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identity}=\\text{persistent invariant causal organization under admissible transport}.}\n","context":0,"kind":"use","source":0},{"line":9,"start":284,"end":11852,"expression":"0.18em] Workspace Theory\\par}\n\\vspace{0.55in}\n{\\Large A Core--Transfer Framework for Distributed Cognition,\\\\ Collaboration, and Epistemic Systems\\par}\n\\vspace{0.8in}\n{\\large Komorebi P.\\par}\n\\vspace{0.65in}\n{\\large\\bfseries Foundations, Dynamics, Agency, and Empirical Certification\\par}\n\\vfill\n{\\color{RuleColor}\\rule{0.70\\textwidth}{0.8pt}\\par}\n\\vspace{0.25in}\n{\\small August 2026\\par}\n\\end{titlepage}\n\\clearpage\n\\pagenumbering{roman}\n\n\\begin{abstract}\nCollaborative reasoning increasingly spans people, computational models, documents, software,\ninstruments, and persistent records. Yet the existence of many interacting components does not,\nby itself, establish a shared workspace, and successful group performance does not show that the\noperative state is genuinely distributed rather than copied, centralized, or merely correlated.\nStratified Relational Workspace Theory develops a formal answer to this problem. A realization\nis represented by heterogeneous local state spaces connected by transfer maps. Its exact shared\ncore is the space of global compatible sections, equivalently the fixed subspace of relational\nholonomy when the transfers are invertible. A transfer Laplacian identifies approximate shared\ndirections and provides spectral stability, gauge covariance, and pathwise leakage bounds.\n\nThe structural theory is joined to a dynamic and causal account. Canonical predictive states\ndetermine whether a proposed workspace retains the conditional law relevant to a declared\nfuture task. Feasible interventions, coalition risk, matched relational nulls, and reconstruction\ntests distinguish causally operative distributed organization from archives, replicated memory,\nand centralized controllers. Further results address persistence under changing realizations,\nstable reconstruction after component loss, identity under replacement, local-to-global gluing,\nhierarchical composition, learned transfer structure, boundary recovery, goal-conditioned action,\nand certificate-directed repair. Because distributedness can be destroyed by merging components\nor manufactured by splitting a centralized state into complementary shares, the theory also treats\ndecomposition as part of the claim. Faithful coarse-graining preserves exact structural cores, while\nfamily-robust certification requires every gate to retain a positive margin across a declared set of\nadmissible decompositions. The same discipline is applied to the complete scientific specification.\nTargets, losses, horizons, interventions, nulls, and thresholds are made explicit, and the theory develops\nspecification envelopes, transport radii, experimental-equivalence classes, partial-identification bounds,\nand uniform finite-sample certification. A certificate is therefore distinguished from a favorable\nanalysis: it may be pointwise valid, specification-robust, sign-identified, or statistically unresolved.\nThe resulting rule is noncompensatory. Structural coherence, predictive alignment, causal efficacy,\nrelational irreducibility, non-collapse, reconstructive continuity, provenance, and self-maintenance are\nseparate claims, and failure of one cannot be hidden by success on another.\n\nExact finite examples show how the same framework applies to human collaboration, human--AI\nresearch loops, and multi-agent control. A review of adjacent work locates the individual\nmathematical ingredients in established traditions including distributed cognition, local systems\nand sheaves, spectral perturbation, sufficient statistics, structural causal models, frame reconstruction,\nsystem identification, and feedback control. The candidate contribution is therefore\nnot a new name for any one of these constructions. It is their integration into a single falsifiable\naccount of persistent relational workspaces and the matched near misses that such an account\nmust reject. The framework does not infer phenomenal consciousness, personhood, or moral\nstatus from coherence, agency, or self-maintenance.\n\\end{abstract}\n\n\\tableofcontents\n\\clearpage\n\\pagenumbering{arabic}\n\n\\section*{Introduction}\n\\addcontentsline{toc}{section}{Introduction}\n\nScientific and technical work rarely resides in a single mind or device. A proof may be divided\namong a conjecture, a critic's counterexample, a notebook, a computer algebra system, and a\nverification script. A human--AI research process may distribute objectives, formal manipulations,\nexecution, and persistent memory across different components. Multi-agent systems divide sensing,\nplanning, communication, and repair in still other ways. In each case, it is natural to speak of a\nshared workspace. The phrase is useful, but without a formal criterion it can conceal several\nfundamentally different organizations.\n\nA common archive is not necessarily a workspace. Nor is a group whose apparent coordination is\nproduced by one complete controller surrounded by decorative channels. Two systems can contain\nthe same local information while differing in how that information is related, transported, and used.\nLikewise, a system can perform well without preserving its operative organization through interruption,\nreplacement, or component loss. The central problem is therefore not simply whether information is\nshared. It is whether there is a persistent relational state whose structure and causal capacities cannot\nbe reduced to any single declared component.\n\nThe philosophical background is supplied by extended and distributed cognition, transactive memory,\ncollective intelligence, and theories of group agency \\cite{Hutchins1995,ClarkChalmers1998,Wegner1987,ListPettit2011}.\nThose traditions motivate the possibility that cognitive organization may span people and artifacts.\nThe theory asks a narrower mathematical question: what would have to be measured and proved\nbefore such an organization should be called a relational workspace? The answer must distinguish\nphysical realization from abstract identity, local representation from globally compatible content,\nand statistical association from causal operation.\n\nThe static theory begins with an interaction graph whose vertices carry heterogeneous local state spaces\nand whose edges carry transfer maps. Each local state is decomposed into a transferable core and a\nsubstrate-specific residual. Exact shared content is represented by global compatible sections. When\ncore transfers are invertible, the same object appears as the fixed subspace of the holonomy representation.\nApproximate shared content is extracted as a low-energy spectral subspace of a transfer Laplacian.\nThis formulation supplies explicit obstruction, stability, gauge, and leakage calculations rather than\nrelying on an informal appeal to shared context.\n\nStructure alone is insufficient. A proposed workspace must also retain the predictive state relevant to\na declared future, respond under feasible intervention, outperform matched relational nulls, and resist\nreduction to an individual node or a local-storage surrogate. Persistence introduces further requirements:\nthe workspace must remain trackable as its realization changes, preserve nontrivial content, support\nreconstruction after loss, and maintain an identity profile under admissible replacement. Local workspaces\nmust glue without hidden incompatibility, and hierarchical organization must not conceal centralization.\nFormation claims are evaluated separately from operational claims so that a scaffolded but functional\nworkspace is not dismissed, while an endogenously learned but causally irrelevant pattern is not promoted.\nA separate decomposition analysis asks which claims survive faithful coarse-graining and plausible\nalternative interface descriptions. This is necessary because arbitrary merging can conceal relationality,\nwhile arbitrary splitting can manufacture it. The analysis is then extended from decomposition to the full\nscientific specification. A workspace claim depends on its declared target, loss, horizon, intervention\nfamily, null panel, and thresholds. The theory therefore distinguishes pointwise validity from robustness\nacross defensible specifications and distinguishes both from identifiability under the experiments that are\nactually available. This prevents a favorable target or threshold from being mistaken for a general result\nand prevents an observationally indistinguishable hidden controller from being ruled out by assertion.