Notation by work
Self-Improving Intelligence
All notation · Glossary · Source coverage
312 symbol spellings in 641 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 312 entries are available below.
BNotation table · 50 passages
Search aliases: B
Notation table · Self_Improving_Intelligence_v1.0.0.tex, line 93
Ξ, B · An evaluation contract and its resource allowance.
Exact mathematical passage
B
B_0Notation table · 18 passages
Search aliases: B_0
Notation table · Self_Improving_Intelligence_v1.0.0.tex, line 95
B_0⊆ X · Possible current states of a finite repair problem.
Exact mathematical passage
B_0\subseteq X
eNotation table · 20 passages
Search aliases: e
Notation table · Self_Improving_Intelligence_v1.0.0.tex, line 96
e, r, P · An experiment, a terminal repair, and an executable diagnostic tree.
Exact mathematical passage
e
FNotation table · 8 passages
Search aliases: F
Notation table · Self_Improving_Intelligence_v1.0.0.tex, line 97
F_h(B_0) · Nondominated execution/storage pairs for successful trees of depth at most h.
Exact mathematical passage
F_h(B_0)
F_hNotation table · 1 passage
Search aliases: F_h
Notation table · Self_Improving_Intelligence_v1.0.0.tex, line 97
F_h(B_0) · Nondominated execution/storage pairs for successful trees of depth at most h.
Exact mathematical passage
F_h(B_0)
hNotation table · 30 passages
Search aliases: h
Notation table · Self_Improving_Intelligence_v1.0.0.tex, line 97
F_h(B_0) · Nondominated execution/storage pairs for successful trees of depth at most h.
Exact mathematical passage
F_h(B_0)
\mathcal KNotation table · 6 passages
Search aliases: K mathcal
Notation table · Self_Improving_Intelligence_v1.0.0.tex, line 99
K, W_h,k · A safe continuation set and states guaranteeing k progress events within h attempts.
Exact mathematical passage
\mathcal K
kNotation table · 26 passages
Search aliases: k
Notation table · Self_Improving_Intelligence_v1.0.0.tex, line 99
K, W_h,k · A safe continuation set and states guaranteeing k progress events within h attempts.
Exact mathematical passage
W_{h,k}
LNotation table · 15 passages
Search aliases: L
Notation table · Self_Improving_Intelligence_v1.0.0.tex, line 92
midrule S, U, L · A target solver, an improvement procedure, and a reusable library.
Exact mathematical passage
L
PNotation table · 16 passages
Search aliases: P
Notation table · Self_Improving_Intelligence_v1.0.0.tex, line 96
e, r, P · An experiment, a terminal repair, and an executable diagnostic tree.
Exact mathematical passage
P
rNotation table · 23 passages
Search aliases: r
Notation table · Self_Improving_Intelligence_v1.0.0.tex, line 96
e, r, P · An experiment, a terminal repair, and an executable diagnostic tree.
Exact mathematical passage
r
\RcalNotation table · 5 passages
Search aliases: Rcal R mathcal
Notation table · Self_Improving_Intelligence_v1.0.0.tex, line 94
R_Ξ(U,L;B) · Expected improvement produced by the procedure, not an oracle over its unselected candidates.
Exact mathematical passage
\Rcal_\Xi(U,L;B)
\Rcal_\XiNotation table · 4 passages
Search aliases: Rcal_Ξ R_Ξ Rcal Xi mathcal Xi
Notation table · Self_Improving_Intelligence_v1.0.0.tex, line 94
R_Ξ(U,L;B) · Expected improvement produced by the procedure, not an oracle over its unselected candidates.
Exact mathematical passage
\Rcal_\Xi(U,L;B)
SNotation table · 8 passages
Search aliases: S
Notation table · Self_Improving_Intelligence_v1.0.0.tex, line 92
midrule S, U, L · A target solver, an improvement procedure, and a reusable library.
Exact mathematical passage
S
UNotation table · 17 passages
Search aliases: U
Notation table · Self_Improving_Intelligence_v1.0.0.tex, line 92
midrule S, U, L · A target solver, an improvement procedure, and a reusable library.
Exact mathematical passage
U
WNotation table · 6 passages
Search aliases: W
Notation table · Self_Improving_Intelligence_v1.0.0.tex, line 99
K, W_h,k · A safe continuation set and states guaranteeing k progress events within h attempts.
Exact mathematical passage
W_{h,k}
W_{h,k}Notation table · 3 passages
Search aliases: W_h,k
Notation table · Self_Improving_Intelligence_v1.0.0.tex, line 99
K, W_h,k · A safe continuation set and states guaranteeing k progress events within h attempts.
Exact mathematical passage
W_{h,k}
XNotation table · 18 passages
Search aliases: X
Notation table · Self_Improving_Intelligence_v1.0.0.tex, line 95
B_0⊆ X · Possible current states of a finite repair problem.
Exact mathematical passage
B_0\subseteq X
\XiNotation table · 9 passages
Search aliases: Ξ Xi
Notation table · Self_Improving_Intelligence_v1.0.0.tex, line 93
Ξ, B · An evaluation contract and its resource allowance.
Exact mathematical passage
\Xi
\sigmaNotation table · 11 passages
Search aliases: σ sigma
Notation table · Self_Improving_Intelligence_v1.0.0.tex, line 98
σ, τ · Correspondences of states and operations in a declared transfer.
Exact mathematical passage
\sigma
\tauNotation table · 12 passages
Search aliases: τ tau
Notation table · Self_Improving_Intelligence_v1.0.0.tex, line 98
σ, τ · Correspondences of states and operations in a declared transfer.
Exact mathematical passage
\tau
\subseteqNotation table · 15 passages
Search aliases: ⊆ subseteq
Notation table · Self_Improving_Intelligence_v1.0.0.tex, line 95
B_0⊆ X · Possible current states of a finite repair problem.
Exact mathematical passage
B_0\subseteq X
\mathcal ADefinition passage · 2 passages
Search aliases: A mathcal
Definition passage: Finite repair experiment model · Self_Improving_Intelligence_v1.0.0.tex, line 271
[Finite repair experiment model] A model has a finite nonempty state set X, a finite experiment set E, and a finite repair set A. Experiment e is allowed on a domain D_e⊆ X, changes the state by f_e:D_e→ X, and reports o_e:D_e→ O_e. Its execution cost c_e≥0 is fixed and known. Repair a has fixed cost c_a≥0 and succeeds on a declared set G_a⊆ X. All costs are in one stated unit. A nonempty set B_0⊆ X records the possible current states.
Exact mathematical passage
\mathcal A
aDefinition passage · 67 passages
Search aliases: a
Definition passage: Finite repair experiment model · Self_Improving_Intelligence_v1.0.0.tex, line 271
[Finite repair experiment model] A model has a finite nonempty state set X, a finite experiment set E, and a finite repair set A. Experiment e is allowed on a domain D_e⊆ X, changes the state by f_e:D_e→ X, and reports o_e:D_e→ O_e. Its execution cost c_e≥0 is fixed and known. Repair a has fixed cost c_a≥0 and succeeds on a declared set G_a⊆ X. All costs are in one stated unit. A nonempty set B_0⊆ X records the possible current states.
Exact mathematical passage
a
cDefinition passage · 15 passages
Search aliases: c
Definition passage: Finite repair experiment model · Self_Improving_Intelligence_v1.0.0.tex, line 271
[Finite repair experiment model] A model has a finite nonempty state set X, a finite experiment set E, and a finite repair set A. Experiment e is allowed on a domain D_e⊆ X, changes the state by f_e:D_e→ X, and reports o_e:D_e→ O_e. Its execution cost c_e≥0 is fixed and known. Repair a has fixed cost c_a≥0 and succeeds on a declared set G_a⊆ X. All costs are in one stated unit. A nonempty set B_0⊆ X records the possible current states.
Exact mathematical passage
c_e\geq0
c_aDefinition passage · 2 passages
Search aliases: c_a
Definition passage: Finite repair experiment model · Self_Improving_Intelligence_v1.0.0.tex, line 271
[Finite repair experiment model] A model has a finite nonempty state set X, a finite experiment set E, and a finite repair set A. Experiment e is allowed on a domain D_e⊆ X, changes the state by f_e:D_e→ X, and reports o_e:D_e→ O_e. Its execution cost c_e≥0 is fixed and known. Repair a has fixed cost c_a≥0 and succeeds on a declared set G_a⊆ X. All costs are in one stated unit. A nonempty set B_0⊆ X records the possible current states.
Exact mathematical passage
c_a\geq0
c_eDefinition passage · 2 passages
Search aliases: c_e
Definition passage: Finite repair experiment model · Self_Improving_Intelligence_v1.0.0.tex, line 271
[Finite repair experiment model] A model has a finite nonempty state set X, a finite experiment set E, and a finite repair set A. Experiment e is allowed on a domain D_e⊆ X, changes the state by f_e:D_e→ X, and reports o_e:D_e→ O_e. Its execution cost c_e≥0 is fixed and known. Repair a has fixed cost c_a≥0 and succeeds on a declared set G_a⊆ X. All costs are in one stated unit. A nonempty set B_0⊆ X records the possible current states.
Exact mathematical passage
c_e\geq0
DDefinition passage · 26 passages
Search aliases: D
Definition passage: Finite repair experiment model · Self_Improving_Intelligence_v1.0.0.tex, line 271
[Finite repair experiment model] A model has a finite nonempty state set X, a finite experiment set E, and a finite repair set A. Experiment e is allowed on a domain D_e⊆ X, changes the state by f_e:D_e→ X, and reports o_e:D_e→ O_e. Its execution cost c_e≥0 is fixed and known. Repair a has fixed cost c_a≥0 and succeeds on a declared set G_a⊆ X. All costs are in one stated unit. A nonempty set B_0⊆ X records the possible current states.
Exact mathematical passage
D_e\subseteq X
D_eDefinition passage · 6 passages
Search aliases: D_e
Definition passage: Finite repair experiment model · Self_Improving_Intelligence_v1.0.0.tex, line 271
[Finite repair experiment model] A model has a finite nonempty state set X, a finite experiment set E, and a finite repair set A. Experiment e is allowed on a domain D_e⊆ X, changes the state by f_e:D_e→ X, and reports o_e:D_e→ O_e. Its execution cost c_e≥0 is fixed and known. Repair a has fixed cost c_a≥0 and succeeds on a declared set G_a⊆ X. All costs are in one stated unit. A nonempty set B_0⊆ X records the possible current states.
Exact mathematical passage
D_e\subseteq X
\mathcal EDefinition passage · 2 passages
Search aliases: E mathcal
Definition passage: Finite repair experiment model · Self_Improving_Intelligence_v1.0.0.tex, line 271
[Finite repair experiment model] A model has a finite nonempty state set X, a finite experiment set E, and a finite repair set A. Experiment e is allowed on a domain D_e⊆ X, changes the state by f_e:D_e→ X, and reports o_e:D_e→ O_e. Its execution cost c_e≥0 is fixed and known. Repair a has fixed cost c_a≥0 and succeeds on a declared set G_a⊆ X. All costs are in one stated unit. A nonempty set B_0⊆ X records the possible current states.
Exact mathematical passage
\mathcal E
fDefinition passage · 14 passages
Search aliases: f
Definition passage: Finite repair experiment model · Self_Improving_Intelligence_v1.0.0.tex, line 271
[Finite repair experiment model] A model has a finite nonempty state set X, a finite experiment set E, and a finite repair set A. Experiment e is allowed on a domain D_e⊆ X, changes the state by f_e:D_e→ X, and reports o_e:D_e→ O_e. Its execution cost c_e≥0 is fixed and known. Repair a has fixed cost c_a≥0 and succeeds on a declared set G_a⊆ X. All costs are in one stated unit. A nonempty set B_0⊆ X records the possible current states.
Exact mathematical passage
f_e:D_e\to X
f_eDefinition passage · 3 passages
Search aliases: f_e
Definition passage: Finite repair experiment model · Self_Improving_Intelligence_v1.0.0.tex, line 271
[Finite repair experiment model] A model has a finite nonempty state set X, a finite experiment set E, and a finite repair set A. Experiment e is allowed on a domain D_e⊆ X, changes the state by f_e:D_e→ X, and reports o_e:D_e→ O_e. Its execution cost c_e≥0 is fixed and known. Repair a has fixed cost c_a≥0 and succeeds on a declared set G_a⊆ X. All costs are in one stated unit. A nonempty set B_0⊆ X records the possible current states.
Exact mathematical passage
f_e:D_e\to X
GDefinition passage · 13 passages
Search aliases: G
Definition passage: Finite repair experiment model · Self_Improving_Intelligence_v1.0.0.tex, line 271
riment model] A model has a finite nonempty state set X, a finite experiment set E, and a finite repair set A. Experiment e is allowed on a domain D_e⊆ X, changes the state by f_e:D_e→ X, and reports o_e:D_e→ O_e. Its execution cost c_e≥0 is fixed and known. Repair a has fixed cost c_a≥0 and succeeds on a declared set G_a⊆ X. All costs are in one stated unit. A nonempty set B_0⊆ X records the possible current states.
Exact mathematical passage
G_a\subseteq X
G_aDefinition passage · 7 passages
Search aliases: G_a
Definition passage: Finite repair experiment model · Self_Improving_Intelligence_v1.0.0.tex, line 271
riment model] A model has a finite nonempty state set X, a finite experiment set E, and a finite repair set A. Experiment e is allowed on a domain D_e⊆ X, changes the state by f_e:D_e→ X, and reports o_e:D_e→ O_e. Its execution cost c_e≥0 is fixed and known. Repair a has fixed cost c_a≥0 and succeeds on a declared set G_a⊆ X. All costs are in one stated unit. A nonempty set B_0⊆ X records the possible current states.
Exact mathematical passage
G_a\subseteq X
\mathcal HDefinition passage · 9 passages
Search aliases: H mathcal
Definition passage: Improvement contract · Self_Improving_Intelligence_v1.0.0.tex, line 173
[Improvement contract] A contract Ξ specifies a distribution H of starting-solver/task pairs, an initial information and library boundary, permitted observations and modifications, a resource allowance B, protected predicates, and an evaluation rule J_τ(S)∈[0,1]. The improvement procedure returns one selected descendant or the unchanged baseline. Selection must use only the admitted development information. The final evaluation is not available as an oracle for choosing retrospectively among all…
Exact mathematical passage
\mathcal H
JDefinition passage · 2 passages
Search aliases: J
Definition passage: Improvement contract · Self_Improving_Intelligence_v1.0.0.tex, line 173
[Improvement contract] A contract Ξ specifies a distribution H of starting-solver/task pairs, an initial information and library boundary, permitted observations and modifications, a resource allowance B, protected predicates, and an evaluation rule J_τ(S)∈[0,1]. The improvement procedure returns one selected descendant or the unchanged baseline. Selection must use only the admitted development information. The final evaluation is not available as an oracle for choosing retrospectively among all…
Exact mathematical passage
J_\tau(S)\in[0,1]
J_\tauDefinition passage · 2 passages
Search aliases: J_τ tau
Definition passage: Improvement contract · Self_Improving_Intelligence_v1.0.0.tex, line 173
[Improvement contract] A contract Ξ specifies a distribution H of starting-solver/task pairs, an initial information and library boundary, permitted observations and modifications, a resource allowance B, protected predicates, and an evaluation rule J_τ(S)∈[0,1]. The improvement procedure returns one selected descendant or the unchanged baseline. Selection must use only the admitted development information. The final evaluation is not available as an oracle for choosing retrospectively among all…
Exact mathematical passage
J_\tau(S)\in[0,1]
ODefinition passage · 2 passages
Search aliases: O
Definition passage: Finite repair experiment model · Self_Improving_Intelligence_v1.0.0.tex, line 271
[Finite repair experiment model] A model has a finite nonempty state set X, a finite experiment set E, and a finite repair set A. Experiment e is allowed on a domain D_e⊆ X, changes the state by f_e:D_e→ X, and reports o_e:D_e→ O_e. Its execution cost c_e≥0 is fixed and known. Repair a has fixed cost c_a≥0 and succeeds on a declared set G_a⊆ X. All costs are in one stated unit. A nonempty set B_0⊆ X records the possible current states.
Exact mathematical passage
o_e:D_e\to O_e
oDefinition passage · 13 passages
Search aliases: o
Definition passage: Finite repair experiment model · Self_Improving_Intelligence_v1.0.0.tex, line 271
[Finite repair experiment model] A model has a finite nonempty state set X, a finite experiment set E, and a finite repair set A. Experiment e is allowed on a domain D_e⊆ X, changes the state by f_e:D_e→ X, and reports o_e:D_e→ O_e. Its execution cost c_e≥0 is fixed and known. Repair a has fixed cost c_a≥0 and succeeds on a declared set G_a⊆ X. All costs are in one stated unit. A nonempty set B_0⊆ X records the possible current states.
Exact mathematical passage
o_e:D_e\to O_e
O_eDefinition passage · 2 passages
Search aliases: O_e
Definition passage: Finite repair experiment model · Self_Improving_Intelligence_v1.0.0.tex, line 271
[Finite repair experiment model] A model has a finite nonempty state set X, a finite experiment set E, and a finite repair set A. Experiment e is allowed on a domain D_e⊆ X, changes the state by f_e:D_e→ X, and reports o_e:D_e→ O_e. Its execution cost c_e≥0 is fixed and known. Repair a has fixed cost c_a≥0 and succeeds on a declared set G_a⊆ X. All costs are in one stated unit. A nonempty set B_0⊆ X records the possible current states.
Exact mathematical passage
o_e:D_e\to O_e
o_eDefinition passage · 3 passages
Search aliases: o_e
Definition passage: Finite repair experiment model · Self_Improving_Intelligence_v1.0.0.tex, line 271
[Finite repair experiment model] A model has a finite nonempty state set X, a finite experiment set E, and a finite repair set A. Experiment e is allowed on a domain D_e⊆ X, changes the state by f_e:D_e→ X, and reports o_e:D_e→ O_e. Its execution cost c_e≥0 is fixed and known. Repair a has fixed cost c_a≥0 and succeeds on a declared set G_a⊆ X. All costs are in one stated unit. A nonempty set B_0⊆ X records the possible current states.
Exact mathematical passage
o_e:D_e\to O_e
\toDefinition passage · 11 passages
Search aliases: → to
Definition passage: Finite repair experiment model · Self_Improving_Intelligence_v1.0.0.tex, line 271
[Finite repair experiment model] A model has a finite nonempty state set X, a finite experiment set E, and a finite repair set A. Experiment e is allowed on a domain D_e⊆ X, changes the state by f_e:D_e→ X, and reports o_e:D_e→ O_e. Its execution cost c_e≥0 is fixed and known. Repair a has fixed cost c_a≥0 and succeeds on a declared set G_a⊆ X. All costs are in one stated unit. A nonempty set B_0⊆ X records the possible current states.
Exact mathematical passage
f_e:D_e\to X
\inDefinition passage · 31 passages
Search aliases: ∈ in
Definition passage: Improvement contract · Self_Improving_Intelligence_v1.0.0.tex, line 173
[Improvement contract] A contract Ξ specifies a distribution H of starting-solver/task pairs, an initial information and library boundary, permitted observations and modifications, a resource allowance B, protected predicates, and an evaluation rule J_τ(S)∈[0,1]. The improvement procedure returns one selected descendant or the unchanged baseline. Selection must use only the admitted development information. The final evaluation is not available as an oracle for choosing retrospectively among all…
Exact mathematical passage
J_\tau(S)\in[0,1]
\geqDefinition passage · 32 passages
Search aliases: ≥ geq
Definition passage: Finite repair experiment model · Self_Improving_Intelligence_v1.0.0.tex, line 271
[Finite repair experiment model] A model has a finite nonempty state set X, a finite experiment set E, and a finite repair set A. Experiment e is allowed on a domain D_e⊆ X, changes the state by f_e:D_e→ X, and reports o_e:D_e→ O_e. Its execution cost c_e≥0 is fixed and known. Repair a has fixed cost c_a≥0 and succeeds on a declared set G_a⊆ X. All costs are in one stated unit. A nonempty set B_0⊆ X records the possible current states.
Exact mathematical passage
c_e\geq0
=Introduction passage · 75 passages
Search aliases: =
Introduction passage · Self_Improving_Intelligence_v1.0.0.tex, line 177
Write U(S,τ,L;B) for the selected result, including the procedure's randomness where applicable. Its productivity is R_Ξ(U,L;B)= E_(S,τ)∼ H [J_τ(U(S,τ,L;B))-J_τ(S)]. Invalid execution and out-of-budget behavior receive the treatment fixed by Ξ, not an exclusion chosen after seeing the outcome. The score may be a primary coordinate of a larger profile. Resource, validity, and group-specific constraints need not be folded into one scalar.
Exact mathematical passage
\label{eq:productivity} \Rcal_\Xi(U,L;B)=\E_{(S,\tau)\sim\mathcal H} \big[J_\tau(U(S,\tau,L;B))-J_\tau(S)\big].
\EIntroduction passage · 4 passages
Search aliases: E mathbb
Introduction passage · Self_Improving_Intelligence_v1.0.0.tex, line 177
Write U(S,τ,L;B) for the selected result, including the procedure's randomness where applicable. Its productivity is R_Ξ(U,L;B)= E_(S,τ)∼ H [J_τ(U(S,τ,L;B))-J_τ(S)]. Invalid execution and out-of-budget behavior receive the treatment fixed by Ξ, not an exclusion chosen after seeing the outcome. The score may be a primary coordinate of a larger profile. Resource, validity, and group-specific constraints need not be folded into one scalar.
Exact mathematical passage
\label{eq:productivity} \Rcal_\Xi(U,L;B)=\E_{(S,\tau)\sim\mathcal H} \big[J_\tau(U(S,\tau,L;B))-J_\tau(S)\big].
\E_{(S,\tau)\sim\mathcal H}Introduction passage · 1 passage
Search aliases: E_(S,τ)∼ H E tau sim mathcal mathbb tau sim mathcal
Introduction passage · Self_Improving_Intelligence_v1.0.0.tex, line 177
Write U(S,τ,L;B) for the selected result, including the procedure's randomness where applicable. Its productivity is R_Ξ(U,L;B)= E_(S,τ)∼ H [J_τ(U(S,τ,L;B))-J_τ(S)]. Invalid execution and out-of-budget behavior receive the treatment fixed by Ξ, not an exclusion chosen after seeing the outcome. The score may be a primary coordinate of a larger profile. Resource, validity, and group-specific constraints need not be folded into one scalar.
Exact mathematical passage
\label{eq:productivity} \Rcal_\Xi(U,L;B)=\E_{(S,\tau)\sim\mathcal H} \big[J_\tau(U(S,\tau,L;B))-J_\tau(S)\big].
