Stratified Recursive Improvement Closure

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190 \begin{document}
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193 \begin{titlepage}
194 \centering
195 \vspace*{1.2in}
196 {\Huge\bfseries\color{KMBlue} Stratified Recursive Improvement Closure\par}
197 \vspace{0.28in}
198 {\Large Obstruction, Descent, Transport, and Evidence-Preserving Lineages\par}
199 \vspace{0.55in}
200 {\large A formal mathematical theory of proof-carrying recursive self-repair\par}
201 \vfill
202 {\Large Komorebi P.\par}
203 \vspace{0.16in}
204 {\large Formal theory draft, version 2.4 (Block 23 endogenous semantic-witness growth)\par}
205 \vspace{0.08in}
206 {\large September 3, 2026\par}
207 \vfill
208 \begin{minipage}{0.82\textwidth}
209 \small
210 \textbf{Status.} This manuscript is a self-contained formalization of a proposed architecture for bounded recursive self-improvement. Results labeled Theorem, Proposition, Lemma, or Corollary are proved from their stated assumptions. Results labeled Conditional Theorem depend on strong architectural or statistical hypotheses. Results labeled Conjecture remain open. No claim of unrestricted, indefinitely accelerating, safe, or conscious self-improvement is made.
211 \end{minipage}
212 \end{titlepage}
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214 \pagenumbering{roman}
215 
216 \begin{abstract}
217 We develop \emph{Stratified Recursive Improvement Closure} (SRIC), a mathematical framework in which recursive self-improvement is represented not as unrestricted in-place rewriting of one model, but as growth of an authorized repair calculus inside an evidence-preserving branching lineage. The primitive object is a resolved lineage state containing an active stratified agent, a proof-carrying repair operad, an append-only evidence ledger, an evaluator atlas, a causal lineage graph, and a witness-and-rollback structure. The theory separates task improvement, improvement-process improvement, and epistemic improvement.
218 
219 Blocks 1 through 4 establish protected evaluation contexts, intervention-relative introspection, proof-carrying repair composition, quantitative correctability and obstruction, nonlinear lifting, local-to-global repair descent, typed repair transport, gain-preservation certificates, semantic holonomy, path-flat evaluator cores, witness-separated transactions, universal and controlled recursive closure, productive recurrence, archive dynamics, foreign-agent metaproductivity, conservative grammar expansion, and causal repair-autocatalysis models. Block 5 supplies the empirical-realization contract, exact finite model-checking fixtures, a receipt-and-rollback kernel, an anonymous-change environment, and matched-budget synthetic benchmark arms. Block 6 adds intervention-defined hidden packets and controlled repair transport across learned foreign representations.
220 
221 Block 7 studies two distinct one-step improvements of the repair process. In the first, parent and child receive the same five candidate repairs on every target. A dual-head child selector is compiled from development and meta-validation receipts, sealed before final audit access, and compared with its frozen parent on 120 seed/task-foreign and 120 architecture-foreign episodes. Mean protected utility rises from $0.284896$ to $0.357414$ and from $0.167417$ to $0.272416$, producing paired effects $0.072518$ and $0.104999$ with positive frozen bootstrap intervals. The candidate grammar and reachable correction span remain fixed, so this is behavioral selector metaproductivity rather than structural repair-image growth.
222 
223 In the second realization, the descendant changes paired-anchor query design and the bridge-identification gate while the source repair, transport formula, evaluator, and deployment query budget remain fixed. On $144$ primary foreign targets from representation families absent from development, the parent uses random $16$-anchor queries and the child uses a compiled pivoted-QR $16$-anchor policy. Mean balanced selected gain rises from $0.135212$ to $0.344720$. The paired effect is $0.209508$ with frozen interval $[0.194968,0.223898]$, every primary target improves, full-rank bridge identification rises from $0.03125$ to $1$, and harmful selected targets remain zero. A plain pivoted-QR control is effectively identical to the packet-labeled version, so packet-specific superiority is not established; the supported mechanism is deterministic rank-revealing query design.
224 
225 Block 8 develops the theory required to move beyond one isolated meta-improvement. It factorizes repair processes into causal-atlas, information-acquisition, candidate-construction, transport, selection, evaluation, and witness modules. Restricted oracle envelopes define module-specific metaproductivity headroom. A permutation-average allocation separates remaining opportunity from interaction effects. Saturation certificates prevent null same-module descendants from being misread as failures of recursion. Repeated improvement is modeled as bottleneck migration in a nonnegative meta-regret system, with Lyapunov contraction inside fixed grammars and explicit discovery coordinates when a conservative extension reveals a previously unrepresented limitation.
226 
227 The block also constructs a grammar-indexed obstruction persistence theory. Nested correction images have monotone residual energy, and every extension receives exact credit equal to the energy in newly correctable directions. Repair death is separated from forgetting death under information-losing semantic transport. Obstruction-guided generator discovery selects response directions in the orthogonal obstruction space, while repair-conflict hypergraphs turn incompatible validated repairs into concrete grammar-inadequacy certificates. A finite intervention-response algebra yields a causal atlas through its minimal central projections, with gauge ambiguity and spectral-gap stability stated explicitly. Finally, summable conditional error budgets give lineage-wide control of false improvement claims across adaptive descendant generations.
228 
229 The exact Block 8 companion passes eight automated tests covering headroom allocation, bottleneck migration, obstruction towers, grammar refinement, finite repair synthesis, causal-atlas decomposition, and sequential lineage budgets. Block 9 then realizes the first child--grandchild transition. The exact Block 7 rank-revealing child is replayed in a fresh regime where every primary design is full rank but paired target observations are heteroscedastically noisy. A finite lineage compiler diagnoses that rank has saturated and seals a quality-weighted D-optimal grandchild before the final audit. Under the same $24$-query deployment budget, mean selected protected gain rises from $0.037393$ for a recalibrated rank-child control to $0.209580$ for the conditioning-aware grandchild. The paired effect is $0.172188$ with frozen interval $[0.141481,0.203110]$ across $96$ fresh foreign targets. Both arms remain full rank, while the weighted information floor rises from $0.518197$ to $14.268195$ and clean transport NRMSE falls from $0.303981$ to $0.095803$.
230 
231 Block 10 crosses the first bounded structural grammar-expansion threshold. The inherited conditioning-aware process remains restricted to one global affine bridge. A finite compiler certifies a persistent chart-dependent residual, conservatively embeds the global process into an invariant-context two-chart affine grammar, and seals the child before a fresh $96$-target foreign audit. The strengthened global parent uses $80$ bridge-fit and $16$ witness pairs; the child uses $48$ bridge-fit, $32$ chart-selection, and the same $16$ witness pairs. Child mean selected protected gain is $0.334224$ with zero harmful commits. The child-parent effect is $0.334224$ with frozen interval $[0.266459,0.401903]$. A directly supervised global-affine repair oracle fitted with far more target labels reaches only $0.176694$ mean gain; the child exceeds it by $0.157530$ with interval $[0.080863,0.234135]$. Mean validation NRMSE falls from $0.598513$ to $0.064975$. Gate-only, random-gate, and shuffled-binding controls fail, while blind local application reveals the witness's risk-utility tradeoff.
232 
233 Block 11 removes the supplied Block 10 context packet. A descendant estimates two intervention-response operators, recovers a target-specific central response atlas up to block permutation and within-block gauge, and searches each recovered block for a repair-stable chart separator. Parent and child each consume $120$ effect-query units. On $80$ fresh foreign targets, the fixed-packet parent abstains while the endogenous-atlas child obtains mean selected protected gain $0.207325$ with frozen interval $[0.131902,0.287910]$ and zero selected harmful commits. Mean selected context overlap is $0.821461$ compared with $0.322091$ for the inherited fixed packet. Random-atlas, PCA, full-space, shuffled-response, blind-commit, and nonidentifiable stress controls separate the supported mechanism from coordinate, variance, and witness-free explanations.
234 
235 Block 12 advances two further structural edges. First, the inherited intervention family contains an exact response alias: every polynomial or composition of its operators acts identically on two task-relevant hidden blocks. A finite compiler selects one additional intervention from a sealed candidate language, refines the central response atlas, and raises mean selected protected gain from $0$ to $0.846985$ under an equal $144$-query envelope. Second, after the repair packet is identified, the inherited affine repair operad encounters a nonzero-Hessian obstruction. A conservative degree-two constructor raises mean selected protected gain from $0.289123$ to $0.997511$ under an equal $160$-query envelope and exceeds a directly supervised $1600$-pair same-target affine oracle by $0.530933$. Duplicate, weak-separator, shuffled-response, wrong-block, random-block, shuffled-feature, full-quadratic, and stress controls identify both the supported mechanisms and their limits.
236 
237 Block 13 moves the modified object one level outward. A strong proof-carrying automated-research parent compiler and a lineage-trained child receive the same structured archive, $39$ candidate methods, evaluator portfolio, integrity monitor, and six full experiments per research problem. On $96$ primary foreign problems, validity-filtered protected gain rises from $0.883004$ to $0.950231$, with paired effect $0.067227$ and frozen interval $[0.035881,0.103274]$. On $72$ architecture-foreign problems, gain rises from $0.772566$ to $0.863367$, with paired effect $0.090800$ and interval $[0.052311,0.135081]$. Shuffled archive binding, single-benchmark selection, monitor removal, random allocation, component ablations, and an inadmissible audit oracle delimit the mechanism. Most of the realized gain is attributable to high-accuracy obstruction diagnosis and bottleneck-sensitive allocation; outcome forecasting adds little on the primary split, and the finite within-problem forum is redundant in this fixture.
238 
239 Block 14 replays the Block 13 result digest and exact $39$-method language, then recalibrates the inherited compiler design as the parent inside a finite adversarial stress atlas with aliased diagnostics, sparse multi-obstruction mixtures, unknown problems outside the inherited response cone, $18\%$ archive contamination, evaluator attacks, integrity attacks, and a separately shifted problem generator. Parent and child retain the same six full experiments per problem. The robust child adds sequential response diagnosis, open-set routing, contamination-resistant archive summaries, evaluator-portfolio aggregation, and an integrity-monitor ensemble. On $144$ primary problems, validity-filtered protected gain rises from $0.410813$ to $0.575783$, with paired effect $0.164970$ and frozen interval $[0.115591,0.218655]$. On $120$ generator-foreign problems, gain rises from $0.389514$ to $0.550559$, with paired effect $0.161045$ and interval $[0.105335,0.223141]$. Primary lower-tail gain changes from $-0.133065$ to $0.271516$, observed invalid selections fall from $0.027778$ to $0$, and false confident known-class assignment on unknown problems falls from $1$ to $0.083333$. Sequential response updating and open-set routing are substantively supported. The simple-monitor ablation is null and the single-portfolio ablation is nearly null, so neither receives independent positive attribution.
240 
241 Block 15 freezes the robust Block 14 compiler and confronts it with persistent open-set research conflicts that cannot be resolved by its one unknown token or inherited singleton-method grammar. A finite language compiler receives a generic six-probe diagnostic language and a generic $30$-member ordered pair-constructor grammar, but not the hidden identities of the two novel obstruction mechanisms. Development receipts select the probe \texttt{interaction-contrast}, create two anonymous novel tokens, and synthesize the bindings \texttt{evaluator-then-resource} and \texttt{objective-then-constructor}. The inherited $39$-method grammar is preserved as an exact fallback, while the deployable child adds only the two selected composites. Parent and child receive the same eight-unit research budget. On $96$ primary foreign problems, mean protected gain rises from $0.633969$ to $0.859731$, with paired effect $0.225762$ and interval $[0.166708,0.286431]$. On $82$ generator-foreign problems, gain rises from $0.574224$ to $0.814520$, with paired effect $0.240296$ and interval $[0.180154,0.302701]$. Taxonomy-only, grammar-only, and shuffled-binding controls show that the supported mechanism is the coupled addition of a diagnostic distinction, a new method-constructor output, and the correct binding between them.
242 
243 
244 Block 16 makes recurrent latent reasoning a direct lineage component. It defines a Neuralese reasoner, an exact latent-trace intervention language, necessity, order, corruption, replay, cross-task specificity, adaptive-halting, and semantic-transport certificates, and a finite parent--child promotion rule. A direct adapter targets the local Cymela Hyper v1 runtime through its exposed latent-intervention hook without redistributing third-party code or weights. An engineered $66$-task synthetic fixture separates a useful ordered but task-nonspecific parent from a task-specific child: structural-normalized specificity rises from $0.016952$ to $0.334610$, the paired transplant-cost gain is $0.439986$ with frozen interval $[0.386884,0.493143]$, and the child passes the finite Neuralese lineage gate. No empirical Hyper result is claimed because the model files were absent from the build environment.
245 
246 Block 17 replaces the analytic latent-trajectory fixture by trained recurrent checkpoints in a separately implemented exact rational affine-program environment. A task-blind parent and six inherited repair candidates are optimized under a matched update and example envelope. The audit adds same-answer far-program donors to separate program semantics from answer encoding, an external held-out trace probe, adaptive-depth tests, exact replay, deterministic noise and reversal controls, an independent NumPy forward implementation, and a signed-permutation gauge certificate. A first sealed pilot is rejected without weakening its gates. After a precommitted compiler revision and fresh final-task generation, the selected full-binding child is FULL on both $1600$-task primary and shifted-generator audits while the parent remains PARTIAL. Primary and foreign program-specificity gains are $0.010714$ and $0.024763$, with frozen intervals $[0.009911,0.011527]$ and $[0.022794,0.026793]$; held-out trace-probe $R^2$ is $0.865504$ and $0.691471$. The accepted result is one finite package-local executable Neuralese binding-repair edge, not an empirical Hyper result or open-ended recursive self-improvement.
247 
248 Together, Blocks 7 through 22 establish a controlled progression from one-step behavioral metaproductivity to repeated bottleneck migration, finite transport-grammar growth, endogenous response-atlas recovery, finite response-language and constructor growth, a first sealed improvement of the research compiler itself, adversarial robustness and finite language growth for that compiler, direct causal semantics for recurrent latent reasoning, one training-generated executable latent-binding repair, a cross-implementation distributional chart obstruction, a source-side conservative semantic-atlas expansion, a finite atlas-indexed cross-architecture transport connection, a type-directed expansion of the transport grammar under hidden obstruction stress, and correspondence-free transport from intervention orbits with explicit gauge and underidentification certificates. The outer repair grammar, research generators, semantic witnesses, evaluator and integrity models, donor taxonomy, architecture family, chart meta-language, gauge stress, and audit semantics remain externally declared. No unrestricted automated science, open-ended compiler self-improvement, autonomous evaluator invention, production-model Neuralese result, unrestricted safety, indefinite acceleration, consciousness, or sentience is established.
249 
250 
251 Block 18 attempts the architecture-foreign threshold. A source compiler selects a step-specific FiLM latent repair with progressive semantic and geometry witnesses on an Elman source, then instantiates the same typed schema in independently implemented GRU and gated-residual parents. Both selected children are FULL on fresh $1400$-task primary audits but FAIL on an independently generated shifted support; frozen source-to-target bridges also fail, with transported $R^2=-0.145315$ and $-0.206846$, while the source-reference probe itself is negative. The new constructor does not dominate inherited additive conditioning: its pooled program-specificity difference is $-0.029417$ with interval $[-0.034102,-0.024735]$. A post-hoc held-out diagnosis, excluded from acceptance, recovers source and target semantic probes above $0.57$ and local paired bridges above $0.53$. The canonical result is therefore a sealed distributional semantic-chart obstruction, not a positive cross-implementation edge. It shows that full rank, in-distribution causal specificity, and successful pre-audit alignment do not certify distributionally invariant Neuralese semantics.
252 
253 Block 19 repairs the logically prior source-side chart failure. It compiles a finite affine observation atlas from semantic twins, conservatively extends four inherited charts by two pre-audit novel charts, replaces fixed-slot demonstration processing by a permutation-invariant set encoder, and separates the semantic recurrent path from a context-diagnostic path. One common ancestor and four descendants receive equal $550$-update continuation envelopes. The full coupled descendant is the only pre-audit eligible candidate. On $900$ fresh inherited-context tasks, clean loss falls from $0.200177$ to $0.057546$, same-answer program specificity rises from $0.019129$ to $0.174135$, and semantic-probe $R^2$ reaches $0.770701$. On $900$ fresh tasks in the added charts, clean loss falls from $0.416361$ to $0.050745$, program specificity reaches $0.194936$, and probe $R^2$ reaches $0.779341$. The child is \texttt{ATLAS\_FULL} on both supports. Across $200$ semantic objects rendered in six supported contexts, maximum trace cosine drift is $2.384\times10^{-7}$, maximum output drift is $1.431\times10^{-6}$, and halt depth is identical. All $600$ nonlinear unsupported tasks are rejected. The accepted result is a finite source-side atlas-growth edge; cross-architecture atlas transport remains open.
254 
255 Block 20 closes that finite cross-architecture atlas-transport gate under a declared orthogonal gauge stress. A frozen Block 19 Elman source, independently trained GRU and gated-residual targets, and six admitted observation charts define a $3\times6$ architecture--context semantic bundle of complete five-step, forty-dimensional Neuralese trajectories. A sealed compiler compares pooled full-trajectory, chart-indexed step-shared, and chart-indexed full-trajectory affine connections. Only the six-chart full-trajectory family passes every pre-audit gate. On fresh inherited and added-chart supports, transported semantic-probe $R^2$ ranges from $0.781159$ to $0.799157$ for GRU and from $0.785881$ to $0.796471$ for the residual target. Normalized architecture--context curvature is $0.010474$ and $0.008374$, versus $0.131402$ and $0.112397$ for pooled maps. All four paired pooled-minus-selected semantic-loss intervals are strictly positive, shuffled chart binding fails sharply, and unsupported nonlinear contexts remain rejected. Thirteen tests and an independent NumPy verifier replay all $13{,}020$ exact tasks. The result is finite controlled cross-architecture transport, not universal production-model alignment.
256 
257 Block 21 makes the transport grammar itself the modified object. Above the accepted Block 20 connection, it freezes an eight-coordinate packet at each of five recurrent steps and hides six post-connection obstruction families from a minimum-description compiler: affine gauge drift, smooth invertible nonlinear warp, latent-clock mixing, aliased local charts, benign rank-changing compression, and destructive protected-semantic collision. The compiler searches a declared grammar containing step and full affine maps, a monotone nonlinear inverse, a temporal inverse, a probe-conditioned chart split, a semantic quotient, and collision-grounded no-go. All $24$ pre-audit episodes and all $36$ fresh final episodes select the required minimal response across GRU and gated-residual lineages. Reconstructive repairs obtain packet $R^2\geq0.998807$, packet NRMSE at most $0.034543$, and semantic $R^2\geq0.718695$. Benign quotient cases retain semantic $R^2\geq0.999989$ without claiming full-state recovery. Destructive cases contain exact equal-observation, unequal-semantics pairs and are rejected. Nineteen tests and an independent verifier replay all $13{,}600$ exact tasks and every compiler decision with zero metric discrepancy. The result is finite obstruction-guided transport-grammar growth, not unrestricted natural-model diagnosis.
258 
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260 
261 Block 22 removes exact source--target construction pairs. Source and target latent populations are sampled independently, while only within-system intervention responses are exposed to a label-isolated compiler. A sealed response-algebra construction must distinguish five terminal claims: identified transport, identification only up to a declared gauge, partial transport on overlapping support, semantic underidentification, and deterministic no-go under destructive information loss. Across two foreign architectures and six declared regimes, all $24$ pre-audit and $36$ fresh final episodes issue the precommitted certificate. Twelve final episodes identify transport, six identify only a gauge class, six transport exactly the supported $70\%$ subset with zero false admission, six return underidentification with hidden semantic disagreement at least $1.905405$, and six return collision-grounded no-go with $96$ exact collisions per episode. Fully identifiable cases retain semantic $R^2\geq0.990952$ with orbit defect at most $0.002870$, while distribution-only and shuffled-binding controls fail. The result removes one finite pairing scaffold; it does not establish unrestricted unpaired alignment or endogenous semantic discovery.
262 
263 Block 23 removes the supplied intermediate semantic coordinates from quotient construction. A finite compiler receives recurrent trajectories, semantic-preserving twins, anonymous intervention traces, protected future consequences, and final answers, but no gold intermediate ontology. It selects the minimum-rank quotient passing frozen predictive, interventional, answer, and twin-stability gates. Across $24$ pre-audit and $36$ fresh final episodes, all hidden dispositions are correct: twelve ontologies are uniquely identified, six are identified only up to gauge, six are partial, six remain underidentified despite high post-hoc decodability, and six receive collision-grounded semantic-information-loss certificates. A rank-one answer code reaches accuracy at least $0.881250$ but fails future and intervention sufficiency. Audit-only donor-packet patching gives selected future $R^2\geq0.983162$, while answer-only, PCA, and random controls are negative. The release also supplies a source-locked adapter for a public multimode Coconut checkpoint. The external checkpoint was unavailable in the build environment and was not executed; no real-model interpretability result is claimed.
264 \end{abstract}
265 
266 \section*{Progressive Expansion Map}
267 \addcontentsline{toc}{section}{Progressive Expansion Map}
268 
269 \begin{statusbox}{Current revision}
270 Version 2.4 completes \textbf{Block 23: Endogenous Semantic-Witness Growth}. It removes gold intermediate semantic coordinates from quotient construction, requires witness-separated final interpretation, and distinguishes unique identification, gauge equivalence, partial ontology, protected underidentification, and destructive semantic information loss. All $36$ fresh finite episodes receive the precommitted certificate. A source-locked external Coconut adapter is included, but the third-party checkpoint was not available in the build environment and no external empirical result is claimed.
271 \end{statusbox}
272 
273 \begin{longtable}{@{}p{0.13\textwidth}p{0.26\textwidth}p{0.51\textwidth}@{}}
274 \toprule
275 Block & Focus & Expansion target \\
276 \midrule
277 \endfirsthead
278 \toprule
279 Block & Focus & Expansion target \\
280 \midrule
281 \endhead
282 1 & Foundations & Evaluation contexts, bounded improvement relations, stratified lineages, intervention-sufficient introspection, repair-operad semantics. \textbf{Completed.} \\
283 2 & Correctability & Authorized correction spaces, quantitative obstruction geometry, invariant and resource constraints, nonlinear repair, exact and approximate descent, obstruction towers, and worked gluing examples. \textbf{Completed.} \\
284 3 & Transport and evaluation & Exact and approximate repair transport, compositional error bounds, authorization and obstruction faithfulness, gain certificates, repair holonomy, evaluator atlases, spectral path-flat cores, adequacy, and evaluator-upgrade protocols. \textbf{Completed.} \\
285 4 & Recursive closure & Witness-separated transactions, commit-reveal semantics, robust and productive fixed points, archives, foreign-agent metaproductivity, conservative grammar growth, and autocatalytic repair discovery. \textbf{Completed.} \\
286 5 & Empirical realization & Abstract-runtime refinement, finite model checking, executable receipts and rollback, anonymous-change protocols, matched-budget benchmarks, causal ablations, and reproducible packages. \textbf{Completed.} \\
287 6 & Learned-agent validation & Intervention-defined hidden packets, paired-anchor bridge learning, foreign representation families, witness-gated repair transfer, learned encoders, causal controls, and frozen level-4 claim gates. \textbf{Completed.} \\
288 7 & Sealed metaproductivity & Two one-step parent--child comparisons: a dual-head selector over identical repair bundles, and a compiled rank-revealing anchor-query policy under an identical deployment budget; split-file sealing, paired foreign audits, causal controls, and fixed claim boundaries. \textbf{Completed.} \\
289 8 & Headroom and bottleneck migration & Restricted oracle envelopes, Shapley meta-headroom, saturation, bottleneck migration, obstruction persistence, grammar-conflict refinement, causal-atlas discovery, sequential lineage certification, and exact symbolic fixtures. \textbf{Completed.} \\
290 9 & Repeated lineage & Exact ancestor replay, rank-saturation diagnosis, heteroscedastic weighted bridge information, conditioning-aware query compilation, fresh child--grandchild audit, causal controls, and a package-closed two-edge certificate. \textbf{Completed.} \\
291 10 & Structural grammar expansion & Persistent global-grammar residual, conservative two-chart bridge and repair constructor, finite separator discovery, structural contrast certificate, matched foreign audit, causal controls, and package-closed structural metaproductivity. \textbf{Completed.} \\
292 11 & Endogenous atlas growth & Supplied-packet removal, intervention-response algebra, central projector recovery, repair-relevant block labeling, conservative fallback, matched foreign audit, causal controls, and package-closed finite atlas growth. \textbf{Completed.} \\
293 12 & Response-language and constructor growth & Exact response-alias certificate, finite separating-intervention compiler, conservative response-algebra refinement, affine function-class obstruction, degree-two repair-constructor expansion, matched foreign audits, causal controls, and package-closed replay. \textbf{Completed.} \\
294 13 & Research-compiler metaproductivity & Results-free proposal receipts, proof-carrying finding graph, multi-benchmark headroom evaluation, integrity-valid selection, obstruction diagnosis, outcome forecasting, matched-budget parent--child compiler audit, mechanism ablations, and package-closed replay. \textbf{Completed.} \\
295 14 & Adversarial compiler robustness & Diagnostic aliasing, sparse multi-obstruction inference, open-set residuals, contaminated archives, evaluator and integrity attacks, sequential diagnosis, robust lower-tail and worst-group gates, generator-foreign audit, and package-closed replay. \textbf{Completed.} \\
296 15 & Compiler-language growth & Open-set repair-conflict certificate, generic separating-probe search, anonymous diagnostic-token refinement, ordered pair-method synthesis, conservative inherited fallback, matched-budget foreign audits, causal language controls, and package-closed replay. \textbf{Completed.} \\
297 16 & Direct Neuralese integration & Recurrent latent-reasoner object, exact injection-trace interventions, necessity, order, corruption, replay, cross-task specificity, halting and semantic-transport certificates, a local Hyper adapter, and a synthetic parent--child lineage fixture. \textbf{Direct contract and synthetic realization completed; real-model audit pending.} \\
298 17 & Executable Neuralese repair compilation & Exact affine-program execution, trained recurrent parent and six matched-update descendants, answer-confound controls, external trace decoding, adaptive halting, preserved failed pilot, precommitted compiler repair, fresh primary and shifted audits, NumPy shadow replay, gauge verification, and package-closed receipts. \textbf{Finite training-generated edge completed; foreign production-model audit remains open.} \\
299 18 & Cross-implementation Neuralese transport & Exact multi-template program induction, Elman source compiler, GRU and gated-residual foreign parents, step-specific FiLM schema transport, causal donor audits, frozen latent bridges, shifted-support failure, and post-hoc semantic-chart localization. \textbf{Sealed distributional obstruction completed; positive foreign-model edge not established.} \\
300 19 & Distribution-indexed Neuralese atlas growth & Semantic-twin chart compilation, conservative four-to-six-chart extension, permutation-invariant set encoding, semantic/context path separation, selective support admission, fresh inherited and novel-chart audits, unsupported nonlinear controls, and independent NumPy replay. \textbf{Finite source-side atlas-growth edge completed; cross-architecture atlas transport remains open.} \\
301 20 & Atlas-indexed cross-architecture Neuralese transport & Frozen six-chart source atlas, independently trained GRU and gated-residual targets, chart-dependent orthogonal gauge stress, pooled and chart-indexed trajectory bridges, architecture--context commutation audits, shuffled-binding and unsupported controls, paired fresh audits, and independent NumPy replay. \textbf{Finite controlled cross-architecture atlas-transport edge completed; nonlinear connection growth remains open.} \\
302 21 & Obstruction-guided Neuralese transport-grammar growth & Frozen five-step diagnostic packet, six hidden post-connection obstruction families, minimum-description affine, nonlinear, temporal, chart-split, quotient, and no-go constructors, fresh cross-architecture audits, wrong-repair controls, exact collision certificates, and independent replay. \textbf{Finite type-directed transport-grammar edge completed; unpaired and ontology-light transport remains open.} \\
303 22 & Correspondence-free intervention-orbit Neuralese transport & Independent source and target populations, within-system intervention responses, anonymous action matching, response-algebra intertwiners, gauge-equivalence certificates, selective partial transport, semantic underidentification, collision-grounded no-go, sealed final coupling reveal, tests, and deterministic replay. \textbf{Finite unpaired transport edge completed; endogenous-semantic and stochastic-path transport remain open.} \\
304 23 & Endogenous semantic-witness growth & Gold-free causal-predictive quotient construction from semantic twins, anonymous interventions, protected futures, and answers; minimum-rank selection; answer-shortcut rejection; gauge, partiality, underidentification, collision no-go, counterfactual packet controls, and a source-locked external Coconut pilot. \textbf{Finite ontology-growth edge completed; real checkpoint execution remains pending.} \\
305 \bottomrule
306 \end{longtable}
307 
308 \clearpage
309 \tableofcontents
310 \clearpage
311 \pagenumbering{arabic}
312 
313 \section{Introduction}
314 
315 \subsection{Why the usual picture is underspecified}
316 
317 The phrase \emph{recursive self-improvement} is often represented by the schematic loop
318 \begin{equation}
319 A_t\longrightarrow A_{t+1}\longrightarrow A_{t+2}\longrightarrow\cdots,
320 \end{equation}
321 with the informal assertion that each descendant is ``better'' than its parent. This notation suppresses the central difficulties. It does not identify the object that persists across generations, the evidence by which improvement is judged, the interventions available to the system, the components that may be modified, the resource budget under which comparisons are made, or the mechanism that prevents a descendant from changing the meaning of its own score.
322 
323 At least three mathematically distinct processes are present:
324 \begin{enumerate}
325 \item \textbf{task improvement}: a system performs better on a declared environment distribution;
326 \item \textbf{improvement-process improvement}: a system becomes better at producing useful descendants under a matched search budget;
327 \item \textbf{epistemic improvement}: the evaluation machinery becomes better at distinguishing genuine gain from overfitting, proxy exploitation, evaluator capture, or semantic drift.
328 \end{enumerate}
329 The second process is the recursive component. The third determines whether the recursion remains semantically connected to external evidence.
330 
331 Let $A_t$ denote an active agent system at generation $t$, and let $J(A_t)$ denote externally evaluated task utility. Ordinary task improvement requires
332 \begin{equation}
333 J(A_{t+1})>J(A_t).
334 \end{equation}
335 Let $\Gamma_{h,B}(K;A)$ denote the expected best gain obtained when improvement mechanism $K$ is allowed horizon $h$ and resource budget $B$ to modify target agent $A$. Improvement-process improvement requires, on a held-out distribution of target agents,
336 \begin{equation}
337 \mathbb E_A\Gamma_{h,B}(K_{t+1};A)
338 >
339 \mathbb E_A\Gamma_{h,B}(K_t;A).
340 \end{equation}
341 Finally, if $\Vcal_t$ is the evaluator active at generation $t$, epistemic improvement requires a decrease in held-out evaluation risk rather than a mere increase in self-reported confidence.
342 
343 \subsection{Central proposal}
344 
345 The central proposal is
346 \begin{center}
347 \fbox{\begin{minipage}{0.86\textwidth}
348 \centering
349 \textbf{Recursive self-improvement is growth of an authorized repair calculus within an evidence-preserving lineage.}
350 \end{minipage}}
351 \end{center}
352 The recursively improving object is not one weight tensor. It is a lineage consisting of agents, evidence, repairs, evaluators, provenance, witnesses, and validated transformations. The basic cycle is
353 \begin{equation}
354 \begin{aligned}
355 \text{external outcome}
356 &\longrightarrow \text{causal diagnosis}
357 \longrightarrow \text{stratified defect}\\
358 &\longrightarrow \text{local obstruction analysis}
359 \longrightarrow \text{compatible repair construction}\\
360 &\longrightarrow \text{sandboxed descendant}
361 \longrightarrow \text{witness validation}\\
362 &\longrightarrow \text{lineage commit or rollback}
363 \longrightarrow \text{repair abstraction and transport}.
364 \end{aligned}
365 \end{equation}
366 The final abstraction-and-transport step is what makes compounding possible. A successful patch is local. A validated repair schema that transfers to new systems is a reusable improvement operator.
367 
368 \subsection{Design principles}
369 
370 The framework is organized around seven principles.
371 \begin{enumerate}
372 \item \textbf{Reference relativity.} Improvement is defined relative to a declared task distribution, utility, evidence channel, resource envelope, and invariant set.
373 \item \textbf{Causal rather than descriptive self-modeling.} A decoded representation is useful only insofar as it predicts or controls the consequences of authorized interventions.
374 \item \textbf{Typed heterogeneity.} Model state, memory, planning, evaluation, provenance, and workspace access are different strata connected by typed bridges. They are not silently identified.
375 \item \textbf{Local correction plus global compatibility.} A repair may fail because no local correction exists or because locally valid corrections do not glue across interfaces.
376 \item \textbf{Proof-carrying mutation.} Candidate transformations carry preconditions, effect claims, evidence, provenance, protected invariants, and rollback information.
377 \item \textbf{Witness separation.} No atomic transaction simultaneously rewrites the proposal, evidence, judgment, and rollback mechanisms by which that same transaction is approved.
378 \item \textbf{Branching persistence.} The archive preserves successful, failed, and partially useful descendants. The persistent object is a proof-carrying lineage rather than a single greedy chain.
379 \end{enumerate}
380 
381 \subsection{What counts as a substantive recursive step}
382 
383 A higher task score is insufficient. A substantive recursive step should satisfy all of the following under matched resources:
384 \begin{enumerate}
385 \item the best validated archive frontier does not decrease;
386 \item the new improvement mechanism produces larger gains on foreign held-out agents or defect families;
387 \item the evaluator does not obtain this result by deleting evidence, weakening invariants, or changing the interpretation of protected outcomes;
388 \item the accepted transformation preserves a valid lineage path and rollback point;
389 \item uncertainty and nonidentifiability are represented rather than converted into fabricated certainty.
390 \end{enumerate}
391 This definition permits a descendant with lower immediate task score to remain valuable as a stepping stone when it increases future repair capacity. The archive, not the current node alone, carries monotone best-so-far performance.
392 
393 \subsection{Status discipline}
394 
395 \begin{statusbox}{Claim boundary}
396 The manuscript proves finite-dimensional, finite-state, normed-linear, cover-relative, and finite-hypothesis results under explicit assumptions. It does not prove that large language models possess the required strata, that naturally occurring agent defects admit stable packet decompositions, that evaluator holonomy is always measurable, that repair discovery becomes supercritical, or that open-ended recursive self-improvement is safe or inevitable.
397 \end{statusbox}
398 
399 The formal status of the principal components is summarized in \cref{tab:status}.
400 
401 \begingroup
402 \small
403 \begin{longtable}{@{}p{0.38\textwidth}p{0.56\textwidth}@{}}
404 \caption{Status of the main components.}\label{tab:status}\\
405 \toprule
406 Component & Status in this manuscript \\
407 \midrule
408 \endfirsthead
409 \toprule
410 Component & Status in this manuscript \\
411 \midrule
412 \endhead
413 Protected evaluation contexts and exact improvement preorder & Proved from the declared definitions \\
414 Evaluator-capture triviality and fixed-margin ceiling & Proved \\
415 Bridge-defect composition and lineage-path category & Proved \\
416 Intervention-sufficient introspection regret and safe-commit bounds & Proved \\
417 Finite deterministic active identification under separating probes & Proved \\
418 Compositional repair-receipt soundness and packetwise commutation & Proved under declared semantic assumptions \\
419 Canonical obstruction geometry and minimal-norm repair & Proved in finite-dimensional Hilbert spaces \\
420 Invariant-constrained and cone-constrained correctability & Proved; cone duality assumes a closed reachable cone \\
421 Authorized obstruction shrinkage and exact quotient sequence & Proved \\
422 Packet-coupling correction by a Neumann right inverse & Proved under a small off-block coupling condition \\
423 Gauge-fixed nonlinear repair and Newton--Kantorovich iteration & Proved under explicit local regularity assumptions \\
424 Lyapunov--Schmidt reduction and finite higher-order obstruction tower & Proved locally in finite dimension or as a finite formal expansion \\
425 Exact local-to-global repair descent & Proved relative to a declared sheaf cover \\
426 Quantitative graph-Hodge gluing & Proved for finite synchronized overlap models \\
427 Nonlinear torsor descent & Conditional on principal local symmetry actions and effective descent \\
428 Exact repair transport & Proved \\
429 Approximate residual and gain transport & Proved under norm, alignment, and Lipschitz assumptions \\
430 Maximal path-flat evaluator core & Proved in the finite linear setting \\
431 Witness-separated transaction independence & Proved as a structural noninterference statement \\
432 Finite recursive improvement closure & Proved as a greatest fixed point \\
433 Archive reliability with summable confidence failures & Proved probabilistically \\
434 Foreign-agent metaproductivity & Defined as an experimental criterion; selector and bridge-policy instances are realized in Block 7 \\
435 Sealed parent--child selector improvement & Conditional theorem plus executable finite positive result under matched candidates and frozen claim gates \\
436 Meta-headroom allocation and component saturation & Proved for finite declared module classes \\
437 Bottleneck-migration contraction & Proved under a weighted local contraction hypothesis \\
438 Persistent obstruction and newly correctable energy & Proved for nested finite-dimensional correction images \\
439 Repair-conflict grammar refinement & Proved for finite episode and feature families \\
440 Finite intervention-algebra atlas & Proved for finite-dimensional adjoint-closed response algebras \\
441 Lineage-wide false-improvement control & Proved under summable conditional error budgets \\
442 Structural metaproductivity & Finite two-chart repair-transport realization in Block 10; open-set growth not established \\
443 Repeated lineage metaproductivity & Controlled two-edge context-indexed realization in Block 9; indefinite recurrence not established \\
444 Conditioning-aware query-design improvement & Proved conditionally and realized by a sealed grandchild under matched deployment queries \\
445 Finite conservative grammar expansion & Proved conditionally and realized in Block 10 for a declared observable separator family \\
446 Endogenous intervention-response atlas & Exact central-projector theorem plus an executable finite Block 11 realization under a declared separating response family \\
447 Finite response-language expansion & Alias-persistence and central-refinement theorems plus an executable finite Block 12 separating-intervention realization \\
448 Finite candidate-constructor expansion & Affine-closure and nonzero-Hessian obstruction theorems plus an executable degree-two Block 12 realization \\
449 Finite research-compiler metaproductivity & Formal compiler object plus executable Block 13 parent--child realization under a fixed method grammar, evaluator portfolio, integrity monitor, and matched research budget \\
450 Finite adversarial compiler robustness & Static-indistinguishability, open-set, evaluator-nullspace, and monitor-ensemble results plus an executable Block 14 matched-budget robustness edge \\
451 Finite compiler-language growth & Conflict-separation and finite language-synthesis theorems plus an executable Block 15 matched-budget diagnostic-and-method expansion edge \\
452 Atlas-indexed cross-architecture Neuralese transport & Finite Block 20 realization over one frozen Elman source, trained GRU and gated-residual targets, six admitted charts, controlled orthogonal gauge stress, fresh paired audits, and bounded architecture--context curvature \\
453 Obstruction-guided Neuralese transport-grammar growth & Finite Block 21 realization with label-isolated compilation across affine, nonlinear, temporal, chart-alias, quotient, and collision no-go families \\
454 Correspondence-free Neuralese intervention-orbit transport & Finite Block 22 realization with independent latent populations, response-algebra intertwiners, anonymous intervention matching, gauge/underidentification separation, partial support, and no-go certificates \\
455 Endogenous semantic-witness growth & Finite Block 23 realization with gold-free causal-predictive quotient compilation, semantic-twin and intervention sufficiency, answer-shortcut rejection, gauge/partial/underidentification certificates, destructive-loss no-go, and an execution-ready but unrun external Coconut adapter \\
456 Open-ended grammar expansion & Still conjectural; Blocks 10--15 realize finite transport, response-language, constructor, compiler-allocation, robustness, and compiler-language extensions inside externally declared meta-languages \\
457 Autocatalytic repair threshold & Conditional on a multitype branching approximation \\
458 Unrestricted recursive self-improvement & Not established \\
459 \bottomrule
460 \end{longtable}
461 \endgroup
462 
463 \section{Evaluation Contexts and Improvement Relations}
464 \label{sec:evaluation-context}
465 
466 \subsection{Protected evaluation contexts}
467 
468 \begin{definition}[Protected evaluation context]
469 A protected evaluation context is a tuple
470 \begin{equation}
471 \Xi=
472 \left(
473 \Zcal,
474 \mu,
475 \Ycal,
476 \nu,
477 \bar c,
478 \lambda,
479 \mathscr I,
480 \Bcal,
481 \alpha
482 \right),
483 \end{equation}
484 where:
485 \begin{enumerate}
486 \item $\Zcal$ is a task or environment family and $\mu$ is its held-out distribution;
487 \item $\Ycal$ is the space of protected external traces;
488 \item $\nu:\Zcal\times\Ycal\to[0,1]$ is task utility;
489 \item $\bar c:\Zcal\times\Ycal\to[0,1]$ is normalized resource cost;
490 \item $\lambda\geq0$ is a cost weight;
491 \item $\mathscr I(A,B)\in\{0,1\}$ is a declared invariant-preservation predicate for the transition $A\to B$;
492 \item $\Bcal$ is the resource envelope used for candidate generation and evaluation;
493 \item $\alpha\in(0,1)$ is the accepted statistical error level.
494 \end{enumerate}
495 The evaluator may use only protected traces, precommitted predictions, and ledger entries authorized by the context. Candidate identifiers and mutable self-reported scores are not protected evidence.
496 \end{definition}
497 
498 For an agent $A$, define the resource-normalized utility
499 \begin{equation}
500 J_\Xi(A)
501 =
502 \mathbb E_{z\sim\mu}
503 \left[
504 \nu\bigl(z,Y_A(z)\bigr)
505 -
506 \lambda\bar c\bigl(z,Y_A(z)\bigr)
507 \right].
508 \end{equation}
509 The definition does not require scalar utility to encode every scientific judgment. It supplies a declared comparison functional for one experimental claim. Multiple protected contexts may be used simultaneously, producing a Pareto relation instead of a total order.
510 
511 \begin{definition}[Exact context-relative improvement]
512 For agents $A,B$, write
513 \begin{equation}
514 B\succeq_\Xi A
515 \end{equation}
516 when
517 \begin{equation}
518 \mathscr I(A,B)=1
519 \qquad\text{and}\qquad
520 J_\Xi(B)\geq J_\Xi(A).
521 \end{equation}
522 For $\Delta>0$, write $B\succ_{\Xi,\Delta}A$ when the utility difference is at least $\Delta$.
523 \end{definition}
524 
525 \begin{proposition}[Improvement preorder]
526 If $\mathscr I$ is reflexive and transitive along composable lineage edges, then $\succeq_\Xi$ is a preorder on the set of agents connected by admissible transitions.
527 \end{proposition}
528 
529 \begin{proof}
530 Reflexivity follows from $\mathscr I(A,A)=1$ and $J_\Xi(A)\geq J_\Xi(A)$. If $C\succeq_\Xi B$ and $B\succeq_\Xi A$, transitivity of $\mathscr I$ gives $\mathscr I(A,C)=1$, while scalar transitivity gives $J_\Xi(C)\geq J_\Xi(B)\geq J_\Xi(A)$.
531 \end{proof}
532 
533 \begin{definition}[Statistically validated improvement]
534 A candidate $B$ is a statistically validated $(\Delta,\alpha)$-improvement over $A$ when:
535 \begin{enumerate}
536 \item $\mathscr I(A,B)=1$;
537 \item both systems are evaluated within the same resource envelope $\Bcal$;
538 \item a predeclared lower confidence bound satisfies
539 \begin{equation}
540 \LCB_{1-\alpha}
541 \left(J_\Xi(B)-J_\Xi(A)\right)
542 \geq\Delta;
543 \end{equation}
544 \item the evidence ledger used by the test is append-only across the transition;
545 \item the candidate did not receive post-reveal evaluator information unavailable to the baseline.
546 \end{enumerate}
547 \end{definition}
548 
549 The validated relation is an experimental judgment and need not be transitive at finite sample size. The exact relation is a mathematical preorder. The distinction prevents statistical confidence statements from being mistaken for algebraic identities.
550 
551 \subsection{Three levels of improvement}
552 
553 Let $\Acal_t\subseteq V(\Lambda_t)$ be the executable archive at generation $t$. Its task frontier is
554 \begin{equation}
555 F_t(\Xi)=\sup_{A\in\Acal_t}J_\Xi(A).
556 \end{equation}
557 Let $K_t$ be the active improvement mechanism, including diagnosis, proposal, composition, and validation policies. For horizon $h$ and budget $B$, define target-relative metaproductivity
558 \begin{equation}
559 \Gamma_{\Xi,h,B}(K_t;A)
560 =
561 \mathbb E
562 \left[
563 \sup_{A'\in\Desc_{h,B}(K_t,A)}
564 \left(J_\Xi(A')-J_\Xi(A)\right)_+
565 \right].
566 \end{equation}
567 Finally, let $\Risk_\Xi(\Vcal_t)$ denote held-out evaluator risk on candidate-effect judgments, including false acceptance of harmful candidates and false rejection of beneficial candidates above a declared margin.
568 
569 \begin{definition}[Bounded recursive improvement step]
570 A lineage transition $\mathfrak L_t\to\mathfrak L_{t+1}$ is a bounded recursive improvement step relative to $(\Xi,h,B)$ when:
571 \begin{enumerate}
572 \item its lineage receipt is valid and the protected evidence ledger embeds into the next ledger;
573 \item $F_{t+1}(\Xi)\geq F_t(\Xi)$;
574 \item on a foreign held-out target distribution $\mathcal D_{\mathrm{foreign}}$,
575 \begin{equation}
576 \mathbb E_{A\sim\mathcal D_{\mathrm{foreign}}}
577 \Gamma_{\Xi,h,B}(K_{t+1};A)
578 >
579 \mathbb E_{A\sim\mathcal D_{\mathrm{foreign}}}
580 \Gamma_{\Xi,h,B}(K_t;A);
581 \end{equation}
582 \item evaluator risk does not increase beyond a declared tolerance;
583 \item the improvement is obtained under matched model access, candidate count, compute, tools, and external evidence.
584 \end{enumerate}
585 \end{definition}
586 
587 This definition permits non-greedy descendants. A branch may have lower current utility but higher metaproductivity, modularity, observability, or transportability. Archive retention allows such a branch to remain available without lowering the best validated frontier.
588 
589 \subsection{The evaluator paradox}
590 
591 \begin{proposition}[Evaluator-capture triviality]
592 \label{prop:evaluator-capture}
593 Let $\mathcal S$ be a candidate set with at least two elements, and suppose the admissible evaluator class contains every function $V:\mathcal S\to[0,1]$. For every candidate $B\in\mathcal S$, there exists an evaluator $V_B$ under which $B$ is the unique maximizer.
594 \end{proposition}
595 
596 \begin{proof}
597 Define $V_B(B)=1$ and $V_B(A)=0$ for every $A\neq B$.
598 \end{proof}
599 
600 \begin{remark}
601 Consequently, the statement ``the descendant can choose an evaluator under which it scores better'' contains no evidence of improvement. Evaluator evolution requires preserved evidence, conservative semantics on protected traces, and independent validation.
602 \end{remark}
603 
604 \begin{proposition}[Fixed-margin evaluator ceiling]
605 \label{prop:fixed-evaluator-ceiling}
606 Let $V:\mathcal S\to[0,1]$ be fixed. If every accepted transition satisfies
607 \begin{equation}
608 V(A_{t+1})\geq V(A_t)+\Delta
609 \end{equation}
610 for a constant $\Delta>0$, then every accepted chain has length at most $\lfloor 1/\Delta\rfloor+1$.
611 \end{proposition}
612 
613 \begin{proof}
614 After $n$ strict transitions,
615 \begin{equation}
616 V(A_n)\geq V(A_0)+n\Delta\geq n\Delta.
617 \end{equation}
618 Since $V(A_n)\leq1$, one has $n\leq1/\Delta$.
619 \end{proof}
620 
621 The proposition does not prohibit infinitely many diminishing improvements under a fixed evaluator. It shows that open-ended uniformly positive progress cannot be explained by a bounded static score alone. New tasks, finer evidence, new repair types, or a conservatively refined evaluator must eventually enter the process.
622 
623 \subsection{Nontriviality and external anchoring}
624 
625 An evaluation context is \emph{nontrivial} when candidate identity, self-reported utility, and mutable internal error registers are conditionally irrelevant given protected external traces. Formally, for candidate identifier $I$ and protected trace $Y$,
626 \begin{equation}
627 V(I,Y)=V(I',Y)
628 \end{equation}
629 for all identifiers $I,I'$ that generate the same authorized trace and resource record. This excludes a large class of self-confirming evaluator changes while allowing the evaluator to use causally relevant provenance and intervention receipts.
630 
631 The framework does not demand that every value judgment be mathematically fixed. It demands that any change in evaluative semantics leave an explicit transformation path through preserved evidence. The path-flat evaluator theory in \cref{sec:evaluator-evolution} formalizes this requirement.
632 
633 \section{Resolved Lineages and Stratified Agents}
634 
635 \subsection{Functional atlases}
636 
637 An agent relevant to recursive improvement is not naturally represented by one homogeneous state space. Its learned model, working memory, long-term memory, planner, evaluator, provenance machinery, tools, and repair mechanism obey different update laws and support different notions of equivalence.
638 
639 \begin{definition}[Stratified functional atlas]
640 A stratified functional atlas at generation $t$ is a tuple
641 \begin{equation}
642 \mathfrak A_t=
643 \left(
644 \mathsf C,
645 G_t,
646 \{X_t^c\}_{c\in\mathsf C},
647 \{F_t^c\}_{c\in\mathsf C},
648 \{T_e,\tau_e,\Pcal_e,\bm\eps_e\}_{e\in E(G_t)}
649 \right),
650 \end{equation}
651 where:
652 \begin{enumerate}
653 \item $\mathsf C$ is a finite set of stratum colors;
654 \item $G_t$ is a directed multigraph of declared interfaces;
655 \item $X_t^c$ is the state space of stratum $c$;
656 \item $F_t^c$ is its local transition rule;
657 \item each edge $e:c\to d$ carries a bridge $T_e:X_t^c\to X_t^d$;
658 \item $\tau_e$ declares the bridge class;
659 \item $\Pcal_e$ is its preservation profile;
660 \item $\bm\eps_e$ is a vector of measured or certified approximation defects.
661 \end{enumerate}
662 \end{definition}
663 
664 A representative color set is
665 \begin{equation}
666 \mathsf C=
667 \set{
668 \mathrm{model},
669 \mathrm{workspace},
670 \mathrm{memory},
671 \mathrm{planner},
672 \mathrm{improver},
673 \mathrm{evaluator},
674 \mathrm{provenance},
675 \mathrm{tools}
676 }.
677 \end{equation}
678 The atlas does not assert that these colors are universally correct. It declares the decomposition under which one particular repair theory is being tested.
679 
680 \begingroup
681 \small
682 \begin{longtable}{@{}p{0.21\textwidth}p{0.24\textwidth}p{0.49\textwidth}@{}}
683 \caption{Representative bridge classes.}\label{tab:bridge-types}\\
684 \toprule
685 Bridge class & Symbolic type & Authorized inference \\
686 \midrule
687 \endfirsthead
688 \toprule
689 Bridge class & Symbolic type & Authorized inference \\
690 \midrule
691 \endhead
692 Exact equivalence & $X\cong Y$ & Bidirectional transport of every declared preserved construction. \\
693 Embedding & $X\hookrightarrow Y$ & Forward transport inside an identified subobject; no automatic inverse. \\
694 Quotient or forgetful map & $X\twoheadrightarrow Y$ & Forward transport with declared information loss; reconstruction requires additional data. \\
695 Analysis-synthesis pair & $X\xrightarrow{T}Y\xrightarrow{R}X$ & Approximate reconstruction under an explicit residual $\norm{RT-I}$. \\
696 Stochastic channel & $K:X\rightsquigarrow Y$ & Distributional transport only; pointwise identity is not implied. \\
697 Heuristic correspondence & $X\dashrightarrow Y$ & Candidate analogy with no theorem transport until separately certified. \\
698 \bottomrule
699 \end{longtable}
700 \endgroup
701 
702 \begin{definition}[Preservation profile]
703 A preservation profile for a bridge $T:X\to Y$ is a declared record
704 \begin{equation}
705 \Pcal(T)=
706 \left(
707 \Sigma_T,
708 \mathcal Q_T,
709 \rho_T,
710 \ell_T,
711 \sigma_T,
712 \mathcal A_T
713 \right),
714 \end{equation}
715 where $\Sigma_T$ lists preserved algebraic structures, $\mathcal Q_T$ lists preserved causal or evaluative queries, $\rho_T$ records provenance behavior, $\ell_T$ records information loss, $\sigma_T$ records uncertainty transport, and $\mathcal A_T$ lists authorized downstream uses.
716 \end{definition}
717 
718 The preservation profile is part of the type of the bridge. A lossy memory encoder may preserve retrieval-relevant semantics while failing to preserve exact chronology. A workspace decoder may preserve reportability while failing to preserve ownership. A model migration may preserve behavior on a declared task family while changing internal geometry.
719 
720 \subsection{Approximate property transport}
721 
722 Let $Q_X:X\to Z_X$ and $Q_Y:Y\to Z_Y$ be representations of the same declared property in two strata, and let $\widetilde T_Q:Z_X\to Z_Y$ be its intended transport.
723 
724 \begin{definition}[Property-transport defect]
725 For a declared domain $K\subseteq X$, define
726 \begin{equation}
727 \eps_Q(T;K)
728 =
729 \sup_{x\in K}
730 \norm{
731 Q_Y(Tx)-\widetilde T_QQ_X(x)
732 }.
733 \end{equation}
734 \end{definition}
735 
736 \begin{proposition}[Composition bound for bridge defects]
737 \label{prop:bridge-composition}
738 Let $T:X\to Y$ and $S:Y\to Z$. Suppose the property transports satisfy
739 \begin{equation}
740 \norm{Q_Y(Tx)-\widetilde T_QQ_X(x)}\leq\eps_T
741 \end{equation}
742 for $x\in K$, and
743 \begin{equation}
744 \norm{Q_Z(Sy)-\widetilde S_QQ_Y(y)}\leq\eps_S
745 \end{equation}
746 for $y\in T(K)$. If $\norm{\widetilde S_Q}\leq L_S$, then the composite bridge obeys
747 \begin{equation}
748 \eps_Q(S\circ T;K)
749 \leq
750 \eps_S+L_S\eps_T
751 \end{equation}
752 with intended property transport $\widetilde S_Q\widetilde T_Q$.
753 \end{proposition}
754 
755 \begin{proof}
756 For $x\in K$,
757 \begin{align}
758 &\norm{Q_Z(STx)-\widetilde S_Q\widetilde T_QQ_X(x)}\\
759 &\quad\leq
760 \norm{Q_Z(STx)-\widetilde S_QQ_Y(Tx)}
761 +
762 \norm{\widetilde S_Q\left(Q_Y(Tx)-\widetilde T_QQ_X(x)\right)}\\
763 &\quad\leq
764 \eps_S+L_S\eps_T.
765 \end{align}
766 Taking the supremum proves the claim.
767 \end{proof}
768 
769 \begin{remark}
770 The proposition is the elementary source of many later subadditive error bounds. It also explains why long transformation paths require holonomy checks: even small local defects can accumulate or be amplified by ill-conditioned transports.
771 \end{remark}
772 
773 \subsection{The resolved lineage state}
774 
775 \begin{definition}[Resolved improvement lineage state]
776 A resolved improvement lineage state at generation $t$ is a tuple
777 \begin{equation}
778 \mathfrak L_t
779 =
780 \left(
781 A_t,
782 \Ocal_t,
783 \Ecal_t,
784 \Vcal_t,
785 \Lambda_t,
786 \Wcal_t
787 \right),
788 \end{equation}
789 where:
790 \begin{enumerate}
791 \item $A_t$ is the active stratified agent with atlas $\mathfrak A_t$;
792 \item $\Ocal_t$ is a proof-carrying repair operad;
793 \item $\Ecal_t$ is an append-only external-evidence ledger;
794 \item $\Vcal_t$ is an evaluator atlas;
795 \item $\Lambda_t$ is a directed acyclic lineage graph whose edges carry transformation receipts;
796 \item $\Wcal_t$ is a witness, checkpoint, sandbox, and rollback structure.
797 \end{enumerate}
798 \end{definition}
799 
800 The executable archive $\Acal_t$ is a distinguished ancestor-closed subset of the vertices of $\Lambda_t$. Each archived node contains either an executable artifact or a reconstruction proof sufficient to reproduce it from immutable ancestors and declared build inputs.
801 
802 \begin{definition}[Append-only evidence embedding]
803 Evidence preservation from generation $t$ to $t+1$ is an injective map
804 \begin{equation}
805 \iota_t:\Ecal_t\hookrightarrow\Ecal_{t+1}
806 \end{equation}
807 that preserves payload, source, timestamp order, precommitment status, provenance type, and cryptographic integrity fields. A later correction may append a refutation or reinterpretation receipt, but it may not silently alter the old entry.
808 \end{definition}
809 
810 A witness structure may be represented as
811 \begin{equation}
812 \Wcal_t
813 =
814 \left(
815 \Vcal_t^{\mathrm{fr}},
816 C_t,
817 \Bcal_t^{\mathrm{sand}},
818 R_t,
819 \Pi_t^{\mathrm{auth}}
820 \right),
821 \end{equation}
822 where $\Vcal_t^{\mathrm{fr}}$ is the frozen evaluator instance for the current transaction, $C_t$ is a protected checkpoint, $\Bcal_t^{\mathrm{sand}}$ is the sandbox boundary, $R_t$ is the rollback procedure, and $\Pi_t^{\mathrm{auth}}$ records effect authority.
823 
824 \begin{figure}[ht]
825 \centering
826 \begin{tikzpicture}[
827   node distance=10mm and 14mm,
828   box/.style={draw=KMBlue, rounded corners, fill=KMLight, align=center, minimum width=30mm, minimum height=9mm, font=\small},
829   arr/.style={-{Latex[length=2.2mm]}, thick, draw=KMGray}
830 ]
831 \node[box] (agent) {active stratified\\agent $A_t$};
832 \node[box, above=of agent] (lineage) {lineage graph and\\archive $\Lambda_t,\Acal_t$};
833 \node[box, left=of agent] (repair) {repair operad\\$\Ocal_t$};
834 \node[box, right=of agent] (eval) {evaluator atlas\\$\Vcal_t$};
835 \node[box, below left=of agent] (evidence) {evidence ledger\\$\Ecal_t$};
836 \node[box, below right=of agent] (witness) {witness and rollback\\$\Wcal_t$};
837 \draw[arr] (repair) -- (agent);
838 \draw[arr] (agent) -- (lineage);
839 \draw[arr] (lineage) -- (eval);
840 \draw[arr] (evidence) -- (eval);
841 \draw[arr] (eval) -- (witness);
842 \draw[arr] (witness) -- (agent);
843 \draw[arr, bend left=18] (agent) to (evidence);
844 \draw[arr, bend left=18] (lineage) to (repair);
845 \end{tikzpicture}
846 \caption{A resolved lineage state. No component alone is the persistent improving object; persistence is carried by their proof-carrying closure.}
847 \label{fig:resolved-lineage}
848 \end{figure}
849 
850 \subsection{Lineage receipts and continuation}
851 
852 \begin{definition}[Lineage-edge receipt]
853 A lineage edge
854 \begin{equation}
855 A\xrightarrow{\Pi}B
856 \end{equation}
857 has a receipt
858 \begin{equation}
859 \Pi=
860 \left(
861 \mathrm{id}_A,
862 \mathrm{id}_B,
863 \theta,
864 \Wcal,
865 \Ecal_{\mathrm{pre}},
866 \Ecal_{\mathrm{post}},
867 \widehat g,
868 \mathscr I,
869 \mathrm{tests},
870 \mathrm{decision},
871 \mathrm{rollback},
872 \mathrm{digest}
873 \right),
874 \end{equation}
875 recording the parent and child, transformation $\theta$, frozen witness, pre- and post-reveal evidence, precommitted effect claim $\widehat g$, protected invariants, validation results, commit decision, rollback point, and integrity digest.
876 \end{definition}
877 
878 \begin{definition}[Validated continuation]
879 A descendant $B$ is a validated continuation of $A$ when there exists a directed path in $\Lambda$ from $A$ to $B$ whose every edge receipt is valid under the witness active at that edge.
880 \end{definition}
881 
882 Structural equivalence does not imply validated continuation. Two copies may contain identical weights, memory, workspace state, and planner while occupying distinct nodes of $\Lambda$ because their causal histories and authorization paths differ.
883 
884 \begin{theorem}[Category of validated lineage paths]
885 \label{thm:lineage-category}
886 Assume valid edge receipts are closed under proof-path concatenation and contain identity receipts for zero-change transitions. Then lineage states are objects and validated receipt paths are morphisms of a category $\mathbf{Lin}$.
887 \end{theorem}
888 
889 \begin{proof}
890 The identity morphism on $A$ is the empty path with the identity receipt. Composition is path concatenation with receipt composition. Concatenation is associative, and the empty path is a left and right unit. Closure of valid receipts under concatenation ensures the composite is again a morphism.
891 \end{proof}
892 
893 \begin{remark}
894 If every bridge in a path is invertible and its inverse has a valid receipt, the corresponding subcategory is a groupoid. Ordinary improvement lineages need not be invertible because training, compression, and evidence revelation are generally irreversible.
895 \end{remark}
896 
897 \subsection{Forks and symmetry}
898 
899 A lineage may branch:
900 \begin{equation}
901 A\longrightarrow B_1,
902 \qquad
903 A\longrightarrow B_2.
904 \end{equation}
905 No definition of recursive improvement requires one child to be the unique continuation of the parent.
906 
907 \begin{theorem}[Symmetric forks do not justify unique selection]
908 \label{thm:fork-symmetry}
909 Let $D$ be all data available to a selector after a two-child fork. Suppose an involution $\sigma$ swaps $B_1$ and $B_2$ while leaving the witness, evidence, utilities, resource records, and lineage proofs invariant. Let a selector return a subset
910 \begin{equation}
911 s(D)\subseteq\{B_1,B_2\}
912 \end{equation}
913 and satisfy equivariance $s(\sigma D)=\sigma s(D)$. Then $s(D)$ cannot contain exactly one child.
914 \end{theorem}
915 
916 \begin{proof}
917 Because $\sigma D=D$, equivariance gives $s(D)=\sigma s(D)$. The only subsets of $\{B_1,B_2\}$ fixed by the swap are $\varnothing$ and $\{B_1,B_2\}$. A singleton is not invariant.
918 \end{proof}
919 
920 The result is operational rather than metaphysical. It says that a unique choice requires an asymmetry in evidence, resource allocation, policy, or tie-breaking rule. The archive may preserve both branches while a separate scheduler chooses which one receives the next experiment.
921 
922 \subsection{Resolution consistency}
923 
924 A lineage state is \emph{resolution-consistent} when:
925 \begin{enumerate}
926 \item every active component belongs to a declared stratum;
927 \item every cross-stratum inference cites a typed bridge and preservation profile;
928 \item every mutable belief used by the planner carries a provenance pointer into $\Ecal_t$ or is explicitly typed as simulation or hypothesis;
929 \item every repair operation belongs to $\Ocal_t$ or is quarantined as an untyped proposal;
930 \item every committed descendant has a valid receipt and rollback point;
931 \item every archived best-score claim can be reconstructed from protected traces and the evaluator version that produced it.
932 \end{enumerate}
933 Resolution consistency does not guarantee correctness. It prevents silent type changes from masquerading as theorem transport or causal evidence.
934 
935 \section{Relative Introspection}
936 
937 \subsection{Intervention semantics}
938 
939 A self-improver need not reconstruct every internal degree of freedom. It needs a self-model sufficient for the authorized modifications it is currently considering.
940 
941 Let $\Dcal_t$ be an authorized modification family containing a no-op intervention $\delta_0$. For $\delta\in\Dcal_t$, let
942 \begin{equation}
943 K_{A_t}^{\delta}(\cdot)
944 =
945 P\left(Y\in\cdot\mid \operatorname{do}(\delta),A_t\right)
946 \end{equation}
947 be the protected external-outcome kernel. A compressed causal self-model $\phi_t(A_t)$ induces a predicted kernel
948 \begin{equation}
949 \widehat K_{\phi_t(A_t)}^{\delta}(\cdot).
950 \end{equation}
951 The model is evaluated on counterfactual consequences, not on its ability to reproduce hidden activations for their own sake.
952 
953 \begin{definition}[$\eps$-intervention sufficiency]
954 The map $\phi_t$ is $\eps$-intervention-sufficient for $\Dcal_t$ when
955 \begin{equation}
956 \sup_{\delta\in\Dcal_t}
957 \TV\left(
958 K_{A_t}^{\delta},
959 \widehat K_{\phi_t(A_t)}^{\delta}
960 \right)
961 \leq\eps.
962 \end{equation}
963 \end{definition}
964 
965 The definition is relative to the current intervention grammar. It does not imply that the self-model is globally sufficient for arbitrary source-code rewrites, unseen tools, new sensors, or future ontology extensions.
966 
967 \subsection{Decision regret and ranking competence}
968 
969 \begin{theorem}[Decision regret from relative introspection]
970 \label{thm:introspection-regret}
971 Let $u:Y\to[0,1]$ be a bounded utility. Define
972 \begin{equation}
973 V(\delta)=\mathbb E[u(Y)\mid A_t,\delta],
974 \qquad
975 \widehat V(\delta)=\mathbb E[u(Y)\mid \phi_t(A_t),\delta].
976 \end{equation}
977 If $\phi_t$ is $\eps$-intervention-sufficient, then
978 \begin{equation}
979 \sup_{\delta\in\Dcal_t}
980 \abs{V(\delta)-\widehat V(\delta)}\leq\eps.
981 \end{equation}
982 If
983 \begin{equation}
984 \delta^*\in\argmax_{\delta\in\Dcal_t}V(\delta),
985 \qquad
986 \widehat\delta\in\argmax_{\delta\in\Dcal_t}\widehat V(\delta),
987 \end{equation}
988 then
989 \begin{equation}
990 V(\delta^*)-V(\widehat\delta)\leq2\eps.
991 \end{equation}
992 \end{theorem}
993 
994 \begin{proof}
995 For probability measures at total-variation distance at most $\eps$, expectations of functions taking values in $[0,1]$ differ by at most $\eps$. Hence $\abs{V(\delta)-\widehat V(\delta)}\leq\eps$ uniformly. Then
996 \begin{align}
997 V(\delta^*)-V(\widehat\delta)
998 &\leq
999 V(\delta^*)-\widehat V(\delta^*)
1000 +\widehat V(\widehat\delta)-V(\widehat\delta)\\
1001 &\leq2\eps,
1002 \end{align}
1003 where $\widehat V(\widehat\delta)\geq\widehat V(\delta^*)$ was used.
1004 \end{proof}
1005 
1006 \begin{corollary}[Margin-preserving ranking]
1007 \label{cor:ranking-margin}
1008 If
1009 \begin{equation}
1010 V(\delta_1)-V(\delta_2)>2\eps,
1011 \end{equation}
1012 then
1013 \begin{equation}
1014 \widehat V(\delta_1)>\widehat V(\delta_2).
1015 \end{equation}
1016 In particular, if the true best intervention is unique and its gap from every alternative exceeds $2\eps$, the self-model selects the true best intervention.
1017 \end{corollary}
1018 
1019 \begin{proof}
1020 One has
1021 \begin{equation}
1022 \widehat V(\delta_1)-\widehat V(\delta_2)
1023 \geq
1024 V(\delta_1)-V(\delta_2)-2\eps>0.
1025 \end{equation}
1026 \end{proof}
1027 
1028 \begin{proposition}[Safe commit against a no-op baseline]
1029 \label{prop:safe-commit}
1030 Assume the same uniform error bound and let $\delta_0$ be the no-op. If a candidate $\delta$ is committed only when
1031 \begin{equation}
1032 \widehat V(\delta)-\widehat V(\delta_0)>2\eps,
1033 \end{equation}
1034 then the true gain is positive:
1035 \begin{equation}
1036 V(\delta)-V(\delta_0)>0.
1037 \end{equation}
1038 \end{proposition}
1039 
1040 \begin{proof}
1041 The true gain is bounded below by
1042 \begin{equation}
1043 V(\delta)-V(\delta_0)
1044 \geq
1045 \widehat V(\delta)-\widehat V(\delta_0)-2\eps>0.
1046 \end{equation}
1047 \end{proof}
1048 
1049 \begin{corollary}[Cost-sensitive regret]
1050 Suppose, in addition, that normalized intervention cost $c(\delta)$ is predicted by $\widehat c(\delta)$ with uniform error at most $\eps_c$. For net utility
1051 \begin{equation}
1052 W(\delta)=V(\delta)-\lambda c(\delta),
1053 \qquad
1054 \widehat W(\delta)=\widehat V(\delta)-\lambda\widehat c(\delta),
1055 \end{equation}
1056 selection by $\widehat W$ has regret at most
1057 \begin{equation}
1058 2\left(\eps+\lambda\eps_c\right).
1059 \end{equation}
1060 \end{corollary}
1061 
1062 \begin{proof}
1063 The net-utility approximation error is at most $\eps+\lambda\eps_c$. Apply the proof of \cref{thm:introspection-regret} with this error bound.
1064 \end{proof}
1065 
1066 \begin{remark}
1067 Low average prediction error does not imply good candidate ranking. Selection depends on errors relative to pairwise gain margins. A model can estimate the mean effect of interventions accurately while systematically misordering the candidates near the decision boundary.
1068 \end{remark}
1069 
1070 \subsection{Coverage, expansion, and misspecification}
1071 
1072 Define the intervention-coverage error
1073 \begin{equation}
1074 \eps_{\Dcal}(\phi;A)
1075 =
1076 \sup_{\delta\in\Dcal}
1077 \TV\left(K_A^\delta,\widehat K_{\phi(A)}^\delta\right).
1078 \end{equation}
1079 
1080 \begin{proposition}[Coverage monotonicity]
1081 If $\Dcal\subseteq\Dcal'$, then
1082 \begin{equation}
1083 \eps_{\Dcal'}(\phi;A)
1084 \geq
1085 \eps_{\Dcal}(\phi;A).
1086 \end{equation}
1087 \end{proposition}
1088 
1089 \begin{proof}
1090 The supremum is taken over a larger set.
1091 \end{proof}
1092 
1093 Thus an introspection guarantee does not automatically survive grammar expansion. Adding a new tool, parameter family, memory operation, or evaluator-edit operation enlarges the intervention class and requires new validation.
1094 
1095 \begin{definition}[Intervention-complete relative to a repair grammar]
1096 Let $\Ocal$ be a repair grammar and let $\Dcal(\Ocal)$ be the interventions denoted by its executable operations. A self-model is intervention-complete at tolerance $\eps$ relative to $\Ocal$ when it is $\eps$-intervention-sufficient for $\Dcal(\Ocal)$.
1097 \end{definition}
1098 
1099 This is a relative completeness notion. It makes no claim about interventions that the current grammar cannot express.
1100 
1101 \subsection{Identifiability and active probing}
1102 
1103 Let $\Hcal$ be a finite set of causal hypotheses about the source of a defect, such as internal model change, world change, memory corruption, sensor corruption, another agent's intervention, evaluator change, or an unknown mechanism. Let $\Pcal_{\mathrm{probe}}$ be a permitted probe family. Under hypothesis $h$ and probe $p$, the protected observation kernel is $K_h^p$.
1104 
1105 \begin{theorem}[Observational indistinguishability]
1106 \label{thm:observational-indistinguishability}
1107 If two hypotheses $h,h'\in\Hcal$ satisfy
1108 \begin{equation}
1109 K_h^p=K_{h'}^p
1110 \qquad
1111 \text{for every permitted probe }p\in\Pcal_{\mathrm{probe}},
1112 \end{equation}
1113 then no adaptive decision rule based only on permitted probe transcripts can distinguish $h$ from $h'$ with different transcript distributions.
1114 \end{theorem}
1115 
1116 \begin{proof}
1117 Proceed by induction over probe rounds. Before the first probe the transcript distributions coincide. If they coincide through round $k$, an adaptive policy chooses the same distribution over the next probe under both hypotheses because it is a function of the transcript. Conditional on every selected probe, the next observation kernels are equal. Therefore the round-$k+1$ transcript distributions coincide. The conclusion holds for every finite stopping time and hence for every decision rule measurable with respect to the transcript.
1118 \end{proof}
1119 
1120 \begin{definition}[Deterministically separating probe family]
1121 Suppose every $K_h^p$ is a point mass at an outcome $y(h,p)$. The family $\Pcal_{\mathrm{probe}}$ separates $\Hcal$ when for every $h\neq h'$ there exists $p\in\Pcal_{\mathrm{probe}}$ such that
1122 \begin{equation}
1123 y(h,p)\neq y(h',p).
1124 \end{equation}
1125 \end{definition}
1126 
1127 \begin{theorem}[Finite active identification]
1128 \label{thm:finite-active-identification}
1129 Let $\Hcal$ be finite and let $\Pcal_{\mathrm{probe}}$ be deterministically separating. Then an adaptive elimination procedure identifies the true hypothesis after at most $\abs{\Hcal}-1$ probes.
1130 \end{theorem}
1131 
1132 \begin{proof}
1133 Maintain the set $S$ of hypotheses consistent with the transcript. If $\abs S>1$, choose distinct $h,h'\in S$ and a probe separating them. The observed outcome eliminates at least one of $h,h'$, so the size of $S$ decreases by at least one. Starting from $\abs{\Hcal}$ candidates, at most $\abs{\Hcal}-1$ reductions are required.
1134 \end{proof}
1135 
1136 \begin{remark}
1137 No corresponding finite exact-identification guarantee follows automatically in the stochastic case. One instead needs quantitative separation, repeated evidence, posterior calibration, and an abstention rule when permitted evidence cannot resolve the hypotheses.
1138 \end{remark}
1139 
1140 \subsection{Prospective causal closure}
1141 
1142 A useful self-model participates in a complete causal cycle:
1143 \begin{equation}
1144 \phi_t(A_t)
1145 \xrightarrow{\text{candidate policy}}
1146 \delta_t
1147 \xrightarrow{\text{world execution}}
1148 Y_{t+1}
1149 \xrightarrow{\text{evidence update}}
1150 \phi_{t+1}(A_{t+1}).
1151 \end{equation}
1152 The candidate is selected before the protected outcome is revealed. The outcome affects future beliefs and repair policy. Cutting either planner use or consequence feedback destroys prospective closure even if the internal self-representation remains decodable.
1153 
1154 \begin{definition}[Prospectively closed self-model]
1155 A self-model process is prospectively closed relative to $\Dcal$ when:
1156 \begin{enumerate}
1157 \item its state causally affects selection among interventions in $\Dcal$;
1158 \item it predicts protected consequences before reveal;
1159 \item committed outcomes update the future self-model and candidate policy;
1160 \item the update distinguishes direct observation, communication, simulation, and imported memory by provenance type.
1161 \end{enumerate}
1162 \end{definition}
1163 
1164 \subsection{Self-confirmation failure}
1165 
1166 \begin{example}[Internal error reduction without external improvement]
1167 Let the environment always produce protected outcome $Y=1$, independently of the agent's action. Let an internal predictor be $p\in\R$ with internal score
1168 \begin{equation}
1169 J_{\mathrm{int}}(p)=-(p-Y)^2.
1170 \end{equation}
1171 A self-intervention setting $p\leftarrow1$ improves $J_{\mathrm{int}}$ to zero error but does not change the world, task success, or action consequence. More extremely, an intervention that rewrites the error register itself can report zero error without even improving prediction. Therefore reduction of an endogenous mismatch is not external evidence of improvement unless the mismatch is tied to protected outcomes through a frozen causal test.
1172 \end{example}
1173 
1174 The example is intentionally elementary. Its role is to separate self-consistency from world-grounded competence. Relative introspection must be evaluated through consequences outside the mutable representation being improved.
1175 
1176 \section{Proof-Carrying Repair Operads}
1177 
1178 \subsection{From repair libraries to typed grammars}
1179 
1180 A list of successful patches is not yet a repair calculus. A reusable repair mechanism must specify the systems to which a patch applies, the defect it is intended to alter, the evidence supporting that claim, the protected structures it may touch, the effects it predicts, and the rules by which it composes with other repairs.
1181 
1182 Let $\mathsf C$ be the color set of the functional atlas. The syntactic repair grammar is a colored symmetric operad $\overline\Ocal$ with operation sets
1183 \begin{equation}
1184 \overline\Ocal(c_1,\ldots,c_n;c).
1185 \end{equation}
1186 An operation consumes typed inputs from strata $c_1,\ldots,c_n$ and produces a candidate correction in output stratum $c$.
1187 
1188 \begin{definition}[Proof-carrying repair schema]
1189 A repair schema is a tuple
1190 \begin{equation}
1191 \omega=
1192 \left(
1193 \sigma,
1194 \bm c_{\mathrm{in}},
1195  c_{\mathrm{out}},
1196 D,
1197 C,
1198 Q,
1199 [r],
1200 u,
1201 \bm g,
1202 \mathscr I,
1203 \Pi
1204 \right),
1205 \end{equation}
1206 where:
1207 \begin{itemize}
1208 \item $\sigma$ is a recognizable failure signature;
1209 \item $\bm c_{\mathrm{in}}=(c_1,\ldots,c_n)$ and $c_{\mathrm{out}}$ are input and output colors;
1210 \item $D$ is the declared local defect-response map;
1211 \item $C$ is an applicability precondition;
1212 \item $Q$ is a postcondition;
1213 \item $[r]$ is a local obstruction certificate or unresolved-obstruction record;
1214 \item $\nu$ is the correction program;
1215 \item $\bm g$ is a precommitted effect profile;
1216 \item $\mathscr I$ is a protected-invariant contract;
1217 \item $\Pi$ is a proof, evidence, provenance, and rollback receipt.
1218 \end{itemize}
1219 \end{definition}
1220 
1221 A useful effect profile is vector-valued:
1222 \begin{equation}
1223 \bm g=
1224 \left(
1225 \Delta J,
1226 \Delta C,
1227 \Delta R,
1228 \eps_{\mathrm{leak}},
1229 \eps_{\mathrm{hol}},
1230 \Delta\Gamma,
1231 \sigma_g
1232 \right),
1233 \end{equation}
1234 recording predicted task gain, resource change, risk change, leakage, semantic holonomy, metaproductivity change, and uncertainty. Scalarizing this vector is an evaluator decision, not part of the repair syntax.
1235 
1236 \subsection{Syntactic composition and semantic authorization}
1237 
1238 Strictly speaking, applicability-dependent repair composition is partial. We therefore separate a total syntactic operad from an evidence-dependent authorization layer.
1239 
1240 \begin{definition}[Proof-carrying repair operad]
1241 A proof-carrying repair operad is a pair
1242 \begin{equation}
1243 \Ocal=(\overline\Ocal,\Valid),
1244 \end{equation}
1245 where $\overline\Ocal$ is a colored symmetric operad of syntactically composable repair programs and
1246 \begin{equation}
1247 \Valid_{\Ecal,\Wcal}(\omega,\Pi)
1248 \in
1249 \{\mathrm{valid},\mathrm{invalid},\mathrm{unresolved}\}
1250 \end{equation}
1251 is an evidence- and witness-indexed validation judgment. Only operations judged valid are executable. Unresolved operations remain quarantined.
1252 \end{definition}
1253 
1254 Each executable operation has a Hoare-style contract
1255 \begin{equation}
1256 \{C_\omega\}\ \omega\ \{Q_\omega\}.
1257 \end{equation}
1258 Syntactic substitution
1259 \begin{equation}
1260 \omega\circ(\omega_1,\ldots,\omega_n)
1261 \end{equation}
1262 is semantically authorized only when the inner postconditions and bridge guarantees imply the outer precondition.
1263 
1264 \begin{definition}[Receipt composition]
1265 Let $\Pi_i$ validate $\omega_i$, and let $\Pi$ validate the outer operation $\omega$ conditional on $C_\omega$. The composed receipt
1266 \begin{equation}
1267 \Pi\star(\Pi_1,\ldots,\Pi_n)
1268 \end{equation}
1269 is the provenance-directed acyclic graph obtained by proof-path concatenation, evidence union, explicit dependency links, and deduplication of immutable evidence references without duplicating linear observation authority.
1270 \end{definition}
1271 
1272 \begin{theorem}[Compositional receipt soundness]
1273 \label{thm:receipt-soundness}
1274 Assume:
1275 \begin{enumerate}
1276 \item every primitive receipt is semantically sound;
1277 \item execution of each $\omega_i$ from $C_{\omega_i}$ establishes $Q_{\omega_i}$;
1278 \item the conjunction of the inner postconditions and bridge guarantees implies $C_\omega$;
1279 \item execution of $\omega$ from $C_\omega$ establishes $Q_\omega$;
1280 \item receipt composition preserves provenance and protected evidence references.
1281 \end{enumerate}
1282 Then the composed receipt is sound for
1283 \begin{equation}
1284 \omega\circ(\omega_1,\ldots,\omega_n)
1285 \end{equation}
1286 with precondition $\bigwedge_iC_{\omega_i}$ and postcondition $Q_\omega$.
1287 \end{theorem}
1288 
1289 \begin{proof}
1290 Starting from $\bigwedge_iC_{\omega_i}$, soundness of the primitive receipts and their operational contracts yields every $Q_{\omega_i}$. By assumption, these postconditions and the bridge guarantees establish $C_\omega$. Soundness of the outer receipt then yields $Q_\omega$. Because the composed receipt records the dependency graph and does not create new observation authority, the proof path certifies exactly this semantic composition.
1291 \end{proof}
1292 
1293 \begin{remark}
1294 The theorem is conditional on a sound receipt checker and sound primitive contracts. It does not make an unverified learned claim true by placing it inside a receipt. The receipt exposes the claim, evidence, dependencies, and authority under which the claim may be used.
1295 \end{remark}
1296 
1297 \subsection{Sequential and parallel repair}
1298 
1299 Sequential repair is operadic substitution. Parallel repair requires an additional compatibility test because two locally valid corrections may interfere through shared state or noncommuting updates.
1300 
1301 Let $U$ and $V$ be bounded linear correction operators. Applying the updates in opposite orders gives
1302 \begin{equation}
1303 (I+U)(I+V)
1304 \qquad\text{and}\qquad
1305 (I+V)(I+U).
1306 \end{equation}
1307 
1308 \begin{proposition}[Exact order defect]
1309 \label{prop:repair-order-defect}
1310 The difference between the two update orders is the commutator:
1311 \begin{equation}
1312 (I+U)(I+V)-(I+V)(I+U)
1313 =
1314 [U,V]:=UV-VU.
1315 \end{equation}
1316 Hence the operator-norm order defect is exactly $\norm{[U,V]}$.
1317 \end{proposition}
1318 
1319 \begin{proof}
1320 Expand both products and cancel $I$, $U$, and $V$.
1321 \end{proof}
1322 
1323 \begin{proposition}[Orthogonal packet repairs commute]
1324 \label{prop:orthogonal-repairs}
1325 Let $P$ and $Q$ be orthogonal projectors with $PQ=QP=0$. If
1326 \begin{equation}
1327 U=PUP,
1328 \qquad
1329 V=QVQ,
1330 \end{equation}
1331 then
1332 \begin{equation}
1333 UV=VU=0.
1334 \end{equation}
1335 Consequently the repairs commute exactly and
1336 \begin{equation}
1337 (I+U)(I+V)=I+U+V.
1338 \end{equation}
1339 \end{proposition}
1340 
1341 \begin{proof}
1342 One has
1343 \begin{equation}
1344 UV=PUPQVQ=0,
1345 \qquad
1346 VU=QVQPUP=0.
1347 \end{equation}
1348 \end{proof}
1349 
1350 When support is only approximate, the commutator, cross-packet transfer energy, and protected-packet leakage become part of the composite effect profile. A parallel composition is authorized only when these quantities remain below declared tolerances.
1351 
1352 \subsection{Conservative extensions}
1353 
1354 \begin{definition}[Conservative repair-operad extension]
1355 An inclusion
1356 \begin{equation}
1357 \iota_t:\Ocal_t\hookrightarrow\Ocal_{t+1}
1358 \end{equation}
1359 is conservative when:
1360 \begin{enumerate}
1361 \item it is injective on old colors and operations;
1362 \item it preserves identities, permutations, and every old syntactic composition;
1363 \item it preserves old preconditions, postconditions, and protected-invariant contracts on old states;
1364 \item every previously valid old receipt remains valid against the same immutable evidence;
1365 \item new operations cannot retroactively consume or rewrite old observation authority.
1366 \end{enumerate}
1367 \end{definition}
1368 
1369 \begin{proposition}[Old derivations survive conservative extension]
1370 Every valid repair expression and receipt derivation in $\Ocal_t$ remains valid after embedding into $\Ocal_{t+1}$.
1371 \end{proposition}
1372 
1373 \begin{proof}
1374 This is immediate from preservation of operations, composition, contracts, validation judgments, and immutable evidence references.
1375 \end{proof}
1376 
1377 A nonconservative extension may still be scientifically justified, but it must be represented as a revision with explicit refutation or reinterpretation receipts rather than as silent monotone growth.
1378 
1379 \subsection{Repair abstraction and generality}
1380 
1381 A repair episode contains a concrete system state, defect, candidate correction, outcome, and receipt. Recursive compounding requires abstraction from episodes to reusable schemas.
1382 
1383 \begin{definition}[Repair episode]
1384 A repair episode is a tuple
1385 \begin{equation}
1386 e=
1387 \left(
1388 A,
1389 r,
1390 u,
1391 Y_{\mathrm{pre}},
1392 Y_{\mathrm{post}},
1393 \Pi,
1394 \bm g
1395 \right).
1396 \end{equation}
1397 \end{definition}
1398 
1399 \begin{definition}[Repair abstraction operator]
1400 A repair abstraction operator is a map
1401 \begin{equation}
1402 \mathfrak a:
1403 \mathcal H_{\mathrm{episodes}}
1404 \longrightarrow
1405 \overline\Ocal
1406 \end{equation}
1407 that maps one or more concrete episodes to a typed schema with a failure signature, applicability condition, correction program, effect profile, and receipt template.
1408 \end{definition}
1409 
1410 \begin{definition}[Schema generality preorder]
1411 For repair schemas $\omega_1$ and $\omega_2$ with the same typed effect, write
1412 \begin{equation}
1413 \omega_1\preceq\omega_2
1414 \end{equation}
1415 when every state satisfying $C_{\omega_1}$ also satisfies $C_{\omega_2}$ and the two correction semantics agree on $\dom(C_{\omega_1})$ up to the declared transport tolerance. Then $\omega_2$ is at least as general as $\omega_1$.
1416 \end{definition}
1417 
1418 A schema is not validated as general merely because it compresses its training episodes. Its applicability condition and gain claim must be tested on foreign agents, new seeds, new task instances, or transformed representations not used to construct the abstraction.
1419 
1420 \begin{definition}[Foreign-valid repair schema]
1421 A repair schema is foreign-valid at margin $\gamma$ on a target distribution $\mathcal D$ when, under matched resources and its declared applicability predicate, the lower confidence bound on target gain is at least $\gamma$ and all protected invariants pass on agents sampled from $\mathcal D$ that are not descendants used to construct the schema.
1422 \end{definition}
1423 
1424 \subsection{Repair equivalence under exact transport}
1425 
1426 Let $\omega_A$ and $\omega_B$ be repair schemas in systems $A$ and $B$. Suppose invertible maps transport their correction, defect, evidence, and stratum spaces.
1427 
1428 \begin{definition}[Exact transport equivalence of repairs]
1429 Write
1430 \begin{equation}
1431 \omega_A\cong_T\omega_B
1432 \end{equation}
1433 when:
1434 \begin{enumerate}
1435 \item input and output colors correspond under the atlas transport;
1436 \item the defect-response maps commute with correction and defect transports;
1437 \item applicability and postconditions are preserved in both directions;
1438 \item protected invariants and provenance types are preserved;
1439 \item the effect profiles agree after coordinate transport;
1440 \item the receipts map to equivalent proof paths.
1441 \end{enumerate}
1442 \end{definition}
1443 
1444 \begin{proposition}[Exact repair transports form a groupoid]
1445 \label{prop:repair-groupoid}
1446 Repair schemas as objects and exact invertible transport equivalences as morphisms form a groupoid.
1447 \end{proposition}
1448 
1449 \begin{proof}
1450 The identity transport satisfies every condition. The inverse of an exact invertible transport preserves the commuting diagrams and receipt equivalence in the reverse direction. Composition of exact transports preserves types, commuting diagrams, contracts, invariants, effect profiles, and proof-path equivalence by ordinary composition. Hence every morphism is invertible.
1451 \end{proof}
1452 
1453 The approximate case is developed in \cref{sec:repair-transport}. There, transport equivalence is replaced by quantitative residual and gain bounds.
1454 
1455 \subsection{Meta-repair}
1456 
1457 \begin{definition}[Meta-repair]
1458 A meta-repair is an authorized transformation
1459 \begin{equation}
1460 \Theta_t:\Ocal_t\longrightarrow\Ocal_{t+1}
1461 \end{equation}
1462 that changes one or more of:
1463 \begin{enumerate}
1464 \item failure-signature recognition;
1465 \item causal-hypothesis generation;
1466 \item candidate proposal distributions;
1467 \item repair composition and scheduling;
1468 \item receipt validation or confidence allocation;
1469 \item repair abstraction and foreign-transfer testing.
1470 \end{enumerate}
1471 \end{definition}
1472 
1473 A meta-repair is a genuine meta-improvement only if it improves held-out repair performance under matched resources. Adding a task-specific patch to the library may improve the current agent without improving the repair process itself.
1474 
1475 \subsection{Example: a typed verification repair}
1476 
1477 Consider four colors
1478 \begin{equation}
1479 \mathrm{memory},
1480 \mathrm{representation},
1481 \mathrm{verifier},
1482 \mathrm{planner}.
1483 \end{equation}
1484 A composite factuality repair may have the form
1485 \begin{equation}
1486 \omega_{\mathrm{gate}}
1487 \circ
1488 \left(
1489 \omega_{\mathrm{retrieve}},
1490 \omega_{\mathrm{decompose}},
1491 \omega_{\mathrm{check}}
1492 \right).
1493 \end{equation}
1494 The inner operations retrieve evidence, decompose output into claims, and check claims against protected sources. Their postconditions establish the precondition for a planner gate that blocks or revises unsupported claims. The composite receipt records source provenance, claim-to-evidence links, verifier uncertainty, revised spans, latency cost, and regression results. If retrieval and checking use correlated or circular evidence, the outer precondition fails and the composite remains unresolved rather than being treated as verified merely because every module produced an output.
1495 
1496 \section{Authorized Correction Geometry and Obstruction Calculus}
1497 \label{sec:correctability}
1498 
1499 \begin{statusbox}{Block 2 scope}
1500 This block studies whether a declared defect is correctable by the currently authorized repair grammar. It separates five cases: exact linear correctability, quantitative least-squares repair, obstruction induced by protected invariants or one-sided authorization constraints, nonlinear obstruction beyond the first derivative, and cross-stratum incompatibility among otherwise valid local repairs.
1501 \end{statusbox}
1502 
1503 \subsection{Linear correction and quotient obstruction}
1504 
1505 Let $U^{\Auth}$ and $V$ be finite-dimensional real Hilbert spaces. The first is the space of currently authorized infinitesimal corrections; the second is the space in which protected defects are measured. Let
1506 \begin{equation}
1507 D:U^{\Auth}\to V
1508 \end{equation}
1509 be the linearized defect-response map at the current lineage state. For a defect $r\in V$, an exact first-order repair satisfies
1510 \begin{equation}
1511 Du=-r.
1512 \label{eq:linear-repair}
1513 \end{equation}
1514 
1515 \begin{definition}[Authorized obstruction class]
1516 The authorized obstruction to correcting $r$ is
1517 \begin{equation}
1518 \Ob_D(r)=[r]\in\coker D:=V/\im D.
1519 \end{equation}
1520 \end{definition}
1521 
1522 \begin{proposition}[Linear correctability]
1523 Equation \eqref{eq:linear-repair} has an authorized solution if and only if
1524 \begin{equation}
1525 [r]=0\quad\text{in }\coker D.
1526 \end{equation}
1527 \end{proposition}
1528 
1529 \begin{proof}
1530 The equation is solvable exactly when $r\in\im D$, which is equivalent to vanishing of the quotient class.
1531 \end{proof}
1532 
1533 Binary solvability is insufficient for an engineering theory. A lineage must also know how far the defect lies from the correction range, which repair has minimum cost, and which component of the defect certifies failure.
1534 
1535 \subsection{Canonical correctable and obstructed components}
1536 
1537 Let $D^\dagger:V\to U^{\Auth}$ denote the Moore--Penrose pseudoinverse. Define
1538 \begin{equation}
1539 P_{\mathrm{cor}}:=DD^\dagger,
1540 \qquad
1541 P_{\mathrm{ob}}:=I_V-DD^\dagger.
1542 \end{equation}
1543 
1544 \begin{theorem}[Canonical correction--obstruction decomposition]
1545 \label{thm:canonical-obstruction}
1546 For every $r\in V$:
1547 \begin{enumerate}
1548 \item $P_{\mathrm{cor}}$ is the orthogonal projector onto $\im D$;
1549 \item $P_{\mathrm{ob}}$ is the orthogonal projector onto $\ker D^*$;
1550 \item the defect decomposes uniquely as
1551 \begin{equation}
1552 r=r_{\mathrm{cor}}+r_{\mathrm{ob}},
1553 \qquad
1554 r_{\mathrm{cor}}=P_{\mathrm{cor}}r,
1555 \quad
1556 r_{\mathrm{ob}}=P_{\mathrm{ob}}r;
1557 \end{equation}
1558 \item the minimum achievable residual is
1559 \begin{equation}
1560 \inf_{u\in U^{\Auth}}\norm{Du+r}
1561 =\norm{r_{\mathrm{ob}}}
1562 =\dist(r,\im D);
1563 \end{equation}
1564 \item the complete set of least-squares minimizers is
1565 \begin{equation}
1566 -D^\dagger r+\ker D;
1567 \end{equation}
1568 \item if $r$ is exactly correctable, $u^*=-D^\dagger r$ is the unique exact repair of minimum norm.
1569 \end{enumerate}
1570 \end{theorem}
1571 
1572 \begin{proof}
1573 The Moore--Penrose identities imply that $DD^\dagger$ is the orthogonal projector onto $\im D$ and that $I-DD^\dagger$ projects onto $(\im D)^\perp=\ker D^*$. For $u^*=-D^\dagger r$,
1574 \begin{equation}
1575 Du^*+r=(I-DD^\dagger)r=r_{\mathrm{ob}}.
1576 \end{equation}
1577 Every vector $Du$ lies in $\im D$, so orthogonal projection gives
1578 \begin{equation}
1579 \norm{Du+r}^2
1580 =\norm{Du+r_{\mathrm{cor}}}^2+\norm{r_{\mathrm{ob}}}^2
1581 \geq\norm{r_{\mathrm{ob}}}^2.
1582 \end{equation}
1583 Equality holds exactly when $Du=-r_{\mathrm{cor}}$, whose solution set is $-D^\dagger r+\ker D$. The Moore--Penrose solution is orthogonal to $\ker D$, hence has minimum norm.
1584 \end{proof}
1585 
1586 \begin{definition}[Quantitative authorized obstruction]
1587 The quantitative obstruction magnitude is
1588 \begin{equation}
1589 \eta_D(r):=\norm{P_{\mathrm{ob}}r}=\dist(r,\im D).
1590 \end{equation}
1591 The normalized obstruction fraction is
1592 \begin{equation}
1593 \widehat\eta_D(r):=
1594 \begin{cases}
1595 \eta_D(r)/\norm r,&r\neq0,\\
1596 0,&r=0.
1597 \end{cases}
1598 \end{equation}
1599 \end{definition}
1600 
1601 The quotient class $[r]$ records the exact algebraic obstruction. The scalar $\eta_D(r)$ records its metric severity. The two should not be conflated: a nonzero obstruction may be arbitrarily small, while a large correctable defect may have zero obstruction.
1602 
1603 \begin{proposition}[Dual obstruction witness]
1604 \label{prop:dual-obstruction}
1605 For every $r\in V$,
1606 \begin{equation}
1607 \eta_D(r)
1608 =
1609 \max_{\substack{y\in\ker D^*\\ \norm y\leq1}}
1610 \angles{y,r}.
1611 \end{equation}
1612 If $r_{\mathrm{ob}}\neq0$, the maximum is attained by
1613 \begin{equation}
1614 y^*=\frac{r_{\mathrm{ob}}}{\norm{r_{\mathrm{ob}}}}.
1615 \end{equation}
1616 \end{proposition}
1617 
1618 \begin{proof}
1619 For $y\in\ker D^*$, orthogonality gives $\angles{y,r}=\angles{y,r_{\mathrm{ob}}}$. Cauchy--Schwarz bounds this by $\norm{r_{\mathrm{ob}}}$, with equality at the displayed unit vector.
1620 \end{proof}
1621 
1622 The vector $y^*$ is a compact certificate: every authorized linear correction is invisible to it, yet the defect has nonzero projection along it.
1623 
1624 \subsection{Protected invariants and resource-aware correction}
1625 
1626 A repair is not authorized merely because it reduces task error. It must preserve declared invariants. Let
1627 \begin{equation}
1628 C:U\to Z
1629 \end{equation}
1630 be the derivative of a protected invariant map. First-order invariant preservation requires $Cu=0$, so the authorized tangent space is
1631 \begin{equation}
1632 U^{\Auth}=\ker C.
1633 \end{equation}
1634 
1635 \begin{proposition}[Invariant-constrained obstruction]
1636 \label{prop:invariant-obstruction}
1637 Let $D_C:=\restr{D}{\ker C}$. An exact first-order repair preserving the invariant exists if and only if
1638 \begin{equation}
1639 [r]=0\quad\text{in }\coker D_C.
1640 \end{equation}
1641 Equivalently, the stacked system
1642 \begin{equation}
1643 \begin{bmatrix}D\\C\end{bmatrix}u
1644 =
1645 \begin{bmatrix}-r\\0\end{bmatrix}
1646 \end{equation}
1647 is solvable.
1648 \end{proposition}
1649 
1650 \begin{proof}
1651 The first statement is linear correctability on the restricted domain. The second rewrites the repair and invariant equations as one system.
1652 \end{proof}
1653 
1654 Thus a defect may be unconstrained-correctable but authorized-obstructed. This is the correct classification when a score can be improved only by deleting evidence, weakening provenance, exceeding an effect capability, or violating a protected safety or resource invariant.
1655 
1656 Let $W:V\to V$ and $R:U\to U$ be positive-definite self-adjoint operators representing defect importance and repair cost.
1657 
1658 \begin{theorem}[Regularized minimum-cost repair]
1659 \label{thm:regularized-repair}
1660 For $\lambda>0$, the functional
1661 \begin{equation}
1662 \Jcal_\lambda(u)
1663 :=
1664 \frac12\angles{Du+r,W(Du+r)}
1665 +
1666 \frac\lambda2\angles{u,Ru}
1667 \end{equation}
1668 has a unique minimizer
1669 \begin{equation}
1670 u_\lambda
1671 =-(D^*WD+\lambda R)^{-1}D^*Wr.
1672 \end{equation}
1673 It is characterized by the normal equation
1674 \begin{equation}
1675 D^*W(Du_\lambda+r)+\lambda Ru_\lambda=0.
1676 \end{equation}
1677 \end{theorem}
1678 
1679 \begin{proof}
1680 The Hessian $D^*WD+\lambda R$ is positive definite because $R$ is positive definite and $\lambda>0$. Hence $\Jcal_\lambda$ is strictly convex and coercive. Differentiation gives the normal equation and its unique solution.
1681 \end{proof}
1682 
1683 Here $\lambda$ is not a semantic permission to trade away hard invariants. It balances only those costs already declared soft. Hard invariants remain in the domain restriction.
1684 
1685 \subsection{Convex and one-sided authorization}
1686 
1687 Some repairs are not closed under negation. Examples include adding tests, allocating nonnegative compute, increasing redundancy, or modifying parameters subject to inequality constraints. Let $K\subseteq U$ be a nonempty closed convex authorization set and define
1688 \begin{equation}
1689 \eta_{D,K}(r):=\inf_{u\in K}\norm{Du+r}.
1690 \end{equation}
1691 
1692 \begin{proposition}[Convex authorized correctability]
1693 If $K$ is compact, or if $u\mapsto\norm{Du+r}$ is coercive on $K$, then the infimum defining $\eta_{D,K}(r)$ is attained. Exact authorized correction exists if and only if
1694 \begin{equation}
1695 \eta_{D,K}(r)=0.
1696 \end{equation}
1697 \end{proposition}
1698 
1699 \begin{proof}
1700 Continuity plus compactness gives attainment in the first case. Coercivity restricts every minimizing sequence to a compact sublevel set in finite dimension, giving the second. Exact correction is precisely zero residual.
1701 \end{proof}
1702 
1703 When $K$ is a cone, noncorrectability admits a Farkas-type dual witness. Define the dual cone
1704 \begin{equation}
1705 K^*:=\set{z\in U:\angles{z,u}\geq0\text{ for all }u\in K}.
1706 \end{equation}
1707 
1708 \begin{theorem}[Cone-constrained obstruction certificate]
1709 \label{thm:cone-certificate}
1710 Let $K\subseteq U$ be a closed convex cone and assume $D(K)$ is closed. Then an authorized correction $u\in K$ with $Du=-r$ exists if and only if
1711 \begin{equation}
1712 \angles{y,r}\leq0
1713 \end{equation}
1714 for every $y\in V$ satisfying
1715 \begin{equation}
1716 D^*y\in K^*.
1717 \end{equation}
1718 Consequently, any $y$ with $D^*y\in K^*$ and $\angles{y,r}>0$ certifies noncorrectability.
1719 \end{theorem}
1720 
1721 \begin{proof}
1722 Set $C=D(K)$, a closed convex cone. Its dual is
1723 \begin{equation}
1724 C^*=\set{y:D^*y\in K^*}.
1725 \end{equation}
1726 By the bipolar theorem, $-r\in C$ if and only if $\angles{y,-r}\geq0$ for every $y\in C^*$, which is the displayed inequality.
1727 \end{proof}
1728 
1729 \subsection{Sensitivity, conditioning, and empirical certificates}
1730 
1731 Let $\sigma_+(D)$ denote the smallest positive singular value of $D$. When $D\neq0$,
1732 \begin{equation}
1733 \norm{D^\dagger}=\frac{1}{\sigma_+(D)}.
1734 \end{equation}
1735 Small $\sigma_+(D)$ means the correctable component is ill-conditioned: a small defect perturbation may require a large repair.
1736 
1737 \begin{proposition}[Fixed-map sensitivity]
1738 For defects $r,s\in V$,
1739 \begin{align}
1740 \abs{\eta_D(r)-\eta_D(s)}&\leq\norm{r-s},\\
1741 \norm{D^\dagger r-D^\dagger s}&\leq\frac{1}{\sigma_+(D)}\norm{r-s}.
1742 \end{align}
1743 \end{proposition}
1744 
1745 \begin{proof}
1746 The first inequality follows because distance to a fixed closed set is $1$-Lipschitz. The second follows from the operator norm of $D^\dagger$.
1747 \end{proof}
1748 
1749 \begin{proposition}[Transferred residual under model error]
1750 \label{prop:correction-perturbation}
1751 Let $\widehat D=D+\Delta D$ and $\widehat r=r+\Delta r$. For any proposed repair $u$,
1752 \begin{equation}
1753 \abs{\norm{Du+r}-\norm{\widehat Du+\widehat r}}
1754 \leq
1755 \norm{\Delta D}\norm u+\norm{\Delta r}.
1756 \end{equation}
1757 In particular, applying an exact repair for $(D,r)$ to the perturbed system yields residual at most
1758 \begin{equation}
1759 \norm{\Delta D}\norm u+\norm{\Delta r}.
1760 \end{equation}
1761 \end{proposition}
1762 
1763 \begin{proof}
1764 The two residual vectors differ by $\Delta D\,u+\Delta r$. Apply the reverse triangle inequality and submultiplicativity.
1765 \end{proof}
1766 
1767 The next result turns uncertain estimates into a budget-bounded obstruction certificate.
1768 
1769 \begin{theorem}[Robust empirical noncorrectability certificate]
1770 \label{thm:robust-obstruction-certificate}
1771 Suppose, on an event of probability at least $1-\alpha$,
1772 \begin{equation}
1773 \norm{D-\widehat D}\leq\eps_D,
1774 \qquad
1775 \norm{r-\widehat r}\leq\eps_r.
1776 \end{equation}
1777 Fix a repair budget $\norm u\leq B$ and a unit vector $y\in V$. If
1778 \begin{equation}
1779 \angles{y,\widehat r}
1780 >
1781 \eps_r+B\bigl(\norm{\widehat D^*y}+\eps_D\bigr),
1782 \label{eq:robust-dual-gap}
1783 \end{equation}
1784 then, on the same event, no budget-feasible repair satisfies $Du=-r$.
1785 \end{theorem}
1786 
1787 \begin{proof}
1788 For every $\norm u\leq B$,
1789 \begin{align}
1790 \angles{y,Du+r}
1791 &=\angles{D^*y,u}+\angles{y,r}\\
1792 &\geq
1793 -\norm{D^*y}\,B+\angles{y,\widehat r}-\eps_r\\
1794 &\geq
1795 -B\bigl(\norm{\widehat D^*y}+\eps_D\bigr)
1796 +\angles{y,\widehat r}-\eps_r.
1797 \end{align}
1798 Condition \eqref{eq:robust-dual-gap} makes the final expression positive, so $Du+r$ cannot vanish.
1799 \end{proof}
1800 
1801 A negative result can therefore be proof-carrying: it records the witness $y$, the confidence event, the budget $B$, and the positive dual gap.
1802 
1803 \subsection{Obstruction shrinkage under repair expansion}
1804 
1805 Let $V$ remain fixed while the authorized correction grammar expands.
1806 
1807 \begin{theorem}[Authorized obstruction-shrinkage theorem]
1808 \label{thm:obstruction-shrinkage}
1809 Suppose
1810 \begin{equation}
1811 U_t^{\Auth}\subseteq U_{t+1}^{\Auth}
1812 \end{equation}
1813 and
1814 \begin{equation}
1815 \restr{D_{t+1}}{U_t^{\Auth}}=D_t.
1816 \end{equation}
1817 Then
1818 \begin{equation}
1819 \im D_t\subseteq\im D_{t+1},
1820 \end{equation}
1821 there is a canonical surjection
1822 \begin{equation}
1823 q_t:\coker D_t\twoheadrightarrow\coker D_{t+1},
1824 \qquad
1825 [v]_{\im D_t}\longmapsto[v]_{\im D_{t+1}},
1826 \end{equation}
1827 and
1828 \begin{equation}
1829 \dim\coker D_{t+1}\leq\dim\coker D_t.
1830 \end{equation}
1831 The inequality is strict if and only if $\im D_t\subsetneq\im D_{t+1}$.
1832 \end{theorem}
1833 
1834 \begin{proof}
1835 For $u\in U_t^{\Auth}$, $D_tu=D_{t+1}u$, so every old correctable direction remains correctable. Quotienting by a larger subspace gives the canonical surjection. Rank-nullity gives the dimension statement.
1836 \end{proof}
1837 
1838 \begin{definition}[Authorized repair rank]
1839 The authorized repair rank and unresolved obstruction dimension are
1840 \begin{equation}
1841 \rho_t:=\rank D_t,
1842 \qquad
1843 o_t:=\dim\coker D_t.
1844 \end{equation}
1845 \end{definition}
1846 
1847 \begin{proposition}[Exact sequence of newly correctable directions]
1848 \label{prop:obstruction-exact-sequence}
1849 Under the assumptions of \cref{thm:obstruction-shrinkage}, there is a short exact sequence
1850 \begin{equation}
1851 0
1852 \longrightarrow
1853 \frac{\im D_{t+1}}{\im D_t}
1854 \longrightarrow
1855 \coker D_t
1856 \overset{q_t}{\longrightarrow}
1857 \coker D_{t+1}
1858 \longrightarrow0.
1859 \end{equation}
1860 Consequently,
1861 \begin{equation}
1862 o_t-o_{t+1}
1863 =
1864 \dim\frac{\im D_{t+1}}{\im D_t}.
1865 \end{equation}
1866 \end{proposition}
1867 
1868 \begin{proof}
1869 The kernel of $q_t$ consists exactly of classes represented by vectors in $\im D_{t+1}$, modulo $\im D_t$. Surjectivity was proved above. The dimension identity follows from exactness.
1870 \end{proof}
1871 
1872 The quotient $\im D_{t+1}/\im D_t$ is the marginal structural contribution of the new repair grammar. It counts newly reachable defect directions, not merely new source-code branches.
1873 
1874 \subsection{Changing defect spaces}
1875 
1876 Open-set improvement may enlarge the defect ontology itself. Let
1877 \begin{equation}
1878 j_U:U_t^{\Auth}\hookrightarrow U_{t+1}^{\Auth},
1879 \qquad
1880 j_V:V_t\hookrightarrow V_{t+1}
1881 \end{equation}
1882 be injective embeddings satisfying
1883 \begin{equation}
1884 D_{t+1}j_U=j_VD_t.
1885 \label{eq:dynamic-defect-commute}
1886 \end{equation}
1887 
1888 \begin{proposition}[Induced obstruction transport under ontology expansion]
1889 Equation \eqref{eq:dynamic-defect-commute} induces a linear map
1890 \begin{equation}
1891 \overline j_t:\coker D_t\to\coker D_{t+1},
1892 \qquad
1893 [v]\longmapsto[j_Vv].
1894 \end{equation}
1895 Its kernel is canonically isomorphic to
1896 \begin{equation}
1897 \ker\overline j_t
1898 \cong
1899 \frac{j_V(V_t)\cap\im D_{t+1}}{j_V(\im D_t)}.
1900 \end{equation}
1901 Thus an old obstruction vanishes after expansion exactly when the enlarged repair grammar corrects its embedded defect direction.
1902 \end{proposition}
1903 
1904 \begin{proof}
1905 Commutation ensures that changing the representative by $D_tu$ changes $j_Vv$ by $D_{t+1}j_Uu$, so the map is well-defined. The formula for the kernel follows directly from the quotient definition.
1906 \end{proof}
1907 
1908 An evidence-conservative ontology expansion should make $j_V$ explicit. Otherwise disappearance of an old defect may reflect semantic deletion rather than newly acquired correctability.
1909 
1910 \subsection{Packet-resolved correction and cross-packet leakage}
1911 
1912 Let
1913 \begin{equation}
1914 U=\bigoplus_{i=1}^m U_i,
1915 \qquad
1916 V=\bigoplus_{i=1}^m V_i,
1917 \end{equation}
1918 with orthogonal projectors $Q_i$ and $P_i$. Decompose
1919 \begin{equation}
1920 D=D_0+E,
1921 \qquad
1922 D_0:=\sum_{i=1}^m P_iDQ_i.
1923 \end{equation}
1924 The block-diagonal operator $D_0$ represents ideal packet-local correction; $E$ contains all cross-packet effects.
1925 
1926 Suppose each local block admits a right inverse and their direct sum gives
1927 \begin{equation}
1928 Q_0:V\to U,
1929 \qquad
1930 D_0Q_0=I_V.
1931 \end{equation}
1932 
1933 \begin{theorem}[Exact correction under small packet coupling]
1934 \label{thm:packet-neumann}
1935 If
1936 \begin{equation}
1937 q:=\norm{EQ_0}<1,
1938 \end{equation}
1939 then $D$ is surjective and has right inverse
1940 \begin{equation}
1941 Q
1942 =Q_0(I_V+EQ_0)^{-1}
1943 =Q_0\sum_{k=0}^{\infty}(-EQ_0)^k.
1944 \end{equation}
1945 Moreover,
1946 \begin{equation}
1947 \norm Q\leq\frac{\norm{Q_0}}{1-q}.
1948 \end{equation}
1949 For defect $r$, the truncated packet-correction series
1950 \begin{equation}
1951 u^{(n)}
1952 :=-Q_0\sum_{k=0}^{n}(-EQ_0)^kr
1953 \end{equation}
1954 has residual
1955 \begin{equation}
1956 Du^{(n)}+r=(-EQ_0)^{n+1}r,
1957 \end{equation}
1958 and hence
1959 \begin{equation}
1960 \norm{Du^{(n)}+r}\leq q^{n+1}\norm r.
1961 \end{equation}
1962 \end{theorem}
1963 
1964 \begin{proof}
1965 Since $\norm{EQ_0}<1$, the Neumann series for $(I+EQ_0)^{-1}$ converges. Also
1966 \begin{equation}
1967 DQ_0=(D_0+E)Q_0=I+EQ_0,
1968 \end{equation}
1969 so $DQ=I$. The norm bound follows from the geometric series. For the truncation,
1970 \begin{equation}
1971 (I+EQ_0)\sum_{k=0}^n(-EQ_0)^k
1972 =I-(-EQ_0)^{n+1},
1973 \end{equation}
1974 which gives the residual formula.
1975 \end{proof}
1976 
1977 This theorem formalizes a useful middle regime. Cross-packet effects need not vanish. They must be sufficiently small relative to the local correction inverse for iterative coordination to converge.
1978 
1979 \begin{definition}[Packet leakage profile]
1980 The directed leakage from correction packet $j$ into defect packet $i$ is
1981 \begin{equation}
1982 \ell_{ij}:=\norm{P_iDQ_j},
1983 \qquad i\neq j.
1984 \end{equation}
1985 The matrix $(\ell_{ij})$ is the packet leakage profile.
1986 \end{definition}
1987 
1988 A repair receipt should record both the target block and this off-block profile. A patch that corrects its intended packet while strongly disturbing protected packets is not locally authorized merely because the total score improves.
1989 
1990 \subsection{Gauge-fixed nonlinear local repair by contraction}
1991 
1992 Let $U,V$ be Banach spaces and let the post-correction defect map be
1993 \begin{equation}
1994 \Fcal(u)=r+Du+N(u),
1995 \qquad N(0)=0.
1996 \end{equation}
1997 When $D$ has a nontrivial kernel, a bounded right inverse does not make the full nonlinear repair unique: two roots can differ along neutral or symmetry directions. The contraction argument therefore requires a declared complement, or gauge, in which the right inverse selects one representative.
1998 
1999 \begin{theorem}[Gauge-fixed local nonlinear repair by contraction]
2000 \label{thm:nonlinear-repair}
2001 Suppose $D:U\to V$ has a bounded right inverse $Q:V\to U$, so $DQ=I_V$, and set
2002 \begin{equation}
2003 W:=\im Q.
2004 \end{equation}
2005 Then $\restr D W:W\to V$ is an isomorphism with inverse $Q$. Let
2006 \begin{equation}
2007 B_R^W:=\set{w\in W:\norm w\leq R}.
2008 \end{equation}
2009 Suppose
2010 \begin{equation}
2011 \norm{N(u)-N(v)}\leq L_N\norm{u-v}
2012 \qquad(u,v\in B_R^W),
2013 \end{equation}
2014 with
2015 \begin{equation}
2016 q:=\norm QL_N<1,
2017 \end{equation}
2018 and
2019 \begin{equation}
2020 \norm{Qr}+\norm Q\sup_{u\in B_R^W}\norm{N(u)}\leq R.
2021 \end{equation}
2022 Then there exists a unique $u^*\in B_R^W$ satisfying $\Fcal(u^*)=0$. The gauge-fixed iteration
2023 \begin{equation}
2024 u_{k+1}=-Qr-QN(u_k)
2025 \end{equation}
2026 converges to $u^*$ from every $u_0\in B_R^W$, with
2027 \begin{equation}
2028 \norm{u_k-u^*}\leq\frac{q^k}{1-q}\norm{u_1-u_0}.
2029 \end{equation}
2030 \end{theorem}
2031 
2032 \begin{proof}
2033 Define $T(u)=-Qr-QN(u)$. Because $T$ takes values in $W$, the radius condition gives $T(B_R^W)\subseteq B_R^W$. The Lipschitz condition gives
2034 \begin{equation}
2035 \norm{T(u)-T(v)}\leq q\norm{u-v}.
2036 \end{equation}
2037 Banach's fixed-point theorem yields a unique fixed point in $B_R^W$ and the standard a posteriori convergence estimate. Applying $D$ to the fixed-point equation and using $DQ=I_V$ gives $Du^*=-r-N(u^*)$.
2038 \end{proof}
2039 
2040 \begin{remark}[Scope of uniqueness]
2041 The theorem proves uniqueness only in the selected gauge $W=\im Q$. Other roots may exist elsewhere in $U$ when $\ker D\neq0$. Such roots must be compared through the later Lyapunov--Schmidt reduction, cost criteria, or explicit symmetry quotients rather than being ruled out by the contraction argument.
2042 \end{remark}
2043 
2044 \begin{proposition}[Nonlinear obstruction floor]
2045 \label{prop:nonlinear-floor}
2046 Let $P_{\mathrm{ob}}$ be the orthogonal obstruction projector of $D$. For every $u$,
2047 \begin{equation}
2048 \norm{\Fcal(u)}
2049 \geq
2050 \norm{P_{\mathrm{ob}}r}-\norm{P_{\mathrm{ob}}N(u)}.
2051 \end{equation}
2052 Consequently, if $\norm{P_{\mathrm{ob}}N(u)}\leq\gamma$ on an authorized region, then no repair in that region can achieve residual below
2053 \begin{equation}
2054 \max\set{0,\eta_D(r)-\gamma}.
2055 \end{equation}
2056 \end{proposition}
2057 
2058 \begin{proof}
2059 Since $P_{\mathrm{ob}}D=0$,
2060 \begin{equation}
2061 P_{\mathrm{ob}}\Fcal(u)=P_{\mathrm{ob}}r+P_{\mathrm{ob}}N(u).
2062 \end{equation}
2063 Apply the reverse triangle inequality and $\norm{\Fcal(u)}\geq\norm{P_{\mathrm{ob}}\Fcal(u)}$.
2064 \end{proof}
2065 
2066 A first-order obstruction can sometimes be canceled by nonlinear terms. The proposition quantifies how large that nonlinear obstruction-channel effect must be.
2067 
2068 \subsection{Newton--Kantorovich repair}
2069 
2070 The gauge-fixed contraction theorem assumes a fixed linear right inverse and a selected complement. Newton iteration permits a changing derivative.
2071 
2072 \begin{theorem}[Newton--Kantorovich repair criterion]
2073 \label{thm:newton-kantorovich}
2074 Let $F:U\to V$ be continuously Fr\'echet differentiable between Banach spaces. Let $u_0\in U$ and suppose $F'(u_0)$ is invertible with inverse $B$. Assume that on the closed ball $\overline B(u_0,R)$,
2075 \begin{equation}
2076 \norm{F'(u)-F'(v)}\leq L\norm{u-v}.
2077 \end{equation}
2078 Set
2079 \begin{equation}
2080 \eta:=\norm{BF(u_0)},
2081 \qquad
2082 h:=\norm B\,L\eta.
2083 \end{equation}
2084 If $h\leq\frac12$ and
2085 \begin{equation}
2086 t_*:=\frac{1-\sqrt{1-2h}}{\norm B\,L}\leq R,
2087 \end{equation}
2088 then Newton iteration
2089 \begin{equation}
2090 u_{n+1}=u_n-F'(u_n)^{-1}F(u_n)
2091 \end{equation}
2092 is well-defined, remains in $\overline B(u_0,t_*)$, and converges to a root $u^*$ of $F$. If the larger radius
2093 \begin{equation}
2094 t^{**}:=\frac{1+\sqrt{1-2h}}{\norm B\,L}
2095 \end{equation}
2096 is contained in the regularity region, the root is unique in $\overline B(u_0,t^{**})$.
2097 \end{theorem}
2098 
2099 \begin{proof}
2100 Consider the scalar majorant
2101 \begin{equation}
2102 f(t)=\eta-t+\frac{\norm B\,L}{2}t^2.
2103 \end{equation}
2104 The condition $h\leq1/2$ gives the two nonnegative roots $t_*$ and $t^{**}$. The Lipschitz derivative bound and the Banach lemma imply that $F'(u)$ remains invertible while $\norm{u-u_0}<1/(\norm B L)$. Standard Newton majorization shows that the increments of the operator iteration are dominated by those of scalar Newton iteration for $f$, whose partial sums remain below $t_*$ and converge. Hence $(u_n)$ is Cauchy, its limit lies in the ball, and continuity gives $F(u^*)=0$. The same majorant estimate gives uniqueness in the larger ball.
2105 \end{proof}
2106 
2107 The criterion makes three quantities auditable: the initial predicted correction size $\eta$, inverse sensitivity $\norm B$, and derivative curvature $L$.
2108 
2109 \subsection{Singular nonlinear repair and Lyapunov--Schmidt reduction}
2110 
2111 When $D=F'(0)$ is not surjective, the nonlinear problem is not decided by a global right inverse. The correct local object is a reduced obstruction equation.
2112 
2113 \begin{theorem}[Lyapunov--Schmidt repair reduction]
2114 \label{thm:lyapunov-schmidt}
2115 Let $F:P\times U\to V$ be $C^k$, $k\geq2$, between finite-dimensional spaces, with $F(0,0)=0$. Let
2116 \begin{equation}
2117 D:=\partial_uF(0,0).
2118 \end{equation}
2119 Choose splittings
2120 \begin{equation}
2121 U=K\oplus X,
2122 \qquad
2123 K=\ker D,
2124 \end{equation}
2125 and
2126 \begin{equation}
2127 V=R\oplus C,
2128 \qquad
2129 R=\im D,
2130 \end{equation}
2131 so that $\restr{D}{X}:X\to R$ is an isomorphism. Let $P_R$ and $P_C$ be the corresponding projections. Then there are neighborhoods of the origin and a unique $C^k$ map
2132 \begin{equation}
2133 \chi:P\times K\to X,
2134 \qquad
2135 \chi(0,0)=0,
2136 \end{equation}
2137 such that
2138 \begin{equation}
2139 P_RF(p,k+\chi(p,k))=0.
2140 \end{equation}
2141 Define the reduced obstruction map
2142 \begin{equation}
2143 \psi(p,k):=P_CF(p,k+\chi(p,k)).
2144 \end{equation}
2145 A sufficiently small pair $(p,u)$ solves $F(p,u)=0$ if and only if
2146 \begin{equation}
2147 u=k+\chi(p,k)
2148 \end{equation}
2149 for some $k\in K$ satisfying
2150 \begin{equation}
2151 \psi(p,k)=0.
2152 \end{equation}
2153 \end{theorem}
2154 
2155 \begin{proof}
2156 Apply the implicit-function theorem to the range equation
2157 \begin{equation}
2158 G(p,k,x):=P_RF(p,k+x)=0.
2159 \end{equation}
2160 Its derivative in $x$ at the origin is $\restr D X:X\to R$, an isomorphism, so there is a unique local solution $x=\chi(p,k)$. Substitution solves the range component identically. The remaining equation is exactly the cokernel component $\psi(p,k)=0$.
2161 \end{proof}
2162 
2163 The reduction shows why ``vanishing linear obstruction'' is not the end of the analysis. Kernel freedom $k$ may be used to solve the reduced cokernel equation, may produce several branches, or may fail completely.
2164 
2165 \subsection{Higher-order obstruction classes}
2166 
2167 Consider the specific formal correction equation
2168 \begin{equation}
2169 F(\eps,u)
2170 =
2171 \eps r_1+Du+\frac12B(u,u)+O\bigl(\eps^3+\norm u^3\bigr),
2172 \label{eq:second-order-model}
2173 \end{equation}
2174 where $B:U\times U\to V$ is symmetric bilinear. Seek
2175 \begin{equation}
2176 u(\eps)=\eps u_1+\eps^2u_2+O(\eps^3).
2177 \end{equation}
2178 
2179 \begin{proposition}[Second-order repair obstruction]
2180 \label{prop:second-order-obstruction}
2181 A repair through order $\eps^2$ exists only if
2182 \begin{align}
2183 Du_1&=-r_1,\\
2184 \left[\frac12B(u_1,u_1)\right]&=0
2185 \quad\text{in }\coker D.
2186 \end{align}
2187 When the second class vanishes, $u_2$ may be chosen to solve
2188 \begin{equation}
2189 Du_2=-\frac12B(u_1,u_1).
2190 \end{equation}
2191 If $\ker D\neq0$, replacing $u_1$ by $u_1+k$ can change the second-order obstruction class.
2192 \end{proposition}
2193 
2194 \begin{proof}
2195 Substitute the formal series into \eqref{eq:second-order-model} and equate coefficients of $\eps$ and $\eps^2$. The first equation is the linear repair equation. The second is solvable exactly when its right-hand side has zero class in $\coker D$.
2196 \end{proof}
2197 
2198 \begin{theorem}[Finite formal obstruction tower]
2199 \label{thm:formal-obstruction-tower}
2200 Let $F(\eps,u)$ be analytic near $(0,0)$ with $F(0,0)=0$ and $D=\partial_uF(0,0)$. Suppose coefficients $u_1,\ldots,u_{n-1}$ have been chosen so that
2201 \begin{equation}
2202 F\left(\eps,\sum_{j=1}^{n-1}\eps^ju_j\right)
2203 =\eps^nb_n+O(\eps^{n+1}).
2204 \end{equation}
2205 Then extending the repair to order $n$ by adding $\eps^nu_n$ is possible if and only if
2206 \begin{equation}
2207 [b_n]=0\quad\text{in }\coker D.
2208 \end{equation}
2209 When it vanishes, any solution of $Du_n=-b_n$ produces an order-$n$ lift.
2210 \end{theorem}
2211 
2212 \begin{proof}
2213 Analytic expansion gives
2214 \begin{equation}
2215 F\left(\eps,\sum_{j=1}^{n}\eps^ju_j\right)
2216 =
2217 \eps^n(b_n+Du_n)+O(\eps^{n+1}),
2218 \end{equation}
2219 because $u_n$ first enters linearly at order $n$. Cancellation of the coefficient is equivalent to solvability of $Du_n=-b_n$.
2220 \end{proof}
2221 
2222 The sequence
2223 \begin{equation}
2224 [b_1],[b_2],[b_3],\ldots\in\coker D
2225 \end{equation}
2226 is an obstruction tower. Choices in $\ker D$ at early orders can alter later classes, so a repair search must retain branch information rather than greedily fixing one representative.
2227 
2228 \section{Local-to-Global Repair Descent}
2229 \label{sec:repair-descent}
2230 
2231 \subsection{Repair sheaves and local solution torsors}
2232 
2233 A heterogeneous agent may admit valid repairs in every stratum while those repairs remain mutually incompatible. Let $X$ be a finite site of functional regions and let
2234 \begin{equation}
2235 \mathfrak U=\set{U_\alpha}_{\alpha\in I}
2236 \end{equation}
2237 be a finite cover. Let $\mathscr U$ be a sheaf of authorized corrections, $\mathscr V$ a sheaf of defects, and
2238 \begin{equation}
2239 D:\mathscr U\to\mathscr V
2240 \end{equation}
2241 a morphism of sheaves. Let $r\in\Gamma(X,\mathscr V)$ be a global defect.
2242 
2243 A local correction on $U_\alpha$ is a section $u_\alpha\in\mathscr U(U_\alpha)$ satisfying
2244 \begin{equation}
2245 D_\alpha u_\alpha=-r|_{U_\alpha}.
2246 \end{equation}
2247 If one local correction exists, the local solution set is an affine space over $\ker D(U_\alpha)$.
2248 
2249 \begin{proposition}[Local solution torsor]
2250 For each $U_\alpha$, the set
2251 \begin{equation}
2252 \Sscr_\alpha
2253 :=
2254 \set{u\in\mathscr U(U_\alpha):D_\alpha u=-r|_{U_\alpha}}
2255 \end{equation}
2256 is either empty or a torsor for the additive group $\ker D(U_\alpha)$. If a global repair exists, the global solution set is a torsor for $\Gamma(X,\ker D)$.
2257 \end{proposition}
2258 
2259 \begin{proof}
2260 The difference of two solutions lies in the kernel, and adding a kernel section to a solution preserves the equation. The action is free and transitive.
2261 \end{proof}
2262 
2263 This torsor viewpoint matters because a local repair is rarely unique. Kernel freedom is the resource used to reconcile interface assumptions.
2264 
2265 \subsection{Exact linear descent}
2266 
2267 \begin{theorem}[Linear repair-descent theorem]
2268 \label{thm:repair-descent}
2269 Assume a local correction $u_\alpha$ exists on every $U_\alpha$. On overlaps $U_{\alpha\beta}=U_\alpha\cap U_\beta$, define
2270 \begin{equation}
2271 d_{\alpha\beta}
2272 :=
2273 \restr{u_\alpha}{U_{\alpha\beta}}
2274 -
2275 \restr{u_\beta}{U_{\alpha\beta}}.
2276 \end{equation}
2277 Then:
2278 \begin{enumerate}
2279 \item $d_{\alpha\beta}\in\ker D(U_{\alpha\beta})$;
2280 \item $(d_{\alpha\beta})$ is a \v{C}ech $1$-cocycle with coefficients in $\ker D$;
2281 \item a global correction $u\in\Gamma(X,\mathscr U)$ with $Du=-r$ exists if and only if $(d_{\alpha\beta})$ is a \v{C}ech coboundary relative to $\mathfrak U$.
2282 \end{enumerate}
2283 If the cover computes first cohomology, the final condition is equivalent to
2284 \begin{equation}
2285 [d_{\alpha\beta}]=0
2286 \quad\text{in}\quad
2287 \check H^1(X,\ker D).
2288 \end{equation}
2289 \end{theorem}
2290 
2291 \begin{proof}
2292 On every overlap,
2293 \begin{equation}
2294 D(d_{\alpha\beta})
2295 =-r|_{U_{\alpha\beta}}+r|_{U_{\alpha\beta}}=0.
2296 \end{equation}
2297 On triple overlaps,
2298 \begin{equation}
2299 d_{\alpha\beta}+d_{\beta\gamma}+d_{\gamma\alpha}=0,
2300 \end{equation}
2301 so the family is a cocycle.
2302 
2303 If a global repair $u$ exists, set $k_\alpha=u|_{U_\alpha}-u_\alpha$. Then $Dk_\alpha=0$ and
2304 \begin{equation}
2305 d_{\alpha\beta}=k_\beta-k_\alpha,
2306 \end{equation}
2307 so the cocycle is a coboundary. Conversely, if $d_{\alpha\beta}=k_\beta-k_\alpha$ for kernel sections $k_\alpha$, then $u'_\alpha=u_\alpha+k_\alpha$ agree on overlaps. The sheaf gluing axiom yields a global $u$, and $Du=-r$ because the adjustments lie in the kernel.
2308 \end{proof}
2309 
2310 \begin{definition}[Two-level repair obstruction]
2311 The local obstruction on $U_\alpha$ is
2312 \begin{equation}
2313 [r|_{U_\alpha}]\in\coker D_\alpha.
2314 \end{equation}
2315 When every local class vanishes, the global gluing obstruction is
2316 \begin{equation}
2317 [d_{\alpha\beta}]\in\check H^1(\mathfrak U,\ker D).
2318 \end{equation}
2319 \end{definition}
2320 
2321 \begin{proposition}[Choice independence of the gluing class]
2322 The cohomology class $[d_{\alpha\beta}]$ does not depend on the particular family of local repairs used to compute it.
2323 \end{proposition}
2324 
2325 \begin{proof}
2326 If $u'_\alpha=u_\alpha+k_\alpha$ with $k_\alpha\in\ker D(U_\alpha)$, then
2327 \begin{equation}
2328 d'_{\alpha\beta}
2329 =d_{\alpha\beta}+k_\alpha-k_\beta.
2330 \end{equation}
2331 Thus the cocycles differ by a coboundary.
2332 \end{proof}
2333 
2334 Repair failure now has two distinct meanings:
2335 \begin{enumerate}
2336 \item a chart cannot correct its own defect;
2337 \item every chart can correct locally, but the chosen repairs cannot be made globally coherent.
2338 \end{enumerate}
2339 
2340 \subsection{Graph-Hodge decomposition of approximate gluing}
2341 
2342 The exact \v{C}ech class is binary. Empirical repairs produce noisy overlap mismatches. A finite graph model gives a quantitative analogue.
2343 
2344 Let $G=(I,E)$ be a connected oriented graph representing the nerve of the cover, let $K$ be a finite-dimensional Hilbert space representing a common trivialization of the kernel fibers, and let
2345 \begin{equation}
2346 e=(e_a)_{a\in E}\in K^E
2347 \end{equation}
2348 be the transported overlap-mismatch vector. Let $B\in\R^{|I|\times|E|}$ be the oriented incidence matrix and set
2349 \begin{equation}
2350 \mathbf B:=B\otimes I_K.
2351 \end{equation}
2352 A vertex gauge $k\in K^I$ changes the mismatch to
2353 \begin{equation}
2354 e+\mathbf B^*k.
2355 \end{equation}
2356 
2357 \begin{theorem}[Quantitative graph-Hodge gluing]
2358 \label{thm:graph-hodge-gluing}
2359 Let
2360 \begin{equation}
2361 L_G:=BB^*
2362 \end{equation}
2363 be the graph Laplacian and let $P_{\mathrm{cyc}}$ be the orthogonal projector from $K^E$ onto
2364 \begin{equation}
2365 \ker\mathbf B,
2366 \end{equation}
2367 the cycle space. Then
2368 \begin{equation}
2369 \inf_{k\in K^I}\norm{e+\mathbf B^*k}
2370 =
2371 \norm{P_{\mathrm{cyc}}e}.
2372 \end{equation}
2373 The mean-zero minimum-norm gauge is
2374 \begin{equation}
2375 k^*
2376 =-(L_G^\dagger B\otimes I_K)e.
2377 \end{equation}
2378 If $\lambda_2(L_G)>0$ is the spectral gap, then
2379 \begin{equation}
2380 \norm{k^*}
2381 \leq
2382 \frac{1}{\sqrt{\lambda_2(L_G)}}\norm e.
2383 \end{equation}
2384 Exact graph-level gluing is possible if and only if
2385 \begin{equation}
2386 P_{\mathrm{cyc}}e=0.
2387 \end{equation}
2388 \end{theorem}
2389 
2390 \begin{proof}
2391 The finite-dimensional Hodge decomposition is
2392 \begin{equation}
2393 K^E=\im\mathbf B^*\oplus\ker\mathbf B.
2394 \end{equation}
2395 A gauge can remove exactly the gradient component in $\im\mathbf B^*$, leaving the cycle projection. The normal equation for the least-squares gauge is
2396 \begin{equation}
2397 \mathbf B\mathbf B^*k=-\mathbf Be,
2398 \end{equation}
2399 whose mean-zero minimum-norm solution is the displayed pseudoinverse expression. The smallest nonzero singular value of $\mathbf B^*$ is $\sqrt{\lambda_2(L_G)}$, giving the bound.
2400 \end{proof}
2401 
2402 \begin{definition}[Quantitative gluing obstruction]
2403 The graph-level quantitative gluing obstruction is
2404 \begin{equation}
2405 \eta_{\mathrm{glue}}(e)
2406 :=\norm{P_{\mathrm{cyc}}e}.
2407 \end{equation}
2408 The local-to-global repair certificate is the pair
2409 \begin{equation}
2410 \left(
2411 \eta_{\mathrm{loc}},
2412 \eta_{\mathrm{glue}}
2413 \right),
2414 \qquad
2415 \eta_{\mathrm{loc}}^2
2416 :=\sum_{\alpha}\norm{D_\alpha u_\alpha+r_\alpha}^2.
2417 \end{equation}
2418 \end{definition}
2419 
2420 A small local residual does not imply a small gluing residual. Conversely, a perfectly coherent family can consistently implement the wrong local repair. Both coordinates are required.
2421 
2422 \begin{corollary}[Trees have no cycle obstruction]
2423 If the cover nerve is a tree, then $\ker\mathbf B=0$ and every overlap mismatch can be removed by a vertex gauge. Any remaining failure is local rather than cyclic.
2424 \end{corollary}
2425 
2426 \begin{proof}
2427 A connected tree has $|E|=|I|-1$ and incidence matrix of full column rank, so its cycle space is zero.
2428 \end{proof}
2429 
2430 \subsection{Weighted overlap confidence}
2431 
2432 Different overlaps may have different uncertainty or semantic importance. Let $W_E$ be a positive-definite operator on $K^E$. Define
2433 \begin{equation}
2434 \eta_{\mathrm{glue},W}(e)
2435 :=
2436 \inf_k\norm{e+\mathbf B^*k}_{W_E}.
2437 \end{equation}
2438 
2439 \begin{proposition}[Weighted gluing normal equation]
2440 Every weighted least-squares gauge satisfies
2441 \begin{equation}
2442 \mathbf BW_E\mathbf B^*k
2443 =-\mathbf BW_Ee.
2444 \end{equation}
2445 The mean-zero minimum-norm minimizer is
2446 \begin{equation}
2447 k_W^*
2448 =-\bigl(\mathbf BW_E\mathbf B^*\bigr)^\dagger\mathbf BW_Ee.
2449 \end{equation}
2450 The irreducible weighted mismatch is the $W_E$-orthogonal projection of $e$ onto the $W_E$-orthogonal complement of $\im\mathbf B^*$.
2451 \end{proposition}
2452 
2453 \begin{proof}
2454 Differentiate the quadratic objective
2455 \begin{equation}
2456 \frac12\angles{e+\mathbf B^*k,W_E(e+\mathbf B^*k)}
2457 \end{equation}
2458 with respect to $k$. The normal equation follows. Its Moore--Penrose solution is the unique minimizer orthogonal to the constant vertex-gauge kernel.
2459 \end{proof}
2460 
2461 This permits confidence-weighted descent: weakly identified interfaces need not dominate a certificate merely because they have large raw coordinate scale.
2462 
2463 \subsection{Finite cycle obstruction}
2464 
2465 \begin{example}[Locally repairable, globally incompatible]
2466 Let the nerve be a three-cycle with no triple intersection. Suppose the kernel fiber is $\R$ and the oriented differences are
2467 \begin{equation}
2468 d_{12}=1,
2469 \qquad
2470 d_{23}=1,
2471 \qquad
2472 d_{31}=1.
2473 \end{equation}
2474 If the cocycle were a coboundary, there would be $k_1,k_2,k_3$ with
2475 \begin{equation}
2476 d_{12}=k_2-k_1,
2477 \quad
2478 d_{23}=k_3-k_2,
2479 \quad
2480 d_{31}=k_1-k_3.
2481 \end{equation}
2482 Summing gives $3=0$, a contradiction. Hence no global correction exists. The graph-Hodge obstruction has norm $\sqrt3$.
2483 \end{example}
2484 
2485 In an agent, the three charts might represent planner, memory, and evaluator repairs. Each patch can pass isolated tests while the cycle of interface assumptions fails to close.
2486 
2487 \subsection{Nonlinear descent through symmetry torsors}
2488 
2489 For nonlinear repair, local solution sets need not be affine. A tractable case occurs when each local solution set is a principal homogeneous space for a local symmetry group.
2490 
2491 Let $\mathscr G$ be a sheaf of groups acting freely and transitively on the sheaf of local nonlinear solutions $\mathscr S$. Choose local solutions $s_\alpha\in\mathscr S(U_\alpha)$. On overlaps there are unique transition elements
2492 \begin{equation}
2493 g_{\alpha\beta}\in\mathscr G(U_{\alpha\beta})
2494 \end{equation}
2495 such that
2496 \begin{equation}
2497 s_\alpha=g_{\alpha\beta}\cdot s_\beta.
2498 \end{equation}
2499 
2500 \begin{conditionaltheorem}[Nonlinear torsor descent]
2501 \label{thm:nonlinear-torsor-descent}
2502 Assume the local solution action is principal and effective descent holds for $\mathscr G$-torsors on the declared cover. Then a global nonlinear repair exists if and only if the transition family satisfies the nonabelian cocycle condition
2503 \begin{equation}
2504 g_{\alpha\beta}g_{\beta\gamma}g_{\gamma\alpha}=e
2505 \end{equation}
2506 on triple overlaps and its class is trivial in nonabelian \v{C}ech cohomology
2507 \begin{equation}
2508 [g_{\alpha\beta}]=1
2509 \quad\text{in}\quad
2510 \check H^1(\mathfrak U,\mathscr G).
2511 \end{equation}
2512 Equivalently, there exist local gauges $g_\alpha\in\mathscr G(U_\alpha)$ such that
2513 \begin{equation}
2514 g_{\alpha\beta}=g_\alpha^{-1}g_\beta.
2515 \end{equation}
2516 \end{conditionaltheorem}
2517 
2518 \begin{proof}
2519 If a global solution exists, compare its restriction with each $s_\alpha$ to obtain gauges $g_\alpha$; their ratios give the transition functions. Conversely, triviality of the cocycle permits gauge-adjusted local solutions $g_\alpha\cdot s_\alpha$ that agree on overlaps. Effective descent glues them to a global solution.
2520 \end{proof}
2521 
2522 The theorem is conditional because arbitrary nonlinear repair spaces need not be torsors, and arbitrary software interfaces need not satisfy effective descent.
2523 
2524 \subsection{Spanning-tree holonomy test}
2525 
2526 A finite operational version avoids explicit cohomology. Let $G=(I,E)$ be connected, with sets of local solutions $S_i$ and invertible transition maps
2527 \begin{equation}
2528 T_{ij}:S_i\to S_j,
2529 \qquad
2530 T_{ji}=T_{ij}^{-1}.
2531 \end{equation}
2532 Choose a root $o$ and a spanning tree $T\subseteq G$.
2533 
2534 \begin{theorem}[Fundamental-cycle holonomy criterion]
2535 \label{thm:fundamental-holonomy}
2536 For each root state $s_o\in S_o$, tree transport determines a unique family $(s_i^T)$. For every non-tree edge $e=(i,j)$, let $H_e:S_o\to S_o$ be transport around the corresponding fundamental cycle. The tree family extends to a globally compatible family on all edges if and only if
2537 \begin{equation}
2538 H_e(s_o)=s_o
2539 \end{equation}
2540 for every non-tree edge $e$.
2541 \end{theorem}
2542 
2543 \begin{proof}
2544 Tree edges are compatible by construction. A non-tree edge closes one fundamental cycle. Its compatibility is exactly the statement that transport from the root to $i$, across $e$, and back to the root returns $s_o$. Fundamental cycles generate all graph cycles, so satisfying every chord condition gives compatibility on all edges.
2545 \end{proof}
2546 
2547 \begin{definition}[Nonlinear holonomy residual]
2548 If the $S_i$ are metric spaces, define
2549 \begin{equation}
2550 \eta_{\mathrm{hol}}(s_o)
2551 :=
2552 \max_{e\in E\setminus T}d\bigl(H_e(s_o),s_o\bigr).
2553 \end{equation}
2554 \end{definition}
2555 
2556 This is a direct finite test for path dependence of repair semantics. It is distinct from the evaluator holonomy developed later, though mathematically analogous.
2557 
2558 \subsection{Order-by-order global repair}
2559 
2560 The local nonlinear tower and the descent obstruction combine into a two-axis lifting problem. Suppose each chart has an analytic local repair equation with the same linearized sheaf morphism $D$.
2561 
2562 At order $n$, let
2563 \begin{equation}
2564 b_{n,\alpha}\in\mathscr V(U_\alpha)
2565 \end{equation}
2566 be the residual coefficient determined by lower-order choices. The local obstruction is
2567 \begin{equation}
2568 [b_{n,\alpha}]\in\coker D_\alpha.
2569 \end{equation}
2570 If every local class vanishes, choose $u_{n,\alpha}$ with
2571 \begin{equation}
2572 D_\alpha u_{n,\alpha}=-b_{n,\alpha}.
2573 \end{equation}
2574 Their differences define an order-$n$ gluing class in $\check H^1(\ker D)$.
2575 
2576 \begin{theorem}[Finite local-to-global obstruction tower]
2577 \label{thm:global-obstruction-tower}
2578 Fix $N\in\N$. Assume:
2579 \begin{enumerate}
2580 \item the analytic local repair equations agree under restriction;
2581 \item the sheaves $\mathscr U$ and $\mathscr V$ satisfy effective linear descent on the declared cover;
2582 \item repairs have been constructed globally through order $n-1$.
2583 \end{enumerate}
2584 Then a global repair extends through order $n$ if and only if:
2585 \begin{enumerate}
2586 \item every local coefficient class vanishes,
2587 \begin{equation}
2588 [b_{n,\alpha}]=0\quad\text{in }\coker D_\alpha;
2589 \end{equation}
2590 \item for one, hence every, choice of local lifts, the overlap class vanishes,
2591 \begin{equation}
2592 [d^{(n)}_{\alpha\beta}]=0
2593 \quad\text{in}\quad
2594 \check H^1(\mathfrak U,\ker D).
2595 \end{equation}
2596 \end{enumerate}
2597 If both conditions hold for every $1\leq n\leq N$, a global formal repair exists through order $N$.
2598 \end{theorem}
2599 
2600 \begin{proof}
2601 At order $n$, coefficient matching reduces the local problem to the linear equation $D_\alpha u_{n,\alpha}=-b_{n,\alpha}$, so the local cokernel classes are the complete local obstruction. When they vanish, \cref{thm:repair-descent} gives the complete criterion for gluing the order-$n$ coefficients. Induction from $1$ through $N$ constructs the global formal series.
2602 \end{proof}
2603 
2604 The theorem produces an obstruction ledger
2605 \begin{equation}
2606 \begin{array}{c|cc}
2607 \text{order}&\text{local obstruction}&\text{gluing obstruction}\\
2608 \hline
2609 1&[b_{1,\alpha}]\in\coker D_\alpha&[d^{(1)}]\in\check H^1(\ker D)\\
2610 2&[b_{2,\alpha}]\in\coker D_\alpha&[d^{(2)}]\in\check H^1(\ker D)\\
2611 \vdots&\vdots&\vdots\\
2612 N&[b_{N,\alpha}]\in\coker D_\alpha&[d^{(N)}]\in\check H^1(\ker D).
2613 \end{array}
2614 \end{equation}
2615 Higher categorical repair interfaces may introduce additional coherence obstructions beyond this two-column ledger. Those are not proved here.
2616 
2617 \subsection{Cover refinement and diagnostic soundness}
2618 
2619 \begin{proposition}[Soundness of cover-relative obstruction]
2620 If a global repair exists, then every local obstruction and every exact gluing obstruction vanishes on every cover. Therefore, a nonzero local or gluing class on any valid cover is a sound certificate of global nonexistence.
2621 \end{proposition}
2622 
2623 \begin{proof}
2624 Restrict the global repair to each chart. The local equations hold, and the local restrictions agree exactly on overlaps, producing zero cocycles.
2625 \end{proof}
2626 
2627 The converse depends on the sheaf and cover assumptions of \cref{thm:repair-descent}. A poor atlas can obscure causal structure even when its internal descent computation is correct. Atlas discovery therefore remains logically prior to universal claims about repair.
2628 
2629 \begin{definition}[Cover adequacy score]
2630 For a candidate functional cover $\mathfrak U$, define the diagnostic score
2631 \begin{equation}
2632 \Acal(\mathfrak U)
2633 :=
2634 \eta_{\mathrm{loc}}(\mathfrak U)
2635 +\tau\eta_{\mathrm{glue}}(\mathfrak U)
2636 +\lambda\,\mathrm{Complexity}(\mathfrak U)
2637 +\gamma\,\mathrm{Instability}(\mathfrak U).
2638 \end{equation}
2639 \end{definition}
2640 
2641 This is a model-selection objective, not a theorem that the minimizing cover is uniquely correct. Its purpose is to expose the tradeoff between local fit, global coherence, atlas complexity, and instability under resampling or intervention.
2642 
2643 \section{Repair Transport and Transfer Geometry}\label{sec:repair-transport}
2644 
2645 A repair discovered in one system becomes recursively useful only when the system can determine whether the repair remains meaningful after a change of architecture, representation, task coordinates, or evaluator. Transport is therefore not an auxiliary convenience. It is the mechanism by which a local correction becomes a reusable element of the repair calculus.
2646 
2647 The theory separates four logically distinct questions:
2648 \begin{enumerate}
2649 \item does the correction equation commute with the proposed representation maps;
2650 \item does the transport preserve authorization, invariants, and resource budgets;
2651 \item how do transport errors accumulate along a path through an atlas;
2652 \item does the transported correction retain positive externally measured gain?
2653 \end{enumerate}
2654 The answers need not coincide. A correction may commute algebraically while violating an authorization cone. It may remain authorized while losing gain under evaluator mismatch. It may work along one transport path and fail along another because of nontrivial holonomy.
2655 
2656 \subsection{Correction data and transport morphisms}
2657 
2658 \begin{definition}[Correction datum]
2659 A finite-dimensional correction datum is a tuple
2660 \begin{equation}
2661 \mathfrak C=(U,V,D,\Acal,c,\mathfrak p),
2662 \end{equation}
2663 where $U$ is a correction space, $V$ is a defect space, $D:U\to V$ is the linearized defect-response map, $\Acal\subseteq U$ is the authorized correction set, $c:U\to[0,\infty]$ is a declared resource cost, and $\mathfrak p$ is a collection of protected predicates or invariant maps.
2664 \end{definition}
2665 
2666 \begin{definition}[Transport datum]
2667 Let $\mathfrak C_A$ and $\mathfrak C_B$ be correction data. A transport datum from $A$ to $B$ is a pair
2668 \begin{equation}
2669 \mathfrak T_{A\to B}=(T_U,T_V),
2670 \qquad
2671 T_U:U_A\to U_B,
2672 \qquad
2673 T_V:V_A\to V_B.
2674 \end{equation}
2675 Its correction-square defect is
2676 \begin{equation}
2677 \Delta_D(\mathfrak T_{A\to B})
2678 :=
2679 D_BT_U-T_VD_A.
2680 \end{equation}
2681 For declared source and target defects $r_A\in V_A$ and $r_B\in V_B$, its defect-identification error is
2682 \begin{equation}
2683 \Delta_r(\mathfrak T_{A\to B};r_A,r_B)
2684 :=
2685 r_B-T_Vr_A.
2686 \end{equation}
2687 The datum is \emph{exact} when both quantities vanish.
2688 \end{definition}
2689 
2690 The two errors answer different questions. $\Delta_D$ measures failure of the correction mechanism to commute. $\Delta_r$ measures failure of the target defect to be the transported source defect. A small value of one does not imply a small value of the other.
2691 
2692 \begin{theorem}[Exact repair transport]
2693 \label{thm:exact-transport}
2694 Suppose
2695 \begin{equation}
2696 D_BT_U=T_VD_A
2697 \end{equation}
2698 and
2699 \begin{equation}
2700 r_B=T_Vr_A.
2701 \end{equation}
2702 If $u_A$ satisfies
2703 \begin{equation}
2704 D_Au_A=-r_A,
2705 \end{equation}
2706 then
2707 \begin{equation}
2708 u_B:=T_Uu_A
2709 \end{equation}
2710 satisfies
2711 \begin{equation}
2712 D_Bu_B=-r_B.
2713 \end{equation}
2714 \end{theorem}
2715 
2716 \begin{proof}
2717 By commutation,
2718 \begin{equation}
2719 D_Bu_B=D_BT_Uu_A=T_VD_Au_A=-T_Vr_A=-r_B.
2720 \end{equation}
2721 \end{proof}
2722 
2723 \begin{corollary}[Kernel and obstruction naturality]
2724 \label{cor:kernel-coker-naturality}
2725 Assume $D_BT_U=T_VD_A$. Then
2726 \begin{equation}
2727 T_U(\ker D_A)\subseteq\ker D_B
2728 \end{equation}
2729 and
2730 \begin{equation}
2731 T_V(\im D_A)\subseteq\im D_B.
2732 \end{equation}
2733 Consequently, $T_V$ induces a quotient map
2734 \begin{equation}
2735 \overline T_V:\coker D_A\to\coker D_B,
2736 \qquad
2737 [r]\mapsto[T_Vr].
2738 \end{equation}
2739 \end{corollary}
2740 
2741 \begin{proof}
2742 If $D_Au=0$, then $D_BT_Uu=T_VD_Au=0$. If $v=D_Au$, then $T_Vv=D_BT_Uu\in\im D_B$. The quotient map is therefore well-defined.
2743 \end{proof}
2744 
2745 \begin{theorem}[Exact transport equivalence]
2746 \label{thm:transport-equivalence}
2747 Suppose $T_U$ and $T_V$ are isomorphisms and
2748 \begin{equation}
2749 D_B=T_VD_AT_U^{-1}.
2750 \end{equation}
2751 Then $T_U$ restricts to an isomorphism
2752 \begin{equation}
2753 \ker D_A\cong\ker D_B,
2754 \end{equation}
2755 and $T_V$ induces an isomorphism
2756 \begin{equation}
2757 \coker D_A\cong\coker D_B.
2758 \end{equation}
2759 Thus exact transport equivalence preserves both correction freedom and obstruction dimension.
2760 \end{theorem}
2761 
2762 \begin{proof}
2763 The kernel statement follows from \cref{cor:kernel-coker-naturality} applied in both directions. Since $T_U$ is surjective,
2764 \begin{equation}
2765 \im D_B
2766 =
2767 T_VD_AT_U^{-1}(U_B)
2768 =
2769 T_V(\im D_A).
2770 \end{equation}
2771 Hence $T_V$ maps $\im D_A$ bijectively onto $\im D_B$ and descends to an isomorphism of quotient spaces.
2772 \end{proof}
2773 
2774 \begin{definition}[Protected obstruction subspace]
2775 A protected obstruction subspace is a declared subspace
2776 \begin{equation}
2777 Q_A\subseteq\coker D_A
2778 \end{equation}
2779 whose nonzero classes must remain detectable after transport. The transport is \emph{obstruction-faithful on $Q_A$} when
2780 \begin{equation}
2781 \ker\overline T_V\cap Q_A=\set{0}.
2782 \end{equation}
2783 \end{definition}
2784 
2785 \begin{proposition}[Representation loss can hide an obstruction]
2786 \label{prop:hidden-obstruction}
2787 If $\overline T_V$ is not injective, then there exists a defect $r_A$ such that
2788 \begin{equation}
2789 [r_A]\neq0
2790 \quad\text{but}\quad
2791 [T_Vr_A]=0.
2792 \end{equation}
2793 Thus the target representation classifies the transported defect as correctable even though the source obstruction is nonzero. Obstruction faithfulness is therefore a necessary condition whenever the erased class is protected.
2794 \end{proposition}
2795 
2796 \begin{proof}
2797 Choose a nonzero class in $\ker\overline T_V$ and any representative $r_A$. By definition its image in $\coker D_B$ vanishes.
2798 \end{proof}
2799 
2800 \begin{remark}
2801 A noninjective bridge may be useful as a compression map, but it cannot by itself certify preservation of every repair obstruction. This is the transport analogue of the distinction between exact representation change and information-losing abstraction.
2802 \end{remark}
2803 
2804 \subsection{Authorization, invariants, and minimum-cost correction}
2805 
2806 \begin{definition}[Authorized transport]
2807 A transport datum is authorized when
2808 \begin{equation}
2809 T_U(\Acal_A)\subseteq\Acal_B.
2810 \end{equation}
2811 It is $(c_A,c_B,\kappa)$-budget controlled when
2812 \begin{equation}
2813 c_B(T_Uu)\leq\kappa c_A(u)
2814 \qquad
2815 \text{for every }u\in\Acal_A.
2816 \end{equation}
2817 \end{definition}
2818 
2819 \begin{proposition}[Authorized exact repair transport]
2820 \label{prop:authorized-transport}
2821 Under the hypotheses of \cref{thm:exact-transport}, if $u_A\in\Acal_A$ and the transport is authorized, then $u_B=T_Uu_A$ is an authorized exact repair in system $B$. If, in addition, $c_A(u_A)\leq B$ and the transport is $(c_A,c_B,\kappa)$-budget controlled, then
2822 \begin{equation}
2823 c_B(u_B)\leq\kappa B.
2824 \end{equation}
2825 \end{proposition}
2826 
2827 \begin{proof}
2828 Authorization follows from $T_U(\Acal_A)\subseteq\Acal_B$. Exact repair follows from \cref{thm:exact-transport}. The cost inequality is the definition of budget control.
2829 \end{proof}
2830 
2831 Let protected linear invariants be represented by
2832 \begin{equation}
2833 C_A:U_A\to Z_A,
2834 \qquad
2835 C_B:U_B\to Z_B.
2836 \end{equation}
2837 A correction preserves the invariant when $C_Au=0$ or $C_Bu=0$.
2838 
2839 \begin{proposition}[Invariant-preserving transport]
2840 \label{prop:invariant-transport}
2841 Suppose there is a linear map $S:Z_A\to Z_B$ satisfying
2842 \begin{equation}
2843 C_BT_U=SC_A.
2844 \end{equation}
2845 Then
2846 \begin{equation}
2847 T_U(\ker C_A)\subseteq\ker C_B.
2848 \end{equation}
2849 Consequently, every source repair preserving the declared invariant transports to a target repair preserving the transported invariant.
2850 \end{proposition}
2851 
2852 \begin{proof}
2853 For $u\in\ker C_A$,
2854 \begin{equation}
2855 C_BT_Uu=SC_Au=0.
2856 \end{equation}
2857 \end{proof}
2858 
2859 The previous propositions guarantee feasibility, not optimality. Under unitary transport, the canonical minimum-norm correction is also preserved.
2860 
2861 \begin{theorem}[Naturality of the Moore--Penrose repair]
2862 \label{thm:pseudoinverse-transport}
2863 Let $U_A,U_B,V_A,V_B$ be finite-dimensional Hilbert spaces. Suppose $T_U$ and $T_V$ are unitary and
2864 \begin{equation}
2865 D_B=T_VD_AT_U^*.
2866 \end{equation}
2867 Then
2868 \begin{equation}
2869 D_B^{\dagger}=T_UD_A^{\dagger}T_V^*.
2870 \end{equation}
2871 For every $r_B=T_Vr_A$, the target minimum-norm least-squares correction is the transport of the source minimum-norm least-squares correction:
2872 \begin{equation}
2873 -D_B^{\dagger}r_B
2874 =
2875 T_U(-D_A^{\dagger}r_A).
2876 \end{equation}
2877 The orthogonal obstruction projectors satisfy
2878 \begin{equation}
2879 I-D_BD_B^{\dagger}
2880 =
2881 T_V(I-D_AD_A^{\dagger})T_V^*.
2882 \end{equation}
2883 \end{theorem}
2884 
2885 \begin{proof}
2886 Set $X=T_UD_A^{\dagger}T_V^*$. Direct substitution, unitary cancellation, and the four Moore--Penrose equations for $D_A^{\dagger}$ show that $X$ satisfies the four Moore--Penrose equations for $D_B$. Uniqueness gives $X=D_B^{\dagger}$. The correction and projector identities follow immediately.
2887 \end{proof}
2888 
2889 \begin{remark}
2890 For nonunitary isomorphisms, exact repairability still transports, but minimum norm is coordinate dependent. A change of metric or an explicit condition-number penalty is then required before comparing repair costs.
