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theta","display":"q_\\theta","kind":"use","occurrences":[27,28]},{"id":"symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"use","occurrences":[59,61,62,69]},{"id":"symbol-b53f9683364fb5fd","tex":"\\R","aliases":"R","display":null,"kind":"use","occurrences":[56]},{"id":"symbol-2171bce1de561fa0","tex":"R^2","aliases":"R^2","display":"R^2","kind":"use","occurrences":[62,69]},{"id":"symbol-1523811833b5e4c0","tex":"R^2_{\\mathrm{future}}","aliases":"R^2_future mathrm","display":"R^2_{\\mathrm{future}}","kind":"use","occurrences":[59,61]},{"id":"symbol-311080e714276ff6","tex":"R^2_{\\mathrm{intervention}}","aliases":"R^2_intervention mathrm","display":"R^2_{\\mathrm{intervention}}","kind":"use","occurrences":[59,61]},{"id":"symbol-9d8a6ecda587180a","tex":"\\R^{80}","aliases":"R^80 R","display":null,"kind":"use","occurrences":[56]},{"id":"symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"use","occurrences":[15,16]},{"id":"symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"use","occurrences":[15,17,18,19,22]},{"id":"symbol-559f90b6098df902","tex":"\\mathrm{twin}","aliases":"twin mathrm","display":"\\mathrm{twin}","kind":"use","occurrences":[28,59]},{"id":"symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"use","occurrences":[41,42,58]},{"id":"symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"use","occurrences":[49,52,55]},{"id":"symbol-2b4af44cb17349bb","tex":"\\theta","aliases":"θ theta","display":"\\theta","kind":"use","occurrences":[27,28]},{"id":"symbol-10ce0200b450ea95","tex":"\\phi","aliases":"ϕ phi","display":"\\phi","kind":"use","occurrences":[16,18,22,23]},{"id":"symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"use","occurrences":[10,15,16,40,49,50,53]},{"id":"symbol-fcc886982cee8077","tex":"\\circ","aliases":"∘ circ","display":"\\circ","kind":"use","occurrences":[18]},{"id":"symbol-c584776d69beecf7","tex":"\\neq","aliases":"≠ neq","display":"\\neq","kind":"use","occurrences":[42,52]},{"id":"symbol-2dd02f064b00c6d5","tex":"\\geq","aliases":"≥ geq","display":"\\geq","kind":"use","occurrences":[59]}],"contexts":["Block 23 separates two questions that must not be collapsed.  The first is theoretical and finite:\n\\begin{equation}\n\\boxed{\n\\begin{gathered}\n\\text{Can a lineage construct a minimal semantic quotient from}\\\\\n\\text{future consequences, interventions, and semantic twins}\\\\\n\\text{without receiving gold intermediate semantic coordinates?}\n\\end{gathered}}\n\\label{eq:b23-central-question}\n\\end{equation}\nThe second is empirical: can the same contract be applied to a third-party latent-reasoning checkpoint?  The released package answers the first question positively inside a controlled finite fixture and supplies a source-locked adapter for the second.  The external checkpoint was not present in the release environment and was not executed.  The two evidential statuses are recorded separately.","Let $H$ be a measurable latent-state space, let $\\Ucal$ be a declared finite intervention language, and let $Y^{u}_{0:r}$ denote the protected future observations, actions, scores, or externally checkable consequences generated after intervention word $u\\in\\Ucal^{\\le r}$.  The semantic object sought in this block is not a linguistic paraphrase of every coordinate.  It is the quotient induced by the distinctions that matter for permitted future experiments.","\\begin{definition}[Causal-predictive equivalence]\nFor $h,h'\\in H$, write\n\\begin{equation}\nh\\sim_{\\Ucal,r}h'\n\\end{equation}\nwhen\n\\begin{equation}\n\\mathcal L\\!\\left(Y^{u}_{0:r}\\mid h\\right)\n=\n\\mathcal L\\!\\left(Y^{u}_{0:r}\\mid h'\\right)\n\\qquad\n\\text{for every }u\\in\\Ucal^{\\le r}.\n\\label{eq:b23-causal-equivalence}\n\\end{equation}\nThe quotient\n\\begin{equation}\nZ_{\\Ucal,r}=H/{\\sim_{\\Ucal,r}}\n\\label{eq:b23-semantic-quotient}\n\\end{equation}\nis the finite-horizon causal-predictive semantic space relative to the declared experiment language.\n\\end{definition}","The definition is intervention relative.  Enlarging $\\Ucal$ or the horizon can refine the quotient.  