\llbracketIntroduction passage · 2 passages
Search aliases: llbracket
Introduction passage · Self_Improving_Intelligence_v1.0.0.tex, line 814
A certified quotient preserves the investigation's evidence. Write llbracket prrbracket(z) for an expression's value in context z. Let n(p) be a deterministic representation produced by finitely many checked substitutions of valid equations. We only require n(p)=n(q) Longrightarrow llbracket prrbracket(z)=llbracket qrrbracket(z) for every admitted z. The reverse implication is unnecessary. Distinct normal forms may denote the same function. Missing such an equivalence wastes an optimization oppo…
Exact mathematical passage
\llbracket p\rrbracket(z)
\LongrightarrowIntroduction passage · 1 passage
Search aliases: Longrightarrow
Introduction passage · Self_Improving_Intelligence_v1.0.0.tex, line 815
A certified quotient preserves the investigation's evidence. Write llbracket prrbracket(z) for an expression's value in context z. Let n(p) be a deterministic representation produced by finitely many checked substitutions of valid equations. We only require n(p)=n(q) Longrightarrow llbracket prrbracket(z)=llbracket qrrbracket(z) for every admitted z. The reverse implication is unnecessary. Distinct normal forms may denote the same function. Missing such an equivalence wastes an optimization oppo…
Exact mathematical passage
n(p)=n(q)\quad\Longrightarrow\quad \llbracket p\rrbracket(z)=\llbracket q\rrbracket(z) \quad\text{for every admitted }z.
nIntroduction passage · 44 passages
Search aliases: n
Introduction passage · Self_Improving_Intelligence_v1.0.0.tex, line 814
A certified quotient preserves the investigation's evidence. Write llbracket prrbracket(z) for an expression's value in context z. Let n(p) be a deterministic representation produced by finitely many checked substitutions of valid equations. We only require n(p)=n(q) Longrightarrow llbracket prrbracket(z)=llbracket qrrbracket(z) for every admitted z. The reverse implication is unnecessary. Distinct normal forms may denote the same function. Missing such an equivalence wastes an optimization oppo…
Exact mathematical passage
n(p)
pIntroduction passage · 22 passages
Search aliases: p
Introduction passage · Self_Improving_Intelligence_v1.0.0.tex, line 814
A certified quotient preserves the investigation's evidence. Write llbracket prrbracket(z) for an expression's value in context z. Let n(p) be a deterministic representation produced by finitely many checked substitutions of valid equations. We only require n(p)=n(q) Longrightarrow llbracket prrbracket(z)=llbracket qrrbracket(z) for every admitted z. The reverse implication is unnecessary. Distinct normal forms may denote the same function. Missing such an equivalence wastes an optimization oppo…
Exact mathematical passage
\llbracket p\rrbracket(z)
qIntroduction passage · 14 passages
Search aliases: q
Introduction passage · Self_Improving_Intelligence_v1.0.0.tex, line 815
A certified quotient preserves the investigation's evidence. Write llbracket prrbracket(z) for an expression's value in context z. Let n(p) be a deterministic representation produced by finitely many checked substitutions of valid equations. We only require n(p)=n(q) Longrightarrow llbracket prrbracket(z)=llbracket qrrbracket(z) for every admitted z. The reverse implication is unnecessary. Distinct normal forms may denote the same function. Missing such an equivalence wastes an optimization oppo…
Exact mathematical passage
n(p)=n(q)\quad\Longrightarrow\quad \llbracket p\rrbracket(z)=\llbracket q\rrbracket(z) \quad\text{for every admitted }z.
\rrbracketIntroduction passage · 2 passages
Search aliases: rrbracket
Introduction passage · Self_Improving_Intelligence_v1.0.0.tex, line 814
A certified quotient preserves the investigation's evidence. Write llbracket prrbracket(z) for an expression's value in context z. Let n(p) be a deterministic representation produced by finitely many checked substitutions of valid equations. We only require n(p)=n(q) Longrightarrow llbracket prrbracket(z)=llbracket qrrbracket(z) for every admitted z. The reverse implication is unnecessary. Distinct normal forms may denote the same function. Missing such an equivalence wastes an optimization oppo…
Exact mathematical passage
\llbracket p\rrbracket(z)
zIntroduction passage · 12 passages
Search aliases: z
Introduction passage · Self_Improving_Intelligence_v1.0.0.tex, line 814
A certified quotient preserves the investigation's evidence. Write llbracket prrbracket(z) for an expression's value in context z. Let n(p) be a deterministic representation produced by finitely many checked substitutions of valid equations. We only require n(p)=n(q) Longrightarrow llbracket prrbracket(z)=llbracket qrrbracket(z) for every admitted z. The reverse implication is unnecessary. Distinct normal forms may denote the same function. Missing such an equivalence wastes an optimization oppo…
Exact mathematical passage
\llbracket p\rrbracket(z)
\simIntroduction passage · 1 passage
Search aliases: ∼ sim
Introduction passage · Self_Improving_Intelligence_v1.0.0.tex, line 177
Write U(S,τ,L;B) for the selected result, including the procedure's randomness where applicable. Its productivity is R_Ξ(U,L;B)= E_(S,τ)∼ H [J_τ(U(S,τ,L;B))-J_τ(S)]. Invalid execution and out-of-budget behavior receive the treatment fixed by Ξ, not an exclusion chosen after seeing the outcome. The score may be a primary coordinate of a larger profile. Resource, validity, and group-specific constraints need not be folded into one scalar.
Exact mathematical passage
\label{eq:productivity} \Rcal_\Xi(U,L;B)=\E_{(S,\tau)\sim\mathcal H} \big[J_\tau(U(S,\tau,L;B))-J_\tau(S)\big].
*Use in source · 6 passages
Search aliases: *
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 381
niform assessment error] If |widehat R_i-R_i|≤ε for every candidate and hatimath maximizes widehat R_i, then R_hatimath≥ max_iR_i-2ε. If a baseline is included and the selected candidate's estimated advantage exceeds 2ε+γ, its true advantage exceeds γ. For a true maximizer i^*, R_hatimath≥widehat R_hatimath-ε ≥widehat R_i^*-ε ≥ R_i^*-2ε. Apply the same two error bounds to the selected candidate and the baseline for the second assertion.
Exact mathematical passage
i^*
+Use in source · 34 passages
Search aliases: +
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 208
The model is fully declared. Setting each known cell as the defect on a calibration copy would permit learning the complete query table in 8(8+3)=88 observation calls. The code instead generates that table from the analytical specification and records it as supplied information. It does not claim to have executed those calibrations or inferred a hidden environment.
Exact mathematical passage
8(8+3)=88
\mathbf1Use in source · 2 passages
Search aliases: 1 mathbf
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 206
A diagnostic query names a subset A and returns 1{j∈ A}. The reference language contains eight singleton queries and three bit-coordinate queries. Repair r_i succeeds exactly when i=j. Because repair is irreversible within the episode, a procedure may commit only when every surviving possibility justifies its choice. Otherwise it may continue investigating or abstain.
Exact mathematical passage
\mathbf 1\{j\in A\}
AUse in source · 15 passages
Search aliases: A
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 206
A diagnostic query names a subset A and returns 1{j∈ A}. The reference language contains eight singleton queries and three bit-coordinate queries. Repair r_i succeeds exactly when i=j. Because repair is irreversible within the episode, a procedure may commit only when every surviving possibility justifies its choice. Otherwise it may continue investigating or abstain.
Exact mathematical passage
A
A'Use in source · 3 passages
Search aliases: A'
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 952
ompilers satisfy the common-domain premises of Theorem~[source reference]. If all stages complete, quotient-guided rule acquisition followed by optimized compilation returns the same learned expression and the same diagnostic controller as ordinary rule acquisition followed by the reference ordered compilation. In a common additive resource model, writing K for extra preparation cost, A,A' for the two acquisition costs, and D,D' for the two compilation costs, the composite saves work exactly whe…
Exact mathematical passage
A,A'
\mathcal A'Use in source · 1 passage
Search aliases: A' mathcal
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 407
Consider source and target finite repair models. Let σ:X→ X' be a bijection, τ: E→ E' a bijection of experiments, ψ: A→ A' a bijection of repairs, and η_e:O_e→ O'_τ(e) a bijection of report labels. Assume σ(D_e) · =D'_τ(e), · σ f_e(x) · =f'_τ(e)σ(x),
Exact mathematical passage
\psi:\mathcal A\to\mathcal A'
a'Use in source · 2 passages
Search aliases: a'
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 981
Suppose a revision replaces a by a'. Mark a vertex directly stale whenever one of its declared component values differs. Let Δ be the union of those vertices and all their descendants in the dependency graph. Equivalently, in a topological order, mark v whenever a|_D_v≠ a'|_D_v or pa(v)∩Δ≠∅. Here equality refers to the declared artifact value, not merely an unchanged version label. This procedure is conservative. A changed premise may leave the conclusion true, but the old evidence is not reused…
Exact mathematical passage
a'
A_0Use in source · 1 passage
Search aliases: A_0
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1401
appendix A complete diagnostic trace. Take n=8 and hidden defect j=5, whose binary digits are 101. One optimal tree uses the three queries A_0={1,3,5,7}, A_1={2,3,6,7}, A_2={4,5,6,7}. The precise order may differ among equally optimal compiled trees. For this order, the supports are {0,…,7}xrightarrow1{1,3,5,7} xrightarrow0{1,5}xrightarrow1{5}. The terminal repair is r_5. The path uses three diagnostic calls and one repair call. The complete tree stores all eight repair leaves and seven query no…
Exact mathematical passage
A_0=\{1,3,5,7\},\qquad A_1=\{2,3,6,7\},\qquad A_2=\{4,5,6,7\}.
A_1Use in source · 1 passage
Search aliases: A_1
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1401
appendix A complete diagnostic trace. Take n=8 and hidden defect j=5, whose binary digits are 101. One optimal tree uses the three queries A_0={1,3,5,7}, A_1={2,3,6,7}, A_2={4,5,6,7}. The precise order may differ among equally optimal compiled trees. For this order, the supports are {0,…,7}xrightarrow1{1,3,5,7} xrightarrow0{1,5}xrightarrow1{5}. The terminal repair is r_5. The path uses three diagnostic calls and one repair call. The complete tree stores all eight repair leaves and seven query no…
Exact mathematical passage
A_0=\{1,3,5,7\},\qquad A_1=\{2,3,6,7\},\qquad A_2=\{4,5,6,7\}.
a_1Use in source · 1 passage
Search aliases: a_1
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 977
A declared dependency graph. Let an artifact state be a=(a_1,…,a_d). Its components may include operation semantics, a grammar and its order, an equation library, experiment tables, repair-success sets, a returned controller, and a resource model. A finite directed acyclic graph has one vertex v for each claimed guarantee. Its incoming vertices pa(v) identify guarantees used as premises. Its direct dependency set D_v⊆{1,…,d} identifies the artifact components used directly. Each vertex carries a…
Exact mathematical passage
a=(a_1,\ldots,a_d)
A_2Use in source · 1 passage
Search aliases: A_2
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1401
appendix A complete diagnostic trace. Take n=8 and hidden defect j=5, whose binary digits are 101. One optimal tree uses the three queries A_0={1,3,5,7}, A_1={2,3,6,7}, A_2={4,5,6,7}. The precise order may differ among equally optimal compiled trees. For this order, the supports are {0,…,7}xrightarrow1{1,3,5,7} xrightarrow0{1,5}xrightarrow1{5}. The terminal repair is r_5. The path uses three diagnostic calls and one repair call. The complete tree stores all eight repair leaves and seven query no…
Exact mathematical passage
A_0=\{1,3,5,7\},\qquad A_1=\{2,3,6,7\},\qquad A_2=\{4,5,6,7\}.
a_dUse in source · 1 passage
Search aliases: a_d
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 977
A declared dependency graph. Let an artifact state be a=(a_1,…,a_d). Its components may include operation semantics, a grammar and its order, an equation library, experiment tables, repair-success sets, a returned controller, and a resource model. A finite directed acyclic graph has one vertex v for each claimed guarantee. Its incoming vertices pa(v) identify guarantees used as premises. Its direct dependency set D_v⊆{1,…,d} identifies the artifact components used directly. Each vertex carries a…
Exact mathematical passage
a=(a_1,\ldots,a_d)
A_rUse in source · 1 passage
Search aliases: A_r
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1413
For a target relabeling σ(i)=3i+1pmod8, the defect has target label σ(5)=0. Each query becomes σ(A_r) and the terminal repair becomes r'_σ(5). Reports and tree shape agree with the source, but the label adapter is part of the transported interface. Applying the old numeric repair label five directly in the target would be incorrect. This is why syntactic resemblance does not replace a semantic map.
Exact mathematical passage
\sigma(A_r)
A_tUse in source · 4 passages
Search aliases: A_t
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1077
[Conditional error budgets across an adaptive lineage] Let H_t-1 be the information available before attempt t. Suppose the event A_t that attempt t accepts a false improvement claim satisfies Pr(A_tmid H_t-1)≤δ_t almost surely, for deterministic δ_t≥0 with ∑_t≥1δ_t≤δ. Then Pr(∃ t≥1:A_t)≤δ. The candidate-generating process and the time at which attempts stop may depend on the earlier history. Taking expectations gives Pr(A_t)≤δ_t. The union bound gives the result for any finite number of attempt…
Exact mathematical passage
A_t
bUse in source · 43 passages
Search aliases: b
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 242
Repair sufficiency is not full identification. Suppose R(θ_1)={a,c} and R(θ_2)={b,c}. The cases are distinct, yet repair c works in both. A method can stop without learning which case occurred. Conversely, pairwise overlap of repair sets does not guarantee one repair for an entire uncertainty set. The three sets {a,b}, {b,c}, and {a,c} intersect pairwise and have empty total intersection. The stopping condition must refer to the whole surviving set.
Exact mathematical passage
\mathcal R(\theta_2)=\{b,c\}
B'_{\tau(e),\eta_e(o)}Use in source · 1 passage
Search aliases: B'_τ(e),η_e(o) tau eta
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 425
e preserved. In particular, F_h(B_0)= F'_h(σ(B_0)). The same conclusion holds for every selected tree, not only a cost-optimal tree. A terminal repair is valid on B_0 exactly when its image is valid on σ(B_0). For an internal experiment, domain preservation gives uniform admissibility. Transition and observation preservation yield σ(B_e,o)=B'_τ(e),η_e(o). Apply induction to every child. The report bijection preserves the number of children, the maximum execution cost, and the sum of stored child…
Exact mathematical passage
\sigma(B_{e,o})=B'_{\tau(e),\eta_e(o)}.
B_{e,o}Use in source · 4 passages
Search aliases: B_e,o
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 275
A chosen experiment must be uniformly allowed: B_0⊆ D_e. After observing o, the possible post-experiment states are B_e,o={f_e(x):x∈ B_0, o_e(x)=o}. Only nonempty branches are retained. A terminal repair is justified when B_0⊆ G_a. If the original state or an earlier obligation matters, it must be retained as an immutable tag in X. Merely changing the current state into a known state is not recovery of its history.
Exact mathematical passage
\label{eq:post} B_{e,o}=\{f_e(x):x\in B_0,\ o_e(x)=o\}.
\bar yUse in source · 3 passages
Search aliases: bar y bar
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1281
[Local operation agreement preserves finite continuation] Let D be invariant under every source operation F_a, let q:D→ Z, and suppose qF_a=G_aq on D for every admitted a. If the required output satisfies y=bar yq on D, then for every finite operation word w, qF_w=G_wq, yF_w=bar yG_wq. The empty word gives the identity. Assume the relation holds for a prefix. Domain invariance allows application of the next local equation, giving qF_aF_w=G_aqF_w=G_aG_wq. Induction establishes the first identity …
Exact mathematical passage
y=\bar yq
CUse in source · 40 passages
Search aliases: C
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 615
ation available inside a search A compiler is considering a binary query on a support of m possible immutable cases. It reads the query's public model rows in a fixed order. After a prefix, a of the inspected cases have report zero and b have report one. There are m-a-b rows left to read. If this query is the first of at most d≥1 remaining queries, either child can contain at most C=2^d-1 cases when distinct cases need distinct repair leaves. The local cardinality premise m≤2C is checked before …
Exact mathematical passage
C=2^{d-1}
C_{\rm execute}Use in source · 1 passage
Search aliases: C_rm execute rm
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1174
Use a resource vector rather than an unexplained token total: C=(C_rm model,C_rm search,C_rm validate,C_rm install, C_rm execute,C_rm store). These coordinates need not share units. Model construction may involve environment observations. Search and validation may involve processor work. Installation may require conversion or compilation. Execution work concerns one deployment. Storage concerns the retained library and complete controller. A scalar comparison needs declared conversion weights or…
Exact mathematical passage
C=(C_{\rm model},C_{\rm search},C_{\rm validate},C_{\rm install}, C_{\rm execute},C_{\rm store}).
C_{\rm install}Use in source · 1 passage
Search aliases: C_rm install rm
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1174
Use a resource vector rather than an unexplained token total: C=(C_rm model,C_rm search,C_rm validate,C_rm install, C_rm execute,C_rm store). These coordinates need not share units. Model construction may involve environment observations. Search and validation may involve processor work. Installation may require conversion or compilation. Execution work concerns one deployment. Storage concerns the retained library and complete controller. A scalar comparison needs declared conversion weights or…
Exact mathematical passage
C=(C_{\rm model},C_{\rm search},C_{\rm validate},C_{\rm install}, C_{\rm execute},C_{\rm store}).
C_{\rm model}Use in source · 1 passage
Search aliases: C_rm model rm
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1174
Use a resource vector rather than an unexplained token total: C=(C_rm model,C_rm search,C_rm validate,C_rm install, C_rm execute,C_rm store). These coordinates need not share units. Model construction may involve environment observations. Search and validation may involve processor work. Installation may require conversion or compilation. Execution work concerns one deployment. Storage concerns the retained library and complete controller. A scalar comparison needs declared conversion weights or…
Exact mathematical passage
C=(C_{\rm model},C_{\rm search},C_{\rm validate},C_{\rm install}, C_{\rm execute},C_{\rm store}).
C_{\rm prepare}Use in source · 2 passages
Search aliases: C_rm prepare rm
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 873
Let C_rm prepare be the complete preparation cost in a declared common resource model, including failed proposals and certificate checking, and let d_t be the subsequent saving on task t after including dispatch. Reuse is beneficial in that model exactly when ∑_t=1^Td_t>C_rm prepare. This is the earlier amortization identity applied to acquisition itself. The reference does not convert unlike counters into a universal wall-clock claim. Even one setup coordinate, 518,140 schema matches, exceeds t…
Exact mathematical passage
C_{\rm prepare}
C_{\rm search}Use in source · 1 passage
Search aliases: C_rm search rm
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1174
Use a resource vector rather than an unexplained token total: C=(C_rm model,C_rm search,C_rm validate,C_rm install, C_rm execute,C_rm store). These coordinates need not share units. Model construction may involve environment observations. Search and validation may involve processor work. Installation may require conversion or compilation. Execution work concerns one deployment. Storage concerns the retained library and complete controller. A scalar comparison needs declared conversion weights or…
Exact mathematical passage
C=(C_{\rm model},C_{\rm search},C_{\rm validate},C_{\rm install}, C_{\rm execute},C_{\rm store}).
C_{\rm store}Use in source · 1 passage
Search aliases: C_rm store rm
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1174
Use a resource vector rather than an unexplained token total: C=(C_rm model,C_rm search,C_rm validate,C_rm install, C_rm execute,C_rm store). These coordinates need not share units. Model construction may involve environment observations. Search and validation may involve processor work. Installation may require conversion or compilation. Execution work concerns one deployment. Storage concerns the retained library and complete controller. A scalar comparison needs declared conversion weights or…
Exact mathematical passage
C=(C_{\rm model},C_{\rm search},C_{\rm validate},C_{\rm install}, C_{\rm execute},C_{\rm store}).
C_{\rm validate}Use in source · 1 passage
Search aliases: C_rm validate rm
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1174
Use a resource vector rather than an unexplained token total: C=(C_rm model,C_rm search,C_rm validate,C_rm install, C_rm execute,C_rm store). These coordinates need not share units. Model construction may involve environment observations. Search and validation may involve processor work. Installation may require conversion or compilation. Execution work concerns one deployment. Storage concerns the retained library and complete controller. A scalar comparison needs declared conversion weights or…
Exact mathematical passage
C=(C_{\rm model},C_{\rm search},C_{\rm validate},C_{\rm install}, C_{\rm execute},C_{\rm store}).
c_tUse in source · 3 passages
Search aliases: c_t
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1260
A simple accounting model can express the additional conditions for acceleration. Let g_t be an expected useful gain and c_t>0 the complete cost of the corresponding discovery and acceptance cycle. Improvement productivity per unit resource is g_t/c_t. Raising g_t alone does not raise that ratio when c_t rises faster. A larger reachable repair set is similarly insufficient when proposal quality falls or validation becomes the limiting cost.
Exact mathematical passage
c_t>0
\mathrm{compare}Use in source · 1 passage
Search aliases: compare mathrm
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 764
If a model-row read has declared price w>0 and a new comparison has price one, the additional per-use saving under this conservative two-unit account is Δ(w)=w (R_full-R_stream)-N_compare. For the 64-case row it is positive when w>14209/8287. At w=1 it is negative. The price is a modeling choice, not a measured statement about Python or a physical machine. Interpreted AST visits, installed code, grammar construction, labels, and development search remain additional recorded coordinates. No weigh…
Exact mathematical passage
\Delta(w)=w\,(R_{\mathrm{full}}-R_{\mathrm{stream}})-N_{\mathrm{compare}}.
\operatorname{conv}Use in source · 1 passage
Search aliases: conv operatorname
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1046
[Resource dominance and declared price uncertainty] For a fixed ledger and workload, w·Δ(n)≥0 for every nonnegative price vector w if and only if Δ(n) is componentwise nonnegative. If admissible prices form the convex hull of a nonempty finite set W_0, then min_w∈conv(W_0)w·Δ(n) =min_w∈ W_0w·Δ(n). Thus strict improvement throughout that price set holds exactly when every listed vertex gives a strictly positive margin. The claims concern the same workflow, workload, and complete additive ledger. …
Exact mathematical passage
\min_{w\in\operatorname{conv}(W_0)}w\cdot\Delta(n) =\min_{w\in W_0}w\cdot\Delta(n).
dUse in source · 33 passages
Search aliases: d
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 453
[The resource price of conservative extension] Suppose an embedding preserves every old procedure's successful behavior and adds at most d≥0 execution work and s≥0 stored records. Every old allowance (b,m) is then feasible in the new language under allowance (b+d,m+s). Feasibility at the unchanged allowance follows only when the embedded procedure fits that allowance. Semantic conservativity alone supplies no such guarantee. Embed a witness for the old allowance. By hypothesis its behavior is pr…
Exact mathematical passage
d\geq0
D'Use in source · 3 passages
Search aliases: D'
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 952
s of Theorem~[source reference]. If all stages complete, quotient-guided rule acquisition followed by optimized compilation returns the same learned expression and the same diagnostic controller as ordinary rule acquisition followed by the reference ordered compilation. In a common additive resource model, writing K for extra preparation cost, A,A' for the two acquisition costs, and D,D' for the two compilation costs, the composite saves work exactly when (A-A')+(D-D')>K. The semantic conclusion…
Exact mathematical passage
D,D'
D'_{\tau(e)}Use in source · 1 passage
Search aliases: D'_τ(e) tau
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 408
Consider source and target finite repair models. Let σ:X→ X' be a bijection, τ: E→ E' a bijection of experiments, ψ: A→ A' a bijection of repairs, and η_e:O_e→ O'_τ(e) a bijection of report labels. Assume σ(D_e) · =D'_τ(e), · σ f_e(x) · =f'_τ(e)σ(x), η_e(o_e(x)) · =o'_τ(e)(σ x), · σ(G_a) · =G'_ψ(a),nonumber and matching execution costs. These assumptions concern supplied semantics. Their discovery or learned approximation is an additional problem.