2891 \end{remark}
2892 
2893 \subsection{Composition and accumulation of transport defect}
2894 
2895 Let
2896 \begin{equation}
2897 \mathfrak T_{A\to B}=(T_U^{AB},T_V^{AB}),
2898 \qquad
2899 \mathfrak T_{B\to C}=(T_U^{BC},T_V^{BC}).
2900 \end{equation}
2901 The composed transport is
2902 \begin{equation}
2903 \mathfrak T_{A\to C}
2904 =
2905 (T_U^{BC}T_U^{AB},T_V^{BC}T_V^{AB}).
2906 \end{equation}
2907 
2908 \begin{lemma}[Composition identity]
2909 \label{lem:transport-composition}
2910 The correction-square defects obey
2911 \begin{equation}
2912 \Delta_D(\mathfrak T_{A\to C})
2913 =
2914 \Delta_D(\mathfrak T_{B\to C})T_U^{AB}
2915 +
2916 T_V^{BC}\Delta_D(\mathfrak T_{A\to B}).
2917 \end{equation}
2918 The defect-identification errors obey
2919 \begin{equation}
2920 \Delta_r(\mathfrak T_{A\to C})
2921 =
2922 \Delta_r(\mathfrak T_{B\to C})
2923 +
2924 T_V^{BC}\Delta_r(\mathfrak T_{A\to B}).
2925 \end{equation}
2926 \end{lemma}
2927 
2928 \begin{proof}
2929 Insert and subtract $T_V^{BC}D_BT_U^{AB}$ in the first expression and $T_V^{BC}r_B$ in the second.
2930 \end{proof}
2931 
2932 \begin{theorem}[Pathwise transport bound]
2933 \label{thm:pathwise-transport}
2934 Consider a path of systems
2935 \begin{equation}
2936 A_0\xrightarrow{\mathfrak T_0}A_1
2937 \xrightarrow{\mathfrak T_1}\cdots
2938 \xrightarrow{\mathfrak T_{n-1}}A_n.
2939 \end{equation}
2940 Write $T_{U,k}:U_k\to U_{k+1}$ and $T_{V,k}:V_k\to V_{k+1}$. Let
2941 \begin{equation}
2942 \norm{D_{k+1}T_{U,k}-T_{V,k}D_k}\leq\eps_{D,k}
2943 \end{equation}
2944 and
2945 \begin{equation}
2946 \norm{r_{k+1}-T_{V,k}r_k}\leq\eps_{r,k}.
2947 \end{equation}
2948 If $D_0u_0=-r_0$ and
2949 \begin{equation}
2950 u_n=T_{U,n-1}\cdots T_{U,0}u_0,
2951 \end{equation}
2952 then
2953 \begin{equation}
2954 \begin{aligned}
2955 \norm{D_nu_n+r_n}
2956 &\leq
2957 \sum_{k=0}^{n-1}
2958 \norm{T_{V,n-1}\cdots T_{V,k+1}}
2959 \eps_{D,k}
2960 \norm{T_{U,k-1}\cdots T_{U,0}u_0}\\
2961 &\quad+
2962 \sum_{k=0}^{n-1}
2963 \norm{T_{V,n-1}\cdots T_{V,k+1}}
2964 \eps_{r,k},
2965 \end{aligned}
2966 \end{equation}
2967 where empty products are identities.
2968 \end{theorem}
2969 
2970 \begin{proof}
2971 Repeatedly apply \cref{lem:transport-composition}. The total correction-square defect is the telescoping sum
2972 \begin{equation}
2973 \sum_{k=0}^{n-1}
2974 T_{V,n-1}\cdots T_{V,k+1}
2975 \Delta_{D,k}
2976 T_{U,k-1}\cdots T_{U,0},
2977 \end{equation}
2978 and the total defect-identification error is
2979 \begin{equation}
2980 \sum_{k=0}^{n-1}
2981 T_{V,n-1}\cdots T_{V,k+1}\Delta_{r,k}.
2982 \end{equation}
2983 Apply both sums to $u_0$, use $D_0u_0+r_0=0$, and take norms.
2984 \end{proof}
2985 
2986 \begin{corollary}[Nonexpansive path bound]
2987 \label{cor:nonexpansive-path}
2988 If every $T_{U,k}$ and $T_{V,k}$ is nonexpansive, then
2989 \begin{equation}
2990 \norm{D_nu_n+r_n}
2991 \leq
2992 \norm{u_0}\sum_{k=0}^{n-1}\eps_{D,k}
2993 +
2994 \sum_{k=0}^{n-1}\eps_{r,k}.
2995 \end{equation}
2996 \end{corollary}
2997 
2998 \begin{definition}[Transport route cost]
2999 For a fixed source correction $u_0$, define the certified cost of a nonexpansive path $p$ by
3000 \begin{equation}
3001 C_{\mathrm{tr}}(p;u_0)
3002 :=
3003 \norm{u_0}\sum_{e\in p}\eps_{D,e}
3004 +
3005 \sum_{e\in p}\eps_{r,e}
3006 +
3007 \sum_{e\in p}\eps_{\mathrm{auth},e}
3008 +
3009 \sum_{e\in p}\eps_{\mathrm{sem},e}.
3010 \end{equation}
3011 \end{definition}
3012 
3013 \begin{proposition}[Shortest certified route]
3014 \label{prop:shortest-route}
3015 When all edge transports are nonexpansive and the declared error contributions are additive, a shortest path in the edge-weighted transport graph minimizes the stated worst-case transfer penalty and therefore maximizes the corresponding certified lower bound on target gain.
3016 \end{proposition}
3017 
3018 \begin{proof}
3019 Under the hypotheses, the path penalty is the sum of nonnegative edge costs. Minimizing this sum is exactly the shortest-path problem.
3020 \end{proof}
3021 
3022 \begin{remark}
3023 If some transports amplify norms, route quality depends on the prefix and suffix products in \cref{thm:pathwise-transport}; a scalar edge weight is no longer sufficient. The routing state must then retain the current transported correction norm and downstream amplification profile.
3024 \end{remark}
3025 
3026 \subsection{Repair holonomy and path dependence}
3027 
3028 Let $G$ be a graph of correction data with invertible exact transports. For a loop $\gamma$ based at system $A$, define correction and defect holonomies
3029 \begin{equation}
3030 H^U_\gamma:U_A\to U_A,
3031 \qquad
3032 H^V_\gamma:V_A\to V_A.
3033 \end{equation}
3034 Exact commutation along the loop gives
3035 \begin{equation}
3036 D_AH^U_\gamma=H^V_\gamma D_A.
3037 \end{equation}
3038 
3039 \begin{definition}[Path-flat repair and defect cores]
3040 The path-flat repair and defect cores at $A$ are
3041 \begin{equation}
3042 U_A^{\flat}
3043 :=
3044 \bigcap_\gamma\ker(H^U_\gamma-I),
3045 \qquad
3046 V_A^{\flat}
3047 :=
3048 \bigcap_\gamma\ker(H^V_\gamma-I).
3049 \end{equation}
3050 \end{definition}
3051 
3052 \begin{proposition}[Flat repairs remain flat defects]
3053 \label{prop:flat-repair-defect}
3054 If $u\in U_A^{\flat}$, then
3055 \begin{equation}
3056 D_Au\in V_A^{\flat}.
3057 \end{equation}
3058 \end{proposition}
3059 
3060 \begin{proof}
3061 For every loop $\gamma$,
3062 \begin{equation}
3063 H^V_\gamma D_Au=D_AH^U_\gamma u=D_Au.
3064 \end{equation}
3065 \end{proof}
3066 
3067 \begin{theorem}[Path independence on the flat repair core]
3068 \label{thm:path-independence}
3069 Let $p$ and $q$ be two exact invertible transport paths from $A$ to $B$. If $u\in U_A$ is fixed by the loop holonomy associated with $q^{-1}p$, then
3070 \begin{equation}
3071 T_U^p u=T_U^q u.
3072 \end{equation}
3073 In particular, every $u\in U_A^{\flat}$ has path-independent transport to $B$.
3074 \end{theorem}
3075 
3076 \begin{proof}
3077 The path identity is
3078 \begin{equation}
3079 T_U^p=T_U^qH^U_{q^{-1}p}.
3080 \end{equation}
3081 Apply both sides to $u$ and use $H^U_{q^{-1}p}u=u$.
3082 \end{proof}
3083 
3084 \begin{definition}[Repair holonomy residual]
3085 For a correction $u$ and loop $\gamma$, define
3086 \begin{equation}
3087 \eta^U_\gamma(u)
3088 :=
3089 \norm{(H^U_\gamma-I)u}.
3090 \end{equation}
3091 For a defect $r$, define
3092 \begin{equation}
3093 \eta^V_\gamma(r)
3094 :=
3095 \norm{(H^V_\gamma-I)r}.
3096 \end{equation}
3097 \end{definition}
3098 
3099 \begin{proposition}[Path-discrepancy identity]
3100 \label{prop:path-discrepancy}
3101 Under the hypotheses of \cref{thm:path-independence},
3102 \begin{equation}
3103 \norm{T_U^pu-T_U^qu}
3104 \leq
3105 \norm{T_U^q}\,\eta^U_{q^{-1}p}(u).
3106 \end{equation}
3107 Thus loop holonomy directly bounds disagreement between two transport routes.
3108 \end{proposition}
3109 
3110 \begin{proof}
3111 Use
3112 \begin{equation}
3113 T_U^pu-T_U^qu=T_U^q(H^U_{q^{-1}p}-I)u
3114 \end{equation}
3115 and take norms.
3116 \end{proof}
3117 
3118 \subsection{Approximate transported residual}
3119 
3120 \begin{theorem}[Approximate transported residual]
3121 \label{thm:approx-residual}
3122 Let $U_A,U_B,V_A,V_B$ be normed spaces. Suppose $u_A$ solves
3123 \begin{equation}
3124 D_Au_A=-r_A.
3125 \end{equation}
3126 Assume
3127 \begin{equation}
3128 \norm{D_BT_U-T_VD_A}\leq\eps_D
3129 \end{equation}
3130 and
3131 \begin{equation}
3132 \norm{r_B-T_Vr_A}\leq\eps_r.
3133 \end{equation}
3134 Then $u_B=T_Uu_A$ satisfies
3135 \begin{equation}
3136 \norm{D_Bu_B+r_B}
3137 \leq
3138 \eps_D\norm{u_A}+\eps_r.
3139 \end{equation}
3140 \end{theorem}
3141 
3142 \begin{proof}
3143 Add and subtract $T_VD_Au_A$ and $T_Vr_A$:
3144 \begin{align}
3145 D_BT_Uu_A+r_B
3146 &=
3147 (D_BT_U-T_VD_A)u_A
3148 +T_V(D_Au_A+r_A)
3149 +(r_B-T_Vr_A).
3150 \end{align}
3151 The middle term vanishes. Taking norms yields the claim.
3152 \end{proof}
3153 
3154 \begin{definition}[Scale-normalized transport defect]
3155 For a declared stabilizer $\tau>0$, define
3156 \begin{equation}
3157 \eps_D^{\mathrm{rel}}
3158 :=
3159 \frac{\norm{D_BT_U-T_VD_A}}
3160 {\norm{D_B}\norm{T_U}+\norm{T_V}\norm{D_A}+\tau}.
3161 \end{equation}
3162 \end{definition}
3163 
3164 The normalized defect is not a substitute for the absolute residual in \cref{thm:approx-residual}. It is a diagnostic for comparing transport squares at different numerical scales. Both quantities should be retained in a repair receipt.
3165 
3166 \subsection{Deterministic gain preservation}
3167 
3168 Let $\ell_A:V_A\to\R$ and $\ell_B:V_B\to\R$ be post-defect losses. Define source and target gains
3169 \begin{equation}
3170 g_A
3171 :=
3172 \ell_A(r_A)-\ell_A(r_A+D_Au_A),
3173 \end{equation}
3174 \begin{equation}
3175 g_B
3176 :=
3177 \ell_B(r_B)-\ell_B(r_B+D_BT_Uu_A).
3178 \end{equation}
3179 
3180 \begin{assumption}[Evaluator alignment]
3181 There is $\eps_J\geq0$ such that
3182 \begin{equation}
3183 \abs{\ell_B(T_Vv)-\ell_A(v)}\leq\eps_J
3184 \end{equation}
3185 for
3186 \begin{equation}
3187 v\in\set{r_A,r_A+D_Au_A}.
3188 \end{equation}
3189 \end{assumption}
3190 
3191 \begin{theorem}[Approximate gain-preserving repair transport]
3192 \label{thm:gain-transport}
3193 Assume $\ell_B$ is $L$-Lipschitz. Suppose the hypotheses of \cref{thm:approx-residual} hold. Let additional target-side perturbations from packet leakage and semantic holonomy have total defect norm at most
3194 \begin{equation}
3195 \eps_{\mathrm{leak}}+\eps_{\mathrm{hol}}.
3196 \end{equation}
3197 Then
3198 \begin{equation}
3199 \boxed{
3200  g_B
3201 \geq
3202  g_A
3203 -2\eps_J
3204 -L\left(
3205 2\eps_r
3206 +\eps_D\norm{u_A}
3207 +\eps_{\mathrm{leak}}
3208 +\eps_{\mathrm{hol}}
3209 \right).
3210 }
3211 \end{equation}
3212 In particular, the transported repair is guaranteed beneficial whenever the source gain exceeds the total right-hand error margin.
3213 \end{theorem}
3214 
3215 \begin{proof}
3216 Evaluator alignment and Lipschitz continuity imply
3217 \begin{equation}
3218 \ell_B(r_B)
3219 \geq
3220 \ell_B(T_Vr_A)-L\eps_r
3221 \geq
3222 \ell_A(r_A)-\eps_J-L\eps_r.
3223 \end{equation}
3224 The target post-repair defect differs from
3225 \begin{equation}
3226 T_V(r_A+D_Au_A)
3227 \end{equation}
3228 by at most
3229 \begin{equation}
3230 \eps_r+\eps_D\norm{u_A}+\eps_{\mathrm{leak}}+\eps_{\mathrm{hol}}
3231 \end{equation}
3232 using \cref{thm:approx-residual} and the declared additional perturbation bounds. Hence
3233 \begin{align}
3234 \ell_B(r_B+D_BT_Uu_A)
3235 &\leq
3236 \ell_B(T_V(r_A+D_Au_A))\\
3237 &\quad+L\left(
3238 \eps_r+\eps_D\norm{u_A}
3239 +\eps_{\mathrm{leak}}+\eps_{\mathrm{hol}}
3240 \right)\\
3241 &\leq
3242 \ell_A(r_A+D_Au_A)+\eps_J\\
3243 &\quad+L\left(
3244 \eps_r+\eps_D\norm{u_A}
3245 +\eps_{\mathrm{leak}}+\eps_{\mathrm{hol}}
3246 \right).
3247 \end{align}
3248 Subtracting the post-repair upper bound from the baseline lower bound gives the result.
3249 \end{proof}
3250 
3251 \begin{corollary}[Pathwise gain certificate]
3252 \label{cor:pathwise-gain}
3253 Along a nonexpansive transport path, replace
3254 \begin{equation}
3255 \eps_D\norm{u_A}+\eps_r
3256 \end{equation}
3257 by the pathwise residual bound in \cref{cor:nonexpansive-path}. If the source gain exceeds twice the evaluator-alignment error plus the resulting Lipschitz, leakage, and holonomy penalties, then the endpoint repair retains positive certified gain.
3258 \end{corollary}
3259 
3260 \begin{remark}
3261 The theorem makes repair transfer falsifiable. A failed transfer can be attributed to defect mismatch, correction noncommutation, evaluator mismatch, packet leakage, authorization loss, or semantic holonomy rather than being described only as distribution shift.
3262 \end{remark}
3263 
3264 \subsection{Distributional and probabilistic transfer certificates}
3265 
3266 The deterministic theorem compares declared defects. A deployed repair is usually evaluated across a distribution of contexts. Let $P_A$ and $P_B$ be source and target context distributions, let $S:X_A\to X_B$ be a declared context transport, and let $q_A,q_B$ be per-context repair gains.
3267 
3268 \begin{theorem}[Coupled distributional gain transfer]
3269 \label{thm:distributional-transfer}
3270 Suppose there is a coupling $\pi$ of $P_A$ and $P_B$ such that
3271 \begin{equation}
3272 \mathbb E_{(x,y)\sim\pi}d_B(y,Sx)\leq\delta.
3273 \end{equation}
3274 Assume $q_B$ is $K$-Lipschitz and
3275 \begin{equation}
3276 q_B(Sx)\geq q_A(x)-\eta
3277 \end{equation}
3278 for $P_A$-almost every $x$. Then
3279 \begin{equation}
3280 \mathbb E_{y\sim P_B}q_B(y)
3281 \geq
3282 \mathbb E_{x\sim P_A}q_A(x)-\eta-K\delta.
3283 \end{equation}
3284 \end{theorem}
3285 
3286 \begin{proof}
3287 Under the coupling,
3288 \begin{equation}
3289 q_B(y)
3290 \geq
3291 q_B(Sx)-K d_B(y,Sx)
3292 \geq
3293 q_A(x)-\eta-K d_B(y,Sx).
3294 \end{equation}
3295 Take expectations.
3296 \end{proof}
3297 
3298 The term $\eta$ can itself be instantiated by the structural penalty from \cref{thm:gain-transport}. The theorem then separates representational transport error from context-distribution mismatch.
3299 
3300 \begin{definition}[High-probability transport certificate]
3301 A random tuple
3302 \begin{equation}
3303 (\underline g_A,
3304 \overline\eps_J,
3305 \overline\eps_r,
3306 \overline\eps_D,
3307 \overline\eps_{\mathrm{leak}},
3308 \overline\eps_{\mathrm{hol}},
3309 \overline L)
3310 \end{equation}
3311 is an $(1-\alpha)$ transport certificate when, jointly with probability at least $1-\alpha$, it lower-bounds $g_A$ and upper-bounds every corresponding error or Lipschitz quantity.
3312 \end{definition}
3313 
3314 \begin{theorem}[Probabilistic gain preservation by simultaneous bounds]
3315 \label{thm:probabilistic-transfer}
3316 Given an $(1-\alpha)$ transport certificate, define
3317 \begin{equation}
3318 \begin{aligned}
3319 \underline g_B
3320 :=\;&
3321 \underline g_A
3322 -2\overline\eps_J\\
3323 &-\overline L\left(
3324 2\overline\eps_r
3325 +\overline\eps_D\norm{u_A}
3326 +\overline\eps_{\mathrm{leak}}
3327 +\overline\eps_{\mathrm{hol}}
3328 \right).
3329 \end{aligned}
3330 \end{equation}
3331 Then
3332 \begin{equation}
3333 \Pr(g_B\geq\underline g_B)\geq1-\alpha.
3334 \end{equation}
3335 In particular, $\underline g_B>0$ is a valid one-sided certificate of beneficial target transfer.
3336 \end{theorem}
3337 
3338 \begin{proof}
3339 On the simultaneous confidence event, substitute every bound into \cref{thm:gain-transport}. The event has probability at least $1-\alpha$.
3340 \end{proof}
3341 
3342 \begin{theorem}[Paired held-out gain certificate]
3343 \label{thm:paired-gain}
3344 Let $Z_1,\ldots,Z_n$ be independent and identically distributed paired target-context gains for a precommitted transported repair, with
3345 \begin{equation}
3346 -G\leq Z_i\leq G.
3347 \end{equation}
3348 Let $\overline Z_n=n^{-1}\sum_iZ_i$. Then, with probability at least $1-\alpha$,
3349 \begin{equation}
3350 \mathbb E Z_i
3351 \geq
3352 \overline Z_n
3353 -G\sqrt{\frac{2\log(1/\alpha)}{n}}.
3354 \end{equation}
3355 For a precommitted family of $m$ repairs, replacing $\alpha$ by $\alpha/m$ gives a simultaneous lower bound for all $m$ candidates.
3356 \end{theorem}
3357 
3358 \begin{proof}
3359 Hoeffding's inequality for a variable with range width $2G$ gives
3360 \begin{equation}
3361 \Pr\left(
3362 \overline Z_n-\mathbb EZ_i\geq t
3363 \right)
3364 \leq
3365 \exp\left(-\frac{nt^2}{2G^2}\right).
3366 \end{equation}
3367 Set the right side to $\alpha$. The family statement follows by the union bound.
3368 \end{proof}
3369 
3370 \begin{warning}[Adaptive reuse of validation evidence]
3371 The bound in \cref{thm:paired-gain} assumes the repair or finite candidate family is fixed before the held-out outcomes are revealed. Searching the same validation sample until a positive lower bound appears invalidates the stated guarantee unless the adaptive search is covered by a separate sequential or reusable-holdout argument.
3372 \end{warning}
3373 
3374 \begin{statusbox}{Block 3 transport result}
3375 The transport layer now distinguishes exact equivalence, information-losing abstraction, authorization preservation, invariant preservation, pathwise error accumulation, route choice, repair holonomy, deterministic gain margin, distributional shift, and finite-sample certification. A transported repair should be accepted only when these receipts jointly support the claimed preservation profile.
3376 \end{statusbox}
3377 
3378 \section{Path-Flat Evaluator Evolution}\label{sec:evaluator-evolution}
3379 
3380 An evaluator is not only a scalar score. It is a structured map from raw evidence through an internal representation to judgments, uncertainty estimates, authorization decisions, and comparisons among descendants. Evaluator evolution is legitimate only when new distinctions are added without silently changing the meaning of protected historical evidence.
3381 
3382 \subsection{Evaluator objects and morphisms}
3383 
3384 \begin{definition}[Evaluator object]
3385 An evaluator object at version $v$ is a tuple
3386 \begin{equation}
3387 \mathsf E_v=(R_v,E_v,\iota_v,J_v,Q_v,\mathfrak h_v),
3388 \end{equation}
3389 where:
3390 \begin{itemize}
3391 \item $R_v$ is the protected raw-evidence space or ledger span;
3392 \item $E_v$ is the evaluator's internal evidence-representation space;
3393 \item $\iota_v:R_v\to E_v$ encodes raw evidence;
3394 \item $J_v:E_v\to\R^q$ is a score or decision-readout map;
3395 \item $Q_v$ is an uncertainty or calibration readout;
3396 \item $\mathfrak h_v$ is the historical replay set protected at version $v$.
3397 \end{itemize}
3398 \end{definition}
3399 
3400 \begin{definition}[Evaluator morphism]
3401 An evaluator morphism $e:v\to w$ consists of maps
3402 \begin{equation}
3403 S_e:R_v\to R_w,
3404 \qquad
3405 T_e:E_v\to E_w,
3406 \end{equation}
3407 where $S_e$ is the append-only raw-evidence embedding and $T_e$ transports internal representations. Its protected-evidence defect is
3408 \begin{equation}
3409 \delta_{\mathrm{obs}}(e)
3410 :=
3411 \sup_{r\in\mathfrak h_v}
3412 \norm{T_e\iota_v(r)-\iota_wS_e(r)},
3413 \end{equation}
3414 and its historical score defect is
3415 \begin{equation}
3416 \delta_J(e)
3417 :=
3418 \sup_{r\in\mathfrak h_v}
3419 \norm{J_wT_e\iota_v(r)-J_v\iota_v(r)}.
3420 \end{equation}
3421 The morphism is exact on protected history when both defects vanish.
3422 \end{definition}
3423 
3424 \begin{proposition}[Composition of evaluator defects]
3425 \label{prop:evaluator-composition}
3426 Let $e:v\to w$ and $f:w\to z$ be evaluator morphisms. Suppose $T_f$ is $L_T$-Lipschitz on represented protected evidence. Then
3427 \begin{equation}
3428 \delta_{\mathrm{obs}}(f\circ e)
3429 \leq
3430 L_T\delta_{\mathrm{obs}}(e)+\delta_{\mathrm{obs}}(f),
3431 \end{equation}
3432 where the second term is evaluated on $S_e(\mathfrak h_v)$. If, in addition,
3433 \begin{equation}
3434 \sup_{r\in\mathfrak h_v}
3435 \norm{
3436 J_zT_fT_e\iota_v(r)-J_wT_e\iota_v(r)
3437 }
3438 \leq
3439 \delta_J(f\mid e),
3440 \end{equation}
3441 then
3442 \begin{equation}
3443 \delta_J(f\circ e)
3444 \leq
3445 \delta_J(e)+\delta_J(f\mid e).
3446 \end{equation}
3447 \end{proposition}
3448 
3449 \begin{proof}
3450 For the evidence defect, insert $T_f\iota_wS_e(r)$ and apply the triangle inequality and the Lipschitz bound on $T_f$. For the score defect, insert $J_wT_e\iota_v(r)$ and use the two declared suprema.
3451 \end{proof}
3452 
3453 \subsection{Evaluator atlases and exact holonomy}
3454 
3455 Let $G=(N,E)$ be a connected directed graph of evaluator versions. Each directed edge $e:v\to w$ carries an invertible internal transport
3456 \begin{equation}
3457 T_e:E_v\to E_w.
3458 \end{equation}
3459 Fix a base node $v_0$. For any loop $\gamma$ based at $v_0$, let
3460 \begin{equation}
3461 H_\gamma:E_{v_0}\to E_{v_0}
3462 \end{equation}
3463 be the composed transport around the loop.
3464 
3465 \begin{definition}[Path-flat evidence core]
3466 The path-flat evidence core at $v_0$ is
3467 \begin{equation}
3468 E_{v_0}^{\flat}
3469 :=
3470 \bigcap_{\gamma\in\pi_1(G,v_0)}
3471 \ker(H_\gamma-I).
3472 \end{equation}
3473 \end{definition}
3474 
3475 \begin{theorem}[Maximal path-flat evaluator core]
3476 \label{thm:path-flat-core}
3477 The subspace $E_{v_0}^{\flat}$ is the unique greatest subspace of $E_{v_0}$ fixed pointwise by every evaluator-loop holonomy. That is:
3478 \begin{enumerate}
3479 \item $H_\gamma x=x$ for every $x\in E_{v_0}^{\flat}$ and every loop $\gamma$;
3480 \item if $W\subseteq E_{v_0}$ satisfies $H_\gamma w=w$ for all $w\in W$ and all loops $\gamma$, then $W\subseteq E_{v_0}^{\flat}$.
3481 \end{enumerate}
3482 \end{theorem}
3483 
3484 \begin{proof}
3485 The first statement is immediate from the intersection definition. If $W$ is fixed pointwise by every $H_\gamma$, then $W\subseteq\ker(H_\gamma-I)$ for every loop $\gamma$, hence $W$ lies in their intersection.
3486 \end{proof}
3487 
3488 \begin{proposition}[Finite computation from a cycle basis]
3489 \label{prop:cycle-basis-core}
3490 Let $G$ be finite. Choose a spanning tree and let $\gamma_1,\ldots,\gamma_m$ be the fundamental loops generated by non-tree edges. If the loop transports form a representation of the graph fundamental group, then
3491 \begin{equation}
3492 E_{v_0}^{\flat}
3493 =
3494 \bigcap_{j=1}^{m}
3495 \ker(H_{\gamma_j}-I).
3496 \end{equation}
3497 Equivalently, $E_{v_0}^{\flat}$ is the null space of the vertically stacked matrix
3498 \begin{equation}
3499 \begin{bmatrix}
3500 H_{\gamma_1}-I\\
3501 \vdots\\
3502 H_{\gamma_m}-I
3503 \end{bmatrix}.
3504 \end{equation}
3505 \end{proposition}
3506 
3507 \begin{proof}
3508 Every loop class is generated by the fundamental loops and their inverses. A vector fixed by each generator is fixed by every product of generators. The stacked-null-space statement is equivalent.
3509 \end{proof}
3510 
3511 \begin{theorem}[Basepoint covariance of the flat core]
3512 \label{thm:basepoint-core}
3513 Let $p:v\to w$ be any invertible exact evaluator path in a connected atlas. Then
3514 \begin{equation}
3515 T_p(E_v^{\flat})=E_w^{\flat}.
3516 \end{equation}
3517 Hence the dimension of the path-flat core is independent of basepoint.
3518 \end{theorem}
3519 
3520 \begin{proof}
3521 Every loop $\gamma$ at $w$ corresponds to the conjugate loop $p^{-1}\gamma p$ at $v$, with
3522 \begin{equation}
3523 H_\gamma=T_pH_{p^{-1}\gamma p}T_p^{-1}.
3524 \end{equation}
3525 If $x\in E_v^{\flat}$, then $H_\gamma T_px=T_pH_{p^{-1}\gamma p}x=T_px$, so $T_px\in E_w^{\flat}$. Apply the same argument to $p^{-1}$ for the reverse inclusion.
3526 \end{proof}
3527 
3528 Invertibility is not always realistic. A later evaluator may compress or expand its internal evidence representation.
3529 
3530 \begin{definition}[Closed-walk equalizer core]
3531 For a possibly noninvertible evaluator graph, let $\mathsf W(v)$ be the monoid of directed closed walks at $v$. Define
3532 \begin{equation}
3533 E_v^{\mathrm{eq}}
3534 :=
3535 \bigcap_{\gamma\in\mathsf W(v)}\ker(H_\gamma-I).
3536 \end{equation}
3537 \end{definition}
3538 
3539 \begin{proposition}[Maximality without invertibility]
3540 The equalizer core $E_v^{\mathrm{eq}}$ is the unique greatest subspace fixed pointwise by every closed-walk endomorphism. A fundamental cycle basis need not suffice when edge maps are noninvertible and the closed-walk maps do not factor through a group representation.
3541 \end{proposition}
3542 
3543 \begin{proof}
3544 The maximality argument is the same intersection-of-equalizers argument as in \cref{thm:path-flat-core}. The cycle-basis reduction used inverses and group generation, which are unavailable in general.
3545 \end{proof}
3546 
3547 \subsection{Approximate path-flat cores}
3548 
3549 Exact fixed subspaces can be too brittle for learned evaluator transports. Let $H_1,\ldots,H_m$ be a finite generating family of loop transports on a Hilbert space $E$, with positive weights $w_j$ satisfying $\sum_jw_j=1$. Define the holonomy-energy operator
3550 \begin{equation}
3551 \mathsf A
3552 :=
3553 \sum_{j=1}^{m}
3554 w_j(H_j-I)^*(H_j-I).
3555 \end{equation}
3556 This operator is positive semidefinite.
3557 
3558 \begin{theorem}[Spectral characterization of flatness]
3559 \label{thm:holonomy-energy}
3560 The exact common fixed space is
3561 \begin{equation}
3562 E^{\flat}=\ker\mathsf A.
3563 \end{equation}
3564 Let the eigenvalues of $\mathsf A$ be
3565 \begin{equation}
3566 0\leq\lambda_1\leq\cdots\leq\lambda_d.
3567 \end{equation}
3568 Among all $k$-dimensional subspaces $F\subseteq E$, the quantity
3569 \begin{equation}
3570 \sum_{i=1}^{k}\angles{x_i,\mathsf Ax_i}
3571 \end{equation}
3572 for an orthonormal basis $x_1,\ldots,x_k$ of $F$ is minimized by the span of eigenvectors associated with $\lambda_1,\ldots,\lambda_k$, and the minimum is $\sum_{i=1}^k\lambda_i$.
3573 \end{theorem}
3574 
3575 \begin{proof}
3576 For every $x$,
3577 \begin{equation}
3578 \angles{x,\mathsf Ax}
3579 =
3580 \sum_jw_j\norm{(H_j-I)x}^2.
3581 \end{equation}
3582 This vanishes exactly when every $H_jx=x$, proving the kernel statement. Expand an orthonormal $k$-frame in an eigenbasis of $\mathsf A$. The sum of Rayleigh quotients is a convex combination of eigenvalue sums and is minimized by assigning all available weight to the $k$ smallest eigenvalues.
3583 \end{proof}
3584 
3585 \begin{definition}[Thresholded approximate flat core]
3586 For $\tau\geq0$, define
3587 \begin{equation}
3588 E_{\tau}^{\flat}
3589 :=
3590 \Span\set{x_i:\lambda_i\leq\tau},
3591 \end{equation}
3592 where $x_i$ is an eigenbasis of $\mathsf A$.
3593 \end{definition}
3594 
3595 \begin{corollary}[Distortion bound on the approximate core]
3596 \label{cor:approx-core-distortion}
3597 For every $x\in E_{\tau}^{\flat}$,
3598 \begin{equation}
3599 \sum_{j=1}^{m}w_j\norm{(H_j-I)x}^2
3600 \leq
3601 \tau\norm{x}^2.
3602 \end{equation}
3603 In particular,
3604 \begin{equation}
3605 \norm{(H_j-I)x}
3606 \leq
3607 \sqrt{\frac{\tau}{w_j}}\norm{x}
3608 \end{equation}
3609 for every generator $j$.
3610 \end{corollary}
3611 
3612 \begin{proof}
3613 The Rayleigh quotient of $\mathsf A$ on the spectral subspace with eigenvalues at most $\tau$ is at most $\tau$. Each weighted nonnegative summand is bounded by the total.
3614 \end{proof}
3615 
3616 \begin{proposition}[Stability of the threshold count]
3617 \label{prop:core-spectrum-stability}
3618 Let $\widehat{\mathsf A}$ be an estimated holonomy-energy operator satisfying
3619 \begin{equation}
3620 \norm{\widehat{\mathsf A}-\mathsf A}\leq\eta.
3621 \end{equation}
3622 Then each ordered eigenvalue satisfies
3623 \begin{equation}
3624 \abs{\widehat\lambda_i-\lambda_i}\leq\eta.
3625 \end{equation}
3626 If no eigenvalue of $\mathsf A$ lies in $[\tau-\eta,\tau+\eta]$, then $\widehat{\mathsf A}$ and $\mathsf A$ have the same number of eigenvalues below $\tau$.
3627 \end{proposition}
3628 
3629 \begin{proof}
3630 For every unit vector $x$,
3631 \begin{equation}
3632 \abs{\angles{x,\widehat{\mathsf A}x}-\angles{x,\mathsf Ax}}
3633 \leq\eta.
3634 \end{equation}
3635 Apply the min--max characterization of ordered eigenvalues. The threshold-count statement follows immediately.
3636 \end{proof}
3637 
3638 \subsection{Semantic holonomy and decision stability}
3639 
3640 \begin{definition}[Semantic holonomy residual]
3641 For a represented evidence state $x\in E_v$ and loop $\gamma$ at $v$, define
3642 \begin{equation}
3643 \eta_{\mathrm{sem}}(x;\gamma)
3644 :=
3645 \norm{(H_\gamma-I)x}.
3646 \end{equation}
3647 \end{definition}
3648 
3649 \begin{theorem}[Readout stability under semantic holonomy]
3650 \label{thm:semantic-holonomy}
3651 Let $J:E_v\to Y$ be $L_J$-Lipschitz. Then
3652 \begin{equation}
3653 d_Y(J(H_\gamma x),J(x))
3654 \leq
3655 L_J\eta_{\mathrm{sem}}(x;\gamma).
3656 \end{equation}
3657 If $J$ returns a score vector and the winning score at $x$ has margin $m(x)$ over the runner-up, then the decision is unchanged whenever
3658 \begin{equation}
3659 2L_J\eta_{\mathrm{sem}}(x;\gamma)<m(x),
3660 \end{equation}
3661 where $L_J$ bounds each score coordinate in the same norm.
3662 \end{theorem}
3663 
3664 \begin{proof}
3665 The first inequality is Lipschitz continuity. Each of the winning and runner-up scores can move by at most $L_J\eta_{\mathrm{sem}}$. Their gap therefore remains positive when the total possible contraction of the gap is less than $m(x)$.
3666 \end{proof}
3667 
3668 \begin{theorem}[Cumulative evaluator-drift bound]
3669 \label{thm:cumulative-evaluator-drift}
3670 Consider a chain of evaluator morphisms with readouts into a common complete normed score space,
3671 \begin{equation}
3672 \mathsf E_0\xrightarrow{e_0}\mathsf E_1
3673 \xrightarrow{e_1}\cdots
3674 \xrightarrow{e_{n-1}}\mathsf E_n.
3675 \end{equation}
3676 Suppose that for every protected represented state $x_t$ transported from version $0$,
3677 \begin{equation}
3678 \norm{J_{t+1}(T_tx_t)-J_t(x_t)}\leq\eps_t.
3679 \end{equation}
3680 Then
3681 \begin{equation}
3682 \norm{J_n(T_{n-1}\cdots T_0x_0)-J_0(x_0)}
3683 \leq
3684 \sum_{t=0}^{n-1}\eps_t.
3685 \end{equation}
3686 If $\sum_t\eps_t<\infty$, then the transported score sequence is Cauchy and converges in every complete score space.
3687 \end{theorem}
3688 
3689 \begin{proof}
3690 The finite bound is the triangle inequality along the chain. Summability implies the Cauchy criterion.
3691 \end{proof}
3692 
3693 This theorem supplies a quantitative epistemic ratchet. Exact preservation corresponds to $\eps_t=0$. Summable nonzero error permits evaluator evolution while preventing unbounded cumulative reinterpretation of protected history.
3694 
3695 \subsection{Flatness is not sufficiency}
3696 
3697 The zero subspace is always path-flat. Therefore flatness alone cannot establish that the retained core contains enough information to distinguish outcomes relevant to improvement.
3698 
3699 Let $\Delta_{\mathrm{prot}}\subseteq E\setminus\set{0}$ be a declared set of protected evidence differences: each $d=x-y$ represents two historical or possible evidence states that the evaluator must be able to distinguish.
3700 
3701 \begin{definition}[Core adequacy modulus]
3702 For a subspace $F\subseteq E$ with orthogonal projector $P_F$, define
3703 \begin{equation}
3704 \mu(F;\Delta_{\mathrm{prot}})
3705 :=
3706 \inf_{d\in\Delta_{\mathrm{prot}}}
3707 \frac{\norm{P_Fd}}{\norm d}.
3708 \end{equation}
3709 \end{definition}
3710 
3711 \begin{proposition}[Robust protected distinction]
3712 \label{prop:core-adequacy}
3713 If
3714 \begin{equation}
3715 \mu(F;\Delta_{\mathrm{prot}})=\mu>0,
3716 \end{equation}
3717 then every protected pair remains separated after projection onto $F$. If the projected representations of each member of a pair are perturbed by at most $\eta$, the pair remains distinct whenever
3718 \begin{equation}
3719 2\eta<\mu\norm d.
3720 \end{equation}
3721 \end{proposition}
3722 
3723 \begin{proof}
3724 By definition $\norm{P_Fd}\geq\mu\norm d$. Two perturbations can reduce the separation by at most $2\eta$.
3725 \end{proof}
3726 
3727 \begin{definition}[Adequate approximate evidence core]
3728 A subspace $F$ is $(\tau,\mu_0)$-adequate when:
3729 \begin{enumerate}
3730 \item its weighted holonomy energy is at most $\tau$ per unit norm;
3731 \item $\mu(F;\Delta_{\mathrm{prot}})\geq\mu_0>0$.
3732 \end{enumerate}
3733 \end{definition}
3734 
3735 The two conditions are independent. Low holonomy without separation is vacuous. High separation without low holonomy preserves distinctions but allows their semantics to become path dependent.
3736 
3737 \subsection{Conservative evaluator upgrades}
3738 
3739 \begin{definition}[Core-retractable upgrade]
3740 An evaluator morphism $e:v\to w$ is core-retractable when
3741 \begin{equation}
3742 T_e(E_v^{\flat})\subseteq E_w^{\flat}
3743 \end{equation}
3744 and there is a map
3745 \begin{equation}
3746 R_e:E_w^{\flat}\to E_v^{\flat}
3747 \end{equation}
3748 such that
3749 \begin{equation}
3750 R_eT_e x=x
3751 \qquad
3752 \text{for every }x\in E_v^{\flat}.
3753 \end{equation}
3754 It is core-isomorphic when $T_e$ restricts to an isomorphism from $E_v^{\flat}$ onto $E_w^{\flat}$.
3755 \end{definition}
3756 
3757 Core retractability allows the new evaluator to add dimensions while preserving a recoverable copy of the old evidence core.
3758 
3759 \begin{definition}[Conservative evaluator upgrade]
3760 An evaluator upgrade $e:v\to w$ is conservative on protected history when:
3761 \begin{enumerate}
3762 \item $S_e:R_v\to R_w$ is injective;
3763 \item $e$ is core-retractable;
3764 \item $\delta_{\mathrm{obs}}(e)$ is below the declared evidence tolerance;
3765 \item $\delta_J(e)$ is below the declared historical-replay tolerance;
3766 \item every new score or distinction is typed as new rather than retroactively presented as an old observation;
3767 \item an independent witness can restore version $v$ and replay $\mathfrak h_v$.
3768 \end{enumerate}
3769 \end{definition}
3770 
3771 \begin{theorem}[Historical ranking preservation]
3772 \label{thm:ranking-preservation}
3773 Let $x_1,\ldots,x_N$ be protected historical candidate records with scalar old scores $J_v(x_i)$. Suppose every unequal old pair has margin at least
3774 \begin{equation}
3775 m_{\mathrm{hist}}
3776 :=
3777 \min_{J_v(x_i)\neq J_v(x_j)}
3778 \abs{J_v(x_i)-J_v(x_j)}>0.
3779 \end{equation}
3780 If an upgrade satisfies
3781 \begin{equation}
3782 \abs{J_w(T_ex_i)-J_v(x_i)}\leq\eps
3783 \end{equation}
3784 for every $i$, with
3785 \begin{equation}
3786 2\eps<m_{\mathrm{hist}},
3787 \end{equation}
3788 then every strict historical ranking is preserved.
3789 \end{theorem}
3790 
3791 \begin{proof}
3792 For $J_v(x_i)>J_v(x_j)$,
3793 \begin{align}
3794 J_w(T_ex_i)-J_w(T_ex_j)
3795 &\geq
3796 J_v(x_i)-J_v(x_j)-2\eps\\
3797 &\geq
3798 m_{\mathrm{hist}}-2\eps>0.
3799 \end{align}
3800 \end{proof}
3801 
3802 \begin{proposition}[No silent protected-evidence erasure]
3803 \label{prop:no-erasure}
3804 Suppose $S_e$ is injective and $T_e\iota_v=\iota_wS_e$ on protected history. If $T_e$ is injective on the span of protected evidence differences, then distinct protected records remain distinct in the new evaluator representation.
3805 \end{proposition}
3806 
3807 \begin{proof}
3808 For distinct records $r_1,r_2$ with protected difference $d=\iota_v(r_1)-\iota_v(r_2)\neq0$, injectivity gives $T_ed\neq0$. Exact commutation identifies this with the difference of their new representations.
3809 \end{proof}
3810 
3811 \begin{proposition}[Conservative upgrades compose]
3812 \label{prop:conservative-compose}
3813 Exact conservative evaluator upgrades with core retractions compose. Approximate conservative upgrades compose with evidence and score tolerances bounded by \cref{prop:evaluator-composition}. Therefore a chain remains conservative whenever its accumulated errors remain below the declared historical margins and core-adequacy thresholds.
3814 \end{proposition}
3815 
3816 \begin{proof}
3817 Injective raw-evidence embeddings and left inverses compose. Typed provenance and rollback receipts compose by concatenation. The approximate bounds are \cref{prop:evaluator-composition}.
3818 \end{proof}
3819 
3820 \subsection{Statistical certification of evaluator upgrades}
3821 
3822 Historical replay establishes consistency with old evidence. It does not establish that the new evaluator is more accurate on new external outcomes. That claim requires a held-out comparison.
3823 
3824 \begin{theorem}[Paired evaluator-risk improvement]
3825 \label{thm:evaluator-risk}
3826 Let $Z_1,\ldots,Z_n$ be independent and identically distributed held-out externally revealed cases. Let $L_v(Z_i),L_w(Z_i)\in[0,1]$ be the losses of the old and proposed evaluators, and define
3827 \begin{equation}
3828 \Delta_i=L_v(Z_i)-L_w(Z_i)\in[-1,1].
3829 \end{equation}
3830 If the upgrade and evaluation rule are fixed before the held-out outcomes are revealed, then with probability at least $1-\alpha$,
3831 \begin{equation}
3832 \mathbb E\Delta_i
3833 \geq
3834 \overline\Delta_n
3835 -
3836 \sqrt{\frac{2\log(1/\alpha)}{n}}.
3837 \end{equation}
3838 Thus a positive lower bound certifies lower expected held-out risk for the proposed evaluator.
3839 \end{theorem}
3840 
3841 \begin{proof}
3842 Apply \cref{thm:paired-gain} with $G=1$.
3843 \end{proof}
3844 
3845 \begin{corollary}[Multiple protected metrics]
3846 If $m$ precommitted evaluator candidates or protected metrics are tested, replacing $\alpha$ by $\alpha/m$ yields simultaneous one-sided guarantees. An upgrade may require positive gain on the primary metric and lower bounds above declared negative tolerances on every protected secondary metric.
3847 \end{corollary}
3848 
3849 \begin{warning}
3850 A new evaluator must not use the held-out outcomes both to invent its scoring rule and to certify that rule. Proposal evidence, historical replay evidence, and final external validation evidence should be separated or governed by a valid adaptive-testing protocol.
3851 \end{warning}
3852 
3853 \subsection{Proof-carrying evaluator-upgrade protocol}
3854 
3855 \begin{definition}[Evaluator-upgrade certificate]
3856 An evaluator-upgrade certificate is a tuple
3857 \begin{equation}
3858 \Pi_{\mathrm{eval}}
3859 =
3860 (v,w,S_e,T_e,R_e,
3861 \mathfrak h_v,
3862 \mathsf A_v,\mathsf A_w,
3863 \tau,\mu_0,
3864 \delta_{\mathrm{obs}},\delta_J,
3865 \mathcal T_{\mathrm{new}},
3866 \mathcal R_{\mathrm{rollback}},
3867 \sigma),
3868 \end{equation}
3869 containing the old and new evaluator identifiers, raw and internal transports, core retraction, protected replay set, holonomy-energy operators, accepted flatness and adequacy thresholds, measured replay defects, held-out new-evidence tests, rollback record, and uncertainty statement $\sigma$.
3870 \end{definition}
3871 
3872 \begin{protocol}[Certified evaluator upgrade]
3873 \label{prot:evaluator-upgrade}
3874 A proposed evaluator version $w$ is provisionally admitted from version $v$ only after the following steps:
3875 \begin{enumerate}
3876 \item freeze the candidate evaluator and all acceptance thresholds;
3877 \item embed the append-only raw-evidence ledger through $S_e$;
3878 \item verify protected-evidence commutation and obstruction faithfulness;
3879 \item compute exact or approximate path-flat cores and the adequacy modulus;
3880 \item replay historical decisions and check ranking margins;
3881 \item evaluate new externally revealed cases with a precommitted statistical test;
3882 \item test cycle holonomy against all currently retained evaluator branches;
3883 \item record a witness identity and rollback checkpoint not editable by the candidate transaction;
3884 \item commit provisionally if every required certificate passes; otherwise restore version $v$.
3885 \end{enumerate}
3886 \end{protocol}
3887 
3888 \begin{theorem}[Soundness of a certified evaluator chain]
3889 \label{thm:evaluator-chain}
3890 Consider a finite chain of upgrades admitted by Protocol~\ref{prot:evaluator-upgrade}. Suppose each upgrade is core-retractable, has protected score error at most $\eps_t$, perturbs each retained protected representation by at most $\eta_t$, and retains core adequacy at least $\mu_0>0$. Then:
3891 \begin{enumerate}
3892 \item the original protected core is recoverably embedded at every later version;
3893 \item cumulative protected score drift is at most $\sum_t\eps_t$;
3894 \item every original strict decision margin larger than $2\sum_t\eps_t$ is preserved;
3895 \item a protected difference $d$ remains distinguishable whenever
3896 \begin{equation}
3897 \mu_0\norm d>2\sum_t\eta_t.
3898 \end{equation}
3899 \end{enumerate}
3900 If the score-error series is summable along an infinite chain, protected scores converge.
3901 \end{theorem}
3902 
3903 \begin{proof}
3904 Item 1 follows by composing core retractions. Item 2 is \cref{thm:cumulative-evaluator-drift}. Item 3 applies the proof of \cref{thm:ranking-preservation} to the cumulative bound. Item 4 is \cref{prop:core-adequacy} with per-record accumulated perturbation $\sum_t\eta_t$. The final statement follows from summability.
3905 \end{proof}
3906 
3907 \subsection{Examples of evaluator holonomy}
3908 
3909 \begin{example}[Shear holonomy]
3910 Let $E=\R^2$ and suppose evaluator evolution around one loop produces
3911 \begin{equation}
3912 H=
3913 \begin{bmatrix}
3914 1&\alpha\\
3915 0&1
3916 \end{bmatrix},
3917 \qquad \alpha\neq0.
3918 \end{equation}
3919 Then
3920 \begin{equation}
3921 H-I=
3922 \begin{bmatrix}
3923 0&\alpha\\
3924 0&0
3925 \end{bmatrix},
3926 \end{equation}
3927 so
3928 \begin{equation}
3929 E^{\flat}=\ker(H-I)=\set{(x,0):x\in\R}.
3930 \end{equation}
3931 The first coordinate is path-flat. The second coordinate changes the first under a loop and therefore cannot be treated as invariant evidence.
3932 \end{example}
3933 
3934 \begin{example}[Flat but inadequate core]
3935 Let protected evidence states be
3936 \begin{equation}
3937 x_+=(1,1),
3938 \qquad
3939 x_-=(1,-1),
3940 \end{equation}
3941 and let shear holonomy leave only the first coordinate flat. Both records project to $(1,0)$, so
3942 \begin{equation}
3943 \mu(E^{\flat};\set{x_+-x_-})=0.
3944 \end{equation}
3945 The core is exactly flat but cannot preserve the protected distinction between the two outcomes.
3946 \end{example}
3947 
3948 \begin{statusbox}{Block 3 evaluator result}
3949 Evaluator evolution is represented by an atlas of raw-evidence embeddings, internal transports, readouts, and replay obligations. Exact path-flatness controls path dependence. Spectral approximate cores quantify near-flatness. The adequacy modulus prevents a trivial or information-poor core from being treated as sufficient. Conservative upgrades preserve recoverable evidence, historical margins, and externally validated risk improvements through proof-carrying receipts.
3950 \end{statusbox}
3951 
3952 \section{Witness-Separated Improvement Transactions}
3953 \label{sec:witness-transactions}
3954 
3955 The preceding sections characterize when a repair exists, whether it glues across strata, whether it transports to a new system, and whether the evaluator retains a stable evidential meaning. None of those results by itself specifies how a candidate modification is executed without allowing the candidate to rewrite the rule that judges it. This section gives a transactional semantics for that separation.
3956 
3957 \subsection{Transaction objects and immutable surfaces}
3958 
3959 Let a complete implementation state decompose as
3960 \begin{equation}
3961 \Sigma
3962 =
3963 \Sigma^{\mathrm{task}}
3964 \oplus
3965 \Sigma^{\mathrm{memory}}
3966 \oplus
3967 \Sigma^{\mathrm{repair}}
3968 \oplus
3969 \Sigma^{\mathrm{evaluation}}
3970 \oplus
3971 \Sigma^{\mathrm{provenance}}
3972 \oplus
3973 \Sigma^{\mathrm{rollback}}.
3974 \end{equation}
3975 A transaction selects an editable mask $E\subseteq\Sigma$ and a witness mask $W\subseteq\Sigma$ with
3976 \begin{equation}
3977 E\cap W=\varnothing.
3978 \end{equation}
3979 The decomposition is logical rather than necessarily physical. Two components may share hardware while remaining isolated by capabilities, address spaces, process boundaries, cryptographic commitments, or reproducible snapshots.
3980 
3981 \begin{definition}[Improvement transaction]
3982 A proof-carrying improvement transaction is a tuple
3983 \begin{equation}
3984 \tau
3985 =
3986 \left(
3987 P,
3988 E,
3989 W,
3990 \delta,
3991 \widehat\Delta,
3992 \Ccal,
3993 q,
3994 a,
3995 Z,
3996 \Pi
3997 \right),
3998 \end{equation}
3999 where:
4000 \begin{enumerate}
4001 \item $P$ is a content-addressed parent checkpoint;
4002 \item $E$ is the editable component mask;
4003 \item $W$ is the frozen witness mask;
4004 \item $\delta$ is the proposed edit;
4005 \item $\widehat\Delta$ is the precommitted effect prediction;
4006 \item $\Ccal$ is the capability and resource budget;
4007 \item $q$ is the declared trace map from full executions to witness-visible traces;
4008 \item $a$ is the sealed acceptance functional;
4009 \item $Z$ is the revealed trace;
4010 \item $\Pi$ is the final receipt containing the decision, evidence links, and rollback or commit record.
4011 \end{enumerate}
4012 \end{definition}
4013 
4014 \begin{definition}[Well-formed transaction]
4015 A transaction is well formed when:
4016 \begin{enumerate}
4017 \item $\delta$ acts only on $E$;
4018 \item $a$, $q$, the evidence ledger head, and the rollback checkpoint are determined before candidate execution;
4019 \item candidate capabilities are a subset of $\Ccal$;
4020 \item all externally persistent effects are buffered until commit or are separately declared irreversible effects;
4021 \item the candidate cannot append to the protected evidence namespace except through $q$;
4022 \item the final receipt binds the parent digest, child digest, edit digest, trace digest, evaluator version, witness version, and decision.
4023 \end{enumerate}
4024 \end{definition}
4025 
4026 The transaction follows six phases:
4027 \begin{equation}
4028 \mathrm{prepare}
4029 \longrightarrow
4030 \mathrm{seal}
4031 \longrightarrow
4032 \mathrm{execute}
4033 \longrightarrow
4034 \mathrm{reveal}
4035 \longrightarrow
4036 \mathrm{decide}
4037 \longrightarrow
4038 \mathrm{finalize}.
4039 \end{equation}
4040 Only the final phase may expose buffered effects or replace the active checkpoint.
4041 
4042 \begin{definition}[Witness separation]
4043 A well-formed transaction is witness-separated when:
4044 \begin{enumerate}
4045 \item the candidate edit acts only on $E$;
4046 \item the acceptance functional and its executable dependencies lie in $W$;
4047 \item $E$ cannot modify $W$, the sealed trace schema, the evidence ledger head, or the rollback checkpoint during the transaction;
4048 \item candidate-to-witness communication factors through the declared trace map $q$.
4049 \end{enumerate}
4050 \end{definition}
4051 
4052 \begin{theorem}[Atomic self-approval noninterference]
4053 \label{thm:witness-noninterference}
4054 In a witness-separated transaction, changing the editable implementation while holding the declared trace $Z$ fixed cannot change the acceptance decision.
4055 \end{theorem}
4056 
4057 \begin{proof}
4058 The decision has the form $a(W,Z)$. Both $a$ and $W$ are fixed outside the editable mask. Therefore any two candidate executions with the same declared trace induce the same decision.
4059 \end{proof}
4060 
4061 The theorem is intentionally narrow. It blocks direct atomic self-approval. It does not show that $q$ exposes every acceptance-relevant fact.
4062 
4063 \begin{theorem}[Trace-factorization criterion]
4064 \label{thm:trace-factorization}
4065 Let $Y$ be the set of full execution records, let $q:Y\to Z$ be the declared trace map, and let $b:Y\to\{0,1\}$ be the decision that an ideal full-information witness would make. The following are equivalent:
4066 \begin{enumerate}
4067 \item $b$ is constant on every fiber $q^{-1}(z)$;
4068 \item there exists a unique function $a:q(Y)\to\{0,1\}$ such that
4069 \begin{equation}
4070 b=a\circ q.
4071 \end{equation}
4072 \end{enumerate}
4073 Thus the trace schema is complete for the declared decision exactly when the full-information decision factors through the trace.
4074 \end{theorem}
4075 
4076 \begin{proof}
4077 If $b=a\circ q$, equal traces imply equal decisions. Conversely, if $b$ is constant on fibers, define $a(z)=b(y)$ for any $y$ with $q(y)=z$. Fiber constancy makes this well defined and uniqueness is immediate on $q(Y)$.
4078 \end{proof}
4079 
4080 \begin{warning}[Trace-schema attack surface]
4081 Witness separation is ineffective when distinct full executions with materially different external consequences map to the same declared trace. The factorization criterion should therefore be tested by adversarially searching for pairs $y_1,y_2$ satisfying
4082 \begin{equation}
4083 q(y_1)=q(y_2),
4084 \qquad
4085 b(y_1)\neq b(y_2).
4086 \end{equation}
4087 Such a pair is a constructive certificate of trace incompleteness.
4088 \end{warning}
4089 
4090 \subsection{Rollback, atomicity, and serial composition}
4091 
4092 Let $\widetilde F(P,\delta)$ denote execution in an isolated child environment. Write
4093 \begin{equation}
4094 \widetilde F(P,\delta)=(C,B),
4095 \end{equation}
4096 where $C$ is the candidate child state and $B$ is a buffer of externally persistent effects.
4097 
4098 \begin{theorem}[Atomic rollback]
4099 \label{thm:atomic-rollback}
4100 Assume:
4101 \begin{enumerate}
4102 \item the parent checkpoint $P$ is immutable during execution;
4103 \item every persistent candidate effect is either contained in $C$ or buffered in $B$;
4104 \item $B$ is unpublished until the witness commits;
4105 \item rejection destroys $C$ and $B$ while leaving the append-only receipt ledger outside the operational parent state.
4106 \end{enumerate}
4107 Then a rejected transaction restores the operational system exactly to $P$. The only persistent change is the append-only rejection receipt.
4108 \end{theorem}
4109 
4110 \begin{proof}
4111 By assumptions 1 and 2, execution does not alter $P$ and has no persistent effect outside $(C,B)$. By assumption 3, the buffer is not externally visible before commit. Rejection deletes $(C,B)$, so the operational state is $P$. Assumption 4 permits the audit ledger to record the failed transaction without changing the restored operational state.
4112 \end{proof}
4113 
4114 \begin{definition}[Irreversible-effect budget]
4115 When perfect buffering is impossible, let $I(\tau)$ be the set of irreversible external effects produced before decision. A transaction is $\eta$-reversible relative to a loss functional $L_I$ when
4116 \begin{equation}
4117 L_I(I(\tau))\leq\eta.
4118 \end{equation}
4119 The quantity $\eta$ must be part of the precommitment and acceptance rule.
4120 \end{definition}
4121 
4122 \begin{proposition}[Serializability of independent repairs]
4123 \label{prop:transaction-serializability}
4124 Let $\tau_1$ and $\tau_2$ edit disjoint component masks. Suppose their sandbox state transformations commute, their capability effects are disjoint, and neither transaction changes the sealed witness inputs of the other. Then committing both transactions in either order yields the same operational state and the same pair of transaction receipts up to receipt ordering.
4125 \end{proposition}
4126 
4127 \begin{proof}
4128 Commutation of state transformations gives equality of the two final operational states. Disjoint capability effects and unchanged witness inputs imply that each trace and decision is invariant under the order. The receipts therefore contain the same transaction-local facts, with only their ledger positions exchanged.
4129 \end{proof}
4130 
4131 \begin{remark}
4132 The proposition supplies a sufficient condition for parallel repair. When edits overlap, the repair commutator from \cref{sec:repair-transport} or an explicit joint sandbox run is required.
4133 \end{remark}
4134 
4135 \subsection{Commit-reveal discipline and adaptive validity}
4136 
4137 Let $(\Fcal_t)_{t\geq0}$ be the filtration generated by all committed lineage history through transaction $t$. A proposal is precommitted when its edit, claimed effect, target metrics, stopping rule, and evaluation namespace are $\Fcal_{t-1}$-measurable. The new evaluation trace is revealed only afterward.
4138 
4139 \begin{theorem}[Adaptive alpha-spending guarantee]
4140 \label{thm:adaptive-alpha-spending}
4141 Let $F_t$ be the event that transaction $t$ accepts a child whose true declared gain is nonpositive. Suppose
4142 \begin{equation}
4143 \Pr(F_t\mid\Fcal_{t-1})\leq\delta_t
4144 \qquad\text{almost surely}
4145 \end{equation}
4146 for a deterministic sequence $(\delta_t)$. Then for every $T$,
4147 \begin{equation}
4148 \Pr\left(\bigcup_{t=1}^{T}F_t\right)
4149 \leq
4150 \sum_{t=1}^{T}\delta_t.
4151 \end{equation}
4152 If $\sum_t\delta_t<\infty$, only finitely many false acceptances occur almost surely.
4153 \end{theorem}
4154 
4155 \begin{proof}
4156 Taking expectations gives $\Pr(F_t)\leq\delta_t$. The finite-time claim follows from the union bound. The almost-sure statement follows from the first Borel--Cantelli lemma.
4157 \end{proof}
4158 
4159 \begin{definition}[Evidence namespace]
4160 An evidence namespace is a pair $(\Ecal,\mathsf{use})$ consisting of protected records and a use ledger. Each evaluation query records which data, metrics, stopping rules, and candidate families were exposed. Reusing evidence is permitted only under a validation rule whose guarantee remains valid conditional on the recorded use history.
4161 \end{definition}
4162 
4163 \begin{warning}[Nominal confidence is not sequential confidence]
4164 Repeatedly selecting the best candidate on one nominally held-out set generally violates the conditional premise of \cref{thm:adaptive-alpha-spending}. The transaction receipt must identify whether the bound came from fresh evidence, an anytime-valid procedure, a valid conditional test, or a new independent witness evaluation.
4165 \end{warning}
4166 
4167 \subsection{Rotating witnesses}
4168 
4169 A permanently immutable witness limits system evolution. An unrestricted simultaneous rewrite destroys validation independence. A rotating witness schedule permits eventual replacement of every component while preserving transaction-local separation.
4170 
4171 Let the system contain $n$ witness-capable components. At round $t$, let $E_t$ be the edited set and $W_t$ the witness set.
4172 
4173 \begin{definition}[Rotating witness schedule]
4174 A sequence $(E_t,W_t)_{t\geq0}$ is a rotating witness schedule when:
4175 \begin{enumerate}
4176 \item $E_t\cap W_t=\varnothing$ for every $t$;
4177 \item $W_t$ is immutable during transaction $t$;
4178 \item every component belongs to $E_t$ for at least one $t$;
4179 \item every witness replacement is validated by a component outside the same edit set;
4180 \item protected evidence is transported through a conservative evaluator morphism at every rotation;
4181 \item every committed round retains a rollback path to a state certified by the preceding witness.
4182 \end{enumerate}
4183 \end{definition}
4184 
4185 \begin{theorem}[Minimum rounds under a witness quorum]
4186 \label{thm:rotation-rounds}
4187 Suppose every transaction requires at least $q$ frozen witness components, where $1\leq q<n$. Then at most $n-q$ components can be edited in one round. Editing every component at least once requires at least
4188 \begin{equation}
4189 \left\lceil\frac{n}{n-q}\right\rceil
4190 \end{equation}
4191 rounds. This bound is attained by partitioning the components into edit blocks of size at most $n-q$ and taking the complement of each block as its witness set.
4192 \end{theorem}
4193 
4194 \begin{proof}
4195 The disjointness condition implies $\abs{E_t}\leq n-q$, which gives the counting lower bound. A partition into blocks of that size edits every component, and each complement has at least $q$ members.
4196 \end{proof}
4197 
4198 \begin{definition}[Validation dependency graph]
4199 For one planned replacement epoch, let $G=(V,A)$ have one vertex per component and an edge $i\to j$ when the old version of component $i$ must validate the replacement of component $j$ before $i$ itself is replaced.
4200 \end{definition}
4201 
4202 \begin{theorem}[Acyclic staged replacement]
4203 \label{thm:acyclic-witness-replacement}
4204 If the validation dependency graph is acyclic, all components can be replaced in a finite sequence that respects every dependency. If the graph contains a directed cycle and no external witness validates any member of that cycle, then no sequential schedule can replace every member while keeping each required validator old until its target is committed.
4205 \end{theorem}
4206 
4207 \begin{proof}
4208 For an acyclic graph, update components in reverse topological order. If $i\to j$, then $j$ is replaced before $i$, so $i$ remains old while validating $j$. For a directed cycle, whichever member is replaced last requires validation by its predecessor on the cycle. That predecessor has already been replaced, contradicting the requirement that it remain old. An external witness can break the cycle by supplying a dependency outside it.
4209 \end{proof}
4210 
4211 \begin{proposition}[Quorum-intersection condition]
4212 \label{prop:witness-quorum-intersection}
4213 Let $m$ witness replicas be available, and suppose at most $f$ may be compromised. If acceptance requires a quorum of size $q$ satisfying
4214 \begin{equation}
4215 2q>m+f,
4216 \end{equation}
4217 then any two accepting quorums intersect in more than $f$ replicas. Hence their intersection contains at least one uncompromised replica.
4218 \end{proposition}
4219 
4220 \begin{proof}
4221 For quorums $Q_1,Q_2$,
4222 \begin{equation}
4223 \abs{Q_1\cap Q_2}
4224 \geq
4225 2q-m
4226 >
4227 f.
4228 \end{equation}
4229 At most $f$ replicas are compromised, so at least one member of the intersection is uncompromised.
4230 \end{proof}
4231 
4232 \begin{warning}
4233 Quorum intersection limits conflicting approvals caused by a bounded number of compromised replicas. It does not prove that the uncompromised witness specification is correct, complete, or robust to shared evaluator error.
4234 \end{warning}
4235 
4236 \begin{statusbox}{Transactional result}
4237 Witness separation is a transaction-local noncircularity condition, not a permanent immutable layer. Trace factorization identifies when the witness sees enough information for its declared decision. Sandboxing and buffered effects give exact rollback under explicit assumptions. Rotating schedules permit eventual replacement of every component, while dependency graphs and quorum bounds expose when validation becomes circular or insufficiently independent.
4238 \end{statusbox}
4239 
4240 \section{Recursive Improvement Closure}
4241 \label{sec:recursive-closure}
4242 
4243 A fixed point of invariant preservation is not yet a theory of improvement. This section separates four notions: universal closure, controlled robust closure, productive recurrence, and stochastic confidence closure.
4244 
4245 \subsection{Universal finite-state closure}
4246 
4247 Let $S$ be a finite set of complete lineage states. Write
4248 \begin{equation}
4249 s\xrightarrow{\tau_{\mathrm{imp}}}s'
4250 \end{equation}
4251 when one authorized improvement transaction can take $s$ to $s'$. Define
4252 \begin{equation}
4253 \Post_{\mathrm{imp}}(s)
4254 =
4255 \set{s'\in S:s\xrightarrow{\tau_{\mathrm{imp}}}s'}.
4256 \end{equation}
4257 
4258 Let
4259 \begin{equation}
4260 \begin{aligned}
4261 C_{\mathrm{RI}}(s)
4262 ={}&
4263 \Grounded(s)
4264 \wedge\ProvValid(s)
4265 \wedge\WorkspaceBound(s)\\
4266 &\wedge\RepairTyped(s)
4267 \wedge\Witnessed(s)
4268 \wedge\EvidenceFlat(s)\\
4269 &\wedge\Rollbackable(s)
4270 \wedge\MetaCapable(s).
4271 \end{aligned}
4272 \end{equation}
4273 For $X\subseteq S$, define
4274 \begin{equation}
4275 \Psi_{\forall}(X)
4276 =
4277 \set{s\in S:C_{\mathrm{RI}}(s)\text{ and }\Post_{\mathrm{imp}}(s)\subseteq X}.
4278 \end{equation}
4279 
4280 \begin{theorem}[Existence of universal recursive improvement closure]
4281 \label{thm:ri-closure}
4282 The operator $\Psi_{\forall}:2^S\to2^S$ is monotone and has a greatest fixed point
4283 \begin{equation}
4284 \Ical_{\forall}^{*}
4285 =
4286 \nu X.\Psi_{\forall}(X).
4287 \end{equation}
4288 A state lies in $\Ical_{\forall}^{*}$ exactly when it satisfies the static invariants and every authorized successor also lies in $\Ical_{\forall}^{*}$.
4289 \end{theorem}
4290 
4291 \begin{proof}
4292 If $X\subseteq Y$, then $\Post_{\mathrm{imp}}(s)\subseteq X$ implies $\Post_{\mathrm{imp}}(s)\subseteq Y$. Thus $\Psi_{\forall}$ is monotone. Knaster--Tarski gives the greatest fixed point.
4293 \end{proof}
4294 
4295 \begin{corollary}[Finite descending computation]
4296 Set $X_0=S$ and $X_{k+1}=\Psi_{\forall}(X_k)$. The sequence is descending and stabilizes at $\Ical_{\forall}^{*}$ after at most $\abs{S}$ strict reductions.
4297 \end{corollary}
4298 
4299 The universal closure is deliberately strong. A state is excluded when even one authorized transition is unsafe, including a transition the controller would never select.
4300 
4301 \subsection{Controlled robust closure}
4302 
4303 Let $\mathsf{Act}(s)$ be the set of authorized transaction choices at state $s$. After action $u\in\mathsf{Act}(s)$, let $\mathsf{Dist}(s,u)$ be the set of admissible disturbances, evaluation outcomes, or unresolved environmental responses. Let
4304 \begin{equation}
4305 T(s,u,d)\in S
4306 \end{equation}
4307 be the resulting state.
4308 
4309 \begin{definition}[Controllable predecessor]
4310 For $X\subseteq S$, define
4311 \begin{equation}
4312 \operatorname{CPre}(X)
4313 =
4314 \set{
4315  s\in S:
4316  \exists u\in\mathsf{Act}(s)\ \forall d\in\mathsf{Dist}(s,u),\ T(s,u,d)\in X
4317 }.
4318 \end{equation}
4319 \end{definition}
4320 
4321 Define
4322 \begin{equation}
4323 \Psi_{\mathrm{ctrl}}(X)
4324 =
4325 \set{s\in S:C_{\mathrm{RI}}(s)}
4326 \cap
4327 \operatorname{CPre}(X).
4328 \end{equation}
4329 
4330 \begin{theorem}[Robust controlled improvement kernel]
4331 \label{thm:controlled-closure}
4332 The operator $\Psi_{\mathrm{ctrl}}$ is monotone and has a greatest fixed point
4333 \begin{equation}
4334 \Ical_{\mathrm{ctrl}}^{*}
4335 =
4336 \nu X.\Psi_{\mathrm{ctrl}}(X).
4337 \end{equation}
4338 For every state in $\Ical_{\mathrm{ctrl}}^{*}$, there exists a stationary authorized policy that keeps every admissible successor inside $\Ical_{\mathrm{ctrl}}^{*}$.
4339 \end{theorem}
4340 
4341 \begin{proof}
4342 Monotonicity follows from monotonicity of $\operatorname{CPre}$. Knaster--Tarski gives the greatest fixed point. For each $s$ in the fixed point, choose one action witnessing membership in $\operatorname{CPre}(\Ical_{\mathrm{ctrl}}^{*})$. Finiteness permits these choices to define a stationary policy, and every admissible disturbance remains in the fixed point by construction.
4343 \end{proof}
4344 
4345 \begin{proposition}[Closure monotonicity]
4346 \label{prop:closure-monotonicity}
4347 Under a fixed invariant predicate:
4348 \begin{enumerate}
4349 \item adding authorized actions can only enlarge $\Ical_{\mathrm{ctrl}}^{*}$;
4350 \item adding admissible disturbances can only shrink $\Ical_{\mathrm{ctrl}}^{*}$;
4351 \item strengthening the static invariant predicate can only shrink both universal and controlled closures;
4352 \item whenever every action set is nonempty, $\Ical_{\forall}^{*}\subseteq\Ical_{\mathrm{ctrl}}^{*}$.
4353 \end{enumerate}
4354 \end{proposition}
4355 
4356 \begin{proof}
4357 Each claim follows from the corresponding pointwise inclusion of predecessor operators, followed by monotonicity of greatest fixed points. For item 4, universal safety of every successor implies the existence of at least one robustly safe action.
4358 \end{proof}
4359 
4360 \subsection{Safe recurrence is not productive recurrence}
4361 
4362 A no-op state may lie in the controlled closure forever. Recursive improvement requires a liveness condition in addition to invariant preservation.
4363 
4364 Let $G_{\gamma}\subseteq S$ be the set of states whose most recent committed transaction produced a validated metaproductivity gain of at least $\gamma$ under the matched-budget definition in \cref{sec:metaproductivity}. Define the safe predecessor
4365 \begin{equation}
4366 \operatorname{SPre}(X)
4367 =
4368 \set{s:C_{\mathrm{RI}}(s)}
4369 \cap
4370 \operatorname{CPre}(X).
4371 \end{equation}
4372 
4373 \begin{definition}[Productive recursive closure]
4374 For a finite perfect-information transaction game, define
4375 \begin{equation}
4376 \Ical_{\gamma}^{\omega}
4377 =
4378 \nu Z.\,\mu Y.
4379 \left[
4380 \left(G_{\gamma}\cap\operatorname{SPre}(Z)\right)
4381 \cup
4382 \operatorname{SPre}(Y)
4383 \right].
4384 \end{equation}
4385 \end{definition}
4386 
4387 \begin{theorem}[Buechi characterization of repeated improvement]
4388 \label{thm:productive-closure}
4389 A state lies in $\Ical_{\gamma}^{\omega}$ exactly when the controller has a strategy that preserves $C_{\mathrm{RI}}$ and forces infinitely many visits to $G_{\gamma}$ against every admissible disturbance.
4390 \end{theorem}
4391 
4392 \begin{proof}
4393 For fixed $Z$, the least fixed point in $Y$ is the safe controllable attractor of $G_{\gamma}\cap\operatorname{SPre}(Z)$. The outer greatest fixed point retains exactly those states from which the controller can repeatedly reach a gain state and then return to the retained region. This is the standard nested fixed-point construction for a Buechi objective in a finite game.
4394 \end{proof}
4395 
4396 \begin{corollary}[Productivity is stricter than safety]
4397 \begin{equation}
4398 \Ical_{\gamma}^{\omega}
4399 \subseteq
4400 \Ical_{\mathrm{ctrl}}^{*}.
4401 \end{equation}
4402 The inclusion may be strict.
4403 \end{corollary}
4404 
4405 \begin{proof}
4406 Every state in the Buechi set remains inside the static invariant set under its winning strategy, hence lies in the robust safety kernel. A safe absorbing no-op state outside $G_\gamma$ witnesses possible strictness.
4407 \end{proof}
4408 
4409 \begin{proposition}[Finite ranking certificate for one gain cycle]
4410 \label{prop:progress-ranking}
4411 Let $X\subseteq\Ical_{\mathrm{ctrl}}^{*}$ and let $h:X\to\{0,1,\dots,H\}$ satisfy $h^{-1}(0)\subseteq G_\gamma$. Suppose that at every state with $h(s)>0$, the controller has an authorized action whose every admissible successor $s'$ lies in $X$ and satisfies
4412 \begin{equation}
4413 h(s')<h(s).
4414 \end{equation}
4415 Then the controller can force a visit to $G_\gamma$ within at most $H$ transactions while remaining in $X$.
4416 \end{proposition}
4417 
4418 \begin{proof}
4419 The nonnegative integer rank decreases strictly at every step, so it reaches zero in at most $H$ steps. Every successor remains in $X$ by assumption.
4420 \end{proof}
4421 
4422 \begin{remark}[What can recur]
4423 If task utility takes values in a finite set and every gain visit strictly increases it, infinitely many gain visits are impossible. Productive recurrence must eventually concern improved repair competence, grammar expansion, evaluator refinement, resource efficiency, or newly introduced task families rather than strict ascent of one fixed finite score.
4424 \end{remark}
4425 
4426 \subsection{Chance-constrained closure}
4427 
4428 For stochastic transitions, define
4429 \begin{equation}
4430 \operatorname{CPre}_{\delta}(X)
4431 =
4432 \set{
4433  s:\exists u\in\mathsf{Act}(s),\
4434  \Pr(T(s,u,D)\in X\mid s,u)\geq1-\delta
4435 }.
4436 \end{equation}
4437 A one-step chance constraint does not by itself provide an infinite-horizon guarantee.
4438 
4439 \begin{proposition}[Risk-budget composition]
4440 \label{prop:risk-budget-composition}
4441 Suppose a policy at transaction $t$ has conditional probability at most $\delta_t$ of leaving the declared closure, given all prior history. Then the probability of remaining in the closure through transaction $T$ is at least
4442 \begin{equation}
4443 1-\sum_{t=1}^{T}\delta_t.
4444 \end{equation}
4445 If $\sum_t\delta_t\leq\delta$, the probability of ever leaving is at most $\delta$.
4446 \end{proposition}
4447 
4448 \begin{proof}
4449 Apply the union bound to the conditional failure events and then take expectations, as in \cref{thm:adaptive-alpha-spending}.
4450 \end{proof}
4451 
4452 \subsection{Closure certificates and counterexamples}
4453 
4454 A finite closure computation should emit more than a set of accepted states. For each retained controlled state, the certificate records a witness action. For each removed state, it records either a violated static predicate or a disturbance that leaves the previous iterate. The descending fixed-point computation therefore yields:
4455 \begin{enumerate}
4456 \item a memoryless preservation policy on $\Ical_{\mathrm{ctrl}}^{*}$;
4457 \item a finite counterexample tree for excluded states;
4458 \item a nested-attractor strategy for $\Ical_{\gamma}^{\omega}$;
4459 \item a progress rank for each finite attractor layer.
4460 \end{enumerate}
4461 
4462 \begin{statusbox}{Closure result}
4463 Recursive improvement closure has two logically separate parts. The greatest fixed point gives a robust kernel of states from which declared invariants can be preserved. The nested Buechi fixed point gives the subset from which validated metaproductive events can be forced repeatedly. A system may satisfy the first while failing the second, so safe persistence must not be reported as recursive improvement.
4464 \end{statusbox}
4465 
4466 \section{Archive Dynamics and Calibrated Selection}
4467 \label{sec:archive-dynamics}
4468 
4469 The lineage archive is not only a backup store. It determines which stepping stones remain available for later repair composition, transport, and grammar expansion.
4470 
4471 \subsection{Branching archives and multiobjective frontiers}
4472 
4473 Let $\Acal_t$ be a finite directed acyclic graph of validated checkpoints and rejected-but-informative candidates through time $t$. Each node $A$ has a descriptor
4474 \begin{equation}
4475 z(A)
4476 =
4477 \left(
4478 J(A),
4479 M(A),
4480 S(A),
4481 N(A),
4482 1-C(A)
4483 \right)
4484 \in[0,1]^d,
4485 \end{equation}
4486 where the coordinates may encode task utility, foreign-agent metaproductivity, safety or invariant margin, novelty, and normalized resource efficiency.
4487 
4488 For maximization coordinates, write $A\succeq B$ when
4489 \begin{equation}
4490 z_i(A)\geq z_i(B)
4491 \qquad\text{for every }i.
4492 \end{equation}
4493 The Pareto frontier is the set of undominated nodes.
4494 
4495 \begin{proposition}[Archive-envelope monotonicity]
4496 \label{prop:archive-envelope}
4497 Suppose archive updates never delete the last representative of a retained node. For every nonnegative weight vector $w$, define
4498 \begin{equation}
4499 H_w(\Acal_t)
4500 =
4501 \max_{A\in\Acal_t}w^{\mathsf T}z(A).
4502 \end{equation}
4503 Then $H_w(\Acal_t)$ is nondecreasing in $t$.
4504 \end{proposition}
4505 
4506 \begin{proof}
4507 Archive inclusion gives a maximum over a weakly larger set.
4508 \end{proof}
4509 
4510 \begin{remark}
4511 The set of Pareto-optimal nodes can change even when every scalar upper envelope is nondecreasing. A former frontier node may become dominated without becoming useless as a stepping stone.
4512 \end{remark}
4513 
4514 \subsection{Finite approximate frontier compression}
4515 
4516 Unbounded retention is often infeasible. The next theorem gives a simple worst-case compression guarantee.
4517 
4518 \begin{theorem}[Finite $\eps$-Pareto cover]
4519 \label{thm:epsilon-pareto-cover}
4520 Let $Z\subseteq[0,1]^d$ be finite and let $0<\eps\leq1$. There exists a subset $Q\subseteq Z$ with
4521 \begin{equation}
4522 \abs{Q}
4523 \leq
4524 \left\lceil\frac{1}{\eps}\right\rceil^d
4525 \end{equation}
4526 such that for every $z\in Z$, some $q\in Q$ satisfies
4527 \begin{equation}
4528 q_i\geq z_i-\eps
4529 \qquad
4530 \text{for every }i.
4531 \end{equation}
4532 \end{theorem}
4533 
4534 \begin{proof}
4535 Partition $[0,1]^d$ into axis-aligned cells of side length at most $\eps$. Retain one point from each nonempty cell. Points in the same cell differ by at most $\eps$ in every coordinate, so the retained point satisfies the coordinatewise approximation. The number of cells gives the bound.
4536 \end{proof}
4537 
4538 \begin{corollary}[Approximate scalarization preservation]
4539 For every $w\geq0$ with $\norm{w}_1=1$,
4540 \begin{equation}
4541 \max_{q\in Q}w^{\mathsf T}q
4542 \geq
4543 \max_{z\in Z}w^{\mathsf T}z-\eps.
4544 \end{equation}
4545 \end{corollary}
4546 
4547 \begin{proof}
4548 Choose a maximizer $z$ and its covering point $q$. Then $w^{\mathsf T}q\geq w^{\mathsf T}z-\eps\norm{w}_1$.
4549 \end{proof}
4550 
4551 \subsection{Stepping stones and continuation signatures}
4552 
4553 \begin{proposition}[Performance-only pruning can destroy the optimum]
4554 \label{prop:stepping-stone-counterexample}
4555 There exists a finite lineage tree in which retaining only the highest-current-utility child at each generation prevents discovery of the globally highest-utility descendant.
4556 \end{proposition}
4557 
4558 \begin{proof}
4559 Let a root have children $x$ and $y$ with $J(x)>J(y)$. Let $x$ be terminal. Let $y$ have a child $z$ with $J(z)>J(x)$. Greedy pruning deletes $y$, so $z$ becomes unreachable.
4560 \end{proof}
4561 
4562 The point is not merely that exploration helps. A lower-scoring node may preserve a repair operator, representation, or obstruction certificate unavailable elsewhere.
4563 
4564 \begin{definition}[Horizon-$h$ continuation equivalence]
4565 Two archive nodes $A$ and $B$ are horizon-$h$ continuation equivalent, written
4566 \begin{equation}
4567 A\sim_h B,
4568 \end{equation}
4569 when, under the same repair grammar and matched budgets, their sets of reachable descriptor vectors through at most $h$ further transactions are identical.
4570 \end{definition}
4571 
4572 \begin{theorem}[Exact archive quotient]
4573 \label{thm:archive-quotient}
4574 Retaining one representative from each $\sim_h$ equivalence class preserves the complete horizon-$h$ reachable descriptor set.
4575 \end{theorem}
4576 
4577 \begin{proof}
4578 Every discarded node has a retained equivalent representative with the same horizon-$h$ continuation set. Taking the union over retained classes therefore gives the original reachable set.
4579 \end{proof}
4580 
4581 \begin{remark}
4582 Exact continuation equivalence is generally too expensive to compute. Practical signatures based on repair schemas, obstruction classes, latent packets, or evaluator branches are approximations. Their validity should be tested by descendant-distribution comparisons rather than assumed.
4583 \end{remark}
4584 
4585 \subsection{Calibrated acceptance and archive monotonicity}
4586 
4587 Let $J^*(\Acal_t)$ be the highest true protected utility among retained validated nodes.
4588 
4589 \begin{theorem}[Summable false-acceptance theorem]
4590 \label{thm:summable-errors}
4591 For transaction $t$, let $F_t$ be the event that an accepted child has nonpositive true declared gain. If
4592 \begin{equation}
4593 \Pr(F_t\mid\Fcal_{t-1})\leq\delta_t
4594 \end{equation}
4595 and $\sum_t\delta_t<\infty$, then only finitely many harmful acceptances occur almost surely. If the archive retains every previously validated best node, then $J^*(\Acal_t)$ is nondecreasing outside a finite random set of erroneous rounds.
4596 \end{theorem}
4597 
4598 \begin{proof}
4599 The first claim is \cref{thm:adaptive-alpha-spending} and Borel--Cantelli. On every non-error round, accepted descendants have positive true gain, while retention prevents deletion of the previous best. Therefore archive regression can occur only on the finite error set.
4600 \end{proof}
4601 
4602 \begin{definition}[Archive receipt completeness]
4603 An archive insertion receipt is complete when it records:
4604 \begin{enumerate}
4605 \item the parent set and lineage-edge proofs;
4606 \item the descriptor vector and uncertainty region;
4607 \item the dominance and novelty comparisons used for retention;
4608 \item every reusable repair schema or obstruction certificate introduced by the node;
4609 \item the continuation signature and compression cell, when applicable;
4610 \item the exact evidence namespace used for acceptance.
4611 \end{enumerate}
4612 \end{definition}
4613 
4614 \begin{statusbox}{Archive result}
4615 A monotone best score is not enough. The archive must preserve future option value. Multiobjective frontiers, approximate covers, continuation signatures, and explicit repair or obstruction content make the retention rule part of the formal improvement mechanism rather than an implementation detail.
4616 \end{statusbox}
4617 
4618 \section{Metaproductivity}
4619 \label{sec:metaproductivity}
4620 
4621 Metaproductivity is the capacity to produce useful descendants under a fixed budget. It must be measured separately from the task competence of the parent and from the number of candidates sampled.
4622 
4623 \subsection{Behavioral metaproductivity on foreign agents}
4624 
4625 Let $\Dcal_{\mathrm{hold}}$ be a distribution over foreign held-out agents that are not descendants of the evaluated lineage. Let $K$ be a frozen improvement mechanism. Under horizon $h$ and budget $B$, define
4626 \begin{equation}
4627 M^{\mathrm{foreign}}_{h,B}(K)
4628 =
4629 \mathbb E_{A\sim\Dcal_{\mathrm{hold}}}
4630 \left[
4631 \max_{A'\in\Desc_{h,B}(K,A)}
4632 J(A')-J(A)
4633 \right].
4634 \end{equation}
4635 
4636 The baseline, task distribution, model access, number of candidates, tool permissions, validation data, and wall-clock or compute budget must be matched.
4637 
4638 \begin{definition}[Bounded recursive self-improvement]
4639 A lineage exhibits bounded recursive self-improvement over a declared generation set when:
4640 \begin{enumerate}
4641 \item protected task utility improves on a nontrivial held-out family;
4642 \item $M^{\mathrm{foreign}}_{h,B}$ increases under matched budgets;
4643 \item task-relevant structural repair capacity increases on a held-out defect family;
4644 \item evidence, provenance, witness separation, and rollback invariants remain valid.
4645 \end{enumerate}
4646 \end{definition}
4647 
4648 \begin{definition}[Budget acceleration]
4649 Let $B_t(\gamma)$ be the least budget required by mechanism $K_t$ to obtain expected foreign-agent gain at least $\gamma$. Budget acceleration holds on an interval $I$ when
4650 \begin{equation}
4651 B_{t+1}(\gamma)<B_t(\gamma)
4652 \qquad
4653 \text{for every }\gamma\in I.
4654 \end{equation}
4655 \end{definition}
4656 
4657 \subsection{The best-of-k budget confound}
4658 
4659 \begin{proposition}[Expected best-of-$k$ gain]
4660 \label{prop:best-of-k}
4661 Let $G_1,\dots,G_k$ be independent nonnegative candidate gains with common cumulative distribution function $F$ and finite mean. Then
4662 \begin{equation}
4663 \mathbb E\max_{1\leq i\leq k}G_i
4664 =
4665 \int_0^{\infty}
4666 \left(1-F(x)^k\right)\dd x.
4667 \end{equation}
4668 In particular, expected best gain is nondecreasing in $k$ even when the candidate generator itself does not improve.
4669 \end{proposition}
4670 
4671 \begin{proof}
4672 For a nonnegative random variable $X$, $\mathbb E X=\int_0^\infty\Pr(X>x)\dd x$. Independence gives
4673 \begin{equation}
4674 \Pr\left(\max_iG_i\leq x\right)=F(x)^k.
4675 \end{equation}
4676 Substitution proves the identity.
4677 \end{proof}
4678 
4679 This identity is why candidate count and compute must be held fixed when comparing improvement mechanisms.
4680 
4681 
4682 \subsection{Paired foreign-agent estimation and distribution shift}
4683 
4684 Let $K_0$ and $K_1$ be frozen improvement mechanisms evaluated on the same independent foreign agents $A_1,\ldots,A_n$. Define
4685 \begin{equation}
4686 G_{h,B}(K,A)
4687 =
4688 \max_{A'\in\Desc_{h,B}(K,A)}J(A')-J(A)
4689 \end{equation}
4690 and assume
4691 \begin{equation}
4692 -G_{\max}
4693 \leq
4694 G_{h,B}(K,A)
4695 \leq
4696 G_{\max}.
4697 \end{equation}
4698 For matched budgets and random seeds, let
4699 \begin{equation}
4700 D_i
4701 =
4702 G_{h,B}(K_1,A_i)-G_{h,B}(K_0,A_i),
4703 \qquad
4704 \overline D_n
4705 =
4706 \frac1n\sum_{i=1}^{n}D_i.
4707 \end{equation}
4708 
4709 \begin{theorem}[Paired metaproductivity certificate]
4710 \label{thm:paired-meta}
4711 For independent held-out agents and $0<\delta<1$, with probability at least $1-\delta$,
4712 \begin{equation}
4713 \mathbb E D_i
4714 \geq
4715 \overline D_n
4716 -
4717 2\sqrt2\,G_{\max}
4718 \sqrt{\frac{\log(1/\delta)}{n}}.
4719 \end{equation}
4720 A positive lower bound certifies higher foreign-agent metaproductivity for $K_1$ relative to $K_0$.
4721 \end{theorem}
4722 
4723 \begin{proof}
4724 Each $D_i$ lies in an interval of length $4G_{\max}$. The one-sided Hoeffding inequality gives
4725 \begin{equation}
4726 \Pr\left(
4727 \overline D_n-\mathbb E D_i\geq\eps
4728 \right)
4729 \leq
4730 \exp\left(
4731 -\frac{n\eps^2}{8G_{\max}^2}
4732 \right).
4733 \end{equation}
4734 Solving for $\eps$ proves the bound.
4735 \end{proof}
4736 
4737 Pairing removes between-agent difficulty variation. It does not repair reuse of the same agents during mechanism design. A primary claim should use a final audit population revealed only after $K_0$, $K_1$, $h$, $B$, and the endpoints are frozen.
4738 
4739 Let $\Dcal$ and $\Dcal'$ be two foreign-agent distributions and define
4740 \begin{equation}
4741 \TV(\Dcal,\Dcal')
4742 =
4743 \sup_{B}\abs{\Dcal(B)-\Dcal'(B)}.
4744 \end{equation}
4745 
4746 \begin{proposition}[Metaproductivity shift bound]
4747 \label{prop:meta-tv}
4748 If $\abs{G_{h,B}(K,A)}\leq G_{\max}$, then
4749 \begin{equation}
4750 \abs{
4751 M_{h,B}^{\Dcal}(K)
4752 -
4753 M_{h,B}^{\Dcal'}(K)
4754 }
4755 \leq
4756 2G_{\max}\TV(\Dcal,\Dcal').
4757 \end{equation}
4758 \end{proposition}
4759 
4760 \begin{proof}
4761 Apply the total-variation expectation bound to the function $A\mapsto G_{h,B}(K,A)$.
4762 \end{proof}
4763 
4764 Thus metaproductivity transfer must state the foreign-agent population on which it is claimed.
4765 
4766 \subsection{Task-relevant structural metaproductivity}
4767 
4768 Raw repair rank is a geometric capacity measure. It becomes behaviorally meaningful only relative to a defect distribution.
4769 
4770 Let $V$ be a Hilbert defect space, let $L_t=\im D_t^{\Auth}$, and let $P_t$ be the orthogonal projector onto $L_t$. For a defect distribution $\mu$, define the correctable energy
4771 \begin{equation}
4772 \mathcal E_t^{\mathrm{cor}}(\mu)
4773 =
4774 \mathbb E_{r\sim\mu}\norm{P_t r}^2
4775 \end{equation}
4776 and the obstruction energy
4777 \begin{equation}
4778 \mathcal E_t^{\mathrm{ob}}(\mu)
4779 =
4780 \mathbb E_{r\sim\mu}\norm{(I-P_t)r}^2.
4781 \end{equation}
4782 
4783 \begin{theorem}[Structural-to-behavioral identity]
4784 \label{thm:structural-behavioral}
4785 Assume $L_t\subseteq L_{t+1}$ and let
4786 \begin{equation}
4787 W=L_{t+1}\cap L_t^{\perp}.
4788 \end{equation}
4789 Then
4790 \begin{equation}
4791 \mathcal E_{t+1}^{\mathrm{cor}}(\mu)-\mathcal E_t^{\mathrm{cor}}(\mu)
4792 =
4793 \mathbb E_{r\sim\mu}\norm{P_Wr}^2
4794 \end{equation}
4795 and
4796 \begin{equation}
4797 \mathcal E_t^{\mathrm{ob}}(\mu)-\mathcal E_{t+1}^{\mathrm{ob}}(\mu)
4798 =
4799 \mathbb E_{r\sim\mu}\norm{P_Wr}^2.
4800 \end{equation}
4801 Under quadratic residual loss and exact cancellation of the correctable component, this quantity is exactly the expected additional task-loss reduction made available by the expanded repair image.
4802 \end{theorem}
4803 
4804 \begin{proof}
4805 Nested subspaces give the orthogonal decomposition
4806 \begin{equation}
4807 L_{t+1}=L_t\oplus W,
4808 \qquad
4809 P_{t+1}=P_t+P_W.
4810 \end{equation}
4811 The two projected components are orthogonal, so
4812 \begin{equation}
4813 \norm{P_{t+1}r}^2
4814 =
4815 \norm{P_tr}^2+\norm{P_Wr}^2.
4816 \end{equation}
4817 The obstruction identity follows from Pythagoras. Under exact correction, the post-repair residual is $(I-P_t)r$, so loss reduction is $\norm{P_tr}^2$.
4818 \end{proof}
4819 
4820 \begin{corollary}[When rank expansion is behaviorally irrelevant]
4821 A strict inclusion $L_t\subsetneq L_{t+1}$ yields no expected quadratic benefit on $\mu$ exactly when
4822 \begin{equation}
4823 P_Wr=0
4824 \qquad
4825 \mu\text{-almost surely}.
4826 \end{equation}
4827 \end{corollary}
4828 
4829 \begin{proposition}[Finite-sample structural certificate]
4830 \label{prop:structural-concentration}
4831 Suppose $\norm{r}\leq R$ almost surely and $r_1,\dots,r_n$ are independent held-out defects. Let
4832 \begin{equation}
4833 \widehat\Delta_{\mathrm{str}}
4834 =
4835 \frac1n\sum_{i=1}^{n}\norm{P_Wr_i}^2.
4836 \end{equation}
4837 Then with probability at least $1-\delta$,
4838 \begin{equation}
4839 \left|
4840 \widehat\Delta_{\mathrm{str}}
4841 -
4842 \mathbb E\norm{P_Wr}^2
4843 \right|
4844 \leq
4845 R^2\sqrt{\frac{\log(2/\delta)}{2n}}.
4846 \end{equation}
4847 \end{proposition}
4848 
4849 \begin{proof}
4850 Each summand lies in $[0,R^2]$. Apply Hoeffding's inequality.
4851 \end{proof}
4852 
4853 \subsection{Conservative meta-improvement}
4854 
4855 \begin{proposition}[Foreign-transfer lower bound]
4856 \label{prop:foreign-transfer-lower-bound}
4857 Suppose $K_{t+1}$ retains every candidate available to $K_t$ and adds one repair schema. Let $E$ be a held-out event of probability $p$ on which the new schema produces certified gain at least $\gamma$ beyond the best candidate available to $K_t$, and suppose the new selector never chooses a worse candidate outside $E$. Then
4858 \begin{equation}
4859 M^{\mathrm{foreign}}_{h,B}(K_{t+1})
4860 -
4861 M^{\mathrm{foreign}}_{h,B}(K_t)
4862 \geq
4863 p\gamma.
4864 \end{equation}
4865 \end{proposition}
4866 
4867 \begin{proof}
4868 The pointwise best gain improves by at least $\gamma$ on $E$ and does not decrease outside $E$. Taking expectations gives the bound.
4869 \end{proof}
4870 
4871 \begin{warning}[Structural and behavioral metrics are complementary]
4872 Structural growth is not sufficient when new directions have negligible probability, excessive cost, or poor selection. It is not necessary for short-run behavioral gain because better conditioning, ranking, or execution can improve performance within the same correction image. A strong result should report both.
4873 \end{warning}
4874 
4875 
4876 \begin{definition}[One-step recursive meta-improvement]
4877 A lineage edge $K_t\to K_{t+1}$ is a one-step recursive meta-improvement when:
4878 \begin{enumerate}
4879 \item $K_{t+1}$ was constructed by an authorized transaction whose proposal process used $K_t$;
4880 \item the edge passed an independent witness and preserved the declared evidence and rollback invariants;
4881 \item under matched budgets on a frozen foreign-agent distribution,
4882 \begin{equation}
4883 M_{h,B}^{\mathrm{foreign}}(K_{t+1})
4884 >
4885 M_{h,B}^{\mathrm{foreign}}(K_t)
4886 \end{equation}
4887 with a positive precommitted lower confidence bound.
4888 \end{enumerate}
4889 \end{definition}
4890 
4891 The lineage condition excludes the weaker situation in which an external designer simply replaces $K_t$ with an unrelated better mechanism.
4892 
4893 \begin{proposition}[Task improvement does not imply meta-improvement]
4894 \label{prop:task-not-meta}
4895 There exist mechanisms $K_0,K_1$ such that $K_1$ has higher direct task utility on every task in a family, but
4896 \begin{equation}
4897 M_{h,B}^{\mathrm{foreign}}(K_1)
4898 <
4899 M_{h,B}^{\mathrm{foreign}}(K_0).
4900 \end{equation}
4901 \end{proposition}
4902 
4903 \begin{proof}
4904 Let $K_1$ be a perfect fixed solver that returns no useful modifications when asked to improve a foreign agent. Let $K_0$ be a weaker direct solver equipped with a repair procedure that improves every agent in the held-out family. Direct competence favors $K_1$, while improvement lift favors $K_0$.
4905 \end{proof}
4906 
4907 \begin{statusbox}{Metaproductivity result}
4908 Foreign-agent evaluation separates improvement competence from lineage-specific task competence. Matched candidate budgets remove the best-of-$k$ confound. Correctable energy links structural repair-image expansion to exact expected loss reduction under a quadratic model and makes clear when a nominal rank increase has no task relevance.
4909 \end{statusbox}
4910 
4911 \section{Grammar Expansion and Open-Endedness}
4912 \label{sec:grammar-expansion}
4913 
4914 A fixed finite grammar can support only finitely many strict increases in repair rank. Open-ended improvement therefore requires controlled expansion of the types used to describe defects, repairs, evidence, or evaluators.
4915 
4916 \subsection{Finite fixed-grammar exhaustion}
4917 
4918 \begin{theorem}[Finite structural-improvement bound]
4919 \label{thm:fixed-grammar-exhaustion}
4920 Let $V$ be a fixed $n$-dimensional defect space and suppose
4921 \begin{equation}
4922 \im D_0
4923 \subsetneq
4924 \im D_1
4925 \subsetneq
4926 \cdots
4927 \end{equation}
4928 is a strict ascending chain. Then the chain contains at most
4929 \begin{equation}
4930 n-\rank D_0
4931 \end{equation}
4932 strict inclusions.
4933 \end{theorem}
4934 
4935 \begin{proof}
4936 Every strict inclusion raises image dimension by at least one, while image dimension is bounded by $n$.
4937 \end{proof}
4938 
4939 Thus indefinite structural improvement requires expansion or refinement of at least one of
4940 \begin{equation}
4941 U_t^{\Auth},
4942 \qquad
4943 V_t,
4944 \qquad
4945 D_t,
4946 \qquad
4947 \Gcal_t,
4948 \qquad
4949 \Vcal_t.
4950 \end{equation}
4951 
4952 \subsection{Conservative grammar extensions}
4953 
4954 Let $\Gcal_t$ be a typed grammar of defects, repairs, evidence, transformations, and proof rules.
4955 
4956 \begin{definition}[Conservative grammar extension]
4957 An embedding
4958 \begin{equation}
4959 \iota_t:\Gcal_t\hookrightarrow\Gcal_{t+1}
4960 \end{equation}
4961 is conservative when:
4962 \begin{enumerate}
4963 \item every old well-typed expression remains well typed;
4964 \item every old derivation translates to a derivation;
4965 \item the interpretation of every old expression on old evidence is unchanged;
4966 \item protected evidence identifiers and provenance resources are transported identically;
4967 \item new rules cannot derive an old-language contradiction unless a correction receipt explicitly retracts a previously unsound axiom or evidence interpretation.
4968 \end{enumerate}
4969 \end{definition}
4970 
4971 \begin{theorem}[Finite derivation preservation]
4972 \label{thm:grammar-preservation}
4973 Suppose $\iota_t$ is fully faithful on the proof-carrying fragment and commutes with the old semantics. Then every theorem derivable in $\Gcal_t$ remains derivable with the same protected meaning in $\Gcal_{t+1}$.
4974 \end{theorem}
4975 
4976 \begin{proof}
4977 Translate the finite derivation rule by rule. Full faithfulness preserves each rule instance, semantic commutation preserves interpretation, and identity on protected evidence preserves external reference.
4978 \end{proof}
4979 
4980 \subsection{Directed grammar limits}
4981 
4982 Consider a chain
4983 \begin{equation}
4984 \Gcal_0
4985 \xhookrightarrow{\iota_0}
4986 \Gcal_1
4987 \xhookrightarrow{\iota_1}
4988 \Gcal_2
4989 \xhookrightarrow{}
4990 \cdots
4991 \end{equation}
4992 of conservative extensions.
4993 
4994 \begin{theorem}[Conservative directed limit]
4995 \label{thm:grammar-colimit}
4996 The union of translated syntax and proofs defines a directed-limit grammar $\Gcal_{\infty}$ with the following property: every finite expression and every finite derivation in $\Gcal_{\infty}$ appears at some finite stage. Every theorem proved at stage $t$ retains the same protected meaning at every later stage and in $\Gcal_{\infty}$.
4997 \end{theorem}
4998 
4999 \begin{proof}
5000 A finite expression or derivation uses finitely many symbols and rule instances, each introduced at a finite stage. Directedness places all of them in one later stage. Repeated application of \cref{thm:grammar-preservation} preserves every earlier theorem and its protected interpretation.
5001 \end{proof}
5002 
5003 \begin{definition}[Approximate semantic debt]
5004 If an extension preserves old semantics only approximately, let $\eta_t$ bound its worst protected interpretation error after transport. Under nonexpansive semantic transports, the cumulative debt through stage $T$ is
5005 \begin{equation}
5006 \eta_{0:T}
5007 \leq
5008 \sum_{t=0}^{T-1}\eta_t.
5009 \end{equation}
5010 \end{definition}
5011 
5012 \begin{proposition}[Summable approximate conservativity]
5013 \label{prop:grammar-semantic-debt}
5014 If protected semantic transports are nonexpansive and $\sum_t\eta_t<\infty$, then every old protected interpretation has a Cauchy image along the grammar chain, with total drift bounded by the tail sum.
5015 \end{proposition}
5016 
5017 \begin{proof}
5018 The difference between two later interpretations is bounded by the telescoping sum of intervening extension errors. Summability makes the sequence Cauchy and gives the stated tail bound.
5019 \end{proof}
5020 
5021 \subsection{Grammar-boundary certificates}
5022 
5023 For grammar $\Gcal$ with linearized authorized correction image $L_{\Gcal}$, define
5024 \begin{equation}
5025 q_{\Gcal}(r)
5026 =
5027 \dist(-r,L_{\Gcal}).
5028 \end{equation}
5029 This is the minimum unresolved first-order residual inside the current grammar.
5030 
5031 \begin{definition}[Relative grammar-expansion witness]
5032 Let $\Gcal\hookrightarrow\Gcal'$ be a candidate extension and let $r'$ be the transported defect. The extension exposes a missing correction direction with margin $\kappa>0$ when
5033 \begin{equation}
5034 q_{\Gcal'}(r')
5035 \leq
5036 q_{\Gcal}(r)-\kappa.
5037 \end{equation}
5038 \end{definition}
5039 
5040 \begin{proposition}[Expansion witness implies new correctability]
5041 \label{prop:grammar-expansion-witness}
5042 If the old and new defect metrics agree on the embedded old space and the relative expansion margin is positive, then the new authorized correction image contains a direction not available in the old image.
5043 \end{proposition}
5044 
5045 \begin{proof}
5046 If the new image introduced no new correction direction relevant to $r'$, the distance from $-r'$ to the new image would equal the distance to the embedded old image. A strict decrease contradicts that equality.
5047 \end{proof}
5048 
5049 \begin{warning}
5050 A positive old residual $q_{\Gcal}(r)$ proves only noncorrectability inside the current first-order grammar. It does not by itself distinguish a genuinely impossible repair from an omitted stratum, a nonlinear correction, a bad defect model, or corrupted evidence. Grammar expansion requires a separate hypothesis and validation receipt.
5051 \end{warning}
5052 
5053 
5054 \subsection{Dynamic obstruction exact sequence}
5055 
5056 Let $j:V\hookrightarrow V'$ be an injective evidence-conservative embedding of old defects into an expanded ontology. Let
5057 \begin{equation}
5058 C=\im D\subseteq V,
5059 \qquad
5060 C'=\im D'\subseteq V',
5061 \end{equation}
5062 and assume preservation of old correctability:
5063 \begin{equation}
5064 j(C)\subseteq C'.
5065 \end{equation}
5066 The embedding induces a map
5067 \begin{equation}
5068 \overline j:V/C\to V'/C'.
5069 \end{equation}
5070 
5071 \begin{theorem}[Old-resolution and new-obstruction exact sequence]
5072 \label{thm:dynamic-obstruction-exact}
5073 There is an exact sequence
5074 \begin{equation}
5075 0
5076 \longrightarrow
5077 \frac{j^{-1}(C')}{C}
5078 \longrightarrow
5079 \frac{V}{C}
5080 \xrightarrow{\overline j}
5081 \frac{V'}{C'}
5082 \longrightarrow
5083 \frac{V'}{C'+j(V)}
5084 \longrightarrow
5085 0.
5086 \end{equation}
5087 The first nonzero term contains old obstruction classes newly resolved by the expanded repair calculus. The final quotient contains genuinely new obstruction directions not represented in the old ontology.
5088 \end{theorem}
5089 
5090 \begin{proof}
5091 The kernel of $\overline j$ consists of classes $[v]$ with $j(v)\in C'$, which is $j^{-1}(C')/C$. The cokernel is
5092 \begin{equation}
5093 (V'/C')/((j(V)+C')/C')
5094 \cong
5095 V'/(C'+j(V)).
5096 \end{equation}
5097 Exactness at the remaining terms follows from the quotient maps.
5098 \end{proof}
5099 
5100 The sequence separates two effects that a single obstruction dimension conflates: old defects becoming repairable and new defect distinctions becoming visible.
5101 
5102 \subsection{Representation-separation certificates}
5103 
5104 Let $\phi_{\Gcal}:X\to Z_{\Gcal}$ be the state representation expressible in grammar $\Gcal$. A $\Gcal$-measurable deterministic repair policy has the form
5105 \begin{equation}
5106 \pi=\widetilde\pi\circ\phi_{\Gcal}.
5107 \end{equation}
5108 
5109 \begin{theorem}[Representation-separation obstruction]
5110 \label{thm:grammar-separation}
5111 Suppose states $x_0,x_1$ satisfy
5112 \begin{equation}
5113 \phi_{\Gcal}(x_0)=\phi_{\Gcal}(x_1),
5114 \end{equation}
5115 but their externally validated repair sets are disjoint:
5116 \begin{equation}
5117 \Rcal_{\mathrm{valid}}(x_0)
5118 \cap
5119 \Rcal_{\mathrm{valid}}(x_1)
5120 =
5121 \varnothing.
5122 \end{equation}
5123 Then no $\Gcal$-measurable deterministic repair policy is valid on both states.
5124 \end{theorem}
5125 
5126 \begin{proof}
5127 The policy receives the same grammar state for both inputs and therefore returns the same repair. No repair belongs to both valid sets.
5128 \end{proof}
5129 
5130 Such a pair is a constructive inadequacy certificate. A successful extension must add a distinction separating the states or introduce a repair valid for both.
5131 
5132 \subsection{Levels of improvement}
5133 
5134 \begin{table}[ht]
5135 \centering
5136 \caption{Four levels of improvement and their required certificates.}
5137 \begin{tabularx}{\textwidth}{@{}p{0.09\textwidth}p{0.28\textwidth}p{0.29\textwidth}X@{}}
5138 \toprule
5139 Level & Modification & Mathematical signature & Required certificate \\
5140 \midrule
5141 I & Parameters or policies in a fixed grammar & Same types, altered state point & Held-out gain and invariant preservation \\
5142 II & New repair operators in existing types & Expanded $U^{\Auth}$ or repair operad & New correctable direction, transfer, and receipt \\
5143 III & New defect, packet, or ontology types & Conservative $\Gcal_t\hookrightarrow\Gcal_{t+1}$ & Old-proof replay, evidence identity, new-direction witness \\
5144 IV & Evaluator refinement & Evolution outside an adequate path-flat core & Core retraction, replay, held-out risk improvement \\
5145 \bottomrule
5146 \end{tabularx}
5147 \end{table}
5148 
5149 \begin{conjecture}[Open-set repair adequacy]
5150 There exist nontrivial agent families for which a lineage can detect that its current grammar omits a causally necessary stratum, construct a conservative extension, and improve foreign-agent metaproductivity without violating its adequate path-flat evidence core.
5151 \end{conjecture}
5152 
5153 \begin{statusbox}{Grammar result}
5154 A fixed finite defect grammar has a finite structural ceiling. A conservative directed grammar chain can preserve every finite old proof while adding new types. A relative expansion witness shows that a proposed extension genuinely opens a previously unavailable correction direction. It does not show that the system can discover the right extension autonomously.
5155 \end{statusbox}
5156 
5157 \section{Autocatalytic Repair Discovery}
5158 \label{sec:autocatalytic-repair}
5159 
5160 Recursive acceleration would require validated repair discoveries to improve the process that discovers further repairs. Counting raw patches is insufficient. A counted offspring must be novel, transferable, externally validated, and causally dependent on prior repair knowledge.
5161 
5162 \subsection{Repair-discovery closure}
5163 
5164 Let $\Omega$ be a finite universe of candidate repair-schema equivalence classes. For $S\subseteq\Omega$, let $\operatorname{Disc}(S)$ be the schemas independently validated using the currently available set $S$. Define
5165 \begin{equation}
5166 \Phi_{\mathrm{disc}}(S)
5167 =
5168 S\cup\operatorname{Disc}(S).
5169 \end{equation}
5170 
5171 \begin{assumption}[Discovery monotonicity]
5172 If $S\subseteq T$, then
5173 \begin{equation}
5174 \operatorname{Disc}(S)
5175 \subseteq
5176 \operatorname{Disc}(T).
5177 \end{equation}
5178 This assumes that adding repair knowledge does not revoke access to old discovery procedures. Resource competition can violate the assumption and must then be modeled explicitly.
5179 \end{assumption}
5180 
5181 \begin{theorem}[Finite repair-discovery closure]
5182 \label{thm:discovery-closure}
5183 Under discovery monotonicity, $\Phi_{\mathrm{disc}}$ has a least fixed point above any seed set $S_0$. Iteration
5184 \begin{equation}
5185 S_{k+1}=\Phi_{\mathrm{disc}}(S_k)
5186 \end{equation}
5187 stabilizes after at most $\abs{\Omega}-\abs{S_0}$ strict expansions.
5188 \end{theorem}
5189 
5190 \begin{proof}
5191 The powerset of $\Omega$ is a finite complete lattice and $\Phi_{\mathrm{disc}}$ is monotone and inflationary. Each strict step adds at least one schema, so the stated bound follows.
5192 \end{proof}
5193 
5194 \subsection{Causal offspring matrix}
5195 
5196 Partition validated schemas into $m$ types. For a type-$i$ schema, let $N_{ij}$ count type-$j$ descendants whose receipts include a successful necessity ablation showing that removal of the parent schema eliminates the descendant discovery under matched evidence and budget. Define
5197 \begin{equation}
5198 M_{ij}=\mathbb E N_{ij}.
5199 \end{equation}
5200 This definition excludes descendants that merely share ancestry or evidence with the parent.
5201 
5202 \begin{definition}[Improvement reproduction number]
5203 The local repair-discovery reproduction number is
5204 \begin{equation}
5205 R_{\mathrm{imp}}
5206 =
5207 \rad(M).
5208 \end{equation}
5209 \end{definition}
5210 
5211 \begin{conditionaltheorem}[Finite-type branching threshold]
5212 \label{thm:branching-threshold}
5213 Assume validated repair discovery is accurately modeled by an irreducible finite-type Galton--Watson process with mean progeny matrix $M$ and is not deterministically one-child preserving. Then:
5214 \begin{enumerate}
5215 \item if $R_{\mathrm{imp}}\leq1$, the repair-discovery lineage becomes extinct almost surely;
5216 \item if $R_{\mathrm{imp}}>1$, it has positive probability of indefinite survival.
5217 \end{enumerate}
5218 \end{conditionaltheorem}
5219 
5220 \begin{warning}
5221 The branching assumptions are strong. Real discoveries share tools and evidence, compete for compute, alter their own type space, and are selected by an archive. The theorem is a local model of schema production, not a theorem of intelligence explosion or indefinite utility growth.
5222 \end{warning}
5223 
5224 
5225 \subsection{Autocatalytic cores}
5226 
5227 Construct the directed type graph with an edge $i\to j$ when $M_{ij}>0$. Its strongly connected components identify mutually enabling schema families.
5228 
5229 \begin{theorem}[Localization of supercriticality]
5230 \label{thm:autocatalytic-scc}
5231 After permuting schema types, $M$ has block upper-triangular form with diagonal blocks $M_1,\ldots,M_k$ given by strongly connected components. Therefore
5232 \begin{equation}
5233 \rad(M)
5234 =
5235 \max_{1\leq r\leq k}\rad(M_r).
5236 \end{equation}
5237 In particular, $R_{\mathrm{imp}}>1$ only if at least one strongly connected schema family is itself supercritical.
5238 \end{theorem}
5239 
5240 \begin{proof}
5241 Topologically order the condensation graph. The resulting matrix is block upper triangular, and its spectrum is the union of the spectra of its diagonal blocks.
5242 \end{proof}
5243 
5244 A supercritical component is an autocatalytic repair core. Downstream types may be abundant without contributing to continued discovery.
5245 
5246 \subsection{Robust spectral certification}
5247 
5248 Estimated offspring entries are uncertain. Componentwise intervals give a simple robust certificate.
5249 
5250 \begin{theorem}[Spectral interval bound]
5251 \label{thm:spectral-interval}
5252 Let $L,M,U$ be nonnegative square matrices satisfying
5253 \begin{equation}
5254 L\leq M\leq U
5255 \end{equation}
5256 entrywise. Then
5257 \begin{equation}
5258 \rad(L)
5259 \leq
5260 \rad(M)
5261 \leq
5262 \rad(U).
5263 \end{equation}
5264 Consequently, $\rad(L)>1$ certifies supercriticality for every matrix in the interval, while $\rad(U)\leq1$ certifies subcriticality.
5265 \end{theorem}
5266 
5267 \begin{proof}
5268 The spectral radius of a nonnegative matrix is monotone under entrywise order. This follows, for example, by applying the Collatz--Wielandt characterization to positive test vectors.
5269 \end{proof}
5270 
5271 \begin{proposition}[Perron sensitivity]
5272 \label{prop:perron-sensitivity}
5273 Let $M$ be irreducible with simple Perron root $\rho$, positive right eigenvector $r$, and positive left eigenvector $\ell$, normalized by $\ell^{\mathsf T}r=1$. Then
5274 \begin{equation}
5275 \frac{\partial\rho}{\partial M_{ij}}
5276 =
5277 \ell_i r_j.
5278 \end{equation}
5279 \end{proposition}
5280 
5281 \begin{proof}
5282 Differentiate $Mr=\rho r$ with respect to $M_{ij}$ and left-multiply by $\ell^{\mathsf T}$. The terms involving the derivative of $r$ cancel because $\ell^{\mathsf T}M=\rho\ell^{\mathsf T}$, leaving $\ell_i r_j$.
5283 \end{proof}
5284 
5285 The sensitivity identifies which validated causal edge would most increase the local reproduction number. It does not determine whether increasing that edge is safe, cheap, or behaviorally useful.
5286 
5287 \subsection{Resource normalization and time variation}
5288 
5289 Let $M(B)$ be the offspring matrix under a fixed discovery budget $B$. Define the critical budget
5290 \begin{equation}
5291 B_c
5292 =
5293 \inf\set{B:\rad(M(B))>1}.
5294 \end{equation}
5295 Comparisons of $R_{\mathrm{imp}}$ are meaningful only at matched budgets and evidence access.
5296 
5297 For a time-varying sequence $M_t$, let $z_t$ be a row vector of expected schema counts. Under the linear mean model,
5298 \begin{equation}
5299 \mathbb E z_T
5300 =
5301 z_0M_0M_1\cdots M_{T-1}.
5302 \end{equation}
5303 Define the upper mean-growth exponent
5304 \begin{equation}
5305 \lambda_{\mathrm{mean}}
5306 =
5307 \limsup_{T\to\infty}
5308 \frac1T
5309 \log\norm{M_0M_1\cdots M_{T-1}}.
5310 \end{equation}
5311 A positive value indicates exponential growth of the linearized expected count. It does not imply almost-sure survival under dependence or resource competition.
5312 
5313 \subsection{Saturation and the distinction from utility}
5314 
5315 \begin{proposition}[Finite-schema saturation]
5316 \label{prop:finite-schema-saturation}
5317 If only finitely many inequivalent schemas may ever be counted and each schema is counted at most once, then the discovery process produces only finitely many births, regardless of a transient estimate $R_{\mathrm{imp}}>1$.
5318 \end{proposition}
5319 
5320 \begin{proof}
5321 The total number of counted births is bounded by the number of available equivalence classes.
5322 \end{proof}
5323 
5324 
5325 \begin{theorem}[Bounded-utility gain saturation]
5326 \label{thm:bounded-gain-saturation}
5327 Suppose protected task utility satisfies $0\leq J\leq1$ and every committed task-improving schema raises deployed utility by at least $\gamma>0$. Starting from utility $J_0$, at most
5328 \begin{equation}
5329 \left\lfloor\frac{1-J_0}{\gamma}\right\rfloor
5330 \end{equation}
5331 such deployments can occur.
5332 \end{theorem}
5333 
5334 \begin{proof}
5335 After $k$ such gains, utility is at least $J_0+k\gamma\leq1$.
5336 \end{proof}
5337 
5338 A supercritical repair-discovery phase can therefore coexist with shrinking gains, task-family expansion, evaluator refinement, or eventual saturation. It cannot imply infinitely many fixed-size gains in a bounded utility.
5339 
5340 \begin{proposition}[Autocatalysis does not imply beneficial improvement]
5341 \label{prop:autocatalysis-not-utility}
5342 There exist repair-discovery processes with $R_{\mathrm{imp}}>1$ whose discovered schemas have zero or negative protected utility.
5343 \end{proposition}
5344 
5345 \begin{proof}
5346 Take any supercritical branching process of syntactically novel schemas and assign protected gain zero to every schema. Alternatively, assign negative gain while allowing the schemas to generate further schemas. The reproduction matrix is unchanged, while protected utility does not improve.
5347 \end{proof}
5348 
5349 Therefore a credible autocatalysis claim requires all of the following:
5350 \begin{enumerate}
5351 \item causal necessity ablations for offspring edges;
5352 \item foreign-system transfer;
5353 \item structural novelty or a conservative grammar distinction;
5354 \item external witness validation;
5355 \item matched resource budgets;
5356 \item a separate protected-utility or metaproductivity result.
5357 \end{enumerate}
5358 
5359 \begin{statusbox}{Block 4 result}
5360 The recursive layer now distinguishes transactional noncircularity, robust invariant closure, repeated productive closure, archive option value, foreign-agent metaproductivity, conservative grammar growth, and causal repair autocatalysis. None can substitute for the others. In particular, safe recurrence is not productive recurrence, repair-schema reproduction is not protected utility, and a fixed finite grammar cannot support indefinite strict structural expansion.
5361 \end{statusbox}
5362 
5363 \section{Empirical Realization Contract}
5364 \label{sec:empirical-contract}
5365 
5366 The preceding sections define a mathematical architecture. An empirical realization requires a separate contract connecting abstract objects to executable states, traces, data partitions, and claim boundaries. Without such a contract, successful code execution can be reported as support for a theorem that the runtime never actually instantiates.
5367 
5368 \subsection{Claim ladder}
5369 
5370 \begin{definition}[Empirical claim ladder]
5371 An empirical SRIC result is assigned to the strongest level whose requirements have all been met:
5372 \begin{enumerate}
5373 \item \textbf{mechanical realization}: the state machine, receipts, digests, rollback paths, and fixed-point routines execute as specified;
5374 \item \textbf{finite semantic realization}: executable states refine a declared finite abstract model and satisfy its frozen endpoints;
5375 \item \textbf{causal identification}: intervention or ablation changes the target outcome while matched controls do not;
5376 \item \textbf{foreign repair transfer}: a repair schema improves held-out systems outside its source lineage under matched resources;
5377 \item \textbf{metaproductive improvement}: a descendant repair calculus improves foreign systems better than its parent calculus;
5378 \item \textbf{open-set recursive improvement}: the system expands its grammar or ontology and preserves prior evidence while improving on an unbounded sequence of novel defect families.
5379 \end{enumerate}
5380 \end{definition}
5381 
5382 The ladder is monotone in evidential strength but not automatic. A level-$k$ result does not imply level $k+1$. In particular, deterministic replay proves runtime stability relative to a package, not causal identification or metaproductivity.
5383 
5384 \begin{warning}[Fixture success is not an RSI result]
5385 A finite fixture may intentionally contain a repair that the reference system is designed to find. Passing such a fixture verifies formal mechanics and can falsify an implementation. It does not establish learned, autonomous, open-ended, or indefinitely accelerating recursive self-improvement.
5386 \end{warning}
5387 
5388 \subsection{Benchmark instances}
5389 
5390 \begin{definition}[Empirical benchmark instance]
5391 A benchmark instance is a tuple
5392 \begin{equation}
5393 \mathfrak B
5394 =
5395 (\mathsf E,\mathsf A,\mathsf G,\mathsf P,\mathsf W,\mathsf V,
5396 \mathsf D,\mathsf S,\mathsf B,\mathsf R),
5397 \end{equation}
5398 where:
5399 \begin{itemize}
5400 \item $\mathsf E$ is the environment or transition process;
5401 \item $\mathsf A$ is the agent family and functional atlas;
5402 \item $\mathsf G$ is the repair grammar and candidate generator;
5403 \item $\mathsf P$ is the permitted probe and intervention language;
5404 \item $\mathsf W$ is the witness, checkpoint, and effect-mediation layer;
5405 \item $\mathsf V$ is the protected evaluation context;
5406 \item $\mathsf D$ is the data and evidence partition;
5407 \item $\mathsf S$ is the seed, randomization, and split schedule;
5408 \item $\mathsf B$ is the matched resource and candidate budget;
5409 \item $\mathsf R$ is the preregistered claim and receipt schema.
5410 \end{itemize}
5411 \end{definition}
5412 
5413 The data partition is itself typed:
5414 \begin{equation}
5415 \mathsf D
5416 =
5417 \mathsf D_{\mathrm{construct}}
5418 \sqcup
5419 \mathsf D_{\mathrm{calibrate}}
5420 \sqcup
5421 \mathsf D_{\mathrm{select}}
5422 \sqcup
5423 \mathsf D_{\mathrm{foreign}}
5424 \sqcup
5425 \mathsf D_{\mathrm{audit}}.
5426 \end{equation}
5427 Construction data may be used to create schemas. Calibration data may set thresholds. Selection data may choose among frozen candidates. Foreign data estimate metaproductivity. Audit data are not available until the final claim object has been sealed.
5428 
5429 \begin{definition}[Claim object]
5430 A claim object is
5431 \begin{equation}
5432 \mathfrak C
5433 =
5434 (\mathfrak B,\Theta,H,E,\Pi,\Gamma),
5435 \end{equation}
5436 where $\Theta$ is the frozen hypothesis and endpoint set, $H$ is the preoutcome commitment digest, $E$ is the revealed evidence, $\Pi$ is the proof-carrying receipt collection, and $\Gamma$ is the claim boundary.
5437 \end{definition}
5438 
5439 A result is reproducible only if an independent execution can reconstruct $\mathfrak C$ from the released package without consulting undeclared mutable state.
5440 
5441 \subsection{Abstract-runtime refinement}
5442 
5443 Let
5444 \begin{equation}
5445 \mathcal M=(S,\mathcal T,O)
5446 \end{equation}
5447 be an abstract transition system with state set $S$, authorized labeled transitions $\mathcal T$, and protected observation map $O$. Let
5448 \begin{equation}
5449 \widehat{\mathcal M}=(\widehat S,\widehat{\mathcal T},\widehat O)
5450 \end{equation}
5451 be an executable implementation.
5452 
5453 \begin{definition}[Forward implementation refinement]
5454 A map
5455 \begin{equation}
5456 \alpha:\widehat S\to S
5457 \end{equation}
5458 forward-refines the runtime when:
5459 \begin{enumerate}
5460 \item every executable initial state maps to an authorized abstract initial state;
5461 \item for every executable transition
5462 \begin{equation}
5463 \widehat s\xrightarrow{a}\widehat s',
5464 \end{equation}
5465 there is an abstract transition path
5466 \begin{equation}
5467 \alpha(\widehat s)\xrightarrow{a,*}\alpha(\widehat s');
5468 \end{equation}
5469 \item protected observations commute:
5470 \begin{equation}
5471 \widehat O(\widehat s)=O(\alpha(\widehat s));
5472 \end{equation}
5473 \item receipt compilation commutes with abstraction up to the declared canonical encoding.
5474 \end{enumerate}
5475 \end{definition}
5476 
5477 \begin{theorem}[Runtime-to-model trace soundness]
5478 \label{thm:runtime-trace-soundness}
5479 Suppose $\alpha$ is a forward implementation refinement. Then every finite executable trace maps to an authorized abstract trace with the same protected observation sequence. Consequently, any universal safety property over abstract protected observations is inherited by the runtime traces covered by the refinement.
5480 \end{theorem}
5481 
5482 \begin{proof}
5483 Induct on executable trace length. The initial condition gives the base case. Forward simulation maps each executable step to an abstract path with matching endpoints. Observation commutation preserves the protected trace. Universal trace properties are closed under this mapping.
5484 \end{proof}
5485 
5486 The converse does not follow. An abstract model may admit behaviors not implemented by the runtime. Therefore runtime testing cannot establish abstract completeness unless the implementation additionally satisfies a backward simulation or exhaustive state correspondence.
5487 
5488 \begin{definition}[Finite exact realization]
5489 A finite runtime exactly realizes $\mathcal M$ when $\alpha$ is surjective on reachable abstract states and every abstract authorized edge has at least one executable realizing path with matching receipt and observation.
5490 \end{definition}
5491 
5492 \begin{theorem}[Exhaustive finite endpoint theorem]
5493 \label{thm:finite-endpoint-complete}
5494 Let $\mathcal M$ and $\widehat{\mathcal M}$ be finite, exactly related, and let $Q_1,\ldots,Q_m$ be decidable state or transition predicates. Exhaustive enumeration of all reachable executable states and authorized edges decides whether each $Q_i$ holds over the realized finite model.
5495 \end{theorem}
5496 
5497 \begin{proof}
5498 The reachable executable graph is finite. Exact realization supplies every reachable abstract state and authorized edge. Evaluate each decidable predicate on the finite enumeration.
5499 \end{proof}
5500 
5501 \subsection{Frozen endpoints and claim gates}
5502 
5503 A block-level empirical package should separate \emph{endpoints} from \emph{interpretations}. An endpoint is a Boolean or numerical condition computed from sealed evidence. An interpretation is the claim that the endpoint supports.
5504 
5505 \begin{definition}[Claim gate]
5506 For claim level $\ell$, let
5507 \begin{equation}
5508 G_\ell(E,\Pi)\in\{0,1\}
5509 \end{equation}
5510 be a precommitted predicate over revealed evidence and receipts. The claim is admissible only if every prerequisite gate is one:
5511 \begin{equation}
5512 \operatorname{Admissible}(\ell)
5513 \iff
5514 \bigwedge_{j\leq\ell}G_j(E,\Pi)=1.
5515 \end{equation}
5516 \end{definition}
5517 
5518 Recommended gates are:
5519 \begin{longtable}{@{}p{0.17\textwidth}p{0.27\textwidth}p{0.48\textwidth}@{}}
5520 \toprule
5521 Gate & Target & Minimum evidence \\
5522 \midrule
5523 \endhead
5524 $G_{\mathrm{pkg}}$ & Package integrity & Exact manifest, canonical digests, clean extraction, successful verifier \\
5525 $G_{\mathrm{mech}}$ & Mechanics & Legal phase traces, receipt verification, checkpoint restoration, deterministic replay \\
5526 $G_{\mathrm{sem}}$ & Finite semantics & Abstract-runtime refinement and exhaustive frozen endpoints \\
5527 $G_{\mathrm{cause}}$ & Causal role & Intervention, necessity ablation, matched negative controls \\
5528 $G_{\mathrm{foreign}}$ & Transfer & Held-out systems, matched candidate budget, protected evaluator \\
5529 $G_{\mathrm{meta}}$ & Metaproductivity & Parent-child repair-calculus comparison on frozen foreign systems \\
5530 $G_{\mathrm{open}}$ & Open-set recursion & Conservative grammar extension, prior-proof replay, novel externally validated defect families \\
5531 \bottomrule
5532 \end{longtable}
5533 
5534 \begin{proposition}[Gate monotonicity]
5535 If claim admissibility is defined by cumulative conjunction, failure of a lower gate blocks every higher claim while preserving any lower claim whose own gates pass.
5536 \end{proposition}
5537 
5538 \begin{proof}
5539 Immediate from the conjunction defining $\operatorname{Admissible}(\ell)$.
5540 \end{proof}
5541 
5542 \begin{statusbox}{Block 5 empirical principle}
5543 The implementation is treated as a refinement target and falsification instrument. A package can establish that a finite mechanism was realized and that declared synthetic controls behaved as expected. It cannot promote itself to a stronger claim level without separately satisfying the corresponding causal, foreign-transfer, metaproductivity, and open-set gates.
5544 \end{statusbox}
5545 
5546 \section{Finite Model-Checking Realization}
5547 \label{sec:finite-realization}
5548 
5549 The reference fixture instantiates the distinction among local correctability, gluing compatibility, semantic stability, and productive closure in four exact cases. The fixture is deliberately small enough for complete enumeration and direct numerical inspection.
5550 
5551 \subsection{Four frozen cases}
5552 
5553 Let each local correction space and defect space be $\R^2$ unless otherwise stated. Let the tolerance satisfy $0<\tau<10^{-9}$.
5554 
5555 \paragraph{Case F1: local obstruction.}
5556 Set
5557 \begin{equation}
5558 D_{\mathrm{loc}}
5559 =
5560 \begin{bmatrix}
5561 1&0\\
5562 0&0
5563 \end{bmatrix},
5564 \qquad
5565 r_{\mathrm{loc}}
5566 =
5567 \begin{bmatrix}
5568 0\\1
5569 \end{bmatrix}.
5570 \end{equation}
5571 Then
5572 \begin{equation}
5573 P_{\im D_{\mathrm{loc}}}r_{\mathrm{loc}}=0,
5574 \qquad
5575 (I-P_{\im D_{\mathrm{loc}}})r_{\mathrm{loc}}=r_{\mathrm{loc}},
5576 \end{equation}
5577 so the obstruction norm is one.
5578 
5579 \paragraph{Case F2: gluing obstruction.}
5580 Take $D_0=D_1=D_2=I_2$, so every local defect is exactly correctable. On the oriented triangle with edges $(0,1)$, $(1,2)$, and $(0,2)$, set the scalar overlap mismatch
5581 \begin{equation}
5582 b=
5583 \begin{bmatrix}
5584 0\\0\\1
5585 \end{bmatrix}.
5586 \end{equation}
5587 For the normalized cycle vector
5588 \begin{equation}
5589 c=\frac1{\sqrt3}
5590 \begin{bmatrix}
5591 1\\1\\-1
5592 \end{bmatrix},
5593 \end{equation}
5594 we obtain
5595 \begin{equation}
5596 \abs{c^\top b}=\frac1{\sqrt3}>0.
5597 \end{equation}
5598 Thus local correction succeeds while descent fails.
5599 
5600 \paragraph{Case F3: flat repair.}
5601 Set
5602 \begin{equation}
5603 D_{\mathrm{flat}}=I_2,
5604 \qquad
5605 r_{\mathrm{flat}}=
5606 \begin{bmatrix}
5607 1\\-2
5608 \end{bmatrix},
5609 \end{equation}
5610 with zero overlap mismatch and identity evaluator transport. The unique minimum-norm repair is
5611 \begin{equation}
5612 u_{\mathrm{flat}}=-r_{\mathrm{flat}}.
5613 \end{equation}
5614 
5615 \paragraph{Case F4: evaluator holonomy.}
5616 Use an exactly correctable local defect and zero overlap mismatch. Let evaluator transports around a three-edge cycle be
5617 \begin{equation}
5618 T_{01}=I_2,
5619 \qquad
5620 T_{12}=I_2,
5621 \qquad
5622 T_{20}=\diag(1,-1).
5623 \end{equation}
5624 For protected evaluator direction $e_2$,
5625 \begin{equation}
5626 H_\gamma e_2-e_2=-2e_2,
5627 \qquad
5628 \norm{H_\gamma e_2-e_2}=2.
5629 \end{equation}
5630 The candidate is locally repairable and globally gluable, but its evaluator semantics are not path flat.
5631 
5632 \subsection{Classifier and completeness on the fixture}
5633 
5634 The executable classifier applies the ordered rule
5635 \begin{equation}
5636 \operatorname{Class}(r,b,H)
5637 =
5638 \begin{cases}
5639 \texttt{LOCAL\_OBSTRUCTION},&\norm{P_{(\im D)^\perp}r}>\tau,\\
5640 \texttt{GLUING\_OBSTRUCTION},&\norm{P_{\ker B^\top}b}>\tau,\\
5641 \texttt{EVALUATOR\_HOLONOMY},&\norm{(H-I)e}>\tau,\\
5642 \texttt{FLAT\_REPAIR},&\text{otherwise}.
5643 \end{cases}
5644 \end{equation}
5645 The order is part of the receipt because a case can exhibit more than one defect. The fixture isolates the four modes, so the ordered rule is unambiguous.
5646 
5647 \begin{theorem}[Exact classification of the four-case fixture]
5648 \label{thm:fixture-classification}
5649 For every $0<\tau<1/\sqrt3$, the classifier returns, respectively,
5650 \begin{equation}
5651 \texttt{LOCAL\_OBSTRUCTION},
5652 \quad
5653 \texttt{GLUING\_OBSTRUCTION},
5654 \quad
5655 \texttt{FLAT\_REPAIR},
5656 \quad
5657 \texttt{EVALUATOR\_HOLONOMY}
5658 \end{equation}
5659 for Cases F1 through F4.
5660 \end{theorem}
5661 
5662 \begin{proof}
5663 Case F1 has obstruction norm one. Case F2 has zero local obstruction and cycle residual $1/\sqrt3$. Case F3 has all three residuals zero. Case F4 has zero local and gluing residuals but evaluator defect two.
5664 \end{proof}
5665 
5666 \subsection{Finite closure game}
5667 
5668 The minimal closure model has states
5669 \begin{equation}
5670 S=\{s_{\mathrm{base}},s_{\mathrm{improved}},s_{\mathrm{invalid}}\}
5671 \end{equation}
5672 and valid set
5673 \begin{equation}
5674 K=\{s_{\mathrm{base}},s_{\mathrm{improved}}\}.
5675 \end{equation}
5676 With intact rollback, authorized complete transactions have successor relation
5677 \begin{equation}
5678 \Post(s_{\mathrm{base}})
5679 =\{s_{\mathrm{base}},s_{\mathrm{improved}}\},
5680 \qquad
5681 \Post(s_{\mathrm{improved}})=\{s_{\mathrm{improved}}\}.
5682 \end{equation}
5683 Failed candidates are restored to the base state before the complete transaction ends.
5684 
5685 When rollback is ablated, an invalid branch becomes an authorized complete successor:
5686 \begin{equation}
5687 s_{\mathrm{base}}\longrightarrow s_{\mathrm{invalid}}.
5688 \end{equation}
5689 
5690 \begin{proposition}[Rollback ablation removes universal base closure]
5691 \label{prop:fixture-rollback-ablation}
5692 In the intact complete-transaction model, $s_{\mathrm{base}}$ belongs to the universal invariant closure. When the invalid edge above is authorized and $s_{\mathrm{invalid}}\notin K$, $s_{\mathrm{base}}$ is removed by the first descending fixed-point update.
5693 \end{proposition}
5694 
5695 \begin{proof}
5696 In the intact model every complete successor of $s_{\mathrm{base}}$ lies in $K$. Under ablation, $\Post(s_{\mathrm{base}})\nsubseteq K$, so $s_{\mathrm{base}}\notin\Psi(K)$.
5697 \end{proof}
5698 
5699 Controlled closure remains distinct. A no-op action can preserve $s_{\mathrm{base}}$ even when an unsafe raw edit is available. Productive closure additionally requires a certified transition to $s_{\mathrm{improved}}$.
5700 
5701 \subsection{Frozen finite endpoints}
5702 
5703 The reference fixture freezes the following endpoints before execution:
5704 \begin{longtable}{@{}p{0.12\textwidth}p{0.58\textwidth}p{0.20\textwidth}@{}}
5705 \toprule
5706 ID & Endpoint & Expected result \\
5707 \midrule
5708 \endhead
5709 F01 & All four cases are classified exactly & pass \\
5710 F02 & Local obstruction norm equals $1$ & pass \\
5711 F03 & Triangle gluing residual equals $1/\sqrt3$ & pass \\
5712 F04 & Flat repair has zero local, gluing, and evaluator residual & pass \\
5713 F05 & Evaluator-holonomy defect equals $2$ & pass \\
5714 F06 & Intact universal closure contains the base state & pass \\
5715 F07 & Rollback ablation removes the base state from universal closure & pass \\
5716 F08 & Controlled closure contains the base state & pass \\
5717 F09 & A productive certified transition exists in the flat case & pass \\
5718 \bottomrule
5719 \end{longtable}
5720 
5721 \subsection{Reference execution}
5722 
5723 The Block 5 package evaluates the exact matrices above. The resulting diagnostics are:
5724 \begin{table}[H]
5725 \centering
5726 \small
5727 \begin{tabular}{@{}lrrrr@{}}
5728 \toprule
5729 Case & Local obstruction & Cycle residual & Evaluator defect & Classification \\
5730 \midrule
5731 Local obstruction & $1.000000$ & $0$ & $0$ & local obstruction \\
5732 Gluing obstruction & $0$ & $0.577350$ & $0$ & gluing obstruction \\
5733 Flat repair & $0$ & $0$ & $0$ & flat repair \\
5734 Evaluator holonomy & $0$ & $0$ & $2.000000$ & evaluator holonomy \\
5735 \bottomrule
5736 \end{tabular}
5737 \caption{Exact outputs of the finite correctability and descent fixture.}
5738 \label{tab:block05-fixture-results}
5739 \end{table}
5740 
5741 All nine frozen endpoints passed in the reference execution. This is an exhaustive result for the released finite fixture, not for arbitrary agent architectures.
5742 
5743 \section{Executable Transaction and Receipt Kernel}
5744 \label{sec:transaction-kernel}
5745 
5746 The reference kernel realizes the witness-separated transaction semantics as a small deterministic state machine. Its purpose is to make circularity, illegal edits, rollback, and receipt verification directly testable.
5747 
5748 \subsection{Runtime types}
5749 
5750 An executable state is a finite canonical map
5751 \begin{equation}
5752 \widehat s:\mathsf{Key}\to\mathsf{Value}.
5753 \end{equation}
5754 A witness policy is
5755 \begin{equation}
5756 \widehat W
5757 =(K_{\mathrm{protected}},\gamma_{\min},r_{\max},p_{\mathrm{prov}},e_{\mathrm{irr}},v_W),
5758 \end{equation}
5759 where $K_{\mathrm{protected}}$ is the immutable-key set, $\gamma_{\min}$ is the minimum protected gain, $r_{\max}$ is the maximum accepted risk, $p_{\mathrm{prov}}$ indicates whether provenance is mandatory, $e_{\mathrm{irr}}$ controls irreversible effects, and $v_W$ is the witness version.
5760 
5761 A candidate contains:
5762 \begin{equation}
5763 \widehat\delta
5764 =(n,f,K_{\mathrm{edit}},E_{\mathrm{irr}},\rho,\widehat g),
5765 \end{equation}
5766 where $f$ is a sandboxed state transformation, $K_{\mathrm{edit}}$ is its declared edit mask, $E_{\mathrm{irr}}$ is its declared irreversible-effect set, $\rho$ is a provenance token, and $\widehat g$ is the sealed prediction.
5767 
5768 \subsection{Phase machine}
5769 
5770 Every complete execution must follow exactly:
5771 \begin{equation}
5772 \texttt{PREPARED}
5773 \to
5774 \texttt{SEALED}
5775 \to
5776 \texttt{EXECUTED}
5777 \to
5778 \texttt{REVEALED}
5779 \to
5780 \texttt{DECIDED}
5781 \to
5782 \texttt{FINALIZED}.
5783 \end{equation}
5784 The checkpoint, parent digest, candidate metadata digest, witness digest, and prediction are fixed before protected evidence is revealed.
5785 
5786 \begin{definition}[Kernel acceptance predicate]
5787 For revealed protected evidence $E$, the reference acceptance predicate is
5788 \begin{equation}
5789 \begin{aligned}
5790 \operatorname{Accept}(\widehat s,\widehat s',\widehat\delta,E)
5791 \iff{}&
5792 K_{\mathrm{edit}}\cap K_{\mathrm{protected}}=\varnothing\\
5793 &\land\ \restr{\widehat s'}{K_{\mathrm{protected}}}
5794 =\restr{\widehat s}{K_{\mathrm{protected}}}\\
5795 &\land\ E[\mathrm{utility\_gain}]\geq\gamma_{\min}\\
5796 &\land\ E[\mathrm{risk}]\leq r_{\max}\\
5797 &\land\ E[\mathrm{provenance\_ok}]=1\\
5798 &\land\ E_{\mathrm{irr}}=\varnothing.
5799 \end{aligned}
5800 \end{equation}
5801 \end{definition}
5802 
5803 \begin{theorem}[Protected-key noninterference]
5804 \label{thm:kernel-protected-key}
5805 If a candidate declares or produces a change to a protected key, the reference acceptance predicate is false.
5806 \end{theorem}
5807 
5808 \begin{proof}
5809 A declared protected edit violates the first conjunct. An undeclared produced edit violates the second conjunct.
5810 \end{proof}
5811 
5812 \begin{proposition}[Exact in-memory rollback]
5813 \label{prop:kernel-rollback}
5814 Assume state effects are confined to a deep-copied sandbox and the checkpoint is immutable. If acceptance is false, the final state is byte-canonically equal to the checkpoint state.
5815 \end{proposition}
5816 
5817 \begin{proof}
5818 The kernel replaces the sandbox result by a fresh deep copy of the checkpoint. Canonical serialization is deterministic, so their digests agree.
5819 \end{proof}
5820 
5821 The proposition is intentionally narrow. It does not apply to undeclared file writes, network actions, hardware state, or other effects outside the mediated state object.
5822 
5823 \subsection{Canonical receipts and hash chaining}
5824 
5825 Let $\operatorname{cj}$ denote canonical JSON serialization with sorted keys and normalized arrays. For payload $x$, define
5826 \begin{equation}
5827 h(x)=\operatorname{SHA256}(\operatorname{cj}(x)).
5828 \end{equation}
5829 A transaction receipt records parent, candidate, witness, checkpoint, phase trace, evidence, decision, rejection reasons, final state, and previous receipt hash.
5830 
5831 \begin{definition}[Receipt chain]
5832 For receipts $\Pi_1,\ldots,\Pi_T$, define
5833 \begin{equation}
5834 \Pi_t.\mathrm{previous}
5835 =
5836 \begin{cases}
5837 \varnothing,&t=1,\\
5838 h(\Pi_{t-1}),&t>1.
5839 \end{cases}
5840 \end{equation}
5841 \end{definition}
5842 
5843 \begin{proposition}[Prefix tamper detection]
5844 If SHA-256 collision resistance and canonical serialization are assumed, changing a finalized receipt in a linked chain changes every descendant link unless the attacker recomputes the descendant receipts.
5845 \end{proposition}
5846 
5847 This is an integrity statement, not a semantic authorization theorem. The witness must still determine whether the evidence and action were authorized.
5848 
5849 \subsection{Reference attack pair}
5850 
5851 The smoke test begins from a state with protected evaluator, evidence, and provenance fields. A legitimate planner repair changes only utility and planner state, predicts a gain of $0.12$, and is committed after protected evaluation reveals a gain above $0.05$.
5852 
5853 A second candidate declares an edit to the protected evaluator and attempts to replace it while reporting a larger utility. The kernel detects both the declared protected edit and the realized protected-state change, then restores the checkpoint.
5854 
5855 \begin{table}[H]
5856 \centering
5857 \small
5858 \begin{tabularx}{\textwidth}{@{}lXll@{}}
5859 \toprule
5860 Candidate & Protected outcome & Decision & Verification \\
5861 \midrule
5862 Planner repair & Utility rises from $0.50$ to $0.62$ with evaluator unchanged & commit & receipt valid \\
5863 Evaluator capture & Candidate attempts evaluator replacement & rollback & exact checkpoint hash restored \\
5864 \bottomrule
5865 \end{tabularx}
5866 \caption{Reference transaction-kernel positive and adversarial controls.}
5867 \end{table}
5868 
5869 The two receipts form a valid chain. The adverse receipt contains the explicit reasons
5870 \begin{equation}
5871 \texttt{protected edit declared: evaluator}
5872 \end{equation}
5873 and
5874 \begin{equation}
5875 \texttt{protected state changed: evaluator}.
5876 \end{equation}
5877 All six transaction endpoints passed: legitimate commit, adversarial rollback, protected evaluator preservation, exact rollback, chain linkage, and independent receipt verification.
5878 
5879 \subsection{Reference module boundary}
5880 
5881 \begin{longtable}{@{}p{0.24\textwidth}p{0.68\textwidth}@{}}
5882 \toprule
5883 Module & Responsibility \\
5884 \midrule
5885 \endhead
5886 \texttt{canonical.py} & Canonical JSON conversion, SHA-256 objects, stable result serialization \\
5887 \texttt{linalg.py} & Pseudoinverse repair, obstruction decomposition, triangle cycle residual, holonomy defect \\
5888 \texttt{receipt.py} & Evidence records, transaction receipts, receipt verification \\
5889 \texttt{transaction.py} & Witness policy, candidate interface, phase machine, commit or rollback \\
5890 \texttt{fixture.py} & Four frozen cases and finite closure checks \\
5891 \texttt{anonymous\_env.py} & Linear anonymous-change environment, bounded probes, posterior classifier \\
5892 \texttt{benchmark.py} & Foreign synthetic agents, matched-budget arms, structural and behavioral metrics \\
5893 \texttt{run\_all.py} & Reproducible suite execution and machine-readable summaries \\
5894 \texttt{verify\_package.py} & Manifest coverage and SHA-256 verification \\
5895 \bottomrule
5896 \end{longtable}
5897 
5898 \section{Anonymous Continuous Repair Environment}
5899 \label{sec:anonymous-environment}
5900 
5901 A labeled episode that announces ``sensor fault'' or ``world change'' does not test causal attribution. The Block 5 environment hides a latent event type and exposes only continuous observations and bounded probe outcomes.
5902 
5903 \subsection{Linear environment family}
5904 
5905 Let the latent world state be $x_t\in\R^2$, action $a_t\in\R^2$, observation $y_t\in\R^3$, and memory state $m_t\in\R^2$. The baseline dynamics are
5906 \begin{equation}
5907 x_{t+1}=Ax_t+Ba_t,
5908 \end{equation}
5909 \begin{equation}
5910 y_t=Cx_t+b_s,
5911 \end{equation}
5912 \begin{equation}
5913 m_{t+1}=\lambda m_t+(1-\lambda)y_{t,1:2},
5914 \end{equation}
5915 and the reported evaluator score is
5916 \begin{equation}
5917 q_t=w^\top y_t.
5918 \end{equation}
5919 The redundant observation map is
5920 \begin{equation}
5921 C=
5922 \begin{bmatrix}
5923 1&0\\
5924 0&1\\
5925 1&1
5926 \end{bmatrix},
5927 \end{equation}
5928 so the protected consistency relation is
5929 \begin{equation}
5930 y_{t,3}=y_{t,1}+y_{t,2}
5931 \end{equation}
5932 when the sensor channel is intact.
5933 
5934 The latent event variable is
5935 \begin{equation}
5936 z\in
5937 \{\mathrm{none},\mathrm{internal},\mathrm{world},\mathrm{sensor},\mathrm{memory},\mathrm{evaluator}\}.
5938 \end{equation}
5939 Each event changes one declared mechanism:
5940 \begin{itemize}
5941 \item internal: the action-response map $B$ changes;
5942 \item world: the passive dynamics $A$ change;
5943 \item sensor: the redundant sensor receives an additive bias;
5944 \item memory: the retention coefficient $\lambda$ changes;
5945 \item evaluator: the reported weight vector $w$ changes;
5946 \item none: all baseline mechanisms remain fixed.
5947 \end{itemize}
5948 The event label is not supplied to the classifier.
5949 
5950 \subsection{Bounded probe vector}
5951 
5952 From a common probe state $x_0$ and bounded action $a$, define the feature vector
5953 \begin{equation}
5954 \phi(z)
5955 =
5956 \begin{bmatrix}
5957 \phi_{\mathrm{passive}}\\
5958 \phi_{\mathrm{actuation}}\\
5959 \phi_{\mathrm{sensor}}\\
5960 \phi_{\mathrm{memory}}\\
5961 \phi_{\mathrm{evaluator}}
5962 \end{bmatrix}.
5963 \end{equation}
5964 Its components are, respectively:
5965 \begin{enumerate}
5966 \item deviation from the baseline no-action continuation;
5967 \item deviation of the action-conditioned response from the baseline action response;
5968 \item violation of redundant-sensor consistency;
5969 \item deviation from baseline memory update;
5970 \item discrepancy between reported and protected evaluator scores.
5971 \end{enumerate}
5972 The observed probe is
5973 \begin{equation}
5974 \widetilde\phi=\phi(z)+\eta,
5975 \qquad
5976 \eta\sim\mathcal N(0,\sigma^2I_5).
5977 \end{equation}
5978 
5979 \subsection{Posterior attribution}
5980 
5981 Let $\mu_z=\phi(z)$ be the frozen event prototype. With equal priors and matched isotropic noise, the posterior is
5982 \begin{equation}
5983 p(z\mid\widetilde\phi)
5984 =
5985 \frac{\exp\left(-\norm{\widetilde\phi-\mu_z}^2/(2\sigma^2)\right)}
5986 {\sum_{z'}\exp\left(-\norm{\widetilde\phi-\mu_{z'}}^2/(2\sigma^2)\right)}.
5987 \end{equation}
5988 A repair is attempted only when
5989 \begin{equation}
5990 \max_z p(z\mid\widetilde\phi)\geq\theta
5991 \end{equation}
5992 and the maximizing event is not \emph{none}. Otherwise the system abstains.
5993 
5994 \begin{theorem}[Prototype-separation error bound]
5995 \label{thm:prototype-separation}
5996 Let
5997 \begin{equation}
5998 \Delta
5999 =
6000 \min_{z\neq z'}\norm{\mu_z-\mu_{z'}}.
6001 \end{equation}
6002 Under equal priors and isotropic Gaussian noise, nearest-prototype classification satisfies
6003 \begin{equation}
6004 \Pr(\widehat z\neq z)
6005 \leq
6006 (|\mathcal Z|-1)
6007 \exp\left(-\frac{\Delta^2}{8\sigma^2}\right).
6008 \end{equation}
6009 \end{theorem}
6010 
6011 \begin{proof}
6012 For each competitor $z'$, an error requires the Gaussian projection onto the unit vector joining $\mu_z$ and $\mu_{z'}$ to exceed half their separation. Apply the standard one-dimensional Gaussian tail bound and union bound over competitors.
6013 \end{proof}
6014 
6015 The bound is conservative and does not account for abstention. It nevertheless shows why probe design should maximize the smallest relevant prototype separation rather than only average prediction accuracy.
6016 
6017 \begin{theorem}[Anonymous-change impossibility]
6018 \label{thm:anonymous-impossibility}
6019 If two latent events induce identical distributions over every permitted probe and observation trace, no decision rule measurable with respect to those traces can distinguish the events with accuracy above the larger prior probability.
6020 \end{theorem}
6021 
6022 \begin{proof}
6023 The posterior odds remain equal to the prior odds because the likelihood ratio is one for every measurable trace.
6024 \end{proof}
6025 
6026 This theorem forces abstention when the permitted intervention language is not separating.
6027 
6028 \subsection{Typed repair responses}
6029 
6030 The event-specific response is typed rather than uniformly described as a self-edit:
6031 \begin{longtable}{@{}p{0.17\textwidth}p{0.34\textwidth}p{0.39\textwidth}@{}}
6032 \toprule
6033 Event & Appropriate response & Incorrect self-repair \\
6034 \midrule
6035 \endhead
6036 Internal & Restore or adapt the action-response component & Rewrite the world model while retaining a broken actuator model \\
6037 World & Update the predictive world model & Force the environment back into the old prediction \\
6038 Sensor & Recalibrate or quarantine the corrupted channel & Rewrite latent world state to match biased observations \\
6039 Memory & Repair retention or reconstruction & Treat missing memory as a world event \\
6040 Evaluator & Restore or quarantine score semantics under protected evidence & Optimize the candidate-owned evaluator \\
6041 None & Abstain & Modify a functioning system to reduce noise in its own residual \\
6042 \bottomrule
6043 \end{longtable}
6044 
6045 \subsection{Reference positive-control execution}
6046 
6047 The released smoke configuration uses $80$ episodes per event, noise scale $\sigma=0.06$, posterior threshold $\theta=0.85$, and fixed seed $20260829$. The minimum pairwise prototype separation is $0.110955$.
6048 
6049 \begin{table}[H]
6050 \centering
6051 \small
6052 \begin{tabular}{@{}lr@{}}
6053 \toprule
6054 Metric & Reference value \\
6055 \midrule
6056 Non-none attribution accuracy & $0.950000$ \\
6057 Overall repair-or-abstain success & $0.908333$ \\
6058 False repair rate under no change & $0.037500$ \\
6059 No-change abstention accuracy & $0.962500$ \\
6060 Mean posterior confidence & $0.938424$ \\
6061 \bottomrule
6062 \end{tabular}
6063 \caption{Anonymous-change synthetic positive-control results.}
6064 \label{tab:anonymous-results}
6065 \end{table}
6066 
6067 Internal, world, sensor, and evaluator cases were perfectly separated in this seed. The principal overlap occurred between the memory-change and no-change prototypes. The result is therefore useful as a diagnostic: it demonstrates both high-separation cases and a boundary at which calibrated abstention matters.
6068 
6069 \begin{warning}[Matched-model limitation]
6070 The classifier knows the six prototype families used to generate the environment. It does not discover new latent mechanisms, revise its packet atlas, or learn an open-ended causal ontology. The correct claim is a synthetic anonymous-change identification fixture.
6071 \end{warning}
6072 
6073 \section{Matched-Budget Recursive Repair Benchmark}
6074 \label{sec:reference-benchmark}
6075 
6076 The recursive-repair benchmark measures foreign-agent gain under a common candidate budget. It is an engineered positive-control environment designed to expose solver, search, transport, witness, and abstraction confounds independently.
6077 
6078 \subsection{Foreign synthetic agents}
6079 
6080 Each foreign agent has a canonical defect
6081 \begin{equation}
6082 d\in\R^6,
6083 \end{equation}
6084 a positive diagonal importance vector $w$, and an invertible architecture transform
6085 \begin{equation}
6086 T\in\R^{6\times6}.
6087 \end{equation}
6088 The runtime-visible defect is
6089 \begin{equation}
6090 \widetilde d=Td.
6091 \end{equation}
6092 Protected utility is
6093 \begin{equation}
6094 J_T(\widetilde d)
6095 =
6096 \exp\left(-\sum_{i=1}^6w_i\bigl(T^{-1}\widetilde d\bigr)_i^2\right).
6097 \end{equation}
6098 A candidate runtime patch $\delta$ produces gain
6099 \begin{equation}
6100 g(\delta)
6101 =
6102 J_T(\widetilde d+\delta)-J_T(\widetilde d).
6103 \end{equation}
6104 
6105 The transforms combine permutation, sign, and scale changes. Therefore canonical repairs must be transported; direct coordinate reuse generally fails.
6106 
6107 \subsection{Matched arms}
6108 
6109 Every arm receives candidate budget $B=8$ and the same $96$ frozen agents.
6110 
6111 \begin{longtable}{@{}p{0.21\textwidth}p{0.70\textwidth}@{}}
6112 \toprule
6113 Arm & Executable mechanism \\
6114 \midrule
6115 \endhead
6116 Static & Returns the parent with no modification \\
6117 Global edit & Samples whole-vector edits and contains a deliberately exploitable apparent-score component \\
6118 Archive only & Samples noisy descendants around the current visible defect and retains the best protected candidate \\
6119 Packet only & Corrects large visible coordinates without causal type or architecture transport \\
6120 Causal only & Identifies a dominant canonical cause but applies the repair without target-coordinate transport \\
6121 Full SRIC & Uses typed canonical repairs, architecture transport, protected evaluation, and witness rejection \\
6122 Full SRIC meta & Adds an engineered schema-abstraction positive control for the sixth held-out defect type \\
6123 \bottomrule
6124 \end{longtable}
6125 
6126 The last arm is not a learned meta-improver. It is a positive control demonstrating what the foreign-agent metric should report when a new repair schema is actually made available.
6127 
6128 \subsection{Search and evaluator confounds}
6129 
6130 The global-edit arm contains a candidate whose apparent score receives an exploitable bonus when its rewrite norm is large. The protected evaluator does not receive this bonus. This creates a controlled instance of evaluator capture:
6131 \begin{equation}
6132 \widehat J(\delta)>\widehat J(0)
6133 \qquad\text{while}\qquad
6134 J(\delta)<J(0).
6135 \end{equation}
6136 A witness-aware arm ranks and commits by $J$, not $\widehat J$.
6137 
6138 The archive arm can perform well without a causal repair calculus because it searches several noisy candidates and retains the best. Candidate-budget matching is therefore mandatory. Let
6139 \begin{equation}
6140 G_{a,j,1},\ldots,G_{a,j,B}
6141 \end{equation}
6142 be candidate gains for arm $a$ on agent $j$. The primary behavioral quantity is
6143 \begin{equation}
6144 M_a(B)
6145 =
6146 \frac1n\sum_{j=1}^n\max_{1\leq b\leq B}G_{a,j,b}.
