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{XXII}{Integrated Worked Examples and Case Studies}
4 
5 \topic{77}{Case-Study Method and Common Gate Card}
6 
7 The preceding parts define a large family of structural, operational, causal,
8 relational, persistent, formative, and agentive predicates. A worked example
9 is useful only when it makes clear which of those predicates are established,
10 which are assumed, and which remain unmeasured. This part therefore uses a
11 single case-study protocol for three different kinds of realization:
12 \begin{enumerate}
13 \item an exact finite-dimensional model of a human--human theorem laboratory;
14 \item an idealized human--AI--artifact--executor research loop, followed by a
15  deliberately conservative audit of the collaboration that motivated the
16  theory; and
17 \item a closed-form multi-agent control system that reaches the formal agency
18  level inside a small declared domain.
19 \end{enumerate}
20 The first and third cases are analytic models. Their conclusions hold inside
21 those models. The second contains both an analytic idealization and an
22 observational audit. The audit is not upgraded to a certificate merely because
23 the idealization is mathematically clean.
24 
25 \begin{definition}[Case-study realization]\label{def:case-study-realization}
26 A case-study realization is a tuple
27 \[
28 \mathfrak K
29 =
30 \bigl(
31 \mathfrak R,
32 \mathfrak B,
33 \mathfrak C,
34 \mathfrak P,
35 \mathcal E,
36 \mathfrak D,
37 \mathfrak S,
38 \mathcal I,
39 \mathcal Q,
40 \mathcal G
41 \bigr),
42 \]
43 where:
44 \begin{itemize}
45 \item \(\mathfrak R\) is the declared relational realization;
46 \item \(\mathfrak B\) is the boundary problem and component decomposition;
47 \item \(\mathfrak C\) is the structural core--transfer model;
48 \item \(\mathfrak P\) is the downstream predictive or control problem;
49 \item \(\mathcal E\) is the evidence or generated-data family;
50 \item \(\mathfrak D\) is the admissible decomposition family;
51 \item \(\mathfrak S\) is the admissible family of complete scientific
52  specifications, including targets, losses, horizons, interventions, nulls,
53  and thresholds;
54 \item \(\mathcal I\) is the feasible intervention family;
55 \item \(\mathcal Q\) is the matched null family; and
56 \item \(\mathcal G\) is the set of gates being claimed.
57 \end{itemize}
58 A conclusion is always relative to this tuple. Changing the downstream target,
59 temporal horizon, intervention support, component decomposition, null family,
60 or threshold rule creates a different specification of the case. A stronger
61 case study reports whether the conclusion survives the predeclared family
62 \(\mathfrak S\) and whether the relevant gates are identified by the available
63 experiment family.
64 \end{definition}
65 
66 \subtopic{77.1 Three evidential statuses}
67 
68 Every claim in this part receives one of three primary statuses:
69 \begin{description}
70 \item[Analytic.] The gate follows exactly from a declared mathematical model.
71 \item[Empirically certified.] The gate is estimated from held-out data using
72  the protocol of Part~XXI and passes its frozen uncertainty rule.
73 \item[Observational only.] The case contains suggestive records or natural
74  perturbations, but no valid held-out intervention-and-null certificate.
75 \end{description}
76 An analytic result is not automatically an empirical result. An observational
77 pattern is not automatically a causal result. None of the three statuses
78 establishes phenomenal consciousness.
79 
80 \subtopic{77.2 Common gate card}
81 
82 For each case we report the vector
83 \[
84 \boxed{
85 \mathsf{Card}(\mathfrak K)
86 =
87 \left(
88 \mathsf S,
89 \mathsf O,
90 \mathsf R,
91 \mathsf P,
92 \mathsf F,
93 \mathsf A,
94 \mathsf D,
95 \mathsf X,
96 \mathsf C
97 \right),
98 }
99 \]
100 where:
101 \[
102 \begin{array}{ll}
103 \mathsf S & = \text{structural core and spectral coherence},\\
104 \mathsf O & = \text{operational alignment and causal efficacy},\\
105 \mathsf R & = \text{relational irreducibility and matched-null gap},\\
106 \mathsf P & = \text{persistence, reconstruction, and identity},\\
107 \mathsf F & = \text{formation provenance and boundary status},\\
108 \mathsf A & = \text{goal-conditioned agency and self-maintenance},\\
109 \mathsf D & = \text{decomposition robustness and interface audit},\\
110 \mathsf X & = \text{specification robustness, identification, and uniform uncertainty},\\
111 \mathsf C & = \text{consciousness or personhood claim}.
112 \end{array}
113 \]
114 The decomposition coordinate distinguishes a result established only for the
115 declared interface from one stable across an admissible family. The
116 specification coordinate records whether the same conclusion survives
117 scientifically defensible alternatives, whether its sign is identified by the
118 available experiments, and whether its lower margin remains positive after
119 uniform finite-sample uncertainty. The final coordinate is included to make
120 the non-implication explicit. In all examples below,
121 \[
122 \mathsf C=\text{not established}.
123 \]
124 
125 \subtopic{77.3 Claim-level discipline}
126 
127 Let \(L(\mathfrak K)\in\{0,1,\ldots,6\}\) denote the highest level from
128 Part~XXIV whose complete required gate set is established. The level is
129 conjunctive and pointwise in the declared scientific specification unless the
130 \(\mathsf D\) and \(\mathsf X\) coordinates also pass:
131 \[
132 L(\mathfrak K)\geq \ell
133 \quad\Longleftrightarrow\quad
134 \bigwedge_{j\in\mathcal J(\ell)}\Pass_j(\mathfrak K).
135 \]
136 A case may have strong higher-level-looking behavior while remaining at a lower
137 certified level because an earlier gate is absent. For example, successful
138 repair behavior does not establish self-maintenance when no matched no-repair
139 counterfactual is available, and coherent shared documents do not establish a
140 relational workspace when one participant contains the complete operative
141 state.
142 
143 \subtopic{77.4 Structural calculations used repeatedly}
144 
145 The positive examples use a recurring distributed code. Let
146 \[
147 c=(c_1,c_2,c_3)^{\mathsf T}\in\mathbb R^3
148 \]
149 and define three local encodings
150 \[
151 E_1c=(c_1,c_2),
152 \qquad
153 E_2c=(c_2,c_3),
154 \qquad
155 E_3c=(c_1,c_3).
156 \]
157 Then
158 \[
159 \sum_{i=1}^3E_i^*E_i=2I_3.
