Stratified Relational Workspace Theory

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1 % This file is part of the modular source for Stratified Relational Workspace Theory.
2 % !TeX root = ../Stratified_Relational_Workspace_Theory.tex
3 \theorypart{XXI}{Empirical Certification and Reporting Protocol}
4 
5 \topic{74}{Required Experiments}
6 
7 A workspace claim is not certified by a single benchmark score. The complete
8 protocol is organized as a sequence of declarations, held-out measurements,
9 implemented interventions, matched nulls, specification sweeps, identification
10 checks, and gate decisions. Every reported
11 result must identify the claim level it supports. Measurements that were not
12 performed remain unmeasured rather than being inferred from neighboring gates.
13 
14 \subtopic{74.1 Claim declaration before evaluation}
15 
16 Before the certification split is opened, record:
17 \begin{enumerate}
18 \item the full reference specification
19  \(\varsigma_0=(\mathcal D,Y^+,\ell,\tau,\mathcal I,\mathcal N,\Theta)\);
20 \item the admissible decomposition family \(\mathfrak D\) and specification
21  family \(\mathfrak S\);
22 \item the target trajectory \(Y^+\), loss, horizon, and environment family;
23 \item the candidate structural workspace extractor;
24 \item the goal family \(\mathcal G\), utility normalization, and feasible
25  action set;
26 \item the intervention families \(\mathcal I_W,\mathcal I_G,\mathcal I_A,
27  \mathcal I_D,\mathcal I_R\);
28 \item the null panel and the local distributions each null must preserve;
29 \item the required certificate level and every numerical threshold;
30 \item the formation, calibration, certification, and stress-test splits;
31 \item the uncertainty procedure and resampling unit; and
32 \item the stopping and model-selection rules;
33 \item the experiment family and model class used for identifiability; and
34 \item the complete specification-selection history and multiplicity rule.
35 \end{enumerate}
36 Changing any of these after observing certification outcomes creates a new
37 analysis that must be reported as exploratory or evaluated on a fresh split.
38 
39 \subtopic{74.2 Data and intervention partition}
40 
41 Use at least four logically distinct partitions:
42 \[
43 \boxed{
44 \mathcal D
45 =
46 \mathcal D_{\mathrm{form}}
47 \sqcup
48 \mathcal D_{\mathrm{cal}}
49 \sqcup
50 \mathcal D_{\mathrm{cert}}
51 \sqcup
52 \mathcal D_{\mathrm{stress}}.
53 }
54 \]
55 Their roles are:
56 \begin{description}
57 \item[Formation.] Learn membership, transfer maps, workspace channels,
58  boundaries, policies, and repair models.
59 \item[Calibration.] Select thresholds, intervention amplitudes, and
60  uncertainty procedures without making the final pass decision.
61 \item[Certification.] Estimate the declared gates once under the frozen
62  analysis plan.
63 \item[Stress test.] Evaluate distribution shift, repeated damage, replacement,
64  and long-horizon persistence.
65 \end{description}
66 When independent partitions are impossible, the report must state the precise
67 cross-fitting or nested-resampling procedure and preserve formation leakage
68 controls.
69 
70 \subtopic{74.3 Structural tests}
71 
72 Report the complete low-energy spectrum relevant to the selected threshold:
73 \[
74 \spec(L_H),
75 \qquad
76  d_\varepsilon,
77 \qquad
78  g_\varepsilon,
79 \qquad
80  \delta_{\mathrm{coh}}.
81 \]
82 At minimum include:
83 \begin{itemize}
84 \item the construction and normalization of \(D_H\);
85 \item edge weights and estimated transfer uncertainties;
86 \item the selected eigenspace dimension and threshold rule;
87 \item the spectral gap with uncertainty or perturbation bound;
88 \item stability under bootstrap, seed, and representation replacement; and
89 \item the exact or approximate kernel residual.
90 \end{itemize}
91 
92 \subtopic{74.4 Transfer and operation tests}
93 
94 Report
95 \[
96 \eta_{\mathrm{int}},
97 \qquad
98 \delta_e^{\mathrm{tr}},
99 \qquad
100 \Omega_\gamma,
101 \qquad
102 \Lambda_\gamma,
103 \]
104 together with:
105 \begin{itemize}
106 \item the paths and cycle basis used;
107 \item path-length dependence of leakage;
108 \item the gauge-alignment procedure;
109 \item held-out transfer-prediction error;
110 \item operation-intertwining defects; and
111 \item the approximate closure bound from Theorem~\(T_{10}\).