A coordinate omitted by every permitted experiment is not identified merely because it exists in an audit-only simulator.","\\begin{proposition}[Causal-predictive sufficiency]\nLet $q:H\\to Z_{\\Ucal,r}$ be the quotient map.  For every $u\\in\\Ucal^{\\le r}$, there exists a Markov kernel $K_u$ such that\n\\begin{equation}\n\\mathcal L\\!\\left(Y^{u}_{0:r}\\mid h\\right)=K_u(q(h)).\n\\end{equation}\nThus $q(h)$ is sufficient for every protected consequence appearing in the equivalence relation.\n\\end{proposition}","\\begin{theorem}[Minimality of the exact semantic quotient]\nSuppose $s:H\\to S$ is sufficient for every protected law in \\cref{eq:b23-causal-equivalence}.  Then there exists a map $\\phi:S\\to Z_{\\Ucal,r}$ on $s(H)$ satisfying\n\\begin{equation}\nq=\\phi\\circ s.\n\\label{eq:b23-minimal-factorization}\n\\end{equation}\nConsequently, any exact sufficient representation refines the causal-predictive quotient.\n\\end{theorem}","\\begin{proof}\nIf $s(h)=s(h')$, sufficiency gives identical protected conditional laws for every declared intervention word.  Hence $h\\sim_{\\Ucal,r}h'$ and $q(h)=q(h')$.  Define $\\phi(s(h))=q(h)$.  The preceding implication makes $\\phi$ well defined.\n\\end{proof}","\\begin{definition}[Semantic twin]\nTwo 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.\n\\end{definition}","For a candidate quotient $q_\\theta$, the empirical twin discrepancy is\n\\begin{equation}\nD_{\\mathrm{twin}}(q_\\theta)\n=\n\\mathbb E\n\\frac{\\norm{q_\\theta(x)-q_\\theta(x')}}\n{\\max\\{\\norm{q_\\theta(x)},\\eps\\}}.\n\\label{eq:b23-twin-drift}\n\\end{equation}\nA candidate is construction-valid only when it jointly predicts future consequences and intervention effects, predicts the externally scored answer, and keeps \\cref{eq:b23-twin-drift} below the declared ceiling.","\\begin{definition}[Witness-separated ontology transaction]\nAn ontology transaction consists of construction evidence $E_{\\mathrm{con}}$, a compiled quotient $q$, a pre-reveal commitment $\\sigma(q)$, and witness evidence $E_{\\mathrm{wit}}$.  It is witness separated when $E_{\\mathrm{wit}}$ is inaccessible to quotient construction and candidate selection, and the final certificate is computed only after $\\sigma(q)$ is fixed.\n\\end{definition}","n{definition}[Witness-separated ontology transaction]\nAn ontology transaction consists of construction evidence $E_{\\mathrm{con}}$, a compiled quotient $q$, a pre-reveal commitment $\\sigma(q)$, and witness evidence $E_{\\mathrm{wit}}$.  It is witness separated when $E_{\\mathrm{wit}}$ is inaccessible to quotient construction and candidate selection, and the final certificate is computed only after $\\sigma(q)$ is fixed.\n\\end{definition}","\\begin{proposition}[Circular witness limitation]\nIf the same target coordinates determine both $q$ and its reported semantic score, arbitrarily high in-sample semantic decodability does not imply out-of-construction semantic validity.\n\\end{proposition}","\\begin{definition}[Protected gauge block]\nA subspace $G\\subseteq Z$ is a protected gauge block when every permitted future and intervention law is invariant under a declared group $\\mathcal G\\leq\\Aut(G)$.  The identified object is then the orbit $[z]_\\mathcal G$, not a preferred basis inside $G$.\n\\end{definition}","\\begin{theorem}[Semantic underidentification]\nSuppose two semantic maps $q_1,q_2:H\\to Z$ induce exactly the same compiler-visible future and intervention laws, but there exists a protected witness $W$ such that\n\\begin{equation}\nW(q_1(h))\\neq W(q_2(h))\n\\end{equation}\nwith 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.\n\\end{theorem}","\\begin{definition}[Partial ontology]\nLet $P\\subseteq Z$ denote the protected semantic coordinates.  A quotient identifies a partial ontology when it is sufficient for a declared subset $P_0\\subsetneq P$, returns calibrated ignorance for $P\\setminus P_0$, and records coverage\n\\begin{equation}\n\\operatorname{cov}(q)=\\frac{\\dim P_0}{\\dim P}.\n\\end{equation}\n\\end{definition}","\\begin{theorem}[Deterministic semantic collision no-go]\nLet $y:H\\to Y$ be the representation exposed to a target-only compiler and let $z:H\\to Z$ be a protected semantic variable.  