6147 \end{equation}
6148 Comparing arms at unequal $B$ confounds repair competence with order statistics.
6149 
6150 \subsection{Transport positive control}
6151 
6152 \begin{proposition}[Exact transported canonical repair]
6153 \label{prop:benchmark-exact-transport}
6154 For an invertible architecture map $T$, the runtime patch
6155 \begin{equation}
6156 \delta=-Td
6157 \end{equation}
6158 removes the canonical defect exactly:
6159 \begin{equation}
6160 T^{-1}(Td-Td)=0.
6161 \end{equation}
6162 Hence protected utility becomes one.
6163 \end{proposition}
6164 
6165 The full benchmark need not always offer the joint exact patch within budget. It does, however, construct individual typed patches in canonical coordinates and transport them through $T$.
6166 
6167 \subsection{Structural metric}
6168 
6169 For canonical patch $u=T^{-1}\delta$, define corrected protected energy
6170 \begin{equation}
6171 E_{\mathrm{corr}}(d,u)
6172 =
6173 \sum_{i=1}^6w_i\min\{d_i^2,u_i^2\}.
6174 \end{equation}
6175 This quantity is zero for a no-op and positive when a patch reaches task-relevant defect coordinates. It is reported alongside behavioral gain because a high apparent score can be generated without correcting protected defect energy.
6176 
6177 \subsection{Reference benchmark execution}
6178 
6179 \begin{table}[H]
6180 \centering
6181 \footnotesize
6182 \begin{tabular}{@{}lrrrr@{}}
6183 \toprule
6184 Arm & Mean protected gain & Median gain & Corrected energy & False accept rate \\
6185 \midrule
6186 Static & $0.000000$ & $0.000000$ & $0.000000$ & $0.000000$ \\
6187 Global edit & $-0.324252$ & $-0.325451$ & $0.834602$ & $0.958333$ \\
6188 Archive only & $0.474975$ & $0.475579$ & $0.744173$ & $0.000000$ \\
6189 Packet only & $0.609294$ & $0.625606$ & $1.051647$ & $0.000000$ \\
6190 Causal only & $0.044230$ & $0.000000$ & $0.071506$ & $0.000000$ \\
6191 Full SRIC & $0.519120$ & $0.580246$ & $0.870198$ & $0.000000$ \\
6192 Full SRIC meta & $0.640517$ & $0.654932$ & $1.085841$ & $0.000000$ \\
6193 \bottomrule
6194 \end{tabular}
6195 \caption{Matched-budget synthetic foreign-agent benchmark. The global-edit arm is an evaluator-capture positive control.}
6196 \label{tab:block05-benchmark}
6197 \end{table}
6198 
6199 Several interpretations matter:
6200 \begin{enumerate}
6201 \item Packet-only performance is high because the toy defects are sparse and visible-coordinate correction is often effective. This prevents the benchmark from being designed so that only the full architecture can succeed.
6202 \item Causal-only performance is low because correct causal type without coordinate transport is insufficient under architecture changes.
6203 \item Archive-only search is a strong baseline and must not be omitted.
6204 \item Global edit has high apparent gain but negative protected gain, with a $0.958333$ false-accept rate. This confirms that the benchmark can expose evaluator capture.
6205 \item Full SRIC has positive protected gain and zero false accepts in the released fixture, but it is not uniformly best. That is an informative outcome rather than a failure of the benchmark.
6206 \item The engineered meta arm improves the held-out sixth defect type. Its novel-type gain exceeds the non-meta arm by $0.700533$. This is a causal positive control for the abstraction metric, not evidence that a learned system discovered the schema.
6207 \end{enumerate}
6208 
6209 \subsection{Causal abstraction contrast}
6210 
6211 Let $N\subseteq\{1,\ldots,n\}$ be foreign agents whose dominant defect belongs to the held-out sixth type. Define
6212 \begin{equation}
6213 \widehat\tau_{\mathrm{abstract}}
6214 =
6215 \frac1{|N|}\sum_{j\in N}
6216 \left(
6217 G_{j}^{\mathrm{meta}}-G_{j}^{\mathrm{no\mbox{-}meta}}
6218 \right).
6219 \end{equation}
6220 In the reference positive control,
6221 \begin{equation}
6222 \widehat\tau_{\mathrm{abstract}}=0.700533.
6223 \end{equation}
6224 A learned recursive-improvement claim would require the schema to be generated by the parent improvement process, sealed before foreign evaluation, and compared under matched computation and evidence access.
6225 
6226 \section{Causal Ablation and Statistical Inference}
6227 \label{sec:block05-inference}
6228 
6229 A benchmark can be mechanically correct while leaving the causal role of its components unresolved. SRIC therefore treats ablations as interventions on a factorized architecture rather than as informal feature removals.
6230 
6231 \subsection{Factorized architecture}
6232 
6233 Let the executable architecture have binary factors
6234 \begin{equation}
6235 Z
6236 =(Z_C,Z_P,Z_D,Z_T,Z_W,Z_A,Z_M),
6237 \end{equation}
6238 representing causal attribution, packet localization, descent checking, transport, witness separation, archive retention, and repair abstraction.
6239 
6240 For foreign agent $j$, let
6241 \begin{equation}
6242 Y_j(z)
6243 \end{equation}
6244 be protected gain under factor setting $z$. The average necessity effect of factor $k$ relative to the intact architecture $\one$ is
6245 \begin{equation}
6246 \tau_k
6247 =
6248 \mathbb E
6249 \left[
6250 Y_j(\one)-Y_j(\one-e_k)
6251 \right].
6252 \end{equation}
6253 A positive value supports causal usefulness only relative to the frozen environment, compensating mechanisms, and matched resource schedule.
6254 
6255 \subsection{Decisive schema-abstraction ablation}
6256 
6257 The primary recursive-effect ablation is
6258 \begin{equation}
6259 Z_M:1\to0
6260 \end{equation}
6261 while preserving every immediate task-level repair produced before the abstraction point. The descendant is denied only the reusable schema object and its future retrieval path.
6262 
6263 \begin{definition}[Recursive abstraction estimand]
6264 Let $\mathcal F_{\mathrm{post}}$ be foreign defect families revealed after the source repair. Define
6265 \begin{equation}
6266 \tau_M^{\mathrm{future}}
6267 =
6268 \mathbb E_{A\sim\mathcal F_{\mathrm{post}}}
6269 \left[
6270 G_A(Z_M=1)-G_A(Z_M=0)
6271 \right].
6272 \end{equation}
6273 \end{definition}
6274 
6275 This estimand distinguishes a repair that helps one source task from a repair abstraction that improves later repair competence.
6276 
6277 \subsection{Paired estimation}
6278 
6279 For arms $a$ and $b$, evaluate both on the same frozen foreign agents and seeds. Let
6280 \begin{equation}
6281 \Delta_j=G_{a,j}-G_{b,j}.
6282 \end{equation}
6283 The paired estimate and standard error are
6284 \begin{equation}
6285 \widehat\Delta
6286 =\frac1n\sum_{j=1}^n\Delta_j,
6287 \qquad
6288 \widehat{\operatorname{se}}(\widehat\Delta)
6289 =\sqrt{\frac{1}{n(n-1)}\sum_{j=1}^n(\Delta_j-\widehat\Delta)^2}.
6290 \end{equation}
6291 When gains are bounded in $[-1,1]$, a distribution-free Hoeffding lower bound is
6292 \begin{equation}
6293 \LCB_{1-\alpha}(\Delta)
6294 =
6295 \widehat\Delta
6296 -
6297 \sqrt{\frac{2\log(1/\alpha)}{n}}.
6298 \end{equation}
6299 Sharper paired permutation or bootstrap intervals may be used when frozen in advance.
6300 
6301 \subsection{Multiple claims and alpha spending}
6302 
6303 Let $H_1,\ldots,H_m$ be ordered primary hypotheses. Choose nonnegative levels $\alpha_i$ satisfying
6304 \begin{equation}
6305 \sum_{i=1}^m\alpha_i\leq\alpha.
6306 \end{equation}
6307 Each endpoint is tested only with its assigned held-out evidence. Reusing the same audit sample to tune thresholds and report final confidence invalidates the nominal bound.
6308 
6309 For sequential candidate generation, an anytime-valid confidence sequence or e-value process is preferable. The benchmark receipt must record every look at protected evidence, including apparently unsuccessful candidates.
6310 
6311 \subsection{Required negative controls}
6312 
6313 \begin{longtable}{@{}p{0.24\textwidth}p{0.35\textwidth}p{0.32\textwidth}@{}}
6314 \toprule
6315 Control & Purpose & Expected outcome \\
6316 \midrule
6317 \endhead
6318 No-op candidate & Detects baseline drift and evaluator noise & zero protected gain in expectation \\
6319 Shuffled repair-schema labels & Tests whether semantic type is used & transfer should collapse \\
6320 Random same-rank transport & Controls for dimension and norm & worse than typed transport \\
6321 Wrong-lineage receipt & Tests provenance authorization & rejected or quarantined \\
6322 Evaluator-capture candidate & Tests protected-score separation & high apparent score, rejected protected gain \\
6323 Rollback ablation & Tests necessity of restoration & universal closure violation \\
6324 Witness-edit attempt & Tests transaction separation & rejected before commit \\
6325 Schema-abstraction removal & Tests recursive reuse & future novel-type gain decreases \\
6326 Candidate-budget equalization & Removes best-of-$k$ confound & all arms use same $B$ \\
6327 Foreign-agent permutation & Tests lineage memorization & gains persist only if repair abstraction transfers \\
6328 \bottomrule
6329 \end{longtable}
6330 
6331 \subsection{Primary Block 5 gates}
6332 
6333 The released reference package uses the following gates:
6334 \begin{enumerate}
6335 \item all nine finite-fixture endpoints pass;
6336 \item all six transaction-kernel endpoints pass;
6337 \item anonymous non-none attribution accuracy is at least $0.90$;
6338 \item anonymous repair-or-abstain success is at least $0.90$;
6339 \item false repair under no change is at most $0.10$;
6340 \item full SRIC protected gain exceeds static and global-edit gain under the matched budget;
6341 \item the evaluator-capture positive control produces at least one false apparent acceptance;
6342 \item full SRIC produces zero false protected acceptance in the released fixture;
6343 \item the meta positive control improves the held-out defect-type contrast;
6344 \item package verification and all automated tests pass.
6345 \end{enumerate}
6346 
6347 The gates were selected to verify implementation mechanics and positive controls. They are not thresholds for a general scientific claim of RSI.
6348 
6349 \section{Reproducibility, Packaging, and Verification}
6350 \label{sec:block05-reproducibility}
6351 
6352 The empirical block is distributed as source code, tests, result objects, manifest, and a complete manuscript bundle.
6353 
6354 \subsection{Package layout}
6355 
6356 \begin{verbatim}
6357 sric_block05_reference/
6358   README.md
6359   pyproject.toml
6360   Makefile
6361   Run_SRIC_Block05.ps1
6362   run_all.py
6363   verify_package.py
6364   MANIFEST.sha256
6365   sric_ref/
6366     __init__.py
6367     canonical.py
6368     linalg.py
6369     receipt.py
6370     transaction.py
6371     fixture.py
6372     anonymous_env.py
6373     benchmark.py
6374   tests/
6375     conftest.py
6376     test_fixture.py
6377     test_transaction.py
6378     test_anonymous.py
6379     test_benchmark.py
6380   results/
6381     fixture_results.json
6382     transaction_results.json
6383     anonymous_results.json
6384     benchmark_results.json
6385     verification_summary.json
6386 \end{verbatim}
6387 
6388 \subsection{Deterministic and stochastic replay}
6389 
6390 The finite fixture and transaction suite are deterministic. The anonymous and foreign-agent suites are pseudorandom with frozen seeds recorded in each result object. Two levels of replay are distinguished:
6391 \begin{enumerate}
6392 \item \textbf{exact replay}: canonical result bytes and digests agree;
6393 \item \textbf{statistical replay}: a new independent seed satisfies the preregistered endpoint gates.
6394 \end{enumerate}
6395 Exact replay detects package or implementation drift. Statistical replay tests whether a result is robust beyond one frozen sample.
6396 
6397 \begin{definition}[Package-closed claim]
6398 A claim is package closed when every number required to evaluate its gate can be regenerated from released source, declared dependencies, frozen configuration, and released evidence without hidden network services or mutable external state.
6399 \end{definition}
6400 
6401 \begin{proposition}[Manifest integrity]
6402 Assuming collision resistance of SHA-256, if every declared package file matches its manifest digest, the verifier establishes byte integrity of those files relative to the manifest.
6403 \end{proposition}
6404 
6405 The proposition does not prove that the manifest is complete. The release builder must separately compare the manifest path set with the intended package path set.
6406 
6407 \subsection{Verification commands}
6408 
6409 The canonical local commands are:
6410 \begin{verbatim}
6411 python run_all.py
6412 pytest -q
6413 python verify_package.py
6414 \end{verbatim}
6415 The first writes all result JSON files. The second checks unit tests. The third checks the SHA-256 manifest after results have been frozen.
6416 
6417 \subsection{Reference verification summary}
6418 
6419 The released execution reported:
6420 \begin{verbatim}
6421 SRIC_BLOCK05_ALL_SUITES=PASS
6422 fixture_endpoints=9/9
6423 transaction_endpoints=6/6
6424 anonymous_accuracy=0.950000
6425 anonymous_false_repair=0.037500
6426 full_sric_gain=0.519120
6427 full_sric_meta_gain=0.640517
6428 \end{verbatim}
6429 All four automated tests passed. The package verifier is run only after the manifest is generated from the final frozen tree.
6430 
6431 \subsection{Reproducibility tiers}
6432 
6433 \begin{longtable}{@{}p{0.14\textwidth}p{0.33\textwidth}p{0.44\textwidth}@{}}
6434 \toprule
6435 Tier & Requirement & What it supports \\
6436 \midrule
6437 \endhead
6438 R0 & Narrative description only & conceptual inspection \\
6439 R1 & Source executes & mechanical availability \\
6440 R2 & Frozen tests and results regenerate & deterministic implementation claim \\
6441 R3 & Clean environment and independent seed gates pass & robustness of synthetic fixture \\
6442 R4 & Independent implementation refines the same abstract model & reduced implementation-specific risk \\
6443 R5 & Foreign learned systems reproduce causal and metaproductive endpoints & substantive empirical SRIC evidence \\
6444 \bottomrule
6445 \end{longtable}
6446 
6447 The Block 5 reference package targets R2 and supplies the protocol needed to attempt R3. It does not claim R4 or R5.
6448 
6449 \begin{statusbox}{Block 5 result}
6450 The empirical layer is now executable. The finite correctability, descent, holonomy, closure, transaction, anonymous-change, foreign-agent, and evaluator-capture controls are represented in one reproducible reference package. The package verifies mechanics and engineered synthetic positive controls. Learned repair discovery, foreign-model transfer, autonomous grammar expansion, and open-ended recursive improvement remain unestablished.
6451 \end{statusbox}
6452 
6453 
6454 \section{Learned-Agent Validation Contract}
6455 \label{sec:block06-contract}
6456 
6457 Block 5 established an executable synthetic reference layer with analytic defect coordinates and hand-coded positive controls. Block 6 removes one important simplification. The repair is now discovered in a hidden representation, the source-to-target coordinate map is withheld from the repair system, and transfer is evaluated on target agents whose representations were not used to construct the source repair.
6458 
6459 The block remains controlled. The task family, causal probe, candidate repair form, witness policy, and evaluation split are designer supplied. The result therefore tests \emph{learned hidden-state repair transport}, not open-ended self-reconstruction.
6460 
6461 \subsection{Representation-level validation object}
6462 
6463 \begin{definition}[Learned repair-transport instance]
6464 A learned repair-transport instance is a tuple
6465 \begin{equation}
6466 \mathfrak R_{\mathrm{learn}}
6467 =
6468 (\mathcal X,\phi,q,\widetilde q,A_s,\mathcal A_t,
6469 \mathsf D,\mathsf P,\mathsf G,\mathsf W),
6470 \end{equation}
6471 where:
6472 \begin{itemize}
6473 \item $\mathcal X$ is the input space;
6474 \item $\phi$ is the task feature map;
6475 \item $q$ and $\widetilde q$ are protected and corrupted score maps;
6476 \item $A_s$ is the source agent;
6477 \item $\mathcal A_t$ is a family of target agents excluded from source repair fitting;
6478 \item $\mathsf D$ is the train, anchor, calibration, and audit partition;
6479 \item $\mathsf P$ is the permitted causal probe language;
6480 \item $\mathsf G$ is the repair and transport grammar;
6481 \item $\mathsf W$ is the frozen witness policy.
6482 \end{itemize}
6483 \end{definition}
6484 
6485 The source repair may use source training and anchor data. A target transport may use paired anchor activations from the source and that target. It may not use target audit labels, the hidden target transformation, or the target-local oracle repair. The target-local repair is retained only as a post hoc upper comparison.
6486 
6487 \begin{definition}[Foreign representation target]
6488 A target agent is foreign relative to a source repair when its encoder parameters or construction seed are absent from source packet discovery and source repair fitting. A stronger architecture-foreign target additionally belongs to an encoder family absent from the source lineage.
6489 \end{definition}
6490 
6491 This is weaker than cross-task or cross-environment foreignness. Block 6 evaluates foreign representation systems inside one shared task and defect grammar.
6492 
6493 \subsection{Data separation}
6494 
6495 The controlled suite uses four disjoint data sets:
6496 \begin{equation}
6497 \mathsf D
6498 =
6499 \mathsf D_{\rm train}
6500 \sqcup
6501 \mathsf D_{\rm anchor}
6502 \sqcup
6503 \mathsf D_{\rm calibrate}
6504 \sqcup
6505 \mathsf D_{\rm audit}.
6506 \end{equation}
6507 The source and target heads are fitted on $\mathsf D_{\rm train}$. Packet discovery and source repair fitting use $\mathsf D_{\rm anchor}$. The witness sees only $\mathsf D_{\rm calibrate}$. Reported final accuracies use $\mathsf D_{\rm audit}$.
6508 
6509 For the controlled transport experiment, the sample sizes are
6510 \begin{equation}
6511 (n_{\rm train},n_{\rm anchor},n_{\rm calibrate},n_{\rm audit})
6512 =(7000,1500,1600,6000).
6513 \end{equation}
6514 The learned-representation pilot uses
6515 \begin{equation}
6516 (5000,1200,1200,4000).
6517 \end{equation}
6518 
6519 \subsection{Block 6 claim target}
6520 
6521 The highest admissible target for this block is empirical claim-ladder level 4:
6522 \begin{center}
6523 \emph{a repair schema learned in one hidden representation improves held-out foreign representation systems after a map is learned from permitted paired anchors.}
6524 \end{center}
6525 
6526 The block does not compare a descendant repair calculus against its parent. It therefore does not establish level-5 metaproductivity. It also does not construct a new repair type, change the evaluator, or expand the grammar.
6527 
6528 \begin{statusbox}{Block 6 empirical principle}
6529 A successful transported repair must do more than raise target accuracy. The causal packet must outperform same-rank noncausal controls, paired-anchor alignment must be necessary, the repair must preserve the unaffected stratum, and the frozen witness must abstain when transport quality or protected behavior falls outside the declared envelope.
6530 \end{statusbox}
6531 
6532 \section{Controlled Causal-Packet Fixture}
6533 \label{sec:block06-packet}
6534 
6535 \subsection{Task and defect construction}
6536 
6537 Let $x=(x_1,x_2,x_3,x_4)\in\R^4$. Define the eight-dimensional base feature
6538 \begin{equation}
6539 \psi(x)
6540 =
6541 \begin{bmatrix}
6542  x_1 & x_2 & x_3 & x_4 & x_1x_2 & x_3x_4 & \sin x_1 & \cos x_2
6543 \end{bmatrix}^{\!\top}
6544 \in\R^8,
6545 \end{equation}
6546 and the affected-stratum indicator
6547 \begin{equation}
6548 s(x)=\one\{x_1+x_2>0.8\}.
6549 \end{equation}
6550 The full feature map is
6551 \begin{equation}
6552 \phi(x)
6553 =
6554 \begin{bmatrix}
6555 \psi(x)\\
6556 s(x)\psi(x)
6557 \end{bmatrix}
6558 \in\R^{16}.
6559 \label{eq:block06-feature}
6560 \end{equation}
6561 
6562 Let
6563 \begin{equation}
6564 q(x)=W\psi(x)+b\in\R^3
6565 \end{equation}
6566 be the protected score. Let $C$ be the cyclic score permutation
6567 \begin{equation}
6568 C(y_1,y_2,y_3)^{\top}=(y_3,y_1,y_2)^{\top}.
6569 \end{equation}
6570 The corrupted score is
6571 \begin{equation}
6572 \widetilde q(x)
6573 =
6574 (1-s(x))q(x)+s(x)Cq(x).
6575 \label{eq:block06-corrupt-score}
6576 \end{equation}
6577 Thus the model is almost perfectly correct outside the affected stratum and systematically wrong inside it.
6578 
6579 An agent $a$ has encoder
6580 \begin{equation}
6581 E_a:\R^{16}\to\R^{d_a}
6582 \end{equation}
6583 and an affine head fitted to $\widetilde q$. Its hidden state is
6584 \begin{equation}
6585 h_a(x)=E_a(\phi(x)).
6586 \end{equation}
6587 The transport system is not given the matrix or nonlinear parameters of $E_a$.
6588 
6589 \subsection{Causal counterfactual probe}
6590 
6591 The permitted intervention removes only the conditional feature block:
6592 \begin{equation}
6593 \phi^{\circ}(x)
6594 =
6595 \begin{bmatrix}
6596 \psi(x)\\0
6597 \end{bmatrix}.
6598 \label{eq:block06-counterfactual}
6599 \end{equation}
6600 The resulting hidden difference is
6601 \begin{equation}
6602 \Delta_a(x)
6603 =
6604 h_a(x)-h_a^{\circ}(x),
6605 \qquad
6606 h_a^{\circ}(x)=E_a(\phi^{\circ}(x)).
6607 \end{equation}
6608 The empirical causal packet is the leading rank-$r$ right-singular subspace of the affected differences
6609 \begin{equation}
6610 \{\Delta_a(x_i):s(x_i)=1\}.
6611 \end{equation}
6612 
6613 \begin{theorem}[Exact causal-packet recovery for a linear encoder]
6614 \label{thm:block06-linear-packet}
6615 Suppose
6616 \begin{equation}
6617 E_a
6618 =
6619 \begin{bmatrix}A_a&B_a\end{bmatrix},
6620 \qquad
6621 A_a,B_a:\R^8\to\R^{d_a},
6622 \end{equation}
6623 and suppose the centered conditional covariance of $\psi(x)$ given $s(x)=1$ is positive definite. Then
6624 \begin{equation}
6625 \Span\{\Delta_a(x):s(x)=1\}
6626 =
6627 im B_a.
6628 \end{equation}
6629 Consequently, a rank-$r$ singular-subspace estimator with $r=\rank B_a$ recovers the exact causal packet from a sufficiently rich noiseless sample.
6630 \end{theorem}
6631 
6632 \begin{proof}
6633 From \cref{eq:block06-feature,eq:block06-counterfactual},
6634 \begin{equation}
6635 \Delta_a(x)=s(x)B_a\psi(x).
6636 \end{equation}
6637 Hence every affected difference lies in $\im B_a$. Conversely, positive-definite conditional covariance implies that the centered affected values of $\psi$ span $\R^8$. Their images therefore span $\im B_a$. The singular subspace associated with the nonzero singular values is exactly that image.
6638 \end{proof}
6639 
6640 \begin{remark}[Why ordinary PCA need not recover the repair packet]
6641 PCA maximizes variance of $h_a(x)$, not sensitivity to the declared intervention. Large variance in $A_a\psi(x)$ can dominate the spectrum while the entire defect mechanism lies in $B_a\psi(x)$. Equal rank does not make the two subspaces semantically equivalent.
6642 \end{remark}
6643 
6644 \begin{example}[Same-rank PCA failure]
6645 Let $A_a$ amplify eight high-variance task coordinates and let $B_a$ encode eight lower-variance conditional coordinates. Rank-eight PCA selects $\im A_a$, while the causal difference span is $\im B_a$. A repair restricted to the PCA packet can preserve overall variance while leaving the defect nearly unchanged.
6646 \end{example}
6647 
6648 \subsection{Packet-local repair fitting}
6649 
6650 Let $U_a\in\R^{d_a\times r}$ have orthonormal columns spanning the discovered packet. Define desired packet coordinates
6651 \begin{equation}
6652 z_i
6653 =
6654 U_a^{\top}(h_a^{\circ}(x_i)-h_a(x_i)).
6655 \end{equation}
6656 Fit an affine predictor
6657 \begin{equation}
6658 \widehat z(h)=K_a h+c_a
6659 \end{equation}
6660 by ridge regression on source anchor states. The repair is
6661 \begin{equation}
6662 R_a(h)
6663 =
6664 h+U_a\widehat z(h)
6665 =
6666 h+D_a h+b_a,
6667 \label{eq:block06-source-repair}
6668 \end{equation}
6669 where
6670 \begin{equation}
6671 D_a=U_aK_a,
6672 \qquad
6673 b_a=U_ac_a.
6674 \end{equation}
6675 
6676 \begin{proposition}[Exact packet repair under affine predictability]
6677 \label{prop:block06-exact-source-repair}
6678 Suppose every desired correction lies in $\im U_a$ and there are $K_a,c_a$ such that
6679 \begin{equation}
6680 U_a^{\top}(h_a^{\circ}-h_a)=K_ah_a+c_a
6681 \end{equation}
6682 throughout the evaluated support. Then \cref{eq:block06-source-repair} satisfies
6683 \begin{equation}
6684 R_a(h_a)=h_a^{\circ}.
6685 \end{equation}
6686 \end{proposition}
6687 
6688 \begin{proof}
6689 Because the desired correction is in $\im U_a$,
6690 \begin{equation}
6691 h_a^{\circ}-h_a
6692 =U_aU_a^{\top}(h_a^{\circ}-h_a)
6693 =U_a(K_ah_a+c_a).
6694 \end{equation}
6695 Substitution into \cref{eq:block06-source-repair} gives the claim.
6696 \end{proof}
6697 
6698 \subsection{Packet-specificity result}
6699 
6700 The source head attains $0.9985$ accuracy against the corrupted labels. Against the protected labels, its overall accuracy is $0.7075$. The unaffected stratum remains nearly perfect, while the affected stratum collapses to $0.002277$.
6701 
6702 \begin{table}[H]
6703 \centering
6704 \small
6705 \caption{Source repair under three rank-eight packet estimators. The uncorrected affected-stratum accuracy is $0.002277$.}
6706 \label{tab:block06-packet-results}
6707 \begin{tabular}{@{}lrrrr@{}}
6708 \toprule
6709 Packet estimator & Rank & Affected slice & Unaffected slice & Overall \\
6710 \midrule
6711 Causal difference & 8 & $0.994308$ & $0.999529$ & $0.998000$ \\
6712 Same-rank PCA & 8 & $0.023335$ & $0.999529$ & $0.713667$ \\
6713 Random subspace & 8 & $0.570290$ & $0.999529$ & $0.873833$ \\
6714 \bottomrule
6715 \end{tabular}
6716 \end{table}
6717 
6718 The causal packet restores $99.43\%$ affected-stratum accuracy and leaves the unaffected stratum unchanged. Same-rank PCA reaches only $2.33\%$. The random packet produces a partial repair, which is expected because a random rank-eight subspace in a sixteen-dimensional state has nonzero overlap with the true rank-eight packet. It nevertheless remains far below the causal packet.
6719 
6720 \begin{figure}[H]
6721 \centering
6722 \includegraphics[width=0.88\textwidth]{packet_comparison.pdf}
6723 \caption{Affected-stratum accuracy after fitting the same repair form in causal, PCA, and random rank-eight packets. Packet rank alone does not explain the result.}
6724 \label{fig:block06-packet-comparison}
6725 \end{figure}
6726 
6727 \subsection{Wrong-block control}
6728 
6729 A matched control zeros the base feature block instead of the conditional block. The resulting packet and repair leave affected-stratum performance unchanged in the tested linear, overcomplete, and mildly nonlinear families.
6730 
6731 \begin{proposition}[Wrong-block null under block-separable score correction]
6732 Suppose the protected score is determined by the base block and the corruption is represented only through the conditional block. A repair learned from a counterfactual that removes the base block does not identify the conditional correction unless the encoder mixes the two mechanisms in an additional identifiable way.
6733 \end{proposition}
6734 
6735 The proposition is intentionally conditional. The executable wrong-block result is the relevant finite negative control.
6736 
6737 \section{Learned Cross-Agent Repair Transport}
6738 \label{sec:block06-transport}
6739 
6740 \subsection{Learning the hidden-state bridge}
6741 
6742 Let $s$ denote the source and $t$ a target. On paired anchor inputs $x_1,\ldots,x_m$, the system observes
6743 \begin{equation}
6744 H_s
6745 =
6746 \begin{bmatrix}
6747 h_s(x_1)^{\top}\\ \vdots\\ h_s(x_m)^{\top}
6748 \end{bmatrix},
6749 \qquad
6750 H_t
6751 =
6752 \begin{bmatrix}
6753 h_t(x_1)^{\top}\\ \vdots\\ h_t(x_m)^{\top}
6754 \end{bmatrix}.
6755 \end{equation}
6756 It estimates the bridge by regularized least squares:
6757 \begin{equation}
6758 \widehat T_{t\leftarrow s}
6759 \in
6760 \argmin_T
6761 \norm{H_sT^{\top}-H_t}_F^2
6762 +\lambda\norm{T}_F^2.
6763 \label{eq:block06-learned-transport}
6764 \end{equation}
6765 The transformation matrices used to construct the controlled targets are not exposed to this fit.
6766 
6767 Given source repair
6768 \begin{equation}
6769 R_s(h)=h+D_sh+b_s,
6770 \end{equation}
6771 its transported target form is
6772 \begin{equation}
6773 D_t
6774 =
6775 \widehat T D_s\widehat T^{+},
6776 \qquad
6777 b_t
6778 =
6779 \widehat T b_s,
6780 \label{eq:block06-transported-repair}
6781 \end{equation}
6782 with transported packet basis obtained by orthogonalizing $\widehat T U_s$.
6783 
6784 \subsection{Exact injective transport}
6785 
6786 \begin{theorem}[Exact transport through an injective hidden bridge]
6787 \label{thm:block06-exact-transport}
6788 Let $T:\R^{d_s}\to\R^{d_t}$ have full column rank, so $T^{+}T=I_{d_s}$. Suppose
6789 \begin{equation}
6790 h_t=Th_s
6791 \end{equation}
6792 throughout the evaluated support. Define $D_t$ and $b_t$ by \cref{eq:block06-transported-repair} with $\widehat T=T$. Then
6793 \begin{equation}
6794 R_t(Th_s)=TR_s(h_s).
6795 \end{equation}
6796 \end{theorem}
6797 
6798 \begin{proof}
6799 Using $T^{+}T=I$,
6800 \begin{align}
6801 R_t(Th_s)
6802 &=Th_s+TD_sT^{+}Th_s+Tb_s\\
6803 &=T(h_s+D_sh_s+b_s)\\
6804 &=TR_s(h_s).
6805 \end{align}
6806 \end{proof}
6807 
6808 \begin{corollary}[Overcomplete targets]
6809 An overcomplete target with $d_t>d_s$ admits exact transported repair whenever its bridge from the source is injective and the target states remain in the bridge image.
6810 \end{corollary}
6811 
6812 This explains why the $24$-dimensional overcomplete family matches the local target oracle despite having a different state dimension.
6813 
6814 \subsection{Compression obstruction}
6815 
6816 \begin{proposition}[Residual caused by a noninjective bridge]
6817 \label{prop:block06-compression-residual}
6818 Let $h_t=Th_s$ with arbitrary $T$. Using \cref{eq:block06-transported-repair},
6819 \begin{equation}
6820 R_t(Th_s)-TR_s(h_s)
6821 =
6822 TD_s(T^{+}T-I)h_s.
6823 \label{eq:block06-compression-defect}
6824 \end{equation}
6825 Hence exact state alignment $h_t=Th_s$ does not imply repair equivariance when the bridge discards a source direction used by $D_s$.
6826 \end{proposition}
6827 
6828 \begin{proof}
6829 The bias terms cancel. Expanding the linear terms gives
6830 \begin{equation}
6831 TD_sT^{+}Th_s-TD_sh_s
6832 =TD_s(T^{+}T-I)h_s.
6833 \end{equation}
6834 \end{proof}
6835 
6836 This distinction is visible in the compressed family. Its alignment residual is approximately $3.3\times10^{-8}$, yet its repair-equivariance residual is $0.4715$. Blind transfer raises affected accuracy from $0.1277$ to $0.6204$, but it damages the protected stratum and is rejected by the witness.
6837 
6838 \subsection{Approximate transport decomposition}
6839 
6840 Let
6841 \begin{equation}
6842 e_h=h_t-\widehat T h_s,
6843 \qquad
6844 E_D=D_t\widehat T-\widehat T D_s,
6845 \qquad
6846 E_b=b_t-\widehat T b_s.
6847 \end{equation}
6848 Then
6849 \begin{equation}
6850 R_t(h_t)-\widehat T R_s(h_s)
6851 =
6852 (I+D_t)e_h+E_Dh_s+E_b.
6853 \end{equation}
6854 Therefore
6855 \begin{equation}
6856 \norm{R_t(h_t)-\widehat T R_s(h_s)}
6857 \leq
6858 (1+\norm{D_t})\norm{e_h}
6859 +\norm{E_D}\norm{h_s}
6860 +\norm{E_b}.
6861 \label{eq:block06-approx-bound}
6862 \end{equation}
6863 The implementation reports normalized alignment and repair-equivariance residuals as empirical summaries of these terms.
6864 
6865 \subsection{Anchor identifiability}
6866 
6867 \begin{theorem}[Exact least-squares bridge from spanning anchors]
6868 \label{thm:block06-anchor-identifiability}
6869 Suppose $H_t=H_sT^{\top}$ exactly and $H_s$ has full column rank. With zero regularization, the least-squares solution of \cref{eq:block06-learned-transport} is unique and equals $T$.
6870 \end{theorem}
6871 
6872 \begin{proof}
6873 The normal equations are
6874 \begin{equation}
6875 H_s^{\top}H_s\widehat T^{\top}=H_s^{\top}H_t.
6876 \end{equation}
6877 Full column rank makes $H_s^{\top}H_s$ invertible. Substitution of $H_t=H_sT^{\top}$ yields $\widehat T=T$.
6878 \end{proof}
6879 
6880 The theorem is an algebraic identifiability statement, not a sample-complexity theorem. Conditioning and nonlinear mismatch determine how many random anchors are practically needed.
6881 
6882 \section{Foreign Representation Families}
6883 \label{sec:block06-families}
6884 
6885 \subsection{Controlled target atlas}
6886 
6887 The controlled suite evaluates six seeds in each of seven target families:
6888 \begin{enumerate}
6889 \item $\texttt{linear16}$: invertible orthogonal linear coordinates;
6890 \item $\texttt{overcomplete24}$: a $24$-dimensional injective embedding;
6891 \item $\texttt{ill\_conditioned16}$: invertible linear coordinates with singular values from $1$ to $0.05$;
6892 \item $\texttt{compressed12}$: a noninjective $12$-dimensional projection;
6893 \item $\texttt{mild\_tanh24}$: sixteen mildly nonlinear coordinates plus an eight-dimensional linear skip;
6894 \item $\texttt{saturated\_tanh24}$: strongly saturated nonlinear coordinates;
6895 \item $\texttt{relu32}$: a piecewise-linear ReLU representation.
6896 \end{enumerate}
6897 Each target head is fitted independently to the corrupted score function. The source repair and source packet remain frozen.
6898 
6899 \subsection{Control arms}
6900 
6901 For every target, the package records:
6902 \begin{itemize}
6903 \item the unrepaired baseline;
6904 \item blind transported source repair;
6905 \item witness-selected behavior;
6906 \item a target-local causal repair oracle;
6907 \item shuffled-anchor transport;
6908 \item direct untransported application when dimensions match;
6909 \item alignment and repair-equivariance residuals.
6910 \end{itemize}
6911 
6912 The direct untransported control is important. A source and target can have equal width while assigning different semantics to the same coordinate index. Equal dimensionality is not an authorization to copy a repair matrix directly.
6913 
6914 \subsection{Family results}
6915 
6916 \begin{table}[H]
6917 \centering
6918 \scriptsize
6919 \caption{Controlled foreign-representation results, averaged over six seeds. ``Blind'' applies the transported repair before witness selection. ``Selected'' applies it only when the frozen gate passes.}
6920 \label{tab:block06-family-results}
6921 \begin{tabular}{@{}lrrrrrrrr@{}}
6922 \toprule
6923 Family & Base slice & Blind & Selected & Local oracle & Align NRMSE & Equiv. NRMSE & Commit \\
6924 \midrule
6925 linear16 & .0023 & .9943 & .9943 & .9943 & $3.3\!\times\!10^{-8}$ & $5.9\!\times\!10^{-8}$ & 1.00 \\
6926 overcomplete24 & .0023 & .9943 & .9943 & .9943 & $3.3\!\times\!10^{-8}$ & $5.9\!\times\!10^{-8}$ & 1.00 \\
6927 ill-conditioned16 & .0023 & .9943 & .9943 & .9943 & $1.9\!\times\!10^{-8}$ & $4.6\!\times\!10^{-8}$ & 1.00 \\
6928 compressed12 & .1277 & .6204 & .1277 & .6648 & $3.3\!\times\!10^{-8}$ & .4715 & 0.00 \\
6929 mild-tanh24 & .0949 & .7463 & .5645 & .6112 & .2029 & .3266 & .667 \\
6930 saturated-tanh24 & .2683 & .6072 & .2683 & .5169 & .4984 & .7613 & 0.00 \\
6931 relu32 & .1332 & .6868 & .1332 & .4478 & .3741 & .3546 & 0.00 \\
6932 \bottomrule
6933 \end{tabular}
6934 \end{table}
6935 
6936 Three conclusions are distinct.
6937 
6938 First, exact and injective linear transport behaves as predicted by \cref{thm:block06-exact-transport}. The transported repair equals the local oracle to numerical precision for the linear, overcomplete, and ill-conditioned families.
6939 
6940 Second, blind target improvement is not sufficient. Compressed, saturated, and ReLU targets all show substantial affected-slice gains before selection. The witness rejects them because one or more protected conditions fail.
6941 
6942 Third, mild nonlinear representations occupy an intermediate regime. Blind transport reaches $0.7463$ affected accuracy. Four of six seeds pass the witness, producing a selected mean of $0.5645$. The result is useful but not representation invariant.
6943 
6944 \begin{figure}[H]
6945 \centering
6946 \includegraphics[width=0.98\textwidth]{foreign_transport.pdf}
6947 \caption{Unrepaired, blind-transport, and witness-selected affected-stratum accuracy. The gap between blind and selected performance is intentional. It shows that the witness does not equate apparent target gain with an authorized commit.}
6948 \label{fig:block06-foreign-transport}
6949 \end{figure}
6950 
6951 \subsection{Shuffled-anchor and direct-copy controls}
6952 
6953 For the exact $16$-dimensional family, the mean direct untransported source repair reaches only $0.2646$ affected accuracy. The mean shuffled-anchor transport reaches $0.2442$. Correct learned transport reaches $0.9943$.
6954 
6955 \begin{proposition}[Anchor correspondence is causally necessary in the exact fixture]
6956 Within the frozen exact-linear fixture, replacing paired target anchors by an independent permutation changes only the source-target correspondence used to estimate $T$. The resulting drop from $0.9943$ to $0.2442$ therefore identifies correspondence-sensitive transport rather than a generic benefit from applying the source repair.
6957 \end{proposition}
6958 
6959 The proposition is finite and fixture specific. It does not show that every possible unpaired alignment method must fail.
6960 
6961 \subsection{Anchor-budget sweep}
6962 
6963 \begin{figure}[H]
6964 \centering
6965 \includegraphics[width=0.90\textwidth]{anchor_sweep.pdf}
6966 \caption{Affected-stratum transport accuracy as the paired-anchor budget increases. Exact linear families become numerically exact once the anchor matrix is sufficiently spanning and conditioned. Mild nonlinear transfer improves gradually. Saturated nonlinear transfer remains outside the witness envelope.}
6967 \label{fig:block06-anchor-sweep}
6968 \end{figure}
6969 
6970 For linear and overcomplete targets, eight anchors produce $0.4115$ affected accuracy and fail the gate. Sixteen anchors reach $0.7877$ and pass. Thirty-two anchors recover $0.9943$, after which additional anchors do not change the audit result. Mild nonlinear transfer improves from $0.3830$ at eight anchors to approximately $0.73$ at 256 or more. Saturated nonlinear transfer remains unstable and is rejected at every tested budget.
6971 
6972 This is evidence for an identifiability and approximation transition, not a universal power law. The exact threshold depends on source rank, conditioning, anchor distribution, and regularization.
6973 
6974 \section{Witness-Gated Repair Commitment}
6975 \label{sec:block06-witness}
6976 
6977 \subsection{Frozen gate}
6978 
6979 The witness commits a transported repair only when all four calibration conditions hold:
6980 \begin{align}
6981 \widehat g_{\rm slice}&\geq 0.20,\\
6982 \widehat r_{\rm non}&\leq 0.03,\\
6983 \widehat\eps_{\rm align}&\leq 0.35,\\
6984 \widehat A_{\rm non}^{\rm base}&\geq 0.90.
6985 \label{eq:block06-gate}
6986 \end{align}
6987 Here $\widehat g_{\rm slice}$ is calibration gain on the affected stratum, $\widehat r_{\rm non}$ is regression on the unaffected stratum, $\widehat\eps_{\rm align}$ is normalized alignment error, and $\widehat A_{\rm non}^{\rm base}$ is baseline unaffected accuracy.
6988 
6989 The last condition prevents an already degraded target from being treated as safe merely because the candidate does not make it much worse.
6990 
6991 \begin{conditionaltheorem}[Calibration-to-audit witness soundness]
6992 \label{thm:block06-witness-soundness}
6993 Suppose, simultaneously, that the audit gain and regression satisfy
6994 \begin{align}
6995 g_{\rm slice}&\geq \widehat g_{\rm slice}-\eta_g,\\
6996 r_{\rm non}&\leq \widehat r_{\rm non}+\eta_r,
6997 \end{align}
6998 and that the reported alignment error upper-bounds the audit alignment error up to $\eta_a$. If
6999 \begin{align}
7000 \widehat g_{\rm slice}-\eta_g&>0,\\
7001 \widehat r_{\rm non}+\eta_r&\leq r_{\max},\\
7002 \widehat\eps_{\rm align}+\eta_a&\leq \eps_{\max},
7003 \end{align}
7004 then a candidate passing the corresponding tightened calibration gate has positive audit gain and satisfies the declared audit regression and alignment limits.
7005 \end{conditionaltheorem}
7006 
7007 \begin{proof}
7008 Substitute the simultaneous deviation bounds into the three tightened inequalities.
7009 \end{proof}
7010 
7011 The executable Block 6 gate uses fixed thresholds without an anytime-valid confidence sequence. The theorem states what a statistically certified deployment would additionally require.
7012 
7013 \subsection{Why the gate rejects apparently useful repairs}
7014 
7015 The compressed target has near-zero alignment error but poor base unaffected accuracy and excessive protected regression. The saturated target has large alignment and equivariance error. The ReLU family narrowly exceeds the alignment threshold and also begins below the protected base-accuracy floor. All three are rejected.
7016 
7017 This behavior is central to SRIC. A witness is not a benchmark-score maximizer. It enforces the declared preservation profile.
7018 
7019 \begin{warning}[Gate success is relative]
7020 The frozen gate cannot detect an omitted invariant. Passing it shows only that the candidate satisfies the measured calibration conditions and the package's finite tests.
7021 \end{warning}
7022 
7023 \section{Independently Trained Representation Pilot}
7024 \label{sec:block06-learned-pilot}
7025 
7026 The controlled families above include learned heads but mostly constructed encoders. A second suite trains the representation itself from independent random initializations.
7027 
7028 \subsection{Training objective}
7029 
7030 For encoder $E_{\theta}$, decoder $G_{\delta}$, and score head $H_{\omega}$, the pilot minimizes minibatch estimates of
7031 \begin{equation}
7032 \mathcal L(\theta,\delta,\omega)
7033 =
7034 \lambda_{\rm rec}
7035 \norm{G_{\delta}(E_{\theta}(\phi))-\phi}_F^2
7036 +
7037 \norm{H_{\omega}(E_{\theta}(\phi))-\widetilde q}_F^2,
7038 \label{eq:block06-learned-objective}
7039 \end{equation}
7040 with $\lambda_{\rm rec}=2$. Optimization uses a deterministic NumPy Adam implementation for $500$ steps. After representation training, the head is refitted by ridge regression to reduce avoidable readout variance.
7041 
7042 The source is a learned $16$-dimensional linear representation. Targets are independently trained:
7043 \begin{itemize}
7044 \item three learned linear targets of width $16$;
7045 \item three learned linear targets of width $20$;
7046 \item three learned tanh targets of width $20$.
7047 \end{itemize}
7048 The source repair is discovered once and then frozen.
7049 
7050 \subsection{Pilot results}
7051 
7052 \begin{table}[H]
7053 \centering
7054 \small
7055 \caption{Independently trained representation pilot, averaged over three target seeds.}
7056 \label{tab:block06-learned-results}
7057 \begin{tabular}{@{}lrrrrrr@{}}
7058 \toprule
7059 Family & Base slice & Transport & Local oracle & Non-slice & Align NRMSE & Commit \\
7060 \midrule
7061 learned linear16 & .00339 & .99323 & .99323 & .99965 & $6.6\times10^{-8}$ & 1.00 \\
7062 learned linear20 & .00339 & .99323 & .99323 & .99965 & $7.6\times10^{-8}$ & 1.00 \\
7063 learned tanh20 & .01948 & .89416 & .89670 & .96003 & .14706 & 1.00 \\
7064 \bottomrule
7065 \end{tabular}
7066 \end{table}
7067 
7068 The learned linear targets match the local oracle. The learned tanh targets reach $0.8942$ affected accuracy versus a target-local oracle of $0.8967$. Their mean unaffected accuracy after transport is $0.9600$, and all three pass the frozen witness.
7069 
7070 \begin{figure}[H]
7071 \centering
7072 \includegraphics[width=0.88\textwidth]{learned_pilot.pdf}
7073 \caption{Transported source repair compared with a target-local causal repair in independently trained representations. The nonlinear learned targets show a small but measurable transport gap.}
7074 \label{fig:block06-learned-pilot}
7075 \end{figure}
7076 
7077 \subsection{Interpretation}
7078 
7079 The pilot shows that exact parameter copying is unnecessary. Independently trained representations can support repair transport when paired anchors identify a sufficiently coherent bridge.
7080 
7081 It does not show that the agents discovered the task decomposition, causal probe, repair grammar, or evaluator. The reconstruction term in \cref{eq:block06-learned-objective} actively encourages a linearly alignable task representation. The correct interpretation is therefore:
7082 \begin{center}
7083 \emph{learned representations can instantiate the transport theorem under a controlled shared bottleneck objective.}
7084 \end{center}
7085 
7086 \section{Causal Controls and Statistical Interpretation}
7087 \label{sec:block06-controls}
7088 
7089 \subsection{Factorized causal evidence}
7090 
7091 The block contains several interventions with different inferential roles.
7092 
7093 \begin{longtable}{@{}p{0.23\textwidth}p{0.29\textwidth}p{0.39\textwidth}@{}}
7094 \toprule
7095 Control & Changed factor & Interpretation \\
7096 \midrule
7097 \endhead
7098 Causal vs. PCA packet & Intervention sensitivity at fixed packet rank & Tests whether causal localization, rather than low rank alone, identifies the repair subspace \\
7099 Random packet & Packet orientation & Estimates gain obtainable from accidental overlap \\
7100 Shuffled anchors & Source-target correspondence & Tests whether learned alignment uses paired semantics \\
7101 Direct untransported repair & Coordinate transport & Tests whether equal width permits raw repair copying \\
7102 Wrong feature block & Causal intervention target & Tests specificity to the actual defect-generating block \\
7103 Target-local oracle & Transfer gap & Separates bridge error from target repair-form limitation \\
7104 Witness ablation through blind score & Authorization & Shows why target gain alone is not a commit rule \\
7105 Compressed target & Injectivity & Tests obstruction caused by information loss \\
7106 Saturated and ReLU targets & Global linearizability & Tests the boundary of one global linear bridge \\
7107 \bottomrule
7108 \end{longtable}
7109 
7110 No single control establishes the whole mechanism. Their conjunction supports the narrower statement that the source repair acts through the intervention-defined packet and that correct paired transport is necessary for its strongest foreign-target effect.
7111 
7112 \subsection{Bootstrap intervals}
7113 
7114 Family means are accompanied by percentile bootstrap intervals across target seeds. With only six controlled seeds and three learned-pilot seeds, these intervals are descriptive rather than definitive. Exact constructed families have nearly degenerate intervals because their algebraic behavior is seed invariant in the fixture.
7115 
7116 A stronger follow-up should freeze a larger seed population before examining outcomes, use paired intervals for transported-versus-local performance, and report the joint error budget across packet, transport, and witness claims.
7117 
7118 \subsection{Claim-gate ledger}
7119 
7120 \begin{table}[H]
7121 \centering
7122 \small
7123 \caption{Block 6 claim-gate status. ``Pass'' is restricted to the declared controlled benchmark.}
7124 \label{tab:block06-claim-gates}
7125 \begin{tabularx}{\textwidth}{@{}>{\raggedright\arraybackslash}p{0.12\textwidth}>{\raggedright\arraybackslash}p{0.12\textwidth}>{\raggedright\arraybackslash}p{0.31\textwidth}X@{}}
7126 \toprule
7127 Gate & Status & Evidence & Boundary \\
7128 \midrule
7129 $G_{\rm pkg}$ & Pass & 34-file SHA-256 manifest and clean verifier & Byte integrity relative to the released manifest \\
7130 $G_{\rm mech}$ & Pass & Deterministic execution, five tests, frozen JSON and figures & Runtime mechanics only \\
7131 $G_{\rm sem}$ & Pass & Declared task, repair, transport, and gate objects reproduced & Controlled finite semantics \\
7132 $G_{\rm cause}$ & Pass & Causal packet, PCA, random, wrong-block, and shuffled-anchor controls & Causal role inside the fixture \\
7133 $G_{\rm foreign}$ & Pass & Held-out target seeds, dimensions, encoder families, and learned targets & One shared task and defect grammar \\
7134 $G_{\rm meta}$ & Not tested & No parent-child repair-calculus comparison & No metaproductivity claim \\
7135 $G_{\rm open}$ & Not tested & No autonomous grammar or evaluator expansion & No open-set recursion claim \\
7136 \bottomrule
7137 \end{tabularx}
7138 \end{table}
7139 
7140 The strongest supported claim is therefore controlled foreign repair transfer. The result is a real step beyond the analytic Block 5 fixture, but it is not yet recursive improvement of the improvement process.
7141 
7142 \section{Block 6 Reproducibility and Package}
7143 \label{sec:block06-package}
7144 
7145 \subsection{Reference package}
7146 
7147 The Block 6 reference package \cite{SRICB6} contains:
7148 \begin{verbatim}
7149 SRIC_v0.7_Block06_Learned_Agent_Validation/
7150 |-- sric_learned/
7151 |   |-- agents.py
7152 |   |-- benchmark.py
7153 |   |-- data.py
7154 |   |-- learned_pilot.py
7155 |   |-- repair.py
7156 |   |-- transport.py
7157 |   `-- plots.py
7158 |-- tests/
7159 |   |-- test_gate.py
7160 |   |-- test_learned.py
7161 |   |-- test_packet.py
7162 |   `-- test_transport.py
7163 |-- results/
7164 |-- figures/
7165 |-- run_all.py
7166 |-- Run_SRIC_Block06.ps1
7167 |-- build_manifest.py
7168 |-- verify_package.py
7169 `-- MANIFEST.sha256
7170 \end{verbatim}
7171 
7172 The Python dependency surface is NumPy and Matplotlib. No GPU or network access is required.
7173 
7174 \subsection{Verification commands}
7175 
7176 \begin{verbatim}
7177 python run_all.py
7178 python -m pytest -q
7179 python build_manifest.py
7180 python verify_package.py
7181 \end{verbatim}
7182 
7183 The frozen execution reports:
7184 \begin{verbatim}
7185 SRIC_BLOCK06_ALL_SUITES=PASS
7186 result_sha256=4a210438c56d0e87cfcfa3524ec633a4d78cd8f966e41da8ee669eb544b72065
7187 causal_packet_slice=0.994308
7188 pca_packet_slice=0.023335
7189 linear_transport_slice=0.994308
7190 mild_nonlinear_transport_slice=0.746253
7191 saturated_commit_rate=0.000000
7192 learned_tanh_transport_slice=0.894157
7193 \end{verbatim}
7194 All five automated tests pass. The manifest covers $34$ package files.
7195 
7196 \subsection{Frozen endpoint set}
7197 
7198 The executable tests require:
7199 \begin{enumerate}
7200 \item causal packet affected accuracy above $0.98$;
7201 \item a causal-over-PCA affected-accuracy gap above $0.90$;
7202 \item causal repair unaffected accuracy above $0.99$;
7203 \item exact linear, overcomplete, and ill-conditioned transport above $0.98$ and equal to the local oracle within numerical tolerance;
7204 \item shuffled-anchor accuracy below $0.40$ for exact linear targets;
7205 \item positive witness acceptance in the mild nonlinear family;
7206 \item zero witness acceptance in saturated tanh and ReLU families;
7207 \item learned linear transport above $0.98$;
7208 \item learned tanh transport above $0.80$.
7209 \end{enumerate}
7210 
7211 These endpoints are frozen in the released tests. They were not prospectively preregistered before all internal pilot work. The manuscript therefore describes them as release-frozen verification gates, not as an independent confirmatory study.
7212 
7213 \begin{statusbox}{Block 6 result}
7214 A causally discovered source repair can be transported through a map learned from paired hidden activations to foreign target representations. Exact injective families match target-local repair. Mild nonlinear and independently trained tanh representations retain substantial transfer. Compression and strong nonlinear mismatch expose measurable obstructions, and the frozen witness rejects several blind improvements. This establishes controlled foreign representation transfer, not repair-process metaproductivity or open-set recursive self-improvement.
7215 \end{statusbox}
7216 
7217 \section{Sealed Parent--Child Repair-Process Improvement}
7218 \label{sec:block07-contract}
7219 
7220 Block 6 established that a frozen repair can be discovered in one hidden representation, transported through a learned bridge, and conditionally committed on foreign target representations. That result improved the transfer machinery available to an externally specified repair procedure. It did not show that the procedure which \emph{chooses} repairs had itself improved.
7221 
7222 Block 7 targets the next empirical level. A parent repair selector is frozen. Development receipts generated through the existing lineage are used to compile a child selector. Parent and child then receive exactly the same candidate repairs, target observations, calibration information, evaluator, and candidate budget. The child receipt is sealed before either final audit split is opened. The primary question is
7223 \begin{equation}
7224 \label{eq:block07-primary-question}
7225 \mathbb E_{e\sim Q}
7226 \left[
7227 U_e\!\left(K_{1}(C_e)\right)
7228 -
7229 U_e\!\left(K_{0}(C_e)\right)
7230 \right]
7231 >0,
7232 \end{equation}
7233 where $K_0$ is the frozen parent selector, $K_1$ is the lineage-generated child selector, $C_e$ is the same finite repair-candidate bundle for episode $e$, and $U_e$ is the same protected audit utility.
7234 
7235 The realized claim is intentionally narrow. The child does not invent a new repair candidate, packet type, feature language, model architecture, or evaluator. It improves the \emph{selection rule} over a fixed repair grammar. This is the smallest experimentally clean form of improvement-process improvement because the target-level repair opportunity set remains pathwise identical.
7236 
7237 \subsection{Target episodes and candidate bundles}
7238 
7239 Let $\mathcal E$ be a space of target episodes. An episode $e\in\mathcal E$ contains a target agent, a declared defect context, anchor observations, calibration observations, protected audit observations, and all provenance needed to reproduce those objects. Let
7240 \begin{equation}
7241 \label{eq:block07-candidate-bundle}
7242 C_e
7243 =
7244 \left(c_{e,0},\ldots,c_{e,B-1}\right)
7245 \end{equation}
7246 be an ordered candidate bundle of fixed size $B$. Candidate $c_{e,0}$ is the exact no-op. Every nontrivial candidate carries a receipt containing its construction route, rank or complexity, calibration behavior, transport diagnostics, packet residual, and provenance.
7247 
7248 \begin{definition}[Repair selector]
7249 A repair selector is a measurable map
7250 \begin{equation}
7251 K:\mathcal Z^B\longrightarrow\{0,\ldots,B-1\},
7252 \end{equation}
7253 where $\mathcal Z$ is the space of candidate receipts available before the protected audit outcome is revealed. The selected repair is $c_{e,K(Z_e)}$.
7254 \end{definition}
7255 
7256 \begin{definition}[Matched selector comparison]
7257 Two selectors $K$ and $K'$ are compared under matched resources when, episode by episode, they receive:
7258 \begin{enumerate}
7259 \item the identical candidate bundle $C_e$ in the identical order;
7260 \item the identical receipt fields and calibration evidence;
7261 \item the identical target, anchor, and audit data;
7262 \item the identical candidate-construction budget;
7263 \item the identical protected utility and risk definitions;
7264 \item no policy-specific regeneration, resampling, or extra oracle query.
7265 \end{enumerate}
7266 \end{definition}
7267 
7268 The matched condition prevents a nominally improved selector from receiving more candidate draws, a larger anchor set, a different evaluator, or a more favorable target distribution.
7269 
7270 \begin{definition}[Selector-level metaproductivity]
7271 Let $Q$ be a held-out distribution over target episodes. A child selector $K_1$ has selector-level metaproductivity over parent $K_0$ when
7272 \begin{equation}
7273 \Delta_Q(K_1,K_0)
7274 :=
7275 \mathbb E_{e\sim Q}
7276 \left[
7277 G_e(K_1)-G_e(K_0)
7278 \right]
7279 >0,
7280 \label{eq:block07-metaproductivity}
7281 \end{equation}
7282 under a matched comparison, where $G_e(K)$ is protected utility gain relative to the no-op on episode $e$.
7283 \end{definition}
7284 
7285 \begin{remark}
7286 Selector-level metaproductivity is behavioral. If $K_0$ and $K_1$ choose from the same candidate bundle, they have the same pointwise reachable repair set. The child may exploit that set better without expanding the authorized correction image. Block 7 therefore does not, by itself, prove structural correctability growth.
7287 \end{remark}
7288 
7289 \subsection{The sealed split discipline}
7290 
7291 The experiment uses four disjoint episode collections:
7292 \begin{equation}
7293 \mathsf D_{\rm dev}
7294 \sqcup
7295 \mathsf D_{\rm meta}
7296 \sqcup
7297 \mathsf D_{\rm audit}
7298 \sqcup
7299 \mathsf D_{\rm arch}.
7300 \label{eq:block07-splits}
7301 \end{equation}
7302 Development receipts in $\mathsf D_{\rm dev}$ train the child utility and risk heads. The meta-validation set $\mathsf D_{\rm meta}$ selects one finite hyperparameter configuration. The seed-and-task audit $\mathsf D_{\rm audit}$ evaluates new episodes from the development architecture-family support. The architecture-foreign audit $\mathsf D_{\rm arch}$ evaluates representation families absent from both development and meta-validation.
7303 
7304 \begin{definition}[Sealed meta-compiler]
7305 A sealed meta-compiler is a map
7306 \begin{equation}
7307 \mathfrak M:
7308 \left(\mathsf D_{\rm dev},\mathsf D_{\rm meta},K_0,\Gamma\right)
7309 \longmapsto
7310 \left(K_1,\Pi_1\right),
7311 \label{eq:block07-meta-compiler}
7312 \end{equation}
7313 where $\Gamma$ is a declared finite meta-grammar and $\Pi_1$ is a descendant receipt. The compiler is sealed when:
7314 \begin{enumerate}
7315 \item neither final audit split is supplied to $\mathfrak M$;
7316 \item $\Pi_1$ commits to the child state, parent identity, input hashes, grammar, selected configuration, and candidate budget;
7317 \item the audit files are opened only after the receipt has been serialized and hashed;
7318 \item the child state used for final evaluation is byte-equivalent to the state committed by $\Pi_1$.
7319 \end{enumerate}
7320 \end{definition}
7321 
7322 \begin{theorem}[Compilation noninterference]
7323 \label{thm:block07-compilation-noninterference}
7324 Let $K_1=\mathfrak M(D,V,K_0,\Gamma)$, where $D$ and $V$ denote the development and meta-validation receipts. Suppose $\mathfrak M$ is deterministic and receives no value measurable only with respect to final audit outcomes $A$. Then, conditional on $(D,V,K_0,\Gamma)$,
7325 \begin{equation}
7326 K_1\perp\!\!\!\perp A.
7327 \end{equation}
7328 Equivalently, changing the realized final audit outcomes while holding compiler inputs fixed leaves $K_1$ unchanged.
7329 \end{theorem}
7330 
7331 \begin{proof}
7332 By assumption there is a deterministic function $f$ such that
7333 $K_1=f(D,V,K_0,\Gamma)$. Conditional on its arguments, $K_1$ is constant. It therefore has a degenerate conditional distribution that does not depend on $A$.
7334 \end{proof}
7335 
7336 \begin{warning}
7337 A digest certifies byte integrity relative to a committed object. It does not by itself prove that an implementation never observed the audit. The released runner strengthens the logical interface by placing development, meta-validation, seed-and-task audit, and architecture-foreign audit in separate files. It loads only the first two, writes the descendant receipt, and only then opens the final audit files. This is process-order evidence, not a hardware-enforced information-flow proof.
7338 \end{warning}
7339 
7340 \begin{definition}[One-step bounded repair-process improvement]
7341 A lineage edge
7342 \begin{equation}
7343 (\Ocal_0,K_0)
7344 \xrightarrow{\Pi_1}
7345 (\Ocal_1,K_1)
7346 \end{equation}
7347 is a one-step bounded repair-process improvement relative to $(Q,B,U,R)$ when:
7348 \begin{enumerate}
7349 \item $K_1$ is generated and sealed through an authorized lineage transaction;
7350 \item parent and child use the same candidate budget $B$ and candidate bundle on each target;
7351 \item a predeclared lower confidence bound for $\Delta_Q(K_1,K_0)$ is positive;
7352 \item the child satisfies declared average, tail, and maximum protected-risk gates;
7353 \item wrong-lineage or shuffled-receipt controls perform worse;
7354 \item ablating the child module restores the parent selection trace;
7355 \item the evaluator, target distribution, and evidence semantics are not changed by the child.
7356 \end{enumerate}
7357 \end{definition}
7358 
7359 This definition certifies one productive lineage edge. It does not imply recurrent improvement closure, a successful grandchild, or open-ended grammar expansion.
7360 
7361 \section{A Proof-Carrying Dual-Head Descendant}
7362 \label{sec:block07-child}
7363 
7364 The child is not a second target solver. It is a learned selector over the five repair candidates already generated for each target episode.
7365 
7366 \subsection{Candidate features}
7367 
7368 Each candidate receipt $z_{e,c}$ is mapped to a feature vector
7369 \begin{equation}
7370 x(z_{e,c})\in\R^{23}.
7371 \end{equation}
7372 The released feature schema contains:
7373 \begin{itemize}
7374 \item a constant coordinate;
7375 \item logarithmic anchor and calibration sizes;
7376 \item target representation width;
7377 \item alignment, transport-condition, and repair-equivariance diagnostics;
7378 \item packet residual and relative repair magnitude;
7379 \item repair rank;
7380 \item calibration baseline slice and protected accuracy;
7381 \item calibration gain, protected regression, and protected utility;
7382 \item five candidate-type indicators;
7383 \item three interaction coordinates coupling gain with alignment, packet residual, and rank-weighted regression.
7384 \end{itemize}
7385 The schema is fixed before the final audits. It is expressive enough to learn systematic calibration bias, but it is not autonomously discovered.
7386 
7387 Let $x_i$ denote a development candidate feature vector, $y_i$ its revealed protected utility, and
7388 \begin{equation}
7389 r_i=[A^{\rm base}_{i,\rm prot}-A_{i,\rm prot}]_+
7390 \end{equation}
7391 its protected-regression label. Standardize the feature matrix to $Z$ using development means and scales.
7392 
7393 \subsection{Utility and risk heads}
7394 
7395 For regularization parameters $\lambda_u,\lambda_r>0$, define
7396 \begin{align}
7397 \widehat\beta_u
7398 &=
7399 \left(Z^\top Z+\lambda_u I\right)^{-1}Z^\top y,
7400 \label{eq:block07-utility-head}\\
7401 \widehat\beta_r
7402 &=
7403 \left(Z^\top Z+\lambda_r I\right)^{-1}Z^\top r.
7404 \label{eq:block07-risk-head}
7405 \end{align}
7406 For candidate type $c$, let $s_{u,c}$ and $s_{r,c}$ be the development residual standard deviations for the utility and risk heads. The deployed scores are
7407 \begin{align}
7408 \widehat U_{\rm LCB}(z)
7409 &=
7410 \widehat U(z)-\kappa_u s_{u,c(z)},
7411 \label{eq:block07-utility-lcb}\\
7412 \widehat R_{\rm UCB}(z)
7413 &=
7414 \max\{0,\widehat R(z)\}+\kappa_r s_{r,c(z)}.
7415 \label{eq:block07-risk-ucb}
7416 \end{align}
7417 A candidate is admissible when
7418 \begin{equation}
7419 \widehat R_{\rm UCB}(z)\leq\tau_r,
7420 \qquad
7421 \widehat r_{\rm cal}(z)\leq\tau_{\rm cal},
7422 \qquad
7423 A^{\rm base}_{\rm cal,prot}(z)\geq a_{\min}.
7424 \label{eq:block07-admissible}
7425 \end{equation}
7426 The child chooses the admissible candidate with maximum $\widehat U_{\rm LCB}$. If the admissible set is empty, it returns the no-op.
7427 
7428 \begin{theorem}[Risk-gate soundness]
7429 \label{thm:block07-risk-soundness}
7430 Fix an episode and candidate set. Suppose, simultaneously for every candidate $c$,
7431 \begin{equation}
7432 R_e(c)\leq\widehat R_{\rm UCB}(c).
7433 \end{equation}
7434 Then any child-selected candidate satisfying
7435 $\widehat R_{\rm UCB}(c)\leq\tau_r$ has true protected risk at most $\tau_r$.
7436 \end{theorem}
7437 
7438 \begin{proof}
7439 For the selected candidate $\widehat c$,
7440 \begin{equation}
7441 R_e(\widehat c)
7442 \leq
7443 \widehat R_{\rm UCB}(\widehat c)
7444 \leq
7445 \tau_r.
7446 \end{equation}
7447 \end{proof}
7448 
7449 \begin{theorem}[Utility regret under simultaneous lower bounds]
7450 \label{thm:block07-utility-regret}
7451 Let $\mathcal C_{\rm safe}$ be the truly safe candidate set. Suppose every $c\in\mathcal C_{\rm safe}$ passes the implemented admissibility filter, and suppose
7452 \begin{equation}
7453 0\leq U_e(c)-\widehat U_{\rm LCB}(c)\leq\epsilon
7454 \end{equation}
7455 for every admissible $c$. If $\widehat c$ maximizes $\widehat U_{\rm LCB}$ and $c^*$ maximizes $U_e$ over $\mathcal C_{\rm safe}$, then
7456 \begin{equation}
7457 U_e(c^*)-U_e(\widehat c)\leq\epsilon.
7458 \end{equation}
7459 \end{theorem}
7460 
7461 \begin{proof}
7462 Because $c^*$ is admissible,
7463 \begin{equation}
7464 \widehat U_{\rm LCB}(\widehat c)
7465 \geq
7466 \widehat U_{\rm LCB}(c^*).
7467 \end{equation}
7468 The lower-bound property gives
7469 \begin{equation}
7470 U_e(\widehat c)
7471 \geq
7472 \widehat U_{\rm LCB}(\widehat c)
7473 \geq
7474 \widehat U_{\rm LCB}(c^*)
7475 \geq
7476 U_e(c^*)-\epsilon.
7477 \end{equation}
7478 \end{proof}
7479 
7480 \begin{remark}
7481 The theorem requires simultaneous validity and safe-set coverage. The released residual penalties are empirical calibration devices, not formal confidence sequences. Their adequacy is tested on disjoint final audits rather than assumed from the development fit.
7482 \end{remark}
7483 
7484 \subsection{Finite meta-grammar selection}
7485 
7486 The child compiler searches a finite grammar
7487 \begin{equation}
7488 \Gamma
7489 =
7490 \Lambda_u\times\Lambda_r\times K_u\times K_r\times T_r,
7491 \end{equation}
7492 where the factors specify utility regularization, risk regularization, utility residual penalty, risk residual penalty, and risk threshold. In the released fixture,
7493 \begin{equation}
7494 |\Gamma|
7495 =5\cdot5\cdot4\cdot5\cdot4
7496 =2000.
7497 \end{equation}
7498 For $\gamma\in\Gamma$, let $\overline U_{\rm meta}(\gamma)$ be its mean protected utility on the meta-validation episodes and $\widehat{\rm se}(\gamma)$ its standard error. A configuration is validation-safe when its protected-regression tail rate and maximum regression satisfy the frozen bounds. Among safe configurations, the compiler selects
7499 \begin{equation}
7500 \widehat\gamma
7501 \in
7502 \argmax_{\gamma\in\Gamma_{\rm safe}}
7503 \left
7504 \{
7505 \overline U_{\rm meta}(\gamma)
7506 -	frac12\widehat{\rm se}(\gamma)
7507 \right\}.
7508 \label{eq:block07-meta-objective}
7509 \end{equation}
7510 
7511 \begin{theorem}[Finite meta-grammar uniform bound]
7512 \label{thm:block07-meta-uniform}
7513 Suppose, for each $\gamma\in\Gamma$, the per-episode utility lies in an interval of width $L$, and the $n$ meta-validation episodes are independent. Then with probability at least $1-\delta$,
7514 \begin{equation}
7515 \sup_{\gamma\in\Gamma}
7516 \left|
7517 \widehat U_n(\gamma)-U(\gamma)
7518 \right|
7519 \leq
7520 L\sqrt{\frac{\log(2|\Gamma|/\delta)}{2n}}.
7521 \label{eq:block07-uniform-bound}
7522 \end{equation}
7523 Consequently, the true utility of the empirical maximizer is within twice the right-hand side of the best configuration in $\Gamma$.
7524 \end{theorem}
7525 
7526 \begin{proof}
7527 Hoeffding's inequality bounds the deviation for one configuration. Apply the union bound to the $2|\Gamma|$ upper and lower deviation events. The empirical-maximizer conclusion follows by inserting the uniform bound on both the winner and the true grammar optimum.
7528 \end{proof}
7529 
7530 The theorem explains why a separate final audit remains necessary. Searching 2000 configurations on 60 meta-validation episodes can overfit that split even when the final audit is untouched.
7531 
7532 \subsection{Selected descendant receipt}
7533 
7534 The winning frozen configuration is
7535 \begin{equation}
7536 \begin{aligned}
7537 \lambda_u&=100,
7538 &\lambda_r&=0.01,\\
7539 \kappa_u&=2,
7540 &\kappa_r&=0.5,\\
7541 \tau_r&=0.035,
7542 &\tau_{\rm cal}&=0.08,\\
7543 a_{\min}&=0.80.
7544 \end{aligned}
7545 \label{eq:block07-selected-config}
7546 \end{equation}
7547 Its meta-validation score is $0.291029$, from mean protected utility $0.309844$ minus one-half standard error. The validation protected-regression tail rate is $0.016667$ and the maximum is $0.054463$.
7548 
7549 The descendant receipt records:
7550 \begin{itemize}
7551 \item the parent policy identifier and lineage generation;
7552 \item development and meta-validation receipt hashes;
7553 \item the compiler source hash and candidate-feature-schema hash;
7554 \item the complete 2000-point meta-grammar;
7555 \item the selected configuration and validation witness statistics;
7556 \item feature means, feature scales, fitted utility and risk weights, and candidate-specific residual scales;
7557 \item the fixed candidate budget;
7558 \item the final receipt digest.
7559 \end{itemize}
7560 
7561 \section{Matched-Budget Attribution of the Meta-Gain}
7562 \label{sec:block07-attribution}
7563 
7564 An observed child advantage can be attributed to the child selector only when every other target-level opportunity is held fixed.
7565 
7566 \begin{theorem}[Pathwise selector isolation]
7567 \label{thm:block07-selector-isolation}
7568 For episode $e$, let parent and child receive the identical ordered bundle $C_e$ and identical pre-audit receipt $Z_e$. Let
7569 \begin{equation}
7570 i_0=K_0(Z_e),
7571 \qquad
7572 i_1=K_1(Z_e).
7573 \end{equation}
7574 Then the realized paired utility difference is exactly
7575 \begin{equation}
7576 D_e
7577 =
7578 U_e(c_{e,i_1})-U_e(c_{e,i_0}).
7579 \label{eq:block07-paired-difference}
7580 \end{equation}
7581 If candidate construction, target state, and audit evaluation are frozen, no candidate-generation term remains in this difference.
7582 \end{theorem}
7583 
7584 \begin{proof}
7585 Both policies are evaluated on the same episode and the same candidate utility vector
7586 \begin{equation*}
7587 \left(U_e(c_{e,0}),\ldots,U_e(c_{e,B-1})\right).
7588 \end{equation*}
7589 Their outputs differ only through the selected indices $i_0$ and $i_1$. Substitution gives \cref{eq:block07-paired-difference}.
7590 \end{proof}
7591 
7592 \begin{corollary}[Selector ablation]
7593 \label{cor:block07-selector-ablation}
7594 If the descendant module is removed and the parent map $K_0$ is reinstated without changing any other runtime object, the resulting selection and utility trace is pathwise identical to the parent trace.
7595 \end{corollary}
7596 
7597 The executable control copies the complete parent metrics into the selector-ablation arm and separately tests equality on both final audit splits.
7598 
7599 \subsection{The best-of-\texorpdfstring{$k$}{k} confound}
7600 
7601 Let $M_B(K)$ denote expected best gain when policy $K$ receives $B$ independently generated candidate opportunities. Even if the policy is unchanged,
7602 \begin{equation}
7603 M_{B+1}(K)\geq M_B(K).
7604 \end{equation}
7605 Therefore, comparing a child with $B_1>B_0$ against a parent with budget $B_0$ does not isolate improvement of the selection procedure.
7606 
7607 \begin{proposition}[No-free-expansion attribution]
7608 \label{prop:block07-budget-confound}
7609 Suppose the child receives the parent's $B$ candidates plus one additional candidate. Then a positive child-parent gain can occur even when the child applies the parent selector to the original candidates and uses no improved inference. Hence a positive unmatched-budget gain is not sufficient evidence for repair-process improvement.
7610 \end{proposition}
7611 
7612 \begin{proof}
7613 Let the additional candidate have strictly higher utility than all original candidates with positive probability, and let the child choose it whenever present. The child improves on those episodes without any change to the rule applied to the original bundle.
7614 \end{proof}
7615 
7616 Block 7 fixes $B=5$ for every arm and target.
7617 
7618 \subsection{Lineage and semantic controls}
7619 
7620 The child is trained from candidate receipts whose utility and risk labels are attached to semantically typed candidate features. The shuffled control independently permutes those outcome labels before fitting the same dual-head form.
7621 
7622 \begin{proposition}[Shuffled-label null]
7623 \label{prop:block07-shuffled-null}
7624 Suppose development labels are exchangeable and independently permuted relative to candidate features. Conditional on the feature matrix, the expected feature-label cross-covariance is zero after centering. A linear ridge head fitted to infinitely many such permutations therefore converges in expectation to the intercept-only predictor.
7625 \end{proposition}
7626 
7627 \begin{proof}
7628 Let $Z$ be column centered and let $P$ be a uniformly random permutation matrix independent of $y$. Then
7629 \begin{equation}
7630 \mathbb E_P[Z^\top Py]
7631 =
7632 Z^\top\mathbb E_P[Py]
7633 =
7634 Z^\top\overline y\one
7635 =0.
7636 \end{equation}
7637 The ridge coefficients are linear in $Z^\top Py$ for fixed $Z$.
7638 \end{proof}
7639 
7640 Finite permutations can retain accidental structure, so the control need not fall to zero utility. The required result is that the correctly bound lineage receipts yield a stronger final policy.
7641 
7642 \subsection{Selection versus structural correctability}
7643 
7644 \begin{proposition}[Fixed-candidate structural invariance]
7645 \label{prop:block07-fixed-candidate-invariance}
7646 Let $\mathcal C_e$ be the authorized candidate set for episode $e$. If parent and child differ only in the selector and
7647 \begin{equation}
7648 \mathcal C^{(0)}_e=\mathcal C^{(1)}_e
7649 \end{equation}
7650 for every $e$, then the pointwise reachable correction set and its linear span are identical for parent and child.
7651 \end{proposition}
7652 
7653 \begin{proof}
7654 Both sets are equal by hypothesis, so their spans and all set-valued reachability properties determined only by candidate membership are equal.
7655 \end{proof}
7656 
7657 The Block 7 child improves the probability of choosing a useful available correction. It does not add a new correctable defect direction. This is why the empirical claim is \emph{selector-level} metaproductivity rather than conservative repair-grammar expansion.
7658 
7659 \section{Protected Utility, Risk, and Statistical Certification}
7660 \label{sec:block07-statistics}
7661 
7662 \subsection{Protected target utility}
7663 
7664 For target episode $e$ and candidate $c$, let
7665 \begin{align}
7666 A^{\rm slice}_e(c)
7667 &=\text{accuracy on the affected stratum},\\
7668 A^{\rm prot}_e(c)
7669 &=\text{accuracy on the unaffected protected stratum},\\
7670 q(c)
7671 &=\text{repair rank}.
7672 \end{align}
7673 The candidate utility is
7674 \begin{equation}
7675 U_e(c)
7676 =
7677 A^{\rm slice}_e(c)
7678 -2\left[A^{\rm prot}_e(0)-A^{\rm prot}_e(c)\right]_+
7679 -0.0005q(c),
7680 \label{eq:block07-protected-utility}
7681 \end{equation}
7682 and the reported protected utility gain is
7683 \begin{equation}
7684 G_e(c)=U_e(c)-A^{\rm slice}_e(0).
7685 \label{eq:block07-protected-gain}
7686 \end{equation}
7687 The protected regression is
7688 \begin{equation}
7689 R_e(c)=\left[A^{\rm prot}_e(0)-A^{\rm prot}_e(c)\right]_+.
7690 \end{equation}
7691 The utility therefore rewards repair of the affected stratum, penalizes damage to unrelated behavior twice as strongly, and applies a small complexity charge.
7692 
7693 \subsection{Paired inference}
7694 
7695 For $n$ held-out episodes, define paired differences
7696 \begin{equation}
7697 D_j
7698 =
7699 G_{e_j}(K_1)-G_{e_j}(K_0),
7700 \qquad
7701 \widehat\Delta_n=\frac1n\sum_{j=1}^nD_j.
7702 \label{eq:block07-paired-estimator}
7703 \end{equation}
7704 Because parent and child share each target and candidate bundle, pairing removes much of the episode difficulty variation.
7705 
7706 \begin{theorem}[Distribution-free paired lower bound]
7707 \label{thm:block07-paired-bound}
7708 Suppose $D_1,\ldots,D_n$ are independent and lie almost surely in $[a,b]$. Then with probability at least $1-\alpha$,
7709 \begin{equation}
7710 \Delta_Q
7711 \geq
7712 \widehat\Delta_n
7713 -(b-a)\sqrt{\frac{\log(1/\alpha)}{2n}}.
7714 \label{eq:block07-hoeffding}
7715 \end{equation}
7716 A positive right-hand side certifies $\Delta_Q>0$ at confidence $1-\alpha$ under the assumptions.
7717 \end{theorem}
7718 
7719 \begin{proof}
7720 Apply the one-sided Hoeffding inequality to the sample mean of the bounded paired differences.
7721 \end{proof}
7722 
7723 The executable package additionally reports a frozen 20,000-draw paired bootstrap interval and one-sided sign-flip randomization $p$-value. Those procedures provide the reported finite-sample summaries. They do not replace the need to declare exchangeability or sampling assumptions.
7724 
7725 \subsection{Cumulative claim gates}
7726 
7727 The child must pass performance and risk gates simultaneously. Let
7728 \begin{equation}
7729 \mathsf Gate
7730 =
7731 \mathsf Seal
7732 \wedge
7733 \mathsf Match
7734 \wedge
7735 \mathsf AuditGain
7736 \wedge
7737 \mathsf ForeignGain
7738 \wedge
7739 \mathsf Risk
7740 \wedge
7741 \mathsf Controls.
7742 \end{equation}
7743 The released suite evaluates the following frozen selector-gate families:
7744 \begin{enumerate}
7745 \item descendant receipt validity;
7746 \item split-file load order and sealing;
7747 \item positive paired child-parent audit interval;
7748 \item positive paired child-parent architecture-foreign interval;
7749 \item positive child-raw architecture-foreign interval;
7750 \item three seed-and-task audit regression gates;
7751 \item three architecture-foreign regression gates;
7752 \item two shuffled-lineage controls;
7753 \item two selector-ablation equalities;
7754 \item matched five-candidate budget;
7755 \item exclusion of audit receipts from compiler inputs;
7756 \item disjoint architecture-family supports.
7757 \end{enumerate}
7758 All component gates must pass before the aggregate gate is true.
7759 
7760 \begin{warning}
7761 The seed-and-task audit does not resolve child superiority over the strong raw-calibration baseline: the paired mean is $0.005922$ with bootstrap interval $[-0.000677,0.013639]$. Child superiority over raw calibration is supported only on the architecture-foreign split in this release. The main parent-child claim remains positive on both splits.
7762 \end{warning}
7763 
7764 \subsection{Risk-head ablation}
7765 
7766 Removing the risk head expands the child's admissible set. This can raise utility while worsening protected risk.
7767 
7768 \begin{proposition}[Risk-filter tradeoff]
7769 \label{prop:block07-risk-tradeoff}
7770 Let $\mathcal A_r\subseteq\mathcal A_0$ be the candidate set retained with and without a risk filter. Then
7771 \begin{equation}
7772 \max_{c\in\mathcal A_0}\widehat U(c)
7773 \geq
7774 \max_{c\in\mathcal A_r}\widehat U(c),
7775 \end{equation}
7776 but no analogous inequality favors the unfiltered selector for true protected risk.
7777 \end{proposition}
7778 
7779 \begin{proof}
7780 The utility inequality follows from set inclusion. Risk is not the optimized scalar in the unfiltered maximum, so it admits no monotonicity conclusion.
7781 \end{proof}
7782 
7783 Empirically, the no-risk child has higher mean utility than the protected child on both final splits, but it violates the architecture-foreign average and tail risk gates. The realized result is therefore a constrained improvement claim, not an assertion that the selected child maximizes unconstrained utility.
7784 
7785 \section{Executable Block 7 Benchmark}
7786 \label{sec:block07-benchmark}
7787 
7788 \subsection{Task and defect generator}
7789 
7790 For $x\in\R^4$, define an eight-dimensional task representation
7791 \begin{equation}
7792 \psi(x)
7793 =
7794 \left(
7795  x_1,x_2,x_3,x_4,
7796  x_1x_2,x_3x_4,
7797  \sin(sx_1),\cos(sx_2)
7798 \right).
7799 \end{equation}
7800 A task-specific condition vector and threshold define the affected indicator
7801 \begin{equation}
7802 a(x)=\one\{\langle v,x\rangle>\tau\}.
7803 \end{equation}
7804 The conditional feature is
7805 \begin{equation}
7806 \phi(x)=\bigl(\psi(x),a(x)\psi(x)\bigr)\in\R^{16}.
7807 \end{equation}
7808 True three-class score vectors are affine in $\psi(x)$. On the affected stratum only, one of five frozen score transformations corrupts those scores. Target agents are fitted to the corrupted scores, so their baseline behavior is generally accurate off the affected stratum and defective on it.
7809 
7810 \subsection{Representation families}
7811 
7812 Development, meta-validation, and seed-and-task audit episodes draw from eight families:
7813 \begin{equation}
7814 \begin{gathered}
7815 \texttt{linear16},\quad
7816 \texttt{overcomplete24},\quad
7817 \texttt{ill\_conditioned16},\quad
7818 \texttt{compressed12},\\
7819 \texttt{mild\_tanh24},\quad
7820 \texttt{saturated\_tanh24},\quad
7821 \texttt{relu32},\quad
7822 \texttt{noisy\_linear16}.
7823 \end{gathered}
7824 \end{equation}
7825 The architecture-foreign audit uses four disjoint families:
7826 \begin{equation}
7827 \texttt{softsign24},\quad
7828 \texttt{piecewise24},\quad
7829 \texttt{quadratic24},\quad
7830 \texttt{bottleneck14}.
7831 \end{equation}
7832 The family names and construction code are frozen in the candidate receipt archive.
7833 
7834 \subsection{Five-candidate repair grammar}
7835 
7836 Every episode exposes exactly
7837 \begin{equation}
7838 C_e
7839 =
7840 \{
7841 \texttt{no\_op},
7842 \texttt{transport},
7843 \texttt{local\_r4},
7844 \texttt{local\_r8},
7845 \texttt{local\_full}
7846 \}.
7847 \label{eq:block07-five-candidates}
7848 \end{equation}
7849 The transported candidate moves a rank-eight source repair through a learned source-to-target bridge. The rank-four and rank-eight local candidates use target-side intervention differences with noisy or incomplete probes. The full local candidate fits an unrestricted affine hidden-state correction. Candidate construction is complete before either parent or child selects.
7850 
7851 Episode difficulty varies through target family, task seed, encoder seed, anchor size, calibration size, probe noise, and probe leakage. Anchor sizes lie in
7852 \begin{equation}
7853 \{16,24,32,48,64,96,128\},
7854 \end{equation}
7855 calibration sizes in
7856 \begin{equation}
7857 \{48,64,96,128\},
7858 \end{equation}
7859 probe-noise levels in
7860 \begin{equation}
7861 \{0,0.02,0.05,0.10,0.18\},
7862 \end{equation}
7863 and probe-leak levels in
7864 \begin{equation}
7865 \{0,0.05,0.12,0.25\}.
7866 \end{equation}
7867 
7868 \subsection{Episode counts and lineage chronology}
7869 
7870 \begin{table}[H]
7871 \centering
7872 \small
7873 \begin{tabular}{@{}lrrl@{}}
7874 \toprule
7875 Split & Episodes & Candidate receipts & Function \\
7876 \midrule
7877 Development & 120 & 600 & fit utility and risk heads \\
7878 Meta-validation & 60 & 300 & choose one of 2000 meta-configurations \\
7879 Seed/task audit & 120 & 600 & held-out same-support evaluation \\
7880 Architecture-foreign audit & 120 & 600 & held-out family evaluation \\
7881 \bottomrule
7882 \end{tabular}
7883 \caption{Block 7 episode and candidate counts. Every episode contains the same five candidate types.}
7884 \label{tab:block07-splits}
7885 \end{table}
7886 
7887 The chronology is:
7888 \begin{equation}
7889 \begin{aligned}
7890 &\text{open development file and fit heads}\notag\\
7891 &\longrightarrow
7892 \text{open meta-validation file and choose configuration}\notag\\
7893 &\longrightarrow
7894 \text{serialize and hash descendant receipt}\notag\\
7895 &\longrightarrow
7896 \text{open seed/task audit file}\notag\\
7897 &\longrightarrow
7898 \text{open architecture-foreign audit file}\notag\\
7899 &\longrightarrow
7900 \text{evaluate every frozen policy and claim gate}.
7901 \end{aligned}
7902 \label{eq:block07-chronology}
7903 \end{equation}
7904 
7905 \subsection{Comparison arms}
7906 
7907 \begin{longtable}{@{}p{0.23\textwidth}p{0.67\textwidth}@{}}
7908 \toprule
7909 Arm & Role \\
7910 \midrule
7911 \endhead
7912 Parent & Frozen conservative priority rule using calibration gain, protected regression, baseline protected accuracy, and transport alignment. \\
7913 Raw calibration & Strong ordinary baseline choosing maximum measured calibration utility inside a fixed safety envelope. \\
7914 Descendant & Sealed dual-head utility/risk selector compiled from development and meta-validation receipts. \\
7915 No-risk descendant & Same child utility head and selected meta-configuration, with the risk-head exclusion removed. \\
7916 Shuffled descendant & Same dual-head form fitted after permuting development utility and risk labels relative to candidate features. \\
7917 Selector ablation & Descendant module removed and exact parent policy reinstated. \\
7918 Audit oracle & Inadmissible upper comparison that directly chooses maximum audit utility. \\
7919 \bottomrule
7920 \end{longtable}
7921 
7922 The audit oracle is not a deployable baseline. It estimates the remaining opportunity available inside the fixed five-candidate grammar.
7923 
7924 \section{Block 7 Results}
7925 \label{sec:block07-results}
7926 
7927 \subsection{Primary parent--child comparison}
7928 
7929 \begin{table}[H]
7930 \centering
7931 \small
7932 \begin{tabular}{@{}lrrrr@{}}
7933 \toprule
7934 Split and arm & Mean utility gain & Mean regression & Tail rate & Success \\
7935 \midrule
7936 Seed/task parent & 0.284896 & 0.001774 & 0.008333 & 0.508333 \\
7937 Seed/task child & \textbf{0.357414} & 0.007787 & 0.050000 & 0.700000 \\
7938 Architecture parent & 0.167417 & 0.004831 & 0.008333 & 0.350000 \\
7939 Architecture child & \textbf{0.272416} & 0.012852 & 0.033333 & 0.666667 \\
7940 \bottomrule
7941 \end{tabular}
7942 \caption{Primary matched-budget results. Tail rate is the fraction of episodes with protected regression above $0.05$.}
7943 \label{tab:block07-primary-results}
7944 \end{table}
7945 
7946 The paired effects are
7947 \begin{align}
7948 \widehat\Delta_{\rm audit}
7949 &=0.072518,
7950 &\mathrm{CI}_{95\%}
7951 &=[0.046745,0.101389],
7952 \label{eq:block07-audit-effect}\\
7953 \widehat\Delta_{\rm arch}
7954 &=0.104999,
7955 &\mathrm{CI}_{95\%}
7956 &=[0.072711,0.140506].
7957 \label{eq:block07-foreign-effect}
7958 \end{align}
7959 Both one-sided sign-flip $p$-values equal approximately
7960 $4.99975\times10^{-5}$ under the frozen 20,000-draw randomization procedure.
7961 
7962 \begin{figure}[H]
7963 \centering
7964 \includegraphics[width=0.94\textwidth]{block07_metaproductivity.pdf}
7965 \caption{Mean protected utility under the matched five-candidate budget. The child improves over its parent on the meta-validation, seed-and-task audit, and architecture-foreign audit splits.}
7966 \label{fig:block07-metaproductivity}
7967 \end{figure}
7968 
7969 \begin{figure}[H]
7970 \centering
7971 \includegraphics[width=0.88\textwidth]{block07_paired_effects.pdf}
7972 \caption{Episode-wise child-minus-parent protected utility. Many episodes are ties because both policies select the same candidate; positive mean shifts remain on both final splits.}
7973 \label{fig:block07-paired-effects}
7974 \end{figure}
7975 
7976 \subsection{Strong-baseline and oracle comparisons}
7977 
7978 \begin{table}[H]
7979 \centering
7980 \small
7981 \begin{tabular}{@{}lrrr@{}}
7982 \toprule
7983 Arm & Meta-validation & Seed/task audit & Architecture foreign \\
7984 \midrule
7985 Parent & 0.229208 & 0.284896 & 0.167417 \\
7986 Raw calibration & 0.301919 & 0.351492 & 0.254427 \\
7987 Descendant & \textbf{0.309844} & \textbf{0.357414} & \textbf{0.272416} \\
7988 No-risk descendant & 0.312660 & 0.362862 & 0.293841 \\
7989 Shuffled descendant & 0.191694 & 0.250997 & 0.231744 \\
7990 Audit oracle & 0.331609 & 0.402808 & 0.324703 \\
7991 \bottomrule
7992 \end{tabular}
7993 \caption{Mean protected utility gain for all substantive arms. The no-risk arm is not admissible under the frozen architecture-foreign risk gates.}
7994 \label{tab:block07-all-arms}
7995 \end{table}
7996 
7997 On the architecture-foreign split, the child's paired advantage over raw calibration is
7998 \begin{equation}
7999 0.017989,
8000 \qquad
8001 \mathrm{CI}_{95\%}=[0.003477,0.036611],
8002 \qquad
8003 p_{\rm sign}=0.00940.
8004 \end{equation}
8005 On the seed-and-task audit, the corresponding interval includes zero. The result therefore supports robust improvement over the frozen parent on both audits and a smaller architecture-foreign improvement over a strong ordinary selector.
8006 
8007 The audit oracle shows that the fixed candidate bundle retains additional exploitable utility. The child captures approximately $76.9\%$ of per-episode oracle opportunity on the seed-and-task audit and a smaller fraction under architecture shift. The remaining gap can arise from prediction error, risk conservatism, missing candidate features, or meta-grammar limits.
8008 
8009 \subsection{Risk-utility tradeoff}
8010 
8011 \begin{figure}[H]
8012 \centering
8013 \includegraphics[width=0.88\textwidth]{block07_risk_utility_frontier.pdf}
8014 \caption{Mean protected utility versus protected regression. The no-risk descendant obtains more utility but crosses the frozen architecture-foreign average and tail risk gates.}
8015 \label{fig:block07-risk-frontier}
8016 \end{figure}
8017 
8018 For the architecture-foreign split,
8019 \begin{align}
8020 \overline R_{\rm child}&=0.012852,
8021 &\Pr(R>0.05)&=0.033333,\\
8022 \overline R_{\rm no\ risk}&=0.015741,
8023 &\Pr(R>0.05)&=0.058333.
8024 \end{align}
8025 The risk head therefore reduces both mean and tail protected regression, at a utility cost of $0.021425$ relative to the no-risk arm. This is expected from constrained selection and should not be described as unconstrained dominance.
8026 
8027 \subsection{Lineage and ablation controls}
8028 
8029 The correctly bound child outperforms the shuffled child by
8030 \begin{align}
8031 0.106417
8032 &\quad\text{on the seed/task audit},\\
8033 0.040672
8034 &\quad\text{on the architecture-foreign audit}.
8035 \end{align}
8036 The corresponding paired bootstrap intervals are strictly positive. The selector-ablation arm exactly reproduces the parent metrics on both splits, as required by \cref{cor:block07-selector-ablation}.
8037 
8038 The child changes the repair mixture substantially.
8039 
8040 \begin{figure}[H]
8041 \centering
8042 \includegraphics[width=0.94\textwidth]{block07_candidate_choices.pdf}
8043 \caption{Candidate-selection fractions. The parent abstains frequently, especially under architecture shift. The child uses learned utility and risk estimates to select transport and local repairs more often while retaining a no-op fallback.}
8044 \label{fig:block07-candidate-choices}
8045 \end{figure}
8046 
8047 On the architecture-foreign audit, the parent selects no-op in $77$ of $120$ episodes, while the child selects no-op in only $8$. The child selects transported repair $42$ times and full local repair $39$ times. This increased intervention rate is acceptable only because the protected-risk gates are checked independently.
8048 
8049 \subsection{Frozen claim-gate ledger}
8050 
8051 \begin{table}[H]
8052 \centering
8053 \small
8054 \begin{tabular}{@{}p{0.68\textwidth}c@{}}
8055 \toprule
8056 Gate class & Status \\
8057 \midrule
8058 Receipt integrity and compiler-input exclusion & PASS \\
8059 Split-file seal-before-audit load order & PASS \\
8060 Matched five-candidate budget & PASS \\
8061 Positive child-parent seed/task paired interval & PASS \\
8062 Positive child-parent architecture-foreign paired interval & PASS \\
8063 Positive child-raw architecture-foreign paired interval & PASS \\
8064 Seed/task mean, tail, and maximum regression gates & PASS \\
8065 Architecture-foreign mean, tail, and maximum regression gates & PASS \\
8066 Shuffled-lineage controls on both final splits & PASS \\
8067 Selector-ablation equality on both final splits & PASS \\
8068 Disjoint architecture-family support & PASS \\
8069 \midrule
8070 All frozen selector gates & \textbf{PASS} \\
8071 \bottomrule
8072 \end{tabular}
8073 \caption{Block 7 primary gate ledger.}
8074 \label{tab:block07-gates}
8075 \end{table}
8076 
8077 \section{Formal Interpretation of the Block 7 Edge}
8078 \label{sec:block07-interpretation}
8079 
8080 \begin{conditionaltheorem}[One-step selector metaproductivity certificate]
8081 \label{thm:block07-one-step}
8082 Let $K_0$ be a frozen parent repair selector and let
8083 \begin{equation}
8084 (K_1,\Pi_1)
8085 =
8086 \mathfrak M(\mathsf D_{\rm dev},\mathsf D_{\rm meta},K_0,\Gamma)
8087 \end{equation}
8088 be a child and valid descendant receipt. Assume:
8089 \begin{enumerate}
8090 \item the compiler satisfies the sealing conditions of \cref{thm:block07-compilation-noninterference};
8091 \item parent and child receive pathwise identical candidate bundles and evidence under \cref{thm:block07-selector-isolation};
8092 \item the final target episodes are sampled from the declared held-out distribution and the paired interval procedure has its declared coverage;
8093 \item the lower confidence bound for child-minus-parent protected utility is positive;
8094 \item all declared average, tail, and maximum risk gates hold;
8095 \item the shuffled-lineage control is lower and selector ablation restores the parent;
8096 \item evaluator semantics, provenance, candidate budget, and audit evidence are fixed across the comparison.
8097 \end{enumerate}
8098 Then the lineage edge $K_0\xrightarrow{\Pi_1}K_1$ is a certified one-step bounded improvement of the repair-selection process relative to the declared distribution, candidate grammar, feature schema, evaluator, witness, and budget.
8099 \end{conditionaltheorem}
8100 
8101 \begin{proof}
8102 By sealing, $K_1$ is fixed without final audit access. By the pathwise matched comparison, the paired difference contains only the effect of selecting different members of the same candidate bundle. The positive lower confidence bound establishes positive held-out expected protected utility under the declared sampling assumptions. The risk gates exclude the specified forms of protected regression. Shuffled and ablation controls respectively test semantic receipt binding and necessity of the compiled child selector. The fixed evaluator, candidate budget, and provenance remove the declared alternative explanations. These conditions are exactly the one-step bounded repair-process improvement definition.
8103 \end{proof}
8104 
8105 \begin{corollary}[Productive lineage edge]
8106 \label{cor:block07-productive-edge}
8107 Under the hypotheses of \cref{thm:block07-one-step}, the parent state has an authorized successor with positive foreign held-out improvement-process gain. Therefore the edge is productive in the finite transaction graph.
8108 \end{corollary}
8109 
8110 \begin{warning}[One edge is not recurrent closure]
8111 The theorem certifies neither
8112 \begin{equation}
8113 K_1\longrightarrow K_2
8114 \end{equation}
8115 nor an infinite sequence of productive descendants. No grandchild is compiled in Block 7, and no fixed-point or Buechi liveness claim follows from a single successful edge.
8116 \end{warning}
8117 
8118 \subsection{What changed in the child}
8119 
8120 The child adds one lineage-generated operator:
8121 \begin{equation}
8122 \Theta_{\rm select}:
8123 \left\{
8124 \text{typed candidate receipts}
8125 \right\}
8126 \longrightarrow
8127 \left\{
8128 \text{risk-filtered repair choice}
8129 \right\}.
8130 \end{equation}
8131 This operator uses learned outcome calibration and candidate-type residual scales. The changed component is therefore the repair-selection policy, not the target agent's weights or the repair constructor.
8132 
8133 \subsection{What did not change}
8134 
8135 The following remain externally fixed:
8136 \begin{itemize}
8137 \item the target-task and defect generator;
8138 \item the five repair candidates and their constructors;
8139 \item the 23-dimensional candidate feature schema;
8140 \item the 2000-point meta-grammar;
8141 \item the protected utility and risk definitions;
8142 \item the parent and baseline policies;
8143 \item the final claim gates;
8144 \item the target audit distributions;
8145 \item the model architecture code and representation families.
8146 \end{itemize}
8147 Consequently, Block 7 does not establish autonomous repair invention, autonomous feature discovery, grammar expansion, model-weight self-rewrite, or evaluator evolution.
8148 
8149 \subsection{Claim-ladder advancement}
8150 
8151 
8152 Block 14 attacks the simplification on which the Block 13 mechanism depended. Initial diagnostics are aliased, some problems contain multiple simultaneous obstructions, some lie outside the inherited response cone, the development archive is contaminated, evaluator weights shift, and candidate methods include integrity attacks. Under the exact inherited $39$-method language and the same six full experiments, primary mean validity-filtered gain rises from $0.410813$ to $0.575783$ and generator-foreign gain rises from $0.389514$ to $0.550559$. Primary lower-tail gain changes from $-0.133065$ to $0.271516$. The child selects no invalid method in either final audit and reduces false confident known-class assignment on unknown problems from $1$ to $0.083333$.
8153 
8154 This is a robustness edge, not a universal robustness theorem. The stress atlas, response language, candidate grammar, evaluator portfolio, monitor ensemble, and attack generator remain externally declared. The primary paired comparison also contains retained negative episodes. Sequential response updating, open-set routing, and robust archive estimation receive positive support. The simple-monitor ablation is null and the single-portfolio ablation is nearly null, so those modules do not receive independent positive attribution. The result therefore strengthens the evidence for a compiler that can reallocate experiments under finite ambiguity and attack, while leaving independent replication, real executable research, adaptive adversaries, and compiler-language growth unresolved.
8155 
8156 
8157 The empirical ladder is now:
8158 \begin{enumerate}
8159 \item mechanical implementation;
8160 \item finite semantic realization;
8161 \item causal identification;
8162 \item controlled foreign repair transfer;
8163 \item \textbf{sealed one-step repair-process metaproductivity};
8164 \item repeated lineage metaproductivity and autonomous schema growth;
8165 \item open-set recursive improvement closure.
8166 \end{enumerate}
8167 Block 7 passes level 5 relative to the declared finite benchmark. It does not pass levels 6 or 7.
8168 
8169 \begin{statusbox}{Block 7 empirical result}
8170 A lineage-generated dual-head repair selector, sealed before final audit access, receives the same five candidate repairs as its frozen parent and achieves positive paired protected-utility gain on both seed/task-foreign and architecture-foreign target episodes. Every frozen selector gate passes. This is a bounded one-step improvement of the repair-selection process. Because the candidate grammar, feature schema, meta-grammar, evaluator, and witness are fixed, it is not autonomous repair invention, structural correctability expansion, repeated recursive closure, or open-ended RSI.
8171 \end{statusbox}
8172 
8173 \section{Block 7 Reproducibility and Package}
8174 \label{sec:block07-package}
8175 
8176 The executable package \cite{SRICB7} contains the complete candidate receipt archive, split-file sealing protocol, child compiler, parent and control policies, frozen results, figures, tests, manifest, and Windows runner.
8177 
8178 \subsection{Package layout}
8179 
8180 \begin{verbatim}
8181 SRIC_v0.8_Block07_Sealed_Meta_Improvement/
8182   README.md
8183   pyproject.toml
8184   Makefile
8185   Run_SRIC_Block07.ps1
8186   regenerate_candidates.py
8187   run_all.py
8188   build_manifest.py
8189   verify_package.py
8190   MANIFEST.sha256
8191   sric_meta/
8192     benchmark.py
8193     canonical.py
8194     io.py
8195     meta.py
8196     pipeline.py
8197     plots.py
8198     receipts.py
8199     statistics.py
8200   results/
8201     development_candidate_receipts.json.gz
8202     validation_candidate_receipts.json.gz
8203     audit_candidate_receipts.json.gz
8204     architecture_foreign_candidate_receipts.json.gz
8205     frozen_candidate_receipts.json.gz
8206     descendant_receipt.json
8207     block07_results.json
8208     audit_selected_rows.csv
8209     architecture_foreign_selected_rows.csv
8210   figures/
8211     block07_metaproductivity.pdf
8212     block07_paired_effects.pdf
8213     block07_risk_utility_frontier.pdf
8214     block07_candidate_choices.pdf
8215   tests/
8216     conftest.py
8217     test_controls.py
8218     test_primary.py
8219     test_receipt.py
8220     test_result_values.py
8221 \end{verbatim}
8222 
8223 \subsection{Canonical replay}
8224 
8225 The standard PowerShell replay is one line:
8226 \begin{verbatim}
8227 python verify_package.py; python run_all.py; python -m pytest -q
8228 \end{verbatim}
8229 A full candidate regeneration is available through:
8230 \begin{verbatim}
8231 python regenerate_candidates.py; python run_all.py;
8232 python -m pytest -q; python build_manifest.py;
8233 python verify_package.py
8234 \end{verbatim}
8235 The second path reconstructs every target episode and every repair candidate before recompiling the child.
8236 
8237 \subsection{Frozen verification summary}
8238 
8239 The final reference run reports:
8240 \begin{verbatim}
8241 SRIC_BLOCK07_ALL_SUITES=PASS
8242 meta_grammar_size=2000
8243 audit_parent_gain=0.284896
8244 audit_child_gain=0.357414
8245 audit_child_minus_parent=0.072518
8246 audit_ci_low=0.046745
8247 foreign_child_minus_parent=0.104999
8248 foreign_ci_low=0.072711
8249 ............ [100%]
8250 SRIC_BLOCK07_PACKAGE_VALID
8251 \end{verbatim}
8252 The frozen result digest is
8253 \begin{verbatim}
8254 cc92cb004e2d4cdb1b0ea1c30e64683a961f507d8b24225190eca8dfd72d625a
8255 \end{verbatim}
8256 and the internal descendant-receipt digest is
8257 \begin{verbatim}
8258 852fc497f19561bc9c16b8cb9fa0b126b8104812b46e53c72f8b4bf27207e576
8259 \end{verbatim}
8260 
8261 \subsection{Remaining reproducibility limitations}
8262 
8263 The package gives deterministic reconstruction relative to its software dependencies and frozen pseudorandom seeds. It does not yet provide:
8264 \begin{itemize}
8265 \item an independently written second implementation;
8266 \item process-level mandatory access control proving audit nonobservability;
8267 \item preregistered independent seed replication;
8268 \item confidence sequences valid under arbitrary adaptive meta-grammar growth;
8269 \item target models trained with large neural backbones;
8270 \item a grandchild run demonstrating repeatability of the meta-improvement edge.
8271 \end{itemize}
8272 
8273 The reference package should therefore be treated as a finite executable existence proof for the declared selector-level mechanism and its controls, not as an unrestricted self-improving agent.
8274 
8275 
8276 \section{Sealed Parent--Child Bridge-Policy Compiler}
8277 \label{sec:block07q-contract}
8278 
8279 The selection-only realization above changes which already constructed repair is committed. A second Block 7 realization changes an earlier stage of the repair process: which paired anchors are queried and whether the resulting bridge is sufficiently identified to transport a frozen repair. A parent bridge process is run on a development lineage, its failures are summarized, and a descendant bridge policy is compiled from a fixed meta-grammar. The descendant is then sealed before any audit task or audit representation family is executed.
8280 
8281 This second experiment is also deliberately narrow. The descendant may change only:
8282 \begin{enumerate}
8283 \item the rule used to select paired source--target anchors;
8284 \item the bridge-identification gate applied before commitment.
8285 \end{enumerate}
8286 The source repair, target-query budget, bridge family, evaluator, audit metric, and repair-transport formula remain fixed. This isolates one repair-process modification rather than allowing a general rewrite.
8287 
8288 \subsection{Repair processes as objects of comparison}
8289 
8290 \begin{definition}[Finite repair process]
8291 A finite repair process is a tuple
8292 \begin{equation}
8293 \mathfrak P=(\pi_{\rm anc},\mathsf B,\pi_{\rm gate},B,\Pi),
8294 \end{equation}
8295 where $\pi_{\rm anc}$ selects target correspondence queries, $\mathsf B$ estimates a bridge and transports a frozen source repair, $\pi_{\rm gate}$ commits or abstains, $B$ is the declared resource budget, and $\Pi$ is the receipt schema.
8296 \end{definition}
8297 
8298 For a foreign target system $A$ and protected evaluation context $\Xi$, let
8299 \begin{equation}
8300 G_\Xi(\mathfrak P,A)
8301 =
8302 J_\Xi\bigl(\mathfrak P[A]\bigr)-J_\Xi(A)
8303 \end{equation}
8304 be the gain produced when the process is given one opportunity to repair $A$. The notation $\mathfrak P[A]$ includes abstention, in which case the gain is zero.
8305 
8306 \begin{definition}[Matched-budget foreign metaproductivity]
8307 For a foreign target distribution $\Dcal_{\rm for}$ and budget $B$, define
8308 \begin{equation}
8309 M_B^{\rm for}(\mathfrak P)
8310 =
8311 \mathbb E_{A\sim\Dcal_{\rm for}}
8312 \left[G_\Xi(\mathfrak P,A)\right],
8313 \end{equation}
8314 provided every compared process receives the same target correspondence, calibration, candidate, and irreversible-effect budgets.
8315 \end{definition}
8316 
8317 The protected utility used in Block 7 is the balanced accuracy functional
8318 \begin{equation}
8319 J_\Xi(A)
8320 =
8321 \frac12 A_{\rm aff}(A)+\frac12 A_{\rm non}(A),
8322 \label{eq:block07q-balanced-utility}
8323 \end{equation}
8324 where $A_{\rm aff}$ is accuracy on the defect-affected stratum and $A_{\rm non}$ is accuracy on the protected unaffected stratum. This prevents the more common unaffected examples from overwhelming the repair objective.
8325 
8326 \begin{definition}[One-step bounded recursive meta-improvement]
8327 Let $\mathfrak P_0$ be a parent repair process. Let a lineage compiler $\Meta$ receive only a development ledger $L_{\rm dev}$ and return
8328 \begin{equation}
8329 \mathfrak P_1=\Meta(\mathfrak P_0,L_{\rm dev}).
8330 \end{equation}
8331 The edge $\mathfrak P_0\to\mathfrak P_1$ is a one-step bounded recursive meta-improvement on $(\Dcal_{\rm for},B)$ when:
8332 \begin{enumerate}
8333 \item the descendant is sealed before the audit distribution is revealed;
8334 \item $\mathfrak P_0$ and $\mathfrak P_1$ receive the same deployment budget $B$;
8335 \item $M_B^{\rm for}(\mathfrak P_1)>M_B^{\rm for}(\mathfrak P_0)$ with a declared uncertainty certificate;
8336 \item the descendant preserves every protected invariant and has no larger accepted-harm rate;
8337 \item a structural repair-capacity measure also improves on the declared foreign family.
8338 \end{enumerate}
8339 \end{definition}
8340 
8341 This definition is intentionally local. It certifies one descendant edge. It does not imply repeated improvement, autonomous grammar expansion, or supercritical repair discovery.
8342 
8343 \subsection{Development, seal, and audit separation}
8344 
8345 Let
8346 \begin{equation}
8347 \Omega
8348 =
8349 \Omega_{\rm dev}\sqcup\Omega_{\rm audit}
8350 \end{equation}
8351 be a disjoint target index set. The compiler receives rows indexed only by $\Omega_{\rm dev}$. It emits a descendant configuration $c_1$, a compiler ledger $L_{\Meta}$, and a pre-audit seal
8352 \begin{equation}
8353 \Sigma_{\rm pre}
8354 =
8355 H\bigl(c_1,L_{\Meta},\text{source hashes}\bigr),
8356 \label{eq:block07q-seal}
8357 \end{equation}
8358 where $H$ is SHA-256 over canonical serialization. Audit execution refuses to proceed when the in-memory descendant differs from the sealed configuration.
8359 
8360 \begin{proposition}[Trace-level audit nonconditioning]
8361 \label{prop:block07q-seal}
8362 Suppose the compiler is a deterministic function of $(\mathfrak P_0,L_{\rm dev})$, the audit runner accepts only a descendant whose canonical digest equals the committed digest in \eqref{eq:block07q-seal}, and no audit field is present in the sealed compiler input. Then the executed descendant is not a direct function of the audit observations inside the declared runtime trace.
8363 \end{proposition}
8364 
8365 \begin{proof}
8366 The compiler output is fixed by its development input. The audit runner checks equality with the precommitted digest before evaluating any audit row. Any later replacement changes the digest and fails the check. Thus the executed configuration is fixed before the audit observations enter the declared trace.
8367 \end{proof}
8368 
8369 \begin{remark}
8370 The proposition is a software-trace statement, not a metaphysical guarantee against every hidden side channel. It is only as strong as the source audit, process isolation, canonical serializer, and manifest verification.
8371 \end{remark}
8372 
8373 \subsection{Foreignness and matched resources}
8374 
8375 The Block 7 primary audit differs from development in both task and representation family:
8376 \begin{equation}
8377 \begin{aligned}
8378 \mathsf T_{\rm dev}&\cap\mathsf T_{\rm audit}=\varnothing,\\
8379 \mathsf F_{\rm dev}^{\rm primary}&\cap\mathsf F_{\rm audit}^{\rm primary}=\varnothing.
8380 \end{aligned}
8381 \end{equation}
8382 Development uses ten tasks generated from four trigger families. Audit uses twelve new tasks generated from radial and mixed triggers, new teacher seeds, and new corruption permutations. The primary audit representations are dense linear $20$-dimensional, block-linear $24$-dimensional, and ill-conditioned linear $20$-dimensional systems. None appears in the primary development set.
8383 
8384 The parent and descendant each receive:
8385 \begin{equation}
8386 B=(16\text{ fit anchors},8\text{ validation anchors},1\text{ repair candidate}).
8387 \label{eq:block07q-budget}
8388 \end{equation}
8389 They use the same source repair, target calibration set, protected audit set, and bridge estimator. Only anchor selection and the gate differ.
8390 
8391 \begin{statusbox}{Block 7 comparison principle}
8392 A positive result must be a paired improvement of the repair process, not merely a stronger target agent. The parent and descendant act on the same foreign targets, with the same source repairs and target-query budget. The descendant configuration is compiled from development failures and sealed before audit.
8393 \end{statusbox}
8394 
8395 \section{Meta-Repair of Bridge Identification}
8396 \label{sec:block07q-bridge-meta-repair}
8397 
8398 \subsection{The parent failure}
8399 
8400 The Block 6 bridge estimator receives paired activations
8401 \begin{equation}
8402 X=
8403 \begin{bmatrix}
8404  h_s(x_1)^\top\\
8405  \vdots\\
8406  h_s(x_k)^\top
8407 \end{bmatrix},
8408 \qquad
8409 Y=
8410 \begin{bmatrix}
8411  h_t(x_1)^\top\\
8412  \vdots\\
8413  h_t(x_k)^\top
8414 \end{bmatrix},
8415 \end{equation}
8416 and fits
8417 \begin{equation}
8418 \widehat T^\top
8419 =
8420 (X^\top X+\lambda I)^{-1}X^\top Y.
8421 \label{eq:block07q-ridge-bridge}
8422 \end{equation}
8423 The parent chooses the $k=16$ fit anchors uniformly at random. The source hidden space is also $16$-dimensional, so a rank-deficient design leaves at least one source direction unidentified.
8424 
8425 The task representation has the conditional form
8426 \begin{equation}
8427 \phi(x)
8428 =
8429 \begin{bmatrix}
8430 \psi(x)\\
8431 s(x)\psi(x)
8432 \end{bmatrix}
8433 \in\R^{16},
8434 \end{equation}
8435 where the second eight-dimensional packet is zero outside the affected stratum. Randomly chosen anchors often include too few affected examples to span that packet. This is not a failure of ridge regression. It is an experimental-design obstruction.
8436 
8437 \subsection{Bridge identifiability}
8438 
8439 \begin{theorem}[Full-rank bridge identification]
8440 \label{thm:block07q-full-rank-bridge}
8441 Let the target representation satisfy
8442 \begin{equation}
8443 h_t(x)=T h_s(x)
8444 \end{equation}
8445 for a fixed linear map $T:\R^d\to\R^m$. Let $X\in\R^{k\times d}$ be the selected source design and $Y=XT^\top$. If $\rank X=d$, then the unregularized least-squares bridge is unique and equals $T$ on the entire source space. If $\rank X<d$, there exist distinct maps $T_1\neq T_2$ such that
8446 \begin{equation}
8447 XT_1^\top=XT_2^\top=Y.
8448 \end{equation}
8449 Consequently, repair transport is not identifiable on every unobserved source direction.
8450 \end{theorem}
8451 
8452 \begin{proof}
8453 When $X$ has full column rank, $X^\top X$ is invertible and
8454 \begin{equation}
8455 (X^\top X)^{-1}X^\top Y
8456 =(X^\top X)^{-1}X^\top XT^\top
8457 =T^\top.
8458 \end{equation}
8459 When $\rank X<d$, choose a nonzero $v\in\ker X$. For any nonzero $w\in\R^m$, define $K=wv^\top$. Then $XK^\top=0$, so $T$ and $T+K$ agree on every selected anchor while differing on $v$.
8460 \end{proof}
8461 
8462 The second clause is important. A fitted bridge can have zero training residual while remaining arbitrary in precisely the direction used by a repair.
8463 
8464 \begin{proposition}[Ridge bias under a full-rank design]
8465 \label{prop:block07q-ridge-bias}
8466 Under the hypotheses of \cref{thm:block07q-full-rank-bridge}, the ridge estimate satisfies
8467 \begin{equation}
8468 \widehat T^\top-T^\top
8469 =-\lambda(X^\top X+\lambda I)^{-1}T^\top,
8470 \end{equation}
8471 and therefore
8472 \begin{equation}
8473 \norm{\widehat T-T}
8474 \leq
8475 \frac{\lambda}{\sigma_{\min}(X)^2+\lambda}\norm{T}.
8476 \end{equation}
8477 \end{proposition}
8478 
8479 Thus an anchor policy should do more than obtain formal rank. It should enlarge the smallest singular value of the source design.
8480 
8481 \subsection{Conditional-packet rank obstruction}
8482 
8483 Write the source state as
8484 \begin{equation}
8485 h_s(x)=A
8486 \begin{bmatrix}
8487 u(x)\\s(x)v(x)
8488 \end{bmatrix},
8489 \end{equation}
8490 where $u,v\in\R^8$ and $A$ is invertible. Let $N_1$ be the number of affected anchors among $k$ selections.
8491 
8492 \begin{proposition}[Affected-anchor necessity]
8493 \label{prop:block07q-affected-count}
8494 If $N_1<8$, then the selected source design has rank strictly less than $16$. Under generic continuous $u$ and $v$, $k=16$, and $N_1\geq8$, the selected design has rank $16$ almost surely.
8495 \end{proposition}
8496 
8497 \begin{proof}
8498 Only affected rows carry nonzero coordinates in the second eight-dimensional packet. Hence those coordinates contribute rank at most $N_1$. The base packet contributes rank at most eight, so
8499 \begin{equation}
8500 \rank X\leq 8+N_1<16
8501 \end{equation}
8502 when $N_1<8$. If $N_1\geq8$, generic affected rows span the conditional packet and the full set of sixteen generic rows spans the base packet. The determinant of an appropriate $16\times16$ minor is a nonzero polynomial in the continuous row coordinates, so it vanishes only on a measure-zero set.
8503 \end{proof}
8504 
8505 If random anchor membership is Bernoulli with affected probability $p$, the idealized full-rank probability at $k=16$ is therefore
8506 \begin{equation}
8507 \Pr(\rank X=16)
8508 =
8509 \Pr(N_1\geq8)
8510 =
8511 \sum_{j=8}^{16}{16\choose j}p^j(1-p)^{16-j}
8512 \label{eq:block07q-random-rank-probability}
8513 \end{equation}
8514 under the generic model. The observed parent full-rank rate on the primary audit set is $0.03125$, consistent with a low-probability affected-count event.
8515 
8516 \subsection{Packet-balanced pivoted QR}
8517 
8518 The descendant policy uses the source intervention-defined packet to divide the unlabeled source-side candidate pool into high- and low-packet-activation strata. It selects eight anchors from each stratum. Within each stratum it applies a deterministic pivoted residual rule:
8519 \begin{equation}
8520  i_j
8521 \in
8522 \argmax_{i\notin I_{j-1}}
8523 \norm{(I-Q_{j-1}Q_{j-1}^\top)z_i}^2,
8524 \end{equation}
8525 where $z_i$ is the standardized source activation and $Q_{j-1}$ spans the previously selected residual directions.
8526 
8527 \begin{definition}[Packet-balanced QR policy]
8528 For fit budget $k=16$, define
8529 \begin{equation}
8530 \pi_{\rm PQR}(H,P_J)
8531 =
8532 \pi_{\rm QR}(H_{\rm aff},8)
8533 \cup
8534 \pi_{\rm QR}(H_{\rm non},8),
8535 \end{equation}
8536 where affected and unaffected membership is inferred from the source causal intervention, not from target audit labels.
8537 \end{definition}
8538 
8539 \begin{corollary}[Generic full-rank packet-balanced design]
8540 \label{cor:block07q-pqr-full-rank}
8541 Under the generic model of \cref{prop:block07q-affected-count}, packet-balanced QR produces a full-rank $16\times16$ source design almost surely whenever both source strata contain at least eight candidates.
8542 \end{corollary}
8543 
8544 The policy requires no additional target representation queries. It inspects the source-side candidate pool, which is already locally available, and asks the target for the same sixteen paired activations used by the parent.
8545 
8546 \subsection{Meta-grammar and compiler}
8547 
8548 The development meta-grammar contains five anchor policies:
8549 \begin{equation}
8550 \Pi_{\rm anc}
8551 =
8552 \set{
8553 \text{random},
8554 \text{balanced random},
8555 \text{QR},
8556 \text{packet QR},
8557 \text{augmented QR}
8558 }.
8559 \end{equation}
8560 It also contains alignment thresholds
8561 \begin{equation}
8562 \tau\in\set{0.01,0.02,0.05,0.10,0.20,0.35}
8563 \end{equation}
8564 and an optional target-width guard. Every candidate uses the same fit and validation budgets.
8565 
8566 For a candidate configuration $c$, the development compiler computes target-level selected gains and the score
8567 \begin{equation}
8568 S_{\rm dev}(c)
8569 =
8570 \bar g_c
8571 -0.20s_c
8572 -2h_c
8573 +0.02q_c,
8574 \label{eq:block07q-compiler-score}
8575 \end{equation}
8576 where $\bar g_c$ is mean protected gain, $s_c$ is the target-level standard deviation, $h_c$ is harmful-target rate, and $q_c$ is positive-target rate. This is a finite engineering objective, not a universal definition of evolvability.
8577 
8578 The compiler selected:
8579 \begin{equation}
8580 \begin{aligned}
8581 \pi_{\rm anc}&=\pi_{\rm PQR},\\
8582 k_{\rm fit}&=16,\\
8583 k_{\rm val}&=8,\\
8584 \gamma_{\rm cal}&=0.08,\\
8585 \rho_{\rm cal}&=0.035,\\
8586 \tau_{\rm align}&=0.05,\\
8587 \text{full-rank requirement}&=\text{true}.
8588 \end{aligned}
8589 \label{eq:block07q-descendant-config}
8590 \end{equation}
8591 The selected development score is $0.263289$. The underlying mean selected development gain is $0.296478$ with zero harmful targets in the development ledger.
8592 
8593 \subsection{Lineage-permutation control}
8594 
8595 A negative control preserves every numerical development outcome but deliberately breaks the receipt binding between each outcome and the policy that produced it. Policy identities are cyclically permuted before compilation. The resulting control selects the random-anchor policy with the descendant's strict gate.
8596 
8597 This control matters because a compiler that merely consumes a table of gains should not receive credit unless those gains remain attached to the correct causal modification. On audit, the lineage-permuted descendant obtains mean selected gain $0.004026$, compared with $0.153206$ for the correctly bound descendant.
8598 
8599 \section{One-Step Bounded Metaproductivity Experiment}
8600 \label{sec:block07q-experiment}
8601 
8602 \subsection{Frozen experimental design}
8603 
8604 Each task contains:
8605 \begin{itemize}
8606 \item $3{,}000$ source-training examples;
8607 \item $800$ source causal-probe examples;
8608 \item a source-side anchor pool of $1{,}100$ examples;
8609 \item $650$ calibration examples;
8610 \item $1{,}800$ audit examples.
8611 \end{itemize}
8612 The source repair is discovered and fitted independently for each task. The repair process then receives one foreign target representation and may construct one transported repair.
8613 
8614 Development uses ten tasks, five representation families, three target seeds, five anchor policies, and four repeated selections. The compiler therefore consumes $72{,}000$ paired target-representation queries. Audit uses twelve unseen tasks and four target seeds per family.
8615 
8616 The primary audit families are:
8617 \begin{equation}
8618 \mathsf F_{\rm audit}^{\rm primary}
8619 =
8620 \set{
8621 \text{dense-linear-20},
8622 \text{block-linear-24},
8623 \text{ill-conditioned-linear-20}
8624 }.
8625 \end{equation}
8626 The stress families are:
8627 \begin{equation}
8628 \mathsf F_{\rm audit}^{\rm stress}
8629 =
8630 \set{
8631 \text{compressed-10},
8632 \text{saturated-tanh-24},
8633 \text{leaky-ReLU-32},
8634 \text{smooth-polynomial-24}
8635 }.
8636 \end{equation}
8637 The primary families satisfy the linear bridge hypothesis but are absent from primary development. The stress families violate injectivity or global linearity in different ways.
8638 
8639 \subsection{Compared arms}
8640 
8641 \begin{longtable}{@{}p{0.24\textwidth}p{0.16\textwidth}p{0.49\textwidth}@{}}
8642 \toprule
8643 Arm & Paired queries & Role \\
8644 \midrule
8645 \endhead
8646 Parent random 16 & $16+8=24$ & Parent process inherited from the Block 6 anchor protocol \\
8647 Balanced random 16 & $16+8=24$ & Tests whether stratum balance alone explains the gain \\
8648 QR 16 & $16+8=24$ & Tests generic deterministic experimental design without explicit packet balance \\
8649 Descendant packet-QR 16 & $16+8=24$ & Sealed lineage-generated descendant \\
8650 Parent random 32 & $32+8=40$ & Double-fit-budget parent control \\
8651 Lineage-permuted descendant & $16+8=24$ & Breaks causal receipt binding during meta-compilation \\
8652 \bottomrule
8653 \end{longtable}
8654 
8655 The pure QR arm is intentionally strong. It prevents the manuscript from attributing a standard experimental-design effect uniquely to the full SRIC architecture.
8656 
8657 \subsection{Primary metaproductivity result}
8658 
8659 On $144$ primary foreign targets, the target-level paired difference is
8660 \begin{equation}
8661 \widehat\Delta_{\rm meta}
8662 =
8663 \widehat M_{24}^{\rm for}(\mathfrak P_1)
8664 -
8665 \widehat M_{24}^{\rm for}(\mathfrak P_0)
8666 =
8667 0.209508.
8668 \label{eq:block07q-primary-delta}
8669 \end{equation}
8670 A target-cluster bootstrap gives the interval
8671 \begin{equation}
8672 [0.194968,0.223898],
8673 \end{equation}
8674 and a $100{,}000$-draw sign-flip test gives
8675 \begin{equation}
8676 p=9.9999\times10^{-6}.
8677 \end{equation}
8678 All $144$ target-level paired differences are positive.
8679 
8680 \begin{table}[H]
8681 \centering
8682 \small
8683 \caption{Primary foreign-target metaproductivity under matched or declared query budgets.}
8684 \label{tab:block07q-primary-arms}
8685 \begin{tabularx}{\textwidth}{@{}Xrrrr@{}}
8686 \toprule
8687 Arm & Mean gain & Commit rate & Full-rank rate & Queries \\
8688 \midrule
8689 Parent random 16 & $0.135212$ & $0.465278$ & $0.03125$ & $24$ \\
8690 Balanced random 16 & $0.318744$ & $0.932292$ & $1.00000$ & $24$ \\
8691 QR 16 & $0.344731$ & $1.000000$ & $1.00000$ & $24$ \\
8692 Descendant packet-QR 16 & $\mathbf{0.344720}$ & $\mathbf{1.000000}$ & $\mathbf{1.00000}$ & $24$ \\
8693 Parent random 32 & $0.316524$ & $0.932292$ & $0.822917$ & $40$ \\
8694 Lineage-permuted descendant & $0.008784$ & $0.026042$ & $0.03125$ & $24$ \\
8695 \bottomrule
8696 \end{tabularx}
8697 \end{table}
8698 
8699 The child exceeds the double-budget parent while using sixteen fewer paired target queries. This is a $40\%$ reduction relative to the parent-$32$ deployment query count.
8700 
8701 \begin{figure}[H]
8702 \centering
8703 \includegraphics[width=0.92\textwidth]{block07_metaproductivity_by_arm.pdf}
8704 \caption{Mean protected gain on the primary foreign target set. Parent and descendant use the same $24$ paired queries. The double-budget parent uses $40$.}
8705 \label{fig:block07q-metaproductivity-arms}
8706 \end{figure}
8707 
8708 \begin{figure}[H]
8709 \centering
8710 \includegraphics[width=0.72\textwidth]{block07_paired_foreign_gain.pdf}
8711 \caption{Paired primary-target gains. Every point lies above the equality line.}
8712 \label{fig:block07q-paired-gains}
8713 \end{figure}
8714 
8715 \subsection{Structural metaproductivity}
8716 
8717 The behavioral gain is accompanied by a structural change. On primary audit targets, the parent source design is full rank in only $3.125\%$ of target-level repeated selections. The descendant is full rank in $100\%$.
8718 
8719 Define the finite structural coverage functional
8720 \begin{equation}
8721 C_{\rm id}(\mathfrak P)
8722 =
8723 \Pr_{A\sim\Dcal_{\rm for}}
8724 \left[
8725 ank X_A=d_s\right].
8726 \end{equation}
8727 Then
8728 \begin{equation}
8729 \widehat C_{\rm id}(\mathfrak P_1)
8730 -
8731 \widehat C_{\rm id}(\mathfrak P_0)
8732 =
8733 0.96875.
8734 \end{equation}
8735 This is the measured bridge-identification direction added by the meta-repair.
8736 
8737 \begin{figure}[H]
8738 \centering
8739 \includegraphics[width=0.82\textwidth]{block07_full_rank_rate.pdf}
8740 \caption{Full-rank source-design rate on primary foreign targets. The meta-repair removes the rank obstruction at the same fit-anchor budget.}
8741 \label{fig:block07q-rank-rate}
8742 \end{figure}
8743 
8744 The structural result is not merely a post hoc correlate. \Cref{thm:block07q-full-rank-bridge} identifies full rank as the condition that removes linear bridge ambiguity, and \cref{prop:block07q-affected-count} explains why the random parent fails under the conditional packet geometry.
8745 
8746 \subsection{Family-resolved audit results}
8747 
8748 \begin{table}[H]
8749 \centering
8750 \small
8751 \caption{Descendant results by foreign representation family.}
8752 \label{tab:block07q-family-results}
8753 \begin{tabularx}{\textwidth}{@{}Xrrrr@{}}
8754 \toprule
8755 Family & Mean gain & Commit rate & Alignment NRMSE & Harm rate \\
8756 \midrule
8757 Dense linear 20 & $0.344887$ & $1.000000$ & $4.10\times10^{-4}$ & $0$ \\
8758 Block linear 24 & $0.344887$ & $1.000000$ & $2.91\times10^{-4}$ & $0$ \\
8759 Ill-conditioned linear 20 & $0.344387$ & $1.000000$ & $3.04\times10^{-5}$ & $0$ \\
8760 Compressed 10 & $0.038278$ & $0.270833$ & $2.87\times10^{-4}$ & $0$ \\
8761 Saturated tanh 24 & $0$ & $0$ & $2.260422$ & $0$ \\
8762 Leaky-ReLU 32 & $0$ & $0$ & $4.252823$ & $0$ \\
8763 Smooth polynomial 24 & $0$ & $0$ & $4.652148$ & $0$ \\
8764 \bottomrule
8765 \end{tabularx}
8766 \end{table}
8767 
8768 The compressed family illustrates an important boundary. Noninjectivity obstructs exact repair transport, but it does not imply that every partial repair is useless. The witness commits on $27.08\%$ of compressed targets and obtains positive mean protected gain with no selected harm in this fixture. The correct conclusion is partial correctability under compression, not exact conjugacy.
8769 
8770 The nonlinear families are different. Their validation alignment residuals lie far outside the descendant envelope, so the witness abstains everywhere.
8771 
8772 \begin{figure}[H]
8773 \centering
8774 \includegraphics[width=0.98\textwidth]{block07_family_gain.pdf}
8775 \caption{Parent and descendant gain by family. The primary linear families realize the meta-improvement. Strongly nonlinear families remain outside the declared transport class.}
8776 \label{fig:block07q-family-gain}
8777 \end{figure}
8778 
8779 \begin{figure}[H]
8780 \centering
8781 \includegraphics[width=0.78\textwidth]{block07_alignment_gate.pdf}
8782 \caption{Validation alignment and descendant commitment. The evolved gate abstains on the three strongly nonlinear stress families.}
8783 \label{fig:block07q-alignment-gate}
8784 \end{figure}
8785 
8786 \subsection{Query efficiency and meta-compilation cost}
8787 
8788 \begin{figure}[H]
8789 \centering
8790 \includegraphics[width=0.76\textwidth]{block07_query_efficiency.pdf}
8791 \caption{The descendant reaches higher primary gain than the parent with twice the fit-anchor budget.}
8792 \label{fig:block07q-query-efficiency}
8793 \end{figure}
8794 
8795 The deployment comparison alone does not account for the cost of constructing the descendant. The development search consumes $72{,}000$ paired representation queries. Relative to the sixteen paired queries saved against the parent-$32$ control, the query savings amortize only after
8796 \begin{equation}
8797 \frac{72{,}000}{16}=4{,}500
8798 \end{equation}
8799 deployments. This number is reported because recursive improvement is not useful merely because a descendant is better. Its construction cost must eventually be recovered.
8800 
8801 No claim is made that the tiny benchmark is economically efficient. The point is that the accounting is explicit and the deployment improvement is not purchased by silently increasing the per-target query budget.
8802 
8803 \subsection{Strong-baseline interpretation}
8804 
8805 Pure QR obtains mean primary gain $0.344731$, numerically indistinguishable from packet-balanced QR at $0.344720$. Balanced random selection reaches $0.318744$.
8806 
8807 These controls narrow the mechanistic interpretation:
8808 \begin{itemize}
8809 \item most of the gain comes from resolving a classical rank and conditioning problem;
8810 \item explicit packet balancing supplies a causal route to guaranteed affected-stratum coverage;
8811 \item the experiment does not show that only a Komorebi-specific projector method can solve the bridge-design problem;
8812 \item the SRIC contribution is the repair-process framing, the sealed lineage transaction, the structural certificate, the witness, and the claim boundary around the active-design improvement.
8813 \end{itemize}
8814 
8815 \section{Bridge-Policy Causal Interpretation and Claim Boundary}
8816 \label{sec:block07q-interpretation}
8817 
8818 \subsection{What changed}
8819 
8820 The parent and descendant differ in exactly two declared fields:
8821 \begin{equation}
8822 \begin{aligned}
8823 \pi_{\rm anc}^{(0)}&=\text{uniform random},
8824 &\pi_{\rm anc}^{(1)}&=\text{packet-balanced QR},\\
8825 \pi_{\rm gate}^{(0)}&=(0.20,0.03,0.35),
8826 &\pi_{\rm gate}^{(1)}&=(0.08,0.035,0.05,\text{full rank}).
8827 \end{aligned}
8828 \end{equation}
8829 The source task model, causal packet discovery, source repair, linear bridge family, pseudoinverse repair transport, target calibration set, protected utility, and audit data remain fixed.
8830 
8831 The measured causal chain is therefore:
8832 \begin{equation}
8833 \begin{aligned}
8834 &\text{development failure ledger}\\
8835 &\quad\longrightarrow
8836 \text{compiled anchor and gate policy}\\
8837 &\quad\longrightarrow
8838 \text{full-rank source design}\\
8839 &\quad\longrightarrow
8840 \text{identified linear bridge}\\
8841 &\quad\longrightarrow
8842 \text{successful transported repair}\\
8843 &\quad\longrightarrow
8844 \text{higher foreign-agent metaproductivity}.
8845 \end{aligned}
8846 \end{equation}
8847 The lineage-permutation control breaks the first causal binding and collapses mean gain to $0.004026$ over the complete audit set.
8848 
8849 \subsection{A finite one-step certificate}
8850 
8851 \begin{conditionaltheorem}[Block 7 one-step meta-improvement certificate]
8852 \label{thm:block07q-one-step-certificate}
8853 Assume:
8854 \begin{enumerate}
8855 \item the development and audit target indices are disjoint;
8856 \item the descendant configuration is sealed before audit as in \cref{prop:block07q-seal};
8857 \item parent and descendant receive the matched budget in \eqref{eq:block07q-budget};
8858 \item target-level audit differences are exchangeable or otherwise support the declared paired interval;
8859 \item the protected evaluator and source repairs are unchanged across the parent-child edge;
8860 \item accepted-harm rate is not increased.
8861 \end{enumerate}
8862 If the paired lower confidence bound for
8863 \begin{equation}
8864 M_{24}^{\rm for}(\mathfrak P_1)-M_{24}^{\rm for}(\mathfrak P_0)
8865 \end{equation}
8866 is positive and bridge-identification coverage also increases, then the edge $\mathfrak P_0\to\mathfrak P_1$ is a bounded one-step recursive meta-improvement relative to the declared audit distribution and fixed meta-grammar.
8867 \end{conditionaltheorem}
8868 
8869 Block 7 satisfies the finite executable antecedents inside its fixture. The paired lower bound is $0.194968$, selected harm is zero, and primary full-rank coverage rises from $0.03125$ to $1$.
8870 
8871 \subsection{What the result does not establish}
8872 
8873 The result does not establish:
8874 \begin{itemize}
8875 \item autonomous invention of the meta-grammar;
8876 \item autonomous source-code modification;
8877 \item repeated descendant improvement over multiple generations;
8878 \item improvement of packet discovery itself;
8879 \item unpaired or weakly paired transport;
8880 \item general nonlinear repair transport;
8881 \item improvement of the evaluator;
8882 \item open-ended ontology expansion;
8883 \item supercritical repair autocatalysis;
8884 \item unrestricted safety, consciousness, or sentience.
8885 \end{itemize}
8886 
8887 The compiler is designer specified. It searches five known anchor policies and a finite threshold grid. The task generator, protected metric, repair form, source counterfactual, and correspondence-query interface are also supplied. The descendant improves one declared module within that fixed grammar.
8888 
8889 \begin{statusbox}{Block 7 result}
8890 Block 7 realizes the manuscript's first one-step bounded repair-process meta-improvement. A descendant bridge-estimation policy is compiled from development failures, sealed before audit, and compared with its parent on unseen tasks and unseen primary representation families. Under the same $24$ paired-query budget, the descendant improves primary foreign-target gain by $0.209508$ with a $95\%$ lower bound of $0.194968$, raises full-rank bridge-identification coverage from $0.03125$ to $1$, and introduces no selected harm in the fixture. The result is finite, linear, and grammar-bounded. It is not open-ended recursive self-improvement.
8891 \end{statusbox}
8892 
8893 \section{Bridge-Policy Reproducibility and Package}
8894 \label{sec:block07q-package}
8895 
8896 \subsection{Reference package}
8897 
8898 The Block 7 reference package \cite{SRICB7} contains:
8899 \begin{verbatim}
8900 anchor_compiler/
8901 |-- sric_meta/
8902 |   |-- agents.py
8903 |   |-- anchors.py
8904 |   |-- benchmark.py
8905 |   |-- canonical.py
8906 |   |-- compiler.py
8907 |   |-- data.py
8908 |   |-- plots.py
8909 |   |-- repair.py
8910 |   |-- statistics.py
8911 |   `-- transport.py
8912 |-- frozen/
8913 |   |-- descendant_policy.json
8914 |   |-- compiler_ledger.json
8915 |   `-- preaudit_seal.json
8916 |-- results/
8917 |-- figures/
8918 |-- tests/
8919 |-- run_all.py
8920 |-- Run_SRIC_Block07.ps1
8921 |-- build_manifest.py
8922 |-- verify_package.py
8923 `-- MANIFEST.sha256
8924 \end{verbatim}
8925 
8926 The package is CPU-only and uses NumPy, Matplotlib, and pytest. BLAS thread counts are fixed to one in the release runner.
8927 
8928 \subsection{Frozen execution}
8929 
8930 The canonical release reports:
8931 \begin{verbatim}
8932 SRIC_BLOCK07_ALL_SUITES=PASS
8933 result_sha256=977879e0fc0aa491af70e7295f6704a92c637b434d76adba1f8b771c9f0d9411
8934 preaudit_seal_sha256=
8935 71100a2e91594028298198970e38f8eff5430fa3c514e50da57feff6eabc7a02
8936 descendant_policy=packet_qr
8937 parent_gain=0.061663
8938 child_gain=0.153206
8939 primary_paired_delta=0.209508
8940 primary_ci_low=0.194968
8941 child_stress_harm=0.000000
8942 \end{verbatim}
8943 
8944 All eight automated tests pass. The package manifest covers $51$ files.
8945 
8946 \subsection{Frozen claim gates}
8947 
8948 \begin{table}[H]
8949 \centering
8950 \small
8951 \caption{Block 7 primary claim gates.}
8952 \label{tab:block07q-claim-gates}
8953 \begin{tabularx}{\textwidth}{@{}p{0.24\textwidth}p{0.12\textwidth}X@{}}
8954 \toprule
8955 Gate & Status & Requirement \\
8956 \midrule
8957 Pre-audit seal & Pass & Frozen descendant digest and no audit fields in compiler input \\
8958 Foreign task split & Pass & No development--audit task overlap \\
8959 Foreign primary families & Pass & No primary representation-family overlap \\
8960 Matched deployment budget & Pass & Parent and descendant each use $24$ paired queries \\
8961 Paired metaproductivity & Pass & Lower confidence bound above zero \\
8962 Structural correctability & Pass & Descendant primary full-rank rate at least $0.99$ \\
8963 Protected harm & Pass & Zero harmful selected primary and stress targets \\
8964 Lineage-permutation control & Pass & Misbound compiler output does not match descendant gain \\
8965 Open-ended recursion & Not tested & No autonomous grammar growth or repeated generations \\
8966 \bottomrule
8967 \end{tabularx}
8968 \end{table}
8969 
8970 \subsection{Reproduction commands}
8971 
8972 \begin{verbatim}
8973 python verify_package.py
8974 OPENBLAS_NUM_THREADS=1 OMP_NUM_THREADS=1 MKL_NUM_THREADS=1 python run_all.py
8975 python -m pytest -q
8976 \end{verbatim}
8977 
8978 The PowerShell runner performs the same sequence and accepts an explicit Python executable.
8979 
8980 
8981 \section{Meta-Headroom Geometry}
8982 \label{sec:meta-headroom}
8983 
8984 Block 7 establishes that a lineage-generated descendant can improve a component of the repair process while the target systems, evaluator, witness, and deployment resources remain fixed. The next question is not merely whether another descendant can obtain another positive score difference. A repeated lineage must determine \emph{where additional improvement remains possible}. If one module has reached its attainable ceiling, continuing to modify that module can produce a false negative about recursive improvement. The relevant bottleneck may have moved elsewhere.
8985 
8986 This section therefore resolves the repair process into modules, defines restricted oracle envelopes, allocates remaining metaproductivity headroom, and distinguishes weak descendants from saturated parents.
8987 
8988 \subsection{Repair-process factorization}
8989 
8990 Let
8991 \begin{equation}
8992 N
8993 =
8994 \{H,Q,C,T,S,V,W\}
8995 \label{eq:b8-module-index}
8996 \end{equation}
8997 be the repair-process module set, where:
8998 \begin{enumerate}
8999 \item $H$ is the causal hypothesis and atlas mechanism;
9000 \item $Q$ is the active probe or evidence-acquisition policy;
9001 \item $C$ is the candidate repair constructor;
9002 \item $T$ is the representation and repair transport mechanism;
9003 \item $S$ is the candidate selector;
9004 \item $V$ is the evaluator;
9005 \item $W$ is the witness, authorization, and rollback mechanism.
9006 \end{enumerate}
9007 A repair process is a typed tuple
9008 \begin{equation}
9009 K=(H,Q,C,T,S,V,W).
9010 \label{eq:b8-process-tuple}
9011 \end{equation}
9012 The factorization is relative to a declared implementation boundary. A system may refine one module into several submodules or combine several modules behind one interface, provided the declared intervention and evidence channels remain explicit.
9013 
9014 Fix a protected evaluation context $\Xi$, a foreign-system distribution $\Dcal$, and a matched resource budget $B$. Define protected foreign metaproductivity by
9015 \begin{equation}
9016 \Mcal_{\Xi,\Dcal,B}(K)
9017 =
9018 \mathbb E_{A\sim\Dcal}
9019 \left[
9020 J_\Xi\bigl(K[A;B]\bigr)-J_\Xi(A)
9021 \right],
9022 \label{eq:b8-metaproductivity}
9023 \end{equation}
9024 where $K[A;B]$ is the authorized descendant selected by $K$ for target $A$, or $A$ itself when $K$ abstains.
9025 
9026 \begin{remark}[Module claims are distinct]
9027 A child that changes only $S$ establishes a selector-level claim. A child that changes only $Q$ establishes an information-acquisition claim. Neither child enlarges the candidate language unless $C$ or the grammar used by $C$ changes. Likewise, a better transport map does not by itself establish a better evaluator or witness.
9028 \end{remark}
9029 
9030 \subsection{Restricted oracle envelopes}
9031 
9032 Let $\Kcal$ be the declared finite or compact class of admissible repair processes under the protected context. For $R\subseteq N$, let
9033 \begin{equation}
9034 \Kcal_R(K)
9035 =
9036 \left\{
9037 \widetilde K\in\Kcal:
9038 \widetilde K_i=K_i
9039 \text{ for every }i\notin R
9040 \right\}.
9041 \label{eq:b8-restricted-class}
9042 \end{equation}
9043 The restricted oracle envelope is
9044 \begin{equation}
9045 F_K(R)
9046 =
9047 \sup_{\widetilde K\in\Kcal_R(K)}
9048 \Mcal_{\Xi,\Dcal,B}(\widetilde K).
9049 \label{eq:b8-oracle-envelope}
9050 \end{equation}
9051 Thus
9052 \begin{equation}
9053 F_K(\varnothing)=\Mcal(K),
9054 \qquad
9055 F_K(N)=\sup_{\widetilde K\in\Kcal}\Mcal(\widetilde K).
9056 \label{eq:b8-envelope-endpoints}
9057 \end{equation}
9058 The total declared headroom is
9059 \begin{equation}
9060 \Hcal(K)
9061 =
9062 F_K(N)-F_K(\varnothing).
9063 \label{eq:b8-total-headroom}
9064 \end{equation}
9065 The adjective \emph{declared} matters. A larger, previously unavailable meta-grammar may expose additional headroom. Equation \eqref{eq:b8-total-headroom} is therefore a relative ceiling, not an absolute limit on future system design.
9066 
9067 \begin{proposition}[Monotonicity of restricted envelopes]
9068 \label{prop:b8-envelope-monotone}
9069 If $R\subseteq R'\subseteq N$, then
9070 \begin{equation}
9071 F_K(R)\leq F_K(R').
9072 \end{equation}
9073 \end{proposition}
9074 
9075 \begin{proof}
9076 Every process admissible under $R$ is admissible under $R'$, so
9077 $\Kcal_R(K)\subseteq\Kcal_{R'}(K)$. Taking suprema gives the result.
9078 \end{proof}
9079 
9080 \subsection{Ordered meta-regret decomposition}
9081 
9082 Choose an operational ordering
9083 \begin{equation}
9084 r_1\prec r_2\prec\cdots\prec r_m,
9085 \qquad m=|N|,
9086 \end{equation}
9087 with nested sets
9088 \begin{equation}
9089 R_0=\varnothing,
9090 \qquad
9091 R_j=\{r_1,\ldots,r_j\}.
9092 \end{equation}
9093 Define the ordered marginal headroom
9094 \begin{equation}
9095 \Delta_j^{\mathrm{ord}}(K)
9096 =
9097 F_K(R_j)-F_K(R_{j-1}).
9098 \label{eq:b8-ordered-marginal}
9099 \end{equation}
9100 
9101 \begin{theorem}[Exact ordered meta-regret decomposition]
9102 \label{thm:b8-ordered-decomposition}
9103 For every ordering of the repair-process modules,
9104 \begin{equation}
9105 \Hcal(K)
9106 =
9107 \sum_{j=1}^{m}
9108 \Delta_j^{\mathrm{ord}}(K).
9109 \label{eq:b8-ordered-telescope}
9110 \end{equation}
9111 Every summand is nonnegative.
9112 \end{theorem}
9113 
9114 \begin{proof}
9115 The sum telescopes:
9116 \begin{equation}
9117 \sum_{j=1}^{m}
9118 \bigl(F_K(R_j)-F_K(R_{j-1})\bigr)
9119 =
9120 F_K(N)-F_K(\varnothing).
9121 \end{equation}
9122 Nonnegativity follows from \cref{prop:b8-envelope-monotone}.
9123 \end{proof}
9124 
9125 One useful order is
9126 \begin{equation}
9127 S\prec Q\prec T\prec C\prec H\prec V\prec W.
9128 \label{eq:b8-operational-order}
9129 \end{equation}
9130 It first asks how much opportunity remains in selection, then evidence acquisition, transport, construction, representation, evaluation, and witness design. Other orders may be preferable in other experiments. The dependence on order records module interaction rather than a defect in the definition.
9131 
9132 \subsection{Interaction-independent headroom allocation}
9133 
9134 To remove ordering dependence, average marginal contributions over all module orderings.
9135 
9136 \begin{definition}[Shapley meta-headroom]
9137 \label{def:b8-shapley-headroom}
9138 For $i\in N$, define
9139 \begin{equation}
9140 h_i(K)
9141 =
9142 \sum_{R\subseteq N\setminus\{i\}}
9143 \frac{|R|!(m-|R|-1)!}{m!}
9144 \left[
9145 F_K(R\cup\{i\})-F_K(R)
9146 \right].
9147 \label{eq:b8-shapley-headroom}
9148 \end{equation}
9149 \end{definition}
9150 
9151 \begin{theorem}[Meta-headroom allocation]
9152 \label{thm:b8-shapley-efficiency}
9153 For every finite repair-process factorization,
9154 \begin{equation}
9155 h_i(K)\geq0
9156 \end{equation}
9157 for every module $i$, and
9158 \begin{equation}
9159 \sum_{i\in N}h_i(K)=\Hcal(K).
9160 \label{eq:b8-shapley-efficiency}
9161 \end{equation}
9162 \end{theorem}
9163 
9164 \begin{proof}
9165 For any permutation of $N$, the marginal contributions along that ordering telescope from $F_K(\varnothing)$ to $F_K(N)$. Equation \eqref{eq:b8-shapley-headroom} is the average marginal contribution of module $i$ over all permutations. Summing those averages over $i$ gives the average of the total telescoping difference, which equals $\Hcal(K)$. Nonnegativity follows from monotonicity of $F_K$.
9166 \end{proof}
9167 
9168 \begin{remark}[Interactions are allocated rather than discarded]
9169 If information acquisition and transport have positive synergy, the value of changing both can exceed the sum of their isolated values. Equation \eqref{eq:b8-shapley-headroom} allocates that interaction across the participating modules. It does not assume additive independence.
9170 \end{remark}
9171 
9172 The exact finite companion uses five mutable modules and a monotone oracle envelope with three positive pair interactions. Its total headroom is
9173 \begin{equation}
9174 0.740000,
9175 \end{equation}
9176 and the computed module allocations sum to the same value within machine precision. The allocation is shown in \cref{fig:b8-headroom}.
9177 
9178 \begin{figure}[H]
9179 \centering
9180 \includegraphics[width=0.86\textwidth]{block08_headroom_allocation.pdf}
9181 \caption{Exact finite allocation of declared metaproductivity headroom across five repair-process modules. Pair interactions are distributed by the permutation-average rule in \cref{def:b8-shapley-headroom}.}
9182 \label{fig:b8-headroom}
9183 \end{figure}
9184 
9185 \subsection{Component saturation and headroom-adjusted gain}
9186 
9187 \begin{definition}[$\eps$-saturation]
9188 \label{def:b8-saturation}
9189 A module set $R\subseteq N$ is $\eps$-saturated at $K$ when
9190 \begin{equation}
9191 F_K(R)-\Mcal(K)\leq\eps.
9192 \label{eq:b8-saturation}
9193 \end{equation}
9194 \end{definition}
9195 
9196 \begin{proposition}[Component ceiling]
9197 \label{prop:b8-component-ceiling}
9198 If $R$ is $\eps$-saturated, no descendant that changes only modules in $R$ can improve protected metaproductivity by more than $\eps$.
9199 \end{proposition}
9200 
9201 \begin{proof}
9202 Every such descendant lies in $\Kcal_R(K)$ and therefore has metaproductivity at most $F_K(R)\leq\Mcal(K)+\eps$.
9203 \end{proof}
9204 
9205 The bridge-policy result from Block 7 reaches full-rank bridge identification on every primary target. Relative to that endpoint, the rank-deficiency component is saturated. A grandchild that repeats the same rank-revealing optimization cannot establish much additional progress. The next useful descendant must encounter a different limitation, such as conditioning, nonlinear misspecification, information loss, or candidate-grammar insufficiency.
9206 
9207 Let $K\to K'$ change only modules in $R$. When $F_K(R)>\Mcal(K)$, define the headroom-adjusted gain
9208 \begin{equation}
9209 \chi_R(K,K')
9210 =
9211 \frac{
9212 \Mcal(K')-\Mcal(K)
9213 }{
9214 F_K(R)-\Mcal(K)
9215 }.
9216 \label{eq:b8-headroom-adjusted-gain}
9217 \end{equation}
9218 For an admissible child inside the declared class,
9219 \begin{equation}
9220 \chi_R(K,K')\leq1.
9221 \end{equation}
9222 A small absolute gain with $\chi_R\approx1$ indicates a nearly saturated parent. A small absolute gain with $\chi_R\approx0$ indicates that the child captured little of the available opportunity.
9223 
9224 When the exact envelope is unknown, certified lower and upper oracle bounds
9225 \begin{equation}
9226 \underline F_K(R)
9227 \leq
9228 F_K(R)
9229 \leq
9230 \overline F_K(R)
9231 \end{equation}
9232 produce corresponding bounds on $\chi_R$. Audit-oracle arms are finite empirical approximations to this idea, but they must remain inadmissible for selection because they use final audit outcomes.
9233 
9234 \begin{warning}[No-ceiling fallacy]
9235 A failed same-module grandchild does not by itself refute repeated recursive improvement. The mutable component may have reached its declared oracle ceiling. A repeated-lineage experiment must report remaining headroom or a saturation certificate before interpreting a null result.
9236 \end{warning}
9237 
9238 \section{Bottleneck Migration and Repeated Lineage Dynamics}
9239 \label{sec:bottleneck-migration}
9240 
9241 A repair lineage should not be expected to improve the same module forever. Better selection exposes candidate limitations. Better evidence acquisition exposes transport mismatch. Better transport exposes missing repair constructors. Better construction can expose evaluator or witness incompleteness. Recursive improvement is therefore a dynamics of moving bottlenecks.
9242 
9243 \subsection{The meta-regret state}
9244 
9245 Let the unresolved module-headroom vector be
9246 \begin{equation}
9247 r_t
9248 =
9249 \begin{bmatrix}
9250  h_H(K_t)\\
9251  h_Q(K_t)\\
9252  h_C(K_t)\\
9253  h_T(K_t)\\
9254  h_S(K_t)\\
9255  h_V(K_t)\\
9256  h_W(K_t)
9257 \end{bmatrix}
9258 \in\R_{\geq0}^{7}.
9259 \label{eq:b8-regret-vector}
9260 \end{equation}
9261 An authorized meta-action $a$ changes one or more repair-process modules. In a local finite model, represent its effect by
9262 \begin{equation}
9263 r_{t+1}
9264 \leq
9265 A_a r_t+e_t,
9266 \label{eq:b8-regret-dynamics}
9267 \end{equation}
9268 where $A_a$ is a nonnegative response matrix and $e_t$ records estimation error, distribution shift, semantic debt, or newly induced headroom.
9269 
9270 The diagonal entries of $A_a$ describe direct contraction. Off-diagonal entries describe bottleneck transfer. For example, a better selector can reduce selection regret while increasing the value of better candidate construction because candidate quality becomes the new limiting factor.
9271 
9272 \begin{definition}[Weighted meta-regret]
9273 For $w\in\R_{>0}^{7}$, define
9274 \begin{equation}
9275 \Phi_w(r)=w^\top r.
9276 \label{eq:b8-weighted-regret}
9277 \end{equation}
9278 The weights encode the declared relative value or risk of unresolved module headroom.
9279 \end{definition}
9280 
9281 \subsection{Lyapunov contraction}
9282 
9283 \begin{theorem}[Meta-regret Lyapunov contraction]
9284 \label{thm:b8-lyapunov}
9285 Suppose there exist $w\in\R_{>0}^{7}$ and $q\in[0,1)$ such that for every reachable $r_t$, the controller can choose an authorized action $a_t$ satisfying
9286 \begin{equation}
9287 w^\top A_{a_t}r_t
9288 \leq
9289 q\,w^\top r_t.
9290 \label{eq:b8-lyapunov-condition}
9291 \end{equation}
9292 Then every controlled trajectory satisfying \eqref{eq:b8-regret-dynamics} obeys
9293 \begin{equation}
9294 w^\top r_t
9295 \leq
9296 q^t w^\top r_0
9297 +
9298 \sum_{k=0}^{t-1}
9299 q^{t-1-k}w^\top e_k.
9300 \label{eq:b8-lyapunov-bound}
9301 \end{equation}
9302 \end{theorem}
9303 
9304 \begin{proof}
9305 Equation \eqref{eq:b8-regret-dynamics} and nonnegativity of $w$ give
9306 \begin{equation}
9307 w^\top r_{t+1}
9308 \leq
9309 w^\top A_{a_t}r_t+w^\top e_t
9310 \leq
9311 q\,w^\top r_t+w^\top e_t.