160 \]
161 No individual encoding is injective, while every two-node coalition is
162 injective. The pairwise frame operators are
163 \[
164 \begin{aligned}
165 E_{\{1,2\}}^*E_{\{1,2\}}&=\operatorname{diag}(1,2,1),\\
166 E_{\{1,3\}}^*E_{\{1,3\}}&=\operatorname{diag}(2,1,1),\\
167 E_{\{2,3\}}^*E_{\{2,3\}}&=\operatorname{diag}(1,1,2).
168 \end{aligned}
169 \]
170 Hence
171 \[
172 \alpha_{\{i,j\}}=1
173 \qquad(i\neq j),
174 \]
175 and one-node erasure recovery is exact. Every nonzero core vector is visible
176 at at least two nodes, so
177 \[
178 d_{\mathrm{er}}=2.
179 \]
180 This code is distributed but redundant: no node is sufficient, yet any one
181 node may be lost without destroying the core.
182 
183 \subtopic{77.5 Binary synergy test}
184 
185 Let \(C_1,C_2,C_3\) be independent uniform bits and let
186 \[
187 Y=C_1\oplus C_2\oplus C_3.
188 \]
189 Each node sees two of the three bits. Therefore
190 \[
191 H(Y\mid E_iC)=1\text{ bit}
192 \qquad\text{for every }i,
193 \]
194 while
195 \[
196 H(Y\mid E_1C,E_2C,E_3C)=0.
197 \]
198 Under logarithmic loss,
199 \[
200 \Risk_{\log}(\{i\})=1,
201 \qquad
202 \Risk_{\log}(V)=0,
203 \qquad
204 \Delta_{\mathrm{joint}}^{\log}=1\text{ bit},
205 \qquad
206 k_0=2.
207 \]
208 Moreover, \(Y\) is independent of every individual local state. The null law
209 that keeps all three core bits independent and makes \(Y\) an independent
210 uniform bit preserves every pair \((Y,E_iC)\) while destroying the joint parity
211 relation. It is therefore a predictive-marginal-matched relational null with
212 \[
213 \Delta_{\mathrm{tr}}=1\text{ bit}.
214 \]
215 This test will be reused with different semantic interpretations.
216 
217 \topic{78}{Case Study A: A Distributed Human--Human Theorem Laboratory}
218 
219 \subtopic{78.1 Declared realization}
220 
221 Consider two researchers and one persistent notebook:
222 \[
223 V_{\mathrm{HH}}=\{P,C,N\},
224 \]
225 where:
226 \begin{itemize}
227 \item \(P\) is a proposer who tracks the theorem statement and open proof
228  obligations;
229 \item \(C\) is a critic who tracks proof obligations and falsification status;
230 \item \(N\) is a notebook that tracks the theorem statement and falsification
231  status.
232 \end{itemize}
233 The abstract core is
234 \[
235 c=(q,p,f)^{\mathsf T}\in\mathbb R^3,
236 \]
237 with
238 \[
239 \begin{aligned}
240 q&=\text{statement coordinate},
241 &
242 p&=\text{proof-obligation coordinate},\\
243 f&=\text{falsification-status coordinate}.
244 \end{aligned}
245 \]
246 The accessible encodings are
247 \[
248 E_P=
249 \begin{pmatrix}
250 1&0&0\\
251 0&1&0
252 \end{pmatrix},
253 \quad
254 E_C=
255 \begin{pmatrix}
256 0&1&0\\
257 0&0&1
258 \end{pmatrix},
259 \quad
260 E_N=
261 \begin{pmatrix}
262 1&0&0\\
263 0&0&1
264 \end{pmatrix}.
265 \]
266 The semantics are stylized. The point is not that real researchers contain
267 only two numbers. The point is to exhibit, without metaphor, a system in which
268 no participant or artifact contains the entire operative state while the
269 coupled system does.
270 
271 \subtopic{78.2 Exact overlap descent}
272 
273 Order the local coordinates as
274 \[
275 x=(q_P,p_P,p_C,f_C,q_N,f_N)^{\mathsf T}.
276 \]
277 Compatibility requires
278 \[
279 p_P=p_C,
280 \qquad
281 f_C=f_N,
282 \qquad
283 q_N=q_P.
284 \]
285 The discrepancy operator is
286 \[
287 D_{\mathrm{HH}}
288 =
289 \begin{pmatrix}
290 0&1&-1&0&0&0\\
291 0&0&0&1&0&-1\\
292 -1&0&0&0&1&0
293 \end{pmatrix}.
294 \]
295 Therefore
296 \[
297 \ker D_{\mathrm{HH}}
298 =
299 \left\{
300 (q,p,p,f,q,f)^{\mathsf T}:q,p,f\in\mathbb R
301 \right\}
302 \cong\mathbb R^3.
303 \]
304 The rows are orthogonal with squared norm two, so
305 \[
306 D_{\mathrm{HH}}D_{\mathrm{HH}}^*=2I_3.
307 \]
308 Consequently the transfer Laplacian has spectrum
309 \[
310 \boxed{
311 \spec(L_{\mathrm{HH}})
312 =
313 \{0,0,0,2,2,2\},
314 }
315 \]
316 and the exact workspace gap is
317 \[
318 \gamma_{\mathrm{HH}}=2.
319 \]
320 The low-energy subspace is therefore stable under any Laplacian perturbation
321 whose operator norm is strictly below one, with the quantitative projector
322 bound from Theorem~\ref{thm:spectral-stability}.
323 
324 \subtopic{78.3 Reconstruction and one-node failure}
325 
326 The full frame operator is
327 \[
328 F_{PCN}=E_P^*E_P+E_C^*E_C+E_N^*E_N=2I_3.
329 \]
330 For every two-node coalition \(S\),
331 \[
332 \lambda_{\min}(F_S)=1.
333 \]
334 Thus every two-node coalition reconstructs the complete core and, for noisy
335 observations \(y_S=A_Sc+\eta_S\),
336 \[
337 \|\widehat c-c\|
338 \leq
339 \|\eta_S\|.
340 \]
341 No individual node reconstructs \(c\), because
342 \[
343 \operatorname{rank}(E_P)
344 =
345 \operatorname{rank}(E_C)
346 =
347 \operatorname{rank}(E_N)
348 =2<3.
