_F} {\max\{\norm{R_u^s}_F,\eps\}}. \label{eq:b22-orbit-defect}
\delta_{\rm rank}Use in source · 1 passage
Search aliases: δ_rm rank delta rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20476
δ_rm rank · Bridge-identification rank defect
Exact mathematical passage
\delta_{\rm rank}
\Delta_tUse in source · 5 passages
Search aliases: Δ_t Delta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10203
Let the true foreign metaproductivity effect be Δ_t = M(K_t+1)-M(K_t). Let L_t be a lower confidence bound computed from fresh audit evidence satisfying Pr(Δ_t<L_tmidF_t) ≤ δ_t. Accept the edge only when L_t>0.
Exact mathematical passage
\Delta_t = \Mcal(K_{t+1})-\Mcal(K_t). \label{eq:b8-edge-effect}
\delta_tUse in source · 18 passages
Search aliases: δ_t delta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1143
A useful self-model participates in a complete causal cycle: ϕ_t(A_t) xrightarrowcandidate policy δ_t xrightarrowworld execution Y_t+1 xrightarrowevidence update ϕ_t+1(A_t+1). The candidate is selected before the protected outcome is revealed. The outcome affects future beliefs and repair policy. Cutting either planner use or consequence feedback destroys prospective closure even if the internal self-representation remains decodable.
Exact mathematical passage
\phi_t(A_t) \xrightarrow{\text{candidate policy}} \delta_t \xrightarrow{\text{world execution}} Y_{t+1} \xrightarrow{\text{evidence update}} \phi_{t+1}(A_{t+1}).
\varepsilonUse in source · 3 passages
Search aliases: ε varepsilon
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11350
1∈ℝ^d+1 for homogeneous affine coordinates. Let c:H_s→{-1,+1} be an observable chart variable. A two-chart affine relation is specified by bar T_-,bar T_+∈ℝ^q×(d+1), with y=bar T_c(h)bar h+ε. Equivalently, writing bar A=fracbar T_++bar T_-2, bar B=fracbar T_+-bar T_-2, we have y=(bar A+c(h)bar B)bar h+ε.
Exact mathematical passage
y=\bar T_{c(h)}\bar h+\varepsilon. \label{eq:b10-chart-map}
\epsilon^{-2}Use in source · 1 passage
Search aliases: ϵ^-2 epsilon
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21772
Let X_ϵ=diag(1,1,ϵ), 0<ϵll1. Then rank X_ϵ=3, but λ_min(X_ϵ^ TX_ϵ)=ϵ^2 and the worst-direction least-squares variance scales as ϵ^-2. A binary rank gate cannot distinguish this design from an orthogonal one.
Exact mathematical passage
\epsilon^{-2}
\epsilon^2Use in source · 1 passage
Search aliases: ϵ^2 epsilon
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21769
Let X_ϵ=diag(1,1,ϵ), 0<ϵll1. Then rank X_ϵ=3, but λ_min(X_ϵ^ TX_ϵ)=ϵ^2 and the worst-direction least-squares variance scales as ϵ^-2. A binary rank gate cannot distinguish this design from an orthogonal one.
Exact mathematical passage
\lambda_{\min}(X_\epsilon^{\mathsf T}X_\epsilon)=\epsilon^2
\epsilon_gUse in source · 3 passages
Search aliases: ϵ_g epsilon
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11601
[Gate misclassification bound] Let g be the correct chart and widehat g an estimated chart. Suppose the two target repair outputs satisfy normR_1(y)-R_0(y)≤ L throughout the evaluated region and Pr(widehat g≠ g)≤ϵ_g. Then EnormR_widehat g(y)-R_g(y)^2 ≤ ϵ_gL^2. If protected utility is L_J-Lipschitz in repaired state, then abs E J(R_widehat g(y))- E J(R_g(y)) ≤ L_JLsqrtϵ_g.
Exact mathematical passage
\Pr(\widehat g\neq g)\leq\epsilon_g.
\epsilon_{\rm archive}Use in source · 1 passage
Search aliases: ϵ_rm archive epsilon rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 14685
The development archive is contaminated at rate ϵ_rm archive=0.18. Contamination can perturb diagnostics, obstruction labels, research outcomes, and response records. The finite candidate grammar contains the exact 39 methods inherited from Block 13: five variants in each of six obstruction-matched families, five generic variants, and the four original integrity and capability controls. Parent and child receive the same canonical candidate order and grammar digest.
Exact mathematical passage
\epsilon_{\rm archive}=0.18.
\zetaUse in source · 2 passages
Search aliases: ζ zeta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16093
Let ϕ(τ)∈ℝ^19 be the padded task serialization, and let e=E_θ(ϕ(τ))∈ℝ^40 be the prompt representation. The latent trace has four states. Given conditioning mode α, define z_0=ζ, z_k+1 = tanh (Rz_k+s_k+C_α(e,m)), 0≤ k<4, where s_k is a learned step embedding. The task-blind parent satisfies C_parent(e,m)=0. The descendants use one of the following conditioning maps: C_depth(e,m) · =D 1_m,
Exact mathematical passage
z_0=\zeta, \qquad z_{k+1} = \tanh\!\left(Rz_k+s_k+C_\alpha(e,m)\right), \qquad 0\leq k<4, \label{eq:b17-trace}
\zeta'Use in source · 1 passage
Search aliases: ζ' zeta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16497
Let P be a signed permutation matrix on the latent space. Define transformed latent parameters ζ'=Pζ, R'=PRP^ T, s_k'=Ps_k, C_α'=PC_α, and transform every latent-consuming linear map by right multiplication with P^ T. Then the transformed model has z_k'=Pz_k and identical external outputs.
Exact mathematical passage
\zeta'=P\zeta, \quad R'=PRP^{\mathsf T}, \quad s_k'=Ps_k, \quad C_\alpha'=PC_\alpha, \label{eq:b17-gauge-trace}
\eta^2Use in source · 1 passage
Search aliases: η^2 eta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10980
Hoeffding's inequality for a variable with range length 2G gives Pr(abswidehatM(K)-M(K)>η) ≤ 2exp(-fracnη^22G^2). Apply a union bound over N configurations and solve for η. The selected-candidate inequality follows by comparing the empirical maximizer with the true maximizer and paying one deviation term for each.
Exact mathematical passage
\Pr\left(\abs{\widehat\Mcal(K)-\Mcal(K)}>\eta\right) \leq 2\exp\left(-\frac{n\eta^2}{2G^2}\right).
\eta^U_{q^{-1}p}Use in source · 1 passage
Search aliases: η^U_q^-1p eta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3102
[Path-discrepancy identity] Under the hypotheses of [source reference], normT_U^pu-T_U^qu ≤ normT_U^q η^U_q^-1p(u). Thus loop holonomy directly bounds disagreement between two transport routes.
Exact mathematical passage
\norm{T_U^pu-T_U^qu} \leq \norm{T_U^q}\,\eta^U_{q^{-1}p}(u).
\eta_aUse in source · 2 passages
Search aliases: η_a eta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6998
r_rm non · ≤ widehat r_rm non+η_r, and that the reported alignment error upper-bounds the audit alignment error up to η_a. If widehat g_rm slice-η_g · >0,
Exact mathematical passage
\eta_a
\eta_gUse in source · 2 passages
Search aliases: η_g eta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6994
[Calibration-to-audit witness soundness] Suppose, simultaneously, that the audit gain and regression satisfy g_rm slice · ≥ widehat g_rm slice-η_g, r_rm non · ≤ widehat r_rm non+η_r, and that the reported alignment error upper-bounds the audit alignment error up to η_a. If widehat g_rm slice-η_g · >0,
Exact mathematical passage
g_{\rm slice}&\geq \widehat g_{\rm slice}-\eta_g,\\ r_{\rm non}&\leq \widehat r_{\rm non}+\eta_r,
\eta_{\mathrm{glue},W}Use in source · 1 passage
Search aliases: η_glue,W eta mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2433
Different overlaps may have different uncertainty or semantic importance. Let W_E be a positive-definite operator on K^E. Define η_glue,W(e) := inf_knorme+ B^*k_W_E.
Exact mathematical passage
\eta_{\mathrm{glue},W}(e) := \inf_k\norm{e+\mathbf B^*k}_{W_E}.
\eta_IUse in source · 2 passages
Search aliases: η_I eta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9752
[One-generator obstruction reduction] If bar v≠0, then η_I'(r)^2 = η_I(r)^2 - fracabsanglesz,bar v^2normbar v^2. If bar v=0, the extension introduces no new first-order correctability.
Exact mathematical passage
\eta_{I'}(r)^2 = \eta_I(r)^2 - \frac{\abs{\angles{z,\bar v}}^2}{\norm{\bar v}^2}. \label{eq:b8-one-generator}
\eta_{I'}Use in source · 2 passages
Search aliases: η_I' eta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9752
[One-generator obstruction reduction] If bar v≠0, then η_I'(r)^2 = η_I(r)^2 - fracabsanglesz,bar v^2normbar v^2. If bar v=0, the extension introduces no new first-order correctability.
Exact mathematical passage
\eta_{I'}(r)^2 = \eta_I(r)^2 - \frac{\abs{\angles{z,\bar v}}^2}{\norm{\bar v}^2}. \label{eq:b8-one-generator}
\eta_rUse in source · 2 passages
Search aliases: η_r eta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6994
[Calibration-to-audit witness soundness] Suppose, simultaneously, that the audit gain and regression satisfy g_rm slice · ≥ widehat g_rm slice-η_g, r_rm non · ≤ widehat r_rm non+η_r, and that the reported alignment error upper-bounds the audit alignment error up to η_a. If widehat g_rm slice-η_g · >0,
Exact mathematical passage
g_{\rm slice}&\geq \widehat g_{\rm slice}-\eta_g,\\ r_{\rm non}&\leq \widehat r_{\rm non}+\eta_r,
\eta_{t+1}Use in source · 1 passage
Search aliases: η_t+1 eta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9549
[Newly correctable energy identity] For nested correction images, η_t(r)^2-η_t+1(r)^2 = normP_I_t+1∩ I_t^⊥r^2.
Exact mathematical passage
\eta_t(r)^2-\eta_{t+1}(r)^2 = \norm{P_{I_{t+1}\cap I_t^\perp}r}^2. \label{eq:b8-new-energy}
\eta_{\mathrm{with}}Use in source · 1 passage
Search aliases: η_with eta mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9846
Define the causal generator contribution Δ_gen = E_Q[η_without(r)^2] - E_Q[η_with(r)^2]. The contribution is admissible only when a matched ablation isolates the generator while holding the remaining grammar, evidence, and selection budget fixed.
Exact mathematical passage
\Delta_{\mathrm{gen}} = E_Q\bigl[\eta_{\mathrm{without}}(r)^2\bigr] - E_Q\bigl[\eta_{\mathrm{with}}(r)^2\bigr]. \label{eq:b8-generator-credit}
\eta_{\mathrm{without}}Use in source · 1 passage
Search aliases: η_without eta mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9846
Define the causal generator contribution Δ_gen = E_Q[η_without(r)^2] - E_Q[η_with(r)^2]. The contribution is admissible only when a matched ablation isolates the generator while holding the remaining grammar, evidence, and selection budget fixed.
Exact mathematical passage
\Delta_{\mathrm{gen}} = E_Q\bigl[\eta_{\mathrm{without}}(r)^2\bigr] - E_Q\bigl[\eta_{\mathrm{with}}(r)^2\bigr]. \label{eq:b8-generator-credit}
\theta_*Use in source · 2 passages
Search aliases: θ_* theta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11726
-0.25 . Hence P_ctxR_s(h)=P_ctxh. The observable chart is generated by a task-specific separator c(h)=1{anglesu_*,P_ctxh>θ_*}. The compiler does not receive u_* or θ_*, but it receives the declared protected packet in which a valid separator may be sought.
Exact mathematical passage
c(h)=\one\{\angles{u_*,P_{\mathrm{ctx}}h}>\theta_*\}.
\theta_{\mathrm{abs}}Use in source · 1 passage
Search aliases: θ_abs theta mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16619
Let the template set be Θ = {θ_aff,θ_abs, θ_quad,θ_hinge}. For integer parameters a,b,c and integer input x, the exact programs are f_aff(x) · =ax+b,
Exact mathematical passage
\Theta = \{\theta_{\mathrm{aff}},\theta_{\mathrm{abs}}, \theta_{\mathrm{quad}},\theta_{\mathrm{hinge}}\}. \label{eq:b18-template-set}
\theta_{\mathrm{aff}}Use in source · 1 passage
Search aliases: θ_aff theta mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16619
Let the template set be Θ = {θ_aff,θ_abs, θ_quad,θ_hinge}. For integer parameters a,b,c and integer input x, the exact programs are f_aff(x) · =ax+b,
Exact mathematical passage
\Theta = \{\theta_{\mathrm{aff}},\theta_{\mathrm{abs}}, \theta_{\mathrm{quad}},\theta_{\mathrm{hinge}}\}. \label{eq:b18-template-set}
\theta_{\mathrm{hinge}}Use in source · 1 passage
Search aliases: θ_hinge theta mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16619
Let the template set be Θ = {θ_aff,θ_abs, θ_quad,θ_hinge}. For integer parameters a,b,c and integer input x, the exact programs are f_aff(x) · =ax+b,
Exact mathematical passage
\Theta = \{\theta_{\mathrm{aff}},\theta_{\mathrm{abs}}, \theta_{\mathrm{quad}},\theta_{\mathrm{hinge}}\}. \label{eq:b18-template-set}
\Theta_kUse in source · 2 passages
Search aliases: Θ_k Theta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11564
In the executable fixture, P_ctxh=(h_1,h_2), and the source repair leaves these coordinates invariant. The initial separator vocabulary is F_X = set 1{anglesu_k,P_ctxh>θ} : 0≤ k<128, θ∈Θ_k(X) , where u_k=(cos(2π k/128),sin(2π k/128)) and Θ_k(X) contains zero and the empirical 0.30,0.40,0.50,0.60,0.70 quantiles of the projected fit states. For every candidate separator, the compiler fits two weighted affine bridges and evaluates held-out chart-selection error. It then performs at most five determ…
Exact mathematical passage
\Fcal_X = \set{ \one\{\angles{u_k,P_{\mathrm{ctx}}h}>\theta\} : 0\leq k<128, \ \theta\in\Theta_k(X) },
\theta_{\mathrm{quad}}Use in source · 1 passage
Search aliases: θ_quad theta mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16619
Let the template set be Θ = {θ_aff,θ_abs, θ_quad,θ_hinge}. For integer parameters a,b,c and integer input x, the exact programs are f_aff(x) · =ax+b,
Exact mathematical passage
\Theta = \{\theta_{\mathrm{aff}},\theta_{\mathrm{abs}}, \theta_{\mathrm{quad}},\theta_{\mathrm{hinge}}\}. \label{eq:b18-template-set}
\Theta_{\rm select}Use in source · 1 passage
Search aliases: Θ_rm select Theta rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8121
The child adds one lineage-generated operator: Θ_rm select: { typed candidate receipts } ⟶ { risk-filtered repair choice }. This operator uses learned outcome calibration and candidate-type residual scales. The changed component is therefore the repair-selection policy, not the target agent's weights or the repair constructor.
Exact mathematical passage
\Theta_{\rm select}: \left\{ \text{typed candidate receipts} \right\} \longrightarrow \left\{ \text{risk-filtered repair choice} \right\}.
\iota_0Use in source · 1 passage
Search aliases: ι_0 iota
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4983
Consider a chain G_0 xhookrightarrowι_0 G_1 xhookrightarrowι_1 G_2 xhookrightarrow ⋯ of conservative extensions.
Exact mathematical passage
\Gcal_0 \xhookrightarrow{\iota_0} \Gcal_1 \xhookrightarrow{\iota_1} \Gcal_2 \xhookrightarrow{} \cdots
\iota_1Use in source · 1 passage
Search aliases: ι_1 iota
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4983
Consider a chain G_0 xhookrightarrowι_0 G_1 xhookrightarrowι_1 G_2 xhookrightarrow ⋯ of conservative extensions.
Exact mathematical passage
\Gcal_0 \xhookrightarrow{\iota_0} \Gcal_1 \xhookrightarrow{\iota_1} \Gcal_2 \xhookrightarrow{} \cdots
\iota_{10\to11}Use in source · 1 passage
Search aliases: ι_10→11 iota to
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12407
[Conservative embedding of the Block 10 grammar] There is an injective embedding ι_10→11:G_10↪G_11 obtained by fixing the atlas output to P_ctx and retaining the inherited separator, bridge, repair, evaluator, and witness objects. Every Block 10 execution has an observationally identical image in G_11.
Exact mathematical passage
\iota_{10\to11}:\Gcal_{10}\hookrightarrow\Gcal_{11}
\iota_gUse in source · 1 passage
Search aliases: ι_g iota
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11400
[Conservative embedding of the global grammar] For every separator g∈F, the map ι_g:G_glob↪G_chart, bar T⟼(g,bar T,bar T) is injective and preserves every global-affine prediction exactly.
Exact mathematical passage
\iota_g:\Gcal_{\mathrm{glob}}\hookrightarrow\Gcal_{\mathrm{chart}}, \qquad \bar T\longmapsto(g,\bar T,\bar T)
\kappa_DUse in source · 1 passage
Search aliases: κ_D kappa
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20512
κ_D · Normalized chart repair-contrast magnitude
Exact mathematical passage
\kappa_D
\kappa_rUse in source · 2 passages
Search aliases: κ_r kappa
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7407
widehatβ_r · = (Z^top Z+λ_r I)^-1Z^top r. For candidate type c, let s_u,c and s_r,c be the development residual standard deviations for the utility and risk heads. The deployed scores are widehat U_rm LCB(z) · = widehat U(z)-κ_u s_u,c(z), widehat R_rm UCB(z) · = max{0,widehat R(z)}+κ_r s_r,c(z). A candidate is admissible when widehat R_rm UCB(z)≤τ_r, widehat r_rm cal(z)≤τ_rm cal, A^rm base_rm cal,prot(z)≥ a_min. The child chooses the admissible candidate with maximum widehat U_rm LCB. If the adm…
Exact mathematical passage
\widehat U_{\rm LCB}(z) &= \widehat U(z)-\kappa_u s_{u,c(z)}, \label{eq:block07-utility-lcb}\\ \widehat R_{\rm UCB}(z) &= \max\{0,\widehat R(z)\}+\kappa_r s_{r,c(z)}. \label{eq:block07-risk-ucb}
\kappa_{\mathrm{square}}Use in source · 1 passage
Search aliases: κ_square kappa mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17934
A candidate is pre-audit eligible only if both foreign target models are ATLAS_FULL and the candidate jointly satisfies R^2_transport · ≥0.75, · R^2_worst chart · ≥0.74, ρ_preserve · ≥0.98, · NRMSE_trace · ≤0.05, κ_square · ≤0.02. · · The selection score is the weakest normalized margin across these gates. This prevents a strong aggregate R^2 from compensating for a failed chart, high curvature, or poor trajectory reconstruction.
Exact mathematical passage
\begin{aligned} R^2_{\mathrm{transport}}&\geq0.75, &\qquad R^2_{\mathrm{worst\ chart}}&\geq0.74,\\ \rho_{\mathrm{preserve}}&\geq0.98, & \operatorname{NRMSE}_{\mathrm{trace}}&\leq0.05,\\ \kappa_{\mathrm{square}}&\leq0.02. && \end{aligned} \label{eq:b20-thresholds}
\kappa_TUse in source · 1 passage
Search aliases: κ_T kappa
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11548
The executable package records a normalized chart-specific bridge contrast κ_T = fracnormbar T_1-bar T_0_F frac12(normbar T_1_F+normbar T_0_F)+10^-12. A positive contrast alone is not a structural certificate. Block 10 combines it with global-oracle exhaustion, local residual reduction, conservative embedding, matched resources, foreign gain, and binding controls.