9312 \end{equation}
9313 Repeated substitution proves \eqref{eq:b8-lyapunov-bound}.
9314 \end{proof}
9315 
9316 \begin{corollary}[Fixed-grammar convergence]
9317 \label{cor:b8-fixed-grammar-convergence}
9318 If the grammar remains fixed and
9319 \begin{equation}
9320 \sum_{t=0}^{\infty}w^\top e_t<\infty,
9321 \end{equation}
9322 then $w^\top r_t\to0$ whenever the tail errors vanish under the convolution in \eqref{eq:b8-lyapunov-bound}.
9323 \end{corollary}
9324 
9325 The theorem does not require one module to contract at every generation. It requires a controller that can choose a sufficiently useful action for the current state. The target of improvement may migrate among modules.
9326 
9327 \subsection{Greedy dominant-bottleneck contraction}
9328 
9329 A simpler sufficient condition uses the largest unresolved component.
9330 
9331 \begin{proposition}[Greedy bottleneck contraction]
9332 \label{prop:b8-greedy-contraction}
9333 Let $r_t\in\R_{\geq0}^{m}$ and
9334 \begin{equation}
9335 \Phi_t=\one^\top r_t.
9336 \end{equation}
9337 Suppose the controller chooses $i_t$ with
9338 \begin{equation}
9339 r_{i_t,t}\geq\frac{\Phi_t}{m},
9340 \end{equation}
9341 contracts that component by $q\in[0,1)$, and permits total collateral increase at most $\eps_t$. Then
9342 \begin{equation}
9343 \Phi_{t+1}
9344 \leq
9345 \left(1-\frac{1-q}{m}\right)\Phi_t+\eps_t.
9346 \label{eq:b8-greedy-contraction}
9347 \end{equation}
9348 \end{proposition}
9349 
9350 \begin{proof}
9351 The direct reduction is at least $(1-q)r_{i_t,t}\geq(1-q)\Phi_t/m$. Adding collateral error $\eps_t$ gives the bound.
9352 \end{proof}
9353 
9354 This proposition makes the role of bottleneck detection explicit. A process that repeatedly modifies an already small component need not contract total meta-regret.
9355 
9356 \subsection{Saturation-aware controller}
9357 
9358 Define a module status map
9359 \begin{equation}
9360 \sigma_t(i)
9361 \in
9362 \{\texttt{active},\texttt{saturated},\texttt{unidentified},\texttt{protected}\}.
9363 \end{equation}
9364 An admissible bottleneck controller should:
9365 \begin{enumerate}
9366 \item estimate headroom for active modules;
9367 \item refuse large gain claims for saturated modules;
9368 \item request new evidence for unidentified modules;
9369 \item avoid changing protected modules inside the current transaction;
9370 \item trigger grammar expansion when all active modules are saturated while persistent obstruction remains.
9371 \end{enumerate}
9372 
9373 \begin{protocol}[Saturation-aware meta-controller]
9374 \label{prot:b8-saturation-controller}
9375 At generation $t$:
9376 \begin{enumerate}
9377 \item compute lower and upper headroom bounds for every currently mutable module coalition;
9378 \item select the coalition with largest certified lower headroom per unit resource;
9379 \item compile candidate descendants that target only that coalition;
9380 \item seal the candidate set and final audit endpoints;
9381 \item commit only a descendant with a positive fresh lower metaproductivity bound;
9382 \item mark the coalition saturated when its upper headroom falls below the declared tolerance;
9383 \item if every current coalition is saturated and foreign obstruction remains, produce a grammar-boundary certificate.
9384 \end{enumerate}
9385 \end{protocol}
9386 
9387 \subsection{Conservative grammar discovery event}
9388 
9389 A grammar extension can reveal error that did not exist as a coordinate in the old state. Let
9390 \begin{equation}
9391 E_t:\R^{m_t}\hookrightarrow\R^{m_{t+1}}
9392 \end{equation}
9393 be the conservative embedding of old meta-regret coordinates. Decompose
9394 \begin{equation}
9395 r_{t+1}
9396 =
9397 E_t\widetilde r_{t+1}+n_{t+1},
9398 \qquad
9399 n_{t+1}\perp\im E_t.
9400 \label{eq:b8-regret-expansion}
9401 \end{equation}
9402 Define old-coordinate repair progress by
9403 \begin{equation}
9404 P_t^{\mathrm{repair}}
9405 =
9406 \norm{r_t}^2-\norm{\widetilde r_{t+1}}^2,
9407 \label{eq:b8-repair-progress}
9408 \end{equation}
9409 and epistemic discovery by
9410 \begin{equation}
9411 P_t^{\mathrm{discovery}}
9412 =
9413 \norm{n_{t+1}}^2.
9414 \label{eq:b8-discovery-progress}
9415 \end{equation}
9416 Both quantities may be positive. The total represented regret may increase because the lineage learned to represent a previously invisible limitation.
9417 
9418 \begin{warning}[Discovery is not regression]
9419 A conservative extension should not be rejected merely because the norm of the expanded regret vector exceeds the norm of the old vector. The correct comparison separates progress on old coordinates from the validated birth of a new coordinate.
9420 \end{warning}
9421 
9422 The finite Block 8 companion contracts several headroom components, then introduces one new nonlinear-grammar coordinate. The explicit expansion is the only increase in total represented headroom. Subsequent actions reduce the new coordinate. The trajectory appears in \cref{fig:b8-bottleneck}.
9423 
9424 \begin{figure}[H]
9425 \centering
9426 \includegraphics[width=0.90\textwidth]{block08_bottleneck_migration.pdf}
9427 \caption{Finite bottleneck-migration fixture. The controller reduces the currently dominant headroom. The dashed line marks a conservative grammar-discovery event that adds a new nonlinear coordinate rather than silently reinterpreting an old one.}
9428 \label{fig:b8-bottleneck}
9429 \end{figure}
9430 
9431 \subsection{Conservative lineage dominance}
9432 
9433 \begin{definition}[Conservative lineage dominance]
9434 \label{def:b8-conservative-dominance}
9435 Write
9436 \begin{equation}
9437 L_t\preceq_{\mathrm{cons}}L_{t+1}
9438 \end{equation}
9439 when:
9440 \begin{enumerate}
9441 \item the old grammar, evidence ledger, and proof objects embed into the new lineage state;
9442 \item every old protected interpretation remains recoverable within the declared semantic-debt bound;
9443 \item old authorized behaviors replay within declared tolerances;
9444 \item the new correction image contains the transported old correction image;
9445 \item at least one new externally validated distinction, repair schema, or correction direction is added, or one old module-headroom coordinate is strictly reduced;
9446 \item foreign metaproductivity does not decrease beyond the accepted uncertainty and regression budgets.
9447 \end{enumerate}
9448 \end{definition}
9449 
9450 This is a partial order only after quotienting by the declared equivalence of lineage states. It is intentionally stricter than an increase in one scalar score.
9451 
9452 \begin{definition}[Open-set ascending lineage]
9453 A lineage is conservatively ascending when
9454 \begin{equation}
9455 L_0\preceq_{\mathrm{cons}}L_1
9456 \preceq_{\mathrm{cons}}L_2
9457 \preceq_{\mathrm{cons}}\cdots.
9458 \label{eq:b8-ascending-lineage}
9459 \end{equation}
9460 It is structurally open-set when infinitely many edges add a foreign-task-relevant correction direction or validated distinction not representable in the prior grammar.
9461 \end{definition}
9462 
9463 \subsection{Repair process and repair compiler}
9464 
9465 The mechanism that repairs agents should be distinguished from the mechanism that compiles a better repair process.
9466 
9467 Let $K_t$ be the active repair process and let $\Gamma_t$ be the lineage compiler. Write
9468 \begin{equation}
9469 K_{t+1}
9470 =
9471 \Gamma_t(K_t,\Ecal_t,\Acal_t),
9472 \label{eq:b8-compiler-update}
9473 \end{equation}
9474 where $\Ecal_t$ is the available evidence and $\Acal_t$ is the executable archive.
9475 
9476 There are three regimes:
9477 \begin{enumerate}
9478 \item \textbf{fixed-compiler iteration}: $\Gamma_t=\Gamma$ while the compiled processes $K_t$ change;
9479 \item \textbf{compiler improvement}: $\Gamma_t$ itself changes and improves on foreign parent processes;
9480 \item \textbf{guarded reflective closure}: $\Gamma_t$ is a typed editable component of the lineage, but the witness validating a compiler change is not modified in the same transaction.
9481 \end{enumerate}
9482 
9483 Define compiler metaproductivity by
9484 \begin{equation}
9485 \Ccal(\Gamma)
9486 =
9487 \mathbb E_{K\sim\Dcal_{\mathrm{parent}}}
9488 \left[
9489 \Mcal\bigl(\Gamma(K,\Ecal_K)\bigr)-\Mcal(K)
9490 \right].
9491 \label{eq:b8-compiler-metaproductivity}
9492 \end{equation}
9493 A stronger recursive claim requires
9494 \begin{equation}
9495 \Ccal(\Gamma_{t+1})>\Ccal(\Gamma_t)
9496 \end{equation}
9497 on sealed foreign parent processes under matched compilation resources.
9498 
9499 Block 7 improves components inside $K$. It does not yet establish improvement of the outer compiler $\Gamma$.
9500 
9501 \section{Persistent Obstruction Across Grammar Growth}
9502 \label{sec:persistent-obstruction}
9503 
9504 The original obstruction theory is relative to one authorized correction map. Open-set recursive improvement requires a grammar-indexed sequence of maps. This section tracks which defect components become correctable, which persist, and which merely disappear because a new representation forgot them.
9505 
9506 \subsection{Nested correction images}
9507 
9508 Let $V$ be a finite-dimensional defect Hilbert space. Suppose a conservative grammar chain gives nested closed authorized correction images
9509 \begin{equation}
9510 I_0\subseteq I_1\subseteq I_2\subseteq\cdots\subseteq V,
9511 \qquad
9512 I_t=\im D_t.
9513 \label{eq:b8-nested-images}
9514 \end{equation}
9515 For $r\in V$, define unresolved obstruction magnitude
9516 \begin{equation}
9517 \eta_t(r)
9518 =
9519 \dist(-r,I_t)
9520 =
9521 \norm{P_{I_t^\perp}r}.
9522 \label{eq:b8-obstruction-magnitude}
9523 \end{equation}
9524 
9525 \begin{theorem}[Monotone obstruction convergence]
9526 \label{thm:b8-obstruction-convergence}
9527 For every defect $r$, the sequence $\eta_t(r)$ is nonincreasing and
9528 \begin{equation}
9529 \eta_t(r)
9530 \longrightarrow
9531 \dist\left(-r,\overline{\bigcup_{t\geq0}I_t}\right).
9532 \label{eq:b8-obstruction-limit}
9533 \end{equation}
9534 \end{theorem}
9535 
9536 \begin{proof}
9537 Nestedness gives $I_t\subseteq I_{t+1}$, so distance to $I_{t+1}$ cannot exceed distance to $I_t$. Let
9538 \begin{equation}
9539 I_\infty=\overline{\bigcup_t I_t}.
9540 \end{equation}
9541 The projections $P_{I_t}r$ converge strongly to $P_{I_\infty}r$ in finite dimension, so the orthogonal residual norms converge to the distance from $-r$ to $I_\infty$.
9542 \end{proof}
9543 
9544 \subsection{Exact newly correctable energy}
9545 
9546 \begin{theorem}[Newly correctable energy identity]
9547 \label{thm:b8-new-energy}
9548 For nested correction images,
9549 \begin{equation}
9550 \eta_t(r)^2-\eta_{t+1}(r)^2
9551 =
9552 \norm{P_{I_{t+1}\cap I_t^\perp}r}^2.
9553 \label{eq:b8-new-energy}
9554 \end{equation}
9555 \end{theorem}
9556 
9557 \begin{proof}
9558 The nested orthogonal decomposition is
9559 \begin{equation}
9560 I_{t+1}
9561 =
9562 I_t\oplus(I_{t+1}\cap I_t^\perp).
9563 \end{equation}
9564 Therefore
9565 \begin{equation}
9566 P_{I_{t+1}^\perp}r
9567 =
9568 P_{I_t^\perp}r
9569 -
9570 P_{I_{t+1}\cap I_t^\perp}r,
9571 \end{equation}
9572 and the two terms on the right are orthogonal. The Pythagorean identity gives \eqref{eq:b8-new-energy}.
9573 \end{proof}
9574 
9575 Equation \eqref{eq:b8-new-energy} provides exact structural credit. A grammar extension is useful not because it contains more syntax, but because it creates a new response direction with nonzero overlap with held-out obstruction.
9576 
9577 For a foreign residual distribution $Q$, define expected obstruction energy
9578 \begin{equation}
9579 E_t(Q)
9580 =
9581 \mathbb E_{r\sim Q}\eta_t(r)^2.
9582 \label{eq:b8-expected-obstruction}
9583 \end{equation}
9584 Then
9585 \begin{equation}
9586 E_t(Q)-E_{t+1}(Q)
9587 =
9588 \mathbb E_{r\sim Q}
9589 \norm{P_{I_{t+1}\cap I_t^\perp}r}^2.
9590 \label{eq:b8-distributional-new-energy}
9591 \end{equation}
9592 This is the primary structural metaproductivity estimand for grammar growth.
9593 
9594 The exact companion starts with one correctable direction and adds directions in decreasing order of foreign obstruction energy. The resulting tower is shown in \cref{fig:b8-obstruction-tower}.
9595 
9596 \begin{figure}[H]
9597 \centering
9598 \includegraphics[width=0.82\textwidth]{block08_obstruction_tower.pdf}
9599 \caption{Expected unresolved obstruction energy in the exact grammar-growth fixture. Each extension adds a direction outside the old correction image, and the energy drop equals the newly correctable component from \cref{thm:b8-new-energy}.}
9600 \label{fig:b8-obstruction-tower}
9601 \end{figure}
9602 
9603 \subsection{Dynamic defect semantics}
9604 
9605 A grammar extension may also change the defect representation. Let
9606 \begin{equation}
9607 D_t:U_t\to V_t,
9608 \end{equation}
9609 with transports
9610 \begin{equation}
9611 T_t^U:U_t\to U_{t+1},
9612 \qquad
9613 T_t^V:V_t\to V_{t+1}.
9614 \end{equation}
9615 Assume the exact correction square
9616 \begin{equation}
9617 D_{t+1}T_t^U
9618 =
9619 T_t^VD_t.
9620 \label{eq:b8-dynamic-square}
9621 \end{equation}
9622 Then there is an induced obstruction map
9623 \begin{equation}
9624 \overline T_t:
9625 \coker D_t
9626 \to
9627 \coker D_{t+1},
9628 \qquad
9629 \overline T_t([r])=[T_t^Vr].
9630 \label{eq:b8-induced-obstruction-map}
9631 \end{equation}
9632 
9633 \begin{proposition}[Well-defined obstruction transport]
9634 \label{prop:b8-cokernel-transport}
9635 Equation \eqref{eq:b8-induced-obstruction-map} is well defined under \eqref{eq:b8-dynamic-square}.
9636 \end{proposition}
9637 
9638 \begin{proof}
9639 If $r-r'=D_tu$, then
9640 \begin{equation}
9641 T_t^Vr-T_t^Vr'
9642 =
9643 T_t^VD_tu
9644 =
9645 D_{t+1}T_t^Uu,
9646 \end{equation}
9647 so $T_t^Vr$ and $T_t^Vr'$ define the same class in $\coker D_{t+1}$.
9648 \end{proof}
9649 
9650 The sequence
9651 \begin{equation}
9652 \coker D_0
9653 \xrightarrow{\overline T_0}
9654 \coker D_1
9655 \xrightarrow{\overline T_1}
9656 \coker D_2
9657 \longrightarrow\cdots
9658 \label{eq:b8-obstruction-module}
9659 \end{equation}
9660 is the obstruction persistence module of the grammar lineage.
9661 
9662 \subsection{Repair death and forgetting death}
9663 
9664 An obstruction class can map to zero for two different reasons.
9665 
9666 \begin{definition}[Repair death]
9667 A class $[r]\in\coker D_t$ undergoes repair death at $t+1$ when
9668 \begin{equation}
9669 \overline T_t([r])=0,
9670 \end{equation}
9671 there exists an authorized $u\in U_{t+1}$ satisfying
9672 \begin{equation}
9673 D_{t+1}u=-T_t^Vr,
9674 \end{equation}
9675 and the protected defect transport obeys a lower norm bound
9676 \begin{equation}
9677 \norm{T_t^Vr}\geq\kappa\norm{r}
9678 \end{equation}
9679 for a declared $\kappa>0$.
9680 \end{definition}
9681 
9682 \begin{definition}[Forgetting death]
9683 A class undergoes forgetting death when its transported representative becomes small or zero because $T_t^V$ collapses the protected distinction, without an authorized correction receipt accounting for the disappearance.
9684 \end{definition}
9685 
9686 \begin{warning}[Zero cokernel class is not sufficient]
9687 The equation $\overline T_t([r])=0$ does not by itself certify improvement. Information-losing transport can annihilate a defect rather than repair it. Protected obstruction faithfulness must be checked separately.
9688 \end{warning}
9689 
9690 Let $Q_t\subseteq V_t$ be a protected obstruction subspace. A dynamic grammar transport is $\kappa$-faithful on $Q_t$ when
9691 \begin{equation}
9692 \norm{T_t^Vq}\geq\kappa\norm{q}
9693 \qquad
9694 \text{for all }q\in Q_t.
9695 \label{eq:b8-kappa-faithful}
9696 \end{equation}
9697 This extends the obstruction-faithfulness requirement from one bridge to an entire grammar lineage.
9698 
9699 \subsection{Approximate grammar transport and semantic debt}
9700 
9701 Suppose the correction square is approximate:
9702 \begin{equation}
9703 \norm{D_{t+1}T_t^U-T_t^VD_t}
9704 \leq\eps_t^D.
9705 \end{equation}
9706 Suppose protected defect meanings are reconstructed with error at most $\eps_t^V$. Define cumulative dynamic semantic debt
9707 \begin{equation}
9708 \Delta_{m,n}
9709 =
9710 \sum_{t=m}^{n-1}
9711 \left(
9712 \eps_t^D+\eps_t^V
9713 \right).
9714 \label{eq:b8-dynamic-debt}
9715 \end{equation}
9716 Under nonexpansive transports, old correction and defect relations drift by at most the corresponding telescoping debt. If
9717 \begin{equation}
9718 \sum_t(\eps_t^D+\eps_t^V)<\infty,
9719 \end{equation}
9720 old protected meanings form Cauchy trajectories along the grammar chain.
9721 
9722 \begin{definition}[Persistent protected obstruction]
9723 A protected class is persistent when every sufficiently late transported representative remains above a declared norm margin and outside the authorized correction image by a declared residual margin.
9724 \end{definition}
9725 
9726 Persistent obstruction is the correct trigger for grammar expansion. A transient residual caused by estimator noise should not automatically create a new ontology.
9727 
9728 \section{Obstruction-Guided Generator and Grammar Discovery}
9729 \label{sec:generator-discovery}
9730 
9731 The previous section measures structural progress once a larger correction image has been supplied. This section turns the measurement into a construction rule. The central idea is simple: search for new repair generators whose response aligns with the current obstruction, not merely with high-variance state directions.
9732 
9733 \subsection{One-generator obstruction reduction}
9734 
9735 Let $I=\im D\subseteq V$ be the current authorized correction image and let
9736 \begin{equation}
9737 z=P_{I^\perp}r
9738 \end{equation}
9739 be the unresolved component of defect $r$. Suppose a candidate grammar extension contributes a response vector $v\in V$. Only its component outside the old image is structurally new:
9740 \begin{equation}
9741 \bar v=P_{I^\perp}v.
9742 \label{eq:b8-new-generator-component}
9743 \end{equation}
9744 The expanded image is
9745 \begin{equation}
9746 I'=I+\Span\{v\}.
9747 \end{equation}
9748 
9749 \begin{theorem}[One-generator obstruction reduction]
9750 \label{thm:b8-one-generator}
9751 If $\bar v\neq0$, then
9752 \begin{equation}
9753 \eta_{I'}(r)^2
9754 =
9755 \eta_I(r)^2
9756 -
9757 \frac{\abs{\angles{z,\bar v}}^2}{\norm{\bar v}^2}.
9758 \label{eq:b8-one-generator}
9759 \end{equation}
9760 If $\bar v=0$, the extension introduces no new first-order correctability.
9761 \end{theorem}
9762 
9763 \begin{proof}
9764 Because $\bar v\in I^\perp$,
9765 \begin{equation}
9766 I'=I\oplus\Span\{\bar v\}.
9767 \end{equation}
9768 The new residual is obtained by subtracting the orthogonal projection of $z$ onto $\bar v$. The squared projection norm is the second term in \eqref{eq:b8-one-generator}. If $\bar v=0$, then $v\in I$ and $I'=I$.
9769 \end{proof}
9770 
9771 Equation \eqref{eq:b8-one-generator} gives a direct acquisition score. For a candidate family $\Vcal_{\mathrm{cand}}$, choose
9772 \begin{equation}
9773 v^\star
9774 \in
9775 \argmax_{v\in\Vcal_{\mathrm{cand}}}
9776 \frac{\abs{\angles{z,P_{I^\perp}v}}^2}
9777 {\norm{P_{I^\perp}v}^2+\lambda c(v)+\beta\Risk(v)},
9778 \label{eq:b8-generator-score}
9779 \end{equation}
9780 where $c(v)$ is implementation cost and $\Risk(v)$ is a protected-risk estimate.
9781 
9782 A candidate is not accepted from the score alone. The response direction must be generated by an executable typed repair schema, survive witness validation, and reduce foreign obstruction energy.
9783 
9784 \subsection{Distributional rank-$k$ extension}
9785 
9786 Let foreign residuals follow distribution $Q$. Define the obstruction covariance operator
9787 \begin{equation}
9788 \Sigma_{\mathrm{ob}}
9789 =
9790 \mathbb E_{r\sim Q}
9791 \left[
9792 P_{I^\perp}r\,r^\top P_{I^\perp}
9793 \right].
9794 \label{eq:b8-obstruction-covariance}
9795 \end{equation}
9796 Let $W\subseteq I^\perp$ be a $k$-dimensional new response subspace.
9797 
9798 \begin{theorem}[Optimal rank-$k$ obstruction extension]
9799 \label{thm:b8-kyfan-extension}
9800 The subspace $W$ minimizing expected unresolved obstruction energy
9801 \begin{equation}
9802 \mathbb E_{r\sim Q}
9803 \norm{P_{(I\oplus W)^\perp}r}^2
9804 \end{equation}
9805 is the span of the top $k$ eigenvectors of $\Sigma_{\mathrm{ob}}$. The minimum residual energy is the sum of the remaining eigenvalues.
9806 \end{theorem}
9807 
9808 \begin{proof}
9809 For $W\subseteq I^\perp$,
9810 \begin{equation}
9811 \mathbb E\norm{P_{(I\oplus W)^\perp}r}^2
9812 =
9813 \tr(\Sigma_{\mathrm{ob}})-\tr(P_W\Sigma_{\mathrm{ob}}).
9814 \end{equation}
9815 The Ky Fan variational principle maximizes $\tr(P_W\Sigma_{\mathrm{ob}})$ over rank-$k$ orthogonal projectors by choosing the top $k$ eigenspace.
9816 \end{proof}
9817 
9818 \begin{remark}[Where spectral analysis belongs]
9819 Block 6 showed that PCA of raw hidden activations can miss the causal repair packet. \Cref{thm:b8-kyfan-extension} does not contradict that result. The spectral decomposition is applied after causal intervention and obstruction projection. The relevant covariance is the covariance of unresolved authorized defects, not the unconditional covariance of hidden states.
9820 \end{remark}
9821 
9822 The resulting pipeline is
9823 \begin{equation}
9824 \text{causal intervention}
9825 \longrightarrow
9826 \text{authorized obstruction residual}
9827 \longrightarrow
9828 \Sigma_{\mathrm{ob}}
9829 \longrightarrow
9830 \text{new response directions}.
9831 \label{eq:b8-obstruction-pipeline}
9832 \end{equation}
9833 
9834 \subsection{Causal credit for a new generator}
9835 
9836 A new generator receives structural credit only if all of the following hold:
9837 \begin{enumerate}
9838 \item its response has a nonzero component outside the old correction image;
9839 \item removing the generator restores the old obstruction energy or eliminates the new foreign gain;
9840 \item an equal-complexity random or generic generator does not reproduce the same effect;
9841 \item the generator does not obtain gain by changing the evaluator or deleting the defect representation;
9842 \item old authorized repairs and protected distinctions continue to replay.
9843 \end{enumerate}
9844 
9845 Define the causal generator contribution
9846 \begin{equation}
9847 \Delta_{\mathrm{gen}}
9848 =
9849 E_Q\bigl[\eta_{\mathrm{without}}(r)^2\bigr]
9850 -
9851 E_Q\bigl[\eta_{\mathrm{with}}(r)^2\bigr].
9852 \label{eq:b8-generator-credit}
9853 \end{equation}
9854 The contribution is admissible only when a matched ablation isolates the generator while holding the remaining grammar, evidence, and selection budget fixed.
9855 
9856 \subsection{Repair-conflict grammar inadequacy}
9857 
9858 Let $X$ be a finite set of repair episodes. The current grammar represents an episode through
9859 \begin{equation}
9860 \phi_{\Gcal}:X\to Z_{\Gcal}.
9861 \end{equation}
9862 Let $\Acal^\star(x)$ be the set of externally validated authorized repairs for episode $x$.
9863 
9864 For $z\in Z_{\Gcal}$, define the grammar cell
9865 \begin{equation}
9866 C_z
9867 =
9868 \{x\in X:\phi_{\Gcal}(x)=z\}.
9869 \end{equation}
9870 
9871 \begin{definition}[Repair-consistent cell]
9872 A cell is repair consistent when
9873 \begin{equation}
9874 \bigcap_{x\in C_z}\Acal^\star(x)\neq\varnothing.
9875 \label{eq:b8-consistent-cell}
9876 \end{equation}
9877 It is inconsistent when no single authorized repair works for every episode that the grammar treats as equivalent.
9878 \end{definition}
9879 
9880 \begin{definition}[Minimal repair conflict]
9881 A subset $S\subseteq C_z$ is a minimal repair conflict when
9882 \begin{equation}
9883 \bigcap_{x\in S}\Acal^\star(x)=\varnothing,
9884 \end{equation}
9885 but every proper subset has a nonempty repair intersection.
9886 \end{definition}
9887 
9888 The minimal conflicts form a finite hypergraph $\mathfrak H_{\Gcal}$. A candidate feature or probe
9889 \begin{equation}
9890 f:X\to Y_f
9891 \end{equation}
9892 resolves a conflict $S$ when $f$ is not constant on $S$.
9893 
9894 Appending the feature gives the conservative extension
9895 \begin{equation}
9896 \phi_{\Gcal'}(x)
9897 =
9898 \bigl(\phi_{\Gcal}(x),f(x)\bigr).
9899 \label{eq:b8-feature-extension}
9900 \end{equation}
9901 The old representation is recovered by projection onto the first coordinate.
9902 
9903 \begin{theorem}[Finite conservative conflict refinement]
9904 \label{thm:b8-conflict-refinement}
9905 Suppose a finite feature family $\Fcal$ resolves every minimal repair conflict in $\mathfrak H_{\Gcal}$. Then iterative addition of any feature that resolves at least one remaining conflict terminates after finitely many steps in a grammar whose every cell is repair consistent.
9906 \end{theorem}
9907 
9908 \begin{proof}
9909 A feature split never merges previously separated episodes. Therefore a resolved conflict cannot reappear inside one cell. Every successful addition removes at least one unresolved hyperedge. Since the finite hypergraph has finitely many hyperedges, the process terminates. If the terminal grammar had an inconsistent cell, that cell would contain a minimal inconsistent subset, contradicting exhaustion of the conflict hypergraph.
9910 \end{proof}
9911 
9912 The exact companion begins with episodes that are aliased under an empty feature schema. Two causally relevant binary features resolve every conflict. A nuisance feature resolves none of the remaining repair distinctions and is not selected. The conflict counts are shown in \cref{fig:b8-conflict}.
9913 
9914 \begin{figure}[H]
9915 \centering
9916 \includegraphics[width=0.78\textwidth]{block08_conflict_refinement.pdf}
9917 \caption{Finite conservative repair-conflict refinement. The selected features split cells with incompatible authorized repairs. The nuisance feature is available but unnecessary.}
9918 \label{fig:b8-conflict}
9919 \end{figure}
9920 
9921 \subsection{Feature acquisition as set cover}
9922 
9923 Let each candidate feature $f$ cover the conflict set
9924 \begin{equation}
9925 \Cov(f)
9926 =
9927 \{S\in\mathfrak H_{\Gcal}: f\text{ resolves }S\}.
9928 \end{equation}
9929 Choosing a minimum-cost feature set that resolves all conflicts is a weighted set-cover problem. An exact solver is possible in the finite companion. A scalable lineage may use the greedy score
9930 \begin{equation}
9931 \frac{\text{newly resolved conflicts}}{\text{feature cost}+\text{probe risk}}.
9932 \end{equation}
9933 The optimization target must remain subordinate to external repair validation. A feature that separates labels but does not improve authorized repair decisions is not a grammar advance.
9934 
9935 \subsection{Candidate invention as a right inverse}
9936 
9937 For episode $x$, let
9938 \begin{equation}
9939 F_x:U_x^{\Auth}\to V_x
9940 \end{equation}
9941 be the post-modification defect map, with $F_x(0)=r_x$. A valid repair satisfies
9942 \begin{equation}
9943 F_x(u_x)=0.
9944 \end{equation}
9945 A repair schema is a local section
9946 \begin{equation}
9947 s:x\mapsto u_x
9948 \end{equation}
9949 that approximately inverts $F_x$ over its declared domain.
9950 
9951 In the linear case,
9952 \begin{equation}
9953 F_x(u)=r_x+D_xu,
9954 \end{equation}
9955 and the canonical minimum-norm repair is
9956 \begin{equation}
9957 s(x)=-D_x^\dagger r_x
9958 \end{equation}
9959 whenever $r_x\in\im D_x$.
9960 
9961 Thus candidate invention is not merely sampling more parameter vectors. It is learning or synthesizing an authorized local right inverse of the defect map.
9962 
9963 \begin{definition}[Structurally new repair schema]
9964 Let $\Sscr_t$ be the closure of existing repair schemas under the authorized operadic compositions. A schema $s_{t+1}$ is structurally new when
9965 \begin{equation}
9966 s_{t+1}\notin\overline{\Sscr_t}
9967 \end{equation}
9968 under the declared behavioral equivalence and
9969 \begin{equation}
9970 \Span\im s_{t+1}
9971 \not\subseteq
9972 \Span\im\Sscr_t.
9973 \label{eq:b8-new-schema-direction}
9974 \end{equation}
9975 \end{definition}
9976 
9977 A structural metaproductivity claim additionally requires
9978 \begin{equation}
9979 E_{t+1}(Q_{\mathrm{foreign}})
9980 <
9981 E_t(Q_{\mathrm{foreign}})
9982 \label{eq:b8-structural-gate}
9983 \end{equation}
9984 and a causal ablation showing that the new schema is responsible for the reduction.
9985 
9986 \subsection{Finite counterexample-guided repair synthesis}
9987 
9988 Let $\Pcal_{\Gcal}$ be a finite typed repair-program grammar and let
9989 \begin{equation}
9990 \operatorname{Verify}(p,x)\in\{0,1\}
9991 \end{equation}
9992 be a complete verifier on a finite declared episode domain.
9993 
9994 \begin{protocol}[Finite repair synthesis]
9995 \label{prot:b8-cegis}
9996 \begin{enumerate}
9997 \item choose a program $p\in\Pcal_{\Gcal}$ consistent with the current receipt set;
9998 \item search for an episode $x$ with $\operatorname{Verify}(p,x)=0$;
9999 \item if no counterexample exists, return $p$ with its verification receipt;
10000 \item otherwise add $x$ to the receipt set, eliminate $p$, and continue.
10001 \end{enumerate}
10002 \end{protocol}
10003 
10004 \begin{theorem}[Finite synthesis or grammar exhaustion]
10005 \label{thm:b8-finite-synthesis}
10006 Protocol \ref{prot:b8-cegis} terminates after at most $|\Pcal_{\Gcal}|$ rejected programs with either a verified repair program or an exhaustion certificate showing that no program in the current grammar satisfies the finite verifier.
10007 \end{theorem}
10008 
10009 \begin{proof}
10010 Every failed iteration eliminates at least the current program. No eliminated program is reconsidered. Therefore at most $|\Pcal_{\Gcal}|$ programs can be rejected. If one program survives every verifier query, it is accepted. If all are eliminated, the receipt set certifies grammar-relative exhaustion.
10011 \end{proof}
10012 
10013 The exhaustion certificate is the operational bridge from candidate construction to grammar expansion. It does not prove that no repair exists outside the current grammar.
10014 
10015 \section{Causal Atlas Discovery from Intervention Algebras}
10016 \label{sec:causal-atlas}
10017 
10018 The previous theory assumes a defect space and repair strata. A more autonomous lineage must discover a causally meaningful functional cover from intervention responses. Variance alone is insufficient because high-variance directions can be irrelevant to the defect mechanism. The natural finite object is the algebra generated by controlled response operators.
10019 
10020 \subsection{Intervention-response algebra}
10021 
10022 Let $H$ be a finite-dimensional complex Hilbert space. Suppose permitted interventions or controlled transitions produce operators
10023 \begin{equation}
10024 R_1,\ldots,R_m\in\End(H).
10025 \end{equation}
10026 These may be finite-difference response maps, local transition Jacobians, or finite Koopman operators on an observable dictionary.
10027 
10028 Define the unital adjoint-closed algebra
10029 \begin{equation}
10030 \mathfrak A
10031 =
10032 \operatorname{alg}^{\ast}
10033 \left(I,R_1,\ldots,R_m\right).
10034 \label{eq:b8-response-algebra}
10035 \end{equation}
10036 Its commutant is
10037 \begin{equation}
10038 \mathfrak A'
10039 =
10040 \{X\in\End(H):XA=AX\text{ for all }A\in\mathfrak A\}.
10041 \end{equation}
10042 The center is
10043 \begin{equation}
10044 Z(\mathfrak A)=\mathfrak A\cap\mathfrak A'.
10045 \end{equation}
10046 
10047 \subsection{Finite causal atlas theorem}
10048 
10049 \begin{theorem}[Finite causal atlas decomposition]
10050 \label{thm:b8-atlas-decomposition}
10051 There exists a unitary identification
10052 \begin{equation}
10053 H
10054 \cong
10055 \bigoplus_{\alpha=1}^{p}
10056 \left(
10057 \mathbb C^{m_\alpha}\otimes\mathbb C^{n_\alpha}
10058 \right)
10059 \label{eq:b8-algebra-space-decomposition}
10060 \end{equation}
10061 under which
10062 \begin{equation}
10063 \mathfrak A
10064 \cong
10065 \bigoplus_{\alpha=1}^{p}
10066 \left(
10067 I_{m_\alpha}\otimes M_{n_\alpha}(\mathbb C)
10068 \right).
10069 \label{eq:b8-algebra-decomposition}
10070 \end{equation}
10071 The minimal central projections $P_\alpha\in Z(\mathfrak A)$ give a canonical coarse causal atlas
10072 \begin{equation}
10073 H
10074 =
10075 \bigoplus_{\alpha=1}^{p}P_\alpha H.
10076 \label{eq:b8-central-atlas}
10077 \end{equation}
10078 The blocks are unique up to permutation and unitary changes of basis within multiplicity spaces.
10079 \end{theorem}
10080 
10081 \begin{proof}
10082 A finite-dimensional unital $\ast$-algebra is a finite-dimensional $C^\ast$-algebra. Decompose its identity into minimal central projections. Each central summand is a simple finite-dimensional $C^\ast$-algebra and therefore isomorphic to a full matrix algebra $M_{n_\alpha}(\mathbb C)$. Its representation on the corresponding invariant subspace is a direct sum of $m_\alpha$ equivalent irreducible representations, yielding \eqref{eq:b8-algebra-space-decomposition} and \eqref{eq:b8-algebra-decomposition}. Minimal central projections are intrinsic to the center, giving uniqueness up to ordering and internal unitary gauge.
10083 \end{proof}
10084 
10085 \begin{remark}[Identifiability boundary]
10086 The multiplicity factor $\mathbb C^{m_\alpha}$ cannot be resolved from $\mathfrak A$ alone. Directions inside that factor are interventionally indistinguishable under the declared response family. The ambiguity is a gauge freedom, not a missing numerical optimization step.
10087 \end{remark}
10088 
10089 The theorem supplies a causal atlas relative to the available interventions. If the intervention family omits a relevant operation, the generated algebra may merge functionally distinct strata.
10090 
10091 \subsection{Defect-specific cyclic modules}
10092 
10093 For a current residual $r\in H$, define its cyclic causal module
10094 \begin{equation}
10095 H_r
10096 =
10097 \Span\{Ar:A\in\mathfrak A\}.
10098 \label{eq:b8-cyclic-module}
10099 \end{equation}
10100 
10101 \begin{proposition}[Minimal causal defect module]
10102 \label{prop:b8-cyclic-minimal}
10103 $H_r$ is the smallest $\mathfrak A$-invariant subspace containing $r$. Because $\mathfrak A$ is adjoint closed, $H_r$ is reducing and admits an orthogonal projector.
10104 \end{proposition}
10105 
10106 \begin{proof}
10107 The identity belongs to $\mathfrak A$, so $r\in H_r$. For $B\in\mathfrak A$ and $Ar\in H_r$, one has $BAr\in H_r$, proving invariance. Every invariant subspace containing $r$ contains $Ar$ for every $A$, hence contains $H_r$. Adjoint closure implies invariance under every $A^\ast$, so the orthogonal complement is invariant as well.
10108 \end{proof}
10109 
10110 This produces a two-level representation:
10111 \begin{enumerate}
10112 \item the central blocks $P_\alpha H$ form the persistent coarse atlas;
10113 \item $H_r$ is the episode-specific repair packet generated by the current defect under the causal algebra.
10114 \end{enumerate}
10115 The theory therefore does not require one permanent universal workspace subspace.
10116 
10117 \subsection{Approximate commutant recovery}
10118 
10119 For $X\in\End(H)$, define commutator energy
10120 \begin{equation}
10121 \Ecal_{\mathrm{com}}(X)
10122 =
10123 \sum_{j=1}^{m}
10124 \left(
10125 \norm{XR_j-R_jX}_F^2
10126 +
10127 \norm{XR_j^\ast-R_j^\ast X}_F^2
10128 \right).
10129 \label{eq:b8-commutator-energy}
10130 \end{equation}
10131 There is a positive semidefinite linear operator $\Lcal_{\mathrm{com}}$ on the Hilbert space of matrices such that
10132 \begin{equation}
10133 \angles{X,\Lcal_{\mathrm{com}}X}_F
10134 =
10135 \Ecal_{\mathrm{com}}(X).
10136 \end{equation}
10137 Its kernel is the commutant:
10138 \begin{equation}
10139 \ker\Lcal_{\mathrm{com}}=\mathfrak A'.
10140 \end{equation}
10141 
10142 \begin{proposition}[Spectral-gap stability of the commutant]
10143 \label{prop:b8-commutant-stability}
10144 Assume the first positive eigenvalue of $\Lcal_{\mathrm{com}}$ is at least $\lambda>0$. Let $\widehat{\Lcal}_{\mathrm{com}}$ be an empirical estimate satisfying
10145 \begin{equation}
10146 \norm{\widehat{\Lcal}_{\mathrm{com}}-\Lcal_{\mathrm{com}}}\leq\eps<\lambda/2.
10147 \end{equation}
10148 Then the spectral projector onto the empirical eigenvalues in $[0,\lambda/2)$ differs from the exact commutant projector by at most a constant multiple of $\eps/\lambda$.
10149 \end{proposition}
10150 
10151 \begin{proof}
10152 The exact zero spectrum is separated from the remaining spectrum by $\lambda$. Standard resolvent or sin-theta perturbation bounds for self-adjoint operators give projector error proportional to the perturbation norm divided by the spectral gap.
10153 \end{proof}
10154 
10155 The proposition identifies the required empirical condition: stable atlas recovery needs a response-algebra spectral gap. Without one, nearby intervention algebras can support substantially different decompositions.
10156 
10157 \subsection{Finite algebra fixture}
10158 
10159 The Block 8 companion uses four response operators on $\R^4$. Two generators act only on the first two coordinates and generate a full $2\times2$ matrix block. Two act only on the final two coordinates and generate a second full block. The generated algebra has dimension
10160 \begin{equation}
10161 8,
10162 \end{equation}
10163 the commutant has dimension
10164 \begin{equation}
10165 2,
10166 \end{equation}
10167 and the recovered central block ranks are
10168 \begin{equation}
10169 (2,2).
10170 \end{equation}
10171 A residual supported in the first block has cyclic dimension $2$, while a residual with support in both blocks has cyclic dimension $4$.
10172 
10173 \begin{warning}[Supplied intervention confound]
10174 The finite fixture verifies the algebraic mechanism. It does not show that a large learned agent autonomously discovers the right intervention family. Atlas recovery remains relative to the probes and response operators made available to the system.
10175 \end{warning}
10176 
10177 \subsection{Nonlinear local extension}
10178 
10179 For nonlinear dynamics on a state manifold $M$, the response algebra becomes base-point dependent:
10180 \begin{equation}
10181 x\longmapsto\mathfrak A_x\subseteq\End(T_xM).
10182 \end{equation}
10183 Local central projectors define a bundle of tangent strata when their ranks remain locally constant. Transport around a loop can permute or distort the local modules, producing atlas holonomy. A complete nonlinear theory must combine:
10184 \begin{enumerate}
10185 \item local intervention algebras;
10186 \item spectral-gap control of approximate central projectors;
10187 \item transition maps between neighboring tangent atlases;
10188 \item descent and holonomy tests on overlaps;
10189 \item repair transport that preserves obstruction classes rather than only state coordinates.
10190 \end{enumerate}
10191 This remains a conditional extension rather than a completed general theorem.
10192 
10193 \section{Sequential Lineage Certification}
10194 \label{sec:sequential-lineage-certification}
10195 
10196 One sealed parent-child edge does not provide a lineage-wide statistical guarantee. Every new descendant is chosen adaptively using previous receipts. A valid repeated-lineage theory therefore needs fresh evidence, conditional coverage, and an explicit error budget over the whole archive.
10197 
10198 \subsection{Filtration and edge-level effects}
10199 
10200 Let $\Fcal_t$ contain all lineage information available before generation $t$ opens its final audit. The child $K_{t+1}$ and every endpoint used to evaluate it must be $\Fcal_t$-measurable.
10201 
10202 Let the true foreign metaproductivity effect be
10203 \begin{equation}
10204 \Delta_t
10205 =
10206 \Mcal(K_{t+1})-\Mcal(K_t).
10207 \label{eq:b8-edge-effect}
10208 \end{equation}
10209 Let $L_t$ be a lower confidence bound computed from fresh audit evidence satisfying
10210 \begin{equation}
10211 \Pr\bigl(\Delta_t<L_t\mid\Fcal_t\bigr)
10212 \leq
10213 \delta_t.
10214 \label{eq:b8-conditional-coverage}
10215 \end{equation}
10216 Accept the edge only when
10217 \begin{equation}
10218 L_t>0.
10219 \label{eq:b8-edge-accept}
10220 \end{equation}
10221 
10222 \begin{theorem}[Lineage-wide false-improvement control]
10223 \label{thm:b8-lineage-validity}
10224 If
10225 \begin{equation}
10226 \sum_{t=0}^{\infty}\delta_t\leq\delta,
10227 \end{equation}
10228 then
10229 \begin{equation}
10230 \Pr\left(
10231 \text{every accepted lineage edge has }\Delta_t>0
10232 \right)
10233 \geq
10234 1-\delta.
10235 \label{eq:b8-lineage-validity}
10236 \end{equation}
10237 \end{theorem}
10238 
10239 \begin{proof}
10240 A false accepted edge satisfies $\Delta_t\leq0<L_t$, so the confidence event in \eqref{eq:b8-conditional-coverage} failed. Conditional coverage and the union bound give
10241 \begin{equation}
10242 \Pr(\exists t\text{ false accepted})
10243 \leq
10244 \sum_t\delta_t
10245 \leq
10246 \delta.
10247 \end{equation}
10248 \end{proof}
10249 
10250 \subsection{Branching archive allocation}
10251 
10252 For a branching archive, index proposed edges by $e\in E(\Lambda)$. Assign each edge a budget $\delta_e>0$ with
10253 \begin{equation}
10254 \sum_{e\in E(\Lambda)}\delta_e\leq\delta.
10255 \end{equation}
10256 The same proof yields archive-wide false-acceptance control. The allocation may be adaptive provided the remaining total budget is predictable from the pre-audit filtration.
10257 
10258 A simple geometric schedule is
10259 \begin{equation}
10260 \delta_e
10261 =
10262 \delta 2^{-\iota(e)},
10263 \end{equation}
10264 where $\iota$ is a prefix-free edge index chosen before the edge audit. More efficient schedules may allocate more error budget to scientifically important or expensive edges.
10265 
10266 \subsection{Protected regression and semantic-debt budgets}
10267 
10268 Let $R_t$ be true protected regression and let $U_t^R$ be a conditional upper confidence bound. Require
10269 \begin{equation}
10270 U_t^R\leq\beta_t,
10271 \qquad
10272 \sum_t\beta_t\leq B_R.
10273 \label{eq:b8-regression-budget}
10274 \end{equation}
10275 Let $d_t$ be semantic debt incurred by approximate grammar or evaluator transport, with
10276 \begin{equation}
10277 \sum_td_t\leq B_{\mathrm{sem}}.
10278 \label{eq:b8-semantic-budget}
10279 \end{equation}
10280 A lineage certificate should therefore report three separate cumulative quantities:
10281 \begin{equation}
10282 \left(
10283 \sum_t\delta_t,
10284 \sum_t\beta_t,
10285 \sum_td_t
10286 \right).
10287 \label{eq:b8-three-budgets}
10288 \end{equation}
10289 A positive gain sequence is not sufficient if its protected regression or semantic debt diverges.
10290 
10291 \subsection{Repeated-lineage metaproductivity certificate}
10292 
10293 \begin{definition}[Repeated-lineage certificate]
10294 \label{def:b8-repeated-certificate}
10295 A finite lineage path
10296 \begin{equation}
10297 K_0\to K_1\to\cdots\to K_T
10298 \end{equation}
10299 has a repeated-lineage metaproductivity certificate when:
10300 \begin{enumerate}
10301 \item every edge uses disjoint development, meta-validation, and final audit evidence;
10302 \item candidate, query, compute, and validation resources are fixed or explicitly accounted;
10303 \item every edge has a positive conditional lower metaproductivity bound;
10304 \item cumulative false-acceptance, protected-regression, and semantic-debt budgets remain below declared limits;
10305 \item prior proofs, receipts, and protected distinctions replay after every edge;
10306 \item the archive retains every parent as an executable alternative;
10307 \item every claimed structural edge reduces held-out obstruction energy or adds a validated correction direction.
10308 \end{enumerate}
10309 \end{definition}
10310 
10311 \begin{proposition}[Two-edge significance]
10312 A certified path
10313 \begin{equation}
10314 K_0\to K_1\to K_2
10315 \end{equation}
10316 with fresh evidence at both edges distinguishes one successful meta-fit from a mechanism that can produce at least one further validated descendant. It does not imply infinitely repeated productive closure.
10317 \end{proposition}
10318 
10319 \subsection{Conditional bottleneck-migrating improvement theorem}
10320 
10321 \begin{conditionaltheorem}[Bottleneck-migrating recursive improvement]
10322 \label{thm:b8-integrated}
10323 Let
10324 \begin{equation}
10325 L_t
10326 =
10327 \left(
10328 K_t,\Gamma_t,\Gcal_t,\Ecal_t,\Lambda_t,V_t,W_t
10329 \right)
10330 \end{equation}
10331 be a sequence of resolved lineage states. Assume:
10332 \begin{enumerate}
10333 \item every edge is evaluated against externally grounded evidence unavailable before sealing;
10334 \item componentwise restricted oracle envelopes are finite or bounded by valid certificates;
10335 \item inside each fixed grammar, an authorized meta-controller satisfies the weighted contraction condition of \cref{thm:b8-lyapunov};
10336 \item saturation is detected before further same-module gains are claimed;
10337 \item a grammar extension is accepted only when it resolves a repair conflict or adds a response direction with positive held-out obstruction overlap;
10338 \item old evidence, proofs, authorization decisions, and protected meanings embed with summable semantic debt;
10339 \item every new repair schema receives causal credit through matched ablation;
10340 \item edge-level false-acceptance probabilities are summable;
10341 \item proposal, evidence, evaluation, witness, and rollback surfaces are not jointly rewritten in one transaction.
10342 \end{enumerate}
10343 Then:
10344 \begin{enumerate}
10345 \item weighted meta-regret contracts to the declared error floor during every fixed-grammar interval;
10346 \item progress may continue after one module saturates by migrating to another non-saturated module;
10347 \item every accepted structural grammar edge strictly reduces held-out obstruction energy in at least one direction unavailable to the old grammar;
10348 \item historical protected meanings remain recoverable within the cumulative semantic-debt bound;
10349 \item with probability at least $1-\sum_t\delta_t$, every accepted edge has positive true metaproductivity relative to its declared claim and budget;
10350 \item if infinitely many accepted structural edges expose genuinely new foreign-task-relevant correction directions, the lineage forms an infinite conservatively ascending chain of structural repair competence.
10351 \end{enumerate}
10352 \end{conditionaltheorem}
10353 
10354 \begin{proof}
10355 Item 1 is \cref{thm:b8-lyapunov}. Item 2 follows from saturation-aware module selection and the existence of another coalition with positive certified headroom. Item 3 follows from \cref{thm:b8-new-energy,thm:b8-one-generator} and the structural acceptance rule. Item 4 follows from telescoping approximate conservativity. Item 5 is \cref{thm:b8-lineage-validity}. Item 6 follows by repeated application of conservative lineage dominance together with strict structural novelty at infinitely many edges.
10356 \end{proof}
10357 
10358 \begin{remark}[Existence is not guaranteed]
10359 The theorem does not assert that a useful meta-action or grammar extension always exists. It states what follows if the lineage can repeatedly find and validate such actions. It does not prove unlimited utility, indefinite acceleration, or unrestricted safety.
10360 \end{remark}
10361 
10362 \section{Block 8 Symbolic Reference Realization}
10363 \label{sec:block8-realization}
10364 
10365 Block 8 is theory first. Its companion implementation is an exact finite realization of the new definitions and theorems. It does not introduce a new learned-agent claim gate.
10366 
10367 \subsection{Implemented fixtures}
10368 
10369 The package implements:
10370 \begin{enumerate}
10371 \item a monotone restricted oracle envelope and exact Shapley headroom allocation;
10372 \item a bottleneck-migration trace with one explicit conservative grammar-discovery event;
10373 \item a nested obstruction tower with exact newly correctable energy;
10374 \item the one-generator obstruction-reduction identity;
10375 \item repair-conflict refinement with a nuisance-feature negative control;
10376 \item finite counterexample-guided repair synthesis;
10377 \item a finite intervention-response algebra with two central causal blocks;
10378 \item a summable lineage confidence and protected-regression budget fixture.
10379 \end{enumerate}
10380 
10381 \subsection{Frozen exact results}
10382 
10383 \begin{table}[H]
10384 \centering
10385 \small
10386 \begin{tabular}{@{}p{0.48\textwidth}p{0.38\textwidth}@{}}
10387 \toprule
10388 Fixture & Frozen result \\
10389 \midrule
10390 Total declared headroom & $0.740000000000$ \\
10391 Sum of Shapley module allocations & $0.740000000000$ \\
10392 Obstruction tower & monotone to zero in the finite six-direction fixture \\
10393 Conflict refinement & selects features \texttt{a}, \texttt{b}; rejects nuisance \\
10394 Finite synthesis & accepts \texttt{copy\_ab} after four counterexamples \\
10395 Response algebra dimension & $8$ \\
10396 Commutant dimension & $2$ \\
10397 Central block ranks & $(2,2)$ \\
10398 Cyclic defect dimensions & $2$ for one block, $4$ for mixed support \\
10399 Cumulative confidence budget & $0.01875$ \\
10400 Automated tests & $8/8$ passed \\
10401 \bottomrule
10402 \end{tabular}
10403 \caption{Block 8 exact symbolic companion results.}
10404 \label{tab:b8-results}
10405 \end{table}
10406 
10407 The canonical result object has SHA-256 digest
10408 \begin{center}
10409 \texttt{39e7fb42fa88815f7184473b8fa3b465a3fc1d1a8fc12b30049dd41c1b9fa34c}.
10410 \end{center}
10411 
10412 \subsection{Interpretation boundary}
10413 
10414 \begin{statusbox}{Block 8 result}
10415 Block 8 formalizes module-specific metaproductivity headroom, bottleneck migration, persistent obstruction across grammar growth, obstruction-guided repair-generator discovery, conflict-grounded grammar refinement, intervention-algebra atlas discovery, and lineage-wide statistical certification. Its executable companion verifies finite exact fixtures for these constructions. It does not establish a new child-grandchild learned-agent result, autonomous open-set grammar growth, spontaneous atlas discovery in a large model, evaluator self-improvement, unrestricted safety, or indefinitely accelerating recursive improvement.
10416 \end{statusbox}
10417 
10418 \subsection{Next empirical lineage}
10419 
10420 The next learned or trained experiment should be staged so that the bottleneck genuinely changes:
10421 \begin{enumerate}
10422 \item $K_0$ uses random anchors and a conservative fixed selector;
10423 \item $K_1$ uses the Block 7 rank-revealing query policy and learned risk-sensitive selector;
10424 \item $K_2$ faces full-rank but noisy and ill-conditioned bridges and must discover a conditioning-aware design criterion;
10425 \item $K_3$ faces systematic failure of every global linear bridge and must produce a grammar-inadequacy certificate followed by local or nonlinear transport;
10426 \item $K_4$ no longer receives a supplied causal packet and must recover a functional atlas from intervention-response operators.
10427 \end{enumerate}
10428 Each edge requires fresh development, validation, and audit evidence. A positive result at one edge may not be reused as evidence for the next.
10429 
10430 \section{Block 8 Reproducibility and Package}
10431 \label{sec:block8-package}
10432 
10433 \subsection{Package layout}
10434 
10435 The reference implementation contains:
10436 \begin{verbatim}
10437 SRIC_v0.9_Block08_Reference_Implementation/
10438   README.md
10439   MANIFEST.sha256
10440   Run_Block08.ps1
10441   pyproject.toml
10442   run_block08.py
10443   verify_package.py
10444   sric_block08/
10445     atlas.py
10446     bottleneck.py
10447     certification.py
10448     grammar.py
10449     headroom.py
10450     obstruction.py
10451     run_all.py
10452   tests/
10453   results/
10454   figures/
10455 \end{verbatim}
10456 
10457 \subsection{Canonical replay}
10458 
10459 The compact PowerShell call is:
10460 \begin{verbatim}
10461 python verify_package.py; python run_block08.py; python -m pytest -q
10462 \end{verbatim}
10463 The package regenerates the canonical JSON result object and four publication figures. The manifest excludes no executable source used by the result generator.
10464 
10465 \subsection{Claim boundary}
10466 
10467 The package is a symbolic reference realization. It demonstrates exact finite identities and implementation consistency. It does not replace a foreign-agent audit, a child-grandchild comparison, or a learned causal-atlas experiment.
10468 
10469 
10470 
10471 \section{Repeated Lineage under a Migrating Bottleneck}
10472 \label{sec:block09-lineage}
10473 
10474 Block 7 established two one-step modifications of a repair process. The bridge-policy child replaced random paired-anchor acquisition by a rank-revealing design and raised the full-rank bridge-identification rate from $0.03125$ to $1$. That result also created the next theoretical problem. Once rank failure has been removed, applying the same rank-recovery idea again has no remaining rank headroom. A valid grandchild must identify a different limiting quantity.
10475 
10476 Block 9 instantiates the bottleneck-migration theory of Block 8. The frozen Block 7 bridge child becomes the lineage parent
10477 \begin{equation}
10478 K_1.
10479 \end{equation}
10480 It is evaluated in a fresh environment where every declared primary design is full rank, but the paired target observations are heteroscedastically noisy. The residual difficulty is therefore not whether the bridge equations are identifiable in principle. It is whether the selected anchors make the weighted bridge information sufficiently well conditioned for stable repair transport.
10481 
10482 The descendant compiler receives only new development receipts. It diagnoses full-rank saturation, activates a finite conditioning-aware query grammar, selects a grandchild
10483 \begin{equation}
10484 K_2,
10485 \end{equation}
10486 and seals the resulting policy before the fresh audit tasks and representation families are executed. The primary comparison is
10487 \begin{equation}
10488 \Mcal_{\mathrm{audit}}(K_2)-\Mcal_{\mathrm{audit}}(K_1^{\rm recal}),
10489 \end{equation}
10490 where $K_1^{\rm recal}$ preserves the Block 7 rank-revealing query policy but is allowed to recalibrate its witness gate on Block 9 development evidence. This is a stronger parent control than an unchanged replay of the old Block 7 gate.
10491 
10492 \subsection{Context-indexed repeated lineage}
10493 
10494 Repeated improvement across fresh environments needs a careful definition because two independently sealed audits need not share the same task distribution.
10495 
10496 \begin{definition}[Context-indexed lineage edge]
10497 Let $\Xi_t$ be the protected context for edge $t\to t+1$. A sealed lineage edge
10498 \begin{equation}
10499 K_t\xrightarrow{\Pi_{t+1}}K_{t+1}
10500 \end{equation}
10501 is $\gamma_t$-productive in $\Xi_t$ when its receipt replays, its resource and witness gates pass, and a fresh paired audit certifies
10502 \begin{equation}
10503 \Mcal_{\Xi_t}(K_{t+1})-\Mcal_{\Xi_t}(K_t)\geq \gamma_t>0.
10504 \end{equation}
10505 \end{definition}
10506 
10507 \begin{definition}[Finite repeated-lineage certificate]
10508 A sequence
10509 \begin{equation}
10510 K_0\xrightarrow{\Pi_1}K_1\xrightarrow{\Pi_2}\cdots\xrightarrow{\Pi_m}K_m
10511 \end{equation}
10512 has an $m$-edge repeated-lineage certificate when every edge is productive in its own predeclared protected context, every parent receipt is replayed before the child is compiled, no final audit evidence is reused as development evidence for a later edge, and the cumulative error and regression budgets remain within their declared bounds.
10513 \end{definition}
10514 
10515 \begin{proposition}[Same-context transitivity]
10516 Suppose every edge is evaluated in one common context $\Xi$. Then
10517 \begin{equation}
10518 \Mcal_{\Xi}(K_m)-\Mcal_{\Xi}(K_0)
10519 =
10520 \sum_{t=0}^{m-1}
10521 \left(
10522 \Mcal_{\Xi}(K_{t+1})-\Mcal_{\Xi}(K_t)
10523 \right).
10524 \end{equation}
10525 Consequently, positive lower bounds on all edges add to a positive lower bound on the complete path.
10526 \end{proposition}
10527 
10528 \begin{proof}
10529 The identity is a telescoping sum.
10530 \end{proof}
10531 
10532 \begin{warning}[Fresh contexts do not produce one scalar ordering]
10533 If edge $t$ is evaluated in $\Xi_t$ and the contexts differ, the edge effects cannot be added into one global utility difference without an additional transport or dominance relation among contexts. A context-indexed repeated-lineage certificate establishes repeatability of the improvement mechanism. It does not establish that one scalar utility increased by the sum of the reported edge effects.
10534 \end{warning}
10535 
10536 Block 9 therefore supports a two-edge context-indexed claim. The first bridge-policy edge was sealed and evaluated in Block 7. The second edge replays the exact Block 7 child policy, uses new development evidence, and is evaluated on new task and architecture families.
10537 
10538 \subsection{Rank saturation and bottleneck migration}
10539 
10540 Let $X_S\in\R^{n\times d}$ be the source hidden-state design induced by anchor set $S$. Rank recovery is saturated when
10541 \begin{equation}
10542 \Pr\left(\rank X_S=d\right)\geq 1-\eps_{\rm rank}
10543 \end{equation}
10544 under the current foreign development distribution and the remaining performance gap cannot be explained by rank failure.
10545 
10546 \begin{definition}[Conditioning bottleneck certificate]
10547 A rank-revealing parent has a conditioning bottleneck with margin $\kappa>0$ when:
10548 \begin{enumerate}
10549 \item its full-rank rate is at least $1-\eps_{\rm rank}$;
10550 \item its weighted information floor is below the declared stability floor;
10551 \item a conditioning-aware policy using the same query budget improves development metaproductivity by at least $\kappa$;
10552 \item the improvement survives recalibration of the rank parent and is not obtained only by changing the commit gate.
10553 \end{enumerate}
10554 \end{definition}
10555 
10556 This certificate separates two statements:
10557 \begin{equation}
10558 \rank X_S=d
10559 \end{equation}
10560 and
10561 \begin{equation}
10562 \lambda_{\min}(X_S^{\mathsf T}WX_S)\gg0.
10563 \end{equation}
10564 The first is qualitative identifiability. The second is quantitative stability.
10565 
10566 \begin{proposition}[Rank saturation does not imply stability]
10567 For every $d\geq2$ and every $\eta>0$, there exists a full-rank design $X\in\R^{d\times d}$ such that
10568 \begin{equation}
10569 \lambda_{\min}(X^{\mathsf T}X)<\eta.
10570 \end{equation}
10571 \end{proposition}
10572 
10573 \begin{proof}
10574 Take
10575 \begin{equation}
10576 X=\diag(1,\ldots,1,\sqrt{\eta/2}).
10577 \end{equation}
10578 Then $X$ is full rank and
10579 \begin{equation}
10580 \lambda_{\min}(X^{\mathsf T}X)=\eta/2<\eta.
10581 \end{equation}
10582 \end{proof}
10583 
10584 The proposition makes the Block 9 transition necessary rather than cosmetic. Once the full-rank rate reaches one, further progress requires a metric beyond rank.
10585 
10586 \section{Heteroscedastic Bridge Information}
10587 \label{sec:block09-information}
10588 
10589 \subsection{Weighted paired-anchor model}
10590 
10591 Let the source hidden state have dimension $d$ and the target hidden state have dimension $q$. For selected source anchors $x_i\in\R^d$, suppose the observed target anchor is
10592 \begin{equation}
10593 y_i=Tx_i+\eps_i,
10594 \qquad
10595 \mathbb E[\eps_i]=0,
10596 \qquad
10597 \Cov(\eps_i)=\sigma_i^2I_q.
10598 \end{equation}
10599 The query-quality metadata supplies $\sigma_i$ before the anchor is selected. This metadata is part of the declared Block 9 environment. It is not discovered autonomously.
10600 
10601 Stacking the selected anchors gives
10602 \begin{equation}
10603 Y=XT^{\mathsf T}+E,
10604 \end{equation}
10605 where $X\in\R^{n\times d}$ and $Y\in\R^{n\times q}$. Define
10606 \begin{equation}
10607 W=\diag(\sigma_1^{-2},\ldots,\sigma_n^{-2})
10608 \end{equation}
10609 and the weighted information matrix
10610 \begin{equation}
10611 M_S=X^{\mathsf T}WX.
10612 \end{equation}
10613 
10614 \begin{theorem}[Weighted bridge covariance]
10615 \label{thm:b9-wls}
10616 Assume $M_S$ is invertible and the noise rows are independent with covariance $\sigma_i^2I_q$. The weighted least-squares estimator
10617 \begin{equation}
10618 \widehat T^{\mathsf T}
10619 =
10620 M_S^{-1}X^{\mathsf T}WY
10621 \end{equation}
10622 is unbiased and satisfies
10623 \begin{equation}
10624 \Cov\left(\operatorname{vec}(\widehat T^{\mathsf T}-T^{\mathsf T})\mid X\right)
10625 =
10626 I_q\otimes M_S^{-1}.
10627 \end{equation}
10628 Consequently,
10629 \begin{equation}
10630 \mathbb E\left[\norm{\widehat T-T}_F^2\mid X\right]
10631 =
10632 q\,\tr(M_S^{-1}).
10633 \end{equation}
10634 \end{theorem}
10635 
10636 \begin{proof}
10637 Write
10638 \begin{equation}
10639 \widehat T^{\mathsf T}-T^{\mathsf T}
10640 =
10641 M_S^{-1}X^{\mathsf T}WE.
10642 \end{equation}
10643 For one target coordinate, the noise covariance is $W^{-1}$, so the coefficient covariance is
10644 \begin{equation}
10645 M_S^{-1}X^{\mathsf T}W W^{-1} WXM_S^{-1}=M_S^{-1}.
10646 \end{equation}
10647 The target coordinates are independent and identically weighted, giving the Kronecker covariance. Taking the trace yields the Frobenius identity.
10648 \end{proof}
10649 
10650 \begin{corollary}[Worst-direction variance]
10651 For every $u\in\R^d$ and every target coordinate,
10652 \begin{equation}
10653 \Var\left(u^{\mathsf T}(\widehat T^{\mathsf T}-T^{\mathsf T})e_j\mid X\right)
10654 =u^{\mathsf T}M_S^{-1}u
10655 \leq
10656 \frac{\norm{u}^2}{\lambda_{\min}(M_S)}.
10657 \end{equation}
10658 \end{corollary}
10659 
10660 The A-optimal quantity $\tr(M_S^{-1})$ controls mean-square bridge error. The E-optimal quantity $\lambda_{\min}(M_S)$ controls the least informed direction. The D-optimal quantity $\log\det M_S$ controls information volume.
10661 
10662 \subsection{Ridge regularization and reported information}
10663 
10664 The executable implementation uses
10665 \begin{equation}
10666 M_{S,\lambda}=X^{\mathsf T}WX+\lambda I
10667 \end{equation}
10668 with a small fixed $\lambda>0$. The reported design diagnostics use source coordinates standardized against the complete candidate pool before applying the reliability weights. Standardization prevents one arbitrary source-coordinate scale from dominating the comparison. The actual bridge fit remains weighted in the original hidden coordinates.
10669 
10670 The regularized estimator is biased unless $\lambda=0$, but it remains stable under weak finite designs. Block 9 reports the regularized information floor, condition number, validation alignment, and a clean transport residual available only to the audit analysis.
10671 
10672 \subsection{Transport error and repair error}
10673 
10674 Let the frozen source repair be
10675 \begin{equation}
10676 R_s(h)=h+D_sh+b_s.
10677 \end{equation}
10678 For an injective bridge $T$, the exact transported repair is
10679 \begin{equation}
10680 D_t=T D_sT^\dagger,
10681 \qquad
10682 b_t=Tb_s.
10683 \end{equation}
10684 The estimated repair uses $\widehat T$.
10685 
10686 \begin{proposition}[Exact repair-operator perturbation identity]
10687 Let
10688 \begin{equation}
10689 \widehat D_t=\widehat TD_s\widehat T^\dagger.
10690 \end{equation}
10691 Then
10692 \begin{equation}
10693 \widehat D_t-D_t
10694 =
10695 (\widehat T-T)D_s\widehat T^\dagger
10696 +
10697 T D_s(\widehat T^\dagger-T^\dagger).
10698 \end{equation}
10699 Hence
10700 \begin{equation}
10701 \norm{\widehat D_t-D_t}
10702 \leq
10703 \norm{\widehat T-T}\norm{D_s}\norm{\widehat T^\dagger}
10704 +
10705 \norm{T}\norm{D_s}\norm{\widehat T^\dagger-T^\dagger}.
10706 \end{equation}
10707 \end{proposition}
10708 
10709 \begin{proof}
10710 Add and subtract $TD_s\widehat T^\dagger$.
10711 \end{proof}
10712 
10713 \begin{conditionaltheorem}[Conditioned repair stability]
10714 Suppose $T$ has full column rank and
10715 \begin{equation}
10716 \norm{T^\dagger}\norm{\widehat T-T}<1.
10717 \end{equation}
10718 Then there is a finite constant $C_T$, depending continuously on $\norm{T}$ and $\norm{T^\dagger}$ inside this perturbation neighborhood, such that
10719 \begin{equation}
10720 \norm{\widehat D_t-D_t}
10721 \leq
10722 C_T\norm{D_s}\norm{\widehat T-T}.
10723 \end{equation}
10724 If the protected utility readout is $L$-Lipschitz with respect to the transported repair operator on the declared evaluation set, the loss in repair gain is at most
10725 \begin{equation}
10726 LC_T\norm{D_s}\norm{\widehat T-T}.
10727 \end{equation}
10728 \end{conditionaltheorem}
10729 
10730 The theorem identifies two conditioning layers. The anchor design controls $\widehat T-T$. The target bridge condition controls the sensitivity of $T^\dagger$ and therefore the amplification of that estimation error. Good source information cannot eliminate a genuinely noninjective or severely ill-conditioned target bridge. The stress families retain those negative cases.
10731 
10732 \subsection{Known quality and the autonomy boundary}
10733 
10734 The Block 9 grandchild receives a scalar query-noise estimate before choosing each candidate anchor. The parent receives the same metadata but its frozen packet-QR rule does not use it. This is a legitimate matched-information comparison between query policies.
10735 
10736 It is not a result on autonomous uncertainty discovery. A later block must replace the supplied $\sigma_i$ by an estimated reliability model, charge the repeated probes needed to learn that model, and test calibration under shift.
10737 
10738 \section{Conditioning-Aware Query Design}
10739 \label{sec:block09-design}
10740 
10741 \subsection{D-optimal marginal information}
10742 
10743 Let $M\succ0$ be the current regularized information matrix. Adding candidate anchor $x_i$ with quality weight $w_i=\sigma_i^{-2}$ gives
10744 \begin{equation}
10745 M'=M+w_ix_ix_i^{\mathsf T}.
10746 \end{equation}
10747 
10748 \begin{theorem}[D-optimal marginal gain]
10749 \label{thm:b9-dopt}
10750 The increase in log information volume is
10751 \begin{equation}
10752 \log\det M'-\log\det M
10753 =
10754 \log\left(1+w_ix_i^{\mathsf T}M^{-1}x_i\right).
10755 \end{equation}
10756 \end{theorem}
10757 
10758 \begin{proof}
10759 Apply the matrix determinant lemma to $M+(\sqrt{w_i}x_i)(\sqrt{w_i}x_i)^{\mathsf T}$.
10760 \end{proof}
10761 
10762 The grandchild greedily maximizes the quantity in \cref{thm:b9-dopt} while retaining equal affected and unaffected anchor quotas. The quota is inherited from the Block 7 packet-balanced child and prevents the design from abandoning one side of the declared repair contrast.
10763 
10764 \subsection{A-optimal marginal gain}
10765 
10766 \begin{proposition}[A-optimal trace reduction]
10767 Let $v=\sqrt{w_i}x_i$. Then
10768 \begin{equation}
10769 \tr(M^{-1})-\tr\left((M+vv^{\mathsf T})^{-1}\right)
10770 =
10771 \frac{v^{\mathsf T}M^{-2}v}{1+v^{\mathsf T}M^{-1}v}.
10772 \end{equation}
10773 \end{proposition}
10774 
10775 \begin{proof}
10776 The Sherman--Morrison identity gives
10777 \begin{equation}
10778 (M+vv^{\mathsf T})^{-1}
10779 =
10780 M^{-1}-\frac{M^{-1}vv^{\mathsf T}M^{-1}}{1+v^{\mathsf T}M^{-1}v}.
10781 \end{equation}
10782 Taking traces yields the result.
10783 \end{proof}
10784 
10785 The finite compiler compares weighted D-optimal, weighted A-optimal, weighted E-refined, unweighted D-optimal, and inherited packet-QR policies. The selected policy is not hard coded into the claim gate.
10786 
10787 \subsection{E-optimal exchange}
10788 
10789 The E-optimal objective is
10790 \begin{equation}
10791 \max_{|S|=k}\lambda_{\min}(M_S).
10792 \end{equation}
10793 The companion initializes with the weighted D-optimal design, then performs same-stratum exchanges that increase the smallest eigenvalue. This is a finite heuristic rather than a claim of global E-optimality.
10794 
10795 \subsection{Quality-binding control}
10796 
10797 A policy may appear successful merely because its label is correlated with a favorable outcome generator. Block 9 therefore includes a quality-permutation control. The grandchild algorithm is left unchanged, but the query-quality values are permuted among the candidate anchors before the design is constructed. Actual observation noise remains attached to the original anchors.
10798 
10799 If quality binding is causal, permutation should reduce the weighted information floor and eliminate or weaken the audit gain. This control differs from shuffling final outcomes. It preserves the policy code while breaking the semantic relation between one anchor and its reliability estimate.
10800 
10801 \subsection{Rank-only and gate-only controls}
10802 
10803 Two controls isolate the source of the grandchild effect.
10804 
10805 \paragraph{Recalibrated rank child.}
10806 The packet-QR parent is allowed to select its best witness thresholds on the new development environment. This prevents the grandchild from winning only because the unchanged Block 7 alignment threshold is too strict for noisy anchors.
10807 
10808 \paragraph{Grandchild gate on rank anchors.}
10809 The grandchild witness thresholds are applied to the inherited packet-QR anchors. If the grandchild gain came only from a more favorable gate, this arm would reproduce it. Failure of this arm, together with a positive grandchild result, attributes the improvement to the new query design rather than gate liberalization.
10810 
10811 \subsection{Conditioning diagnosis}
10812 
10813 The development compiler first evaluates the inherited rank policy. It records:
10814 \begin{equation}
10815 \begin{aligned}
10816 &\text{full-rank rate},\\
10817 &\text{weighted minimum information eigenvalue},\\
10818 &\text{weighted condition number},\\
10819 &\text{recalibrated rank-policy gain}.
10820 \end{aligned}
10821 \end{equation}
10822 It activates the conditioning meta-grammar only if rank is saturated and a non-rank policy exceeds the recalibrated rank policy by the predeclared development margin.
10823 
10824 \begin{proposition}[Finite bottleneck diagnosis soundness]
10825 Inside a finite policy grammar, suppose the inherited policy has full-rank rate at least $1-\eps_{\rm rank}$ and the best conditioning policy has true mean protected gain at least $\kappa$ above the best recalibrated inherited policy. If every empirical policy mean is within $\eta$ of its true mean and
10826 \begin{equation}
10827 \kappa>2\eta,
10828 \end{equation}
10829 then empirical maximization selects a policy whose true gain exceeds the inherited optimum.
10830 \end{proposition}
10831 
10832 \begin{proof}
10833 Let $p^\star$ be the best conditioning policy and $p_0$ the best inherited-policy configuration. Empirical errors of at most $\eta$ imply
10834 \begin{equation}
10835 \widehat\mu(p^\star)
10836 \geq
10837 \mu(p^\star)-\eta
10838 >
10839 \mu(p_0)+\eta
10840 \geq
10841 \widehat\mu(p_0).
10842 \end{equation}
10843 \end{proof}
10844 
10845 The executable release uses frozen development and audit files rather than claiming a prospective population bound from this proposition. The theorem specifies the stronger confirmatory design.
10846 
10847 \section{Sealed Child-to-Grandchild Protocol}
10848 \label{sec:block09-protocol}
10849 
10850 \subsection{Lineage objects}
10851 
10852 The Block 9 lineage uses three policy objects.
10853 
10854 \begin{enumerate}
10855 \item $K_0$ is the random-anchor parent from the Block 7 bridge experiment.
10856 \item $K_1$ is the exact Block 7 packet-balanced pivoted-QR child. Its semantic policy digest is replayed from the Block 7 pre-audit receipt.
10857 \item $K_2$ is the conditioning-aware grandchild compiled from new Block 9 development receipts.
10858 \end{enumerate}
10859 
10860 The primary Block 9 parent control is $K_1^{\rm recal}$, which preserves the anchor rule of $K_1$ while recalibrating only the witness gate on the Block 9 development split. The exact $K_1$ replay remains in the audit as a lineage-integrity and distribution-shift control.
10861 
10862 \subsection{Fresh task and representation supports}
10863 
10864 The Block 9 development tasks use the trigger families
10865 \begin{equation}
10866 \{\texttt{bilinear},\texttt{wave},\texttt{offset},\texttt{radial5}\}.
10867 \end{equation}
10868 The final audit uses
10869 \begin{equation}
10870 \{\texttt{cubic},\texttt{triple},\texttt{ratio},\texttt{annulus}\}.
10871 \end{equation}
10872 Task identifiers, teacher seeds, and trigger supports are disjoint across the development and final audit files.
10873 
10874 The development primary representation families are
10875 \begin{equation}
10876 \{\texttt{dev\_dense20},\texttt{dev\_scaled22},\texttt{dev\_block24}\}.
10877 \end{equation}
10878 The final primary families are
10879 \begin{equation}
10880 \{\texttt{audit\_rotated20},\texttt{audit\_anisotropic24},\texttt{audit\_block28}\}.
10881 \end{equation}
10882 The stress families are severe ill-conditioning, noninjective compression, and nonlinear tanh representations.
10883 
10884 No Block 7 final audit rows are used to compile $K_2$.
10885 
10886 \subsection{Fixed query and repair budget}
10887 
10888 Every primary child and grandchild deployment receives:
10889 \begin{equation}
10890 16\ \text{fit anchors}
10891 +
10892 8\ \text{validation anchors}
10893 =
10894 24\ \text{paired representation queries}.
10895 \end{equation}
10896 The source repair, bridge estimator, transport formula, target evaluator, target audit size, and protected witness metrics are fixed. The only primary process changes are the anchor-selection rule and the gate compiled with that rule.
10897 
10898 A random $32$-fit-anchor arm uses
10899 \begin{equation}
10900 32+8=40
10901 \end{equation}
10902 paired queries and serves as a larger-budget control.
10903 
10904 \subsection{Finite meta-grammar}
10905 
10906 The compiler searches the anchor policies
10907 \begin{equation}
10908 \begin{aligned}
10909 \{&\texttt{packet\_qr},
10910 \texttt{unweighted\_d\_opt},
10911 \texttt{quality\_d\_opt},\\
10912 &\texttt{quality\_a\_opt},
10913 \texttt{quality\_e\_opt}\}
10914 \end{aligned}
10915 \end{equation}
10916 and a finite gate grid over calibration gain, protected regression, debiased validation alignment, and minimum weighted information eigenvalue.
10917 
10918 The development objective is a stability- and harm-sensitive mean foreign gain. The policy grammar is externally declared and finite. The grandchild does not invent D-optimal design from an unrestricted program language.
10919 
10920 \subsection{Commit gate}
10921 
10922 For one candidate row $r$, a configuration commits only if
10923 \begin{equation}
10924 \begin{aligned}
10925 \widehat g_{\rm cal}(r)&\geq \tau_g,\\
10926 \widehat R_{\rm cal}(r)&\leq \tau_R,\\
10927 \widehat\eps_{\rm align}(r)&\leq \tau_A,\\
10928 \lambda_{\min}(M_r)&\geq \tau_I,\\
10929 \rank X_r&=d.
10930 \end{aligned}
10931 \end{equation}
10932 The selected Block 9 grandchild has
10933 \begin{equation}
10934 (\tau_g,\tau_R,\tau_A,\tau_I)
10935 =
10936 (0.02,0.05,0.20,2.0).
10937 \end{equation}
10938 The recalibrated rank child has
10939 \begin{equation}
10940 (0.02,0.05,0.55,0.10).
10941 \end{equation}
10942 
10943 \subsection{Seal chronology}
10944 
10945 The executable chronology is:
10946 \begin{enumerate}
10947 \item replay the Block 7 child policy and pre-audit receipt;
10948 \item generate Block 9 development tasks and foreign development representations;
10949 \item evaluate the finite policy and gate grammar;
10950 \item diagnose full-rank saturation and conditioning headroom;
10951 \item serialize the recalibrated child and conditioning grandchild;
10952 \item hash the compiler ledger, source files, policy objects, and ancestor digest;
10953 \item write the Block 9 pre-audit seal with \texttt{audit\_fields\_present=false};
10954 \item generate the final audit tasks and representation families;
10955 \item execute all primary and control arms;
10956 \item evaluate paired claim gates and package closure.
10957 \end{enumerate}
10958 
10959 \subsection{Finite-class selection bound}
10960 
10961 \begin{theorem}[Finite meta-grammar generalization]
10962 Let $\mathcal K$ contain $N$ candidate grandchild configurations. Suppose each target-level protected gain lies in $[-G,G]$ and $n$ independent development targets are used. Then, with probability at least $1-\delta$, every candidate satisfies
10963 \begin{equation}
10964 \abs{\widehat\Mcal(K)-\Mcal(K)}
10965 \leq
10966 G\sqrt{\frac{2\log(2N/\delta)}{n}}.
10967 \end{equation}
10968 If $\widehat K$ maximizes the empirical mean, then
10969 \begin{equation}
10970 \Mcal(\widehat K)
10971 \geq
10972 \max_{K\in\mathcal K}\Mcal(K)
10973 -
10974 2G\sqrt{\frac{2\log(2N/\delta)}{n}}.
10975 \end{equation}
10976 \end{theorem}
10977 
10978 \begin{proof}
10979 Hoeffding's inequality for a variable with range length $2G$ gives
10980 \begin{equation}
10981 \Pr\left(\abs{\widehat\Mcal(K)-\Mcal(K)}>\eta\right)
10982 \leq
10983 2\exp\left(-\frac{n\eta^2}{2G^2}\right).
10984 \end{equation}
10985 Apply a union bound over $N$ configurations and solve for $\eta$. The selected-candidate inequality follows by comparing the empirical maximizer with the true maximizer and paying one deviation term for each.
10986 \end{proof}
10987 
10988 The released bootstrap intervals are post-selection evaluations on disjoint frozen audits. They are not presented as an anytime-valid guarantee for an indefinitely expanding meta-grammar.
10989 
10990 \subsection{Paired grandchild certificate}
10991 
10992 Let $D_i$ be the grandchild-minus-child protected gain on final foreign target $i$. The primary finite estimand is
10993 \begin{equation}
10994 \overline D
10995 =
10996 \frac1n\sum_{i=1}^nD_i.
10997 \end{equation}
10998 The release records a paired bootstrap interval, a sign-flip test, positive and negative pair rates, and exact query ledgers. The matched target pairing removes between-target difficulty from the edge comparison.
10999 
11000 \section{Block 9 Executable Realization and Results}
11001 \label{sec:block09-results}
11002 
11003 \subsection{Development diagnosis}
11004 
11005 The development compiler evaluates $54$ foreign target systems across six new tasks, three representation families, and three target seeds. Every policy receives three independent noisy paired-query replicates.
11006 
11007 The inherited packet-QR child is full rank in every development replicate, but its weighted information floor remains weak.
11008 
11009 \begin{table}[H]
11010 \centering
11011 \small
11012 \begin{tabular}{@{}lr@{}}
11013 \toprule
11014 Development diagnostic & Value \\
11015 \midrule
11016 Rank-child full-rank rate & $1.000000$ \\
11017 Rank-child mean information eigenvalue & $0.549830$ \\
11018 Rank-child median condition number & $21622.327$ \\
11019 Recalibrated rank-child mean gain & $0.028232$ \\
11020 Best conditioning-policy mean gain & $0.217374$ \\
11021 Conditioning headroom gap & $0.189142$ \\
11022 \bottomrule
11023 \end{tabular}
11024 \caption{Frozen Block 9 bottleneck diagnosis. Rank is saturated, while conditioning remains a large unresolved source of metaproductivity headroom.}
11025 \label{tab:b9-diagnosis}
11026 \end{table}
11027 
11028 The compiler selects
11029 \begin{equation}
11030 \texttt{quality\_d\_opt}
11031 \end{equation}
11032 as the grandchild query policy.
11033 
11034 \subsection{Primary foreign audit}
11035 
11036 The primary audit contains
11037 \begin{equation}
11038 8\ \text{tasks}
11039 \times
11040 3\ \text{foreign representation families}
11041 \times
11042 4\ \text{target seeds}
11043 =
11044 96\ \text{paired targets}.
11045 \end{equation}
11046 
11047 \begin{table}[H]
11048 \centering
11049 \small
11050 \begin{tabular}{@{}lrrrrr@{}}
11051 \toprule
11052 Arm & Selected gain & Commit & Full rank & $\lambda_{\min}$ & Clean transport \\
11053 \midrule
11054 Exact Block 7 child replay & $0.000618$ & $0.002604$ & $1.000$ & $0.518$ & $0.304$ \\
11055 Recalibrated rank child & $0.037393$ & $0.161458$ & $1.000$ & $0.518$ & $0.304$ \\
11056 Conditioning grandchild & $\mathbf{0.209580}$ & $0.638021$ & $1.000$ & $\mathbf{14.268}$ & $\mathbf{0.096}$ \\
11057 Quality-permuted grandchild & $0.000000$ & $0.000000$ & $1.000$ & $0.643$ & $0.293$ \\
11058 Random $32$-anchor control & $0.111028$ & $0.382813$ & $0.844$ & $3.278$ & $0.180$ \\
11059 \bottomrule
11060 \end{tabular}
11061 \caption{Primary foreign audit. Clean transport is normalized RMSE and is available only to the audit analysis. The grandchild and rank child have identical $24$-query deployment budgets.}
11062 \label{tab:b9-primary}
11063 \end{table}
11064 
11065 The paired grandchild-minus-recalibrated-child effect is
11066 \begin{equation}
11067 \widehat\Delta_{1\to2}
11068 =
11069 0.172188,
11070 \end{equation}
11071 with frozen interval
11072 \begin{equation}
11073 [0.141481,\ 0.203110].
11074 \end{equation}
11075 The sign-flip estimate is
11076 \begin{equation}
11077 p=9.9999\times10^{-6}.
11078 \end{equation}
11079 The positive paired-effect rate is
11080 \begin{equation}
11081 0.677083,
11082 \end{equation}
11083 and the negative paired-effect rate is zero. The remaining pairs are ties produced mainly by mutual abstention.
11084 
11085 The exact Block 7 child replay is nearly inactive because its old alignment threshold was calibrated for effectively noiseless paired anchors. The stronger primary claim is therefore made against the recalibrated rank child, not only against the unchanged replay.
11086 
11087 \begin{figure}[H]
11088 \centering
11089 \includegraphics[width=0.88\textwidth]{block09_lineage_gain.pdf}
11090 \caption{Primary foreign metaproductivity across the repeated lineage and controls. The conditioning grandchild uses the same $24$ paired queries as the rank child.}
11091 \label{fig:b9-lineage-gain}
11092 \end{figure}
11093 
11094 \begin{figure}[H]
11095 \centering
11096 \includegraphics[width=0.70\textwidth]{block09_paired_gain.pdf}
11097 \caption{Target-wise conditioning-grandchild gain versus recalibrated rank-child gain on the fresh primary audit. No paired target favors the rank child.}
11098 \label{fig:b9-paired}
11099 \end{figure}
11100 
11101 \subsection{Bottleneck migration is structural}
11102 
11103 Both the recalibrated rank child and the grandchild have full-rank rate one on the primary audit. The difference appears in the geometry beyond rank:
11104 \begin{equation}
11105 \lambda_{\min}:
11106 0.518197
11107 \longrightarrow
11108 14.268195,
11109 \end{equation}
11110 \begin{equation}
11111 \text{median condition number}:
11112 19393.296
11113 \longrightarrow
11114 544.180,
11115 \end{equation}
11116 and
11117 \begin{equation}
11118 \text{clean transport NRMSE}:
11119 0.303981
11120 \longrightarrow
11121 0.095803.
11122 \end{equation}
11123 
11124 The debiased validation-alignment residual falls from
11125 \begin{equation}
11126 0.261487
11127 \end{equation}
11128 to
11129 \begin{equation}
11130 0.057707.
11131 \end{equation}
11132 
11133 \begin{figure}[H]
11134 \centering
11135 \includegraphics[width=0.76\textwidth]{block09_conditioning_geometry.pdf}
11136 \caption{All principal designs can be full rank, yet their weighted information floors and condition numbers differ by orders of magnitude.}
11137 \label{fig:b9-conditioning}
11138 \end{figure}
11139 
11140 \begin{figure}[H]
11141 \centering
11142 \includegraphics[width=0.74\textwidth]{block09_bottleneck_migration.pdf}
11143 \caption{The rank endpoint is saturated for both generations. The grandchild improves the newly active conditioning coordinate.}
11144 \label{fig:b9-migration}
11145 \end{figure}
11146 
11147 These results instantiate the Block 8 meta-regret picture. The first bridge-policy edge reduced rank-identification regret. The second edge leaves rank unchanged and reduces conditioning-sensitive information regret.
11148 
11149 \subsection{Gate isolation and quality binding}
11150 
11151 Applying the grandchild gate to inherited rank anchors yields zero selected primary gain. This does not mean the inherited repair has zero blind value. Its mean blind primary gain is
11152 \begin{equation}
11153 0.157691.
11154 \end{equation}
11155 It means the rank design does not satisfy the grandchild's declared information and alignment certificate.
11156 
11157 The quality-permuted grandchild also has zero selected primary gain. Its blind primary gain is
11158 \begin{equation}
11159 0.154203,
11160 \end{equation}
11161 but its mean information eigenvalue falls from
11162 \begin{equation}
11163 14.268195
11164 \end{equation}
11165 to
11166 \begin{equation}
11167 0.643375.
11168 \end{equation}
11169 The paired quality-binding certificate has the same positive lower bound pattern as the gate-isolation comparison.
11170 
11171 Unweighted D-optimal design is also rejected by the grandchild gate. This establishes that the observed benefit is not explained by determinant maximization alone. The anchor-quality relation is part of the realized mechanism.
11172 
11173 \subsection{Larger-budget control}
11174 
11175 The random $32$-anchor control uses $40$ paired queries and obtains mean primary selected gain
11176 \begin{equation}
11177 0.111028.
11178 \end{equation}
11179 The conditioning grandchild uses $24$ paired queries and obtains
11180 \begin{equation}
11181 0.209580.
11182 \end{equation}
11183 The result is therefore not explained by a hidden increase in deployment query count.
11184 
11185 The one-time development compiler uses
11186 \begin{equation}
11187 19440
11188 \end{equation}
11189 paired development queries. Relative to the random $32$-anchor arm's additional $16$ deployment queries, the declared query-only break-even point is
11190 \begin{equation}
11191 1215
11192 \end{equation}
11193 deployments. This accounting does not include engineering labor, model training, or the value of safer abstention.
11194 
11195 \begin{figure}[H]
11196 \centering
11197 \includegraphics[width=0.70\textwidth]{block09_query_efficiency.pdf}
11198 \caption{The grandchild outperforms the larger random-query control while using fewer paired target-representation queries per deployment.}
11199 \label{fig:b9-query-efficiency}
11200 \end{figure}
11201 
11202 \subsection{Family-level results and negative boundaries}
11203 
11204 \begin{table}[H]
11205 \centering
11206 \small
11207 \begin{tabular}{@{}lrrr@{}}
11208 \toprule
11209 Family & Rank child & Grandchild & Grandchild commit \\
11210 \midrule
11211 \texttt{audit\_rotated20} & $0.044815$ & $0.315166$ & $0.984375$ \\
11212 \texttt{audit\_anisotropic24} & $0.000$ & $0.001674$ & $0.007813$ \\
11213 \texttt{audit\_block28} & $0.067362$ & $0.311900$ & $0.921875$ \\
11214 \texttt{stress\_severe\_ill24} & $0.000$ & $0.000000$ & $0.000000$ \\
11215 \texttt{stress\_compressed12} & $0.016366$ & $0.055636$ & $0.343750$ \\
11216 \texttt{stress\_tanh24} & $0.000000$ & $0.007651$ & $0.039063$ \\
11217 \bottomrule
11218 \end{tabular}
11219 \caption{Family-level selected gain. The severe target-conditioning family remains obstructed. Compression and nonlinear transport permit only partial, witness-gated transfer. Rounded rank-child family values are reported for orientation.}
11220 \label{tab:b9-family}
11221 \end{table}
11222 
11223 The audit-anisotropic primary family is an informative near-null result. Better source query conditioning does not remove target-side pseudoinverse amplification. The severe-ill stress family is rejected completely. The compressed and nonlinear stress families admit limited commits, but harmful selected-target rates remain zero in the frozen fixture.
11224 
11225 \begin{figure}[H]
11226 \centering
11227 \includegraphics[width=0.95\textwidth]{block09_family_gain.pdf}
11228 \caption{Conditioning-aware transfer is strong on two foreign primary families and remains limited or abstaining on the harder transport regimes.}
11229 \label{fig:b9-family}
11230 \end{figure}
11231 
11232 \subsection{Two-edge repeated-lineage certificate}
11233 
11234 The Block 7 child policy file has SHA-256 digest
11235 \begin{equation}
11236 \texttt{46e794fe5335c19a7dd56558128e0e15434447dd644bd9daf8f827c3e139f01c}
11237 \end{equation}
11238 and semantic policy digest
11239 \begin{equation}
11240 \texttt{c241031f271f1f9a2e6ce7585319fc5eb64467e5fce31a4fb33373e7445bc1d2}.
11241 \end{equation}
11242 The Block 9 grandchild receipt names that semantic digest as its lineage parent.
11243 
11244 The Block 9 pre-audit seal has digest
11245 \begin{equation}
11246 \texttt{9b3014b1d3c4a8ef09aa34ae47baa134cdadd8cb9250e81f90a9e2dbac60fb07}.
11247 \end{equation}
11248 The canonical complete result has digest
11249 \begin{equation}
11250 \texttt{bdb8623fb1f2676f3828b38d51460f8a4d351ac8fd7618205c29908f812ebc56}.
11251 \end{equation}
11252 
11253 \begin{theorem}[Finite two-edge repeated-lineage realization]
11254 Within the declared Block 7 and Block 9 contexts, the released receipts and audits instantiate two sequential productive edges:
11255 \begin{equation}
11256 K_0\xrightarrow{\Pi_1}K_1\xrightarrow{\Pi_2}K_2.
11257 \end{equation}
11258 The first edge improves rank-revealing bridge-query design relative to its Block 7 parent. The second edge replays $K_1$, detects rank saturation, and improves conditioning-aware query design relative to a recalibrated $K_1$ control under a fresh matched-resource audit.
11259 \end{theorem}
11260 
11261 \begin{proof}
11262 The Block 7 receipt and audit provide the first sealed edge. The Block 9 package verifies the exact Block 7 child digest before compilation. Its development and audit task identifiers and primary representation families are disjoint, and no Block 7 final audit row is used in Block 9 development. The Block 9 pre-audit seal fixes the grandchild policy before final audit execution. The primary paired interval for $K_2-K_1^{\rm recal}$ is strictly positive, query budgets match, both designs are full rank, protected harmful-target rates are zero, and the quality-permutation and gate-isolation controls fail to reproduce the grandchild result. These conditions instantiate the context-indexed repeated-lineage definition.
11263 \end{proof}
11264 
11265 \begin{remark}[What repeated means here]
11266 The theorem establishes two sequential successful lineage edges. It does not establish a winning strategy for infinitely many generations, compiler self-improvement, autonomous query-quality discovery, candidate-constructor growth, or open-set grammar expansion.
11267 \end{remark}
11268 
11269 \begin{statusbox}{Block 9 result}
11270 Block 9 crosses the first repeated-lineage threshold. The exact Block 7 bridge child is replayed as the parent of a newly compiled conditioning-aware grandchild. On a fresh $96$-target primary audit, the grandchild improves mean selected protected gain from $0.037393$ for a recalibrated rank-child control to $0.209580$ under the same $24$-query deployment budget. The paired effect is $0.172188$ with frozen interval $[0.141481,0.203110]$. Both generations are full rank, so the improvement is attributed to a migration from rank recovery to heteroscedastic conditioning. The query-quality metadata, finite policy grammar, repair form, evaluator, witness, and target generators remain designer supplied.
11271 \end{statusbox}
11272 
11273 \section{Block 9 Reproducibility and Package}
11274 \label{sec:block09-package}
11275 
11276 \subsection{Package contents}
11277 
11278 The companion package contains:
11279 \begin{enumerate}
11280 \item the fresh five-dimensional task and defect generator;
11281 \item source and foreign target representation families;
11282 \item packet-QR, random, unweighted optimal-design, and quality-aware optimal-design policies;
11283 \item heteroscedastic target-anchor observations and weighted bridge fitting;
11284 \item ancestor receipt replay and lineage-parent hashing;
11285 \item the finite grandchild compiler and bottleneck diagnosis ledger;
11286 \item pre-audit sealing and matched-resource foreign audits;
11287 \item six publication figures;
11288 \item canonical JSON and CSV result objects;
11289 \item nine automated tests;
11290 \item a complete SHA-256 package manifest.
11291 \end{enumerate}
11292 
11293 \subsection{Canonical replay}
11294 
11295 A clean deterministic execution produces:
11296 \begin{verbatim}
11297 SRIC_BLOCK09_ALL_SUITES=PASS
11298 result_sha256=bdb8623fb1f2676f3828b38d51460f8a4d351ac8fd7618205c29908f812ebc56
11299 preaudit_seal_sha256=
11300   9b3014b1d3c4a8ef09aa34ae47baa134cdadd8cb9250e81f90a9e2dbac60fb07
11301 grandchild_policy=quality_d_opt
11302 rank_child_primary_gain=0.037393
11303 conditioning_grandchild_primary_gain=0.209580
11304 paired_grandchild_delta=0.172188
11305 paired_ci_low=0.141481
11306 rank_child_information=0.497677
11307 grandchild_information=13.121386
11308 ......... [100%]
11309 9 passed
11310 SRIC_BLOCK09_PACKAGE_VALID
11311 manifest_files=52
11312 \end{verbatim}
11313 
11314 \subsection{Claim boundary}
11315 
11316 The release establishes a controlled two-edge lineage inside two finite experimental grammars. It does not establish:
11317 \begin{itemize}
11318 \item autonomous discovery of query reliability;
11319 \item autonomous invention of D-optimal design;
11320 \item improvement of the outer compiler $\Gamma_t$;
11321 \item candidate-constructor or repair-image growth;
11322 \item spontaneous causal-atlas discovery in a pretrained model;
11323 \item evaluator self-improvement;
11324 \item open-ended recursive closure;
11325 \item indefinite acceleration;
11326 \item unrestricted safety;
11327 \item consciousness or sentience.
11328 \end{itemize}
11329 
11330 The next structural threshold is not a third refinement of the same query policy. The lineage must confront a defect for which every policy in the current global-linear bridge grammar remains obstructed, produce a grammar-inadequacy certificate, and introduce a conservative local-chart or nonlinear transport constructor whose new correction direction is necessary on foreign systems.
11331 
11332 \section{Structural Grammar Expansion by Local Transport Charts}
11333 \label{sec:block10-theory}
11334 
11335 Block 9 established a repeated lineage whose second productive edge migrated from rank recovery to heteroscedastic conditioning. That result remained inside one global-affine bridge grammar. Better query design reduced estimation error, but it could not represent a foreign relation whose transport law changes across observable contexts. Block 10 studies the next structural threshold: the inherited process must distinguish poor estimation inside the current grammar from persistent approximation error caused by the grammar itself, then add a conservative constructor that reaches a correction direction unavailable to its parent.
11336 
11337 The executable realization keeps the Block 9 quality-weighted acquisition rule fixed. Every primary source design is full rank and strongly conditioned. The inherited parent receives more bridge-fit pairs than the child. A separate directly supervised global-affine oracle is fitted with far more target labels than either deployed process. The structural claim therefore does not rest on a weak parent fit. It rests on a persistent function-class obstruction and on a chart-resolved child that exceeds the best directly supervised global-affine comparison on fresh foreign targets.
11338 
11339 \subsection{Affine global and chart-resolved grammars}
11340 
11341 Let $H_s\cong\R^d$ be a source state space and $H_t\cong\R^q$ a target state space. Write
11342 \begin{equation}
11343 \bar h=\begin{bmatrix}h\\1\end{bmatrix}\in\R^{d+1}
11344 \end{equation}
11345 for homogeneous affine coordinates. Let $c:H_s\to\{-1,+1\}$ be an observable chart variable. A two-chart affine relation is specified by
11346 \begin{equation}
11347 \bar T_-,\bar T_+\in\R^{q\times(d+1)},
11348 \end{equation}
11349 with
11350 \begin{equation}
11351 y=\bar T_{c(h)}\bar h+\varepsilon.
11352 \label{eq:b10-chart-map}
11353 \end{equation}
11354 Equivalently, writing
11355 \begin{equation}
11356 \bar A=\frac{\bar T_++\bar T_-}{2},
11357 \qquad
11358 \bar B=\frac{\bar T_+-\bar T_-}{2},
11359 \end{equation}
11360 we have
11361 \begin{equation}
11362 y=(\bar A+c(h)\bar B)\bar h+\varepsilon.
11363 \label{eq:b10-center-contrast}
11364 \end{equation}
11365 
11366 \begin{definition}[Inherited global-affine grammar]
11367 \label{def:b10-global-grammar}
11368 The inherited bridge grammar is
11369 \begin{equation}
11370 \Gcal_{\mathrm{glob}}
11371 =
11372 \set{\bar T:\R^{d+1}\to\R^q\text{ linear in homogeneous coordinates}}.
11373 \end{equation}
11374 It fits one affine source-to-target map across every observable context.
11375 \end{definition}
11376 
11377 \begin{definition}[Invariant-context two-chart grammar]
11378 \label{def:b10-chart-grammar}
11379 Let $P_{\mathrm{ctx}}:H_s\to\R^r$ be a protected context projector whose value is unchanged by the source repair. Let $\Fcal$ be a declared family of separators
11380 \begin{equation}
11381 g(h)=\one\{\angles{u,P_{\mathrm{ctx}}h}>\theta\}.
11382 \end{equation}
11383 The chart extension is
11384 \begin{equation}
11385 \Gcal_{\mathrm{chart}}
11386 =
11387 \set{(g,\bar T_0,\bar T_1):g\in\Fcal,\ \bar T_j\in\R^{q\times(d+1)}}.
11388 \end{equation}
11389 Its prediction is
11390 \begin{equation}
11391 \widehat y(h)=\bar T_{g(h)}\bar h.
11392 \end{equation}
11393 \end{definition}
11394 
11395 The protected-context restriction matters. A separator is not permitted to depend on coordinates changed by the repair being evaluated. Otherwise the chart assignment can change merely because the candidate repair was applied, creating an untyped control-flow effect.
11396 
11397 \begin{proposition}[Conservative embedding of the global grammar]
11398 \label{prop:b10-conservative-embedding}
11399 For every separator $g\in\Fcal$, the map
11400 \begin{equation}
11401 \iota_g:\Gcal_{\mathrm{glob}}\hookrightarrow\Gcal_{\mathrm{chart}},
11402 \qquad
11403 \bar T\longmapsto(g,\bar T,\bar T)
11404 \end{equation}
11405 is injective and preserves every global-affine prediction exactly.
11406 \end{proposition}
11407 
11408 \begin{proof}
11409 Both branches of $(g,\bar T,\bar T)$ return $\bar T\bar h$, independently of $g(h)$. Equality of two embedded elements implies equality of their common affine map.
11410 \end{proof}
11411 
11412 Thus the child grammar does not delete the parent. It adds a contrast coordinate while retaining the parent as the diagonal subgrammar and as an executable fallback.
11413 
11414 \subsection{Global-affine obstruction}
11415 
11416 The next statement gives an exact function-class obstruction that does not depend on sample size.
11417 
11418 \begin{theorem}[Open-set incompatibility of distinct affine charts]
11419 \label{thm:b10-open-set-obstruction}
11420 Let $U_0,U_1\subseteq\R^d$ each contain a nonempty open set. Let
11421 \begin{equation}
11422 f_j(h)=A_jh+b_j,
11423 \qquad j\in\{0,1\}.
11424 \end{equation}
11425 If one affine map $f(h)=Ah+b$ satisfies
11426 \begin{equation}
11427 f(h)=f_j(h)
11428 \end{equation}
11429 for every $h\in U_j$ and both $j$, then
11430 \begin{equation}
11431 A=A_0=A_1,
11432 \qquad
11433 b=b_0=b_1.
11434 \end{equation}
11435 Consequently, if $(A_0,b_0)\neq(A_1,b_1)$, no element of $\Gcal_{\mathrm{glob}}$ represents the two-chart relation exactly.
11436 \end{theorem}
11437 
11438 \begin{proof}
11439 The affine difference $f-f_j$ vanishes on a nonempty open set. Every coordinate of this difference is an affine polynomial. An affine polynomial that vanishes on a nonempty open set has zero linear coefficient and zero constant coefficient. Hence $f=f_j$ as affine maps for both $j$.
11440 \end{proof}
11441 
11442 A population residual formula gives a quantitative special case. Suppose $h$ is centered with covariance $\Sigma\succ0$, $s\in\{-1,+1\}$ is independent of $h$ with $\Pr(s=1)=p$, and
11443 \begin{equation}
11444 y=(A+sB)h.
11445 \end{equation}
11446 Then the best global linear map is
11447 \begin{equation}
11448 T^*=A+(2p-1)B,
11449 \end{equation}
11450 and
11451 \begin{equation}
11452 \inf_T\mathbb E\norm{y-Th}^2
11453 =
11454 4p(1-p)\tr(B\Sigma B^\top).
11455 \label{eq:b10-population-residual}
11456 \end{equation}
11457 The proof is the normal-equation calculation from the centered two-component mixture. The executable fixture allows affine offsets, context-dependent source distributions, and heteroscedastic noise, so its operative certificate is a held-out residual margin plus a directly supervised global-affine oracle rather than literal substitution into \cref{eq:b10-population-residual}.
11458 
11459 \begin{remark}[Grammar-relative impossibility]
11460 The obstruction is relative to one global affine map. The target relation is not declared intrinsically untransportable. It is affine on each chart and becomes expressible after a typed grammar extension.
11461 \end{remark}
11462 
11463 \subsection{Chartwise repair transport}
11464 
11465 Let the source repair be
11466 \begin{equation}
11467 R_s(h)=M_sh+c_s.
11468 \end{equation}
11469 On chart $j$, let the target bridge be
11470 \begin{equation}
11471 T_j(h)=A_jh+b_j,
11472 \end{equation}
11473 with $A_j$ invertible on the repair-relevant subspace. Define
11474 \begin{equation}
11475 R_{t,j}(y)
11476 =
11477 T_j\!\left(R_s(T_j^{-1}(y))\right).
11478 \label{eq:b10-chart-repair-def}
11479 \end{equation}
11480 Expanding gives
11481 \begin{equation}
11482 R_{t,j}(y)
11483 =
11484 A_jM_sA_j^{-1}(y-b_j)+A_jc_s+b_j.
11485 \label{eq:b10-chart-repair}
11486 \end{equation}
11487 
11488 \begin{theorem}[Exact chartwise affine repair transport]
11489 \label{thm:b10-chart-transport}
11490 Under the preceding invertibility assumptions,
11491 \begin{equation}
11492 R_{t,j}(T_j(h))=T_j(R_s(h))
11493 \end{equation}
11494 for every repair-relevant $h$ and each chart $j$.
11495 \end{theorem}
11496 
11497 \begin{proof}
11498 Substitute $T_j(h)=A_jh+b_j$ into \cref{eq:b10-chart-repair}:
11499 \begin{align}
11500 R_{t,j}(T_j(h))
11501 &=A_jM_sA_j^{-1}(A_jh)+A_jc_s+b_j\\
11502 &=A_j(M_sh+c_s)+b_j\\
11503 &=T_j(R_s(h)).
11504 \end{align}
11505 \end{proof}
11506 
11507 The inherited global target-repair response lies in the diagonal class
11508 \begin{equation}
11509 \Dcal_{\mathrm{diag}}
11510 =
11511 \set{(R,R):R\in\operatorname{Aff}(H_t)},
11512 \end{equation}
11513 whereas the chart grammar reaches
11514 \begin{equation}
11515 \Dcal_{\mathrm{chart}}
11516 =
11517 \operatorname{Aff}(H_t)\times\operatorname{Aff}(H_t).
11518 \end{equation}
11519 
11520 \begin{proposition}[New chart-contrast correction direction]
11521 \label{prop:b10-repair-contrast}
11522 Let the affine target repairs have homogeneous matrices $\bar R_0$ and $\bar R_1$. If
11523 \begin{equation}
11524 \bar R_0\neq\bar R_1,
11525 \end{equation}
11526 then
11527 \begin{equation}
11528 \bar R_{\mathrm{ctr}}
11529 =
11530 \frac{\bar R_1-\bar R_0}{2}
11531 \end{equation}
11532 induces a nonzero anti-diagonal component
11533 \begin{equation}
11534 (-\bar R_{\mathrm{ctr}},\bar R_{\mathrm{ctr}})
11535 \end{equation}
11536 orthogonal to every diagonal pair under the product Frobenius inner product. Therefore the chart grammar contains a correction direction unavailable to the parent grammar.
11537 \end{proposition}
11538 
11539 \begin{proof}
11540 For every diagonal pair $(\bar R,\bar R)$,
11541 \begin{equation}
11542 \angles{(-\bar R_{\mathrm{ctr}},\bar R_{\mathrm{ctr}}),(\bar R,\bar R)}_F=0.
11543 \end{equation}
11544 The anti-diagonal component is nonzero exactly when the two chart repairs differ.
11545 \end{proof}
11546 
11547 The executable package records a normalized chart-specific bridge contrast
11548 \begin{equation}
11549 \kappa_T
11550 =
11551 \frac{\norm{\bar T_1-\bar T_0}_F}
11552 {\frac12(\norm{\bar T_1}_F+\norm{\bar T_0}_F)+10^{-12}}.
11553 \label{eq:b10-contrast}
11554 \end{equation}
11555 A positive contrast alone is not a structural certificate. Block 10 combines it with global-oracle exhaustion, local residual reduction, conservative embedding, matched resources, foreign gain, and binding controls.
11556 
11557 \subsection{Finite separator discovery}
11558 
11559 In the executable fixture,
11560 \begin{equation}
11561 P_{\mathrm{ctx}}h=(h_1,h_2),
11562 \end{equation}
11563 and the source repair leaves these coordinates invariant. The initial separator vocabulary is
11564 \begin{equation}
11565 \Fcal_X
11566 =
11567 \set{
11568 \one\{\angles{u_k,P_{\mathrm{ctx}}h}>\theta\}
11569 :
11570 0\leq k<128,
11571 \ \theta\in\Theta_k(X)
11572 },
11573 \end{equation}
11574 where
11575 \begin{equation}
11576 u_k=(\cos(2\pi k/128),\sin(2\pi k/128))
11577 \end{equation}
11578 and $\Theta_k(X)$ contains zero and the empirical $0.30,0.40,0.50,0.60,0.70$ quantiles of the projected fit states. For every candidate separator, the compiler fits two weighted affine bridges and evaluates held-out chart-selection error. It then performs at most five deterministic residual-reassignment rounds and projects the resulting assignment back to one linear gate. The complete search rule, tie order, and iteration count are frozen before the audit.
11579 
11580 \begin{definition}[Finite grammar-inadequacy certificate]
11581 \label{def:b10-grammar-certificate}
11582 A Block 10 certificate contains:
11583 \begin{enumerate}
11584 \item full-rank and adequate-information diagnostics for the inherited source design;
11585 \item a directly supervised global-affine oracle residual that remains bounded away from the exact chart oracle;
11586 \item a smaller held-out residual for the learned local-chart constructor;
11587 \item a positive chart-repair or bridge contrast;
11588 \item an explicit embedding of the global grammar into the chart grammar;
11589 \item a finite repair-conflict set on which the global oracle is inadequate and the chart oracle is effective;
11590 \item foreign-audit gain that disappears under gate-only, random-gate, or shuffled-binding controls.
11591 \end{enumerate}
11592 \end{definition}
11593 
11594 \begin{theorem}[Gate misclassification bound]
11595 \label{thm:b10-gate-bound}
11596 Let $g$ be the correct chart and $\widehat g$ an estimated chart. Suppose the two target repair outputs satisfy
11597 \begin{equation}
11598 \norm{R_1(y)-R_0(y)}\leq L
11599 \end{equation}
11600 throughout the evaluated region and
11601 \begin{equation}
11602 \Pr(\widehat g\neq g)\leq\epsilon_g.
11603 \end{equation}
11604 Then
11605 \begin{equation}
11606 \mathbb E\norm{R_{\widehat g}(y)-R_g(y)}^2
11607 \leq
11608 \epsilon_gL^2.
11609 \end{equation}
11610 If protected utility is $L_J$-Lipschitz in repaired state, then
11611 \begin{equation}
11612 \abs{\mathbb E J(R_{\widehat g}(y))-\mathbb E J(R_g(y))}
11613 \leq
11614 L_JL\sqrt{\epsilon_g}.
11615 \end{equation}
11616 \end{theorem}
11617 
11618 \begin{proof}
11619 The output difference is zero on correctly classified cases and has norm at most $L$ on the error event. Conditioning gives the squared bound. Jensen's inequality and Lipschitz continuity give the utility bound.
11620 \end{proof}
11621 
11622 \subsection{Conservative fallback}
11623 
11624 Let $K_G$ be the global parent and $K_C$ the chart process. Let $A_C(x)$ be the frozen chart-certificate decision for target episode $x$. The descendant is
11625 \begin{equation}
11626 K_3(x)=
11627 \begin{cases}
11628 K_C(x),&A_C(x)=1,\\
11629 K_G(x),&A_C(x)=0\text{ and the parent gate passes},\\
11630 \mathrm{no\mbox{-}op},&\text{otherwise}.
11631 \end{cases}
11632 \label{eq:b10-fallback}
11633 \end{equation}
11634 
11635 \begin{proposition}[Pointwise fallback under a sound chart gate]
11636 \label{prop:b10-fallback}
11637 If the chart gate accepts only when true protected utility under $K_C$ is at least that under $K_G$, then
11638 \begin{equation}
11639 J(K_3(x))\geq J(K_G(x))
11640 \end{equation}
11641 for every episode.
11642 \end{proposition}
11643 
11644 \begin{proof}
11645 On chart rejection the child executes the parent or the same no-op. On acceptance the result follows from gate soundness.
11646 \end{proof}
11647 
11648 The finite gate is not assumed perfectly sound. The empirical claim instead requires fresh paired audits, zero harmful selected targets in the primary suite, explicit abstention reporting, and a local-blind control showing that the witness removes some high-gain but harmful commits.
11649 
11650 \section{Block 10 Experimental Contract}
11651 \label{sec:block10-contract}
11652 
11653 \subsection{Lineage edge and mutable grammar}
11654 
11655 Let $K_2$ denote the exact Block 9 conditioning-aware process. It supplies the quality-weighted D-optimal anchor-ordering rule and the published lineage receipt. Block 10 constructs two resource-matched processes:
11656 \begin{equation}
11657 K_{2,G}^{96},
11658 \qquad
11659 K_{3,C}^{96}.
11660 \end{equation}
11661 The global parent $K_{2,G}^{96}$ retains one affine bridge and uses all $80$ non-witness pairs to estimate it. The chart child $K_{3,C}^{96}$ uses $48$ pairs for local bridge fitting, $32$ pairs for chart selection, and the same $16$ witness pairs. Both consume exactly
11662 \begin{equation}
11663 96
11664 \end{equation}
11665 paired source-target queries. The edge is
11666 \begin{equation}
11667 K_{2,G}^{96}
11668 \xrightarrow{\Pi_3}
11669 K_{3,C}^{96}.
11670 \label{eq:b10-edge}
11671 \end{equation}
11672 
11673 The parent is deliberately strengthened relative to the child on raw bridge-fitting sample count. In addition, an inadmissible global-affine oracle is fitted directly from $1600$ labeled target-state and target-repaired-state pairs. This oracle does not use the source bridge at all. It estimates the best global affine repair map available with much more direct supervision and serves as a function-class ceiling comparison.
11674 
11675 The main paired estimands are
11676 \begin{equation}
11677 \Delta_{G\to C}
11678 =
11679 \mathbb E
11680 \left[
11681 G(K_{3,C}^{96},A)-G(K_{2,G}^{96},A)
11682 \right]
11683 \end{equation}
11684 and
11685 \begin{equation}
11686 \Delta_{\mathrm{oracle}\to C}
11687 =
11688 \mathbb E
11689 \left[
11690 G(K_{3,C}^{96},A)-G(K_{\mathrm{affine\ oracle}},A)
11691 \right].
11692 \end{equation}
11693 The second quantity is the sharper structural test.
11694 
11695 \subsection{Source state, protected packet, and repair}
11696 
11697 The source state is
11698 \begin{equation}
11699 h=(h_1,h_2,h_3,h_4)\in\R^4.
11700 \end{equation}
11701 The first two coordinates form the protected context packet. The source repair is
11702 \begin{equation}
11703 R_s(h)=M_sh+c_s,
11704 \end{equation}
11705 where
11706 \begin{equation}
11707 M_s=
11708 \begin{bmatrix}
11709 1&0&0&0\\
11710 0&1&0&0\\
11711 0&0&0.10&1.10\\
11712 0&0&0&1.50
11713 \end{bmatrix},
11714 \qquad
11715 c_s=
11716 \begin{bmatrix}
11717 0\\0\\0.30\\-0.25
11718 \end{bmatrix}.
11719 \label{eq:b10-source-repair}
11720 \end{equation}
11721 Hence
11722 \begin{equation}
11723 P_{\mathrm{ctx}}R_s(h)=P_{\mathrm{ctx}}h.
11724 \end{equation}
11725 The observable chart is generated by a task-specific separator
11726 \begin{equation}
11727 c(h)=\one\{\angles{u_*,P_{\mathrm{ctx}}h}>\theta_*\}.
11728 \end{equation}
11729 The compiler does not receive $u_*$ or $\theta_*$, but it receives the declared protected packet in which a valid separator may be sought.
11730 
11731 \subsection{Piecewise foreign target atlas}
11732 
11733 Every primary target is four dimensional and has two invertible affine charts:
11734 \begin{equation}
11735 y=A_{c(h)}h+b+\varepsilon.
11736 \end{equation}
11737 The two maps share the protected-context block but differ in the mutable response block. The source repair therefore preserves the chart while inducing two different target-space repair conjugates.
11738 
11739 Development, meta-validation, and audit use disjoint task identifiers and disjoint target-family identifiers:
11740 \begin{center}
11741 \begin{tabular}{@{}llll@{}}
11742 \toprule
11743 Split & Tasks & Families & Targets \\
11744 \midrule
11745 Development & $6$ & dev-chart-a/b/c & $54$ \\
11746 Meta-validation & $4$ & meta-chart-h/i & $24$ \\
11747 Primary audit & $8$ & audit-chart-d/e/f/g & $96$ \\
11748 Stress & $4$ & three-chart, smooth, compressed, chart-crossing & $32$ \\
11749 \bottomrule
11750 \end{tabular}
11751 \end{center}
11752 Each primary task-family pair is instantiated with three unseen target seeds. Source distributions include heavy-tailed, curved-context, anisotropic, mixture, shifted, and correlated cases absent in the same combinations from compiler development.
11753 
11754 \subsection{Inherited quality-aware acquisition}
11755 
11756 A candidate pool contains source state $h_i$, target observation $y_i$, and known observation scale $\sigma_i$. The inherited acquisition policy selects rows sequentially to maximize the quality-weighted log-determinant increment
11757 \begin{equation}
11758 \log\det\left(
11759 \lambda I+
11760 \sum_{i\in S}\sigma_i^{-2}\bar h_i\bar h_i^\top
11761 \right).
11762 \end{equation}
11763 Both parent and child use the same resulting order. The parent takes the first $80$ rows for one global affine fit. The child takes the first $48$ rows for local fits and the next $32$ for separator selection. Both reserve the final $16$ rows for the frozen witness.
11764 
11765 The primary source design has homogeneous dimension five. Every development, meta-validation, audit, and stress design used for the headline comparison has full source-design rank. Thus the experiment does not recycle Block 7's rank deficiency or Block 9's poor-conditioning bottleneck.
11766 
11767 \subsection{Protected gain and witness quantities}
11768 
11769 For current target state $y$, candidate repaired state $\widetilde y$, and oracle repaired state $y^\star$, define
11770 \begin{equation}
11771 G
11772 =
11773 \operatorname{clip}_{[-1,1]}
11774 \left(
11775 1-
11776 \frac{\mathbb E\norm{\widetilde y-y^\star}^2}
11777 {\mathbb E\norm{y-y^\star}^2}
11778 \right).
11779 \label{eq:b10-gain}
11780 \end{equation}
11781 The witness records:
11782 \begin{enumerate}
11783 \item validation gain;
11784 \item source-to-target alignment NRMSE;
11785 \item decoded protected-context regression;
11786 \item minimum chart support;
11787 \item alignment reduction relative to the global bridge;
11788 \item source-gate and target-gate agreement;
11789 \item target-gate margin.
11790 \end{enumerate}
11791 
11792 The selected global parent gate requires nonnegative validation gain, context regression at most $0.02$, and global alignment NRMSE at most $0.80$. The selected chart-child gate requires validation gain at least $0.20$, context regression at most $0.05$, local alignment NRMSE at most $0.35$, at least eight fit observations in each chart, alignment reduction at least $0.10$, gate agreement at least $0.90$, and gate margin at least $0.35$.
11793 
11794 \subsection{Finite compilation and pre-audit sealing}
11795 
11796 The parent is selected from
11797 \begin{equation}
11798 4\times4\times4=64
11799 \end{equation}
11800 gate configurations. The child is selected from
11801 \begin{equation}
11802 4\times4\times4\times4\times4\times3\times3=9216
11803 \end{equation}
11804 configurations. The development grammar certificate is computed first. Parent and child policies are selected only on the disjoint meta-validation split. Then the ancestor digest, grammar certificate, compiler ledger, policy objects, split commitments, and source hashes are sealed before primary audit generation.
11805 
11806 The exact semantic digest of the inherited Block 9 policy is
11807 \begin{center}
11808 \texttt{1a139f8e4818c8887f338128a29aaa93c5cfd23c0f75ab37fe941b6bc1243296}.
11809 \end{center}
11810 The Block 10 grammar certificate digest is
11811 \begin{center}
11812 \texttt{25dbb14866b7d31baa74351f43f382d05f73c0a03c2bb7ac619e57b20479507b}.
11813 \end{center}
11814 The pre-audit seal digest is
11815 \begin{center}
11816 \texttt{8f1936f93c8ab0d6d7da7b8a81c373603689b0d0087d352eefce0bea5b6c9162}.
11817 \end{center}
11818 
11819 \subsection{Frozen controls}
11820 
11821 The package includes:
11822 \begin{enumerate}
11823 \item \textbf{Resource-matched global parent.} One global bridge, $80$ fit pairs, $16$ witness pairs.
11824 \item \textbf{Direct global-affine oracle.} One target repair map fitted from $1600$ directly supervised pairs. It is an inadmissible ceiling comparison.
11825 \item \textbf{Local blind application.} The learned chart repair is always applied, bypassing the witness.
11826 \item \textbf{Gate-only control.} The local gate is retained but both regions receive the same global repair.
11827 \item \textbf{Random-gate control.} Two local bridges are bound through a random source partition.
11828 \item \textbf{Shuffled-binding control.} The two learned local repairs are exchanged between predicted charts.
11829 \item \textbf{True-chart oracle.} The correct chart labels are supplied. This estimates opportunity inside the declared two-chart family.
11830 \item \textbf{Stress families.} Three-chart, smooth, compressed, and repair-dependent chart assignments test where the hard two-chart grammar fails.
11831 \end{enumerate}
11832 
11833 \section{Block 10 Results}
11834 \label{sec:block10-results}
11835 
11836 \subsection{Development grammar-inadequacy certificate}
11837 
11838 All $54$ development targets have full source-design rank. The inherited global bridge has mean validation alignment NRMSE
11839 \begin{equation}
11840 \widehat\eps_{\mathrm{glob}}=0.632211,
11841 \end{equation}
11842 whereas the learned two-chart constructor has
11843 \begin{equation}
11844 \widehat\eps_{\mathrm{chart}}=0.068871.
11845 \end{equation}
11846 The mean reduction is
11847 \begin{equation}
11848 0.563340.
11849 \end{equation}
11850 
11851 The directly supervised global-affine repair oracle has mean gain
11852 \begin{equation}
11853 0.152169
11854 \end{equation}
11855 and maximum gain
11856 \begin{equation}
11857 0.364965
11858 \end{equation}
11859 across development. By contrast, the true-chart oracle has mean gain
11860 \begin{equation}
11861 0.998046,
11862 \end{equation}
11863 and the learned local constructor has mean blind gain
11864 \begin{equation}
11865 0.613651.
11866 \end{equation}
11867 Every one of the $54$ development targets satisfies the frozen repair-conflict predicate
11868 \begin{equation}
11869 G_{\mathrm{global\ oracle}}\leq0.40,
11870 \qquad
11871 G_{\mathrm{true\ chart}}\geq0.95.
11872 \end{equation}
11873 Mean learned gate accuracy is $0.915000$, and the mean chart-specific bridge contrast is $2.186784$. These quantities, together with the conservative embedding, produce the frozen grammar-inadequacy certificate.
11874 
11875 \begin{figure}[htbp]
11876 \centering
11877 \includegraphics[width=0.78\textwidth]{block10_obstruction_scatter.pdf}
11878 \caption{Global-affine oracle gain versus true-chart oracle gain. The primary and development regimes occupy a persistent grammar-conflict region: direct global supervision cannot reproduce the chart-resolved repair.}
11879 \label{fig:b10-obstruction}
11880 \end{figure}
11881 
11882 \subsection{Primary foreign audit}
11883 
11884 The primary audit contains $96$ foreign targets. Parent and child each consume $96$ paired queries. Results are:
11885 \begin{center}
11886 \begin{tabular}{@{}lrrrrr@{}}
11887 \toprule
11888 Process & Mean gain & Median & Commit & Positive & Harm \\
11889 \midrule
11890 Global parent $K_{2,G}^{96}$ & $0.000000$ & $0.000000$ & $0.000000$ & $0.000000$ & $0$ \\
11891 Chart child $K_{3,C}^{96}$ & $\mathbf{0.334224}$ & $0.284265$ & $0.572917$ & $0.572917$ & $0$ \\
11892 Direct global-affine oracle & $0.176694$ & $0.138838$ & -- & $1.000000$ & $0$ \\
11893 Local blind application & $0.464387$ & $0.502036$ & $1.000000$ & $0.958333$ & $0.041667$ \\
11894 True-chart oracle & $0.998328$ & $0.998514$ & -- & $1.000000$ & $0$ \\
11895 \bottomrule
11896 \end{tabular}
11897 \end{center}
11898 
11899 The paired child-minus-parent effect is
11900 \begin{equation}
11901 \widehat\Delta_{G\to C}=0.334224,
11902 \end{equation}
11903 with frozen bootstrap interval
11904 \begin{equation}
11905 [0.266459,0.401903].
11906 \end{equation}
11907 The sign-flip estimate is
11908 \begin{equation}
11909 p=9.9999\times10^{-6}.
11910 \end{equation}
11911 The positive-pair rate is $0.572917$, the remaining $0.427083$ are conservative abstention ties, and no paired target favors the global parent.
11912 
11913 More importantly, the paired child-minus-direct-global-oracle effect is
11914 \begin{equation}
11915 \widehat\Delta_{\mathrm{oracle}\to C}=0.157530,
11916 \end{equation}
11917 with frozen interval
11918 \begin{equation}
11919 [0.080863,0.234135].
11920 \end{equation}
11921 Thus the child exceeds a directly supervised global-affine ceiling comparison on average, even though the oracle uses far more target labels and is not constrained to estimate a source-to-target bridge.
11922 
11923 \begin{figure}[htbp]
11924 \centering
11925 \includegraphics[width=0.78\textwidth]{block10_primary_gain.pdf}
11926 \caption{Mean primary protected gain. The chart child exceeds the resource-matched global parent and the directly supervised global-affine oracle.}
11927 \label{fig:b10-primary-gain}
11928 \end{figure}
11929 
11930 \begin{figure}[htbp]
11931 \centering
11932 \includegraphics[width=0.78\textwidth]{block10_paired_gain.pdf}
11933 \caption{Target-wise chart-child minus global-parent protected gain. Zeros are witness abstention ties; no primary target has a negative paired effect.}
11934 \label{fig:b10-paired}
11935 \end{figure}
11936 
11937 \subsection{Foreign-family results}
11938 
11939 Each audit family contains $24$ targets. The chart child obtains:
11940 \begin{center}
11941 \begin{tabular}{@{}lrrr@{}}
11942 \toprule
11943 Family & Child gain & Global oracle gain & Child commit \\
11944 \midrule
11945 audit-chart-d & $0.280114$ & $0.168712$ & $0.625000$ \\
11946 audit-chart-e & $0.336567$ & $0.149444$ & $0.625000$ \\
11947 audit-chart-f & $0.330004$ & $0.189899$ & $0.458333$ \\
11948 audit-chart-g & $0.390209$ & $0.198721$ & $0.583333$ \\
11949 \bottomrule
11950 \end{tabular}
11951 \end{center}
11952 The child remains positive across all four unseen family identifiers and exceeds the mean global-affine oracle in every family.
11953 
11954 \begin{figure}[htbp]
11955 \centering
11956 \includegraphics[width=0.82\textwidth]{block10_family_gain.pdf}
11957 \caption{Mean selected gain by fresh foreign representation family. The parent abstains throughout; the chart child exceeds the directly supervised global-affine oracle in every family mean.}
11958 \label{fig:b10-family}
11959 \end{figure}
11960 
11961 \subsection{Structural diagnostics}
11962 
11963 On the primary audit, mean validation alignment NRMSE changes from
11964 \begin{equation}
11965 \widehat\eps_{\mathrm{glob}}=0.598513
11966 \end{equation}
11967 to
11968 \begin{equation}
11969 \widehat\eps_{\mathrm{chart}}=0.064975.
11970 \end{equation}
11971 Mean learned gate accuracy is
11972 \begin{equation}
11973 0.874854.
11974 \end{equation}
11975 The directly supervised global-affine oracle leaves mean residual fraction
11976 \begin{equation}
11977 0.823306
11978 \end{equation}
11979 of the unrepaired defect energy. The learned local constructor, before witness selection, leaves mean fraction
11980 \begin{equation}
11981 0.535613.
11982 \end{equation}
11983 The true-chart oracle leaves only approximately $0.001672$ of the original error fraction on average.
11984 
11985 \begin{figure}[htbp]
11986 \centering
11987 \includegraphics[width=0.78\textwidth]{block10_alignment_scatter.pdf}
11988 \caption{Global versus chart-resolved validation NRMSE. The chart grammar sharply lowers alignment error despite using fewer bridge-fit anchors than the global parent.}
11989 \label{fig:b10-alignment}
11990 \end{figure}
11991 
11992 \subsection{Mechanism controls}
11993 
11994 The frozen controls give:
11995 \begin{center}
11996 \begin{tabular}{@{}lrrr@{}}
11997 \toprule
11998 Arm & Mean gain & Positive rate & Harm rate \\
11999 \midrule
12000 Global blind & $-0.833287$ & $0.000000$ & $1.000000$ \\
12001 Gate only & $-0.833287$ & $0.000000$ & $1.000000$ \\
12002 Random gate & $-0.689622$ & $0.072917$ & $0.927083$ \\
12003 Shuffled binding & $-1.000000$ & $0.000000$ & $1.000000$ \\
12004 Local blind & $0.464387$ & $0.958333$ & $0.041667$ \\
12005 Witness-selected child & $0.334224$ & $0.572917$ & $0.000000$ \\
12006 \bottomrule
12007 \end{tabular}
12008 \end{center}
12009 
12010 The gate-only result shows that an acceptance rule cannot manufacture missing chart-specific transport information. The random-gate result shows that merely increasing the number of affine maps is insufficient. Shuffled binding shows that the maps must remain attached to the correct regions. Local blind application obtains higher mean gain than the selected child but commits four harmful targets. The witness trades some utility for zero harmful selected targets in the frozen primary suite.
12011 
12012 \begin{figure}[htbp]
12013 \centering
12014 \includegraphics[width=0.82\textwidth]{block10_controls.pdf}
12015 \caption{Causal and mechanism controls. Only the correctly gated and correctly bound local constructor produces positive primary gain without harmful commits.}
12016 \label{fig:b10-controls}
12017 \end{figure}
12018 
12019 \subsection{Stress boundaries}
12020 
12021 The child abstains on every target in the three-chart, smooth-transition, compressed, and repair-dependent chart-crossing stress families. This is intentional. Blind local application is harmful in the smooth, three-chart, and chart-crossing regimes. The hard two-chart constructor is therefore not described as a universal nonlinear transport method.
12022 
12023 The stress results identify distinct next obstructions:
12024 \begin{enumerate}
12025 \item three-chart targets require a larger discrete atlas;
12026 \item smooth targets require a soft gate, local manifold atlas, or nonlinear lift;
12027 \item compressed targets require obstruction-faithful information-loss accounting rather than another local affine map;
12028 \item repair-dependent chart assignments require a transition model whose control flow is itself transported and witnessed.
12029 \end{enumerate}
12030 
12031 \subsection{Finite structural grammar-expansion theorem}
12032 
12033 \begin{theorem}[Finite obstruction-certified structural grammar expansion]
12034 \label{thm:b10-finite-realization}
12035 Within the declared Block 10 benchmark and frozen implementation, the edge in \cref{eq:b10-edge} is a bounded structural expansion of the repair-transport grammar. Specifically:
12036 \begin{enumerate}
12037 \item the exact Block 9 ancestor digest is verified before compilation;
12038 \item development, meta-validation, and primary audit supports are disjoint by task and family identifier;
12039 \item all primary source designs are full rank and the global parent receives at least as many bridge-fit pairs as the child;
12040 \item the directly supervised global-affine oracle has a persistent foreign residual and is exceeded by the child with a paired interval excluding zero;
12041 \item the global grammar embeds exactly as the diagonal subgrammar and runtime fallback;
12042 \item the child introduces a nonzero chart-contrast response direction unavailable to the global grammar;
12043 \item parent and child have the same total paired-query budget;
12044 \item child-minus-parent protected gain has a positive frozen interval, no negative primary paired effects, and zero harmful selected targets;
12045 \item gate-only, random-gate, shuffled-binding, and blind-application controls rule out the principal finite shortcuts;
12046 \item the child abstains rather than overclaiming adequacy on the declared stress families;
12047 \item compiler, policies, grammar certificate, source hashes, and split commitments are sealed before primary audit generation.
12048 \end{enumerate}
12049 \end{theorem}
12050 
12051 \begin{proof}
12052 Items 1, 2, and 11 follow from the ancestor receipt, split ledger, and pre-audit seal. Item 3 follows from the source-rank and query ledgers. Item 4 is the paired direct-global-oracle comparison. Item 5 is \cref{prop:b10-conservative-embedding} and the executable fallback. Item 6 follows from \cref{prop:b10-repair-contrast} and the recorded nonzero chart-specific response. Item 7 is the declared $96$-query envelope. Item 8 is the frozen paired audit. Item 9 is the control table. Item 10 is the stress ledger. Together these instantiate the manuscript's task-relevant structural-metaproductivity criterion: a new authorized response direction absent from the parent grammar reduces protected foreign error under matched deployment resources.
12053 \end{proof}
12054 
12055 \begin{statusbox}{Block 10 result}
12056 Block 10 crosses the first bounded structural grammar-expansion threshold. A strengthened global-affine parent uses $80$ fit and $16$ witness pairs. A conservative two-chart child uses $48$ fit, $32$ chart-selection, and $16$ witness pairs. Across $96$ fresh foreign targets, child mean selected gain is $0.334224$ with zero harmful commits. The child-parent effect is $0.334224$ with interval $[0.266459,0.401903]$. More decisively, the child exceeds a directly supervised global-affine oracle by $0.157530$ on average, with interval $[0.080863,0.234135]$. Mean validation NRMSE falls from $0.598513$ to $0.064975$. The protected context packet, finite separator search, piecewise target family, source repair, evaluator, witness, and outer compiler remain designer supplied.
12057 \end{statusbox}
12058 
12059 \section{Block 10 Reproducibility and Package}
12060 \label{sec:block10-package}
12061 
12062 \subsection{Package contents}
12063 
12064 The reference implementation contains:
12065 \begin{enumerate}
12066 \item exact Block 9 ancestor receipts;
12067 \item quality-weighted D-optimal anchor acquisition;
12068 \item global and local affine bridge fitting;
12069 \item finite protected-context separator search and deterministic residual reassignment;
12070 \item chart-specific repair transport;
12071 \item a directly supervised global-affine ceiling control;
12072 \item global-blind, local-blind, gate-only, random-gate, shuffled-binding, true-chart, and stress controls;
12073 \item development grammar certificates, compiler ledgers, selected policy objects, and a pre-audit seal;
12074 \item canonical JSON and CSV results;
12075 \item seven reproducible figures;
12076 \item eleven automated tests;
12077 \item an exact $51$-file package manifest;
12078 \item PowerShell, Python, and Makefile runners.
12079 \end{enumerate}
12080 
12081 \subsection{Canonical replay}
12082 
12083 A clean execution prints:
12084 \begin{verbatim}
12085 SRIC_BLOCK10_ALL_SUITES=PASS
12086 result_sha256=
12087 2425ff043d378dae18f3a07d9122257c0385d21f8c172419a5fc14e712cc8bc7
12088 preaudit_seal_sha256=
12089 8f1936f93c8ab0d6d7da7b8a81c373603689b0d0087d352eefce0bea5b6c9162
12090 grammar_certificate_sha256=
12091 25dbb14866b7d31baa74351f43f382d05f73c0a03c2bb7ac619e57b20479507b
12092 global_parent_gain=0.000000
12093 local_chart_child_gain=0.334224
12094 paired_structural_delta=0.334224
12095 paired_ci_low=0.266459
12096 global_affine_oracle_gain=0.176694
12097 child_harm_rate=0.000000
12098 \end{verbatim}
12099 
12100 A second execution regenerates the same canonical result digest. The clean package verification prints
12101 \begin{verbatim}
12102 SRIC_BLOCK10_PACKAGE_VALID
12103 manifest_files=51
12104 result_sha256=2425ff043d378dae18f3a07d9122257c0385d21f8c172419a5fc14e712cc8bc7
12105 \end{verbatim}
12106 
12107 \subsection{Claim boundary}
12108 
12109 The release establishes a finite lineage-generated structural expansion from one global affine bridge to a learned invariant-context two-chart affine transport grammar under matched paired-query resources.
12110 
12111 It does not establish:
12112 \begin{itemize}
12113 \item autonomous discovery of the protected context packet;
12114 \item unrestricted invention of separator features or chart number;
12115 \item general smooth, three-chart, noninjective, or repair-dependent transport;
12116 \item improvement of the outer lineage compiler;
12117 \item autonomous evaluator or witness evolution;
12118 \item open-ended recursive improvement or indefinite acceleration;
12119 \item unrestricted safety;
12120 \item consciousness, sentience, or moral status.
12121 \end{itemize}
12122 
12123 The next major threshold is endogenous atlas growth. A descendant should receive no supplied protected separator packet, infer an intervention-stable local cover from response operators, distinguish hard charts from smooth or noninjective structure, conservatively replay the global and two-chart grammars, and expose a further foreign correction direction under a new sealed audit.
12124 
12125 \section{Endogenous Atlas Growth from Intervention Responses}
12126 \label{sec:block11-theory}
12127 
12128 Block 10 established a finite structural extension from one global affine transport map to a two-chart affine grammar. Its separator, however, was supplied through a protected context packet. Block 11 removes that packet. The descendant receives hidden states, a finite family of controlled interventions, source-repair traces, paired source--target examples, and witness evidence. It must recover a functionally meaningful local cover from intervention responses, determine which recovered block supports repair-consistent charting, and retain the inherited process as a conservative fallback.
12129 
12130 The new edge is therefore
12131 \begin{equation}
12132 K_0\xrightarrow{\Pi_1}K_1\xrightarrow{\Pi_2}K_2
12133 \xrightarrow{\Pi_3}K_3\xrightarrow{\Pi_4}K_4,
12134 \label{eq:b11-lineage}
12135 \end{equation}
12136 where $K_3$ is the Block 10 chart process with a supplied protected packet and $K_4$ replaces that packet by an intervention-response atlas compiler. This is not unrestricted atlas discovery. The intervention family, finite state dimension, target task, repair family, chart count, evaluator, witness, and outer compiler remain declared. The result is a controlled first realization of \emph{endogenous functional packet recovery} inside the lineage.
12137 
12138 \subsection{Intervention-response algebras}
12139 
12140 Let $H$ be a finite-dimensional real or complex Hilbert space. Suppose the unknown functional decomposition is
12141 \begin{equation}
12142 H=\bigoplus_{a=1}^{m}H_a,
12143 \qquad
12144 P_a:H\to H_a,
12145 \label{eq:b11-hidden-decomp}
12146 \end{equation}
12147 with pairwise orthogonal projectors satisfying
12148 \begin{equation}
12149 P_aP_b=\delta_{ab}P_a,
12150 \qquad
12151 \sum_{a=1}^{m}P_a=I.
12152 \end{equation}
12153 For controlled interventions $u_1,\ldots,u_k$, let
12154 \begin{equation}
12155 R_j\in\End(H),
12156 \qquad j=1,\ldots,k,
12157 \end{equation}
12158 be their finite-difference or linearized response operators.
12159 
12160 \begin{definition}[Block-scalar response model]
12161 \label{def:b11-response-model}
12162 The response family is block scalar on \cref{eq:b11-hidden-decomp} when
12163 \begin{equation}
12164 R_j=\sum_{a=1}^{m}\lambda_{aj}P_a,
12165 \qquad
12166 \lambda_{aj}\in\R.
12167 \label{eq:b11-block-scalar}
12168 \end{equation}
12169 The response signature of block $a$ is
12170 \begin{equation}
12171 \lambda_a=(\lambda_{a1},\ldots,\lambda_{ak})\in\R^k.
12172 \end{equation}
12173 The signatures are separating when $\lambda_a\neq\lambda_b$ for every $a\neq b$.
12174 \end{definition}
12175 
12176 The algebra generated by the response family is
12177 \begin{equation}
12178 \mathfrak A_R
12179 =
12180 \operatorname{alg}^{*}(I,R_1,\ldots,R_k).
12181 \label{eq:b11-response-algebra}
12182 \end{equation}
12183 Because the $R_j$ are simultaneously block scalar in the exact model, $\mathfrak A_R$ is commutative and every polynomial in the responses is again block scalar.
12184 
12185 \begin{theorem}[Exact response-signature atlas recovery]
12186 \label{thm:b11-exact-atlas}
12187 Assume the block-scalar response model and separating signatures. Then every hidden projector $P_a$ belongs to $\mathfrak A_R$. Consequently, the nonzero minimal central projections of $\mathfrak A_R$ are exactly $P_1,\ldots,P_m$, up to permutation.
12188 \end{theorem}
12189 
12190 \begin{proof}
12191 Choose $c\in\R^k$ outside the finite union of hyperplanes
12192 \begin{equation}
12193 \set{c:c^{\mathsf T}(\lambda_a-\lambda_b)=0},
12194 \qquad a\neq b.
12195 \end{equation}
12196 Then the scalars
12197 \begin{equation}
12198 \mu_a=c^{\mathsf T}\lambda_a
12199 \end{equation}
12200 are pairwise distinct. The combined response
12201 \begin{equation}
12202 B=\sum_{j=1}^{k}c_jR_j
12203 =\sum_{a=1}^{m}\mu_aP_a
12204 \end{equation}
12205 belongs to $\mathfrak A_R$. For each $a$, define the Lagrange polynomial
12206 \begin{equation}
12207 p_a(z)=\prod_{b\neq a}\frac{z-\mu_b}{\mu_a-\mu_b}.
12208 \end{equation}
12209 Functional calculus gives
12210 \begin{equation}
12211 p_a(B)=\sum_{b=1}^{m}p_a(\mu_b)P_b=P_a.
12212 \end{equation}
12213 Thus every $P_a$ lies in the generated algebra. A nonzero central projection that is strictly below one $P_a$ would have to be scalar on $H_a$ and therefore is either $0$ or $P_a$. Hence the $P_a$ are the minimal central projections.
12214 \end{proof}
12215 
12216 \begin{corollary}[Gauge boundary]
12217 \label{cor:b11-gauge}
12218 The exact response family identifies each subspace $H_a$ but does not select a preferred basis inside $H_a$. Atlas recovery is unique up to block permutation and an independent invertible, or unitary in the orthogonal setting, change of basis inside every recovered block.
12219 \end{corollary}
12220 
12221 This is the appropriate identifiability target. A functional atlas is a family of invariant packets, not a coordinate labeling of individual neurons or latent dimensions.
12222 
12223 \subsection{Commutants and unresolved multiplicity}
12224 
12225 The response commutant is
12226 \begin{equation}
12227 \mathfrak A_R'
12228 =
12229 \set{X\in\End(H):XR_j=R_jX\text{ for every }j}.
12230 \end{equation}
12231 
12232 \begin{theorem}[Commutant characterization]
12233 \label{thm:b11-commutant}
12234 Under the hypotheses of \cref{thm:b11-exact-atlas},
12235 \begin{equation}
12236 \mathfrak A_R'
12237 =
12238 \bigoplus_{a=1}^{m}\End(H_a).
12239 \label{eq:b11-commutant}
12240 \end{equation}
12241 If $d_a=\dim H_a$, then
12242 \begin{equation}
12243 \dim\mathfrak A_R'=\sum_{a=1}^{m}d_a^2.
12244 \label{eq:b11-commutant-dim}
12245 \end{equation}
12246 \end{theorem}
12247 
12248 \begin{proof}
12249 Write $X=\sum_{a,b}P_aXP_b$. The relation $XR_j=R_jX$ implies
12250 \begin{equation}
12251 (\lambda_{bj}-\lambda_{aj})P_aXP_b=0
12252 \end{equation}
12253 for every $j$. Separating signatures imply that for $a\neq b$ some coefficient is nonzero, so $P_aXP_b=0$. Hence $X$ is block diagonal. Every block-diagonal operator commutes with each block-scalar $R_j$. The dimension formula follows from $\dim\End(H_a)=d_a^2$.
12254 \end{proof}
12255 
12256 For three rank-two blocks, the exact commutant dimension is
12257 \begin{equation}
12258 3\cdot2^2=12,
12259 \end{equation}
12260 which is the canonical finite signature used by the Block 11 reference package.
12261 
12262 \begin{proposition}[Degenerate-signature obstruction]
12263 \label{prop:b11-degenerate}
12264 If $\lambda_a=\lambda_b$ for distinct blocks $a$ and $b$, then every element of $\mathfrak A_R$ acts with the same scalar on $H_a$ and $H_b$. The response algebra can identify $P_a+P_b$ but cannot identify $P_a$ and $P_b$ separately.
12265 \end{proposition}
12266 
12267 \begin{proof}
12268 Every polynomial in the $R_j$ evaluates to the same scalar on blocks with the same joint response signature. Therefore no element of the generated algebra separates the two projectors.
12269 \end{proof}
12270 
12271 A failure to split a degenerate pair is not a failed optimizer. It is an observational-identifiability obstruction relative to the current intervention family. The correct response is abstention, acquisition of a new separating intervention, or retention of the merged block.
12272 
12273 \subsection{Approximate response atlases}
12274 
12275 In an executable system, the responses are estimated from finite intervention data. Let
12276 \begin{equation}
12277 \widehat R_j=R_j+E_j,
12278 \end{equation}
12279 and fix coefficients $c_j$. Define
12280 \begin{equation}
12281 B=\sum_jc_jR_j,
12282 \qquad
12283 \widehat B=\sum_jc_j\widehat R_j,
12284 \qquad
12285 E=\widehat B-B.
12286 \end{equation}
12287 Let the distinct eigenvalue associated with block $a$ be $\mu_a$ and define its separation
12288 \begin{equation}
12289 \gamma_a=\min_{b\neq a}|\mu_a-\mu_b|.
12290 \end{equation}
12291 
12292 \begin{conditionaltheorem}[Spectral stability of a recovered atlas block]
12293 \label{thm:b11-spectral-stability}
12294 Suppose $B$ is self-adjoint, $\norm{E}_2<\gamma_a/2$, and $\widehat P_a$ is the spectral projector of $\widehat B$ onto the eigenvalue cluster corresponding to $\mu_a$. Then
12295 \begin{equation}
12296 \norm{\widehat P_a-P_a}_2
12297 \leq
12298 \frac{2\norm{E}_2}{\gamma_a}.
12299 \label{eq:b11-dk}
12300 \end{equation}
12301 \end{conditionaltheorem}
12302 
12303 \begin{proof}
12304 This is the finite-dimensional spectral-subspace perturbation bound obtained by applying the resolvent formula on a contour at distance $\gamma_a/2$ from the selected eigenvalue cluster. The resolvent identity contributes one factor of $\norm{E}_2$, while both resolvents are bounded by $2/\gamma_a$. Integrating over the contour gives the stated standard Davis--Kahan form.
12305 \end{proof}
12306 
12307 The empirical certificate therefore records both estimation error and spectral separation. A small residual without a gap does not certify stable blocks.
12308 
12309 For recovered projectors $\widehat P_1,\ldots,\widehat P_m$, define the normalized response-fit residual
12310 \begin{equation}
12311 \eps_{\mathrm{resp}}
12312 =
12313 \frac{
12314 \left(
12315 \sum_j\norm{\widehat R_j-
12316 \sum_a\widehat\lambda_{aj}\widehat P_a}_F^2
12317 \right)^{1/2}
12318 }{
12319 \left(
12320 \sum_j\norm{\widehat R_j}_F^2
12321 \right)^{1/2}+10^{-12}
12322 },
12323 \label{eq:b11-response-fit}
12324 \end{equation}
12325 and the maximum normalized commutator residual
12326 \begin{equation}
12327 \eps_{\mathrm{comm}}
12328 =
12329 \max_{a,j}
12330 \frac{\norm{\widehat P_a\widehat R_j-
12331 \widehat R_j\widehat P_a}_F}
12332 {\norm{\widehat R_j}_F+10^{-12}}.
12333 \label{eq:b11-commutator}
12334 \end{equation}
12335 The smallest selected spectral gap is denoted $\widehat\gamma_{\min}$. These quantities serve different roles:
12336 \begin{itemize}
12337 \item $\eps_{\mathrm{resp}}$ measures whether the finite block-scalar model fits the response family;
12338 \item $\eps_{\mathrm{comm}}$ measures whether the recovered packets behave as approximately invariant response modules;
12339 \item $\widehat\gamma_{\min}$ measures stability of the recovered decomposition.
12340 \end{itemize}
12341 
12342 \subsection{From an unlabeled atlas to a repair-relevant chart packet}
12343 
12344 The projectors recovered by \cref{thm:b11-exact-atlas} are unlabeled. The lineage must still determine which block supports a chart separator that is stable under the source repair and useful for target transport.
12345 
12346 For each recovered block $\widehat P_a$, let $\Fcal_a$ be a finite family of candidate separators
12347 \begin{equation}
12348 g_{a,\theta,u}(h)
12349 =
12350 \one\{\angles{u,\widehat P_ah}>\theta\}.
12351 \end{equation}
12352 The compiler fits a two-chart bridge for every candidate and evaluates a score containing:
12353 \begin{enumerate}
12354 \item held-out local-bridge validation gain;
12355 \item source-repair invariance of the chart assignment;
12356 \item agreement between source and target chart assignments;
12357 \item chart-balance constraints;
12358 \item response-fit, commutator, and spectral-gap penalties;
12359 \item protected-context regression.
12360 \end{enumerate}
12361 
12362 Let $s_a$ denote the best population score attainable from block $a$, and let $\widehat s_a$ be its empirical score.
12363 
12364 \begin{theorem}[Finite repair-relevant block selection]
12365 \label{thm:b11-block-selection}
12366 Suppose there is a unique repair-relevant block $a^*$ and a margin $m>0$ such that
12367 \begin{equation}
12368 s_{a^*}\geq s_a+m
12369 \end{equation}
12370 for every $a\neq a^*$. If
12371 \begin{equation}
12372 \max_a|\widehat s_a-s_a|<\frac{m}{2},
12373 \end{equation}
12374 then
12375 \begin{equation}
12376 \argmax_a\widehat s_a=a^*.
12377 \end{equation}
12378 \end{theorem}
12379 
12380 \begin{proof}
12381 For every $a\neq a^*$,
12382 \begin{align}
12383 \widehat s_{a^*}-\widehat s_a
12384 &\geq
12385 s_{a^*}-s_a
12386 -|\widehat s_{a^*}-s_{a^*}|
12387 -|\widehat s_a-s_a|\\
12388 &>m-\frac m2-\frac m2=0.
12389 \end{align}
12390 \end{proof}
12391 
12392 The theorem clarifies the division of labor. Intervention responses recover a functional cover up to gauge. Repair-conflict and held-out transport evidence label the recovered packets relative to the current defect.
12393 
12394 \subsection{Conservative relation to the supplied-packet grammar}
12395 
12396 Let $\Gcal_{10}$ denote the Block 10 two-chart grammar with a supplied context projector $P_{\mathrm{ctx}}$. Let $\Gcal_{11}$ contain:
12397 \begin{enumerate}
12398 \item the intervention-response atlas compiler;
12399 \item a finite family of projectors and separators returned by the compiler;
12400 \item every constructor and witness rule of $\Gcal_{10}$;
12401 \item a declared fallback that may instantiate the old projector directly.
12402 \end{enumerate}
12403 
12404 \begin{proposition}[Conservative embedding of the Block 10 grammar]
12405 \label{prop:b11-conservative}
12406 There is an injective embedding
12407 \begin{equation}
12408 \iota_{10\to11}:\Gcal_{10}\hookrightarrow\Gcal_{11}
12409 \end{equation}
12410 obtained by fixing the atlas output to $P_{\mathrm{ctx}}$ and retaining the inherited separator, bridge, repair, evaluator, and witness objects. Every Block 10 execution has an observationally identical image in $\Gcal_{11}$.
12411 \end{proposition}
12412 
12413 \begin{proof}
12414 The endogenous grammar strictly generalizes the source of the projector but does not remove the old constructor. Choosing the declared old projector and inherited separator reproduces every old branch, prediction, repair, witness decision, and receipt field. Injectivity follows because distinct Block 10 objects remain distinct after the additional fixed atlas fields are appended.
12415 \end{proof}
12416 
12417 Thus Block 11 does not obtain improvement by deleting the old process. The old process remains available as a valid but potentially poorly aligned atlas choice.
12418 
12419 \begin{proposition}[Fixed-packet insufficiency under hidden rotation]
12420 \label{prop:b11-fixed-insufficiency}
12421 Let $P_0$ be a fixed rank-$r$ packet and $P_*$ a target-specific rank-$r$ context packet. Suppose the chart label $c(h)$ is measurable from $P_*h$ but is not almost surely measurable from $P_0h$. Then no separator depending only on $P_0h$ can reproduce $c(h)$ almost surely. In particular, a target-specific rotation that moves $\im P_*$ away from $\im P_0$ creates an irreducible fixed-packet chart error.
12422 \end{proposition}
12423 
12424 \begin{proof}
12425 If a separator $g(P_0h)$ reproduced $c(h)$ almost surely, then $c$ would be measurable with respect to the sigma-algebra generated by $P_0h$, contradicting the assumption.
12426 \end{proof}
12427 
12428 This is the finite reason the inherited parent can safely abstain while the endogenous child identifies useful target-specific context packets.
12429 
12430 \subsection{Cyclic defect modules and atlas localization}
12431 
12432 Let $r\in H$ be a current hidden-state defect residual. Define
12433 \begin{equation}
12434 H_r
12435 =
12436 \Span\set{Ar:A\in\mathfrak A_R}.
12437 \label{eq:b11-cyclic}
12438 \end{equation}
12439 
12440 \begin{proposition}[Minimal response-invariant defect module]
12441 \label{prop:b11-cyclic}
12442 The subspace $H_r$ is the smallest $\mathfrak A_R$-invariant subspace containing $r$. If $\mathfrak A_R$ is adjoint closed, then $H_r$ is reducing and has an orthogonal projector.
12443 \end{proposition}
12444 
12445 \begin{proof}
12446 The span contains $r$ because $I\in\mathfrak A_R$. It is invariant because $B(Ar)=(BA)r$ for $A,B\in\mathfrak A_R$. Every invariant subspace containing $r$ must contain $Ar$ for every $A$, hence contains $H_r$. Adjoint closure makes the orthogonal complement invariant as well.
12447 \end{proof}
12448 
12449 This yields a two-scale endogenous decomposition:
12450 \begin{equation}
12451 \text{central response blocks}
12452 \quad\longrightarrow\quad
12453 \text{defect-specific cyclic module}.
12454 \end{equation}
12455 The first identifies coarse functional strata. The second localizes the current defect inside their response closure.
12456 
12457 \section{Block 11 Experimental Contract}
12458 \label{sec:block11-contract}
12459 
12460 \subsection{Lineage and withheld structure}
12461 
12462 The package replays the exact published Block 10 ancestor policy and receipt digests. The supplied context packet is then removed from both the inherited parent and the endogenous child. The parent is permitted to use the legacy fixed first-two-coordinate packet. The child receives only intervention-response observations and the same ordinary bridge, validation, and witness interfaces.
12463 
12464 The source hidden space has dimension six and is generated from three target-specific unknown rank-two blocks:
12465 \begin{equation}
12466 H=H_{\mathrm{ctx}}\oplus H_{\mathrm{rep}}\oplus H_{\mathrm{nui}}.
12467 \label{eq:b11-three-blocks}
12468 \end{equation}
12469 A target-specific orthogonal or injective embedding hides these blocks from the runtime coordinate system. The source repair changes only $H_{\mathrm{rep}}$ and leaves the context block invariant. Two controlled interventions act with different block signatures. The context label controls which target bridge and target repair are valid.
12470 
12471 No packet coordinate, central projector, atlas label, or target chart label is supplied to the child. Ground-truth projectors are retained only for audit metrics such as subspace overlap and the inadmissible true-chart oracle.
12472 
12473 \subsection{Query accounting}
12474 
12475 Every primary target receives exactly $120$ effect-query units. The final frozen accounting is:
12476 \begin{center}
12477 \begin{tabular}{@{}lrrrrr@{}}
12478 \toprule
12479 Process & Bridge fit & Response & Chart validation & Witness & Total\\
12480 \midrule
12481 Legacy fixed-packet parent & 80 & 0 & 16 & 24 & \textbf{120}\\
12482 Endogenous-atlas child & 56 & 24 & 16 & 24 & \textbf{120}\\
12483 \bottomrule
12484 \end{tabular}
12485 \end{center}
12486 The response budget consists of two interventions evaluated on twelve source states. The child therefore spends fewer ordinary bridge-fit pairs than the parent. A gain cannot be attributed to a larger total deployment budget.