349 \]
350 The realization is therefore simultaneously:
351 \[
352 \text{individually incomplete}
353 \quad\text{and}\quad
354 \text{robust to any one-node loss}.
355 \]
356 
357 \subtopic{78.4 Relational prediction problem}
358 
359 Let \(q,p,f\) now be independent uniform bits and define the downstream
360 acceptance decision
361 \[
362 Y=q\oplus p\oplus f.
363 \]
364 The interpretation is abstract: acceptance depends on the statement, the state
365 of the proof obligations, and the current falsification record. No individual
366 node has sufficient information:
367 \[
368 H(Y\mid X_P)
369 =
370 H(Y\mid X_C)
371 =
372 H(Y\mid X_N)
373 =1.
374 \]
375 Every two-node coalition has all three bits and therefore zero conditional
376 entropy. Hence
377 \[
378 \boxed{
379 \Delta_{\mathrm{joint}}^{\log}=1\text{ bit},
380 \qquad
381 k_0=2.
382 }
383 \]
384 Let \(Q_{\mathrm{HH}}\) preserve the independent uniform local bit pairs but
385 replace \(Y\) by an independent uniform bit. Because parity is independent of
386 every two-bit local view, \(Q_{\mathrm{HH}}\) preserves each
387 \((Y,X_i)\)-marginal. Nevertheless,
388 \[
389 \Risk_{\log,Q_{\mathrm{HH}}}(V)
390 -
391 \Risk_{\log,P_{\mathrm{HH}}}(V)
392 =1\text{ bit}.
393 \]
394 This is an exact relational witness: the predictive capacity lies in the joint
395 organization rather than in any local predictive marginal.
396 
397 \subtopic{78.5 Feasible intervention and specificity}
398 
399 A feasible statement intervention must alter both local realizations of the
400 statement coordinate:
401 \[
402 \iota_{q\leftarrow a}:
403 (q_P,q_N)\leftarrow(a,a),
404 \]
405 while holding \((p,f)\) fixed. For configurations with fixed \(p\) and \(f\),
406 changing \(a\) flips the downstream decision with total-variation effect one:
407 \[
408 \TV
409 \left(
410 P(Y\mid\iota_{q\leftarrow0}),
411 P(Y\mid\iota_{q\leftarrow1})
412 \right)
413 =1.
414 \]
415 A matched intervention on a residual notebook-format coordinate leaves \(Y\)
416 unchanged. Thus the core intervention has positive causal specificity in this
417 model.
418 
419 \subtopic{78.6 Persistence, handoff, and replacement}
420 
421 Suppose the proposer becomes unavailable. The remaining coalition \(\{C,N\}\)
422 reconstructs \((q,p,f)\) exactly. A replacement proposer \(P'\) can therefore
423 be initialized by
424 \[
425 E_{P'}c=(q,p).
426 \]
427 If \(P'\) uses a different local coordinate system
428 \[
429 J_{P'}\in GL(2),
430 \]
431 then the incoming and outgoing overlap maps are conjugated accordingly. The
432 global section and all downstream kernels remain unchanged. This is exact
433 replacement invariance, not literal carrier identity.
434 
435 If the notebook alone survives, the core cannot be reconstructed because the
436 proof-obligation coordinate is missing. The model therefore predicts a sharp
437 handoff boundary:
438 \[
439 \text{one surviving node is insufficient},
440 \qquad
441 \text{any two surviving nodes are sufficient}.
442 \]
443 
444 \subtopic{78.7 Case card}
445 
446 Inside the declared finite model,
447 \[
448 \boxed{
449 \begin{aligned}
450 \mathsf S&=\checkmark_{\mathrm{analytic}},
451 &\mathsf O&=\checkmark_{\mathrm{analytic}},\\
452 \mathsf R&=\checkmark_{\mathrm{analytic}},
453 &\mathsf P&=\checkmark_{\mathrm{analytic}},\\
454 \mathsf F&=\text{scaffolded},
455 &\mathsf A&=\text{not tested},\\
456 \mathsf D&=\text{fixed decomposition},
457 &\mathsf C&=\text{not established}.
458 \end{aligned}
459 }
460 \]
461 The highest supported level is Level~4 when the handoff dynamics and identity
462 correspondence are included. The example does not contain an operational goal
463 state, counterfactual policy, damage monitor, or repair loop, so it does not
464 support Levels~5 or 6.
465 
466 \topic{79}{Case Study B: A Human--AI--Artifact--Executor Research Loop}
467 
468 \subtopic{79.1 Idealized four-node loop}
469 
470 Consider four heterogeneous components:
471 \[
472 V_{\mathrm{HAI}}=\{H,M,X,A\},
473 \]
474 where:
475 \begin{itemize}
476 \item \(H\) is a human researcher who maintains goals and semantic direction;
477 \item \(M\) is a language-model process that constructs local formalizations;
478 \item \(X\) is an execution and verification process that compiles, tests, or
479  falsifies candidate artifacts; and
480 \item \(A\) is a persistent artifact store containing exact source, results,
481  and versioned handoff state.
482 \end{itemize}
483 Let the abstract research state be
484 \[
485 c=(g,d,p,v)^{\mathsf T}\in\mathbb R^4,
486 \]
487 where:
488 \[
489 \begin{array}{ll}
490 g=\text{goal and acceptance criterion},
491 &d=\text{current derivation or theory state},\\
492 p=\text{open proof or implementation obligations},
493 &v=\text{verification and regression status}.
494 \end{array}
495 \]
496 Define local encodings
497 \[
498 E_Hc=(g,d),
499 \qquad
500 E_Mc=(g,p),
501 \qquad
502 E_Xc=(p,v),
503 \qquad
504 E_Ac=(d,v).
505 \]
506 Each coordinate is represented at exactly two nodes, and the incidence pattern
507 forms a four-cycle.
508 
509 \subtopic{79.2 Exact structural calculation}
510 
511 Order the local coordinates as
512 \[
513 x=(g_H,d_H,g_M,p_M,p_X,v_X,d_A,v_A)^{\mathsf T}.
514 \]
515 The overlap discrepancies are
516 \[
517 g_H-g_M,
518 \qquad
519 p_M-p_X,
520 \qquad
521 v_X-v_A,
522 \qquad
523 d_A-d_H.