Exact mathematical passage
\kappa_T = \frac{\norm{\bar T_1-\bar T_0}_F} {\frac12(\norm{\bar T_1}_F+\norm{\bar T_0}_F)+10^{-12}}. \label{eq:b10-contrast}
\kappa_uUse in source · 2 passages
Search aliases: κ_u kappa
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7407
widehatβ_r · = (Z^top Z+λ_r I)^-1Z^top r. For candidate type c, let s_u,c and s_r,c be the development residual standard deviations for the utility and risk heads. The deployed scores are widehat U_rm LCB(z) · = widehat U(z)-κ_u s_u,c(z), widehat R_rm UCB(z) · = max{0,widehat R(z)}+κ_r s_r,c(z). A candidate is admissible when widehat R_rm UCB(z)≤τ_r, widehat r_rm cal(z)≤τ_rm cal, A^rm base_rm cal,prot(z)≥ a_min. The child chooses the admissible candidate with maximum widehat U_rm LCB. If the adm…
Exact mathematical passage
\widehat U_{\rm LCB}(z) &= \widehat U(z)-\kappa_u s_{u,c(z)}, \label{eq:block07-utility-lcb}\\ \widehat R_{\rm UCB}(z) &= \max\{0,\widehat R(z)\}+\kappa_r s_{r,c(z)}. \label{eq:block07-risk-ucb}
\lambda_1Use in source · 2 passages
Search aliases: λ_1 lambda
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3565
[Spectral characterization of flatness] The exact common fixed space is E^flat=ker A. Let the eigenvalues of A be 0≤λ_1≤⋯≤λ_d. Among all k-dimensional subspaces F⊆ E, the quantity ∑_i=1^kanglesx_i, Ax_i for an orthonormal basis x_1,…,x_k of F is minimized by the span of eigenvectors associated with λ_1,…,λ_k, and the minimum is ∑_i=1^kλ_i.
Exact mathematical passage
0\leq\lambda_1\leq\cdots\leq\lambda_d.
\lambda_{1j}Use in source · 1 passage
Search aliases: λ_1j lambda
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 19616
The executable fixture uses five commuting symmetric response operators. In latent semantic coordinates they are diagonal: D_u=diag(λ_u1,…,λ_ud), and each system exposes an orthogonal gauge Q_a: R_u^a=Q_aD_uQ_a^top. The joint response signature of semantic coordinate j is λ_j=(λ_1j,…,λ_mj).
Exact mathematical passage
\lambda_j=(\lambda_{1j},\ldots,\lambda_{mj}).
\lambda_{ai_1}Use in source · 1 passage
Search aliases: λ_ai_1 lambda
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12928
Every element of A(R) is a finite linear combination of words in the generators. On H_a, a word R_i_1⋯ R_i_ℓ acts by the scalar λ_ai_1⋯λ_ai_ℓ. Equality of the signature vectors makes the same scalar act on H_b. Linear combinations preserve the equality. A central idempotent separating the blocks would act as 1 on one and 0 on the other, contradicting the first conclusion.
Exact mathematical passage
\lambda_{ai_1}\cdots\lambda_{ai_\ell}
\lambda_{ai_\ell}Use in source · 1 passage
Search aliases: λ_ai_ℓ lambda ell
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12928
Every element of A(R) is a finite linear combination of words in the generators. On H_a, a word R_i_1⋯ R_i_ℓ acts by the scalar λ_ai_1⋯λ_ai_ℓ. Equality of the signature vectors makes the same scalar act on H_b. Linear combinations preserve the equality. A central idempotent separating the blocks would act as 1 on one and 0 on the other, contradicting the first conclusion.
Exact mathematical passage
\lambda_{ai_1}\cdots\lambda_{ai_\ell}
\lambda_{bj}Use in source · 1 passage
Search aliases: λ_bj lambda
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12250
Write X=∑_a,bP_aXP_b. The relation XR_j=R_jX implies (λ_bj-λ_aj)P_aXP_b=0 for every j. Separating signatures imply that for a≠ b some coefficient is nonzero, so P_aXP_b=0. Hence X is block diagonal. Every block-diagonal operator commutes with each block-scalar R_j. The dimension formula follows from dimEnd(H_a)=d_a^2.
Exact mathematical passage
(\lambda_{bj}-\lambda_{aj})P_aXP_b=0
\lambda_cUse in source · 1 passage
Search aliases: λ_c lambda
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13888
A simple acquisition rule is A_i = widehat g_i -λ_rwidehat r_i +λ_uwidehatσ_i +λ_h p_θ(h_imid x_τ) -λ_c C_i, where h_i is the method family, p_θ is an obstruction-class posterior, widehatσ_i is a declared uncertainty bonus, and C_i is experimental cost.
Exact mathematical passage
A_i = \widehat g_i -\lambda_r\widehat r_i +\lambda_u\widehat\sigma_i +\lambda_h p_\theta(h_i\mid x_\tau) -\lambda_c C_i, \label{eq:b13-acquisition}
\lambda_{c}Use in source · 1 passage
Search aliases: λ_c lambda
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13011
Let mathscr S be a finite typed candidate-intervention grammar. For candidate S∈mathscr S, define a development score Score_resp(S) = widehat M_foreign(S) -λ_cC(S) -λ_commε_comm(S) -λ_fitε_fit(S) -λ_riskwidehat R(S). The compiler selects a candidate only after verifying that it adds a nontrivial response-signature distinction and preserves the inherited response language as an executable subgrammar.
Exact mathematical passage
\operatorname{Score}_{\mathrm{resp}}(S) = \widehat M_{\mathrm{foreign}}(S) -\lambda_{c}C(S) -\lambda_{\mathrm{comm}}\eps_{\mathrm{comm}}(S) -\lambda_{\mathrm{fit}}\eps_{\mathrm{fit}}(S) -\lambda_{\mathrm{risk}}\widehat R(S). \label{eq:b12-response-score}
\lambda_{\mathrm{comm}}Use in source · 1 passage
Search aliases: λ_comm lambda mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13011
Let mathscr S be a finite typed candidate-intervention grammar. For candidate S∈mathscr S, define a development score Score_resp(S) = widehat M_foreign(S) -λ_cC(S) -λ_commε_comm(S) -λ_fitε_fit(S) -λ_riskwidehat R(S). The compiler selects a candidate only after verifying that it adds a nontrivial response-signature distinction and preserves the inherited response language as an executable subgrammar.
Exact mathematical passage
\operatorname{Score}_{\mathrm{resp}}(S) = \widehat M_{\mathrm{foreign}}(S) -\lambda_{c}C(S) -\lambda_{\mathrm{comm}}\eps_{\mathrm{comm}}(S) -\lambda_{\mathrm{fit}}\eps_{\mathrm{fit}}(S) -\lambda_{\mathrm{risk}}\widehat R(S). \label{eq:b12-response-score}
\lambda_dUse in source · 1 passage
Search aliases: λ_d lambda
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3565
[Spectral characterization of flatness] The exact common fixed space is E^flat=ker A. Let the eigenvalues of A be 0≤λ_1≤⋯≤λ_d. Among all k-dimensional subspaces F⊆ E, the quantity ∑_i=1^kanglesx_i, Ax_i for an orthonormal basis x_1,…,x_k of F is minimized by the span of eigenvectors associated with λ_1,…,λ_k, and the minimum is ∑_i=1^kλ_i.
Exact mathematical passage
0\leq\lambda_1\leq\cdots\leq\lambda_d.
\lambda_{\mathrm{fit}}Use in source · 1 passage
Search aliases: λ_fit lambda mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13011
Let mathscr S be a finite typed candidate-intervention grammar. For candidate S∈mathscr S, define a development score Score_resp(S) = widehat M_foreign(S) -λ_cC(S) -λ_commε_comm(S) -λ_fitε_fit(S) -λ_riskwidehat R(S). The compiler selects a candidate only after verifying that it adds a nontrivial response-signature distinction and preserves the inherited response language as an executable subgrammar.
Exact mathematical passage
\operatorname{Score}_{\mathrm{resp}}(S) = \widehat M_{\mathrm{foreign}}(S) -\lambda_{c}C(S) -\lambda_{\mathrm{comm}}\eps_{\mathrm{comm}}(S) -\lambda_{\mathrm{fit}}\eps_{\mathrm{fit}}(S) -\lambda_{\mathrm{risk}}\widehat R(S). \label{eq:b12-response-score}
\lambda_hUse in source · 1 passage
Search aliases: λ_h lambda
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13888
A simple acquisition rule is A_i = widehat g_i -λ_rwidehat r_i +λ_uwidehatσ_i +λ_h p_θ(h_imid x_τ) -λ_c C_i, where h_i is the method family, p_θ is an obstruction-class posterior, widehatσ_i is a declared uncertainty bonus, and C_i is experimental cost.
Exact mathematical passage
A_i = \widehat g_i -\lambda_r\widehat r_i +\lambda_u\widehat\sigma_i +\lambda_h p_\theta(h_i\mid x_\tau) -\lambda_c C_i, \label{eq:b13-acquisition}
\lambda_jUse in source · 2 passages
Search aliases: λ_j lambda
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15790
For fixed maximum depth K_max, let λ_t(x)=Λ(z_t(x)). Define cumulative halt mass M_t(x) = ∑_j=1^t λ_j(x) ∏_q<j(1-λ_q(x)), and dynamic depth T(x)=min{t:M_t(x)≥τ}, with declared threshold τ. Block 16 does not infer adaptive computation from the existence of a halt head. It requires a difficulty variable d(x) fixed before result access and tests both ρ_Spearman(d(x),T(x))>0 and nondegenerate prompt variation in the halt schedules. A constant depth can coexist with fully task-specific Neuralese; con…
Exact mathematical passage
M_t(x) = \sum_{j=1}^{t} \lambda_j(x) \prod_{q<j}(1-\lambda_q(x)),
\lambda_kUse in source · 1 passage
Search aliases: λ_k lambda
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3572
ation E^flat=ker A. Let the eigenvalues of A be 0≤λ_1≤⋯≤λ_d. Among all k-dimensional subspaces F⊆ E, the quantity ∑_i=1^kanglesx_i, Ax_i for an orthonormal basis x_1,…,x_k of F is minimized by the span of eigenvectors associated with λ_1,…,λ_k, and the minimum is ∑_i=1^kλ_i.
Exact mathematical passage
\lambda_1,\ldots,\lambda_k
\lambda_{\mathrm{mean}}Use in source · 2 passages
Search aliases: λ_mean lambda mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5304
For a time-varying sequence M_t, let z_t be a row vector of expected schema counts. Under the linear mean model, E z_T = z_0M_0M_1⋯ M_T-1. Define the upper mean-growth exponent λ_mean = limsup_T→∞ frac1T lognormM_0M_1⋯ M_T-1. A positive value indicates exponential growth of the linearized expected count. It does not imply almost-sure survival under dependence or resource competition.
Exact mathematical passage
\lambda_{\mathrm{mean}} = \limsup_{T\to\infty} \frac1T \log\norm{M_0M_1\cdots M_{T-1}}.
\lambda_{\min}Use in source · 16 passages
Search aliases: λ_min lambda min
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10561
This certificate separates two statements: rank X_S=d and λ_min(X_S^ TWX_S)gg0. The first is qualitative identifiability. The second is quantitative stability.
Exact mathematical passage
\lambda_{\min}(X_S^{\mathsf T}WX_S)\gg0.
\lambda_{mj}Use in source · 1 passage
Search aliases: λ_mj lambda
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 19616
The executable fixture uses five commuting symmetric response operators. In latent semantic coordinates they are diagonal: D_u=diag(λ_u1,…,λ_ud), and each system exposes an orthogonal gauge Q_a: R_u^a=Q_aD_uQ_a^top. The joint response signature of semantic coordinate j is λ_j=(λ_1j,…,λ_mj).
Exact mathematical passage
\lambda_j=(\lambda_{1j},\ldots,\lambda_{mj}).
\lambda_qUse in source · 1 passage
Search aliases: λ_q lambda
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15790
For fixed maximum depth K_max, let λ_t(x)=Λ(z_t(x)). Define cumulative halt mass M_t(x) = ∑_j=1^t λ_j(x) ∏_q<j(1-λ_q(x)), and dynamic depth T(x)=min{t:M_t(x)≥τ}, with declared threshold τ. Block 16 does not infer adaptive computation from the existence of a halt head. It requires a difficulty variable d(x) fixed before result access and tests both ρ_Spearman(d(x),T(x))>0 and nondegenerate prompt variation in the halt schedules. A constant depth can coexist with fully task-specific Neuralese; con…
Exact mathematical passage
M_t(x) = \sum_{j=1}^{t} \lambda_j(x) \prod_{q<j}(1-\lambda_q(x)),
\Lambda_rUse in source · 1 passage
Search aliases: Λ_r Lambda
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7487
The child compiler searches a finite grammar Γ = Λ_u×Λ_r× K_u× K_r× T_r, where the factors specify utility regularization, risk regularization, utility residual penalty, risk residual penalty, and risk threshold. In the released fixture, |Γ| =5·5·4·5·4 =2000. For γ∈Γ, let overline U_rm meta(γ) be its mean protected utility on the meta-validation episodes and widehatrm se(γ) its standard error. A configuration is validation-safe when its protected-regression tail rate and maximum regression satis…
Exact mathematical passage
\Gamma = \Lambda_u\times\Lambda_r\times K_u\times K_r\times T_r,
\lambda_rUse in source · 4 passages
Search aliases: λ_r lambda
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7395
For regularization parameters λ_u,λ_r>0, define widehatβ_u · = (Z^top Z+λ_u I)^-1Z^top y,
Exact mathematical passage
\lambda_u,\lambda_r>0
\lambda_{\mathrm{risk}}Use in source · 1 passage
Search aliases: λ_risk lambda mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13011
Let mathscr S be a finite typed candidate-intervention grammar. For candidate S∈mathscr S, define a development score Score_resp(S) = widehat M_foreign(S) -λ_cC(S) -λ_commε_comm(S) -λ_fitε_fit(S) -λ_riskwidehat R(S). The compiler selects a candidate only after verifying that it adds a nontrivial response-signature distinction and preserves the inherited response language as an executable subgrammar.
Exact mathematical passage
\operatorname{Score}_{\mathrm{resp}}(S) = \widehat M_{\mathrm{foreign}}(S) -\lambda_{c}C(S) -\lambda_{\mathrm{comm}}\eps_{\mathrm{comm}}(S) -\lambda_{\mathrm{fit}}\eps_{\mathrm{fit}}(S) -\lambda_{\mathrm{risk}}\widehat R(S). \label{eq:b12-response-score}
\lambda_{\rm rec}Use in source · 2 passages
Search aliases: λ_rm rec lambda rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7031
For encoder E_θ, decoder G_δ, and score head H_ω, the pilot minimizes minibatch estimates of L(θ,δ,ω) = λ_rm rec normG_δ(E_θ(ϕ))-ϕ_F^2 + normH_ω(E_θ(ϕ))-widetilde q_F^2, with λ_rm rec=2. Optimization uses a deterministic NumPy Adam implementation for 500 steps. After representation training, the head is refitted by ridge regression to reduce avoidable readout variance.
Exact mathematical passage
\mathcal L(\theta,\delta,\omega) = \lambda_{\rm rec} \norm{G_{\delta}(E_{\theta}(\phi))-\phi}_F^2 + \norm{H_{\omega}(E_{\theta}(\phi))-\widetilde q}_F^2, \label{eq:block06-learned-objective}
\lambda_tUse in source · 1 passage
Search aliases: λ_t lambda
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15789
For fixed maximum depth K_max, let λ_t(x)=Λ(z_t(x)). Define cumulative halt mass M_t(x) = ∑_j=1^t λ_j(x) ∏_q<j(1-λ_q(x)), and dynamic depth T(x)=min{t:M_t(x)≥τ}, with declared threshold τ. Block 16 does not infer adaptive computation from the existence of a halt head. It requires a difficulty variable d(x) fixed before result access and tests both ρ_Spearman(d(x),T(x))>0 and nondegenerate prompt variation in the halt schedules. A constant depth can coexist with fully task-specific Neuralese; con…
Exact mathematical passage
\lambda_t(x)=\Lambda(z_t(x))
\Lambda_uUse in source · 1 passage
Search aliases: Λ_u Lambda
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7487
The child compiler searches a finite grammar Γ = Λ_u×Λ_r× K_u× K_r× T_r, where the factors specify utility regularization, risk regularization, utility residual penalty, risk residual penalty, and risk threshold. In the released fixture, |Γ| =5·5·4·5·4 =2000. For γ∈Γ, let overline U_rm meta(γ) be its mean protected utility on the meta-validation episodes and widehatrm se(γ) its standard error. A configuration is validation-safe when its protected-regression tail rate and maximum regression satis…
Exact mathematical passage
\Gamma = \Lambda_u\times\Lambda_r\times K_u\times K_r\times T_r,
\lambda_uUse in source · 4 passages
Search aliases: λ_u lambda
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7395
For regularization parameters λ_u,λ_r>0, define widehatβ_u · = (Z^top Z+λ_u I)^-1Z^top y,
Exact mathematical passage
\lambda_u,\lambda_r>0
\lambda_{u1}Use in source · 1 passage
Search aliases: λ_u1 lambda
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 19608
The executable fixture uses five commuting symmetric response operators. In latent semantic coordinates they are diagonal: D_u=diag(λ_u1,…,λ_ud), and each system exposes an orthogonal gauge Q_a: R_u^a=Q_aD_uQ_a^top. The joint response signature of semantic coordinate j is λ_j=(λ_1j,…,λ_mj).
Exact mathematical passage
D_u=\operatorname{diag}(\lambda_{u1},\ldots,\lambda_{ud}),
\lambda_{ud}Use in source · 1 passage
Search aliases: λ_ud lambda
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 19608
The executable fixture uses five commuting symmetric response operators. In latent semantic coordinates they are diagonal: D_u=diag(λ_u1,…,λ_ud), and each system exposes an orthogonal gauge Q_a: R_u^a=Q_aD_uQ_a^top. The joint response signature of semantic coordinate j is λ_j=(λ_1j,…,λ_mj).
Exact mathematical passage
D_u=\operatorname{diag}(\lambda_{u1},\ldots,\lambda_{ud}),
\mu_AUse in source · 3 passages
Search aliases: μ_A mu
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16893
[In-distribution bridge success does not imply foreign success] For every finite dimension d≥1 and every K>0, there exist two context distributions μ_A,μ_B, full-rank source and target trace covariances under both distributions, and fixed linear maps P,B such that semantic prediction and transport are exact under μ_A while transported squared error under μ_B exceeds K.
Exact mathematical passage
\mu_A,\mu_B
\mu_aUse in source · 6 passages
Search aliases: μ_a mu
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12197
Choose c∈ℝ^k outside the finite union of hyperplanes setc:c^ T(λ_a-λ_b)=0, a≠ b. Then the scalars μ_a=c^ Tλ_a are pairwise distinct. The combined response B=∑_j=1^kc_jR_j =∑_a=1^mμ_aP_a belongs to A_R. For each a, define the Lagrange polynomial p_a(z)=∏_b≠ afracz-μ_bμ_a-μ_b. Functional calculus gives p_a(B)=∑_b=1^mp_a(μ_b)P_b=P_a. Thus every P_a lies in the generated algebra. A nonzero central projection that is strictly below one P_a would have to be scalar on H_a and therefore is either 0 or P…
Exact mathematical passage
\mu_a=c^{\mathsf T}\lambda_a
\mu_BUse in source · 3 passages
Search aliases: μ_B mu
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16893
[In-distribution bridge success does not imply foreign success] For every finite dimension d≥1 and every K>0, there exist two context distributions μ_A,μ_B, full-rank source and target trace covariances under both distributions, and fixed linear maps P,B such that semantic prediction and transport are exact under μ_A while transported squared error under μ_B exceeds K.