112 \end{itemize}
113 
114 \subtopic{74.5 Predictive alignment and coalition tests}
115 
116 Report
117 \[
118 \delta_{\mathrm{align}}^{\mathcal I},
119 \qquad
120 \{\Delta_A^\ell\}_{A\subseteq V},
121 \qquad
122  k_\varepsilon,
123 \qquad
124 \delta_A^Z.
125 \]
126 When the number of coalitions is too large for exhaustive evaluation, state the
127 sampling design, include every singleton and every leave-one-out coalition,
128 and report the most favorable coalition found for the reducibility null. The
129 claim must not be based only on average coalition performance.
130 
131 \subtopic{74.6 Causal and reentry tests}
132 
133 Report
134 \[
135 \eta_{\mathrm{causal}}^\nu,
136 \qquad
137 \Delta_{\mathrm{specific}}(r),
138 \qquad
139 \delta_{\mathrm{med}},
140 \qquad
141 \eta_{\mathrm{rec}}(\tau),
142 \qquad
143 \eta_{\mathrm{loop}}(\tau).
144 \]
145 Every causal estimate must be paired with:
146 \begin{enumerate}
147 \item a concrete mechanism-level implementation;
148 \item a manipulation check;
149 \item matched background variables;
150 \item declared direct paths that were blocked or preserved;
151 \item intervention magnitude and cost;
152 \item an off-workspace matched intervention; and
153 \item the downstream outcome window.
154 \end{enumerate}
155 
156 \subtopic{74.7 Persistence, reconstruction, and replacement tests}
157 
158 Report
159 \[
160 d_{\mathrm{id}}(W_t,W_{t+\tau}),
161 \qquad
162 \Rec_k,
163 \qquad
164  e_t,
165 \qquad
166 \alpha^{(k)},
167 \qquad
168 \mu_t^{\mathrm{keep}}.
169 \]
170 The report must separate:
171 \begin{itemize}
172 \item subspace tracking from state tracking;
173 \item coherence from non-collapse;
174 \item reconstruction from mere prediction;
175 \item repair from successor-workspace creation; and
176 \item structural replacement from causal and reconstructive replacement.
177 \end{itemize}
178 
179 \subtopic{74.8 Formation-provenance tests}
180 
181 Report
182 \[
183 \mathbf E_{\mathrm{form}},
184 \qquad
185 \delta^{\mathrm{scaf}},
186 \qquad
187 \delta^{\mathrm{leak}},
188 \qquad
189 \Delta_q^{\mathrm{hist}}.
190 \]
191 Include the external scaffold, information available to the learner, history
192 shuffles, cross-scaffold retraining, and every route by which certification
193 labels or target boundaries could have entered formation.
194 
195 \subtopic{74.9 Learned-structure and boundary tests}
196 
197 Report
198 \[
199 \|\widehat H-H_*\|,
200 \qquad
201 \Delta_J,
202 \qquad
203 \Delta_t^B,
204 \qquad
205 \COut_\tau(\widehat S),
206 \qquad
207 \CIn_\tau(\widehat S).
208 \]
209 Boundary reports must include the admissible candidate family, full-system and
210 empty-set controls, singleton and centralized controls, score components,
211 margin to the runner-up, and temporal stability under held-out environment
212 drift.
213 
214 \subtopic{74.10 Goal and counterfactual-policy tests}
215 
216 Report the agency coordinates
217 \[
218 \eta_G(\tau),
219 \qquad
220 \delta_G^{\mathrm{cue}}(\tau),
221 \qquad
222 \eta_{G\to A},
223 \qquad
224 \delta_Q,
225 \qquad
226 \eta_{V\to A},
227 \qquad
228 \eta_{A\to Y},
229 \qquad
230  k_{\mathrm{ctl},\varepsilon}.
231 \]
232 The required experiments include:
233 \begin{enumerate}
234 \item cue-controlled goal interventions;
235 \item goal-channel ablation and permutation;
236 \item counterfactual-value calibration against action interventions;
237 \item value-label permutation among feasible actions;
238 \item critic or value-channel blocking;
239 \item action replay under held-out perturbations;
240 \item policy regret or value-gap estimation; and
241 \item coalition reconstruction of the complete control state.