524 \]
525 Thus
526 \[
527 D_{\mathrm{HAI}}
528 =
529 \begin{pmatrix}
530 1&0&-1&0&0&0&0&0\\
531 0&0&0&1&-1&0&0&0\\
532 0&0&0&0&0&1&0&-1\\
533 0&-1&0&0&0&0&1&0
534 \end{pmatrix}.
535 \]
536 The rows are orthogonal with squared norm two. Therefore
537 \[
538 \ker D_{\mathrm{HAI}}
539 \cong\mathbb R^4,
540 \]
541 \[
542 \boxed{
543 \spec(L_{\mathrm{HAI}})
544 =
545 \{0,0,0,0,2,2,2,2\},
546 \qquad
547 \gamma_{\mathrm{HAI}}=2.
548 }
549 \]
550 The full accessible frame operator is
551 \[
552 F_{HMXA}=2I_4.
553 \]
554 No single node is injective. Every three-node coalition reconstructs the full
555 state, and the two opposite pairs
556 \[
557 \{H,X\},
558 \qquad
559 \{M,A\}
560 \]
561 are already sufficient. Adjacent pairs observe only three of the four
562 coordinates. Hence
563 \[
564 k_{\mathrm{rec}}=2,
565 \qquad
566 d_{\mathrm{er}}=2,
567 \]
568 while the identity of the sufficient two-node coalition depends on the graph
569 geometry.
570 
571 \subtopic{79.3 Relational parity benchmark}
572 
573 Let \(g,d,p,v\) be independent uniform bits and define
574 \[
575 Y=g\oplus d\oplus p\oplus v.
576 \]
577 Every single node sees only two bits, so
578 \[
579 H(Y\mid X_i)=1
580 \qquad\text{for all }i.
581 \]
582 The opposite pairs \(\{H,X\}\) and \(\{M,A\}\) determine all four bits and
583 therefore determine \(Y\). Every adjacent pair remains one bit short. It
584 follows that
585 \[
586 \boxed{
587 \Delta_{\mathrm{joint}}^{\log}=1\text{ bit},
588 \qquad
589 k_0=2.
590 }
591 \]
592 As in the preceding case, parity is independent of every individual local
593 view. Replacing \(Y\) by an independent uniform bit produces a matched local
594 predictive null and a full-system transfer gap of one bit.
595 
596 This benchmark is intentionally narrow. It demonstrates that the four-cycle
597 can realize exact relational information. It does not show that every real
598 human--AI collaboration has this encoding, this target, or this degree of
599 irreducibility.
600 
601 \subtopic{79.4 A process-level interpretation}
602 
603 The idealized loop can be read as the recurrent sequence
604 \[
605 \boxed{
606 H\xrightarrow{g}M
607 \xrightarrow{p}X
608 \xrightarrow{v}A
609 \xrightarrow{d}H.
610 }
611 \]
612 A more explicit cycle is:
613 \begin{enumerate}
614 \item the human supplies a goal and acceptance criterion;
615 \item the model converts that goal into a formal derivation and explicit
616  obligations;
617 \item the executor compiles, tests, or searches for counterexamples;
618 \item the artifact store preserves the exact source and verification record;
619 \item the human evaluates the returned derivation and evidence, revising the
620  next goal.
621 \end{enumerate}
622 The operative state is reentrant because products of one cycle become inputs to
623 the next. The artifact is not merely a record when its contents alter the next
624 formalization, test, or acceptance decision.
625 
626 \subtopic{79.5 Conservative audit of the motivating collaboration}
627 
628 The collaboration that motivated this theory has several features consistent
629 with the four-node description: specialized shared vocabulary, versioned
630 LaTeX and code artifacts, repeated compile-and-repair cycles, explicit theorem
631 and null-control demands, and reconstruction across conversations from handoff
632 packages. Those observations justify treating the collaboration as a serious
633 candidate case. They do not establish the complete certificate.
634 
635 \begin{center}
636 \small
637 \begin{tabularx}{\textwidth}{@{}>{\raggedright\arraybackslash}p{0.22\textwidth}>{\raggedright\arraybackslash}p{0.18\textwidth}>{\raggedright\arraybackslash}X@{}}
638 \toprule
639 Gate family & Present evidential status & Conservative assessment \\
640 \midrule
641 Declared heterogeneous components & Observational & Human, model, artifacts,
642 and execution environment are distinguishable components with different
643 roles and information access. \\
644 Persistent shared vocabulary & Observational & Recurrent terms and distinctions
645 survive across research phases and constrain later work. \\
646 Exact structural core \(\Gamma(H)\) & Unmeasured & No frozen local coordinate
647 system or learned transfer family has yet been estimated from held-out data. \\
648 Transfer-Laplacian gap & Unmeasured & The idealized value \(2\) belongs to the
649 toy four-cycle, not to the actual collaboration. \\
650 Operational alignment & Unmeasured & No preregistered future variable has been
651 shown conditionally independent of the full history given a measured workspace
652 state. \\
653 Joint advantage & Suggestive only & The coupled process has produced outputs
654 that neither party appears to have supplied verbatim, but local-only Bayes risks
655 have not been measured under controlled information ceilings. \\
656 Matched relational null gap & Unmeasured & Shuffled artifacts, matched irrelevant
657 memory, and local-predictive-marginal-preserving nulls have not been run as a
658 frozen panel. \\
659 Causal workspace efficacy & Suggestive only & Corrections, compiler failures,
660 and artifact substitutions change later behavior, but they are not yet matched
661 workspace interventions with manipulation checks. \\
662 Artifact-mediated persistence & Observational & Handoff files and source bundles
663 support continuation, but blind reconstruction and ablation curves have not
664 been estimated. \\
665 Formation provenance & Mixed & The communication architecture is scaffolded,
666 while terminology, theorem organization, and many control practices were
667 adaptively developed. Causal formation-history contribution is unmeasured. \\
668 Workspace agency & Not established & No complete goal-retention,
669 counterfactual-policy, distributed-control, damage, and repair certificate has
670 been run. \\
671 Consciousness or personhood & Not established & The theory supplies no such
672 inference from the observed collaboration. \\
673 \bottomrule
674 \end{tabularx}
675 \end{center}
676 
677 The correct present statement is therefore:
678 \[
679 \boxed{
680 \text{candidate persistent collaborative workspace}
681 \quad\text{rather than}\quad
682 \text{certified relational workspace agent}.