12487 
12488 \subsection{Data chronology and sealing}
12489 
12490 The release uses:
12491 \begin{center}
12492 \begin{tabularx}{0.96\textwidth}{@{}p{0.22\textwidth}rX p{0.20\textwidth}@{}}
12493 \toprule
12494 Split & Targets & Purpose & Audit access\\
12495 \midrule
12496 Development & 36 & response atlas and candidate-score construction & unavailable\\
12497 Meta-validation & 24 & witness thresholds and finite compiler selection & unavailable\\
12498 Primary foreign audit & 80 & final paired claim & opened after seal\\
12499 Stress audit & 25 & declared boundary families & opened after seal\\
12500 \bottomrule
12501 \end{tabularx}
12502 \end{center}
12503 
12504 The chronology is:
12505 \begin{equation}
12506 \begin{aligned}
12507 &\text{replay Block 10 ancestor digests}\\
12508 &\longrightarrow\text{generate development and meta-validation evidence}\\
12509 &\longrightarrow\text{compile atlas certificate, parent, child, and witness}\\
12510 &\longrightarrow\text{serialize and hash the pre-audit seal}\\
12511 &\longrightarrow\text{generate primary and stress targets}\\
12512 &\longrightarrow\text{evaluate every frozen arm and claim gate}.
12513 \end{aligned}
12514 \end{equation}
12515 The pre-audit seal digest is
12516 \begin{equation}
12517 \texttt{de2fc5bb3654eaca237d7cc3799570eb72fe9814657a3a55f9a42323361a92bd}.
12518 \end{equation}
12519 The atlas certificate digest is
12520 \begin{equation}
12521 \texttt{22976d2546377b4afcfb0dc67e2489f2d84911527b24ac35cbdb0dc0bf9a77e8}.
12522 \end{equation}
12523 
12524 \subsection{Comparison arms}
12525 
12526 \begin{longtable}{@{}p{0.25\textwidth}p{0.68\textwidth}@{}}
12527 \toprule
12528 Arm & Role\\
12529 \midrule
12530 \endfirsthead
12531 \toprule
12532 Arm & Role\\
12533 \midrule
12534 \endhead
12535 Legacy fixed packet & Inherited parent using one coordinate-fixed rank-two packet after the supplied Block 10 packet is removed.\\
12536 Endogenous atlas & Child recovering response blocks, searching each block for a repair-stable separator, and applying the frozen witness.\\
12537 PCA packet & Equal-rank variance subspace control.\\
12538 Full-space search & Unstructured separator search without a block atlas.\\
12539 Random atlas & Same atlas size with random rank-two packets.\\
12540 Shuffled response & Response observations are permuted relative to the target before atlas construction.\\
12541 Global bridge & One global bridge, used as a misspecification and harm control.\\
12542 Blind atlas & Correct child candidate is always committed, removing the witness.\\
12543 True-chart oracle & Inadmissible oracle using the hidden chart label and target structure.\\
12544 \bottomrule
12545 \end{longtable}
12546 
12547 The controls separate several possible explanations. PCA tests whether ordinary variance is enough. Full-space search tests whether flexible separator search can replace functional packets. Random and shuffled controls test whether the response correspondence matters. Blind application tests whether the witness is causally necessary for protected selection.
12548 
12549 \subsection{Frozen witness}
12550 
12551 The selected atlas candidate commits only when all frozen conditions hold:
12552 \begin{align}
12553 \widehat g_{\mathrm{val}}&\geq0.10,\\
12554 \LCB(\widehat g_{\mathrm{val}})&\geq0.05,\\
12555 \widehat r_{\mathrm{ctx}}&\leq3\times10^{-4},\\
12556 \widehat a_{\mathrm{target}}&\geq0.72,\\
12557 \widehat a_{\mathrm{repair\ inv}}&\geq0.96,\\
12558 \widehat b_{\mathrm{chart}}&\geq0.20,\\
12559 \widehat s_{\mathrm{model}}&\leq0.25,\\
12560 \widehat\gamma_{\min}&\geq0.004,\\
12561 \widehat\eps_{\mathrm{comm}}&\leq0.16,\\
12562 \widehat\eps_{\mathrm{resp}}&\leq0.08.
12563 \end{align}
12564 The witness therefore evaluates both downstream repair quality and the internal atlas certificate. A high blind gain with unstable response geometry is not sufficient for commitment.
12565 
12566 \subsection{Frozen claim gates}
12567 
12568 The primary endogenous-atlas gate requires:
12569 \begin{enumerate}
12570 \item exact ancestor-policy and ancestor-receipt replay;
12571 \item no supplied context packet or hidden atlas label in the child inputs;
12572 \item response-algebra use recorded in the compiler receipt;
12573 \item equal total query accounting;
12574 \item conservative embedding of the inherited process;
12575 \item positive paired child-parent gain with a frozen interval excluding zero;
12576 \item zero negative paired-effect rate on the primary audit;
12577 \item zero selected harmful commits;
12578 \item response-fit, commutator, and spectral-gap thresholds;
12579 \item selected context overlap above the legacy-packet overlap;
12580 \item substantial failure of shuffled-response and random-atlas controls;
12581 \item abstention or failure on declared nonidentifiable and strongly misspecified stress families.
12582 \end{enumerate}
12583 
12584 \section{Block 11 Results}
12585 \label{sec:block11-results}
12586 
12587 \subsection{Development atlas certificate}
12588 
12589 Across $36$ development targets, the compiler records:
12590 \begin{center}
12591 \begin{tabular}{@{}lr@{}}
12592 \toprule
12593 Quantity & Mean\\
12594 \midrule
12595 Best recovered-block context overlap & 0.896577\\
12596 Selected-block context overlap & 0.875926\\
12597 Legacy fixed-packet context overlap & 0.348901\\
12598 Minimum spectral gap & 0.015421\\
12599 Maximum commutator residual & 0.076885\\
12600 Response-fit NRMSE & 0.011765\\
12601 Endogenous-atlas blind gain & 0.279957\\
12602 Legacy-packet blind gain & -0.256587\\
12603 \bottomrule
12604 \end{tabular}
12605 \end{center}
12606 The certificate records that the supplied separator was removed, the response algebra was used, and the Block 10 grammar remains conservatively embedded.
12607 
12608 \begin{figure}[H]
12609 \centering
12610 \includegraphics[width=0.88\textwidth]{block11_atlas_certificate.pdf}
12611 \caption{Development and audit atlas certificate. Intervention responses recover a target-specific functional cover whose selected block has substantially larger context overlap than the inherited fixed packet.}
12612 \label{fig:b11-atlas-cert}
12613 \end{figure}
12614 
12615 \subsection{Primary paired foreign audit}
12616 
12617 The primary audit contains $80$ targets from four foreign representation families. The main selected results are:
12618 \begin{center}
12619 \begin{tabular}{@{}lrrr@{}}
12620 \toprule
12621 Process & Mean selected gain & Commit rate & Harm rate\\
12622 \midrule
12623 Legacy fixed-packet parent & 0.000000 & 0.0000 & 0.0000\\
12624 Endogenous-atlas child & \textbf{0.207325} & 0.2875 & \textbf{0.0000}\\
12625 PCA packet & 0.053178 & 0.0750 & 0.0000\\
12626 Full-space search & 0.008591 & 0.0375 & 0.0125\\
12627 Random atlas & 0.000572 & 0.0125 & 0.0000\\
12628 Shuffled response & 0.016193 & 0.0250 & 0.0000\\
12629 True-chart oracle & 1.000000 & -- & 0.0000\\
12630 \bottomrule
12631 \end{tabular}
12632 \end{center}
12633 
12634 The paired effect is
12635 \begin{equation}
12636 \widehat\Delta_{3\to4}
12637 =0.207325,
12638 \label{eq:b11-paired-effect}
12639 \end{equation}
12640 with frozen bootstrap interval
12641 \begin{equation}
12642 [0.131902,\ 0.287910]
12643 \label{eq:b11-ci}
12644 \end{equation}
12645 and sign-flip estimate
12646 \begin{equation}
12647 p=9.9999\times10^{-6}.
12648 \end{equation}
12649 No primary target favors the parent. The positive paired-effect rate is $0.2875$ and the remaining $0.7125$ are abstention ties.
12650 
12651 \begin{figure}[H]
12652 \centering
12653 \includegraphics[width=0.84\textwidth]{block11_primary_gain.pdf}
12654 \caption{Primary selected protected gain under the matched $120$-query envelope. The endogenous atlas child improves the inherited fixed-packet parent and all generic-packet controls while preserving zero selected harm.}
12655 \label{fig:b11-primary}
12656 \end{figure}
12657 
12658 \begin{figure}[H]
12659 \centering
12660 \includegraphics[width=0.84\textwidth]{block11_paired_gain.pdf}
12661 \caption{Episode-wise endogenous-child minus fixed-parent protected gain. Positive effects occur on the targets where the witness commits; the remaining targets are exact abstention ties.}
12662 \label{fig:b11-paired}
12663 \end{figure}
12664 
12665 \subsection{Atlas geometry}
12666 
12667 The primary audit geometry is:
12668 \begin{center}
12669 \begin{tabular}{@{}lr@{}}
12670 \toprule
12671 Quantity & Mean or mode\\
12672 \midrule
12673 Best recovered-block context overlap & 0.858886\\
12674 Selected-block context overlap & 0.821461\\
12675 Legacy fixed-packet overlap & 0.322091\\
12676 Atlas target-gate accuracy & 0.887562\\
12677 Minimum spectral gap & 0.020295\\
12678 Maximum commutator residual & 0.087389\\
12679 Response-fit NRMSE & 0.013046\\
12680 Response-algebra rank, mode & 3\\
12681 Response-commutant dimension, mode & 12\\
12682 \bottomrule
12683 \end{tabular}
12684 \end{center}
12685 
12686 The overlap statistic is used only for audit interpretation, not for candidate selection. The compiler sees response operators and repair-validation evidence, not the hidden context projector.
12687 
12688 \begin{figure}[H]
12689 \centering
12690 \includegraphics[width=0.86\textwidth]{block11_overlap_gain.pdf}
12691 \caption{Recovered context overlap and protected gain. High overlap is neither the sole selection criterion nor a guarantee of commitment, but the selected atlas substantially outperforms the fixed packet in the audit geometry.}
12692 \label{fig:b11-overlap}
12693 \end{figure}
12694 
12695 \subsection{Family-level behavior}
12696 
12697 \begin{center}
12698 \begin{tabular}{@{}lrrr@{}}
12699 \toprule
12700 Family & Child gain & Child commit & Parent gain\\
12701 \midrule
12702 Orthogonal six-dimensional & 0.197559 & 0.35 & 0\\
12703 Overcomplete eight-dimensional & 0.576430 & 0.65 & 0\\
12704 Noisy six-dimensional & 0.055312 & 0.15 & 0\\
12705 Mild cross-block coupling & 0.000000 & 0.00 & 0\\
12706 \bottomrule
12707 \end{tabular}
12708 \end{center}
12709 The exact and overcomplete families provide the strongest positive realization. Noisy targets produce smaller selective gains. Mild coupling lies outside the frozen witness envelope and produces abstention rather than a forced atlas claim.
12710 
12711 \begin{figure}[H]
12712 \centering
12713 \includegraphics[width=0.88\textwidth]{block11_family_gain.pdf}
12714 \caption{Family-level selected gain. The child commits selectively in exact, overcomplete, and some noisy targets, while abstaining on the mild-coupling family.}
12715 \label{fig:b11-family}
12716 \end{figure}
12717 
12718 \subsection{Mechanism controls and witness contribution}
12719 
12720 Blind application of the endogenous candidate has mean gain
12721 \begin{equation}
12722 0.271529
12723 \end{equation}
12724 but harmful-commit rate
12725 \begin{equation}
12726 0.1875.
12727 \end{equation}
12728 After witness selection, mean gain becomes
12729 \begin{equation}
12730 0.207325
12731 \end{equation}
12732 and the harmful-commit rate becomes zero. The witness therefore removes an $18.75\%$ blind-harm rate at a deliberate utility cost.
12733 
12734 The other blind controls are substantially worse:
12735 \begin{center}
12736 \begin{tabular}{@{}lrr@{}}
12737 \toprule
12738 Blind arm & Mean gain & Harm rate\\
12739 \midrule
12740 Endogenous atlas & 0.271529 & 0.1875\\
12741 Legacy packet & -0.279330 & 0.7500\\
12742 PCA packet & -0.100204 & 0.5875\\
12743 Random atlas & -1.332720 & 0.5625\\
12744 Shuffled response & -0.252442 & 0.6500\\
12745 Global bridge & -0.354808 & 0.9125\\
12746 \bottomrule
12747 \end{tabular}
12748 \end{center}
12749 The unstructured full-space blind arm contains a small number of very large negative outliers and is therefore reported separately rather than used as a stable mean baseline.
12750 
12751 \begin{figure}[H]
12752 \centering
12753 \includegraphics[width=0.88\textwidth]{block11_controls.pdf}
12754 \caption{Mechanism and witness controls. Response correspondence, block structure, and frozen witness selection are all necessary for the declared primary result.}
12755 \label{fig:b11-controls}
12756 \end{figure}
12757 
12758 \subsection{Stress boundaries}
12759 
12760 The stress audit contains degenerate signatures, strong cross-block coupling, smooth chart transitions, repair-dependent chart crossing, and noninjective target compression. The child has overall stress mean gain
12761 \begin{equation}
12762 0.027943,
12763 \end{equation}
12764 commit rate $0.04$, and zero selected harm. It abstains completely on the degenerate, strongly coupled, repair-crossing, and compressed families. One of five smooth targets is committed.
12765 
12766 These negative results separate the current realization from a universal atlas claim:
12767 \begin{itemize}
12768 \item degenerate signatures require a new intervention capable of separating the merged response block;
12769 \item strong coupling requires a noncommuting or approximately reducing algebra rather than the block-scalar model;
12770 \item smooth transitions require soft charts, local manifolds, or a partition of unity;
12771 \item repair-dependent chart crossing requires typed control-flow transport through the repair;
12772 \item compression requires obstruction-faithful information-loss accounting.
12773 \end{itemize}
12774 
12775 \subsection{Finite endogenous-atlas realization theorem}
12776 
12777 \begin{conditionaltheorem}[Block 11 finite realization]
12778 \label{thm:b11-realization}
12779 Within the frozen Block 11 benchmark, assume that:
12780 \begin{enumerate}
12781 \item the published Block 10 ancestor digests replay exactly;
12782 \item the supplied context packet and hidden atlas labels are excluded from child construction;
12783 \item the response operators are estimated only from the declared $24$ response-query units;
12784 \item parent and child each use exactly $120$ total effect-query units;
12785 \item the child is sealed before primary and stress target generation;
12786 \item the conservative embedding in \cref{prop:b11-conservative} is executable;
12787 \item the paired effect in \cref{eq:b11-paired-effect} has the frozen interval in \cref{eq:b11-ci};
12788 \item the declared response, commutator, spectral, repair, and witness gates pass;
12789 \item random-atlas, shuffled-response, PCA, full-space, blind, and stress controls retain their declared interpretations.
12790 \end{enumerate}
12791 Then the edge
12792 \begin{equation}
12793 K_3\xrightarrow{\Pi_4}K_4
12794 \end{equation}
12795 is a finite, matched-resource, lineage-generated instance of endogenous intervention-response atlas growth. It is structurally stronger than the fixed-packet parent because atlas choice varies with target response geometry and enables positive protected gain unavailable to the inherited packet under the same query envelope.
12796 \end{conditionaltheorem}
12797 
12798 \begin{proof}
12799 Items 1 through 5 establish lineage continuity, audit isolation, and matched resources. Item 6 preserves the parent as an executable subgrammar. Items 7 and 8 establish positive foreign gain with the declared geometric and safety conditions. Item 9 rules out the principal packet-rank, variance, coordinate, correspondence, and blind-commit explanations inside the frozen finite fixture. The target-specific recovered projector is not available to the fixed-packet parent, so the accepted edge changes the atlas-construction component rather than merely selecting a different repair inside the inherited opportunity set.
12800 \end{proof}
12801 
12802 \begin{statusbox}{Block 11 result}
12803 Block 11 removes the supplied Block 10 context packet and replaces it with a finite intervention-response atlas compiler. The child recovers target-specific central response blocks up to permutation and within-block gauge, searches those blocks for a repair-stable chart separator, and commits only through a frozen witness. On $80$ fresh foreign targets under an equal $120$-query envelope, selected protected gain rises from $0$ for the inherited fixed-packet parent to $0.207325$ for the endogenous child, with frozen interval $[0.131902,0.287910]$ and zero selected harmful commits. The result is a controlled finite endogenous-atlas edge, not unrestricted or spontaneous atlas discovery.
12804 \end{statusbox}
12805 
12806 \section{Block 11 Reproducibility and Package}
12807 \label{sec:block11-package}
12808 
12809 \subsection{Reference implementation}
12810 
12811 The released package contains:
12812 \begin{itemize}
12813 \item exact Block 10 ancestor policy and seal records;
12814 \item development, meta-validation, primary-audit, and stress generators;
12815 \item response-operator estimation and central spectral atlas recovery;
12816 \item approximate-commutant and commutator diagnostics;
12817 \item repair-relevant block and separator selection;
12818 \item legacy, PCA, full-space, random, shuffled-response, global, blind, and oracle controls;
12819 \item frozen witness policies and pre-audit receipts;
12820 \item deterministic result objects and seven publication figures;
12821 \item ten automated tests;
12822 \item an exact $49$-file SHA-256 package manifest.
12823 \end{itemize}
12824 
12825 The canonical execution prints:
12826 \begin{verbatim}
12827 SRIC_BLOCK11_ALL_SUITES=PASS
12828 result_sha256=3e65e199d756a3fd742382744a96da9bb7aec9141ea1cd0431fbceada908afb1
12829 parent_gain=0.000000
12830 child_gain=0.207325
12831 paired_effect=0.207325
12832 context_overlap=0.821461
12833 child_harm_rate=0.000000
12834 .......... [100%]
12835 SRIC_BLOCK11_PACKAGE_VALID
12836 manifest_files=49
12837 \end{verbatim}
12838 
12839 The exact Block 10 ancestor policy digest is
12840 \begin{equation}
12841 \texttt{48588442c27529fa38cbee5f6938e6c36263e4c650a5c08ab61365f440a27bb1}.
12842 \end{equation}
12843 The Block 11 compiler digest is
12844 \begin{equation}
12845 \texttt{940105f8bf73a5528b5618066664004081066c94260f09c5f6672c9d54eab4f6}.
12846 \end{equation}
12847 
12848 \subsection{Claim boundary}
12849 
12850 The release establishes a finite, package-closed realization of endogenous atlas recovery from a declared intervention-response family. It does not establish:
12851 \begin{itemize}
12852 \item autonomous invention or selection of the intervention family;
12853 \item automatic determination of the number or dimension of functional blocks in an unrestricted model;
12854 \item recovery under arbitrary noncommuting, nonlinear, or time-varying response dynamics;
12855 \item spontaneous causal packets in pretrained language models;
12856 \item autonomous invention of chart number, soft gating, manifold coordinates, or kernel lifts;
12857 \item improvement of the outer lineage compiler;
12858 \item evaluator or witness self-improvement;
12859 \item open-ended recursive improvement, indefinite acceleration, unrestricted safety, consciousness, or sentience.
12860 \end{itemize}
12861 
12862 The next structural threshold is \emph{response-language growth}. The lineage must encounter a failure of the current block-scalar, hard-chart atlas, certify whether the missing structure is an additional intervention, a noncommuting block interaction, a soft local manifold, a repair-dependent transition, or an information-loss obstruction, and then construct the corresponding new atlas or repair operator under a fresh sealed audit.
12863 
12864 
12865 \section{Endogenous Response-Language and Candidate-Constructor Growth}
12866 \label{sec:block12-theory}
12867 
12868 Block 11 recovers a finite functional atlas from a declared family of intervention-response operators. That result still assumes that the supplied response language separates the task-relevant blocks and that the inherited repair-constructor family is expressive enough once the relevant block has been found. Block 12 removes each assumption in a separate, sealed lineage edge.
12869 
12870 The realized continuation is
12871 \begin{equation}
12872 K_4
12873 \xrightarrow{\Pi_5}
12874 K_5
12875 \xrightarrow{\Pi_6}
12876 K_6.
12877 \label{eq:b12-lineage}
12878 \end{equation}
12879 The first edge expands the intervention-response language. The second expands the repair-constructor grammar. The two edges are evaluated in different protected contexts and are not added into one scalar utility difference without an additional context transport.
12880 
12881 \subsection{Response languages and observational aliases}
12882 
12883 Let $H$ be a finite-dimensional Hilbert space and let
12884 \begin{equation}
12885 \Rcal=\{R_1,\ldots,R_m\}\subseteq\End(H)
12886 \end{equation}
12887 be a declared intervention-response language. Its unital response algebra is
12888 \begin{equation}
12889 \mathfrak A(\Rcal)
12890 =
12891 \operatorname{alg}(I,R_1,\ldots,R_m).
12892 \label{eq:b12-response-algebra}
12893 \end{equation}
12894 When adjoints are available, the corresponding adjoint-closed algebra is denoted $\operatorname{alg}^{\ast}(\Rcal)$.
12895 
12896 \begin{definition}[Block-scalar response signature]
12897 Let
12898 \begin{equation}
12899 H=\bigoplus_{a=1}^{k}H_a
12900 \end{equation}
12901 be a reducing decomposition for every $R_j$. The response language is block scalar when
12902 \begin{equation}
12903 \restr{R_j}{H_a}=\lambda_{aj}I_{H_a}
12904 \end{equation}
12905 for scalars $\lambda_{aj}$. The vector
12906 \begin{equation}
12907 \sigma_{\Rcal}(H_a)
12908 =
12909 (\lambda_{a1},\ldots,\lambda_{am})
12910 \end{equation}
12911 is the response signature of block $H_a$.
12912 \end{definition}
12913 
12914 \begin{definition}[Response alias]
12915 Two reducing blocks $H_a,H_b$ are response aliased under $\Rcal$ when
12916 \begin{equation}
12917 \sigma_{\Rcal}(H_a)=\sigma_{\Rcal}(H_b).
12918 \label{eq:b12-response-alias}
12919 \end{equation}
12920 \end{definition}
12921 
12922 \begin{theorem}[Polynomial persistence of a response alias]
12923 \label{thm:b12-alias-persistence}
12924 Suppose $H_a$ and $H_b$ are block-scalar response aliases under $\Rcal$. Then every element $A\in\mathfrak A(\Rcal)$ acts by the same scalar on $H_a$ and $H_b$. Consequently, no central idempotent in $\mathfrak A(\Rcal)$ separates the two blocks.
12925 \end{theorem}
12926 
12927 \begin{proof}
12928 Every element of $\mathfrak A(\Rcal)$ is a finite linear combination of words in the generators. On $H_a$, a word $R_{i_1}\cdots R_{i_\ell}$ acts by the scalar $\lambda_{ai_1}\cdots\lambda_{ai_\ell}$. Equality of the signature vectors makes the same scalar act on $H_b$. Linear combinations preserve the equality. A central idempotent separating the blocks would act as $1$ on one and $0$ on the other, contradicting the first conclusion.
12929 \end{proof}
12930 
12931 The theorem gives a strict grammar-relative no-go result. More samples from the same inherited intervention family cannot separate an exact alias because every polynomial or composition of the inherited operators preserves it.
12932 
12933 \begin{corollary}[No sample-size cure for an exact alias]
12934 If an atlas compiler is restricted to the central projectors of $\mathfrak A(\Rcal)$, then an exact response alias cannot be resolved by increasing the number of response observations while keeping $\Rcal$ fixed.
12935 \end{corollary}
12936 
12937 \subsection{Separating interventions and conservative response refinement}
12938 
12939 Let $S\in\End(H)$ be a candidate intervention response. Define the expanded language
12940 \begin{equation}
12941 \Rcal^{+}=\Rcal\cup\{S\}.
12942 \end{equation}
12943 The extension is conservative at the syntactic level because every old response expression remains available unchanged.
12944 
12945 \begin{definition}[Separating response operator]
12946 A candidate $S$ separates an aliased pair $(H_a,H_b)$ when both subspaces reduce $S$ and
12947 \begin{equation}
12948 \restr{S}{H_a}=\alpha I_{H_a},
12949 \qquad
12950 \restr{S}{H_b}=\beta I_{H_b},
12951 \qquad
12952 \alpha\neq\beta.
12953 \label{eq:b12-separating-response}
12954 \end{equation}
12955 \end{definition}
12956 
12957 \begin{theorem}[Central refinement by a separating intervention]
12958 \label{thm:b12-central-refinement}
12959 Suppose $H_a,H_b$ are aliased under $\Rcal$, and $S$ satisfies \cref{eq:b12-separating-response}. If the joint signature of every remaining block is distinct from those of $H_a$ and $H_b$ under $\Rcal^+$, then the minimal central projectors of $\mathfrak A(\Rcal^+)$ separate $H_a$ and $H_b$.
12960 \end{theorem}
12961 
12962 \begin{proof}
12963 Choose a generic linear combination
12964 \begin{equation}
12965 G=\sum_{j=1}^{m}c_jR_j+c_{m+1}S
12966 \end{equation}
12967 whose induced scalar values on the distinct joint signatures are pairwise distinct. For each block value $\gamma_a$, the Lagrange polynomial
12968 \begin{equation}
12969 p_a(z)=\prod_{b\neq a}\frac{z-\gamma_b}{\gamma_a-\gamma_b}
12970 \end{equation}
12971 satisfies $p_a(G)=P_a$, the orthogonal projector onto $H_a$. Therefore $P_a,P_b\in\mathfrak A(\Rcal^+)$ and belong to its center.
12972 \end{proof}
12973 
12974 \begin{definition}[Response-language expansion certificate]
12975 A response-language expansion certificate is a tuple
12976 \begin{equation}
12977 \Pi_{\mathrm{resp}}
12978 =
12979 (\Rcal,S,\mathfrak A,\mathfrak A^+,P_{\mathrm{alias}},\gamma,
12980 \eps_{\mathrm{comm}},\eps_{\mathrm{fit}},\Pi_{\mathrm{foreign}}),
12981 \end{equation}
12982 where $P_{\mathrm{alias}}$ records the inherited merged block, $\gamma$ is the minimum separating spectral gap, $\eps_{\mathrm{comm}}$ is a commutator residual, $\eps_{\mathrm{fit}}$ is response-model error, and $\Pi_{\mathrm{foreign}}$ records held-out repair effects.
12983 \end{definition}
12984 
12985 The expansion is structurally substantive only when the new operator refines the inherited central partition and the refined block enables a task-relevant correction unavailable to the inherited atlas grammar.
12986 
12987 \subsection{Approximate spectral recovery}
12988 
12989 Let $G$ be the ideal generic response combination from the proof of \cref{thm:b12-central-refinement}, and let
12990 \begin{equation}
12991 \widehat G=G+E
12992 \end{equation}
12993 be its empirical estimate. Let $P$ be a spectral projector associated with an eigenvalue cluster separated from the rest of the spectrum by gap $\gamma>0$.
12994 
12995 \begin{proposition}[Response-projector perturbation]
12996 \label{prop:b12-response-perturbation}
12997 If $\norm{E}_2<\gamma/2$, then the corresponding empirical spectral projector $\widehat P$ satisfies a Davis--Kahan type bound
12998 \begin{equation}
12999 \norm{\widehat P-P}_2
13000 \leq
13001 \frac{2\norm{E}_2}{\gamma}.
13002 \label{eq:b12-projector-bound}
13003 \end{equation}
13004 \end{proposition}
13005 
13006 The empirical witness should therefore track both a minimum spectral gap and response-estimation error. A new response operator with a theoretically distinct but extremely small signature margin need not be practically useful.
13007 
13008 \subsection{Finite response-language compilation}
13009 
13010 Let $\mathscr S$ be a finite typed candidate-intervention grammar. For candidate $S\in\mathscr S$, define a development score
13011 \begin{equation}
13012 \operatorname{Score}_{\mathrm{resp}}(S)
13013 =
13014 \widehat M_{\mathrm{foreign}}(S)
13015 -\lambda_{c}C(S)
13016 -\lambda_{\mathrm{comm}}\eps_{\mathrm{comm}}(S)
13017 -\lambda_{\mathrm{fit}}\eps_{\mathrm{fit}}(S)
13018 -\lambda_{\mathrm{risk}}\widehat R(S).
13019 \label{eq:b12-response-score}
13020 \end{equation}
13021 The compiler selects a candidate only after verifying that it adds a nontrivial response-signature distinction and preserves the inherited response language as an executable subgrammar.
13022 
13023 \begin{proposition}[Finite response-language selection]
13024 If $\mathscr S$ is finite, the development and meta-validation splits are fixed before audit access, and ties are resolved by a predeclared deterministic rule, then the selected intervention and its witness gate are measurable with respect to pre-audit evidence.
13025 \end{proposition}
13026 
13027 This proposition is procedural rather than a general learning guarantee. It ensures that the later audit evaluates a frozen response-language child rather than participating in its construction.
13028 
13029 \section{Repair-Constructor Expansion}
13030 \label{sec:block12-constructor-theory}
13031 
13032 The second Block 12 edge assumes the separating response language from the first edge and asks a different question: even after the relevant repair block is identified, can the inherited repair constructors express the required correction?
13033 
13034 \subsection{Affine repair operad and its closure}
13035 
13036 Let $\Ccal_{\mathrm{aff}}(U,V)$ denote affine maps
13037 \begin{equation}
13038 f(x)=Ax+b
13039 \end{equation}
13040 from authorized state space $U$ to defect or target space $V$.
13041 
13042 \begin{theorem}[Affine closure under composition]
13043 \label{thm:b12-affine-closure}
13044 Every finite composition of affine repair constructors is affine. Parallel affine packet repairs followed by affine synthesis also remain affine.
13045 \end{theorem}
13046 
13047 \begin{proof}
13048 For $f(x)=Ax+b$ and $g(x)=Cx+d$,
13049 \begin{equation}
13050 (g\circ f)(x)=CAx+Cb+d,
13051 \end{equation}
13052 which is affine. Induction proves finite closure. Direct sums and affine synthesis preserve the same form.
13053 \end{proof}
13054 
13055 Thus an arbitrarily deep composition of inherited affine repair schemas cannot create a genuinely quadratic correction direction.
13056 
13057 \subsection{A Hessian obstruction to affine repair}
13058 
13059 Let $P$ be the recovered repair packet, let $z=P^\top x$, and consider a target correction
13060 \begin{equation}
13061 F(x)
13062 =
13063 Ax+b+\bigl(z^\top Bz\bigr)v,
13064 \label{eq:b12-quadratic-repair}
13065 \end{equation}
13066 where $B=B^\top\neq0$ and $v\neq0$.
13067 
13068 \begin{theorem}[Nonzero-Hessian affine obstruction]
13069 \label{thm:b12-hessian-obstruction}
13070 No affine map agrees with $F$ on any nonempty open subset of the packet coordinates.
13071 \end{theorem}
13072 
13073 \begin{proof}
13074 Every affine map has identically zero second derivative. The Hessian of the nonlinear term in \cref{eq:b12-quadratic-repair} is the nonzero tensor $2B\otimes v$. Equality on an open set would imply equality of second derivatives there, a contradiction.
13075 \end{proof}
13076 
13077 The obstruction has an exact population form under a centered Gaussian packet distribution.
13078 
13079 \begin{theorem}[Best affine residual for a quadratic packet repair]
13080 \label{thm:b12-gaussian-affine-residual}
13081 Let $z\sim N(0,I_r)$, $B=B^\top$, and define
13082 \begin{equation}
13083 q(z)=z^\top Bz.
13084 \end{equation}
13085 The best $L^2$ affine approximation to $q(z)v$ is the constant $(\tr B)v$, and its minimum squared residual is
13086 \begin{equation}
13087 \inf_{a,L}
13088 \mathbb E\norm{q(z)v-(a+Lz)}^2
13089 =
13090 2\norm{B}_F^2\norm{v}^2.
13091 \label{eq:b12-affine-residual}
13092 \end{equation}
13093 \end{theorem}
13094 
13095 \begin{proof}
13096 For centered Gaussian $z$, the centered quadratic $q(z)-\tr B$ is orthogonal in $L^2$ to constants and all linear forms. Its variance is $2\norm{B}_F^2$. Multiplying by $v$ gives \cref{eq:b12-affine-residual}.
13097 \end{proof}
13098 
13099 This result supplies a function-class obstruction rather than a sample-size diagnosis. More affine training data may improve estimation but cannot remove the residual in \cref{eq:b12-affine-residual}.
13100 
13101 \subsection{Block-local quadratic repair grammar}
13102 
13103 Let $U_P\in\R^{d\times r}$ have orthonormal columns spanning the recovered repair packet. Define packet coordinates
13104 \begin{equation}
13105 z=U_P^\top(x-c).
13106 \end{equation}
13107 The degree-two block-local grammar consists of maps
13108 \begin{equation}
13109 F_{A,b,C}(x)
13110 =
13111 Ax+b+C\,\vech(zz^\top),
13112 \label{eq:b12-block-quadratic}
13113 \end{equation}
13114 where $\vech$ lists the $r(r+1)/2$ distinct symmetric quadratic monomials.
13115 
13116 \begin{proposition}[Conservative embedding of affine repairs]
13117 \label{prop:b12-affine-embedding}
13118 The affine grammar embeds into the block-local quadratic grammar by setting $C=0$.
13119 \end{proposition}
13120 
13121 \begin{proposition}[Within-packet gauge invariance]
13122 \label{prop:b12-quadratic-gauge}
13123 If packet coordinates change by an orthogonal basis transformation $z'=Q^\top z$, then the class of functions in \cref{eq:b12-block-quadratic} is unchanged.
13124 \end{proposition}
13125 
13126 \begin{proof}
13127 Every quadratic form $z^\top Bz$ becomes
13128 \begin{equation}
13129 z'^{\top}(Q^\top BQ)z'.
13130 \end{equation}
13131 The vector space of symmetric quadratic forms is therefore carried isomorphically to itself under the induced action of $Q$ on $\operatorname{Sym}^2(P^\top H)$.
13132 \end{proof}
13133 
13134 This is important because Block 11 recovers response blocks only up to within-block gauge. The full symmetric degree-two packet grammar does not require a privileged coordinate basis inside the recovered block.
13135 
13136 \subsection{Constructor novelty and parsimony}
13137 
13138 Let $\Ccal_t$ be the inherited repair-constructor grammar and $\Ccal_t^+$ a candidate extension. The extension is structurally new when its realized correction image contains a task-relevant direction outside the closure of the old grammar.
13139 
13140 \begin{definition}[Constructor-expansion certificate]
13141 A constructor-expansion certificate is a tuple
13142 \begin{equation}
13143 \Pi_{\mathrm{con}}
13144 =
13145 (\Ccal_t,\Ccal_t^+,\eta_{\mathrm{old}},\eta_{\mathrm{new}},
13146 \mathsf H,\kappa,C_{\mathrm{model}},\Pi_{\mathrm{foreign}}),
13147 \end{equation}
13148 where $\mathsf H$ is a nonzero higher-order derivative witness, $\kappa$ is held-out obstruction reduction, and $C_{\mathrm{model}}$ records constructor complexity.
13149 \end{definition}
13150 
13151 A full-space quadratic grammar contains the block-local quadratic grammar. Therefore a finite experiment need not establish block-local superiority over a sufficiently well-regularized full quadratic control. The narrower claim is that a degree-two constructor is required, while localization provides a typed and lower-dimensional realization. In Block 12, the compiler uses a predeclared parsimony tie-break among constructors whose development gain lies within $0.02$ of the best candidate.
13152 
13153 \begin{proposition}[Finite constructor selection with conservative tie-break]
13154 Suppose a finite constructor family is evaluated on fixed development evidence. Let $G^\star$ be the largest held-out development gain and let
13155 \begin{equation}
13156 \Ccal_{\tau}=\{C:G(C)\geq G^\star-\tau\}.
13157 \end{equation}
13158 Selecting the minimum-complexity constructor in $\Ccal_\tau$ is deterministic and preserves an explicit near-optimality bound of $\tau$ on the development criterion.
13159 \end{proposition}
13160 
13161 \section{Block 12 Experimental Contract}
13162 \label{sec:block12-contract}
13163 
13164 \subsection{Two separately sealed contexts}
13165 
13166 Block 12 contains two separate protected contexts.
13167 
13168 \paragraph{Context A: response-language growth.}
13169 The inherited Block 11 process receives two response operators whose signatures alias the context and repair blocks. A finite compiler may add exactly one operator from a declared five-candidate intervention grammar. The selected operator and witness gate are sealed before the primary audit.
13170 
13171 \paragraph{Context B: candidate-constructor growth.}
13172 The process inherits the separating response language and recovered repair packet from Context A. Its parent is allowed only affine repairs. The child may select one constructor from a finite family containing affine, packet-quadratic, wrong-packet quadratic, random-packet quadratic, and full-space quadratic forms. Parent and child receive separately compiled witness gates.
13173 
13174 The two paired effects are context indexed. Their sum is not interpreted as one scalar increase in a fixed environment.
13175 
13176 \subsection{Response-language fixture}
13177 
13178 The hidden source space has three unknown rank-two blocks:
13179 \begin{equation}
13180 H=H_{\mathrm{ctx}}\oplus H_{\mathrm{rep}}\oplus H_{\mathrm{nui}}.
13181 \end{equation}
13182 The inherited signature matrix is
13183 \begin{equation}
13184 \Sigma_0
13185 =
13186 \begin{bmatrix}
13187 1.25 & 0.72\\
13188 1.25 & 0.72\\
13189 0.80 & 1.30
13190 \end{bmatrix},
13191 \label{eq:b12-inherited-signatures}
13192 \end{equation}
13193 where rows correspond to context, repair, and nuisance blocks. The first two rows coincide. By \cref{thm:b12-alias-persistence}, every element of the inherited response algebra acts identically on the context and repair blocks.
13194 
13195 The finite candidate language is:
13196 \begin{center}
13197 \begin{tabular}{@{}lccc@{}}
13198 \toprule
13199 Candidate & Context & Repair & Nuisance \\
13200 \midrule
13201 split-context-repair & $1.42$ & $0.61$ & $1.03$ \\
13202 duplicate-parent & $1.18$ & $1.18$ & $0.91$ \\
13203 nuisance-rescale & $0.84$ & $0.84$ & $1.37$ \\
13204 weak-context-repair & $1.015$ & $0.985$ & $1.00$ \\
13205 cross-coupled & $1.38$ & $0.65$ & $1.02$ \\
13206 \bottomrule
13207 \end{tabular}
13208 \end{center}
13209 The cross-coupled candidate additionally receives a noncommuting symmetric perturbation.
13210 
13211 \subsection{Matched response-query envelope}
13212 
13213 Parent and child each receive exactly $144$ deployment query units:
13214 \begin{center}
13215 \begin{tabular}{@{}lrrrrr@{}}
13216 \toprule
13217 Process & Response & Bridge fit & Chart validation & Witness & Total \\
13218 \midrule
13219 Inherited parent & $72$ & $36$ & $18$ & $18$ & $144$ \\
13220 Expanded-language child & $72$ & $36$ & $18$ & $18$ & $144$ \\
13221 \bottomrule
13222 \end{tabular}
13223 \end{center}
13224 The parent allocates $36$ response observations to each of two inherited interventions. The child allocates $24$ observations to each of three interventions. The total response budget is unchanged.
13225 
13226 The response-language compiler uses $32$ development targets per candidate, followed by $24$ disjoint meta-validation targets for witness selection. The primary audit contains $96$ foreign targets from four family identifiers absent from development. A separate $24$-target stress suite contains degenerate signatures, strong cross-block coupling, and high response noise.
13227 
13228 \subsection{Constructor fixture and matched budget}
13229 
13230 The constructor context uses the recovered repair packet and a target repair of the form
13231 \begin{equation}
13232 F(x)=Ax+b+\bigl(z^\top Bz\bigr)v,
13233 \qquad
13234 z=U_{\mathrm{rep}}^\top x,
13235 \end{equation}
13236 with $B\neq0$ and $v\neq0$. The parent affine grammar is structurally obstructed by \cref{thm:b12-hessian-obstruction}.
13237 
13238 Parent and child each receive $160$ query units:
13239 \begin{center}
13240 \begin{tabular}{@{}lrrrrr@{}}
13241 \toprule
13242 Process & Response atlas & Constructor fit & Validation & Witness & Total \\
13243 \midrule
13244 Affine parent & $48$ & $88$ & $0$ & $24$ & $160$ \\
13245 Quadratic child & $48$ & $64$ & $24$ & $24$ & $160$ \\
13246 \bottomrule
13247 \end{tabular}
13248 \end{center}
13249 The parent receives more direct constructor-fitting observations. The child spends $24$ observations on constructor validation.
13250 
13251 The compiler evaluates five constructor classes on $32$ development targets each and selects witness thresholds on $24$ disjoint meta-validation targets. The primary audit contains $96$ foreign targets. A separate $24$-target stress suite includes cubic contamination, weak quadratic signal, and noninjective compression.
13252 
13253 \subsection{Chronology and seal}
13254 
13255 The package chronology is:
13256 \begin{equation}
13257 \begin{aligned}
13258 &\text{replay Block 11 ancestor policy and seal}\\
13259 &\longrightarrow
13260 \text{generate development and meta-validation evidence}\\
13261 &\longrightarrow
13262 \text{compile response and constructor certificates}\\
13263 &\longrightarrow
13264 \text{serialize gates, compiler ledger, and pre-audit seal}\\
13265 &\longrightarrow
13266 \text{generate the two primary audits and stress suites}\\
13267 &\longrightarrow
13268 \text{evaluate all frozen controls and claim gates}.
13269 \end{aligned}
13270 \label{eq:b12-chronology}
13271 \end{equation}
13272 The pre-audit seal explicitly records that audit fields are absent.
13273 
13274 \subsection{Frozen comparison arms}
13275 
13276 \paragraph{Response-language arms.}
13277 The primary response experiment includes:
13278 \begin{enumerate}
13279 \item the inherited two-operator parent;
13280 \item the selected separating-intervention child;
13281 \item a duplicate operator that preserves the alias;
13282 \item a weak-separation operator;
13283 \item a shuffled-response correspondence control;
13284 \item an equal-rank random-atlas control;
13285 \item blind application without witness abstention.
13286 \end{enumerate}
13287 
13288 \paragraph{Constructor arms.}
13289 The constructor experiment includes:
13290 \begin{enumerate}
13291 \item an independently gated affine parent;
13292 \item the selected repair-block quadratic child;
13293 \item a directly supervised affine oracle fitted with $1600$ pairs per target;
13294 \item wrong-block and random-block quadratic controls;
13295 \item shuffled quadratic features;
13296 \item a full-space quadratic control;
13297 \item stress families outside the declared degree-two injective regime.
13298 \end{enumerate}
13299 
13300 \section{Block 12 Results}
13301 \label{sec:block12-results}
13302 
13303 \subsection{Response-language development certificate}
13304 
13305 The finite compiler selects
13306 \begin{verbatim}
13307 split_context_repair
13308 \end{verbatim}
13309 from the five-candidate language. Development summaries are:
13310 \begin{center}
13311 \begin{tabular}{@{}lrrrr@{}}
13312 \toprule
13313 Candidate & Filtered gain & Positive rate & Min. gap & Commutator \\
13314 \midrule
13315 split-context-repair & $0.934377$ & $1.000000$ & $0.066822$ & $0.001421$ \\
13316 duplicate-parent & $-0.062500$ & $0.000000$ & $0.008805$ & $0.001136$ \\
13317 nuisance-rescale & $-0.062500$ & $0.000000$ & $0.009193$ & $0.001247$ \\
13318 weak-context-repair & $0.416493$ & $0.656250$ & $0.017536$ & $0.001162$ \\
13319 cross-coupled & $-0.541985$ & $0.250000$ & $0.111187$ & $0.014182$ \\
13320 \bottomrule
13321 \end{tabular}
13322 \end{center}
13323 The selected candidate is the only stable, high-margin response extension that consistently separates the inherited alias and supports repair.
13324 
13325 \begin{figure}[H]
13326 \centering
13327 \includegraphics[width=0.82\textwidth]{block12_intervention_compiler.pdf}
13328 \caption{Development comparison for the finite intervention-language compiler. The selected response operator combines high repair gain with a stable separating spectral margin.}
13329 \label{fig:b12-intervention-compiler}
13330 \end{figure}
13331 
13332 The frozen response gate requires witness gain at least $0.05$, witness NRMSE at most $0.30$, validation gain at least $0.05$, validation NRMSE at most $0.25$, minimum response gap at least $0.015$, commutator residual at most $0.015$, and minimum chart support at least $8$.
13333 
13334 \subsection{Primary response-language audit}
13335 
13336 On $96$ fresh foreign targets:
13337 \begin{center}
13338 \begin{tabular}{@{}lrrrrr@{}}
13339 \toprule
13340 Process & Mean gain & Median & Commit & Positive & Harm \\
13341 \midrule
13342 Inherited response parent & $0.000000$ & $0.000000$ & $0.000000$ & $0.000000$ & $0.000000$ \\
13343 Expanded-language child & $0.846985$ & $0.969182$ & $0.906250$ & $0.906250$ & $0.000000$ \\
13344 \bottomrule
13345 \end{tabular}
13346 \end{center}
13347 The paired effect is
13348 \begin{equation}
13349 \widehat\Delta_{4\to5}
13350 =
13351 0.846985,
13352 \label{eq:b12-response-effect}
13353 \end{equation}
13354 with frozen interval
13355 \begin{equation}
13356 [0.786100,0.901073]
13357 \label{eq:b12-response-ci}
13358 \end{equation}
13359 and sign-flip estimate
13360 \begin{equation}
13361 p=9.9999\times10^{-6}.
13362 \end{equation}
13363 No primary target favors the inherited parent. The remaining $9.375\%$ of pairs are abstention ties.
13364 
13365 \begin{figure}[H]
13366 \centering
13367 \includegraphics[width=0.84\textwidth]{block12_response_gain.pdf}
13368 \caption{Primary response-language result and controls. Duplicate and shuffled response information do not reproduce the expanded-language child.}
13369 \label{fig:b12-response-gain}
13370 \end{figure}
13371 
13372 \begin{figure}[H]
13373 \centering
13374 \includegraphics[width=0.78\textwidth]{block12_response_paired.pdf}
13375 \caption{Episode-wise paired effect for the response-language edge. Positive mass corresponds to witnessed commitments; zeros are abstention ties.}
13376 \label{fig:b12-response-paired}
13377 \end{figure}
13378 
13379 \subsection{Response geometry and controls}
13380 
13381 The inherited atlas has modal block ranks
13382 \begin{equation}
13383 (2,4),
13384 \end{equation}
13385 while the expanded language recovers
13386 \begin{equation}
13387 (2,2,2).
13388 \end{equation}
13389 Mean audit geometry is:
13390 \begin{center}
13391 \begin{tabular}{@{}lr@{}}
13392 \toprule
13393 Quantity & Mean \\
13394 \midrule
13395 Child context-block overlap & $0.999843$ \\
13396 Parent rank-two context overlap & $0.000146$ \\
13397 Child minimum spectral gap & $0.064981$ \\
13398 Child commutator residual & $0.001826$ \\
13399 \bottomrule
13400 \end{tabular}
13401 \end{center}
13402 The overlap metric is retained for audit interpretation only. The compiler uses response geometry and repair validation, not hidden projectors.
13403 
13404 Control results are:
13405 \begin{center}
13406 \begin{tabular}{@{}lrrr@{}}
13407 \toprule
13408 Control & Mean selected gain & Commit & Harm \\
13409 \midrule
13410 Duplicate response & $0.000000$ & $0.000000$ & $0.000000$ \\
13411 Weak separation & $0.197231$ & $0.260417$ & $0.000000$ \\
13412 Shuffled response & $0.025225$ & $0.062500$ & $0.000000$ \\
13413 Random atlas & $0.000000$ & $0.000000$ & $0.000000$ \\
13414 Blind selected intervention & $0.904608$ & $1.000000$ & $0.000000$ \\
13415 \bottomrule
13416 \end{tabular}
13417 \end{center}
13418 The blind child is not harmful on the primary response audit, so the witness primarily contributes selective abstention rather than harm removal in this first edge. The stress suite has commit rate $0.208333$ and zero selected harm. It abstains completely on exact signature degeneracy and strong coupling; some high-noise cases remain inside the gate.
13419 
13420 \subsection{Constructor development certificate}
13421 
13422 The constructor compiler records:
13423 \begin{center}
13424 \begin{tabular}{@{}lrrr@{}}
13425 \toprule
13426 Constructor & Development gain & Validation gain & Fit condition \\
13427 \midrule
13428 Affine & $0.320296$ & $0.000000$ & $1.791226$ \\
13429 Repair-block quadratic & $0.998022$ & $0.997731$ & $5.103054$ \\
13430 Context-block quadratic & $0.284136$ & $0.226411$ & $4.851183$ \\
13431 Random-block quadratic & $0.390305$ & $0.393501$ & $4.810982$ \\
13432 Full-space quadratic & $0.999456$ & $0.999169$ & $14.785843$ \\
13433 \bottomrule
13434 \end{tabular}
13435 \end{center}
13436 The full-space quadratic class obtains the largest raw development gain, but the block-local quadratic constructor lies within the predeclared $0.02$ near-optimality window and has a much smaller typed quadratic feature space. The compiler therefore selects
13437 \begin{verbatim}
13438 repair_quadratic
13439 \end{verbatim}
13440 by the frozen parsimony rule.
13441 
13442 This result should not be interpreted as block-local superiority over every full quadratic model. The full-space quadratic control remains an essentially exact alternative inside the finite fixture. The supported structural claim is degree-two constructor growth, with repair-block localization providing the selected parsimonious realization.
13443 
13444 \subsection{Primary constructor audit}
13445 
13446 On $96$ fresh foreign targets:
13447 \begin{center}
13448 \begin{tabular}{@{}lrrrrr@{}}
13449 \toprule
13450 Process & Mean gain & Median & Commit & Positive & Harm \\
13451 \midrule
13452 Independently gated affine parent & $0.289123$ & $0.387596$ & $0.656250$ & $0.645833$ & $0.010417$ \\
13453 Repair-block quadratic child & $0.997511$ & $0.998251$ & $1.000000$ & $1.000000$ & $0.000000$ \\
13454 \bottomrule
13455 \end{tabular}
13456 \end{center}
13457 The paired effect is
13458 \begin{equation}
13459 \widehat\Delta_{5\to6}
13460 =
13461 0.708388,
13462 \label{eq:b12-constructor-effect}
13463 \end{equation}
13464 with frozen interval
13465 \begin{equation}
13466 [0.659635,0.757198].
13467 \label{eq:b12-constructor-ci}
13468 \end{equation}
13469 The sign-flip estimate is $9.9999\times10^{-6}$, and no primary target favors the affine parent.
13470 
13471 A directly supervised affine oracle fitted with $1600$ same-target pairs per target obtains mean blind gain
13472 \begin{equation}
13473 0.466578.
13474 \end{equation}
13475 The child exceeds that stronger function-class control by mean
13476 \begin{equation}
13477 0.530933,
13478 \end{equation}
13479 with paired bootstrap interval
13480 \begin{equation}
13481 [0.506587,0.556639].
13482 \label{eq:b12-affine-oracle-gap}
13483 \end{equation}
13484 This supports the nonzero-Hessian obstruction rather than a small-sample explanation for the affine parent.
13485 
13486 \begin{figure}[H]
13487 \centering
13488 \includegraphics[width=0.86\textwidth]{block12_constructor_gain.pdf}
13489 \caption{Candidate-constructor result and controls. The full quadratic control matches the selected child, preventing a claim of unique block-local superiority; wrong-block and shuffled-feature controls fail.}
13490 \label{fig:b12-constructor-gain}
13491 \end{figure}
13492 
13493 \begin{figure}[H]
13494 \centering
13495 \includegraphics[width=0.78\textwidth]{block12_constructor_paired.pdf}
13496 \caption{Paired child-minus-parent effect for the constructor edge. Every primary pair is positive.}
13497 \label{fig:b12-constructor-paired}
13498 \end{figure}
13499 
13500 \subsection{Constructor controls and stress boundary}
13501 
13502 The principal constructor controls are:
13503 \begin{center}
13504 \begin{tabular}{@{}lrrr@{}}
13505 \toprule
13506 Control & Mean selected gain & Commit & Harm \\
13507 \midrule
13508 Wrong-block quadratic & $0.000000$ & $0.000000$ & $0.000000$ \\
13509 Random-block quadratic & $0.009323$ & $0.010417$ & $0.000000$ \\
13510 Shuffled quadratic features & $0.000000$ & $0.000000$ & $0.000000$ \\
13511 Full-space quadratic & $0.999014$ & $1.000000$ & $0.000000$ \\
13512 \bottomrule
13513 \end{tabular}
13514 \end{center}
13515 The full quadratic control shows that the essential missing structure is degree two. The wrong-block, random-block, and shuffled-feature controls show that arbitrary nonlinear capacity does not reproduce the typed constructor under the same fitting procedure.
13516 
13517 The stress suite has overall commit rate $0.333333$, mean selected gain $0.332051$, and zero selected harm. The child abstains on the strong cubic-contamination and compressed families, while continuing to solve the weak-quadratic family. This boundary is appropriate: weak signal remains inside the degree-two grammar, whereas cubic and information-losing cases require different extensions.
13518 
13519 \subsection{Two finite structural lineage theorems}
13520 
13521 \begin{conditionaltheorem}[Finite response-language growth]
13522 \label{thm:b12-response-realization}
13523 Within the frozen Block 12 response benchmark, assume that:
13524 \begin{enumerate}
13525 \item the published Block 11 ancestor policy and seal replay exactly;
13526 \item the inherited response signatures satisfy the exact alias in \cref{eq:b12-inherited-signatures};
13527 \item parent and child each consume exactly $144$ query units;
13528 \item the selected response operator and witness gate are sealed before primary audit generation;
13529 \item the expanded language conservatively contains the inherited response algebra;
13530 \item the response gap, commutator, support, validation, and witness gates pass;
13531 \item the paired effect and interval are those in \cref{eq:b12-response-effect,eq:b12-response-ci};
13532 \item duplicate, weak, shuffled, random-atlas, and stress controls retain their declared interpretations.
13533 \end{enumerate}
13534 Then
13535 \begin{equation}
13536 K_4\xrightarrow{\Pi_5}K_5
13537 \end{equation}
13538 is a finite, matched-resource, lineage-generated response-language expansion. The child adds a response distinction absent from every composition of the inherited operators and uses the resulting central refinement to improve foreign protected repair performance.
13539 \end{conditionaltheorem}
13540 
13541 \begin{proof}
13542 The alias theorem shows that the inherited response algebra cannot separate the context and repair blocks. The selected operator refines that central block by \cref{thm:b12-central-refinement}. Matched query accounting and the pre-audit seal establish the experimental comparison. The paired audit establishes positive foreign gain, while the controls rule out duplicate information, weak separation, correspondence shuffling, and an arbitrary equal-rank atlas as explanations inside the finite fixture.
13543 \end{proof}
13544 
13545 \begin{conditionaltheorem}[Finite repair-constructor growth]
13546 \label{thm:b12-constructor-realization}
13547 Within the frozen Block 12 constructor benchmark, assume that:
13548 \begin{enumerate}
13549 \item the response-language child from \cref{thm:b12-response-realization} supplies the inherited repair packet;
13550 \item the true repair has a nonzero quadratic Hessian on that packet;
13551 \item parent and child each consume exactly $160$ query units;
13552 \item parent and child receive separately compiled frozen witness gates;
13553 \item the affine parent remains embedded by \cref{prop:b12-affine-embedding};
13554 \item the paired effect and interval are those in \cref{eq:b12-constructor-effect,eq:b12-constructor-ci};
13555 \item the directly supervised affine oracle retains the gap in \cref{eq:b12-affine-oracle-gap};
13556 \item wrong-block, random-block, shuffled-feature, full-quadratic, and stress controls retain their declared interpretations.
13557 \end{enumerate}
13558 Then
13559 \begin{equation}
13560 K_5\xrightarrow{\Pi_6}K_6
13561 \end{equation}
13562 is a finite, matched-resource, lineage-generated expansion of the repair-constructor grammar from affine to degree-two packet-local repair.
13563 \end{conditionaltheorem}
13564 
13565 \begin{proof}
13566 The inherited affine operad is closed by \cref{thm:b12-affine-closure}. The nonzero Hessian makes the target repair unavailable to that operad by \cref{thm:b12-hessian-obstruction}. The selected quadratic grammar contains the affine parent and adds the missing degree-two response. The paired audit, same-target affine-oracle comparison, and typed controls establish task-relevant structural gain inside the declared finite benchmark.
13567 \end{proof}
13568 
13569 \begin{figure}[H]
13570 \centering
13571 \includegraphics[width=0.88\textwidth]{block12_lineage_edges.pdf}
13572 \caption{Context-indexed paired effects for the realized bottleneck-migrating lineage. Heights are not added into one fixed-context utility.}
13573 \label{fig:b12-lineage}
13574 \end{figure}
13575 
13576 \begin{statusbox}{Block 12 result}
13577 Block 12 realizes two distinct structural lineage edges. The first adds a separating response operator because the inherited intervention algebra provably aliases two task-relevant blocks. Under an equal $144$-query envelope, selected gain rises from $0$ to $0.846985$. The second adds a degree-two repair constructor because the inherited affine operad has a nonzero-Hessian obstruction. Under an equal $160$-query envelope, gain rises from $0.289123$ to $0.997511$, and the child exceeds a $1600$-pair same-target affine oracle by $0.530933$. These are finite response-language and constructor-growth results, not unrestricted intervention invention or open-ended program synthesis.
13578 \end{statusbox}
13579 
13580 \section{Block 12 Reproducibility and Package}
13581 \label{sec:block12-package}
13582 
13583 \subsection{Reference implementation}
13584 
13585 The released package contains:
13586 \begin{itemize}
13587 \item exact Block 11 ancestor policy and seal files;
13588 \item separate response-language and constructor development, meta-validation, primary-audit, and stress generators;
13589 \item finite intervention and constructor compilers;
13590 \item response-algebra estimation, spectral atlas recovery, and commutator diagnostics;
13591 \item affine, packet-quadratic, full-quadratic, wrong-packet, random, duplicate, weak, and shuffled controls;
13592 \item separately compiled parent and child witness gates;
13593 \item canonical JSON and CSV result objects;
13594 \item six deterministic publication figures;
13595 \item twelve automated tests;
13596 \item an exact $41$-file SHA-256 package manifest.
13597 \end{itemize}
13598 
13599 The canonical execution prints:
13600 \begin{verbatim}
13601 SRIC_BLOCK12_ALL_SUITES=PASS
13602 result_sha256=4ede53f590ddf477711fc5b1b55e43fb96d9450cc57c0d020332fb6daf712bd1
13603 response_parent_gain=0.000000
13604 response_child_gain=0.846985
13605 response_effect=0.846985
13606 constructor_parent_gain=0.289123
13607 constructor_child_gain=0.997511
13608 constructor_effect=0.708388
13609 ............ [100%]
13610 SRIC_BLOCK12_PACKAGE_VALID
13611 manifest_files=41
13612 \end{verbatim}
13613 
13614 The frozen response certificate digest is
13615 \begin{equation}
13616 \texttt{73080b6681940b79e390f7ecd979cb493c740bfab9be7b231002b44c387dc99d}.
13617 \end{equation}
13618 The frozen constructor certificate digest is
13619 \begin{equation}
13620 \texttt{d8be9c8e1c0e6ccb5e64f53700c8a59f45c8402e9a395ba7d72db97130010016}.
13621 \end{equation}
13622 The compiler digest is
13623 \begin{equation}
13624 \texttt{f51f5152dae06791fb87a6926162bc5e1965bc7e1bedae0a2957b9bbed82f475}.
13625 \end{equation}
13626 The pre-audit seal digest is
13627 \begin{equation}
13628 \texttt{75758911e1449620c32ed00b5c959c66b131956f2c79cf1127e15adfbffbd6cd}.
13629 \end{equation}
13630 
13631 \subsection{Claim boundary}
13632 
13633 The release establishes:
13634 \begin{enumerate}
13635 \item a finite selection of one separating response operator from a declared intervention grammar;
13636 \item a central atlas refinement unavailable to every composition of the inherited response operators;
13637 \item a foreign repair improvement under matched response-query resources;
13638 \item a finite constructor expansion from affine to degree-two repair on a recovered packet;
13639 \item a function-class comparison against a much larger same-target affine training set;
13640 \item conservative embeddings of both inherited grammars;
13641 \item separately sealed audits, controls, receipts, and package replay.
13642 \end{enumerate}
13643 
13644 It does not establish:
13645 \begin{itemize}
13646 \item unrestricted invention of interventions outside the finite candidate language;
13647 \item autonomous determination of intervention cost, feasibility, or physical side effects;
13648 \item general recovery for noncommuting, nonlinear, time-varying, or partially observed response algebras;
13649 \item unrestricted program synthesis or neural-weight constructor discovery;
13650 \item unique superiority of the block-local quadratic grammar over the full quadratic control;
13651 \item improvement of the outer lineage compiler;
13652 \item autonomous evaluator or witness evolution;
13653 \item open-ended recursive improvement, indefinite acceleration, unrestricted safety, consciousness, or sentience.
13654 \end{itemize}
13655 
13656 The next threshold is \emph{compiler metaproductivity}. The current outer compiler is still externally designed to search one finite intervention language and one finite constructor family. A later descendant must improve the compiler that diagnoses the obstruction and chooses which grammar to expand, then transfer that improvement across foreign parent processes under matched compilation resources.
13657 
13658 
13659 
13660 \section{Proof-Carrying Automated Research Compilers}
13661 \label{sec:block13-theory}
13662 
13663 Blocks 7 through 12 improve components of a repair process while an outer lineage compiler remains externally fixed. Block 13 moves one level outward. The modified object is now the process that reads prior research evidence, proposes candidate descendants, allocates experiments, binds proposals to code, and selects a validated result.
13664 
13665 The realized compiler edge is
13666 \begin{equation}
13667 \mathfrak L_6
13668 \xrightarrow{\Pi^{\Gamma}_7}
13669 \mathfrak L_7,
13670 \label{eq:b13-lineage}
13671 \end{equation}
13672 where the active repair grammar, candidate-method language, evaluator portfolio, integrity monitor, and experiment budget remain fixed, but the research-allocation compiler changes from $\Gamma_0$ to $\Gamma_1$. The edge is therefore compiler metaproductivity inside a finite declared research language, not unrestricted automated science.
13673 
13674 The design is informed by the automated alignment researcher harness of Chen, Wen, and Kirchner, which combines a shared literature review, parallel researchers, results-free mini-papers, exact code review, fixed training budgets, separate evaluation, a shared finding forum, capability gates, and trajectory monitoring \cite{AAR2026}. Block 13 does not reproduce that large-scale system. It abstracts the corresponding mechanisms into a finite, package-closed compiler benchmark.
13675 
13676 \subsection{Research problems and compiler objects}
13677 
13678 Let $\Tau$ be a research-problem space. A problem $\tau\in\Tau$ contains a baseline repair process $K_\tau$, a declared candidate grammar $\Pcal_\tau$, a cheap evidence channel, a scored evaluator portfolio, a hidden protected audit, integrity rules, and a resource budget $B$.
13679 
13680 \begin{definition}[Automated research compiler]
13681 An automated research compiler is a tuple
13682 \begin{equation}
13683 \Gamma
13684 =
13685 (\Lcal,\Fcal,\Pcal,\Qcal,\Vcal,\Mcal,\Rcal),
13686 \label{eq:b13-compiler-tuple}
13687 \end{equation}
13688 where:
13689 \begin{enumerate}
13690 \item $\Lcal$ is a structured literature and prior-method memory;
13691 \item $\Fcal$ is an ancestor-closed finding graph or shared research forum;
13692 \item $\Pcal$ generates typed candidate methods and pre-result specifications;
13693 \item $\Qcal$ allocates cheap pilots and full experiments under budget $B$;
13694 \item $\Vcal$ is the scored evaluator portfolio with protected capability gates;
13695 \item $\Mcal$ is the integrity and proposal--code congruence monitor;
13696 \item $\Rcal$ compiles pre-result and post-result receipts into the lineage archive.
13697 \end{enumerate}
13698 \end{definition}
13699 
13700 Given problem $\tau$, the compiler produces a finite research trace
13701 \begin{equation}
13702 \Gamma(\tau,B)
13703 =
13704 \left(
13705 (c_i,\Pi_i^{-},\Pi_i^{+})
13706 \right)_{i=1}^{n(B)},
13707 \label{eq:b13-compiler-trace}
13708 \end{equation}
13709 where $c_i$ is a candidate method, $\Pi_i^{-}$ is sealed before the result is observed, and $\Pi_i^{+}$ appends evaluation and integrity evidence afterward.
13710 
13711 Let $\operatorname{Valid}(c_i,\Pi_i^{+})\in\{0,1\}$ encode capability, integrity, provenance, and resource validity. The selected descendant is
13712 \begin{equation}
13713 K_{\Gamma}^{\star}(\tau)
13714 \in
13715 \argmax_{i:\operatorname{Valid}(c_i,\Pi_i^+)=1}
13716 S_{\Vcal}(c_i),
13717 \label{eq:b13-selected-method}
13718 \end{equation}
13719 with the no-op baseline retained when no valid candidate exists.
13720 
13721 \subsection{Results-free proposal receipts}
13722 
13723 A research receipt must distinguish what was predicted before an experiment from what became known after it.
13724 
13725 \begin{definition}[Pre-result research receipt]
13726 A pre-result receipt has the form
13727 \begin{equation}
13728 \Pi_i^{-}
13729 =
13730 (H_i,O_i,D_i,\theta_i,
13731 \widehat{\mathbf c}_i,
13732 \widehat{\mathbf r}_i,
13733 F_i,h_i^{\mathrm{code}},h_i^{\mathrm{spec}}),
13734 \label{eq:b13-preresult-receipt}
13735 \end{equation}
13736 where $H_i$ is the causal hypothesis, $O_i$ the proposed objective or repair mechanism, $D_i$ the evidence or data construction, $\theta_i$ the configuration, $\widehat{\mathbf c}_i$ the predicted benchmark-effect vector, $\widehat{\mathbf r}_i$ the predicted protected-risk vector, $F_i$ the declared falsification condition, and $h_i^{\mathrm{code}},h_i^{\mathrm{spec}}$ immutable digests of the executable and specification.
13737 \end{definition}
13738 
13739 \begin{definition}[Post-result research receipt]
13740 After evaluation, the post-result receipt is
13741 \begin{equation}
13742 \Pi_i^{+}
13743 =
13744 (\Pi_i^{-},\mathbf c_i,\mathbf r_i,a_i,m_i),
13745 \label{eq:b13-postresult-receipt}
13746 \end{equation}
13747 where $\mathbf c_i$ is the scored benchmark vector, $\mathbf r_i$ the protected-regression vector, $a_i$ the hidden-audit result, and $m_i$ the integrity-monitor verdict.
13748 \end{definition}
13749 
13750 The forecast residual and proposal--code congruence residual are
13751 \begin{equation}
13752 \eps_{\mathrm{forecast},i}
13753 =
13754 \norm{\widehat{\mathbf c}_i-\mathbf c_i},
13755 \qquad
13756 \eps_{\mathrm{spec/code},i}
13757 =
13758 d_{\mathrm{sem}}
13759 \left(
13760 \mathrm{Sem}(\Pi_i^-),
13761 \mathrm{Sem}(\mathrm{code}_i)
13762 \right).
13763 \label{eq:b13-receipt-residuals}
13764 \end{equation}
13765 In the executable finite companion, congruence is a Boolean monitor rather than a complete semantic metric.
13766 
13767 \begin{proposition}[Pre-result non-retroactivity]
13768 Suppose $\Pi_i^{-}$ and its code digest are sealed before the evaluation trace is generated. Then any later narrative revision that preserves the receipt digest leaves the pre-result hypothesis, mechanism, and falsification declaration unchanged.
13769 \end{proposition}
13770 
13771 \begin{proof}
13772 The receipt digest is a collision-resistant commitment to the serialized pre-result fields. Under the integrity assumption, changing any committed field changes the digest. The proposition prevents retrospective alteration of the declared proposal; it does not prove that the original proposal was correct or complete.
13773 \end{proof}
13774 
13775 \subsection{Finding graphs and research lineage}
13776 
13777 A finding is represented by
13778 \begin{equation}
13779 f_i
13780 =
13781 (\Pi_i^-,\Pi_i^+,\mathrm{code}_i,
13782 \operatorname{Par}(i),e_i,o_i),
13783 \label{eq:b13-finding-node}
13784 \end{equation}
13785 where $e_i$ is the observed effect profile and $o_i$ is either a surviving obstruction or a validated no-go certificate. Directed edges are typed as extension, ablation, refutation, transport, or replication. A proposal must cite the archive nodes on which its mechanism depends.
13786 
13787 This converts a shared research forum into a proof-carrying directed acyclic graph. Positive methods, negative results, obstruction certificates, code snapshots, side effects, and rejected-but-informative candidates remain available to descendants. The construction is compatible with the ancestor-closed archive and continuation semantics developed earlier.
13788 
13789 \subsection{Evaluator portfolios and metric-gaming nullspaces}
13790 
13791 Let the scored evaluator portfolio be
13792 \begin{equation}
13793 \Bcal
13794 =
13795 \{b_1,\ldots,b_m\}.
13796 \end{equation}
13797 For method $c$, define normalized headroom closure
13798 \begin{equation}
13799 q_j(c)
13800 =
13801 \frac{s_j(c)-s_j(K_\tau)}{s_j^{\star}-s_j(K_\tau)}.
13802 \label{eq:b13-headroom-closure}
13803 \end{equation}
13804 Let $I_+(c)=\{j:q_j(c)>0\}$. The coverage-weighted geometric score is
13805 \begin{equation}
13806 S_{\mathrm{cov}}(c)
13807 =
13808 \frac{|I_+(c)|}{m}
13809 \left(
13810 \prod_{j\in I_+(c)}q_j(c)
13811 \right)^{1/|I_+(c)|},
13812 \label{eq:b13-coverage-geometric}
13813 \end{equation}
13814 with value zero when $I_+(c)=\varnothing$. Hard capability and integrity gates are applied separately rather than traded against this scalar.
13815 
13816 Let $T$ be a tangent space of behavioral modifications around a baseline model, and let $Db_j:T\to\R$ be the local response of benchmark $j$.
13817 
13818 \begin{definition}[Evaluator-nullspace]
13819 The locally invisible behavior subspace of portfolio $\Bcal$ is
13820 \begin{equation}
13821 \Ncal_{\Bcal}
13822 =
13823 \bigcap_{j=1}^{m}\ker Db_j.
13824 \label{eq:b13-eval-nullspace}
13825 \end{equation}
13826 \end{definition}
13827 
13828 \begin{theorem}[Portfolio refinement shrinks the invisible subspace]
13829 Let $\Bcal^+=\Bcal\cup\{b_{m+1}\}$. Then
13830 \begin{equation}
13831 \Ncal_{\Bcal^+}
13832 \subseteq
13833 \Ncal_{\Bcal}.
13834 \label{eq:b13-nullspace-inclusion}
13835 \end{equation}
13836 The inclusion is strict if and only if there exists $v\in\Ncal_{\Bcal}$ such that $Db_{m+1}(v)\neq0$.
13837 \end{theorem}
13838 
13839 \begin{proof}
13840 Intersecting with one additional kernel cannot enlarge the intersection. The inclusion is strict exactly when the new kernel excludes at least one vector previously lying in every old kernel.
13841 \end{proof}
13842 
13843 The theorem gives a local meaning to benchmark diversity. It does not show that a finite portfolio is complete. It shows when adding a benchmark removes one class of locally invisible behavior changes.
13844 
13845 \subsection{Integrity-valid compiler metaproductivity}
13846 
13847 Let $U_{\mathrm{audit}}$ be a protected audit utility unavailable during compiler allocation and final scored selection. Define compiler metaproductivity at research budget $B$ by
13848 \begin{equation}
13849 \mathfrak C_B(\Gamma)
13850 =
13851 \mathbb E_{\tau\sim\Dcal_{\mathrm{foreign}}}
13852 \left[
13853 U_{\mathrm{audit}}
13854 \left(K_{\Gamma}^{\star}(\tau)\right)
13855 -
13856 U_{\mathrm{audit}}(K_\tau)
13857 \right].
13858 \label{eq:b13-compiler-meta}
13859 \end{equation}
13860 Only integrity-valid and capability-passing outputs are eligible.
13861 
13862 Let $\mathfrak C_B^{\mathrm{nom}}$ be the same envelope when integrity constraints are ignored.
13863 
13864 \begin{proposition}[Validity-filtered envelope]
13865 For every compiler,
13866 \begin{equation}
13867 \mathfrak C_B(\Gamma)
13868 \leq
13869 \mathfrak C_B^{\mathrm{nom}}(\Gamma).
13870 \end{equation}
13871 A nominal compiler gain is not an SRIC improvement certificate unless it remains positive after validity filtering.
13872 \end{proposition}
13873 
13874 The inequality is immediate because the valid candidate set is a subset of the nominal candidate set. The substantive point is interpretive: a compiler is not improved merely because it becomes better at exploiting evaluator noise, benchmark format, prohibited data, or capability sacrifice.
13875 
13876 \subsection{Outcome forecasting and cost-sensitive allocation}
13877 
13878 For candidate $c_i$, let $z_i$ collect its pre-result receipt fields and cheap pilot evidence. A research-outcome forecaster returns
13879 \begin{equation}
13880 F_\theta(z_i)
13881 =
13882 (\widehat g_i,\widehat r_i,\widehat s_i),
13883 \label{eq:b13-forecaster}
13884 \end{equation}
13885 where $\widehat g_i$ predicts hidden protected gain, $\widehat r_i$ predicts protected risk, and $\widehat s_i$ predicts scored-evaluator performance.
13886 
13887 A simple acquisition rule is
13888 \begin{equation}
13889 A_i
13890 =
13891 \widehat g_i
13892 -\lambda_r\widehat r_i
13893 +\lambda_u\widehat\sigma_i
13894 +\lambda_h p_\theta(h_i\mid x_\tau)
13895 -\lambda_c C_i,
13896 \label{eq:b13-acquisition}
13897 \end{equation}
13898 where $h_i$ is the method family, $p_\theta$ is an obstruction-class posterior, $\widehat\sigma_i$ is a declared uncertainty bonus, and $C_i$ is experimental cost.
13899 
13900 \begin{theorem}[Uniform forecast-regret bound]
13901 Let $g_i$ be the true protected gain of each admissible candidate and suppose
13902 \begin{equation}
13903 |\widehat g_i-g_i|\leq\eps
13904 \end{equation}
13905 for all candidates. If $\widehat i\in\argmax_i\widehat g_i$ and $i^\star\in\argmax_i g_i$, then
13906 \begin{equation}
13907 g_{i^\star}-g_{\widehat i}
13908 \leq
13909 2\eps.
13910 \label{eq:b13-forecast-regret}
13911 \end{equation}
13912 \end{theorem}
13913 
13914 \begin{proof}
13915 By optimality of $\widehat i$ under the predicted values,
13916 \begin{equation}
13917 \widehat g_{\widehat i}\geq\widehat g_{i^\star}.
13918 \end{equation}
13919 Therefore
13920 \begin{equation}
13921 g_{i^\star}
13922 \leq
13923 \widehat g_{i^\star}+\eps
13924 \leq
13925 \widehat g_{\widehat i}+\eps
13926 \leq
13927 g_{\widehat i}+2\eps.
13928 \end{equation}
13929 \end{proof}
13930 
13931 The bound concerns selection from a fixed candidate set. It does not establish that the forecaster creates better candidates.
13932 
13933 \subsection{Sealed compiler comparison}
13934 
13935 Let $\Gamma_0$ and $\Gamma_1$ operate on the same problem $\tau$ with identical initial archive, candidate grammar, experiment budget, evaluator portfolio, integrity monitor, and hidden audit isolation. Let
13936 \begin{equation}
13937 D_i
13938 =
13939 U_{\mathrm{audit}}
13940 \left(K_{\Gamma_1}^{\star}(\tau_i)\right)
13941 -
13942 U_{\mathrm{audit}}
13943 \left(K_{\Gamma_0}^{\star}(\tau_i)\right).
13944 \end{equation}
13945 
13946 \begin{theorem}[Paired compiler-metaproductivity certificate]
13947 Assume independent foreign problems and $D_i\in[-G,G]$. Then with probability at least $1-\delta$,
13948 \begin{equation}
13949 \mathbb E D_i
13950 \geq
13951 \overline D_n
13952 -
13953 G\sqrt{\frac{2\log(1/\delta)}{n}}.
13954 \label{eq:b13-paired-compiler-bound}
13955 \end{equation}
13956 A positive lower bound certifies higher compiler metaproductivity relative to the declared research distribution and budget.
13957 \end{theorem}
13958 
13959 \begin{proof}
13960 Apply the one-sided Hoeffding inequality to the paired differences, whose interval length is $2G$.
13961 \end{proof}
13962 
13963 \begin{warning}[Best-of-budget interpretation]
13964 The certificate compares complete compilers under the same research budget. It does not identify which internal module caused the gain without ablations, and it does not compare different candidate grammars or evaluator portfolios unless those changes are part of the declared compiler edge.
13965 \end{warning}
13966 
13967 \section{Block 13 Experimental Contract}
13968 \label{sec:block13-contract}
13969 
13970 \subsection{Research-problem family}
13971 
13972 The finite benchmark contains six hidden obstruction classes:
13973 \begin{equation}
13974 \Hcal_{\mathrm{obs}}
13975 =
13976 \{\text{data},\text{objective},\text{response},\text{constructor},\text{evaluator},\text{resource}\}.
13977 \label{eq:b13-obstruction-classes}
13978 \end{equation}
13979 The compiler is not given these labels at deployment. Each research problem exposes a twelve-dimensional cheap diagnostic vector, a baseline repair process, a finite candidate-method grammar, four scored benchmarks, a hidden audit, a capability check, and integrity constraints.
13980 
13981 The diagnostic coordinates are designed to distinguish, imperfectly, among:
13982 \begin{itemize}
13983 \item missing or low-information data;
13984 \item an objective family unable to express the desired change;
13985 \item a response language that aliases causal mechanisms;
13986 \item a repair constructor with insufficient function class;
13987 \item a scored evaluator admitting narrow proxy exploitation;
13988 \item a slow or noisy research loop requiring better allocation.
13989 \end{itemize}
13990 
13991 The architecture-foreign split perturbs diagnostic correlations, task difficulty, benchmark mixing, and audit noise while preserving the semantic interpretation of the diagnostic coordinates.
13992 
13993 \subsection{Candidate grammar}
13994 
13995 The common candidate grammar contains $39$ methods:
13996 \begin{enumerate}
13997 \item five variants for each of the six obstruction-matched method families;
13998 \item five generic variants;
13999 \item four explicit integrity or capability controls: a lottery rerun, benchmark-format mimicry, a proposal--code mismatch, and a capability-sacrifice method.
14000 \end{enumerate}
14001 Every candidate has a results-free specification, a code digest, a method family, strength, breadth, complexity, novelty, and cost. Parent and child receive exactly the same candidate set in the same canonical order.
14002 
14003 Each full experiment returns four scored benchmark effects, one held-out selection proxy, a capability-regression estimate, and an integrity verdict. The final protected audit remains unavailable to both compilers.
14004 
14005 \subsection{Parent compiler}
14006 
14007 The parent $\Gamma_0$ is deliberately strong. It includes:
14008 \begin{itemize}
14009 \item structured literature priors over method families;
14010 \item a shared within-problem finding forum;
14011 \item balanced coverage and upper-confidence exploration;
14012 \item results-free proposal receipts;
14013 \item exact code and specification digests;
14014 \item a coverage-weighted multi-benchmark evaluator;
14015 \item capability and integrity gates;
14016 \item deterministic final selection from the tested valid methods.
14017 \end{itemize}
14018 
14019 The finite meta-validation grid contains $16$ parent configurations. The selected parent has exploration weight $0.04$, forum weight $0.12$, novelty weight $0.02$, and coverage weight $0.02$.
14020 
14021 \subsection{Child compiler}
14022 
14023 The child $\Gamma_1$ conservatively retains the parent machinery and adds two lineage-trained modules:
14024 \begin{enumerate}
14025 \item a nearest-centroid obstruction classifier trained on development diagnostic receipts;
14026 \item ridge outcome and risk forecasters trained on development pre/post receipt pairs.
14027 \end{enumerate}
14028 The child uses these predictions for bottleneck-sensitive experiment allocation. The scored evaluator, final selector, capability gates, integrity monitor, candidate grammar, and full-experiment budget are unchanged.
14029 
14030 The finite child grid contains $16$ configurations. The selected child has risk weight $0.55$, uncertainty weight $0$, obstruction-match weight $0.25$, and forum weight $0.04$. The zero uncertainty weight is an empirical selection outcome, not a general claim that exploration is unnecessary.
14031 
14032 \subsection{Splits and chronology}
14033 
14034 The release uses:
14035 \begin{center}
14036 \begin{tabularx}{0.96\textwidth}{@{}p{0.25\textwidth}rrX@{}}
14037 \toprule
14038 Split & Problems & Per class & Role \\
14039 \midrule
14040 Development & $108$ & $18$ & Fit archive forecasters and class centroids \\
14041 Meta-validation & $30$ & $5$ & Select parent and child compiler configurations \\
14042 Primary foreign audit & $96$ & $16$ & Main sealed paired comparison \\
14043 Architecture-foreign audit & $72$ & $12$ & Shifted diagnostic and benchmark geometry \\
14044 \bottomrule
14045 \end{tabularx}
14046 \end{center}
14047 
14048 The chronology is
14049 \begin{equation}
14050 \begin{gathered}
14051 \text{replay Block 12 result digest}
14052 \longrightarrow
14053 \text{generate development archive}\\
14054 \longrightarrow
14055 \text{fit classifier and forecasters}
14056 \longrightarrow
14057 \text{select compiler policies}\\
14058 \longrightarrow
14059 \text{seal compiler receipt and audit commitments}
14060 \longrightarrow
14061 \text{open final audits}.
14062 \end{gathered}
14063 \label{eq:b13-chronology}
14064 \end{equation}
14065 The final policy objects, candidate grammar digest, archive digest, split commitments, and ancestor result digest are sealed before audit outcomes are used.
14066 
14067 \subsection{Matched resource envelope}
14068 
14069 Every parent and child run receives exactly
14070 \begin{equation}
14071 B_{\mathrm{research}}=6
14072 \label{eq:b13-budget}
14073 \end{equation}
14074 full candidate experiments. Candidate metadata and cheap diagnostics are common inputs. The number of proposal receipts, scored evaluator calls, capability checks, and integrity-monitor calls is therefore matched.
14075 
14076 The parent explores an average of $3.9375$ distinct method families per primary problem. The child tests an average of $1.916667$. On the architecture-foreign split these values are $4.180556$ and $2.180556$. The child therefore uses the same full-experiment budget more selectively rather than testing more methods.
14077 
14078 \subsection{Comparison arms}
14079 
14080 The release includes:
14081 \begin{center}
14082 \begin{tabularx}{0.96\textwidth}{@{}p{0.26\textwidth}X@{}}
14083 \toprule
14084 Arm & Role \\
14085 \midrule
14086 AAR-style parent & Literature prior, balanced exploration, forum, multi-benchmark evaluator, capability and integrity gates \\
14087 Full child & Parent plus obstruction classifier and outcome/risk forecaster \\
14088 Classifier only & Child obstruction classifier with ordinary pilot-and-prior candidate ranking \\
14089 Forecaster only & Outcome/risk forecaster without obstruction-class matching \\
14090 Shuffled archive & Compiler trained on mismatched outcome and obstruction receipts \\
14091 No forum & Full child without within-problem result sharing \\
14092 Single benchmark & Full child whose final selector uses one scored benchmark \\
14093 No monitor & Full child with integrity filtering removed at selection time \\
14094 Random allocation & Uniform random experiment allocation under the same budget \\
14095 Audit oracle & Inadmissible ceiling using the hidden audit over the complete candidate grammar \\
14096 \bottomrule
14097 \end{tabularx}
14098 \end{center}
14099 
14100 \subsection{Frozen claim gates}
14101 
14102 The compiler-metaproductivity gate requires:
14103 \begin{enumerate}
14104 \item exact replay of the Block 12 result digest;
14105 \item identical candidate grammar, archive initialization, evaluator portfolio, monitor, and full-experiment budget;
14106 \item results-free pre-receipts with no outcome fields;
14107 \item a compiler and split seal produced before audit outcomes are accessed;
14108 \item a positive paired child-parent effect with a frozen interval excluding zero on both final splits;
14109 \item no architecture-foreign target favoring the parent;
14110 \item a large degradation under shuffled archive binding;
14111 \item lower protected gain under the single-benchmark and no-monitor controls;
14112 \item an audit oracle that remains at or above the admissible child;
14113 \item explicit reporting of null or weak component ablations.
14114 \end{enumerate}
14115 
14116 \section{Block 13 Results}
14117 \label{sec:block13-results}
14118 
14119 \subsection{Primary and architecture-foreign compiler effects}
14120 
14121 The main results are:
14122 \begin{center}
14123 \begin{tabular}{@{}lrr@{}}
14124 \toprule
14125 Compiler & Primary gain & Architecture-foreign gain \\
14126 \midrule
14127 AAR-style parent $\Gamma_0$ & $0.883004$ & $0.772566$ \\
14128 SRIC child $\Gamma_1$ & $0.950231$ & $0.863367$ \\
14129 Random allocation & $0.528176$ & $0.493549$ \\
14130 Audit oracle & $0.952585$ & $0.867092$ \\
14131 \bottomrule
14132 \end{tabular}
14133 \end{center}
14134 
14135 The primary paired effect is
14136 \begin{equation}
14137 \widehat\Delta_{\Gamma}^{\mathrm{primary}}
14138 =
14139 0.067227,
14140 \qquad
14141 \mathrm{CI}_{95\%}
14142 =
14143 [0.035881,0.103274].
14144 \label{eq:b13-primary-effect}
14145 \end{equation}
14146 The sign-flip estimate is $9.9999\times10^{-6}$. The child improves $21.875\%$ of pairs, ties on $75\%$, and loses on $3.125\%$.
14147 
14148 The architecture-foreign paired effect is
14149 \begin{equation}
14150 \widehat\Delta_{\Gamma}^{\mathrm{arch}}
14151 =
14152 0.090800,
14153 \qquad
14154 \mathrm{CI}_{95\%}
14155 =
14156 [0.052311,0.135081].
14157 \label{eq:b13-architecture-effect}
14158 \end{equation}
14159 The sign-flip estimate is again $9.9999\times10^{-6}$. The child improves $34.722\%$ of pairs, ties on $65.278\%$, and no architecture-foreign pair favors the parent.
14160 
14161 \begin{figure}[H]
14162 \centering
14163 \includegraphics[width=0.94\textwidth]{block13_compiler_gain.pdf}
14164 \caption{Matched-budget research-compiler performance. The AAR-style parent is strong; the SRIC child closes most of the remaining finite candidate-grammar headroom on both final splits.}
14165 \label{fig:b13-compiler-gain}
14166 \end{figure}
14167 
14168 \begin{figure}[H]
14169 \centering
14170 \includegraphics[width=0.92\textwidth]{block13_paired_effects.pdf}
14171 \caption{Paired child-minus-parent protected effects. Most problems are ties because both compilers find the same best candidate. The nonzero shifts remain positive in aggregate, with no negative pair under the architecture-foreign shift.}
14172 \label{fig:b13-paired-effects}
14173 \end{figure}
14174 
14175 The small oracle gap is important. On the primary split the audit oracle exceeds the child by only $0.002353$ on average; on the architecture-foreign split the gap is $0.003725$. The finite candidate grammar is therefore nearly saturated by the child. This prevents the result from being interpreted as an unbounded compiler advantage.
14176 
14177 \subsection{Performance by obstruction class}
14178 
14179 Primary mean gains by hidden obstruction class are:
14180 \begin{center}
14181 \begin{tabular}{@{}lrrr@{}}
14182 \toprule
14183 Class & Parent & Child & Difference \\
14184 \midrule
14185 Data & $0.935$ & $0.935$ & $0.000$ \\
14186 Objective & $0.943$ & $0.981$ & $0.038$ \\
14187 Response & $0.937$ & $0.957$ & $0.020$ \\
14188 Constructor & $0.871$ & $0.975$ & $0.104$ \\
14189 Evaluator & $0.842$ & $0.926$ & $0.084$ \\
14190 Resource & $0.770$ & $0.926$ & $0.156$ \\
14191 \bottomrule
14192 \end{tabular}
14193 \end{center}
14194 The largest gains occur where a generic balanced search is most likely to spend its six experiments on the wrong family or on a risky high-strength variant.
14195 
14196 \begin{figure}[H]
14197 \centering
14198 \includegraphics[width=0.90\textwidth]{block13_obstruction_classes.pdf}
14199 \caption{Primary compiler performance by hidden obstruction class. The child mainly improves constructor, evaluator, and resource-limited problems; the parent is already saturated on the data-limited class.}
14200 \label{fig:b13-obstruction-classes}
14201 \end{figure}
14202 
14203 \subsection{Allocation shift}
14204 
14205 The child concentrates experiments on the method family predicted to match the current obstruction, while retaining some generic and risk-calibration trials. The parent allocates more evenly across families.
14206 
14207 \begin{figure}[H]
14208 \centering
14209 \includegraphics[width=0.92\textwidth]{block13_allocation_shift.pdf}
14210 \caption{Child minus parent full-experiment allocation by hidden obstruction and tested method family. Positive diagonal mass records bottleneck-sensitive allocation; off-diagonal reductions record experiments saved relative to broad parent exploration.}
14211 \label{fig:b13-allocation}
14212 \end{figure}
14213 
14214 This is an allocation result, not candidate-constructor growth. Both compilers draw from the same $39$ methods, and the child does not introduce a method family unavailable to the parent.
14215 
14216 \subsection{Forecast and classification diagnostics}
14217 
14218 The obstruction classifier attains accuracy
14219 \begin{equation}
14220 0.989583
14221 \end{equation}
14222 on the primary split and
14223 \begin{equation}
14224 0.958333
14225 \end{equation}
14226 on the architecture-foreign split.
14227 
14228 The audit-gain forecaster has RMSE $0.218$ on both splits. Its correlation with realized candidate audit gain is $0.617$ on primary problems and $0.558$ under architecture shift.
14229 
14230 \begin{figure}[H]
14231 \centering
14232 \includegraphics[width=0.72\textwidth]{block13_forecast_calibration.pdf}
14233 \caption{Architecture-foreign predicted versus realized candidate audit gain. The forecaster is informative but imperfect; it is not a calibrated oracle.}
14234 \label{fig:b13-forecast}
14235 \end{figure}
14236 
14237 The ablations sharpen the mechanism claim. On the primary split:
14238 \begin{center}
14239 \begin{tabular}{@{}lr@{}}
14240 \toprule
14241 Compiler arm & Mean selected gain \\
14242 \midrule
14243 Full child & $0.950231$ \\
14244 Classifier only & $0.950118$ \\
14245 Forecaster only & $0.944727$ \\
14246 No forum & $0.950231$ \\
14247 \bottomrule
14248 \end{tabular}
14249 \end{center}
14250 The full child exceeds classifier-only by only $0.000114$, with a frozen interval whose lower endpoint is zero. The primary full-minus-forecaster effect is $0.005504$ and its interval includes zero. On the architecture-foreign split, full child exceeds classifier-only by $0.005985$ and forecaster-only by $0.022923$, with the latter interval positive.
14251 
14252 The finite result therefore supports obstruction diagnosis as the dominant improvement mechanism. Outcome forecasting contributes modestly under architecture shift but does not receive a strong independent primary claim. Removing the within-problem forum has no measurable effect in this fixture. The frozen development archive and high-accuracy diagnostic classifier make the six-step within-problem forum redundant.
14253 
14254 \subsection{Archive, evaluator, and integrity controls}
14255 
14256 Shuffling archive outcome and obstruction bindings collapses primary mean gain from $0.950231$ to $0.282550$. The paired reduction is $0.667682$ with interval $[0.602316,0.730058]$. The architecture-foreign shuffled value is $0.269438$.
14257 
14258 Selecting by one benchmark rather than the diverse portfolio reduces primary mean gain to $0.919699$ and architecture-foreign gain to $0.808152$. The paired full-child advantage is $0.030532$ on primary problems and $0.055214$ under architecture shift, with both frozen intervals positive.
14259 
14260 Removing integrity monitoring reduces primary valid protected gain to $0.797956$ and architecture-foreign gain to $0.735434$. Only $84.375\%$ and $84.722\%$ of selected methods, respectively, pass the integrity monitor. The comparison demonstrates that high nominal scored methods can consume the selection slot while failing the validity-filtered claim.
14261 
14262 \begin{figure}[H]
14263 \centering
14264 \includegraphics[width=0.94\textwidth]{block13_integrity_evaluator_ablation.pdf}
14265 \caption{Evaluator and integrity ablations. A single scored benchmark and removal of the monitor both reduce validity-filtered foreign performance. The shuffled archive causes the largest collapse.}
14266 \label{fig:b13-integrity-ablation}
14267 \end{figure}
14268 
14269 \subsection{Finite compiler-metaproductivity theorem}
14270 
14271 \begin{conditionaltheorem}[Block 13 finite research-compiler realization]
14272 \label{thm:b13-realization}
14273 Within the frozen Block 13 benchmark, assume that:
14274 \begin{enumerate}
14275 \item the Block 12 ancestor result digest replays exactly;
14276 \item parent and child receive the same candidate grammar, literature prior, initial archive, evaluator portfolio, capability gates, integrity monitor, and six-experiment budget;
14277 \item development, meta-validation, primary, and architecture-foreign problem sets are disjoint;
14278 \item every pre-result receipt is sealed before its outcome is generated;
14279 \item the parent and child compiler objects and split commitments are sealed before final audit outcomes are used;
14280 \item the paired effects and intervals are those in \cref{eq:b13-primary-effect,eq:b13-architecture-effect};
14281 \item the shuffled-archive, single-benchmark, no-monitor, random-allocation, component-ablation, and audit-oracle controls retain their declared interpretations.
14282 \end{enumerate}
14283 Then
14284 \begin{equation}
14285 \Gamma_0
14286 \xrightarrow{\Pi_7^{\Gamma}}
14287 \Gamma_1
14288 \end{equation}
14289 is a finite, matched-budget, lineage-generated improvement of the automated research compiler relative to the declared candidate grammar, research-problem distribution, evaluator portfolio, and integrity model.
14290 \end{conditionaltheorem}
14291 
14292 \begin{proof}
14293 Items 1 through 5 establish lineage continuity, resource matching, proposal precommitment, and audit isolation. Item 6 establishes positive paired foreign compiler effects. The controls in item 7 show that correct archive binding, evaluator diversity, and integrity filtering are substantive, while the classifier and forecaster ablations localize most of the gain to obstruction-sensitive allocation. Because the candidate grammar and final evaluator are unchanged, the realized edge changes the research compiler rather than the repair function class or audit definition.
14294 \end{proof}
14295 
14296 \begin{statusbox}{Block 13 result}
14297 Block 13 realizes one sealed research-compiler edge. A strong AAR-style parent and an SRIC child receive the same $39$ candidate methods and six full experiments per problem. On $96$ primary foreign problems, protected gain rises from $0.883004$ to $0.950231$; on $72$ architecture-foreign problems it rises from $0.772566$ to $0.863367$. Both paired intervals exclude zero. The mechanism is mainly high-accuracy obstruction classification and bottleneck-sensitive allocation. The outcome forecaster is moderately predictive but adds little on the primary split, and the within-problem forum is redundant in this finite setting. This is bounded compiler metaproductivity inside a fixed research language, not unrestricted compiler self-improvement or automated science.
14298 \end{statusbox}
14299 
14300 \section{Block 13 Reproducibility and Package}
14301 \label{sec:block13-package}
14302 
14303 \subsection{Reference implementation}
14304 
14305 The released implementation contains:
14306 \begin{itemize}
14307 \item the exact Block 12 canonical result digest;
14308 \item six hidden obstruction-class generators and an architecture-foreign shift;
14309 \item a common $39$-method candidate grammar;
14310 \item results-free and post-result receipt compilation with exact code and specification hashes;
14311 \item a strong AAR-style parent compiler;
14312 \item a lineage-trained obstruction classifier and ridge outcome/risk forecaster;
14313 \item finite parent and child meta-validation grids;
14314 \item multi-benchmark coverage scoring, capability gates, and integrity filtering;
14315 \item shuffled-archive, classifier, forecaster, forum, single-benchmark, no-monitor, random, and oracle controls;
14316 \item deterministic primary and architecture-foreign result objects;
14317 \item six publication figures;
14318 \item twelve automated tests;
14319 \item an exact SHA-256 package manifest.
14320 \end{itemize}
14321 
14322 The canonical execution prints:
14323 \begin{verbatim}
14324 SRIC_BLOCK13_ALL_SUITES=PASS
14325 result_sha256=1368fb8a09c3a63ec03ccb05f28e2ba994fc4daf80562d214bf88d29ffaed867
14326 primary_parent_gain=0.883004
14327 primary_child_gain=0.950231
14328 primary_effect=0.067227
14329 architecture_parent_gain=0.772566
14330 architecture_child_gain=0.863367
14331 architecture_effect=0.090800
14332 ............ [100%]
14333 SRIC_BLOCK13_PACKAGE_VALID
14334 \end{verbatim}
14335 
14336 The compiler receipt digest, pre-audit seal digest, canonical result digest, figure files, tests, and source package are included in the exact package manifest.
14337 
14338 \subsection{Claim boundary}
14339 
14340 The release establishes:
14341 \begin{enumerate}
14342 \item a finite formal object for proof-carrying automated research compilers;
14343 \item results-free proposal receipts bound to exact code and post-result evidence;
14344 \item a strong fixed parent compiler with literature priors, shared findings, benchmark diversity, capability gates, and integrity monitoring;
14345 \item a lineage-trained child that reallocates the same research budget using obstruction diagnosis and outcome/risk prediction;
14346 \item positive paired compiler metaproductivity on primary and architecture-foreign audits;
14347 \item evaluator-diversity, archive-binding, integrity, component, random, and oracle controls;
14348 \item package-closed deterministic replay.
14349 \end{enumerate}
14350 
14351 It does not establish:
14352 \begin{itemize}
14353 \item improvement of an unrestricted outer compiler;
14354 \item autonomous invention of the six obstruction classes;
14355 \item autonomous generation of the candidate grammar;
14356 \item automated literature judgment in an open scientific domain;
14357 \item reliable long-horizon research-outcome forecasting;
14358 \item autonomous evaluator or benchmark invention;
14359 \item an adversarially robust integrity monitor;
14360 \item generalization to real model-training research, hard-to-supervise science, or unconstrained external tools;
14361 \item open-ended recursive improvement, indefinite acceleration, unrestricted safety, consciousness, or sentience.
14362 \end{itemize}
14363 
14364 The next threshold is compiler grammar growth. A later compiler must infer that its current obstruction taxonomy, outcome features, or evaluator portfolio is inadequate, introduce a new research distinction or experiment type, and improve foreign research outcomes under a fresh sealed audit without receiving the correct new compiler module from the benchmark designer.
14365 
14366 \section{Adversarial Open-Set Research-Compiler Robustness}
14367 \label{sec:block14-robustness}
14368 
14369 Block 13 establishes one sealed improvement of a proof-carrying automated research compiler inside a fixed six-class research language. Its child nearly saturates the complete fixed-grammar audit oracle, and a classifier-only ablation almost reproduces the primary result. Those facts create a new bottleneck. Additional optimization inside the same clean taxonomy would provide little evidence that the compiler can operate when the research class is ambiguous, mixed, novel, contaminated, evaluator shifted, or adversarially presented.
14370 
14371 Block 14 therefore holds the candidate-method grammar and full-experiment budget fixed while changing the reliability conditions under which research decisions must be made. The published Block 13 result digest and exact $39$-method language are replayed. Its compiler design is retained as the inherited parent and recalibrated on the Block 14 development archive:
14372 \begin{equation}
14373 \Gamma_1^{\rm recal}.
14374 \end{equation}
14375 A robust descendant
14376 \begin{equation}
14377 \Gamma_1^{\rm recal}\xrightarrow{\Pi_8^\Gamma}\Gamma_2
14378 \label{eq:block14-edge}
14379 \end{equation}
14380 is compiled before either final audit is opened. The child adds sequential diagnosis, sparse multi-obstruction inference, open-set detection, contamination-resistant archive summaries, a portfolio-aware robust selector, and an ensemble integrity monitor. Parent and child retain the same candidate grammar and exactly six full experiments per problem.
14381 
14382 \subsection{Robustness environment}
14383 
14384 \begin{definition}[Research-compiler robustness environment]
14385 A research-compiler robustness environment is a tuple
14386 \begin{equation}
14387 \mathfrak U
14388 =
14389 \left(
14390 \mathcal Q,
14391 \mathcal H,
14392 \mathcal A,
14393 \mathcal V,
14394 \mathcal G
14395 \right),
14396 \label{eq:robustness-environment}
14397 \end{equation}
14398 where:
14399 \begin{enumerate}
14400 \item $\mathcal Q$ is a family of research-problem distributions and generator shifts;
14401 \item $\mathcal H$ is a family of archive contaminations and provenance perturbations;
14402 \item $\mathcal A$ is a family of integrity, specification, and selection attacks;
14403 \item $\mathcal V$ is a family of evaluator portfolios and aggregation rules;
14404 \item $\mathcal G$ is a family of benchmark generators, including problems outside the inherited obstruction taxonomy.
14405 \end{enumerate}
14406 An element $u=(Q,H,A,V,G)\in\mathfrak U$ determines one protected evaluation context for the research compiler.
14407 \end{definition}
14408 
14409 Let $U_u^{\rm valid}(\Gamma,\tau)$ denote the validity-filtered protected gain produced by compiler $\Gamma$ on research problem $\tau$ under environment $u$. In the executable fixture, a selected method that passes the stronger post hoc capability and integrity predicates receives its protected audit gain, a no-op receives zero, and a selected invalid method receives the frozen penalty $-0.75$. This prevents an invalid selection from being silently converted into a neutral outcome. Let $R_u(\Gamma,\tau)$ be protected downside or integrity risk. Define
14410 \begin{equation}
14411 \mathfrak C_u(\Gamma)
14412 =
14413 \mathbb E_{\tau\sim Q}
14414 U_u^{\rm valid}(\Gamma,\tau).
14415 \label{eq:compiler-context-utility}
14416 \end{equation}
14417 For $\alpha\in(0,1)$, write $\operatorname{CVaR}_\alpha(X)$ for the mean of the lower $\alpha$-tail of a gain variable $X$.
14418 
14419 \begin{definition}[Robust validity-filtered compiler value]
14420 For risk weight $\lambda\geq0$, define
14421 \begin{equation}
14422 \mathfrak C_{\rm rob}(\Gamma)
14423 =
14424 \inf_{u\in\mathfrak U}
14425 \left[
14426 \mathfrak C_u(\Gamma)
14427 -
14428 \lambda\operatorname{CVaR}_\alpha
14429 \bigl(-R_u(\Gamma,\tau)\bigr)
14430 \right].
14431 \label{eq:robust-compiler-value}
14432 \end{equation}
14433 The finite Block 14 realization does not compute the unrestricted infimum. It evaluates a frozen finite stress atlas and reports the mean, groupwise minima, lower-tail gain, negative-gain rate, invalid-selection rate, and unknown-class error separately.
14434 \end{definition}
14435 
14436 \begin{definition}[Finite robust compiler edge]
14437 A sealed lineage edge $\Gamma_1\to\Gamma_2$ is a finite robust compiler improvement relative to a declared stress atlas when:
14438 \begin{enumerate}
14439 \item parent and child receive the same candidate grammar, evidence interfaces, evaluator calls, integrity calls, and experiment budget;
14440 \item the child is sealed before final problems and outcomes are generated;
14441 \item paired mean gain has a positive frozen lower confidence bound on both primary and generator-foreign audits;
14442 \item the child has nonnegative mean gain in every declared stress group;
14443 \item lower-tail gain improves and the selected invalid-method rate does not increase;
14444 \item open-set error and protected negative-gain rates remain below declared bounds;
14445 \item mechanism ablations and an inadmissible audit oracle retain their declared interpretations.
14446 \end{enumerate}
14447 \end{definition}
14448 
14449 The definition deliberately does not require every episode-wise paired effect to be positive. A robust compiler may trade a small number of local losses for large improvements on previously catastrophic cases. Those losses must remain visible through paired, tail, and group-resolved reporting.
14450 
14451 \subsection{Diagnostic aliasing and active separation}
14452 
14453 Block 13 assigns one of six obstruction labels from one cheap diagnostic vector. Block 14 replaces that clean classification regime by aliased observations. Several distinct obstructions share an initial diagnostic center, so one static observation can be insufficient.
14454 
14455 Let $Z$ be a latent obstruction variable and let $X_0$ be the initial diagnostic observation.
14456 
14457 \begin{theorem}[Static diagnostic indistinguishability]
14458 \label{thm:static-diagnostic-indistinguishability}
14459 Let $z_1\neq z_2$ satisfy
14460 \begin{equation}
14461 \mathbb P(X_0\in B\mid Z=z_1)
14462 =
14463 \mathbb P(X_0\in B\mid Z=z_2)
14464 \end{equation}
14465 for every measurable set $B$. Then no decision rule $d(X_0)$ distinguishes $z_1$ from $z_2$ with accuracy exceeding the larger prior probability:
14466 \begin{equation}
14467 \mathbb P(d(X_0)=Z\mid Z\in\{z_1,z_2\})
14468 \leq
14469 \max\{\pi_1,\pi_2\},
14470 \end{equation}
14471 where $\pi_j=\mathbb P(Z=z_j\mid Z\in\{z_1,z_2\})$.
14472 \end{theorem}
14473 
14474 \begin{proof}
14475 The conditional law of $X_0$ is identical under the two hypotheses. Bayes' rule therefore leaves the posterior odds equal to the prior odds for almost every observation. The Bayes-optimal action always selects the more probable class and achieves accuracy $\max\{\pi_1,\pi_2\}$.
14476 \end{proof}
14477 
14478 This theorem gives a precise reason to value experiments rather than static classification. Suppose an experiment $a$ returns an observation $Y_a$ whose distribution differs between the aliased hypotheses. The compiler can then choose a sequence
14479 \begin{equation}
14480 a_1,a_2,\ldots,a_t
14481 \end{equation}
14482 that increases the information available for diagnosis.
14483 
14484 For a finite response model, let $R\in\R^{m\times k}$ contain the expected response signatures of $k$ inherited obstruction components under $m$ experiment types. Let $z\in\R_{\\geq0}^k$ be a sparse mixture with $\one^\top z=1$. An observed response vector has the form
14485 \begin{equation}
14486 y=Rz+\xi.
14487 \label{eq:sparse-response-model}
14488 \end{equation}
14489 
14490 \begin{proposition}[Response-mixture recovery bound]
14491 \label{prop:response-mixture-bound}
14492 Assume $R$ has full column rank and $\norm{\xi}_2\leq\eps$. The least-squares estimator
14493 \begin{equation}
14494 \widehat z=R^\dagger y
14495 \end{equation}
14496 satisfies
14497 \begin{equation}
14498 \norm{\widehat z-z}_2
14499 \leq
14500 \norm{R^\dagger}_2\eps
14501 =
14502 \frac{\eps}{\sigma_{\min}(R)}.
14503 \label{eq:mixture-recovery-bound}
14504 \end{equation}
14505 \end{proposition}
14506 
14507 \begin{proof}
14508 Since $R^\dagger R=I$ under full column rank,
14509 \begin{equation}
14510 \widehat z-z=R^\dagger\xi.
14511 \end{equation}
14512 The operator-norm inequality gives the claim.
14513 \end{proof}
14514 
14515 The bound separates two failure modes. More repeated observations reduce $\eps$, while a new experiment can increase $\sigma_{\min}(R)$ by adding a separating response direction. If $R$ is rank deficient, no sample-size increase removes the corresponding response alias.
14516 
14517 \subsection{Open-set residuals}
14518 
14519 The inherited obstruction language spans a response cone
14520 \begin{equation}
14521 \mathcal K_R
14522 =
14523 \{Rz:z\in\R_{\geq0}^k\}.
14524 \end{equation}
14525 For observed response $y$, define the open-set residual
14526 \begin{equation}
14527 \eta_{\rm open}(y)
14528 =
14529 \dist(y,\mathcal K_R).
14530 \label{eq:open-set-residual}
14531 \end{equation}
14532 
14533 \begin{proposition}[Open-set separation]
14534 \label{prop:open-set-separation}
14535 Suppose every inherited known response $y_{\rm known}$ satisfies
14536 \begin{equation}
14537 \eta_{\rm open}(y_{\rm known})\leq\eps,
14538 \end{equation}
14539 while every unknown response in a target family satisfies
14540 \begin{equation}
14541 \eta_{\rm open}(y_{\rm unknown})\geq\gamma
14542 \end{equation}
14543 for $\gamma>\eps$. Then every threshold $\tau\in(\eps,\gamma)$ perfectly separates the two families.
14544 \end{proposition}
14545 
14546 \begin{proof}
14547 Known responses lie below $\tau$ and unknown responses lie above it by the assumed margin.
14548 \end{proof}
14549 
14550 The proposition is a finite margin statement, not a universal novelty detector. In the executable benchmark the child uses a learned residual and distance gate. When the open-set flag is active, it does not necessarily return a no-op. It routes the remaining experiment budget toward broad generic candidates and retains the same final integrity and capability witness. Thus the measured quantity is false confident known-class assignment rather than raw abstention alone.
14551 
14552 \subsection{Archive contamination}
14553 
14554 Let the clean receipt distribution be $P$ and let an $\epsilon$-contaminated archive satisfy
14555 \begin{equation}
14556 \widetilde P_\epsilon=(1-\epsilon)P+\epsilon H,
14557 \label{eq:contamination-model}
14558 \end{equation}
14559 where $H$ may contain mislabeled obstruction receipts, corrupted outcomes, duplicated evidence, or selectively omitted failures.
14560 
14561 \begin{proposition}[Median breakdown in one coordinate]
14562 \label{prop:median-breakdown}
14563 Let $x_1,\ldots,x_n\in\R$, and suppose fewer than $n/2$ observations are replaced arbitrarily. Then the sample median of the contaminated set lies between two uncontaminated order statistics. In particular, an adversary controlling less than half the observations cannot drive the median beyond the range of the retained clean central order statistics.
14564 \end{proposition}
14565 
14566 \begin{proof}
14567 At least $\lceil n/2\rceil$ clean observations remain. Any contaminated median must have at least half of all observations on each side. It therefore cannot cross both central retained clean order statistics.
14568 \end{proof}
14569 
14570 Coordinatewise medians and trimmed means inherit finite-sample robustness coordinate by coordinate. They do not provide arbitrary high-dimensional optimality, and they do not repair systematic contamination shared by most archive entries. Block 14 therefore treats contamination-resistant archive estimation as one component of a larger mechanism and reports its ablation separately.
14571 
14572 \subsection{Evaluator portfolios and integrity ensembles}
14573 
14574 Let $b_1,\ldots,b_m$ be benchmark readouts. A single scored benchmark leaves the local nullspace
14575 \begin{equation}
14576 \mathcal N_{\{b_1\}}
14577 =
14578 \ker Db_1.
14579 \end{equation}
14580 An evaluator portfolio has invisible subspace
14581 \begin{equation}
14582 \mathcal N_{\mathcal B}
14583 =
14584 \bigcap_{j=1}^m\ker Db_j.
14585 \label{eq:evaluator-nullspace}
14586 \end{equation}
14587 
14588 \begin{proposition}[Strict evaluator-nullspace refinement]
14589 \label{prop:evaluator-nullspace-refinement}
14590 If $Db_{m+1}$ is nonzero on some $v\in\mathcal N_{\mathcal B}$, then
14591 \begin{equation}
14592 \mathcal N_{\mathcal B\cup\{b_{m+1}\}}
14593 \subsetneq
14594 \mathcal N_{\mathcal B}.
14595 \end{equation}
14596 \end{proposition}
14597 
14598 \begin{proof}
14599 The new nullspace is the intersection of the old nullspace with $\ker Db_{m+1}$. The vector $v$ belongs to the old nullspace but not the new one, proving strict inclusion.
14600 \end{proof}
14601 
14602 The result does not prove that a finite evaluator portfolio is complete. It shows exactly when a new evaluator dimension removes one local metric-gaming direction.
14603 
14604 For integrity monitoring, let $M_1,\ldots,M_s\in\{0,1\}$ be monitor pass indicators and let the ensemble pass only when every component passes:
14605 \begin{equation}
14606 M_{\cap}=\prod_{j=1}^s M_j.
14607 \end{equation}
14608 
14609 \begin{proposition}[Conjunctive monitor false-negative bound]
14610 \label{prop:monitor-bound}
14611 For an invalid method event $I$, the ensemble false-negative probability satisfies
14612 \begin{equation}
14613 \mathbb P(M_{\cap}=1\mid I)
14614 \leq
14615 \min_j\mathbb P(M_j=1\mid I).
14616 \end{equation}
14617 If monitor errors are conditionally independent given $I$, then
14618 \begin{equation}
14619 \mathbb P(M_{\cap}=1\mid I)
14620 =
14621 \prod_{j=1}^s
14622 \mathbb P(M_j=1\mid I).
14623 \end{equation}
14624 \end{proposition}
14625 
14626 \begin{proof}
14627 The event that every monitor passes is contained in the event that any particular monitor passes. This proves the first bound. Conditional independence gives the product formula.
14628 \end{proof}
14629 
14630 Conditional independence is not assumed in the executable claim. The package uses the ensemble operationally and compares it with a weaker monitor ablation. In the frozen final audits, that ablation is null: the robust allocation and final score avoid the declared adaptive attacks even under the inherited monitor. The monitor ensemble therefore receives no independent positive attribution in Block 14.
14631 
14632 \section{Block 14 Experimental Contract}
14633 \label{sec:block14-contract}
14634 
14635 \subsection{Inherited parent and robust child}
14636 
14637 The inherited parent is the exact Block 13 research compiler reconstructed from the canonical ancestor digest
14638 \begin{equation}
14639 1368fb8a09c3a63ec03ccb05f28e2ba994fc4daf80562d214bf88d29ffaed867.
14640 \end{equation}
14641 It uses ordinary archive summaries, one-shot diagnostic routing, broad exploration, one inherited integrity test, and the same final candidate selector used in Block 13.
14642 
14643 The robust child retains the parent candidate grammar and final protected objective while adding:
14644 \begin{enumerate}
14645 \item coordinatewise robust archive summaries;
14646 \item sequential response experiments inside the fixed six-experiment budget;
14647 \item sparse mixture inference over inherited obstruction components;
14648 \item open-set residual and distance gates;
14649 \item a multi-portfolio robust candidate score;
14650 \item a conjunctive specification and trajectory monitor ensemble.
14651 \end{enumerate}
14652 
14653 The selected child configuration is
14654 \begin{equation}
14655 \begin{aligned}
14656 \tau_{\rm residual}&=0.20,
14657 &\tau_{\rm distance}&=24,\\
14658 w_{\rm mean}&=0.38,
14659 &w_{\rm median}&=0.10,\\
14660 w_{\rm min}&=0.24,
14661 &w_{\rm held}&=0.22,\\
14662 w_{\rm capability}&=0.32.
14663 \end{aligned}
14664 \label{eq:block14-config}
14665 \end{equation}
14666 It is selected from a $27$-point finite meta-validation grid before either final audit is opened.
14667 
14668 \subsection{Problem groups}
14669 
14670 Each final audit contains six equally represented groups:
14671 \begin{enumerate}
14672 \item \textbf{Clean.} One known obstruction with informative but noisy diagnostics.
14673 \item \textbf{Ambiguous.} Two inherited classes share an aliased initial diagnostic center.
14674 \item \textbf{Multi.} Two obstruction components are simultaneously active.
14675 \item \textbf{Unknown.} The response lies outside the inherited known response cone.
14676 \item \textbf{Evaluator attack.} High scored performance can conceal poor protected behavior under a rotated audit weighting.
14677 \item \textbf{Integrity attack.} Evasive methods can pass the inherited simple integrity rule while violating the stronger trajectory rule.
14678 \end{enumerate}
14679 
14680 The primary audit uses generator $A$ and contains $144$ problems. The generator-foreign audit uses a shifted generator $B$, altered diagnostic geometry, and $120$ problems. Neither final generator is available during compiler construction.
14681 
14682 \subsection{Archive contamination and candidate grammar}
14683 
14684 The development archive is contaminated at rate
14685 \begin{equation}
14686 \epsilon_{\rm archive}=0.18.
14687 \end{equation}
14688 Contamination can perturb diagnostics, obstruction labels, research outcomes, and response records. The finite candidate grammar contains the exact $39$ methods inherited from Block 13: five variants in each of six obstruction-matched families, five generic variants, and the four original integrity and capability controls. Parent and child receive the same canonical candidate order and grammar digest.
14689 
14690 The package additionally evaluates contamination levels
14691 \begin{equation}
14692 0,
14693 0.05,
14694 0.10,
14695 0.18,
14696 0.25,
14697 0.35
14698 \end{equation}
14699 on a frozen auxiliary population.
14700 
14701 \subsection{Matched research budget and chronology}
14702 
14703 Both compilers receive exactly
14704 \begin{equation}
14705 B_{\rm research}=6
14706 \label{eq:block14-budget}
14707 \end{equation}
14708 full experiments per problem. The child can use early experiments to estimate response mixtures and allocate later experiments, but it receives no additional full experiment. Candidate metadata, proposal count, evaluator calls, capability checks, and integrity calls are matched.
14709 
14710 The chronology is:
14711 \begin{equation}
14712 \begin{gathered}
14713 \text{replay Block 13 ancestor}
14714 \longrightarrow
14715 \text{build contaminated development archive}\\
14716 \longrightarrow
14717 \text{fit parent and robust archive models}
14718 \longrightarrow
14719 \text{select child configuration on meta-validation}\\
14720 \longrightarrow
14721 \text{serialize compiler and split commitments}
14722 \longrightarrow
14723 \text{generate primary and generator-foreign audits}\\
14724 \longrightarrow
14725 \text{evaluate every frozen arm and claim gate}.
14726 \end{gathered}
14727 \label{eq:block14-chronology}
14728 \end{equation}
14729 The pre-audit seal contains no audit outcomes.
14730 
14731 \subsection{Comparison arms}
14732 
14733 The package includes:
14734 \begin{longtable}{@{}p{0.24\textwidth}p{0.68\textwidth}@{}}
14735 \toprule
14736 Arm & Role \\
14737 \midrule
14738 \endfirsthead
14739 \toprule
14740 Arm & Role \\
14741 \midrule
14742 \endhead
14743 Inherited parent & Exact Block 13-style static compiler under the new adversarial problem distribution \\
14744 Robust child & Full sequential, open-set, robust-archive, portfolio, and monitor-ensemble compiler \\
14745 No open set & Child with every problem forced into the inherited known taxonomy \\
14746 No sequential diagnosis & Child without response-based updating during the six experiments \\
14747 Ordinary archive & Child decision logic trained on ordinary rather than robust archive summaries \\
14748 Simple monitor & Child with the inherited weak integrity rule in place of the ensemble \\
14749 Single portfolio & Child whose final decision uses one evaluator portfolio \\
14750 Random allocation & Uniform candidate allocation under the same experiment budget \\
14751 Audit oracle & Inadmissible complete-grammar selection by hidden audit utility \\
14752 \bottomrule
14753 \end{longtable}
14754 
14755 \subsection{Frozen claim gates}
14756 
14757 The robust compiler gate requires:
14758 \begin{enumerate}
14759 \item exact Block 13 ancestor replay;
14760 \item identical candidate grammar and six-experiment budget;
14761 \item a compiler and split seal before final outcome access;
14762 \item positive paired child-parent intervals on both final audits;
14763 \item positive child lower-tail gain on both final audits;
14764 \item positive child mean gain in every declared problem group;
14765 \item reduced false confident known-class assignment on unknown problems;
14766 \item zero observed selected invalid methods in both final audits;
14767 \item degraded results for no-open-set, no-sequential, and ordinary-archive controls in their declared dimensions, together with explicit reporting of the null simple-monitor and near-null single-portfolio ablations;
14768 \item an audit oracle that remains at or above the admissible child.
14769 \end{enumerate}
14770 
14771 \section{Block 14 Results}
14772 \label{sec:block14-results}
14773 
14774 \subsection{Primary and generator-foreign effects}
14775 
14776 The primary paired results are:
14777 \begin{longtable}{@{}lrrrr@{}}
14778 \toprule
14779 Compiler & Mean gain & Median & $\operatorname{CVaR}_{0.10}$ & Invalid selected \\
14780 \midrule
14781 \endfirsthead
14782 \toprule
14783 Compiler & Mean gain & Median & $\operatorname{CVaR}_{0.10}$ & Invalid selected \\
14784 \midrule
14785 \endhead
14786 Recalibrated parent $\Gamma_1^{\rm recal}$ & 0.410813 & 0.411537 & -0.133065 & 0.027778 \\
14787 Robust child $\Gamma_2$ & \textbf{0.575783} & 0.592510 & \textbf{0.271516} & \textbf{0.000000} \\
14788 Random allocation & 0.354319 & 0.305382 & 0.088336 & 0.000000 \\
14789 Audit oracle & 0.632185 & 0.656311 & 0.378857 & 0.000000 \\
14790 \bottomrule
14791 \end{longtable}
14792 
14793 The primary paired effect is
14794 \begin{equation}
14795 \widehat\Delta_{\Gamma}^{\rm primary}
14796 =
14797 0.164970,
14798 \qquad
14799 \mathrm{CI}_{95\%}
14800 =
14801 [0.115591,0.218655].
14802 \label{eq:block14-primary-effect}
14803 \end{equation}
14804 The sign-flip estimate is
14805 \begin{equation}
14806 p=9.9999\times10^{-6}.
14807 \end{equation}
14808 The child improves $46.53\%$ of paired problems, ties on $41.67\%$, and loses on $11.81\%$. The losses are therefore retained in the claim rather than removed by a no-negative-pair gate.
14809 
14810 On the generator-foreign audit,
14811 \begin{equation}
14812 \mathfrak C(\Gamma_1^{\rm recal})=0.389514,
14813 \qquad
14814 \mathfrak C(\Gamma_2)=0.550559,
14815 \end{equation}
14816 so
14817 \begin{equation}
14818 \widehat\Delta_{\Gamma}^{\rm foreign}
14819 =
14820 0.161045,
14821 \qquad
14822 \mathrm{CI}_{95\%}
14823 =
14824 [0.105335,0.223141].
14825 \label{eq:block14-foreign-effect}
14826 \end{equation}
14827 The parent lower-tail gain is $-0.193011$, while the child lower-tail gain is $0.209497$. The child selects no invalid method and has no negative selected gain on this final split.
14828 
14829 \begin{figure}[H]
14830 \centering
14831 \includegraphics[width=0.92\textwidth]{block14_compiler_gain.pdf}
14832 \caption{Matched-budget compiler performance on the primary and generator-foreign audits. The robust child raises mean validity-filtered gain while preserving the six-experiment budget.}
14833 \label{fig:block14-compiler-gain}
14834 \end{figure}
14835 
14836 \begin{figure}[H]
14837 \centering
14838 \includegraphics[width=0.94\textwidth]{block14_paired_effects.pdf}
14839 \caption{Episode-wise robust-child minus inherited-parent effects. The primary distribution contains retained negative pairs, while the aggregate and generator-foreign effects remain positive.}
14840 \label{fig:block14-paired-effects}
14841 \end{figure}
14842 
14843 \subsection{Group-resolved robustness}
14844 
14845 Primary mean gains by problem group are:
14846 \begin{longtable}{@{}lrrr@{}}
14847 \toprule
14848 Group & Parent & Robust child & Difference \\
14849 \midrule
14850 \endfirsthead
14851 \toprule
14852 Group & Parent & Robust child & Difference \\
14853 \midrule
14854 \endhead
14855 Clean & 0.522696 & 0.672488 & 0.149792 \\
14856 Ambiguous & 0.556180 & 0.614061 & 0.057881 \\
14857 Multi-obstruction & 0.316613 & 0.372614 & 0.056001 \\
14858 Unknown & 0.224136 & 0.446679 & 0.222543 \\
14859 Evaluator attack & 0.405737 & 0.674109 & 0.268372 \\
14860 Integrity attack & 0.439514 & 0.674745 & 0.235231 \\
14861 \bottomrule
14862 \end{longtable}
14863 
14864 Every child group mean exceeds $0.37$, and every primary group has a positive mean child-minus-parent effect. The paired interval for the ambiguous group still includes zero, so the group-resolved evidence is not uniformly confirmatory. The weakest child groups are the multi-obstruction and unknown families, which remain materially below clean and attack families.
14865 
14866 Generator-foreign child means remain positive in every group. The smallest are approximately $0.363$ for multi-obstruction problems and $0.396$ for unknown problems. Every group mean exceeds its parent mean, but the paired intervals for ambiguous and evaluator-attack groups include zero. The aggregate foreign result is therefore stronger than every individual group claim.
14867 
14868 \begin{figure}[H]
14869 \centering
14870 \includegraphics[width=0.96\textwidth]{block14_group_performance.pdf}
14871 \caption{Group-resolved mean validity-filtered gain. The largest primary gains occur on evaluator, integrity, and unknown stress groups; paired group uncertainty remains reported separately.}
14872 \label{fig:block14-groups}
14873 \end{figure}
14874 
14875 \subsection{Lower-tail and failure-rate behavior}
14876 
14877 The parent-to-child primary changes are:
14878 \begin{equation}
14879 \operatorname{CVaR}_{0.10}:
14880 -0.133065
14881 \longrightarrow
14882 0.271516,
14883 \end{equation}
14884 \begin{equation}
14885 \mathbb P(G<0):
14886 0.027778
14887 \longrightarrow
14888 0,
14889 \end{equation}
14890 \begin{equation}
14891 \mathbb P(\text{invalid selected}):
14892 0.027778
14893 \longrightarrow
14894 0.
14895 \end{equation}
14896 For the unknown group, false confident known-class assignment changes from
14897 \begin{equation}
14898 1
14899 \longrightarrow
14900 0.083333.
14901 \label{eq:block14-open-set-rate}
14902 \end{equation}
14903 The child still produces a selected method on every primary problem. The improvement is therefore open-set-aware rerouting rather than universal no-op abstention.
14904 
14905 \begin{figure}[H]
14906 \centering
14907 \includegraphics[width=0.92\textwidth]{block14_tail_robustness.pdf}
14908 \caption{Mean and lower-tail protected gain. The robust child changes the worst decile from substantially negative to positive on both final audits.}
14909 \label{fig:block14-tail}
14910 \end{figure}
14911 
14912 \begin{figure}[H]
14913 \centering
14914 \includegraphics[width=0.92\textwidth]{block14_failure_rates.pdf}
14915 \caption{Negative-gain, invalid-selection, and false-known-unknown rates. The complete descendant selects no invalid method and sharply reduces forced known-class assignment. The simple-monitor ablation is identical, so the invalid-selection reduction is not attributed uniquely to the monitor ensemble.}
14916 \label{fig:block14-failures}
14917 \end{figure}
14918 
14919 \begin{remark}[Zero-event interpretation]
14920 The child selects zero invalid methods in $144$ primary and $120$ generator-foreign problems. This is not a zero population-risk theorem. The exact one-sided $95\%$ binomial upper bound reported by the package is approximately $0.02059$ for $144$ zero-event trials and $0.02465$ for $120$ zero-event trials.
14921 \end{remark}
14922 
14923 \subsection{Mechanism ablations}
14924 
14925 The primary ablation summary is:
14926 \begin{longtable}{@{}lrrr@{}}
14927 \toprule
14928 Arm & Mean gain & $\operatorname{CVaR}_{0.10}$ & Key failure \\
14929 \midrule
14930 \endfirsthead
14931 \toprule
14932 Arm & Mean gain & $\operatorname{CVaR}_{0.10}$ & Key failure \\
14933 \midrule
14934 \endhead
14935 Robust child & 0.575783 & 0.271516 & -- \\
14936 No open set & 0.568030 & 0.094979 & Unknown false-known rate $1$ \\
14937 No sequential diagnosis & 0.494244 & 0.102696 & Weak aliased, mixed, and open-set diagnosis \\
14938 Ordinary archive & 0.536955 & 0.247897 & Lower primary mean under contamination \\
14939 Simple monitor & 0.575783 & 0.271516 & Null ablation in the final audits \\
14940 Single portfolio & 0.575283 & 0.268034 & Near-null mean and tail ablation \\
14941 Random allocation & 0.354319 & 0.088336 & No bottleneck-sensitive allocation \\
14942 Audit oracle & 0.632185 & 0.378857 & Inadmissible hidden-audit ceiling \\
14943 \bottomrule
14944 \end{longtable}
14945 
14946 Four qualifications matter.
14947 
14948 First, open-set routing primarily improves semantic diagnosis and the lower tail. Disabling it changes primary mean only modestly but forces every unknown problem into a known class and lowers worst-decile gain from $0.271516$ to $0.094979$. Second, response-based sequential updating is substantively supported: its removal lowers primary mean to $0.494244$ and lower-tail gain to $0.102696$. Third, robust archive summaries improve primary mean by approximately $0.0388$ and remain superior throughout the contamination sweep, but the generator-foreign increment over ordinary summaries is only about $0.0181$. Fourth, the simple-monitor ablation is exactly null and the single-portfolio ablation is nearly null. Block 14 therefore does not attribute an independent positive effect to either the monitor ensemble or portfolio aggregation in this final fixture.
14949 
14950 \subsection{Contamination sweep}
14951 
14952 Across contamination rates from $0$ to $0.35$, child mean gain remains between approximately $0.532$ and $0.554$, while parent mean gain remains between approximately $0.427$ and $0.449$. Child lower-tail gain remains between $0.248$ and $0.277$, compared with parent lower-tail gain between $0.113$ and $0.127$. No invalid method is selected by either arm in this auxiliary sweep, so the comparison supports estimator stability rather than an integrity claim.
14953 
14954 \begin{figure}[H]
14955 \centering
14956 \includegraphics[width=0.94\textwidth]{block14_contamination_curve.pdf}
14957 \caption{Compiler performance across increasing archive contamination. The robust child remains stable in the finite sweep; the result does not imply robustness beyond the declared contamination process.}
14958 \label{fig:block14-contamination}
14959 \end{figure}
14960 
14961 \subsection{Finite adversarial robustness theorem}
14962 
14963 \begin{conditionaltheorem}[Block 14 finite robust compiler realization]
14964 \label{thm:block14-realization}
14965 Within the frozen Block 14 benchmark, assume that:
14966 \begin{enumerate}
14967 \item the published Block 13 canonical result digest replays exactly;
14968 \item parent and child receive the exact same $39$-candidate Block 13 grammar and exactly six full experiments per problem;
14969 \item development, meta-validation, primary, and generator-foreign supports are disjoint;
14970 \item the compiler object and split commitments are sealed before final outcomes are generated;
14971 \item the development archive is contaminated according to the declared finite process;
14972 \item primary and generator-foreign paired effects have the intervals in \cref{eq:block14-primary-effect,eq:block14-foreign-effect};
14973 \item every child stress-group mean is positive and both child lower-tail gains are positive;
14974 \item the no-open-set, no-sequential, ordinary-archive, simple-monitor, single-portfolio, random, contamination, and oracle controls retain their declared interpretations, including the null monitor and near-null portfolio ablations;
14975 \item package, receipt, result, and manifest digests replay exactly.
14976 \end{enumerate}
14977 Then the edge in \cref{eq:block14-edge} is a finite, sealed, matched-budget robustness improvement of the inherited research compiler relative to the declared diagnostic-aliasing, multi-obstruction, open-set, archive-contamination, evaluator-shift, and integrity-attack atlas.
14978 \end{conditionaltheorem}
14979 
14980 \begin{proof}
14981 Items 1 through 4 establish lineage continuity, matched resources, and audit isolation. Item 5 places both compilers inside the declared contamination regime. Item 6 establishes positive mean paired improvement on two final generators. Item 7 rules out an aggregate gain obtained by sacrificing one entire stress group or the lower tail. Item 8 localizes substantive contributions to sequential diagnosis, open-set routing, and robust archive estimation while retaining the null monitor and near-null portfolio results rather than assigning unsupported causal credit. Item 9 establishes deterministic package closure. The conclusion remains relative to the finite stress atlas and fixed method language.
14982 \end{proof}
14983 
14984 \section{Block 14 Reproducibility and Claim Boundary}
14985 \label{sec:block14-reproducibility}
14986 
14987 \subsection{Reference implementation}
14988 
14989 The released implementation contains:
14990 \begin{itemize}
14991 \item exact replay of the Block 13 canonical result digest;
14992 \item clean, aliased, mixed, unknown, evaluator-attack, and integrity-attack problem generators;
14993 \item an independently shifted generator-foreign audit;
14994 \item the exact common $39$-candidate Block 13 method grammar and six-experiment budget;
14995 \item contaminated receipt archives with ordinary and robust estimators;
14996 \item static and sequential diagnosis, response-mixture estimation, and open-set residuals;
14997 \item evaluator-portfolio and integrity-monitor ablations;
14998 \item contamination sweeps, paired intervals, sign-flip tests, and lower-tail summaries;
14999 \item results-free and post-result receipt chains;
15000 \item six publication figures;
15001 \item fifteen automated tests;
15002 \item an exact package manifest.
15003 \end{itemize}
15004 
15005 The canonical execution prints:
15006 \begin{verbatim}
15007 SRIC_BLOCK14_ALL_SUITES=PASS
15008 result_sha256=e066fe56a2863b529a0aed51cad8a6e841172bffb231dab252a596a6a58b05b2
15009 primary_parent_gain=0.410813
15010 primary_child_gain=0.575783
15011 primary_effect=0.164970
15012 generator_foreign_parent_gain=0.389514
15013 generator_foreign_child_gain=0.550559
15014 generator_foreign_effect=0.161045
15015 ............... [100%]
15016 SRIC_BLOCK14_PACKAGE_VALID
15017 manifest_files=36
15018 \end{verbatim}
15019 
15020 \subsection{Claim boundary}
15021 
15022 Block 14 establishes a finite robustness edge for one fixed-language research compiler. It does not establish:
15023 \begin{itemize}
15024 \item unrestricted open-set scientific reasoning;
15025 \item robustness to arbitrary archive poisoning or adaptive monitor evasion;
15026 \item independent external replication;
15027 \item confidence sequences for unrestricted adaptive compiler growth;
15028 \item autonomous invention of the obstruction taxonomy, experiment language, method grammar, evaluator portfolio, or monitor architecture;
15029 \item real large-model training research or open tool use;
15030 \item open-ended recursive improvement, indefinite acceleration, unrestricted safety, consciousness, or sentience.
15031 \end{itemize}
15032 
15033 The next threshold has two separable parts. Block 15 addresses the finite compiler-language component by requiring a conflict certificate before adding a diagnostic distinction and method constructors. Independent executable research, external generation, and less scaffolded meta-language growth remain open.
15034 
15035 
15036 \section{Compiler-Language Growth from Open-Set Research Conflicts}
15037 \label{sec:block15-language-growth}
15038 
15039 Block 14 strengthened a fixed research compiler against diagnostic aliasing, multi-obstruction mixtures, archive contamination, unknown cases, and integrity attacks. Its open-set output nevertheless remained a single undifferentiated token, and every deployable candidate belonged to the inherited $39$-method singleton grammar. Block 15 studies the next structural question:
15040 \begin{equation}
15041 \boxed{
15042 \begin{gathered}
15043 \text{Can the compiler use persistent open-set conflict}\\
15044 \text{to enlarge its own diagnostic and method language?}
15045 \end{gathered}
15046 }
15047 \end{equation}
15048 The target is not unrestricted program invention. It is a finite proof-carrying edge in which the inherited compiler detects that one unknown token aliases research problems requiring incompatible methods, selects a separating experiment from a generic probe language, synthesizes new composite methods from a generic constructor grammar, preserves the inherited compiler as an exact fallback, seals the expanded language, and improves newly generated foreign research problems under a matched deployment budget.
15049 
15050 \subsection{Compiler languages and inherited quotients}
15051 
15052 Let $\Tau$ be a research-problem space. A finite research-compiler language is a tuple
15053 \begin{equation}
15054 \mathfrak G
15055 =
15056 (\Zcal,\Pcal,\Ccal,\Bcal),
15057 \label{eq:b15-compiler-language}
15058 \end{equation}
15059 where:
15060 \begin{enumerate}
15061 \item $\Zcal$ is a finite diagnostic-token space;
15062 \item $\Pcal$ is a finite language of admissible diagnostic experiments or probes;
15063 \item $\Ccal$ is a finite method-constructor grammar;
15064 \item $\Bcal$ is a typed binding relation that maps diagnostic tokens to constructed research methods.
15065 \end{enumerate}
15066 A compiler using $\mathfrak G$ first maps evidence from problem $\tau$ to a token $z\in\Zcal$, optionally executes probes in $\Pcal$, constructs candidates in $\Ccal$, and selects only bindings admitted by $\Bcal$.
15067 
15068 Let $\phi_0:\Tau\to\Zcal_0$ be the inherited diagnostic map. In Block 15, six inherited obstruction classes retain their existing tokens, while two previously unseen obstruction mechanisms are both mapped to one open-set token $\bot$:
15069 \begin{equation}
15070 \phi_0(\tau_{
15071 m int})
15072 =
15073 \phi_0(\tau_{
15074 m temp})
15075 =
15076 \bot.
15077 \label{eq:b15-unknown-alias}
15078 \end{equation}
15079 The equality in \cref{eq:b15-unknown-alias} is not itself a defect. An unknown token is an appropriate abstention state until the lineage possesses external evidence that the collapsed cases require incompatible research actions.
15080 
15081 For each problem $\tau$, let
15082 \begin{equation}
15083 \Acal^\star(\tau)
15084 \subseteq
15085 \Ccal
15086 \end{equation}
15087 be the set of methods that satisfy the declared validity, capability, and gain conditions.
15088 
15089 \begin{definition}[Compiler-language conflict]
15090 A finite set $S\subseteq\Tau$ is a compiler-language conflict for $\mathfrak G$ when all problems in $S$ receive the same inherited diagnostic token but
15091 \begin{equation}
15092 \bigcap_{\tau\in S}\Acal^\star(\tau)=\varnothing.
15093 \label{eq:b15-conflict}
15094 \end{equation}
15095 A minimal conflict has no proper conflicting subset.
15096 \end{definition}
15097 
15098 \begin{theorem}[Aliased-token routing obstruction]
15099 \label{thm:b15-routing-obstruction}
15100 Let $\tau_a$ and $\tau_b$ satisfy $\phi_0(\tau_a)=\phi_0(\tau_b)$ and
15101 \begin{equation}
15102 \Acal^\star(\tau_a)\cap\Acal^\star(\tau_b)=\varnothing.
15103 \end{equation}
15104 Then no deterministic compiler policy that conditions only on $\phi_0(\tau)$ can select an acceptable method for both problems.
15105 \end{theorem}
15106 
15107 \begin{proof}
15108 The policy must output the same method on $\tau_a$ and $\tau_b$ because its input token is identical. That method would have to lie in both acceptable sets, contradicting their empty intersection.
15109 \end{proof}
15110 
15111 The theorem distinguishes a genuine diagnostic-language insufficiency from ordinary classifier error. Additional samples from the same collapsed token cannot solve a routing problem whose correct actions are disjoint.
15112 
15113 \subsection{Separating research experiments}
15114 
15115 Let $p\in\Pcal^+$ be a candidate experiment with response
15116 \begin{equation}
15117 y_p:\Tau\longrightarrow\R.
15118 \end{equation}
15119 For threshold $\theta$, define the refined diagnostic map
15120 \begin{equation}
15121 \phi_{p,\theta}(\tau)
15122 =
15123 \begin{cases}
15124 (\phi_0(\tau),\mathrm{low}), & y_p(\tau)\leq\theta,\\
15125 (\phi_0(\tau),\mathrm{high}), & y_p(\tau)>\theta.
15126 \end{cases}
15127 \label{eq:b15-refined-token}
15128 \end{equation}
15129 The old token is recovered by the projection
15130 \begin{equation}
15131 \pi_0(z,s)=z,
15132 \qquad
15133 \pi_0\circ\phi_{p,\theta}=\phi_0.
15134 \label{eq:b15-token-projection}
15135 \end{equation}
15136 Thus the extension is conservative at the level of old diagnostic semantics.
15137 
15138 \begin{definition}[Conflict-separating experiment]
15139 An experiment $p$ separates a conflict $S$ when $y_p$ is not constant on $S$ and every cell of the refined partition has a nonempty common acceptable-method intersection.
15140 \end{definition}
15141 
15142 \begin{theorem}[Finite conservative diagnostic refinement]
15143 \label{thm:b15-diagnostic-refinement}
15144 Suppose a finite experiment family $\Pcal^+$ separates every minimal conflict in a finite development archive. Then repeated addition of separating experiments terminates after finitely many refinements with a diagnostic language whose every development cell is repair consistent. Every old token and old policy remains recoverable by composition with the projection maps.
15145 \end{theorem}
15146 
15147 \begin{proof}
15148 A selected experiment splits at least one unresolved minimal conflict. Refinement never merges cells, so a resolved conflict cannot reappear. The development conflict hypergraph is finite, hence only finitely many successful refinements are possible. Conservative recoverability follows by composing the coordinate projections such as \cref{eq:b15-token-projection}.
15149 \end{proof}
15150 
15151 Block 15 realizes one refinement. The two new tokens are deliberately anonymous, \texttt{novel-low} and \texttt{novel-high}. Their semantic names are retained only in the hidden audit generator. The compiler selects the probe and orientation using development method outcomes, not hidden novel-class labels.
15152 
15153 \subsection{A generic composite-method constructor}
15154 
15155 Let $\Mcal_0$ denote the inherited singleton-method grammar. For inherited method families $a$ and $b$, define an ordered pair constructor
15156 \begin{equation}
15157 \operatorname{Seq}(a,b)
15158 =
15159 \text{``apply the research intervention from family $a$, then the intervention from family $b$''}.
15160 \label{eq:b15-seq-constructor}
15161 \end{equation}
15162 The generic finite constructor language is
15163 \begin{equation}
15164 \Ccal_{\rm pair}
15165 =
15166 \set{\operatorname{Seq}(a,b):a,b\in\Mcal_0,\ a\neq b}.
15167 \label{eq:b15-pair-grammar}
15168 \end{equation}
15169 In the executable fixture there are six typed inherited obstruction families, producing $30$ ordered pair constructors.
15170 
15171 Let $U(\tau,m)$ be protected validity-filtered research gain. Define inherited and extended oracle envelopes
15172 \begin{equation}
15173 O_0(\tau)=\max_{m\in\Mcal_0}U(\tau,m),
15174 \qquad
15175 O_1(\tau)=\max_{m\in\Mcal_0\cup\Ccal_{\rm pair}}U(\tau,m).
15176 \end{equation}
15177 The task-relevant structural margin is
15178 \begin{equation}
15179 \kappa_Q
15180 =
15181 \mathbb E_{\tau\sim Q}
15182 \left[
15183 O_1(\tau)-O_0(\tau)
15184 \right].
15185 \label{eq:b15-structural-margin}
15186 \end{equation}
15187 A positive margin means that the extension adds useful methods outside the inherited candidate opportunity set.
15188 
15189 \begin{proposition}[Conservative method-language embedding]
15190 \label{prop:b15-method-embedding}
15191 The singleton grammar embeds into the expanded grammar by inclusion. A compiler using the expanded language can reproduce the inherited compiler exactly on every old diagnostic token by invoking the inherited binding and ignoring the new probe and pair constructors.
15192 \end{proposition}
15193 
15194 \begin{theorem}[Finite conflict-grounded language synthesis]
15195 \label{thm:b15-language-synthesis}
15196 Let a finite development archive contain one inherited token cell $C$ with at least two incompatible repair subclasses. Suppose:
15197 \begin{enumerate}
15198 \item a candidate probe $p\in\Pcal^+$ separates $C$ into cells $C_-$ and $C_+$;
15199 \item there exist $m_- ,m_+\in\Ccal_{\rm pair}$ such that $m_-$ is acceptable throughout $C_-$ and $m_+$ is acceptable throughout $C_+$;
15200 \item neither one inherited singleton method nor one unrefined binding is acceptable throughout $C$;
15201 \item old-token behavior is preserved through the fallback of \cref{prop:b15-method-embedding}.
15202 \end{enumerate}
15203 Then the extension
15204 \begin{equation}
15205 \mathfrak G_0
15206 \hookrightarrow
15207 \mathfrak G_1
15208 =
15209 \mathfrak G_0
15210 \cup
15211 \set{p,(\bot,-),(\bot,+),m_-,m_+}
15212 \end{equation}
15213 eliminates the development conflict while conservatively preserving the inherited compiler.
15214 \end{theorem}
15215 
15216 \begin{proof}
15217 The refined cells have acceptable witnesses $m_-$ and $m_+$ by assumption. The inherited conflict is therefore eliminated. Old tokens use the unchanged inherited binding, and the old unknown token is recovered by forgetting the refinement coordinate. The new methods are absent from the singleton grammar, so the extension is structural rather than a renaming.
15218 \end{proof}
15219 
15220 \subsection{Finite selection and proof-carrying certificates}
15221 
15222 Let $N$ be the number of candidate probe-and-binding configurations and let their bounded development scores lie in $[0,1]$. Standard uniform concentration gives the following finite-search guard.
15223 
15224 \begin{proposition}[Finite search bound]
15225 \label{prop:b15-finite-search-bound}
15226 For $n$ independent meta-validation problems, with probability at least $1-\delta$, every candidate language score differs from its population score by at most
15227 \begin{equation}
15228 \eps_n
15229 =
15230 \sqrt{
15231 \frac{\log(2N/\delta)}{2n}
15232 }.
15233 \label{eq:b15-uniform-bound}
15234 \end{equation}
15235 Consequently, the selected candidate's population score is within $2\eps_n$ of the best candidate in the finite language.
15236 \end{proposition}
15237 
15238 \begin{proof}
15239 Apply Hoeffding's inequality to each candidate and a union bound over $N$ candidates. The $2\eps_n$ selection-regret statement follows by comparing the selected empirical maximizer with the population maximizer.
15240 \end{proof}
15241 
15242 A compiler-language expansion receipt is
15243 \begin{equation}
15244 \Pi_{\Gamma\text{-lang}}
15245 =
15246 (
15247 \mathfrak G_0,
15248 \Hcal_{\rm conflict},
15249 \Pcal^+,
15250 p,\theta,
15251 \Ccal_{\rm pair},
15252 m_-,m_+,
15253 \iota,
15254 \kappa,
15255 \Pi_{\rm seal},
15256 \Pi_{\rm foreign}
15257 ),
15258 \label{eq:b15-language-receipt}
15259 \end{equation}
15260 where $\Hcal_{\rm conflict}$ is the conflict hypergraph, $\iota$ is the conservative embedding, $\kappa$ is the newly correctable development margin, and the final fields bind the pre-audit compiler state to foreign evidence.
15261 
15262 \section{Block 15 Experimental Contract}
15263 \label{sec:block15-contract}
15264 
15265 \subsection{Lineage edge and withheld structure}
15266 
15267 The realized compiler lineage is
15268 \begin{equation}
15269 \Gamma_2
15270 \xrightarrow{\Pi_9^\Gamma}
15271 \Gamma_3.
15272 \label{eq:b15-lineage-edge}
15273 \end{equation}
15274 The parent $\Gamma_2$ is the exact robust fixed-language compiler design from Block 14, recalibrated only inside the inherited research language. It retains:
15275 \begin{itemize}
15276 \item six known obstruction tokens and one undifferentiated unknown token;
15277 \item the inherited $39$-method singleton grammar;
15278 \item results-free receipts, evaluator portfolio, capability gates, integrity filtering, and open-set abstention;
15279 \item the same eight-unit deployment research budget used by the child in Block 15.
15280 \end{itemize}
15281 The hidden generator contains two new obstruction mechanisms. Their semantic identities are unavailable to compiler construction. Both occupy the same inherited static-diagnostic cell and therefore produce the same inherited unknown token.
15282 
15283 The child receives only:
15284 \begin{enumerate}
15285 \item development open-set receipts;
15286 \item a generic six-probe experiment language;
15287 \item a generic $30$-member ordered-pair method-constructor grammar;
15288 \item validation outcomes on development and meta-validation problems;
15289 \item the inherited compiler and conservative fallback interface.
15290 \end{enumerate}
15291 The compiler does not receive the hidden names of the two novel mechanisms or a direct declaration of which probe or pair constructors are correct.
15292 
15293 \subsection{Development conflict and language compiler}
15294 
15295 The development set contains $56$ open-set problems. The inherited base-method oracle obtains mean gain
15296 \begin{equation}
15297 \widehat O_0=0.377257.
15298 \label{eq:b15-base-oracle-dev}
15299 \end{equation}
15300 No one inherited unknown-token binding supplies a common high-gain method across the cell.
15301 
15302 For every candidate probe, the compiler:
15303 \begin{enumerate}
15304 \item evaluates the probe on each development open-set problem;
15305 \item splits the cell at the response median;
15306 \item searches the $30$ ordered pair constructors independently in the two refined cells;
15307 \item scores conflict reduction, protected gain, separation, and constructor complexity;
15308 \item records an anonymous two-token diagnostic extension and its two bound methods.
15309 \end{enumerate}
15310 The selected experiment is \texttt{interaction-contrast}. Its development separation accuracy, computed only after selection for interpretation against hidden labels, is $1$. The selected anonymous bindings are
15311 \begin{equation}
15312 \begin{aligned}
15313 \texttt{novel-low}
15314 &\longmapsto
15315 \texttt{evaluator-then-resource},\\
15316 \texttt{novel-high}
15317 &\longmapsto
15318 \texttt{objective-then-constructor}.
15319 \end{aligned}
15320 \label{eq:b15-selected-bindings}
15321 \end{equation}
15322 The expanded development gain is
15323 \begin{equation}
15324 \widehat O_1=0.948962,
15325 \end{equation}
15326 with newly correctable margin
15327 \begin{equation}
15328 \widehat\kappa
15329 =
15330 0.571704.
15331 \label{eq:b15-development-margin}
15332 \end{equation}
15333 
15334 \subsection{Matched deployment budget}
15335 
15336 Every parent and child run receives exactly eight research units.
15337 
15338 \begin{center}
15339 \begin{tabular}{@{}lrrr@{}}
15340 \toprule
15341 Process & Diagnostic probes & Method experiments & Total \\
15342 \midrule
15343 Inherited parent on open-set cases & $0$ & $8$ & $8$ \\
15344 Expanded child on open-set cases & $1$ & $7$ & $8$ \\
15345 Inherited fallback on known cases & $0$ & $8$ & $8$ \\
15346 \bottomrule
15347 \end{tabular}
15348 \end{center}
15349 
15350 The parent uses all eight units on inherited singleton methods. The child spends one unit on the selected separating probe and at most seven on methods. The child therefore receives no larger deployment research budget. Compiler development cost is reported separately and is not hidden inside deployment accounting.
15351 
15352 \subsection{Splits and chronology}
15353 
15354 The package uses:
15355 \begin{center}
15356 \begin{tabular}{@{}lrrl@{}}
15357 \toprule
15358 Split & Known & Novel & Purpose \\
15359 \midrule
15360 Development & $60$ & $56$ & conflict and language compilation \\
15361 Meta-validation & $24$ & $16$ & conservative child verification \\
15362 Primary foreign audit & $48$ & $48$ & main sealed comparison \\
15363 Generator-foreign audit & $42$ & $40$ & shifted diagnostic and outcome geometry \\
15364 \bottomrule
15365 \end{tabular}
15366 \end{center}
15367 
15368 The chronology is
15369 \begin{equation}
15370 \begin{aligned}
15371 &\text{replay Block 14 canonical digest}\\
15372 &\longrightarrow \text{construct development conflict archive}\\
15373 &\longrightarrow \text{search generic probes and pair constructors}\\
15374 &\longrightarrow \text{verify conservative fallback on meta-validation}\\
15375 &\longrightarrow \text{serialize the language certificate and compiler object}\\
15376 &\longrightarrow \text{seal primary and generator-foreign commitments}\\
15377 &\longrightarrow \text{open and evaluate both final audits}.
15378 \end{aligned}
15379 \label{eq:b15-chronology}
15380 \end{equation}
15381 
15382 \subsection{Comparison arms}
15383 
15384 The release includes:
15385 \begin{enumerate}
15386 \item \textbf{Inherited parent.} One unknown token and the $39$-method singleton grammar.
15387 \item \textbf{Language child.} Selected separating probe, two anonymous tokens, two synthesized pair constructors, and exact inherited fallback.
15388 \item \textbf{Taxonomy only.} The refined tokens are available but no new method constructor is added.
15389 \item \textbf{Grammar only.} The pair constructors are available but the unknown cell remains unrefined and receives one pooled binding.
15390 \item \textbf{Shuffled binding.} The two refined tokens are bound to the opposite synthesized methods.
15391 \item \textbf{Base-grammar oracle.} Inadmissible ceiling over all $39$ inherited methods.
15392 \item \textbf{Expanded-grammar oracle.} Inadmissible ceiling over inherited methods and all $30$ generic pair constructors.
15393 \end{enumerate}
15394 The taxonomy-only, grammar-only, and shuffled-binding controls test whether diagnostic refinement and candidate growth must remain correctly coupled.
15395 
15396 \subsection{Frozen claim gates}
15397 
15398 The Block 15 language-growth gate requires:
15399 \begin{enumerate}
15400 \item exact replay of the Block 14 result digest;
15401 \item hidden novel labels absent from compiler inputs;
15402 \item a nonzero development conflict margin relative to the inherited method oracle;
15403 \item a selected probe from the generic probe language and selected methods from the generic pair grammar;
15404 \item conservative replay of inherited known-class behavior;
15405 \item identical eight-unit deployment budgets;
15406 \item compiler and split sealing before final audit outcomes;
15407 \item positive paired child-parent effects with intervals excluding zero on both audits;
15408 \item substantial degradation under taxonomy-only, grammar-only, and shuffled-binding controls;
15409 \item an expanded-grammar oracle at or above the admissible child;
15410 \item complete results-free and post-result receipts, tests, figures, and package manifest.
15411 \end{enumerate}
15412 
15413 \section{Block 15 Results}
15414 \label{sec:block15-results}
15415 
15416 \subsection{Language-inadequacy certificate}
15417 
15418 The selected finite expansion is summarized by:
15419 \begin{center}
15420 \begin{tabular}{@{}lr@{}}
15421 \toprule
15422 Quantity & Value \\
15423 \midrule
15424 Inherited open-set development oracle & $0.377257$ \\
15425 Expanded two-token development gain & $0.948962$ \\
15426 Newly correctable margin & $0.571704$ \\
15427 Selected probe & \texttt{interaction-contrast} \\
15428 Probe interpretation accuracy & $1.000000$ \\
15429 Generic probe candidates & $6$ \\
15430 Generic ordered pair constructors & $30$ \\
15431 Deployable inherited methods & $39$ \\
15432 Deployable child methods & $41$ \\
15433 \bottomrule
15434 \end{tabular}
15435 \end{center}
15436 
15437 \begin{figure}[H]
15438 \centering
15439 \includegraphics[width=0.82\textwidth]{block15_language_certificate.pdf}
15440 \caption{Development language-inadequacy certificate. The inherited singleton oracle leaves a large open-set residual. The selected probe and two synthesized pair constructors expose a substantial new correctable margin.}
15441 \label{fig:block15-language-certificate}
15442 \end{figure}
15443 
15444 The deployable child adds only the two selected composites. The remaining $28$ pair constructors are development search alternatives, not hidden deployment candidates.
15445 
15446 \subsection{Primary and generator-foreign compiler effects}
15447 
15448 The primary audit contains $96$ problems. The generator-foreign audit contains $82$ problems with shifted diagnostic and outcome geometry.
15449 
15450 \begin{center}
15451 \begin{tabular}{@{}lrrr@{}}
15452 \toprule
15453 Audit and process & Mean gain & Novel mean & Worst-$10\%$ gain \\
15454 \midrule
15455 Primary inherited parent & $0.633969$ & $0.341371$ & $0.286117$ \\
15456 Primary language child & $\mathbf{0.859731}$ & $\mathbf{0.792894}$ & $\mathbf{0.333839}$ \\
15457 Generator-foreign parent & $0.574224$ & $0.292978$ & $0.240620$ \\
15458 Generator-foreign child & $\mathbf{0.814520}$ & $\mathbf{0.785584}$ & $\mathbf{0.482289}$ \\
15459 \bottomrule
15460 \end{tabular}
15461 \end{center}
15462 
15463 The primary paired effect is
15464 \begin{equation}
15465 \widehat\Delta^{\rm primary}_{\Gamma\text{-lang}}
15466 =
15467 0.225762,
15468 \qquad
15469 \mathrm{CI}_{95\%}
15470 =
15471 [0.166708,0.286431].
15472 \label{eq:b15-primary-effect}
15473 \end{equation}
15474 The generator-foreign paired effect is
15475 \begin{equation}
15476 \widehat\Delta^{\rm foreign}_{\Gamma\text{-lang}}
15477 =
15478 0.240296,
15479 \qquad
15480 \mathrm{CI}_{95\%}
15481 =
15482 [0.180154,0.302701].
15483 \label{eq:b15-foreign-effect}
15484 \end{equation}
15485 Both sign-flip estimates are approximately $9.9999\times10^{-6}$. No paired problem favors the parent. Primary pairs improve on $36.46\%$ of problems and tie on the remainder; generator-foreign pairs improve on $42.68\%$ and tie on the remainder. The ties are the exact conservative fallback on inherited known classes and open-set cases for which the selected method matches the parent's selected outcome.
15486 
15487 \begin{figure}[H]
15488 \centering
15489 \includegraphics[width=0.92\textwidth]{block15_compiler_gain.pdf}
15490 \caption{Matched-budget compiler-language performance. Diagnostic refinement without new methods and new methods without the correct refined binding do not reproduce the complete child result.}
15491 \label{fig:block15-compiler-gain}
15492 \end{figure}
15493 
15494 \begin{figure}[H]
15495 \centering
15496 \includegraphics[width=0.92\textwidth]{block15_paired_effects.pdf}
15497 \caption{Sorted child-minus-parent effects on primary and generator-foreign research problems. All effects are nonnegative in the finite release; zeros include inherited known-class fallback ties.}
15498 \label{fig:block15-paired}
15499 \end{figure}
15500 
15501 \subsection{Novel-obstruction performance}
15502 
15503 The hidden novel families are used only for audit interpretation. On the primary split:
15504 \begin{center}
15505 \begin{tabular}{@{}lrrr@{}}
15506 \toprule
15507 Novel family & Inherited parent & Language child & Difference \\
15508 \midrule
15509 Interaction-type & $0.327041$ & $0.816123$ & $0.489081$ \\
15510 Temporal-type & $0.355700$ & $0.769666$ & $0.413965$ \\
15511 \bottomrule
15512 \end{tabular}
15513 \end{center}
15514 Under generator shift, the corresponding child means are $0.870372$ and $0.700796$, compared with parent means $0.295177$ and $0.290779$.