Exact mathematical passage
\mu_A,\mu_B
\mu_bUse in source · 3 passages
Search aliases: μ_b mu
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12206
of hyperplanes setc:c^ T(λ_a-λ_b)=0, a≠ b. Then the scalars μ_a=c^ Tλ_a are pairwise distinct. The combined response B=∑_j=1^kc_jR_j =∑_a=1^mμ_aP_a belongs to A_R. For each a, define the Lagrange polynomial p_a(z)=∏_b≠ afracz-μ_bμ_a-μ_b. Functional calculus gives p_a(B)=∑_b=1^mp_a(μ_b)P_b=P_a. Thus every P_a lies in the generated algebra. A nonzero central projection that is strictly below one P_a would have to be scalar on H_a and therefore is either 0 or P_a. Hence the P_a are the minimal cent…
Exact mathematical passage
p_a(z)=\prod_{b\neq a}\frac{z-\mu_b}{\mu_a-\mu_b}.
\mu_zUse in source · 5 passages
Search aliases: μ_z mu
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5981
Let μ_z=ϕ(z) be the frozen event prototype. With equal priors and matched isotropic noise, the posterior is p(zmidwidetildeϕ) = fracexp(-normwidetildeϕ-μ_z^2/(2σ^2)) ∑_z'exp(-normwidetildeϕ-μ_z'^2/(2σ^2)). A repair is attempted only when max_z p(zmidwidetildeϕ)≥θ and the maximizing event is not none. Otherwise the system abstains.
Exact mathematical passage
\mu_z=\phi(z)
\mu_{z'}Use in source · 3 passages
Search aliases: μ_z' mu
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5982
Let μ_z=ϕ(z) be the frozen event prototype. With equal priors and matched isotropic noise, the posterior is p(zmidwidetildeϕ) = fracexp(-normwidetildeϕ-μ_z^2/(2σ^2)) ∑_z'exp(-normwidetildeϕ-μ_z'^2/(2σ^2)). A repair is attempted only when max_z p(zmidwidetildeϕ)≥θ and the maximizing event is not none. Otherwise the system abstains.
Exact mathematical passage
p(z\mid\widetilde\phi) = \frac{\exp\left(-\norm{\widetilde\phi-\mu_z}^2/(2\sigma^2)\right)} {\sum_{z'}\exp\left(-\norm{\widetilde\phi-\mu_{z'}}^2/(2\sigma^2)\right)}.
\xiUse in source · 7 passages
Search aliases: ξ xi
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 14485
geq0^k be a sparse mixture with 1^top z=1. An observed response vector has the form y=Rz+ξ.
Exact mathematical passage
y=Rz+\xi. \label{eq:sparse-response-model}
\Xi_0Use in source · 3 passages
Search aliases: Ξ_0 Xi
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21802
Let edge K_0→ K_1 improve a bridge task distribution Ξ_0, while K_1→ K_2 improves a heteroscedastic distribution Ξ_1. Positive effects in both contexts do not imply M_Ξ_0(K_2)>M_Ξ_0(K_0) or any additive path gain unless Ξ_0 and Ξ_1 share a common utility or a validated context transport.
Exact mathematical passage
\Xi_0
\Xi_1Use in source · 2 passages
Search aliases: Ξ_1 Xi
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21802
Let edge K_0→ K_1 improve a bridge task distribution Ξ_0, while K_1→ K_2 improves a heteroscedastic distribution Ξ_1. Positive effects in both contexts do not imply M_Ξ_0(K_2)>M_Ξ_0(K_0) or any additive path gain unless Ξ_0 and Ξ_1 share a common utility or a validated context transport.
Exact mathematical passage
\Xi_1
\Pi^{\Gamma}_7Use in source · 1 passage
Search aliases: Π^Γ_7 Pi Gamma
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13666
The realized compiler edge is L_6 xrightarrowΠ^Γ_7 L_7, where the active repair grammar, candidate-method language, evaluator portfolio, integrity monitor, and experiment budget remain fixed, but the research-allocation compiler changes from Γ_0 to Γ_1. The edge is therefore compiler metaproductivity inside a finite declared research language, not unrestricted automated science.
Exact mathematical passage
\mathfrak L_6 \xrightarrow{\Pi^{\Gamma}_7} \mathfrak L_7, \label{eq:b13-lineage}
\pi_0Use in source · 1 passage
Search aliases: π_0 pi
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15130
(ϕ_0(τ),high), · y_p(τ)>θ. The old token is recovered by the projection π_0(z,s)=z, π_0∘ϕ_p,θ=ϕ_0. Thus the extension is conservative at the level of old diagnostic semantics.
Exact mathematical passage
\pi_0(z,s)=z, \qquad \pi_0\circ\phi_{p,\theta}=\phi_0. \label{eq:b15-token-projection}
\Pi_{16}Use in source · 1 passage
Search aliases: Π_16 Pi
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15868
child-minus-parent transplant-cost interval has positive lower endpoint; item child clean loss does not regress beyond a predeclared tolerance; item every claimed packet or transport passes its own held-out causal certificate; item model, task, audit, and result digests are committed in an append-only receipt chain before promotion. Then the accepted transition L_t^NL xrightarrowΠ_16 L_t+1^NL is a finite evidence-preserving improvement of the Neuralese reasoning component. If the child-producing…
Exact mathematical passage
\mathfrak L_t^{\mathrm{NL}} \xrightarrow{\Pi_{16}} \mathfrak L_{t+1}^{\mathrm{NL}}
\Pi_{17}Use in source · 2 passages
Search aliases: Π_17 Pi
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16551
ain their declared interpretations; item the child is FULL and adaptive on both fresh final supports, while the paired effects in [source reference] have positive frozen intervals; item the independent NumPy, receipt, witness, and gauge checks pass within their declared numerical tolerances. Then the accepted transition L_16^NL xrightarrowΠ_17 L_17^NL is a finite, evidence-preserving, training-generated repair of the Neuralese binding component on the declared affine-program environment and gene…
Exact mathematical passage
\mathfrak L_{16}^{\mathrm{NL}} \xrightarrow{\Pi_{17}} \mathfrak L_{17}^{\mathrm{NL}} \label{eq:b17-lineage-edge}
\Pi_{18}Use in source · 2 passages
Search aliases: Π_18 Pi
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 18984
al receipt is frozen, a separate held-out split diagnosis refits only shift-local readouts and bridges. Source and target probes recover R^2 between 0.574 and 0.601, paired bridges recover 0.539 to 0.555, and shuffled bridges remain negative. Thus the next missing object is a context-indexed or invariant semantic atlas with abstention, not another static full-rank bridge. No transition Π_18 is accepted in this revision.
Exact mathematical passage
\Pi_{18}
\Pi_{19}Use in source · 2 passages
Search aliases: Π_19 Pi
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17579
item the admission results in [source reference] hold; item set-only, chart-only, shuffled-binding, no-abstention, and always-abstain controls retain their declared interpretations; item the automated tests, independent environment replay, independent chart reconstruction, independent NumPy forward pass, receipts, and package manifest verify. Then the edge K_18 xrightarrowΠ_19 K_19 is a finite, conservative, source-side Neuralese semantic-atlas expansion. It resolves two observation contexts abs…
Exact mathematical passage
K_{18} \xrightarrow{\Pi_{19}} K_{19}
\pi_2Use in source · 2 passages
Search aliases: π_2 pi
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 14466
[Static diagnostic indistinguishability] Let z_1≠ z_2 satisfy P(X_0∈ Bmid Z=z_1) = P(X_0∈ Bmid Z=z_2) for every measurable set B. Then no decision rule d(X_0) distinguishes z_1 from z_2 with accuracy exceeding the larger prior probability: P(d(X_0)=Zmid Z∈{z_1,z_2}) ≤ max{π_1,π_2}, where π_j= P(Z=z_jmid Z∈{z_1,z_2}).
Exact mathematical passage
\mathbb P(d(X_0)=Z\mid Z\in\{z_1,z_2\}) \leq \max\{\pi_1,\pi_2\},
\Pi_{20}Use in source · 1 passage
Search aliases: Π_20 Pi
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20344
Block 20 accepts the finite transport continuation L_19^NL xrightarrowΠ_20 L_20^NL, where the added object is an atlas-indexed connection carrying complete Neuralese trajectories from GRU and gated-residual fibers into the frozen Elman source fibers. The edge preserves the six-chart atlas, passes fresh semantic and curvature audits, and rejects unsupported contexts. Its scope remains the declared finite architecture, chart, gauge, and affine transport grammars.
Exact mathematical passage
\mathfrak L_{19}^{\mathrm{NL}} \xrightarrow{\Pi_{20}} \mathfrak L_{20}^{\mathrm{NL}},
\Pi_{21}Use in source · 1 passage
Search aliases: Π_21 Pi
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20352
Block 21 accepts the finite grammar continuation L_20^NL xrightarrowΠ_21 L_21^NL, where the transport process gains a typed response grammar over affine drift, nonlinear warp, latent-clock mixing, chart aliasing, benign semantic quotient, and destructive semantic loss. The edge preserves every inherited valid repair and accepts only the minimum-description candidate that passes the protected witness.
Exact mathematical passage
\mathfrak L_{20}^{\mathrm{NL}} \xrightarrow{\Pi_{21}} \mathfrak L_{21}^{\mathrm{NL}},
\Pi_3Use in source · 5 passages
Search aliases: Π_3 Pi
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11666
K_3,C^96. The global parent K_2,G^96 retains one affine bridge and uses all 80 non-witness pairs to estimate it. The chart child K_3,C^96 uses 48 pairs for local bridge fitting, 32 pairs for chart selection, and the same 16 witness pairs. Both consume exactly 96 paired source-target queries. The edge is K_2,G^96 xrightarrowΠ_3 K_3,C^96.
Exact mathematical passage
K_{2,G}^{96} \xrightarrow{\Pi_3} K_{3,C}^{96}. \label{eq:b10-edge}
\Pi_4Use in source · 5 passages
Search aliases: Π_4 Pi
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12131
The new edge is therefore K_0xrightarrowΠ_1K_1xrightarrowΠ_2K_2 xrightarrowΠ_3K_3xrightarrowΠ_4K_4, where K_3 is the Block 10 chart process with a supplied protected packet and K_4 replaces that packet by an intervention-response atlas compiler. This is not unrestricted atlas discovery. The intervention family, finite state dimension, target task, repair family, chart count, evaluator, witness, and outer compiler remain declared. The result is a controlled first realization of endogenous functio…
Exact mathematical passage
K_0\xrightarrow{\Pi_1}K_1\xrightarrow{\Pi_2}K_2 \xrightarrow{\Pi_3}K_3\xrightarrow{\Pi_4}K_4, \label{eq:b11-lineage}
\Pi_5Use in source · 3 passages
Search aliases: Π_5 Pi
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12871
The realized continuation is K_4 xrightarrowΠ_5 K_5 xrightarrowΠ_6 K_6. The first edge expands the intervention-response language. The second expands the repair-constructor grammar. The two edges are evaluated in different protected contexts and are not added into one scalar utility difference without an additional context transport.
Exact mathematical passage
K_4 \xrightarrow{\Pi_5} K_5 \xrightarrow{\Pi_6} K_6. \label{eq:b12-lineage}
\Pi_6Use in source · 3 passages
Search aliases: Π_6 Pi
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12871
The realized continuation is K_4 xrightarrowΠ_5 K_5 xrightarrowΠ_6 K_6. The first edge expands the intervention-response language. The second expands the repair-constructor grammar. The two edges are evaluated in different protected contexts and are not added into one scalar utility difference without an additional context transport.
Exact mathematical passage
K_4 \xrightarrow{\Pi_5} K_5 \xrightarrow{\Pi_6} K_6. \label{eq:b12-lineage}
\Pi_8^\GammaUse in source · 1 passage
Search aliases: Π_8^Γ Pi Gamma
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 14376
d full-experiment budget fixed while changing the reliability conditions under which research decisions must be made. The published Block 13 result digest and exact 39-method language are replayed. Its compiler design is retained as the inherited parent and recalibrated on the Block 14 development archive: Γ_1^rm recal. A robust descendant Γ_1^rm recalxrightarrowΠ_8^ΓΓ_2 is compiled before either final audit is opened. The child adds sequential diagnosis, sparse multi-obstruction inference, open…
Exact mathematical passage
\Gamma_1^{\rm recal}\xrightarrow{\Pi_8^\Gamma}\Gamma_2 \label{eq:block14-edge}
\Pi_9^{\Gamma}Use in source · 1 passage
Search aliases: Π_9^Γ Pi Gamma
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20305
Block 15 adds the first finite compiler-language edge, Γ_2xrightarrowΠ_9^ΓΓ_3. The inherited compiler retains one undifferentiated unknown token and a singleton-method grammar. Development repair conflicts select a separating probe, two anonymous refined tokens, and two ordered pair-method constructors from generic finite meta-languages. The old compiler remains an exact fallback. Under equal eight-unit research budgets, primary gain rises from 0.633969 to 0.859731 and generator-foreign gain ris…
Exact mathematical passage
\Gamma_2\xrightarrow{\Pi_9^{\Gamma}}\Gamma_3.
\pi_jUse in source · 1 passage
Search aliases: π_j pi
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 14471
nostic-indistinguishability Let z_1≠ z_2 satisfy P(X_0∈ Bmid Z=z_1) = P(X_0∈ Bmid Z=z_2) for every measurable set B. Then no decision rule d(X_0) distinguishes z_1 from z_2 with accuracy exceeding the larger prior probability: P(d(X_0)=Zmid Z∈{z_1,z_2}) ≤ max{π_1,π_2}, where π_j= P(Z=z_jmid Z∈{z_1,z_2}).
Exact mathematical passage
\pi_j=\mathbb P(Z=z_j\mid Z\in\{z_1,z_2\})
\Pi_{\rm anc}Use in source · 1 passage
Search aliases: Π_rm anc Pi rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8549
The development meta-grammar contains five anchor policies: Π_rm anc = set random, balanced random, QR, packet QR, augmented QR . It also contains alignment thresholds τ∈set0.01,0.02,0.05,0.10,0.20,0.35 and an optional target-width guard. Every candidate uses the same fit and validation budgets.
Exact mathematical passage
\Pi_{\rm anc} = \set{ \text{random}, \text{balanced random}, \text{QR}, \text{packet QR}, \text{augmented QR} }.
\pi_{\rm anc}^{(0)}Use in source · 1 passage
Search aliases: π_rm anc^(0) pi rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8821
The parent and descendant differ in exactly two declared fields: π_rm anc^(0) · =uniform random, · π_rm anc^(1) · =packet-balanced QR, π_rm gate^(0) · =(0.20,0.03,0.35), · π_rm gate^(1) · =(0.08,0.035,0.05,full rank). The source task model, causal packet discovery, source repair, linear bridge family, pseudoinverse repair transport, target calibration set, protected utility, and audit data remain fixed.
Exact mathematical passage
\begin{aligned} \pi_{\rm anc}^{(0)}&=\text{uniform random}, &\pi_{\rm anc}^{(1)}&=\text{packet-balanced QR},\\ \pi_{\rm gate}^{(0)}&=(0.20,0.03,0.35), &\pi_{\rm gate}^{(1)}&=(0.08,0.035,0.05,\text{full rank}). \end{aligned}
\pi_{\rm anc}^{(1)}Use in source · 1 passage
Search aliases: π_rm anc^(1) pi rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8821
The parent and descendant differ in exactly two declared fields: π_rm anc^(0) · =uniform random, · π_rm anc^(1) · =packet-balanced QR, π_rm gate^(0) · =(0.20,0.03,0.35), · π_rm gate^(1) · =(0.08,0.035,0.05,full rank). The source task model, causal packet discovery, source repair, linear bridge family, pseudoinverse repair transport, target calibration set, protected utility, and audit data remain fixed.
Exact mathematical passage
\begin{aligned} \pi_{\rm anc}^{(0)}&=\text{uniform random}, &\pi_{\rm anc}^{(1)}&=\text{packet-balanced QR},\\ \pi_{\rm gate}^{(0)}&=(0.20,0.03,0.35), &\pi_{\rm gate}^{(1)}&=(0.08,0.035,0.05,\text{full rank}). \end{aligned}
\Pi_{\rm foreign}Use in source · 1 passage
Search aliases: Π_rm foreign Pi rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15243
A compiler-language expansion receipt is Π_Γ-lang = ( G_0, H_rm conflict, P^+, p,θ, C_rm pair, m_-,m_+, ι, κ, Π_rm seal, Π_rm foreign ), where H_rm conflict is the conflict hypergraph, ι is the conservative embedding, κ is the newly correctable development margin, and the final fields bind the pre-audit compiler state to foreign evidence.
Exact mathematical passage
\Pi_{\Gamma\text{-lang}} = ( \mathfrak G_0, \Hcal_{\rm conflict}, \Pcal^+, p,\theta, \Ccal_{\rm pair}, m_-,m_+, \iota, \kappa, \Pi_{\rm seal}, \Pi_{\rm foreign} ), \label{eq:b15-language-receipt}
\pi_{\rm gate}^{(0)}Use in source · 1 passage
Search aliases: π_rm gate^(0) pi rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8821
The parent and descendant differ in exactly two declared fields: π_rm anc^(0) · =uniform random, · π_rm anc^(1) · =packet-balanced QR, π_rm gate^(0) · =(0.20,0.03,0.35), · π_rm gate^(1) · =(0.08,0.035,0.05,full rank). The source task model, causal packet discovery, source repair, linear bridge family, pseudoinverse repair transport, target calibration set, protected utility, and audit data remain fixed.
Exact mathematical passage
\begin{aligned} \pi_{\rm anc}^{(0)}&=\text{uniform random}, &\pi_{\rm anc}^{(1)}&=\text{packet-balanced QR},\\ \pi_{\rm gate}^{(0)}&=(0.20,0.03,0.35), &\pi_{\rm gate}^{(1)}&=(0.08,0.035,0.05,\text{full rank}). \end{aligned}
\pi_{\rm gate}^{(1)}Use in source · 1 passage
Search aliases: π_rm gate^(1) pi rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8821
The parent and descendant differ in exactly two declared fields: π_rm anc^(0) · =uniform random, · π_rm anc^(1) · =packet-balanced QR, π_rm gate^(0) · =(0.20,0.03,0.35), · π_rm gate^(1) · =(0.08,0.035,0.05,full rank). The source task model, causal packet discovery, source repair, linear bridge family, pseudoinverse repair transport, target calibration set, protected utility, and audit data remain fixed.
Exact mathematical passage
\begin{aligned} \pi_{\rm anc}^{(0)}&=\text{uniform random}, &\pi_{\rm anc}^{(1)}&=\text{packet-balanced QR},\\ \pi_{\rm gate}^{(0)}&=(0.20,0.03,0.35), &\pi_{\rm gate}^{(1)}&=(0.08,0.035,0.05,\text{full rank}). \end{aligned}
\Pi_{\rm seal}Use in source · 1 passage
Search aliases: Π_rm seal Pi rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15243
A compiler-language expansion receipt is Π_Γ-lang = ( G_0, H_rm conflict, P^+, p,θ, C_rm pair, m_-,m_+, ι, κ, Π_rm seal, Π_rm foreign ), where H_rm conflict is the conflict hypergraph, ι is the conservative embedding, κ is the newly correctable development margin, and the final fields bind the pre-audit compiler state to foreign evidence.
