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alpha","display":"\\bigoplus_{\\alpha=1}^{p}","kind":"use","occurrences":[2874,2875,2877]},{"id":"symbol-a56a245d78482165","tex":"\\otimes","aliases":"⊗ otimes","display":"\\otimes","kind":"use","occurrences":[842,850,2874,2875,3012,3670,5599]},{"id":"symbol-28fddfd6ee2f43f3","tex":"\\bigcap_{j=1}^m","aliases":"⋂_j=1^m bigcap","display":"\\bigcap_{j=1}^m","kind":"use","occurrences":[4154]},{"id":"symbol-036096cd7e9ecc74","tex":"\\bigcup","aliases":"⋃ bigcup","display":"\\bigcup","kind":"use","occurrences":[1422]},{"id":"symbol-490d7f1b2a39a789","tex":"\\bigcup_{t=1}^{T}","aliases":"⋃_t=1^T bigcup","display":"\\bigcup_{t=1}^{T}","kind":"use","occurrences":[1422]},{"id":"symbol-437de0b3cc0e5a5f","tex":"\\langle","aliases":"⟨ langle","display":"\\langle","kind":"use","occurrences":[2370]},{"id":"symbol-7b52d298f1764e49","tex":"\\rangle","aliases":"⟩ rangle","display":"\\rangle","kind":"use","occurrences":[2370]}],"contexts":["wo 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.","e-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.","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.","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.","ld 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.","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.","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.","t 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.","ed-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.","idge-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.","he 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.","$ 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.","eign 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.","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 $0.095803$.","nsition. 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 $0.095803$.","eplayed 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 $0.095803$.","aired 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 $0.095803$.","teroscedastically 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 $0.095803$.","nite 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 $0.095803$.","d 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 $0.095803$.","inal 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 $0.095803$.","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 $0.095803$.","elected 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 $0.095803$.","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 wi","rst 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.","al 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.","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.","ioning-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.","cted 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.","ies 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.","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.","iant-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.","bal 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.","ss 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.","$48$ 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.","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.",". 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.","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.","es 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.","rators, 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.","in-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.","hes 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.","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.","ock 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.","r 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.","e 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.","ntervention 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.","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.","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.","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.","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-sensi","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 redu","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.","s 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.","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.","arrying 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.","eage-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.","hild 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.","d 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.","e 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.","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","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 chang","tside 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…","luator 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.",", 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.","rately 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.","r. 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.","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.","bust 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.","s 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.",", 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.","amination-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.","ggregation, 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.","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.","roblems, 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.","ity-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.","fect $0.164970$ 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.",".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.","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","eceives 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.","ment 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.","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.","nymous 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.","size 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.","t{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.","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.","erited $39$-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.","thod 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.","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.","he 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.","s 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.","antic-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.","t 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.","otion 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.","gets 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.","swer 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.","dependent 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.","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.","ned-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.","ificate. 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.","thout weakening its gates. After a precommitted compiler revision and fresh final-task generation, the selected full-binding child is FULL on both $1600$-task primary and shifted-generator audits while the parent remains PARTIAL. Primary and foreign program-specificity gains are $0.010714$ and $0.024763$, with frozen intervals $[0.009911,0.011527]$ and $[0.022794,0.026793]$; held-out trace-probe $R^2$ is $0.865504$ and $0.691471$. The accepted result is one finite package-local executable Neuralese binding-repair edge, not an empirical Hyper result or open-ended recursive self-improvement.","kening 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.","s. 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.","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-d","r 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.","cific 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.","ual 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.","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.","get 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.","d $R^2=-0.145315$ 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.","ogically 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 d…","onservatively 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-…","t 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 …",", 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.","mutation-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.","iant 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.","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.","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.","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.","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.","ceive 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.","velopes. 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.","led 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.","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.","text 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.","ty 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.","mantic-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.","asks 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.","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, sh","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.","6$ 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.","e--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.","f 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.","e-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.","d 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.","ry, 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.","tep-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.","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.","$ 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.","ler: 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.","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.","ollision. 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.","ed 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.","p 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.","emporal 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.","oss 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.","vention 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.","re 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.","ared 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.","estructive 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.","ures 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.","des 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.","ve 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.","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","ed 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 w…","nd 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.\n\\end{abstract}",": 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.\n\\end{abstract}","\\begin{statusbox}{Current revision}\nVersion 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.\n\\end{statusbox}","The phrase \\emph{recursive self-improvement} is often represented by the schematic loop\n\\begin{equation}\nA_t\\longrightarrow A_{t+1}\\longrightarrow A_{t+2}\\longrightarrow\\cdots,\n\\end{equation}\nwith 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.","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\n\\begin{equation}\nJ(A_{t+1})>J(A_t).\n\\end{equation}\nLet $\\Gamma_{h,B}(K;A)$ 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,\n\\begin{equation}\n\\mathbb E_A\\Gamma_{h,B}(K_{t+1};A)\n>\n\\mathbb E_A\\Gamma_{h,B}(K_t;A).\n\\end{equation}\nFinally, if $\\Vcal_t$ is the evaluator active at generation $t$, epistemic improvement requires a decrease in held-out evaluation risk rather than","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\n\\begin{equation}\nJ(A_{t+1})>J(A_t).\n\\end{equation}\nLet $\\Gamma_{h,B}(K;A)$ 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,\n\\begin{equation}\n\\mathbb E_A\\Gamma_{h,B}(K_{t+1};A)\n>\n\\mathbb E_A\\Gamma_{h,B}(K_t;A).\n\\end{equation}\nFinally, 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.","denote externally evaluated task utility. Ordinary task improvement requires\n\\begin{equation}\nJ(A_{t+1})>J(A_t).\n\\end{equation}\nLet $\\Gamma_{h,B}(K;A)$ 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,\n\\begin{equation}\n\\mathbb E_A\\Gamma_{h,B}(K_{t+1};A)\n>\n\\mathbb E_A\\Gamma_{h,B}(K_t;A).\n\\end{equation}\nFinally, 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.","{t+1})>J(A_t).\n\\end{equation}\nLet $\\Gamma_{h,B}(K;A)$ 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,\n\\begin{equation}\n\\mathbb E_A\\Gamma_{h,B}(K_{t+1};A)\n>\n\\mathbb E_A\\Gamma_{h,B}(K_t;A).\n\\end{equation}\nFinally, 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.","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,\n\\begin{equation}\n\\mathbb E_A\\Gamma_{h,B}(K_{t+1};A)\n>\n\\mathbb E_A\\Gamma_{h,B}(K_t;A).\n\\end{equation}\nFinally, 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.","inipage}{0.86\\textwidth}\n\\centering\n\\textbf{Recursive self-improvement is growth of an authorized repair calculus within an evidence-preserving lineage.}\n\\end{minipage}}\n\\end{center}\nThe 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\n\\begin{equation}\n\\begin{aligned}\n\\text{external outcome}\n&\\longrightarrow \\text{causal diagnosis}\n\\longrightarrow \\text{stratified defect}\\\\\n&\\longrightarrow \\text{local obstruction analysis}\n\\longrightarrow \\text{compatible repair construction}\\\\\n&\\longrightarrow \\text{sandboxed descendant}\n\\longrightarrow \\text{witness validation}\\\\\n&\\longrightarrow \\text{lineage commit or rollback}\n\\longrightarrow \\text{repair abstraction and transport}.\n\\end{aligned}\n\\end{equation}\nThe 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.","\\begin{definition}[Protected evaluation context]\nA protected evaluation context is a tuple\n\\begin{equation}\n\\Xi=\n\\left(\n\\Zcal,\n\\mu,\n\\Ycal,\n\\nu,\n\\bar c,\n\\lambda,\n\\mathscr I,\n\\Bcal,\n\\alpha\n\\right),\n\\end{equation}\nwhere:\n\\begin{enumerate}\n\\item $\\Zcal$ is a task or environment family and $\\mu$ is its held-out distribution;\n\\item $\\Ycal$ is the space of protected external traces;\n\\item $\\nu:\\Zcal\\times\\Ycal\\to[0,1]$ is task utility;\n\\item $\\bar c:\\Zcal\\times\\Ycal\\to[0,1]$ is normalized resource cost;\n\\item $\\lambda\\geq0$ is a cost weight;\n\\item $\\mathscr I(A,B)\\in\\{0,1\\}$ is a declared invariant-preservation predicate for the transition $A\\to B$;\n\\item $\\Bcal$ is the resource envelope used for candidate generation and evaluation;\n\\item $\\alpha\\in(0,1)$ is the accepted statistical error level.\n\\end{enumerate}\nThe evaluator may use only protected traces, precommitted predictions, and ledge","\\begin{definition}[Protected evaluation context]\nA protected evaluation context is a tuple\n\\begin{equation}\n\\Xi=\n\\left(\n\\Zcal,\n\\mu,\n\\Ycal,\n\\nu,\n\\bar c,\n\\lambda,\n\\mathscr I,\n\\Bcal,\n\\alpha\n\\right),\n\\end{equation}\nwhere:\n\\begin{enumerate}\n\\item $\\Zcal$ is a task or environment family and $\\mu$ is its held-out distribution;\n\\item $\\Ycal$ is the space of protected external traces;\n\\item $\\nu:\\Zcal\\times\\Ycal\\to[0,1]$ is task utility;\n\\item $\\bar c:\\Zcal\\times\\Ycal\\to[0,1]$ is normalized resource cost;\n\\item $\\lambda\\geq0$ is a cost weight;\n\\item $\\mathscr I(A,B)\\in\\{0,1\\}$ is a declared invariant-preservation predicate for the transition $A\\to B$;\n\\item $\\Bcal$ is the resource envelope used for candidate generation and evaluation;\n\\item $\\alpha\\in(0,1)$ is the accepted statistical error level.\n\\end{enumerate}\nThe evaluator may use only protected traces, precommitted predictions, and ledger entries authorized by the context. Candidate ident","\\begin{definition}[Protected evaluation context]\nA protected evaluation context is a tuple\n\\begin{equation}\n\\Xi=\n\\left(\n\\Zcal,\n\\mu,\n\\Ycal,\n\\nu,\n\\bar c,\n\\lambda,\n\\mathscr I,\n\\Bcal,\n\\alpha\n\\right),\n\\end{equation}\nwhere:\n\\begin{enumerate}\n\\item $\\Zcal$ is a task or environment family and $\\mu$ is its held-out distribution;\n\\item $\\Ycal$ is the space of protected external traces;\n\\item $\\nu:\\Zcal\\times\\Ycal\\to[0,1]$ is task utility;\n\\item $\\bar c:\\Zcal\\times\\Ycal\\to[0,1]$ is normalized resource cost;\n\\item $\\lambda\\geq0$ is a cost weight;\n\\item $\\mathscr I(A,B)\\in\\{0,1\\}$ is a declared invariant-preservation predicate for the transition $A\\to B$;\n\\item $\\Bcal$ is the resource envelope used for candidate generation and evaluation;\n\\item $\\alpha\\in(0,1)$ is the accepted statistical error level.\n\\end{enumerate}\nThe evaluator may use only protected traces, precommitted predictions, and ledger entries authorized by the context. Candidate identifiers and mutable self-reported scores ar","\\begin{definition}[Protected evaluation context]\nA protected evaluation context is a tuple\n\\begin{equation}\n\\Xi=\n\\left(\n\\Zcal,\n\\mu,\n\\Ycal,\n\\nu,\n\\bar c,\n\\lambda,\n\\mathscr I,\n\\Bcal,\n\\alpha\n\\right),\n\\end{equation}\nwhere:\n\\begin{enumerate}\n\\item $\\Zcal$ is a task or environment family and $\\mu$ is its held-out distribution;\n\\item $\\Ycal$ is the space of protected external traces;\n\\item $\\nu:\\Zcal\\times\\Ycal\\to[0,1]$ is task utility;\n\\item $\\bar c:\\Zcal\\times\\Ycal\\to[0,1]$ is normalized resource cost;\n\\item $\\lambda\\geq0$ is a cost weight;\n\\item $\\mathscr I(A,B)\\in\\{0,1\\}$ is a declared invariant-preservation predicate for the transition $A\\to B$;\n\\item $\\Bcal$ is the resource envelope used for candidate generation and evaluation;\n\\item $\\alpha\\in(0,1)$ is the accepted statistical error level.\n\\end{enumerate}\nThe 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.\n\\end{definition}","context]\nA protected evaluation context is a tuple\n\\begin{equation}\n\\Xi=\n\\left(\n\\Zcal,\n\\mu,\n\\Ycal,\n\\nu,\n\\bar c,\n\\lambda,\n\\mathscr I,\n\\Bcal,\n\\alpha\n\\right),\n\\end{equation}\nwhere:\n\\begin{enumerate}\n\\item $\\Zcal$ is a task or environment family and $\\mu$ is its held-out distribution;\n\\item $\\Ycal$ is the space of protected external traces;\n\\item $\\nu:\\Zcal\\times\\Ycal\\to[0,1]$ is task utility;\n\\item $\\bar c:\\Zcal\\times\\Ycal\\to[0,1]$ is normalized resource cost;\n\\item $\\lambda\\geq0$ is a cost weight;\n\\item $\\mathscr I(A,B)\\in\\{0,1\\}$ is a declared invariant-preservation predicate for the transition $A\\to B$;\n\\item $\\Bcal$ is the resource envelope used for candidate generation and evaluation;\n\\item $\\alpha\\in(0,1)$ is the accepted statistical error level.\n\\end{enumerate}\nThe 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.\n\\end{definition}","Xi=\n\\left(\n\\Zcal,\n\\mu,\n\\Ycal,\n\\nu,\n\\bar c,\n\\lambda,\n\\mathscr I,\n\\Bcal,\n\\alpha\n\\right),\n\\end{equation}\nwhere:\n\\begin{enumerate}\n\\item $\\Zcal$ is a task or environment family and $\\mu$ is its held-out distribution;\n\\item $\\Ycal$ is the space of protected external traces;\n\\item $\\nu:\\Zcal\\times\\Ycal\\to[0,1]$ is task utility;\n\\item $\\bar c:\\Zcal\\times\\Ycal\\to[0,1]$ is normalized resource cost;\n\\item $\\lambda\\geq0$ is a cost weight;\n\\item $\\mathscr I(A,B)\\in\\{0,1\\}$ is a declared invariant-preservation predicate for the transition $A\\to B$;\n\\item $\\Bcal$ is the resource envelope used for candidate generation and evaluation;\n\\item $\\alpha\\in(0,1)$ is the accepted statistical error level.\n\\end{enumerate}\nThe 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.\n\\end{definition}","c,\n\\lambda,\n\\mathscr I,\n\\Bcal,\n\\alpha\n\\right),\n\\end{equation}\nwhere:\n\\begin{enumerate}\n\\item $\\Zcal$ is a task or environment family and $\\mu$ is its held-out distribution;\n\\item $\\Ycal$ is the space of protected external traces;\n\\item $\\nu:\\Zcal\\times\\Ycal\\to[0,1]$ is task utility;\n\\item $\\bar c:\\Zcal\\times\\Ycal\\to[0,1]$ is normalized resource cost;\n\\item $\\lambda\\geq0$ is a cost weight;\n\\item $\\mathscr I(A,B)\\in\\{0,1\\}$ is a declared invariant-preservation predicate for the transition $A\\to B$;\n\\item $\\Bcal$ is the resource envelope used for candidate generation and evaluation;\n\\item $\\alpha\\in(0,1)$ is the accepted statistical error level.\n\\end{enumerate}\nThe 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.\n\\end{definition}","$\\Zcal$ is a task or environment family and $\\mu$ is its held-out distribution;\n\\item $\\Ycal$ is the space of protected external traces;\n\\item $\\nu:\\Zcal\\times\\Ycal\\to[0,1]$ is task utility;\n\\item $\\bar c:\\Zcal\\times\\Ycal\\to[0,1]$ is normalized resource cost;\n\\item $\\lambda\\geq0$ is a cost weight;\n\\item $\\mathscr I(A,B)\\in\\{0,1\\}$ is a declared invariant-preservation predicate for the transition $A\\to B$;\n\\item $\\Bcal$ is the resource envelope used for candidate generation and evaluation;\n\\item $\\alpha\\in(0,1)$ is the accepted statistical error level.\n\\end{enumerate}\nThe 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.\n\\end{definition}","k or environment family and $\\mu$ is its held-out distribution;\n\\item $\\Ycal$ is the space of protected external traces;\n\\item $\\nu:\\Zcal\\times\\Ycal\\to[0,1]$ is task utility;\n\\item $\\bar c:\\Zcal\\times\\Ycal\\to[0,1]$ is normalized resource cost;\n\\item $\\lambda\\geq0$ is a cost weight;\n\\item $\\mathscr I(A,B)\\in\\{0,1\\}$ is a declared invariant-preservation predicate for the transition $A\\to B$;\n\\item $\\Bcal$ is the resource envelope used for candidate generation and evaluation;\n\\item $\\alpha\\in(0,1)$ is the accepted statistical error level.\n\\end{enumerate}\nThe 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.\n\\end{definition}","space of protected external traces;\n\\item $\\nu:\\Zcal\\times\\Ycal\\to[0,1]$ is task utility;\n\\item $\\bar c:\\Zcal\\times\\Ycal\\to[0,1]$ is normalized resource cost;\n\\item $\\lambda\\geq0$ is a cost weight;\n\\item $\\mathscr I(A,B)\\in\\{0,1\\}$ is a declared invariant-preservation predicate for the transition $A\\to B$;\n\\item $\\Bcal$ is the resource envelope used for candidate generation and evaluation;\n\\item $\\alpha\\in(0,1)$ is the accepted statistical error level.\n\\end{enumerate}\nThe 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.\n\\end{definition}","For an agent $A$, define the resource-normalized utility\n\\begin{equation}\nJ_\\Xi(A)\n=\n\\mathbb E_{z\\sim\\mu}\n\\left[\n\\nu\\bigl(z,Y_A(z)\\bigr)\n-\n\\lambda\\bar c\\bigl(z,Y_A(z)\\bigr)\n\\right].\n\\end{equation}\nThe 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.","\\begin{definition}[Exact context-relative improvement]\nFor agents $A,B$, write\n\\begin{equation}\nB\\succeq_\\Xi A\n\\end{equation}\nwhen\n\\begin{equation}\n\\mathscr I(A,B)=1\n\\qquad\\text{and}\\qquad\nJ_\\Xi(B)\\geq J_\\Xi(A).\n\\end{equation}\nFor $\\Delta>0$, write $B\\succ_{\\Xi,\\Delta}A$ when the utility difference is at least $\\Delta$.\n\\end{definition}","\\begin{proposition}[Improvement preorder]\nIf $\\mathscr I$ is reflexive and transitive along composable lineage edges, then $\\succeq_\\Xi$ is a preorder on the set of agents connected by admissible transitions.\n\\end{proposition}","\\begin{proof}\nReflexivity 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 $J_\\Xi(C)\\geq J_\\Xi(B)\\geq J_\\Xi(A)$.\n\\end{proof}","\\begin{definition}[Statistically validated improvement]\nA candidate $B$ is a statistically validated $(\\Delta,\\alpha)$-improvement over $A$ when:\n\\begin{enumerate}\n\\item $\\mathscr I(A,B)=1$;\n\\item both systems are evaluated within the same resource envelope $\\Bcal$;\n\\item a predeclared lower confidence bound satisfies\n\\begin{equation}\n\\LCB_{1-\\alpha}\n\\left(J_\\Xi(B)-J_\\Xi(A)\\right)\n\\geq\\Delta;\n\\end{equation}\n\\item the evidence ledger used by the test is append-only across the transition;\n\\item the candidate did not receive post-reveal evaluator information unavailable to the baseline.\n\\end{enumerate}\n\\end{definition}","Let $\\Acal_t\\subseteq V(\\Lambda_t)$ be the executable archive at generation $t$. Its task frontier is\n\\begin{equation}\nF_t(\\Xi)=\\sup_{A\\in\\Acal_t}J_\\Xi(A).\n\\end{equation}\nLet $K_t$ be the active improvement mechanism, including diagnosis, proposal, composition, and validation policies. For horizon $h$ and budget $B$, define target-relative metaproductivity\n\\begin{equation}\n\\Gamma_{\\Xi,h,B}(K_t;A)\n=\n\\mathbb E\n\\left[\n\\sup_{A'\\in\\Desc_{h,B}(K_t,A)}\n\\left(J_\\Xi(A')-J_\\Xi(A)\\right)_+\n\\right].\n\\end{equation}\nFinally, 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.","(\\Xi)=\\sup_{A\\in\\Acal_t}J_\\Xi(A).\n\\end{equation}\nLet $K_t$ be the active improvement mechanism, including diagnosis, proposal, composition, and validation policies. For horizon $h$ and budget $B$, define target-relative metaproductivity\n\\begin{equation}\n\\Gamma_{\\Xi,h,B}(K_t;A)\n=\n\\mathbb E\n\\left[\n\\sup_{A'\\in\\Desc_{h,B}(K_t,A)}\n\\left(J_\\Xi(A')-J_\\Xi(A)\\right)_+\n\\right].\n\\end{equation}\nFinally, 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.","\\begin{definition}[Bounded recursive improvement step]\nA lineage transition $\\mathfrak L_t\\to\\mathfrak L_{t+1}$ is a bounded recursive improvement step relative to $(\\Xi,h,B)$ when:\n\\begin{enumerate}\n\\item its lineage receipt is valid and the protected evidence ledger embeds into the next ledger;\n\\item $F_{t+1}(\\Xi)\\geq F_t(\\Xi)$;\n\\item on a foreign held-out target distribution $\\mathcal D_{\\mathrm{foreign}}$,\n\\begin{equation}\n\\mathbb E_{A\\sim\\mathcal D_{\\mathrm{foreign}}}\n\\Gamma_{\\Xi,h,B}(K_{t+1};A)\n>\n\\mathbb E_{A\\sim\\mathcal D_{\\mathrm{foreign}}}\n\\Gamma_{\\Xi,h,B}(K_t;A);\n\\end{equation}\n\\item evaluator risk does not increase beyond a declared tolerance;\n\\item the improvement is obtained under matched model access, candidate count, compute, tools, and external evidence.\n\\end{enumerate}\n\\end{definiti","\\begin{definition}[Bounded recursive improvement step]\nA lineage transition $\\mathfrak L_t\\to\\mathfrak L_{t+1}$ is a bounded recursive improvement step relative to $(\\Xi,h,B)$ when:\n\\begin{enumerate}\n\\item its lineage receipt is valid and the protected evidence ledger embeds into the next ledger;\n\\item $F_{t+1}(\\Xi)\\geq F_t(\\Xi)$;\n\\item on a foreign held-out target distribution $\\mathcal D_{\\mathrm{foreign}}$,\n\\begin{equation}\n\\mathbb E_{A\\sim\\mathcal D_{\\mathrm{foreign}}}\n\\Gamma_{\\Xi,h,B}(K_{t+1};A)\n>\n\\mathbb E_{A\\sim\\mathcal D_{\\mathrm{foreign}}}\n\\Gamma_{\\Xi,h,B}(K_t;A);\n\\end{equation}\n\\item evaluator risk does not increase beyond a declared tolerance;\n\\item the improvement is obtained under matched model access, candidate count, compute, tools, and external evidence.\n\\end{enumerate}\n\\end{definition}","ursive improvement step]\nA lineage transition $\\mathfrak L_t\\to\\mathfrak L_{t+1}$ is a bounded recursive improvement step relative to $(\\Xi,h,B)$ when:\n\\begin{enumerate}\n\\item its lineage receipt is valid and the protected evidence ledger embeds into the next ledger;\n\\item $F_{t+1}(\\Xi)\\geq F_t(\\Xi)$;\n\\item on a foreign held-out target distribution $\\mathcal D_{\\mathrm{foreign}}$,\n\\begin{equation}\n\\mathbb E_{A\\sim\\mathcal D_{\\mathrm{foreign}}}\n\\Gamma_{\\Xi,h,B}(K_{t+1};A)\n>\n\\mathbb E_{A\\sim\\mathcal D_{\\mathrm{foreign}}}\n\\Gamma_{\\Xi,h,B}(K_t;A);\n\\end{equation}\n\\item evaluator risk does not increase beyond a declared tolerance;\n\\item the improvement is obtained under matched model access, candidate count, compute, tools, and external evidence.\n\\end{enumerate}\n\\end{definition}","\\begin{proposition}[Evaluator-capture triviality]\n\\label{prop:evaluator-capture}\nLet $\\mathcal S$ 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.\n\\end{proposition}","\\begin{proof}\nDefine $V_B(B)=1$ and $V_B(A)=0$ for every $A\\neq B$.\n\\end{proof}","\\begin{proposition}[Fixed-margin evaluator ceiling]\n\\label{prop:fixed-evaluator-ceiling}\nLet $V:\\mathcal S\\to[0,1]$ be fixed. If every accepted transition satisfies\n\\begin{equation}\nV(A_{t+1})\\geq V(A_t)+\\Delta\n\\end{equation}\nfor a constant $\\Delta>0$, then every accepted chain has length at most $\\lfloor 1/\\Delta\\rfloor+1$.\n\\end{proposition}","\\begin{proof}\nAfter $n$ strict transitions,\n\\begin{equation}\nV(A_n)\\geq V(A_0)+n\\Delta\\geq n\\Delta.\n\\end{equation}\nSince $V(A_n)\\leq1$, one has $n\\leq1/\\Delta$.\n\\end{proof}","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 $Y$,\n\\begin{equation}\nV(I,Y)=V(I',Y)\n\\end{equation}\nfor 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.","\\begin{definition}[Stratified functional atlas]\nA stratified functional atlas at generation $t$ is a tuple\n\\begin{equation}\n\\mathfrak A_t=\n\\left(\n\\mathsf C,\nG_t,\n\\{X_t^c\\}_{c\\in\\mathsf C},\n\\{F_t^c\\}_{c\\in\\mathsf C},\n\\{T_e,\\tau_e,\\Pcal_e,\\bm\\eps_e\\}_{e\\in E(G_t)}\n\\right),\n\\end{equation}\nwhere:\n\\begin{enumerate}\n\\item $\\mathsf C$ is a finite set of stratum colors;\n\\item $G_t$ is a directed multigraph of declared interfaces;\n\\item $X_t^c$ is the state space of stratum $c$;\n\\item $F_t^c$ is its local transition rule;\n\\item each edge $e:c\\to d$ carries a bridge $T_e:X_t^c\\to X_t^d$;\n\\item $\\tau_e$ declares the bridge class;\n\\item $\\Pcal_e$ is its preservation profile;\n\\item $\\bm\\eps_e$ is a vector of measured or certified approximation defects.\n\\end{enumerate}\n\\end{definition}","ctional atlas]\nA stratified functional atlas at generation $t$ is a tuple\n\\begin{equation}\n\\mathfrak A_t=\n\\left(\n\\mathsf C,\nG_t,\n\\{X_t^c\\}_{c\\in\\mathsf C},\n\\{F_t^c\\}_{c\\in\\mathsf C},\n\\{T_e,\\tau_e,\\Pcal_e,\\bm\\eps_e\\}_{e\\in E(G_t)}\n\\right),\n\\end{equation}\nwhere:\n\\begin{enumerate}\n\\item $\\mathsf C$ is a finite set of stratum colors;\n\\item $G_t$ is a directed multigraph of declared interfaces;\n\\item $X_t^c$ is the state space of stratum $c$;\n\\item $F_t^c$ is its local transition rule;\n\\item each edge $e:c\\to d$ carries a bridge $T_e:X_t^c\\to X_t^d$;\n\\item $\\tau_e$ declares the bridge class;\n\\item $\\Pcal_e$ is its preservation profile;\n\\item $\\bm\\eps_e$ is a vector of measured or certified approximation defects.\n\\end{enumerate}\n\\end{definition}","atlas at generation $t$ is a tuple\n\\begin{equation}\n\\mathfrak A_t=\n\\left(\n\\mathsf C,\nG_t,\n\\{X_t^c\\}_{c\\in\\mathsf C},\n\\{F_t^c\\}_{c\\in\\mathsf C},\n\\{T_e,\\tau_e,\\Pcal_e,\\bm\\eps_e\\}_{e\\in E(G_t)}\n\\right),\n\\end{equation}\nwhere:\n\\begin{enumerate}\n\\item $\\mathsf C$ is a finite set of stratum colors;\n\\item $G_t$ is a directed multigraph of declared interfaces;\n\\item $X_t^c$ is the state space of stratum $c$;\n\\item $F_t^c$ is its local transition rule;\n\\item each edge $e:c\\to d$ carries a bridge $T_e:X_t^c\\to X_t^d$;\n\\item $\\tau_e$ declares the bridge class;\n\\item $\\Pcal_e$ is its preservation profile;\n\\item $\\bm\\eps_e$ is a vector of measured or certified approximation defects.\n\\end{enumerate}\n\\end{definition}","eneration $t$ is a tuple\n\\begin{equation}\n\\mathfrak A_t=\n\\left(\n\\mathsf C,\nG_t,\n\\{X_t^c\\}_{c\\in\\mathsf C},\n\\{F_t^c\\}_{c\\in\\mathsf C},\n\\{T_e,\\tau_e,\\Pcal_e,\\bm\\eps_e\\}_{e\\in E(G_t)}\n\\right),\n\\end{equation}\nwhere:\n\\begin{enumerate}\n\\item $\\mathsf C$ is a finite set of stratum colors;\n\\item $G_t$ is a directed multigraph of declared interfaces;\n\\item $X_t^c$ is the state space of stratum $c$;\n\\item $F_t^c$ is its local transition rule;\n\\item each edge $e:c\\to d$ carries a bridge $T_e:X_t^c\\to X_t^d$;\n\\item $\\tau_e$ declares the bridge class;\n\\item $\\Pcal_e$ is its preservation profile;\n\\item $\\bm\\eps_e$ is a vector of measured or certified approximation defects.\n\\end{enumerate}\n\\end{definition}","t=\n\\left(\n\\mathsf C,\nG_t,\n\\{X_t^c\\}_{c\\in\\mathsf C},\n\\{F_t^c\\}_{c\\in\\mathsf C},\n\\{T_e,\\tau_e,\\Pcal_e,\\bm\\eps_e\\}_{e\\in E(G_t)}\n\\right),\n\\end{equation}\nwhere:\n\\begin{enumerate}\n\\item $\\mathsf C$ is a finite set of stratum colors;\n\\item $G_t$ is a directed multigraph of declared interfaces;\n\\item $X_t^c$ is the state space of stratum $c$;\n\\item $F_t^c$ is its local transition rule;\n\\item each edge $e:c\\to d$ carries a bridge $T_e:X_t^c\\to X_t^d$;\n\\item $\\tau_e$ declares the bridge class;\n\\item $\\Pcal_e$ is its preservation profile;\n\\item $\\bm\\eps_e$ is a vector of measured or certified approximation defects.\n\\end{enumerate}\n\\end{definition}","X_t^c\\}_{c\\in\\mathsf C},\n\\{F_t^c\\}_{c\\in\\mathsf C},\n\\{T_e,\\tau_e,\\Pcal_e,\\bm\\eps_e\\}_{e\\in E(G_t)}\n\\right),\n\\end{equation}\nwhere:\n\\begin{enumerate}\n\\item $\\mathsf C$ is a finite set of stratum colors;\n\\item $G_t$ is a directed multigraph of declared interfaces;\n\\item $X_t^c$ is the state space of stratum $c$;\n\\item $F_t^c$ is its local transition rule;\n\\item each edge $e:c\\to d$ carries a bridge $T_e:X_t^c\\to X_t^d$;\n\\item $\\tau_e$ declares the bridge class;\n\\item $\\Pcal_e$ is its preservation profile;\n\\item $\\bm\\eps_e$ is a vector of measured or certified approximation defects.\n\\end{enumerate}\n\\end{definition}","_t^c\\}_{c\\in\\mathsf C},\n\\{T_e,\\tau_e,\\Pcal_e,\\bm\\eps_e\\}_{e\\in E(G_t)}\n\\right),\n\\end{equation}\nwhere:\n\\begin{enumerate}\n\\item $\\mathsf C$ is a finite set of stratum colors;\n\\item $G_t$ is a directed multigraph of declared interfaces;\n\\item $X_t^c$ is the state space of stratum $c$;\n\\item $F_t^c$ is its local transition rule;\n\\item each edge $e:c\\to d$ carries a bridge $T_e:X_t^c\\to X_t^d$;\n\\item $\\tau_e$ declares the bridge class;\n\\item $\\Pcal_e$ is its preservation profile;\n\\item $\\bm\\eps_e$ is a vector of measured or certified approximation defects.\n\\end{enumerate}\n\\end{definition}","_e,\\bm\\eps_e\\}_{e\\in E(G_t)}\n\\right),\n\\end{equation}\nwhere:\n\\begin{enumerate}\n\\item $\\mathsf C$ is a finite set of stratum colors;\n\\item $G_t$ is a directed multigraph of declared interfaces;\n\\item $X_t^c$ is the state space of stratum $c$;\n\\item $F_t^c$ is its local transition rule;\n\\item each edge $e:c\\to d$ carries a bridge $T_e:X_t^c\\to X_t^d$;\n\\item $\\tau_e$ declares the bridge class;\n\\item $\\Pcal_e$ is its preservation profile;\n\\item $\\bm\\eps_e$ is a vector of measured or certified approximation defects.\n\\end{enumerate}\n\\end{definition}","uation}\nwhere:\n\\begin{enumerate}\n\\item $\\mathsf C$ is a finite set of stratum colors;\n\\item $G_t$ is a directed multigraph of declared interfaces;\n\\item $X_t^c$ is the state space of stratum $c$;\n\\item $F_t^c$ is its local transition rule;\n\\item each edge $e:c\\to d$ carries a bridge $T_e:X_t^c\\to X_t^d$;\n\\item $\\tau_e$ declares the bridge class;\n\\item $\\Pcal_e$ is its preservation profile;\n\\item $\\bm\\eps_e$ is a vector of measured or certified approximation defects.\n\\end{enumerate}\n\\end{definition}","A representative color set is\n\\begin{equation}\n\\mathsf C=\n\\set{\n\\mathrm{model},\n\\mathrm{workspace},\n\\mathrm{memory},\n\\mathrm{planner},\n\\mathrm{improver},\n\\mathrm{evaluator},\n\\mathrm{provenance},\n\\mathrm{tools}\n}.\n\\end{equation}\nThe atlas does not assert that these colors are universally correct. It declares the decomposition under which one particular repair theory is being tested.","\\midrule\n\\endhead\nExact equivalence & $X\\cong Y$ & Bidirectional transport of every declared preserved construction.","Embedding & $X\\hookrightarrow Y$ & Forward transport inside an identified subobject; no automatic inverse.","Quotient or forgetful map & $X\\twoheadrightarrow Y$ & Forward transport with declared information loss; reconstruction requires additional data.","Analysis-synthesis pair & $X\\xrightarrow{T}Y\\xrightarrow{R}X$ & Approximate reconstruction under an explicit residual $\\norm{RT-I}$.","Stochastic channel & $K:X\\rightsquigarrow Y$ & Distributional transport only; pointwise identity is not implied.","Heuristic correspondence & $X\\dashrightarrow Y$ & Candidate analogy with no theorem transport until separately certified.","\\begin{definition}[Preservation profile]\nA preservation profile for a bridge $T:X\\to Y$ is a declared record\n\\begin{equation}\n\\Pcal(T)=\n\\left(\n\\Sigma_T,\n\\mathcal Q_T,\n\\rho_T,\n\\ell_T,\n\\sigma_T,\n\\mathcal A_T\n\\right),\n\\end{equation}\nwhere $\\Sigma_T$ 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.\n\\end{definition}","reservation profile]\nA preservation profile for a bridge $T:X\\to Y$ is a declared record\n\\begin{equation}\n\\Pcal(T)=\n\\left(\n\\Sigma_T,\n\\mathcal Q_T,\n\\rho_T,\n\\ell_T,\n\\sigma_T,\n\\mathcal A_T\n\\right),\n\\end{equation}\nwhere $\\Sigma_T$ 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.\n\\end{definition}","r a bridge $T:X\\to Y$ is a declared record\n\\begin{equation}\n\\Pcal(T)=\n\\left(\n\\Sigma_T,\n\\mathcal Q_T,\n\\rho_T,\n\\ell_T,\n\\sigma_T,\n\\mathcal A_T\n\\right),\n\\end{equation}\nwhere $\\Sigma_T$ 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.\n\\end{definition}","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.","\\begin{definition}[Property-transport defect]\nFor a declared domain $K\\subseteq X$, define\n\\begin{equation}\n\\eps_Q(T;K)\n=\n\\sup_{x\\in K}\n\\norm{\nQ_Y(Tx)-\\widetilde T_QQ_X(x)\n}.\n\\end{equation}\n\\end{definition}","\\begin{proposition}[Composition bound for bridge defects]\n\\label{prop:bridge-composition}\nLet $T:X\\to Y$ and $S:Y\\to Z$. Suppose the property transports satisfy\n\\begin{equation}\n\\norm{Q_Y(Tx)-\\widetilde T_QQ_X(x)}\\leq\\eps_T\n\\end{equation}\nfor $x\\in K$, and\n\\begin{equation}\n\\norm{Q_Z(Sy)-\\widetilde S_QQ_Y(y)}\\leq\\eps_S\n\\end{equation}\nfor $y\\in T(K)$. If $\\norm{\\widetilde S_Q}\\leq L_S$, then the composite bridge obeys\n\\begin{equation}\n\\eps_Q(S\\circ T;K)\n\\leq\n\\eps_S+L_S\\eps_T\n\\end{equation}\nwith intended property transport $\\widetilde S_Q\\widetilde T_Q$.\n\\end{proposition}","d for bridge defects]\n\\label{prop:bridge-composition}\nLet $T:X\\to Y$ and $S:Y\\to Z$. Suppose the property transports satisfy\n\\begin{equation}\n\\norm{Q_Y(Tx)-\\widetilde T_QQ_X(x)}\\leq\\eps_T\n\\end{equation}\nfor $x\\in K$, and\n\\begin{equation}\n\\norm{Q_Z(Sy)-\\widetilde S_QQ_Y(y)}\\leq\\eps_S\n\\end{equation}\nfor $y\\in T(K)$. If $\\norm{\\widetilde S_Q}\\leq L_S$, then the composite bridge obeys\n\\begin{equation}\n\\eps_Q(S\\circ T;K)\n\\leq\n\\eps_S+L_S\\eps_T\n\\end{equation}\nwith intended property transport $\\widetilde S_Q\\widetilde T_Q$.\n\\end{proposition}","e the property transports satisfy\n\\begin{equation}\n\\norm{Q_Y(Tx)-\\widetilde T_QQ_X(x)}\\leq\\eps_T\n\\end{equation}\nfor $x\\in K$, and\n\\begin{equation}\n\\norm{Q_Z(Sy)-\\widetilde S_QQ_Y(y)}\\leq\\eps_S\n\\end{equation}\nfor $y\\in T(K)$. If $\\norm{\\widetilde S_Q}\\leq L_S$, then the composite bridge obeys\n\\begin{equation}\n\\eps_Q(S\\circ T;K)\n\\leq\n\\eps_S+L_S\\eps_T\n\\end{equation}\nwith intended property transport $\\widetilde S_Q\\widetilde T_Q$.\n\\end{proposition}","\\begin{proof}\nFor $x\\in K$,\n\\begin{align}\n&\\norm{Q_Z(STx)-\\widetilde S_Q\\widetilde T_QQ_X(x)}","\\begin{proof}\nFor $x\\in K$,\n\\begin{align}\n&\\norm{Q_Z(STx)-\\widetilde S_Q\\widetilde T_QQ_X(x)}\\\\\n&\\quad\\leq\n\\norm{Q_Z(STx)-\\widetilde S_QQ_Y(Tx)}\n+\n\\norm{\\widetilde S_Q\\left(Q_Y(Tx)-\\widetilde T_QQ_X(x)\\right)}\\\\\n&\\quad\\leq\n\\eps_S+L_S\\eps_T.\n\\end{align}\nTaking the supremum proves the claim.\n\\end{proof}","\\begin{definition}[Resolved improvement lineage state]\nA resolved improvement lineage state at generation $t$ is a tuple\n\\begin{equation}\n\\mathfrak L_t\n=\n\\left(\nA_t,\n\\Ocal_t,\n\\Ecal_t,\n\\Vcal_t,\n\\Lambda_t,\n\\Wcal_t\n\\right),\n\\end{equation}\nwhere:\n\\begin{enumerate}\n\\item $A_t$ is the active stratified agent with atlas $\\mathfrak A_t$;\n\\item $\\Ocal_t$ is a proof-carrying repair operad;\n\\item $\\Ecal_t$ is an append-only external-evidence ledger;\n\\item $\\Vcal_t$ is an evaluator atlas;\n\\item $\\Lambda_t$ is a directed acyclic lineage graph whose edges carry transformation receipts;\n\\item $\\Wcal_t$ is a witness, checkpoint, sandbox, and rollback structure.\n\\end{enumerate}\n\\end{definition}","ate]\nA resolved improvement lineage state at generation $t$ is a tuple\n\\begin{equation}\n\\mathfrak L_t\n=\n\\left(\nA_t,\n\\Ocal_t,\n\\Ecal_t,\n\\Vcal_t,\n\\Lambda_t,\n\\Wcal_t\n\\right),\n\\end{equation}\nwhere:\n\\begin{enumerate}\n\\item $A_t$ is the active stratified agent with atlas $\\mathfrak A_t$;\n\\item $\\Ocal_t$ is a proof-carrying repair operad;\n\\item $\\Ecal_t$ is an append-only external-evidence ledger;\n\\item $\\Vcal_t$ is an evaluator atlas;\n\\item $\\Lambda_t$ is a directed acyclic lineage graph whose edges carry transformation receipts;\n\\item $\\Wcal_t$ is a witness, checkpoint, sandbox, and rollback structure.\n\\end{enumerate}\n\\end{definition}","te at generation $t$ is a tuple\n\\begin{equation}\n\\mathfrak L_t\n=\n\\left(\nA_t,\n\\Ocal_t,\n\\Ecal_t,\n\\Vcal_t,\n\\Lambda_t,\n\\Wcal_t\n\\right),\n\\end{equation}\nwhere:\n\\begin{enumerate}\n\\item $A_t$ is the active stratified agent with atlas $\\mathfrak A_t$;\n\\item $\\Ocal_t$ is a proof-carrying repair operad;\n\\item $\\Ecal_t$ is an append-only external-evidence ledger;\n\\item $\\Vcal_t$ is an evaluator atlas;\n\\item $\\Lambda_t$ is a directed acyclic lineage graph whose edges carry transformation receipts;\n\\item $\\Wcal_t$ is a witness, checkpoint, sandbox, and rollback structure.\n\\end{enumerate}\n\\end{definition}","cal_t,\n\\Lambda_t,\n\\Wcal_t\n\\right),\n\\end{equation}\nwhere:\n\\begin{enumerate}\n\\item $A_t$ is the active stratified agent with atlas $\\mathfrak A_t$;\n\\item $\\Ocal_t$ is a proof-carrying repair operad;\n\\item $\\Ecal_t$ is an append-only external-evidence ledger;\n\\item $\\Vcal_t$ is an evaluator atlas;\n\\item $\\Lambda_t$ is a directed acyclic lineage graph whose edges carry transformation receipts;\n\\item $\\Wcal_t$ is a witness, checkpoint, sandbox, and rollback structure.\n\\end{enumerate}\n\\end{definition}","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.","\\begin{definition}[Append-only evidence embedding]\nEvidence preservation from generation $t$ to $t+1$ is an injective map\n\\begin{equation}\n\\iota_t:\\Ecal_t\\hookrightarrow\\Ecal_{t+1}\n\\end{equation}\nthat 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.\n\\end{definition}","A witness structure may be represented as\n\\begin{equation}\n\\Wcal_t\n=\n\\left(\n\\Vcal_t^{\\mathrm{fr}},\nC_t,\n\\Bcal_t^{\\mathrm{sand}},\nR_t,\n\\Pi_t^{\\mathrm{auth}}\n\\right),\n\\end{equation}\nwhere $\\Vcal_t^{\\mathrm{fr}}$ 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.","agent $A_t$};\n\\node[box, above=of agent] (lineage) {lineage graph and","archive $\\Lambda_t,\\Acal_t$};\n\\node[box, left=of agent] (repair) {repair operad","$\\Ocal_t$};\n\\node[box, right=of agent] (eval) {evaluator atlas","$\\Vcal_t$};\n\\node[box, below left=of agent] (evidence) {evidence ledger","$\\Ecal_t$};\n\\node[box, below right=of agent] (witness) {witness and rollback","$\\Wcal_t$};\n\\draw[arr] (repair) -- (agent);\n\\draw[arr] (agent) -- (lineage);\n\\draw[arr] (lineage) -- (eval);\n\\draw[arr] (evidence) -- (eval);\n\\draw[arr] (eval) -- (witness);\n\\draw[arr] (witness) -- (agent);\n\\draw[arr, bend left=18] (agent) to (evidence);\n\\draw[arr, bend left=18] (lineage) to (repair);\n\\end{tikzpicture}\n\\caption{A resolved lineage state. No component alone is the persistent improving object; persistence is carried by their proof-carrying closure.}\n\\label{fig:resolved-lineage}\n\\end{figure}","\\begin{definition}[Lineage-edge receipt]\nA lineage edge\n\\begin{equation}\nA\\xrightarrow{\\Pi}B\n\\end{equation}\nhas a receipt\n\\begin{equation}\n\\Pi=\n\\left(\n\\mathrm{id}_A,\n\\mathrm{id}_B,\n\\theta,\n\\Wcal,\n\\Ecal_{\\mathrm{pre}},\n\\Ecal_{\\mathrm{post}},\n\\widehat g,\n\\mathscr I,\n\\mathrm{tests},\n\\mathrm{decision},\n\\mathrm{rollback},\n\\mathrm{digest}\n\\right),\n\\end{equation}\nrecording 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.\n\\end{definition}","definition}[Lineage-edge receipt]\nA lineage edge\n\\begin{equation}\nA\\xrightarrow{\\Pi}B\n\\end{equation}\nhas a receipt\n\\begin{equation}\n\\Pi=\n\\left(\n\\mathrm{id}_A,\n\\mathrm{id}_B,\n\\theta,\n\\Wcal,\n\\Ecal_{\\mathrm{pre}},\n\\Ecal_{\\mathrm{post}},\n\\widehat g,\n\\mathscr I,\n\\mathrm{tests},\n\\mathrm{decision},\n\\mathrm{rollback},\n\\mathrm{digest}\n\\right),\n\\end{equation}\nrecording 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.\n\\end{definition}","}B\n\\end{equation}\nhas a receipt\n\\begin{equation}\n\\Pi=\n\\left(\n\\mathrm{id}_A,\n\\mathrm{id}_B,\n\\theta,\n\\Wcal,\n\\Ecal_{\\mathrm{pre}},\n\\Ecal_{\\mathrm{post}},\n\\widehat g,\n\\mathscr I,\n\\mathrm{tests},\n\\mathrm{decision},\n\\mathrm{rollback},\n\\mathrm{digest}\n\\right),\n\\end{equation}\nrecording 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.\n\\end{definition}","\\begin{definition}[Validated continuation]\nA 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.\n\\end{definition}","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.","\\begin{theorem}[Category of validated lineage paths]\n\\label{thm:lineage-category}\nAssume 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 $\\mathbf{Lin}$.\n\\end{theorem}","\\begin{proof}\nThe 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.\n\\end{proof}","A lineage may branch:\n\\begin{equation}\nA\\longrightarrow B_1,\n\\qquad\nA\\longrightarrow B_2.\n\\end{equation}\nNo definition of recursive improvement requires one child to be the unique continuation of the parent.","\\begin{theorem}[Symmetric forks do not justify unique selection]\n\\label{thm:fork-symmetry}\nLet $D$ 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\n\\begin{equation}\ns(D)\\subseteq\\{B_1,B_2\\}\n\\end{equation}\nand satisfy equivariance $s(\\sigma D)=\\sigma s(D)$. Then $s(D)$ cannot contain exactly one child.\n\\end{theorem}","ric forks do not justify unique selection]\n\\label{thm:fork-symmetry}\nLet $D$ 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\n\\begin{equation}\ns(D)\\subseteq\\{B_1,B_2\\}\n\\end{equation}\nand satisfy equivariance $s(\\sigma D)=\\sigma s(D)$. Then $s(D)$ cannot contain exactly one child.\n\\end{theorem}","selection]\n\\label{thm:fork-symmetry}\nLet $D$ 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\n\\begin{equation}\ns(D)\\subseteq\\{B_1,B_2\\}\n\\end{equation}\nand satisfy equivariance $s(\\sigma D)=\\sigma s(D)$. Then $s(D)$ cannot contain exactly one child.\n\\end{theorem}","\\begin{proof}\nBecause $\\sigma D=D$, 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.\n\\end{proof}","A lineage state is \\emph{resolution-consistent} when:\n\\begin{enumerate}\n\\item every active component belongs to a declared stratum;\n\\item every cross-stratum inference cites a typed bridge and preservation profile;\n\\item every mutable belief used by the planner carries a provenance pointer into $\\Ecal_t$ or is explicitly typed as simulation or hypothesis;\n\\item every repair operation belongs to $\\Ocal_t$ or is quarantined as an untyped proposal;\n\\item every committed descendant has a valid receipt and rollback point;\n\\item every archived best-score claim can be reconstructed from protected traces and the evaluator version that produced it.\n\\end{enumerate}\nResolution consistency does not guarantee correctness. It prevents silent type changes from masquerading as theorem transport or causal evidence.","Let $\\Dcal_t$ be an authorized modification family containing a no-op intervention $\\delta_0$. For $\\delta\\in\\Dcal_t$, let\n\\begin{equation}\nK_{A_t}^{\\delta}(\\cdot)\n=\nP\\left(Y\\in\\cdot\\mid \\operatorname{do}(\\delta),A_t\\right)\n\\end{equation}\nbe the protected external-outcome kernel. A compressed causal self-model $\\phi_t(A_t)$ induces a predicted kernel\n\\begin{equation}\n\\widehat K_{\\phi_t(A_t)}^{\\delta}(\\cdot).\n\\end{equation}\nThe model is evaluated on counterfactual consequences, not on its ability to reproduce hidden activations for their own sake.","\\begin{definition}[$\\eps$-intervention sufficiency]\nThe map $\\phi_t$ is $\\eps$-intervention-sufficient for $\\Dcal_t$ when\n\\begin{equation}\n\\sup_{\\delta\\in\\Dcal_t}\n\\TV\\left(\nK_{A_t}^{\\delta},\n\\widehat K_{\\phi_t(A_t)}^{\\delta}\n\\right)\n\\leq\\eps.\n\\end{equation}\n\\end{definition}","\\begin{theorem}[Decision regret from relative introspection]\n\\label{thm:introspection-regret}\nLet $u:Y\\to[0,1]$ be a bounded utility. Define\n\\begin{equation}\nV(\\delta)=\\mathbb E[u(Y)\\mid A_t,\\delta],\n\\qquad\n\\widehat V(\\delta)=\\mathbb E[u(Y)\\mid \\phi_t(A_t),\\delta].\n\\end{equation}\nIf $\\phi_t$ is $\\eps$-intervention-sufficient, then\n\\begin{equation}\n\\sup_{\\delta\\in\\Dcal_t}\n\\abs{V(\\delta)-\\widehat V(\\delta)}\\leq\\eps.\n\\end{equation}\nIf\n\\begin{equation}\n\\delta^*\\in\\argmax_{\\delta\\in\\Dcal_t}V(\\delta),\n\\qquad\n\\widehat\\delta\\in\\argmax_{\\delta\\in\\Dcal_t}\\widehat V(\\delta),\n\\end{equation}\nthen\n\\begin{equation}\nV(\\delta^*)-V(\\widehat\\delta)\\leq2\\eps.\n\\end{equation}\n\\end{theorem}","pection]\n\\label{thm:introspection-regret}\nLet $u:Y\\to[0,1]$ be a bounded utility. Define\n\\begin{equation}\nV(\\delta)=\\mathbb E[u(Y)\\mid A_t,\\delta],\n\\qquad\n\\widehat V(\\delta)=\\mathbb E[u(Y)\\mid \\phi_t(A_t),\\delta].\n\\end{equation}\nIf $\\phi_t$ is $\\eps$-intervention-sufficient, then\n\\begin{equation}\n\\sup_{\\delta\\in\\Dcal_t}\n\\abs{V(\\delta)-\\widehat V(\\delta)}\\leq\\eps.\n\\end{equation}\nIf\n\\begin{equation}\n\\delta^*\\in\\argmax_{\\delta\\in\\Dcal_t}V(\\delta),\n\\qquad\n\\widehat\\delta\\in\\argmax_{\\delta\\in\\Dcal_t}\\widehat V(\\delta),\n\\end{equation}\nthen\n\\begin{equation}\nV(\\delta^*)-V(\\widehat\\delta)\\leq2\\eps.\n\\end{equation}\n\\end{theorem}","\\widehat V(\\delta)=\\mathbb E[u(Y)\\mid \\phi_t(A_t),\\delta].\n\\end{equation}\nIf $\\phi_t$ is $\\eps$-intervention-sufficient, then\n\\begin{equation}\n\\sup_{\\delta\\in\\Dcal_t}\n\\abs{V(\\delta)-\\widehat V(\\delta)}\\leq\\eps.\n\\end{equation}\nIf\n\\begin{equation}\n\\delta^*\\in\\argmax_{\\delta\\in\\Dcal_t}V(\\delta),\n\\qquad\n\\widehat\\delta\\in\\argmax_{\\delta\\in\\Dcal_t}\\widehat V(\\delta),\n\\end{equation}\nthen\n\\begin{equation}\nV(\\delta^*)-V(\\widehat\\delta)\\leq2\\eps.\n\\end{equation}\n\\end{theorem}","\\begin{proof}\nFor 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\n\\begin{align}\nV(\\delta^*)-V(\\widehat\\delta)\n&\\leq\nV(\\delta^*)-\\widehat V(\\delta^*)\n+\\widehat V(\\widehat\\delta)-V(\\widehat\\delta)","\\begin{proof}\nFor 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\n\\begin{align}\nV(\\delta^*)-V(\\widehat\\delta)\n&\\leq\nV(\\delta^*)-\\widehat V(\\delta^*)\n+\\widehat V(\\widehat\\delta)-V(\\widehat\\delta)\\\\\n&\\leq2\\eps,\n\\end{align}\nwhere $\\widehat V(\\widehat\\delta)\\geq\\widehat V(\\delta^*)$ was used.\n\\end{proof}","&\\leq2\\eps,\n\\end{align}\nwhere $\\widehat V(\\widehat\\delta)\\geq\\widehat V(\\delta^*)$ was used.\n\\end{proof}","\\begin{corollary}[Margin-preserving ranking]\n\\label{cor:ranking-margin}\nIf\n\\begin{equation}\nV(\\delta_1)-V(\\delta_2)>2\\eps,\n\\end{equation}\nthen\n\\begin{equation}\n\\widehat V(\\delta_1)>\\widehat V(\\delta_2).\n\\end{equation}\nIn 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.\n\\end{corollary}","\\begin{proof}\nOne has\n\\begin{equation}\n\\widehat V(\\delta_1)-\\widehat V(\\delta_2)\n\\geq\nV(\\delta_1)-V(\\delta_2)-2\\eps>0.\n\\end{equation}\n\\end{proof}","\\begin{proposition}[Safe commit against a no-op baseline]\n\\label{prop:safe-commit}\nAssume the same uniform error bound and let $\\delta_0$ be the no-op. If a candidate $\\delta$ is committed only when\n\\begin{equation}\n\\widehat V(\\delta)-\\widehat V(\\delta_0)>2\\eps,\n\\end{equation}\nthen the true gain is positive:\n\\begin{equation}\nV(\\delta)-V(\\delta_0)>0.\n\\end{equation}\n\\end{proposition}","\\begin{proof}\nThe true gain is bounded below by\n\\begin{equation}\nV(\\delta)-V(\\delta_0)\n\\geq\n\\widehat V(\\delta)-\\widehat V(\\delta_0)-2\\eps>0.\n\\end{equation}\n\\end{proof}","\\begin{corollary}[Cost-sensitive regret]\nSuppose, in addition, that normalized intervention cost $c(\\delta)$ is predicted by $\\widehat c(\\delta)$ with uniform error at most $\\eps_c$. For net utility\n\\begin{equation}\nW(\\delta)=V(\\delta)-\\lambda c(\\delta),\n\\qquad\n\\widehat W(\\delta)=\\widehat V(\\delta)-\\lambda\\widehat c(\\delta),\n\\end{equation}\nselection by $\\widehat W$ has regret at most\n\\begin{equation}\n2\\left(\\eps+\\lambda\\eps_c\\right).\n\\end{equation}\n\\end{corollary}","gin{corollary}[Cost-sensitive regret]\nSuppose, in addition, that normalized intervention cost $c(\\delta)$ is predicted by $\\widehat c(\\delta)$ with uniform error at most $\\eps_c$. For net utility\n\\begin{equation}\nW(\\delta)=V(\\delta)-\\lambda c(\\delta),\n\\qquad\n\\widehat W(\\delta)=\\widehat V(\\delta)-\\lambda\\widehat c(\\delta),\n\\end{equation}\nselection by $\\widehat W$ has regret at most\n\\begin{equation}\n2\\left(\\eps+\\lambda\\eps_c\\right).\n\\end{equation}\n\\end{corollary}","\\begin{proof}\nThe net-utility approximation error is at most $\\eps+\\lambda\\eps_c$. Apply the proof of \\cref{thm:introspection-regret} with this error bound.\n\\end{proof}","Define the intervention-coverage error\n\\begin{equation}\n\\eps_{\\Dcal}(\\phi;A)\n=\n\\sup_{\\delta\\in\\Dcal}\n\\TV\\left(K_A^\\delta,\\widehat K_{\\phi(A)}^\\delta\\right).\n\\end{equation}","\\begin{proposition}[Coverage monotonicity]\nIf $\\Dcal\\subseteq\\Dcal'$, then\n\\begin{equation}\n\\eps_{\\Dcal'}(\\phi;A)\n\\geq\n\\eps_{\\Dcal}(\\phi;A).\n\\end{equation}\n\\end{proposition}","\\begin{definition}[Intervention-complete relative to a repair grammar]\nLet $\\Ocal$ 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 $\\Dcal(\\Ocal)$.\n\\end{definition}","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 $K_h^p$.","\\begin{theorem}[Observational indistinguishability]\n\\label{thm:observational-indistinguishability}\nIf two hypotheses $h,h'\\in\\Hcal$ satisfy\n\\begin{equation}\nK_h^p=K_{h'}^p\n\\qquad\n\\text{for every permitted probe }p\\in\\Pcal_{\\mathrm{probe}},\n\\end{equation}\nthen no adaptive decision rule based only on permitted probe transcripts can distinguish $h$ from $h'$ with different transcript distributions.\n\\end{theorem}","\\begin{proof}\nProceed 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.\n\\end{proof}","\\begin{definition}[Deterministically separating probe family]\nSuppose 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\n\\begin{equation}\ny(h,p)\\neq y(h',p).\n\\end{equation}\n\\end{definition}","\\begin{theorem}[Finite active identification]\n\\label{thm:finite-active-identification}\nLet $\\Hcal$ 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.\n\\end{theorem}","\\begin{proof}\nMaintain 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.\n\\end{proof}","A useful self-model participates in a complete causal cycle:\n\\begin{equation}\n\\phi_t(A_t)\n\\xrightarrow{\\text{candidate policy}}\n\\delta_t\n\\xrightarrow{\\text{world execution}}\nY_{t+1}\n\\xrightarrow{\\text{evidence update}}\n\\phi_{t+1}(A_{t+1}).\n\\end{equation}\nThe 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.","\\begin{definition}[Prospectively closed self-model]\nA self-model process is prospectively closed relative to $\\Dcal$ when:\n\\begin{enumerate}\n\\item its state causally affects selection among interventions in $\\Dcal$;\n\\item it predicts protected consequences before reveal;\n\\item committed outcomes update the future self-model and candidate policy;\n\\item the update distinguishes direct observation, communication, simulation, and imported memory by provenance type.\n\\end{enumerate}\n\\end{definition}","\\begin{example}[Internal error reduction without external improvement]\nLet 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\n\\begin{equation}\nJ_{\\mathrm{int}}(p)=-(p-Y)^2.\n\\end{equation}\nA 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.\n\\end{example}","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\n\\begin{equation}\n\\overline\\Ocal(c_1,\\ldots,c_n;c).\n\\end{equation}\nAn operation consumes typed inputs from strata $c_1,\\ldots,c_n$ and produces a candidate correction in output stratum $c$.","\\begin{definition}[Proof-carrying repair schema]\nA repair schema is a tuple\n\\begin{equation}\n\\omega=\n\\left(\n\\sigma,\n\\bm c_{\\mathrm{in}},\n c_{\\mathrm{out}},\nD,\nC,\nQ,\n[r],\nu,\n\\bm g,\n\\mathscr I,\n\\Pi\n\\right),\n\\end{equation}\nwhere:\n\\begin{itemize}\n\\item $\\sigma$ is a recognizable failure signature;\n\\item $\\bm c_{\\mathrm{in}}=(c_1,\\ldots,c_n)$ and $c_{\\mathrm{out}}$ are input and output colors;\n\\item $D$ is the declared local defect-response map;\n\\item $C$ is an applicability precondition;\n\\item $Q$ is a postcondition;\n\\item $[r]$ is a local obstruction certificate or unresolved-obstruction record;\n\\item $\\nu$ is the correction program;\n\\item $\\bm g$ is a precommitted effect profile;\n\\item $\\mathscr I$ is a protected-invariant contract;\n\\item $\\Pi$ is a proof, evidence, provenance, and rollback receipt.\n\\end{itemize}\n\\end{definition}","epair schema is a tuple\n\\begin{equation}\n\\omega=\n\\left(\n\\sigma,\n\\bm c_{\\mathrm{in}},\n c_{\\mathrm{out}},\nD,\nC,\nQ,\n[r],\nu,\n\\bm g,\n\\mathscr I,\n\\Pi\n\\right),\n\\end{equation}\nwhere:\n\\begin{itemize}\n\\item $\\sigma$ is a recognizable failure signature;\n\\item $\\bm c_{\\mathrm{in}}=(c_1,\\ldots,c_n)$ and $c_{\\mathrm{out}}$ are input and output colors;\n\\item $D$ is the declared local defect-response map;\n\\item $C$ is an applicability precondition;\n\\item $Q$ is a postcondition;\n\\item $[r]$ is a local obstruction certificate or unresolved-obstruction record;\n\\item $\\nu$ is the correction program;\n\\item $\\bm g$ is a precommitted effect profile;\n\\item $\\mathscr I$ is a protected-invariant contract;\n\\item $\\Pi$ is a proof, evidence, provenance, and rollback receipt.\n\\end{itemize}\n\\end{definition}","ega=\n\\left(\n\\sigma,\n\\bm c_{\\mathrm{in}},\n c_{\\mathrm{out}},\nD,\nC,\nQ,\n[r],\nu,\n\\bm g,\n\\mathscr I,\n\\Pi\n\\right),\n\\end{equation}\nwhere:\n\\begin{itemize}\n\\item $\\sigma$ is a recognizable failure signature;\n\\item $\\bm c_{\\mathrm{in}}=(c_1,\\ldots,c_n)$ and $c_{\\mathrm{out}}$ are input and output colors;\n\\item $D$ is the declared local defect-response map;\n\\item $C$ is an applicability precondition;\n\\item $Q$ is a postcondition;\n\\item $[r]$ is a local obstruction certificate or unresolved-obstruction record;\n\\item $\\nu$ is the correction program;\n\\item $\\bm g$ is a precommitted effect profile;\n\\item $\\mathscr I$ is a protected-invariant contract;\n\\item $\\Pi$ is a proof, evidence, provenance, and rollback receipt.\n\\end{itemize}\n\\end{definition}","thrm{in}},\n c_{\\mathrm{out}},\nD,\nC,\nQ,\n[r],\nu,\n\\bm g,\n\\mathscr I,\n\\Pi\n\\right),\n\\end{equation}\nwhere:\n\\begin{itemize}\n\\item $\\sigma$ is a recognizable failure signature;\n\\item $\\bm c_{\\mathrm{in}}=(c_1,\\ldots,c_n)$ and $c_{\\mathrm{out}}$ are input and output colors;\n\\item $D$ is the declared local defect-response map;\n\\item $C$ is an applicability precondition;\n\\item $Q$ is a postcondition;\n\\item $[r]$ is a local obstruction certificate or unresolved-obstruction record;\n\\item $\\nu$ is the correction program;\n\\item $\\bm g$ is a precommitted effect profile;\n\\item $\\mathscr I$ is a protected-invariant contract;\n\\item $\\Pi$ is a proof, evidence, provenance, and rollback receipt.\n\\end{itemize}\n\\end{definition}","nd{equation}\nwhere:\n\\begin{itemize}\n\\item $\\sigma$ is a recognizable failure signature;\n\\item $\\bm c_{\\mathrm{in}}=(c_1,\\ldots,c_n)$ and $c_{\\mathrm{out}}$ are input and output colors;\n\\item $D$ is the declared local defect-response map;\n\\item $C$ is an applicability precondition;\n\\item $Q$ is a postcondition;\n\\item $[r]$ is a local obstruction certificate or unresolved-obstruction record;\n\\item $\\nu$ is the correction program;\n\\item $\\bm g$ is a precommitted effect profile;\n\\item $\\mathscr I$ is a protected-invariant contract;\n\\item $\\Pi$ is a proof, evidence, provenance, and rollback receipt.\n\\end{itemize}\n\\end{definition}","em $\\sigma$ is a recognizable failure signature;\n\\item $\\bm c_{\\mathrm{in}}=(c_1,\\ldots,c_n)$ and $c_{\\mathrm{out}}$ are input and output colors;\n\\item $D$ is the declared local defect-response map;\n\\item $C$ is an applicability precondition;\n\\item $Q$ is a postcondition;\n\\item $[r]$ is a local obstruction certificate or unresolved-obstruction record;\n\\item $\\nu$ is the correction program;\n\\item $\\bm g$ is a precommitted effect profile;\n\\item $\\mathscr I$ is a protected-invariant contract;\n\\item $\\Pi$ is a proof, evidence, provenance, and rollback receipt.\n\\end{itemize}\n\\end{definition}","\\item $\\bm c_{\\mathrm{in}}=(c_1,\\ldots,c_n)$ and $c_{\\mathrm{out}}$ are input and output colors;\n\\item $D$ is the declared local defect-response map;\n\\item $C$ is an applicability precondition;\n\\item $Q$ is a postcondition;\n\\item $[r]$ is a local obstruction certificate or unresolved-obstruction record;\n\\item $\\nu$ is the correction program;\n\\item $\\bm g$ is a precommitted effect profile;\n\\item $\\mathscr I$ is a protected-invariant contract;\n\\item $\\Pi$ is a proof, evidence, provenance, and rollback receipt.\n\\end{itemize}\n\\end{definition}","\\mathrm{out}}$ are input and output colors;\n\\item $D$ is the declared local defect-response map;\n\\item $C$ is an applicability precondition;\n\\item $Q$ is a postcondition;\n\\item $[r]$ is a local obstruction certificate or unresolved-obstruction record;\n\\item $\\nu$ is the correction program;\n\\item $\\bm g$ is a precommitted effect profile;\n\\item $\\mathscr I$ is a protected-invariant contract;\n\\item $\\Pi$ is a proof, evidence, provenance, and rollback receipt.\n\\end{itemize}\n\\end{definition}","A useful effect profile is vector-valued:\n\\begin{equation}\n\\bm g=\n\\left(\n\\Delta J,\n\\Delta C,\n\\Delta R,\n\\eps_{\\mathrm{leak}},\n\\eps_{\\mathrm{hol}},\n\\Delta\\Gamma,\n\\sigma_g\n\\right),\n\\end{equation}\nrecording 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.","\\begin{definition}[Proof-carrying repair operad]\nA proof-carrying repair operad is a pair\n\\begin{equation}\n\\Ocal=(\\overline\\Ocal,\\Valid),\n\\end{equation}\nwhere $\\overline\\Ocal$ is a colored symmetric operad of syntactically composable repair programs and\n\\begin{equation}\n\\Valid_{\\Ecal,\\Wcal}(\\omega,\\Pi)\n\\in\n\\{\\mathrm{valid},\\mathrm{invalid},\\mathrm{unresolved}\\}\n\\end{equation}\nis an evidence- and witness-indexed validation judgment. Only operations judged valid are executable. Unresolved operations remain quarantined.\n\\end{definition}","Each executable operation has a Hoare-style contract\n\\begin{equation}\n\\{C_\\omega\\}\\ \\omega\\ \\{Q_\\omega\\}.\n\\end{equation}\nSyntactic substitution\n\\begin{equation}\n\\omega\\circ(\\omega_1,\\ldots,\\omega_n)\n\\end{equation}\nis semantically authorized only when the inner postconditions and bridge guarantees imply the outer precondition.","\\begin{definition}[Receipt composition]\nLet $\\Pi_i$ validate $\\omega_i$, and let $\\Pi$ validate the outer operation $\\omega$ conditional on $C_\\omega$. The composed receipt\n\\begin{equation}\n\\Pi\\star(\\Pi_1,\\ldots,\\Pi_n)\n\\end{equation}\nis 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.\n\\end{definition}","\\begin{theorem}[Compositional receipt soundness]\n\\label{thm:receipt-soundness}\nAssume:\n\\begin{enumerate}\n\\item every primitive receipt is semantically sound;\n\\item execution of each $\\omega_i$ from $C_{\\omega_i}$ establishes $Q_{\\omega_i}$;\n\\item the conjunction of the inner postconditions and bridge guarantees implies $C_\\omega$;\n\\item execution of $\\omega$ from $C_\\omega$ establishes $Q_\\omega$;\n\\item receipt composition preserves provenance and protected evidence references.\n\\end{enumerate}\nThen the composed receipt is sound for\n\\begin{equation}\n\\omega\\circ(\\omega_1,\\ldots,\\omega_n)\n\\end{equation}\nwith precondition $\\bigwedge_iC_{\\omega_i}$ and postcondition $Q_\\omega$.\n\\end{theorem}","nd;\n\\item execution of each $\\omega_i$ from $C_{\\omega_i}$ establishes $Q_{\\omega_i}$;\n\\item the conjunction of the inner postconditions and bridge guarantees implies $C_\\omega$;\n\\item execution of $\\omega$ from $C_\\omega$ establishes $Q_\\omega$;\n\\item receipt composition preserves provenance and protected evidence references.\n\\end{enumerate}\nThen the composed receipt is sound for\n\\begin{equation}\n\\omega\\circ(\\omega_1,\\ldots,\\omega_n)\n\\end{equation}\nwith precondition $\\bigwedge_iC_{\\omega_i}$ and postcondition $Q_\\omega$.\n\\end{theorem}","_{\\omega_i}$;\n\\item the conjunction of the inner postconditions and bridge guarantees implies $C_\\omega$;\n\\item execution of $\\omega$ from $C_\\omega$ establishes $Q_\\omega$;\n\\item receipt composition preserves provenance and protected evidence references.\n\\end{enumerate}\nThen the composed receipt is sound for\n\\begin{equation}\n\\omega\\circ(\\omega_1,\\ldots,\\omega_n)\n\\end{equation}\nwith precondition $\\bigwedge_iC_{\\omega_i}$ and postcondition $Q_\\omega$.\n\\end{theorem}","nner postconditions and bridge guarantees implies $C_\\omega$;\n\\item execution of $\\omega$ from $C_\\omega$ establishes $Q_\\omega$;\n\\item receipt composition preserves provenance and protected evidence references.\n\\end{enumerate}\nThen the composed receipt is sound for\n\\begin{equation}\n\\omega\\circ(\\omega_1,\\ldots,\\omega_n)\n\\end{equation}\nwith precondition $\\bigwedge_iC_{\\omega_i}$ and postcondition $Q_\\omega$.\n\\end{theorem}","\\begin{proof}\nStarting 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.\n\\end{proof}","Let $U$ and $V$ be bounded linear correction operators. Applying the updates in opposite orders gives\n\\begin{equation}\n(I+U)(I+V)\n\\qquad\\text{and}\\qquad\n(I+V)(I+U).\n\\end{equation}","\\begin{proposition}[Exact order defect]\n\\label{prop:repair-order-defect}\nThe difference between the two update orders is the commutator:\n\\begin{equation}\n(I+U)(I+V)-(I+V)(I+U)\n=\n[U,V]:=UV-VU.\n\\end{equation}\nHence the operator-norm order defect is exactly $\\norm{[U,V]}$.\n\\end{proposition}","\\begin{proof}\nExpand both products and cancel $I$, $U$, and $V$.\n\\end{proof}","\\begin{proposition}[Orthogonal packet repairs commute]\n\\label{prop:orthogonal-repairs}\nLet $P$ and $Q$ be orthogonal projectors with $PQ=QP=0$. If\n\\begin{equation}\nU=PUP,\n\\qquad\nV=QVQ,\n\\end{equation}\nthen\n\\begin{equation}\nUV=VU=0.\n\\end{equation}\nConsequently the repairs commute exactly and\n\\begin{equation}\n(I+U)(I+V)=I+U+V.\n\\end{equation}\n\\end{proposition}","\\begin{proof}\nOne has\n\\begin{equation}\nUV=PUPQVQ=0,\n\\qquad\nVU=QVQPUP=0.\n\\end{equation}\n\\end{proof}","\\begin{definition}[Conservative repair-operad extension]\nAn inclusion\n\\begin{equation}\n\\iota_t:\\Ocal_t\\hookrightarrow\\Ocal_{t+1}\n\\end{equation}\nis conservative when:\n\\begin{enumerate}\n\\item it is injective on old colors and operations;\n\\item it preserves identities, permutations, and every old syntactic composition;\n\\item it preserves old preconditions, postconditions, and protected-invariant contracts on old states;\n\\item every previously valid old receipt remains valid against the same immutable evidence;\n\\item new operations cannot retroactively consume or rewrite old observation authority.\n\\end{enumerate}\n\\end{definition}","\\begin{proposition}[Old derivations survive conservative extension]\nEvery valid repair expression and receipt derivation in $\\Ocal_t$ remains valid after embedding into $\\Ocal_{t+1}$.\n\\end{proposition}","\\begin{definition}[Repair episode]\nA repair episode is a tuple\n\\begin{equation}\ne=\n\\left(\nA,\nr,\nu,\nY_{\\mathrm{pre}},\nY_{\\mathrm{post}},\n\\Pi,\n\\bm g\n\\right).\n\\end{equation}\n\\end{definition}","\\begin{definition}[Repair abstraction operator]\nA repair abstraction operator is a map\n\\begin{equation}\n\\mathfrak a:\n\\mathcal H_{\\mathrm{episodes}}\n\\longrightarrow\n\\overline\\Ocal\n\\end{equation}\nthat maps one or more concrete episodes to a typed schema with a failure signature, applicability condition, correction program, effect profile, and receipt template.\n\\end{definition}","\\begin{definition}[Schema generality preorder]\nFor repair schemas $\\omega_1$ and $\\omega_2$ with the same typed effect, write\n\\begin{equation}\n\\omega_1\\preceq\\omega_2\n\\end{equation}\nwhen 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 $\\omega_1$.\n\\end{definition}","begin{definition}[Schema generality preorder]\nFor repair schemas $\\omega_1$ and $\\omega_2$ with the same typed effect, write\n\\begin{equation}\n\\omega_1\\preceq\\omega_2\n\\end{equation}\nwhen 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 $\\omega_1$.\n\\end{definition}","\\begin{definition}[Foreign-valid repair schema]\nA 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.\n\\end{definition}","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.","\\begin{definition}[Exact transport equivalence of repairs]\nWrite\n\\begin{equation}\n\\omega_A\\cong_T\\omega_B\n\\end{equation}\nwhen:\n\\begin{enumerate}\n\\item input and output colors correspond under the atlas transport;\n\\item the defect-response maps commute with correction and defect transports;\n\\item applicability and postconditions are preserved in both directions;\n\\item protected invariants and provenance types are preserved;\n\\item the effect profiles agree after coordinate transport;\n\\item the receipts map to equivalent proof paths.\n\\end{enumerate}\n\\end{definition}","\\begin{definition}[Meta-repair]\nA meta-repair is an authorized transformation\n\\begin{equation}\n\\Theta_t:\\Ocal_t\\longrightarrow\\Ocal_{t+1}\n\\end{equation}\nthat changes one or more of:\n\\begin{enumerate}\n\\item failure-signature recognition;\n\\item causal-hypothesis generation;\n\\item candidate proposal distributions;\n\\item repair composition and scheduling;\n\\item receipt validation or confidence allocation;\n\\item repair abstraction and foreign-transfer testing.\n\\end{enumerate}\n\\end{definition}","Consider four colors\n\\begin{equation}\n\\mathrm{memory},\n\\mathrm{representation},\n\\mathrm{verifier},\n\\mathrm{planner}.\n\\end{equation}\nA composite factuality repair may have the form\n\\begin{equation}\n\\omega_{\\mathrm{gate}}\n\\circ\n\\left(\n\\omega_{\\mathrm{retrieve}},\n\\omega_{\\mathrm{decompose}},\n\\omega_{\\mathrm{check}}\n\\right).\n\\end{equation}\nThe 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 compos","Consider four colors\n\\begin{equation}\n\\mathrm{memory},\n\\mathrm{representation},\n\\mathrm{verifier},\n\\mathrm{planner}.\n\\end{equation}\nA composite factuality repair may have the form\n\\begin{equation}\n\\omega_{\\mathrm{gate}}\n\\circ\n\\left(\n\\omega_{\\mathrm{retrieve}},\n\\omega_{\\mathrm{decompose}},\n\\omega_{\\mathrm{check}}\n\\right).\n\\end{equation}\nThe 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.","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\n\\begin{equation}\nD:U^{\\Auth}\\to V\n\\end{equation}\nbe the linearized defect-response map at the current lineage state. For a defect $r\\in V$, an exact first-order repair satisfies\n\\begin{equation}\nDu=-r.\n\\label{eq:linear-repair}\n\\end{equation}","$U^{\\Auth}$ 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\n\\begin{equation}\nD:U^{\\Auth}\\to V\n\\end{equation}\nbe the linearized defect-response map at the current lineage state. For a defect $r\\in V$, an exact first-order repair satisfies\n\\begin{equation}\nDu=-r.\n\\label{eq:linear-repair}\n\\end{equation}","\\begin{definition}[Authorized obstruction class]\nThe authorized obstruction to correcting $r$ is\n\\begin{equation}\n\\Ob_D(r)=[r]\\in\\coker D:=V/\\im D.\n\\end{equation}\n\\end{definition}","\\begin{proposition}[Linear correctability]\nEquation \\eqref{eq:linear-repair} has an authorized solution if and only if\n\\begin{equation}\n[r]=0\\quad\\text{in }\\coker D.\n\\end{equation}\n\\end{proposition}","\\begin{proof}\nThe equation is solvable exactly when $r\\in\\im D$, which is equivalent to vanishing of the quotient class.\n\\end{proof}","Let $D^\\dagger:V\\to U^{\\Auth}$ denote the Moore--Penrose pseudoinverse. Define\n\\begin{equation}\nP_{\\mathrm{cor}}:=DD^\\dagger,\n\\qquad\nP_{\\mathrm{ob}}:=I_V-DD^\\dagger.\n\\end{equation}","\\begin{theorem}[Canonical correction--obstruction decomposition]\n\\label{thm:canonical-obstruction}\nFor every $r\\in V$:\n\\begin{enumerate}\n\\item $P_{\\mathrm{cor}}$ is the orthogonal projector onto $\\im D$;\n\\item $P_{\\mathrm{ob}}$ is the orthogonal projector onto $\\ker D^*$;\n\\item the defect decomposes uniquely as\n\\begin{equation}\nr=r_{\\mathrm{cor}}+r_{\\mathrm{ob}},\n\\qquad\nr_{\\mathrm{cor}}=P_{\\mathrm{cor}}r,\n\\quad\nr_{\\mathrm{ob}}=P_{\\mathrm{ob}}r;\n\\end{equation}\n\\item the minimum achievable residual is\n\\begin{equation}\n\\inf_{u\\in U^{\\Auth}}\\norm{Du+r}\n=\\norm{r_{\\mathrm{ob}}}\n=\\dist(r,\\im D);\n\\end{equation}\n\\item the complete set of least-squares minimizers is\n\\begin{equation}\n-D^\\dagger r+\\ker D;\n\\end{equation}\n\\item if $r$ is exactly correctable, $u^*=-D^\\dagger r$ is the unique exact repair of minimum norm","\\begin{theorem}[Canonical correction--obstruction decomposition]\n\\label{thm:canonical-obstruction}\nFor every $r\\in V$:\n\\begin{enumerate}\n\\item $P_{\\mathrm{cor}}$ is the orthogonal projector onto $\\im D$;\n\\item $P_{\\mathrm{ob}}$ is the orthogonal projector onto $\\ker D^*$;\n\\item the defect decomposes uniquely as\n\\begin{equation}\nr=r_{\\mathrm{cor}}+r_{\\mathrm{ob}},\n\\qquad\nr_{\\mathrm{cor}}=P_{\\mathrm{cor}}r,\n\\quad\nr_{\\mathrm{ob}}=P_{\\mathrm{ob}}r;\n\\end{equation}\n\\item the minimum achievable residual is\n\\begin{equation}\n\\inf_{u\\in U^{\\Auth}}\\norm{Du+r}\n=\\norm{r_{\\mathrm{ob}}}\n=\\dist(r,\\im D);\n\\end{equation}\n\\item the complete set of least-squares minimizers is\n\\begin{equation}\n-D^\\dagger r+\\ker D;\n\\end{equation}\n\\item if $r$ is exactly correctable, $u^*=-D^\\dagger r$ is the unique exact repair of minimum norm.\n\\end{enumerate}\n\\end{theorem}","egin{enumerate}\n\\item $P_{\\mathrm{cor}}$ is the orthogonal projector onto $\\im D$;\n\\item $P_{\\mathrm{ob}}$ is the orthogonal projector onto $\\ker D^*$;\n\\item the defect decomposes uniquely as\n\\begin{equation}\nr=r_{\\mathrm{cor}}+r_{\\mathrm{ob}},\n\\qquad\nr_{\\mathrm{cor}}=P_{\\mathrm{cor}}r,\n\\quad\nr_{\\mathrm{ob}}=P_{\\mathrm{ob}}r;\n\\end{equation}\n\\item the minimum achievable residual is\n\\begin{equation}\n\\inf_{u\\in U^{\\Auth}}\\norm{Du+r}\n=\\norm{r_{\\mathrm{ob}}}\n=\\dist(r,\\im D);\n\\end{equation}\n\\item the complete set of least-squares minimizers is\n\\begin{equation}\n-D^\\dagger r+\\ker D;\n\\end{equation}\n\\item if $r$ is exactly correctable, $u^*=-D^\\dagger r$ is the unique exact repair of minimum norm.\n\\end{enumerate}\n\\end{theorem}","e defect decomposes uniquely as\n\\begin{equation}\nr=r_{\\mathrm{cor}}+r_{\\mathrm{ob}},\n\\qquad\nr_{\\mathrm{cor}}=P_{\\mathrm{cor}}r,\n\\quad\nr_{\\mathrm{ob}}=P_{\\mathrm{ob}}r;\n\\end{equation}\n\\item the minimum achievable residual is\n\\begin{equation}\n\\inf_{u\\in U^{\\Auth}}\\norm{Du+r}\n=\\norm{r_{\\mathrm{ob}}}\n=\\dist(r,\\im D);\n\\end{equation}\n\\item the complete set of least-squares minimizers is\n\\begin{equation}\n-D^\\dagger r+\\ker D;\n\\end{equation}\n\\item if $r$ is exactly correctable, $u^*=-D^\\dagger r$ is the unique exact repair of minimum norm.\n\\end{enumerate}\n\\end{theorem}","}\nr=r_{\\mathrm{cor}}+r_{\\mathrm{ob}},\n\\qquad\nr_{\\mathrm{cor}}=P_{\\mathrm{cor}}r,\n\\quad\nr_{\\mathrm{ob}}=P_{\\mathrm{ob}}r;\n\\end{equation}\n\\item the minimum achievable residual is\n\\begin{equation}\n\\inf_{u\\in U^{\\Auth}}\\norm{Du+r}\n=\\norm{r_{\\mathrm{ob}}}\n=\\dist(r,\\im D);\n\\end{equation}\n\\item the complete set of least-squares minimizers is\n\\begin{equation}\n-D^\\dagger r+\\ker D;\n\\end{equation}\n\\item if $r$ is exactly correctable, $u^*=-D^\\dagger r$ is the unique exact repair of minimum norm.\n\\end{enumerate}\n\\end{theorem}","hrm{ob}},\n\\qquad\nr_{\\mathrm{cor}}=P_{\\mathrm{cor}}r,\n\\quad\nr_{\\mathrm{ob}}=P_{\\mathrm{ob}}r;\n\\end{equation}\n\\item the minimum achievable residual is\n\\begin{equation}\n\\inf_{u\\in U^{\\Auth}}\\norm{Du+r}\n=\\norm{r_{\\mathrm{ob}}}\n=\\dist(r,\\im D);\n\\end{equation}\n\\item the complete set of least-squares minimizers is\n\\begin{equation}\n-D^\\dagger r+\\ker D;\n\\end{equation}\n\\item if $r$ is exactly correctable, $u^*=-D^\\dagger r$ is the unique exact repair of minimum norm.\n\\end{enumerate}\n\\end{theorem}","\\begin{proof}\nThe 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 $u^*=-D^\\dagger r$,\n\\begin{equation}\nDu^*+r=(I-DD^\\dagger)r=r_{\\mathrm{ob}}.\n\\end{equation}\nEvery vector $Du$ lies in $\\im D$, so orthogonal projection gives\n\\begin{equation}\n\\norm{Du+r}^2\n=\\norm{Du+r_{\\mathrm{cor}}}^2+\\norm{r_{\\mathrm{ob}}}^2\n\\geq\\norm{r_{\\mathrm{ob}}}^2.\n\\end{equation}\nEquality 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.\n\\end{proof}","onto $\\im D$ and that $I-DD^\\dagger$ projects onto $(\\im D)^\\perp=\\ker D^*$. For $u^*=-D^\\dagger r$,\n\\begin{equation}\nDu^*+r=(I-DD^\\dagger)r=r_{\\mathrm{ob}}.\n\\end{equation}\nEvery vector $Du$ lies in $\\im D$, so orthogonal projection gives\n\\begin{equation}\n\\norm{Du+r}^2\n=\\norm{Du+r_{\\mathrm{cor}}}^2+\\norm{r_{\\mathrm{ob}}}^2\n\\geq\\norm{r_{\\mathrm{ob}}}^2.\n\\end{equation}\nEquality 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.\n\\end{proof}","onto $(\\im D)^\\perp=\\ker D^*$. For $u^*=-D^\\dagger r$,\n\\begin{equation}\nDu^*+r=(I-DD^\\dagger)r=r_{\\mathrm{ob}}.\n\\end{equation}\nEvery vector $Du$ lies in $\\im D$, so orthogonal projection gives\n\\begin{equation}\n\\norm{Du+r}^2\n=\\norm{Du+r_{\\mathrm{cor}}}^2+\\norm{r_{\\mathrm{ob}}}^2\n\\geq\\norm{r_{\\mathrm{ob}}}^2.\n\\end{equation}\nEquality 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.\n\\end{proof}","ion}\nDu^*+r=(I-DD^\\dagger)r=r_{\\mathrm{ob}}.\n\\end{equation}\nEvery vector $Du$ lies in $\\im D$, so orthogonal projection gives\n\\begin{equation}\n\\norm{Du+r}^2\n=\\norm{Du+r_{\\mathrm{cor}}}^2+\\norm{r_{\\mathrm{ob}}}^2\n\\geq\\norm{r_{\\mathrm{ob}}}^2.\n\\end{equation}\nEquality 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.\n\\end{proof}","\\begin{definition}[Quantitative authorized obstruction]\nThe quantitative obstruction magnitude is\n\\begin{equation}\n\\eta_D(r):=\\norm{P_{\\mathrm{ob}}r}=\\dist(r,\\im D).\n\\end{equation}\nThe normalized obstruction fraction is\n\\begin{equation}\n\\widehat\\eta_D(r):=\n\\begin{cases}\n\\eta_D(r)/\\norm r,&r\\neq0,","\\begin{definition}[Quantitative authorized obstruction]\nThe quantitative obstruction magnitude is\n\\begin{equation}\n\\eta_D(r):=\\norm{P_{\\mathrm{ob}}r}=\\dist(r,\\im D).\n\\end{equation}\nThe normalized obstruction fraction is\n\\begin{equation}\n\\widehat\\eta_D(r):=\n\\begin{cases}\n\\eta_D(r)/\\norm r,&r\\neq0,\\\\\n0,&r=0.\n\\end{cases}\n\\end{equation}\n\\end{definition}","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.","\\begin{proposition}[Dual obstruction witness]\n\\label{prop:dual-obstruction}\nFor every $r\\in V$,\n\\begin{equation}\n\\eta_D(r)\n=\n\\max_{\\substack{y\\in\\ker D^*","\\begin{proposition}[Dual obstruction witness]\n\\label{prop:dual-obstruction}\nFor every $r\\in V$,\n\\begin{equation}\n\\eta_D(r)\n=\n\\max_{\\substack{y\\in\\ker D^*\\\\ \\norm y\\leq1}}\n\\angles{y,r}.\n\\end{equation}\nIf $r_{\\mathrm{ob}}\\neq0$, the maximum is attained by\n\\begin{equation}\ny^*=\\frac{r_{\\mathrm{ob}}}{\\norm{r_{\\mathrm{ob}}}}.\n\\end{equation}\n\\end{proposition}","\\norm y\\leq1}}\n\\angles{y,r}.\n\\end{equation}\nIf $r_{\\mathrm{ob}}\\neq0$, the maximum is attained by\n\\begin{equation}\ny^*=\\frac{r_{\\mathrm{ob}}}{\\norm{r_{\\mathrm{ob}}}}.\n\\end{equation}\n\\end{proposition}","\\begin{proof}\nFor $y\\in\\ker D^*$, 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.\n\\end{proof}","The vector $y^*$ is a compact certificate: every authorized linear correction is invisible to it, yet the defect has nonzero projection along it.","A repair is not authorized merely because it reduces task error. It must preserve declared invariants. Let\n\\begin{equation}\nC:U\\to Z\n\\end{equation}\nbe the derivative of a protected invariant map. First-order invariant preservation requires $Cu=0$, so the authorized tangent space is\n\\begin{equation}\nU^{\\Auth}=\\ker C.\n\\end{equation}","\\begin{proposition}[Invariant-constrained obstruction]\n\\label{prop:invariant-obstruction}\nLet $D_C:=\\restr{D}{\\ker C}$. An exact first-order repair preserving the invariant exists if and only if\n\\begin{equation}\n[r]=0\\quad\\text{in }\\coker D_C.\n\\end{equation}\nEquivalently, the stacked system\n\\begin{equation}\n\\begin{bmatrix}D","\\begin{proposition}[Invariant-constrained obstruction]\n\\label{prop:invariant-obstruction}\nLet $D_C:=\\restr{D}{\\ker C}$. An exact first-order repair preserving the invariant exists if and only if\n\\begin{equation}\n[r]=0\\quad\\text{in }\\coker D_C.\n\\end{equation}\nEquivalently, the stacked system\n\\begin{equation}\n\\begin{bmatrix}D\\\\C\\end{bmatrix}u\n=\n\\begin{bmatrix}-r\\\\0\\end{bmatrix}\n\\end{equation}\nis solvable.\n\\end{proposition}","Let $W:V\\to V$ and $R:U\\to U$ be positive-definite self-adjoint operators representing defect importance and repair cost.","\\begin{theorem}[Regularized minimum-cost repair]\n\\label{thm:regularized-repair}\nFor $\\lambda>0$, the functional\n\\begin{equation}\n\\Jcal_\\lambda(u)\n:=\n\\frac12\\angles{Du+r,W(Du+r)}\n+\n\\frac\\lambda2\\angles{u,Ru}\n\\end{equation}\nhas a unique minimizer\n\\begin{equation}\nu_\\lambda\n=-(D^*WD+\\lambda R)^{-1}D^*Wr.\n\\end{equation}\nIt is characterized by the normal equation\n\\begin{equation}\nD^*W(Du_\\lambda+r)+\\lambda Ru_\\lambda=0.\n\\end{equation}\n\\end{theorem}","\\begin{proof}\nThe 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.\n\\end{proof}","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.","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\n\\begin{equation}\n\\eta_{D,K}(r):=\\inf_{u\\in K}\\norm{Du+r}.\n\\end{equation}","\\begin{proposition}[Convex authorized correctability]\nIf $K$ 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\n\\begin{equation}\n\\eta_{D,K}(r)=0.\n\\end{equation}\n\\end{proposition}","When $K$ is a cone, noncorrectability admits a Farkas-type dual witness. Define the dual cone\n\\begin{equation}\nK^*:=\\set{z\\in U:\\angles{z,u}\\geq0\\text{ for all }u\\in K}.\n\\end{equation}","\\begin{theorem}[Cone-constrained obstruction certificate]\n\\label{thm:cone-certificate}\nLet $K\\subseteq U$ 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\n\\begin{equation}\n\\angles{y,r}\\leq0\n\\end{equation}\nfor every $y\\in V$ satisfying\n\\begin{equation}\nD^*y\\in K^*.\n\\end{equation}\nConsequently, any $y$ with $D^*y\\in K^*$ and $\\angles{y,r}>0$ certifies noncorrectability.\n\\end{theorem}","in{theorem}[Cone-constrained obstruction certificate]\n\\label{thm:cone-certificate}\nLet $K\\subseteq U$ 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\n\\begin{equation}\n\\angles{y,r}\\leq0\n\\end{equation}\nfor every $y\\in V$ satisfying\n\\begin{equation}\nD^*y\\in K^*.\n\\end{equation}\nConsequently, any $y$ with $D^*y\\in K^*$ and $\\angles{y,r}>0$ certifies noncorrectability.\n\\end{theorem}","\\begin{proof}\nSet $C=D(K)$, a closed convex cone. Its dual is\n\\begin{equation}\nC^*=\\set{y:D^*y\\in K^*}.\n\\end{equation}\nBy the bipolar theorem, $-r\\in C$ if and only if $\\angles{y,-r}\\geq0$ for every $y\\in C^*$, which is the displayed inequality.\n\\end{proof}","Let $\\sigma_+(D)$ denote the smallest positive singular value of $D$. When $D\\neq0$,\n\\begin{equation}\n\\norm{D^\\dagger}=\\frac{1}{\\sigma_+(D)}.\n\\end{equation}\nSmall $\\sigma_+(D)$ means the correctable component is ill-conditioned: a small defect perturbation may require a large repair.","\\begin{proposition}[Fixed-map sensitivity]\nFor defects $r,s\\in V$,\n\\begin{align}\n\\abs{\\eta_D(r)-\\eta_D(s)}&\\leq\\norm{r-s},","\\begin{proposition}[Fixed-map sensitivity]\nFor defects $r,s\\in V$,\n\\begin{align}\n\\abs{\\eta_D(r)-\\eta_D(s)}&\\leq\\norm{r-s},\\\\\n\\norm{D^\\dagger r-D^\\dagger s}&\\leq\\frac{1}{\\sigma_+(D)}\\norm{r-s}.\n\\end{align}\n\\end{proposition}","\\begin{proof}\nThe first inequality follows because distance to a fixed closed set is $1$-Lipschitz. The second follows from the operator norm of $D^\\dagger$.\n\\end{proof}","\\begin{proposition}[Transferred residual under model error]\n\\label{prop:correction-perturbation}\nLet $\\widehat D=D+\\Delta D$ and $\\widehat r=r+\\Delta r$. For any proposed repair $u$,\n\\begin{equation}\n\\abs{\\norm{Du+r}-\\norm{\\widehat Du+\\widehat r}}\n\\leq\n\\norm{\\Delta D}\\norm u+\\norm{\\Delta r}.\n\\end{equation}\nIn particular, applying an exact repair for $(D,r)$ to the perturbed system yields residual at most\n\\begin{equation}\n\\norm{\\Delta D}\\norm u+\\norm{\\Delta r}.\n\\end{equation}\n\\end{proposition}","sferred residual under model error]\n\\label{prop:correction-perturbation}\nLet $\\widehat D=D+\\Delta D$ and $\\widehat r=r+\\Delta r$. For any proposed repair $u$,\n\\begin{equation}\n\\abs{\\norm{Du+r}-\\norm{\\widehat Du+\\widehat r}}\n\\leq\n\\norm{\\Delta D}\\norm u+\\norm{\\Delta r}.\n\\end{equation}\nIn particular, applying an exact repair for $(D,r)$ to the perturbed system yields residual at most\n\\begin{equation}\n\\norm{\\Delta D}\\norm u+\\norm{\\Delta r}.\n\\end{equation}\n\\end{proposition}","\\begin{proof}\nThe two residual vectors differ by $\\Delta D\\,u+\\Delta r$. Apply the reverse triangle inequality and submultiplicativity.\n\\end{proof}","\\begin{theorem}[Robust empirical noncorrectability certificate]\n\\label{thm:robust-obstruction-certificate}\nSuppose, on an event of probability at least $1-\\alpha$,\n\\begin{equation}\n\\norm{D-\\widehat D}\\leq\\eps_D,\n\\qquad\n\\norm{r-\\widehat r}\\leq\\eps_r.\n\\end{equation}\nFix a repair budget $\\norm u\\leq B$ and a unit vector $y\\in V$. If\n\\begin{equation}\n\\angles{y,\\widehat r}\n>\n\\eps_r+B\\bigl(\\norm{\\widehat D^*y}+\\eps_D\\bigr),\n\\label{eq:robust-dual-gap}\n\\end{equation}\nthen, on the same event, no budget-feasible repair satisfies $Du=-r$.\n\\end{theorem}","t of probability at least $1-\\alpha$,\n\\begin{equation}\n\\norm{D-\\widehat D}\\leq\\eps_D,\n\\qquad\n\\norm{r-\\widehat r}\\leq\\eps_r.\n\\end{equation}\nFix a repair budget $\\norm u\\leq B$ and a unit vector $y\\in V$. If\n\\begin{equation}\n\\angles{y,\\widehat r}\n>\n\\eps_r+B\\bigl(\\norm{\\widehat D^*y}+\\eps_D\\bigr),\n\\label{eq:robust-dual-gap}\n\\end{equation}\nthen, on the same event, no budget-feasible repair satisfies $Du=-r$.\n\\end{theorem}","\\begin{proof}\nFor every $\\norm u\\leq B$,\n\\begin{align}\n\\angles{y,Du+r}\n&=\\angles{D^*y,u}+\\angles{y,r}","\\begin{proof}\nFor every $\\norm u\\leq B$,\n\\begin{align}\n\\angles{y,Du+r}\n&=\\angles{D^*y,u}+\\angles{y,r}\\\\\n&\\geq\n-\\norm{D^*y}\\,B+\\angles{y,\\widehat r}-\\eps_r\\\\\n&\\geq\n-B\\bigl(\\norm{\\widehat D^*y}+\\eps_D\\bigr)\n+\\angles{y,\\widehat r}-\\eps_r.\n\\end{align}\nCondition \\eqref{eq:robust-dual-gap} makes the final expression positive, so $Du+r$ cannot vanish.\n\\end{proof}","&\\geq\n-B\\bigl(\\norm{\\widehat D^*y}+\\eps_D\\bigr)\n+\\angles{y,\\widehat r}-\\eps_r.\n\\end{align}\nCondition \\eqref{eq:robust-dual-gap} makes the final expression positive, so $Du+r$ cannot vanish.\n\\end{proof}","A negative result can therefore be proof-carrying: it records the witness $y$, the confidence event, the budget $B$, and the positive dual gap.","Let $V$ remain fixed while the authorized correction grammar expands.","\\begin{theorem}[Authorized obstruction-shrinkage theorem]\n\\label{thm:obstruction-shrinkage}\nSuppose\n\\begin{equation}\nU_t^{\\Auth}\\subseteq U_{t+1}^{\\Auth}\n\\end{equation}\nand\n\\begin{equation}\n\\restr{D_{t+1}}{U_t^{\\Auth}}=D_t.\n\\end{equation}\nThen\n\\begin{equation}\n\\im D_t\\subseteq\\im D_{t+1},\n\\end{equation}\nthere is a canonical surjection\n\\begin{equation}\nq_t:\\coker D_t\\twoheadrightarrow\\coker D_{t+1},\n\\qquad\n[v]_{\\im D_t}\\longmapsto[v]_{\\im D_{t+1}},\n\\end{equation}\nand\n\\begin{equation}\n\\dim\\coker D_{t+1}\\leq\\dim\\coker D_t.\n\\end{equation}\nThe inequality is strict if and only if $\\im D_t\\subsetneq\\im D_{t+1}$.\n\\end{theorem}","age}\nSuppose\n\\begin{equation}\nU_t^{\\Auth}\\subseteq U_{t+1}^{\\Auth}\n\\end{equation}\nand\n\\begin{equation}\n\\restr{D_{t+1}}{U_t^{\\Auth}}=D_t.\n\\end{equation}\nThen\n\\begin{equation}\n\\im D_t\\subseteq\\im D_{t+1},\n\\end{equation}\nthere is a canonical surjection\n\\begin{equation}\nq_t:\\coker D_t\\twoheadrightarrow\\coker D_{t+1},\n\\qquad\n[v]_{\\im D_t}\\longmapsto[v]_{\\im D_{t+1}},\n\\end{equation}\nand\n\\begin{equation}\n\\dim\\coker D_{t+1}\\leq\\dim\\coker D_t.\n\\end{equation}\nThe inequality is strict if and only if $\\im D_t\\subsetneq\\im D_{t+1}$.\n\\end{theorem}","uation}\n\\restr{D_{t+1}}{U_t^{\\Auth}}=D_t.\n\\end{equation}\nThen\n\\begin{equation}\n\\im D_t\\subseteq\\im D_{t+1},\n\\end{equation}\nthere is a canonical surjection\n\\begin{equation}\nq_t:\\coker D_t\\twoheadrightarrow\\coker D_{t+1},\n\\qquad\n[v]_{\\im D_t}\\longmapsto[v]_{\\im D_{t+1}},\n\\end{equation}\nand\n\\begin{equation}\n\\dim\\coker D_{t+1}\\leq\\dim\\coker D_t.\n\\end{equation}\nThe inequality is strict if and only if $\\im D_t\\subsetneq\\im D_{t+1}$.\n\\end{theorem}","\\begin{proof}\nFor $u\\in U_t^{\\Auth}$, $D_tu=D_{t+1}u$, so every old correctable direction remains correctable. Quotienting by a larger subspace gives the canonical surjection. Rank-nullity gives the dimension statement.\n\\end{proof}","\\begin{definition}[Authorized repair rank]\nThe authorized repair rank and unresolved obstruction dimension are\n\\begin{equation}\n\\rho_t:=\\rank D_t,\n\\qquad\no_t:=\\dim\\coker D_t.\n\\end{equation}\n\\end{definition}","\\begin{proposition}[Exact sequence of newly correctable directions]\n\\label{prop:obstruction-exact-sequence}\nUnder the assumptions of \\cref{thm:obstruction-shrinkage}, there is a short exact sequence\n\\begin{equation}\n0\n\\longrightarrow\n\\frac{\\im D_{t+1}}{\\im D_t}\n\\longrightarrow\n\\coker D_t\n\\overset{q_t}{\\longrightarrow}\n\\coker D_{t+1}\n\\longrightarrow0.\n\\end{equation}\nConsequently,\n\\begin{equation}\no_t-o_{t+1}\n=\n\\dim\\frac{\\im D_{t+1}}{\\im D_t}.\n\\end{equation}\n\\end{proposition}","\\begin{proof}\nThe 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.\n\\end{proof}","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.","Open-set improvement may enlarge the defect ontology itself. Let\n\\begin{equation}\nj_U:U_t^{\\Auth}\\hookrightarrow U_{t+1}^{\\Auth},\n\\qquad\nj_V:V_t\\hookrightarrow V_{t+1}\n\\end{equation}\nbe injective embeddings satisfying\n\\begin{equation}\nD_{t+1}j_U=j_VD_t.\n\\label{eq:dynamic-defect-commute}\n\\end{equation}","\\begin{proposition}[Induced obstruction transport under ontology expansion]\nEquation \\eqref{eq:dynamic-defect-commute} induces a linear map\n\\begin{equation}\n\\overline j_t:\\coker D_t\\to\\coker D_{t+1},\n\\qquad\n[v]\\longmapsto[j_Vv].\n\\end{equation}\nIts kernel is canonically isomorphic to\n\\begin{equation}\n\\ker\\overline j_t\n\\cong\n\\frac{j_V(V_t)\\cap\\im D_{t+1}}{j_V(\\im D_t)}.\n\\end{equation}\nThus an old obstruction vanishes after expansion exactly when the enlarged repair grammar corrects its embedded defect direction.\n\\end{proposition}","\\begin{proof}\nCommutation 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.\n\\end{proof}","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.","Let\n\\begin{equation}\nU=\\bigoplus_{i=1}^m U_i,\n\\qquad\nV=\\bigoplus_{i=1}^m V_i,\n\\end{equation}\nwith orthogonal projectors $Q_i$ and $P_i$. Decompose\n\\begin{equation}\nD=D_0+E,\n\\qquad\nD_0:=\\sum_{i=1}^m P_iDQ_i.\n\\end{equation}\nThe block-diagonal operator $D_0$ represents ideal packet-local correction; $E$ contains all cross-packet effects.","Suppose each local block admits a right inverse and their direct sum gives\n\\begin{equation}\nQ_0:V\\to U,\n\\qquad\nD_0Q_0=I_V.\n\\end{equation}","\\begin{theorem}[Exact correction under small packet coupling]\n\\label{thm:packet-neumann}\nIf\n\\begin{equation}\nq:=\\norm{EQ_0}<1,\n\\end{equation}\nthen $D$ is surjective and has right inverse\n\\begin{equation}\nQ\n=Q_0(I_V+EQ_0)^{-1}\n=Q_0\\sum_{k=0}^{\\infty}(-EQ_0)^k.\n\\end{equation}\nMoreover,\n\\begin{equation}\n\\norm Q\\leq\\frac{\\norm{Q_0}}{1-q}.\n\\end{equation}\nFor defect $r$, the truncated packet-correction series\n\\begin{equation}\nu^{(n)}\n:=-Q_0\\sum_{k=0}^{n}(-EQ_0)^kr\n\\end{equation}\nhas residual\n\\begin{equation}\nDu^{(n)}+r=(-EQ_0)^{n+1}r,\n\\end{equation}\nand hence\n\\begin{equation}\n\\norm{Du^{(n)}+r}\\leq q^{n+1}\\norm r.\n\\end{equation}\n\\end{theorem}","orrection under small packet coupling]\n\\label{thm:packet-neumann}\nIf\n\\begin{equation}\nq:=\\norm{EQ_0}<1,\n\\end{equation}\nthen $D$ is surjective and has right inverse\n\\begin{equation}\nQ\n=Q_0(I_V+EQ_0)^{-1}\n=Q_0\\sum_{k=0}^{\\infty}(-EQ_0)^k.\n\\end{equation}\nMoreover,\n\\begin{equation}\n\\norm Q\\leq\\frac{\\norm{Q_0}}{1-q}.\n\\end{equation}\nFor defect $r$, the truncated packet-correction series\n\\begin{equation}\nu^{(n)}\n:=-Q_0\\sum_{k=0}^{n}(-EQ_0)^kr\n\\end{equation}\nhas residual\n\\begin{equation}\nDu^{(n)}+r=(-EQ_0)^{n+1}r,\n\\end{equation}\nand hence\n\\begin{equation}\n\\norm{Du^{(n)}+r}\\leq q^{n+1}\\norm r.\n\\end{equation}\n\\end{theorem}","}\nq:=\\norm{EQ_0}<1,\n\\end{equation}\nthen $D$ is surjective and has right inverse\n\\begin{equation}\nQ\n=Q_0(I_V+EQ_0)^{-1}\n=Q_0\\sum_{k=0}^{\\infty}(-EQ_0)^k.\n\\end{equation}\nMoreover,\n\\begin{equation}\n\\norm Q\\leq\\frac{\\norm{Q_0}}{1-q}.\n\\end{equation}\nFor defect $r$, the truncated packet-correction series\n\\begin{equation}\nu^{(n)}\n:=-Q_0\\sum_{k=0}^{n}(-EQ_0)^kr\n\\end{equation}\nhas residual\n\\begin{equation}\nDu^{(n)}+r=(-EQ_0)^{n+1}r,\n\\end{equation}\nand hence\n\\begin{equation}\n\\norm{Du^{(n)}+r}\\leq q^{n+1}\\norm r.\n\\end{equation}\n\\end{theorem}","ht inverse\n\\begin{equation}\nQ\n=Q_0(I_V+EQ_0)^{-1}\n=Q_0\\sum_{k=0}^{\\infty}(-EQ_0)^k.\n\\end{equation}\nMoreover,\n\\begin{equation}\n\\norm Q\\leq\\frac{\\norm{Q_0}}{1-q}.\n\\end{equation}\nFor defect $r$, the truncated packet-correction series\n\\begin{equation}\nu^{(n)}\n:=-Q_0\\sum_{k=0}^{n}(-EQ_0)^kr\n\\end{equation}\nhas residual\n\\begin{equation}\nDu^{(n)}+r=(-EQ_0)^{n+1}r,\n\\end{equation}\nand hence\n\\begin{equation}\n\\norm{Du^{(n)}+r}\\leq q^{n+1}\\norm r.\n\\end{equation}\n\\end{theorem}","\\begin{proof}\nSince $\\norm{EQ_0}<1$, the Neumann series for $(I+EQ_0)^{-1}$ converges. Also\n\\begin{equation}\nDQ_0=(D_0+E)Q_0=I+EQ_0,\n\\end{equation}\nso $DQ=I$. The norm bound follows from the geometric series. For the truncation,\n\\begin{equation}\n(I+EQ_0)\\sum_{k=0}^n(-EQ_0)^k\n=I-(-EQ_0)^{n+1},\n\\end{equation}\nwhich gives the residual formula.\n\\end{proof}","\\begin{definition}[Packet leakage profile]\nThe directed leakage from correction packet $j$ into defect packet $i$ is\n\\begin{equation}\n\\ell_{ij}:=\\norm{P_iDQ_j},\n\\qquad i\\neq j.\n\\end{equation}\nThe matrix $(\\ell_{ij})$ is the packet leakage profile.\n\\end{definition}","Let $U,V$ be Banach spaces and let the post-correction defect map be\n\\begin{equation}\n\\Fcal(u)=r+Du+N(u),\n\\qquad N(0)=0.\n\\end{equation}\nWhen $D$ 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.","\\begin{theorem}[Gauge-fixed local nonlinear repair by contraction]\n\\label{thm:nonlinear-repair}\nSuppose $D:U\\to V$ has a bounded right inverse $Q:V\\to U$, so $DQ=I_V$, and set\n\\begin{equation}\nW:=\\im Q.\n\\end{equation}\nThen $\\restr D W:W\\to V$ is an isomorphism with inverse $Q$. Let\n\\begin{equation}\nB_R^W:=\\set{w\\in W:\\norm w\\leq R}.\n\\end{equation}\nSuppose\n\\begin{equation}\n\\norm{N(u)-N(v)}\\leq L_N\\norm{u-v}\n\\qquad(u,v\\in B_R^W),\n\\end{equation}\nwith\n\\begin{equation}\nq:=\\norm QL_N<1,\n\\end{equation}\nand\n\\begin{equation}\n\\norm{Qr}+\\norm Q\\sup_{u\\in B_R^W}\\norm{N(u)}\\leq R.\n\\end{equation}\nThen there exists a unique $u^*\\in B_R^W$ satisfying $\\Fcal(u^*)=0$. The gauge-fixed iteration\n\\begin{equation}\nu_{k+1}=-Qr-QN(u_k)\n\\end{equation}\nconverges to $u^*$ from every $u_0\\in B_R^W$, with\n\\begin{equation}\n\\norm{u_","\\begin{theorem}[Gauge-fixed local nonlinear repair by contraction]\n\\label{thm:nonlinear-repair}\nSuppose $D:U\\to V$ has a bounded right inverse $Q:V\\to U$, so $DQ=I_V$, and set\n\\begin{equation}\nW:=\\im Q.\n\\end{equation}\nThen $\\restr D W:W\\to V$ is an isomorphism with inverse $Q$. Let\n\\begin{equation}\nB_R^W:=\\set{w\\in W:\\norm w\\leq R}.\n\\end{equation}\nSuppose\n\\begin{equation}\n\\norm{N(u)-N(v)}\\leq L_N\\norm{u-v}\n\\qquad(u,v\\in B_R^W),\n\\end{equation}\nwith\n\\begin{equation}\nq:=\\norm QL_N<1,\n\\end{equation}\nand\n\\begin{equation}\n\\norm{Qr}+\\norm Q\\sup_{u\\in B_R^W}\\norm{N(u)}\\leq R.\n\\end{equation}\nThen there exists a unique $u^*\\in B_R^W$ satisfying $\\Fcal(u^*)=0$. The gauge-fixed iteration\n\\begin{equation}\nu_{k+1}=-Qr-QN(u_k)\n\\end{equation}\nconverges to $u^*$ from every $u_0\\in B_R^W$, with\n\\begin{equation}\n\\norm{u_k-u^*}\\leq\\frac{q^k}{1-q}\\norm{u_1-u_0}","\\begin{theorem}[Gauge-fixed local nonlinear repair by contraction]\n\\label{thm:nonlinear-repair}\nSuppose $D:U\\to V$ has a bounded right inverse $Q:V\\to U$, so $DQ=I_V$, and set\n\\begin{equation}\nW:=\\im Q.\n\\end{equation}\nThen $\\restr D W:W\\to V$ is an isomorphism with inverse $Q$. Let\n\\begin{equation}\nB_R^W:=\\set{w\\in W:\\norm w\\leq R}.\n\\end{equation}\nSuppose\n\\begin{equation}\n\\norm{N(u)-N(v)}\\leq L_N\\norm{u-v}\n\\qquad(u,v\\in B_R^W),\n\\end{equation}\nwith\n\\begin{equation}\nq:=\\norm QL_N<1,\n\\end{equation}\nand\n\\begin{equation}\n\\norm{Qr}+\\norm Q\\sup_{u\\in B_R^W}\\norm{N(u)}\\leq R.\n\\end{equation}\nThen there exists a unique $u^*\\in B_R^W$ satisfying $\\Fcal(u^*)=0$. The gauge-fixed iteration\n\\begin{equation}\nu_{k+1}=-Qr-QN(u_k)\n\\end{equation}\nconverges to $u^*$ from every $u_0\\in B_R^W$, with\n\\begin{equation}\n\\norm{u_k-u^*}\\leq\\frac{q^k}{1-q}\\norm{u_1-u_0}.\n\\end{equati","\\begin{theorem}[Gauge-fixed local nonlinear repair by contraction]\n\\label{thm:nonlinear-repair}\nSuppose $D:U\\to V$ has a bounded right inverse $Q:V\\to U$, so $DQ=I_V$, and set\n\\begin{equation}\nW:=\\im Q.\n\\end{equation}\nThen $\\restr D W:W\\to V$ is an isomorphism with inverse $Q$. Let\n\\begin{equation}\nB_R^W:=\\set{w\\in W:\\norm w\\leq R}.\n\\end{equation}\nSuppose\n\\begin{equation}\n\\norm{N(u)-N(v)}\\leq L_N\\norm{u-v}\n\\qquad(u,v\\in B_R^W),\n\\end{equation}\nwith\n\\begin{equation}\nq:=\\norm QL_N<1,\n\\end{equation}\nand\n\\begin{equation}\n\\norm{Qr}+\\norm Q\\sup_{u\\in B_R^W}\\norm{N(u)}\\leq R.\n\\end{equation}\nThen there exists a unique $u^*\\in B_R^W$ satisfying $\\Fcal(u^*)=0$. The gauge-fixed iteration\n\\begin{equation}\nu_{k+1}=-Qr-QN(u_k)\n\\end{equation}\nconverges to $u^*$ from every $u_0\\in B_R^W$, with\n\\begin{equation}\n\\norm{u_k-u^*}\\leq\\frac{q^k}{1-q}\\norm{u_1-u_0}.\n\\end{equation}\n\\end{theorem}","label{thm:nonlinear-repair}\nSuppose $D:U\\to V$ has a bounded right inverse $Q:V\\to U$, so $DQ=I_V$, and set\n\\begin{equation}\nW:=\\im Q.\n\\end{equation}\nThen $\\restr D W:W\\to V$ is an isomorphism with inverse $Q$. Let\n\\begin{equation}\nB_R^W:=\\set{w\\in W:\\norm w\\leq R}.\n\\end{equation}\nSuppose\n\\begin{equation}\n\\norm{N(u)-N(v)}\\leq L_N\\norm{u-v}\n\\qquad(u,v\\in B_R^W),\n\\end{equation}\nwith\n\\begin{equation}\nq:=\\norm QL_N<1,\n\\end{equation}\nand\n\\begin{equation}\n\\norm{Qr}+\\norm Q\\sup_{u\\in B_R^W}\\norm{N(u)}\\leq R.\n\\end{equation}\nThen there exists a unique $u^*\\in B_R^W$ satisfying $\\Fcal(u^*)=0$. The gauge-fixed iteration\n\\begin{equation}\nu_{k+1}=-Qr-QN(u_k)\n\\end{equation}\nconverges to $u^*$ from every $u_0\\in B_R^W$, with\n\\begin{equation}\n\\norm{u_k-u^*}\\leq\\frac{q^k}{1-q}\\norm{u_1-u_0}.\n\\end{equation}\n\\end{theorem}","bounded right inverse $Q:V\\to U$, so $DQ=I_V$, and set\n\\begin{equation}\nW:=\\im Q.\n\\end{equation}\nThen $\\restr D W:W\\to V$ is an isomorphism with inverse $Q$. Let\n\\begin{equation}\nB_R^W:=\\set{w\\in W:\\norm w\\leq R}.\n\\end{equation}\nSuppose\n\\begin{equation}\n\\norm{N(u)-N(v)}\\leq L_N\\norm{u-v}\n\\qquad(u,v\\in B_R^W),\n\\end{equation}\nwith\n\\begin{equation}\nq:=\\norm QL_N<1,\n\\end{equation}\nand\n\\begin{equation}\n\\norm{Qr}+\\norm Q\\sup_{u\\in B_R^W}\\norm{N(u)}\\leq R.\n\\end{equation}\nThen there exists a unique $u^*\\in B_R^W$ satisfying $\\Fcal(u^*)=0$. The gauge-fixed iteration\n\\begin{equation}\nu_{k+1}=-Qr-QN(u_k)\n\\end{equation}\nconverges to $u^*$ from every $u_0\\in B_R^W$, with\n\\begin{equation}\n\\norm{u_k-u^*}\\leq\\frac{q^k}{1-q}\\norm{u_1-u_0}.\n\\end{equation}\n\\end{theorem}","Then $\\restr D W:W\\to V$ is an isomorphism with inverse $Q$. Let\n\\begin{equation}\nB_R^W:=\\set{w\\in W:\\norm w\\leq R}.\n\\end{equation}\nSuppose\n\\begin{equation}\n\\norm{N(u)-N(v)}\\leq L_N\\norm{u-v}\n\\qquad(u,v\\in B_R^W),\n\\end{equation}\nwith\n\\begin{equation}\nq:=\\norm QL_N<1,\n\\end{equation}\nand\n\\begin{equation}\n\\norm{Qr}+\\norm Q\\sup_{u\\in B_R^W}\\norm{N(u)}\\leq R.\n\\end{equation}\nThen there exists a unique $u^*\\in B_R^W$ satisfying $\\Fcal(u^*)=0$. The gauge-fixed iteration\n\\begin{equation}\nu_{k+1}=-Qr-QN(u_k)\n\\end{equation}\nconverges to $u^*$ from every $u_0\\in B_R^W$, with\n\\begin{equation}\n\\norm{u_k-u^*}\\leq\\frac{q^k}{1-q}\\norm{u_1-u_0}.\n\\end{equation}\n\\end{theorem}","s an isomorphism with inverse $Q$. Let\n\\begin{equation}\nB_R^W:=\\set{w\\in W:\\norm w\\leq R}.\n\\end{equation}\nSuppose\n\\begin{equation}\n\\norm{N(u)-N(v)}\\leq L_N\\norm{u-v}\n\\qquad(u,v\\in B_R^W),\n\\end{equation}\nwith\n\\begin{equation}\nq:=\\norm QL_N<1,\n\\end{equation}\nand\n\\begin{equation}\n\\norm{Qr}+\\norm Q\\sup_{u\\in B_R^W}\\norm{N(u)}\\leq R.\n\\end{equation}\nThen there exists a unique $u^*\\in B_R^W$ satisfying $\\Fcal(u^*)=0$. The gauge-fixed iteration\n\\begin{equation}\nu_{k+1}=-Qr-QN(u_k)\n\\end{equation}\nconverges to $u^*$ from every $u_0\\in B_R^W$, with\n\\begin{equation}\n\\norm{u_k-u^*}\\leq\\frac{q^k}{1-q}\\norm{u_1-u_0}.\n\\end{equation}\n\\end{theorem}","_R^W:=\\set{w\\in W:\\norm w\\leq R}.\n\\end{equation}\nSuppose\n\\begin{equation}\n\\norm{N(u)-N(v)}\\leq L_N\\norm{u-v}\n\\qquad(u,v\\in B_R^W),\n\\end{equation}\nwith\n\\begin{equation}\nq:=\\norm QL_N<1,\n\\end{equation}\nand\n\\begin{equation}\n\\norm{Qr}+\\norm Q\\sup_{u\\in B_R^W}\\norm{N(u)}\\leq R.\n\\end{equation}\nThen there exists a unique $u^*\\in B_R^W$ satisfying $\\Fcal(u^*)=0$. The gauge-fixed iteration\n\\begin{equation}\nu_{k+1}=-Qr-QN(u_k)\n\\end{equation}\nconverges to $u^*$ from every $u_0\\in B_R^W$, with\n\\begin{equation}\n\\norm{u_k-u^*}\\leq\\frac{q^k}{1-q}\\norm{u_1-u_0}.\n\\end{equation}\n\\end{theorem}","uppose\n\\begin{equation}\n\\norm{N(u)-N(v)}\\leq L_N\\norm{u-v}\n\\qquad(u,v\\in B_R^W),\n\\end{equation}\nwith\n\\begin{equation}\nq:=\\norm QL_N<1,\n\\end{equation}\nand\n\\begin{equation}\n\\norm{Qr}+\\norm Q\\sup_{u\\in B_R^W}\\norm{N(u)}\\leq R.\n\\end{equation}\nThen there exists a unique $u^*\\in B_R^W$ satisfying $\\Fcal(u^*)=0$. The gauge-fixed iteration\n\\begin{equation}\nu_{k+1}=-Qr-QN(u_k)\n\\end{equation}\nconverges to $u^*$ from every $u_0\\in B_R^W$, with\n\\begin{equation}\n\\norm{u_k-u^*}\\leq\\frac{q^k}{1-q}\\norm{u_1-u_0}.\n\\end{equation}\n\\end{theorem}","ation}\n\\norm{N(u)-N(v)}\\leq L_N\\norm{u-v}\n\\qquad(u,v\\in B_R^W),\n\\end{equation}\nwith\n\\begin{equation}\nq:=\\norm QL_N<1,\n\\end{equation}\nand\n\\begin{equation}\n\\norm{Qr}+\\norm Q\\sup_{u\\in B_R^W}\\norm{N(u)}\\leq R.\n\\end{equation}\nThen there exists a unique $u^*\\in B_R^W$ satisfying $\\Fcal(u^*)=0$. The gauge-fixed iteration\n\\begin{equation}\nu_{k+1}=-Qr-QN(u_k)\n\\end{equation}\nconverges to $u^*$ from every $u_0\\in B_R^W$, with\n\\begin{equation}\n\\norm{u_k-u^*}\\leq\\frac{q^k}{1-q}\\norm{u_1-u_0}.\n\\end{equation}\n\\end{theorem}","{u-v}\n\\qquad(u,v\\in B_R^W),\n\\end{equation}\nwith\n\\begin{equation}\nq:=\\norm QL_N<1,\n\\end{equation}\nand\n\\begin{equation}\n\\norm{Qr}+\\norm Q\\sup_{u\\in B_R^W}\\norm{N(u)}\\leq R.\n\\end{equation}\nThen there exists a unique $u^*\\in B_R^W$ satisfying $\\Fcal(u^*)=0$. The gauge-fixed iteration\n\\begin{equation}\nu_{k+1}=-Qr-QN(u_k)\n\\end{equation}\nconverges to $u^*$ from every $u_0\\in B_R^W$, with\n\\begin{equation}\n\\norm{u_k-u^*}\\leq\\frac{q^k}{1-q}\\norm{u_1-u_0}.\n\\end{equation}\n\\end{theorem}","\\begin{proof}\nDefine $T(u)=-Qr-QN(u)$. Because $T$ takes values in $W$, the radius condition gives $T(B_R^W)\\subseteq B_R^W$. The Lipschitz condition gives\n\\begin{equation}\n\\norm{T(u)-T(v)}\\leq q\\norm{u-v}.\n\\end{equation}\nBanach'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 $Du^*=-r-N(u^*)$.\n\\end{proof}","oof}\nDefine $T(u)=-Qr-QN(u)$. Because $T$ takes values in $W$, the radius condition gives $T(B_R^W)\\subseteq B_R^W$. The Lipschitz condition gives\n\\begin{equation}\n\\norm{T(u)-T(v)}\\leq q\\norm{u-v}.\n\\end{equation}\nBanach'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 $Du^*=-r-N(u^*)$.\n\\end{proof}","\\begin{remark}[Scope of uniqueness]\nThe 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.\n\\end{remark}","\\begin{proposition}[Nonlinear obstruction floor]\n\\label{prop:nonlinear-floor}\nLet $P_{\\mathrm{ob}}$ be the orthogonal obstruction projector of $D$. For every $u$,\n\\begin{equation}\n\\norm{\\Fcal(u)}\n\\geq\n\\norm{P_{\\mathrm{ob}}r}-\\norm{P_{\\mathrm{ob}}N(u)}.\n\\end{equation}\nConsequently, if $\\norm{P_{\\mathrm{ob}}N(u)}\\leq\\gamma$ on an authorized region, then no repair in that region can achieve residual below\n\\begin{equation}\n\\max\\set{0,\\eta_D(r)-\\gamma}.\n\\end{equation}\n\\end{proposition}","nlinear obstruction floor]\n\\label{prop:nonlinear-floor}\nLet $P_{\\mathrm{ob}}$ be the orthogonal obstruction projector of $D$. For every $u$,\n\\begin{equation}\n\\norm{\\Fcal(u)}\n\\geq\n\\norm{P_{\\mathrm{ob}}r}-\\norm{P_{\\mathrm{ob}}N(u)}.\n\\end{equation}\nConsequently, if $\\norm{P_{\\mathrm{ob}}N(u)}\\leq\\gamma$ on an authorized region, then no repair in that region can achieve residual below\n\\begin{equation}\n\\max\\set{0,\\eta_D(r)-\\gamma}.\n\\end{equation}\n\\end{proposition}","\\begin{proof}\nSince $P_{\\mathrm{ob}}D=0$,\n\\begin{equation}\nP_{\\mathrm{ob}}\\Fcal(u)=P_{\\mathrm{ob}}r+P_{\\mathrm{ob}}N(u).\n\\end{equation}\nApply the reverse triangle inequality and $\\norm{\\Fcal(u)}\\geq\\norm{P_{\\mathrm{ob}}\\Fcal(u)}$.\n\\end{proof}","\\begin{theorem}[Newton--Kantorovich repair criterion]\n\\label{thm:newton-kantorovich}\nLet $F:U\\to V$ 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 $\\overline B(u_0,R)$,\n\\begin{equation}\n\\norm{F'(u)-F'(v)}\\leq L\\norm{u-v}.\n\\end{equation}\nSet\n\\begin{equation}\n\\eta:=\\norm{BF(u_0)},\n\\qquad\nh:=\\norm B\\,L\\eta.\n\\end{equation}\nIf $h\\leq\\frac12$ and\n\\begin{equation}\nt_*:=\\frac{1-\\sqrt{1-2h}}{\\norm B\\,L}\\leq R,\n\\end{equation}\nthen Newton iteration\n\\begin{equation}\nu_{n+1}=u_n-F'(u_n)^{-1}F(u_n)\n\\end{equation}\nis well-defined, remains in $\\overline B(u_0,t_*)$, and converges to a root $u^*$ of $F$. If the larger radius\n\\begin{equation}\nt^{**}:=\\frac{1+\\sqrt{1-2h}}{\\norm B\\,L}\n\\end{eq","\\begin{theorem}[Newton--Kantorovich repair criterion]\n\\label{thm:newton-kantorovich}\nLet $F:U\\to V$ 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 $\\overline B(u_0,R)$,\n\\begin{equation}\n\\norm{F'(u)-F'(v)}\\leq L\\norm{u-v}.\n\\end{equation}\nSet\n\\begin{equation}\n\\eta:=\\norm{BF(u_0)},\n\\qquad\nh:=\\norm B\\,L\\eta.\n\\end{equation}\nIf $h\\leq\\frac12$ and\n\\begin{equation}\nt_*:=\\frac{1-\\sqrt{1-2h}}{\\norm B\\,L}\\leq R,\n\\end{equation}\nthen Newton iteration\n\\begin{equation}\nu_{n+1}=u_n-F'(u_n)^{-1}F(u_n)\n\\end{equation}\nis well-defined, remains in $\\overline B(u_0,t_*)$, and converges to a root $u^*$ of $F$. If the larger radius\n\\begin{equation}\nt^{**}:=\\frac{1+\\sqrt{1-2h}}{\\norm B\\,L}\n\\end{equation}\nis contained in the regularity region, the root is unique in $\\overline","\\begin{theorem}[Newton--Kantorovich repair criterion]\n\\label{thm:newton-kantorovich}\nLet $F:U\\to V$ 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 $\\overline B(u_0,R)$,\n\\begin{equation}\n\\norm{F'(u)-F'(v)}\\leq L\\norm{u-v}.\n\\end{equation}\nSet\n\\begin{equation}\n\\eta:=\\norm{BF(u_0)},\n\\qquad\nh:=\\norm B\\,L\\eta.\n\\end{equation}\nIf $h\\leq\\frac12$ and\n\\begin{equation}\nt_*:=\\frac{1-\\sqrt{1-2h}}{\\norm B\\,L}\\leq R,\n\\end{equation}\nthen Newton iteration\n\\begin{equation}\nu_{n+1}=u_n-F'(u_n)^{-1}F(u_n)\n\\end{equation}\nis well-defined, remains in $\\overline B(u_0,t_*)$, and converges to a root $u^*$ of $F$. If the larger radius\n\\begin{equation}\nt^{**}:=\\frac{1+\\sqrt{1-2h}}{\\norm B\\,L}\n\\end{equation}\nis contained in the regularity region, the root is unique in $\\overline B(u_0,t^{**})$.\n\\end{","\\begin{theorem}[Newton--Kantorovich repair criterion]\n\\label{thm:newton-kantorovich}\nLet $F:U\\to V$ 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 $\\overline B(u_0,R)$,\n\\begin{equation}\n\\norm{F'(u)-F'(v)}\\leq L\\norm{u-v}.\n\\end{equation}\nSet\n\\begin{equation}\n\\eta:=\\norm{BF(u_0)},\n\\qquad\nh:=\\norm B\\,L\\eta.\n\\end{equation}\nIf $h\\leq\\frac12$ and\n\\begin{equation}\nt_*:=\\frac{1-\\sqrt{1-2h}}{\\norm B\\,L}\\leq R,\n\\end{equation}\nthen Newton iteration\n\\begin{equation}\nu_{n+1}=u_n-F'(u_n)^{-1}F(u_n)\n\\end{equation}\nis well-defined, remains in $\\overline B(u_0,t_*)$, and converges to a root $u^*$ of $F$. If the larger radius\n\\begin{equation}\nt^{**}:=\\frac{1+\\sqrt{1-2h}}{\\norm B\\,L}\n\\end{equation}\nis contained in the regularity region, the root is unique in $\\overline B(u_0,t^{**})$.\n\\end{theorem}","criterion]\n\\label{thm:newton-kantorovich}\nLet $F:U\\to V$ 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 $\\overline B(u_0,R)$,\n\\begin{equation}\n\\norm{F'(u)-F'(v)}\\leq L\\norm{u-v}.\n\\end{equation}\nSet\n\\begin{equation}\n\\eta:=\\norm{BF(u_0)},\n\\qquad\nh:=\\norm B\\,L\\eta.\n\\end{equation}\nIf $h\\leq\\frac12$ and\n\\begin{equation}\nt_*:=\\frac{1-\\sqrt{1-2h}}{\\norm B\\,L}\\leq R,\n\\end{equation}\nthen Newton iteration\n\\begin{equation}\nu_{n+1}=u_n-F'(u_n)^{-1}F(u_n)\n\\end{equation}\nis well-defined, remains in $\\overline B(u_0,t_*)$, and converges to a root $u^*$ of $F$. If the larger radius\n\\begin{equation}\nt^{**}:=\\frac{1+\\sqrt{1-2h}}{\\norm B\\,L}\n\\end{equation}\nis contained in the regularity region, the root is unique in $\\overline B(u_0,t^{**})$.\n\\end{theorem}","orovich}\nLet $F:U\\to V$ 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 $\\overline B(u_0,R)$,\n\\begin{equation}\n\\norm{F'(u)-F'(v)}\\leq L\\norm{u-v}.\n\\end{equation}\nSet\n\\begin{equation}\n\\eta:=\\norm{BF(u_0)},\n\\qquad\nh:=\\norm B\\,L\\eta.\n\\end{equation}\nIf $h\\leq\\frac12$ and\n\\begin{equation}\nt_*:=\\frac{1-\\sqrt{1-2h}}{\\norm B\\,L}\\leq R,\n\\end{equation}\nthen Newton iteration\n\\begin{equation}\nu_{n+1}=u_n-F'(u_n)^{-1}F(u_n)\n\\end{equation}\nis well-defined, remains in $\\overline B(u_0,t_*)$, and converges to a root $u^*$ of $F$. If the larger radius\n\\begin{equation}\nt^{**}:=\\frac{1+\\sqrt{1-2h}}{\\norm B\\,L}\n\\end{equation}\nis contained in the regularity region, the root is unique in $\\overline B(u_0,t^{**})$.\n\\end{theorem}","U$ and suppose $F'(u_0)$ is invertible with inverse $B$. Assume that on the closed ball $\\overline B(u_0,R)$,\n\\begin{equation}\n\\norm{F'(u)-F'(v)}\\leq L\\norm{u-v}.\n\\end{equation}\nSet\n\\begin{equation}\n\\eta:=\\norm{BF(u_0)},\n\\qquad\nh:=\\norm B\\,L\\eta.\n\\end{equation}\nIf $h\\leq\\frac12$ and\n\\begin{equation}\nt_*:=\\frac{1-\\sqrt{1-2h}}{\\norm B\\,L}\\leq R,\n\\end{equation}\nthen Newton iteration\n\\begin{equation}\nu_{n+1}=u_n-F'(u_n)^{-1}F(u_n)\n\\end{equation}\nis well-defined, remains in $\\overline B(u_0,t_*)$, and converges to a root $u^*$ of $F$. If the larger radius\n\\begin{equation}\nt^{**}:=\\frac{1+\\sqrt{1-2h}}{\\norm B\\,L}\n\\end{equation}\nis contained in the regularity region, the root is unique in $\\overline B(u_0,t^{**})$.\n\\end{theorem}","closed ball $\\overline B(u_0,R)$,\n\\begin{equation}\n\\norm{F'(u)-F'(v)}\\leq L\\norm{u-v}.\n\\end{equation}\nSet\n\\begin{equation}\n\\eta:=\\norm{BF(u_0)},\n\\qquad\nh:=\\norm B\\,L\\eta.\n\\end{equation}\nIf $h\\leq\\frac12$ and\n\\begin{equation}\nt_*:=\\frac{1-\\sqrt{1-2h}}{\\norm B\\,L}\\leq R,\n\\end{equation}\nthen Newton iteration\n\\begin{equation}\nu_{n+1}=u_n-F'(u_n)^{-1}F(u_n)\n\\end{equation}\nis well-defined, remains in $\\overline B(u_0,t_*)$, and converges to a root $u^*$ of $F$. If the larger radius\n\\begin{equation}\nt^{**}:=\\frac{1+\\sqrt{1-2h}}{\\norm B\\,L}\n\\end{equation}\nis contained in the regularity region, the root is unique in $\\overline B(u_0,t^{**})$.\n\\end{theorem}","on}\n\\norm{F'(u)-F'(v)}\\leq L\\norm{u-v}.\n\\end{equation}\nSet\n\\begin{equation}\n\\eta:=\\norm{BF(u_0)},\n\\qquad\nh:=\\norm B\\,L\\eta.\n\\end{equation}\nIf $h\\leq\\frac12$ and\n\\begin{equation}\nt_*:=\\frac{1-\\sqrt{1-2h}}{\\norm B\\,L}\\leq R,\n\\end{equation}\nthen Newton iteration\n\\begin{equation}\nu_{n+1}=u_n-F'(u_n)^{-1}F(u_n)\n\\end{equation}\nis well-defined, remains in $\\overline B(u_0,t_*)$, and converges to a root $u^*$ of $F$. If the larger radius\n\\begin{equation}\nt^{**}:=\\frac{1+\\sqrt{1-2h}}{\\norm B\\,L}\n\\end{equation}\nis contained in the regularity region, the root is unique in $\\overline B(u_0,t^{**})$.\n\\end{theorem}","{F'(u)-F'(v)}\\leq L\\norm{u-v}.\n\\end{equation}\nSet\n\\begin{equation}\n\\eta:=\\norm{BF(u_0)},\n\\qquad\nh:=\\norm B\\,L\\eta.\n\\end{equation}\nIf $h\\leq\\frac12$ and\n\\begin{equation}\nt_*:=\\frac{1-\\sqrt{1-2h}}{\\norm B\\,L}\\leq R,\n\\end{equation}\nthen Newton iteration\n\\begin{equation}\nu_{n+1}=u_n-F'(u_n)^{-1}F(u_n)\n\\end{equation}\nis well-defined, remains in $\\overline B(u_0,t_*)$, and converges to a root $u^*$ of $F$. If the larger radius\n\\begin{equation}\nt^{**}:=\\frac{1+\\sqrt{1-2h}}{\\norm B\\,L}\n\\end{equation}\nis contained in the regularity region, the root is unique in $\\overline B(u_0,t^{**})$.\n\\end{theorem}","ion}\nSet\n\\begin{equation}\n\\eta:=\\norm{BF(u_0)},\n\\qquad\nh:=\\norm B\\,L\\eta.\n\\end{equation}\nIf $h\\leq\\frac12$ and\n\\begin{equation}\nt_*:=\\frac{1-\\sqrt{1-2h}}{\\norm B\\,L}\\leq R,\n\\end{equation}\nthen Newton iteration\n\\begin{equation}\nu_{n+1}=u_n-F'(u_n)^{-1}F(u_n)\n\\end{equation}\nis well-defined, remains in $\\overline B(u_0,t_*)$, and converges to a root $u^*$ of $F$. If the larger radius\n\\begin{equation}\nt^{**}:=\\frac{1+\\sqrt{1-2h}}{\\norm B\\,L}\n\\end{equation}\nis contained in the regularity region, the root is unique in $\\overline B(u_0,t^{**})$.\n\\end{theorem}","quation}\nt_*:=\\frac{1-\\sqrt{1-2h}}{\\norm B\\,L}\\leq R,\n\\end{equation}\nthen Newton iteration\n\\begin{equation}\nu_{n+1}=u_n-F'(u_n)^{-1}F(u_n)\n\\end{equation}\nis well-defined, remains in $\\overline B(u_0,t_*)$, and converges to a root $u^*$ of $F$. If the larger radius\n\\begin{equation}\nt^{**}:=\\frac{1+\\sqrt{1-2h}}{\\norm B\\,L}\n\\end{equation}\nis contained in the regularity region, the root is unique in $\\overline B(u_0,t^{**})$.\n\\end{theorem}","\\begin{proof}\nConsider the scalar majorant\n\\begin{equation}\nf(t)=\\eta-t+\\frac{\\norm B\\,L}{2}t^2.\n\\end{equation}\nThe 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.\n\\end{proof}","quation}\nf(t)=\\eta-t+\\frac{\\norm B\\,L}{2}t^2.\n\\end{equation}\nThe 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.\n\\end{proof}","}{2}t^2.\n\\end{equation}\nThe 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.\n\\end{proof}","e 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.\n\\end{proof}","{**}$. 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.\n\\end{proof}","The criterion makes three quantities auditable: the initial predicted correction size $\\eta$, inverse sensitivity $\\norm B$, and derivative curvature $L$.","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.","\\begin{theorem}[Lyapunov--Schmidt repair reduction]\n\\label{thm:lyapunov-schmidt}\nLet $F:P\\times U\\to V$ be $C^k$, $k\\geq2$, between finite-dimensional spaces, with $F(0,0)=0$. Let\n\\begin{equation}\nD:=\\partial_uF(0,0).\n\\end{equation}\nChoose splittings\n\\begin{equation}\nU=K\\oplus X,\n\\qquad\nK=\\ker D,\n\\end{equation}\nand\n\\begin{equation}\nV=R\\oplus C,\n\\qquad\nR=\\im D,\n\\end{equation}\nso 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\n\\begin{equation}\n\\chi:P\\times K\\to X,\n\\qquad\n\\chi(0,0)=0,\n\\end{equation}\nsuch that\n\\begin{equation}\nP_RF(p,k+\\chi(p,k))=0.\n\\end{equation}\nDefine the reduced obstruction map\n\\begin{equation}\n\\psi(p,k):=P_CF(p,k+\\chi(p,k)).\n\\end{equation}\nA sufficiently sma","\\begin{theorem}[Lyapunov--Schmidt repair reduction]\n\\label{thm:lyapunov-schmidt}\nLet $F:P\\times U\\to V$ be $C^k$, $k\\geq2$, between finite-dimensional spaces, with $F(0,0)=0$. Let\n\\begin{equation}\nD:=\\partial_uF(0,0).\n\\end{equation}\nChoose splittings\n\\begin{equation}\nU=K\\oplus X,\n\\qquad\nK=\\ker D,\n\\end{equation}\nand\n\\begin{equation}\nV=R\\oplus C,\n\\qquad\nR=\\im D,\n\\end{equation}\nso 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\n\\begin{equation}\n\\chi:P\\times K\\to X,\n\\qquad\n\\chi(0,0)=0,\n\\end{equation}\nsuch that\n\\begin{equation}\nP_RF(p,k+\\chi(p,k))=0.\n\\end{equation}\nDefine the reduced obstruction map\n\\begin{equation}\n\\psi(p,k):=P_CF(p,k+\\chi(p,k)).\n\\end{equation}\nA sufficiently small pair $","\\begin{theorem}[Lyapunov--Schmidt repair reduction]\n\\label{thm:lyapunov-schmidt}\nLet $F:P\\times U\\to V$ be $C^k$, $k\\geq2$, between finite-dimensional spaces, with $F(0,0)=0$. Let\n\\begin{equation}\nD:=\\partial_uF(0,0).\n\\end{equation}\nChoose splittings\n\\begin{equation}\nU=K\\oplus X,\n\\qquad\nK=\\ker D,\n\\end{equation}\nand\n\\begin{equation}\nV=R\\oplus C,\n\\qquad\nR=\\im D,\n\\end{equation}\nso 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\n\\begin{equation}\n\\chi:P\\times K\\to X,\n\\qquad\n\\chi(0,0)=0,\n\\end{equation}\nsuch that\n\\begin{equation}\nP_RF(p,k+\\chi(p,k))=0.\n\\end{equation}\nDefine the reduced obstruction map\n\\begin{equation}\n\\psi(p,k):=P_CF(p,k+\\chi(p,k)).\n\\end{equation}\nA sufficiently small pair $(p,u)$ sol","\\begin{theorem}[Lyapunov--Schmidt repair reduction]\n\\label{thm:lyapunov-schmidt}\nLet $F:P\\times U\\to V$ be $C^k$, $k\\geq2$, between finite-dimensional spaces, with $F(0,0)=0$. Let\n\\begin{equation}\nD:=\\partial_uF(0,0).