Exact mathematical passage
\label{eq:transport} \sigma(D_e)&=D'_{\tau(e)}, & \sigma f_e(x)&=f'_{\tau(e)}\sigma(x),\\ \eta_e(o_e(x))&=o'_{\tau(e)}(\sigma x), & \sigma(G_a)&=G'_{\psi(a)},\nonumber
D_0Use in source · 3 passages
Search aliases: D_0
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 572
lowdown from a certified two-worker race] Suppose a parent and a shortcut worker can be paused after each unit of a common work measure and resumed without losing progress. Each may announce completion only after producing a result certified for the same target. Alternate one unit of each worker and stop at the first certified completion. If their standalone completion costs are D_0,D_1∈ N_>0∪{∞} and at least one is finite, at most 2min{D_0,D_1} worker units are executed. If each dispatched unit…
Exact mathematical passage
D_0,D_1\in\mathbb N_{>0}\cup\{\infty\}
d_0Use in source · 3 passages
Search aliases: d_0
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 696
The same search with fewer model-row reads. The reference compiler first checks the immutable, unit-cost, binary-query and singleton-repair scope of Theorem~[source reference]. It then conducts an ordered depth-first feasibility search at the information lower depth d_0=lceillog_2|B_0|rceil. The full-scan version reads every current row of a candidate query before rejecting an overfull child. The streaming version consults the certified prefix rule after each read. Both use the same query order,…
Exact mathematical passage
d_0=\lceil\log_2|B_0|\rceil
D_1Use in source · 3 passages
Search aliases: D_1
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 572
lowdown from a certified two-worker race] Suppose a parent and a shortcut worker can be paused after each unit of a common work measure and resumed without losing progress. Each may announce completion only after producing a result certified for the same target. Alternate one unit of each worker and stop at the first certified completion. If their standalone completion costs are D_0,D_1∈ N_>0∪{∞} and at least one is finite, at most 2min{D_0,D_1} worker units are executed. If each dispatched unit…
Exact mathematical passage
D_0,D_1\in\mathbb N_{>0}\cup\{\infty\}
D_e^\thetaUse in source · 2 passages
Search aliases: D_e^θ theta
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 322
A useful first extension retains several explicit possibilities rather than replacing uncertainty by one convenient estimate. Let Θ be a finite nonempty set of candidate models with a common state set X, experiment names, repair names, and nonnegative costs. Model θ specifies D_e^θ, f_e^θ, o_e^θ, and G_a^θ. The unknown model is fixed during an investigation. The initial information is a nonempty set B⊆Θ× X. The procedure observes reports, not the model label or the hidden physical state.
Exact mathematical passage
D_e^\theta
d_tUse in source · 2 passages
Search aliases: d_t
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 873
Let C_rm prepare be the complete preparation cost in a declared common resource model, including failed proposals and certificate checking, and let d_t be the subsequent saving on task t after including dispatch. Reuse is beneficial in that model exactly when ∑_t=1^Td_t>C_rm prepare. This is the earlier amortization identity applied to acquisition itself. The reference does not convert unlike counters into a universal wall-clock claim. Even one setup coordinate, 518,140 schema matches, exceeds t…
Exact mathematical passage
d_t
D_vUse in source · 3 passages
Search aliases: D_v
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 977
state be a=(a_1,…,a_d). Its components may include operation semantics, a grammar and its order, an equation library, experiment tables, repair-success sets, a returned controller, and a resource model. A finite directed acyclic graph has one vertex v for each claimed guarantee. Its incoming vertices pa(v) identify guarantees used as premises. Its direct dependency set D_v⊆{1,…,d} identifies the artifact components used directly. Each vertex carries a witness w_v and a local verification rule.
Exact mathematical passage
D_v\subseteq\{1,\ldots,d\}
EUse in source · 9 passages
Search aliases: E
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 196
What persists across a recursive step. A lineage state can be written schematically as Z_t=(S_t,U_t,L_t,E_t,Ξ_t,ρ_t), where E_t records evidence and rejected alternatives, and ρ_t records remaining resources and scope. A proposed modification may target any admitted subset. A comparison across different Ξ_t needs a declared bridge. It cannot be justified merely because both scores are called accuracy.
Exact mathematical passage
Z_t=(S_t,U_t,L_t,E_t,\Xi_t,\rho_t),
\mathcal E'Use in source · 1 passage
Search aliases: E' mathcal
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 407
Consider source and target finite repair models. Let σ:X→ X' be a bijection, τ: E→ E' a bijection of experiments, ψ: A→ A' a bijection of repairs, and η_e:O_e→ O'_τ(e) a bijection of report labels. Assume σ(D_e) · =D'_τ(e), · σ f_e(x) · =f'_τ(e)σ(x),
Exact mathematical passage
\tau:\mathcal E\to\mathcal E'
E_0Use in source · 1 passage
Search aliases: E_0
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1130
Choose a nonnegative bet λ_t, measurable from H_t-1, with λ_t≤1/(1+γ). Define E_0=1, E_t=E_t-1(1+λ_t(Y_t-γ)). The variable E_t is an evidence process, not a posterior probability that the candidate is better. Predictability means choosing the bet before observing the score to which it is applied.
Exact mathematical passage
\label{eq:anytime-capital} E_0=1,\qquad E_t=E_{t-1}\bigl(1+\lambda_t(Y_t-\gamma)\bigr).
E_0^{(j)}Use in source · 1 passage
Search aliases: E_0^(j)
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1164
Evidence across recursive generations. For attempt j, condition on the complete earlier lineage, freeze its candidate, and initialize a fresh valid audit at E_0^(j)=1 with error allowance δ_j. If ∑_jδ_j≤δ, Proposition~[source reference] bounds the probability of ever accepting a false margin claim by δ. Each audit can stop sequentially, and the next candidate may depend on completed earlier audits. The premise remains conditional validity for each actual attempt.
Exact mathematical passage
E_0^{(j)}=1
\E_{P_0}Use in source · 1 passage
Search aliases: E_P_0 E mathbb
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 230
ry decision: report case zero if the repair belongs to R(0), and case one otherwise. Success in case zero implies a correct decision. Success in case one also implies a correct decision because the two repair sets are disjoint. Thus repair success is at most the success of this binary test. For a randomized test with probability f(o)∈[0,1] of reporting zero, the latter success is tfrac12(1+ E_P_0f- E_P_1f) ≤tfrac12(1+d_TV(P_0,P_1)). The inequality follows by summing the positive differences of t…
Exact mathematical passage
\tfrac12\big(1+\E_{P_0}f-\E_{P_1}f\big) \leq\tfrac12(1+\TV(P_0,P_1)).
\E_{P_1}Use in source · 1 passage
Search aliases: E_P_1 E mathbb
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 230
ry decision: report case zero if the repair belongs to R(0), and case one otherwise. Success in case zero implies a correct decision. Success in case one also implies a correct decision because the two repair sets are disjoint. Thus repair success is at most the success of this binary test. For a randomized test with probability f(o)∈[0,1] of reporting zero, the latter success is tfrac12(1+ E_P_0f- E_P_1f) ≤tfrac12(1+d_TV(P_0,P_1)). The inequality follows by summing the positive differences of t…
Exact mathematical passage
\tfrac12\big(1+\E_{P_0}f-\E_{P_1}f\big) \leq\tfrac12(1+\TV(P_0,P_1)).
E_tUse in source · 8 passages
Search aliases: E_t
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 196
What persists across a recursive step. A lineage state can be written schematically as Z_t=(S_t,U_t,L_t,E_t,Ξ_t,ρ_t), where E_t records evidence and rejected alternatives, and ρ_t records remaining resources and scope. A proposed modification may target any admitted subset. A comparison across different Ξ_t needs a declared bridge. It cannot be justified merely because both scores are called accuracy.
Exact mathematical passage
Z_t=(S_t,U_t,L_t,E_t,\Xi_t,\rho_t),
E_{t-1}Use in source · 2 passages
Search aliases: E_t-1
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1130
Choose a nonnegative bet λ_t, measurable from H_t-1, with λ_t≤1/(1+γ). Define E_0=1, E_t=E_t-1(1+λ_t(Y_t-γ)). The variable E_t is an evidence process, not a posterior probability that the candidate is better. Predictability means choosing the bet before observing the score to which it is applied.
Exact mathematical passage
\label{eq:anytime-capital} E_0=1,\qquad E_t=E_{t-1}\bigl(1+\lambda_t(Y_t-\gamma)\bigr).
\expUse in source · 1 passage
Search aliases: exp
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1096
One sufficient fresh-audit construction. Freeze a candidate and its comparator using the prior history. Draw n_t fresh evaluation cases conditionally independently from the declared distribution and record paired score differences Y_i∈[-1,1]. Let Δ_t be their conditional mean and widehatΔ_t their average. The bounded-variable concentration inequality gives Pr(widehatΔ_t-Δ_t≥ rmid H_t-1) ≤ exp(-n_tr^2/2). Consequently, accepting a claim Δ_t≥γ only when widehatΔ_t-sqrt2log(1/δ_t)/n_t≥γ meets the p…
Exact mathematical passage
\Pr\left(\widehat\Delta_t-\Delta_t\geq r\mid\mathcal H_{t-1}\right) \leq \exp(-n_tr^2/2).
\mathcal FUse in source · 3 passages
Search aliases: F mathcal
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 284
For horizon h, let F_h(B_0) be the nondominated attainable pairs (w,m). A pair dominates another when it is no larger in either coordinate and strictly smaller in at least one. Let T(B_0)={(c_a,1):B_0⊆ G_a}. Then F_0(B_0) · =Pareto T(B_0),
Exact mathematical passage
\mathcal F_h(B_0)
\mathcal F'_hUse in source · 1 passage
Search aliases: F'_h mathcal
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 418
[Exact transport of diagnostic procedures] Under~[source reference], relabeling every experiment, branch, and repair transports successful diagnostic trees bijectively between the source support B_0 and target support σ(B_0). Execution cost, complete tree storage, and depth are preserved. In particular, F_h(B_0)= F'_h(σ(B_0)). The same conclusion holds for every selected tree, not only a cost-optimal tree. A terminal repair is valid on B_0 exactly when its image is valid on σ(B_0). For an intern…
Exact mathematical passage
\mathcal F_h(B_0)=\mathcal F'_h(\sigma(B_0)).
f'_{\tau(e)}Use in source · 1 passage
Search aliases: f'_τ(e) tau
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 408
Consider source and target finite repair models. Let σ:X→ X' be a bijection, τ: E→ E' a bijection of experiments, ψ: A→ A' a bijection of repairs, and η_e:O_e→ O'_τ(e) a bijection of report labels. Assume σ(D_e) · =D'_τ(e), · σ f_e(x) · =f'_τ(e)σ(x), η_e(o_e(x)) · =o'_τ(e)(σ x), · σ(G_a) · =G'_ψ(a),nonumber and matching execution costs. These assumptions concern supplied semantics. Their discovery or learned approximation is an additional problem.
Exact mathematical passage
\label{eq:transport} \sigma(D_e)&=D'_{\tau(e)}, & \sigma f_e(x)&=f'_{\tau(e)}\sigma(x),\\ \eta_e(o_e(x))&=o'_{\tau(e)}(\sigma x), & \sigma(G_a)&=G'_{\psi(a)},\nonumber
\mathcal F_0Use in source · 1 passage
Search aliases: F_0 mathcal
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 285
For horizon h, let F_h(B_0) be the nondominated attainable pairs (w,m). A pair dominates another when it is no larger in either coordinate and strictly smaller in at least one. Let T(B_0)={(c_a,1):B_0⊆ G_a}. Then F_0(B_0) · =Pareto T(B_0), F_h(B_0) · =Pareto(T(B_0)∪ ⋃_e:B_0⊆ D_e {(c_e+max_o w_o, 1+∑_o m_o): (w_o,m_o)∈ F_h-1(B_e,o) ∀ o}). nonumber An empty child frontier makes that experiment unavailable for a successful complete tree. An empty union is not a zero-cost solution.
Exact mathematical passage
\label{eq:pareto} \mathcal F_0(B_0)&=\Pareto T(B_0),\\ \mathcal F_h(B_0)&=\Pareto\left(T(B_0)\cup \bigcup_{e:B_0\subseteq D_e} \left\{\left(c_e+\max_o w_o,\ 1+\sum_o m_o\right): (w_o,m_o)\in\mathcal F_{h-1}(B_{e,o})\ \forall o\right\}\right). \nonumber
F_aUse in source · 3 passages
Search aliases: F_a
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1277
[Local operation agreement preserves finite continuation] Let D be invariant under every source operation F_a, let q:D→ Z, and suppose qF_a=G_aq on D for every admitted a. If the required output satisfies y=bar yq on D, then for every finite operation word w, qF_w=G_wq, yF_w=bar yG_wq. The empty word gives the identity. Assume the relation holds for a prefix. Domain invariance allows application of the next local equation, giving qF_aF_w=G_aqF_w=G_aG_wq. Induction establishes the first identity …
Exact mathematical passage
F_a
f_e^\thetaUse in source · 3 passages
Search aliases: f_e^θ theta
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 322
A useful first extension retains several explicit possibilities rather than replacing uncertainty by one convenient estimate. Let Θ be a finite nonempty set of candidate models with a common state set X, experiment names, repair names, and nonnegative costs. Model θ specifies D_e^θ, f_e^θ, o_e^θ, and G_a^θ. The unknown model is fixed during an investigation. The initial information is a nonempty set B⊆Θ× X. The procedure observes reports, not the model label or the hidden physical state.
Exact mathematical passage
f_e^\theta
\mathcal F_hUse in source · 3 passages
Search aliases: F_h mathcal
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 284
For horizon h, let F_h(B_0) be the nondominated attainable pairs (w,m). A pair dominates another when it is no larger in either coordinate and strictly smaller in at least one. Let T(B_0)={(c_a,1):B_0⊆ G_a}. Then F_0(B_0) · =Pareto T(B_0),
Exact mathematical passage
\mathcal F_h(B_0)
\mathcal F_{h-1}Use in source · 1 passage
Search aliases: F_h-1 mathcal
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 285
For horizon h, let F_h(B_0) be the nondominated attainable pairs (w,m). A pair dominates another when it is no larger in either coordinate and strictly smaller in at least one. Let T(B_0)={(c_a,1):B_0⊆ G_a}. Then F_0(B_0) · =Pareto T(B_0), F_h(B_0) · =Pareto(T(B_0)∪ ⋃_e:B_0⊆ D_e {(c_e+max_o w_o, 1+∑_o m_o): (w_o,m_o)∈ F_h-1(B_e,o) ∀ o}). nonumber An empty child frontier makes that experiment unavailable for a successful complete tree. An empty union is not a zero-cost solution.
Exact mathematical passage
\label{eq:pareto} \mathcal F_0(B_0)&=\Pareto T(B_0),\\ \mathcal F_h(B_0)&=\Pareto\left(T(B_0)\cup \bigcup_{e:B_0\subseteq D_e} \left\{\left(c_e+\max_o w_o,\ 1+\sum_o m_o\right): (w_o,m_o)\in\mathcal F_{h-1}(B_{e,o})\ \forall o\right\}\right). \nonumber
F_uUse in source · 3 passages
Search aliases: F_u
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 247
Repairing a failed representation. Let X be a finite set with deterministic operations F_u:X→ X and output labels q(x). Begin with a partition P_0 that preserves both the old retained labels and q. Define the next partition by splitting a block whenever two states in it have successors in different current blocks under some permitted operation. Equivalently, x≡_P_t+1y Longleftrightarrow x≡_P_ty and F_u(x)≡_P_tF_u(y) for every u. The construction uses a complete transition description. Having a f…
Exact mathematical passage
F_u:X\to X
F_wUse in source · 2 passages
Search aliases: F_w
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1282
[Local operation agreement preserves finite continuation] Let D be invariant under every source operation F_a, let q:D→ Z, and suppose qF_a=G_aq on D for every admitted a. If the required output satisfies y=bar yq on D, then for every finite operation word w, qF_w=G_wq, yF_w=bar yG_wq. The empty word gives the identity. Assume the relation holds for a prefix. Domain invariance allows application of the next local equation, giving qF_aF_w=G_aqF_w=G_aG_wq. Induction establishes the first identity …
Exact mathematical passage
qF_w=G_wq,\qquad yF_w=\bar yG_wq.
\mathrm{full}Use in source · 1 passage
Search aliases: full mathrm
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 764
If a model-row read has declared price w>0 and a new comparison has price one, the additional per-use saving under this conservative two-unit account is Δ(w)=w (R_full-R_stream)-N_compare. For the 64-case row it is positive when w>14209/8287. At w=1 it is negative. The price is a modeling choice, not a measured statement about Python or a physical machine. Interpreted AST visits, installed code, grammar construction, labels, and development search remain additional recorded coordinates. No weigh…
Exact mathematical passage
\Delta(w)=w\,(R_{\mathrm{full}}-R_{\mathrm{stream}})-N_{\mathrm{compare}}.
\mathcal GUse in source · 9 passages
Search aliases: G mathcal
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 393
For a fixed budget, let R^*_ G be the largest productivity among procedures expressible in a declared language G, with installed data and interpreter costs held fixed. The gap R^*_ G-R(U) is search or selection headroom within that language. Comparing R^*_ G' with R^*_ G concerns expressibility and resources. Neither upper envelope is automatically available to the learner.
Exact mathematical passage
R^*_{\mathcal G}
gUse in source · 10 passages
Search aliases: g
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 921
Paired repairs change the task, not just its notation. Consider 2g possible hidden cases divided into g pairs, where g is a power of two. One repair succeeds on both members of each pair. The query language contains equality questions and bits of the pair index. The greedy weight constructor gives weight one to the first member of each pair and zero to the second, with total weight g. A depth log_2g tree is both necessary and sufficient, so the diagnostic-plus-repair work is 1+log_2g and a full …
Exact mathematical passage
2g
\mathcal G'Use in source · 1 passage
Search aliases: G' mathcal
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 393
For a fixed budget, let R^*_ G be the largest productivity among procedures expressible in a declared language G, with installed data and interpreter costs held fixed. The gap R^*_ G-R(U) is search or selection headroom within that language. Comparing R^*_ G' with R^*_ G concerns expressibility and resources. Neither upper envelope is automatically available to the learner.
Exact mathematical passage
R^*_{\mathcal G'}
G'_{\psi(a)}Use in source · 1 passage
Search aliases: G'_ψ(a) psi
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 408
Consider source and target finite repair models. Let σ:X→ X' be a bijection, τ: E→ E' a bijection of experiments, ψ: A→ A' a bijection of repairs, and η_e:O_e→ O'_τ(e) a bijection of report labels. Assume σ(D_e) · =D'_τ(e), · σ f_e(x) · =f'_τ(e)σ(x), η_e(o_e(x)) · =o'_τ(e)(σ x), · σ(G_a) · =G'_ψ(a),nonumber and matching execution costs. These assumptions concern supplied semantics. Their discovery or learned approximation is an additional problem.
Exact mathematical passage
\label{eq:transport} \sigma(D_e)&=D'_{\tau(e)}, & \sigma f_e(x)&=f'_{\tau(e)}\sigma(x),\\ \eta_e(o_e(x))&=o'_{\tau(e)}(\sigma x), & \sigma(G_a)&=G'_{\psi(a)},\nonumber
G_a^\thetaUse in source · 3 passages
Search aliases: G_a^θ theta
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 322
A useful first extension retains several explicit possibilities rather than replacing uncertainty by one convenient estimate. Let Θ be a finite nonempty set of candidate models with a common state set X, experiment names, repair names, and nonnegative costs. Model θ specifies D_e^θ, f_e^θ, o_e^θ, and G_a^θ. The unknown model is fixed during an investigation. The initial information is a nonempty set B⊆Θ× X. The procedure observes reports, not the model label or the hidden physical state.
Exact mathematical passage
G_a^\theta
G_rUse in source · 1 passage
Search aliases: G_r
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 722
Let B be a finite nonempty immutable-case support and let G_r⊆ B be each repair's success set restricted to it. Define κ(B) to be the smallest number of repair-success sets whose union covers B. Set κ(B)=∞ when no such cover exists. This number concerns the repaired objective, not the number of hidden states. Computing it can itself be expensive.
Exact mathematical passage
G_r\subseteq B
g_tUse in source · 3 passages
Search aliases: g_t
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1260
A simple accounting model can express the additional conditions for acceleration. Let g_t be an expected useful gain and c_t>0 the complete cost of the corresponding discovery and acceptance cycle. Improvement productivity per unit resource is g_t/c_t. Raising g_t alone does not raise that ratio when c_t rises faster. A larger reachable repair set is similarly insufficient when proposal quality falls or validation becomes the limiting cost.
Exact mathematical passage
g_t
G_wUse in source · 2 passages
Search aliases: G_w
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1282
[Local operation agreement preserves finite continuation] Let D be invariant under every source operation F_a, let q:D→ Z, and suppose qF_a=G_aq on D for every admitted a. If the required output satisfies y=bar yq on D, then for every finite operation word w, qF_w=G_wq, yF_w=bar yG_wq. The empty word gives the identity. Assume the relation holds for a prefix. Domain invariance allows application of the next local equation, giving qF_aF_w=G_aqF_w=G_aG_wq. Induction establishes the first identity …
Exact mathematical passage
qF_w=G_wq,\qquad yF_w=\bar yG_wq.
G_{\theta,a}Use in source · 1 passage
Search aliases: G_θ,a theta
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1270
What would learn. Let r_ϕ encode the retained state, G_θ,a describe proposed operations, π_ω choose observations or experiments, and p_ψ propose candidate programs or parameter updates. A development episode can modify ϕ,θ,ω,ψ through a declared learning rule. A persistent program library can also change by adding, replacing, or removing checked operators. Temporary deliberation without a retained modification is not called a new learned generation here.
Exact mathematical passage
G_{\theta,a}
\GcalUse in source · 2 passages
Search aliases: Gcal G mathcal
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 793
Four objects and two meanings of success. Let G=(p_0,…,p_N-1) be a finite ordered grammar of pure, total expressions. A target task supplies contexts z∈ D and desired values f(z). An ordinary learner uses counterexamples to find the first grammar expression that agrees with all of D. A separate equational learner observes expressions on a development probe set and proposes equations. A checker admits only equations supported by a derivation in the declared arithmetic theory.
Exact mathematical passage
\Gcal=(p_0,\ldots,p_{N-1})
\mathcal H_{t-1}Use in source · 7 passages
Search aliases: H_t-1 mathcal
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1077
[Conditional error budgets across an adaptive lineage] Let H_t-1 be the information available before attempt t. Suppose the event A_t that attempt t accepts a false improvement claim satisfies Pr(A_tmid H_t-1)≤δ_t almost surely, for deterministic δ_t≥0 with ∑_t≥1δ_t≤δ. Then Pr(∃ t≥1:A_t)≤δ. The candidate-generating process and the time at which attempts stop may depend on the earlier history. Taking expectations gives Pr(A_t)≤δ_t. The union bound gives the result for any finite number of attempt…
Exact mathematical passage
\mathcal H_{t-1}
\hat\imathUse in source · 3 passages
Search aliases: hatimath hat imath
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 374
[Selection under uniform assessment error] If |widehat R_i-R_i|≤ε for every candidate and hatimath maximizes widehat R_i, then R_hatimath≥ max_iR_i-2ε. If a baseline is included and the selected candidate's estimated advantage exceeds 2ε+γ, its true advantage exceeds γ. For a true maximizer i^*, R_hatimath≥widehat R_hatimath-ε ≥widehat R_i^*-ε ≥ R_i^*-2ε. Apply the same two error bounds to the selected candidate and the baseline for the second assertion.
