Notation by work
SRIC: endogenous semantic witness
All notation · Glossary · Source coverage
78 symbol spellings in 79 mathematical passages across 5 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 78 entries are available below.
=Definition passage · 12 passages
Search aliases: =
Definition passage: Causal-predictive equivalence · block23_section_body.tex, line 28
[Causal-predictive equivalence] For h,h'∈ H, write h∼_Ucal,rh' when L (Y^u_0:rmid h) = L (Y^u_0:rmid h') for every u∈Ucal^≤ r. The quotient Z_Ucal,r=H/∼_Ucal,r is the finite-horizon causal-predictive semantic space relative to the declared experiment language.
Exact mathematical passage
\mathcal L\!\left(Y^{u}_{0:r}\mid h\right) = \mathcal L\!\left(Y^{u}_{0:r}\mid h'\right) \qquad \text{for every }u\in\Ucal^{\le r}. \label{eq:b23-causal-equivalence}
\AutDefinition passage · 1 passage
Search aliases: Aut
Definition passage: Protected gauge block · block23_section_body.tex, line 121
[Protected gauge block] A subspace G⊆ Z is a protected gauge block when every permitted future and intervention law is invariant under a declared group G≤Aut(G). The identified object is then the orbit [z]_ G, not a preferred basis inside G.
Exact mathematical passage
\mathcal G\leq\Aut(G)
\mathrm{con}Definition passage · 1 passage
Search aliases: con mathrm
Definition passage: Witness-separated ontology transaction · block23_section_body.tex, line 103
[Witness-separated ontology transaction] An ontology transaction consists of construction evidence E_con, a compiled quotient q, a pre-reveal commitment σ(q), and witness evidence E_wit. It is witness separated when E_wit is inaccessible to quotient construction and candidate selection, and the final certificate is computed only after σ(q) is fixed.
Exact mathematical passage
E_{\mathrm{con}}
\operatorname{cov}Definition passage · 1 passage
Search aliases: cov operatorname
Definition passage: Partial ontology · block23_section_body.tex, line 142
[Partial ontology] Let P⊆ Z denote the protected semantic coordinates. A quotient identifies a partial ontology when it is sufficient for a declared subset P_0subsetneq P, returns calibrated ignorance for P∖ P_0, and records coverage cov(q)=fracdim P_0dim P.
Exact mathematical passage
\operatorname{cov}(q)=\frac{\dim P_0}{\dim P}.
\dimDefinition passage · 1 passage
Search aliases: dim
Definition passage: Partial ontology · block23_section_body.tex, line 142
[Partial ontology] Let P⊆ Z denote the protected semantic coordinates. A quotient identifies a partial ontology when it is sufficient for a declared subset P_0subsetneq P, returns calibrated ignorance for P∖ P_0, and records coverage cov(q)=fracdim P_0dim P.
Exact mathematical passage
\operatorname{cov}(q)=\frac{\dim P_0}{\dim P}.
EDefinition passage · 3 passages
Search aliases: E
Definition passage: Witness-separated ontology transaction · block23_section_body.tex, line 103
[Witness-separated ontology transaction] An ontology transaction consists of construction evidence E_con, a compiled quotient q, a pre-reveal commitment σ(q), and witness evidence E_wit. It is witness separated when E_wit is inaccessible to quotient construction and candidate selection, and the final certificate is computed only after σ(q) is fixed.
Exact mathematical passage
E_{\mathrm{con}}
E_{\mathrm{con}}Definition passage · 1 passage
Search aliases: E_con mathrm
Definition passage: Witness-separated ontology transaction · block23_section_body.tex, line 103
[Witness-separated ontology transaction] An ontology transaction consists of construction evidence E_con, a compiled quotient q, a pre-reveal commitment σ(q), and witness evidence E_wit. It is witness separated when E_wit is inaccessible to quotient construction and candidate selection, and the final certificate is computed only after σ(q) is fixed.
Exact mathematical passage
E_{\mathrm{con}}
E_{\mathrm{wit}}Definition passage · 2 passages
Search aliases: E_wit mathrm
Definition passage: Witness-separated ontology transaction · block23_section_body.tex, line 103
[Witness-separated ontology transaction] An ontology transaction consists of construction evidence E_con, a compiled quotient q, a pre-reveal commitment σ(q), and witness evidence E_wit. It is witness separated when E_wit is inaccessible to quotient construction and candidate selection, and the final certificate is computed only after σ(q) is fixed.
Exact mathematical passage
E_{\mathrm{wit}}
GDefinition passage · 3 passages
Search aliases: G
Definition passage: Protected gauge block · block23_section_body.tex, line 121
[Protected gauge block] A subspace G⊆ Z is a protected gauge block when every permitted future and intervention law is invariant under a declared group G≤Aut(G). The identified object is then the orbit [z]_ G, not a preferred basis inside G.
Exact mathematical passage
G\subseteq Z
\mathcal GDefinition passage · 2 passages
Search aliases: G mathcal
Definition passage: Protected gauge block · block23_section_body.tex, line 121
[Protected gauge block] A subspace G⊆ Z is a protected gauge block when every permitted future and intervention law is invariant under a declared group G≤Aut(G). The identified object is then the orbit [z]_ G, not a preferred basis inside G.
Exact mathematical passage
\mathcal G\leq\Aut(G)
HDefinition passage · 9 passages
Search aliases: H
Definition passage: Causal-predictive equivalence · block23_section_body.tex, line 23
[Causal-predictive equivalence] For h,h'∈ H, write h∼_Ucal,rh' when L (Y^u_0:rmid h) = L (Y^u_0:rmid h') for every u∈Ucal^≤ r. The quotient Z_Ucal,r=H/∼_Ucal,r is the finite-horizon causal-predictive semantic space relative to the declared experiment language.