Exact mathematical passage
\Pi_{\Gamma\text{-lang}} = ( \mathfrak G_0, \Hcal_{\rm conflict}, \Pcal^+, p,\theta, \Ccal_{\rm pair}, m_-,m_+, \iota, \kappa, \Pi_{\rm seal}, \Pi_{\rm foreign} ), \label{eq:b15-language-receipt}
\Pi_t^{\mathrm{auth}}Use in source · 2 passages
Search aliases: Π_t^auth Pi mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 811
A witness structure may be represented as W_t = ( V_t^fr, C_t, B_t^sand, R_t, Π_t^auth ), where V_t^fr is the frozen evaluator instance for the current transaction, C_t is a protected checkpoint, B_t^sand is the sandbox boundary, R_t is the rollback procedure, and Π_t^auth records effect authority.
Exact mathematical passage
\Wcal_t = \left( \Vcal_t^{\mathrm{fr}}, C_t, \Bcal_t^{\mathrm{sand}}, R_t, \Pi_t^{\mathrm{auth}} \right),
\rho_{\mathrm{halt}}Use in source · 1 passage
Search aliases: ρ_halt rho mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16303
positive interval and fraction gates, local continuity, donor coverage, adaptive halting. Among eligible candidates it maximizes S(α) = · min{ fracR^2_probe0.65, fracQ_prog0.15, fracQ_ans0.20 } +0.20R^2_probe · -4 MSE_clean +0.02ρ_halt -Penalty_dynamic. The minimum term makes the weakest normalized semantic gate the principal bottleneck. The compiler seal commits the selected candidate, all candidate and parent state hashes, training receipts, meta-validation task digest, exact thresholds, selec…
Exact mathematical passage
\begin{split} S(\alpha) ={}& \min\left\{ \frac{R^2_{\mathrm{probe}}}{0.65}, \frac{Q_{\mathrm{prog}}}{0.15}, \frac{Q_{\mathrm{ans}}}{0.20} \right\} +0.20R^2_{\mathrm{probe}}\\ &-4\,\mathrm{MSE}_{\mathrm{clean}} +0.02\rho_{\mathrm{halt}} -\mathsf{Penalty}_{\mathrm{dynamic}}. \end{split} \label{eq:b17-selection-score}
\rho_{\mathrm{preserve}}Use in source · 1 passage
Search aliases: ρ_preserve rho mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17934
A candidate is pre-audit eligible only if both foreign target models are ATLAS_FULL and the candidate jointly satisfies R^2_transport · ≥0.75, · R^2_worst chart · ≥0.74, ρ_preserve · ≥0.98, · NRMSE_trace · ≤0.05, κ_square · ≤0.02. · · The selection score is the weakest normalized margin across these gates. This prevents a strong aggregate R^2 from compensating for a failed chart, high curvature, or poor trajectory reconstruction.
Exact mathematical passage
\begin{aligned} R^2_{\mathrm{transport}}&\geq0.75, &\qquad R^2_{\mathrm{worst\ chart}}&\geq0.74,\\ \rho_{\mathrm{preserve}}&\geq0.98, & \operatorname{NRMSE}_{\mathrm{trace}}&\leq0.05,\\ \kappa_{\mathrm{square}}&\leq0.02. && \end{aligned} \label{eq:b20-thresholds}
\rho_{\rm cal}Use in source · 1 passage
Search aliases: ρ_rm cal rho rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8579
The compiler selected: π_rm anc · =π_rm PQR, k_rm fit · =16, k_rm val · =8, γ_rm cal · =0.08, ρ_rm cal · =0.035, τ_rm align · =0.05, full-rank requirement · =true. The selected development score is 0.263289. The underlying mean selected development gain is 0.296478 with zero harmful targets in the development ledger.
Exact mathematical passage
\begin{aligned} \pi_{\rm anc}&=\pi_{\rm PQR},\\ k_{\rm fit}&=16,\\ k_{\rm val}&=8,\\ \gamma_{\rm cal}&=0.08,\\ \rho_{\rm cal}&=0.035,\\ \tau_{\rm align}&=0.05,\\ \text{full-rank requirement}&=\text{true}. \end{aligned} \label{eq:block07q-descendant-config}
\rho_{\mathrm{Spearman}}Use in source · 1 passage
Search aliases: ρ_Spearman rho mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15802
halt mass M_t(x) = ∑_j=1^t λ_j(x) ∏_q<j(1-λ_q(x)), and dynamic depth T(x)=min{t:M_t(x)≥τ}, with declared threshold τ. Block 16 does not infer adaptive computation from the existence of a halt head. It requires a difficulty variable d(x) fixed before result access and tests both ρ_Spearman(d(x),T(x))>0 and nondegenerate prompt variation in the halt schedules. A constant depth can coexist with fully task-specific Neuralese; conversely, variable depth can be driven by nuisance prompt features. The …
Exact mathematical passage
\rho_{\mathrm{Spearman}}(d(x),T(x))>0
\sigma^{-1}Use in source · 2 passages
Search aliases: σ^-1 sigma
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 19583
[Intertwiner uniqueness modulo the response commutant] Let T_1,T_2:H_t→ H_s be invertible and satisfy T_iR_v^t=R_σ^-1(v)^sT_i for every v∈ V. Then S=T_2^-1T_1 belongs to A_t'. Hence exact intervention-orbit data identifies a transport at most up to the response commutant.
Exact mathematical passage
T_iR_v^t=R_{\sigma^{-1}(v)}^sT_i
\sigma^2Use in source · 3 passages
Search aliases: σ^2 sigma
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5973
nts are, respectively: item deviation from the baseline no-action continuation; item deviation of the action-conditioned response from the baseline action response; item violation of redundant-sensor consistency; item deviation from baseline memory update; item discrepancy between reported and protected evaluator scores. The observed probe is widetildeϕ=ϕ(z)+η, η∼ N(0,σ^2I_5).
Exact mathematical passage
\widetilde\phi=\phi(z)+\eta, \qquad \eta\sim\mathcal N(0,\sigma^2I_5).
\Sigma^{\mathrm{evaluation}}Use in source · 1 passage
Search aliases: Σ^evaluation Sigma mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3960
Let a complete implementation state decompose as Σ = Σ^task ⊕ Σ^memory ⊕ Σ^repair ⊕ Σ^evaluation ⊕ Σ^provenance ⊕ Σ^rollback. A transaction selects an editable mask E⊆Σ and a witness mask W⊆Σ with E∩ W=∅. The decomposition is logical rather than necessarily physical. Two components may share hardware while remaining isolated by capabilities, address spaces, process boundaries, cryptographic commitments, or reproducible snapshots.
Exact mathematical passage
\Sigma = \Sigma^{\mathrm{task}} \oplus \Sigma^{\mathrm{memory}} \oplus \Sigma^{\mathrm{repair}} \oplus \Sigma^{\mathrm{evaluation}} \oplus \Sigma^{\mathrm{provenance}} \oplus \Sigma^{\mathrm{rollback}}.
\Sigma^{\mathrm{memory}}Use in source · 1 passage
Search aliases: Σ^memory Sigma mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3960
Let a complete implementation state decompose as Σ = Σ^task ⊕ Σ^memory ⊕ Σ^repair ⊕ Σ^evaluation ⊕ Σ^provenance ⊕ Σ^rollback. A transaction selects an editable mask E⊆Σ and a witness mask W⊆Σ with E∩ W=∅. The decomposition is logical rather than necessarily physical. Two components may share hardware while remaining isolated by capabilities, address spaces, process boundaries, cryptographic commitments, or reproducible snapshots.
Exact mathematical passage
\Sigma = \Sigma^{\mathrm{task}} \oplus \Sigma^{\mathrm{memory}} \oplus \Sigma^{\mathrm{repair}} \oplus \Sigma^{\mathrm{evaluation}} \oplus \Sigma^{\mathrm{provenance}} \oplus \Sigma^{\mathrm{rollback}}.
\Sigma^{\mathrm{provenance}}Use in source · 1 passage
Search aliases: Σ^provenance Sigma mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3960
Let a complete implementation state decompose as Σ = Σ^task ⊕ Σ^memory ⊕ Σ^repair ⊕ Σ^evaluation ⊕ Σ^provenance ⊕ Σ^rollback. A transaction selects an editable mask E⊆Σ and a witness mask W⊆Σ with E∩ W=∅. The decomposition is logical rather than necessarily physical. Two components may share hardware while remaining isolated by capabilities, address spaces, process boundaries, cryptographic commitments, or reproducible snapshots.
Exact mathematical passage
\Sigma = \Sigma^{\mathrm{task}} \oplus \Sigma^{\mathrm{memory}} \oplus \Sigma^{\mathrm{repair}} \oplus \Sigma^{\mathrm{evaluation}} \oplus \Sigma^{\mathrm{provenance}} \oplus \Sigma^{\mathrm{rollback}}.
\Sigma^{\mathrm{repair}}Use in source · 1 passage
Search aliases: Σ^repair Sigma mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3960
Let a complete implementation state decompose as Σ = Σ^task ⊕ Σ^memory ⊕ Σ^repair ⊕ Σ^evaluation ⊕ Σ^provenance ⊕ Σ^rollback. A transaction selects an editable mask E⊆Σ and a witness mask W⊆Σ with E∩ W=∅. The decomposition is logical rather than necessarily physical. Two components may share hardware while remaining isolated by capabilities, address spaces, process boundaries, cryptographic commitments, or reproducible snapshots.
Exact mathematical passage
\Sigma = \Sigma^{\mathrm{task}} \oplus \Sigma^{\mathrm{memory}} \oplus \Sigma^{\mathrm{repair}} \oplus \Sigma^{\mathrm{evaluation}} \oplus \Sigma^{\mathrm{provenance}} \oplus \Sigma^{\mathrm{rollback}}.
\Sigma^{\mathrm{rollback}}Use in source · 1 passage
Search aliases: Σ^rollback Sigma mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3960
Let a complete implementation state decompose as Σ = Σ^task ⊕ Σ^memory ⊕ Σ^repair ⊕ Σ^evaluation ⊕ Σ^provenance ⊕ Σ^rollback. A transaction selects an editable mask E⊆Σ and a witness mask W⊆Σ with E∩ W=∅. The decomposition is logical rather than necessarily physical. Two components may share hardware while remaining isolated by capabilities, address spaces, process boundaries, cryptographic commitments, or reproducible snapshots.
Exact mathematical passage
\Sigma = \Sigma^{\mathrm{task}} \oplus \Sigma^{\mathrm{memory}} \oplus \Sigma^{\mathrm{repair}} \oplus \Sigma^{\mathrm{evaluation}} \oplus \Sigma^{\mathrm{provenance}} \oplus \Sigma^{\mathrm{rollback}}.
\Sigma^{\mathrm{task}}Use in source · 1 passage
Search aliases: Σ^task Sigma mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3960
Let a complete implementation state decompose as Σ = Σ^task ⊕ Σ^memory ⊕ Σ^repair ⊕ Σ^evaluation ⊕ Σ^provenance ⊕ Σ^rollback. A transaction selects an editable mask E⊆Σ and a witness mask W⊆Σ with E∩ W=∅. The decomposition is logical rather than necessarily physical. Two components may share hardware while remaining isolated by capabilities, address spaces, process boundaries, cryptographic commitments, or reproducible snapshots.
Exact mathematical passage
\Sigma = \Sigma^{\mathrm{task}} \oplus \Sigma^{\mathrm{memory}} \oplus \Sigma^{\mathrm{repair}} \oplus \Sigma^{\mathrm{evaluation}} \oplus \Sigma^{\mathrm{provenance}} \oplus \Sigma^{\mathrm{rollback}}.
\Sigma_0Use in source · 1 passage
Search aliases: Σ_0 Sigma
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13183
The hidden source space has three unknown rank-two blocks: H=H_ctx⊕ H_rep⊕ H_nui. The inherited signature matrix is Σ_0 = 1.25 · 0.72 1.25 · 0.72 0.80 · 1.30 , where rows correspond to context, repair, and nuisance blocks. The first two rows coincide. By [source reference], every element of the inherited response algebra acts identically on the context and repair blocks.
Exact mathematical passage
\Sigma_0 = \begin{bmatrix} 1.25 & 0.72\\ 1.25 & 0.72\\ 0.80 & 1.30 \end{bmatrix}, \label{eq:b12-inherited-signatures}
\sigma_1^{-2}Use in source · 1 passage
Search aliases: σ_1^-2 sigma
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10606
Stacking the selected anchors gives Y=XT^ T+E, where X∈ℝ^n× d and Y∈ℝ^n× q. Define W=diag(σ_1^-2,…,σ_n^-2) and the weighted information matrix M_S=X^ TWX.
Exact mathematical passage
W=\diag(\sigma_1^{-2},\ldots,\sigma_n^{-2})
\sigma_dUse in source · 1 passage
Search aliases: σ_d sigma
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17210
[Nearest-signature route stability] Let the minimum pairwise normalized signature separation of an atlas be Δ_A = min_c≠ dd_sig(σ_c,σ_d). If an observed signature from chart c has normalized perturbation at most δ and 2δ<Δ_A, nearest-signature routing returns c. If additionally δ≤τ_adm, the route is admitted.
Exact mathematical passage
\Delta_{\Acal} = \min_{c\neq d}d_{\mathrm{sig}}(\sigma_c,\sigma_d).
\sigma_gUse in source · 1 passage
Search aliases: σ_g sigma
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1222
A useful effect profile is vector-valued: g= ( Δ J, Δ C, Δ R, ε_leak, ε_hol, ΔΓ, σ_g ), recording predicted task gain, resource change, risk change, leakage, semantic holonomy, metaproductivity change, and uncertainty. Scalarizing this vector is an evaluator decision, not part of the repair syntax.
Exact mathematical passage
\bm g= \left( \Delta J, \Delta C, \Delta R, \eps_{\mathrm{leak}}, \eps_{\mathrm{hol}}, \Delta\Gamma, \sigma_g \right),
\sigma_iUse in source · 4 passages
Search aliases: σ_i sigma
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10599
Let the source hidden state have dimension d and the target hidden state have dimension q. For selected source anchors x_i∈ℝ^d, suppose the observed target anchor is y_i=Tx_i+ε_i, E[ε_i]=0, Cov(ε_i)=σ_i^2I_q. The query-quality metadata supplies σ_i before the anchor is selected. This metadata is part of the declared Block 9 environment. It is not discovered autonomously.
Exact mathematical passage
\sigma_i
\sigma_i^{-2}Use in source · 2 passages
Search aliases: σ_i^-2 sigma
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10743
Let Msucc0 be the current regularized information matrix. Adding candidate anchor x_i with quality weight w_i=σ_i^-2 gives M'=M+w_ix_ix_i^ T.
Exact mathematical passage
w_i=\sigma_i^{-2}
\sigma_i^2Use in source · 3 passages
Search aliases: σ_i^2 sigma
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10592
Let the source hidden state have dimension d and the target hidden state have dimension q. For selected source anchors x_i∈ℝ^d, suppose the observed target anchor is y_i=Tx_i+ε_i, E[ε_i]=0, Cov(ε_i)=σ_i^2I_q. The query-quality metadata supplies σ_i before the anchor is selected. This metadata is part of the declared Block 9 environment. It is not discovered autonomously.
Exact mathematical passage
y_i=Tx_i+\eps_i, \qquad \mathbb E[\eps_i]=0, \qquad \Cov(\eps_i)=\sigma_i^2I_q.
\sigma_{\min}Use in source · 4 passages
Search aliases: σ_min sigma min
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8472
[Ridge bias under a full-rank design] Under the hypotheses of [source reference], the ridge estimate satisfies widehat T^top-T^top =-λ(X^top X+λ I)^-1T^top, and therefore normwidehat T-T ≤ fracλσ_min(X)^2+λnormT.
Exact mathematical passage
\norm{\widehat T-T} \leq \frac{\lambda}{\sigma_{\min}(X)^2+\lambda}\norm{T}.
\sigma_n^{-2}Use in source · 1 passage
Search aliases: σ_n^-2 sigma
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10606
Stacking the selected anchors gives Y=XT^ T+E, where X∈ℝ^n× d and Y∈ℝ^n× q. Define W=diag(σ_1^-2,…,σ_n^-2) and the weighted information matrix M_S=X^ TWX.
Exact mathematical passage
W=\diag(\sigma_1^{-2},\ldots,\sigma_n^{-2})
\Sigma_{\mathrm{ob}}Use in source · 6 passages
Search aliases: Σ_ob Sigma mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9787
Let foreign residuals follow distribution Q. Define the obstruction covariance operator Σ_ob = E_r∼ Q [ P_I^⊥r r^top P_I^⊥ ]. Let W⊆ I^⊥ be a k-dimensional new response subspace.
Exact mathematical passage
\Sigma_{\mathrm{ob}} = \mathbb E_{r\sim Q} \left[ P_{I^\perp}r\,r^\top P_{I^\perp} \right]. \label{eq:b8-obstruction-covariance}
\Sigma_{\rm pre}Use in source · 1 passage
Search aliases: Σ_rm pre Sigma rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8352
Let Ω = Ω_rm devsqcupΩ_rm audit be a disjoint target index set. The compiler receives rows indexed only by Ω_rm dev. It emits a descendant configuration c_1, a compiler ledger L_Meta, and a pre-audit seal Σ_rm pre = H(c_1,L_Meta,source hashes), where H is SHA-256 over canonical serialization. Audit execution refuses to proceed when the in-memory descendant differs from the sealed configuration.
Exact mathematical passage
\Sigma_{\rm pre} = H\bigl(c_1,L_{\Meta},\text{source hashes}\bigr), \label{eq:block07q-seal}
\sigma_tUse in source · 1 passage
Search aliases: σ_t sigma
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9359
Define a module status map σ_t(i) ∈ {active,saturated,unidentified,protected}. An admissible bottleneck controller should: item estimate headroom for active modules; item refuse large gain claims for saturated modules; item request new evidence for unidentified modules; item avoid changing protected modules inside the current transaction; item trigger grammar expansion when all active modules are saturated while persistent obstruction remains.
Exact mathematical passage
\sigma_t(i) \in \{\texttt{active},\texttt{saturated},\texttt{unidentified},\texttt{protected}\}.
\tau'Use in source · 2 passages
Search aliases: τ' tau
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16172
Let d_prog(τ,τ') be a normalized distance between (A_m,B_m) and (A'_m',B'_m'), and let d_ans(τ,τ') = absy_m/4-y'_m'/4. For a recipient task τ_i, define four donor classes: item SAP: same-answer, far-program, approximately depth-matched donors; item FAD: far-answer, depth-matched donors; item SN: semantic-neighbor donors; item RMD: random depth-matched donors. Three donors per available class are selected by deterministic task-identifier hashing. Each audit records donor coverage and the realized…
Exact mathematical passage
d_{\mathrm{prog}}(\tau,\tau')
\tau_{
m int}Use in source · 1 passage
Search aliases: τ_ m int tau
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15069
Let ϕ_0:T→Z_0 be the inherited diagnostic map. In Block 15, six inherited obstruction classes retain their existing tokens, while two previously unseen obstruction mechanisms are both mapped to one open-set token bot: ϕ_0(τ_ m int) = ϕ_0(τ_ m temp) = bot. The equality in [source reference] is not itself a defect. An unknown token is an appropriate abstention state until the lineage possesses external evidence that the collapsed cases require incompatible research actions.
Exact mathematical passage
\phi_0(\tau_{ m int}) = \phi_0(\tau_{ m temp}) = \bot. \label{eq:b15-unknown-alias}
\tau_{
m temp}Use in source · 1 passage
Search aliases: τ_ m temp tau
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15069
Let ϕ_0:T→Z_0 be the inherited diagnostic map. In Block 15, six inherited obstruction classes retain their existing tokens, while two previously unseen obstruction mechanisms are both mapped to one open-set token bot: ϕ_0(τ_ m int) = ϕ_0(τ_ m temp) = bot. The equality in [source reference] is not itself a defect. An unknown token is an appropriate abstention state until the lineage possesses external evidence that the collapsed cases require incompatible research actions.