\n\\end{equation}\nChoose splittings\n\\begin{equation}\nU=K\\oplus X,\n\\qquad\nK=\\ker D,\n\\end{equation}\nand\n\\begin{equation}\nV=R\\oplus C,\n\\qquad\nR=\\im D,\n\\end{equation}\nso 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\n\\begin{equation}\n\\chi:P\\times K\\to X,\n\\qquad\n\\chi(0,0)=0,\n\\end{equation}\nsuch that\n\\begin{equation}\nP_RF(p,k+\\chi(p,k))=0.\n\\end{equation}\nDefine the reduced obstruction map\n\\begin{equation}\n\\psi(p,k):=P_CF(p,k+\\chi(p,k)).\n\\end{equation}\nA sufficiently small pair $(p,u)$ solves $F(p,u)=0$ if and only if\n\\begin{equation}\nu=k+\\","\\begin{theorem}[Lyapunov--Schmidt repair reduction]\n\\label{thm:lyapunov-schmidt}\nLet $F:P\\times U\\to V$ be $C^k$, $k\\geq2$, between finite-dimensional spaces, with $F(0,0)=0$. Let\n\\begin{equation}\nD:=\\partial_uF(0,0).\n\\end{equation}\nChoose splittings\n\\begin{equation}\nU=K\\oplus X,\n\\qquad\nK=\\ker D,\n\\end{equation}\nand\n\\begin{equation}\nV=R\\oplus C,\n\\qquad\nR=\\im D,\n\\end{equation}\nso 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\n\\begin{equation}\n\\chi:P\\times K\\to X,\n\\qquad\n\\chi(0,0)=0,\n\\end{equation}\nsuch that\n\\begin{equation}\nP_RF(p,k+\\chi(p,k))=0.\n\\end{equation}\nDefine the reduced obstruction map\n\\begin{equation}\n\\psi(p,k):=P_CF(p,k+\\chi(p,k)).\n\\end{equation}\nA sufficiently small pair $(p,u)$ solves $F(p,u)=0$ if and only if\n\\begin{equation}\nu=k+\\chi(p,k)\n\\end{equation}\nfor some $k\\in K$ sat","\\begin{theorem}[Lyapunov--Schmidt repair reduction]\n\\label{thm:lyapunov-schmidt}\nLet $F:P\\times U\\to V$ be $C^k$, $k\\geq2$, between finite-dimensional spaces, with $F(0,0)=0$. Let\n\\begin{equation}\nD:=\\partial_uF(0,0).\n\\end{equation}\nChoose splittings\n\\begin{equation}\nU=K\\oplus X,\n\\qquad\nK=\\ker D,\n\\end{equation}\nand\n\\begin{equation}\nV=R\\oplus C,\n\\qquad\nR=\\im D,\n\\end{equation}\nso 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\n\\begin{equation}\n\\chi:P\\times K\\to X,\n\\qquad\n\\chi(0,0)=0,\n\\end{equation}\nsuch that\n\\begin{equation}\nP_RF(p,k+\\chi(p,k))=0.\n\\end{equation}\nDefine the reduced obstruction map\n\\begin{equation}\n\\psi(p,k):=P_CF(p,k+\\chi(p,k)).\n\\end{equation}\nA sufficiently small pair $(p,u)$ solves $F(p,u)=0$ if and only if\n\\begin{equation}\nu=k+\\chi(p,k)\n\\end{equation}\nfor some $k\\in K$ satisfying\n\\begin{equation}\n\\psi(p,k)=0.\n\\end{equation}\n\\end{theorem}","t repair reduction]\n\\label{thm:lyapunov-schmidt}\nLet $F:P\\times U\\to V$ be $C^k$, $k\\geq2$, between finite-dimensional spaces, with $F(0,0)=0$. Let\n\\begin{equation}\nD:=\\partial_uF(0,0).\n\\end{equation}\nChoose splittings\n\\begin{equation}\nU=K\\oplus X,\n\\qquad\nK=\\ker D,\n\\end{equation}\nand\n\\begin{equation}\nV=R\\oplus C,\n\\qquad\nR=\\im D,\n\\end{equation}\nso 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\n\\begin{equation}\n\\chi:P\\times K\\to X,\n\\qquad\n\\chi(0,0)=0,\n\\end{equation}\nsuch that\n\\begin{equation}\nP_RF(p,k+\\chi(p,k))=0.\n\\end{equation}\nDefine the reduced obstruction map\n\\begin{equation}\n\\psi(p,k):=P_CF(p,k+\\chi(p,k)).\n\\end{equation}\nA sufficiently small pair $(p,u)$ solves $F(p,u)=0$ if and only if\n\\begin{equation}\nu=k+\\chi(p,k)\n\\end{equation}\nfor some $k\\in K$ satisfying\n\\begin{equation}\n\\psi(p,k)=0.\n\\end{equation}\n\\end{theorem}","eduction]\n\\label{thm:lyapunov-schmidt}\nLet $F:P\\times U\\to V$ be $C^k$, $k\\geq2$, between finite-dimensional spaces, with $F(0,0)=0$. Let\n\\begin{equation}\nD:=\\partial_uF(0,0).\n\\end{equation}\nChoose splittings\n\\begin{equation}\nU=K\\oplus X,\n\\qquad\nK=\\ker D,\n\\end{equation}\nand\n\\begin{equation}\nV=R\\oplus C,\n\\qquad\nR=\\im D,\n\\end{equation}\nso 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\n\\begin{equation}\n\\chi:P\\times K\\to X,\n\\qquad\n\\chi(0,0)=0,\n\\end{equation}\nsuch that\n\\begin{equation}\nP_RF(p,k+\\chi(p,k))=0.\n\\end{equation}\nDefine the reduced obstruction map\n\\begin{equation}\n\\psi(p,k):=P_CF(p,k+\\chi(p,k)).\n\\end{equation}\nA sufficiently small pair $(p,u)$ solves $F(p,u)=0$ if and only if\n\\begin{equation}\nu=k+\\chi(p,k)\n\\end{equation}\nfor some $k\\in K$ satisfying\n\\begin{equation}\n\\psi(p,k)=0.\n\\end{equation}\n\\end{theorem}","-dimensional spaces, with $F(0,0)=0$. Let\n\\begin{equation}\nD:=\\partial_uF(0,0).\n\\end{equation}\nChoose splittings\n\\begin{equation}\nU=K\\oplus X,\n\\qquad\nK=\\ker D,\n\\end{equation}\nand\n\\begin{equation}\nV=R\\oplus C,\n\\qquad\nR=\\im D,\n\\end{equation}\nso 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\n\\begin{equation}\n\\chi:P\\times K\\to X,\n\\qquad\n\\chi(0,0)=0,\n\\end{equation}\nsuch that\n\\begin{equation}\nP_RF(p,k+\\chi(p,k))=0.\n\\end{equation}\nDefine the reduced obstruction map\n\\begin{equation}\n\\psi(p,k):=P_CF(p,k+\\chi(p,k)).\n\\end{equation}\nA sufficiently small pair $(p,u)$ solves $F(p,u)=0$ if and only if\n\\begin{equation}\nu=k+\\chi(p,k)\n\\end{equation}\nfor some $k\\in K$ satisfying\n\\begin{equation}\n\\psi(p,k)=0.\n\\end{equation}\n\\end{theorem}","$F(0,0)=0$. Let\n\\begin{equation}\nD:=\\partial_uF(0,0).\n\\end{equation}\nChoose splittings\n\\begin{equation}\nU=K\\oplus X,\n\\qquad\nK=\\ker D,\n\\end{equation}\nand\n\\begin{equation}\nV=R\\oplus C,\n\\qquad\nR=\\im D,\n\\end{equation}\nso 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\n\\begin{equation}\n\\chi:P\\times K\\to X,\n\\qquad\n\\chi(0,0)=0,\n\\end{equation}\nsuch that\n\\begin{equation}\nP_RF(p,k+\\chi(p,k))=0.\n\\end{equation}\nDefine the reduced obstruction map\n\\begin{equation}\n\\psi(p,k):=P_CF(p,k+\\chi(p,k)).\n\\end{equation}\nA sufficiently small pair $(p,u)$ solves $F(p,u)=0$ if and only if\n\\begin{equation}\nu=k+\\chi(p,k)\n\\end{equation}\nfor some $k\\in K$ satisfying\n\\begin{equation}\n\\psi(p,k)=0.\n\\end{equation}\n\\end{theorem}","ings\n\\begin{equation}\nU=K\\oplus X,\n\\qquad\nK=\\ker D,\n\\end{equation}\nand\n\\begin{equation}\nV=R\\oplus C,\n\\qquad\nR=\\im D,\n\\end{equation}\nso 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\n\\begin{equation}\n\\chi:P\\times K\\to X,\n\\qquad\n\\chi(0,0)=0,\n\\end{equation}\nsuch that\n\\begin{equation}\nP_RF(p,k+\\chi(p,k))=0.\n\\end{equation}\nDefine the reduced obstruction map\n\\begin{equation}\n\\psi(p,k):=P_CF(p,k+\\chi(p,k)).\n\\end{equation}\nA sufficiently small pair $(p,u)$ solves $F(p,u)=0$ if and only if\n\\begin{equation}\nu=k+\\chi(p,k)\n\\end{equation}\nfor some $k\\in K$ satisfying\n\\begin{equation}\n\\psi(p,k)=0.\n\\end{equation}\n\\end{theorem}","R\\oplus C,\n\\qquad\nR=\\im D,\n\\end{equation}\nso 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\n\\begin{equation}\n\\chi:P\\times K\\to X,\n\\qquad\n\\chi(0,0)=0,\n\\end{equation}\nsuch that\n\\begin{equation}\nP_RF(p,k+\\chi(p,k))=0.\n\\end{equation}\nDefine the reduced obstruction map\n\\begin{equation}\n\\psi(p,k):=P_CF(p,k+\\chi(p,k)).\n\\end{equation}\nA sufficiently small pair $(p,u)$ solves $F(p,u)=0$ if and only if\n\\begin{equation}\nu=k+\\chi(p,k)\n\\end{equation}\nfor some $k\\in K$ satisfying\n\\begin{equation}\n\\psi(p,k)=0.\n\\end{equation}\n\\end{theorem}","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\n\\begin{equation}\n\\chi:P\\times K\\to X,\n\\qquad\n\\chi(0,0)=0,\n\\end{equation}\nsuch that\n\\begin{equation}\nP_RF(p,k+\\chi(p,k))=0.\n\\end{equation}\nDefine the reduced obstruction map\n\\begin{equation}\n\\psi(p,k):=P_CF(p,k+\\chi(p,k)).\n\\end{equation}\nA sufficiently small pair $(p,u)$ solves $F(p,u)=0$ if and only if\n\\begin{equation}\nu=k+\\chi(p,k)\n\\end{equation}\nfor some $k\\in K$ satisfying\n\\begin{equation}\n\\psi(p,k)=0.\n\\end{equation}\n\\end{theorem}","Let $P_R$ and $P_C$ be the corresponding projections. Then there are neighborhoods of the origin and a unique $C^k$ map\n\\begin{equation}\n\\chi:P\\times K\\to X,\n\\qquad\n\\chi(0,0)=0,\n\\end{equation}\nsuch that\n\\begin{equation}\nP_RF(p,k+\\chi(p,k))=0.\n\\end{equation}\nDefine the reduced obstruction map\n\\begin{equation}\n\\psi(p,k):=P_CF(p,k+\\chi(p,k)).\n\\end{equation}\nA sufficiently small pair $(p,u)$ solves $F(p,u)=0$ if and only if\n\\begin{equation}\nu=k+\\chi(p,k)\n\\end{equation}\nfor some $k\\in K$ satisfying\n\\begin{equation}\n\\psi(p,k)=0.\n\\end{equation}\n\\end{theorem}","projections. Then there are neighborhoods of the origin and a unique $C^k$ map\n\\begin{equation}\n\\chi:P\\times K\\to X,\n\\qquad\n\\chi(0,0)=0,\n\\end{equation}\nsuch that\n\\begin{equation}\nP_RF(p,k+\\chi(p,k))=0.\n\\end{equation}\nDefine the reduced obstruction map\n\\begin{equation}\n\\psi(p,k):=P_CF(p,k+\\chi(p,k)).\n\\end{equation}\nA sufficiently small pair $(p,u)$ solves $F(p,u)=0$ if and only if\n\\begin{equation}\nu=k+\\chi(p,k)\n\\end{equation}\nfor some $k\\in K$ satisfying\n\\begin{equation}\n\\psi(p,k)=0.\n\\end{equation}\n\\end{theorem}","ds of the origin and a unique $C^k$ map\n\\begin{equation}\n\\chi:P\\times K\\to X,\n\\qquad\n\\chi(0,0)=0,\n\\end{equation}\nsuch that\n\\begin{equation}\nP_RF(p,k+\\chi(p,k))=0.\n\\end{equation}\nDefine the reduced obstruction map\n\\begin{equation}\n\\psi(p,k):=P_CF(p,k+\\chi(p,k)).\n\\end{equation}\nA sufficiently small pair $(p,u)$ solves $F(p,u)=0$ if and only if\n\\begin{equation}\nu=k+\\chi(p,k)\n\\end{equation}\nfor some $k\\in K$ satisfying\n\\begin{equation}\n\\psi(p,k)=0.\n\\end{equation}\n\\end{theorem}","map\n\\begin{equation}\n\\chi:P\\times K\\to X,\n\\qquad\n\\chi(0,0)=0,\n\\end{equation}\nsuch that\n\\begin{equation}\nP_RF(p,k+\\chi(p,k))=0.\n\\end{equation}\nDefine the reduced obstruction map\n\\begin{equation}\n\\psi(p,k):=P_CF(p,k+\\chi(p,k)).\n\\end{equation}\nA sufficiently small pair $(p,u)$ solves $F(p,u)=0$ if and only if\n\\begin{equation}\nu=k+\\chi(p,k)\n\\end{equation}\nfor some $k\\in K$ satisfying\n\\begin{equation}\n\\psi(p,k)=0.\n\\end{equation}\n\\end{theorem}","\\begin{proof}\nApply the implicit-function theorem to the range equation\n\\begin{equation}\nG(p,k,x):=P_RF(p,k+x)=0.\n\\end{equation}\nIts 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 $\\psi(p,k)=0$.\n\\end{proof}","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.","Consider the specific formal correction equation\n\\begin{equation}\nF(\\eps,u)\n=\n\\eps r_1+Du+\\frac12B(u,u)+O\\bigl(\\eps^3+\\norm u^3\\bigr),\n\\label{eq:second-order-model}\n\\end{equation}\nwhere $B:U\\times U\\to V$ is symmetric bilinear. Seek\n\\begin{equation}\nu(\\eps)=\\eps u_1+\\eps^2u_2+O(\\eps^3).\n\\end{equation}","\\begin{proposition}[Second-order repair obstruction]\n\\label{prop:second-order-obstruction}\nA repair through order $\\eps^2$ exists only if\n\\begin{align}\nDu_1&=-r_1,","\\begin{proposition}[Second-order repair obstruction]\n\\label{prop:second-order-obstruction}\nA repair through order $\\eps^2$ exists only if\n\\begin{align}\nDu_1&=-r_1,\\\\\n\\left[\\frac12B(u_1,u_1)\\right]&=0\n\\quad\\text{in }\\coker D.\n\\end{align}\nWhen the second class vanishes, $u_2$ may be chosen to solve\n\\begin{equation}\nDu_2=-\\frac12B(u_1,u_1).\n\\end{equation}\nIf $\\ker D\\neq0$, replacing $u_1$ by $u_1+k$ can change the second-order obstruction class.\n\\end{proposition}","\\left[\\frac12B(u_1,u_1)\\right]&=0\n\\quad\\text{in }\\coker D.\n\\end{align}\nWhen the second class vanishes, $u_2$ may be chosen to solve\n\\begin{equation}\nDu_2=-\\frac12B(u_1,u_1).\n\\end{equation}\nIf $\\ker D\\neq0$, replacing $u_1$ by $u_1+k$ can change the second-order obstruction class.\n\\end{proposition}","\\begin{proof}\nSubstitute 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 $\\coker D$.\n\\end{proof}","\\begin{theorem}[Finite formal obstruction tower]\n\\label{thm:formal-obstruction-tower}\nLet $F(\\eps,u)$ 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\n\\begin{equation}\nF\\left(\\eps,\\sum_{j=1}^{n-1}\\eps^ju_j\\right)\n=\\eps^nb_n+O(\\eps^{n+1}).\n\\end{equation}\nThen extending the repair to order $n$ by adding $\\eps^nu_n$ is possible if and only if\n\\begin{equation}\n[b_n]=0\\quad\\text{in }\\coker D.\n\\end{equation}\nWhen it vanishes, any solution of $Du_n=-b_n$ produces an order-$n$ lift.\n\\end{theorem}","ower]\n\\label{thm:formal-obstruction-tower}\nLet $F(\\eps,u)$ 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\n\\begin{equation}\nF\\left(\\eps,\\sum_{j=1}^{n-1}\\eps^ju_j\\right)\n=\\eps^nb_n+O(\\eps^{n+1}).\n\\end{equation}\nThen extending the repair to order $n$ by adding $\\eps^nu_n$ is possible if and only if\n\\begin{equation}\n[b_n]=0\\quad\\text{in }\\coker D.\n\\end{equation}\nWhen it vanishes, any solution of $Du_n=-b_n$ produces an order-$n$ lift.\n\\end{theorem}","with $F(0,0)=0$ and $D=\\partial_uF(0,0)$. Suppose coefficients $u_1,\\ldots,u_{n-1}$ have been chosen so that\n\\begin{equation}\nF\\left(\\eps,\\sum_{j=1}^{n-1}\\eps^ju_j\\right)\n=\\eps^nb_n+O(\\eps^{n+1}).\n\\end{equation}\nThen extending the repair to order $n$ by adding $\\eps^nu_n$ is possible if and only if\n\\begin{equation}\n[b_n]=0\\quad\\text{in }\\coker D.\n\\end{equation}\nWhen it vanishes, any solution of $Du_n=-b_n$ produces an order-$n$ lift.\n\\end{theorem}","al_uF(0,0)$. Suppose coefficients $u_1,\\ldots,u_{n-1}$ have been chosen so that\n\\begin{equation}\nF\\left(\\eps,\\sum_{j=1}^{n-1}\\eps^ju_j\\right)\n=\\eps^nb_n+O(\\eps^{n+1}).\n\\end{equation}\nThen extending the repair to order $n$ by adding $\\eps^nu_n$ is possible if and only if\n\\begin{equation}\n[b_n]=0\\quad\\text{in }\\coker D.\n\\end{equation}\nWhen it vanishes, any solution of $Du_n=-b_n$ produces an order-$n$ lift.\n\\end{theorem}","\\begin{proof}\nAnalytic expansion gives\n\\begin{equation}\nF\\left(\\eps,\\sum_{j=1}^{n}\\eps^ju_j\\right)\n=\n\\eps^n(b_n+Du_n)+O(\\eps^{n+1}),\n\\end{equation}\nbecause $u_n$ first enters linearly at order $n$. Cancellation of the coefficient is equivalent to solvability of $Du_n=-b_n$.\n\\end{proof}","The sequence\n\\begin{equation}\n[b_1],[b_2],[b_3],\\ldots\\in\\coker D\n\\end{equation}\nis 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.","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\n\\begin{equation}\n\\mathfrak U=\\set{U_\\alpha}_{\\alpha\\in I}\n\\end{equation}\nbe a finite cover. Let $\\mathscr U$ be a sheaf of authorized corrections, $\\mathscr V$ a sheaf of defects, and\n\\begin{equation}\nD:\\mathscr U\\to\\mathscr V\n\\end{equation}\na morphism of sheaves. Let $r\\in\\Gamma(X,\\mathscr V)$ be a global defect.","repairs in every stratum while those repairs remain mutually incompatible. Let $X$ be a finite site of functional regions and let\n\\begin{equation}\n\\mathfrak U=\\set{U_\\alpha}_{\\alpha\\in I}\n\\end{equation}\nbe a finite cover. Let $\\mathscr U$ be a sheaf of authorized corrections, $\\mathscr V$ a sheaf of defects, and\n\\begin{equation}\nD:\\mathscr U\\to\\mathscr V\n\\end{equation}\na morphism of sheaves. Let $r\\in\\Gamma(X,\\mathscr V)$ be a global defect.","A local correction on $U_\\alpha$ is a section $u_\\alpha\\in\\mathscr U(U_\\alpha)$ satisfying\n\\begin{equation}\nD_\\alpha u_\\alpha=-r|_{U_\\alpha}.\n\\end{equation}\nIf one local correction exists, the local solution set is an affine space over $\\ker D(U_\\alpha)$.","\\begin{proposition}[Local solution torsor]\nFor each $U_\\alpha$, the set\n\\begin{equation}\n\\Sscr_\\alpha\n:=\n\\set{u\\in\\mathscr U(U_\\alpha):D_\\alpha u=-r|_{U_\\alpha}}\n\\end{equation}\nis 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 $\\Gamma(X,\\ker D)$.\n\\end{proposition}","\\begin{theorem}[Linear repair-descent theorem]\n\\label{thm:repair-descent}\nAssume a local correction $u_\\alpha$ exists on every $U_\\alpha$. On overlaps $U_{\\alpha\\beta}=U_\\alpha\\cap U_\\beta$, define\n\\begin{equation}\nd_{\\alpha\\beta}\n:=\n\\restr{u_\\alpha}{U_{\\alpha\\beta}}\n-\n\\restr{u_\\beta}{U_{\\alpha\\beta}}.\n\\end{equation}\nThen:\n\\begin{enumerate}\n\\item $d_{\\alpha\\beta}\\in\\ker D(U_{\\alpha\\beta})$;\n\\item $(d_{\\alpha\\beta})$ is a \\v{C}ech $1$-cocycle with coefficients in $\\ker D$;\n\\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 $\\mathfrak U$.\n\\end{enumerate}\nIf the cover computes first cohomology, the final condition is equivalent to\n\\begin{equation}\n[d_{\\alpha\\beta}]=0\n\\quad\\text{in}\\quad\n\\check H^1(X,\\ker D).","\\begin{theorem}[Linear repair-descent theorem]\n\\label{thm:repair-descent}\nAssume a local correction $u_\\alpha$ exists on every $U_\\alpha$. On overlaps $U_{\\alpha\\beta}=U_\\alpha\\cap U_\\beta$, define\n\\begin{equation}\nd_{\\alpha\\beta}\n:=\n\\restr{u_\\alpha}{U_{\\alpha\\beta}}\n-\n\\restr{u_\\beta}{U_{\\alpha\\beta}}.\n\\end{equation}\nThen:\n\\begin{enumerate}\n\\item $d_{\\alpha\\beta}\\in\\ker D(U_{\\alpha\\beta})$;\n\\item $(d_{\\alpha\\beta})$ is a \\v{C}ech $1$-cocycle with coefficients in $\\ker D$;\n\\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 $\\mathfrak U$.\n\\end{enumerate}\nIf the cover computes first cohomology, the final condition is equivalent to\n\\begin{equation}\n[d_{\\alpha\\beta}]=0\n\\quad\\text{in}\\quad\n\\check H^1(X,\\ker D).\n\\end{equation}\n\\end{theorem","\\begin{theorem}[Linear repair-descent theorem]\n\\label{thm:repair-descent}\nAssume a local correction $u_\\alpha$ exists on every $U_\\alpha$. On overlaps $U_{\\alpha\\beta}=U_\\alpha\\cap U_\\beta$, define\n\\begin{equation}\nd_{\\alpha\\beta}\n:=\n\\restr{u_\\alpha}{U_{\\alpha\\beta}}\n-\n\\restr{u_\\beta}{U_{\\alpha\\beta}}.\n\\end{equation}\nThen:\n\\begin{enumerate}\n\\item $d_{\\alpha\\beta}\\in\\ker D(U_{\\alpha\\beta})$;\n\\item $(d_{\\alpha\\beta})$ is a \\v{C}ech $1$-cocycle with coefficients in $\\ker D$;\n\\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 $\\mathfrak U$.\n\\end{enumerate}\nIf the cover computes first cohomology, the final condition is equivalent to\n\\begin{equation}\n[d_{\\alpha\\beta}]=0\n\\quad\\text{in}\\quad\n\\check H^1(X,\\ker D).\n\\end{equation}\n\\end{theorem}","begin{theorem}[Linear repair-descent theorem]\n\\label{thm:repair-descent}\nAssume a local correction $u_\\alpha$ exists on every $U_\\alpha$. On overlaps $U_{\\alpha\\beta}=U_\\alpha\\cap U_\\beta$, define\n\\begin{equation}\nd_{\\alpha\\beta}\n:=\n\\restr{u_\\alpha}{U_{\\alpha\\beta}}\n-\n\\restr{u_\\beta}{U_{\\alpha\\beta}}.\n\\end{equation}\nThen:\n\\begin{enumerate}\n\\item $d_{\\alpha\\beta}\\in\\ker D(U_{\\alpha\\beta})$;\n\\item $(d_{\\alpha\\beta})$ is a \\v{C}ech $1$-cocycle with coefficients in $\\ker D$;\n\\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 $\\mathfrak U$.\n\\end{enumerate}\nIf the cover computes first cohomology, the final condition is equivalent to\n\\begin{equation}\n[d_{\\alpha\\beta}]=0\n\\quad\\text{in}\\quad\n\\check H^1(X,\\ker D).\n\\end{equation}\n\\end{theorem}","nt theorem]\n\\label{thm:repair-descent}\nAssume a local correction $u_\\alpha$ exists on every $U_\\alpha$. On overlaps $U_{\\alpha\\beta}=U_\\alpha\\cap U_\\beta$, define\n\\begin{equation}\nd_{\\alpha\\beta}\n:=\n\\restr{u_\\alpha}{U_{\\alpha\\beta}}\n-\n\\restr{u_\\beta}{U_{\\alpha\\beta}}.\n\\end{equation}\nThen:\n\\begin{enumerate}\n\\item $d_{\\alpha\\beta}\\in\\ker D(U_{\\alpha\\beta})$;\n\\item $(d_{\\alpha\\beta})$ is a \\v{C}ech $1$-cocycle with coefficients in $\\ker D$;\n\\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 $\\mathfrak U$.\n\\end{enumerate}\nIf the cover computes first cohomology, the final condition is equivalent to\n\\begin{equation}\n[d_{\\alpha\\beta}]=0\n\\quad\\text{in}\\quad\n\\check H^1(X,\\ker D).\n\\end{equation}\n\\end{theorem}","cent}\nAssume a local correction $u_\\alpha$ exists on every $U_\\alpha$. On overlaps $U_{\\alpha\\beta}=U_\\alpha\\cap U_\\beta$, define\n\\begin{equation}\nd_{\\alpha\\beta}\n:=\n\\restr{u_\\alpha}{U_{\\alpha\\beta}}\n-\n\\restr{u_\\beta}{U_{\\alpha\\beta}}.\n\\end{equation}\nThen:\n\\begin{enumerate}\n\\item $d_{\\alpha\\beta}\\in\\ker D(U_{\\alpha\\beta})$;\n\\item $(d_{\\alpha\\beta})$ is a \\v{C}ech $1$-cocycle with coefficients in $\\ker D$;\n\\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 $\\mathfrak U$.\n\\end{enumerate}\nIf the cover computes first cohomology, the final condition is equivalent to\n\\begin{equation}\n[d_{\\alpha\\beta}]=0\n\\quad\\text{in}\\quad\n\\check H^1(X,\\ker D).\n\\end{equation}\n\\end{theorem}","alpha$ exists on every $U_\\alpha$. On overlaps $U_{\\alpha\\beta}=U_\\alpha\\cap U_\\beta$, define\n\\begin{equation}\nd_{\\alpha\\beta}\n:=\n\\restr{u_\\alpha}{U_{\\alpha\\beta}}\n-\n\\restr{u_\\beta}{U_{\\alpha\\beta}}.\n\\end{equation}\nThen:\n\\begin{enumerate}\n\\item $d_{\\alpha\\beta}\\in\\ker D(U_{\\alpha\\beta})$;\n\\item $(d_{\\alpha\\beta})$ is a \\v{C}ech $1$-cocycle with coefficients in $\\ker D$;\n\\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 $\\mathfrak U$.\n\\end{enumerate}\nIf the cover computes first cohomology, the final condition is equivalent to\n\\begin{equation}\n[d_{\\alpha\\beta}]=0\n\\quad\\text{in}\\quad\n\\check H^1(X,\\ker D).\n\\end{equation}\n\\end{theorem}","$. On overlaps $U_{\\alpha\\beta}=U_\\alpha\\cap U_\\beta$, define\n\\begin{equation}\nd_{\\alpha\\beta}\n:=\n\\restr{u_\\alpha}{U_{\\alpha\\beta}}\n-\n\\restr{u_\\beta}{U_{\\alpha\\beta}}.\n\\end{equation}\nThen:\n\\begin{enumerate}\n\\item $d_{\\alpha\\beta}\\in\\ker D(U_{\\alpha\\beta})$;\n\\item $(d_{\\alpha\\beta})$ is a \\v{C}ech $1$-cocycle with coefficients in $\\ker D$;\n\\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 $\\mathfrak U$.\n\\end{enumerate}\nIf the cover computes first cohomology, the final condition is equivalent to\n\\begin{equation}\n[d_{\\alpha\\beta}]=0\n\\quad\\text{in}\\quad\n\\check H^1(X,\\ker D).\n\\end{equation}\n\\end{theorem}","}=U_\\alpha\\cap U_\\beta$, define\n\\begin{equation}\nd_{\\alpha\\beta}\n:=\n\\restr{u_\\alpha}{U_{\\alpha\\beta}}\n-\n\\restr{u_\\beta}{U_{\\alpha\\beta}}.\n\\end{equation}\nThen:\n\\begin{enumerate}\n\\item $d_{\\alpha\\beta}\\in\\ker D(U_{\\alpha\\beta})$;\n\\item $(d_{\\alpha\\beta})$ is a \\v{C}ech $1$-cocycle with coefficients in $\\ker D$;\n\\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 $\\mathfrak U$.\n\\end{enumerate}\nIf the cover computes first cohomology, the final condition is equivalent to\n\\begin{equation}\n[d_{\\alpha\\beta}]=0\n\\quad\\text{in}\\quad\n\\check H^1(X,\\ker D).\n\\end{equation}\n\\end{theorem}","a\\beta}\n:=\n\\restr{u_\\alpha}{U_{\\alpha\\beta}}\n-\n\\restr{u_\\beta}{U_{\\alpha\\beta}}.\n\\end{equation}\nThen:\n\\begin{enumerate}\n\\item $d_{\\alpha\\beta}\\in\\ker D(U_{\\alpha\\beta})$;\n\\item $(d_{\\alpha\\beta})$ is a \\v{C}ech $1$-cocycle with coefficients in $\\ker D$;\n\\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 $\\mathfrak U$.\n\\end{enumerate}\nIf the cover computes first cohomology, the final condition is equivalent to\n\\begin{equation}\n[d_{\\alpha\\beta}]=0\n\\quad\\text{in}\\quad\n\\check H^1(X,\\ker D).\n\\end{equation}\n\\end{theorem}","em $d_{\\alpha\\beta}\\in\\ker D(U_{\\alpha\\beta})$;\n\\item $(d_{\\alpha\\beta})$ is a \\v{C}ech $1$-cocycle with coefficients in $\\ker D$;\n\\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 $\\mathfrak U$.\n\\end{enumerate}\nIf the cover computes first cohomology, the final condition is equivalent to\n\\begin{equation}\n[d_{\\alpha\\beta}]=0\n\\quad\\text{in}\\quad\n\\check H^1(X,\\ker D).\n\\end{equation}\n\\end{theorem}","\\begin{proof}\nOn every overlap,\n\\begin{equation}\nD(d_{\\alpha\\beta})\n=-r|_{U_{\\alpha\\beta}}+r|_{U_{\\alpha\\beta}}=0.\n\\end{equation}\nOn triple overlaps,\n\\begin{equation}\nd_{\\alpha\\beta}+d_{\\beta\\gamma}+d_{\\gamma\\alpha}=0,\n\\end{equation}\nso the family is a cocycle.","If a global repair $u$ exists, set $k_\\alpha=u|_{U_\\alpha}-u_\\alpha$. Then $Dk_\\alpha=0$ and\n\\begin{equation}\nd_{\\alpha\\beta}=k_\\beta-k_\\alpha,\n\\end{equation}\nso 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.\n\\end{proof}","\\begin{definition}[Two-level repair obstruction]\nThe local obstruction on $U_\\alpha$ is\n\\begin{equation}\n[r|_{U_\\alpha}]\\in\\coker D_\\alpha.\n\\end{equation}\nWhen every local class vanishes, the global gluing obstruction is\n\\begin{equation}\n[d_{\\alpha\\beta}]\\in\\check H^1(\\mathfrak U,\\ker D).\n\\end{equation}\n\\end{definition}","\\begin{proposition}[Choice independence of the gluing class]\nThe cohomology class $[d_{\\alpha\\beta}]$ does not depend on the particular family of local repairs used to compute it.\n\\end{proposition}","\\begin{proof}\nIf $u'_\\alpha=u_\\alpha+k_\\alpha$ with $k_\\alpha\\in\\ker D(U_\\alpha)$, then\n\\begin{equation}\nd'_{\\alpha\\beta}\n=d_{\\alpha\\beta}+k_\\alpha-k_\\beta.\n\\end{equation}\nThus the cocycles differ by a coboundary.\n\\end{proof}","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\n\\begin{equation}\ne=(e_a)_{a\\in E}\\in K^E\n\\end{equation}\nbe the transported overlap-mismatch vector. Let $B\\in\\R^{|I|\\times|E|}$ be the oriented incidence matrix and set\n\\begin{equation}\n\\mathbf B:=B\\otimes I_K.\n\\end{equation}\nA vertex gauge $k\\in K^I$ changes the mismatch to\n\\begin{equation}\ne+\\mathbf B^*k.\n\\end{equation}","d graph representing the nerve of the cover, let $K$ be a finite-dimensional Hilbert space representing a common trivialization of the kernel fibers, and let\n\\begin{equation}\ne=(e_a)_{a\\in E}\\in K^E\n\\end{equation}\nbe the transported overlap-mismatch vector. Let $B\\in\\R^{|I|\\times|E|}$ be the oriented incidence matrix and set\n\\begin{equation}\n\\mathbf B:=B\\otimes I_K.\n\\end{equation}\nA vertex gauge $k\\in K^I$ changes the mismatch to\n\\begin{equation}\ne+\\mathbf B^*k.\n\\end{equation}","K$ be a finite-dimensional Hilbert space representing a common trivialization of the kernel fibers, and let\n\\begin{equation}\ne=(e_a)_{a\\in E}\\in K^E\n\\end{equation}\nbe the transported overlap-mismatch vector. Let $B\\in\\R^{|I|\\times|E|}$ be the oriented incidence matrix and set\n\\begin{equation}\n\\mathbf B:=B\\otimes I_K.\n\\end{equation}\nA vertex gauge $k\\in K^I$ changes the mismatch to\n\\begin{equation}\ne+\\mathbf B^*k.\n\\end{equation}","\\begin{theorem}[Quantitative graph-Hodge gluing]\n\\label{thm:graph-hodge-gluing}\nLet\n\\begin{equation}\nL_G:=BB^*\n\\end{equation}\nbe the graph Laplacian and let $P_{\\mathrm{cyc}}$ be the orthogonal projector from $K^E$ onto\n\\begin{equation}\n\\ker\\mathbf B,\n\\end{equation}\nthe cycle space. Then\n\\begin{equation}\n\\inf_{k\\in K^I}\\norm{e+\\mathbf B^*k}\n=\n\\norm{P_{\\mathrm{cyc}}e}.\n\\end{equation}\nThe mean-zero minimum-norm gauge is\n\\begin{equation}\nk^*\n=-(L_G^\\dagger B\\otimes I_K)e.\n\\end{equation}\nIf $\\lambda_2(L_G)>0$ is the spectral gap, then\n\\begin{equation}\n\\norm{k^*}\n\\leq\n\\frac{1}{\\sqrt{\\lambda_2(L_G)}}\\norm e.\n\\end{equation}\nExact graph-level gluing is possible if and only if\n\\begin{equation}\nP_{\\mathrm{cyc}}e=0.\n\\end{equation}\n\\end{theorem}","ge gluing]\n\\label{thm:graph-hodge-gluing}\nLet\n\\begin{equation}\nL_G:=BB^*\n\\end{equation}\nbe the graph Laplacian and let $P_{\\mathrm{cyc}}$ be the orthogonal projector from $K^E$ onto\n\\begin{equation}\n\\ker\\mathbf B,\n\\end{equation}\nthe cycle space. Then\n\\begin{equation}\n\\inf_{k\\in K^I}\\norm{e+\\mathbf B^*k}\n=\n\\norm{P_{\\mathrm{cyc}}e}.\n\\end{equation}\nThe mean-zero minimum-norm gauge is\n\\begin{equation}\nk^*\n=-(L_G^\\dagger B\\otimes I_K)e.\n\\end{equation}\nIf $\\lambda_2(L_G)>0$ is the spectral gap, then\n\\begin{equation}\n\\norm{k^*}\n\\leq\n\\frac{1}{\\sqrt{\\lambda_2(L_G)}}\\norm e.\n\\end{equation}\nExact graph-level gluing is possible if and only if\n\\begin{equation}\nP_{\\mathrm{cyc}}e=0.\n\\end{equation}\n\\end{theorem}","uation}\nL_G:=BB^*\n\\end{equation}\nbe the graph Laplacian and let $P_{\\mathrm{cyc}}$ be the orthogonal projector from $K^E$ onto\n\\begin{equation}\n\\ker\\mathbf B,\n\\end{equation}\nthe cycle space. Then\n\\begin{equation}\n\\inf_{k\\in K^I}\\norm{e+\\mathbf B^*k}\n=\n\\norm{P_{\\mathrm{cyc}}e}.\n\\end{equation}\nThe mean-zero minimum-norm gauge is\n\\begin{equation}\nk^*\n=-(L_G^\\dagger B\\otimes I_K)e.\n\\end{equation}\nIf $\\lambda_2(L_G)>0$ is the spectral gap, then\n\\begin{equation}\n\\norm{k^*}\n\\leq\n\\frac{1}{\\sqrt{\\lambda_2(L_G)}}\\norm e.\n\\end{equation}\nExact graph-level gluing is possible if and only if\n\\begin{equation}\nP_{\\mathrm{cyc}}e=0.\n\\end{equation}\n\\end{theorem}","let $P_{\\mathrm{cyc}}$ be the orthogonal projector from $K^E$ onto\n\\begin{equation}\n\\ker\\mathbf B,\n\\end{equation}\nthe cycle space. Then\n\\begin{equation}\n\\inf_{k\\in K^I}\\norm{e+\\mathbf B^*k}\n=\n\\norm{P_{\\mathrm{cyc}}e}.\n\\end{equation}\nThe mean-zero minimum-norm gauge is\n\\begin{equation}\nk^*\n=-(L_G^\\dagger B\\otimes I_K)e.\n\\end{equation}\nIf $\\lambda_2(L_G)>0$ is the spectral gap, then\n\\begin{equation}\n\\norm{k^*}\n\\leq\n\\frac{1}{\\sqrt{\\lambda_2(L_G)}}\\norm e.\n\\end{equation}\nExact graph-level gluing is possible if and only if\n\\begin{equation}\nP_{\\mathrm{cyc}}e=0.\n\\end{equation}\n\\end{theorem}","in{equation}\n\\inf_{k\\in K^I}\\norm{e+\\mathbf B^*k}\n=\n\\norm{P_{\\mathrm{cyc}}e}.\n\\end{equation}\nThe mean-zero minimum-norm gauge is\n\\begin{equation}\nk^*\n=-(L_G^\\dagger B\\otimes I_K)e.\n\\end{equation}\nIf $\\lambda_2(L_G)>0$ is the spectral gap, then\n\\begin{equation}\n\\norm{k^*}\n\\leq\n\\frac{1}{\\sqrt{\\lambda_2(L_G)}}\\norm e.\n\\end{equation}\nExact graph-level gluing is possible if and only if\n\\begin{equation}\nP_{\\mathrm{cyc}}e=0.\n\\end{equation}\n\\end{theorem}","\\begin{proof}\nThe finite-dimensional Hodge decomposition is\n\\begin{equation}\nK^E=\\im\\mathbf B^*\\oplus\\ker\\mathbf B.\n\\end{equation}\nA gauge can remove exactly the gradient component in $\\im\\mathbf B^*$, leaving the cycle projection. The normal equation for the least-squares gauge is\n\\begin{equation}\n\\mathbf B\\mathbf B^*k=-\\mathbf Be,\n\\end{equation}\nwhose 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.\n\\end{proof}","tion}\nK^E=\\im\\mathbf B^*\\oplus\\ker\\mathbf B.\n\\end{equation}\nA gauge can remove exactly the gradient component in $\\im\\mathbf B^*$, leaving the cycle projection. The normal equation for the least-squares gauge is\n\\begin{equation}\n\\mathbf B\\mathbf B^*k=-\\mathbf Be,\n\\end{equation}\nwhose 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.\n\\end{proof}","hbf B^*\\oplus\\ker\\mathbf B.\n\\end{equation}\nA gauge can remove exactly the gradient component in $\\im\\mathbf B^*$, leaving the cycle projection. The normal equation for the least-squares gauge is\n\\begin{equation}\n\\mathbf B\\mathbf B^*k=-\\mathbf Be,\n\\end{equation}\nwhose 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.\n\\end{proof}","\\begin{definition}[Quantitative gluing obstruction]\nThe graph-level quantitative gluing obstruction is\n\\begin{equation}\n\\eta_{\\mathrm{glue}}(e)\n:=\\norm{P_{\\mathrm{cyc}}e}.\n\\end{equation}\nThe local-to-global repair certificate is the pair\n\\begin{equation}\n\\left(\n\\eta_{\\mathrm{loc}},\n\\eta_{\\mathrm{glue}}\n\\right),\n\\qquad\n\\eta_{\\mathrm{loc}}^2\n:=\\sum_{\\alpha}\\norm{D_\\alpha u_\\alpha+r_\\alpha}^2.\n\\end{equation}\n\\end{definition}","\\begin{corollary}[Trees have no cycle obstruction]\nIf 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.\n\\end{corollary}","\\begin{proof}\nA connected tree has $|E|=|I|-1$ and incidence matrix of full column rank, so its cycle space is zero.\n\\end{proof}","Different overlaps may have different uncertainty or semantic importance. Let $W_E$ be a positive-definite operator on $K^E$. Define\n\\begin{equation}\n\\eta_{\\mathrm{glue},W}(e)\n:=\n\\inf_k\\norm{e+\\mathbf B^*k}_{W_E}.\n\\end{equation}","\\begin{proposition}[Weighted gluing normal equation]\nEvery weighted least-squares gauge satisfies\n\\begin{equation}\n\\mathbf BW_E\\mathbf B^*k\n=-\\mathbf BW_Ee.\n\\end{equation}\nThe mean-zero minimum-norm minimizer is\n\\begin{equation}\nk_W^*\n=-\\bigl(\\mathbf BW_E\\mathbf B^*\\bigr)^\\dagger\\mathbf BW_Ee.\n\\end{equation}\nThe irreducible weighted mismatch is the $W_E$-orthogonal projection of $e$ onto the $W_E$-orthogonal complement of $\\im\\mathbf B^*$.\n\\end{proposition}","d gluing normal equation]\nEvery weighted least-squares gauge satisfies\n\\begin{equation}\n\\mathbf BW_E\\mathbf B^*k\n=-\\mathbf BW_Ee.\n\\end{equation}\nThe mean-zero minimum-norm minimizer is\n\\begin{equation}\nk_W^*\n=-\\bigl(\\mathbf BW_E\\mathbf B^*\\bigr)^\\dagger\\mathbf BW_Ee.\n\\end{equation}\nThe irreducible weighted mismatch is the $W_E$-orthogonal projection of $e$ onto the $W_E$-orthogonal complement of $\\im\\mathbf B^*$.\n\\end{proposition}","\\begin{proof}\nDifferentiate the quadratic objective\n\\begin{equation}\n\\frac12\\angles{e+\\mathbf B^*k,W_E(e+\\mathbf B^*k)}\n\\end{equation}\nwith respect to $k$. The normal equation follows. Its Moore--Penrose solution is the unique minimizer orthogonal to the constant vertex-gauge kernel.\n\\end{proof}","\\begin{example}[Locally repairable, globally incompatible]\nLet the nerve be a three-cycle with no triple intersection. Suppose the kernel fiber is $\\R$ and the oriented differences are\n\\begin{equation}\nd_{12}=1,\n\\qquad\nd_{23}=1,\n\\qquad\nd_{31}=1.\n\\end{equation}\nIf the cocycle were a coboundary, there would be $k_1,k_2,k_3$ with\n\\begin{equation}\nd_{12}=k_2-k_1,\n\\quad\nd_{23}=k_3-k_2,\n\\quad\nd_{31}=k_1-k_3.\n\\end{equation}\nSumming gives $3=0$, a contradiction. Hence no global correction exists. The graph-Hodge obstruction has norm $\\sqrt3$.\n\\end{example}","globally incompatible]\nLet the nerve be a three-cycle with no triple intersection. Suppose the kernel fiber is $\\R$ and the oriented differences are\n\\begin{equation}\nd_{12}=1,\n\\qquad\nd_{23}=1,\n\\qquad\nd_{31}=1.\n\\end{equation}\nIf the cocycle were a coboundary, there would be $k_1,k_2,k_3$ with\n\\begin{equation}\nd_{12}=k_2-k_1,\n\\quad\nd_{23}=k_3-k_2,\n\\quad\nd_{31}=k_1-k_3.\n\\end{equation}\nSumming gives $3=0$, a contradiction. Hence no global correction exists. The graph-Hodge obstruction has norm $\\sqrt3$.\n\\end{example}","ernel fiber is $\\R$ and the oriented differences are\n\\begin{equation}\nd_{12}=1,\n\\qquad\nd_{23}=1,\n\\qquad\nd_{31}=1.\n\\end{equation}\nIf the cocycle were a coboundary, there would be $k_1,k_2,k_3$ with\n\\begin{equation}\nd_{12}=k_2-k_1,\n\\quad\nd_{23}=k_3-k_2,\n\\quad\nd_{31}=k_1-k_3.\n\\end{equation}\nSumming gives $3=0$, a contradiction. Hence no global correction exists. The graph-Hodge obstruction has norm $\\sqrt3$.\n\\end{example}","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\n\\begin{equation}\ng_{\\alpha\\beta}\\in\\mathscr G(U_{\\alpha\\beta})\n\\end{equation}\nsuch that\n\\begin{equation}\ns_\\alpha=g_{\\alpha\\beta}\\cdot s_\\beta.\n\\end{equation}","\\begin{conditionaltheorem}[Nonlinear torsor descent]\n\\label{thm:nonlinear-torsor-descent}\nAssume 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\n\\begin{equation}\ng_{\\alpha\\beta}g_{\\beta\\gamma}g_{\\gamma\\alpha}=e\n\\end{equation}\non triple overlaps and its class is trivial in nonabelian \\v{C}ech cohomology\n\\begin{equation}\n[g_{\\alpha\\beta}]=1\n\\quad\\text{in}\\quad\n\\check H^1(\\mathfrak U,\\mathscr G).\n\\end{equation}\nEquivalently, there exist local gauges $g_\\alpha\\in\\mathscr G(U_\\alpha)$ such that\n\\begin{equation}\ng_{\\alpha\\beta}=g_\\alpha^{-1}g_\\beta.\n\\end{equation}\n\\end{conditionaltheorem}","l 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\n\\begin{equation}\ng_{\\alpha\\beta}g_{\\beta\\gamma}g_{\\gamma\\alpha}=e\n\\end{equation}\non triple overlaps and its class is trivial in nonabelian \\v{C}ech cohomology\n\\begin{equation}\n[g_{\\alpha\\beta}]=1\n\\quad\\text{in}\\quad\n\\check H^1(\\mathfrak U,\\mathscr G).\n\\end{equation}\nEquivalently, there exist local gauges $g_\\alpha\\in\\mathscr G(U_\\alpha)$ such that\n\\begin{equation}\ng_{\\alpha\\beta}=g_\\alpha^{-1}g_\\beta.\n\\end{equation}\n\\end{conditionaltheorem}","epair exists if and only if the transition family satisfies the nonabelian cocycle condition\n\\begin{equation}\ng_{\\alpha\\beta}g_{\\beta\\gamma}g_{\\gamma\\alpha}=e\n\\end{equation}\non triple overlaps and its class is trivial in nonabelian \\v{C}ech cohomology\n\\begin{equation}\n[g_{\\alpha\\beta}]=1\n\\quad\\text{in}\\quad\n\\check H^1(\\mathfrak U,\\mathscr G).\n\\end{equation}\nEquivalently, there exist local gauges $g_\\alpha\\in\\mathscr G(U_\\alpha)$ such that\n\\begin{equation}\ng_{\\alpha\\beta}=g_\\alpha^{-1}g_\\beta.\n\\end{equation}\n\\end{conditionaltheorem}","the nonabelian cocycle condition\n\\begin{equation}\ng_{\\alpha\\beta}g_{\\beta\\gamma}g_{\\gamma\\alpha}=e\n\\end{equation}\non triple overlaps and its class is trivial in nonabelian \\v{C}ech cohomology\n\\begin{equation}\n[g_{\\alpha\\beta}]=1\n\\quad\\text{in}\\quad\n\\check H^1(\\mathfrak U,\\mathscr G).\n\\end{equation}\nEquivalently, there exist local gauges $g_\\alpha\\in\\mathscr G(U_\\alpha)$ such that\n\\begin{equation}\ng_{\\alpha\\beta}=g_\\alpha^{-1}g_\\beta.\n\\end{equation}\n\\end{conditionaltheorem}","\\begin{proof}\nIf 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.\n\\end{proof}","A finite operational version avoids explicit cohomology. Let $G=(I,E)$ be connected, with sets of local solutions $S_i$ and invertible transition maps\n\\begin{equation}\nT_{ij}:S_i\\to S_j,\n\\qquad\nT_{ji}=T_{ij}^{-1}.\n\\end{equation}\nChoose a root $o$ and a spanning tree $T\\subseteq G$.","\\begin{theorem}[Fundamental-cycle holonomy criterion]\n\\label{thm:fundamental-holonomy}\nFor 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\n\\begin{equation}\nH_e(s_o)=s_o\n\\end{equation}\nfor every non-tree edge $e$.\n\\end{theorem}","y criterion]\n\\label{thm:fundamental-holonomy}\nFor 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\n\\begin{equation}\nH_e(s_o)=s_o\n\\end{equation}\nfor every non-tree edge $e$.\n\\end{theorem}","\\begin{proof}\nTree 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.\n\\end{proof}","\\begin{definition}[Nonlinear holonomy residual]\nIf the $S_i$ are metric spaces, define\n\\begin{equation}\n\\eta_{\\mathrm{hol}}(s_o)\n:=\n\\max_{e\\in E\\setminus T}d\\bigl(H_e(s_o),s_o\\bigr).\n\\end{equation}\n\\end{definition}","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 $D$.","At order $n$, let\n\\begin{equation}\nb_{n,\\alpha}\\in\\mathscr V(U_\\alpha)\n\\end{equation}\nbe the residual coefficient determined by lower-order choices. The local obstruction is\n\\begin{equation}\n[b_{n,\\alpha}]\\in\\coker D_\\alpha.\n\\end{equation}\nIf every local class vanishes, choose $u_{n,\\alpha}$ with\n\\begin{equation}\nD_\\alpha u_{n,\\alpha}=-b_{n,\\alpha}.\n\\end{equation}\nTheir differences define an order-$n$ gluing class in $\\check H^1(\\ker D)$.","order $n$, let\n\\begin{equation}\nb_{n,\\alpha}\\in\\mathscr V(U_\\alpha)\n\\end{equation}\nbe the residual coefficient determined by lower-order choices. The local obstruction is\n\\begin{equation}\n[b_{n,\\alpha}]\\in\\coker D_\\alpha.\n\\end{equation}\nIf every local class vanishes, choose $u_{n,\\alpha}$ with\n\\begin{equation}\nD_\\alpha u_{n,\\alpha}=-b_{n,\\alpha}.\n\\end{equation}\nTheir differences define an order-$n$ gluing class in $\\check H^1(\\ker D)$.","in{equation}\nb_{n,\\alpha}\\in\\mathscr V(U_\\alpha)\n\\end{equation}\nbe the residual coefficient determined by lower-order choices. The local obstruction is\n\\begin{equation}\n[b_{n,\\alpha}]\\in\\coker D_\\alpha.\n\\end{equation}\nIf every local class vanishes, choose $u_{n,\\alpha}$ with\n\\begin{equation}\nD_\\alpha u_{n,\\alpha}=-b_{n,\\alpha}.\n\\end{equation}\nTheir differences define an order-$n$ gluing class in $\\check H^1(\\ker D)$.","\\begin{theorem}[Finite local-to-global obstruction tower]\n\\label{thm:global-obstruction-tower}\nFix $N\\in\\N$. Assume:\n\\begin{enumerate}\n\\item the analytic local repair equations agree under restriction;\n\\item the sheaves $\\mathscr U$ and $\\mathscr V$ satisfy effective linear descent on the declared cover;\n\\item repairs have been constructed globally through order $n-1$.\n\\end{enumerate}\nThen a global repair extends through order $n$ if and only if:\n\\begin{enumerate}\n\\item every local coefficient class vanishes,\n\\begin{equation}\n[b_{n,\\alpha}]=0\\quad\\text{in }\\coker D_\\alpha;\n\\end{equation}\n\\item for one, hence every, choice of local lifts, the overlap class vanishes,\n\\begin{equation}\n[d^{(n)}_{\\alpha\\beta}]=0\n\\quad\\text{in}\\quad\n\\check H^1(\\mathfrak U,\\ker D).\n\\end{equation}\n\\end{enumerate}\nIf bot","\\begin{theorem}[Finite local-to-global obstruction tower]\n\\label{thm:global-obstruction-tower}\nFix $N\\in\\N$. Assume:\n\\begin{enumerate}\n\\item the analytic local repair equations agree under restriction;\n\\item the sheaves $\\mathscr U$ and $\\mathscr V$ satisfy effective linear descent on the declared cover;\n\\item repairs have been constructed globally through order $n-1$.\n\\end{enumerate}\nThen a global repair extends through order $n$ if and only if:\n\\begin{enumerate}\n\\item every local coefficient class vanishes,\n\\begin{equation}\n[b_{n,\\alpha}]=0\\quad\\text{in }\\coker D_\\alpha;\n\\end{equation}\n\\item for one, hence every, choice of local lifts, the overlap class vanishes,\n\\begin{equation}\n[d^{(n)}_{\\alpha\\beta}]=0\n\\quad\\text{in}\\quad\n\\check H^1(\\mathfrak U,\\ker D).\n\\end{equation}\n\\end{enumerate}\nIf both conditions hold for every $1\\leq n\\leq N$, a global formal repair exists through order $N$.\n\\end{theorem}","global obstruction tower]\n\\label{thm:global-obstruction-tower}\nFix $N\\in\\N$. Assume:\n\\begin{enumerate}\n\\item the analytic local repair equations agree under restriction;\n\\item the sheaves $\\mathscr U$ and $\\mathscr V$ satisfy effective linear descent on the declared cover;\n\\item repairs have been constructed globally through order $n-1$.\n\\end{enumerate}\nThen a global repair extends through order $n$ if and only if:\n\\begin{enumerate}\n\\item every local coefficient class vanishes,\n\\begin{equation}\n[b_{n,\\alpha}]=0\\quad\\text{in }\\coker D_\\alpha;\n\\end{equation}\n\\item for one, hence every, choice of local lifts, the overlap class vanishes,\n\\begin{equation}\n[d^{(n)}_{\\alpha\\beta}]=0\n\\quad\\text{in}\\quad\n\\check H^1(\\mathfrak U,\\ker D).\n\\end{equation}\n\\end{enumerate}\nIf both conditions hold for every $1\\leq n\\leq N$, a global formal repair exists through order $N$.\n\\end{theorem}","te}\n\\item the analytic local repair equations agree under restriction;\n\\item the sheaves $\\mathscr U$ and $\\mathscr V$ satisfy effective linear descent on the declared cover;\n\\item repairs have been constructed globally through order $n-1$.\n\\end{enumerate}\nThen a global repair extends through order $n$ if and only if:\n\\begin{enumerate}\n\\item every local coefficient class vanishes,\n\\begin{equation}\n[b_{n,\\alpha}]=0\\quad\\text{in }\\coker D_\\alpha;\n\\end{equation}\n\\item for one, hence every, choice of local lifts, the overlap class vanishes,\n\\begin{equation}\n[d^{(n)}_{\\alpha\\beta}]=0\n\\quad\\text{in}\\quad\n\\check H^1(\\mathfrak U,\\ker D).\n\\end{equation}\n\\end{enumerate}\nIf both conditions hold for every $1\\leq n\\leq N$, a global formal repair exists through order $N$.\n\\end{theorem}","declared cover;\n\\item repairs have been constructed globally through order $n-1$.\n\\end{enumerate}\nThen a global repair extends through order $n$ if and only if:\n\\begin{enumerate}\n\\item every local coefficient class vanishes,\n\\begin{equation}\n[b_{n,\\alpha}]=0\\quad\\text{in }\\coker D_\\alpha;\n\\end{equation}\n\\item for one, hence every, choice of local lifts, the overlap class vanishes,\n\\begin{equation}\n[d^{(n)}_{\\alpha\\beta}]=0\n\\quad\\text{in}\\quad\n\\check H^1(\\mathfrak U,\\ker D).\n\\end{equation}\n\\end{enumerate}\nIf both conditions hold for every $1\\leq n\\leq N$, a global formal repair exists through order $N$.\n\\end{theorem}","if and only if:\n\\begin{enumerate}\n\\item every local coefficient class vanishes,\n\\begin{equation}\n[b_{n,\\alpha}]=0\\quad\\text{in }\\coker D_\\alpha;\n\\end{equation}\n\\item for one, hence every, choice of local lifts, the overlap class vanishes,\n\\begin{equation}\n[d^{(n)}_{\\alpha\\beta}]=0\n\\quad\\text{in}\\quad\n\\check H^1(\\mathfrak U,\\ker D).\n\\end{equation}\n\\end{enumerate}\nIf both conditions hold for every $1\\leq n\\leq N$, a global formal repair exists through order $N$.\n\\end{theorem}","nt class vanishes,\n\\begin{equation}\n[b_{n,\\alpha}]=0\\quad\\text{in }\\coker D_\\alpha;\n\\end{equation}\n\\item for one, hence every, choice of local lifts, the overlap class vanishes,\n\\begin{equation}\n[d^{(n)}_{\\alpha\\beta}]=0\n\\quad\\text{in}\\quad\n\\check H^1(\\mathfrak U,\\ker D).\n\\end{equation}\n\\end{enumerate}\nIf both conditions hold for every $1\\leq n\\leq N$, a global formal repair exists through order $N$.\n\\end{theorem}","\\begin{proof}\nAt 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.\n\\end{proof}","The theorem produces an obstruction ledger\n\\begin{equation}\n\\begin{array}{c|cc}\n\\text{order}&\\text{local obstruction}&\\text{gluing obstruction}\\\\\n\\hline\n1&[b_{1,\\alpha}]\\in\\coker D_\\alpha&[d^{(1)}]\\in\\check H^1(\\ker D)\\\\\n2&[b_{2,\\alpha}]\\in\\coker D_\\alpha&[d^{(2)}]\\in\\check H^1(\\ker D)\\\\\n\\vdots&\\vdots&\\vdots\\\\\nN&[b_{N,\\alpha}]\\in\\coker D_\\alpha&[d^{(N)}]\\in\\check H^1(\\ker D).\n\\end{array}\n\\end{equation}\nHigher categorical repair interfaces may introduce additional coherence obstructions beyond this two-column ledger. Those are not proved here.","\\begin{definition}[Cover adequacy score]\nFor a candidate functional cover $\\mathfrak U$, define the diagnostic score\n\\begin{equation}\n\\Acal(\\mathfrak U)\n:=\n\\eta_{\\mathrm{loc}}(\\mathfrak U)\n+\\tau\\eta_{\\mathrm{glue}}(\\mathfrak U)\n+\\lambda\\,\\mathrm{Complexity}(\\mathfrak U)\n+\\gamma\\,\\mathrm{Instability}(\\mathfrak U).\n\\end{equation}\n\\end{definition}","\\begin{definition}[Correction datum]\nA finite-dimensional correction datum is a tuple\n\\begin{equation}\n\\mathfrak C=(U,V,D,\\Acal,c,\\mathfrak p),\n\\end{equation}\nwhere $U$ 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.\n\\end{definition}","\\begin{definition}[Transport datum]\nLet $\\mathfrak C_A$ and $\\mathfrak C_B$ be correction data. A transport datum from $A$ to $B$ is a pair\n\\begin{equation}\n\\mathfrak T_{A\\to B}=(T_U,T_V),\n\\qquad\nT_U:U_A\\to U_B,\n\\qquad\nT_V:V_A\\to V_B.\n\\end{equation}\nIts correction-square defect is\n\\begin{equation}\n\\Delta_D(\\mathfrak T_{A\\to B})\n:=\nD_BT_U-T_VD_A.\n\\end{equation}\nFor declared source and target defects $r_A\\in V_A$ and $r_B\\in V_B$, its defect-identification error is\n\\begin{equation}\n\\Delta_r(\\mathfrak T_{A\\to B};r_A,r_B)\n:=\nr_B-T_Vr_A.\n\\end{equation}\nThe datum is \\emph{exact} when both quantities vanish.\n\\end{definition}","gin{definition}[Transport datum]\nLet $\\mathfrak C_A$ and $\\mathfrak C_B$ be correction data. A transport datum from $A$ to $B$ is a pair\n\\begin{equation}\n\\mathfrak T_{A\\to B}=(T_U,T_V),\n\\qquad\nT_U:U_A\\to U_B,\n\\qquad\nT_V:V_A\\to V_B.\n\\end{equation}\nIts correction-square defect is\n\\begin{equation}\n\\Delta_D(\\mathfrak T_{A\\to B})\n:=\nD_BT_U-T_VD_A.\n\\end{equation}\nFor declared source and target defects $r_A\\in V_A$ and $r_B\\in V_B$, its defect-identification error is\n\\begin{equation}\n\\Delta_r(\\mathfrak T_{A\\to B};r_A,r_B)\n:=\nr_B-T_Vr_A.\n\\end{equation}\nThe datum is \\emph{exact} when both quantities vanish.\n\\end{definition}","ransport datum]\nLet $\\mathfrak C_A$ and $\\mathfrak C_B$ be correction data. A transport datum from $A$ to $B$ is a pair\n\\begin{equation}\n\\mathfrak T_{A\\to B}=(T_U,T_V),\n\\qquad\nT_U:U_A\\to U_B,\n\\qquad\nT_V:V_A\\to V_B.\n\\end{equation}\nIts correction-square defect is\n\\begin{equation}\n\\Delta_D(\\mathfrak T_{A\\to B})\n:=\nD_BT_U-T_VD_A.\n\\end{equation}\nFor declared source and target defects $r_A\\in V_A$ and $r_B\\in V_B$, its defect-identification error is\n\\begin{equation}\n\\Delta_r(\\mathfrak T_{A\\to B};r_A,r_B)\n:=\nr_B-T_Vr_A.\n\\end{equation}\nThe datum is \\emph{exact} when both quantities vanish.\n\\end{definition}","ction data. A transport datum from $A$ to $B$ is a pair\n\\begin{equation}\n\\mathfrak T_{A\\to B}=(T_U,T_V),\n\\qquad\nT_U:U_A\\to U_B,\n\\qquad\nT_V:V_A\\to V_B.\n\\end{equation}\nIts correction-square defect is\n\\begin{equation}\n\\Delta_D(\\mathfrak T_{A\\to B})\n:=\nD_BT_U-T_VD_A.\n\\end{equation}\nFor declared source and target defects $r_A\\in V_A$ and $r_B\\in V_B$, its defect-identification error is\n\\begin{equation}\n\\Delta_r(\\mathfrak T_{A\\to B};r_A,r_B)\n:=\nr_B-T_Vr_A.\n\\end{equation}\nThe datum is \\emph{exact} when both quantities vanish.\n\\end{definition}","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.","\\begin{theorem}[Exact repair transport]\n\\label{thm:exact-transport}\nSuppose\n\\begin{equation}\nD_BT_U=T_VD_A\n\\end{equation}\nand\n\\begin{equation}\nr_B=T_Vr_A.\n\\end{equation}\nIf $u_A$ satisfies\n\\begin{equation}\nD_Au_A=-r_A,\n\\end{equation}\nthen\n\\begin{equation}\nu_B:=T_Uu_A\n\\end{equation}\nsatisfies\n\\begin{equation}\nD_Bu_B=-r_B.\n\\end{equation}\n\\end{theorem}","\\begin{proof}\nBy commutation,\n\\begin{equation}\nD_Bu_B=D_BT_Uu_A=T_VD_Au_A=-T_Vr_A=-r_B.\n\\end{equation}\n\\end{proof}","\\begin{corollary}[Kernel and obstruction naturality]\n\\label{cor:kernel-coker-naturality}\nAssume $D_BT_U=T_VD_A$. Then\n\\begin{equation}\nT_U(\\ker D_A)\\subseteq\\ker D_B\n\\end{equation}\nand\n\\begin{equation}\nT_V(\\im D_A)\\subseteq\\im D_B.\n\\end{equation}\nConsequently, $T_V$ induces a quotient map\n\\begin{equation}\n\\overline T_V:\\coker D_A\\to\\coker D_B,\n\\qquad\n[r]\\mapsto[T_Vr].\n\\end{equation}\n\\end{corollary}","\\begin{proof}\nIf $D_Au=0$, 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.\n\\end{proof}","\\begin{theorem}[Exact transport equivalence]\n\\label{thm:transport-equivalence}\nSuppose $T_U$ and $T_V$ are isomorphisms and\n\\begin{equation}\nD_B=T_VD_AT_U^{-1}.\n\\end{equation}\nThen $T_U$ restricts to an isomorphism\n\\begin{equation}\n\\ker D_A\\cong\\ker D_B,\n\\end{equation}\nand $T_V$ induces an isomorphism\n\\begin{equation}\n\\coker D_A\\cong\\coker D_B.\n\\end{equation}\nThus exact transport equivalence preserves both correction freedom and obstruction dimension.\n\\end{theorem}","\\begin{proof}\nThe kernel statement follows from \\cref{cor:kernel-coker-naturality} applied in both directions. Since $T_U$ is surjective,\n\\begin{equation}\n\\im D_B\n=\nT_VD_AT_U^{-1}(U_B)\n=\nT_V(\\im D_A).\n\\end{equation}\nHence $T_V$ maps $\\im D_A$ bijectively onto $\\im D_B$ and descends to an isomorphism of quotient spaces.\n\\end{proof}","\\begin{definition}[Protected obstruction subspace]\nA protected obstruction subspace is a declared subspace\n\\begin{equation}\nQ_A\\subseteq\\coker D_A\n\\end{equation}\nwhose nonzero classes must remain detectable after transport. The transport is \\emph{obstruction-faithful on $Q_A$} when\n\\begin{equation}\n\\ker\\overline T_V\\cap Q_A=\\set{0}.\n\\end{equation}\n\\end{definition}","\\begin{proposition}[Representation loss can hide an obstruction]\n\\label{prop:hidden-obstruction}\nIf $\\overline T_V$ is not injective, then there exists a defect $r_A$ such that\n\\begin{equation}\n[r_A]\\neq0\n\\quad\\text{but}\\quad\n[T_Vr_A]=0.\n\\end{equation}\nThus 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.\n\\end{proposition}","\\begin{proof}\nChoose a nonzero class in $\\ker\\overline T_V$ and any representative $r_A$. By definition its image in $\\coker D_B$ vanishes.\n\\end{proof}","\\begin{definition}[Authorized transport]\nA transport datum is authorized when\n\\begin{equation}\nT_U(\\Acal_A)\\subseteq\\Acal_B.\n\\end{equation}\nIt is $(c_A,c_B,\\kappa)$-budget controlled when\n\\begin{equation}\nc_B(T_Uu)\\leq\\kappa c_A(u)\n\\qquad\n\\text{for every }u\\in\\Acal_A.\n\\end{equation}\n\\end{definition}","\\begin{proposition}[Authorized exact repair transport]\n\\label{prop:authorized-transport}\nUnder 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\n\\begin{equation}\nc_B(u_B)\\leq\\kappa B.\n\\end{equation}\n\\end{proposition}","\\begin{proof}\nAuthorization 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.\n\\end{proof}","Let protected linear invariants be represented by\n\\begin{equation}\nC_A:U_A\\to Z_A,\n\\qquad\nC_B:U_B\\to Z_B.\n\\end{equation}\nA correction preserves the invariant when $C_Au=0$ or $C_Bu=0$.","\\begin{proposition}[Invariant-preserving transport]\n\\label{prop:invariant-transport}\nSuppose there is a linear map $S:Z_A\\to Z_B$ satisfying\n\\begin{equation}\nC_BT_U=SC_A.\n\\end{equation}\nThen\n\\begin{equation}\nT_U(\\ker C_A)\\subseteq\\ker C_B.\n\\end{equation}\nConsequently, every source repair preserving the declared invariant transports to a target repair preserving the transported invariant.\n\\end{proposition}","\\begin{proof}\nFor $u\\in\\ker C_A$,\n\\begin{equation}\nC_BT_Uu=SC_Au=0.\n\\end{equation}\n\\end{proof}","\\begin{theorem}[Naturality of the Moore--Penrose repair]\n\\label{thm:pseudoinverse-transport}\nLet $U_A,U_B,V_A,V_B$ be finite-dimensional Hilbert spaces. Suppose $T_U$ and $T_V$ are unitary and\n\\begin{equation}\nD_B=T_VD_AT_U^*.\n\\end{equation}\nThen\n\\begin{equation}\nD_B^{\\dagger}=T_UD_A^{\\dagger}T_V^*.\n\\end{equation}\nFor every $r_B=T_Vr_A$, the target minimum-norm least-squares correction is the transport of the source minimum-norm least-squares correction:\n\\begin{equation}\n-D_B^{\\dagger}r_B\n=\nT_U(-D_A^{\\dagger}r_A).\n\\end{equation}\nThe orthogonal obstruction projectors satisfy\n\\begin{equation}\nI-D_BD_B^{\\dagger}\n=\nT_V(I-D_AD_A^{\\dagger})T_V^*.\n\\end{equation}\n\\end{theorem}","nverse-transport}\nLet $U_A,U_B,V_A,V_B$ be finite-dimensional Hilbert spaces. Suppose $T_U$ and $T_V$ are unitary and\n\\begin{equation}\nD_B=T_VD_AT_U^*.\n\\end{equation}\nThen\n\\begin{equation}\nD_B^{\\dagger}=T_UD_A^{\\dagger}T_V^*.\n\\end{equation}\nFor every $r_B=T_Vr_A$, the target minimum-norm least-squares correction is the transport of the source minimum-norm least-squares correction:\n\\begin{equation}\n-D_B^{\\dagger}r_B\n=\nT_U(-D_A^{\\dagger}r_A).\n\\end{equation}\nThe orthogonal obstruction projectors satisfy\n\\begin{equation}\nI-D_BD_B^{\\dagger}\n=\nT_V(I-D_AD_A^{\\dagger})T_V^*.\n\\end{equation}\n\\end{theorem}","in{equation}\nD_B=T_VD_AT_U^*.\n\\end{equation}\nThen\n\\begin{equation}\nD_B^{\\dagger}=T_UD_A^{\\dagger}T_V^*.\n\\end{equation}\nFor every $r_B=T_Vr_A$, the target minimum-norm least-squares correction is the transport of the source minimum-norm least-squares correction:\n\\begin{equation}\n-D_B^{\\dagger}r_B\n=\nT_U(-D_A^{\\dagger}r_A).\n\\end{equation}\nThe orthogonal obstruction projectors satisfy\n\\begin{equation}\nI-D_BD_B^{\\dagger}\n=\nT_V(I-D_AD_A^{\\dagger})T_V^*.\n\\end{equation}\n\\end{theorem}","\\begin{proof}\nSet $X=T_UD_A^{\\dagger}T_V^*$. 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.\n\\end{proof}","Let\n\\begin{equation}\n\\mathfrak T_{A\\to B}=(T_U^{AB},T_V^{AB}),\n\\qquad\n\\mathfrak T_{B\\to C}=(T_U^{BC},T_V^{BC}).\n\\end{equation}\nThe composed transport is\n\\begin{equation}\n\\mathfrak T_{A\\to C}\n=\n(T_U^{BC}T_U^{AB},T_V^{BC}T_V^{AB}).\n\\end{equation}","\\begin{lemma}[Composition identity]\n\\label{lem:transport-composition}\nThe correction-square defects obey\n\\begin{equation}\n\\Delta_D(\\mathfrak T_{A\\to C})\n=\n\\Delta_D(\\mathfrak T_{B\\to C})T_U^{AB}\n+\nT_V^{BC}\\Delta_D(\\mathfrak T_{A\\to B}).\n\\end{equation}\nThe defect-identification errors obey\n\\begin{equation}\n\\Delta_r(\\mathfrak T_{A\\to C})\n=\n\\Delta_r(\\mathfrak T_{B\\to C})\n+\nT_V^{BC}\\Delta_r(\\mathfrak T_{A\\to B}).\n\\end{equation}\n\\end{lemma}","\\begin{proof}\nInsert and subtract $T_V^{BC}D_BT_U^{AB}$ in the first expression and $T_V^{BC}r_B$ in the second.\n\\end{proof}","\\begin{theorem}[Pathwise transport bound]\n\\label{thm:pathwise-transport}\nConsider a path of systems\n\\begin{equation}\nA_0\\xrightarrow{\\mathfrak T_0}A_1\n\\xrightarrow{\\mathfrak T_1}\\cdots\n\\xrightarrow{\\mathfrak T_{n-1}}A_n.\n\\end{equation}\nWrite $T_{U,k}:U_k\\to U_{k+1}$ and $T_{V,k}:V_k\\to V_{k+1}$. Let\n\\begin{equation}\n\\norm{D_{k+1}T_{U,k}-T_{V,k}D_k}\\leq\\eps_{D,k}\n\\end{equation}\nand\n\\begin{equation}\n\\norm{r_{k+1}-T_{V,k}r_k}\\leq\\eps_{r,k}.\n\\end{equation}\nIf $D_0u_0=-r_0$ and\n\\begin{equation}\nu_n=T_{U,n-1}\\cdots T_{U,0}u_0,\n\\end{equation}\nthen\n\\begin{equation}\n\\begin{aligned}\n\\norm{D_nu_n+r_n}\n&\\leq\n\\sum_{k=0}^{n-1}\n\\norm{T_{V,n-1}\\cdots T_{V,k+1}}\n\\eps_{D,k}\n\\norm{T_{U,k-1}\\cdots T_{U,0}u_0}","-transport}\nConsider a path of systems\n\\begin{equation}\nA_0\\xrightarrow{\\mathfrak T_0}A_1\n\\xrightarrow{\\mathfrak T_1}\\cdots\n\\xrightarrow{\\mathfrak T_{n-1}}A_n.\n\\end{equation}\nWrite $T_{U,k}:U_k\\to U_{k+1}$ and $T_{V,k}:V_k\\to V_{k+1}$. Let\n\\begin{equation}\n\\norm{D_{k+1}T_{U,k}-T_{V,k}D_k}\\leq\\eps_{D,k}\n\\end{equation}\nand\n\\begin{equation}\n\\norm{r_{k+1}-T_{V,k}r_k}\\leq\\eps_{r,k}.\n\\end{equation}\nIf $D_0u_0=-r_0$ and\n\\begin{equation}\nu_n=T_{U,n-1}\\cdots T_{U,0}u_0,\n\\end{equation}\nthen\n\\begin{equation}\n\\begin{aligned}\n\\norm{D_nu_n+r_n}\n&\\leq\n\\sum_{k=0}^{n-1}\n\\norm{T_{V,n-1}\\cdots T_{V,k+1}}\n\\eps_{D,k}\n\\norm{T_{U,k-1}\\cdots T_{U,0}u_0}","stems\n\\begin{equation}\nA_0\\xrightarrow{\\mathfrak T_0}A_1\n\\xrightarrow{\\mathfrak T_1}\\cdots\n\\xrightarrow{\\mathfrak T_{n-1}}A_n.\n\\end{equation}\nWrite $T_{U,k}:U_k\\to U_{k+1}$ and $T_{V,k}:V_k\\to V_{k+1}$. Let\n\\begin{equation}\n\\norm{D_{k+1}T_{U,k}-T_{V,k}D_k}\\leq\\eps_{D,k}\n\\end{equation}\nand\n\\begin{equation}\n\\norm{r_{k+1}-T_{V,k}r_k}\\leq\\eps_{r,k}.\n\\end{equation}\nIf $D_0u_0=-r_0$ and\n\\begin{equation}\nu_n=T_{U,n-1}\\cdots T_{U,0}u_0,\n\\end{equation}\nthen\n\\begin{equation}\n\\begin{aligned}\n\\norm{D_nu_n+r_n}\n&\\leq\n\\sum_{k=0}^{n-1}\n\\norm{T_{V,n-1}\\cdots T_{V,k+1}}\n\\eps_{D,k}\n\\norm{T_{U,k-1}\\cdots T_{U,0}u_0}","{\\mathfrak T_1}\\cdots\n\\xrightarrow{\\mathfrak T_{n-1}}A_n.\n\\end{equation}\nWrite $T_{U,k}:U_k\\to U_{k+1}$ and $T_{V,k}:V_k\\to V_{k+1}$. Let\n\\begin{equation}\n\\norm{D_{k+1}T_{U,k}-T_{V,k}D_k}\\leq\\eps_{D,k}\n\\end{equation}\nand\n\\begin{equation}\n\\norm{r_{k+1}-T_{V,k}r_k}\\leq\\eps_{r,k}.\n\\end{equation}\nIf $D_0u_0=-r_0$ and\n\\begin{equation}\nu_n=T_{U,n-1}\\cdots T_{U,0}u_0,\n\\end{equation}\nthen\n\\begin{equation}\n\\begin{aligned}\n\\norm{D_nu_n+r_n}\n&\\leq\n\\sum_{k=0}^{n-1}\n\\norm{T_{V,n-1}\\cdots T_{V,k+1}}\n\\eps_{D,k}\n\\norm{T_{U,k-1}\\cdots T_{U,0}u_0}\\\\\n&\\quad+\n\\sum_{k=0}^{n-1}\n\\norm{T_{V,n-1}\\cdots T_{V,k+1}}\n\\eps_{r,k},\n\\end{aligned}\n\\end{equation}\nwhere empty products are identities.\n\\end{theorem}","\\begin{proof}\nRepeatedly apply \\cref{lem:transport-composition}. The total correction-square defect is the telescoping sum\n\\begin{equation}\n\\sum_{k=0}^{n-1}\nT_{V,n-1}\\cdots T_{V,k+1}\n\\Delta_{D,k}\nT_{U,k-1}\\cdots T_{U,0},\n\\end{equation}\nand the total defect-identification error is\n\\begin{equation}\n\\sum_{k=0}^{n-1}\nT_{V,n-1}\\cdots T_{V,k+1}\\Delta_{r,k}.\n\\end{equation}\nApply both sums to $u_0$, use $D_0u_0+r_0=0$, and take norms.\n\\end{proof}","\\begin{corollary}[Nonexpansive path bound]\n\\label{cor:nonexpansive-path}\nIf every $T_{U,k}$ and $T_{V,k}$ is nonexpansive, then\n\\begin{equation}\n\\norm{D_nu_n+r_n}\n\\leq\n\\norm{u_0}\\sum_{k=0}^{n-1}\\eps_{D,k}\n+\n\\sum_{k=0}^{n-1}\\eps_{r,k}.\n\\end{equation}\n\\end{corollary}","\\begin{definition}[Transport route cost]\nFor a fixed source correction $u_0$, define the certified cost of a nonexpansive path $p$ by\n\\begin{equation}\nC_{\\mathrm{tr}}(p;u_0)\n:=\n\\norm{u_0}\\sum_{e\\in p}\\eps_{D,e}\n+\n\\sum_{e\\in p}\\eps_{r,e}\n+\n\\sum_{e\\in p}\\eps_{\\mathrm{auth},e}\n+\n\\sum_{e\\in p}\\eps_{\\mathrm{sem},e}.\n\\end{equation}\n\\end{definition}","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\n\\begin{equation}\nH^U_\\gamma:U_A\\to U_A,\n\\qquad\nH^V_\\gamma:V_A\\to V_A.\n\\end{equation}\nExact commutation along the loop gives\n\\begin{equation}\nD_AH^U_\\gamma=H^V_\\gamma D_A.\n\\end{equation}","\\begin{definition}[Path-flat repair and defect cores]\nThe path-flat repair and defect cores at $A$ are\n\\begin{equation}\nU_A^{\\flat}\n:=\n\\bigcap_\\gamma\\ker(H^U_\\gamma-I),\n\\qquad\nV_A^{\\flat}\n:=\n\\bigcap_\\gamma\\ker(H^V_\\gamma-I).\n\\end{equation}\n\\end{definition}","\\begin{proposition}[Flat repairs remain flat defects]\n\\label{prop:flat-repair-defect}\nIf $u\\in U_A^{\\flat}$, then\n\\begin{equation}\nD_Au\\in V_A^{\\flat}.\n\\end{equation}\n\\end{proposition}","\\begin{proof}\nFor every loop $\\gamma$,\n\\begin{equation}\nH^V_\\gamma D_Au=D_AH^U_\\gamma u=D_Au.\n\\end{equation}\n\\end{proof}","\\begin{theorem}[Path independence on the flat repair core]\n\\label{thm:path-independence}\nLet $p$ 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\n\\begin{equation}\nT_U^p u=T_U^q u.\n\\end{equation}\nIn particular, every $u\\in U_A^{\\flat}$ has path-independent transport to $B$.\n\\end{theorem}","\\begin{proof}\nThe path identity is\n\\begin{equation}\nT_U^p=T_U^qH^U_{q^{-1}p}.\n\\end{equation}\nApply both sides to $u$ and use $H^U_{q^{-1}p}u=u$.\n\\end{proof}","\\begin{definition}[Repair holonomy residual]\nFor a correction $u$ and loop $\\gamma$, define\n\\begin{equation}\n\\eta^U_\\gamma(u)\n:=\n\\norm{(H^U_\\gamma-I)u}.\n\\end{equation}\nFor a defect $r$, define\n\\begin{equation}\n\\eta^V_\\gamma(r)\n:=\n\\norm{(H^V_\\gamma-I)r}.\n\\end{equation}\n\\end{definition}","\\begin{proposition}[Path-discrepancy identity]\n\\label{prop:path-discrepancy}\nUnder the hypotheses of \\cref{thm:path-independence},\n\\begin{equation}\n\\norm{T_U^pu-T_U^qu}\n\\leq\n\\norm{T_U^q}\\,\\eta^U_{q^{-1}p}(u).\n\\end{equation}\nThus loop holonomy directly bounds disagreement between two transport routes.\n\\end{proposition}","\\begin{proof}\nUse\n\\begin{equation}\nT_U^pu-T_U^qu=T_U^q(H^U_{q^{-1}p}-I)u\n\\end{equation}\nand take norms.\n\\end{proof}","\\begin{theorem}[Approximate transported residual]\n\\label{thm:approx-residual}\nLet $U_A,U_B,V_A,V_B$ be normed spaces. Suppose $u_A$ solves\n\\begin{equation}\nD_Au_A=-r_A.\n\\end{equation}\nAssume\n\\begin{equation}\n\\norm{D_BT_U-T_VD_A}\\leq\\eps_D\n\\end{equation}\nand\n\\begin{equation}\n\\norm{r_B-T_Vr_A}\\leq\\eps_r.\n\\end{equation}\nThen $u_B=T_Uu_A$ satisfies\n\\begin{equation}\n\\norm{D_Bu_B+r_B}\n\\leq\n\\eps_D\\norm{u_A}+\\eps_r.\n\\end{equation}\n\\end{theorem}","\\begin{proof}\nAdd and subtract $T_VD_Au_A$ and $T_Vr_A$:\n\\begin{align}\nD_BT_Uu_A+r_B\n&=\n(D_BT_U-T_VD_A)u_A\n+T_V(D_Au_A+r_A)\n+(r_B-T_Vr_A).\n\\end{align}\nThe middle term vanishes. Taking norms yields the claim.\n\\end{proof}","\\begin{definition}[Scale-normalized transport defect]\nFor a declared stabilizer $\\tau>0$, define\n\\begin{equation}\n\\eps_D^{\\mathrm{rel}}\n:=\n\\frac{\\norm{D_BT_U-T_VD_A}}\n{\\norm{D_B}\\norm{T_U}+\\norm{T_V}\\norm{D_A}+\\tau}.\n\\end{equation}\n\\end{definition}","Let $\\ell_A:V_A\\to\\R$ and $\\ell_B:V_B\\to\\R$ be post-defect losses. Define source and target gains\n\\begin{equation}\ng_A\n:=\n\\ell_A(r_A)-\\ell_A(r_A+D_Au_A),\n\\end{equation}\n\\begin{equation}\ng_B\n:=\n\\ell_B(r_B)-\\ell_B(r_B+D_BT_Uu_A).\n\\end{equation}","\\begin{assumption}[Evaluator alignment]\nThere is $\\eps_J\\geq0$ such that\n\\begin{equation}\n\\abs{\\ell_B(T_Vv)-\\ell_A(v)}\\leq\\eps_J\n\\end{equation}\nfor\n\\begin{equation}\nv\\in\\set{r_A,r_A+D_Au_A}.\n\\end{equation}\n\\end{assumption}","\\begin{theorem}[Approximate gain-preserving repair transport]\n\\label{thm:gain-transport}\nAssume $\\ell_B$ 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\n\\begin{equation}\n\\eps_{\\mathrm{leak}}+\\eps_{\\mathrm{hol}}.\n\\end{equation}\nThen\n\\begin{equation}\n\\boxed{\n g_B\n\\geq\n g_A\n-2\\eps_J\n-L\\left(\n2\\eps_r\n+\\eps_D\\norm{u_A}\n+\\eps_{\\mathrm{leak}}\n+\\eps_{\\mathrm{hol}}\n\\right).\n}\n\\end{equation}\nIn particular, the transported repair is guaranteed beneficial whenever the source gain exceeds the total right-hand error margin.\n\\end{theorem}","\\begin{proof}\nEvaluator alignment and Lipschitz continuity imply\n\\begin{equation}\n\\ell_B(r_B)\n\\geq\n\\ell_B(T_Vr_A)-L\\eps_r\n\\geq\n\\ell_A(r_A)-\\eps_J-L\\eps_r.\n\\end{equation}\nThe target post-repair defect differs from\n\\begin{equation}\nT_V(r_A+D_Au_A)\n\\end{equation}\nby at most\n\\begin{equation}\n\\eps_r+\\eps_D\\norm{u_A}+\\eps_{\\mathrm{leak}}+\\eps_{\\mathrm{hol}}\n\\end{equation}\nusing \\cref{thm:approx-residual} and the declared additional perturbation bounds. Hence\n\\begin{align}\n\\ell_B(r_B+D_BT_Uu_A)\n&\\leq\n\\ell_B(T_V(r_A+D_Au_A))","n{equation}\n\\ell_B(r_B)\n\\geq\n\\ell_B(T_Vr_A)-L\\eps_r\n\\geq\n\\ell_A(r_A)-\\eps_J-L\\eps_r.\n\\end{equation}\nThe target post-repair defect differs from\n\\begin{equation}\nT_V(r_A+D_Au_A)\n\\end{equation}\nby at most\n\\begin{equation}\n\\eps_r+\\eps_D\\norm{u_A}+\\eps_{\\mathrm{leak}}+\\eps_{\\mathrm{hol}}\n\\end{equation}\nusing \\cref{thm:approx-residual} and the declared additional perturbation bounds. Hence\n\\begin{align}\n\\ell_B(r_B+D_BT_Uu_A)\n&\\leq\n\\ell_B(T_V(r_A+D_Au_A))\\\\\n&\\quad+L\\left(\n\\eps_r+\\eps_D\\norm{u_A}\n+\\eps_{\\mathrm{leak}}+\\eps_{\\mathrm{hol}}\n\\right)\\\\\n&\\leq\n\\ell_A(r_A+D_Au_A)+\\eps_J\\\\\n&\\quad+L\\left(\n\\eps_r+\\eps_D\\norm{u_A}\n+\\eps_{\\mathrm{leak}}+\\eps_{\\mathrm{hol}}\n\\right).\n\\end{align}\nSubtracting the post-repair upper bound from the baseline lower bound gives the result.\n\\end{proof}","\\begin{corollary}[Pathwise gain certificate]\n\\label{cor:pathwise-gain}\nAlong a nonexpansive transport path, replace\n\\begin{equation}\n\\eps_D\\norm{u_A}+\\eps_r\n\\end{equation}\nby 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.\n\\end{corollary}","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.","\\begin{theorem}[Coupled distributional gain transfer]\n\\label{thm:distributional-transfer}\nSuppose there is a coupling $\\pi$ of $P_A$ and $P_B$ such that\n\\begin{equation}\n\\mathbb E_{(x,y)\\sim\\pi}d_B(y,Sx)\\leq\\delta.\n\\end{equation}\nAssume $q_B$ is $K$-Lipschitz and\n\\begin{equation}\nq_B(Sx)\\geq q_A(x)-\\eta\n\\end{equation}\nfor $P_A$-almost every $x$. Then\n\\begin{equation}\n\\mathbb E_{y\\sim P_B}q_B(y)\n\\geq\n\\mathbb E_{x\\sim P_A}q_A(x)-\\eta-K\\delta.\n\\end{equation}\n\\end{theorem}","\\begin{proof}\nUnder the coupling,\n\\begin{equation}\nq_B(y)\n\\geq\nq_B(Sx)-K d_B(y,Sx)\n\\geq\nq_A(x)-\\eta-K d_B(y,Sx).\n\\end{equation}\nTake expectations.\n\\end{proof}","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.","\\begin{definition}[High-probability transport certificate]\nA random tuple\n\\begin{equation}\n(\\underline g_A,\n\\overline\\eps_J,\n\\overline\\eps_r,\n\\overline\\eps_D,\n\\overline\\eps_{\\mathrm{leak}},\n\\overline\\eps_{\\mathrm{hol}},\n\\overline L)\n\\end{equation}\nis 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.\n\\end{definition}","\\begin{theorem}[Probabilistic gain preservation by simultaneous bounds]\n\\label{thm:probabilistic-transfer}\nGiven an $(1-\\alpha)$ transport certificate, define\n\\begin{equation}\n\\begin{aligned}\n\\underline g_B\n:=\\;&\n\\underline g_A\n-2\\overline\\eps_J","\\begin{theorem}[Probabilistic gain preservation by simultaneous bounds]\n\\label{thm:probabilistic-transfer}\nGiven an $(1-\\alpha)$ transport certificate, define\n\\begin{equation}\n\\begin{aligned}\n\\underline g_B\n:=\\;&\n\\underline g_A\n-2\\overline\\eps_J\\\\\n&-\\overline L\\left(\n2\\overline\\eps_r\n+\\overline\\eps_D\\norm{u_A}\n+\\overline\\eps_{\\mathrm{leak}}\n+\\overline\\eps_{\\mathrm{hol}}\n\\right).\n\\end{aligned}\n\\end{equation}\nThen\n\\begin{equation}\n\\Pr(g_B\\geq\\underline g_B)\\geq1-\\alpha.\n\\end{equation}\nIn particular, $\\underline g_B>0$ is a valid one-sided certificate of beneficial target transfer.\n\\end{theorem}","&-\\overline L\\left(\n2\\overline\\eps_r\n+\\overline\\eps_D\\norm{u_A}\n+\\overline\\eps_{\\mathrm{leak}}\n+\\overline\\eps_{\\mathrm{hol}}\n\\right).\n\\end{aligned}\n\\end{equation}\nThen\n\\begin{equation}\n\\Pr(g_B\\geq\\underline g_B)\\geq1-\\alpha.\n\\end{equation}\nIn particular, $\\underline g_B>0$ is a valid one-sided certificate of beneficial target transfer.\n\\end{theorem}","\\begin{proof}\nOn the simultaneous confidence event, substitute every bound into \\cref{thm:gain-transport}. The event has probability at least $1-\\alpha$.\n\\end{proof}","\\begin{theorem}[Paired held-out gain certificate]\n\\label{thm:paired-gain}\nLet $Z_1,\\ldots,Z_n$ be independent and identically distributed paired target-context gains for a precommitted transported repair, with\n\\begin{equation}\n-G\\leq Z_i\\leq G.\n\\end{equation}\nLet $\\overline Z_n=n^{-1}\\sum_iZ_i$. Then, with probability at least $1-\\alpha$,\n\\begin{equation}\n\\mathbb E Z_i\n\\geq\n\\overline Z_n\n-G\\sqrt{\\frac{2\\log(1/\\alpha)}{n}}.\n\\end{equation}\nFor a precommitted family of $m$ repairs, replacing $\\alpha$ by $\\alpha/m$ gives a simultaneous lower bound for all $m$ candidates.\n\\end{theorem}","}\nLet $Z_1,\\ldots,Z_n$ be independent and identically distributed paired target-context gains for a precommitted transported repair, with\n\\begin{equation}\n-G\\leq Z_i\\leq G.\n\\end{equation}\nLet $\\overline Z_n=n^{-1}\\sum_iZ_i$. Then, with probability at least $1-\\alpha$,\n\\begin{equation}\n\\mathbb E Z_i\n\\geq\n\\overline Z_n\n-G\\sqrt{\\frac{2\\log(1/\\alpha)}{n}}.\n\\end{equation}\nFor a precommitted family of $m$ repairs, replacing $\\alpha$ by $\\alpha/m$ gives a simultaneous lower bound for all $m$ candidates.\n\\end{theorem}","be independent and identically distributed paired target-context gains for a precommitted transported repair, with\n\\begin{equation}\n-G\\leq Z_i\\leq G.\n\\end{equation}\nLet $\\overline Z_n=n^{-1}\\sum_iZ_i$. Then, with probability at least $1-\\alpha$,\n\\begin{equation}\n\\mathbb E Z_i\n\\geq\n\\overline Z_n\n-G\\sqrt{\\frac{2\\log(1/\\alpha)}{n}}.\n\\end{equation}\nFor a precommitted family of $m$ repairs, replacing $\\alpha$ by $\\alpha/m$ gives a simultaneous lower bound for all $m$ candidates.\n\\end{theorem}","nt and identically distributed paired target-context gains for a precommitted transported repair, with\n\\begin{equation}\n-G\\leq Z_i\\leq G.\n\\end{equation}\nLet $\\overline Z_n=n^{-1}\\sum_iZ_i$. Then, with probability at least $1-\\alpha$,\n\\begin{equation}\n\\mathbb E Z_i\n\\geq\n\\overline Z_n\n-G\\sqrt{\\frac{2\\log(1/\\alpha)}{n}}.\n\\end{equation}\nFor a precommitted family of $m$ repairs, replacing $\\alpha$ by $\\alpha/m$ gives a simultaneous lower bound for all $m$ candidates.\n\\end{theorem}","gains for a precommitted transported repair, with\n\\begin{equation}\n-G\\leq Z_i\\leq G.\n\\end{equation}\nLet $\\overline Z_n=n^{-1}\\sum_iZ_i$. Then, with probability at least $1-\\alpha$,\n\\begin{equation}\n\\mathbb E Z_i\n\\geq\n\\overline Z_n\n-G\\sqrt{\\frac{2\\log(1/\\alpha)}{n}}.\n\\end{equation}\nFor a precommitted family of $m$ repairs, replacing $\\alpha$ by $\\alpha/m$ gives a simultaneous lower bound for all $m$ candidates.\n\\end{theorem}","\\begin{proof}\nHoeffding's inequality for a variable with range width $2G$ gives\n\\begin{equation}\n\\Pr\\left(\n\\overline Z_n-\\mathbb EZ_i\\geq t\n\\right)\n\\leq\n\\exp\\left(-\\frac{nt^2}{2G^2}\\right).\n\\end{equation}\nSet the right side to $\\alpha$. The family statement follows by the union bound.\n\\end{proof}","\\begin{definition}[Evaluator object]\nAn evaluator object at version $v$ is a tuple\n\\begin{equation}\n\\mathsf E_v=(R_v,E_v,\\iota_v,J_v,Q_v,\\mathfrak h_v),\n\\end{equation}\nwhere:\n\\begin{itemize}\n\\item $R_v$ is the protected raw-evidence space or ledger span;\n\\item $E_v$ is the evaluator's internal evidence-representation space;\n\\item $\\iota_v:R_v\\to E_v$ encodes raw evidence;\n\\item $J_v:E_v\\to\\R^q$ is a score or decision-readout map;\n\\item $Q_v$ is an uncertainty or calibration readout;\n\\item $\\mathfrak h_v$ is the historical replay set protected at version $v$.\n\\end{itemize}\n\\end{definition}","valuator object at version $v$ is a tuple\n\\begin{equation}\n\\mathsf E_v=(R_v,E_v,\\iota_v,J_v,Q_v,\\mathfrak h_v),\n\\end{equation}\nwhere:\n\\begin{itemize}\n\\item $R_v$ is the protected raw-evidence space or ledger span;\n\\item $E_v$ is the evaluator's internal evidence-representation space;\n\\item $\\iota_v:R_v\\to E_v$ encodes raw evidence;\n\\item $J_v:E_v\\to\\R^q$ is a score or decision-readout map;\n\\item $Q_v$ is an uncertainty or calibration readout;\n\\item $\\mathfrak h_v$ is the historical replay set protected at version $v$.\n\\end{itemize}\n\\end{definition}","ion}\n\\mathsf E_v=(R_v,E_v,\\iota_v,J_v,Q_v,\\mathfrak h_v),\n\\end{equation}\nwhere:\n\\begin{itemize}\n\\item $R_v$ is the protected raw-evidence space or ledger span;\n\\item $E_v$ is the evaluator's internal evidence-representation space;\n\\item $\\iota_v:R_v\\to E_v$ encodes raw evidence;\n\\item $J_v:E_v\\to\\R^q$ is a score or decision-readout map;\n\\item $Q_v$ is an uncertainty or calibration readout;\n\\item $\\mathfrak h_v$ is the historical replay set protected at version $v$.\n\\end{itemize}\n\\end{definition}","ation}\nwhere:\n\\begin{itemize}\n\\item $R_v$ is the protected raw-evidence space or ledger span;\n\\item $E_v$ is the evaluator's internal evidence-representation space;\n\\item $\\iota_v:R_v\\to E_v$ encodes raw evidence;\n\\item $J_v:E_v\\to\\R^q$ is a score or decision-readout map;\n\\item $Q_v$ is an uncertainty or calibration readout;\n\\item $\\mathfrak h_v$ is the historical replay set protected at version $v$.\n\\end{itemize}\n\\end{definition}","\\begin{definition}[Evaluator morphism]\nAn evaluator morphism $e:v\\to w$ consists of maps\n\\begin{equation}\nS_e:R_v\\to R_w,\n\\qquad\nT_e:E_v\\to E_w,\n\\end{equation}\nwhere $S_e$ is the append-only raw-evidence embedding and $T_e$ transports internal representations. Its protected-evidence defect is\n\\begin{equation}\n\\delta_{\\mathrm{obs}}(e)\n:=\n\\sup_{r\\in\\mathfrak h_v}\n\\norm{T_e\\iota_v(r)-\\iota_wS_e(r)},\n\\end{equation}\nand its historical score defect is\n\\begin{equation}\n\\delta_J(e)\n:=\n\\sup_{r\\in\\mathfrak h_v}\n\\norm{J_wT_e\\iota_v(r)-J_v\\iota_v(r)}.\n\\end{equation}\nThe morphism is exact on protected history when both defects vanish.\n\\end{definition}","to w$ consists of maps\n\\begin{equation}\nS_e:R_v\\to R_w,\n\\qquad\nT_e:E_v\\to E_w,\n\\end{equation}\nwhere $S_e$ is the append-only raw-evidence embedding and $T_e$ transports internal representations. Its protected-evidence defect is\n\\begin{equation}\n\\delta_{\\mathrm{obs}}(e)\n:=\n\\sup_{r\\in\\mathfrak h_v}\n\\norm{T_e\\iota_v(r)-\\iota_wS_e(r)},\n\\end{equation}\nand its historical score defect is\n\\begin{equation}\n\\delta_J(e)\n:=\n\\sup_{r\\in\\mathfrak h_v}\n\\norm{J_wT_e\\iota_v(r)-J_v\\iota_v(r)}.\n\\end{equation}\nThe morphism is exact on protected history when both defects vanish.\n\\end{definition}","\\begin{proposition}[Composition of evaluator defects]\n\\label{prop:evaluator-composition}\nLet $e:v\\to w$ and $f:w\\to z$ be evaluator morphisms. Suppose $T_f$ is $L_T$-Lipschitz on represented protected evidence. Then\n\\begin{equation}\n\\delta_{\\mathrm{obs}}(f\\circ e)\n\\leq\nL_T\\delta_{\\mathrm{obs}}(e)+\\delta_{\\mathrm{obs}}(f),\n\\end{equation}\nwhere the second term is evaluated on $S_e(\\mathfrak h_v)$. If, in addition,\n\\begin{equation}\n\\sup_{r\\in\\mathfrak h_v}\n\\norm{\nJ_zT_fT_e\\iota_v(r)-J_wT_e\\iota_v(r)\n}\n\\leq\n\\delta_J(f\\mid e),\n\\end{equation}\nthen\n\\begin{equation}\n\\delta_J(f\\circ e)\n\\leq\n\\delta_J(e)+\\delta_J(f\\mid e).\n\\end{equation}\n\\end{proposition}","of evaluator defects]\n\\label{prop:evaluator-composition}\nLet $e:v\\to w$ and $f:w\\to z$ be evaluator morphisms. Suppose $T_f$ is $L_T$-Lipschitz on represented protected evidence. Then\n\\begin{equation}\n\\delta_{\\mathrm{obs}}(f\\circ e)\n\\leq\nL_T\\delta_{\\mathrm{obs}}(e)+\\delta_{\\mathrm{obs}}(f),\n\\end{equation}\nwhere the second term is evaluated on $S_e(\\mathfrak h_v)$. If, in addition,\n\\begin{equation}\n\\sup_{r\\in\\mathfrak h_v}\n\\norm{\nJ_zT_fT_e\\iota_v(r)-J_wT_e\\iota_v(r)\n}\n\\leq\n\\delta_J(f\\mid e),\n\\end{equation}\nthen\n\\begin{equation}\n\\delta_J(f\\circ e)\n\\leq\n\\delta_J(e)+\\delta_J(f\\mid e).\n\\end{equation}\n\\end{proposition}","$-Lipschitz on represented protected evidence. Then\n\\begin{equation}\n\\delta_{\\mathrm{obs}}(f\\circ e)\n\\leq\nL_T\\delta_{\\mathrm{obs}}(e)+\\delta_{\\mathrm{obs}}(f),\n\\end{equation}\nwhere the second term is evaluated on $S_e(\\mathfrak h_v)$. If, in addition,\n\\begin{equation}\n\\sup_{r\\in\\mathfrak h_v}\n\\norm{\nJ_zT_fT_e\\iota_v(r)-J_wT_e\\iota_v(r)\n}\n\\leq\n\\delta_J(f\\mid e),\n\\end{equation}\nthen\n\\begin{equation}\n\\delta_J(f\\circ e)\n\\leq\n\\delta_J(e)+\\delta_J(f\\mid e).\n\\end{equation}\n\\end{proposition}","\\begin{proof}\nFor 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.\n\\end{proof}","Let $G=(N,E)$ be a connected directed graph of evaluator versions. Each directed edge $e:v\\to w$ carries an invertible internal transport\n\\begin{equation}\nT_e:E_v\\to E_w.\n\\end{equation}\nFix a base node $v_0$. For any loop $\\gamma$ based at $v_0$, let\n\\begin{equation}\nH_\\gamma:E_{v_0}\\to E_{v_0}\n\\end{equation}\nbe the composed transport around the loop.","\\begin{definition}[Path-flat evidence core]\nThe path-flat evidence core at $v_0$ is\n\\begin{equation}\nE_{v_0}^{\\flat}\n:=\n\\bigcap_{\\gamma\\in\\pi_1(G,v_0)}\n\\ker(H_\\gamma-I).\n\\end{equation}\n\\end{definition}","\\begin{theorem}[Maximal path-flat evaluator core]\n\\label{thm:path-flat-core}\nThe subspace $E_{v_0}^{\\flat}$ is the unique greatest subspace of $E_{v_0}$ fixed pointwise by every evaluator-loop holonomy. That is:\n\\begin{enumerate}\n\\item $H_\\gamma x=x$ for every $x\\in E_{v_0}^{\\flat}$ and every loop $\\gamma$;\n\\item if $W\\subseteq E_{v_0}$ satisfies $H_\\gamma w=w$ for all $w\\in W$ and all loops $\\gamma$, then $W\\subseteq E_{v_0}^{\\flat}$.\n\\end{enumerate}\n\\end{theorem}","rem}[Maximal path-flat evaluator core]\n\\label{thm:path-flat-core}\nThe subspace $E_{v_0}^{\\flat}$ is the unique greatest subspace of $E_{v_0}$ fixed pointwise by every evaluator-loop holonomy. That is:\n\\begin{enumerate}\n\\item $H_\\gamma x=x$ for every $x\\in E_{v_0}^{\\flat}$ and every loop $\\gamma$;\n\\item if $W\\subseteq E_{v_0}$ satisfies $H_\\gamma w=w$ for all $w\\in W$ and all loops $\\gamma$, then $W\\subseteq E_{v_0}^{\\flat}$.\n\\end{enumerate}\n\\end{theorem}","\\begin{proof}\nThe 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.\n\\end{proof}","\\begin{proposition}[Finite computation from a cycle basis]\n\\label{prop:cycle-basis-core}\nLet $G$ 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\n\\begin{equation}\nE_{v_0}^{\\flat}\n=\n\\bigcap_{j=1}^{m}\n\\ker(H_{\\gamma_j}-I).\n\\end{equation}\nEquivalently, $E_{v_0}^{\\flat}$ is the null space of the vertically stacked matrix\n\\begin{equation}\n\\begin{bmatrix}\nH_{\\gamma_1}-I","proposition}[Finite computation from a cycle basis]\n\\label{prop:cycle-basis-core}\nLet $G$ 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\n\\begin{equation}\nE_{v_0}^{\\flat}\n=\n\\bigcap_{j=1}^{m}\n\\ker(H_{\\gamma_j}-I).\n\\end{equation}\nEquivalently, $E_{v_0}^{\\flat}$ is the null space of the vertically stacked matrix\n\\begin{equation}\n\\begin{bmatrix}\nH_{\\gamma_1}-I","t $G$ 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\n\\begin{equation}\nE_{v_0}^{\\flat}\n=\n\\bigcap_{j=1}^{m}\n\\ker(H_{\\gamma_j}-I).\n\\end{equation}\nEquivalently, $E_{v_0}^{\\flat}$ is the null space of the vertically stacked matrix\n\\begin{equation}\n\\begin{bmatrix}\nH_{\\gamma_1}-I\\\\\n\\vdots\\\\\nH_{\\gamma_m}-I\n\\end{bmatrix}.\n\\end{equation}\n\\end{proposition}","\\begin{theorem}[Basepoint covariance of the flat core]\n\\label{thm:basepoint-core}\nLet $p:v\\to w$ be any invertible exact evaluator path in a connected atlas. Then\n\\begin{equation}\nT_p(E_v^{\\flat})=E_w^{\\flat}.\n\\end{equation}\nHence the dimension of the path-flat core is independent of basepoint.\n\\end{theorem}","\\begin{proof}\nEvery loop $\\gamma$ at $w$ corresponds to the conjugate loop $p^{-1}\\gamma p$ at $v$, with\n\\begin{equation}\nH_\\gamma=T_pH_{p^{-1}\\gamma p}T_p^{-1}.\n\\end{equation}\nIf $x\\in E_v^{\\flat}$, 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.\n\\end{proof}","\\begin{definition}[Closed-walk equalizer core]\nFor a possibly noninvertible evaluator graph, let $\\mathsf W(v)$ be the monoid of directed closed walks at $v$. Define\n\\begin{equation}\nE_v^{\\mathrm{eq}}\n:=\n\\bigcap_{\\gamma\\in\\mathsf W(v)}\\ker(H_\\gamma-I).\n\\end{equation}\n\\end{definition}","\\begin{proposition}[Maximality without invertibility]\nThe 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.\n\\end{proposition}","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\n\\begin{equation}\n\\mathsf A\n:=\n\\sum_{j=1}^{m}\nw_j(H_j-I)^*(H_j-I).\n\\end{equation}\nThis operator is positive semidefinite.","\\begin{theorem}[Spectral characterization of flatness]\n\\label{thm:holonomy-energy}\nThe exact common fixed space is\n\\begin{equation}\nE^{\\flat}=\\ker\\mathsf A.\n\\end{equation}\nLet the eigenvalues of $\\mathsf A$ be\n\\begin{equation}\n0\\leq\\lambda_1\\leq\\cdots\\leq\\lambda_d.\n\\end{equation}\nAmong all $k$-dimensional subspaces $F\\subseteq E$, the quantity\n\\begin{equation}\n\\sum_{i=1}^{k}\\angles{x_i,\\mathsf Ax_i}\n\\end{equation}\nfor 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 $\\sum_{i=1}^k\\lambda_i$.\n\\end{theorem}","flatness]\n\\label{thm:holonomy-energy}\nThe exact common fixed space is\n\\begin{equation}\nE^{\\flat}=\\ker\\mathsf A.\n\\end{equation}\nLet the eigenvalues of $\\mathsf A$ be\n\\begin{equation}\n0\\leq\\lambda_1\\leq\\cdots\\leq\\lambda_d.\n\\end{equation}\nAmong all $k$-dimensional subspaces $F\\subseteq E$, the quantity\n\\begin{equation}\n\\sum_{i=1}^{k}\\angles{x_i,\\mathsf Ax_i}\n\\end{equation}\nfor 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 $\\sum_{i=1}^k\\lambda_i$.\n\\end{theorem}","m:holonomy-energy}\nThe exact common fixed space is\n\\begin{equation}\nE^{\\flat}=\\ker\\mathsf A.\n\\end{equation}\nLet the eigenvalues of $\\mathsf A$ be\n\\begin{equation}\n0\\leq\\lambda_1\\leq\\cdots\\leq\\lambda_d.\n\\end{equation}\nAmong all $k$-dimensional subspaces $F\\subseteq E$, the quantity\n\\begin{equation}\n\\sum_{i=1}^{k}\\angles{x_i,\\mathsf Ax_i}\n\\end{equation}\nfor 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 $\\sum_{i=1}^k\\lambda_i$.\n\\end{theorem}","ation}\nE^{\\flat}=\\ker\\mathsf A.\n\\end{equation}\nLet the eigenvalues of $\\mathsf A$ be\n\\begin{equation}\n0\\leq\\lambda_1\\leq\\cdots\\leq\\lambda_d.\n\\end{equation}\nAmong all $k$-dimensional subspaces $F\\subseteq E$, the quantity\n\\begin{equation}\n\\sum_{i=1}^{k}\\angles{x_i,\\mathsf Ax_i}\n\\end{equation}\nfor 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 $\\sum_{i=1}^k\\lambda_i$.\n\\end{theorem}","t the eigenvalues of $\\mathsf A$ be\n\\begin{equation}\n0\\leq\\lambda_1\\leq\\cdots\\leq\\lambda_d.\n\\end{equation}\nAmong all $k$-dimensional subspaces $F\\subseteq E$, the quantity\n\\begin{equation}\n\\sum_{i=1}^{k}\\angles{x_i,\\mathsf Ax_i}\n\\end{equation}\nfor 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 $\\sum_{i=1}^k\\lambda_i$.\n\\end{theorem}","\\begin{proof}\nFor every $x$,\n\\begin{equation}\n\\angles{x,\\mathsf Ax}\n=\n\\sum_jw_j\\norm{(H_j-I)x}^2.\n\\end{equation}\nThis 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.\n\\end{proof}","\\begin{definition}[Thresholded approximate flat core]\nFor $\\tau\\geq0$, define\n\\begin{equation}\nE_{\\tau}^{\\flat}\n:=\n\\Span\\set{x_i:\\lambda_i\\leq\\tau},\n\\end{equation}\nwhere $x_i$ is an eigenbasis of $\\mathsf A$.\n\\end{definition}","\\begin{corollary}[Distortion bound on the approximate core]\n\\label{cor:approx-core-distortion}\nFor every $x\\in E_{\\tau}^{\\flat}$,\n\\begin{equation}\n\\sum_{j=1}^{m}w_j\\norm{(H_j-I)x}^2\n\\leq\n\\tau\\norm{x}^2.\n\\end{equation}\nIn particular,\n\\begin{equation}\n\\norm{(H_j-I)x}\n\\leq\n\\sqrt{\\frac{\\tau}{w_j}}\\norm{x}\n\\end{equation}\nfor every generator $j$.\n\\end{corollary}","\\begin{proof}\nThe 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.\n\\end{proof}","\\begin{proposition}[Stability of the threshold count]\n\\label{prop:core-spectrum-stability}\nLet $\\widehat{\\mathsf A}$ be an estimated holonomy-energy operator satisfying\n\\begin{equation}\n\\norm{\\widehat{\\mathsf A}-\\mathsf A}\\leq\\eta.\n\\end{equation}\nThen each ordered eigenvalue satisfies\n\\begin{equation}\n\\abs{\\widehat\\lambda_i-\\lambda_i}\\leq\\eta.\n\\end{equation}\nIf 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 $\\tau$.\n\\end{proposition}","egin{proposition}[Stability of the threshold count]\n\\label{prop:core-spectrum-stability}\nLet $\\widehat{\\mathsf A}$ be an estimated holonomy-energy operator satisfying\n\\begin{equation}\n\\norm{\\widehat{\\mathsf A}-\\mathsf A}\\leq\\eta.\n\\end{equation}\nThen each ordered eigenvalue satisfies\n\\begin{equation}\n\\abs{\\widehat\\lambda_i-\\lambda_i}\\leq\\eta.\n\\end{equation}\nIf 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 $\\tau$.\n\\end{proposition}","the threshold count]\n\\label{prop:core-spectrum-stability}\nLet $\\widehat{\\mathsf A}$ be an estimated holonomy-energy operator satisfying\n\\begin{equation}\n\\norm{\\widehat{\\mathsf A}-\\mathsf A}\\leq\\eta.\n\\end{equation}\nThen each ordered eigenvalue satisfies\n\\begin{equation}\n\\abs{\\widehat\\lambda_i-\\lambda_i}\\leq\\eta.\n\\end{equation}\nIf 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 $\\tau$.\n\\end{proposition}","el{prop:core-spectrum-stability}\nLet $\\widehat{\\mathsf A}$ be an estimated holonomy-energy operator satisfying\n\\begin{equation}\n\\norm{\\widehat{\\mathsf A}-\\mathsf A}\\leq\\eta.\n\\end{equation}\nThen each ordered eigenvalue satisfies\n\\begin{equation}\n\\abs{\\widehat\\lambda_i-\\lambda_i}\\leq\\eta.\n\\end{equation}\nIf 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 $\\tau$.\n\\end{proposition}","A}$ be an estimated holonomy-energy operator satisfying\n\\begin{equation}\n\\norm{\\widehat{\\mathsf A}-\\mathsf A}\\leq\\eta.\n\\end{equation}\nThen each ordered eigenvalue satisfies\n\\begin{equation}\n\\abs{\\widehat\\lambda_i-\\lambda_i}\\leq\\eta.\n\\end{equation}\nIf 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 $\\tau$.\n\\end{proposition}","\\begin{proof}\nFor every unit vector $x$,\n\\begin{equation}\n\\abs{\\angles{x,\\widehat{\\mathsf A}x}-\\angles{x,\\mathsf Ax}}\n\\leq\\eta.\n\\end{equation}\nApply the min--max characterization of ordered eigenvalues. The threshold-count statement follows immediately.\n\\end{proof}","\\begin{definition}[Semantic holonomy residual]\nFor a represented evidence state $x\\in E_v$ and loop $\\gamma$ at $v$, define\n\\begin{equation}\n\\eta_{\\mathrm{sem}}(x;\\gamma)\n:=\n\\norm{(H_\\gamma-I)x}.\n\\end{equation}\n\\end{definition}","\\begin{theorem}[Readout stability under semantic holonomy]\n\\label{thm:semantic-holonomy}\nLet $J:E_v\\to Y$ be $L_J$-Lipschitz. Then\n\\begin{equation}\nd_Y(J(H_\\gamma x),J(x))\n\\leq\nL_J\\eta_{\\mathrm{sem}}(x;\\gamma).\n\\end{equation}\nIf $J$ returns a score vector and the winning score at $x$ has margin $m(x)$ over the runner-up, then the decision is unchanged whenever\n\\begin{equation}\n2L_J\\eta_{\\mathrm{sem}}(x;\\gamma)<m(x),\n\\end{equation}\nwhere $L_J$ bounds each score coordinate in the same norm.\n\\end{theorem}","mantic holonomy]\n\\label{thm:semantic-holonomy}\nLet $J:E_v\\to Y$ be $L_J$-Lipschitz. Then\n\\begin{equation}\nd_Y(J(H_\\gamma x),J(x))\n\\leq\nL_J\\eta_{\\mathrm{sem}}(x;\\gamma).\n\\end{equation}\nIf $J$ returns a score vector and the winning score at $x$ has margin $m(x)$ over the runner-up, then the decision is unchanged whenever\n\\begin{equation}\n2L_J\\eta_{\\mathrm{sem}}(x;\\gamma)<m(x),\n\\end{equation}\nwhere $L_J$ bounds each score coordinate in the same norm.\n\\end{theorem}","\\begin{proof}\nThe 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 $m(x)$.\n\\end{proof}","\\begin{theorem}[Cumulative evaluator-drift bound]\n\\label{thm:cumulative-evaluator-drift}\nConsider a chain of evaluator morphisms with readouts into a common complete normed score space,\n\\begin{equation}\n\\mathsf E_0\\xrightarrow{e_0}\\mathsf E_1\n\\xrightarrow{e_1}\\cdots\n\\xrightarrow{e_{n-1}}\\mathsf E_n.\n\\end{equation}\nSuppose that for every protected represented state $x_t$ transported from version $0$,\n\\begin{equation}\n\\norm{J_{t+1}(T_tx_t)-J_t(x_t)}\\leq\\eps_t.\n\\end{equation}\nThen\n\\begin{equation}\n\\norm{J_n(T_{n-1}\\cdots T_0x_0)-J_0(x_0)}\n\\leq\n\\sum_{t=0}^{n-1}\\eps_t.\n\\end{equation}\nIf $\\sum_t\\eps_t<\\infty$, then the transported score sequence is Cauchy and converges in every complete score space.\n\\end{theorem}","ulative evaluator-drift bound]\n\\label{thm:cumulative-evaluator-drift}\nConsider a chain of evaluator morphisms with readouts into a common complete normed score space,\n\\begin{equation}\n\\mathsf E_0\\xrightarrow{e_0}\\mathsf E_1\n\\xrightarrow{e_1}\\cdots\n\\xrightarrow{e_{n-1}}\\mathsf E_n.\n\\end{equation}\nSuppose that for every protected represented state $x_t$ transported from version $0$,\n\\begin{equation}\n\\norm{J_{t+1}(T_tx_t)-J_t(x_t)}\\leq\\eps_t.\n\\end{equation}\nThen\n\\begin{equation}\n\\norm{J_n(T_{n-1}\\cdots T_0x_0)-J_0(x_0)}\n\\leq\n\\sum_{t=0}^{n-1}\\eps_t.\n\\end{equation}\nIf $\\sum_t\\eps_t<\\infty$, then the transported score sequence is Cauchy and converges in every complete score space.\n\\end{theorem}","chain of evaluator morphisms with readouts into a common complete normed score space,\n\\begin{equation}\n\\mathsf E_0\\xrightarrow{e_0}\\mathsf E_1\n\\xrightarrow{e_1}\\cdots\n\\xrightarrow{e_{n-1}}\\mathsf E_n.\n\\end{equation}\nSuppose that for every protected represented state $x_t$ transported from version $0$,\n\\begin{equation}\n\\norm{J_{t+1}(T_tx_t)-J_t(x_t)}\\leq\\eps_t.\n\\end{equation}\nThen\n\\begin{equation}\n\\norm{J_n(T_{n-1}\\cdots T_0x_0)-J_0(x_0)}\n\\leq\n\\sum_{t=0}^{n-1}\\eps_t.\n\\end{equation}\nIf $\\sum_t\\eps_t<\\infty$, then the transported score sequence is Cauchy and converges in every complete score space.\n\\end{theorem}","in{equation}\n\\mathsf E_0\\xrightarrow{e_0}\\mathsf E_1\n\\xrightarrow{e_1}\\cdots\n\\xrightarrow{e_{n-1}}\\mathsf E_n.\n\\end{equation}\nSuppose that for every protected represented state $x_t$ transported from version $0$,\n\\begin{equation}\n\\norm{J_{t+1}(T_tx_t)-J_t(x_t)}\\leq\\eps_t.\n\\end{equation}\nThen\n\\begin{equation}\n\\norm{J_n(T_{n-1}\\cdots T_0x_0)-J_0(x_0)}\n\\leq\n\\sum_{t=0}^{n-1}\\eps_t.\n\\end{equation}\nIf $\\sum_t\\eps_t<\\infty$, then the transported score sequence is Cauchy and converges in every complete score space.\n\\end{theorem}","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.","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.","\\begin{definition}[Core adequacy modulus]\nFor a subspace $F\\subseteq E$ with orthogonal projector $P_F$, define\n\\begin{equation}\n\\mu(F;\\Delta_{\\mathrm{prot}})\n:=\n\\inf_{d\\in\\Delta_{\\mathrm{prot}}}\n\\frac{\\norm{P_Fd}}{\\norm d}.\n\\end{equation}\n\\end{definition}","\\begin{proposition}[Robust protected distinction]\n\\label{prop:core-adequacy}\nIf\n\\begin{equation}\n\\mu(F;\\Delta_{\\mathrm{prot}})=\\mu>0,\n\\end{equation}\nthen 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\n\\begin{equation}\n2\\eta<\\mu\\norm d.\n\\end{equation}\n\\end{proposition}","\\begin{proof}\nBy definition $\\norm{P_Fd}\\geq\\mu\\norm d$. Two perturbations can reduce the separation by at most $2\\eta$.\n\\end{proof}","\\begin{definition}[Adequate approximate evidence core]\nA subspace $F$ is $(\\tau,\\mu_0)$-adequate when:\n\\begin{enumerate}\n\\item its weighted holonomy energy is at most $\\tau$ per unit norm;\n\\item $\\mu(F;\\Delta_{\\mathrm{prot}})\\geq\\mu_0>0$.\n\\end{enumerate}\n\\end{definition}","\\begin{definition}[Core-retractable upgrade]\nAn evaluator morphism $e:v\\to w$ is core-retractable when\n\\begin{equation}\nT_e(E_v^{\\flat})\\subseteq E_w^{\\flat}\n\\end{equation}\nand there is a map\n\\begin{equation}\nR_e:E_w^{\\flat}\\to E_v^{\\flat}\n\\end{equation}\nsuch that\n\\begin{equation}\nR_eT_e x=x\n\\qquad\n\\text{for every }x\\in E_v^{\\flat}.\n\\end{equation}\nIt is core-isomorphic when $T_e$ restricts to an isomorphism from $E_v^{\\flat}$ onto $E_w^{\\flat}$.\n\\end{definition}","}[Core-retractable upgrade]\nAn evaluator morphism $e:v\\to w$ is core-retractable when\n\\begin{equation}\nT_e(E_v^{\\flat})\\subseteq E_w^{\\flat}\n\\end{equation}\nand there is a map\n\\begin{equation}\nR_e:E_w^{\\flat}\\to E_v^{\\flat}\n\\end{equation}\nsuch that\n\\begin{equation}\nR_eT_e x=x\n\\qquad\n\\text{for every }x\\in E_v^{\\flat}.\n\\end{equation}\nIt is core-isomorphic when $T_e$ restricts to an isomorphism from $E_v^{\\flat}$ onto $E_w^{\\flat}$.\n\\end{definition}","upgrade]\nAn evaluator morphism $e:v\\to w$ is core-retractable when\n\\begin{equation}\nT_e(E_v^{\\flat})\\subseteq E_w^{\\flat}\n\\end{equation}\nand there is a map\n\\begin{equation}\nR_e:E_w^{\\flat}\\to E_v^{\\flat}\n\\end{equation}\nsuch that\n\\begin{equation}\nR_eT_e x=x\n\\qquad\n\\text{for every }x\\in E_v^{\\flat}.\n\\end{equation}\nIt is core-isomorphic when $T_e$ restricts to an isomorphism from $E_v^{\\flat}$ onto $E_w^{\\flat}$.\n\\end{definition}","\\begin{definition}[Conservative evaluator upgrade]\nAn evaluator upgrade $e:v\\to w$ is conservative on protected history when:\n\\begin{enumerate}\n\\item $S_e:R_v\\to R_w$ is injective;\n\\item $e$ is core-retractable;\n\\item $\\delta_{\\mathrm{obs}}(e)$ is below the declared evidence tolerance;\n\\item $\\delta_J(e)$ is below the declared historical-replay tolerance;\n\\item every new score or distinction is typed as new rather than retroactively presented as an old observation;\n\\item an independent witness can restore version $v$ and replay $\\mathfrak h_v$.\n\\end{enumerate}\n\\end{definition}","when:\n\\begin{enumerate}\n\\item $S_e:R_v\\to R_w$ is injective;\n\\item $e$ is core-retractable;\n\\item $\\delta_{\\mathrm{obs}}(e)$ is below the declared evidence tolerance;\n\\item $\\delta_J(e)$ is below the declared historical-replay tolerance;\n\\item every new score or distinction is typed as new rather than retroactively presented as an old observation;\n\\item an independent witness can restore version $v$ and replay $\\mathfrak h_v$.\n\\end{enumerate}\n\\end{definition}","umerate}\n\\item $S_e:R_v\\to R_w$ is injective;\n\\item $e$ is core-retractable;\n\\item $\\delta_{\\mathrm{obs}}(e)$ is below the declared evidence tolerance;\n\\item $\\delta_J(e)$ is below the declared historical-replay tolerance;\n\\item every new score or distinction is typed as new rather than retroactively presented as an old observation;\n\\item an independent witness can restore version $v$ and replay $\\mathfrak h_v$.\n\\end{enumerate}\n\\end{definition}","\\begin{theorem}[Historical ranking preservation]\n\\label{thm:ranking-preservation}\nLet $x_1,\\ldots,x_N$ be protected historical candidate records with scalar old scores $J_v(x_i)$. Suppose every unequal old pair has margin at least\n\\begin{equation}\nm_{\\mathrm{hist}}\n:=\n\\min_{J_v(x_i)\\neq J_v(x_j)}\n\\abs{J_v(x_i)-J_v(x_j)}>0.\n\\end{equation}\nIf an upgrade satisfies\n\\begin{equation}\n\\abs{J_w(T_ex_i)-J_v(x_i)}\\leq\\eps\n\\end{equation}\nfor every $i$, with\n\\begin{equation}\n2\\eps<m_{\\mathrm{hist}},\n\\end{equation}\nthen every strict historical ranking is preserved.\n\\end{theorem}","ation]\n\\label{thm:ranking-preservation}\nLet $x_1,\\ldots,x_N$ be protected historical candidate records with scalar old scores $J_v(x_i)$. Suppose every unequal old pair has margin at least\n\\begin{equation}\nm_{\\mathrm{hist}}\n:=\n\\min_{J_v(x_i)\\neq J_v(x_j)}\n\\abs{J_v(x_i)-J_v(x_j)}>0.\n\\end{equation}\nIf an upgrade satisfies\n\\begin{equation}\n\\abs{J_w(T_ex_i)-J_v(x_i)}\\leq\\eps\n\\end{equation}\nfor every $i$, with\n\\begin{equation}\n2\\eps<m_{\\mathrm{hist}},\n\\end{equation}\nthen every strict historical ranking is preserved.\n\\end{theorem}","-preservation}\nLet $x_1,\\ldots,x_N$ be protected historical candidate records with scalar old scores $J_v(x_i)$. Suppose every unequal old pair has margin at least\n\\begin{equation}\nm_{\\mathrm{hist}}\n:=\n\\min_{J_v(x_i)\\neq J_v(x_j)}\n\\abs{J_v(x_i)-J_v(x_j)}>0.\n\\end{equation}\nIf an upgrade satisfies\n\\begin{equation}\n\\abs{J_w(T_ex_i)-J_v(x_i)}\\leq\\eps\n\\end{equation}\nfor every $i$, with\n\\begin{equation}\n2\\eps<m_{\\mathrm{hist}},\n\\end{equation}\nthen every strict historical ranking is preserved.\n\\end{theorem}","\\begin{proof}\nFor $J_v(x_i)>J_v(x_j)$,\n\\begin{align}\nJ_w(T_ex_i)-J_w(T_ex_j)\n&\\geq\nJ_v(x_i)-J_v(x_j)-2\\eps","\\begin{proof}\nFor $J_v(x_i)>J_v(x_j)$,\n\\begin{align}\nJ_w(T_ex_i)-J_w(T_ex_j)\n&\\geq\nJ_v(x_i)-J_v(x_j)-2\\eps\\\\\n&\\geq\nm_{\\mathrm{hist}}-2\\eps>0.\n\\end{align}\n\\end{proof}","\\begin{proposition}[No silent protected-evidence erasure]\n\\label{prop:no-erasure}\nSuppose $S_e$ 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.\n\\end{proposition}","\\begin{proof}\nFor 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.\n\\end{proof}","\\begin{theorem}[Paired evaluator-risk improvement]\n\\label{thm:evaluator-risk}\nLet $Z_1,\\ldots,Z_n$ 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\n\\begin{equation}\n\\Delta_i=L_v(Z_i)-L_w(Z_i)\\in[-1,1].\n\\end{equation}\nIf the upgrade and evaluation rule are fixed before the held-out outcomes are revealed, then with probability at least $1-\\alpha$,\n\\begin{equation}\n\\mathbb E\\Delta_i\n\\geq\n\\overline\\Delta_n\n-\n\\sqrt{\\frac{2\\log(1/\\alpha)}{n}}.\n\\end{equation}\nThus a positive lower bound certifies lower expected held-out risk for the proposed evaluator.\n\\end{theorem}","m:evaluator-risk}\nLet $Z_1,\\ldots,Z_n$ 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\n\\begin{equation}\n\\Delta_i=L_v(Z_i)-L_w(Z_i)\\in[-1,1].\n\\end{equation}\nIf the upgrade and evaluation rule are fixed before the held-out outcomes are revealed, then with probability at least $1-\\alpha$,\n\\begin{equation}\n\\mathbb E\\Delta_i\n\\geq\n\\overline\\Delta_n\n-\n\\sqrt{\\frac{2\\log(1/\\alpha)}{n}}.\n\\end{equation}\nThus a positive lower bound certifies lower expected held-out risk for the proposed evaluator.\n\\end{theorem}","\\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\n\\begin{equation}\n\\Delta_i=L_v(Z_i)-L_w(Z_i)\\in[-1,1].\n\\end{equation}\nIf the upgrade and evaluation rule are fixed before the held-out outcomes are revealed, then with probability at least $1-\\alpha$,\n\\begin{equation}\n\\mathbb E\\Delta_i\n\\geq\n\\overline\\Delta_n\n-\n\\sqrt{\\frac{2\\log(1/\\alpha)}{n}}.\n\\end{equation}\nThus a positive lower bound certifies lower expected held-out risk for the proposed evaluator.\n\\end{theorem}","\\begin{proof}\nApply \\cref{thm:paired-gain} with $G=1$.\n\\end{proof}","\\begin{corollary}[Multiple protected metrics]\nIf $m$ 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.\n\\end{corollary}","\\begin{definition}[Evaluator-upgrade certificate]\nAn evaluator-upgrade certificate is a tuple\n\\begin{equation}\n\\Pi_{\\mathrm{eval}}\n=\n(v,w,S_e,T_e,R_e,\n\\mathfrak h_v,\n\\mathsf A_v,\\mathsf A_w,\n\\tau,\\mu_0,\n\\delta_{\\mathrm{obs}},\\delta_J,\n\\mathcal T_{\\mathrm{new}},\n\\mathcal R_{\\mathrm{rollback}},\n\\sigma),\n\\end{equation}\ncontaining 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 $\\sigma$.\n\\end{definition}","delta_{\\mathrm{obs}},\\delta_J,\n\\mathcal T_{\\mathrm{new}},\n\\mathcal R_{\\mathrm{rollback}},\n\\sigma),\n\\end{equation}\ncontaining 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 $\\sigma$.\n\\end{definition}","\\begin{protocol}[Certified evaluator upgrade]\n\\label{prot:evaluator-upgrade}\nA proposed evaluator version $w$ is provisionally admitted from version $v$ only after the following steps:\n\\begin{enumerate}\n\\item freeze the candidate evaluator and all acceptance thresholds;\n\\item embed the append-only raw-evidence ledger through $S_e$;\n\\item verify protected-evidence commutation and obstruction faithfulness;\n\\item compute exact or approximate path-flat cores and the adequacy modulus;\n\\item replay historical decisions and check ranking margins;\n\\item evaluate new externally revealed cases with a precommitted statistical test;\n\\item test cycle holonomy against all currently retained evaluator branches;\n\\item record a witness identity and rollback checkpoint not editable by the candidate transaction;\n\\it","\\begin{protocol}[Certified evaluator upgrade]\n\\label{prot:evaluator-upgrade}\nA proposed evaluator version $w$ is provisionally admitted from version $v$ only after the following steps:\n\\begin{enumerate}\n\\item freeze the candidate evaluator and all acceptance thresholds;\n\\item embed the append-only raw-evidence ledger through $S_e$;\n\\item verify protected-evidence commutation and obstruction faithfulness;\n\\item compute exact or approximate path-flat cores and the adequacy modulus;\n\\item replay historical decisions and check ranking margins;\n\\item evaluate new externally revealed cases with a precommitted statistical test;\n\\item test cycle holonomy against all currently retained evaluator branches;\n\\item record a witness identity and rollback checkpoint not editable by the candidate transaction;\n\\item commit provisionally if every required c","\\begin{protocol}[Certified evaluator upgrade]\n\\label{prot:evaluator-upgrade}\nA proposed evaluator version $w$ is provisionally admitted from version $v$ only after the following steps:\n\\begin{enumerate}\n\\item freeze the candidate evaluator and all acceptance thresholds;\n\\item embed the append-only raw-evidence ledger through $S_e$;\n\\item verify protected-evidence commutation and obstruction faithfulness;\n\\item compute exact or approximate path-flat cores and the adequacy modulus;\n\\item replay historical decisions and check ranking margins;\n\\item evaluate new externally revealed cases with a precommitted statistical test;\n\\item test cycle holonomy against all currently retained evaluator branches;\n\\item record a witness identity and rollback checkpoint not editable by the candidate transaction;\n\\item commit provisionally if every required certificate passes; otherwise restore version $v$.\n\\end{enumerate}\n\\end{protocol}","historical decisions and check ranking margins;\n\\item evaluate new externally revealed cases with a precommitted statistical test;\n\\item test cycle holonomy against all currently retained evaluator branches;\n\\item record a witness identity and rollback checkpoint not editable by the candidate transaction;\n\\item commit provisionally if every required certificate passes; otherwise restore version $v$.\n\\end{enumerate}\n\\end{protocol}","\\begin{theorem}[Soundness of a certified evaluator chain]\n\\label{thm:evaluator-chain}\nConsider 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:\n\\begin{enumerate}\n\\item the original protected core is recoverably embedded at every later version;\n\\item cumulative protected score drift is at most $\\sum_t\\eps_t$;\n\\item every original strict decision margin larger than $2\\sum_t\\eps_t$ is preserved;\n\\item a protected difference $d$ remains distinguishable whenever\n\\begin{equation}\n\\mu_0\\norm d>2\\sum_t\\eta_t.\n\\end{equation}\n\\end{enumerate}\nIf the score-error series is summable along an infinite chain, protected scores converge.\n\\end{theorem}","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:\n\\begin{enumerate}\n\\item the original protected core is recoverably embedded at every later version;\n\\item cumulative protected score drift is at most $\\sum_t\\eps_t$;\n\\item every original strict decision margin larger than $2\\sum_t\\eps_t$ is preserved;\n\\item a protected difference $d$ remains distinguishable whenever\n\\begin{equation}\n\\mu_0\\norm d>2\\sum_t\\eta_t.\n\\end{equation}\n\\end{enumerate}\nIf the score-error series is summable along an infinite chain, protected scores converge.\n\\end{theorem}","etractable, 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:\n\\begin{enumerate}\n\\item the original protected core is recoverably embedded at every later version;\n\\item cumulative protected score drift is at most $\\sum_t\\eps_t$;\n\\item every original strict decision margin larger than $2\\sum_t\\eps_t$ is preserved;\n\\item a protected difference $d$ remains distinguishable whenever\n\\begin{equation}\n\\mu_0\\norm d>2\\sum_t\\eta_t.\n\\end{equation}\n\\end{enumerate}\nIf the score-error series is summable along an infinite chain, protected scores converge.\n\\end{theorem}","turbs each retained protected representation by at most $\\eta_t$, and retains core adequacy at least $\\mu_0>0$. Then:\n\\begin{enumerate}\n\\item the original protected core is recoverably embedded at every later version;\n\\item cumulative protected score drift is at most $\\sum_t\\eps_t$;\n\\item every original strict decision margin larger than $2\\sum_t\\eps_t$ is preserved;\n\\item a protected difference $d$ remains distinguishable whenever\n\\begin{equation}\n\\mu_0\\norm d>2\\sum_t\\eta_t.\n\\end{equation}\n\\end{enumerate}\nIf the score-error series is summable along an infinite chain, protected scores converge.\n\\end{theorem}","ost $\\eta_t$, and retains core adequacy at least $\\mu_0>0$. Then:\n\\begin{enumerate}\n\\item the original protected core is recoverably embedded at every later version;\n\\item cumulative protected score drift is at most $\\sum_t\\eps_t$;\n\\item every original strict decision margin larger than $2\\sum_t\\eps_t$ is preserved;\n\\item a protected difference $d$ remains distinguishable whenever\n\\begin{equation}\n\\mu_0\\norm d>2\\sum_t\\eta_t.\n\\end{equation}\n\\end{enumerate}\nIf the score-error series is summable along an infinite chain, protected scores converge.\n\\end{theorem}","\\begin{proof}\nItem 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.\n\\end{proof}","\\begin{example}[Shear holonomy]\nLet $E=\\R^2$ and suppose evaluator evolution around one loop produces\n\\begin{equation}\nH=\n\\begin{bmatrix}\n1&\\alpha","\\begin{example}[Shear holonomy]\nLet $E=\\R^2$ and suppose evaluator evolution around one loop produces\n\\begin{equation}\nH=\n\\begin{bmatrix}\n1&\\alpha\\\\\n0&1\n\\end{bmatrix},\n\\qquad \\alpha\\neq0.\n\\end{equation}\nThen\n\\begin{equation}\nH-I=\n\\begin{bmatrix}\n0&\\alpha","0&1\n\\end{bmatrix},\n\\qquad \\alpha\\neq0.\n\\end{equation}\nThen\n\\begin{equation}\nH-I=\n\\begin{bmatrix}\n0&\\alpha\\\\\n0&0\n\\end{bmatrix},\n\\end{equation}\nso\n\\begin{equation}\nE^{\\flat}=\\ker(H-I)=\\set{(x,0):x\\in\\R}.\n\\end{equation}\nThe first coordinate is path-flat. The second coordinate changes the first under a loop and therefore cannot be treated as invariant evidence.\n\\end{example}","0&0\n\\end{bmatrix},\n\\end{equation}\nso\n\\begin{equation}\nE^{\\flat}=\\ker(H-I)=\\set{(x,0):x\\in\\R}.\n\\end{equation}\nThe first coordinate is path-flat. The second coordinate changes the first under a loop and therefore cannot be treated as invariant evidence.\n\\end{example}","\\begin{example}[Flat but inadequate core]\nLet protected evidence states be\n\\begin{equation}\nx_+=(1,1),\n\\qquad\nx_-=(1,-1),\n\\end{equation}\nand let shear holonomy leave only the first coordinate flat. Both records project to $(1,0)$, so\n\\begin{equation}\n\\mu(E^{\\flat};\\set{x_+-x_-})=0.\n\\end{equation}\nThe core is exactly flat but cannot preserve the protected distinction between the two outcomes.\n\\end{example}","Let a complete implementation state decompose as\n\\begin{equation}\n\\Sigma\n=\n\\Sigma^{\\mathrm{task}}\n\\oplus\n\\Sigma^{\\mathrm{memory}}\n\\oplus\n\\Sigma^{\\mathrm{repair}}\n\\oplus\n\\Sigma^{\\mathrm{evaluation}}\n\\oplus\n\\Sigma^{\\mathrm{provenance}}\n\\oplus\n\\Sigma^{\\mathrm{rollback}}.\n\\end{equation}\nA transaction selects an editable mask $E\\subseteq\\Sigma$ and a witness mask $W\\subseteq\\Sigma$ with\n\\begin{equation}\nE\\cap W=\\varnothing.\n\\end{equation}\nThe 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.","et a complete implementation state decompose as\n\\begin{equation}\n\\Sigma\n=\n\\Sigma^{\\mathrm{task}}\n\\oplus\n\\Sigma^{\\mathrm{memory}}\n\\oplus\n\\Sigma^{\\mathrm{repair}}\n\\oplus\n\\Sigma^{\\mathrm{evaluation}}\n\\oplus\n\\Sigma^{\\mathrm{provenance}}\n\\oplus\n\\Sigma^{\\mathrm{rollback}}.\n\\end{equation}\nA transaction selects an editable mask $E\\subseteq\\Sigma$ and a witness mask $W\\subseteq\\Sigma$ with\n\\begin{equation}\nE\\cap W=\\varnothing.\n\\end{equation}\nThe 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.","\\begin{definition}[Improvement transaction]\nA proof-carrying improvement transaction is a tuple\n\\begin{equation}\n\\tau\n=\n\\left(\nP,\nE,\nW,\n\\delta,\n\\widehat\\Delta,\n\\Ccal,\nq,\na,\nZ,\n\\Pi\n\\right),\n\\end{equation}\nwhere:\n\\begin{enumerate}\n\\item $P$ is a content-addressed parent checkpoint;\n\\item $E$ is the editable component mask;\n\\item $W$ is the frozen witness mask;\n\\item $\\delta$ is the proposed edit;\n\\item $\\widehat\\Delta$ is the precommitted effect prediction;\n\\item $\\Ccal$ is the capability and resource budget;\n\\item $q$ is the declared trace map from full executions to witness-visible traces;\n\\item $a$ is the sealed acceptance functional;\n\\item $Z$ is the revealed trace;\n\\item $\\Pi$ is the final receipt containing the decision, evidence links, and rollback or commit record.\n\\end{enumerate}\n\\end{definition}","n{definition}[Improvement transaction]\nA proof-carrying improvement transaction is a tuple\n\\begin{equation}\n\\tau\n=\n\\left(\nP,\nE,\nW,\n\\delta,\n\\widehat\\Delta,\n\\Ccal,\nq,\na,\nZ,\n\\Pi\n\\right),\n\\end{equation}\nwhere:\n\\begin{enumerate}\n\\item $P$ is a content-addressed parent checkpoint;\n\\item $E$ is the editable component mask;\n\\item $W$ is the frozen witness mask;\n\\item $\\delta$ is the proposed edit;\n\\item $\\widehat\\Delta$ is the precommitted effect prediction;\n\\item $\\Ccal$ is the capability and resource budget;\n\\item $q$ is the declared trace map from full executions to witness-visible traces;\n\\item $a$ is the sealed acceptance functional;\n\\item $Z$ is the revealed trace;\n\\item $\\Pi$ is the final receipt containing the decision, evidence links, and rollback or commit record.\n\\end{enumerate}\n\\end{definition}","ement transaction is a tuple\n\\begin{equation}\n\\tau\n=\n\\left(\nP,\nE,\nW,\n\\delta,\n\\widehat\\Delta,\n\\Ccal,\nq,\na,\nZ,\n\\Pi\n\\right),\n\\end{equation}\nwhere:\n\\begin{enumerate}\n\\item $P$ is a content-addressed parent checkpoint;\n\\item $E$ is the editable component mask;\n\\item $W$ is the frozen witness mask;\n\\item $\\delta$ is the proposed edit;\n\\item $\\widehat\\Delta$ is the precommitted effect prediction;\n\\item $\\Ccal$ is the capability and resource budget;\n\\item $q$ is the declared trace map from full executions to witness-visible traces;\n\\item $a$ is the sealed acceptance functional;\n\\item $Z$ is the revealed trace;\n\\item $\\Pi$ is the final receipt containing the decision, evidence links, and rollback or commit record.\n\\end{enumerate}\n\\end{definition}","\\left(\nP,\nE,\nW,\n\\delta,\n\\widehat\\Delta,\n\\Ccal,\nq,\na,\nZ,\n\\Pi\n\\right),\n\\end{equation}\nwhere:\n\\begin{enumerate}\n\\item $P$ is a content-addressed parent checkpoint;\n\\item $E$ is the editable component mask;\n\\item $W$ is the frozen witness mask;\n\\item $\\delta$ is the proposed edit;\n\\item $\\widehat\\Delta$ is the precommitted effect prediction;\n\\item $\\Ccal$ is the capability and resource budget;\n\\item $q$ is the declared trace map from full executions to witness-visible traces;\n\\item $a$ is the sealed acceptance functional;\n\\item $Z$ is the revealed trace;\n\\item $\\Pi$ is the final receipt containing the decision, evidence links, and rollback or commit record.\n\\end{enumerate}\n\\end{definition}","where:\n\\begin{enumerate}\n\\item $P$ is a content-addressed parent checkpoint;\n\\item $E$ is the editable component mask;\n\\item $W$ is the frozen witness mask;\n\\item $\\delta$ is the proposed edit;\n\\item $\\widehat\\Delta$ is the precommitted effect prediction;\n\\item $\\Ccal$ is the capability and resource budget;\n\\item $q$ is the declared trace map from full executions to witness-visible traces;\n\\item $a$ is the sealed acceptance functional;\n\\item $Z$ is the revealed trace;\n\\item $\\Pi$ is the final receipt containing the decision, evidence links, and rollback or commit record.\n\\end{enumerate}\n\\end{definition}","-addressed parent checkpoint;\n\\item $E$ is the editable component mask;\n\\item $W$ is the frozen witness mask;\n\\item $\\delta$ is the proposed edit;\n\\item $\\widehat\\Delta$ is the precommitted effect prediction;\n\\item $\\Ccal$ is the capability and resource budget;\n\\item $q$ is the declared trace map from full executions to witness-visible traces;\n\\item $a$ is the sealed acceptance functional;\n\\item $Z$ is the revealed trace;\n\\item $\\Pi$ is the final receipt containing the decision, evidence links, and rollback or commit record.\n\\end{enumerate}\n\\end{definition}","em $E$ is the editable component mask;\n\\item $W$ is the frozen witness mask;\n\\item $\\delta$ is the proposed edit;\n\\item $\\widehat\\Delta$ is the precommitted effect prediction;\n\\item $\\Ccal$ is the capability and resource budget;\n\\item $q$ is the declared trace map from full executions to witness-visible traces;\n\\item $a$ is the sealed acceptance functional;\n\\item $Z$ is the revealed trace;\n\\item $\\Pi$ is the final receipt containing the decision, evidence links, and rollback or commit record.\n\\end{enumerate}\n\\end{definition}","\\begin{definition}[Well-formed transaction]\nA transaction is well formed when:\n\\begin{enumerate}\n\\item $\\delta$ acts only on $E$;\n\\item $a$, $q$, the evidence ledger head, and the rollback checkpoint are determined before candidate execution;\n\\item candidate capabilities are a subset of $\\Ccal$;\n\\item all externally persistent effects are buffered until commit or are separately declared irreversible effects;\n\\item the candidate cannot append to the protected evidence namespace except through $q$;\n\\item the final receipt binds the parent digest, child digest, edit digest, trace digest, evaluator version, witness version, and decision.\n\\end{enumerate}\n\\end{definition}","item $\\delta$ acts only on $E$;\n\\item $a$, $q$, the evidence ledger head, and the rollback checkpoint are determined before candidate execution;\n\\item candidate capabilities are a subset of $\\Ccal$;\n\\item all externally persistent effects are buffered until commit or are separately declared irreversible effects;\n\\item the candidate cannot append to the protected evidence namespace except through $q$;\n\\item the final receipt binds the parent digest, child digest, edit digest, trace digest, evaluator version, witness version, and decision.\n\\end{enumerate}\n\\end{definition}","The transaction follows six phases:\n\\begin{equation}\n\\mathrm{prepare}\n\\longrightarrow\n\\mathrm{seal}\n\\longrightarrow\n\\mathrm{execute}\n\\longrightarrow\n\\mathrm{reveal}\n\\longrightarrow\n\\mathrm{decide}\n\\longrightarrow\n\\mathrm{finalize}.\n\\end{equation}\nOnly the final phase may expose buffered effects or replace the active checkpoint.","\\begin{definition}[Witness separation]\nA well-formed transaction is witness-separated when:\n\\begin{enumerate}\n\\item the candidate edit acts only on $E$;\n\\item the acceptance functional and its executable dependencies lie in $W$;\n\\item $E$ cannot modify $W$, the sealed trace schema, the evidence ledger head, or the rollback checkpoint during the transaction;\n\\item candidate-to-witness communication factors through the declared trace map $q$.\n\\end{enumerate}\n\\end{definition}","well-formed transaction is witness-separated when:\n\\begin{enumerate}\n\\item the candidate edit acts only on $E$;\n\\item the acceptance functional and its executable dependencies lie in $W$;\n\\item $E$ cannot modify $W$, the sealed trace schema, the evidence ledger head, or the rollback checkpoint during the transaction;\n\\item candidate-to-witness communication factors through the declared trace map $q$.\n\\end{enumerate}\n\\end{definition}","\\begin{theorem}[Atomic self-approval noninterference]\n\\label{thm:witness-noninterference}\nIn a witness-separated transaction, changing the editable implementation while holding the declared trace $Z$ fixed cannot change the acceptance decision.\n\\end{theorem}","\\begin{proof}\nThe 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.\n\\end{proof}","The theorem is intentionally narrow. It blocks direct atomic self-approval. It does not show that $q$ exposes every acceptance-relevant fact.","\\begin{theorem}[Trace-factorization criterion]\n\\label{thm:trace-factorization}\nLet $Y$ 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:\n\\begin{enumerate}\n\\item $b$ is constant on every fiber $q^{-1}(z)$;\n\\item there exists a unique function $a:q(Y)\\to\\{0,1\\}$ such that\n\\begin{equation}\nb=a\\circ q.\n\\end{equation}\n\\end{enumerate}\nThus the trace schema is complete for the declared decision exactly when the full-information decision factors through the trace.\n\\end{theorem}","iterion]\n\\label{thm:trace-factorization}\nLet $Y$ 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:\n\\begin{enumerate}\n\\item $b$ is constant on every fiber $q^{-1}(z)$;\n\\item there exists a unique function $a:q(Y)\\to\\{0,1\\}$ such that\n\\begin{equation}\nb=a\\circ q.\n\\end{equation}\n\\end{enumerate}\nThus the trace schema is complete for the declared decision exactly when the full-information decision factors through the trace.\n\\end{theorem}","\\begin{proof}\nIf $b=a\\circ q$, 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 $q(Y)$.\n\\end{proof}","\\begin{warning}[Trace-schema attack surface]\nWitness 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\n\\begin{equation}\nq(y_1)=q(y_2),\n\\qquad\nb(y_1)\\neq b(y_2).\n\\end{equation}\nSuch a pair is a constructive certificate of trace incompleteness.\n\\end{warning}","Let $\\widetilde F(P,\\delta)$ denote execution in an isolated child environment. Write\n\\begin{equation}\n\\widetilde F(P,\\delta)=(C,B),\n\\end{equation}\nwhere $C$ is the candidate child state and $B$ is a buffer of externally persistent effects.","\\begin{theorem}[Atomic rollback]\n\\label{thm:atomic-rollback}\nAssume:\n\\begin{enumerate}\n\\item the parent checkpoint $P$ is immutable during execution;\n\\item every persistent candidate effect is either contained in $C$ or buffered in $B$;\n\\item $B$ is unpublished until the witness commits;\n\\item rejection destroys $C$ and $B$ while leaving the append-only receipt ledger outside the operational parent state.\n\\end{enumerate}\nThen a rejected transaction restores the operational system exactly to $P$. The only persistent change is the append-only rejection receipt.\n\\end{theorem}","parent checkpoint $P$ is immutable during execution;\n\\item every persistent candidate effect is either contained in $C$ or buffered in $B$;\n\\item $B$ is unpublished until the witness commits;\n\\item rejection destroys $C$ and $B$ while leaving the append-only receipt ledger outside the operational parent state.\n\\end{enumerate}\nThen a rejected transaction restores the operational system exactly to $P$. The only persistent change is the append-only rejection receipt.\n\\end{theorem}","\\begin{proof}\nBy 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.\n\\end{proof}","\\begin{definition}[Irreversible-effect budget]\nWhen 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\n\\begin{equation}\nL_I(I(\\tau))\\leq\\eta.