Exact mathematical passage
h,h'\in H
hDefinition passage · 13 passages
Search aliases: h
Definition passage: Causal-predictive equivalence · block23_section_body.tex, line 23
[Causal-predictive equivalence] For h,h'∈ H, write h∼_Ucal,rh' when L (Y^u_0:rmid h) = L (Y^u_0:rmid h') for every u∈Ucal^≤ r. The quotient Z_Ucal,r=H/∼_Ucal,r is the finite-horizon causal-predictive semantic space relative to the declared experiment language.
Exact mathematical passage
h,h'\in H
h'Definition passage · 6 passages
Search aliases: h'
Definition passage: Causal-predictive equivalence · block23_section_body.tex, line 23
[Causal-predictive equivalence] For h,h'∈ H, write h∼_Ucal,rh' when L (Y^u_0:rmid h) = L (Y^u_0:rmid h') for every u∈Ucal^≤ r. The quotient Z_Ucal,r=H/∼_Ucal,r is the finite-horizon causal-predictive semantic space relative to the declared experiment language.
Exact mathematical passage
h,h'\in H
\mathcal LDefinition passage · 2 passages
Search aliases: L mathcal
Definition passage: Causal-predictive equivalence · block23_section_body.tex, line 28
[Causal-predictive equivalence] For h,h'∈ H, write h∼_Ucal,rh' when L (Y^u_0:rmid h) = L (Y^u_0:rmid h') for every u∈Ucal^≤ r. The quotient Z_Ucal,r=H/∼_Ucal,r is the finite-horizon causal-predictive semantic space relative to the declared experiment language.
Exact mathematical passage
\mathcal L\!\left(Y^{u}_{0:r}\mid h\right) = \mathcal L\!\left(Y^{u}_{0:r}\mid h'\right) \qquad \text{for every }u\in\Ucal^{\le r}. \label{eq:b23-causal-equivalence}
\midDefinition passage · 2 passages
Search aliases: mid
Definition passage: Causal-predictive equivalence · block23_section_body.tex, line 28
[Causal-predictive equivalence] For h,h'∈ H, write h∼_Ucal,rh' when L (Y^u_0:rmid h) = L (Y^u_0:rmid h') for every u∈Ucal^≤ r. The quotient Z_Ucal,r=H/∼_Ucal,r is the finite-horizon causal-predictive semantic space relative to the declared experiment language.
Exact mathematical passage
\mathcal L\!\left(Y^{u}_{0:r}\mid h\right) = \mathcal L\!\left(Y^{u}_{0:r}\mid h'\right) \qquad \text{for every }u\in\Ucal^{\le r}. \label{eq:b23-causal-equivalence}
PDefinition passage · 4 passages
Search aliases: P
Definition passage: Partial ontology · block23_section_body.tex, line 141
[Partial ontology] Let P⊆ Z denote the protected semantic coordinates. A quotient identifies a partial ontology when it is sufficient for a declared subset P_0subsetneq P, returns calibrated ignorance for P∖ P_0, and records coverage cov(q)=fracdim P_0dim P.
Exact mathematical passage
P\subseteq Z
P_0Definition passage · 3 passages
Search aliases: P_0
Definition passage: Partial ontology · block23_section_body.tex, line 141
[Partial ontology] Let P⊆ Z denote the protected semantic coordinates. A quotient identifies a partial ontology when it is sufficient for a declared subset P_0subsetneq P, returns calibrated ignorance for P∖ P_0, and records coverage cov(q)=fracdim P_0dim P.
Exact mathematical passage
P_0\subsetneq P
qDefinition passage · 17 passages
Search aliases: q
Definition passage: Witness-separated ontology transaction · block23_section_body.tex, line 103
[Witness-separated ontology transaction] An ontology transaction consists of construction evidence E_con, a compiled quotient q, a pre-reveal commitment σ(q), and witness evidence E_wit. It is witness separated when E_wit is inaccessible to quotient construction and candidate selection, and the final certificate is computed only after σ(q) is fixed.
Exact mathematical passage
q
rDefinition passage · 10 passages
Search aliases: r
Definition passage: Causal-predictive equivalence · block23_section_body.tex, line 24
[Causal-predictive equivalence] For h,h'∈ H, write h∼_Ucal,rh' when L (Y^u_0:rmid h) = L (Y^u_0:rmid h') for every u∈Ucal^≤ r. The quotient Z_Ucal,r=H/∼_Ucal,r is the finite-horizon causal-predictive semantic space relative to the declared experiment language.
Exact mathematical passage
h\sim_{\Ucal,r}h'
\subsetneqDefinition passage · 1 passage
Search aliases: subsetneq
Definition passage: Partial ontology · block23_section_body.tex, line 141
[Partial ontology] Let P⊆ Z denote the protected semantic coordinates. A quotient identifies a partial ontology when it is sufficient for a declared subset P_0subsetneq P, returns calibrated ignorance for P∖ P_0, and records coverage cov(q)=fracdim P_0dim P.
Exact mathematical passage
P_0\subsetneq P
uDefinition passage · 6 passages
Search aliases: u
Definition passage: Causal-predictive equivalence · block23_section_body.tex, line 28
[Causal-predictive equivalence] For h,h'∈ H, write h∼_Ucal,rh' when L (Y^u_0:rmid h) = L (Y^u_0:rmid h') for every u∈Ucal^≤ r. The quotient Z_Ucal,r=H/∼_Ucal,r is the finite-horizon causal-predictive semantic space relative to the declared experiment language.
Exact mathematical passage
\mathcal L\!\left(Y^{u}_{0:r}\mid h\right) = \mathcal L\!\left(Y^{u}_{0:r}\mid h'\right) \qquad \text{for every }u\in\Ucal^{\le r}. \label{eq:b23-causal-equivalence}
\UcalDefinition passage · 11 passages
Search aliases: Ucal
Definition passage: Causal-predictive equivalence · block23_section_body.tex, line 24
[Causal-predictive equivalence] For h,h'∈ H, write h∼_Ucal,rh' when L (Y^u_0:rmid h) = L (Y^u_0:rmid h') for every u∈Ucal^≤ r. The quotient Z_Ucal,r=H/∼_Ucal,r is the finite-horizon causal-predictive semantic space relative to the declared experiment language.