Exact mathematical passage
\hat\imath
iUse in source · 35 passages
Search aliases: i
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 206
A diagnostic query names a subset A and returns 1{j∈ A}. The reference language contains eight singleton queries and three bit-coordinate queries. Repair r_i succeeds exactly when i=j. Because repair is irreversible within the episode, a procedure may commit only when every surviving possibility justifies its choice. Otherwise it may continue investigating or abstain.
Exact mathematical passage
r_i
i^*Use in source · 2 passages
Search aliases: i^*
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 381
niform assessment error] If |widehat R_i-R_i|≤ε for every candidate and hatimath maximizes widehat R_i, then R_hatimath≥ max_iR_i-2ε. If a baseline is included and the selected candidate's estimated advantage exceeds 2ε+γ, its true advantage exceeds γ. For a true maximizer i^*, R_hatimath≥widehat R_hatimath-ε ≥widehat R_i^*-ε ≥ R_i^*-2ε. Apply the same two error bounds to the selected candidate and the baseline for the second assertion.
Exact mathematical passage
i^*
i_1Use in source · 3 passages
Search aliases: i_1
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 483
Every case produces one deterministic query history. Two different cases cannot terminate at the same repair leaf because no repair succeeds in both. Thus there are at least n nonempty leaves. A tree of binary query depth d has at most 2^d leaves, so d≥lceillog_2 nrceil. Every path also executes a unit-cost repair. For storage, let l be the number of leaves and let i_1,i_2 count unary and binary internal nodes. Counting edges in a finite rooted tree gives i_1+2i_2=i_1+i_2+l-1, hence i_2=l-1. Tot…
Exact mathematical passage
i_1,i_2
i_2Use in source · 3 passages
Search aliases: i_2
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 483
Every case produces one deterministic query history. Two different cases cannot terminate at the same repair leaf because no repair succeeds in both. Thus there are at least n nonempty leaves. A tree of binary query depth d has at most 2^d leaves, so d≥lceillog_2 nrceil. Every path also executes a unit-cost repair. For storage, let l be the number of leaves and let i_1,i_2 count unary and binary internal nodes. Counting edges in a finite rooted tree gives i_1+2i_2=i_1+i_2+l-1, hence i_2=l-1. Tot…
Exact mathematical passage
i_1,i_2
jUse in source · 17 passages
Search aliases: j
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 204
A small world with reusable diagnostic structure. Consider eight cells with exactly one latent defect, indexed by j∈{0,…,7}. The solver must repair that cell while preserving the others. Its old representation retains only the number of defects, so every episode initially looks identical.
Exact mathematical passage
j\in\{0,\ldots,7\}
KUse in source · 7 passages
Search aliases: K
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 952
and the original and optimized compilers satisfy the common-domain premises of Theorem~[source reference]. If all stages complete, quotient-guided rule acquisition followed by optimized compilation returns the same learned expression and the same diagnostic controller as ordinary rule acquisition followed by the reference ordered compilation. In a common additive resource model, writing K for extra preparation cost, A,A' for the two acquisition costs, and D,D' for the two compilation costs, the …
Exact mathematical passage
K
K_0Use in source · 1 passage
Search aliases: K_0
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1208
Define K_0=Safe, K_n+1={z∈ K_n:∃ a∈ A(z), Post(z,a)⊆ K_n}. The descending finite sequence reaches a fixed point K. It is the set from which some policy keeps every successor safe indefinitely. This can include doing nothing forever. Safety alone is not recursive improvement.
Exact mathematical passage
K_0=\Safe,\qquad K_{n+1}=\{z\in K_n:\exists a\in A(z),\ \Post(z,a)\subseteq K_n\}.
K_nUse in source · 1 passage
Search aliases: K_n
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1208
Define K_0=Safe, K_n+1={z∈ K_n:∃ a∈ A(z), Post(z,a)⊆ K_n}. The descending finite sequence reaches a fixed point K. It is the set from which some policy keeps every successor safe indefinitely. This can include doing nothing forever. Safety alone is not recursive improvement.
Exact mathematical passage
K_0=\Safe,\qquad K_{n+1}=\{z\in K_n:\exists a\in A(z),\ \Post(z,a)\subseteq K_n\}.
K_{n+1}Use in source · 1 passage
Search aliases: K_n+1
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1208
Define K_0=Safe, K_n+1={z∈ K_n:∃ a∈ A(z), Post(z,a)⊆ K_n}. The descending finite sequence reaches a fixed point K. It is the set from which some policy keeps every successor safe indefinitely. This can include doing nothing forever. Safety alone is not recursive improvement.
Exact mathematical passage
K_0=\Safe,\qquad K_{n+1}=\{z\in K_n:\exists a\in A(z),\ \Post(z,a)\subseteq K_n\}.
lUse in source · 5 passages
Search aliases: l
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 483
exhaustive synthesis. Every case produces one deterministic query history. Two different cases cannot terminate at the same repair leaf because no repair succeeds in both. Thus there are at least n nonempty leaves. A tree of binary query depth d has at most 2^d leaves, so d≥lceillog_2 nrceil. Every path also executes a unit-cost repair. For storage, let l be the number of leaves and let i_1,i_2 count unary and binary internal nodes. Counting edges in a finite rooted tree gives i_1+2i_2=i_1+i_2+l…
Exact mathematical passage
l
L'Use in source · 3 passages
Search aliases: L'
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 186
Fixed-library and full-system comparisons. A comparison of R_Ξ(U',L;B) with R_Ξ(U,L;B) isolates process change at a common library. A comparison with (U',L') measures the combined descendant. Both are useful, but they answer different questions. A crossed comparison of (U,L), (U',L), (U,L'), and (U',L') can expose interaction between procedure and library. There is no general additive decomposition that makes their contributions independent.
Exact mathematical passage
(U',L')
L_tUse in source · 1 passage
Search aliases: L_t
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 196
What persists across a recursive step. A lineage state can be written schematically as Z_t=(S_t,U_t,L_t,E_t,Ξ_t,ρ_t), where E_t records evidence and rejected alternatives, and ρ_t records remaining resources and scope. A proposed modification may target any admitted subset. A comparison across different Ξ_t needs a declared bridge. It cannot be justified merely because both scores are called accuracy.
Exact mathematical passage
Z_t=(S_t,U_t,L_t,E_t,\Xi_t,\rho_t),
\lceilUse in source · 3 passages
Search aliases: lceil
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 477
[Checked lower-bound closure of a diagnostic search] Every successful controller in this scope has worst-case execution work at least 1+lceillog_2 nrceil and storage at least 2n-1. If an independently checked candidate attains both bounds and respects the requested horizon, its cost pair is the entire nondominated frontier. Returning this candidate is therefore equivalent in the protected work/storage objective to completing the exhaustive synthesis. Every case produces one deterministic query h…
Exact mathematical passage
1+\lceil\log_2 n\rceil
\lfloorUse in source · 1 passage
Search aliases: lfloor
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1249
[A fixed-margin ceiling on a fixed score] Suppose a lineage has a fixed measure R_t∈[a,b], every accepted progress event increases it by at least γ>0, and all other transitions are nondecreasing in that same measure. Then the number of progress events is at most lfloorfracb-aγrfloor. An infinite sequence of positive increments is compatible with bounded total improvement when no common positive lower bound is imposed. After k progress events, telescoping and nondecrease give R_t≥ R_0+kγ≥ a+kγ. S…
Exact mathematical passage
\left\lfloor\frac{b-a}{\gamma}\right\rfloor.
\logUse in source · 8 passages
Search aliases: log
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 477
[Checked lower-bound closure of a diagnostic search] Every successful controller in this scope has worst-case execution work at least 1+lceillog_2 nrceil and storage at least 2n-1. If an independently checked candidate attains both bounds and respects the requested horizon, its cost pair is the entire nondominated frontier. Returning this candidate is therefore equivalent in the protected work/storage objective to completing the exhaustive synthesis. Every case produces one deterministic query h…
Exact mathematical passage
1+\lceil\log_2 n\rceil
\log_2Use in source · 7 passages
Search aliases: log_2 log
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 477
[Checked lower-bound closure of a diagnostic search] Every successful controller in this scope has worst-case execution work at least 1+lceillog_2 nrceil and storage at least 2n-1. If an independently checked candidate attains both bounds and respects the requested horizon, its cost pair is the entire nondominated frontier. Returning this candidate is therefore equivalent in the protected work/storage objective to completing the exhaustive synthesis. Every case produces one deterministic query h…
Exact mathematical passage
1+\lceil\log_2 n\rceil
\LongleftrightarrowUse in source · 1 passage
Search aliases: Longleftrightarrow
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 248
Repairing a failed representation. Let X be a finite set with deterministic operations F_u:X→ X and output labels q(x). Begin with a partition P_0 that preserves both the old retained labels and q. Define the next partition by splitting a block whenever two states in it have successors in different current blocks under some permitted operation. Equivalently, x≡_P_t+1y Longleftrightarrow x≡_P_ty and F_u(x)≡_P_tF_u(y) for every u. The construction uses a complete transition description. Having a f…
Exact mathematical passage
x\equiv_{P_{t+1}}y \quad\Longleftrightarrow\quad x\equiv_{P_t}y\ \text{ and }\ F_u(x)\equiv_{P_t}F_u(y)\ \text{for every }u.
MUse in source · 1 passage
Search aliases: M
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 282
Let W(P,B_0) be the largest execution cost of a complete path, including its terminal repair. Let M(P) be the number of stored tree records. These costs are not independently minimized. A single tree has to realize both.
Exact mathematical passage
M(P)
mUse in source · 27 passages
Search aliases: m
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 284
For horizon h, let F_h(B_0) be the nondominated attainable pairs (w,m). A pair dominates another when it is no larger in either coordinate and strictly smaller in at least one. Let T(B_0)={(c_a,1):B_0⊆ G_a}. Then F_0(B_0) · =Pareto T(B_0),
Exact mathematical passage
(w,m)
m_oUse in source · 1 passage
Search aliases: m_o
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 285
For horizon h, let F_h(B_0) be the nondominated attainable pairs (w,m). A pair dominates another when it is no larger in either coordinate and strictly smaller in at least one. Let T(B_0)={(c_a,1):B_0⊆ G_a}. Then F_0(B_0) · =Pareto T(B_0), F_h(B_0) · =Pareto(T(B_0)∪ ⋃_e:B_0⊆ D_e {(c_e+max_o w_o, 1+∑_o m_o): (w_o,m_o)∈ F_h-1(B_e,o) ∀ o}). nonumber An empty child frontier makes that experiment unavailable for a successful complete tree. An empty union is not a zero-cost solution.
Exact mathematical passage
\label{eq:pareto} \mathcal F_0(B_0)&=\Pareto T(B_0),\\ \mathcal F_h(B_0)&=\Pareto\left(T(B_0)\cup \bigcup_{e:B_0\subseteq D_e} \left\{\left(c_e+\max_o w_o,\ 1+\sum_o m_o\right): (w_o,m_o)\in\mathcal F_{h-1}(B_{e,o})\ \forall o\right\}\right). \nonumber
\mathbinUse in source · 2 passages
Search aliases: mathbin
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 883
[The same operator spelling with changed semantics] The identity x+0=x is valid for ordinary addition. Suppose a revised interpreter uses the same printed symbol but executes amathbin⊕b=a+b+1. Then xmathbin⊕0≠ x. A cache certified for the former interpreter cannot be imported merely because its expression strings still parse. A semantic identifier helps detect ordinary version mismatch, but an identifier falsely attached to a changed implementation is not itself proof of faithful execution.
Exact mathematical passage
a\mathbin{\oplus}b=a+b+1
\maxUse in source · 14 passages
Search aliases: max
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 285
For horizon h, let F_h(B_0) be the nondominated attainable pairs (w,m). A pair dominates another when it is no larger in either coordinate and strictly smaller in at least one. Let T(B_0)={(c_a,1):B_0⊆ G_a}. Then F_0(B_0) · =Pareto T(B_0), F_h(B_0) · =Pareto(T(B_0)∪ ⋃_e:B_0⊆ D_e {(c_e+max_o w_o, 1+∑_o m_o): (w_o,m_o)∈ F_h-1(B_e,o) ∀ o}). nonumber An empty child frontier makes that experiment unavailable for a successful complete tree. An empty union is not a zero-cost solution.
Exact mathematical passage
\label{eq:pareto} \mathcal F_0(B_0)&=\Pareto T(B_0),\\ \mathcal F_h(B_0)&=\Pareto\left(T(B_0)\cup \bigcup_{e:B_0\subseteq D_e} \left\{\left(c_e+\max_o w_o,\ 1+\sum_o m_o\right): (w_o,m_o)\in\mathcal F_{h-1}(B_{e,o})\ \forall o\right\}\right). \nonumber
\max_iUse in source · 1 passage
Search aliases: max_i max
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 375
[Selection under uniform assessment error] If |widehat R_i-R_i|≤ε for every candidate and hatimath maximizes widehat R_i, then R_hatimath≥ max_iR_i-2ε. If a baseline is included and the selected candidate's estimated advantage exceeds 2ε+γ, its true advantage exceeds γ. For a true maximizer i^*, R_hatimath≥widehat R_hatimath-ε ≥widehat R_i^*-ε ≥ R_i^*-2ε. Apply the same two error bounds to the selected candidate and the baseline for the second assertion.
Exact mathematical passage
R_{\hat\imath}\geq \max_iR_i-2\varepsilon.
\max_oUse in source · 1 passage
Search aliases: max_o max
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 285
For horizon h, let F_h(B_0) be the nondominated attainable pairs (w,m). A pair dominates another when it is no larger in either coordinate and strictly smaller in at least one. Let T(B_0)={(c_a,1):B_0⊆ G_a}. Then F_0(B_0) · =Pareto T(B_0), F_h(B_0) · =Pareto(T(B_0)∪ ⋃_e:B_0⊆ D_e {(c_e+max_o w_o, 1+∑_o m_o): (w_o,m_o)∈ F_h-1(B_e,o) ∀ o}). nonumber An empty child frontier makes that experiment unavailable for a successful complete tree. An empty union is not a zero-cost solution.
Exact mathematical passage
\label{eq:pareto} \mathcal F_0(B_0)&=\Pareto T(B_0),\\ \mathcal F_h(B_0)&=\Pareto\left(T(B_0)\cup \bigcup_{e:B_0\subseteq D_e} \left\{\left(c_e+\max_o w_o,\ 1+\sum_o m_o\right): (w_o,m_o)\in\mathcal F_{h-1}(B_{e,o})\ \forall o\right\}\right). \nonumber
\midUse in source · 4 passages
Search aliases: mid
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1078
[Conditional error budgets across an adaptive lineage] Let H_t-1 be the information available before attempt t. Suppose the event A_t that attempt t accepts a false improvement claim satisfies Pr(A_tmid H_t-1)≤δ_t almost surely, for deterministic δ_t≥0 with ∑_t≥1δ_t≤δ. Then Pr(∃ t≥1:A_t)≤δ. The candidate-generating process and the time at which attempts stop may depend on the earlier history. Taking expectations gives Pr(A_t)≤δ_t. The union bound gives the result for any finite number of attempt…
Exact mathematical passage
\Pr(A_t\mid\mathcal H_{t-1})\leq\delta_t \quad\text{almost surely},
\minUse in source · 7 passages
Search aliases: min
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 572
nd a shortcut worker can be paused after each unit of a common work measure and resumed without losing progress. Each may announce completion only after producing a result certified for the same target. Alternate one unit of each worker and stop at the first certified completion. If their standalone completion costs are D_0,D_1∈ N_>0∪{∞} and at least one is finite, at most 2min{D_0,D_1} worker units are executed. If each dispatched unit costs at most an additional c≥0 scheduler units in the same…
Exact mathematical passage
2\min\{D_0,D_1\}
\min_{w\in W_0}Use in source · 1 passage
Search aliases: min_w∈ W_0 min in
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1046
[Resource dominance and declared price uncertainty] For a fixed ledger and workload, w·Δ(n)≥0 for every nonnegative price vector w if and only if Δ(n) is componentwise nonnegative. If admissible prices form the convex hull of a nonempty finite set W_0, then min_w∈conv(W_0)w·Δ(n) =min_w∈ W_0w·Δ(n). Thus strict improvement throughout that price set holds exactly when every listed vertex gives a strictly positive margin. The claims concern the same workflow, workload, and complete additive ledger. …
Exact mathematical passage
\min_{w\in\operatorname{conv}(W_0)}w\cdot\Delta(n) =\min_{w\in W_0}w\cdot\Delta(n).
\min_{w\in\operatorname{conv}(W_0)}Use in source · 1 passage
Search aliases: min_w∈conv(W_0) min in operatorname
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1046
[Resource dominance and declared price uncertainty] For a fixed ledger and workload, w·Δ(n)≥0 for every nonnegative price vector w if and only if Δ(n) is componentwise nonnegative. If admissible prices form the convex hull of a nonempty finite set W_0, then min_w∈conv(W_0)w·Δ(n) =min_w∈ W_0w·Δ(n). Thus strict improvement throughout that price set holds exactly when every listed vertex gives a strictly positive margin. The claims concern the same workflow, workload, and complete additive ledger. …
Exact mathematical passage
\min_{w\in\operatorname{conv}(W_0)}w\cdot\Delta(n) =\min_{w\in W_0}w\cdot\Delta(n).
NUse in source · 19 passages
Search aliases: N
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 371
For a fixed finite collection {U_1,…,U_N}, suppose a declared assessment gives estimates widehat R_i of true productivities R_i. This assessment can be exact on a fully specified finite distribution or statistical under additional assumptions. Choosing a candidate must use the assessment actually available at selection time, not a hidden final score inspected afterward.
Exact mathematical passage
\{U_1,\ldots,U_N\}
\mathbb NUse in source · 1 passage
Search aliases: N mathbb
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 572
lowdown from a certified two-worker race] Suppose a parent and a shortcut worker can be paused after each unit of a common work measure and resumed without losing progress. Each may announce completion only after producing a result certified for the same target. Alternate one unit of each worker and stop at the first certified completion. If their standalone completion costs are D_0,D_1∈ N_>0∪{∞} and at least one is finite, at most 2min{D_0,D_1} worker units are executed. If each dispatched unit…
Exact mathematical passage
D_0,D_1\in\mathbb N_{>0}\cup\{\infty\}
\mathbb N_{>0}Use in source · 1 passage
Search aliases: N_>0 mathbb
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 572
lowdown from a certified two-worker race] Suppose a parent and a shortcut worker can be paused after each unit of a common work measure and resumed without losing progress. Each may announce completion only after producing a result certified for the same target. Alternate one unit of each worker and stop at the first certified completion. If their standalone completion costs are D_0,D_1∈ N_>0∪{∞} and at least one is finite, at most 2min{D_0,D_1} worker units are executed. If each dispatched unit…
Exact mathematical passage
D_0,D_1\in\mathbb N_{>0}\cup\{\infty\}
N_{\mathrm{compare}}Use in source · 1 passage
Search aliases: N_compare mathrm
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 764
If a model-row read has declared price w>0 and a new comparison has price one, the additional per-use saving under this conservative two-unit account is Δ(w)=w (R_full-R_stream)-N_compare. For the 64-case row it is positive when w>14209/8287. At w=1 it is negative. The price is a modeling choice, not a measured statement about Python or a physical machine. Interpreted AST visits, installed code, grammar construction, labels, and development search remain additional recorded coordinates. No weigh…
Exact mathematical passage
\Delta(w)=w\,(R_{\mathrm{full}}-R_{\mathrm{stream}})-N_{\mathrm{compare}}.
n_iUse in source · 2 passages
Search aliases: n_i
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1038
Which gains survive a change of resource prices? Fix an additive ledger with d≥1 named event types. Let K∈ R_≥0^d be extra setup, let s_i∈ R^d be the old-minus-new per-use event vector for workload type i, and let n_i be its nonnegative integer multiplicity. The net event saving is Δ(n)=∑_i n_i s_i-K. A price vector w∈ R_≥0^d specifies conversions into one declared additive resource measure. Its priced saving is w·Δ(n). The prices are modeling assumptions, not learned hardware laws inferred from…
Exact mathematical passage
n_i
n_tUse in source · 5 passages
Search aliases: n_t
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1095
One sufficient fresh-audit construction. Freeze a candidate and its comparator using the prior history. Draw n_t fresh evaluation cases conditionally independently from the declared distribution and record paired score differences Y_i∈[-1,1]. Let Δ_t be their conditional mean and widehatΔ_t their average. The bounded-variable concentration inequality gives Pr(widehatΔ_t-Δ_t≥ rmid H_t-1) ≤ exp(-n_tr^2/2). Consequently, accepting a claim Δ_t≥γ only when widehatΔ_t-sqrt2log(1/δ_t)/n_t≥γ meets the p…
Exact mathematical passage
n_t
O'_{\tau(e)}Use in source · 1 passage
Search aliases: O'_τ(e) tau
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 407
Consider source and target finite repair models. Let σ:X→ X' be a bijection, τ: E→ E' a bijection of experiments, ψ: A→ A' a bijection of repairs, and η_e:O_e→ O'_τ(e) a bijection of report labels. Assume σ(D_e) · =D'_τ(e), · σ f_e(x) · =f'_τ(e)σ(x),
Exact mathematical passage
\eta_e:O_e\to O'_{\tau(e)}
o'_{\tau(e)}Use in source · 1 passage
Search aliases: o'_τ(e) tau
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 408
Consider source and target finite repair models. Let σ:X→ X' be a bijection, τ: E→ E' a bijection of experiments, ψ: A→ A' a bijection of repairs, and η_e:O_e→ O'_τ(e) a bijection of report labels. Assume σ(D_e) · =D'_τ(e), · σ f_e(x) · =f'_τ(e)σ(x), η_e(o_e(x)) · =o'_τ(e)(σ x), · σ(G_a) · =G'_ψ(a),nonumber and matching execution costs. These assumptions concern supplied semantics. Their discovery or learned approximation is an additional problem.
Exact mathematical passage
\label{eq:transport} \sigma(D_e)&=D'_{\tau(e)}, & \sigma f_e(x)&=f'_{\tau(e)}\sigma(x),\\ \eta_e(o_e(x))&=o'_{\tau(e)}(\sigma x), & \sigma(G_a)&=G'_{\psi(a)},\nonumber
o_e^\thetaUse in source · 3 passages
Search aliases: o_e^θ theta
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 322
A useful first extension retains several explicit possibilities rather than replacing uncertainty by one convenient estimate. Let Θ be a finite nonempty set of candidate models with a common state set X, experiment names, repair names, and nonnegative costs. Model θ specifies D_e^θ, f_e^θ, o_e^θ, and G_a^θ. The unknown model is fixed during an investigation. The initial information is a nonempty set B⊆Θ× X. The procedure observes reports, not the model label or the hidden physical state.