\n\nThe theory extends to domain-relative agency only after these lower-level conditions are in place. Goals\nmust persist without repeated external rewriting, counterfactual values must guide feasible actions, damage\nmust be detected through the workspace, and repairs must improve a matched no-repair trajectory while\npreserving identity. These criteria define an engineering notion of workspace agency. They neither assert\nnor deny phenomenal consciousness. Consciousness, personhood, and moral status remain separate questions\nthat require independent evidence.\n\nMost of the mathematical tools used here are established. Inverse limits, holonomy, connection Laplacians,\nperturbation theory, sufficient statistics, causal models, frame reconstruction, system identification, and\nfeedback control all have substantial prior literatures. The contribution sought here is integrative: a\nnoncompensatory framework that places these tools in one common architecture and specifies the near-miss\nsystems that each layer must reject. Exact examples provide calibration, while the motivating human--AI\ncollaboration is treated conservatively as a candidate case rather than as a certified instance.\n\nThe paper proceeds from exact structure to approximate structure, dynamic operation, relational\nirreducibility, persistence, reconstruction, identity, gluing, formation, boundary discovery, agency,\ndecomposition robustness, specification robustness, identifiability, and empirical certification. It then\ngives worked cases, situates the framework relative to adjacent work, and closes with its philosophical\nconsequences, limitations, and open problems.\n\n\\section*{Notation and Conventions}\n\\addcontentsline{toc}{section}{Notation and Conventions}\n\nA concrete realization is denoted by \\(\\mathfrak R\\). Its interaction graph is \\(G=(V,E,s,t)\\), with local\nstate spaces \\(X_i\\) at vertices and transfer maps \\(T_e:X_{s(e)}\\to X_{t(e)}\\) on directed edges. When a\ncore--residual decomposition is used, \\(X_i=C_i\\oplus R_i\\), and \\(H_e\\) denotes the induced core transfer.\nThe exact structural workspace is the compatible-section space \\(\\Gamma(H)\\); the transfer discrepancy is\n\\(D_H\\), the transfer Laplacian is \\(L_H=D_H^*D_H\\), and \\(\\Gamma_\\varepsilon(H)\\) denotes a selected\nlow-energy workspace subspace. Dynamic workspace states are written \\(W_t\\), while \\(Z_t\\) denotes the\ncanonical predictive state associated with a declared future variable \\(Y_t^+\\).\n\nUnless stated otherwise, the structural results use finite-dimensional real or complex Hilbert spaces and\nfinite graphs. Probabilistic statements are made on standard Borel spaces when regular conditional laws are\nrequired. All approximate conclusions are indexed by explicit tolerances, spectral gaps, intervention\nfamilies, loss functions, or null classes. A claim of relational irreducibility is always relative to a declared\ndecomposition of the realization into components. The symbol \\(\\mathcal D\\) denotes one such\ndecomposition, and \\(\\mathfrak D\\) denotes an admissible family used for robustness analysis. 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V_{\\mathcal D}}","kind":"use","occurrences":[1870]},{"id":"symbol-f8430698aa9f66f9","tex":"\\min_{t0.\n\\]\nThey may encode channel reliability, frequency, precision, experimental\nimportance, or inverse noise variance. Exact compatibility is independent of\npositive rescaling of these weights, while approximate spectra are not.","\\begin{example}[Human--model--artifact graph]\\label{ex:hma-graph}\nThe motivating three-node system is represented by\n\\[\nV=\\{h,m,a\\},\n\\qquad\nE=\\{h\\to m,m\\to a,a\\to h\\}.\n\\]\nAdditional edges may encode direct retrieval \\(a\\to m\\), human revision\n\\(h\\to a\\), or independent verification. The graph records permitted\ntranslations, not merely physical communication links.\n\\end{example}","\\begin{definition}[Admissible local decomposition]\\label{def:core-residual}\nFor every vertex \\(i\\), choose an orthogonal decomposition\n\\[\nX_i=C_i\\oplus R_i,\n\\]\nwith projectors\n\\[\nP_i:X_i\\to C_i,\n\\qquad\nQ_i=I-P_i:X_i\\to R_i.\n\\]\nThe subspace \\(C_i\\) is the candidate locally realized transferable core. The\nsubspace \\(R_i\\) contains substrate-specific, context-specific, or otherwise\nnoncore degrees of freedom relative to the present analysis.\n\\end{definition}","Relative to these decompositions, every edge map has the block form\n\\[\nT_e=\n\\begin{pmatrix}\nH_e & U_e\\\\\nV_e & K_e\n\\end{pmatrix},\n\\]\nwhere\n\\[\nH_e=P_{t(e)}T_eP_{s(e)}:C_{s(e)}\\to C_{t(e)},\n\\]\n\\[\nU_e=P_{t(e)}T_eQ_{s(e)}:R_{s(e)}\\to C_{t(e)},\n\\]\n\\[\nV_e=Q_{t(e)}T_eP_{s(e)}:C_{s(e)}\\to R_{t(e)},\n\\]\n\\[\nK_e=Q_{t(e)}T_eQ_{s(e)}:R_{s(e)}\\to R_{t(e)}.\n\\]","V_e & K_e\n\\end{pmatrix},\n\\]\nwhere\n\\[\nH_e=P_{t(e)}T_eP_{s(e)}:C_{s(e)}\\to C_{t(e)},\n\\]\n\\[\nU_e=P_{t(e)}T_eQ_{s(e)}:R_{s(e)}\\to C_{t(e)},\n\\]\n\\[\nV_e=Q_{t(e)}T_eP_{s(e)}:C_{s(e)}\\to R_{t(e)},\n\\]\n\\[\nK_e=Q_{t(e)}T_eQ_{s(e)}:R_{s(e)}\\to R_{t(e)}.\n\\]","\\textbf{Core preservation} means\n\\[\nV_e=0.\n\\]\nA core input cannot leak into the target residual.","\\textbf{Core autonomy} means\n\\[\nU_e=V_e=0.\n\\]\nCore and residual evolve as separate transfer systems.","\\textbf{Core leakage}, \\textbf{residual contamination}, and \\textbf{total\ntransfer defect} are measured by\n\\[\n\\lambda_e^{\\mathrm{out}}=\\|V_e\\|,\n\\qquad\n\\lambda_e^{\\mathrm{in}}=\\|U_e\\|,\n\\]\n\\[\n\\delta_e^{\\mathrm{tr}}\n=\n\\|U_e\\|+\\|V_e\\|.\n\\]\nThese quantities depend on the chosen norms and projectors. Their empirical\nmeaning must therefore be tied to a declared intervention and prediction\nprotocol.","For composable edges \\(e:i\\to j\\) and \\(f:j\\to k\\), direct block\nmultiplication gives\n\\[\nT_fT_e\n=\n\\begin{pmatrix}\nH_fH_e+U_fV_e & H_fU_e+U_fK_e","For composable edges \\(e:i\\to j\\) and \\(f:j\\to k\\), direct block\nmultiplication gives\n\\[\nT_fT_e\n=\n\\begin{pmatrix}\nH_fH_e+U_fV_e & H_fU_e+U_fK_e\\\\\nV_fH_e+K_fV_e & V_fU_e+K_fK_e\n\\end{pmatrix}.\n\\]\nThe induced core-to-core block is therefore\n\\[\nP_kT_fT_eP_i\n=\nH_fH_e+U_fV_e.\n\\]\nThe term \\(U_fV_e\\) is a residual-mediated correction: information leaves the\ncandidate core at the intermediate node and later re-enters it. Consequently,\nthe simple path product \\(H_fH_e\\) faithfully represents full core transport\nonly under a preservation condition such as \\(V_e=0\\), or under an explicit\nmodel that includes these mediated corrections.","V_fH_e+K_fV_e & V_fU_e+K_fK_e\n\\end{pmatrix}.\n\\]\nThe induced core-to-core block is therefore\n\\[\nP_kT_fT_eP_i\n=\nH_fH_e+U_fV_e.\n\\]\nThe term \\(U_fV_e\\) is a residual-mediated correction: information leaves the\ncandidate core at the intermediate node and later re-enters it. Consequently,\nthe simple path product \\(H_fH_e\\) faithfully represents full core transport\nonly under a preservation condition such as \\(V_e=0\\), or under an explicit\nmodel that includes these mediated corrections.","K_fV_e & V_fU_e+K_fK_e\n\\end{pmatrix}.\n\\]\nThe induced core-to-core block is therefore\n\\[\nP_kT_fT_eP_i\n=\nH_fH_e+U_fV_e.\n\\]\nThe term \\(U_fV_e\\) is a residual-mediated correction: information leaves the\ncandidate core at the intermediate node and later re-enters it. Consequently,\nthe simple path product \\(H_fH_e\\) faithfully represents full core transport\nonly under a preservation condition such as \\(V_e=0\\), or under an explicit\nmodel that includes these mediated corrections.","\\begin{proposition}[Pathwise core preservation]\\label{prop:pathwise-preservation}\nSuppose \\(V_e=0\\) for every edge in a directed path\n\\(\\gamma=e_m\\cdots e_1\\). Then\n\\[\nT_\\gamma(C_{s(\\gamma)})\\subseteq C_{t(\\gamma)}\n\\]\nand\n\\[\nP_{t(\\gamma)}T_\\gamma P_{s(\\gamma)}\n=\nH_{e_m}\\cdots H_{e_1}.\n\\]\n\\end{proposition}","\\begin{proof}\nFor a single edge, \\(V_e=0\\) means\n\\[\nT_e(c,0)=(H_ec,0),\n\\]\nso core states remain in the core. Inductively, if the first \\(r\\) transfers\nmap \\(c\\) to \\((H_{e_r}\\cdots H_{e_1}c,0)\\), applying \\(T_{e_{r+1}}\\) gives\n\\[\n(H_{e_{r+1}}H_{e_r}\\cdots H_{e_1}c,0).