Exact mathematical passage
h\sim_{\Ucal,r}h'
\Ucal^{\le r}Definition passage · 3 passages
Search aliases: Ucal^≤ r Ucal le
Definition passage: Causal-predictive equivalence · block23_section_body.tex, line 28
[Causal-predictive equivalence] For h,h'∈ H, write h∼_Ucal,rh' when L (Y^u_0:rmid h) = L (Y^u_0:rmid h') for every u∈Ucal^≤ r. The quotient Z_Ucal,r=H/∼_Ucal,r is the finite-horizon causal-predictive semantic space relative to the declared experiment language.
Exact mathematical passage
\mathcal L\!\left(Y^{u}_{0:r}\mid h\right) = \mathcal L\!\left(Y^{u}_{0:r}\mid h'\right) \qquad \text{for every }u\in\Ucal^{\le r}. \label{eq:b23-causal-equivalence}
\mathrm{wit}Definition passage · 2 passages
Search aliases: wit mathrm
Definition passage: Witness-separated ontology transaction · block23_section_body.tex, line 103
[Witness-separated ontology transaction] An ontology transaction consists of construction evidence E_con, a compiled quotient q, a pre-reveal commitment σ(q), and witness evidence E_wit. It is witness separated when E_wit is inaccessible to quotient construction and candidate selection, and the final certificate is computed only after σ(q) is fixed.
Exact mathematical passage
E_{\mathrm{wit}}
xDefinition passage · 3 passages
Search aliases: x
Definition passage: Semantic twin · block23_section_body.tex, line 78
[Semantic twin] Two observed trajectories x and x' are semantic twins relative to Ucal when they arise from the same protected task and have the same protected future and intervention laws, while nuisance coordinates, rendering variables, or architecture-specific gauges may differ.
Exact mathematical passage
x
x'Definition passage · 2 passages
Search aliases: x'
Definition passage: Semantic twin · block23_section_body.tex, line 78
[Semantic twin] Two observed trajectories x and x' are semantic twins relative to Ucal when they arise from the same protected task and have the same protected future and intervention laws, while nuisance coordinates, rendering variables, or architecture-specific gauges may differ.
Exact mathematical passage
x'
YDefinition passage · 5 passages
Search aliases: Y
Definition passage: Causal-predictive equivalence · block23_section_body.tex, line 28
[Causal-predictive equivalence] For h,h'∈ H, write h∼_Ucal,rh' when L (Y^u_0:rmid h) = L (Y^u_0:rmid h') for every u∈Ucal^≤ r. The quotient Z_Ucal,r=H/∼_Ucal,r is the finite-horizon causal-predictive semantic space relative to the declared experiment language.
Exact mathematical passage
\mathcal L\!\left(Y^{u}_{0:r}\mid h\right) = \mathcal L\!\left(Y^{u}_{0:r}\mid h'\right) \qquad \text{for every }u\in\Ucal^{\le r}. \label{eq:b23-causal-equivalence}
Y^{u}_{0:r}Definition passage · 3 passages
Search aliases: Y^u_0:r
Definition passage: Causal-predictive equivalence · block23_section_body.tex, line 28
[Causal-predictive equivalence] For h,h'∈ H, write h∼_Ucal,rh' when L (Y^u_0:rmid h) = L (Y^u_0:rmid h') for every u∈Ucal^≤ r. The quotient Z_Ucal,r=H/∼_Ucal,r is the finite-horizon causal-predictive semantic space relative to the declared experiment language.
Exact mathematical passage
\mathcal L\!\left(Y^{u}_{0:r}\mid h\right) = \mathcal L\!\left(Y^{u}_{0:r}\mid h'\right) \qquad \text{for every }u\in\Ucal^{\le r}. \label{eq:b23-causal-equivalence}
ZDefinition passage · 8 passages
Search aliases: Z
Definition passage: Causal-predictive equivalence · block23_section_body.tex, line 37
[Causal-predictive equivalence] For h,h'∈ H, write h∼_Ucal,rh' when L (Y^u_0:rmid h) = L (Y^u_0:rmid h') for every u∈Ucal^≤ r. The quotient Z_Ucal,r=H/∼_Ucal,r is the finite-horizon causal-predictive semantic space relative to the declared experiment language.
Exact mathematical passage
Z_{\Ucal,r}=H/{\sim_{\Ucal,r}} \label{eq:b23-semantic-quotient}
zDefinition passage · 4 passages
Search aliases: z
Definition passage: Protected gauge block · block23_section_body.tex, line 121
[Protected gauge block] A subspace G⊆ Z is a protected gauge block when every permitted future and intervention law is invariant under a declared group G≤Aut(G). The identified object is then the orbit [z]_ G, not a preferred basis inside G.
Exact mathematical passage
[z]_\mathcal G
Z_{\Ucal,r}Definition passage · 3 passages
Search aliases: Z_Ucal,r Ucal
Definition passage: Causal-predictive equivalence · block23_section_body.tex, line 37
[Causal-predictive equivalence] For h,h'∈ H, write h∼_Ucal,rh' when L (Y^u_0:rmid h) = L (Y^u_0:rmid h') for every u∈Ucal^≤ r. The quotient Z_Ucal,r=H/∼_Ucal,r is the finite-horizon causal-predictive semantic space relative to the declared experiment language.