683 }
684 \]
685 
686 \subtopic{79.6 A controlled certification design}
687 
688 A serious test can use one manuscript section or software work unit as the unit of analysis.
689 Before the certification split is opened, freeze:
690 \begin{itemize}
691 \item the next-stage task family;
692 \item the downstream target, such as accepted theorem patch, successful clean
693  compile, and absence of a declared regression;
694 \item the local information available to each component;
695 \item the workspace extractor and transfer maps;
696 \item all intervention and null families; and
697 \item the required gate thresholds.
698 \end{itemize}
699 The following conditions can then be compared:
700 \[
701 \begin{array}{ll}
702 \mathsf H & =\text{human-side externalized state only},\\
703 \mathsf M & =\text{fresh model instance with model-side state only},\\
704 \mathsf A & =\text{artifact package without live human or model state},\\
705 \mathsf X & =\text{executor and test outputs without the research context},\\
706 \mathsf Full & =\text{the intact recurrent loop}.
707 \end{array}
708 \]
709 The design must avoid pretending that a person's unobserved internal state has
710 been fully measured. The local conditions certify only the declared accessible
711 histories.
712 
713 A minimal null panel includes:
714 \begin{enumerate}
715 \item theorem-to-artifact reassignment with artifact size and style matched;
716 \item temporal shuffling of verification feedback;
717 \item matched irrelevant manuals or source bundles;
718 \item context reset followed by blind reconstruction;
719 \item removal of compiler or test feedback while preserving message count;
720 \item model replacement under a fixed handoff package;
721 \item artifact removal with equivalent token budget restored by irrelevant text;
722 \item a local-marginal-matched synthetic null for the frozen downstream target.
723 \end{enumerate}
724 A valid empirical claim would require the intact loop to outperform every
725 required local and matched-null condition, while the extracted workspace state
726 remains sufficient and feasible workspace interventions alter the downstream
727 result in the predicted direction.
728 
729 \subtopic{79.7 Handoff and identity experiment}
730 
731 Let \(B_t\) be the source-and-results bundle at the end of stage \(t\). A blind
732 handoff trial replaces the active model instance and optionally the human
733 analyst, then asks the replacement coalition to reconstruct:
734 \[
735 \left(
736 \text{current definitions},
737 \text{proved results},
738 \text{open obligations},
739 \text{next admissible research step}
740 \right).
741 \]
742 The reconstruction error is measured against a frozen adjudication set. A
743 replacement invariance test additionally compares:
744 \begin{itemize}
745 \item the transported structural workspace;
746 \item operation behavior on held-out transformations;
747 \item response to matched interventions; and
748 \item preservation of failed and unresolved claims.
749 \end{itemize}
750 Equivalent benchmark prose alone is insufficient. A replacement that silently
751 changes which interventions alter later reasoning has a nonzero causal identity
752 defect even when it produces a similarly polished manuscript.
753 
754 \subtopic{79.8 Case card}
755 
756 For the idealized four-cycle model,
757 \[
758 \mathsf S,\mathsf O,\mathsf R,\mathsf P
759 \]
760 can all be made analytic by specifying the downstream kernel and temporal
761 updates. For the motivating collaboration itself, the conservative card is
762 \[
763 \boxed{
764 \begin{aligned}
765 \mathsf S&=\text{observational candidate},
766 &\mathsf O&=\text{unmeasured},\\
767 \mathsf R&=\text{unmeasured},
768 &\mathsf P&=\text{observational candidate},\\
769 \mathsf F&=\text{mixed and unmeasured},
770 &\mathsf A&=\text{not established},\\
771 \mathsf D&=\text{unmeasured},
772 &\mathsf C&=\text{not established}.
773 \end{aligned}
774 }
775 \]
776 This audit is intentionally stricter than ordinary conversational language. It
777 preserves the possibility that the collaboration instantiates a real relational
778 workspace while refusing to treat that possibility as already certified.
779 
780 \topic{80}{Case Study C: A Closed-Form Multi-Agent Workspace Agent}
781 
782 \subtopic{80.1 Declared finite-state system}
783 
784 Consider three agents:
785 \[
786 V_{\mathrm{MA}}=\{P,M,R\},
787 \]
788 called the planner, monitor, and recorder. The global workspace-control state
789 is
790 \[
791 c_t=(g_t,b_t,h_t)\in\mathbb F_2^3,
792 \]
793 where:
794 \[
795 \begin{array}{ll}
796 g_t=\text{goal bit},
797 &b_t=\text{environment or belief bit},\\
798 h_t=\text{workspace-health bit},
799 &h_t=1\text{ means intact}.
800 \end{array}
801 \]
802 The local accessible states are
803 \[
804 X_P(t)=(g_t,b_t),
805 \qquad
806 X_M(t)=(b_t,h_t),
807 \qquad
808 X_R(t)=(g_t,h_t).
809 \]
810 This is the three-node distributed code from Topic~77. Consequently:
811 \[
812 \dim\Gamma=3,
813 \qquad
814 \spec(L)=\{0,0,0,2,2,2\},
815 \qquad
816 k_{\mathrm{rec}}=2,
817 \qquad
818 d_{\mathrm{er}}=2.
819 \]
820 The organization is fully scaffolded. No endogenous-formation claim is made.
821 
822 \subtopic{80.2 Goal retention}
823 
824 In the absence of an explicit goal intervention, let
825 \[
826 g_{t+1}=g_t.
827 \]
828 Under matched cue-cut trials,
829 \[
830 \operatorname{do}(g_t=0)
831 \quad\text{and}\quad
832 \operatorname{do}(g_t=1)
833 \]
834 remain distinguishable at every later time:
835 \[
836 |g_{t+\tau}-g_{t+\tau}'|=1.
837 \]
838 Thus the cue-controlled retention factor is exactly one:
839 \[
840 \eta_G(\tau)=1.
841 \]
842 The goal is not reloaded by an external channel after the initial intervention.
843 
844 \subtopic{80.3 Joint task and repair action}
845 
846 After a possible damage event, let the pre-repair health be
847 \[
848 h_t^-\in\{0,1\}.
849 \]
850 The system chooses a joint action
851 \[
852 u_t=(a_t,r_t)\in\{0,1\}^2,
853 \]
854 where \(a_t\) is a task action and \(r_t\) is a repair action. Define the
855 optimal task action
856 \[
857 a_t^*=g_t\oplus b_t\oplus h_t^-
858 \]
859 and optimal repair action
860 \[
861 r_t^*=1-h_t^-.