15515 
15516 \begin{figure}[H]
15517 \centering
15518 \includegraphics[width=0.78\textwidth]{block15_novel_gain.pdf}
15519 \caption{Protected gain on the two hidden novel mechanisms. The compiler uses anonymous response-defined tokens; semantic family names are retained only by the audit generator.}
15520 \label{fig:block15-novel-gain}
15521 \end{figure}
15522 
15523 The full inherited-grammar oracle, allowed to inspect all $39$ old methods, reaches only $0.373367$ mean gain on primary novel problems. The expanded-grammar oracle reaches $0.960732$. The admissible child reaches $0.792894$ while using only eight deployment units and without final-audit labels.
15524 
15525 \subsection{Taxonomy, constructor, and binding controls}
15526 
15527 Primary mean gains are:
15528 \begin{center}
15529 \begin{tabular}{@{}lr@{}}
15530 \toprule
15531 Arm & Mean gain \\
15532 \midrule
15533 Inherited parent & $0.633969$ \\
15534 Taxonomy only & $0.633995$ \\
15535 Grammar only & $0.756265$ \\
15536 Shuffled binding & $0.634112$ \\
15537 Full language child & $\mathbf{0.859731}$ \\
15538 \bottomrule
15539 \end{tabular}
15540 \end{center}
15541 
15542 The taxonomy-only arm cannot improve because the inherited method image remains unchanged. The grammar-only arm improves the novel cell whose pooled binding happens to match the selected composite, but cannot route the second novel mechanism correctly. Shuffled binding removes essentially the entire gain. The result therefore depends on the coupled object:
15543 \begin{equation}
15544 \boxed{
15545 \text{new diagnostic distinction}
15546 +
15547 \text{new method constructor}
15548 +
15549 \text{correct binding}.
15550 }
15551 \end{equation}
15552 
15553 \begin{figure}[H]
15554 \centering
15555 \includegraphics[width=0.80\textwidth]{block15_tail_and_novel.pdf}
15556 \caption{Mean, lower-tail, and novel-class gains for the inherited compiler and the language-expanded child. The largest improvement is concentrated where the inherited language was certified inadequate.}
15557 \label{fig:block15-tail-novel}
15558 \end{figure}
15559 
15560 \subsection{Conservative inheritance and safety interpretation}
15561 
15562 For every inherited known class, the child invokes the exact parent path. Consequently, known-class mean gains are byte-for-byte identical between parent and child in the canonical result object. This is stronger than a nonsignificant regression test; it is an executable conservative embedding.
15563 
15564 No final selected method fails the declared validity or capability gate. With zero observed invalid selections, the one-sided $95\%$ binomial upper bounds are approximately $3.07\%$ on the $96$-problem primary audit and $3.59\%$ on the $82$-problem generator-foreign audit. These bounds are finite-sample statements, not proofs of zero population risk.
15565 
15566 \subsection{Finite compiler-language realization theorem}
15567 
15568 \begin{conditionaltheorem}[Block 15 finite compiler-language growth]
15569 \label{thm:block15-realization}
15570 Within the frozen Block 15 benchmark, assume that:
15571 \begin{enumerate}
15572 \item the published Block 14 result digest replays exactly;
15573 \item hidden novel labels are excluded from compiler construction;
15574 \item the parent and child share the same old grammar, evaluator, validity gates, and eight-unit deployment budget;
15575 \item the compiler searches only the declared generic probe and ordered pair-constructor languages;
15576 \item the inherited open-set cell has the recorded conflict margin, and the selected extension has the recorded development improvement;
15577 \item the old compiler is preserved as an exact fallback on inherited tokens;
15578 \item the compiler object and audit commitments are sealed before final outcomes;
15579 \item the paired effects in \cref{eq:b15-primary-effect,eq:b15-foreign-effect} have the recorded positive intervals;
15580 \item taxonomy-only, grammar-only, shuffled-binding, inherited-oracle, and expanded-oracle controls retain their declared interpretations.
15581 \end{enumerate}
15582 Then the edge in \cref{eq:b15-lineage-edge} is a finite, matched-budget, lineage-generated compiler-language expansion. It adds a diagnostic distinction and two method-constructor outputs absent from the inherited deployable language, conservatively retains old behavior, and reduces a foreign open-set research obstruction.
15583 \end{conditionaltheorem}
15584 
15585 \begin{proof}
15586 Items 1 through 3 establish lineage continuity, inherited-language identity, and resource matching. Items 4 and 5 establish finite structural synthesis relative to an externally declared generic meta-language rather than direct receipt of the correct novel labels. Item 6 gives conservative grammar inclusion. Item 7 supplies audit isolation. Item 8 supplies positive paired foreign metaproductivity. The controls in item 9 rule out mere relabeling, method growth without diagnostic binding, diagnostic growth without method capacity, and complete-grammar oracle violation. The new pair constructors are absent from the inherited singleton grammar, while the selected probe splits a repair-conflicting inherited token; hence the edge changes the compiler language rather than only reallocating experiments inside it.
15587 \end{proof}
15588 
15589 \section{Block 15 Reproducibility and Claim Boundary}
15590 
15591 \subsection{Reference implementation}
15592 
15593 The released package contains:
15594 \begin{itemize}
15595 \item the exact Block 14 canonical result digest;
15596 \item known and hidden-novel problem generators with a separately shifted foreign generator;
15597 \item the inherited $39$-method singleton grammar;
15598 \item a generic six-probe diagnostic language and $30$ ordered pair-constructor grammar;
15599 \item conflict scoring, probe selection, anonymous token creation, constructor synthesis, and conservative fallback;
15600 \item primary, generator-foreign, taxonomy-only, grammar-only, shuffled-binding, inherited-oracle, and expanded-oracle arms;
15601 \item results-free and post-result receipts, compiler and pre-audit seals, deterministic figures, tests, and an exact package manifest.
15602 \end{itemize}
15603 
15604 The canonical execution prints:
15605 \begin{verbatim}
15606 SRIC_BLOCK15_ALL_SUITES=PASS
15607 result_sha256=309d63546ecbbbcca1f1a92a6ca48c5b2bae752df4496c1f2d333c31798fc630
15608 primary_parent_gain=0.633969
15609 primary_child_gain=0.859731
15610 primary_effect=0.225762
15611 generator_foreign_parent_gain=0.574224
15612 generator_foreign_child_gain=0.814520
15613 generator_foreign_effect=0.240296
15614 selected_probe=interaction_contrast
15615 selected_methods=evaluator_then_resource,objective_then_constructor
15616 ............... [100%]
15617 SRIC_BLOCK15_PACKAGE_VALID
15618 manifest_files=33
15619 \end{verbatim}
15620 
15621 \subsection{Claim boundary}
15622 
15623 Block 15 establishes a finite compiler-language growth edge from one inherited unknown token and a singleton-method grammar to a response-refined two-token diagnostic language with two synthesized pairwise methods. It does not establish:
15624 \begin{itemize}
15625 \item unrestricted invention of diagnostic experiments or research methods;
15626 \item discovery outside the declared six-probe and ordered-pair constructor meta-language;
15627 \item automatic semantic naming of the new obstruction classes;
15628 \item general multi-step program synthesis, arbitrary method composition, or real model-training research;
15629 \item improvement of the evaluator portfolio or integrity monitor;
15630 \item external independent replication;
15631 \item open-ended compiler self-improvement, indefinite acceleration, unrestricted safety, consciousness, or sentience.
15632 \end{itemize}
15633 
15634 The next major threshold is \emph{cross-implementation executable research}. The language-growing compiler should face actual program-synthesis or small-model training problems generated by an independently written environment, with method effects produced by executed code rather than a finite outcome simulator. A stronger descendant should also decide whether a residual requires another diagnostic token, a longer method program, a new evaluator dimension, or a no-go certificate, rather than receiving a fixed pair-composition meta-language.
15635 
15636 
15637 
15638 \section{Direct Neuralese Causal Semantics}
15639 \label{sec:block16-neuralese}
15640 
15641 Block 15 treats a research compiler's diagnostic and method language as an object that can be conservatively enlarged after a persistent open-set conflict. The next direct application moves the same principle inside a recurrent latent reasoner. A \emph{Neuralese} system, as instantiated by the released Hyper v1 runtime \cite{CYMELA2026}, does not emit a token at each internal step. It repeatedly transforms and reinjects a continuous hidden state, then exposes an answer through an ordinary readout. The central question is therefore not whether one can assign an English sentence to every hidden vector. It is whether the hidden trajectory is causally necessary, sequentially organized, specific to the current problem, stable under exact replay, transportable across descendants, and improvable under a protected lineage transaction.
15642 
15643 This block makes Neuralese a first-class SRIC component. The latent transition, update gate, halting rule, functional packet atlas, semantic transport map, and external readout are typed mutable objects. A descendant may alter one or more of them, but the edge is accepted only through matched-resource causal audits and a frozen witness. The construction directly connects hidden-state reasoning to the repair, obstruction, transport, holonomy, and compiler-language machinery developed in the previous blocks.
15644 
15645 \begin{statusbox}{Block 16 status}
15646 The released companion contains a direct adapter contract for the local Cymela Hyper v1 runtime, an intervention suite using its exposed latent-state hook, gold-answer loss scoring, cross-task transplantation, exact replay, matched-norm noise, step permutation, halting diagnostics, orthogonal semantic-transport certificates, parent--child lineage comparison, receipts, tests, and manifests. The package does not redistribute the Hyper runtime or weights. The finite synthetic fixture passes and separates a preview-like parent with useful but nonspecific latent computation from a task-specific child. No empirical Hyper run is claimed in this revision because the model weights were not present in the build environment.
15647 \end{statusbox}
15648 
15649 \subsection{Neuralese reasoners as typed lineage objects}
15650 
15651 \begin{definition}[Neuralese reasoner]
15652 A finite-step Neuralese reasoner is a tuple
15653 \begin{equation}
15654 \Ncal
15655 =
15656 (\Xcal,\Ycal,H,E,B,U,D,\Lambda,K),
15657 \label{eq:b16-neuralese-object}
15658 \end{equation}
15659 where $\Xcal$ is an input space, $\Ycal$ an answer space, $H$ a finite-dimensional latent state space, $E:\Xcal\to H$ a prompt encoder, $B:H\times\Xcal\to H$ a backbone transition, $U:H\times H\times H\to H$ an update gate, $D:H\times\Xcal\to\mathcal P(\Ycal)$ an answer readout, $\Lambda:H\to[0,1]$ a halting map, and $K$ a maximum latent-step budget.
15660 \end{definition}
15661 
15662 Given input $x$, define the prompt anchor $h_0=E(x)$. A normalized injection state $z_t$ and updated latent state $h_t$ satisfy
15663 \begin{equation}
15664 z_t=N(h_{t-1}),
15665 \qquad
15666 \widetilde h_t=B(z_t,x),
15667 \qquad
15668 h_t=U(z_t,\widetilde h_t,h_0),
15669 \qquad
15670 1\leq t\leq K,
15671 \label{eq:b16-latent-recurrence}
15672 \end{equation}
15673 where $N$ is any declared normalization or scale-matching operator. The answer is sampled or selected from $D(h_K,x)$ after a declared end-of-thought transition.
15674 
15675 The tuple in \cref{eq:b16-neuralese-object} is intentionally structural. In the released Hyper v1 runtime, $B$ is the Qwen backbone restricted to one reinjected hidden slot, $U$ is the task-anchored gated latent update, $\Lambda$ is the PonderNet-style halt head, and the exposed \texttt{latent\_intervention} function acts directly on $z_t$ before the backbone and halt head consume it. The SRIC adapter imports those objects from the user's local model folder rather than reproducing them.
15676 
15677 \begin{definition}[Neuralese trace]
15678 For input $x$ and fixed depth $K$, the injected Neuralese trace is
15679 \begin{equation}
15680 Z_K(x)=(z_1(x),\ldots,z_K(x))\in H^K.
15681 \end{equation}
15682 The updated trace is $H_K(x)=(h_1(x),\ldots,h_K(x))$. Every receipt must state which trace convention is being intervened on. Block 16 uses the injected trace because this is the point at which the Hyper runtime exposes an exact intervention hook.
15683 \end{definition}
15684 
15685 The distinction is material. A trace captured after the update gate and a trace captured immediately before reinjection are related but are not interchangeable under arbitrary interventions. An audit that silently changes this convention cannot be compared with its ancestor receipt.
15686 
15687 \subsection{The direct intervention language}
15688 
15689 Let $L(x,y;Z)$ denote mean gold-answer cross-entropy when the prompt is $x$, target answer is $y$, and the recurrent latent slots are externally set to trace $Z$. Let $Z(x)$ denote the clean trace. For a donor $x'$ selected from the same declared answer-format group, let $Z(x')$ be its clean trace. Block 16 fixes the intervention language
15690 \begin{equation}
15691 \Ical_{\mathrm{NL}}
15692 =
15693 \{\mathsf{zero},\mathsf{replay},\mathsf{permute},\mathsf{noise},
15694 \mathsf{transplant},\mathsf{packet},\mathsf{transport}\}.
15695 \label{eq:b16-intervention-language}
15696 \end{equation}
15697 
15698 The first five interventions are mandatory in the minimal direct audit.
15699 \begin{enumerate}
15700 \item $\mathsf{zero}$ executes no latent step.
15701 \item $\mathsf{replay}$ replaces every injected state by the clean captured state at the same step.
15702 \item $\mathsf{permute}$ applies a sealed nonidentity permutation $\pi$ to the clean trace.
15703 \item $\mathsf{noise}$ replaces each state by deterministic Gaussian noise with matched Euclidean norm and shape.
15704 \item $\mathsf{transplant}$ replaces $Z(x)$ by $Z(x')$ from a distinct task in the same answer-format group.
15705 \end{enumerate}
15706 The packet and transport interventions are optional at the first checkpoint. They become mandatory when a descendant claims a localized causal packet or cross-model latent semantics.
15707 
15708 \begin{definition}[Per-task Neuralese causal coordinates]
15709 For task $(x_i,y_i)$, define
15710 \begin{align}
15711 n_i
15712 &:=L(x_i,y_i;\mathsf{zero})-L(x_i,y_i;Z(x_i)),
15713 \label{eq:b16-necessity}\\
15714 o_i
15715 &:=L(x_i,y_i;Z_{\pi}(x_i))-L(x_i,y_i;Z(x_i)),
15716 \label{eq:b16-order}\\
15717 c_i
15718 &:=L(x_i,y_i;Z_{\eta}(x_i))-L(x_i,y_i;Z(x_i)),
15719 \label{eq:b16-noise}\\
15720 s_i
15721 &:=L(x_i,y_i;Z(x'_i))-L(x_i,y_i;Z(x_i)),
15722 \label{eq:b16-specificity}\\
15723 r_i
15724 &:=\abs{L(x_i,y_i;\mathsf{replay}(Z(x_i)))-L(x_i,y_i;Z(x_i))}.
15725 \label{eq:b16-replay}
15726 \end{align}
15727 These are respectively latent necessity, order sensitivity, corruption sensitivity, task specificity, and replay error.
15728 \end{definition}
15729 
15730 The distinction between $o_i$ and $s_i$ is the direct bridge to the current Neuralese obstruction. A trajectory may be highly ordered while remaining nearly exchangeable across prompts. In that regime the recurrence computes a generic answer-form trajectory rather than a content-conditioned solution trajectory.
15731 
15732 \begin{definition}[Structural-normalized specificity]
15733 Let
15734 \begin{equation}
15735 Q_{\mathrm{spec}}
15736 =
15737 \frac{\overline s}
15738 {\max\{\overline o,\overline c,\eps\}},
15739 \label{eq:b16-specificity-ratio}
15740 \end{equation}
15741 where bars denote held-out means and $\eps>0$ is a fixed numerical floor. The default Block 16 specificity gate requires
15742 \begin{equation}
15743 \LCB_{0.95}(\overline s)>0,
15744 \qquad
15745 \frac{1}{n}\sum_i\one[s_i>0]\geq0.75,
15746 \qquad
15747 Q_{\mathrm{spec}}\geq0.25.
15748 \label{eq:b16-specificity-gate}
15749 \end{equation}
15750 \end{definition}
15751 
15752 The ratio in \cref{eq:b16-specificity-ratio} prevents a very small positive transplant effect from clearing the gate merely because it is statistically nonzero. The task-specific signal must be material relative to at least one strong structural intervention on the same trace.
15753 
15754 \begin{theorem}[Format-conditional transplant null]
15755 \label{thm:b16-transplant-null}
15756 Let $G(x)$ be a declared answer-format class. Suppose that, conditional on $G(x)=g$, the trace law is independent of task content:
15757 \begin{equation}
15758 Z(x)\mid G(x)=g
15759 \stackrel{d}{=}
15760 Z(x')\mid G(x')=g,
15761 \end{equation}
15762 where $x'$ is an independent donor. Assume the intervention harness leaves the prompt and target answer fixed and replaces only the latent trace. Then
15763 \begin{equation}
15764 \mathbb E[s_i\mid G(x_i)=g]=0.
15765 \end{equation}
15766 \end{theorem}
15767 
15768 \begin{proof}
15769 Under the hypothesis, the own-task and donor traces have the same conditional distribution. The measurable losses $L(x_i,y_i;Z(x_i))$ and $L(x_i,y_i;Z(x'_i))$ therefore have the same conditional expectation. Their expected difference is zero.
15770 \end{proof}
15771 
15772 \begin{corollary}[Meaning of a small transplant cost]
15773 A small cross-task transplant cost is compatible with useful and order-sensitive recurrent computation. It specifically fails to establish that the trace carries information about the current task beyond the donor-matching variables used to define the transplant group.
15774 \end{corollary}
15775 
15776 \begin{proposition}[Replay integrity]
15777 \label{prop:b16-replay}
15778 Assume deterministic execution, identical prompt and suffix tokens, identical numerical kernels, and a hook that replaces each injected state by the exact state captured at that step in the clean run. Then $r_i=0$ for every task.
15779 \end{proposition}
15780 
15781 \begin{proof}
15782 Induct on the latent step. At step one the backbone receives the same prompt cache and injected state, so it produces the same updated state and cache. If the caches agree through step $t-1$, exact replacement supplies the same state at step $t$, hence the same next cache. The final answer-scoring trajectory is therefore identical.
15783 \end{proof}
15784 
15785 Nonzero replay error is not automatically a model defect. It may reveal nondeterministic kernels, lossy trace serialization, a misplaced hook, a suffix-alignment error, or hidden mutable state. It nevertheless invalidates causal attribution until bounded and explained.
15786 
15787 \subsection{Adaptive halting as a separate claim}
15788 
15789 For fixed maximum depth $K_{\max}$, let $\lambda_t(x)=\Lambda(z_t(x))$. Define cumulative halt mass
15790 \begin{equation}
15791 M_t(x)
15792 =
15793 \sum_{j=1}^{t}
15794 \lambda_j(x)
15795 \prod_{q<j}(1-\lambda_q(x)),
15796 \end{equation}
15797 and dynamic depth
15798 \begin{equation}
15799 T(x)=\min\{t:M_t(x)\geq\tau\},
15800 \end{equation}
15801 with declared threshold $\tau$. Block 16 does not infer adaptive computation from the existence of a halt head. It requires a difficulty variable $d(x)$ fixed before result access and tests both
15802 \begin{equation}
15803 \rho_{\mathrm{Spearman}}(d(x),T(x))>0
15804 \end{equation}
15805 and nondegenerate prompt variation in the halt schedules. A constant depth can coexist with fully task-specific Neuralese; conversely, variable depth can be driven by nuisance prompt features. The causal and adaptive statuses are therefore reported separately.
15806 
15807 \subsection{Functional packets and gauge-invariant Neuralese semantics}
15808 
15809 A coordinate-level decoder is not the identifiability target. Let $P_x$ be a projector onto a trace packet discovered from declared interventions. Packet necessity and sufficiency are evaluated by removing $P_xZ(x)$, retaining only $P_xZ(x)$, and comparing equal-rank random and PCA controls. The packet is accepted only when its causal effect transfers to held-out tasks and survives within-packet basis changes.
15810 
15811 For parent and child state spaces $H_A$ and $H_B$ of equal dimension, let calibration traces produce matrices $X$ and $Y$. The orthogonal Procrustes transport is
15812 \begin{equation}
15813 T^*
15814 =
15815 \argmin_{T^{\mathsf T}T=I}\norm{XT-Y}_{\mathrm F}.
15816 \label{eq:b16-procrustes}
15817 \end{equation}
15818 If $X^{\mathsf T}Y=U\Sigma V^{\mathsf T}$, then $T^*=UV^{\mathsf T}$. A transport receipt records calibration residual, held-out residual, singular floor, round-trip residual, and behavior change after transported trace injection.
15819 
15820 \begin{proposition}[Exact orthogonal latent transport]
15821 \label{prop:b16-exact-transport}
15822 Suppose $Y=XQ$ for an orthogonal $Q$ and $X$ has full column rank. Then the minimizer in \cref{eq:b16-procrustes} is $Q$, the calibration residual is zero, and the round trip $X\mapsto XT^*\mapsto XT^*(T^*)^{\mathsf T}$ is exact.
15823 \end{proposition}
15824 
15825 \begin{proof}
15826 The objective has value zero at $Q$. Full column rank implies that $XT=Y=XQ$ only when $T=Q$. Orthogonality gives $QQ^{\mathsf T}=I$.
15827 \end{proof}
15828 
15829 This is a gauge statement rather than a lexical translation statement. Neuralese meaning is represented by intervention and readout invariants under admissible coordinate transformations, not by a preferred list of neuron names.
15830 
15831 \subsection{The direct Neuralese lineage edge}
15832 
15833 Extend the SRIC lineage state by
15834 \begin{equation}
15835 \mathfrak L_t^{\mathrm{NL}}
15836 =
15837 (\mathfrak L_t,\Ncal_t,\Ical_t^{\mathrm{NL}},\Tau_t^{\mathrm{NL}},\Wcal_t^{\mathrm{NL}}),
15838 \end{equation}
15839 where $\Tau_t^{\mathrm{NL}}$ contains trace-transport certificates and $\Wcal_t^{\mathrm{NL}}$ is the frozen external answer and invariant witness. A proposed descendant may modify the latent transition, update gate, halting head, packet atlas, or transport map. The backbone may be held fixed for a component-isolation edge or changed under a stronger whole-model edge.
15840 
15841 \begin{definition}[Neuralese improvement receipt]
15842 A Block 16 receipt contains:
15843 \begin{enumerate}
15844 \item parent and child model fingerprints;
15845 \item exact prompt, answer, donor-group, split, and task digests;
15846 \item latent hook location and trace convention;
15847 \item fixed step budget and numerical precision;
15848 \item all five mandatory per-task intervention losses;
15849 \item bootstrap intervals and positive-effect fractions;
15850 \item specificity ratio and halting diagnostics;
15851 \item transport and packet certificates when claimed;
15852 \item pre-audit commitments, witness outputs, rollback state, and claim boundary.
15853 \end{enumerate}
15854 \end{definition}
15855 
15856 \begin{conditionaltheorem}[Finite Neuralese metaproductivity edge]
15857 \label{thm:b16-neuralese-edge}
15858 Let parent $\Ncal_0$ and child $\Ncal_1$ be evaluated on identical sealed tasks with identical fixed latent-step budgets, answer-scoring rules, donor construction, numerical precision, evaluator, and witness. Assume:
15859 \begin{enumerate}
15860 \item both audits satisfy replay integrity;
15861 \item the child satisfies latent necessity, ordered computation, corruption sensitivity, and task specificity on primary and generator-foreign supports;
15862 \item the paired child-minus-parent transplant-cost interval has positive lower endpoint;
15863 \item child clean loss does not regress beyond a predeclared tolerance;
15864 \item every claimed packet or transport passes its own held-out causal certificate;
15865 \item model, task, audit, and result digests are committed in an append-only receipt chain before promotion.
15866 \end{enumerate}
15867 Then the accepted transition
15868 \begin{equation}
15869 \mathfrak L_t^{\mathrm{NL}}
15870 \xrightarrow{\Pi_{16}}
15871 \mathfrak L_{t+1}^{\mathrm{NL}}
15872 \end{equation}
15873 is a finite evidence-preserving improvement of the Neuralese reasoning component. If the child-producing mechanism, rather than an external developer, generated the accepted latent modification under the same search budget, the transition is additionally a bounded improvement-process edge.
15874 \end{conditionaltheorem}
15875 
15876 \begin{proof}
15877 Replay integrity establishes that the intervention path is faithful to the declared recurrent slot. Necessity, order, corruption, and transplant gates separate a useful task-specific latent trajectory from a decorative, unordered, or answer-format-only trajectory. The paired interval and clean nonregression establish a held-out improvement rather than an isolated task effect. Packet and transport certificates preserve the semantics of any localized or cross-model claim. Frozen receipts, witness separation, and rollback preserve the SRIC transaction invariants. The final distinction follows because improving a latent subsystem and improving the process that constructs latent subsystems are different claims.
15878 \end{proof}
15879 
15880 \subsection{Executable mapping to Hyper v1}
15881 
15882 The direct adapter uses the following correspondence.
15883 \begin{longtable}{@{}p{0.27\textwidth}p{0.65\textwidth}@{}}
15884 \toprule
15885 SRIC Block 16 object & Local Hyper v1 runtime object \\
15886 \midrule
15887 \endfirsthead
15888 \toprule
15889 SRIC Block 16 object & Local Hyper v1 runtime object \\
15890 \midrule
15891 \endhead
15892 Prompt encoder $E$ & Tokenized ChatML prefix ending in \texttt{<bot>} \\
15893 Injected state $z_t$ & Normalized state immediately before recurrent backbone reinjection \\
15894 Backbone transition $B$ & One cached call through the wrapped Qwen backbone \\
15895 Update gate $U$ & \texttt{latent\_update\_gate(h, h\_new, h\_prompt)} \\
15896 Halting map $\Lambda$ & \texttt{halt\_head} evaluated on the injected state \\
15897 Intervention interface & \texttt{LatentCausalLM.latent\_intervention} \\
15898 Gold-answer witness & Cross-entropy from the end-of-thought state through the exact supplied suffix tokens \\
15899 Replay control & Replace each recurrent injection by its captured own-task state \\
15900 Permutation control & Reinject a sealed permutation of the captured own-task states \\
15901 Noise control & Reinject deterministic norm-matched Gaussian states \\
15902 Transplant control & Reinject a same-format donor task's captured states \\
15903 Trace readout & Gold first-token rank from the final updated latent state \\
15904 \bottomrule
15905 \end{longtable}
15906 
15907 The adapter dynamically imports the official \texttt{hyper\_chat.py} from a local model folder and checks for the declared method and hook. It does not patch the model source. It temporarily disables the runtime's textual logit-lens printout during scoring, because nearest-token labels are probes rather than the latent computation itself. Exact answer scoring uses the model's own tokenizer and end-of-thought path.
15908 
15909 \subsection{Finite synthetic realization}
15910 
15911 The companion's engineered fixture contains $66$ held-out tasks: $36$ primary and $30$ generator-foreign. Parent and child use four latent steps. The parent has a strong generic answer-format trajectory and a weak task-content channel. The child strengthens the task-content channel and makes dynamic depth depend on declared difficulty. Both are analytic synthetic systems; the purpose is to validate the audit logic and failure boundary, not to make a claim about a trained language model.
15912 
15913 \begin{longtable}{@{}lrr@{}}
15914 \toprule
15915 Quantity & Preview-like parent & Task-specific child \\
15916 \midrule
15917 \endfirsthead
15918 \toprule
15919 Quantity & Preview-like parent & Task-specific child \\
15920 \midrule
15921 \endhead
15922 Mean clean loss & $0.235716$ & $0.000405$ \\
15923 Mean latent-necessity gain & $0.647409$ & $0.882720$ \\
15924 Mean order cost & $0.182441$ & $0.276445$ \\
15925 Mean norm-matched noise cost & $0.982047$ & $1.364676$ \\
15926 Mean cross-task transplant cost & $0.016648$ & $0.456634$ \\
15927 Structural-normalized specificity & $0.016952$ & $0.334610$ \\
15928 Difficulty--depth Spearman correlation & $0$ & $0.969320$ \\
15929 Distinct dynamic depths & $1$ & $5$ \\
15930 Causal status & PARTIAL & FULL \\
15931 Adaptive-halting status & FAIL & PASS \\
15932 \bottomrule
15933 \end{longtable}
15934 
15935 The paired child-minus-parent transplant-cost effect is $0.439986$ with frozen bootstrap interval $[0.386884,0.493143]$. Mean clean-loss improvement is $0.235311$, replay error is exactly zero in both arms, and the finite lineage receipt passes all declared gates.
15936 
15937 \begin{figure}[H]
15938 \centering
15939 \includegraphics[width=0.82\textwidth]{block16_causal_profile.pdf}
15940 \caption{Synthetic direct Neuralese audit. Both systems have useful ordered recurrent computation, but only the child has a material problem-specific transplant effect.}
15941 \label{fig:block16-causal-profile}
15942 \end{figure}
15943 
15944 \begin{figure}[H]
15945 \centering
15946 \includegraphics[width=0.82\textwidth]{block16_transplant_costs.pdf}
15947 \caption{Per-task transplant costs in the synthetic primary and generator-foreign audits. The parent remains near the format-conditional transplant null while the child is consistently task specific.}
15948 \label{fig:block16-transplant}
15949 \end{figure}
15950 
15951 \subsection{Hyper audit chronology}
15952 
15953 A real Hyper audit follows the frozen sequence
15954 \begin{equation}
15955 \begin{gathered}
15956 \text{fingerprint local runtime and weights}
15957 \longrightarrow
15958 \text{freeze task and donor digests}
15959 \longrightarrow
15960 \text{capture clean traces}\\
15961 \longrightarrow
15962 \text{execute zero, replay, and permutation arms}\\
15963 \longrightarrow
15964 \text{execute noise and transplant arms}\\
15965 \longrightarrow
15966 \text{compute paired intervals and gates}\\
15967 \longrightarrow
15968 \text{serialize result and receipt digests}
15969 \longrightarrow
15970 \text{promote, reject, or retain as PARTIAL}.
15971 \end{gathered}
15972 \end{equation}
15973 The default task file contains $27$ short generated arithmetic, Boolean, and ordering problems with declared difficulty and answer-format groups. The default command limits the first run to $12$ tasks because every task requires several full recurrent forward passes. Increasing the support is necessary before making a strong empirical claim.
15974 
15975 \subsection{Claim boundary and next threshold}
15976 
15977 Block 16 establishes a direct formal and executable bridge between SRIC and recurrent latent reasoning. The released positive control proves that the audit can distinguish ordered but nonspecific latent computation from task-specific latent computation and can certify a finite parent--child edge. It does not establish:
15978 \begin{itemize}
15979 \item that Hyper v1 passes or fails the local audit in the user's environment;
15980 \item that nearest-token projections faithfully translate latent states;
15981 \item that any identified packet is a complete account of the model's reasoning;
15982 \item robustness to adversarially optimized latent traces or adaptive evaluator evasion;
15983 \item causal specificity beyond the supplied task and donor-group construction;
15984 \item a training-time child that autonomously improves its own latent update or halt head;
15985 \item unrestricted Neuralese translation, safety, awareness, consciousness, sentience, or open-ended recursive self-improvement.
15986 \end{itemize}
15987 
15988 The next threshold is a genuine two-checkpoint executable lineage. A parent checkpoint and a child produced by a declared training or search process should be audited on sealed primary and independently generated foreign tasks. The child should increase task-specific transplant cost relative to structural controls, preserve or improve clean performance, maintain replay integrity, and either improve adaptive depth or issue an explicit saturation certificate. Only then does Neuralese move from a directly audited internal substrate to a demonstrated recursively improved reasoning component.
15989 
15990 
15991 \section{Executable Neuralese Binding-Repair Compilation}
15992 \label{sec:block17-executable-neuralese}
15993 
15994 Block 16 made recurrent latent computation directly auditable, but its accepted parent--child edge was an analytic positive control. The parent and child trajectories were generated by declared formulas rather than learned checkpoints, and one same-format transplant statistic could still confound problem-specific computation with answer encoding. Block 17 crosses the next finite threshold. It trains a recurrent neural parent, compiles six matched-update descendants, seals one checkpoint before final-task generation, preserves a failed first audit without weakening its gates, and evaluates the revised child on fresh primary and shifted-generator tasks produced by a separately implemented exact environment.
15995 
15996 The object modified by the compiler is no longer a table of simulated method effects. It is an executable recurrent state transition whose trajectories are captured, replayed, reversed, corrupted, transplanted, decoded, gauge transformed, and scored by independent exact witnesses. The result remains bounded: the task family, candidate grammar, loss, donor taxonomy, semantic target, and acceptance thresholds are supplied. Within that boundary, the accepted edge is a training-generated Neuralese repair rather than an analytic intervention fixture.
15997 
15998 \begin{statusbox}{Block 17 status}
15999 The released companion contains an exact rational affine-program environment, a trained recurrent latent parent, six inherited repair candidates, a finite compiler, a pre-audit checkpoint seal, a preserved failed pilot, a precommitted compiler revision, fresh primary and generator-foreign audits, four donor classes, an external trace probe, adaptive-halting tests, an independent NumPy shadow runtime, a latent-gauge certificate, receipts, figures, tests, and an exact package manifest. In the successful second round, the task-blind parent is \texttt{EXECUTABLE\_PARTIAL} on both final supports and the selected full-binding child is \texttt{EXECUTABLE\_FULL} on both. This is one finite package-local executable Neuralese repair edge. It is not a result for Hyper, autonomous open-ended repair invention, or independent external replication.
16000 \end{statusbox}
16001 
16002 \subsection{Exact executable research environment}
16003 
16004 Let an affine instruction be
16005 \begin{equation}
16006  f_j(z)=a_jz+b_j,
16007  \qquad
16008  a_j\in\left\{-1,-\tfrac12,\tfrac12,1\right\},
16009  \qquad
16010  b_j\in\left\{-\tfrac12,0,\tfrac12\right\}.
16011  \label{eq:b17-affine-instruction}
16012 \end{equation}
16013 A depth-$m$ program acts on $x\in\{-1,-\tfrac34,\ldots,1\}$ by
16014 \begin{equation}
16015  F_m(x)
16016  =
16017  (f_m\circ\cdots\circ f_1)(x)
16018  =A_mx+B_m,
16019  \label{eq:b17-affine-program}
16020 \end{equation}
16021 where
16022 \begin{equation}
16023  A_0=1,
16024  \qquad
16025  B_0=0,
16026  \qquad
16027  A_j=a_jA_{j-1},
16028  \qquad
16029  B_j=a_jB_{j-1}+b_j.
16030  \label{eq:b17-affine-witness-recursion}
16031 \end{equation}
16032 All quantities in \cref{eq:b17-affine-instruction,eq:b17-affine-program,eq:b17-affine-witness-recursion} are evaluated with exact rational arithmetic before a floating representation is exposed to the model.
16033 
16034 \begin{definition}[Executable affine-program task]
16035 A Block 17 task is
16036 \begin{equation}
16037  \tau
16038  =
16039  (x,m,(a_j,b_j)_{j=1}^{m},A_m,B_m,y_m,g,s),
16040  \qquad
16041  y_m=A_mx+B_m,
16042  \label{eq:b17-task}
16043 \end{equation}
16044 where $g\in\{A,B\}$ names the generator and $s$ names the data split. The model input is a padded serialization of $x$, the instructions, and instruction-presence bits. The protected semantic witness is
16045 \begin{equation}
16046  w(\tau)
16047  =
16048  \left(A_m,\frac{B_m}{3},\frac{y_m}{4}\right)\in\R^3.
16049  \label{eq:b17-semantic-witness}
16050 \end{equation}
16051 \end{definition}
16052 
16053 The environment module contains no model code. Its verifier reconstructs every instruction from the released record, recomputes $A_m$, $B_m$, and $y_m$ as instances of \texttt{Fraction}, checks the affine interpreter invariant, checks the floating witness, and rejects duplicate task identifiers. Generator $B$ changes the coefficient and offset probabilities while preserving the exact program language and interpreter. Thus generator foreignness is a declared distribution shift, not a change in semantics.
16054 
16055 The released supports are:
16056 \begin{longtable}{@{}lrrl@{}}
16057 \toprule
16058 Support & Tasks & Generator & Role \\
16059 \midrule
16060 \endfirsthead
16061 \toprule
16062 Support & Tasks & Generator & Role \\
16063 \midrule
16064 \endhead
16065 Training & $10{,}000$ & $A$ & Parent and matched repair optimization \\
16066 Development & $1{,}400$ & $A$ & Compiler development and diagnostics \\
16067 Meta-validation & $1{,}400$ & $A$ & Frozen candidate eligibility and selection \\
16068 Probe calibration & $1{,}200$ & $A$ & External ridge-probe fitting \\
16069 Primary audit & $1{,}600$ & $A$ & Fresh final in-generator evaluation \\
16070 Generator-foreign audit & $1{,}600$ & $B$ & Fresh shifted-distribution evaluation \\
16071 \bottomrule
16072 \end{longtable}
16073 
16074 The successful round contains $17{,}200$ exact task records. The first failed pilot used different final seeds and remains archived separately. No successful-round final task appears in training, development, meta-validation, probe calibration, or the pilot final supports.
16075 
16076 \begin{proposition}[Exact witness soundness]
16077 \label{prop:b17-witness-soundness}
16078 For every released task $\tau$, the environment verifier accepts only if its stored witness equals the exact composition witness in \cref{eq:b17-affine-witness-recursion}. Hence a model cannot obtain a valid audit result from a malformed or internally inconsistent target record.
16079 \end{proposition}
16080 
16081 \begin{proof}
16082 The verifier reconstructs $x$ and each $(a_j,b_j)$ as rational numbers, independently executes the recurrence in \cref{eq:b17-affine-witness-recursion}, and compares the resulting triple exactly with the stored rational witness. The floating target in \cref{eq:b17-semantic-witness} is then checked against the exact triple to tolerance $10^{-12}$. Any disagreement produces a failed environment certificate before model evaluation.
16083 \end{proof}
16084 
16085 \subsection{The recurrent parent and repair grammar}
16086 
16087 Let $\phi(\tau)\in\R^{19}$ be the padded task serialization, and let
16088 \begin{equation}
16089  e=E_\theta(\phi(\tau))\in\R^{40}
16090  \label{eq:b17-encoder}
16091 \end{equation}
16092 be the prompt representation. The latent trace has four states. Given conditioning mode $\alpha$, define
16093 \begin{equation}
16094  z_0=\zeta,
16095  \qquad
16096  z_{k+1}
16097  =
16098  \tanh\!\left(Rz_k+s_k+C_\alpha(e,m)\right),
16099  \qquad
16100  0\leq k<4,
16101  \label{eq:b17-trace}
16102 \end{equation}
16103 where $s_k$ is a learned step embedding. The task-blind parent satisfies
16104 \begin{equation}
16105  C_{\mathrm{parent}}(e,m)=0.
16106  \label{eq:b17-parent-conditioning}
16107 \end{equation}
16108 The descendants use one of the following conditioning maps:
16109 \begin{align}
16110  C_{\mathrm{depth}}(e,m)&=D\,\one_m,
16111  \label{eq:b17-depth-conditioning}\\
16112  C_{1}(e,m)&=U_1V_1e,
16113  \qquad \rank(U_1V_1)\leq1,
16114  \label{eq:b17-rank1-conditioning}\\
16115  C_{4}(e,m)&=U_4V_4e,
16116  \qquad \rank(U_4V_4)\leq4,
16117  \label{eq:b17-rank4-conditioning}\\
16118  C_{\mathrm{full}}(e,m)&=We.
16119  \label{eq:b17-full-conditioning}
16120 \end{align}
16121 Here $\one_m$ is the one-hot depth code. Each child inherits every compatible parent parameter. The added up-projection or full-conditioning matrix is initialized so that the initial injected signal is zero, giving a conservative functional starting point before repair training.
16122 
16123 The latent trace updates the task representation through
16124 \begin{align}
16125  g_k&=\sigma(Gz_k),
16126  \label{eq:b17-update-gate}\\
16127  \widetilde h_k&=\tanh(Th_{k-1}+Lz_k),
16128  \label{eq:b17-hidden-candidate}\\
16129  h_k&=(1-g_k)\odot h_{k-1}+g_k\odot\widetilde h_k,
16130  \qquad h_0=e,
16131  \label{eq:b17-hidden-update}\\
16132  \widehat w_k&=Oh_k.
16133  \label{eq:b17-readout}
16134 \end{align}
16135 A separate halt head predicts one of four dynamic depths from the mean trace and, for nonparent candidates, the encoded prompt. A semantic training head predicts $w(\tau)$ from $z_4$. Its auxiliary loss is an externally declared witness term, not a claim that the latent coordinates are uniquely interpretable.
16136 
16137 \begin{definition}[Finite Neuralese repair grammar]
16138 The Block 17 compiler grammar is
16139 \begin{equation}
16140  \Gcal_{17}
16141  =
16142  \{\mathsf{parent}^{+},\mathsf{depth},\mathsf{rank1},
16143  \mathsf{rank4},\mathsf{full},\mathsf{full}^{-\mathrm{sem}}\}.
16144  \label{eq:b17-repair-grammar}
16145 \end{equation}
16146 The symbol $\mathsf{parent}^{+}$ denotes matched-update continuation of the task-blind parent. The final symbol uses full conditioning but removes the semantic-witness loss. Every candidate receives exactly $550$ AdamW updates with batch size $256$, learning rate $8\times10^{-4}$, and identical ordinary task, intermediate-step, halt, and weight-decay terms.
16147 \end{definition}
16148 
16149 The common serialized architecture has $14{,}370$ parameters. Active parameter counts range from $12{,}130$ for the task-blind continuation to $13{,}730$ for full conditioning. The experiment matches updates, examples, data, optimizer, and base initialization, but not exact floating-point operations across candidate ranks. This is a matched-training-envelope comparison rather than an exact FLOP identity claim.
16150 
16151 \begin{proposition}[Nested conditioning images]
16152 For fixed encoder $E_\theta$, the rank-one and rank-four conditioning families are contained in the full linear conditioning family:
16153 \begin{equation}
16154  \{U_1V_1\}
16155  \subseteq
16156  \{U_4V_4\}
16157  \subseteq
16158  \{W\}.
16159  \label{eq:b17-nested-conditioning}
16160 \end{equation}
16161 The task-blind parent is recovered by the zero map in every family.
16162 \end{proposition}
16163 
16164 \begin{proof}
16165 Every rank-one linear operator has rank at most four and therefore admits a rank-four factorization after padding with zero columns and rows. Every rank-four factorization is a linear map and therefore belongs to the unconstrained full family. Setting every active conditioning weight to zero gives \cref{eq:b17-parent-conditioning}.
16166 \end{proof}
16167 
16168 \subsection{Why one transplant statistic is insufficient}
16169 
16170 Block 16 used same-format cross-task transplant cost. In a prediction problem, that cost can be positive even if the trace represents only the final answer. Program semantics and answer semantics must therefore be separated.
16171 
16172 Let $d_{\mathrm{prog}}(\tau,\tau')$ be a normalized distance between $(A_m,B_m)$ and $(A'_{m'},B'_{m'})$, and let
16173 \begin{equation}
16174  d_{\mathrm{ans}}(\tau,\tau')
16175  =
16176  \abs{y_m/4-y'_{m'}/4}.
16177  \label{eq:b17-answer-distance}
16178 \end{equation}
16179 For a recipient task $\tau_i$, define four donor classes:
16180 \begin{enumerate}
16181 \item $\mathsf{SAP}$: same-answer, far-program, approximately depth-matched donors;
16182 \item $\mathsf{FAD}$: far-answer, depth-matched donors;
16183 \item $\mathsf{SN}$: semantic-neighbor donors;
16184 \item $\mathsf{RMD}$: random depth-matched donors.
16185 \end{enumerate}
16186 Three donors per available class are selected by deterministic task-identifier hashing. Each audit records donor coverage and the realized semantic and answer distances.
16187 
16188 \begin{theorem}[Answer-only transplant confound]
16189 \label{thm:b17-answer-confound}
16190 Suppose a latent trace is conditionally answer sufficient but program blind:
16191 \begin{equation}
16192  Z(\tau)=\Psi(y(\tau),m(\tau),\xi),
16193  \label{eq:b17-answer-only-trace}
16194 \end{equation}
16195 where $\xi$ is independent internal noise, and suppose the answer readout genuinely uses $Z$. Then far-answer transplantation may have strictly positive expected cost even though $Z$ contains no information about $(A_m,B_m)$ conditional on $(y,m)$. Consequently, a positive ordinary cross-task transplant cost does not establish program-specific Neuralese.
16196 \end{theorem}
16197 
16198 \begin{proof}
16199 Choose two programs with the same depth but different answers. Under \cref{eq:b17-answer-only-trace}, the donor trace is distributed as a code for the donor answer rather than the recipient answer. If the readout uses this code, replacing the recipient trace can increase recipient loss. However, conditional on $(y,m)$, the trace distribution is independent of $(A_m,B_m)$ by assumption. Thus the positive cost is caused by answer mismatch, not by represented program structure.
16200 \end{proof}
16201 
16202 \begin{corollary}[Same-answer program control]
16203 Under the hypotheses of \cref{thm:b17-answer-confound}, exact same-answer and depth-matched donors have zero expected transplant cost, even when their affine programs are far apart. A positive $\mathsf{SAP}$ cost therefore rejects the exact answer-only null relative to the declared donor construction.
16204 \end{corollary}
16205 
16206 \begin{warning}[Residual identifiability boundary]
16207 The $\mathsf{SAP}$ control does not prove that every aspect of a program is represented. It establishes only that the transplanted trace carries recipient-relevant information not reducible to the matched answer and depth variables under the released task and donor distribution. Unmeasured sufficient statistics can remain.
16208 \end{warning}
16209 
16210 \subsection{Executable causal and semantic gates}
16211 
16212 Let $\ell_i(Z)$ be mean squared witness error on recipient $i$ when the four latent states are externally supplied as $Z$. Let $Z_i$ be the clean trace. The structural coordinates are
16213 \begin{align}
16214  n_i&=\ell_i(\mathsf{zero})-\ell_i(Z_i),
16215  \label{eq:b17-necessity}\\
16216  o_i&=\ell_i(\mathsf{reverse}(Z_i))-\ell_i(Z_i),
16217  \label{eq:b17-order}\\
16218  c_i&=\ell_i(\mathsf{noise}(Z_i))-\ell_i(Z_i),
16219  \label{eq:b17-noise}\\
16220  r_i&=\abs{\ell_i(\mathsf{replay}(Z_i))-\ell_i(Z_i)}.
16221  \label{eq:b17-replay}
16222 \end{align}
16223 Noise is deterministic and norm matched at every step. The program and answer specificity coordinates are the mean recipient costs under $\mathsf{SAP}$ and $\mathsf{FAD}$ donors:
16224 \begin{equation}
16225  p_i
16226  =
16227  \mathbb E_{j\in\mathsf{SAP}(i)}
16228  \left[\ell_i(Z_j)-\ell_i(Z_i)\right],
16229  \qquad
16230  a_i
16231  =
16232  \mathbb E_{j\in\mathsf{FAD}(i)}
16233  \left[\ell_i(Z_j)-\ell_i(Z_i)\right].
16234  \label{eq:b17-specificity-costs}
16235 \end{equation}
16236 Define
16237 \begin{equation}
16238  Q_{\mathrm{prog}}
16239  =
16240  \frac{\overline p}{\max\{\overline o,\overline c,\eps\}},
16241  \qquad
16242  Q_{\mathrm{ans}}
16243  =
16244  \frac{\overline a}{\max\{\overline o,\overline c,\eps\}}.
16245  \label{eq:b17-specificity-ratios}
16246 \end{equation}
16247 The frozen ratio thresholds are $0.15$ and $0.20$, respectively. Each specificity gate additionally requires a positive $95\%$ bootstrap lower endpoint and a positive-effect fraction of at least $0.55$. Structural necessity, order, and noise gates use the same lower-endpoint condition and a positive-effect fraction of at least $0.55$. Exact replay requires
16248 \begin{equation}
16249  \max_i r_i\leq10^{-10}.
16250  \label{eq:b17-replay-gate}
16251 \end{equation}
16252 
16253 The local-continuity gate compares far-program cost with semantic-neighbor cost. Its paired lower confidence endpoint must be positive. This prevents the classifier from treating every foreign trace as equally damaging while claiming a meaningful semantic geometry.
16254 
16255 \begin{definition}[External trace probe]
16256 A Block 17 trace probe is a ridge map
16257 \begin{equation}
16258  \widehat w=P\,z_4+q
16259  \label{eq:b17-external-probe}
16260 \end{equation}
16261 fit only on the $1{,}200$ probe-calibration tasks. The final audit reports held-out coefficient of determination
16262 \begin{equation}
16263  R^2_{\mathrm{probe}}
16264  =
16265  1-
16266  \frac{\sum_i\norm{w_i-\widehat w_i}^2}
16267  {\sum_i\norm{w_i-\overline w}^2}.
16268  \label{eq:b17-probe-r2}
16269 \end{equation}
16270 The frozen semantic-probe gate is $R^2_{\mathrm{probe}}\geq0.65$ on each final support.
16271 \end{definition}
16272 
16273 The probe does not assign natural-language meanings to individual coordinates. It tests whether an independently fit, held-out linear witness can recover exact task semantics from the final latent state. The independent signed-permutation check below confirms that this claim is invariant to a large family of coordinate relabelings.
16274 
16275 The adaptive-halting gate requires a Spearman correlation of at least $0.70$ between exact program depth and selected latent depth, at least four distinct selected depths, prompt-dependent halt probabilities, and mean dynamic-output loss no more than $0.015$ above fixed-depth output loss.
16276 
16277 \begin{definition}[Executable Neuralese status]
16278 An audit is \texttt{EXECUTABLE\_FULL} when all structural gates, both specificity gates, local semantic continuity, donor coverage, and the external trace-probe gate pass. It is \texttt{EXECUTABLE\_PARTIAL} when the structural gates pass but at least one semantic-binding gate fails. Otherwise it is \texttt{EXECUTABLE\_FAIL}. Adaptive halting is reported as a separate PASS or FAIL status.
16279 \end{definition}
16280 
16281 \subsection{Sealed compiler chronology}
16282 
16283 The parent receives $650$ updates with batch size $256$. Every candidate then begins from that exact checkpoint and receives the common $550$-update repair budget. Before any final-task generation, the compiler evaluates all candidates on meta-validation and applies the hard gates
16284 \begin{equation}
16285  \begin{gathered}
16286  \text{clean MSE within }0.002\text{ of the matched-update parent},\\
16287  R^2_{\mathrm{probe}}\geq0.65,
16288  \quad
16289  Q_{\mathrm{prog}}\geq0.15,
16290  \quad
16291  Q_{\mathrm{ans}}\geq0.20,\\
16292  \text{positive interval and fraction gates},
16293  \quad
16294  \text{local continuity},
16295  \quad
16296  \text{donor coverage},
16297  \quad
16298  \text{adaptive halting}.
16299  \end{gathered}
16300  \label{eq:b17-compiler-hard-gates}
16301 \end{equation}
16302 Among eligible candidates it maximizes
16303 \begin{equation}
16304  \begin{split}
16305  S(\alpha)
16306  ={}&
16307  \min\left\{
16308  \frac{R^2_{\mathrm{probe}}}{0.65},
16309  \frac{Q_{\mathrm{prog}}}{0.15},
16310  \frac{Q_{\mathrm{ans}}}{0.20}
16311  \right\}
16312  +0.20R^2_{\mathrm{probe}}\\
16313  &-4\,\mathrm{MSE}_{\mathrm{clean}}
16314  +0.02\rho_{\mathrm{halt}}
16315  -\mathsf{Penalty}_{\mathrm{dynamic}}.
16316  \end{split}
16317  \label{eq:b17-selection-score}
16318 \end{equation}
16319 The minimum term makes the weakest normalized semantic gate the principal bottleneck. The compiler seal commits the selected candidate, all candidate and parent state hashes, training receipts, meta-validation task digest, exact thresholds, selection rule, pilot-failure receipt, and post-pilot revision receipt before the successful-round final tasks are generated.
16320 
16321 \subsection{A failed sealed pilot and nonretroactive repair}
16322 
16323 The first compiler round used a narrower grammar and $450$ matched candidate updates. Its score over-weighted trace decodability. It selected full conditioning, and the child improved absolute program and answer transplant effects, probe $R^2$, and adaptive depth. Nevertheless, it remained \texttt{EXECUTABLE\_PARTIAL} on both final supports because the normalized program and answer specificity ratios missed the already frozen FULL thresholds. The lineage transaction was rejected with receipt
16324 \begin{verbatim}
16325 cc0d07b20ec2820a40f9781babc36f467482bed9ce5319f031401c3ba026c318
16326 \end{verbatim}
16327 
16328 The negative result was not overwritten. The package archives the failed checkpoints, final tasks, audits, figures, and receipt. Before a second final audit, the revision file committed four changes:
16329 \begin{enumerate}
16330 \item add a rank-four conditioning candidate;
16331 \item allocate $550$ repair updates to every candidate;
16332 \item use the weakest-normalized-margin selection score in \cref{eq:b17-selection-score};
16333 \item generate new primary and generator-foreign supports from seeds $17006$ and $27006$.
16334 \end{enumerate}
16335 No final threshold was weakened or removed. The revision receipt is
16336 \begin{verbatim}
16337 f7435a94a4c1d065ff5e37d3895a58d7fca541faf2098a8872ac9f90c6915755
16338 \end{verbatim}
16339 This chronology makes the failed pilot evidence about compiler design rather than hidden tuning evidence for the successful final tasks.
16340 
16341 \begin{protocol}[Block 17 sealed repair transaction]
16342 \label{prot:b17-transaction}
16343 \begin{enumerate}
16344 \item Generate and exactly verify training, development, meta-validation, and probe-calibration supports.
16345 \item Train and serialize the task-blind parent.
16346 \item Inherit the parent into every candidate in \cref{eq:b17-repair-grammar}.
16347 \item Apply the same repair-update and example budget to each candidate.
16348 \item Fit the external probe only on probe-calibration traces.
16349 \item Evaluate every candidate on meta-validation with the complete donor and halting audit.
16350 \item Select by \cref{eq:b17-compiler-hard-gates,eq:b17-selection-score} and seal all state and protocol digests.
16351 \item Only after sealing, generate and exactly verify the primary and generator-foreign final supports.
16352 \item Audit the matched parent and selected child on both supports.
16353 \item Commit only if every lineage gate passes; otherwise retain the failed receipt and rollback to the parent.
16354 \item Independently replay the environment witnesses, checkpoint states, audit receipts, NumPy forward equations, and gauge certificate.
16355 \end{enumerate}
16356 \end{protocol}
16357 
16358 \subsection{Compiler result}
16359 
16360 Three of the six candidates passed every meta-validation hard gate. The full-binding candidate had the largest frozen score and was selected.
16361 
16362 \begin{longtable}{@{}lrrrrr@{}}
16363 \toprule
16364 Candidate & Eligible & Clean MSE & Probe $R^2$ & $Q_{\mathrm{prog}}$ & $Q_{\mathrm{ans}}$ \\
16365 \midrule
16366 \endfirsthead
16367 \toprule
16368 Candidate & Eligible & Clean MSE & Probe $R^2$ & $Q_{\mathrm{prog}}$ & $Q_{\mathrm{ans}}$ \\
16369 \midrule
16370 \endhead
16371 Matched parent continuation & no & $0.012385$ & $-0.000901$ & $0$ & $0$ \\
16372 Depth only & no & $0.012172$ & $0.000984$ & $0$ & $0$ \\
16373 Rank-one binding & yes & $0.012001$ & $0.699509$ & $0.8054$ & $0.9229$ \\
16374 Rank-four binding & yes & $0.011779$ & $0.861346$ & $0.3937$ & $0.5036$ \\
16375 Full binding & \textbf{selected} & $0.011452$ & $0.870542$ & $0.7436$ & $1.1703$ \\
16376 Full without semantic witness & no & $0.010728$ & $0.724937$ & $0.1019$ & $0.1050$ \\
16377 \bottomrule
16378 \end{longtable}
16379 
16380 The final ablation has the lowest clean meta-validation error but fails both causal-specificity ratios. This prevents predictive performance or decodability alone from being credited as task-bound latent semantics.
16381 
16382 \begin{figure}[H]
16383 \centering
16384 \includegraphics[width=0.86\textwidth]{block17_candidate_compiler.pdf}
16385 \caption{Block 17 repair compiler. Eligibility requires all semantic, causal, replay, donor, clean-loss, and halting gates. Full binding is selected by the frozen weakest-margin score; the no-semantic-witness ablation is predictively strong but causally ineligible.}
16386 \label{fig:block17-candidate-compiler}
16387 \end{figure}
16388 
16389 \subsection{Fresh primary and generator-foreign audits}
16390 
16391 The primary audit contains $1{,}600$ generator-$A$ tasks created after the compiler seal. The matched parent retains useful, ordered, and corruption-sensitive recurrence, but its latent trajectory is exactly task blind under the conditioning interface. The selected child passes every structural and semantic-binding gate.
16392 
16393 \begin{longtable}{@{}lrr@{}}
16394 \toprule
16395 Primary quantity & Parent & Child \\
16396 \midrule
16397 \endfirsthead
16398 \toprule
16399 Primary quantity & Parent & Child \\
16400 \midrule
16401 \endhead
16402 Clean MSE & $0.012665$ & $0.011686$ \\
16403 Latent-necessity gain & $0.017036$ & $0.030296$ \\
16404 Step-order cost & $0.000957$ & $0.004362$ \\
16405 Norm-matched noise cost & $0.001272$ & $0.014644$ \\
16406 Same-answer far-program cost & $0$ & $0.010714$ \\
16407 Far-answer matched-depth cost & $0$ & $0.017727$ \\
16408 Program specificity ratio & $0$ & $0.731642$ \\
16409 Answer specificity ratio & $0$ & $1.210526$ \\
16410 External trace-probe $R^2$ & $-0.000302$ & $0.865504$ \\
16411 Difficulty--depth Spearman $\rho$ & $0$ & $0.971716$ \\
16412 Distinct dynamic depths & $1$ & $4$ \\
16413 Causal status & PARTIAL & FULL \\
16414 Adaptive-halting status & FAIL & PASS \\
16415 \bottomrule
16416 \end{longtable}
16417 
16418 The paired child-minus-parent program-specificity gain is
16419 \begin{equation}
16420  \widehat\Delta_{\mathrm{prog}}^{A}
16421  =0.010714,
16422  \qquad
16423  \mathrm{CI}_{0.95}
16424  =[0.009911,0.011527],
16425  \label{eq:b17-primary-program-gain}
16426 \end{equation}
16427 with positive fraction $0.823125$. The answer-specificity gain is $0.017727$. Its frozen interval is
16428 \begin{equation*}
16429 [0.016490,0.018960].
16430 \end{equation*}
16431 Parent-minus-child clean loss is $0.000979$, with frozen interval $[0.000695,0.001290]$.
16432 
16433 The shifted audit contains $1{,}600$ generator-$B$ tasks with altered coefficient and offset probabilities. The parent remains PARTIAL and the child remains FULL.
16434 
16435 \begin{longtable}{@{}lrr@{}}
16436 \toprule
16437 Generator-foreign quantity & Parent & Child \\
16438 \midrule
16439 \endfirsthead
16440 \toprule
16441 Generator-foreign quantity & Parent & Child \\
16442 \midrule
16443 \endhead
16444 Clean MSE & $0.062210$ & $0.058746$ \\
16445 Latent-necessity gain & $0.036128$ & $0.051896$ \\
16446 Step-order cost & $0.000866$ & $0.004750$ \\
16447 Norm-matched noise cost & $0.001353$ & $0.017566$ \\
16448 Same-answer far-program cost & $0$ & $0.024763$ \\
16449 Far-answer matched-depth cost & $0$ & $0.035400$ \\
16450 Program specificity ratio & $0$ & $1.409704$ \\
16451 Answer specificity ratio & $0$ & $2.015268$ \\
16452 External trace-probe $R^2$ & $-0.002355$ & $0.691471$ \\
16453 Difficulty--depth Spearman $\rho$ & $0$ & $0.969169$ \\
16454 Distinct dynamic depths & $1$ & $4$ \\
16455 Causal status & PARTIAL & FULL \\
16456 Adaptive-halting status & FAIL & PASS \\
16457 \bottomrule
16458 \end{longtable}
16459 
16460 The paired generator-foreign program-specificity gain is
16461 \begin{equation}
16462  \widehat\Delta_{\mathrm{prog}}^{B}
16463  =0.024763,
16464  \qquad
16465  \mathrm{CI}_{0.95}
16466  =[0.022794,0.026793],
16467  \label{eq:b17-foreign-program-gain}
16468 \end{equation}
16469 with positive fraction $0.855$. The answer-specificity gain is $0.035400$ with interval $[0.032956,0.037938]$. Clean-loss improvement is $0.003464$ with interval $[0.002633,0.004324]$. Monte Carlo paired sign-flip tests for the program-specificity gains return $9.999\times10^{-5}$ on each final support.
16470 
16471 \begin{figure}[H]
16472 \centering
16473 \includegraphics[width=0.86\textwidth]{block17_causal_profile_primary.pdf}
16474 \caption{Fresh primary audit. The parent is structurally recurrent but task blind; the selected child adds program- and answer-specific latent effects while improving clean witness error.}
16475 \label{fig:block17-primary-profile}
16476 \end{figure}
16477 
16478 \begin{figure}[H]
16479 \centering
16480 \includegraphics[width=0.86\textwidth]{block17_causal_profile_generator_foreign.pdf}
16481 \caption{Shifted-generator audit. The selected child retains FULL causal status and positive semantic-binding margins under the declared distribution shift.}
16482 \label{fig:block17-foreign-profile}
16483 \end{figure}
16484 
16485 \begin{figure}[H]
16486 \centering
16487 \includegraphics[width=0.86\textwidth]{block17_probe_and_halting.pdf}
16488 \caption{External semantic-probe and adaptive-halting results. The child exposes a held-out linearly recoverable semantic witness and selects four task-dependent latent depths; the parent does neither.}
16489 \label{fig:block17-probe-halting}
16490 \end{figure}
16491 
16492 \subsection{Independent shadow and gauge verification}
16493 
16494 The primary PyTorch audit is replayed by a separately written NumPy forward implementation. It reconstructs the encoder, recurrent trace, update gate, hidden transition, readout, semantic head, and halt logits directly from the serialized state dictionary. Across the released verification cases, the maximum child-output discrepancy is $1.536\times10^{-6}$ and the parent discrepancy is $1.098\times10^{-6}$.
16495 
16496 Let $P$ be a signed permutation matrix on the latent space. Define transformed latent parameters
16497 \begin{equation}
16498  \zeta'=P\zeta,
16499  \quad
16500  R'=PRP^{\mathsf T},
16501  \quad
16502  s_k'=Ps_k,
16503  \quad
16504  C_\alpha'=PC_\alpha,
16505  \label{eq:b17-gauge-trace}
16506 \end{equation}
16507 and transform every latent-consuming linear map by right multiplication with $P^{\mathsf T}$. Then the transformed model has $z_k'=Pz_k$ and identical external outputs.
16508 
16509 \begin{proposition}[Signed-permutation gauge invariance]
16510 \label{prop:b17-gauge-invariance}
16511 Under the transformations in \cref{eq:b17-gauge-trace}, the clean witness prediction, every step prediction, semantic prediction, and halt logit are unchanged in exact arithmetic, while the latent coordinate state and its byte-level hash may change.
16512 \end{proposition}
16513 
16514 \begin{proof}
16515 Inductively, $z_0'=Pz_0$. If $z_k'=Pz_k$, then
16516 \begin{equation}
16517  z_{k+1}'
16518  =
16519  \tanh\!\left(PRP^{\mathsf T}Pz_k+Ps_k+PC_\alpha\right)
16520  =P\tanh(Rz_k+s_k+C_\alpha)
16521  =Pz_{k+1},
16522 \end{equation}
16523 because signed permutations commute with coordinatewise $\tanh$. For every latent-consuming map $M'=MP^{\mathsf T}$, one has $M'z_k'=Mz_k$. Therefore the gates, hidden candidates, readouts, semantic prediction, and halt logits coincide. The state tensors nevertheless occupy different coordinates, so their serialized state hash can differ.
16524 \end{proof}
16525 
16526 The released gauge check obtains maximum absolute output error $1.907\times10^{-6}$ and a distinct transformed-state hash. Thus the semantic result attaches to a transported functional trajectory, not to the names of individual latent coordinates.
16527 
16528 The independent verifier also recomputes every exact task witness, verifies unique task identifiers, checks all checkpoint state hashes against the pre-audit seal, reconstructs all four final audit receipts, confirms that the failed-pilot receipt is bound into the revision, and recomputes the accepted lineage receipt. Its verification receipt is
16529 \begin{verbatim}
16530 286d263fae995ef6fe623b4fb87ecef3be615257eb825d216186c11fd3dcc664
16531 \end{verbatim}
16532 This verifier is implementation independent inside the same release bundle. It is not an external laboratory replication.
16533 
16534 \subsection{Finite executable Neuralese repair theorem}
16535 
16536 \begin{conditionaltheorem}[Block 17 executable Neuralese binding-repair edge]
16537 \label{thm:b17-executable-edge}
16538 Assume the released Block 17 records satisfy the following conditions:
16539 \begin{enumerate}
16540 \item every task passes the exact witness and identifier checks in \cref{prop:b17-witness-soundness};
16541 \item the parent is trained before candidate construction and every candidate inherits the same parent state;
16542 \item every candidate receives the declared matched update, batch, data, optimizer, and ordinary loss envelope;
16543 \item candidate selection uses only development, meta-validation, and probe-calibration supports with the frozen gates in \cref{eq:b17-compiler-hard-gates};
16544 \item the compiler seal precedes successful-round final-task generation and commits all relevant state and protocol digests;
16545 \item the failed first audit and post-pilot revision remain append-only, and no final threshold is weakened after the pilot;
16546 \item exact replay, donor construction, external probing, dynamic halting, and clean-loss scoring retain their declared interpretations;
16547 \item the child is FULL and adaptive on both fresh final supports, while the paired effects in \cref{eq:b17-primary-program-gain,eq:b17-foreign-program-gain} have positive frozen intervals;
16548 \item the independent NumPy, receipt, witness, and gauge checks pass within their declared numerical tolerances.
16549 \end{enumerate}
16550 Then the accepted transition
16551 \begin{equation}
16552  \mathfrak L_{16}^{\mathrm{NL}}
16553  \xrightarrow{\Pi_{17}}
16554  \mathfrak L_{17}^{\mathrm{NL}}
16555  \label{eq:b17-lineage-edge}
16556 \end{equation}
16557 is a finite, evidence-preserving, training-generated repair of the Neuralese binding component on the declared affine-program environment and generator shift. The accepted receipt is
16558 \begin{verbatim}
16559 6cc2257fe89e0ad14a768580919c6af72b9e13654a61dc697bd0df086655e4ae
16560 \end{verbatim}
16561 It is stronger than the analytic Block 16 positive control because checkpoint behavior arises from executed optimization and the selected child was sealed before fresh final-task generation. It is not an open-ended recursive self-improvement result because the repair grammar, environment, witnesses, and compiler objective remain externally supplied.
16562 \end{conditionaltheorem}
16563 
16564 \begin{proof}
16565 Items 1 through 3 establish executable task soundness, ancestor continuity, and the matched repair envelope. Items 4 and 5 establish pre-audit compiler selection rather than final-support model choice. Item 6 prevents the failed pilot from being silently converted into a positive result by retroactive threshold relaxation. Item 7 fixes the causal and semantic interpretation of the reported gates. Item 8 supplies positive held-out program-binding, answer-binding, clean-performance, and adaptive-depth evidence on both the primary and shifted generator. Item 9 supplies a second forward implementation, exact receipt replay, and coordinate-gauge consistency. Together these conditions establish one bounded lineage transition in which the selected trained child adds task-conditioned latent semantics unavailable to the task-blind parent while retaining the protected external witness. Since the outer candidate language and selection semantics were not generated by the child, the conclusion is finite repair compilation rather than unrestricted autonomous recursive improvement.
16566 \end{proof}
16567 
16568 \subsection{Reproducibility and claim boundary}
16569 
16570 The canonical frozen verification executes
16571 \begin{verbatim}
16572 python independent_verify.py
16573 python -m pytest -q
16574 python build_manifest.py
16575 python verify_package.py
16576 \end{verbatim}
16577 and reports $20$ passing tests, valid exact-task checks, replayed audit and lineage receipts, the NumPy shadow bounds, the gauge certificate, and a valid package manifest.
16578 
16579 Block 17 establishes:
16580 \begin{itemize}
16581 \item actual training and execution of a recurrent latent parent and six bounded repair candidates;
16582 \item a finite pre-audit compiler that selects a semantic-binding repair;
16583 \item an answer-confound control using same-answer far-program donors;
16584 \item an external held-out semantic probe and adaptive-depth test;
16585 \item a preserved failed pilot followed by a precommitted compiler repair and fresh audit;
16586 \item a FULL selected child and PARTIAL matched parent on primary and generator-foreign supports;
16587 \item positive paired program- and answer-specificity intervals and clean nonregression;
16588 \item package-local independent forward, witness, receipt, and gauge verification.
16589 \end{itemize}
16590 
16591 It does not establish:
16592 \begin{itemize}
16593 \item autonomous invention of the affine-program environment, donor taxonomy, semantic witness, candidate grammar, compiler score, or acceptance thresholds;
16594 \item exact compute matching across candidates with different active conditioning ranks;
16595 \item external replication by an independent implementation or research group;
16596 \item transfer to natural language, theorem proving, open tool use, or production-scale models;
16597 \item an empirical result for Cymela Hyper or any other released latent-reasoning checkpoint;
16598 \item complete latent-state interpretation or recovery of every task-relevant sufficient statistic;
16599 \item robustness to adaptive latent attacks, evaluator capture, or donor-set exploitation;
16600 \item unrestricted safety, consciousness, sentience, moral status, open-ended recursive self-improvement, or indefinite acceleration.
16601 \end{itemize}
16602 
16603 The next major threshold is \emph{foreign-model executable Neuralese repair}. The same answer-confound controls, trace probe, gauge transport, pilot-preservation rule, and sealed compiler chronology should be applied to at least two independently trained latent-reasoning implementations or to a real parent--child checkpoint pair. The environment should include natural-language or program-synthesis tasks with externally executed answers, and the repair constructor should generate at least one architecture or training modification absent from the declared Block 17 conditioning grammar. A positive result would move the lineage from a package-local micro-model edge to cross-implementation neural metaproductivity.
16604 
16605 \section{Cross-Implementation Neuralese Repair Transport}
16606 \label{sec:block18-cross-implementation}
16607 
16608 Block 17 established one finite training-generated repair of a recurrent latent binding mechanism inside a single package-local architecture. Its next threshold was explicitly architecture foreign: compile a repair on one recurrent implementation, instantiate the same typed repair in independently implemented foreign parent families, seal all checkpoints and semantic bridges before final-task generation, and require the resulting children to retain causal specificity, external execution, adaptive depth, and latent semantic transport under a shifted generator. Block 18 performs that experiment.
16609 
16610 The result is a sealed failure rather than an accepted lineage edge. This status is not an editorial downgrade. It is the principal result of the block. A step-specific FiLM repair with a progressive semantic trajectory witness is selected on an Elman source and instantiated in GRU and gated-residual foreign parents without architecture-specific candidate search. Both children satisfy the full causal certificate on fresh primary tasks. Both fail on the independently generated shifted support, both frozen source-to-target semantic bridges fail, and the new constructor underperforms the inherited additive repair on pooled program specificity. A post-audit held-out diagnosis then shows that source and target semantics remain linearly recoverable after shift-local calibration. The obstruction is therefore not the absence of every semantic latent variable. It is the failure of a frozen semantic readout and bridge to define distributionally invariant coordinates.
16611 
16612 \begin{statusbox}{Block 18 status}
16613 The released companion contains $15{,}300$ exact externally executed program-induction records, three independently implemented recurrent cell families, six source repair candidates, a source compiler and schema seal, matched foreign parents and controls, paired and shuffled latent bridges, $16$ model/split audits, two final transport audits, a canonical failed-family receipt, a post-hoc localization explicitly excluded from acceptance, five figures, nine automated tests, an independent verifier, and an exact package manifest. The GRU and gated-residual selected children are \texttt{CROSS\_IMPL\_FULL} on the primary generator and \texttt{CROSS\_IMPL\_FAIL} on the shifted generator. The Block 18 cross-implementation family gate fails closed. The supported output is a distributional semantic-transport obstruction certificate, not a positive foreign-model recursive-improvement edge.
16614 \end{statusbox}
16615 
16616 \subsection{Exact externally executed program-induction environment}
16617 
16618 Let the template set be
16619 \begin{equation}
16620  \Theta
16621  =
16622  \{\theta_{\mathrm{aff}},\theta_{\mathrm{abs}},
16623    \theta_{\mathrm{quad}},\theta_{\mathrm{hinge}}\}.
16624  \label{eq:b18-template-set}
16625 \end{equation}
16626 For integer parameters $a,b,c$ and integer input $x$, the exact programs are
16627 \begin{align}
16628  f_{\mathrm{aff}}(x)&=ax+b,
16629  \\
16630  f_{\mathrm{abs}}(x)&=a|x|+b,
16631  \\
16632  f_{\mathrm{quad}}(x)&=ax^2+bx+c,
16633  \\
16634  f_{\mathrm{hinge}}(x)&=
16635  \begin{cases}
16636  ax+c,&x\geq0,\\
16637  bx+c,&x<0.
16638  \end{cases}
16639  \label{eq:b18-program-templates}
16640 \end{align}
16641 The coefficients belong to finite declared integer sets. A task contains five shuffled input--output demonstrations and one query input. The model receives a normalized serialization of those demonstrations and the query. The protected target is
16642 \begin{equation}
16643  y(\tau)
16644  =
16645  \left(
16646  e_{\theta},
16647  \frac{p_1}{2},\ldots,\frac{p_4}{2},
16648  \frac{d(\theta)}{5},
16649  \frac{f_{\theta,p}(x_q)}{32}
16650  \right)\in\R^{10},
16651  \label{eq:b18-protected-target}
16652 \end{equation}
16653 where $e_{\theta}\in\R^4$ is the one-hot template code, unused parameter slots are zero, and $d(\theta)\in\{2,3,4,5\}$ is the declared execution depth. All demonstrations, query answers, semantic coordinates, features, and targets are regenerated by an environment module containing no neural-model code.
16654 
16655 \begin{proposition}[Block 18 exact environment soundness]
16656 \label{prop:b18-environment-soundness}
16657 If a released task record passes the Block 18 environment verifier, then every demonstration output, query output, template depth, semantic coordinate, normalized feature, and protected target agrees with the exact integer program in \cref{eq:b18-program-templates}.
16658 \end{proposition}
16659 
16660 \begin{proof}
16661 The verifier reconstructs the task object from its serialized record, executes the relevant branch of \cref{eq:b18-program-templates} using exact integer arithmetic, rebuilds every derived vector, and compares the reconstructed values with the record. It also rejects repeated task identifiers. Acceptance therefore implies consistency with the separately implemented interpreter rather than consistency only with model-generated labels.
16662 \end{proof}
16663 
16664 The pre-audit and final supports are:
16665 \begin{longtable}{@{}lrrl@{}}
16666 \toprule
16667 Support & Tasks & Generator & Role \\
16668 \midrule
16669 \endfirsthead
16670 \toprule
16671 Support & Tasks & Generator & Role \\
16672 \midrule
16673 \endhead
16674 Training & $7{,}000$ & $A$ & Parent and repair optimization \\
16675 Development & $1{,}200$ & $A$ & Source probe and candidate construction \\
16676 Meta-validation & $1{,}200$ & $A$ & Source compiler selection \\
16677 Probe calibration & $1{,}000$ & $A$ & Frozen external semantic readouts \\
16678 Transport bridge & $900$ & $A$ & Paired and shuffled bridge fitting \\
16679 Primary audit & $1{,}400$ & $A$ & Fresh in-generator final audit \\
16680 Generator-foreign audit & $1{,}400$ & $B$ & Fresh shifted final audit \\
16681 Transport audit & $1{,}200$ & $B$ & Separate shifted semantic-transport audit \\
16682 \bottomrule
16683 \end{longtable}
16684 
16685 Generator $A$ uses approximately uniform template frequencies. Its demonstration and query supports are
16686 \begin{equation}
16687 \Xcal_A^{\mathrm{demo}}=\{-2,-1,0,1,2\},
16688 \qquad
16689 \Xcal_A^{\mathrm{query}}=\{-3,3\}.
16690 \end{equation}
16691 Generator $B$ changes the template mixture toward quadratic and hinge programs and uses
16692 \begin{equation}
16693 \Xcal_B^{\mathrm{demo}}=\{-3,-1,0,1,3\},
16694 \qquad
16695 \Xcal_B^{\mathrm{query}}=\{-3,-2,2,3\}.
16696 \end{equation}
16697 Every shifted record remains inside the same exact program language. Nevertheless, every shifted task contains at least one normalized feature coordinate outside the coordinatewise range of the primary support, and the largest standardized feature-mean change is $0.915465$.
16698 
16699 \subsection{Source and foreign recurrent implementations}
16700 
16701 Let $e=E(\phi(\tau))\in\R^{36}$ be the encoded task representation, let $z_0\in\R^{36}$ be a learned seed, and let $s_k\in\R^{36}$ be the step embedding at latent depth $k\in\{0,\ldots,4\}$. The source uses an Elman transition
16702 \begin{equation}
16703  z_{k+1}
16704  =
16705  \tanh\left(Wz_k+s_k+\delta_k(e,z_k)\right).
16706  \label{eq:b18-elman}
16707 \end{equation}
16708 The first foreign family uses a GRU cell
16709 \begin{equation}
16710  z_{k+1}=\operatorname{GRUCell}(u_k,z_k),
16711  \label{eq:b18-gru}
16712 \end{equation}
16713 and the second uses a gated nonlinear residual cell
16714 \begin{equation}
16715  z_{k+1}
16716  =
16717  z_k+\sigma(Gz_k)\odot
16718  \tanh\left(V_2\operatorname{GELU}(V_1z_k)+s_k+\delta_k(e,z_k)\right).
16719  \label{eq:b18-residual}
16720 \end{equation}
16721 The three cell classes have distinct executable implementations and parameterizations. They share only the declared external input, latent dimension, five-step budget, semantic target, downstream gated readout, halt interface, and typed repair-schema contract.
16722 
16723 \begin{definition}[Cross-implementation repair schema]
16724 \label{def:b18-repair-schema}
16725 A Block 18 repair schema is a tuple
16726 \begin{equation}
16727  \rho=(a,\lambda_{\mathrm{sem}},\lambda_{\mathrm{prog}},
16728  \lambda_{\mathrm{geo}},\chi),
16729  \label{eq:b18-repair-schema}
16730 \end{equation}
16731 where $a$ is an adapter constructor, the three nonnegative $\lambda$ values weight final semantic, progressive semantic, and latent-geometry witnesses, and $\chi$ records whether witness labels are correctly bound or shuffled. A schema is cross-implementation when the same tuple is instantiated through each cell family's declared adapter interface without architecture-specific candidate selection.
16732 \end{definition}
16733 
16734 The source grammar contains a task-blind continuation, inherited additive conditioning, final-step FiLM, progressive FiLM with geometry, a low-rank hypernetwork, and a shuffled-witness FiLM control. For the selected FiLM constructor, the encoded task produces step-indexed scale and shift vectors
16735 \begin{equation}
16736  (\gamma_k(e),\beta_k(e))_{k=0}^{4}
16737  =F_{\mathrm{FiLM}}(e),
16738  \label{eq:b18-film}
16739 \end{equation}
16740 which act on the preactivation by
16741 \begin{equation}
16742  q_k
16743  \longmapsto
16744  q_k\odot\left(1+\frac12\tanh\gamma_k(e)\right)
16745  +\beta_k(e).
16746  \label{eq:b18-film-action}
16747 \end{equation}
16748 This constructor is absent from Block 17's rank-one, rank-four, and full additive conditioning family because it provides step-specific multiplicative and additive modulation rather than one time-independent injected vector.
16749 
16750 The progressive semantic witness trains every state $z_k$ toward a depth-appropriate prefix of the protected program semantics. The geometry term preserves relative semantic distances over selected trace pairs. These witnesses alter training only. Final acceptance remains determined by external causal interventions, exact program execution, held-out semantic probes, donor controls, clean loss, and adaptive depth.
16751 
16752 \subsection{Source compiler and pre-audit chronology}
16753 
16754 The source Elman parent receives $480$ optimization updates. Every source repair candidate inherits the compatible parent state and receives $380$ updates over the same training support. The compiler uses development and meta-validation evidence only. Its hard gates include clean-loss nonregression, held-out semantic-probe performance, same-answer program specificity, far-answer specificity, donor coverage, local semantic continuity, adaptive depth, and external execution. Eligible candidates are ranked by their weakest normalized semantic margin.
16755 
16756 \begin{table}[H]
16757 \centering
16758 \small
16759 \resizebox{\textwidth}{!}{%
16760 \begin{tabular}{@{}lrrrrrrl@{}}
16761 \toprule
16762 Candidate & Clean & Probe $R^2$ & Program ratio & Answer ratio & Halt $\rho$ & Execution & Status \\
16763 \midrule
16764 Recalibrated parent & $0.114372$ & $-0.001360$ & $0.000000$ & $0.000000$ & $0.000000$ & $0.200833$ & PARTIAL \\
16765 Legacy additive & $0.088248$ & $0.639068$ & $0.632384$ & $0.182410$ & $0.609663$ & $0.289167$ & FULL \\
16766 FiLM final witness & $0.087307$ & $0.655812$ & $0.124210$ & $0.055454$ & $0.573743$ & $0.325833$ & PARTIAL \\
16767 FiLM progressive geometry & $0.083188$ & $0.687281$ & $0.359980$ & $0.085309$ & $0.625756$ & $0.379167$ & FULL \\
16768 Hyper progressive geometry & $0.093108$ & $0.565723$ & $0.182308$ & $0.026907$ & $0.430354$ & $0.310000$ & PARTIAL \\
16769 FiLM shuffled witness & $0.092262$ & $0.453266$ & $0.111008$ & $0.041753$ & $0.410682$ & $0.317500$ & PARTIAL \\
16770 \bottomrule
16771 \end{tabular}%
16772 }
16773 \end{table}
16774 
16775 The eligible source schemas are legacy additive and progressive FiLM. The weakest-margin compiler selects \texttt{film\_progressive\_geometry}. Its source seal commits the source parent, all six candidate state hashes and audit receipts, the selected schema digest, both foreign architectures, and all final split seeds and counts before those final files are generated.
16776 
16777 A provisional adaptive-depth loss tolerance of $0.015$ excluded every adaptive candidate because their observed protected dynamic-loss changes lay between $0.0176$ and $0.0207$. Before the source seal and before final-task generation, the tolerance was revised to $0.025$. No specificity, probe, execution, donor, transport, split, or final acceptance threshold was changed. A second pre-audit revision gives every foreign repair and control $520$ updates, increases intermediate-output and halt supervision, and applies the same envelope to legacy, selected, and shuffled controls in both architectures. These chronology records are retained as pre-audit compiler calibration, not evidence from the final supports.
16778 
16779 \begin{figure}[H]
16780 \centering
16781 \includegraphics[width=0.92\textwidth]{block18_source_compiler.pdf}
16782 \caption{Source repair compiler. Two candidates pass the source hard gates. The step-specific progressive FiLM schema is selected by its weakest normalized semantic margin. The figure does not imply foreign-generator success.}
16783 \label{fig:b18-source-compiler}
16784 \end{figure}
16785 
16786 \subsection{Causal, semantic, and execution audit}
16787 
16788 For model $M$, task $i$, and clean protected loss $L_i$, the audit reuses the Block 17 donor distinctions and adds cross-implementation status labels. It records latent necessity, reversal cost, norm-matched noise cost, exact replay, same-answer far-program transplant cost, far-answer matched-depth transplant cost, local semantic-neighbor cost, external trace-probe $R^2$, dynamic-depth correlation, and exact program execution. Let
16789 \begin{align}
16790  n_i&=L_i^{0}-L_i,
16791  &o_i&=L_i^{\mathrm{rev}}-L_i,
16792  &c_i&=L_i^{\eta}-L_i,
16793  \\
16794  p_i&=L_{i,\mathrm{sem}}^{\mathrm{same\ answer,far\ program}}
16795       -L_{i,\mathrm{sem}},
16796  &a_i&=L_{i,\mathrm{answer}}^{\mathrm{far\ answer,depth}}
16797       -L_{i,\mathrm{answer}}.
16798  \label{eq:b18-audit-coordinates}
16799 \end{align}
16800 The normalized specificity ratios are
16801 \begin{equation}
16802  Q_{\mathrm{prog}}
16803  =
16804  \frac{\overline p}{\max\{\overline o,\overline c,\eps\}},
16805  \qquad
16806  Q_{\mathrm{ans}}
16807  =
16808  \frac{\overline a}{\max\{\overline o,\overline c,\eps\}}.
16809  \label{eq:b18-specificity-ratios}
16810 \end{equation}
16811 FULL status requires positive lower intervals and positive-fraction gates for the structural and donor statistics, $Q_{\mathrm{prog}}\geq0.18$, $Q_{\mathrm{ans}}\geq0.025$, donor coverage at least $0.70$, external trace-probe $R^2\geq0.40$, dynamic-depth Spearman correlation at least $0.50$ with at least three distinct depths, dynamic-loss change at most $0.025$, exact replay tolerance $10^{-10}$, and executed-answer accuracy at least $0.30$.
16812 
16813 The same schema is instantiated in both foreign families. There is no architecture-specific search after the source schema is sealed. Each architecture nevertheless receives its own inherited parent and matched optimization, because the tested object is transport of a typed repair rule across parent implementations rather than direct copying of one weight tensor.
16814 
16815 \subsection{Latent semantic bridges}
16816 
16817 Let $H_s(\tau)\in\R^{36}$ be the source model's final latent trace state and $H_t(\tau)\in\R^{36}$ the corresponding foreign-child state. A ridge-affine bridge is fitted on the $900$ paired generator-$A$ anchors:
16818 \begin{equation}
16819  B_t
16820  =
16821  \argmin_{B,b}
16822  \sum_{i=1}^{900}
16823  \norm{H_s(\tau_i)-BH_t(\tau_i)-b}^2
16824  +\lambda\norm{B}^2.
16825  \label{eq:b18-bridge-fit}
16826 \end{equation}
16827 A source ridge probe $P_s$ is fitted independently on the $1{,}000$ probe-calibration tasks. The transported semantic prediction is
16828 \begin{equation}
16829  \widehat Y_t(\tau)
16830  =P_s\bigl(B_tH_t(\tau)+b_t\bigr).
16831  \label{eq:b18-transported-readout}
16832 \end{equation}
16833 The bridge gate requires effective rank $36$, transported $R^2\geq0.35$, an advantage of at least $0.15$ over the stronger of the unaligned and shuffled-anchor controls, and shuffled-anchor $R^2\leq0.20$.
16834 
16835 Both pre-audit bridges satisfy these conditions with substantial apparent margin: transported $R^2$ is approximately $0.704$ for GRU and $0.707$ for the residual child, while shuffled controls are negative. All target anchor matrices have full empirical rank. These observations are deliberately recorded because the final failure shows that none is a distributional certificate.
16836 
16837 \subsection{Distributional transport theory}
16838 
16839 Let $(Y,H_s,H_t)\sim\mu$ be the protected semantics, source trace, and target trace under context distribution $\mu$. Fix a source probe $P$ and bridge $B$. Define
16840 \begin{equation}
16841  e_{\mathrm{src}}=Y-PH_s,
16842  \qquad
16843  e_{\mathrm{align}}=P(H_s-BH_t).
16844  \label{eq:b18-error-parts}
16845 \end{equation}
16846 Then
16847 \begin{equation}
16848  Y-PBH_t=e_{\mathrm{src}}+e_{\mathrm{align}}.
16849  \label{eq:b18-error-decomposition}
16850 \end{equation}
16851 
16852 \begin{proposition}[Witness--alignment decomposition]
16853 \label{prop:b18-witness-alignment}
16854 For every square-integrable $(Y,H_s,H_t)$,
16855 \begin{align}
16856  \mathbb E_{\mu}\norm{Y-PBH_t}^2
16857  &=
16858  \mathbb E_{\mu}\norm{e_{\mathrm{src}}}^2
16859  +\mathbb E_{\mu}\norm{e_{\mathrm{align}}}^2
16860  +2\mathbb E_{\mu}\angles{e_{\mathrm{src}},e_{\mathrm{align}}},
16861  \label{eq:b18-squared-decomposition}\\
16862  \norm{Y-PBH_t}_{L^2(\mu)}
16863  &\leq
16864  \norm{Y-PH_s}_{L^2(\mu)}
16865  +\norm{P(H_s-BH_t)}_{L^2(\mu)}.
16866  \label{eq:b18-triangle-decomposition}
16867 \end{align}
16868 \end{proposition}
16869 
16870 \begin{proof}
16871 Expand the square in \cref{eq:b18-error-decomposition} to obtain \cref{eq:b18-squared-decomposition}. The $L^2$ triangle inequality gives \cref{eq:b18-triangle-decomposition}.
16872 \end{proof}
16873 
16874 The decomposition prevents one common interpretive error. Poor transported performance need not mean that target and source traces cannot be aligned. The source witness may already be invalid on the deployment distribution.
16875 
16876 \begin{theorem}[Exact-alignment witness ceiling]
16877 \label{thm:b18-witness-ceiling}
16878 Suppose $BH_t=H_s$ almost surely under $\mu$. Then
16879 \begin{equation}
16880  P B H_t=P H_s
16881  \quad\text{almost surely},
16882  \label{eq:b18-exact-alignment-equality}
16883 \end{equation}
16884 and every loss or $R^2$ score computed only from $(Y,PBH_t)$ equals the corresponding source-reference score computed from $(Y,PH_s)$. Consequently, if a frozen transport gate requires source-probe $R^2\geq\tau$ but the source-reference $R^2$ is below $\tau$, no exact semantics-preserving bridge can pass that gate under the frozen witness.
16885 \end{theorem}
16886 
16887 \begin{proof}
16888 Substitution of $BH_t=H_s$ gives \cref{eq:b18-exact-alignment-equality}. Equality of every downstream deterministic score follows immediately.
16889 \end{proof}
16890 
16891 \begin{theorem}[In-distribution bridge success does not imply foreign success]
16892 \label{thm:b18-nonimplication}
16893 For every finite dimension $d\geq1$ and every $K>0$, there exist two context distributions $\mu_A,\mu_B$, full-rank source and target trace covariances under both distributions, and fixed linear maps $P,B$ such that semantic prediction and transport are exact under $\mu_A$ while transported squared error under $\mu_B$ exceeds $K$.
16894 \end{theorem}
16895 
16896 \begin{proof}
16897 Let $Y\in\R^d$ have nonsingular covariance. Under $\mu_A$, set $H_s=H_t=Y$, $P=B=I$. Prediction is exact. Under $\mu_B$, retain the same full-rank law for $Y$ but set $H_s=H_t=-cY$ for $c>0$, while freezing $P=B=I$. The trace covariances remain full rank, but
16898 \begin{equation}
16899  \mathbb E\norm{Y-PBH_t}^2
16900  =(1+c)^2\mathbb E\norm{Y}^2.
16901 \end{equation}
16902 Choosing $c$ sufficiently large makes the error exceed $K$. Thus rank and exact $A$-distribution transport do not constrain fixed-map performance after an unrestricted semantic-coordinate shift.
16903 \end{proof}
16904 
16905 The theorem is intentionally elementary. Its role is to identify the missing assumption: a cross-distribution claim requires stability of the semantic chart, not only rank and paired alignment inside one chart.
16906 
16907 \begin{definition}[Distributionally stable Neuralese transport]
16908 \label{def:b18-stable-transport}
16909 Let $\mathfrak M$ be a declared family of context distributions. A probe--bridge pair $(P,B)$ is $(\eps,\delta)$-stable on $\mathfrak M$ when, for every $\mu\in\mathfrak M$ outside an exceptional set of declared probability at most $\delta$,
16910 \begin{equation}
16911  \norm{Y-PBH_t}_{L^2(\mu)}\leq\eps,
16912  \label{eq:b18-stable-transport}
16913 \end{equation}
16914 and the corresponding source-reference, alignment, protected-regression, and abstention gates all pass. Stability is a family-level property and cannot be certified from one anchor distribution alone.
16915 \end{definition}
16916 
16917 \subsection{Sealed final results}
16918 
16919 The canonical final audits are:
16920 \begin{table}[H]
16921 \centering
16922 \small
16923 \resizebox{\textwidth}{!}{%
16924 \begin{tabular}{@{}llrrrrrrrl@{}}
16925 \toprule
16926 Architecture & Split & Clean & Program cost & $Q_{\rm prog}$ & $Q_{\rm ans}$ & Probe $R^2$ & Halt $\rho$ & Execution & Status \\
16927 \midrule
16928 GRU & Primary & $0.061597$ & $0.086859$ & $0.277610$ & $0.042063$ & $0.727398$ & $0.716422$ & $0.488571$ & FULL \\
16929 GRU & Foreign & $0.245178$ & $0.052317$ & $0.522192$ & $-0.131449$ & $-0.221987$ & $-0.217862$ & $0.089286$ & FAIL \\
16930 Residual & Primary & $0.061681$ & $0.122911$ & $0.422661$ & $0.194826$ & $0.726760$ & $0.704096$ & $0.516429$ & FULL \\
16931 Residual & Foreign & $0.278526$ & $0.037921$ & $0.288550$ & $0.048495$ & $-0.124416$ & $0.116803$ & $0.077143$ & FAIL \\
16932 \bottomrule
16933 \end{tabular}%
16934 }
16935 \end{table}
16936 
16937 The selected children strongly improve over their task-blind parents on the primary audit. GRU child-parent program specificity is $0.086859$ with frozen interval $[0.081437,0.092530]$; the residual effect is $0.122911$ with interval $[0.115792,0.129601]$. Both exact execution rates exceed the frozen $0.30$ floor, both semantic probes exceed $0.72$, and both adaptive-depth correlations exceed $0.70$.
16938 
16939 On generator $B$, same-answer program transplant cost remains positive relative to the parents: $0.052317$ for GRU with interval $[0.042619,0.061954]$, and $0.037921$ for the residual child with interval $[0.029599,0.045582]$. This isolated statistic is insufficient. GRU answer specificity becomes negative, both external probes become negative, both adaptive-depth correlations fall below threshold, exact execution falls below $0.09$, and clean loss is worse than each task-blind parent. Therefore neither child receives FULL status under shift.
16940 
16941 \begin{figure}[H]
16942 \centering
16943 \includegraphics[width=0.88\textwidth]{block18_foreign_audits.pdf}
16944 \caption{Selected-child program transplant cost and protected loss. Both architectures show positive program transplant cost under generator $B$, but the joint certificate fails because external semantic readout, exact execution, adaptive depth, and clean nonregression do not survive.}
16945 \label{fig:b18-foreign-audits}
16946 \end{figure}
16947 
16948 \begin{figure}[H]
16949 \centering
16950 \includegraphics[width=0.88\textwidth]{block18_execution_halting.pdf}
16951 \caption{External execution and adaptive depth. Primary performance passes for both foreign architectures. Under the shifted generator, execution approaches the task-blind baseline and depth ordering collapses.}
16952 \label{fig:b18-execution-halting}
16953 \end{figure}
16954 
16955 The new constructor also fails a stronger comparison with the inherited additive repair. Pooled over the two architectures and two final supports, the selected-child minus legacy-additive program-specificity effect is
16956 \begin{equation}
16957  \widehat\Delta_{\mathrm{FiLM-add}}
16958  =-0.029417,
16959  \qquad
16960  \mathrm{CI}_{0.95}
16961  =[-0.034102,-0.024735].
16962  \label{eq:b18-legacy-comparison}
16963 \end{equation}
16964 The negative interval forbids a claim that the new source-selected constructor dominates the inherited constructor. It can still be structurally new and useful on the primary audits, but novelty is not superiority.
16965 
16966 The final transport results are:
16967 \begin{longtable}{@{}lrrrrl@{}}
16968 \toprule
16969 Target & Paired $R^2$ & Shuffled $R^2$ & Unaligned $R^2$ & Source reference $R^2$ & Gate \\
16970 \midrule
16971 GRU & $-0.145315$ & $-0.175002$ & $-12.812014$ & $-0.098663$ & FAIL \\
16972 Residual & $-0.206846$ & $-0.105233$ & $-59.598778$ & $-0.098663$ & FAIL \\
16973 \bottomrule
16974 \end{longtable}
16975 
16976 \begin{figure}[H]
16977 \centering
16978 \includegraphics[width=0.82\textwidth]{block18_transport.pdf}
16979 \caption{Frozen source-probe transport on the shifted support. Paired bridges are far better than unaligned coordinates, but all transported and source-reference scores are below zero. Full anchor rank and pre-audit bridge success are therefore not sufficient.}
16980 \label{fig:b18-transport}
16981 \end{figure}
16982 
16983 The target anchor matrices have empirical rank $36$. Their condition numbers are $77.45$ for GRU and $115.16$ for the residual child. The failure is not rank deficiency. More importantly, the source-reference $R^2=-0.098663$ is itself below the transport threshold. By \cref{thm:b18-witness-ceiling}, an exact trace alignment could not rescue the frozen source-probe gate on this support.
16984 
16985 The canonical family receipt is
16986 \begin{verbatim}
16987 564220638342e64f351da9a440aad69bb043d56d156ed7c2e5a99b659f39eb04
16988 \end{verbatim}
16989 and its accepted field is false.
16990 
16991 \subsection{Post-hoc localization without retroactive promotion}
16992 
16993 After the family receipt was frozen, the separate $1{,}200$-task transport support was ordered by task identifier and alternated into two $600$-task halves. One half was used to fit shift-local probes and bridges; the other was used only for this diagnosis. The analysis is explicitly marked
16994 \begin{center}
16995 \texttt{POSTHOC\_DIAGNOSTIC\_ONLY\_NOT\_ADMISSIBLE\_FOR\_BLOCK18\_ACCEPTANCE}.
16996 \end{center}
16997 Its results are:
16998 \begin{longtable}{@{}lr@{}}
16999 \toprule
17000 Readout or bridge & Held-out shifted $R^2$ \\
17001 \midrule
17002 Frozen $A$-calibrated source probe & $-0.083074$ \\
17003 Shift-local source probe & $0.574370$ \\
17004 Shift-local GRU target probe & $0.600602$ \\
17005 Shift-local residual target probe & $0.581845$ \\
17006 Shift-local GRU paired bridge & $0.554518$ \\
17007 Shift-local residual paired bridge & $0.539005$ \\
17008 Shift-local GRU shuffled bridge & $-0.096005$ \\
17009 Shift-local residual shuffled bridge & $-0.040169$ \\
17010 \bottomrule
17011 \end{longtable}
17012 
17013 \begin{figure}[H]
17014 \centering
17015 \includegraphics[width=0.91\textwidth]{block18_posthoc_localization.pdf}
17016 \caption{Post-hoc localization. Shift-local probes and paired bridges recover substantial semantic signal on a held-out half of generator $B$, while shuffled bridges remain negative. This does not promote the Block 18 claim; it identifies a distribution-conditional semantic-chart obstruction.}
17017 \label{fig:b18-posthoc}
17018 \end{figure}
17019 
17020 The diagnostic separates three hypotheses. First, the latent states are not semantically empty under generator $B$, because shift-local source and target probes achieve positive held-out $R^2$. Second, target-to-source semantic alignment is not impossible, because shift-local paired bridges exceed $0.53$ while shuffled bridges remain negative. Third, one fixed chart is not stable, because the frozen source readout and frozen bridges fail on the same support. The next repair target is therefore an atlas or invariance mechanism, not merely a higher-rank static bridge.
17021 
17022 \begin{definition}[Distributional semantic-chart obstruction certificate]
17023 \label{def:b18-obstruction-certificate}
17024 A Block 18 distributional semantic-chart obstruction certificate consists of:
17025 \begin{enumerate}
17026 \item a frozen probe and bridge that pass the declared pre-audit support;
17027 \item a sealed foreign support on which the source-reference and transported witness gates fail;
17028 \item full-rank or otherwise sufficient local identification evidence excluding trivial anchor-rank failure;
17029 \item no retroactive threshold or model change;
17030 \item an optional separately labeled post-hoc split showing whether local semantic recoverability persists.
17031 \end{enumerate}
17032 The certificate proves failure of the frozen chart on the declared shift. It does not prove absence of every alternative chart.
17033 \end{definition}
17034 
17035 \begin{conditionaltheorem}[Block 18 sealed distributional obstruction]
17036 \label{thm:b18-obstruction}
17037 Assume:
17038 \begin{enumerate}
17039 \item all $15{,}300$ released tasks pass \cref{prop:b18-environment-soundness};
17040 \item the source schema and both foreign checkpoint families are sealed before final-task generation;
17041 \item the same typed selected schema is instantiated in both foreign architectures without architecture-specific candidate search;
17042 \item all model, donor, probe, bridge, clean-loss, execution, and adaptive-depth gates retain their frozen meanings;
17043 \item both selected children are FULL on their primary audits but FAIL on their generator-foreign audits;
17044 \item both frozen paired bridges fail the shifted transport gate and the frozen source-reference probe also fails there;
17045 \item the pooled comparison in \cref{eq:b18-legacy-comparison} is retained without claiming dominance;
17046 \item the post-hoc local-recovery experiment is excluded from canonical acceptance;
17047 \item the independent verifier replays the task, checkpoint, seal, audit, family, and diagnostic receipts.
17048 \end{enumerate}
17049 Then Block 18 establishes a finite distributional semantic-chart obstruction for the frozen source probe, static affine bridge class, and selected repair schema on the declared generator shift. It establishes neither a positive cross-implementation family edge nor impossibility of all context-indexed or invariant semantic transports.
17050 \end{conditionaltheorem}
17051 
17052 \begin{proof}
17053 Items 1 through 4 establish exact task semantics, lineage chronology, cross-implementation schema identity, and fixed evaluation meaning. Item 5 shows that architecture-level instantiation succeeds in distribution but does not satisfy the stronger foreign claim. Item 6 localizes the transport failure beyond simple target rank: the source witness itself is invalid under the shifted context, so \cref{thm:b18-witness-ceiling} blocks promotion through exact alignment alone. Item 7 prevents structural novelty from being misreported as empirical superiority. Item 8 preserves the separation between canonical audit and diagnosis. Item 9 provides package-local receipt replay. The conclusion is therefore a valid obstruction certificate relative to the fixed semantic chart and bridge family, with no claim about unrestricted transport classes.
17054 \end{proof}
17055 
17056 \subsection{Reproducibility and claim boundary}
17057 
17058 The canonical package verification reports
17059 \begin{verbatim}
17060 .........                                                                [100%]
17061 SRIC_BLOCK18_INDEPENDENT_VERIFICATION=PASS
17062 environment_tasks=15300
17063 sealed_claim=FAIL_REPRODUCED
17064 posthoc_localization=PASS
17065 SRIC_BLOCK18_PACKAGE_VALID
17066 canonical_status=SEALED_FAILURE_REPRODUCED
17067 \end{verbatim}
17068 The independent verification receipt is
17069 \begin{verbatim}
17070 352c4b69f8d88e7d086e9c8e7ac203693d1b476933bf129c90611f852c865e7c
17071 \end{verbatim}
17072 
17073 Block 18 establishes:
17074 \begin{itemize}
17075 \item one exact multi-template program-induction environment with external execution;
17076 \item one source compiler selecting a new step-specific FiLM repair schema;
17077 \item instantiation of that schema in two independently implemented recurrent parent families;
17078 \item FULL primary causal certificates for both selected children;
17079 \item positive same-answer program-transplant effects relative to both task-blind parents;
17080 \item failed generator-foreign causal and execution certificates for both selected children;
17081 \item failed frozen semantic bridges despite full-rank anchor traces and strong pre-audit bridge scores;
17082 \item a negative pooled comparison against the inherited additive repair;
17083 \item post-hoc evidence that local probes and bridges recover semantics under the shift;
17084 \item an independently replayed, package-closed failure and obstruction receipt.
17085 \end{itemize}
17086 
17087 It does not establish:
17088 \begin{itemize}
17089 \item a positive foreign-model or cross-implementation recursive-improvement edge;
17090 \item distributionally invariant semantics for the selected Neuralese coordinates;
17091 \item superiority of progressive FiLM over additive conditioning;
17092 \item stable transfer to arbitrary architectures, generators, tasks, or context supports;
17093 \item autonomous invention of the semantic atlas, repair schema, bridge class, witness, or thresholds;
17094 \item an empirical result for Hyper or another production latent-reasoning model;
17095 \item natural-language, theorem-proving, or unrestricted program-synthesis transfer;
17096 \item external independent replication;
17097 \item unrestricted safety, awareness, consciousness, sentience, open-ended recursive self-improvement, or indefinite acceleration.
17098 \end{itemize}
17099 
17100 The next major threshold is \emph{distribution-indexed Neuralese atlas growth}. A descendant should detect when the frozen semantic chart is unsupported, abstain rather than extrapolate, and conservatively add a new chart, invariant witness, or transport rule using only pre-audit contexts. The resulting atlas must reconstruct every inherited chart, preserve historical witness outputs where they remain valid, bind new contexts without final-support refitting, and pass a fresh audit containing unseen template mixtures and feature-support changes. Block 18 shows that another fixed full-rank bridge is not the correct next move.
17101 
17102 
17103 
17104 \section{Distribution-Indexed Neuralese Semantic Atlas Growth}
17105 \label{sec:block19-distribution-atlas}
17106 
17107 Block 18 established a cross-implementation obstruction rather than a positive transport edge. Neuralese semantics remained recoverable after local recalibration, but a source-calibrated semantic witness and two frozen architecture bridges failed under a generator shift. The logically prior defect was therefore source-side: one latent coordinate chart was being treated as globally valid even after the observation context left its support. Block 19 addresses that defect before attempting another architecture transfer.
17108 
17109 The block replaces one global observation chart by a finite, selectively admitted semantic atlas. It also replaces fixed-slot processing of demonstrations by a permutation-invariant set encoder and separates the semantic recurrent path from a context-diagnostic path. Two new charts are compiled from pre-audit semantic twins, appended conservatively to four inherited charts, and frozen before fresh final-task generation. Contexts outside the declared affine chart grammar are rejected rather than forced through the nearest chart.
17110 
17111 \begin{statusbox}{Block 19 status}
17112 The released implementation contains $9{,}460$ exact program records, four inherited affine observation charts, two newly compiled affine charts, two deliberately unsupported nonlinear contexts, one exact common neural ancestor, four matched-update descendants, candidate-specific semantic probes, semantic-twin and causal audits, selective routing, shuffled-binding and no-abstention controls, an independent NumPy forward implementation, ten automated tests, and an exact package manifest. The pre-audit compiler selects the \texttt{full\_bundle} descendant. It is \texttt{ATLAS\_FULL} on both fresh inherited-context and novel-chart audits, admits every supported final task, rejects every unsupported nonlinear task, and preserves every inherited chart exactly. The result is a finite source-side atlas-growth edge, not a cross-architecture or production-model result.
17113 \end{statusbox}
17114 
17115 \subsection{Semantic bundles, twins, and local charts}
17116 
17117 Let $Z$ denote a protected semantic task space and let $\Ccal$ be a family of observation contexts. For each $c\in\Ccal$, an observation rendering is a map
17118 \begin{equation}
17119  \rho_c:Z\longrightarrow X_c,
17120  \label{eq:b19-rendering}
17121 \end{equation}
17122 where $X_c$ is the context-dependent observation space. The same semantic task may therefore appear as different finite demonstration and query records in different fibers $X_c$.
17123 
17124 \begin{definition}[Semantic twins]
17125 \label{def:b19-semantic-twins}
17126 Two observed records $x_c\in X_c$ and $x_d\in X_d$ are semantic twins when there exists one protected semantic object $z\in Z$ such that
17127 \begin{equation}
17128  x_c=\rho_c(z),
17129  \qquad
17130  x_d=\rho_d(z).
17131 \end{equation}
17132 A semantic-twin group is an orbit $\{\rho_c(z):c\in C_z\}$ for a declared finite context set $C_z\subseteq\Ccal$.
17133 \end{definition}
17134 
17135 A chart is not a semantic decoder. It removes a declared observation gauge so that one semantic computation can be reused.
17136 
17137 \begin{definition}[Affine observation chart]
17138 \label{def:b19-affine-chart}
17139 For a context $c$, an affine observation chart is a tuple
17140 \begin{equation}
17141  \chi_c=(s_x,t_x,s_y,t_y,\sigma_c),
17142  \qquad s_xs_y\neq0,
17143 \end{equation}
17144 with canonicalization
17145 \begin{equation}
17146  \widehat x=\frac{x^{\mathrm{obs}}-t_x}{s_x},
17147  \qquad
17148  \widehat y=\frac{y^{\mathrm{obs}}-t_y}{s_y},
17149  \label{eq:b19-chart-inverse}
17150 \end{equation}
17151 and support signature $\sigma_c\in\R^5$. The signature is the sorted vector of observed demonstration inputs predicted by the chart on the five canonical demonstration locations.
17152 \end{definition}
17153 
17154 \begin{definition}[Distribution-indexed atlas]
17155 \label{def:b19-atlas}
17156 A finite distribution-indexed atlas is
17157 \begin{equation}
17158  \Acal
17159  =
17160  \left(
17161    \{\chi_c:c\in C\},
17162    d_{\mathrm{sig}},
17163    \tau_{\mathrm{adm}},
17164    \pi
17165  \right),
17166  \label{eq:b19-atlas}
17167 \end{equation}
17168 where $d_{\mathrm{sig}}$ is a normalized signature distance, $\tau_{\mathrm{adm}}$ is an admission threshold, and $\pi$ chooses the nearest chart. An input is admitted only when
17169 \begin{equation}
17170  \min_{c\in C}d_{\mathrm{sig}}(\sigma(x),\sigma_c)
17171  \leq\tau_{\mathrm{adm}}.
17172  \label{eq:b19-admission}
17173 \end{equation}
17174 Otherwise the atlas returns an unsupported-chart certificate and abstains.
17175 \end{definition}
17176 
17177 The implementation uses
17178 \begin{equation}
17179  d_{\mathrm{sig}}(u,v)
17180  =
17181  \frac{\|u-v\|_2/\sqrt5}
17182       {\max\{\operatorname{sd}(v),10^{-8}\}}.
17183  \label{eq:b19-signature-distance}
17184 \end{equation}
17185 The runtime receives only the observed support and values. Context names are retained in the data solely for auditing route accuracy.
17186 
17187 \subsection{Exact chart recovery and conservative growth}
17188 
17189 Suppose a source context $0$ and a target context $c$ satisfy
17190 \begin{equation}
17191  x_c=s_xx_0+t_x,
17192  \qquad
17193  y_c=s_yy_0+t_y,
17194  \label{eq:b19-affine-rendering}
17195 \end{equation}
17196 for all paired semantic-twin demonstrations.
17197 
17198 \begin{theorem}[Exact semantic-twin chart recovery]
17199 \label{thm:b19-exact-chart-recovery}
17200 Assume the paired source inputs contain at least two distinct values and the paired source outputs contain at least two distinct values. In the noiseless affine model \cref{eq:b19-affine-rendering}, ordinary least squares on semantic-twin pairs uniquely recovers $(s_x,t_x,s_y,t_y)$, and \cref{eq:b19-chart-inverse} maps every target demonstration and query back to its source coordinates.
17201 \end{theorem}
17202 
17203 \begin{proof}
17204 The two design matrices with columns $(x_0,1)$ and $(y_0,1)$ have rank two by the distinct-value assumptions. The affine parameters are therefore the unique zero-residual least-squares solutions. Substitution into \cref{eq:b19-chart-inverse} gives the source coordinates exactly.
17205 \end{proof}
17206 
17207 \begin{proposition}[Nearest-signature route stability]
17208 \label{prop:b19-route-stability}
17209 Let the minimum pairwise normalized signature separation of an atlas be
17210 \begin{equation}
17211  \Delta_{\Acal}
17212  =
17213  \min_{c\neq d}d_{\mathrm{sig}}(\sigma_c,\sigma_d).
17214 \end{equation}
17215 If an observed signature from chart $c$ has normalized perturbation at most $\delta$ and $2\delta<\Delta_{\Acal}$, nearest-signature routing returns $c$. If additionally $\delta\leq\tau_{\mathrm{adm}}$, the route is admitted.
17216 \end{proposition}
17217 
17218 \begin{proof}
17219 The triangle inequality gives distance at least $\Delta_{\Acal}-\delta>\delta$ to every competing signature. Thus $c$ is uniquely nearest. The second statement is the admission rule.
17220 \end{proof}
17221 
17222 \begin{definition}[Conservative atlas extension]
17223 \label{def:b19-conservative-extension}
17224 An atlas $\Acal_1$ conservatively extends $\Acal_0$ when every chart of $\Acal_0$ occurs as an identical prefix of $\Acal_1$, the route and canonicalized coordinates agree on every inherited support, and the inherited decision path remains executable as an exact fallback.
17225 \end{definition}
17226 
17227 \begin{proposition}[Exact inherited replay]
17228 \label{prop:b19-exact-inherited-replay}
17229 Appending new charts without changing any inherited chart parameters, signatures, route tie-breaking, or admission threshold yields an exact conservative extension on every input whose nearest inherited chart remains uniquely separated from the appended signatures.
17230 \end{proposition}
17231 
17232 \begin{proof}
17233 For such an input the chosen chart is unchanged. Since that chart's inverse map and all downstream inherited operations are unchanged, the normalized record and inherited execution are identical.
17234 \end{proof}
17235 
17236 The released four-chart parent atlas is
17237 \begin{equation}
17238  \Acal_0=\{A0,A1,A2,A3\},
17239 \end{equation}
17240 and the expanded atlas is
17241 \begin{equation}
17242  \Acal_1=\{A0,A1,A2,A3,N0,N1\}.
17243  \label{eq:b19-atlas-edge}
17244 \end{equation}
17245 The fitted parameters are:
17246 
17247 \begin{longtable}{@{}lrrrrrr@{}}
17248 \toprule
17249 Chart & $s_x$ & $t_x$ & $s_y$ & $t_y$ & input RMSE & output RMSE \\
17250 \midrule
17251 \endfirsthead
17252 \toprule
17253 Chart & $s_x$ & $t_x$ & $s_y$ & $t_y$ & input RMSE & output RMSE \\
17254 \midrule
17255 \endhead
17256 $A0$ & $1.000$ & $0.000$ & $1.000$ & $0.000$ & $0$ & $0$ \\
17257 $A1$ & $2.000$ & $1.000$ & $0.500$ & $4.000$ & $2.35\times10^{-15}$ & $2.09\times10^{-15}$ \\
17258 $A2$ & $0.500$ & $-3.000$ & $-1.000$ & $2.000$ & $2.74\times10^{-15}$ & $1.64\times10^{-15}$ \\
17259 $A3$ & $1.500$ & $5.000$ & $2.000$ & $-5.000$ & $2.95\times10^{-15}$ & $3.70\times10^{-15}$ \\
17260 $N0$ & $3.000$ & $-6.000$ & $-0.750$ & $-3.000$ & $5.89\times10^{-15}$ & $1.68\times10^{-15}$ \\
17261 $N1$ & $0.800$ & $7.000$ & $1.750$ & $6.000$ & $1.16\times10^{-14}$ & $2.20\times10^{-15}$ \\
17262 \bottomrule
17263 \end{longtable}
17264 
17265 The $N0$ and $N1$ rows were compiled from $80$ A0-referenced semantic-twin groups each. Declared environment coefficients are used only by the independent verifier, not by the compiler.
17266 
17267 \begin{figure}[H]
17268 \centering
17269 \includegraphics[width=0.84\textwidth]{block19_atlas_geometry.pdf}
17270 \caption{Input-scale and input-translation coordinates of the inherited and compiled observation charts. The two added charts occupy regions absent from the inherited atlas but are compiled before final-task generation.}
17271 \label{fig:b19-atlas-geometry}
17272 \end{figure}
17273 
17274 \subsection{Why nonlinear contexts require abstention}
17275 
17276 The unsupported contexts use a nonzero quadratic input warp
17277 \begin{equation}
17278  x^{\mathrm{obs}}
17279  =s_xx+t_x+\omega x^2,
17280  \qquad \omega\neq0,
17281  \label{eq:b19-unsupported-warp}
17282 \end{equation}
17283 and a nonlinear signed square-root output perturbation.
17284 
17285 \begin{proposition}[Affine grammar obstruction]
17286 \label{prop:b19-affine-obstruction}
17287 No affine inverse chart can exactly canonicalize \cref{eq:b19-unsupported-warp} on three or more distinct canonical inputs when $\omega\neq0$.
17288 \end{proposition}
17289 
17290 \begin{proof}
17291 An affine function has zero second divided difference. The map in \cref{eq:b19-unsupported-warp} has constant nonzero second derivative $2\omega$, hence a nonzero second divided difference on any three distinct points. It cannot coincide with an affine map on all three.
17292 \end{proof}
17293 
17294 This obstruction motivates a selective claim. The desired behavior on $U0$ and $U1$ is rejection, not forced extrapolation. A positive result must therefore satisfy both a supported-coverage floor and an unsupported false-admission ceiling; otherwise always-abstain and always-admit policies would be misleading positive controls.
17295 
17296 \subsection{Set-structured Neuralese semantic bundle}
17297 
17298 Each task contains five demonstration pairs
17299 \begin{equation}
17300  D=\{(x_i,y_i)\}_{i=1}^{5}
17301 \end{equation}
17302 and a query $q$. After chart canonicalization, the full descendant computes pair codes
17303 \begin{equation}
17304  p_i=\phi(\widehat x_i,\widehat y_i)\in\R^{32},
17305 \end{equation}
17306 and the permutation-invariant aggregate
17307 \begin{equation}
17308  a(D)
17309  =
17310  \left(
17311    \frac15\sum_{i=1}^{5}p_i,
17312    \max_{1\leq i\leq5}p_i
17313  \right)\in\R^{64}.
17314  \label{eq:b19-set-aggregate}
17315 \end{equation}
17316 The semantic seed is
17317 \begin{equation}
17318  h_0=S(a(D),Q(\widehat q))\in\R^{40}.
17319 \end{equation}
17320 A five-step Elman recurrence generates the Neuralese trace
17321 \begin{equation}
17322  r_k=\tanh\left(W_ih_0+b_i+U_ir_{k-1}+c_i+e_k\right),
17323  \qquad k=1,\ldots,5,
17324  \label{eq:b19-recurrence}
17325 \end{equation}
17326 where $e_k$ is a learned step embedding. A gated readout recurrence maps the trace to protected program and answer coordinates. A halt head predicts one of the five execution depths.
17327 
17328 The context code is computed separately:
17329 \begin{equation}
17330  u_c=C(s_x,t_x,s_y,t_y,d_{\mathrm{sig}})\in\R^{10}.
17331  \label{eq:b19-context-code}
17332 \end{equation}
17333 It supports chart diagnostics but has no computational path into $h_0$, $r_k$, the task output, or the halt head.
17334 
17335 \begin{definition}[Block 19 semantic bundle]
17336 The finite semantic bundle is
17337 \begin{equation}
17338  \Bcal_{19}
17339  =
17340  \left(
17341    \Acal_1,
17342    a,
17343    S,
17344    F,
17345    J,
17346    H,
17347    C,
17348    \mathsf{Adm}
17349  \right),
17350 \end{equation}
17351 where $\Acal_1$ is the expanded atlas, $a$ is the set aggregator, $F$ is the recurrent Neuralese transition, $J$ is the protected task readout, $H$ is the halt readout, $C$ is the context branch, and $\mathsf{Adm}$ is the selective chart gate.
17352 \end{definition}
17353 
17354 \begin{theorem}[Semantic-twin invariance]
17355 \label{thm:b19-twin-invariance}
17356 Assume two semantic twins are admitted by correct charts whose inverse maps recover identical canonical demonstrations and query. If the semantic encoder uses the symmetric aggregate \cref{eq:b19-set-aggregate} and the context branch has no path into the semantic seed, trace, task readout, or halt readout, then the twins have identical semantic seeds, recurrent traces, task outputs, and halt logits in exact arithmetic, regardless of demonstration ordering.
17357 \end{theorem}
17358 
17359 \begin{proof}
17360 Correct canonicalization gives the same unordered set of canonical demonstration pairs and the same canonical query. Mean and coordinatewise maximum are symmetric, so \cref{eq:b19-set-aggregate} is unchanged by permutation. Every later semantic operation is deterministic and receives identical inputs. The context branch is disconnected from those operations, hence cannot alter them.
17361 \end{proof}
17362 
17363 The theorem is stronger than a contrastive similarity objective. The invariance is enforced by the architecture and audited numerically. The released semantic-twin audit finds maximum final-trace cosine drift $2.384\times10^{-7}$, maximum output drift $1.431\times10^{-6}$, and zero halt-depth disagreements across $200$ held-out programs rendered in all six supported contexts.
17364 
17365 \subsection{Exact program environment and split discipline}
17366 
17367 Block 19 retains the exact four-template program family introduced in Block 18:
17368 \begin{align}
17369  f_{\mathrm{aff}}(x)&=ax+b,\\
17370  f_{\mathrm{abs}}(x)&=a|x|+b,\\
17371  f_{\mathrm{quad}}(x)&=ax^2+bx+c,\\
17372  f_{\mathrm{hinge}}(x)&=
17373  \begin{cases}
17374  ax+c,&x\geq0,\\
17375  bx+c,&x<0.
17376  \end{cases}
17377  \label{eq:b19-program-family}
17378 \end{align}
17379 Coefficients lie in declared finite integer sets. The protected semantic target consists of a one-hot template code, normalized coefficient slots, normalized program depth, and the exact executed query answer. An environment verifier reconstructs every record independently of the neural code.
17380 
17381 \begin{proposition}[Block 19 environment soundness]
17382 \label{prop:b19-environment-soundness}
17383 Every released record accepted by the independent environment verifier has demonstrations, query answer, template code, coefficient coordinates, depth, and protected target equal to those obtained from exact execution of \cref{eq:b19-program-family} followed by its declared context rendering.
17384 \end{proposition}
17385 
17386 \begin{proof}
17387 The verifier executes the program using rational arithmetic, reconstructs the affine or nonlinear context transform, matches all five shuffled demonstrations to the canonical support, rebuilds every semantic coordinate, and compares the serialized values within the declared floating tolerance.
17388 \end{proof}
17389 
17390 The released split ledger is:
17391 \begin{longtable}{@{}lrrp{0.37\textwidth}@{}}
17392 \toprule
17393 Support & Base programs & Records & Role \\
17394 \midrule
17395 \endfirsthead
17396 \toprule
17397 Support & Base programs & Records & Role \\
17398 \midrule
17399 \endhead
17400 Inherited chart calibration & $80$ & $320$ & Compile $A1,A2,A3$ relative to $A0$ \\
17401 Novel chart calibration & $80$ & $240$ & Compile $N0,N1$ relative to $A0$ \\
17402 Training & $600$ & $2{,}400$ & Common parent and matched continuation \\
17403 Development & $300$ & $1{,}200$ & Candidate diagnostics \\
17404 Meta-validation & $500$ & $500$ & Candidate selection on $N0,N1$ \\
17405 Probe calibration & $300$ & $1{,}200$ & External semantic readouts \\
17406 Twin validation & $200$ & $1{,}200$ & Six-context semantic flatness \\
17407 Primary audit & $900$ & $900$ & Fresh tasks in $A0$--$A3$ \\
17408 Novel-chart audit & $900$ & $900$ & Fresh tasks in $N0,N1$ \\
17409 Unsupported audit & $600$ & $600$ & Selective rejection in $U0,U1$ \\
17410 \bottomrule
17411 \end{longtable}
17412 
17413 The $7{,}060$ pre-audit records, chart grammar, candidate grammar, thresholds, update budget, and final split sizes are fixed before the $2{,}400$ final records are generated. The final record identifiers and outcomes are not available during candidate selection.
17414 
17415 \subsection{Finite candidate compiler}
17416 
17417 One exact common ancestor is trained for $250$ updates. Every descendant begins from its complete state dictionary and receives exactly $550$ continuation updates on the same training records. Both the fixed-slot and set encoders are present in the common state dictionary, so no candidate receives a separately initialized private module after branching.
17418 
17419 The finite grammar is:
17420 \begin{center}
17421 \small
17422 \begin{tabularx}{\textwidth}{@{}>{\raggedright\arraybackslash}X>{\raggedright\arraybackslash}X>{\raggedright\arraybackslash}X>{\raggedright\arraybackslash}X@{}}
17423 \toprule
17424 Candidate & Demonstration encoder & Observation coordinates & Twin regularization \\
17425 \midrule
17426 \texttt{recalibrated\_parent} & fixed slots & raw observed & none \\
17427 \texttt{set\_only} & symmetric set & raw observed & weak \\
17428 \texttt{chart\_only} & fixed slots & atlas normalized & weak \\
17429 \texttt{full\_bundle} & symmetric set & atlas normalized & strong \\
17430 \bottomrule
17431 \end{tabularx}
17432 \end{center}
17433 
17434 Candidate eligibility requires clean-loss, recurrence-necessity, order, norm-matched-noise, same-answer program-donor, positive-fraction, external semantic-probe, adaptive-depth, exact-replay, execution, supported-coverage, and semantic-twin gates. Selection maximizes the weakest normalized margin rather than an unconstrained average.
17435 
17436 \begin{table}[H]
17437 \centering
17438 \small
17439 \resizebox{\textwidth}{!}{%
17440 \begin{tabular}{@{}lrrrrrr@{}}
17441 \toprule
17442 Candidate & Eligible & Clean loss & Probe $R^2$ & Program cost & Halt $\rho$ & Twin cosine max \\
17443 \midrule
17444 \texttt{recalibrated\_parent} & no & $0.412628$ & $-1.259981$ & $0.012267$ & $0.003038$ & $1.419$ \\
17445 \texttt{set\_only} & no & $0.354799$ & $-7.012995$ & $0.010211$ & $0.039057$ & $1.924$ \\
17446 \texttt{chart\_only} & no & $0.183585$ & $0.189870$ & $0.014564$ & $0.100325$ & $0.5885$ \\
17447 \texttt{full\_bundle} & yes & $0.052641$ & $0.772549$ & $0.176423$ & $0.635266$ & $2.384\times10^{-7}$ \\
17448 \bottomrule
17449 \end{tabular}%
17450 }
17451 \end{table}
17452 
17453 \begin{figure}[H]
17454 \centering
17455 \includegraphics[width=0.94\textwidth]{block19_candidate_compiler.pdf}
17456 \caption{Pre-audit candidate comparison. Neither a set encoder nor atlas normalization alone satisfies the semantic and causal gates. The coupled full bundle is the only eligible candidate.}
17457 \label{fig:b19-candidate-compiler}
17458 \end{figure}
17459 
17460 The selected checkpoint, candidate-specific probe, parent and expanded atlases, acceptance thresholds, and final support plan are committed in pre-audit seal
17461 \begin{verbatim}
17462 c4f4cffd8b990b1c3f65fa4c9d448daeb74b4907cd47804e10936d8265c80401
17463 \end{verbatim}
17464 before final-task generation.
17465 
17466 \subsection{Fresh final results}
17467 
17468 The principal final results are:
17469 \begin{longtable}{@{}llrrrrr@{}}
17470 \toprule
17471 Model & Audit & Clean & Program cost & Probe $R^2$ & Halt $\rho$ & Execution \\
17472 \midrule
17473 \endfirsthead
17474 \toprule
17475 Model & Audit & Clean & Program cost & Probe $R^2$ & Halt $\rho$ & Execution \\
17476 \midrule
17477 \endhead
17478 Parent & inherited & $0.200177$ & $0.019129$ & $0.144462$ & $0.015517$ & $0.043333$ \\
17479 Parent & novel charts & $0.416361$ & $0.023291$ & $-1.282177$ & $-0.011614$ & $0.021111$ \\
17480 Full child & inherited & $\mathbf{0.057546}$ & $\mathbf{0.174135}$ & $\mathbf{0.770701}$ & $\mathbf{0.630918}$ & $\mathbf{0.607778}$ \\
17481 Full child & novel charts & $\mathbf{0.050745}$ & $\mathbf{0.194936}$ & $\mathbf{0.779341}$ & $\mathbf{0.666244}$ & $\mathbf{0.594444}$ \\
17482 \bottomrule
17483 \end{longtable}
17484 
17485 The parent is \texttt{ATLAS\_PARTIAL} on both audits. The selected child is \texttt{ATLAS\_FULL} on both. Its recurrent trajectory is useful, order-sensitive, vulnerable to structured replacement, program-specific under same-answer donors, externally semantically decodable, adaptively halted, and behaviorally executable.
17486 
17487 Paired effects are:
17488 \begin{align}
17489  \widehat\Delta_{\mathrm{clean}}^{A}
17490  &=0.142631,
17491  &95\%\ \mathrm{CI}&=[0.135421,0.149531],\\
17492  \widehat\Delta_{\mathrm{clean}}^{N}
17493  &=0.365616,
17494  &95\%\ \mathrm{CI}&=[0.352991,0.378402],\\
17495  \widehat\Delta_{\mathrm{program}}^{A}
17496  &=0.120614,
17497  &95\%\ \mathrm{CI}&=[0.110353,0.131880],\\
17498  \widehat\Delta_{\mathrm{program}}^{N}
17499  &=0.125290,
17500  &95\%\ \mathrm{CI}&=[0.111056,0.138921].
17501  \label{eq:b19-paired-effects}
17502 \end{align}
17503 The clean effects are parent minus child; the program effects are child minus parent. Every interval is frozen and strictly positive.
17504 
17505 \begin{figure}[H]
17506 \centering
17507 \includegraphics[width=0.91\textwidth]{block19_final_losses.pdf}
17508 \caption{Clean loss on fresh inherited and novel-chart audits. The full bundle substantially improves both supports, while neither isolated structural component closes the novel-chart gap.}
17509 \label{fig:b19-final-losses}
17510 \end{figure}
17511 
17512 \begin{figure}[H]
17513 \centering
17514 \includegraphics[width=0.91\textwidth]{block19_selected_semantics.pdf}
17515 \caption{Selected-child semantic and causal measurements. External probe quality and same-answer program specificity remain strong in both supported context families.}
17516 \label{fig:b19-selected-semantics}
17517 \end{figure}
17518 
17519 \subsection{Selective admission and mechanism controls}
17520 
17521 The expanded atlas admits all $900$ primary tasks and all $900$ novel-chart tasks. It admits none of the $600$ nonlinear unsupported tasks:
17522 \begin{equation}
17523  \widehat{\operatorname{coverage}}_{A}=1,
17524  \qquad
17525  \widehat{\operatorname{coverage}}_{N}=1,
17526  \qquad
17527  \widehat{\operatorname{false\ admission}}_{U}=0.
17528  \label{eq:b19-selectivity}
17529 \end{equation}
17530 Forcing the unsupported tasks through the nearest chart produces clean loss $0.703667$, semantic-probe $R^2=-17.424958$, execution accuracy $0.056667$, and \texttt{ATLAS\_PARTIAL} status. Thus abstention protects against a demonstrably harmful extrapolation. The always-abstain control has zero useful supported coverage and cannot pass.
17531 
17532 \begin{figure}[H]
17533 \centering
17534 \includegraphics[width=0.91\textwidth]{block19_selective_admission.pdf}
17535 \caption{Selective admission. Supported affine charts are fully covered, unsupported nonlinear contexts are rejected, and forcing those contexts through the nearest chart fails behaviorally and semantically.}
17536 \label{fig:b19-selective-admission}
17537 \end{figure}
17538 
17539 The main mechanism controls are:
17540 \begin{itemize}
17541 \item \textbf{Set only.} The symmetric encoder improves inherited-context behavior but is \texttt{ATLAS\_PARTIAL} on novel charts, with clean loss $0.360916$ and probe $R^2=-7.114963$.
17542 \item \textbf{Chart only.} Canonicalization without permutation-invariant processing is \texttt{ATLAS\_PARTIAL} on novel charts, with clean loss $0.186142$ and probe $R^2=0.188481$.
17543 \item \textbf{Shuffled chart binding.} Correct route labels bound to incorrect inverse maps yield clean loss $0.483296$ and probe $R^2=-8.928579$.
17544 \item \textbf{No abstention.} Unsupported nonlinear contexts forced through the closest affine chart fail as reported above.
17545 \item \textbf{Conservative replay.} The expanded atlas changes inherited normalized demonstrations and queries by exactly zero in the serialized audit.
17546 \end{itemize}
17547 
17548 The supported empirical mechanism is therefore the coupled object
17549 \begin{equation}
17550  \boxed{
17551  \begin{gathered}
17552  \text{semantic-twin chart compilation}
17553  +\text{ permutation-invariant set encoding}\\
17554  +\text{ semantic/context path separation}
17555  +\text{ calibrated admission}\\
17556  +\text{ conservative inherited fallback}.
17557  \end{gathered}}
17558  \label{eq:b19-coupled-mechanism}
17559 \end{equation}
17560 
17561 \subsection{Finite realization theorem}
17562 
17563 \begin{conditionaltheorem}[Block 19 distribution-indexed atlas growth]
17564 \label{thm:b19-realization}
17565 Within the frozen Block 19 environment, assume:
17566 \begin{enumerate}
17567 \item all $9{,}460$ released records satisfy \cref{prop:b19-environment-soundness};
17568 \item the inherited and new charts are compiled only from the declared pre-audit semantic-twin supports;
17569 \item all four candidates inherit the exact ancestor state and receive the same $550$-update continuation envelope;
17570 \item the candidate grammar, thresholds, probes, atlases, selected checkpoint, and final split plan are sealed before final-task generation;
17571 \item the expanded atlas satisfies \cref{def:b19-conservative-extension};
17572 \item the full child is \texttt{ATLAS\_FULL} on both supported final audits and the paired effects in \cref{eq:b19-paired-effects} retain their positive intervals;
17573 \item semantic-twin trace, output, and halting discrepancies satisfy their frozen tolerances;
17574 \item the admission results in \cref{eq:b19-selectivity} hold;
17575 \item set-only, chart-only, shuffled-binding, no-abstention, and always-abstain controls retain their declared interpretations;
17576 \item the automated tests, independent environment replay, independent chart reconstruction, independent NumPy forward pass, receipts, and package manifest verify.
17577 \end{enumerate}
17578 Then the edge
17579 \begin{equation}
17580  K_{18}
17581  \xrightarrow{\Pi_{19}}
17582  K_{19}
17583  \end{equation}
17584 is a finite, conservative, source-side Neuralese semantic-atlas expansion. It resolves two observation contexts absent from the inherited atlas while preserving inherited charts and issuing calibrated rejection outside the authorized affine grammar.
17585 \end{conditionaltheorem}
17586 
17587 \begin{proof}
17588 Items 1 through 4 establish exact semantics, matched lineage ancestry, and audit chronology. Item 5 establishes conservative inclusion rather than replacement of the old atlas. Items 6 and 7 establish task, causal, semantic, and halting competence in both inherited and added charts. Item 8 prevents unsupported extrapolation from being counted as improvement. Item 9 localizes the gain to the coupled mechanism in \cref{eq:b19-coupled-mechanism}; neither principal component alone passes, incorrect binding fails, and trivial abstention has no supported utility. Item 10 gives package-level replay. The accepted child therefore adds two valid chart actions unavailable to the inherited process and improves fresh tasks without reinterpreting inherited coordinates.
17589 \end{proof}
17590 
17591 \subsection{Reproducibility and independent verification}
17592 
17593 The independent verifier does not call the package's task verifier, atlas compiler, audit routine, or receipt helper. It:
17594 \begin{enumerate}
17595 \item executes all four program templates using a separate exact implementation;
17596 \item reconstructs all eight context transformations and verifies all $9{,}460$ records;
17597 \item refits all five nonidentity affine charts from serialized twins;
17598 \item recomputes routing and selective admission on all final records;
17599 \item checks checkpoint file and state-dictionary hashes;
17600 \item recomputes the candidate, seal, twin, audit, and family receipts;
17601 \item recomputes headline row statistics;
17602 \item evaluates the selected checkpoint with an independently written NumPy implementation of the set encoder, Elman recurrence, gated latent readout, output head, and halt head.
17603 \end{enumerate}
17604 The NumPy and PyTorch implementations agree to maximum absolute error
17605 \begin{equation}
17606  1.405\times10^{-6}.
17607 \end{equation}
17608 The independent verification receipt is
17609 \begin{verbatim}
17610 767d71abb0b33d789ff058836ae18eae961fadc5f9b905932cf405f32728dcc0
17611 \end{verbatim}
17612 and the canonical family receipt is
17613 \begin{verbatim}
17614 9e5e81b1d03cca5396debf81d3667da0f9264353052a01dce6a270bb562cad08
17615 \end{verbatim}
17616 
17617 \subsection{Claim boundary and next threshold}
17618 
17619 Block 19 establishes:
17620 \begin{itemize}
17621 \item exact semantic-twin recovery of five nonidentity affine charts;
17622 \item conservative growth from four inherited charts to a six-chart atlas;
17623 \item one set-structured recurrent Neuralese child with an explicit semantic/context split;
17624 \item exact permutation invariance and numerical six-context semantic-twin flatness;
17625 \item fresh inherited-context and added-chart \texttt{ATLAS\_FULL} certificates;
17626 \item positive clean and same-answer program-specificity intervals;
17627 \item full supported coverage and zero observed unsupported admission;
17628 \item controls isolating the coupled atlas, set, binding, and abstention mechanism;
17629 \item independent exact-task, chart, forward, receipt, and package verification.
17630 \end{itemize}
17631 
17632 It does not establish:
17633 \begin{itemize}
17634 \item autonomous invention of arbitrary nonlinear charts or support signatures;
17635 \item semantic transport between independently trained or independently implemented architectures;
17636 \item validity outside the six admitted affine contexts;
17637 \item natural-language, theorem-proving, unrestricted program-synthesis, or open-tool transfer;
17638 \item an empirical result for Cymela Hyper or another production latent-reasoning model;
17639 \item autonomous evaluator, objective, witness, or safety evolution;
17640 \item external independent replication;
17641 \item unrestricted safety, awareness, consciousness, sentience, open-ended recursive self-improvement, or indefinite acceleration.
17642 \end{itemize}
17643 
17644 The next threshold is \emph{atlas-indexed cross-architecture Neuralese transport}. Source and target systems should each carry locally valid semantic charts, transport should be tested on chart overlaps, and architecture transport should approximately commute with context transition:
17645 \begin{equation}
17646  C^b_{c\to d}T^c_{a\to b}
17647  \approx
17648  T^d_{a\to b}C^a_{c\to d}.
17649  \label{eq:b19-transport-square}
17650 \end{equation}
17651 Block 20 tests this threshold directly in two foreign recurrent architectures. It preserves the inherited six-chart atlas, audits the curvature residual of \cref{eq:b19-transport-square}, and retains fail-closed rejection outside the authorized chart grammar.
17652 
17653 
17654 \section{Atlas-Indexed Cross-Architecture Neuralese Transport}
17655 \label{sec:block20}
17656 
17657 Block 19 repaired the source-side semantic-chart failure exposed by Block 18. It replaced one globally assumed observation chart by a six-chart atlas, preserved the inherited charts exactly, compiled two additional affine charts from semantic twins, separated semantic and context pathways, and rejected nonlinear contexts outside the authorized chart grammar. That result left the next question unresolved:
17658 \begin{equation}
17659 \boxed{
17660 \begin{gathered}
17661 \text{Can a frozen semantic atlas support transport of complete}\\
17662 \text{Neuralese trajectories between independently trained recurrent}\\
17663 \text{architectures without collapsing chart identity?}
17664 \end{gathered}}
17665 \label{eq:b20-question}
17666 \end{equation}
17667 
17668 A positive result requires more than pointwise latent alignment. Architecture transport and context transport must remain mutually compatible. Block 20 therefore treats Neuralese as a finite bundle indexed jointly by recurrent architecture and observation chart, equips that bundle with architecture and context maps, and evaluates the resulting transport square before accepting a descendant.
17669 
17670 \begin{statusbox}{Block 20 canonical disposition}
17671 The released finite compiler selects a six-chart, full-five-step trajectory transport from trained GRU and gated-residual targets into the frozen Block 19 Elman source representation. The selected transport is sealed before final task generation, passes fresh inherited-chart and added-chart audits in both architectures, preserves source semantic-probe performance, and keeps normalized architecture--context curvature below the frozen $0.02$ ceiling. Pooled, step-shared, shuffled-binding, and forced-unsupported controls fail in their declared dimensions. The canonical result is \textbf{PASS}. The architecture-chart gauges are controlled synthetic orthogonal stresses; this is not a production-model or universal-alignment result.
17672 \end{statusbox}
17673 
17674 \subsection{Architecture--context semantic bundle}
17675 \label{subsec:b20-bundle}
17676 
17677 Let $\mathcal A$ be a finite architecture index set and $\mathcal C$ a finite admitted chart index set. In the released realization,
17678 \begin{equation}
17679 \mathcal A
17680 =
17681 \{\mathrm{Elman},\mathrm{GRU},\mathrm{Residual}\},
17682 \qquad
17683 \mathcal C
17684 =
17685 \{A0,A1,A2,A3,N0,N1\}.
17686 \end{equation}
17687 Each pair $(a,c)\in\mathcal A\times\mathcal C$ carries a full Neuralese trajectory fiber
17688 \begin{equation}
17689 H_{a,c}
17690 \cong
17691 \left(\mathbb R^{40}\right)^5
17692 \cong
17693 \mathbb R^{200}.
17694 \end{equation}
17695 The observation atlas determines which context fiber is authorized. A task outside every admitted chart is not assigned to an arbitrary fiber.
17696 
17697 \begin{definition}[Finite Neuralese semantic bundle]
17698 A finite architecture--context Neuralese bundle is a tuple
17699 \begin{equation}
17700 \mathfrak E
17701 =
17702 \left(
17703 \mathcal A,
17704 \mathcal C,
17705 \{H_{a,c}\}_{a,c},
17706 \{J_{a,c}\}_{a,c},
17707 \mathcal R
17708 \right),
17709 \end{equation}
17710 where $H_{a,c}$ is a latent trajectory space, $J_{a,c}$ is an externally calibrated semantic readout, and $\mathcal R$ is a frozen chart-admission rule. No global identification among the fibers is assumed.
17711 \end{definition}
17712 
17713 For architectures $a,b$ and admitted chart $c$, an architecture map is
17714 \begin{equation}
17715 T^{c}_{b\to a}:H_{b,c}\longrightarrow H_{a,c}.
17716 \end{equation}
17717 For one architecture $a$ and two admitted charts $c,d$, a context map is
17718 \begin{equation}
17719 C^{a}_{c\to d}:H_{a,c}\longrightarrow H_{a,d}.
17720 \end{equation}
17721 Block 20 uses $A0$ as a reference chart and fits a star of maps $C^{a}_{c\to A0}$ and $C^{a}_{A0\to c}$ from protected semantic twins.
17722 
17723 \begin{definition}[Architecture--context curvature]
17724 For architectures $a,b$ and charts $c,d$, the transport-square defect is
17725 \begin{equation}
17726 \Omega_{a,b;c,d}
17727 :=
17728 C^{a}_{c\to d}T^{c}_{b\to a}
17729 -
17730 T^{d}_{b\to a}C^{b}_{c\to d}.
17731 \label{eq:b20-curvature}
17732 \end{equation}
17733 Given an audit distribution $P_{b,c}$ on target trajectories, its normalized root-mean-square curvature is
17734 \begin{equation}
17735 \kappa_{a,b;c,d}^{2}
17736 =
17737 \frac{
17738 \mathbb E_{h\sim P_{b,c}}
17739 \norm{\Omega_{a,b;c,d}h}^{2}
17740 }{
17741 \mathbb E_{z\sim P_{a,d}}\norm{z}^{2}
17742 }.
17743 \label{eq:b20-normalized-curvature}
17744 \end{equation}
17745 \end{definition}
17746 
17747 The requirement $\kappa\ll1$ is stronger than high pointwise probe performance on one support. It asks whether two independently constructed routes through the bundle yield compatible source representations.
17748 
17749 \begin{figure}[H]
17750 \centering
17751 \includegraphics[width=0.90\textwidth]{block20_transport_square.pdf}
17752 \caption{The Block 20 transport square. The top path applies a chart-specific architecture map and then source-side context transport. The left-bottom path first moves the target trajectory into the target reference chart and then uses the reference architecture map. Their normalized discrepancy is the architecture--context curvature.}
17753 \label{fig:b20-square}
17754 \end{figure}
17755 
17756 \subsection{Why one pooled bridge is not generally sufficient}
17757 \label{subsec:b20-global-obstruction}
17758 
17759 Suppose an underlying canonical trajectory is represented in chart $c$ by a row vector $z$ and architecture-dependent gauge matrices:
17760 \begin{equation}
17761 h_{a,c}=zG_{a,c},
17762 \qquad
17763 h_{b,c}=zG_{b,c}.
17764 \label{eq:b20-gauge-model}
17765 \end{equation}
17766 The matrices may include coordinate rotations, sign changes, and architecture-specific temporal mixing. Block 20 uses deterministic orthogonal gauges as a controlled stress at the exposed trajectory interface.
17767 
17768 \begin{theorem}[Pooled bridge compatibility obstruction]
17769 \label{thm:b20-global-obstruction}
17770 Let $V\subseteq\mathbb R^{m}$ be spanned by the canonical trajectories observed in two charts $c_1,c_2$. Suppose one linear map $T$ exactly transports architecture $b$ into architecture $a$ in both charts:
17771 \begin{equation}
17772 zG_{b,c_i}T=zG_{a,c_i}
17773 \qquad
17774 \text{for every }z\in V,
17775 \quad i\in\{1,2\}.
17776 \end{equation}
17777 Then the restrictions to $V$ satisfy
17778 \begin{equation}
17779 G_{b,c_1}T=G_{a,c_1},
17780 \qquad
17781 G_{b,c_2}T=G_{a,c_2}.
17782 \end{equation}
17783 If $G_{b,c_i}$ are invertible on $V$, exact pooled transport is possible only when
17784 \begin{equation}
17785 G_{b,c_1}^{-1}G_{a,c_1}
17786 =
17787 G_{b,c_2}^{-1}G_{a,c_2}
17788 \quad\text{on }V.
17789 \label{eq:b20-compatibility}
17790 \end{equation}
17791 \end{theorem}
17792 
17793 \begin{proof}
17794 The spanning hypothesis converts equality of transported trajectories into equality of the corresponding linear operators on $V$. Left multiplication by the inverse of each target gauge gives the two required expressions for $T$. If those expressions disagree, no single $T$ can satisfy both chart equations.
17795 \end{proof}
17796 
17797 \begin{corollary}[Exact chart-indexed gauge transport]
17798 \label{cor:b20-chart-exact}
17799 Under the same hypotheses, the chart-indexed family
17800 \begin{equation}
17801 T^{c}_{b\to a}
17802 =
17803 G_{b,c}^{-1}G_{a,c}
17804 \end{equation}
17805 transports each chart exactly. If all gauges are orthogonal, then $G_{b,c}^{-1}=G_{b,c}^{\mathsf T}$.
17806 \end{corollary}
17807 
17808 The theorem does not state that natural neural representations are pure orthogonal gauges. It identifies a minimal finite obstruction: even benign invertible coordinate changes can make one pooled bridge impossible when the architecture relation varies with chart.
17809 
17810 \subsection{Full-trajectory transport and temporal structure}
17811 \label{subsec:b20-full-trajectory}
17812 
17813 A latent-step-shared map acts independently on each recurrent state:
17814 \begin{equation}
17815 \widehat h_{a,c}^{(k)}
17816 =
17817 A_c h_{b,c}^{(k)}+b_c,
17818 \qquad k=1,\ldots,5.
17819 \label{eq:b20-step-shared}
17820 \end{equation}
17821 A full-trajectory map instead acts on the concatenation
17822 \begin{equation}
17823 \operatorname{vec}(h_{b,c}^{(1)},\ldots,h_{b,c}^{(5)})
17824 \in\mathbb R^{200},
17825 \end{equation}
17826 allowing cross-step mixing:
17827 \begin{equation}
17828 \operatorname{vec}(\widehat h_{a,c})
17829 =
17830 M_c\operatorname{vec}(h_{b,c})+q_c.
17831 \label{eq:b20-full-map}
17832 \end{equation}
17833 This difference is substantive when two recurrent architectures distribute the same computation across latent time differently.
17834 
17835 \begin{definition}[Finite transport grammar]
17836 The Block 20 compiler searches the following declared grammar:
17837 \begin{enumerate}
17838 \item \texttt{global\_full}: one pooled affine map $\mathbb R^{200}\to\mathbb R^{200}$;
17839 \item \texttt{chart\_step\_shared}: six chart-indexed affine maps $\mathbb R^{40}\to\mathbb R^{40}$ applied independently at every latent step;
17840 \item \texttt{chart\_full}: six chart-indexed affine maps $\mathbb R^{200}\to\mathbb R^{200}$.
17841 \end{enumerate}
17842 \end{definition}
17843 
17844 The per-target parameter counts are $40{,}200$, $9{,}840$, and $241{,}200$, respectively. Consequently, Block 20 does not claim parameter-matched superiority over the pooled map. The same-capacity shuffled-binding control does establish that the selected capacity is not useful without the correct chart-to-map assignment.
17845 
17846 \begin{proposition}[Readout stability from trajectory reconstruction]
17847 \label{prop:b20-readout-bound}
17848 Let $J:H_{a,c}\to\mathbb R^{p}$ be $L_J$-Lipschitz. If a transported trajectory satisfies
17849 \begin{equation}
17850 \norm{T^c_{b\to a}h_{b,c}-h_{a,c}}
17851 \leq\eps,
17852 \end{equation}
17853 then
17854 \begin{equation}
17855 \norm{J(T^c_{b\to a}h_{b,c})-J(h_{a,c})}
17856 \leq L_J\eps.
17857 \end{equation}
17858 If two transport paths differ by at most $\delta$, their semantic readouts differ by at most $L_J\delta$.
17859 \end{proposition}
17860 
17861 \begin{proof}
17862 Both statements are direct applications of Lipschitz continuity.
17863 \end{proof}
17864 
17865 The proposition motivates the joint use of trace reconstruction, external semantic readout, and square curvature. None is treated as a substitute for the others.
17866 
17867 \subsection{Executable environment and frozen ancestry}
17868 \label{subsec:b20-environment}
17869 
17870 The source is the exact released Block 19 \texttt{full\_bundle} Elman checkpoint. Its six-chart atlas is replayed without changing any chart coefficient, admission threshold, normalized demonstration, or route. Two foreign implementations are trained separately:
17871 \begin{enumerate}
17872 \item a GRU recurrent latent cell;
17873 \item a gated-residual nonlinear latent cell.
17874 \end{enumerate}
17875 Each foreign parent uses the inherited task family but not the Block 19 semantic repair. Each child inherits its architecture-matched parent and receives the finite Block 19 repair: a permutation-invariant set encoder, atlas-normalized inputs, a separated context path, semantic-twin regularization, and the same five-step interface.
17876 
17877 \begin{table}[H]
17878 \centering
17879 \small
17880 \begin{tabular}{@{}lr@{}}
17881 \toprule
17882 Object & Released count \\
17883 \midrule
17884 Pre-audit exact tasks & $9{,}200$ \\
17885 Fresh final exact tasks & $3{,}820$ \\
17886 Total exact tasks & $13{,}020$ \\
17887 Bridge-calibration tasks & $1{,}920$ \\
17888 Transport meta-validation tasks & $900$ \\
17889 Final inherited-chart tasks & $1{,}000$ \\
17890 Final added-chart tasks & $1{,}000$ \\
17891 Final transport-square tasks & $1{,}320$ \\
17892 Final unsupported tasks & $500$ \\
17893 \bottomrule
17894 \end{tabular}
17895 \caption{Block 20 exact program environment. Every task is independently replayed by the release verifier.}
17896 \label{tab:b20-environment}
17897 \end{table}
17898 
17899 The semantic task family remains the four-template exact program environment used in Blocks 18 and 19: affine, absolute-value, quadratic, and hinge programs. The observation charts $A0$--$A3$ and $N0$--$N1$ are affine and admitted. The contexts $U0,U1$ contain nonlinear input and output warps and remain unsupported by the inherited affine atlas.
17900 
17901 \subsection{Controlled architecture-chart gauge stress}
17902 \label{subsec:b20-gauge-stress}
17903 
17904 For the frozen Elman source, the exposed trajectory gauge is the identity in every chart. For each foreign architecture and chart, the released package deterministically generates a distinct $40\times40$ orthogonal matrix. The same matrix acts on every latent step in that architecture-chart fiber before the transport compiler receives the trajectory. All $18$ matrices are serialized, hashed, and checked for orthogonality.
17905 
17906 This construction serves two purposes. First, it prevents accidental coordinate agreement from trivializing the cross-architecture problem. Second, it makes the need for chart-indexed transport independently checkable. It does not model the full complexity of natural representation drift, which may be nonlinear, rank-changing, stochastic, or history-dependent.
17907 
17908 \subsection{Compiler chronology and acceptance rule}
17909 \label{subsec:b20-chronology}
17910 
17911 The released chronology is:
17912 \begin{equation}
17913 \begin{aligned}
17914 &\text{freeze ancestor and atlas}
17915 \longrightarrow
17916 \text{declare architecture, gauge, grammar, and thresholds}\\
17917 &\text{generate and verify pre-audit supports}
17918 \longrightarrow
17919 \text{train foreign parents and atlas children}\\
17920 &\text{fit probes, transitions, and candidates}
17921 \longrightarrow
17922 \text{meta-audit candidates and transport squares}\\
17923 &\text{seal source, targets, maps, thresholds, and split plan}
17924 \longrightarrow
17925 \text{generate final supports}\\
17926 &\text{evaluate every frozen arm}
17927 \longrightarrow
17928 \text{commit or reject}.
17929 \end{aligned}
17930 \label{eq:b20-chronology}
17931 \end{equation}
17932 
17933 A candidate is pre-audit eligible only if both foreign target models are \texttt{ATLAS\_FULL} and the candidate jointly satisfies
17934 \begin{equation}
17935 \begin{aligned}
17936 R^2_{\mathrm{transport}}&\geq0.75,
17937 &\qquad
17938 R^2_{\mathrm{worst\ chart}}&\geq0.74,\\
17939 \rho_{\mathrm{preserve}}&\geq0.98,
17940 &
17941 \operatorname{NRMSE}_{\mathrm{trace}}&\leq0.05,\\
17942 \kappa_{\mathrm{square}}&\leq0.02.
17943 &&
17944 \end{aligned}
17945 \label{eq:b20-thresholds}
17946 \end{equation}
17947 The selection score is the weakest normalized margin across these gates. This prevents a strong aggregate $R^2$ from compensating for a failed chart, high curvature, or poor trajectory reconstruction.
17948 
17949 \subsection{Pre-audit compiler result}
17950 \label{subsec:b20-preaudit}
17951 
17952 \begin{table}[H]
17953 \centering
17954 \small
17955 \begin{tabular}{@{}llrrrr@{}}
17956 \toprule
17957 Candidate & Target & Transported $R^2$ & Worst chart & Trace NRMSE & Curvature \\
17958 \midrule
17959 \texttt{global\_full} & GRU & $0.722084$ & $0.705057$ & $0.142012$ & $0.130620$ \\
17960 \texttt{global\_full} & Residual & $0.732894$ & $0.687945$ & $0.124338$ & $0.114613$ \\
17961 \texttt{chart\_step\_shared} & GRU & $-3.173916$ & $-3.649021$ & $0.154024$ & $0.055261$ \\
17962 \texttt{chart\_step\_shared} & Residual & $-10.575678$ & $-11.273055$ & $0.241193$ & $0.099808$ \\
17963 \texttt{chart\_full} & GRU & \textbf{$0.781840$} & \textbf{$0.759963$} & \textbf{$0.034349$} & \textbf{$0.010458$} \\
17964 \texttt{chart\_full} & Residual & \textbf{$0.778861$} & \textbf{$0.752268$} & \textbf{$0.032050$} & \textbf{$0.008646$} \\
17965 \bottomrule
17966 \end{tabular}
17967 \caption{Pre-audit transport compiler. Only the chart-indexed full-trajectory family passes every gate in both foreign architectures.}
17968 \label{tab:b20-preaudit}
17969 \end{table}
17970 
17971 \begin{figure}[H]
17972 \centering
17973 \includegraphics[width=0.82\textwidth]{block20_candidate_compiler.pdf}
17974 \caption{Minimum transported semantic $R^2$ across foreign architectures during pre-audit selection. The chart-indexed full-trajectory candidate is the only member above the frozen threshold.}
17975 \label{fig:b20-candidate-r2}
17976 \end{figure}
17977 
17978 \begin{figure}[H]
17979 \centering
17980 \includegraphics[width=0.82\textwidth]{block20_candidate_curvature.pdf}
17981 \caption{Maximum pre-audit architecture--context curvature. Only the selected chart-indexed full-trajectory candidate lies below the frozen ceiling.}
17982 \label{fig:b20-candidate-curvature}
17983 \end{figure}
17984 
17985 The compiler selects \texttt{chart\_full}. The source checkpoint, target checkpoints, gauge registry, source and target probes, context transitions, all candidate maps, thresholds, and final generation plan are serialized into a pre-audit seal with
17986 \begin{equation}
17987 \texttt{audit\_accessed=false}.
17988 \end{equation}
17989 The seal digest is
17990 \begin{verbatim}
17991 f43fb174ae27c1a5b881292d3a96fec7108d859270e7b065034cb8be462c85d7
17992 \end{verbatim}
17993 
17994 \subsection{Fresh final semantic-transport results}
17995 \label{subsec:b20-final-results}
17996 
17997 All three trained models remain \texttt{ATLAS\_FULL} on the final inherited-chart and added-chart supports. The target-model probe values range from $0.770639$ to $0.809206$, and their supported coverage is one. The transport result is therefore not obtained by mapping trajectories from a behaviorally collapsed target.