242 \end{enumerate}
243 
244 \subtopic{74.11 Damage and repair tests}
245 
246 For every damage family, report
247 \[
248 \eta_{D\to M},
249 \qquad
250 \eta_{M\to R},
251 \qquad
252 \eta_{R\to v},
253 \qquad
254 \Delta_{\mathrm{repair}}^{\mathrm{spec}},
255 \qquad
256 \delta_{\mathrm{rep-id}},
257 \qquad
258 \sup_t e_t.
259 \]
260 The repeated-disturbance experiment must contain:
261 \begin{enumerate}
262 \item a no-damage baseline;
263 \item damage with repair disabled;
264 \item damage with the intact repair loop;
265 \item monitor ablation;
266 \item cost-matched random repair;
267 \item off-workspace repair;
268 \item repeated and compound damage; and
269 \item a held-out disturbance distribution.
270 \end{enumerate}
271 
272 \topic{75}{Required Null Families}
273 
274 The core null panel is:
275 \[
276 \begin{aligned}
277 N_1&=\text{shuffled transfer maps with validated local matching},\\
278 N_2&=\text{shuffled cross-node temporal correspondence},\\
279 N_3&=\text{matched-capacity local memory},\\
280 N_4&=\text{matched-size irrelevant artifact},\\
281 N_5&=\text{participant ablation},\\
282 N_6&=\text{artifact ablation},\\
283 N_7&=\text{workspace implementation intervention},\\
284 N_8&=\text{matched off-workspace intervention},\\
285 N_9&=\text{representation replacement},\\
286 N_{10}&=\text{context reset and reconstruction},\\
287 N_{11}&=\text{matched formation-history shuffle},\\
288 N_{12}&=\text{cross-scaffold re-encoding and retraining},\\
289 N_{13}&=\text{boundary-label and certification leakage audit},\\
290 N_{14}&=\text{empty, full, singleton, and centralized boundary nulls},\\
291 N_{15}&=\text{held-out transfer and membership re-estimation},\\
292 N_{16}&=\text{boundary damage, use, and repair intervention},\\
293 N_{17}&=\text{goal-code permutation with cues and actions matched},\\
294 N_{18}&=\text{persistent external-cue rewrite with internal retention cut},\\
295 N_{19}&=\text{counterfactual-value label permutation},\\
296 N_{20}&=\text{critic or value-channel ablation},\\
297 N_{21}&=\text{open-loop action replay under held-out perturbation},\\
298 N_{22}&=\text{monitor-blind cost-matched random repair},\\
299 N_{23}&=\text{off-workspace repair with matched intervention norm},\\
300 N_{24}&=\text{identity-breaking reconstruction with matched task score},\\
301 N_{25}&=\text{merge control for components sharing an uninterruptible controller},\\
302 N_{26}&=\text{secret-sharing split of a known centralized controller},\\
303 N_{27}&=\text{hidden shared-memory and common-cue exposure test},\\
304 N_{28}&=\text{frozen certificate sweep across the admissible decomposition family},\\
305 N_{29}&=\text{post hoc target-selection audit},\\
306 N_{30}&=\text{horizon cherry-picking audit},\\
307 N_{31}&=\text{loss and utility substitution audit},\\
308 N_{32}&=\text{restricted intervention-family audit},\\
309 N_{33}&=\text{threshold and gate-set tuning audit},\\
310 N_{34}&=\text{frozen certificate sweep across the admissible specification family}.
311 \end{aligned}
312 \]
313 
314 \subtopic{75.1 Null validity record}
315 
316 Each observational null must state:
317 \begin{itemize}
318 \item the local distributions it preserves;
319 \item the relational dependency it destroys;
320 \item the distance between intact and null local marginals;
321 \item the capacity, parameter count, and information budget; and
322 \item whether null fitting used the same data budget as the intact model.
323 \end{itemize}
324 Each causal null must state:
325 \begin{itemize}
326 \item the manipulated mechanism;
327 \item the matched background and intervention cost;
328 \item the intended causal path destroyed or preserved;
329 \item the manipulation check; and
330 \item unintended direct effects detected during validation.
331 \end{itemize}
332 A failed manipulation check invalidates the null. It does not count as evidence
333 for the workspace claim.