\n\\end{equation}\nThe quantity $\\eta$ must be part of the precommitment and acceptance rule.\n\\end{definition}","\\begin{proposition}[Serializability of independent repairs]\n\\label{prop:transaction-serializability}\nLet $\\tau_1$ 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.\n\\end{proposition}","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.","\\begin{theorem}[Adaptive alpha-spending guarantee]\n\\label{thm:adaptive-alpha-spending}\nLet $F_t$ be the event that transaction $t$ accepts a child whose true declared gain is nonpositive. Suppose\n\\begin{equation}\n\\Pr(F_t\\mid\\Fcal_{t-1})\\leq\\delta_t\n\\qquad\\text{almost surely}\n\\end{equation}\nfor a deterministic sequence $(\\delta_t)$. Then for every $T$,\n\\begin{equation}\n\\Pr\\left(\\bigcup_{t=1}^{T}F_t\\right)\n\\leq\n\\sum_{t=1}^{T}\\delta_t.\n\\end{equation}\nIf $\\sum_t\\delta_t<\\infty$, only finitely many false acceptances occur almost surely.\n\\end{theorem}","l{thm:adaptive-alpha-spending}\nLet $F_t$ be the event that transaction $t$ accepts a child whose true declared gain is nonpositive. Suppose\n\\begin{equation}\n\\Pr(F_t\\mid\\Fcal_{t-1})\\leq\\delta_t\n\\qquad\\text{almost surely}\n\\end{equation}\nfor a deterministic sequence $(\\delta_t)$. Then for every $T$,\n\\begin{equation}\n\\Pr\\left(\\bigcup_{t=1}^{T}F_t\\right)\n\\leq\n\\sum_{t=1}^{T}\\delta_t.\n\\end{equation}\nIf $\\sum_t\\delta_t<\\infty$, only finitely many false acceptances occur almost surely.\n\\end{theorem}","\\begin{proof}\nTaking 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.\n\\end{proof}","\\begin{definition}[Evidence namespace]\nAn 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.\n\\end{definition}","Let the system contain $n$ witness-capable components. At round $t$, let $E_t$ be the edited set and $W_t$ the witness set.","\\begin{definition}[Rotating witness schedule]\nA sequence $(E_t,W_t)_{t\\geq0}$ is a rotating witness schedule when:\n\\begin{enumerate}\n\\item $E_t\\cap W_t=\\varnothing$ for every $t$;\n\\item $W_t$ is immutable during transaction $t$;\n\\item every component belongs to $E_t$ for at least one $t$;\n\\item every witness replacement is validated by a component outside the same edit set;\n\\item protected evidence is transported through a conservative evaluator morphism at every rotation;\n\\item every committed round retains a rollback path to a state certified by the preceding witness.\n\\end{enumerate}\n\\end{definition}","\\begin{theorem}[Minimum rounds under a witness quorum]\n\\label{thm:rotation-rounds}\nSuppose 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\n\\begin{equation}\n\\left\\lceil\\frac{n}{n-q}\\right\\rceil\n\\end{equation}\nrounds. 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.\n\\end{theorem}","label{thm:rotation-rounds}\nSuppose 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\n\\begin{equation}\n\\left\\lceil\\frac{n}{n-q}\\right\\rceil\n\\end{equation}\nrounds. 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.\n\\end{theorem}","\\begin{proof}\nThe 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.\n\\end{proof}","\\begin{definition}[Validation dependency graph]\nFor 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.\n\\end{definition}","\\begin{proof}\nFor 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.\n\\end{proof}","\\begin{proposition}[Quorum-intersection condition]\n\\label{prop:witness-quorum-intersection}\nLet $m$ witness replicas be available, and suppose at most $f$ may be compromised. If acceptance requires a quorum of size $q$ satisfying\n\\begin{equation}\n2q>m+f,\n\\end{equation}\nthen any two accepting quorums intersect in more than $f$ replicas. Hence their intersection contains at least one uncompromised replica.\n\\end{proposition}","\\begin{proof}\nFor quorums $Q_1,Q_2$,\n\\begin{equation}\n\\abs{Q_1\\cap Q_2}\n\\geq\n2q-m\n>\nf.\n\\end{equation}\nAt most $f$ replicas are compromised, so at least one member of the intersection is uncompromised.\n\\end{proof}","Let $S$ be a finite set of complete lineage states. Write\n\\begin{equation}\ns\\xrightarrow{\\tau_{\\mathrm{imp}}}s'\n\\end{equation}\nwhen one authorized improvement transaction can take $s$ to $s'$. Define\n\\begin{equation}\n\\Post_{\\mathrm{imp}}(s)\n=\n\\set{s'\\in S:s\\xrightarrow{\\tau_{\\mathrm{imp}}}s'}.\n\\end{equation}","Let\n\\begin{equation}\n\\begin{aligned}\nC_{\\mathrm{RI}}(s)\n={}&\n\\Grounded(s)\n\\wedge\\ProvValid(s)\n\\wedge\\WorkspaceBound(s)\\\\\n&\\wedge\\RepairTyped(s)\n\\wedge\\Witnessed(s)\n\\wedge\\EvidenceFlat(s)\\\\\n&\\wedge\\Rollbackable(s)\n\\wedge\\MetaCapable(s).\n\\end{aligned}\n\\end{equation}\nFor $X\\subseteq S$, define\n\\begin{equation}\n\\Psi_{\\forall}(X)\n=\n\\set{s\\in S:C_{\\mathrm{RI}}(s)\\text{ and }\\Post_{\\mathrm{imp}}(s)\\subseteq X}.\n\\end{equation}","&\\wedge\\Rollbackable(s)\n\\wedge\\MetaCapable(s).\n\\end{aligned}\n\\end{equation}\nFor $X\\subseteq S$, define\n\\begin{equation}\n\\Psi_{\\forall}(X)\n=\n\\set{s\\in S:C_{\\mathrm{RI}}(s)\\text{ and }\\Post_{\\mathrm{imp}}(s)\\subseteq X}.\n\\end{equation}","\\begin{theorem}[Existence of universal recursive improvement closure]\n\\label{thm:ri-closure}\nThe operator $\\Psi_{\\forall}:2^S\\to2^S$ is monotone and has a greatest fixed point\n\\begin{equation}\n\\Ical_{\\forall}^{*}\n=\n\\nu X.\\Psi_{\\forall}(X).\n\\end{equation}\nA state lies in $\\Ical_{\\forall}^{*}$ exactly when it satisfies the static invariants and every authorized successor also lies in $\\Ical_{\\forall}^{*}$.\n\\end{theorem}","\\begin{proof}\nIf $X\\subseteq Y$, 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.\n\\end{proof}","\\begin{corollary}[Finite descending computation]\nSet $X_0=S$ and $X_{k+1}=\\Psi_{\\forall}(X_k)$. The sequence is descending and stabilizes at $\\Ical_{\\forall}^{*}$ after at most $\\abs{S}$ strict reductions.\n\\end{corollary}","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\n\\begin{equation}\nT(s,u,d)\\in S\n\\end{equation}\nbe the resulting state.","\\begin{definition}[Controllable predecessor]\nFor $X\\subseteq S$, define\n\\begin{equation}\n\\operatorname{CPre}(X)\n=\n\\set{\n s\\in S:\n \\exists u\\in\\mathsf{Act}(s)\\ \\forall d\\in\\mathsf{Dist}(s,u),\\ T(s,u,d)\\in X\n}.\n\\end{equation}\n\\end{definition}","Define\n\\begin{equation}\n\\Psi_{\\mathrm{ctrl}}(X)\n=\n\\set{s\\in S:C_{\\mathrm{RI}}(s)}\n\\cap\n\\operatorname{CPre}(X).\n\\end{equation}","\\begin{theorem}[Robust controlled improvement kernel]\n\\label{thm:controlled-closure}\nThe operator $\\Psi_{\\mathrm{ctrl}}$ is monotone and has a greatest fixed point\n\\begin{equation}\n\\Ical_{\\mathrm{ctrl}}^{*}\n=\n\\nu X.\\Psi_{\\mathrm{ctrl}}(X).\n\\end{equation}\nFor every state in $\\Ical_{\\mathrm{ctrl}}^{*}$, there exists a stationary authorized policy that keeps every admissible successor inside $\\Ical_{\\mathrm{ctrl}}^{*}$.\n\\end{theorem}","\\begin{proof}\nMonotonicity 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.\n\\end{proof}","\\begin{proposition}[Closure monotonicity]\n\\label{prop:closure-monotonicity}\nUnder a fixed invariant predicate:\n\\begin{enumerate}\n\\item adding authorized actions can only enlarge $\\Ical_{\\mathrm{ctrl}}^{*}$;\n\\item adding admissible disturbances can only shrink $\\Ical_{\\mathrm{ctrl}}^{*}$;\n\\item strengthening the static invariant predicate can only shrink both universal and controlled closures;\n\\item whenever every action set is nonempty, $\\Ical_{\\forall}^{*}\\subseteq\\Ical_{\\mathrm{ctrl}}^{*}$.\n\\end{enumerate}\n\\end{proposition}","\\label{prop:closure-monotonicity}\nUnder a fixed invariant predicate:\n\\begin{enumerate}\n\\item adding authorized actions can only enlarge $\\Ical_{\\mathrm{ctrl}}^{*}$;\n\\item adding admissible disturbances can only shrink $\\Ical_{\\mathrm{ctrl}}^{*}$;\n\\item strengthening the static invariant predicate can only shrink both universal and controlled closures;\n\\item whenever every action set is nonempty, $\\Ical_{\\forall}^{*}\\subseteq\\Ical_{\\mathrm{ctrl}}^{*}$.\n\\end{enumerate}\n\\end{proposition}","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\n\\begin{equation}\n\\operatorname{SPre}(X)\n=\n\\set{s:C_{\\mathrm{RI}}(s)}\n\\cap\n\\operatorname{CPre}(X).\n\\end{equation}","\\begin{definition}[Productive recursive closure]\nFor a finite perfect-information transaction game, define\n\\begin{equation}\n\\Ical_{\\gamma}^{\\omega}\n=\n\\nu Z.\\,\\mu Y.\n\\left[\n\\left(G_{\\gamma}\\cap\\operatorname{SPre}(Z)\\right)\n\\cup\n\\operatorname{SPre}(Y)\n\\right].\n\\end{equation}\n\\end{definition}","\\begin{theorem}[Buechi characterization of repeated improvement]\n\\label{thm:productive-closure}\nA 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.\n\\end{theorem}","\\begin{proof}\nFor 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.\n\\end{proof}","\\begin{corollary}[Productivity is stricter than safety]\n\\begin{equation}\n\\Ical_{\\gamma}^{\\omega}\n\\subseteq\n\\Ical_{\\mathrm{ctrl}}^{*}.\n\\end{equation}\nThe inclusion may be strict.\n\\end{corollary}","\\begin{proof}\nEvery 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.\n\\end{proof}","\\begin{proposition}[Finite ranking certificate for one gain cycle]\n\\label{prop:progress-ranking}\nLet $X\\subseteq\\Ical_{\\mathrm{ctrl}}^{*}$ 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\n\\begin{equation}\nh(s')<h(s).\n\\end{equation}\nThen the controller can force a visit to $G_\\gamma$ within at most $H$ transactions while remaining in $X$.\n\\end{proposition}","ate for one gain cycle]\n\\label{prop:progress-ranking}\nLet $X\\subseteq\\Ical_{\\mathrm{ctrl}}^{*}$ 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\n\\begin{equation}\nh(s')<h(s).\n\\end{equation}\nThen the controller can force a visit to $G_\\gamma$ within at most $H$ transactions while remaining in $X$.\n\\end{proposition}","abel{prop:progress-ranking}\nLet $X\\subseteq\\Ical_{\\mathrm{ctrl}}^{*}$ 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\n\\begin{equation}\nh(s')<h(s).\n\\end{equation}\nThen the controller can force a visit to $G_\\gamma$ within at most $H$ transactions while remaining in $X$.\n\\end{proposition}","ubseteq\\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\n\\begin{equation}\nh(s')<h(s).\n\\end{equation}\nThen the controller can force a visit to $G_\\gamma$ within at most $H$ transactions while remaining in $X$.\n\\end{proposition}","\\begin{proof}\nThe nonnegative integer rank decreases strictly at every step, so it reaches zero in at most $H$ steps. Every successor remains in $X$ by assumption.\n\\end{proof}","For stochastic transitions, define\n\\begin{equation}\n\\operatorname{CPre}_{\\delta}(X)\n=\n\\set{\n s:\\exists u\\in\\mathsf{Act}(s),\\\n \\Pr(T(s,u,D)\\in X\\mid s,u)\\geq1-\\delta\n}.\n\\end{equation}\nA one-step chance constraint does not by itself provide an infinite-horizon guarantee.","\\begin{proposition}[Risk-budget composition]\n\\label{prop:risk-budget-composition}\nSuppose 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\n\\begin{equation}\n1-\\sum_{t=1}^{T}\\delta_t.\n\\end{equation}\nIf $\\sum_t\\delta_t\\leq\\delta$, the probability of ever leaving is at most $\\delta$.\n\\end{proposition}","on]\n\\label{prop:risk-budget-composition}\nSuppose 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\n\\begin{equation}\n1-\\sum_{t=1}^{T}\\delta_t.\n\\end{equation}\nIf $\\sum_t\\delta_t\\leq\\delta$, the probability of ever leaving is at most $\\delta$.\n\\end{proposition}","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:\n\\begin{enumerate}\n\\item a memoryless preservation policy on $\\Ical_{\\mathrm{ctrl}}^{*}$;\n\\item a finite counterexample tree for excluded states;\n\\item a nested-attractor strategy for $\\Ical_{\\gamma}^{\\omega}$;\n\\item a progress rank for each finite attractor layer.\n\\end{enumerate}","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:\n\\begin{enumerate}\n\\item a memoryless preservation policy on $\\Ical_{\\mathrm{ctrl}}^{*}$;\n\\item a finite counterexample tree for excluded states;\n\\item a nested-attractor strategy for $\\Ical_{\\gamma}^{\\omega}$;\n\\item a progress rank for each finite attractor layer.\n\\end{enumerate}","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\n\\begin{equation}\nz(A)\n=\n\\left(\nJ(A),\nM(A),\nS(A),\nN(A),\n1-C(A)\n\\right)\n\\in[0,1]^d,\n\\end{equation}\nwhere the coordinates may encode task utility, foreign-agent metaproductivity, safety or invariant margin, novelty, and normalized resource efficiency.","For maximization coordinates, write $A\\succeq B$ when\n\\begin{equation}\nz_i(A)\\geq z_i(B)\n\\qquad\\text{for every }i.\n\\end{equation}\nThe Pareto frontier is the set of undominated nodes.","\\begin{proposition}[Archive-envelope monotonicity]\n\\label{prop:archive-envelope}\nSuppose archive updates never delete the last representative of a retained node. For every nonnegative weight vector $w$, define\n\\begin{equation}\nH_w(\\Acal_t)\n=\n\\max_{A\\in\\Acal_t}w^{\\mathsf T}z(A).\n\\end{equation}\nThen $H_w(\\Acal_t)$ is nondecreasing in $t$.\n\\end{proposition}","\\begin{theorem}[Finite $\\eps$-Pareto cover]\n\\label{thm:epsilon-pareto-cover}\nLet $Z\\subseteq[0,1]^d$ be finite and let $0<\\eps\\leq1$. There exists a subset $Q\\subseteq Z$ with\n\\begin{equation}\n\\abs{Q}\n\\leq\n\\left\\lceil\\frac{1}{\\eps}\\right\\rceil^d\n\\end{equation}\nsuch that for every $z\\in Z$, some $q\\in Q$ satisfies\n\\begin{equation}\nq_i\\geq z_i-\\eps\n\\qquad\n\\text{for every }i.\n\\end{equation}\n\\end{theorem}","\\begin{proof}\nPartition $[0,1]^d$ 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.\n\\end{proof}","\\begin{corollary}[Approximate scalarization preservation]\nFor every $w\\geq0$ with $\\norm{w}_1=1$,\n\\begin{equation}\n\\max_{q\\in Q}w^{\\mathsf T}q\n\\geq\n\\max_{z\\in Z}w^{\\mathsf T}z-\\eps.\n\\end{equation}\n\\end{corollary}","\\begin{proof}\nChoose a maximizer $z$ and its covering point $q$. Then $w^{\\mathsf T}q\\geq w^{\\mathsf T}z-\\eps\\norm{w}_1$.\n\\end{proof}","\\begin{proof}\nLet 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.\n\\end{proof}","\\begin{definition}[Horizon-$h$ continuation equivalence]\nTwo archive nodes $A$ and $B$ are horizon-$h$ continuation equivalent, written\n\\begin{equation}\nA\\sim_h B,\n\\end{equation}\nwhen, under the same repair grammar and matched budgets, their sets of reachable descriptor vectors through at most $h$ further transactions are identical.\n\\end{definition}","\\begin{theorem}[Exact archive quotient]\n\\label{thm:archive-quotient}\nRetaining one representative from each $\\sim_h$ equivalence class preserves the complete horizon-$h$ reachable descriptor set.\n\\end{theorem}","\\begin{proof}\nEvery 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.\n\\end{proof}","Let $J^*(\\Acal_t)$ be the highest true protected utility among retained validated nodes.","\\begin{theorem}[Summable false-acceptance theorem]\n\\label{thm:summable-errors}\nFor transaction $t$, let $F_t$ be the event that an accepted child has nonpositive true declared gain. If\n\\begin{equation}\n\\Pr(F_t\\mid\\Fcal_{t-1})\\leq\\delta_t\n\\end{equation}\nand $\\sum_t\\delta_t<\\infty$, 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.\n\\end{theorem}","}[Summable false-acceptance theorem]\n\\label{thm:summable-errors}\nFor transaction $t$, let $F_t$ be the event that an accepted child has nonpositive true declared gain. If\n\\begin{equation}\n\\Pr(F_t\\mid\\Fcal_{t-1})\\leq\\delta_t\n\\end{equation}\nand $\\sum_t\\delta_t<\\infty$, 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.\n\\end{theorem}","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\n\\begin{equation}\nM^{\\mathrm{foreign}}_{h,B}(K)\n=\n\\mathbb E_{A\\sim\\Dcal_{\\mathrm{hold}}}\n\\left[\n\\max_{A'\\in\\Desc_{h,B}(K,A)}\nJ(A')-J(A)\n\\right].\n\\end{equation}","\\begin{definition}[Bounded recursive self-improvement]\nA lineage exhibits bounded recursive self-improvement over a declared generation set when:\n\\begin{enumerate}\n\\item protected task utility improves on a nontrivial held-out family;\n\\item $M^{\\mathrm{foreign}}_{h,B}$ increases under matched budgets;\n\\item task-relevant structural repair capacity increases on a held-out defect family;\n\\item evidence, provenance, witness separation, and rollback invariants remain valid.\n\\end{enumerate}\n\\end{definition}","\\begin{definition}[Budget acceleration]\nLet $B_t(\\gamma)$ 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\n\\begin{equation}\nB_{t+1}(\\gamma)<B_t(\\gamma)\n\\qquad\n\\text{for every }\\gamma\\in I.\n\\end{equation}\n\\end{definition}","\\begin{proposition}[Expected best-of-$k$ gain]\n\\label{prop:best-of-k}\nLet $G_1,\\dots,G_k$ be independent nonnegative candidate gains with common cumulative distribution function $F$ and finite mean. Then\n\\begin{equation}\n\\mathbb E\\max_{1\\leq i\\leq k}G_i\n=\n\\int_0^{\\infty}\n\\left(1-F(x)^k\\right)\\dd x.\n\\end{equation}\nIn particular, expected best gain is nondecreasing in $k$ even when the candidate generator itself does not improve.\n\\end{proposition}","\\begin{proof}\nFor a nonnegative random variable $X$, $\\mathbb E X=\\int_0^\\infty\\Pr(X>x)\\dd x$. Independence gives\n\\begin{equation}\n\\Pr\\left(\\max_iG_i\\leq x\\right)=F(x)^k.\n\\end{equation}\nSubstitution proves the identity.\n\\end{proof}","Let $K_0$ and $K_1$ be frozen improvement mechanisms evaluated on the same independent foreign agents $A_1,\\ldots,A_n$. Define\n\\begin{equation}\nG_{h,B}(K,A)\n=\n\\max_{A'\\in\\Desc_{h,B}(K,A)}J(A')-J(A)\n\\end{equation}\nand assume\n\\begin{equation}\n-G_{\\max}\n\\leq\nG_{h,B}(K,A)\n\\leq\nG_{\\max}.\n\\end{equation}\nFor matched budgets and random seeds, let\n\\begin{equation}\nD_i\n=\nG_{h,B}(K_1,A_i)-G_{h,B}(K_0,A_i),\n\\qquad\n\\overline D_n\n=\n\\frac1n\\sum_{i=1}^{n}D_i.\n\\end{equation}","\\begin{theorem}[Paired metaproductivity certificate]\n\\label{thm:paired-meta}\nFor independent held-out agents and $0<\\delta<1$, with probability at least $1-\\delta$,\n\\begin{equation}\n\\mathbb E D_i\n\\geq\n\\overline D_n\n-\n2\\sqrt2\\,G_{\\max}\n\\sqrt{\\frac{\\log(1/\\delta)}{n}}.\n\\end{equation}\nA positive lower bound certifies higher foreign-agent metaproductivity for $K_1$ relative to $K_0$.\n\\end{theorem}","\\begin{proof}\nEach $D_i$ lies in an interval of length $4G_{\\max}$. The one-sided Hoeffding inequality gives\n\\begin{equation}\n\\Pr\\left(\n\\overline D_n-\\mathbb E D_i\\geq\\eps\n\\right)\n\\leq\n\\exp\\left(\n-\\frac{n\\eps^2}{8G_{\\max}^2}\n\\right).\n\\end{equation}\nSolving for $\\eps$ proves the bound.\n\\end{proof}","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.","Let $\\Dcal$ and $\\Dcal'$ be two foreign-agent distributions and define\n\\begin{equation}\n\\TV(\\Dcal,\\Dcal')\n=\n\\sup_{B}\\abs{\\Dcal(B)-\\Dcal'(B)}.\n\\end{equation}","\\begin{proposition}[Metaproductivity shift bound]\n\\label{prop:meta-tv}\nIf $\\abs{G_{h,B}(K,A)}\\leq G_{\\max}$, then\n\\begin{equation}\n\\abs{\nM_{h,B}^{\\Dcal}(K)\n-\nM_{h,B}^{\\Dcal'}(K)\n}\n\\leq\n2G_{\\max}\\TV(\\Dcal,\\Dcal').\n\\end{equation}\n\\end{proposition}","\\begin{proof}\nApply the total-variation expectation bound to the function $A\\mapsto G_{h,B}(K,A)$.\n\\end{proof}","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\n\\begin{equation}\n\\mathcal E_t^{\\mathrm{cor}}(\\mu)\n=\n\\mathbb E_{r\\sim\\mu}\\norm{P_t r}^2\n\\end{equation}\nand the obstruction energy\n\\begin{equation}\n\\mathcal E_t^{\\mathrm{ob}}(\\mu)\n=\n\\mathbb E_{r\\sim\\mu}\\norm{(I-P_t)r}^2.\n\\end{equation}","\\begin{theorem}[Structural-to-behavioral identity]\n\\label{thm:structural-behavioral}\nAssume $L_t\\subseteq L_{t+1}$ and let\n\\begin{equation}\nW=L_{t+1}\\cap L_t^{\\perp}.\n\\end{equation}\nThen\n\\begin{equation}\n\\mathcal E_{t+1}^{\\mathrm{cor}}(\\mu)-\\mathcal E_t^{\\mathrm{cor}}(\\mu)\n=\n\\mathbb E_{r\\sim\\mu}\\norm{P_Wr}^2\n\\end{equation}\nand\n\\begin{equation}\n\\mathcal E_t^{\\mathrm{ob}}(\\mu)-\\mathcal E_{t+1}^{\\mathrm{ob}}(\\mu)\n=\n\\mathbb E_{r\\sim\\mu}\\norm{P_Wr}^2.\n\\end{equation}\nUnder 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.\n\\end{theorem}","\\begin{proof}\nNested subspaces give the orthogonal decomposition\n\\begin{equation}\nL_{t+1}=L_t\\oplus W,\n\\qquad\nP_{t+1}=P_t+P_W.\n\\end{equation}\nThe two projected components are orthogonal, so\n\\begin{equation}\n\\norm{P_{t+1}r}^2\n=\n\\norm{P_tr}^2+\\norm{P_Wr}^2.\n\\end{equation}\nThe obstruction identity follows from Pythagoras. Under exact correction, the post-repair residual is $(I-P_t)r$, so loss reduction is $\\norm{P_tr}^2$.\n\\end{proof}","proof}\nNested subspaces give the orthogonal decomposition\n\\begin{equation}\nL_{t+1}=L_t\\oplus W,\n\\qquad\nP_{t+1}=P_t+P_W.\n\\end{equation}\nThe two projected components are orthogonal, so\n\\begin{equation}\n\\norm{P_{t+1}r}^2\n=\n\\norm{P_tr}^2+\\norm{P_Wr}^2.\n\\end{equation}\nThe obstruction identity follows from Pythagoras. Under exact correction, the post-repair residual is $(I-P_t)r$, so loss reduction is $\\norm{P_tr}^2$.\n\\end{proof}","\\begin{corollary}[When rank expansion is behaviorally irrelevant]\nA strict inclusion $L_t\\subsetneq L_{t+1}$ yields no expected quadratic benefit on $\\mu$ exactly when\n\\begin{equation}\nP_Wr=0\n\\qquad\n\\mu\\text{-almost surely}.\n\\end{equation}\n\\end{corollary}","\\begin{proposition}[Finite-sample structural certificate]\n\\label{prop:structural-concentration}\nSuppose $\\norm{r}\\leq R$ almost surely and $r_1,\\dots,r_n$ are independent held-out defects. Let\n\\begin{equation}\n\\widehat\\Delta_{\\mathrm{str}}\n=\n\\frac1n\\sum_{i=1}^{n}\\norm{P_Wr_i}^2.\n\\end{equation}\nThen with probability at least $1-\\delta$,\n\\begin{equation}\n\\left|\n\\widehat\\Delta_{\\mathrm{str}}\n-\n\\mathbb E\\norm{P_Wr}^2\n\\right|\n\\leq\nR^2\\sqrt{\\frac{\\log(2/\\delta)}{2n}}.\n\\end{equation}\n\\end{proposition}","\\begin{proof}\nEach summand lies in $[0,R^2]$. Apply Hoeffding's inequality.\n\\end{proof}","\\begin{proposition}[Foreign-transfer lower bound]\n\\label{prop:foreign-transfer-lower-bound}\nSuppose $K_{t+1}$ 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\n\\begin{equation}\nM^{\\mathrm{foreign}}_{h,B}(K_{t+1})\n-\nM^{\\mathrm{foreign}}_{h,B}(K_t)\n\\geq\np\\gamma.\n\\end{equation}\n\\end{proposition}","position}[Foreign-transfer lower bound]\n\\label{prop:foreign-transfer-lower-bound}\nSuppose $K_{t+1}$ 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\n\\begin{equation}\nM^{\\mathrm{foreign}}_{h,B}(K_{t+1})\n-\nM^{\\mathrm{foreign}}_{h,B}(K_t)\n\\geq\np\\gamma.\n\\end{equation}\n\\end{proposition}","r lower bound]\n\\label{prop:foreign-transfer-lower-bound}\nSuppose $K_{t+1}$ 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\n\\begin{equation}\nM^{\\mathrm{foreign}}_{h,B}(K_{t+1})\n-\nM^{\\mathrm{foreign}}_{h,B}(K_t)\n\\geq\np\\gamma.\n\\end{equation}\n\\end{proposition}","\\begin{proof}\nThe pointwise best gain improves by at least $\\gamma$ on $E$ and does not decrease outside $E$. Taking expectations gives the bound.\n\\end{proof}","\\begin{definition}[One-step recursive meta-improvement]\nA lineage edge $K_t\\to K_{t+1}$ is a one-step recursive meta-improvement when:\n\\begin{enumerate}\n\\item $K_{t+1}$ was constructed by an authorized transaction whose proposal process used $K_t$;\n\\item the edge passed an independent witness and preserved the declared evidence and rollback invariants;\n\\item under matched budgets on a frozen foreign-agent distribution,\n\\begin{equation}\nM_{h,B}^{\\mathrm{foreign}}(K_{t+1})\n>\nM_{h,B}^{\\mathrm{foreign}}(K_t)\n\\end{equation}\nwith a positive precommitted lower confidence bound.\n\\end{enumerate}\n\\end{definition}","eta-improvement]\nA lineage edge $K_t\\to K_{t+1}$ is a one-step recursive meta-improvement when:\n\\begin{enumerate}\n\\item $K_{t+1}$ was constructed by an authorized transaction whose proposal process used $K_t$;\n\\item the edge passed an independent witness and preserved the declared evidence and rollback invariants;\n\\item under matched budgets on a frozen foreign-agent distribution,\n\\begin{equation}\nM_{h,B}^{\\mathrm{foreign}}(K_{t+1})\n>\nM_{h,B}^{\\mathrm{foreign}}(K_t)\n\\end{equation}\nwith a positive precommitted lower confidence bound.\n\\end{enumerate}\n\\end{definition}","The lineage condition excludes the weaker situation in which an external designer simply replaces $K_t$ with an unrelated better mechanism.","\\begin{proposition}[Task improvement does not imply meta-improvement]\n\\label{prop:task-not-meta}\nThere exist mechanisms $K_0,K_1$ such that $K_1$ has higher direct task utility on every task in a family, but\n\\begin{equation}\nM_{h,B}^{\\mathrm{foreign}}(K_1)\n<\nM_{h,B}^{\\mathrm{foreign}}(K_0).\n\\end{equation}\n\\end{proposition}","\\begin{proof}\nLet $K_1$ 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 $K_0$.\n\\end{proof}","\\begin{statusbox}{Metaproductivity result}\nForeign-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.\n\\end{statusbox}","\\begin{theorem}[Finite structural-improvement bound]\n\\label{thm:fixed-grammar-exhaustion}\nLet $V$ be a fixed $n$-dimensional defect space and suppose\n\\begin{equation}\n\\im D_0\n\\subsetneq\n\\im D_1\n\\subsetneq\n\\cdots\n\\end{equation}\nis a strict ascending chain. Then the chain contains at most\n\\begin{equation}\nn-\\rank D_0\n\\end{equation}\nstrict inclusions.\n\\end{theorem}","\\begin{proof}\nEvery strict inclusion raises image dimension by at least one, while image dimension is bounded by $n$.\n\\end{proof}","Thus indefinite structural improvement requires expansion or refinement of at least one of\n\\begin{equation}\nU_t^{\\Auth},\n\\qquad\nV_t,\n\\qquad\nD_t,\n\\qquad\n\\Gcal_t,\n\\qquad\n\\Vcal_t.\n\\end{equation}","Let $\\Gcal_t$ be a typed grammar of defects, repairs, evidence, transformations, and proof rules.","\\begin{definition}[Conservative grammar extension]\nAn embedding\n\\begin{equation}\n\\iota_t:\\Gcal_t\\hookrightarrow\\Gcal_{t+1}\n\\end{equation}\nis conservative when:\n\\begin{enumerate}\n\\item every old well-typed expression remains well typed;\n\\item every old derivation translates to a derivation;\n\\item the interpretation of every old expression on old evidence is unchanged;\n\\item protected evidence identifiers and provenance resources are transported identically;\n\\item new rules cannot derive an old-language contradiction unless a correction receipt explicitly retracts a previously unsound axiom or evidence interpretation.\n\\end{enumerate}\n\\end{definition}","\\begin{theorem}[Finite derivation preservation]\n\\label{thm:grammar-preservation}\nSuppose $\\iota_t$ 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 $\\Gcal_{t+1}$.\n\\end{theorem}","Consider a chain\n\\begin{equation}\n\\Gcal_0\n\\xhookrightarrow{\\iota_0}\n\\Gcal_1\n\\xhookrightarrow{\\iota_1}\n\\Gcal_2\n\\xhookrightarrow{}\n\\cdots\n\\end{equation}\nof conservative extensions.","\\begin{theorem}[Conservative directed limit]\n\\label{thm:grammar-colimit}\nThe 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 $\\Gcal_{\\infty}$.\n\\end{theorem}","\\begin{definition}[Approximate semantic debt]\nIf 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\n\\begin{equation}\n\\eta_{0:T}\n\\leq\n\\sum_{t=0}^{T-1}\\eta_t.\n\\end{equation}\n\\end{definition}","\\begin{proposition}[Summable approximate conservativity]\n\\label{prop:grammar-semantic-debt}\nIf 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.\n\\end{proposition}","For grammar $\\Gcal$ with linearized authorized correction image $L_{\\Gcal}$, define\n\\begin{equation}\nq_{\\Gcal}(r)\n=\n\\dist(-r,L_{\\Gcal}).\n\\end{equation}\nThis is the minimum unresolved first-order residual inside the current grammar.","\\begin{definition}[Relative grammar-expansion witness]\nLet $\\Gcal\\hookrightarrow\\Gcal'$ be a candidate extension and let $r'$ be the transported defect. The extension exposes a missing correction direction with margin $\\kappa>0$ when\n\\begin{equation}\nq_{\\Gcal'}(r')\n\\leq\nq_{\\Gcal}(r)-\\kappa.\n\\end{equation}\n\\end{definition}","\\begin{proof}\nIf 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.\n\\end{proof}","\\begin{warning}\nA 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.\n\\end{warning}","Let $j:V\\hookrightarrow V'$ be an injective evidence-conservative embedding of old defects into an expanded ontology. Let\n\\begin{equation}\nC=\\im D\\subseteq V,\n\\qquad\nC'=\\im D'\\subseteq V',\n\\end{equation}\nand assume preservation of old correctability:\n\\begin{equation}\nj(C)\\subseteq C'.\n\\end{equation}\nThe embedding induces a map\n\\begin{equation}\n\\overline j:V/C\\to V'/C'.\n\\end{equation}","\\begin{theorem}[Old-resolution and new-obstruction exact sequence]\n\\label{thm:dynamic-obstruction-exact}\nThere is an exact sequence\n\\begin{equation}\n0\n\\longrightarrow\n\\frac{j^{-1}(C')}{C}\n\\longrightarrow\n\\frac{V}{C}\n\\xrightarrow{\\overline j}\n\\frac{V'}{C'}\n\\longrightarrow\n\\frac{V'}{C'+j(V)}\n\\longrightarrow\n0.\n\\end{equation}\nThe 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.\n\\end{theorem}","\\begin{proof}\nThe kernel of $\\overline j$ consists of classes $[v]$ with $j(v)\\in C'$, which is $j^{-1}(C')/C$. The cokernel is\n\\begin{equation}\n(V'/C')/((j(V)+C')/C')\n\\cong\nV'/(C'+j(V)).\n\\end{equation}\nExactness at the remaining terms follows from the quotient maps.\n\\end{proof}","Let $\\phi_{\\Gcal}:X\\to Z_{\\Gcal}$ be the state representation expressible in grammar $\\Gcal$. A $\\Gcal$-measurable deterministic repair policy has the form\n\\begin{equation}\n\\pi=\\widetilde\\pi\\circ\\phi_{\\Gcal}.\n\\end{equation}","\\begin{theorem}[Representation-separation obstruction]\n\\label{thm:grammar-separation}\nSuppose states $x_0,x_1$ satisfy\n\\begin{equation}\n\\phi_{\\Gcal}(x_0)=\\phi_{\\Gcal}(x_1),\n\\end{equation}\nbut their externally validated repair sets are disjoint:\n\\begin{equation}\n\\Rcal_{\\mathrm{valid}}(x_0)\n\\cap\n\\Rcal_{\\mathrm{valid}}(x_1)\n=\n\\varnothing.\n\\end{equation}\nThen no $\\Gcal$-measurable deterministic repair policy is valid on both states.\n\\end{theorem}","II & New repair operators in existing types & Expanded $U^{\\Auth}$ or repair operad & New correctable direction, transfer, and receipt","III & New defect, packet, or ontology types & Conservative $\\Gcal_t\\hookrightarrow\\Gcal_{t+1}$ & Old-proof replay, evidence identity, new-direction witness","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\n\\begin{equation}\n\\Phi_{\\mathrm{disc}}(S)\n=\nS\\cup\\operatorname{Disc}(S).\n\\end{equation}","\\begin{assumption}[Discovery monotonicity]\nIf $S\\subseteq T$, then\n\\begin{equation}\n\\operatorname{Disc}(S)\n\\subseteq\n\\operatorname{Disc}(T).\n\\end{equation}\nThis 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.\n\\end{assumption}","\\begin{theorem}[Finite repair-discovery closure]\n\\label{thm:discovery-closure}\nUnder discovery monotonicity, $\\Phi_{\\mathrm{disc}}$ has a least fixed point above any seed set $S_0$. Iteration\n\\begin{equation}\nS_{k+1}=\\Phi_{\\mathrm{disc}}(S_k)\n\\end{equation}\nstabilizes after at most $\\abs{\\Omega}-\\abs{S_0}$ strict expansions.\n\\end{theorem}","\\begin{proof}\nThe 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.\n\\end{proof}","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\n\\begin{equation}\nM_{ij}=\\mathbb E N_{ij}.\n\\end{equation}\nThis definition excludes descendants that merely share ancestry or evidence with the parent.","\\begin{definition}[Improvement reproduction number]\nThe local repair-discovery reproduction number is\n\\begin{equation}\nR_{\\mathrm{imp}}\n=\n\\rad(M).\n\\end{equation}\n\\end{definition}","\\begin{conditionaltheorem}[Finite-type branching threshold]\n\\label{thm:branching-threshold}\nAssume 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:\n\\begin{enumerate}\n\\item if $R_{\\mathrm{imp}}\\leq1$, the repair-discovery lineage becomes extinct almost surely;\n\\item if $R_{\\mathrm{imp}}>1$, it has positive probability of indefinite survival.\n\\end{enumerate}\n\\end{conditionaltheorem}","nditionaltheorem}[Finite-type branching threshold]\n\\label{thm:branching-threshold}\nAssume 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:\n\\begin{enumerate}\n\\item if $R_{\\mathrm{imp}}\\leq1$, the repair-discovery lineage becomes extinct almost surely;\n\\item if $R_{\\mathrm{imp}}>1$, it has positive probability of indefinite survival.\n\\end{enumerate}\n\\end{conditionaltheorem}","Construct the directed type graph with an edge $i\\to j$ when $M_{ij}>0$. Its strongly connected components identify mutually enabling schema families.","\\begin{theorem}[Localization of supercriticality]\n\\label{thm:autocatalytic-scc}\nAfter permuting schema types, $M$ has block upper-triangular form with diagonal blocks $M_1,\\ldots,M_k$ given by strongly connected components. Therefore\n\\begin{equation}\n\\rad(M)\n=\n\\max_{1\\leq r\\leq k}\\rad(M_r).\n\\end{equation}\nIn particular, $R_{\\mathrm{imp}}>1$ only if at least one strongly connected schema family is itself supercritical.\n\\end{theorem}","\\begin{theorem}[Spectral interval bound]\n\\label{thm:spectral-interval}\nLet $L,M,U$ be nonnegative square matrices satisfying\n\\begin{equation}\nL\\leq M\\leq U\n\\end{equation}\nentrywise. Then\n\\begin{equation}\n\\rad(L)\n\\leq\n\\rad(M)\n\\leq\n\\rad(U).\n\\end{equation}\nConsequently, $\\rad(L)>1$ certifies supercriticality for every matrix in the interval, while $\\rad(U)\\leq1$ certifies subcriticality.\n\\end{theorem}","\\begin{proposition}[Perron sensitivity]\n\\label{prop:perron-sensitivity}\nLet $M$ be irreducible with simple Perron root $\\rho$, positive right eigenvector $r$, and positive left eigenvector $\\ell$, normalized by $\\ell^{\\mathsf T}r=1$. Then\n\\begin{equation}\n\\frac{\\partial\\rho}{\\partial M_{ij}}\n=\n\\ell_i r_j.\n\\end{equation}\n\\end{proposition}","\\begin{proof}\nDifferentiate $Mr=\\rho r$ 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 $\\ell_i r_j$.\n\\end{proof}","Let $M(B)$ be the offspring matrix under a fixed discovery budget $B$. Define the critical budget\n\\begin{equation}\nB_c\n=\n\\inf\\set{B:\\rad(M(B))>1}.\n\\end{equation}\nComparisons of $R_{\\mathrm{imp}}$ are meaningful only at matched budgets and evidence access.","For a time-varying sequence $M_t$, let $z_t$ be a row vector of expected schema counts. Under the linear mean model,\n\\begin{equation}\n\\mathbb E z_T\n=\nz_0M_0M_1\\cdots M_{T-1}.\n\\end{equation}\nDefine the upper mean-growth exponent\n\\begin{equation}\n\\lambda_{\\mathrm{mean}}\n=\n\\limsup_{T\\to\\infty}\n\\frac1T\n\\log\\norm{M_0M_1\\cdots M_{T-1}}.\n\\end{equation}\nA positive value indicates exponential growth of the linearized expected count. It does not imply almost-sure survival under dependence or resource competition.","\\begin{proposition}[Finite-schema saturation]\n\\label{prop:finite-schema-saturation}\nIf 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 $R_{\\mathrm{imp}}>1$.\n\\end{proposition}","\\begin{theorem}[Bounded-utility gain saturation]\n\\label{thm:bounded-gain-saturation}\nSuppose 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\n\\begin{equation}\n\\left\\lfloor\\frac{1-J_0}{\\gamma}\\right\\rfloor\n\\end{equation}\nsuch deployments can occur.\n\\end{theorem}","\\begin{proof}\nAfter $k$ such gains, utility is at least $J_0+k\\gamma\\leq1$.\n\\end{proof}","\\begin{proposition}[Autocatalysis does not imply beneficial improvement]\n\\label{prop:autocatalysis-not-utility}\nThere exist repair-discovery processes with $R_{\\mathrm{imp}}>1$ whose discovered schemas have zero or negative protected utility.\n\\end{proposition}","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.","\\begin{definition}[Empirical benchmark instance]\nA benchmark instance is a tuple\n\\begin{equation}\n\\mathfrak B\n=\n(\\mathsf E,\\mathsf A,\\mathsf G,\\mathsf P,\\mathsf W,\\mathsf V,\n\\mathsf D,\\mathsf S,\\mathsf B,\\mathsf R),\n\\end{equation}\nwhere:\n\\begin{itemize}\n\\item $\\mathsf E$ is the environment or transition process;\n\\item $\\mathsf A$ is the agent family and functional atlas;\n\\item $\\mathsf G$ is the repair grammar and candidate generator;\n\\item $\\mathsf P$ is the permitted probe and intervention language;\n\\item $\\mathsf W$ is the witness, checkpoint, and effect-mediation layer;\n\\item $\\mathsf V$ is the protected evaluation context;\n\\item $\\mathsf D$ is the data and evidence partition;\n\\item $\\mathsf S$ is the seed, randomization, and split schedule;\n\\item $\\mathsf B$ is the matched resource and candidate budget;\n\\item $\\mathsf R$ is the preregistered claim and receipt schema.\n\\end{itemize}\n\\end{definition}","e]\nA benchmark instance is a tuple\n\\begin{equation}\n\\mathfrak B\n=\n(\\mathsf E,\\mathsf A,\\mathsf G,\\mathsf P,\\mathsf W,\\mathsf V,\n\\mathsf D,\\mathsf S,\\mathsf B,\\mathsf R),\n\\end{equation}\nwhere:\n\\begin{itemize}\n\\item $\\mathsf E$ is the environment or transition process;\n\\item $\\mathsf A$ is the agent family and functional atlas;\n\\item $\\mathsf G$ is the repair grammar and candidate generator;\n\\item $\\mathsf P$ is the permitted probe and intervention language;\n\\item $\\mathsf W$ is the witness, checkpoint, and effect-mediation layer;\n\\item $\\mathsf V$ is the protected evaluation context;\n\\item $\\mathsf D$ is the data and evidence partition;\n\\item $\\mathsf S$ is the seed, randomization, and split schedule;\n\\item $\\mathsf B$ is the matched resource and candidate budget;\n\\item $\\mathsf R$ is the preregistered claim and receipt schema.\n\\end{itemize}\n\\end{definition}","mathsf E,\\mathsf A,\\mathsf G,\\mathsf P,\\mathsf W,\\mathsf V,\n\\mathsf D,\\mathsf S,\\mathsf B,\\mathsf R),\n\\end{equation}\nwhere:\n\\begin{itemize}\n\\item $\\mathsf E$ is the environment or transition process;\n\\item $\\mathsf A$ is the agent family and functional atlas;\n\\item $\\mathsf G$ is the repair grammar and candidate generator;\n\\item $\\mathsf P$ is the permitted probe and intervention language;\n\\item $\\mathsf W$ is the witness, checkpoint, and effect-mediation layer;\n\\item $\\mathsf V$ is the protected evaluation context;\n\\item $\\mathsf D$ is the data and evidence partition;\n\\item $\\mathsf S$ is the seed, randomization, and split schedule;\n\\item $\\mathsf B$ is the matched resource and candidate budget;\n\\item $\\mathsf R$ is the preregistered claim and receipt schema.\n\\end{itemize}\n\\end{definition}","hsf S,\\mathsf B,\\mathsf R),\n\\end{equation}\nwhere:\n\\begin{itemize}\n\\item $\\mathsf E$ is the environment or transition process;\n\\item $\\mathsf A$ is the agent family and functional atlas;\n\\item $\\mathsf G$ is the repair grammar and candidate generator;\n\\item $\\mathsf P$ is the permitted probe and intervention language;\n\\item $\\mathsf W$ is the witness, checkpoint, and effect-mediation layer;\n\\item $\\mathsf V$ is the protected evaluation context;\n\\item $\\mathsf D$ is the data and evidence partition;\n\\item $\\mathsf S$ is the seed, randomization, and split schedule;\n\\item $\\mathsf B$ is the matched resource and candidate budget;\n\\item $\\mathsf R$ is the preregistered claim and receipt schema.\n\\end{itemize}\n\\end{definition}","n{itemize}\n\\item $\\mathsf E$ is the environment or transition process;\n\\item $\\mathsf A$ is the agent family and functional atlas;\n\\item $\\mathsf G$ is the repair grammar and candidate generator;\n\\item $\\mathsf P$ is the permitted probe and intervention language;\n\\item $\\mathsf W$ is the witness, checkpoint, and effect-mediation layer;\n\\item $\\mathsf V$ is the protected evaluation context;\n\\item $\\mathsf D$ is the data and evidence partition;\n\\item $\\mathsf S$ is the seed, randomization, and split schedule;\n\\item $\\mathsf B$ is the matched resource and candidate budget;\n\\item $\\mathsf R$ is the preregistered claim and receipt schema.\n\\end{itemize}\n\\end{definition}","nsition process;\n\\item $\\mathsf A$ is the agent family and functional atlas;\n\\item $\\mathsf G$ is the repair grammar and candidate generator;\n\\item $\\mathsf P$ is the permitted probe and intervention language;\n\\item $\\mathsf W$ is the witness, checkpoint, and effect-mediation layer;\n\\item $\\mathsf V$ is the protected evaluation context;\n\\item $\\mathsf D$ is the data and evidence partition;\n\\item $\\mathsf S$ is the seed, randomization, and split schedule;\n\\item $\\mathsf B$ is the matched resource and candidate budget;\n\\item $\\mathsf R$ is the preregistered claim and receipt schema.\n\\end{itemize}\n\\end{definition}","nal atlas;\n\\item $\\mathsf G$ is the repair grammar and candidate generator;\n\\item $\\mathsf P$ is the permitted probe and intervention language;\n\\item $\\mathsf W$ is the witness, checkpoint, and effect-mediation layer;\n\\item $\\mathsf V$ is the protected evaluation context;\n\\item $\\mathsf D$ is the data and evidence partition;\n\\item $\\mathsf S$ is the seed, randomization, and split schedule;\n\\item $\\mathsf B$ is the matched resource and candidate budget;\n\\item $\\mathsf R$ is the preregistered claim and receipt schema.\n\\end{itemize}\n\\end{definition}","generator;\n\\item $\\mathsf P$ is the permitted probe and intervention language;\n\\item $\\mathsf W$ is the witness, checkpoint, and effect-mediation layer;\n\\item $\\mathsf V$ is the protected evaluation context;\n\\item $\\mathsf D$ is the data and evidence partition;\n\\item $\\mathsf S$ is the seed, randomization, and split schedule;\n\\item $\\mathsf B$ is the matched resource and candidate budget;\n\\item $\\mathsf R$ is the preregistered claim and receipt schema.\n\\end{itemize}\n\\end{definition}","The data partition is itself typed:\n\\begin{equation}\n\\mathsf D\n=\n\\mathsf D_{\\mathrm{construct}}\n\\sqcup\n\\mathsf D_{\\mathrm{calibrate}}\n\\sqcup\n\\mathsf D_{\\mathrm{select}}\n\\sqcup\n\\mathsf D_{\\mathrm{foreign}}\n\\sqcup\n\\mathsf D_{\\mathrm{audit}}.\n\\end{equation}\nConstruction 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.","\\begin{definition}[Claim object]\nA claim object is\n\\begin{equation}\n\\mathfrak C\n=\n(\\mathfrak B,\\Theta,H,E,\\Pi,\\Gamma),\n\\end{equation}\nwhere $\\Theta$ 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.\n\\end{definition}","A result is reproducible only if an independent execution can reconstruct $\\mathfrak C$ from the released package without consulting undeclared mutable state.","Let\n\\begin{equation}\n\\mathcal M=(S,\\mathcal T,O)\n\\end{equation}\nbe an abstract transition system with state set $S$, authorized labeled transitions $\\mathcal T$, and protected observation map $O$. Let\n\\begin{equation}\n\\widehat{\\mathcal M}=(\\widehat S,\\widehat{\\mathcal T},\\widehat O)\n\\end{equation}\nbe an executable implementation.","\\begin{definition}[Forward implementation refinement]\nA map\n\\begin{equation}\n\\alpha:\\widehat S\\to S\n\\end{equation}\nforward-refines the runtime when:\n\\begin{enumerate}\n\\item every executable initial state maps to an authorized abstract initial state;\n\\item for every executable transition\n\\begin{equation}\n\\widehat s\\xrightarrow{a}\\widehat s',\n\\end{equation}\nthere is an abstract transition path\n\\begin{equation}\n\\alpha(\\widehat s)\\xrightarrow{a,*}\\alpha(\\widehat s');\n\\end{equation}\n\\item protected observations commute:\n\\begin{equation}\n\\widehat O(\\widehat s)=O(\\alpha(\\widehat s));\n\\end{equation}\n\\item receipt compilation commutes with abstraction up to the declared canonical encoding.\n\\end{enumerate}\n\\end{definition}","nition}[Forward implementation refinement]\nA map\n\\begin{equation}\n\\alpha:\\widehat S\\to S\n\\end{equation}\nforward-refines the runtime when:\n\\begin{enumerate}\n\\item every executable initial state maps to an authorized abstract initial state;\n\\item for every executable transition\n\\begin{equation}\n\\widehat s\\xrightarrow{a}\\widehat s',\n\\end{equation}\nthere is an abstract transition path\n\\begin{equation}\n\\alpha(\\widehat s)\\xrightarrow{a,*}\\alpha(\\widehat s');\n\\end{equation}\n\\item protected observations commute:\n\\begin{equation}\n\\widehat O(\\widehat s)=O(\\alpha(\\widehat s));\n\\end{equation}\n\\item receipt compilation commutes with abstraction up to the declared canonical encoding.\n\\end{enumerate}\n\\end{definition}","ntime when:\n\\begin{enumerate}\n\\item every executable initial state maps to an authorized abstract initial state;\n\\item for every executable transition\n\\begin{equation}\n\\widehat s\\xrightarrow{a}\\widehat s',\n\\end{equation}\nthere is an abstract transition path\n\\begin{equation}\n\\alpha(\\widehat s)\\xrightarrow{a,*}\\alpha(\\widehat s');\n\\end{equation}\n\\item protected observations commute:\n\\begin{equation}\n\\widehat O(\\widehat s)=O(\\alpha(\\widehat s));\n\\end{equation}\n\\item receipt compilation commutes with abstraction up to the declared canonical encoding.\n\\end{enumerate}\n\\end{definition}","\\begin{theorem}[Runtime-to-model trace soundness]\n\\label{thm:runtime-trace-soundness}\nSuppose $\\alpha$ 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.\n\\end{theorem}","\\begin{definition}[Finite exact realization]\nA 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.\n\\end{definition}","\\begin{theorem}[Exhaustive finite endpoint theorem]\n\\label{thm:finite-endpoint-complete}\nLet $\\mathcal M$ 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.\n\\end{theorem}","\\begin{definition}[Claim gate]\nFor claim level $\\ell$, let\n\\begin{equation}\nG_\\ell(E,\\Pi)\\in\\{0,1\\}\n\\end{equation}\nbe a precommitted predicate over revealed evidence and receipts. The claim is admissible only if every prerequisite gate is one:\n\\begin{equation}\n\\operatorname{Admissible}(\\ell)\n\\iff\n\\bigwedge_{j\\leq\\ell}G_j(E,\\Pi)=1.\n\\end{equation}\n\\end{definition}","\\midrule\n\\endhead\n$G_{\\mathrm{pkg}}$ & Package integrity & Exact manifest, canonical digests, clean extraction, successful verifier","$G_{\\mathrm{mech}}$ & Mechanics & Legal phase traces, receipt verification, checkpoint restoration, deterministic replay","$G_{\\mathrm{sem}}$ & Finite semantics & Abstract-runtime refinement and exhaustive frozen endpoints","$G_{\\mathrm{cause}}$ & Causal role & Intervention, necessity ablation, matched negative controls","$G_{\\mathrm{foreign}}$ & Transfer & Held-out systems, matched candidate budget, protected evaluator","$G_{\\mathrm{meta}}$ & Metaproductivity & Parent-child repair-calculus comparison on frozen foreign systems","$G_{\\mathrm{open}}$ & Open-set recursion & Conservative grammar extension, prior-proof replay, novel externally validated defect families","\\begin{proof}\nImmediate from the conjunction defining $\\operatorname{Admissible}(\\ell)$.\n\\end{proof}","Let each local correction space and defect space be $\\R^2$ unless otherwise stated. Let the tolerance satisfy $0<\\tau<10^{-9}$.","\\paragraph{Case F1: local obstruction.}\nSet\n\\begin{equation}\nD_{\\mathrm{loc}}\n=\n\\begin{bmatrix}\n1&0\\\\\n0&0\n\\end{bmatrix},\n\\qquad\nr_{\\mathrm{loc}}\n=\n\\begin{bmatrix}\n0\\\\1\n\\end{bmatrix}.\n\\end{equation}\nThen\n\\begin{equation}\nP_{\\im D_{\\mathrm{loc}}}r_{\\mathrm{loc}}=0,\n\\qquad\n(I-P_{\\im D_{\\mathrm{loc}}})r_{\\mathrm{loc}}=r_{\\mathrm{loc}},\n\\end{equation}\nso the obstruction norm is one.","1\n\\end{bmatrix}.\n\\end{equation}\nThen\n\\begin{equation}\nP_{\\im D_{\\mathrm{loc}}}r_{\\mathrm{loc}}=0,\n\\qquad\n(I-P_{\\im D_{\\mathrm{loc}}})r_{\\mathrm{loc}}=r_{\\mathrm{loc}},\n\\end{equation}\nso the obstruction norm is one.","\\paragraph{Case F2: gluing obstruction.}\nTake $D_0=D_1=D_2=I_2$, 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\n\\begin{equation}\nb=\n\\begin{bmatrix}\n0","\\paragraph{Case F2: gluing obstruction.}\nTake $D_0=D_1=D_2=I_2$, 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\n\\begin{equation}\nb=\n\\begin{bmatrix}\n0\\\\0\\\\1\n\\end{bmatrix}.\n\\end{equation}\nFor the normalized cycle vector\n\\begin{equation}\nc=\\frac1{\\sqrt3}\n\\begin{bmatrix}\n1","1\n\\end{bmatrix}.\n\\end{equation}\nFor the normalized cycle vector\n\\begin{equation}\nc=\\frac1{\\sqrt3}\n\\begin{bmatrix}\n1\\\\1\\\\-1\n\\end{bmatrix},\n\\end{equation}\nwe obtain\n\\begin{equation}\n\\abs{c^\\top b}=\\frac1{\\sqrt3}>0.\n\\end{equation}\nThus local correction succeeds while descent fails.","-1\n\\end{bmatrix},\n\\end{equation}\nwe obtain\n\\begin{equation}\n\\abs{c^\\top b}=\\frac1{\\sqrt3}>0.\n\\end{equation}\nThus local correction succeeds while descent fails.","\\paragraph{Case F3: flat repair.}\nSet\n\\begin{equation}\nD_{\\mathrm{flat}}=I_2,\n\\qquad\nr_{\\mathrm{flat}}=\n\\begin{bmatrix}\n1\\\\-2\n\\end{bmatrix},\n\\end{equation}\nwith zero overlap mismatch and identity evaluator transport. The unique minimum-norm repair is\n\\begin{equation}\nu_{\\mathrm{flat}}=-r_{\\mathrm{flat}}.\n\\end{equation}","-2\n\\end{bmatrix},\n\\end{equation}\nwith zero overlap mismatch and identity evaluator transport. The unique minimum-norm repair is\n\\begin{equation}\nu_{\\mathrm{flat}}=-r_{\\mathrm{flat}}.\n\\end{equation}","\\paragraph{Case F4: evaluator holonomy.}\nUse an exactly correctable local defect and zero overlap mismatch. Let evaluator transports around a three-edge cycle be\n\\begin{equation}\nT_{01}=I_2,\n\\qquad\nT_{12}=I_2,\n\\qquad\nT_{20}=\\diag(1,-1).\n\\end{equation}\nFor protected evaluator direction $e_2$,\n\\begin{equation}\nH_\\gamma e_2-e_2=-2e_2,\n\\qquad\n\\norm{H_\\gamma e_2-e_2}=2.\n\\end{equation}\nThe candidate is locally repairable and globally gluable, but its evaluator semantics are not path flat.","The executable classifier applies the ordered rule\n\\begin{equation}\n\\operatorname{Class}(r,b,H)\n=\n\\begin{cases}\n\\texttt{LOCAL\\_OBSTRUCTION},&\\norm{P_{(\\im D)^\\perp}r}>\\tau,\\\\\n\\texttt{GLUING\\_OBSTRUCTION},&\\norm{P_{\\ker B^\\top}b}>\\tau,\\\\\n\\texttt{EVALUATOR\\_HOLONOMY},&\\norm{(H-I)e}>\\tau,\\\\\n\\texttt{FLAT\\_REPAIR},&\\text{otherwise}.\n\\end{cases}\n\\end{equation}\nThe 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.","\\begin{theorem}[Exact classification of the four-case fixture]\n\\label{thm:fixture-classification}\nFor every $0<\\tau<1/\\sqrt3$, the classifier returns, respectively,\n\\begin{equation}\n\\texttt{LOCAL\\_OBSTRUCTION},\n\\quad\n\\texttt{GLUING\\_OBSTRUCTION},\n\\quad\n\\texttt{FLAT\\_REPAIR},\n\\quad\n\\texttt{EVALUATOR\\_HOLONOMY}\n\\end{equation}\nfor Cases F1 through F4.\n\\end{theorem}","\\begin{proof}\nCase 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.\n\\end{proof}","The minimal closure model has states\n\\begin{equation}\nS=\\{s_{\\mathrm{base}},s_{\\mathrm{improved}},s_{\\mathrm{invalid}}\\}\n\\end{equation}\nand valid set\n\\begin{equation}\nK=\\{s_{\\mathrm{base}},s_{\\mathrm{improved}}\\}.\n\\end{equation}\nWith intact rollback, authorized complete transactions have successor relation\n\\begin{equation}\n\\Post(s_{\\mathrm{base}})\n=\\{s_{\\mathrm{base}},s_{\\mathrm{improved}}\\},\n\\qquad\n\\Post(s_{\\mathrm{improved}})=\\{s_{\\mathrm{improved}}\\}.\n\\end{equation}\nFailed candidates are restored to the base state before the complete transaction ends.","When rollback is ablated, an invalid branch becomes an authorized complete successor:\n\\begin{equation}\ns_{\\mathrm{base}}\\longrightarrow s_{\\mathrm{invalid}}.\n\\end{equation}","\\begin{proposition}[Rollback ablation removes universal base closure]\n\\label{prop:fixture-rollback-ablation}\nIn 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.\n\\end{proposition}","\\begin{proof}\nIn the intact model every complete successor of $s_{\\mathrm{base}}$ lies in $K$. Under ablation, $\\Post(s_{\\mathrm{base}})\\nsubseteq K$, so $s_{\\mathrm{base}}\\notin\\Psi(K)$.\n\\end{proof}","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 $s_{\\mathrm{improved}}$.","F02 & Local obstruction norm equals $1$ & pass","F03 & Triangle gluing residual equals $1/\\sqrt3$ & pass","F05 & Evaluator-holonomy defect equals $2$ & pass","\\midrule\nLocal obstruction & $1.000000$ & $0$ & $0$ & local obstruction","Gluing obstruction & $0$ & $0.577350$ & $0$ & gluing obstruction","Flat repair & $0$ & $0$ & $0$ & flat repair","Evaluator holonomy & $0$ & $0$ & $2.000000$ & evaluator holonomy","An executable state is a finite canonical map\n\\begin{equation}\n\\widehat s:\\mathsf{Key}\\to\\mathsf{Value}.\n\\end{equation}\nA witness policy is\n\\begin{equation}\n\\widehat W\n=(K_{\\mathrm{protected}},\\gamma_{\\min},r_{\\max},p_{\\mathrm{prov}},e_{\\mathrm{irr}},v_W),\n\\end{equation}\nwhere $K_{\\mathrm{protected}}$ 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.","e is a finite canonical map\n\\begin{equation}\n\\widehat s:\\mathsf{Key}\\to\\mathsf{Value}.\n\\end{equation}\nA witness policy is\n\\begin{equation}\n\\widehat W\n=(K_{\\mathrm{protected}},\\gamma_{\\min},r_{\\max},p_{\\mathrm{prov}},e_{\\mathrm{irr}},v_W),\n\\end{equation}\nwhere $K_{\\mathrm{protected}}$ 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.","{Key}\\to\\mathsf{Value}.\n\\end{equation}\nA witness policy is\n\\begin{equation}\n\\widehat W\n=(K_{\\mathrm{protected}},\\gamma_{\\min},r_{\\max},p_{\\mathrm{prov}},e_{\\mathrm{irr}},v_W),\n\\end{equation}\nwhere $K_{\\mathrm{protected}}$ 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.","y is\n\\begin{equation}\n\\widehat W\n=(K_{\\mathrm{protected}},\\gamma_{\\min},r_{\\max},p_{\\mathrm{prov}},e_{\\mathrm{irr}},v_W),\n\\end{equation}\nwhere $K_{\\mathrm{protected}}$ 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.","A candidate contains:\n\\begin{equation}\n\\widehat\\delta\n=(n,f,K_{\\mathrm{edit}},E_{\\mathrm{irr}},\\rho,\\widehat g),\n\\end{equation}\nwhere $f$ 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.","Every complete execution must follow exactly:\n\\begin{equation}\n\\texttt{PREPARED}\n\\to\n\\texttt{SEALED}\n\\to\n\\texttt{EXECUTED}\n\\to\n\\texttt{REVEALED}\n\\to\n\\texttt{DECIDED}\n\\to\n\\texttt{FINALIZED}.\n\\end{equation}\nThe checkpoint, parent digest, candidate metadata digest, witness digest, and prediction are fixed before protected evidence is revealed.","\\begin{definition}[Kernel acceptance predicate]\nFor revealed protected evidence $E$, the reference acceptance predicate is\n\\begin{equation}\n\\begin{aligned}\n\\operatorname{Accept}(\\widehat s,\\widehat s',\\widehat\\delta,E)\n\\iff{}&\nK_{\\mathrm{edit}}\\cap K_{\\mathrm{protected}}=\\varnothing","\\begin{definition}[Kernel acceptance predicate]\nFor revealed protected evidence $E$, the reference acceptance predicate is\n\\begin{equation}\n\\begin{aligned}\n\\operatorname{Accept}(\\widehat s,\\widehat s',\\widehat\\delta,E)\n\\iff{}&\nK_{\\mathrm{edit}}\\cap K_{\\mathrm{protected}}=\\varnothing\\\\\n&\\land\\ \\restr{\\widehat s'}{K_{\\mathrm{protected}}}\n=\\restr{\\widehat s}{K_{\\mathrm{protected}}}\\\\\n&\\land\\ E[\\mathrm{utility\\_gain}]\\geq\\gamma_{\\min}\\\\\n&\\land\\ E[\\mathrm{risk}]\\leq r_{\\max}\\\\\n&\\land\\ E[\\mathrm{provenance\\_ok}]=1\\\\\n&\\land\\ E_{\\mathrm{irr}}=\\varnothing.\n\\end{aligned}\n\\end{equation}\n\\end{definition}","Let $\\operatorname{cj}$ denote canonical JSON serialization with sorted keys and normalized arrays. For payload $x$, define\n\\begin{equation}\nh(x)=\\operatorname{SHA256}(\\operatorname{cj}(x)).\n\\end{equation}\nA transaction receipt records parent, candidate, witness, checkpoint, phase trace, evidence, decision, rejection reasons, final state, and previous receipt hash.","\\begin{definition}[Receipt chain]\nFor receipts $\\Pi_1,\\ldots,\\Pi_T$, define\n\\begin{equation}\n\\Pi_t.\\mathrm{previous}\n=\n\\begin{cases}\n\\varnothing,&t=1,","\\begin{definition}[Receipt chain]\nFor receipts $\\Pi_1,\\ldots,\\Pi_T$, define\n\\begin{equation}\n\\Pi_t.\\mathrm{previous}\n=\n\\begin{cases}\n\\varnothing,&t=1,\\\\\nh(\\Pi_{t-1}),&t>1.\n\\end{cases}\n\\end{equation}\n\\end{definition}","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 $0.05$.","\\midrule\nPlanner repair & Utility rises from $0.50$ to $0.62$ with evaluator unchanged & commit & receipt valid","The two receipts form a valid chain. The adverse receipt contains the explicit reasons\n\\begin{equation}\n\\texttt{protected edit declared: evaluator}\n\\end{equation}\nand\n\\begin{equation}\n\\texttt{protected state changed: evaluator}.\n\\end{equation}\nAll six transaction endpoints passed: legitimate commit, adversarial rollback, protected evaluator preservation, exact rollback, chain linkage, and independent receipt verification.","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\n\\begin{equation}\nx_{t+1}=Ax_t+Ba_t,\n\\end{equation}\n\\begin{equation}\ny_t=Cx_t+b_s,\n\\end{equation}\n\\begin{equation}\nm_{t+1}=\\lambda m_t+(1-\\lambda)y_{t,1:2},\n\\end{equation}\nand the reported evaluator score is\n\\begin{equation}\nq_t=w^\\top y_t.\n\\end{equation}\nThe redundant observation map is\n\\begin{equation}\nC=\n\\begin{bmatrix}\n1&0","a_t\\in\\R^2$, observation $y_t\\in\\R^3$, and memory state $m_t\\in\\R^2$. The baseline dynamics are\n\\begin{equation}\nx_{t+1}=Ax_t+Ba_t,\n\\end{equation}\n\\begin{equation}\ny_t=Cx_t+b_s,\n\\end{equation}\n\\begin{equation}\nm_{t+1}=\\lambda m_t+(1-\\lambda)y_{t,1:2},\n\\end{equation}\nand the reported evaluator score is\n\\begin{equation}\nq_t=w^\\top y_t.\n\\end{equation}\nThe redundant observation map is\n\\begin{equation}\nC=\n\\begin{bmatrix}\n1&0\\\\\n0&1\\\\\n1&1\n\\end{bmatrix},\n\\end{equation}\nso the protected consistency relation is\n\\begin{equation}\ny_{t,3}=y_{t,1}+y_{t,2}\n\\end{equation}\nwhen the sensor channel is intact.","1&1\n\\end{bmatrix},\n\\end{equation}\nso the protected consistency relation is\n\\begin{equation}\ny_{t,3}=y_{t,1}+y_{t,2}\n\\end{equation}\nwhen the sensor channel is intact.","The latent event variable is\n\\begin{equation}\nz\\in\n\\{\\mathrm{none},\\mathrm{internal},\\mathrm{world},\\mathrm{sensor},\\mathrm{memory},\\mathrm{evaluator}\\}.\n\\end{equation}\nEach event changes one declared mechanism:\n\\begin{itemize}\n\\item internal: the action-response map $B$ changes;\n\\item world: the passive dynamics $A$ change;\n\\item sensor: the redundant sensor receives an additive bias;\n\\item memory: the retention coefficient $\\lambda$ changes;\n\\item evaluator: the reported weight vector $w$ changes;\n\\item none: all baseline mechanisms remain fixed.\n\\end{itemize}\nThe event label is not supplied to the classifier.","begin{equation}\nz\\in\n\\{\\mathrm{none},\\mathrm{internal},\\mathrm{world},\\mathrm{sensor},\\mathrm{memory},\\mathrm{evaluator}\\}.\n\\end{equation}\nEach event changes one declared mechanism:\n\\begin{itemize}\n\\item internal: the action-response map $B$ changes;\n\\item world: the passive dynamics $A$ change;\n\\item sensor: the redundant sensor receives an additive bias;\n\\item memory: the retention coefficient $\\lambda$ changes;\n\\item evaluator: the reported weight vector $w$ changes;\n\\item none: all baseline mechanisms remain fixed.\n\\end{itemize}\nThe event label is not supplied to the classifier.","world},\\mathrm{sensor},\\mathrm{memory},\\mathrm{evaluator}\\}.\n\\end{equation}\nEach event changes one declared mechanism:\n\\begin{itemize}\n\\item internal: the action-response map $B$ changes;\n\\item world: the passive dynamics $A$ change;\n\\item sensor: the redundant sensor receives an additive bias;\n\\item memory: the retention coefficient $\\lambda$ changes;\n\\item evaluator: the reported weight vector $w$ changes;\n\\item none: all baseline mechanisms remain fixed.\n\\end{itemize}\nThe event label is not supplied to the classifier.","From a common probe state $x_0$ and bounded action $a$, define the feature vector\n\\begin{equation}\n\\phi(z)\n=\n\\begin{bmatrix}\n\\phi_{\\mathrm{passive}}","From a common probe state $x_0$ and bounded action $a$, define the feature vector\n\\begin{equation}\n\\phi(z)\n=\n\\begin{bmatrix}\n\\phi_{\\mathrm{passive}}\\\\\n\\phi_{\\mathrm{actuation}}\\\\\n\\phi_{\\mathrm{sensor}}\\\\\n\\phi_{\\mathrm{memory}}\\\\\n\\phi_{\\mathrm{evaluator}}\n\\end{bmatrix}.\n\\end{equation}\nIts components are, respectively:\n\\begin{enumerate}\n\\item deviation from the baseline no-action continuation;\n\\item deviation of the action-conditioned response from the baseline action response;\n\\item violation of redundant-sensor consistency;\n\\item deviation from baseline memory update;\n\\item discrepancy between reported and protected evaluator scores.\n\\end{enumerate}\nThe observed probe is\n\\begin{equation}\n\\widetilde\\phi=\\phi(z)+\\eta,\n\\qquad\n\\eta\\sim\\mathcal N(0,\\sigma^2I_5).\n\\end{equation}","nts are, respectively:\n\\begin{enumerate}\n\\item deviation from the baseline no-action continuation;\n\\item deviation of the action-conditioned response from the baseline action response;\n\\item violation of redundant-sensor consistency;\n\\item deviation from baseline memory update;\n\\item discrepancy between reported and protected evaluator scores.\n\\end{enumerate}\nThe observed probe is\n\\begin{equation}\n\\widetilde\\phi=\\phi(z)+\\eta,\n\\qquad\n\\eta\\sim\\mathcal N(0,\\sigma^2I_5).\n\\end{equation}","Let $\\mu_z=\\phi(z)$ be the frozen event prototype. With equal priors and matched isotropic noise, the posterior is\n\\begin{equation}\np(z\\mid\\widetilde\\phi)\n=\n\\frac{\\exp\\left(-\\norm{\\widetilde\\phi-\\mu_z}^2/(2\\sigma^2)\\right)}\n{\\sum_{z'}\\exp\\left(-\\norm{\\widetilde\\phi-\\mu_{z'}}^2/(2\\sigma^2)\\right)}.\n\\end{equation}\nA repair is attempted only when\n\\begin{equation}\n\\max_z p(z\\mid\\widetilde\\phi)\\geq\\theta\n\\end{equation}\nand the maximizing event is not \\emph{none}. Otherwise the system abstains.","\\begin{theorem}[Prototype-separation error bound]\n\\label{thm:prototype-separation}\nLet\n\\begin{equation}\n\\Delta\n=\n\\min_{z\\neq z'}\\norm{\\mu_z-\\mu_{z'}}.\n\\end{equation}\nUnder equal priors and isotropic Gaussian noise, nearest-prototype classification satisfies\n\\begin{equation}\n\\Pr(\\widehat z\\neq z)\n\\leq\n(|\\mathcal Z|-1)\n\\exp\\left(-\\frac{\\Delta^2}{8\\sigma^2}\\right).\n\\end{equation}\n\\end{theorem}","\\begin{proof}\nFor 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.\n\\end{proof}","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 $0.110955$.","\\midrule\nNon-none attribution accuracy & $0.950000$","Overall repair-or-abstain success & $0.908333$","False repair rate under no change & $0.037500$","No-change abstention accuracy & $0.962500$","Mean posterior confidence & $0.938424$","Each foreign agent has a canonical defect\n\\begin{equation}\nd\\in\\R^6,\n\\end{equation}\na positive diagonal importance vector $w$, and an invertible architecture transform\n\\begin{equation}\nT\\in\\R^{6\\times6}.\n\\end{equation}\nThe runtime-visible defect is\n\\begin{equation}\n\\widetilde d=Td.\n\\end{equation}\nProtected utility is\n\\begin{equation}\nJ_T(\\widetilde d)\n=\n\\exp\\left(-\\sum_{i=1}^6w_i\\bigl(T^{-1}\\widetilde d\\bigr)_i^2\\right).\n\\end{equation}\nA candidate runtime patch $\\delta$ produces gain\n\\begin{equation}\ng(\\delta)\n=\nJ_T(\\widetilde d+\\delta)-J_T(\\widetilde d).\n\\end{equation}",",\n\\end{equation}\na positive diagonal importance vector $w$, and an invertible architecture transform\n\\begin{equation}\nT\\in\\R^{6\\times6}.\n\\end{equation}\nThe runtime-visible defect is\n\\begin{equation}\n\\widetilde d=Td.\n\\end{equation}\nProtected utility is\n\\begin{equation}\nJ_T(\\widetilde d)\n=\n\\exp\\left(-\\sum_{i=1}^6w_i\\bigl(T^{-1}\\widetilde d\\bigr)_i^2\\right).\n\\end{equation}\nA candidate runtime patch $\\delta$ produces gain\n\\begin{equation}\ng(\\delta)\n=\nJ_T(\\widetilde d+\\delta)-J_T(\\widetilde d).\n\\end{equation}","mportance vector $w$, and an invertible architecture transform\n\\begin{equation}\nT\\in\\R^{6\\times6}.\n\\end{equation}\nThe runtime-visible defect is\n\\begin{equation}\n\\widetilde d=Td.\n\\end{equation}\nProtected utility is\n\\begin{equation}\nJ_T(\\widetilde d)\n=\n\\exp\\left(-\\sum_{i=1}^6w_i\\bigl(T^{-1}\\widetilde d\\bigr)_i^2\\right).\n\\end{equation}\nA candidate runtime patch $\\delta$ produces gain\n\\begin{equation}\ng(\\delta)\n=\nJ_T(\\widetilde d+\\delta)-J_T(\\widetilde d).\n\\end{equation}","Every arm receives candidate budget $B=8$ and the same $96$ frozen agents.","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:\n\\begin{equation}\n\\widehat J(\\delta)>\\widehat J(0)\n\\qquad\\text{while}\\qquad\nJ(\\delta)<J(0).\n\\end{equation}\nA witness-aware arm ranks and commits by $J$, not $\\widehat J$.","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\n\\begin{equation}\nG_{a,j,1},\\ldots,G_{a,j,B}\n\\end{equation}\nbe candidate gains for arm $a$ on agent $j$. The primary behavioral quantity is\n\\begin{equation}\nM_a(B)\n=\n\\frac1n\\sum_{j=1}^n\\max_{1\\leq b\\leq B}G_{a,j,b}.\n\\end{equation}\nComparing arms at unequal $B$ confounds repair competence with order statistics.","ausal repair calculus because it searches several noisy candidates and retains the best. Candidate-budget matching is therefore mandatory. Let\n\\begin{equation}\nG_{a,j,1},\\ldots,G_{a,j,B}\n\\end{equation}\nbe candidate gains for arm $a$ on agent $j$. The primary behavioral quantity is\n\\begin{equation}\nM_a(B)\n=\n\\frac1n\\sum_{j=1}^n\\max_{1\\leq b\\leq B}G_{a,j,b}.\n\\end{equation}\nComparing arms at unequal $B$ confounds repair competence with order statistics.","\\begin{proposition}[Exact transported canonical repair]\n\\label{prop:benchmark-exact-transport}\nFor an invertible architecture map $T$, the runtime patch\n\\begin{equation}\n\\delta=-Td\n\\end{equation}\nremoves the canonical defect exactly:\n\\begin{equation}\nT^{-1}(Td-Td)=0.\n\\end{equation}\nHence protected utility becomes one.\n\\end{proposition}","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 $T$.","For canonical patch $u=T^{-1}\\delta$, define corrected protected energy\n\\begin{equation}\nE_{\\mathrm{corr}}(d,u)\n=\n\\sum_{i=1}^6w_i\\min\\{d_i^2,u_i^2\\}.\n\\end{equation}\nThis 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.","\\midrule\nStatic & $0.000000$ & $0.000000$ & $0.000000$ & $0.000000$","Global edit & $-0.324252$ & $-0.325451$ & $0.834602$ & $0.958333$","Archive only & $0.474975$ & $0.475579$ & $0.744173$ & $0.000000$","Packet only & $0.609294$ & $0.625606$ & $1.051647$ & $0.000000$","Causal only & $0.044230$ & $0.000000$ & $0.071506$ & $0.000000$","Full SRIC & $0.519120$ & $0.580246$ & $0.870198$ & $0.000000$","Full SRIC meta & $0.640517$ & $0.654932$ & $1.085841$ & $0.000000$","ften effective. This prevents the benchmark from being designed so that only the full architecture can succeed.\n\\item Causal-only performance is low because correct causal type without coordinate transport is insufficient under architecture changes.\n\\item Archive-only search is a strong baseline and must not be omitted.\n\\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.\n\\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.\n\\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.\n\\end{enumerate}","333$ false-accept rate. This confirms that the benchmark can expose evaluator capture.\n\\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.\n\\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.\n\\end{enumerate}","Let $N\\subseteq\\{1,\\ldots,n\\}$ be foreign agents whose dominant defect belongs to the held-out sixth type. Define\n\\begin{equation}\n\\widehat\\tau_{\\mathrm{abstract}}\n=\n\\frac1{|N|}\\sum_{j\\in N}\n\\left(\nG_{j}^{\\mathrm{meta}}-G_{j}^{\\mathrm{no\\mbox{-}meta}}\n\\right).\n\\end{equation}\nIn the reference positive control,\n\\begin{equation}\n\\widehat\\tau_{\\mathrm{abstract}}=0.700533.\n\\end{equation}\nA 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.","Let the executable architecture have binary factors\n\\begin{equation}\nZ\n=(Z_C,Z_P,Z_D,Z_T,Z_W,Z_A,Z_M),\n\\end{equation}\nrepresenting causal attribution, packet localization, descent checking, transport, witness separation, archive retention, and repair abstraction.","For foreign agent $j$, let\n\\begin{equation}\nY_j(z)\n\\end{equation}\nbe protected gain under factor setting $z$. The average necessity effect of factor $k$ relative to the intact architecture $\\one$ is\n\\begin{equation}\n\\tau_k\n=\n\\mathbb E\n\\left[\nY_j(\\one)-Y_j(\\one-e_k)\n\\right].\n\\end{equation}\nA positive value supports causal usefulness only relative to the frozen environment, compensating mechanisms, and matched resource schedule.","The primary recursive-effect ablation is\n\\begin{equation}\nZ_M:1\\to0\n\\end{equation}\nwhile 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.","\\begin{definition}[Recursive abstraction estimand]\nLet $\\mathcal F_{\\mathrm{post}}$ be foreign defect families revealed after the source repair. Define\n\\begin{equation}\n\\tau_M^{\\mathrm{future}}\n=\n\\mathbb E_{A\\sim\\mathcal F_{\\mathrm{post}}}\n\\left[\nG_A(Z_M=1)-G_A(Z_M=0)\n\\right].\n\\end{equation}\n\\end{definition}","For arms $a$ and $b$, evaluate both on the same frozen foreign agents and seeds. Let\n\\begin{equation}\n\\Delta_j=G_{a,j}-G_{b,j}.\n\\end{equation}\nThe paired estimate and standard error are\n\\begin{equation}\n\\widehat\\Delta\n=\\frac1n\\sum_{j=1}^n\\Delta_j,\n\\qquad\n\\widehat{\\operatorname{se}}(\\widehat\\Delta)\n=\\sqrt{\\frac{1}{n(n-1)}\\sum_{j=1}^n(\\Delta_j-\\widehat\\Delta)^2}.\n\\end{equation}\nWhen gains are bounded in $[-1,1]$, a distribution-free Hoeffding lower bound is\n\\begin{equation}\n\\LCB_{1-\\alpha}(\\Delta)\n=\n\\widehat\\Delta\n-\n\\sqrt{\\frac{2\\log(1/\\alpha)}{n}}.\n\\end{equation}\nSharper paired permutation or bootstrap intervals may be used when frozen in advance.","ms $a$ and $b$, evaluate both on the same frozen foreign agents and seeds. Let\n\\begin{equation}\n\\Delta_j=G_{a,j}-G_{b,j}.\n\\end{equation}\nThe paired estimate and standard error are\n\\begin{equation}\n\\widehat\\Delta\n=\\frac1n\\sum_{j=1}^n\\Delta_j,\n\\qquad\n\\widehat{\\operatorname{se}}(\\widehat\\Delta)\n=\\sqrt{\\frac{1}{n(n-1)}\\sum_{j=1}^n(\\Delta_j-\\widehat\\Delta)^2}.\n\\end{equation}\nWhen gains are bounded in $[-1,1]$, a distribution-free Hoeffding lower bound is\n\\begin{equation}\n\\LCB_{1-\\alpha}(\\Delta)\n=\n\\widehat\\Delta\n-\n\\sqrt{\\frac{2\\log(1/\\alpha)}{n}}.\n\\end{equation}\nSharper paired permutation or bootstrap intervals may be used when frozen in advance.","eds. Let\n\\begin{equation}\n\\Delta_j=G_{a,j}-G_{b,j}.\n\\end{equation}\nThe paired estimate and standard error are\n\\begin{equation}\n\\widehat\\Delta\n=\\frac1n\\sum_{j=1}^n\\Delta_j,\n\\qquad\n\\widehat{\\operatorname{se}}(\\widehat\\Delta)\n=\\sqrt{\\frac{1}{n(n-1)}\\sum_{j=1}^n(\\Delta_j-\\widehat\\Delta)^2}.\n\\end{equation}\nWhen gains are bounded in $[-1,1]$, a distribution-free Hoeffding lower bound is\n\\begin{equation}\n\\LCB_{1-\\alpha}(\\Delta)\n=\n\\widehat\\Delta\n-\n\\sqrt{\\frac{2\\log(1/\\alpha)}{n}}.\n\\end{equation}\nSharper paired permutation or bootstrap intervals may be used when frozen in advance.","Let $H_1,\\ldots,H_m$ be ordered primary hypotheses. Choose nonnegative levels $\\alpha_i$ satisfying\n\\begin{equation}\n\\sum_{i=1}^m\\alpha_i\\leq\\alpha.\n\\end{equation}\nEach 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.","Candidate-budget equalization & Removes best-of-$k$ confound & all arms use same $B$","The released reference package uses the following gates:\n\\begin{enumerate}\n\\item all nine finite-fixture endpoints pass;\n\\item all six transaction-kernel endpoints pass;\n\\item anonymous non-none attribution accuracy is at least $0.90$;\n\\item anonymous repair-or-abstain success is at least $0.90$;\n\\item false repair under no change is at most $0.10$;\n\\item full SRIC protected gain exceeds static and global-edit gain under the matched budget;\n\\item the evaluator-capture positive control produces at least one false apparent acceptance;\n\\item full SRIC produces zero false protected acceptance in the released fixture;\n\\item the meta positive control improves the held-out defect-type contrast;\n\\item package verification and all automated tests pass.\n\\end{enumerate}","\\begin{definition}[Learned repair-transport instance]\nA learned repair-transport instance is a tuple\n\\begin{equation}\n\\mathfrak R_{\\mathrm{learn}}\n=\n(\\mathcal X,\\phi,q,\\widetilde q,A_s,\\mathcal A_t,\n\\mathsf D,\\mathsf P,\\mathsf G,\\mathsf W),\n\\end{equation}\nwhere:\n\\begin{itemize}\n\\item $\\mathcal X$ is the input space;\n\\item $\\phi$ is the task feature map;\n\\item $q$ and $\\widetilde q$ are protected and corrupted score maps;\n\\item $A_s$ is the source agent;\n\\item $\\mathcal A_t$ is a family of target agents excluded from source repair fitting;\n\\item $\\mathsf D$ is the train, anchor, calibration, and audit partition;\n\\item $\\mathsf P$ is the permitted causal probe language;\n\\item $\\mathsf G$ is the repair and transport grammar;\n\\item $\\mathsf W$ is the frozen witness policy.\n\\end{itemize}\n\\end{definition}","r-transport instance]\nA learned repair-transport instance is a tuple\n\\begin{equation}\n\\mathfrak R_{\\mathrm{learn}}\n=\n(\\mathcal X,\\phi,q,\\widetilde q,A_s,\\mathcal A_t,\n\\mathsf D,\\mathsf P,\\mathsf G,\\mathsf W),\n\\end{equation}\nwhere:\n\\begin{itemize}\n\\item $\\mathcal X$ is the input space;\n\\item $\\phi$ is the task feature map;\n\\item $q$ and $\\widetilde q$ are protected and corrupted score maps;\n\\item $A_s$ is the source agent;\n\\item $\\mathcal A_t$ is a family of target agents excluded from source repair fitting;\n\\item $\\mathsf D$ is the train, anchor, calibration, and audit partition;\n\\item $\\mathsf P$ is the permitted causal probe language;\n\\item $\\mathsf G$ is the repair and transport grammar;\n\\item $\\mathsf W$ is the frozen witness policy.\n\\end{itemize}\n\\end{definition}","epair-transport instance is a tuple\n\\begin{equation}\n\\mathfrak R_{\\mathrm{learn}}\n=\n(\\mathcal X,\\phi,q,\\widetilde q,A_s,\\mathcal A_t,\n\\mathsf D,\\mathsf P,\\mathsf G,\\mathsf W),\n\\end{equation}\nwhere:\n\\begin{itemize}\n\\item $\\mathcal X$ is the input space;\n\\item $\\phi$ is the task feature map;\n\\item $q$ and $\\widetilde q$ are protected and corrupted score maps;\n\\item $A_s$ is the source agent;\n\\item $\\mathcal A_t$ is a family of target agents excluded from source repair fitting;\n\\item $\\mathsf D$ is the train, anchor, calibration, and audit partition;\n\\item $\\mathsf P$ is the permitted causal probe language;\n\\item $\\mathsf G$ is the repair and transport grammar;\n\\item $\\mathsf W$ is the frozen witness policy.\n\\end{itemize}\n\\end{definition}","athcal X,\\phi,q,\\widetilde q,A_s,\\mathcal A_t,\n\\mathsf D,\\mathsf P,\\mathsf G,\\mathsf W),\n\\end{equation}\nwhere:\n\\begin{itemize}\n\\item $\\mathcal X$ is the input space;\n\\item $\\phi$ is the task feature map;\n\\item $q$ and $\\widetilde q$ are protected and corrupted score maps;\n\\item $A_s$ is the source agent;\n\\item $\\mathcal A_t$ is a family of target agents excluded from source repair fitting;\n\\item $\\mathsf D$ is the train, anchor, calibration, and audit partition;\n\\item $\\mathsf P$ is the permitted causal probe language;\n\\item $\\mathsf G$ is the repair and transport grammar;\n\\item $\\mathsf W$ is the frozen witness policy.\n\\end{itemize}\n\\end{definition}","G,\\mathsf W),\n\\end{equation}\nwhere:\n\\begin{itemize}\n\\item $\\mathcal X$ is the input space;\n\\item $\\phi$ is the task feature map;\n\\item $q$ and $\\widetilde q$ are protected and corrupted score maps;\n\\item $A_s$ is the source agent;\n\\item $\\mathcal A_t$ is a family of target agents excluded from source repair fitting;\n\\item $\\mathsf D$ is the train, anchor, calibration, and audit partition;\n\\item $\\mathsf P$ is the permitted causal probe language;\n\\item $\\mathsf G$ is the repair and transport grammar;\n\\item $\\mathsf W$ is the frozen witness policy.\n\\end{itemize}\n\\end{definition}","$\\mathcal X$ is the input space;\n\\item $\\phi$ is the task feature map;\n\\item $q$ and $\\widetilde q$ are protected and corrupted score maps;\n\\item $A_s$ is the source agent;\n\\item $\\mathcal A_t$ is a family of target agents excluded from source repair fitting;\n\\item $\\mathsf D$ is the train, anchor, calibration, and audit partition;\n\\item $\\mathsf P$ is the permitted causal probe language;\n\\item $\\mathsf G$ is the repair and transport grammar;\n\\item $\\mathsf W$ is the frozen witness policy.\n\\end{itemize}\n\\end{definition}","ask feature map;\n\\item $q$ and $\\widetilde q$ are protected and corrupted score maps;\n\\item $A_s$ is the source agent;\n\\item $\\mathcal A_t$ is a family of target agents excluded from source repair fitting;\n\\item $\\mathsf D$ is the train, anchor, calibration, and audit partition;\n\\item $\\mathsf P$ is the permitted causal probe language;\n\\item $\\mathsf G$ is the repair and transport grammar;\n\\item $\\mathsf W$ is the frozen witness policy.\n\\end{itemize}\n\\end{definition}","The controlled suite uses four disjoint data sets:\n\\begin{equation}\n\\mathsf D\n=\n\\mathsf D_{\\rm train}\n\\sqcup\n\\mathsf D_{\\rm anchor}\n\\sqcup\n\\mathsf D_{\\rm calibrate}\n\\sqcup\n\\mathsf D_{\\rm audit}.\n\\end{equation}\nThe 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 $\\mathsf D_{\\rm audit}$.","isjoint data sets:\n\\begin{equation}\n\\mathsf D\n=\n\\mathsf D_{\\rm train}\n\\sqcup\n\\mathsf D_{\\rm anchor}\n\\sqcup\n\\mathsf D_{\\rm calibrate}\n\\sqcup\n\\mathsf D_{\\rm audit}.\n\\end{equation}\nThe 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 $\\mathsf D_{\\rm audit}$.","For the controlled transport experiment, the sample sizes are\n\\begin{equation}\n(n_{\\rm train},n_{\\rm anchor},n_{\\rm calibrate},n_{\\rm audit})\n=(7000,1500,1600,6000).\n\\end{equation}\nThe learned-representation pilot uses\n\\begin{equation}\n(5000,1200,1200,4000).\n\\end{equation}","Let $x=(x_1,x_2,x_3,x_4)\\in\\R^4$. Define the eight-dimensional base feature\n\\begin{equation}\n\\psi(x)\n=\n\\begin{bmatrix}\n x_1 & x_2 & x_3 & x_4 & x_1x_2 & x_3x_4 & \\sin x_1 & \\cos x_2\n\\end{bmatrix}^{\\!\\top}\n\\in\\R^8,\n\\end{equation}\nand the affected-stratum indicator\n\\begin{equation}\ns(x)=\\one\\{x_1+x_2>0.8\\}.\n\\end{equation}\nThe full feature map is\n\\begin{equation}\n\\phi(x)\n=\n\\begin{bmatrix}\n\\psi(x)","Let $x=(x_1,x_2,x_3,x_4)\\in\\R^4$. Define the eight-dimensional base feature\n\\begin{equation}\n\\psi(x)\n=\n\\begin{bmatrix}\n x_1 & x_2 & x_3 & x_4 & x_1x_2 & x_3x_4 & \\sin x_1 & \\cos x_2\n\\end{bmatrix}^{\\!\\top}\n\\in\\R^8,\n\\end{equation}\nand the affected-stratum indicator\n\\begin{equation}\ns(x)=\\one\\{x_1+x_2>0.8\\}.\n\\end{equation}\nThe full feature map is\n\\begin{equation}\n\\phi(x)\n=\n\\begin{bmatrix}\n\\psi(x)\\\\\ns(x)\\psi(x)\n\\end{bmatrix}\n\\in\\R^{16}.\n\\label{eq:block06-feature}\n\\end{equation}","Let\n\\begin{equation}\nq(x)=W\\psi(x)+b\\in\\R^3\n\\end{equation}\nbe the protected score. Let $C$ be the cyclic score permutation\n\\begin{equation}\nC(y_1,y_2,y_3)^{\\top}=(y_3,y_1,y_2)^{\\top}.\n\\end{equation}\nThe corrupted score is\n\\begin{equation}\n\\widetilde q(x)\n=\n(1-s(x))q(x)+s(x)Cq(x).\n\\label{eq:block06-corrupt-score}\n\\end{equation}\nThus the model is almost perfectly correct outside the affected stratum and systematically wrong inside it.","An agent $a$ has encoder\n\\begin{equation}\nE_a:\\R^{16}\\to\\R^{d_a}\n\\end{equation}\nand an affine head fitted to $\\widetilde q$. Its hidden state is\n\\begin{equation}\nh_a(x)=E_a(\\phi(x)).\n\\end{equation}\nThe transport system is not given the matrix or nonlinear parameters of $E_a$.","The permitted intervention removes only the conditional feature block:\n\\begin{equation}\n\\phi^{\\circ}(x)\n=\n\\begin{bmatrix}\n\\psi(x)\\\\0\n\\end{bmatrix}.\n\\label{eq:block06-counterfactual}\n\\end{equation}\nThe resulting hidden difference is\n\\begin{equation}\n\\Delta_a(x)\n=\nh_a(x)-h_a^{\\circ}(x),\n\\qquad\nh_a^{\\circ}(x)=E_a(\\phi^{\\circ}(x)).\n\\end{equation}\nThe empirical causal packet is the leading rank-$r$ right-singular subspace of the affected differences\n\\begin{equation}\n\\{\\Delta_a(x_i):s(x_i)=1\\}.\n\\end{equation}","0\n\\end{bmatrix}.\n\\label{eq:block06-counterfactual}\n\\end{equation}\nThe resulting hidden difference is\n\\begin{equation}\n\\Delta_a(x)\n=\nh_a(x)-h_a^{\\circ}(x),\n\\qquad\nh_a^{\\circ}(x)=E_a(\\phi^{\\circ}(x)).\n\\end{equation}\nThe empirical causal packet is the leading rank-$r$ right-singular subspace of the affected differences\n\\begin{equation}\n\\{\\Delta_a(x_i):s(x_i)=1\\}.\n\\end{equation}","\\begin{theorem}[Exact causal-packet recovery for a linear encoder]\n\\label{thm:block06-linear-packet}\nSuppose\n\\begin{equation}\nE_a\n=\n\\begin{bmatrix}A_a&B_a\\end{bmatrix},\n\\qquad\nA_a,B_a:\\R^8\\to\\R^{d_a},\n\\end{equation}\nand suppose the centered conditional covariance of $\\psi(x)$ given $s(x)=1$ is positive definite. Then\n\\begin{equation}\n\\Span\\{\\Delta_a(x):s(x)=1\\}\n=\nim B_a.\n\\end{equation}\nConsequently, a rank-$r$ singular-subspace estimator with $r=\\rank B_a$ recovers the exact causal packet from a sufficiently rich noiseless sample.\n\\end{theorem}","rem}[Exact causal-packet recovery for a linear encoder]\n\\label{thm:block06-linear-packet}\nSuppose\n\\begin{equation}\nE_a\n=\n\\begin{bmatrix}A_a&B_a\\end{bmatrix},\n\\qquad\nA_a,B_a:\\R^8\\to\\R^{d_a},\n\\end{equation}\nand suppose the centered conditional covariance of $\\psi(x)$ given $s(x)=1$ is positive definite. Then\n\\begin{equation}\n\\Span\\{\\Delta_a(x):s(x)=1\\}\n=\nim B_a.\n\\end{equation}\nConsequently, a rank-$r$ singular-subspace estimator with $r=\\rank B_a$ recovers the exact causal packet from a sufficiently rich noiseless sample.\n\\end{theorem}","a linear encoder]\n\\label{thm:block06-linear-packet}\nSuppose\n\\begin{equation}\nE_a\n=\n\\begin{bmatrix}A_a&B_a\\end{bmatrix},\n\\qquad\nA_a,B_a:\\R^8\\to\\R^{d_a},\n\\end{equation}\nand suppose the centered conditional covariance of $\\psi(x)$ given $s(x)=1$ is positive definite. Then\n\\begin{equation}\n\\Span\\{\\Delta_a(x):s(x)=1\\}\n=\nim B_a.\n\\end{equation}\nConsequently, a rank-$r$ singular-subspace estimator with $r=\\rank B_a$ recovers the exact causal packet from a sufficiently rich noiseless sample.\n\\end{theorem}","\\begin{proof}\nFrom \\cref{eq:block06-feature,eq:block06-counterfactual},\n\\begin{equation}\n\\Delta_a(x)=s(x)B_a\\psi(x).\n\\end{equation}\nHence 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.\n\\end{proof}","\\begin{remark}[Why ordinary PCA need not recover the repair packet]\nPCA 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.\n\\end{remark}","\\begin{example}[Same-rank PCA failure]\nLet $A_a$ 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.\n\\end{example}","Let $U_a\\in\\R^{d_a\\times r}$ have orthonormal columns spanning the discovered packet. Define desired packet coordinates\n\\begin{equation}\nz_i\n=\nU_a^{\\top}(h_a^{\\circ}(x_i)-h_a(x_i)).\n\\end{equation}\nFit an affine predictor\n\\begin{equation}\n\\widehat z(h)=K_a h+c_a\n\\end{equation}\nby ridge regression on source anchor states. The repair is\n\\begin{equation}\nR_a(h)\n=\nh+U_a\\widehat z(h)\n=\nh+D_a h+b_a,\n\\label{eq:block06-source-repair}\n\\end{equation}\nwhere\n\\begin{equation}\nD_a=U_aK_a,\n\\qquad\nb_a=U_ac_a.\n\\end{equation}","discovered packet. Define desired packet coordinates\n\\begin{equation}\nz_i\n=\nU_a^{\\top}(h_a^{\\circ}(x_i)-h_a(x_i)).\n\\end{equation}\nFit an affine predictor\n\\begin{equation}\n\\widehat z(h)=K_a h+c_a\n\\end{equation}\nby ridge regression on source anchor states. The repair is\n\\begin{equation}\nR_a(h)\n=\nh+U_a\\widehat z(h)\n=\nh+D_a h+b_a,\n\\label{eq:block06-source-repair}\n\\end{equation}\nwhere\n\\begin{equation}\nD_a=U_aK_a,\n\\qquad\nb_a=U_ac_a.\n\\end{equation}","\\begin{proposition}[Exact packet repair under affine predictability]\n\\label{prop:block06-exact-source-repair}\nSuppose every desired correction lies in $\\im U_a$ and there are $K_a,c_a$ such that\n\\begin{equation}\nU_a^{\\top}(h_a^{\\circ}-h_a)=K_ah_a+c_a\n\\end{equation}\nthroughout the evaluated support. Then \\cref{eq:block06-source-repair} satisfies\n\\begin{equation}\nR_a(h_a)=h_a^{\\circ}.\n\\end{equation}\n\\end{proposition}","\\begin{proof}\nBecause the desired correction is in $\\im U_a$,\n\\begin{equation}\nh_a^{\\circ}-h_a\n=U_aU_a^{\\top}(h_a^{\\circ}-h_a)\n=U_a(K_ah_a+c_a).\n\\end{equation}\nSubstitution into \\cref{eq:block06-source-repair} gives the claim.\n\\end{proof}","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 $0.002277$.","\\begin{table}[H]\n\\centering\n\\small\n\\caption{Source repair under three rank-eight packet estimators. The uncorrected affected-stratum accuracy is $0.002277$.}\n\\label{tab:block06-packet-results}\n\\begin{tabular}{@{}lrrrr@{}}\n\\toprule\nPacket estimator & Rank & Affected slice & Unaffected slice & Overall","\\midrule\nCausal difference & 8 & $0.994308$ & $0.999529$ & $0.998000$","Same-rank PCA & 8 & $0.023335$ & $0.999529$ & $0.713667$","Random subspace & 8 & $0.570290$ & $0.999529$ & $0.873833$","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.","Let $s$ denote the source and $t$ a target. On paired anchor inputs $x_1,\\ldots,x_m$, the system observes\n\\begin{equation}\nH_s\n=\n\\begin{bmatrix}\nh_s(x_1)^{\\top}","Let $s$ denote the source and $t$ a target. On paired anchor inputs $x_1,\\ldots,x_m$, the system observes\n\\begin{equation}\nH_s\n=\n\\begin{bmatrix}\nh_s(x_1)^{\\top}\\\\ \\vdots\\\\ h_s(x_m)^{\\top}\n\\end{bmatrix},\n\\qquad\nH_t\n=\n\\begin{bmatrix}\nh_t(x_1)^{\\top}\\\\ \\vdots\\\\ h_t(x_m)^{\\top}\n\\end{bmatrix}.\n\\end{equation}\nIt estimates the bridge by regularized least squares:\n\\begin{equation}\n\\widehat T_{t\\leftarrow s}\n\\in\n\\argmin_T\n\\norm{H_sT^{\\top}-H_t}_F^2\n+\\lambda\\norm{T}_F^2.\n\\label{eq:block06-learned-transport}\n\\end{equation}\nThe transformation matrices used to construct the controlled targets are not exposed to this fit.","h_t(x_m)^{\\top}\n\\end{bmatrix}.\n\\end{equation}\nIt estimates the bridge by regularized least squares:\n\\begin{equation}\n\\widehat T_{t\\leftarrow s}\n\\in\n\\argmin_T\n\\norm{H_sT^{\\top}-H_t}_F^2\n+\\lambda\\norm{T}_F^2.\n\\label{eq:block06-learned-transport}\n\\end{equation}\nThe transformation matrices used to construct the controlled targets are not exposed to this fit.","Given source repair\n\\begin{equation}\nR_s(h)=h+D_sh+b_s,\n\\end{equation}\nits transported target form is\n\\begin{equation}\nD_t\n=\n\\widehat T D_s\\widehat T^{+},\n\\qquad\nb_t\n=\n\\widehat T b_s,\n\\label{eq:block06-transported-repair}\n\\end{equation}\nwith transported packet basis obtained by orthogonalizing $\\widehat T U_s$.","\\begin{theorem}[Exact transport through an injective hidden bridge]\n\\label{thm:block06-exact-transport}\nLet $T:\\R^{d_s}\\to\\R^{d_t}$ have full column rank, so $T^{+}T=I_{d_s}$. Suppose\n\\begin{equation}\nh_t=Th_s\n\\end{equation}\nthroughout the evaluated support. Define $D_t$ and $b_t$ by \\cref{eq:block06-transported-repair} with $\\widehat T=T$. Then\n\\begin{equation}\nR_t(Th_s)=TR_s(h_s).\n\\end{equation}\n\\end{theorem}","\\begin{proof}\nUsing $T^{+}T=I$,\n\\begin{align}\nR_t(Th_s)\n&=Th_s+TD_sT^{+}Th_s+Tb_s","\\begin{proof}\nUsing $T^{+}T=I$,\n\\begin{align}\nR_t(Th_s)\n&=Th_s+TD_sT^{+}Th_s+Tb_s\\\\\n&=T(h_s+D_sh_s+b_s)\\\\\n&=TR_s(h_s).\n\\end{align}\n\\end{proof}","\\begin{corollary}[Overcomplete targets]\nAn 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.\n\\end{corollary}","This explains why the $24$-dimensional overcomplete family matches the local target oracle despite having a different state dimension.","\\begin{proposition}[Residual caused by a noninjective bridge]\n\\label{prop:block06-compression-residual}\nLet $h_t=Th_s$ with arbitrary $T$. Using \\cref{eq:block06-transported-repair},\n\\begin{equation}\nR_t(Th_s)-TR_s(h_s)\n=\nTD_s(T^{+}T-I)h_s.\n\\label{eq:block06-compression-defect}\n\\end{equation}\nHence exact state alignment $h_t=Th_s$ does not imply repair equivariance when the bridge discards a source direction used by $D_s$.\n\\end{proposition}","esidual caused by a noninjective bridge]\n\\label{prop:block06-compression-residual}\nLet $h_t=Th_s$ with arbitrary $T$. Using \\cref{eq:block06-transported-repair},\n\\begin{equation}\nR_t(Th_s)-TR_s(h_s)\n=\nTD_s(T^{+}T-I)h_s.\n\\label{eq:block06-compression-defect}\n\\end{equation}\nHence exact state alignment $h_t=Th_s$ does not imply repair equivariance when the bridge discards a source direction used by $D_s$.\n\\end{proposition}","\\begin{proof}\nThe bias terms cancel. Expanding the linear terms gives\n\\begin{equation}\nTD_sT^{+}Th_s-TD_sh_s\n=TD_s(T^{+}T-I)h_s.\n\\end{equation}\n\\end{proof}","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.","Let\n\\begin{equation}\ne_h=h_t-\\widehat T h_s,\n\\qquad\nE_D=D_t\\widehat T-\\widehat T D_s,\n\\qquad\nE_b=b_t-\\widehat T b_s.\n\\end{equation}\nThen\n\\begin{equation}\nR_t(h_t)-\\widehat T R_s(h_s)\n=\n(I+D_t)e_h+E_Dh_s+E_b.\n\\end{equation}\nTherefore\n\\begin{equation}\n\\norm{R_t(h_t)-\\widehat T R_s(h_s)}\n\\leq\n(1+\\norm{D_t})\\norm{e_h}\n+\\norm{E_D}\\norm{h_s}\n+\\norm{E_b}.\n\\label{eq:block06-approx-bound}\n\\end{equation}\nThe implementation reports normalized alignment and repair-equivariance residuals as empirical summaries of these terms.","\\begin{theorem}[Exact least-squares bridge from spanning anchors]\n\\label{thm:block06-anchor-identifiability}\nSuppose $H_t=H_sT^{\\top}$ 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 $T$.\n\\end{theorem}","\\begin{proof}\nThe normal equations are\n\\begin{equation}\nH_s^{\\top}H_s\\widehat T^{\\top}=H_s^{\\top}H_t.\n\\end{equation}\nFull column rank makes $H_s^{\\top}H_s$ invertible. Substitution of $H_t=H_sT^{\\top}$ yields $\\widehat T=T$.\n\\end{proof}","The controlled suite evaluates six seeds in each of seven target families:\n\\begin{enumerate}\n\\item $\\texttt{linear16}$: invertible orthogonal linear coordinates;\n\\item $\\texttt{overcomplete24}$: a $24$-dimensional injective embedding;\n\\item $\\texttt{ill\\_conditioned16}$: invertible linear coordinates with singular values from $1$ to $0.05$;\n\\item $\\texttt{compressed12}$: a noninjective $12$-dimensional projection;\n\\item $\\texttt{mild\\_tanh24}$: sixteen mildly nonlinear coordinates plus an eight-dimensional linear skip;\n\\item $\\texttt{saturated\\_tanh24}$: strongly saturated nonlinear coordinates;\n\\item $\\texttt{relu32}$: a piecewise-linear ReLU representation.\n\\end{enumerate}\nEach target head is fitted independently to the corrupted score function. The source repair and source packet remain frozen.","uates six seeds in each of seven target families:\n\\begin{enumerate}\n\\item $\\texttt{linear16}$: invertible orthogonal linear coordinates;\n\\item $\\texttt{overcomplete24}$: a $24$-dimensional injective embedding;\n\\item $\\texttt{ill\\_conditioned16}$: invertible linear coordinates with singular values from $1$ to $0.05$;\n\\item $\\texttt{compressed12}$: a noninjective $12$-dimensional projection;\n\\item $\\texttt{mild\\_tanh24}$: sixteen mildly nonlinear coordinates plus an eight-dimensional linear skip;\n\\item $\\texttt{saturated\\_tanh24}$: strongly saturated nonlinear coordinates;\n\\item $\\texttt{relu32}$: a piecewise-linear ReLU representation.\n\\end{enumerate}\nEach target head is fitted independently to the corrupted score function. The source repair and source packet remain frozen.","rthogonal linear coordinates;\n\\item $\\texttt{overcomplete24}$: a $24$-dimensional injective embedding;\n\\item $\\texttt{ill\\_conditioned16}$: invertible linear coordinates with singular values from $1$ to $0.05$;\n\\item $\\texttt{compressed12}$: a noninjective $12$-dimensional projection;\n\\item $\\texttt{mild\\_tanh24}$: sixteen mildly nonlinear coordinates plus an eight-dimensional linear skip;\n\\item $\\texttt{saturated\\_tanh24}$: strongly saturated nonlinear coordinates;\n\\item $\\texttt{relu32}$: a piecewise-linear ReLU representation.\n\\end{enumerate}\nEach target head is fitted independently to the corrupted score function. The source repair and source packet remain frozen.","nal injective embedding;\n\\item $\\texttt{ill\\_conditioned16}$: invertible linear coordinates with singular values from $1$ to $0.05$;\n\\item $\\texttt{compressed12}$: a noninjective $12$-dimensional projection;\n\\item $\\texttt{mild\\_tanh24}$: sixteen mildly nonlinear coordinates plus an eight-dimensional linear skip;\n\\item $\\texttt{saturated\\_tanh24}$: strongly saturated nonlinear coordinates;\n\\item $\\texttt{relu32}$: a piecewise-linear ReLU representation.\n\\end{enumerate}\nEach target head is fitted independently to the corrupted score function. The source repair and source packet remain frozen.","\\midrule\nlinear16 & .0023 & .9943 & .9943 & .9943 & $3.3\\!\\times\\!10^{-8}$ & $5.9\\!\\times\\!10^{-8}$ & 1.00","overcomplete24 & .0023 & .9943 & .9943 & .9943 & $3.3\\!\\times\\!10^{-8}$ & $5.9\\!\\times\\!10^{-8}$ & 1.00","ill-conditioned16 & .0023 & .9943 & .9943 & .9943 & $1.9\\!\\times\\!10^{-8}$ & $4.6\\!\\times\\!10^{-8}$ & 1.00","compressed12 & .1277 & .6204 & .1277 & .6648 & $3.3\\!\\times\\!10^{-8}$ & .4715 & 0.00","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.","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 $0.9943$.","\\begin{proposition}[Anchor correspondence is causally necessary in the exact fixture]\nWithin 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.\n\\end{proposition}","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.","The witness commits a transported repair only when all four calibration conditions hold:\n\\begin{align}\n\\widehat g_{\\rm slice}&\\geq 0.20,\\\\\n\\widehat r_{\\rm non}&\\leq 0.03,\\\\\n\\widehat\\eps_{\\rm align}&\\leq 0.35,\\\\\n\\widehat A_{\\rm non}^{\\rm base}&\\geq 0.90.\n\\label{eq:block06-gate}\n\\end{align}\nHere $\\widehat g_{\\rm slice}$ 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.","\\widehat A_{\\rm non}^{\\rm base}&\\geq 0.90.\n\\label{eq:block06-gate}\n\\end{align}\nHere $\\widehat g_{\\rm slice}$ 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.","\\begin{conditionaltheorem}[Calibration-to-audit witness soundness]\n\\label{thm:block06-witness-soundness}\nSuppose, simultaneously, that the audit gain and regression satisfy\n\\begin{align}\ng_{\\rm slice}&\\geq \\widehat g_{\\rm slice}-\\eta_g,\\\\\nr_{\\rm non}&\\leq \\widehat r_{\\rm non}+\\eta_r,\n\\end{align}\nand that the reported alignment error upper-bounds the audit alignment error up to $\\eta_a$. If\n\\begin{align}\n\\widehat g_{\\rm slice}-\\eta_g&>0,","r_{\\rm non}&\\leq \\widehat r_{\\rm non}+\\eta_r,\n\\end{align}\nand that the reported alignment error upper-bounds the audit alignment error up to $\\eta_a$. If\n\\begin{align}\n\\widehat g_{\\rm slice}-\\eta_g&>0,","r_{\\rm non}&\\leq \\widehat r_{\\rm non}+\\eta_r,\n\\end{align}\nand that the reported alignment error upper-bounds the audit alignment error up to $\\eta_a$. If\n\\begin{align}\n\\widehat g_{\\rm slice}-\\eta_g&>0,\\\\\n\\widehat r_{\\rm non}+\\eta_r&\\leq r_{\\max},\\\\\n\\widehat\\eps_{\\rm align}+\\eta_a&\\leq \\eps_{\\max},\n\\end{align}\nthen a candidate passing the corresponding tightened calibration gate has positive audit gain and satisfies the declared audit regression and alignment limits.\n\\end{conditionaltheorem}","For encoder $E_{\\theta}$, decoder $G_{\\delta}$, and score head $H_{\\omega}$, the pilot minimizes minibatch estimates of\n\\begin{equation}\n\\mathcal L(\\theta,\\delta,\\omega)\n=\n\\lambda_{\\rm rec}\n\\norm{G_{\\delta}(E_{\\theta}(\\phi))-\\phi}_F^2\n+\n\\norm{H_{\\omega}(E_{\\theta}(\\phi))-\\widetilde q}_F^2,\n\\label{eq:block06-learned-objective}\n\\end{equation}\nwith $\\lambda_{\\rm rec}=2$. 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.","_{\\delta}$, and score head $H_{\\omega}$, the pilot minimizes minibatch estimates of\n\\begin{equation}\n\\mathcal L(\\theta,\\delta,\\omega)\n=\n\\lambda_{\\rm rec}\n\\norm{G_{\\delta}(E_{\\theta}(\\phi))-\\phi}_F^2\n+\n\\norm{H_{\\omega}(E_{\\theta}(\\phi))-\\widetilde q}_F^2,\n\\label{eq:block06-learned-objective}\n\\end{equation}\nwith $\\lambda_{\\rm rec}=2$. 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.","The source is a learned $16$-dimensional linear representation. Targets are independently trained:\n\\begin{itemize}\n\\item three learned linear targets of width $16$;\n\\item three learned linear targets of width $20$;\n\\item three learned tanh targets of width $20$.\n\\end{itemize}\nThe source repair is discovered once and then frozen.","\\midrule\nlearned linear16 & .00339 & .99323 & .99323 & .99965 & $6.6\\times10^{-8}$ & 1.00","learned linear20 & .00339 & .99323 & .99323 & .99965 & $7.6\\times10^{-8}$ & 1.00","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.","\\midrule\n$G_{\\rm pkg}$ & Pass & 34-file SHA-256 manifest and clean verifier & Byte integrity relative to the released manifest","$G_{\\rm mech}$ & Pass & Deterministic execution, five tests, frozen JSON and figures & Runtime mechanics only","$G_{\\rm sem}$ & Pass & Declared task, repair, transport, and gate objects reproduced & Controlled finite semantics","$G_{\\rm cause}$ & Pass & Causal packet, PCA, random, wrong-block, and shuffled-anchor controls & Causal role inside the fixture","$G_{\\rm foreign}$ & Pass & Held-out target seeds, dimensions, encoder families, and learned targets & One shared task and defect grammar","$G_{\\rm meta}$ & Not tested & No parent-child repair-calculus comparison & No metaproductivity claim","$G_{\\rm open}$ & Not tested & No autonomous grammar or evaluator expansion & No open-set recursion claim","on reports:\n\\begin{verbatim}\nSRIC_BLOCK06_ALL_SUITES=PASS\nresult_sha256=4a210438c56d0e87cfcfa3524ec633a4d78cd8f966e41da8ee669eb544b72065\ncausal_packet_slice=0.994308\npca_packet_slice=0.023335\nlinear_transport_slice=0.994308\nmild_nonlinear_transport_slice=0.746253\nsaturated_commit_rate=0.000000\nlearned_tanh_transport_slice=0.894157\n\\end{verbatim}\nAll five automated tests pass. The manifest covers $34$ package files.","The executable tests require:\n\\begin{enumerate}\n\\item causal packet affected accuracy above $0.98$;\n\\item a causal-over-PCA affected-accuracy gap above $0.90$;\n\\item causal repair unaffected accuracy above $0.99$;\n\\item exact linear, overcomplete, and ill-conditioned transport above $0.98$ and equal to the local oracle within numerical tolerance;\n\\item shuffled-anchor accuracy below $0.40$ for exact linear targets;\n\\item positive witness acceptance in the mild nonlinear family;\n\\item zero witness acceptance in saturated tanh and ReLU families;\n\\item learned linear transport above $0.98$;\n\\item learned tanh transport above $0.80$.\n\\end{enumerate}","ed accuracy above $0.99$;\n\\item exact linear, overcomplete, and ill-conditioned transport above $0.98$ and equal to the local oracle within numerical tolerance;\n\\item shuffled-anchor accuracy below $0.40$ for exact linear targets;\n\\item positive witness acceptance in the mild nonlinear family;\n\\item zero witness acceptance in saturated tanh and ReLU families;\n\\item learned linear transport above $0.98$;\n\\item learned tanh transport above $0.80$.\n\\end{enumerate}","r, overcomplete, and ill-conditioned transport above $0.98$ and equal to the local oracle within numerical tolerance;\n\\item shuffled-anchor accuracy below $0.40$ for exact linear targets;\n\\item positive witness acceptance in the mild nonlinear family;\n\\item zero witness acceptance in saturated tanh and ReLU families;\n\\item learned linear transport above $0.98$;\n\\item learned tanh transport above $0.80$.\n\\end{enumerate}","al level. A parent repair selector is frozen. Development receipts generated through the existing lineage are used to compile a child selector. Parent and child then receive exactly the same candidate repairs, target observations, calibration information, evaluator, and candidate budget. The child receipt is sealed before either final audit split is opened. The primary question is\n\\begin{equation}\n\\label{eq:block07-primary-question}\n\\mathbb E_{e\\sim Q}\n\\left[\nU_e\\!\\left(K_{1}(C_e)\\right)\n-\nU_e\\!\\left(K_{0}(C_e)\\right)\n\\right]\n>0,\n\\end{equation}\nwhere $K_0$ 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.","ld 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\n\\begin{equation}\n\\label{eq:block07-primary-question}\n\\mathbb E_{e\\sim Q}\n\\left[\nU_e\\!\\left(K_{1}(C_e)\\right)\n-\nU_e\\!\\left(K_{0}(C_e)\\right)\n\\right]\n>0,\n\\end{equation}\nwhere $K_0$ 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.","idate 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\n\\begin{equation}\n\\label{eq:block07-primary-question}\n\\mathbb E_{e\\sim Q}\n\\left[\nU_e\\!\\left(K_{1}(C_e)\\right)\n-\nU_e\\!\\left(K_{0}(C_e)\\right)\n\\right]\n>0,\n\\end{equation}\nwhere $K_0$ 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.","information, evaluator, and candidate budget. The child receipt is sealed before either final audit split is opened. The primary question is\n\\begin{equation}\n\\label{eq:block07-primary-question}\n\\mathbb E_{e\\sim Q}\n\\left[\nU_e\\!\\left(K_{1}(C_e)\\right)\n-\nU_e\\!\\left(K_{0}(C_e)\\right)\n\\right]\n>0,\n\\end{equation}\nwhere $K_0$ 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.","ipt is sealed before either final audit split is opened. The primary question is\n\\begin{equation}\n\\label{eq:block07-primary-question}\n\\mathbb E_{e\\sim Q}\n\\left[\nU_e\\!\\left(K_{1}(C_e)\\right)\n-\nU_e\\!\\left(K_{0}(C_e)\\right)\n\\right]\n>0,\n\\end{equation}\nwhere $K_0$ 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.","aled before either final audit split is opened. The primary question is\n\\begin{equation}\n\\label{eq:block07-primary-question}\n\\mathbb E_{e\\sim Q}\n\\left[\nU_e\\!\\left(K_{1}(C_e)\\right)\n-\nU_e\\!\\left(K_{0}(C_e)\\right)\n\\right]\n>0,\n\\end{equation}\nwhere $K_0$ 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.","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\n\\begin{equation}\n\\label{eq:block07-candidate-bundle}\nC_e\n=\n\\left(c_{e,0},\\ldots,c_{e,B-1}\\right)\n\\end{equation}\nbe 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.","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\n\\begin{equation}\n\\label{eq:block07-candidate-bundle}\nC_e\n=\n\\left(c_{e,0},\\ldots,c_{e,B-1}\\right)\n\\end{equation}\nbe 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.","t 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\n\\begin{equation}\n\\label{eq:block07-candidate-bundle}\nC_e\n=\n\\left(c_{e,0},\\ldots,c_{e,B-1}\\right)\n\\end{equation}\nbe 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.","\\begin{definition}[Repair selector]\nA repair selector is a measurable map\n\\begin{equation}\nK:\\mathcal Z^B\\longrightarrow\\{0,\\ldots,B-1\\},\n\\end{equation}\nwhere $\\mathcal Z$ is the space of candidate receipts available before the protected audit outcome is revealed. The selected repair is $c_{e,K(Z_e)}$.\n\\end{definition}","\\begin{definition}[Matched selector comparison]\nTwo selectors $K$ and $K'$ are compared under matched resources when, episode by episode, they receive:\n\\begin{enumerate}\n\\item the identical candidate bundle $C_e$ in the identical order;\n\\item the identical receipt fields and calibration evidence;\n\\item the identical target, anchor, and audit data;\n\\item the identical candidate-construction budget;\n\\item the identical protected utility and risk definitions;\n\\item no policy-specific regeneration, resampling, or extra oracle query.\n\\end{enumerate}\n\\end{definition}","\\begin{definition}[Selector-level metaproductivity]\nLet $Q$ be a held-out distribution over target episodes. A child selector $K_1$ has selector-level metaproductivity over parent $K_0$ when\n\\begin{equation}\n\\Delta_Q(K_1,K_0)\n:=\n\\mathbb E_{e\\sim Q}\n\\left[\nG_e(K_1)-G_e(K_0)\n\\right]\n>0,\n\\label{eq:block07-metaproductivity}\n\\end{equation}\nunder a matched comparison, where $G_e(K)$ is protected utility gain relative to the no-op on episode $e$.\n\\end{definition}","oductivity]\nLet $Q$ be a held-out distribution over target episodes. A child selector $K_1$ has selector-level metaproductivity over parent $K_0$ when\n\\begin{equation}\n\\Delta_Q(K_1,K_0)\n:=\n\\mathbb E_{e\\sim Q}\n\\left[\nG_e(K_1)-G_e(K_0)\n\\right]\n>0,\n\\label{eq:block07-metaproductivity}\n\\end{equation}\nunder a matched comparison, where $G_e(K)$ is protected utility gain relative to the no-op on episode $e$.\n\\end{definition}","\\begin{remark}\nSelector-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.\n\\end{remark}","The experiment uses four disjoint episode collections:\n\\begin{equation}\n\\mathsf D_{\\rm dev}\n\\sqcup\n\\mathsf D_{\\rm meta}\n\\sqcup\n\\mathsf D_{\\rm audit}\n\\sqcup\n\\mathsf D_{\\rm arch}.\n\\label{eq:block07-splits}\n\\end{equation}\nDevelopment 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.","isjoint episode collections:\n\\begin{equation}\n\\mathsf D_{\\rm dev}\n\\sqcup\n\\mathsf D_{\\rm meta}\n\\sqcup\n\\mathsf D_{\\rm audit}\n\\sqcup\n\\mathsf D_{\\rm arch}.\n\\label{eq:block07-splits}\n\\end{equation}\nDevelopment 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.","p\n\\mathsf D_{\\rm arch}.\n\\label{eq:block07-splits}\n\\end{equation}\nDevelopment 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.","\\begin{definition}[Sealed meta-compiler]\nA sealed meta-compiler is a map\n\\begin{equation}\n\\mathfrak M:\n\\left(\\mathsf D_{\\rm dev},\\mathsf D_{\\rm meta},K_0,\\Gamma\\right)\n\\longmapsto\n\\left(K_1,\\Pi_1\\right),\n\\label{eq:block07-meta-compiler}\n\\end{equation}\nwhere $\\Gamma$ is a declared finite meta-grammar and $\\Pi_1$ is a descendant receipt. The compiler is sealed when:\n\\begin{enumerate}\n\\item neither final audit split is supplied to $\\mathfrak M$;\n\\item $\\Pi_1$ commits to the child state, parent identity, input hashes, grammar, selected configuration, and candidate budget;\n\\item the audit files are opened only after the receipt has been serialized and hashed;\n\\item the child state used for final evaluation is byte-equivalent to the state committed by $\\Pi_1$.\n\\end{enumerate}\n\\end{definition}","mpiler]\nA sealed meta-compiler is a map\n\\begin{equation}\n\\mathfrak M:\n\\left(\\mathsf D_{\\rm dev},\\mathsf D_{\\rm meta},K_0,\\Gamma\\right)\n\\longmapsto\n\\left(K_1,\\Pi_1\\right),\n\\label{eq:block07-meta-compiler}\n\\end{equation}\nwhere $\\Gamma$ is a declared finite meta-grammar and $\\Pi_1$ is a descendant receipt. The compiler is sealed when:\n\\begin{enumerate}\n\\item neither final audit split is supplied to $\\mathfrak M$;\n\\item $\\Pi_1$ commits to the child state, parent identity, input hashes, grammar, selected configuration, and candidate budget;\n\\item the audit files are opened only after the receipt has been serialized and hashed;\n\\item the child state used for final evaluation is byte-equivalent to the state committed by $\\Pi_1$.\n\\end{enumerate}\n\\end{definition}","-compiler is a map\n\\begin{equation}\n\\mathfrak M:\n\\left(\\mathsf D_{\\rm dev},\\mathsf D_{\\rm meta},K_0,\\Gamma\\right)\n\\longmapsto\n\\left(K_1,\\Pi_1\\right),\n\\label{eq:block07-meta-compiler}\n\\end{equation}\nwhere $\\Gamma$ is a declared finite meta-grammar and $\\Pi_1$ is a descendant receipt. The compiler is sealed when:\n\\begin{enumerate}\n\\item neither final audit split is supplied to $\\mathfrak M$;\n\\item $\\Pi_1$ commits to the child state, parent identity, input hashes, grammar, selected configuration, and candidate budget;\n\\item the audit files are opened only after the receipt has been serialized and hashed;\n\\item the child state used for final evaluation is byte-equivalent to the state committed by $\\Pi_1$.\n\\end{enumerate}\n\\end{definition}","led when:\n\\begin{enumerate}\n\\item neither final audit split is supplied to $\\mathfrak M$;\n\\item $\\Pi_1$ commits to the child state, parent identity, input hashes, grammar, selected configuration, and candidate budget;\n\\item the audit files are opened only after the receipt has been serialized and hashed;\n\\item the child state used for final evaluation is byte-equivalent to the state committed by $\\Pi_1$.\n\\end{enumerate}\n\\end{definition}","\\begin{theorem}[Compilation noninterference]\n\\label{thm:block07-compilation-noninterference}\nLet $K_1=\\mathfrak M(D,V,K_0,\\Gamma)$, 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 $(D,V,K_0,\\Gamma)$,\n\\begin{equation}\nK_1\\perp\\!\\!\\!\\perp A.\n\\end{equation}\nEquivalently, changing the realized final audit outcomes while holding compiler inputs fixed leaves $K_1$ unchanged.\n\\end{theorem}","_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 $(D,V,K_0,\\Gamma)$,\n\\begin{equation}\nK_1\\perp\\!\\!\\!\\perp A.\n\\end{equation}\nEquivalently, changing the realized final audit outcomes while holding compiler inputs fixed leaves $K_1$ unchanged.\n\\end{theorem}","\\begin{proof}\nBy assumption there is a deterministic function $f$ such that\n$K_1=f(D,V,K_0,\\Gamma)$. Conditional on its arguments, $K_1$ is constant. It therefore has a degenerate conditional distribution that does not depend on $A$.\n\\end{proof}","\\begin{definition}[One-step bounded repair-process improvement]\nA lineage edge\n\\begin{equation}\n(\\Ocal_0,K_0)\n\\xrightarrow{\\Pi_1}\n(\\Ocal_1,K_1)\n\\end{equation}\nis a one-step bounded repair-process improvement relative to $(Q,B,U,R)$ when:\n\\begin{enumerate}\n\\item $K_1$ is generated and sealed through an authorized lineage transaction;\n\\item parent and child use the same candidate budget $B$ and candidate bundle on each target;\n\\item a predeclared lower confidence bound for $\\Delta_Q(K_1,K_0)$ is positive;\n\\item the child satisfies declared average, tail, and maximum protected-risk gates;\n\\item wrong-lineage or shuffled-receipt controls perform worse;\n\\item ablating the child module restores the parent selection trace;\n\\item the evaluator, target distribution, and evidence semantics are not changed by the child.\n\\end{enumerate}\n\\end{de","\\begin{definition}[One-step bounded repair-process improvement]\nA lineage edge\n\\begin{equation}\n(\\Ocal_0,K_0)\n\\xrightarrow{\\Pi_1}\n(\\Ocal_1,K_1)\n\\end{equation}\nis a one-step bounded repair-process improvement relative to $(Q,B,U,R)$ when:\n\\begin{enumerate}\n\\item $K_1$ is generated and sealed through an authorized lineage transaction;\n\\item parent and child use the same candidate budget $B$ and candidate bundle on each target;\n\\item a predeclared lower confidence bound for $\\Delta_Q(K_1,K_0)$ is positive;\n\\item the child satisfies declared average, tail, and maximum protected-risk gates;\n\\item wrong-lineage or shuffled-receipt controls perform worse;\n\\item ablating the child module restores the parent selection trace;\n\\item the evaluator, target distribution, and evidence semantics are not changed by the child.\n\\end{enumerate}\n\\end{definition}","e\n\\begin{equation}\n(\\Ocal_0,K_0)\n\\xrightarrow{\\Pi_1}\n(\\Ocal_1,K_1)\n\\end{equation}\nis a one-step bounded repair-process improvement relative to $(Q,B,U,R)$ when:\n\\begin{enumerate}\n\\item $K_1$ is generated and sealed through an authorized lineage transaction;\n\\item parent and child use the same candidate budget $B$ and candidate bundle on each target;\n\\item a predeclared lower confidence bound for $\\Delta_Q(K_1,K_0)$ is positive;\n\\item the child satisfies declared average, tail, and maximum protected-risk gates;\n\\item wrong-lineage or shuffled-receipt controls perform worse;\n\\item ablating the child module restores the parent selection trace;\n\\item the evaluator, target distribution, and evidence semantics are not changed by the child.\n\\end{enumerate}\n\\end{definition}","Each candidate receipt $z_{e,c}$ is mapped to a feature vector\n\\begin{equation}\nx(z_{e,c})\\in\\R^{23}.\n\\end{equation}\nThe released feature schema contains:\n\\begin{itemize}\n\\item a constant coordinate;\n\\item logarithmic anchor and calibration sizes;\n\\item target representation width;\n\\item alignment, transport-condition, and repair-equivariance diagnostics;\n\\item packet residual and relative repair magnitude;\n\\item repair rank;\n\\item calibration baseline slice and protected accuracy;\n\\item calibration gain, protected regression, and protected utility;\n\\item five candidate-type indicators;\n\\item three interaction coordinates coupling gain with alignment, packet residual, and rank-weighted regression.\n\\end{itemize}\nThe schema","Each candidate receipt $z_{e,c}$ is mapped to a feature vector\n\\begin{equation}\nx(z_{e,c})\\in\\R^{23}.\n\\end{equation}\nThe released feature schema contains:\n\\begin{itemize}\n\\item a constant coordinate;\n\\item logarithmic anchor and calibration sizes;\n\\item target representation width;\n\\item alignment, transport-condition, and repair-equivariance diagnostics;\n\\item packet residual and relative repair magnitude;\n\\item repair rank;\n\\item calibration baseline slice and protected accuracy;\n\\item calibration gain, protected regression, and protected utility;\n\\item five candidate-type indicators;\n\\item three interaction coordinates coupling gain with alignment, packet residual, and rank-weighted regression.\n\\end{itemize}\nThe schema is fixed before the final audits. It is expressive enough to learn sys","Let $x_i$ denote a development candidate feature vector, $y_i$ its revealed protected utility, and\n\\begin{equation}\nr_i=[A^{\\rm base}_{i,\\rm prot}-A_{i,\\rm prot}]_+\n\\end{equation}\nits protected-regression label. Standardize the feature matrix to $Z$ using development means and scales.","For regularization parameters $\\lambda_u,\\lambda_r>0$, define\n\\begin{align}\n\\widehat\\beta_u\n&=\n\\left(Z^\\top Z+\\lambda_u I\\right)^{-1}Z^\\top y,\n\\label{eq:block07-utility-head}","For regularization parameters $\\lambda_u,\\lambda_r>0$, define\n\\begin{align}\n\\widehat\\beta_u\n&=\n\\left(Z^\\top Z+\\lambda_u I\\right)^{-1}Z^\\top y,\n\\label{eq:block07-utility-head}\\\\\n\\widehat\\beta_r\n&=\n\\left(Z^\\top Z+\\lambda_r I\\right)^{-1}Z^\\top r.\n\\label{eq:block07-risk-head}\n\\end{align}\nFor 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\n\\begin{align}\n\\widehat U_{\\rm LCB}(z)\n&=\n\\widehat U(z)-\\kappa_u s_{u,c(z)},\n\\label{eq:block07-utility-lcb}","\\widehat\\beta_r\n&=\n\\left(Z^\\top Z+\\lambda_r I\\right)^{-1}Z^\\top r.\n\\label{eq:block07-risk-head}\n\\end{align}\nFor 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\n\\begin{align}\n\\widehat U_{\\rm LCB}(z)\n&=\n\\widehat U(z)-\\kappa_u s_{u,c(z)},\n\\label{eq:block07-utility-lcb}","\\widehat\\beta_r\n&=\n\\left(Z^\\top Z+\\lambda_r I\\right)^{-1}Z^\\top r.\n\\label{eq:block07-risk-head}\n\\end{align}\nFor 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\n\\begin{align}\n\\widehat U_{\\rm LCB}(z)\n&=\n\\widehat U(z)-\\kappa_u s_{u,c(z)},\n\\label{eq:block07-utility-lcb}\\\\\n\\widehat R_{\\rm UCB}(z)\n&=\n\\max\\{0,\\widehat R(z)\\}+\\kappa_r s_{r,c(z)}.\n\\label{eq:block07-risk-ucb}\n\\end{align}\nA candidate is admissible when\n\\begin{equation}\n\\widehat R_{\\rm UCB}(z)\\leq\\tau_r,\n\\qquad\n\\widehat r_{\\rm cal}(z)\\leq\\tau_{\\rm cal},\n\\qquad\nA^{\\rm base}_{\\rm cal,prot}(z)\\geq a_{\\min}.\n\\label{eq:block07-admissible}\n\\end{equation}\nThe child chooses the admissible candidate with maximum $\\widehat U_{\\rm LCB}$. If the admissible set is empty, it returns the no-op.","\\widehat R_{\\rm UCB}(z)\n&=\n\\max\\{0,\\widehat R(z)\\}+\\kappa_r s_{r,c(z)}.\n\\label{eq:block07-risk-ucb}\n\\end{align}\nA candidate is admissible when\n\\begin{equation}\n\\widehat R_{\\rm UCB}(z)\\leq\\tau_r,\n\\qquad\n\\widehat r_{\\rm cal}(z)\\leq\\tau_{\\rm cal},\n\\qquad\nA^{\\rm base}_{\\rm cal,prot}(z)\\geq a_{\\min}.\n\\label{eq:block07-admissible}\n\\end{equation}\nThe child chooses the admissible candidate with maximum $\\widehat U_{\\rm LCB}$. If the admissible set is empty, it returns the no-op.","\\begin{theorem}[Risk-gate soundness]\n\\label{thm:block07-risk-soundness}\nFix an episode and candidate set. Suppose, simultaneously for every candidate $c$,\n\\begin{equation}\nR_e(c)\\leq\\widehat R_{\\rm UCB}(c).\n\\end{equation}\nThen any child-selected candidate satisfying\n$\\widehat R_{\\rm UCB}(c)\\leq\\tau_r$ has true protected risk at most $\\tau_r$.\n\\end{theorem}","\\begin{proof}\nFor the selected candidate $\\widehat c$,\n\\begin{equation}\nR_e(\\widehat c)\n\\leq\n\\widehat R_{\\rm UCB}(\\widehat c)\n\\leq\n\\tau_r.\n\\end{equation}\n\\end{proof}","\\begin{theorem}[Utility regret under simultaneous lower bounds]\n\\label{thm:block07-utility-regret}\nLet $\\mathcal C_{\\rm safe}$ be the truly safe candidate set. Suppose every $c\\in\\mathcal C_{\\rm safe}$ passes the implemented admissibility filter, and suppose\n\\begin{equation}\n0\\leq U_e(c)-\\widehat U_{\\rm LCB}(c)\\leq\\epsilon\n\\end{equation}\nfor every admissible $c$. If $\\widehat c$ maximizes $\\widehat U_{\\rm LCB}$ and $c^*$ maximizes $U_e$ over $\\mathcal C_{\\rm safe}$, then\n\\begin{equation}\nU_e(c^*)-U_e(\\widehat c)\\leq\\epsilon.\n\\end{equation}\n\\end{theorem}","ity regret under simultaneous lower bounds]\n\\label{thm:block07-utility-regret}\nLet $\\mathcal C_{\\rm safe}$ be the truly safe candidate set. Suppose every $c\\in\\mathcal C_{\\rm safe}$ passes the implemented admissibility filter, and suppose\n\\begin{equation}\n0\\leq U_e(c)-\\widehat U_{\\rm LCB}(c)\\leq\\epsilon\n\\end{equation}\nfor every admissible $c$. If $\\widehat c$ maximizes $\\widehat U_{\\rm LCB}$ and $c^*$ maximizes $U_e$ over $\\mathcal C_{\\rm safe}$, then\n\\begin{equation}\nU_e(c^*)-U_e(\\widehat c)\\leq\\epsilon.\n\\end{equation}\n\\end{theorem}","simultaneous lower bounds]\n\\label{thm:block07-utility-regret}\nLet $\\mathcal C_{\\rm safe}$ be the truly safe candidate set. Suppose every $c\\in\\mathcal C_{\\rm safe}$ passes the implemented admissibility filter, and suppose\n\\begin{equation}\n0\\leq U_e(c)-\\widehat U_{\\rm LCB}(c)\\leq\\epsilon\n\\end{equation}\nfor every admissible $c$. If $\\widehat c$ maximizes $\\widehat U_{\\rm LCB}$ and $c^*$ maximizes $U_e$ over $\\mathcal C_{\\rm safe}$, then\n\\begin{equation}\nU_e(c^*)-U_e(\\widehat c)\\leq\\epsilon.\n\\end{equation}\n\\end{theorem}","us lower bounds]\n\\label{thm:block07-utility-regret}\nLet $\\mathcal C_{\\rm safe}$ be the truly safe candidate set. Suppose every $c\\in\\mathcal C_{\\rm safe}$ passes the implemented admissibility filter, and suppose\n\\begin{equation}\n0\\leq U_e(c)-\\widehat U_{\\rm LCB}(c)\\leq\\epsilon\n\\end{equation}\nfor every admissible $c$. If $\\widehat c$ maximizes $\\widehat U_{\\rm LCB}$ and $c^*$ maximizes $U_e$ over $\\mathcal C_{\\rm safe}$, then\n\\begin{equation}\nU_e(c^*)-U_e(\\widehat c)\\leq\\epsilon.\n\\end{equation}\n\\end{theorem}","egret}\nLet $\\mathcal C_{\\rm safe}$ be the truly safe candidate set. Suppose every $c\\in\\mathcal C_{\\rm safe}$ passes the implemented admissibility filter, and suppose\n\\begin{equation}\n0\\leq U_e(c)-\\widehat U_{\\rm LCB}(c)\\leq\\epsilon\n\\end{equation}\nfor every admissible $c$. If $\\widehat c$ maximizes $\\widehat U_{\\rm LCB}$ and $c^*$ maximizes $U_e$ over $\\mathcal C_{\\rm safe}$, then\n\\begin{equation}\nU_e(c^*)-U_e(\\widehat c)\\leq\\epsilon.\n\\end{equation}\n\\end{theorem}","\\begin{proof}\nBecause $c^*$ is admissible,\n\\begin{equation}\n\\widehat U_{\\rm LCB}(\\widehat c)\n\\geq\n\\widehat U_{\\rm LCB}(c^*).\n\\end{equation}\nThe lower-bound property gives\n\\begin{equation}\nU_e(\\widehat c)\n\\geq\n\\widehat U_{\\rm LCB}(\\widehat c)\n\\geq\n\\widehat U_{\\rm LCB}(c^*)\n\\geq\nU_e(c^*)-\\epsilon.