Exact mathematical passage
\phi_0(\tau_{ m int}) = \phi_0(\tau_{ m temp}) = \bot. \label{eq:b15-unknown-alias}
\tau_1Use in source · 1 passage
Search aliases: τ_1 tau
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4124
[Serializability of independent repairs] Let τ_1 and τ_2 edit disjoint component masks. Suppose their sandbox state transformations commute, their capability effects are disjoint, and neither transaction changes the sealed witness inputs of the other. Then committing both transactions in either order yields the same operational state and the same pair of transaction receipts up to receipt ordering.
Exact mathematical passage
\tau_1
\tau_2Use in source · 1 passage
Search aliases: τ_2 tau
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4124
[Serializability of independent repairs] Let τ_1 and τ_2 edit disjoint component masks. Suppose their sandbox state transformations commute, their capability effects are disjoint, and neither transaction changes the sealed witness inputs of the other. Then committing both transactions in either order yields the same operational state and the same pair of transaction receipts up to receipt ordering.
Exact mathematical passage
\tau_2
\tau_AUse in source · 2 passages
Search aliases: τ_A tau
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10923
For one candidate row r, a configuration commits only if widehat g_rm cal(r) · ≥ τ_g, widehat R_rm cal(r) · ≤ τ_R, widehatε_rm align(r) · ≤ τ_A, λ_min(M_r) · ≥ τ_I, rank X_r · =d. The selected Block 9 grandchild has (τ_g,τ_R,τ_A,τ_I) = (0.02,0.05,0.20,2.0). The recalibrated rank child has (0.02,0.05,0.55,0.10).
Exact mathematical passage
\begin{aligned} \widehat g_{\rm cal}(r)&\geq \tau_g,\\ \widehat R_{\rm cal}(r)&\leq \tau_R,\\ \widehat\eps_{\rm align}(r)&\leq \tau_A,\\ \lambda_{\min}(M_r)&\geq \tau_I,\\ \rank X_r&=d. \end{aligned}
\tau_aUse in source · 4 passages
Search aliases: τ_a tau
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15100
[Aliased-token routing obstruction] Let τ_a and τ_b satisfy ϕ_0(τ_a)=ϕ_0(τ_b) and A^⋆(τ_a)∩A^⋆(τ_b)=∅. Then no deterministic compiler policy that conditions only on ϕ_0(τ) can select an acceptable method for both problems.
Exact mathematical passage
\tau_a
\tau_bUse in source · 4 passages
Search aliases: τ_b tau
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15100
[Aliased-token routing obstruction] Let τ_a and τ_b satisfy ϕ_0(τ_a)=ϕ_0(τ_b) and A^⋆(τ_a)∩A^⋆(τ_b)=∅. Then no deterministic compiler policy that conditions only on ϕ_0(τ) can select an acceptable method for both problems.
Exact mathematical passage
\tau_b
\tau_gUse in source · 2 passages
Search aliases: τ_g tau
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10923
For one candidate row r, a configuration commits only if widehat g_rm cal(r) · ≥ τ_g, widehat R_rm cal(r) · ≤ τ_R, widehatε_rm align(r) · ≤ τ_A, λ_min(M_r) · ≥ τ_I, rank X_r · =d. The selected Block 9 grandchild has (τ_g,τ_R,τ_A,τ_I) = (0.02,0.05,0.20,2.0). The recalibrated rank child has (0.02,0.05,0.55,0.10).
Exact mathematical passage
\begin{aligned} \widehat g_{\rm cal}(r)&\geq \tau_g,\\ \widehat R_{\rm cal}(r)&\leq \tau_R,\\ \widehat\eps_{\rm align}(r)&\leq \tau_A,\\ \lambda_{\min}(M_r)&\geq \tau_I,\\ \rank X_r&=d. \end{aligned}
\tau_IUse in source · 2 passages
Search aliases: τ_I tau
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10923
For one candidate row r, a configuration commits only if widehat g_rm cal(r) · ≥ τ_g, widehat R_rm cal(r) · ≤ τ_R, widehatε_rm align(r) · ≤ τ_A, λ_min(M_r) · ≥ τ_I, rank X_r · =d. The selected Block 9 grandchild has (τ_g,τ_R,τ_A,τ_I) = (0.02,0.05,0.20,2.0). The recalibrated rank child has (0.02,0.05,0.55,0.10).
Exact mathematical passage
\begin{aligned} \widehat g_{\rm cal}(r)&\geq \tau_g,\\ \widehat R_{\rm cal}(r)&\leq \tau_R,\\ \widehat\eps_{\rm align}(r)&\leq \tau_A,\\ \lambda_{\min}(M_r)&\geq \tau_I,\\ \rank X_r&=d. \end{aligned}
\tau_iUse in source · 3 passages
Search aliases: τ_i tau
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13936
Let Γ_0 and Γ_1 operate on the same problem τ with identical initial archive, candidate grammar, experiment budget, evaluator portfolio, integrity monitor, and hidden audit isolation. Let D_i = U_audit (K_Γ_1^⋆(τ_i)) - U_audit (K_Γ_0^⋆(τ_i)).
Exact mathematical passage
D_i = U_{\mathrm{audit}} \left(K_{\Gamma_1}^{\star}(\tau_i)\right) - U_{\mathrm{audit}} \left(K_{\Gamma_0}^{\star}(\tau_i)\right).
\tau_{\mathrm{imp}}Use in source · 2 passages
Search aliases: τ_imp tau mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4248
Let S be a finite set of complete lineage states. Write sxrightarrowτ_imps' when one authorized improvement transaction can take s to s'. Define Post_imp(s) = sets'∈ S:sxrightarrowτ_imps'.
Exact mathematical passage
s\xrightarrow{\tau_{\mathrm{imp}}}s'
\tau_kUse in source · 1 passage
Search aliases: τ_k tau
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6245
For foreign agent j, let Y_j(z) be protected gain under factor setting z. The average necessity effect of factor k relative to the intact architecture 1 is τ_k = E [ Y_j(1)-Y_j(1-e_k) ]. A positive value supports causal usefulness only relative to the frozen environment, compensating mechanisms, and matched resource schedule.
Exact mathematical passage
\tau_k = \mathbb E \left[ Y_j(\one)-Y_j(\one-e_k) \right].
\tau_RUse in source · 2 passages
Search aliases: τ_R tau
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10923
For one candidate row r, a configuration commits only if widehat g_rm cal(r) · ≥ τ_g, widehat R_rm cal(r) · ≤ τ_R, widehatε_rm align(r) · ≤ τ_A, λ_min(M_r) · ≥ τ_I, rank X_r · =d. The selected Block 9 grandchild has (τ_g,τ_R,τ_A,τ_I) = (0.02,0.05,0.20,2.0). The recalibrated rank child has (0.02,0.05,0.55,0.10).
Exact mathematical passage
\begin{aligned} \widehat g_{\rm cal}(r)&\geq \tau_g,\\ \widehat R_{\rm cal}(r)&\leq \tau_R,\\ \widehat\eps_{\rm align}(r)&\leq \tau_A,\\ \lambda_{\min}(M_r)&\geq \tau_I,\\ \rank X_r&=d. \end{aligned}
\tau_rUse in source · 6 passages
Search aliases: τ_r tau
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7418
widehat R_rm UCB(z) · = max{0,widehat R(z)}+κ_r s_r,c(z). A candidate is admissible when widehat R_rm UCB(z)≤τ_r, widehat r_rm cal(z)≤τ_rm cal, A^rm base_rm cal,prot(z)≥ a_min. The child chooses the admissible candidate with maximum widehat U_rm LCB. If the admissible set is empty, it returns the no-op.
Exact mathematical passage
\widehat R_{\rm UCB}(z)\leq\tau_r, \qquad \widehat r_{\rm cal}(z)\leq\tau_{\rm cal}, \qquad A^{\rm base}_{\rm cal,prot}(z)\geq a_{\min}. \label{eq:block07-admissible}
\tau_{\rm align}Use in source · 1 passage
Search aliases: τ_rm align tau rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8579
The compiler selected: π_rm anc · =π_rm PQR, k_rm fit · =16, k_rm val · =8, γ_rm cal · =0.08, ρ_rm cal · =0.035, τ_rm align · =0.05, full-rank requirement · =true. The selected development score is 0.263289. The underlying mean selected development gain is 0.296478 with zero harmful targets in the development ledger.
Exact mathematical passage
\begin{aligned} \pi_{\rm anc}&=\pi_{\rm PQR},\\ k_{\rm fit}&=16,\\ k_{\rm val}&=8,\\ \gamma_{\rm cal}&=0.08,\\ \rho_{\rm cal}&=0.035,\\ \tau_{\rm align}&=0.05,\\ \text{full-rank requirement}&=\text{true}. \end{aligned} \label{eq:block07q-descendant-config}
\tau_{\rm cal}Use in source · 2 passages
Search aliases: τ_rm cal tau rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7418
widehat R_rm UCB(z) · = max{0,widehat R(z)}+κ_r s_r,c(z). A candidate is admissible when widehat R_rm UCB(z)≤τ_r, widehat r_rm cal(z)≤τ_rm cal, A^rm base_rm cal,prot(z)≥ a_min. The child chooses the admissible candidate with maximum widehat U_rm LCB. If the admissible set is empty, it returns the no-op.
Exact mathematical passage
\widehat R_{\rm UCB}(z)\leq\tau_r, \qquad \widehat r_{\rm cal}(z)\leq\tau_{\rm cal}, \qquad A^{\rm base}_{\rm cal,prot}(z)\geq a_{\min}. \label{eq:block07-admissible}
\tau_{\rm distance}Use in source · 1 passage
Search aliases: τ_rm distance tau rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 14654
The selected child configuration is τ_rm residual · =0.20, · τ_rm distance · =24, w_rm mean · =0.38, · w_rm median · =0.10, w_rm min · =0.24, · w_rm held · =0.22, w_rm capability · =0.32. It is selected from a 27-point finite meta-validation grid before either final audit is opened.
Exact mathematical passage
\begin{aligned} \tau_{\rm residual}&=0.20, &\tau_{\rm distance}&=24,\\ w_{\rm mean}&=0.38, &w_{\rm median}&=0.10,\\ w_{\rm min}&=0.24, &w_{\rm held}&=0.22,\\ w_{\rm capability}&=0.32. \end{aligned} \label{eq:block14-config}
\tau_{\rm residual}Use in source · 1 passage
Search aliases: τ_rm residual tau rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 14654
The selected child configuration is τ_rm residual · =0.20, · τ_rm distance · =24, w_rm mean · =0.38, · w_rm median · =0.10, w_rm min · =0.24, · w_rm held · =0.22, w_rm capability · =0.32. It is selected from a 27-point finite meta-validation grid before either final audit is opened.
Exact mathematical passage
\begin{aligned} \tau_{\rm residual}&=0.20, &\tau_{\rm distance}&=24,\\ w_{\rm mean}&=0.38, &w_{\rm median}&=0.10,\\ w_{\rm min}&=0.24, &w_{\rm held}&=0.22,\\ w_{\rm capability}&=0.32. \end{aligned} \label{eq:block14-config}
\phi^{-1}_{\alpha}Use in source · 1 passage
Search aliases: ϕ^-1_α phi alpha
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 18278
The nonlinear family applies Y_t = fracsinh (α(X_tA+b))α, α>0, coordinatewise. The declared inverse constructor first applies ϕ^-1_α(y) = fracarsinh(α y)α and then fits a step-affine map.
Exact mathematical passage
\phi^{-1}_{\alpha}(y) = \frac{\operatorname{arsinh}(\alpha y)}{\alpha}
\phi^{\circ}Use in source · 4 passages
Search aliases: ϕ^∘ phi circ
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6592
The permitted intervention removes only the conditional feature block: ϕ^∘(x) = ψ(x) 0 . The resulting hidden difference is Δ_a(x) = h_a(x)-h_a^∘(x), h_a^∘(x)=E_a(ϕ^∘(x)). The empirical causal packet is the leading rank-r right-singular subspace of the affected differences {Δ_a(x_i):s(x_i)=1}.
Exact mathematical passage
\phi^{\circ}(x) = \begin{bmatrix} \psi(x)\\0 \end{bmatrix}. \label{eq:block06-counterfactual}
\phi_{\mathrm{actuation}}Use in source · 1 passage
Search aliases: ϕ_actuation phi mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5953
From a common probe state x_0 and bounded action a, define the feature vector ϕ(z) = ϕ_passive ϕ_actuation ϕ_sensor ϕ_memory ϕ_evaluator . Its components are, respectively: item deviation from the baseline no-action continuation; item deviation of the action-conditioned response from the baseline action response; item violation of redundant-sensor consistency; item deviation from baseline memory update; item discrepancy between reported and protected evaluator scores. The observed probe is widet…
Exact mathematical passage
\phi(z) = \begin{bmatrix} \phi_{\mathrm{passive}}\\ \phi_{\mathrm{actuation}}\\ \phi_{\mathrm{sensor}}\\ \phi_{\mathrm{memory}}\\ \phi_{\mathrm{evaluator}} \end{bmatrix}.
\Phi_{\mathrm{disc}}Use in source · 5 passages
Search aliases: Φ_disc Phi mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5165
Let Ω be a finite universe of candidate repair-schema equivalence classes. For S⊆Ω, let Disc(S) be the schemas independently validated using the currently available set S. Define Φ_disc(S) = S∪Disc(S).
Exact mathematical passage
\Phi_{\mathrm{disc}}(S) = S\cup\operatorname{Disc}(S).
\phi_{\mathrm{evaluator}}Use in source · 1 passage
Search aliases: ϕ_evaluator phi mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5953
From a common probe state x_0 and bounded action a, define the feature vector ϕ(z) = ϕ_passive ϕ_actuation ϕ_sensor ϕ_memory ϕ_evaluator . Its components are, respectively: item deviation from the baseline no-action continuation; item deviation of the action-conditioned response from the baseline action response; item violation of redundant-sensor consistency; item deviation from baseline memory update; item discrepancy between reported and protected evaluator scores. The observed probe is widet…
Exact mathematical passage
\phi(z) = \begin{bmatrix} \phi_{\mathrm{passive}}\\ \phi_{\mathrm{actuation}}\\ \phi_{\mathrm{sensor}}\\ \phi_{\mathrm{memory}}\\ \phi_{\mathrm{evaluator}} \end{bmatrix}.
\phi_{\Gcal}Use in source · 7 passages
Search aliases: ϕ_Gcal ϕ_G phi Gcal phi mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5104
Let ϕ_G:X→ Z_G be the state representation expressible in grammar G. A G-measurable deterministic repair policy has the form π=widetildeπ∘ϕ_G.
Exact mathematical passage
\phi_{\Gcal}:X\to Z_{\Gcal}
\phi_{\Gcal'}Use in source · 2 passages
Search aliases: ϕ_Gcal' ϕ_G' phi Gcal phi mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9895
Appending the feature gives the conservative extension ϕ_G'(x) = (ϕ_G(x),f(x)). The old representation is recovered by projection onto the first coordinate.
Exact mathematical passage
\phi_{\Gcal'}(x) = \bigl(\phi_{\Gcal}(x),f(x)\bigr). \label{eq:b8-feature-extension}
\phi_{\mathrm{memory}}Use in source · 1 passage
Search aliases: ϕ_memory phi mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5953
From a common probe state x_0 and bounded action a, define the feature vector ϕ(z) = ϕ_passive ϕ_actuation ϕ_sensor ϕ_memory ϕ_evaluator . Its components are, respectively: item deviation from the baseline no-action continuation; item deviation of the action-conditioned response from the baseline action response; item violation of redundant-sensor consistency; item deviation from baseline memory update; item discrepancy between reported and protected evaluator scores. The observed probe is widet…
Exact mathematical passage
\phi(z) = \begin{bmatrix} \phi_{\mathrm{passive}}\\ \phi_{\mathrm{actuation}}\\ \phi_{\mathrm{sensor}}\\ \phi_{\mathrm{memory}}\\ \phi_{\mathrm{evaluator}} \end{bmatrix}.
\phi_{p,\theta}Use in source · 2 passages
Search aliases: ϕ_p,θ phi theta
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15120
Let p∈P^+ be a candidate experiment with response y_p:T⟶ℝ. For threshold θ, define the refined diagnostic map ϕ_p,θ(τ) = (ϕ_0(τ),low), · y_p(τ)≤θ, (ϕ_0(τ),high), · y_p(τ)>θ. The old token is recovered by the projection π_0(z,s)=z, π_0∘ϕ_p,θ=ϕ_0. Thus the extension is conservative at the level of old diagnostic semantics.
Exact mathematical passage
\phi_{p,\theta}(\tau) = \begin{cases} (\phi_0(\tau),\mathrm{low}), & y_p(\tau)\leq\theta,\\ (\phi_0(\tau),\mathrm{high}), & y_p(\tau)>\theta. \end{cases} \label{eq:b15-refined-token}
\phi_{\mathrm{passive}}Use in source · 1 passage
Search aliases: ϕ_passive phi mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5953
From a common probe state x_0 and bounded action a, define the feature vector ϕ(z) = ϕ_passive ϕ_actuation ϕ_sensor ϕ_memory ϕ_evaluator . Its components are, respectively: item deviation from the baseline no-action continuation; item deviation of the action-conditioned response from the baseline action response; item violation of redundant-sensor consistency; item deviation from baseline memory update; item discrepancy between reported and protected evaluator scores. The observed probe is widet…
Exact mathematical passage
\phi(z) = \begin{bmatrix} \phi_{\mathrm{passive}}\\ \phi_{\mathrm{actuation}}\\ \phi_{\mathrm{sensor}}\\ \phi_{\mathrm{memory}}\\ \phi_{\mathrm{evaluator}} \end{bmatrix}.
\phi_{\mathrm{sensor}}Use in source · 1 passage
Search aliases: ϕ_sensor phi mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5953
From a common probe state x_0 and bounded action a, define the feature vector ϕ(z) = ϕ_passive ϕ_actuation ϕ_sensor ϕ_memory ϕ_evaluator . Its components are, respectively: item deviation from the baseline no-action continuation; item deviation of the action-conditioned response from the baseline action response; item violation of redundant-sensor consistency; item deviation from baseline memory update; item discrepancy between reported and protected evaluator scores. The observed probe is widet…
Exact mathematical passage
\phi(z) = \begin{bmatrix} \phi_{\mathrm{passive}}\\ \phi_{\mathrm{actuation}}\\ \phi_{\mathrm{sensor}}\\ \phi_{\mathrm{memory}}\\ \phi_{\mathrm{evaluator}} \end{bmatrix}.
\Phi_tUse in source · 4 passages
Search aliases: Φ_t Phi
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9334
[Greedy bottleneck contraction] Let r_t∈ℝ_≥0^m and Φ_t=1^top r_t. Suppose the controller chooses i_t with r_i_t,t≥fracΦ_tm, contracts that component by q∈[0,1), and permits total collateral increase at most ε_t. Then Φ_t+1 ≤ (1-frac1-qm)Φ_t+ε_t.
Exact mathematical passage
\Phi_t=\one^\top r_t.
\Phi_{t+1}Use in source · 1 passage
Search aliases: Φ_t+1 Phi
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9342
[Greedy bottleneck contraction] Let r_t∈ℝ_≥0^m and Φ_t=1^top r_t. Suppose the controller chooses i_t with r_i_t,t≥fracΦ_tm, contracts that component by q∈[0,1), and permits total collateral increase at most ε_t. Then Φ_t+1 ≤ (1-frac1-qm)Φ_t+ε_t.
Exact mathematical passage
\Phi_{t+1} \leq \left(1-\frac{1-q}{m}\right)\Phi_t+\eps_t. \label{eq:b8-greedy-contraction}
\phi_{t+1}Use in source · 1 passage
Search aliases: ϕ_t+1 phi
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1143
A useful self-model participates in a complete causal cycle: ϕ_t(A_t) xrightarrowcandidate policy δ_t xrightarrowworld execution Y_t+1 xrightarrowevidence update ϕ_t+1(A_t+1). The candidate is selected before the protected outcome is revealed. The outcome affects future beliefs and repair policy. Cutting either planner use or consequence feedback destroys prospective closure even if the internal self-representation remains decodable.