334 
335 \subtopic{75.2 Claim-specific null subsets}
336 
337 A successful claim requires
338 \[
339 \Pass(N_1,\ldots,N_{28})
340 \]
341 only for the subset relevant to its stated level. The subset must be frozen
342 before certification. Failed applicable nulls, invalid nulls, and unmeasured
343 nulls are retained separately in the final report. They must not be merged
344 into a generic ``not significant'' category.
345 
346 \topic{76}{Certification Vector and Complete Reporting Templates}
347 
348 \subtopic{76.1 Extended certification vector}
349 
350 Use a vector rather than one aggregate score:
351 \[
352 \boxed{
353 \begin{aligned}
354 \mathcal C_{\mathrm{full}}(W)
355 =\bigl(&
356  d_\varepsilon,
357  g_\varepsilon,
358  \delta_{\mathrm{coh}},
359  \delta_{\mathrm{align}}^{\mathcal I},
360  \Delta_{\mathrm{joint}}^\ell,
361  k_\varepsilon,\\
362 &\boldsymbol\Delta_{\mathrm{tr}}(\mathcal Q),
363  \eta_{\mathrm{causal}}^\nu,
364  \Delta_{\mathrm{specific}}(r),
365  \eta_{\mathrm{loop}}(\tau),\\
366 &d_{\mathrm{id}},
367  \Rec_k,
368  \underline{\mathbf m}_{\mathfrak D},
369  \boldsymbol\Delta_{\mathrm{dec}},
370  \mathbf E_{\mathrm{form}},
371  \delta^{\mathrm{scaf}},
372  \delta^{\mathrm{leak}},\\
373 &\Delta_J,
374  \Delta_t^B,
375  \COut_\tau(\widehat S),
376  \CIn_\tau(\widehat S),\\
377 &\eta_G(\tau),
378  \delta_G^{\mathrm{cue}}(\tau),
379  \eta_{G\to A},
380  \delta_Q,
381  \eta_{V\to A},
382  \eta_{A\to Y},\\
383 &\eta_{D\to M},
384  \eta_{M\to R},
385  \eta_{R\to v},
386  \Delta_{\mathrm{repair}}^{\mathrm{spec}},
387  \delta_{\mathrm{rep-id}},
388  \sup_t e_t
389 \bigr).
390 \end{aligned}
391 }
392 \]
393 A claim passes only when every required coordinate lies in its predeclared
394 admissible region. No weighted average may compensate for a failed required
395 gate.
396 
397 \subtopic{76.2 Claim card template}
398 
399 \begin{longtable}{@{}>{\raggedright\arraybackslash}p{0.29\textwidth}>{\raggedright\arraybackslash}p{0.65\textwidth}@{}}
400 \toprule
401 Field & Required entry \\
402 \midrule
403 \endfirsthead
404 \toprule
405 Field & Required entry \\
406 \midrule
407 \endhead
408 Claim identifier & Unique versioned identifier tied to code and result
409 artifacts. \\
410 Claim level & Archive, shared representation, operational workspace,
411 relational workspace, persistent workspace, self-maintaining workspace, or
412 workspace agent. \\
413 Reference decomposition & Nodes, artifacts, channels, local-access maps, and coalition universe. \\
414 Admissible decomposition family & Plausible merges, splits, shared-resource assignments, exclusions, and audit rationale. \\
415 Target and horizon & \(Y^+\), loss, utility, prediction horizon, control
416 horizon, and maintenance horizon. \\
417 Environment scope & Training, certification, and stress-test environment
418 families. \\
419 Workspace extractor & Frozen construction of \(D_H\), \(L_H\), spectral
420 threshold, and state readout. \\
421 Goal and policy & Goal codes, utility normalization, action set, and policy
422 class. \\
423 Boundary & Candidate family, score, admissibility constraints, and selected
424 boundary. \\
425 Interventions & Feasible workspace, goal, action, damage, and repair
426 implementations. \\
427 Null panel & Applicable \(N_j\), preservation targets, and fit budgets. \\
428 Thresholds & Every gate direction, numerical cutoff, and uncertainty rule. \\
429 Data split & Formation, calibration, certification, stress, and cross-fitting
430 logic. \\
431 Stopping rule & Number of seeds, runs, interventions, and any sequential
432 stopping criterion. \\
433 Status & Confirmatory, exploratory, replication, or audit. \\
434 \bottomrule
435 \end{longtable}
436 
437 \subtopic{76.3 Gate-estimate table template}
438 
439 For every required coordinate, include one row with the following fields:
440 