\n\\]\nThe conclusion follows after \\(m\\) steps.\n\\end{proof}","\\begin{corollary}[Autonomous splitting]\\label{cor:autonomous-splitting}\nIf \\(U_e=V_e=0\\) for every edge, then the complete diagram decomposes as the\ndirect sum of two diagrams,\n\\[\n\\mathcal X\\cong H\\oplus K,\n\\]\nand every path transfer satisfies\n\\[\nT_\\gamma\n=\n\\begin{pmatrix}\nH_\\gamma&0","\\begin{corollary}[Autonomous splitting]\\label{cor:autonomous-splitting}\nIf \\(U_e=V_e=0\\) for every edge, then the complete diagram decomposes as the\ndirect sum of two diagrams,\n\\[\n\\mathcal X\\cong H\\oplus K,\n\\]\nand every path transfer satisfies\n\\[\nT_\\gamma\n=\n\\begin{pmatrix}\nH_\\gamma&0\\\\\n0&K_\\gamma\n\\end{pmatrix}.\n\\]\n\\end{corollary}","\\begin{definition}[Exact core-transfer system]\\label{def:exact-core-transfer}\nAn exact core-transfer system on \\(G\\) consists of:\n\\begin{enumerate}\n\\item decompositions \\(X_i=C_i\\oplus R_i\\);\n\\item core-preserving edge maps \\(V_e=0\\); and\n\\item invertible core blocks \\(H_e\\in GL(C_{s(e)},C_{t(e)})\\) whenever\nholonomy is invoked.\n\\end{enumerate}\n\\end{definition}","The requirement \\(V_e=0\\) concerns the relation between the selected core and\nthe full transfer. The requirement that \\(H_e\\) be invertible concerns the\nability to compare core representations along reversed paths. They answer\ndifferent questions and should not be conflated.","\\begin{definition}[Operation algebra]\\label{def:operation-algebra}\nLet \\(\\mathbb F\\in\\{\\mathbb R,\\mathbb C\\}\\). An operation algebra\n\\(\\mathcal A\\) is a unital associative \\(\\mathbb F\\)-algebra whose elements\nrepresent abstract transformations, inferential rules, questions, or\nprocedures that can be realized in multiple strata.\n\\end{definition}","Every core space carries a unital representation\n\\[\n\\rho_i:\\mathcal A\\longrightarrow\\End(C_i).\n\\]\nFor \\(a\\in\\mathcal A\\), the operator \\(\\rho_i(a)\\) is the realization of the\nsame abstract operation in local coordinates.","\\begin{definition}[Exact operation-compatible transfer]\\label{def:intertwiner}\nA core transfer \\(H_e:C_i\\to C_j\\) is operation compatible when it is an\nintertwiner:\n\\[\n\\boxed{\nH_e\\rho_i(a)=\\rho_j(a)H_e\n}\n\\qquad\n\\forall a\\in\\mathcal A.\n\\]\nEquivalently, the diagram\n\\[\n\\begin{array}{ccc}\nC_i&\\xrightarrow{\\rho_i(a)}&C_i","\\begin{definition}[Exact operation-compatible transfer]\\label{def:intertwiner}\nA core transfer \\(H_e:C_i\\to C_j\\) is operation compatible when it is an\nintertwiner:\n\\[\n\\boxed{\nH_e\\rho_i(a)=\\rho_j(a)H_e\n}\n\\qquad\n\\forall a\\in\\mathcal A.\n\\]\nEquivalently, the diagram\n\\[\n\\begin{array}{ccc}\nC_i&\\xrightarrow{\\rho_i(a)}&C_i\\\\\n\\downarrow H_e&&\\downarrow H_e\\\\\nC_j&\\xrightarrow{\\rho_j(a)}&C_j\n\\end{array}\n\\]\ncommutes for every \\(a\\).\n\\end{definition}","C_j&\\xrightarrow{\\rho_j(a)}&C_j\n\\end{array}\n\\]\ncommutes for every \\(a\\).\n\\end{definition}","\\begin{proposition}[Path composition of intertwiners]\\label{prop:path-intertwiner}\nIf every edge of a path \\(\\gamma:i\\to j\\) intertwines the local\n\\(\\mathcal A\\)-representations, then\n\\[\nH_\\gamma\\rho_i(a)=\\rho_j(a)H_\\gamma\n\\qquad\n\\forall a\\in\\mathcal A.\n\\]\n\\end{proposition}","\\begin{proof}\nFor composable intertwiners \\(H_e:C_i\\to C_j\\) and\n\\(H_f:C_j\\to C_k\\),\n\\[\nH_fH_e\\rho_i(a)\n=H_f\\rho_j(a)H_e\n=\\rho_k(a)H_fH_e.\n\\]\nInduction over path length proves the claim.\n\\end{proof}","\\begin{corollary}[Holonomy commutes with abstract operations]\\label{cor:holonomy-commutes}\nFor every loop \\(\\gamma:o\\to o\\),\n\\[\nH_\\gamma\\rho_o(a)=\\rho_o(a)H_\\gamma.\n\\]\nHence\n\\[\n\\rho_o(\\mathcal A)\n\\subseteq\n\\operatorname{Comm}(\\Hol_o),\n\\]\nwhere \\(\\operatorname{Comm}(\\Hol_o)\\) is the commutant of the holonomy\nrepresentation.\n\\end{corollary}","The edgewise defect is\n\\[\n\\Xi_e(a)\n=\nH_e\\rho_{s(e)}(a)-\\rho_{t(e)}(a)H_e.\n\\]\nFor a bounded generating family \\(\\mathcal G\\subseteq\\mathcal A\\), define\n\\[\n\\eta_{\\mathrm{int}}(\\mathcal G)\n=\n\\sup_{a\\in\\mathcal G}\n\\sum_{e\\in E}w_e\\|\\Xi_e(a)\\|^2.\n\\]\nWhen \\(\\mathcal A\\) is normed and its unit ball is appropriate, one may take\n\\[\n\\mathcal G=\\{a:\\|a\\|_{\\mathcal A}\\leq1\\}.\n\\]\nExact representational universality means\n\\[\n\\eta_{\\mathrm{int}}=0,\n\\]\nwhile approximate universality means\n\\[\n\\eta_{\\mathrm{int}}\\leq\\varepsilon_{\\mathrm{int}}.\n\\]\nThe use of a generating family prevents the definition from depending on an\nunbounded rescaling of algebra elements.","efect is\n\\[\n\\Xi_e(a)\n=\nH_e\\rho_{s(e)}(a)-\\rho_{t(e)}(a)H_e.\n\\]\nFor a bounded generating family \\(\\mathcal G\\subseteq\\mathcal A\\), define\n\\[\n\\eta_{\\mathrm{int}}(\\mathcal G)\n=\n\\sup_{a\\in\\mathcal G}\n\\sum_{e\\in E}w_e\\|\\Xi_e(a)\\|^2.\n\\]\nWhen \\(\\mathcal A\\) is normed and its unit ball is appropriate, one may take\n\\[\n\\mathcal G=\\{a:\\|a\\|_{\\mathcal A}\\leq1\\}.\n\\]\nExact representational universality means\n\\[\n\\eta_{\\mathrm{int}}=0,\n\\]\nwhile approximate universality means\n\\[\n\\eta_{\\mathrm{int}}\\leq\\varepsilon_{\\mathrm{int}}.\n\\]\nThe use of a generating family prevents the definition from depending on an\nunbounded rescaling of algebra elements.","a bounded generating family \\(\\mathcal G\\subseteq\\mathcal A\\), define\n\\[\n\\eta_{\\mathrm{int}}(\\mathcal G)\n=\n\\sup_{a\\in\\mathcal G}\n\\sum_{e\\in E}w_e\\|\\Xi_e(a)\\|^2.\n\\]\nWhen \\(\\mathcal A\\) is normed and its unit ball is appropriate, one may take\n\\[\n\\mathcal G=\\{a:\\|a\\|_{\\mathcal A}\\leq1\\}.\n\\]\nExact representational universality means\n\\[\n\\eta_{\\mathrm{int}}=0,\n\\]\nwhile approximate universality means\n\\[\n\\eta_{\\mathrm{int}}\\leq\\varepsilon_{\\mathrm{int}}.\n\\]\nThe use of a generating family prevents the definition from depending on an\nunbounded rescaling of algebra elements.","\\begin{example}[Equivalent operations in different coordinates]\\label{ex:coordinate-intertwiner}\nLet \\(C_1=C_2=\\mathbb R^2\\), choose \\(S\\in GL_2(\\mathbb R)\\), and set\n\\[\nH_{12}=S,\n\\qquad\n\\rho_2(a)=S\\rho_1(a)S^{-1}.\n\\]\nThen\n\\[\nH_{12}\\rho_1(a)\n=S\\rho_1(a)\n=\\rho_2(a)S\n=\\rho_2(a)H_{12}.\n\\]\nThe local matrices can look different while realizing the same abstract\noperation. This is the elementary model of representation change used\nthroughout the theory.\n\\end{example}","\\begin{definition}[Global compatible section]\\label{def:global-section}\nThe exact structural workspace is\n\\[\n\\boxed{\n\\Gamma(H)\n=\n\\left\\{\n(c_i)_{i\\in V}\\in\\bigoplus_{i\\in V}C_i:\n\\ c_{t(e)}=H_ec_{s(e)}\\ \\forall e\\in E\n\\right\\}.\n}\n\\]\nAn element \\(c=(c_i)\\in\\Gamma(H)\\) is a global compatible section.\n\\end{definition}","Equivalently,\n\\[\n\\Gamma(H)=\\varprojlim_{\\Path(G)}C_i.\n\\]\nA section is not a single vector copied verbatim at every vertex. It is one\nabstract element represented by a family of local vectors that agree under the\nspecified translation maps.","A nontrivial exact structural workspace exists when\n\\[\n\\dim\\Gamma(H)>0.\n\\]\nIts exact dimension is\n\\[\nd_{\\Workspace}=\\dim\\Gamma(H).\n\\]","Fix a root \\(o\\). The evaluation map is\n\\[\n\\ev_o:\\Gamma(H)\\to C_o,\n\\qquad\n\\ev_o((c_i))=c_o.\n\\]","\\begin{lemma}[Root determination]\\label{lem:root-determination}\nIf \\(|G|\\) is connected and every core transfer is invertible, then\n\\(\\ev_o\\) is injective. Consequently,\n\\[\n\\dim\\Gamma(H)\\leq\\dim C_o.\n\\]\n\\end{lemma}","\\begin{proof}\nChoose for each vertex \\(i\\) a path \\(\\tau_i:o\\to i\\). Compatibility gives\n\\[\nc_i=H_{\\tau_i}c_o.\n\\]\nThus a section with \\(c_o=0\\) vanishes at every vertex. Hence \\(\\ev_o\\) is\ninjective.\n\\end{proof}","\\begin{lemma}[Tree propagation]\\label{lem:tree-propagation}\nIf \\(|G|\\) is a tree and every \\(H_e\\) is invertible, then every\n\\(x\\in C_o\\) extends uniquely to a global compatible section.\n\\end{lemma}","\\begin{proof}\nA tree contains a unique undirected path \\(\\tau_i:o\\to i\\) for every vertex.\nDefine \\(c_i=H_{\\tau_i}x\\). Because every edge belongs to these unique path\nrelations, the family is compatible. Uniqueness follows from\nLemma~\\ref{lem:root-determination}.