Exact mathematical passage
\phi_t(A_t) \xrightarrow{\text{candidate policy}} \delta_t \xrightarrow{\text{world execution}} Y_{t+1} \xrightarrow{\text{evidence update}} \phi_{t+1}(A_{t+1}).
\chi_RUse in source · 6 passages
Search aliases: χ_R chi
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9208
Let K→ K' change only modules in R. When F_K(R)>M(K), define the headroom-adjusted gain χ_R(K,K') = frac M(K')-M(K) F_K(R)-M(K) . For an admissible child inside the declared class, χ_R(K,K')≤1. A small absolute gain with χ_R≈1 indicates a nearly saturated parent. A small absolute gain with χ_R≈0 indicates that the child captured little of the available opportunity.
Exact mathematical passage
\chi_R(K,K') = \frac{ \Mcal(K')-\Mcal(K) }{ F_K(R)-\Mcal(K) }. \label{eq:b8-headroom-adjusted-gain}
\PsiUse in source · 10 passages
Search aliases: Ψ Psi
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4274
· wedgeRollbackable(s) wedgeMetaCapable(s). For X⊆ S, define Ψ_∀(X) = sets∈ S:C_RI(s) and Post_imp(s)⊆ X.
Exact mathematical passage
\Psi_{\forall}(X) = \set{s\in S:C_{\mathrm{RI}}(s)\text{ and }\Post_{\mathrm{imp}}(s)\subseteq X}.
\psiUse in source · 20 passages
Search aliases: ψ psi
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2142
R⊕ C, R=im D, so that restrDX:X→ R is an isomorphism. Let P_R and P_C be the corresponding projections. Then there are neighborhoods of the origin and a unique C^k map χ:P× K→ X, χ(0,0)=0, such that P_RF(p,k+χ(p,k))=0. Define the reduced obstruction map ψ(p,k):=P_CF(p,k+χ(p,k)). A sufficiently small pair (p,u) solves F(p,u)=0 if and only if u=k+χ(p,k) for some k∈ K satisfying ψ(p,k)=0.
Exact mathematical passage
\psi(p,k):=P_CF(p,k+\chi(p,k)).
\Psi_{\mathrm{ctrl}}Use in source · 3 passages
Search aliases: Ψ_ctrl Psi mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4322
Define Ψ_ctrl(X) = sets∈ S:C_RI(s) ∩ CPre(X).
Exact mathematical passage
\Psi_{\mathrm{ctrl}}(X) = \set{s\in S:C_{\mathrm{RI}}(s)} \cap \operatorname{CPre}(X).
\Psi_{\forall}Use in source · 5 passages
Search aliases: Ψ_∀ Psi forall
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4274
· wedgeRollbackable(s) wedgeMetaCapable(s). For X⊆ S, define Ψ_∀(X) = sets∈ S:C_RI(s) and Post_imp(s)⊆ X.
Exact mathematical passage
\Psi_{\forall}(X) = \set{s\in S:C_{\mathrm{RI}}(s)\text{ and }\Post_{\mathrm{imp}}(s)\subseteq X}.
\omega_0Use in source · 2 passages
Search aliases: ω_0 omega
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21028
Suppose a sandbox returns only the scalar task score q(ω)=accuracy(ω). Execution ω_0 reaches accuracy 0.9 using authorized retrieval. Execution ω_1 reaches the same accuracy after importing a prohibited hidden answer key. Then q(ω_0)=q(ω_1)=0.9, but the ideal full-information decisions differ. By [source reference], no witness using only accuracy can accept the first and reject the second. Adding signed tool-call provenance or a complete effect receipt refines the trace and can separate the exec…
Exact mathematical passage
\omega_0
\omega_{CH}Use in source · 1 passage
Search aliases: ω_CH omega
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21495
Consider five mutable modules: N_5 = {S,Q,T,C,H}, representing selection, information acquisition, transport, candidate construction, and causal-atlas structure. Let F_K(∅)=0.20. Assign singleton contributions (0.08,0.12,0.10,0.16,0.14) and pair synergies ω_QT=0.05, ω_CH=0.06, ω_SC=0.03. Then F_K(N_5)=0.94, H(K)=0.74. Every pair synergy is divided equally between its two participants by the permutation-average rule. Therefore h_S · =0.08+frac0.032=0.095,
Exact mathematical passage
\omega_{QT}=0.05, \qquad \omega_{CH}=0.06, \qquad \omega_{SC}=0.03.
\omega_{\mathrm{check}}Use in source · 1 passage
Search aliases: ω_check omega mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1485
Consider four colors memory, representation, verifier, planner. A composite factuality repair may have the form ω_gate ∘ ( ω_retrieve, ω_decompose, ω_check ). The inner operations retrieve evidence, decompose output into claims, and check claims against protected sources. Their postconditions establish the precondition for a planner gate that blocks or revises unsupported claims. The composite receipt records source provenance, claim-to-evidence links, verifier uncertainty, revised spans, latenc…
Exact mathematical passage
\omega_{\mathrm{gate}} \circ \left( \omega_{\mathrm{retrieve}}, \omega_{\mathrm{decompose}}, \omega_{\mathrm{check}} \right).
\omega_{\mathrm{decompose}}Use in source · 1 passage
Search aliases: ω_decompose omega mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1485
Consider four colors memory, representation, verifier, planner. A composite factuality repair may have the form ω_gate ∘ ( ω_retrieve, ω_decompose, ω_check ). The inner operations retrieve evidence, decompose output into claims, and check claims against protected sources. Their postconditions establish the precondition for a planner gate that blocks or revises unsupported claims. The composite receipt records source provenance, claim-to-evidence links, verifier uncertainty, revised spans, latenc…
Exact mathematical passage
\omega_{\mathrm{gate}} \circ \left( \omega_{\mathrm{retrieve}}, \omega_{\mathrm{decompose}}, \omega_{\mathrm{check}} \right).
\omega_{\mathrm{gate}}Use in source · 1 passage
Search aliases: ω_gate omega mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1485
Consider four colors memory, representation, verifier, planner. A composite factuality repair may have the form ω_gate ∘ ( ω_retrieve, ω_decompose, ω_check ). The inner operations retrieve evidence, decompose output into claims, and check claims against protected sources. Their postconditions establish the precondition for a planner gate that blocks or revises unsupported claims. The composite receipt records source provenance, claim-to-evidence links, verifier uncertainty, revised spans, latenc…
Exact mathematical passage
\omega_{\mathrm{gate}} \circ \left( \omega_{\mathrm{retrieve}}, \omega_{\mathrm{decompose}}, \omega_{\mathrm{check}} \right).
\omega_nUse in source · 2 passages
Search aliases: ω_n omega
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1259
Each executable operation has a Hoare-style contract {C_ω} ω {Q_ω}. Syntactic substitution ω∘(ω_1,…,ω_n) is semantically authorized only when the inner postconditions and bridge guarantees imply the outer precondition.
Exact mathematical passage
\omega\circ(\omega_1,\ldots,\omega_n)
\omega_{QT}Use in source · 1 passage
Search aliases: ω_QT omega
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21495
Consider five mutable modules: N_5 = {S,Q,T,C,H}, representing selection, information acquisition, transport, candidate construction, and causal-atlas structure. Let F_K(∅)=0.20. Assign singleton contributions (0.08,0.12,0.10,0.16,0.14) and pair synergies ω_QT=0.05, ω_CH=0.06, ω_SC=0.03. Then F_K(N_5)=0.94, H(K)=0.74. Every pair synergy is divided equally between its two participants by the permutation-average rule. Therefore h_S · =0.08+frac0.032=0.095,
Exact mathematical passage
\omega_{QT}=0.05, \qquad \omega_{CH}=0.06, \qquad \omega_{SC}=0.03.
\omega_{\mathrm{retrieve}}Use in source · 1 passage
Search aliases: ω_retrieve omega mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1485
Consider four colors memory, representation, verifier, planner. A composite factuality repair may have the form ω_gate ∘ ( ω_retrieve, ω_decompose, ω_check ). The inner operations retrieve evidence, decompose output into claims, and check claims against protected sources. Their postconditions establish the precondition for a planner gate that blocks or revises unsupported claims. The composite receipt records source provenance, claim-to-evidence links, verifier uncertainty, revised spans, latenc…
Exact mathematical passage
\omega_{\mathrm{gate}} \circ \left( \omega_{\mathrm{retrieve}}, \omega_{\mathrm{decompose}}, \omega_{\mathrm{check}} \right).
\Omega_{\rm audit}Use in source · 1 passage
Search aliases: Ω_rm audit Omega rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8346
Let Ω = Ω_rm devsqcupΩ_rm audit be a disjoint target index set. The compiler receives rows indexed only by Ω_rm dev. It emits a descendant configuration c_1, a compiler ledger L_Meta, and a pre-audit seal Σ_rm pre = H(c_1,L_Meta,source hashes), where H is SHA-256 over canonical serialization. Audit execution refuses to proceed when the in-memory descendant differs from the sealed configuration.
Exact mathematical passage
\Omega = \Omega_{\rm dev}\sqcup\Omega_{\rm audit}
\Omega_{\rm dev}Use in source · 2 passages
Search aliases: Ω_rm dev Omega rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8346
Let Ω = Ω_rm devsqcupΩ_rm audit be a disjoint target index set. The compiler receives rows indexed only by Ω_rm dev. It emits a descendant configuration c_1, a compiler ledger L_Meta, and a pre-audit seal Σ_rm pre = H(c_1,L_Meta,source hashes), where H is SHA-256 over canonical serialization. Audit execution refuses to proceed when the in-memory descendant differs from the sealed configuration.
Exact mathematical passage
\Omega = \Omega_{\rm dev}\sqcup\Omega_{\rm audit}
\omega_{SC}Use in source · 1 passage
Search aliases: ω_SC omega
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21495
Consider five mutable modules: N_5 = {S,Q,T,C,H}, representing selection, information acquisition, transport, candidate construction, and causal-atlas structure. Let F_K(∅)=0.20. Assign singleton contributions (0.08,0.12,0.10,0.16,0.14) and pair synergies ω_QT=0.05, ω_CH=0.06, ω_SC=0.03. Then F_K(N_5)=0.94, H(K)=0.74. Every pair synergy is divided equally between its two participants by the permutation-average rule. Therefore h_S · =0.08+frac0.032=0.095,
Exact mathematical passage
\omega_{QT}=0.05, \qquad \omega_{CH}=0.06, \qquad \omega_{SC}=0.03.
\dotsUse in source · 3 passages
Search aliases: … dots
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4411
[Finite ranking certificate for one gain cycle] Let X⊆I_ctrl^* and let h:X→{0,1,…,H} satisfy h^-1(0)⊆ G_γ. Suppose that at every state with h(s)>0, the controller has an authorized action whose every admissible successor s' lies in X and satisfies h(s')<h(s). Then the controller can force a visit to G_γ within at most H transactions while remaining in X.
Exact mathematical passage
h:X\to\{0,1,\dots,H\}
\ell^{\mathsf T}Use in source · 3 passages
Search aliases: ℓ^ T ell mathsf
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5273
[Perron sensitivity] Let M be irreducible with simple Perron root ρ, positive right eigenvector r, and positive left eigenvector ℓ, normalized by ℓ^ Tr=1. Then frac∂ρ∂ M_ij = ℓ_i r_j.
Exact mathematical passage
\ell^{\mathsf T}r=1
\ell_AUse in source · 5 passages
Search aliases: ℓ_A ell
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3168
Let ℓ_A:V_A→ℝ and ℓ_B:V_B→ℝ be post-defect losses. Define source and target gains g_A := ℓ_A(r_A)-ℓ_A(r_A+D_Au_A), g_B := ℓ_B(r_B)-ℓ_B(r_B+D_BT_Uu_A).
Exact mathematical passage
\ell_A:V_A\to\R
\ell_BUse in source · 6 passages
Search aliases: ℓ_B ell
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3168
Let ℓ_A:V_A→ℝ and ℓ_B:V_B→ℝ be post-defect losses. Define source and target gains g_A := ℓ_A(r_A)-ℓ_A(r_A+D_Au_A), g_B := ℓ_B(r_B)-ℓ_B(r_B+D_BT_Uu_A).
Exact mathematical passage
\ell_B:V_B\to\R
\ell_{\mathrm{chart}}Use in source · 1 passage
Search aliases: ℓ_chart ell mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 18043
For every final task, let ℓ_global and ℓ_chart be the source-probe semantic losses after pooled and selected transport. The paired effect is Δ_i = ℓ_global,i - ℓ_chart,i. Positive values favor the selected chart-indexed map.
Exact mathematical passage
\ell_{\mathrm{chart}}
\ell_{\mathrm{chart},i}Use in source · 1 passage
Search aliases: ℓ_chart,i ell mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 18044
For every final task, let ℓ_global and ℓ_chart be the source-probe semantic losses after pooled and selected transport. The paired effect is Δ_i = ℓ_global,i - ℓ_chart,i. Positive values favor the selected chart-indexed map.
Exact mathematical passage
\Delta_i = \ell_{\mathrm{global},i} - \ell_{\mathrm{chart},i}.
\ell_{\mathrm{global}}Use in source · 1 passage
Search aliases: ℓ_global ell mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 18043
For every final task, let ℓ_global and ℓ_chart be the source-probe semantic losses after pooled and selected transport. The paired effect is Δ_i = ℓ_global,i - ℓ_chart,i. Positive values favor the selected chart-indexed map.
Exact mathematical passage
\ell_{\mathrm{global}}
\ell_{\mathrm{global},i}Use in source · 1 passage
Search aliases: ℓ_global,i ell mathrm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 18044
For every final task, let ℓ_global and ℓ_chart be the source-probe semantic losses after pooled and selected transport. The paired effect is Δ_i = ℓ_global,i - ℓ_chart,i. Positive values favor the selected chart-indexed map.
Exact mathematical passage
\Delta_i = \ell_{\mathrm{global},i} - \ell_{\mathrm{chart},i}.
\ell_iUse in source · 5 passages
Search aliases: ℓ_i ell
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5274
[Perron sensitivity] Let M be irreducible with simple Perron root ρ, positive right eigenvector r, and positive left eigenvector ℓ, normalized by ℓ^ Tr=1. Then frac∂ρ∂ M_ij = ℓ_i r_j.
Exact mathematical passage
\frac{\partial\rho}{\partial M_{ij}} = \ell_i r_j.
\ell_{\rm selected,i}Use in source · 1 passage
Search aliases: ℓ_rm selected,i ell rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 18506
For nonlinear, temporal, and chart-alias episodes, the inherited step-affine map is retained as a wrong-repair control. The paired effect is Δ_i = ℓ_rm step affine,i - ℓ_rm selected,i. Every frozen bootstrap interval is strictly positive.
Exact mathematical passage
\Delta_i = \ell_{\rm step\ affine,i} - \ell_{\rm selected,i}.
\ell_{\rm step\ affine,i}Use in source · 1 passage
Search aliases: ℓ_rm step affine,i ell rm
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 18506
For nonlinear, temporal, and chart-alias episodes, the inherited step-affine map is retained as a wrong-repair control. The paired effect is Δ_i = ℓ_rm step affine,i - ℓ_rm selected,i. Every frozen bootstrap interval is strictly positive.
Exact mathematical passage
\Delta_i = \ell_{\rm step\ affine,i} - \ell_{\rm selected,i}.
\leftarrowUse in source · 3 passages
Search aliases: ← leftarrow
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1171
[Internal error reduction without external improvement] Let the environment always produce protected outcome Y=1, independently of the agent's action. Let an internal predictor be p∈ℝ with internal score J_int(p)=-(p-Y)^2. A self-intervention setting p←1 improves J_int to zero error but does not change the world, task success, or action consequence. More extremely, an intervention that rewrites the error register itself can report zero error without even improving prediction. Therefore reduction…
Exact mathematical passage
p\leftarrow1
\twoheadrightarrowUse in source · 2 passages
Search aliases: ↠ twoheadrightarrow
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 694
Quotient or forgetful map · X↠ Y · Forward transport with declared information loss; reconstruction requires additional data.
Exact mathematical passage
X\twoheadrightarrow Y
\mapstoUse in source · 7 passages
Search aliases: ↦ mapsto
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1693
[Convex authorized correctability] If K is compact, or if u↦normDu+r is coercive on K, then the infimum defining η_D,K(r) is attained. Exact authorized correction exists if and only if η_D,K(r)=0.
Exact mathematical passage
u\mapsto\norm{Du+r}
\RightarrowUse in source · 1 passage
Search aliases: ⇒ Rightarrow
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21345
The example separates two predicates: ImprovesSlice(R_t) not⇒ AuthorizedCommit(R_t).
Exact mathematical passage
\mathrm{ImprovesSlice}(R_t) \not\Rightarrow \mathrm{AuthorizedCommit}(R_t).
\partialUse in source · 4 passages
Search aliases: ∂ partial
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2116
[Lyapunov--Schmidt repair reduction] Let F:P× U→ V be C^k, k≥2, between finite-dimensional spaces, with F(0,0)=0. Let D:=∂_uF(0,0). Choose splittings U=K⊕ X, K=ker D, and V=R⊕ C, R=im D, so that restrDX:X→ R is an isomorphism. Let P_R and P_C be the corresponding projections. Then there are neighborhoods of the origin and a unique C^k map χ:P× K→ X, χ(0,0)=0, such that P_RF(p,k+χ(p,k))=0. Define the reduced obstruction map ψ(p,k):=P_CF(p,k+χ(p,k)). A sufficiently small pair (p,u) solves F(p,u)=0…
Exact mathematical passage
D:=\partial_uF(0,0).
\partial_{(x,y)}Use in source · 1 passage
Search aliases: ∂_(x,y) partial
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 20775
x^2 . At (ε,x,y)=(0,0,0), D=∂_(x,y)F = 1 · 0 0 · 0 . The first-order defect r_1=e_1 is correctable by choosing x_1=-1. However, the order-ε^2 term in the second coordinate is 1, which represents a nonzero class in coker D. Indeed, the exact first equation forces x=-ε, and then the second equation becomes ε^2=0. Thus no nonzero small ε admits a repair despite vanishing of the first-order obstruction.
Exact mathematical passage
D=\partial_{(x,y)}F = \begin{bmatrix} 1&0\\0&0 \end{bmatrix}.
\partial_uUse in source · 2 passages
Search aliases: ∂_u partial
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2116
[Lyapunov--Schmidt repair reduction] Let F:P× U→ V be C^k, k≥2, between finite-dimensional spaces, with F(0,0)=0. Let D:=∂_uF(0,0). Choose splittings U=K⊕ X, K=ker D, and V=R⊕ C, R=im D, so that restrDX:X→ R is an isomorphism. Let P_R and P_C be the corresponding projections. Then there are neighborhoods of the origin and a unique C^k map χ:P× K→ X, χ(0,0)=0, such that P_RF(p,k+χ(p,k))=0. Define the reduced obstruction map ψ(p,k):=P_CF(p,k+χ(p,k)). A sufficiently small pair (p,u) solves F(p,u)=0…
Exact mathematical passage
D:=\partial_uF(0,0).
\prodUse in source · 6 passages
Search aliases: ∏ prod
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12206
of hyperplanes setc:c^ T(λ_a-λ_b)=0, a≠ b. Then the scalars μ_a=c^ Tλ_a are pairwise distinct. The combined response B=∑_j=1^kc_jR_j =∑_a=1^mμ_aP_a belongs to A_R. For each a, define the Lagrange polynomial p_a(z)=∏_b≠ afracz-μ_bμ_a-μ_b. Functional calculus gives p_a(B)=∑_b=1^mp_a(μ_b)P_b=P_a. Thus every P_a lies in the generated algebra. A nonzero central projection that is strictly below one P_a would have to be scalar on H_a and therefore is either 0 or P_a. Hence the P_a are the minimal cent…
Exact mathematical passage
p_a(z)=\prod_{b\neq a}\frac{z-\mu_b}{\mu_a-\mu_b}.