441 \begin{longtable}{@{}>{\raggedright\arraybackslash}p{0.20\textwidth}>{\raggedright\arraybackslash}p{0.73\textwidth}@{}}
442 \toprule
443 Field & Required entry \\
444 \midrule
445 \endfirsthead
446 \toprule
447 Field & Required entry \\
448 \midrule
449 \endhead
450 Gate & Formal symbol and verbal interpretation. \\
451 Estimator & Exact statistic, normalization, and software implementation. \\
452 Evaluation unit & Episode, participant, graph, seed, environment, or other
453 independent unit. \\
454 Point estimate & Estimate on the frozen certification split. \\
455 Uncertainty & Interval, bound, or posterior region with declared coverage or
456 credibility. \\
457 Threshold & Predeclared pass region. \\
458 Manipulation status & Passed, failed, or not applicable. \\
459 Decision & Pass, fail, invalid, underpowered, or unmeasured. \\
460 Sensitivity & Result under reasonable alternative normalizations and
461 resampling choices. \\
462 Artifact reference & File, table, raw-result key, and code commit. \\
463 \bottomrule
464 \end{longtable}
465 
466 \subtopic{76.4 Intervention ledger template}
467 
468 \begin{longtable}{@{}p{0.20\textwidth}p{0.73\textwidth}@{}}
469 \toprule
470 Field & Required entry \\
471 \midrule
472 \endfirsthead
473 \toprule
474 Field & Required entry \\
475 \midrule
476 \endhead
477 Intervention ID & Unique identifier and randomization seed. \\
478 Target mechanism & Workspace coordinate, goal code, value channel, action,
479 boundary, monitor, or repair channel. \\
480 Implementation & Exact code path, hook, artifact edit, or mechanism change. \\
481 Dose & Norm, duration, timing, and support of the intervention. \\
482 Background match & Variables fixed, blocked, randomized, or balanced. \\
483 Excluded paths & Direct paths intentionally removed from the causal contrast. \\
484 Manipulation check & Evidence that the intended mechanism changed by the
485 specified amount. \\
486 Collateral effects & Changes outside the target mechanism. \\
487 Outcome & Downstream variable and evaluation window. \\
488 Effect estimate & Point estimate, uncertainty, and sign convention. \\
489 Cost & Compute, latency, information, action, or structural cost. \\
490 Validity status & Valid, invalid, ambiguous, or unmeasured. \\
491 \bottomrule
492 \end{longtable}
493 
494 \subtopic{76.5 Null ledger template}
495 
496 \begin{longtable}{@{}p{0.20\textwidth}p{0.73\textwidth}@{}}
497 \toprule
498 Field & Required entry \\
499 \midrule
500 \endfirsthead
501 \toprule
502 Field & Required entry \\
503 \midrule
504 \endhead
505 Null ID & One of \(N_1,\ldots,N_{34}\) or a declared extension. \\
506 Claim attacked & The exact explanation the null is designed to preserve or
507 destroy. \\
508 Matched quantities & Local marginals, capacity, training budget, action cost,
509 intervention norm, or other controlled quantities. \\
510 Destroyed relation & Edge alignment, temporal order, value ranking, goal
511 identity, monitor mediation, or another relational feature. \\
512 Fit procedure & Data, optimization, seeds, and hyperparameter budget. \\
513 Validity metric & Distance from the intended matched quantities. \\
514 Primary contrast & Intact minus null effect with uncertainty. \\
515 Disposition & Rejects null, fails to reject, invalid null, or underpowered. \\
516 Failure notes & Any mismatch or collateral mechanism change. \\
517 \bottomrule
518 \end{longtable}
519 
520 \subtopic{76.6 Agency and maintenance report template}
521 
522 The agency section must provide the following matrix:
523 
524 \begin{center}
525 \small
526 \begin{tabularx}{\textwidth}{@{}p{0.22\textwidth}p{0.23\textwidth}X@{}}
527 \toprule
528 Component & Primary metric & Required contrast \\
529 \midrule
530 Goal semantics & Held-out \(Q_t^g\) ordering & Distinct goals reverse or alter
531 at least one declared action ordering. \\