Exact mathematical passage
Z_{\Ucal,r}=H/{\sim_{\Ucal,r}} \label{eq:b23-semantic-quotient}
\sigmaDefinition passage · 2 passages
Search aliases: σ sigma
Definition passage: Witness-separated ontology transaction · block23_section_body.tex, line 103
[Witness-separated ontology transaction] An ontology transaction consists of construction evidence E_con, a compiled quotient q, a pre-reveal commitment σ(q), and witness evidence E_wit. It is witness separated when E_wit is inaccessible to quotient construction and candidate selection, and the final certificate is computed only after σ(q) is fixed.
Exact mathematical passage
\sigma(q)
\inDefinition passage · 4 passages
Search aliases: ∈ in
Definition passage: Causal-predictive equivalence · block23_section_body.tex, line 23
[Causal-predictive equivalence] For h,h'∈ H, write h∼_Ucal,rh' when L (Y^u_0:rmid h) = L (Y^u_0:rmid h') for every u∈Ucal^≤ r. The quotient Z_Ucal,r=H/∼_Ucal,r is the finite-horizon causal-predictive semantic space relative to the declared experiment language.
Exact mathematical passage
h,h'\in H
\setminusDefinition passage · 1 passage
Search aliases: ∖ setminus
Definition passage: Partial ontology · block23_section_body.tex, line 141
[Partial ontology] Let P⊆ Z denote the protected semantic coordinates. A quotient identifies a partial ontology when it is sufficient for a declared subset P_0subsetneq P, returns calibrated ignorance for P∖ P_0, and records coverage cov(q)=fracdim P_0dim P.
Exact mathematical passage
P\setminus P_0
\simDefinition passage · 3 passages
Search aliases: ∼ sim
Definition passage: Causal-predictive equivalence · block23_section_body.tex, line 24
[Causal-predictive equivalence] For h,h'∈ H, write h∼_Ucal,rh' when L (Y^u_0:rmid h) = L (Y^u_0:rmid h') for every u∈Ucal^≤ r. The quotient Z_Ucal,r=H/∼_Ucal,r is the finite-horizon causal-predictive semantic space relative to the declared experiment language.
Exact mathematical passage
h\sim_{\Ucal,r}h'
\sim_{\Ucal,r}Definition passage · 3 passages
Search aliases: ∼_Ucal,r sim Ucal
Definition passage: Causal-predictive equivalence · block23_section_body.tex, line 24
[Causal-predictive equivalence] For h,h'∈ H, write h∼_Ucal,rh' when L (Y^u_0:rmid h) = L (Y^u_0:rmid h') for every u∈Ucal^≤ r. The quotient Z_Ucal,r=H/∼_Ucal,r is the finite-horizon causal-predictive semantic space relative to the declared experiment language.
Exact mathematical passage
h\sim_{\Ucal,r}h'
\leDefinition passage · 3 passages
Search aliases: ≤ le
Definition passage: Causal-predictive equivalence · block23_section_body.tex, line 28
[Causal-predictive equivalence] For h,h'∈ H, write h∼_Ucal,rh' when L (Y^u_0:rmid h) = L (Y^u_0:rmid h') for every u∈Ucal^≤ r. The quotient Z_Ucal,r=H/∼_Ucal,r is the finite-horizon causal-predictive semantic space relative to the declared experiment language.
Exact mathematical passage
\mathcal L\!\left(Y^{u}_{0:r}\mid h\right) = \mathcal L\!\left(Y^{u}_{0:r}\mid h'\right) \qquad \text{for every }u\in\Ucal^{\le r}. \label{eq:b23-causal-equivalence}
\leqDefinition passage · 3 passages
Search aliases: ≤ leq
Definition passage: Protected gauge block · block23_section_body.tex, line 121
[Protected gauge block] A subspace G⊆ Z is a protected gauge block when every permitted future and intervention law is invariant under a declared group G≤Aut(G). The identified object is then the orbit [z]_ G, not a preferred basis inside G.
Exact mathematical passage
\mathcal G\leq\Aut(G)
\subseteqDefinition passage · 2 passages
Search aliases: ⊆ subseteq
Definition passage: Protected gauge block · block23_section_body.tex, line 121
[Protected gauge block] A subspace G⊆ Z is a protected gauge block when every permitted future and intervention law is invariant under a declared group G≤Aut(G). The identified object is then the orbit [z]_ G, not a preferred basis inside G.
Exact mathematical passage
G\subseteq Z
\operatorname{Acc}Use in source · 1 passage
Search aliases: Acc operatorname
Use in source · block23_section_body.tex, line 186
full identity, equal-rank random }. For each nontrivial constructor, ranks one through six are considered. The full identity is an oversized control. A projection W is valid only when, on both recurrent systems, R^2_future · ≥0.82, R^2_intervention · ≥0.90, Acc_answer · ≥0.78, D_twin · ≤0.22. Among valid candidates, the compiler minimizes rank and then follows the frozen method priority. No hidden semantic score enters the choice.
Exact mathematical passage
\begin{aligned} R^2_{\mathrm{future}}&\geq0.82,\\ R^2_{\mathrm{intervention}}&\geq0.90,\\ \operatorname{Acc}_{\mathrm{answer}}&\geq0.78,\\ D_{\mathrm{twin}}&\leq0.22. \end{aligned} \label{eq:b23-visible-gates}
\operatorname{Acc}_{\mathrm{answer}}Use in source · 1 passage
Search aliases: Acc_answer operatorname mathrm
Use in source · block23_section_body.tex, line 186
full identity, equal-rank random }. For each nontrivial constructor, ranks one through six are considered. The full identity is an oversized control. A projection W is valid only when, on both recurrent systems, R^2_future · ≥0.82, R^2_intervention · ≥0.90, Acc_answer · ≥0.78, D_twin · ≤0.22. Among valid candidates, the compiler minimizes rank and then follows the frozen method priority. No hidden semantic score enters the choice.