\prod_{b\neq a}Use in source · 2 passages
Search aliases: ∏_b≠ a prod neq
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12206
of hyperplanes setc:c^ T(λ_a-λ_b)=0, a≠ b. Then the scalars μ_a=c^ Tλ_a are pairwise distinct. The combined response B=∑_j=1^kc_jR_j =∑_a=1^mμ_aP_a belongs to A_R. For each a, define the Lagrange polynomial p_a(z)=∏_b≠ afracz-μ_bμ_a-μ_b. Functional calculus gives p_a(B)=∑_b=1^mp_a(μ_b)P_b=P_a. Thus every P_a lies in the generated algebra. A nonzero central projection that is strictly below one P_a would have to be scalar on H_a and therefore is either 0 or P_a. Hence the P_a are the minimal cent…
Exact mathematical passage
p_a(z)=\prod_{b\neq a}\frac{z-\mu_b}{\mu_a-\mu_b}.
\prod_{j=1}^sUse in source · 2 passages
Search aliases: ∏_j=1^s prod
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 14605
For integrity monitoring, let M_1,…,M_s∈{0,1} be monitor pass indicators and let the ensemble pass only when every component passes: M_∩=∏_j=1^s M_j.
Exact mathematical passage
M_{\cap}=\prod_{j=1}^s M_j.
\prod_{j\in I_+(c)}Use in source · 1 passage
Search aliases: ∏_j∈ I_+(c) prod in
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 13805
Let the scored evaluator portfolio be B = {b_1,…,b_m}. For method c, define normalized headroom closure q_j(c) = fracs_j(c)-s_j(K_τ)s_j^⋆-s_j(K_τ). Let I_+(c)={j:q_j(c)>0}. The coverage-weighted geometric score is S_cov(c) = frac|I_+(c)|m ( ∏_j∈ I_+(c)q_j(c) )^1/|I_+(c)|, with value zero when I_+(c)=∅. Hard capability and integrity gates are applied separately rather than traded against this scalar.
Exact mathematical passage
S_{\mathrm{cov}}(c) = \frac{|I_+(c)|}{m} \left( \prod_{j\in I_+(c)}q_j(c) \right)^{1/|I_+(c)|}, \label{eq:b13-coverage-geometric}
\prod_{q<j}Use in source · 1 passage
Search aliases: ∏_q<j prod
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15790
For fixed maximum depth K_max, let λ_t(x)=Λ(z_t(x)). Define cumulative halt mass M_t(x) = ∑_j=1^t λ_j(x) ∏_q<j(1-λ_q(x)), and dynamic depth T(x)=min{t:M_t(x)≥τ}, with declared threshold τ. Block 16 does not infer adaptive computation from the existence of a halt head. It requires a difficulty variable d(x) fixed before result access and tests both ρ_Spearman(d(x),T(x))>0 and nondegenerate prompt variation in the halt schedules. A constant depth can coexist with fully task-specific Neuralese; con…
Exact mathematical passage
M_t(x) = \sum_{j=1}^{t} \lambda_j(x) \prod_{q<j}(1-\lambda_q(x)),
\sum_aUse in source · 1 passage
Search aliases: ∑_a sum
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12310
For recovered projectors widehat P_1,…,widehat P_m, define the normalized response-fit residual ε_resp = frac ( ∑_jnormwidehat R_j- ∑_awidehatλ_ajwidehat P_a_F^2 )^1/2 ( ∑_jnormwidehat R_j_F^2 )^1/2+10^-12 , and the maximum normalized commutator residual ε_comm = max_a,j fracnormwidehat P_awidehat R_j- widehat R_jwidehat P_a_F normwidehat R_j_F+10^-12. The smallest selected spectral gap is denoted widehatγ_min. These quantities serve different roles: item ε_resp measures whether the finite block…
Exact mathematical passage
\eps_{\mathrm{resp}} = \frac{ \left( \sum_j\norm{\widehat R_j- \sum_a\widehat\lambda_{aj}\widehat P_a}_F^2 \right)^{1/2} }{ \left( \sum_j\norm{\widehat R_j}_F^2 \right)^{1/2}+10^{-12} }, \label{eq:b11-response-fit}
\sum_{a,b}Use in source · 1 passage
Search aliases: ∑_a,b sum
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12249
Write X=∑_a,bP_aXP_b. The relation XR_j=R_jX implies (λ_bj-λ_aj)P_aXP_b=0 for every j. Separating signatures imply that for a≠ b some coefficient is nonzero, so P_aXP_b=0. Hence X is block diagonal. Every block-diagonal operator commutes with each block-scalar R_j. The dimension formula follows from dimEnd(H_a)=d_a^2.
Exact mathematical passage
X=\sum_{a,b}P_aXP_b
\sum_{b=1}^{m}Use in source · 1 passage
Search aliases: ∑_b=1^m sum
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12210
Then the scalars μ_a=c^ Tλ_a are pairwise distinct. The combined response B=∑_j=1^kc_jR_j =∑_a=1^mμ_aP_a belongs to A_R. For each a, define the Lagrange polynomial p_a(z)=∏_b≠ afracz-μ_bμ_a-μ_b. Functional calculus gives p_a(B)=∑_b=1^mp_a(μ_b)P_b=P_a. Thus every P_a lies in the generated algebra. A nonzero central projection that is strictly below one P_a would have to be scalar on H_a and therefore is either 0 or P_a. Hence the P_a are the minimal central projections.
Exact mathematical passage
p_a(B)=\sum_{b=1}^{m}p_a(\mu_b)P_b=P_a.
\sum_{e\in E(\Lambda)}Use in source · 1 passage
Search aliases: ∑_e∈ E(Λ) sum in Lambda
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10253
For a branching archive, index proposed edges by e∈ E(Λ). Assign each edge a budget δ_e>0 with ∑_e∈ E(Λ)δ_e≤δ. The same proof yields archive-wide false-acceptance control. The allocation may be adaptive provided the remaining total budget is predictable from the pre-audit filtration.
Exact mathematical passage
\sum_{e\in E(\Lambda)}\delta_e\leq\delta.
\sum_{i=1}^{5}Use in source · 1 passage
Search aliases: ∑_i=1^5 sum
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 17307
Each task contains five demonstration pairs D={(x_i,y_i)}_i=1^5 and a query q. After chart canonicalization, the full descendant computes pair codes p_i=ϕ(widehat x_i,widehat y_i)∈ℝ^32, and the permutation-invariant aggregate a(D) = ( frac15∑_i=1^5p_i, max_1≤ i≤5p_i )∈ℝ^64. The semantic seed is h_0=S(a(D),Q(widehat q))∈ℝ^40. A five-step Elman recurrence generates the Neuralese trace r_k=tanh(W_ih_0+b_i+U_ir_k-1+c_i+e_k), k=1,…,5, where e_k is a learned step embedding. A gated readout recurrence …
Exact mathematical passage
a(D) = \left( \frac15\sum_{i=1}^{5}p_i, \max_{1\leq i\leq5}p_i \right)\in\R^{64}. \label{eq:b19-set-aggregate}
\sum_{i=1}^6Use in source · 2 passages
Search aliases: ∑_i=1^6 sum
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6093
Each foreign agent has a canonical defect d∈ℝ^6, a positive diagonal importance vector w, and an invertible architecture transform T∈ℝ^6×6. The runtime-visible defect is widetilde d=Td. Protected utility is J_T(widetilde d) = exp(-∑_i=1^6w_i(T^-1widetilde d)_i^2). A candidate runtime patch δ produces gain g(δ) = J_T(widetilde d+δ)-J_T(widetilde d).
Exact mathematical passage
J_T(\widetilde d) = \exp\left(-\sum_{i=1}^6w_i\bigl(T^{-1}\widetilde d\bigr)_i^2\right).
\sum_{i=1}^{900}Use in source · 1 passage
Search aliases: ∑_i=1^900 sum
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 16818
Let H_s(τ)∈ℝ^36 be the source model's final latent trace state and H_t(τ)∈ℝ^36 the corresponding foreign-child state. A ridge-affine bridge is fitted on the 900 paired generator-A anchors: B_t = arg min_B,b ∑_i=1^900 normH_s(τ_i)-BH_t(τ_i)-b^2 +λnormB^2. A source ridge probe P_s is fitted independently on the 1,000 probe-calibration tasks. The transported semantic prediction is widehat Y_t(τ) =P_s(B_tH_t(τ)+b_t). The bridge gate requires effective rank 36, transported R^2≥0.35, an advantage of a…
Exact mathematical passage
B_t = \argmin_{B,b} \sum_{i=1}^{900} \norm{H_s(\tau_i)-BH_t(\tau_i)-b}^2 +\lambda\norm{B}^2. \label{eq:b18-bridge-fit}
\sum_{i=1}^kUse in source · 1 passage
Search aliases: ∑_i=1^k sum
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3572
t the eigenvalues of A be 0≤λ_1≤⋯≤λ_d. Among all k-dimensional subspaces F⊆ E, the quantity ∑_i=1^kanglesx_i, Ax_i for an orthonormal basis x_1,…,x_k of F is minimized by the span of eigenvectors associated with λ_1,…,λ_k, and the minimum is ∑_i=1^kλ_i.
Exact mathematical passage
\sum_{i=1}^k\lambda_i
\sum_{i=1}^{k}Use in source · 1 passage
Search aliases: ∑_i=1^k sum
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3569
[Spectral characterization of flatness] The exact common fixed space is E^flat=ker A. Let the eigenvalues of A be 0≤λ_1≤⋯≤λ_d. Among all k-dimensional subspaces F⊆ E, the quantity ∑_i=1^kanglesx_i, Ax_i for an orthonormal basis x_1,…,x_k of F is minimized by the span of eigenvectors associated with λ_1,…,λ_k, and the minimum is ∑_i=1^kλ_i.
Exact mathematical passage
\sum_{i=1}^{k}\angles{x_i,\mathsf Ax_i}
\sum_{i=1}^mUse in source · 2 passages
Search aliases: ∑_i=1^m sum
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1919
Let U=⊕_i=1^m U_i, V=⊕_i=1^m V_i, with orthogonal projectors Q_i and P_i. Decompose D=D_0+E, D_0:=∑_i=1^m P_iDQ_i. The block-diagonal operator D_0 represents ideal packet-local correction; E contains all cross-packet effects.
Exact mathematical passage
D=D_0+E, \qquad D_0:=\sum_{i=1}^m P_iDQ_i.
\sum_{i=1}^nUse in source · 1 passage
Search aliases: ∑_i=1^n sum
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10993
Let D_i be the grandchild-minus-child protected gain on final foreign target i. The primary finite estimand is overline D = frac1n∑_i=1^nD_i. The release records a paired bootstrap interval, a sign-flip test, positive and negative pair rates, and exact query ledgers. The matched target pairing removes between-target difficulty from the edge comparison.
Exact mathematical passage
\overline D = \frac1n\sum_{i=1}^nD_i.
\sum_{i=1}^{n}Use in source · 2 passages
Search aliases: ∑_i=1^n sum
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4699
Let K_0 and K_1 be frozen improvement mechanisms evaluated on the same independent foreign agents A_1,…,A_n. Define G_h,B(K,A) = max_A'∈Desc_h,B(K,A)J(A')-J(A) and assume -G_max ≤ G_h,B(K,A) ≤ G_max. For matched budgets and random seeds, let D_i = G_h,B(K_1,A_i)-G_h,B(K_0,A_i), overline D_n = frac1n∑_i=1^nD_i.
Exact mathematical passage
D_i = G_{h,B}(K_1,A_i)-G_{h,B}(K_0,A_i), \qquad \overline D_n = \frac1n\sum_{i=1}^{n}D_i.
\sum_{i\in N}Use in source · 1 passage
Search aliases: ∑_i∈ N sum in
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9158
[Meta-headroom allocation] For every finite repair-process factorization, h_i(K)≥0 for every module i, and ∑_i∈ Nh_i(K)=H(K).
Exact mathematical passage
\sum_{i\in N}h_i(K)=\Hcal(K). \label{eq:b8-shapley-efficiency}
\sum_{i\in S}Use in source · 1 passage
Search aliases: ∑_i∈ S sum in
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 11757
A candidate pool contains source state h_i, target observation y_i, and known observation scale σ_i. The inherited acquisition policy selects rows sequentially to maximize the quality-weighted log-determinant increment logdet( λ I+ ∑_i∈ Sσ_i^-2bar h_ibar h_i^top ). Both parent and child use the same resulting order. The parent takes the first 80 rows for one global affine fit. The child takes the first 48 rows for local fits and the next 32 for separator selection. Both reserve the final 16 rows…
Exact mathematical passage
\log\det\left( \lambda I+ \sum_{i\in S}\sigma_i^{-2}\bar h_i\bar h_i^\top \right).
\sum_jUse in source · 4 passages
Search aliases: ∑_j sum
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3549
Exact fixed subspaces can be too brittle for learned evaluator transports. Let H_1,…,H_m be a finite generating family of loop transports on a Hilbert space E, with positive weights w_j satisfying ∑_jw_j=1. Define the holonomy-energy operator A := ∑_j=1^m w_j(H_j-I)^*(H_j-I). This operator is positive semidefinite.
Exact mathematical passage
\sum_jw_j=1
\sum_{j=1}^{k}Use in source · 1 passage
Search aliases: ∑_j=1^k sum
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12201
Choose c∈ℝ^k outside the finite union of hyperplanes setc:c^ T(λ_a-λ_b)=0, a≠ b. Then the scalars μ_a=c^ Tλ_a are pairwise distinct. The combined response B=∑_j=1^kc_jR_j =∑_a=1^mμ_aP_a belongs to A_R. For each a, define the Lagrange polynomial p_a(z)=∏_b≠ afracz-μ_bμ_a-μ_b. Functional calculus gives p_a(B)=∑_b=1^mp_a(μ_b)P_b=P_a. Thus every P_a lies in the generated algebra. A nonzero central projection that is strictly below one P_a would have to be scalar on H_a and therefore is either 0 or P…
Exact mathematical passage
B=\sum_{j=1}^{k}c_jR_j =\sum_{a=1}^{m}\mu_aP_a
\sum_{j=1}^{m}Use in source · 6 passages
Search aliases: ∑_j=1^m sum
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3550
Exact fixed subspaces can be too brittle for learned evaluator transports. Let H_1,…,H_m be a finite generating family of loop transports on a Hilbert space E, with positive weights w_j satisfying ∑_jw_j=1. Define the holonomy-energy operator A := ∑_j=1^m w_j(H_j-I)^*(H_j-I). This operator is positive semidefinite.
Exact mathematical passage
\mathsf A := \sum_{j=1}^{m} w_j(H_j-I)^*(H_j-I).
\sum_{j=1}^nUse in source · 3 passages
Search aliases: ∑_j=1^n sum
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6143
The archive arm can perform well without a causal repair calculus because it searches several noisy candidates and retains the best. Candidate-budget matching is therefore mandatory. Let G_a,j,1,…,G_a,j,B be candidate gains for arm a on agent j. The primary behavioral quantity is M_a(B) = frac1n∑_j=1^nmax_1≤ b≤ BG_a,j,b. Comparing arms at unequal B confounds repair competence with order statistics.
Exact mathematical passage
M_a(B) = \frac1n\sum_{j=1}^n\max_{1\leq b\leq B}G_{a,j,b}.
\sum_{j=1}^{n}Use in source · 1 passage
Search aliases: ∑_j=1^n sum
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2214
Analytic expansion gives F(ε,∑_j=1^nε^ju_j) = ε^n(b_n+Du_n)+O(ε^n+1), because u_n first enters linearly at order n. Cancellation of the coefficient is equivalent to solvability of Du_n=-b_n.
Exact mathematical passage
F\left(\eps,\sum_{j=1}^{n}\eps^ju_j\right) = \eps^n(b_n+Du_n)+O(\eps^{n+1}),
\sum_{j=1}^{n-1}Use in source · 1 passage
Search aliases: ∑_j=1^n-1 sum
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2201
[Finite formal obstruction tower] Let F(ε,u) be analytic near (0,0) with F(0,0)=0 and D=∂_uF(0,0). Suppose coefficients u_1,…,u_n-1 have been chosen so that F(ε,∑_j=1^n-1ε^ju_j) =ε^nb_n+O(ε^n+1). Then extending the repair to order n by adding ε^nu_n is possible if and only if [b_n]=0 in coker D. When it vanishes, any solution of Du_n=-b_n produces an order-n lift.
Exact mathematical passage
F\left(\eps,\sum_{j=1}^{n-1}\eps^ju_j\right) =\eps^nb_n+O(\eps^{n+1}).
\sum_{j=1}^{t}Use in source · 1 passage
Search aliases: ∑_j=1^t sum
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 15790
For fixed maximum depth K_max, let λ_t(x)=Λ(z_t(x)). Define cumulative halt mass M_t(x) = ∑_j=1^t λ_j(x) ∏_q<j(1-λ_q(x)), and dynamic depth T(x)=min{t:M_t(x)≥τ}, with declared threshold τ. Block 16 does not infer adaptive computation from the existence of a halt head. It requires a difficulty variable d(x) fixed before result access and tests both ρ_Spearman(d(x),T(x))>0 and nondegenerate prompt variation in the halt schedules. A constant depth can coexist with fully task-specific Neuralese; con…
Exact mathematical passage
M_t(x) = \sum_{j=1}^{t} \lambda_j(x) \prod_{q<j}(1-\lambda_q(x)),
\sum_{j=8}^{16}Use in source · 1 passage
Search aliases: ∑_j=8^16 sum
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 8506
If random anchor membership is Bernoulli with affected probability p, the idealized full-rank probability at k=16 is therefore Pr(rank X=16) = Pr(N_1≥8) = ∑_j=8^1616choose jp^j(1-p)^16-j under the generic model. The observed parent full-rank rate on the primary audit set is 0.03125, consistent with a low-probability affected-count event.
Exact mathematical passage
\Pr(\rank X=16) = \Pr(N_1\geq8) = \sum_{j=8}^{16}{16\choose j}p^j(1-p)^{16-j} \label{eq:block07q-random-rank-probability}
\sum_{j\in N}Use in source · 1 passage
Search aliases: ∑_j∈ N sum in
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 6212
Let N⊆{1,…,n} be foreign agents whose dominant defect belongs to the held-out sixth type. Define widehatτ_abstract = frac1|N|∑_j∈ N ( G_j^meta-G_j^no-meta ). In the reference positive control, widehatτ_abstract=0.700533. A learned recursive-improvement claim would require the schema to be generated by the parent improvement process, sealed before foreign evaluation, and compared under matched computation and evidence access.
Exact mathematical passage
\widehat\tau_{\mathrm{abstract}} = \frac1{|N|}\sum_{j\in N} \left( G_{j}^{\mathrm{meta}}-G_{j}^{\mathrm{no\mbox{-}meta}} \right).
\sum_{k=0}^nUse in source · 1 passage
Search aliases: ∑_k=0^n sum
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1970
Since normEQ_0<1, the Neumann series for (I+EQ_0)^-1 converges. Also DQ_0=(D_0+E)Q_0=I+EQ_0, so DQ=I. The norm bound follows from the geometric series. For the truncation, (I+EQ_0)∑_k=0^n(-EQ_0)^k =I-(-EQ_0)^n+1, which gives the residual formula.