\n\\end{proof}","Fix a root \\(o\\in V\\). For every loop class\n\\[\n[\\gamma]\\in\\pi_1(|G|,o),\n\\]\ndefine the holonomy\n\\[\n\\Hol([\\gamma])=H_\\gamma\\in GL(C_o).\n\\]\nFunctoriality gives a group representation\n\\[\n\\operatorname{hol}_o:\n\\pi_1(|G|,o)\\longrightarrow GL(C_o).\n\\]\nIts image is the holonomy group\n\\[\n\\Hol_o=\\operatorname{im}(\\operatorname{hol}_o).\n\\]","The obstruction associated with a loop is\n\\[\n\\Omega_\\gamma=H_\\gamma-I_{C_o}.\n\\]\nThe maximal path-independent root core is\n\\[\n\\boxed{\nC_*\n=\n\\bigcap_{\\gamma:o\\to o}\\ker(H_\\gamma-I).\n}\n\\]\nEquivalently,\n\\[\nC_*=C_o^{\\Hol_o}=\\operatorname{Fix}(\\Hol_o).\n\\]","Choose a spanning tree \\(\\mathcal T\\subseteq|G|\\). Let\n\\[\n\\tau_i:o\\to i\n\\]\nbe the unique tree path. Every non-tree edge \\(e:i\\to j\\) determines a\nfundamental loop\n\\[\n\\gamma_e=\\tau_j^{-1}e\\tau_i\n\\]\nwith transfer\n\\[\nH_{\\gamma_e}=H_{\\tau_j}^{-1}H_eH_{\\tau_i}.\n\\]\nThe fundamental group of a finite graph is generated freely by these cotree\nloops. Hence it is enough to impose fixed-point conditions for a finite set of\ngenerators.","\\begin{lemma}[Finite obstruction set]\\label{lem:finite-obstruction}\nLet \\(e_1,\\ldots,e_r\\) be the non-tree edges of a connected finite graph.\nThen\n\\[\nC_*\n=\n\\bigcap_{q=1}^{r}\n\\ker(H_{\\gamma_{e_q}}-I).\n\\]\n\\end{lemma}","\\begin{proof}\nA vector fixed by every generator is fixed by each inverse generator and by\nevery finite product of generators and inverses. Those products represent all\nelements of \\(\\pi_1(|G|,o)\\). The reverse inclusion is immediate.\n\\end{proof}","The number of independent graph cycles is\n\\[\nr=|E_{\\mathrm{und}}|-|V|+1\n\\]\nfor a connected underlying graph, after treating parallel edges with their\nmultiplicity.","\\begin{theorem}[Limit--holonomy equivalence]\\label{thm:limit-holonomy}\nLet \\(|G|\\) be connected and let\n\\[\nH:\\Pi_1(G)\\to\\mathbf{FinHilb}^{\\times}\n\\]\nbe a finite-dimensional local system. For any root \\(o\\in V\\), evaluation at\nthe root induces a canonical vector-space isomorphism\n\\[\n\\boxed{\n\\ev_o:\\Gamma(H)\n\\xrightarrow{\\ \\cong\\ }\nC_o^{\\Hol_o}\n=\n\\bigcap_{\\gamma:o\\to o}\\ker(H_\\gamma-I).\n}\n\\]\nConsequently,\n\\[\n\\Gamma(H)\\neq\\{0\\}\n\\]\nif and only if the holonomy representation contains a nonzero trivial\nsubrepresentation.\n\\end{theorem}","heorem}[Limit--holonomy equivalence]\\label{thm:limit-holonomy}\nLet \\(|G|\\) be connected and let\n\\[\nH:\\Pi_1(G)\\to\\mathbf{FinHilb}^{\\times}\n\\]\nbe a finite-dimensional local system. For any root \\(o\\in V\\), evaluation at\nthe root induces a canonical vector-space isomorphism\n\\[\n\\boxed{\n\\ev_o:\\Gamma(H)\n\\xrightarrow{\\ \\cong\\ }\nC_o^{\\Hol_o}\n=\n\\bigcap_{\\gamma:o\\to o}\\ker(H_\\gamma-I).\n}\n\\]\nConsequently,\n\\[\n\\Gamma(H)\\neq\\{0\\}\n\\]\nif and only if the holonomy representation contains a nonzero trivial\nsubrepresentation.\n\\end{theorem}","\\begin{proof}\nLet \\(c=(c_i)\\in\\Gamma(H)\\). For any loop \\(\\gamma:o\\to o\\), compatibility\nalong the loop gives\n\\[\nH_\\gamma c_o=c_o.\n\\]\nTherefore \\(\\ev_o(c)=c_o\\in C_o^{\\Hol_o}\\). By\nLemma~\\ref{lem:root-determination}, \\(\\ev_o\\) is injective.","Conversely, let \\(x\\in C_o^{\\Hol_o}\\). Choose a path\n\\(\\tau_i:o\\to i\\) for each vertex and define\n\\[\nc_i=H_{\\tau_i}x.\n\\]\nThis definition is independent of the chosen path. Indeed, if\n\\(\\sigma_i:o\\to i\\) is another path, then\n\\[\n\\sigma_i^{-1}\\tau_i:o\\to o\n\\]\nis a loop, and invariance of \\(x\\) gives\n\\[\nH_{\\sigma_i}^{-1}H_{\\tau_i}x=x.\n\\]\nMultiplying by \\(H_{\\sigma_i}\\) yields\n\\[\nH_{\\tau_i}x=H_{\\sigma_i}x.\n\\]\nFor any edge \\(e:i\\to j\\), the paths \\(e\\tau_i\\) and \\(\\tau_j\\) both run\nfrom \\(o\\) to \\(j\\). Path independence therefore gives\n\\[\nH_ec_i\n=H_eH_{\\tau_i}x\n=H_{\\tau_j}x\n=c_j.\n\\]\nThus \\((c_i)\\) is a global compatible section and \\(\\ev_o((c_i))=x\\).\nHence \\(\\ev_o\\) is surjective and therefore an isomorphism.\n\\end{proof}","_o^{\\Hol_o}\\). Choose a path\n\\(\\tau_i:o\\to i\\) for each vertex and define\n\\[\nc_i=H_{\\tau_i}x.\n\\]\nThis definition is independent of the chosen path. Indeed, if\n\\(\\sigma_i:o\\to i\\) is another path, then\n\\[\n\\sigma_i^{-1}\\tau_i:o\\to o\n\\]\nis a loop, and invariance of \\(x\\) gives\n\\[\nH_{\\sigma_i}^{-1}H_{\\tau_i}x=x.\n\\]\nMultiplying by \\(H_{\\sigma_i}\\) yields\n\\[\nH_{\\tau_i}x=H_{\\sigma_i}x.\n\\]\nFor any edge \\(e:i\\to j\\), the paths \\(e\\tau_i\\) and \\(\\tau_j\\) both run\nfrom \\(o\\) to \\(j\\). Path independence therefore gives\n\\[\nH_ec_i\n=H_eH_{\\tau_i}x\n=H_{\\tau_j}x\n=c_j.\n\\]\nThus \\((c_i)\\) is a global compatible section and \\(\\ev_o((c_i))=x\\).\nHence \\(\\ev_o\\) is surjective and therefore an isomorphism.\n\\end{proof}","path\n\\(\\tau_i:o\\to i\\) for each vertex and define\n\\[\nc_i=H_{\\tau_i}x.\n\\]\nThis definition is independent of the chosen path. Indeed, if\n\\(\\sigma_i:o\\to i\\) is another path, then\n\\[\n\\sigma_i^{-1}\\tau_i:o\\to o\n\\]\nis a loop, and invariance of \\(x\\) gives\n\\[\nH_{\\sigma_i}^{-1}H_{\\tau_i}x=x.\n\\]\nMultiplying by \\(H_{\\sigma_i}\\) yields\n\\[\nH_{\\tau_i}x=H_{\\sigma_i}x.\n\\]\nFor any edge \\(e:i\\to j\\), the paths \\(e\\tau_i\\) and \\(\\tau_j\\) both run\nfrom \\(o\\) to \\(j\\). Path independence therefore gives\n\\[\nH_ec_i\n=H_eH_{\\tau_i}x\n=H_{\\tau_j}x\n=c_j.\n\\]\nThus \\((c_i)\\) is a global compatible section and \\(\\ev_o((c_i))=x\\).\nHence \\(\\ev_o\\) is surjective and therefore an isomorphism.\n\\end{proof}","to i\\) for each vertex and define\n\\[\nc_i=H_{\\tau_i}x.\n\\]\nThis definition is independent of the chosen path. Indeed, if\n\\(\\sigma_i:o\\to i\\) is another path, then\n\\[\n\\sigma_i^{-1}\\tau_i:o\\to o\n\\]\nis a loop, and invariance of \\(x\\) gives\n\\[\nH_{\\sigma_i}^{-1}H_{\\tau_i}x=x.\n\\]\nMultiplying by \\(H_{\\sigma_i}\\) yields\n\\[\nH_{\\tau_i}x=H_{\\sigma_i}x.\n\\]\nFor any edge \\(e:i\\to j\\), the paths \\(e\\tau_i\\) and \\(\\tau_j\\) both run\nfrom \\(o\\) to \\(j\\). Path independence therefore gives\n\\[\nH_ec_i\n=H_eH_{\\tau_i}x\n=H_{\\tau_j}x\n=c_j.\n\\]\nThus \\((c_i)\\) is a global compatible section and \\(\\ev_o((c_i))=x\\).\nHence \\(\\ev_o\\) is surjective and therefore an isomorphism.\n\\end{proof}","d define\n\\[\nc_i=H_{\\tau_i}x.\n\\]\nThis definition is independent of the chosen path. Indeed, if\n\\(\\sigma_i:o\\to i\\) is another path, then\n\\[\n\\sigma_i^{-1}\\tau_i:o\\to o\n\\]\nis a loop, and invariance of \\(x\\) gives\n\\[\nH_{\\sigma_i}^{-1}H_{\\tau_i}x=x.\n\\]\nMultiplying by \\(H_{\\sigma_i}\\) yields\n\\[\nH_{\\tau_i}x=H_{\\sigma_i}x.\n\\]\nFor any edge \\(e:i\\to j\\), the paths \\(e\\tau_i\\) and \\(\\tau_j\\) both run\nfrom \\(o\\) to \\(j\\). Path independence therefore gives\n\\[\nH_ec_i\n=H_eH_{\\tau_i}x\n=H_{\\tau_j}x\n=c_j.\n\\]\nThus \\((c_i)\\) is a global compatible section and \\(\\ev_o((c_i))=x\\).\nHence \\(\\ev_o\\) is surjective and therefore an isomorphism.\n\\end{proof}","\\[\nc_i=H_{\\tau_i}x.\n\\]\nThis definition is independent of the chosen path. Indeed, if\n\\(\\sigma_i:o\\to i\\) is another path, then\n\\[\n\\sigma_i^{-1}\\tau_i:o\\to o\n\\]\nis a loop, and invariance of \\(x\\) gives\n\\[\nH_{\\sigma_i}^{-1}H_{\\tau_i}x=x.\n\\]\nMultiplying by \\(H_{\\sigma_i}\\) yields\n\\[\nH_{\\tau_i}x=H_{\\sigma_i}x.\n\\]\nFor any edge \\(e:i\\to j\\), the paths \\(e\\tau_i\\) and \\(\\tau_j\\) both run\nfrom \\(o\\) to \\(j\\). Path independence therefore gives\n\\[\nH_ec_i\n=H_eH_{\\tau_i}x\n=H_{\\tau_j}x\n=c_j.\n\\]\nThus \\((c_i)\\) is a global compatible section and \\(\\ev_o((c_i))=x\\).\nHence \\(\\ev_o\\) is surjective and therefore an isomorphism.\n\\end{proof}","independent of the chosen path. Indeed, if\n\\(\\sigma_i:o\\to i\\) is another path, then\n\\[\n\\sigma_i^{-1}\\tau_i:o\\to o\n\\]\nis a loop, and invariance of \\(x\\) gives\n\\[\nH_{\\sigma_i}^{-1}H_{\\tau_i}x=x.\n\\]\nMultiplying by \\(H_{\\sigma_i}\\) yields\n\\[\nH_{\\tau_i}x=H_{\\sigma_i}x.\n\\]\nFor any edge \\(e:i\\to j\\), the paths \\(e\\tau_i\\) and \\(\\tau_j\\) both run\nfrom \\(o\\) to \\(j\\). Path independence therefore gives\n\\[\nH_ec_i\n=H_eH_{\\tau_i}x\n=H_{\\tau_j}x\n=c_j.\n\\]\nThus \\((c_i)\\) is a global compatible section and \\(\\ev_o((c_i))=x\\).\nHence \\(\\ev_o\\) is surjective and therefore an isomorphism.