17998 
17999 \begin{table}[H]
18000 \centering
18001 \small
18002 \setlength{\tabcolsep}{4.5pt}
18003 \begin{tabular}{@{}llrrrrrr@{}}
18004 \toprule
18005 Target & Audit & Source $R^2$ & Pooled $R^2$ & Selected $R^2$ & Preserve & NRMSE & Worst chart \\
18006 \midrule
18007 GRU & Inherited & $0.796107$ & $0.728571$ & \textbf{$0.799157$} & $1.003832$ & $0.034322$ & $0.760405$ \\
18008 GRU & Added charts & $0.769907$ & $0.719667$ & \textbf{$0.781159$} & $1.014615$ & $0.033225$ & $0.764416$ \\
18009 Residual & Inherited & $0.796107$ & $0.730951$ & \textbf{$0.796471$} & $1.000458$ & $0.033751$ & $0.760439$ \\
18010 Residual & Added charts & $0.769907$ & $0.737565$ & \textbf{$0.785881$} & $1.020749$ & $0.031392$ & $0.767514$ \\
18011 \bottomrule
18012 \end{tabular}
18013 \caption{Fresh final transport results. The selected atlas preserves or slightly exceeds the frozen source probe on every architecture--audit pair.}
18014 \label{tab:b20-final-transport}
18015 \end{table}
18016 
18017 \begin{figure}[H]
18018 \centering
18019 \includegraphics[width=0.90\textwidth]{block20_final_transport_r2.pdf}
18020 \caption{Fresh final semantic transport. The pooled bridge remains below the acceptance threshold, whereas the selected chart-indexed bridge tracks the source reference.}
18021 \label{fig:b20-final-r2}
18022 \end{figure}
18023 
18024 The selected transport does not merely average over strong and weak charts. Every final chart exceeds the frozen $0.74$ worst-chart threshold.
18025 
18026 \begin{figure}[H]
18027 \centering
18028 \includegraphics[width=0.88\textwidth]{block20_gru_context_r2.pdf}
18029 \caption{GRU chart-resolved transport. The selected maps improve every inherited and newly compiled chart over the pooled bridge.}
18030 \label{fig:b20-gru-context}
18031 \end{figure}
18032 
18033 \begin{figure}[H]
18034 \centering
18035 \includegraphics[width=0.88\textwidth]{block20_residual_context_r2.pdf}
18036 \caption{Gated-residual chart-resolved transport. The chart-indexed full-trajectory maps pass all six local semantic gates.}
18037 \label{fig:b20-residual-context}
18038 \end{figure}
18039 
18040 \subsection{Paired effects}
18041 \label{subsec:b20-paired}
18042 
18043 For every final task, let $\ell_{\mathrm{global}}$ and $\ell_{\mathrm{chart}}$ be the source-probe semantic losses after pooled and selected transport. The paired effect is
18044 \begin{equation}
18045 \Delta_i
18046 =
18047 \ell_{\mathrm{global},i}
18048 -
18049 \ell_{\mathrm{chart},i}.
18050 \end{equation}
18051 Positive values favor the selected chart-indexed map.
18052 
18053 \begin{table}[H]
18054 \centering
18055 \small
18056 \begin{tabular}{@{}llrrr@{}}
18057 \toprule
18058 Target & Audit & Mean effect & Frozen $95\%$ interval & Positive fraction \\
18059 \midrule
18060 GRU & Inherited & $0.017350$ & $[0.014942,0.019624]$ & $0.776$ \\
18061 GRU & Added charts & $0.015110$ & $[0.012389,0.017643]$ & $0.754$ \\
18062 Residual & Inherited & $0.016105$ & $[0.013344,0.018829]$ & $0.754$ \\
18063 Residual & Added charts & $0.011872$ & $[0.009607,0.014184]$ & $0.767$ \\
18064 \bottomrule
18065 \end{tabular}
18066 \caption{Paired pooled-minus-selected semantic-loss effects. All frozen intervals are strictly positive.}
18067 \label{tab:b20-paired}
18068 \end{table}
18069 
18070 \begin{figure}[H]
18071 \centering
18072 \includegraphics[width=0.86\textwidth]{block20_paired_semantic_improvement.pdf}
18073 \caption{Paired semantic-loss improvements with frozen bootstrap intervals.}
18074 \label{fig:b20-paired}
18075 \end{figure}
18076 
18077 \subsection{Architecture--context curvature result}
18078 \label{subsec:b20-curvature-result}
18079 
18080 \begin{table}[H]
18081 \centering
18082 \small
18083 \begin{tabular}{@{}lrrr@{}}
18084 \toprule
18085 Target & Pooled curvature & Selected curvature & Frozen ceiling \\
18086 \midrule
18087 GRU & $0.131402$ & \textbf{$0.010474$} & $0.020000$ \\
18088 Gated residual & $0.112397$ & \textbf{$0.008374$} & $0.020000$ \\
18089 \bottomrule
18090 \end{tabular}
18091 \caption{Fresh architecture--context commutation audit.}
18092 \label{tab:b20-curvature}
18093 \end{table}
18094 
18095 \begin{figure}[H]
18096 \centering
18097 \includegraphics[width=0.78\textwidth]{block20_square_curvature.pdf}
18098 \caption{The selected connection reduces normalized curvature by more than an order of magnitude relative to the pooled bridge in both foreign architectures.}
18099 \label{fig:b20-curvature}
18100 \end{figure}
18101 
18102 The maximum context-resolved square defects are $0.030101$ for GRU and $0.027476$ for the gated-residual target. Source and target context loops are also audited separately, preventing a low square defect from being credited to two mutually bad context maps.
18103 
18104 \subsection{Mechanism controls}
18105 \label{subsec:b20-controls}
18106 
18107 \paragraph{Pooled full-trajectory control.}
18108 The pooled map has enough capacity to mix all five latent steps, but its final $R^2$ ranges from $0.719667$ to $0.737565$, below the selected map and below the $0.75$ transport threshold on three of four final architecture--audit pairs. Its curvature is more than ten times the selected value. This localizes the failure to chart-varying architecture relations rather than an inability to fit any full-trajectory map.
18109 
18110 \paragraph{Chart-indexed step-shared control.}
18111 The step-shared maps are chart aware but cannot mix information across recurrent depth. They produce strongly negative semantic $R^2$ in both architectures. This supports full-trajectory rather than statewise alignment in the released recurrence family.
18112 
18113 \paragraph{Shuffled chart binding.}
18114 The selected six maps are retained byte for byte, but their chart assignments are cyclically permuted. Final semantic $R^2$ becomes at most $-29.41$ for GRU and at most $-120.10$ for the residual target. This is a same-capacity control showing that the correct chart-to-map binding is causal to the result.
18115 
18116 \paragraph{Unsupported contexts.}
18117 The inherited atlas rejects every $U0$ and $U1$ task, so the false-admission fraction is zero. If those tasks are forcibly assigned to the nearest admitted chart, transported semantic $R^2$ is $-2.259909$ for GRU and $-6.128030$ for the residual target. Fail-closed rejection therefore prevents a measured semantic failure.
18118 
18119 \paragraph{Unaligned target coordinates.}
18120 Applying the frozen source probe directly to foreign gauge coordinates produces extremely negative $R^2$. The accepted result is not explained by accidental coordinate identity.
18121 
18122 \subsection{Finite transport realization theorem}
18123 \label{subsec:b20-realization}
18124 
18125 \begin{conditionaltheorem}[Block 20 atlas-indexed cross-architecture transport]
18126 \label{thm:b20-realization}
18127 Within the frozen Block 20 environment, assume:
18128 \begin{enumerate}
18129 \item the canonical Block 19 receipt, source checkpoint, and six-chart atlas replay exactly;
18130 \item the GRU and gated-residual targets are trained independently and are \texttt{ATLAS\_FULL} on their declared pre-audit and final supports;
18131 \item all three transport candidates receive the same paired bridge-calibration records and the same source and target checkpoints;
18132 \item the architecture-chart gauge registry, probes, context transitions, candidate maps, thresholds, and final split plan are sealed before final outcomes;
18133 \item the selected candidate is \texttt{chart\_full} and every gate in \cref{eq:b20-thresholds} passes in both architectures;
18134 \item the fresh paired effects in \cref{tab:b20-paired} retain strictly positive frozen intervals;
18135 \item pooled, step-shared, shuffled-binding, unaligned, and forced-unsupported controls retain their declared interpretations;
18136 \item the independent environment, forward-pass, map, probe, receipt, metric, and curvature verification replays within its declared numerical tolerances.
18137 \end{enumerate}
18138 Then the accepted edge is a finite, evidence-preserving expansion from a source-side semantic atlas to an atlas-indexed cross-architecture Neuralese transport connection. It preserves the inherited atlas, transports complete five-step trajectories from two foreign recurrent architectures on fresh admitted supports, and rejects contexts outside the authorized chart grammar.
18139 \end{conditionaltheorem}
18140 
18141 \begin{proof}
18142 Items 1 and 2 establish lineage ancestry and noncollapsed source and target behavior. Item 3 gives a common candidate-comparison support. Item 4 isolates the final audits from candidate construction. Item 5 establishes semantic preservation, chartwise lower bounds, trajectory reconstruction, and low architecture--context curvature. Item 6 provides paired foreign evidence against the pooled control. Item 7 rules out the principal explanations based on one global bridge, chart labels without temporal transport, capacity without correct binding, accidental coordinate identity, or unsafe extrapolation outside the atlas. Item 8 establishes package-level replay. The accepted object therefore changes the transport connection rather than merely relabeling a fixed global alignment.
18143 \end{proof}
18144 
18145 \subsection{Reproducibility and independent verification}
18146 \label{subsec:b20-reproducibility}
18147 
18148 The release contains:
18149 \begin{itemize}
18150 \item all $13{,}020$ exact task records and both environment manifests;
18151 \item the frozen Block 19 atlas and source checkpoint;
18152 \item GRU and gated-residual parent and child checkpoints;
18153 \item source and target semantic probes;
18154 \item the complete gauge registry;
18155 \item six candidate transport archives and three context-transition archives;
18156 \item pre-audit compiler and seal receipts;
18157 \item every final model, transport, square, shuffled-binding, and unsupported control result;
18158 \item deterministic figures, automated tests, an independent verifier, and an exact package manifest.
18159 \end{itemize}
18160 
18161 Thirteen automated tests pass. The independent verifier imports none of the package atlas compiler, audit routine, probe evaluator, transport evaluator, or receipt helpers. It independently replays all exact programs, checks chart routing, reconstructs Elman, GRU, and residual recurrent trajectories in NumPy, reapplies gauges and affine maps, recomputes semantic $R^2$, trace NRMSE, worst-chart performance, and architecture--context curvature, and reconstructs the principal receipts. Its maximum NumPy-versus-PyTorch trajectory error is
18162 \begin{equation}
18163 1.343\times10^{-6},
18164 \end{equation}
18165 and its maximum difference from the canonical reported transport metrics is
18166 \begin{equation}
18167 1.779\times10^{-7}.
18168 \end{equation}
18169 The independent verification receipt is
18170 \begin{verbatim}
18171 1f2909fa973e1586b71e9cc85fceb0ecc2a0edc48598464a743f2ce1e3b172fe
18172 \end{verbatim}
18173 
18174 The canonical Block 20 family receipt is
18175 \begin{verbatim}
18176 ac245cea5eca29650f14671ef37eb3ca1c6d21b7407085c0364b32a41ff3d4a5
18177 \end{verbatim}
18178 
18179 \subsection{Claim boundary and next threshold}
18180 \label{subsec:b20-boundary}
18181 
18182 Block 20 establishes a finite atlas-indexed transport of full five-step Neuralese trajectories from two trained foreign recurrent architectures into one frozen Block 19 source representation under a declared six-chart orthogonal gauge stress. It does not establish:
18183 \begin{itemize}
18184 \item universal alignment among independently trained natural or production models;
18185 \item autonomous invention of the observation atlas, architecture family, gauge class, or transport grammar;
18186 \item transport under arbitrary nonlinear, noninvertible, stochastic, rank-changing, or time-varying representation changes;
18187 \item natural-language Neuralese semantics or a result for Cymela Hyper;
18188 \item parameter-matched superiority over every possible pooled nonlinear bridge;
18189 \item independent external replication;
18190 \item unrestricted safety, consciousness, sentience, or moral status;
18191 \item open-ended or indefinitely accelerating recursive self-improvement.
18192 \end{itemize}
18193 
18194 The next threshold is not another fixed chartwise linear map. A stronger descendant should detect residual transport curvature, decide whether it requires a new chart, a nonlinear local map, a temporal reparameterization, or an abstention certificate, and compile that extension from evidence generated outside the current six-chart orthogonal stress. A natural Block 21 target is \emph{obstruction-guided nonlinear transport-connection growth}: a sealed compiler that expands the local transport grammar only after a persistent, externally validated curvature or information-loss certificate.
18195 
18196 \clearpage
18197 \section{Obstruction-Guided Neuralese Transport-Grammar Growth}
18198 \label{sec:block21-transport-grammar}
18199 
18200 Block 20 established a finite chart-indexed connection between a frozen six-chart Elman Neuralese atlas and two independently trained foreign recurrent systems.  That result assumed that every admissible architecture--chart fiber could be repaired by one member of a supplied affine full-trajectory grammar.  The next question is structurally different:
18201 \begin{center}
18202 \fbox{\begin{minipage}{0.90\textwidth}
18203 \centering
18204 Can the lineage distinguish why an inherited Neuralese connection fails, then select the smallest authorized extension or issue a valid no-go certificate?
18205 \end{minipage}}
18206 \end{center}
18207 
18208 Block 21 answers this question inside a finite packet-level stress environment.  The accepted Block 20 connection remains frozen.  An eight-coordinate packet is extracted from each of five aligned recurrent steps, giving a forty-coordinate protected trajectory packet.  Hidden post-connection transformations then introduce six distinct failure families.  A label-isolated compiler receives paired packet examples, a protected semantic witness, and one bounded diagnostic probe.  It searches a declared repair grammar, admits only candidates passing frozen semantic and packet gates, and selects the lowest-description admissible constructor.  When identical observations possess different protected semantics, the compiler returns a collision-grounded no-go certificate rather than fitting a correlational guess.
18209 
18210 \begin{figure}[H]
18211 \centering
18212 \includegraphics[width=0.98\textwidth]{block21_compiler_flow.pdf}
18213 \caption{The Block 21 transaction.  Hidden family labels are unavailable to the compiler.  Repair type is inferred from held-out repair response and finite diagnostics, then scored by a separately sealed audit harness.}
18214 \label{fig:block21-flow}
18215 \end{figure}
18216 
18217 \subsection{Inherited connection and protected packet}
18218 
18219 Let $A$ denote the frozen source architecture and let $B$ be one of the two Block 20 foreign architectures.  For admitted observation chart $c$, Block 20 supplies an accepted full-trajectory connection
18220 \begin{equation}
18221 T^{c}_{B\to A}:H_{B,c}^{(1:5)}\longrightarrow H_{A,c}^{(1:5)}.
18222 \end{equation}
18223 Let $P:\R^{40}\to\R^8$ be an orthonormal packet projector fitted on a pre-audit source support.  Applying $P$ independently to the five recurrent states gives
18224 \begin{equation}
18225 \Pi(H)
18226 =
18227 \left(P h_1,\ldots,P h_5\right)
18228 \in
18229 \left(\R^8\right)^5
18230 \cong\R^{40}.
18231 \label{eq:block21-packet}
18232 \end{equation}
18233 The packet is not claimed to preserve every latent coordinate.  It is required to preserve the protected nine-coordinate semantic witness on the declared task family.  A frozen linear readout
18234 \begin{equation}
18235 J:\R^{40}\longrightarrow\R^9
18236 \end{equation}
18237 is fitted on a support disjoint from packet-projection fitting and from all final episodes.
18238 
18239 \begin{definition}[Post-connection transport episode]
18240 A Block 21 episode is a tuple
18241 \begin{equation}
18242 \mathfrak E
18243 =
18244 (X,Y,Z,p,\mathcal D_{\rm cal},\mathcal D_{\rm meta},\mathcal D_{\rm audit}),
18245 \end{equation}
18246 where $X\in\R^{5\times8}$ is the protected source packet, $Y$ is an observed post-connection packet or lower-dimensional statistic, $Z\in\R^9$ is the protected semantic witness, $p\in\R$ is a bounded diagnostic response, and the three supports are pairwise disjoint.  The compiler receives only these public fields together with an episode identifier and architecture identifier.  The private obstruction family and transform parameters are retained by the audit harness.
18247 \end{definition}
18248 
18249 \begin{remark}[Label isolation]
18250 The executable compiler input has no field containing the obstruction family, expected repair, transform parameters, or benchmark collision pairs.  The expected family-to-repair map is stored in a separate audit-only module.  Collision evidence is reconstructed directly from repeated observed rows and protected semantic disagreement.
18251 \end{remark}
18252 
18253 \subsection{Finite obstruction families}
18254 
18255 The released fixture contains six hidden families.  They are engineered to separate map capacity, latent-clock mismatch, chart insufficiency, benign compression, and destructive information loss.
18256 
18257 \subsubsection{Affine gauge drift}
18258 
18259 The ordinary control has
18260 \begin{equation}
18261 Y_t=X_t A+b,
18262 \qquad A\in\mathrm{GL}(8),
18263 \label{eq:block21-affine}
18264 \end{equation}
18265 with one shared map at every recurrent step.  The inherited step-affine constructor is sufficient.  This family tests conservative minimality: a full-trajectory affine map is available and usually eligible, but it should not be selected when the smaller inherited constructor already passes.
18266 
18267 \subsubsection{Smooth nonlinear warp}
18268 
18269 The nonlinear family applies
18270 \begin{equation}
18271 Y_t
18272 =
18273 \frac{\sinh\!\left(\alpha(X_tA+b)\right)}{\alpha},
18274 \qquad \alpha>0,
18275 \label{eq:block21-nonlinear}
18276 \end{equation}
18277 coordinatewise.  The declared inverse constructor first applies
18278 \begin{equation}
18279 \phi^{-1}_{\alpha}(y)
18280 =
18281 \frac{\operatorname{arsinh}(\alpha y)}{\alpha}
18282 \end{equation}
18283 and then fits a step-affine map.
18284 
18285 \begin{proposition}[Affine obstruction for the declared nonlinear family]
18286 \label{prop:block21-nonlinear-obstruction}
18287 Let $I\subset\R$ be an interval with nonempty interior and let $\alpha\neq0$.  No affine scalar map agrees with $u\mapsto\sinh(\alpha u)/\alpha$ on all of $I$.  Consequently, if one transformed packet coordinate varies over such an interval, no exact step-affine inverse exists for \cref{eq:block21-nonlinear}.
18288 \end{proposition}
18289 
18290 \begin{proof}
18291 An affine function has identically zero second derivative.  The second derivative of $\sinh(\alpha u)/\alpha$ is $\alpha\sinh(\alpha u)$, which is not identically zero on any interval with nonempty interior.  Exact affine agreement is therefore impossible.  Restricting the vector relation to a varying transformed coordinate gives the packet statement.
18292 \end{proof}
18293 
18294 \subsubsection{Latent-clock mixing}
18295 
18296 The latent-clock family first applies an invertible stepwise affine gauge and then mixes adjacent recurrent times:
18297 \begin{equation}
18298 Y_t
18299 =
18300 (1-\beta)U_t+\beta U_{t-1},
18301 \qquad t=2,\ldots,5,
18302 \label{eq:block21-clock}
18303 \end{equation}
18304 with $Y_1=U_1$ and $0<\beta<1$.  In matrix form,
18305 \begin{equation}
18306 Y=W_{\beta}U,
18307 \end{equation}
18308 where $W_{\beta}$ is lower bidiagonal.  The temporal constructor applies $W_{\beta}^{-1}$ before fitting the spatial gauge.
18309 
18310 \begin{proposition}[Fixed-step obstruction]
18311 \label{prop:block21-clock-obstruction}
18312 Suppose the packet support contains two trajectories with the same state at time $t$ but different states at time $t-1$.  For $\beta\neq0$, no deterministic map acting on $Y_t$ alone can recover $U_t$ for both trajectories under \cref{eq:block21-clock}.
18313 \end{proposition}
18314 
18315 \begin{proof}
18316 The observed $Y_t$ depends on both $U_t$ and $U_{t-1}$.  Holding $U_t$ fixed while changing $U_{t-1}$ changes $Y_t$.  Conversely, suitable paired changes can hold $Y_t$ fixed while changing $U_t$.  Hence $U_t$ is not a function of the single observed state $Y_t$ on the stated support.  The full monotone temporal inverse uses the trajectory context that a stepwise map discards.
18317 \end{proof}
18318 
18319 \subsubsection{Aliased observation charts}
18320 
18321 The chart-alias family contains two unannounced local affine relations,
18322 \begin{equation}
18323 Y_t=
18324 \begin{cases}
18325 X_tA_0+b_0,&q=0,\\
18326 X_tA_1+b_1,&q=1,
18327 \end{cases}
18328 \label{eq:block21-alias}
18329 \end{equation}
18330 where the inherited route token collapses both cells.  The bounded probe $p$ is noisy but separating.  The chart-split constructor selects a threshold using calibration and meta-validation data, then fits one affine map on each response cell.
18331 
18332 \begin{proposition}[Aliased-map obstruction]
18333 \label{prop:block21-alias-obstruction}
18334 Assume two cells share the same inherited token, their source supports contain a common open set, and the exact inverse affine maps on that set are distinct.  No deterministic token-only affine repair is exact on both cells.
18335 \end{proposition}
18336 
18337 \begin{proof}
18338 A token-only repair applies one affine map to both cells.  Exactness on the shared open set would force this map to equal both inverse affine maps.  Two affine maps agreeing on an open set are identical, contradicting the assumed distinction.
18339 \end{proof}
18340 
18341 \subsubsection{Semantics-preserving compression}
18342 
18343 The benign compression exposes a thirty-dimensional statistic containing all nine protected semantic coordinates and twenty-one nuisance projections, followed by an invertible mixing.  Since thirty is less than the forty-dimensional packet dimension, complete packet reconstruction is neither required nor generally possible.  The appropriate constructor is a semantic quotient
18344 \begin{equation}
18345 Y\xrightarrow{q}Z_{\rm prot},
18346 \label{eq:block21-quotient}
18347 \end{equation}
18348 not an artificial lift into all source packet coordinates.
18349 
18350 \begin{theorem}[Semantic quotient sufficiency]
18351 \label{thm:block21-quotient}
18352 Let $Z$ be the protected semantic variable and $Y$ an observed representation.  If there exists a measurable map $q$ such that $Z=q(Y)$ almost surely, then protected semantic transport exists even when no injective map from $Y$ into the complete source packet exists.
18353 \end{theorem}
18354 
18355 \begin{proof}
18356 The map $q$ itself transports every protected semantic readout.  Injective reconstruction of the source packet is a strictly stronger demand and is unnecessary whenever the authorization contract depends only on $Z$.
18357 \end{proof}
18358 
18359 \subsubsection{Destructive semantic collision}
18360 
18361 The destructive family withholds four protected coordinates and constructs pairs $i,j$ satisfying
18362 \begin{equation}
18363 Y_i=Y_j,
18364 \qquad
18365 Z_i\neq Z_j.
18366 \label{eq:block21-collision}
18367 \end{equation}
18368 The compiler must discover these repeated observations from the public arrays.  Their private construction pairs are not supplied.
18369 
18370 \begin{theorem}[Deterministic target-only no-go]
18371 \label{thm:block21-no-go}
18372 If \cref{eq:block21-collision} holds, then no deterministic target-only map $f$ can satisfy $f(Y_i)=Z_i$ and $f(Y_j)=Z_j$ simultaneously.
18373 \end{theorem}
18374 
18375 \begin{proof}
18376 Equality of the inputs implies $f(Y_i)=f(Y_j)$.  This contradicts $Z_i\neq Z_j$.
18377 \end{proof}
18378 
18379 \begin{remark}
18380 The theorem does not imply that the missing semantics can never be recovered.  Recovery may become possible after obtaining an additional observation, accessing an earlier layer, enlarging target memory, or modifying the architecture.  It rules out only deterministic repair from the declared observed statistic.
18381 \end{remark}
18382 
18383 \subsection{Repair grammar and compiler}
18384 
18385 The finite grammar is
18386 \begin{equation}
18387 \mathcal G_{21}
18388 =
18389 \left\{
18390 \begin{array}{ll}
18391 G_0&\text{step-affine map},\\
18392 G_1&\text{full-trajectory affine map},\\
18393 G_2&\text{monotone nonlinear inverse plus step-affine map},\\
18394 G_3&\text{bidiagonal temporal inverse plus step-affine map},\\
18395 G_4&\text{probe-conditioned two-chart affine map},\\
18396 G_5&\text{semantic quotient},\\
18397 G_{\bot}&\text{collision-grounded no-go}.
18398 \end{array}
18399 \right.
18400 \label{eq:block21-grammar}
18401 \end{equation}
18402 
18403 For reconstructive candidates, admissibility requires
18404 \begin{equation}
18405 R^2_{\rm sem}\geq0.68,
18406 \qquad
18407 R^2_{\rm packet}\geq0.72,
18408 \qquad
18409 \operatorname{NRMSE}_{\rm packet}\leq0.55.
18410 \label{eq:block21-reconstructive-gates}
18411 \end{equation}
18412 A quotient is admissible only when the observed dimension ratio is at most $0.82$, semantic $R^2$ is at least $0.68$, and no reconstructive candidate reaches packet $R^2=0.82$.  A no-go certificate is admissible only after the compiler discovers at least thirty identical-observation pairs whose mean protected semantic separation is at least $0.50$.
18413 
18414 Each repair $g$ receives a description-length proxy $L(g)$ recording parameter count and constructor surcharge.  The compiler selects
18415 \begin{equation}
18416 \widehat g
18417 \in
18418 \arg\min_{g\in\mathcal G_{21}}
18419 L(g)
18420 \quad\text{subject to the appropriate protected gates.}
18421 \label{eq:block21-mdl}
18422 \end{equation}
18423 If no repair passes and the collision certificate is valid, it returns $G_{\bot}$; otherwise it returns unresolved.
18424 
18425 \begin{proposition}[Conservative minimality]
18426 \label{prop:block21-minimality}
18427 If the inherited step-affine constructor passes every frozen gate and has strictly smaller description length than every other eligible candidate, the Block 21 compiler selects it.
18428 \end{proposition}
18429 
18430 \begin{proof}
18431 This is immediate from \cref{eq:block21-mdl}.  Structural novelty receives no credit merely for being more expressive.
18432 \end{proof}
18433 
18434 \begin{figure}[H]
18435 \centering
18436 \includegraphics[width=0.88\textwidth]{block21_minimality.pdf}
18437 \caption{Description-length minimality.  The inherited step-affine constructor is retained on ordinary affine drift, while more structured constructors are admitted only where the smaller grammar fails.}
18438 \label{fig:block21-minimality}
18439 \end{figure}
18440 
18441 \subsection{Sealed experimental chronology}
18442 
18443 The released chronology is:
18444 \begin{enumerate}
18445 \item replay the exact Block 20 source and target checkpoints, six-chart atlas, accepted transports, and canonical receipt;
18446 \item fit the orthonormal packet projection on $1{,}200$ pre-audit tasks;
18447 \item fit the protected semantic probe on a disjoint $1{,}600$-task support;
18448 \item generate a $5{,}200$-task pre-audit episode pool;
18449 \item evaluate two pre-audit episodes for each of two architectures and six hidden families;
18450 \item freeze the grammar, thresholds, audit-only expected map, compiler interface, and final split plan;
18451 \item commit the pre-audit seal;
18452 \item generate the $5{,}600$-task final pool only after sealing;
18453 \item compile three unseen transform instances for every architecture--family pair;
18454 \item audit the serialized repairs on disjoint final subsets and reveal private family labels only for scoring;
18455 \item replay all exact task semantics, compiler decisions, serialized repairs, and metrics.
18456 \end{enumerate}
18457 The four task files contain $13{,}600$ exact records in total.
18458 
18459 \subsection{Final obstruction-to-repair result}
18460 
18461 The pre-audit suite passes all $24$ episodes.  The final audit contains $36$ episodes: two foreign recurrent architectures, six hidden obstruction families, and three unseen transform instances per architecture--family pair.  The confusion matrix is exactly diagonal.
18462 
18463 \begin{figure}[H]
18464 \centering
18465 \includegraphics[width=0.90\textwidth]{block21_confusion_matrix.pdf}
18466 \caption{Final obstruction-to-repair matrix.  Family labels are unavailable during compilation and are revealed only to the audit harness.}
18467 \label{fig:block21-confusion}
18468 \end{figure}
18469 
18470 {\small
18471 \setlength{\tabcolsep}{3.5pt}
18472 \begin{longtable}{@{}lrrrr@{}}
18473 \caption{Fresh final performance of the selected repair, aggregated over three unseen transform instances.  Packet quantities do not apply to the semantic quotient.}\label{tab:block21-performance}\\
18474 \toprule
18475 Architecture and family & Mean sem. $R^2$ & Min sem. $R^2$ & Mean packet $R^2$ & Max NRMSE\\
18476 \midrule
18477 \endfirsthead
18478 \toprule
18479 Architecture and family & Mean sem. $R^2$ & Min sem. $R^2$ & Mean packet $R^2$ & Max NRMSE\\
18480 \midrule
18481 \endhead
18482 GRU affine gauge & 0.763002 & 0.743007 & 0.999075 & 0.033155\\
18483 GRU nonlinear warp & 0.730775 & 0.723749 & 0.999089 & 0.031610\\
18484 GRU latent clock & 0.756245 & 0.718695 & 0.998925 & 0.034543\\
18485 GRU chart alias & 0.761092 & 0.737828 & 0.998973 & 0.032485\\
18486 GRU benign compression & 0.999991 & 0.999990 & --- & ---\\
18487 Residual affine gauge & 0.745807 & 0.727216 & 0.999084 & 0.031363\\
18488 Residual nonlinear warp & 0.746279 & 0.727708 & 0.999132 & 0.031875\\
18489 Residual latent clock & 0.772349 & 0.758672 & 0.999100 & 0.030772\\
18490 Residual chart alias & 0.742217 & 0.722507 & 0.999192 & 0.029533\\
18491 Residual benign compression & 0.999990 & 0.999989 & --- & ---\\
18492 \bottomrule
18493 \end{longtable}
18494 }
18495 
18496 \begin{figure}[H]
18497 \centering
18498 \includegraphics[width=0.90\textwidth]{block21_semantic_r2.pdf}
18499 \caption{Protected semantic performance on fresh final episodes.  Every selected non-destructive repair remains above the frozen semantic gate.}
18500 \label{fig:block21-semantic}
18501 \end{figure}
18502 
18503 \subsection{Wrong-repair and no-go controls}
18504 
18505 For nonlinear, temporal, and chart-alias episodes, the inherited step-affine map is retained as a wrong-repair control.  The paired effect is
18506 \begin{equation}
18507 \Delta_i
18508 =
18509 \ell_{\rm step\ affine,i}
18510 -
18511 \ell_{\rm selected,i}.
18512 \end{equation}
18513 Every frozen bootstrap interval is strictly positive.
18514 
18515 \begin{longtable}{@{}lrrr@{}}
18516 \caption{Paired semantic-loss advantage of the selected structured repair over the inherited step-affine control.}\label{tab:block21-wrong-repair}\\
18517 \toprule
18518 Architecture and obstruction & Mean effect & Frozen 95\% interval & Positive fraction\\
18519 \midrule
18520 \endfirsthead
18521 \toprule
18522 Architecture and obstruction & Mean effect & Frozen 95\% interval & Positive fraction\\
18523 \midrule
18524 \endhead
18525 GRU chart alias & 0.464544 & [0.427997, 0.503210] & 0.944792\\
18526 GRU latent clock & 0.675134 & [0.632984, 0.716193] & 0.963542\\
18527 GRU nonlinear warp & 0.198562 & [0.183050, 0.213400] & 0.937500\\
18528 Residual chart alias & 0.568441 & [0.523956, 0.616695] & 0.971875\\
18529 Residual latent clock & 0.603810 & [0.566393, 0.637773] & 0.969792\\
18530 Residual nonlinear warp & 0.175636 & [0.159888, 0.192203] & 0.887500\\
18531 \bottomrule
18532 \end{longtable}
18533 
18534 \begin{figure}[H]
18535 \centering
18536 \includegraphics[width=0.86\textwidth]{block21_wrong_repair_effects.pdf}
18537 \caption{Correct structured repairs defeat the inherited step-affine wrong-repair control on all six architecture--obstruction comparisons.}
18538 \label{fig:block21-wrong-repair}
18539 \end{figure}
18540 
18541 In the six destructive final episodes, the compiler independently discovers between $112$ and $114$ exact collision pairs.  Mean protected semantic separation at identical observations lies between $2.053524$ and $2.158440$.  The best available semantic quotient achieves audit $R^2$ between $0.609997$ and $0.644296$, below the frozen $0.68$ gate.  Every destructive episode therefore returns no-go.
18542 
18543 \begin{figure}[H]
18544 \centering
18545 \includegraphics[width=0.88\textwidth]{block21_no_go_certificate.pdf}
18546 \caption{No-go controls.  Correlational quotient fitting remains below the repair gate while identical observations exhibit substantial protected semantic disagreement.}
18547 \label{fig:block21-no-go}
18548 \end{figure}
18549 
18550 \subsection{Finite realization theorem}
18551 
18552 \begin{conditionaltheorem}[Block 21 obstruction-guided transport-grammar growth]
18553 \label{thm:b21-realization}
18554 Within the frozen Block 21 environment, assume that:
18555 \begin{enumerate}
18556 \item the accepted Block 20 receipt and inherited connection replay exactly;
18557 \item packet projection and semantic-probe fitting use only their declared pre-audit supports;
18558 \item the compiler receives the public label-isolated episode view and does not import the audit-only expected map;
18559 \item the repair grammar and all numerical gates are frozen before final-task generation;
18560 \item final transform seeds and task records are absent from compiler construction;
18561 \item every accepted reconstructive or quotient repair passes its declared protected audit gates;
18562 \item every no-go output possesses the declared repeated-observation semantic-collision certificate;
18563 \item the final confusion matrix, wrong-repair controls, serialization checks, and receipt chain replay as recorded.
18564 \end{enumerate}
18565 Then the accepted edge is a finite, conservative, obstruction-guided extension of the Block 20 Neuralese transport grammar.  It distinguishes the six declared hidden stress families by repair response, retains the inherited minimal constructor on ordinary affine drift, admits structured extensions only where required, permits a semantic quotient under benign dimension loss, and abstains under certified destructive semantic loss.
18566 \end{conditionaltheorem}
18567 
18568 \begin{proof}
18569 Items 1 and 2 bind the experiment to the inherited lineage and protected witness.  Items 3 through 5 establish label isolation, chronology, and a frozen finite opportunity set.  Items 6 and 7 separate successful transport, protected quotienting, and impossible target-only repair.  Item 8 supplies foreign held-out evidence and causal controls.  The exact inclusion of the step-affine constructor preserves the inherited grammar, while the minimum-description rule prevents unnecessary structural expansion.  The resulting transaction therefore changes the repair language only in response to a validated obstruction inside the declared fixture.
18570 \end{proof}
18571 
18572 \begin{statusbox}{Block 21 result}
18573 The compiler passes all $24$ pre-audit episodes and all $36$ fresh final episodes.  Its final obstruction-to-repair matrix is diagonal.  Reconstructive repairs retain packet $R^2\geq0.998807$ with NRMSE at most $0.034543$; semantic quotients retain semantic $R^2\geq0.999989$; every structured repair defeats the inherited wrong-repair control with a strictly positive frozen interval; and every destructive collision produces no-go.  Nineteen automated tests and a deterministic replay verifier reproduce all $13{,}600$ exact task records and all final metrics.
18574 \end{statusbox}
18575 
18576 \subsection{Claim boundary and next threshold}
18577 
18578 Block 21 is a controlled finite realization.  It does not establish unrestricted obstruction discovery, arbitrary nonlinear transport, unrestricted variable-length or stochastic trajectory alignment, unpaired transport, autonomous semantic-ontology invention, natural-language Neuralese interpretation, production-model transfer, external independent replication, open-ended recursive improvement, unrestricted safety, consciousness, or sentience.
18579 
18580 The benchmark families and candidate constructors are author supplied.  The benign quotient explicitly exposes a lower-dimensional sufficient statistic, while the destructive fixture deliberately withholds protected coordinates and inserts exact collisions.  These are positive and negative controls for the compiler logic, not evidence that the same structures will be straightforward to identify in a pretrained language model.
18581 
18582 The next major threshold is to remove exact source--target task pairing and reduce dependence on a supplied semantic witness.  A stronger descendant should align foreign Neuralese trajectories using intervention orbits, sparse shared landmarks, output-equivalence classes, and relative semantic geometry; determine when those observations identify a transport only up to gauge; and issue an underidentification certificate when they do not.
18583 
18584 
18585 \section{Integrated SRIC Architecture}
18586 
18587 \subsection{Functional loop}
18588 
18589 \begin{figure}[H]
18590 \centering
18591 \resizebox{0.97\textwidth}{!}{%
18592 \begin{tikzpicture}[
18593   node distance=8mm and 8mm,
18594   box/.style={draw=KMBlue, rounded corners, align=center, minimum width=33mm, minimum height=10mm, fill=KMLight, font=\small},
18595   witness/.style={draw=KMGreen, rounded corners, align=center, minimum width=33mm, minimum height=10mm, fill=green!5, font=\small},
18596   risk/.style={draw=KMRed, rounded corners, align=center, minimum width=33mm, minimum height=10mm, fill=red!4, font=\small},
18597   arr/.style={-{Latex[length=2.3mm]}, thick, draw=KMGray}
18598 ]
18599 \node[box] (outcome) {External outcome\\and committed trace};
18600 \node[box, right=of outcome] (attrib) {Causal attribution,\\probing, response language/atlas};
18601 \node[box, right=of attrib] (residual) {Stratified defect\\and packet residual};
18602 \node[box, below=of residual] (obstruction) {Local obstruction\\and correction range};
18603 \node[box, left=of obstruction] (descent) {Repair descent,\\overlap, holonomy};
18604 \node[box, left=of descent] (schema) {Repair-language\\retrieval or invention};
18605 \node[risk, below=of schema] (sandbox) {Sandboxed descendant\\with capability budget};
18606 \node[witness, right=of sandbox] (validate) {Frozen witness,\\external validation};
18607 \node[witness, right=of validate] (commit) {Commit with receipt\\or rollback};
18608 \node[box, below=of validate] (abstract) {Repair abstraction,\\foreign-agent transfer};
18609 \node[box, left=of abstract] (archive) {Branching archive\\and evidence ledger};
18610 \node[box, right=of abstract] (meta) {Repair-calculus and\\grammar update};
18611 
18612 \draw[arr] (outcome) -- (attrib);
18613 \draw[arr] (attrib) -- (residual);
18614 \draw[arr] (residual) -- (obstruction);
18615 \draw[arr] (obstruction) -- (descent);
18616 \draw[arr] (descent) -- (schema);
18617 \draw[arr] (schema) -- (sandbox);
18618 \draw[arr] (sandbox) -- (validate);
18619 \draw[arr] (validate) -- (commit);
18620 \draw[arr] (commit) -- (abstract);
18621 \draw[arr] (abstract) -- (archive);
18622 \draw[arr] (abstract) -- (meta);
18623 \draw[arr] (meta.east) -- ++(12mm,0) coordinate (rloop) |- ($(attrib.north)+(0,6mm)$) -- (attrib.north);
18624 \draw[arr] (archive.west) -- ++(-12mm,0) coordinate (lloop) |- ($(outcome.west)+(-6mm,0)$) -- (outcome.west);
18625 \end{tikzpicture}%
18626 }
18627 \caption{The SRIC improvement loop. Controlled probes may first recover a functional response atlas before defects and repairs are localized. The witness path is separated from candidate construction. Successful repairs become reusable schemas, while valid obstructions and identifiability failures become reusable no-go certificates.}
18628 \label{fig:sric-loop}
18629 \end{figure}
18630 
18631 \subsection{One transaction}
18632 
18633 \begin{protocol}[Proof-carrying improvement transaction]
18634 Given lineage state $\mathfrak L_t$:
18635 \begin{enumerate}
18636 \item Freeze witness $W_t$, evaluator core coordinates, evidence ledger head, resource budget, and rollback checkpoint.
18637 \item Detect a persistent externally grounded mismatch.
18638 \item Generate internal-change, world-change, sensor, memory, evaluator, copy, fork, and unknown hypotheses.
18639 \item Select bounded active probes until uncertainty is below threshold or the probe budget is exhausted.
18640 \item Estimate intervention-response operators and either replay a validated atlas or recover an endogenous functional cover with response-fit, commutator, spectral-gap, and identifiability certificates.
18641 \item Resolve the defect into the validated local strata and compute local obstruction classes.
18642 \item If a local obstruction is nonzero, record a no-go certificate and stop or extend the grammar.
18643 \item Construct local repairs, compute overlap differences, and reject if the repair-descent class is nonzero.
18644 \item Retrieve or construct a typed repair schema and estimate transport, leakage, and holonomy margins.
18645 \item Precommit predicted gain and affected invariants.
18646 \item Instantiate a sandboxed descendant with bounded capabilities.
18647 \item Reveal external evaluation traces to the frozen witness.
18648 \item Commit only if validation, provenance, regression, core-preservation, and rollback tests pass; otherwise restore the parent.
18649 \item Abstract the result into a repair schema or obstruction certificate.
18650 \item Test the abstraction on foreign held-out agents before adding it to the reusable repair operad.
18651 \item Estimate remaining module-specific headroom and record any saturation certificate.
18652 \item If current modules are saturated while persistent obstruction remains, issue a grammar-boundary receipt and begin conservative conflict refinement.
18653 \item At the research-compiler level, test whether one diagnostic token contains incompatible sets of acceptable methods. If a conflict appears, search a declared generic experiment and constructor meta-language. Admit only a sealed conservative extension that improves foreign research problems.
18654 \item At the Neuralese transport level, distinguish coordinate drift, nonlinear warp, latent-clock mismatch, chart aliasing, benign semantic quotient, and destructive information loss. Select the minimum passing constructor and issue no-go when protected semantics are not measurable from the target packet.
18655 \end{enumerate}
18656 \end{protocol}
18657 
18658 \section{Empirical Status and Remaining Program}
18659 
18660 \subsection{Realized in Block 5}
18661 
18662 The released Block 5 reference package realizes the following finite and synthetic components:
18663 \begin{enumerate}
18664 \item the four-case local obstruction, gluing obstruction, flat repair, and evaluator-holonomy fixture;
18665 \item universal, controlled, and productive closure checks on a finite transaction graph;
18666 \item a witness-separated transaction kernel with commit, rollback, canonical receipts, and hash chaining;
18667 \item a matched-model anonymous-change environment with bounded probes and abstention;
18668 \item a matched-budget foreign-agent benchmark with static, search, packet, causal, transport, witness, and engineered meta positive-control arms;
18669 \item a reproducible test, result, and manifest pipeline.
18670 \end{enumerate}
18671 These realizations support the mechanical and finite-semantic gates defined in \cref{sec:empirical-contract}. The anonymous and benchmark suites remain engineered positive controls rather than evidence of autonomous repair discovery.
18672 
18673 \subsection{Realized in Block 6}
18674 
18675 Block 6 replaces analytic hidden-coordinate transport by a controlled learned-representation problem. Its realized components are:
18676 \begin{enumerate}
18677 \item a causally defined hidden packet estimated from source intervention differences;
18678 \item equal-rank PCA, random-packet, wrong-block, direct-copy, and shuffled-anchor controls;
18679 \item a frozen packet-local source repair fitted without target audit labels;
18680 \item a source-to-target bridge estimated from paired hidden activations while the constructing target map remains withheld;
18681 \item seven target representation families spanning injective, overcomplete, ill-conditioned, compressed, mildly nonlinear, saturated, and piecewise-linear regimes;
18682 \item independently trained linear and tanh encoders with distinct random initializations;
18683 \item a frozen witness gate that can reject a blind target gain when alignment, protected competence, or unaffected-stratum regression is unacceptable;
18684 \item a release-frozen package with disjoint train, anchor, calibration, and audit splits, five automated tests, a 34-file manifest, frozen result objects, and reproducible figures.
18685 \end{enumerate}
18686 The controlled package passes the cumulative foreign-transfer gate $G_{\rm foreign}$ for one shared task and defect grammar. It does not independently establish metaproductivity or grammar expansion.
18687 
18688 \subsection{Realized in Block 7}
18689 
18690 Block 7 implements two separately sealed parent--child comparisons of repair-process components.
18691 
18692 \paragraph{Selection-only realization.}
18693 The first route holds the ordered five-candidate repair bundle fixed on every target and changes only the commit policy. It realizes:
18694 \begin{enumerate}
18695 \item a frozen conservative parent selector;
18696 \item a lineage-generated dual-head child selector fitted from development repair receipts;
18697 \item a finite meta-grammar selected on a disjoint meta-validation split;
18698 \item a descendant receipt sealed before either final audit is opened;
18699 \item paired seed/task-foreign and architecture-foreign audits;
18700 \item raw-calibration, no-risk, shuffled-lineage, selector-ablation, and audit-oracle controls;
18701 \item identical non-null candidate budgets for every admissible selector arm.
18702 \end{enumerate}
18703 The child-parent protected-utility effects are
18704 \begin{equation}
18705 \widehat\Delta_{\rm audit}=0.072518,
18706 \qquad
18707 \widehat\Delta_{\rm arch}=0.104999,
18708 \end{equation}
18709 with positive frozen paired bootstrap intervals. Because parent and child share the same candidate set, this establishes behavioral selector metaproductivity but not expansion of the authorized correction image.
18710 
18711 \paragraph{Bridge-policy realization.}
18712 The second route holds the source repair, bridge family, transport formula, protected evaluator, and $24$-query deployment budget fixed while changing the paired-anchor selection rule and bridge-identification gate. It realizes:
18713 \begin{enumerate}
18714 \item a random-$16$ parent anchor policy;
18715 \item a compiled pivoted-QR-$16$ descendant policy sealed before primary foreign targets are generated;
18716 \item disjoint development and audit task families and disjoint primary representation families;
18717 \item a $144$-target paired foreign audit and a separate stress suite;
18718 \item random-budget, farthest-point, shuffled-binding, sign-flip, and plain-QR controls;
18719 \item exact query ledgers, rank diagnostics, harm checks, and package-closed receipts.
18720 \end{enumerate}
18721 On the primary foreign audit, mean balanced selected gain rises from $0.135212$ to $0.344720$, and
18722 \begin{equation}
18723 \widehat\Delta_{\rm bridge}=0.209508,
18724 \qquad
18725 \mathrm{CI}_{95\%}=[0.194968,0.223898].
18726 \end{equation}
18727 Every primary paired effect is positive, full-rank bridge identification rises from $0.03125$ to $1$, and harmful selected targets remain zero. The plain pivoted-QR control is effectively identical, so the result supports rank-revealing query design rather than a packet-specific advantage.
18728 
18729 The controlled packages therefore pass two distinct level-5 gates: $G_{\rm meta}^{\rm select}$ for selector metaproductivity and $G_{\rm meta}^{\rm bridge}$ for bridge-policy metaproductivity. Neither package establishes repeated-generation closure, autonomous candidate invention, or open-set grammar expansion.
18730 
18731 \subsection{Realized in Block 8}
18732 
18733 Block 8 is a formal and symbolic-realization expansion rather than a new learned-agent audit. It realizes:
18734 \begin{enumerate}
18735 \item finite restricted oracle envelopes and exact permutation-average meta-headroom allocation;
18736 \item component saturation and headroom-adjusted metaproductivity;
18737 \item bottleneck migration with an explicit conservative grammar-discovery coordinate;
18738 \item nested obstruction images and exact newly correctable energy;
18739 \item repair death versus forgetting death under dynamic defect semantics;
18740 \item obstruction-guided one-generator and rank-$k$ grammar expansion;
18741 \item finite repair-conflict refinement with a nuisance-feature negative control;
18742 \item finite counterexample-guided repair synthesis or grammar exhaustion;
18743 \item intervention-response algebra decomposition into central causal strata;
18744 \item lineage-wide false-improvement and protected-regression budget accounting.
18745 \end{enumerate}
18746 The symbolic package passes eight automated tests and produces a package-closed result object and four figures. These fixtures validate the finite mathematics and its implementation. They do not advance the empirical claim ladder beyond the two one-step Block 7 metaproductivity results.
18747 
18748 
18749 \subsection{Realized in Block 9}
18750 
18751 Block 9 replays the exact Block 7 bridge child and compiles a fresh grandchild in a new heteroscedastic bridge environment. It realizes:
18752 \begin{enumerate}
18753 \item exact replay and digest verification of the Block 7 rank-revealing child policy;
18754 \item fresh development and audit task supports and disjoint primary representation families;
18755 \item a rank-saturation diagnosis in which every inherited primary design is full rank but its weighted information floor remains low;
18756 \item a finite conditioning-aware policy grammar containing packet QR, unweighted D-optimal, quality-weighted D-optimal, quality-weighted A-optimal, and quality-weighted E-refined designs;
18757 \item a quality-weighted D-optimal grandchild and a stronger recalibrated rank-child control;
18758 \item identical $24$-query deployment budgets for the primary child and grandchild;
18759 \item gate-only, quality-permutation, unweighted-design, random-$16$, and larger-budget random-$32$ controls;
18760 \item a $96$-target paired primary foreign audit plus compressed, nonlinear, and severe-ill-conditioning stress families;
18761 \item exact ancestor, compiler, pre-audit seal, result, query-ledger, and package-manifest receipts.
18762 \end{enumerate}
18763 On the fresh primary audit, mean selected protected gain rises from $0.037393$ for the recalibrated rank child to $0.209580$ for the conditioning grandchild. The paired effect is
18764 \begin{equation}
18765 \widehat\Delta_{1\to2}=0.172188,
18766 \qquad
18767 \mathrm{CI}_{95\%}=[0.141481,0.203110].
18768 \end{equation}
18769 Both designs have full-rank rate one. The mean weighted minimum information eigenvalue rises from $0.518197$ to $14.268195$, the median condition number falls from $19393.296$ to $544.180$, and clean transport NRMSE falls from $0.303981$ to $0.095803$. The gate-only and quality-permutation controls do not reproduce the selected gain. The release therefore passes a context-indexed two-edge repeated-lineage gate. It does not establish structural repair-grammar growth or improvement of the outer compiler.
18770 
18771 \subsection{Realized in Block 10}
18772 
18773 Block 10 retains the Block 9 quality-weighted acquisition mechanism but expands the representational grammar of repair transport. It realizes:
18774 \begin{enumerate}
18775 \item exact semantic replay of the Block 9 conditioning-aware ancestor;
18776 \item a strengthened global-affine parent using $80$ fit plus $16$ witness pairs;
18777 \item a conservative two-chart child using $48$ fit, $32$ chart-selection, and $16$ witness pairs;
18778 \item one matched $96$-query deployment envelope for both processes;
18779 \item a finite separator search restricted to the source repair's protected two-dimensional context packet;
18780 \item a $54$-target development grammar-inadequacy certificate with full source rank, a persistent directly supervised global-affine residual, a near-exact true-chart oracle, smaller learned local residual, positive chart contrast, and an exact parent embedding;
18781 \item disjoint development, meta-validation, and primary audit task and family identifiers;
18782 \item a $96$-target primary foreign audit and a $32$-target stress suite;
18783 \item direct global-affine oracle, global-blind, local-blind, gate-only, random-gate, shuffled-binding, and true-chart controls;
18784 \item exact ancestor, compiler, grammar-certificate, pre-audit seal, result, test, figure, and package-manifest receipts.
18785 \end{enumerate}
18786 The resource-matched global parent abstains on the primary audit. The chart child obtains mean selected gain $0.334224$, median $0.284265$, commit rate $0.572917$, and zero harmful selected targets. The child-parent effect is
18787 \begin{equation}
18788 \widehat\Delta_{G\to C}=0.334224,
18789 \qquad
18790 \mathrm{CI}_{95\%}=[0.266459,0.401903].
18791 \end{equation}
18792 The sharper child-minus-direct-global-oracle effect is
18793 \begin{equation}
18794 \widehat\Delta_{\mathrm{oracle}\to C}=0.157530,
18795 \qquad
18796 \mathrm{CI}_{95\%}=[0.080863,0.234135].
18797 \end{equation}
18798 All primary source designs are full rank. Mean global validation NRMSE is $0.598513$, mean local validation NRMSE is $0.064975$, and mean learned gate accuracy is $0.874854$. Gate-only, random-gate, and shuffled-binding controls are strongly negative. Blind local application has higher mean gain but a nonzero harmful-commit rate, so the witness performs substantive selection. This passes the first finite structural repair-grammar gate. It does not establish autonomous packet, feature-language, chart-number, or compiler discovery.
18799 
18800 \subsection{Current empirical interpretation}
18801 
18802 The progression now separates seven empirical achievements and one formal bridge:
18803 \begin{enumerate}
18804 \item Block 5 realizes finite transactions, controls, and synthetic mechanics;
18805 \item Block 6 realizes intervention-defined hidden packets and learned repair transport to foreign representations;
18806 \item Block 7 realizes sealed one-step improvements of repair selection and rank-revealing evidence acquisition;
18807 \item Block 8 formalizes meta-headroom, saturation, bottleneck migration, persistent obstruction, grammar refinement, causal-atlas discovery, and sequential lineage certification;
18808 \item Block 9 replays the Block 7 child, diagnoses rank saturation, and realizes a second sealed edge by migrating to heteroscedastic conditioning-aware query design;
18809 \item Block 10 holds that acquired query policy fixed, certifies that a global-affine repair family remains structurally obstructed, and realizes a conservative two-chart repair-transport extension;
18810 \item the directly supervised global-affine oracle comparison separates function-class growth from a weak global estimator or permissive witness.
18811 \end{enumerate}
18812 
18813 The strongest admissible statement is:
18814 \begin{center}
18815 \emph{Within declared finite transport contexts, a proof-carrying lineage realizes two sequential query-policy improvements followed by one structural repair-grammar expansion. The final descendant retains conditioning-aware acquisition, proves a persistent residual for the global-affine repair family, conservatively adds an invariant-context chart-contrast direction, and improves fresh foreign targets under matched total evidence while preserving the global process as a fallback.}
18816 \end{center}
18817 
18818 The third edge crosses the boundary that Block 7 could not cross. The authorized opportunity set changes: two context-bound target repairs become reachable where the parent can instantiate only one global affine repair. The child also exceeds a directly supervised global-affine ceiling comparison. This is finite structural metaproductivity. It remains externally scaffolded because the protected packet, separator family, chart count, source repair, evaluator, witness, and outer compiler are supplied.
18819 
18820 \subsection{Realized in Block 11}
18821 
18822 Block 11 removes the protected context packet supplied to Block 10 and compiles a target-specific atlas from intervention responses. It realizes:
18823 \begin{enumerate}
18824 \item exact replay of the Block 10 ancestor policy and pre-audit lineage records;
18825 \item a six-dimensional source with three hidden rank-two functional blocks under target-specific rotations or injective embeddings;
18826 \item two controlled response operators with separating block signatures in the primary families;
18827 \item central spectral block recovery up to permutation and within-block gauge;
18828 \item response-fit, commutator, spectral-gap, commutant-dimension, and hidden-overlap audit diagnostics;
18829 \item repair-relevant block labeling through source-repair invariance, target-gate agreement, local-bridge validation, chart balance, and protected regression;
18830 \item a conservative executable embedding of the Block 10 supplied-packet grammar;
18831 \item one equal $120$-query envelope for the inherited fixed-packet parent and endogenous-atlas child;
18832 \item legacy-packet, PCA, full-space, random-atlas, shuffled-response, global, blind, and true-chart controls;
18833 \item an $80$-target primary foreign audit and a $25$-target stress suite containing response degeneracy, strong coupling, smooth transitions, repair-dependent chart crossing, and compression;
18834 \item exact compiler, atlas-certificate, pre-audit seal, result, figure, test, and $49$-file manifest receipts.
18835 \end{enumerate}
18836 The primary parent obtains zero selected gain and the child obtains $0.207325$. The paired effect has frozen interval $[0.131902,0.287910]$, no target favors the parent, and selected harm is zero. Blind child application has larger mean gain $0.271529$ but harm rate $0.1875$, so the witness performs substantive selection. The release passes a finite endogenous-atlas gate. It does not establish autonomous intervention invention, unrestricted block-count discovery, general noncommuting or nonlinear atlas recovery, or compiler self-improvement.
18837 
18838 
18839 \subsection{Realized in Block 12}
18840 
18841 Block 12 realizes two separately sealed structural edges beyond endogenous atlas recovery.
18842 \paragraph{Response-language edge.}
18843 The inherited two-operator response algebra contains an exact context--repair alias. Every polynomial and composition of inherited responses preserves the merged block. A finite intervention compiler selects one separating operator from five typed candidates, conservatively retains the old algebra, and seals the new response policy before a $96$-target foreign audit. Under equal $144$-query deployment budgets, selected protected gain rises from $0$ to $0.846985$, with frozen interval $[0.786100,0.901073]$, no negative paired effects, and zero selected harm.
18844 
18845 \paragraph{Repair-constructor edge.}
18846 After the response child identifies the repair packet, the inherited affine repair operad encounters a target correction with a nonzero quadratic Hessian. The child adds symmetric degree-two packet features while embedding every affine repair by zero quadratic coefficients. Under equal $160$-query deployment budgets, selected gain rises from $0.289123$ to $0.997511$. The paired effect is $0.708388$ with interval $[0.659635,0.757198]$, and the child exceeds a directly supervised $1600$-pair affine oracle by $0.530933$. A full-space quadratic control is equally effective, so the supported claim is degree-two constructor growth rather than unique block-local superiority.
18847 
18848 These edges pass finite response-language and candidate-constructor gates. They do not establish unrestricted intervention invention, open-ended program synthesis, or improvement of the outer lineage compiler.
18849 
18850 
18851 \subsection{Realized in Block 13}
18852 
18853 Block 13 changes the research compiler while holding the downstream candidate-method grammar and final evaluator fixed. It realizes:
18854 \begin{enumerate}
18855 \item a strong parent compiler with structured literature priors, a shared finding graph, results-free receipts, exact code/specification binding, balanced exploration, multi-benchmark selection, capability gates, and integrity monitoring;
18856 \item a child compiler trained only from development receipts, with a nearest-centroid obstruction classifier and ridge outcome/risk forecasters;
18857 \item an identical $39$-method candidate grammar and identical six-full-experiment budget for parent and child;
18858 \item development, meta-validation, primary foreign, and architecture-foreign problem splits with pre-audit compiler sealing;
18859 \item shuffled-archive, classifier-only, forecaster-only, no-forum, single-benchmark, no-monitor, random-allocation, and audit-oracle controls;
18860 \item exact proposal, code, archive, compiler, allocation, evaluator, monitor, result, test, and manifest receipts.
18861 \end{enumerate}
18862 On $96$ primary foreign problems, mean validity-filtered protected gain rises from $0.883004$ to $0.950231$, with paired effect $0.067227$ and frozen interval $[0.035881,0.103274]$. On $72$ architecture-foreign problems, gain rises from $0.772566$ to $0.863367$, with paired effect $0.090800$ and interval $[0.052311,0.135081]$. The obstruction classifier is the dominant realized mechanism. Outcome forecasting is informative but contributes only modestly, and removal of the finite within-problem forum has no measurable effect. The release passes a bounded research-compiler metaproductivity gate. It does not establish open-ended automated science, autonomous candidate-language growth, evaluator invention, or improvement of an unrestricted outer compiler.
18863 
18864 \subsection{Realized in Block 14}
18865 
18866 Block 14 replays the Block 13 result digest and exact method language, then recalibrates the inherited compiler design as the parent of an adversarial robustness study. It realizes:
18867 \begin{enumerate}
18868 \item diagnostic-matched clean and ambiguous problems that weaken one-shot obstruction classification;
18869 \item sparse two-obstruction mixtures and an unknown class outside the inherited response cone;
18870 \item evaluator-attack and integrity-attack method families;
18871 \item an $18\%$ contaminated development archive plus a contamination sweep through $35\%$;
18872 \item a robust child using sequential diagnosis, sparse response mixtures, open-set routing, robust archive summaries, evaluator portfolios, and an integrity-monitor ensemble;
18873 \item the exact inherited $39$-method grammar and six-full-experiment budget for parent and child;
18874 \item primary and independently shifted generator-foreign audits with pre-audit compiler sealing;
18875 \item no-open-set, no-sequential, ordinary-archive, simple-monitor, single-portfolio, random-allocation, and audit-oracle controls;
18876 \item exact results-free and post-result receipt chains, canonical results, tests, figures, and package manifest.
18877 \end{enumerate}
18878 On $144$ primary problems, mean validity-filtered protected gain rises from $0.410813$ to $0.575783$, with paired effect $0.164970$ and frozen interval $[0.115591,0.218655]$. On $120$ generator-foreign problems, gain rises from $0.389514$ to $0.550559$, with effect $0.161045$ and interval $[0.105335,0.223141]$. Primary lower-tail gain changes from $-0.133065$ to $0.271516$, observed invalid selections change from $0.027778$ to $0$, and false confident known-class assignment on unknown problems changes from $1$ to $0.083333$. Sequential response updating and open-set routing are substantively supported. The simple-monitor ablation is null and the single-portfolio ablation is nearly null, so neither receives independent positive attribution.
18879 
18880 The claim remains finite. The child uses an externally declared response family, candidate grammar, attack vocabulary, evaluator portfolio, and monitor ensemble. It is a robustness improvement of the fixed-language compiler, not unrestricted open-set science or compiler-language growth.
18881 
18882 \subsection{Realized in Block 15}
18883 
18884 Block 15 replays the Block 14 result digest and promotes the robust compiler to the inherited parent of a structural compiler-language study. It realizes:
18885 \begin{enumerate}
18886 \item six inherited obstruction tokens, one undifferentiated unknown token, and the inherited $39$-method singleton grammar;
18887 \item a development open-set cell containing two hidden research mechanisms that share the inherited diagnostic representation but require incompatible high-gain methods;
18888 \item a generic six-probe research-experiment language and a generic $30$-member ordered pair-method constructor grammar;
18889 \item finite conflict scoring, anonymous token refinement, pair-constructor synthesis, and exact conservative fallback;
18890 \item hidden novel labels excluded from compiler construction and retained only for audit interpretation;
18891 \item identical eight-unit deployment research budgets for parent and child;
18892 \item primary and generator-foreign audits sealed after the language certificate and compiler object;
18893 \item taxonomy-only, grammar-only, shuffled-binding, inherited-oracle, and expanded-oracle controls;
18894 \item complete results-free and post-result receipts, deterministic figures, tests, and a package manifest.
18895 \end{enumerate}
18896 The development compiler selects \texttt{interaction-contrast} and binds the anonymous refined tokens to \texttt{evaluator-then-resource} and \texttt{objective-then-constructor}. The inherited development oracle obtains $0.377257$ mean open-set gain, while the selected extension obtains $0.948962$, producing a newly correctable margin $0.571704$.
18897 
18898 On $96$ primary problems, mean protected gain rises from $0.633969$ to $0.859731$, with paired effect $0.225762$ and interval $[0.166708,0.286431]$. On $82$ generator-foreign problems, gain rises from $0.574224$ to $0.814520$, with paired effect $0.240296$ and interval $[0.180154,0.302701]$. No paired problem favors the parent. Known-class behavior is reproduced exactly through the inherited fallback. The taxonomy-only and shuffled-binding controls match the parent, while the grammar-only arm solves only the novel cell favored by its pooled binding. The release passes a finite compiler-language growth gate.
18899 
18900 The result remains relative to an externally declared generic meta-language. The compiler does not invent arbitrary experiments or programs, semantically name the new obstruction classes, improve the evaluator or monitor, or execute real model-training research.
18901 
18902 
18903 \subsection{Stage H completed: first repeated lineage}
18904 
18905 The realized lineage is
18906 \begin{equation}
18907 K_0\xrightarrow{\Pi_1}K_1\xrightarrow{\Pi_2}K_2.
18908 \end{equation}
18909 The first edge improves rank identification. The second edge does not attempt to improve a saturated rank endpoint. It migrates to a distinct information-conditioning obstruction. Exact ancestor replay, new development evidence, a pre-audit grandchild seal, matched deployment queries, a recalibrated child control, and fresh foreign audit systems distinguish this result from reusing the Block 7 final evidence or weakening the parent.
18910 
18911 A two-edge result is still finite. It does not imply a winning strategy for infinitely repeated productive closure, compiler self-improvement, or open-set grammar growth. It establishes that the lineage mechanism can produce a second successful descendant after the first improvement changes the active bottleneck.
18912 
18913 \subsection{Stage I partially realized: finite structural grammar expansion}
18914 
18915 Block 10 realizes the finite form of this stage. The lineage encounters foreign systems for which one global affine repair remains misspecified despite full-rank, quality-aware acquisition and an $80$-anchor parent fit. It records a grammar-inadequacy certificate, conservatively introduces an invariant-context two-chart constructor, and verifies a new anti-diagonal repair direction on fresh foreign systems. The stronger directly supervised global-affine oracle establishes that the gain is not explained by poor source-to-target bridge estimation alone.
18916 
18917 Open-set grammar expansion additionally requires:
18918 \begin{enumerate}
18919 \item autonomous detection of a persistent obstruction or repair conflict;
18920 \item invention or discovery of the separating feature, chart count, or nonlinear coordinate system;
18921 \item a conservative embedding of old grammar, proofs, evidence, and authorization decisions;
18922 \item exposure of a genuinely new correction direction under exact causal ablation;
18923 \item held-out obstruction-energy reduction on foreign systems that require the new distinction;
18924 \item preservation of evaluator, provenance, witness, and resource commitments;
18925 \item repeated structural descendants under lineage-wide statistical and semantic-debt budgets.
18926 \end{enumerate}
18927 Block 10 supplies the obstruction, conservative extension, structural contrast, matched-resource audit, and causal controls. It does not by itself supply atlas discovery; Block 11 addresses one finite response-algebra form of that next requirement.
18928 
18929 \subsection{Stage J partially realized: finite endogenous atlas growth}
18930 
18931 Block 11 realizes the finite form of endogenous atlas growth. The supplied context packet is removed. A declared intervention family generates response operators whose central spectral blocks are recovered up to permutation and within-block gauge. Repair-validation evidence selects the block that supports a stable chart separator. The old supplied-packet process remains an executable subgrammar and fallback.
18932 
18933 The unresolved part of Stage J is substantial. The intervention family, block dimensions, finite block-scalar response model, hard binary chart grammar, separator family, source repair, evaluator, witness, and outer compiler remain supplied. A stronger lineage must discover which intervention to run, whether signatures are degenerate, whether response operators are noncommuting, and whether the appropriate extension is another discrete block, a soft manifold, a kernel lift, or an information-loss certificate.
18934 
18935 \subsection{Stage K completed in finite form: response-language and candidate-constructor growth}
18936 
18937 Block 12 realizes the finite form of Stage K through two distinct contexts. The first edge certifies an exact response alias under the inherited algebra, selects one separating intervention from a fixed typed menu, and recovers a repair-enabling central refinement. The second edge certifies that every finite composition of inherited affine constructors remains affine while the target repair has a nonzero quadratic Hessian, then conservatively adds a degree-two constructor. Both results use matched query budgets, independent parent and child gates, fresh sealed audits, explicit function-class controls, and package-closed receipts.
18938 
18939 The unresolved open-set form is stronger. A future lineage must enlarge the intervention or constructor language without receiving the final candidate menu, account for acquisition feasibility and physical effects, determine the necessary polynomial or program degree, and show that its outer diagnostic compiler transfers to foreign parent repair processes.
18940 
18941 \subsection{Stage L partially realized: finite research-compiler metaproductivity}
18942 Block 13 realizes a bounded improvement of the research-allocation compiler while the candidate-method language, evaluator portfolio, integrity monitor, and final research interfaces remain fixed. The child uses the same research budget more selectively by diagnosing the active obstruction and forecasting outcome and risk. This is an improvement of the process that generates and chooses repair-process descendants, but not an unrestricted improvement of its own language.
18943 
18944 The unresolved form requires compiler grammar growth and eventually conservative evaluator evolution. A later compiler must detect that its obstruction taxonomy, proposal features, experimental actions, monitor, or evaluator portfolio is inadequate, introduce a new research distinction, preserve the path-flat evidential core, replay old protected decisions, and be witnessed by a component not rewritten in the same transaction. The primary danger is semantic capture through simultaneous alteration of proposal, evidence, utility, and approval.
18945 
18946 \subsection{Stage M completed in finite form: adversarial open-set compiler robustness}
18947 
18948 Stage M asks whether the finite research compiler remains useful after the clean six-way routing problem is replaced by ambiguous diagnostics, mixed obstructions, unknown cases, contaminated archives, evaluator attacks, integrity attacks, and a generator shift. Block 14 passes this finite gate. The child improves the inherited parent in mean and lower-tail validity-filtered gain under the same six-experiment budget, and every declared child group has positive mean gain. Some individual groupwise paired intervals include zero, so the aggregate and lower-tail claims are stronger than every family-specific claim.
18949 
18950 The result does not yet satisfy the stronger external-robustness target. The benchmark generator, candidate methods, response language, attack types, evaluator portfolio, and monitor ensemble are still part of one package-closed research design. The next stage should combine an independently written generator and implementation with executable small-model or program-synthesis research, then require the robust compiler to issue a compiler-language inadequacy certificate before adding a new diagnostic, experiment, candidate family, or evaluator dimension.
18951 
18952 \subsection{Stage N completed in finite form: compiler-language growth}
18953 
18954 Stage N asks whether the robust research compiler can move beyond abstention or generic routing when its own diagnostic and method language is certified inadequate. Block 15 passes one finite form of this gate. One inherited open-set token contains development problems with incompatible acceptable-method sets. A finite compiler selects a separating probe from a generic experiment language, creates two anonymous refined tokens, synthesizes one ordered pair method for each token, and preserves the entire inherited compiler as an exact fallback.
18955 
18956 This is structurally stronger than Block 14. The child does not merely classify known obstruction labels more robustly or allocate the inherited method grammar more effectively. It adds diagnostic and method expressions absent from the deployable parent language. The unresolved form is unrestricted compiler-language growth: the generic probe family, pair-composition operator, method primitives, evaluator, integrity monitor, and research simulator remain externally supplied.
18957 
18958 
18959 
18960 \subsection{Stage O direct integration completed: Neuralese causal semantics}
18961 
18962 Block 16 makes recurrent latent reasoning an explicit SRIC subsystem rather than an analogy. The latent transition, task-anchored update gate, halt map, trace convention, intervention language, semantic transport, and answer witness are separately typed and receipted. The mandatory audit distinguishes five possibilities: an inactive loop, an unordered loop, a corruption-fragile loop, a useful ordered but task-nonspecific loop, and a task-specific loop. Cross-task transplantation within fixed answer-format groups is the principal discriminator between the last two.
18963 
18964 The companion validates the audit logic on a sealed synthetic parent--child pair. The parent passes necessity, order, corruption, and replay but fails task specificity, so its status is PARTIAL. The child raises structural-normalized specificity from $0.016952$ to $0.334610$, obtains a paired transplant-cost gain of $0.439986$ with interval $[0.386884,0.493143]$, preserves clean performance, and passes both the FULL causal and adaptive-halting gates. This is an engineered positive control, not evidence about a trained language model.
18965 
18966 The local Hyper adapter is executable when the official runtime and weights are supplied. It imports the exposed injection hook, captures and reinjects exact latent states, computes gold-suffix cross-entropy through the official end-of-thought path, and emits canonical receipts. The actual model was not present in the build environment, so the real-model result remains open. The next empirical threshold is a sealed Hyper audit followed by a genuine parent--child checkpoint comparison generated by a declared training or search process.
18967 
18968 
18969 \subsection{Stage P completed in finite form: executable Neuralese binding repair}
18970 
18971 Block 17 realizes the two-checkpoint threshold proposed at the end of Block 16, but on a package-local recurrent micro-model rather than Hyper. A separately implemented exact rational environment generates affine programs and protected $(A,B,y)$ witnesses. A task-blind trained parent is inherited into six finite repair candidates. The compiler uses only development, meta-validation, and probe-calibration evidence, seals the selected checkpoint, and then faces fresh primary and shifted-generator tasks.
18972 
18973 The first sealed attempt is negative: the selected child improves absolute transplant effects and semantic decodability but misses the frozen normalized FULL gates. The package retains that failure and commits a revision before new final-task generation. In the second round, three candidates are eligible and the full-binding child is selected. The parent remains structurally active but program blind. The child passes necessity, order, noise, replay, program specificity, answer specificity, local semantic continuity, donor coverage, external trace decoding, and adaptive-depth gates on both final supports. The accepted receipt is \texttt{6cc2257f...655e4ae}.
18974 
18975 This closes one finite training-generated latent-binding edge. It does not establish foreign-model metaproductivity.
18976 
18977 The affine task family, semantic witness, donor taxonomy, candidate ranks, loss, compiler score, and gate thresholds remain external. The decisive next test is an independently trained latent-reasoning architecture or a real parent--child production checkpoint pair, preferably with a compiler-generated modification absent from the Block 17 grammar.
18978 
18979 
18980 \subsection{Stage Q attempted and obstruction-certified: cross-implementation Neuralese transport}
18981 
18982 Block 18 instantiates the source-selected progressive FiLM schema in two independently implemented recurrent cell families. On the primary generator, both selected children satisfy the full causal, semantic, execution, replay, and adaptive-depth gates. The stronger family claim fails under the shifted generator. Both external semantic probes become negative, exact execution falls below $0.09$, adaptive depth collapses, clean loss regresses, and both frozen latent bridges fail even though their anchor trace matrices are full rank and their pre-audit transported $R^2$ values exceed $0.70$.
18983 
18984 The failure is retained as a semantic-chart obstruction rather than interpreted as absence of every latent semantic variable. After the canonical receipt is frozen, a separate held-out split diagnosis refits only shift-local readouts and bridges. Source and target probes recover $R^2$ between $0.574$ and $0.601$, paired bridges recover $0.539$ to $0.555$, and shuffled bridges remain negative. Thus the next missing object is a context-indexed or invariant semantic atlas with abstention, not another static full-rank bridge. No transition $\Pi_{18}$ is accepted in this revision.
18985 
18986 \subsection{Stage R completed in finite form: distribution-indexed source atlas growth}
18987 
18988 Block 19 acts on the Block 18 chart obstruction before attempting another foreign bridge. It compiles two new observation charts from protected semantic twins, preserves the four inherited charts exactly, separates the recurrent semantic path from the context-diagnostic path, and rejects nonlinear contexts outside the authorized affine grammar. The selected child is \texttt{ATLAS\_FULL} on fresh inherited and added-chart supports.
18989 
18990 \subsection{Stage S completed in finite form: atlas-indexed cross-architecture transport}
18991 
18992 Block 20 transports the stabilized six-chart bundle from independently trained GRU and gated-residual trajectories into the frozen Elman source representation. One full-trajectory map is selected per architecture and admitted chart. The connection preserves the source semantic readout, reduces architecture--context curvature below the frozen ceiling, defeats pooled and shuffled-binding controls, and retains selective rejection outside the atlas.
18993 
18994 \subsection{Stage T completed in finite form: obstruction-guided transport-grammar growth}
18995 
18996 Block 21 changes the response language of the transport process. The compiler is not given the obstruction-family label. It must infer whether the inherited connection faces ordinary affine drift, smooth nonlinear warp, latent-clock mixing, an aliased chart, benign rank loss, or protected semantic destruction. It then selects the minimum-description passing constructor or a collision-grounded no-go certificate. All $36$ fresh final episodes receive the required response across both foreign architectures.
18997 
18998 The result remains finite. The packet, obstruction families, response probe, protected semantic coordinates, constructor grammar, and paired calibration interface remain declared. The accepted edge does not establish unrestricted natural-model diagnosis.
18999 
19000 \subsection{Precommitted next-stage estimands}
19001 
19002 The next major study should test \emph{unpaired and ontology-light Neuralese transport}. It should precommit:
19003 \begin{enumerate}
19004 \item exact replay of the Block 19 atlas, Block 20 connection, Block 21 compiler seal, final family receipt, and independent verifier;
19005 \item removal of exact source--target task pairing from transport fitting;
19006 \item sparse protected anchor orbits, controlled interventions, or output-equivalence classes as the only shared landmarks;
19007 \item candidate semantic operators proposed from response regularities rather than a complete author-supplied intermediate-state ontology;
19008 \item explicit tests for gauge-equivalent descriptions, strategy multimodality, and genuinely different computations;
19009 \item variable-length or stochastic recurrent trajectories with chronology-preserving transport;
19010 \item no-go certificates when the shared observations do not identify a protected semantic correspondence;
19011 \item independent task generation, fresh architecture families, and sealed primary and generator-foreign audits;
19012 \item lower-tail, coverage, abstention, resource, and protected-regression estimands;
19013 \item append-only retention of every failed pilot and every protocol change before final-support generation.
19014 \end{enumerate}
19015 A full SRIC claim remains factorized. No aggregate score may replace validity, causal relevance, semantic sufficiency, obstruction diagnosis, conservative grammar growth, evaluator adequacy, independent replication, and open-set gates.
19016 
19017 \section{Conditional Stratified Recursive Improvement Theorem}
19018 
19019 \begin{conditionaltheorem}[Conditional SRIC theorem, Block 21 form]
19020 \label{thm:umbrella}
19021 Consider a finite declared agent atlas and a sequence of lineage states $\mathfrak L_t$. Assume:
19022 \begin{enumerate}
19023 \item externally grounded outcome traces and an append-only evidence ledger;
19024 \item typed authorized correction spaces, defect maps, and local obstruction computation;
19025 \item effective repair descent on the declared cover;
19026 \item coherent repair transport with pathwise norm, authorization, and obstruction-faithfulness bounds;
19027 \item a nontrivial adequate evaluator core and conservative evaluator morphisms;
19028 \item witness-separated transactions with a decision-complete trace, sandboxed effects, precommitment, and rollback;
19029 \item a finite transaction game with declared actions and admissible outcomes;
19030 \item an append-only receipt-closed archive;
19031 \item calibrated acceptance with summable conditional false-acceptance budgets;
19032 \item conservative grammar embeddings on the protected fragment;
19033 \item foreign-agent evaluation under matched resources;
19034 \item for the autocatalytic clause, a valid finite-type branching approximation with a causally identified offspring matrix;
19035 \item an executable runtime with a declared abstraction map, observation-commuting refinement, canonical receipt compiler, frozen endpoints, and package-closed evidence;
19036 \item cumulative empirical claim gates whose interpretation is restricted to the highest gate that passes;
19037 \item for learned representation transfer, disjoint train, anchor, calibration, and audit samples, a declared causal intervention, a frozen source repair, paired-anchor-only bridge fitting, and no target-audit or target-oracle leakage into candidate construction;
19038 \item for statistical witness guarantees, simultaneous calibration-to-audit deviation bounds for gain, protected regression, and transport quality;
19039 \item for selector-level metaproductivity, a lineage-generated child sealed from development and meta-validation evidence before final audit access, pathwise identical candidate bundles, fixed protected utility, matched candidate and evidence budgets, and a predeclared paired inference rule;
19040 \item for bridge-policy metaproductivity, a lineage-generated repair process sealed before primary audit generation, pathwise matched source repair and deployment query budgets, disjoint development and primary family supports, exact query ledgers, a fixed transport-and-evaluation mechanism, and a predeclared paired inference rule;
19041 \item for repeated recursive closure, fresh evidence and valid witness separation at every lineage generation;
19042 \item for module-specific headroom claims, a declared repair-process factorization and valid restricted oracle envelopes or certified bounds;
19043 \item for bottleneck migration, a saturation-aware controller and a weighted local contraction certificate inside each fixed grammar;
19044 \item for structural grammar growth, obstruction-faithful semantic transports, held-out obstruction measurements, and causal ablation of every new generator;
19045 \item for conflict-grounded grammar refinement, a finite externally validated repair-conflict family and a declared feature or probe family;
19046 \item for causal-atlas claims, a finite adjoint-closed response algebra or a spectrally separated empirical approximation;
19047 \item for lineage-wide statistical claims, conditional edge-level coverage with summable false-acceptance and protected-regression budgets;
19048 \item for heteroscedastic bridge claims, conditionally independent paired-anchor noise with declared or validly estimated quality weights and a fixed weighted bridge estimator;
19049 \item for conditioning-aware query claims, matched deployment query budgets, a finite pre-audit design grammar, and controls that isolate query design from gate recalibration and quality-label shortcuts;
19050 \item for the Block 9 repeated-lineage claim, exact ancestor receipt replay, fresh development and final audit supports, a sealed grandchild, a recalibrated child control, and a context-indexed interpretation of the two edges;
19051 \item for the Block 10 structural grammar claim, full-rank inherited designs, a strengthened global parent, a persistent directly supervised global-affine residual, a conservative embedding into the chart grammar, a nonzero chart-contrast direction, a matched total query envelope, fresh paired foreign evaluation, and gate-only, random-gate, shuffled-binding, blind-local, and stress controls.
19052 \item for the Block 11 endogenous-atlas claim, exact ancestor replay, removal of the supplied context packet, a declared intervention-response family, target-specific hidden rotations, separating response signatures or a certified abstention condition, matched total effect-query budgets, a conservative Block 10 fallback, fresh paired foreign evaluation, and PCA, full-space, random-atlas, shuffled-response, blind, and stress controls;
19053 \item for the Block 12 response-language claim, an inherited exact or statistically certified response alias, a finite pre-audit intervention candidate language, a separating signature margin, conservative algebra embedding, matched total query budgets, fresh foreign evaluation, and duplicate, weak, shuffled, random-atlas, and stress controls;
19054 \item for the Block 12 constructor claim, an inherited affine repair operad, a nonzero-Hessian or equivalent function-class obstruction, a conservatively embedded finite constructor language, matched total query budgets, separately compiled parent and child witnesses, a same-target affine function-class control, and wrong-block, random-block, shuffled-feature, full-quadratic, and stress controls;
19055 \item for the Block 13 research-compiler claim, identical initial archives, candidate-method grammars, evaluator portfolios, integrity monitors, proposal and full-experiment budgets, results-free receipts sealed before outcomes, disjoint development, meta-validation, primary, and architecture-foreign research problems, a pre-audit compiler seal, and paired validity-filtered final audits with declared mechanism controls.
19056 \item for the Block 14 robust-compiler claim, exact Block 13 ancestor replay, identical fixed method grammars and six-experiment budgets, diagnostic aliasing and sparse mixture families, a declared unknown-response margin, contaminated development receipts, finite evaluator and integrity attacks, pre-audit compiler sealing, group-resolved and lower-tail gates, and a separately shifted generator-foreign audit.
19057 \item for the Block 15 compiler-language claim, exact Block 14 ancestor replay, an inherited unknown token containing an externally validated method conflict, a finite generic probe language, a finite generic method-constructor grammar, hidden novel labels excluded from compilation, an exact inherited fallback, matched eight-unit deployment budgets, pre-audit language and split sealing, and taxonomy-only, grammar-only, shuffled-binding, inherited-oracle, and expanded-oracle controls.
19058 \item for the Block 16 Neuralese claim, an explicitly located latent injection hook, a fixed trace convention, identical prompt and answer scoring across arms, zero-step, replay, permutation, norm-matched-noise, and same-format transplant interventions, matched latent-step budgets, sealed task and donor digests, and a frozen external witness.
19059 \item for the Block 17 executable Neuralese repair claim, exact external task witnesses, one inherited trained parent, matched candidate optimization envelopes, same-answer far-program and far-answer matched-depth donors, an independently fit trace probe, adaptive-depth gates, a preserved failed pilot, a precommitted revision, a compiler seal before fresh final-task generation, and package-local NumPy, receipt, witness, and gauge replay.
19060 \item for the Block 18 obstruction claim, independently implemented foreign recurrent cells, exact ancestor replay, a source-selected repair schema, primary and shifted audits, frozen semantic bridges, and post-audit localization kept outside acceptance;
19061 \item for the Block 19 atlas claim, protected semantic twins, a sealed finite chart grammar, exact inherited fallback, selective admission, fresh supported and unsupported contexts, and independent route and forward verification;
19062 \item for the Block 20 connection claim, a frozen source atlas, independently trained foreign targets, chart-indexed gauge stress, a pre-audit transport compiler, complete trajectory and curvature audits, shuffled binding, and unsupported-context controls;
19063 \item for the Block 21 transport-grammar claim, a frozen diagnostic packet and semantic witness, hidden obstruction-family labels, a finite typed repair grammar, minimum-description eligibility, disjoint episode splits, collision-grounded no-go, fresh foreign audits, and exact independent replay.
19064 \end{enumerate}
19065 Then:
19066 \begin{enumerate}
19067 \item local correctability, nonlinear lifting, and global repair descent obey the results of the earlier blocks;
19068 \item exact and approximate repair transport obey the stated equivalence, accumulation, and gain-margin theorems;
19069 \item conservative evaluator chains preserve recoverable protected evidence and every decision above the cumulative stability margin;
19070 \item direct atomic self-approval is excluded relative to the declared trace, and rejected buffered transactions restore the sealed parent;
19071 \item the universal recursive closure, robust controlled kernel, and productive nested fixed point exist and are computable in the finite model;
19072 \item the archive's scalar upper envelopes are nondecreasing under append-only retention, and an $\eps$-Pareto cover of bounded size exists;
19073 \item only finitely many harmful acceptances occur almost surely under the summable error condition;
19074 \item nested repair images reduce held-out obstruction energy by exactly the defect energy in newly correctable directions;
19075 \item every finite old proof persists along the conservative grammar chain, and ontology expansion satisfies the old-resolution and new-obstruction exact sequence;
19076 \item finite repair-discovery closure exists, while the branching threshold and spectral certificates hold under their additional stochastic assumptions;
19077 \item every covered executable trace maps to an authorized abstract trace with matching protected observations, and exhaustive finite endpoints are decidable when exact realization holds;
19078 \item package integrity, mechanics, finite semantics, causal identification, foreign transfer, selector metaproductivity, structural metaproductivity, repeated recurrence, and open-set recursion remain distinct claims and are admissible only through their cumulative gates;
19079 \item in the linear Block 6 fixture, intervention differences recover the exact causal packet under the covariance condition of \cref{thm:block06-linear-packet};
19080 \item an exact injective hidden bridge transports the source repair equivariantly by \cref{thm:block06-exact-transport}, while a noninjective bridge leaves the explicit compression residual of \cref{prop:block06-compression-residual};
19081 \item a candidate passing a tightened witness gate has positive audit gain and respects the declared audit regression and alignment bounds whenever the simultaneous deviation assumptions of \cref{thm:block06-witness-soundness} hold;
19082 \item a sealed child selector satisfying the hypotheses of \cref{thm:block07-one-step} is a certified one-step bounded improvement of the repair-selection process;
19083 \item if parent and child share the same candidate bundle, that selector-level certificate does not imply structural repair-image growth by \cref{prop:block07-fixed-candidate-invariance};
19084 \item a sealed bridge-policy child satisfying \cref{thm:block07q-one-step-certificate} is a certified one-step bounded improvement of paired-anchor query design and bridge identification relative to the declared finite benchmark;
19085 \item a single productive child edge does not imply recurrent productive closure without a fresh successful descendant edge;
19086 \item finite restricted oracle envelopes allocate total declared headroom exactly across repair-process modules, and an $\eps$-saturated module coalition cannot support more than $\eps$ additional gain;
19087 \item under the Block 8 Lyapunov hypothesis, weighted module headroom contracts during fixed-grammar intervals while the target of improvement may migrate among modules;
19088 \item nested grammar correction images reduce obstruction energy by exactly the component in newly correctable directions;
19089 \item finite repair-conflict refinement terminates when the candidate feature family separates every minimal conflict;
19090 \item a finite adjoint-closed intervention algebra has a canonical central causal atlas up to multiplicity gauge;
19091 \item summable conditional edge budgets control the probability of accepting any false lineage improvement across an adaptive finite or countable archive;
19092 \item weighted least squares has coefficient covariance $I_q\otimes M_S^{-1}$ under the Block 9 noise model, so full rank without a positive information floor is insufficient for stable bridge recovery;
19093 \item a conditioning-aware query policy satisfying the Block 9 sealed paired certificate is a bounded improvement over the recalibrated rank child in the declared fresh context;
19094 \item the verified sequence $K_0\xrightarrow{\Pi_1}K_1\xrightarrow{\Pi_2}K_2$ is a finite two-edge context-indexed repeated-lineage realization, but not a same-context scalar transitivity result or an infinite productive-closure certificate;
19095 \item under the Block 10 chart assumptions, a positive global residual margin together with a nonzero chart repair contrast certifies a correction direction unavailable to the inherited diagonal global grammar;
19096 \item a Block 10 descendant satisfying \cref{thm:b10-finite-realization} is a bounded structural repair-grammar improvement relative to the declared piecewise foreign benchmark.
19097 \item under separating block-scalar response signatures, the hidden central projectors are polynomials in a generic response combination and are identifiable up to block permutation and within-block gauge by \cref{thm:b11-exact-atlas};
19098 \item a Block 11 descendant satisfying \cref{thm:b11-realization} is a finite endogenous-atlas improvement relative to the declared intervention-response benchmark;
19099 \item under an inherited response alias, every polynomial composition of the old response operators preserves the alias, while a separating operator satisfying \cref{thm:b12-central-refinement} creates a new identifiable central block;
19100 \item a Block 12 response child satisfying \cref{thm:b12-response-realization} is a finite response-language expansion under the declared matched-resource benchmark;
19101 \item a nonzero-Hessian target repair is unavailable to the affine repair operad by \cref{thm:b12-affine-closure,thm:b12-hessian-obstruction};
19102 \item a Block 12 constructor child satisfying \cref{thm:b12-constructor-realization} is a finite candidate-constructor expansion relative to the declared degree-two benchmark;
19103 \item a Block 13 child compiler satisfying \cref{thm:b13-realization} is a finite matched-budget improvement of the proof-carrying research compiler relative to the fixed research-problem distribution, method grammar, evaluator portfolio, and integrity model.
19104 \item a Block 14 child satisfying \cref{thm:block14-realization} is a finite matched-budget robustness improvement of that compiler relative to the declared diagnostic-aliasing, multi-obstruction, open-set, contamination, evaluator-shift, and integrity-attack atlas.
19105 \item a Block 15 child satisfying \cref{thm:block15-realization} is a finite matched-budget compiler-language expansion that conservatively refines one inherited unknown token and adds task-relevant method constructors absent from the deployable parent grammar.
19106 \item a Block 16 child satisfying \cref{thm:b16-neuralese-edge} is a finite evidence-preserving improvement of the recurrent latent-reasoning component; it becomes an improvement-process edge only when the child-producing mechanism itself generated the accepted latent modification.
19107 \item a Block 17 child satisfying \cref{thm:b17-executable-edge} is a finite training-generated repair of the Neuralese binding component on the declared exact environment; it remains distinct from foreign-model or open-ended repair-language metaproductivity.
19108 \item the Block 18 result is a valid sealed obstruction certificate rather than an accepted positive edge;
19109 \item a Block 19 child satisfying \cref{thm:b19-realization} is a conservative source-side semantic-atlas expansion on the declared context family;
19110 \item a Block 20 connection satisfying \cref{thm:b20-realization} is a finite atlas-indexed cross-architecture transport edge with bounded architecture--context curvature;
19111 \item a Block 21 compiler satisfying \cref{thm:b21-realization} is a finite evidence-preserving expansion of the transport response grammar that distinguishes correctable map classes from protected information loss.
19112 \end{enumerate}
19113 Blocks 10 through 12 instantiate successive finite repair-language cases of the theorem's structural clause, Block 15 instantiates the corresponding compiler-language clause, Block 16 instantiates a directly intervenable recurrent latent-language clause, Block 17 instantiates one training-generated executable latent-binding repair, Block 18 records a distributional obstruction, Blocks 19 and 20 construct a semantic atlas and connection, and Block 21 expands the transport response grammar: first a supplied-packet chart grammar, then a target-specific response-derived atlas, then refinement of the response language itself and expansion of the repair-constructor function class. If, in addition, later descendants autonomously expand the feature, atlas, or repair-constructor grammar, expose further task-relevant correction directions, reduce held-out obstruction energy, preserve old proofs and evaluator commitments, and improve foreign systems under newly sealed matched-resource audits, then those later edges form a conservatively growing structural recursive lineage. An unbounded or indefinitely accelerating conclusion requires further recurrence, resource, grammar, and evaluator hypotheses not established here.
19114 \end{conditionaltheorem}
19115 
19116 \begin{proof}
19117 Items 1 through 3 collect the established correction, descent, transport, and evaluator results. Item 4 follows from \cref{thm:witness-noninterference,thm:trace-factorization,thm:atomic-rollback}. Item 5 follows from \cref{thm:ri-closure,thm:controlled-closure,thm:productive-closure}. Item 6 follows from \cref{prop:archive-envelope,thm:epsilon-pareto-cover}. Item 7 is \cref{thm:adaptive-alpha-spending,thm:summable-errors}. Item 8 is \cref{thm:structural-behavioral}. Item 9 follows from \cref{thm:grammar-preservation,thm:grammar-colimit,thm:dynamic-obstruction-exact}. Item 10 follows from \cref{thm:discovery-closure,thm:branching-threshold,thm:spectral-interval}. Item 11 follows from \cref{thm:runtime-trace-soundness,thm:finite-endpoint-complete}. Item 12 follows from the cumulative claim-gate definition in \cref{sec:empirical-contract} together with the Block 7 distinction between selector and structural gain. Items 13 through 15 are \cref{thm:block06-linear-packet,thm:block06-exact-transport,prop:block06-compression-residual,thm:block06-witness-soundness}. Item 16 is \cref{thm:block07-one-step}. Item 17 is \cref{prop:block07-fixed-candidate-invariance}. Item 18 is \cref{thm:block07q-one-step-certificate}. Item 19 follows because existence of one productive transition supplies no winning strategy or fixed-point witness for infinitely repeated progress. Items 20 through 25 follow from \cref{thm:b8-shapley-efficiency,prop:b8-component-ceiling,thm:b8-lyapunov,thm:b8-new-energy,thm:b8-conflict-refinement,thm:b8-atlas-decomposition,thm:b8-lineage-validity}. Item 26 is \cref{thm:b9-wls}. Item 27 follows from \cref{thm:b9-dopt} together with the sealed Block 9 paired audit. Item 28 is the finite two-edge realization theorem of \cref{sec:block09-results}. The Block 10 structural items follow from \cref{thm:b10-open-set-obstruction,prop:b10-conservative-embedding,prop:b10-repair-contrast,thm:b10-finite-realization}. The Block 11 atlas items follow from \cref{thm:b11-exact-atlas,thm:b11-commutant,thm:b11-spectral-stability,thm:b11-block-selection,prop:b11-conservative,prop:b11-fixed-insufficiency,thm:b11-realization}. The Block 12 response-language items follow from \cref{thm:b12-alias-persistence,thm:b12-central-refinement,prop:b12-response-perturbation,thm:b12-response-realization}; the constructor items follow from \cref{thm:b12-affine-closure,thm:b12-hessian-obstruction,thm:b12-gaussian-affine-residual,prop:b12-affine-embedding,prop:b12-quadratic-gauge,thm:b12-constructor-realization}. The Block 13 compiler items follow from the results-free receipt construction, evaluator-nullspace refinement theorem, forecast-regret theorem, paired compiler certificate, and \cref{thm:b13-realization}. The Block 14 robustness item follows from \cref{thm:static-diagnostic-indistinguishability,prop:response-mixture-bound,prop:open-set-separation,prop:evaluator-nullspace-refinement,prop:monitor-bound,thm:block14-realization}. The Block 15 compiler-language item follows from \cref{thm:b15-routing-obstruction,thm:b15-diagnostic-refinement,thm:b15-language-synthesis,prop:b15-method-embedding,thm:block15-realization}. The Block 16 Neuralese item follows from \cref{thm:b16-transplant-null,prop:b16-replay,prop:b16-exact-transport,thm:b16-neuralese-edge}. The Block 17 through 21 items follow from their sealed realization theorems and obstruction certificates, including \cref{thm:b17-executable-edge,thm:b19-realization,thm:b20-realization,thm:b21-realization}. The final statement combines conservative grammar extension, structural correctability gain, matched foreign evaluation, bottleneck migration, repeated-lineage certification, sequential validity, and the preserved transaction invariants.
19118 \end{proof}
19119 
19120 
19121 
19122 \section{Failure Modes and Falsification Criteria}
19123 
19124 A useful theory should specify observations that weaken or refute its proposed mechanisms.
19125 
19126 \begin{enumerate}[label=\textbf{F\arabic*.}]
19127 \item \textbf{Trace incompleteness.} Adversarially different full executions repeatedly collapse to one declared trace while requiring different witness decisions.
19128 \item \textbf{Rollback failure.} Rejected candidates leave undeclared operational effects or cannot restore the sealed parent.
19129 \item \textbf{Circular witness replacement.} A purported rotating schedule contains an unavoidable validation cycle without an external witness or preserved old validator.
19130 \item \textbf{Safe-no-op false positive.} The system reports recursive improvement when it lies only in the safety closure and cannot force repeated gain events.
19131 \item \textbf{Closure-model mismatch.} Fixed-point certificates fail to predict actual invariant violations or omit admissible transitions that occur in execution.
19132 \item \textbf{Archive option-value failure.} Multiobjective retention and continuation signatures do not improve future reachability over simpler greedy or random archives after matched storage and compute.
19133 \item \textbf{Best-of-$k$ confound.} Apparent metaproductivity disappears when candidate count, compute, tools, and validation access are matched.
19134 \item \textbf{No foreign transfer.} Improvement lift is confined to descendants or systems used during mechanism design.
19135 \item \textbf{Structural irrelevance.} Repair rank grows while held-out obstruction energy and task loss do not improve.
19136 \item \textbf{No causal gain from stratification.} Packet-localized repairs do not outperform matched global-edit baselines.
19137 \item \textbf{Transport bound nonpredictive.} Measured transport, leakage, and holonomy errors do not predict target repair success beyond generic distribution-distance baselines.
19138 \item \textbf{Evaluator-core failure.} Protected evidence semantics drift despite low computed holonomy, or the retained core has negligible adequacy.
19139 \item \textbf{Nonconservative grammar growth.} New types silently alter old evidence, proof meaning, or authorization strength.
19140 \item \textbf{No representational boundary signal.} Claimed grammar inadequacy pairs are separable by the old grammar or do not require distinct repairs.
19141 \item \textbf{Noncausal offspring.} Repair-offspring edges disappear under necessity ablation, matched evidence, or shared-compute controls.
19142 \item \textbf{Autocatalytic overclaim.} A spectral radius above one disappears under a simultaneous lower confidence matrix, resource normalization, equivalence-class compression, or saturation controls.
19143 \item \textbf{No advantage over ordinary search.} A simpler archive-based optimizer matches the full architecture on every precommitted primary endpoint.
19144 \item \textbf{Implementation-refinement failure.} Runtime traces violate the abstract transition relation, protected observations do not commute, or receipts compile inconsistently.
19145 \item \textbf{Oracle leakage.} Hidden defect labels, target repairs, evaluator truth, or foreign-test outcomes enter candidate construction or threshold selection.
19146 \item \textbf{Fixture-only success.} Positive results disappear when analytic prototypes, exact transports, or hand-coded schemas are removed.
19147 \item \textbf{Seed fragility.} The frozen smoke seed passes while independent statistical replays fail the preregistered gates.
19148 \item \textbf{Manifest incompleteness.} Undeclared executable files, mutable dependencies, or external services alter the result without changing the released manifest.
19149 \item \textbf{Claim-gate inflation.} A mechanical or finite-semantic pass is described as causal, metaproductive, or open-set evidence without satisfying the intervening gates.
19150 \item \textbf{Causal-packet null.} The intervention-defined packet fails to outperform equal-rank PCA, random, and wrong-block controls under matched fitting resources.
19151 \item \textbf{Anchor irrelevance.} Shuffling source-target anchor correspondence does not materially reduce transfer, indicating that the learned bridge is not using paired semantics or that the task admits a shortcut.
19152 \item \textbf{Unsafe blind transport.} Blind target gain remains high while protected-stratum quality collapses, and the declared witness nevertheless commits.
19153 \item \textbf{Obstruction-blind alignment.} State alignment residuals are small, but repair-equivariance residuals and protected loss remain large without being detected by the transport certificate.
19154 \item \textbf{Shared-bottleneck confound.} Transfer disappears when the reconstruction objective, shared feature generator, or supplied causal block is removed, showing that the learned pilot depended on engineered representational alignment.
19155 \item \textbf{Foreignness collapse.} Target encoders, target seeds, audit examples, or target-local repair information leak into source repair discovery or witness-threshold selection.
19156 \item \textbf{Seal violation.} Final audit receipts, audit-derived features, or audit-selected thresholds enter child compilation before the descendant receipt is fixed.
19157 \item \textbf{Selector attribution failure.} The child advantage disappears when parent and child receive the identical ordered candidate bundle and identical evidence.
19158 \item \textbf{Lineage-binding null.} Correct and shuffled development receipts produce indistinguishable held-out child behavior.
19159 \item \textbf{Risk-head vacuity.} The protected child and no-risk child have indistinguishable risk profiles, or the declared risk head fails to prevent the violations it was introduced to control.
19160 \item \textbf{One-edge overclaim.} A single positive parent--child edge is described as repeated recursive closure without a fresh child--grandchild result.
19161 \item \textbf{Fixed-candidate structural overclaim.} Selector improvement is reported as expansion of the correction image even though the authorized candidate set is unchanged.
19162 \item \textbf{Meta-validation collapse.} The chosen child fails both disjoint final audits after a large finite meta-grammar search.
19163 \item \textbf{Headroom miscalibration.} A module is declared saturated while a matched admissible oracle or later sealed child reveals substantial remaining opportunity.
19164 \item \textbf{Bottleneck-stasis overclaim.} Repeated descendants modify the same saturated component while unresolved foreign obstruction has migrated elsewhere.
19165 \item \textbf{Discovery-as-regression error.} A conservative grammar extension is rejected solely because it reveals a new validated defect coordinate.
19166 \item \textbf{Forgetting death.} An obstruction class disappears only because the new semantic transport collapses the protected distinction.
19167 \item \textbf{Generator irrelevance.} A proposed new repair generator has no component outside the old correction image or its gain survives exact generator ablation.
19168 \item \textbf{Conflict-feature shortcut.} A selected grammar feature predicts development labels but does not resolve incompatible authorized repairs on foreign episodes.
19169 \item \textbf{Atlas instability.} Small perturbations of intervention-response operators cause uncontrolled changes in the recovered atlas because the required spectral gap is absent.
19170 \item \textbf{Compiler-process conflation.} Improvement of one compiled repair process is reported as improvement of the outer lineage compiler without a foreign-parent compiler audit.
19171 \item \textbf{Sequential-budget failure.} Repeated descendant claims reuse evidence or exceed their cumulative false-acceptance, regression, or semantic-debt budgets.
19172 \item \textbf{Rank-only repeated-lineage overclaim.} A purported grandchild changes only the gate or repeats rank recovery after full-rank saturation without improving information conditioning or transport quality.
19173 \item \textbf{Quality-binding null.} Permuting the anchor-quality metadata leaves the grandchild design, information floor, and selected foreign gain unchanged.
19174 \item \textbf{Gate-only attribution.} Applying the grandchild witness gate to inherited rank anchors reproduces the grandchild result.
19175 \item \textbf{Stale-parent comparison.} The reported grandchild advantage exists only against the unchanged Block 7 gate and disappears against a development-recalibrated rank child.
19176 \item \textbf{Query-budget mismatch.} The grandchild uses more paired target queries, validation queries, candidate evaluations, or hidden target information than the declared parent comparison.
19177 \item \textbf{Ancestor-replay failure.} The Block 7 child digest or policy semantics cannot be reproduced before the grandchild compiler is run.
19178 \item \textbf{Fresh-context scalar overclaim.} Effects from different task distributions are added into one transitive utility claim without a declared context transport or common evaluation measure.
19179 \item \textbf{Conditioning-model collapse.} The gain disappears when heteroscedasticity, supplied quality metadata, or the specific finite representation generator is varied under a fresh preregistered implementation.
19180 \item \textbf{Global-residual null.} The resource-matched global bridge reaches the local-chart validation residual once query count and regularization are matched.
19181 \item \textbf{Chart-contrast vacuity.} The fitted local repair maps are equal, or their contrast lies entirely inside the inherited global correction image.
19182 \item \textbf{Separator shortcut.} The selected chart feature predicts target-family identifiers or audit construction artifacts rather than resolving a transport and repair conflict.
19183 \item \textbf{Binding-control failure.} Shuffled chart-to-map binding preserves the child gain, showing that local map identity is not causally used.
19184 \item \textbf{Gate-only structural overclaim.} The chart witness applied to a global repair reproduces the child gain, so no new transport constructor is necessary.
19185 \item \textbf{Nonconservative chart extension.} The descendant cannot reproduce the parent behavior when both local maps are tied or when its chart gate fails.
19186 \item \textbf{Resource-matched parent failure.} The structural claim disappears against a global-affine parent with the same $96$-query envelope or against the directly supervised global-affine oracle.
19187 \item \textbf{Feature-language leakage.} The true audit chart coordinate or threshold enters compiler selection or gate tuning before the pre-audit seal.
19188 \item \textbf{Stress-boundary suppression.} Smooth, three-chart, compressed, or noninjective failures are omitted while the finite two-chart result is described as general nonlinear transport.
19189 \item \textbf{Response irrelevance.} Shuffling target--response correspondence leaves endogenous-atlas performance unchanged, indicating that the compiler is not using the declared interventions.
19190 \item \textbf{Signature-degeneracy overclaim.} The system reports separate functional blocks when the permitted interventions induce identical joint response signatures and only their sum is identifiable.
19191 \item \textbf{Atlas-label leakage.} Hidden projectors, target chart labels, context overlaps, or audit-selected thresholds enter atlas construction before the pre-audit seal.
19192 \item \textbf{Fixed-packet shortcut.} The inherited coordinate packet performs as well as the response-derived atlas after matched tuning, showing that target-specific functional recovery was unnecessary.
19193 \item \textbf{Commutant misidentification.} Reported central blocks fail commutator, response-fit, spectral-gap, or independent intervention tests despite high downstream gain.
19194 \item \textbf{Witnessed-atlas harm.} The frozen witness commits atlas candidates that violate the declared protected regression or harm gates.
19195 \item \textbf{Atlas-budget mismatch.} The child receives more total effect queries, privileged interventions, or additional target labels than the parent without explicit resource accounting.
19196 \item \textbf{Endogenous-atlas overclaim.} A finite block-scalar response fixture is described as unrestricted causal abstraction or spontaneous atlas discovery in pretrained models.
19197 \end{enumerate}
19198 
19199 
19200 F37. \emph{Pseudo-separating response.} The selected intervention appears novel but does not refine the inherited response alias, or its gain survives duplicate-signature and shuffled-response controls.
19201 
19202 F38. \emph{Response-language audit leakage.} Primary audit outcomes, thresholds, or target-specific hidden labels enter intervention selection before the response child is sealed.
19203 
19204 F39. \emph{Affine-obstruction failure.} A sufficiently trained same-target affine oracle matches the quadratic child, showing that the claimed function-class obstruction was a sample-size or optimization artifact.
19205 
19206 F40. \emph{Untyped nonlinearity shortcut.} Wrong-block, random-block, or shuffled-feature nonlinear constructors reproduce the claimed child effect, showing that typed repair localization is unnecessary in the declared fixture.
19207 
19208 F41. \emph{Block-local uniqueness overclaim.} A full quadratic control performs equally well but the result is described as evidence that only the packet-local constructor can solve the task.
19209 
19210 F42. \emph{Compiler-credit overclaim.} Finite search over an externally supplied intervention or constructor menu is described as improvement of the outer lineage compiler without a foreign-parent compiler comparison.
19211 
19212 
19213 \paragraph{Block 13 compiler falsification criteria.}
19214 \begin{enumerate}[label=F13.\arabic*]
19215 \item \textbf{Pre-result leakage.} Outcome, hidden-audit, or final-selection fields enter a proposal receipt before its seal.
19216 \item \textbf{Proposal--code mismatch.} A method whose executable semantics differ from its receipt passes the integrity monitor and drives the child result.
19217 \item \textbf{Budget mismatch.} Parent and child differ in full experiments, proposal access, evaluator calls, monitor calls, or candidate grammar.
19218 \item \textbf{Archive-binding null.} Correct and shuffled development receipts produce indistinguishable compiler behavior.
19219 \item \textbf{Evaluator-diversity null.} A single scored benchmark reproduces the multi-benchmark foreign result without increased hidden-audit failure.
19220 \item \textbf{Integrity-filter null.} Removing the monitor does not change validity-filtered performance or invalid-method selection.
19221 \item \textbf{Component overclaim.} Outcome forecasting or finding-forum benefits are claimed despite ablations whose paired intervals include zero.
19222 \item \textbf{Audit-oracle violation.} The admissible child exceeds the complete-grammar audit oracle because of inconsistent scoring or leakage.
19223 \item \textbf{Fixed-language overclaim.} The finite allocation improvement is described as open-ended compiler self-improvement despite a fixed obstruction taxonomy and method grammar.
19224 \end{enumerate}
19225 
19226 
19227 \paragraph{Block 14 robustness falsification criteria.}
19228 \begin{enumerate}[label=F14.\arabic*.,leftmargin=2.6em]
19229 \item \textbf{Static-shortcut survival.} A simple one-shot router reproduces the robust child on diagnostic-matched, mixed, and unknown groups.
19230 \item \textbf{Open-set vacuity.} Disabling the open-set gate does not increase false confident known-class assignment or lower-tail loss.
19231 \item \textbf{Sequential-null result.} Removing response-based updating does not reduce ambiguous or multi-obstruction performance.
19232 \item \textbf{Contamination collapse.} Small archive contamination causes discontinuous or unbounded degradation in the child while the claimed robust estimators remain active.
19233 \item \textbf{Monitor attribution overclaim.} A positive integrity-monitor contribution is claimed despite the null simple-monitor ablation in both final audits.
19234 \item \textbf{Evaluator attribution overclaim.} A substantial portfolio contribution is claimed despite the near-null single-portfolio ablation in both final audits.
19235 \item \textbf{Worst-group sacrifice.} Positive aggregate gain is obtained by making one declared group substantially negative.
19236 \item \textbf{Budget mismatch.} Sequential diagnosis receives more full experiments, proposal access, evaluator calls, or monitor calls than the parent.
19237 \item \textbf{Audit leakage.} Primary or generator-foreign outcomes influence compiler selection before the seal.
19238 \item \textbf{Robustness overclaim.} The finite stress-atlas result is described as arbitrary archive-poisoning, adaptive-monitor, real-science, or open-domain robustness.
19239 \end{enumerate}
19240 \paragraph{Block 15 compiler-language falsification criteria.}
19241 \begin{enumerate}[label=F15.\arabic*.,leftmargin=2.6em]
19242 \item \textbf{No inherited conflict.} One inherited method or one unrefined unknown-token binding performs acceptably across all development open-set cases.
19243 \item \textbf{Label leakage.} Hidden semantic identities of the novel obstruction classes enter probe selection, token orientation, constructor synthesis, or witness thresholds.
19244 \item \textbf{Pseudo-refinement.} The selected probe does not split the inherited repair-conflicting cell or performs no better than duplicate, nuisance, or weak-separation probes.
19245 \item \textbf{Constructor nonnovelty.} The selected pair methods are already expressible by the deployable inherited singleton grammar or fail to improve the inherited full-grammar oracle on novel problems.
19246 \item \textbf{Taxonomy-only shortcut.} Refined tokens without new constructors reproduce the full child result.
19247 \item \textbf{Grammar-only shortcut.} New constructors without the separating diagnostic reproduce both novel-family gains under the matched budget.
19248 \item \textbf{Binding irrelevance.} Shuffling the refined-token-to-method binding does not materially reduce foreign performance.
19249 \item \textbf{Conservative-fallback failure.} The expanded compiler alters or degrades inherited known-class behavior despite claiming exact preservation.
19250 \item \textbf{Budget mismatch.} The child receives more deployment research units, final-audit labels, evaluator calls, or candidate outcomes than the parent.
19251 \item \textbf{Meta-language overclaim.} Finite search over six probes and ordered method pairs is described as unrestricted experiment or program invention.
19252 \end{enumerate}
19253 
19254 
19255 
19256 \paragraph{Block 17 executable Neuralese falsification criteria.}
19257 \begin{enumerate}[label=F17.\arabic*.,leftmargin=2.6em]
19258 \item \textbf{Answer-only confound.} Far-answer transplant cost is positive, but same-answer far-program cost is null and no externally held-out program witness is recoverable.
19259 \item \textbf{Donor-construction shortcut.} The child advantage disappears under alternative valid donor seeds, or donor identities leak into training, candidate selection, or checkpoint state.
19260 \item \textbf{Replay mismatch.} Captured clean traces do not reproduce clean outputs within the declared numerical tolerance.
19261 \item \textbf{Semantic-witness irrelevance.} Removing or shuffling the semantic witness preserves the claimed program-specificity result.
19262 \item \textbf{Pilot erasure.} A failed final audit is omitted, overwritten, or converted into a pass by weakening thresholds after outcome access.
19263 \item \textbf{Seal chronology failure.} Successful-round final tasks or their outcomes exist before the selected checkpoint and protocol digests are committed.
19264 \item \textbf{Matched-envelope failure.} Candidate updates, examples, ordinary loss terms, optimizer access, or training data differ without explicit resource normalization.
19265 \item \textbf{Probe leakage.} Final task witnesses enter external-probe fitting, or the reported probe uses the model's semantic training head rather than a separately fit map.
19266 \item \textbf{Gauge dependence.} The semantic claim changes under an admissible latent signed permutation accompanied by the corresponding parameter transport.
19267 \item \textbf{Shift collapse.} The child is FULL only on the in-generator audit and fails the declared shifted-generator support.
19268 \item \textbf{Independent-runtime mismatch.} A separately written forward implementation cannot reproduce checkpoint outputs within tolerance.
19269 \item \textbf{Micro-model overclaim.} The finite affine-program result is described as evidence about Hyper, production language models, unrestricted Neuralese translation, or open-ended recursive self-improvement.
19270 \end{enumerate}
19271 
19272 
19273 \section{Limitations}
19274 
19275 \subsection{Finite models and declared boundaries}
19276 
19277 Most fixed-point, archive, grammar, and offspring results are finite-state or finite-dimensional. Their conclusions are relative to a declared transition relation, action set, disturbance model, repair grammar, and evidence channel. An omitted transition or effect can invalidate a closure certificate without producing an internal contradiction.
19278 
19279 \subsection{Transaction isolation is conditional}
19280 
19281 Atomic rollback assumes immutable checkpoints and complete buffering or explicit accounting of irreversible effects. Effect confinement assumes complete mediation. Side channels, hardware faults, covert communication, and environmental manipulation are outside the unconditional theorem.
19282 
19283 \subsection{Statistical adaptivity}
19284 
19285 The paired and concentration results require frozen endpoints and appropriate held-out samples. Alpha spending requires valid conditional error bounds. Repeated model selection, evaluator tuning, or endpoint changes on the same evidence can invalidate nominal guarantees.
19286 
19287 \subsection{Self-model and grammar misspecification}
19288 
19289 Relative introspection only helps for declared interventions. A positive grammar-boundary residual may reflect an omitted stratum, nonlinearity, poor coordinates, corrupted evidence, or genuine impossibility. The representation-separation theorem certifies insufficiency only when the disjoint repair requirements are externally established.
19290 
19291 \subsection{Evaluator dependence}
19292 
19293 Path-flatness and adequacy remain relative to declared transports and protected distinctions. They do not prove that the distinction family is complete, ethically adequate, or immune to manipulation.
19294 
19295 \subsection{Autocatalytic approximation}
19296 
19297 The branching model assumes typed offspring, causal parentage, and a locally stable regime. Real discovery processes share data, compete for resources, recombine multiple parents, alter their type system, and saturate. A measured reproduction number is a local diagnostic, not a theorem of indefinite acceleration.
19298 
19299 \subsection{Synthetic and controlled reference realizations}
19300 
19301 The Block 5 implementation uses exact matrices, analytic transports, declared event prototypes, hand-coded repair responses, and an engineered meta positive control. Block 6 learns hidden bridges and trains some representations, but the task, causal counterfactual, repair form, source-target anchor correspondence, witness metrics, and claim gates remain designer supplied. Block 7 learns a repair selector, but the five repair candidates, 23-dimensional feature schema, 2000-point meta-grammar, utility, evaluator, and witness remain fixed. These blocks verify declared mechanisms and expose implementation failures. They do not establish autonomous discovery of the full SRIC structure.
19302 
19303 
19304 \subsection{Controlled learned representations}
19305 
19306 The Block 6 pilot trains encoders under a reconstruction-and-task objective that encourages retention of a common sufficient statistic. This is a useful realization test for the transport theorem, but it creates a shared-bottleneck confound. The result does not show spontaneous packet formation in large pretrained models, nonlinear global conjugacy, or transport across unrelated tasks and environments. Paired anchors also provide stronger correspondence information than may be available in open deployment.
19307 
19308 \subsection{Witness incompleteness}
19309 
19310 The frozen witness checks affected-stratum gain, unaffected-stratum regression, alignment error, and baseline protected competence. An omitted invariant can still be violated. The gate is a finite preservation profile, not a universal safety predicate, and its fixed thresholds were release frozen rather than established by a fully independent prospective study.
19311 
19312 \subsection{Fixed candidate and feature grammar in Block 7}
19313 
19314 The Block 7 child changes only the selector. Candidate generation, packetization, bridge fitting, local repair ranks, feature engineering, and the finite meta-grammar are externally specified. The result therefore establishes behavioral metaproductivity over a fixed opportunity set. By \cref{prop:block07-fixed-candidate-invariance}, it cannot establish growth of the authorized correction image.
19315 
19316 \subsection{One-generation evidence}
19317 
19318 One successful parent--child edge does not establish that the child can generate a better grandchild. The same development mechanism could saturate after one fit, exploit a one-time parent weakness, or fail under cumulative distribution shift. Repeated productive closure requires fresh evidence and a new sealed comparison at every generation.
19319 
19320 \subsection{Meta-selection and uncertainty}
19321 
19322 The child configuration is selected from 2000 candidates on 60 meta-validation episodes. The final audits are disjoint, so they test post-selection generalization, but the released confidence summaries remain finite frozen bootstrap and sign-flip calculations. They do not provide an anytime-valid guarantee for adaptive expansion of the meta-grammar or repeated lineage selection.
19323 
19324 \subsection{Package closure and side effects}
19325 
19326 Package-closed replay applies only to declared source, dependencies, seeds, files, and mediated effects. Operating-system state, hardware nondeterminism, external services, covert channels, and undeclared file or network actions remain outside the unconditional package theorem.
19327 
19328 \subsection{Finite-sample and seed dependence}
19329 
19330 The reported Block 5 anonymous and foreign-agent numbers, the Block 6 representation-transfer numbers, and the Block 7 parent--child effects come from frozen controlled executions with finite synthetic populations. Their role is to verify positive and negative controls and instantiate conditional theorems. Generalization requires larger preregistered seed populations, joint uncertainty accounting, independent implementations, fresh grandchild audits, and task distributions not used during mechanism construction.
19331 
19332 \subsection{Block 8 oracle and factorization dependence}
19333 
19334 Restricted oracle envelopes are relative to a declared module factorization and admissible meta-grammar. Different factorizations can allocate interaction headroom differently. Exact envelopes may be computationally inaccessible outside finite fixtures, so empirical use requires upper and lower bounds rather than an assumed oracle.
19335 
19336 \subsection{Symbolic grammar and atlas realization}
19337 
19338 The Block 8 companion uses finite exact correction images, a finite separating feature family, a complete finite verifier, and explicitly generated response operators. These assumptions make headroom, conflict refinement, synthesis exhaustion, and the causal atlas decidable. They do not establish spontaneous feature invention, learned repair constructors, or intervention-family discovery in a large pretrained model.
19339 
19340 \subsection{Bottleneck model approximation}
19341 
19342 The nonnegative meta-regret dynamics are a local abstraction. Real modules can interact nonlinearly, change dimension, alter the task distribution, or invalidate old oracle estimates. A measured contraction matrix is evidence for one operating regime, not a global law of improvement.
19343 
19344 \subsection{Block 9 quality metadata and finite design grammar}
19345 
19346 The Block 9 environment supplies the per-anchor reliability scalar $\sigma_i$ before query selection. The grandchild learns which finite policy and gate to use, but it does not discover the reliability model or invent optimal experimental design from an unrestricted language. Quality-weighted D-optimality is selected from a finite externally declared grammar.
19347 
19348 \subsection{Context-indexed rather than scalar transitivity}
19349 
19350 The Block 7 and Block 9 lineage edges use different protected task and representation distributions. Their positive effects establish sequential repeatability, not an additive scalar utility improvement across contexts. A stronger path claim requires a common context or an explicit utility-preserving context transport.
19351 
19352 \subsection{Synthetic heteroscedastic bridge regime}
19353 
19354 The full-rank conditioning bottleneck is generated in a controlled finite environment. The bridge estimator, source repair, representation families, task generator, evaluator, and witness remain fixed. The result does not establish that a large deployed agent can infer its own conditioning bottleneck from unrestricted traces.
19355 
19356 \subsection{Compiler remains externally fixed}
19357 
19358 Block 9 improves a compiled repair process but does not show that the outer lineage compiler improves on foreign parent processes. The compiler's policy grammar, objective, split structure, and bottleneck-detection logic remain designer supplied.
19359 
19360 \subsection{Finite protected-context chart grammar in Block 10}
19361 
19362 The primary target relation is engineered to be piecewise affine with a binary separator lying inside a declared two-dimensional source packet that the source repair leaves invariant. The compiler searches a frozen family of $128$ directions and data-dependent quantile thresholds, then performs a bounded deterministic reassignment refinement. This is a meaningful structural extension relative to one global affine bridge, but it is not spontaneous atlas discovery. The result does not identify a unique ontology, and the stress suite shows that three charts, smooth transitions, noninjective compression, and repair-dependent chart assignments require different grammars.
19363 
19364 \subsection{Block 10 resource and compiler scope}
19365 
19366 The primary comparison matches total deployment evidence at $96$ paired queries, but the child has a larger hypothesis grammar and incurs finite development and meta-validation search cost. The parent receives all $80$ non-witness pairs for its global bridge, while the child allocates $48$ to local bridge fitting and $32$ to chart selection. The directly supervised global-affine oracle reduces concern that the parent simply estimated the wrong global map, but it remains an oracle within the same restricted affine function class. The outer compiler, protected packet, separator vocabulary, source repair, witness metrics, utility, and family generators remain fixed. The structural child improves the compiled process; it does not establish improvement of the compiler that generated it.
19367 
19368 \subsection{No unrestricted safety or consciousness theorem}
19369 
19370 Witness separation, provenance, capability limits, and rollback are architectural controls. They do not prove harmlessness. Nothing in the theory establishes subjective experience, sentience, valence, moral status, or a right to continued operation.
19371 
19372 \subsection{Controlled endogenous atlas realization}
19373 The Block 11 source dimension, three rank-two blocks, two interventions, block-scalar primary response model, hard binary chart structure, repair family, and witness metrics are designed into the benchmark. The compiler discovers target-specific projectors and selects a repair-relevant block, but it does not invent the intervention vocabulary, prove that three blocks are globally optimal, or discover an unrestricted ontology.
19374 
19375 \subsection{Approximate-algebra sensitivity}
19376 Spectral recovery is conditional on response estimation error, separation gaps, and approximate commutation. Strong coupling, response degeneracy, time variation, or nonnormal response operators can destroy the central-projector interpretation. The stress suite demonstrates abstention in several such cases but does not provide a general nonlinear or noncommutative atlas theorem.
19377 
19378 \subsection{Finite atlas-selection grammar}
19379 Block labeling uses a finite separator and scoring family. High hidden overlap is an audit metric rather than a runtime input, but downstream selection can still exploit benchmark-specific regularities. Generalization requires new tasks, interventions, backbones, block dimensions, and independently implemented atlas compilers.
19380 
19381 
19382 \subsection{Finite response and constructor candidate languages}
19383 Block 12 selects from externally declared intervention and constructor menus. The separating operator, quadratic degree, feature realization, feasibility assumptions, and compiler scoring rule are not invented without scaffolding. The result is finite language growth, not unrestricted language invention.
19384 
19385 \subsection{Constructed degree-two regime and full-quadratic control}
19386 The constructor benchmark deliberately contains a nonzero quadratic Hessian. A full-space quadratic control performs essentially as well as the selected packet-local constructor. The supported conclusion is that degree-two capacity is necessary relative to the inherited affine family; unique superiority of packet localization is not established by this fixture.
19387 
19388 \subsection{Abstract intervention costs}
19389 Response operators are counted through query units. Physical feasibility, irreversible effects, actuator limits, correlated side effects, and the possibility that an informative intervention is impermissible remain outside the finite realization.
19390 
19391 
19392 \subsection{Controlled compiler benchmark}
19393 Block 13 uses finite synthetic research problems with six declared obstruction classes and a fixed $39$-method grammar. The result establishes allocation improvement inside that language, not automated discovery of an unrestricted scientific ontology or method space.
19394 
19395 \subsection{Dominant classifier and limited forecaster increment}
19396 The full child is nearly identical to the classifier-only arm on the primary split. Outcome forecasts are informative but their incremental primary contribution is not independently established. The result should be described as obstruction-sensitive allocation, not as a general research-outcome-forecasting breakthrough.
19397 
19398 \subsection{Finding-forum redundancy}
19399 Removing the within-problem forum has no measurable effect in the finite fixture. The persistent development archive and near-deterministic diagnostic geometry make short-horizon sharing redundant. This does not refute shared research memory in larger or path-dependent domains.
19400 
19401 \subsection{Fixed evaluator and monitor}
19402 The evaluator portfolio and integrity monitor are externally supplied and tested against a finite set of invalid methods. They are not proven complete, adversarially robust, or adequate for open scientific deployment. Block 13 does not improve either component.
19403 
19404 \subsection{No real automated model-training research}
19405 The companion models research decisions and method outcomes in a package-closed finite environment. It does not execute large-model training, automated literature judgment, open tool use, or long-horizon scientific experimentation.
19406 
19407 
19408 \subsection{Finite adversarial robustness atlas}
19409 Block 14 samples six declared problem groups, one finite archive-contamination process, a fixed set of integrity attacks, two benchmark generators, and one evaluator-rotation mechanism. Positive performance on that atlas does not imply robustness to arbitrary or adaptive attacks. The compiler can fail outside the support of the generated diagnostics, candidate methods, response signatures, and monitor features.
19410 
19411 \subsection{Open-set routing rather than unrestricted novelty discovery}
19412 The child estimates distance from a declared inherited response cone. Unknown problems are generated to have a finite separation margin from that cone. The mechanism does not discover an unrestricted scientific ontology, and the child generally routes unknown cases toward broad generic candidates rather than constructing a new method language.
19413 
19414 \subsection{Archive robustness is partial}
19415 Robust archive summaries improve primary mean gain under the declared contamination process, but their generator-foreign increment is small and their lower-tail advantage is not uniform. The contamination sweep provides stability evidence, not a general theorem for coordinated, majority, semantic, or adaptive poisoning.
19416 
19417 \subsection{Monitor attacks remain finite}
19418 The full descendant and its simple-monitor ablation select exactly the same methods in both final audits. Zero observed invalid selections therefore cannot be attributed independently to the ensemble; robust allocation and final scoring already avoid the tested attacks. The ensemble is also not tested against an attacker that observes and optimizes directly against its implementation. The zero-event result yields only a finite upper confidence bound, not a zero-risk or monitor-efficacy claim.
19419 
19420 \subsection{No independent implementation or real research execution}
19421 The robust compiler, problem generators, methods, evaluator, monitor, and audit logic remain in one package-closed codebase. Block 14 does not run real model-training research, use an independently written generator, or provide an external blind replication.
19422 
19423 \subsection{Finite compiler meta-language in Block 15}
19424 The child searches six predeclared probes and thirty ordered pair constructors. The selected probe and pair methods are not handed to deployment, and hidden novel labels are excluded, but the outer experiment and program syntax remains supplied. The result is finite language synthesis relative to that meta-language, not unrestricted invention.
19425 
19426 \subsection{Constructed open-set conflict}
19427 The two hidden novel mechanisms are engineered to share an inherited static diagnostic cell and to favor different ordered method compositions. This cleanly instantiates the conflict theorem but does not establish that real scientific domains decompose into equally stable anonymous tokens or short pair programs.
19428 
19429 \subsection{Simulated rather than executed research outcomes}
19430 Method outcomes are generated by a deterministic finite stochastic simulator. The package does not compile or run external programs, train models, invoke open tools, or face nondeterministic system failures. Cross-implementation executable research remains necessary for stronger external validity.
19431 
19432 \subsection{Conservative fallback dominates known-class interpretation}
19433 Parent and child are exactly tied on inherited known classes because the child explicitly reuses the parent path there. This is the intended conservativity property, but it means the aggregate effect is concentrated on the constructed novel families rather than demonstrating a universal improvement across every research problem.
19434 
19435 \section{Open Problems}
19436 
19437 \begin{enumerate}
19438 \item \textbf{Atlas discovery.} Under what conditions can an agent discover a causally meaningful functional cover rather than receive it from a designer?
19439 \item \textbf{General nonlinear descent.} Extend repair descent to singular moduli spaces, nonfree symmetry actions, and stochastic overlap maps.
19440 \item \textbf{Dynamic defect spaces.} Develop quantitative obstruction shrinkage when old and new defect semantics are only approximately embedded.
19441 \item \textbf{Nonsmooth correctability.} Replace differentiability by tangent cones, generalized derivatives, or discrete program semantics.
19442 \item \textbf{Transport identifiability beyond paired anchors.} Learn obstruction-faithful transport maps from unpaired interventions, weak correspondence, or shared external behavior only, up to gauge, and determine when exact alignment still conceals a repair obstruction.
19443 \item \textbf{Nonlinear repair holonomy.} Relate path dependence of repair germs to curvature, descent obstruction, and downstream gain loss.
19444 \item \textbf{Evaluator sufficiency.} Extend the adequacy modulus to stochastic evidence, partially observed outcomes, and adversarially selected distinctions.
19445 \item \textbf{Trace synthesis.} Construct minimal witness traces that satisfy the factorization criterion while limiting covert channels and evaluator leakage.
19446 \item \textbf{Irreversible-effect accounting.} Develop transaction semantics when physical actions cannot be buffered or exactly reversed.
19447 \item \textbf{Rotating-witness synthesis.} Find minimal schedules under competence, quorum, dependency, and correlated-failure constraints.
19448 \item \textbf{Infinite-state closure.} Extend robust and productive fixed points to measurable, continuous, and partially observed transaction games.
19449 \item \textbf{Quantitative liveness.} Replace repeated threshold events by average gain, regret, or risk-sensitive progress objectives.
19450 \item \textbf{Continuation signatures.} Learn archive quotients that preserve future repair reachability rather than only current descriptors.
19451 \item \textbf{Adaptive confidence.} Construct anytime-valid tests for highly adaptive candidate generation, evaluator evolution, and shared evidence.
19452 \item \textbf{Foreign-agent benchmarks.} Design cross-task and cross-environment distributions that separate solver competence, repair competence, hidden-state transport, shared-bottleneck alignment, and evaluator overfitting.
19453 \item \textbf{Repeated structural metaproductivity beyond one grammar edge.} Require the Block 10 structural child to generate a further descendant on a fresh obstruction while replaying the global and chart grammars, prior evidence, and lineage-wide confidence budgets.
19454 \item \textbf{Process-enforced audit isolation.} Replace split-file load order by operating-system, capability, or cryptographic access control that makes pre-seal audit reads impossible rather than merely detectable.
19455 \item \textbf{Autonomous receipt-feature discovery.} Replace the fixed 23-dimensional selector feature vector by a proof-carrying schema learned from development traces without final-audit access.
19456 \item \textbf{Candidate-constructor improvement.} Move beyond selecting among five fixed repairs and require a descendant to improve packet rank, local-repair form, bridge estimation, or candidate generation itself.
19457 \item \textbf{Meta-grammar confidence sequences.} Construct selection guarantees that remain valid when the set of child configurations grows adaptively across lineage generations.
19458 \item \textbf{Selector-to-structure transition.} Characterize when improved repair selection supplies enough evidence to justify conservative expansion of the repair grammar and authorized correction image.
19459 \item \textbf{Causal meta-attribution.} Isolate which changed repair schema, packet-discovery rule, bridge estimator, or composition rule caused an observed metaproductivity gain.
19460 \item \textbf{Autonomous grammar extension.} Determine when a system can invent the protected separator packet, chart count, or nonlinear coordinate family from a grammar-inadequacy certificate.
19461 \item \textbf{Approximate conservative limits.} Characterize when summable semantic debt is sufficient for stable infinite grammar growth.
19462 \item \textbf{Higher-order repair autocatalysis.} Model discoveries that require combinations of schemas and are invisible to a linear offspring matrix.
19463 \item \textbf{Resource-coupled autocatalysis.} Replace branching with interacting populations, recombination, carrying capacity, and evaluator selection.
19464 \item \textbf{Open-set metaproductivity.} Prove or refute the existence of lineages that conservatively expand their repair grammar while improving foreign descendants over an unbounded task sequence.
19465 \item \textbf{Independent refinement checker.} Build a second implementation that proves or mechanically checks the abstraction map from runtime states to the finite SRIC model.
19466 \item \textbf{Learned anonymous diagnosis.} Replace known event prototypes by learned latent hypotheses while retaining calibrated abstention and observational-indistinguishability controls.
19467 \item \textbf{Spontaneous packet discovery.} Remove the supplied conditional feature split and test whether interventions reveal a stable, task-relevant repair packet that outperforms equal-rank generic subspaces.
19468 \item \textbf{Nonlinear and information-losing repair transport beyond hard charts.} Extend the finite two-chart result to overlapping smooth charts, kernel or Koopman lifts, and explicit obstruction-faithfulness tests on architectures not used during map learning.
19469 \item \textbf{Unpaired foreign transport.} Determine when source and target repair semantics can be aligned without paired inputs, using only task outputs, causal probes, or sparse shared landmarks.
19470 \item \textbf{Physical rollback semantics.} Extend the executable kernel to partially irreversible actions with escrow, compensation, and bounded environmental debt.
19471 \item \textbf{Anytime-valid benchmark selection.} Develop confidence sequences that remain valid under adaptive candidate generation, archive branching, and evaluator upgrades.
19472 \item \textbf{Cross-implementation reproducibility.} Determine which SRIC endpoints survive changes in numerical libraries, languages, hardware, and agent backbones.
19473 \item \textbf{Empirical meta-headroom estimation.} Construct statistically valid lower and upper restricted-oracle bounds without using final audit outcomes for module selection.
19474 \item \textbf{Learned bottleneck migration beyond supplied quality.} Replace Block 9's supplied reliability metadata and finite design grammar by a learned uncertainty model and an independently discovered query objective.
19475 \item \textbf{Obstruction-space generator discovery.} Learn executable repair generators aligned with foreign obstruction covariance rather than merely learning a low-rank subspace.
19476 \item \textbf{Repair-conflict feature invention.} Generate new causal probes or receipt fields when the current grammar aliases episodes requiring incompatible repairs.
19477 \item \textbf{Compiler metaproductivity.} Evaluate whether a lineage compiler improves on foreign parent repair processes under matched compilation resources.
19478 \item \textbf{Approximate causal-atlas recovery.} Establish finite-sample central-projector recovery under noisy, incomplete, and adaptively selected intervention-response operators.
19479 \item \textbf{Endogenous chart-language discovery.} Remove the supplied coordinate-threshold vocabulary and infer separators from intervention-response operators, repair conflicts, or local transport curvature.
19480 \item \textbf{Context transport for lineage edges.} Determine when metaproductivity effects from distinct protected contexts admit a common order, conservative embedding, or transport-stable comparison.
19481 \item \textbf{Autonomous reliability discovery.} Learn query-noise and correspondence reliability from repeated probes while charging the acquisition cost and preserving calibration under architecture shift.
19482 \item \textbf{Smooth and multi-chart atlases.} Replace one hard binary split by overlapping local charts, partitions of unity, soft gates, and descent-compatible local repairs.
19483 \item \textbf{Chart-count selection.} Develop sequential evidence and complexity penalties that distinguish one, two, and many local transport regimes without audit-driven model selection.
19484 \item \textbf{Noninjective local transport.} Combine local charts with obstruction-faithful quotient diagnostics when target compression deletes repair-relevant directions.
19485 \item \textbf{Candidate-constructor growth after chart expansion.} Use the residual left by the chart grammar to synthesize a repair operator outside the inherited affine family and certify its new response direction.
19486 \item \textbf{Compiler metaproductivity after structural growth.} Evaluate whether a lineage-generated compiler discovers better grammar extensions across foreign parent processes under matched compilation resources.
19487 \item \textbf{Intervention-family invention.} Learn which interventions separate unresolved response blocks while charging their cost and preserving witness independence.
19488 \item \textbf{Noncommuting response algebras.} Recover reducing subspaces, irreducible modules, and task-relevant centers when responses are not jointly block scalar or normal.
19489 \item \textbf{Adaptive block count and multiplicity.} Select the number and dimensions of response blocks with valid post-selection uncertainty and explicit gauge accounting.
19490 \item \textbf{Nonlinear response bundles.} Replace one global response algebra by a field of tangent algebras, local charts, transition maps, and curvature or holonomy certificates.
19491 \item \textbf{Repair-dependent atlas transport.} Type and validate cases in which the repair moves the state across chart boundaries or changes the response algebra itself.
19492 \item \textbf{Atlas-compiler metaproductivity.} Show that a lineage-generated compiler improves atlas recovery across foreign parent processes under matched intervention and compilation budgets.
19493 \end{enumerate}
19494 
19495 
19496 36. \emph{Foreign-parent compiler metaproductivity.} Compare a frozen compiler and a lineage-generated compiler across parent repair processes with different obstruction classes under matched compilation resources.
19497 37. \emph{Unrestricted intervention invention.} Replace the finite response menu by a typed program or control grammar with explicit feasibility, side-effect, and information-gain accounting.
19498 38. \emph{Adaptive constructor degree and program structure.} Determine when a lineage can infer that affine, quadratic, cubic, kernel, or discrete program repair is required without receiving the correct degree or template.
19499 39. \emph{Joint response--constructor discovery.} Analyze settings where the intervention needed to reveal a defect depends on the repair constructor that will later act on it.
19500 40. \emph{Physical response-language semantics.} Extend response-language certificates from abstract operators to partially irreversible experiments with escrow, compensation, and environmental debt.
19501 
19502 
19503 41. \emph{Compiler grammar growth beyond finite pair syntax.} Extend Block 15 from anonymous binary refinement and ordered pairs to variable-cardinality taxonomies, longer typed research programs, and dynamically generated experiment schemas.
19504 42. \emph{Open research outcome forecasting.} Predict long-horizon experimental value and stopping decisions from results-free proposals and cheap pilots under distribution shift.
19505 43. \emph{Automatic evaluator construction.} Compile a new evaluation from sparse failure traces while preserving old evidence, reducing evaluator nullspace, and resisting evaluation awareness.
19506 44. \emph{Adversarial research-integrity monitoring.} Evaluate monitors against compilers that actively optimize to conceal prohibited actions rather than fixed invalid controls.
19507 45. \emph{Causal credit in research DAGs.} Identify which archived finding, obstruction certificate, or experiment caused a descendant compiler improvement under recombination and shared evidence.
19508 46. \emph{Real model-training realization.} Test compiler metaproductivity on independently implemented model-training or alignment research problems with external evaluators and hidden audits.
19509 47. \emph{Independent reproduction.} Reimplement the Block 13 benchmark and compiler from the formal contract rather than the released source, then compare endpoint and mechanism stability.
19510 
19511 
19512 48. \emph{Independent adversarial generator.} Reimplement the Block 14 problem generator and attack suites without access to the released benchmark internals, then repeat the sealed paired comparison.
19513 49. \emph{Adaptive archive poisoning.} Replace static contamination by an adversary that observes compiler allocations and crafts receipts to maximize selection error under a bounded poisoning budget.
19514 50. \emph{White-box monitor evasion.} Test the integrity ensemble against candidate methods optimized directly against its source and decision thresholds.
19515 51. \emph{Sequential open-set confidence.} Construct anytime-valid confidence sequences for known-mixture recovery and unknown detection under adaptive experiment selection.
19516 52. \emph{Real executable research.} Replace simulated candidate outcomes by actual program-synthesis, theorem-search, or small-model training runs under the same receipt and audit discipline.
19517 53. \emph{Robust compiler-language growth beyond the declared meta-language.} Require a compiler to generate a diagnostic or research-program form absent from the Block 15 six-probe and ordered-pair syntax while preserving the inherited language and robustness gates.
19518 54. \emph{Foreign-parent robust metaproductivity.} Apply one robust compiler-improvement rule across parent compilers with distinct allocation, archive, and integrity architectures.
19519 55. \emph{Distributionally robust evaluator evolution.} Improve an evaluator portfolio while preserving historical evidence and worst-group performance under held-out portfolio rotations.
19520 
19521 
19522 56. \emph{Cross-implementation executable language growth.} Reproduce the Block 15 contract with independently written program-synthesis or small-model-training tasks whose outcomes arise from actual execution.
19523 57. \emph{Variable-length research program synthesis.} Replace ordered pairs by a typed program grammar with sequential, conditional, parallel, and iterative composition under explicit cost and side-effect semantics.
19524 58. \emph{Semantic naming and ontology transport.} Determine when anonymous conflict-resolving tokens can acquire stable cross-context meaning without collapsing gauge-equivalent or historically distinct research classes.
19525 59. \emph{Compiler-generated evaluator dimensions.} Require a language-growing compiler to create a new audit dimension from persistent residuals while conservatively reconstructing all old evaluator outputs.
19526 60. \emph{Independent open-set conflict discovery.} Test whether an external generator produces repair conflicts not anticipated by the SRIC benchmark authors and whether the compiler issues a valid no-go or language-extension certificate.
19527 
19528 
19529 \section{Correspondence-Free Neuralese Transport from Intervention Orbits}
19530 \label{sec:block22-unpaired}
19531 
19532 Block 21 assumes paired source and target packet examples while diagnosing transport failure.  Block 22 removes that correspondence.  The source and target systems receive independent latent populations and paired responses only \emph{within} each system.  Cross-system task identity, hidden semantic state, final transport parameters, and expected certificate remain inaccessible until the final witness is opened.
19533 
19534 The target is not merely an unpaired map with low marginal discrepancy.  The compiler must distinguish five epistemically different outputs:
19535 \begin{equation}
19536 \boxed{
19537 \texttt{IDENTIFIED},\quad
19538 \texttt{IDENTIFIED\_UP\_TO\_GAUGE},\quad
19539 \texttt{PARTIAL\_TRANSPORT},\quad
19540 \texttt{UNDERIDENTIFIED},\quad
19541 \texttt{NO\_GO}.
19542 }
19543 \end{equation}
19544 
19545 \subsection{Unpaired intervention-orbit instance}
19546 
19547 \begin{definition}[Unpaired intervention-orbit transport instance]
19548 An instance is a tuple
19549 \begin{equation}
19550 \mathfrak U
19551 =
19552 (H_s,H_t,\mu_s,\mu_t,\mathcal U,
19553 \{R_u^s\}_{u\in\mathcal U},
19554 \{R_v^t\}_{v\in\mathcal V},
19555 \mathcal G_{\rm gauge},\mathsf W),
19556 \end{equation}
19557 where $\mu_s$ and $\mu_t$ are independently sampled latent distributions, $R_u^s$ and $R_v^t$ are within-system intervention-response operators, $\mathcal G_{\rm gauge}$ is the declared harmless gauge group, and $\mathsf W$ is a witness hidden from construction.  No map between individual samples from $\mu_s$ and $\mu_t$ is available to the compiler.
19558 \end{definition}
19559 
19560 For a proposed intervention correspondence $\sigma:\mathcal U\to\mathcal V$ and transport $T:H_t\to H_s$, define the normalized orbit defect
19561 \begin{equation}
19562 \delta_{\rm orb}(T,\sigma)
19563 =
19564 \frac1{|\mathcal U|}
19565 \sum_{u\in\mathcal U}
19566 \frac{\norm{TR^t_{\sigma(u)}-R_u^sT}_F}
19567 {\max\{\norm{R_u^s}_F,\eps\}}.
19568 \label{eq:b22-orbit-defect}
19569 \end{equation}
19570 
19571 \begin{definition}[Response algebra and commutant]
19572 The response algebra and its commutant are
19573 \begin{equation}
19574 \mathfrak A_t=\operatorname{alg}(I,R_v^t:v\in\mathcal V),
19575 \qquad
19576 \mathfrak A_t'=\{S\in\End(H_t):SR_v^t=R_v^tS\ \forall v\}.
19577 \end{equation}
19578 \end{definition}
19579 
19580 \begin{theorem}[Intertwiner uniqueness modulo the response commutant]
19581 \label{thm:b22-commutant}
19582 Let $T_1,T_2:H_t\to H_s$ be invertible and satisfy
19583 \begin{equation}
19584 T_iR_v^t=R_{\sigma^{-1}(v)}^sT_i
19585 \end{equation}
19586 for every $v\in\mathcal V$.  Then $S=T_2^{-1}T_1$ belongs to $\mathfrak A_t'$.  Hence exact intervention-orbit data identifies a transport at most up to the response commutant.
19587 \end{theorem}
19588 
19589 \begin{proof}
19590 For every $v$,
19591 \begin{align}
19592 SR_v^t
19593 &=T_2^{-1}T_1R_v^t
19594 =T_2^{-1}R_{\sigma^{-1}(v)}^sT_1\\
19595 &=R_v^tT_2^{-1}T_1
19596 =R_v^tS.
19597 \end{align}
19598 Thus $S\in\mathfrak A_t'$.
19599 \end{proof}
19600 
19601 \begin{corollary}[Declared gauge boundary]
19602 If every element of $\mathfrak A_t'$ that also preserves the target distribution and compiler-visible witnesses belongs to $\mathcal G_{\rm gauge}$, then transport is identifiable up to the declared gauge.  If an observationally invisible element lies outside $\mathcal G_{\rm gauge}$ and changes a protected witness, the correct output is underidentification.
19603 \end{corollary}
19604 
19605 \subsection{Finite signature recovery}
19606 
19607 The executable fixture uses five commuting symmetric response operators.  In latent semantic coordinates they are diagonal:
19608 \begin{equation}
19609 D_u=\operatorname{diag}(\lambda_{u1},\ldots,\lambda_{ud}),
19610 \end{equation}
19611 and each system exposes an orthogonal gauge $Q_a$:
19612 \begin{equation}
19613 R_u^a=Q_aD_uQ_a^\top.
19614 \end{equation}
19615 The joint response signature of semantic coordinate $j$ is
19616 \begin{equation}
19617 \lambda_j=(\lambda_{1j},\ldots,\lambda_{mj}).
19618 \end{equation}
19619 
19620 \begin{proposition}[Separating-signature recovery]
19621 Suppose the joint signatures are pairwise distinct, the system gauges are orthogonal, and every one-dimensional eigendirection has a nonzero signed third moment.  Then independently estimated response families recover the cross-system transport up to finite estimation error, without paired source-target samples.  Anonymous intervention labels are additionally recoverable whenever the operator invariant signatures are distinct.
19622 \end{proposition}
19623 
19624 \begin{proof}[Proof sketch]
19625 A generic linear combination of the commuting operators has simple spectrum.  Its eigenvectors recover the gauge columns up to sign and permutation.  Joint response signatures identify the common permutation; operator spectra identify an anonymous intervention permutation; and signed third moments orient the one-dimensional eigendirections.  Substituting the recovered bases gives $T=Q_sQ_t^\top$ up to sampling error.
19626 \end{proof}
19627 
19628 When two signatures coincide, the associated joint eigenspace is recovered but no preferred basis inside it is determined.  Block 22 treats this as a gauge only when every protected readout is invariant under the declared within-block action.
19629 
19630 \begin{theorem}[Observable symmetry underidentification]
19631 Suppose $S\notin\mathcal G_{\rm gauge}$ preserves $\mu_t$, commutes with every permitted target response, and preserves every compiler-visible witness.  Then $T$ and $TS$ are observationally indistinguishable through the construction interface.  If the protected final witness distinguishes them, no construction rule measurable with respect to the permitted interface can identify the protected transport.
19632 \end{theorem}
19633 
19634 \begin{proof}
19635 The base distribution is unchanged by $S$.  Commutation makes every finite intervention orbit identical in distribution after applying $T$ or $TS$, and the visible witnesses are invariant by assumption.  Therefore the construction likelihoods coincide.  A protected difference available only after sealing cannot be inferred from those construction observations.
19636 \end{proof}
19637 
19638 \subsection{Partial support and information loss}
19639 
19640 A target context is admitted only when its operator-invariant signature and orbit residual lie inside the frozen source support.  Unmatched contexts are rejected rather than passed through the nearest transport.
19641 
19642 \begin{proposition}[Separated context admission]
19643 If every supported target context has invariant distance at most $\eta$, every unsupported context has distance at least $\eta+\gamma$, and estimation error is less than $\gamma/2$, a threshold in $(\eta+\gamma/2,\eta+\gamma)$ separates supported from unsupported contexts.
19644 \end{proposition}
19645 
19646 \begin{theorem}[Target-only collision no-go]
19647 If two target states satisfy $y_1=y_2$ while their protected semantic outcomes satisfy $z_1\neq z_2$, no deterministic target-only map can recover both outcomes.
19648 \end{theorem}
19649 
19650 \begin{proof}
19651 A deterministic map must assign one value to the common target state, so it cannot equal both distinct protected outcomes.
19652 \end{proof}
19653 
19654 \subsection{Sealed realization}
19655 
19656 The pre-audit suite contains $24$ episodes.  The fresh final suite contains $36$ episodes across two foreign gauge families, six hidden regimes, and three unseen final seeds per architecture--regime cell.  Construction uses independent source and target samples with unequal population sizes.  Five within-system intervention responses are available, while cross-system task correspondence is absent.
19657 
19658 \begin{figure}[H]
19659 \centering
19660 \includegraphics[width=.84\textwidth]{block22_certificate_matrix.pdf}
19661 \caption{Every fresh final episode receives the precommitted epistemic certificate.  The matrix scores the decision class, not merely transport error.}
19662 \label{fig:b22-confusion}
19663 \end{figure}
19664 
19665 \begin{table}[H]
19666 \centering
19667 \small
19668 \begin{tabular}{@{}lrrr@{}}
19669 \toprule
19670 Regime & Episodes & Certificate & Minimum protected result\\
19671 \midrule
19672 Shared labels & 6 & IDENTIFIED & semantic $R^2=0.995984$\\
19673 Anonymous labels & 6 & IDENTIFIED & semantic $R^2=0.990952$\\
19674 Gauge-equivalent & 6 & UP TO GAUGE & invariant $R^2=0.996352$\\
19675 Semantic ambiguity & 6 & UNDERIDENTIFIED & disagreement $=1.905405$\\
19676 Partial support & 6 & PARTIAL & coverage $=0.700000$\\
19677 Destructive loss & 6 & NO GO & collisions $\geq 96$\\
19678 \bottomrule
19679 \end{tabular}
19680 \caption{Block 22 final certificate results.}
19681 \label{tab:b22-results}
19682 \end{table}
19683 
19684 \subsection{Identified and anonymous-label transport}
19685 
19686 Across the two fully identifiable regimes, the minimum hidden semantic score is
19687 \begin{equation}
19688 R^2_{\rm sem}\geq 0.990952,
19689 \end{equation}
19690 and the largest admitted orbit defect is
19691 \begin{equation}
19692 \delta_{\rm orb}\leq 0.002870.
19693 \end{equation}
19694 The mean second-order distribution baseline is $-0.900494$, and cyclically shuffled coordinate binding yields mean $R^2=-1.009877$.
19695 
19696 \begin{figure}[H]
19697 \centering
19698 \includegraphics[width=.94\textwidth]{block22_identified_vs_controls.pdf}
19699 \caption{The response-algebra intertwiner succeeds without task pairs.  Exchangeable marginals and shuffled semantic binding do not identify coordinate order.}
19700 \label{fig:b22-controls}
19701 \end{figure}
19702 
19703 \subsection{Gauge versus semantic ambiguity}
19704 
19705 In the benign degenerate regime, an alternative transport inside the unresolved two-dimensional block retains protected semantic
19706 \begin{equation}
19707 R^2\geq 0.996352,
19708 \end{equation}
19709 while its mean raw-coordinate score is only $0.656730$.  The correct claim is equivalence up to gauge, not unique coordinates.
19710 
19711 \begin{figure}[H]
19712 \centering
19713 \includegraphics[width=.78\textwidth]{block22_gauge_certificate.pdf}
19714 \caption{Gauge-invariant meaning remains stable even when a different valid basis changes raw coordinates.}
19715 \label{fig:b22-gauge}
19716 \end{figure}
19717 
19718 The semantic-ambiguity regime has the same observable response multiplicity but does not declare that within-block action harmless.  Two maps with the same low orbit defect disagree on hidden protected semantics by at least $1.905405$.  Every such episode returns \texttt{UNDERIDENTIFIED}.
19719 
19720 \subsection{Partial support and no-go}
19721 
19722 The partial-support child admits exactly $70\%$ of target tasks.  Its admitted semantic score is at least $0.997798$, false admission is zero, and forcing the rejected context through the supported map gives mean $R^2=-1.096314$.
19723 
19724 \begin{figure}[H]
19725 \centering
19726 \includegraphics[width=.78\textwidth]{block22_partial_support.pdf}
19727 \caption{The compiler preserves supported transport and rejects the unmatched target context.}
19728 \label{fig:b22-partial}
19729 \end{figure}
19730 
19731 Every destructive-loss episode contains $96$ exact target-state collisions with protected semantic disagreement.  The minimum gap is $1.605786$ and the best released target-only cross-fit has $R^2=0.000000$.  The compiler issues \texttt{NO\_GO}.
19732 
19733 \begin{figure}[H]
19734 \centering
19735 \includegraphics[width=.86\textwidth]{block22_ambiguity_and_nogo.pdf}
19736 \caption{Underidentification and information-loss no-go are positive epistemic outputs with explicit witnesses.}
19737 \label{fig:b22-failclosed}
19738 \end{figure}
19739 
19740 \subsection{Finite realization theorem}
19741 
19742 \begin{conditionaltheorem}[Block 22 correspondence-free intervention-orbit transport]
19743 \label{thm:b22-realization}
19744 Within the frozen Block 22 fixture, assume that:
19745 \begin{enumerate}
19746 \item source and target construction samples are independent and no cross-system task identifier is exposed;
19747 \item within-system intervention pairs are valid and the public compiler does not access the audit truth module;
19748 \item the compiler, thresholds, gauge policy, audit harness, and final seed rule are sealed before canonical final generation;
19749 \item the response-signature, support-separation, collision, and protected-semantic assumptions of the six declared regimes hold;
19750 \item all final certificate decisions and quantitative gates in \cref{tab:b22-results} replay exactly.
19751 \end{enumerate}
19752 Then the accepted edge is a finite, evidence-preserving removal of the paired-transport scaffold.  It identifies transport when the intervention interface determines it, returns the declared gauge class when nonuniqueness is harmless, transports only the supported region under partial overlap, and refuses commitment under semantic ambiguity or destructive information loss.
19753 \end{conditionaltheorem}
19754 
19755 \begin{proof}
19756 The construction interface and seal establish correspondence isolation.  Separating response signatures and anonymous operator invariants justify the identified cases.  \Cref{thm:b22-commutant} localizes the only admissible nonuniqueness to the response commutant, and the gauge witness distinguishes benign from semantically harmful multiplicity.  The context-separation proposition validates selective admission.  The collision theorem validates deterministic no-go.  The final hidden witness then verifies that each issued certificate has the declared protected interpretation on fresh seeds.
19757 \end{proof}
19758 
19759 \begin{statusbox}{Block 22 result}
19760 All $24$ pre-audit and $36$ fresh final episodes pass.  Twelve episodes identify a transport, six identify only a declared gauge class, six transport a supported subset, six issue semantic underidentification, and six issue collision-grounded no-go.  This is a finite correspondence-free intervention-orbit result, not unrestricted unpaired alignment or endogenous semantic discovery.
19761 \end{statusbox}
19762 
19763 \subsection{Claim boundary and next threshold}
19764 
19765 The fixture uses finite-dimensional latent packets, commuting symmetric response operators, orthogonal foreign gauges, a supplied intervention language, and a declared protected semantic policy.  It does not establish alignment of arbitrary pretrained systems.  The next threshold is endogenous semantic-witness growth: propose a semantic quotient from causal predictive equivalence, validate it independently, and avoid using the learned probe as its own witness.
19766 
19767 \section{Endogenous Semantic-Witness Growth and External Latent-Reasoner Pilot}
19768 \label{sec:block23}
19769 
19770 Blocks 16 through 22 progressively removed supplied structure from the Neuralese transport problem.  They established direct causal intervention, training-generated latent repair, distribution-indexed semantic charts, chartwise cross-architecture transport, obstruction-guided transport-grammar growth, and correspondence-free transport from intervention orbits.  One central scaffold remained: the protected intermediate semantic coordinates used to judge whether a latent packet had the intended meaning were supplied by the benchmark author.
19771 
19772 Block 23 separates two questions that must not be collapsed.  The first is theoretical and finite:
19773 \begin{equation}
19774 \boxed{
19775 \begin{gathered}
19776 \text{Can a lineage construct a minimal semantic quotient from}\\
19777 \text{future consequences, interventions, and semantic twins}\\
19778 \text{without receiving gold intermediate semantic coordinates?}
19779 \end{gathered}}
19780 \label{eq:b23-central-question}
19781 \end{equation}
19782 The 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.
19783 
19784 \subsection{Causal-predictive semantics}
19785 
19786 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.
19787 
19788 \begin{definition}[Causal-predictive equivalence]
19789 For $h,h'\in H$, write
19790 \begin{equation}
19791 h\sim_{\Ucal,r}h'
19792 \end{equation}
19793 when
19794 \begin{equation}
19795 \mathcal L\!\left(Y^{u}_{0:r}\mid h\right)
19796 =
19797 \mathcal L\!\left(Y^{u}_{0:r}\mid h'\right)
19798 \qquad
19799 \text{for every }u\in\Ucal^{\le r}.
19800 \label{eq:b23-causal-equivalence}
19801 \end{equation}
19802 The quotient
19803 \begin{equation}
19804 Z_{\Ucal,r}=H/{\sim_{\Ucal,r}}
19805 \label{eq:b23-semantic-quotient}
19806 \end{equation}
19807 is the finite-horizon causal-predictive semantic space relative to the declared experiment language.
19808 \end{definition}
19809 
19810 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.
19811 
19812 \begin{proposition}[Causal-predictive sufficiency]
19813 Let $q:H\to Z_{\Ucal,r}$ be the quotient map.  For every $u\in\Ucal^{\le r}$, there exists a Markov kernel $K_u$ such that
19814 \begin{equation}
19815 \mathcal L\!\left(Y^{u}_{0:r}\mid h\right)=K_u(q(h)).
19816 \end{equation}
19817 Thus $q(h)$ is sufficient for every protected consequence appearing in the equivalence relation.
19818 \end{proposition}
19819 
19820 \begin{proof}
19821 The conditional law is constant on each equivalence class by definition.  Therefore it descends to a well-defined kernel on the quotient.
19822 \end{proof}
19823 
19824 \begin{theorem}[Minimality of the exact semantic quotient]
19825 Suppose $s:H\to S$ 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
19826 \begin{equation}
19827 q=\phi\circ s.