Exact mathematical passage
\begin{aligned} R^2_{\mathrm{future}}&\geq0.82,\\ R^2_{\mathrm{intervention}}&\geq0.90,\\ \operatorname{Acc}_{\mathrm{answer}}&\geq0.78,\\ D_{\mathrm{twin}}&\leq0.22. \end{aligned} \label{eq:b23-visible-gates}
\mathrm{answer}Use in source · 1 passage
Search aliases: answer mathrm
Use in source · block23_section_body.tex, line 186
full identity, equal-rank random }. For each nontrivial constructor, ranks one through six are considered. The full identity is an oversized control. A projection W is valid only when, on both recurrent systems, R^2_future · ≥0.82, R^2_intervention · ≥0.90, Acc_answer · ≥0.78, D_twin · ≤0.22. Among valid candidates, the compiler minimizes rank and then follows the frozen method priority. No hidden semantic score enters the choice.
Exact mathematical passage
\begin{aligned} R^2_{\mathrm{future}}&\geq0.82,\\ R^2_{\mathrm{intervention}}&\geq0.90,\\ \operatorname{Acc}_{\mathrm{answer}}&\geq0.78,\\ D_{\mathrm{twin}}&\leq0.22. \end{aligned} \label{eq:b23-visible-gates}
DUse in source · 2 passages
Search aliases: D
Use in source · block23_section_body.tex, line 82
For a candidate quotient q_θ, the empirical twin discrepancy is D_twin(q_θ) = E fracnormq_θ(x)-q_θ(x') max{normq_θ(x),eps}. A candidate is construction-valid only when it jointly predicts future consequences and intervention effects, predicts the externally scored answer, and keeps [source reference] below the declared ceiling.
Exact mathematical passage
D_{\mathrm{twin}}(q_\theta) = \mathbb E \frac{\norm{q_\theta(x)-q_\theta(x')}} {\max\{\norm{q_\theta(x)},\eps\}}. \label{eq:b23-twin-drift}
D_{\mathrm{twin}}Use in source · 2 passages
Search aliases: D_twin mathrm
Use in source · block23_section_body.tex, line 82
For a candidate quotient q_θ, the empirical twin discrepancy is D_twin(q_θ) = E fracnormq_θ(x)-q_θ(x') max{normq_θ(x),eps}. A candidate is construction-valid only when it jointly predicts future consequences and intervention effects, predicts the externally scored answer, and keeps [source reference] below the declared ceiling.
Exact mathematical passage
D_{\mathrm{twin}}(q_\theta) = \mathbb E \frac{\norm{q_\theta(x)-q_\theta(x')}} {\max\{\norm{q_\theta(x)},\eps\}}. \label{eq:b23-twin-drift}
\mathbb EUse in source · 1 passage
Search aliases: E mathbb
Use in source · block23_section_body.tex, line 82
For a candidate quotient q_θ, the empirical twin discrepancy is D_twin(q_θ) = E fracnormq_θ(x)-q_θ(x') max{normq_θ(x),eps}. A candidate is construction-valid only when it jointly predicts future consequences and intervention effects, predicts the externally scored answer, and keeps [source reference] below the declared ceiling.
Exact mathematical passage
D_{\mathrm{twin}}(q_\theta) = \mathbb E \frac{\norm{q_\theta(x)-q_\theta(x')}} {\max\{\norm{q_\theta(x)},\eps\}}. \label{eq:b23-twin-drift}
\epsUse in source · 1 passage
Search aliases: eps
Use in source · block23_section_body.tex, line 82
For a candidate quotient q_θ, the empirical twin discrepancy is D_twin(q_θ) = E fracnormq_θ(x)-q_θ(x') max{normq_θ(x),eps}. A candidate is construction-valid only when it jointly predicts future consequences and intervention effects, predicts the externally scored answer, and keeps [source reference] below the declared ceiling.
Exact mathematical passage
D_{\mathrm{twin}}(q_\theta) = \mathbb E \frac{\norm{q_\theta(x)-q_\theta(x')}} {\max\{\norm{q_\theta(x)},\eps\}}. \label{eq:b23-twin-drift}
fUse in source · 2 passages
Search aliases: f
Use in source · block23_section_body.tex, line 159
[Deterministic semantic collision no-go] Let y:H→ Y be the representation exposed to a target-only compiler and let z:H→ Z be a protected semantic variable. If there exist h_1,h_2 such that y(h_1)=y(h_2), z(h_1)≠ z(h_2), then no deterministic map f:Y→ Z recovers z on both states.
Exact mathematical passage
f:Y\to Z
\mathrm{future}Use in source · 2 passages
Search aliases: future mathrm
Use in source · block23_section_body.tex, line 186
full identity, equal-rank random }. For each nontrivial constructor, ranks one through six are considered. The full identity is an oversized control. A projection W is valid only when, on both recurrent systems, R^2_future · ≥0.82, R^2_intervention · ≥0.90, Acc_answer · ≥0.78, D_twin · ≤0.22. Among valid candidates, the compiler minimizes rank and then follows the frozen method priority. No hidden semantic score enters the choice.