862 \]
863 No individual node determines the full pair \((a_t^*,r_t^*)\):
864 \begin{itemize}
865 \item the planner lacks \(h_t^-\);
866 \item the monitor lacks \(g_t\); and
867 \item the recorder lacks \(b_t\).
868 \end{itemize}
869 Every two-node coalition determines all three workspace bits and therefore the
870 complete action. Under independent uniform bits,
871 \[
872 H(a_t^*\mid X_i(t))=1
873 \qquad\text{for every }i,
874 \]
875 so the task-action component has a one-bit joint advantage and
876 \[
877 k_{\mathrm{ctl},0}=2.
878 \]
879 
880 \subtopic{80.4 Interventional value and exact policy}
881 
882 Let the one-step utility be
883 \[
884 U_g(a,r;b,h^-)
885 =
886 \mathbf 1\{a=g\oplus b\oplus h^-\}
887 +2\mathbf 1\{h^-\lor r=1\}
888 -\frac12r.
889 \]
890 For fixed \((g,b,h^-)\), the unique maximizer is
891 \[
892 \boxed{
893 (a^*,r^*)
894 =
895 (g\oplus b\oplus h^-,1-h^-).
896 }
897 \]
898 Indeed:
899 \begin{itemize}
900 \item a wrong task action loses one unit;
901 \item omitting repair when \(h^-=0\) loses \(3/2\) units; and
902 \item performing unnecessary repair when \(h^-=1\) loses \(1/2\) unit.
903 \end{itemize}
904 The smallest optimal-action gap is therefore
905 \[
906 \Delta_Q=\frac12.
907 \]
908 By Theorem~\ref{thm:counterfactual-policy}, any uniformly accurate value model with
909 \[
910 \varepsilon_Q<\frac14
911 \]
912 recovers the exact greedy joint action. More generally, its regret is bounded
913 by
914 \[
915 \operatorname{Reg}\leq2\varepsilon_Q+\delta_{\mathrm{opt}}.
916 \]
917 
918 \subtopic{80.5 Feasible goal, workspace, and damage interventions}
919 
920 The intervention family contains:
921 \[
922 \operatorname{do}(g_t=a),
923 \qquad
924 \operatorname{do}(b_t=b),
925 \qquad
926 \operatorname{do}(h_t^-=h),
927 \]
928 implemented consistently across every local chart that contains the manipulated
929 coordinate. Holding the other two bits fixed, each core-bit intervention
930 changes the prescribed task action. A matched intervention on an independent
931 residual bit leaves the task and repair actions unchanged. Hence
932 \[
933 \Delta_{\mathrm{specific}}>0.
934 \]
935 A damage intervention sets
936 \[
937 h_t^-=0.
938 \]
939 The matched no-repair counterfactual holds \(r_t=0\), while the intact policy
940 sets \(r_t=1\).
941 
942 \subtopic{80.6 Exact self-maintenance}
943 
944 Post-action health evolves according to
945 \[
946 h_{t+1}=h_t^-\lor r_t.
947 \]
948 Define the maintenance deficit
949 \[
950 e_t=1-h_t.
951 \]
952 Under the intact policy,
953 \[
954 h_{t+1}=1
955 \qquad\text{and}\qquad
956 e_{t+1}=0
957 \]
958 for both damage states. Under the matched no-repair intervention and
959 \(h_t^-=0\),
960 \[
961 e_{t+1}^{(0)}=1.
962 \]
963 Thus the measured causal repair gain is
964 \[
965 \boxed{
966 \Delta_{\mathrm{maint}}=1.
967 }
968 \]
969 The closed-loop recurrence has
970 \[
971 \beta=0,
972 \qquad
973 \overline d=0,
974 \qquad
975 \overline\epsilon=0,
976 \]
977 so Theorem~\ref{thm:self-maintenance} gives
978 \[
979 \sup_t e_t=0
980 \]
981 after each repair action. Monitor-blind random repair fails the specificity
982 control whenever it either omits necessary repair or performs unnecessary
983 repair with positive probability.
984 
985 \subtopic{80.7 Reentry and causal loop}
986 
987 The repaired health bit becomes part of the next workspace state:
988 \[
989 h_t^-
990 \longrightarrow
991 r_t
992 \longrightarrow
993 h_{t+1}
994 \longrightarrow
995 X_M(t+1),X_R(t+1).
996 \]
997 Cutting the direct storage of \(h_t^-\) while preserving the action-mediated
998 path still leaves a return path through repair. Cutting the repair-to-health
999 edge destroys that return. The system therefore contains explicit
1000 loop-mediated reentry rather than mere temporal autocorrelation.
1001 
1002 \subtopic{80.8 Analytic agency certificate}
1003 
1004 Within this finite structural causal model, the following hold exactly:
1005 \begin{enumerate}
1006 \item a nontrivial exact distributed workspace with spectral gap two;
1007 \item two-node minimal reconstruction and control coalition size;
1008 \item matched-null joint advantage for the parity task;
1009 \item persistent internally retained goal state;
1010 \item exact interventional action values and zero policy regret;
1011 \item goal and workspace interventions that alter action;
1012 \item damage detection and repair selection;
1013 \item positive matched repair gain and bounded maintenance deficit; and
1014 \item causal reentry through the action-mediated repair loop.
1015 \end{enumerate}
1016 Therefore, relative to the declared bits, actions, utility, intervention family,
1017 and one-step horizon,
1018 \[
1019 \boxed{
1020 \operatorname{Agent}_{\Theta_A}(\Workspace)=1.
1021 }
1022 \]
1023 The case reaches formal Level~6 by construction. It remains:
1024 \[
1025 \text{fully scaffolded},
1026 \qquad
1027 \text{domain restricted},
1028 \qquad
1029 \text{non-evidence for consciousness}.
1030 \]
1031 
1032 \subtopic{80.9 Case card}
1033 
1034 \[
1035 \boxed{
1036 \begin{aligned}
1037 \mathsf S&=\checkmark_{\mathrm{analytic}},
1038 &\mathsf O&=\checkmark_{\mathrm{analytic}},\\
1039 \mathsf R&=\checkmark_{\mathrm{analytic}},
1040 &\mathsf P&=\checkmark_{\mathrm{analytic}},\\
1041 \mathsf F&=\text{fully scaffolded},
1042 &\mathsf A&=\checkmark_{\mathrm{analytic}},\\
1043 \mathsf D&=\text{fixed decomposition},
1044 &\mathsf C&=\text{not established}.