\n\\end{equation}\n\\end{proof}","The child compiler searches a finite grammar\n\\begin{equation}\n\\Gamma\n=\n\\Lambda_u\\times\\Lambda_r\\times K_u\\times K_r\\times T_r,\n\\end{equation}\nwhere the factors specify utility regularization, risk regularization, utility residual penalty, risk residual penalty, and risk threshold. In the released fixture,\n\\begin{equation}\n|\\Gamma|\n=5\\cdot5\\cdot4\\cdot5\\cdot4\n=2000.\n\\end{equation}\nFor $\\gamma\\in\\Gamma$, 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\n\\begin{equation}\n\\widehat\\gamma\n\\in\n\\argmax_{\\gamma\\in\\Gamma_{\\rm safe}}\n\\left\n\\{\n\\overline U_{\\rm","The child compiler searches a finite grammar\n\\begin{equation}\n\\Gamma\n=\n\\Lambda_u\\times\\Lambda_r\\times K_u\\times K_r\\times T_r,\n\\end{equation}\nwhere the factors specify utility regularization, risk regularization, utility residual penalty, risk residual penalty, and risk threshold. In the released fixture,\n\\begin{equation}\n|\\Gamma|\n=5\\cdot5\\cdot4\\cdot5\\cdot4\n=2000.\n\\end{equation}\nFor $\\gamma\\in\\Gamma$, 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\n\\begin{equation}\n\\widehat\\gamma\n\\in\n\\argmax_{\\gamma\\in\\Gamma_{\\rm safe}}\n\\left\n\\{\n\\overline U_{\\rm meta}(\\gamma)\n-\tfrac12\\widehat{\\rm se}(\\gamma)\n\\right\\}.\n\\label{eq:block07-meta-objective}\n\\end{equation}","compiler searches a finite grammar\n\\begin{equation}\n\\Gamma\n=\n\\Lambda_u\\times\\Lambda_r\\times K_u\\times K_r\\times T_r,\n\\end{equation}\nwhere the factors specify utility regularization, risk regularization, utility residual penalty, risk residual penalty, and risk threshold. In the released fixture,\n\\begin{equation}\n|\\Gamma|\n=5\\cdot5\\cdot4\\cdot5\\cdot4\n=2000.\n\\end{equation}\nFor $\\gamma\\in\\Gamma$, 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\n\\begin{equation}\n\\widehat\\gamma\n\\in\n\\argmax_{\\gamma\\in\\Gamma_{\\rm safe}}\n\\left\n\\{\n\\overline U_{\\rm meta}(\\gamma)\n-\tfrac12\\widehat{\\rm se}(\\gamma)\n\\right\\}.\n\\label{eq:block07-meta-objective}\n\\end{equation}","es K_r\\times T_r,\n\\end{equation}\nwhere the factors specify utility regularization, risk regularization, utility residual penalty, risk residual penalty, and risk threshold. In the released fixture,\n\\begin{equation}\n|\\Gamma|\n=5\\cdot5\\cdot4\\cdot5\\cdot4\n=2000.\n\\end{equation}\nFor $\\gamma\\in\\Gamma$, 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\n\\begin{equation}\n\\widehat\\gamma\n\\in\n\\argmax_{\\gamma\\in\\Gamma_{\\rm safe}}\n\\left\n\\{\n\\overline U_{\\rm meta}(\\gamma)\n-\tfrac12\\widehat{\\rm se}(\\gamma)\n\\right\\}.\n\\label{eq:block07-meta-objective}\n\\end{equation}","ot4\\cdot5\\cdot4\n=2000.\n\\end{equation}\nFor $\\gamma\\in\\Gamma$, 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\n\\begin{equation}\n\\widehat\\gamma\n\\in\n\\argmax_{\\gamma\\in\\Gamma_{\\rm safe}}\n\\left\n\\{\n\\overline U_{\\rm meta}(\\gamma)\n-\tfrac12\\widehat{\\rm se}(\\gamma)\n\\right\\}.\n\\label{eq:block07-meta-objective}\n\\end{equation}","\\begin{theorem}[Finite meta-grammar uniform bound]\n\\label{thm:block07-meta-uniform}\nSuppose, 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 $1-\\delta$,\n\\begin{equation}\n\\sup_{\\gamma\\in\\Gamma}\n\\left|\n\\widehat U_n(\\gamma)-U(\\gamma)\n\\right|\n\\leq\nL\\sqrt{\\frac{\\log(2|\\Gamma|/\\delta)}{2n}}.\n\\label{eq:block07-uniform-bound}\n\\end{equation}\nConsequently, the true utility of the empirical maximizer is within twice the right-hand side of the best configuration in $\\Gamma$.\n\\end{theorem}","the $n$ meta-validation episodes are independent. Then with probability at least $1-\\delta$,\n\\begin{equation}\n\\sup_{\\gamma\\in\\Gamma}\n\\left|\n\\widehat U_n(\\gamma)-U(\\gamma)\n\\right|\n\\leq\nL\\sqrt{\\frac{\\log(2|\\Gamma|/\\delta)}{2n}}.\n\\label{eq:block07-uniform-bound}\n\\end{equation}\nConsequently, the true utility of the empirical maximizer is within twice the right-hand side of the best configuration in $\\Gamma$.\n\\end{theorem}","\\begin{proof}\nHoeffding'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.\n\\end{proof}","The winning frozen configuration is\n\\begin{equation}\n\\begin{aligned}\n\\lambda_u&=100,\n&\\lambda_r&=0.01,\\\\\n\\kappa_u&=2,\n&\\kappa_r&=0.5,\\\\\n\\tau_r&=0.035,\n&\\tau_{\\rm cal}&=0.08,\\\\\na_{\\min}&=0.80.\n\\end{aligned}\n\\label{eq:block07-selected-config}\n\\end{equation}\nIts 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 $0.054463$.","a_{\\min}&=0.80.\n\\end{aligned}\n\\label{eq:block07-selected-config}\n\\end{equation}\nIts 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 $0.054463$.","\\begin{theorem}[Pathwise selector isolation]\n\\label{thm:block07-selector-isolation}\nFor episode $e$, let parent and child receive the identical ordered bundle $C_e$ and identical pre-audit receipt $Z_e$. Let\n\\begin{equation}\ni_0=K_0(Z_e),\n\\qquad\ni_1=K_1(Z_e).\n\\end{equation}\nThen the realized paired utility difference is exactly\n\\begin{equation}\nD_e\n=\nU_e(c_{e,i_1})-U_e(c_{e,i_0}).\n\\label{eq:block07-paired-difference}\n\\end{equation}\nIf candidate construction, target state, and audit evaluation are frozen, no candidate-generation term remains in this difference.\n\\end{theorem}","\\begin{proof}\nBoth policies are evaluated on the same episode and the same candidate utility vector\n\\begin{equation*}\n\\left(U_e(c_{e,0}),\\ldots,U_e(c_{e,B-1})\\right).\n\\end{equation*}\nTheir outputs differ only through the selected indices $i_0$ and $i_1$. Substitution gives \\cref{eq:block07-paired-difference}.\n\\end{proof}","\\begin{corollary}[Selector ablation]\n\\label{cor:block07-selector-ablation}\nIf 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.\n\\end{corollary}","\\subsection{The best-of-\\texorpdfstring{$k$}{k} confound}","Let $M_B(K)$ denote expected best gain when policy $K$ receives $B$ independently generated candidate opportunities. Even if the policy is unchanged,\n\\begin{equation}\nM_{B+1}(K)\\geq M_B(K).\n\\end{equation}\nTherefore, comparing a child with $B_1>B_0$ against a parent with budget $B_0$ does not isolate improvement of the selection procedure.","\\begin{proposition}[No-free-expansion attribution]\n\\label{prop:block07-budget-confound}\nSuppose 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.\n\\end{proposition}","Block 7 fixes $B=5$ for every arm and target.","\\begin{proof}\nLet $Z$ be column centered and let $P$ be a uniformly random permutation matrix independent of $y$. Then\n\\begin{equation}\n\\mathbb E_P[Z^\\top Py]\n=\nZ^\\top\\mathbb E_P[Py]\n=\nZ^\\top\\overline y\\one\n=0.\n\\end{equation}\nThe ridge coefficients are linear in $Z^\\top Py$ for fixed $Z$.\n\\end{proof}","\\begin{proposition}[Fixed-candidate structural invariance]\n\\label{prop:block07-fixed-candidate-invariance}\nLet $\\mathcal C_e$ be the authorized candidate set for episode $e$. If parent and child differ only in the selector and\n\\begin{equation}\n\\mathcal C^{(0)}_e=\\mathcal C^{(1)}_e\n\\end{equation}\nfor every $e$, then the pointwise reachable correction set and its linear span are identical for parent and child.\n\\end{proposition}","For target episode $e$ and candidate $c$, let\n\\begin{align}\nA^{\\rm slice}_e(c)\n&=\\text{accuracy on the affected stratum},","For target episode $e$ and candidate $c$, let\n\\begin{align}\nA^{\\rm slice}_e(c)\n&=\\text{accuracy on the affected stratum},\\\\\nA^{\\rm prot}_e(c)\n&=\\text{accuracy on the unaffected protected stratum},\\\\\nq(c)\n&=\\text{repair rank}.\n\\end{align}\nThe candidate utility is\n\\begin{equation}\nU_e(c)\n=\nA^{\\rm slice}_e(c)\n-2\\left[A^{\\rm prot}_e(0)-A^{\\rm prot}_e(c)\\right]_+\n-0.0005q(c),\n\\label{eq:block07-protected-utility}\n\\end{equation}\nand the reported protected utility gain is\n\\begin{equation}\nG_e(c)=U_e(c)-A^{\\rm slice}_e(0).\n\\label{eq:block07-protected-gain}\n\\end{equation}\nThe protected regression is\n\\begin{equation}\nR_e(c)=\\left[A^{\\rm prot}_e(0)-A^{\\rm prot}_e(c)\\right]_+.\n\\end{equation}\nThe utility therefore rewards repair of the affected stratum, penalizes damage to unrelated behavior twice as strongly, and applies a small complexity charge.","q(c)\n&=\\text{repair rank}.\n\\end{align}\nThe candidate utility is\n\\begin{equation}\nU_e(c)\n=\nA^{\\rm slice}_e(c)\n-2\\left[A^{\\rm prot}_e(0)-A^{\\rm prot}_e(c)\\right]_+\n-0.0005q(c),\n\\label{eq:block07-protected-utility}\n\\end{equation}\nand the reported protected utility gain is\n\\begin{equation}\nG_e(c)=U_e(c)-A^{\\rm slice}_e(0).\n\\label{eq:block07-protected-gain}\n\\end{equation}\nThe protected regression is\n\\begin{equation}\nR_e(c)=\\left[A^{\\rm prot}_e(0)-A^{\\rm prot}_e(c)\\right]_+.\n\\end{equation}\nThe utility therefore rewards repair of the affected stratum, penalizes damage to unrelated behavior twice as strongly, and applies a small complexity charge.","epair rank}.\n\\end{align}\nThe candidate utility is\n\\begin{equation}\nU_e(c)\n=\nA^{\\rm slice}_e(c)\n-2\\left[A^{\\rm prot}_e(0)-A^{\\rm prot}_e(c)\\right]_+\n-0.0005q(c),\n\\label{eq:block07-protected-utility}\n\\end{equation}\nand the reported protected utility gain is\n\\begin{equation}\nG_e(c)=U_e(c)-A^{\\rm slice}_e(0).\n\\label{eq:block07-protected-gain}\n\\end{equation}\nThe protected regression is\n\\begin{equation}\nR_e(c)=\\left[A^{\\rm prot}_e(0)-A^{\\rm prot}_e(c)\\right]_+.\n\\end{equation}\nThe utility therefore rewards repair of the affected stratum, penalizes damage to unrelated behavior twice as strongly, and applies a small complexity charge.","For $n$ held-out episodes, define paired differences\n\\begin{equation}\nD_j\n=\nG_{e_j}(K_1)-G_{e_j}(K_0),\n\\qquad\n\\widehat\\Delta_n=\\frac1n\\sum_{j=1}^nD_j.\n\\label{eq:block07-paired-estimator}\n\\end{equation}\nBecause parent and child share each target and candidate bundle, pairing removes much of the episode difficulty variation.","\\begin{theorem}[Distribution-free paired lower bound]\n\\label{thm:block07-paired-bound}\nSuppose $D_1,\\ldots,D_n$ are independent and lie almost surely in $[a,b]$. Then with probability at least $1-\\alpha$,\n\\begin{equation}\n\\Delta_Q\n\\geq\n\\widehat\\Delta_n\n-(b-a)\\sqrt{\\frac{\\log(1/\\alpha)}{2n}}.\n\\label{eq:block07-hoeffding}\n\\end{equation}\nA positive right-hand side certifies $\\Delta_Q>0$ at confidence $1-\\alpha$ under the assumptions.\n\\end{theorem}","egin{theorem}[Distribution-free paired lower bound]\n\\label{thm:block07-paired-bound}\nSuppose $D_1,\\ldots,D_n$ are independent and lie almost surely in $[a,b]$. Then with probability at least $1-\\alpha$,\n\\begin{equation}\n\\Delta_Q\n\\geq\n\\widehat\\Delta_n\n-(b-a)\\sqrt{\\frac{\\log(1/\\alpha)}{2n}}.\n\\label{eq:block07-hoeffding}\n\\end{equation}\nA positive right-hand side certifies $\\Delta_Q>0$ at confidence $1-\\alpha$ under the assumptions.\n\\end{theorem}","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.","The child must pass performance and risk gates simultaneously. Let\n\\begin{equation}\n\\mathsf Gate\n=\n\\mathsf Seal\n\\wedge\n\\mathsf Match\n\\wedge\n\\mathsf AuditGain\n\\wedge\n\\mathsf ForeignGain\n\\wedge\n\\mathsf Risk\n\\wedge\n\\mathsf Controls.\n\\end{equation}\nThe released suite evaluates the following frozen selector-gate families:\n\\begin{enumerate}\n\\item descendant receipt validity;\n\\item split-file load order and sealing;\n\\item positive paired child-parent audit interval;\n\\item positive paired child-parent architecture-foreign interval;\n\\item positive child-raw architecture-foreign interval;\n\\item three seed-and-task audit regression gates;\n\\item three architecture-foreign regression gates;\n\\item two shuffled-lineage controls;\n\\item two selector-ablation equalities;\n\\item matched five-candidate budget;\n\\item exclusion of audit receipts from compiler inputs;\n\\item disjoint architecture-family supports.\n\\end{enumerate}\nAll componen","\\begin{warning}\nThe 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.\n\\end{warning}","\\begin{proposition}[Risk-filter tradeoff]\n\\label{prop:block07-risk-tradeoff}\nLet $\\mathcal A_r\\subseteq\\mathcal A_0$ be the candidate set retained with and without a risk filter. Then\n\\begin{equation}\n\\max_{c\\in\\mathcal A_0}\\widehat U(c)\n\\geq\n\\max_{c\\in\\mathcal A_r}\\widehat U(c),\n\\end{equation}\nbut no analogous inequality favors the unfiltered selector for true protected risk.\n\\end{proposition}","For $x\\in\\R^4$, define an eight-dimensional task representation\n\\begin{equation}\n\\psi(x)\n=\n\\left(\n x_1,x_2,x_3,x_4,\n x_1x_2,x_3x_4,\n \\sin(sx_1),\\cos(sx_2)\n\\right).\n\\end{equation}\nA task-specific condition vector and threshold define the affected indicator\n\\begin{equation}\na(x)=\\one\\{\\langle v,x\\rangle>\\tau\\}.\n\\end{equation}\nThe conditional feature is\n\\begin{equation}\n\\phi(x)=\\bigl(\\psi(x),a(x)\\psi(x)\\bigr)\\in\\R^{16}.\n\\end{equation}\nTrue 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","For $x\\in\\R^4$, define an eight-dimensional task representation\n\\begin{equation}\n\\psi(x)\n=\n\\left(\n x_1,x_2,x_3,x_4,\n x_1x_2,x_3x_4,\n \\sin(sx_1),\\cos(sx_2)\n\\right).\n\\end{equation}\nA task-specific condition vector and threshold define the affected indicator\n\\begin{equation}\na(x)=\\one\\{\\langle v,x\\rangle>\\tau\\}.\n\\end{equation}\nThe conditional feature is\n\\begin{equation}\n\\phi(x)=\\bigl(\\psi(x),a(x)\\psi(x)\\bigr)\\in\\R^{16}.\n\\end{equation}\nTrue 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.","psi(x)\n=\n\\left(\n x_1,x_2,x_3,x_4,\n x_1x_2,x_3x_4,\n \\sin(sx_1),\\cos(sx_2)\n\\right).\n\\end{equation}\nA task-specific condition vector and threshold define the affected indicator\n\\begin{equation}\na(x)=\\one\\{\\langle v,x\\rangle>\\tau\\}.\n\\end{equation}\nThe conditional feature is\n\\begin{equation}\n\\phi(x)=\\bigl(\\psi(x),a(x)\\psi(x)\\bigr)\\in\\R^{16}.\n\\end{equation}\nTrue 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.","Development, meta-validation, and seed-and-task audit episodes draw from eight families:\n\\begin{equation}\n\\begin{gathered}\n\\texttt{linear16},\\quad\n\\texttt{overcomplete24},\\quad\n\\texttt{ill\\_conditioned16},\\quad\n\\texttt{compressed12},\\\\\n\\texttt{mild\\_tanh24},\\quad\n\\texttt{saturated\\_tanh24},\\quad\n\\texttt{relu32},\\quad\n\\texttt{noisy\\_linear16}.\n\\end{gathered}\n\\end{equation}\nThe architecture-foreign audit uses four disjoint families:\n\\begin{equation}\n\\texttt{softsign24},\\quad\n\\texttt{piecewise24},\\quad\n\\texttt{quadratic24},\\quad\n\\texttt{bottleneck14}.\n\\end{equation}\nThe family names and construction code are frozen in the candidate receipt archive.","\\texttt{mild\\_tanh24},\\quad\n\\texttt{saturated\\_tanh24},\\quad\n\\texttt{relu32},\\quad\n\\texttt{noisy\\_linear16}.\n\\end{gathered}\n\\end{equation}\nThe architecture-foreign audit uses four disjoint families:\n\\begin{equation}\n\\texttt{softsign24},\\quad\n\\texttt{piecewise24},\\quad\n\\texttt{quadratic24},\\quad\n\\texttt{bottleneck14}.\n\\end{equation}\nThe family names and construction code are frozen in the candidate receipt archive.","Every episode exposes exactly\n\\begin{equation}\nC_e\n=\n\\{\n\\texttt{no\\_op},\n\\texttt{transport},\n\\texttt{local\\_r4},\n\\texttt{local\\_r8},\n\\texttt{local\\_full}\n\\}.\n\\label{eq:block07-five-candidates}\n\\end{equation}\nThe 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.","Episode difficulty varies through target family, task seed, encoder seed, anchor size, calibration size, probe noise, and probe leakage. Anchor sizes lie in\n\\begin{equation}\n\\{16,24,32,48,64,96,128\\},\n\\end{equation}\ncalibration sizes in\n\\begin{equation}\n\\{48,64,96,128\\},\n\\end{equation}\nprobe-noise levels in\n\\begin{equation}\n\\{0,0.02,0.05,0.10,0.18\\},\n\\end{equation}\nand probe-leak levels in\n\\begin{equation}\n\\{0,0.05,0.12,0.25\\}.\n\\end{equation}","ifficulty varies through target family, task seed, encoder seed, anchor size, calibration size, probe noise, and probe leakage. Anchor sizes lie in\n\\begin{equation}\n\\{16,24,32,48,64,96,128\\},\n\\end{equation}\ncalibration sizes in\n\\begin{equation}\n\\{48,64,96,128\\},\n\\end{equation}\nprobe-noise levels in\n\\begin{equation}\n\\{0,0.02,0.05,0.10,0.18\\},\n\\end{equation}\nand probe-leak levels in\n\\begin{equation}\n\\{0,0.05,0.12,0.25\\}.\n\\end{equation}","The chronology is:\n\\begin{equation}\n\\begin{aligned}\n&\\text{open development file and fit heads}\\notag\\\\\n&\\longrightarrow\n\\text{open meta-validation file and choose configuration}\\notag\\\\\n&\\longrightarrow\n\\text{serialize and hash descendant receipt}\\notag\\\\\n&\\longrightarrow\n\\text{open seed/task audit file}\\notag\\\\\n&\\longrightarrow\n\\text{open architecture-foreign audit file}\\notag\\\\\n&\\longrightarrow\n\\text{evaluate every frozen policy and claim gate}.\n\\end{aligned}\n\\label{eq:block07-chronology}\n\\end{equation}","\\bottomrule\n\\end{tabular}\n\\caption{Primary matched-budget results. Tail rate is the fraction of episodes with protected regression above $0.05$.}\n\\label{tab:block07-primary-results}\n\\end{table}","The paired effects are\n\\begin{align}\n\\widehat\\Delta_{\\rm audit}\n&=0.072518,\n&\\mathrm{CI}_{95\\%}\n&=[0.046745,0.101389],\n\\label{eq:block07-audit-effect}\\\\\n\\widehat\\Delta_{\\rm arch}\n&=0.104999,\n&\\mathrm{CI}_{95\\%}\n&=[0.072711,0.140506].\n\\label{eq:block07-foreign-effect}\n\\end{align}\nBoth one-sided sign-flip $p$-values equal approximately\n$4.99975\\times10^{-5}$ under the frozen 20,000-draw randomization procedure.","\\widehat\\Delta_{\\rm arch}\n&=0.104999,\n&\\mathrm{CI}_{95\\%}\n&=[0.072711,0.140506].\n\\label{eq:block07-foreign-effect}\n\\end{align}\nBoth one-sided sign-flip $p$-values equal approximately\n$4.99975\\times10^{-5}$ under the frozen 20,000-draw randomization procedure.","On the architecture-foreign split, the child's paired advantage over raw calibration is\n\\begin{equation}\n0.017989,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.003477,0.036611],\n\\qquad\np_{\\rm sign}=0.00940.\n\\end{equation}\nOn 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.","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.","For the architecture-foreign split,\n\\begin{align}\n\\overline R_{\\rm child}&=0.012852,\n&\\Pr(R>0.05)&=0.033333,\\\\\n\\overline R_{\\rm no\\ risk}&=0.015741,\n&\\Pr(R>0.05)&=0.058333.\n\\end{align}\nThe 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.","\\overline R_{\\rm no\\ risk}&=0.015741,\n&\\Pr(R>0.05)&=0.058333.\n\\end{align}\nThe 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.","The correctly bound child outperforms the shuffled child by\n\\begin{align}\n0.106417\n&\\quad\\text{on the seed/task audit},\\\\\n0.040672\n&\\quad\\text{on the architecture-foreign audit}.\n\\end{align}\nThe 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}.","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.","\\begin{conditionaltheorem}[One-step selector metaproductivity certificate]\n\\label{thm:block07-one-step}\nLet $K_0$ be a frozen parent repair selector and let\n\\begin{equation}\n(K_1,\\Pi_1)\n=\n\\mathfrak M(\\mathsf D_{\\rm dev},\\mathsf D_{\\rm meta},K_0,\\Gamma)\n\\end{equation}\nbe a child and valid descendant receipt. Assume:\n\\begin{enumerate}\n\\item the compiler satisfies the sealing conditions of \\cref{thm:block07-compilation-noninterference};\n\\item parent and child receive pathwise identical candidate bundles and evidence under \\cref{thm:block07-selector-isolation};\n\\item the final target episodes are sampled from the declared held-out distribution and the paired interval procedure has its declared coverage;\n\\item the lower confidence bound for child-minus-parent protected utility is positive;\n\\item all declar","\\begin{conditionaltheorem}[One-step selector metaproductivity certificate]\n\\label{thm:block07-one-step}\nLet $K_0$ be a frozen parent repair selector and let\n\\begin{equation}\n(K_1,\\Pi_1)\n=\n\\mathfrak M(\\mathsf D_{\\rm dev},\\mathsf D_{\\rm meta},K_0,\\Gamma)\n\\end{equation}\nbe a child and valid descendant receipt. Assume:\n\\begin{enumerate}\n\\item the compiler satisfies the sealing conditions of \\cref{thm:block07-compilation-noninterference};\n\\item parent and child receive pathwise identical candidate bundles and evidence under \\cref{thm:block07-selector-isolation};\n\\item the final target episodes are sampled from the declared held-out distribution and the paired interval procedure has its declared coverage;\n\\item the lower confidence bound for child-minus-parent protected utility is positive;\n\\item all declared average, tail, and maximum risk gates hold;\n\\item the shuffled-lineage control is lower and selector ablation restores the parent;\n\\item e","ared coverage;\n\\item the lower confidence bound for child-minus-parent protected utility is positive;\n\\item all declared average, tail, and maximum risk gates hold;\n\\item the shuffled-lineage control is lower and selector ablation restores the parent;\n\\item evaluator semantics, provenance, candidate budget, and audit evidence are fixed across the comparison.\n\\end{enumerate}\nThen 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.\n\\end{conditionaltheorem}","\\begin{proof}\nBy 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.\n\\end{p","\\begin{warning}[One edge is not recurrent closure]\nThe theorem certifies neither\n\\begin{equation}\nK_1\\longrightarrow K_2\n\\end{equation}\nnor 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.\n\\end{warning}","The child adds one lineage-generated operator:\n\\begin{equation}\n\\Theta_{\\rm select}:\n\\left\\{\n\\text{typed candidate receipts}\n\\right\\}\n\\longrightarrow\n\\left\\{\n\\text{risk-filtered repair choice}\n\\right\\}.\n\\end{equation}\nThis 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.","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 $0.083333$.","chanism 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 $0.083333$.","ed. 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 $0.083333$.","s 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 $0.083333$.","iple 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 $0.083333$.","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 $0.083333$.","esponse 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 $0.083333$.","nherited $39$-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 $0.083333$.","d $39$-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 $0.083333$.","\\begin{definition}[Finite repair process]\nA finite repair process is a tuple\n\\begin{equation}\n\\mathfrak P=(\\pi_{\\rm anc},\\mathsf B,\\pi_{\\rm gate},B,\\Pi),\n\\end{equation}\nwhere $\\pi_{\\rm anc}$ 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.\n\\end{definition}","For a foreign target system $A$ and protected evaluation context $\\Xi$, let\n\\begin{equation}\nG_\\Xi(\\mathfrak P,A)\n=\nJ_\\Xi\\bigl(\\mathfrak P[A]\\bigr)-J_\\Xi(A)\n\\end{equation}\nbe 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.","\\begin{definition}[Matched-budget foreign metaproductivity]\nFor a foreign target distribution $\\Dcal_{\\rm for}$ and budget $B$, define\n\\begin{equation}\nM_B^{\\rm for}(\\mathfrak P)\n=\n\\mathbb E_{A\\sim\\Dcal_{\\rm for}}\n\\left[G_\\Xi(\\mathfrak P,A)\\right],\n\\end{equation}\nprovided every compared process receives the same target correspondence, calibration, candidate, and irreversible-effect budgets.\n\\end{definition}","The protected utility used in Block 7 is the balanced accuracy functional\n\\begin{equation}\nJ_\\Xi(A)\n=\n\\frac12 A_{\\rm aff}(A)+\\frac12 A_{\\rm non}(A),\n\\label{eq:block07q-balanced-utility}\n\\end{equation}\nwhere $A_{\\rm aff}$ 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.","\\begin{definition}[One-step bounded recursive meta-improvement]\nLet $\\mathfrak P_0$ be a parent repair process. Let a lineage compiler $\\Meta$ receive only a development ledger $L_{\\rm dev}$ and return\n\\begin{equation}\n\\mathfrak P_1=\\Meta(\\mathfrak P_0,L_{\\rm dev}).\n\\end{equation}\nThe edge $\\mathfrak P_0\\to\\mathfrak P_1$ is a one-step bounded recursive meta-improvement on $(\\Dcal_{\\rm for},B)$ when:\n\\begin{enumerate}\n\\item the descendant is sealed before the audit distribution is revealed;\n\\item $\\mathfrak P_0$ and $\\mathfrak P_1$ receive the same deployment budget $B$;\n\\item $M_B^{\\rm for}(\\mathfrak P_1)>M_B^{\\rm for}(\\mathfrak P_0)$ with a declared uncertainty certificate;\n\\item the descendant preserves every protected invariant and has no larger accepted-harm rate;\n\\it","\\begin{definition}[One-step bounded recursive meta-improvement]\nLet $\\mathfrak P_0$ be a parent repair process. Let a lineage compiler $\\Meta$ receive only a development ledger $L_{\\rm dev}$ and return\n\\begin{equation}\n\\mathfrak P_1=\\Meta(\\mathfrak P_0,L_{\\rm dev}).\n\\end{equation}\nThe edge $\\mathfrak P_0\\to\\mathfrak P_1$ is a one-step bounded recursive meta-improvement on $(\\Dcal_{\\rm for},B)$ when:\n\\begin{enumerate}\n\\item the descendant is sealed before the audit distribution is revealed;\n\\item $\\mathfrak P_0$ and $\\mathfrak P_1$ receive the same deployment budget $B$;\n\\item $M_B^{\\rm for}(\\mathfrak P_1)>M_B^{\\rm for}(\\mathfrak P_0)$ with a declared uncertainty certificate;\n\\item the descendant preserves every protected invariant and has no larger accepted-harm rate;\n\\item a structural repair-capacity measure also improves on th","\\begin{definition}[One-step bounded recursive meta-improvement]\nLet $\\mathfrak P_0$ be a parent repair process. Let a lineage compiler $\\Meta$ receive only a development ledger $L_{\\rm dev}$ and return\n\\begin{equation}\n\\mathfrak P_1=\\Meta(\\mathfrak P_0,L_{\\rm dev}).\n\\end{equation}\nThe edge $\\mathfrak P_0\\to\\mathfrak P_1$ is a one-step bounded recursive meta-improvement on $(\\Dcal_{\\rm for},B)$ when:\n\\begin{enumerate}\n\\item the descendant is sealed before the audit distribution is revealed;\n\\item $\\mathfrak P_0$ and $\\mathfrak P_1$ receive the same deployment budget $B$;\n\\item $M_B^{\\rm for}(\\mathfrak P_1)>M_B^{\\rm for}(\\mathfrak P_0)$ with a declared uncertainty certificate;\n\\item the descendant preserves every protected invariant and has no larger accepted-harm rate;\n\\item a structural repair-capacity measure also improves on the declared foreign family.\n\\end{enumerate}\n\\end{","\\begin{definition}[One-step bounded recursive meta-improvement]\nLet $\\mathfrak P_0$ be a parent repair process. Let a lineage compiler $\\Meta$ receive only a development ledger $L_{\\rm dev}$ and return\n\\begin{equation}\n\\mathfrak P_1=\\Meta(\\mathfrak P_0,L_{\\rm dev}).\n\\end{equation}\nThe edge $\\mathfrak P_0\\to\\mathfrak P_1$ is a one-step bounded recursive meta-improvement on $(\\Dcal_{\\rm for},B)$ when:\n\\begin{enumerate}\n\\item the descendant is sealed before the audit distribution is revealed;\n\\item $\\mathfrak P_0$ and $\\mathfrak P_1$ receive the same deployment budget $B$;\n\\item $M_B^{\\rm for}(\\mathfrak P_1)>M_B^{\\rm for}(\\mathfrak P_0)$ with a declared uncertainty certificate;\n\\item the descendant preserves every protected invariant and has no larger accepted-harm rate;\n\\item a structural repair-capacity measure also improves on the declared foreign family.\n\\end{enumerate}\n\\end{definition}","process. Let a lineage compiler $\\Meta$ receive only a development ledger $L_{\\rm dev}$ and return\n\\begin{equation}\n\\mathfrak P_1=\\Meta(\\mathfrak P_0,L_{\\rm dev}).\n\\end{equation}\nThe edge $\\mathfrak P_0\\to\\mathfrak P_1$ is a one-step bounded recursive meta-improvement on $(\\Dcal_{\\rm for},B)$ when:\n\\begin{enumerate}\n\\item the descendant is sealed before the audit distribution is revealed;\n\\item $\\mathfrak P_0$ and $\\mathfrak P_1$ receive the same deployment budget $B$;\n\\item $M_B^{\\rm for}(\\mathfrak P_1)>M_B^{\\rm for}(\\mathfrak P_0)$ with a declared uncertainty certificate;\n\\item the descendant preserves every protected invariant and has no larger accepted-harm rate;\n\\item a structural repair-capacity measure also improves on the declared foreign family.\n\\end{enumerate}\n\\end{definition}","age compiler $\\Meta$ receive only a development ledger $L_{\\rm dev}$ and return\n\\begin{equation}\n\\mathfrak P_1=\\Meta(\\mathfrak P_0,L_{\\rm dev}).\n\\end{equation}\nThe edge $\\mathfrak P_0\\to\\mathfrak P_1$ is a one-step bounded recursive meta-improvement on $(\\Dcal_{\\rm for},B)$ when:\n\\begin{enumerate}\n\\item the descendant is sealed before the audit distribution is revealed;\n\\item $\\mathfrak P_0$ and $\\mathfrak P_1$ receive the same deployment budget $B$;\n\\item $M_B^{\\rm for}(\\mathfrak P_1)>M_B^{\\rm for}(\\mathfrak P_0)$ with a declared uncertainty certificate;\n\\item the descendant preserves every protected invariant and has no larger accepted-harm rate;\n\\item a structural repair-capacity measure also improves on the declared foreign family.\n\\end{enumerate}\n\\end{definition}","ger $L_{\\rm dev}$ and return\n\\begin{equation}\n\\mathfrak P_1=\\Meta(\\mathfrak P_0,L_{\\rm dev}).\n\\end{equation}\nThe edge $\\mathfrak P_0\\to\\mathfrak P_1$ is a one-step bounded recursive meta-improvement on $(\\Dcal_{\\rm for},B)$ when:\n\\begin{enumerate}\n\\item the descendant is sealed before the audit distribution is revealed;\n\\item $\\mathfrak P_0$ and $\\mathfrak P_1$ receive the same deployment budget $B$;\n\\item $M_B^{\\rm for}(\\mathfrak P_1)>M_B^{\\rm for}(\\mathfrak P_0)$ with a declared uncertainty certificate;\n\\item the descendant preserves every protected invariant and has no larger accepted-harm rate;\n\\item a structural repair-capacity measure also improves on the declared foreign family.\n\\end{enumerate}\n\\end{definition}","dev}$ and return\n\\begin{equation}\n\\mathfrak P_1=\\Meta(\\mathfrak P_0,L_{\\rm dev}).\n\\end{equation}\nThe edge $\\mathfrak P_0\\to\\mathfrak P_1$ is a one-step bounded recursive meta-improvement on $(\\Dcal_{\\rm for},B)$ when:\n\\begin{enumerate}\n\\item the descendant is sealed before the audit distribution is revealed;\n\\item $\\mathfrak P_0$ and $\\mathfrak P_1$ receive the same deployment budget $B$;\n\\item $M_B^{\\rm for}(\\mathfrak P_1)>M_B^{\\rm for}(\\mathfrak P_0)$ with a declared uncertainty certificate;\n\\item the descendant preserves every protected invariant and has no larger accepted-harm rate;\n\\item a structural repair-capacity measure also improves on the declared foreign family.\n\\end{enumerate}\n\\end{definition}","Let\n\\begin{equation}\n\\Omega\n=\n\\Omega_{\\rm dev}\\sqcup\\Omega_{\\rm audit}\n\\end{equation}\nbe 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\n\\begin{equation}\n\\Sigma_{\\rm pre}\n=\nH\\bigl(c_1,L_{\\Meta},\\text{source hashes}\\bigr),\n\\label{eq:block07q-seal}\n\\end{equation}\nwhere $H$ is SHA-256 over canonical serialization. Audit execution refuses to proceed when the in-memory descendant differs from the sealed configuration.","in{equation}\n\\Omega\n=\n\\Omega_{\\rm dev}\\sqcup\\Omega_{\\rm audit}\n\\end{equation}\nbe 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\n\\begin{equation}\n\\Sigma_{\\rm pre}\n=\nH\\bigl(c_1,L_{\\Meta},\\text{source hashes}\\bigr),\n\\label{eq:block07q-seal}\n\\end{equation}\nwhere $H$ is SHA-256 over canonical serialization. Audit execution refuses to proceed when the in-memory descendant differs from the sealed configuration.","\\begin{proposition}[Trace-level audit nonconditioning]\n\\label{prop:block07q-seal}\nSuppose 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.\n\\end{proposition}","The Block 7 primary audit differs from development in both task and representation family:\n\\begin{equation}\n\\begin{aligned}\n\\mathsf T_{\\rm dev}&\\cap\\mathsf T_{\\rm audit}=\\varnothing,\\\\\n\\mathsf F_{\\rm dev}^{\\rm primary}&\\cap\\mathsf F_{\\rm audit}^{\\rm primary}=\\varnothing.\n\\end{aligned}\n\\end{equation}\nDevelopment 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.","\\mathsf F_{\\rm dev}^{\\rm primary}&\\cap\\mathsf F_{\\rm audit}^{\\rm primary}=\\varnothing.\n\\end{aligned}\n\\end{equation}\nDevelopment 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.","y}&\\cap\\mathsf F_{\\rm audit}^{\\rm primary}=\\varnothing.\n\\end{aligned}\n\\end{equation}\nDevelopment 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.","The parent and descendant each receive:\n\\begin{equation}\nB=(16\\text{ fit anchors},8\\text{ validation anchors},1\\text{ repair candidate}).\n\\label{eq:block07q-budget}\n\\end{equation}\nThey use the same source repair, target calibration set, protected audit set, and bridge estimator. Only anchor selection and the gate differ.","The Block 6 bridge estimator receives paired activations\n\\begin{equation}\nX=\n\\begin{bmatrix}\n h_s(x_1)^\\top\\\\\n \\vdots\\\\\n h_s(x_k)^\\top\n\\end{bmatrix},\n\\qquad\nY=\n\\begin{bmatrix}\n h_t(x_1)^\\top\\\\\n \\vdots\\\\\n h_t(x_k)^\\top\n\\end{bmatrix},\n\\end{equation}\nand fits\n\\begin{equation}\n\\widehat T^\\top\n=\n(X^\\top X+\\lambda I)^{-1}X^\\top Y.\n\\label{eq:block07q-ridge-bridge}\n\\end{equation}\nThe 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.","h_t(x_k)^\\top\n\\end{bmatrix},\n\\end{equation}\nand fits\n\\begin{equation}\n\\widehat T^\\top\n=\n(X^\\top X+\\lambda I)^{-1}X^\\top Y.\n\\label{eq:block07q-ridge-bridge}\n\\end{equation}\nThe 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.","The task representation has the conditional form\n\\begin{equation}\n\\phi(x)\n=\n\\begin{bmatrix}\n\\psi(x)\\\\\ns(x)\\psi(x)\n\\end{bmatrix}\n\\in\\R^{16},\n\\end{equation}\nwhere 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.","\\begin{theorem}[Full-rank bridge identification]\n\\label{thm:block07q-full-rank-bridge}\nLet the target representation satisfy\n\\begin{equation}\nh_t(x)=T h_s(x)\n\\end{equation}\nfor 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\n\\begin{equation}\nXT_1^\\top=XT_2^\\top=Y.\n\\end{equation}\nConsequently, repair transport is not identifiable on every unobserved source direction.\n\\end{theorem}","egin{theorem}[Full-rank bridge identification]\n\\label{thm:block07q-full-rank-bridge}\nLet the target representation satisfy\n\\begin{equation}\nh_t(x)=T h_s(x)\n\\end{equation}\nfor 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\n\\begin{equation}\nXT_1^\\top=XT_2^\\top=Y.\n\\end{equation}\nConsequently, repair transport is not identifiable on every unobserved source direction.\n\\end{theorem}","cation]\n\\label{thm:block07q-full-rank-bridge}\nLet the target representation satisfy\n\\begin{equation}\nh_t(x)=T h_s(x)\n\\end{equation}\nfor 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\n\\begin{equation}\nXT_1^\\top=XT_2^\\top=Y.\n\\end{equation}\nConsequently, repair transport is not identifiable on every unobserved source direction.\n\\end{theorem}","ridge}\nLet the target representation satisfy\n\\begin{equation}\nh_t(x)=T h_s(x)\n\\end{equation}\nfor 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\n\\begin{equation}\nXT_1^\\top=XT_2^\\top=Y.\n\\end{equation}\nConsequently, repair transport is not identifiable on every unobserved source direction.\n\\end{theorem}","\\begin{proof}\nWhen $X$ has full column rank, $X^\\top X$ is invertible and\n\\begin{equation}\n(X^\\top X)^{-1}X^\\top Y\n=(X^\\top X)^{-1}X^\\top XT^\\top\n=T^\\top.\n\\end{equation}\nWhen $\\rank X<d$, 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 $v$.\n\\end{proof}","\\begin{proposition}[Ridge bias under a full-rank design]\n\\label{prop:block07q-ridge-bias}\nUnder the hypotheses of \\cref{thm:block07q-full-rank-bridge}, the ridge estimate satisfies\n\\begin{equation}\n\\widehat T^\\top-T^\\top\n=-\\lambda(X^\\top X+\\lambda I)^{-1}T^\\top,\n\\end{equation}\nand therefore\n\\begin{equation}\n\\norm{\\widehat T-T}\n\\leq\n\\frac{\\lambda}{\\sigma_{\\min}(X)^2+\\lambda}\\norm{T}.\n\\end{equation}\n\\end{proposition}","Write the source state as\n\\begin{equation}\nh_s(x)=A\n\\begin{bmatrix}\nu(x)\\\\s(x)v(x)\n\\end{bmatrix},\n\\end{equation}\nwhere $u,v\\in\\R^8$ and $A$ is invertible. Let $N_1$ be the number of affected anchors among $k$ selections.","s(x)v(x)\n\\end{bmatrix},\n\\end{equation}\nwhere $u,v\\in\\R^8$ and $A$ is invertible. Let $N_1$ be the number of affected anchors among $k$ selections.","\\begin{proposition}[Affected-anchor necessity]\n\\label{prop:block07q-affected-count}\nIf $N_1<8$, 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.\n\\end{proposition}","\\begin{proof}\nOnly 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\n\\begin{equation}\n\\rank X\\leq 8+N_1<16\n\\end{equation}\nwhen $N_1<8$. 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.\n\\end{proof}","ry 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\n\\begin{equation}\n\\rank X\\leq 8+N_1<16\n\\end{equation}\nwhen $N_1<8$. 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.\n\\end{proof}","If random anchor membership is Bernoulli with affected probability $p$, the idealized full-rank probability at $k=16$ is therefore\n\\begin{equation}\n\\Pr(\\rank X=16)\n=\n\\Pr(N_1\\geq8)\n=\n\\sum_{j=8}^{16}{16\\choose j}p^j(1-p)^{16-j}\n\\label{eq:block07q-random-rank-probability}\n\\end{equation}\nunder 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.","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:\n\\begin{equation}\n i_j\n\\in\n\\argmax_{i\\notin I_{j-1}}\n\\norm{(I-Q_{j-1}Q_{j-1}^\\top)z_i}^2,\n\\end{equation}\nwhere $z_i$ is the standardized source activation and $Q_{j-1}$ spans the previously selected residual directions.","ce 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:\n\\begin{equation}\n i_j\n\\in\n\\argmax_{i\\notin I_{j-1}}\n\\norm{(I-Q_{j-1}Q_{j-1}^\\top)z_i}^2,\n\\end{equation}\nwhere $z_i$ is the standardized source activation and $Q_{j-1}$ spans the previously selected residual directions.","\\begin{definition}[Packet-balanced QR policy]\nFor fit budget $k=16$, define\n\\begin{equation}\n\\pi_{\\rm PQR}(H,P_J)\n=\n\\pi_{\\rm QR}(H_{\\rm aff},8)\n\\cup\n\\pi_{\\rm QR}(H_{\\rm non},8),\n\\end{equation}\nwhere affected and unaffected membership is inferred from the source causal intervention, not from target audit labels.\n\\end{definition}","\\begin{corollary}[Generic full-rank packet-balanced design]\n\\label{cor:block07q-pqr-full-rank}\nUnder 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.\n\\end{corollary}","The development meta-grammar contains five anchor policies:\n\\begin{equation}\n\\Pi_{\\rm anc}\n=\n\\set{\n\\text{random},\n\\text{balanced random},\n\\text{QR},\n\\text{packet QR},\n\\text{augmented QR}\n}.\n\\end{equation}\nIt also contains alignment thresholds\n\\begin{equation}\n\\tau\\in\\set{0.01,0.02,0.05,0.10,0.20,0.35}\n\\end{equation}\nand an optional target-width guard. Every candidate uses the same fit and validation budgets.","For a candidate configuration $c$, the development compiler computes target-level selected gains and the score\n\\begin{equation}\nS_{\\rm dev}(c)\n=\n\\bar g_c\n-0.20s_c\n-2h_c\n+0.02q_c,\n\\label{eq:block07q-compiler-score}\n\\end{equation}\nwhere $\\bar g_c$ 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.","The compiler selected:\n\\begin{equation}\n\\begin{aligned}\n\\pi_{\\rm anc}&=\\pi_{\\rm PQR},\\\\\nk_{\\rm fit}&=16,\\\\\nk_{\\rm val}&=8,\\\\\n\\gamma_{\\rm cal}&=0.08,\\\\\n\\rho_{\\rm cal}&=0.035,\\\\\n\\tau_{\\rm align}&=0.05,\\\\\n\\text{full-rank requirement}&=\\text{true}.\n\\end{aligned}\n\\label{eq:block07q-descendant-config}\n\\end{equation}\nThe selected development score is $0.263289$. The underlying mean selected development gain is $0.296478$ with zero harmful targets in the development ledger.","\\text{full-rank requirement}&=\\text{true}.\n\\end{aligned}\n\\label{eq:block07q-descendant-config}\n\\end{equation}\nThe selected development score is $0.263289$. The underlying mean selected development gain is $0.296478$ with zero harmful targets in the development ledger.","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.","Each task contains:\n\\begin{itemize}\n\\item $3{,}000$ source-training examples;\n\\item $800$ source causal-probe examples;\n\\item a source-side anchor pool of $1{,}100$ examples;\n\\item $650$ calibration examples;\n\\item $1{,}800$ audit examples.\n\\end{itemize}\nThe 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.","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.","The primary audit families are:\n\\begin{equation}\n\\mathsf F_{\\rm audit}^{\\rm primary}\n=\n\\set{\n\\text{dense-linear-20},\n\\text{block-linear-24},\n\\text{ill-conditioned-linear-20}\n}.\n\\end{equation}\nThe stress families are:\n\\begin{equation}\n\\mathsf F_{\\rm audit}^{\\rm stress}\n=\n\\set{\n\\text{compressed-10},\n\\text{saturated-tanh-24},\n\\text{leaky-ReLU-32},\n\\text{smooth-polynomial-24}\n}.\n\\end{equation}\nThe primary families satisfy the linear bridge hypothesis but are absent from primary development. The stress families violate injectivity or global linearity in different ways.","\\midrule\n\\endhead\nParent random 16 & $16+8=24$ & Parent process inherited from the Block 6 anchor protocol","Balanced random 16 & $16+8=24$ & Tests whether stratum balance alone explains the gain","QR 16 & $16+8=24$ & Tests generic deterministic experimental design without explicit packet balance","Descendant packet-QR 16 & $16+8=24$ & Sealed lineage-generated descendant","Parent random 32 & $32+8=40$ & Double-fit-budget parent control","Lineage-permuted descendant & $16+8=24$ & Breaks causal receipt binding during meta-compilation","On $144$ primary foreign targets, the target-level paired difference is\n\\begin{equation}\n\\widehat\\Delta_{\\rm meta}\n=\n\\widehat M_{24}^{\\rm for}(\\mathfrak P_1)\n-\n\\widehat M_{24}^{\\rm for}(\\mathfrak P_0)\n=\n0.209508.\n\\label{eq:block07q-primary-delta}\n\\end{equation}\nA target-cluster bootstrap gives the interval\n\\begin{equation}\n[0.194968,0.223898],\n\\end{equation}\nand a $100{,}000$-draw sign-flip test gives\n\\begin{equation}\np=9.9999\\times10^{-6}.\n\\end{equation}\nAll $144$ target-level paired differences are positive.","ign targets, the target-level paired difference is\n\\begin{equation}\n\\widehat\\Delta_{\\rm meta}\n=\n\\widehat M_{24}^{\\rm for}(\\mathfrak P_1)\n-\n\\widehat M_{24}^{\\rm for}(\\mathfrak P_0)\n=\n0.209508.\n\\label{eq:block07q-primary-delta}\n\\end{equation}\nA target-cluster bootstrap gives the interval\n\\begin{equation}\n[0.194968,0.223898],\n\\end{equation}\nand a $100{,}000$-draw sign-flip test gives\n\\begin{equation}\np=9.9999\\times10^{-6}.\n\\end{equation}\nAll $144$ target-level paired differences are positive.","nce is\n\\begin{equation}\n\\widehat\\Delta_{\\rm meta}\n=\n\\widehat M_{24}^{\\rm for}(\\mathfrak P_1)\n-\n\\widehat M_{24}^{\\rm for}(\\mathfrak P_0)\n=\n0.209508.\n\\label{eq:block07q-primary-delta}\n\\end{equation}\nA target-cluster bootstrap gives the interval\n\\begin{equation}\n[0.194968,0.223898],\n\\end{equation}\nand a $100{,}000$-draw sign-flip test gives\n\\begin{equation}\np=9.9999\\times10^{-6}.\n\\end{equation}\nAll $144$ target-level paired differences are positive.","\\midrule\nParent random 16 & $0.135212$ & $0.465278$ & $0.03125$ & $24$","Balanced random 16 & $0.318744$ & $0.932292$ & $1.00000$ & $24$","QR 16 & $0.344731$ & $1.000000$ & $1.00000$ & $24$","Descendant packet-QR 16 & $\\mathbf{0.344720}$ & $\\mathbf{1.000000}$ & $\\mathbf{1.00000}$ & $24$","Parent random 32 & $0.316524$ & $0.932292$ & $0.822917$ & $40$","Lineage-permuted descendant & $0.008784$ & $0.026042$ & $0.03125$ & $24$","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.","\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.92\\textwidth]{block07_metaproductivity_by_arm.pdf}\n\\caption{Mean protected gain on the primary foreign target set. Parent and descendant use the same $24$ paired queries. The double-budget parent uses $40$.}\n\\label{fig:block07q-metaproductivity-arms}\n\\end{figure}","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 $100\\%$.","Define the finite structural coverage functional\n\\begin{equation}\nC_{\\rm id}(\\mathfrak P)\n=\n\\Pr_{A\\sim\\Dcal_{\\rm for}}\n\\left[\nank X_A=d_s\\right].\n\\end{equation}\nThen\n\\begin{equation}\n\\widehat C_{\\rm id}(\\mathfrak P_1)\n-\n\\widehat C_{\\rm id}(\\mathfrak P_0)\n=\n0.96875.\n\\end{equation}\nThis is the measured bridge-identification direction added by the meta-repair.","\\midrule\nDense linear 20 & $0.344887$ & $1.000000$ & $4.10\\times10^{-4}$ & $0$","Block linear 24 & $0.344887$ & $1.000000$ & $2.91\\times10^{-4}$ & $0$","Ill-conditioned linear 20 & $0.344387$ & $1.000000$ & $3.04\\times10^{-5}$ & $0$","Compressed 10 & $0.038278$ & $0.270833$ & $2.87\\times10^{-4}$ & $0$","Saturated tanh 24 & $0$ & $0$ & $2.260422$ & $0$","Leaky-ReLU 32 & $0$ & $0$ & $4.252823$ & $0$","Smooth polynomial 24 & $0$ & $0$ & $4.652148$ & $0$","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.","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\n\\begin{equation}\n\\frac{72{,}000}{16}=4{,}500\n\\end{equation}\ndeployments. This number is reported because recursive improvement is not useful merely because a descendant is better. Its construction cost must eventually be recovered.","Pure QR obtains mean primary gain $0.344731$, numerically indistinguishable from packet-balanced QR at $0.344720$. Balanced random selection reaches $0.318744$.","The parent and descendant differ in exactly two declared fields:\n\\begin{equation}\n\\begin{aligned}\n\\pi_{\\rm anc}^{(0)}&=\\text{uniform random},\n&\\pi_{\\rm anc}^{(1)}&=\\text{packet-balanced QR},\\\\\n\\pi_{\\rm gate}^{(0)}&=(0.20,0.03,0.35),\n&\\pi_{\\rm gate}^{(1)}&=(0.08,0.035,0.05,\\text{full rank}).\n\\end{aligned}\n\\end{equation}\nThe source task model, causal packet discovery, source repair, linear bridge family, pseudoinverse repair transport, target calibration set, protected utility, and audit data remain fixed.","The measured causal chain is therefore:\n\\begin{equation}\n\\begin{aligned}\n&\\text{development failure ledger}\\\\\n&\\quad\\longrightarrow\n\\text{compiled anchor and gate policy}\\\\\n&\\quad\\longrightarrow\n\\text{full-rank source design}\\\\\n&\\quad\\longrightarrow\n\\text{identified linear bridge}\\\\\n&\\quad\\longrightarrow\n\\text{successful transported repair}\\\\\n&\\quad\\longrightarrow\n\\text{higher foreign-agent metaproductivity}.\n\\end{aligned}\n\\end{equation}\nThe lineage-permutation control breaks the first causal binding and collapses mean gain to $0.004026$ over the complete audit set.","&\\quad\\longrightarrow\n\\text{higher foreign-agent metaproductivity}.\n\\end{aligned}\n\\end{equation}\nThe lineage-permutation control breaks the first causal binding and collapses mean gain to $0.004026$ over the complete audit set.","\\item parent and descendant receive the matched budget in \\eqref{eq:block07q-budget};\n\\item target-level audit differences are exchangeable or otherwise support the declared paired interval;\n\\item the protected evaluator and source repairs are unchanged across the parent-child edge;\n\\item accepted-harm rate is not increased.\n\\end{enumerate}\nIf the paired lower confidence bound for\n\\begin{equation}\nM_{24}^{\\rm for}(\\mathfrak P_1)-M_{24}^{\\rm for}(\\mathfrak P_0)\n\\end{equation}\nis 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.\n\\end{conditionaltheorem}","rt the declared paired interval;\n\\item the protected evaluator and source repairs are unchanged across the parent-child edge;\n\\item accepted-harm rate is not increased.\n\\end{enumerate}\nIf the paired lower confidence bound for\n\\begin{equation}\nM_{24}^{\\rm for}(\\mathfrak P_1)-M_{24}^{\\rm for}(\\mathfrak P_0)\n\\end{equation}\nis 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.\n\\end{conditionaltheorem}","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 $1$.","\\begin{statusbox}{Block 7 result}\nBlock 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.\n\\end{statusbox}","tatusbox}{Block 7 result}\nBlock 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.\n\\end{statusbox}","result}\nBlock 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.\n\\end{statusbox}","es 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.\n\\end{statusbox}","provement. 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.\n\\end{statusbox}","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.\n\\end{statusbox}","All eight automated tests pass. The package manifest covers $51$ files.","Matched deployment budget & Pass & Parent and descendant each use $24$ paired queries","Structural correctability & Pass & Descendant primary full-rank rate at least $0.99$","Let\n\\begin{equation}\nN\n=\n\\{H,Q,C,T,S,V,W\\}\n\\label{eq:b8-module-index}\n\\end{equation}\nbe the repair-process module set, where:\n\\begin{enumerate}\n\\item $H$ is the causal hypothesis and atlas mechanism;\n\\item $Q$ is the active probe or evidence-acquisition policy;\n\\item $C$ is the candidate repair constructor;\n\\item $T$ is the representation and repair transport mechanism;\n\\item $S$ is the candidate selector;\n\\item $V$ is the evaluator;\n\\item $W$ is the witness, authorization, and rollback mechanism.\n\\end{enumerate}\nA repair process is a typed tuple\n\\begin{equation}\nK=(H,Q,C,T,S,V,W).\n\\label{eq:b8-process-tuple}\n\\end{equation}\nThe factorization is relative to a declared implementation boundary. A system may refine one module into several submodules or combine sev","Let\n\\begin{equation}\nN\n=\n\\{H,Q,C,T,S,V,W\\}\n\\label{eq:b8-module-index}\n\\end{equation}\nbe the repair-process module set, where:\n\\begin{enumerate}\n\\item $H$ is the causal hypothesis and atlas mechanism;\n\\item $Q$ is the active probe or evidence-acquisition policy;\n\\item $C$ is the candidate repair constructor;\n\\item $T$ is the representation and repair transport mechanism;\n\\item $S$ is the candidate selector;\n\\item $V$ is the evaluator;\n\\item $W$ is the witness, authorization, and rollback mechanism.\n\\end{enumerate}\nA repair process is a typed tuple\n\\begin{equation}\nK=(H,Q,C,T,S,V,W).\n\\label{eq:b8-process-tuple}\n\\end{equation}\nThe 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","Let\n\\begin{equation}\nN\n=\n\\{H,Q,C,T,S,V,W\\}\n\\label{eq:b8-module-index}\n\\end{equation}\nbe the repair-process module set, where:\n\\begin{enumerate}\n\\item $H$ is the causal hypothesis and atlas mechanism;\n\\item $Q$ is the active probe or evidence-acquisition policy;\n\\item $C$ is the candidate repair constructor;\n\\item $T$ is the representation and repair transport mechanism;\n\\item $S$ is the candidate selector;\n\\item $V$ is the evaluator;\n\\item $W$ is the witness, authorization, and rollback mechanism.\n\\end{enumerate}\nA repair process is a typed tuple\n\\begin{equation}\nK=(H,Q,C,T,S,V,W).\n\\label{eq:b8-process-tuple}\n\\end{equation}\nThe 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.","on}\nN\n=\n\\{H,Q,C,T,S,V,W\\}\n\\label{eq:b8-module-index}\n\\end{equation}\nbe the repair-process module set, where:\n\\begin{enumerate}\n\\item $H$ is the causal hypothesis and atlas mechanism;\n\\item $Q$ is the active probe or evidence-acquisition policy;\n\\item $C$ is the candidate repair constructor;\n\\item $T$ is the representation and repair transport mechanism;\n\\item $S$ is the candidate selector;\n\\item $V$ is the evaluator;\n\\item $W$ is the witness, authorization, and rollback mechanism.\n\\end{enumerate}\nA repair process is a typed tuple\n\\begin{equation}\nK=(H,Q,C,T,S,V,W).\n\\label{eq:b8-process-tuple}\n\\end{equation}\nThe 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.","abel{eq:b8-module-index}\n\\end{equation}\nbe the repair-process module set, where:\n\\begin{enumerate}\n\\item $H$ is the causal hypothesis and atlas mechanism;\n\\item $Q$ is the active probe or evidence-acquisition policy;\n\\item $C$ is the candidate repair constructor;\n\\item $T$ is the representation and repair transport mechanism;\n\\item $S$ is the candidate selector;\n\\item $V$ is the evaluator;\n\\item $W$ is the witness, authorization, and rollback mechanism.\n\\end{enumerate}\nA repair process is a typed tuple\n\\begin{equation}\nK=(H,Q,C,T,S,V,W).\n\\label{eq:b8-process-tuple}\n\\end{equation}\nThe 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.","ypothesis and atlas mechanism;\n\\item $Q$ is the active probe or evidence-acquisition policy;\n\\item $C$ is the candidate repair constructor;\n\\item $T$ is the representation and repair transport mechanism;\n\\item $S$ is the candidate selector;\n\\item $V$ is the evaluator;\n\\item $W$ is the witness, authorization, and rollback mechanism.\n\\end{enumerate}\nA repair process is a typed tuple\n\\begin{equation}\nK=(H,Q,C,T,S,V,W).\n\\label{eq:b8-process-tuple}\n\\end{equation}\nThe 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.","Fix a protected evaluation context $\\Xi$, a foreign-system distribution $\\Dcal$, and a matched resource budget $B$. Define protected foreign metaproductivity by\n\\begin{equation}\n\\Mcal_{\\Xi,\\Dcal,B}(K)\n=\n\\mathbb E_{A\\sim\\Dcal}\n\\left[\nJ_\\Xi\\bigl(K[A;B]\\bigr)-J_\\Xi(A)\n\\right],\n\\label{eq:b8-metaproductivity}\n\\end{equation}\nwhere $K[A;B]$ is the authorized descendant selected by $K$ for target $A$, or $A$ itself when $K$ abstains.","ix a protected evaluation context $\\Xi$, a foreign-system distribution $\\Dcal$, and a matched resource budget $B$. Define protected foreign metaproductivity by\n\\begin{equation}\n\\Mcal_{\\Xi,\\Dcal,B}(K)\n=\n\\mathbb E_{A\\sim\\Dcal}\n\\left[\nJ_\\Xi\\bigl(K[A;B]\\bigr)-J_\\Xi(A)\n\\right],\n\\label{eq:b8-metaproductivity}\n\\end{equation}\nwhere $K[A;B]$ is the authorized descendant selected by $K$ for target $A$, or $A$ itself when $K$ abstains.","valuation context $\\Xi$, a foreign-system distribution $\\Dcal$, and a matched resource budget $B$. Define protected foreign metaproductivity by\n\\begin{equation}\n\\Mcal_{\\Xi,\\Dcal,B}(K)\n=\n\\mathbb E_{A\\sim\\Dcal}\n\\left[\nJ_\\Xi\\bigl(K[A;B]\\bigr)-J_\\Xi(A)\n\\right],\n\\label{eq:b8-metaproductivity}\n\\end{equation}\nwhere $K[A;B]$ is the authorized descendant selected by $K$ for target $A$, or $A$ itself when $K$ abstains.","\\begin{remark}[Module claims are distinct]\nA 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.\n\\end{remark}","Let $\\Kcal$ be the declared finite or compact class of admissible repair processes under the protected context. For $R\\subseteq N$, let\n\\begin{equation}\n\\Kcal_R(K)\n=\n\\left\\{\n\\widetilde K\\in\\Kcal:\n\\widetilde K_i=K_i\n\\text{ for every }i\\notin R\n\\right\\}.\n\\label{eq:b8-restricted-class}\n\\end{equation}\nThe restricted oracle envelope is\n\\begin{equation}\nF_K(R)\n=\n\\sup_{\\widetilde K\\in\\Kcal_R(K)}\n\\Mcal_{\\Xi,\\Dcal,B}(\\widetilde K).\n\\label{eq:b8-oracle-envelope}\n\\end{equation}\nThus\n\\begin{equation}\nF_K(\\varnothing)=\\Mcal(K),\n\\qquad\nF_K(N)=\\sup_{\\widetilde K\\in\\Kcal}\\Mcal(\\widetilde K).\n\\label{eq:b8-envelope-endpoints}\n\\end{equation}\nThe total declared headroom is\n\\begin{equation}\n\\Hcal(K)\n=\nF_K(N)-F_K(\\varnothi","Let $\\Kcal$ be the declared finite or compact class of admissible repair processes under the protected context. For $R\\subseteq N$, let\n\\begin{equation}\n\\Kcal_R(K)\n=\n\\left\\{\n\\widetilde K\\in\\Kcal:\n\\widetilde K_i=K_i\n\\text{ for every }i\\notin R\n\\right\\}.\n\\label{eq:b8-restricted-class}\n\\end{equation}\nThe restricted oracle envelope is\n\\begin{equation}\nF_K(R)\n=\n\\sup_{\\widetilde K\\in\\Kcal_R(K)}\n\\Mcal_{\\Xi,\\Dcal,B}(\\widetilde K).\n\\label{eq:b8-oracle-envelope}\n\\end{equation}\nThus\n\\begin{equation}\nF_K(\\varnothing)=\\Mcal(K),\n\\qquad\nF_K(N)=\\sup_{\\widetilde K\\in\\Kcal}\\Mcal(\\widetilde K).\n\\label{eq:b8-envelope-endpoints}\n\\end{equation}\nThe total declared headroom is\n\\begin{equation}\n\\Hcal(K)\n=\nF_K(N)-F_K(\\varnothing).\n\\label{eq:b8-total-headroom}\n\\end{equation}\nThe adjective \\emph{declared} matters. A larger, previously unavailabl","Let $\\Kcal$ be the declared finite or compact class of admissible repair processes under the protected context. For $R\\subseteq N$, let\n\\begin{equation}\n\\Kcal_R(K)\n=\n\\left\\{\n\\widetilde K\\in\\Kcal:\n\\widetilde K_i=K_i\n\\text{ for every }i\\notin R\n\\right\\}.\n\\label{eq:b8-restricted-class}\n\\end{equation}\nThe restricted oracle envelope is\n\\begin{equation}\nF_K(R)\n=\n\\sup_{\\widetilde K\\in\\Kcal_R(K)}\n\\Mcal_{\\Xi,\\Dcal,B}(\\widetilde K).\n\\label{eq:b8-oracle-envelope}\n\\end{equation}\nThus\n\\begin{equation}\nF_K(\\varnothing)=\\Mcal(K),\n\\qquad\nF_K(N)=\\sup_{\\widetilde K\\in\\Kcal}\\Mcal(\\widetilde K).\n\\label{eq:b8-envelope-endpoints}\n\\end{equation}\nThe total declared headroom is\n\\begin{equation}\n\\Hcal(K)\n=\nF_K(N)-F_K(\\varnothing).\n\\label{eq:b8-total-headroom}\n\\end{equation}\nThe 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 syste","Let $\\Kcal$ be the declared finite or compact class of admissible repair processes under the protected context. For $R\\subseteq N$, let\n\\begin{equation}\n\\Kcal_R(K)\n=\n\\left\\{\n\\widetilde K\\in\\Kcal:\n\\widetilde K_i=K_i\n\\text{ for every }i\\notin R\n\\right\\}.\n\\label{eq:b8-restricted-class}\n\\end{equation}\nThe restricted oracle envelope is\n\\begin{equation}\nF_K(R)\n=\n\\sup_{\\widetilde K\\in\\Kcal_R(K)}\n\\Mcal_{\\Xi,\\Dcal,B}(\\widetilde K).\n\\label{eq:b8-oracle-envelope}\n\\end{equation}\nThus\n\\begin{equation}\nF_K(\\varnothing)=\\Mcal(K),\n\\qquad\nF_K(N)=\\sup_{\\widetilde K\\in\\Kcal}\\Mcal(\\widetilde K).\n\\label{eq:b8-envelope-endpoints}\n\\end{equation}\nThe total declared headroom is\n\\begin{equation}\n\\Hcal(K)\n=\nF_K(N)-F_K(\\varnothing).\n\\label{eq:b8-total-headroom}\n\\end{equation}\nThe 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.","protected context. For $R\\subseteq N$, let\n\\begin{equation}\n\\Kcal_R(K)\n=\n\\left\\{\n\\widetilde K\\in\\Kcal:\n\\widetilde K_i=K_i\n\\text{ for every }i\\notin R\n\\right\\}.\n\\label{eq:b8-restricted-class}\n\\end{equation}\nThe restricted oracle envelope is\n\\begin{equation}\nF_K(R)\n=\n\\sup_{\\widetilde K\\in\\Kcal_R(K)}\n\\Mcal_{\\Xi,\\Dcal,B}(\\widetilde K).\n\\label{eq:b8-oracle-envelope}\n\\end{equation}\nThus\n\\begin{equation}\nF_K(\\varnothing)=\\Mcal(K),\n\\qquad\nF_K(N)=\\sup_{\\widetilde K\\in\\Kcal}\\Mcal(\\widetilde K).\n\\label{eq:b8-envelope-endpoints}\n\\end{equation}\nThe total declared headroom is\n\\begin{equation}\n\\Hcal(K)\n=\nF_K(N)-F_K(\\varnothing).\n\\label{eq:b8-total-headroom}\n\\end{equation}\nThe 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.","lass}\n\\end{equation}\nThe restricted oracle envelope is\n\\begin{equation}\nF_K(R)\n=\n\\sup_{\\widetilde K\\in\\Kcal_R(K)}\n\\Mcal_{\\Xi,\\Dcal,B}(\\widetilde K).\n\\label{eq:b8-oracle-envelope}\n\\end{equation}\nThus\n\\begin{equation}\nF_K(\\varnothing)=\\Mcal(K),\n\\qquad\nF_K(N)=\\sup_{\\widetilde K\\in\\Kcal}\\Mcal(\\widetilde K).\n\\label{eq:b8-envelope-endpoints}\n\\end{equation}\nThe total declared headroom is\n\\begin{equation}\n\\Hcal(K)\n=\nF_K(N)-F_K(\\varnothing).\n\\label{eq:b8-total-headroom}\n\\end{equation}\nThe 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.","\\begin{proposition}[Monotonicity of restricted envelopes]\n\\label{prop:b8-envelope-monotone}\nIf $R\\subseteq R'\\subseteq N$, then\n\\begin{equation}\nF_K(R)\\leq F_K(R').\n\\end{equation}\n\\end{proposition}","\\begin{proof}\nEvery process admissible under $R$ is admissible under $R'$, so\n$\\Kcal_R(K)\\subseteq\\Kcal_{R'}(K)$. Taking suprema gives the result.\n\\end{proof}","Choose an operational ordering\n\\begin{equation}\nr_1\\prec r_2\\prec\\cdots\\prec r_m,\n\\qquad m=|N|,\n\\end{equation}\nwith nested sets\n\\begin{equation}\nR_0=\\varnothing,\n\\qquad\nR_j=\\{r_1,\\ldots,r_j\\}.\n\\end{equation}\nDefine the ordered marginal headroom\n\\begin{equation}\n\\Delta_j^{\\mathrm{ord}}(K)\n=\nF_K(R_j)-F_K(R_{j-1}).\n\\label{eq:b8-ordered-marginal}\n\\end{equation}","\\begin{theorem}[Exact ordered meta-regret decomposition]\n\\label{thm:b8-ordered-decomposition}\nFor every ordering of the repair-process modules,\n\\begin{equation}\n\\Hcal(K)\n=\n\\sum_{j=1}^{m}\n\\Delta_j^{\\mathrm{ord}}(K).\n\\label{eq:b8-ordered-telescope}\n\\end{equation}\nEvery summand is nonnegative.\n\\end{theorem}","\\begin{proof}\nThe sum telescopes:\n\\begin{equation}\n\\sum_{j=1}^{m}\n\\bigl(F_K(R_j)-F_K(R_{j-1})\\bigr)\n=\nF_K(N)-F_K(\\varnothing).\n\\end{equation}\nNonnegativity follows from \\cref{prop:b8-envelope-monotone}.\n\\end{proof}","One useful order is\n\\begin{equation}\nS\\prec Q\\prec T\\prec C\\prec H\\prec V\\prec W.\n\\label{eq:b8-operational-order}\n\\end{equation}\nIt 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.","\\begin{definition}[Shapley meta-headroom]\n\\label{def:b8-shapley-headroom}\nFor $i\\in N$, define\n\\begin{equation}\nh_i(K)\n=\n\\sum_{R\\subseteq N\\setminus\\{i\\}}\n\\frac{|R|!(m-|R|-1)!}{m!}\n\\left[\nF_K(R\\cup\\{i\\})-F_K(R)\n\\right].\n\\label{eq:b8-shapley-headroom}\n\\end{equation}\n\\end{definition}","\\begin{theorem}[Meta-headroom allocation]\n\\label{thm:b8-shapley-efficiency}\nFor every finite repair-process factorization,\n\\begin{equation}\nh_i(K)\\geq0\n\\end{equation}\nfor every module $i$, and\n\\begin{equation}\n\\sum_{i\\in N}h_i(K)=\\Hcal(K).\n\\label{eq:b8-shapley-efficiency}\n\\end{equation}\n\\end{theorem}","\\begin{proof}\nFor 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 $F_K$.\n\\end{proof}","{proof}\nFor 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 $F_K$.\n\\end{proof}","The exact finite companion uses five mutable modules and a monotone oracle envelope with three positive pair interactions. Its total headroom is\n\\begin{equation}\n0.740000,\n\\end{equation}\nand the computed module allocations sum to the same value within machine precision. The allocation is shown in \\cref{fig:b8-headroom}.","\\begin{definition}[$\\eps$-saturation]\n\\label{def:b8-saturation}\nA module set $R\\subseteq N$ is $\\eps$-saturated at $K$ when\n\\begin{equation}\nF_K(R)-\\Mcal(K)\\leq\\eps.\n\\label{eq:b8-saturation}\n\\end{equation}\n\\end{definition}","\\begin{proposition}[Component ceiling]\n\\label{prop:b8-component-ceiling}\nIf $R$ is $\\eps$-saturated, no descendant that changes only modules in $R$ can improve protected metaproductivity by more than $\\eps$.\n\\end{proposition}","\\begin{proof}\nEvery such descendant lies in $\\Kcal_R(K)$ and therefore has metaproductivity at most $F_K(R)\\leq\\Mcal(K)+\\eps$.\n\\end{proof}","Let $K\\to K'$ change only modules in $R$. When $F_K(R)>\\Mcal(K)$, define the headroom-adjusted gain\n\\begin{equation}\n\\chi_R(K,K')\n=\n\\frac{\n\\Mcal(K')-\\Mcal(K)\n}{\nF_K(R)-\\Mcal(K)\n}.\n\\label{eq:b8-headroom-adjusted-gain}\n\\end{equation}\nFor an admissible child inside the declared class,\n\\begin{equation}\n\\chi_R(K,K')\\leq1.\n\\end{equation}\nA 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.","hen $F_K(R)>\\Mcal(K)$, define the headroom-adjusted gain\n\\begin{equation}\n\\chi_R(K,K')\n=\n\\frac{\n\\Mcal(K')-\\Mcal(K)\n}{\nF_K(R)-\\Mcal(K)\n}.\n\\label{eq:b8-headroom-adjusted-gain}\n\\end{equation}\nFor an admissible child inside the declared class,\n\\begin{equation}\n\\chi_R(K,K')\\leq1.\n\\end{equation}\nA 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.","When the exact envelope is unknown, certified lower and upper oracle bounds\n\\begin{equation}\n\\underline F_K(R)\n\\leq\nF_K(R)\n\\leq\n\\overline F_K(R)\n\\end{equation}\nproduce 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.","Let the unresolved module-headroom vector be\n\\begin{equation}\nr_t\n=\n\\begin{bmatrix}\n h_H(K_t)\\\\\n h_Q(K_t)\\\\\n h_C(K_t)\\\\\n h_T(K_t)\\\\\n h_S(K_t)\\\\\n h_V(K_t)\\\\\n h_W(K_t)\n\\end{bmatrix}\n\\in\\R_{\\geq0}^{7}.\n\\label{eq:b8-regret-vector}\n\\end{equation}\nAn authorized meta-action $a$ changes one or more repair-process modules. In a local finite model, represent its effect by\n\\begin{equation}\nr_{t+1}\n\\leq\nA_a r_t+e_t,\n\\label{eq:b8-regret-dynamics}\n\\end{equation}\nwhere $A_a$ is a nonnegative response matrix and $e_t$ records estimation error, distribution shift, semantic debt, or newly induced headroom.","h_W(K_t)\n\\end{bmatrix}\n\\in\\R_{\\geq0}^{7}.\n\\label{eq:b8-regret-vector}\n\\end{equation}\nAn authorized meta-action $a$ changes one or more repair-process modules. In a local finite model, represent its effect by\n\\begin{equation}\nr_{t+1}\n\\leq\nA_a r_t+e_t,\n\\label{eq:b8-regret-dynamics}\n\\end{equation}\nwhere $A_a$ is a nonnegative response matrix and $e_t$ records estimation error, distribution shift, semantic debt, or newly induced headroom.","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.","\\begin{definition}[Weighted meta-regret]\nFor $w\\in\\R_{>0}^{7}$, define\n\\begin{equation}\n\\Phi_w(r)=w^\\top r.\n\\label{eq:b8-weighted-regret}\n\\end{equation}\nThe weights encode the declared relative value or risk of unresolved module headroom.\n\\end{definition}","\\begin{theorem}[Meta-regret Lyapunov contraction]\n\\label{thm:b8-lyapunov}\nSuppose 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\n\\begin{equation}\nw^\\top A_{a_t}r_t\n\\leq\nq\\,w^\\top r_t.\n\\label{eq:b8-lyapunov-condition}\n\\end{equation}\nThen every controlled trajectory satisfying \\eqref{eq:b8-regret-dynamics} obeys\n\\begin{equation}\nw^\\top r_t\n\\leq\nq^t w^\\top r_0\n+\n\\sum_{k=0}^{t-1}\nq^{t-1-k}w^\\top e_k.\n\\label{eq:b8-lyapunov-bound}\n\\end{equation}\n\\end{theorem}","Lyapunov contraction]\n\\label{thm:b8-lyapunov}\nSuppose 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\n\\begin{equation}\nw^\\top A_{a_t}r_t\n\\leq\nq\\,w^\\top r_t.\n\\label{eq:b8-lyapunov-condition}\n\\end{equation}\nThen every controlled trajectory satisfying \\eqref{eq:b8-regret-dynamics} obeys\n\\begin{equation}\nw^\\top r_t\n\\leq\nq^t w^\\top r_0\n+\n\\sum_{k=0}^{t-1}\nq^{t-1-k}w^\\top e_k.\n\\label{eq:b8-lyapunov-bound}\n\\end{equation}\n\\end{theorem}","\\begin{proof}\nEquation \\eqref{eq:b8-regret-dynamics} and nonnegativity of $w$ give\n\\begin{equation}\nw^\\top r_{t+1}\n\\leq\nw^\\top A_{a_t}r_t+w^\\top e_t\n\\leq\nq\\,w^\\top r_t+w^\\top e_t.\n\\end{equation}\nRepeated substitution proves \\eqref{eq:b8-lyapunov-bound}.\n\\end{proof}","\\begin{corollary}[Fixed-grammar convergence]\n\\label{cor:b8-fixed-grammar-convergence}\nIf the grammar remains fixed and\n\\begin{equation}\n\\sum_{t=0}^{\\infty}w^\\top e_t<\\infty,\n\\end{equation}\nthen $w^\\top r_t\\to0$ whenever the tail errors vanish under the convolution in \\eqref{eq:b8-lyapunov-bound}.\n\\end{corollary}","\\begin{proposition}[Greedy bottleneck contraction]\n\\label{prop:b8-greedy-contraction}\nLet $r_t\\in\\R_{\\geq0}^{m}$ and\n\\begin{equation}\n\\Phi_t=\\one^\\top r_t.\n\\end{equation}\nSuppose the controller chooses $i_t$ with\n\\begin{equation}\nr_{i_t,t}\\geq\\frac{\\Phi_t}{m},\n\\end{equation}\ncontracts that component by $q\\in[0,1)$, and permits total collateral increase at most $\\eps_t$. Then\n\\begin{equation}\n\\Phi_{t+1}\n\\leq\n\\left(1-\\frac{1-q}{m}\\right)\\Phi_t+\\eps_t.\n\\label{eq:b8-greedy-contraction}\n\\end{equation}\n\\end{proposition}","\\begin{proof}\nThe 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.\n\\end{proof}","Define a module status map\n\\begin{equation}\n\\sigma_t(i)\n\\in\n\\{\\texttt{active},\\texttt{saturated},\\texttt{unidentified},\\texttt{protected}\\}.\n\\end{equation}\nAn admissible bottleneck controller should:\n\\begin{enumerate}\n\\item estimate headroom for active modules;\n\\item refuse large gain claims for saturated modules;\n\\item request new evidence for unidentified modules;\n\\item avoid changing protected modules inside the current transaction;\n\\item trigger grammar expansion when all active modules are saturated while persistent obstruction remains.\n\\end{enumerate}","\\begin{protocol}[Saturation-aware meta-controller]\n\\label{prot:b8-saturation-controller}\nAt generation $t$:\n\\begin{enumerate}\n\\item compute lower and upper headroom bounds for every currently mutable module coalition;\n\\item select the coalition with largest certified lower headroom per unit resource;\n\\item compile candidate descendants that target only that coalition;\n\\item seal the candidate set and final audit endpoints;\n\\item commit only a descendant with a positive fresh lower metaproductivity bound;\n\\item mark the coalition saturated when its upper headroom falls below the declared tolerance;\n\\item if every current coalition is saturated and foreign obstruction remains, produce a grammar-boundary certificate.\n\\end{enumerate}\n\\end{protocol}","A grammar extension can reveal error that did not exist as a coordinate in the old state. Let\n\\begin{equation}\nE_t:\\R^{m_t}\\hookrightarrow\\R^{m_{t+1}}\n\\end{equation}\nbe the conservative embedding of old meta-regret coordinates. Decompose\n\\begin{equation}\nr_{t+1}\n=\nE_t\\widetilde r_{t+1}+n_{t+1},\n\\qquad\nn_{t+1}\\perp\\im E_t.\n\\label{eq:b8-regret-expansion}\n\\end{equation}\nDefine old-coordinate repair progress by\n\\begin{equation}\nP_t^{\\mathrm{repair}}\n=\n\\norm{r_t}^2-\\norm{\\widetilde r_{t+1}}^2,\n\\label{eq:b8-repair-progress}\n\\end{equation}\nand epistemic discovery by\n\\begin{equation}\nP_t^{\\mathrm{discovery}}\n=\n\\norm{n_{t+1}}^2.\n\\label{eq:b8-discovery-progress}\n\\end{equation}\nBoth quantities may be positive. The total represented regret may increase because the lineage learned to represent a previously invisible limitation.","eal error that did not exist as a coordinate in the old state. Let\n\\begin{equation}\nE_t:\\R^{m_t}\\hookrightarrow\\R^{m_{t+1}}\n\\end{equation}\nbe the conservative embedding of old meta-regret coordinates. Decompose\n\\begin{equation}\nr_{t+1}\n=\nE_t\\widetilde r_{t+1}+n_{t+1},\n\\qquad\nn_{t+1}\\perp\\im E_t.\n\\label{eq:b8-regret-expansion}\n\\end{equation}\nDefine old-coordinate repair progress by\n\\begin{equation}\nP_t^{\\mathrm{repair}}\n=\n\\norm{r_t}^2-\\norm{\\widetilde r_{t+1}}^2,\n\\label{eq:b8-repair-progress}\n\\end{equation}\nand epistemic discovery by\n\\begin{equation}\nP_t^{\\mathrm{discovery}}\n=\n\\norm{n_{t+1}}^2.\n\\label{eq:b8-discovery-progress}\n\\end{equation}\nBoth quantities may be positive. The total represented regret may increase because the lineage learned to represent a previously invisible limitation.","ive embedding of old meta-regret coordinates. Decompose\n\\begin{equation}\nr_{t+1}\n=\nE_t\\widetilde r_{t+1}+n_{t+1},\n\\qquad\nn_{t+1}\\perp\\im E_t.\n\\label{eq:b8-regret-expansion}\n\\end{equation}\nDefine old-coordinate repair progress by\n\\begin{equation}\nP_t^{\\mathrm{repair}}\n=\n\\norm{r_t}^2-\\norm{\\widetilde r_{t+1}}^2,\n\\label{eq:b8-repair-progress}\n\\end{equation}\nand epistemic discovery by\n\\begin{equation}\nP_t^{\\mathrm{discovery}}\n=\n\\norm{n_{t+1}}^2.\n\\label{eq:b8-discovery-progress}\n\\end{equation}\nBoth quantities may be positive. The total represented regret may increase because the lineage learned to represent a previously invisible limitation.","\\begin{definition}[Conservative lineage dominance]\n\\label{def:b8-conservative-dominance}\nWrite\n\\begin{equation}\nL_t\\preceq_{\\mathrm{cons}}L_{t+1}\n\\end{equation}\nwhen:\n\\begin{enumerate}\n\\item the old grammar, evidence ledger, and proof objects embed into the new lineage state;\n\\item every old protected interpretation remains recoverable within the declared semantic-debt bound;\n\\item old authorized behaviors replay within declared tolerances;\n\\item the new correction image contains the transported old correction image;\n\\item at least one new externally validated distinction, repair schema, or correction direction is added, or one old module-headroom coordinate is strictly reduced;\n\\item foreign metaproductivity does not decrease beyond the accepted uncertainty and regression budgets.\n\\end{enumerate}\n\\end{definition}","\\begin{definition}[Open-set ascending lineage]\nA lineage is conservatively ascending when\n\\begin{equation}\nL_0\\preceq_{\\mathrm{cons}}L_1\n\\preceq_{\\mathrm{cons}}L_2\n\\preceq_{\\mathrm{cons}}\\cdots.\n\\label{eq:b8-ascending-lineage}\n\\end{equation}\nIt is structurally open-set when infinitely many edges add a foreign-task-relevant correction direction or validated distinction not representable in the prior grammar.\n\\end{definition}","Let $K_t$ be the active repair process and let $\\Gamma_t$ be the lineage compiler. Write\n\\begin{equation}\nK_{t+1}\n=\n\\Gamma_t(K_t,\\Ecal_t,\\Acal_t),\n\\label{eq:b8-compiler-update}\n\\end{equation}\nwhere $\\Ecal_t$ is the available evidence and $\\Acal_t$ is the executable archive.","There are three regimes:\n\\begin{enumerate}\n\\item \\textbf{fixed-compiler iteration}: $\\Gamma_t=\\Gamma$ while the compiled processes $K_t$ change;\n\\item \\textbf{compiler improvement}: $\\Gamma_t$ itself changes and improves on foreign parent processes;\n\\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.\n\\end{enumerate}","Define compiler metaproductivity by\n\\begin{equation}\n\\Ccal(\\Gamma)\n=\n\\mathbb E_{K\\sim\\Dcal_{\\mathrm{parent}}}\n\\left[\n\\Mcal\\bigl(\\Gamma(K,\\Ecal_K)\\bigr)-\\Mcal(K)\n\\right].\n\\label{eq:b8-compiler-metaproductivity}\n\\end{equation}\nA stronger recursive claim requires\n\\begin{equation}\n\\Ccal(\\Gamma_{t+1})>\\Ccal(\\Gamma_t)\n\\end{equation}\non sealed foreign parent processes under matched compilation resources.","Block 7 improves components inside $K$. It does not yet establish improvement of the outer compiler $\\Gamma$.","Let $V$ be a finite-dimensional defect Hilbert space. Suppose a conservative grammar chain gives nested closed authorized correction images\n\\begin{equation}\nI_0\\subseteq I_1\\subseteq I_2\\subseteq\\cdots\\subseteq V,\n\\qquad\nI_t=\\im D_t.\n\\label{eq:b8-nested-images}\n\\end{equation}\nFor $r\\in V$, define unresolved obstruction magnitude\n\\begin{equation}\n\\eta_t(r)\n=\n\\dist(-r,I_t)\n=\n\\norm{P_{I_t^\\perp}r}.\n\\label{eq:b8-obstruction-magnitude}\n\\end{equation}","\\begin{theorem}[Monotone obstruction convergence]\n\\label{thm:b8-obstruction-convergence}\nFor every defect $r$, the sequence $\\eta_t(r)$ is nonincreasing and\n\\begin{equation}\n\\eta_t(r)\n\\longrightarrow\n\\dist\\left(-r,\\overline{\\bigcup_{t\\geq0}I_t}\\right).\n\\label{eq:b8-obstruction-limit}\n\\end{equation}\n\\end{theorem}","\\begin{proof}\nNestedness gives $I_t\\subseteq I_{t+1}$, so distance to $I_{t+1}$ cannot exceed distance to $I_t$. Let\n\\begin{equation}\nI_\\infty=\\overline{\\bigcup_t I_t}.\n\\end{equation}\nThe 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 $I_\\infty$.\n\\end{proof}","\\begin{theorem}[Newly correctable energy identity]\n\\label{thm:b8-new-energy}\nFor nested correction images,\n\\begin{equation}\n\\eta_t(r)^2-\\eta_{t+1}(r)^2\n=\n\\norm{P_{I_{t+1}\\cap I_t^\\perp}r}^2.\n\\label{eq:b8-new-energy}\n\\end{equation}\n\\end{theorem}","\\begin{proof}\nThe nested orthogonal decomposition is\n\\begin{equation}\nI_{t+1}\n=\nI_t\\oplus(I_{t+1}\\cap I_t^\\perp).\n\\end{equation}\nTherefore\n\\begin{equation}\nP_{I_{t+1}^\\perp}r\n=\nP_{I_t^\\perp}r\n-\nP_{I_{t+1}\\cap I_t^\\perp}r,\n\\end{equation}\nand the two terms on the right are orthogonal. The Pythagorean identity gives \\eqref{eq:b8-new-energy}.\n\\end{proof}","For a foreign residual distribution $Q$, define expected obstruction energy\n\\begin{equation}\nE_t(Q)\n=\n\\mathbb E_{r\\sim Q}\\eta_t(r)^2.\n\\label{eq:b8-expected-obstruction}\n\\end{equation}\nThen\n\\begin{equation}\nE_t(Q)-E_{t+1}(Q)\n=\n\\mathbb E_{r\\sim Q}\n\\norm{P_{I_{t+1}\\cap I_t^\\perp}r}^2.\n\\label{eq:b8-distributional-new-energy}\n\\end{equation}\nThis is the primary structural metaproductivity estimand for grammar growth.","A grammar extension may also change the defect representation. Let\n\\begin{equation}\nD_t:U_t\\to V_t,\n\\end{equation}\nwith transports\n\\begin{equation}\nT_t^U:U_t\\to U_{t+1},\n\\qquad\nT_t^V:V_t\\to V_{t+1}.\n\\end{equation}\nAssume the exact correction square\n\\begin{equation}\nD_{t+1}T_t^U\n=\nT_t^VD_t.\n\\label{eq:b8-dynamic-square}\n\\end{equation}\nThen there is an induced obstruction map\n\\begin{equation}\n\\overline T_t:\n\\coker D_t\n\\to\n\\coker D_{t+1},\n\\qquad\n\\overline T_t([r])=[T_t^Vr].\n\\label{eq:b8-induced-obstruction-map}\n\\end{equation}","\\begin{proof}\nIf $r-r'=D_tu$, then\n\\begin{equation}\nT_t^Vr-T_t^Vr'\n=\nT_t^VD_tu\n=\nD_{t+1}T_t^Uu,\n\\end{equation}\nso $T_t^Vr$ and $T_t^Vr'$ define the same class in $\\coker D_{t+1}$.\n\\end{proof}","The sequence\n\\begin{equation}\n\\coker D_0\n\\xrightarrow{\\overline T_0}\n\\coker D_1\n\\xrightarrow{\\overline T_1}\n\\coker D_2\n\\longrightarrow\\cdots\n\\label{eq:b8-obstruction-module}\n\\end{equation}\nis the obstruction persistence module of the grammar lineage.","\\begin{definition}[Repair death]\nA class $[r]\\in\\coker D_t$ undergoes repair death at $t+1$ when\n\\begin{equation}\n\\overline T_t([r])=0,\n\\end{equation}\nthere exists an authorized $u\\in U_{t+1}$ satisfying\n\\begin{equation}\nD_{t+1}u=-T_t^Vr,\n\\end{equation}\nand the protected defect transport obeys a lower norm bound\n\\begin{equation}\n\\norm{T_t^Vr}\\geq\\kappa\\norm{r}\n\\end{equation}\nfor a declared $\\kappa>0$.\n\\end{definition}","\\begin{definition}[Forgetting death]\nA 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.\n\\end{definition}","\\begin{warning}[Zero cokernel class is not sufficient]\nThe 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.\n\\end{warning}","Let $Q_t\\subseteq V_t$ be a protected obstruction subspace. A dynamic grammar transport is $\\kappa$-faithful on $Q_t$ when\n\\begin{equation}\n\\norm{T_t^Vq}\\geq\\kappa\\norm{q}\n\\qquad\n\\text{for all }q\\in Q_t.\n\\label{eq:b8-kappa-faithful}\n\\end{equation}\nThis extends the obstruction-faithfulness requirement from one bridge to an entire grammar lineage.","Suppose the correction square is approximate:\n\\begin{equation}\n\\norm{D_{t+1}T_t^U-T_t^VD_t}\n\\leq\\eps_t^D.\n\\end{equation}\nSuppose protected defect meanings are reconstructed with error at most $\\eps_t^V$. Define cumulative dynamic semantic debt\n\\begin{equation}\n\\Delta_{m,n}\n=\n\\sum_{t=m}^{n-1}\n\\left(\n\\eps_t^D+\\eps_t^V\n\\right).\n\\label{eq:b8-dynamic-debt}\n\\end{equation}\nUnder nonexpansive transports, old correction and defect relations drift by at most the corresponding telescoping debt. If\n\\begin{equation}\n\\sum_t(\\eps_t^D+\\eps_t^V)<\\infty,\n\\end{equation}\nold protected meanings form Cauchy trajectories along the grammar chain.","nd{equation}\nSuppose protected defect meanings are reconstructed with error at most $\\eps_t^V$. Define cumulative dynamic semantic debt\n\\begin{equation}\n\\Delta_{m,n}\n=\n\\sum_{t=m}^{n-1}\n\\left(\n\\eps_t^D+\\eps_t^V\n\\right).\n\\label{eq:b8-dynamic-debt}\n\\end{equation}\nUnder nonexpansive transports, old correction and defect relations drift by at most the corresponding telescoping debt. If\n\\begin{equation}\n\\sum_t(\\eps_t^D+\\eps_t^V)<\\infty,\n\\end{equation}\nold protected meanings form Cauchy trajectories along the grammar chain.","Let $I=\\im D\\subseteq V$ be the current authorized correction image and let\n\\begin{equation}\nz=P_{I^\\perp}r\n\\end{equation}\nbe 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:\n\\begin{equation}\n\\bar v=P_{I^\\perp}v.\n\\label{eq:b8-new-generator-component}\n\\end{equation}\nThe expanded image is\n\\begin{equation}\nI'=I+\\Span\\{v\\}.\n\\end{equation}","rent authorized correction image and let\n\\begin{equation}\nz=P_{I^\\perp}r\n\\end{equation}\nbe 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:\n\\begin{equation}\n\\bar v=P_{I^\\perp}v.\n\\label{eq:b8-new-generator-component}\n\\end{equation}\nThe expanded image is\n\\begin{equation}\nI'=I+\\Span\\{v\\}.\n\\end{equation}","\\begin{theorem}[One-generator obstruction reduction]\n\\label{thm:b8-one-generator}\nIf $\\bar v\\neq0$, then\n\\begin{equation}\n\\eta_{I'}(r)^2\n=\n\\eta_I(r)^2\n-\n\\frac{\\abs{\\angles{z,\\bar v}}^2}{\\norm{\\bar v}^2}.\n\\label{eq:b8-one-generator}\n\\end{equation}\nIf $\\bar v=0$, the extension introduces no new first-order correctability.\n\\end{theorem}","\\begin{proof}\nBecause $\\bar v\\in I^\\perp$,\n\\begin{equation}\nI'=I\\oplus\\Span\\{\\bar v\\}.\n\\end{equation}\nThe 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 $I'=I$.\n\\end{proof}","Equation \\eqref{eq:b8-one-generator} gives a direct acquisition score. For a candidate family $\\Vcal_{\\mathrm{cand}}$, choose\n\\begin{equation}\nv^\\star\n\\in\n\\argmax_{v\\in\\Vcal_{\\mathrm{cand}}}\n\\frac{\\abs{\\angles{z,P_{I^\\perp}v}}^2}\n{\\norm{P_{I^\\perp}v}^2+\\lambda c(v)+\\beta\\Risk(v)},\n\\label{eq:b8-generator-score}\n\\end{equation}\nwhere $c(v)$ is implementation cost and $\\Risk(v)$ is a protected-risk estimate.","\\subsection{Distributional rank-$k$ extension}","Let foreign residuals follow distribution $Q$. Define the obstruction covariance operator\n\\begin{equation}\n\\Sigma_{\\mathrm{ob}}\n=\n\\mathbb E_{r\\sim Q}\n\\left[\nP_{I^\\perp}r\\,r^\\top P_{I^\\perp}\n\\right].\n\\label{eq:b8-obstruction-covariance}\n\\end{equation}\nLet $W\\subseteq I^\\perp$ be a $k$-dimensional new response subspace.","\\begin{theorem}[Optimal rank-$k$ obstruction extension]\n\\label{thm:b8-kyfan-extension}\nThe subspace $W$ minimizing expected unresolved obstruction energy\n\\begin{equation}\n\\mathbb E_{r\\sim Q}\n\\norm{P_{(I\\oplus W)^\\perp}r}^2\n\\end{equation}\nis the span of the top $k$ eigenvectors of $\\Sigma_{\\mathrm{ob}}$. The minimum residual energy is the sum of the remaining eigenvalues.\n\\end{theorem}","\\begin{proof}\nFor $W\\subseteq I^\\perp$,\n\\begin{equation}\n\\mathbb E\\norm{P_{(I\\oplus W)^\\perp}r}^2\n=\n\\tr(\\Sigma_{\\mathrm{ob}})-\\tr(P_W\\Sigma_{\\mathrm{ob}}).\n\\end{equation}\nThe Ky Fan variational principle maximizes $\\tr(P_W\\Sigma_{\\mathrm{ob}})$ over rank-$k$ orthogonal projectors by choosing the top $k$ eigenspace.\n\\end{proof}","The resulting pipeline is\n\\begin{equation}\n\\text{causal intervention}\n\\longrightarrow\n\\text{authorized obstruction residual}\n\\longrightarrow\n\\Sigma_{\\mathrm{ob}}\n\\longrightarrow\n\\text{new response directions}.\n\\label{eq:b8-obstruction-pipeline}\n\\end{equation}","Define the causal generator contribution\n\\begin{equation}\n\\Delta_{\\mathrm{gen}}\n=\nE_Q\\bigl[\\eta_{\\mathrm{without}}(r)^2\\bigr]\n-\nE_Q\\bigl[\\eta_{\\mathrm{with}}(r)^2\\bigr].\n\\label{eq:b8-generator-credit}\n\\end{equation}\nThe contribution is admissible only when a matched ablation isolates the generator while holding the remaining grammar, evidence, and selection budget fixed.","Let $X$ be a finite set of repair episodes. The current grammar represents an episode through\n\\begin{equation}\n\\phi_{\\Gcal}:X\\to Z_{\\Gcal}.\n\\end{equation}\nLet $\\Acal^\\star(x)$ be the set of externally validated authorized repairs for episode $x$.","For $z\\in Z_{\\Gcal}$, define the grammar cell\n\\begin{equation}\nC_z\n=\n\\{x\\in X:\\phi_{\\Gcal}(x)=z\\}.\n\\end{equation}","\\begin{definition}[Repair-consistent cell]\nA cell is repair consistent when\n\\begin{equation}\n\\bigcap_{x\\in C_z}\\Acal^\\star(x)\\neq\\varnothing.\n\\label{eq:b8-consistent-cell}\n\\end{equation}\nIt is inconsistent when no single authorized repair works for every episode that the grammar treats as equivalent.\n\\end{definition}","\\begin{definition}[Minimal repair conflict]\nA subset $S\\subseteq C_z$ is a minimal repair conflict when\n\\begin{equation}\n\\bigcap_{x\\in S}\\Acal^\\star(x)=\\varnothing,\n\\end{equation}\nbut every proper subset has a nonempty repair intersection.\n\\end{definition}","The minimal conflicts form a finite hypergraph $\\mathfrak H_{\\Gcal}$. A candidate feature or probe\n\\begin{equation}\nf:X\\to Y_f\n\\end{equation}\nresolves a conflict $S$ when $f$ is not constant on $S$.","Appending the feature gives the conservative extension\n\\begin{equation}\n\\phi_{\\Gcal'}(x)\n=\n\\bigl(\\phi_{\\Gcal}(x),f(x)\\bigr).\n\\label{eq:b8-feature-extension}\n\\end{equation}\nThe old representation is recovered by projection onto the first coordinate.","\\begin{theorem}[Finite conservative conflict refinement]\n\\label{thm:b8-conflict-refinement}\nSuppose 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.\n\\end{theorem}","Let each candidate feature $f$ cover the conflict set\n\\begin{equation}\n\\Cov(f)\n=\n\\{S\\in\\mathfrak H_{\\Gcal}: f\\text{ resolves }S\\}.\n\\end{equation}\nChoosing 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\n\\begin{equation}\n\\frac{\\text{newly resolved conflicts}}{\\text{feature cost}+\\text{probe risk}}.\n\\end{equation}\nThe 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.","For episode $x$, let\n\\begin{equation}\nF_x:U_x^{\\Auth}\\to V_x\n\\end{equation}\nbe the post-modification defect map, with $F_x(0)=r_x$. A valid repair satisfies\n\\begin{equation}\nF_x(u_x)=0.\n\\end{equation}\nA repair schema is a local section\n\\begin{equation}\ns:x\\mapsto u_x\n\\end{equation}\nthat approximately inverts $F_x$ over its declared domain.","In the linear case,\n\\begin{equation}\nF_x(u)=r_x+D_xu,\n\\end{equation}\nand the canonical minimum-norm repair is\n\\begin{equation}\ns(x)=-D_x^\\dagger r_x\n\\end{equation}\nwhenever $r_x\\in\\im D_x$.","\\begin{definition}[Structurally new repair schema]\nLet $\\Sscr_t$ be the closure of existing repair schemas under the authorized operadic compositions. A schema $s_{t+1}$ is structurally new when\n\\begin{equation}\ns_{t+1}\\notin\\overline{\\Sscr_t}\n\\end{equation}\nunder the declared behavioral equivalence and\n\\begin{equation}\n\\Span\\im s_{t+1}\n\\not\\subseteq\n\\Span\\im\\Sscr_t.\n\\label{eq:b8-new-schema-direction}\n\\end{equation}\n\\end{definition}","A structural metaproductivity claim additionally requires\n\\begin{equation}\nE_{t+1}(Q_{\\mathrm{foreign}})\n<\nE_t(Q_{\\mathrm{foreign}})\n\\label{eq:b8-structural-gate}\n\\end{equation}\nand a causal ablation showing that the new schema is responsible for the reduction.","Let $\\Pcal_{\\Gcal}$ be a finite typed repair-program grammar and let\n\\begin{equation}\n\\operatorname{Verify}(p,x)\\in\\{0,1\\}\n\\end{equation}\nbe a complete verifier on a finite declared episode domain.","\\begin{protocol}[Finite repair synthesis]\n\\label{prot:b8-cegis}\n\\begin{enumerate}\n\\item choose a program $p\\in\\Pcal_{\\Gcal}$ consistent with the current receipt set;\n\\item search for an episode $x$ with $\\operatorname{Verify}(p,x)=0$;\n\\item if no counterexample exists, return $p$ with its verification receipt;\n\\item otherwise add $x$ to the receipt set, eliminate $p$, and continue.\n\\end{enumerate}\n\\end{protocol}","\\begin{theorem}[Finite synthesis or grammar exhaustion]\n\\label{thm:b8-finite-synthesis}\nProtocol \\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.\n\\end{theorem}","\\begin{proof}\nEvery 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.\n\\end{proof}","Let $H$ be a finite-dimensional complex Hilbert space. Suppose permitted interventions or controlled transitions produce operators\n\\begin{equation}\nR_1,\\ldots,R_m\\in\\End(H).\n\\end{equation}\nThese may be finite-difference response maps, local transition Jacobians, or finite Koopman operators on an observable dictionary.","Define the unital adjoint-closed algebra\n\\begin{equation}\n\\mathfrak A\n=\n\\operatorname{alg}^{\\ast}\n\\left(I,R_1,\\ldots,R_m\\right).\n\\label{eq:b8-response-algebra}\n\\end{equation}\nIts commutant is\n\\begin{equation}\n\\mathfrak A'\n=\n\\{X\\in\\End(H):XA=AX\\text{ for all }A\\in\\mathfrak A\\}.\n\\end{equation}\nThe center is\n\\begin{equation}\nZ(\\mathfrak A)=\\mathfrak A\\cap\\mathfrak A'.\n\\end{equation}","\\begin{theorem}[Finite causal atlas decomposition]\n\\label{thm:b8-atlas-decomposition}\nThere exists a unitary identification\n\\begin{equation}\nH\n\\cong\n\\bigoplus_{\\alpha=1}^{p}\n\\left(\n\\mathbb C^{m_\\alpha}\\otimes\\mathbb C^{n_\\alpha}\n\\right)\n\\label{eq:b8-algebra-space-decomposition}\n\\end{equation}\nunder which\n\\begin{equation}\n\\mathfrak A\n\\cong\n\\bigoplus_{\\alpha=1}^{p}\n\\left(\nI_{m_\\alpha}\\otimes M_{n_\\alpha}(\\mathbb C)\n\\right).\n\\label{eq:b8-algebra-decomposition}\n\\end{equation}\nThe minimal central projections $P_\\alpha\\in Z(\\mathfrak A)$ give a canonical coarse causal atlas\n\\begin{equation}\nH\n=\n\\bigoplus_{\\alpha=1}^{p}P_\\alpha H.\n\\label{eq:b8-central-atlas}\n\\end{equation}\nThe blocks are unique up to permutation and unitary changes of basis within multiplicity spaces.\n\\end{theorem}","dentification\n\\begin{equation}\nH\n\\cong\n\\bigoplus_{\\alpha=1}^{p}\n\\left(\n\\mathbb C^{m_\\alpha}\\otimes\\mathbb C^{n_\\alpha}\n\\right)\n\\label{eq:b8-algebra-space-decomposition}\n\\end{equation}\nunder which\n\\begin{equation}\n\\mathfrak A\n\\cong\n\\bigoplus_{\\alpha=1}^{p}\n\\left(\nI_{m_\\alpha}\\otimes M_{n_\\alpha}(\\mathbb C)\n\\right).\n\\label{eq:b8-algebra-decomposition}\n\\end{equation}\nThe minimal central projections $P_\\alpha\\in Z(\\mathfrak A)$ give a canonical coarse causal atlas\n\\begin{equation}\nH\n=\n\\bigoplus_{\\alpha=1}^{p}P_\\alpha H.\n\\label{eq:b8-central-atlas}\n\\end{equation}\nThe blocks are unique up to permutation and unitary changes of basis within multiplicity spaces.\n\\end{theorem}","{m_\\alpha}\\otimes\\mathbb C^{n_\\alpha}\n\\right)\n\\label{eq:b8-algebra-space-decomposition}\n\\end{equation}\nunder which\n\\begin{equation}\n\\mathfrak A\n\\cong\n\\bigoplus_{\\alpha=1}^{p}\n\\left(\nI_{m_\\alpha}\\otimes M_{n_\\alpha}(\\mathbb C)\n\\right).\n\\label{eq:b8-algebra-decomposition}\n\\end{equation}\nThe minimal central projections $P_\\alpha\\in Z(\\mathfrak A)$ give a canonical coarse causal atlas\n\\begin{equation}\nH\n=\n\\bigoplus_{\\alpha=1}^{p}P_\\alpha H.\n\\label{eq:b8-central-atlas}\n\\end{equation}\nThe blocks are unique up to permutation and unitary changes of basis within multiplicity spaces.\n\\end{theorem}","\\begin{proof}\nA 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.\n\\end{proof}","\\begin{remark}[Identifiability boundary]\nThe 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.\n\\end{remark}","For a current residual $r\\in H$, define its cyclic causal module\n\\begin{equation}\nH_r\n=\n\\Span\\{Ar:A\\in\\mathfrak A\\}.\n\\label{eq:b8-cyclic-module}\n\\end{equation}","\\begin{proposition}[Minimal causal defect module]\n\\label{prop:b8-cyclic-minimal}\n$H_r$ is the smallest $\\mathfrak A$-invariant subspace containing $r$. Because $\\mathfrak A$ is adjoint closed, $H_r$ is reducing and admits an orthogonal projector.\n\\end{proposition}","\\begin{proof}\nThe 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.\n\\end{proof}","This produces a two-level representation:\n\\begin{enumerate}\n\\item the central blocks $P_\\alpha H$ form the persistent coarse atlas;\n\\item $H_r$ is the episode-specific repair packet generated by the current defect under the causal algebra.\n\\end{enumerate}\nThe theory therefore does not require one permanent universal workspace subspace.","For $X\\in\\End(H)$, define commutator energy\n\\begin{equation}\n\\Ecal_{\\mathrm{com}}(X)\n=\n\\sum_{j=1}^{m}\n\\left(\n\\norm{XR_j-R_jX}_F^2\n+\n\\norm{XR_j^\\ast-R_j^\\ast X}_F^2\n\\right).\n\\label{eq:b8-commutator-energy}\n\\end{equation}\nThere is a positive semidefinite linear operator $\\Lcal_{\\mathrm{com}}$ on the Hilbert space of matrices such that\n\\begin{equation}\n\\angles{X,\\Lcal_{\\mathrm{com}}X}_F\n=\n\\Ecal_{\\mathrm{com}}(X).\n\\end{equation}\nIts kernel is the commutant:\n\\begin{equation}\n\\ker\\Lcal_{\\mathrm{com}}=\\mathfrak A'.\n\\end{equation}","m{com}}(X)\n=\n\\sum_{j=1}^{m}\n\\left(\n\\norm{XR_j-R_jX}_F^2\n+\n\\norm{XR_j^\\ast-R_j^\\ast X}_F^2\n\\right).\n\\label{eq:b8-commutator-energy}\n\\end{equation}\nThere is a positive semidefinite linear operator $\\Lcal_{\\mathrm{com}}$ on the Hilbert space of matrices such that\n\\begin{equation}\n\\angles{X,\\Lcal_{\\mathrm{com}}X}_F\n=\n\\Ecal_{\\mathrm{com}}(X).\n\\end{equation}\nIts kernel is the commutant:\n\\begin{equation}\n\\ker\\Lcal_{\\mathrm{com}}=\\mathfrak A'.\n\\end{equation}","\\begin{proposition}[Spectral-gap stability of the commutant]\n\\label{prop:b8-commutant-stability}\nAssume the first positive eigenvalue of $\\Lcal_{\\mathrm{com}}$ is at least $\\lambda>0$. Let $\\widehat{\\Lcal}_{\\mathrm{com}}$ be an empirical estimate satisfying\n\\begin{equation}\n\\norm{\\widehat{\\Lcal}_{\\mathrm{com}}-\\Lcal_{\\mathrm{com}}}\\leq\\eps<\\lambda/2.\n\\end{equation}\nThen the spectral projector onto the empirical eigenvalues in $[0,\\lambda/2)$ differs from the exact commutant projector by at most a constant multiple of $\\eps/\\lambda$.\n\\end{proposition}","p stability of the commutant]\n\\label{prop:b8-commutant-stability}\nAssume the first positive eigenvalue of $\\Lcal_{\\mathrm{com}}$ is at least $\\lambda>0$. Let $\\widehat{\\Lcal}_{\\mathrm{com}}$ be an empirical estimate satisfying\n\\begin{equation}\n\\norm{\\widehat{\\Lcal}_{\\mathrm{com}}-\\Lcal_{\\mathrm{com}}}\\leq\\eps<\\lambda/2.\n\\end{equation}\nThen the spectral projector onto the empirical eigenvalues in $[0,\\lambda/2)$ differs from the exact commutant projector by at most a constant multiple of $\\eps/\\lambda$.\n\\end{proposition}","igenvalue of $\\Lcal_{\\mathrm{com}}$ is at least $\\lambda>0$. Let $\\widehat{\\Lcal}_{\\mathrm{com}}$ be an empirical estimate satisfying\n\\begin{equation}\n\\norm{\\widehat{\\Lcal}_{\\mathrm{com}}-\\Lcal_{\\mathrm{com}}}\\leq\\eps<\\lambda/2.\n\\end{equation}\nThen the spectral projector onto the empirical eigenvalues in $[0,\\lambda/2)$ differs from the exact commutant projector by at most a constant multiple of $\\eps/\\lambda$.\n\\end{proposition}","\\begin{proof}\nThe 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.\n\\end{proof}","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\n\\begin{equation}\n8,\n\\end{equation}\nthe commutant has dimension\n\\begin{equation}\n2,\n\\end{equation}\nand the recovered central block ranks are\n\\begin{equation}\n(2,2).\n\\end{equation}\nA residual supported in the first block has cyclic dimension $2$, while a residual with support in both blocks has cyclic dimension $4$.","our 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\n\\begin{equation}\n8,\n\\end{equation}\nthe commutant has dimension\n\\begin{equation}\n2,\n\\end{equation}\nand the recovered central block ranks are\n\\begin{equation}\n(2,2).\n\\end{equation}\nA residual supported in the first block has cyclic dimension $2$, while a residual with support in both blocks has cyclic dimension $4$.","s 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\n\\begin{equation}\n8,\n\\end{equation}\nthe commutant has dimension\n\\begin{equation}\n2,\n\\end{equation}\nand the recovered central block ranks are\n\\begin{equation}\n(2,2).\n\\end{equation}\nA residual supported in the first block has cyclic dimension $2$, while a residual with support in both blocks has cyclic dimension $4$.","l two coordinates and generate a second full block. The generated algebra has dimension\n\\begin{equation}\n8,\n\\end{equation}\nthe commutant has dimension\n\\begin{equation}\n2,\n\\end{equation}\nand the recovered central block ranks are\n\\begin{equation}\n(2,2).\n\\end{equation}\nA residual supported in the first block has cyclic dimension $2$, while a residual with support in both blocks has cyclic dimension $4$.","For nonlinear dynamics on a state manifold $M$, the response algebra becomes base-point dependent:\n\\begin{equation}\nx\\longmapsto\\mathfrak A_x\\subseteq\\End(T_xM).\n\\end{equation}\nLocal 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:\n\\begin{enumerate}\n\\item local intervention algebras;\n\\item spectral-gap control of approximate central projectors;\n\\item transition maps between neighboring tangent atlases;\n\\item descent and holonomy tests on overlaps;\n\\item repair transport that preserves obstruction classes rather than only state coordinates.\n\\end{enumerate}\nThi","For nonlinear dynamics on a state manifold $M$, the response algebra becomes base-point dependent:\n\\begin{equation}\nx\\longmapsto\\mathfrak A_x\\subseteq\\End(T_xM).\n\\end{equation}\nLocal 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:\n\\begin{enumerate}\n\\item local intervention algebras;\n\\item spectral-gap control of approximate central projectors;\n\\item transition maps between neighboring tangent atlases;\n\\item descent and holonomy tests on overlaps;\n\\item repair transport that preserves obstruction classes rather than only state coordinates.\n\\end{enumerate}\nThis remains a conditional extension rather than a completed general theorem.","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.","Let the true foreign metaproductivity effect be\n\\begin{equation}\n\\Delta_t\n=\n\\Mcal(K_{t+1})-\\Mcal(K_t).\n\\label{eq:b8-edge-effect}\n\\end{equation}\nLet $L_t$ be a lower confidence bound computed from fresh audit evidence satisfying\n\\begin{equation}\n\\Pr\\bigl(\\Delta_t<L_t\\mid\\Fcal_t\\bigr)\n\\leq\n\\delta_t.\n\\label{eq:b8-conditional-coverage}\n\\end{equation}\nAccept the edge only when\n\\begin{equation}\nL_t>0.\n\\label{eq:b8-edge-accept}\n\\end{equation}","\\begin{theorem}[Lineage-wide false-improvement control]\n\\label{thm:b8-lineage-validity}\nIf\n\\begin{equation}\n\\sum_{t=0}^{\\infty}\\delta_t\\leq\\delta,\n\\end{equation}\nthen\n\\begin{equation}\n\\Pr\\left(\n\\text{every accepted lineage edge has }\\Delta_t>0\n\\right)\n\\geq\n1-\\delta.\n\\label{eq:b8-lineage-validity}\n\\end{equation}\n\\end{theorem}","\\begin{proof}\nA 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\n\\begin{equation}\n\\Pr(\\exists t\\text{ false accepted})\n\\leq\n\\sum_t\\delta_t\n\\leq\n\\delta.\n\\end{equation}\n\\end{proof}","For a branching archive, index proposed edges by $e\\in E(\\Lambda)$. Assign each edge a budget $\\delta_e>0$ with\n\\begin{equation}\n\\sum_{e\\in E(\\Lambda)}\\delta_e\\leq\\delta.\n\\end{equation}\nThe 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.","A simple geometric schedule is\n\\begin{equation}\n\\delta_e\n=\n\\delta 2^{-\\iota(e)},\n\\end{equation}\nwhere $\\iota$ 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.","Let $R_t$ be true protected regression and let $U_t^R$ be a conditional upper confidence bound. Require\n\\begin{equation}\nU_t^R\\leq\\beta_t,\n\\qquad\n\\sum_t\\beta_t\\leq B_R.\n\\label{eq:b8-regression-budget}\n\\end{equation}\nLet $d_t$ be semantic debt incurred by approximate grammar or evaluator transport, with\n\\begin{equation}\n\\sum_td_t\\leq B_{\\mathrm{sem}}.\n\\label{eq:b8-semantic-budget}\n\\end{equation}\nA lineage certificate should therefore report three separate cumulative quantities:\n\\begin{equation}\n\\left(\n\\sum_t\\delta_t,\n\\sum_t\\beta_t,\n\\sum_td_t\n\\right).\n\\label{eq:b8-three-budgets}\n\\end{equation}\nA positive gain sequence is not sufficient if its protected regression or semantic debt diverges.","quire\n\\begin{equation}\nU_t^R\\leq\\beta_t,\n\\qquad\n\\sum_t\\beta_t\\leq B_R.\n\\label{eq:b8-regression-budget}\n\\end{equation}\nLet $d_t$ be semantic debt incurred by approximate grammar or evaluator transport, with\n\\begin{equation}\n\\sum_td_t\\leq B_{\\mathrm{sem}}.\n\\label{eq:b8-semantic-budget}\n\\end{equation}\nA lineage certificate should therefore report three separate cumulative quantities:\n\\begin{equation}\n\\left(\n\\sum_t\\delta_t,\n\\sum_t\\beta_t,\n\\sum_td_t\n\\right).\n\\label{eq:b8-three-budgets}\n\\end{equation}\nA positive gain sequence is not sufficient if its protected regression or semantic debt diverges.","\\begin{definition}[Repeated-lineage certificate]\n\\label{def:b8-repeated-certificate}\nA finite lineage path\n\\begin{equation}\nK_0\\to K_1\\to\\cdots\\to K_T\n\\end{equation}\nhas a repeated-lineage metaproductivity certificate when:\n\\begin{enumerate}\n\\item every edge uses disjoint development, meta-validation, and final audit evidence;\n\\item candidate, query, compute, and validation resources are fixed or explicitly accounted;\n\\item every edge has a positive conditional lower metaproductivity bound;\n\\item cumulative false-acceptance, protected-regression, and semantic-debt budgets remain below declared limits;\n\\item prior proofs, receipts, and protected distinctions replay after every edge;\n\\item the archive retains every parent as an executable alternative;\n\\item every claimed structural edge reduces held-out obstruction energy or adds a validated","\\begin{proposition}[Two-edge significance]\nA certified path\n\\begin{equation}\nK_0\\to K_1\\to K_2\n\\end{equation}\nwith 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.\n\\end{proposition}","\\begin{conditionaltheorem}[Bottleneck-migrating recursive improvement]\n\\label{thm:b8-integrated}\nLet\n\\begin{equation}\nL_t\n=\n\\left(\nK_t,\\Gamma_t,\\Gcal_t,\\Ecal_t,\\Lambda_t,V_t,W_t\n\\right)\n\\end{equation}\nbe a sequence of resolved lineage states. Assume:\n\\begin{enumerate}\n\\item every edge is evaluated against externally grounded evidence unavailable before sealing;\n\\item componentwise restricted oracle envelopes are finite or bounded by valid certificates;\n\\item inside each fixed grammar, an authorized meta-controller satisfies the weighted contraction condition of \\cref{thm:b8-lyapunov};\n\\item saturation is detected before further same-module gains are claimed;\n\\item a grammar extension is accepted only when it resolves a repair conflict or adds a response direction with positive held-out obstruction overlap;\n\\item old evidence, proofs, authorization decisions, and protected m","d-grammar interval;\n\\item progress may continue after one module saturates by migrating to another non-saturated module;\n\\item every accepted structural grammar edge strictly reduces held-out obstruction energy in at least one direction unavailable to the old grammar;\n\\item historical protected meanings remain recoverable within the cumulative semantic-debt bound;\n\\item with probability at least $1-\\sum_t\\delta_t$, every accepted edge has positive true metaproductivity relative to its declared claim and budget;\n\\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.\n\\end{enumerate}\n\\end{conditionaltheorem}","\\midrule\nTotal declared headroom & $0.740000000000$","Sum of Shapley module allocations & $0.740000000000$","Response algebra dimension & $8$","Commutant dimension & $2$","Central block ranks & $(2,2)$","Cyclic defect dimensions & $2$ for one block, $4$ for mixed support","Cumulative confidence budget & $0.01875$","Automated tests & $8/8$ passed","The next learned or trained experiment should be staged so that the bottleneck genuinely changes:\n\\begin{enumerate}\n\\item $K_0$ uses random anchors and a conservative fixed selector;\n\\item $K_1$ uses the Block 7 rank-revealing query policy and learned risk-sensitive selector;\n\\item $K_2$ faces full-rank but noisy and ill-conditioned bridges and must discover a conditioning-aware design criterion;\n\\item $K_3$ faces systematic failure of every global linear bridge and must produce a grammar-inadequacy certificate followed by local or nonlinear transport;\n\\item $K_4$ no longer receives a supplied causal packet and must recover a functional atlas from intervention-response operators.\n\\end{enumerate}\nEach edge requires fresh development, validation, and audit evidence. A positive result at one edge may not be reused as","The next learned or trained experiment should be staged so that the bottleneck genuinely changes:\n\\begin{enumerate}\n\\item $K_0$ uses random anchors and a conservative fixed selector;\n\\item $K_1$ uses the Block 7 rank-revealing query policy and learned risk-sensitive selector;\n\\item $K_2$ faces full-rank but noisy and ill-conditioned bridges and must discover a conditioning-aware design criterion;\n\\item $K_3$ faces systematic failure of every global linear bridge and must produce a grammar-inadequacy certificate followed by local or nonlinear transport;\n\\item $K_4$ no longer receives a supplied causal packet and must recover a functional atlas from intervention-response operators.\n\\end{enumerate}\nEach edge requires fresh development, validation, and audit evidence. A positive result at one edge may not be reused as evidence for the next.","t learned or trained experiment should be staged so that the bottleneck genuinely changes:\n\\begin{enumerate}\n\\item $K_0$ uses random anchors and a conservative fixed selector;\n\\item $K_1$ uses the Block 7 rank-revealing query policy and learned risk-sensitive selector;\n\\item $K_2$ faces full-rank but noisy and ill-conditioned bridges and must discover a conditioning-aware design criterion;\n\\item $K_3$ faces systematic failure of every global linear bridge and must produce a grammar-inadequacy certificate followed by local or nonlinear transport;\n\\item $K_4$ no longer receives a supplied causal packet and must recover a functional atlas from intervention-response operators.\n\\end{enumerate}\nEach edge requires fresh development, validation, and audit evidence. A positive result at one edge may not be reused as evidence for the next.","fixed selector;\n\\item $K_1$ uses the Block 7 rank-revealing query policy and learned risk-sensitive selector;\n\\item $K_2$ faces full-rank but noisy and ill-conditioned bridges and must discover a conditioning-aware design criterion;\n\\item $K_3$ faces systematic failure of every global linear bridge and must produce a grammar-inadequacy certificate followed by local or nonlinear transport;\n\\item $K_4$ no longer receives a supplied causal packet and must recover a functional atlas from intervention-response operators.\n\\end{enumerate}\nEach edge requires fresh development, validation, and audit evidence. A positive result at one edge may not be reused as evidence for the next.","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.","Block 9 instantiates the bottleneck-migration theory of Block 8. The frozen Block 7 bridge child becomes the lineage parent\n\\begin{equation}\nK_1.\n\\end{equation}\nIt 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.","The descendant compiler receives only new development receipts. It diagnoses full-rank saturation, activates a finite conditioning-aware query grammar, selects a grandchild\n\\begin{equation}\nK_2,\n\\end{equation}\nand seals the resulting policy before the fresh audit tasks and representation families are executed. The primary comparison is\n\\begin{equation}\n\\Mcal_{\\mathrm{audit}}(K_2)-\\Mcal_{\\mathrm{audit}}(K_1^{\\rm recal}),\n\\end{equation}\nwhere $K_1^{\\rm recal}$ 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.","lopment receipts. It diagnoses full-rank saturation, activates a finite conditioning-aware query grammar, selects a grandchild\n\\begin{equation}\nK_2,\n\\end{equation}\nand seals the resulting policy before the fresh audit tasks and representation families are executed. The primary comparison is\n\\begin{equation}\n\\Mcal_{\\mathrm{audit}}(K_2)-\\Mcal_{\\mathrm{audit}}(K_1^{\\rm recal}),\n\\end{equation}\nwhere $K_1^{\\rm recal}$ 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.","\\begin{definition}[Context-indexed lineage edge]\nLet $\\Xi_t$ be the protected context for edge $t\\to t+1$. A sealed lineage edge\n\\begin{equation}\nK_t\\xrightarrow{\\Pi_{t+1}}K_{t+1}\n\\end{equation}\nis $\\gamma_t$-productive in $\\Xi_t$ when its receipt replays, its resource and witness gates pass, and a fresh paired audit certifies\n\\begin{equation}\n\\Mcal_{\\Xi_t}(K_{t+1})-\\Mcal_{\\Xi_t}(K_t)\\geq \\gamma_t>0.\n\\end{equation}\n\\end{definition}","\\begin{definition}[Finite repeated-lineage certificate]\nA sequence\n\\begin{equation}\nK_0\\xrightarrow{\\Pi_1}K_1\\xrightarrow{\\Pi_2}\\cdots\\xrightarrow{\\Pi_m}K_m\n\\end{equation}\nhas 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.\n\\end{definition}","\\begin{proposition}[Same-context transitivity]\nSuppose every edge is evaluated in one common context $\\Xi$. Then\n\\begin{equation}\n\\Mcal_{\\Xi}(K_m)-\\Mcal_{\\Xi}(K_0)\n=\n\\sum_{t=0}^{m-1}\n\\left(\n\\Mcal_{\\Xi}(K_{t+1})-\\Mcal_{\\Xi}(K_t)\n\\right).\n\\end{equation}\nConsequently, positive lower bounds on all edges add to a positive lower bound on the complete path.\n\\end{proposition}","\\begin{warning}[Fresh contexts do not produce one scalar ordering]\nIf 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.\n\\end{warning}","Let $X_S\\in\\R^{n\\times d}$ be the source hidden-state design induced by anchor set $S$. Rank recovery is saturated when\n\\begin{equation}\n\\Pr\\left(\\rank X_S=d\\right)\\geq 1-\\eps_{\\rm rank}\n\\end{equation}\nunder the current foreign development distribution and the remaining performance gap cannot be explained by rank failure.","\\begin{definition}[Conditioning bottleneck certificate]\nA rank-revealing parent has a conditioning bottleneck with margin $\\kappa>0$ when:\n\\begin{enumerate}\n\\item its full-rank rate is at least $1-\\eps_{\\rm rank}$;\n\\item its weighted information floor is below the declared stability floor;\n\\item a conditioning-aware policy using the same query budget improves development metaproductivity by at least $\\kappa$;\n\\item the improvement survives recalibration of the rank parent and is not obtained only by changing the commit gate.\n\\end{enumerate}\n\\end{definition}","in{definition}[Conditioning bottleneck certificate]\nA rank-revealing parent has a conditioning bottleneck with margin $\\kappa>0$ when:\n\\begin{enumerate}\n\\item its full-rank rate is at least $1-\\eps_{\\rm rank}$;\n\\item its weighted information floor is below the declared stability floor;\n\\item a conditioning-aware policy using the same query budget improves development metaproductivity by at least $\\kappa$;\n\\item the improvement survives recalibration of the rank parent and is not obtained only by changing the commit gate.\n\\end{enumerate}\n\\end{definition}","This certificate separates two statements:\n\\begin{equation}\n\\rank X_S=d\n\\end{equation}\nand\n\\begin{equation}\n\\lambda_{\\min}(X_S^{\\mathsf T}WX_S)\\gg0.\n\\end{equation}\nThe first is qualitative identifiability. The second is quantitative stability.","\\begin{proposition}[Rank saturation does not imply stability]\nFor every $d\\geq2$ and every $\\eta>0$, there exists a full-rank design $X\\in\\R^{d\\times d}$ such that\n\\begin{equation}\n\\lambda_{\\min}(X^{\\mathsf T}X)<\\eta.\n\\end{equation}\n\\end{proposition}","\\begin{proof}\nTake\n\\begin{equation}\nX=\\diag(1,\\ldots,1,\\sqrt{\\eta/2}).\n\\end{equation}\nThen $X$ is full rank and\n\\begin{equation}\n\\lambda_{\\min}(X^{\\mathsf T}X)=\\eta/2<\\eta.\n\\end{equation}\n\\end{proof}","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\n\\begin{equation}\ny_i=Tx_i+\\eps_i,\n\\qquad\n\\mathbb E[\\eps_i]=0,\n\\qquad\n\\Cov(\\eps_i)=\\sigma_i^2I_q.\n\\end{equation}\nThe 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.","Stacking the selected anchors gives\n\\begin{equation}\nY=XT^{\\mathsf T}+E,\n\\end{equation}\nwhere $X\\in\\R^{n\\times d}$ and $Y\\in\\R^{n\\times q}$. Define\n\\begin{equation}\nW=\\diag(\\sigma_1^{-2},\\ldots,\\sigma_n^{-2})\n\\end{equation}\nand the weighted information matrix\n\\begin{equation}\nM_S=X^{\\mathsf T}WX.\n\\end{equation}","\\begin{theorem}[Weighted bridge covariance]\n\\label{thm:b9-wls}\nAssume $M_S$ is invertible and the noise rows are independent with covariance $\\sigma_i^2I_q$. The weighted least-squares estimator\n\\begin{equation}\n\\widehat T^{\\mathsf T}\n=\nM_S^{-1}X^{\\mathsf T}WY\n\\end{equation}\nis unbiased and satisfies\n\\begin{equation}\n\\Cov\\left(\\operatorname{vec}(\\widehat T^{\\mathsf T}-T^{\\mathsf T})\\mid X\\right)\n=\nI_q\\otimes M_S^{-1}.\n\\end{equation}\nConsequently,\n\\begin{equation}\n\\mathbb E\\left[\\norm{\\widehat T-T}_F^2\\mid X\\right]\n=\nq\\,\\tr(M_S^{-1}).\n\\end{equation}\n\\end{theorem}","me $M_S$ is invertible and the noise rows are independent with covariance $\\sigma_i^2I_q$. The weighted least-squares estimator\n\\begin{equation}\n\\widehat T^{\\mathsf T}\n=\nM_S^{-1}X^{\\mathsf T}WY\n\\end{equation}\nis unbiased and satisfies\n\\begin{equation}\n\\Cov\\left(\\operatorname{vec}(\\widehat T^{\\mathsf T}-T^{\\mathsf T})\\mid X\\right)\n=\nI_q\\otimes M_S^{-1}.\n\\end{equation}\nConsequently,\n\\begin{equation}\n\\mathbb E\\left[\\norm{\\widehat T-T}_F^2\\mid X\\right]\n=\nq\\,\\tr(M_S^{-1}).\n\\end{equation}\n\\end{theorem}","\\begin{proof}\nWrite\n\\begin{equation}\n\\widehat T^{\\mathsf T}-T^{\\mathsf T}\n=\nM_S^{-1}X^{\\mathsf T}WE.\n\\end{equation}\nFor one target coordinate, the noise covariance is $W^{-1}$, so the coefficient covariance is\n\\begin{equation}\nM_S^{-1}X^{\\mathsf T}W W^{-1} WXM_S^{-1}=M_S^{-1}.\n\\end{equation}\nThe target coordinates are independent and identically weighted, giving the Kronecker covariance. Taking the trace yields the Frobenius identity.\n\\end{proof}","\\begin{corollary}[Worst-direction variance]\nFor every $u\\in\\R^d$ and every target coordinate,\n\\begin{equation}\n\\Var\\left(u^{\\mathsf T}(\\widehat T^{\\mathsf T}-T^{\\mathsf T})e_j\\mid X\\right)\n=u^{\\mathsf T}M_S^{-1}u\n\\leq\n\\frac{\\norm{u}^2}{\\lambda_{\\min}(M_S)}.\n\\end{equation}\n\\end{corollary}","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.","The executable implementation uses\n\\begin{equation}\nM_{S,\\lambda}=X^{\\mathsf T}WX+\\lambda I\n\\end{equation}\nwith 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.","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.","Let the frozen source repair be\n\\begin{equation}\nR_s(h)=h+D_sh+b_s.\n\\end{equation}\nFor an injective bridge $T$, the exact transported repair is\n\\begin{equation}\nD_t=T D_sT^\\dagger,\n\\qquad\nb_t=Tb_s.\n\\end{equation}\nThe estimated repair uses $\\widehat T$.","\\begin{proposition}[Exact repair-operator perturbation identity]\nLet\n\\begin{equation}\n\\widehat D_t=\\widehat TD_s\\widehat T^\\dagger.\n\\end{equation}\nThen\n\\begin{equation}\n\\widehat D_t-D_t\n=\n(\\widehat T-T)D_s\\widehat T^\\dagger\n+\nT D_s(\\widehat T^\\dagger-T^\\dagger).\n\\end{equation}\nHence\n\\begin{equation}\n\\norm{\\widehat D_t-D_t}\n\\leq\n\\norm{\\widehat T-T}\\norm{D_s}\\norm{\\widehat T^\\dagger}\n+\n\\norm{T}\\norm{D_s}\\norm{\\widehat T^\\dagger-T^\\dagger}.\n\\end{equation}\n\\end{proposition}","\\begin{proof}\nAdd and subtract $TD_s\\widehat T^\\dagger$.\n\\end{proof}","\\begin{conditionaltheorem}[Conditioned repair stability]\nSuppose $T$ has full column rank and\n\\begin{equation}\n\\norm{T^\\dagger}\\norm{\\widehat T-T}<1.\n\\end{equation}\nThen there is a finite constant $C_T$, depending continuously on $\\norm{T}$ and $\\norm{T^\\dagger}$ inside this perturbation neighborhood, such that\n\\begin{equation}\n\\norm{\\widehat D_t-D_t}\n\\leq\nC_T\\norm{D_s}\\norm{\\widehat T-T}.\n\\end{equation}\nIf 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\n\\begin{equation}\nLC_T\\norm{D_s}\\norm{\\widehat T-T}.\n\\end{equation}\n\\end{conditionaltheorem}","stability]\nSuppose $T$ has full column rank and\n\\begin{equation}\n\\norm{T^\\dagger}\\norm{\\widehat T-T}<1.\n\\end{equation}\nThen there is a finite constant $C_T$, depending continuously on $\\norm{T}$ and $\\norm{T^\\dagger}$ inside this perturbation neighborhood, such that\n\\begin{equation}\n\\norm{\\widehat D_t-D_t}\n\\leq\nC_T\\norm{D_s}\\norm{\\widehat T-T}.\n\\end{equation}\nIf 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\n\\begin{equation}\nLC_T\\norm{D_s}\\norm{\\widehat T-T}.\n\\end{equation}\n\\end{conditionaltheorem}","onstant $C_T$, depending continuously on $\\norm{T}$ and $\\norm{T^\\dagger}$ inside this perturbation neighborhood, such that\n\\begin{equation}\n\\norm{\\widehat D_t-D_t}\n\\leq\nC_T\\norm{D_s}\\norm{\\widehat T-T}.\n\\end{equation}\nIf 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\n\\begin{equation}\nLC_T\\norm{D_s}\\norm{\\widehat T-T}.\n\\end{equation}\n\\end{conditionaltheorem}","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.","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.","Let $M\\succ0$ be the current regularized information matrix. Adding candidate anchor $x_i$ with quality weight $w_i=\\sigma_i^{-2}$ gives\n\\begin{equation}\nM'=M+w_ix_ix_i^{\\mathsf T}.\n\\end{equation}","\\begin{theorem}[D-optimal marginal gain]\n\\label{thm:b9-dopt}\nThe increase in log information volume is\n\\begin{equation}\n\\log\\det M'-\\log\\det M\n=\n\\log\\left(1+w_ix_i^{\\mathsf T}M^{-1}x_i\\right).\n\\end{equation}\n\\end{theorem}","\\begin{proof}\nApply the matrix determinant lemma to $M+(\\sqrt{w_i}x_i)(\\sqrt{w_i}x_i)^{\\mathsf T}$.\n\\end{proof}","\\begin{proposition}[A-optimal trace reduction]\nLet $v=\\sqrt{w_i}x_i$. Then\n\\begin{equation}\n\\tr(M^{-1})-\\tr\\left((M+vv^{\\mathsf T})^{-1}\\right)\n=\n\\frac{v^{\\mathsf T}M^{-2}v}{1+v^{\\mathsf T}M^{-1}v}.\n\\end{equation}\n\\end{proposition}","\\begin{proof}\nThe Sherman--Morrison identity gives\n\\begin{equation}\n(M+vv^{\\mathsf T})^{-1}\n=\nM^{-1}-\\frac{M^{-1}vv^{\\mathsf T}M^{-1}}{1+v^{\\mathsf T}M^{-1}v}.\n\\end{equation}\nTaking traces yields the result.\n\\end{proof}","The E-optimal objective is\n\\begin{equation}\n\\max_{|S|=k}\\lambda_{\\min}(M_S).\n\\end{equation}\nThe 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.","The development compiler first evaluates the inherited rank policy. It records:\n\\begin{equation}\n\\begin{aligned}\n&\\text{full-rank rate},\\\\\n&\\text{weighted minimum information eigenvalue},\\\\\n&\\text{weighted condition number},\\\\\n&\\text{recalibrated rank-policy gain}.\n\\end{aligned}\n\\end{equation}\nIt 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.","\\begin{proposition}[Finite bottleneck diagnosis soundness]\nInside 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\n\\begin{equation}\n\\kappa>2\\eta,\n\\end{equation}\nthen empirical maximization selects a policy whose true gain exceeds the inherited optimum.\n\\end{proposition}","\\begin{proof}\nLet $p^\\star$ be the best conditioning policy and $p_0$ the best inherited-policy configuration. Empirical errors of at most $\\eta$ imply\n\\begin{equation}\n\\widehat\\mu(p^\\star)\n\\geq\n\\mu(p^\\star)-\\eta\n>\n\\mu(p_0)+\\eta\n\\geq\n\\widehat\\mu(p_0).\n\\end{equation}\n\\end{proof}","\\begin{enumerate}\n\\item $K_0$ is the random-anchor parent from the Block 7 bridge experiment.\n\\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.\n\\item $K_2$ is the conditioning-aware grandchild compiled from new Block 9 development receipts.\n\\end{enumerate}","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.","The Block 9 development tasks use the trigger families\n\\begin{equation}\n\\{\\texttt{bilinear},\\texttt{wave},\\texttt{offset},\\texttt{radial5}\\}.\n\\end{equation}\nThe final audit uses\n\\begin{equation}\n\\{\\texttt{cubic},\\texttt{triple},\\texttt{ratio},\\texttt{annulus}\\}.\n\\end{equation}\nTask identifiers, teacher seeds, and trigger supports are disjoint across the development and final audit files.","The development primary representation families are\n\\begin{equation}\n\\{\\texttt{dev\\_dense20},\\texttt{dev\\_scaled22},\\texttt{dev\\_block24}\\}.\n\\end{equation}\nThe final primary families are\n\\begin{equation}\n\\{\\texttt{audit\\_rotated20},\\texttt{audit\\_anisotropic24},\\texttt{audit\\_block28}\\}.\n\\end{equation}\nThe stress families are severe ill-conditioning, noninjective compression, and nonlinear tanh representations.","No Block 7 final audit rows are used to compile $K_2$.","Every primary child and grandchild deployment receives:\n\\begin{equation}\n16\\ \\text{fit anchors}\n+\n8\\ \\text{validation anchors}\n=\n24\\ \\text{paired representation queries}.