\n\\end{proof}","i:o\\to i\\) is another path, then\n\\[\n\\sigma_i^{-1}\\tau_i:o\\to o\n\\]\nis a loop, and invariance of \\(x\\) gives\n\\[\nH_{\\sigma_i}^{-1}H_{\\tau_i}x=x.\n\\]\nMultiplying by \\(H_{\\sigma_i}\\) yields\n\\[\nH_{\\tau_i}x=H_{\\sigma_i}x.\n\\]\nFor any edge \\(e:i\\to j\\), the paths \\(e\\tau_i\\) and \\(\\tau_j\\) both run\nfrom \\(o\\) to \\(j\\). Path independence therefore gives\n\\[\nH_ec_i\n=H_eH_{\\tau_i}x\n=H_{\\tau_j}x\n=c_j.\n\\]\nThus \\((c_i)\\) is a global compatible section and \\(\\ev_o((c_i))=x\\).\nHence \\(\\ev_o\\) is surjective and therefore an isomorphism.\n\\end{proof}","{-1}\\tau_i:o\\to o\n\\]\nis a loop, and invariance of \\(x\\) gives\n\\[\nH_{\\sigma_i}^{-1}H_{\\tau_i}x=x.\n\\]\nMultiplying by \\(H_{\\sigma_i}\\) yields\n\\[\nH_{\\tau_i}x=H_{\\sigma_i}x.\n\\]\nFor any edge \\(e:i\\to j\\), the paths \\(e\\tau_i\\) and \\(\\tau_j\\) both run\nfrom \\(o\\) to \\(j\\). Path independence therefore gives\n\\[\nH_ec_i\n=H_eH_{\\tau_i}x\n=H_{\\tau_j}x\n=c_j.\n\\]\nThus \\((c_i)\\) is a global compatible section and \\(\\ev_o((c_i))=x\\).\nHence \\(\\ev_o\\) is surjective and therefore an isomorphism.\n\\end{proof}","loop, and invariance of \\(x\\) gives\n\\[\nH_{\\sigma_i}^{-1}H_{\\tau_i}x=x.\n\\]\nMultiplying by \\(H_{\\sigma_i}\\) yields\n\\[\nH_{\\tau_i}x=H_{\\sigma_i}x.\n\\]\nFor any edge \\(e:i\\to j\\), the paths \\(e\\tau_i\\) and \\(\\tau_j\\) both run\nfrom \\(o\\) to \\(j\\). Path independence therefore gives\n\\[\nH_ec_i\n=H_eH_{\\tau_i}x\n=H_{\\tau_j}x\n=c_j.\n\\]\nThus \\((c_i)\\) is a global compatible section and \\(\\ev_o((c_i))=x\\).\nHence \\(\\ev_o\\) is surjective and therefore an isomorphism.\n\\end{proof}","\\begin{corollary}[Tree case]\\label{cor:tree-core}\nIf \\(|G|\\) is a tree, then \\(\\Hol_o=\\{I\\}\\) and\n\\[\n\\Gamma(H)\\cong C_o.\n\\]\nIn particular,\n\\[\nd_{\\Workspace}=\\dim C_o.\n\\]\n\\end{corollary}","\\begin{corollary}[Finite matrix test]\\label{cor:finite-matrix-test}\nLet \\(\\gamma_1,\\ldots,\\gamma_r\\) be fundamental cycle generators and suppose\n\\(\\dim C_o=d\\). Define the stacked obstruction matrix\n\\[\n\\mathcal M\n=\n\\begin{pmatrix}\nH_{\\gamma_1}-I","\\begin{corollary}[Finite matrix test]\\label{cor:finite-matrix-test}\nLet \\(\\gamma_1,\\ldots,\\gamma_r\\) be fundamental cycle generators and suppose\n\\(\\dim C_o=d\\). Define the stacked obstruction matrix\n\\[\n\\mathcal M\n=\n\\begin{pmatrix}\nH_{\\gamma_1}-I\\\\\n\\vdots\\\\\nH_{\\gamma_r}-I\n\\end{pmatrix}.\n\\]\nThen\n\\[\n\\boxed{\nC_*=\\ker\\mathcal M,\n\\qquad\nd_{\\Workspace}=d-\\operatorname{rank}(\\mathcal M).\n}\n\\]\n\\end{corollary}","H_{\\gamma_r}-I\n\\end{pmatrix}.\n\\]\nThen\n\\[\n\\boxed{\nC_*=\\ker\\mathcal M,\n\\qquad\nd_{\\Workspace}=d-\\operatorname{rank}(\\mathcal M).\n}\n\\]\n\\end{corollary}","\\begin{corollary}[Averaging projector for finite holonomy]\\label{cor:averaging-projector}\nIf \\(\\Hol_o\\) is finite and acts unitarily, then\n\\[\nP_*=\n\\frac{1}{|\\Hol_o|}\n\\sum_{h\\in\\Hol_o}h\n\\]\nis the orthogonal projector onto \\(C_*\\).\n\\end{corollary}","\\begin{proof}\nFor every \\(g\\in\\Hol_o\\), left multiplication permutes the group elements, so\n\\(gP_*=P_*\\). Thus \\(\\operatorname{ran}P_*\\subseteq C_*\\). If\n\\(x\\in C_*\\), then \\(P_*x=x\\), so \\(C_*\\subseteq\\operatorname{ran}P_*\\).\nUnitarity gives \\(P_*^*=P_*\\), and the range characterization implies\n\\(P_*^2=P_*\\).\n\\end{proof}","\\begin{example}[One-dimensional core surviving a contradictory cycle]\\label{ex:reflection-cycle}\nLet a three-node cycle have \\(C_1=C_2=C_3=\\mathbb R^2\\) and\n\\[\nH_{12}=I,\n\\qquad\nH_{23}=I,\n\\qquad\nH_{31}=J,\n\\qquad\nJ=\n\\begin{pmatrix}\n1&0","\\begin{example}[One-dimensional core surviving a contradictory cycle]\\label{ex:reflection-cycle}\nLet a three-node cycle have \\(C_1=C_2=C_3=\\mathbb R^2\\) and\n\\[\nH_{12}=I,\n\\qquad\nH_{23}=I,\n\\qquad\nH_{31}=J,\n\\qquad\nJ=\n\\begin{pmatrix}\n1&0\\\\\n0&-1\n\\end{pmatrix}.\n\\]\nThe cycle holonomy is \\(J\\), hence\n\\[\nC_*=\\ker(J-I)=\\operatorname{span}\\{(1,0)^\\top\\}.\n\\]\nThe first coordinate circulates consistently through all three strata. The\nsecond returns with its sign reversed and therefore cannot define a nonzero\nexact shared section. Thus\n\\[\nd_{\\Workspace}=1.\n\\]\n\\end{example}","0&-1\n\\end{pmatrix}.\n\\]\nThe cycle holonomy is \\(J\\), hence\n\\[\nC_*=\\ker(J-I)=\\operatorname{span}\\{(1,0)^\\top\\}.\n\\]\nThe first coordinate circulates consistently through all three strata. The\nsecond returns with its sign reversed and therefore cannot define a nonzero\nexact shared section. Thus\n\\[\nd_{\\Workspace}=1.\n\\]\n\\end{example}","\\begin{example}[Local compatibility without a global nonzero core]\\label{ex:rotation-obstruction}\nLet the only nontrivial cycle holonomy on \\(\\mathbb R^2\\) be a rotation\n\\[\nR_\\theta=\n\\begin{pmatrix}\n\\cos\\theta&-\\sin\\theta","\\begin{example}[Local compatibility without a global nonzero core]\\label{ex:rotation-obstruction}\nLet the only nontrivial cycle holonomy on \\(\\mathbb R^2\\) be a rotation\n\\[\nR_\\theta=\n\\begin{pmatrix}\n\\cos\\theta&-\\sin\\theta\\\\\n\\sin\\theta&\\cos\\theta\n\\end{pmatrix}\n\\]\nwith \\(\\theta\\notin2\\pi\\mathbb Z\\). Then\n\\[\n\\ker(R_\\theta-I)=\\{0\\}.\n\\]\nEvery edge map is individually invertible and well-defined, yet the complete\ncycle admits no nonzero exact shared core. Pairwise translatability is\ntherefore insufficient for global path independence.\n\\end{example}","\\sin\\theta&\\cos\\theta\n\\end{pmatrix}\n\\]\nwith \\(\\theta\\notin2\\pi\\mathbb Z\\). Then\n\\[\n\\ker(R_\\theta-I)=\\{0\\}.\n\\]\nEvery edge map is individually invertible and well-defined, yet the complete\ncycle admits no nonzero exact shared core. Pairwise translatability is\ntherefore insufficient for global path independence.\n\\end{example}","\\begin{theorem}[Global operation closure]\\label{thm:operator-closure}\nSuppose every core transfer intertwines the representations of\n\\(\\mathcal A\\):\n\\[\nH_e\\rho_{s(e)}(a)=\\rho_{t(e)}(a)H_e\n\\qquad\n\\forall e\\in E,\\ a\\in\\mathcal A.\n\\]\nThen \\(\\Gamma(H)\\) is an \\(\\mathcal A\\)-module under the componentwise action\n\\[\n\\rho_\\Gamma(a)(c_i)_{i\\in V}\n=\n(\\rho_i(a)c_i)_{i\\in V}.\n\\]\nEquivalently, the holonomy-fixed root core is invariant:\n\\[\n\\boxed{\n\\rho_o(a)C_*\\subseteq C_*\n\\qquad\n\\forall a\\in\\mathcal A.\n}\n\\]\n\\end{theorem}","on closure]\\label{thm:operator-closure}\nSuppose every core transfer intertwines the representations of\n\\(\\mathcal A\\):\n\\[\nH_e\\rho_{s(e)}(a)=\\rho_{t(e)}(a)H_e\n\\qquad\n\\forall e\\in E,\\ a\\in\\mathcal A.\n\\]\nThen \\(\\Gamma(H)\\) is an \\(\\mathcal A\\)-module under the componentwise action\n\\[\n\\rho_\\Gamma(a)(c_i)_{i\\in V}\n=\n(\\rho_i(a)c_i)_{i\\in V}.\n\\]\nEquivalently, the holonomy-fixed root core is invariant:\n\\[\n\\boxed{\n\\rho_o(a)C_*\\subseteq C_*\n\\qquad\n\\forall a\\in\\mathcal A.\n}\n\\]\n\\end{theorem}","\\begin{proof}\nLet \\(c=(c_i)\\in\\Gamma(H)\\). For every edge \\(e:i\\to j\\),\n\\[\nH_e\\rho_i(a)c_i\n=\n\\rho_j(a)H_ec_i\n=\n\\rho_j(a)c_j.\n\\]\nTherefore the family \\((\\rho_i(a)c_i)\\) is compatible and belongs to\n\\(\\Gamma(H)\\). The representation identities on each \\(C_i\\) hold\ncomponentwise, so \\(\\rho_\\Gamma\\) is an \\(\\mathcal A\\)-representation.","Under the isomorphism of Theorem~\\ref{thm:limit-holonomy}, this action is\nrepresented at the root by \\(\\rho_o\\). Alternatively, if \\(x\\in C_*\\), then\nCorollary~\\ref{cor:holonomy-commutes} gives, for every loop \\(\\gamma\\),\n\\[\nH_\\gamma\\rho_o(a)x\n=\n\\rho_o(a)H_\\gamma x\n=\n\\rho_o(a)x.\n\\]\nHence \\(\\rho_o(a)x\\in C_*\\).\n\\end{proof}","\\begin{corollary}[Inherited algebraic structure]\\label{cor:inherited-structure}\nAny algebraic relation satisfied by \\(\\rho_i(\\mathcal A)\\) is inherited by the\nglobal action. In particular, if \\(a,b\\in\\mathcal A\\), then\n\\[\n\\rho_\\Gamma(ab)=\\rho_\\Gamma(a)\\rho_\\Gamma(b),\n\\qquad\n\\rho_\\Gamma(1)=I_{\\Gamma(H)}.\n\\]\n\\end{corollary}","\\begin{definition}[Natural workspace operators]\\label{def:workspace-operator-algebra}\nDefine\n\\[\n\\boxed{\n\\mathcal O_{\\Workspace}\n=\n\\left\\{\n(A_i)_{i\\in V}:\nA_{t(e)}H_e=H_eA_{s(e)}\\ \\forall e\n\\right\\}.\n}\n\\]\nCategorically,\n\\[\n\\mathcal O_{\\Workspace}=\\Nat(H,H),\n\\]\nthe algebra of natural endomorphisms of the local system.