532 Goal retention & \(\eta_G(\tau)\) & Goal intervention with future cues matched
533 or removed. \\
534 Goal use & \(\eta_{G\to A}\) & Goal intervention versus goal-label replay. \\
535 Counterfactual accuracy & \(\delta_Q\) & Internal value versus external action
536 interventions. \\
537 Value use & \(\eta_{V\to A}\) & Value-label permutation and value-channel
538 ablation. \\
539 Outcome control & \(\eta_{A\to Y}\) & Feasible action intervention. \\
540 Damage detection & \(\eta_{D\to M}\) & Damage versus matched no-damage. \\
541 Repair mediation & \(\eta_{M\to R}\) & Monitor intervention and monitor
542 ablation. \\
543 Restoration & \(\eta_{R\to v}\) & Repair versus matched no-repair. \\
544 Repair specificity & \(\Delta_{\mathrm{repair}}^{\mathrm{spec}}\) & Intended
545 repair versus cost-matched generic action. \\
546 Identity preservation & \(\delta_{\mathrm{rep-id}}\) & Pre-damage workspace
547 versus post-repair workspace. \\
548 Long-run maintenance & \(\sup_t e_t\) & Repeated held-out disturbances. \\
549 Relational location & \(k_{\mathrm{ctl},\varepsilon}\) & Full system versus
550 all single nodes and matched local nulls. \\
551 \bottomrule
552 \end{tabularx}
553 \end{center}
554 
555 \subtopic{76.7 Uncertainty and multiplicity}
556 
557 The independent resampling unit must match the data-generating process. Token,
558 time-step, or edge resampling is invalid when dependence is organized at the
559 episode, participant, graph, or environment level. Recommended procedures
560 include:
561 \begin{itemize}
562 \item environment- or episode-level bootstrap;
563 \item repeated independent training seeds;
564 \item cluster-robust intervals for repeated observations;
565 \item permutation tests that preserve the declared local structure;
566 \item concentration bounds for uniform finite candidate families; and
567 \item stress-test intervals over held-out environment families.
568 \end{itemize}
569 
570 Pass decisions should use one-sided uncertainty bounds. For a lower-bounded
571 gate \(\theta\geq\tau\), pass only when the lower confidence bound exceeds
572 \(\tau\). For an upper-bounded defect \(\delta\leq\varepsilon\), pass only when
573 the upper confidence bound is below \(\varepsilon\).
574 
575 When a conclusion ranges over \(\mathfrak S\), the uncertainty rule must be
576 uniform over every gate and specification included in the claim. Report the
577 pointwise interval, the family-wide lower envelope, the covering or
578 multiplicity correction, and the identification status. A narrow pointwise
579 interval does not establish a robust result when model or specification
580 uncertainty remains wide.
581 
582 \subtopic{76.8 \(T_{24}\): Gatewise certification validity }
583 
584 \begin{theorem}[Intersection-gate false-certification control]
585 \label{thm:gatewise-validity}
586 Let a certificate require all gates \(G_1,\ldots,G_m\) to pass. For each gate
587 \(j\), let \(H_{0j}\) denote the null that the required property is absent, and
588 assume the predeclared test satisfies
589 \[
590 \sup_{P\in H_{0j}}
591 \mathbb P_P(G_j\text{ passes})
592 \leq
593 \alpha_j.
594 \]
595 The global null is
596 \[
597 H_0
598 =
599 \bigcup_{j=1}^m H_{0j},
600 \]
601 meaning that at least one required property is absent. If the certificate is
602 declared only when every gate passes, then
603 \[
604 \boxed{
605 \sup_{P\in H_0}
606 \mathbb P_P(\text{certificate passes})
607 \leq
608 \max_{1\leq j\leq m}\alpha_j.
609 }
610 \]
611 No independence assumption among gate tests is required.
612 \end{theorem}
613 
614 \begin{proof}
615 For any \(P\in H_0\), choose an index \(j_P\) such that \(P\in H_{0j_P}\).
616 The event that every gate passes is contained in the event that gate \(j_P\)
617 passes. Hence
618 \[
619 \mathbb P_P(\text{certificate passes})
620 \leq
621 \mathbb P_P(G_{j_P}\text{ passes})
622 \leq
623 \alpha_{j_P}
624 \leq
625 \max_j\alpha_j.