\n\\end{equation}\nThe 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.","A random $32$-fit-anchor arm uses\n\\begin{equation}\n32+8=40\n\\end{equation}\npaired queries and serves as a larger-budget control.","The compiler searches the anchor policies\n\\begin{equation}\n\\begin{aligned}\n\\{&\\texttt{packet\\_qr},\n\\texttt{unweighted\\_d\\_opt},\n\\texttt{quality\\_d\\_opt},\\\\\n&\\texttt{quality\\_a\\_opt},\n\\texttt{quality\\_e\\_opt}\\}\n\\end{aligned}\n\\end{equation}\nand a finite gate grid over calibration gain, protected regression, debiased validation alignment, and minimum weighted information eigenvalue.","For one candidate row $r$, a configuration commits only if\n\\begin{equation}\n\\begin{aligned}\n\\widehat g_{\\rm cal}(r)&\\geq \\tau_g,","For one candidate row $r$, a configuration commits only if\n\\begin{equation}\n\\begin{aligned}\n\\widehat g_{\\rm cal}(r)&\\geq \\tau_g,\\\\\n\\widehat R_{\\rm cal}(r)&\\leq \\tau_R,\\\\\n\\widehat\\eps_{\\rm align}(r)&\\leq \\tau_A,\\\\\n\\lambda_{\\min}(M_r)&\\geq \\tau_I,\\\\\n\\rank X_r&=d.\n\\end{aligned}\n\\end{equation}\nThe selected Block 9 grandchild has\n\\begin{equation}\n(\\tau_g,\\tau_R,\\tau_A,\\tau_I)\n=\n(0.02,0.05,0.20,2.0).\n\\end{equation}\nThe recalibrated rank child has\n\\begin{equation}\n(0.02,0.05,0.55,0.10).\n\\end{equation}","\\rank X_r&=d.\n\\end{aligned}\n\\end{equation}\nThe selected Block 9 grandchild has\n\\begin{equation}\n(\\tau_g,\\tau_R,\\tau_A,\\tau_I)\n=\n(0.02,0.05,0.20,2.0).\n\\end{equation}\nThe recalibrated rank child has\n\\begin{equation}\n(0.02,0.05,0.55,0.10).\n\\end{equation}","\\begin{theorem}[Finite meta-grammar generalization]\nLet $\\mathcal K$ 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\n\\begin{equation}\n\\abs{\\widehat\\Mcal(K)-\\Mcal(K)}\n\\leq\nG\\sqrt{\\frac{2\\log(2N/\\delta)}{n}}.\n\\end{equation}\nIf $\\widehat K$ maximizes the empirical mean, then\n\\begin{equation}\n\\Mcal(\\widehat K)\n\\geq\n\\max_{K\\in\\mathcal K}\\Mcal(K)\n-\n2G\\sqrt{\\frac{2\\log(2N/\\delta)}{n}}.\n\\end{equation}\n\\end{theorem}","n{theorem}[Finite meta-grammar generalization]\nLet $\\mathcal K$ 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\n\\begin{equation}\n\\abs{\\widehat\\Mcal(K)-\\Mcal(K)}\n\\leq\nG\\sqrt{\\frac{2\\log(2N/\\delta)}{n}}.\n\\end{equation}\nIf $\\widehat K$ maximizes the empirical mean, then\n\\begin{equation}\n\\Mcal(\\widehat K)\n\\geq\n\\max_{K\\in\\mathcal K}\\Mcal(K)\n-\n2G\\sqrt{\\frac{2\\log(2N/\\delta)}{n}}.\n\\end{equation}\n\\end{theorem}","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\n\\begin{equation}\n\\abs{\\widehat\\Mcal(K)-\\Mcal(K)}\n\\leq\nG\\sqrt{\\frac{2\\log(2N/\\delta)}{n}}.\n\\end{equation}\nIf $\\widehat K$ maximizes the empirical mean, then\n\\begin{equation}\n\\Mcal(\\widehat K)\n\\geq\n\\max_{K\\in\\mathcal K}\\Mcal(K)\n-\n2G\\sqrt{\\frac{2\\log(2N/\\delta)}{n}}.\n\\end{equation}\n\\end{theorem}","\\begin{proof}\nHoeffding's inequality for a variable with range length $2G$ gives\n\\begin{equation}\n\\Pr\\left(\\abs{\\widehat\\Mcal(K)-\\Mcal(K)}>\\eta\\right)\n\\leq\n2\\exp\\left(-\\frac{n\\eta^2}{2G^2}\\right).\n\\end{equation}\nApply 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.\n\\end{proof}","Let $D_i$ be the grandchild-minus-child protected gain on final foreign target $i$. The primary finite estimand is\n\\begin{equation}\n\\overline D\n=\n\\frac1n\\sum_{i=1}^nD_i.\n\\end{equation}\nThe 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.","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.","\\midrule\nRank-child full-rank rate & $1.000000$","Rank-child mean information eigenvalue & $0.549830$","Rank-child median condition number & $21622.327$","Recalibrated rank-child mean gain & $0.028232$","Best conditioning-policy mean gain & $0.217374$","Conditioning headroom gap & $0.189142$","The compiler selects\n\\begin{equation}\n\\texttt{quality\\_d\\_opt}\n\\end{equation}\nas the grandchild query policy.","The primary audit contains\n\\begin{equation}\n8\\ \\text{tasks}\n\\times\n3\\ \\text{foreign representation families}\n\\times\n4\\ \\text{target seeds}\n=\n96\\ \\text{paired targets}.\n\\end{equation}","\\begin{table}[H]\n\\centering\n\\small\n\\begin{tabular}{@{}lrrrrr@{}}\n\\toprule\nArm & Selected gain & Commit & Full rank & $\\lambda_{\\min}$ & Clean transport","\\midrule\nExact Block 7 child replay & $0.000618$ & $0.002604$ & $1.000$ & $0.518$ & $0.304$","Recalibrated rank child & $0.037393$ & $0.161458$ & $1.000$ & $0.518$ & $0.304$","Conditioning grandchild & $\\mathbf{0.209580}$ & $0.638021$ & $1.000$ & $\\mathbf{14.268}$ & $\\mathbf{0.096}$","Quality-permuted grandchild & $0.000000$ & $0.000000$ & $1.000$ & $0.643$ & $0.293$","Random $32$-anchor control & $0.111028$ & $0.382813$ & $0.844$ & $3.278$ & $0.180$","\\bottomrule\n\\end{tabular}\n\\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.}\n\\label{tab:b9-primary}\n\\end{table}","The paired grandchild-minus-recalibrated-child effect is\n\\begin{equation}\n\\widehat\\Delta_{1\\to2}\n=\n0.172188,\n\\end{equation}\nwith frozen interval\n\\begin{equation}\n[0.141481,\\ 0.203110].\n\\end{equation}\nThe sign-flip estimate is\n\\begin{equation}\np=9.9999\\times10^{-6}.\n\\end{equation}\nThe positive paired-effect rate is\n\\begin{equation}\n0.677083,\n\\end{equation}\nand the negative paired-effect rate is zero. The remaining pairs are ties produced mainly by mutual abstention.","\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.88\\textwidth]{block09_lineage_gain.pdf}\n\\caption{Primary foreign metaproductivity across the repeated lineage and controls. The conditioning grandchild uses the same $24$ paired queries as the rank child.}\n\\label{fig:b9-lineage-gain}\n\\end{figure}","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:\n\\begin{equation}\n\\lambda_{\\min}:\n0.518197\n\\longrightarrow\n14.268195,\n\\end{equation}\n\\begin{equation}\n\\text{median condition number}:\n19393.296\n\\longrightarrow\n544.180,\n\\end{equation}\nand\n\\begin{equation}\n\\text{clean transport NRMSE}:\n0.303981\n\\longrightarrow\n0.095803.\n\\end{equation}","The debiased validation-alignment residual falls from\n\\begin{equation}\n0.261487\n\\end{equation}\nto\n\\begin{equation}\n0.057707.\n\\end{equation}","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\n\\begin{equation}\n0.157691.\n\\end{equation}\nIt means the rank design does not satisfy the grandchild's declared information and alignment certificate.","The quality-permuted grandchild also has zero selected primary gain. Its blind primary gain is\n\\begin{equation}\n0.154203,\n\\end{equation}\nbut its mean information eigenvalue falls from\n\\begin{equation}\n14.268195\n\\end{equation}\nto\n\\begin{equation}\n0.643375.\n\\end{equation}\nThe paired quality-binding certificate has the same positive lower bound pattern as the gate-isolation comparison.","The random $32$-anchor control uses $40$ paired queries and obtains mean primary selected gain\n\\begin{equation}\n0.111028.\n\\end{equation}\nThe conditioning grandchild uses $24$ paired queries and obtains\n\\begin{equation}\n0.209580.\n\\end{equation}\nThe result is therefore not explained by a hidden increase in deployment query count.","The one-time development compiler uses\n\\begin{equation}\n19440\n\\end{equation}\npaired development queries. Relative to the random $32$-anchor arm's additional $16$ deployment queries, the declared query-only break-even point is\n\\begin{equation}\n1215\n\\end{equation}\ndeployments. This accounting does not include engineering labor, model training, or the value of safer abstention.","\\midrule\n\\texttt{audit\\_rotated20} & $0.044815$ & $0.315166$ & $0.984375$","\\texttt{audit\\_anisotropic24} & $0.000$ & $0.001674$ & $0.007813$","\\texttt{audit\\_block28} & $0.067362$ & $0.311900$ & $0.921875$","\\texttt{stress\\_severe\\_ill24} & $0.000$ & $0.000000$ & $0.000000$","\\texttt{stress\\_compressed12} & $0.016366$ & $0.055636$ & $0.343750$","\\texttt{stress\\_tanh24} & $0.000000$ & $0.007651$ & $0.039063$","The Block 7 child policy file has SHA-256 digest\n\\begin{equation}\n\\texttt{46e794fe5335c19a7dd56558128e0e15434447dd644bd9daf8f827c3e139f01c}\n\\end{equation}\nand semantic policy digest\n\\begin{equation}\n\\texttt{c241031f271f1f9a2e6ce7585319fc5eb64467e5fce31a4fb33373e7445bc1d2}.\n\\end{equation}\nThe Block 9 grandchild receipt names that semantic digest as its lineage parent.","The Block 9 pre-audit seal has digest\n\\begin{equation}\n\\texttt{9b3014b1d3c4a8ef09aa34ae47baa134cdadd8cb9250e81f90a9e2dbac60fb07}.\n\\end{equation}\nThe canonical complete result has digest\n\\begin{equation}\n\\texttt{bdb8623fb1f2676f3828b38d51460f8a4d351ac8fd7618205c29908f812ebc56}.\n\\end{equation}","\\begin{theorem}[Finite two-edge repeated-lineage realization]\nWithin the declared Block 7 and Block 9 contexts, the released receipts and audits instantiate two sequential productive edges:\n\\begin{equation}\nK_0\\xrightarrow{\\Pi_1}K_1\\xrightarrow{\\Pi_2}K_2.\n\\end{equation}\nThe 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.\n\\end{theorem}","nd Block 9 contexts, the released receipts and audits instantiate two sequential productive edges:\n\\begin{equation}\nK_0\\xrightarrow{\\Pi_1}K_1\\xrightarrow{\\Pi_2}K_2.\n\\end{equation}\nThe 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.\n\\end{theorem}","eipt 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.\n\\end{proof}","\\begin{statusbox}{Block 9 result}\nBlock 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.\n\\end{statusbox}","k 9 result}\nBlock 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.\n\\end{statusbox}","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.\n\\end{statusbox}","The release establishes a controlled two-edge lineage inside two finite experimental grammars. It does not establish:\n\\begin{itemize}\n\\item autonomous discovery of query reliability;\n\\item autonomous invention of D-optimal design;\n\\item improvement of the outer compiler $\\Gamma_t$;\n\\item candidate-constructor or repair-image growth;\n\\item spontaneous causal-atlas discovery in a pretrained model;\n\\item evaluator self-improvement;\n\\item open-ended recursive closure;\n\\item indefinite acceleration;\n\\item unrestricted safety;\n\\item consciousness or sentience.\n\\end{itemize}","Let $H_s\\cong\\R^d$ be a source state space and $H_t\\cong\\R^q$ a target state space. Write\n\\begin{equation}\n\\bar h=\\begin{bmatrix}h","Let $H_s\\cong\\R^d$ be a source state space and $H_t\\cong\\R^q$ a target state space. Write\n\\begin{equation}\n\\bar h=\\begin{bmatrix}h\\\\1\\end{bmatrix}\\in\\R^{d+1}\n\\end{equation}\nfor homogeneous affine coordinates. Let $c:H_s\\to\\{-1,+1\\}$ be an observable chart variable. A two-chart affine relation is specified by\n\\begin{equation}\n\\bar T_-,\\bar T_+\\in\\R^{q\\times(d+1)},\n\\end{equation}\nwith\n\\begin{equation}\ny=\\bar T_{c(h)}\\bar h+\\varepsilon.\n\\label{eq:b10-chart-map}\n\\end{equation}\nEquivalently, writing\n\\begin{equation}\n\\bar A=\\frac{\\bar T_++\\bar T_-}{2},\n\\qquad\n\\bar B=\\frac{\\bar T_+-\\bar T_-}{2},\n\\end{equation}\nwe have\n\\begin{equation}\ny=(\\bar A+c(h)\\bar B)\\bar h+\\varepsilon.\n\\label{eq:b10-center-contrast}\n\\end{equation}","1\\end{bmatrix}\\in\\R^{d+1}\n\\end{equation}\nfor homogeneous affine coordinates. Let $c:H_s\\to\\{-1,+1\\}$ be an observable chart variable. A two-chart affine relation is specified by\n\\begin{equation}\n\\bar T_-,\\bar T_+\\in\\R^{q\\times(d+1)},\n\\end{equation}\nwith\n\\begin{equation}\ny=\\bar T_{c(h)}\\bar h+\\varepsilon.\n\\label{eq:b10-chart-map}\n\\end{equation}\nEquivalently, writing\n\\begin{equation}\n\\bar A=\\frac{\\bar T_++\\bar T_-}{2},\n\\qquad\n\\bar B=\\frac{\\bar T_+-\\bar T_-}{2},\n\\end{equation}\nwe have\n\\begin{equation}\ny=(\\bar A+c(h)\\bar B)\\bar h+\\varepsilon.\n\\label{eq:b10-center-contrast}\n\\end{equation}","an observable chart variable. A two-chart affine relation is specified by\n\\begin{equation}\n\\bar T_-,\\bar T_+\\in\\R^{q\\times(d+1)},\n\\end{equation}\nwith\n\\begin{equation}\ny=\\bar T_{c(h)}\\bar h+\\varepsilon.\n\\label{eq:b10-chart-map}\n\\end{equation}\nEquivalently, writing\n\\begin{equation}\n\\bar A=\\frac{\\bar T_++\\bar T_-}{2},\n\\qquad\n\\bar B=\\frac{\\bar T_+-\\bar T_-}{2},\n\\end{equation}\nwe have\n\\begin{equation}\ny=(\\bar A+c(h)\\bar B)\\bar h+\\varepsilon.\n\\label{eq:b10-center-contrast}\n\\end{equation}","\\begin{definition}[Inherited global-affine grammar]\n\\label{def:b10-global-grammar}\nThe inherited bridge grammar is\n\\begin{equation}\n\\Gcal_{\\mathrm{glob}}\n=\n\\set{\\bar T:\\R^{d+1}\\to\\R^q\\text{ linear in homogeneous coordinates}}.\n\\end{equation}\nIt fits one affine source-to-target map across every observable context.\n\\end{definition}","\\begin{definition}[Invariant-context two-chart grammar]\n\\label{def:b10-chart-grammar}\nLet $P_{\\mathrm{ctx}}:H_s\\to\\R^r$ be a protected context projector whose value is unchanged by the source repair. Let $\\Fcal$ be a declared family of separators\n\\begin{equation}\ng(h)=\\one\\{\\angles{u,P_{\\mathrm{ctx}}h}>\\theta\\}.\n\\end{equation}\nThe chart extension is\n\\begin{equation}\n\\Gcal_{\\mathrm{chart}}\n=\n\\set{(g,\\bar T_0,\\bar T_1):g\\in\\Fcal,\\ \\bar T_j\\in\\R^{q\\times(d+1)}}.\n\\end{equation}\nIts prediction is\n\\begin{equation}\n\\widehat y(h)=\\bar T_{g(h)}\\bar h.\n\\end{equation}\n\\end{definition}","o\\R^r$ be a protected context projector whose value is unchanged by the source repair. Let $\\Fcal$ be a declared family of separators\n\\begin{equation}\ng(h)=\\one\\{\\angles{u,P_{\\mathrm{ctx}}h}>\\theta\\}.\n\\end{equation}\nThe chart extension is\n\\begin{equation}\n\\Gcal_{\\mathrm{chart}}\n=\n\\set{(g,\\bar T_0,\\bar T_1):g\\in\\Fcal,\\ \\bar T_j\\in\\R^{q\\times(d+1)}}.\n\\end{equation}\nIts prediction is\n\\begin{equation}\n\\widehat y(h)=\\bar T_{g(h)}\\bar h.\n\\end{equation}\n\\end{definition}","\\begin{proposition}[Conservative embedding of the global grammar]\n\\label{prop:b10-conservative-embedding}\nFor every separator $g\\in\\Fcal$, the map\n\\begin{equation}\n\\iota_g:\\Gcal_{\\mathrm{glob}}\\hookrightarrow\\Gcal_{\\mathrm{chart}},\n\\qquad\n\\bar T\\longmapsto(g,\\bar T,\\bar T)\n\\end{equation}\nis injective and preserves every global-affine prediction exactly.\n\\end{proposition}","\\begin{proof}\nBoth 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.\n\\end{proof}","\\begin{theorem}[Open-set incompatibility of distinct affine charts]\n\\label{thm:b10-open-set-obstruction}\nLet $U_0,U_1\\subseteq\\R^d$ each contain a nonempty open set. Let\n\\begin{equation}\nf_j(h)=A_jh+b_j,\n\\qquad j\\in\\{0,1\\}.\n\\end{equation}\nIf one affine map $f(h)=Ah+b$ satisfies\n\\begin{equation}\nf(h)=f_j(h)\n\\end{equation}\nfor every $h\\in U_j$ and both $j$, then\n\\begin{equation}\nA=A_0=A_1,\n\\qquad\nb=b_0=b_1.\n\\end{equation}\nConsequently, if $(A_0,b_0)\\neq(A_1,b_1)$, no element of $\\Gcal_{\\mathrm{glob}}$ represents the two-chart relation exactly.\n\\end{theorem}","f distinct affine charts]\n\\label{thm:b10-open-set-obstruction}\nLet $U_0,U_1\\subseteq\\R^d$ each contain a nonempty open set. Let\n\\begin{equation}\nf_j(h)=A_jh+b_j,\n\\qquad j\\in\\{0,1\\}.\n\\end{equation}\nIf one affine map $f(h)=Ah+b$ satisfies\n\\begin{equation}\nf(h)=f_j(h)\n\\end{equation}\nfor every $h\\in U_j$ and both $j$, then\n\\begin{equation}\nA=A_0=A_1,\n\\qquad\nb=b_0=b_1.\n\\end{equation}\nConsequently, if $(A_0,b_0)\\neq(A_1,b_1)$, no element of $\\Gcal_{\\mathrm{glob}}$ represents the two-chart relation exactly.\n\\end{theorem}","-open-set-obstruction}\nLet $U_0,U_1\\subseteq\\R^d$ each contain a nonempty open set. Let\n\\begin{equation}\nf_j(h)=A_jh+b_j,\n\\qquad j\\in\\{0,1\\}.\n\\end{equation}\nIf one affine map $f(h)=Ah+b$ satisfies\n\\begin{equation}\nf(h)=f_j(h)\n\\end{equation}\nfor every $h\\in U_j$ and both $j$, then\n\\begin{equation}\nA=A_0=A_1,\n\\qquad\nb=b_0=b_1.\n\\end{equation}\nConsequently, if $(A_0,b_0)\\neq(A_1,b_1)$, no element of $\\Gcal_{\\mathrm{glob}}$ represents the two-chart relation exactly.\n\\end{theorem}","\\begin{proof}\nThe 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 $j$.\n\\end{proof}","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\n\\begin{equation}\ny=(A+sB)h.\n\\end{equation}\nThen the best global linear map is\n\\begin{equation}\nT^*=A+(2p-1)B,\n\\end{equation}\nand\n\\begin{equation}\n\\inf_T\\mathbb E\\norm{y-Th}^2\n=\n4p(1-p)\\tr(B\\Sigma B^\\top).\n\\label{eq:b10-population-residual}\n\\end{equation}\nThe 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","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\n\\begin{equation}\ny=(A+sB)h.\n\\end{equation}\nThen the best global linear map is\n\\begin{equation}\nT^*=A+(2p-1)B,\n\\end{equation}\nand\n\\begin{equation}\n\\inf_T\\mathbb E\\norm{y-Th}^2\n=\n4p(1-p)\\tr(B\\Sigma B^\\top).\n\\label{eq:b10-population-residual}\n\\end{equation}\nThe 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}.","Let the source repair be\n\\begin{equation}\nR_s(h)=M_sh+c_s.\n\\end{equation}\nOn chart $j$, let the target bridge be\n\\begin{equation}\nT_j(h)=A_jh+b_j,\n\\end{equation}\nwith $A_j$ invertible on the repair-relevant subspace. Define\n\\begin{equation}\nR_{t,j}(y)\n=\nT_j\\!\\left(R_s(T_j^{-1}(y))\\right).\n\\label{eq:b10-chart-repair-def}\n\\end{equation}\nExpanding gives\n\\begin{equation}\nR_{t,j}(y)\n=\nA_jM_sA_j^{-1}(y-b_j)+A_jc_s+b_j.\n\\label{eq:b10-chart-repair}\n\\end{equation}","\\begin{theorem}[Exact chartwise affine repair transport]\n\\label{thm:b10-chart-transport}\nUnder the preceding invertibility assumptions,\n\\begin{equation}\nR_{t,j}(T_j(h))=T_j(R_s(h))\n\\end{equation}\nfor every repair-relevant $h$ and each chart $j$.\n\\end{theorem}","\\begin{proof}\nSubstitute $T_j(h)=A_jh+b_j$ into \\cref{eq:b10-chart-repair}:\n\\begin{align}\nR_{t,j}(T_j(h))\n&=A_jM_sA_j^{-1}(A_jh)+A_jc_s+b_j","\\begin{proof}\nSubstitute $T_j(h)=A_jh+b_j$ into \\cref{eq:b10-chart-repair}:\n\\begin{align}\nR_{t,j}(T_j(h))\n&=A_jM_sA_j^{-1}(A_jh)+A_jc_s+b_j\\\\\n&=A_j(M_sh+c_s)+b_j\\\\\n&=T_j(R_s(h)).\n\\end{align}\n\\end{proof}","The inherited global target-repair response lies in the diagonal class\n\\begin{equation}\n\\Dcal_{\\mathrm{diag}}\n=\n\\set{(R,R):R\\in\\operatorname{Aff}(H_t)},\n\\end{equation}\nwhereas the chart grammar reaches\n\\begin{equation}\n\\Dcal_{\\mathrm{chart}}\n=\n\\operatorname{Aff}(H_t)\\times\\operatorname{Aff}(H_t).\n\\end{equation}","\\begin{proposition}[New chart-contrast correction direction]\n\\label{prop:b10-repair-contrast}\nLet the affine target repairs have homogeneous matrices $\\bar R_0$ and $\\bar R_1$. If\n\\begin{equation}\n\\bar R_0\\neq\\bar R_1,\n\\end{equation}\nthen\n\\begin{equation}\n\\bar R_{\\mathrm{ctr}}\n=\n\\frac{\\bar R_1-\\bar R_0}{2}\n\\end{equation}\ninduces a nonzero anti-diagonal component\n\\begin{equation}\n(-\\bar R_{\\mathrm{ctr}},\\bar R_{\\mathrm{ctr}})\n\\end{equation}\northogonal to every diagonal pair under the product Frobenius inner product. Therefore the chart grammar contains a correction direction unavailable to the parent grammar.\n\\end{proposition}","\\begin{proof}\nFor every diagonal pair $(\\bar R,\\bar R)$,\n\\begin{equation}\n\\angles{(-\\bar R_{\\mathrm{ctr}},\\bar R_{\\mathrm{ctr}}),(\\bar R,\\bar R)}_F=0.\n\\end{equation}\nThe anti-diagonal component is nonzero exactly when the two chart repairs differ.\n\\end{proof}","The executable package records a normalized chart-specific bridge contrast\n\\begin{equation}\n\\kappa_T\n=\n\\frac{\\norm{\\bar T_1-\\bar T_0}_F}\n{\\frac12(\\norm{\\bar T_1}_F+\\norm{\\bar T_0}_F)+10^{-12}}.\n\\label{eq:b10-contrast}\n\\end{equation}\nA 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.","In the executable fixture,\n\\begin{equation}\nP_{\\mathrm{ctx}}h=(h_1,h_2),\n\\end{equation}\nand the source repair leaves these coordinates invariant. The initial separator vocabulary is\n\\begin{equation}\n\\Fcal_X\n=\n\\set{\n\\one\\{\\angles{u_k,P_{\\mathrm{ctx}}h}>\\theta\\}\n:\n0\\leq k<128,\n\\ \\theta\\in\\Theta_k(X)\n},\n\\end{equation}\nwhere\n\\begin{equation}\nu_k=(\\cos(2\\pi k/128),\\sin(2\\pi k/128))\n\\end{equation}\nand $\\Theta_k(X)$ 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","In the executable fixture,\n\\begin{equation}\nP_{\\mathrm{ctx}}h=(h_1,h_2),\n\\end{equation}\nand the source repair leaves these coordinates invariant. The initial separator vocabulary is\n\\begin{equation}\n\\Fcal_X\n=\n\\set{\n\\one\\{\\angles{u_k,P_{\\mathrm{ctx}}h}>\\theta\\}\n:\n0\\leq k<128,\n\\ \\theta\\in\\Theta_k(X)\n},\n\\end{equation}\nwhere\n\\begin{equation}\nu_k=(\\cos(2\\pi k/128),\\sin(2\\pi k/128))\n\\end{equation}\nand $\\Theta_k(X)$ 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.","{\\mathrm{ctx}}h=(h_1,h_2),\n\\end{equation}\nand the source repair leaves these coordinates invariant. The initial separator vocabulary is\n\\begin{equation}\n\\Fcal_X\n=\n\\set{\n\\one\\{\\angles{u_k,P_{\\mathrm{ctx}}h}>\\theta\\}\n:\n0\\leq k<128,\n\\ \\theta\\in\\Theta_k(X)\n},\n\\end{equation}\nwhere\n\\begin{equation}\nu_k=(\\cos(2\\pi k/128),\\sin(2\\pi k/128))\n\\end{equation}\nand $\\Theta_k(X)$ 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.","\\begin{theorem}[Gate misclassification bound]\n\\label{thm:b10-gate-bound}\nLet $g$ be the correct chart and $\\widehat g$ an estimated chart. Suppose the two target repair outputs satisfy\n\\begin{equation}\n\\norm{R_1(y)-R_0(y)}\\leq L\n\\end{equation}\nthroughout the evaluated region and\n\\begin{equation}\n\\Pr(\\widehat g\\neq g)\\leq\\epsilon_g.\n\\end{equation}\nThen\n\\begin{equation}\n\\mathbb E\\norm{R_{\\widehat g}(y)-R_g(y)}^2\n\\leq\n\\epsilon_gL^2.\n\\end{equation}\nIf protected utility is $L_J$-Lipschitz in repaired state, then\n\\begin{equation}\n\\abs{\\mathbb E J(R_{\\widehat g}(y))-\\mathbb E J(R_g(y))}\n\\leq\nL_JL\\sqrt{\\epsilon_g}.\n\\end{equation}\n\\end{theorem}","et $g$ be the correct chart and $\\widehat g$ an estimated chart. Suppose the two target repair outputs satisfy\n\\begin{equation}\n\\norm{R_1(y)-R_0(y)}\\leq L\n\\end{equation}\nthroughout the evaluated region and\n\\begin{equation}\n\\Pr(\\widehat g\\neq g)\\leq\\epsilon_g.\n\\end{equation}\nThen\n\\begin{equation}\n\\mathbb E\\norm{R_{\\widehat g}(y)-R_g(y)}^2\n\\leq\n\\epsilon_gL^2.\n\\end{equation}\nIf protected utility is $L_J$-Lipschitz in repaired state, then\n\\begin{equation}\n\\abs{\\mathbb E J(R_{\\widehat g}(y))-\\mathbb E J(R_g(y))}\n\\leq\nL_JL\\sqrt{\\epsilon_g}.\n\\end{equation}\n\\end{theorem}","ed chart. Suppose the two target repair outputs satisfy\n\\begin{equation}\n\\norm{R_1(y)-R_0(y)}\\leq L\n\\end{equation}\nthroughout the evaluated region and\n\\begin{equation}\n\\Pr(\\widehat g\\neq g)\\leq\\epsilon_g.\n\\end{equation}\nThen\n\\begin{equation}\n\\mathbb E\\norm{R_{\\widehat g}(y)-R_g(y)}^2\n\\leq\n\\epsilon_gL^2.\n\\end{equation}\nIf protected utility is $L_J$-Lipschitz in repaired state, then\n\\begin{equation}\n\\abs{\\mathbb E J(R_{\\widehat g}(y))-\\mathbb E J(R_g(y))}\n\\leq\nL_JL\\sqrt{\\epsilon_g}.\n\\end{equation}\n\\end{theorem}","\\begin{proof}\nThe 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.\n\\end{proof}","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\n\\begin{equation}\nK_3(x)=\n\\begin{cases}\nK_C(x),&A_C(x)=1,","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\n\\begin{equation}\nK_3(x)=\n\\begin{cases}\nK_C(x),&A_C(x)=1,\\\\\nK_G(x),&A_C(x)=0\\text{ and the parent gate passes},\\\\\n\\mathrm{no\\mbox{-}op},&\\text{otherwise}.\n\\end{cases}\n\\label{eq:b10-fallback}\n\\end{equation}","\\begin{proposition}[Pointwise fallback under a sound chart gate]\n\\label{prop:b10-fallback}\nIf the chart gate accepts only when true protected utility under $K_C$ is at least that under $K_G$, then\n\\begin{equation}\nJ(K_3(x))\\geq J(K_G(x))\n\\end{equation}\nfor every episode.\n\\end{proposition}","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:\n\\begin{equation}\nK_{2,G}^{96},\n\\qquad\nK_{3,C}^{96}.\n\\end{equation}\nThe 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\n\\begin{equation}\n96\n\\end{equation}\npaired source-target queries. The edge is\n\\begin{equation}\nK_{2,G}^{96}\n\\xrightarrow{\\Pi_3}\nK_{3,C}^{96}.\n\\label{eq:b10-ed","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:\n\\begin{equation}\nK_{2,G}^{96},\n\\qquad\nK_{3,C}^{96}.\n\\end{equation}\nThe 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\n\\begin{equation}\n96\n\\end{equation}\npaired source-target queries. The edge is\n\\begin{equation}\nK_{2,G}^{96}\n\\xrightarrow{\\Pi_3}\nK_{3,C}^{96}.\n\\label{eq:b10-edge}\n\\end{equation}","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:\n\\begin{equation}\nK_{2,G}^{96},\n\\qquad\nK_{3,C}^{96}.\n\\end{equation}\nThe 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\n\\begin{equation}\n96\n\\end{equation}\npaired source-target queries. The edge is\n\\begin{equation}\nK_{2,G}^{96}\n\\xrightarrow{\\Pi_3}\nK_{3,C}^{96}.\n\\label{eq:b10-edge}\n\\end{equation}","ock 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:\n\\begin{equation}\nK_{2,G}^{96},\n\\qquad\nK_{3,C}^{96}.\n\\end{equation}\nThe 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\n\\begin{equation}\n96\n\\end{equation}\npaired source-target queries. The edge is\n\\begin{equation}\nK_{2,G}^{96}\n\\xrightarrow{\\Pi_3}\nK_{3,C}^{96}.\n\\label{eq:b10-edge}\n\\end{equation}","supplies the quality-weighted D-optimal anchor-ordering rule and the published lineage receipt. Block 10 constructs two resource-matched processes:\n\\begin{equation}\nK_{2,G}^{96},\n\\qquad\nK_{3,C}^{96}.\n\\end{equation}\nThe 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\n\\begin{equation}\n96\n\\end{equation}\npaired source-target queries. The edge is\n\\begin{equation}\nK_{2,G}^{96}\n\\xrightarrow{\\Pi_3}\nK_{3,C}^{96}.\n\\label{eq:b10-edge}\n\\end{equation}","r-ordering rule and the published lineage receipt. Block 10 constructs two resource-matched processes:\n\\begin{equation}\nK_{2,G}^{96},\n\\qquad\nK_{3,C}^{96}.\n\\end{equation}\nThe 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\n\\begin{equation}\n96\n\\end{equation}\npaired source-target queries. The edge is\n\\begin{equation}\nK_{2,G}^{96}\n\\xrightarrow{\\Pi_3}\nK_{3,C}^{96}.\n\\label{eq:b10-edge}\n\\end{equation}","10 constructs two resource-matched processes:\n\\begin{equation}\nK_{2,G}^{96},\n\\qquad\nK_{3,C}^{96}.\n\\end{equation}\nThe 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\n\\begin{equation}\n96\n\\end{equation}\npaired source-target queries. The edge is\n\\begin{equation}\nK_{2,G}^{96}\n\\xrightarrow{\\Pi_3}\nK_{3,C}^{96}.\n\\label{eq:b10-edge}\n\\end{equation}","\\qquad\nK_{3,C}^{96}.\n\\end{equation}\nThe 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\n\\begin{equation}\n96\n\\end{equation}\npaired source-target queries. The edge is\n\\begin{equation}\nK_{2,G}^{96}\n\\xrightarrow{\\Pi_3}\nK_{3,C}^{96}.\n\\label{eq:b10-edge}\n\\end{equation}","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.","The main paired estimands are\n\\begin{equation}\n\\Delta_{G\\to C}\n=\n\\mathbb E\n\\left[\nG(K_{3,C}^{96},A)-G(K_{2,G}^{96},A)\n\\right]\n\\end{equation}\nand\n\\begin{equation}\n\\Delta_{\\mathrm{oracle}\\to C}\n=\n\\mathbb E\n\\left[\nG(K_{3,C}^{96},A)-G(K_{\\mathrm{affine\\ oracle}},A)\n\\right].\n\\end{equation}\nThe second quantity is the sharper structural test.","The source state is\n\\begin{equation}\nh=(h_1,h_2,h_3,h_4)\\in\\R^4.\n\\end{equation}\nThe first two coordinates form the protected context packet. The source repair is\n\\begin{equation}\nR_s(h)=M_sh+c_s,\n\\end{equation}\nwhere\n\\begin{equation}\nM_s=\n\\begin{bmatrix}\n1&0&0&0","The source state is\n\\begin{equation}\nh=(h_1,h_2,h_3,h_4)\\in\\R^4.\n\\end{equation}\nThe first two coordinates form the protected context packet. The source repair is\n\\begin{equation}\nR_s(h)=M_sh+c_s,\n\\end{equation}\nwhere\n\\begin{equation}\nM_s=\n\\begin{bmatrix}\n1&0&0&0\\\\\n0&1&0&0\\\\\n0&0&0.10&1.10\\\\\n0&0&0&1.50\n\\end{bmatrix},\n\\qquad\nc_s=\n\\begin{bmatrix}\n0\\\\0\\\\0.30\\\\-0.25\n\\end{bmatrix}.\n\\label{eq:b10-source-repair}\n\\end{equation}\nHence\n\\begin{equation}\nP_{\\mathrm{ctx}}R_s(h)=P_{\\mathrm{ctx}}h.\n\\end{equation}\nThe observable chart is generated by a task-specific separator\n\\begin{equation}\nc(h)=\\one\\{\\angles{u_*,P_{\\mathrm{ctx}}h}>\\theta_*\\}.\n\\end{equation}\nThe compiler does not receive $u_*$ or $\\theta_*$, but it receives the declared protected packet in which a valid separator may be sought.","-0.25\n\\end{bmatrix}.\n\\label{eq:b10-source-repair}\n\\end{equation}\nHence\n\\begin{equation}\nP_{\\mathrm{ctx}}R_s(h)=P_{\\mathrm{ctx}}h.\n\\end{equation}\nThe observable chart is generated by a task-specific separator\n\\begin{equation}\nc(h)=\\one\\{\\angles{u_*,P_{\\mathrm{ctx}}h}>\\theta_*\\}.\n\\end{equation}\nThe compiler does not receive $u_*$ or $\\theta_*$, but it receives the declared protected packet in which a valid separator may be sought.","Every primary target is four dimensional and has two invertible affine charts:\n\\begin{equation}\ny=A_{c(h)}h+b+\\varepsilon.\n\\end{equation}\nThe 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.","\\midrule\nDevelopment & $6$ & dev-chart-a/b/c & $54$","Meta-validation & $4$ & meta-chart-h/i & $24$","Primary audit & $8$ & audit-chart-d/e/f/g & $96$","Stress & $4$ & three-chart, smooth, compressed, chart-crossing & $32$","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\n\\begin{equation}\n\\log\\det\\left(\n\\lambda I+\n\\sum_{i\\in S}\\sigma_i^{-2}\\bar h_i\\bar h_i^\\top\n\\right).\n\\end{equation}\nBoth 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.","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\n\\begin{equation}\n\\log\\det\\left(\n\\lambda I+\n\\sum_{i\\in S}\\sigma_i^{-2}\\bar h_i\\bar h_i^\\top\n\\right).\n\\end{equation}\nBoth 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.","ation scale $\\sigma_i$. The inherited acquisition policy selects rows sequentially to maximize the quality-weighted log-determinant increment\n\\begin{equation}\n\\log\\det\\left(\n\\lambda I+\n\\sum_{i\\in S}\\sigma_i^{-2}\\bar h_i\\bar h_i^\\top\n\\right).\n\\end{equation}\nBoth 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.","acquisition policy selects rows sequentially to maximize the quality-weighted log-determinant increment\n\\begin{equation}\n\\log\\det\\left(\n\\lambda I+\n\\sum_{i\\in S}\\sigma_i^{-2}\\bar h_i\\bar h_i^\\top\n\\right).\n\\end{equation}\nBoth 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.","ize the quality-weighted log-determinant increment\n\\begin{equation}\n\\log\\det\\left(\n\\lambda I+\n\\sum_{i\\in S}\\sigma_i^{-2}\\bar h_i\\bar h_i^\\top\n\\right).\n\\end{equation}\nBoth 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.","For current target state $y$, candidate repaired state $\\widetilde y$, and oracle repaired state $y^\\star$, define\n\\begin{equation}\nG\n=\n\\operatorname{clip}_{[-1,1]}\n\\left(\n1-\n\\frac{\\mathbb E\\norm{\\widetilde y-y^\\star}^2}\n{\\mathbb E\\norm{y-y^\\star}^2}\n\\right).\n\\label{eq:b10-gain}\n\\end{equation}\nThe witness records:\n\\begin{enumerate}\n\\item validation gain;\n\\item source-to-target alignment NRMSE;\n\\item decoded protected-context regression;\n\\item minimum chart support;\n\\item alignment reduction relative to the global bridge;\n\\item source-gate and target-gate agreement;\n\\item target-gate margin.\n\\end{enumerate}","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 $0.35$.","e 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 $0.35$.","uires 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 $0.35$.","The parent is selected from\n\\begin{equation}\n4\\times4\\times4=64\n\\end{equation}\ngate configurations. The child is selected from\n\\begin{equation}\n4\\times4\\times4\\times4\\times4\\times3\\times3=9216\n\\end{equation}\nconfigurations. 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.","The package includes:\n\\begin{enumerate}\n\\item \\textbf{Resource-matched global parent.} One global bridge, $80$ fit pairs, $16$ witness pairs.\n\\item \\textbf{Direct global-affine oracle.} One target repair map fitted from $1600$ directly supervised pairs. It is an inadmissible ceiling comparison.\n\\item \\textbf{Local blind application.} The learned chart repair is always applied, bypassing the witness.\n\\item \\textbf{Gate-only control.} The local gate is retained but both regions receive the same global repair.\n\\item \\textbf{Random-gate control.} Two local bridges are bound through a random source partition.\n\\item \\textbf{Shuffled-binding control.} The two learned local repairs are exchanged between predicted charts.\n\\item \\textbf{True-chart oracle.} The correct chart labels are supplied. This estimate","The package includes:\n\\begin{enumerate}\n\\item \\textbf{Resource-matched global parent.} One global bridge, $80$ fit pairs, $16$ witness pairs.\n\\item \\textbf{Direct global-affine oracle.} One target repair map fitted from $1600$ directly supervised pairs. It is an inadmissible ceiling comparison.\n\\item \\textbf{Local blind application.} The learned chart repair is always applied, bypassing the witness.\n\\item \\textbf{Gate-only control.} The local gate is retained but both regions receive the same global repair.\n\\item \\textbf{Random-gate control.} Two local bridges are bound through a random source partition.\n\\item \\textbf{Shuffled-binding control.} The two learned local repairs are exchanged between predicted charts.\n\\item \\textbf{True-chart oracle.} The correct chart labels are supplied. This estimates opportunity in","The package includes:\n\\begin{enumerate}\n\\item \\textbf{Resource-matched global parent.} One global bridge, $80$ fit pairs, $16$ witness pairs.\n\\item \\textbf{Direct global-affine oracle.} One target repair map fitted from $1600$ directly supervised pairs. It is an inadmissible ceiling comparison.\n\\item \\textbf{Local blind application.} The learned chart repair is always applied, bypassing the witness.\n\\item \\textbf{Gate-only control.} The local gate is retained but both regions receive the same global repair.\n\\item \\textbf{Random-gate control.} Two local bridges are bound through a random source partition.\n\\item \\textbf{Shuffled-binding control.} The two learned local repairs are exchanged between predicted charts.\n\\item \\textbf{True-chart oracle.} The correct chart labels are supplied. This estimates opportunity inside the declared two-chart family.\n\\item \\textbf{Stress families.} Three-chart, smooth, compressed,","All $54$ development targets have full source-design rank. The inherited global bridge has mean validation alignment NRMSE\n\\begin{equation}\n\\widehat\\eps_{\\mathrm{glob}}=0.632211,\n\\end{equation}\nwhereas the learned two-chart constructor has\n\\begin{equation}\n\\widehat\\eps_{\\mathrm{chart}}=0.068871.\n\\end{equation}\nThe mean reduction is\n\\begin{equation}\n0.563340.\n\\end{equation}","The directly supervised global-affine repair oracle has mean gain\n\\begin{equation}\n0.152169\n\\end{equation}\nand maximum gain\n\\begin{equation}\n0.364965\n\\end{equation}\nacross development. By contrast, the true-chart oracle has mean gain\n\\begin{equation}\n0.998046,\n\\end{equation}\nand the learned local constructor has mean blind gain\n\\begin{equation}\n0.613651.\n\\end{equation}\nEvery one of the $54$ development targets satisfies the frozen repair-conflict predicate\n\\begin{equation}\nG_{\\mathrm{global\\ oracle}}\\leq0.40,\n\\qquad\nG_{\\mathrm{true\\ chart}}\\geq0.95.\n\\end{equation}\nMean 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.","tion}\n0.152169\n\\end{equation}\nand maximum gain\n\\begin{equation}\n0.364965\n\\end{equation}\nacross development. By contrast, the true-chart oracle has mean gain\n\\begin{equation}\n0.998046,\n\\end{equation}\nand the learned local constructor has mean blind gain\n\\begin{equation}\n0.613651.\n\\end{equation}\nEvery one of the $54$ development targets satisfies the frozen repair-conflict predicate\n\\begin{equation}\nG_{\\mathrm{global\\ oracle}}\\leq0.40,\n\\qquad\nG_{\\mathrm{true\\ chart}}\\geq0.95.\n\\end{equation}\nMean 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.","true-chart oracle has mean gain\n\\begin{equation}\n0.998046,\n\\end{equation}\nand the learned local constructor has mean blind gain\n\\begin{equation}\n0.613651.\n\\end{equation}\nEvery one of the $54$ development targets satisfies the frozen repair-conflict predicate\n\\begin{equation}\nG_{\\mathrm{global\\ oracle}}\\leq0.40,\n\\qquad\nG_{\\mathrm{true\\ chart}}\\geq0.95.\n\\end{equation}\nMean 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.","\\end{equation}\nand the learned local constructor has mean blind gain\n\\begin{equation}\n0.613651.\n\\end{equation}\nEvery one of the $54$ development targets satisfies the frozen repair-conflict predicate\n\\begin{equation}\nG_{\\mathrm{global\\ oracle}}\\leq0.40,\n\\qquad\nG_{\\mathrm{true\\ chart}}\\geq0.95.\n\\end{equation}\nMean 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.","The primary audit contains $96$ foreign targets. Parent and child each consume $96$ paired queries. Results are:\n\\begin{center}\n\\begin{tabular}{@{}lrrrrr@{}}\n\\toprule\nProcess & Mean gain & Median & Commit & Positive & Harm","\\midrule\nGlobal parent $K_{2,G}^{96}$ & $0.000000$ & $0.000000$ & $0.000000$ & $0.000000$ & $0$","Chart child $K_{3,C}^{96}$ & $\\mathbf{0.334224}$ & $0.284265$ & $0.572917$ & $0.572917$ & $0$","Direct global-affine oracle & $0.176694$ & $0.138838$ & -- & $1.000000$ & $0$","Local blind application & $0.464387$ & $0.502036$ & $1.000000$ & $0.958333$ & $0.041667$","True-chart oracle & $0.998328$ & $0.998514$ & -- & $1.000000$ & $0$","The paired child-minus-parent effect is\n\\begin{equation}\n\\widehat\\Delta_{G\\to C}=0.334224,\n\\end{equation}\nwith frozen bootstrap interval\n\\begin{equation}\n[0.266459,0.401903].\n\\end{equation}\nThe sign-flip estimate is\n\\begin{equation}\np=9.9999\\times10^{-6}.\n\\end{equation}\nThe positive-pair rate is $0.572917$, the remaining $0.427083$ are conservative abstention ties, and no paired target favors the global parent.","More importantly, the paired child-minus-direct-global-oracle effect is\n\\begin{equation}\n\\widehat\\Delta_{\\mathrm{oracle}\\to C}=0.157530,\n\\end{equation}\nwith frozen interval\n\\begin{equation}\n[0.080863,0.234135].\n\\end{equation}\nThus 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.","Each audit family contains $24$ targets. The chart child obtains:\n\\begin{center}\n\\begin{tabular}{@{}lrrr@{}}\n\\toprule\nFamily & Child gain & Global oracle gain & Child commit","\\midrule\naudit-chart-d & $0.280114$ & $0.168712$ & $0.625000$","audit-chart-e & $0.336567$ & $0.149444$ & $0.625000$","audit-chart-f & $0.330004$ & $0.189899$ & $0.458333$","audit-chart-g & $0.390209$ & $0.198721$ & $0.583333$","On the primary audit, mean validation alignment NRMSE changes from\n\\begin{equation}\n\\widehat\\eps_{\\mathrm{glob}}=0.598513\n\\end{equation}\nto\n\\begin{equation}\n\\widehat\\eps_{\\mathrm{chart}}=0.064975.\n\\end{equation}\nMean learned gate accuracy is\n\\begin{equation}\n0.874854.\n\\end{equation}\nThe directly supervised global-affine oracle leaves mean residual fraction\n\\begin{equation}\n0.823306\n\\end{equation}\nof the unrepaired defect energy. The learned local constructor, before witness selection, leaves mean fraction\n\\begin{equation}\n0.535613.\n\\end{equation}\nThe true-chart oracle leaves only approximately $0.001672$ of the original error fraction on average.","equation}\nto\n\\begin{equation}\n\\widehat\\eps_{\\mathrm{chart}}=0.064975.\n\\end{equation}\nMean learned gate accuracy is\n\\begin{equation}\n0.874854.\n\\end{equation}\nThe directly supervised global-affine oracle leaves mean residual fraction\n\\begin{equation}\n0.823306\n\\end{equation}\nof the unrepaired defect energy. The learned local constructor, before witness selection, leaves mean fraction\n\\begin{equation}\n0.535613.\n\\end{equation}\nThe true-chart oracle leaves only approximately $0.001672$ of the original error fraction on average.","equation}\nMean learned gate accuracy is\n\\begin{equation}\n0.874854.\n\\end{equation}\nThe directly supervised global-affine oracle leaves mean residual fraction\n\\begin{equation}\n0.823306\n\\end{equation}\nof the unrepaired defect energy. The learned local constructor, before witness selection, leaves mean fraction\n\\begin{equation}\n0.535613.\n\\end{equation}\nThe true-chart oracle leaves only approximately $0.001672$ of the original error fraction on average.","\\midrule\nGlobal blind & $-0.833287$ & $0.000000$ & $1.000000$","Gate only & $-0.833287$ & $0.000000$ & $1.000000$","Random gate & $-0.689622$ & $0.072917$ & $0.927083$","Shuffled binding & $-1.000000$ & $0.000000$ & $1.000000$","Local blind & $0.464387$ & $0.958333$ & $0.041667$","Witness-selected child & $0.334224$ & $0.572917$ & $0.000000$","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.\n\\end{proof}","\\begin{statusbox}{Block 10 result}\nBlock 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.\n\\end{statusbox}","n{statusbox}{Block 10 result}\nBlock 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.\n\\end{statusbox}","ult}\nBlock 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.\n\\end{statusbox}","fine 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.\n\\end{statusbox}","tness 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.\n\\end{statusbox}","t, $32$ 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.\n\\end{statusbox}","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.\n\\end{statusbox}","ransport;\n\\item a directly supervised global-affine ceiling control;\n\\item global-blind, local-blind, gate-only, random-gate, shuffled-binding, true-chart, and stress controls;\n\\item development grammar certificates, compiler ledgers, selected policy objects, and a pre-audit seal;\n\\item canonical JSON and CSV results;\n\\item seven reproducible figures;\n\\item eleven automated tests;\n\\item an exact $51$-file package manifest;\n\\item PowerShell, Python, and Makefile runners.\n\\end{enumerate}","The new edge is therefore\n\\begin{equation}\nK_0\\xrightarrow{\\Pi_1}K_1\\xrightarrow{\\Pi_2}K_2\n\\xrightarrow{\\Pi_3}K_3\\xrightarrow{\\Pi_4}K_4,\n\\label{eq:b11-lineage}\n\\end{equation}\nwhere $K_3$ 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.","Let $H$ be a finite-dimensional real or complex Hilbert space. Suppose the unknown functional decomposition is\n\\begin{equation}\nH=\\bigoplus_{a=1}^{m}H_a,\n\\qquad\nP_a:H\\to H_a,\n\\label{eq:b11-hidden-decomp}\n\\end{equation}\nwith pairwise orthogonal projectors satisfying\n\\begin{equation}\nP_aP_b=\\delta_{ab}P_a,\n\\qquad\n\\sum_{a=1}^{m}P_a=I.\n\\end{equation}\nFor controlled interventions $u_1,\\ldots,u_k$, let\n\\begin{equation}\nR_j\\in\\End(H),\n\\qquad j=1,\\ldots,k,\n\\end{equation}\nbe their finite-difference or linearized response operators.","ite-dimensional real or complex Hilbert space. Suppose the unknown functional decomposition is\n\\begin{equation}\nH=\\bigoplus_{a=1}^{m}H_a,\n\\qquad\nP_a:H\\to H_a,\n\\label{eq:b11-hidden-decomp}\n\\end{equation}\nwith pairwise orthogonal projectors satisfying\n\\begin{equation}\nP_aP_b=\\delta_{ab}P_a,\n\\qquad\n\\sum_{a=1}^{m}P_a=I.\n\\end{equation}\nFor controlled interventions $u_1,\\ldots,u_k$, let\n\\begin{equation}\nR_j\\in\\End(H),\n\\qquad j=1,\\ldots,k,\n\\end{equation}\nbe their finite-difference or linearized response operators.","\\begin{definition}[Block-scalar response model]\n\\label{def:b11-response-model}\nThe response family is block scalar on \\cref{eq:b11-hidden-decomp} when\n\\begin{equation}\nR_j=\\sum_{a=1}^{m}\\lambda_{aj}P_a,\n\\qquad\n\\lambda_{aj}\\in\\R.\n\\label{eq:b11-block-scalar}\n\\end{equation}\nThe response signature of block $a$ is\n\\begin{equation}\n\\lambda_a=(\\lambda_{a1},\\ldots,\\lambda_{ak})\\in\\R^k.\n\\end{equation}\nThe signatures are separating when $\\lambda_a\\neq\\lambda_b$ for every $a\\neq b$.\n\\end{definition}","response model]\n\\label{def:b11-response-model}\nThe response family is block scalar on \\cref{eq:b11-hidden-decomp} when\n\\begin{equation}\nR_j=\\sum_{a=1}^{m}\\lambda_{aj}P_a,\n\\qquad\n\\lambda_{aj}\\in\\R.\n\\label{eq:b11-block-scalar}\n\\end{equation}\nThe response signature of block $a$ is\n\\begin{equation}\n\\lambda_a=(\\lambda_{a1},\\ldots,\\lambda_{ak})\\in\\R^k.\n\\end{equation}\nThe signatures are separating when $\\lambda_a\\neq\\lambda_b$ for every $a\\neq b$.\n\\end{definition}","onse-model}\nThe response family is block scalar on \\cref{eq:b11-hidden-decomp} when\n\\begin{equation}\nR_j=\\sum_{a=1}^{m}\\lambda_{aj}P_a,\n\\qquad\n\\lambda_{aj}\\in\\R.\n\\label{eq:b11-block-scalar}\n\\end{equation}\nThe response signature of block $a$ is\n\\begin{equation}\n\\lambda_a=(\\lambda_{a1},\\ldots,\\lambda_{ak})\\in\\R^k.\n\\end{equation}\nThe signatures are separating when $\\lambda_a\\neq\\lambda_b$ for every $a\\neq b$.\n\\end{definition}","The algebra generated by the response family is\n\\begin{equation}\n\\mathfrak A_R\n=\n\\operatorname{alg}^{*}(I,R_1,\\ldots,R_k).\n\\label{eq:b11-response-algebra}\n\\end{equation}\nBecause 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.","\\begin{theorem}[Exact response-signature atlas recovery]\n\\label{thm:b11-exact-atlas}\nAssume 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.\n\\end{theorem}","\\begin{proof}\nChoose $c\\in\\R^k$ outside the finite union of hyperplanes\n\\begin{equation}\n\\set{c:c^{\\mathsf T}(\\lambda_a-\\lambda_b)=0},\n\\qquad a\\neq b.\n\\end{equation}\nThen the scalars\n\\begin{equation}\n\\mu_a=c^{\\mathsf T}\\lambda_a\n\\end{equation}\nare pairwise distinct. The combined response\n\\begin{equation}\nB=\\sum_{j=1}^{k}c_jR_j\n=\\sum_{a=1}^{m}\\mu_aP_a\n\\end{equation}\nbelongs to $\\mathfrak A_R$. For each $a$, define the Lagrange polynomial\n\\begin{equation}\np_a(z)=\\prod_{b\\neq a}\\frac{z-\\mu_b}{\\mu_a-\\mu_b}.\n\\end{equation}\nFunctional calculus gives\n\\begin{equation}\np_a(B)=\\sum_{b=1}^{m}p_a(\\mu_b)P_b=P_a.\n\\end{equation}\nThus every $P_a$ lies in the generated algebra. A nonzero central projection that is strictly below one $P_a","\\begin{proof}\nChoose $c\\in\\R^k$ outside the finite union of hyperplanes\n\\begin{equation}\n\\set{c:c^{\\mathsf T}(\\lambda_a-\\lambda_b)=0},\n\\qquad a\\neq b.\n\\end{equation}\nThen the scalars\n\\begin{equation}\n\\mu_a=c^{\\mathsf T}\\lambda_a\n\\end{equation}\nare pairwise distinct. The combined response\n\\begin{equation}\nB=\\sum_{j=1}^{k}c_jR_j\n=\\sum_{a=1}^{m}\\mu_aP_a\n\\end{equation}\nbelongs to $\\mathfrak A_R$. For each $a$, define the Lagrange polynomial\n\\begin{equation}\np_a(z)=\\prod_{b\\neq a}\\frac{z-\\mu_b}{\\mu_a-\\mu_b}.\n\\end{equation}\nFunctional calculus gives\n\\begin{equation}\np_a(B)=\\sum_{b=1}^{m}p_a(\\mu_b)P_b=P_a.\n\\end{equation}\nThus 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 projecti","\\begin{proof}\nChoose $c\\in\\R^k$ outside the finite union of hyperplanes\n\\begin{equation}\n\\set{c:c^{\\mathsf T}(\\lambda_a-\\lambda_b)=0},\n\\qquad a\\neq b.\n\\end{equation}\nThen the scalars\n\\begin{equation}\n\\mu_a=c^{\\mathsf T}\\lambda_a\n\\end{equation}\nare pairwise distinct. The combined response\n\\begin{equation}\nB=\\sum_{j=1}^{k}c_jR_j\n=\\sum_{a=1}^{m}\\mu_aP_a\n\\end{equation}\nbelongs to $\\mathfrak A_R$. For each $a$, define the Lagrange polynomial\n\\begin{equation}\np_a(z)=\\prod_{b\\neq a}\\frac{z-\\mu_b}{\\mu_a-\\mu_b}.\n\\end{equation}\nFunctional calculus gives\n\\begin{equation}\np_a(B)=\\sum_{b=1}^{m}p_a(\\mu_b)P_b=P_a.\n\\end{equation}\nThus 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.\n\\end{proof}","{proof}\nChoose $c\\in\\R^k$ outside the finite union of hyperplanes\n\\begin{equation}\n\\set{c:c^{\\mathsf T}(\\lambda_a-\\lambda_b)=0},\n\\qquad a\\neq b.\n\\end{equation}\nThen the scalars\n\\begin{equation}\n\\mu_a=c^{\\mathsf T}\\lambda_a\n\\end{equation}\nare pairwise distinct. The combined response\n\\begin{equation}\nB=\\sum_{j=1}^{k}c_jR_j\n=\\sum_{a=1}^{m}\\mu_aP_a\n\\end{equation}\nbelongs to $\\mathfrak A_R$. For each $a$, define the Lagrange polynomial\n\\begin{equation}\np_a(z)=\\prod_{b\\neq a}\\frac{z-\\mu_b}{\\mu_a-\\mu_b}.\n\\end{equation}\nFunctional calculus gives\n\\begin{equation}\np_a(B)=\\sum_{b=1}^{m}p_a(\\mu_b)P_b=P_a.\n\\end{equation}\nThus 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.\n\\end{proof}","of hyperplanes\n\\begin{equation}\n\\set{c:c^{\\mathsf T}(\\lambda_a-\\lambda_b)=0},\n\\qquad a\\neq b.\n\\end{equation}\nThen the scalars\n\\begin{equation}\n\\mu_a=c^{\\mathsf T}\\lambda_a\n\\end{equation}\nare pairwise distinct. The combined response\n\\begin{equation}\nB=\\sum_{j=1}^{k}c_jR_j\n=\\sum_{a=1}^{m}\\mu_aP_a\n\\end{equation}\nbelongs to $\\mathfrak A_R$. For each $a$, define the Lagrange polynomial\n\\begin{equation}\np_a(z)=\\prod_{b\\neq a}\\frac{z-\\mu_b}{\\mu_a-\\mu_b}.\n\\end{equation}\nFunctional calculus gives\n\\begin{equation}\np_a(B)=\\sum_{b=1}^{m}p_a(\\mu_b)P_b=P_a.\n\\end{equation}\nThus 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.\n\\end{proof}","Then the scalars\n\\begin{equation}\n\\mu_a=c^{\\mathsf T}\\lambda_a\n\\end{equation}\nare pairwise distinct. The combined response\n\\begin{equation}\nB=\\sum_{j=1}^{k}c_jR_j\n=\\sum_{a=1}^{m}\\mu_aP_a\n\\end{equation}\nbelongs to $\\mathfrak A_R$. For each $a$, define the Lagrange polynomial\n\\begin{equation}\np_a(z)=\\prod_{b\\neq a}\\frac{z-\\mu_b}{\\mu_a-\\mu_b}.\n\\end{equation}\nFunctional calculus gives\n\\begin{equation}\np_a(B)=\\sum_{b=1}^{m}p_a(\\mu_b)P_b=P_a.\n\\end{equation}\nThus 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.\n\\end{proof}","equation}\nare pairwise distinct. The combined response\n\\begin{equation}\nB=\\sum_{j=1}^{k}c_jR_j\n=\\sum_{a=1}^{m}\\mu_aP_a\n\\end{equation}\nbelongs to $\\mathfrak A_R$. For each $a$, define the Lagrange polynomial\n\\begin{equation}\np_a(z)=\\prod_{b\\neq a}\\frac{z-\\mu_b}{\\mu_a-\\mu_b}.\n\\end{equation}\nFunctional calculus gives\n\\begin{equation}\np_a(B)=\\sum_{b=1}^{m}p_a(\\mu_b)P_b=P_a.\n\\end{equation}\nThus 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.\n\\end{proof}","j\n=\\sum_{a=1}^{m}\\mu_aP_a\n\\end{equation}\nbelongs to $\\mathfrak A_R$. For each $a$, define the Lagrange polynomial\n\\begin{equation}\np_a(z)=\\prod_{b\\neq a}\\frac{z-\\mu_b}{\\mu_a-\\mu_b}.\n\\end{equation}\nFunctional calculus gives\n\\begin{equation}\np_a(B)=\\sum_{b=1}^{m}p_a(\\mu_b)P_b=P_a.\n\\end{equation}\nThus 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.\n\\end{proof}","uation}\nbelongs to $\\mathfrak A_R$. For each $a$, define the Lagrange polynomial\n\\begin{equation}\np_a(z)=\\prod_{b\\neq a}\\frac{z-\\mu_b}{\\mu_a-\\mu_b}.\n\\end{equation}\nFunctional calculus gives\n\\begin{equation}\np_a(B)=\\sum_{b=1}^{m}p_a(\\mu_b)P_b=P_a.\n\\end{equation}\nThus 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.\n\\end{proof}","A_R$. For each $a$, define the Lagrange polynomial\n\\begin{equation}\np_a(z)=\\prod_{b\\neq a}\\frac{z-\\mu_b}{\\mu_a-\\mu_b}.\n\\end{equation}\nFunctional calculus gives\n\\begin{equation}\np_a(B)=\\sum_{b=1}^{m}p_a(\\mu_b)P_b=P_a.\n\\end{equation}\nThus 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.\n\\end{proof}","or each $a$, define the Lagrange polynomial\n\\begin{equation}\np_a(z)=\\prod_{b\\neq a}\\frac{z-\\mu_b}{\\mu_a-\\mu_b}.\n\\end{equation}\nFunctional calculus gives\n\\begin{equation}\np_a(B)=\\sum_{b=1}^{m}p_a(\\mu_b)P_b=P_a.\n\\end{equation}\nThus 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.\n\\end{proof}","ne the Lagrange polynomial\n\\begin{equation}\np_a(z)=\\prod_{b\\neq a}\\frac{z-\\mu_b}{\\mu_a-\\mu_b}.\n\\end{equation}\nFunctional calculus gives\n\\begin{equation}\np_a(B)=\\sum_{b=1}^{m}p_a(\\mu_b)P_b=P_a.\n\\end{equation}\nThus 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.\n\\end{proof}","\\begin{corollary}[Gauge boundary]\n\\label{cor:b11-gauge}\nThe 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.\n\\end{corollary}","The response commutant is\n\\begin{equation}\n\\mathfrak A_R'\n=\n\\set{X\\in\\End(H):XR_j=R_jX\\text{ for every }j}.\n\\end{equation}","\\begin{theorem}[Commutant characterization]\n\\label{thm:b11-commutant}\nUnder the hypotheses of \\cref{thm:b11-exact-atlas},\n\\begin{equation}\n\\mathfrak A_R'\n=\n\\bigoplus_{a=1}^{m}\\End(H_a).\n\\label{eq:b11-commutant}\n\\end{equation}\nIf $d_a=\\dim H_a$, then\n\\begin{equation}\n\\dim\\mathfrak A_R'=\\sum_{a=1}^{m}d_a^2.\n\\label{eq:b11-commutant-dim}\n\\end{equation}\n\\end{theorem}","\\begin{proof}\nWrite $X=\\sum_{a,b}P_aXP_b$. The relation $XR_j=R_jX$ implies\n\\begin{equation}\n(\\lambda_{bj}-\\lambda_{aj})P_aXP_b=0\n\\end{equation}\nfor 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 $\\dim\\End(H_a)=d_a^2$.\n\\end{proof}","For three rank-two blocks, the exact commutant dimension is\n\\begin{equation}\n3\\cdot2^2=12,\n\\end{equation}\nwhich is the canonical finite signature used by the Block 11 reference package.","\\begin{proposition}[Degenerate-signature obstruction]\n\\label{prop:b11-degenerate}\nIf $\\lambda_a=\\lambda_b$ 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.\n\\end{proposition}","\\begin{proof}\nEvery 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.\n\\end{proof}","In an executable system, the responses are estimated from finite intervention data. Let\n\\begin{equation}\n\\widehat R_j=R_j+E_j,\n\\end{equation}\nand fix coefficients $c_j$. Define\n\\begin{equation}\nB=\\sum_jc_jR_j,\n\\qquad\n\\widehat B=\\sum_jc_j\\widehat R_j,\n\\qquad\nE=\\widehat B-B.\n\\end{equation}\nLet the distinct eigenvalue associated with block $a$ be $\\mu_a$ and define its separation\n\\begin{equation}\n\\gamma_a=\\min_{b\\neq a}|\\mu_a-\\mu_b|.\n\\end{equation}","\\begin{conditionaltheorem}[Spectral stability of a recovered atlas block]\n\\label{thm:b11-spectral-stability}\nSuppose $B$ 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\n\\begin{equation}\n\\norm{\\widehat P_a-P_a}_2\n\\leq\n\\frac{2\\norm{E}_2}{\\gamma_a}.\n\\label{eq:b11-dk}\n\\end{equation}\n\\end{conditionaltheorem}","\\begin{proof}\nThis 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.\n\\end{proof}","For recovered projectors $\\widehat P_1,\\ldots,\\widehat P_m$, define the normalized response-fit residual\n\\begin{equation}\n\\eps_{\\mathrm{resp}}\n=\n\\frac{\n\\left(\n\\sum_j\\norm{\\widehat R_j-\n\\sum_a\\widehat\\lambda_{aj}\\widehat P_a}_F^2\n\\right)^{1/2}\n}{\n\\left(\n\\sum_j\\norm{\\widehat R_j}_F^2\n\\right)^{1/2}+10^{-12}\n},\n\\label{eq:b11-response-fit}\n\\end{equation}\nand the maximum normalized commutator residual\n\\begin{equation}\n\\eps_{\\mathrm{comm}}\n=\n\\max_{a,j}\n\\frac{\\norm{\\widehat P_a\\widehat R_j-\n\\widehat R_j\\widehat P_a}_F}\n{\\norm{\\widehat R_j}_F+10^{-12}}.\n\\label{eq:b11-commutator}\n\\end{equation}\nThe smallest selected spectral gap is denoted $\\widehat\\gamma_{\\min}$. These quantities serve different roles:\n\\begin{itemize}\n\\item $\\eps_{\\mathrm{resp}}$ measures w","For recovered projectors $\\widehat P_1,\\ldots,\\widehat P_m$, define the normalized response-fit residual\n\\begin{equation}\n\\eps_{\\mathrm{resp}}\n=\n\\frac{\n\\left(\n\\sum_j\\norm{\\widehat R_j-\n\\sum_a\\widehat\\lambda_{aj}\\widehat P_a}_F^2\n\\right)^{1/2}\n}{\n\\left(\n\\sum_j\\norm{\\widehat R_j}_F^2\n\\right)^{1/2}+10^{-12}\n},\n\\label{eq:b11-response-fit}\n\\end{equation}\nand the maximum normalized commutator residual\n\\begin{equation}\n\\eps_{\\mathrm{comm}}\n=\n\\max_{a,j}\n\\frac{\\norm{\\widehat P_a\\widehat R_j-\n\\widehat R_j\\widehat P_a}_F}\n{\\norm{\\widehat R_j}_F+10^{-12}}.\n\\label{eq:b11-commutator}\n\\end{equation}\nThe smallest selected spectral gap is denoted $\\widehat\\gamma_{\\min}$. These quantities serve different roles:\n\\begin{itemize}\n\\item $\\eps_{\\mathrm{resp}}$ measures whether the finite block-scalar model fits the response family;\n\\item $\\eps_{\\mathrm{comm}}$ measures whether the recovered packets behave as approximately invariant response modules;\n\\item $\\widehat\\gamma_{\\min}$ measures stability of the recovered decomposition.\n\\end{itemize}","rojectors $\\widehat P_1,\\ldots,\\widehat P_m$, define the normalized response-fit residual\n\\begin{equation}\n\\eps_{\\mathrm{resp}}\n=\n\\frac{\n\\left(\n\\sum_j\\norm{\\widehat R_j-\n\\sum_a\\widehat\\lambda_{aj}\\widehat P_a}_F^2\n\\right)^{1/2}\n}{\n\\left(\n\\sum_j\\norm{\\widehat R_j}_F^2\n\\right)^{1/2}+10^{-12}\n},\n\\label{eq:b11-response-fit}\n\\end{equation}\nand the maximum normalized commutator residual\n\\begin{equation}\n\\eps_{\\mathrm{comm}}\n=\n\\max_{a,j}\n\\frac{\\norm{\\widehat P_a\\widehat R_j-\n\\widehat R_j\\widehat P_a}_F}\n{\\norm{\\widehat R_j}_F+10^{-12}}.\n\\label{eq:b11-commutator}\n\\end{equation}\nThe smallest selected spectral gap is denoted $\\widehat\\gamma_{\\min}$. These quantities serve different roles:\n\\begin{itemize}\n\\item $\\eps_{\\mathrm{resp}}$ measures whether the finite block-scalar model fits the response family;\n\\item $\\eps_{\\mathrm{comm}}$ measures whether the recovered packets behave as approximately invariant response modules;\n\\item $\\widehat\\gamma_{\\min}$ measures stability of the recovered decomposition.\n\\end{itemize}","/2}\n}{\n\\left(\n\\sum_j\\norm{\\widehat R_j}_F^2\n\\right)^{1/2}+10^{-12}\n},\n\\label{eq:b11-response-fit}\n\\end{equation}\nand the maximum normalized commutator residual\n\\begin{equation}\n\\eps_{\\mathrm{comm}}\n=\n\\max_{a,j}\n\\frac{\\norm{\\widehat P_a\\widehat R_j-\n\\widehat R_j\\widehat P_a}_F}\n{\\norm{\\widehat R_j}_F+10^{-12}}.\n\\label{eq:b11-commutator}\n\\end{equation}\nThe smallest selected spectral gap is denoted $\\widehat\\gamma_{\\min}$. These quantities serve different roles:\n\\begin{itemize}\n\\item $\\eps_{\\mathrm{resp}}$ measures whether the finite block-scalar model fits the response family;\n\\item $\\eps_{\\mathrm{comm}}$ measures whether the recovered packets behave as approximately invariant response modules;\n\\item $\\widehat\\gamma_{\\min}$ measures stability of the recovered decomposition.\n\\end{itemize}","ponse-fit}\n\\end{equation}\nand the maximum normalized commutator residual\n\\begin{equation}\n\\eps_{\\mathrm{comm}}\n=\n\\max_{a,j}\n\\frac{\\norm{\\widehat P_a\\widehat R_j-\n\\widehat R_j\\widehat P_a}_F}\n{\\norm{\\widehat R_j}_F+10^{-12}}.\n\\label{eq:b11-commutator}\n\\end{equation}\nThe smallest selected spectral gap is denoted $\\widehat\\gamma_{\\min}$. These quantities serve different roles:\n\\begin{itemize}\n\\item $\\eps_{\\mathrm{resp}}$ measures whether the finite block-scalar model fits the response family;\n\\item $\\eps_{\\mathrm{comm}}$ measures whether the recovered packets behave as approximately invariant response modules;\n\\item $\\widehat\\gamma_{\\min}$ measures stability of the recovered decomposition.\n\\end{itemize}","m{comm}}\n=\n\\max_{a,j}\n\\frac{\\norm{\\widehat P_a\\widehat R_j-\n\\widehat R_j\\widehat P_a}_F}\n{\\norm{\\widehat R_j}_F+10^{-12}}.\n\\label{eq:b11-commutator}\n\\end{equation}\nThe smallest selected spectral gap is denoted $\\widehat\\gamma_{\\min}$. These quantities serve different roles:\n\\begin{itemize}\n\\item $\\eps_{\\mathrm{resp}}$ measures whether the finite block-scalar model fits the response family;\n\\item $\\eps_{\\mathrm{comm}}$ measures whether the recovered packets behave as approximately invariant response modules;\n\\item $\\widehat\\gamma_{\\min}$ measures stability of the recovered decomposition.\n\\end{itemize}","}.\n\\label{eq:b11-commutator}\n\\end{equation}\nThe smallest selected spectral gap is denoted $\\widehat\\gamma_{\\min}$. These quantities serve different roles:\n\\begin{itemize}\n\\item $\\eps_{\\mathrm{resp}}$ measures whether the finite block-scalar model fits the response family;\n\\item $\\eps_{\\mathrm{comm}}$ measures whether the recovered packets behave as approximately invariant response modules;\n\\item $\\widehat\\gamma_{\\min}$ measures stability of the recovered decomposition.\n\\end{itemize}","For each recovered block $\\widehat P_a$, let $\\Fcal_a$ be a finite family of candidate separators\n\\begin{equation}\ng_{a,\\theta,u}(h)\n=\n\\one\\{\\angles{u,\\widehat P_ah}>\\theta\\}.\n\\end{equation}\nThe compiler fits a two-chart bridge for every candidate and evaluates a score containing:\n\\begin{enumerate}\n\\item held-out local-bridge validation gain;\n\\item source-repair invariance of the chart assignment;\n\\item agreement between source and target chart assignments;\n\\item chart-balance constraints;\n\\item response-fit, commutator, and spectral-gap penalties;\n\\item protected-context regression.\n\\end{enumerate}","Let $s_a$ denote the best population score attainable from block $a$, and let $\\widehat s_a$ be its empirical score.","\\begin{theorem}[Finite repair-relevant block selection]\n\\label{thm:b11-block-selection}\nSuppose there is a unique repair-relevant block $a^*$ and a margin $m>0$ such that\n\\begin{equation}\ns_{a^*}\\geq s_a+m\n\\end{equation}\nfor every $a\\neq a^*$. If\n\\begin{equation}\n\\max_a|\\widehat s_a-s_a|<\\frac{m}{2},\n\\end{equation}\nthen\n\\begin{equation}\n\\argmax_a\\widehat s_a=a^*.\n\\end{equation}\n\\end{theorem}","\\begin{proof}\nFor every $a\\neq a^*$,\n\\begin{align}\n\\widehat s_{a^*}-\\widehat s_a\n&\\geq\ns_{a^*}-s_a\n-|\\widehat s_{a^*}-s_{a^*}|\n-|\\widehat s_a-s_a|","\\begin{proof}\nFor every $a\\neq a^*$,\n\\begin{align}\n\\widehat s_{a^*}-\\widehat s_a\n&\\geq\ns_{a^*}-s_a\n-|\\widehat s_{a^*}-s_{a^*}|\n-|\\widehat s_a-s_a|\\\\\n&>m-\\frac m2-\\frac m2=0.\n\\end{align}\n\\end{proof}","Let $\\Gcal_{10}$ denote the Block 10 two-chart grammar with a supplied context projector $P_{\\mathrm{ctx}}$. Let $\\Gcal_{11}$ contain:\n\\begin{enumerate}\n\\item the intervention-response atlas compiler;\n\\item a finite family of projectors and separators returned by the compiler;\n\\item every constructor and witness rule of $\\Gcal_{10}$;\n\\item a declared fallback that may instantiate the old projector directly.\n\\end{enumerate}","\\begin{proposition}[Conservative embedding of the Block 10 grammar]\n\\label{prop:b11-conservative}\nThere is an injective embedding\n\\begin{equation}\n\\iota_{10\\to11}:\\Gcal_{10}\\hookrightarrow\\Gcal_{11}\n\\end{equation}\nobtained 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 $\\Gcal_{11}$.\n\\end{proposition}","ative embedding of the Block 10 grammar]\n\\label{prop:b11-conservative}\nThere is an injective embedding\n\\begin{equation}\n\\iota_{10\\to11}:\\Gcal_{10}\\hookrightarrow\\Gcal_{11}\n\\end{equation}\nobtained 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 $\\Gcal_{11}$.\n\\end{proposition}","\\begin{proposition}[Fixed-packet insufficiency under hidden rotation]\n\\label{prop:b11-fixed-insufficiency}\nLet $P_0$ 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.\n\\end{proposition}","ufficiency under hidden rotation]\n\\label{prop:b11-fixed-insufficiency}\nLet $P_0$ 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.\n\\end{proposition}","den rotation]\n\\label{prop:b11-fixed-insufficiency}\nLet $P_0$ 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.\n\\end{proposition}","\\begin{proof}\nIf 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.\n\\end{proof}","Let $r\\in H$ be a current hidden-state defect residual. Define\n\\begin{equation}\nH_r\n=\n\\Span\\set{Ar:A\\in\\mathfrak A_R}.\n\\label{eq:b11-cyclic}\n\\end{equation}","\\begin{proposition}[Minimal response-invariant defect module]\n\\label{prop:b11-cyclic}\nThe 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.\n\\end{proposition}","\\begin{proof}\nThe 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.\n\\end{proof}","This yields a two-scale endogenous decomposition:\n\\begin{equation}\n\\text{central response blocks}\n\\quad\\longrightarrow\\quad\n\\text{defect-specific cyclic module}.\n\\end{equation}\nThe first identifies coarse functional strata. The second localizes the current defect inside their response closure.","The source hidden space has dimension six and is generated from three target-specific unknown rank-two blocks:\n\\begin{equation}\nH=H_{\\mathrm{ctx}}\\oplus H_{\\mathrm{rep}}\\oplus H_{\\mathrm{nui}}.\n\\label{eq:b11-three-blocks}\n\\end{equation}\nA 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.","Every primary target receives exactly $120$ effect-query units. The final frozen accounting is:\n\\begin{center}\n\\begin{tabular}{@{}lrrrrr@{}}\n\\toprule\nProcess & Bridge fit & Response & Chart validation & Witness & Total","The chronology is:\n\\begin{equation}\n\\begin{aligned}\n&\\text{replay Block 10 ancestor digests}\\\\\n&\\longrightarrow\\text{generate development and meta-validation evidence}\\\\\n&\\longrightarrow\\text{compile atlas certificate, parent, child, and witness}\\\\\n&\\longrightarrow\\text{serialize and hash the pre-audit seal}\\\\\n&\\longrightarrow\\text{generate primary and stress targets}\\\\\n&\\longrightarrow\\text{evaluate every frozen arm and claim gate}.\n\\end{aligned}\n\\end{equation}\nThe pre-audit seal digest is\n\\begin{equation}\n\\texttt{de2fc5bb3654eaca237d7cc3799570eb72fe9814657a3a55f9a42323361a92bd}.\n\\end{equation}\nThe atlas certificate digest is\n\\begin{equation}\n\\texttt{22976d2546377b4afcfb0dc67e2489f2d84911527b24ac35cbdb0dc0bf9a77e8}.\n\\end{equation}","&\\longrightarrow\\text{evaluate every frozen arm and claim gate}.\n\\end{aligned}\n\\end{equation}\nThe pre-audit seal digest is\n\\begin{equation}\n\\texttt{de2fc5bb3654eaca237d7cc3799570eb72fe9814657a3a55f9a42323361a92bd}.\n\\end{equation}\nThe atlas certificate digest is\n\\begin{equation}\n\\texttt{22976d2546377b4afcfb0dc67e2489f2d84911527b24ac35cbdb0dc0bf9a77e8}.\n\\end{equation}","The selected atlas candidate commits only when all frozen conditions hold:\n\\begin{align}\n\\widehat g_{\\mathrm{val}}&\\geq0.10,\\\\\n\\LCB(\\widehat g_{\\mathrm{val}})&\\geq0.05,\\\\\n\\widehat r_{\\mathrm{ctx}}&\\leq3\\times10^{-4},\\\\\n\\widehat a_{\\mathrm{target}}&\\geq0.72,\\\\\n\\widehat a_{\\mathrm{repair\\ inv}}&\\geq0.96,\\\\\n\\widehat b_{\\mathrm{chart}}&\\geq0.20,\\\\\n\\widehat s_{\\mathrm{model}}&\\leq0.25,\\\\\n\\widehat\\gamma_{\\min}&\\geq0.004,\\\\\n\\widehat\\eps_{\\mathrm{comm}}&\\leq0.16,\\\\\n\\widehat\\eps_{\\mathrm{resp}}&\\leq0.08.\n\\end{align}\nThe 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.","Across $36$ development targets, the compiler records:\n\\begin{center}\n\\begin{tabular}{@{}lr@{}}\n\\toprule\nQuantity & Mean","The primary audit contains $80$ targets from four foreign representation families. The main selected results are:\n\\begin{center}\n\\begin{tabular}{@{}lrrr@{}}\n\\toprule\nProcess & Mean selected gain & Commit rate & Harm rate","The paired effect is\n\\begin{equation}\n\\widehat\\Delta_{3\\to4}\n=0.207325,\n\\label{eq:b11-paired-effect}\n\\end{equation}\nwith frozen bootstrap interval\n\\begin{equation}\n[0.131902,\\ 0.287910]\n\\label{eq:b11-ci}\n\\end{equation}\nand sign-flip estimate\n\\begin{equation}\np=9.9999\\times10^{-6}.\n\\end{equation}\nNo primary target favors the parent. The positive paired-effect rate is $0.2875$ and the remaining $0.7125$ are abstention ties.","\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.84\\textwidth]{block11_primary_gain.pdf}\n\\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.}\n\\label{fig:b11-primary}\n\\end{figure}","Blind application of the endogenous candidate has mean gain\n\\begin{equation}\n0.271529\n\\end{equation}\nbut harmful-commit rate\n\\begin{equation}\n0.1875.\n\\end{equation}\nAfter witness selection, mean gain becomes\n\\begin{equation}\n0.207325\n\\end{equation}\nand the harmful-commit rate becomes zero. The witness therefore removes an $18.75\\%$ blind-harm rate at a deliberate utility cost.","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\n\\begin{equation}\n0.027943,\n\\end{equation}\ncommit 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.","\\begin{conditionaltheorem}[Block 11 finite realization]\n\\label{thm:b11-realization}\nWithin the frozen Block 11 benchmark, assume that:\n\\begin{enumerate}\n\\item the published Block 10 ancestor digests replay exactly;\n\\item the supplied context packet and hidden atlas labels are excluded from child construction;\n\\item the response operators are estimated only from the declared $24$ response-query units;\n\\item parent and child each use exactly $120$ total effect-query units;\n\\item the child is sealed before primary and stress target generation;\n\\item the conservative embedding in \\cref{prop:b11-conservative} is executable;\n\\item the paired effect in \\cref{eq:b11-paired-effect} has the frozen interval in \\cref{eq:b11-ci};\n\\item the declared response, commutator, spectral, repair, and witness gates pass;\n\\item random-atlas, shuffled-response, PCA, full-space, blind, and stress controls retain their declared interpretations.\n\\end{enumerate}\nThen the edge\n\\begin{equation}\nK_3\\xrightarrow{\\Pi_4}K_4\n\\end{equation}\nis a finite, matched-resource, lineage-generated instance of","alization]\n\\label{thm:b11-realization}\nWithin the frozen Block 11 benchmark, assume that:\n\\begin{enumerate}\n\\item the published Block 10 ancestor digests replay exactly;\n\\item the supplied context packet and hidden atlas labels are excluded from child construction;\n\\item the response operators are estimated only from the declared $24$ response-query units;\n\\item parent and child each use exactly $120$ total effect-query units;\n\\item the child is sealed before primary and stress target generation;\n\\item the conservative embedding in \\cref{prop:b11-conservative} is executable;\n\\item the paired effect in \\cref{eq:b11-paired-effect} has the frozen interval in \\cref{eq:b11-ci};\n\\item the declared response, commutator, spectral, repair, and witness gates pass;\n\\item random-atlas, shuffled-response, PCA, full-space, blind, and stress controls retain their declared interpretations.\n\\end{enumerate}\nThen the edge\n\\begin{equation}\nK_3\\xrightarrow{\\Pi_4}K_4\n\\end{equation}\nis a finite, matched-resource, lineage-generated instance of endogenous intervention-response atlas growth. It is structurall…","g in \\cref{prop:b11-conservative} is executable;\n\\item the paired effect in \\cref{eq:b11-paired-effect} has the frozen interval in \\cref{eq:b11-ci};\n\\item the declared response, commutator, spectral, repair, and witness gates pass;\n\\item random-atlas, shuffled-response, PCA, full-space, blind, and stress controls retain their declared interpretations.\n\\end{enumerate}\nThen the edge\n\\begin{equation}\nK_3\\xrightarrow{\\Pi_4}K_4\n\\end{equation}\nis 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.\n\\end{conditionaltheorem}","\\begin{statusbox}{Block 11 result}\nBlock 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.\n\\end{statusbox}","{statusbox}{Block 11 result}\nBlock 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.\n\\end{statusbox}","d 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.\n\\end{statusbox}","h 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.\n\\end{statusbox}","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.\n\\end{statusbox}","central spectral atlas recovery;\n\\item approximate-commutant and commutator diagnostics;\n\\item repair-relevant block and separator selection;\n\\item legacy, PCA, full-space, random, shuffled-response, global, blind, and oracle controls;\n\\item frozen witness policies and pre-audit receipts;\n\\item deterministic result objects and seven publication figures;\n\\item ten automated tests;\n\\item an exact $49$-file SHA-256 package manifest.\n\\end{itemize}","The exact Block 10 ancestor policy digest is\n\\begin{equation}\n\\texttt{48588442c27529fa38cbee5f6938e6c36263e4c650a5c08ab61365f440a27bb1}.\n\\end{equation}\nThe Block 11 compiler digest is\n\\begin{equation}\n\\texttt{940105f8bf73a5528b5618066664004081066c94260f09c5f6672c9d54eab4f6}.\n\\end{equation}","The realized continuation is\n\\begin{equation}\nK_4\n\\xrightarrow{\\Pi_5}\nK_5\n\\xrightarrow{\\Pi_6}\nK_6.\n\\label{eq:b12-lineage}\n\\end{equation}\nThe 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.","Let $H$ be a finite-dimensional Hilbert space and let\n\\begin{equation}\n\\Rcal=\\{R_1,\\ldots,R_m\\}\\subseteq\\End(H)\n\\end{equation}\nbe a declared intervention-response language. Its unital response algebra is\n\\begin{equation}\n\\mathfrak A(\\Rcal)\n=\n\\operatorname{alg}(I,R_1,\\ldots,R_m).\n\\label{eq:b12-response-algebra}\n\\end{equation}\nWhen adjoints are available, the corresponding adjoint-closed algebra is denoted $\\operatorname{alg}^{\\ast}(\\Rcal)$.","e a finite-dimensional Hilbert space and let\n\\begin{equation}\n\\Rcal=\\{R_1,\\ldots,R_m\\}\\subseteq\\End(H)\n\\end{equation}\nbe a declared intervention-response language. Its unital response algebra is\n\\begin{equation}\n\\mathfrak A(\\Rcal)\n=\n\\operatorname{alg}(I,R_1,\\ldots,R_m).\n\\label{eq:b12-response-algebra}\n\\end{equation}\nWhen adjoints are available, the corresponding adjoint-closed algebra is denoted $\\operatorname{alg}^{\\ast}(\\Rcal)$.","\\begin{definition}[Block-scalar response signature]\nLet\n\\begin{equation}\nH=\\bigoplus_{a=1}^{k}H_a\n\\end{equation}\nbe a reducing decomposition for every $R_j$. The response language is block scalar when\n\\begin{equation}\n\\restr{R_j}{H_a}=\\lambda_{aj}I_{H_a}\n\\end{equation}\nfor scalars $\\lambda_{aj}$. The vector\n\\begin{equation}\n\\sigma_{\\Rcal}(H_a)\n=\n(\\lambda_{a1},\\ldots,\\lambda_{am})\n\\end{equation}\nis the response signature of block $H_a$.\n\\end{definition}","sponse signature]\nLet\n\\begin{equation}\nH=\\bigoplus_{a=1}^{k}H_a\n\\end{equation}\nbe a reducing decomposition for every $R_j$. The response language is block scalar when\n\\begin{equation}\n\\restr{R_j}{H_a}=\\lambda_{aj}I_{H_a}\n\\end{equation}\nfor scalars $\\lambda_{aj}$. The vector\n\\begin{equation}\n\\sigma_{\\Rcal}(H_a)\n=\n(\\lambda_{a1},\\ldots,\\lambda_{am})\n\\end{equation}\nis the response signature of block $H_a$.\n\\end{definition}","\\begin{definition}[Response alias]\nTwo reducing blocks $H_a,H_b$ are response aliased under $\\Rcal$ when\n\\begin{equation}\n\\sigma_{\\Rcal}(H_a)=\\sigma_{\\Rcal}(H_b).\n\\label{eq:b12-response-alias}\n\\end{equation}\n\\end{definition}","\\begin{theorem}[Polynomial persistence of a response alias]\n\\label{thm:b12-alias-persistence}\nSuppose $H_a$ 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.\n\\end{theorem}","\\begin{proof}\nEvery 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.\n\\end{proof}","\\begin{corollary}[No sample-size cure for an exact alias]\nIf 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.\n\\end{corollary}","Let $S\\in\\End(H)$ be a candidate intervention response. Define the expanded language\n\\begin{equation}\n\\Rcal^{+}=\\Rcal\\cup\\{S\\}.\n\\end{equation}\nThe extension is conservative at the syntactic level because every old response expression remains available unchanged.","\\begin{definition}[Separating response operator]\nA candidate $S$ separates an aliased pair $(H_a,H_b)$ when both subspaces reduce $S$ and\n\\begin{equation}\n\\restr{S}{H_a}=\\alpha I_{H_a},\n\\qquad\n\\restr{S}{H_b}=\\beta I_{H_b},\n\\qquad\n\\alpha\\neq\\beta.\n\\label{eq:b12-separating-response}\n\\end{equation}\n\\end{definition}","\\begin{theorem}[Central refinement by a separating intervention]\n\\label{thm:b12-central-refinement}\nSuppose $H_a,H_b$ 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 $H_b$.\n\\end{theorem}","\\begin{proof}\nChoose a generic linear combination\n\\begin{equation}\nG=\\sum_{j=1}^{m}c_jR_j+c_{m+1}S\n\\end{equation}\nwhose induced scalar values on the distinct joint signatures are pairwise distinct. For each block value $\\gamma_a$, the Lagrange polynomial\n\\begin{equation}\np_a(z)=\\prod_{b\\neq a}\\frac{z-\\gamma_b}{\\gamma_a-\\gamma_b}\n\\end{equation}\nsatisfies $p_a(G)=P_a$, the orthogonal projector onto $H_a$. Therefore $P_a,P_b\\in\\mathfrak A(\\Rcal^+)$ and belong to its center.\n\\end{proof}","begin{proof}\nChoose a generic linear combination\n\\begin{equation}\nG=\\sum_{j=1}^{m}c_jR_j+c_{m+1}S\n\\end{equation}\nwhose induced scalar values on the distinct joint signatures are pairwise distinct. For each block value $\\gamma_a$, the Lagrange polynomial\n\\begin{equation}\np_a(z)=\\prod_{b\\neq a}\\frac{z-\\gamma_b}{\\gamma_a-\\gamma_b}\n\\end{equation}\nsatisfies $p_a(G)=P_a$, the orthogonal projector onto $H_a$. Therefore $P_a,P_b\\in\\mathfrak A(\\Rcal^+)$ and belong to its center.\n\\end{proof}","se a generic linear combination\n\\begin{equation}\nG=\\sum_{j=1}^{m}c_jR_j+c_{m+1}S\n\\end{equation}\nwhose induced scalar values on the distinct joint signatures are pairwise distinct. For each block value $\\gamma_a$, the Lagrange polynomial\n\\begin{equation}\np_a(z)=\\prod_{b\\neq a}\\frac{z-\\gamma_b}{\\gamma_a-\\gamma_b}\n\\end{equation}\nsatisfies $p_a(G)=P_a$, the orthogonal projector onto $H_a$. Therefore $P_a,P_b\\in\\mathfrak A(\\Rcal^+)$ and belong to its center.\n\\end{proof}","\\begin{definition}[Response-language expansion certificate]\nA response-language expansion certificate is a tuple\n\\begin{equation}\n\\Pi_{\\mathrm{resp}}\n=\n(\\Rcal,S,\\mathfrak A,\\mathfrak A^+,P_{\\mathrm{alias}},\\gamma,\n\\eps_{\\mathrm{comm}},\\eps_{\\mathrm{fit}},\\Pi_{\\mathrm{foreign}}),\n\\end{equation}\nwhere $P_{\\mathrm{alias}}$ 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.\n\\end{definition}","efinition}[Response-language expansion certificate]\nA response-language expansion certificate is a tuple\n\\begin{equation}\n\\Pi_{\\mathrm{resp}}\n=\n(\\Rcal,S,\\mathfrak A,\\mathfrak A^+,P_{\\mathrm{alias}},\\gamma,\n\\eps_{\\mathrm{comm}},\\eps_{\\mathrm{fit}},\\Pi_{\\mathrm{foreign}}),\n\\end{equation}\nwhere $P_{\\mathrm{alias}}$ 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.\n\\end{definition}","e]\nA response-language expansion certificate is a tuple\n\\begin{equation}\n\\Pi_{\\mathrm{resp}}\n=\n(\\Rcal,S,\\mathfrak A,\\mathfrak A^+,P_{\\mathrm{alias}},\\gamma,\n\\eps_{\\mathrm{comm}},\\eps_{\\mathrm{fit}},\\Pi_{\\mathrm{foreign}}),\n\\end{equation}\nwhere $P_{\\mathrm{alias}}$ 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.\n\\end{definition}","uple\n\\begin{equation}\n\\Pi_{\\mathrm{resp}}\n=\n(\\Rcal,S,\\mathfrak A,\\mathfrak A^+,P_{\\mathrm{alias}},\\gamma,\n\\eps_{\\mathrm{comm}},\\eps_{\\mathrm{fit}},\\Pi_{\\mathrm{foreign}}),\n\\end{equation}\nwhere $P_{\\mathrm{alias}}$ 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.\n\\end{definition}","Let $G$ be the ideal generic response combination from the proof of \\cref{thm:b12-central-refinement}, and let\n\\begin{equation}\n\\widehat G=G+E\n\\end{equation}\nbe its empirical estimate. Let $P$ be a spectral projector associated with an eigenvalue cluster separated from the rest of the spectrum by gap $\\gamma>0$.","\\begin{proposition}[Response-projector perturbation]\n\\label{prop:b12-response-perturbation}\nIf $\\norm{E}_2<\\gamma/2$, then the corresponding empirical spectral projector $\\widehat P$ satisfies a Davis--Kahan type bound\n\\begin{equation}\n\\norm{\\widehat P-P}_2\n\\leq\n\\frac{2\\norm{E}_2}{\\gamma}.\n\\label{eq:b12-projector-bound}\n\\end{equation}\n\\end{proposition}","Let $\\mathscr S$ be a finite typed candidate-intervention grammar. For candidate $S\\in\\mathscr S$, define a development score\n\\begin{equation}\n\\operatorname{Score}_{\\mathrm{resp}}(S)\n=\n\\widehat M_{\\mathrm{foreign}}(S)\n-\\lambda_{c}C(S)\n-\\lambda_{\\mathrm{comm}}\\eps_{\\mathrm{comm}}(S)\n-\\lambda_{\\mathrm{fit}}\\eps_{\\mathrm{fit}}(S)\n-\\lambda_{\\mathrm{risk}}\\widehat R(S).\n\\label{eq:b12-response-score}\n\\end{equation}\nThe 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.","\\begin{proposition}[Finite response-language selection]\nIf $\\mathscr S$ 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.\n\\end{proposition}","Let $\\Ccal_{\\mathrm{aff}}(U,V)$ denote affine maps\n\\begin{equation}\nf(x)=Ax+b\n\\end{equation}\nfrom authorized state space $U$ to defect or target space $V$.","\\begin{proof}\nFor $f(x)=Ax+b$ and $g(x)=Cx+d$,\n\\begin{equation}\n(g\\circ f)(x)=CAx+Cb+d,\n\\end{equation}\nwhich is affine. Induction proves finite closure. Direct sums and affine synthesis preserve the same form.\n\\end{proof}","Let $P$ be the recovered repair packet, let $z=P^\\top x$, and consider a target correction\n\\begin{equation}\nF(x)\n=\nAx+b+\\bigl(z^\\top Bz\\bigr)v,\n\\label{eq:b12-quadratic-repair}\n\\end{equation}\nwhere $B=B^\\top\\neq0$ and $v\\neq0$.","\\begin{theorem}[Nonzero-Hessian affine obstruction]\n\\label{thm:b12-hessian-obstruction}\nNo affine map agrees with $F$ on any nonempty open subset of the packet coordinates.\n\\end{theorem}","\\begin{proof}\nEvery 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.\n\\end{proof}","\\begin{theorem}[Best affine residual for a quadratic packet repair]\n\\label{thm:b12-gaussian-affine-residual}\nLet $z\\sim N(0,I_r)$, $B=B^\\top$, and define\n\\begin{equation}\nq(z)=z^\\top Bz.\n\\end{equation}\nThe best $L^2$ affine approximation to $q(z)v$ is the constant $(\\tr B)v$, and its minimum squared residual is\n\\begin{equation}\n\\inf_{a,L}\n\\mathbb E\\norm{q(z)v-(a+Lz)}^2\n=\n2\\norm{B}_F^2\\norm{v}^2.\n\\label{eq:b12-affine-residual}\n\\end{equation}\n\\end{theorem}","\\begin{proof}\nFor 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}.\n\\end{proof}","Let $U_P\\in\\R^{d\\times r}$ have orthonormal columns spanning the recovered repair packet. Define packet coordinates\n\\begin{equation}\nz=U_P^\\top(x-c).\n\\end{equation}\nThe degree-two block-local grammar consists of maps\n\\begin{equation}\nF_{A,b,C}(x)\n=\nAx+b+C\\,\\vech(zz^\\top),\n\\label{eq:b12-block-quadratic}\n\\end{equation}\nwhere $\\vech$ lists the $r(r+1)/2$ distinct symmetric quadratic monomials.","\\begin{proposition}[Conservative embedding of affine repairs]\n\\label{prop:b12-affine-embedding}\nThe affine grammar embeds into the block-local quadratic grammar by setting $C=0$.\n\\end{proposition}","\\begin{proposition}[Within-packet gauge invariance]\n\\label{prop:b12-quadratic-gauge}\nIf 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.\n\\end{proposition}","\\begin{proof}\nEvery quadratic form $z^\\top Bz$ becomes\n\\begin{equation}\nz'^{\\top}(Q^\\top BQ)z'.\n\\end{equation}\nThe vector space of symmetric quadratic forms is therefore carried isomorphically to itself under the induced action of $Q$ on $\\operatorname{Sym}^2(P^\\top H)$.\n\\end{proof}","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.","\\begin{definition}[Constructor-expansion certificate]\nA constructor-expansion certificate is a tuple\n\\begin{equation}\n\\Pi_{\\mathrm{con}}\n=\n(\\Ccal_t,\\Ccal_t^+,\\eta_{\\mathrm{old}},\\eta_{\\mathrm{new}},\n\\mathsf H,\\kappa,C_{\\mathrm{model}},\\Pi_{\\mathrm{foreign}}),\n\\end{equation}\nwhere $\\mathsf H$ is a nonzero higher-order derivative witness, $\\kappa$ is held-out obstruction reduction, and $C_{\\mathrm{model}}$ records constructor complexity.\n\\end{definition}","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.","\\begin{proposition}[Finite constructor selection with conservative tie-break]\nSuppose a finite constructor family is evaluated on fixed development evidence. Let $G^\\star$ be the largest held-out development gain and let\n\\begin{equation}\n\\Ccal_{\\tau}=\\{C:G(C)\\geq G^\\star-\\tau\\}.\n\\end{equation}\nSelecting the minimum-complexity constructor in $\\Ccal_\\tau$ is deterministic and preserves an explicit near-optimality bound of $\\tau$ on the development criterion.\n\\end{proposition}","e constructor selection with conservative tie-break]\nSuppose a finite constructor family is evaluated on fixed development evidence. Let $G^\\star$ be the largest held-out development gain and let\n\\begin{equation}\n\\Ccal_{\\tau}=\\{C:G(C)\\geq G^\\star-\\tau\\}.\n\\end{equation}\nSelecting the minimum-complexity constructor in $\\Ccal_\\tau$ is deterministic and preserves an explicit near-optimality bound of $\\tau$ on the development criterion.\n\\end{proposition}","The hidden source space has three unknown rank-two blocks:\n\\begin{equation}\nH=H_{\\mathrm{ctx}}\\oplus H_{\\mathrm{rep}}\\oplus H_{\\mathrm{nui}}.\n\\end{equation}\nThe inherited signature matrix is\n\\begin{equation}\n\\Sigma_0\n=\n\\begin{bmatrix}\n1.25 & 0.72","The hidden source space has three unknown rank-two blocks:\n\\begin{equation}\nH=H_{\\mathrm{ctx}}\\oplus H_{\\mathrm{rep}}\\oplus H_{\\mathrm{nui}}.\n\\end{equation}\nThe inherited signature matrix is\n\\begin{equation}\n\\Sigma_0\n=\n\\begin{bmatrix}\n1.25 & 0.72\\\\\n1.25 & 0.72\\\\\n0.80 & 1.30\n\\end{bmatrix},\n\\label{eq:b12-inherited-signatures}\n\\end{equation}\nwhere 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.","\\midrule\nsplit-context-repair & $1.42$ & $0.61$ & $1.03$","duplicate-parent & $1.18$ & $1.18$ & $0.91$","nuisance-rescale & $0.84$ & $0.84$ & $1.37$","weak-context-repair & $1.015$ & $0.985$ & $1.00$","cross-coupled & $1.38$ & $0.65$ & $1.02$","Parent and child each receive exactly $144$ deployment query units:\n\\begin{center}\n\\begin{tabular}{@{}lrrrrr@{}}\n\\toprule\nProcess & Response & Bridge fit & Chart validation & Witness & Total","\\midrule\nInherited parent & $72$ & $36$ & $18$ & $18$ & $144$","Expanded-language child & $72$ & $36$ & $18$ & $18$ & $144$","\\bottomrule\n\\end{tabular}\n\\end{center}\nThe 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.","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.","The constructor context uses the recovered repair packet and a target repair of the form\n\\begin{equation}\nF(x)=Ax+b+\\bigl(z^\\top Bz\\bigr)v,\n\\qquad\nz=U_{\\mathrm{rep}}^\\top x,\n\\end{equation}\nwith $B\\neq0$ and $v\\neq0$. The parent affine grammar is structurally obstructed by \\cref{thm:b12-hessian-obstruction}.","Parent and child each receive $160$ query units:\n\\begin{center}\n\\begin{tabular}{@{}lrrrrr@{}}\n\\toprule\nProcess & Response atlas & Constructor fit & Validation & Witness & Total","\\midrule\nAffine parent & $48$ & $88$ & $0$ & $24$ & $160$","Quadratic child & $48$ & $64$ & $24$ & $24$ & $160$","\\bottomrule\n\\end{tabular}\n\\end{center}\nThe parent receives more direct constructor-fitting observations. The child spends $24$ observations on constructor validation.","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.","The package chronology is:\n\\begin{equation}\n\\begin{aligned}\n&\\text{replay Block 11 ancestor policy and seal}\\\\\n&\\longrightarrow\n\\text{generate development and meta-validation evidence}\\\\\n&\\longrightarrow\n\\text{compile response and constructor certificates}\\\\\n&\\longrightarrow\n\\text{serialize gates, compiler ledger, and pre-audit seal}\\\\\n&\\longrightarrow\n\\text{generate the two primary audits and stress suites}\\\\\n&\\longrightarrow\n\\text{evaluate all frozen controls and claim gates}.\n\\end{aligned}\n\\label{eq:b12-chronology}\n\\end{equation}\nThe pre-audit seal explicitly records that audit fields are absent.","\\paragraph{Constructor arms.}\nThe constructor experiment includes:\n\\begin{enumerate}\n\\item an independently gated affine parent;\n\\item the selected repair-block quadratic child;\n\\item a directly supervised affine oracle fitted with $1600$ pairs per target;\n\\item wrong-block and random-block quadratic controls;\n\\item shuffled quadratic features;\n\\item a full-space quadratic control;\n\\item stress families outside the declared degree-two injective regime.\n\\end{enumerate}","\\midrule\nsplit-context-repair & $0.934377$ & $1.000000$ & $0.066822$ & $0.001421$","duplicate-parent & $-0.062500$ & $0.000000$ & $0.008805$ & $0.001136$","nuisance-rescale & $-0.062500$ & $0.000000$ & $0.009193$ & $0.001247$","weak-context-repair & $0.416493$ & $0.656250$ & $0.017536$ & $0.001162$","cross-coupled & $-0.541985$ & $0.250000$ & $0.111187$ & $0.014182$","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 $8$.","On $96$ fresh foreign targets:\n\\begin{center}\n\\begin{tabular}{@{}lrrrrr@{}}\n\\toprule\nProcess & Mean gain & Median & Commit & Positive & Harm","\\midrule\nInherited response parent & $0.000000$ & $0.000000$ & $0.000000$ & $0.000000$ & $0.000000$","Expanded-language child & $0.846985$ & $0.969182$ & $0.906250$ & $0.906250$ & $0.000000$","\\bottomrule\n\\end{tabular}\n\\end{center}\nThe paired effect is\n\\begin{equation}\n\\widehat\\Delta_{4\\to5}\n=\n0.846985,\n\\label{eq:b12-response-effect}\n\\end{equation}\nwith frozen interval\n\\begin{equation}\n[0.786100,0.901073]\n\\label{eq:b12-response-ci}\n\\end{equation}\nand sign-flip estimate\n\\begin{equation}\np=9.9999\\times10^{-6}.\n\\end{equation}\nNo primary target favors the inherited parent. The remaining $9.375\\%$ of pairs are abstention ties.","The inherited atlas has modal block ranks\n\\begin{equation}\n(2,4),\n\\end{equation}\nwhile the expanded language recovers\n\\begin{equation}\n(2,2,2).\n\\end{equation}\nMean audit geometry is:\n\\begin{center}\n\\begin{tabular}{@{}lr@{}}\n\\toprule\nQuantity & Mean","\\midrule\nChild context-block overlap & $0.999843$","Parent rank-two context overlap & $0.000146$","Child minimum spectral gap & $0.064981$","Child commutator residual & $0.001826$","\\midrule\nDuplicate response & $0.000000$ & $0.000000$ & $0.000000$","Weak separation & $0.197231$ & $0.260417$ & $0.000000$","Shuffled response & $0.025225$ & $0.062500$ & $0.000000$","Random atlas & $0.000000$ & $0.000000$ & $0.000000$","Blind selected intervention & $0.904608$ & $1.000000$ & $0.000000$","\\bottomrule\n\\end{tabular}\n\\end{center}\nThe 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.","\\midrule\nAffine & $0.320296$ & $0.000000$ & $1.791226$","Repair-block quadratic & $0.998022$ & $0.997731$ & $5.103054$","Context-block quadratic & $0.284136$ & $0.226411$ & $4.851183$","Random-block quadratic & $0.390305$ & $0.393501$ & $4.810982$","Full-space quadratic & $0.999456$ & $0.999169$ & $14.785843$","\\bottomrule\n\\end{tabular}\n\\end{center}\nThe 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\n\\begin{verbatim}\nrepair_quadratic\n\\end{verbatim}\nby the frozen parsimony rule.","\\midrule\nIndependently gated affine parent & $0.289123$ & $0.387596$ & $0.656250$ & $0.645833$ & $0.010417$","Repair-block quadratic child & $0.997511$ & $0.998251$ & $1.000000$ & $1.000000$ & $0.000000$","\\bottomrule\n\\end{tabular}\n\\end{center}\nThe paired effect is\n\\begin{equation}\n\\widehat\\Delta_{5\\to6}\n=\n0.708388,\n\\label{eq:b12-constructor-effect}\n\\end{equation}\nwith frozen interval\n\\begin{equation}\n[0.659635,0.757198].\n\\label{eq:b12-constructor-ci}\n\\end{equation}\nThe sign-flip estimate is $9.9999\\times10^{-6}$, and no primary target favors the affine parent.","A directly supervised affine oracle fitted with $1600$ same-target pairs per target obtains mean blind gain\n\\begin{equation}\n0.466578.\n\\end{equation}\nThe child exceeds that stronger function-class control by mean\n\\begin{equation}\n0.530933,\n\\end{equation}\nwith paired bootstrap interval\n\\begin{equation}\n[0.506587,0.556639].\n\\label{eq:b12-affine-oracle-gap}\n\\end{equation}\nThis supports the nonzero-Hessian obstruction rather than a small-sample explanation for the affine parent.","\\midrule\nWrong-block quadratic & $0.000000$ & $0.000000$ & $0.000000$","Random-block quadratic & $0.009323$ & $0.010417$ & $0.000000$","Shuffled quadratic features & $0.000000$ & $0.000000$ & $0.000000$","Full-space quadratic & $0.999014$ & $1.000000$ & $0.000000$","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.","\\begin{conditionaltheorem}[Finite response-language growth]\n\\label{thm:b12-response-realization}\nWithin the frozen Block 12 response benchmark, assume that:\n\\begin{enumerate}\n\\item the published Block 11 ancestor policy and seal replay exactly;\n\\item the inherited response signatures satisfy the exact alias in \\cref{eq:b12-inherited-signatures};\n\\item parent and child each consume exactly $144$ query units;\n\\item the selected response operator and witness gate are sealed before primary audit generation;\n\\item the expanded language conservatively contains the inherited response algebra;\n\\item the response gap, commutator, support, validation, and witness gates pass;\n\\item the paired effect and interval are those in \\cref{eq:b12-response-effect,eq:b12-response-ci};\n\\item duplicate, weak, shuffled, random-atlas, and stress controls retain their declared interpretations.\n\\end{enumerate}\nThen\n\\begin{equation}\nK_4\\xrightarrow{\\Pi_5}K_5\n\\end{equation}\nis a finite, matched-resource, lineage-generated response-language expansion. The child adds a response distinction absent from every com","e expanded language conservatively contains the inherited response algebra;\n\\item the response gap, commutator, support, validation, and witness gates pass;\n\\item the paired effect and interval are those in \\cref{eq:b12-response-effect,eq:b12-response-ci};\n\\item duplicate, weak, shuffled, random-atlas, and stress controls retain their declared interpretations.\n\\end{enumerate}\nThen\n\\begin{equation}\nK_4\\xrightarrow{\\Pi_5}K_5\n\\end{equation}\nis 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.\n\\end{conditionaltheorem}","ditionaltheorem}[Finite repair-constructor growth]\n\\label{thm:b12-constructor-realization}\nWithin the frozen Block 12 constructor benchmark, assume that:\n\\begin{enumerate}\n\\item the response-language child from \\cref{thm:b12-response-realization} supplies the inherited repair packet;\n\\item the true repair has a nonzero quadratic Hessian on that packet;\n\\item parent and child each consume exactly $160$ query units;\n\\item parent and child receive separately compiled frozen witness gates;\n\\item the affine parent remains embedded by \\cref{prop:b12-affine-embedding};\n\\item the paired effect and interval are those in \\cref{eq:b12-constructor-effect,eq:b12-constructor-ci};\n\\item the directly supervised affine oracle retains the gap in \\cref{eq:b12-affine-oracle-gap};\n\\item wrong-block, random-block, shuffled-feature, full-quadratic, and stress controls retain their declared interpretations.\n\\end{enumerate}\nThen\n\\begin{equation}\nK_5\\xrightarrow{\\Pi_6}K_6\n\\end{equation}\nis a finite, matched-resource, lineage-generated expansion of the repair-constructor grammar from affine to degree-two packe…","\\cref{prop:b12-affine-embedding};\n\\item the paired effect and interval are those in \\cref{eq:b12-constructor-effect,eq:b12-constructor-ci};\n\\item the directly supervised affine oracle retains the gap in \\cref{eq:b12-affine-oracle-gap};\n\\item wrong-block, random-block, shuffled-feature, full-quadratic, and stress controls retain their declared interpretations.\n\\end{enumerate}\nThen\n\\begin{equation}\nK_5\\xrightarrow{\\Pi_6}K_6\n\\end{equation}\nis a finite, matched-resource, lineage-generated expansion of the repair-constructor grammar from affine to degree-two packet-local repair.\n\\end{conditionaltheorem}","\\begin{statusbox}{Block 12 result}\nBlock 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.\n\\end{statusbox}","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.\n\\end{statusbox}","ineage 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.\n\\end{statusbox}","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.\n\\end{statusbox}","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.\n\\end{statusbox}","ion 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.\n\\end{statusbox}","pilers;\n\\item response-algebra estimation, spectral atlas recovery, and commutator diagnostics;\n\\item affine, packet-quadratic, full-quadratic, wrong-packet, random, duplicate, weak, and shuffled controls;\n\\item separately compiled parent and child witness gates;\n\\item canonical JSON and CSV result objects;\n\\item six deterministic publication figures;\n\\item twelve automated tests;\n\\item an exact $41$-file SHA-256 package manifest.\n\\end{itemize}","The frozen response certificate digest is\n\\begin{equation}\n\\texttt{73080b6681940b79e390f7ecd979cb493c740bfab9be7b231002b44c387dc99d}.\n\\end{equation}\nThe frozen constructor certificate digest is\n\\begin{equation}\n\\texttt{d8be9c8e1c0e6ccb5e64f53700c8a59f45c8402e9a395ba7d72db97130010016}.\n\\end{equation}\nThe compiler digest is\n\\begin{equation}\n\\texttt{f51f5152dae06791fb87a6926162bc5e1965bc7e1bedae0a2957b9bbed82f475}.\n\\end{equation}\nThe pre-audit seal digest is\n\\begin{equation}\n\\texttt{75758911e1449620c32ed00b5c959c66b131956f2c79cf1127e15adfbffbd6cd}.\n\\end{equation}","1940b79e390f7ecd979cb493c740bfab9be7b231002b44c387dc99d}.\n\\end{equation}\nThe frozen constructor certificate digest is\n\\begin{equation}\n\\texttt{d8be9c8e1c0e6ccb5e64f53700c8a59f45c8402e9a395ba7d72db97130010016}.\n\\end{equation}\nThe compiler digest is\n\\begin{equation}\n\\texttt{f51f5152dae06791fb87a6926162bc5e1965bc7e1bedae0a2957b9bbed82f475}.\n\\end{equation}\nThe pre-audit seal digest is\n\\begin{equation}\n\\texttt{75758911e1449620c32ed00b5c959c66b131956f2c79cf1127e15adfbffbd6cd}.\n\\end{equation}","The realized compiler edge is\n\\begin{equation}\n\\mathfrak L_6\n\\xrightarrow{\\Pi^{\\Gamma}_7}\n\\mathfrak L_7,\n\\label{eq:b13-lineage}\n\\end{equation}\nwhere 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.","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 $B$.","\\begin{definition}[Automated research compiler]\nAn automated research compiler is a tuple\n\\begin{equation}\n\\Gamma\n=\n(\\Lcal,\\Fcal,\\Pcal,\\Qcal,\\Vcal,\\Mcal,\\Rcal),\n\\label{eq:b13-compiler-tuple}\n\\end{equation}\nwhere:\n\\begin{enumerate}\n\\item $\\Lcal$ is a structured literature and prior-method memory;\n\\item $\\Fcal$ is an ancestor-closed finding graph or shared research forum;\n\\item $\\Pcal$ generates typed candidate methods and pre-result specifications;\n\\item $\\Qcal$ allocates cheap pilots and full experiments under budget $B$;\n\\item $\\Vcal$ is the scored evaluator portfolio with protected capability gates;\n\\item $\\Mcal$ is the integrity and proposal--code congruence monitor;\n\\item $\\Rcal$ compiles pre-result and post-result receipts into the lineage archive.\n\\end{enumerate}\n\\end{definition}","d research compiler is a tuple\n\\begin{equation}\n\\Gamma\n=\n(\\Lcal,\\Fcal,\\Pcal,\\Qcal,\\Vcal,\\Mcal,\\Rcal),\n\\label{eq:b13-compiler-tuple}\n\\end{equation}\nwhere:\n\\begin{enumerate}\n\\item $\\Lcal$ is a structured literature and prior-method memory;\n\\item $\\Fcal$ is an ancestor-closed finding graph or shared research forum;\n\\item $\\Pcal$ generates typed candidate methods and pre-result specifications;\n\\item $\\Qcal$ allocates cheap pilots and full experiments under budget $B$;\n\\item $\\Vcal$ is the scored evaluator portfolio with protected capability gates;\n\\item $\\Mcal$ is the integrity and proposal--code congruence monitor;\n\\item $\\Rcal$ compiles pre-result and post-result receipts into the lineage archive.\n\\end{enumerate}\n\\end{definition}","Fcal,\\Pcal,\\Qcal,\\Vcal,\\Mcal,\\Rcal),\n\\label{eq:b13-compiler-tuple}\n\\end{equation}\nwhere:\n\\begin{enumerate}\n\\item $\\Lcal$ is a structured literature and prior-method memory;\n\\item $\\Fcal$ is an ancestor-closed finding graph or shared research forum;\n\\item $\\Pcal$ generates typed candidate methods and pre-result specifications;\n\\item $\\Qcal$ allocates cheap pilots and full experiments under budget $B$;\n\\item $\\Vcal$ is the scored evaluator portfolio with protected capability gates;\n\\item $\\Mcal$ is the integrity and proposal--code congruence monitor;\n\\item $\\Rcal$ compiles pre-result and post-result receipts into the lineage archive.\n\\end{enumerate}\n\\end{definition}","\\Qcal,\\Vcal,\\Mcal,\\Rcal),\n\\label{eq:b13-compiler-tuple}\n\\end{equation}\nwhere:\n\\begin{enumerate}\n\\item $\\Lcal$ is a structured literature and prior-method memory;\n\\item $\\Fcal$ is an ancestor-closed finding graph or shared research forum;\n\\item $\\Pcal$ generates typed candidate methods and pre-result specifications;\n\\item $\\Qcal$ allocates cheap pilots and full experiments under budget $B$;\n\\item $\\Vcal$ is the scored evaluator portfolio with protected capability gates;\n\\item $\\Mcal$ is the integrity and proposal--code congruence monitor;\n\\item $\\Rcal$ compiles pre-result and post-result receipts into the lineage archive.\n\\end{enumerate}\n\\end{definition}","gin{enumerate}\n\\item $\\Lcal$ is a structured literature and prior-method memory;\n\\item $\\Fcal$ is an ancestor-closed finding graph or shared research forum;\n\\item $\\Pcal$ generates typed candidate methods and pre-result specifications;\n\\item $\\Qcal$ allocates cheap pilots and full experiments under budget $B$;\n\\item $\\Vcal$ is the scored evaluator portfolio with protected capability gates;\n\\item $\\Mcal$ is the integrity and proposal--code congruence monitor;\n\\item $\\Rcal$ compiles pre-result and post-result receipts into the lineage archive.\n\\end{enumerate}\n\\end{definition}","od memory;\n\\item $\\Fcal$ is an ancestor-closed finding graph or shared research forum;\n\\item $\\Pcal$ generates typed candidate methods and pre-result specifications;\n\\item $\\Qcal$ allocates cheap pilots and full experiments under budget $B$;\n\\item $\\Vcal$ is the scored evaluator portfolio with protected capability gates;\n\\item $\\Mcal$ is the integrity and proposal--code congruence monitor;\n\\item $\\Rcal$ compiles pre-result and post-result receipts into the lineage archive.\n\\end{enumerate}\n\\end{definition}","Given problem $\\tau$, the compiler produces a finite research trace\n\\begin{equation}\n\\Gamma(\\tau,B)\n=\n\\left(\n(c_i,\\Pi_i^{-},\\Pi_i^{+})\n\\right)_{i=1}^{n(B)},\n\\label{eq:b13-compiler-trace}\n\\end{equation}\nwhere $c_i$ is a candidate method, $\\Pi_i^{-}$ is sealed before the result is observed, and $\\Pi_i^{+}$ appends evaluation and integrity evidence afterward.","Let $\\operatorname{Valid}(c_i,\\Pi_i^{+})\\in\\{0,1\\}$ encode capability, integrity, provenance, and resource validity. The selected descendant is\n\\begin{equation}\nK_{\\Gamma}^{\\star}(\\tau)\n\\in\n\\argmax_{i:\\operatorname{Valid}(c_i,\\Pi_i^+)=1}\nS_{\\Vcal}(c_i),\n\\label{eq:b13-selected-method}\n\\end{equation}\nwith the no-op baseline retained when no valid candidate exists.","\\begin{definition}[Pre-result research receipt]\nA pre-result receipt has the form\n\\begin{equation}\n\\Pi_i^{-}\n=\n(H_i,O_i,D_i,\\theta_i,\n\\widehat{\\mathbf c}_i,\n\\widehat{\\mathbf r}_i,\nF_i,h_i^{\\mathrm{code}},h_i^{\\mathrm{spec}}),\n\\label{eq:b13-preresult-receipt}\n\\end{equation}\nwhere $H_i$ 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.\n\\end{definition}","in{definition}[Pre-result research receipt]\nA pre-result receipt has the form\n\\begin{equation}\n\\Pi_i^{-}\n=\n(H_i,O_i,D_i,\\theta_i,\n\\widehat{\\mathbf c}_i,\n\\widehat{\\mathbf r}_i,\nF_i,h_i^{\\mathrm{code}},h_i^{\\mathrm{spec}}),\n\\label{eq:b13-preresult-receipt}\n\\end{equation}\nwhere $H_i$ 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.\n\\end{definition}","arch receipt]\nA pre-result receipt has the form\n\\begin{equation}\n\\Pi_i^{-}\n=\n(H_i,O_i,D_i,\\theta_i,\n\\widehat{\\mathbf c}_i,\n\\widehat{\\mathbf r}_i,\nF_i,h_i^{\\mathrm{code}},h_i^{\\mathrm{spec}}),\n\\label{eq:b13-preresult-receipt}\n\\end{equation}\nwhere $H_i$ 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.\n\\end{definition}","}\n\\Pi_i^{-}\n=\n(H_i,O_i,D_i,\\theta_i,\n\\widehat{\\mathbf c}_i,\n\\widehat{\\mathbf r}_i,\nF_i,h_i^{\\mathrm{code}},h_i^{\\mathrm{spec}}),\n\\label{eq:b13-preresult-receipt}\n\\end{equation}\nwhere $H_i$ 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.\n\\end{definition}","widehat{\\mathbf r}_i,\nF_i,h_i^{\\mathrm{code}},h_i^{\\mathrm{spec}}),\n\\label{eq:b13-preresult-receipt}\n\\end{equation}\nwhere $H_i$ 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.\n\\end{definition}","i^{\\mathrm{spec}}),\n\\label{eq:b13-preresult-receipt}\n\\end{equation}\nwhere $H_i$ 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.\n\\end{definition}","\\begin{definition}[Post-result research receipt]\nAfter evaluation, the post-result receipt is\n\\begin{equation}\n\\Pi_i^{+}\n=\n(\\Pi_i^{-},\\mathbf c_i,\\mathbf r_i,a_i,m_i),\n\\label{eq:b13-postresult-receipt}\n\\end{equation}\nwhere $\\mathbf c_i$ 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.\n\\end{definition}","The forecast residual and proposal--code congruence residual are\n\\begin{equation}\n\\eps_{\\mathrm{forecast},i}\n=\n\\norm{\\widehat{\\mathbf c}_i-\\mathbf c_i},\n\\qquad\n\\eps_{\\mathrm{spec/code},i}\n=\nd_{\\mathrm{sem}}\n\\left(\n\\mathrm{Sem}(\\Pi_i^-),\n\\mathrm{Sem}(\\mathrm{code}_i)\n\\right).\n\\label{eq:b13-receipt-residuals}\n\\end{equation}\nIn the executable finite companion, congruence is a Boolean monitor rather than a complete semantic metric.","\\begin{proposition}[Pre-result non-retroactivity]\nSuppose $\\Pi_i^{-}$ 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.\n\\end{proposition}","A finding is represented by\n\\begin{equation}\nf_i\n=\n(\\Pi_i^-,\\Pi_i^+,\\mathrm{code}_i,\n\\operatorname{Par}(i),e_i,o_i),\n\\label{eq:b13-finding-node}\n\\end{equation}\nwhere $e_i$ 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.","Let the scored evaluator portfolio be\n\\begin{equation}\n\\Bcal\n=\n\\{b_1,\\ldots,b_m\\}.\n\\end{equation}\nFor method $c$, define normalized headroom closure\n\\begin{equation}\nq_j(c)\n=\n\\frac{s_j(c)-s_j(K_\\tau)}{s_j^{\\star}-s_j(K_\\tau)}.\n\\label{eq:b13-headroom-closure}\n\\end{equation}\nLet $I_+(c)=\\{j:q_j(c)>0\\}$. The coverage-weighted geometric score is\n\\begin{equation}\nS_{\\mathrm{cov}}(c)\n=\n\\frac{|I_+(c)|}{m}\n\\left(\n\\prod_{j\\in I_+(c)}q_j(c)\n\\right)^{1/|I_+(c)|},\n\\label{eq:b13-coverage-geometric}\n\\end{equation}\nwith value zero when $I_+(c)=\\varnothing$. Hard capability and integrity gates are applied separately rather than traded against this scalar.","zed headroom closure\n\\begin{equation}\nq_j(c)\n=\n\\frac{s_j(c)-s_j(K_\\tau)}{s_j^{\\star}-s_j(K_\\tau)}.\n\\label{eq:b13-headroom-closure}\n\\end{equation}\nLet $I_+(c)=\\{j:q_j(c)>0\\}$. The coverage-weighted geometric score is\n\\begin{equation}\nS_{\\mathrm{cov}}(c)\n=\n\\frac{|I_+(c)|}{m}\n\\left(\n\\prod_{j\\in I_+(c)}q_j(c)\n\\right)^{1/|I_+(c)|},\n\\label{eq:b13-coverage-geometric}\n\\end{equation}\nwith value zero when $I_+(c)=\\varnothing$. Hard capability and integrity gates are applied separately rather than traded against this scalar.","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 $j$.","\\begin{definition}[Evaluator-nullspace]\nThe locally invisible behavior subspace of portfolio $\\Bcal$ is\n\\begin{equation}\n\\Ncal_{\\Bcal}\n=\n\\bigcap_{j=1}^{m}\\ker Db_j.\n\\label{eq:b13-eval-nullspace}\n\\end{equation}\n\\end{definition}","\\begin{theorem}[Portfolio refinement shrinks the invisible subspace]\nLet $\\Bcal^+=\\Bcal\\cup\\{b_{m+1}\\}$. Then\n\\begin{equation}\n\\Ncal_{\\Bcal^+}\n\\subseteq\n\\Ncal_{\\Bcal}.\n\\label{eq:b13-nullspace-inclusion}\n\\end{equation}\nThe inclusion is strict if and only if there exists $v\\in\\Ncal_{\\Bcal}$ such that $Db_{m+1}(v)\\neq0$.\n\\end{theorem}","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\n\\begin{equation}\n\\mathfrak C_B(\\Gamma)\n=\n\\mathbb E_{\\tau\\sim\\Dcal_{\\mathrm{foreign}}}\n\\left[\nU_{\\mathrm{audit}}\n\\left(K_{\\Gamma}^{\\star}(\\tau)\\right)\n-\nU_{\\mathrm{audit}}(K_\\tau)\n\\right].\n\\label{eq:b13-compiler-meta}\n\\end{equation}\nOnly integrity-valid and capability-passing outputs are eligible.","Let $\\mathfrak C_B^{\\mathrm{nom}}$ be the same envelope when integrity constraints are ignored.","\\begin{proposition}[Validity-filtered envelope]\nFor every compiler,\n\\begin{equation}\n\\mathfrak C_B(\\Gamma)\n\\leq\n\\mathfrak C_B^{\\mathrm{nom}}(\\Gamma).\n\\end{equation}\nA nominal compiler gain is not an SRIC improvement certificate unless it remains positive after validity filtering.\n\\end{proposition}","For candidate $c_i$, let $z_i$ collect its pre-result receipt fields and cheap pilot evidence. A research-outcome forecaster returns\n\\begin{equation}\nF_\\theta(z_i)\n=\n(\\widehat g_i,\\widehat r_i,\\widehat s_i),\n\\label{eq:b13-forecaster}\n\\end{equation}\nwhere $\\widehat g_i$ predicts hidden protected gain, $\\widehat r_i$ predicts protected risk, and $\\widehat s_i$ predicts scored-evaluator performance.","A simple acquisition rule is\n\\begin{equation}\nA_i\n=\n\\widehat g_i\n-\\lambda_r\\widehat r_i\n+\\lambda_u\\widehat\\sigma_i\n+\\lambda_h p_\\theta(h_i\\mid x_\\tau)\n-\\lambda_c C_i,\n\\label{eq:b13-acquisition}\n\\end{equation}\nwhere $h_i$ 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.","\\begin{theorem}[Uniform forecast-regret bound]\nLet $g_i$ be the true protected gain of each admissible candidate and suppose\n\\begin{equation}\n|\\widehat g_i-g_i|\\leq\\eps\n\\end{equation}\nfor all candidates. If $\\widehat i\\in\\argmax_i\\widehat g_i$ and $i^\\star\\in\\argmax_i g_i$, then\n\\begin{equation}\ng_{i^\\star}-g_{\\widehat i}\n\\leq\n2\\eps.\n\\label{eq:b13-forecast-regret}\n\\end{equation}\n\\end{theorem}","\\begin{proof}\nBy optimality of $\\widehat i$ under the predicted values,\n\\begin{equation}\n\\widehat g_{\\widehat i}\\geq\\widehat g_{i^\\star}.\n\\end{equation}\nTherefore\n\\begin{equation}\ng_{i^\\star}\n\\leq\n\\widehat g_{i^\\star}+\\eps\n\\leq\n\\widehat g_{\\widehat i}+\\eps\n\\leq\ng_{\\widehat i}+2\\eps.\n\\end{equation}\n\\end{proof}","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\n\\begin{equation}\nD_i\n=\nU_{\\mathrm{audit}}\n\\left(K_{\\Gamma_1}^{\\star}(\\tau_i)\\right)\n-\nU_{\\mathrm{audit}}\n\\left(K_{\\Gamma_0}^{\\star}(\\tau_i)\\right).\n\\end{equation}","\\begin{theorem}[Paired compiler-metaproductivity certificate]\nAssume independent foreign problems and $D_i\\in[-G,G]$. Then with probability at least $1-\\delta$,\n\\begin{equation}\n\\mathbb E D_i\n\\geq\n\\overline D_n\n-\nG\\sqrt{\\frac{2\\log(1/\\delta)}{n}}.\n\\label{eq:b13-paired-compiler-bound}\n\\end{equation}\nA positive lower bound certifies higher compiler metaproductivity relative to the declared research distribution and budget.\n\\end{theorem}","\\begin{proof}\nApply the one-sided Hoeffding inequality to the paired differences, whose interval length is $2G$.\n\\end{proof}","The finite benchmark contains six hidden obstruction classes:\n\\begin{equation}\n\\Hcal_{\\mathrm{obs}}\n=\n\\{\\text{data},\\text{objective},\\text{response},\\text{constructor},\\text{evaluator},\\text{resource}\\}.\n\\label{eq:b13-obstruction-classes}\n\\end{equation}\nThe 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.","The common candidate grammar contains $39$ methods:\n\\begin{enumerate}\n\\item five variants for each of the six obstruction-matched method families;\n\\item five generic variants;\n\\item four explicit integrity or capability controls: a lottery rerun, benchmark-format mimicry, a proposal--code mismatch, and a capability-sacrifice method.\n\\end{enumerate}\nEvery 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.","The parent $\\Gamma_0$ is deliberately strong. It includes:\n\\begin{itemize}\n\\item structured literature priors over method families;\n\\item a shared within-problem finding forum;\n\\item balanced coverage and upper-confidence exploration;\n\\item results-free proposal receipts;\n\\item exact code and specification digests;\n\\item a coverage-weighted multi-benchmark evaluator;\n\\item capability and integrity gates;\n\\item deterministic final selection from the tested valid methods.\n\\end{itemize}","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 $0.02$.","The child $\\Gamma_1$ conservatively retains the parent machinery and adds two lineage-trained modules:\n\\begin{enumerate}\n\\item a nearest-centroid obstruction classifier trained on development diagnostic receipts;\n\\item ridge outcome and risk forecasters trained on development pre/post receipt pairs.\n\\end{enumerate}\nThe 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.","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.","\\midrule\nDevelopment & $108$ & $18$ & Fit archive forecasters and class centroids","Meta-validation & $30$ & $5$ & Select parent and child compiler configurations","Primary foreign audit & $96$ & $16$ & Main sealed paired comparison","Architecture-foreign audit & $72$ & $12$ & Shifted diagnostic and benchmark geometry","The chronology is\n\\begin{equation}\n\\begin{gathered}\n\\text{replay Block 12 result digest}\n\\longrightarrow\n\\text{generate development archive}\\\\\n\\longrightarrow\n\\text{fit classifier and forecasters}\n\\longrightarrow\n\\text{select compiler policies}\\\\\n\\longrightarrow\n\\text{seal compiler receipt and audit commitments}\n\\longrightarrow\n\\text{open final audits}.\n\\end{gathered}\n\\label{eq:b13-chronology}\n\\end{equation}\nThe final policy objects, candidate grammar digest, archive digest, split commitments, and ancestor result digest are sealed before audit outcomes are used.","Every parent and child run receives exactly\n\\begin{equation}\nB_{\\mathrm{research}}=6\n\\label{eq:b13-budget}\n\\end{equation}\nfull 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.","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.","\\midrule\nAAR-style parent $\\Gamma_0$ & $0.883004$ & $0.772566$","SRIC child $\\Gamma_1$ & $0.950231$ & $0.863367$","Random allocation & $0.528176$ & $0.493549$","Audit oracle & $0.952585$ & $0.867092$","The primary paired effect is\n\\begin{equation}\n\\widehat\\Delta_{\\Gamma}^{\\mathrm{primary}}\n=\n0.067227,\n\\qquad\n\\mathrm{CI}_{95\\%}\n=\n[0.035881,0.103274].\n\\label{eq:b13-primary-effect}\n\\end{equation}\nThe sign-flip estimate is $9.9999\\times10^{-6}$. The child improves $21.875\\%$ of pairs, ties on $75\\%$, and loses on $3.125\\%$.","The architecture-foreign paired effect is\n\\begin{equation}\n\\widehat\\Delta_{\\Gamma}^{\\mathrm{arch}}\n=\n0.090800,\n\\qquad\n\\mathrm{CI}_{95\\%}\n=\n[0.052311,0.135081].\n\\label{eq:b13-architecture-effect}\n\\end{equation}\nThe 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.","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.","\\midrule\nData & $0.935$ & $0.935$ & $0.000$","Objective & $0.943$ & $0.981$ & $0.038$","Response & $0.937$ & $0.957$ & $0.020$","Constructor & $0.871$ & $0.975$ & $0.104$","Evaluator & $0.842$ & $0.926$ & $0.084$","Resource & $0.770$ & $0.926$ & $0.156$","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.","The obstruction classifier attains accuracy\n\\begin{equation}\n0.989583\n\\end{equation}\non the primary split and\n\\begin{equation}\n0.958333\n\\end{equation}\non the architecture-foreign split.","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.","\\midrule\nFull child & $0.950231$","Classifier only & $0.950118$","Forecaster only & $0.944727$","No forum & $0.950231$","\\bottomrule\n\\end{tabular}\n\\end{center}\nThe 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.","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 $0.269438$.","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.","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.","ild compiler objects and split commitments are sealed before final audit outcomes are used;\n\\item the paired effects and intervals are those in \\cref{eq:b13-primary-effect,eq:b13-architecture-effect};\n\\item the shuffled-archive, single-benchmark, no-monitor, random-allocation, component-ablation, and audit-oracle controls retain their declared interpretations.\n\\end{enumerate}\nThen\n\\begin{equation}\n\\Gamma_0\n\\xrightarrow{\\Pi_7^{\\Gamma}}\n\\Gamma_1\n\\end{equation}\nis 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.\n\\end{conditionaltheorem}","\\begin{statusbox}{Block 13 result}\nBlock 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.\n\\end{statusbox}","The released implementation contains:\n\\begin{itemize}\n\\item the exact Block 12 canonical result digest;\n\\item six hidden obstruction-class generators and an architecture-foreign shift;\n\\item a common $39$-method candidate grammar;\n\\item results-free and post-result receipt compilation with exact code and specification hashes;\n\\item a strong AAR-style parent compiler;\n\\item a lineage-trained obstruction classifier and ridge outcome/risk forecaster;\n\\item finite parent and child meta-validation grids;\n\\item multi-benchmark coverage scoring, capability gates, and integrity filtering;\n\\item shuffled-archive, classifier, forecaster, forum, single-benchmark, no-monitor, random, and oracle controls;\n\\item deterministic primary and architecture-foreign result objects;\n\\item six publication figures;\n\\item twelve automated tests;\n\\item an exact SHA-256 package manifest.\n\\end{itemize}","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:\n\\begin{equation}\n\\Gamma_1^{\\rm recal}.\n\\end{equation}\nA robust descendant\n\\begin{equation}\n\\Gamma_1^{\\rm recal}\\xrightarrow{\\Pi_8^\\Gamma}\\Gamma_2\n\\label{eq:block14-edge}\n\\end{equation}\nis 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.","d full-experiment budget fixed while changing the reliability conditions under which research decisions must be made. The published Block 13 result digest and exact $39$-method language are replayed. Its compiler design is retained as the inherited parent and recalibrated on the Block 14 development archive:\n\\begin{equation}\n\\Gamma_1^{\\rm recal}.\n\\end{equation}\nA robust descendant\n\\begin{equation}\n\\Gamma_1^{\\rm recal}\\xrightarrow{\\Pi_8^\\Gamma}\\Gamma_2\n\\label{eq:block14-edge}\n\\end{equation}\nis 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.","\\begin{definition}[Research-compiler robustness environment]\nA research-compiler robustness environment is a tuple\n\\begin{equation}\n\\mathfrak U\n=\n\\left(\n\\mathcal Q,\n\\mathcal H,\n\\mathcal A,\n\\mathcal V,\n\\mathcal G\n\\right),\n\\label{eq:robustness-environment}\n\\end{equation}\nwhere:\n\\begin{enumerate}\n\\item $\\mathcal Q$ is a family of research-problem distributions and generator shifts;\n\\item $\\mathcal H$ is a family of archive contaminations and provenance perturbations;\n\\item $\\mathcal A$ is a family of integrity, specification, and selection attacks;\n\\item $\\mathcal V$ is a family of evaluator portfolios and aggregation rules;\n\\item $\\mathcal G$ is a family of benchmark generators, including problems outside the inherited obstruction taxonomy.\n\\end{enumerate}\nAn element $u=(Q,H,A,V,G)\\in\\mathfrak U$ determines one protected evaluation context for the research compiler.\n\\end{definition}","iler robustness environment is a tuple\n\\begin{equation}\n\\mathfrak U\n=\n\\left(\n\\mathcal Q,\n\\mathcal H,\n\\mathcal A,\n\\mathcal V,\n\\mathcal G\n\\right),\n\\label{eq:robustness-environment}\n\\end{equation}\nwhere:\n\\begin{enumerate}\n\\item $\\mathcal Q$ is a family of research-problem distributions and generator shifts;\n\\item $\\mathcal H$ is a family of archive contaminations and provenance perturbations;\n\\item $\\mathcal A$ is a family of integrity, specification, and selection attacks;\n\\item $\\mathcal V$ is a family of evaluator portfolios and aggregation rules;\n\\item $\\mathcal G$ is a family of benchmark generators, including problems outside the inherited obstruction taxonomy.\n\\end{enumerate}\nAn element $u=(Q,H,A,V,G)\\in\\mathfrak U$ determines one protected evaluation context for the research compiler.\n\\end{definition}","al Q,\n\\mathcal H,\n\\mathcal A,\n\\mathcal V,\n\\mathcal G\n\\right),\n\\label{eq:robustness-environment}\n\\end{equation}\nwhere:\n\\begin{enumerate}\n\\item $\\mathcal Q$ is a family of research-problem distributions and generator shifts;\n\\item $\\mathcal H$ is a family of archive contaminations and provenance perturbations;\n\\item $\\mathcal A$ is a family of integrity, specification, and selection attacks;\n\\item $\\mathcal V$ is a family of evaluator portfolios and aggregation rules;\n\\item $\\mathcal G$ is a family of benchmark generators, including problems outside the inherited obstruction taxonomy.