Exact mathematical passage
\begin{aligned} R^2_{\mathrm{future}}&\geq0.82,\\ R^2_{\mathrm{intervention}}&\geq0.90,\\ \operatorname{Acc}_{\mathrm{answer}}&\geq0.78,\\ D_{\mathrm{twin}}&\leq0.22. \end{aligned} \label{eq:b23-visible-gates}
\GcalUse in source · 1 passage
Search aliases: Gcal
Use in source · block23_section_body.tex, line 173
The finite grammar is Gcal_23 = { l answer-only, PCA, predictive, intervention-response, joint minimal, full identity, equal-rank random }. For each nontrivial constructor, ranks one through six are considered. The full identity is an oversized control. A projection W is valid only when, on both recurrent systems, R^2_future · ≥0.82,
Exact mathematical passage
\Gcal_{23} = \left\{ \begin{array}{l} \text{answer-only},\ \text{PCA},\ \text{predictive},\ \\ \text{intervention-response},\ \text{joint minimal},\\ \text{full identity},\ \text{equal-rank random} \end{array} \right\}. \label{eq:b23-grammar}
\Gcal_{23}Use in source · 1 passage
Search aliases: Gcal_23 Gcal
Use in source · block23_section_body.tex, line 173
The finite grammar is Gcal_23 = { l answer-only, PCA, predictive, intervention-response, joint minimal, full identity, equal-rank random }. For each nontrivial constructor, ranks one through six are considered. The full identity is an oversized control. A projection W is valid only when, on both recurrent systems, R^2_future · ≥0.82,
Exact mathematical passage
\Gcal_{23} = \left\{ \begin{array}{l} \text{answer-only},\ \text{PCA},\ \text{predictive},\ \\ \text{intervention-response},\ \text{joint minimal},\\ \text{full identity},\ \text{equal-rank random} \end{array} \right\}. \label{eq:b23-grammar}
h_1Use in source · 3 passages
Search aliases: h_1
Use in source · block23_section_body.tex, line 152
[Deterministic semantic collision no-go] Let y:H→ Y be the representation exposed to a target-only compiler and let z:H→ Z be a protected semantic variable. If there exist h_1,h_2 such that y(h_1)=y(h_2), z(h_1)≠ z(h_2), then no deterministic map f:Y→ Z recovers z on both states.
Exact mathematical passage
h_1,h_2
h_2Use in source · 3 passages
Search aliases: h_2
Use in source · block23_section_body.tex, line 152
[Deterministic semantic collision no-go] Let y:H→ Y be the representation exposed to a target-only compiler and let z:H→ Z be a protected semantic variable. If there exist h_1,h_2 such that y(h_1)=y(h_2), z(h_1)≠ z(h_2), then no deterministic map f:Y→ Z recovers z on both states.
Exact mathematical passage
h_1,h_2
\mathrm{intervention}Use in source · 2 passages
Search aliases: intervention mathrm
Use in source · block23_section_body.tex, line 186
full identity, equal-rank random }. For each nontrivial constructor, ranks one through six are considered. The full identity is an oversized control. A projection W is valid only when, on both recurrent systems, R^2_future · ≥0.82, R^2_intervention · ≥0.90, Acc_answer · ≥0.78, D_twin · ≤0.22. Among valid candidates, the compiler minimizes rank and then follows the frozen method priority. No hidden semantic score enters the choice.
Exact mathematical passage
\begin{aligned} R^2_{\mathrm{future}}&\geq0.82,\\ R^2_{\mathrm{intervention}}&\geq0.90,\\ \operatorname{Acc}_{\mathrm{answer}}&\geq0.78,\\ D_{\mathrm{twin}}&\leq0.22. \end{aligned} \label{eq:b23-visible-gates}
KUse in source · 2 passages
Search aliases: K
Use in source · block23_section_body.tex, line 47
[Causal-predictive sufficiency] Let q:H→ Z_Ucal,r be the quotient map. For every u∈Ucal^≤ r, there exists a Markov kernel K_u such that L (Y^u_0:rmid h)=K_u(q(h)). Thus q(h) is sufficient for every protected consequence appearing in the equivalence relation.
Exact mathematical passage
K_u
K_uUse in source · 2 passages
Search aliases: K_u
Use in source · block23_section_body.tex, line 47
[Causal-predictive sufficiency] Let q:H→ Z_Ucal,r be the quotient map. For every u∈Ucal^≤ r, there exists a Markov kernel K_u such that L (Y^u_0:rmid h)=K_u(q(h)). Thus q(h) is sufficient for every protected consequence appearing in the equivalence relation.
Exact mathematical passage
K_u
lUse in source · 1 passage
Search aliases: l
Use in source · block23_section_body.tex, line 173
The finite grammar is Gcal_23 = { l answer-only, PCA, predictive, intervention-response, joint minimal, full identity, equal-rank random }. For each nontrivial constructor, ranks one through six are considered. The full identity is an oversized control. A projection W is valid only when, on both recurrent systems, R^2_future · ≥0.82,
Exact mathematical passage
\Gcal_{23} = \left\{ \begin{array}{l} \text{answer-only},\ \text{PCA},\ \text{predictive},\ \\ \text{intervention-response},\ \text{joint minimal},\\ \text{full identity},\ \text{equal-rank random} \end{array} \right\}. \label{eq:b23-grammar}
\maxUse in source · 1 passage
Search aliases: max
Use in source · block23_section_body.tex, line 82
For a candidate quotient q_θ, the empirical twin discrepancy is D_twin(q_θ) = E fracnormq_θ(x)-q_θ(x') max{normq_θ(x),eps}. A candidate is construction-valid only when it jointly predicts future consequences and intervention effects, predicts the externally scored answer, and keeps [source reference] below the declared ceiling.
Exact mathematical passage
D_{\mathrm{twin}}(q_\theta) = \mathbb E \frac{\norm{q_\theta(x)-q_\theta(x')}} {\max\{\norm{q_\theta(x)},\eps\}}. \label{eq:b23-twin-drift}
\normUse in source · 1 passage
Search aliases: norm
Use in source · block23_section_body.tex, line 82
For a candidate quotient q_θ, the empirical twin discrepancy is D_twin(q_θ) = E fracnormq_θ(x)-q_θ(x') max{normq_θ(x),eps}. A candidate is construction-valid only when it jointly predicts future consequences and intervention effects, predicts the externally scored answer, and keeps [source reference] below the declared ceiling.