Exact mathematical passage
o_e^\theta
\mathcal PUse in source · 7 passages
Search aliases: P mathcal
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 623
[Feature sufficiency for a search decision] Let P be a finite set of search contexts, let r: P→ Z be their available feature map, and let ℓ: P→{0,1} be the required decision label. A deterministic rule f:Z→{0,1} satisfying fr=ℓ exists if and only if ℓ is constant on each nonempty fibre of r. A pair in one fibre with different labels rules out every such rule, irrespective of its program length. If f exists, equal features have the same output under f and hence the same label. Conversely, assign …
Exact mathematical passage
\mathcal P
P_0Use in source · 8 passages
Search aliases: P_0
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 222
[Feedback indistinguishability limits repair] Suppose two cases have disjoint successful-repair sets and induce transcript laws P_0,P_1 under a fixed admitted investigation protocol. A repair selector observes only the transcript and independent private randomness. Under equal prior probabilities, its average probability of successful repair is at most frac1+d_TV(P_0,P_1)2. In particular, identical transcript laws rule out certain success in both cases. From the selected repair construct a binar…
Exact mathematical passage
P_0,P_1
p_0Use in source · 1 passage
Search aliases: p_0
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 793
Four objects and two meanings of success. Let G=(p_0,…,p_N-1) be a finite ordered grammar of pure, total expressions. A target task supplies contexts z∈ D and desired values f(z). An ordinary learner uses counterexamples to find the first grammar expression that agrees with all of D. A separate equational learner observes expressions on a development probe set and proposes equations. A checker admits only equations supported by a derivation in the declared arithmetic theory.
Exact mathematical passage
\Gcal=(p_0,\ldots,p_{N-1})
P_1Use in source · 3 passages
Search aliases: P_1
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 222
[Feedback indistinguishability limits repair] Suppose two cases have disjoint successful-repair sets and induce transcript laws P_0,P_1 under a fixed admitted investigation protocol. A repair selector observes only the transcript and independent private randomness. Under equal prior probabilities, its average probability of successful repair is at most frac1+d_TV(P_0,P_1)2. In particular, identical transcript laws rule out certain success in both cases. From the selected repair construct a binar…
Exact mathematical passage
P_0,P_1
p_iUse in source · 8 passages
Search aliases: p_i
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 823
[Sound ordered quotienting from checked equations] Assume the expression language is pure and total, every admitted schema equation is valid for all well-typed substitutions in the common semantic domain, and each normalization terminates after valid contextual replacements. For each p_i, keep its earliest grammar representative ρ(i)=min{j:n(p_j)=n(p_i)}. Then ρ(i)≤ i, p_i and p_ρ(i) agree on every admitted context, and every pointwise specification realizable in G is realizable in its retained …
Exact mathematical passage
p_i
p_jUse in source · 4 passages
Search aliases: p_j
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 824
[Sound ordered quotienting from checked equations] Assume the expression language is pure and total, every admitted schema equation is valid for all well-typed substitutions in the common semantic domain, and each normalization terminates after valid contextual replacements. For each p_i, keep its earliest grammar representative ρ(i)=min{j:n(p_j)=n(p_i)}. Then ρ(i)≤ i, p_i and p_ρ(i) agree on every admitted context, and every pointwise specification realizable in G is realizable in its retained …
Exact mathematical passage
\rho(i)=\min\{j:n(p_j)=n(p_i)\}.
p_{N-1}Use in source · 1 passage
Search aliases: p_N-1
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 793
Four objects and two meanings of success. Let G=(p_0,…,p_N-1) be a finite ordered grammar of pure, total expressions. A target task supplies contexts z∈ D and desired values f(z). An ordinary learner uses counterexamples to find the first grammar expression that agrees with all of D. A separate equational learner observes expressions on a development probe set and proposes equations. A checker admits only equations supported by a derivation in the declared arithmetic theory.
Exact mathematical passage
\Gcal=(p_0,\ldots,p_{N-1})
P_tUse in source · 4 passages
Search aliases: P_t
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 248
Repairing a failed representation. Let X be a finite set with deterministic operations F_u:X→ X and output labels q(x). Begin with a partition P_0 that preserves both the old retained labels and q. Define the next partition by splitting a block whenever two states in it have successors in different current blocks under some permitted operation. Equivalently, x≡_P_t+1y Longleftrightarrow x≡_P_ty and F_u(x)≡_P_tF_u(y) for every u. The construction uses a complete transition description. Having a f…
Exact mathematical passage
x\equiv_{P_{t+1}}y \quad\Longleftrightarrow\quad x\equiv_{P_t}y\ \text{ and }\ F_u(x)\equiv_{P_t}F_u(y)\ \text{for every }u.
P_{t+1}Use in source · 2 passages
Search aliases: P_t+1
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 248
Repairing a failed representation. Let X be a finite set with deterministic operations F_u:X→ X and output labels q(x). Begin with a partition P_0 that preserves both the old retained labels and q. Define the next partition by splitting a block whenever two states in it have successors in different current blocks under some permitted operation. Equivalently, x≡_P_t+1y Longleftrightarrow x≡_P_ty and F_u(x)≡_P_tF_u(y) for every u. The construction uses a complete transition description. Having a f…
Exact mathematical passage
x\equiv_{P_{t+1}}y \quad\Longleftrightarrow\quad x\equiv_{P_t}y\ \text{ and }\ F_u(x)\equiv_{P_t}F_u(y)\ \text{for every }u.
p_{\rho(i)}Use in source · 1 passage
Search aliases: p_ρ(i) rho
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 827
cked equations] Assume the expression language is pure and total, every admitted schema equation is valid for all well-typed substitutions in the common semantic domain, and each normalization terminates after valid contextual replacements. For each p_i, keep its earliest grammar representative ρ(i)=min{j:n(p_j)=n(p_i)}. Then ρ(i)≤ i, p_i and p_ρ(i) agree on every admitted context, and every pointwise specification realizable in G is realizable in its retained representatives. No confluence or c…
Exact mathematical passage
p_{\rho(i)}
p_\psiUse in source · 1 passage
Search aliases: p_ψ psi
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1270
What would learn. Let r_ϕ encode the retained state, G_θ,a describe proposed operations, π_ω choose observations or experiments, and p_ψ propose candidate programs or parameter updates. A development episode can modify ϕ,θ,ω,ψ through a declared learning rule. A persistent program library can also change by adding, replacing, or removing checked operators. Temporary deliberation without a retained modification is not called a new learned generation here.
Exact mathematical passage
p_\psi
\operatorname{pa}Use in source · 2 passages
Search aliases: pa operatorname
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 977
A declared dependency graph. Let an artifact state be a=(a_1,…,a_d). Its components may include operation semantics, a grammar and its order, an equation library, experiment tables, repair-success sets, a returned controller, and a resource model. A finite directed acyclic graph has one vertex v for each claimed guarantee. Its incoming vertices pa(v) identify guarantees used as premises. Its direct dependency set D_v⊆{1,…,d} identifies the artifact components used directly. Each vertex carries a…
Exact mathematical passage
\operatorname{pa}(v)
\ParetoUse in source · 1 passage
Search aliases: Pareto operatorname
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 285
For horizon h, let F_h(B_0) be the nondominated attainable pairs (w,m). A pair dominates another when it is no larger in either coordinate and strictly smaller in at least one. Let T(B_0)={(c_a,1):B_0⊆ G_a}. Then F_0(B_0) · =Pareto T(B_0), F_h(B_0) · =Pareto(T(B_0)∪ ⋃_e:B_0⊆ D_e {(c_e+max_o w_o, 1+∑_o m_o): (w_o,m_o)∈ F_h-1(B_e,o) ∀ o}). nonumber An empty child frontier makes that experiment unavailable for a successful complete tree. An empty union is not a zero-cost solution.
Exact mathematical passage
\label{eq:pareto} \mathcal F_0(B_0)&=\Pareto T(B_0),\\ \mathcal F_h(B_0)&=\Pareto\left(T(B_0)\cup \bigcup_{e:B_0\subseteq D_e} \left\{\left(c_e+\max_o w_o,\ 1+\sum_o m_o\right): (w_o,m_o)\in\mathcal F_{h-1}(B_{e,o})\ \forall o\right\}\right). \nonumber
\pmodUse in source · 1 passage
Search aliases: pmod
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1413
For a target relabeling σ(i)=3i+1pmod8, the defect has target label σ(5)=0. Each query becomes σ(A_r) and the terminal repair becomes r'_σ(5). Reports and tree shape agree with the source, but the label adapter is part of the transported interface. Applying the old numeric repair label five directly in the target would be incorrect. This is why syntactic resemblance does not replace a semantic map.
Exact mathematical passage
\sigma(i)=3i+1\pmod8
\PostUse in source · 3 passages
Search aliases: Post operatorname
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1205
Each z has a finite set of authorized attempts A(z). An attempt has a nonempty set Post(z,a) of possible successor configurations. This can represent all outcomes of a stochastic proposal or an abstract uncertainty set. A safe set Safe⊆ Z encodes the declared semantic, evidence, authority, and resource conditions. A pause or terminal state may have a zero-progress self-loop. Actual expensive attempts must debit resources in Z, or the model must explicitly state their replenishment.
Exact mathematical passage
\Post(z,a)
\PrUse in source · 6 passages
Search aliases: Pr
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1078
[Conditional error budgets across an adaptive lineage] Let H_t-1 be the information available before attempt t. Suppose the event A_t that attempt t accepts a false improvement claim satisfies Pr(A_tmid H_t-1)≤δ_t almost surely, for deterministic δ_t≥0 with ∑_t≥1δ_t≤δ. Then Pr(∃ t≥1:A_t)≤δ. The candidate-generating process and the time at which attempts stop may depend on the earlier history. Taking expectations gives Pr(A_t)≤δ_t. The union bound gives the result for any finite number of attempt…
Exact mathematical passage
\Pr(A_t\mid\mathcal H_{t-1})\leq\delta_t \quad\text{almost surely},
RUse in source · 18 passages
Search aliases: R
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 260
es every finite continuation of the retained labels and outputs. A strict refinement increases the number of nonempty blocks, which is bounded by |X|. At a fixed point, equivalent states have equivalent successors, so the quotient operations are well-defined. Induction on word length then preserves all retained labels after every finite word. For minimality, let R be any stable partition refining P_0. If R refines P_t, stability of R implies that R-equivalent states have successors in the same P…
Exact mathematical passage
R
\mathbb RUse in source · 3 passages
Search aliases: R mathbb
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1038
Which gains survive a change of resource prices? Fix an additive ledger with d≥1 named event types. Let K∈ R_≥0^d be extra setup, let s_i∈ R^d be the old-minus-new per-use event vector for workload type i, and let n_i be its nonnegative integer multiplicity. The net event saving is Δ(n)=∑_i n_i s_i-K. A price vector w∈ R_≥0^d specifies conversions into one declared additive resource measure. Its priced saving is w·Δ(n). The prices are modeling assumptions, not learned hardware laws inferred from…
Exact mathematical passage
K\in\mathbb R_{\ge0}^d
\mathcal RUse in source · 4 passages
Search aliases: R mathcal
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 219
Let a hidden case θ determine a set R(θ) of successful repairs. The set may have several members. Diagnosis need only obtain enough information to choose a repair in the appropriate set. It need not identify every hidden property of θ.
Exact mathematical passage
\mathcal R(\theta)
r'_{\sigma(5)}Use in source · 1 passage
Search aliases: r'_σ(5) sigma
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1413
For a target relabeling σ(i)=3i+1pmod8, the defect has target label σ(5)=0. Each query becomes σ(A_r) and the terminal repair becomes r'_σ(5). Reports and tree shape agree with the source, but the label adapter is part of the transported interface. Applying the old numeric repair label five directly in the target would be incorrect. This is why syntactic resemblance does not replace a semantic map.
Exact mathematical passage
r'_{\sigma(5)}
R^*_{\mathcal G}Use in source · 3 passages
Search aliases: R^*_ G mathcal
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 393
For a fixed budget, let R^*_ G be the largest productivity among procedures expressible in a declared language G, with installed data and interpreter costs held fixed. The gap R^*_ G-R(U) is search or selection headroom within that language. Comparing R^*_ G' with R^*_ G concerns expressibility and resources. Neither upper envelope is automatically available to the learner.
Exact mathematical passage
R^*_{\mathcal G}
R^*_{\mathcal G'}Use in source · 1 passage
Search aliases: R^*_ G' mathcal
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 393
For a fixed budget, let R^*_ G be the largest productivity among procedures expressible in a declared language G, with installed data and interpreter costs held fixed. The gap R^*_ G-R(U) is search or selection headroom within that language. Comparing R^*_ G' with R^*_ G concerns expressibility and resources. Neither upper envelope is automatically available to the learner.
Exact mathematical passage
R^*_{\mathcal G'}
r^2Use in source · 1 passage
Search aliases: r^2
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1096
One sufficient fresh-audit construction. Freeze a candidate and its comparator using the prior history. Draw n_t fresh evaluation cases conditionally independently from the declared distribution and record paired score differences Y_i∈[-1,1]. Let Δ_t be their conditional mean and widehatΔ_t their average. The bounded-variable concentration inequality gives Pr(widehatΔ_t-Δ_t≥ rmid H_t-1) ≤ exp(-n_tr^2/2). Consequently, accepting a claim Δ_t≥γ only when widehatΔ_t-sqrt2log(1/δ_t)/n_t≥γ meets the p…
Exact mathematical passage
\Pr\left(\widehat\Delta_t-\Delta_t\geq r\mid\mathcal H_{t-1}\right) \leq \exp(-n_tr^2/2).
\mathbb R^dUse in source · 1 passage
Search aliases: R^d mathbb
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1038
Which gains survive a change of resource prices? Fix an additive ledger with d≥1 named event types. Let K∈ R_≥0^d be extra setup, let s_i∈ R^d be the old-minus-new per-use event vector for workload type i, and let n_i be its nonnegative integer multiplicity. The net event saving is Δ(n)=∑_i n_i s_i-K. A price vector w∈ R_≥0^d specifies conversions into one declared additive resource measure. Its priced saving is w·Δ(n). The prices are modeling assumptions, not learned hardware laws inferred from…
Exact mathematical passage
s_i\in\mathbb R^d
R_0Use in source · 1 passage
Search aliases: R_0
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1255
ma>0, and all other transitions are nondecreasing in that same measure. Then the number of progress events is at most lfloorfracb-aγrfloor. An infinite sequence of positive increments is compatible with bounded total improvement when no common positive lower bound is imposed. After k progress events, telescoping and nondecrease give R_t≥ R_0+kγ≥ a+kγ. Since R_t≤ b, the bound follows. For the second assertion, start at zero and use increments 2^-t for t≥1. Every increment is positive and their pa…
Exact mathematical passage
R_t\geq R_0+k\gamma\geq a+k\gamma
r_5Use in source · 1 passage
Search aliases: r_5
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1409
p:trace Take n=8 and hidden defect j=5, whose binary digits are 101. One optimal tree uses the three queries A_0={1,3,5,7}, A_1={2,3,6,7}, A_2={4,5,6,7}. The precise order may differ among equally optimal compiled trees. For this order, the supports are {0,…,7}xrightarrow1{1,3,5,7} xrightarrow0{1,5}xrightarrow1{5}. The terminal repair is r_5. The path uses three diagnostic calls and one repair call. The complete tree stores all eight repair leaves and seven query nodes, not merely this path.
Exact mathematical passage
r_5
R_{\mathrm{full}}Use in source · 1 passage
Search aliases: R_full mathrm
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 764
If a model-row read has declared price w>0 and a new comparison has price one, the additional per-use saving under this conservative two-unit account is Δ(w)=w (R_full-R_stream)-N_compare. For the 64-case row it is positive when w>14209/8287. At w=1 it is negative. The price is a modeling choice, not a measured statement about Python or a physical machine. Interpreted AST visits, installed code, grammar construction, labels, and development search remain additional recorded coordinates. No weigh…
Exact mathematical passage
\Delta(w)=w\,(R_{\mathrm{full}}-R_{\mathrm{stream}})-N_{\mathrm{compare}}.
R_{\hat\imath}Use in source · 2 passages
Search aliases: R_hatimath hat imath
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 375
[Selection under uniform assessment error] If |widehat R_i-R_i|≤ε for every candidate and hatimath maximizes widehat R_i, then R_hatimath≥ max_iR_i-2ε. If a baseline is included and the selected candidate's estimated advantage exceeds 2ε+γ, its true advantage exceeds γ. For a true maximizer i^*, R_hatimath≥widehat R_hatimath-ε ≥widehat R_i^*-ε ≥ R_i^*-2ε. Apply the same two error bounds to the selected candidate and the baseline for the second assertion.
Exact mathematical passage
R_{\hat\imath}\geq \max_iR_i-2\varepsilon.
R_iUse in source · 3 passages
Search aliases: R_i
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 371
For a fixed finite collection {U_1,…,U_N}, suppose a declared assessment gives estimates widehat R_i of true productivities R_i. This assessment can be exact on a fully specified finite distribution or statistical under additional assumptions. Choosing a candidate must use the assessment actually available at selection time, not a hidden final score inspected afterward.
Exact mathematical passage
R_i
r_iUse in source · 2 passages
Search aliases: r_i
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 206
A diagnostic query names a subset A and returns 1{j∈ A}. The reference language contains eight singleton queries and three bit-coordinate queries. Repair r_i succeeds exactly when i=j. Because repair is irreversible within the episode, a procedure may commit only when every surviving possibility justifies its choice. Otherwise it may continue investigating or abstain.
Exact mathematical passage
r_i
R_{i^*}Use in source · 1 passage
Search aliases: R_i^*
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 382
assessment error] If |widehat R_i-R_i|≤ε for every candidate and hatimath maximizes widehat R_i, then R_hatimath≥ max_iR_i-2ε. If a baseline is included and the selected candidate's estimated advantage exceeds 2ε+γ, its true advantage exceeds γ. For a true maximizer i^*, R_hatimath≥widehat R_hatimath-ε ≥widehat R_i^*-ε ≥ R_i^*-2ε. Apply the same two error bounds to the selected candidate and the baseline for the second assertion.
Exact mathematical passage
R_{\hat\imath}\geq\widehat R_{\hat\imath}-\varepsilon \geq\widehat R_{i^*}-\varepsilon \geq R_{i^*}-2\varepsilon
r_{o\oplus\theta}Use in source · 1 passage
Search aliases: r_o⊕θ oplus theta
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 352
Add a permitted reference measurement that reports θ, leaves x unchanged, and costs two. The procedure can calibrate, make the ordinary query, and choose r_o⊕θ. Its worst-case work is four and its complete binary tree has seven records. Either measurement alone leaves both repairs possible. The two measurements are therefore both necessary in this example, regardless of their order. The executable reference construction returns exactly (4,7) at horizons at least two.
Exact mathematical passage
r_{o\oplus\theta}
R_{\mathrm{shortcut}}Use in source · 1 passage
Search aliases: R_shortcut mathrm
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 509
total unit-cost binary immutable queries, and the balanced proposer produces a perfect depth-k tree. In the stipulated implementation, the semantic guard reads every query row once, the proposer scans all q queries at each internal support, and verification performs one support reconstruction and one complete case replay. The total number of transition/report record reads is R_shortcut=nq(k+1)+2nk. The usual hidden-cell family with all singleton queries and k coordinate queries satisfies the pre…
Exact mathematical passage
\label{eq:compiler-reads} R_{\mathrm{shortcut}}=nq(k+1)+2nk.
R_{\mathrm{stream}}Use in source · 1 passage
Search aliases: R_stream mathrm
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 764
If a model-row read has declared price w>0 and a new comparison has price one, the additional per-use saving under this conservative two-unit account is Δ(w)=w (R_full-R_stream)-N_compare. For the 64-case row it is positive when w>14209/8287. At w=1 it is negative. The price is a modeling choice, not a measured statement about Python or a physical machine. Interpreted AST visits, installed code, grammar construction, labels, and development search remain additional recorded coordinates. No weigh…
Exact mathematical passage
\Delta(w)=w\,(R_{\mathrm{full}}-R_{\mathrm{stream}})-N_{\mathrm{compare}}.
R_tUse in source · 3 passages
Search aliases: R_t
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1248
[A fixed-margin ceiling on a fixed score] Suppose a lineage has a fixed measure R_t∈[a,b], every accepted progress event increases it by at least γ>0, and all other transitions are nondecreasing in that same measure. Then the number of progress events is at most lfloorfracb-aγrfloor. An infinite sequence of positive increments is compatible with bounded total improvement when no common positive lower bound is imposed. After k progress events, telescoping and nondecrease give R_t≥ R_0+kγ≥ a+kγ. S…
Exact mathematical passage
R_t\in[a,b]
r_\phiUse in source · 1 passage
Search aliases: r_ϕ phi
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1270
What would learn. Let r_ϕ encode the retained state, G_θ,a describe proposed operations, π_ω choose observations or experiments, and p_ψ propose candidate programs or parameter updates. A development episode can modify ϕ,θ,ω,ψ through a declared learning rule. A persistent program library can also change by adding, replacing, or removing checked operators. Temporary deliberation without a retained modification is not called a new learned generation here.
Exact mathematical passage
r_\phi
\mathbb R_{\ge0}^dUse in source · 2 passages
Search aliases: R_≥0^d mathbb ge
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1038
Which gains survive a change of resource prices? Fix an additive ledger with d≥1 named event types. Let K∈ R_≥0^d be extra setup, let s_i∈ R^d be the old-minus-new per-use event vector for workload type i, and let n_i be its nonnegative integer multiplicity. The net event saving is Δ(n)=∑_i n_i s_i-K. A price vector w∈ R_≥0^d specifies conversions into one declared additive resource measure. Its priced saving is w·Δ(n). The prices are modeling assumptions, not learned hardware laws inferred from…
Exact mathematical passage
K\in\mathbb R_{\ge0}^d
\Rcal'Use in source · 1 passage
Search aliases: Rcal' R' Rcal mathcal
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 435
Approximate performance transfer. Suppose a source comparison gives productivity advantage Δ to U' over U. Let widetilde U',widetilde U be actual transported target procedures. If each procedure's productivity differs from its source value by at most ε, then R'(widetilde U')- R'(widetilde U) ≥Δ-2ε. This follows by subtracting the two inequalities. A common source of the premise is an exact task correspondence plus a bounded evaluation-law shift. For any fixed gain function in [-1,1], changing th…
Exact mathematical passage
\label{eq:gaintransport} \Rcal'(\widetilde U')-\Rcal'(\widetilde U) \geq\Delta-2\varepsilon.
\rceilUse in source · 3 passages
Search aliases: rceil
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 477
[Checked lower-bound closure of a diagnostic search] Every successful controller in this scope has worst-case execution work at least 1+lceillog_2 nrceil and storage at least 2n-1. If an independently checked candidate attains both bounds and respects the requested horizon, its cost pair is the entire nondominated frontier. Returning this candidate is therefore equivalent in the protected work/storage objective to completing the exhaustive synthesis. Every case produces one deterministic query h…
Exact mathematical passage
1+\lceil\log_2 n\rceil
\rfloorUse in source · 1 passage
Search aliases: rfloor
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1249
[A fixed-margin ceiling on a fixed score] Suppose a lineage has a fixed measure R_t∈[a,b], every accepted progress event increases it by at least γ>0, and all other transitions are nondecreasing in that same measure. Then the number of progress events is at most lfloorfracb-aγrfloor. An infinite sequence of positive increments is compatible with bounded total improvement when no common positive lower bound is imposed. After k progress events, telescoping and nondecrease give R_t≥ R_0+kγ≥ a+kγ. S…
Exact mathematical passage
\left\lfloor\frac{b-a}{\gamma}\right\rfloor.