\n\\end{enumerate}\nAn element $u=(Q,H,A,V,G)\\in\\mathfrak U$ determines one protected evaluation context for the research compiler.\n\\end{definition}","ness-environment}\n\\end{equation}\nwhere:\n\\begin{enumerate}\n\\item $\\mathcal Q$ is a family of research-problem distributions and generator shifts;\n\\item $\\mathcal H$ is a family of archive contaminations and provenance perturbations;\n\\item $\\mathcal A$ is a family of integrity, specification, and selection attacks;\n\\item $\\mathcal V$ is a family of evaluator portfolios and aggregation rules;\n\\item $\\mathcal G$ is a family of benchmark generators, including problems outside the inherited obstruction taxonomy.\n\\end{enumerate}\nAn element $u=(Q,H,A,V,G)\\in\\mathfrak U$ determines one protected evaluation context for the research compiler.\n\\end{definition}","fts;\n\\item $\\mathcal H$ is a family of archive contaminations and provenance perturbations;\n\\item $\\mathcal A$ is a family of integrity, specification, and selection attacks;\n\\item $\\mathcal V$ is a family of evaluator portfolios and aggregation rules;\n\\item $\\mathcal G$ is a family of benchmark generators, including problems outside the inherited obstruction taxonomy.\n\\end{enumerate}\nAn element $u=(Q,H,A,V,G)\\in\\mathfrak U$ determines one protected evaluation context for the research compiler.\n\\end{definition}","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\n\\begin{equation}\n\\mathfrak C_u(\\Gamma)\n=\n\\mathbb E_{\\tau\\sim Q}\nU_u^{\\rm valid}(\\Gamma,\\tau).\n\\label{eq:compiler-context-utility}\n\\end{equation}\nFor $\\alpha\\in(0,1)$, write $","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\n\\begin{equation}\n\\mathfrak C_u(\\Gamma)\n=\n\\mathbb E_{\\tau\\sim Q}\nU_u^{\\rm valid}(\\Gamma,\\tau).\n\\label{eq:compiler-context-utility}\n\\end{equation}\nFor $\\alpha\\in(0,1)$, write $\\operatorname{CVaR}_\\alpha(X)$ for the mean of the lower $\\alpha$-tail of","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\n\\begin{equation}\n\\mathfrak C_u(\\Gamma)\n=\n\\mathbb E_{\\tau\\sim Q}\nU_u^{\\rm valid}(\\Gamma,\\tau).\n\\label{eq:compiler-context-utility}\n\\end{equation}\nFor $\\alpha\\in(0,1)$, write $\\operatorname{CVaR}_\\alpha(X)$ for the mean of the lower $\\alpha$-tail of a gain variable $X$.","mpiler $\\Gamma$ 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\n\\begin{equation}\n\\mathfrak C_u(\\Gamma)\n=\n\\mathbb E_{\\tau\\sim Q}\nU_u^{\\rm valid}(\\Gamma,\\tau).\n\\label{eq:compiler-context-utility}\n\\end{equation}\nFor $\\alpha\\in(0,1)$, write $\\operatorname{CVaR}_\\alpha(X)$ for the mean of the lower $\\alpha$-tail of a gain variable $X$.","e fixture, a selected method that passes the stronger post hoc capability and integrity predicates receives its protected audit gain, a no-op receives zero, and a selected invalid method receives the frozen penalty $-0.75$. This prevents an invalid selection from being silently converted into a neutral outcome. Let $R_u(\\Gamma,\\tau)$ be protected downside or integrity risk. Define\n\\begin{equation}\n\\mathfrak C_u(\\Gamma)\n=\n\\mathbb E_{\\tau\\sim Q}\nU_u^{\\rm valid}(\\Gamma,\\tau).\n\\label{eq:compiler-context-utility}\n\\end{equation}\nFor $\\alpha\\in(0,1)$, write $\\operatorname{CVaR}_\\alpha(X)$ for the mean of the lower $\\alpha$-tail of a gain variable $X$.","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\n\\begin{equation}\n\\mathfrak C_u(\\Gamma)\n=\n\\mathbb E_{\\tau\\sim Q}\nU_u^{\\rm valid}(\\Gamma,\\tau).\n\\label{eq:compiler-context-utility}\n\\end{equation}\nFor $\\alpha\\in(0,1)$, write $\\operatorname{CVaR}_\\alpha(X)$ for the mean of the lower $\\alpha$-tail of a gain variable $X$.","nd 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\n\\begin{equation}\n\\mathfrak C_u(\\Gamma)\n=\n\\mathbb E_{\\tau\\sim Q}\nU_u^{\\rm valid}(\\Gamma,\\tau).\n\\label{eq:compiler-context-utility}\n\\end{equation}\nFor $\\alpha\\in(0,1)$, write $\\operatorname{CVaR}_\\alpha(X)$ for the mean of the lower $\\alpha$-tail of a gain variable $X$.","-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\n\\begin{equation}\n\\mathfrak C_u(\\Gamma)\n=\n\\mathbb E_{\\tau\\sim Q}\nU_u^{\\rm valid}(\\Gamma,\\tau).\n\\label{eq:compiler-context-utility}\n\\end{equation}\nFor $\\alpha\\in(0,1)$, write $\\operatorname{CVaR}_\\alpha(X)$ for the mean of the lower $\\alpha$-tail of a gain variable $X$.","selection from being silently converted into a neutral outcome. Let $R_u(\\Gamma,\\tau)$ be protected downside or integrity risk. Define\n\\begin{equation}\n\\mathfrak C_u(\\Gamma)\n=\n\\mathbb E_{\\tau\\sim Q}\nU_u^{\\rm valid}(\\Gamma,\\tau).\n\\label{eq:compiler-context-utility}\n\\end{equation}\nFor $\\alpha\\in(0,1)$, write $\\operatorname{CVaR}_\\alpha(X)$ for the mean of the lower $\\alpha$-tail of a gain variable $X$.","\\begin{definition}[Robust validity-filtered compiler value]\nFor risk weight $\\lambda\\geq0$, define\n\\begin{equation}\n\\mathfrak C_{\\rm rob}(\\Gamma)\n=\n\\inf_{u\\in\\mathfrak U}\n\\left[\n\\mathfrak C_u(\\Gamma)\n-\n\\lambda\\operatorname{CVaR}_\\alpha\n\\bigl(-R_u(\\Gamma,\\tau)\\bigr)\n\\right].\n\\label{eq:robust-compiler-value}\n\\end{equation}\nThe 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.\n\\end{definition}","\\begin{definition}[Finite robust compiler edge]\nA sealed lineage edge $\\Gamma_1\\to\\Gamma_2$ is a finite robust compiler improvement relative to a declared stress atlas when:\n\\begin{enumerate}\n\\item parent and child receive the same candidate grammar, evidence interfaces, evaluator calls, integrity calls, and experiment budget;\n\\item the child is sealed before final problems and outcomes are generated;\n\\item paired mean gain has a positive frozen lower confidence bound on both primary and generator-foreign audits;\n\\item the child has nonnegative mean gain in every declared stress group;\n\\item lower-tail gain improves and the selected invalid-method rate does not increase;\n\\item open-set error and protected negative-gain rates remain below declared bounds;\n\\item mechanism ablations","Let $Z$ be a latent obstruction variable and let $X_0$ be the initial diagnostic observation.","\\begin{theorem}[Static diagnostic indistinguishability]\n\\label{thm:static-diagnostic-indistinguishability}\nLet $z_1\\neq z_2$ satisfy\n\\begin{equation}\n\\mathbb P(X_0\\in B\\mid Z=z_1)\n=\n\\mathbb P(X_0\\in B\\mid Z=z_2)\n\\end{equation}\nfor every measurable set $B$. Then no decision rule $d(X_0)$ distinguishes $z_1$ from $z_2$ with accuracy exceeding the larger prior probability:\n\\begin{equation}\n\\mathbb P(d(X_0)=Z\\mid Z\\in\\{z_1,z_2\\})\n\\leq\n\\max\\{\\pi_1,\\pi_2\\},\n\\end{equation}\nwhere $\\pi_j=\\mathbb P(Z=z_j\\mid Z\\in\\{z_1,z_2\\})$.\n\\end{theorem}","nostic-indistinguishability}\nLet $z_1\\neq z_2$ satisfy\n\\begin{equation}\n\\mathbb P(X_0\\in B\\mid Z=z_1)\n=\n\\mathbb P(X_0\\in B\\mid Z=z_2)\n\\end{equation}\nfor every measurable set $B$. Then no decision rule $d(X_0)$ distinguishes $z_1$ from $z_2$ with accuracy exceeding the larger prior probability:\n\\begin{equation}\n\\mathbb P(d(X_0)=Z\\mid Z\\in\\{z_1,z_2\\})\n\\leq\n\\max\\{\\pi_1,\\pi_2\\},\n\\end{equation}\nwhere $\\pi_j=\\mathbb P(Z=z_j\\mid Z\\in\\{z_1,z_2\\})$.\n\\end{theorem}","\\begin{proof}\nThe 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 $\\max\\{\\pi_1,\\pi_2\\}$.\n\\end{proof}","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\n\\begin{equation}\na_1,a_2,\\ldots,a_t\n\\end{equation}\nthat increases the information available for diagnosis.","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_{","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\n\\begin{equation}\ny=Rz+\\xi.\n\\label{eq:sparse-response-model}\n\\end{equation}","geq0}^k$ be a sparse mixture with $\\one^\\top z=1$. An observed response vector has the form\n\\begin{equation}\ny=Rz+\\xi.\n\\label{eq:sparse-response-model}\n\\end{equation}","\\begin{proposition}[Response-mixture recovery bound]\n\\label{prop:response-mixture-bound}\nAssume $R$ has full column rank and $\\norm{\\xi}_2\\leq\\eps$. The least-squares estimator\n\\begin{equation}\n\\widehat z=R^\\dagger y\n\\end{equation}\nsatisfies\n\\begin{equation}\n\\norm{\\widehat z-z}_2\n\\leq\n\\norm{R^\\dagger}_2\\eps\n=\n\\frac{\\eps}{\\sigma_{\\min}(R)}.\n\\label{eq:mixture-recovery-bound}\n\\end{equation}\n\\end{proposition}","\\begin{proof}\nSince $R^\\dagger R=I$ under full column rank,\n\\begin{equation}\n\\widehat z-z=R^\\dagger\\xi.\n\\end{equation}\nThe operator-norm inequality gives the claim.\n\\end{proof}","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.","The inherited obstruction language spans a response cone\n\\begin{equation}\n\\mathcal K_R\n=\n\\{Rz:z\\in\\R_{\\geq0}^k\\}.\n\\end{equation}\nFor observed response $y$, define the open-set residual\n\\begin{equation}\n\\eta_{\\rm open}(y)\n=\n\\dist(y,\\mathcal K_R).\n\\label{eq:open-set-residual}\n\\end{equation}","\\begin{proposition}[Open-set separation]\n\\label{prop:open-set-separation}\nSuppose every inherited known response $y_{\\rm known}$ satisfies\n\\begin{equation}\n\\eta_{\\rm open}(y_{\\rm known})\\leq\\eps,\n\\end{equation}\nwhile every unknown response in a target family satisfies\n\\begin{equation}\n\\eta_{\\rm open}(y_{\\rm unknown})\\geq\\gamma\n\\end{equation}\nfor $\\gamma>\\eps$. Then every threshold $\\tau\\in(\\eps,\\gamma)$ perfectly separates the two families.\n\\end{proposition}","\\begin{proof}\nKnown responses lie below $\\tau$ and unknown responses lie above it by the assumed margin.\n\\end{proof}","Let the clean receipt distribution be $P$ and let an $\\epsilon$-contaminated archive satisfy\n\\begin{equation}\n\\widetilde P_\\epsilon=(1-\\epsilon)P+\\epsilon H,\n\\label{eq:contamination-model}\n\\end{equation}\nwhere $H$ may contain mislabeled obstruction receipts, corrupted outcomes, duplicated evidence, or selectively omitted failures.","\\begin{proposition}[Median breakdown in one coordinate]\n\\label{prop:median-breakdown}\nLet $x_1,\\ldots,x_n\\in\\R$, 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.\n\\end{proposition}","\\begin{proof}\nAt 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.\n\\end{proof}","Let $b_1,\\ldots,b_m$ be benchmark readouts. A single scored benchmark leaves the local nullspace\n\\begin{equation}\n\\mathcal N_{\\{b_1\\}}\n=\n\\ker Db_1.\n\\end{equation}\nAn evaluator portfolio has invisible subspace\n\\begin{equation}\n\\mathcal N_{\\mathcal B}\n=\n\\bigcap_{j=1}^m\\ker Db_j.\n\\label{eq:evaluator-nullspace}\n\\end{equation}","\\begin{proposition}[Strict evaluator-nullspace refinement]\n\\label{prop:evaluator-nullspace-refinement}\nIf $Db_{m+1}$ is nonzero on some $v\\in\\mathcal N_{\\mathcal B}$, then\n\\begin{equation}\n\\mathcal N_{\\mathcal B\\cup\\{b_{m+1}\\}}\n\\subsetneq\n\\mathcal N_{\\mathcal B}.\n\\end{equation}\n\\end{proposition}","\\begin{proof}\nThe 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.\n\\end{proof}","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:\n\\begin{equation}\nM_{\\cap}=\\prod_{j=1}^s M_j.\n\\end{equation}","\\begin{proposition}[Conjunctive monitor false-negative bound]\n\\label{prop:monitor-bound}\nFor an invalid method event $I$, the ensemble false-negative probability satisfies\n\\begin{equation}\n\\mathbb P(M_{\\cap}=1\\mid I)\n\\leq\n\\min_j\\mathbb P(M_j=1\\mid I).\n\\end{equation}\nIf monitor errors are conditionally independent given $I$, then\n\\begin{equation}\n\\mathbb P(M_{\\cap}=1\\mid I)\n=\n\\prod_{j=1}^s\n\\mathbb P(M_j=1\\mid I).\n\\end{equation}\n\\end{proposition}","The inherited parent is the exact Block 13 research compiler reconstructed from the canonical ancestor digest\n\\begin{equation}\n1368fb8a09c3a63ec03ccb05f28e2ba994fc4daf80562d214bf88d29ffaed867.\n\\end{equation}\nIt uses ordinary archive summaries, one-shot diagnostic routing, broad exploration, one inherited integrity test, and the same final candidate selector used in Block 13.","The selected child configuration is\n\\begin{equation}\n\\begin{aligned}\n\\tau_{\\rm residual}&=0.20,\n&\\tau_{\\rm distance}&=24,\\\\\nw_{\\rm mean}&=0.38,\n&w_{\\rm median}&=0.10,\\\\\nw_{\\rm min}&=0.24,\n&w_{\\rm held}&=0.22,\\\\\nw_{\\rm capability}&=0.32.\n\\end{aligned}\n\\label{eq:block14-config}\n\\end{equation}\nIt is selected from a $27$-point finite meta-validation grid before either final audit is opened.","w_{\\rm capability}&=0.32.\n\\end{aligned}\n\\label{eq:block14-config}\n\\end{equation}\nIt is selected from a $27$-point finite meta-validation grid before either final audit is opened.","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.","The development archive is contaminated at rate\n\\begin{equation}\n\\epsilon_{\\rm archive}=0.18.\n\\end{equation}\nContamination 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.","The package additionally evaluates contamination levels\n\\begin{equation}\n0,\n0.05,\n0.10,\n0.18,\n0.25,\n0.35\n\\end{equation}\non a frozen auxiliary population.","Both compilers receive exactly\n\\begin{equation}\nB_{\\rm research}=6\n\\label{eq:block14-budget}\n\\end{equation}\nfull 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.","The chronology is:\n\\begin{equation}\n\\begin{gathered}\n\\text{replay Block 13 ancestor}\n\\longrightarrow\n\\text{build contaminated development archive}\\\\\n\\longrightarrow\n\\text{fit parent and robust archive models}\n\\longrightarrow\n\\text{select child configuration on meta-validation}\\\\\n\\longrightarrow\n\\text{serialize compiler and split commitments}\n\\longrightarrow\n\\text{generate primary and generator-foreign audits}\\\\\n\\longrightarrow\n\\text{evaluate every frozen arm and claim gate}.\n\\end{gathered}\n\\label{eq:block14-chronology}\n\\end{equation}\nThe pre-audit seal contains no audit outcomes.","The primary paired results are:\n\\begin{longtable}{@{}lrrrr@{}}\n\\toprule\nCompiler & Mean gain & Median & $\\operatorname{CVaR}_{0.10}$ & Invalid selected","\\midrule\n\\endfirsthead\n\\toprule\nCompiler & Mean gain & Median & $\\operatorname{CVaR}_{0.10}$ & Invalid selected","\\midrule\n\\endhead\nRecalibrated parent $\\Gamma_1^{\\rm recal}$ & 0.410813 & 0.411537 & -0.133065 & 0.027778","Robust child $\\Gamma_2$ & \\textbf{0.575783} & 0.592510 & \\textbf{0.271516} & \\textbf{0.000000}","The primary paired effect is\n\\begin{equation}\n\\widehat\\Delta_{\\Gamma}^{\\rm primary}\n=\n0.164970,\n\\qquad\n\\mathrm{CI}_{95\\%}\n=\n[0.115591,0.218655].\n\\label{eq:block14-primary-effect}\n\\end{equation}\nThe sign-flip estimate is\n\\begin{equation}\np=9.9999\\times10^{-6}.\n\\end{equation}\nThe 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.","On the generator-foreign audit,\n\\begin{equation}\n\\mathfrak C(\\Gamma_1^{\\rm recal})=0.389514,\n\\qquad\n\\mathfrak C(\\Gamma_2)=0.550559,\n\\end{equation}\nso\n\\begin{equation}\n\\widehat\\Delta_{\\Gamma}^{\\rm foreign}\n=\n0.161045,\n\\qquad\n\\mathrm{CI}_{95\\%}\n=\n[0.105335,0.223141].\n\\label{eq:block14-foreign-effect}\n\\end{equation}\nThe 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.","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.","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.","The parent-to-child primary changes are:\n\\begin{equation}\n\\operatorname{CVaR}_{0.10}:\n-0.133065\n\\longrightarrow\n0.271516,\n\\end{equation}\n\\begin{equation}\n\\mathbb P(G<0):\n0.027778\n\\longrightarrow\n0,\n\\end{equation}\n\\begin{equation}\n\\mathbb P(\\text{invalid selected}):\n0.027778\n\\longrightarrow\n0.\n\\end{equation}\nFor the unknown group, false confident known-class assignment changes from\n\\begin{equation}\n1\n\\longrightarrow\n0.083333.\n\\label{eq:block14-open-set-rate}\n\\end{equation}\nThe child still produces a selected method on every primary problem. The improvement is therefore open-set-aware rerouting rather than universal no-op abstention.","\\begin{remark}[Zero-event interpretation]\nThe 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.\n\\end{remark}","The primary ablation summary is:\n\\begin{longtable}{@{}lrrr@{}}\n\\toprule\nArm & Mean gain & $\\operatorname{CVaR}_{0.10}$ & Key failure","\\midrule\n\\endfirsthead\n\\toprule\nArm & Mean gain & $\\operatorname{CVaR}_{0.10}$ & Key failure","No open set & 0.568030 & 0.094979 & Unknown false-known rate $1$","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.","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.","e 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.","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.","\\begin{conditionaltheorem}[Block 14 finite robust compiler realization]\n\\label{thm:block14-realization}\nWithin the frozen Block 14 benchmark, assume that:\n\\begin{enumerate}\n\\item the published Block 13 canonical result digest replays exactly;\n\\item parent and child receive the exact same $39$-candidate Block 13 grammar and exactly six full experiments per problem;\n\\item development, meta-validation, primary, and generator-foreign supports are disjoint;\n\\item the compiler object and split commitments are sealed before final outcomes are generated;\n\\item the development archive is contaminated according to the declared finite process;\n\\item primary and generator-foreign paired effects have the intervals in \\cref{eq:block14-primary-effect,eq:block14-foreign-effect};\n\\item every child stress-group mean is positive and both child lower-tail gains are positive;\n\\item the no-open-set, no-sequential, ordinary-archive, simple-monitor, single-portfolio, random, contamination, and oracle","The released implementation contains:\n\\begin{itemize}\n\\item exact replay of the Block 13 canonical result digest;\n\\item clean, aliased, mixed, unknown, evaluator-attack, and integrity-attack problem generators;\n\\item an independently shifted generator-foreign audit;\n\\item the exact common $39$-candidate Block 13 method grammar and six-experiment budget;\n\\item contaminated receipt archives with ordinary and robust estimators;\n\\item static and sequential diagnosis, response-mixture estimation, and open-set residuals;\n\\item evaluator-portfolio and integrity-monitor ablations;\n\\item contamination sweeps, paired intervals, sign-flip tests, and lower-tail summaries;\n\\item results-free and post-result receipt chains;\n\\item six publication figures;\n\\item fifteen automated tests;\n\\item an exact package manifest.\n\\end{itemize}","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:\n\\begin{equation}\n\\boxed{\n\\begin{gathered}\n\\text{Can the compiler use persistent open-set conflict}","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:\n\\begin{equation}\n\\boxed{\n\\begin{gathered}\n\\text{Can the compiler use persistent open-set conflict}\\\\\n\\text{to enlarge its own diagnostic and method language?}\n\\end{gathered}\n}\n\\end{equation}\nThe 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.","Let $\\Tau$ be a research-problem space. A finite research-compiler language is a tuple\n\\begin{equation}\n\\mathfrak G\n=\n(\\Zcal,\\Pcal,\\Ccal,\\Bcal),\n\\label{eq:b15-compiler-language}\n\\end{equation}\nwhere:\n\\begin{enumerate}\n\\item $\\Zcal$ is a finite diagnostic-token space;\n\\item $\\Pcal$ is a finite language of admissible diagnostic experiments or probes;\n\\item $\\Ccal$ is a finite method-constructor grammar;\n\\item $\\Bcal$ is a typed binding relation that maps diagnostic tokens to constructed research methods.\n\\end{enumerate}\nA 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","Let $\\Tau$ be a research-problem space. A finite research-compiler language is a tuple\n\\begin{equation}\n\\mathfrak G\n=\n(\\Zcal,\\Pcal,\\Ccal,\\Bcal),\n\\label{eq:b15-compiler-language}\n\\end{equation}\nwhere:\n\\begin{enumerate}\n\\item $\\Zcal$ is a finite diagnostic-token space;\n\\item $\\Pcal$ is a finite language of admissible diagnostic experiments or probes;\n\\item $\\Ccal$ is a finite method-constructor grammar;\n\\item $\\Bcal$ is a typed binding relation that maps diagnostic tokens to constructed research methods.\n\\end{enumerate}\nA 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 $\\Bcal$.","e a research-problem space. A finite research-compiler language is a tuple\n\\begin{equation}\n\\mathfrak G\n=\n(\\Zcal,\\Pcal,\\Ccal,\\Bcal),\n\\label{eq:b15-compiler-language}\n\\end{equation}\nwhere:\n\\begin{enumerate}\n\\item $\\Zcal$ is a finite diagnostic-token space;\n\\item $\\Pcal$ is a finite language of admissible diagnostic experiments or probes;\n\\item $\\Ccal$ is a finite method-constructor grammar;\n\\item $\\Bcal$ is a typed binding relation that maps diagnostic tokens to constructed research methods.\n\\end{enumerate}\nA 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 $\\Bcal$.","),\n\\label{eq:b15-compiler-language}\n\\end{equation}\nwhere:\n\\begin{enumerate}\n\\item $\\Zcal$ is a finite diagnostic-token space;\n\\item $\\Pcal$ is a finite language of admissible diagnostic experiments or probes;\n\\item $\\Ccal$ is a finite method-constructor grammar;\n\\item $\\Bcal$ is a typed binding relation that maps diagnostic tokens to constructed research methods.\n\\end{enumerate}\nA 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 $\\Bcal$.","on}\nwhere:\n\\begin{enumerate}\n\\item $\\Zcal$ is a finite diagnostic-token space;\n\\item $\\Pcal$ is a finite language of admissible diagnostic experiments or probes;\n\\item $\\Ccal$ is a finite method-constructor grammar;\n\\item $\\Bcal$ is a typed binding relation that maps diagnostic tokens to constructed research methods.\n\\end{enumerate}\nA 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 $\\Bcal$.","enumerate}\n\\item $\\Zcal$ is a finite diagnostic-token space;\n\\item $\\Pcal$ is a finite language of admissible diagnostic experiments or probes;\n\\item $\\Ccal$ is a finite method-constructor grammar;\n\\item $\\Bcal$ is a typed binding relation that maps diagnostic tokens to constructed research methods.\n\\end{enumerate}\nA 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 $\\Bcal$.","stic-token space;\n\\item $\\Pcal$ is a finite language of admissible diagnostic experiments or probes;\n\\item $\\Ccal$ is a finite method-constructor grammar;\n\\item $\\Bcal$ is a typed binding relation that maps diagnostic tokens to constructed research methods.\n\\end{enumerate}\nA 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 $\\Bcal$.","a finite language of admissible diagnostic experiments or probes;\n\\item $\\Ccal$ is a finite method-constructor grammar;\n\\item $\\Bcal$ is a typed binding relation that maps diagnostic tokens to constructed research methods.\n\\end{enumerate}\nA 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 $\\Bcal$.","eriments or probes;\n\\item $\\Ccal$ is a finite method-constructor grammar;\n\\item $\\Bcal$ is a typed binding relation that maps diagnostic tokens to constructed research methods.\n\\end{enumerate}\nA 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 $\\Bcal$.","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 $\\bot$:\n\\begin{equation}\n\\phi_0(\\tau_{\nm int})\n=\n\\phi_0(\\tau_{\nm temp})\n=\n\\bot.\n\\label{eq:b15-unknown-alias}\n\\end{equation}\nThe 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.","For each problem $\\tau$, let\n\\begin{equation}\n\\Acal^\\star(\\tau)\n\\subseteq\n\\Ccal\n\\end{equation}\nbe the set of methods that satisfy the declared validity, capability, and gain conditions.","\\begin{definition}[Compiler-language conflict]\nA 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\n\\begin{equation}\n\\bigcap_{\\tau\\in S}\\Acal^\\star(\\tau)=\\varnothing.\n\\label{eq:b15-conflict}\n\\end{equation}\nA minimal conflict has no proper conflicting subset.\n\\end{definition}","\\begin{theorem}[Aliased-token routing obstruction]\n\\label{thm:b15-routing-obstruction}\nLet $\\tau_a$ and $\\tau_b$ satisfy $\\phi_0(\\tau_a)=\\phi_0(\\tau_b)$ and\n\\begin{equation}\n\\Acal^\\star(\\tau_a)\\cap\\Acal^\\star(\\tau_b)=\\varnothing.\n\\end{equation}\nThen no deterministic compiler policy that conditions only on $\\phi_0(\\tau)$ can select an acceptable method for both problems.\n\\end{theorem}","\\begin{proof}\nThe 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.\n\\end{proof}","Let $p\\in\\Pcal^+$ be a candidate experiment with response\n\\begin{equation}\ny_p:\\Tau\\longrightarrow\\R.\n\\end{equation}\nFor threshold $\\theta$, define the refined diagnostic map\n\\begin{equation}\n\\phi_{p,\\theta}(\\tau)\n=\n\\begin{cases}\n(\\phi_0(\\tau),\\mathrm{low}), & y_p(\\tau)\\leq\\theta,","Let $p\\in\\Pcal^+$ be a candidate experiment with response\n\\begin{equation}\ny_p:\\Tau\\longrightarrow\\R.\n\\end{equation}\nFor threshold $\\theta$, define the refined diagnostic map\n\\begin{equation}\n\\phi_{p,\\theta}(\\tau)\n=\n\\begin{cases}\n(\\phi_0(\\tau),\\mathrm{low}), & y_p(\\tau)\\leq\\theta,\\\\\n(\\phi_0(\\tau),\\mathrm{high}), & y_p(\\tau)>\\theta.\n\\end{cases}\n\\label{eq:b15-refined-token}\n\\end{equation}\nThe old token is recovered by the projection\n\\begin{equation}\n\\pi_0(z,s)=z,\n\\qquad\n\\pi_0\\circ\\phi_{p,\\theta}=\\phi_0.\n\\label{eq:b15-token-projection}\n\\end{equation}\nThus the extension is conservative at the level of old diagnostic semantics.","(\\phi_0(\\tau),\\mathrm{high}), & y_p(\\tau)>\\theta.\n\\end{cases}\n\\label{eq:b15-refined-token}\n\\end{equation}\nThe old token is recovered by the projection\n\\begin{equation}\n\\pi_0(z,s)=z,\n\\qquad\n\\pi_0\\circ\\phi_{p,\\theta}=\\phi_0.\n\\label{eq:b15-token-projection}\n\\end{equation}\nThus the extension is conservative at the level of old diagnostic semantics.","\\begin{definition}[Conflict-separating experiment]\nAn 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.\n\\end{definition}","\\begin{theorem}[Finite conservative diagnostic refinement]\n\\label{thm:b15-diagnostic-refinement}\nSuppose 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.\n\\end{theorem}","Let $\\Mcal_0$ denote the inherited singleton-method grammar. For inherited method families $a$ and $b$, define an ordered pair constructor\n\\begin{equation}\n\\operatorname{Seq}(a,b)\n=\n\\text{``apply the research intervention from family $a$, then the intervention from family $b$''}.\n\\label{eq:b15-seq-constructor}\n\\end{equation}\nThe generic finite constructor language is\n\\begin{equation}\n\\Ccal_{\\rm pair}\n=\n\\set{\\operatorname{Seq}(a,b):a,b\\in\\Mcal_0,\\ a\\neq b}.\n\\label{eq:b15-pair-grammar}\n\\end{equation}\nIn the executable fixture there are six typed inherited obstruction families, producing $30$ ordered pair constructors.","ly the research intervention from family $a$, then the intervention from family $b$''}.\n\\label{eq:b15-seq-constructor}\n\\end{equation}\nThe generic finite constructor language is\n\\begin{equation}\n\\Ccal_{\\rm pair}\n=\n\\set{\\operatorname{Seq}(a,b):a,b\\in\\Mcal_0,\\ a\\neq b}.\n\\label{eq:b15-pair-grammar}\n\\end{equation}\nIn the executable fixture there are six typed inherited obstruction families, producing $30$ ordered pair constructors.","Let $U(\\tau,m)$ be protected validity-filtered research gain. Define inherited and extended oracle envelopes\n\\begin{equation}\nO_0(\\tau)=\\max_{m\\in\\Mcal_0}U(\\tau,m),\n\\qquad\nO_1(\\tau)=\\max_{m\\in\\Mcal_0\\cup\\Ccal_{\\rm pair}}U(\\tau,m).\n\\end{equation}\nThe task-relevant structural margin is\n\\begin{equation}\n\\kappa_Q\n=\n\\mathbb E_{\\tau\\sim Q}\n\\left[\nO_1(\\tau)-O_0(\\tau)\n\\right].\n\\label{eq:b15-structural-margin}\n\\end{equation}\nA positive margin means that the extension adds useful methods outside the inherited candidate opportunity set.","\\begin{theorem}[Finite conflict-grounded language synthesis]\n\\label{thm:b15-language-synthesis}\nLet a finite development archive contain one inherited token cell $C$ with at least two incompatible repair subclasses. Suppose:\n\\begin{enumerate}\n\\item a candidate probe $p\\in\\Pcal^+$ separates $C$ into cells $C_-$ and $C_+$;\n\\item there exist $m_- ,m_+\\in\\Ccal_{\\rm pair}$ such that $m_-$ is acceptable throughout $C_-$ and $m_+$ is acceptable throughout $C_+$;\n\\item neither one inherited singleton method nor one unrefined binding is acceptable throughout $C$;\n\\item old-token behavior is preserved through the fallback of \\cref{prop:b15-method-embedding}.\n\\end{enumerate}\nThen the extension\n\\begin{equation}\n\\mathfrak G_0\n\\hookrightarrow\n\\mathfrak G_1\n=\n\\mathfrak G_0\n\\cup\n\\set{p,(\\bot,-),(\\bot,+),m_-,m_+}\n\\end{equation}\neliminates the development conflict while","\\begin{theorem}[Finite conflict-grounded language synthesis]\n\\label{thm:b15-language-synthesis}\nLet a finite development archive contain one inherited token cell $C$ with at least two incompatible repair subclasses. Suppose:\n\\begin{enumerate}\n\\item a candidate probe $p\\in\\Pcal^+$ separates $C$ into cells $C_-$ and $C_+$;\n\\item there exist $m_- ,m_+\\in\\Ccal_{\\rm pair}$ such that $m_-$ is acceptable throughout $C_-$ and $m_+$ is acceptable throughout $C_+$;\n\\item neither one inherited singleton method nor one unrefined binding is acceptable throughout $C$;\n\\item old-token behavior is preserved through the fallback of \\cref{prop:b15-method-embedding}.\n\\end{enumerate}\nThen the extension\n\\begin{equation}\n\\mathfrak G_0\n\\hookrightarrow\n\\mathfrak G_1\n=\n\\mathfrak G_0\n\\cup\n\\set{p,(\\bot,-),(\\bot,+),m_-,m_+}\n\\end{equation}\neliminates the development conflict while conservatively preserving the inherited compiler.\n\\end{theorem}","m}[Finite conflict-grounded language synthesis]\n\\label{thm:b15-language-synthesis}\nLet a finite development archive contain one inherited token cell $C$ with at least two incompatible repair subclasses. Suppose:\n\\begin{enumerate}\n\\item a candidate probe $p\\in\\Pcal^+$ separates $C$ into cells $C_-$ and $C_+$;\n\\item there exist $m_- ,m_+\\in\\Ccal_{\\rm pair}$ such that $m_-$ is acceptable throughout $C_-$ and $m_+$ is acceptable throughout $C_+$;\n\\item neither one inherited singleton method nor one unrefined binding is acceptable throughout $C$;\n\\item old-token behavior is preserved through the fallback of \\cref{prop:b15-method-embedding}.\n\\end{enumerate}\nThen the extension\n\\begin{equation}\n\\mathfrak G_0\n\\hookrightarrow\n\\mathfrak G_1\n=\n\\mathfrak G_0\n\\cup\n\\set{p,(\\bot,-),(\\bot,+),m_-,m_+}\n\\end{equation}\neliminates the development conflict while conservatively preserving the inherited compiler.\n\\end{theorem}","conflict-grounded language synthesis]\n\\label{thm:b15-language-synthesis}\nLet a finite development archive contain one inherited token cell $C$ with at least two incompatible repair subclasses. Suppose:\n\\begin{enumerate}\n\\item a candidate probe $p\\in\\Pcal^+$ separates $C$ into cells $C_-$ and $C_+$;\n\\item there exist $m_- ,m_+\\in\\Ccal_{\\rm pair}$ such that $m_-$ is acceptable throughout $C_-$ and $m_+$ is acceptable throughout $C_+$;\n\\item neither one inherited singleton method nor one unrefined binding is acceptable throughout $C$;\n\\item old-token behavior is preserved through the fallback of \\cref{prop:b15-method-embedding}.\n\\end{enumerate}\nThen the extension\n\\begin{equation}\n\\mathfrak G_0\n\\hookrightarrow\n\\mathfrak G_1\n=\n\\mathfrak G_0\n\\cup\n\\set{p,(\\bot,-),(\\bot,+),m_-,m_+}\n\\end{equation}\neliminates the development conflict while conservatively preserving the inherited compiler.\n\\end{theorem}","hesis]\n\\label{thm:b15-language-synthesis}\nLet a finite development archive contain one inherited token cell $C$ with at least two incompatible repair subclasses. Suppose:\n\\begin{enumerate}\n\\item a candidate probe $p\\in\\Pcal^+$ separates $C$ into cells $C_-$ and $C_+$;\n\\item there exist $m_- ,m_+\\in\\Ccal_{\\rm pair}$ such that $m_-$ is acceptable throughout $C_-$ and $m_+$ is acceptable throughout $C_+$;\n\\item neither one inherited singleton method nor one unrefined binding is acceptable throughout $C$;\n\\item old-token behavior is preserved through the fallback of \\cref{prop:b15-method-embedding}.\n\\end{enumerate}\nThen the extension\n\\begin{equation}\n\\mathfrak G_0\n\\hookrightarrow\n\\mathfrak G_1\n=\n\\mathfrak G_0\n\\cup\n\\set{p,(\\bot,-),(\\bot,+),m_-,m_+}\n\\end{equation}\neliminates the development conflict while conservatively preserving the inherited compiler.\n\\end{theorem}","cell $C$ with at least two incompatible repair subclasses. Suppose:\n\\begin{enumerate}\n\\item a candidate probe $p\\in\\Pcal^+$ separates $C$ into cells $C_-$ and $C_+$;\n\\item there exist $m_- ,m_+\\in\\Ccal_{\\rm pair}$ such that $m_-$ is acceptable throughout $C_-$ and $m_+$ is acceptable throughout $C_+$;\n\\item neither one inherited singleton method nor one unrefined binding is acceptable throughout $C$;\n\\item old-token behavior is preserved through the fallback of \\cref{prop:b15-method-embedding}.\n\\end{enumerate}\nThen the extension\n\\begin{equation}\n\\mathfrak G_0\n\\hookrightarrow\n\\mathfrak G_1\n=\n\\mathfrak G_0\n\\cup\n\\set{p,(\\bot,-),(\\bot,+),m_-,m_+}\n\\end{equation}\neliminates the development conflict while conservatively preserving the inherited compiler.\n\\end{theorem}","_-$ and $C_+$;\n\\item there exist $m_- ,m_+\\in\\Ccal_{\\rm pair}$ such that $m_-$ is acceptable throughout $C_-$ and $m_+$ is acceptable throughout $C_+$;\n\\item neither one inherited singleton method nor one unrefined binding is acceptable throughout $C$;\n\\item old-token behavior is preserved through the fallback of \\cref{prop:b15-method-embedding}.\n\\end{enumerate}\nThen the extension\n\\begin{equation}\n\\mathfrak G_0\n\\hookrightarrow\n\\mathfrak G_1\n=\n\\mathfrak G_0\n\\cup\n\\set{p,(\\bot,-),(\\bot,+),m_-,m_+}\n\\end{equation}\neliminates the development conflict while conservatively preserving the inherited compiler.\n\\end{theorem}","\\begin{proof}\nThe 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.\n\\end{proof}","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.","\\begin{proposition}[Finite search bound]\n\\label{prop:b15-finite-search-bound}\nFor $n$ independent meta-validation problems, with probability at least $1-\\delta$, every candidate language score differs from its population score by at most\n\\begin{equation}\n\\eps_n\n=\n\\sqrt{\n\\frac{\\log(2N/\\delta)}{2n}\n}.\n\\label{eq:b15-uniform-bound}\n\\end{equation}\nConsequently, the selected candidate's population score is within $2\\eps_n$ of the best candidate in the finite language.\n\\end{proposition}","sition}[Finite search bound]\n\\label{prop:b15-finite-search-bound}\nFor $n$ independent meta-validation problems, with probability at least $1-\\delta$, every candidate language score differs from its population score by at most\n\\begin{equation}\n\\eps_n\n=\n\\sqrt{\n\\frac{\\log(2N/\\delta)}{2n}\n}.\n\\label{eq:b15-uniform-bound}\n\\end{equation}\nConsequently, the selected candidate's population score is within $2\\eps_n$ of the best candidate in the finite language.\n\\end{proposition}","\\begin{proof}\nApply 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.\n\\end{proof}","A compiler-language expansion receipt is\n\\begin{equation}\n\\Pi_{\\Gamma\\text{-lang}}\n=\n(\n\\mathfrak G_0,\n\\Hcal_{\\rm conflict},\n\\Pcal^+,\np,\\theta,\n\\Ccal_{\\rm pair},\nm_-,m_+,\n\\iota,\n\\kappa,\n\\Pi_{\\rm seal},\n\\Pi_{\\rm foreign}\n),\n\\label{eq:b15-language-receipt}\n\\end{equation}\nwhere $\\Hcal_{\\rm conflict}$ 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.","The realized compiler lineage is\n\\begin{equation}\n\\Gamma_2\n\\xrightarrow{\\Pi_9^\\Gamma}\n\\Gamma_3.\n\\label{eq:b15-lineage-edge}\n\\end{equation}\nThe parent $\\Gamma_2$ is the exact robust fixed-language compiler design from Block 14, recalibrated only inside the inherited research language. It retains:\n\\begin{itemize}\n\\item six known obstruction tokens and one undifferentiated unknown token;\n\\item the inherited $39$-method singleton grammar;\n\\item results-free receipts, evaluator portfolio, capability gates, integrity filtering, and open-set abstention;\n\\item the same eight-unit deployment research budget used by the child in Block 15.\n\\end{itemize}\nThe hidden generator contains two new obstruction mechanisms. Their semantic identities are unavailable to compiler construction. Both occupy the same inherited static-diagn","The realized compiler lineage is\n\\begin{equation}\n\\Gamma_2\n\\xrightarrow{\\Pi_9^\\Gamma}\n\\Gamma_3.\n\\label{eq:b15-lineage-edge}\n\\end{equation}\nThe parent $\\Gamma_2$ is the exact robust fixed-language compiler design from Block 14, recalibrated only inside the inherited research language. It retains:\n\\begin{itemize}\n\\item six known obstruction tokens and one undifferentiated unknown token;\n\\item the inherited $39$-method singleton grammar;\n\\item results-free receipts, evaluator portfolio, capability gates, integrity filtering, and open-set abstention;\n\\item the same eight-unit deployment research budget used by the child in Block 15.\n\\end{itemize}\nThe 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 th","zed compiler lineage is\n\\begin{equation}\n\\Gamma_2\n\\xrightarrow{\\Pi_9^\\Gamma}\n\\Gamma_3.\n\\label{eq:b15-lineage-edge}\n\\end{equation}\nThe parent $\\Gamma_2$ is the exact robust fixed-language compiler design from Block 14, recalibrated only inside the inherited research language. It retains:\n\\begin{itemize}\n\\item six known obstruction tokens and one undifferentiated unknown token;\n\\item the inherited $39$-method singleton grammar;\n\\item results-free receipts, evaluator portfolio, capability gates, integrity filtering, and open-set abstention;\n\\item the same eight-unit deployment research budget used by the child in Block 15.\n\\end{itemize}\nThe 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.","The child receives only:\n\\begin{enumerate}\n\\item development open-set receipts;\n\\item a generic six-probe experiment language;\n\\item a generic $30$-member ordered-pair method-constructor grammar;\n\\item validation outcomes on development and meta-validation problems;\n\\item the inherited compiler and conservative fallback interface.\n\\end{enumerate}\nThe compiler does not receive the hidden names of the two novel mechanisms or a direct declaration of which probe or pair constructors are correct.","The development set contains $56$ open-set problems. The inherited base-method oracle obtains mean gain\n\\begin{equation}\n\\widehat O_0=0.377257.\n\\label{eq:b15-base-oracle-dev}\n\\end{equation}\nNo one inherited unknown-token binding supplies a common high-gain method across the cell.","For every candidate probe, the compiler:\n\\begin{enumerate}\n\\item evaluates the probe on each development open-set problem;\n\\item splits the cell at the response median;\n\\item searches the $30$ ordered pair constructors independently in the two refined cells;\n\\item scores conflict reduction, protected gain, separation, and constructor complexity;\n\\item records an anonymous two-token diagnostic extension and its two bound methods.\n\\end{enumerate}\nThe 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\n\\begin{equation}\n\\begin{aligned}\n\\texttt{novel-low}\n&\\longmapsto\n\\texttt{evaluator-then-resource},","ndependently in the two refined cells;\n\\item scores conflict reduction, protected gain, separation, and constructor complexity;\n\\item records an anonymous two-token diagnostic extension and its two bound methods.\n\\end{enumerate}\nThe 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\n\\begin{equation}\n\\begin{aligned}\n\\texttt{novel-low}\n&\\longmapsto\n\\texttt{evaluator-then-resource},","lict reduction, protected gain, separation, and constructor complexity;\n\\item records an anonymous two-token diagnostic extension and its two bound methods.\n\\end{enumerate}\nThe 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\n\\begin{equation}\n\\begin{aligned}\n\\texttt{novel-low}\n&\\longmapsto\n\\texttt{evaluator-then-resource},\\\\\n\\texttt{novel-high}\n&\\longmapsto\n\\texttt{objective-then-constructor}.\n\\end{aligned}\n\\label{eq:b15-selected-bindings}\n\\end{equation}\nThe expanded development gain is\n\\begin{equation}\n\\widehat O_1=0.948962,\n\\end{equation}\nwith newly correctable margin\n\\begin{equation}\n\\widehat\\kappa\n=\n0.571704.\n\\label{eq:b15-development-margin}\n\\end{equation}","\\texttt{novel-high}\n&\\longmapsto\n\\texttt{objective-then-constructor}.\n\\end{aligned}\n\\label{eq:b15-selected-bindings}\n\\end{equation}\nThe expanded development gain is\n\\begin{equation}\n\\widehat O_1=0.948962,\n\\end{equation}\nwith newly correctable margin\n\\begin{equation}\n\\widehat\\kappa\n=\n0.571704.\n\\label{eq:b15-development-margin}\n\\end{equation}","\\midrule\nInherited parent on open-set cases & $0$ & $8$ & $8$","Expanded child on open-set cases & $1$ & $7$ & $8$","Inherited fallback on known cases & $0$ & $8$ & $8$","\\midrule\nDevelopment & $60$ & $56$ & conflict and language compilation","Meta-validation & $24$ & $16$ & conservative child verification","Primary foreign audit & $48$ & $48$ & main sealed comparison","Generator-foreign audit & $42$ & $40$ & shifted diagnostic and outcome geometry","The chronology is\n\\begin{equation}\n\\begin{aligned}\n&\\text{replay Block 14 canonical digest}\\\\\n&\\longrightarrow \\text{construct development conflict archive}\\\\\n&\\longrightarrow \\text{search generic probes and pair constructors}\\\\\n&\\longrightarrow \\text{verify conservative fallback on meta-validation}\\\\\n&\\longrightarrow \\text{serialize the language certificate and compiler object}\\\\\n&\\longrightarrow \\text{seal primary and generator-foreign commitments}\\\\\n&\\longrightarrow \\text{open and evaluate both final audits}.\n\\end{aligned}\n\\label{eq:b15-chronology}\n\\end{equation}","The release includes:\n\\begin{enumerate}\n\\item \\textbf{Inherited parent.} One unknown token and the $39$-method singleton grammar.\n\\item \\textbf{Language child.} Selected separating probe, two anonymous tokens, two synthesized pair constructors, and exact inherited fallback.\n\\item \\textbf{Taxonomy only.} The refined tokens are available but no new method constructor is added.\n\\item \\textbf{Grammar only.} The pair constructors are available but the unknown cell remains unrefined and receives one pooled binding.\n\\item \\textbf{Shuffled binding.} The two refined tokens are bound to the opposite synthesized methods.\n\\item \\textbf{Base-grammar oracle.} Inadmissible ceiling over all $39$ inherited methods.\n\\item \\textbf{Expanded-grammar oracle.} Inadmissible ceiling over inherited methods and all $3","tbf{Taxonomy only.} The refined tokens are available but no new method constructor is added.\n\\item \\textbf{Grammar only.} The pair constructors are available but the unknown cell remains unrefined and receives one pooled binding.\n\\item \\textbf{Shuffled binding.} The two refined tokens are bound to the opposite synthesized methods.\n\\item \\textbf{Base-grammar oracle.} Inadmissible ceiling over all $39$ inherited methods.\n\\item \\textbf{Expanded-grammar oracle.} Inadmissible ceiling over inherited methods and all $30$ generic pair constructors.\n\\end{enumerate}\nThe taxonomy-only, grammar-only, and shuffled-binding controls test whether diagnostic refinement and candidate growth must remain correctly coupled.","nly.} The pair constructors are available but the unknown cell remains unrefined and receives one pooled binding.\n\\item \\textbf{Shuffled binding.} The two refined tokens are bound to the opposite synthesized methods.\n\\item \\textbf{Base-grammar oracle.} Inadmissible ceiling over all $39$ inherited methods.\n\\item \\textbf{Expanded-grammar oracle.} Inadmissible ceiling over inherited methods and all $30$ generic pair constructors.\n\\end{enumerate}\nThe taxonomy-only, grammar-only, and shuffled-binding controls test whether diagnostic refinement and candidate growth must remain correctly coupled.","\\midrule\nInherited open-set development oracle & $0.377257$","Expanded two-token development gain & $0.948962$","Newly correctable margin & $0.571704$","Probe interpretation accuracy & $1.000000$","Generic probe candidates & $6$","Generic ordered pair constructors & $30$","Deployable inherited methods & $39$","Deployable child methods & $41$","The deployable child adds only the two selected composites. The remaining $28$ pair constructors are development search alternatives, not hidden deployment candidates.","The primary audit contains $96$ problems. The generator-foreign audit contains $82$ problems with shifted diagnostic and outcome geometry.","\\begin{center}\n\\begin{tabular}{@{}lrrr@{}}\n\\toprule\nAudit and process & Mean gain & Novel mean & Worst-$10\\%$ gain","\\midrule\nPrimary inherited parent & $0.633969$ & $0.341371$ & $0.286117$","Primary language child & $\\mathbf{0.859731}$ & $\\mathbf{0.792894}$ & $\\mathbf{0.333839}$","Generator-foreign parent & $0.574224$ & $0.292978$ & $0.240620$","Generator-foreign child & $\\mathbf{0.814520}$ & $\\mathbf{0.785584}$ & $\\mathbf{0.482289}$","The primary paired effect is\n\\begin{equation}\n\\widehat\\Delta^{\\rm primary}_{\\Gamma\\text{-lang}}\n=\n0.225762,\n\\qquad\n\\mathrm{CI}_{95\\%}\n=\n[0.166708,0.286431].\n\\label{eq:b15-primary-effect}\n\\end{equation}\nThe generator-foreign paired effect is\n\\begin{equation}\n\\widehat\\Delta^{\\rm foreign}_{\\Gamma\\text{-lang}}\n=\n0.240296,\n\\qquad\n\\mathrm{CI}_{95\\%}\n=\n[0.180154,0.302701].\n\\label{eq:b15-foreign-effect}\n\\end{equation}\nBoth 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.","ta^{\\rm primary}_{\\Gamma\\text{-lang}}\n=\n0.225762,\n\\qquad\n\\mathrm{CI}_{95\\%}\n=\n[0.166708,0.286431].\n\\label{eq:b15-primary-effect}\n\\end{equation}\nThe generator-foreign paired effect is\n\\begin{equation}\n\\widehat\\Delta^{\\rm foreign}_{\\Gamma\\text{-lang}}\n=\n0.240296,\n\\qquad\n\\mathrm{CI}_{95\\%}\n=\n[0.180154,0.302701].\n\\label{eq:b15-foreign-effect}\n\\end{equation}\nBoth 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.","08,0.286431].\n\\label{eq:b15-primary-effect}\n\\end{equation}\nThe generator-foreign paired effect is\n\\begin{equation}\n\\widehat\\Delta^{\\rm foreign}_{\\Gamma\\text{-lang}}\n=\n0.240296,\n\\qquad\n\\mathrm{CI}_{95\\%}\n=\n[0.180154,0.302701].\n\\label{eq:b15-foreign-effect}\n\\end{equation}\nBoth 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.","ired effect is\n\\begin{equation}\n\\widehat\\Delta^{\\rm foreign}_{\\Gamma\\text{-lang}}\n=\n0.240296,\n\\qquad\n\\mathrm{CI}_{95\\%}\n=\n[0.180154,0.302701].\n\\label{eq:b15-foreign-effect}\n\\end{equation}\nBoth 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.","\\midrule\nInteraction-type & $0.327041$ & $0.816123$ & $0.489081$","Temporal-type & $0.355700$ & $0.769666$ & $0.413965$","\\bottomrule\n\\end{tabular}\n\\end{center}\nUnder generator shift, the corresponding child means are $0.870372$ and $0.700796$, compared with parent means $0.295177$ and $0.290779$.","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.","\\midrule\nInherited parent & $0.633969$","Taxonomy only & $0.633995$","Grammar only & $0.756265$","Shuffled binding & $0.634112$","Full language child & $\\mathbf{0.859731}$","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:\n\\begin{equation}\n\\boxed{\n\\text{new diagnostic distinction}\n+\n\\text{new method constructor}\n+\n\\text{correct binding}.\n}\n\\end{equation}","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.","The released package contains:\n\\begin{itemize}\n\\item the exact Block 14 canonical result digest;\n\\item known and hidden-novel problem generators with a separately shifted foreign generator;\n\\item the inherited $39$-method singleton grammar;\n\\item a generic six-probe diagnostic language and $30$ ordered pair-constructor grammar;\n\\item conflict scoring, probe selection, anonymous token creation, constructor synthesis, and conservative fallback;\n\\item primary, generator-foreign, taxonomy-only, grammar-only, shuffled-binding, inherited-oracle, and expanded-oracle arms;\n\\item results-free and post-result receipts, compiler and pre-audit seals, deterministic figures, tests, and an exact package manifest.\n\\end{itemize}","\\begin{definition}[Neuralese reasoner]\nA finite-step Neuralese reasoner is a tuple\n\\begin{equation}\n\\Ncal\n=\n(\\Xcal,\\Ycal,H,E,B,U,D,\\Lambda,K),\n\\label{eq:b16-neuralese-object}\n\\end{equation}\nwhere $\\Xcal$ 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.\n\\end{definition}","on}[Neuralese reasoner]\nA finite-step Neuralese reasoner is a tuple\n\\begin{equation}\n\\Ncal\n=\n(\\Xcal,\\Ycal,H,E,B,U,D,\\Lambda,K),\n\\label{eq:b16-neuralese-object}\n\\end{equation}\nwhere $\\Xcal$ 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.\n\\end{definition}","is a tuple\n\\begin{equation}\n\\Ncal\n=\n(\\Xcal,\\Ycal,H,E,B,U,D,\\Lambda,K),\n\\label{eq:b16-neuralese-object}\n\\end{equation}\nwhere $\\Xcal$ 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.\n\\end{definition}","Xcal,\\Ycal,H,E,B,U,D,\\Lambda,K),\n\\label{eq:b16-neuralese-object}\n\\end{equation}\nwhere $\\Xcal$ 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.\n\\end{definition}","Given input $x$, define the prompt anchor $h_0=E(x)$. A normalized injection state $z_t$ and updated latent state $h_t$ satisfy\n\\begin{equation}\nz_t=N(h_{t-1}),\n\\qquad\n\\widetilde h_t=B(z_t,x),\n\\qquad\nh_t=U(z_t,\\widetilde h_t,h_0),\n\\qquad\n1\\leq t\\leq K,\n\\label{eq:b16-latent-recurrence}\n\\end{equation}\nwhere $N$ 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.","ut $x$, define the prompt anchor $h_0=E(x)$. A normalized injection state $z_t$ and updated latent state $h_t$ satisfy\n\\begin{equation}\nz_t=N(h_{t-1}),\n\\qquad\n\\widetilde h_t=B(z_t,x),\n\\qquad\nh_t=U(z_t,\\widetilde h_t,h_0),\n\\qquad\n1\\leq t\\leq K,\n\\label{eq:b16-latent-recurrence}\n\\end{equation}\nwhere $N$ 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.","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.","\\begin{definition}[Neuralese trace]\nFor input $x$ and fixed depth $K$, the injected Neuralese trace is\n\\begin{equation}\nZ_K(x)=(z_1(x),\\ldots,z_K(x))\\in H^K.\n\\end{equation}\nThe 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.\n\\end{definition}","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\n\\begin{equation}\n\\Ical_{\\mathrm{NL}}\n=\n\\{\\mathsf{zero},\\mathsf{replay},\\mathsf{permute},\\mathsf{noise},\n\\mathsf{transplant},\\mathsf{packet},\\mathsf{transport}\\}.\n\\label{eq:b16-intervention-language}\n\\end{equation}","The first five interventions are mandatory in the minimal direct audit.\n\\begin{enumerate}\n\\item $\\mathsf{zero}$ executes no latent step.\n\\item $\\mathsf{replay}$ replaces every injected state by the clean captured state at the same step.\n\\item $\\mathsf{permute}$ applies a sealed nonidentity permutation $\\pi$ to the clean trace.\n\\item $\\mathsf{noise}$ replaces each state by deterministic Gaussian noise with matched Euclidean norm and shape.\n\\item $\\mathsf{transplant}$ replaces $Z(x)$ by $Z(x')$ from a distinct task in the same answer-format group.\n\\end{enumerate}\nThe 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.","minimal direct audit.\n\\begin{enumerate}\n\\item $\\mathsf{zero}$ executes no latent step.\n\\item $\\mathsf{replay}$ replaces every injected state by the clean captured state at the same step.\n\\item $\\mathsf{permute}$ applies a sealed nonidentity permutation $\\pi$ to the clean trace.\n\\item $\\mathsf{noise}$ replaces each state by deterministic Gaussian noise with matched Euclidean norm and shape.\n\\item $\\mathsf{transplant}$ replaces $Z(x)$ by $Z(x')$ from a distinct task in the same answer-format group.\n\\end{enumerate}\nThe 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.","umerate}\n\\item $\\mathsf{zero}$ executes no latent step.\n\\item $\\mathsf{replay}$ replaces every injected state by the clean captured state at the same step.\n\\item $\\mathsf{permute}$ applies a sealed nonidentity permutation $\\pi$ to the clean trace.\n\\item $\\mathsf{noise}$ replaces each state by deterministic Gaussian noise with matched Euclidean norm and shape.\n\\item $\\mathsf{transplant}$ replaces $Z(x)$ by $Z(x')$ from a distinct task in the same answer-format group.\n\\end{enumerate}\nThe 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.","item $\\mathsf{zero}$ executes no latent step.\n\\item $\\mathsf{replay}$ replaces every injected state by the clean captured state at the same step.\n\\item $\\mathsf{permute}$ applies a sealed nonidentity permutation $\\pi$ to the clean trace.\n\\item $\\mathsf{noise}$ replaces each state by deterministic Gaussian noise with matched Euclidean norm and shape.\n\\item $\\mathsf{transplant}$ replaces $Z(x)$ by $Z(x')$ from a distinct task in the same answer-format group.\n\\end{enumerate}\nThe 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.","\\begin{definition}[Per-task Neuralese causal coordinates]\nFor task $(x_i,y_i)$, define\n\\begin{align}\nn_i\n&:=L(x_i,y_i;\\mathsf{zero})-L(x_i,y_i;Z(x_i)),\n\\label{eq:b16-necessity}","\\begin{definition}[Per-task Neuralese causal coordinates]\nFor task $(x_i,y_i)$, define\n\\begin{align}\nn_i\n&:=L(x_i,y_i;\\mathsf{zero})-L(x_i,y_i;Z(x_i)),\n\\label{eq:b16-necessity}\\\\\no_i\n&:=L(x_i,y_i;Z_{\\pi}(x_i))-L(x_i,y_i;Z(x_i)),\n\\label{eq:b16-order}\\\\\nc_i\n&:=L(x_i,y_i;Z_{\\eta}(x_i))-L(x_i,y_i;Z(x_i)),\n\\label{eq:b16-noise}\\\\\ns_i\n&:=L(x_i,y_i;Z(x'_i))-L(x_i,y_i;Z(x_i)),\n\\label{eq:b16-specificity}\\\\\nr_i\n&:=\\abs{L(x_i,y_i;\\mathsf{replay}(Z(x_i)))-L(x_i,y_i;Z(x_i))}.\n\\label{eq:b16-replay}\n\\end{align}\nThese are respectively latent necessity, order sensitivity, corruption sensitivity, task specificity, and replay error.\n\\end{definition}","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.","\\begin{definition}[Structural-normalized specificity]\nLet\n\\begin{equation}\nQ_{\\mathrm{spec}}\n=\n\\frac{\\overline s}\n{\\max\\{\\overline o,\\overline c,\\eps\\}},\n\\label{eq:b16-specificity-ratio}\n\\end{equation}\nwhere bars denote held-out means and $\\eps>0$ is a fixed numerical floor. The default Block 16 specificity gate requires\n\\begin{equation}\n\\LCB_{0.95}(\\overline s)>0,\n\\qquad\n\\frac{1}{n}\\sum_i\\one[s_i>0]\\geq0.75,\n\\qquad\nQ_{\\mathrm{spec}}\\geq0.25.\n\\label{eq:b16-specificity-gate}\n\\end{equation}\n\\end{definition}","\\begin{theorem}[Format-conditional transplant null]\n\\label{thm:b16-transplant-null}\nLet $G(x)$ be a declared answer-format class. Suppose that, conditional on $G(x)=g$, the trace law is independent of task content:\n\\begin{equation}\nZ(x)\\mid G(x)=g\n\\stackrel{d}{=}\nZ(x')\\mid G(x')=g,\n\\end{equation}\nwhere $x'$ is an independent donor. Assume the intervention harness leaves the prompt and target answer fixed and replaces only the latent trace. Then\n\\begin{equation}\n\\mathbb E[s_i\\mid G(x_i)=g]=0.\n\\end{equation}\n\\end{theorem}","6-transplant-null}\nLet $G(x)$ be a declared answer-format class. Suppose that, conditional on $G(x)=g$, the trace law is independent of task content:\n\\begin{equation}\nZ(x)\\mid G(x)=g\n\\stackrel{d}{=}\nZ(x')\\mid G(x')=g,\n\\end{equation}\nwhere $x'$ is an independent donor. Assume the intervention harness leaves the prompt and target answer fixed and replaces only the latent trace. Then\n\\begin{equation}\n\\mathbb E[s_i\\mid G(x_i)=g]=0.\n\\end{equation}\n\\end{theorem}","\\begin{proof}\nUnder 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.\n\\end{proof}","\\begin{proposition}[Replay integrity]\n\\label{prop:b16-replay}\nAssume 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.\n\\end{proposition}","\\begin{proof}\nInduct 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.\n\\end{proof}","For fixed maximum depth $K_{\\max}$, let $\\lambda_t(x)=\\Lambda(z_t(x))$. Define cumulative halt mass\n\\begin{equation}\nM_t(x)\n=\n\\sum_{j=1}^{t}\n\\lambda_j(x)\n\\prod_{q<j}(1-\\lambda_q(x)),\n\\end{equation}\nand dynamic depth\n\\begin{equation}\nT(x)=\\min\\{t:M_t(x)\\geq\\tau\\},\n\\end{equation}\nwith 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\n\\begin{equation}\n\\rho_{\\mathrm{Spearman}}(d(x),T(x))>0\n\\end{equation}\nand 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 featur","For fixed maximum depth $K_{\\max}$, let $\\lambda_t(x)=\\Lambda(z_t(x))$. Define cumulative halt mass\n\\begin{equation}\nM_t(x)\n=\n\\sum_{j=1}^{t}\n\\lambda_j(x)\n\\prod_{q<j}(1-\\lambda_q(x)),\n\\end{equation}\nand dynamic depth\n\\begin{equation}\nT(x)=\\min\\{t:M_t(x)\\geq\\tau\\},\n\\end{equation}\nwith 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\n\\begin{equation}\n\\rho_{\\mathrm{Spearman}}(d(x),T(x))>0\n\\end{equation}\nand 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","For fixed maximum depth $K_{\\max}$, let $\\lambda_t(x)=\\Lambda(z_t(x))$. Define cumulative halt mass\n\\begin{equation}\nM_t(x)\n=\n\\sum_{j=1}^{t}\n\\lambda_j(x)\n\\prod_{q<j}(1-\\lambda_q(x)),\n\\end{equation}\nand dynamic depth\n\\begin{equation}\nT(x)=\\min\\{t:M_t(x)\\geq\\tau\\},\n\\end{equation}\nwith 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\n\\begin{equation}\n\\rho_{\\mathrm{Spearman}}(d(x),T(x))>0\n\\end{equation}\nand 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.","_{\\max}$, let $\\lambda_t(x)=\\Lambda(z_t(x))$. Define cumulative halt mass\n\\begin{equation}\nM_t(x)\n=\n\\sum_{j=1}^{t}\n\\lambda_j(x)\n\\prod_{q<j}(1-\\lambda_q(x)),\n\\end{equation}\nand dynamic depth\n\\begin{equation}\nT(x)=\\min\\{t:M_t(x)\\geq\\tau\\},\n\\end{equation}\nwith 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\n\\begin{equation}\n\\rho_{\\mathrm{Spearman}}(d(x),T(x))>0\n\\end{equation}\nand 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.","halt mass\n\\begin{equation}\nM_t(x)\n=\n\\sum_{j=1}^{t}\n\\lambda_j(x)\n\\prod_{q<j}(1-\\lambda_q(x)),\n\\end{equation}\nand dynamic depth\n\\begin{equation}\nT(x)=\\min\\{t:M_t(x)\\geq\\tau\\},\n\\end{equation}\nwith 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\n\\begin{equation}\n\\rho_{\\mathrm{Spearman}}(d(x),T(x))>0\n\\end{equation}\nand 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.","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.","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\n\\begin{equation}\nT^*\n=\n\\argmin_{T^{\\mathsf T}T=I}\\norm{XT-Y}_{\\mathrm F}.\n\\label{eq:b16-procrustes}\n\\end{equation}\nIf $X^{\\mathsf T}Y=U\\Sigma V^{\\mathsf T}$, 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.","\\begin{proposition}[Exact orthogonal latent transport]\n\\label{prop:b16-exact-transport}\nSuppose $Y=XQ$ 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.\n\\end{proposition}","\\begin{proof}\nThe objective has value zero at $Q$. Full column rank implies that $XT=Y=XQ$ only when $T=Q$. Orthogonality gives $QQ^{\\mathsf T}=I$.\n\\end{proof}","Extend the SRIC lineage state by\n\\begin{equation}\n\\mathfrak L_t^{\\mathrm{NL}}\n=\n(\\mathfrak L_t,\\Ncal_t,\\Ical_t^{\\mathrm{NL}},\\Tau_t^{\\mathrm{NL}},\\Wcal_t^{\\mathrm{NL}}),\n\\end{equation}\nwhere $\\Tau_t^{\\mathrm{NL}}$ 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.","\\begin{conditionaltheorem}[Finite Neuralese metaproductivity edge]\n\\label{thm:b16-neuralese-edge}\nLet 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:\n\\begin{enumerate}\n\\item both audits satisfy replay integrity;\n\\item the child satisfies latent necessity, ordered computation, corruption sensitivity, and task specificity on primary and generator-foreign supports;\n\\item the paired child-minus-parent transplant-cost interval has positive lower endpoint;\n\\item child clean loss does not regress beyond a predeclared tolerance;\n\\item every claimed packet or transport passes its own held-out causal certificate;\n\\item model, task, audit, and result dige","\\begin{conditionaltheorem}[Finite Neuralese metaproductivity edge]\n\\label{thm:b16-neuralese-edge}\nLet 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:\n\\begin{enumerate}\n\\item both audits satisfy replay integrity;\n\\item the child satisfies latent necessity, ordered computation, corruption sensitivity, and task specificity on primary and generator-foreign supports;\n\\item the paired child-minus-parent transplant-cost interval has positive lower endpoint;\n\\item child clean loss does not regress beyond a predeclared tolerance;\n\\item every claimed packet or transport passes its own held-out causal certificate;\n\\item model, task, audit, and result digests are committed in","child-minus-parent transplant-cost interval has positive lower endpoint;\n\\item child clean loss does not regress beyond a predeclared tolerance;\n\\item every claimed packet or transport passes its own held-out causal certificate;\n\\item model, task, audit, and result digests are committed in an append-only receipt chain before promotion.\n\\end{enumerate}\nThen the accepted transition\n\\begin{equation}\n\\mathfrak L_t^{\\mathrm{NL}}\n\\xrightarrow{\\Pi_{16}}\n\\mathfrak L_{t+1}^{\\mathrm{NL}}\n\\end{equation}\nis 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.\n\\end{conditionaltheorem}","\\midrule\n\\endhead\nPrompt encoder $E$ & Tokenized ChatML prefix ending in \\texttt{<bot>}","Injected state $z_t$ & Normalized state immediately before recurrent backbone reinjection","Backbone transition $B$ & One cached call through the wrapped Qwen backbone","Update gate $U$ & \\texttt{latent\\_update\\_gate(h, h\\_new, h\\_prompt)}","Halting map $\\Lambda$ & \\texttt{halt\\_head} evaluated on the injected state","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.","\\midrule\n\\endhead\nMean clean loss & $0.235716$ & $0.000405$","Mean latent-necessity gain & $0.647409$ & $0.882720$","Mean order cost & $0.182441$ & $0.276445$","Mean norm-matched noise cost & $0.982047$ & $1.364676$","Mean cross-task transplant cost & $0.016648$ & $0.456634$","Structural-normalized specificity & $0.016952$ & $0.334610$","Difficulty--depth Spearman correlation & $0$ & $0.969320$","Distinct dynamic depths & $1$ & $5$","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.","A real Hyper audit follows the frozen sequence\n\\begin{equation}\n\\begin{gathered}\n\\text{fingerprint local runtime and weights}\n\\longrightarrow\n\\text{freeze task and donor digests}\n\\longrightarrow\n\\text{capture clean traces}\\\\\n\\longrightarrow\n\\text{execute zero, replay, and permutation arms}\\\\\n\\longrightarrow\n\\text{execute noise and transplant arms}\\\\\n\\longrightarrow\n\\text{compute paired intervals and gates}\\\\\n\\longrightarrow\n\\text{serialize result and receipt digests}\n\\longrightarrow\n\\text{promote, reject, or retain as PARTIAL}.\n\\end{gathered}\n\\end{equation}\nThe 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.","\\longrightarrow\n\\text{serialize result and receipt digests}\n\\longrightarrow\n\\text{promote, reject, or retain as PARTIAL}.\n\\end{gathered}\n\\end{equation}\nThe 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.","Let an affine instruction be\n\\begin{equation}\n f_j(z)=a_jz+b_j,\n \\qquad\n a_j\\in\\left\\{-1,-\\tfrac12,\\tfrac12,1\\right\\},\n \\qquad\n b_j\\in\\left\\{-\\tfrac12,0,\\tfrac12\\right\\}.\n \\label{eq:b17-affine-instruction}\n\\end{equation}\nA depth-$m$ program acts on $x\\in\\{-1,-\\tfrac34,\\ldots,1\\}$ by\n\\begin{equation}\n F_m(x)\n =\n (f_m\\circ\\cdots\\circ f_1)(x)\n =A_mx+B_m,\n \\label{eq:b17-affine-program}\n\\end{equation}\nwhere\n\\begin{equation}\n A_0=1,\n \\qquad\n B_0=0,\n \\qquad\n A_j=a_jA_{j-1},\n \\qquad\n B_j=a_jB_{j-1}+b_j.\n \\label{eq:b17-affine-witness-recursion}\n\\end{equation}\nAll 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.","ion be\n\\begin{equation}\n f_j(z)=a_jz+b_j,\n \\qquad\n a_j\\in\\left\\{-1,-\\tfrac12,\\tfrac12,1\\right\\},\n \\qquad\n b_j\\in\\left\\{-\\tfrac12,0,\\tfrac12\\right\\}.\n \\label{eq:b17-affine-instruction}\n\\end{equation}\nA depth-$m$ program acts on $x\\in\\{-1,-\\tfrac34,\\ldots,1\\}$ by\n\\begin{equation}\n F_m(x)\n =\n (f_m\\circ\\cdots\\circ f_1)(x)\n =A_mx+B_m,\n \\label{eq:b17-affine-program}\n\\end{equation}\nwhere\n\\begin{equation}\n A_0=1,\n \\qquad\n B_0=0,\n \\qquad\n A_j=a_jA_{j-1},\n \\qquad\n B_j=a_jB_{j-1}+b_j.\n \\label{eq:b17-affine-witness-recursion}\n\\end{equation}\nAll 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.","\\begin{definition}[Executable affine-program task]\nA Block 17 task is\n\\begin{equation}\n \\tau\n =\n (x,m,(a_j,b_j)_{j=1}^{m},A_m,B_m,y_m,g,s),\n \\qquad\n y_m=A_mx+B_m,\n \\label{eq:b17-task}\n\\end{equation}\nwhere $g\\in\\{A,B\\}$ 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\n\\begin{equation}\n w(\\tau)\n =\n \\left(A_m,\\frac{B_m}{3},\\frac{y_m}{4}\\right)\\in\\R^3.\n \\label{eq:b17-semantic-witness}\n\\end{equation}\n\\end{definition}","[Executable affine-program task]\nA Block 17 task is\n\\begin{equation}\n \\tau\n =\n (x,m,(a_j,b_j)_{j=1}^{m},A_m,B_m,y_m,g,s),\n \\qquad\n y_m=A_mx+B_m,\n \\label{eq:b17-task}\n\\end{equation}\nwhere $g\\in\\{A,B\\}$ 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\n\\begin{equation}\n w(\\tau)\n =\n \\left(A_m,\\frac{B_m}{3},\\frac{y_m}{4}\\right)\\in\\R^3.\n \\label{eq:b17-semantic-witness}\n\\end{equation}\n\\end{definition}","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.","\\midrule\n\\endhead\nTraining & $10{,}000$ & $A$ & Parent and matched repair optimization","Development & $1{,}400$ & $A$ & Compiler development and diagnostics","Meta-validation & $1{,}400$ & $A$ & Frozen candidate eligibility and selection","Probe calibration & $1{,}200$ & $A$ & External ridge-probe fitting","Primary audit & $1{,}600$ & $A$ & Fresh final in-generator evaluation","Generator-foreign audit & $1{,}600$ & $B$ & Fresh shifted-distribution evaluation","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.","\\begin{proposition}[Exact witness soundness]\n\\label{prop:b17-witness-soundness}\nFor 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.\n\\end{proposition}","\\begin{proof}\nThe 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.\n\\end{proof}","Let $\\phi(\\tau)\\in\\R^{19}$ be the padded task serialization, and let\n\\begin{equation}\n e=E_\\theta(\\phi(\\tau))\\in\\R^{40}\n \\label{eq:b17-encoder}\n\\end{equation}\nbe the prompt representation. The latent trace has four states. Given conditioning mode $\\alpha$, define\n\\begin{equation}\n z_0=\\zeta,\n \\qquad\n z_{k+1}\n =\n \\tanh\\!\\left(Rz_k+s_k+C_\\alpha(e,m)\\right),\n \\qquad\n 0\\leq k<4,\n \\label{eq:b17-trace}\n\\end{equation}\nwhere $s_k$ is a learned step embedding. The task-blind parent satisfies\n\\begin{equation}\n C_{\\mathrm{parent}}(e,m)=0.\n \\label{eq:b17-parent-conditioning}\n\\end{equation}\nThe descendants use one of the following conditioning maps:\n\\begin{align}\n C_{\\mathrm{depth}}(e,m)&=D\\,\\one_m,\n \\label{eq:b17-depth-conditio","Let $\\phi(\\tau)\\in\\R^{19}$ be the padded task serialization, and let\n\\begin{equation}\n e=E_\\theta(\\phi(\\tau))\\in\\R^{40}\n \\label{eq:b17-encoder}\n\\end{equation}\nbe the prompt representation. The latent trace has four states. Given conditioning mode $\\alpha$, define\n\\begin{equation}\n z_0=\\zeta,\n \\qquad\n z_{k+1}\n =\n \\tanh\\!\\left(Rz_k+s_k+C_\\alpha(e,m)\\right),\n \\qquad\n 0\\leq k<4,\n \\label{eq:b17-trace}\n\\end{equation}\nwhere $s_k$ is a learned step embedding. The task-blind parent satisfies\n\\begin{equation}\n C_{\\mathrm{parent}}(e,m)=0.\n \\label{eq:b17-parent-conditioning}\n\\end{equation}\nThe descendants use one of the following conditioning maps:\n\\begin{align}\n C_{\\mathrm{depth}}(e,m)&=D\\,\\one_m,\n \\label{eq:b17-depth-conditioning}","19}$ be the padded task serialization, and let\n\\begin{equation}\n e=E_\\theta(\\phi(\\tau))\\in\\R^{40}\n \\label{eq:b17-encoder}\n\\end{equation}\nbe the prompt representation. The latent trace has four states. Given conditioning mode $\\alpha$, define\n\\begin{equation}\n z_0=\\zeta,\n \\qquad\n z_{k+1}\n =\n \\tanh\\!\\left(Rz_k+s_k+C_\\alpha(e,m)\\right),\n \\qquad\n 0\\leq k<4,\n \\label{eq:b17-trace}\n\\end{equation}\nwhere $s_k$ is a learned step embedding. The task-blind parent satisfies\n\\begin{equation}\n C_{\\mathrm{parent}}(e,m)=0.\n \\label{eq:b17-parent-conditioning}\n\\end{equation}\nThe descendants use one of the following conditioning maps:\n\\begin{align}\n C_{\\mathrm{depth}}(e,m)&=D\\,\\one_m,\n \\label{eq:b17-depth-conditioning}","tau))\\in\\R^{40}\n \\label{eq:b17-encoder}\n\\end{equation}\nbe the prompt representation. The latent trace has four states. Given conditioning mode $\\alpha$, define\n\\begin{equation}\n z_0=\\zeta,\n \\qquad\n z_{k+1}\n =\n \\tanh\\!\\left(Rz_k+s_k+C_\\alpha(e,m)\\right),\n \\qquad\n 0\\leq k<4,\n \\label{eq:b17-trace}\n\\end{equation}\nwhere $s_k$ is a learned step embedding. The task-blind parent satisfies\n\\begin{equation}\n C_{\\mathrm{parent}}(e,m)=0.\n \\label{eq:b17-parent-conditioning}\n\\end{equation}\nThe descendants use one of the following conditioning maps:\n\\begin{align}\n C_{\\mathrm{depth}}(e,m)&=D\\,\\one_m,\n \\label{eq:b17-depth-conditioning}","efine\n\\begin{equation}\n z_0=\\zeta,\n \\qquad\n z_{k+1}\n =\n \\tanh\\!\\left(Rz_k+s_k+C_\\alpha(e,m)\\right),\n \\qquad\n 0\\leq k<4,\n \\label{eq:b17-trace}\n\\end{equation}\nwhere $s_k$ is a learned step embedding. The task-blind parent satisfies\n\\begin{equation}\n C_{\\mathrm{parent}}(e,m)=0.\n \\label{eq:b17-parent-conditioning}\n\\end{equation}\nThe descendants use one of the following conditioning maps:\n\\begin{align}\n C_{\\mathrm{depth}}(e,m)&=D\\,\\one_m,\n \\label{eq:b17-depth-conditioning}\\\\\n C_{1}(e,m)&=U_1V_1e,\n \\qquad \\rank(U_1V_1)\\leq1,\n \\label{eq:b17-rank1-conditioning}\\\\\n C_{4}(e,m)&=U_4V_4e,\n \\qquad \\rank(U_4V_4)\\leq4,\n \\label{eq:b17-rank4-conditioning}\\\\\n C_{\\mathrm{full}}(e,m)&=We.\n \\label{eq:b17-full-conditioning}\n\\end{align}\nHere $\\one_m$ 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.","C_{\\mathrm{full}}(e,m)&=We.\n \\label{eq:b17-full-conditioning}\n\\end{align}\nHere $\\one_m$ 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.","The latent trace updates the task representation through\n\\begin{align}\n g_k&=\\sigma(Gz_k),\n \\label{eq:b17-update-gate}\\\\\n \\widetilde h_k&=\\tanh(Th_{k-1}+Lz_k),\n \\label{eq:b17-hidden-candidate}\\\\\n h_k&=(1-g_k)\\odot h_{k-1}+g_k\\odot\\widetilde h_k,\n \\qquad h_0=e,\n \\label{eq:b17-hidden-update}\\\\\n \\widehat w_k&=Oh_k.\n \\label{eq:b17-readout}\n\\end{align}\nA 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.","\\widehat w_k&=Oh_k.\n \\label{eq:b17-readout}\n\\end{align}\nA 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.","\\begin{definition}[Finite Neuralese repair grammar]\nThe Block 17 compiler grammar is\n\\begin{equation}\n \\Gcal_{17}\n =\n \\{\\mathsf{parent}^{+},\\mathsf{depth},\\mathsf{rank1},\n \\mathsf{rank4},\\mathsf{full},\\mathsf{full}^{-\\mathrm{sem}}\\}.\n \\label{eq:b17-repair-grammar}\n\\end{equation}\nThe 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.\n\\end{definition}","gin{equation}\n \\Gcal_{17}\n =\n \\{\\mathsf{parent}^{+},\\mathsf{depth},\\mathsf{rank1},\n \\mathsf{rank4},\\mathsf{full},\\mathsf{full}^{-\\mathrm{sem}}\\}.\n \\label{eq:b17-repair-grammar}\n\\end{equation}\nThe 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.\n\\end{definition}","hsf{parent}^{+},\\mathsf{depth},\\mathsf{rank1},\n \\mathsf{rank4},\\mathsf{full},\\mathsf{full}^{-\\mathrm{sem}}\\}.\n \\label{eq:b17-repair-grammar}\n\\end{equation}\nThe 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.\n\\end{definition}","sf{depth},\\mathsf{rank1},\n \\mathsf{rank4},\\mathsf{full},\\mathsf{full}^{-\\mathrm{sem}}\\}.\n \\label{eq:b17-repair-grammar}\n\\end{equation}\nThe 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.\n\\end{definition}","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.","\\begin{proposition}[Nested conditioning images]\nFor fixed encoder $E_\\theta$, the rank-one and rank-four conditioning families are contained in the full linear conditioning family:\n\\begin{equation}\n \\{U_1V_1\\}\n \\subseteq\n \\{U_4V_4\\}\n \\subseteq\n \\{W\\}.\n \\label{eq:b17-nested-conditioning}\n\\end{equation}\nThe task-blind parent is recovered by the zero map in every family.\n\\end{proposition}","Let $d_{\\mathrm{prog}}(\\tau,\\tau')$ be a normalized distance between $(A_m,B_m)$ and $(A'_{m'},B'_{m'})$, and let\n\\begin{equation}\n d_{\\mathrm{ans}}(\\tau,\\tau')\n =\n \\abs{y_m/4-y'_{m'}/4}.\n \\label{eq:b17-answer-distance}\n\\end{equation}\nFor a recipient task $\\tau_i$, define four donor classes:\n\\begin{enumerate}\n\\item $\\mathsf{SAP}$: same-answer, far-program, approximately depth-matched donors;\n\\item $\\mathsf{FAD}$: far-answer, depth-matched donors;\n\\item $\\mathsf{SN}$: semantic-neighbor donors;\n\\item $\\mathsf{RMD}$: random depth-matched donors.\n\\end{enumerate}\nThree donors per available class are selected by deterministic task-identifier hashing. Each audit records donor coverage and the realized semantic and answer distances.","t $d_{\\mathrm{prog}}(\\tau,\\tau')$ be a normalized distance between $(A_m,B_m)$ and $(A'_{m'},B'_{m'})$, and let\n\\begin{equation}\n d_{\\mathrm{ans}}(\\tau,\\tau')\n =\n \\abs{y_m/4-y'_{m'}/4}.\n \\label{eq:b17-answer-distance}\n\\end{equation}\nFor a recipient task $\\tau_i$, define four donor classes:\n\\begin{enumerate}\n\\item $\\mathsf{SAP}$: same-answer, far-program, approximately depth-matched donors;\n\\item $\\mathsf{FAD}$: far-answer, depth-matched donors;\n\\item $\\mathsf{SN}$: semantic-neighbor donors;\n\\item $\\mathsf{RMD}$: random depth-matched donors.\n\\end{enumerate}\nThree donors per available class are selected by deterministic task-identifier hashing. Each audit records donor coverage and the realized semantic and answer distances.","ce between $(A_m,B_m)$ and $(A'_{m'},B'_{m'})$, and let\n\\begin{equation}\n d_{\\mathrm{ans}}(\\tau,\\tau')\n =\n \\abs{y_m/4-y'_{m'}/4}.\n \\label{eq:b17-answer-distance}\n\\end{equation}\nFor a recipient task $\\tau_i$, define four donor classes:\n\\begin{enumerate}\n\\item $\\mathsf{SAP}$: same-answer, far-program, approximately depth-matched donors;\n\\item $\\mathsf{FAD}$: far-answer, depth-matched donors;\n\\item $\\mathsf{SN}$: semantic-neighbor donors;\n\\item $\\mathsf{RMD}$: random depth-matched donors.\n\\end{enumerate}\nThree donors per available class are selected by deterministic task-identifier hashing. Each audit records donor coverage and the realized semantic and answer distances.","and let\n\\begin{equation}\n d_{\\mathrm{ans}}(\\tau,\\tau')\n =\n \\abs{y_m/4-y'_{m'}/4}.\n \\label{eq:b17-answer-distance}\n\\end{equation}\nFor a recipient task $\\tau_i$, define four donor classes:\n\\begin{enumerate}\n\\item $\\mathsf{SAP}$: same-answer, far-program, approximately depth-matched donors;\n\\item $\\mathsf{FAD}$: far-answer, depth-matched donors;\n\\item $\\mathsf{SN}$: semantic-neighbor donors;\n\\item $\\mathsf{RMD}$: random depth-matched donors.\n\\end{enumerate}\nThree donors per available class are selected by deterministic task-identifier hashing. Each audit records donor coverage and the realized semantic and answer distances.","\\begin{theorem}[Answer-only transplant confound]\n\\label{thm:b17-answer-confound}\nSuppose a latent trace is conditionally answer sufficient but program blind:\n\\begin{equation}\n Z(\\tau)=\\Psi(y(\\tau),m(\\tau),\\xi),\n \\label{eq:b17-answer-only-trace}\n\\end{equation}\nwhere $\\xi$ 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.\n\\end{theorem}","onfound]\n\\label{thm:b17-answer-confound}\nSuppose a latent trace is conditionally answer sufficient but program blind:\n\\begin{equation}\n Z(\\tau)=\\Psi(y(\\tau),m(\\tau),\\xi),\n \\label{eq:b17-answer-only-trace}\n\\end{equation}\nwhere $\\xi$ 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.\n\\end{theorem}","found}\nSuppose a latent trace is conditionally answer sufficient but program blind:\n\\begin{equation}\n Z(\\tau)=\\Psi(y(\\tau),m(\\tau),\\xi),\n \\label{eq:b17-answer-only-trace}\n\\end{equation}\nwhere $\\xi$ 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.\n\\end{theorem}","ce is conditionally answer sufficient but program blind:\n\\begin{equation}\n Z(\\tau)=\\Psi(y(\\tau),m(\\tau),\\xi),\n \\label{eq:b17-answer-only-trace}\n\\end{equation}\nwhere $\\xi$ 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.\n\\end{theorem}","\\begin{proof}\nChoose 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.\n\\end{proof}","\\begin{corollary}[Same-answer program control]\nUnder 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.\n\\end{corollary}","\\begin{warning}[Residual identifiability boundary]\nThe $\\mathsf{SAP}$ 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.\n\\end{warning}","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\n\\begin{align}\n n_i&=\\ell_i(\\mathsf{zero})-\\ell_i(Z_i),\n \\label{eq:b17-necessity}","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\n\\begin{align}\n n_i&=\\ell_i(\\mathsf{zero})-\\ell_i(Z_i),\n \\label{eq:b17-necessity}\\\\\n o_i&=\\ell_i(\\mathsf{reverse}(Z_i))-\\ell_i(Z_i),\n \\label{eq:b17-order}\\\\\n c_i&=\\ell_i(\\mathsf{noise}(Z_i))-\\ell_i(Z_i),\n \\label{eq:b17-noise}\\\\\n r_i&=\\abs{\\ell_i(\\mathsf{replay}(Z_i))-\\ell_i(Z_i)}.\n \\label{eq:b17-replay}\n\\end{align}\nNoise 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:\n\\begin{equation}\n p_i\n =\n \\mathbb E_{j\\in\\mathsf{SAP}(i)}\n \\left[\\ell_i(Z_j)-\\ell_i(Z_i)\\right],\n \\qquad\n a_i\n =\n \\mathbb E_{j\\in\\mathsf{FAD}(i)}\n \\left[\\ell_i(Z_j)-\\ell_i(Z_i)\\right].\n \\label{eq:b17-specificity-costs}\n\\end{equation}\nDefine\n\\begin{equation}\n Q_{\\mathrm{prog}}\n =\n \\frac{\\overline p}{\\max\\{\\overline o,\\overline c,\\eps\\}},\n \\qquad\n Q_{\\mathrm{ans}}\n =\n \\frac{\\overline a}{\\max\\{\\overline o,\\overline c,\\ep…","r_i&=\\abs{\\ell_i(\\mathsf{replay}(Z_i))-\\ell_i(Z_i)}.\n \\label{eq:b17-replay}\n\\end{align}\nNoise 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:\n\\begin{equation}\n p_i\n =\n \\mathbb E_{j\\in\\mathsf{SAP}(i)}\n \\left[\\ell_i(Z_j)-\\ell_i(Z_i)\\right],\n \\qquad\n a_i\n =\n \\mathbb E_{j\\in\\mathsf{FAD}(i)}\n \\left[\\ell_i(Z_j)-\\ell_i(Z_i)\\right].\n \\label{eq:b17-specificity-costs}\n\\end{equation}\nDefine\n\\begin{equation}\n Q_{\\mathrm{prog}}\n =\n \\frac{\\overline p}{\\max\\{\\overline o,\\overline c,\\eps\\}},\n \\qquad\n Q_{\\mathrm{ans}}\n =\n \\frac{\\overline a}{\\max\\{\\overline o,\\overline c,\\eps\\}}.\n \\label{eq:b17-specificity-ratios}\n\\end{equation}\nThe 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","r_i&=\\abs{\\ell_i(\\mathsf{replay}(Z_i))-\\ell_i(Z_i)}.\n \\label{eq:b17-replay}\n\\end{align}\nNoise 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:\n\\begin{equation}\n p_i\n =\n \\mathbb E_{j\\in\\mathsf{SAP}(i)}\n \\left[\\ell_i(Z_j)-\\ell_i(Z_i)\\right],\n \\qquad\n a_i\n =\n \\mathbb E_{j\\in\\mathsf{FAD}(i)}\n \\left[\\ell_i(Z_j)-\\ell_i(Z_i)\\right].\n \\label{eq:b17-specificity-costs}\n\\end{equation}\nDefine\n\\begin{equation}\n Q_{\\mathrm{prog}}\n =\n \\frac{\\overline p}{\\max\\{\\overline o,\\overline c,\\eps\\}},\n \\qquad\n Q_{\\mathrm{ans}}\n =\n \\frac{\\overline a}{\\max\\{\\overline o,\\overline c,\\eps\\}}.\n \\label{eq:b17-specificity-ratios}\n\\end{equation}\nThe 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","r_i&=\\abs{\\ell_i(\\mathsf{replay}(Z_i))-\\ell_i(Z_i)}.\n \\label{eq:b17-replay}\n\\end{align}\nNoise 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:\n\\begin{equation}\n p_i\n =\n \\mathbb E_{j\\in\\mathsf{SAP}(i)}\n \\left[\\ell_i(Z_j)-\\ell_i(Z_i)\\right],\n \\qquad\n a_i\n =\n \\mathbb E_{j\\in\\mathsf{FAD}(i)}\n \\left[\\ell_i(Z_j)-\\ell_i(Z_i)\\right].\n \\label{eq:b17-specificity-costs}\n\\end{equation}\nDefine\n\\begin{equation}\n Q_{\\mathrm{prog}}\n =\n \\frac{\\overline p}{\\max\\{\\overline o,\\overline c,\\eps\\}},\n \\qquad\n Q_{\\mathrm{ans}}\n =\n \\frac{\\overline a}{\\max\\{\\overline o,\\overline c,\\eps\\}}.\n \\label{eq:b17-specificity-ratios}\n\\end{equation}\nThe 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 requir…","hed at every step. The program and answer specificity coordinates are the mean recipient costs under $\\mathsf{SAP}$ and $\\mathsf{FAD}$ donors:\n\\begin{equation}\n p_i\n =\n \\mathbb E_{j\\in\\mathsf{SAP}(i)}\n \\left[\\ell_i(Z_j)-\\ell_i(Z_i)\\right],\n \\qquad\n a_i\n =\n \\mathbb E_{j\\in\\mathsf{FAD}(i)}\n \\left[\\ell_i(Z_j)-\\ell_i(Z_i)\\right].\n \\label{eq:b17-specificity-costs}\n\\end{equation}\nDefine\n\\begin{equation}\n Q_{\\mathrm{prog}}\n =\n \\frac{\\overline p}{\\max\\{\\overline o,\\overline c,\\eps\\}},\n \\qquad\n Q_{\\mathrm{ans}}\n =\n \\frac{\\overline a}{\\max\\{\\overline o,\\overline c,\\eps\\}}.\n \\label{eq:b17-specificity-ratios}\n\\end{equation}\nThe 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\n\\begin{equation}\n \\max_i r_i\\leq10^{-10}.\n \\label{eq:b17-replay-gate}\n\\end{equation}","=\n \\mathbb E_{j\\in\\mathsf{FAD}(i)}\n \\left[\\ell_i(Z_j)-\\ell_i(Z_i)\\right].\n \\label{eq:b17-specificity-costs}\n\\end{equation}\nDefine\n\\begin{equation}\n Q_{\\mathrm{prog}}\n =\n \\frac{\\overline p}{\\max\\{\\overline o,\\overline c,\\eps\\}},\n \\qquad\n Q_{\\mathrm{ans}}\n =\n \\frac{\\overline a}{\\max\\{\\overline o,\\overline c,\\eps\\}}.\n \\label{eq:b17-specificity-ratios}\n\\end{equation}\nThe 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\n\\begin{equation}\n \\max_i r_i\\leq10^{-10}.\n \\label{eq:b17-replay-gate}\n\\end{equation}","E_{j\\in\\mathsf{FAD}(i)}\n \\left[\\ell_i(Z_j)-\\ell_i(Z_i)\\right].\n \\label{eq:b17-specificity-costs}\n\\end{equation}\nDefine\n\\begin{equation}\n Q_{\\mathrm{prog}}\n =\n \\frac{\\overline p}{\\max\\{\\overline o,\\overline c,\\eps\\}},\n \\qquad\n Q_{\\mathrm{ans}}\n =\n \\frac{\\overline a}{\\max\\{\\overline o,\\overline c,\\eps\\}}.\n \\label{eq:b17-specificity-ratios}\n\\end{equation}\nThe 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\n\\begin{equation}\n \\max_i r_i\\leq10^{-10}.\n \\label{eq:b17-replay-gate}\n\\end{equation}","7-specificity-costs}\n\\end{equation}\nDefine\n\\begin{equation}\n Q_{\\mathrm{prog}}\n =\n \\frac{\\overline p}{\\max\\{\\overline o,\\overline c,\\eps\\}},\n \\qquad\n Q_{\\mathrm{ans}}\n =\n \\frac{\\overline a}{\\max\\{\\overline o,\\overline c,\\eps\\}}.\n \\label{eq:b17-specificity-ratios}\n\\end{equation}\nThe 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\n\\begin{equation}\n \\max_i r_i\\leq10^{-10}.\n \\label{eq:b17-replay-gate}\n\\end{equation}","g}}\n =\n \\frac{\\overline p}{\\max\\{\\overline o,\\overline c,\\eps\\}},\n \\qquad\n Q_{\\mathrm{ans}}\n =\n \\frac{\\overline a}{\\max\\{\\overline o,\\overline c,\\eps\\}}.\n \\label{eq:b17-specificity-ratios}\n\\end{equation}\nThe 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\n\\begin{equation}\n \\max_i r_i\\leq10^{-10}.\n \\label{eq:b17-replay-gate}\n\\end{equation}","overline c,\\eps\\}}.\n \\label{eq:b17-specificity-ratios}\n\\end{equation}\nThe 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\n\\begin{equation}\n \\max_i r_i\\leq10^{-10}.\n \\label{eq:b17-replay-gate}\n\\end{equation}","y-ratios}\n\\end{equation}\nThe 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\n\\begin{equation}\n \\max_i r_i\\leq10^{-10}.\n \\label{eq:b17-replay-gate}\n\\end{equation}","\\begin{definition}[External trace probe]\nA Block 17 trace probe is a ridge map\n\\begin{equation}\n \\widehat w=P\\,z_4+q\n \\label{eq:b17-external-probe}\n\\end{equation}\nfit only on the $1{,}200$ probe-calibration tasks. The final audit reports held-out coefficient of determination\n\\begin{equation}\n R^2_{\\mathrm{probe}}\n =\n 1-\n \\frac{\\sum_i\\norm{w_i-\\widehat w_i}^2}\n {\\sum_i\\norm{w_i-\\overline w}^2}.\n \\label{eq:b17-probe-r2}\n\\end{equation}\nThe frozen semantic-probe gate is $R^2_{\\mathrm{probe}}\\geq0.65$ on each final support.\n\\end{definition}","ge map\n\\begin{equation}\n \\widehat w=P\\,z_4+q\n \\label{eq:b17-external-probe}\n\\end{equation}\nfit only on the $1{,}200$ probe-calibration tasks. The final audit reports held-out coefficient of determination\n\\begin{equation}\n R^2_{\\mathrm{probe}}\n =\n 1-\n \\frac{\\sum_i\\norm{w_i-\\widehat w_i}^2}\n {\\sum_i\\norm{w_i-\\overline w}^2}.\n \\label{eq:b17-probe-r2}\n\\end{equation}\nThe frozen semantic-probe gate is $R^2_{\\mathrm{probe}}\\geq0.65$ on each final support.\n\\end{definition}","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.","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\n\\begin{equation}\n \\begin{gathered}\n \\text{clean MSE within }0.002\\text{ of the matched-update parent},","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\n\\begin{equation}\n \\begin{gathered}\n \\text{clean MSE within }0.002\\text{ of the matched-update parent},\\\\\n R^2_{\\mathrm{probe}}\\geq0.65,\n \\quad\n Q_{\\mathrm{prog}}\\geq0.15,\n \\quad\n Q_{\\mathrm{ans}}\\geq0.20,\\\\\n \\text{positive interval and fraction gates},\n \\quad\n \\text{local continuity},\n \\quad\n \\text{donor coverage},\n \\quad\n \\text{adaptive halting}.\n \\end{gathered}\n \\label{eq:b17-compiler-hard-gates}\n\\end{equation}\nAmong eligible candidates it maximizes\n\\begin{equation}\n \\begin{split}\n S(\\alpha)\n ={}&\n \\min\\left\\{\n \\frac{R^2_{\\mathrm{probe}}}{0.65},\n \\frac{Q_{\\mathrm{prog}}}{0.15},\n \\frac{Q_{\\mathrm{ans}}}{0.20}\n \\right\\}\n +0.20R^2_{\\mathrm{probe}}","\\text{positive interval and fraction gates},\n \\quad\n \\text{local continuity},\n \\quad\n \\text{donor coverage},\n \\quad\n \\text{adaptive halting}.\n \\end{gathered}\n \\label{eq:b17-compiler-hard-gates}\n\\end{equation}\nAmong eligible candidates it maximizes\n\\begin{equation}\n \\begin{split}\n S(\\alpha)\n ={}&\n \\min\\left\\{\n \\frac{R^2_{\\mathrm{probe}}}{0.65},\n \\frac{Q_{\\mathrm{prog}}}{0.15},\n \\frac{Q_{\\mathrm{ans}}}{0.20}\n \\right\\}\n +0.20R^2_{\\mathrm{probe}}\\\\\n &-4\\,\\mathrm{MSE}_{\\mathrm{clean}}\n +0.02\\rho_{\\mathrm{halt}}\n -\\mathsf{Penalty}_{\\mathrm{dynamic}}.\n \\end{split}\n \\label{eq:b17-selection-score}\n\\end{equation}\nThe 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.","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\n\\begin{verbatim}\ncc0d07b20ec2820a40f9781babc36f467482bed9ce5319f031401c3ba026c318\n\\end{verbatim}","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:\n\\begin{enumerate}\n\\item add a rank-four conditioning candidate;\n\\item allocate $550$ repair updates to every candidate;\n\\item use the weakest-normalized-margin selection score in \\cref{eq:b17-selection-score};\n\\item generate new primary and generator-foreign supports from seeds $17006$ and $27006$.\n\\end{enumerate}\nNo final threshold was weakened or removed. The revision receipt is\n\\begin{verbatim}\nf7435a94a4c1d065ff5e37d3895a58d7fca541faf2098a8872ac9f90c6915755\n\\end{verbatim}\nThis chronology makes the failed pilot evidence about compiler design rather than hidden tuning evidence for the successful final tasks.","ints, final tasks, audits, figures, and receipt. Before a second final audit, the revision file committed four changes:\n\\begin{enumerate}\n\\item add a rank-four conditioning candidate;\n\\item allocate $550$ repair updates to every candidate;\n\\item use the weakest-normalized-margin selection score in \\cref{eq:b17-selection-score};\n\\item generate new primary and generator-foreign supports from seeds $17006$ and $27006$.\n\\end{enumerate}\nNo final threshold was weakened or removed. The revision receipt is\n\\begin{verbatim}\nf7435a94a4c1d065ff5e37d3895a58d7fca541faf2098a8872ac9f90c6915755\n\\end{verbatim}\nThis chronology makes the failed pilot evidence about compiler design rather than hidden tuning evidence for the successful final tasks.","tasks, audits, figures, and receipt. Before a second final audit, the revision file committed four changes:\n\\begin{enumerate}\n\\item add a rank-four conditioning candidate;\n\\item allocate $550$ repair updates to every candidate;\n\\item use the weakest-normalized-margin selection score in \\cref{eq:b17-selection-score};\n\\item generate new primary and generator-foreign supports from seeds $17006$ and $27006$.\n\\end{enumerate}\nNo final threshold was weakened or removed. The revision receipt is\n\\begin{verbatim}\nf7435a94a4c1d065ff5e37d3895a58d7fca541faf2098a8872ac9f90c6915755\n\\end{verbatim}\nThis chronology makes the failed pilot evidence about compiler design rather than hidden tuning evidence for the successful final tasks.","\\begin{longtable}{@{}lrrrrr@{}}\n\\toprule\nCandidate & Eligible & Clean MSE & Probe $R^2$ & $Q_{\\mathrm{prog}}$ & $Q_{\\mathrm{ans}}$","\\midrule\n\\endfirsthead\n\\toprule\nCandidate & Eligible & Clean MSE & Probe $R^2$ & $Q_{\\mathrm{prog}}$ & $Q_{\\mathrm{ans}}$","\\midrule\n\\endhead\nMatched parent continuation & no & $0.012385$ & $-0.000901$ & $0$ & $0$","Depth only & no & $0.012172$ & $0.000984$ & $0$ & $0$","Rank-one binding & yes & $0.012001$ & $0.699509$ & $0.8054$ & $0.9229$","Rank-four binding & yes & $0.011779$ & $0.861346$ & $0.3937$ & $0.5036$","Full binding & \\textbf{selected} & $0.011452$ & $0.870542$ & $0.7436$ & $1.1703$","Full without semantic witness & no & $0.010728$ & $0.724937$ & $0.1019$ & $0.1050$","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.","\\midrule\n\\endhead\nClean MSE & $0.012665$ & $0.011686$","Latent-necessity gain & $0.017036$ & $0.030296$","Step-order cost & $0.000957$ & $0.004362$","Norm-matched noise cost & $0.001272$ & $0.014644$","Same-answer far-program cost & $0$ & $0.010714$","Far-answer matched-depth cost & $0$ & $0.017727$","Program specificity ratio & $0$ & $0.731642$","Answer specificity ratio & $0$ & $1.210526$","External trace-probe $R^2$ & $-0.000302$ & $0.865504$","Difficulty--depth Spearman $\\rho$ & $0$ & $0.971716$","Distinct dynamic depths & $1$ & $4$","The paired child-minus-parent program-specificity gain is\n\\begin{equation}\n \\widehat\\Delta_{\\mathrm{prog}}^{A}\n =0.010714,\n \\qquad\n \\mathrm{CI}_{0.95}\n =[0.009911,0.011527],\n \\label{eq:b17-primary-program-gain}\n\\end{equation}\nwith positive fraction $0.823125$. The answer-specificity gain is $0.017727$. Its frozen interval is\n\\begin{equation*}\n[0.016490,0.018960].\n\\end{equation*}\nParent-minus-child clean loss is $0.000979$, with frozen interval $[0.000695,0.001290]$.","-minus-parent program-specificity gain is\n\\begin{equation}\n \\widehat\\Delta_{\\mathrm{prog}}^{A}\n =0.010714,\n \\qquad\n \\mathrm{CI}_{0.95}\n =[0.009911,0.011527],\n \\label{eq:b17-primary-program-gain}\n\\end{equation}\nwith positive fraction $0.823125$. The answer-specificity gain is $0.017727$. Its frozen interval is\n\\begin{equation*}\n[0.016490,0.018960].\n\\end{equation*}\nParent-minus-child clean loss is $0.000979$, with frozen interval $[0.000695,0.001290]$.","gain is\n\\begin{equation}\n \\widehat\\Delta_{\\mathrm{prog}}^{A}\n =0.010714,\n \\qquad\n \\mathrm{CI}_{0.95}\n =[0.009911,0.011527],\n \\label{eq:b17-primary-program-gain}\n\\end{equation}\nwith positive fraction $0.823125$. The answer-specificity gain is $0.017727$. Its frozen interval is\n\\begin{equation*}\n[0.016490,0.018960].\n\\end{equation*}\nParent-minus-child clean loss is $0.000979$, with frozen interval $[0.000695,0.001290]$.","The shifted audit contains $1{,}600$ generator-$B$ tasks with altered coefficient and offset probabilities. The parent remains PARTIAL and the child remains FULL.","\\midrule\n\\endhead\nClean MSE & $0.062210$ & $0.058746$","Latent-necessity gain & $0.036128$ & $0.051896$","Step-order cost & $0.000866$ & $0.004750$","Norm-matched noise cost & $0.001353$ & $0.017566$","Same-answer far-program cost & $0$ & $0.024763$","Far-answer matched-depth cost & $0$ & $0.035400$","Program specificity ratio & $0$ & $1.409704$","Answer specificity ratio & $0$ & $2.015268$","External trace-probe $R^2$ & $-0.002355$ & $0.691471$","Difficulty--depth Spearman $\\rho$ & $0$ & $0.969169$","The paired generator-foreign program-specificity gain is\n\\begin{equation}\n \\widehat\\Delta_{\\mathrm{prog}}^{B}\n =0.024763,\n \\qquad\n \\mathrm{CI}_{0.95}\n =[0.022794,0.026793],\n \\label{eq:b17-foreign-program-gain}\n\\end{equation}\nwith 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.","ta_{\\mathrm{prog}}^{B}\n =0.024763,\n \\qquad\n \\mathrm{CI}_{0.95}\n =[0.022794,0.026793],\n \\label{eq:b17-foreign-program-gain}\n\\end{equation}\nwith 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.","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 $1.098\\times10^{-6}$.","Let $P$ be a signed permutation matrix on the latent space. Define transformed latent parameters\n\\begin{equation}\n \\zeta'=P\\zeta,\n \\quad\n R'=PRP^{\\mathsf T},\n \\quad\n s_k'=Ps_k,\n \\quad\n C_\\alpha'=PC_\\alpha,\n \\label{eq:b17-gauge-trace}\n\\end{equation}\nand 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.","\\begin{proof}\nInductively, $z_0'=Pz_0$. If $z_k'=Pz_k$, then\n\\begin{equation}\n z_{k+1}'\n =\n \\tanh\\!\\left(PRP^{\\mathsf T}Pz_k+Ps_k+PC_\\alpha\\right)\n =P\\tanh(Rz_k+s_k+C_\\alpha)\n =Pz_{k+1},\n\\end{equation}\nbecause 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.\n\\end{proof}","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.","ain their declared interpretations;\n\\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;\n\\item the independent NumPy, receipt, witness, and gauge checks pass within their declared numerical tolerances.\n\\end{enumerate}\nThen the accepted transition\n\\begin{equation}\n \\mathfrak L_{16}^{\\mathrm{NL}}\n \\xrightarrow{\\Pi_{17}}\n \\mathfrak L_{17}^{\\mathrm{NL}}\n \\label{eq:b17-lineage-edge}\n\\end{equation}\nis 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\n\\begin{verbatim}\n6cc2257fe89e0ad14a768580919c6af72b9e13654a61dc697bd0df086655e4ae\n\\end{verbatim}\nIt 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, environmen…","The canonical frozen verification executes\n\\begin{verbatim}\npython independent_verify.py\npython -m pytest -q\npython build_manifest.py\npython verify_package.py\n\\end{verbatim}\nand 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.","\\begin{statusbox}{Block 18 status}\nThe 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 fail","\\begin{statusbox}{Block 18 status}\nThe 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.\n\\end{statusbox}","Let the template set be\n\\begin{equation}\n \\Theta\n =\n \\{\\theta_{\\mathrm{aff}},\\theta_{\\mathrm{abs}},\n   \\theta_{\\mathrm{quad}},\\theta_{\\mathrm{hinge}}\\}.\n \\label{eq:b18-template-set}\n\\end{equation}\nFor integer parameters $a,b,c$ and integer input $x$, the exact programs are\n\\begin{align}\n f_{\\mathrm{aff}}(x)&=ax+b,","Let the template set be\n\\begin{equation}\n \\Theta\n =\n \\{\\theta_{\\mathrm{aff}},\\theta_{\\mathrm{abs}},\n   \\theta_{\\mathrm{quad}},\\theta_{\\mathrm{hinge}}\\}.\n \\label{eq:b18-template-set}\n\\end{equation}\nFor integer parameters $a,b,c$ and integer input $x$, the exact programs are\n\\begin{align}\n f_{\\mathrm{aff}}(x)&=ax+b,\n \\\\\n f_{\\mathrm{abs}}(x)&=a|x|+b,\n \\\\\n f_{\\mathrm{quad}}(x)&=ax^2+bx+c,\n \\\\\n f_{\\mathrm{hinge}}(x)&=\n \\begin{cases}\n ax+c,&x\\geq0,\\\\\n bx+c,&x<0.\n \\end{cases}\n \\label{eq:b18-program-templates}\n\\end{align}\nThe 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\n\\begin{equation}\n y(\\tau)\n =\n \\left(\n e_{\\theta},\n \\frac{p_1}{2},\\ldots,\\frac{p_4}{2},\n \\frac{d(\\theta)}{5},\n \\frac{f_{\\theta,p}(x_q)}{32}\n \\right)\\in\\R^{10},\n \\label{eq:b18-protected-target}\n\\end{equation}\nwhere $e_{\\theta}\\in\\R^4$ is the one-hot template code, unused parameter slots are zero, and $d(\\theta)\\in\\{2,3,4,5\\}$ is the d…","bx+c,&x<0.\n \\end{cases}\n \\label{eq:b18-program-templates}\n\\end{align}\nThe 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\n\\begin{equation}\n y(\\tau)\n =\n \\left(\n e_{\\theta},\n \\frac{p_1}{2},\\ldots,\\frac{p_4}{2},\n \\frac{d(\\theta)}{5},\n \\frac{f_{\\theta,p}(x_q)}{32}\n \\right)\\in\\R^{10},\n \\label{eq:b18-protected-target}\n\\end{equation}\nwhere $e_{\\theta}\\in\\R^4$ 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.","ask 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\n\\begin{equation}\n y(\\tau)\n =\n \\left(\n e_{\\theta},\n \\frac{p_1}{2},\\ldots,\\frac{p_4}{2},\n \\frac{d(\\theta)}{5},\n \\frac{f_{\\theta,p}(x_q)}{32}\n \\right)\\in\\R^{10},\n \\label{eq:b18-protected-target}\n\\end{equation}\nwhere $e_{\\theta}\\in\\R^4$ 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.","receives a normalized serialization of those demonstrations and the query. The protected target is\n\\begin{equation}\n y(\\tau)\n =\n \\left(\n e_{\\theta},\n \\frac{p_1}{2},\\ldots,\\frac{p_4}{2},\n \\frac{d(\\theta)}{5},\n \\frac{f_{\\theta,p}(x_q)}{32}\n \\right)\\in\\R^{10},\n \\label{eq:b18-protected-target}\n\\end{equation}\nwhere $e_{\\theta}\\in\\R^4$ 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.","\\midrule\n\\endhead\nTraining & $7{,}000$ & $A$ & Parent and repair optimization","Development & $1{,}200$ & $A$ & Source probe and candidate construction","Meta-validation & $1{,}200$ & $A$ & Source compiler selection","Probe calibration & $1{,}000$ & $A$ & Frozen external semantic readouts","Transport bridge & $900$ & $A$ & Paired and shuffled bridge fitting","Primary audit & $1{,}400$ & $A$ & Fresh in-generator final audit","Generator-foreign audit & $1{,}400$ & $B$ & Fresh shifted final audit","Transport audit & $1{,}200$ & $B$ & Separate shifted semantic-transport audit","Generator $A$ uses approximately uniform template frequencies. Its demonstration and query supports are\n\\begin{equation}\n\\Xcal_A^{\\mathrm{demo}}=\\{-2,-1,0,1,2\\},\n\\qquad\n\\Xcal_A^{\\mathrm{query}}=\\{-3,3\\}.\n\\end{equation}\nGenerator $B$ changes the template mixture toward quadratic and hinge programs and uses\n\\begin{equation}\n\\Xcal_B^{\\mathrm{demo}}=\\{-3,-1,0,1,3\\},\n\\qquad\n\\Xcal_B^{\\mathrm{query}}=\\{-3,-2,2,3\\}.\n\\end{equation}\nEvery 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 $0.915465$.","ograms and uses\n\\begin{equation}\n\\Xcal_B^{\\mathrm{demo}}=\\{-3,-1,0,1,3\\},\n\\qquad\n\\Xcal_B^{\\mathrm{query}}=\\{-3,-2,2,3\\}.\n\\end{equation}\nEvery 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 $0.915465$.","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\n\\begin{equation}\n z_{k+1}\n =\n \\tanh\\left(Wz_k+s_k+\\delta_k(e,z_k)\\right).\n \\label{eq:b18-elman}\n\\end{equation}\nThe first foreign family uses a GRU cell\n\\begin{equation}\n z_{k+1}=\\operatorname{GRUCell}(u_k,z_k),\n \\label{eq:b18-gru}\n\\end{equation}\nand the second uses a gated nonlinear residual cell\n\\begin{equation}\n z_{k+1}\n =\n z_k+\\sigma(Gz_k)\\odot\n \\tanh\\left(V_2\\operatorname{GELU}(V_1z_k)+s_k+\\delta_k(e,z_k)\\right).\n \\label{eq:b18-residual}\n\\end{equation}\nThe three cell classes have distinct execu","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\n\\begin{equation}\n z_{k+1}\n =\n \\tanh\\left(Wz_k+s_k+\\delta_k(e,z_k)\\right).\n \\label{eq:b18-elman}\n\\end{equation}\nThe first foreign family uses a GRU cell\n\\begin{equation}\n z_{k+1}=\\operatorname{GRUCell}(u_k,z_k),\n \\label{eq:b18-gru}\n\\end{equation}\nand the second uses a gated nonlinear residual cell\n\\begin{equation}\n z_{k+1}\n =\n z_k+\\sigma(Gz_k)\\odot\n \\tanh\\left(V_2\\operatorname{GELU}(V_1z_k)+s_k+\\delta_k(e,z_k)\\right).\n \\label{eq:b18-residual}\n\\end{equation}\nThe three cell classes have distinct executable implementations and parameterizations. They share","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\n\\begin{equation}\n z_{k+1}\n =\n \\tanh\\left(Wz_k+s_k+\\delta_k(e,z_k)\\right).\n \\label{eq:b18-elman}\n\\end{equation}\nThe first foreign family uses a GRU cell\n\\begin{equation}\n z_{k+1}=\\operatorname{GRUCell}(u_k,z_k),\n \\label{eq:b18-gru}\n\\end{equation}\nand the second uses a gated nonlinear residual cell\n\\begin{equation}\n z_{k+1}\n =\n z_k+\\sigma(Gz_k)\\odot\n \\tanh\\left(V_2\\operatorname{GELU}(V_1z_k)+s_k+\\delta_k(e,z_k)\\right).\n \\label{eq:b18-residual}\n\\end{equation}\nThe three cell classes have distinct executable implementations and parameterizations. They share only the declared external input, latent di","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\n\\begin{equation}\n z_{k+1}\n =\n \\tanh\\left(Wz_k+s_k+\\delta_k(e,z_k)\\right).\n \\label{eq:b18-elman}\n\\end{equation}\nThe first foreign family uses a GRU cell\n\\begin{equation}\n z_{k+1}=\\operatorname{GRUCell}(u_k,z_k),\n \\label{eq:b18-gru}\n\\end{equation}\nand the second uses a gated nonlinear residual cell\n\\begin{equation}\n z_{k+1}\n =\n z_k+\\sigma(Gz_k)\\odot\n \\tanh\\left(V_2\\operatorname{GELU}(V_1z_k)+s_k+\\delta_k(e,z_k)\\right).\n \\label{eq:b18-residual}\n\\end{equation}\nThe three cell classes have distinct executable implementations and parameterizations. They share only the declared external input, latent dimension, five-step budget, semantic target, downstream gate","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\n\\begin{equation}\n z_{k+1}\n =\n \\tanh\\left(Wz_k+s_k+\\delta_k(e,z_k)\\right).\n \\label{eq:b18-elman}\n\\end{equation}\nThe first foreign family uses a GRU cell\n\\begin{equation}\n z_{k+1}=\\operatorname{GRUCell}(u_k,z_k),\n \\label{eq:b18-gru}\n\\end{equation}\nand the second uses a gated nonlinear residual cell\n\\begin{equation}\n z_{k+1}\n =\n z_k+\\sigma(Gz_k)\\odot\n \\tanh\\left(V_2\\operatorname{GELU}(V_1z_k)+s_k+\\delta_k(e,z_k)\\right).\n \\label{eq:b18-residual}\n\\end{equation}\nThe 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.","p embedding at latent depth $k\\in\\{0,\\ldots,4\\}$. The source uses an Elman transition\n\\begin{equation}\n z_{k+1}\n =\n \\tanh\\left(Wz_k+s_k+\\delta_k(e,z_k)\\right).\n \\label{eq:b18-elman}\n\\end{equation}\nThe first foreign family uses a GRU cell\n\\begin{equation}\n z_{k+1}=\\operatorname{GRUCell}(u_k,z_k),\n \\label{eq:b18-gru}\n\\end{equation}\nand the second uses a gated nonlinear residual cell\n\\begin{equation}\n z_{k+1}\n =\n z_k+\\sigma(Gz_k)\\odot\n \\tanh\\left(V_2\\operatorname{GELU}(V_1z_k)+s_k+\\delta_k(e,z_k)\\right).\n \\label{eq:b18-residual}\n\\end{equation}\nThe 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.","\\begin{definition}[Cross-implementation repair schema]\n\\label{def:b18-repair-schema}\nA Block 18 repair schema is a tuple\n\\begin{equation}\n \\rho=(a,\\lambda_{\\mathrm{sem}},\\lambda_{\\mathrm{prog}},\n \\lambda_{\\mathrm{geo}},\\chi),\n \\label{eq:b18-repair-schema}\n\\end{equation}\nwhere $a$ 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.\n\\end{definition}","lementation repair schema]\n\\label{def:b18-repair-schema}\nA Block 18 repair schema is a tuple\n\\begin{equation}\n \\rho=(a,\\lambda_{\\mathrm{sem}},\\lambda_{\\mathrm{prog}},\n \\lambda_{\\mathrm{geo}},\\chi),\n \\label{eq:b18-repair-schema}\n\\end{equation}\nwhere $a$ 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.\n\\end{definition}","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\n\\begin{equation}\n (\\gamma_k(e),\\beta_k(e))_{k=0}^{4}\n =F_{\\mathrm{FiLM}}(e),\n \\label{eq:b18-film}\n\\end{equation}\nwhich act on the preactivation by\n\\begin{equation}\n q_k\n \\longmapsto\n q_k\\odot\\left(1+\\frac12\\tanh\\gamma_k(e)\\right)\n +\\beta_k(e).\n \\label{eq:b18-film-action}\n\\end{equation}\nThis 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.","ited additive conditioning, final-step FiLM, progressive FiLM with geometry, a low-rank hypernetwork, and a shuffled-witness FiLM control. For the selected FiLM constructor, the encoded task produces step-indexed scale and shift vectors\n\\begin{equation}\n (\\gamma_k(e),\\beta_k(e))_{k=0}^{4}\n =F_{\\mathrm{FiLM}}(e),\n \\label{eq:b18-film}\n\\end{equation}\nwhich act on the preactivation by\n\\begin{equation}\n q_k\n \\longmapsto\n q_k\\odot\\left(1+\\frac12\\tanh\\gamma_k(e)\\right)\n +\\beta_k(e).\n \\label{eq:b18-film-action}\n\\end{equation}\nThis 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.","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.","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.","\\begin{table}[H]\n\\centering\n\\small\n\\resizebox{\\textwidth}{!}{%\n\\begin{tabular}{@{}lrrrrrrl@{}}\n\\toprule\nCandidate & Clean & Probe $R^2$ & Program ratio & Answer ratio & Halt $\\rho$ & Execution & Status","\\midrule\nRecalibrated parent & $0.114372$ & $-0.001360$ & $0.000000$ & $0.000000$ & $0.000000$ & $0.200833$ & PARTIAL","Legacy additive & $0.088248$ & $0.639068$ & $0.632384$ & $0.182410$ & $0.609663$ & $0.289167$ & FULL","FiLM final witness & $0.087307$ & $0.655812$ & $0.124210$ & $0.055454$ & $0.573743$ & $0.325833$ & PARTIAL","FiLM progressive geometry & $0.083188$ & $0.687281$ & $0.359980$ & $0.085309$ & $0.625756$ & $0.379167$ & FULL","Hyper progressive geometry & $0.093108$ & $0.565723$ & $0.182308$ & $0.026907$ & $0.430354$ & $0.310000$ & PARTIAL","FiLM shuffled witness & $0.092262$ & $0.453266$ & $0.111008$ & $0.041753$ & $0.410682$ & $0.317500$ & PARTIAL","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.","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.","For model $M$, task $i$, and clean protected loss $L_i$, the audit reuses the Block 17 donor distinctions and adds cross-implementation status labels. It records latent necessity, reversal cost, norm-matched noise cost, exact replay, same-answer far-program transplant cost, far-answer matched-depth transplant cost, local semantic-neighbor cost, external trace-probe $R^2$, dynamic-depth correlation, and exact program execution. Let\n\\begin{align}\n n_i&=L_i^{0}-L_i,\n &o_i&=L_i^{\\mathrm{rev}}-L_i,\n &c_i&=L_i^{\\eta}-L_i,","s $L_i$, the audit reuses the Block 17 donor distinctions and adds cross-implementation status labels. It records latent necessity, reversal cost, norm-matched noise cost, exact replay, same-answer far-program transplant cost, far-answer matched-depth transplant cost, local semantic-neighbor cost, external trace-probe $R^2$, dynamic-depth correlation, and exact program execution. Let\n\\begin{align}\n n_i&=L_i^{0}-L_i,\n &o_i&=L_i^{\\mathrm{rev}}-L_i,\n &c_i&=L_i^{\\eta}-L_i,\n \\\\\n p_i&=L_{i,\\mathrm{sem}}^{\\mathrm{same\\ answer,far\\ program}}\n      -L_{i,\\mathrm{sem}},\n &a_i&=L_{i,\\mathrm{answer}}^{\\mathrm{far\\ answer,depth}}\n      -L_{i,\\mathrm{answer}}.\n \\label{eq:b18-audit-coordinates}\n\\end{align}\nThe normalized specificity ratios are\n\\begin{equation}\n Q_{\\mathrm{prog}}\n =\n \\frac{\\overline p}{\\max\\{\\overline o,\\overline c,\\eps\\}},\n \\qquad\n Q_{\\mathrm{ans}}\n =\n \\frac{\\overline a}{\\max\\{\\overline o,\\overline c,\\eps\\}}.\n \\label{eq:b18-specificity-ratios}\n\\end{equation}\nFULL status requires positive lower intervals and positive-fraction gates for the structural and donor statistics, $Q_{\\mathr…","p_i&=L_{i,\\mathrm{sem}}^{\\mathrm{same\\ answer,far\\ program}}\n      -L_{i,\\mathrm{sem}},\n &a_i&=L_{i,\\mathrm{answer}}^{\\mathrm{far\\ answer,depth}}\n      -L_{i,\\mathrm{answer}}.\n \\label{eq:b18-audit-coordinates}\n\\end{align}\nThe normalized specificity ratios are\n\\begin{equation}\n Q_{\\mathrm{prog}}\n =\n \\frac{\\overline p}{\\max\\{\\overline o,\\overline c,\\eps\\}},\n \\qquad\n Q_{\\mathrm{ans}}\n =\n \\frac{\\overline a}{\\max\\{\\overline o,\\overline c,\\eps\\}}.\n \\label{eq:b18-specificity-ratios}\n\\end{equation}\nFULL 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 $0.30$.","nd{align}\nThe normalized specificity ratios are\n\\begin{equation}\n Q_{\\mathrm{prog}}\n =\n \\frac{\\overline p}{\\max\\{\\overline o,\\overline c,\\eps\\}},\n \\qquad\n Q_{\\mathrm{ans}}\n =\n \\frac{\\overline a}{\\max\\{\\overline o,\\overline c,\\eps\\}}.\n \\label{eq:b18-specificity-ratios}\n\\end{equation}\nFULL 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 $0.30$.","ificity ratios are\n\\begin{equation}\n Q_{\\mathrm{prog}}\n =\n \\frac{\\overline p}{\\max\\{\\overline o,\\overline c,\\eps\\}},\n \\qquad\n Q_{\\mathrm{ans}}\n =\n \\frac{\\overline a}{\\max\\{\\overline o,\\overline c,\\eps\\}}.\n \\label{eq:b18-specificity-ratios}\n\\end{equation}\nFULL 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 $0.30$.","}\n =\n \\frac{\\overline p}{\\max\\{\\overline o,\\overline c,\\eps\\}},\n \\qquad\n Q_{\\mathrm{ans}}\n =\n \\frac{\\overline a}{\\max\\{\\overline o,\\overline c,\\eps\\}}.\n \\label{eq:b18-specificity-ratios}\n\\end{equation}\nFULL 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 $0.30$.","\\{\\overline o,\\overline c,\\eps\\}},\n \\qquad\n Q_{\\mathrm{ans}}\n =\n \\frac{\\overline a}{\\max\\{\\overline o,\\overline c,\\eps\\}}.\n \\label{eq:b18-specificity-ratios}\n\\end{equation}\nFULL 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 $0.30$.","}\n =\n \\frac{\\overline a}{\\max\\{\\overline o,\\overline c,\\eps\\}}.\n \\label{eq:b18-specificity-ratios}\n\\end{equation}\nFULL 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 $0.30$.","eq:b18-specificity-ratios}\n\\end{equation}\nFULL 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 $0.30$.","equation}\nFULL 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 $0.30$.","rvals 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 $0.30$.","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:\n\\begin{equation}\n B_t\n =\n \\argmin_{B,b}\n \\sum_{i=1}^{900}\n \\norm{H_s(\\tau_i)-BH_t(\\tau_i)-b}^2\n +\\lambda\\norm{B}^2.\n \\label{eq:b18-bridge-fit}\n\\end{equation}\nA source ridge probe $P_s$ is fitted independently on the $1{,}000$ probe-calibration tasks. The transported semantic prediction is\n\\begin{equation}\n \\widehat Y_t(\\tau)\n =P_s\\bigl(B_tH_t(\\tau)+b_t\\bigr).\n \\label{eq:b18-transported-readout}\n\\end{equation}\nThe bridge gate requires effective rank $36$, transported $R^2\\geq0.35$, an advantage of at least","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:\n\\begin{equation}\n B_t\n =\n \\argmin_{B,b}\n \\sum_{i=1}^{900}\n \\norm{H_s(\\tau_i)-BH_t(\\tau_i)-b}^2\n +\\lambda\\norm{B}^2.\n \\label{eq:b18-bridge-fit}\n\\end{equation}\nA source ridge probe $P_s$ is fitted independently on the $1{,}000$ probe-calibration tasks. The transported semantic prediction is\n\\begin{equation}\n \\widehat Y_t(\\tau)\n =P_s\\bigl(B_tH_t(\\tau)+b_t\\bigr).\n \\label{eq:b18-transported-readout}\n\\end{equation}\nThe 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, a","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:\n\\begin{equation}\n B_t\n =\n \\argmin_{B,b}\n \\sum_{i=1}^{900}\n \\norm{H_s(\\tau_i)-BH_t(\\tau_i)-b}^2\n +\\lambda\\norm{B}^2.\n \\label{eq:b18-bridge-fit}\n\\end{equation}\nA source ridge probe $P_s$ is fitted independently on the $1{,}000$ probe-calibration tasks. The transported semantic prediction is\n\\begin{equation}\n \\widehat Y_t(\\tau)\n =P_s\\bigl(B_tH_t(\\tau)+b_t\\bigr).\n \\label{eq:b18-transported-readout}\n\\end{equation}\nThe 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 $R^2\\leq0.20$.","he 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:\n\\begin{equation}\n B_t\n =\n \\argmin_{B,b}\n \\sum_{i=1}^{900}\n \\norm{H_s(\\tau_i)-BH_t(\\tau_i)-b}^2\n +\\lambda\\norm{B}^2.\n \\label{eq:b18-bridge-fit}\n\\end{equation}\nA source ridge probe $P_s$ is fitted independently on the $1{,}000$ probe-calibration tasks. The transported semantic prediction is\n\\begin{equation}\n \\widehat Y_t(\\tau)\n =P_s\\bigl(B_tH_t(\\tau)+b_t\\bigr).\n \\label{eq:b18-transported-readout}\n\\end{equation}\nThe 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 $R^2\\leq0.20$.","reign-child state. A ridge-affine bridge is fitted on the $900$ paired generator-$A$ anchors:\n\\begin{equation}\n B_t\n =\n \\argmin_{B,b}\n \\sum_{i=1}^{900}\n \\norm{H_s(\\tau_i)-BH_t(\\tau_i)-b}^2\n +\\lambda\\norm{B}^2.\n \\label{eq:b18-bridge-fit}\n\\end{equation}\nA source ridge probe $P_s$ is fitted independently on the $1{,}000$ probe-calibration tasks. The transported semantic prediction is\n\\begin{equation}\n \\widehat Y_t(\\tau)\n =P_s\\bigl(B_tH_t(\\tau)+b_t\\bigr).\n \\label{eq:b18-transported-readout}\n\\end{equation}\nThe 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 $R^2\\leq0.20$.","00}\n \\norm{H_s(\\tau_i)-BH_t(\\tau_i)-b}^2\n +\\lambda\\norm{B}^2.\n \\label{eq:b18-bridge-fit}\n\\end{equation}\nA source ridge probe $P_s$ is fitted independently on the $1{,}000$ probe-calibration tasks. The transported semantic prediction is\n\\begin{equation}\n \\widehat Y_t(\\tau)\n =P_s\\bigl(B_tH_t(\\tau)+b_t\\bigr).\n \\label{eq:b18-transported-readout}\n\\end{equation}\nThe 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 $R^2\\leq0.20$.","u_i)-BH_t(\\tau_i)-b}^2\n +\\lambda\\norm{B}^2.\n \\label{eq:b18-bridge-fit}\n\\end{equation}\nA source ridge probe $P_s$ is fitted independently on the $1{,}000$ probe-calibration tasks. The transported semantic prediction is\n\\begin{equation}\n \\widehat Y_t(\\tau)\n =P_s\\bigl(B_tH_t(\\tau)+b_t\\bigr).\n \\label{eq:b18-transported-readout}\n\\end{equation}\nThe 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 $R^2\\leq0.20$.","^2.\n \\label{eq:b18-bridge-fit}\n\\end{equation}\nA source ridge probe $P_s$ is fitted independently on the $1{,}000$ probe-calibration tasks. The transported semantic prediction is\n\\begin{equation}\n \\widehat Y_t(\\tau)\n =P_s\\bigl(B_tH_t(\\tau)+b_t\\bigr).\n \\label{eq:b18-transported-readout}\n\\end{equation}\nThe 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 $R^2\\leq0.20$.","ntly on the $1{,}000$ probe-calibration tasks. The transported semantic prediction is\n\\begin{equation}\n \\widehat Y_t(\\tau)\n =P_s\\bigl(B_tH_t(\\tau)+b_t\\bigr).\n \\label{eq:b18-transported-readout}\n\\end{equation}\nThe 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 $R^2\\leq0.20$.","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.","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\n\\begin{equation}\n e_{\\mathrm{src}}=Y-PH_s,\n \\qquad\n e_{\\mathrm{align}}=P(H_s-BH_t).\n \\label{eq:b18-error-parts}\n\\end{equation}\nThen\n\\begin{equation}\n Y-PBH_t=e_{\\mathrm{src}}+e_{\\mathrm{align}}.\n \\label{eq:b18-error-decomposition}\n\\end{equation}","\\begin{proposition}[Witness--alignment decomposition]\n\\label{prop:b18-witness-alignment}\nFor every square-integrable $(Y,H_s,H_t)$,\n\\begin{align}\n \\mathbb E_{\\mu}\\norm{Y-PBH_t}^2\n &=\n \\mathbb E_{\\mu}\\norm{e_{\\mathrm{src}}}^2\n +\\mathbb E_{\\mu}\\norm{e_{\\mathrm{align}}}^2\n +2\\mathbb E_{\\mu}\\angles{e_{\\mathrm{src}},e_{\\mathrm{align}}},\n \\label{eq:b18-squared-decomposition}","\\begin{proposition}[Witness--alignment decomposition]\n\\label{prop:b18-witness-alignment}\nFor every square-integrable $(Y,H_s,H_t)$,\n\\begin{align}\n \\mathbb E_{\\mu}\\norm{Y-PBH_t}^2\n &=\n \\mathbb E_{\\mu}\\norm{e_{\\mathrm{src}}}^2\n +\\mathbb E_{\\mu}\\norm{e_{\\mathrm{align}}}^2\n +2\\mathbb E_{\\mu}\\angles{e_{\\mathrm{src}},e_{\\mathrm{align}}},\n \\label{eq:b18-squared-decomposition}\\\\\n \\norm{Y-PBH_t}_{L^2(\\mu)}\n &\\leq\n \\norm{Y-PH_s}_{L^2(\\mu)}\n +\\norm{P(H_s-BH_t)}_{L^2(\\mu)}.\n \\label{eq:b18-triangle-decomposition}\n\\end{align}\n\\end{proposition}","\\begin{proof}\nExpand 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}.\n\\end{proof}","\\begin{theorem}[Exact-alignment witness ceiling]\n\\label{thm:b18-witness-ceiling}\nSuppose $BH_t=H_s$ almost surely under $\\mu$. Then\n\\begin{equation}\n P B H_t=P H_s\n \\quad\\text{almost surely},\n \\label{eq:b18-exact-alignment-equality}\n\\end{equation}\nand 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.\n\\end{theorem}","]\n\\label{thm:b18-witness-ceiling}\nSuppose $BH_t=H_s$ almost surely under $\\mu$. Then\n\\begin{equation}\n P B H_t=P H_s\n \\quad\\text{almost surely},\n \\label{eq:b18-exact-alignment-equality}\n\\end{equation}\nand 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.\n\\end{theorem}","se $BH_t=H_s$ almost surely under $\\mu$. Then\n\\begin{equation}\n P B H_t=P H_s\n \\quad\\text{almost surely},\n \\label{eq:b18-exact-alignment-equality}\n\\end{equation}\nand 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.\n\\end{theorem}","lmost surely under $\\mu$. Then\n\\begin{equation}\n P B H_t=P H_s\n \\quad\\text{almost surely},\n \\label{eq:b18-exact-alignment-equality}\n\\end{equation}\nand 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.\n\\end{theorem}","\\begin{proof}\nSubstitution of $BH_t=H_s$ gives \\cref{eq:b18-exact-alignment-equality}. Equality of every downstream deterministic score follows immediately.\n\\end{proof}","\\begin{theorem}[In-distribution bridge success does not imply foreign success]\n\\label{thm:b18-nonimplication}\nFor 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 $K$.\n\\end{theorem}","ribution bridge success does not imply foreign success]\n\\label{thm:b18-nonimplication}\nFor 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 $K$.\n\\end{theorem}","success does not imply foreign success]\n\\label{thm:b18-nonimplication}\nFor 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 $K$.\n\\end{theorem}","\\begin{proof}\nLet $Y\\in\\R^d$ 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\n\\begin{equation}\n \\mathbb E\\norm{Y-PBH_t}^2\n =(1+c)^2\\mathbb E\\norm{Y}^2.\n\\end{equation}\nChoosing $c$ 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.\n\\end{proof}","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\n\\begin{equation}\n \\mathbb E\\norm{Y-PBH_t}^2\n =(1+c)^2\\mathbb E\\norm{Y}^2.\n\\end{equation}\nChoosing $c$ 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.\n\\end{proof}","r 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\n\\begin{equation}\n \\mathbb E\\norm{Y-PBH_t}^2\n =(1+c)^2\\mathbb E\\norm{Y}^2.\n\\end{equation}\nChoosing $c$ 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.\n\\end{proof}","\\begin{definition}[Distributionally stable Neuralese transport]\n\\label{def:b18-stable-transport}\nLet $\\mathfrak M$ 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 $\\delta$,\n\\begin{equation}\n \\norm{Y-PBH_t}_{L^2(\\mu)}\\leq\\eps,\n \\label{eq:b18-stable-transport}\n\\end{equation}\nand 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.\n\\end{definition}","The canonical final audits are:\n\\begin{table}[H]\n\\centering\n\\small\n\\resizebox{\\textwidth}{!}{%\n\\begin{tabular}{@{}llrrrrrrrl@{}}\n\\toprule\nArchitecture & Split & Clean & Program cost & $Q_{\\rm prog}$ & $Q_{\\rm ans}$ & Probe $R^2$ & Halt $\\rho$ & Execution & Status","\\midrule\nGRU & Primary & $0.061597$ & $0.086859$ & $0.277610$ & $0.042063$ & $0.727398$ & $0.716422$ & $0.488571$ & FULL","GRU & Foreign & $0.245178$ & $0.052317$ & $0.522192$ & $-0.131449$ & $-0.221987$ & $-0.217862$ & $0.089286$ & FAIL","Residual & Primary & $0.061681$ & $0.122911$ & $0.422661$ & $0.194826$ & $0.726760$ & $0.704096$ & $0.516429$ & FULL","Residual & Foreign & $0.278526$ & $0.037921$ & $0.288550$ & $0.048495$ & $-0.124416$ & $0.116803$ & $0.077143$ & FAIL","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 $0.70$.","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.","rogram 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.","\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.88\\textwidth]{block18_foreign_audits.pdf}\n\\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.}\n\\label{fig:b18-foreign-audits}\n\\end{figure}","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\n\\begin{equation}\n \\widehat\\Delta_{\\mathrm{FiLM-add}}\n =-0.029417,\n \\qquad\n \\mathrm{CI}_{0.95}\n =[-0.034102,-0.024735].