Exact mathematical passage
D_{\mathrm{twin}}(q_\theta) = \mathbb E \frac{\norm{q_\theta(x)-q_\theta(x')}} {\max\{\norm{q_\theta(x)},\eps\}}. \label{eq:b23-twin-drift}
q_1Use in source · 3 passages
Search aliases: q_1
Use in source · block23_section_body.tex, line 125
[Semantic underidentification] Suppose two semantic maps q_1,q_2:H→ Z induce exactly the same compiler-visible future and intervention laws, but there exists a protected witness W such that W(q_1(h))≠ W(q_2(h)) with positive probability. No compiler measurable with respect to the visible evidence can validly choose between q_1 and q_2 with probability exceeding the larger prior probability of the two alternatives.
Exact mathematical passage
q_1,q_2:H\to Z
q_2Use in source · 3 passages
Search aliases: q_2
Use in source · block23_section_body.tex, line 125
[Semantic underidentification] Suppose two semantic maps q_1,q_2:H→ Z induce exactly the same compiler-visible future and intervention laws, but there exists a protected witness W such that W(q_1(h))≠ W(q_2(h)) with positive probability. No compiler measurable with respect to the visible evidence can validly choose between q_1 and q_2 with probability exceeding the larger prior probability of the two alternatives.
Exact mathematical passage
q_1,q_2:H\to Z
q_\thetaUse in source · 2 passages
Search aliases: q_θ theta
Use in source · block23_section_body.tex, line 81
For a candidate quotient q_θ, the empirical twin discrepancy is D_twin(q_θ) = E fracnormq_θ(x)-q_θ(x') max{normq_θ(x),eps}. A candidate is construction-valid only when it jointly predicts future consequences and intervention effects, predicts the externally scored answer, and keeps [source reference] below the declared ceiling.
Exact mathematical passage
q_\theta
RUse in source · 4 passages
Search aliases: R
Use in source · block23_section_body.tex, line 186
full identity, equal-rank random }. For each nontrivial constructor, ranks one through six are considered. The full identity is an oversized control. A projection W is valid only when, on both recurrent systems, R^2_future · ≥0.82, R^2_intervention · ≥0.90, Acc_answer · ≥0.78, D_twin · ≤0.22. Among valid candidates, the compiler minimizes rank and then follows the frozen method priority. No hidden semantic score enters the choice.
Exact mathematical passage
\begin{aligned} R^2_{\mathrm{future}}&\geq0.82,\\ R^2_{\mathrm{intervention}}&\geq0.90,\\ \operatorname{Acc}_{\mathrm{answer}}&\geq0.78,\\ D_{\mathrm{twin}}&\leq0.22. \end{aligned} \label{eq:b23-visible-gates}
\RUse in source · 1 passage
Search aliases: R
Use in source · block23_section_body.tex, line 170
Each episode contains two independently gauged recurrent systems. A trajectory has five latent steps, four task-relevant semantic coordinates, and twelve nuisance coordinates per step. The observed flattened trajectory therefore lies in R^80. The semantic dynamics, gauges, nuisance processes, and hidden audit coordinates are not passed to the quotient compiler.
Exact mathematical passage
\R^{80}
R^2Use in source · 2 passages
Search aliases: R^2
Use in source · block23_section_body.tex, line 286
Across uniquely identified regimes, hidden semantic reconstruction is numerically exact in the released fixture. In the gauge regime, minimum gauge-invariant R^2 is 0.995522, while the minimum recovered semantic-subspace overlap is 0.991563. In the semantic-ambiguity regime, the minimum protected disagreement between two observationally equivalent orientations is 0.781682. Although either orientation is post-hoc decodable, the correct disposition remains underidentified.
Exact mathematical passage
R^2
R^2_{\mathrm{future}}Use in source · 2 passages
Search aliases: R^2_future mathrm
Use in source · block23_section_body.tex, line 186
full identity, equal-rank random }. For each nontrivial constructor, ranks one through six are considered. The full identity is an oversized control. A projection W is valid only when, on both recurrent systems, R^2_future · ≥0.82, R^2_intervention · ≥0.90, Acc_answer · ≥0.78, D_twin · ≤0.22. Among valid candidates, the compiler minimizes rank and then follows the frozen method priority. No hidden semantic score enters the choice.
Exact mathematical passage
\begin{aligned} R^2_{\mathrm{future}}&\geq0.82,\\ R^2_{\mathrm{intervention}}&\geq0.90,\\ \operatorname{Acc}_{\mathrm{answer}}&\geq0.78,\\ D_{\mathrm{twin}}&\leq0.22. \end{aligned} \label{eq:b23-visible-gates}
R^2_{\mathrm{intervention}}Use in source · 2 passages
Search aliases: R^2_intervention mathrm
Use in source · block23_section_body.tex, line 186
full identity, equal-rank random }. For each nontrivial constructor, ranks one through six are considered. The full identity is an oversized control. A projection W is valid only when, on both recurrent systems, R^2_future · ≥0.82, R^2_intervention · ≥0.90, Acc_answer · ≥0.78, D_twin · ≤0.22. Among valid candidates, the compiler minimizes rank and then follows the frozen method priority. No hidden semantic score enters the choice.
Exact mathematical passage
\begin{aligned} R^2_{\mathrm{future}}&\geq0.82,\\ R^2_{\mathrm{intervention}}&\geq0.90,\\ \operatorname{Acc}_{\mathrm{answer}}&\geq0.78,\\ D_{\mathrm{twin}}&\leq0.22. \end{aligned} \label{eq:b23-visible-gates}
\R^{80}Use in source · 1 passage
Search aliases: R^80 R
Use in source · block23_section_body.tex, line 170
Each episode contains two independently gauged recurrent systems. A trajectory has five latent steps, four task-relevant semantic coordinates, and twelve nuisance coordinates per step. The observed flattened trajectory therefore lies in R^80. The semantic dynamics, gauges, nuisance processes, and hidden audit coordinates are not passed to the quotient compiler.