\n\\end{definition}","Pointwise addition, scalar multiplication, and composition make\n\\(\\mathcal O_{\\Workspace}\\) a unital associative algebra. Each\n\\(A=(A_i)\\) acts on global sections by\n\\[\n\\pi_{\\Workspace}(A)(c_i)=(A_ic_i).\n\\]\nNaturality guarantees that \\((A_ic_i)\\) is again compatible.","\\begin{theorem}[Commutant characterization]\\label{thm:commutant-characterization}\nLet \\(|G|\\) be connected and let \\(H\\) be an invertible local system. Root\nevaluation induces an algebra isomorphism\n\\[\n\\boxed{\n\\Phi_o:\\mathcal O_{\\Workspace}\n\\xrightarrow{\\ \\cong\\ }\n\\operatorname{Comm}(\\Hol_o),\n\\qquad\n\\Phi_o((A_i))=A_o,\n}\n\\]\nwhere\n\\[\n\\operatorname{Comm}(\\Hol_o)\n=\n\\{B\\in\\End(C_o):Bh=hB\\ \\forall h\\in\\Hol_o\\}.\n\\]\n\\end{theorem}","\\begin{proof}\nIf \\((A_i)\\in\\mathcal O_{\\Workspace}\\), naturality along every loop gives\n\\[\nA_oH_\\gamma=H_\\gamma A_o.\n\\]\nThus \\(A_o\\in\\operatorname{Comm}(\\Hol_o)\\), and \\(\\Phi_o\\) is a well-defined\nalgebra homomorphism.","Choose paths \\(\\tau_i:o\\to i\\). Naturality along \\(\\tau_i\\) gives\n\\[\nA_iH_{\\tau_i}=H_{\\tau_i}A_o,\n\\]\nso\n\\[\nA_i=H_{\\tau_i}A_oH_{\\tau_i}^{-1}.\n\\]\nTherefore \\(A_o\\) determines every \\(A_i\\), and \\(\\Phi_o\\) is injective.","Conversely, take \\(B\\in\\operatorname{Comm}(\\Hol_o)\\) and define\n\\[\nA_i=H_{\\tau_i}BH_{\\tau_i}^{-1}.\n\\]\nThis is independent of the chosen root path: changing \\(\\tau_i\\) multiplies\nit by a loop holonomy that commutes with \\(B\\). For an edge \\(e:i\\to j\\), the\noperator\n\\[\nH_{\\tau_j}^{-1}H_eH_{\\tau_i}\n\\]\nis a root holonomy. Commutation with \\(B\\) rearranges to\n\\[\nA_jH_e=H_eA_i.\n\\]\nHence \\((A_i)\\) is natural and \\(\\Phi_o((A_i))=B\\). Thus \\(\\Phi_o\\) is\nsurjective.\n\\end{proof}","take \\(B\\in\\operatorname{Comm}(\\Hol_o)\\) and define\n\\[\nA_i=H_{\\tau_i}BH_{\\tau_i}^{-1}.\n\\]\nThis is independent of the chosen root path: changing \\(\\tau_i\\) multiplies\nit by a loop holonomy that commutes with \\(B\\). For an edge \\(e:i\\to j\\), the\noperator\n\\[\nH_{\\tau_j}^{-1}H_eH_{\\tau_i}\n\\]\nis a root holonomy. Commutation with \\(B\\) rearranges to\n\\[\nA_jH_e=H_eA_i.\n\\]\nHence \\((A_i)\\) is natural and \\(\\Phi_o((A_i))=B\\). Thus \\(\\Phi_o\\) is\nsurjective.\n\\end{proof}","Comm}(\\Hol_o)\\) and define\n\\[\nA_i=H_{\\tau_i}BH_{\\tau_i}^{-1}.\n\\]\nThis is independent of the chosen root path: changing \\(\\tau_i\\) multiplies\nit by a loop holonomy that commutes with \\(B\\). For an edge \\(e:i\\to j\\), the\noperator\n\\[\nH_{\\tau_j}^{-1}H_eH_{\\tau_i}\n\\]\nis a root holonomy. Commutation with \\(B\\) rearranges to\n\\[\nA_jH_e=H_eA_i.\n\\]\nHence \\((A_i)\\) is natural and \\(\\Phi_o((A_i))=B\\). Thus \\(\\Phi_o\\) is\nsurjective.\n\\end{proof}","The structural workspace is therefore more accurately represented by the\ntriple\n\\[\n\\boxed{\n\\Workspace_{\\mathrm{static}}\n=\n\\bigl(\n\\Gamma(H),\n\\mathcal O_{\\Workspace},\n\\pi_{\\Workspace}\n\\bigr),\n}\n\\]\nnot merely by the vector space \\(\\Gamma(H)\\).","Some natural operators may vanish on every global section while acting\nnontrivially on locally available but globally inconsistent directions. Define\nthe annihilator\n\\[\n\\operatorname{Ann}(\\Gamma(H))\n=\n\\{A\\in\\mathcal O_{\\Workspace}:\n\\pi_{\\Workspace}(A)=0\\}.\n\\]\nThe effective workspace algebra is\n\\[\n\\boxed{\n\\mathcal O_{\\Workspace}^{\\mathrm{eff}}\n=\n\\mathcal O_{\\Workspace}/\\operatorname{Ann}(\\Gamma(H)).\n}\n\\]\nIts action on \\(\\Gamma(H)\\) is faithful by construction.","may vanish on every global section while acting\nnontrivially on locally available but globally inconsistent directions. Define\nthe annihilator\n\\[\n\\operatorname{Ann}(\\Gamma(H))\n=\n\\{A\\in\\mathcal O_{\\Workspace}:\n\\pi_{\\Workspace}(A)=0\\}.\n\\]\nThe effective workspace algebra is\n\\[\n\\boxed{\n\\mathcal O_{\\Workspace}^{\\mathrm{eff}}\n=\n\\mathcal O_{\\Workspace}/\\operatorname{Ann}(\\Gamma(H)).\n}\n\\]\nIts action on \\(\\Gamma(H)\\) is faithful by construction.","\\begin{example}[Operators in the reflection-cycle workspace]\\label{ex:reflection-operators}\nFor Example~\\ref{ex:reflection-cycle}, the holonomy is\n\\[\nJ=\\operatorname{diag}(1,-1).\n\\]\nIts commutant consists of diagonal matrices:\n\\[\n\\operatorname{Comm}(J)\n=\n\\left\\{\n\\begin{pmatrix}a&0","\\begin{example}[Operators in the reflection-cycle workspace]\\label{ex:reflection-operators}\nFor Example~\\ref{ex:reflection-cycle}, the holonomy is\n\\[\nJ=\\operatorname{diag}(1,-1).\n\\]\nIts commutant consists of diagonal matrices:\n\\[\n\\operatorname{Comm}(J)\n=\n\\left\\{\n\\begin{pmatrix}a&0\\\\0&b\\end{pmatrix}:a,b\\in\\mathbb R\n\\right\\}.\n\\]\nThe exact workspace core is the first coordinate. Consequently, only the\nparameter \\(a\\) acts effectively on \\(\\Gamma(H)\\); the \\(b\\)-direction belongs\nto the annihilator of the global core. Hence\n\\[\n\\mathcal O_{\\Workspace}^{\\mathrm{eff}}\\cong\\mathbb R.\n\\]\n\\end{example}","0&b\\end{pmatrix}:a,b\\in\\mathbb R\n\\right\\}.\n\\]\nThe exact workspace core is the first coordinate. Consequently, only the\nparameter \\(a\\) acts effectively on \\(\\Gamma(H)\\); the \\(b\\)-direction belongs\nto the annihilator of the global core. Hence\n\\[\n\\mathcal O_{\\Workspace}^{\\mathrm{eff}}\\cong\\mathbb R.\n\\]\n\\end{example}","Parts~I--III now establish three exact facts:\n\\[\n\\begin{aligned}\n&\\text{global shared content is the holonomy-fixed core},\\\\\n&\\text{universally transported operations preserve that core},\\\\\n&\\text{natural workspace operations are the commutant of holonomy}.\n\\end{aligned}\n\\]\nThese statements do not yet prove that the shared core is robust, causally\nused, irreducibly distributed, or temporally persistent. Those are separate\nquestions developed in the approximate and dynamic theory that follows.","rfectly under every\ntransfer. Approximate theory asks three more realistic questions:\n\\begin{enumerate}\n\\item how strongly a candidate state violates edgewise compatibility;\n\\item whether a low-defect subspace is spectrally separated from incoherent\n directions; and\n\\item how that subspace changes when transfers, metrics, or operations are\n perturbed.\n\\end{enumerate}\nThroughout Parts~IV--V, set\n\\[\n\\mathcal C_V=\\bigoplus_{i\\in V}C_i,\n\\qquad\n\\mathcal C_E=\\bigoplus_{e\\in E}C_{t(e)},\n\\]\nwith their Hilbert direct-sum inner products, and retain strictly positive\nweights \\(w_e\\).","\\item whether a low-defect subspace is spectrally separated from incoherent\n directions; and\n\\item how that subspace changes when transfers, metrics, or operations are\n perturbed.\n\\end{enumerate}\nThroughout Parts~IV--V, set\n\\[\n\\mathcal C_V=\\bigoplus_{i\\in V}C_i,\n\\qquad\n\\mathcal C_E=\\bigoplus_{e\\in E}C_{t(e)},\n\\]\nwith their Hilbert direct-sum inner products, and retain strictly positive\nweights \\(w_e\\).","\\begin{definition}[Transfer discrepancy]\\label{def:transfer-discrepancy}\nThe transfer discrepancy operator is\n\\[\nD_H:\\mathcal C_V\\longrightarrow\\mathcal C_E,\n\\qquad\n(D_Hc)_e\n=\n\\sqrt{w_e}\\bigl(c_{t(e)}-H_ec_{s(e)}\\bigr).\n\\]\nIts squared norm is the coherence energy\n\\[\n\\boxed{\n\\mathcal E_H(c)\n=\n\\|D_Hc\\|^2\n=\n\\sum_{e\\in E}w_e\n\\left\\|c_{t(e)}-H_ec_{s(e)}\\right\\|^2.\n}\n\\]\nThe transfer Laplacian is\n\\[\n\\boxed{L_H=D_H^*D_H.}\n\\]\n\\end{definition}","The operator \\(D_H\\) is an incidence-type derivative twisted by the edge\ntransfers. It compares a state already present at the target with the state\npredicted there by translation from the source. A direction can therefore\nhave low energy even when its coordinates differ substantially across nodes,\nprovided those coordinates differ exactly as the transfer maps prescribe.","For \\(z=(z_e)_{e\\in E}\\in\\mathcal C_E\\), the adjoint is\n\\[\n(D_H^*z)_i\n=\n\\sum_{e:t(e)=i}\\sqrt{w_e}\\,z_e\n-\n\\sum_{e:s(e)=i}\\sqrt{w_e}\\,H_e^*z_e.\n\\]\nConsequently,\n\\[\n\\boxed{\n\\begin{aligned}\n(L_Hc)_i\n={}&\n\\sum_{e:t(e)=i}w_e\\bigl(c_i-H_ec_{s(e)}\\bigr)","For \\(z=(z_e)_{e\\in E}\\in\\mathcal C_E\\), the adjoint is\n\\[\n(D_H^*z)_i\n=\n\\sum_{e:t(e)=i}\\sqrt{w_e}\\,z_e\n-\n\\sum_{e:s(e)=i}\\sqrt{w_e}\\,H_e^*z_e.\n\\]\nConsequently,\n\\[\n\\boxed{\n\\begin{aligned}\n(L_Hc)_i\n={}&\n\\sum_{e:t(e)=i}w_e\\bigl(c_i-H_ec_{s(e)}\\bigr)\\\\\n&+\n\\sum_{e:s(e)=i}w_eH_e^*\n\\bigl(H_ec_i-c_{t(e)}\\bigr).