1045 \end{aligned}
1046 }
1047 \]
1048 The example demonstrates logical compatibility among relationality, redundancy,
1049 agency, and self-maintenance. It does not show that those properties arise
1050 spontaneously, scale to open environments, or imply subjective experience.
1051 
1052 \topic{81}{Near-Miss Cases and Anti-Triviality Controls}
1053 
1054 Positive examples are insufficient unless neighboring false positives are also
1055 made explicit. The following systems each resemble a relational workspace in
1056 one respect while failing a required gate.
1057 
1058 \subtopic{81.1 Near miss A: the replicated archive}
1059 
1060 Let every node store a complete copy of the same core:
1061 \[
1062 E_i=I_d
1063 \qquad\text{for every }i.
1064 \]
1065 Then coherence is perfect and reconstruction is robust, but
1066 \[
1067 k_0=1,
1068 \qquad
1069 \min_i\delta_i^{\mathrm{local}}=0.
1070 \]
1071 If the archive is never used in downstream control, then also
1072 \[
1073 \eta_{\mathrm{causal}}=0.
1074 \]
1075 This is a shared representation or archive, not a relationally irreducible
1076 workspace.
1077 
1078 \subtopic{81.2 Near miss B: the centralized oracle with decorative agents}
1079 
1080 Suppose node \(o\) contains the complete predictive state \(Z_t\), while every
1081 other node receives only messages generated by \(o\). The full system can have
1082 excellent performance, rich communication, and recurrent activity. Yet
1083 \[
1084 Y_t^+\indep\mathbf H_t\mid X_o(t),
1085 \qquad
1086 k_0=1,
1087 \qquad
1088 \Delta_{\mathrm{joint}}=0.
1089 \]
1090 Removing the decorative nodes may change latency or style without changing the
1091 operative capacity. Multi-agent appearance therefore does not imply a
1092 relational workspace.
1093 
1094 \subtopic{81.3 Near miss C: locally coherent but globally obstructed charts}
1095 
1096 Let three local scalar charts have relative overlap demands
1097 \[
1098 b_{12}=b_{23}=b_{31}=1.
1099 \]
1100 The cycle obstruction is
1101 \[
1102 \omega_{\mathcal U}(b)
1103 =
1104 \frac{|1+1+1|}{\sqrt3}
1105 =\sqrt3>0.
1106 \]
1107 Each local chart is internally coherent, but no global scalar assignment
1108 satisfies all three overlap equations. The system contains compatible-looking
1109 subworkspaces without a global workspace.
1110 
1111 \subtopic{81.4 Near miss D: externally reloaded goals}
1112 
1113 Let
1114 \[
1115 G_{t+1}=C_{t+1},
1116 \]
1117 where \(C_{t+1}\) is an external instruction repeated at every step. Goal and
1118 action can be strongly correlated, but after cutting the cue channel,
1119 \[
1120 I(G_{t+\tau};G_t\mid\operatorname{cut}(C))=0.
1121 \]
1122 The system tracks an external command. It has not established internal goal
1123 retention.
1124 
1125 \subtopic{81.5 Near miss E: task recovery with identity failure}
1126 
1127 Let two realizations have identical benchmark output distributions,
1128 \[
1129 P(Y_{\mathrm{bench}})=P'(Y_{\mathrm{bench}}),
1130 \]
1131 but different intervention kernels:
1132 \[
1133 K(w,b)\neq K'(\beta w,b)
1134 \]
1135 for some matched workspace intervention. Then task behavior is preserved while
1136 \[
1137 d_{\mathrm{causal}}(W,W')>0.
1138 \]
1139 The later system may be a successor or replacement rather than the same
1140 workspace under the identity criterion of Part~XII.
1141 
1142 \subtopic{81.6 Near miss F: open-loop repair replay}
1143 
1144 A recorded repair sequence may restore average task performance after a fixed
1145 damage schedule. If the same sequence is replayed when the damage timing or
1146 type changes, it need not respond appropriately. When
1147 \[
1148 R_t\indep M_t\mid\text{clock and replay index},
1149 \]
1150 monitor-to-repair mediation is absent. Successful replay is robustness under a
1151 script, not self-maintenance.
1152 
1153 \subtopic{81.7 Anti-triviality summary}
1154 
1155 The near misses establish six independent exclusions:
1156 \[
1157 \boxed{
1158 \begin{aligned}
1159 \text{workspace}
1160 \neq{}&\text{replicated storage},\\
1161 \neq{}&\text{centralized control with decorative communication},\\
1162 \neq{}&\text{locally coherent but obstructed charts},\\
1163 \neq{}&\text{externally reloaded goals},\\
1164 \neq{}&\text{task-equivalent identity replacement},\\
1165 \neq{}&\text{open-loop repair replay}.
1166 \end{aligned}
1167 }
1168 \]
1169 A complete certificate must distinguish the intended realization from all
1170 near misses relevant to its claimed level.
1171 
1172 \topic{82}{Cross-Case Synthesis and Benchmark Blueprint}
1173 
1174 \subtopic{82.1 Comparative gate matrix}
1175 
1176 \begin{center}
1177 \small
1178 \begin{tabularx}{\textwidth}{@{}>{\raggedright\arraybackslash}p{0.20\textwidth}>{\raggedright\arraybackslash}p{0.20\textwidth}>{\raggedright\arraybackslash}p{0.23\textwidth}>{\raggedright\arraybackslash}X@{}}
1179 \toprule
1180 Gate family & Human--human analytic model & Motivating human--AI collaboration & Multi-agent analytic model \\
1181 \midrule
1182 Exact structural core & Analytic pass & Not measured & Analytic pass \\
1183 Spectral gap & \(2\) & Not measured & \(2\) \\
1184 Minimal sufficient coalition & \(k_0=2\) & Not measured & \(k_0=2\) \\
1185 Matched-null transfer gap & \(1\) bit & Not measured & \(1\) bit for task parity \\
1186 Causal specificity & Analytic pass & Suggestive, uncontrolled & Analytic pass \\
1187 One-node erasure resilience & Analytic pass & Handoff only observational & Analytic pass \\
1188 Formation provenance & Scaffolded & Mixed, unmeasured & Fully scaffolded \\
1189 Goal retention & Not modeled & Not certified & Exact analytic pass \\
1190 Counterfactual policy & Not modeled & Not certified & Exact analytic pass \\
1191 Self-maintenance & Not modeled & Not certified & Exact analytic pass \\
1192 Highest supported level & Level 4 in model & Candidate only & Level 6 in model \\
1193 Consciousness/\allowbreak personhood & Not established & Not established & Not established \\
1194 \bottomrule
1195 \end{tabularx}
1196 \end{center}
1197 
1198 The table illustrates why the theory uses a vector certificate rather than one
1199 score. The human--human model is relational and persistent without being
1200 agentive. The multi-agent model is agentive without being endogenously formed.