Exact mathematical passage
(I+EQ_0)\sum_{k=0}^n(-EQ_0)^k =I-(-EQ_0)^{n+1},
\sum_{k=0}^{n}Use in source · 1 passage
Search aliases: ∑_k=0^n sum
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1950
orrection under small packet coupling] If q:=normEQ_0<1, then D is surjective and has right inverse Q =Q_0(I_V+EQ_0)^-1 =Q_0∑_k=0^∞(-EQ_0)^k. Moreover, norm Q≤fracnormQ_01-q. For defect r, the truncated packet-correction series u^(n) :=-Q_0∑_k=0^n(-EQ_0)^kr has residual Du^(n)+r=(-EQ_0)^n+1r, and hence normDu^(n)+r≤ q^n+1norm r.
Exact mathematical passage
u^{(n)} :=-Q_0\sum_{k=0}^{n}(-EQ_0)^kr
\sum_{k=0}^{n-1}Use in source · 4 passages
Search aliases: ∑_k=0^n-1 sum
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2953
T_1⋯ xrightarrow T_n-1A_n. Write T_U,k:U_k→ U_k+1 and T_V,k:V_k→ V_k+1. Let normD_k+1T_U,k-T_V,kD_k≤ε_D,k and normr_k+1-T_V,kr_k≤ε_r,k. If D_0u_0=-r_0 and u_n=T_U,n-1⋯ T_U,0u_0, then normD_nu_n+r_n · ≤ ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_D,k normT_U,k-1⋯ T_U,0u_0 · + ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_r,k, where empty products are identities.
Exact mathematical passage
\begin{aligned} \norm{D_nu_n+r_n} &\leq \sum_{k=0}^{n-1} \norm{T_{V,n-1}\cdots T_{V,k+1}} \eps_{D,k} \norm{T_{U,k-1}\cdots T_{U,0}u_0}\\ &\quad+ \sum_{k=0}^{n-1} \norm{T_{V,n-1}\cdots T_{V,k+1}} \eps_{r,k}, \end{aligned}
\sum_{k=0}^{t-1}Use in source · 1 passage
Search aliases: ∑_k=0^t-1 sum
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9293
Lyapunov contraction] Suppose there exist w∈ℝ_>0^7 and q∈[0,1) such that for every reachable r_t, the controller can choose an authorized action a_t satisfying w^top A_a_tr_t ≤ q w^top r_t. Then every controlled trajectory satisfying [source reference] obeys w^top r_t ≤ q^t w^top r_0 + ∑_k=0^t-1 q^t-1-kw^top e_k.
Exact mathematical passage
w^\top r_t \leq q^t w^\top r_0 + \sum_{k=0}^{t-1} q^{t-1-k}w^\top e_k. \label{eq:b8-lyapunov-bound}
\sum_{k=0}^{\infty}Use in source · 1 passage
Search aliases: ∑_k=0^∞ sum infty
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1940
[Exact correction under small packet coupling] If q:=normEQ_0<1, then D is surjective and has right inverse Q =Q_0(I_V+EQ_0)^-1 =Q_0∑_k=0^∞(-EQ_0)^k. Moreover, norm Q≤fracnormQ_01-q. For defect r, the truncated packet-correction series u^(n) :=-Q_0∑_k=0^n(-EQ_0)^kr has residual Du^(n)+r=(-EQ_0)^n+1r, and hence normDu^(n)+r≤ q^n+1norm r.
Exact mathematical passage
Q =Q_0(I_V+EQ_0)^{-1} =Q_0\sum_{k=0}^{\infty}(-EQ_0)^k.
\sum_tUse in source · 15 passages
Search aliases: ∑_t sum
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3686
inequation E_0xrightarrowe_0 E_1 xrightarrowe_1⋯ xrightarrowe_n-1 E_n. Suppose that for every protected represented state x_t transported from version 0, normJ_t+1(T_tx_t)-J_t(x_t)≤ε_t. Then normJ_n(T_n-1⋯ T_0x_0)-J_0(x_0) ≤ ∑_t=0^n-1ε_t. If ∑_tε_t<∞, then the transported score sequence is Cauchy and converges in every complete score space.
Exact mathematical passage
\sum_t\eps_t<\infty
\sum_{t=0}^{3}Use in source · 1 passage
Search aliases: ∑_t=0^3 sum
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 21721
Suppose four accepted edges use conditional false-acceptance budgets 0.0100, 0.0050, 0.0025, 0.00125. Then ∑_t=0^3δ_t=0.01875. By [source reference], the probability that any accepted edge has nonpositive true metaproductivity is at most 0.01875, assuming every conditional bound is valid.
Exact mathematical passage
\sum_{t=0}^{3}\delta_t=0.01875.
\sum_{t=0}^{m-1}Use in source · 1 passage
Search aliases: ∑_t=0^m-1 sum
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10517
[Same-context transitivity] Suppose every edge is evaluated in one common context Ξ. Then M_Ξ(K_m)-M_Ξ(K_0) = ∑_t=0^m-1 ( M_Ξ(K_t+1)-M_Ξ(K_t) ). Consequently, positive lower bounds on all edges add to a positive lower bound on the complete path.
Exact mathematical passage
\Mcal_{\Xi}(K_m)-\Mcal_{\Xi}(K_0) = \sum_{t=0}^{m-1} \left( \Mcal_{\Xi}(K_{t+1})-\Mcal_{\Xi}(K_t) \right).
\sum_{t=0}^{n-1}Use in source · 1 passage
Search aliases: ∑_t=0^n-1 sum
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 3681
chain of evaluator morphisms with readouts into a common complete normed score space, E_0xrightarrowe_0 E_1 xrightarrowe_1⋯ xrightarrowe_n-1 E_n. Suppose that for every protected represented state x_t transported from version 0, normJ_t+1(T_tx_t)-J_t(x_t)≤ε_t. Then normJ_n(T_n-1⋯ T_0x_0)-J_0(x_0) ≤ ∑_t=0^n-1ε_t. If ∑_tε_t<∞, then the transported score sequence is Cauchy and converges in every complete score space.
Exact mathematical passage
\norm{J_n(T_{n-1}\cdots T_0x_0)-J_0(x_0)} \leq \sum_{t=0}^{n-1}\eps_t.
\sum_{t=0}^{\infty}Use in source · 2 passages
Search aliases: ∑_t=0^∞ sum infty
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9319
[Fixed-grammar convergence] If the grammar remains fixed and ∑_t=0^∞w^top e_t<∞, then w^top r_t→0 whenever the tail errors vanish under the convolution in [source reference].
Exact mathematical passage
\sum_{t=0}^{\infty}w^\top e_t<\infty,
\sum_{t=1}^{T}Use in source · 2 passages
Search aliases: ∑_t=1^T sum
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4147
[Adaptive alpha-spending guarantee] Let F_t be the event that transaction t accepts a child whose true declared gain is nonpositive. Suppose Pr(F_tmidF_t-1)≤δ_t almost surely for a deterministic sequence (δ_t). Then for every T, Pr(⋃_t=1^TF_t) ≤ ∑_t=1^Tδ_t. If ∑_tδ_t<∞, only finitely many false acceptances occur almost surely.
Exact mathematical passage
\Pr\left(\bigcup_{t=1}^{T}F_t\right) \leq \sum_{t=1}^{T}\delta_t.
\sum_{t=m}^{n-1}Use in source · 1 passage
Search aliases: ∑_t=m^n-1 sum
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9707
Suppose the correction square is approximate: normD_t+1T_t^U-T_t^VD_t ≤ε_t^D. Suppose protected defect meanings are reconstructed with error at most ε_t^V. Define cumulative dynamic semantic debt Δ_m,n = ∑_t=m^n-1 ( ε_t^D+ε_t^V ). Under nonexpansive transports, old correction and defect relations drift by at most the corresponding telescoping debt. If ∑_t(ε_t^D+ε_t^V)<∞, old protected meanings form Cauchy trajectories along the grammar chain.
Exact mathematical passage
\Delta_{m,n} = \sum_{t=m}^{n-1} \left( \eps_t^D+\eps_t^V \right). \label{eq:b8-dynamic-debt}
\sum_{u\in\mathcal U}Use in source · 1 passage
Search aliases: ∑_u∈ U sum in mathcal
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 19561
For a proposed intervention correspondence σ: U→ V and transport T:H_t→ H_s, define the normalized orbit defect δ_rm orb(T,σ) = frac1| U| ∑_u∈ U fracnormTR^t_σ(u)-R_u^sT_F max{normR_u^s_F,ε}.
Exact mathematical passage
\delta_{\rm orb}(T,\sigma) = \frac1{|\mathcal U|} \sum_{u\in\mathcal U} \frac{\norm{TR^t_{\sigma(u)}-R_u^sT}_F} {\max\{\norm{R_u^s}_F,\eps\}}. \label{eq:b22-orbit-defect}
\sum_{z'}Use in source · 1 passage
Search aliases: ∑_z' sum
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 5982
Let μ_z=ϕ(z) be the frozen event prototype. With equal priors and matched isotropic noise, the posterior is p(zmidwidetildeϕ) = fracexp(-normwidetildeϕ-μ_z^2/(2σ^2)) ∑_z'exp(-normwidetildeϕ-μ_z'^2/(2σ^2)). A repair is attempted only when max_z p(zmidwidetildeϕ)≥θ and the maximizing event is not none. Otherwise the system abstains.
Exact mathematical passage
p(z\mid\widetilde\phi) = \frac{\exp\left(-\norm{\widetilde\phi-\mu_z}^2/(2\sigma^2)\right)} {\sum_{z'}\exp\left(-\norm{\widetilde\phi-\mu_{z'}}^2/(2\sigma^2)\right)}.
\astUse in source · 7 passages
Search aliases: ∗ ast
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10029
Define the unital adjoint-closed algebra A = alg^∗ (I,R_1,…,R_m). Its commutant is A' = {X∈End(H):XA=AX for all A∈ A}. The center is Z( A)= A∩ A'.
Exact mathematical passage
\mathfrak A = \operatorname{alg}^{\ast} \left(I,R_1,\ldots,R_m\right). \label{eq:b8-response-algebra}
\circUse in source · 23 passages
Search aliases: ∘ circ
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 747
d for bridge defects] Let T:X→ Y and S:Y→ Z. Suppose the property transports satisfy normQ_Y(Tx)-widetilde T_QQ_X(x)≤ε_T for x∈ K, and normQ_Z(Sy)-widetilde S_QQ_Y(y)≤ε_S for y∈ T(K). If normwidetilde S_Q≤ L_S, then the composite bridge obeys ε_Q(S∘ T;K) ≤ ε_S+L_Sε_T with intended property transport widetilde S_Qwidetilde T_Q.
Exact mathematical passage
\eps_Q(S\circ T;K) \leq \eps_S+L_S\eps_T
\intUse in source · 2 passages
Search aliases: ∫ int
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4662
[Expected best-of-k gain] Let G_1,…,G_k be independent nonnegative candidate gains with common cumulative distribution function F and finite mean. Then Emax_1≤ i≤ kG_i = ∫_0^∞ (1-F(x)^k) d x. In particular, expected best gain is nondecreasing in k even when the candidate generator itself does not improve.
Exact mathematical passage
\mathbb E\max_{1\leq i\leq k}G_i = \int_0^{\infty} \left(1-F(x)^k\right)\dd x.
\int_0^\inftyUse in source · 1 passage
Search aliases: ∫_0^∞ int infty
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4672
For a nonnegative random variable X, E X=∫_0^∞Pr(X>x) d x. Independence gives Pr(max_iG_i≤ x)=F(x)^k. Substitution proves the identity.
Exact mathematical passage
\mathbb E X=\int_0^\infty\Pr(X>x)\dd x
\int_0^{\infty}Use in source · 1 passage
Search aliases: ∫_0^∞ int infty
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4662
[Expected best-of-k gain] Let G_1,…,G_k be independent nonnegative candidate gains with common cumulative distribution function F and finite mean. Then Emax_1≤ i≤ kG_i = ∫_0^∞ (1-F(x)^k) d x. In particular, expected best gain is nondecreasing in k even when the candidate generator itself does not improve.
Exact mathematical passage
\mathbb E\max_{1\leq i\leq k}G_i = \int_0^{\infty} \left(1-F(x)^k\right)\dd x.
\approxUse in source · 10 passages
Search aliases: ≈ approx
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 9222
Let K→ K' change only modules in R. When F_K(R)>M(K), define the headroom-adjusted gain χ_R(K,K') = frac M(K')-M(K) F_K(R)-M(K) . For an admissible child inside the declared class, χ_R(K,K')≤1. A small absolute gain with χ_R≈1 indicates a nearly saturated parent. A small absolute gain with χ_R≈0 indicates that the child captured little of the available opportunity.
Exact mathematical passage
\chi_R\approx1
\subsetUse in source · 2 passages
Search aliases: ⊂ subset
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 18287
[Affine obstruction for the declared nonlinear family] Let I⊂ℝ be an interval with nonempty interior and let α≠0. No affine scalar map agrees with u↦sinh(α u)/α on all of I. Consequently, if one transformed packet coordinate varies over such an interval, no exact step-affine inverse exists for [source reference].
Exact mathematical passage
I\subset\R
\oplusUse in source · 17 passages
Search aliases: ⊕ oplus
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2120
[Lyapunov--Schmidt repair reduction] Let F:P× U→ V be C^k, k≥2, between finite-dimensional spaces, with F(0,0)=0. Let D:=∂_uF(0,0). Choose splittings U=K⊕ X, K=ker D, and V=R⊕ C, R=im D, so that restrDX:X→ R is an isomorphism. Let P_R and P_C be the corresponding projections. Then there are neighborhoods of the origin and a unique C^k map χ:P× K→ X, χ(0,0)=0, such that P_RF(p,k+χ(p,k))=0. Define the reduced obstruction map ψ(p,k):=P_CF(p,k+χ(p,k)). A sufficiently small pair (p,u) solves F(p,u)=0…
Exact mathematical passage
U=K\oplus X, \qquad K=\ker D,
\bigoplus_{a=1}^{m}Use in source · 2 passages
Search aliases: ⊕_a=1^m bigoplus
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 12141
Let H be a finite-dimensional real or complex Hilbert space. Suppose the unknown functional decomposition is H=⊕_a=1^mH_a, P_a:H→ H_a, with pairwise orthogonal projectors satisfying P_aP_b=δ_abP_a, ∑_a=1^mP_a=I. For controlled interventions u_1,…,u_k, let R_j∈End(H), j=1,…,k, be their finite-difference or linearized response operators.
Exact mathematical passage
H=\bigoplus_{a=1}^{m}H_a, \qquad P_a:H\to H_a, \label{eq:b11-hidden-decomp}
\bigoplus_{i=1}^mUse in source · 1 passage
Search aliases: ⊕_i=1^m bigoplus
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 1913
Let U=⊕_i=1^m U_i, V=⊕_i=1^m V_i, with orthogonal projectors Q_i and P_i. Decompose D=D_0+E, D_0:=∑_i=1^m P_iDQ_i. The block-diagonal operator D_0 represents ideal packet-local correction; E contains all cross-packet effects.
Exact mathematical passage
U=\bigoplus_{i=1}^m U_i, \qquad V=\bigoplus_{i=1}^m V_i,
\bigoplus_{\alpha=1}^{p}Use in source · 3 passages
Search aliases: ⊕_α=1^p bigoplus alpha
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 10052
[Finite causal atlas decomposition] There exists a unitary identification H ≅ ⊕_α=1^p ( C^m_α⊗ C^n_α ) under which A ≅ ⊕_α=1^p ( I_m_α⊗ M_n_α( C) ). The minimal central projections P_α∈ Z( A) give a canonical coarse causal atlas H = ⊕_α=1^pP_α H. The blocks are unique up to permutation and unitary changes of basis within multiplicity spaces.
Exact mathematical passage
H \cong \bigoplus_{\alpha=1}^{p} \left( \mathbb C^{m_\alpha}\otimes\mathbb C^{n_\alpha} \right) \label{eq:b8-algebra-space-decomposition}
\otimesUse in source · 7 passages
Search aliases: ⊗ otimes
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 2349
Let G=(I,E) be a connected oriented graph representing the nerve of the cover, let K be a finite-dimensional Hilbert space representing a common trivialization of the kernel fibers, and let e=(e_a)_a∈ E∈ K^E be the transported overlap-mismatch vector. Let B∈ℝ^|I|×|E| be the oriented incidence matrix and set B:=B⊗ I_K. A vertex gauge k∈ K^I changes the mismatch to e+ B^*k.
Exact mathematical passage
\mathbf B:=B\otimes I_K.
\bigcap_{j=1}^mUse in source · 1 passage
Search aliases: ⋂_j=1^m bigcap
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 14581
Let b_1,…,b_m be benchmark readouts. A single scored benchmark leaves the local nullspace N_{b_1} = ker Db_1. An evaluator portfolio has invisible subspace N_ B = ⋂_j=1^mker Db_j.
Exact mathematical passage
\mathcal N_{\mathcal B} = \bigcap_{j=1}^m\ker Db_j. \label{eq:evaluator-nullspace}
\bigcupUse in source · 1 passage
Search aliases: ⋃ bigcup
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4147
[Adaptive alpha-spending guarantee] Let F_t be the event that transaction t accepts a child whose true declared gain is nonpositive. Suppose Pr(F_tmidF_t-1)≤δ_t almost surely for a deterministic sequence (δ_t). Then for every T, Pr(⋃_t=1^TF_t) ≤ ∑_t=1^Tδ_t. If ∑_tδ_t<∞, only finitely many false acceptances occur almost surely.
Exact mathematical passage
\Pr\left(\bigcup_{t=1}^{T}F_t\right) \leq \sum_{t=1}^{T}\delta_t.
\bigcup_{t=1}^{T}Use in source · 1 passage
Search aliases: ⋃_t=1^T bigcup
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 4147
[Adaptive alpha-spending guarantee] Let F_t be the event that transaction t accepts a child whose true declared gain is nonpositive. Suppose Pr(F_tmidF_t-1)≤δ_t almost surely for a deterministic sequence (δ_t). Then for every T, Pr(⋃_t=1^TF_t) ≤ ∑_t=1^Tδ_t. If ∑_tδ_t<∞, only finitely many false acceptances occur almost surely.
Exact mathematical passage
\Pr\left(\bigcup_{t=1}^{T}F_t\right) \leq \sum_{t=1}^{T}\delta_t.
\langleUse in source · 1 passage
Search aliases: ⟨ langle
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7801
For x∈ℝ^4, define an eight-dimensional task representation ψ(x) = ( x_1,x_2,x_3,x_4, x_1x_2,x_3x_4, sin(sx_1),cos(sx_2) ). A task-specific condition vector and threshold define the affected indicator a(x)=1{⟨ v,x⟩>τ}. The conditional feature is ϕ(x)=(ψ(x),a(x)ψ(x))∈ℝ^16. True three-class score vectors are affine in ψ(x). On the affected stratum only, one of five frozen score transformations corrupts those scores. Target agents are fitted to the corrupted scores, so their baseline behavior is gen…
Exact mathematical passage
a(x)=\one\{\langle v,x\rangle>\tau\}.
\rangleUse in source · 1 passage
Search aliases: ⟩ rangle
Use in source · Stratified_Recursive_Improvement_Closure_v2.4_Block23.tex, line 7801
For x∈ℝ^4, define an eight-dimensional task representation ψ(x) = ( x_1,x_2,x_3,x_4, x_1x_2,x_3x_4, sin(sx_1),cos(sx_2) ). A task-specific condition vector and threshold define the affected indicator a(x)=1{⟨ v,x⟩>τ}. The conditional feature is ϕ(x)=(ψ(x),a(x)ψ(x))∈ℝ^16. True three-class score vectors are affine in ψ(x). On the affected stratum only, one of five frozen score transformations corrupts those scores. Target agents are fitted to the corrupted scores, so their baseline behavior is gen…
Exact mathematical passage
a(x)=\one\{\langle v,x\rangle>\tau\}.
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