If there exist $h_1,h_2$ such that\n\\begin{equation}\ny(h_1)=y(h_2),\n\\qquad\nz(h_1)\\neq z(h_2),\n\\label{eq:b23-collision}\n\\end{equation}\nthen no deterministic map $f:Y\\to Z$ recovers $z$ on both states.\n\\end{theorem}","\\begin{proof}\nThe two equal inputs force $f(y(h_1))=f(y(h_2))$, contradicting the unequal protected targets in \\cref{eq:b23-collision}.\n\\end{proof}","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.","The finite grammar is\n\\begin{equation}\n\\Gcal_{23}\n=\n\\left\\{\n\\begin{array}{l}\n\\text{answer-only},\\ \\text{PCA},\\ \\text{predictive},\\ \\\\\n\\text{intervention-response},\\ \\text{joint minimal},\\\\\n\\text{full identity},\\ \\text{equal-rank random}\n\\end{array}\n\\right\\}.\n\\label{eq:b23-grammar}\n\\end{equation}\nFor 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,\n\\begin{equation}\n\\begin{aligned}\nR^2_{\\mathrm{future}}&\\geq0.82,","\\text{full identity},\\ \\text{equal-rank random}\n\\end{array}\n\\right\\}.\n\\label{eq:b23-grammar}\n\\end{equation}\nFor 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,\n\\begin{equation}\n\\begin{aligned}\nR^2_{\\mathrm{future}}&\\geq0.82,","\\text{full identity},\\ \\text{equal-rank random}\n\\end{array}\n\\right\\}.\n\\label{eq:b23-grammar}\n\\end{equation}\nFor 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,\n\\begin{equation}\n\\begin{aligned}\nR^2_{\\mathrm{future}}&\\geq0.82,\\\\\nR^2_{\\mathrm{intervention}}&\\geq0.90,\\\\\n\\operatorname{Acc}_{\\mathrm{answer}}&\\geq0.78,\\\\\nD_{\\mathrm{twin}}&\\leq0.22.\n\\end{aligned}\n\\label{eq:b23-visible-gates}\n\\end{equation}\nAmong valid candidates, the compiler minimizes rank and then follows the frozen method priority.  No hidden semantic score enters the choice.","In the answer-shortcut episodes, the rank-one answer projection achieved held-out answer accuracy at least $0.881250$.  Nevertheless, its best future-consequence and intervention-effect scores were only\n\\begin{equation}\nR^2_{\\mathrm{future}}\\leq0.313061,\n\\qquad\nR^2_{\\mathrm{intervention}}\\leq0.357401.\n\\end{equation}\nIt failed every final construction decision.  This separates answer extraction from semantic-state recovery.","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.","The partial regime identifies three of four protected directions, giving coverage $0.75$.  The fourth direction is not assigned a fabricated value.  Every destructive episode contains $64$ exact collision pairs; the minimum disagreement in externally protected future consequences is $1.736704$.  The collision witness never receives a hidden intermediate semantic coordinate.","After quotient commitment, a donor quotient replaces the selected packet in a new recurrent state.  The protected future and intervention consequences are recomputed through an audit-only runtime decoder rather than the probe used during construction.  The minimum selected-packet donor future $R^2$ is $0.983162$.  The maximum corresponding values for answer-only, PCA, and random controls are $-0.161100$, $-0.706230$, and $-0.703696$.","tient commitment, a donor quotient replaces the selected packet in a new recurrent state.  The protected future and intervention consequences are recomputed through an audit-only runtime decoder rather than the probe used during construction.  The minimum selected-packet donor future $R^2$ is $0.983162$.  The maximum corresponding values for answer-only, PCA, and random controls are $-0.161100$, $-0.706230$, and $-0.703696$.","a donor quotient replaces the selected packet in a new recurrent state.  The protected future and intervention consequences are recomputed through an audit-only runtime decoder rather than the probe used during construction.  The minimum selected-packet donor future $R^2$ is $0.983162$.  The maximum corresponding values for answer-only, PCA, and random controls are $-0.161100$, $-0.706230$, and $-0.703696$.","A deterministic fake runtime tests protocol plumbing but is excluded from the external empirical claim.  Across $72$ fixture tasks it yields exact replay error zero, mean zero-trace cost $0.533811$, permutation cost $0.038959$, norm-matched-noise cost $0.459498$, same-answer different-program transplant cost $0.311093$, and semantic-twin transplant cost zero."]}