\rmUse in source · 3 passages
Search aliases: rm
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 873
Let C_rm prepare be the complete preparation cost in a declared common resource model, including failed proposals and certificate checking, and let d_t be the subsequent saving on task t after including dispatch. Reuse is beneficial in that model exactly when ∑_t=1^Td_t>C_rm prepare. This is the earlier amortization identity applied to acquisition itself. The reference does not convert unlike counters into a universal wall-clock claim. Even one setup coordinate, 518,140 schema matches, exceeds t…
Exact mathematical passage
C_{\rm prepare}
sUse in source · 5 passages
Search aliases: s
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 453
[The resource price of conservative extension] Suppose an embedding preserves every old procedure's successful behavior and adds at most d≥0 execution work and s≥0 stored records. Every old allowance (b,m) is then feasible in the new language under allowance (b+d,m+s). Feasibility at the unchanged allowance follows only when the embedded procedure fits that allowance. Semantic conservativity alone supplies no such guarantee. Embed a witness for the old allowance. By hypothesis its behavior is pr…
Exact mathematical passage
s\geq0
s_iUse in source · 2 passages
Search aliases: s_i
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1038
Which gains survive a change of resource prices? Fix an additive ledger with d≥1 named event types. Let K∈ R_≥0^d be extra setup, let s_i∈ R^d be the old-minus-new per-use event vector for workload type i, and let n_i be its nonnegative integer multiplicity. The net event saving is Δ(n)=∑_i n_i s_i-K. A price vector w∈ R_≥0^d specifies conversions into one declared additive resource measure. Its priced saving is w·Δ(n). The prices are modeling assumptions, not learned hardware laws inferred from…
Exact mathematical passage
s_i\in\mathbb R^d
S_rUse in source · 2 passages
Search aliases: S_r
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 889
Let X be a finite immutable state set and let S_r⊆ X be the success set of repair r. Choose nonnegative rational weights w_x satisfying ∑_x∈ S_rw_x≤1 for every permitted repair r. This is a feasible dual weight assignment for unweighted set cover. We use only its elementary weak-duality property, not an assumption that the best assignment has been found~[source reference].
Exact mathematical passage
S_r\subseteq X
S_tUse in source · 1 passage
Search aliases: S_t
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 196
What persists across a recursive step. A lineage state can be written schematically as Z_t=(S_t,U_t,L_t,E_t,Ξ_t,ρ_t), where E_t records evidence and rejected alternatives, and ρ_t records remaining resources and scope. A proposed modification may target any admitted subset. A comparison across different Ξ_t needs a declared bridge. It cannot be justified merely because both scores are called accuracy.
Exact mathematical passage
Z_t=(S_t,U_t,L_t,E_t,\Xi_t,\rho_t),
\SafeUse in source · 3 passages
Search aliases: Safe mathsf
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1205
Each z has a finite set of authorized attempts A(z). An attempt has a nonempty set Post(z,a) of possible successor configurations. This can represent all outcomes of a stochastic proposal or an abstract uncertainty set. A safe set Safe⊆ Z encodes the declared semantic, evidence, authority, and resource conditions. A pause or terminal state may have a zero-progress self-loop. Actual expensive attempts must debit resources in Z, or the model must explicitly state their replenishment.
Exact mathematical passage
\Safe\subseteq Z
\mathrm{shortcut}Use in source · 1 passage
Search aliases: shortcut mathrm
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 509
total unit-cost binary immutable queries, and the balanced proposer produces a perfect depth-k tree. In the stipulated implementation, the semantic guard reads every query row once, the proposer scans all q queries at each internal support, and verification performs one support reconstruction and one complete case replay. The total number of transition/report record reads is R_shortcut=nq(k+1)+2nk. The usual hidden-cell family with all singleton queries and k coordinate queries satisfies the pre…
Exact mathematical passage
\label{eq:compiler-reads} R_{\mathrm{shortcut}}=nq(k+1)+2nk.
\mathrm{stream}Use in source · 1 passage
Search aliases: stream mathrm
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 764
If a model-row read has declared price w>0 and a new comparison has price one, the additional per-use saving under this conservative two-unit account is Δ(w)=w (R_full-R_stream)-N_compare. For the 64-case row it is positive when w>14209/8287. At w=1 it is negative. The price is a modeling choice, not a measured statement about Python or a physical machine. Interpreted AST visits, installed code, grammar construction, labels, and development search remain additional recorded coordinates. No weigh…
Exact mathematical passage
\Delta(w)=w\,(R_{\mathrm{full}}-R_{\mathrm{stream}})-N_{\mathrm{compare}}.
\supUse in source · 1 passage
Search aliases: sup
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1137
[An anytime-valid paired improvement audit] Under~[source reference] and the stated predictable bet constraints, E_t is a nonnegative supermartingale. For every δ∈(0,1), Pr(sup_t≥0E_t≥1/δ)≤δ. Consequently an audit may reject the stated null at its first crossing of 1/δ, while choosing its stopping time from the observed audit history. The result does not require a predetermined number of cases or independence beyond the conditional-mean premise. Since Y_t-γ≥-(1+γ), every factor in~[source refere…
Exact mathematical passage
\Pr\left(\sup_{t\geq0}E_t\geq1/\delta\right)\leq\delta.
\sup_{t\geq0}Use in source · 1 passage
Search aliases: sup_t≥0 sup geq
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1137
[An anytime-valid paired improvement audit] Under~[source reference] and the stated predictable bet constraints, E_t is a nonnegative supermartingale. For every δ∈(0,1), Pr(sup_t≥0E_t≥1/δ)≤δ. Consequently an audit may reject the stated null at its first crossing of 1/δ, while choosing its stopping time from the observed audit history. The result does not require a predetermined number of cases or independence beyond the conditional-mean premise. Since Y_t-γ≥-(1+γ), every factor in~[source refere…
Exact mathematical passage
\Pr\left(\sup_{t\geq0}E_t\geq1/\delta\right)\leq\delta.
TUse in source · 4 passages
Search aliases: T
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 284
For horizon h, let F_h(B_0) be the nondominated attainable pairs (w,m). A pair dominates another when it is no larger in either coordinate and strictly smaller in at least one. Let T(B_0)={(c_a,1):B_0⊆ G_a}. Then F_0(B_0) · =Pareto T(B_0),
Exact mathematical passage
T(B_0)=\{(c_a,1):B_0\subseteq G_a\}
tUse in source · 57 passages
Search aliases: t
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 196
What persists across a recursive step. A lineage state can be written schematically as Z_t=(S_t,U_t,L_t,E_t,Ξ_t,ρ_t), where E_t records evidence and rejected alternatives, and ρ_t records remaining resources and scope. A proposed modification may target any admitted subset. A comparison across different Ξ_t needs a declared bridge. It cannot be justified merely because both scores are called accuracy.
Exact mathematical passage
Z_t=(S_t,U_t,L_t,E_t,\Xi_t,\rho_t),
\TVUse in source · 2 passages
Search aliases: TV d_TV mathrm
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 223
roposition[Feedback indistinguishability limits repair] Suppose two cases have disjoint successful-repair sets and induce transcript laws P_0,P_1 under a fixed admitted investigation protocol. A repair selector observes only the transcript and independent private randomness. Under equal prior probabilities, its average probability of successful repair is at most frac1+d_TV(P_0,P_1)2. In particular, identical transcript laws rule out certain success in both cases. From the selected repair constru…
Exact mathematical passage
\frac{1+\TV(P_0,P_1)}2.
uUse in source · 7 passages
Search aliases: u
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 247
Repairing a failed representation. Let X be a finite set with deterministic operations F_u:X→ X and output labels q(x). Begin with a partition P_0 that preserves both the old retained labels and q. Define the next partition by splitting a block whenever two states in it have successors in different current blocks under some permitted operation. Equivalently, x≡_P_t+1y Longleftrightarrow x≡_P_ty and F_u(x)≡_P_tF_u(y) for every u. The construction uses a complete transition description. Having a f…
Exact mathematical passage
F_u:X\to X
U'Use in source · 5 passages
Search aliases: U'
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 186
Fixed-library and full-system comparisons. A comparison of R_Ξ(U',L;B) with R_Ξ(U,L;B) isolates process change at a common library. A comparison with (U',L') measures the combined descendant. Both are useful, but they answer different questions. A crossed comparison of (U,L), (U',L), (U,L'), and (U',L') can expose interaction between procedure and library. There is no general additive decomposition that makes their contributions independent.
Exact mathematical passage
\Rcal_\Xi(U',L;B)
U_1Use in source · 1 passage
Search aliases: U_1
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 371
For a fixed finite collection {U_1,…,U_N}, suppose a declared assessment gives estimates widehat R_i of true productivities R_i. This assessment can be exact on a fully specified finite distribution or statistical under additional assumptions. Choosing a candidate must use the assessment actually available at selection time, not a hidden final score inspected afterward.
Exact mathematical passage
\{U_1,\ldots,U_N\}
U_NUse in source · 1 passage
Search aliases: U_N
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 371
For a fixed finite collection {U_1,…,U_N}, suppose a declared assessment gives estimates widehat R_i of true productivities R_i. This assessment can be exact on a fully specified finite distribution or statistical under additional assumptions. Choosing a candidate must use the assessment actually available at selection time, not a hidden final score inspected afterward.
Exact mathematical passage
\{U_1,\ldots,U_N\}
U_tUse in source · 1 passage
Search aliases: U_t
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 196
What persists across a recursive step. A lineage state can be written schematically as Z_t=(S_t,U_t,L_t,E_t,Ξ_t,ρ_t), where E_t records evidence and rejected alternatives, and ρ_t records remaining resources and scope. A proposed modification may target any admitted subset. A comparison across different Ξ_t needs a declared bridge. It cannot be justified merely because both scores are called accuracy.
Exact mathematical passage
Z_t=(S_t,U_t,L_t,E_t,\Xi_t,\rho_t),
vUse in source · 10 passages
Search aliases: v
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 977
A declared dependency graph. Let an artifact state be a=(a_1,…,a_d). Its components may include operation semantics, a grammar and its order, an equation library, experiment tables, repair-success sets, a returned controller, and a resource model. A finite directed acyclic graph has one vertex v for each claimed guarantee. Its incoming vertices pa(v) identify guarantees used as premises. Its direct dependency set D_v⊆{1,…,d} identifies the artifact components used directly. Each vertex carries a…
Exact mathematical passage
v
wUse in source · 21 passages
Search aliases: w
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 284
For horizon h, let F_h(B_0) be the nondominated attainable pairs (w,m). A pair dominates another when it is no larger in either coordinate and strictly smaller in at least one. Let T(B_0)={(c_a,1):B_0⊆ G_a}. Then F_0(B_0) · =Pareto T(B_0),
Exact mathematical passage
(w,m)
W_0Use in source · 2 passages
Search aliases: W_0
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1045
[Resource dominance and declared price uncertainty] For a fixed ledger and workload, w·Δ(n)≥0 for every nonnegative price vector w if and only if Δ(n) is componentwise nonnegative. If admissible prices form the convex hull of a nonempty finite set W_0, then min_w∈conv(W_0)w·Δ(n) =min_w∈ W_0w·Δ(n). Thus strict improvement throughout that price set holds exactly when every listed vertex gives a strictly positive margin. The claims concern the same workflow, workload, and complete additive ledger. …
Exact mathematical passage
W_0
W_{0,0}Use in source · 1 passage
Search aliases: W_0,0
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1215
For each transition, a progress mark p(z,a,z')∈{0,1} is supplied by a checked criterion. A mark of one means the specified process-improvement claim has been established under the same comparison. It is not assigned merely because a file changed. Define W_0,0 · = K, · W_0,k · =∅ (k>0), W_h,k · ={z∈ K:∃ a∈ A(z), Post(z,a)⊆ K,nonumber [-2pt] · z'∈ W_h-1,max{0,k-p(z,a,z')} ∀ z'∈Post(z,a)}.nonumber Here h is the number of remaining attempts and k the number of still-required progress events. A pause…
Exact mathematical passage
\label{eq:progress} W_{0,0}&=\mathcal K, & W_{0,k}&=\varnothing\quad(k>0),\\ W_{h,k}&=\{z\in\mathcal K:\exists a\in A(z),\ \Post(z,a)\subseteq\mathcal K,\nonumber\\[-2pt] &\hspace{34mm} z'\in W_{h-1,\max\{0,k-p(z,a,z')\}} \quad\forall z'\in\Post(z,a)\}.\nonumber
W_{0,k}Use in source · 1 passage
Search aliases: W_0,k
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1215
For each transition, a progress mark p(z,a,z')∈{0,1} is supplied by a checked criterion. A mark of one means the specified process-improvement claim has been established under the same comparison. It is not assigned merely because a file changed. Define W_0,0 · = K, · W_0,k · =∅ (k>0), W_h,k · ={z∈ K:∃ a∈ A(z), Post(z,a)⊆ K,nonumber [-2pt] · z'∈ W_h-1,max{0,k-p(z,a,z')} ∀ z'∈Post(z,a)}.nonumber Here h is the number of remaining attempts and k the number of still-required progress events. A pause…
Exact mathematical passage
\label{eq:progress} W_{0,0}&=\mathcal K, & W_{0,k}&=\varnothing\quad(k>0),\\ W_{h,k}&=\{z\in\mathcal K:\exists a\in A(z),\ \Post(z,a)\subseteq\mathcal K,\nonumber\\[-2pt] &\hspace{34mm} z'\in W_{h-1,\max\{0,k-p(z,a,z')\}} \quad\forall z'\in\Post(z,a)\}.\nonumber
w_1Use in source · 1 passage
Search aliases: w_1
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 916
For example, on three cases with success sets {1,2}, {2,3}, and {1,3}, the ordered greedy constructor returns total weight one. The feasible fractional assignment w_1=w_2=w_3=1/2 has total 3/2 and proves that at least two leaves are required. Thus the supplied constructor does not optimize the bound. Both assignments are valid and neither creates a missing experiment.
Exact mathematical passage
w_1=w_2=w_3=1/2
w_2Use in source · 1 passage
Search aliases: w_2
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 916
For example, on three cases with success sets {1,2}, {2,3}, and {1,3}, the ordered greedy constructor returns total weight one. The feasible fractional assignment w_1=w_2=w_3=1/2 has total 3/2 and proves that at least two leaves are required. Thus the supplied constructor does not optimize the bound. Both assignments are valid and neither creates a missing experiment.
Exact mathematical passage
w_1=w_2=w_3=1/2
w_3Use in source · 1 passage
Search aliases: w_3
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 916
For example, on three cases with success sets {1,2}, {2,3}, and {1,3}, the ordered greedy constructor returns total weight one. The feasible fractional assignment w_1=w_2=w_3=1/2 has total 3/2 and proves that at least two leaves are required. Thus the supplied constructor does not optimize the bound. Both assignments are valid and neither creates a missing experiment.
Exact mathematical passage
w_1=w_2=w_3=1/2
W_{h-1,\max\{0,k-p(z,a,z')\}}Use in source · 1 passage
Search aliases: W_h-1,max{0,k-p(z,a,z')} max
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1215
For each transition, a progress mark p(z,a,z')∈{0,1} is supplied by a checked criterion. A mark of one means the specified process-improvement claim has been established under the same comparison. It is not assigned merely because a file changed. Define W_0,0 · = K, · W_0,k · =∅ (k>0), W_h,k · ={z∈ K:∃ a∈ A(z), Post(z,a)⊆ K,nonumber [-2pt] · z'∈ W_h-1,max{0,k-p(z,a,z')} ∀ z'∈Post(z,a)}.nonumber Here h is the number of remaining attempts and k the number of still-required progress events. A pause…
Exact mathematical passage
\label{eq:progress} W_{0,0}&=\mathcal K, & W_{0,k}&=\varnothing\quad(k>0),\\ W_{h,k}&=\{z\in\mathcal K:\exists a\in A(z),\ \Post(z,a)\subseteq\mathcal K,\nonumber\\[-2pt] &\hspace{34mm} z'\in W_{h-1,\max\{0,k-p(z,a,z')\}} \quad\forall z'\in\Post(z,a)\}.\nonumber
w_oUse in source · 1 passage
Search aliases: w_o
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 285
For horizon h, let F_h(B_0) be the nondominated attainable pairs (w,m). A pair dominates another when it is no larger in either coordinate and strictly smaller in at least one. Let T(B_0)={(c_a,1):B_0⊆ G_a}. Then F_0(B_0) · =Pareto T(B_0), F_h(B_0) · =Pareto(T(B_0)∪ ⋃_e:B_0⊆ D_e {(c_e+max_o w_o, 1+∑_o m_o): (w_o,m_o)∈ F_h-1(B_e,o) ∀ o}). nonumber An empty child frontier makes that experiment unavailable for a successful complete tree. An empty union is not a zero-cost solution.
Exact mathematical passage
\label{eq:pareto} \mathcal F_0(B_0)&=\Pareto T(B_0),\\ \mathcal F_h(B_0)&=\Pareto\left(T(B_0)\cup \bigcup_{e:B_0\subseteq D_e} \left\{\left(c_e+\max_o w_o,\ 1+\sum_o m_o\right): (w_o,m_o)\in\mathcal F_{h-1}(B_{e,o})\ \forall o\right\}\right). \nonumber
w_vUse in source · 1 passage
Search aliases: w_v
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 977
quation library, experiment tables, repair-success sets, a returned controller, and a resource model. A finite directed acyclic graph has one vertex v for each claimed guarantee. Its incoming vertices pa(v) identify guarantees used as premises. Its direct dependency set D_v⊆{1,…,d} identifies the artifact components used directly. Each vertex carries a witness w_v and a local verification rule.
Exact mathematical passage
w_v
w_xUse in source · 3 passages
Search aliases: w_x
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 889
Let X be a finite immutable state set and let S_r⊆ X be the success set of repair r. Choose nonnegative rational weights w_x satisfying ∑_x∈ S_rw_x≤1 for every permitted repair r. This is a feasible dual weight assignment for unweighted set cover. We use only its elementary weak-duality property, not an assumption that the best assignment has been found~[source reference].
Exact mathematical passage
w_x
\wedgeUse in source · 1 passage
Search aliases: wedge
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1149
e conditional-mean premise. Since Y_t-γ≥-(1+γ), every factor in~[source reference] is nonnegative. Conditional expectation and predictability give E[E_tmid H_t-1] =E_t-1(1+λ_t( E[Y_tmid H_t-1]-γ)) ≤ E_t-1. Let τ be the first threshold crossing. For a finite N, the stopped process at τwedge N still has expected value at most one, by induction on the stopped conditional expectations. On {τ≤ N} its value is at least 1/δ, while it is nonnegative otherwise. Therefore Pr(τ≤ N)≤δ. Letting N increase pr…
Exact mathematical passage
\tau\wedge N
\widehat RUse in source · 4 passages
Search aliases: widehat R widehat
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 371
For a fixed finite collection {U_1,…,U_N}, suppose a declared assessment gives estimates widehat R_i of true productivities R_i. This assessment can be exact on a fully specified finite distribution or statistical under additional assumptions. Choosing a candidate must use the assessment actually available at selection time, not a hidden final score inspected afterward.
Exact mathematical passage
\widehat R_i
\widehat R_{\hat\imath}Use in source · 1 passage
Search aliases: widehat R_hatimath widehat hat imath
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 382
assessment error] If |widehat R_i-R_i|≤ε for every candidate and hatimath maximizes widehat R_i, then R_hatimath≥ max_iR_i-2ε. If a baseline is included and the selected candidate's estimated advantage exceeds 2ε+γ, its true advantage exceeds γ. For a true maximizer i^*, R_hatimath≥widehat R_hatimath-ε ≥widehat R_i^*-ε ≥ R_i^*-2ε. Apply the same two error bounds to the selected candidate and the baseline for the second assertion.
Exact mathematical passage
R_{\hat\imath}\geq\widehat R_{\hat\imath}-\varepsilon \geq\widehat R_{i^*}-\varepsilon \geq R_{i^*}-2\varepsilon
\widehat R_iUse in source · 3 passages
Search aliases: widehat R_i widehat
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 371
For a fixed finite collection {U_1,…,U_N}, suppose a declared assessment gives estimates widehat R_i of true productivities R_i. This assessment can be exact on a fully specified finite distribution or statistical under additional assumptions. Choosing a candidate must use the assessment actually available at selection time, not a hidden final score inspected afterward.
Exact mathematical passage
\widehat R_i
\widehat R_{i^*}Use in source · 1 passage
Search aliases: widehat R_i^* widehat
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 382
assessment error] If |widehat R_i-R_i|≤ε for every candidate and hatimath maximizes widehat R_i, then R_hatimath≥ max_iR_i-2ε. If a baseline is included and the selected candidate's estimated advantage exceeds 2ε+γ, its true advantage exceeds γ. For a true maximizer i^*, R_hatimath≥widehat R_hatimath-ε ≥widehat R_i^*-ε ≥ R_i^*-2ε. Apply the same two error bounds to the selected candidate and the baseline for the second assertion.