\n\\end{aligned}\n}\n\\]\nThis formula makes the two contributions at a vertex explicit: incoming\nprediction error and adjoint-propagated outgoing error.","\\begin{proposition}[Positivity and variational identity]\\label{prop:laplacian-positive}\nThe transfer Laplacian is self-adjoint and positive semidefinite. For every\n\\(c\\in\\mathcal C_V\\),\n\\[\n\\langle c,L_Hc\\rangle=\\mathcal E_H(c)\\geq0.\n\\]\n\\end{proposition}","\\begin{proof}\nSelf-adjointness and positivity follow from \\(L_H=D_H^*D_H\\). The displayed\nidentity is\n\\[\n\\langle c,D_H^*D_Hc\\rangle\n=\n\\langle D_Hc,D_Hc\\rangle.\n\\]\n\\end{proof}","\\begin{remark}[Weight dependence]\nReplacing \\(w_e\\) by another positive family does not alter \\(\\ker D_H\\), but\nit generally changes every positive eigenvalue and therefore changes the\napproximate geometry. Edge weights are part of the metric model, not part of\nthe exact compatibility relation.\n\\end{remark}","\\begin{theorem}[Transfer-Laplacian kernel theorem]\\label{thm:laplacian-kernel}\nFor an arbitrary finite directed core diagram, without assuming invertibility,\n\\[\n\\boxed{\n\\ker L_H=\\ker D_H=\\Gamma(H).\n}\n\\]\nThus the exact compatible sections are precisely the zero-energy modes of the\ntransfer Laplacian.\n\\end{theorem}","\\begin{proof}\nIf \\(D_Hc=0\\), then \\(L_Hc=D_H^*D_Hc=0\\). Conversely, if \\(L_Hc=0\\), then\n\\[\n0=\\langle c,L_Hc\\rangle=\\|D_Hc\\|^2,\n\\]\nso \\(D_Hc=0\\). Finally, \\(D_Hc=0\\) if and only if\n\\[\nc_{t(e)}=H_ec_{s(e)}\n\\]\nfor every edge, which is exactly the definition of \\(\\Gamma(H)\\).\n\\end{proof}","\\begin{corollary}[Finite rank computation]\\label{cor:workspace-rank-D}\nLet\n\\[\nn_C=\\sum_{i\\in V}\\dim C_i.\n\\]\nThen\n\\[\n\\boxed{\n\\dim\\Gamma(H)=n_C-\\operatorname{rank}(D_H).\n}\n\\]\nHence the exact workspace dimension can be obtained by a single rank or\nsingular-value computation, even when holonomy is unavailable because some\ntransfers are noninvertible.\n\\end{corollary}","In the local-system regime, Theorems~\\ref{thm:limit-holonomy} and\n\\ref{thm:laplacian-kernel} give two equivalent constructions:\n\\[\n\\Gamma(H)\n\\cong\nC_o^{\\Hol_o}\n\\qquad\\text{and}\\qquad\n\\Gamma(H)=\\ker L_H.\n\\]\nThe first reduces the problem to cycle invariants in a root fiber. The second\nworks directly on all vertex coordinates and extends to noninvertible diagrams.\nThe two computations therefore serve different practical regimes rather than\ncompeting definitions.","Let\n\\[\n0\\leq\\lambda_1\\leq\\lambda_2\\leq\\cdots\\leq\\lambda_N,\n\\qquad\nN=\\dim\\mathcal C_V,\n\\]\nbe the eigenvalues of \\(L_H\\), repeated with multiplicity, and let\n\\(P_I(L_H)\\) denote the spectral projector associated with an interval \\(I\\).","\\begin{definition}[Threshold workspace]\\label{def:threshold-workspace}\nFor \\(\\varepsilon\\geq0\\), define\n\\[\n\\boxed{\n\\Gamma_\\varepsilon(H)\n=\n\\ran P_{[0,\\varepsilon]}(L_H).\n}\n\\]\nIts dimension is\n\\[\nd_\\varepsilon\n=\n\\#\\{k:\\lambda_k\\leq\\varepsilon\\}.\n\\]\nWhen \\(00.\n\\]\nThe index-based form is convenient for perturbation theorems because the\nworkspace rank is declared explicitly.","\\begin{proposition}[Min--max characterization]\\label{prop:minmax}\nFor \\(1\\leq k\\leq N\\),\n\\[\n\\lambda_k\n=\n\\min_{\\substack{S\\subseteq\\mathcal C_V","\\begin{proposition}[Min--max characterization]\\label{prop:minmax}\nFor \\(1\\leq k\\leq N\\),\n\\[\n\\lambda_k\n=\n\\min_{\\substack{S\\subseteq\\mathcal C_V\\\\\\dim S=k}}\n\\ \\max_{\\substack{c\\in S\\\\\\|c\\|=1}}\n\\mathcal E_H(c).\n\\]\nConsequently, the span of the first \\(k\\) eigenvectors minimizes the worst-case\ncoherence energy among all \\(k\\)-dimensional candidate subspaces.\n\\end{proposition}","|c\\|=1}}\n\\mathcal E_H(c).\n\\]\nConsequently, the span of the first \\(k\\) eigenvectors minimizes the worst-case\ncoherence energy among all \\(k\\)-dimensional candidate subspaces.\n\\end{proposition}","\\begin{proof}\nExpand every vector in an orthonormal eigenbasis of \\(L_H\\). The span of the\nfirst \\(k\\) eigenvectors has maximal Rayleigh quotient \\(\\lambda_k\\). Every\n\\(k\\)-dimensional subspace intersects the orthogonal complement of the first\n\\(k-1\\) eigenvectors nontrivially, and any unit vector in that intersection has\nRayleigh quotient at least \\(\\lambda_k\\).\n\\end{proof}","The spectrum supports three distinct interpretations:\n\\[\n\\begin{array}{rcl}\n\\lambda_k=0\n&\\Longleftrightarrow&\n\\text{exact compatible direction},\\\\\n0<\\lambda_k\\ll g_d\n&\\Longrightarrow&\n\\text{approximately transported direction},\\\\\n\\lambda_k\\approx\\lambda_{k+1}\n&\\Longrightarrow&\n\\text{rank selection is unstable at that boundary}.\n\\end{array}\n\\]\nA low eigenvalue can still be scientifically spurious if it arises from a\npoorly chosen weight, a nearly disconnected graph, or a noncausal encoding.\nThe causal and null gates in later parts remain necessary.","For \\(c\\neq0\\), define the Rayleigh defect\n\\[\n\\delta_{\\mathrm{coh}}(c)\n=\n\\frac{\\mathcal E_H(c)}{\\|c\\|^2}.\n\\]\nFor a nonzero subspace \\(W\\subseteq\\mathcal C_V\\), define\n\\[\n\\delta_{\\mathrm{coh}}(W)\n=\n\\sup_{\\substack{c\\in W","For \\(c\\neq0\\), define the Rayleigh defect\n\\[\n\\delta_{\\mathrm{coh}}(c)\n=\n\\frac{\\mathcal E_H(c)}{\\|c\\|^2}.\n\\]\nFor a nonzero subspace \\(W\\subseteq\\mathcal C_V\\), define\n\\[\n\\delta_{\\mathrm{coh}}(W)\n=\n\\sup_{\\substack{c\\in W\\\\\\|c\\|=1}}\n\\mathcal E_H(c).\n\\]\nIf \\(W\\) is \\(L_H\\)-invariant, then\n\\[\n\\delta_{\\mathrm{coh}}(W)\n=\n\\|L_H|_W\\|.\n\\]\nIn particular,\n\\[\n\\delta_{\\mathrm{coh}}(\\Gamma_\\varepsilon(H))\n\\leq\\varepsilon.\n\\]","|c\\|=1}}\n\\mathcal E_H(c).\n\\]\nIf \\(W\\) is \\(L_H\\)-invariant, then\n\\[\n\\delta_{\\mathrm{coh}}(W)\n=\n\\|L_H|_W\\|.\n\\]\nIn particular,\n\\[\n\\delta_{\\mathrm{coh}}(\\Gamma_\\varepsilon(H))\n\\leq\\varepsilon.\n\\]","\\begin{theorem}[Coherence-to-core distance estimate]\\label{thm:poincare-workspace}\nLet\n\\[\nW_0=\\Gamma(H)=\\ker L_H,\n\\qquad\nr=\\dim W_0,\n\\]\nand suppose \\(r0.\n\\]\nThen every \\(c\\in\\mathcal C_V\\) satisfies\n\\[\n\\boxed{\n\\operatorname{dist}(c,W_0)^2\n\\leq\n\\frac{\\mathcal E_H(c)}{\\gamma_H}.\n}\n\\]\nMoreover,\n\\[\n\\mathcal E_H(c)\n\\leq\n\\|L_H\\|\\operatorname{dist}(c,W_0)^2.\n\\]\n\\end{theorem}","\\begin{proof}\nWrite \\(c=P_0c+Q_0c\\), where \\(P_0\\) projects onto \\(W_0\\) and\n\\(Q_0=I-P_0\\). Since \\(L_HP_0=0\\),\n\\[\n\\mathcal E_H(c)\n=\n\\langle Q_0c,L_HQ_0c\\rangle.\n\\]\nThe spectrum of \\(L_H\\) on \\(W_0^\\perp\\) lies in\n\\([\\gamma_H,\\|L_H\\|]\\). Hence\n\\[\n\\gamma_H\\|Q_0c\\|^2\n\\leq\n\\mathcal E_H(c)\n\\leq\n\\|L_H\\|\\|Q_0c\\|^2.\n\\]\nFinally, \\(\\|Q_0c\\|=\\operatorname{dist}(c,W_0)\\).\n\\end{proof}","Let \\(C_1=C_2=\\mathbb R\\), use unit weights, and take two directed transfers\n\\[\nH_{12}=a,\n\\qquad\nH_{21}=b.\n\\]\nThen\n\\[\nD_H\n=\n\\begin{pmatrix}\n-a&1","Let \\(C_1=C_2=\\mathbb R\\), use unit weights, and take two directed transfers\n\\[\nH_{12}=a,\n\\qquad\nH_{21}=b.\n\\]\nThen\n\\[\nD_H\n=\n\\begin{pmatrix}\n-a&1\\\\\n1&-b\n\\end{pmatrix},\n\\qquad\nL_H\n=\n\\begin{pmatrix}\na^2+1&-(a+b)\\\\\n-(a+b)&b^2+1\n\\end{pmatrix}.\n\\]\nWriting\n\\[\n\\tau=a^2+b^2+2,\n\\]\nthe eigenvalues are\n\\[\n\\lambda_\\pm\n=\n\\frac{1}{2}\n\\left(\n\\tau\\pm\\sqrt{\\tau^2-4(ab-1)^2}\n\\right),\n\\]\nand\n\\[\n\\det L_H=(ab-1)^2.\n\\]\nThus a nonzero exact shared core exists precisely when\n\\[\nab=1,\n\\]\nwhich is exactly the trivial-holonomy condition. Away from exact consistency,\n\\[\n\\lambda_-\\lambda_+=(ab-1)^2.\n\\]\nWhen \\(|ab-1|\\) is small while \\(\\lambda_+\\) stays bounded away from zero, the\nsoft workspace eigenvalue is quadratic in the cycle mismatch:\n\\[\n\\lambda_-\n=\n\\frac{(ab-1)^2}{\\lambda_+}.\n\\]\nThis example connects approximate spectral coherence directly to approximate\nholonomy closure.","-(a+b)&b^2+1\n\\end{pmatrix}.\n\\]\nWriting\n\\[\n\\tau=a^2+b^2+2,\n\\]\nthe eigenvalues are\n\\[\n\\lambda_\\pm\n=\n\\frac{1}{2}\n\\left(\n\\tau\\pm\\sqrt{\\tau^2-4(ab-1)^2}\n\\right),\n\\]\nand\n\\[\n\\det L_H=(ab-1)^2.\n\\]\nThus a nonzero exact shared core exists precisely when\n\\[\nab=1,\n\\]\nwhich is exactly the trivial-holonomy condition. Away from exact consistency,\n\\[\n\\lambda_-\\lambda_+=(ab-1)^2.\n\\]\nWhen \\(|ab-1|\\) is small while \\(\\lambda_+\\) stays bounded away from zero, the\nsoft workspace eigenvalue is quadratic in the cycle mismatch:\n\\[\n\\lambda_-\n=\n\\frac{(ab-1)^2}{\\lambda_+}.