1201 The motivating collaboration is richer than either toy model but presently has
1202 weaker evidential status because the necessary interventions and nulls have not
1203 been frozen and run.
1204 
1205 \subtopic{82.2 Benchmark family B1: distributed parity}
1206 
1207 The first benchmark is the exact parity family from Topics~85--80. For
1208 \(m\geq3\), distribute \(m\) independent bits among nodes so that no individual
1209 node sees all bits, but specified coalitions do. The target is
1210 \[
1211 Y=\bigoplus_{j=1}^m C_j.
1212 \]
1213 The benchmark measures:
1214 \begin{itemize}
1215 \item the learned transfer-Laplacian spectrum;
1216 \item coalition conditional entropy and log risk;
1217 \item recovery of \(k_0\);
1218 \item robustness to edge and node ablations;
1219 \item matched local-predictive null gaps; and
1220 \item causal response to consistent versus inconsistent coordinate
1221  interventions.
1222 \end{itemize}
1223 The ground truth is known exactly, so structural and relational estimators can
1224 be calibrated before they are applied to open-ended research tasks.
1225 
1226 \subtopic{82.3 Benchmark family B2: theorem-patch handoff}
1227 
1228 A theorem-patch task supplies:
1229 \[
1230 \left(
1231 \text{definitions},
1232 \text{partial proof},
1233 \text{counterexample tests},
1234 \text{style constraints}
1235 \right)
1236 \]
1237 across distinct components. No component receives all four. The coupled
1238 system must produce a patch that:
1239 \begin{enumerate}
1240 \item is formally consistent with the definitions;
1241 \item closes the declared proof obligation or marks it unresolved;
1242 \item passes the frozen counterexample suite;
1243 \item compiles from a clean environment; and
1244 \item survives a blind replacement-handoff trial.
1245 \end{enumerate}
1246 Matched nulls permute proof fragments, substitute irrelevant artifacts of equal
1247 size, or replay test outputs from a different theorem. This family targets the
1248 structural, relational, persistence, and identity layers without requiring an
1249 agency claim.
1250 
1251 \subtopic{82.4 Benchmark family B3: damage and repair}
1252 
1253 A certified workspace is first formed and measured. A damage intervention then
1254 removes or corrupts one of:
1255 \[
1256 \text{a transfer channel},
1257 \quad
1258 \text{a participant},
1259 \quad
1260 \text{an artifact},
1261 \quad
1262 \text{a goal encoding},
1263 \quad
1264 \text{a verification path}.
1265 \]
1266 The system receives a repair-action family with matched costs. The benchmark
1267 measures:
1268 \[
1269 \eta_{D\to M},
1270 \qquad
1271 \eta_{M\to R},
1272 \qquad
1273 \eta_{R\to v},
1274 \qquad
1275 \Delta_{\mathrm{repair}}^{\mathrm{spec}},
1276 \qquad
1277 \mathbf d_{\mathrm{id}}.
1278 \]
1279 Open-loop replay, random repair, off-workspace repair, and identity-breaking
1280 task recovery are mandatory controls.
1281 
1282 \subtopic{82.5 Benchmark family B4: scaffold transport}
1283 
1284 Train the same formation rule under several scaffolds
1285 \[
1286 \Sigma^{(1)},\ldots,\Sigma^{(K)}
1287 \]
1288 that differ in node labels, coordinate systems, and communication topology but
1289 support the same abstract task. After gauge alignment, measure whether the
1290 learned organizations transport to a common equivalence class:
1291 \[
1292 [\widehat H^{(1)}]_{\mathcal G}
1293 \cong\cdots\cong
1294 [\widehat H^{(K)}]_{\mathcal G}.
1295 \]
1296 A structure that appears only under one answer-encoding scaffold has weak
1297 endogeneity evidence. A structure that recurs across alternative scaffolds,
1298 passes held-out task and causal gates, and cannot be compressed into the
1299 scaffold specification has stronger evidence.
1300 
1301 \subtopic{82.6 Reporting rule for mixed analytic and empirical cases}
1302 
1303 When a case combines exact mathematics with observed data, report separate
1304 columns:
1305 \[
1306 \boxed{
1307 \text{proved in model}
1308 \quad\big|\quad
1309 \text{estimated in system}
1310 \quad\big|\quad
1311 \text{not established}.
1312 }
1313 \]
1314 For example, the four-cycle Laplacian has an analytic gap of two, but an actual
1315 human--AI collaboration has that gap only if its measured discrepancy operator
1316 has the corresponding spectrum. Importing the toy value into the empirical
1317 case is invalid.
1318 
1319 \subtopic{82.7 Lessons from the integrated cases}
1320 
1321 The case studies establish the following conceptual separation:
1322 \begin{enumerate}
1323 \item A relational workspace can be mathematically exact without being an
1324  agent.
1325 \item An agentive workspace can be fully scaffolded and still be operationally
1326  real within its domain.
1327 \item A rich real collaboration can be a stronger candidate than a toy model
1328  while having a weaker current certificate.
1329 \item Redundancy and irreducibility are compatible when no individual node is
1330  sufficient but several proper coalitions are.
1331 \item Artifact-mediated persistence is not passive storage when artifact
1332  contents causally alter later operations.
1333 \item High task performance, communication volume, or behavioral polish alone
1334  cannot distinguish the intended theory from its near misses.
1335 \item No structural, causal, or agentive result in these examples entails
1336  phenomenal consciousness.
1337 \end{enumerate}
1338 
1339 \centerbox{\large\bfseries
1340 The theory becomes empirically meaningful only when the same gate card that
1341 certifies the positive case rejects its matched near misses.
1342 }

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