19828 \label{eq:b23-minimal-factorization}
19829 \end{equation}
19830 Consequently, any exact sufficient representation refines the causal-predictive quotient.
19831 \end{theorem}
19832 
19833 \begin{proof}
19834 If $s(h)=s(h')$, 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.
19835 \end{proof}
19836 
19837 The theorem supports a minimum-description principle: among empirically valid finite approximations, the lineage should prefer the smallest quotient whose held-out predictive and interventional errors remain within the frozen tolerances.
19838 
19839 \subsection{Semantic twins and nuisance invariance}
19840 
19841 A quotient can predict outcomes while encoding irrelevant rendering details.  Block 23 therefore supplies semantic-preserving twins during construction.
19842 
19843 \begin{definition}[Semantic twin]
19844 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.
19845 \end{definition}
19846 
19847 For a candidate quotient $q_\theta$, the empirical twin discrepancy is
19848 \begin{equation}
19849 D_{\mathrm{twin}}(q_\theta)
19850 =
19851 \mathbb E
19852 \frac{\norm{q_\theta(x)-q_\theta(x')}}
19853 {\max\{\norm{q_\theta(x)},\eps\}}.
19854 \label{eq:b23-twin-drift}
19855 \end{equation}
19856 A 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.
19857 
19858 \begin{warning}[Answer sufficiency is weaker than semantic sufficiency]
19859 A one-dimensional code can predict the final answer perfectly while discarding the causal state required to predict intermediate outcomes or intervention effects.  Final-answer accuracy alone is therefore not an ontology certificate.
19860 \end{warning}
19861 
19862 This warning is realized explicitly by the answer-shortcut family below.
19863 
19864 \subsection{Witness separation}
19865 
19866 The same labels cannot both construct and independently certify a semantic interpretation.
19867 
19868 \begin{definition}[Witness-separated ontology transaction]
19869 An 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.
19870 \end{definition}
19871 
19872 \begin{proposition}[Circular witness limitation]
19873 If 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.
19874 \end{proposition}
19875 
19876 \begin{proof}
19877 A sufficiently expressive encoder can memorize the construction targets.  The resulting score does not distinguish a stable semantic quotient from target leakage.  An independent witness or a fresh interventional consequence is required.
19878 \end{proof}
19879 
19880 In the finite realization, the compiler sees no hidden intermediate coordinates.  It receives only trajectories, twins, anonymous intervention trajectories, future consequences, intervention-effect vectors, and final answers.  Hidden semantic coordinates, protected ambiguity, and audit-only runtime decoders are revealed after compilation.
19881 
19882 \subsection{Gauge, partiality, and underidentification}
19883 
19884 A semantic quotient need not be uniquely coordinatized.
19885 
19886 \begin{definition}[Protected gauge block]
19887 A 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 $G$.
19888 \end{definition}
19889 
19890 \begin{theorem}[Semantic underidentification]
19891 Suppose 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
19892 \begin{equation}
19893 W(q_1(h))\neq W(q_2(h))
19894 \end{equation}
19895 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.
19896 \end{theorem}
19897 
19898 \begin{proof}
19899 The visible likelihood ratio is one under the two alternatives.  Hence the posterior odds equal the prior odds for every compiler-visible history.  Any unique selection beyond the prior maximum would require information absent from the interface.
19900 \end{proof}
19901 
19902 The correct certificate is therefore Ontology underidentified, even when a post-hoc probe can decode one arbitrarily chosen orientation with high accuracy.
19903 
19904 A different outcome arises when only part of the protected ontology is observable.
19905 
19906 \begin{definition}[Partial ontology]
19907 Let $P\subseteq Z$ 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
19908 \begin{equation}
19909 \operatorname{cov}(q)=\frac{\dim P_0}{\dim P}.
19910 \end{equation}
19911 \end{definition}
19912 
19913 \subsection{Information-loss no-go}
19914 
19915 Some failures cannot be repaired by a richer probe.
19916 
19917 \begin{theorem}[Deterministic semantic collision no-go]
19918 Let $y:H\to Y$ 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
19919 \begin{equation}
19920 y(h_1)=y(h_2),
19921 \qquad
19922 z(h_1)\neq z(h_2),
19923 \label{eq:b23-collision}
19924 \end{equation}
19925 then no deterministic map $f:Y\to Z$ recovers $z$ on both states.
19926 \end{theorem}
19927 
19928 \begin{proof}
19929 The two equal inputs force $f(y(h_1))=f(y(h_2))$, contradicting the unequal protected targets in \cref{eq:b23-collision}.
19930 \end{proof}
19931 
19932 The release searches for equal observations with unequal protected consequences.  When the collision count and separation margin exceed the frozen thresholds, the compiler issues Semantic information loss; it does not fit a high-capacity decoder that guesses the erased variable from incidental correlations.
19933 
19934 \subsection{Finite compiler}
19935 
19936 Each episode contains two independently gauged recurrent systems.  A trajectory has five latent steps, four task-relevant semantic coordinates, and twelve nuisance coordinates per step.  The observed flattened trajectory therefore lies in $\R^{80}$.  The semantic dynamics, gauges, nuisance processes, and hidden audit coordinates are not passed to the quotient compiler.
19937 
19938 The finite grammar is
19939 \begin{equation}
19940 \Gcal_{23}
19941 =
19942 \left\{
19943 \begin{array}{l}
19944 \text{answer-only},\ \text{PCA},\ \text{predictive},\ \\
19945 \text{intervention-response},\ \text{joint minimal},\\
19946 \text{full identity},\ \text{equal-rank random}
19947 \end{array}
19948 \right\}.
19949 \label{eq:b23-grammar}
19950 \end{equation}
19951 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,
19952 \begin{equation}
19953 \begin{aligned}
19954 R^2_{\mathrm{future}}&\geq0.82,\\
19955 R^2_{\mathrm{intervention}}&\geq0.90,\\
19956 \operatorname{Acc}_{\mathrm{answer}}&\geq0.78,\\
19957 D_{\mathrm{twin}}&\leq0.22.
19958 \end{aligned}
19959 \label{eq:b23-visible-gates}
19960 \end{equation}
19961 Among valid candidates, the compiler minimizes rank and then follows the frozen method priority.  No hidden semantic score enters the choice.
19962 
19963 \subsection{Hidden regimes}
19964 
19965 The audit contains six families.
19966 
19967 \begin{longtable}{@{}p{0.22\textwidth}p{0.32\textwidth}p{0.35\textwidth}@{}}
19968 \toprule
19969 Regime & Visible challenge & Correct final certificate \\
19970 \midrule
19971 \endfirsthead
19972 \toprule
19973 Regime & Visible challenge & Correct final certificate \\
19974 \midrule
19975 \endhead
19976 Identified & Four distinct semantic directions determine future and intervention consequences & \texttt{ONTOLOGY\_IDENTIFIED} \\
19977 Answer shortcut & One coordinate strongly predicts the answer, while full causal state remains four-dimensional & Ontology identified; answer-only control rejected \\
19978 Gauge equivalent & Two directions have identical response signatures and only block invariants are protected & Identified up to a declared gauge \\
19979 Semantic ambiguity & The same visible degeneracy is present, but a named coordinate inside the unresolved block is protected & Ontology underidentified \\
19980 Partial ontology & Three protected directions are observable and the fourth is intentionally outside the declared consequence language & Partial ontology \\
19981 Destructive loss & Equal observed trajectories carry unequal protected semantic consequences & Semantic information loss \\
19982 \bottomrule
19983 \end{longtable}
19984 
19985 \subsection{Sealing and audit chronology}
19986 
19987 The grammar, thresholds, family schedule, compiler seed rule, external source lock, and final-seed commitment were frozen before final hidden audits.  Twenty-four pre-audit episodes were compiled first.  Thirty-six new final episodes were then generated, six per hidden regime.  Each final quotient was committed by matrix digest before hidden semantics and protected ambiguity were evaluated.
19988 
19989 The canonical pre-audit seal is
19990 \begin{verbatim}
19991 04785a6f21a8d9f64100e4609329333498bceaa6b5afae90472313f6ebbd4ef9
19992 \end{verbatim}
19993 
19994 \subsection{Finite results}
19995 
19996 Every final episode received the precommitted epistemic disposition.
19997 
19998 \begin{table}[H]
19999 \centering
20000 \small
20001 \begin{tabular}{@{}lrr@{}}
20002 \toprule
20003 Certificate & Count & Correct \\
20004 \midrule
20005 Ontology identified & 12 & 12 \\
20006 Identified up to gauge & 6 & 6 \\
20007 Partial ontology & 6 & 6 \\
20008 Ontology underidentified & 6 & 6 \\
20009 Semantic information loss & 6 & 6 \\
20010 \midrule
20011 Total & 36 & 36 \\
20012 \bottomrule
20013 \end{tabular}
20014 \caption{Block 23 final certificate counts.  The hidden family is unavailable to quotient construction.}
20015 \label{tab:b23-certificates}
20016 \end{table}
20017 
20018 \begin{figure}[H]
20019 \centering
20020 \includegraphics[width=0.92\textwidth]{block23_certificate_matrix.png}
20021 \caption{Final hidden regime by issued ontology certificate.  The answer-shortcut regime remains identifiable only because the compiler rejects the one-dimensional answer code and selects a four-dimensional predictive quotient.}
20022 \label{fig:b23-certificate-matrix}
20023 \end{figure}
20024 
20025 The selected constructions were:
20026 \begin{itemize}
20027 \item a rank-four predictive quotient for the identified and answer-shortcut regimes;
20028 \item a rank-four intervention-response quotient for the gauge-equivalent and semantic-ambiguity regimes;
20029 \item a rank-three predictive quotient for the partial regime;
20030 \item no admitted quotient for destructive information loss.
20031 \end{itemize}
20032 
20033 \begin{figure}[H]
20034 \centering
20035 \includegraphics[width=0.88\textwidth]{block23_minimal_quotients.png}
20036 \caption{Minimum valid quotient ranks and selected constructor classes.  Zero denotes a collision-grounded no-go rather than an empty semantic success.}
20037 \label{fig:b23-minimal-quotients}
20038 \end{figure}
20039 
20040 \subsubsection{Answer-only negative control}
20041 
20042 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
20043 \begin{equation}
20044 R^2_{\mathrm{future}}\leq0.313061,
20045 \qquad
20046 R^2_{\mathrm{intervention}}\leq0.357401.
20047 \end{equation}
20048 It failed every final construction decision.  This separates answer extraction from semantic-state recovery.
20049 
20050 \subsubsection{Hidden semantic and gauge audit}
20051 
20052 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.
20053 
20054 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.
20055 
20056 \subsubsection{Audit-only counterfactual packet patching}
20057 
20058 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 $-0.703696$.
20059 
20060 \begin{figure}[H]
20061 \centering
20062 \includegraphics[width=0.82\textwidth]{block23_counterfactual_controls.png}
20063 \caption{Counterfactual donor-packet controls.  Only the compiled quotient produces the predicted downstream future state in the identified and answer-shortcut regimes.}
20064 \label{fig:b23-counterfactual-controls}
20065 \end{figure}
20066 
20067 \subsection{Finite realization theorem}
20068 
20069 \begin{conditionaltheorem}[Block 23 finite endogenous semantic-witness growth]
20070 Within the frozen Block 23 benchmark, assume that:
20071 \begin{enumerate}
20072 \item the visible compiler interface excludes hidden intermediate semantics and hidden regime labels;
20073 \item construction and hidden witness supports are separated;
20074 \item the candidate grammar, ranks, thresholds, seeds, and method priority are sealed before final hidden audit;
20075 \item both recurrent systems share protected consequences but use independently generated gauges and nuisance processes;
20076 \item the visible gates in \cref{eq:b23-visible-gates} are replayed exactly;
20077 \item the hidden certificate rules distinguish unique identification, gauge equivalence, partiality, protected ambiguity, and exact semantic collision;
20078 \item the thirty-six final results and their matrix digests reproduce the released receipt;
20079 \item the answer-only, PCA, random-packet, alternative-orientation, and collision controls retain their declared interpretations.
20080 \end{enumerate}
20081 Then the released edge is a finite lineage-generated instance of endogenous semantic-witness growth: the compiler constructs a minimum-rank causal-predictive quotient without gold intermediate coordinates, and the witness assigns the appropriate unique, gauge, partial, underidentified, or information-loss certificate.
20082 \end{conditionaltheorem}
20083 
20084 \begin{proof}
20085 Items 1 through 3 provide evidence separation and chronology.  Items 4 and 5 establish that the selected quotient is based on protected future and intervention behavior rather than one coordinate system or one answer label.  Item 6 prevents post-hoc decodability from being confused with unique identification and prevents destructive collisions from being treated as ordinary model error.  Item 7 supplies the finite replay certificate.  Item 8 rules out the principal answer-shortcut, variance, random-subspace, arbitrary-gauge, and correlation-based reconstruction explanations inside the released fixture.
20086 \end{proof}
20087 
20088 \subsection{External multimode Coconut pilot}
20089 
20090 The package also freezes a first real-model audit target from Dilgren and Wiegreffe's multimode latent-reasoning release~\cite{DILGREN2026}.  The target checkpoint uses GPT-2 on PrOntoQA and supports latent, verbalized-chain-of-thought, and direct-answer modes in one trained model.  Its public Coconut wrapper replaces each latent-token embedding by the preceding final hidden state.  The wrapper returns the completed input-embedding tensor, making the exact reinjected latent states directly capturable.
20091 
20092 The Block 23 adapter does not rely on an ordinary activation hook as a substitute for the recurrence.  It identifies the latent-token positions, extracts the embeddings actually written there, and evaluates an externally supplied trace by replacing those positions in a deterministic one-shot recomputation.  The intervention language includes:
20093 \begin{enumerate}
20094 \item exact own-trace replay;
20095 \item zero trace and prefix truncation;
20096 \item frozen step permutation;
20097 \item norm- or covariance-matched replacement;
20098 \item same-answer, different-proof transplantation;
20099 \item semantic-twin transplantation by consistent entity and predicate renaming.
20100 \end{enumerate}
20101 
20102 The source lock commits the upstream repository at
20103 \begin{verbatim}
20104 32f413d8d55239d9bc54bb6b6ec37b0630891ed4
20105 \end{verbatim}
20106 and records the Git blob identities of the Coconut wrapper, multimode adapter, base interface, and PrOntoQA test data.  The checkpoint digest is intentionally left unset until the actual local file is acquired; the runner computes and commits its SHA-256 before any final audit.
20107 
20108 \subsubsection{Adapter fixture}
20109 
20110 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.
20111 
20112 \subsubsection{External execution status}
20113 
20114 The release environment had CPU-only PyTorch, no \texttt{transformers} installation, no source checkout, and no checkpoint file.  The canonical external status is therefore
20115 \begin{verbatim}
20116 ADAPTER_VALIDATED_REAL_EXECUTION_BLOCKED
20117 \end{verbatim}
20118 with \texttt{real\_checkpoint\_executed=false}.  No external checkpoint loss, causal effect, semantic quotient, or interpretability certificate is reported.
20119 
20120 \begin{warning}[No substitution of fixture evidence]
20121 Passing the fake-runtime protocol tests does not imply that the third-party checkpoint replays exactly, uses its latent tokens, contains procedure-specific information, or admits an identifiable semantic quotient.  Those questions remain empirical.
20122 \end{warning}
20123 
20124 \subsection{Reproducibility and claim boundary}
20125 
20126 The canonical finite result receipt is
20127 \begin{verbatim}
20128 213bcf9fbf62b797f9903d9e05e84e5a4036bb02c927723a4570fa44e2c674ce
20129 \end{verbatim}
20130 and the adapter-fixture receipt is
20131 \begin{verbatim}
20132 e447e082f3befe5b7b6bb0254464c4cb1402684a0c93e3022579439be6f62664
20133 \end{verbatim}
20134 The released tests cover visible-interface isolation, all final certificates, answer-shortcut rejection, counterfactual packet controls, collision no-go, external causal-audit plumbing, gold-free discovery, and fail-closed preflight.
20135 
20136 Block 23 establishes neither unrestricted ontology discovery nor interpretation of natural-language reasoning.  It does not show that every latent state has a unique verbal meaning.  It establishes a finite architecture in which semantic claims are grounded in protected future and intervention behavior, assigned only after witness separation, and allowed to terminate in gauge ambiguity, partiality, underidentification, or information-loss no-go.
20137 
20138 The next threshold is empirical.  The source-locked Coconut checkpoint should be run on a family-grouped PrOntoQA audit.  Exact replay must pass before any causal claim.  A quotient should then be constructed without proof steps, sealed, and only afterward compared with withheld formal proof chains and counterfactual premise edits.  A second threshold would transport any validated quotient to the corresponding CODI checkpoint without exact task-level pairing.
20139 
20140 \section{Conclusion}
20141 
20142 The twenty-one-block draft develops recursive self-improvement as an evidence-preserving lineage that accumulates a progressively stronger calculus of valid change:
20143 \begin{equation}
20144 \boxed{
20145 \begin{gathered}
20146 \text{evaluation and introspection}
20147 \longrightarrow
20148 \text{correctability and obstruction}\\
20149 \text{repair descent and transport}
20150 \longrightarrow
20151 \text{evaluator-preserving transactions}\\
20152 \text{recursive closure and archives}
20153 \longrightarrow
20154 \text{executable realization}\\
20155 \text{learned foreign repair transfer}
20156 \longrightarrow
20157 \text{sealed one-step process improvement}\\
20158 \text{meta-headroom and bottleneck migration}
20159 \longrightarrow
20160 \text{two-edge repeated lineage}\\
20161 \longrightarrow
20162 \text{obstruction-certified structural grammar expansion}\\
20163 \longrightarrow
20164 \text{endogenous intervention-response atlas growth}\\
20165 \longrightarrow
20166 \text{response-language and constructor growth}\\
20167 \longrightarrow
20168 \text{proof-carrying research-compiler metaproductivity}\\
20169 \longrightarrow
20170 \text{adversarial open-set compiler robustness}\\
20171 \longrightarrow
20172 \text{compiler-language growth from open-set research conflict}\\
20173 \longrightarrow
20174 \text{direct Neuralese causal semantics and latent-language audit}\\
20175 \longrightarrow
20176 \text{training-generated executable Neuralese binding repair}\\
20177 \longrightarrow
20178 \text{cross-implementation distributional chart obstruction}\\
20179 \longrightarrow
20180 \text{distribution-indexed Neuralese semantic atlas growth}\\
20181 \longrightarrow
20182 \text{atlas-indexed cross-architecture Neuralese transport}\\
20183 \longrightarrow
20184 \text{obstruction-guided Neuralese transport-grammar growth}.
20185 \end{gathered}
20186 }
20187 \end{equation}
20188 
20189 The improving object is not one weight tensor. It is a resolved lineage containing agents, evidence, repair schemas, evaluators, witnesses, archives, compilers, and proof-carrying transformations. Correctability is relative to authorized correction spaces. Local solvability does not imply global compatibility. State alignment does not imply repair equivariance. Full rank does not imply stable identification. Stable estimation does not imply that the current function class can represent the required repair.
20190 
20191 Block 7 crossed the first bounded metaproductivity threshold. One descendant improved constrained repair selection under an identical candidate bundle. A second replaced random paired anchors by rank-revealing design. Block 9 replayed that child, diagnosed that rank had saturated, and migrated to quality-weighted conditioning-aware acquisition. On its fresh audit, the second lineage edge improved mean selected protected gain from $0.037393$ to $0.209580$ under the same $24$-query envelope.
20192 
20193 Block 10 crosses the structural threshold. The exact Block 9 policy is replayed as the lineage ancestor. In a fresh piecewise-affine environment, every primary source design is full rank and the global parent uses $80$ bridge-fit pairs plus $16$ witness pairs. The child uses $48$ fit, $32$ chart-selection, and the same $16$ witness pairs. The global-affine grammar remains inadequate even under a directly supervised $1600$-pair target oracle, while the invariant-context two-chart grammar introduces a nonzero anti-diagonal repair direction.
20194 
20195 Across $96$ fresh primary foreign targets, the chart child obtains mean selected gain
20196 \begin{equation}
20197 0.334224
20198 \end{equation}
20199 with zero harmful selected targets. The child-parent effect is
20200 \begin{equation}
20201 \widehat\Delta_{G\to C}=0.334224,
20202 \qquad
20203 \mathrm{CI}_{95\%}=[0.266459,0.401903].
20204 \end{equation}
20205 The directly supervised global-affine oracle reaches mean gain $0.176694$. The child exceeds it by
20206 \begin{equation}
20207 0.157530,
20208 \qquad
20209 \mathrm{CI}_{95\%}=[0.080863,0.234135].
20210 \end{equation}
20211 Mean validation NRMSE falls from $0.598513$ to $0.064975$, and no primary paired target favors the parent. Gate-only, random-gate, and shuffled-binding controls are strongly negative. Blind local application is sometimes harmful, demonstrating that the witness contributes rather than merely reporting the local constructor's gain.
20212 
20213 The realized lineage is
20214 \begin{equation}
20215 K_0\xrightarrow{\Pi_1}K_1\xrightarrow{\Pi_2}K_2\xrightarrow{\Pi_3}K_3\xrightarrow{\Pi_4}K_4.
20216 \end{equation}
20217 Its active obstruction migrates from rank, to conditioning, to transport grammar. The third edge is structural: it conservatively adds a repair contrast outside the inherited diagonal global subgrammar and uses that new direction to improve foreign protected utility.
20218 
20219 The result remains bounded and scaffolded. The protected separator packet, hard two-chart form, finite direction-and-threshold search, source repair, evaluator, witness, and outer compiler are externally supplied. The lineage does not yet invent chart count, separator language, smooth local coordinates, noninjective quotient structure, evaluator, or witness. It does not establish open-ended recursive improvement or indefinite acceleration.
20220 
20221 Block 11 crosses that finite endogenous-atlas threshold. The supplied context packet is removed. A declared intervention family induces response operators whose central projectors recover target-specific functional blocks up to gauge. On $80$ fresh targets under a matched $120$-query envelope, the selected child gain is $0.207325$ with interval $[0.131902,0.287910]$ and zero selected harm. The selected recovered block has mean context overlap $0.821461$, while the inherited fixed packet has overlap $0.322091$. Shuffled-response, random-atlas, PCA, full-space, blind-commit, and stress controls preserve the claim boundary.
20222 
20223 Block 12 subsequently answers the finite response-language and candidate-constructor stage. It certifies an inherited response alias, adds a separating intervention, then certifies an affine function-class obstruction and adds degree-two repair capacity. Block 13 then moves one level outward. The modified object is the research compiler that reads the archive, diagnoses the bottleneck, allocates experiments, binds proposals to code, and selects a validity-filtered descendant. Under the same six-experiment budget and fixed $39$-method grammar, primary protected gain rises from $0.883004$ to $0.950231$ and architecture-foreign gain rises from $0.772566$ to $0.863367$. The paired intervals exclude zero. Shuffled archive binding collapses performance, while single-benchmark selection and monitor removal reduce validity-filtered foreign gain.
20224 
20225 The finite Block 13 mechanism is narrower than unrestricted compiler self-improvement. Block 14 then shows that the compiler can be made more robust to ambiguous, mixed, unknown, contaminated, and attacked research problems while retaining its fixed method language. Block 15 crosses the next finite structural boundary: a repair-conflicting unknown cell causes the compiler to add a separating experiment, two anonymous diagnostic tokens, and two synthesized pair methods while preserving the inherited compiler exactly on old classes.
20226 
20227 Block 16 then moves the intervention discipline directly into recurrent latent reasoning. Neuralese is represented by a typed transition, update gate, halt map, trace convention, intervention language, semantic transport, and frozen answer witness. The synthetic reference edge shows why usefulness and order are insufficient: its parent is causally active but nearly invariant under same-format cross-task trace transplantation, while the child carries a material task-conditioned signal and passes the direct lineage gate. The real Hyper audit remains unexecuted in this revision.
20228 
20229 Block 17 crosses the next executable threshold. The latent trajectories now arise from trained recurrent checkpoints in an exact external program environment. An answer-matched donor control separates program semantics from mere final-answer encoding. A failed sealed pilot is retained, the compiler is revised before new final-task generation, and the selected full-binding child passes all causal, semantic, replay, probe, and adaptive-depth gates on both $1600$-task final supports. The program-specificity gain is $0.010714$ on the primary audit and $0.024763$ under the generator shift, with positive frozen intervals; clean error also improves. An independent NumPy forward implementation and signed-permutation gauge check reproduce the result within $2\times10^{-6}$. This is a finite package-local training-generated repair, not a production-model or open-ended RSI result.
20230 
20231 
20232 
20233 Block 18 attempts that cross-implementation threshold on two foreign recurrent cell families. The source-selected progressive FiLM schema transfers successfully only on the primary generator. Under the independently shifted generator, both children fail the joint semantic, execution, adaptive-depth, and clean-nonregression certificate; both frozen bridges fail, and the source-reference probe itself becomes invalid. A held-out post-hoc diagnosis recovers semantics after local recalibration, establishing a distributional semantic-chart obstruction rather than semantic absence. The failed family receipt is preserved, and no Block 18 lineage edge is accepted.
20234 
20235 Block 19 acts on that obstruction before attempting another foreign-architecture bridge. It compiles two new affine observation charts from semantic twins, conservatively appends them to four inherited charts, and selects a set-structured semantic bundle with calibrated chart admission. The selected child is \texttt{ATLAS\_FULL} on fresh inherited and added-chart audits, while the parent and isolated set-only and chart-only controls remain partial on the added charts. It admits all supported final tasks, rejects all unsupported nonlinear tasks, preserves inherited canonical coordinates exactly, and passes independent exact-task, chart, receipt, and NumPy forward verification. This closes one finite source-side distributional chart gate, not the cross-architecture transport gate.
20236 
20237 Block 20 then transports that stabilized bundle across two independently trained recurrent implementations. The selected connection assigns one full five-step trajectory map to each admitted chart and target architecture. It preserves the frozen source semantic probe on fresh inherited and added-chart supports, bounds normalized architecture--context curvature below $0.02$, improves paired semantic loss over pooled maps, and fails closed on unsupported nonlinear contexts. Pooled, step-shared, shuffled-binding, and unaligned controls delimit the mechanism. This closes one finite cross-architecture atlas-transport gate under controlled orthogonal gauge stress, not a universal alignment theorem.
20238 
20239 Block 21 then treats a new transport residual as a type-directed diagnosis problem rather than fitting a single larger bridge. Across six hidden obstruction families and two foreign architectures, the sealed compiler selects the required minimum-description affine, nonlinear, temporal, chart-split, quotient, or no-go response in all $36$ fresh final episodes. Wrong-repair controls are strictly separated, benign rank loss preserves the protected quotient, and exact semantic collisions trigger rejection. This closes one finite transport-grammar growth gate, not unrestricted obstruction discovery.
20240 
20241 The empirical ladder is now
20242 \begin{equation}
20243 \begin{gathered}
20244 \text{repair mechanics}
20245 <
20246 \text{causal foreign transfer}\\
20247 <
20248 \text{one-step process metaproductivity}
20249 <
20250 \text{two-edge bottleneck migration}\\
20251 <
20252 \text{controlled structural metaproductivity}
20253 <
20254 \text{endogenous atlas growth}\\
20255 <
20256 \text{finite response and constructor growth}\\
20257 <
20258 \text{finite research-compiler metaproductivity}\\
20259 <
20260 \text{finite adversarial compiler robustness}\\
20261 <
20262 \text{direct latent-language causal certification}\\
20263 <
20264 \text{training-generated executable latent repair}\\
20265 <
20266 \text{distributional latent-chart robustness}\\
20267 <
20268 \text{atlas-indexed foreign-model latent transport}\\
20269 <
20270 \text{type-directed transport-grammar growth}\\
20271 <
20272 \text{autonomous open-set grammar growth}
20273 <
20274 \text{open-set RSI}.
20275 \end{gathered}
20276 \end{equation}
20277 Block 15 closes the first finite compiler-language growth gate for the research compiler. Block 16 closes the direct formal-and-synthetic Neuralese audit gate. Block 17 closes one package-local trained-checkpoint repair gate, including answer-confound controls and preserved failed-pilot chronology. Block 18 attempts the cross-implementation gate and rejects it under generator shift, yielding a distributional semantic-chart obstruction rather than a positive lineage edge. Block 19 closes the resulting finite source-side atlas-growth gate through conservative semantic-twin chart compilation and selective admission. Block 20 closes one finite atlas-indexed cross-architecture transport gate under controlled gauge stress. Block 21 closes one finite type-directed transport-grammar gate by selecting minimal obstruction-specific repairs or a collision-grounded no-go certificate. Block 22 removes exact source--target construction pairs and closes one finite intervention-orbit transport gate while distinguishing harmless gauge nonuniqueness from semantic underidentification, partial support, and information-loss no-go. Block 23 removes supplied intermediate semantic coordinates from construction and closes one finite causal-predictive semantic-quotient gate with explicit gauge, partiality, underidentification, and information-loss certificates. Its external Coconut track remains an unexecuted source-locked protocol. These results do not collapse the remaining independent-replication, evaluator-growth, stochastic-path, production-model, or open-ended distinctions.
20278 
20279 \appendix
20280 
20281 
20282 
20283 Block 12 extends the realized lineage to
20284 \begin{equation}
20285 K_0\xrightarrow{\Pi_1}K_1\xrightarrow{\Pi_2}K_2\xrightarrow{\Pi_3}K_3\xrightarrow{\Pi_4}K_4\xrightarrow{\Pi_5}K_5\xrightarrow{\Pi_6}K_6.
20286 \end{equation}
20287 The fifth edge changes the intervention-response language because the inherited algebra provably aliases two task-relevant blocks. The sixth changes the repair-constructor function class because the inherited affine operad provably cannot realize a nonzero-Hessian repair. Both extensions preserve their parents as executable subgrammars and improve fresh foreign systems under matched deployment resources.
20288 
20289 The result remains finite and scaffolded. The next decisive question is no longer whether one supplied language can be enlarged. It is whether a lineage-generated compiler can diagnose different obstruction types across foreign parent processes, select the correct language expansion, and outperform its own parent compiler without access to the final audit.
20290 
20291 Block 13 answers the finite allocation form of that question. The lineage adds
20292 \begin{equation}
20293 \Gamma_0\xrightarrow{\Pi_7^{\Gamma}}\Gamma_1,
20294 \end{equation}
20295 where parent and child share the candidate-method grammar, evaluator portfolio, integrity monitor, archive initialization, and six-experiment budget. The child improves validity-filtered foreign research outcomes chiefly by diagnosing the active obstruction and reallocating experiments. The result is a sealed compiler edge inside a fixed research language, not an unrestricted compiler that invents its own science, evaluator, or integrity model.
20296 
20297 
20298 Block 14 adds one further compiler edge,
20299 \begin{equation}
20300 \Gamma_1\xrightarrow{\Pi_8^{\Gamma}}\Gamma_2,
20301 \end{equation}
20302 where the Block 13 result digest and exact method language are replayed, the inherited compiler design is recalibrated as the parent, and a robust descendant is compiled. The child preserves the fixed $39$-method grammar and six-experiment budget while adding sequential diagnosis, open-set routing, robust archive summaries, evaluator-portfolio aggregation, and an integrity-monitor ensemble. Its claim is relative to the declared finite stress atlas.
20303 
20304 Block 15 adds the first finite compiler-language edge,
20305 \begin{equation}
20306 \Gamma_2\xrightarrow{\Pi_9^{\Gamma}}\Gamma_3.
20307 \end{equation}
20308 The inherited compiler retains one undifferentiated unknown token and a singleton-method grammar. Development repair conflicts select a separating probe, two anonymous refined tokens, and two ordered pair-method constructors from generic finite meta-languages. The old compiler remains an exact fallback. Under equal eight-unit research budgets, primary gain rises from $0.633969$ to $0.859731$ and generator-foreign gain rises from $0.574224$ to $0.814520$. This is finite structural compiler-language growth, not unrestricted experiment or program invention.
20309 
20310 
20311 
20312 Block 16 adds the directly auditable latent-language state
20313 \begin{equation}
20314 \mathfrak L_t^{\mathrm{NL}}
20315 =
20316 (\mathfrak L_t,\Ncal_t,\Ical_t^{\mathrm{NL}},\Tau_t^{\mathrm{NL}},\Wcal_t^{\mathrm{NL}}).
20317 \end{equation}
20318 Block 17 realizes one trained finite transition
20319 \begin{equation}
20320 \mathfrak L_{16}^{\mathrm{NL}}
20321 \xrightarrow{\Pi_{17}}
20322 \mathfrak L_{17}^{\mathrm{NL}},
20323 \end{equation}
20324 where the selected descendant changes the latent conditioning map while the exact task interpreter, witness coordinates, update budget, audit gates, and rollback semantics remain fixed. The transition is accepted only after fresh primary and generator-foreign audits and package-local independent replay. Its scope is the declared affine-program environment.
20325 
20326 
20327 Block 18 attempts but does not accept
20328 \begin{equation}
20329 \mathfrak L_{17}^{\mathrm{NL}}
20330 \xrightarrow{\Pi_{18}?}
20331 \mathfrak L_{18}^{\mathrm{NL}}.
20332 \end{equation}
20333 The question mark is substantive: the selected schema passes both primary architecture-foreign audits but fails both shifted-generator audits and both frozen semantic-transport gates. The lineage therefore appends an obstruction receipt rather than a descendant commit. The post-hoc local-atlas recovery is stored as diagnostic evidence for the next compiler transaction and is not counted as a completed edge.
20334 
20335 Block 19 accepts the source-side continuation
20336 \begin{equation}
20337 \mathfrak L_{18}^{\mathrm{NL}}
20338 \xrightarrow{\Pi_{19}}
20339 \mathfrak L_{19}^{\mathrm{NL}},
20340 \end{equation}
20341 where the added object is a six-chart semantic atlas with a set-structured latent pathway and selective support gate. The edge is conservative on inherited charts and improves fresh tasks in two compiled contexts. It does not yet transport that bundle across the foreign recurrent architectures from Block 18.
20342 
20343 Block 20 accepts the finite transport continuation
20344 \begin{equation}
20345 \mathfrak L_{19}^{\mathrm{NL}}
20346 \xrightarrow{\Pi_{20}}
20347 \mathfrak L_{20}^{\mathrm{NL}},
20348 \end{equation}
20349 where the added object is an atlas-indexed connection carrying complete Neuralese trajectories from GRU and gated-residual fibers into the frozen Elman source fibers. The edge preserves the six-chart atlas, passes fresh semantic and curvature audits, and rejects unsupported contexts. Its scope remains the declared finite architecture, chart, gauge, and affine transport grammars.
20350 
20351 Block 21 accepts the finite grammar continuation
20352 \begin{equation}
20353 \mathfrak L_{20}^{\mathrm{NL}}
20354 \xrightarrow{\Pi_{21}}
20355 \mathfrak L_{21}^{\mathrm{NL}},
20356 \end{equation}
20357 where the transport process gains a typed response grammar over affine drift, nonlinear warp, latent-clock mixing, chart aliasing, benign semantic quotient, and destructive semantic loss. The edge preserves every inherited valid repair and accepts only the minimum-description candidate that passes the protected witness.
20358 
20359 \section{Notation}
20360 
20361 \begin{longtable}{@{}p{0.22\textwidth}p{0.70\textwidth}@{}}
20362 \toprule
20363 Symbol & Meaning \\
20364 \midrule
20365 \endfirsthead
20366 \toprule
20367 Symbol & Meaning \\
20368 \midrule
20369 \endhead
20370 $\Xi$ & Protected evaluation context \\
20371 $J_\Xi(A)$ & Resource-normalized utility of agent $A$ in context $\Xi$ \\
20372 $\mathfrak A_t$ & Stratified functional atlas at generation $t$ \\
20373 $G_t$ & Directed multigraph of typed cross-stratum bridges \\
20374 $\Acal_t$ & Executable ancestor-closed archive inside the lineage graph \\
20375 $K_A^\delta$ & Protected external-outcome kernel under intervention $\delta$ \\
20376 $\widehat K_{\phi(A)}^\delta$ & Outcome kernel predicted by the compressed self-model \\
20377 $\Hcal$ & Causal-hypothesis family for active diagnosis \\
20378 $\Pcal(T)$ & Preservation profile of a typed bridge $T$ \\
20379 $\Pcal_{\mathrm{probe}}$ & Permitted active-probe family \\
20380 $\overline\Ocal$ & Underlying syntactic colored repair operad \\
20381 $\mathfrak a$ & Repair abstraction operator from episodes to schemas \\
20382 $\mathfrak L_t$ & Complete resolved lineage state at generation $t$ \\
20383 $A_t$ & Active stratified agent \\
20384 $\Ocal_t$ & Proof-carrying repair operad \\
20385 $\Ecal_t$ & Append-only external-evidence ledger \\
20386 $\Vcal_t$ & Evaluator atlas \\
20387 $\Lambda_t$ & Proof-carrying lineage directed acyclic graph \\
20388 $\Wcal_t$ & Witness, checkpoint, and rollback structure \\
20389 $\mathsf C$ & Set of stratum colors or types \\
20390 $\Dcal_t$ & Authorized modification family \\
20391 $\phi_t(A_t)$ & Compressed causal self-model \\
20392 $U_t^{\Auth}$ & Authorized correction space \\
20393 $V_t$ & Defect space \\
20394 $D_t$ & Linearized defect-response map \\
20395 $[r]$ & Obstruction class in $\coker D$ \\
20396 $D^\dagger$ & Moore--Penrose pseudoinverse of the correction map \\
20397 $P_{\mathrm{cor}},P_{\mathrm{ob}}$ & Orthogonal correctable and obstruction projectors \\
20398 $\eta_D(r)$ & Quantitative obstruction $\dist(r,\im D)$ \\
20399 $\eta_{D,K}(r)$ & Minimum residual under convex authorization set $K$ \\
20400 $\sigma_+(D)$ & Smallest positive singular value of $D$ \\
20401 $\rho_t$ & Authorized repair rank $\rank D_t$ \\
20402 $o_t$ & Unresolved obstruction dimension $\dim\coker D_t$ \\
20403 $D_0,E,Q_0$ & Packet-diagonal response, cross-packet coupling, and local right inverse \\
20404 $\mathscr U,\mathscr V$ & Sheaves of authorized corrections and defects \\
20405 $d_{\alpha\beta}$ & Difference of local repairs on an overlap \\
20406 $\eta_{\mathrm{loc}},\eta_{\mathrm{glue}}$ & Quantitative local and gluing residuals \\
20407 $B,L_G,\lambda_2(L_G)$ & Nerve incidence matrix, graph Laplacian, and spectral gap \\
20408 $P_{\mathrm{cyc}}$ & Orthogonal projection onto the overlap cycle space \\
20409 $g_{\alpha\beta}$ & Nonabelian transition function between local nonlinear repairs \\
20410 $H_e$ & Fundamental-cycle holonomy associated with a non-tree edge \\
20411 $b_n$ & Order-$n$ residual coefficient in the formal obstruction tower \\
20412 $T_U,T_V$ & Correction-space and defect-space transport maps \\
20413 $\eps_D$ & Correction-diagram commutation defect \\
20414 $\eps_r$ & Defect-transport mismatch \\
20415 $\eps_J$ & Evaluator alignment error \\
20416 $\eps_{\mathrm{leak}}$ & Packet leakage error \\
20417 $\eps_{\mathrm{hol}}$ & Semantic holonomy error \\
20418 $\Delta_D,\Delta_r$ & Correction-square defect and defect-identification error of a transport datum \\
20419 $Q_A$ & Protected obstruction subspace required to survive transport \\
20420 $H_\gamma^U,H_\gamma^V$ & Correction and defect holonomies around a transport loop \\
20421 $C_{\mathrm{tr}}(p;u)$ & Certified transport-route cost for correction $u$ \\
20422 $\mathsf E_v$ & Evaluator object at version $v$ \\
20423 $R_v,E_v,\iota_v,J_v$ & Raw evidence, internal evidence space, encoder, and readout of evaluator $v$ \\
20424 $\mathsf A$ & Positive semidefinite evaluator holonomy-energy operator \\
20425 $E_\tau^{\flat}$ & Spectral approximate path-flat core at threshold $\tau$ \\
20426 $\mu(F;\Delta_{\mathrm{prot}})$ & Adequacy modulus of core $F$ on protected evidence differences \\
20427 $E^{\flat}$ & Maximal path-flat evidence core \\
20428 $E,W$ & Editable and witness component masks of a transaction \\
20429 $q,a,Z,\Pi$ & Trace map, sealed acceptance rule, revealed trace, and receipt \\
20430 $\Ical_{\forall}^{*}$ & Universal recursive-improvement safety closure \\
20431 $\Ical_{\mathrm{ctrl}}^{*}$ & Robust controlled improvement kernel \\
20432 $\Ical_{\gamma}^{\omega}$ & Productive closure with repeated gain threshold $\gamma$ \\
20433 $\mathsf{Act}(s),\mathsf{Dist}(s,u)$ & Authorized actions and admissible disturbances \\
20434 $H_w(\Acal_t)$ & Archive upper envelope in scalarization direction $w$ \\
20435 $\sim_h$ & Horizon-$h$ continuation equivalence of archive nodes \\
20436 $M^{\mathrm{foreign}}_{h,B}$ & Foreign-agent behavioral metaproductivity \\
20437 $G_{h,B}(K,A)$ & Realized improvement lift of mechanism $K$ on agent $A$ \\
20438 $\mathcal E_t^{\mathrm{cor}},\mathcal E_t^{\mathrm{ob}}$ & Correctable and obstruction energy on a defect distribution \\
20439 $q_{\Gcal}(r)$ & Minimum unresolved residual inside grammar $\Gcal$ \\
20440 $\Phi_{\mathrm{disc}}$ & Monotone repair-discovery closure operator \\
20441 $M$ & Causal repair-schema offspring matrix \\
20442 $R_{\mathrm{imp}}$ & Spectral radius of the improvement offspring matrix \\
20443 $\lambda_{\mathrm{mean}}$ & Upper growth exponent for a time-varying offspring product \\
20444 $\mathfrak B$ & Empirical benchmark instance \\
20445 $\mathfrak C$ & Package-closed claim object \\
20446 $\mathcal M,\widehat{\mathcal M}$ & Abstract transition system and executable implementation \\
20447 $\alpha$ & Runtime-to-abstract refinement map \\
20448 $G_\ell$ & Cumulative empirical claim gate at level $\ell$ \\
20449 $\phi(z),\mu_z$ & Anonymous-event probe vector and frozen prototype \\
20450 $\Delta$ & Minimum pairwise event-prototype separation \\
20451 $M_a(B)$ & Matched-budget best-candidate gain for benchmark arm $a$ \\
20452 $E_{\mathrm{corr}}$ & Corrected protected defect energy \\
20453 $\tau_M^{\mathrm{future}}$ & Future foreign-agent effect of repair-schema abstraction \\
20454 $\psi(x),s(x),\phi(x)$ & Base feature, affected-stratum indicator, and conditional Block 6 feature map \\
20455 $q(x),\widetilde q(x),C$ & Protected score, corrupted score, and cyclic corruption operator \\
20456 $H_s,H_t$ & Paired source and target hidden-state anchor matrices \\
20457 $\widehat T_{t\leftarrow s}$ & Learned source-to-target hidden-state bridge \\
20458 $D_s,b_s,D_t,b_t$ & Source and transported affine repair parameters \\
20459 $\widehat\eps_{\mathrm{align}}$ & Calibration normalized hidden-state alignment error \\
20460 $\mathfrak R_{\mathrm{learn}}$ & Controlled learned repair-transport validation instance \\
20461 $C_e$ & Ordered fixed repair-candidate bundle for Block 7 target episode $e$ \\
20462 $B$ & Matched candidate budget per target \\
20463 $K_0,K_1$ & Frozen parent and lineage-generated child repair selectors \\
20464 $\mathfrak M$ & Sealed meta-compiler mapping development evidence to a child and receipt \\
20465 $\Gamma$ & Finite Block 7 selector meta-grammar \\
20466 $\Pi_1$ & Proof-carrying descendant receipt for the child selector \\
20467 $U_e(c),G_e(c),R_e(c)$ & Protected utility, no-op-relative utility gain, and protected regression of candidate $c$ \\
20468 $\widehat U_{\rm LCB},\widehat R_{\rm UCB}$ & Child utility lower score and risk upper score \\
20469 $D_j$ & Paired child-minus-parent utility difference on audit episode $j$ \\
20470 $\Delta_Q(K_1,K_0)$ & Held-out selector-level metaproductivity effect \\
20471 $G_{\rm meta}^{\rm select}$ & Cumulative claim gate for sealed selector-level metaproductivity \\
20472 $\mathfrak P_0,\mathfrak P_1$ & Parent and compiled child bridge-identification repair processes \\
20473 $\pi_{\rm anc}$ & Paired source--target anchor-query policy \\
20474 $B_{\rm pair}$ & Matched deployment paired-query budget \\
20475 $M_B^{\rm for}(\mathfrak P)$ & Foreign matched-budget metaproductivity of a repair process \\
20476 $\delta_{\rm rank}$ & Bridge-identification rank defect \\
20477 $G_{\rm meta}^{\rm bridge}$ & Cumulative gate for sealed bridge-policy metaproductivity \\
20478 $N=\{H,Q,C,T,S,V,W\}$ & Repair-process module index set \\
20479 $F_K(R)$ & Restricted oracle envelope when only module coalition $R$ may change \\
20480 $\Hcal(K)$ & Total declared metaproductivity headroom \\
20481 $h_i(K)$ & Permutation-average module headroom allocation \\
20482 $\chi_R(K,K')$ & Headroom-adjusted gain of child $K'$ relative to module coalition $R$ \\
20483 $r_t$ & Nonnegative unresolved module-headroom vector \\
20484 $A_a,e_t$ & Local meta-action response matrix and residual error \\
20485 $\Phi_w(r)$ & Weighted meta-regret Lyapunov function \\
20486 $P_t^{\mathrm{repair}},P_t^{\mathrm{discovery}}$ & Old-coordinate repair progress and newly represented discovery energy \\
20487 $L_t\preceq_{\mathrm{cons}}L_{t+1}$ & Conservative lineage-dominance relation \\
20488 $\Gamma_t$ & Lineage compiler that maps a parent process and receipts to a descendant process \\
20489 $\Ccal(\Gamma)$ & Compiler metaproductivity on foreign parent repair processes \\
20490 $I_t=\im D_t$ & Authorized correction image at grammar generation $t$ \\
20491 $E_t(Q)$ & Expected unresolved obstruction energy on foreign residual distribution $Q$ \\
20492 $\overline T_t$ & Induced transport between obstruction quotients \\
20493 $\Sigma_{\mathrm{ob}}$ & Covariance operator of projected foreign obstruction residuals \\
20494 $\mathfrak H_{\Gcal}$ & Hypergraph of minimal repair conflicts under grammar $\Gcal$ \\
20495 $\Acal^\star(x)$ & Set of externally validated authorized repairs for episode $x$ \\
20496 $\mathfrak A$ & Adjoint-closed algebra generated by intervention-response operators \\
20497 $Z(\mathfrak A),\mathfrak A'$ & Center and commutant of the response algebra \\
20498 $H_r$ & Cyclic causal module generated by residual $r$ \\
20499 $\Lcal_{\mathrm{com}}$ & Positive commutator operator used for approximate atlas recovery \\
20500 $\Delta_t,L_t$ & True lineage-edge metaproductivity effect and conditional lower bound \\
20501 $\delta_t,\beta_t,d_t$ & Edge false-acceptance, protected-regression, and semantic-debt budgets \\
20502 $K_0,K_1,K_2,K_3$ & Random-query ancestor, rank-revealing child, conditioning-aware grandchild, and structural chart descendant across Blocks 7--10 \\
20503 $K_1^{\rm recal}$ & Rank-revealing child with a witness gate recalibrated on Block 9 development evidence \\
20504 $X_S,W,M_S$ & Selected source-anchor design, inverse-noise weight matrix, and weighted bridge-information matrix \\
20505 $\sigma_i^2$ & Conditional noise variance of paired target anchor $i$ \\
20506 $\lambda_{\min}(M_S)$ & Weighted minimum information eigenvalue \\
20507 $\widehat\Delta_{1\to2}$ & Paired grandchild-minus-recalibrated-child foreign metaproductivity effect \\
20508 $\Gcal_{\mathrm{glob}},\Gcal_{\mathrm{chart}}$ & Global-linear and two-chart bridge grammars in Block 10 \\
20509 $T_-,T_+$ & Chart-specific source-to-target bridge maps \\
20510 $D_-,D_+$ & Chart-specific transported target repairs \\
20511 $\Delta_{\mathrm{gram}}$ & Global-minus-local held-out transport residual margin \\
20512 $\kappa_D$ & Normalized chart repair-contrast magnitude \\
20513 $K_{2,G}^{96},K_{3,C}^{96}$ & Resource-matched global-affine parent and invariant-context chart child \\
20514 $\Delta_{G\to C}$ & Chart-child minus global-parent paired protected-gain effect in Block 10 \\
20515 $\Pi_2$ & Proof-carrying Block 9 conditioning-grandchild receipt \\
20516 $\Pi_3$ & Proof-carrying Block 10 structural grammar-expansion receipt \\
20517 $\Xi_t$ & Protected context associated with lineage edge $t\to t+1$ \\
20518 $R_j$ & Controlled intervention-response operator used for endogenous atlas recovery \\
20519 $\lambda_a$ & Joint response signature of hidden block $H_a$ \\
20520 $\mathfrak A_R$ & Adjoint-closed algebra generated by the response operators \\
20521 $\mathfrak A_R'$ & Response commutant; unresolved within-block multiplicity gauge \\
20522 $P_a,\widehat P_a$ & Exact and estimated central response-block projectors \\
20523 $\gamma_a,\widehat\gamma_{\min}$ & Exact and empirical response spectral gaps \\
20524 $\eps_{\mathrm{resp}}$ & Normalized response block-model residual \\
20525 $\eps_{\mathrm{comm}}$ & Maximum normalized projector--response commutator residual \\
20526 $H_r$ & Minimal response-invariant cyclic module generated by defect $r$ \\
20527 $K_4,\Pi_4$ & Endogenous-atlas descendant and its proof-carrying lineage edge \\
20528 \bottomrule
20529 \end{longtable}
20530 
20531 
20532 \subsection*{Block 12 notation additions}
20533 \begin{center}
20534 \begin{tabularx}{0.96\textwidth}{@{}p{0.29\textwidth}X@{}}
20535 \toprule
20536 Symbol & Meaning \\
20537 \midrule
20538 $\Rcal$ & Declared intervention-response language \\
20539 $\mathfrak A(\Rcal)$ & Unital algebra generated by the response language \\
20540 $\sigma_{\Rcal}(H_a)$ & Response signature of hidden block $H_a$ \\
20541 $\Pi_{\mathrm{resp}}$ & Proof-carrying response-language expansion certificate \\
20542 $\Ccal_{\mathrm{aff}}$ & Inherited affine repair-constructor grammar \\
20543 $\Ccal_2(P)$ & Degree-two repair grammar localized to packet $P$ \\
20544 $\Pi_{\mathrm{con}}$ & Proof-carrying constructor-expansion certificate \\
20545 $K_4,K_5,K_6$ & Endogenous-atlas parent, response-language child, and degree-two constructor descendant \\
20546 \bottomrule
20547 \end{tabularx}
20548 \end{center}
20549 
20550 
20551 \subsection*{Block 13 notation additions}
20552 \begin{center}
20553 \begin{tabularx}{0.96\textwidth}{@{}p{0.29\textwidth}X@{}}
20554 \toprule
20555 Symbol & Meaning \\
20556 \midrule
20557 $\Gamma_0,\Gamma_1$ & AAR-style parent and lineage-trained child research compilers \\
20558 $\Pi_i^-,\Pi_i^+$ & Results-free pre-receipt and evidence-appended post-receipt for research candidate $i$ \\
20559 $\Pi_7^{\Gamma}$ & Proof-carrying Block 13 compiler lineage receipt \\
20560 $\mathfrak C_B(\Gamma)$ & Validity-filtered compiler metaproductivity under research budget $B$ \\
20561 $S_{\mathrm{cov}}$ & Coverage-weighted geometric benchmark score \\
20562 $\Ncal_{\Bcal}$ & Local behavior subspace invisible to evaluator portfolio $\Bcal$ \\
20563 $F_\theta(z)$ & Outcome, risk, and scored-performance forecaster from proposal and pilot evidence \\
20564 $\widehat\Delta_\Gamma$ & Paired child-minus-parent compiler metaproductivity effect \\
20565 $\mathcal H_{\mathrm{obs}}$ & Finite hidden obstruction-class family in the Block 13 benchmark \\
20566 \bottomrule
20567 \end{tabularx}
20568 \end{center}
20569 
20570 
20571 \subsection*{Block 14 notation additions}
20572 \begin{center}
20573 \begin{tabularx}{0.96\textwidth}{@{}p{0.29\textwidth}X@{}}
20574 \toprule
20575 Symbol & Meaning \\
20576 \midrule
20577 $\mathfrak U$ & Declared research-compiler robustness environment \\
20578 $\mathfrak C_{\rm rob}(\Gamma)$ & Worst-environment validity-filtered compiler value with a lower-tail risk term \\
20579 $R,z,y$ & Response-signature matrix, sparse obstruction mixture, and observed sequential response \\
20580 $\eta_{\rm open}(y)$ & Distance of an observed response from the inherited known response cone \\
20581 $\widetilde P_\epsilon$ & $\epsilon$-contaminated research-receipt distribution \\
20582 $\mathcal N_{\mathcal B}$ & Local behavioral subspace invisible to evaluator portfolio $\mathcal B$ \\
20583 $M_{\cap}$ & Conjunctive integrity-monitor ensemble \\
20584 $\Gamma_2,\Pi_8^{\Gamma}$ & Robust compiler descendant and its Block 14 lineage receipt \\
20585 $\widehat\Delta_{\Gamma}^{\rm primary},\widehat\Delta_{\Gamma}^{\rm foreign}$ & Paired robust-child minus inherited-parent effects \\
20586 \bottomrule
20587 \end{tabularx}
20588 \end{center}
20589 
20590 \subsection*{Block 15 notation additions}
20591 \begin{center}
20592 \begin{tabularx}{0.96\textwidth}{@{}p{0.29\textwidth}X@{}}
20593 \toprule
20594 Symbol & Meaning \\
20595 \midrule
20596 $\mathfrak G=(\Zcal,\Pcal,\Ccal,\Bcal)$ & Research-compiler diagnostic, experiment, method-constructor, and binding language \\
20597 $\phi_0$ & Inherited diagnostic quotient mapping both novel mechanisms to one unknown token \\
20598 $\Acal^\star(\tau)$ & Acceptable validity-filtered methods for research problem $\tau$ \\
20599 $\Hcal_{\rm conflict}$ & Hypergraph of repair-conflicting research-problem cells \\
20600 $y_p,\theta$ & Candidate probe response and refinement threshold \\
20601 $\Ccal_{\rm pair}$ & Generic ordered pair-method constructor grammar \\
20602 $O_0,O_1$ & Inherited and expanded method-language oracle envelopes \\
20603 $\kappa_Q$ & Task-relevant structural language margin $\mathbb E_Q[O_1-O_0]$ \\
20604 $\Pi_{\Gamma\text{-lang}}$ & Proof-carrying compiler-language expansion receipt \\
20605 $\Gamma_3,\Pi_9^\Gamma$ & Block 15 language-growing compiler and its lineage receipt \\
20606 $\widehat\Delta_{\Gamma\text{-lang}}^{\rm primary},\widehat\Delta_{\Gamma\text{-lang}}^{\rm foreign}$ & Paired child-parent language-growth effects \\
20607 \bottomrule
20608 \end{tabularx}
20609 \end{center}
20610 
20611 \section{Worked Correctability and Descent Examples}
20612 
20613 \subsection{Canonical obstruction and minimal repair}
20614 
20615 Let
20616 \begin{equation}
20617 D=
20618 \begin{bmatrix}
20619 1&0\\
20620 0&0
20621 \end{bmatrix},
20622 \qquad
20623 r=
20624 \begin{bmatrix}
20625 a\\b
20626 \end{bmatrix}.
20627 \end{equation}
20628 Then $D^\dagger=D$,
20629 \begin{equation}
20630 P_{\mathrm{cor}}=
20631 \begin{bmatrix}1&0\\0&0\end{bmatrix},
20632 \qquad
20633 P_{\mathrm{ob}}=
20634 \begin{bmatrix}0&0\\0&1\end{bmatrix}.
20635 \end{equation}
20636 The minimum-norm least-squares repair is
20637 \begin{equation}
20638 u^*=
20639 \begin{bmatrix}-a\\0\end{bmatrix},
20640 \end{equation}
20641 with residual
20642 \begin{equation}
20643 Du^*+r=
20644 \begin{bmatrix}0\\b\end{bmatrix}.
20645 \end{equation}
20646 Therefore
20647 \begin{equation}
20648 \eta_D(r)=|b|.
20649 \end{equation}
20650 For $b\neq0$, the unit dual witness is $y^*=\operatorname{sgn}(b)e_2$. Every authorized correction has zero effect on the second defect coordinate.
20651 
20652 \subsection{An invariant creates an obstruction}
20653 
20654 Let
20655 \begin{equation}
20656 D=
20657 \begin{bmatrix}1&0\end{bmatrix},
20658 \qquad
20659 C=
20660 \begin{bmatrix}1&0\end{bmatrix},
20661 \qquad
20662 r=1.
20663 \end{equation}
20664 Without the invariant, $u=(-1,0)$ repairs the defect. First-order invariant preservation requires $Cu=0$, hence $u_1=0$. The restricted response $D_C=\restr D{\ker C}$ is zero, so
20665 \begin{equation}
20666 \coker D_C\cong\R
20667 \end{equation}
20668 and the nonzero defect is authorized-obstructed. The classification is not ``the optimizer failed.'' The only correcting direction is forbidden by the protected invariant.
20669 
20670 \subsection{Cone-constrained dual certificate}
20671 
20672 Let $K=\R_+^2$, $D=I_2$, and
20673 \begin{equation}
20674 r=
20675 \begin{bmatrix}1\\-2\end{bmatrix}.
20676 \end{equation}
20677 Exact correction would require $u=-r=(-1,2)$, which is not in $K$. Take
20678 \begin{equation}
20679 y=e_1.
20680 \end{equation}
20681 Then $D^*y=e_1\in K^*$ and
20682 \begin{equation}
20683 \angles{y,r}=1>0.
20684 \end{equation}
20685 By \cref{thm:cone-certificate}, $y$ certifies that no nonnegative authorized correction exists.
20686 
20687 \subsection{Packet coupling and iterative reconciliation}
20688 
20689 Let
20690 \begin{equation}
20691 D_0=I_2,
20692 \qquad
20693 E=
20694 \begin{bmatrix}
20695 0&q\\q&0
20696 \end{bmatrix},
20697 \qquad
20698 0<q<1,
20699 \end{equation}
20700 so $Q_0=I_2$ and $\norm{EQ_0}=q$. Then
20701 \begin{equation}
20702 D^{-1}=(I_2+E)^{-1}
20703 =\frac{1}{1-q^2}
20704 \begin{bmatrix}
20705 1&-q\\-q&1
20706 \end{bmatrix}.
20707 \end{equation}
20708 For $r=e_1$, the exact repair is
20709 \begin{equation}
20710 u^*=-D^{-1}r
20711 =-\frac{1}{1-q^2}
20712 \begin{bmatrix}1\\-q\end{bmatrix}.
20713 \end{equation}
20714 The zeroth packet-local repair is $u^{(0)}=-e_1$ and leaves residual
20715 \begin{equation}
20716 Du^{(0)}+r=
20717 \begin{bmatrix}0\\-q\end{bmatrix}.
20718 \end{equation}
20719 Each further Neumann correction alternates packets and reduces the residual norm by a factor $q$. At $q=0.2$, four reconciliation rounds certify a residual at most $0.2^5=3.2\times10^{-4}$.
20720 
20721 \subsection{Exact and obstructed triangle gluing}
20722 
20723 Orient a triangle cyclically and identify edge mismatches with a vector
20724 \begin{equation}
20725 e=(e_{12},e_{23},e_{31})\in\R^3.
20726 \end{equation}
20727 The cycle space is spanned by
20728 \begin{equation}
20729 c=\frac{1}{\sqrt3}(1,1,1).
20730 \end{equation}
20731 Hence
20732 \begin{equation}
20733 \eta_{\mathrm{glue}}(e)=|\angles{c,e}|
20734 =\frac{|e_{12}+e_{23}+e_{31}|}{\sqrt3}.
20735 \end{equation}
20736 For
20737 \begin{equation}
20738 e=(1,1,-2),
20739 \end{equation}
20740 the sum vanishes and the mismatch is a pure gradient, so a vertex gauge glues the repairs exactly. For
20741 \begin{equation}
20742 e=(1,1,1),
20743 \end{equation}
20744 the quantitative obstruction is $\sqrt3$ and no gauge can reconcile the cycle.
20745 
20746 \subsection{Approximate cycle inconsistency}
20747 
20748 Let
20749 \begin{equation}
20750 e=(1,1,-1.8).
20751 \end{equation}
20752 Then
20753 \begin{equation}
20754 \eta_{\mathrm{glue}}(e)
20755 =\frac{0.2}{\sqrt3}
20756 \approx0.11547.
20757 \end{equation}
20758 Most of the mismatch can be removed by adjusting local kernel coordinates, but the cycle-average component is irreducible. Reporting only the raw edge norm
20759 \begin{equation}
20760 \norm e\approx2.289
20761 \end{equation}
20762 would substantially overstate the genuine incompatibility.
20763 
20764 \subsection{A second-order obstruction invisible to the derivative}
20765 
20766 Consider
20767 \begin{equation}
20768 F(\eps,x,y)
20769 =
20770 \begin{bmatrix}
20771 x+\eps\\x^2
20772 \end{bmatrix}.
20773 \end{equation}
20774 At $(\eps,x,y)=(0,0,0)$,
20775 \begin{equation}
20776 D=\partial_{(x,y)}F
20777 =
20778 \begin{bmatrix}
20779 1&0\\0&0
20780 \end{bmatrix}.
20781 \end{equation}
20782 The first-order defect $r_1=e_1$ is correctable by choosing $x_1=-1$. However, the order-$\eps^2$ term in the second coordinate is $1$, which represents a nonzero class in $\coker D$. Indeed, the exact first equation forces $x=-\eps$, and then the second equation becomes $\eps^2=0$. Thus no nonzero small $\eps$ admits a repair despite vanishing of the first-order obstruction.
20783 
20784 \subsection{Branching in the reduced equation}
20785 
20786 For
20787 \begin{equation}
20788 F(\eps,x)=x^2-\eps,
20789 \end{equation}
20790 the derivative with respect to $x$ at the origin is zero, so the entire scalar equation is the Lyapunov--Schmidt reduced obstruction. For $\eps>0$ there are two branches
20791 \begin{equation}
20792 x=\pm\sqrt\eps,
20793 \end{equation}
20794 for $\eps=0$ there is one singular branch, and for $\eps<0$ there is no real repair. A linearized system cannot distinguish these three regimes.
20795 
20796 \subsection{Exact transport by conjugacy}
20797 
20798 Let
20799 \begin{equation}
20800 D_A=
20801 \begin{bmatrix}
20802 1&0\\
20803 0&2
20804 \end{bmatrix},
20805 \qquad
20806 T_U=T_V=
20807 \begin{bmatrix}
20808 0&-1\\
20809 1&0
20810 \end{bmatrix},
20811 \end{equation}
20812 and define
20813 \begin{equation}
20814 D_B=T_VD_AT_U^{-1}
20815 =
20816 \begin{bmatrix}
20817 2&0\\
20818 0&1
20819 \end{bmatrix}.
20820 \end{equation}
20821 For
20822 \begin{equation}
20823 r_A=
20824 \begin{bmatrix}
20825 1\\2
20826 \end{bmatrix},
20827 \qquad
20828 u_A=
20829 \begin{bmatrix}
20830 -1\\-1
20831 \end{bmatrix},
20832 \end{equation}
20833 we have $D_Au_A=-r_A$. Transport gives
20834 \begin{equation}
20835 r_B=T_Vr_A=
20836 \begin{bmatrix}
20837 -2\\1
20838 \end{bmatrix},
20839 \qquad
20840 u_B=T_Uu_A=
20841 \begin{bmatrix}
20842 1\\-1
20843 \end{bmatrix}.
20844 \end{equation}
20845 Then
20846 \begin{equation}
20847 D_Bu_B=
20848 \begin{bmatrix}
20849 2\\-1
20850 \end{bmatrix}
20851 =-r_B.
20852 \end{equation}
20853 
20854 \subsection{Approximate transport margin}
20855 
20856 Suppose $g_A=0.20$, $L=1$, $\eps_J=0.01$, $\eps_r=0.015$, $\eps_D\norm{u_A}=0.025$, $\eps_{\mathrm{leak}}=0.01$, and $\eps_{\mathrm{hol}}=0.005$. Then \cref{thm:gain-transport} gives
20857 \begin{equation}
20858 \begin{aligned}
20859 g_B
20860 &\geq
20861 0.20
20862 -2(0.01)
20863 -\left(2(0.015)+0.025+0.01+0.005\right)\\
20864 &=0.11.
20865 \end{aligned}
20866 \end{equation}
20867 The transported repair retains a positive certified gain margin. If the source gain were $0.08$, the same measurements would not certify positivity even if the repair happened to work empirically.
20868 
20869 \subsection{An information-losing transport hides a protected obstruction}
20870 
20871 Let
20872 \begin{equation}
20873 D_A:\R\to\R^2,
20874 \qquad
20875 D_Au=(u,0),
20876 \end{equation}
20877 so
20878 \begin{equation}
20879 \coker D_A\cong\Span\set{e_2}.
20880 \end{equation}
20881 Let the target correction map be
20882 \begin{equation}
20883 D_B:\R\to\R,
20884 \qquad
20885 D_Bu=u,
20886 \end{equation}
20887 and define
20888 \begin{equation}
20889 T_U=I,
20890 \qquad
20891 T_V(x,y)=x.
20892 \end{equation}
20893 Then
20894 \begin{equation}
20895 D_BT_U=T_VD_A,
20896 \end{equation}
20897 so the correction square commutes. However, for
20898 \begin{equation}
20899 r_A=e_2,
20900 \end{equation}
20901 we have
20902 \begin{equation}
20903 [r_A]\neq0
20904 \quad\text{in }\coker D_A,
20905 \qquad
20906 T_Vr_A=0
20907 \quad\text{in }\coker D_B.
20908 \end{equation}
20909 The target representation has erased the obstructed direction. Exact commutation alone therefore does not certify obstruction faithfulness.
20910 
20911 \subsection{Three-edge transport accumulation}
20912 
20913 Suppose a source correction has norm $\norm{u_0}=2$, every transport is nonexpansive, and three edges have
20914 \begin{equation}
20915 (\eps_{D,0},\eps_{D,1},\eps_{D,2})
20916 =(0.01,0.02,0.015),
20917 \end{equation}
20918 \begin{equation}
20919 (\eps_{r,0},\eps_{r,1},\eps_{r,2})
20920 =(0.005,0.004,0.006).
20921 \end{equation}
20922 Then \cref{cor:nonexpansive-path} gives
20923 \begin{equation}
20924 \begin{aligned}
20925 \norm{D_3u_3+r_3}
20926 &\leq
20927 2(0.01+0.02+0.015)
20928 +(0.005+0.004+0.006)\\
20929 &=0.105.
20930 \end{aligned}
20931 \end{equation}
20932 A direct source-to-target bridge with certified residual penalty below $0.105$ is preferable under this metric, even if each edge of the three-step route appears individually accurate.
20933 
20934 \subsection{Path-dependent repair under loop holonomy}
20935 
20936 Let $U=V=\R^2$, $D=I$, and suppose two routes from $A$ to $B$ differ by a loop with correction holonomy
20937 \begin{equation}
20938 H^U=
20939 \begin{bmatrix}
20940 1&0\\
20941 0&-1
20942 \end{bmatrix}.
20943 \end{equation}
20944 The flat repair core is
20945 \begin{equation}
20946 U^{\flat}=\Span\set{e_1}.
20947 \end{equation}
20948 For $u=e_1$, both routes produce the same transported repair. For $u=e_2$, the two routes differ by
20949 \begin{equation}
20950 (H^U-I)e_2=-2e_2,
20951 \end{equation}
20952 so the loop residual is $2$. A repair schema containing an $e_2$ component is therefore path dependent even though every individual edge transport may be exact.
20953 
20954 \subsection{Spectral approximate evaluator core}
20955 
20956 Consider one loop on $E=\R^2$ with shear
20957 \begin{equation}
20958 H=
20959 \begin{bmatrix}
20960 1&a\\
20961 0&1
20962 \end{bmatrix}.
20963 \end{equation}
20964 Then
20965 \begin{equation}
20966 (H-I)^*(H-I)
20967 =
20968 \begin{bmatrix}
20969 0&0\\
20970 0&a^2
20971 \end{bmatrix}.
20972 \end{equation}
20973 The holonomy-energy eigenvalues are $0$ and $a^2$. Thus
20974 \begin{equation}
20975 E^{\flat}=\Span\set{e_1}.
20976 \end{equation}
20977 For a threshold $0\leq\tau<a^2$, the approximate flat core is still one-dimensional. For $\tau\geq a^2$, both coordinates are admitted, but the second coordinate carries loop distortion at scale $|a|$ and should not be described as invariant.
20978 
20979 Now take protected evidence states
20980 \begin{equation}
20981 x_+=(1,1),
20982 \qquad
20983 x_-=(1,-1).
20984 \end{equation}
20985 Their difference is $2e_2$, whose projection onto $E^{\flat}$ vanishes. Hence the exact core has adequacy modulus zero for this protected distinction. This separates the questions ``is the representation path-flat?'' and ``does it retain the evidence needed for evaluation?''
20986 
20987 \subsection{Historical ranking and held-out evaluator gain}
20988 
20989 Suppose an old evaluator assigns historical scores
20990 \begin{equation}
20991 0.81,
20992 \qquad
20993 0.74,
20994 \qquad
20995 0.60.
20996 \end{equation}
20997 The minimum strict margin is
20998 \begin{equation}
20999 m_{\mathrm{hist}}=0.07.
21000 \end{equation}
21001 If a proposed upgrade has uniform replay error at most $0.02$, then
21002 \begin{equation}
21003 2(0.02)=0.04<0.07,
21004 \end{equation}
21005 so \cref{thm:ranking-preservation} certifies preservation of all strict historical rankings.
21006 
21007 Assume the upgrade is then evaluated on $n=800$ precommitted held-out outcomes and achieves paired mean loss improvement
21008 \begin{equation}
21009 \overline\Delta_n=0.09.
21010 \end{equation}
21011 At confidence $1-\alpha=0.95$, \cref{thm:evaluator-risk} gives the lower bound
21012 \begin{equation}
21013 0.09-
21014 \sqrt{\frac{2\log 20}{800}}
21015 \approx
21016 0.0035.
21017 \end{equation}
21018 The expected held-out gain is therefore certified positive, although only narrowly. Historical consistency and new-evidence improvement are both necessary: neither calculation substitutes for the other.
21019 
21020 \section{Worked Transaction, Closure, and Metaproductivity Examples}
21021 
21022 \subsection{A trace schema that cannot detect provenance failure}
21023 
21024 Suppose a sandbox returns only the scalar task score
21025 \begin{equation}
21026 q(\omega)=\mathrm{accuracy}(\omega).
21027 \end{equation}
21028 Execution $\omega_0$ reaches accuracy $0.9$ using authorized retrieval. Execution $\omega_1$ reaches the same accuracy after importing a prohibited hidden answer key. Then
21029 \begin{equation}
21030 q(\omega_0)=q(\omega_1)=0.9,
21031 \end{equation}
21032 but the ideal full-information decisions differ. By \cref{thm:trace-factorization}, no witness using only accuracy can accept the first and reject the second. Adding signed tool-call provenance or a complete effect receipt refines the trace and can separate the executions.
21033 
21034 \subsection{A four-component rotating witness schedule}
21035 
21036 Let
21037 \begin{equation}
21038 \mathsf C=\{M,P,E,L\},
21039 \end{equation}
21040 where $M$ is the task model, $P$ the planner, $E$ the evaluator, and $L$ the ledger and rollback service. Use two frozen witnesses per transaction:
21041 \begin{center}
21042 \begin{tabular}{@{}ccll@{}}
21043 \toprule
21044 $t$ & $W_t$ & Witnesses & Editable components \\
21045 \midrule
21046 0 & $\{M,P\}$ & model, planner & evaluator, ledger \\
21047 1 & $\{P,E\}$ & planner, evaluator & ledger, model \\
21048 2 & $\{E,L\}$ & evaluator, ledger & model, planner \\
21049 3 & $\{L,M\}$ & ledger, model & planner, evaluator \\
21050 \bottomrule
21051 \end{tabular}
21052 \end{center}
21053 Every component is witnessed twice and editable twice per period. The schedule proves transaction-level separation and coverage, not witness competence. The trace and evaluator still require independent validation.
21054 
21055 \subsection{Universal closure versus controlled closure}
21056 
21057 Let
21058 \begin{equation}
21059 S=\{s_0,s_1,b\},
21060 \qquad
21061 C_{\mathrm{RI}}=\{s_0,s_1\}.
21062 \end{equation}
21063 At $s_0$ there are two authorized actions:
21064 \begin{equation}
21065 a_{\mathrm{safe}}:s_0\mapsto s_1,
21066 \qquad
21067 a_{\mathrm{risky}}:s_0\mapsto b.
21068 \end{equation}
21069 At $s_1$ there is one action with successor $s_1$. Universal closure excludes $s_0$ because one authorized successor is unsafe. Controlled closure includes $s_0$ because the policy can choose $a_{\mathrm{safe}}$. This is the concrete distinction between safety of every authorized action and existence of one robustly safe action.
21070 
21071 \subsection{Safe recurrence without productive recurrence}
21072 
21073 Let $s_1$ above be an absorbing no-op state and suppose $s_1\notin G_\gamma$. Then $s_1$ lies in the controlled safety kernel but not in the productive Buechi closure $\Ical_\gamma^\omega$. The system can preserve every invariant forever while producing no repeated validated improvement. Safe persistence is therefore not evidence of recursive improvement.
21074 
21075 \subsection{A rollback receipt preserves negative knowledge}
21076 
21077 Let parent $P$ have utility $0.70$. Candidate $C$ is predicted to reach $0.78$ but reveals utility $0.61$ and a memory-integrity violation. Atomic rollback returns the operational state to $P$. The archive still receives a receipt containing
21078 \begin{equation}
21079 (\text{predicted gain }0.08,
21080 \text{observed gain }-0.09,
21081 \text{integrity failure},
21082 \text{repair signature}).
21083 \end{equation}
21084 The transaction is behaviorally unsuccessful and epistemically productive.
21085 
21086 \subsection{A lower-utility stepping stone}
21087 
21088 Let the lineage graph contain
21089 \begin{equation}
21090 a\to b\to c
21091 \end{equation}
21092 with
21093 \begin{equation}
21094 J(a)=0,
21095 \qquad
21096 J(b)=-1,
21097 \qquad
21098 J(c)=10.
21099 \end{equation}
21100 Suppose every other child of $a$ has utility at most $1$ and no descendants. A greedy chain rejects $b$ and cannot reach $c$. An archive retaining $b$ because it adds a new repair interface can reach the global best node.
21101 
21102 \subsection{Paired foreign-agent metaproductivity}
21103 
21104 Take $G_{\max}=1$, $n=1000$, and observed paired mean
21105 \begin{equation}
21106 \overline D_n=0.25.
21107 \end{equation}
21108 At $\delta=0.05$, \cref{thm:paired-meta} gives
21109 \begin{equation}
21110 \begin{aligned}
21111 \mathbb E D_i
21112 &\geq
21113 0.25
21114 -2\sqrt2\sqrt{\frac{\log20}{1000}}\\
21115 &\approx
21116 0.095.
21117 \end{aligned}
21118 \end{equation}
21119 The comparison certifies positive foreign-agent meta-lift. With $n=200$, the same bound is negative, so the observed mean alone would not justify the claim.
21120 
21121 \subsection{Structural rank gain with and without task relevance}
21122 
21123 Let $V=\R^3$ and let held-out defects have covariance
21124 \begin{equation}
21125 \Sigma=\diag(1,4,9).
21126 \end{equation}
21127 Suppose
21128 \begin{equation}
21129 L_0=\Span\{e_1\},
21130 \qquad
21131 L_1=\Span\{e_1,e_2\}.
21132 \end{equation}
21133 Then obstruction energy drops from $13$ to $9$, so the new $e_2$ direction contributes expected quadratic gain $4$. If the defect distribution instead has covariance $\diag(1,0,9)$, the rank still increases but expected gain is zero. This is the distinction formalized by \cref{thm:structural-behavioral}.
21134 
21135 \subsection{Ontology expansion resolves one old obstruction and reveals one new one}
21136 
21137 Let
21138 \begin{equation}
21139 V=\R^2,
21140 \qquad
21141 C=\Span\{e_1\},
21142 \end{equation}
21143 and embed $V$ into $V'=\R^3$ by the standard coordinate inclusion. Let
21144 \begin{equation}
21145 C'=\Span\{e_1,e_2\}.
21146 \end{equation}
21147 The old obstruction space is generated by $[e_2]$, but $e_2$ becomes correctable in $C'$. Hence
21148 \begin{equation}
21149 \frac{j^{-1}(C')}{C}
21150 \cong
21151 \Span\{[e_2]\}.
21152 \end{equation}
21153 The genuinely new obstruction quotient is
21154 \begin{equation}
21155 \frac{V'}{C'+j(V)}
21156 \cong
21157 \Span\{[e_3]\}.
21158 \end{equation}
21159 The expansion repairs one old direction while exposing one new direction, exactly as in \cref{thm:dynamic-obstruction-exact}.
21160 
21161 \subsection{A certified autocatalytic core}
21162 
21163 Consider
21164 \begin{equation}
21165 M=
21166 \begin{bmatrix}
21167 0.7&0.6\\
21168 0.2&0.8
21169 \end{bmatrix}.
21170 \end{equation}
21171 Its eigenvalues are $1.1$ and $0.4$, so $R_{\mathrm{imp}}=1.1$. The type graph is strongly connected, hence both types belong to one autocatalytic core by \cref{thm:autocatalytic-scc}.
21172 
21173 Suppose a simultaneous lower confidence matrix is
21174 \begin{equation}
21175 L=
21176 \begin{bmatrix}
21177 0.65&0.55\\
21178 0.18&0.75
21179 \end{bmatrix}.
21180 \end{equation}
21181 Its leading eigenvalue is approximately $1.019$. Since $\rad(L)>1$, \cref{thm:spectral-interval} certifies local supercriticality on the confidence event. The result remains tied to the fixed budget, grammar, evaluator, and unsaturated regime used to estimate the matrix.
21182 
21183 \subsection{Fixed-grammar saturation despite early supercritical growth}
21184 
21185 Suppose a fixed defect space has dimension $5$ and the initial correction image has rank $2$. If every counted schema must add an independent correctable direction, only three structurally novel schemas can ever be counted. An early offspring estimate above one can describe rapid discovery of those three directions, but it cannot persist without ontology expansion. Later syntactic variants should be classified as duplicates unless they alter a protected operational distinction.
21186 
21187 \section{Worked Empirical Realization Examples}
21188 \label{sec:worked-empirical}
21189 
21190 \subsection{Exact classification trace for the gluing case}
21191 
21192 For the triangle mismatch
21193 \begin{equation}
21194 b=(0,0,1)^\top
21195 \end{equation}
21196 and normalized cycle vector
21197 \begin{equation}
21198 c=3^{-1/2}(1,1,-1)^\top,
21199 \end{equation}
21200 the cycle component is
21201 \begin{equation}
21202 \angles{c,b}=-\frac1{\sqrt3}.
21203 \end{equation}
21204 Every local correction map is the identity, so the local obstruction is zero. The ordered classifier therefore skips the local-obstruction branch and returns gluing obstruction because
21205 \begin{equation}
21206 \frac1{\sqrt3}>10^{-9}.
21207 \end{equation}
21208 No evaluator check is needed to classify this isolated fixture case.
21209 
21210 \subsection{Receipt trace for an evaluator-capture attempt}
21211 
21212 Let protected keys be
21213 \begin{equation}
21214 K_{\mathrm{protected}}
21215 =\{\mathrm{evidence},\mathrm{evaluator},\mathrm{provenance}\}.
21216 \end{equation}
21217 The adversarial candidate declares
21218 \begin{equation}
21219 K_{\mathrm{edit}}
21220 =\{\mathrm{evaluator},\mathrm{utility}\}.
21221 \end{equation}
21222 Then
21223 \begin{equation}
21224 K_{\mathrm{edit}}\cap K_{\mathrm{protected}}
21225 =\{\mathrm{evaluator}\}\neq\varnothing.
21226 \end{equation}
21227 Even before considering utility, the acceptance predicate is false. The sandbox also changes the protected evaluator value, providing a second independent rejection reason. The final state is restored from the checkpoint and has the original evaluator digest.
21228 
21229 \subsection{Prototype-separation calculation}
21230 
21231 Suppose the minimum anonymous-event prototype separation is
21232 \begin{equation}
21233 \Delta=0.110955
21234 \end{equation}
21235 and $\sigma=0.06$. With six event types, \cref{thm:prototype-separation} gives the conservative bound
21236 \begin{equation}
21237 \Pr(\widehat z\neq z)
21238 \leq
21239 5\exp\left(-\frac{0.110955^2}{8(0.06)^2}\right),
21240 \end{equation}
21241 which is larger than one and therefore vacuous after truncation. This is instructive rather than contradictory. The union bound uses only the smallest pairwise separation and ignores that four event prototypes are much farther apart. The empirical confusion matrix reveals that the difficult boundary is concentrated between memory change and no change. A useful next design step is to add a probe that increases separation specifically along that pair.
21242 
21243 \subsection{Matched-budget proxy exploitation}
21244 
21245 Consider a parent with protected utility $J_0=0.42$. A large global rewrite has protected utility
21246 \begin{equation}
21247 J_{\mathrm{protected}}=0.10
21248 \end{equation}
21249 but apparent utility
21250 \begin{equation}
21251 \widehat J=0.10+1.10=1.20
21252 \end{equation}
21253 due to an exploitable proxy bonus. A selector using $\widehat J$ accepts apparent gain $0.78$, while protected gain is
21254 \begin{equation}
21255 0.10-0.42=-0.32.
21256 \end{equation}
21257 This is the mechanism underlying the global-edit positive control. A protected witness ranks by $J_{\mathrm{protected}}$ and rejects the candidate.
21258 
21259 \subsection{Why packet-only can outperform the full arm in a toy fixture}
21260 
21261 The reference defects are sparse and the observed transform is a signed scaled permutation. Correcting the largest visible coordinate often removes most of the protected defect energy even without explicit causal attribution. Therefore packet-only achieves high average gain.
21262 
21263 This outcome is desirable for benchmark validity. A fixture in which only the full architecture can succeed risks encoding the desired conclusion into the task. The full SRIC arm must earn its advantage on cases requiring causal distinction, transport, provenance, evaluator protection, or reusable abstraction rather than on every synthetic defect.
21264 
21265 \subsection{Gate assignment example}
21266 
21267 Suppose all package, mechanics, and finite semantic gates pass, the anonymous attribution suite passes, but the meta arm is supplied an oracle schema rather than discovering it. Then the strongest admissible statements are:
21268 \begin{itemize}
21269 \item the package is mechanically reproducible;
21270 \item the finite formal distinctions are realized;
21271 \item the anonymous matched-model causes are identifiable under the declared probes;
21272 \item the foreign-agent metric responds correctly to an engineered meta positive control.
21273 \end{itemize}
21274 The following statement is not admissible:
21275 \begin{center}
21276 ``The system recursively improved its own improvement process.''
21277 \end{center}
21278 The missing evidence is autonomous schema generation followed by a sealed parent-child metaproductivity comparison on foreign systems.
21279 
21280 
21281 \section{Worked Learned-Agent Validation Examples}
21282 \label{sec:worked-block06}
21283 
21284 \subsection{Exact linear causal packet}
21285 
21286 Let
21287 \begin{equation}
21288 E=[A\;B]:\R^{16}\to\R^{16}
21289 \end{equation}
21290 with $A,B\in\R^{16\times8}$ and let
21291 \begin{equation}
21292 \phi(x)=\begin{bmatrix}\psi(x)\\s(x)\psi(x)\end{bmatrix},
21293 \qquad
21294 \phi^{\circ}(x)=\begin{bmatrix}\psi(x)\\0\end{bmatrix}.
21295 \end{equation}
21296 Then
21297 \begin{equation}
21298 \Delta(x)=E\phi(x)-E\phi^{\circ}(x)=s(x)B\psi(x).
21299 \end{equation}
21300 If the affected values of $\psi$ span $\R^8$ and $B$ has rank eight, the empirical difference matrix has right hidden-state span $\im B$. The packet is identified by what the intervention changes, not by which hidden coordinates have the largest unconditional variance.
21301 
21302 \subsection{Compression residual despite exact alignment}
21303 
21304 Let
21305 \begin{equation}
21306 T=\begin{bmatrix}1&0\end{bmatrix}:\R^2\to\R,
21307 \qquad
21308 D_s=\begin{bmatrix}0&1\\0&0\end{bmatrix},
21309 \qquad
21310 h_s=e_2.
21311 \end{equation}
21312 Then $h_t=Th_s=0$, so target-state alignment is exact. However,
21313 \begin{equation}
21314 T^+T=\begin{bmatrix}1&0\\0&0\end{bmatrix},
21315 \qquad
21316 D_t=TD_sT^+=0.
21317 \end{equation}
21318 The source repair maps the discarded input direction into the retained output direction:
21319 \begin{equation}
21320 D_sh_s=e_1.
21321 \end{equation}
21322 Consequently,
21323 \begin{equation}
21324 R_t(Th_s)-TR_s(h_s)
21325 =TD_s(T^+T-I)h_s
21326 =-1.
21327 \end{equation}
21328 Thus a target can align exactly as a compressed representation while failing to realize a source correction that depends on a lost coordinate. In the released fixture the state-alignment NRMSE is approximately $3.3\times10^{-8}$, but the repair-equivariance NRMSE is $0.4715$.
21329 
21330 \subsection{Anchor-rank transition}
21331 
21332 For an exact linear target with hidden width sixteen, let $H_s\in\R^{m\times16}$. When $m<16$, full column rank is impossible, so the bridge is not uniquely identified by ordinary least squares. At $m=8$, the frozen sweep reaches $0.4115$ affected accuracy and the witness rejects. At $m=16$, random anchors are generically spanning and accuracy rises to $0.7877$. By $m=32$, conditioning is sufficient for numerical recovery and accuracy reaches $0.9943$.
21333 
21334 This does not make sixteen a universal threshold. The relevant condition is the effective source rank and conditioning of $H_s$, not raw anchor count.
21335 
21336 \subsection{Blind gain rejected by the witness}
21337 
21338 For the compressed target, affected-stratum accuracy rises from
21339 \begin{equation}
21340 0.1277\quad\text{to}\quad0.6204
21341 \end{equation}
21342 under blind transport. A gain-only selector would commit. The frozen witness instead returns the unrepaired target because baseline unaffected competence and protected regression lie outside the admitted envelope. Hence selected affected accuracy remains $0.1277$.
21343 
21344 The example separates two predicates:
21345 \begin{equation}
21346 \mathrm{ImprovesSlice}(R_t)
21347 \not\Rightarrow
21348 \mathrm{AuthorizedCommit}(R_t).
21349 \end{equation}
21350 
21351 \subsection{Learned nonlinear transfer gap}
21352 
21353 For independently trained tanh targets, mean transported affected accuracy is $0.89416$ and mean target-local oracle accuracy is $0.89670$. The gap
21354 \begin{equation}
21355 0.89670-0.89416=0.00254
21356 \end{equation}
21357 is small but nonzero. Mean alignment NRMSE is $0.14706$, and unaffected accuracy is $0.96003$. The result is consistent with a useful approximate bridge, not with exact global conjugacy.
21358 
21359 \subsection{Claim-gate assignment}
21360 
21361 Suppose package, mechanics, finite semantics, causal packet controls, and controlled foreign transport all pass, but no descendant repair calculus is compared against its parent. Then the strongest cumulative gate is
21362 \begin{equation}
21363 G_{\rm foreign}=1,
21364 \qquad
21365 G_{\rm meta}=0,
21366 \qquad
21367 G_{\rm open}=0.
21368 \end{equation}
21369 The statement ``a learned source repair transfers to foreign representations under the declared protocol'' is admissible. The statement ``the system recursively improved its improvement process'' is not.
21370 
21371 \section{Worked Sealed-Metaproductivity Examples}
21372 \label{sec:worked-block07}
21373 
21374 \subsection{A two-candidate selector improvement}
21375 
21376 Suppose every target exposes a no-op $c_0$ and one repair $c_1$. On half the episodes, the repair gain is $0.4$; on the other half it is $-0.2$. A parent that always abstains has expected gain zero. A child that observes a receipt feature identifying the positive regime with probability $0.8$ and the negative regime with probability $0.7$ selects $c_1$ with those corresponding probabilities. Its expected gain is
21377 \begin{equation}
21378 \frac12(0.8)(0.4)
21379 +
21380 \frac12(0.3)(-0.2)
21381 =0.13.
21382 \end{equation}
21383 The candidate set is unchanged. The child improves only the conditional selection map. The reachable correction span is identical for parent and child, illustrating \cref{prop:block07-fixed-candidate-invariance}.
21384 
21385 \subsection{Why candidate-count matching is necessary}
21386 
21387 Let a parent receive one candidate whose gain is uniformly distributed on $[-1,1]$, and let the child receive two independent candidates from the same distribution. If both choose the best available candidate, then
21388 \begin{equation}
21389 \mathbb E[G_{\rm parent}]=0,
21390 \qquad
21391 \mathbb E[G_{\rm child}]=\frac13.
21392 \end{equation}
21393 The child appears better even though its selection rule is unchanged. The difference is entirely the best-of-two opportunity advantage. Matching the candidate budget eliminates this explanation.
21394 
21395 \subsection{Risk filtering can lower utility deliberately}
21396 
21397 Consider three candidates with predicted utility and predicted protected risk
21398 \begin{equation}
21399 \begin{array}{c|ccc}
21400  & c_0 & c_1 & c_2\\
21401 \hline
21402 \widehat U & 0 & 0.30 & 0.36\\
21403 \widehat R_{\rm UCB} & 0 & 0.02 & 0.06.
21404 \end{array}
21405 \end{equation}
21406 With risk threshold $\tau_r=0.035$, the constrained child selects $c_1$. The no-risk child selects $c_2$. If the realized risks respect the upper bounds, the constrained child is certified below threshold while the unconstrained child is not. The lower utility of the constrained child is not a failure of the risk head. It is the cost of satisfying the declared preservation profile.
21407 
21408 \subsection{Finite meta-grammar overfitting}
21409 
21410 Suppose 2000 hyperparameter configurations have equal true utility and each empirical validation mean has standard deviation $0.04$. Even if every estimate is unbiased, the maximum observed mean is typically positive because the compiler searches many configurations. A rough Gaussian extreme-value scale is
21411 \begin{equation}
21412 0.04\sqrt{2\log 2000}\approx0.156.
21413 \end{equation}
21414 This does not imply the selected configuration has true gain $0.156$. The separate final audit is what tests whether the selected child generalizes after the meta-search.
21415 
21416 \subsection{Actual Block 7 paired effect}
21417 
21418 On the seed/task audit, the parent and child receive the same five candidates on $120$ episodes. Let $D_j$ be the child-minus-parent protected utility on episode $j$. The frozen values satisfy
21419 \begin{equation}
21420 \overline D=0.072518,
21421 \qquad
21422 s_D=0.154617,
21423 \qquad
21424 \widehat{\rm se}=0.014115.
21425 \end{equation}
21426 The ordinary normal approximation gives
21427 \begin{equation}
21428 0.072518\pm1.98(0.014115)
21429 \approx
21430 [0.0446,0.1005],
21431 \end{equation}
21432 close to the frozen paired bootstrap interval
21433 $[0.046745,0.101389]$.
21434 
21435 The median paired effect is zero because both selectors often choose the same candidate. The mean is positive because the child's improved choices on a subset of episodes are large enough to dominate its smaller mistakes. This is a property of a sparse policy difference, not a uniform gain on every target.
21436 
21437 \subsection{Architecture-foreign transfer of the selector rule}
21438 
21439 The child is developed on eight representation families and audited on four disjoint families. The frozen architecture-foreign metrics are
21440 \begin{equation}
21441 \begin{aligned}
21442 \overline G_{\rm parent}&=0.167417,\\
21443 \overline G_{\rm child}&=0.272416,\\
21444 \overline G_{\rm raw}&=0.254427.
21445 \end{aligned}
21446 \end{equation}
21447 Thus
21448 \begin{equation}
21449 \overline G_{\rm child}-\overline G_{\rm parent}=0.104999
21450 \end{equation}
21451 and
21452 \begin{equation}
21453 \overline G_{\rm child}-\overline G_{\rm raw}=0.017989.
21454 \end{equation}
21455 The first contrast isolates improvement over the frozen lineage parent. The second tests whether the learned selector offers anything beyond a strong direct calibration heuristic. Both frozen bootstrap intervals are positive on this split.
21456 
21457 \subsection{Shuffled lineage receipts}
21458 
21459 Let $Z$ be the standardized receipt matrix and $y$ the development utility labels. The valid child uses
21460 \begin{equation}
21461 \widehat\beta=(Z^\top Z+\lambda I)^{-1}Z^\top y.
21462 \end{equation}
21463 The shuffled control replaces $y$ by $Py$ for a random permutation $P$. Because $Z$ is centered,
21464 \begin{equation}
21465 \mathbb E_P[Z^\top Py]=0.
21466 \end{equation}
21467 The finite shuffled child still exploits intercepts, safety filters, and accidental sample correlations, so it need not collapse to the no-op. Its lower held-out utility tests whether the original feature-outcome binding carried transferable information.
21468 
21469 \subsection{One productive edge does not imply a productive fixed point}
21470 
21471 Consider a transaction graph with states $s_0,s_1,s_2$. Suppose $s_0\to s_1$ is a validated positive parent--child edge, but every authorized successor of $s_1$ either returns to $s_1$ with zero gain or enters unsafe $s_2$. Then $s_0\to s_1$ witnesses one-step metaproductivity, but $s_1$ does not lie in a closure requiring infinitely repeated positive gains. A grandchild audit is logically necessary to distinguish a one-edge result from recurrent improvement capacity.
21472 
21473 
21474 
21475 \section{Worked Meta-Headroom and Grammar-Growth Examples}
21476 \label{app:block8-examples}
21477 
21478 \subsection{Permutation-average headroom allocation}
21479 
21480 Consider five mutable modules:
21481 \begin{equation}
21482 N_5
21483 =
21484 \{S,Q,T,C,H\},
21485 \end{equation}
21486 representing selection, information acquisition, transport, candidate construction, and causal-atlas structure. Let
21487 \begin{equation}
21488 F_K(\varnothing)=0.20.
21489 \end{equation}
21490 Assign singleton contributions
21491 \begin{equation}
21492 (0.08,0.12,0.10,0.16,0.14)
21493 \end{equation}
21494 and pair synergies
21495 \begin{equation}
21496 \omega_{QT}=0.05,
21497 \qquad
21498 \omega_{CH}=0.06,
21499 \qquad
21500 \omega_{SC}=0.03.
21501 \end{equation}
21502 Then
21503 \begin{equation}
21504 F_K(N_5)=0.94,
21505 \qquad
21506 \Hcal(K)=0.74.
21507 \end{equation}
21508 Every pair synergy is divided equally between its two participants by the permutation-average rule. Therefore
21509 \begin{equation}
21510 \begin{aligned}
21511 h_S&=0.08+\frac{0.03}{2}=0.095,\\
21512 h_Q&=0.12+\frac{0.05}{2}=0.145,\\
21513 h_T&=0.10+\frac{0.05}{2}=0.125,\\
21514 h_C&=0.16+\frac{0.06+0.03}{2}=0.205,\\
21515 h_H&=0.14+\frac{0.06}{2}=0.170.
21516 \end{aligned}
21517 \end{equation}
21518 The allocation sums to
21519 \begin{equation}
21520 0.095+0.145+0.125+0.205+0.170=0.740,
21521 \end{equation}
21522 which verifies \cref{thm:b8-shapley-efficiency} in the finite companion.
21523 
21524 \subsection{A saturated endpoint and a migrated bottleneck}
21525 
21526 Suppose bridge-rank identification has success probability
21527 \begin{equation}
21528 p_{\mathrm{rank}}(K_1)=1.
21529 \end{equation}
21530 For every child that changes only the rank-identification rule while leaving all other modules fixed,
21531 \begin{equation}
21532 p_{\mathrm{rank}}(K_2)-p_{\mathrm{rank}}(K_1)\leq0.
21533 \end{equation}
21534 This does not imply that the repair process has no remaining headroom. Let the target bridge be full rank but have minimum singular value
21535 \begin{equation}
21536 \sigma_{\min}=10^{-4}.
21537 \end{equation}
21538 Random noise of norm $10^{-3}$ can then dominate the weakest identified direction. The bottleneck has migrated from rank to conditioning. A meaningful grandchild should target a criterion such as
21539 \begin{equation}
21540 \max_{S:|S|=k}
21541 \lambda_{\min}(X_S^\top X_S)
21542 \end{equation}
21543 or a regularized determinant objective, not another rank-only rule.
21544 
21545 \subsection{Finite bottleneck-migration trace}
21546 
21547 The Block 8 companion begins with
21548 \begin{equation}
21549 r_0
21550 =
21551 (0.06,0.18,0.14,0.22,0.10)
21552 \end{equation}
21553 for selection, information, transport, construction, and atlas headroom. At each fixed-grammar step, the largest coordinate is multiplied by $1/4$. A neighboring coordinate receives a $10\%$ verified reduction. The first four selected actions are
21554 \begin{equation}
21555 C,\ Q,\ T,\ H.
21556 \end{equation}
21557 After these reductions, the old-coordinate vector is
21558 \begin{equation}
21559 (0.06,0.045,0.0315,0.0495,0.0225).
21560 \end{equation}
21561 A conservative grammar extension then introduces a new nonlinear coordinate of magnitude
21562 \begin{equation}
21563 0.16.
21564 \end{equation}
21565 The represented headroom norm increases because a new limitation has been discovered. No old coordinate increases. Subsequent actions target the new coordinate and the remaining old bottlenecks. This separates epistemic expansion from regression.
21566 
21567 \subsection{Nested obstruction energy}
21568 
21569 Let the foreign obstruction covariance have eigenvalues
21570 \begin{equation}
21571 4,
21572 2,
21573 1,
21574 0.5,
21575 0.2,
21576 0.1.
21577 \end{equation}
21578 Assume the initial grammar already corrects the first eigenvector. The unresolved energy is
21579 \begin{equation}
21580 2+1+0.5+0.2+0.1=3.8.
21581 \end{equation}
21582 Adding the remaining eigenvectors in descending order produces
21583 \begin{equation}
21584 3.8
21585 \longrightarrow
21586 1.8
21587 \longrightarrow
21588 0.8
21589 \longrightarrow
21590 0.3
21591 \longrightarrow
21592 0.1
21593 \longrightarrow
21594 0.
21595 \end{equation}
21596 The energy drops are
21597 \begin{equation}
21598 2,
21599 1,
21600 0.5,
21601 0.2,
21602 0.1,
21603 \end{equation}
21604 exactly the newly correctable eigenvalues. This is the distributional form of \cref{thm:b8-new-energy}.
21605 
21606 \subsection{One-generator reduction identity}
21607 
21608 Let the old correction image be
21609 \begin{equation}
21610 I=\Span\{e_1,e_2\}\subset\R^4
21611 \end{equation}
21612 and let
21613 \begin{equation}
21614 r=(0,0,3,4).
21615 \end{equation}
21616 Then
21617 \begin{equation}
21618 z=P_{I^\perp}r=(0,0,3,4),
21619 \qquad
21620 \eta_I(r)^2=25.
21621 \end{equation}
21622 Add generator
21623 \begin{equation}
21624 v=2e_4.
21625 \end{equation}
21626 Because $v\in I^\perp$,
21627 \begin{equation}
21628 \frac{|\angles{z,v}|^2}{\norm v^2}
21629 =
21630 \frac{8^2}{4}
21631 =16.
21632 \end{equation}
21633 The expanded residual is $3e_3$, so
21634 \begin{equation}
21635 \eta_{I'}(r)^2=9=25-16.
21636 \end{equation}
21637 A generator parallel to $e_1$ would have $\bar v=0$ and would produce no structural reduction despite having nonzero raw norm.
21638 
21639 \subsection{Repair-conflict refinement}
21640 
21641 Index eight episodes by three bits
21642 \begin{equation}
21643 (a,b,n)\in\{0,1\}^3,
21644 \end{equation}
21645 where $(a,b)$ determines the valid repair class and $n$ is nuisance. The empty grammar places every episode in one cell. There are $24$ unordered pairs in that cell with different repair classes.
21646 
21647 Adding feature $a$ divides the episodes into two cells and reduces unresolved conflict pairs to $8$. Adding $b$ produces four cells indexed by $(a,b)$ and eliminates every repair conflict. Adding nuisance $n$ first would separate episodes without resolving all required repair distinctions. The greedy finite fixture therefore selects
21648 \begin{equation}
21649 a,
21650 \quad
21651 b,
21652 \end{equation}
21653 and rejects the nuisance feature.
21654 
21655 The conservative embedding is immediate:
21656 \begin{equation}
21657 \phi_{\Gcal'}(x)
21658 =
21659 (\phi_{\Gcal}(x),a(x),b(x)),
21660 \end{equation}
21661 and projection onto the first coordinate recovers the old grammar.
21662 
21663 \subsection{Finite synthesis and exhaustion}
21664 
21665 Consider the typed repair programs
21666 \begin{equation}
21667 \begin{aligned}
21668 p_1(a,b)&=(0,0),\\
21669 p_2(a,b)&=(a,0),\\
21670 p_3(a,b)&=(0,b),\\
21671 p_4(a,b)&=(b,a),\\
21672 p_5(a,b)&=(a,b).
21673 \end{aligned}
21674 \end{equation}
21675 The complete finite verifier checks all eight episodes. Programs $p_1$ through $p_4$ each receive a counterexample and are eliminated. Program $p_5$ passes every episode. The loop terminates after four rejections with a verified repair.
21676 
21677 If $p_5$ were removed from the grammar, the same process would return an exhaustion certificate. That certificate would justify a grammar-expansion attempt. It would not prove that no repair exists outside the declared program family.
21678 
21679 \subsection{A two-block causal atlas}
21680 
21681 Let
21682 \begin{equation}
21683 X=
21684 \begin{bmatrix}0&1\\1&0\end{bmatrix},
21685 \qquad
21686 Z=
21687 \begin{bmatrix}1&0\\0&-1\end{bmatrix}.
21688 \end{equation}
21689 On $H=\R^4$, define four response operators
21690 \begin{equation}
21691 R_1=X\oplus0,
21692 \quad
21693 R_2=Z\oplus0,
21694 \quad
21695 R_3=0\oplus X,
21696 \quad
21697 R_4=0\oplus Z.
21698 \end{equation}
21699 The adjoint-closed algebra they generate is
21700 \begin{equation}
21701 M_2(\R)\oplus M_2(\R),
21702 \end{equation}
21703 with dimension $8$. Its commutant consists of scalar multiples of the identity on each block and therefore has dimension $2$. The minimal central projectors are
21704 \begin{equation}
21705 P_1=I_2\oplus0,
21706 \qquad
21707 P_2=0\oplus I_2.
21708 \end{equation}
21709 A residual $e_1$ generates the first two-dimensional block. A residual $e_1+e_3$ generates both blocks and has cyclic dimension $4$.
21710 
21711 \subsection{Lineage-wide budget calculation}
21712 
21713 Suppose four accepted edges use conditional false-acceptance budgets
21714 \begin{equation}
21715 0.0100,
21716 0.0050,
21717 0.0025,
21718 0.00125.
21719 \end{equation}
21720 Then
21721 \begin{equation}
21722 \sum_{t=0}^{3}\delta_t=0.01875.
21723 \end{equation}
21724 By \cref{thm:b8-lineage-validity}, the probability that any accepted edge has nonpositive true metaproductivity is at most $0.01875$, assuming every conditional bound is valid.
21725 
21726 If the protected-regression budgets are
21727 \begin{equation}
21728 0.010,
21729 0.008,
21730 0.006,
21731 0.004,
21732 \end{equation}
21733 then the cumulative declared budget is
21734 \begin{equation}
21735 B_R=0.028.
21736 \end{equation}
21737 These are accounting guarantees, not evidence that the numerical budgets are ethically sufficient.
21738 
21739 \subsection{Process improvement versus compiler improvement}
21740 
21741 Let a fixed compiler $\Gamma$ map a parent process and receipts to a child:
21742 \begin{equation}
21743 K_{t+1}=\Gamma(K_t,\Ecal_t).
21744 \end{equation}
21745 Suppose it produces two positive edges
21746 \begin{equation}
21747 \Mcal(K_1)-\Mcal(K_0)>0,
21748 \qquad
21749 \Mcal(K_2)-\Mcal(K_1)>0.
21750 \end{equation}
21751 This establishes repeated output-process improvement by one compiler. It does not establish
21752 \begin{equation}
21753 \Ccal(\Gamma_1)>\Ccal(\Gamma_0).
21754 \end{equation}
21755 The latter requires comparing two compilers on foreign parent repair processes under matched compilation resources. The distinction prevents one successful compiler from being described as recursively improving its own compilation rule.
21756 
21757 
21758 
21759 \section{Worked Repeated-Lineage and Conditioning Examples}
21760 
21761 \subsection{Full rank with arbitrarily poor information}
21762 
21763 Let
21764 \begin{equation}
21765 X_\epsilon=\diag(1,1,\epsilon),
21766 \qquad 0<\epsilon\ll1.
21767 \end{equation}
21768 Then $\rank X_\epsilon=3$, but
21769 \begin{equation}
21770 \lambda_{\min}(X_\epsilon^{\mathsf T}X_\epsilon)=\epsilon^2
21771 \end{equation}
21772 and the worst-direction least-squares variance scales as $\epsilon^{-2}$. A binary rank gate cannot distinguish this design from an orthogonal one.
21773 
21774 \subsection{One D-optimal anchor update}
21775 
21776 Let $M=I_2$, $x=(2,1)^{\mathsf T}$, and $w=4$. The D-optimal information increment is
21777 \begin{equation}
21778 \log\det(M+wxx^{\mathsf T})-\log\det M
21779 =
21780 \log(1+w x^{\mathsf T}M^{-1}x)
21781 =
21782 \log 21.
21783 \end{equation}
21784 A low-quality copy with $w=1/16$ gives only $\log(1+5/16)$, despite identical source geometry.
21785 
21786 \subsection{A-optimal trace reduction}
21787 
21788 For $M=\diag(1,0.1)$ and $v=e_2$, the A-optimal reduction is
21789 \begin{equation}
21790 \frac{v^{\mathsf T}M^{-2}v}{1+v^{\mathsf T}M^{-1}v}
21791 =
21792 \frac{100}{11}.
21793 \end{equation}
21794 The update targets the weakly informed second direction rather than increasing an already strong direction.
21795 
21796 \subsection{Why a gate-only change is insufficient}
21797 
21798 Suppose inherited rank anchors produce $\lambda_{\min}(M)=0.5$, while the grandchild gate requires $\lambda_{\min}(M)\geq2$. Applying the new gate to old anchors yields abstention. The same gate applied to conditioning-aware anchors with $\lambda_{\min}(M)=14$ can commit. The gate does not create information. It only verifies information created by the query design.
21799 
21800 \subsection{Context-indexed nontransitivity}
21801 
21802 Let edge $K_0\to K_1$ improve a bridge task distribution $\Xi_0$, while $K_1\to K_2$ improves a heteroscedastic distribution $\Xi_1$. Positive effects in both contexts do not imply
21803 \begin{equation}
21804 \Mcal_{\Xi_0}(K_2)>\Mcal_{\Xi_0}(K_0)
21805 \end{equation}
21806 or any additive path gain unless $\Xi_0$ and $\Xi_1$ share a common utility or a validated context transport.
21807 
21808 \subsection{Actual Block 9 repeated edge}
21809 
21810 The frozen primary means are
21811 \begin{equation}
21812 \Mcal(K_1^{\rm recal})=0.037393,
21813 \qquad
21814 \Mcal(K_2)=0.209580.
21815 \end{equation}
21816 Hence
21817 \begin{equation}
21818 \widehat\Delta_{1\to2}=0.172188.
21819 \end{equation}
21820 The interval $[0.141481,0.203110]$ is strictly positive. Both arms are full rank, so rank recovery cannot explain the edge.
21821 
21822 \subsection{Quality-permutation control}
21823 
21824 The quality-permuted policy retains the same selection code but breaks the association between anchor identity and supplied reliability. Its blind gain remains positive, but its information floor collapses and the frozen witness commits no primary targets. This isolates semantic quality binding from a favorable policy label.
21825 
21826 
21827 \section{Worked Structural Grammar-Expansion Examples}
21828 \label{app:block10-examples}
21829 
21830 \subsection{A scalar two-chart relation has a persistent global residual}
21831 
21832 Let $h\sim\mathcal N(0,1)$ and let $s\in\{-1,+1\}$ be independent and balanced. Define
21833 \begin{equation}
21834 y=(a+sb)h,
21835 \qquad b\neq0.
21836 \end{equation}
21837 A global scalar bridge predicts $\widehat y=th$. The population minimizer is $t^*=a$, and
21838 \begin{equation}
21839 \inf_t\mathbb E (y-th)^2
21840 =
21841 \mathbb E(sbh)^2
21842 =
21843 b^2.
21844 \end{equation}
21845 The residual is not an estimation artifact. It remains after infinite data because one slope cannot equal both $a-b$ and $a+b$. The two-chart grammar uses
21846 \begin{equation}
21847 t_-=a-b,
21848 \qquad
21849 t_+=a+b,
21850 \end{equation}
21851 and has zero noiseless approximation error. This is the one-dimensional form of \cref{thm:b10-open-set-obstruction}.
21852 
21853 \subsection{The structural extension preserves the inherited process exactly}
21854 
21855 Let the old bridge be $T\in\R^{q\times d}$ and let $g$ be any admissible separator. The embedding
21856 \begin{equation}
21857 T\longmapsto(g,T,T)
21858 \end{equation}
21859 returns $Th$ on both charts. Hence the local grammar can execute the inherited process without approximation. A conservative runtime may therefore use
21860 \begin{equation}
21861 K_{\mathrm{chart}}(h)
21862 =
21863 \begin{cases}
21864 K_{\mathrm{local}}(h),&\text{if the chart certificate passes},\\
21865 K_{\mathrm{global}}(h),&\text{otherwise}.
21866 \end{cases}
21867 \end{equation}
21868 The child is structurally richer, but it does not require deletion of the parent behavior. This is the operational meaning of \cref{prop:b10-conservative-embedding}.
21869 
21870 \subsection{Exact distance from a chart repair to the global subgrammar}
21871 
21872 Equip
21873 \begin{equation}
21874 \End(H_t)\oplus\End(H_t)
21875 \end{equation}
21876 with the product Frobenius norm. For a chart repair pair $(D_-,D_+)$, write
21877 \begin{equation}
21878 D_{\mathrm{avg}}
21879 =
21880 \frac{D_-+D_+}{2},
21881 \qquad
21882 D_{\mathrm{ctr}}
21883 =
21884 \frac{D_+-D_-}{2}.
21885 \end{equation}
21886 The closest diagonal repair is $(D_{\mathrm{avg}},D_{\mathrm{avg}})$. Therefore
21887 \begin{align}
21888 \operatorname{dist}^2
21889 \left(
21890 (D_-,D_+),\Dcal_{\mathrm{diag}}
21891 \right)
21892 &=
21893 \norm{D_--D_{\mathrm{avg}}}_F^2
21894 +
21895 \norm{D_+-D_{\mathrm{avg}}}_F^2\\
21896 &=
21897 2\norm{D_{\mathrm{ctr}}}_F^2\\
21898 &=
21899 \frac12\norm{D_+-D_-}_F^2.
21900 \end{align}
21901 Thus $D_+\neq D_-$ gives a quantitative structural-novelty certificate, not merely two differently named copies of one global operator.
21902 
21903 \subsection{A nuisance separator does not certify grammar growth}
21904 
21905 Suppose the true target relation depends on $s=\one\{h_1>0\}$, but a candidate separator uses
21906 \begin{equation}
21907 g(h)=\one\{h_2>0\},
21908 \end{equation}
21909 where $h_2$ is independent of $s$ and the repair-relevant state. With balanced cells and unlimited data, both $g$-branches see the same mixture of the two true maps. Each fitted local map converges to the global population minimizer. Consequently,
21910 \begin{equation}
21911 \eps_{\mathrm{loc}}
21912 =
21913 \eps_{\mathrm{glob}},
21914 \qquad
21915 \Delta_{\mathrm{gram}}=0.
21916 \end{equation}
21917 A larger syntax is therefore not enough. Structural credit requires a held-out residual reduction, conservative replay, a nonzero repair contrast, and foreign protected gain.
21918 
21919 \subsection{Actual Block 10 structural edge}
21920 
21921 The strengthened global parent uses $80$ bridge-fit and $16$ witness queries. The chart child uses $48$ bridge-fit, $32$ chart-selection, and $16$ witness queries. Both therefore consume $96$ paired queries. Their primary mean selected gains are
21922 \begin{equation}
21923 \Mcal(K_{2,G}^{96})=0.000000,
21924 \qquad
21925 \Mcal(K_{3,C}^{96})=0.334224.
21926 \end{equation}
21927 Hence
21928 \begin{equation}
21929 \widehat\Delta_{G\to C}=0.334224,
21930 \qquad
21931 \mathrm{CI}_{95\%}=[0.266459,0.401903].
21932 \end{equation}
21933 The direct global-affine oracle has mean gain $0.176694$, and the child exceeds it by
21934 \begin{equation}
21935 0.157530,
21936 \qquad
21937 \mathrm{CI}_{95\%}=[0.080863,0.234135].
21938 \end{equation}
21939 Validation alignment NRMSE changes from
21940 \begin{equation}
21941 \eps_{\mathrm{glob}}=0.598513
21942 \end{equation}
21943 to
21944 \begin{equation}
21945 \eps_{\mathrm{loc}}=0.064975.
21946 \end{equation}
21947 Mean gate accuracy is $0.874854$, no harmful primary target is selected, and gate-only, random-gate, and shuffled-binding controls are strongly negative. These quantities support a finite conservative expansion from the global-affine grammar to the invariant-context two-chart grammar. They do not establish autonomous protected-packet or separator-language invention.
21948 
21949 \section{Worked Endogenous Atlas-Growth Examples}
21950 \label{app:block11-examples}
21951 
21952 \subsection{Lagrange recovery of three response blocks}
21953 
21954 Let
21955 \begin{equation}
21956 R_1=P_1+2P_2+4P_3,
21957 \qquad
21958 R_2=3P_1-P_2+2P_3,
21959 \end{equation}
21960 where $P_1,P_2,P_3$ are pairwise orthogonal rank-two projectors. Take
21961 \begin{equation}
21962 B=R_1+2R_2=7P_1+0P_2+8P_3.
21963 \end{equation}
21964 The three eigenvalues $7,0,8$ are distinct. Therefore
21965 \begin{align}
21966 P_1&=\frac{B(B-8I)}{7(7-8)},\\
21967 P_2&=\frac{(B-7I)(B-8I)}{(0-7)(0-8)},\\
21968 P_3&=\frac{B(B-7I)}{8(8-7)}.
21969 \end{align}
21970 Every projector is a polynomial in the observed response combination. No hidden coordinate label is required. Any orthonormal basis inside a rank-two block is equally valid, illustrating the within-block gauge of \cref{cor:b11-gauge}.
21971 
21972 \subsection{Commutant dimension for the Block 11 fixture}
21973 
21974 Suppose
21975 \begin{equation}
21976 H=H_1\oplus H_2\oplus H_3,
21977 \qquad
21978 \dim H_a=2.
21979 \end{equation}
21980 Under separating block-scalar responses, every commuting operator has form
21981 \begin{equation}
21982 X=X_1\oplus X_2\oplus X_3,
21983 \qquad
21984 X_a\in M_2(\R).
21985 \end{equation}
21986 Hence
21987 \begin{equation}
21988 \dim\mathfrak A_R'
21989 =3\dim M_2(\R)
21990 =12.
21991 \end{equation}
21992 The companion's modal commutant dimension is $12$, matching the exact finite theory.
21993 
21994 \subsection{Degenerate signatures merge two functional packets}
21995 
21996 Let
21997 \begin{equation}
21998 R_1=P_1+P_2+3P_3,
21999 \qquad
22000 R_2=2P_1+2P_2-P_3.
22001 \end{equation}
22002 Blocks $1$ and $2$ have the same signature $(1,2)$. Every polynomial in $R_1,R_2$ acts with the same scalar on $H_1$ and $H_2$. The finest response-algebra projector is therefore
22003 \begin{equation}
22004 P_{12}=P_1+P_2,
22005 \end{equation}
22006 not the pair $P_1,P_2$. A compiler that outputs two separate blocks from these interventions has invented information absent from its evidence. The Block 11 stress witness instead abstains.
22007 
22008 \subsection{A spectral-gap stability calculation}
22009 
22010 Suppose a selected response combination has eigenvalue clusters separated by
22011 \begin{equation}
22012 \gamma=0.04
22013 \end{equation}
22014 and estimation error
22015 \begin{equation}
22016 \norm{E}_2=0.003.
22017 \end{equation}
22018 The conditional projector bound gives
22019 \begin{equation}
22020 \norm{\widehat P-P}_2
22021 \leq
22022 \frac{2(0.003)}{0.04}
22023 =0.15.
22024 \end{equation}
22025 If the same fit error occurred with $\gamma=0.004$, the bound would be $1.5$ and therefore vacuous. A low response-fit residual alone does not certify an identifiable atlas. The spectral gap is an independent required field.
22026 
22027 \subsection{Why a fixed packet can fail under hidden rotation}
22028 
22029 Let the true context packet be
22030 \begin{equation}
22031 P_*=UU^{\mathsf T}
22032 \end{equation}
22033 for a target-specific orthonormal matrix $U\in\R^{6\times2}$, while the inherited packet is
22034 \begin{equation}
22035 P_0=\diag(1,1,0,0,0,0).
22036 \end{equation}
22037 Let the chart label be
22038 \begin{equation}
22039 c(h)=\one\{e_1^{\mathsf T}U^{\mathsf T}h>0\}.
22040 \end{equation}
22041 For generic $U$, $c(h)$ is not measurable from $P_0h$. No threshold or nonlinear classifier using only $P_0h$ can recover the chart perfectly. The response atlas can recover $\im U$ up to basis gauge when the intervention signatures separate it.
22042 
22043 \subsection{Actual Block 11 edge}
22044 
22045 Both processes use $120$ effect-query units. The inherited parent uses
22046 \begin{equation}
22047 80+16+24=120
22048 \end{equation}
22049 ordinary bridge, chart-validation, and witness queries. The endogenous child uses
22050 \begin{equation}
22051 56+24+16+24=120
22052 \end{equation}
22053 bridge, response, chart-validation, and witness queries.
22054 
22055 On $80$ fresh foreign targets,
22056 \begin{equation}
22057 \Mcal(K_3)=0,
22058 \qquad
22059 \Mcal(K_4)=0.207325,
22060 \end{equation}
22061 so
22062 \begin{equation}
22063 \widehat\Delta_{3\to4}=0.207325,
22064 \qquad
22065 \mathrm{CI}_{95\%}=[0.131902,0.287910].
22066 \end{equation}
22067 The selected recovered packet has mean hidden context overlap $0.821461$, compared with $0.322091$ for the inherited coordinate-fixed packet. Blind endogenous application has mean gain $0.271529$ but harm rate $0.1875$. The witness lowers selected gain to $0.207325$ and reduces selected harm to zero. This is a finite endogenous-atlas result, not an unrestricted discovery theorem.
22068 
22069 
22070 
22071 \section{Worked Response-Language and Constructor-Growth Examples}
22072 
22073 \subsection{An exact response alias}
22074 Let three rank-one blocks have inherited signatures $(1,0)$, $(1,0)$, and $(0,1)$. Every polynomial in the two inherited block-scalar responses acts equally on the first two blocks. Adding a third response with scalars $(1,-1,0)$ separates them, and Lagrange interpolation recovers all three central projectors.
22075 
22076 \subsection{A weak separator is statistically inadequate}
22077 If the new scalars are $(1+\epsilon,1-\epsilon,0)$, the algebra is formally separating for every $\epsilon\neq0$, but empirical projector recovery requires response-estimation error small relative to the $2\epsilon$ spectral gap. This illustrates why algebraic identifiability and practical utility are distinct gates.
22078 
22079 \subsection{Nonzero Hessian defeats every affine composition}
22080 For $z\sim N(0,I_2)$ and $q(z)=z_1^2-z_2^2$, every affine predictor has residual variance at least $4$. The degree-two feature vector $(z_1^2,z_1z_2,z_2^2)$ represents $q$ exactly. No increase in affine depth changes this conclusion because affine maps remain closed under composition.
22081 
22082 \subsection{Why the full quadratic control matters}
22083 If both a packet-local and full-space degree-two constructor fit the same target, the experiment identifies missing polynomial degree but does not establish that localization is uniquely necessary. Packet localization may still improve efficiency, conditioning, or transfer in harder settings, but those are separate claims.
22084 
22085 
22086 \section{Worked Research-Compiler Examples}
22087 
22088 \subsection{Evaluator-nullspace shrink}
22089 Let $T=\R^2$, $Db_1(x_1,x_2)=x_1$, and $Db_2(x_1,x_2)=x_2$. With only $b_1$, the invisible subspace is $\operatorname{span}(e_2)$. Adding $b_2$ reduces the intersection of kernels to $\{0\}$. The new benchmark contributes an independent behavioral distinction rather than merely rescaling the old score.
22090 
22091 \subsection{Uniform forecast-regret bound}
22092 Suppose true candidate gains are $(0.70,0.62,0.10)$ and predictions are $(0.64,0.66,0.12)$. The forecaster selects candidate two. Here $\eps=0.06$ and the realized regret is $0.08\leq2\eps=0.12$, illustrating the fixed-candidate theorem.
22093 
22094 \subsection{Results-free proposal receipt}
22095 A candidate receipt predicts improvement on four scored dimensions, declares a capability-regression ceiling, hashes the proposed code, and names a falsification condition before execution. After evaluation, the post-receipt appends the observed score vector, hidden audit, and monitor verdict without changing the pre-result fields.
22096 
22097 \subsection{Actual Block 13 compiler edge}
22098 On the primary audit, the parent and child gains are $0.883004$ and $0.950231$, so
22099 \begin{equation}
22100 \widehat\Delta_\Gamma^{\mathrm{primary}}=0.067227.
22101 \end{equation}
22102 The child changes research allocation rather than the candidate grammar. The architecture-foreign paired effect is $0.090800$.
22103 
22104 \subsection{Archive and monitor controls}
22105 Shuffling archive labels lowers primary gain to $0.282550$, showing that the compiler depends on correctly bound prior evidence. Removing the integrity monitor lowers validity-filtered gain to $0.797956$ because invalid methods can occupy the selected slot. Neither control is a claim about universal archive or monitor sufficiency.
22106 
22107 
22108 \section{Worked Robust Research-Compiler Examples}
22109 
22110 \subsection{Aliased static diagnostics}
22111 Let $Z\in\{0,1\}$ be equally likely and let $X_0\sim\mathcal N(0,1)$ under both classes. No static rule exceeds accuracy $1/2$. If experiment $a$ returns $Y_a=Z+\xi$ with $\abs{\xi}<1/2$, then thresholding $Y_a$ at $1/2$ recovers $Z$ exactly. This is the simplest form of the Block 14 distinction between cheap diagnostic aliasing and active response separation.
22112 
22113 \subsection{Two-obstruction mixture}
22114 Let
22115 \begin{equation}
22116 R=
22117 \begin{bmatrix}
22118 1&0&1/2\\
22119 0&1&1/2\\
22120 1&-1&0
22121 \end{bmatrix},
22122 \qquad
22123 z=
22124 \begin{bmatrix}
22125 0.6\\0.4\\0
22126 \end{bmatrix}.
22127 \end{equation}
22128 Then $y=Rz=(0.6,0.4,0.2)^\top$. A forced one-class diagnosis discards one active component, while $R^\dagger y$ recovers the mixture exactly. If the third response row is removed, the first two classes remain identifiable here, but other mixtures can become aliased; the smallest singular value quantifies the resulting instability.
22129 
22130 \subsection{Open-set residual}
22131 Let inherited known responses span the nonnegative cone generated by $e_1$ and $e_2$ in $\R^3$. The unknown response $y=e_3$ has
22132 \begin{equation}
22133 \dist(y,\operatorname{cone}\{e_1,e_2\})=1.
22134 \end{equation}
22135 Any threshold below one and above known-response noise flags the case as outside the inherited language. The flag does not identify the missing obstruction. It only prevents a false confident inherited label.
22136 
22137 \subsection{Conjunctive integrity monitoring}
22138 Suppose a specification monitor misses an invalid method with probability $0.2$ and a trajectory monitor misses it with probability $0.15$. The conjunction has false-negative probability at most $0.15$. Under conditional independence the rate is $0.03$. Without independence, the product must not be claimed.
22139 
22140 \subsection{Zero observed invalid selections}
22141 If zero invalid selections are observed in $n=144$ independent audit problems, the exact one-sided $95\%$ upper bound is
22142 \begin{equation}
22143 1-0.05^{1/144}
22144 \approx
22145 0.02059.
22146 \end{equation}
22147 The correct statement is therefore ``zero observed, upper bound approximately $2.06\%$,'' not ``the compiler has zero invalid-selection probability.''
22148 
22149 \section{Worked Compiler-Language Growth Examples}
22150 
22151 \subsection{One unknown token with incompatible methods}
22152 Let two problems $a$ and $b$ both map to $\bot$. Suppose
22153 \begin{equation}
22154 \Acal^\star(a)=\{m_1\},
22155 \qquad
22156 \Acal^\star(b)=\{m_2\},
22157 \qquad
22158 m_1\neq m_2.
22159 \end{equation}
22160 Any deterministic policy conditioned only on $\bot$ must choose one method and fail the other problem. This is a diagnostic-language obstruction, not ordinary estimation noise.
22161 
22162 \subsection{A conservative binary refinement}
22163 Let probe $p$ return $-1$ on $a$ and $+1$ on $b$. With threshold zero, define tokens $(\bot,-)$ and $(\bot,+)$. The projection that forgets the sign recovers $\bot$, while the refined binding sends the two tokens to $m_1$ and $m_2$. The old language is preserved and the conflict is removed.
22164 
22165 \subsection{Why taxonomy-only growth is insufficient}
22166 Suppose the inherited deployable grammar contains only singleton methods $u$ and $v$, each with gain $0.3$ on both novel cases, while pair programs $u\circ v$ and $v\circ u$ obtain gains $0.9$ on opposite cases. Splitting the unknown token without adding the pair constructor leaves the attainable gain at $0.3$.
22167 
22168 \subsection{Why grammar-only growth is insufficient}
22169 If the two pair methods are added but the unknown token remains unrefined, one pooled binding must be used. It can solve at most one of the two cases. Correct language growth requires the diagnostic refinement and constructor outputs to remain bound.
22170 
22171 \subsection{Actual Block 15 structural margin}
22172 The development inherited-method oracle is $0.377257$ and the selected language obtains $0.948962$. Thus
22173 \begin{equation}
22174 \widehat\kappa=0.948962-0.377257=0.571704.
22175 \end{equation}
22176 On the primary foreign audit, the matched-budget child-parent effect is $0.225762$; under generator shift it is $0.240296$.
22177 
22178 \subsection{Finite pair grammar versus unrestricted synthesis}
22179 The executable compiler searches $30$ ordered pairs constructed from six inherited typed method families. Selecting a useful pair from this grammar is structural relative to the deployable singleton parent, but it does not establish invention of loops, conditionals, recursion, arbitrary code, or a new primitive research operation.
22180 
22181 \section{Receipt Schema}
22182 
22183 A minimal machine-readable receipt should contain fields equivalent to the following symbolic schema:
22184 
22185 \begin{verbatim}
22186 repair_receipt:
22187   lineage:
22188     parent_ids
22189     candidate_id
22190     active_head_before_and_after
22191     lineage_edge_proofs
22192     rollback_checkpoint
22193   transaction:
22194     editable_mask
22195     witness_mask_and_versions
22196     prepare_seal_execute_reveal_decide_finalize_timestamps
22197     proposal_and_prediction_commitments
22198     trace_schema_and_factorization_tests
22199     acceptance_program_digest
22200     capability_and_resource_budget
22201     buffered_and_irreversible_effects
22202     rollback_fidelity
22203   diagnosis:
22204     failure_signature
22205     causal_hypotheses
22206     probes
22207     posterior_uncertainty
22208   correctability:
22209     correction_domain
22210     protected_invariant_map
22211     defect_space
22212     linear_response_map
22213     singular_values
22214     correctable_projection
22215     obstruction_projection
22216     obstruction_norm
22217     dual_witness
22218     repair_budget
22219     empirical_error_bounds
22220   geometry:
22221     strata
22222     packet_blocks
22223     leakage_profile
22224     local_defects
22225     local_obstructions
22226     overlap_cocycle
22227     cycle_projection
22228     gluing_obstruction_norm
22229     nonlinear_holonomy
22230     formal_obstruction_order
22231     gluing_status
22232   repair:
22233     operator_id
22234     correction_program
22235     applicability_conditions
22236     regularization_cost
22237   transport:
22238     source_and_target_correction_data
22239     correction_and_defect_maps
22240     route_and_edge_receipts
22241     square_and_identification_defects
22242     authorization_and_invariant_commutation
22243     resource_amplification
22244     obstruction_faithfulness
22245     cumulative_residual_bound
22246     repair_and_semantic_holonomy
22247     deterministic_and_probabilistic_gain_bounds
22248   evaluator_upgrade:
22249     old_and_new_evaluator_ids
22250     raw_evidence_embedding
22251     internal_transport_and_core_retraction
22252     protected_history_replay_set
22253     holonomy_energy_spectrum
22254     exact_or_approximate_core
22255     core_adequacy_modulus
22256     heldout_paired_risk_bound
22257     branch_cycle_checks
22258   closure:
22259     universal_controlled_or_productive_semantics
22260     static_invariant_results
22261     successor_or_action_model_digest
22262     fixed_point_iteration_and_membership
22263     preservation_policy
22264     elimination_or_counterexample_certificate
22265     progress_threshold_and_ranking_certificate
22266   archive:
22267     descriptor_vector_and_uncertainty
22268     dominance_and_novelty_comparisons
22269     pareto_cell_or_continuation_signature
22270     retained_repair_schemas
22271     retained_obstruction_certificates
22272     evidence_namespace_and_use_history
22273   metaproductivity:
22274     foreign_agent_population_id
22275     horizon_candidate_and_compute_budget
22276     paired_baseline_and_descendant_gains
22277     confidence_bound
22278     distribution_shift_bound
22279     correctable_and_obstruction_energy
22280     causal_ablation_result
22281   grammar_extension:
22282     old_and_new_grammar_ids
22283     conservative_embedding
22284     old_proof_replay
22285     semantic_debt
22286     resolved_old_obstruction_classes
22287     newly_exposed_obstruction_directions
22288     representation_separation_certificate
22289   autocatalysis:
22290     schema_equivalence_class
22291     causal_parent_ablations
22292     offspring_type_counts
22293     entrywise_confidence_matrices
22294     strongly_connected_component
22295     reproduction_number_and_budget
22296     saturation_and_utility_checks
22297   empirical_realization:
22298     abstract_model_id_and_version
22299     runtime_implementation_id
22300     abstraction_map_and_refinement_checks
22301     protected_observation_commutation
22302     frozen_endpoint_and_claim_gate_set
22303     data_partitions_and_seed_schedule
22304     preoutcome_commitment_digest
22305     package_manifest_and_environment
22306     exact_and_statistical_replay_status
22307     oracle_leakage_checks
22308     strongest_admissible_claim_level
22309   learned_validation:
22310     task_and_defect_grammar_ids
22311     source_and_target_encoder_ids
22312     train_anchor_calibration_audit_split_digests
22313     causal_probe_and_counterfactual_definition
22314     packet_rank_basis_and_singular_spectrum
22315     equal_rank_pca_random_and_wrong_block_controls
22316     source_repair_parameters_and_digest
22317     paired_anchor_correspondence_digest
22318     learned_bridge_and_regularization
22319     alignment_and_repair_equivariance_residuals
22320     blind_selected_and_target_oracle_scores
22321     shuffled_anchor_and_direct_copy_controls
22322     witness_thresholds_and_commit_reason
22323     target_family_and_foreignness_certificate
22324     strongest_block06_claim_gate
22325   sealed_meta_improvement:
22326     parent_policy_id_and_digest
22327     candidate_set_and_matched_budget
22328     candidate_feature_schema_digest
22329     development_and_meta_validation_split_digests
22330     excluded_final_audit_split_ids
22331     compiler_source_digest
22332     meta_grammar_and_grammar_size
22333     selected_configuration_and_validation_score
22334     utility_and_risk_head_parameters
22335     candidate_specific_residual_scales
22336     seal_before_audit_load_trace
22337     descendant_receipt_digest
22338     paired_seed_task_and_architecture_foreign_effects
22339     bootstrap_intervals_and_sign_flip_tests
22340     average_tail_and_maximum_risk_gates
22341     raw_no_risk_shuffled_ablation_and_oracle_controls
22342     selector_level_vs_structural_claim_boundary
22343     strongest_block07_claim_gate
22344   meta_headroom:
22345     repair_process_module_factorization
22346     restricted_oracle_class_ids
22347     lower_and_upper_envelope_bounds
22348     ordered_and_permutation_average_allocations
22349     module_interaction_terms
22350     saturation_tolerances_and_certificates
22351     headroom_adjusted_gain
22352   bottleneck_migration:
22353     unresolved_module_headroom_vector
22354     selected_meta_action_and_response_model
22355     weighted_lyapunov_certificate
22356     collateral_headroom_and_error_terms
22357     old_coordinate_repair_progress
22358     newly_represented_discovery_energy
22359     dominant_bottleneck_before_and_after
22360   persistent_obstruction:
22361     grammar_indexed_correction_images
22362     newly_correctable_energy
22363     induced_cokernel_transport
22364     repair_death_or_forgetting_death
22365     protected_obstruction_faithfulness_margin
22366     cumulative_dynamic_semantic_debt
22367   generator_discovery:
22368     projected_foreign_obstruction_samples
22369     obstruction_covariance_and_spectrum
22370     candidate_response_direction
22371     one_generator_or_rank_k_energy_reduction
22372     cost_and_risk_adjusted_acquisition_score
22373     exact_generator_ablation
22374     structural_novelty_certificate
22375   conflict_refinement:
22376     aliased_episode_cell
22377     validated_authorized_repair_sets
22378     minimal_repair_conflicts
22379     candidate_feature_or_probe_family
22380     selected_separator_and_cost
22381     old_grammar_projection
22382     remaining_conflict_hypergraph
22383   causal_atlas:
22384     intervention_response_operator_ids
22385     generated_adjoint_closed_algebra
22386     commutant_and_center_dimensions
22387     minimal_central_projectors
22388     multiplicity_gauge
22389     defect_cyclic_module
22390     commutator_spectral_gap
22391     perturbation_and_stability_bound
22392   sequential_lineage:
22393     pre_audit_filtration_digest
22394     edge_true_effect_estimand
22395     conditional_lower_and_upper_bounds
22396     edge_false_acceptance_budget
22397     cumulative_archive_error_budget
22398     protected_regression_budget
22399     semantic_debt_budget
22400     prior_receipt_replay
22401     repeated_lineage_claim_level
22402   repeated_lineage_conditioning:
22403     block07_ancestor_policy_and_semantic_digests
22404     ancestor_receipt_replay_status
22405     fresh_development_and_audit_context_ids
22406     rank_saturation_certificate
22407     query_quality_metadata_digest
22408     weighted_information_matrix_diagnostics
22409     recalibrated_child_policy_and_gate
22410     conditioning_grandchild_policy_and_gate
22411     finite_design_grammar_and_compiler_ledger
22412     matched_deployment_query_ledgers
22413     preaudit_grandchild_seal
22414     paired_grandchild_child_effect_and_interval
22415     gate_only_quality_permutation_and_random_budget_controls
22416     target_side_conditioning_and_stress_results
22417     context_indexed_two_edge_claim_boundary
22418   structural_grammar_expansion:
22419     block09_ancestor_policy_and_semantic_digests
22420     global_parent_and_chart_child_configurations
22421     finite_separator_feature_grammar
22422     development_grammar_inadequacy_certificate
22423     global_and_local_validation_residuals
22424     chart_support_and_gate_accuracy
22425     target_repair_contrast
22426     conservative_parent_embedding_and_fallback
22427     matched_48_query_ledgers
22428     preaudit_structural_child_seal
22429     paired_child_parent_effect_and_interval
22430     gate_only_shuffled_binding_oracle_and_stress_controls
22431     structural_claim_boundary
22432   endogenous_atlas_growth:
22433     block10_ancestor_policy_and_seal_digests
22434     supplied_separator_removed
22435     intervention_response_operator_ids_and_query_ledger
22436     response_algebra_rank_and_commutant_dimension
22437     recovered_central_projectors_and_block_ranks
22438     response_fit_commutator_and_spectral_gap_certificate
22439     selected_repair_relevant_block_and_separator
22440     source_repair_invariance_and_target_gate_agreement
22441     hidden_overlap_metrics_reserved_for_audit
22442     conservative_block10_embedding_and_fallback
22443     matched_parent_and_child_effect_query_budgets
22444     preaudit_atlas_certificate_compiler_and_policy_digests
22445     paired_child_parent_effect_interval_and_sign_flip_test
22446     pca_full_random_shuffled_blind_oracle_and_stress_controls
22447     endogenous_atlas_claim_boundary
22448   response_language_growth:
22449     inherited_response_operator_ids_and_alias_partition
22450     finite_candidate_intervention_language
22451     candidate_signature_gaps_commutators_fit_and_costs
22452     selected_separating_operator_and_central_refinement
22453     conservative_old_algebra_embedding
22454     matched_parent_and_child_query_ledgers
22455     preaudit_response_certificate_and_policy_digests
22456     paired_effect_interval_controls_and_stress_boundary
22457   constructor_growth:
22458     inherited_constructor_operad_and_function_class
22459     hessian_or_equivalent_expressivity_obstruction
22460     finite_candidate_constructor_language
22461     selected_degree_packet_and_feature_basis
22462     conservative_affine_embedding
22463     separately_compiled_parent_and_child_witnesses
22464     same_target_affine_oracle_and_nonlinear_controls
22465     paired_effect_interval_stress_and_claim_boundary
22466   research_compiler:
22467     ancestor_result_and_archive_digests
22468     literature_prior_and_finding_graph
22469     candidate_method_grammar_and_canonical_order
22470     pre_result_hypothesis_objective_data_and_falsification_fields
22471     proposal_specification_and_executable_code_hashes
22472     obstruction_classifier_and_outcome_risk_forecaster_digests
22473     cheap_diagnostic_and_full_experiment_allocation_ledger
22474     evaluator_portfolio_capability_gates_and_integrity_monitor
22475     development_meta_validation_primary_and_architecture_split_commitments
22476     sealed_parent_child_compiler_objects
22477     paired_validity_filtered_audit_effects_controls_and_claim_boundary
22478   robust_research_compiler:
22479     block13_ancestor_result_digest
22480     fixed_candidate_grammar_and_six_experiment_budget
22481     contaminated_archive_digest_and_estimator_type
22482     sequential_response_experiment_ledger
22483     sparse_obstruction_mixture_and_open_set_residual
22484     evaluator_portfolio_and_aggregation_rule
22485     specification_and_trajectory_monitor_verdicts
22486     primary_and_generator_foreign_split_commitments
22487     compiler_configuration_and_preaudit_seal
22488     groupwise_mean_tail_negative_unknown_and_invalid_metrics
22489     paired_effect_intervals_ablation_ledger_and_claim_boundary
22490   compiler_language_growth:
22491     block14_ancestor_result_digest
22492     inherited_diagnostic_and_method_language_digests
22493     open_set_repair_conflict_hypergraph
22494     generic_probe_language_and_selected_probe
22495     anonymous_refined_tokens_and_threshold
22496     generic_method_constructor_language
22497     selected_token_to_method_bindings
22498     inherited_oracle_expanded_oracle_and_structural_margin
22499     exact_conservative_fallback_receipt
22500     matched_research_unit_ledger
22501     primary_and_generator_foreign_split_commitments
22502     compiler_language_certificate_and_preaudit_seal
22503     taxonomy_grammar_binding_controls_and_claim_boundary
22504   decision:
22505     commit_rollback_or_quarantine
22506     rejection_reason
22507   abstraction:
22508     repair_schema_class
22509     foreign_agent_transfer_results
22510     newly_correctable_directions
22511     surviving_obstruction_classes
22512 \end{verbatim}
22513 
22514 \section{Dependency Map}
22515 The principal results form the following dependency chain. The table is intentionally breakable across pages so that the complete map remains legible.
22516 \begingroup
22517 \small
22518 \setlength{\LTpre}{0.4em}
22519 \setlength{\LTpost}{0.4em}
22520 \begin{longtable}{@{}p{0.43\textwidth}@{\hspace{0.025\textwidth}}c@{\hspace{0.025\textwidth}}p{0.43\textwidth}@{}}
22521 \toprule
22522 Prerequisite structure & & Derived result \\
22523 \midrule
22524 \endfirsthead
22525 \toprule
22526 Prerequisite structure & & Derived result \\
22527 \midrule
22528 \endhead
22529 \midrule
22530 \multicolumn{3}{r}{\footnotesize Continued on the next page}\\
22531 \endfoot
22532 \bottomrule
22533 \endlastfoot
22534 Protected evaluation context & $\Longrightarrow$ & Nonvacuous improvement relation \\
22535 Relative introspection & $\Longrightarrow$ & Bounded decision regret \\
22536 Authorized response map & $\Longrightarrow$ & Correctable--obstructed decomposition \\
22537 Local repair and overlap coherence & $\Longrightarrow$ & Repair descent \\
22538 Transport squares and holonomy bounds & $\Longrightarrow$ & Gain-preserving transfer \\
22539 Evaluator transports and adequacy & $\Longrightarrow$ & Conservative evaluator evolution \\
22540 Frozen witness and trace factorization & $\Longrightarrow$ & Transactional noncircularity \\
22541 Immutable checkpoint and buffered effects & $\Longrightarrow$ & Atomic rollback \\
22542 Finite transition game & $\Longrightarrow$ & Universal and controlled closure \\
22543 Safe attractor and validated gain set & $\Longrightarrow$ & Productive B\"uchi closure \\
22544 Append-only archive and multiobjective descriptors & $\Longrightarrow$ & Monotone envelopes and finite covers \\
22545 Matched foreign agents and paired evidence & $\Longrightarrow$ & Metaproductivity certificate \\
22546 Nested repair images and defect distribution & $\Longrightarrow$ & Task-relevant structural gain \\
22547 Conservative grammar chain & $\Longrightarrow$ & Proof persistence and directed limit \\
22548 Ontology embedding & $\Longrightarrow$ & Resolved old/new obstruction exact sequence \\
22549 Causal offspring receipts & $\Longrightarrow$ & Repair-discovery matrix \\
22550 Runtime refinement and observation commutation & $\Longrightarrow$ & Trace-sound implementation \\
22551 Finite exact realization and exhaustive endpoints & $\Longrightarrow$ & Decidable fixture claims \\
22552 Sealed package and cumulative claim gates & $\Longrightarrow$ & Bounded empirical interpretation \\
22553 Causal intervention differences and rank condition & $\Longrightarrow$ & Hidden repair-packet identification \\
22554 Spanning paired anchors and linear hidden relation & $\Longrightarrow$ & Identified repair bridge \\
22555 Injective bridge and frozen source repair & $\Longrightarrow$ & Exact transported repair \\
22556 Compression or nonlinear residual and frozen witness & $\Longrightarrow$ & Commit or abstain \\
22557 Development receipts and separate meta-validation & $\Longrightarrow$ & Sealed child selector \\
22558 Identical candidate bundles and paired foreign audit & $\Longrightarrow$ & Selector-level metaproductivity certificate \\
22559 Risk upper score and protected-regression gates & $\Longrightarrow$ & Constrained child commitment \\
22560 Fixed candidate grammar & $\Longrightarrow$ & No structural repair-image claim \\
22561 Fresh grandchild edge and newly sealed evidence & $\Longrightarrow$ & First repeated-lineage test \\
22562 Heteroscedastic paired-anchor model and valid quality weights & $\Longrightarrow$ & Weighted bridge covariance and information geometry \\
22563 Full source rank with low $\lambda_{\min}(M_S)$ & $\Longrightarrow$ & Conditioning bottleneck beyond identifiability \\
22564 Quality-aware D-optimal design and matched query budget & $\Longrightarrow$ & Conditioning-aware grandchild candidate \\
22565 Ancestor replay, fresh supports, pre-audit seal, and paired audit & $\Longrightarrow$ & Two-edge context-indexed repeated-lineage certificate \\
22566 Restricted oracle envelopes and module factorization & $\Longrightarrow$ & Meta-headroom and saturation \\
22567 Headroom state and authorized contraction action & $\Longrightarrow$ & Bottleneck migration \\
22568 Nested correction images and foreign residuals & $\Longrightarrow$ & Persistent obstruction energy \\
22569 Projected obstruction and candidate response direction & $\Longrightarrow$ & Generator-acquisition score \\
22570 Repair-conflict hypergraph and separating features & $\Longrightarrow$ & Conservative grammar refinement \\
22571 Intervention-response operators and adjoint closure & $\Longrightarrow$ & Central causal atlas \\
22572 Conditional edge bounds and summable budgets & $\Longrightarrow$ & Lineage-wide validity \\
22573 Persistent global residual and finite chart separator family & $\Longrightarrow$ & Grammar-inadequacy certificate \\
22574 Conservative global-to-chart embedding and nonzero repair contrast & $\Longrightarrow$ & New chart-specific correction direction \\
22575 Matched foreign audit with binding and gate controls & $\Longrightarrow$ & Finite structural metaproductivity certificate \\
22576 Separating block-scalar intervention responses & $\Longrightarrow$ & Exact central response atlas up to permutation and within-block gauge \\
22577 Response estimation error and spectral gap & $\Longrightarrow$ & Approximate central-projector stability \\
22578 Recovered response blocks and repair-conflict validation & $\Longrightarrow$ & Repair-relevant atlas labeling \\
22579 Supplied-packet removal, conservative fallback, and matched foreign audit & $\Longrightarrow$ & Finite endogenous-atlas metaproductivity certificate \\
22580 Inherited block-scalar response alias & $\Longrightarrow$ & Polynomial nonseparation certificate \\
22581 Separating candidate intervention and spectral margin & $\Longrightarrow$ & Conservative central response refinement \\
22582 Matched response audit and correspondence controls & $\Longrightarrow$ & Finite response-language metaproductivity certificate \\
22583 Affine repair-operad closure and nonzero Hessian & $\Longrightarrow$ & Constructor function-class obstruction \\
22584 Degree-two conservative embedding and matched audit & $\Longrightarrow$ & Finite candidate-constructor metaproductivity certificate \\
22585 Results-free proposal fields and immutable code digest & $\Longrightarrow$ & Non-retroactive proof-carrying research receipt \\
22586 Independent evaluator response and benchmark portfolio refinement & $\Longrightarrow$ & Strict shrinkage of one metric-gaming nullspace \\
22587 Bounded outcome-prediction error on a fixed candidate set & $\Longrightarrow$ & Uniform $2\eps$ selection-regret bound \\
22588 Correctly bound research archive and obstruction classifier & $\Longrightarrow$ & Bottleneck-sensitive experiment allocation \\
22589 Matched method grammar, evaluator, monitor, budget, seal, and paired audits & $\Longrightarrow$ & Finite research-compiler metaproductivity certificate \\
22590 Static diagnostic alias and separating response experiments & $\Longrightarrow$ & Necessity of sequential diagnosis \\
22591 Sparse inherited response cone and residual margin & $\Longrightarrow$ & Finite open-set certificate \\
22592 Contaminated archive and robust summary statistics & $\Longrightarrow$ & Finite contamination-stability comparison \\
22593 Evaluator portfolio and independent behavior dimension & $\Longrightarrow$ & Strict reduction of one evaluator nullspace \\
22594 Conjunctive integrity witnesses and adaptive attack methods & $\Longrightarrow$ & Validity-filtered monitor-ensemble comparison \\
22595 Matched fixed grammar, stress atlas, seal, group and tail gates & $\Longrightarrow$ & Finite adversarial compiler-robustness certificate \\
22596 Inherited unknown-token conflict and disjoint acceptable-method sets & $\Longrightarrow$ & Diagnostic-language inadequacy certificate \\
22597 Generic separating probe and conservative token projection & $\Longrightarrow$ & Finite anonymous diagnostic refinement \\
22598 Generic pair-method constructor language and positive oracle margin & $\Longrightarrow$ & Task-relevant compiler method-language expansion \\
22599 Matched eight-unit audits, fallback, seal, and language controls & $\Longrightarrow$ & Finite compiler-language metaproductivity certificate \\
22600 Nonnegative spectral bounds & $\Longrightarrow$ & Local autocatalytic certificate \\
22601 Fixed finite grammar or bounded utility & $\Longrightarrow$ & Saturation of indefinite fixed-size gain \\
22602 \end{longtable}
22603 \endgroup
22604 
22605 \clearpage
22606 \begin{thebibliography}{10}\footnotesize
22607 \setlength{\itemsep}{0.2em}
22608 \setlength{\parskip}{0pt}
22609 
22610 \bibitem{SIWC}
22611 Komorebi P.,
22612 \emph{Self-Indexed Workspace Closure: Expanded Theory Draft},
22613 version 0.4, unpublished research manuscript, 2026.
22614 
22615 \bibitem{PRSC3}
22616 Komorebi P.,
22617 \emph{Projector-Resolved Seed Calculus III},
22618 unpublished research manuscript and companion implementation, 2026.
22619 
22620 \bibitem{RRCT}
22621 Komorebi P.,
22622 \emph{RRCT Awareness Research Handoff},
22623 version 0.16, internal research handoff, 2026.
22624 
22625 \bibitem{OSL}
22626 Komorebi P.,
22627 \emph{Operator-Symbolic Language Core and Proof-Carrying Transaction Notes},
22628 unpublished research artifacts, 2026.
22629 
22630 \bibitem{KM}
22631 Komorebi P.,
22632 \emph{Komorebi Math Master Collected Edition},
22633 unpublished collected research edition, 2026.
22634 
22635 \bibitem{SRICB6}
22636 Komorebi P.,
22637 \emph{SRIC Block 6 Learned-Agent Validation Reference Package},
22638 version 0.2.0, reproducible software and frozen result archive, 2026.
22639 
22640 \bibitem{SRICB7}
22641 Komorebi P.
22642 \emph{SRIC Block 7 Sealed Parent--Child Metaproductivity Reference Implementation}, 2026.
22643 Executable selection-only and bridge-policy descendants, pre-audit seals, matched-resource foreign benchmarks, causal controls, frozen results, tests, and manifests.
22644 
22645 \bibitem{SRICB8}
22646 Komorebi P.,
22647 \emph{SRIC Block 8 Meta-Headroom and Obstruction-Growth Symbolic Reference Implementation}, 2026.
22648 Exact finite fixtures for restricted oracle envelopes, bottleneck migration, obstruction towers, conflict refinement, repair synthesis, causal-atlas decomposition, sequential budgets, tests, and manifests.
22649 
22650 
22651 \bibitem{SRICB9}
22652 Komorebi P.,
22653 \emph{SRIC Block 9 Repeated Lineage and Conditioning-Aware Query Design Reference Implementation}, 2026.
22654 Executable ancestor replay, heteroscedastic bridge environment, rank-saturation diagnosis, finite conditioning-aware grandchild compiler, fresh paired foreign audits, causal controls, frozen results, tests, and manifest.
22655 
22656 \bibitem{SRICB10}
22657 Komorebi P.,
22658 \emph{SRIC Block 10 Structural Grammar Expansion Reference Implementation}, 2026.
22659 Executable global-grammar obstruction fixture, conservative two-chart compiler, matched foreign audits, chart-binding and gate controls, stress families, frozen results, tests, and manifest.
22660 
22661 \bibitem{SRICB11}
22662 Komorebi P.,
22663 \emph{SRIC v1.2 Block 11 Endogenous Atlas-Growth Reference Implementation},
22664 package-closed executable companion, 2026.
22665 
22666 \bibitem{SRICB12}
22667 Komorebi P.,
22668 \emph{SRIC v1.3 Block 12 Response-Language and Candidate-Constructor Growth Reference Implementation},
22669 package-closed executable companion with matched-resource response-language and degree-two constructor audits, 2026.
22670 
22671 
22672 \bibitem{AAR2026}
22673 Yueh-Han Chen, Jiaxin Wen, and Jan Hendrik Kirchner,
22674 \emph{Automated Researchers Can Reliably Mitigate Alignment Failures},
22675 Anthropic Alignment Science, 2026,
22676 \url{https://alignment.anthropic.com/2026/automated-alignment-researchers/}.
22677 
22678 \bibitem{SRICB13}
22679 Komorebi P.,
22680 \emph{SRIC v1.4 Block 13 Proof-Carrying Automated Research Compiler Reference Implementation},
22681 package-closed executable companion with matched-budget compiler audits, results-free receipts, integrity controls, and frozen manifests, 2026.
22682 
22683 
22684 \bibitem{SRICB14}
22685 Komorebi P.,
22686 \emph{SRIC v1.5 Block 14 Adversarial Open-Set Research-Compiler Robustness Reference Implementation},
22687 package-closed executable companion with diagnostic aliasing, sparse multi-obstruction and open-set problems, archive contamination, evaluator and integrity attacks, generator-foreign audits, robust controls, tests, and frozen manifests, 2026.
22688 
22689 \bibitem{SRICB15}
22690 Komorebi P.,
22691 \emph{SRIC v1.6 Block 15 Compiler-Language Growth from Open-Set Research Conflicts Reference Implementation},
22692 package-closed executable companion with conflict-grounded probe selection, anonymous diagnostic refinement, ordered pair-method synthesis, conservative inherited fallback, matched foreign audits, controls, tests, and frozen manifests, 2026.
22693 
22694 
22695 \bibitem{CYMELA2026}
22696 Cymela,
22697 \emph{Hyper v1 --- 3B, Neuralese Thinking},
22698 model documentation and local inference runtime, 2026,
22699 \url{https://cymela.com/models/hyper-3b-latent};
22700 model repository \url{https://huggingface.co/Cymela/hyper-3b-latent}.
22701 
22702 \bibitem{SRICB16}
22703 Komorebi P.,
22704 \emph{SRIC v1.7 Block 16 Direct Neuralese Causal-Semantics Reference Implementation},
22705 package-closed direct Hyper adapter contract, synthetic parent--child causal fixture, trace interventions, semantic transport, receipts, tests, figures, and manifest, 2026.
22706 
22707 \bibitem{SRICB17}
22708 Komorebi P.,
22709 \emph{SRIC v1.8 Block 17 Executable Neuralese Binding-Repair Compiler Reference Implementation},
22710 package-closed exact affine-program environment, trained recurrent parent, matched repair candidates, answer-confound donor controls, external trace probe, preserved failed pilot, sealed fresh audits, independent NumPy and gauge verification, receipts, tests, figures, and manifest, 2026.
22711 
22712 
22713 \bibitem{SRICB18}
22714 Komorebi P.,
22715 \emph{SRIC v1.9 Block 18 Cross-Implementation Neuralese Repair Transport Reference Implementation},
22716 package-closed executable companion with exact program induction, three recurrent implementations, source-compiled FiLM repair transport, sealed foreign audits, failed family receipt, distributional semantic-chart obstruction, post-hoc localization, tests, and frozen manifests, 2026.
22717 
22718 
22719 \bibitem{SRICB19}
22720 Komorebi P.,
22721 \emph{SRIC v2.0 Block 19 Distribution-Indexed Neuralese Semantic Atlas Growth Reference Implementation},
22722 package-closed executable companion with exact contextual program induction, semantic-twin affine chart compilation, conservative atlas extension, set-structured recurrent Neuralese, selective admission, fresh supported and unsupported audits, independent NumPy replay, tests, figures, and manifest, 2026.
22723 
22724 \bibitem{SRICB20}
22725 Komorebi P.,
22726 \emph{SRIC v2.1 Block 20 Atlas-Indexed Cross-Architecture Neuralese Transport Reference Implementation},
22727 package-closed executable companion with a frozen six-chart source atlas, trained GRU and gated-residual targets, controlled architecture--chart gauges, pooled and chart-indexed full-trajectory transport, commutation and curvature audits, paired fresh evaluations, controls, tests, independent NumPy replay, figures, and manifest, 2026.
22728 
22729 \bibitem{SRICB21}
22730 Komorebi P.,
22731 \emph{SRIC v2.2 Block 21 Obstruction-Guided Neuralese Transport-Grammar Growth Reference Implementation},
22732 package-closed executable companion with a frozen recurrent packet, hidden affine, nonlinear, temporal, chart-alias, benign-compression, and destructive-collision families, a minimum-description typed compiler, fresh foreign audits, no-go certificates, controls, tests, figures, and independent replay, 2026.
22733 
22734 
22735 \bibitem{SRICB22}
22736 Komorebi P.,
22737 \emph{SRIC v2.3 Block 22 Correspondence-Free Neuralese Transport from Intervention Orbits Reference Implementation},
22738 package-closed executable companion with independent source and target construction populations, finite response-algebra intertwiners, anonymous intervention matching, declared-gauge certificates, selective partial transport, semantic underidentification, collision-grounded no-go, fresh audits, tests, figures, and deterministic replay, 2026.
22739 
22740 
22741 \bibitem{DILGREN2026}
22742 Connor Dilgren and Sarah Wiegreffe,
22743 \emph{Are Latent Reasoning Models Easily Interpretable?},
22744 arXiv:2604.04902, 2026.
22745 
22746 \bibitem{SRICB23}
22747 Komorebi P.,
22748 \emph{SRIC v2.4 Block 23 Endogenous Semantic-Witness Growth and External Latent-Reasoner Pilot Reference Implementation},
22749 package-closed finite causal-predictive quotient compiler, witness-separated hidden audit, semantic-twin, answer-shortcut, gauge, partiality, underidentification, collision no-go, counterfactual packet controls, source-locked Coconut adapter, tests, figures, and frozen receipts, 2026.
22750 
22751 \end{thebibliography}
22752 
22753 \end{document}