Exact mathematical passage
\R^{80}
SUse in source · 2 passages
Search aliases: S
Use in source · block23_section_body.tex, line 59
[Minimality of the exact semantic quotient] Suppose s:H→ S is sufficient for every protected law in [source reference]. Then there exists a map ϕ:S→ Z_Ucal,r on s(H) satisfying q=ϕ∘ s. Consequently, any exact sufficient representation refines the causal-predictive quotient.
Exact mathematical passage
s:H\to S
sUse in source · 5 passages
Search aliases: s
Use in source · block23_section_body.tex, line 59
[Minimality of the exact semantic quotient] Suppose s:H→ S is sufficient for every protected law in [source reference]. Then there exists a map ϕ:S→ Z_Ucal,r on s(H) satisfying q=ϕ∘ s. Consequently, any exact sufficient representation refines the causal-predictive quotient.
Exact mathematical passage
s:H\to S
\mathrm{twin}Use in source · 2 passages
Search aliases: twin mathrm
Use in source · block23_section_body.tex, line 82
For a candidate quotient q_θ, the empirical twin discrepancy is D_twin(q_θ) = E fracnormq_θ(x)-q_θ(x') max{normq_θ(x),eps}. A candidate is construction-valid only when it jointly predicts future consequences and intervention effects, predicts the externally scored answer, and keeps [source reference] below the declared ceiling.
Exact mathematical passage
D_{\mathrm{twin}}(q_\theta) = \mathbb E \frac{\norm{q_\theta(x)-q_\theta(x')}} {\max\{\norm{q_\theta(x)},\eps\}}. \label{eq:b23-twin-drift}
WUse in source · 3 passages
Search aliases: W
Use in source · block23_section_body.tex, line 125
[Semantic underidentification] Suppose two semantic maps q_1,q_2:H→ Z induce exactly the same compiler-visible future and intervention laws, but there exists a protected witness W such that W(q_1(h))≠ W(q_2(h)) with positive probability. No compiler measurable with respect to the visible evidence can validly choose between q_1 and q_2 with probability exceeding the larger prior probability of the two alternatives.
Exact mathematical passage
W
yUse in source · 3 passages
Search aliases: y
Use in source · block23_section_body.tex, line 152
[Deterministic semantic collision no-go] Let y:H→ Y be the representation exposed to a target-only compiler and let z:H→ Z be a protected semantic variable. If there exist h_1,h_2 such that y(h_1)=y(h_2), z(h_1)≠ z(h_2), then no deterministic map f:Y→ Z recovers z on both states.
Exact mathematical passage
y:H\to Y
\thetaUse in source · 2 passages
Search aliases: θ theta
Use in source · block23_section_body.tex, line 81
For a candidate quotient q_θ, the empirical twin discrepancy is D_twin(q_θ) = E fracnormq_θ(x)-q_θ(x') max{normq_θ(x),eps}. A candidate is construction-valid only when it jointly predicts future consequences and intervention effects, predicts the externally scored answer, and keeps [source reference] below the declared ceiling.
Exact mathematical passage
q_\theta
\phiUse in source · 4 passages
Search aliases: ϕ phi
Use in source · block23_section_body.tex, line 59
[Minimality of the exact semantic quotient] Suppose s:H→ S is sufficient for every protected law in [source reference]. Then there exists a map ϕ:S→ Z_Ucal,r on s(H) satisfying q=ϕ∘ s. Consequently, any exact sufficient representation refines the causal-predictive quotient.
Exact mathematical passage
\phi:S\to Z_{\Ucal,r}
\toUse in source · 7 passages
Search aliases: → to
Use in source · block23_section_body.tex, line 47
[Causal-predictive sufficiency] Let q:H→ Z_Ucal,r be the quotient map. For every u∈Ucal^≤ r, there exists a Markov kernel K_u such that L (Y^u_0:rmid h)=K_u(q(h)). Thus q(h) is sufficient for every protected consequence appearing in the equivalence relation.
Exact mathematical passage
q:H\to Z_{\Ucal,r}
\circUse in source · 1 passage
Search aliases: ∘ circ
Use in source · block23_section_body.tex, line 60
[Minimality of the exact semantic quotient] Suppose s:H→ S is sufficient for every protected law in [source reference]. Then there exists a map ϕ:S→ Z_Ucal,r on s(H) satisfying q=ϕ∘ s. Consequently, any exact sufficient representation refines the causal-predictive quotient.
Exact mathematical passage
q=\phi\circ s. \label{eq:b23-minimal-factorization}
\neqUse in source · 2 passages
Search aliases: ≠ neq
Use in source · block23_section_body.tex, line 126
[Semantic underidentification] Suppose two semantic maps q_1,q_2:H→ Z induce exactly the same compiler-visible future and intervention laws, but there exists a protected witness W such that W(q_1(h))≠ W(q_2(h)) with positive probability. No compiler measurable with respect to the visible evidence can validly choose between q_1 and q_2 with probability exceeding the larger prior probability of the two alternatives.
Exact mathematical passage
W(q_1(h))\neq W(q_2(h))
\geqUse in source · 1 passage
Search aliases: ≥ geq
Use in source · block23_section_body.tex, line 186
full identity, equal-rank random }. For each nontrivial constructor, ranks one through six are considered. The full identity is an oversized control. A projection W is valid only when, on both recurrent systems, R^2_future · ≥0.82, R^2_intervention · ≥0.90, Acc_answer · ≥0.78, D_twin · ≤0.22. Among valid candidates, the compiler minimizes rank and then follows the frozen method priority. No hidden semantic score enters the choice.
Exact mathematical passage
\begin{aligned} R^2_{\mathrm{future}}&\geq0.82,\\ R^2_{\mathrm{intervention}}&\geq0.90,\\ \operatorname{Acc}_{\mathrm{answer}}&\geq0.78,\\ D_{\mathrm{twin}}&\leq0.22. \end{aligned} \label{eq:b23-visible-gates}
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