Exact mathematical passage
R_{\hat\imath}\geq\widehat R_{\hat\imath}-\varepsilon \geq\widehat R_{i^*}-\varepsilon \geq R_{i^*}-2\varepsilon
\widehat\DeltaUse in source · 3 passages
Search aliases: widehatΔ widehat Delta
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1095
One sufficient fresh-audit construction. Freeze a candidate and its comparator using the prior history. Draw n_t fresh evaluation cases conditionally independently from the declared distribution and record paired score differences Y_i∈[-1,1]. Let Δ_t be their conditional mean and widehatΔ_t their average. The bounded-variable concentration inequality gives Pr(widehatΔ_t-Δ_t≥ rmid H_t-1) ≤ exp(-n_tr^2/2). Consequently, accepting a claim Δ_t≥γ only when widehatΔ_t-sqrt2log(1/δ_t)/n_t≥γ meets the p…
Exact mathematical passage
\widehat\Delta_t
\widehat\Delta_tUse in source · 3 passages
Search aliases: widehatΔ_t widehat Delta
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1095
One sufficient fresh-audit construction. Freeze a candidate and its comparator using the prior history. Draw n_t fresh evaluation cases conditionally independently from the declared distribution and record paired score differences Y_i∈[-1,1]. Let Δ_t be their conditional mean and widehatΔ_t their average. The bounded-variable concentration inequality gives Pr(widehatΔ_t-Δ_t≥ rmid H_t-1) ≤ exp(-n_tr^2/2). Consequently, accepting a claim Δ_t≥γ only when widehatΔ_t-sqrt2log(1/δ_t)/n_t≥γ meets the p…
Exact mathematical passage
\widehat\Delta_t
\widetilde fUse in source · 1 passage
Search aliases: widetilde f widetilde
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 334
[Repair synthesis under a fixed unknown model] Construct a finite repair model on widetilde X=Θ× X by widetilde f_e(θ,x)=(θ,f_e^θ(x)), widetilde o_e(θ,x)=o_e^θ(x), widetilde G_a={(θ,x):x∈ G_a^θ}, with the corresponding tagged domains and the common action costs. Applying Theorem~[source reference] to this model gives exactly the nondominated work/storage pairs of report-dependent trees that succeed for every initial pair in B, within the declared horizon. Every retained tree is valid for the act…
Exact mathematical passage
\widetilde f_e(\theta,x)=(\theta,f_e^\theta(x)),\quad \widetilde o_e(\theta,x)=o_e^\theta(x),\quad \widetilde G_a=\{(\theta,x):x\in G_a^\theta\},
\widetilde f_eUse in source · 1 passage
Search aliases: widetilde f_e widetilde
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 334
[Repair synthesis under a fixed unknown model] Construct a finite repair model on widetilde X=Θ× X by widetilde f_e(θ,x)=(θ,f_e^θ(x)), widetilde o_e(θ,x)=o_e^θ(x), widetilde G_a={(θ,x):x∈ G_a^θ}, with the corresponding tagged domains and the common action costs. Applying Theorem~[source reference] to this model gives exactly the nondominated work/storage pairs of report-dependent trees that succeed for every initial pair in B, within the declared horizon. Every retained tree is valid for the act…
Exact mathematical passage
\widetilde f_e(\theta,x)=(\theta,f_e^\theta(x)),\quad \widetilde o_e(\theta,x)=o_e^\theta(x),\quad \widetilde G_a=\{(\theta,x):x\in G_a^\theta\},
\widetilde GUse in source · 1 passage
Search aliases: widetilde G widetilde
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 334
[Repair synthesis under a fixed unknown model] Construct a finite repair model on widetilde X=Θ× X by widetilde f_e(θ,x)=(θ,f_e^θ(x)), widetilde o_e(θ,x)=o_e^θ(x), widetilde G_a={(θ,x):x∈ G_a^θ}, with the corresponding tagged domains and the common action costs. Applying Theorem~[source reference] to this model gives exactly the nondominated work/storage pairs of report-dependent trees that succeed for every initial pair in B, within the declared horizon. Every retained tree is valid for the act…
Exact mathematical passage
\widetilde f_e(\theta,x)=(\theta,f_e^\theta(x)),\quad \widetilde o_e(\theta,x)=o_e^\theta(x),\quad \widetilde G_a=\{(\theta,x):x\in G_a^\theta\},
\widetilde G_aUse in source · 1 passage
Search aliases: widetilde G_a widetilde
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 334
[Repair synthesis under a fixed unknown model] Construct a finite repair model on widetilde X=Θ× X by widetilde f_e(θ,x)=(θ,f_e^θ(x)), widetilde o_e(θ,x)=o_e^θ(x), widetilde G_a={(θ,x):x∈ G_a^θ}, with the corresponding tagged domains and the common action costs. Applying Theorem~[source reference] to this model gives exactly the nondominated work/storage pairs of report-dependent trees that succeed for every initial pair in B, within the declared horizon. Every retained tree is valid for the act…
Exact mathematical passage
\widetilde f_e(\theta,x)=(\theta,f_e^\theta(x)),\quad \widetilde o_e(\theta,x)=o_e^\theta(x),\quad \widetilde G_a=\{(\theta,x):x\in G_a^\theta\},
\widetilde oUse in source · 1 passage
Search aliases: widetilde o widetilde
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 334
[Repair synthesis under a fixed unknown model] Construct a finite repair model on widetilde X=Θ× X by widetilde f_e(θ,x)=(θ,f_e^θ(x)), widetilde o_e(θ,x)=o_e^θ(x), widetilde G_a={(θ,x):x∈ G_a^θ}, with the corresponding tagged domains and the common action costs. Applying Theorem~[source reference] to this model gives exactly the nondominated work/storage pairs of report-dependent trees that succeed for every initial pair in B, within the declared horizon. Every retained tree is valid for the act…
Exact mathematical passage
\widetilde f_e(\theta,x)=(\theta,f_e^\theta(x)),\quad \widetilde o_e(\theta,x)=o_e^\theta(x),\quad \widetilde G_a=\{(\theta,x):x\in G_a^\theta\},
\widetilde o_eUse in source · 1 passage
Search aliases: widetilde o_e widetilde
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 334
[Repair synthesis under a fixed unknown model] Construct a finite repair model on widetilde X=Θ× X by widetilde f_e(θ,x)=(θ,f_e^θ(x)), widetilde o_e(θ,x)=o_e^θ(x), widetilde G_a={(θ,x):x∈ G_a^θ}, with the corresponding tagged domains and the common action costs. Applying Theorem~[source reference] to this model gives exactly the nondominated work/storage pairs of report-dependent trees that succeed for every initial pair in B, within the declared horizon. Every retained tree is valid for the act…
Exact mathematical passage
\widetilde f_e(\theta,x)=(\theta,f_e^\theta(x)),\quad \widetilde o_e(\theta,x)=o_e^\theta(x),\quad \widetilde G_a=\{(\theta,x):x\in G_a^\theta\},
\widetilde UUse in source · 2 passages
Search aliases: widetilde U widetilde
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 434
Approximate performance transfer. Suppose a source comparison gives productivity advantage Δ to U' over U. Let widetilde U',widetilde U be actual transported target procedures. If each procedure's productivity differs from its source value by at most ε, then R'(widetilde U')- R'(widetilde U) ≥Δ-2ε. This follows by subtracting the two inequalities. A common source of the premise is an exact task correspondence plus a bounded evaluation-law shift. For any fixed gain function in [-1,1], changing th…
Exact mathematical passage
\widetilde U',\widetilde U
\widetilde U'Use in source · 2 passages
Search aliases: widetilde U' widetilde
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 434
Approximate performance transfer. Suppose a source comparison gives productivity advantage Δ to U' over U. Let widetilde U',widetilde U be actual transported target procedures. If each procedure's productivity differs from its source value by at most ε, then R'(widetilde U')- R'(widetilde U) ≥Δ-2ε. This follows by subtracting the two inequalities. A common source of the premise is an exact task correspondence plus a bounded evaluation-law shift. For any fixed gain function in [-1,1], changing th…
Exact mathematical passage
\widetilde U',\widetilde U
\widetilde XUse in source · 1 passage
Search aliases: widetilde X widetilde
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 333
[Repair synthesis under a fixed unknown model] Construct a finite repair model on widetilde X=Θ× X by widetilde f_e(θ,x)=(θ,f_e^θ(x)), widetilde o_e(θ,x)=o_e^θ(x), widetilde G_a={(θ,x):x∈ G_a^θ}, with the corresponding tagged domains and the common action costs. Applying Theorem~[source reference] to this model gives exactly the nondominated work/storage pairs of report-dependent trees that succeed for every initial pair in B, within the declared horizon. Every retained tree is valid for the act…
Exact mathematical passage
\widetilde X=\Theta\times X
xUse in source · 26 passages
Search aliases: x
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 247
Repairing a failed representation. Let X be a finite set with deterministic operations F_u:X→ X and output labels q(x). Begin with a partition P_0 that preserves both the old retained labels and q. Define the next partition by splitting a block whenever two states in it have successors in different current blocks under some permitted operation. Equivalently, x≡_P_t+1y Longleftrightarrow x≡_P_ty and F_u(x)≡_P_tF_u(y) for every u. The construction uses a complete transition description. Having a f…
Exact mathematical passage
q(x)
X'Use in source · 1 passage
Search aliases: X'
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 407
Consider source and target finite repair models. Let σ:X→ X' be a bijection, τ: E→ E' a bijection of experiments, ψ: A→ A' a bijection of repairs, and η_e:O_e→ O'_τ(e) a bijection of report labels. Assume σ(D_e) · =D'_τ(e), · σ f_e(x) · =f'_τ(e)σ(x),
Exact mathematical passage
\sigma:X\to X'
\xrightarrowUse in source · 1 passage
Search aliases: xrightarrow
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1405
appendix A complete diagnostic trace. Take n=8 and hidden defect j=5, whose binary digits are 101. One optimal tree uses the three queries A_0={1,3,5,7}, A_1={2,3,6,7}, A_2={4,5,6,7}. The precise order may differ among equally optimal compiled trees. For this order, the supports are {0,…,7}xrightarrow1{1,3,5,7} xrightarrow0{1,5}xrightarrow1{5}. The terminal repair is r_5. The path uses three diagnostic calls and one repair call. The complete tree stores all eight repair leaves and seven query no…
Exact mathematical passage
\{0,\ldots,7\}\xrightarrow{1}\{1,3,5,7\} \xrightarrow{0}\{1,5\}\xrightarrow{1}\{5\}.
YUse in source · 7 passages
Search aliases: Y
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1095
One sufficient fresh-audit construction. Freeze a candidate and its comparator using the prior history. Draw n_t fresh evaluation cases conditionally independently from the declared distribution and record paired score differences Y_i∈[-1,1]. Let Δ_t be their conditional mean and widehatΔ_t their average. The bounded-variable concentration inequality gives Pr(widehatΔ_t-Δ_t≥ rmid H_t-1) ≤ exp(-n_tr^2/2). Consequently, accepting a claim Δ_t≥γ only when widehatΔ_t-sqrt2log(1/δ_t)/n_t≥γ meets the p…
Exact mathematical passage
Y_i\in[-1,1]
yUse in source · 5 passages
Search aliases: y
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 248
Repairing a failed representation. Let X be a finite set with deterministic operations F_u:X→ X and output labels q(x). Begin with a partition P_0 that preserves both the old retained labels and q. Define the next partition by splitting a block whenever two states in it have successors in different current blocks under some permitted operation. Equivalently, x≡_P_t+1y Longleftrightarrow x≡_P_ty and F_u(x)≡_P_tF_u(y) for every u. The construction uses a complete transition description. Having a f…
Exact mathematical passage
x\equiv_{P_{t+1}}y \quad\Longleftrightarrow\quad x\equiv_{P_t}y\ \text{ and }\ F_u(x)\equiv_{P_t}F_u(y)\ \text{for every }u.
Y_iUse in source · 1 passage
Search aliases: Y_i
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1095
One sufficient fresh-audit construction. Freeze a candidate and its comparator using the prior history. Draw n_t fresh evaluation cases conditionally independently from the declared distribution and record paired score differences Y_i∈[-1,1]. Let Δ_t be their conditional mean and widehatΔ_t their average. The bounded-variable concentration inequality gives Pr(widehatΔ_t-Δ_t≥ rmid H_t-1) ≤ exp(-n_tr^2/2). Consequently, accepting a claim Δ_t≥γ only when widehatΔ_t-sqrt2log(1/δ_t)/n_t≥γ meets the p…
Exact mathematical passage
Y_i\in[-1,1]
Y_tUse in source · 6 passages
Search aliases: Y_t
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1120
We give a small time-uniform alternative whose finite arithmetic is directly inspectable. Before an audit begins, freeze the candidate procedure, its comparator, their installed information, the task law, the common resource cap, and a paired score Y_t∈[-1,1]. Timeouts and invalid outcomes must have a stipulated score rather than disappear from the sample. Fix a proposed improvement margin γ∈[0,1).
Exact mathematical passage
Y_t\in[-1,1]
ZUse in source · 7 passages
Search aliases: Z
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 196
What persists across a recursive step. A lineage state can be written schematically as Z_t=(S_t,U_t,L_t,E_t,Ξ_t,ρ_t), where E_t records evidence and rejected alternatives, and ρ_t records remaining resources and scope. A proposed modification may target any admitted subset. A comparison across different Ξ_t needs a declared bridge. It cannot be justified merely because both scores are called accuracy.
Exact mathematical passage
Z_t=(S_t,U_t,L_t,E_t,\Xi_t,\rho_t),
z'Use in source · 2 passages
Search aliases: z'
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1214
For each transition, a progress mark p(z,a,z')∈{0,1} is supplied by a checked criterion. A mark of one means the specified process-improvement claim has been established under the same comparison. It is not assigned merely because a file changed. Define W_0,0 · = K, · W_0,k · =∅ (k>0),
Exact mathematical passage
p(z,a,z')\in\{0,1\}
Z_tUse in source · 1 passage
Search aliases: Z_t
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 196
What persists across a recursive step. A lineage state can be written schematically as Z_t=(S_t,U_t,L_t,E_t,Ξ_t,ρ_t), where E_t records evidence and rejected alternatives, and ρ_t records remaining resources and scope. A proposed modification may target any admitted subset. A comparison across different Ξ_t needs a declared bridge. It cannot be justified merely because both scores are called accuracy.
Exact mathematical passage
Z_t=(S_t,U_t,L_t,E_t,\Xi_t,\rho_t),
|Use in source · 9 passages
Search aliases: |
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 257
[Least stable refinement of a retained representation] The splitting construction stabilizes after at most |X|-|P_0| strict refinements. Its final partition is the coarsest refinement of P_0 on which every F_u has a well-defined quotient operation. It preserves every finite continuation of the retained labels and outputs. A strict refinement increases the number of nonempty blocks, which is bounded by |X|. At a fixed point, equivalent states have equivalent successors, so the quotient operations…
Exact mathematical passage
|X|-|P_0|
|_{D_v}Use in source · 1 passage
Search aliases: |_D_v
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 982
Suppose a revision replaces a by a'. Mark a vertex directly stale whenever one of its declared component values differs. Let Δ be the union of those vertices and all their descendants in the dependency graph. Equivalently, in a topological order, mark v whenever a|_D_v≠ a'|_D_v or pa(v)∩Δ≠∅. Here equality refers to the declared artifact value, not merely an unchanged version label. This procedure is conservative. A changed premise may leave the conclusion true, but the old evidence is not reused…
Exact mathematical passage
a|_{D_v}\ne a'|_{D_v} \quad\text{or}\quad \operatorname{pa}(v)\cap\Delta\ne\varnothing.
\cdotUse in source · 3 passages
Search aliases: · cdot
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1042
be extra setup, let s_i∈ R^d be the old-minus-new per-use event vector for workload type i, and let n_i be its nonnegative integer multiplicity. The net event saving is Δ(n)=∑_i n_i s_i-K. A price vector w∈ R_≥0^d specifies conversions into one declared additive resource measure. Its priced saving is w·Δ(n). The prices are modeling assumptions, not learned hardware laws inferred from the event counts.
Exact mathematical passage
w\cdot\Delta(n)
\timesUse in source · 2 passages
Search aliases: × times
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 322
t possibilities rather than replacing uncertainty by one convenient estimate. Let Θ be a finite nonempty set of candidate models with a common state set X, experiment names, repair names, and nonnegative costs. Model θ specifies D_e^θ, f_e^θ, o_e^θ, and G_a^θ. The unknown model is fixed during an investigation. The initial information is a nonempty set B⊆Θ× X. The procedure observes reports, not the model label or the hidden physical state.
Exact mathematical passage
B\subseteq\Theta\times X
\gammaUse in source · 15 passages
Search aliases: γ gamma
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 378
[Selection under uniform assessment error] If |widehat R_i-R_i|≤ε for every candidate and hatimath maximizes widehat R_i, then R_hatimath≥ max_iR_i-2ε. If a baseline is included and the selected candidate's estimated advantage exceeds 2ε+γ, its true advantage exceeds γ. For a true maximizer i^*, R_hatimath≥widehat R_hatimath-ε ≥widehat R_i^*-ε ≥ R_i^*-2ε. Apply the same two error bounds to the selected candidate and the baseline for the second assertion.
Exact mathematical passage
2\varepsilon+\gamma
\DeltaUse in source · 19 passages
Search aliases: Δ Delta
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 434
Approximate performance transfer. Suppose a source comparison gives productivity advantage Δ to U' over U. Let widetilde U',widetilde U be actual transported target procedures. If each procedure's productivity differs from its source value by at most ε, then R'(widetilde U')- R'(widetilde U) ≥Δ-2ε. This follows by subtracting the two inequalities. A common source of the premise is an exact task correspondence plus a bounded evaluation-law shift. For any fixed gain function in [-1,1], changing th…
Exact mathematical passage
\Delta
\deltaUse in source · 18 passages
Search aliases: δ delta
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 439
ource value by at most ε, then R'(widetilde U')- R'(widetilde U) ≥Δ-2ε. This follows by subtracting the two inequalities. A common source of the premise is an exact task correspondence plus a bounded evaluation-law shift. For any fixed gain function in [-1,1], changing the task law by total variation δ changes its expectation by at most 2δ. A further pointwise gain discrepancy η therefore gives ε=η+2δ.
Exact mathematical passage
\delta
\delta_jUse in source · 2 passages
Search aliases: δ_j delta
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1164
Evidence across recursive generations. For attempt j, condition on the complete earlier lineage, freeze its candidate, and initialize a fresh valid audit at E_0^(j)=1 with error allowance δ_j. If ∑_jδ_j≤δ, Proposition~[source reference] bounds the probability of ever accepting a false margin claim by δ. Each audit can stop sequentially, and the next candidate may depend on completed earlier audits. The premise remains conditional validity for each actual attempt.
Exact mathematical passage
\delta_j
\Delta_tUse in source · 3 passages
Search aliases: Δ_t Delta
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1095
One sufficient fresh-audit construction. Freeze a candidate and its comparator using the prior history. Draw n_t fresh evaluation cases conditionally independently from the declared distribution and record paired score differences Y_i∈[-1,1]. Let Δ_t be their conditional mean and widehatΔ_t their average. The bounded-variable concentration inequality gives Pr(widehatΔ_t-Δ_t≥ rmid H_t-1) ≤ exp(-n_tr^2/2). Consequently, accepting a claim Δ_t≥γ only when widehatΔ_t-sqrt2log(1/δ_t)/n_t≥γ meets the p…
Exact mathematical passage
\Delta_t
\delta_tUse in source · 6 passages
Search aliases: δ_t delta
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1078
[Conditional error budgets across an adaptive lineage] Let H_t-1 be the information available before attempt t. Suppose the event A_t that attempt t accepts a false improvement claim satisfies Pr(A_tmid H_t-1)≤δ_t almost surely, for deterministic δ_t≥0 with ∑_t≥1δ_t≤δ. Then Pr(∃ t≥1:A_t)≤δ. The candidate-generating process and the time at which attempts stop may depend on the earlier history. Taking expectations gives Pr(A_t)≤δ_t. The union bound gives the result for any finite number of attempt…
Exact mathematical passage
\Pr(A_t\mid\mathcal H_{t-1})\leq\delta_t \quad\text{almost surely},
\varepsilonUse in source · 7 passages
Search aliases: ε varepsilon
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 374
[Selection under uniform assessment error] If |widehat R_i-R_i|≤ε for every candidate and hatimath maximizes widehat R_i, then R_hatimath≥ max_iR_i-2ε. If a baseline is included and the selected candidate's estimated advantage exceeds 2ε+γ, its true advantage exceeds γ. For a true maximizer i^*, R_hatimath≥widehat R_hatimath-ε ≥widehat R_i^*-ε ≥ R_i^*-2ε. Apply the same two error bounds to the selected candidate and the baseline for the second assertion.
Exact mathematical passage
|\widehat R_i-R_i|\leq\varepsilon
\etaUse in source · 5 passages
Search aliases: η eta
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 407
Consider source and target finite repair models. Let σ:X→ X' be a bijection, τ: E→ E' a bijection of experiments, ψ: A→ A' a bijection of repairs, and η_e:O_e→ O'_τ(e) a bijection of report labels. Assume σ(D_e) · =D'_τ(e), · σ f_e(x) · =f'_τ(e)σ(x),
Exact mathematical passage
\eta_e:O_e\to O'_{\tau(e)}
\eta_eUse in source · 3 passages
Search aliases: η_e eta
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 407
Consider source and target finite repair models. Let σ:X→ X' be a bijection, τ: E→ E' a bijection of experiments, ψ: A→ A' a bijection of repairs, and η_e:O_e→ O'_τ(e) a bijection of report labels. Assume σ(D_e) · =D'_τ(e), · σ f_e(x) · =f'_τ(e)σ(x),
Exact mathematical passage
\eta_e:O_e\to O'_{\tau(e)}
\ThetaUse in source · 4 passages
Search aliases: Θ Theta
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 322
A useful first extension retains several explicit possibilities rather than replacing uncertainty by one convenient estimate. Let Θ be a finite nonempty set of candidate models with a common state set X, experiment names, repair names, and nonnegative costs. Model θ specifies D_e^θ, f_e^θ, o_e^θ, and G_a^θ. The unknown model is fixed during an investigation. The initial information is a nonempty set B⊆Θ× X. The procedure observes reports, not the model label or the hidden physical state.
Exact mathematical passage
\Theta
\thetaUse in source · 22 passages
Search aliases: θ theta
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 219
Let a hidden case θ determine a set R(θ) of successful repairs. The set may have several members. Diagnosis need only obtain enough information to choose a repair in the appropriate set. It need not identify every hidden property of θ.
Exact mathematical passage
\theta
\theta_1Use in source · 1 passage
Search aliases: θ_1 theta
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 242
Repair sufficiency is not full identification. Suppose R(θ_1)={a,c} and R(θ_2)={b,c}. The cases are distinct, yet repair c works in both. A method can stop without learning which case occurred. Conversely, pairwise overlap of repair sets does not guarantee one repair for an entire uncertainty set. The three sets {a,b}, {b,c}, and {a,c} intersect pairwise and have empty total intersection. The stopping condition must refer to the whole surviving set.
Exact mathematical passage
\mathcal R(\theta_1)=\{a,c\}
\theta_2Use in source · 1 passage
Search aliases: θ_2 theta
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 242
Repair sufficiency is not full identification. Suppose R(θ_1)={a,c} and R(θ_2)={b,c}. The cases are distinct, yet repair c works in both. A method can stop without learning which case occurred. Conversely, pairwise overlap of repair sets does not guarantee one repair for an entire uncertainty set. The three sets {a,b}, {b,c}, and {a,c} intersect pairwise and have empty total intersection. The stopping condition must refer to the whole surviving set.
Exact mathematical passage
\mathcal R(\theta_2)=\{b,c\}
\kappaUse in source · 9 passages
Search aliases: κ kappa
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 722
Let B be a finite nonempty immutable-case support and let G_r⊆ B be each repair's success set restricted to it. Define κ(B) to be the smallest number of repair-success sets whose union covers B. Set κ(B)=∞ when no such cover exists. This number concerns the repaired objective, not the number of hidden states. Computing it can itself be expensive.
Exact mathematical passage
\kappa(B)
\lambdaUse in source · 5 passages
Search aliases: λ lambda
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1129
Choose a nonnegative bet λ_t, measurable from H_t-1, with λ_t≤1/(1+γ). Define E_0=1, E_t=E_t-1(1+λ_t(Y_t-γ)). The variable E_t is an evidence process, not a posterior probability that the candidate is better. Predictability means choosing the bet before observing the score to which it is applied.
Exact mathematical passage
\lambda_t
\lambda_tUse in source · 5 passages
Search aliases: λ_t lambda
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1129
Choose a nonnegative bet λ_t, measurable from H_t-1, with λ_t≤1/(1+γ). Define E_0=1, E_t=E_t-1(1+λ_t(Y_t-γ)). The variable E_t is an evidence process, not a posterior probability that the candidate is better. Predictability means choosing the bet before observing the score to which it is applied.
Exact mathematical passage
\lambda_t
\Xi_tUse in source · 2 passages
Search aliases: Ξ_t Xi
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 196
What persists across a recursive step. A lineage state can be written schematically as Z_t=(S_t,U_t,L_t,E_t,Ξ_t,ρ_t), where E_t records evidence and rejected alternatives, and ρ_t records remaining resources and scope. A proposed modification may target any admitted subset. A comparison across different Ξ_t needs a declared bridge. It cannot be justified merely because both scores are called accuracy.