626 \]
627 Taking the supremum over \(P\in H_0\) proves the claim.
628 \end{proof}
629 
630 The theorem is an intersection-union guarantee. It applies only when the gate
631 family, thresholds, and tests are fixed before certification and each gate test
632 is uniformly valid for its null. Adaptive gate deletion, selective reporting,
633 or repeated reuse of the certification split invalidates the bound.
634 
635 \subtopic{76.9 Failure taxonomy}
636 
637 Every gate and null receives exactly one primary status:
638 \begin{description}
639 \item[Pass.] The frozen estimate and uncertainty rule satisfy the threshold.
640 \item[Fail.] The gate was validly measured and did not satisfy the threshold.
641 \item[Invalid intervention.] The manipulation check or background match
642  failed.
643 \item[Invalid null.] The null did not preserve its required local quantities
644  or exceeded its fit budget.
645 \item[Underpowered.] The uncertainty interval crosses the threshold by more
646  than the declared indifference region.
647 \item[Unmeasured.] No valid estimate was obtained.
648 \item[Leakage.] Formation or model selection used certification information.
649 \item[Exploratory.] The metric, threshold, or analysis was selected after
650  inspecting certification outcomes.
651 \end{description}
652 Only \emph{Pass} counts toward the formal certificate. ``Underpowered'' and
653 ``unmeasured'' are not failures of the underlying theory, but they are failures
654 to establish the corresponding empirical claim.
655 
656 \subtopic{76.10 Machine-readable result record}
657 
658 The machine-readable record should contain at least the following top-level
659 objects:
660 \[
661 \boxed{
662 \begin{aligned}
663 \mathsf{ResultRecord}
664 =\{&
665 \mathsf{claim},
666 \mathsf{specification},
667 \mathsf{specification\_family},
668 \mathsf{decomposition},
669 \mathsf{data\_splits},
670 \mathsf{formation},\\
671 &\mathsf{workspace\_extractor},
672 \mathsf{goals},
673 \mathsf{actions},
674 \mathsf{interventions},\\
675 &\mathsf{nulls},
676 \mathsf{gates},
677 \mathsf{identified\_sets},
678 \mathsf{specification\_envelopes},
679 \mathsf{uncertainty},
680 \mathsf{failures},
681 \mathsf{artifacts}
682 \}.
683 \end{aligned}
684 }
685 \]
686 Each gate object records its symbol, estimator, direction, threshold, point
687 estimate, interval, split, seed set, status, and artifact reference. Each
688 intervention and null object records its manipulation check and validity
689 status. The schema must preserve failed and invalid entries rather than only
690 the passing subset.
691 
692 \subtopic{76.11 Minimum reproducibility package}
693 
694 A complete certification package contains:
695 \begin{enumerate}
696 \item the frozen claim card;
697 \item source code and environment lock file;
698 \item exact data or deterministic data-generation procedure;
699 \item formation, calibration, certification, and stress-test manifests;
700 \item learned transfer, workspace, policy, monitor, and repair parameters;
701 \item raw per-run and per-intervention results;
702 \item the intervention and null ledgers;
703 \item the complete gate table, including failures;
704 \item figures and tables regenerated from raw results;
705 \item random seeds and hardware or numerical-precision information;
706 \item checksums for every included artifact; and
707 \item a single verification command that reproduces package integrity and the
708  reported summary tables.
709 \end{enumerate}
710 
711 \subtopic{76.12 Decision rule}
712 
713 Let \(\mathcal J(L)\) be the required gate set for claim level \(L\). Then
714 \[
715 \boxed{
716 \operatorname{Certify}_L(W;\varsigma)
717 =
718 \bigwedge_{j\in\mathcal J(L,\varsigma)}
719 \operatorname{Pass}_j(W;\varsigma).
720 }
721 \]
722 For a workspace-agent claim, the required set includes every structural,
723 operational, relational, persistent, goal, counterfactual-control, and
724 self-maintenance gate declared in Parts~IX and XVI. A specification-robust
725 claim additionally requires the uniform rule of Part~XVIII across
726 \(\mathfrak S\). Failure of an earlier level
727 prevents certification of a later level, even when later-level behavior appears
728 strong on a task benchmark.

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