\n \\label{eq:b18-legacy-comparison}\n\\end{equation}\nThe 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.","The final transport results are:\n\\begin{longtable}{@{}lrrrrl@{}}\n\\toprule\nTarget & Paired $R^2$ & Shuffled $R^2$ & Unaligned $R^2$ & Source reference $R^2$ & Gate","\\midrule\nGRU & $-0.145315$ & $-0.175002$ & $-12.812014$ & $-0.098663$ & FAIL","Residual & $-0.206846$ & $-0.105233$ & $-59.598778$ & $-0.098663$ & FAIL","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.","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\n\\begin{center}\n\\texttt{POSTHOC\\_DIAGNOSTIC\\_ONLY\\_NOT\\_ADMISSIBLE\\_FOR\\_BLOCK18\\_ACCEPTANCE}.\n\\end{center}\nIts results are:\n\\begin{longtable}{@{}lr@{}}\n\\toprule\nReadout or bridge & Held-out shifted $R^2$","red 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\n\\begin{center}\n\\texttt{POSTHOC\\_DIAGNOSTIC\\_ONLY\\_NOT\\_ADMISSIBLE\\_FOR\\_BLOCK18\\_ACCEPTANCE}.\n\\end{center}\nIts results are:\n\\begin{longtable}{@{}lr@{}}\n\\toprule\nReadout or bridge & Held-out shifted $R^2$","\\midrule\nFrozen $A$-calibrated source probe & $-0.083074$","Shift-local source probe & $0.574370$","Shift-local GRU target probe & $0.600602$","Shift-local residual target probe & $0.581845$","Shift-local GRU paired bridge & $0.554518$","Shift-local residual paired bridge & $0.539005$","Shift-local GRU shuffled bridge & $-0.096005$","Shift-local residual shuffled bridge & $-0.040169$","\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.91\\textwidth]{block18_posthoc_localization.pdf}\n\\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.}\n\\label{fig:b18-posthoc}\n\\end{figure}","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.","\\begin{conditionaltheorem}[Block 18 sealed distributional obstruction]\n\\label{thm:b18-obstruction}\nAssume:\n\\begin{enumerate}\n\\item all $15{,}300$ released tasks pass \\cref{prop:b18-environment-soundness};\n\\item the source schema and both foreign checkpoint families are sealed before final-task generation;\n\\item the same typed selected schema is instantiated in both foreign architectures without architecture-specific candidate search;\n\\item all model, donor, probe, bridge, clean-loss, execution, and adaptive-depth gates retain their frozen meanings;\n\\item both selected children are FULL on their primary audits but FAIL on their generator-foreign audits;\n\\item both frozen paired bridges fail the shifted transport gate and the frozen source-reference probe also fails there;\n\\item the pooled comparison in \\cref{eq:b18-legacy-comparison}","\\begin{statusbox}{Block 19 status}\nThe 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 inhe","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\n\\begin{equation}\n \\rho_c:Z\\longrightarrow X_c,\n \\label{eq:b19-rendering}\n\\end{equation}\nwhere $X_c$ is the context-dependent observation space. The same semantic task may therefore appear as different finite demonstration and query records in different fibers $X_c$.","protected semantic task space and let $\\Ccal$ be a family of observation contexts. For each $c\\in\\Ccal$, an observation rendering is a map\n\\begin{equation}\n \\rho_c:Z\\longrightarrow X_c,\n \\label{eq:b19-rendering}\n\\end{equation}\nwhere $X_c$ is the context-dependent observation space. The same semantic task may therefore appear as different finite demonstration and query records in different fibers $X_c$.","\\begin{definition}[Semantic twins]\n\\label{def:b19-semantic-twins}\nTwo 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\n\\begin{equation}\n x_c=\\rho_c(z),\n \\qquad\n x_d=\\rho_d(z).\n\\end{equation}\nA semantic-twin group is an orbit $\\{\\rho_c(z):c\\in C_z\\}$ for a declared finite context set $C_z\\subseteq\\Ccal$.\n\\end{definition}","\\begin{definition}[Affine observation chart]\n\\label{def:b19-affine-chart}\nFor a context $c$, an affine observation chart is a tuple\n\\begin{equation}\n \\chi_c=(s_x,t_x,s_y,t_y,\\sigma_c),\n \\qquad s_xs_y\\neq0,\n\\end{equation}\nwith canonicalization\n\\begin{equation}\n \\widehat x=\\frac{x^{\\mathrm{obs}}-t_x}{s_x},\n \\qquad\n \\widehat y=\\frac{y^{\\mathrm{obs}}-t_y}{s_y},\n \\label{eq:b19-chart-inverse}\n\\end{equation}\nand 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.\n\\end{definition}","servation chart]\n\\label{def:b19-affine-chart}\nFor a context $c$, an affine observation chart is a tuple\n\\begin{equation}\n \\chi_c=(s_x,t_x,s_y,t_y,\\sigma_c),\n \\qquad s_xs_y\\neq0,\n\\end{equation}\nwith canonicalization\n\\begin{equation}\n \\widehat x=\\frac{x^{\\mathrm{obs}}-t_x}{s_x},\n \\qquad\n \\widehat y=\\frac{y^{\\mathrm{obs}}-t_y}{s_y},\n \\label{eq:b19-chart-inverse}\n\\end{equation}\nand 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.\n\\end{definition}","\\begin{definition}[Distribution-indexed atlas]\n\\label{def:b19-atlas}\nA finite distribution-indexed atlas is\n\\begin{equation}\n \\Acal\n =\n \\left(\n   \\{\\chi_c:c\\in C\\},\n   d_{\\mathrm{sig}},\n   \\tau_{\\mathrm{adm}},\n   \\pi\n \\right),\n \\label{eq:b19-atlas}\n\\end{equation}\nwhere $d_{\\mathrm{sig}}$ is a normalized signature distance, $\\tau_{\\mathrm{adm}}$ is an admission threshold, and $\\pi$ chooses the nearest chart. An input is admitted only when\n\\begin{equation}\n \\min_{c\\in C}d_{\\mathrm{sig}}(\\sigma(x),\\sigma_c)\n \\leq\\tau_{\\mathrm{adm}}.\n \\label{eq:b19-admission}\n\\end{equation}\nOtherwise the atlas returns an unsupported-chart certificate and abstains.\n\\end{definition}","b19-atlas}\nA finite distribution-indexed atlas is\n\\begin{equation}\n \\Acal\n =\n \\left(\n   \\{\\chi_c:c\\in C\\},\n   d_{\\mathrm{sig}},\n   \\tau_{\\mathrm{adm}},\n   \\pi\n \\right),\n \\label{eq:b19-atlas}\n\\end{equation}\nwhere $d_{\\mathrm{sig}}$ is a normalized signature distance, $\\tau_{\\mathrm{adm}}$ is an admission threshold, and $\\pi$ chooses the nearest chart. An input is admitted only when\n\\begin{equation}\n \\min_{c\\in C}d_{\\mathrm{sig}}(\\sigma(x),\\sigma_c)\n \\leq\\tau_{\\mathrm{adm}}.\n \\label{eq:b19-admission}\n\\end{equation}\nOtherwise the atlas returns an unsupported-chart certificate and abstains.\n\\end{definition}","The implementation uses\n\\begin{equation}\n d_{\\mathrm{sig}}(u,v)\n =\n \\frac{\\|u-v\\|_2/\\sqrt5}\n      {\\max\\{\\operatorname{sd}(v),10^{-8}\\}}.\n \\label{eq:b19-signature-distance}\n\\end{equation}\nThe runtime receives only the observed support and values. Context names are retained in the data solely for auditing route accuracy.","Suppose a source context $0$ and a target context $c$ satisfy\n\\begin{equation}\n x_c=s_xx_0+t_x,\n \\qquad\n y_c=s_yy_0+t_y,\n \\label{eq:b19-affine-rendering}\n\\end{equation}\nfor all paired semantic-twin demonstrations.","\\begin{theorem}[Exact semantic-twin chart recovery]\n\\label{thm:b19-exact-chart-recovery}\nAssume 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.\n\\end{theorem}","\\begin{proof}\nThe 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.\n\\end{proof}","\\begin{proposition}[Nearest-signature route stability]\n\\label{prop:b19-route-stability}\nLet the minimum pairwise normalized signature separation of an atlas be\n\\begin{equation}\n \\Delta_{\\Acal}\n =\n \\min_{c\\neq d}d_{\\mathrm{sig}}(\\sigma_c,\\sigma_d).\n\\end{equation}\nIf 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.\n\\end{proposition}","ition}[Nearest-signature route stability]\n\\label{prop:b19-route-stability}\nLet the minimum pairwise normalized signature separation of an atlas be\n\\begin{equation}\n \\Delta_{\\Acal}\n =\n \\min_{c\\neq d}d_{\\mathrm{sig}}(\\sigma_c,\\sigma_d).\n\\end{equation}\nIf 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.\n\\end{proposition}","ure route stability]\n\\label{prop:b19-route-stability}\nLet the minimum pairwise normalized signature separation of an atlas be\n\\begin{equation}\n \\Delta_{\\Acal}\n =\n \\min_{c\\neq d}d_{\\mathrm{sig}}(\\sigma_c,\\sigma_d).\n\\end{equation}\nIf 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.\n\\end{proposition}","\\begin{proof}\nThe 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.\n\\end{proof}","\\begin{definition}[Conservative atlas extension]\n\\label{def:b19-conservative-extension}\nAn 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.\n\\end{definition}","The released four-chart parent atlas is\n\\begin{equation}\n \\Acal_0=\\{A0,A1,A2,A3\\},\n\\end{equation}\nand the expanded atlas is\n\\begin{equation}\n \\Acal_1=\\{A0,A1,A2,A3,N0,N1\\}.\n \\label{eq:b19-atlas-edge}\n\\end{equation}\nThe fitted parameters are:","\\begin{longtable}{@{}lrrrrrr@{}}\n\\toprule\nChart & $s_x$ & $t_x$ & $s_y$ & $t_y$ & input RMSE & output RMSE","\\midrule\n\\endfirsthead\n\\toprule\nChart & $s_x$ & $t_x$ & $s_y$ & $t_y$ & input RMSE & output RMSE","\\midrule\n\\endhead\n$A0$ & $1.000$ & $0.000$ & $1.000$ & $0.000$ & $0$ & $0$","$A1$ & $2.000$ & $1.000$ & $0.500$ & $4.000$ & $2.35\\times10^{-15}$ & $2.09\\times10^{-15}$","$A2$ & $0.500$ & $-3.000$ & $-1.000$ & $2.000$ & $2.74\\times10^{-15}$ & $1.64\\times10^{-15}$","$A3$ & $1.500$ & $5.000$ & $2.000$ & $-5.000$ & $2.95\\times10^{-15}$ & $3.70\\times10^{-15}$","$N0$ & $3.000$ & $-6.000$ & $-0.750$ & $-3.000$ & $5.89\\times10^{-15}$ & $1.68\\times10^{-15}$","$N1$ & $0.800$ & $7.000$ & $1.750$ & $6.000$ & $1.16\\times10^{-14}$ & $2.20\\times10^{-15}$","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.","The unsupported contexts use a nonzero quadratic input warp\n\\begin{equation}\n x^{\\mathrm{obs}}\n =s_xx+t_x+\\omega x^2,\n \\qquad \\omega\\neq0,\n \\label{eq:b19-unsupported-warp}\n\\end{equation}\nand a nonlinear signed square-root output perturbation.","\\begin{proposition}[Affine grammar obstruction]\n\\label{prop:b19-affine-obstruction}\nNo affine inverse chart can exactly canonicalize \\cref{eq:b19-unsupported-warp} on three or more distinct canonical inputs when $\\omega\\neq0$.\n\\end{proposition}","\\begin{proof}\nAn 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.\n\\end{proof}","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.","Each task contains five demonstration pairs\n\\begin{equation}\n D=\\{(x_i,y_i)\\}_{i=1}^{5}\n\\end{equation}\nand a query $q$. After chart canonicalization, the full descendant computes pair codes\n\\begin{equation}\n p_i=\\phi(\\widehat x_i,\\widehat y_i)\\in\\R^{32},\n\\end{equation}\nand the permutation-invariant aggregate\n\\begin{equation}\n a(D)\n =\n \\left(\n   \\frac15\\sum_{i=1}^{5}p_i,\n   \\max_{1\\leq i\\leq5}p_i\n \\right)\\in\\R^{64}.\n \\label{eq:b19-set-aggregate}\n\\end{equation}\nThe semantic seed is\n\\begin{equation}\n h_0=S(a(D),Q(\\widehat q))\\in\\R^{40}.\n\\end{equation}\nA five-step Elman recurrence generates the Neuralese trace\n\\begin{equation}\n r_k=\\tanh\\left(W_ih_0+b_i+U_ir_{k-1}+c_i+e_k\\right),\n \\qquad k=1,\\ldots,5,\n \\label{eq:b19-recurrence}\n\\end{equation}\nwhere $e_k$ is a learned step embedding","Each task contains five demonstration pairs\n\\begin{equation}\n D=\\{(x_i,y_i)\\}_{i=1}^{5}\n\\end{equation}\nand a query $q$. After chart canonicalization, the full descendant computes pair codes\n\\begin{equation}\n p_i=\\phi(\\widehat x_i,\\widehat y_i)\\in\\R^{32},\n\\end{equation}\nand the permutation-invariant aggregate\n\\begin{equation}\n a(D)\n =\n \\left(\n   \\frac15\\sum_{i=1}^{5}p_i,\n   \\max_{1\\leq i\\leq5}p_i\n \\right)\\in\\R^{64}.\n \\label{eq:b19-set-aggregate}\n\\end{equation}\nThe semantic seed is\n\\begin{equation}\n h_0=S(a(D),Q(\\widehat q))\\in\\R^{40}.\n\\end{equation}\nA five-step Elman recurrence generates the Neuralese trace\n\\begin{equation}\n r_k=\\tanh\\left(W_ih_0+b_i+U_ir_{k-1}+c_i+e_k\\right),\n \\qquad k=1,\\ldots,5,\n \\label{eq:b19-recurrence}\n\\end{equation}\nwhere $e_k$ is a learned step embedding. A gated readout recurrence","Each task contains five demonstration pairs\n\\begin{equation}\n D=\\{(x_i,y_i)\\}_{i=1}^{5}\n\\end{equation}\nand a query $q$. After chart canonicalization, the full descendant computes pair codes\n\\begin{equation}\n p_i=\\phi(\\widehat x_i,\\widehat y_i)\\in\\R^{32},\n\\end{equation}\nand the permutation-invariant aggregate\n\\begin{equation}\n a(D)\n =\n \\left(\n   \\frac15\\sum_{i=1}^{5}p_i,\n   \\max_{1\\leq i\\leq5}p_i\n \\right)\\in\\R^{64}.\n \\label{eq:b19-set-aggregate}\n\\end{equation}\nThe semantic seed is\n\\begin{equation}\n h_0=S(a(D),Q(\\widehat q))\\in\\R^{40}.\n\\end{equation}\nA five-step Elman recurrence generates the Neuralese trace\n\\begin{equation}\n r_k=\\tanh\\left(W_ih_0+b_i+U_ir_{k-1}+c_i+e_k\\right),\n \\qquad k=1,\\ldots,5,\n \\label{eq:b19-recurrence}\n\\end{equation}\nwhere $e_k$ 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.","}\nand a query $q$. After chart canonicalization, the full descendant computes pair codes\n\\begin{equation}\n p_i=\\phi(\\widehat x_i,\\widehat y_i)\\in\\R^{32},\n\\end{equation}\nand the permutation-invariant aggregate\n\\begin{equation}\n a(D)\n =\n \\left(\n   \\frac15\\sum_{i=1}^{5}p_i,\n   \\max_{1\\leq i\\leq5}p_i\n \\right)\\in\\R^{64}.\n \\label{eq:b19-set-aggregate}\n\\end{equation}\nThe semantic seed is\n\\begin{equation}\n h_0=S(a(D),Q(\\widehat q))\\in\\R^{40}.\n\\end{equation}\nA five-step Elman recurrence generates the Neuralese trace\n\\begin{equation}\n r_k=\\tanh\\left(W_ih_0+b_i+U_ir_{k-1}+c_i+e_k\\right),\n \\qquad k=1,\\ldots,5,\n \\label{eq:b19-recurrence}\n\\end{equation}\nwhere $e_k$ 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.","\\widehat y_i)\\in\\R^{32},\n\\end{equation}\nand the permutation-invariant aggregate\n\\begin{equation}\n a(D)\n =\n \\left(\n   \\frac15\\sum_{i=1}^{5}p_i,\n   \\max_{1\\leq i\\leq5}p_i\n \\right)\\in\\R^{64}.\n \\label{eq:b19-set-aggregate}\n\\end{equation}\nThe semantic seed is\n\\begin{equation}\n h_0=S(a(D),Q(\\widehat q))\\in\\R^{40}.\n\\end{equation}\nA five-step Elman recurrence generates the Neuralese trace\n\\begin{equation}\n r_k=\\tanh\\left(W_ih_0+b_i+U_ir_{k-1}+c_i+e_k\\right),\n \\qquad k=1,\\ldots,5,\n \\label{eq:b19-recurrence}\n\\end{equation}\nwhere $e_k$ 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.","um_{i=1}^{5}p_i,\n   \\max_{1\\leq i\\leq5}p_i\n \\right)\\in\\R^{64}.\n \\label{eq:b19-set-aggregate}\n\\end{equation}\nThe semantic seed is\n\\begin{equation}\n h_0=S(a(D),Q(\\widehat q))\\in\\R^{40}.\n\\end{equation}\nA five-step Elman recurrence generates the Neuralese trace\n\\begin{equation}\n r_k=\\tanh\\left(W_ih_0+b_i+U_ir_{k-1}+c_i+e_k\\right),\n \\qquad k=1,\\ldots,5,\n \\label{eq:b19-recurrence}\n\\end{equation}\nwhere $e_k$ 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.","The context code is computed separately:\n\\begin{equation}\n u_c=C(s_x,t_x,s_y,t_y,d_{\\mathrm{sig}})\\in\\R^{10}.\n \\label{eq:b19-context-code}\n\\end{equation}\nIt supports chart diagnostics but has no computational path into $h_0$, $r_k$, the task output, or the halt head.","\\begin{definition}[Block 19 semantic bundle]\nThe finite semantic bundle is\n\\begin{equation}\n \\Bcal_{19}\n =\n \\left(\n   \\Acal_1,\n   a,\n   S,\n   F,\n   J,\n   H,\n   C,\n   \\mathsf{Adm}\n \\right),\n\\end{equation}\nwhere $\\Acal_1$ 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.\n\\end{definition}","n{definition}[Block 19 semantic bundle]\nThe finite semantic bundle is\n\\begin{equation}\n \\Bcal_{19}\n =\n \\left(\n   \\Acal_1,\n   a,\n   S,\n   F,\n   J,\n   H,\n   C,\n   \\mathsf{Adm}\n \\right),\n\\end{equation}\nwhere $\\Acal_1$ 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.\n\\end{definition}","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.","Block 19 retains the exact four-template program family introduced in Block 18:\n\\begin{align}\n f_{\\mathrm{aff}}(x)&=ax+b,\\\\\n f_{\\mathrm{abs}}(x)&=a|x|+b,\\\\\n f_{\\mathrm{quad}}(x)&=ax^2+bx+c,\\\\\n f_{\\mathrm{hinge}}(x)&=\n \\begin{cases}\n ax+c,&x\\geq0,\\\\\n bx+c,&x<0.\n \\end{cases}\n \\label{eq:b19-program-family}\n\\end{align}\nCoefficients 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.","\\midrule\n\\endhead\nInherited chart calibration & $80$ & $320$ & Compile $A1,A2,A3$ relative to $A0$","Novel chart calibration & $80$ & $240$ & Compile $N0,N1$ relative to $A0$","Training & $600$ & $2{,}400$ & Common parent and matched continuation","Development & $300$ & $1{,}200$ & Candidate diagnostics","Meta-validation & $500$ & $500$ & Candidate selection on $N0,N1$","Probe calibration & $300$ & $1{,}200$ & External semantic readouts","Twin validation & $200$ & $1{,}200$ & Six-context semantic flatness","Primary audit & $900$ & $900$ & Fresh tasks in $A0$--$A3$","Novel-chart audit & $900$ & $900$ & Fresh tasks in $N0,N1$","Unsupported audit & $600$ & $600$ & Selective rejection in $U0,U1$","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.","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.","\\begin{table}[H]\n\\centering\n\\small\n\\resizebox{\\textwidth}{!}{%\n\\begin{tabular}{@{}lrrrrrr@{}}\n\\toprule\nCandidate & Eligible & Clean loss & Probe $R^2$ & Program cost & Halt $\\rho$ & Twin cosine max","\\midrule\n\\texttt{recalibrated\\_parent} & no & $0.412628$ & $-1.259981$ & $0.012267$ & $0.003038$ & $1.419$","\\texttt{set\\_only} & no & $0.354799$ & $-7.012995$ & $0.010211$ & $0.039057$ & $1.924$","\\texttt{chart\\_only} & no & $0.183585$ & $0.189870$ & $0.014564$ & $0.100325$ & $0.5885$","\\texttt{full\\_bundle} & yes & $0.052641$ & $0.772549$ & $0.176423$ & $0.635266$ & $2.384\\times10^{-7}$","The principal final results are:\n\\begin{longtable}{@{}llrrrrr@{}}\n\\toprule\nModel & Audit & Clean & Program cost & Probe $R^2$ & Halt $\\rho$ & Execution","\\midrule\n\\endfirsthead\n\\toprule\nModel & Audit & Clean & Program cost & Probe $R^2$ & Halt $\\rho$ & Execution","\\midrule\n\\endhead\nParent & inherited & $0.200177$ & $0.019129$ & $0.144462$ & $0.015517$ & $0.043333$","Parent & novel charts & $0.416361$ & $0.023291$ & $-1.282177$ & $-0.011614$ & $0.021111$","Full child & inherited & $\\mathbf{0.057546}$ & $\\mathbf{0.174135}$ & $\\mathbf{0.770701}$ & $\\mathbf{0.630918}$ & $\\mathbf{0.607778}$","Full child & novel charts & $\\mathbf{0.050745}$ & $\\mathbf{0.194936}$ & $\\mathbf{0.779341}$ & $\\mathbf{0.666244}$ & $\\mathbf{0.594444}$","Paired effects are:\n\\begin{align}\n \\widehat\\Delta_{\\mathrm{clean}}^{A}\n &=0.142631,\n &95\\%\\ \\mathrm{CI}&=[0.135421,0.149531],\\\\\n \\widehat\\Delta_{\\mathrm{clean}}^{N}\n &=0.365616,\n &95\\%\\ \\mathrm{CI}&=[0.352991,0.378402],\\\\\n \\widehat\\Delta_{\\mathrm{program}}^{A}\n &=0.120614,\n &95\\%\\ \\mathrm{CI}&=[0.110353,0.131880],\\\\\n \\widehat\\Delta_{\\mathrm{program}}^{N}\n &=0.125290,\n &95\\%\\ \\mathrm{CI}&=[0.111056,0.138921].\n \\label{eq:b19-paired-effects}\n\\end{align}\nThe clean effects are parent minus child; the program effects are child minus parent. Every interval is frozen and strictly positive.","The expanded atlas admits all $900$ primary tasks and all $900$ novel-chart tasks. It admits none of the $600$ nonlinear unsupported tasks:\n\\begin{equation}\n \\widehat{\\operatorname{coverage}}_{A}=1,\n \\qquad\n \\widehat{\\operatorname{coverage}}_{N}=1,\n \\qquad\n \\widehat{\\operatorname{false\\ admission}}_{U}=0.\n \\label{eq:b19-selectivity}\n\\end{equation}\nForcing 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.","ll $900$ primary tasks and all $900$ novel-chart tasks. It admits none of the $600$ nonlinear unsupported tasks:\n\\begin{equation}\n \\widehat{\\operatorname{coverage}}_{A}=1,\n \\qquad\n \\widehat{\\operatorname{coverage}}_{N}=1,\n \\qquad\n \\widehat{\\operatorname{false\\ admission}}_{U}=0.\n \\label{eq:b19-selectivity}\n\\end{equation}\nForcing 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.","all $900$ novel-chart tasks. It admits none of the $600$ nonlinear unsupported tasks:\n\\begin{equation}\n \\widehat{\\operatorname{coverage}}_{A}=1,\n \\qquad\n \\widehat{\\operatorname{coverage}}_{N}=1,\n \\qquad\n \\widehat{\\operatorname{false\\ admission}}_{U}=0.\n \\label{eq:b19-selectivity}\n\\end{equation}\nForcing 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.","s none of the $600$ nonlinear unsupported tasks:\n\\begin{equation}\n \\widehat{\\operatorname{coverage}}_{A}=1,\n \\qquad\n \\widehat{\\operatorname{coverage}}_{N}=1,\n \\qquad\n \\widehat{\\operatorname{false\\ admission}}_{U}=0.\n \\label{eq:b19-selectivity}\n\\end{equation}\nForcing 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.","The main mechanism controls are:\n\\begin{itemize}\n\\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 $R^2=-7.114963$.\n\\item \\textbf{Chart only.} Canonicalization without permutation-invariant processing is \\texttt{ATLAS\\_PARTIAL} on novel charts, with clean loss $0.186142$ and probe $R^2=0.188481$.\n\\item \\textbf{Shuffled chart binding.} Correct route labels bound to incorrect inverse maps yield clean loss $0.483296$ and probe $R^2=-8.928579$.\n\\item \\textbf{No abstention.} Unsupported nonlinear contexts forced through the closest affine chart fail as reported above.\n\\item \\textbf{Conservative replay.} The expanded atlas changes inherited normalized demonstrations and queries by exactly zero in the serialized audit.\n\\end{itemize}","he main mechanism controls are:\n\\begin{itemize}\n\\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 $R^2=-7.114963$.\n\\item \\textbf{Chart only.} Canonicalization without permutation-invariant processing is \\texttt{ATLAS\\_PARTIAL} on novel charts, with clean loss $0.186142$ and probe $R^2=0.188481$.\n\\item \\textbf{Shuffled chart binding.} Correct route labels bound to incorrect inverse maps yield clean loss $0.483296$ and probe $R^2=-8.928579$.\n\\item \\textbf{No abstention.} Unsupported nonlinear contexts forced through the closest affine chart fail as reported above.\n\\item \\textbf{Conservative replay.} The expanded atlas changes inherited normalized demonstrations and queries by exactly zero in the serialized audit.\n\\end{itemize}","avior but is \\texttt{ATLAS\\_PARTIAL} on novel charts, with clean loss $0.360916$ and probe $R^2=-7.114963$.\n\\item \\textbf{Chart only.} Canonicalization without permutation-invariant processing is \\texttt{ATLAS\\_PARTIAL} on novel charts, with clean loss $0.186142$ and probe $R^2=0.188481$.\n\\item \\textbf{Shuffled chart binding.} Correct route labels bound to incorrect inverse maps yield clean loss $0.483296$ and probe $R^2=-8.928579$.\n\\item \\textbf{No abstention.} Unsupported nonlinear contexts forced through the closest affine chart fail as reported above.\n\\item \\textbf{Conservative replay.} The expanded atlas changes inherited normalized demonstrations and queries by exactly zero in the serialized audit.\n\\end{itemize}","ATLAS\\_PARTIAL} on novel charts, with clean loss $0.360916$ and probe $R^2=-7.114963$.\n\\item \\textbf{Chart only.} Canonicalization without permutation-invariant processing is \\texttt{ATLAS\\_PARTIAL} on novel charts, with clean loss $0.186142$ and probe $R^2=0.188481$.\n\\item \\textbf{Shuffled chart binding.} Correct route labels bound to incorrect inverse maps yield clean loss $0.483296$ and probe $R^2=-8.928579$.\n\\item \\textbf{No abstention.} Unsupported nonlinear contexts forced through the closest affine chart fail as reported above.\n\\item \\textbf{Conservative replay.} The expanded atlas changes inherited normalized demonstrations and queries by exactly zero in the serialized audit.\n\\end{itemize}","The supported empirical mechanism is therefore the coupled object\n\\begin{equation}\n \\boxed{\n \\begin{gathered}\n \\text{semantic-twin chart compilation}\n +\\text{ permutation-invariant set encoding}\\\\\n +\\text{ semantic/context path separation}\n +\\text{ calibrated admission}\\\\\n +\\text{ conservative inherited fallback}.\n \\end{gathered}}\n \\label{eq:b19-coupled-mechanism}\n\\end{equation}","\\begin{conditionaltheorem}[Block 19 distribution-indexed atlas growth]\n\\label{thm:b19-realization}\nWithin the frozen Block 19 environment, assume:\n\\begin{enumerate}\n\\item all $9{,}460$ released records satisfy \\cref{prop:b19-environment-soundness};\n\\item the inherited and new charts are compiled only from the declared pre-audit semantic-twin supports;\n\\item all four candidates inherit the exact ancestor state and receive the same $550$-update continuation envelope;\n\\item the candidate grammar, thresholds, probes, atlases, selected checkpoint, and final split plan are sealed before final-task generation;\n\\item the expanded atlas satisfies \\cref{def:b19-conservative-extension};\n\\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;\n\\item semantic-twin trace, output,","distribution-indexed atlas growth]\n\\label{thm:b19-realization}\nWithin the frozen Block 19 environment, assume:\n\\begin{enumerate}\n\\item all $9{,}460$ released records satisfy \\cref{prop:b19-environment-soundness};\n\\item the inherited and new charts are compiled only from the declared pre-audit semantic-twin supports;\n\\item all four candidates inherit the exact ancestor state and receive the same $550$-update continuation envelope;\n\\item the candidate grammar, thresholds, probes, atlases, selected checkpoint, and final split plan are sealed before final-task generation;\n\\item the expanded atlas satisfies \\cref{def:b19-conservative-extension};\n\\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;\n\\item semantic-twin trace, output, and halting discrepancies satisfy their frozen tolerances;\n\\item the admission results in \\cref{eq:b19-selectivity} hold;\n\\item set-only, chart-only, shuffled-binding, no-abstention, and always-abstain controls retain their declared interpretations;\n\\i…","item the admission results in \\cref{eq:b19-selectivity} hold;\n\\item set-only, chart-only, shuffled-binding, no-abstention, and always-abstain controls retain their declared interpretations;\n\\item the automated tests, independent environment replay, independent chart reconstruction, independent NumPy forward pass, receipts, and package manifest verify.\n\\end{enumerate}\nThen the edge\n\\begin{equation}\n K_{18}\n \\xrightarrow{\\Pi_{19}}\n K_{19}\n \\end{equation}\nis 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.\n\\end{conditionaltheorem}","The independent verifier does not call the package's task verifier, atlas compiler, audit routine, or receipt helper. It:\n\\begin{enumerate}\n\\item executes all four program templates using a separate exact implementation;\n\\item reconstructs all eight context transformations and verifies all $9{,}460$ records;\n\\item refits all five nonidentity affine charts from serialized twins;\n\\item recomputes routing and selective admission on all final records;\n\\item checks checkpoint file and state-dictionary hashes;\n\\item recomputes the candidate, seal, twin, audit, and family receipts;\n\\item recomputes headline row statistics;\n\\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.\n\\end{enumerate}\nThe NumPy and PyTorch implementations agree to maximum absolute error\n\\begin{equation}\n 1.405\\times10^{-6}.\n\\end{equation}\nThe independent verification receipt is\n\\begin{verbatim}","hes;\n\\item recomputes the candidate, seal, twin, audit, and family receipts;\n\\item recomputes headline row statistics;\n\\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.\n\\end{enumerate}\nThe NumPy and PyTorch implementations agree to maximum absolute error\n\\begin{equation}\n 1.405\\times10^{-6}.\n\\end{equation}\nThe independent verification receipt is\n\\begin{verbatim}\n767d71abb0b33d789ff058836ae18eae961fadc5f9b905932cf405f32728dcc0\n\\end{verbatim}\nand the canonical family receipt is\n\\begin{verbatim}\n9e5e81b1d03cca5396debf81d3667da0f9264353052a01dce6a270bb562cad08\n\\end{verbatim}","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:\n\\begin{equation}\n C^b_{c\\to d}T^c_{a\\to b}\n \\approx\n T^d_{a\\to b}C^a_{c\\to d}.\n \\label{eq:b19-transport-square}\n\\end{equation}\nBlock 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.","urce-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:\n\\begin{equation}\n\\boxed{\n\\begin{gathered}\n\\text{Can a frozen semantic atlas support transport of complete}\\\\\n\\text{Neuralese trajectories between independently trained recurrent}\\\\\n\\text{architectures without collapsing chart identity?}\n\\end{gathered}}\n\\label{eq:b20-question}\n\\end{equation}","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.\n\\end{statusbox}","Let $\\mathcal A$ be a finite architecture index set and $\\mathcal C$ a finite admitted chart index set. In the released realization,\n\\begin{equation}\n\\mathcal A\n=\n\\{\\mathrm{Elman},\\mathrm{GRU},\\mathrm{Residual}\\},\n\\qquad\n\\mathcal C\n=\n\\{A0,A1,A2,A3,N0,N1\\}.\n\\end{equation}\nEach pair $(a,c)\\in\\mathcal A\\times\\mathcal C$ carries a full Neuralese trajectory fiber\n\\begin{equation}\nH_{a,c}\n\\cong\n\\left(\\mathbb R^{40}\\right)^5\n\\cong\n\\mathbb R^{200}.\n\\end{equation}\nThe observation atlas determines which context fiber is authorized. A task outside every admitted chart is not assigned to an arbitrary fiber.","\\begin{definition}[Finite Neuralese semantic bundle]\nA finite architecture--context Neuralese bundle is a tuple\n\\begin{equation}\n\\mathfrak E\n=\n\\left(\n\\mathcal A,\n\\mathcal C,\n\\{H_{a,c}\\}_{a,c},\n\\{J_{a,c}\\}_{a,c},\n\\mathcal R\n\\right),\n\\end{equation}\nwhere $H_{a,c}$ 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.\n\\end{definition}","For architectures $a,b$ and admitted chart $c$, an architecture map is\n\\begin{equation}\nT^{c}_{b\\to a}:H_{b,c}\\longrightarrow H_{a,c}.\n\\end{equation}\nFor one architecture $a$ and two admitted charts $c,d$, a context map is\n\\begin{equation}\nC^{a}_{c\\to d}:H_{a,c}\\longrightarrow H_{a,d}.\n\\end{equation}\nBlock 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.","\\begin{definition}[Architecture--context curvature]\nFor architectures $a,b$ and charts $c,d$, the transport-square defect is\n\\begin{equation}\n\\Omega_{a,b;c,d}\n:=\nC^{a}_{c\\to d}T^{c}_{b\\to a}\n-\nT^{d}_{b\\to a}C^{b}_{c\\to d}.\n\\label{eq:b20-curvature}\n\\end{equation}\nGiven an audit distribution $P_{b,c}$ on target trajectories, its normalized root-mean-square curvature is\n\\begin{equation}\n\\kappa_{a,b;c,d}^{2}\n=\n\\frac{\n\\mathbb E_{h\\sim P_{b,c}}\n\\norm{\\Omega_{a,b;c,d}h}^{2}\n}{\n\\mathbb E_{z\\sim P_{a,d}}\\norm{z}^{2}\n}.\n\\label{eq:b20-normalized-curvature}\n\\end{equation}\n\\end{definition}","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.","Suppose an underlying canonical trajectory is represented in chart $c$ by a row vector $z$ and architecture-dependent gauge matrices:\n\\begin{equation}\nh_{a,c}=zG_{a,c},\n\\qquad\nh_{b,c}=zG_{b,c}.\n\\label{eq:b20-gauge-model}\n\\end{equation}\nThe 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.","\\begin{theorem}[Pooled bridge compatibility obstruction]\n\\label{thm:b20-global-obstruction}\nLet $V\\subseteq\\mathbb R^{m}$ 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:\n\\begin{equation}\nzG_{b,c_i}T=zG_{a,c_i}\n\\qquad\n\\text{for every }z\\in V,\n\\quad i\\in\\{1,2\\}.\n\\end{equation}\nThen the restrictions to $V$ satisfy\n\\begin{equation}\nG_{b,c_1}T=G_{a,c_1},\n\\qquad\nG_{b,c_2}T=G_{a,c_2}.\n\\end{equation}\nIf $G_{b,c_i}$ are invertible on $V$, exact pooled transport is possible only when\n\\begin{equation}\nG_{b,c_1}^{-1}G_{a,c_1}\n=\nG_{b,c_2}^{-1}G_{a,c_2}\n\\quad\\text{on }V.\n\\label{eq:b20-compatibility}\n\\end{equation}\n\\end{theorem}","compatibility obstruction]\n\\label{thm:b20-global-obstruction}\nLet $V\\subseteq\\mathbb R^{m}$ 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:\n\\begin{equation}\nzG_{b,c_i}T=zG_{a,c_i}\n\\qquad\n\\text{for every }z\\in V,\n\\quad i\\in\\{1,2\\}.\n\\end{equation}\nThen the restrictions to $V$ satisfy\n\\begin{equation}\nG_{b,c_1}T=G_{a,c_1},\n\\qquad\nG_{b,c_2}T=G_{a,c_2}.\n\\end{equation}\nIf $G_{b,c_i}$ are invertible on $V$, exact pooled transport is possible only when\n\\begin{equation}\nG_{b,c_1}^{-1}G_{a,c_1}\n=\nG_{b,c_2}^{-1}G_{a,c_2}\n\\quad\\text{on }V.\n\\label{eq:b20-compatibility}\n\\end{equation}\n\\end{theorem}","\\label{thm:b20-global-obstruction}\nLet $V\\subseteq\\mathbb R^{m}$ 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:\n\\begin{equation}\nzG_{b,c_i}T=zG_{a,c_i}\n\\qquad\n\\text{for every }z\\in V,\n\\quad i\\in\\{1,2\\}.\n\\end{equation}\nThen the restrictions to $V$ satisfy\n\\begin{equation}\nG_{b,c_1}T=G_{a,c_1},\n\\qquad\nG_{b,c_2}T=G_{a,c_2}.\n\\end{equation}\nIf $G_{b,c_i}$ are invertible on $V$, exact pooled transport is possible only when\n\\begin{equation}\nG_{b,c_1}^{-1}G_{a,c_1}\n=\nG_{b,c_2}^{-1}G_{a,c_2}\n\\quad\\text{on }V.\n\\label{eq:b20-compatibility}\n\\end{equation}\n\\end{theorem}","nned 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:\n\\begin{equation}\nzG_{b,c_i}T=zG_{a,c_i}\n\\qquad\n\\text{for every }z\\in V,\n\\quad i\\in\\{1,2\\}.\n\\end{equation}\nThen the restrictions to $V$ satisfy\n\\begin{equation}\nG_{b,c_1}T=G_{a,c_1},\n\\qquad\nG_{b,c_2}T=G_{a,c_2}.\n\\end{equation}\nIf $G_{b,c_i}$ are invertible on $V$, exact pooled transport is possible only when\n\\begin{equation}\nG_{b,c_1}^{-1}G_{a,c_1}\n=\nG_{b,c_2}^{-1}G_{a,c_2}\n\\quad\\text{on }V.\n\\label{eq:b20-compatibility}\n\\end{equation}\n\\end{theorem}","ries observed in two charts $c_1,c_2$. Suppose one linear map $T$ exactly transports architecture $b$ into architecture $a$ in both charts:\n\\begin{equation}\nzG_{b,c_i}T=zG_{a,c_i}\n\\qquad\n\\text{for every }z\\in V,\n\\quad i\\in\\{1,2\\}.\n\\end{equation}\nThen the restrictions to $V$ satisfy\n\\begin{equation}\nG_{b,c_1}T=G_{a,c_1},\n\\qquad\nG_{b,c_2}T=G_{a,c_2}.\n\\end{equation}\nIf $G_{b,c_i}$ are invertible on $V$, exact pooled transport is possible only when\n\\begin{equation}\nG_{b,c_1}^{-1}G_{a,c_1}\n=\nG_{b,c_2}^{-1}G_{a,c_2}\n\\quad\\text{on }V.\n\\label{eq:b20-compatibility}\n\\end{equation}\n\\end{theorem}","exactly transports architecture $b$ into architecture $a$ in both charts:\n\\begin{equation}\nzG_{b,c_i}T=zG_{a,c_i}\n\\qquad\n\\text{for every }z\\in V,\n\\quad i\\in\\{1,2\\}.\n\\end{equation}\nThen the restrictions to $V$ satisfy\n\\begin{equation}\nG_{b,c_1}T=G_{a,c_1},\n\\qquad\nG_{b,c_2}T=G_{a,c_2}.\n\\end{equation}\nIf $G_{b,c_i}$ are invertible on $V$, exact pooled transport is possible only when\n\\begin{equation}\nG_{b,c_1}^{-1}G_{a,c_1}\n=\nG_{b,c_2}^{-1}G_{a,c_2}\n\\quad\\text{on }V.\n\\label{eq:b20-compatibility}\n\\end{equation}\n\\end{theorem}","\\begin{proof}\nThe 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.\n\\end{proof}","\\begin{corollary}[Exact chart-indexed gauge transport]\n\\label{cor:b20-chart-exact}\nUnder the same hypotheses, the chart-indexed family\n\\begin{equation}\nT^{c}_{b\\to a}\n=\nG_{b,c}^{-1}G_{a,c}\n\\end{equation}\ntransports each chart exactly. If all gauges are orthogonal, then $G_{b,c}^{-1}=G_{b,c}^{\\mathsf T}$.\n\\end{corollary}","A latent-step-shared map acts independently on each recurrent state:\n\\begin{equation}\n\\widehat h_{a,c}^{(k)}\n=\nA_c h_{b,c}^{(k)}+b_c,\n\\qquad k=1,\\ldots,5.\n\\label{eq:b20-step-shared}\n\\end{equation}\nA full-trajectory map instead acts on the concatenation\n\\begin{equation}\n\\operatorname{vec}(h_{b,c}^{(1)},\\ldots,h_{b,c}^{(5)})\n\\in\\mathbb R^{200},\n\\end{equation}\nallowing cross-step mixing:\n\\begin{equation}\n\\operatorname{vec}(\\widehat h_{a,c})\n=\nM_c\\operatorname{vec}(h_{b,c})+q_c.\n\\label{eq:b20-full-map}\n\\end{equation}\nThis difference is substantive when two recurrent architectures distribute the same computation across latent time differently.","tent-step-shared map acts independently on each recurrent state:\n\\begin{equation}\n\\widehat h_{a,c}^{(k)}\n=\nA_c h_{b,c}^{(k)}+b_c,\n\\qquad k=1,\\ldots,5.\n\\label{eq:b20-step-shared}\n\\end{equation}\nA full-trajectory map instead acts on the concatenation\n\\begin{equation}\n\\operatorname{vec}(h_{b,c}^{(1)},\\ldots,h_{b,c}^{(5)})\n\\in\\mathbb R^{200},\n\\end{equation}\nallowing cross-step mixing:\n\\begin{equation}\n\\operatorname{vec}(\\widehat h_{a,c})\n=\nM_c\\operatorname{vec}(h_{b,c})+q_c.\n\\label{eq:b20-full-map}\n\\end{equation}\nThis difference is substantive when two recurrent architectures distribute the same computation across latent time differently.","\\begin{definition}[Finite transport grammar]\nThe Block 20 compiler searches the following declared grammar:\n\\begin{enumerate}\n\\item \\texttt{global\\_full}: one pooled affine map $\\mathbb R^{200}\\to\\mathbb R^{200}$;\n\\item \\texttt{chart\\_step\\_shared}: six chart-indexed affine maps $\\mathbb R^{40}\\to\\mathbb R^{40}$ applied independently at every latent step;\n\\item \\texttt{chart\\_full}: six chart-indexed affine maps $\\mathbb R^{200}\\to\\mathbb R^{200}$.\n\\end{enumerate}\n\\end{definition}","}[Finite transport grammar]\nThe Block 20 compiler searches the following declared grammar:\n\\begin{enumerate}\n\\item \\texttt{global\\_full}: one pooled affine map $\\mathbb R^{200}\\to\\mathbb R^{200}$;\n\\item \\texttt{chart\\_step\\_shared}: six chart-indexed affine maps $\\mathbb R^{40}\\to\\mathbb R^{40}$ applied independently at every latent step;\n\\item \\texttt{chart\\_full}: six chart-indexed affine maps $\\mathbb R^{200}\\to\\mathbb R^{200}$.\n\\end{enumerate}\n\\end{definition}","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.","\\begin{proposition}[Readout stability from trajectory reconstruction]\n\\label{prop:b20-readout-bound}\nLet $J:H_{a,c}\\to\\mathbb R^{p}$ be $L_J$-Lipschitz. If a transported trajectory satisfies\n\\begin{equation}\n\\norm{T^c_{b\\to a}h_{b,c}-h_{a,c}}\n\\leq\\eps,\n\\end{equation}\nthen\n\\begin{equation}\n\\norm{J(T^c_{b\\to a}h_{b,c})-J(h_{a,c})}\n\\leq L_J\\eps.\n\\end{equation}\nIf two transport paths differ by at most $\\delta$, their semantic readouts differ by at most $L_J\\delta$.\n\\end{proposition}","egin{proposition}[Readout stability from trajectory reconstruction]\n\\label{prop:b20-readout-bound}\nLet $J:H_{a,c}\\to\\mathbb R^{p}$ be $L_J$-Lipschitz. If a transported trajectory satisfies\n\\begin{equation}\n\\norm{T^c_{b\\to a}h_{b,c}-h_{a,c}}\n\\leq\\eps,\n\\end{equation}\nthen\n\\begin{equation}\n\\norm{J(T^c_{b\\to a}h_{b,c})-J(h_{a,c})}\n\\leq L_J\\eps.\n\\end{equation}\nIf two transport paths differ by at most $\\delta$, their semantic readouts differ by at most $L_J\\delta$.\n\\end{proposition}","reconstruction]\n\\label{prop:b20-readout-bound}\nLet $J:H_{a,c}\\to\\mathbb R^{p}$ be $L_J$-Lipschitz. If a transported trajectory satisfies\n\\begin{equation}\n\\norm{T^c_{b\\to a}h_{b,c}-h_{a,c}}\n\\leq\\eps,\n\\end{equation}\nthen\n\\begin{equation}\n\\norm{J(T^c_{b\\to a}h_{b,c})-J(h_{a,c})}\n\\leq L_J\\eps.\n\\end{equation}\nIf two transport paths differ by at most $\\delta$, their semantic readouts differ by at most $L_J\\delta$.\n\\end{proposition}","\\midrule\nPre-audit exact tasks & $9{,}200$","Fresh final exact tasks & $3{,}820$","Total exact tasks & $13{,}020$","Bridge-calibration tasks & $1{,}920$","Transport meta-validation tasks & $900$","Final inherited-chart tasks & $1{,}000$","Final added-chart tasks & $1{,}000$","Final transport-square tasks & $1{,}320$","Final unsupported tasks & $500$","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.","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.","The released chronology is:\n\\begin{equation}\n\\begin{aligned}\n&\\text{freeze ancestor and atlas}\n\\longrightarrow\n\\text{declare architecture, gauge, grammar, and thresholds}\\\\\n&\\text{generate and verify pre-audit supports}\n\\longrightarrow\n\\text{train foreign parents and atlas children}\\\\\n&\\text{fit probes, transitions, and candidates}\n\\longrightarrow\n\\text{meta-audit candidates and transport squares}\\\\\n&\\text{seal source, targets, maps, thresholds, and split plan}\n\\longrightarrow\n\\text{generate final supports}\\\\\n&\\text{evaluate every frozen arm}\n\\longrightarrow\n\\text{commit or reject}.\n\\end{aligned}\n\\label{eq:b20-chronology}\n\\end{equation}","A candidate is pre-audit eligible only if both foreign target models are \\texttt{ATLAS\\_FULL} and the candidate jointly satisfies\n\\begin{equation}\n\\begin{aligned}\nR^2_{\\mathrm{transport}}&\\geq0.75,\n&\\qquad\nR^2_{\\mathrm{worst\\ chart}}&\\geq0.74,\\\\\n\\rho_{\\mathrm{preserve}}&\\geq0.98,\n&\n\\operatorname{NRMSE}_{\\mathrm{trace}}&\\leq0.05,\\\\\n\\kappa_{\\mathrm{square}}&\\leq0.02.\n&&\n\\end{aligned}\n\\label{eq:b20-thresholds}\n\\end{equation}\nThe 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.","\\kappa_{\\mathrm{square}}&\\leq0.02.\n&&\n\\end{aligned}\n\\label{eq:b20-thresholds}\n\\end{equation}\nThe 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.","\\begin{table}[H]\n\\centering\n\\small\n\\begin{tabular}{@{}llrrrr@{}}\n\\toprule\nCandidate & Target & Transported $R^2$ & Worst chart & Trace NRMSE & Curvature","\\midrule\n\\texttt{global\\_full} & GRU & $0.722084$ & $0.705057$ & $0.142012$ & $0.130620$","\\texttt{global\\_full} & Residual & $0.732894$ & $0.687945$ & $0.124338$ & $0.114613$","\\texttt{chart\\_step\\_shared} & GRU & $-3.173916$ & $-3.649021$ & $0.154024$ & $0.055261$","\\texttt{chart\\_step\\_shared} & Residual & $-10.575678$ & $-11.273055$ & $0.241193$ & $0.099808$","\\texttt{chart\\_full} & GRU & \\textbf{$0.781840$} & \\textbf{$0.759963$} & \\textbf{$0.034349$} & \\textbf{$0.010458$}","\\texttt{chart\\_full} & Residual & \\textbf{$0.778861$} & \\textbf{$0.752268$} & \\textbf{$0.032050$} & \\textbf{$0.008646$}","\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.82\\textwidth]{block20_candidate_compiler.pdf}\n\\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.}\n\\label{fig:b20-candidate-r2}\n\\end{figure}","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\n\\begin{equation}\n\\texttt{audit\\_accessed=false}.\n\\end{equation}\nThe seal digest is\n\\begin{verbatim}\nf43fb174ae27c1a5b881292d3a96fec7108d859270e7b065034cb8be462c85d7\n\\end{verbatim}","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.","\\begin{table}[H]\n\\centering\n\\small\n\\setlength{\\tabcolsep}{4.5pt}\n\\begin{tabular}{@{}llrrrrrr@{}}\n\\toprule\nTarget & Audit & Source $R^2$ & Pooled $R^2$ & Selected $R^2$ & Preserve & NRMSE & Worst chart","\\midrule\nGRU & Inherited & $0.796107$ & $0.728571$ & \\textbf{$0.799157$} & $1.003832$ & $0.034322$ & $0.760405$","GRU & Added charts & $0.769907$ & $0.719667$ & \\textbf{$0.781159$} & $1.014615$ & $0.033225$ & $0.764416$","Residual & Inherited & $0.796107$ & $0.730951$ & \\textbf{$0.796471$} & $1.000458$ & $0.033751$ & $0.760439$","Residual & Added charts & $0.769907$ & $0.737565$ & \\textbf{$0.785881$} & $1.020749$ & $0.031392$ & $0.767514$","The selected transport does not merely average over strong and weak charts. Every final chart exceeds the frozen $0.74$ worst-chart threshold.","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\n\\begin{equation}\n\\Delta_i\n=\n\\ell_{\\mathrm{global},i}\n-\n\\ell_{\\mathrm{chart},i}.\n\\end{equation}\nPositive values favor the selected chart-indexed map.","\\begin{table}[H]\n\\centering\n\\small\n\\begin{tabular}{@{}llrrr@{}}\n\\toprule\nTarget & Audit & Mean effect & Frozen $95\\%$ interval & Positive fraction","\\midrule\nGRU & Inherited & $0.017350$ & $[0.014942,0.019624]$ & $0.776$","GRU & Added charts & $0.015110$ & $[0.012389,0.017643]$ & $0.754$","Residual & Inherited & $0.016105$ & $[0.013344,0.018829]$ & $0.754$","Residual & Added charts & $0.011872$ & $[0.009607,0.014184]$ & $0.767$","\\midrule\nGRU & $0.131402$ & \\textbf{$0.010474$} & $0.020000$","Gated residual & $0.112397$ & \\textbf{$0.008374$} & $0.020000$","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.","\\paragraph{Pooled full-trajectory control.}\nThe 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.","\\paragraph{Chart-indexed step-shared control.}\nThe 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.","\\paragraph{Shuffled chart binding.}\nThe 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.","\\paragraph{Unsupported contexts.}\nThe 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.","\\paragraph{Unaligned target coordinates.}\nApplying the frozen source probe directly to foreign gauge coordinates produces extremely negative $R^2$. The accepted result is not explained by accidental coordinate identity.","The release contains:\n\\begin{itemize}\n\\item all $13{,}020$ exact task records and both environment manifests;\n\\item the frozen Block 19 atlas and source checkpoint;\n\\item GRU and gated-residual parent and child checkpoints;\n\\item source and target semantic probes;\n\\item the complete gauge registry;\n\\item six candidate transport archives and three context-transition archives;\n\\item pre-audit compiler and seal receipts;\n\\item every final model, transport, square, shuffled-binding, and unsupported control result;\n\\item deterministic figures, automated tests, an independent verifier, and an exact package manifest.\n\\end{itemize}","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\n\\begin{equation}\n1.343\\times10^{-6},\n\\end{equation}\nand its maximum difference from the canonical reported transport metrics is\n\\begin{equation}\n1.779\\times10^{-7}.\n\\end{equation}\nThe independent verification receipt is\n\\begin{verbatim}\n1f2909fa973e1586b71e9cc85fceb0ecc2a0edc48598464a743f2ce1e3b172fe\n\\end{verbatim}","eceipt 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\n\\begin{equation}\n1.343\\times10^{-6},\n\\end{equation}\nand its maximum difference from the canonical reported transport metrics is\n\\begin{equation}\n1.779\\times10^{-7}.\n\\end{equation}\nThe independent verification receipt is\n\\begin{verbatim}\n1f2909fa973e1586b71e9cc85fceb0ecc2a0edc48598464a743f2ce1e3b172fe\n\\end{verbatim}","nt 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\n\\begin{equation}\n1.343\\times10^{-6},\n\\end{equation}\nand its maximum difference from the canonical reported transport metrics is\n\\begin{equation}\n1.779\\times10^{-7}.\n\\end{equation}\nThe independent verification receipt is\n\\begin{verbatim}\n1f2909fa973e1586b71e9cc85fceb0ecc2a0edc48598464a743f2ce1e3b172fe\n\\end{verbatim}","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\n\\begin{equation}\nT^{c}_{B\\to A}:H_{B,c}^{(1:5)}\\longrightarrow H_{A,c}^{(1:5)}.\n\\end{equation}\nLet $P:\\R^{40}\\to\\R^8$ be an orthonormal packet projector fitted on a pre-audit source support.  Applying $P$ independently to the five recurrent states gives\n\\begin{equation}\n\\Pi(H)\n=\n\\left(P h_1,\\ldots,P h_5\\right)\n\\in\n\\left(\\R^8\\right)^5\n\\cong\\R^{40}.\n\\label{eq:block21-packet}\n\\end{equation}\nThe packet is not claimed to preserve every latent coordinate.  It is required to preserve the protected nine-coo","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\n\\begin{equation}\nT^{c}_{B\\to A}:H_{B,c}^{(1:5)}\\longrightarrow H_{A,c}^{(1:5)}.\n\\end{equation}\nLet $P:\\R^{40}\\to\\R^8$ be an orthonormal packet projector fitted on a pre-audit source support.  Applying $P$ independently to the five recurrent states gives\n\\begin{equation}\n\\Pi(H)\n=\n\\left(P h_1,\\ldots,P h_5\\right)\n\\in\n\\left(\\R^8\\right)^5\n\\cong\\R^{40}.\n\\label{eq:block21-packet}\n\\end{equation}\nThe packet is not claimed to preserve every latent coordinate.  It is required to preserve the protected nine-coordinate semantic witness on the declared task fami","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\n\\begin{equation}\nT^{c}_{B\\to A}:H_{B,c}^{(1:5)}\\longrightarrow H_{A,c}^{(1:5)}.\n\\end{equation}\nLet $P:\\R^{40}\\to\\R^8$ be an orthonormal packet projector fitted on a pre-audit source support.  Applying $P$ independently to the five recurrent states gives\n\\begin{equation}\n\\Pi(H)\n=\n\\left(P h_1,\\ldots,P h_5\\right)\n\\in\n\\left(\\R^8\\right)^5\n\\cong\\R^{40}.\n\\label{eq:block21-packet}\n\\end{equation}\nThe 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\n\\begin{equation}\nJ:\\R^{40}\\longrightarrow\\R^9\n\\end{equati","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\n\\begin{equation}\nT^{c}_{B\\to A}:H_{B,c}^{(1:5)}\\longrightarrow H_{A,c}^{(1:5)}.\n\\end{equation}\nLet $P:\\R^{40}\\to\\R^8$ be an orthonormal packet projector fitted on a pre-audit source support.  Applying $P$ independently to the five recurrent states gives\n\\begin{equation}\n\\Pi(H)\n=\n\\left(P h_1,\\ldots,P h_5\\right)\n\\in\n\\left(\\R^8\\right)^5\n\\cong\\R^{40}.\n\\label{eq:block21-packet}\n\\end{equation}\nThe 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\n\\begin{equation}\nJ:\\R^{40}\\longrightarrow\\R^9\n\\end{equation}\nis fitted on a support disjoint from packet-projection fitting and from all final episodes.","$A$ 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\n\\begin{equation}\nT^{c}_{B\\to A}:H_{B,c}^{(1:5)}\\longrightarrow H_{A,c}^{(1:5)}.\n\\end{equation}\nLet $P:\\R^{40}\\to\\R^8$ be an orthonormal packet projector fitted on a pre-audit source support.  Applying $P$ independently to the five recurrent states gives\n\\begin{equation}\n\\Pi(H)\n=\n\\left(P h_1,\\ldots,P h_5\\right)\n\\in\n\\left(\\R^8\\right)^5\n\\cong\\R^{40}.\n\\label{eq:block21-packet}\n\\end{equation}\nThe 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\n\\begin{equation}\nJ:\\R^{40}\\longrightarrow\\R^9\n\\end{equation}\nis fitted on a support disjoint from packet-projection fitting and from all final episodes.","two Block 20 foreign architectures.  For admitted observation chart $c$, Block 20 supplies an accepted full-trajectory connection\n\\begin{equation}\nT^{c}_{B\\to A}:H_{B,c}^{(1:5)}\\longrightarrow H_{A,c}^{(1:5)}.\n\\end{equation}\nLet $P:\\R^{40}\\to\\R^8$ be an orthonormal packet projector fitted on a pre-audit source support.  Applying $P$ independently to the five recurrent states gives\n\\begin{equation}\n\\Pi(H)\n=\n\\left(P h_1,\\ldots,P h_5\\right)\n\\in\n\\left(\\R^8\\right)^5\n\\cong\\R^{40}.\n\\label{eq:block21-packet}\n\\end{equation}\nThe 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\n\\begin{equation}\nJ:\\R^{40}\\longrightarrow\\R^9\n\\end{equation}\nis fitted on a support disjoint from packet-projection fitting and from all final episodes.","g $P$ independently to the five recurrent states gives\n\\begin{equation}\n\\Pi(H)\n=\n\\left(P h_1,\\ldots,P h_5\\right)\n\\in\n\\left(\\R^8\\right)^5\n\\cong\\R^{40}.\n\\label{eq:block21-packet}\n\\end{equation}\nThe 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\n\\begin{equation}\nJ:\\R^{40}\\longrightarrow\\R^9\n\\end{equation}\nis fitted on a support disjoint from packet-projection fitting and from all final episodes.","\\begin{definition}[Post-connection transport episode]\nA Block 21 episode is a tuple\n\\begin{equation}\n\\mathfrak E\n=\n(X,Y,Z,p,\\mathcal D_{\\rm cal},\\mathcal D_{\\rm meta},\\mathcal D_{\\rm audit}),\n\\end{equation}\nwhere $X\\in\\R^{5\\times8}$ 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.\n\\end{definition}","The ordinary control has\n\\begin{equation}\nY_t=X_t A+b,\n\\qquad A\\in\\mathrm{GL}(8),\n\\label{eq:block21-affine}\n\\end{equation}\nwith 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.","The nonlinear family applies\n\\begin{equation}\nY_t\n=\n\\frac{\\sinh\\!\\left(\\alpha(X_tA+b)\\right)}{\\alpha},\n\\qquad \\alpha>0,\n\\label{eq:block21-nonlinear}\n\\end{equation}\ncoordinatewise.  The declared inverse constructor first applies\n\\begin{equation}\n\\phi^{-1}_{\\alpha}(y)\n=\n\\frac{\\operatorname{arsinh}(\\alpha y)}{\\alpha}\n\\end{equation}\nand then fits a step-affine map.","\\begin{proposition}[Affine obstruction for the declared nonlinear family]\n\\label{prop:block21-nonlinear-obstruction}\nLet $I\\subset\\R$ 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}.\n\\end{proposition}","\\begin{proof}\nAn 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.\n\\end{proof}","The latent-clock family first applies an invertible stepwise affine gauge and then mixes adjacent recurrent times:\n\\begin{equation}\nY_t\n=\n(1-\\beta)U_t+\\beta U_{t-1},\n\\qquad t=2,\\ldots,5,\n\\label{eq:block21-clock}\n\\end{equation}\nwith $Y_1=U_1$ and $0<\\beta<1$.  In matrix form,\n\\begin{equation}\nY=W_{\\beta}U,\n\\end{equation}\nwhere $W_{\\beta}$ is lower bidiagonal.  The temporal constructor applies $W_{\\beta}^{-1}$ before fitting the spatial gauge.","\\begin{proposition}[Fixed-step obstruction]\n\\label{prop:block21-clock-obstruction}\nSuppose 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}.\n\\end{proposition}","\\begin{proof}\nThe 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.\n\\end{proof}","The chart-alias family contains two unannounced local affine relations,\n\\begin{equation}\nY_t=\n\\begin{cases}\nX_tA_0+b_0,&q=0,\\\\\nX_tA_1+b_1,&q=1,\n\\end{cases}\n\\label{eq:block21-alias}\n\\end{equation}\nwhere 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.","X_tA_1+b_1,&q=1,\n\\end{cases}\n\\label{eq:block21-alias}\n\\end{equation}\nwhere 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.","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\n\\begin{equation}\nY\\xrightarrow{q}Z_{\\rm prot},\n\\label{eq:block21-quotient}\n\\end{equation}\nnot an artificial lift into all source packet coordinates.","\\begin{theorem}[Semantic quotient sufficiency]\n\\label{thm:block21-quotient}\nLet $Z$ 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.\n\\end{theorem}","\\begin{proof}\nThe 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 $Z$.\n\\end{proof}","The destructive family withholds four protected coordinates and constructs pairs $i,j$ satisfying\n\\begin{equation}\nY_i=Y_j,\n\\qquad\nZ_i\\neq Z_j.\n\\label{eq:block21-collision}\n\\end{equation}\nThe compiler must discover these repeated observations from the public arrays.  Their private construction pairs are not supplied.","\\begin{theorem}[Deterministic target-only no-go]\n\\label{thm:block21-no-go}\nIf \\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.\n\\end{theorem}","\\begin{proof}\nEquality of the inputs implies $f(Y_i)=f(Y_j)$.  This contradicts $Z_i\\neq Z_j$.\n\\end{proof}","The finite grammar is\n\\begin{equation}\n\\mathcal G_{21}\n=\n\\left\\{\n\\begin{array}{ll}\nG_0&\\text{step-affine map},\\\\\nG_1&\\text{full-trajectory affine map},\\\\\nG_2&\\text{monotone nonlinear inverse plus step-affine map},\\\\\nG_3&\\text{bidiagonal temporal inverse plus step-affine map},\\\\\nG_4&\\text{probe-conditioned two-chart affine map},\\\\\nG_5&\\text{semantic quotient},\\\\\nG_{\\bot}&\\text{collision-grounded no-go}.\n\\end{array}\n\\right.\n\\label{eq:block21-grammar}\n\\end{equation}","For reconstructive candidates, admissibility requires\n\\begin{equation}\nR^2_{\\rm sem}\\geq0.68,\n\\qquad\nR^2_{\\rm packet}\\geq0.72,\n\\qquad\n\\operatorname{NRMSE}_{\\rm packet}\\leq0.55.\n\\label{eq:block21-reconstructive-gates}\n\\end{equation}\nA 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 $0.50$.","\\label{eq:block21-reconstructive-gates}\n\\end{equation}\nA 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 $0.50$.","Each repair $g$ receives a description-length proxy $L(g)$ recording parameter count and constructor surcharge.  The compiler selects\n\\begin{equation}\n\\widehat g\n\\in\n\\arg\\min_{g\\in\\mathcal G_{21}}\nL(g)\n\\quad\\text{subject to the appropriate protected gates.}\n\\label{eq:block21-mdl}\n\\end{equation}\nIf no repair passes and the collision certificate is valid, it returns $G_{\\bot}$; otherwise it returns unresolved.","The released chronology is:\n\\begin{enumerate}\n\\item replay the exact Block 20 source and target checkpoints, six-chart atlas, accepted transports, and canonical receipt;\n\\item fit the orthonormal packet projection on $1{,}200$ pre-audit tasks;\n\\item fit the protected semantic probe on a disjoint $1{,}600$-task support;\n\\item generate a $5{,}200$-task pre-audit episode pool;\n\\item evaluate two pre-audit episodes for each of two architectures and six hidden families;\n\\item freeze the grammar, thresholds, audit-only expected map, compiler interface, and final split plan;\n\\item commit the pre-audit seal;\n\\item generate the $5{,}600$-task final pool only after sealing;\n\\item compile three unseen transform instances for every architecture--family pair;\n\\item audit the serialized repairs on disjoint final subsets and reveal private family labels only for scoring;\n\\item replay all exact task semantics, compiler decision","The released chronology is:\n\\begin{enumerate}\n\\item replay the exact Block 20 source and target checkpoints, six-chart atlas, accepted transports, and canonical receipt;\n\\item fit the orthonormal packet projection on $1{,}200$ pre-audit tasks;\n\\item fit the protected semantic probe on a disjoint $1{,}600$-task support;\n\\item generate a $5{,}200$-task pre-audit episode pool;\n\\item evaluate two pre-audit episodes for each of two architectures and six hidden families;\n\\item freeze the grammar, thresholds, audit-only expected map, compiler interface, and final split plan;\n\\item commit the pre-audit seal;\n\\item generate the $5{,}600$-task final pool only after sealing;\n\\item compile three unseen transform instances for every architecture--family pair;\n\\item audit the serialized repairs on disjoint final subsets and reveal private family labels only for scoring;\n\\item replay all exact task semantics, compiler decisions, serialized repairs, and metrics.\n\\end{enumerate}\nThe four task files contain","The released chronology is:\n\\begin{enumerate}\n\\item replay the exact Block 20 source and target checkpoints, six-chart atlas, accepted transports, and canonical receipt;\n\\item fit the orthonormal packet projection on $1{,}200$ pre-audit tasks;\n\\item fit the protected semantic probe on a disjoint $1{,}600$-task support;\n\\item generate a $5{,}200$-task pre-audit episode pool;\n\\item evaluate two pre-audit episodes for each of two architectures and six hidden families;\n\\item freeze the grammar, thresholds, audit-only expected map, compiler interface, and final split plan;\n\\item commit the pre-audit seal;\n\\item generate the $5{,}600$-task final pool only after sealing;\n\\item compile three unseen transform instances for every architecture--family pair;\n\\item audit the serialized repairs on disjoint final subsets and reveal private family labels only for scoring;\n\\item replay all exact task semantics, compiler decisions, serialized repairs, and metrics.\n\\end{enumerate}\nThe four task files contain $13{,}600$ exact records in total.","re-audit tasks;\n\\item fit the protected semantic probe on a disjoint $1{,}600$-task support;\n\\item generate a $5{,}200$-task pre-audit episode pool;\n\\item evaluate two pre-audit episodes for each of two architectures and six hidden families;\n\\item freeze the grammar, thresholds, audit-only expected map, compiler interface, and final split plan;\n\\item commit the pre-audit seal;\n\\item generate the $5{,}600$-task final pool only after sealing;\n\\item compile three unseen transform instances for every architecture--family pair;\n\\item audit the serialized repairs on disjoint final subsets and reveal private family labels only for scoring;\n\\item replay all exact task semantics, compiler decisions, serialized repairs, and metrics.\n\\end{enumerate}\nThe four task files contain $13{,}600$ exact records in total.",";\n\\item generate the $5{,}600$-task final pool only after sealing;\n\\item compile three unseen transform instances for every architecture--family pair;\n\\item audit the serialized repairs on disjoint final subsets and reveal private family labels only for scoring;\n\\item replay all exact task semantics, compiler decisions, serialized repairs, and metrics.\n\\end{enumerate}\nThe four task files contain $13{,}600$ exact records in total.","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.","\\toprule\nArchitecture and family & Mean sem. $R^2$ & Min sem. $R^2$ & Mean packet $R^2$ & Max NRMSE","\\midrule\n\\endfirsthead\n\\toprule\nArchitecture and family & Mean sem. $R^2$ & Min sem. $R^2$ & Mean packet $R^2$ & Max NRMSE","For nonlinear, temporal, and chart-alias episodes, the inherited step-affine map is retained as a wrong-repair control.  The paired effect is\n\\begin{equation}\n\\Delta_i\n=\n\\ell_{\\rm step\\ affine,i}\n-\n\\ell_{\\rm selected,i}.\n\\end{equation}\nEvery frozen bootstrap interval is strictly positive.","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.","\\begin{statusbox}{Block 21 result}\nThe 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.\n\\end{statusbox}","pair 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.\n\\end{statusbox}",") -- (residual);\n\\draw[arr] (residual) -- (obstruction);\n\\draw[arr] (obstruction) -- (descent);\n\\draw[arr] (descent) -- (schema);\n\\draw[arr] (schema) -- (sandbox);\n\\draw[arr] (sandbox) -- (validate);\n\\draw[arr] (validate) -- (commit);\n\\draw[arr] (commit) -- (abstract);\n\\draw[arr] (abstract) -- (archive);\n\\draw[arr] (abstract) -- (meta);\n\\draw[arr] (meta.east) -- ++(12mm,0) coordinate (rloop) |- ($(attrib.north)+(0,6mm)$) -- (attrib.north);\n\\draw[arr] (archive.west) -- ++(-12mm,0) coordinate (lloop) |- ($(outcome.west)+(-6mm,0)$) -- (outcome.west);\n\\end{tikzpicture}%\n}\n\\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.}\n\\label{fig:sric-loop}\n\\end{figure}","escent) -- (schema);\n\\draw[arr] (schema) -- (sandbox);\n\\draw[arr] (sandbox) -- (validate);\n\\draw[arr] (validate) -- (commit);\n\\draw[arr] (commit) -- (abstract);\n\\draw[arr] (abstract) -- (archive);\n\\draw[arr] (abstract) -- (meta);\n\\draw[arr] (meta.east) -- ++(12mm,0) coordinate (rloop) |- ($(attrib.north)+(0,6mm)$) -- (attrib.north);\n\\draw[arr] (archive.west) -- ++(-12mm,0) coordinate (lloop) |- ($(outcome.west)+(-6mm,0)$) -- (outcome.west);\n\\end{tikzpicture}%\n}\n\\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.}\n\\label{fig:sric-loop}\n\\end{figure}","\\begin{protocol}[Proof-carrying improvement transaction]\nGiven lineage state $\\mathfrak L_t$:\n\\begin{enumerate}\n\\item Freeze witness $W_t$, evaluator core coordinates, evidence ledger head, resource budget, and rollback checkpoint.\n\\item Detect a persistent externally grounded mismatch.\n\\item Generate internal-change, world-change, sensor, memory, evaluator, copy, fork, and unknown hypotheses.\n\\item Select bounded active probes until uncertainty is below threshold or the probe budget is exhausted.\n\\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.\n\\item Resolve the defect into the validated local strata and compute local obstructi","\\begin{protocol}[Proof-carrying improvement transaction]\nGiven lineage state $\\mathfrak L_t$:\n\\begin{enumerate}\n\\item Freeze witness $W_t$, evaluator core coordinates, evidence ledger head, resource budget, and rollback checkpoint.\n\\item Detect a persistent externally grounded mismatch.\n\\item Generate internal-change, world-change, sensor, memory, evaluator, copy, fork, and unknown hypotheses.\n\\item Select bounded active probes until uncertainty is below threshold or the probe budget is exhausted.\n\\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.\n\\item Resolve the defect into the validated local strata and compute local obstruction classes.\n\\item If a local obstruction is no","ness gate that can reject a blind target gain when alignment, protected competence, or unaffected-stratum regression is unacceptable;\n\\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.\n\\end{enumerate}\nThe 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.","tion split;\n\\item a descendant receipt sealed before either final audit is opened;\n\\item paired seed/task-foreign and architecture-foreign audits;\n\\item raw-calibration, no-risk, shuffled-lineage, selector-ablation, and audit-oracle controls;\n\\item identical non-null candidate budgets for every admissible selector arm.\n\\end{enumerate}\nThe child-parent protected-utility effects are\n\\begin{equation}\n\\widehat\\Delta_{\\rm audit}=0.072518,\n\\qquad\n\\widehat\\Delta_{\\rm arch}=0.104999,\n\\end{equation}\nwith 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.","\\paragraph{Bridge-policy realization.}\nThe 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:\n\\begin{enumerate}\n\\item a random-$16$ parent anchor policy;\n\\item a compiled pivoted-QR-$16$ descendant policy sealed before primary foreign targets are generated;\n\\item disjoint development and audit task families and disjoint primary representation families;\n\\item a $144$-target paired foreign audit and a separate stress suite;\n\\item random-budget, farthest-point, shuffled-binding, sign-flip, and plain-QR controls;\n\\item exact query ledgers, rank diagnostics, harm checks, and package-closed receipts.\n\\end{enumerate}\nOn the primary foreign audit, mean balanced selecte","\\paragraph{Bridge-policy realization.}\nThe 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:\n\\begin{enumerate}\n\\item a random-$16$ parent anchor policy;\n\\item a compiled pivoted-QR-$16$ descendant policy sealed before primary foreign targets are generated;\n\\item disjoint development and audit task families and disjoint primary representation families;\n\\item a $144$-target paired foreign audit and a separate stress suite;\n\\item random-budget, farthest-point, shuffled-binding, sign-flip, and plain-QR controls;\n\\item exact query ledgers, rank diagnostics, harm checks, and package-closed receipts.\n\\end{enumerate}\nOn the primary foreign audit, mean balanced selected gain rises from $0.135212$ to $0.344720$, and\n\\begin{equation}\n\\widehat\\Delta_{\\rm bridge}=0.209508,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.194968,0.223898].\n\\end{equati","\\paragraph{Bridge-policy realization.}\nThe 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:\n\\begin{enumerate}\n\\item a random-$16$ parent anchor policy;\n\\item a compiled pivoted-QR-$16$ descendant policy sealed before primary foreign targets are generated;\n\\item disjoint development and audit task families and disjoint primary representation families;\n\\item a $144$-target paired foreign audit and a separate stress suite;\n\\item random-budget, farthest-point, shuffled-binding, sign-flip, and plain-QR controls;\n\\item exact query ledgers, rank diagnostics, harm checks, and package-closed receipts.\n\\end{enumerate}\nOn the primary foreign audit, mean balanced selected gain rises from $0.135212$ to $0.344720$, and\n\\begin{equation}\n\\widehat\\Delta_{\\rm bridge}=0.209508,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.194968,0.223898].\n\\end{equation}\nEvery primary paired effect is positive, full-rank","d $24$-query deployment budget fixed while changing the paired-anchor selection rule and bridge-identification gate. It realizes:\n\\begin{enumerate}\n\\item a random-$16$ parent anchor policy;\n\\item a compiled pivoted-QR-$16$ descendant policy sealed before primary foreign targets are generated;\n\\item disjoint development and audit task families and disjoint primary representation families;\n\\item a $144$-target paired foreign audit and a separate stress suite;\n\\item random-budget, farthest-point, shuffled-binding, sign-flip, and plain-QR controls;\n\\item exact query ledgers, rank diagnostics, harm checks, and package-closed receipts.\n\\end{enumerate}\nOn the primary foreign audit, mean balanced selected gain rises from $0.135212$ to $0.344720$, and\n\\begin{equation}\n\\widehat\\Delta_{\\rm bridge}=0.209508,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.194968,0.223898].\n\\end{equation}\nEvery 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-rev…","audit task families and disjoint primary representation families;\n\\item a $144$-target paired foreign audit and a separate stress suite;\n\\item random-budget, farthest-point, shuffled-binding, sign-flip, and plain-QR controls;\n\\item exact query ledgers, rank diagnostics, harm checks, and package-closed receipts.\n\\end{enumerate}\nOn the primary foreign audit, mean balanced selected gain rises from $0.135212$ to $0.344720$, and\n\\begin{equation}\n\\widehat\\Delta_{\\rm bridge}=0.209508,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.194968,0.223898].\n\\end{equation}\nEvery 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.","milies and disjoint primary representation families;\n\\item a $144$-target paired foreign audit and a separate stress suite;\n\\item random-budget, farthest-point, shuffled-binding, sign-flip, and plain-QR controls;\n\\item exact query ledgers, rank diagnostics, harm checks, and package-closed receipts.\n\\end{enumerate}\nOn the primary foreign audit, mean balanced selected gain rises from $0.135212$ to $0.344720$, and\n\\begin{equation}\n\\widehat\\Delta_{\\rm bridge}=0.209508,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.194968,0.223898].\n\\end{equation}\nEvery 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.","resentation families;\n\\item a $144$-target paired foreign audit and a separate stress suite;\n\\item random-budget, farthest-point, shuffled-binding, sign-flip, and plain-QR controls;\n\\item exact query ledgers, rank diagnostics, harm checks, and package-closed receipts.\n\\end{enumerate}\nOn the primary foreign audit, mean balanced selected gain rises from $0.135212$ to $0.344720$, and\n\\begin{equation}\n\\widehat\\Delta_{\\rm bridge}=0.209508,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.194968,0.223898].\n\\end{equation}\nEvery 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.","m exact query ledgers, rank diagnostics, harm checks, and package-closed receipts.\n\\end{enumerate}\nOn the primary foreign audit, mean balanced selected gain rises from $0.135212$ to $0.344720$, and\n\\begin{equation}\n\\widehat\\Delta_{\\rm bridge}=0.209508,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.194968,0.223898].\n\\end{equation}\nEvery 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.","ledgers, rank diagnostics, harm checks, and package-closed receipts.\n\\end{enumerate}\nOn the primary foreign audit, mean balanced selected gain rises from $0.135212$ to $0.344720$, and\n\\begin{equation}\n\\widehat\\Delta_{\\rm bridge}=0.209508,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.194968,0.223898].\n\\end{equation}\nEvery 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.","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.","d exact permutation-average meta-headroom allocation;\n\\item component saturation and headroom-adjusted metaproductivity;\n\\item bottleneck migration with an explicit conservative grammar-discovery coordinate;\n\\item nested obstruction images and exact newly correctable energy;\n\\item repair death versus forgetting death under dynamic defect semantics;\n\\item obstruction-guided one-generator and rank-$k$ grammar expansion;\n\\item finite repair-conflict refinement with a nuisance-feature negative control;\n\\item finite counterexample-guided repair synthesis or grammar exhaustion;\n\\item intervention-response algebra decomposition into central causal strata;\n\\item lineage-wide false-improvement and protected-regression budget accounting.\n\\end{enumerate}\nThe 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.","n which every inherited primary design is full rank but its weighted information floor remains low;\n\\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;\n\\item a quality-weighted D-optimal grandchild and a stronger recalibrated rank-child control;\n\\item identical $24$-query deployment budgets for the primary child and grandchild;\n\\item gate-only, quality-permutation, unweighted-design, random-$16$, and larger-budget random-$32$ controls;\n\\item a $96$-target paired primary foreign audit plus compressed, nonlinear, and severe-ill-conditioning stress families;\n\\item exact ancestor, compiler, pre-audit seal, result, query-ledger, and package-manifest receipts.\n\\end{enumerate}\nOn 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\n\\begin{equation}\n\\widehat\\Delta_{1\\to2}=0.172188,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.141481,0.203110].\n\\end{equ…","e policy grammar containing packet QR, unweighted D-optimal, quality-weighted D-optimal, quality-weighted A-optimal, and quality-weighted E-refined designs;\n\\item a quality-weighted D-optimal grandchild and a stronger recalibrated rank-child control;\n\\item identical $24$-query deployment budgets for the primary child and grandchild;\n\\item gate-only, quality-permutation, unweighted-design, random-$16$, and larger-budget random-$32$ controls;\n\\item a $96$-target paired primary foreign audit plus compressed, nonlinear, and severe-ill-conditioning stress families;\n\\item exact ancestor, compiler, pre-audit seal, result, query-ledger, and package-manifest receipts.\n\\end{enumerate}\nOn 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\n\\begin{equation}\n\\widehat\\Delta_{1\\to2}=0.172188,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.141481,0.203110].\n\\end{equation}\nBoth designs have full-rank rate one. The mean weighted minimum information eigenvalue rises from $0.518197$ to $14.268195$, …","ket QR, unweighted D-optimal, quality-weighted D-optimal, quality-weighted A-optimal, and quality-weighted E-refined designs;\n\\item a quality-weighted D-optimal grandchild and a stronger recalibrated rank-child control;\n\\item identical $24$-query deployment budgets for the primary child and grandchild;\n\\item gate-only, quality-permutation, unweighted-design, random-$16$, and larger-budget random-$32$ controls;\n\\item a $96$-target paired primary foreign audit plus compressed, nonlinear, and severe-ill-conditioning stress families;\n\\item exact ancestor, compiler, pre-audit seal, result, query-ledger, and package-manifest receipts.\n\\end{enumerate}\nOn 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\n\\begin{equation}\n\\widehat\\Delta_{1\\to2}=0.172188,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.141481,0.203110].\n\\end{equation}\nBoth designs have full-rank rate one. The mean weighted minimum information eigenvalue rises from $0.518197$ to $14.268195$, the median condition number fal…","timal, quality-weighted D-optimal, quality-weighted A-optimal, and quality-weighted E-refined designs;\n\\item a quality-weighted D-optimal grandchild and a stronger recalibrated rank-child control;\n\\item identical $24$-query deployment budgets for the primary child and grandchild;\n\\item gate-only, quality-permutation, unweighted-design, random-$16$, and larger-budget random-$32$ controls;\n\\item a $96$-target paired primary foreign audit plus compressed, nonlinear, and severe-ill-conditioning stress families;\n\\item exact ancestor, compiler, pre-audit seal, result, query-ledger, and package-manifest receipts.\n\\end{enumerate}\nOn 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\n\\begin{equation}\n\\widehat\\Delta_{1\\to2}=0.172188,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.141481,0.203110].\n\\end{equation}\nBoth 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 …","uality-permutation, unweighted-design, random-$16$, and larger-budget random-$32$ controls;\n\\item a $96$-target paired primary foreign audit plus compressed, nonlinear, and severe-ill-conditioning stress families;\n\\item exact ancestor, compiler, pre-audit seal, result, query-ledger, and package-manifest receipts.\n\\end{enumerate}\nOn 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\n\\begin{equation}\n\\widehat\\Delta_{1\\to2}=0.172188,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.141481,0.203110].\n\\end{equation}\nBoth 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 impro…","$16$, and larger-budget random-$32$ controls;\n\\item a $96$-target paired primary foreign audit plus compressed, nonlinear, and severe-ill-conditioning stress families;\n\\item exact ancestor, compiler, pre-audit seal, result, query-ledger, and package-manifest receipts.\n\\end{enumerate}\nOn 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\n\\begin{equation}\n\\widehat\\Delta_{1\\to2}=0.172188,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.141481,0.203110].\n\\end{equation}\nBoth 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.","foreign audit plus compressed, nonlinear, and severe-ill-conditioning stress families;\n\\item exact ancestor, compiler, pre-audit seal, result, query-ledger, and package-manifest receipts.\n\\end{enumerate}\nOn 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\n\\begin{equation}\n\\widehat\\Delta_{1\\to2}=0.172188,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.141481,0.203110].\n\\end{equation}\nBoth 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.","numerate}\nOn 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\n\\begin{equation}\n\\widehat\\Delta_{1\\to2}=0.172188,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.141481,0.203110].\n\\end{equation}\nBoth 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.","he 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\n\\begin{equation}\n\\widehat\\Delta_{1\\to2}=0.172188,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.141481,0.203110].\n\\end{equation}\nBoth 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.","rises from $0.037393$ for the recalibrated rank child to $0.209580$ for the conditioning grandchild. The paired effect is\n\\begin{equation}\n\\widehat\\Delta_{1\\to2}=0.172188,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.141481,0.203110].\n\\end{equation}\nBoth 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.","037393$ for the recalibrated rank child to $0.209580$ for the conditioning grandchild. The paired effect is\n\\begin{equation}\n\\widehat\\Delta_{1\\to2}=0.172188,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.141481,0.203110].\n\\end{equation}\nBoth 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.","9580$ for the conditioning grandchild. The paired effect is\n\\begin{equation}\n\\widehat\\Delta_{1\\to2}=0.172188,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.141481,0.203110].\n\\end{equation}\nBoth 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.","conditioning grandchild. The paired effect is\n\\begin{equation}\n\\widehat\\Delta_{1\\to2}=0.172188,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.141481,0.203110].\n\\end{equation}\nBoth 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.","Block 10 retains the Block 9 quality-weighted acquisition mechanism but expands the representational grammar of repair transport. It realizes:\n\\begin{enumerate}\n\\item exact semantic replay of the Block 9 conditioning-aware ancestor;\n\\item a strengthened global-affine parent using $80$ fit plus $16$ witness pairs;\n\\item a conservative two-chart child using $48$ fit, $32$ chart-selection, and $16$ witness pairs;\n\\item one matched $96$-query deployment envelope for both processes;\n\\item a finite separator search restricted to the source repair's protected two-dimensional context packet;\n\\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;\n\\item disjoint development, meta-validation, and primary audit task and family identifiers;\n\\item a $96$-target primary foreign au","Block 10 retains the Block 9 quality-weighted acquisition mechanism but expands the representational grammar of repair transport. It realizes:\n\\begin{enumerate}\n\\item exact semantic replay of the Block 9 conditioning-aware ancestor;\n\\item a strengthened global-affine parent using $80$ fit plus $16$ witness pairs;\n\\item a conservative two-chart child using $48$ fit, $32$ chart-selection, and $16$ witness pairs;\n\\item one matched $96$-query deployment envelope for both processes;\n\\item a finite separator search restricted to the source repair's protected two-dimensional context packet;\n\\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;\n\\item disjoint development, meta-validation, and primary audit task and family identifiers;\n\\item a $96$-target primary foreign audit and a $32$","Block 10 retains the Block 9 quality-weighted acquisition mechanism but expands the representational grammar of repair transport. It realizes:\n\\begin{enumerate}\n\\item exact semantic replay of the Block 9 conditioning-aware ancestor;\n\\item a strengthened global-affine parent using $80$ fit plus $16$ witness pairs;\n\\item a conservative two-chart child using $48$ fit, $32$ chart-selection, and $16$ witness pairs;\n\\item one matched $96$-query deployment envelope for both processes;\n\\item a finite separator search restricted to the source repair's protected two-dimensional context packet;\n\\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;\n\\item disjoint development, meta-validation, and primary audit task and family identifiers;\n\\item a $96$-target primary foreign audit and a $32$-target stress suite;\n\\item direct global-affine oracle, global","Block 10 retains the Block 9 quality-weighted acquisition mechanism but expands the representational grammar of repair transport. It realizes:\n\\begin{enumerate}\n\\item exact semantic replay of the Block 9 conditioning-aware ancestor;\n\\item a strengthened global-affine parent using $80$ fit plus $16$ witness pairs;\n\\item a conservative two-chart child using $48$ fit, $32$ chart-selection, and $16$ witness pairs;\n\\item one matched $96$-query deployment envelope for both processes;\n\\item a finite separator search restricted to the source repair's protected two-dimensional context packet;\n\\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;\n\\item disjoint development, meta-validation, and primary audit task and family identifiers;\n\\item a $96$-target primary foreign audit and a $32$-target stress suite;\n\\item direct global-affine oracle, global-blind, lo","Block 10 retains the Block 9 quality-weighted acquisition mechanism but expands the representational grammar of repair transport. It realizes:\n\\begin{enumerate}\n\\item exact semantic replay of the Block 9 conditioning-aware ancestor;\n\\item a strengthened global-affine parent using $80$ fit plus $16$ witness pairs;\n\\item a conservative two-chart child using $48$ fit, $32$ chart-selection, and $16$ witness pairs;\n\\item one matched $96$-query deployment envelope for both processes;\n\\item a finite separator search restricted to the source repair's protected two-dimensional context packet;\n\\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;\n\\item disjoint development, meta-validation, and primary audit task and family identifiers;\n\\item a $96$-target primary foreign audit and a $32$-target stress suite;\n\\item direct global-affine oracle, global-blind, local-blind, gate-only, rand","ity-weighted acquisition mechanism but expands the representational grammar of repair transport. It realizes:\n\\begin{enumerate}\n\\item exact semantic replay of the Block 9 conditioning-aware ancestor;\n\\item a strengthened global-affine parent using $80$ fit plus $16$ witness pairs;\n\\item a conservative two-chart child using $48$ fit, $32$ chart-selection, and $16$ witness pairs;\n\\item one matched $96$-query deployment envelope for both processes;\n\\item a finite separator search restricted to the source repair's protected two-dimensional context packet;\n\\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;\n\\item disjoint development, meta-validation, and primary audit task and family identifiers;\n\\item a $96$-target primary foreign audit and a $32$-target stress suite;\n\\item direct global-affine oracle, global-blind, local-blind, gate-only, random-gate, shuffled-binding, and true-…","k 9 conditioning-aware ancestor;\n\\item a strengthened global-affine parent using $80$ fit plus $16$ witness pairs;\n\\item a conservative two-chart child using $48$ fit, $32$ chart-selection, and $16$ witness pairs;\n\\item one matched $96$-query deployment envelope for both processes;\n\\item a finite separator search restricted to the source repair's protected two-dimensional context packet;\n\\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;\n\\item disjoint development, meta-validation, and primary audit task and family identifiers;\n\\item a $96$-target primary foreign audit and a $32$-target stress suite;\n\\item direct global-affine oracle, global-blind, local-blind, gate-only, random-gate, shuffled-binding, and true-chart controls;\n\\item exact ancestor, compiler, grammar-certificate, pre-audit seal, result, test, figure, and package-manifest receipts.\n\\end{enumerate}\nThe resource-…","ted two-dimensional context packet;\n\\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;\n\\item disjoint development, meta-validation, and primary audit task and family identifiers;\n\\item a $96$-target primary foreign audit and a $32$-target stress suite;\n\\item direct global-affine oracle, global-blind, local-blind, gate-only, random-gate, shuffled-binding, and true-chart controls;\n\\item exact ancestor, compiler, grammar-certificate, pre-audit seal, result, test, figure, and package-manifest receipts.\n\\end{enumerate}\nThe 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\n\\begin{equation}\n\\widehat\\Delta_{G\\to C}=0.334224,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.266459,0.401903].\n\\end{equation}\nThe sharper child-minus-direc…","m 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;\n\\item disjoint development, meta-validation, and primary audit task and family identifiers;\n\\item a $96$-target primary foreign audit and a $32$-target stress suite;\n\\item direct global-affine oracle, global-blind, local-blind, gate-only, random-gate, shuffled-binding, and true-chart controls;\n\\item exact ancestor, compiler, grammar-certificate, pre-audit seal, result, test, figure, and package-manifest receipts.\n\\end{enumerate}\nThe 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\n\\begin{equation}\n\\widehat\\Delta_{G\\to C}=0.334224,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.266459,0.401903].\n\\end{equation}\nThe sharper child-minus-direct-global-oracle effect is\n\\begin{equatio…","$-target stress suite;\n\\item direct global-affine oracle, global-blind, local-blind, gate-only, random-gate, shuffled-binding, and true-chart controls;\n\\item exact ancestor, compiler, grammar-certificate, pre-audit seal, result, test, figure, and package-manifest receipts.\n\\end{enumerate}\nThe 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\n\\begin{equation}\n\\widehat\\Delta_{G\\to C}=0.334224,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.266459,0.401903].\n\\end{equation}\nThe sharper child-minus-direct-global-oracle effect is\n\\begin{equation}\n\\widehat\\Delta_{\\mathrm{oracle}\\to C}=0.157530,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.080863,0.234135].\n\\end{equation}\nAll 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 me…","te;\n\\item direct global-affine oracle, global-blind, local-blind, gate-only, random-gate, shuffled-binding, and true-chart controls;\n\\item exact ancestor, compiler, grammar-certificate, pre-audit seal, result, test, figure, and package-manifest receipts.\n\\end{enumerate}\nThe 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\n\\begin{equation}\n\\widehat\\Delta_{G\\to C}=0.334224,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.266459,0.401903].\n\\end{equation}\nThe sharper child-minus-direct-global-oracle effect is\n\\begin{equation}\n\\widehat\\Delta_{\\mathrm{oracle}\\to C}=0.157530,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.080863,0.234135].\n\\end{equation}\nAll 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 nonze…","affine oracle, global-blind, local-blind, gate-only, random-gate, shuffled-binding, and true-chart controls;\n\\item exact ancestor, compiler, grammar-certificate, pre-audit seal, result, test, figure, and package-manifest receipts.\n\\end{enumerate}\nThe 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\n\\begin{equation}\n\\widehat\\Delta_{G\\to C}=0.334224,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.266459,0.401903].\n\\end{equation}\nThe sharper child-minus-direct-global-oracle effect is\n\\begin{equation}\n\\widehat\\Delta_{\\mathrm{oracle}\\to C}=0.157530,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.080863,0.234135].\n\\end{equation}\nAll 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, …","rue-chart controls;\n\\item exact ancestor, compiler, grammar-certificate, pre-audit seal, result, test, figure, and package-manifest receipts.\n\\end{enumerate}\nThe 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\n\\begin{equation}\n\\widehat\\Delta_{G\\to C}=0.334224,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.266459,0.401903].\n\\end{equation}\nThe sharper child-minus-direct-global-oracle effect is\n\\begin{equation}\n\\widehat\\Delta_{\\mathrm{oracle}\\to C}=0.157530,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.080863,0.234135].\n\\end{equation}\nAll 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 re…","ce-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\n\\begin{equation}\n\\widehat\\Delta_{G\\to C}=0.334224,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.266459,0.401903].\n\\end{equation}\nThe sharper child-minus-direct-global-oracle effect is\n\\begin{equation}\n\\widehat\\Delta_{\\mathrm{oracle}\\to C}=0.157530,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.080863,0.234135].\n\\end{equation}\nAll 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.","s. The child-parent effect is\n\\begin{equation}\n\\widehat\\Delta_{G\\to C}=0.334224,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.266459,0.401903].\n\\end{equation}\nThe sharper child-minus-direct-global-oracle effect is\n\\begin{equation}\n\\widehat\\Delta_{\\mathrm{oracle}\\to C}=0.157530,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.080863,0.234135].\n\\end{equation}\nAll 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.","on}\n\\widehat\\Delta_{G\\to C}=0.334224,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.266459,0.401903].\n\\end{equation}\nThe sharper child-minus-direct-global-oracle effect is\n\\begin{equation}\n\\widehat\\Delta_{\\mathrm{oracle}\\to C}=0.157530,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.080863,0.234135].\n\\end{equation}\nAll 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.","mathrm{CI}_{95\\%}=[0.266459,0.401903].\n\\end{equation}\nThe sharper child-minus-direct-global-oracle effect is\n\\begin{equation}\n\\widehat\\Delta_{\\mathrm{oracle}\\to C}=0.157530,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.080863,0.234135].\n\\end{equation}\nAll 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.","ermutation and within-block gauge;\n\\item response-fit, commutator, spectral-gap, commutant-dimension, and hidden-overlap audit diagnostics;\n\\item repair-relevant block labeling through source-repair invariance, target-gate agreement, local-bridge validation, chart balance, and protected regression;\n\\item a conservative executable embedding of the Block 10 supplied-packet grammar;\n\\item one equal $120$-query envelope for the inherited fixed-packet parent and endogenous-atlas child;\n\\item legacy-packet, PCA, full-space, random-atlas, shuffled-response, global, blind, and true-chart controls;\n\\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;\n\\item exact compiler, atlas-certificate, pre-audit seal, result, figure, test, and $49$-file manifest receipts.\n\\end{enumerate}\nThe 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…","ce, target-gate agreement, local-bridge validation, chart balance, and protected regression;\n\\item a conservative executable embedding of the Block 10 supplied-packet grammar;\n\\item one equal $120$-query envelope for the inherited fixed-packet parent and endogenous-atlas child;\n\\item legacy-packet, PCA, full-space, random-atlas, shuffled-response, global, blind, and true-chart controls;\n\\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;\n\\item exact compiler, atlas-certificate, pre-audit seal, result, figure, test, and $49$-file manifest receipts.\n\\end{enumerate}\nThe 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 es…","validation, chart balance, and protected regression;\n\\item a conservative executable embedding of the Block 10 supplied-packet grammar;\n\\item one equal $120$-query envelope for the inherited fixed-packet parent and endogenous-atlas child;\n\\item legacy-packet, PCA, full-space, random-atlas, shuffled-response, global, blind, and true-chart controls;\n\\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;\n\\item exact compiler, atlas-certificate, pre-audit seal, result, figure, test, and $49$-file manifest receipts.\n\\end{enumerate}\nThe 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 inventio…","tlas child;\n\\item legacy-packet, PCA, full-space, random-atlas, shuffled-response, global, blind, and true-chart controls;\n\\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;\n\\item exact compiler, atlas-certificate, pre-audit seal, result, figure, test, and $49$-file manifest receipts.\n\\end{enumerate}\nThe 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.","controls;\n\\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;\n\\item exact compiler, atlas-certificate, pre-audit seal, result, figure, test, and $49$-file manifest receipts.\n\\end{enumerate}\nThe 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.","it and a $25$-target stress suite containing response degeneracy, strong coupling, smooth transitions, repair-dependent chart crossing, and compression;\n\\item exact compiler, atlas-certificate, pre-audit seal, result, figure, test, and $49$-file manifest receipts.\n\\end{enumerate}\nThe 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.","t crossing, and compression;\n\\item exact compiler, atlas-certificate, pre-audit seal, result, figure, test, and $49$-file manifest receipts.\n\\end{enumerate}\nThe 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.","on;\n\\item exact compiler, atlas-certificate, pre-audit seal, result, figure, test, and $49$-file manifest receipts.\n\\end{enumerate}\nThe 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.","ous atlas recovery.\n\\paragraph{Response-language edge.}\nThe 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.","-language edge.}\nThe 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.","ns 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.","xact 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.","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.","\\paragraph{Repair-constructor edge.}\nAfter 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.","edge.}\nAfter 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.","e 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.","perad 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.","rection 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.","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;\n\\item a child compiler trained only from development receipts, with a nearest-centroid obstruction classifier and ridge outcome/risk forecasters;\n\\item an identical $39$-method candidate grammar and identical six-full-experiment budget for parent and child;\n\\item development, meta-validation, primary foreign, and architecture-foreign problem splits with pre-audit compiler sealing;\n\\item shuffled-archive, classifier-only, forecaster-only, no-forum, single-benchmark, no-monitor, random-allocation, and audit-oracle controls;\n\\item exact proposal, code, archive, compiler, allocation, evaluator, monitor, result, test, and manifest receipts.\n\\end{enumerate}\nOn $96$ 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-for…","development, meta-validation, primary foreign, and architecture-foreign problem splits with pre-audit compiler sealing;\n\\item shuffled-archive, classifier-only, forecaster-only, no-forum, single-benchmark, no-monitor, random-allocation, and audit-oracle controls;\n\\item exact proposal, code, archive, compiler, allocation, evaluator, monitor, result, test, and manifest receipts.\n\\end{enumerate}\nOn $96$ 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 inv…","splits with pre-audit compiler sealing;\n\\item shuffled-archive, classifier-only, forecaster-only, no-forum, single-benchmark, no-monitor, random-allocation, and audit-oracle controls;\n\\item exact proposal, code, archive, compiler, allocation, evaluator, monitor, result, test, and manifest receipts.\n\\end{enumerate}\nOn $96$ 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.","e-audit compiler sealing;\n\\item shuffled-archive, classifier-only, forecaster-only, no-forum, single-benchmark, no-monitor, random-allocation, and audit-oracle controls;\n\\item exact proposal, code, archive, compiler, allocation, evaluator, monitor, result, test, and manifest receipts.\n\\end{enumerate}\nOn $96$ 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.","shuffled-archive, classifier-only, forecaster-only, no-forum, single-benchmark, no-monitor, random-allocation, and audit-oracle controls;\n\\item exact proposal, code, archive, compiler, allocation, evaluator, monitor, result, test, and manifest receipts.\n\\end{enumerate}\nOn $96$ 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.","nly, forecaster-only, no-forum, single-benchmark, no-monitor, random-allocation, and audit-oracle controls;\n\\item exact proposal, code, archive, compiler, allocation, evaluator, monitor, result, test, and manifest receipts.\n\\end{enumerate}\nOn $96$ 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.","orum, single-benchmark, no-monitor, random-allocation, and audit-oracle controls;\n\\item exact proposal, code, archive, compiler, allocation, evaluator, monitor, result, test, and manifest receipts.\n\\end{enumerate}\nOn $96$ 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.","n, and audit-oracle controls;\n\\item exact proposal, code, archive, compiler, allocation, evaluator, monitor, result, test, and manifest receipts.\n\\end{enumerate}\nOn $96$ 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.","racle controls;\n\\item exact proposal, code, archive, compiler, allocation, evaluator, monitor, result, test, and manifest receipts.\n\\end{enumerate}\nOn $96$ 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.","posal, code, archive, compiler, allocation, evaluator, monitor, result, test, and manifest receipts.\n\\end{enumerate}\nOn $96$ 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.","mpiler, allocation, evaluator, monitor, result, test, and manifest receipts.\n\\end{enumerate}\nOn $96$ 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.","then recalibrates the inherited compiler design as the parent of an adversarial robustness study. It realizes:\n\\begin{enumerate}\n\\item diagnostic-matched clean and ambiguous problems that weaken one-shot obstruction classification;\n\\item sparse two-obstruction mixtures and an unknown class outside the inherited response cone;\n\\item evaluator-attack and integrity-attack method families;\n\\item an $18\\%$ contaminated development archive plus a contamination sweep through $35\\%$;\n\\item a robust child using sequential diagnosis, sparse response mixtures, open-set routing, robust archive summaries, evaluator portfolios, and an integrity-monitor ensemble;\n\\item the exact inherited $39$-method grammar and six-full-experiment budget for parent and child;\n\\item primary and independently shifted generator-foreign audits with pre-audit compiler sealing;\n\\item no-open-set, no-sequential, ordinary-archive, simple-monitor, single-portfolio, random-allocation, and audit-oracle controls;\n\\item exact results-free and post-result receipt chains, canonical results, tests, figures, and package manifest.\n…","arial robustness study. It realizes:\n\\begin{enumerate}\n\\item diagnostic-matched clean and ambiguous problems that weaken one-shot obstruction classification;\n\\item sparse two-obstruction mixtures and an unknown class outside the inherited response cone;\n\\item evaluator-attack and integrity-attack method families;\n\\item an $18\\%$ contaminated development archive plus a contamination sweep through $35\\%$;\n\\item a robust child using sequential diagnosis, sparse response mixtures, open-set routing, robust archive summaries, evaluator portfolios, and an integrity-monitor ensemble;\n\\item the exact inherited $39$-method grammar and six-full-experiment budget for parent and child;\n\\item primary and independently shifted generator-foreign audits with pre-audit compiler sealing;\n\\item no-open-set, no-sequential, ordinary-archive, simple-monitor, single-portfolio, random-allocation, and audit-oracle controls;\n\\item exact results-free and post-result receipt chains, canonical results, tests, figures, and package manifest.\n\\end{enumerate}\nOn $144$ primary problems, mean validity-filtered protecte…","class outside the inherited response cone;\n\\item evaluator-attack and integrity-attack method families;\n\\item an $18\\%$ contaminated development archive plus a contamination sweep through $35\\%$;\n\\item a robust child using sequential diagnosis, sparse response mixtures, open-set routing, robust archive summaries, evaluator portfolios, and an integrity-monitor ensemble;\n\\item the exact inherited $39$-method grammar and six-full-experiment budget for parent and child;\n\\item primary and independently shifted generator-foreign audits with pre-audit compiler sealing;\n\\item no-open-set, no-sequential, ordinary-archive, simple-monitor, single-portfolio, random-allocation, and audit-oracle controls;\n\\item exact results-free and post-result receipt chains, canonical results, tests, figures, and package manifest.\n\\end{enumerate}\nOn $144$ 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.…","riment budget for parent and child;\n\\item primary and independently shifted generator-foreign audits with pre-audit compiler sealing;\n\\item no-open-set, no-sequential, ordinary-archive, simple-monitor, single-portfolio, random-allocation, and audit-oracle controls;\n\\item exact results-free and post-result receipt chains, canonical results, tests, figures, and package manifest.\n\\end{enumerate}\nOn $144$ 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 neith…","ed generator-foreign audits with pre-audit compiler sealing;\n\\item no-open-set, no-sequential, ordinary-archive, simple-monitor, single-portfolio, random-allocation, and audit-oracle controls;\n\\item exact results-free and post-result receipt chains, canonical results, tests, figures, and package manifest.\n\\end{enumerate}\nOn $144$ 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.","oreign audits with pre-audit compiler sealing;\n\\item no-open-set, no-sequential, ordinary-archive, simple-monitor, single-portfolio, random-allocation, and audit-oracle controls;\n\\item exact results-free and post-result receipt chains, canonical results, tests, figures, and package manifest.\n\\end{enumerate}\nOn $144$ 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.","mpiler sealing;\n\\item no-open-set, no-sequential, ordinary-archive, simple-monitor, single-portfolio, random-allocation, and audit-oracle controls;\n\\item exact results-free and post-result receipt chains, canonical results, tests, figures, and package manifest.\n\\end{enumerate}\nOn $144$ 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.","et, no-sequential, ordinary-archive, simple-monitor, single-portfolio, random-allocation, and audit-oracle controls;\n\\item exact results-free and post-result receipt chains, canonical results, tests, figures, and package manifest.\n\\end{enumerate}\nOn $144$ 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.","y-archive, simple-monitor, single-portfolio, random-allocation, and audit-oracle controls;\n\\item exact results-free and post-result receipt chains, canonical results, tests, figures, and package manifest.\n\\end{enumerate}\nOn $144$ 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.","m-allocation, and audit-oracle controls;\n\\item exact results-free and post-result receipt chains, canonical results, tests, figures, and package manifest.\n\\end{enumerate}\nOn $144$ 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.","and audit-oracle controls;\n\\item exact results-free and post-result receipt chains, canonical results, tests, figures, and package manifest.\n\\end{enumerate}\nOn $144$ 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.","s;\n\\item exact results-free and post-result receipt chains, canonical results, tests, figures, and package manifest.\n\\end{enumerate}\nOn $144$ 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.","ree and post-result receipt chains, canonical results, tests, figures, and package manifest.\n\\end{enumerate}\nOn $144$ 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.",", figures, and package manifest.\n\\end{enumerate}\nOn $144$ 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.","package manifest.\n\\end{enumerate}\nOn $144$ 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.","roblems, 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.","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.","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.","$0.164970$ 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.","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:\n\\begin{enumerate}\n\\item six inherited obstruction tokens, one undifferentiated unknown token, and the inherited $39$-method singleton grammar;\n\\item a development open-set cell containing two hidden research mechanisms that share the inherited diagnostic representation but require incompatible high-gain methods;\n\\item a generic six-probe research-experiment language and a generic $30$-member ordered pair-method constructor grammar;\n\\item finite conflict scoring, anonymous token refinement, pair-constructor synthesis, and exact conservative fallback;\n\\item hidden novel labels excluded from compiler construction and retained only for audit interpretation;\n\\item identical eight-unit deployment research budgets for parent and child;\n\\item primary and generator-foreign audits sealed after the language certifi","y. It realizes:\n\\begin{enumerate}\n\\item six inherited obstruction tokens, one undifferentiated unknown token, and the inherited $39$-method singleton grammar;\n\\item a development open-set cell containing two hidden research mechanisms that share the inherited diagnostic representation but require incompatible high-gain methods;\n\\item a generic six-probe research-experiment language and a generic $30$-member ordered pair-method constructor grammar;\n\\item finite conflict scoring, anonymous token refinement, pair-constructor synthesis, and exact conservative fallback;\n\\item hidden novel labels excluded from compiler construction and retained only for audit interpretation;\n\\item identical eight-unit deployment research budgets for parent and child;\n\\item primary and generator-foreign audits sealed after the language certificate and compiler object;\n\\item taxonomy-only, grammar-only, shuffled-binding, inherited-oracle, and expanded-oracle controls;\n\\item complete results-free and post-result receipts, deterministic figures, tests, and a package manifest.\n\\end{enumerate}\nThe development co…","nding, inherited-oracle, and expanded-oracle controls;\n\\item complete results-free and post-result receipts, deterministic figures, tests, and a package manifest.\n\\end{enumerate}\nThe 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 $0.571704$.","e results-free and post-result receipts, deterministic figures, tests, and a package manifest.\n\\end{enumerate}\nThe 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 $0.571704$.","istic figures, tests, and a package manifest.\n\\end{enumerate}\nThe 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 $0.571704$.","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.","The realized lineage is\n\\begin{equation}\nK_0\\xrightarrow{\\Pi_1}K_1\\xrightarrow{\\Pi_2}K_2.\n\\end{equation}\nThe 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.","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.","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.","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.","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 $0.70$.","dren 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 $0.70$.","l 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 $0.70$.","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.","al receipt is frozen, a separate held-out split diagnosis refits only shift-local readouts and bridges. Source and target probes recover $R^2$ between $0.574$ and $0.601$, paired bridges recover $0.539$ to $0.555$, and shuffled bridges remain negative. Thus the next missing object is a context-indexed or invariant semantic atlas with abstention, not another static full-rank bridge. No transition $\\Pi_{18}$ is accepted in this revision.","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.","\\begin{conditionaltheorem}[Conditional SRIC theorem, Block 21 form]\n\\label{thm:umbrella}\nConsider a finite declared agent atlas and a sequence of lineage states $\\mathfrak L_t$. Assume:\n\\begin{enumerate}\n\\item externally grounded outcome traces and an append-only evidence ledger;\n\\item typed authorized correction spaces, defect maps, and local obstruction computation;\n\\item effective repair descent on the declared cover;\n\\item coherent repair transport with pathwise norm, authorization, and obstruction-faithfulness bounds;\n\\item a nontrivial adequate evaluator core and conservative evaluator morphisms;\n\\item witness-separated transactions with a decision-complete trace, sandboxed effects, precommitment, and rollback;\n\\item a finite transaction game with declared actions and admissible outcomes;\n\\item an append-only receipt-closed archive;\n\\item calibrated accepta","tability margin;\n\\item direct atomic self-approval is excluded relative to the declared trace, and rejected buffered transactions restore the sealed parent;\n\\item the universal recursive closure, robust controlled kernel, and productive nested fixed point exist and are computable in the finite model;\n\\item the archive's scalar upper envelopes are nondecreasing under append-only retention, and an $\\eps$-Pareto cover of bounded size exists;\n\\item only finitely many harmful acceptances occur almost surely under the summable error condition;\n\\item nested repair images reduce held-out obstruction energy by exactly the defect energy in newly correctable directions;\n\\item every finite old proof persists along the conservative grammar chain, and ontology expansion satisfies the old-resolution and new-obstruction exact sequence;\n\\item finite repair-discovery closure exists, while the branching threshold and spectral certificates hold under their additional stochastic assumptions;\n\\item every covered executable trace maps to an authorized abstract trace with matching protected observations, an…","rtificate} is a certified one-step bounded improvement of paired-anchor query design and bridge identification relative to the declared finite benchmark;\n\\item a single productive child edge does not imply recurrent productive closure without a fresh successful descendant edge;\n\\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;\n\\item under the Block 8 Lyapunov hypothesis, weighted module headroom contracts during fixed-grammar intervals while the target of improvement may migrate among modules;\n\\item nested grammar correction images reduce obstruction energy by exactly the component in newly correctable directions;\n\\item finite repair-conflict refinement terminates when the candidate feature family separates every minimal conflict;\n\\item a finite adjoint-closed intervention algebra has a canonical central causal atlas up to multiplicity gauge;\n\\item summable conditional edge budgets control the probability of accepting any false linea…","aired-anchor query design and bridge identification relative to the declared finite benchmark;\n\\item a single productive child edge does not imply recurrent productive closure without a fresh successful descendant edge;\n\\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;\n\\item under the Block 8 Lyapunov hypothesis, weighted module headroom contracts during fixed-grammar intervals while the target of improvement may migrate among modules;\n\\item nested grammar correction images reduce obstruction energy by exactly the component in newly correctable directions;\n\\item finite repair-conflict refinement terminates when the candidate feature family separates every minimal conflict;\n\\item a finite adjoint-closed intervention algebra has a canonical central causal atlas up to multiplicity gauge;\n\\item summable conditional edge budgets control the probability of accepting any false lineage improvement across an adaptive finite or countable archi…","nates when the candidate feature family separates every minimal conflict;\n\\item a finite adjoint-closed intervention algebra has a canonical central causal atlas up to multiplicity gauge;\n\\item summable conditional edge budgets control the probability of accepting any false lineage improvement across an adaptive finite or countable archive;\n\\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;\n\\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;\n\\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;\n\\item under the Block 10 chart assumptions, a positive global residual margin together with a nonzero chart repair contrast certifies …","hted 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;\n\\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;\n\\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;\n\\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;\n\\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.\n\\item under separating block-scalar response signatures, the hidden central projectors are polynom…","ents.\n\\item \\textbf{Closure-model mismatch.} Fixed-point certificates fail to predict actual invariant violations or omit admissible transitions that occur in execution.\n\\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.\n\\item \\textbf{Best-of-$k$ confound.} Apparent metaproductivity disappears when candidate count, compute, tools, and validation access are matched.\n\\item \\textbf{No foreign transfer.} Improvement lift is confined to descendants or systems used during mechanism design.\n\\item \\textbf{Structural irrelevance.} Repair rank grows while held-out obstruction energy and task loss do not improve.\n\\item \\textbf{No causal gain from stratification.} Packet-localized repairs do not outperform matched global-edit baselines.\n\\item \\textbf{Transport bound nonpredictive.} Measured transport, leakage, and holonomy errors do not predict target repair success beyond generic distribution-distance baselines.\n\\item \\textbf{Evaluator-core …","chart witness applied to a global repair reproduces the child gain, so no new transport constructor is necessary.\n\\item \\textbf{Nonconservative chart extension.} The descendant cannot reproduce the parent behavior when both local maps are tied or when its chart gate fails.\n\\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.\n\\item \\textbf{Feature-language leakage.} The true audit chart coordinate or threshold enters compiler selection or gate tuning before the pre-audit seal.\n\\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.\n\\item \\textbf{Response irrelevance.} Shuffling target--response correspondence leaves endogenous-atlas performance unchanged, indicating that the compiler is not using the declared interventions.\n\\item \\textbf{Signature-degeneracy overclaim.} The system reports separate functi…","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.","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.","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 th","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.","\\subsection{Controlled compiler benchmark}\nBlock 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.","The target is not merely an unpaired map with low marginal discrepancy.  The compiler must distinguish five epistemically different outputs:\n\\begin{equation}\n\\boxed{\n\\texttt{IDENTIFIED},\\quad\n\\texttt{IDENTIFIED\\_UP\\_TO\\_GAUGE},\\quad\n\\texttt{PARTIAL\\_TRANSPORT},\\quad\n\\texttt{UNDERIDENTIFIED},\\quad\n\\texttt{NO\\_GO}.\n}\n\\end{equation}","\\begin{definition}[Unpaired intervention-orbit transport instance]\nAn instance is a tuple\n\\begin{equation}\n\\mathfrak U\n=\n(H_s,H_t,\\mu_s,\\mu_t,\\mathcal U,\n\\{R_u^s\\}_{u\\in\\mathcal U},\n\\{R_v^t\\}_{v\\in\\mathcal V},\n\\mathcal G_{\\rm gauge},\\mathsf W),\n\\end{equation}\nwhere $\\mu_s$ 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.\n\\end{definition}","{definition}[Unpaired intervention-orbit transport instance]\nAn instance is a tuple\n\\begin{equation}\n\\mathfrak U\n=\n(H_s,H_t,\\mu_s,\\mu_t,\\mathcal U,\n\\{R_u^s\\}_{u\\in\\mathcal U},\n\\{R_v^t\\}_{v\\in\\mathcal V},\n\\mathcal G_{\\rm gauge},\\mathsf W),\n\\end{equation}\nwhere $\\mu_s$ 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.\n\\end{definition}","tance is a tuple\n\\begin{equation}\n\\mathfrak U\n=\n(H_s,H_t,\\mu_s,\\mu_t,\\mathcal U,\n\\{R_u^s\\}_{u\\in\\mathcal U},\n\\{R_v^t\\}_{v\\in\\mathcal V},\n\\mathcal G_{\\rm gauge},\\mathsf W),\n\\end{equation}\nwhere $\\mu_s$ 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.\n\\end{definition}","{u\\in\\mathcal U},\n\\{R_v^t\\}_{v\\in\\mathcal V},\n\\mathcal G_{\\rm gauge},\\mathsf W),\n\\end{equation}\nwhere $\\mu_s$ 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.\n\\end{definition}","l U},\n\\{R_v^t\\}_{v\\in\\mathcal V},\n\\mathcal G_{\\rm gauge},\\mathsf W),\n\\end{equation}\nwhere $\\mu_s$ 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.\n\\end{definition}","For a proposed intervention correspondence $\\sigma:\\mathcal U\\to\\mathcal V$ and transport $T:H_t\\to H_s$, define the normalized orbit defect\n\\begin{equation}\n\\delta_{\\rm orb}(T,\\sigma)\n=\n\\frac1{|\\mathcal U|}\n\\sum_{u\\in\\mathcal U}\n\\frac{\\norm{TR^t_{\\sigma(u)}-R_u^sT}_F}\n{\\max\\{\\norm{R_u^s}_F,\\eps\\}}.\n\\label{eq:b22-orbit-defect}\n\\end{equation}","\\begin{definition}[Response algebra and commutant]\nThe response algebra and its commutant are\n\\begin{equation}\n\\mathfrak A_t=\\operatorname{alg}(I,R_v^t:v\\in\\mathcal V),\n\\qquad\n\\mathfrak A_t'=\\{S\\in\\End(H_t):SR_v^t=R_v^tS\\ \\forall v\\}.\n\\end{equation}\n\\end{definition}","\\begin{theorem}[Intertwiner uniqueness modulo the response commutant]\n\\label{thm:b22-commutant}\nLet $T_1,T_2:H_t\\to H_s$ be invertible and satisfy\n\\begin{equation}\nT_iR_v^t=R_{\\sigma^{-1}(v)}^sT_i\n\\end{equation}\nfor 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.\n\\end{theorem}","\\begin{proof}\nFor every $v$,\n\\begin{align}\nSR_v^t\n&=T_2^{-1}T_1R_v^t\n=T_2^{-1}R_{\\sigma^{-1}(v)}^sT_1","\\begin{proof}\nFor every $v$,\n\\begin{align}\nSR_v^t\n&=T_2^{-1}T_1R_v^t\n=T_2^{-1}R_{\\sigma^{-1}(v)}^sT_1\\\\\n&=R_v^tT_2^{-1}T_1\n=R_v^tS.\n\\end{align}\nThus $S\\in\\mathfrak A_t'$.\n\\end{proof}","&=R_v^tT_2^{-1}T_1\n=R_v^tS.\n\\end{align}\nThus $S\\in\\mathfrak A_t'$.\n\\end{proof}","\\begin{corollary}[Declared gauge boundary]\nIf 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.\n\\end{corollary}","The executable fixture uses five commuting symmetric response operators.  In latent semantic coordinates they are diagonal:\n\\begin{equation}\nD_u=\\operatorname{diag}(\\lambda_{u1},\\ldots,\\lambda_{ud}),\n\\end{equation}\nand each system exposes an orthogonal gauge $Q_a$:\n\\begin{equation}\nR_u^a=Q_aD_uQ_a^\\top.\n\\end{equation}\nThe joint response signature of semantic coordinate $j$ is\n\\begin{equation}\n\\lambda_j=(\\lambda_{1j},\\ldots,\\lambda_{mj}).\n\\end{equation}","}[Proof sketch]\nA 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.\n\\end{proof}","\\begin{theorem}[Observable symmetry underidentification]\nSuppose $S\\notin\\mathcal G_{\\rm gauge}$ 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.