Exact mathematical passage
Z_t=(S_t,U_t,L_t,E_t,\Xi_t,\rho_t),
\piUse in source · 1 passage
Search aliases: π pi
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1270
What would learn. Let r_ϕ encode the retained state, G_θ,a describe proposed operations, π_ω choose observations or experiments, and p_ψ propose candidate programs or parameter updates. A development episode can modify ϕ,θ,ω,ψ through a declared learning rule. A persistent program library can also change by adding, replacing, or removing checked operators. Temporary deliberation without a retained modification is not called a new learned generation here.
Exact mathematical passage
\pi_\omega
\pi_\omegaUse in source · 1 passage
Search aliases: π_ω pi omega
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1270
What would learn. Let r_ϕ encode the retained state, G_θ,a describe proposed operations, π_ω choose observations or experiments, and p_ψ propose candidate programs or parameter updates. A development episode can modify ϕ,θ,ω,ψ through a declared learning rule. A persistent program library can also change by adding, replacing, or removing checked operators. Temporary deliberation without a retained modification is not called a new learned generation here.
Exact mathematical passage
\pi_\omega
\rhoUse in source · 8 passages
Search aliases: ρ rho
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 196
What persists across a recursive step. A lineage state can be written schematically as Z_t=(S_t,U_t,L_t,E_t,Ξ_t,ρ_t), where E_t records evidence and rejected alternatives, and ρ_t records remaining resources and scope. A proposed modification may target any admitted subset. A comparison across different Ξ_t needs a declared bridge. It cannot be justified merely because both scores are called accuracy.
Exact mathematical passage
Z_t=(S_t,U_t,L_t,E_t,\Xi_t,\rho_t),
\rho_tUse in source · 2 passages
Search aliases: ρ_t rho
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 196
What persists across a recursive step. A lineage state can be written schematically as Z_t=(S_t,U_t,L_t,E_t,Ξ_t,ρ_t), where E_t records evidence and rejected alternatives, and ρ_t records remaining resources and scope. A proposed modification may target any admitted subset. A comparison across different Ξ_t needs a declared bridge. It cannot be justified merely because both scores are called accuracy.
Exact mathematical passage
Z_t=(S_t,U_t,L_t,E_t,\Xi_t,\rho_t),
\phiUse in source · 2 passages
Search aliases: ϕ phi
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1270
What would learn. Let r_ϕ encode the retained state, G_θ,a describe proposed operations, π_ω choose observations or experiments, and p_ψ propose candidate programs or parameter updates. A development episode can modify ϕ,θ,ω,ψ through a declared learning rule. A persistent program library can also change by adding, replacing, or removing checked operators. Temporary deliberation without a retained modification is not called a new learned generation here.
Exact mathematical passage
r_\phi
\psiUse in source · 4 passages
Search aliases: ψ psi
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 407
Consider source and target finite repair models. Let σ:X→ X' be a bijection, τ: E→ E' a bijection of experiments, ψ: A→ A' a bijection of repairs, and η_e:O_e→ O'_τ(e) a bijection of report labels. Assume σ(D_e) · =D'_τ(e), · σ f_e(x) · =f'_τ(e)σ(x),
Exact mathematical passage
\psi:\mathcal A\to\mathcal A'
\omegaUse in source · 2 passages
Search aliases: ω omega
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1270
What would learn. Let r_ϕ encode the retained state, G_θ,a describe proposed operations, π_ω choose observations or experiments, and p_ψ propose candidate programs or parameter updates. A development episode can modify ϕ,θ,ω,ψ through a declared learning rule. A persistent program library can also change by adding, replacing, or removing checked operators. Temporary deliberation without a retained modification is not called a new learned generation here.
Exact mathematical passage
\pi_\omega
\ldotsUse in source · 6 passages
Search aliases: … ldots
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 204
A small world with reusable diagnostic structure. Consider eight cells with exactly one latent defect, indexed by j∈{0,…,7}. The solver must repair that cell while preserving the others. Its old representation retains only the number of defects, so every episode initially looks identical.
Exact mathematical passage
j\in\{0,\ldots,7\}
\ellUse in source · 6 passages
Search aliases: ℓ ell
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 623
[Feature sufficiency for a search decision] Let P be a finite set of search contexts, let r: P→ Z be their available feature map, and let ℓ: P→{0,1} be the required decision label. A deterministic rule f:Z→{0,1} satisfying fr=ℓ exists if and only if ℓ is constant on each nonempty fibre of r. A pair in one fibre with different labels rules out every such rule, irrespective of its program length. If f exists, equal features have the same output under f and hence the same label. Conversely, assign …
Exact mathematical passage
\ell:\mathcal P\to\{0,1\}
\forallUse in source · 2 passages
Search aliases: ∀ forall
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 285
For horizon h, let F_h(B_0) be the nondominated attainable pairs (w,m). A pair dominates another when it is no larger in either coordinate and strictly smaller in at least one. Let T(B_0)={(c_a,1):B_0⊆ G_a}. Then F_0(B_0) · =Pareto T(B_0), F_h(B_0) · =Pareto(T(B_0)∪ ⋃_e:B_0⊆ D_e {(c_e+max_o w_o, 1+∑_o m_o): (w_o,m_o)∈ F_h-1(B_e,o) ∀ o}). nonumber An empty child frontier makes that experiment unavailable for a successful complete tree. An empty union is not a zero-cost solution.
Exact mathematical passage
\label{eq:pareto} \mathcal F_0(B_0)&=\Pareto T(B_0),\\ \mathcal F_h(B_0)&=\Pareto\left(T(B_0)\cup \bigcup_{e:B_0\subseteq D_e} \left\{\left(c_e+\max_o w_o,\ 1+\sum_o m_o\right): (w_o,m_o)\in\mathcal F_{h-1}(B_{e,o})\ \forall o\right\}\right). \nonumber
\existsUse in source · 3 passages
Search aliases: ∃ exists
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1083
proposition[Conditional error budgets across an adaptive lineage] Let H_t-1 be the information available before attempt t. Suppose the event A_t that attempt t accepts a false improvement claim satisfies Pr(A_tmid H_t-1)≤δ_t almost surely, for deterministic δ_t≥0 with ∑_t≥1δ_t≤δ. Then Pr(∃ t≥1:A_t)≤δ. The candidate-generating process and the time at which attempts stop may depend on the earlier history. Taking expectations gives Pr(A_t)≤δ_t. The union bound gives the result for any finite number…
Exact mathematical passage
\Pr\big(\exists t\geq1:A_t\big)\leq\delta.
\varnothingUse in source · 2 passages
Search aliases: ∅ varnothing
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 982
Suppose a revision replaces a by a'. Mark a vertex directly stale whenever one of its declared component values differs. Let Δ be the union of those vertices and all their descendants in the dependency graph. Equivalently, in a topological order, mark v whenever a|_D_v≠ a'|_D_v or pa(v)∩Δ≠∅. Here equality refers to the declared artifact value, not merely an unchanged version label. This procedure is conservative. A changed premise may leave the conclusion true, but the old evidence is not reused…
Exact mathematical passage
a|_{D_v}\ne a'|_{D_v} \quad\text{or}\quad \operatorname{pa}(v)\cap\Delta\ne\varnothing.
\sumUse in source · 7 passages
Search aliases: ∑ sum
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 285
For horizon h, let F_h(B_0) be the nondominated attainable pairs (w,m). A pair dominates another when it is no larger in either coordinate and strictly smaller in at least one. Let T(B_0)={(c_a,1):B_0⊆ G_a}. Then F_0(B_0) · =Pareto T(B_0), F_h(B_0) · =Pareto(T(B_0)∪ ⋃_e:B_0⊆ D_e {(c_e+max_o w_o, 1+∑_o m_o): (w_o,m_o)∈ F_h-1(B_e,o) ∀ o}). nonumber An empty child frontier makes that experiment unavailable for a successful complete tree. An empty union is not a zero-cost solution.
Exact mathematical passage
\label{eq:pareto} \mathcal F_0(B_0)&=\Pareto T(B_0),\\ \mathcal F_h(B_0)&=\Pareto\left(T(B_0)\cup \bigcup_{e:B_0\subseteq D_e} \left\{\left(c_e+\max_o w_o,\ 1+\sum_o m_o\right): (w_o,m_o)\in\mathcal F_{h-1}(B_{e,o})\ \forall o\right\}\right). \nonumber
\sum_iUse in source · 1 passage
Search aliases: ∑_i sum
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1039
Which gains survive a change of resource prices? Fix an additive ledger with d≥1 named event types. Let K∈ R_≥0^d be extra setup, let s_i∈ R^d be the old-minus-new per-use event vector for workload type i, and let n_i be its nonnegative integer multiplicity. The net event saving is Δ(n)=∑_i n_i s_i-K. A price vector w∈ R_≥0^d specifies conversions into one declared additive resource measure. Its priced saving is w·Δ(n). The prices are modeling assumptions, not learned hardware laws inferred from…
Exact mathematical passage
\label{eq:vector-saving} \Delta(n)=\sum_i n_i s_i-K.
\sum_jUse in source · 1 passage
Search aliases: ∑_j sum
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1164
Evidence across recursive generations. For attempt j, condition on the complete earlier lineage, freeze its candidate, and initialize a fresh valid audit at E_0^(j)=1 with error allowance δ_j. If ∑_jδ_j≤δ, Proposition~[source reference] bounds the probability of ever accepting a false margin claim by δ. Each audit can stop sequentially, and the next candidate may depend on completed earlier audits. The premise remains conditional validity for each actual attempt.
Exact mathematical passage
\sum_j\delta_j\leq\delta
\sum_oUse in source · 1 passage
Search aliases: ∑_o sum
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 285
For horizon h, let F_h(B_0) be the nondominated attainable pairs (w,m). A pair dominates another when it is no larger in either coordinate and strictly smaller in at least one. Let T(B_0)={(c_a,1):B_0⊆ G_a}. Then F_0(B_0) · =Pareto T(B_0), F_h(B_0) · =Pareto(T(B_0)∪ ⋃_e:B_0⊆ D_e {(c_e+max_o w_o, 1+∑_o m_o): (w_o,m_o)∈ F_h-1(B_e,o) ∀ o}). nonumber An empty child frontier makes that experiment unavailable for a successful complete tree. An empty union is not a zero-cost solution.
Exact mathematical passage
\label{eq:pareto} \mathcal F_0(B_0)&=\Pareto T(B_0),\\ \mathcal F_h(B_0)&=\Pareto\left(T(B_0)\cup \bigcup_{e:B_0\subseteq D_e} \left\{\left(c_e+\max_o w_o,\ 1+\sum_o m_o\right): (w_o,m_o)\in\mathcal F_{h-1}(B_{e,o})\ \forall o\right\}\right). \nonumber
\sum_{t=1}^{T}Use in source · 1 passage
Search aliases: ∑_t=1^T sum
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 874
Let C_rm prepare be the complete preparation cost in a declared common resource model, including failed proposals and certificate checking, and let d_t be the subsequent saving on task t after including dispatch. Reuse is beneficial in that model exactly when ∑_t=1^Td_t>C_rm prepare. This is the earlier amortization identity applied to acquisition itself. The reference does not convert unlike counters into a universal wall-clock claim. Even one setup coordinate, 518,140 schema matches, exceeds t…
Exact mathematical passage
\sum_{t=1}^{T}d_t>C_{\rm prepare}.
\sum_{t\geq1}Use in source · 1 passage
Search aliases: ∑_t≥1 sum geq
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 1082
[Conditional error budgets across an adaptive lineage] Let H_t-1 be the information available before attempt t. Suppose the event A_t that attempt t accepts a false improvement claim satisfies Pr(A_tmid H_t-1)≤δ_t almost surely, for deterministic δ_t≥0 with ∑_t≥1δ_t≤δ. Then Pr(∃ t≥1:A_t)≤δ. The candidate-generating process and the time at which attempts stop may depend on the earlier history. Taking expectations gives Pr(A_t)≤δ_t. The union bound gives the result for any finite number of attempt…
Exact mathematical passage
\sum_{t\geq1}\delta_t\leq\delta
\sum_{x\in B}Use in source · 1 passage
Search aliases: ∑_x∈ B sum in
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 897
[Repair-relative weight and prefix obstruction] Suppose a total immutable deterministic binary diagnostic tree succeeds on nonempty support B and has L reachable leaves and maximum query depth h. Under~[source reference], ∑_x∈ Bw_x≤ L≤2^h. At a candidate query root with h≥1 remaining query levels, if the cases already read into either outcome branch have weight greater than 2^h-1, no assignment of its unread reports can make that branch repairable within the remaining depth. This implication is …
Exact mathematical passage
\sum_{x\in B}w_x\le L\le2^h.
\sum_{x\in S_r}Use in source · 1 passage
Search aliases: ∑_x∈ S_r sum in
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 890
Let X be a finite immutable state set and let S_r⊆ X be the success set of repair r. Choose nonnegative rational weights w_x satisfying ∑_x∈ S_rw_x≤1 for every permitted repair r. This is a feasible dual weight assignment for unweighted set cover. We use only its elementary weak-duality property, not an assumption that the best assignment has been found~[source reference].
Exact mathematical passage
\label{eq:repair-weights} \sum_{x\in S_r}w_x\le1\quad\text{for every permitted repair }r.
\inftyUse in source · 2 passages
Search aliases: ∞ infty
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 572
lowdown from a certified two-worker race] Suppose a parent and a shortcut worker can be paused after each unit of a common work measure and resumed without losing progress. Each may announce completion only after producing a result certified for the same target. Alternate one unit of each worker and stop at the first certified completion. If their standalone completion costs are D_0,D_1∈ N_>0∪{∞} and at least one is finite, at most 2min{D_0,D_1} worker units are executed. If each dispatched unit…
Exact mathematical passage
D_0,D_1\in\mathbb N_{>0}\cup\{\infty\}
\capUse in source · 1 passage
Search aliases: ∩ cap
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 982
Suppose a revision replaces a by a'. Mark a vertex directly stale whenever one of its declared component values differs. Let Δ be the union of those vertices and all their descendants in the dependency graph. Equivalently, in a topological order, mark v whenever a|_D_v≠ a'|_D_v or pa(v)∩Δ≠∅. Here equality refers to the declared artifact value, not merely an unchanged version label. This procedure is conservative. A changed premise may leave the conclusion true, but the old evidence is not reused…
Exact mathematical passage
a|_{D_v}\ne a'|_{D_v} \quad\text{or}\quad \operatorname{pa}(v)\cap\Delta\ne\varnothing.
\cupUse in source · 2 passages
Search aliases: ∪ cup
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 285
For horizon h, let F_h(B_0) be the nondominated attainable pairs (w,m). A pair dominates another when it is no larger in either coordinate and strictly smaller in at least one. Let T(B_0)={(c_a,1):B_0⊆ G_a}. Then F_0(B_0) · =Pareto T(B_0), F_h(B_0) · =Pareto(T(B_0)∪ ⋃_e:B_0⊆ D_e {(c_e+max_o w_o, 1+∑_o m_o): (w_o,m_o)∈ F_h-1(B_e,o) ∀ o}). nonumber An empty child frontier makes that experiment unavailable for a successful complete tree. An empty union is not a zero-cost solution.
Exact mathematical passage
\label{eq:pareto} \mathcal F_0(B_0)&=\Pareto T(B_0),\\ \mathcal F_h(B_0)&=\Pareto\left(T(B_0)\cup \bigcup_{e:B_0\subseteq D_e} \left\{\left(c_e+\max_o w_o,\ 1+\sum_o m_o\right): (w_o,m_o)\in\mathcal F_{h-1}(B_{e,o})\ \forall o\right\}\right). \nonumber
\neUse in source · 2 passages
Search aliases: ≠ ne
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 883
[The same operator spelling with changed semantics] The identity x+0=x is valid for ordinary addition. Suppose a revised interpreter uses the same printed symbol but executes amathbin⊕b=a+b+1. Then xmathbin⊕0≠ x. A cache certified for the former interpreter cannot be imported merely because its expression strings still parse. A semantic identifier helps detect ordinary version mismatch, but an identifier falsely attached to a changed implementation is not itself proof of faithful execution.
Exact mathematical passage
x\mathbin{\oplus}0\ne x
\equivUse in source · 1 passage
Search aliases: ≡ equiv
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 248
Repairing a failed representation. Let X be a finite set with deterministic operations F_u:X→ X and output labels q(x). Begin with a partition P_0 that preserves both the old retained labels and q. Define the next partition by splitting a block whenever two states in it have successors in different current blocks under some permitted operation. Equivalently, x≡_P_t+1y Longleftrightarrow x≡_P_ty and F_u(x)≡_P_tF_u(y) for every u. The construction uses a complete transition description. Having a f…
Exact mathematical passage
x\equiv_{P_{t+1}}y \quad\Longleftrightarrow\quad x\equiv_{P_t}y\ \text{ and }\ F_u(x)\equiv_{P_t}F_u(y)\ \text{for every }u.
\equiv_{P_t}Use in source · 1 passage
Search aliases: ≡_P_t equiv
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 248
Repairing a failed representation. Let X be a finite set with deterministic operations F_u:X→ X and output labels q(x). Begin with a partition P_0 that preserves both the old retained labels and q. Define the next partition by splitting a block whenever two states in it have successors in different current blocks under some permitted operation. Equivalently, x≡_P_t+1y Longleftrightarrow x≡_P_ty and F_u(x)≡_P_tF_u(y) for every u. The construction uses a complete transition description. Having a f…
Exact mathematical passage
x\equiv_{P_{t+1}}y \quad\Longleftrightarrow\quad x\equiv_{P_t}y\ \text{ and }\ F_u(x)\equiv_{P_t}F_u(y)\ \text{for every }u.
\equiv_{P_{t+1}}Use in source · 1 passage
Search aliases: ≡_P_t+1 equiv
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 248
Repairing a failed representation. Let X be a finite set with deterministic operations F_u:X→ X and output labels q(x). Begin with a partition P_0 that preserves both the old retained labels and q. Define the next partition by splitting a block whenever two states in it have successors in different current blocks under some permitted operation. Equivalently, x≡_P_t+1y Longleftrightarrow x≡_P_ty and F_u(x)≡_P_tF_u(y) for every u. The construction uses a complete transition description. Having a f…
Exact mathematical passage
x\equiv_{P_{t+1}}y \quad\Longleftrightarrow\quad x\equiv_{P_t}y\ \text{ and }\ F_u(x)\equiv_{P_t}F_u(y)\ \text{for every }u.
\leUse in source · 5 passages
Search aliases: ≤ le
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 798
Learning equations without confusing probes with proofs. The reference language uses exact integers, the total operations +, min, max, and the comparisons <, ≤, >, ≥, and =. A finite schema language contains 135 terms built from variables x,y, small integer literals, Boolean literals, and at most one binary operation. The proposer evaluates those schemas on the nine contexts (x,y)∈{1,2,3}^2.
Exact mathematical passage
\le
\leqUse in source · 26 passages
Search aliases: ≤ leq
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 230
ry decision: report case zero if the repair belongs to R(0), and case one otherwise. Success in case zero implies a correct decision. Success in case one also implies a correct decision because the two repair sets are disjoint. Thus repair success is at most the success of this binary test. For a randomized test with probability f(o)∈[0,1] of reporting zero, the latter success is tfrac12(1+ E_P_0f- E_P_1f) ≤tfrac12(1+d_TV(P_0,P_1)). The inequality follows by summing the positive differences of t…
Exact mathematical passage
\tfrac12\big(1+\E_{P_0}f-\E_{P_1}f\big) \leq\tfrac12(1+\TV(P_0,P_1)).
\geUse in source · 6 passages
Search aliases: ≥ ge
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 798
Learning equations without confusing probes with proofs. The reference language uses exact integers, the total operations +, min, max, and the comparisons <, ≤, >, ≥, and =. A finite schema language contains 135 terms built from variables x,y, small integer literals, Boolean literals, and at most one binary operation. The proposer evaluates those schemas on the nine contexts (x,y)∈{1,2,3}^2.
Exact mathematical passage
\ge
\oplusUse in source · 4 passages
Search aliases: ⊕ oplus
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 349
One hidden bit and two sensor conventions. [Known-model repair without a common unknown-model policy] Let x∈{0,1} be the cell requiring repair and let θ∈{0,1} be a fixed sensor convention. A nondestructive query reports o=x⊕θ at cost one. Repair r_i succeeds exactly when x=i and costs one. If θ is known, one query followed by the appropriate repair always succeeds. If all four (θ,x) pairs are initially possible, report zero leaves (0,0) and (1,1), while report one leaves (0,1) and (1,0). Repeati…
Exact mathematical passage
o=x\oplus\theta
\bigcupUse in source · 1 passage
Search aliases: ⋃ bigcup
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 285
For horizon h, let F_h(B_0) be the nondominated attainable pairs (w,m). A pair dominates another when it is no larger in either coordinate and strictly smaller in at least one. Let T(B_0)={(c_a,1):B_0⊆ G_a}. Then F_0(B_0) · =Pareto T(B_0), F_h(B_0) · =Pareto(T(B_0)∪ ⋃_e:B_0⊆ D_e {(c_e+max_o w_o, 1+∑_o m_o): (w_o,m_o)∈ F_h-1(B_e,o) ∀ o}). nonumber An empty child frontier makes that experiment unavailable for a successful complete tree. An empty union is not a zero-cost solution.
Exact mathematical passage
\label{eq:pareto} \mathcal F_0(B_0)&=\Pareto T(B_0),\\ \mathcal F_h(B_0)&=\Pareto\left(T(B_0)\cup \bigcup_{e:B_0\subseteq D_e} \left\{\left(c_e+\max_o w_o,\ 1+\sum_o m_o\right): (w_o,m_o)\in\mathcal F_{h-1}(B_{e,o})\ \forall o\right\}\right). \nonumber
\bigcup_{e:B_0\subseteq D_e}Use in source · 1 passage
Search aliases: ⋃_e:B_0⊆ D_e bigcup subseteq
Use in source · Self_Improving_Intelligence_v1.0.0.tex, line 285
For horizon h, let F_h(B_0) be the nondominated attainable pairs (w,m). A pair dominates another when it is no larger in either coordinate and strictly smaller in at least one. Let T(B_0)={(c_a,1):B_0⊆ G_a}. Then F_0(B_0) · =Pareto T(B_0), F_h(B_0) · =Pareto(T(B_0)∪ ⋃_e:B_0⊆ D_e {(c_e+max_o w_o, 1+∑_o m_o): (w_o,m_o)∈ F_h-1(B_e,o) ∀ o}). nonumber An empty child frontier makes that experiment unavailable for a successful complete tree. An empty union is not a zero-cost solution.
Exact mathematical passage
\label{eq:pareto} \mathcal F_0(B_0)&=\Pareto T(B_0),\\ \mathcal F_h(B_0)&=\Pareto\left(T(B_0)\cup \bigcup_{e:B_0\subseteq D_e} \left\{\left(c_e+\max_o w_o,\ 1+\sum_o m_o\right): (w_o,m_o)\in\mathcal F_{h-1}(B_{e,o})\ \forall o\right\}\right). \nonumber
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