\n\\end{theorem}","\\begin{proof}\nThe 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.\n\\end{proof}","\\begin{proposition}[Separated context admission]\nIf 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.\n\\end{proposition}","\\begin{theorem}[Target-only collision no-go]\nIf 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.\n\\end{theorem}","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.","\\midrule\nShared labels & 6 & IDENTIFIED & semantic $R^2=0.995984$","Anonymous labels & 6 & IDENTIFIED & semantic $R^2=0.990952$","Gauge-equivalent & 6 & UP TO GAUGE & invariant $R^2=0.996352$","Semantic ambiguity & 6 & UNDERIDENTIFIED & disagreement $=1.905405$","Partial support & 6 & PARTIAL & coverage $=0.700000$","Destructive loss & 6 & NO GO & collisions $\\geq 96$","Across the two fully identifiable regimes, the minimum hidden semantic score is\n\\begin{equation}\nR^2_{\\rm sem}\\geq 0.990952,\n\\end{equation}\nand the largest admitted orbit defect is\n\\begin{equation}\n\\delta_{\\rm orb}\\leq 0.002870.\n\\end{equation}\nThe mean second-order distribution baseline is $-0.900494$, and cyclically shuffled coordinate binding yields mean $R^2=-1.009877$.","In the benign degenerate regime, an alternative transport inside the unresolved two-dimensional block retains protected semantic\n\\begin{equation}\nR^2\\geq 0.996352,\n\\end{equation}\nwhile its mean raw-coordinate score is only $0.656730$.  The correct claim is equivalence up to gauge, not unique coordinates.","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}.","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 $R^2=-1.096314$.","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}.","\\begin{statusbox}{Block 22 result}\nAll $24$ 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.\n\\end{statusbox}","Block 23 separates two questions that must not be collapsed.  The first is theoretical and finite:\n\\begin{equation}\n\\boxed{\n\\begin{gathered}\n\\text{Can a lineage construct a minimal semantic quotient from}\\\\\n\\text{future consequences, interventions, and semantic twins}\\\\\n\\text{without receiving gold intermediate semantic coordinates?}\n\\end{gathered}}\n\\label{eq:b23-central-question}\n\\end{equation}\nThe second is empirical: can the same contract be applied to a third-party latent-reasoning checkpoint?  The released package answers the first question positively inside a controlled finite fixture and supplies a source-locked adapter for the second.  The external checkpoint was not present in the release environment and was not executed.  The two evidential statuses are recorded separately.","Let $H$ be a measurable latent-state space, let $\\Ucal$ be a declared finite intervention language, and let $Y^{u}_{0:r}$ denote the protected future observations, actions, scores, or externally checkable consequences generated after intervention word $u\\in\\Ucal^{\\le r}$.  The semantic object sought in this block is not a linguistic paraphrase of every coordinate.  It is the quotient induced by the distinctions that matter for permitted future experiments.","\\begin{definition}[Causal-predictive equivalence]\nFor $h,h'\\in H$, write\n\\begin{equation}\nh\\sim_{\\Ucal,r}h'\n\\end{equation}\nwhen\n\\begin{equation}\n\\mathcal L\\!\\left(Y^{u}_{0:r}\\mid h\\right)\n=\n\\mathcal L\\!\\left(Y^{u}_{0:r}\\mid h'\\right)\n\\qquad\n\\text{for every }u\\in\\Ucal^{\\le r}.\n\\label{eq:b23-causal-equivalence}\n\\end{equation}\nThe quotient\n\\begin{equation}\nZ_{\\Ucal,r}=H/{\\sim_{\\Ucal,r}}\n\\label{eq:b23-semantic-quotient}\n\\end{equation}\nis the finite-horizon causal-predictive semantic space relative to the declared experiment language.\n\\end{definition}","The definition is intervention relative.  Enlarging $\\Ucal$ or the horizon can refine the quotient.  A coordinate omitted by every permitted experiment is not identified merely because it exists in an audit-only simulator.","\\begin{proposition}[Causal-predictive sufficiency]\nLet $q:H\\to Z_{\\Ucal,r}$ be the quotient map.  For every $u\\in\\Ucal^{\\le r}$, there exists a Markov kernel $K_u$ such that\n\\begin{equation}\n\\mathcal L\\!\\left(Y^{u}_{0:r}\\mid h\\right)=K_u(q(h)).\n\\end{equation}\nThus $q(h)$ is sufficient for every protected consequence appearing in the equivalence relation.\n\\end{proposition}","\\begin{theorem}[Minimality of the exact semantic quotient]\nSuppose $s:H\\to S$ is sufficient for every protected law in \\cref{eq:b23-causal-equivalence}.  Then there exists a map $\\phi:S\\to Z_{\\Ucal,r}$ on $s(H)$ satisfying\n\\begin{equation}\nq=\\phi\\circ s.\n\\label{eq:b23-minimal-factorization}\n\\end{equation}\nConsequently, any exact sufficient representation refines the causal-predictive quotient.\n\\end{theorem}","\\begin{proof}\nIf $s(h)=s(h')$, sufficiency gives identical protected conditional laws for every declared intervention word.  Hence $h\\sim_{\\Ucal,r}h'$ and $q(h)=q(h')$.  Define $\\phi(s(h))=q(h)$.  The preceding implication makes $\\phi$ well defined.\n\\end{proof}","\\begin{definition}[Semantic twin]\nTwo observed trajectories $x$ and $x'$ are semantic twins relative to $\\Ucal$ when they arise from the same protected task and have the same protected future and intervention laws, while nuisance coordinates, rendering variables, or architecture-specific gauges may differ.\n\\end{definition}","For a candidate quotient $q_\\theta$, the empirical twin discrepancy is\n\\begin{equation}\nD_{\\mathrm{twin}}(q_\\theta)\n=\n\\mathbb E\n\\frac{\\norm{q_\\theta(x)-q_\\theta(x')}}\n{\\max\\{\\norm{q_\\theta(x)},\\eps\\}}.\n\\label{eq:b23-twin-drift}\n\\end{equation}\nA candidate is construction-valid only when it jointly predicts future consequences and intervention effects, predicts the externally scored answer, and keeps \\cref{eq:b23-twin-drift} below the declared ceiling.","\\begin{definition}[Witness-separated ontology transaction]\nAn ontology transaction consists of construction evidence $E_{\\mathrm{con}}$, a compiled quotient $q$, a pre-reveal commitment $\\sigma(q)$, and witness evidence $E_{\\mathrm{wit}}$.  It is witness separated when $E_{\\mathrm{wit}}$ is inaccessible to quotient construction and candidate selection, and the final certificate is computed only after $\\sigma(q)$ is fixed.\n\\end{definition}","n{definition}[Witness-separated ontology transaction]\nAn ontology transaction consists of construction evidence $E_{\\mathrm{con}}$, a compiled quotient $q$, a pre-reveal commitment $\\sigma(q)$, and witness evidence $E_{\\mathrm{wit}}$.  It is witness separated when $E_{\\mathrm{wit}}$ is inaccessible to quotient construction and candidate selection, and the final certificate is computed only after $\\sigma(q)$ is fixed.\n\\end{definition}","\\begin{proposition}[Circular witness limitation]\nIf the same target coordinates determine both $q$ and its reported semantic score, arbitrarily high in-sample semantic decodability does not imply out-of-construction semantic validity.\n\\end{proposition}","\\begin{definition}[Protected gauge block]\nA subspace $G\\subseteq Z$ is a protected gauge block when every permitted future and intervention law is invariant under a declared group $\\mathcal G\\leq\\Aut(G)$.  The identified object is then the orbit $[z]_\\mathcal G$, not a preferred basis inside $G$.\n\\end{definition}","\\begin{theorem}[Semantic underidentification]\nSuppose two semantic maps $q_1,q_2:H\\to Z$ induce exactly the same compiler-visible future and intervention laws, but there exists a protected witness $W$ such that\n\\begin{equation}\nW(q_1(h))\\neq W(q_2(h))\n\\end{equation}\nwith positive probability.  No compiler measurable with respect to the visible evidence can validly choose between $q_1$ and $q_2$ with probability exceeding the larger prior probability of the two alternatives.\n\\end{theorem}","\\begin{definition}[Partial ontology]\nLet $P\\subseteq Z$ denote the protected semantic coordinates.  A quotient identifies a partial ontology when it is sufficient for a declared subset $P_0\\subsetneq P$, returns calibrated ignorance for $P\\setminus P_0$, and records coverage\n\\begin{equation}\n\\operatorname{cov}(q)=\\frac{\\dim P_0}{\\dim P}.\n\\end{equation}\n\\end{definition}","\\begin{theorem}[Deterministic semantic collision no-go]\nLet $y:H\\to Y$ be the representation exposed to a target-only compiler and let $z:H\\to Z$ be a protected semantic variable.  If there exist $h_1,h_2$ such that\n\\begin{equation}\ny(h_1)=y(h_2),\n\\qquad\nz(h_1)\\neq z(h_2),\n\\label{eq:b23-collision}\n\\end{equation}\nthen no deterministic map $f:Y\\to Z$ recovers $z$ on both states.\n\\end{theorem}","\\begin{proof}\nThe two equal inputs force $f(y(h_1))=f(y(h_2))$, contradicting the unequal protected targets in \\cref{eq:b23-collision}.\n\\end{proof}","Each episode contains two independently gauged recurrent systems.  A trajectory has five latent steps, four task-relevant semantic coordinates, and twelve nuisance coordinates per step.  The observed flattened trajectory therefore lies in $\\R^{80}$.  The semantic dynamics, gauges, nuisance processes, and hidden audit coordinates are not passed to the quotient compiler.","The finite grammar is\n\\begin{equation}\n\\Gcal_{23}\n=\n\\left\\{\n\\begin{array}{l}\n\\text{answer-only},\\ \\text{PCA},\\ \\text{predictive},\\ \\\\\n\\text{intervention-response},\\ \\text{joint minimal},\\\\\n\\text{full identity},\\ \\text{equal-rank random}\n\\end{array}\n\\right\\}.\n\\label{eq:b23-grammar}\n\\end{equation}\nFor each nontrivial constructor, ranks one through six are considered.  The full identity is an oversized control.  A projection $W$ is valid only when, on both recurrent systems,\n\\begin{equation}\n\\begin{aligned}\nR^2_{\\mathrm{future}}&\\geq0.82,","\\text{full identity},\\ \\text{equal-rank random}\n\\end{array}\n\\right\\}.\n\\label{eq:b23-grammar}\n\\end{equation}\nFor each nontrivial constructor, ranks one through six are considered.  The full identity is an oversized control.  A projection $W$ is valid only when, on both recurrent systems,\n\\begin{equation}\n\\begin{aligned}\nR^2_{\\mathrm{future}}&\\geq0.82,","\\text{full identity},\\ \\text{equal-rank random}\n\\end{array}\n\\right\\}.\n\\label{eq:b23-grammar}\n\\end{equation}\nFor each nontrivial constructor, ranks one through six are considered.  The full identity is an oversized control.  A projection $W$ is valid only when, on both recurrent systems,\n\\begin{equation}\n\\begin{aligned}\nR^2_{\\mathrm{future}}&\\geq0.82,\\\\\nR^2_{\\mathrm{intervention}}&\\geq0.90,\\\\\n\\operatorname{Acc}_{\\mathrm{answer}}&\\geq0.78,\\\\\nD_{\\mathrm{twin}}&\\leq0.22.\n\\end{aligned}\n\\label{eq:b23-visible-gates}\n\\end{equation}\nAmong valid candidates, the compiler minimizes rank and then follows the frozen method priority.  No hidden semantic score enters the choice.","In the answer-shortcut episodes, the rank-one answer projection achieved held-out answer accuracy at least $0.881250$.  Nevertheless, its best future-consequence and intervention-effect scores were only\n\\begin{equation}\nR^2_{\\mathrm{future}}\\leq0.313061,\n\\qquad\nR^2_{\\mathrm{intervention}}\\leq0.357401.\n\\end{equation}\nIt failed every final construction decision.  This separates answer extraction from semantic-state recovery.","Across uniquely identified regimes, hidden semantic reconstruction is numerically exact in the released fixture.  In the gauge regime, minimum gauge-invariant $R^2$ is $0.995522$, while the minimum recovered semantic-subspace overlap is $0.991563$.  In the semantic-ambiguity regime, the minimum protected disagreement between two observationally equivalent orientations is $0.781682$.  Although either orientation is post-hoc decodable, the correct disposition remains underidentified.","The partial regime identifies three of four protected directions, giving coverage $0.75$.  The fourth direction is not assigned a fabricated value.  Every destructive episode contains $64$ exact collision pairs; the minimum disagreement in externally protected future consequences is $1.736704$.  The collision witness never receives a hidden intermediate semantic coordinate.","After quotient commitment, a donor quotient replaces the selected packet in a new recurrent state.  The protected future and intervention consequences are recomputed through an audit-only runtime decoder rather than the probe used during construction.  The minimum selected-packet donor future $R^2$ is $0.983162$.  The maximum corresponding values for answer-only, PCA, and random controls are $-0.161100$, $-0.706230$, and $-0.703696$.","tient commitment, a donor quotient replaces the selected packet in a new recurrent state.  The protected future and intervention consequences are recomputed through an audit-only runtime decoder rather than the probe used during construction.  The minimum selected-packet donor future $R^2$ is $0.983162$.  The maximum corresponding values for answer-only, PCA, and random controls are $-0.161100$, $-0.706230$, and $-0.703696$.","a donor quotient replaces the selected packet in a new recurrent state.  The protected future and intervention consequences are recomputed through an audit-only runtime decoder rather than the probe used during construction.  The minimum selected-packet donor future $R^2$ is $0.983162$.  The maximum corresponding values for answer-only, PCA, and random controls are $-0.161100$, $-0.706230$, and $-0.703696$.","A deterministic fake runtime tests protocol plumbing but is excluded from the external empirical claim.  Across $72$ fixture tasks it yields exact replay error zero, mean zero-trace cost $0.533811$, permutation cost $0.038959$, norm-matched-noise cost $0.459498$, same-answer different-program transplant cost $0.311093$, and semantic-twin transplant cost zero.","The twenty-one-block draft develops recursive self-improvement as an evidence-preserving lineage that accumulates a progressively stronger calculus of valid change:\n\\begin{equation}\n\\boxed{\n\\begin{gathered}\n\\text{evaluation and introspection}\n\\longrightarrow\n\\text{correctability and obstruction}\\\\\n\\text{repair descent and transport}\n\\longrightarrow\n\\text{evaluator-preserving transactions}\\\\\n\\text{recursive closure and archives}\n\\longrightarrow\n\\text{executable realization}\\\\\n\\text{learned foreign repair transfer}\n\\longrightarrow\n\\text{sealed one-step process improvement}\\\\\n\\text{meta-headroom and bottleneck migration}\n\\longrightarrow\n\\text{two-edge repeated lineage}\\\\\n\\longrightarrow\n\\text{obstruction-certified structural grammar expansion}\\\\\n\\longrightarrow\n\\text{endogenous intervention-response atlas growth}\\\\\n\\longrightarrow\n\\text{response-language and constructor growth}\\\\\n\\longrightarrow\n\\text{proof-carrying research-compiler metaproductivity}\\\\\n\\longrightarrow\n\\text{adversarial open-set compiler robustness}\\\\\n\\longrightarrow\n\\text{compiler-language growth from open-set research…","etaproductivity 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.","y 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.","t 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.","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.","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.","Across $96$ fresh primary foreign targets, the chart child obtains mean selected gain\n\\begin{equation}\n0.334224\n\\end{equation}\nwith zero harmful selected targets. The child-parent effect is\n\\begin{equation}\n\\widehat\\Delta_{G\\to C}=0.334224,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.266459,0.401903].\n\\end{equation}\nThe directly supervised global-affine oracle reaches mean gain $0.176694$. The child exceeds it by\n\\begin{equation}\n0.157530,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.080863,0.234135].\n\\end{equation}\nMean 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 sometime","Across $96$ fresh primary foreign targets, the chart child obtains mean selected gain\n\\begin{equation}\n0.334224\n\\end{equation}\nwith zero harmful selected targets. The child-parent effect is\n\\begin{equation}\n\\widehat\\Delta_{G\\to C}=0.334224,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.266459,0.401903].\n\\end{equation}\nThe directly supervised global-affine oracle reaches mean gain $0.176694$. The child exceeds it by\n\\begin{equation}\n0.157530,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.080863,0.234135].\n\\end{equation}\nMean 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 construct","Across $96$ fresh primary foreign targets, the chart child obtains mean selected gain\n\\begin{equation}\n0.334224\n\\end{equation}\nwith zero harmful selected targets. The child-parent effect is\n\\begin{equation}\n\\widehat\\Delta_{G\\to C}=0.334224,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.266459,0.401903].\n\\end{equation}\nThe directly supervised global-affine oracle reaches mean gain $0.176694$. The child exceeds it by\n\\begin{equation}\n0.157530,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.080863,0.234135].\n\\end{equation}\nMean 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.","primary foreign targets, the chart child obtains mean selected gain\n\\begin{equation}\n0.334224\n\\end{equation}\nwith zero harmful selected targets. The child-parent effect is\n\\begin{equation}\n\\widehat\\Delta_{G\\to C}=0.334224,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.266459,0.401903].\n\\end{equation}\nThe directly supervised global-affine oracle reaches mean gain $0.176694$. The child exceeds it by\n\\begin{equation}\n0.157530,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.080863,0.234135].\n\\end{equation}\nMean 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.","}\nwith zero harmful selected targets. The child-parent effect is\n\\begin{equation}\n\\widehat\\Delta_{G\\to C}=0.334224,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.266459,0.401903].\n\\end{equation}\nThe directly supervised global-affine oracle reaches mean gain $0.176694$. The child exceeds it by\n\\begin{equation}\n0.157530,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.080863,0.234135].\n\\end{equation}\nMean 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.","rmful selected targets. The child-parent effect is\n\\begin{equation}\n\\widehat\\Delta_{G\\to C}=0.334224,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.266459,0.401903].\n\\end{equation}\nThe directly supervised global-affine oracle reaches mean gain $0.176694$. The child exceeds it by\n\\begin{equation}\n0.157530,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.080863,0.234135].\n\\end{equation}\nMean 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.","The realized lineage is\n\\begin{equation}\nK_0\\xrightarrow{\\Pi_1}K_1\\xrightarrow{\\Pi_2}K_2\\xrightarrow{\\Pi_3}K_3\\xrightarrow{\\Pi_4}K_4.\n\\end{equation}\nIts 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.","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.","reshold. 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.","ed 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.","ias, 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.","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.","n-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.","ity. 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.","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.","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.","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.","nt. 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.","ed 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.","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.","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.","The empirical ladder is now\n\\begin{equation}\n\\begin{gathered}\n\\text{repair mechanics}\n<\n\\text{causal foreign transfer}\\\\\n<\n\\text{one-step process metaproductivity}\n<\n\\text{two-edge bottleneck migration}\\\\\n<\n\\text{controlled structural metaproductivity}\n<\n\\text{endogenous atlas growth}\\\\\n<\n\\text{finite response and constructor growth}\\\\\n<\n\\text{finite research-compiler metaproductivity}\\\\\n<\n\\text{finite adversarial compiler robustness}\\\\\n<\n\\text{direct latent-language causal certification}\\\\\n<\n\\text{training-generated executable latent repair}\\\\\n<\n\\text{distributional latent-chart robustness}\\\\\n<\n\\text{atlas-indexed foreign-model latent transport}\\\\\n<\n\\text{type-directed transport-grammar growth}\\\\\n<\n\\text{autonomous open-set grammar growth}\n<\n\\text{open-set RSI}.\n\\end{gathered}\n\\end{equation}\nBlock 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 chronolo…","Block 12 extends the realized lineage to\n\\begin{equation}\nK_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.\n\\end{equation}\nThe 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.","Block 13 answers the finite allocation form of that question. The lineage adds\n\\begin{equation}\n\\Gamma_0\\xrightarrow{\\Pi_7^{\\Gamma}}\\Gamma_1,\n\\end{equation}\nwhere 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.","Block 14 adds one further compiler edge,\n\\begin{equation}\n\\Gamma_1\\xrightarrow{\\Pi_8^{\\Gamma}}\\Gamma_2,\n\\end{equation}\nwhere 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.","Block 15 adds the first finite compiler-language edge,\n\\begin{equation}\n\\Gamma_2\\xrightarrow{\\Pi_9^{\\Gamma}}\\Gamma_3.\n\\end{equation}\nThe 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.","ma}}\\Gamma_3.\n\\end{equation}\nThe 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.","\\end{equation}\nThe 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.","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.","ted 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.","Block 16 adds the directly auditable latent-language state\n\\begin{equation}\n\\mathfrak L_t^{\\mathrm{NL}}\n=\n(\\mathfrak L_t,\\Ncal_t,\\Ical_t^{\\mathrm{NL}},\\Tau_t^{\\mathrm{NL}},\\Wcal_t^{\\mathrm{NL}}).\n\\end{equation}\nBlock 17 realizes one trained finite transition\n\\begin{equation}\n\\mathfrak L_{16}^{\\mathrm{NL}}\n\\xrightarrow{\\Pi_{17}}\n\\mathfrak L_{17}^{\\mathrm{NL}},\n\\end{equation}\nwhere 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.","Block 18 attempts but does not accept\n\\begin{equation}\n\\mathfrak L_{17}^{\\mathrm{NL}}\n\\xrightarrow{\\Pi_{18}?}\n\\mathfrak L_{18}^{\\mathrm{NL}}.\n\\end{equation}\nThe 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.","Block 19 accepts the source-side continuation\n\\begin{equation}\n\\mathfrak L_{18}^{\\mathrm{NL}}\n\\xrightarrow{\\Pi_{19}}\n\\mathfrak L_{19}^{\\mathrm{NL}},\n\\end{equation}\nwhere 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.","Block 20 accepts the finite transport continuation\n\\begin{equation}\n\\mathfrak L_{19}^{\\mathrm{NL}}\n\\xrightarrow{\\Pi_{20}}\n\\mathfrak L_{20}^{\\mathrm{NL}},\n\\end{equation}\nwhere 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.","Block 21 accepts the finite grammar continuation\n\\begin{equation}\n\\mathfrak L_{20}^{\\mathrm{NL}}\n\\xrightarrow{\\Pi_{21}}\n\\mathfrak L_{21}^{\\mathrm{NL}},\n\\end{equation}\nwhere 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.","\\midrule\n\\endhead\n$\\Xi$ & Protected evaluation context","$J_\\Xi(A)$ & Resource-normalized utility of agent $A$ in context $\\Xi$","$\\mathfrak A_t$ & Stratified functional atlas at generation $t$","$G_t$ & Directed multigraph of typed cross-stratum bridges","$\\Acal_t$ & Executable ancestor-closed archive inside the lineage graph","$K_A^\\delta$ & Protected external-outcome kernel under intervention $\\delta$","$\\widehat K_{\\phi(A)}^\\delta$ & Outcome kernel predicted by the compressed self-model","$\\Hcal$ & Causal-hypothesis family for active diagnosis","$\\Pcal(T)$ & Preservation profile of a typed bridge $T$","$\\Pcal_{\\mathrm{probe}}$ & Permitted active-probe family","$\\overline\\Ocal$ & Underlying syntactic colored repair operad","$\\mathfrak a$ & Repair abstraction operator from episodes to schemas","$\\mathfrak L_t$ & Complete resolved lineage state at generation $t$","$A_t$ & Active stratified agent","$\\Ocal_t$ & Proof-carrying repair operad","$\\Ecal_t$ & Append-only external-evidence ledger","$\\Vcal_t$ & Evaluator atlas","$\\Lambda_t$ & Proof-carrying lineage directed acyclic graph","$\\Wcal_t$ & Witness, checkpoint, and rollback structure","$\\mathsf C$ & Set of stratum colors or types","$\\Dcal_t$ & Authorized modification family","$\\phi_t(A_t)$ & Compressed causal self-model","$U_t^{\\Auth}$ & Authorized correction space","$V_t$ & Defect space","$D_t$ & Linearized defect-response map","$[r]$ & Obstruction class in $\\coker D$","$D^\\dagger$ & Moore--Penrose pseudoinverse of the correction map","$P_{\\mathrm{cor}},P_{\\mathrm{ob}}$ & Orthogonal correctable and obstruction projectors","$\\eta_D(r)$ & Quantitative obstruction $\\dist(r,\\im D)$","$\\eta_{D,K}(r)$ & Minimum residual under convex authorization set $K$","$\\sigma_+(D)$ & Smallest positive singular value of $D$","$\\rho_t$ & Authorized repair rank $\\rank D_t$","$o_t$ & Unresolved obstruction dimension $\\dim\\coker D_t$","$D_0,E,Q_0$ & Packet-diagonal response, cross-packet coupling, and local right inverse","$\\mathscr U,\\mathscr V$ & Sheaves of authorized corrections and defects","$d_{\\alpha\\beta}$ & Difference of local repairs on an overlap","$\\eta_{\\mathrm{loc}},\\eta_{\\mathrm{glue}}$ & Quantitative local and gluing residuals","$B,L_G,\\lambda_2(L_G)$ & Nerve incidence matrix, graph Laplacian, and spectral gap","$P_{\\mathrm{cyc}}$ & Orthogonal projection onto the overlap cycle space","$g_{\\alpha\\beta}$ & Nonabelian transition function between local nonlinear repairs","$H_e$ & Fundamental-cycle holonomy associated with a non-tree edge","$b_n$ & Order-$n$ residual coefficient in the formal obstruction tower","$T_U,T_V$ & Correction-space and defect-space transport maps","$\\eps_D$ & Correction-diagram commutation defect","$\\eps_r$ & Defect-transport mismatch","$\\eps_J$ & Evaluator alignment error","$\\eps_{\\mathrm{leak}}$ & Packet leakage error","$\\eps_{\\mathrm{hol}}$ & Semantic holonomy error","$\\Delta_D,\\Delta_r$ & Correction-square defect and defect-identification error of a transport datum","$Q_A$ & Protected obstruction subspace required to survive transport","$H_\\gamma^U,H_\\gamma^V$ & Correction and defect holonomies around a transport loop","$C_{\\mathrm{tr}}(p;u)$ & Certified transport-route cost for correction $u$","$\\mathsf E_v$ & Evaluator object at version $v$","$R_v,E_v,\\iota_v,J_v$ & Raw evidence, internal evidence space, encoder, and readout of evaluator $v$","$\\mathsf A$ & Positive semidefinite evaluator holonomy-energy operator","$E_\\tau^{\\flat}$ & Spectral approximate path-flat core at threshold $\\tau$","$\\mu(F;\\Delta_{\\mathrm{prot}})$ & Adequacy modulus of core $F$ on protected evidence differences","$E^{\\flat}$ & Maximal path-flat evidence core","$E,W$ & Editable and witness component masks of a transaction","$q,a,Z,\\Pi$ & Trace map, sealed acceptance rule, revealed trace, and receipt","$\\Ical_{\\forall}^{*}$ & Universal recursive-improvement safety closure","$\\Ical_{\\mathrm{ctrl}}^{*}$ & Robust controlled improvement kernel","$\\Ical_{\\gamma}^{\\omega}$ & Productive closure with repeated gain threshold $\\gamma$","$\\mathsf{Act}(s),\\mathsf{Dist}(s,u)$ & Authorized actions and admissible disturbances","$H_w(\\Acal_t)$ & Archive upper envelope in scalarization direction $w$","$\\sim_h$ & Horizon-$h$ continuation equivalence of archive nodes","$M^{\\mathrm{foreign}}_{h,B}$ & Foreign-agent behavioral metaproductivity","$G_{h,B}(K,A)$ & Realized improvement lift of mechanism $K$ on agent $A$","$\\mathcal E_t^{\\mathrm{cor}},\\mathcal E_t^{\\mathrm{ob}}$ & Correctable and obstruction energy on a defect distribution","$q_{\\Gcal}(r)$ & Minimum unresolved residual inside grammar $\\Gcal$","$\\Phi_{\\mathrm{disc}}$ & Monotone repair-discovery closure operator","$M$ & Causal repair-schema offspring matrix","$R_{\\mathrm{imp}}$ & Spectral radius of the improvement offspring matrix","$\\lambda_{\\mathrm{mean}}$ & Upper growth exponent for a time-varying offspring product","$\\mathfrak B$ & Empirical benchmark instance","$\\mathfrak C$ & Package-closed claim object","$\\mathcal M,\\widehat{\\mathcal M}$ & Abstract transition system and executable implementation","$\\alpha$ & Runtime-to-abstract refinement map","$G_\\ell$ & Cumulative empirical claim gate at level $\\ell$","$\\phi(z),\\mu_z$ & Anonymous-event probe vector and frozen prototype","$\\Delta$ & Minimum pairwise event-prototype separation","$M_a(B)$ & Matched-budget best-candidate gain for benchmark arm $a$","$E_{\\mathrm{corr}}$ & Corrected protected defect energy","$\\tau_M^{\\mathrm{future}}$ & Future foreign-agent effect of repair-schema abstraction","$\\psi(x),s(x),\\phi(x)$ & Base feature, affected-stratum indicator, and conditional Block 6 feature map","$q(x),\\widetilde q(x),C$ & Protected score, corrupted score, and cyclic corruption operator","$H_s,H_t$ & Paired source and target hidden-state anchor matrices","$\\widehat T_{t\\leftarrow s}$ & Learned source-to-target hidden-state bridge","$D_s,b_s,D_t,b_t$ & Source and transported affine repair parameters","$\\widehat\\eps_{\\mathrm{align}}$ & Calibration normalized hidden-state alignment error","$\\mathfrak R_{\\mathrm{learn}}$ & Controlled learned repair-transport validation instance","$C_e$ & Ordered fixed repair-candidate bundle for Block 7 target episode $e$","$B$ & Matched candidate budget per target","$K_0,K_1$ & Frozen parent and lineage-generated child repair selectors","$\\mathfrak M$ & Sealed meta-compiler mapping development evidence to a child and receipt","$\\Gamma$ & Finite Block 7 selector meta-grammar","$\\Pi_1$ & Proof-carrying descendant receipt for the child selector","$U_e(c),G_e(c),R_e(c)$ & Protected utility, no-op-relative utility gain, and protected regression of candidate $c$","$\\widehat U_{\\rm LCB},\\widehat R_{\\rm UCB}$ & Child utility lower score and risk upper score","$D_j$ & Paired child-minus-parent utility difference on audit episode $j$","$\\Delta_Q(K_1,K_0)$ & Held-out selector-level metaproductivity effect","$G_{\\rm meta}^{\\rm select}$ & Cumulative claim gate for sealed selector-level metaproductivity","$\\mathfrak P_0,\\mathfrak P_1$ & Parent and compiled child bridge-identification repair processes","$\\pi_{\\rm anc}$ & Paired source--target anchor-query policy","$B_{\\rm pair}$ & Matched deployment paired-query budget","$M_B^{\\rm for}(\\mathfrak P)$ & Foreign matched-budget metaproductivity of a repair process","$\\delta_{\\rm rank}$ & Bridge-identification rank defect","$G_{\\rm meta}^{\\rm bridge}$ & Cumulative gate for sealed bridge-policy metaproductivity","$N=\\{H,Q,C,T,S,V,W\\}$ & Repair-process module index set","$F_K(R)$ & Restricted oracle envelope when only module coalition $R$ may change","$\\Hcal(K)$ & Total declared metaproductivity headroom","$h_i(K)$ & Permutation-average module headroom allocation","$\\chi_R(K,K')$ & Headroom-adjusted gain of child $K'$ relative to module coalition $R$","$r_t$ & Nonnegative unresolved module-headroom vector","$A_a,e_t$ & Local meta-action response matrix and residual error","$\\Phi_w(r)$ & Weighted meta-regret Lyapunov function","$P_t^{\\mathrm{repair}},P_t^{\\mathrm{discovery}}$ & Old-coordinate repair progress and newly represented discovery energy","$L_t\\preceq_{\\mathrm{cons}}L_{t+1}$ & Conservative lineage-dominance relation","$\\Gamma_t$ & Lineage compiler that maps a parent process and receipts to a descendant process","$\\Ccal(\\Gamma)$ & Compiler metaproductivity on foreign parent repair processes","$I_t=\\im D_t$ & Authorized correction image at grammar generation $t$","$E_t(Q)$ & Expected unresolved obstruction energy on foreign residual distribution $Q$","$\\overline T_t$ & Induced transport between obstruction quotients","$\\Sigma_{\\mathrm{ob}}$ & Covariance operator of projected foreign obstruction residuals","$\\mathfrak H_{\\Gcal}$ & Hypergraph of minimal repair conflicts under grammar $\\Gcal$","$\\Acal^\\star(x)$ & Set of externally validated authorized repairs for episode $x$","$\\mathfrak A$ & Adjoint-closed algebra generated by intervention-response operators","$Z(\\mathfrak A),\\mathfrak A'$ & Center and commutant of the response algebra","$H_r$ & Cyclic causal module generated by residual $r$","$\\Lcal_{\\mathrm{com}}$ & Positive commutator operator used for approximate atlas recovery","$\\Delta_t,L_t$ & True lineage-edge metaproductivity effect and conditional lower bound","$\\delta_t,\\beta_t,d_t$ & Edge false-acceptance, protected-regression, and semantic-debt budgets","$K_0,K_1,K_2,K_3$ & Random-query ancestor, rank-revealing child, conditioning-aware grandchild, and structural chart descendant across Blocks 7--10","$K_1^{\\rm recal}$ & Rank-revealing child with a witness gate recalibrated on Block 9 development evidence","$X_S,W,M_S$ & Selected source-anchor design, inverse-noise weight matrix, and weighted bridge-information matrix","$\\sigma_i^2$ & Conditional noise variance of paired target anchor $i$","$\\lambda_{\\min}(M_S)$ & Weighted minimum information eigenvalue","$\\widehat\\Delta_{1\\to2}$ & Paired grandchild-minus-recalibrated-child foreign metaproductivity effect","$\\Gcal_{\\mathrm{glob}},\\Gcal_{\\mathrm{chart}}$ & Global-linear and two-chart bridge grammars in Block 10","$T_-,T_+$ & Chart-specific source-to-target bridge maps","$D_-,D_+$ & Chart-specific transported target repairs","$\\Delta_{\\mathrm{gram}}$ & Global-minus-local held-out transport residual margin","$\\kappa_D$ & Normalized chart repair-contrast magnitude","$K_{2,G}^{96},K_{3,C}^{96}$ & Resource-matched global-affine parent and invariant-context chart child","$\\Delta_{G\\to C}$ & Chart-child minus global-parent paired protected-gain effect in Block 10","$\\Pi_2$ & Proof-carrying Block 9 conditioning-grandchild receipt","$\\Pi_3$ & Proof-carrying Block 10 structural grammar-expansion receipt","$\\Xi_t$ & Protected context associated with lineage edge $t\\to t+1$","$R_j$ & Controlled intervention-response operator used for endogenous atlas recovery","$\\lambda_a$ & Joint response signature of hidden block $H_a$","$\\mathfrak A_R$ & Adjoint-closed algebra generated by the response operators","$\\mathfrak A_R'$ & Response commutant; unresolved within-block multiplicity gauge","$P_a,\\widehat P_a$ & Exact and estimated central response-block projectors","$\\gamma_a,\\widehat\\gamma_{\\min}$ & Exact and empirical response spectral gaps","$\\eps_{\\mathrm{resp}}$ & Normalized response block-model residual","$\\eps_{\\mathrm{comm}}$ & Maximum normalized projector--response commutator residual","$H_r$ & Minimal response-invariant cyclic module generated by defect $r$","$K_4,\\Pi_4$ & Endogenous-atlas descendant and its proof-carrying lineage edge","\\midrule\n$\\Rcal$ & Declared intervention-response language","$\\mathfrak A(\\Rcal)$ & Unital algebra generated by the response language","$\\sigma_{\\Rcal}(H_a)$ & Response signature of hidden block $H_a$","$\\Pi_{\\mathrm{resp}}$ & Proof-carrying response-language expansion certificate","$\\Ccal_{\\mathrm{aff}}$ & Inherited affine repair-constructor grammar","$\\Ccal_2(P)$ & Degree-two repair grammar localized to packet $P$","$\\Pi_{\\mathrm{con}}$ & Proof-carrying constructor-expansion certificate","$K_4,K_5,K_6$ & Endogenous-atlas parent, response-language child, and degree-two constructor descendant","\\midrule\n$\\Gamma_0,\\Gamma_1$ & AAR-style parent and lineage-trained child research compilers","$\\Pi_i^-,\\Pi_i^+$ & Results-free pre-receipt and evidence-appended post-receipt for research candidate $i$","$\\Pi_7^{\\Gamma}$ & Proof-carrying Block 13 compiler lineage receipt","$\\mathfrak C_B(\\Gamma)$ & Validity-filtered compiler metaproductivity under research budget $B$","$S_{\\mathrm{cov}}$ & Coverage-weighted geometric benchmark score","$\\Ncal_{\\Bcal}$ & Local behavior subspace invisible to evaluator portfolio $\\Bcal$","$F_\\theta(z)$ & Outcome, risk, and scored-performance forecaster from proposal and pilot evidence","$\\widehat\\Delta_\\Gamma$ & Paired child-minus-parent compiler metaproductivity effect","$\\mathcal H_{\\mathrm{obs}}$ & Finite hidden obstruction-class family in the Block 13 benchmark","\\midrule\n$\\mathfrak U$ & Declared research-compiler robustness environment","$\\mathfrak C_{\\rm rob}(\\Gamma)$ & Worst-environment validity-filtered compiler value with a lower-tail risk term","$R,z,y$ & Response-signature matrix, sparse obstruction mixture, and observed sequential response","$\\eta_{\\rm open}(y)$ & Distance of an observed response from the inherited known response cone","$\\widetilde P_\\epsilon$ & $\\epsilon$-contaminated research-receipt distribution","$\\mathcal N_{\\mathcal B}$ & Local behavioral subspace invisible to evaluator portfolio $\\mathcal B$","$M_{\\cap}$ & Conjunctive integrity-monitor ensemble","$\\Gamma_2,\\Pi_8^{\\Gamma}$ & Robust compiler descendant and its Block 14 lineage receipt","$\\widehat\\Delta_{\\Gamma}^{\\rm primary},\\widehat\\Delta_{\\Gamma}^{\\rm foreign}$ & Paired robust-child minus inherited-parent effects","\\midrule\n$\\mathfrak G=(\\Zcal,\\Pcal,\\Ccal,\\Bcal)$ & Research-compiler diagnostic, experiment, method-constructor, and binding language","$\\phi_0$ & Inherited diagnostic quotient mapping both novel mechanisms to one unknown token","$\\Acal^\\star(\\tau)$ & Acceptable validity-filtered methods for research problem $\\tau$","$\\Hcal_{\\rm conflict}$ & Hypergraph of repair-conflicting research-problem cells","$y_p,\\theta$ & Candidate probe response and refinement threshold","$\\Ccal_{\\rm pair}$ & Generic ordered pair-method constructor grammar","$O_0,O_1$ & Inherited and expanded method-language oracle envelopes","$\\kappa_Q$ & Task-relevant structural language margin $\\mathbb E_Q[O_1-O_0]$","$\\Pi_{\\Gamma\\text{-lang}}$ & Proof-carrying compiler-language expansion receipt","$\\Gamma_3,\\Pi_9^\\Gamma$ & Block 15 language-growing compiler and its lineage receipt","$\\widehat\\Delta_{\\Gamma\\text{-lang}}^{\\rm primary},\\widehat\\Delta_{\\Gamma\\text{-lang}}^{\\rm foreign}$ & Paired child-parent language-growth effects","Let\n\\begin{equation}\nD=\n\\begin{bmatrix}\n1&0\\\\\n0&0\n\\end{bmatrix},\n\\qquad\nr=\n\\begin{bmatrix}\na\\\\b\n\\end{bmatrix}.\n\\end{equation}\nThen $D^\\dagger=D$,\n\\begin{equation}\nP_{\\mathrm{cor}}=\n\\begin{bmatrix}1&0","b\n\\end{bmatrix}.\n\\end{equation}\nThen $D^\\dagger=D$,\n\\begin{equation}\nP_{\\mathrm{cor}}=\n\\begin{bmatrix}1&0","b\n\\end{bmatrix}.\n\\end{equation}\nThen $D^\\dagger=D$,\n\\begin{equation}\nP_{\\mathrm{cor}}=\n\\begin{bmatrix}1&0\\\\0&0\\end{bmatrix},\n\\qquad\nP_{\\mathrm{ob}}=\n\\begin{bmatrix}0&0\\\\0&1\\end{bmatrix}.\n\\end{equation}\nThe minimum-norm least-squares repair is\n\\begin{equation}\nu^*=\n\\begin{bmatrix}-a","0&1\\end{bmatrix}.\n\\end{equation}\nThe minimum-norm least-squares repair is\n\\begin{equation}\nu^*=\n\\begin{bmatrix}-a\\\\0\\end{bmatrix},\n\\end{equation}\nwith residual\n\\begin{equation}\nDu^*+r=\n\\begin{bmatrix}0","0\\end{bmatrix},\n\\end{equation}\nwith residual\n\\begin{equation}\nDu^*+r=\n\\begin{bmatrix}0\\\\b\\end{bmatrix}.\n\\end{equation}\nTherefore\n\\begin{equation}\n\\eta_D(r)=|b|.\n\\end{equation}\nFor $b\\neq0$, the unit dual witness is $y^*=\\operatorname{sgn}(b)e_2$. Every authorized correction has zero effect on the second defect coordinate.","b\\end{bmatrix}.\n\\end{equation}\nTherefore\n\\begin{equation}\n\\eta_D(r)=|b|.\n\\end{equation}\nFor $b\\neq0$, the unit dual witness is $y^*=\\operatorname{sgn}(b)e_2$. Every authorized correction has zero effect on the second defect coordinate.","Let\n\\begin{equation}\nD=\n\\begin{bmatrix}1&0\\end{bmatrix},\n\\qquad\nC=\n\\begin{bmatrix}1&0\\end{bmatrix},\n\\qquad\nr=1.\n\\end{equation}\nWithout 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\n\\begin{equation}\n\\coker D_C\\cong\\R\n\\end{equation}\nand the nonzero defect is authorized-obstructed. The classification is not ``the optimizer failed.'' The only correcting direction is forbidden by the protected invariant.","Let $K=\\R_+^2$, $D=I_2$, and\n\\begin{equation}\nr=\n\\begin{bmatrix}1","Let $K=\\R_+^2$, $D=I_2$, and\n\\begin{equation}\nr=\n\\begin{bmatrix}1\\\\-2\\end{bmatrix}.\n\\end{equation}\nExact correction would require $u=-r=(-1,2)$, which is not in $K$. Take\n\\begin{equation}\ny=e_1.\n\\end{equation}\nThen $D^*y=e_1\\in K^*$ and\n\\begin{equation}\n\\angles{y,r}=1>0.\n\\end{equation}\nBy \\cref{thm:cone-certificate}, $y$ certifies that no nonnegative authorized correction exists.","-2\\end{bmatrix}.\n\\end{equation}\nExact correction would require $u=-r=(-1,2)$, which is not in $K$. Take\n\\begin{equation}\ny=e_1.\n\\end{equation}\nThen $D^*y=e_1\\in K^*$ and\n\\begin{equation}\n\\angles{y,r}=1>0.\n\\end{equation}\nBy \\cref{thm:cone-certificate}, $y$ certifies that no nonnegative authorized correction exists.","Let\n\\begin{equation}\nD_0=I_2,\n\\qquad\nE=\n\\begin{bmatrix}\n0&q\\\\q&0\n\\end{bmatrix},\n\\qquad\n0<q<1,\n\\end{equation}\nso $Q_0=I_2$ and $\\norm{EQ_0}=q$. Then\n\\begin{equation}\nD^{-1}=(I_2+E)^{-1}\n=\\frac{1}{1-q^2}\n\\begin{bmatrix}\n1&-q","q&0\n\\end{bmatrix},\n\\qquad\n0<q<1,\n\\end{equation}\nso $Q_0=I_2$ and $\\norm{EQ_0}=q$. Then\n\\begin{equation}\nD^{-1}=(I_2+E)^{-1}\n=\\frac{1}{1-q^2}\n\\begin{bmatrix}\n1&-q","q&0\n\\end{bmatrix},\n\\qquad\n0<q<1,\n\\end{equation}\nso $Q_0=I_2$ and $\\norm{EQ_0}=q$. Then\n\\begin{equation}\nD^{-1}=(I_2+E)^{-1}\n=\\frac{1}{1-q^2}\n\\begin{bmatrix}\n1&-q\\\\-q&1\n\\end{bmatrix}.\n\\end{equation}\nFor $r=e_1$, the exact repair is\n\\begin{equation}\nu^*=-D^{-1}r\n=-\\frac{1}{1-q^2}\n\\begin{bmatrix}1","-q&1\n\\end{bmatrix}.\n\\end{equation}\nFor $r=e_1$, the exact repair is\n\\begin{equation}\nu^*=-D^{-1}r\n=-\\frac{1}{1-q^2}\n\\begin{bmatrix}1","-q&1\n\\end{bmatrix}.\n\\end{equation}\nFor $r=e_1$, the exact repair is\n\\begin{equation}\nu^*=-D^{-1}r\n=-\\frac{1}{1-q^2}\n\\begin{bmatrix}1\\\\-q\\end{bmatrix}.\n\\end{equation}\nThe zeroth packet-local repair is $u^{(0)}=-e_1$ and leaves residual\n\\begin{equation}\nDu^{(0)}+r=\n\\begin{bmatrix}0","-q\\end{bmatrix}.\n\\end{equation}\nThe zeroth packet-local repair is $u^{(0)}=-e_1$ and leaves residual\n\\begin{equation}\nDu^{(0)}+r=\n\\begin{bmatrix}0","-q\\end{bmatrix}.\n\\end{equation}\nThe zeroth packet-local repair is $u^{(0)}=-e_1$ and leaves residual\n\\begin{equation}\nDu^{(0)}+r=\n\\begin{bmatrix}0\\\\-q\\end{bmatrix}.\n\\end{equation}\nEach 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 $0.2^5=3.2\\times10^{-4}$.","-q\\end{bmatrix}.\n\\end{equation}\nEach 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 $0.2^5=3.2\\times10^{-4}$.","Orient a triangle cyclically and identify edge mismatches with a vector\n\\begin{equation}\ne=(e_{12},e_{23},e_{31})\\in\\R^3.\n\\end{equation}\nThe cycle space is spanned by\n\\begin{equation}\nc=\\frac{1}{\\sqrt3}(1,1,1).\n\\end{equation}\nHence\n\\begin{equation}\n\\eta_{\\mathrm{glue}}(e)=|\\angles{c,e}|\n=\\frac{|e_{12}+e_{23}+e_{31}|}{\\sqrt3}.\n\\end{equation}\nFor\n\\begin{equation}\ne=(1,1,-2),\n\\end{equation}\nthe sum vanishes and the mismatch is a pure gradient, so a vertex gauge glues the repairs exactly. For\n\\begin{equation}\ne=(1,1,1),\n\\end{equation}\nthe quantitative obstruction is $\\sqrt3$ and no gauge can reconcile the cycle.","1})\\in\\R^3.\n\\end{equation}\nThe cycle space is spanned by\n\\begin{equation}\nc=\\frac{1}{\\sqrt3}(1,1,1).\n\\end{equation}\nHence\n\\begin{equation}\n\\eta_{\\mathrm{glue}}(e)=|\\angles{c,e}|\n=\\frac{|e_{12}+e_{23}+e_{31}|}{\\sqrt3}.\n\\end{equation}\nFor\n\\begin{equation}\ne=(1,1,-2),\n\\end{equation}\nthe sum vanishes and the mismatch is a pure gradient, so a vertex gauge glues the repairs exactly. For\n\\begin{equation}\ne=(1,1,1),\n\\end{equation}\nthe quantitative obstruction is $\\sqrt3$ and no gauge can reconcile the cycle.","gin{equation}\nc=\\frac{1}{\\sqrt3}(1,1,1).\n\\end{equation}\nHence\n\\begin{equation}\n\\eta_{\\mathrm{glue}}(e)=|\\angles{c,e}|\n=\\frac{|e_{12}+e_{23}+e_{31}|}{\\sqrt3}.\n\\end{equation}\nFor\n\\begin{equation}\ne=(1,1,-2),\n\\end{equation}\nthe sum vanishes and the mismatch is a pure gradient, so a vertex gauge glues the repairs exactly. For\n\\begin{equation}\ne=(1,1,1),\n\\end{equation}\nthe quantitative obstruction is $\\sqrt3$ and no gauge can reconcile the cycle.","Let\n\\begin{equation}\ne=(1,1,-1.8).\n\\end{equation}\nThen\n\\begin{equation}\n\\eta_{\\mathrm{glue}}(e)\n=\\frac{0.2}{\\sqrt3}\n\\approx0.11547.\n\\end{equation}\nMost of the mismatch can be removed by adjusting local kernel coordinates, but the cycle-average component is irreducible. Reporting only the raw edge norm\n\\begin{equation}\n\\norm e\\approx2.289\n\\end{equation}\nwould substantially overstate the genuine incompatibility.","Consider\n\\begin{equation}\nF(\\eps,x,y)\n=\n\\begin{bmatrix}\nx+\\eps\\\\x^2\n\\end{bmatrix}.\n\\end{equation}\nAt $(\\eps,x,y)=(0,0,0)$,\n\\begin{equation}\nD=\\partial_{(x,y)}F\n=\n\\begin{bmatrix}\n1&0","x^2\n\\end{bmatrix}.\n\\end{equation}\nAt $(\\eps,x,y)=(0,0,0)$,\n\\begin{equation}\nD=\\partial_{(x,y)}F\n=\n\\begin{bmatrix}\n1&0","x^2\n\\end{bmatrix}.\n\\end{equation}\nAt $(\\eps,x,y)=(0,0,0)$,\n\\begin{equation}\nD=\\partial_{(x,y)}F\n=\n\\begin{bmatrix}\n1&0\\\\0&0\n\\end{bmatrix}.\n\\end{equation}\nThe 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.","0&0\n\\end{bmatrix}.\n\\end{equation}\nThe 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.","For\n\\begin{equation}\nF(\\eps,x)=x^2-\\eps,\n\\end{equation}\nthe 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\n\\begin{equation}\nx=\\pm\\sqrt\\eps,\n\\end{equation}\nfor $\\eps=0$ there is one singular branch, and for $\\eps<0$ there is no real repair. A linearized system cannot distinguish these three regimes.","Let\n\\begin{equation}\nD_A=\n\\begin{bmatrix}\n1&0\\\\\n0&2\n\\end{bmatrix},\n\\qquad\nT_U=T_V=\n\\begin{bmatrix}\n0&-1\\\\\n1&0\n\\end{bmatrix},\n\\end{equation}\nand define\n\\begin{equation}\nD_B=T_VD_AT_U^{-1}\n=\n\\begin{bmatrix}\n2&0","1&0\n\\end{bmatrix},\n\\end{equation}\nand define\n\\begin{equation}\nD_B=T_VD_AT_U^{-1}\n=\n\\begin{bmatrix}\n2&0\\\\\n0&1\n\\end{bmatrix}.\n\\end{equation}\nFor\n\\begin{equation}\nr_A=\n\\begin{bmatrix}\n1","0&1\n\\end{bmatrix}.\n\\end{equation}\nFor\n\\begin{equation}\nr_A=\n\\begin{bmatrix}\n1\\\\2\n\\end{bmatrix},\n\\qquad\nu_A=\n\\begin{bmatrix}\n-1\\\\-1\n\\end{bmatrix},\n\\end{equation}\nwe have $D_Au_A=-r_A$. Transport gives\n\\begin{equation}\nr_B=T_Vr_A=\n\\begin{bmatrix}\n-2","-1\n\\end{bmatrix},\n\\end{equation}\nwe have $D_Au_A=-r_A$. Transport gives\n\\begin{equation}\nr_B=T_Vr_A=\n\\begin{bmatrix}\n-2","-1\n\\end{bmatrix},\n\\end{equation}\nwe have $D_Au_A=-r_A$. Transport gives\n\\begin{equation}\nr_B=T_Vr_A=\n\\begin{bmatrix}\n-2\\\\1\n\\end{bmatrix},\n\\qquad\nu_B=T_Uu_A=\n\\begin{bmatrix}\n1\\\\-1\n\\end{bmatrix}.\n\\end{equation}\nThen\n\\begin{equation}\nD_Bu_B=\n\\begin{bmatrix}\n2","-1\n\\end{bmatrix}.\n\\end{equation}\nThen\n\\begin{equation}\nD_Bu_B=\n\\begin{bmatrix}\n2\\\\-1\n\\end{bmatrix}\n=-r_B.\n\\end{equation}","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\n\\begin{equation}\n\\begin{aligned}\ng_B\n&\\geq\n0.20\n-2(0.01)\n-\\left(2(0.015)+0.025+0.01+0.005\\right)","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\n\\begin{equation}\n\\begin{aligned}\ng_B\n&\\geq\n0.20\n-2(0.01)\n-\\left(2(0.015)+0.025+0.01+0.005\\right)\\\\\n&=0.11.\n\\end{aligned}\n\\end{equation}\nThe 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.","&=0.11.\n\\end{aligned}\n\\end{equation}\nThe 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.","Let\n\\begin{equation}\nD_A:\\R\\to\\R^2,\n\\qquad\nD_Au=(u,0),\n\\end{equation}\nso\n\\begin{equation}\n\\coker D_A\\cong\\Span\\set{e_2}.\n\\end{equation}\nLet the target correction map be\n\\begin{equation}\nD_B:\\R\\to\\R,\n\\qquad\nD_Bu=u,\n\\end{equation}\nand define\n\\begin{equation}\nT_U=I,\n\\qquad\nT_V(x,y)=x.\n\\end{equation}\nThen\n\\begin{equation}\nD_BT_U=T_VD_A,\n\\end{equation}\nso the correction square commutes. However, for\n\\begin{equation}\nr_A=e_2,\n\\end{equation}\nwe have\n\\begin{equation}\n[r_A]\\neq0\n\\quad\\text{in }\\coker D_A,\n\\qquad\nT_Vr_A=0\n\\quad\\text{in }\\coker D_B.\n\\end{equation}\nThe target representation has erased the obstructed direction. Exact commutation alone therefore does not certify obstruction faithfulness.","ation}\nD_A:\\R\\to\\R^2,\n\\qquad\nD_Au=(u,0),\n\\end{equation}\nso\n\\begin{equation}\n\\coker D_A\\cong\\Span\\set{e_2}.\n\\end{equation}\nLet the target correction map be\n\\begin{equation}\nD_B:\\R\\to\\R,\n\\qquad\nD_Bu=u,\n\\end{equation}\nand define\n\\begin{equation}\nT_U=I,\n\\qquad\nT_V(x,y)=x.\n\\end{equation}\nThen\n\\begin{equation}\nD_BT_U=T_VD_A,\n\\end{equation}\nso the correction square commutes. However, for\n\\begin{equation}\nr_A=e_2,\n\\end{equation}\nwe have\n\\begin{equation}\n[r_A]\\neq0\n\\quad\\text{in }\\coker D_A,\n\\qquad\nT_Vr_A=0\n\\quad\\text{in }\\coker D_B.\n\\end{equation}\nThe target representation has erased the obstructed direction. Exact commutation alone therefore does not certify obstruction faithfulness.","ation}\nso\n\\begin{equation}\n\\coker D_A\\cong\\Span\\set{e_2}.\n\\end{equation}\nLet the target correction map be\n\\begin{equation}\nD_B:\\R\\to\\R,\n\\qquad\nD_Bu=u,\n\\end{equation}\nand define\n\\begin{equation}\nT_U=I,\n\\qquad\nT_V(x,y)=x.\n\\end{equation}\nThen\n\\begin{equation}\nD_BT_U=T_VD_A,\n\\end{equation}\nso the correction square commutes. However, for\n\\begin{equation}\nr_A=e_2,\n\\end{equation}\nwe have\n\\begin{equation}\n[r_A]\\neq0\n\\quad\\text{in }\\coker D_A,\n\\qquad\nT_Vr_A=0\n\\quad\\text{in }\\coker D_B.\n\\end{equation}\nThe target representation has erased the obstructed direction. Exact commutation alone therefore does not certify obstruction faithfulness.","Suppose a source correction has norm $\\norm{u_0}=2$, every transport is nonexpansive, and three edges have\n\\begin{equation}\n(\\eps_{D,0},\\eps_{D,1},\\eps_{D,2})\n=(0.01,0.02,0.015),\n\\end{equation}\n\\begin{equation}\n(\\eps_{r,0},\\eps_{r,1},\\eps_{r,2})\n=(0.005,0.004,0.006).\n\\end{equation}\nThen \\cref{cor:nonexpansive-path} gives\n\\begin{equation}\n\\begin{aligned}\n\\norm{D_3u_3+r_3}\n&\\leq\n2(0.01+0.02+0.015)\n+(0.005+0.004+0.006)","Suppose a source correction has norm $\\norm{u_0}=2$, every transport is nonexpansive, and three edges have\n\\begin{equation}\n(\\eps_{D,0},\\eps_{D,1},\\eps_{D,2})\n=(0.01,0.02,0.015),\n\\end{equation}\n\\begin{equation}\n(\\eps_{r,0},\\eps_{r,1},\\eps_{r,2})\n=(0.005,0.004,0.006).\n\\end{equation}\nThen \\cref{cor:nonexpansive-path} gives\n\\begin{equation}\n\\begin{aligned}\n\\norm{D_3u_3+r_3}\n&\\leq\n2(0.01+0.02+0.015)\n+(0.005+0.004+0.006)\\\\\n&=0.105.\n\\end{aligned}\n\\end{equation}\nA 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.","&=0.105.\n\\end{aligned}\n\\end{equation}\nA 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.","Let $U=V=\\R^2$, $D=I$, and suppose two routes from $A$ to $B$ differ by a loop with correction holonomy\n\\begin{equation}\nH^U=\n\\begin{bmatrix}\n1&0","Let $U=V=\\R^2$, $D=I$, and suppose two routes from $A$ to $B$ differ by a loop with correction holonomy\n\\begin{equation}\nH^U=\n\\begin{bmatrix}\n1&0\\\\\n0&-1\n\\end{bmatrix}.\n\\end{equation}\nThe flat repair core is\n\\begin{equation}\nU^{\\flat}=\\Span\\set{e_1}.\n\\end{equation}\nFor $u=e_1$, both routes produce the same transported repair. For $u=e_2$, the two routes differ by\n\\begin{equation}\n(H^U-I)e_2=-2e_2,\n\\end{equation}\nso 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.","0&-1\n\\end{bmatrix}.\n\\end{equation}\nThe flat repair core is\n\\begin{equation}\nU^{\\flat}=\\Span\\set{e_1}.\n\\end{equation}\nFor $u=e_1$, both routes produce the same transported repair. For $u=e_2$, the two routes differ by\n\\begin{equation}\n(H^U-I)e_2=-2e_2,\n\\end{equation}\nso 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.","Consider one loop on $E=\\R^2$ with shear\n\\begin{equation}\nH=\n\\begin{bmatrix}\n1&a","Consider one loop on $E=\\R^2$ with shear\n\\begin{equation}\nH=\n\\begin{bmatrix}\n1&a\\\\\n0&1\n\\end{bmatrix}.\n\\end{equation}\nThen\n\\begin{equation}\n(H-I)^*(H-I)\n=\n\\begin{bmatrix}\n0&0","0&1\n\\end{bmatrix}.\n\\end{equation}\nThen\n\\begin{equation}\n(H-I)^*(H-I)\n=\n\\begin{bmatrix}\n0&0\\\\\n0&a^2\n\\end{bmatrix}.\n\\end{equation}\nThe holonomy-energy eigenvalues are $0$ and $a^2$. Thus\n\\begin{equation}\nE^{\\flat}=\\Span\\set{e_1}.\n\\end{equation}\nFor 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.","0&a^2\n\\end{bmatrix}.\n\\end{equation}\nThe holonomy-energy eigenvalues are $0$ and $a^2$. Thus\n\\begin{equation}\nE^{\\flat}=\\Span\\set{e_1}.\n\\end{equation}\nFor 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.","Now take protected evidence states\n\\begin{equation}\nx_+=(1,1),\n\\qquad\nx_-=(1,-1).\n\\end{equation}\nTheir 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?''","Suppose an old evaluator assigns historical scores\n\\begin{equation}\n0.81,\n\\qquad\n0.74,\n\\qquad\n0.60.\n\\end{equation}\nThe minimum strict margin is\n\\begin{equation}\nm_{\\mathrm{hist}}=0.07.\n\\end{equation}\nIf a proposed upgrade has uniform replay error at most $0.02$, then\n\\begin{equation}\n2(0.02)=0.04<0.07,\n\\end{equation}\nso \\cref{thm:ranking-preservation} certifies preservation of all strict historical rankings.","Assume the upgrade is then evaluated on $n=800$ precommitted held-out outcomes and achieves paired mean loss improvement\n\\begin{equation}\n\\overline\\Delta_n=0.09.\n\\end{equation}\nAt confidence $1-\\alpha=0.95$, \\cref{thm:evaluator-risk} gives the lower bound\n\\begin{equation}\n0.09-\n\\sqrt{\\frac{2\\log 20}{800}}\n\\approx\n0.0035.\n\\end{equation}\nThe 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.","Suppose a sandbox returns only the scalar task score\n\\begin{equation}\nq(\\omega)=\\mathrm{accuracy}(\\omega).\n\\end{equation}\nExecution $\\omega_0$ reaches accuracy $0.9$ using authorized retrieval. Execution $\\omega_1$ reaches the same accuracy after importing a prohibited hidden answer key. Then\n\\begin{equation}\nq(\\omega_0)=q(\\omega_1)=0.9,\n\\end{equation}\nbut 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.","Let\n\\begin{equation}\n\\mathsf C=\\{M,P,E,L\\},\n\\end{equation}\nwhere $M$ is the task model, $P$ the planner, $E$ the evaluator, and $L$ the ledger and rollback service. Use two frozen witnesses per transaction:\n\\begin{center}\n\\begin{tabular}{@{}ccll@{}}\n\\toprule\n$t$ & $W_t$ & Witnesses & Editable components","\\midrule\n0 & $\\{M,P\\}$ & model, planner & evaluator, ledger","1 & $\\{P,E\\}$ & planner, evaluator & ledger, model","2 & $\\{E,L\\}$ & evaluator, ledger & model, planner","3 & $\\{L,M\\}$ & ledger, model & planner, evaluator","Let\n\\begin{equation}\nS=\\{s_0,s_1,b\\},\n\\qquad\nC_{\\mathrm{RI}}=\\{s_0,s_1\\}.\n\\end{equation}\nAt $s_0$ there are two authorized actions:\n\\begin{equation}\na_{\\mathrm{safe}}:s_0\\mapsto s_1,\n\\qquad\na_{\\mathrm{risky}}:s_0\\mapsto b.\n\\end{equation}\nAt $s_1$ 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.","1,b\\},\n\\qquad\nC_{\\mathrm{RI}}=\\{s_0,s_1\\}.\n\\end{equation}\nAt $s_0$ there are two authorized actions:\n\\begin{equation}\na_{\\mathrm{safe}}:s_0\\mapsto s_1,\n\\qquad\na_{\\mathrm{risky}}:s_0\\mapsto b.\n\\end{equation}\nAt $s_1$ 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.","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.","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\n\\begin{equation}\n(\\text{predicted gain }0.08,\n\\text{observed gain }-0.09,\n\\text{integrity failure},\n\\text{repair signature}).\n\\end{equation}\nThe transaction is behaviorally unsuccessful and epistemically productive.","Let the lineage graph contain\n\\begin{equation}\na\\to b\\to c\n\\end{equation}\nwith\n\\begin{equation}\nJ(a)=0,\n\\qquad\nJ(b)=-1,\n\\qquad\nJ(c)=10.\n\\end{equation}\nSuppose 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.","Take $G_{\\max}=1$, $n=1000$, and observed paired mean\n\\begin{equation}\n\\overline D_n=0.25.\n\\end{equation}\nAt $\\delta=0.05$, \\cref{thm:paired-meta} gives\n\\begin{equation}\n\\begin{aligned}\n\\mathbb E D_i\n&\\geq\n0.25\n-2\\sqrt2\\sqrt{\\frac{\\log20}{1000}}","Take $G_{\\max}=1$, $n=1000$, and observed paired mean\n\\begin{equation}\n\\overline D_n=0.25.\n\\end{equation}\nAt $\\delta=0.05$, \\cref{thm:paired-meta} gives\n\\begin{equation}\n\\begin{aligned}\n\\mathbb E D_i\n&\\geq\n0.25\n-2\\sqrt2\\sqrt{\\frac{\\log20}{1000}}\\\\\n&\\approx\n0.095.\n\\end{aligned}\n\\end{equation}\nThe 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.","&\\approx\n0.095.\n\\end{aligned}\n\\end{equation}\nThe 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.","Let $V=\\R^3$ and let held-out defects have covariance\n\\begin{equation}\n\\Sigma=\\diag(1,4,9).\n\\end{equation}\nSuppose\n\\begin{equation}\nL_0=\\Span\\{e_1\\},\n\\qquad\nL_1=\\Span\\{e_1,e_2\\}.\n\\end{equation}\nThen 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}.","Let\n\\begin{equation}\nV=\\R^2,\n\\qquad\nC=\\Span\\{e_1\\},\n\\end{equation}\nand embed $V$ into $V'=\\R^3$ by the standard coordinate inclusion. Let\n\\begin{equation}\nC'=\\Span\\{e_1,e_2\\}.\n\\end{equation}\nThe old obstruction space is generated by $[e_2]$, but $e_2$ becomes correctable in $C'$. Hence\n\\begin{equation}\n\\frac{j^{-1}(C')}{C}\n\\cong\n\\Span\\{[e_2]\\}.\n\\end{equation}\nThe genuinely new obstruction quotient is\n\\begin{equation}\n\\frac{V'}{C'+j(V)}\n\\cong\n\\Span\\{[e_3]\\}.\n\\end{equation}\nThe expansion repairs one old direction while exposing one new direction, exactly as in \\cref{thm:dynamic-obstruction-exact}.","V=\\R^2,\n\\qquad\nC=\\Span\\{e_1\\},\n\\end{equation}\nand embed $V$ into $V'=\\R^3$ by the standard coordinate inclusion. Let\n\\begin{equation}\nC'=\\Span\\{e_1,e_2\\}.\n\\end{equation}\nThe old obstruction space is generated by $[e_2]$, but $e_2$ becomes correctable in $C'$. Hence\n\\begin{equation}\n\\frac{j^{-1}(C')}{C}\n\\cong\n\\Span\\{[e_2]\\}.\n\\end{equation}\nThe genuinely new obstruction quotient is\n\\begin{equation}\n\\frac{V'}{C'+j(V)}\n\\cong\n\\Span\\{[e_3]\\}.\n\\end{equation}\nThe expansion repairs one old direction while exposing one new direction, exactly as in \\cref{thm:dynamic-obstruction-exact}.","Consider\n\\begin{equation}\nM=\n\\begin{bmatrix}\n0.7&0.6\\\\\n0.2&0.8\n\\end{bmatrix}.\n\\end{equation}\nIts 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}.","0.2&0.8\n\\end{bmatrix}.\n\\end{equation}\nIts 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}.","Suppose a simultaneous lower confidence matrix is\n\\begin{equation}\nL=\n\\begin{bmatrix}\n0.65&0.55\\\\\n0.18&0.75\n\\end{bmatrix}.\n\\end{equation}\nIts 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.","0.18&0.75\n\\end{bmatrix}.\n\\end{equation}\nIts 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.","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.","For the triangle mismatch\n\\begin{equation}\nb=(0,0,1)^\\top\n\\end{equation}\nand normalized cycle vector\n\\begin{equation}\nc=3^{-1/2}(1,1,-1)^\\top,\n\\end{equation}\nthe cycle component is\n\\begin{equation}\n\\angles{c,b}=-\\frac1{\\sqrt3}.\n\\end{equation}\nEvery 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\n\\begin{equation}\n\\frac1{\\sqrt3}>10^{-9}.\n\\end{equation}\nNo evaluator check is needed to classify this isolated fixture case.","=(0,0,1)^\\top\n\\end{equation}\nand normalized cycle vector\n\\begin{equation}\nc=3^{-1/2}(1,1,-1)^\\top,\n\\end{equation}\nthe cycle component is\n\\begin{equation}\n\\angles{c,b}=-\\frac1{\\sqrt3}.\n\\end{equation}\nEvery 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\n\\begin{equation}\n\\frac1{\\sqrt3}>10^{-9}.\n\\end{equation}\nNo evaluator check is needed to classify this isolated fixture case.","Let protected keys be\n\\begin{equation}\nK_{\\mathrm{protected}}\n=\\{\\mathrm{evidence},\\mathrm{evaluator},\\mathrm{provenance}\\}.\n\\end{equation}\nThe adversarial candidate declares\n\\begin{equation}\nK_{\\mathrm{edit}}\n=\\{\\mathrm{evaluator},\\mathrm{utility}\\}.\n\\end{equation}\nThen\n\\begin{equation}\nK_{\\mathrm{edit}}\\cap K_{\\mathrm{protected}}\n=\\{\\mathrm{evaluator}\\}\\neq\\varnothing.\n\\end{equation}\nEven 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.","Suppose the minimum anonymous-event prototype separation is\n\\begin{equation}\n\\Delta=0.110955\n\\end{equation}\nand $\\sigma=0.06$. With six event types, \\cref{thm:prototype-separation} gives the conservative bound\n\\begin{equation}\n\\Pr(\\widehat z\\neq z)\n\\leq\n5\\exp\\left(-\\frac{0.110955^2}{8(0.06)^2}\\right),\n\\end{equation}\nwhich 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.","Consider a parent with protected utility $J_0=0.42$. A large global rewrite has protected utility\n\\begin{equation}\nJ_{\\mathrm{protected}}=0.10\n\\end{equation}\nbut apparent utility\n\\begin{equation}\n\\widehat J=0.10+1.10=1.20\n\\end{equation}\ndue to an exploitable proxy bonus. A selector using $\\widehat J$ accepts apparent gain $0.78$, while protected gain is\n\\begin{equation}\n0.10-0.42=-0.32.\n\\end{equation}\nThis is the mechanism underlying the global-edit positive control. A protected witness ranks by $J_{\\mathrm{protected}}$ and rejects the candidate.","in{equation}\nJ_{\\mathrm{protected}}=0.10\n\\end{equation}\nbut apparent utility\n\\begin{equation}\n\\widehat J=0.10+1.10=1.20\n\\end{equation}\ndue to an exploitable proxy bonus. A selector using $\\widehat J$ accepts apparent gain $0.78$, while protected gain is\n\\begin{equation}\n0.10-0.42=-0.32.\n\\end{equation}\nThis is the mechanism underlying the global-edit positive control. A protected witness ranks by $J_{\\mathrm{protected}}$ and rejects the candidate.","Let\n\\begin{equation}\nE=[A\\;B]:\\R^{16}\\to\\R^{16}\n\\end{equation}\nwith $A,B\\in\\R^{16\\times8}$ and let\n\\begin{equation}\n\\phi(x)=\\begin{bmatrix}\\psi(x)","Let\n\\begin{equation}\nE=[A\\;B]:\\R^{16}\\to\\R^{16}\n\\end{equation}\nwith $A,B\\in\\R^{16\\times8}$ and let\n\\begin{equation}\n\\phi(x)=\\begin{bmatrix}\\psi(x)\\\\s(x)\\psi(x)\\end{bmatrix},\n\\qquad\n\\phi^{\\circ}(x)=\\begin{bmatrix}\\psi(x)\\\\0\\end{bmatrix}.\n\\end{equation}\nThen\n\\begin{equation}\n\\Delta(x)=E\\phi(x)-E\\phi^{\\circ}(x)=s(x)B\\psi(x).\n\\end{equation}\nIf 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.","0\\end{bmatrix}.\n\\end{equation}\nThen\n\\begin{equation}\n\\Delta(x)=E\\phi(x)-E\\phi^{\\circ}(x)=s(x)B\\psi(x).\n\\end{equation}\nIf 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.","Let\n\\begin{equation}\nT=\\begin{bmatrix}1&0\\end{bmatrix}:\\R^2\\to\\R,\n\\qquad\nD_s=\\begin{bmatrix}0&1\\\\0&0\\end{bmatrix},\n\\qquad\nh_s=e_2.\n\\end{equation}\nThen $h_t=Th_s=0$, so target-state alignment is exact. However,\n\\begin{equation}\nT^+T=\\begin{bmatrix}1&0","0&0\\end{bmatrix},\n\\qquad\nh_s=e_2.\n\\end{equation}\nThen $h_t=Th_s=0$, so target-state alignment is exact. However,\n\\begin{equation}\nT^+T=\\begin{bmatrix}1&0","0&0\\end{bmatrix},\n\\qquad\nh_s=e_2.\n\\end{equation}\nThen $h_t=Th_s=0$, so target-state alignment is exact. However,\n\\begin{equation}\nT^+T=\\begin{bmatrix}1&0\\\\0&0\\end{bmatrix},\n\\qquad\nD_t=TD_sT^+=0.\n\\end{equation}\nThe source repair maps the discarded input direction into the retained output direction:\n\\begin{equation}\nD_sh_s=e_1.\n\\end{equation}\nConsequently,\n\\begin{equation}\nR_t(Th_s)-TR_s(h_s)\n=TD_s(T^+T-I)h_s\n=-1.\n\\end{equation}\nThus 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 $0.4715$.","0&0\\end{bmatrix},\n\\qquad\nD_t=TD_sT^+=0.\n\\end{equation}\nThe source repair maps the discarded input direction into the retained output direction:\n\\begin{equation}\nD_sh_s=e_1.\n\\end{equation}\nConsequently,\n\\begin{equation}\nR_t(Th_s)-TR_s(h_s)\n=TD_s(T^+T-I)h_s\n=-1.\n\\end{equation}\nThus 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 $0.4715$.","rded input direction into the retained output direction:\n\\begin{equation}\nD_sh_s=e_1.\n\\end{equation}\nConsequently,\n\\begin{equation}\nR_t(Th_s)-TR_s(h_s)\n=TD_s(T^+T-I)h_s\n=-1.\n\\end{equation}\nThus 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 $0.4715$.","\\begin{equation}\nD_sh_s=e_1.\n\\end{equation}\nConsequently,\n\\begin{equation}\nR_t(Th_s)-TR_s(h_s)\n=TD_s(T^+T-I)h_s\n=-1.\n\\end{equation}\nThus 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 $0.4715$.","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 $0.9943$.","teen, 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 $0.9943$.","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.","For the compressed target, affected-stratum accuracy rises from\n\\begin{equation}\n0.1277\\quad\\text{to}\\quad0.6204\n\\end{equation}\nunder 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 $0.1277$.","The example separates two predicates:\n\\begin{equation}\n\\mathrm{ImprovesSlice}(R_t)\n\\not\\Rightarrow\n\\mathrm{AuthorizedCommit}(R_t).\n\\end{equation}","For independently trained tanh targets, mean transported affected accuracy is $0.89416$ and mean target-local oracle accuracy is $0.89670$. The gap\n\\begin{equation}\n0.89670-0.89416=0.00254\n\\end{equation}\nis 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.","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\n\\begin{equation}\nG_{\\rm foreign}=1,\n\\qquad\nG_{\\rm meta}=0,\n\\qquad\nG_{\\rm open}=0.\n\\end{equation}\nThe 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.","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\n\\begin{equation}\n\\frac12(0.8)(0.4)\n+\n\\frac12(0.3)(-0.2)\n=0.13.\n\\end{equation}\nThe 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}.","o-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\n\\begin{equation}\n\\frac12(0.8)(0.4)\n+\n\\frac12(0.3)(-0.2)\n=0.13.\n\\end{equation}\nThe 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}.","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\n\\begin{equation}\n\\mathbb E[G_{\\rm parent}]=0,\n\\qquad\n\\mathbb E[G_{\\rm child}]=\\frac13.\n\\end{equation}\nThe 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.","Consider three candidates with predicted utility and predicted protected risk\n\\begin{equation}\n\\begin{array}{c|ccc}\n & c_0 & c_1 & c_2\\\\\n\\hline\n\\widehat U & 0 & 0.30 & 0.36\\\\\n\\widehat R_{\\rm UCB} & 0 & 0.02 & 0.06.\n\\end{array}\n\\end{equation}\nWith 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.","\\widehat R_{\\rm UCB} & 0 & 0.02 & 0.06.\n\\end{array}\n\\end{equation}\nWith 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.","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\n\\begin{equation}\n0.04\\sqrt{2\\log 2000}\\approx0.156.\n\\end{equation}\nThis 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.","tions 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\n\\begin{equation}\n0.04\\sqrt{2\\log 2000}\\approx0.156.\n\\end{equation}\nThis 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.","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\n\\begin{equation}\n\\overline D=0.072518,\n\\qquad\ns_D=0.154617,\n\\qquad\n\\widehat{\\rm se}=0.014115.\n\\end{equation}\nThe ordinary normal approximation gives\n\\begin{equation}\n0.072518\\pm1.98(0.014115)\n\\approx\n[0.0446,0.1005],\n\\end{equation}\nclose to the frozen paired bootstrap interval\n$[0.046745,0.101389]$.","dates on $120$ episodes. Let $D_j$ be the child-minus-parent protected utility on episode $j$. The frozen values satisfy\n\\begin{equation}\n\\overline D=0.072518,\n\\qquad\ns_D=0.154617,\n\\qquad\n\\widehat{\\rm se}=0.014115.\n\\end{equation}\nThe ordinary normal approximation gives\n\\begin{equation}\n0.072518\\pm1.98(0.014115)\n\\approx\n[0.0446,0.1005],\n\\end{equation}\nclose to the frozen paired bootstrap interval\n$[0.046745,0.101389]$.","The child is developed on eight representation families and audited on four disjoint families. The frozen architecture-foreign metrics are\n\\begin{equation}\n\\begin{aligned}\n\\overline G_{\\rm parent}&=0.167417,\\\\\n\\overline G_{\\rm child}&=0.272416,\\\\\n\\overline G_{\\rm raw}&=0.254427.\n\\end{aligned}\n\\end{equation}\nThus\n\\begin{equation}\n\\overline G_{\\rm child}-\\overline G_{\\rm parent}=0.104999\n\\end{equation}\nand\n\\begin{equation}\n\\overline G_{\\rm child}-\\overline G_{\\rm raw}=0.017989.\n\\end{equation}\nThe 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.","\\overline G_{\\rm raw}&=0.254427.\n\\end{aligned}\n\\end{equation}\nThus\n\\begin{equation}\n\\overline G_{\\rm child}-\\overline G_{\\rm parent}=0.104999\n\\end{equation}\nand\n\\begin{equation}\n\\overline G_{\\rm child}-\\overline G_{\\rm raw}=0.017989.\n\\end{equation}\nThe 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.","Let $Z$ be the standardized receipt matrix and $y$ the development utility labels. The valid child uses\n\\begin{equation}\n\\widehat\\beta=(Z^\\top Z+\\lambda I)^{-1}Z^\\top y.\n\\end{equation}\nThe shuffled control replaces $y$ by $Py$ for a random permutation $P$. Because $Z$ is centered,\n\\begin{equation}\n\\mathbb E_P[Z^\\top Py]=0.\n\\end{equation}\nThe 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.","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.","Consider five mutable modules:\n\\begin{equation}\nN_5\n=\n\\{S,Q,T,C,H\\},\n\\end{equation}\nrepresenting selection, information acquisition, transport, candidate construction, and causal-atlas structure. Let\n\\begin{equation}\nF_K(\\varnothing)=0.20.\n\\end{equation}\nAssign singleton contributions\n\\begin{equation}\n(0.08,0.12,0.10,0.16,0.14)\n\\end{equation}\nand pair synergies\n\\begin{equation}\n\\omega_{QT}=0.05,\n\\qquad\n\\omega_{CH}=0.06,\n\\qquad\n\\omega_{SC}=0.03.\n\\end{equation}\nThen\n\\begin{equation}\nF_K(N_5)=0.94,\n\\qquad\n\\Hcal(K)=0.74.\n\\end{equation}\nEvery pair synergy is divided equally between its two participants by the permutation-average rule. Therefore\n\\begin{equation}\n\\begin{aligned}\nh_S&=0.08+\\frac{0.03}{2}=0.095,","epresenting selection, information acquisition, transport, candidate construction, and causal-atlas structure. Let\n\\begin{equation}\nF_K(\\varnothing)=0.20.\n\\end{equation}\nAssign singleton contributions\n\\begin{equation}\n(0.08,0.12,0.10,0.16,0.14)\n\\end{equation}\nand pair synergies\n\\begin{equation}\n\\omega_{QT}=0.05,\n\\qquad\n\\omega_{CH}=0.06,\n\\qquad\n\\omega_{SC}=0.03.\n\\end{equation}\nThen\n\\begin{equation}\nF_K(N_5)=0.94,\n\\qquad\n\\Hcal(K)=0.74.\n\\end{equation}\nEvery pair synergy is divided equally between its two participants by the permutation-average rule. Therefore\n\\begin{equation}\n\\begin{aligned}\nh_S&=0.08+\\frac{0.03}{2}=0.095,","ngleton contributions\n\\begin{equation}\n(0.08,0.12,0.10,0.16,0.14)\n\\end{equation}\nand pair synergies\n\\begin{equation}\n\\omega_{QT}=0.05,\n\\qquad\n\\omega_{CH}=0.06,\n\\qquad\n\\omega_{SC}=0.03.\n\\end{equation}\nThen\n\\begin{equation}\nF_K(N_5)=0.94,\n\\qquad\n\\Hcal(K)=0.74.\n\\end{equation}\nEvery pair synergy is divided equally between its two participants by the permutation-average rule. Therefore\n\\begin{equation}\n\\begin{aligned}\nh_S&=0.08+\\frac{0.03}{2}=0.095,\\\\\nh_Q&=0.12+\\frac{0.05}{2}=0.145,\\\\\nh_T&=0.10+\\frac{0.05}{2}=0.125,\\\\\nh_C&=0.16+\\frac{0.06+0.03}{2}=0.205,\\\\\nh_H&=0.14+\\frac{0.06}{2}=0.170.\n\\end{aligned}\n\\end{equation}\nThe allocation sums to\n\\begin{equation}\n0.095+0.145+0.125+0.205+0.170=0.740,\n\\end{equation}\nwhich verifies \\cref{thm:b8-shapley-efficiency} in the finite companion.","h_H&=0.14+\\frac{0.06}{2}=0.170.\n\\end{aligned}\n\\end{equation}\nThe allocation sums to\n\\begin{equation}\n0.095+0.145+0.125+0.205+0.170=0.740,\n\\end{equation}\nwhich verifies \\cref{thm:b8-shapley-efficiency} in the finite companion.","Suppose bridge-rank identification has success probability\n\\begin{equation}\np_{\\mathrm{rank}}(K_1)=1.\n\\end{equation}\nFor every child that changes only the rank-identification rule while leaving all other modules fixed,\n\\begin{equation}\np_{\\mathrm{rank}}(K_2)-p_{\\mathrm{rank}}(K_1)\\leq0.\n\\end{equation}\nThis does not imply that the repair process has no remaining headroom. Let the target bridge be full rank but have minimum singular value\n\\begin{equation}\n\\sigma_{\\min}=10^{-4}.\n\\end{equation}\nRandom 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\n\\begin{equation}\n\\max_{S:|S|=k}\n\\lambda_{\\min}(X_S^\\top X_S)\n\\end{equation}\nor a regularized determinant objec","Suppose bridge-rank identification has success probability\n\\begin{equation}\np_{\\mathrm{rank}}(K_1)=1.\n\\end{equation}\nFor every child that changes only the rank-identification rule while leaving all other modules fixed,\n\\begin{equation}\np_{\\mathrm{rank}}(K_2)-p_{\\mathrm{rank}}(K_1)\\leq0.\n\\end{equation}\nThis does not imply that the repair process has no remaining headroom. Let the target bridge be full rank but have minimum singular value\n\\begin{equation}\n\\sigma_{\\min}=10^{-4}.\n\\end{equation}\nRandom 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\n\\begin{equation}\n\\max_{S:|S|=k}\n\\lambda_{\\min}(X_S^\\top X_S)\n\\end{equation}\nor a regularized determinant objective, not another rank-only rule.","y\n\\begin{equation}\np_{\\mathrm{rank}}(K_1)=1.\n\\end{equation}\nFor every child that changes only the rank-identification rule while leaving all other modules fixed,\n\\begin{equation}\np_{\\mathrm{rank}}(K_2)-p_{\\mathrm{rank}}(K_1)\\leq0.\n\\end{equation}\nThis does not imply that the repair process has no remaining headroom. Let the target bridge be full rank but have minimum singular value\n\\begin{equation}\n\\sigma_{\\min}=10^{-4}.\n\\end{equation}\nRandom 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\n\\begin{equation}\n\\max_{S:|S|=k}\n\\lambda_{\\min}(X_S^\\top X_S)\n\\end{equation}\nor a regularized determinant objective, not another rank-only rule.","or every child that changes only the rank-identification rule while leaving all other modules fixed,\n\\begin{equation}\np_{\\mathrm{rank}}(K_2)-p_{\\mathrm{rank}}(K_1)\\leq0.\n\\end{equation}\nThis does not imply that the repair process has no remaining headroom. Let the target bridge be full rank but have minimum singular value\n\\begin{equation}\n\\sigma_{\\min}=10^{-4}.\n\\end{equation}\nRandom 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\n\\begin{equation}\n\\max_{S:|S|=k}\n\\lambda_{\\min}(X_S^\\top X_S)\n\\end{equation}\nor a regularized determinant objective, not another rank-only rule.","does not imply that the repair process has no remaining headroom. Let the target bridge be full rank but have minimum singular value\n\\begin{equation}\n\\sigma_{\\min}=10^{-4}.\n\\end{equation}\nRandom 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\n\\begin{equation}\n\\max_{S:|S|=k}\n\\lambda_{\\min}(X_S^\\top X_S)\n\\end{equation}\nor a regularized determinant objective, not another rank-only rule.","The Block 8 companion begins with\n\\begin{equation}\nr_0\n=\n(0.06,0.18,0.14,0.22,0.10)\n\\end{equation}\nfor 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\n\\begin{equation}\nC,\\ Q,\\ T,\\ H.\n\\end{equation}\nAfter these reductions, the old-coordinate vector is\n\\begin{equation}\n(0.06,0.045,0.0315,0.0495,0.0225).\n\\end{equation}\nA conservative grammar extension then introduces a new nonlinear coordinate of magnitude\n\\begin{equation}\n0.16.\n\\end{equation}\nThe represented headroom norm increases because a new limitation has been discovered. No old coordinate increases. Subsequent actions target the","The Block 8 companion begins with\n\\begin{equation}\nr_0\n=\n(0.06,0.18,0.14,0.22,0.10)\n\\end{equation}\nfor 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\n\\begin{equation}\nC,\\ Q,\\ T,\\ H.\n\\end{equation}\nAfter these reductions, the old-coordinate vector is\n\\begin{equation}\n(0.06,0.045,0.0315,0.0495,0.0225).\n\\end{equation}\nA conservative grammar extension then introduces a new nonlinear coordinate of magnitude\n\\begin{equation}\n0.16.\n\\end{equation}\nThe 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.",",0.18,0.14,0.22,0.10)\n\\end{equation}\nfor 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\n\\begin{equation}\nC,\\ Q,\\ T,\\ H.\n\\end{equation}\nAfter these reductions, the old-coordinate vector is\n\\begin{equation}\n(0.06,0.045,0.0315,0.0495,0.0225).\n\\end{equation}\nA conservative grammar extension then introduces a new nonlinear coordinate of magnitude\n\\begin{equation}\n0.16.\n\\end{equation}\nThe 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.","nate is multiplied by $1/4$. A neighboring coordinate receives a $10\\%$ verified reduction. The first four selected actions are\n\\begin{equation}\nC,\\ Q,\\ T,\\ H.\n\\end{equation}\nAfter these reductions, the old-coordinate vector is\n\\begin{equation}\n(0.06,0.045,0.0315,0.0495,0.0225).\n\\end{equation}\nA conservative grammar extension then introduces a new nonlinear coordinate of magnitude\n\\begin{equation}\n0.16.\n\\end{equation}\nThe 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.","Let the foreign obstruction covariance have eigenvalues\n\\begin{equation}\n4,\n2,\n1,\n0.5,\n0.2,\n0.1.\n\\end{equation}\nAssume the initial grammar already corrects the first eigenvector. The unresolved energy is\n\\begin{equation}\n2+1+0.5+0.2+0.1=3.8.\n\\end{equation}\nAdding the remaining eigenvectors in descending order produces\n\\begin{equation}\n3.8\n\\longrightarrow\n1.8\n\\longrightarrow\n0.8\n\\longrightarrow\n0.3\n\\longrightarrow\n0.1\n\\longrightarrow\n0.\n\\end{equation}\nThe energy drops are\n\\begin{equation}\n2,\n1,\n0.5,\n0.2,\n0.1,\n\\end{equation}\nexactly the newly correctable eigenvalues. This is the distributional form of \\cref{thm:b8-new-energy}.","0.1.\n\\end{equation}\nAssume the initial grammar already corrects the first eigenvector. The unresolved energy is\n\\begin{equation}\n2+1+0.5+0.2+0.1=3.8.\n\\end{equation}\nAdding the remaining eigenvectors in descending order produces\n\\begin{equation}\n3.8\n\\longrightarrow\n1.8\n\\longrightarrow\n0.8\n\\longrightarrow\n0.3\n\\longrightarrow\n0.1\n\\longrightarrow\n0.\n\\end{equation}\nThe energy drops are\n\\begin{equation}\n2,\n1,\n0.5,\n0.2,\n0.1,\n\\end{equation}\nexactly the newly correctable eigenvalues. This is the distributional form of \\cref{thm:b8-new-energy}.","Let the old correction image be\n\\begin{equation}\nI=\\Span\\{e_1,e_2\\}\\subset\\R^4\n\\end{equation}\nand let\n\\begin{equation}\nr=(0,0,3,4).\n\\end{equation}\nThen\n\\begin{equation}\nz=P_{I^\\perp}r=(0,0,3,4),\n\\qquad\n\\eta_I(r)^2=25.\n\\end{equation}\nAdd generator\n\\begin{equation}\nv=2e_4.\n\\end{equation}\nBecause $v\\in I^\\perp$,\n\\begin{equation}\n\\frac{|\\angles{z,v}|^2}{\\norm v^2}\n=\n\\frac{8^2}{4}\n=16.\n\\end{equation}\nThe expanded residual is $3e_3$, so\n\\begin{equation}\n\\eta_{I'}(r)^2=9=25-16.\n\\end{equation}\nA generator parallel to $e_1$ would have $\\bar v=0$ and would produce no structural reduction despite having nonzero raw norm.","age be\n\\begin{equation}\nI=\\Span\\{e_1,e_2\\}\\subset\\R^4\n\\end{equation}\nand let\n\\begin{equation}\nr=(0,0,3,4).\n\\end{equation}\nThen\n\\begin{equation}\nz=P_{I^\\perp}r=(0,0,3,4),\n\\qquad\n\\eta_I(r)^2=25.\n\\end{equation}\nAdd generator\n\\begin{equation}\nv=2e_4.\n\\end{equation}\nBecause $v\\in I^\\perp$,\n\\begin{equation}\n\\frac{|\\angles{z,v}|^2}{\\norm v^2}\n=\n\\frac{8^2}{4}\n=16.\n\\end{equation}\nThe expanded residual is $3e_3$, so\n\\begin{equation}\n\\eta_{I'}(r)^2=9=25-16.\n\\end{equation}\nA generator parallel to $e_1$ would have $\\bar v=0$ and would produce no structural reduction despite having nonzero raw norm.","\\Span\\{e_1,e_2\\}\\subset\\R^4\n\\end{equation}\nand let\n\\begin{equation}\nr=(0,0,3,4).\n\\end{equation}\nThen\n\\begin{equation}\nz=P_{I^\\perp}r=(0,0,3,4),\n\\qquad\n\\eta_I(r)^2=25.\n\\end{equation}\nAdd generator\n\\begin{equation}\nv=2e_4.\n\\end{equation}\nBecause $v\\in I^\\perp$,\n\\begin{equation}\n\\frac{|\\angles{z,v}|^2}{\\norm v^2}\n=\n\\frac{8^2}{4}\n=16.\n\\end{equation}\nThe expanded residual is $3e_3$, so\n\\begin{equation}\n\\eta_{I'}(r)^2=9=25-16.\n\\end{equation}\nA generator parallel to $e_1$ would have $\\bar v=0$ and would produce no structural reduction despite having nonzero raw norm.","n}\nr=(0,0,3,4).\n\\end{equation}\nThen\n\\begin{equation}\nz=P_{I^\\perp}r=(0,0,3,4),\n\\qquad\n\\eta_I(r)^2=25.\n\\end{equation}\nAdd generator\n\\begin{equation}\nv=2e_4.\n\\end{equation}\nBecause $v\\in I^\\perp$,\n\\begin{equation}\n\\frac{|\\angles{z,v}|^2}{\\norm v^2}\n=\n\\frac{8^2}{4}\n=16.\n\\end{equation}\nThe expanded residual is $3e_3$, so\n\\begin{equation}\n\\eta_{I'}(r)^2=9=25-16.\n\\end{equation}\nA generator parallel to $e_1$ would have $\\bar v=0$ and would produce no structural reduction despite having nonzero raw norm.","end{equation}\nThen\n\\begin{equation}\nz=P_{I^\\perp}r=(0,0,3,4),\n\\qquad\n\\eta_I(r)^2=25.\n\\end{equation}\nAdd generator\n\\begin{equation}\nv=2e_4.\n\\end{equation}\nBecause $v\\in I^\\perp$,\n\\begin{equation}\n\\frac{|\\angles{z,v}|^2}{\\norm v^2}\n=\n\\frac{8^2}{4}\n=16.\n\\end{equation}\nThe expanded residual is $3e_3$, so\n\\begin{equation}\n\\eta_{I'}(r)^2=9=25-16.\n\\end{equation}\nA generator parallel to $e_1$ would have $\\bar v=0$ and would produce no structural reduction despite having nonzero raw norm.","Index eight episodes by three bits\n\\begin{equation}\n(a,b,n)\\in\\{0,1\\}^3,\n\\end{equation}\nwhere $(a,b)$ 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.","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\n\\begin{equation}\na,\n\\quad\nb,\n\\end{equation}\nand rejects the nuisance feature.","The conservative embedding is immediate:\n\\begin{equation}\n\\phi_{\\Gcal'}(x)\n=\n(\\phi_{\\Gcal}(x),a(x),b(x)),\n\\end{equation}\nand projection onto the first coordinate recovers the old grammar.","Consider the typed repair programs\n\\begin{equation}\n\\begin{aligned}\np_1(a,b)&=(0,0),\\\\\np_2(a,b)&=(a,0),\\\\\np_3(a,b)&=(0,b),\\\\\np_4(a,b)&=(b,a),\\\\\np_5(a,b)&=(a,b).\n\\end{aligned}\n\\end{equation}\nThe 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.","p_5(a,b)&=(a,b).\n\\end{aligned}\n\\end{equation}\nThe 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.","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.","Let\n\\begin{equation}\nX=\n\\begin{bmatrix}0&1\\\\1&0\\end{bmatrix},\n\\qquad\nZ=\n\\begin{bmatrix}1&0\\\\0&-1\\end{bmatrix}.\n\\end{equation}\nOn $H=\\R^4$, define four response operators\n\\begin{equation}\nR_1=X\\oplus0,\n\\quad\nR_2=Z\\oplus0,\n\\quad\nR_3=0\\oplus X,\n\\quad\nR_4=0\\oplus Z.\n\\end{equation}\nThe adjoint-closed algebra they generate is\n\\begin{equation}\nM_2(\\R)\\oplus M_2(\\R),\n\\end{equation}\nwith dimension $8$. Its commutant consists of scalar multiples of the identity on each block and therefore has dimension $2$. The minimal central projectors are\n\\begin{equation}\nP_1=I_2\\oplus0,\n\\qquad\nP_2=0\\oplus I_2.\n\\end{equation}\nA residual $e_1$ generates the first two-dimensional block. A residual $e_1+e_3$ generates both blocks and has cyclic dimension $4$.","0&-1\\end{bmatrix}.\n\\end{equation}\nOn $H=\\R^4$, define four response operators\n\\begin{equation}\nR_1=X\\oplus0,\n\\quad\nR_2=Z\\oplus0,\n\\quad\nR_3=0\\oplus X,\n\\quad\nR_4=0\\oplus Z.\n\\end{equation}\nThe adjoint-closed algebra they generate is\n\\begin{equation}\nM_2(\\R)\\oplus M_2(\\R),\n\\end{equation}\nwith dimension $8$. Its commutant consists of scalar multiples of the identity on each block and therefore has dimension $2$. The minimal central projectors are\n\\begin{equation}\nP_1=I_2\\oplus0,\n\\qquad\nP_2=0\\oplus I_2.\n\\end{equation}\nA residual $e_1$ generates the first two-dimensional block. A residual $e_1+e_3$ generates both blocks and has cyclic dimension $4$.","d{bmatrix}.\n\\end{equation}\nOn $H=\\R^4$, define four response operators\n\\begin{equation}\nR_1=X\\oplus0,\n\\quad\nR_2=Z\\oplus0,\n\\quad\nR_3=0\\oplus X,\n\\quad\nR_4=0\\oplus Z.\n\\end{equation}\nThe adjoint-closed algebra they generate is\n\\begin{equation}\nM_2(\\R)\\oplus M_2(\\R),\n\\end{equation}\nwith dimension $8$. Its commutant consists of scalar multiples of the identity on each block and therefore has dimension $2$. The minimal central projectors are\n\\begin{equation}\nP_1=I_2\\oplus0,\n\\qquad\nP_2=0\\oplus I_2.\n\\end{equation}\nA residual $e_1$ generates the first two-dimensional block. A residual $e_1+e_3$ generates both blocks and has cyclic dimension $4$.","ponse operators\n\\begin{equation}\nR_1=X\\oplus0,\n\\quad\nR_2=Z\\oplus0,\n\\quad\nR_3=0\\oplus X,\n\\quad\nR_4=0\\oplus Z.\n\\end{equation}\nThe adjoint-closed algebra they generate is\n\\begin{equation}\nM_2(\\R)\\oplus M_2(\\R),\n\\end{equation}\nwith dimension $8$. Its commutant consists of scalar multiples of the identity on each block and therefore has dimension $2$. The minimal central projectors are\n\\begin{equation}\nP_1=I_2\\oplus0,\n\\qquad\nP_2=0\\oplus I_2.\n\\end{equation}\nA residual $e_1$ generates the first two-dimensional block. A residual $e_1+e_3$ generates both blocks and has cyclic dimension $4$.","quad\nR_3=0\\oplus X,\n\\quad\nR_4=0\\oplus Z.\n\\end{equation}\nThe adjoint-closed algebra they generate is\n\\begin{equation}\nM_2(\\R)\\oplus M_2(\\R),\n\\end{equation}\nwith dimension $8$. Its commutant consists of scalar multiples of the identity on each block and therefore has dimension $2$. The minimal central projectors are\n\\begin{equation}\nP_1=I_2\\oplus0,\n\\qquad\nP_2=0\\oplus I_2.\n\\end{equation}\nA residual $e_1$ generates the first two-dimensional block. A residual $e_1+e_3$ generates both blocks and has cyclic dimension $4$.","adjoint-closed algebra they generate is\n\\begin{equation}\nM_2(\\R)\\oplus M_2(\\R),\n\\end{equation}\nwith dimension $8$. Its commutant consists of scalar multiples of the identity on each block and therefore has dimension $2$. The minimal central projectors are\n\\begin{equation}\nP_1=I_2\\oplus0,\n\\qquad\nP_2=0\\oplus I_2.\n\\end{equation}\nA residual $e_1$ generates the first two-dimensional block. A residual $e_1+e_3$ generates both blocks and has cyclic dimension $4$.","M_2(\\R)\\oplus M_2(\\R),\n\\end{equation}\nwith dimension $8$. Its commutant consists of scalar multiples of the identity on each block and therefore has dimension $2$. The minimal central projectors are\n\\begin{equation}\nP_1=I_2\\oplus0,\n\\qquad\nP_2=0\\oplus I_2.\n\\end{equation}\nA residual $e_1$ generates the first two-dimensional block. A residual $e_1+e_3$ generates both blocks and has cyclic dimension $4$.","Suppose four accepted edges use conditional false-acceptance budgets\n\\begin{equation}\n0.0100,\n0.0050,\n0.0025,\n0.00125.\n\\end{equation}\nThen\n\\begin{equation}\n\\sum_{t=0}^{3}\\delta_t=0.01875.\n\\end{equation}\nBy \\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.","If the protected-regression budgets are\n\\begin{equation}\n0.010,\n0.008,\n0.006,\n0.004,\n\\end{equation}\nthen the cumulative declared budget is\n\\begin{equation}\nB_R=0.028.\n\\end{equation}\nThese are accounting guarantees, not evidence that the numerical budgets are ethically sufficient.","Let a fixed compiler $\\Gamma$ map a parent process and receipts to a child:\n\\begin{equation}\nK_{t+1}=\\Gamma(K_t,\\Ecal_t).\n\\end{equation}\nSuppose it produces two positive edges\n\\begin{equation}\n\\Mcal(K_1)-\\Mcal(K_0)>0,\n\\qquad\n\\Mcal(K_2)-\\Mcal(K_1)>0.\n\\end{equation}\nThis establishes repeated output-process improvement by one compiler. It does not establish\n\\begin{equation}\n\\Ccal(\\Gamma_1)>\\Ccal(\\Gamma_0).\n\\end{equation}\nThe 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.","Let\n\\begin{equation}\nX_\\epsilon=\\diag(1,1,\\epsilon),\n\\qquad 0<\\epsilon\\ll1.\n\\end{equation}\nThen $\\rank X_\\epsilon=3$, but\n\\begin{equation}\n\\lambda_{\\min}(X_\\epsilon^{\\mathsf T}X_\\epsilon)=\\epsilon^2\n\\end{equation}\nand the worst-direction least-squares variance scales as $\\epsilon^{-2}$. A binary rank gate cannot distinguish this design from an orthogonal one.","Let $M=I_2$, $x=(2,1)^{\\mathsf T}$, and $w=4$. The D-optimal information increment is\n\\begin{equation}\n\\log\\det(M+wxx^{\\mathsf T})-\\log\\det M\n=\n\\log(1+w x^{\\mathsf T}M^{-1}x)\n=\n\\log 21.\n\\end{equation}\nA low-quality copy with $w=1/16$ gives only $\\log(1+5/16)$, despite identical source geometry.","For $M=\\diag(1,0.1)$ and $v=e_2$, the A-optimal reduction is\n\\begin{equation}\n\\frac{v^{\\mathsf T}M^{-2}v}{1+v^{\\mathsf T}M^{-1}v}\n=\n\\frac{100}{11}.\n\\end{equation}\nThe update targets the weakly informed second direction rather than increasing an already strong direction.","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.","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\n\\begin{equation}\n\\Mcal_{\\Xi_0}(K_2)>\\Mcal_{\\Xi_0}(K_0)\n\\end{equation}\nor any additive path gain unless $\\Xi_0$ and $\\Xi_1$ share a common utility or a validated context transport.","The frozen primary means are\n\\begin{equation}\n\\Mcal(K_1^{\\rm recal})=0.037393,\n\\qquad\n\\Mcal(K_2)=0.209580.\n\\end{equation}\nHence\n\\begin{equation}\n\\widehat\\Delta_{1\\to2}=0.172188.\n\\end{equation}\nThe interval $[0.141481,0.203110]$ is strictly positive. Both arms are full rank, so rank recovery cannot explain the edge.","Let $h\\sim\\mathcal N(0,1)$ and let $s\\in\\{-1,+1\\}$ be independent and balanced. Define\n\\begin{equation}\ny=(a+sb)h,\n\\qquad b\\neq0.\n\\end{equation}\nA global scalar bridge predicts $\\widehat y=th$. The population minimizer is $t^*=a$, and\n\\begin{equation}\n\\inf_t\\mathbb E (y-th)^2\n=\n\\mathbb E(sbh)^2\n=\nb^2.\n\\end{equation}\nThe 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\n\\begin{equation}\nt_-=a-b,\n\\qquad\nt_+=a+b,\n\\end{equation}\nand has zero noiseless approximation error. This is the one-dimensional form of \\cref{thm:b10-open-set-obstruction}.","let $s\\in\\{-1,+1\\}$ be independent and balanced. Define\n\\begin{equation}\ny=(a+sb)h,\n\\qquad b\\neq0.\n\\end{equation}\nA global scalar bridge predicts $\\widehat y=th$. The population minimizer is $t^*=a$, and\n\\begin{equation}\n\\inf_t\\mathbb E (y-th)^2\n=\n\\mathbb E(sbh)^2\n=\nb^2.\n\\end{equation}\nThe 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\n\\begin{equation}\nt_-=a-b,\n\\qquad\nt_+=a+b,\n\\end{equation}\nand has zero noiseless approximation error. This is the one-dimensional form of \\cref{thm:b10-open-set-obstruction}.","\\{-1,+1\\}$ be independent and balanced. Define\n\\begin{equation}\ny=(a+sb)h,\n\\qquad b\\neq0.\n\\end{equation}\nA global scalar bridge predicts $\\widehat y=th$. The population minimizer is $t^*=a$, and\n\\begin{equation}\n\\inf_t\\mathbb E (y-th)^2\n=\n\\mathbb E(sbh)^2\n=\nb^2.\n\\end{equation}\nThe 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\n\\begin{equation}\nt_-=a-b,\n\\qquad\nt_+=a+b,\n\\end{equation}\nand has zero noiseless approximation error. This is the one-dimensional form of \\cref{thm:b10-open-set-obstruction}.","egin{equation}\ny=(a+sb)h,\n\\qquad b\\neq0.\n\\end{equation}\nA global scalar bridge predicts $\\widehat y=th$. The population minimizer is $t^*=a$, and\n\\begin{equation}\n\\inf_t\\mathbb E (y-th)^2\n=\n\\mathbb E(sbh)^2\n=\nb^2.\n\\end{equation}\nThe 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\n\\begin{equation}\nt_-=a-b,\n\\qquad\nt_+=a+b,\n\\end{equation}\nand has zero noiseless approximation error. This is the one-dimensional form of \\cref{thm:b10-open-set-obstruction}.","Let the old bridge be $T\\in\\R^{q\\times d}$ and let $g$ be any admissible separator. The embedding\n\\begin{equation}\nT\\longmapsto(g,T,T)\n\\end{equation}\nreturns $Th$ on both charts. Hence the local grammar can execute the inherited process without approximation. A conservative runtime may therefore use\n\\begin{equation}\nK_{\\mathrm{chart}}(h)\n=\n\\begin{cases}\nK_{\\mathrm{local}}(h),&\\text{if the chart certificate passes},","Let the old bridge be $T\\in\\R^{q\\times d}$ and let $g$ be any admissible separator. The embedding\n\\begin{equation}\nT\\longmapsto(g,T,T)\n\\end{equation}\nreturns $Th$ on both charts. Hence the local grammar can execute the inherited process without approximation. A conservative runtime may therefore use\n\\begin{equation}\nK_{\\mathrm{chart}}(h)\n=\n\\begin{cases}\nK_{\\mathrm{local}}(h),&\\text{if the chart certificate passes},\\\\\nK_{\\mathrm{global}}(h),&\\text{otherwise}.\n\\end{cases}\n\\end{equation}\nThe 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}.","Equip\n\\begin{equation}\n\\End(H_t)\\oplus\\End(H_t)\n\\end{equation}\nwith the product Frobenius norm. For a chart repair pair $(D_-,D_+)$, write\n\\begin{equation}\nD_{\\mathrm{avg}}\n=\n\\frac{D_-+D_+}{2},\n\\qquad\nD_{\\mathrm{ctr}}\n=\n\\frac{D_+-D_-}{2}.\n\\end{equation}\nThe closest diagonal repair is $(D_{\\mathrm{avg}},D_{\\mathrm{avg}})$. Therefore\n\\begin{align}\n\\operatorname{dist}^2\n\\left(\n(D_-,D_+),\\Dcal_{\\mathrm{diag}}\n\\right)\n&=\n\\norm{D_--D_{\\mathrm{avg}}}_F^2\n+\n\\norm{D_+-D_{\\mathrm{avg}}}_F^2","Equip\n\\begin{equation}\n\\End(H_t)\\oplus\\End(H_t)\n\\end{equation}\nwith the product Frobenius norm. For a chart repair pair $(D_-,D_+)$, write\n\\begin{equation}\nD_{\\mathrm{avg}}\n=\n\\frac{D_-+D_+}{2},\n\\qquad\nD_{\\mathrm{ctr}}\n=\n\\frac{D_+-D_-}{2}.\n\\end{equation}\nThe closest diagonal repair is $(D_{\\mathrm{avg}},D_{\\mathrm{avg}})$. Therefore\n\\begin{align}\n\\operatorname{dist}^2\n\\left(\n(D_-,D_+),\\Dcal_{\\mathrm{diag}}\n\\right)\n&=\n\\norm{D_--D_{\\mathrm{avg}}}_F^2\n+\n\\norm{D_+-D_{\\mathrm{avg}}}_F^2\\\\\n&=\n2\\norm{D_{\\mathrm{ctr}}}_F^2\\\\\n&=\n\\frac12\\norm{D_+-D_-}_F^2.\n\\end{align}\nThus $D_+\\neq D_-$ gives a quantitative structural-novelty certificate, not merely two differently named copies of one global operator.","&=\n\\frac12\\norm{D_+-D_-}_F^2.\n\\end{align}\nThus $D_+\\neq D_-$ gives a quantitative structural-novelty certificate, not merely two differently named copies of one global operator.","Suppose the true target relation depends on $s=\\one\\{h_1>0\\}$, but a candidate separator uses\n\\begin{equation}\ng(h)=\\one\\{h_2>0\\},\n\\end{equation}\nwhere $h_2$ 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,\n\\begin{equation}\n\\eps_{\\mathrm{loc}}\n=\n\\eps_{\\mathrm{glob}},\n\\qquad\n\\Delta_{\\mathrm{gram}}=0.\n\\end{equation}\nA larger syntax is therefore not enough. Structural credit requires a held-out residual reduction, conservative replay, a nonzero repair contrast, and foreign protected gain.","e true target relation depends on $s=\\one\\{h_1>0\\}$, but a candidate separator uses\n\\begin{equation}\ng(h)=\\one\\{h_2>0\\},\n\\end{equation}\nwhere $h_2$ 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,\n\\begin{equation}\n\\eps_{\\mathrm{loc}}\n=\n\\eps_{\\mathrm{glob}},\n\\qquad\n\\Delta_{\\mathrm{gram}}=0.\n\\end{equation}\nA larger syntax is therefore not enough. Structural credit requires a held-out residual reduction, conservative replay, a nonzero repair contrast, and foreign protected gain.","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\n\\begin{equation}\n\\Mcal(K_{2,G}^{96})=0.000000,\n\\qquad\n\\Mcal(K_{3,C}^{96})=0.334224.\n\\end{equation}\nHence\n\\begin{equation}\n\\widehat\\Delta_{G\\to C}=0.334224,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.266459,0.401903].\n\\end{equation}\nThe direct global-affine oracle has mean gain $0.176694$, and the child exceeds it by\n\\begin{equation}\n0.157530,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.080863,0.234135].\n\\end{equation}\nValidation alignment NRMSE changes from\n\\begin{equation}\n\\eps_{\\mathrm{glob}}=0.598513\n\\end{equation","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\n\\begin{equation}\n\\Mcal(K_{2,G}^{96})=0.000000,\n\\qquad\n\\Mcal(K_{3,C}^{96})=0.334224.\n\\end{equation}\nHence\n\\begin{equation}\n\\widehat\\Delta_{G\\to C}=0.334224,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.266459,0.401903].\n\\end{equation}\nThe direct global-affine oracle has mean gain $0.176694$, and the child exceeds it by\n\\begin{equation}\n0.157530,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.080863,0.234135].\n\\end{equation}\nValidation alignment NRMSE changes from\n\\begin{equation}\n\\eps_{\\mathrm{glob}}=0.598513\n\\end{equation}\nto\n\\begin{equation","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\n\\begin{equation}\n\\Mcal(K_{2,G}^{96})=0.000000,\n\\qquad\n\\Mcal(K_{3,C}^{96})=0.334224.\n\\end{equation}\nHence\n\\begin{equation}\n\\widehat\\Delta_{G\\to C}=0.334224,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.266459,0.401903].\n\\end{equation}\nThe direct global-affine oracle has mean gain $0.176694$, and the child exceeds it by\n\\begin{equation}\n0.157530,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.080863,0.234135].\n\\end{equation}\nValidation alignment NRMSE changes from\n\\begin{equation}\n\\eps_{\\mathrm{glob}}=0.598513\n\\end{equation}\nto\n\\begin{equation}\n\\eps_{\\mathrm{loc}}=0.064975.\n\\end{equati","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\n\\begin{equation}\n\\Mcal(K_{2,G}^{96})=0.000000,\n\\qquad\n\\Mcal(K_{3,C}^{96})=0.334224.\n\\end{equation}\nHence\n\\begin{equation}\n\\widehat\\Delta_{G\\to C}=0.334224,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.266459,0.401903].\n\\end{equation}\nThe direct global-affine oracle has mean gain $0.176694$, and the child exceeds it by\n\\begin{equation}\n0.157530,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.080863,0.234135].\n\\end{equation}\nValidation alignment NRMSE changes from\n\\begin{equation}\n\\eps_{\\mathrm{glob}}=0.598513\n\\end{equation}\nto\n\\begin{equation}\n\\eps_{\\mathrm{loc}}=0.064975.\n\\end{equation}\nMean gate acc","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\n\\begin{equation}\n\\Mcal(K_{2,G}^{96})=0.000000,\n\\qquad\n\\Mcal(K_{3,C}^{96})=0.334224.\n\\end{equation}\nHence\n\\begin{equation}\n\\widehat\\Delta_{G\\to C}=0.334224,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.266459,0.401903].\n\\end{equation}\nThe direct global-affine oracle has mean gain $0.176694$, and the child exceeds it by\n\\begin{equation}\n0.157530,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.080863,0.234135].\n\\end{equation}\nValidation alignment NRMSE changes from\n\\begin{equation}\n\\eps_{\\mathrm{glob}}=0.598513\n\\end{equation}\nto\n\\begin{equation}\n\\eps_{\\mathrm{loc}}=0.064975.\n\\end{equation}\nMean gate accuracy is $0.874854$, no ha","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\n\\begin{equation}\n\\Mcal(K_{2,G}^{96})=0.000000,\n\\qquad\n\\Mcal(K_{3,C}^{96})=0.334224.\n\\end{equation}\nHence\n\\begin{equation}\n\\widehat\\Delta_{G\\to C}=0.334224,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.266459,0.401903].\n\\end{equation}\nThe direct global-affine oracle has mean gain $0.176694$, and the child exceeds it by\n\\begin{equation}\n0.157530,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.080863,0.234135].\n\\end{equation}\nValidation alignment NRMSE changes from\n\\begin{equation}\n\\eps_{\\mathrm{glob}}=0.598513\n\\end{equation}\nto\n\\begin{equation}\n\\eps_{\\mathrm{loc}}=0.064975.\n\\end{equation}\nMean gate accuracy is $0.874854$, no harmful primary target is selected, and gate-on","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\n\\begin{equation}\n\\Mcal(K_{2,G}^{96})=0.000000,\n\\qquad\n\\Mcal(K_{3,C}^{96})=0.334224.\n\\end{equation}\nHence\n\\begin{equation}\n\\widehat\\Delta_{G\\to C}=0.334224,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.266459,0.401903].\n\\end{equation}\nThe direct global-affine oracle has mean gain $0.176694$, and the child exceeds it by\n\\begin{equation}\n0.157530,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.080863,0.234135].\n\\end{equation}\nValidation alignment NRMSE changes from\n\\begin{equation}\n\\eps_{\\mathrm{glob}}=0.598513\n\\end{equation}\nto\n\\begin{equation}\n\\eps_{\\mathrm{loc}}=0.064975.\n\\end{equation}\nMean 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 glob","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\n\\begin{equation}\n\\Mcal(K_{2,G}^{96})=0.000000,\n\\qquad\n\\Mcal(K_{3,C}^{96})=0.334224.\n\\end{equation}\nHence\n\\begin{equation}\n\\widehat\\Delta_{G\\to C}=0.334224,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.266459,0.401903].\n\\end{equation}\nThe direct global-affine oracle has mean gain $0.176694$, and the child exceeds it by\n\\begin{equation}\n0.157530,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.080863,0.234135].\n\\end{equation}\nValidation alignment NRMSE changes from\n\\begin{equation}\n\\eps_{\\mathrm{glob}}=0.598513\n\\end{equation}\nto\n\\begin{equation}\n\\eps_{\\mathrm{loc}}=0.064975.\n\\end{equation}\nMean 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 …","fit, $32$ chart-selection, and $16$ witness queries. Both therefore consume $96$ paired queries. Their primary mean selected gains are\n\\begin{equation}\n\\Mcal(K_{2,G}^{96})=0.000000,\n\\qquad\n\\Mcal(K_{3,C}^{96})=0.334224.\n\\end{equation}\nHence\n\\begin{equation}\n\\widehat\\Delta_{G\\to C}=0.334224,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.266459,0.401903].\n\\end{equation}\nThe direct global-affine oracle has mean gain $0.176694$, and the child exceeds it by\n\\begin{equation}\n0.157530,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.080863,0.234135].\n\\end{equation}\nValidation alignment NRMSE changes from\n\\begin{equation}\n\\eps_{\\mathrm{glob}}=0.598513\n\\end{equation}\nto\n\\begin{equation}\n\\eps_{\\mathrm{loc}}=0.064975.\n\\end{equation}\nMean 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.","th therefore consume $96$ paired queries. Their primary mean selected gains are\n\\begin{equation}\n\\Mcal(K_{2,G}^{96})=0.000000,\n\\qquad\n\\Mcal(K_{3,C}^{96})=0.334224.\n\\end{equation}\nHence\n\\begin{equation}\n\\widehat\\Delta_{G\\to C}=0.334224,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.266459,0.401903].\n\\end{equation}\nThe direct global-affine oracle has mean gain $0.176694$, and the child exceeds it by\n\\begin{equation}\n0.157530,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.080863,0.234135].\n\\end{equation}\nValidation alignment NRMSE changes from\n\\begin{equation}\n\\eps_{\\mathrm{glob}}=0.598513\n\\end{equation}\nto\n\\begin{equation}\n\\eps_{\\mathrm{loc}}=0.064975.\n\\end{equation}\nMean 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.","quad\n\\Mcal(K_{3,C}^{96})=0.334224.\n\\end{equation}\nHence\n\\begin{equation}\n\\widehat\\Delta_{G\\to C}=0.334224,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.266459,0.401903].\n\\end{equation}\nThe direct global-affine oracle has mean gain $0.176694$, and the child exceeds it by\n\\begin{equation}\n0.157530,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.080863,0.234135].\n\\end{equation}\nValidation alignment NRMSE changes from\n\\begin{equation}\n\\eps_{\\mathrm{glob}}=0.598513\n\\end{equation}\nto\n\\begin{equation}\n\\eps_{\\mathrm{loc}}=0.064975.\n\\end{equation}\nMean 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.","uation}\n\\widehat\\Delta_{G\\to C}=0.334224,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.266459,0.401903].\n\\end{equation}\nThe direct global-affine oracle has mean gain $0.176694$, and the child exceeds it by\n\\begin{equation}\n0.157530,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.080863,0.234135].\n\\end{equation}\nValidation alignment NRMSE changes from\n\\begin{equation}\n\\eps_{\\mathrm{glob}}=0.598513\n\\end{equation}\nto\n\\begin{equation}\n\\eps_{\\mathrm{loc}}=0.064975.\n\\end{equation}\nMean 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.","0.266459,0.401903].\n\\end{equation}\nThe direct global-affine oracle has mean gain $0.176694$, and the child exceeds it by\n\\begin{equation}\n0.157530,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.080863,0.234135].\n\\end{equation}\nValidation alignment NRMSE changes from\n\\begin{equation}\n\\eps_{\\mathrm{glob}}=0.598513\n\\end{equation}\nto\n\\begin{equation}\n\\eps_{\\mathrm{loc}}=0.064975.\n\\end{equation}\nMean 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.","Let\n\\begin{equation}\nR_1=P_1+2P_2+4P_3,\n\\qquad\nR_2=3P_1-P_2+2P_3,\n\\end{equation}\nwhere $P_1,P_2,P_3$ are pairwise orthogonal rank-two projectors. Take\n\\begin{equation}\nB=R_1+2R_2=7P_1+0P_2+8P_3.\n\\end{equation}\nThe three eigenvalues $7,0,8$ are distinct. Therefore\n\\begin{align}\nP_1&=\\frac{B(B-8I)}{7(7-8)},","Let\n\\begin{equation}\nR_1=P_1+2P_2+4P_3,\n\\qquad\nR_2=3P_1-P_2+2P_3,\n\\end{equation}\nwhere $P_1,P_2,P_3$ are pairwise orthogonal rank-two projectors. Take\n\\begin{equation}\nB=R_1+2R_2=7P_1+0P_2+8P_3.\n\\end{equation}\nThe three eigenvalues $7,0,8$ are distinct. Therefore\n\\begin{align}\nP_1&=\\frac{B(B-8I)}{7(7-8)},\\\\\nP_2&=\\frac{(B-7I)(B-8I)}{(0-7)(0-8)},\\\\\nP_3&=\\frac{B(B-7I)}{8(8-7)}.\n\\end{align}\nEvery 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}.","Suppose\n\\begin{equation}\nH=H_1\\oplus H_2\\oplus H_3,\n\\qquad\n\\dim H_a=2.\n\\end{equation}\nUnder separating block-scalar responses, every commuting operator has form\n\\begin{equation}\nX=X_1\\oplus X_2\\oplus X_3,\n\\qquad\nX_a\\in M_2(\\R).\n\\end{equation}\nHence\n\\begin{equation}\n\\dim\\mathfrak A_R'\n=3\\dim M_2(\\R)\n=12.\n\\end{equation}\nThe companion's modal commutant dimension is $12$, matching the exact finite theory.","Let\n\\begin{equation}\nR_1=P_1+P_2+3P_3,\n\\qquad\nR_2=2P_1+2P_2-P_3.\n\\end{equation}\nBlocks $1$ 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\n\\begin{equation}\nP_{12}=P_1+P_2,\n\\end{equation}\nnot 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.","Suppose a selected response combination has eigenvalue clusters separated by\n\\begin{equation}\n\\gamma=0.04\n\\end{equation}\nand estimation error\n\\begin{equation}\n\\norm{E}_2=0.003.\n\\end{equation}\nThe conditional projector bound gives\n\\begin{equation}\n\\norm{\\widehat P-P}_2\n\\leq\n\\frac{2(0.003)}{0.04}\n=0.15.\n\\end{equation}\nIf 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.","Let the true context packet be\n\\begin{equation}\nP_*=UU^{\\mathsf T}\n\\end{equation}\nfor a target-specific orthonormal matrix $U\\in\\R^{6\\times2}$, while the inherited packet is\n\\begin{equation}\nP_0=\\diag(1,1,0,0,0,0).\n\\end{equation}\nLet the chart label be\n\\begin{equation}\nc(h)=\\one\\{e_1^{\\mathsf T}U^{\\mathsf T}h>0\\}.\n\\end{equation}\nFor 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.","in{equation}\nP_*=UU^{\\mathsf T}\n\\end{equation}\nfor a target-specific orthonormal matrix $U\\in\\R^{6\\times2}$, while the inherited packet is\n\\begin{equation}\nP_0=\\diag(1,1,0,0,0,0).\n\\end{equation}\nLet the chart label be\n\\begin{equation}\nc(h)=\\one\\{e_1^{\\mathsf T}U^{\\mathsf T}h>0\\}.\n\\end{equation}\nFor 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.","thonormal matrix $U\\in\\R^{6\\times2}$, while the inherited packet is\n\\begin{equation}\nP_0=\\diag(1,1,0,0,0,0).\n\\end{equation}\nLet the chart label be\n\\begin{equation}\nc(h)=\\one\\{e_1^{\\mathsf T}U^{\\mathsf T}h>0\\}.\n\\end{equation}\nFor 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.","Both processes use $120$ effect-query units. The inherited parent uses\n\\begin{equation}\n80+16+24=120\n\\end{equation}\nordinary bridge, chart-validation, and witness queries. The endogenous child uses\n\\begin{equation}\n56+24+16+24=120\n\\end{equation}\nbridge, response, chart-validation, and witness queries.","On $80$ fresh foreign targets,\n\\begin{equation}\n\\Mcal(K_3)=0,\n\\qquad\n\\Mcal(K_4)=0.207325,\n\\end{equation}\nso\n\\begin{equation}\n\\widehat\\Delta_{3\\to4}=0.207325,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.131902,0.287910].\n\\end{equation}\nThe 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.","$ fresh foreign targets,\n\\begin{equation}\n\\Mcal(K_3)=0,\n\\qquad\n\\Mcal(K_4)=0.207325,\n\\end{equation}\nso\n\\begin{equation}\n\\widehat\\Delta_{3\\to4}=0.207325,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.131902,0.287910].\n\\end{equation}\nThe 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.","\\begin{equation}\n\\Mcal(K_3)=0,\n\\qquad\n\\Mcal(K_4)=0.207325,\n\\end{equation}\nso\n\\begin{equation}\n\\widehat\\Delta_{3\\to4}=0.207325,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.131902,0.287910].\n\\end{equation}\nThe 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.","4)=0.207325,\n\\end{equation}\nso\n\\begin{equation}\n\\widehat\\Delta_{3\\to4}=0.207325,\n\\qquad\n\\mathrm{CI}_{95\\%}=[0.131902,0.287910].\n\\end{equation}\nThe 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.","\\subsection{An exact response alias}\nLet 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.","\\subsection{A weak separator is statistically inadequate}\nIf 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.","\\subsection{Nonzero Hessian defeats every affine composition}\nFor $z\\sim N(0,I_2)$ 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.","\\subsection{Evaluator-nullspace shrink}\nLet $T=\\R^2$, $Db_1(x_1,x_2)=x_1$, 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.","\\subsection{Uniform forecast-regret bound}\nSuppose 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.","\\subsection{Actual Block 13 compiler edge}\nOn the primary audit, the parent and child gains are $0.883004$ and $0.950231$, so\n\\begin{equation}\n\\widehat\\Delta_\\Gamma^{\\mathrm{primary}}=0.067227.\n\\end{equation}\nThe child changes research allocation rather than the candidate grammar. The architecture-foreign paired effect is $0.090800$.","\\subsection{Archive and monitor controls}\nShuffling 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.","\\subsection{Aliased static diagnostics}\nLet $Z\\in\\{0,1\\}$ 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.","\\subsection{Two-obstruction mixture}\nLet\n\\begin{equation}\nR=\n\\begin{bmatrix}\n1&0&1/2\\\\\n0&1&1/2\\\\\n1&-1&0\n\\end{bmatrix},\n\\qquad\nz=\n\\begin{bmatrix}\n0.6\\\\0.4\\\\0\n\\end{bmatrix}.\n\\end{equation}\nThen $y=Rz=(0.6,0.4,0.2)^\\top$. 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.","0\n\\end{bmatrix}.\n\\end{equation}\nThen $y=Rz=(0.6,0.4,0.2)^\\top$. 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.","\\subsection{Open-set residual}\nLet inherited known responses span the nonnegative cone generated by $e_1$ and $e_2$ in $\\R^3$. The unknown response $y=e_3$ has\n\\begin{equation}\n\\dist(y,\\operatorname{cone}\\{e_1,e_2\\})=1.\n\\end{equation}\nAny 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.","\\subsection{Conjunctive integrity monitoring}\nSuppose 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.","\\subsection{Zero observed invalid selections}\nIf zero invalid selections are observed in $n=144$ independent audit problems, the exact one-sided $95\\%$ upper bound is\n\\begin{equation}\n1-0.05^{1/144}\n\\approx\n0.02059.\n\\end{equation}\nThe correct statement is therefore ``zero observed, upper bound approximately $2.06\\%$,'' not ``the compiler has zero invalid-selection probability.''","\\subsection{One unknown token with incompatible methods}\nLet two problems $a$ and $b$ both map to $\\bot$. Suppose\n\\begin{equation}\n\\Acal^\\star(a)=\\{m_1\\},\n\\qquad\n\\Acal^\\star(b)=\\{m_2\\},\n\\qquad\nm_1\\neq m_2.\n\\end{equation}\nAny 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.","\\subsection{A conservative binary refinement}\nLet 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.","\\subsection{Why taxonomy-only growth is insufficient}\nSuppose 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 $0.3$.","\\subsection{Actual Block 15 structural margin}\nThe development inherited-method oracle is $0.377257$ and the selected language obtains $0.948962$. Thus\n\\begin{equation}\n\\widehat\\kappa=0.948962-0.377257=0.571704.\n\\end{equation}\nOn the primary foreign audit, the matched-budget child-parent effect is $0.225762$; under generator shift it is $0.240296$.","\\subsection{Finite pair grammar versus unrestricted synthesis}\nThe 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.","\\endfoot\n\\bottomrule\n\\endlastfoot\nProtected evaluation context & $\\Longrightarrow$ & Nonvacuous improvement relation","Relative introspection & $\\Longrightarrow$ & Bounded decision regret","Authorized response map & $\\Longrightarrow$ & Correctable--obstructed decomposition","Local repair and overlap coherence & $\\Longrightarrow$ & Repair descent","Transport squares and holonomy bounds & $\\Longrightarrow$ & Gain-preserving transfer","Evaluator transports and adequacy & $\\Longrightarrow$ & Conservative evaluator evolution","Frozen witness and trace factorization & $\\Longrightarrow$ & Transactional noncircularity","Immutable checkpoint and buffered effects & $\\Longrightarrow$ & Atomic rollback","Finite transition game & $\\Longrightarrow$ & Universal and controlled closure","Safe attractor and validated gain set & $\\Longrightarrow$ & Productive B\\\"uchi closure","Append-only archive and multiobjective descriptors & $\\Longrightarrow$ & Monotone envelopes and finite covers","Matched foreign agents and paired evidence & $\\Longrightarrow$ & Metaproductivity certificate","Nested repair images and defect distribution & $\\Longrightarrow$ & Task-relevant structural gain","Conservative grammar chain & $\\Longrightarrow$ & Proof persistence and directed limit","Ontology embedding & $\\Longrightarrow$ & Resolved old/new obstruction exact sequence","Causal offspring receipts & $\\Longrightarrow$ & Repair-discovery matrix","Runtime refinement and observation commutation & $\\Longrightarrow$ & Trace-sound implementation","Finite exact realization and exhaustive endpoints & $\\Longrightarrow$ & Decidable fixture claims","Sealed package and cumulative claim gates & $\\Longrightarrow$ & Bounded empirical interpretation","Causal intervention differences and rank condition & $\\Longrightarrow$ & Hidden repair-packet identification","Spanning paired anchors and linear hidden relation & $\\Longrightarrow$ & Identified repair bridge","Injective bridge and frozen source repair & $\\Longrightarrow$ & Exact transported repair","Compression or nonlinear residual and frozen witness & $\\Longrightarrow$ & Commit or abstain","Development receipts and separate meta-validation & $\\Longrightarrow$ & Sealed child selector","Identical candidate bundles and paired foreign audit & $\\Longrightarrow$ & Selector-level metaproductivity certificate","Risk upper score and protected-regression gates & $\\Longrightarrow$ & Constrained child commitment","Fixed candidate grammar & $\\Longrightarrow$ & No structural repair-image claim","Fresh grandchild edge and newly sealed evidence & $\\Longrightarrow$ & First repeated-lineage test","Heteroscedastic paired-anchor model and valid quality weights & $\\Longrightarrow$ & Weighted bridge covariance and information geometry","Full source rank with low $\\lambda_{\\min}(M_S)$ & $\\Longrightarrow$ & Conditioning bottleneck beyond identifiability","Quality-aware D-optimal design and matched query budget & $\\Longrightarrow$ & Conditioning-aware grandchild candidate","Ancestor replay, fresh supports, pre-audit seal, and paired audit & $\\Longrightarrow$ & Two-edge context-indexed repeated-lineage certificate","Restricted oracle envelopes and module factorization & $\\Longrightarrow$ & Meta-headroom and saturation","Headroom state and authorized contraction action & $\\Longrightarrow$ & Bottleneck migration","Nested correction images and foreign residuals & $\\Longrightarrow$ & Persistent obstruction energy","Projected obstruction and candidate response direction & $\\Longrightarrow$ & Generator-acquisition score","Repair-conflict hypergraph and separating features & $\\Longrightarrow$ & Conservative grammar refinement","Intervention-response operators and adjoint closure & $\\Longrightarrow$ & Central causal atlas","Conditional edge bounds and summable budgets & $\\Longrightarrow$ & Lineage-wide validity","Persistent global residual and finite chart separator family & $\\Longrightarrow$ & Grammar-inadequacy certificate","Conservative global-to-chart embedding and nonzero repair contrast & $\\Longrightarrow$ & New chart-specific correction direction","Matched foreign audit with binding and gate controls & $\\Longrightarrow$ & Finite structural metaproductivity certificate","Separating block-scalar intervention responses & $\\Longrightarrow$ & Exact central response atlas up to permutation and within-block gauge","Response estimation error and spectral gap & $\\Longrightarrow$ & Approximate central-projector stability","Recovered response blocks and repair-conflict validation & $\\Longrightarrow$ & Repair-relevant atlas labeling","Supplied-packet removal, conservative fallback, and matched foreign audit & $\\Longrightarrow$ & Finite endogenous-atlas metaproductivity certificate","Inherited block-scalar response alias & $\\Longrightarrow$ & Polynomial nonseparation certificate","Separating candidate intervention and spectral margin & $\\Longrightarrow$ & Conservative central response refinement","Matched response audit and correspondence controls & $\\Longrightarrow$ & Finite response-language metaproductivity certificate","Affine repair-operad closure and nonzero Hessian & $\\Longrightarrow$ & Constructor function-class obstruction","Degree-two conservative embedding and matched audit & $\\Longrightarrow$ & Finite candidate-constructor metaproductivity certificate","Results-free proposal fields and immutable code digest & $\\Longrightarrow$ & Non-retroactive proof-carrying research receipt","Independent evaluator response and benchmark portfolio refinement & $\\Longrightarrow$ & Strict shrinkage of one metric-gaming nullspace","Bounded outcome-prediction error on a fixed candidate set & $\\Longrightarrow$ & Uniform $2\\eps$ selection-regret bound","Correctly bound research archive and obstruction classifier & $\\Longrightarrow$ & Bottleneck-sensitive experiment allocation","Matched method grammar, evaluator, monitor, budget, seal, and paired audits & $\\Longrightarrow$ & Finite research-compiler metaproductivity certificate","Static diagnostic alias and separating response experiments & $\\Longrightarrow$ & Necessity of sequential diagnosis","Sparse inherited response cone and residual margin & $\\Longrightarrow$ & Finite open-set certificate","Contaminated archive and robust summary statistics & $\\Longrightarrow$ & Finite contamination-stability comparison","Evaluator portfolio and independent behavior dimension & $\\Longrightarrow$ & Strict reduction of one evaluator nullspace","Conjunctive integrity witnesses and adaptive attack methods & $\\Longrightarrow$ & Validity-filtered monitor-ensemble comparison","Matched fixed grammar, stress atlas, seal, group and tail gates & $\\Longrightarrow$ & Finite adversarial compiler-robustness certificate","Inherited unknown-token conflict and disjoint acceptable-method sets & $\\Longrightarrow$ & Diagnostic-language inadequacy certificate","Generic separating probe and conservative token projection & $\\Longrightarrow$ & Finite anonymous diagnostic refinement","Generic pair-method constructor language and positive oracle margin & $\\Longrightarrow$ & Task-relevant compiler method-language expansion","Matched eight-unit audits, fallback, seal, and language controls & $\\Longrightarrow$ & Finite compiler-language metaproductivity certificate","Nonnegative spectral bounds & $\\Longrightarrow$ & Local autocatalytic certificate","Fixed finite grammar or bounded utility & $\\Longrightarrow$ & Saturation of indefinite fixed-size gain"]}