\n\\]\nThis example connects approximate spectral coherence directly to approximate\nholonomy closure.","a+b)&b^2+1\n\\end{pmatrix}.\n\\]\nWriting\n\\[\n\\tau=a^2+b^2+2,\n\\]\nthe eigenvalues are\n\\[\n\\lambda_\\pm\n=\n\\frac{1}{2}\n\\left(\n\\tau\\pm\\sqrt{\\tau^2-4(ab-1)^2}\n\\right),\n\\]\nand\n\\[\n\\det L_H=(ab-1)^2.\n\\]\nThus a nonzero exact shared core exists precisely when\n\\[\nab=1,\n\\]\nwhich is exactly the trivial-holonomy condition. Away from exact consistency,\n\\[\n\\lambda_-\\lambda_+=(ab-1)^2.\n\\]\nWhen \\(|ab-1|\\) is small while \\(\\lambda_+\\) stays bounded away from zero, the\nsoft workspace eigenvalue is quadratic in the cycle mismatch:\n\\[\n\\lambda_-\n=\n\\frac{(ab-1)^2}{\\lambda_+}.\n\\]\nThis example connects approximate spectral coherence directly to approximate\nholonomy closure.","eft(\n\\tau\\pm\\sqrt{\\tau^2-4(ab-1)^2}\n\\right),\n\\]\nand\n\\[\n\\det L_H=(ab-1)^2.\n\\]\nThus a nonzero exact shared core exists precisely when\n\\[\nab=1,\n\\]\nwhich is exactly the trivial-holonomy condition. Away from exact consistency,\n\\[\n\\lambda_-\\lambda_+=(ab-1)^2.\n\\]\nWhen \\(|ab-1|\\) is small while \\(\\lambda_+\\) stays bounded away from zero, the\nsoft workspace eigenvalue is quadratic in the cycle mismatch:\n\\[\n\\lambda_-\n=\n\\frac{(ab-1)^2}{\\lambda_+}.\n\\]\nThis example connects approximate spectral coherence directly to approximate\nholonomy closure.","\\begin{lemma}[Separated spectral subspaces]\\label{lem:separated-subspaces}\nLet \\(A\\) and \\(B=A+E\\) be self-adjoint operators. Let \\(P\\) be an\n\\(A\\)-spectral projector whose eigenvalues lie in \\(( -\\infty,a]\\), and let\n\\(\\widetilde Q\\) be a \\(B\\)-spectral projector whose eigenvalues lie in\n\\([b,\\infty)\\), with \\(b>a\\). Then\n\\[\n\\|P\\widetilde Q\\|\n\\leq\n\\frac{\\|E\\|}{b-a}.\n\\]\n\\end{lemma}","\\begin{proof}\nOn \\(\\ran P\\) and \\(\\ran\\widetilde Q\\), set\n\\[\nA_P=PAP|_{\\ran P},\n\\qquad\nB_Q=\\widetilde QB\\widetilde Q|_{\\ran\\widetilde Q}.\n\\]\nThe operator \\(X=P\\widetilde Q\\) satisfies the Sylvester equation\n\\[\nA_PX-XB_Q=-PE\\widetilde Q.\n\\]\nLet \\(\\mu=(a+b)/2\\). Since\n\\[\n\\|e^{t(A_P-\\mu I)}\\|\\leq e^{-(b-a)t/2},\n\\qquad\n\\|e^{-t(B_Q-\\mu I)}\\|\\leq e^{-(b-a)t/2},\n\\]\nthe inverse Sylvester operator has the convergent integral representation\n\\[\nX\n=\n\\int_0^\\infty\n e^{t(A_P-\\mu I)}\n (PE\\widetilde Q)\n e^{-t(B_Q-\\mu I)}\\,\\dd t.\n\\]\nTaking norms and integrating gives the result.\n\\end{proof}","n\\widetilde Q\\), set\n\\[\nA_P=PAP|_{\\ran P},\n\\qquad\nB_Q=\\widetilde QB\\widetilde Q|_{\\ran\\widetilde Q}.\n\\]\nThe operator \\(X=P\\widetilde Q\\) satisfies the Sylvester equation\n\\[\nA_PX-XB_Q=-PE\\widetilde Q.\n\\]\nLet \\(\\mu=(a+b)/2\\). Since\n\\[\n\\|e^{t(A_P-\\mu I)}\\|\\leq e^{-(b-a)t/2},\n\\qquad\n\\|e^{-t(B_Q-\\mu I)}\\|\\leq e^{-(b-a)t/2},\n\\]\nthe inverse Sylvester operator has the convergent integral representation\n\\[\nX\n=\n\\int_0^\\infty\n e^{t(A_P-\\mu I)}\n (PE\\widetilde Q)\n e^{-t(B_Q-\\mu I)}\\,\\dd t.\n\\]\nTaking norms and integrating gives the result.\n\\end{proof}","\\begin{theorem}[Spectral workspace stability]\\label{thm:spectral-stability}\nLet \\(L\\) and \\(\\widetilde L=L+E\\) be self-adjoint positive semidefinite\noperators on the same \\(N\\)-dimensional space. Suppose\n\\[\n\\lambda_d(L)<\\lambda_{d+1}(L),\n\\qquad\ng_d=\\lambda_{d+1}(L)-\\lambda_d(L)>0,\n\\]\nand\n\\[\n\\eta=\\|E\\|<\\frac{g_d}{2}.\n\\]\nLet \\(P_d\\) and \\(\\widetilde P_d\\) project onto the first \\(d\\) eigenvectors\nof \\(L\\) and \\(\\widetilde L\\), respectively. Then the rank-\\(d\\) cluster\nremains separated and\n\\[\n\\boxed{\n\\|P_d-\\widetilde P_d\\|\n\\leq\n\\frac{\\eta}{g_d-\\eta}\n\\leq\n\\frac{2\\eta}{g_d}.\n}\n\\]\n\\end{theorem}","gin{theorem}[Spectral workspace stability]\\label{thm:spectral-stability}\nLet \\(L\\) and \\(\\widetilde L=L+E\\) be self-adjoint positive semidefinite\noperators on the same \\(N\\)-dimensional space. Suppose\n\\[\n\\lambda_d(L)<\\lambda_{d+1}(L),\n\\qquad\ng_d=\\lambda_{d+1}(L)-\\lambda_d(L)>0,\n\\]\nand\n\\[\n\\eta=\\|E\\|<\\frac{g_d}{2}.\n\\]\nLet \\(P_d\\) and \\(\\widetilde P_d\\) project onto the first \\(d\\) eigenvectors\nof \\(L\\) and \\(\\widetilde L\\), respectively. Then the rank-\\(d\\) cluster\nremains separated and\n\\[\n\\boxed{\n\\|P_d-\\widetilde P_d\\|\n\\leq\n\\frac{\\eta}{g_d-\\eta}\n\\leq\n\\frac{2\\eta}{g_d}.\n}\n\\]\n\\end{theorem}","m}[Spectral workspace stability]\\label{thm:spectral-stability}\nLet \\(L\\) and \\(\\widetilde L=L+E\\) be self-adjoint positive semidefinite\noperators on the same \\(N\\)-dimensional space. Suppose\n\\[\n\\lambda_d(L)<\\lambda_{d+1}(L),\n\\qquad\ng_d=\\lambda_{d+1}(L)-\\lambda_d(L)>0,\n\\]\nand\n\\[\n\\eta=\\|E\\|<\\frac{g_d}{2}.\n\\]\nLet \\(P_d\\) and \\(\\widetilde P_d\\) project onto the first \\(d\\) eigenvectors\nof \\(L\\) and \\(\\widetilde L\\), respectively. Then the rank-\\(d\\) cluster\nremains separated and\n\\[\n\\boxed{\n\\|P_d-\\widetilde P_d\\|\n\\leq\n\\frac{\\eta}{g_d-\\eta}\n\\leq\n\\frac{2\\eta}{g_d}.\n}\n\\]\n\\end{theorem}","ctral-stability}\nLet \\(L\\) and \\(\\widetilde L=L+E\\) be self-adjoint positive semidefinite\noperators on the same \\(N\\)-dimensional space. Suppose\n\\[\n\\lambda_d(L)<\\lambda_{d+1}(L),\n\\qquad\ng_d=\\lambda_{d+1}(L)-\\lambda_d(L)>0,\n\\]\nand\n\\[\n\\eta=\\|E\\|<\\frac{g_d}{2}.\n\\]\nLet \\(P_d\\) and \\(\\widetilde P_d\\) project onto the first \\(d\\) eigenvectors\nof \\(L\\) and \\(\\widetilde L\\), respectively. Then the rank-\\(d\\) cluster\nremains separated and\n\\[\n\\boxed{\n\\|P_d-\\widetilde P_d\\|\n\\leq\n\\frac{\\eta}{g_d-\\eta}\n\\leq\n\\frac{2\\eta}{g_d}.\n}\n\\]\n\\end{theorem}","detilde L=L+E\\) be self-adjoint positive semidefinite\noperators on the same \\(N\\)-dimensional space. Suppose\n\\[\n\\lambda_d(L)<\\lambda_{d+1}(L),\n\\qquad\ng_d=\\lambda_{d+1}(L)-\\lambda_d(L)>0,\n\\]\nand\n\\[\n\\eta=\\|E\\|<\\frac{g_d}{2}.\n\\]\nLet \\(P_d\\) and \\(\\widetilde P_d\\) project onto the first \\(d\\) eigenvectors\nof \\(L\\) and \\(\\widetilde L\\), respectively. Then the rank-\\(d\\) cluster\nremains separated and\n\\[\n\\boxed{\n\\|P_d-\\widetilde P_d\\|\n\\leq\n\\frac{\\eta}{g_d-\\eta}\n\\leq\n\\frac{2\\eta}{g_d}.\n}\n\\]\n\\end{theorem}","\\begin{proof}\nThe min--max characterization gives the eigenvalue perturbation bound\n\\[\n|\\lambda_k(\\widetilde L)-\\lambda_k(L)|\\leq\\eta.\n\\]\nHence\n\\[\n\\lambda_{d+1}(\\widetilde L)-\\lambda_d(L)\n\\geq g_d-\\eta\n\\]\nand\n\\[\n\\lambda_{d+1}(L)-\\lambda_d(\\widetilde L)\n\\geq g_d-\\eta.\n\\]\nApply Lemma~\\ref{lem:separated-subspaces} first to\n\\(P_d(I-\\widetilde P_d)\\) and then, with the roles of \\(L\\) and\n\\(\\widetilde L\\) exchanged, to \\((I-P_d)\\widetilde P_d\\). Orthogonal\nprojections of equal rank satisfy\n\\[\n\\|P_d-\\widetilde P_d\\|\n=\n\\max\\left\\{\n\\|P_d(I-\\widetilde P_d)\\|,\n\\|(I-P_d)\\widetilde P_d\\|\n\\right\\}.\n\\]\nThis yields the first bound. The second follows from \\(\\eta0,\n\\]\nand let \\(Q_d=I-P_d\\). Then every \\(a\\in\\mathcal A\\) satisfies\n\\[\n\\boxed{\n\\|Q_d\\rho_V(a)P_d\\|\n\\leq\n\\frac{\n\\|\\rho_E(a)\\|\\sqrt{\\varepsilon}+\\beta(a)\n}{\\sqrt{\\mu}}.\n}\n\\]\n\\end{theorem}","\\begin{proof}\nFor \\(x\\in\\ran P_d\\),\n\\[\n\\|D_Hx\\|^2\n=\\langle x,L_Hx\\rangle\n\\leq\\varepsilon\\|x\\|^2.\n\\]\nThe commutator identity gives\n\\[\n\\|D_H\\rho_V(a)x\\|\n\\leq\n\\|\\rho_E(a)\\|\\sqrt{\\varepsilon}\\|x\\|\n+\n\\beta(a)\\|x\\|.\n\\]\nBecause \\(P_d\\) and \\(Q_d\\) reduce \\(L_H\\), the two energy components are\northogonal and\n\\[\n\\|D_HQ_dy\\|\\leq\\|D_Hy\\|.\n\\]\nOn \\(\\ran Q_d\\), the spectrum of \\(L_H\\) is bounded below by \\(\\mu\\), so\n\\[\n\\sqrt{\\mu}\\|Q_dy\\|\n\\leq\n\\|D_HQ_dy\\|.\n\\]\nApply these inequalities to \\(y=\\rho_V(a)x\\), then take the supremum over unit\n\\(x\\in\\ran P_d\\).\n\\end{proof}","roof}\nFor \\(x\\in\\ran P_d\\),\n\\[\n\\|D_Hx\\|^2\n=\\langle x,L_Hx\\rangle\n\\leq\\varepsilon\\|x\\|^2.\n\\]\nThe commutator identity gives\n\\[\n\\|D_H\\rho_V(a)x\\|\n\\leq\n\\|\\rho_E(a)\\|\\sqrt{\\varepsilon}\\|x\\|\n+\n\\beta(a)\\|x\\|.\n\\]\nBecause \\(P_d\\) and \\(Q_d\\) reduce \\(L_H\\), the two energy components are\northogonal and\n\\[\n\\|D_HQ_dy\\|\\leq\\|D_Hy\\|.\n\\]\nOn \\(\\ran Q_d\\), the spectrum of \\(L_H\\) is bounded below by \\(\\mu\\), so\n\\[\n\\sqrt{\\mu}\\|Q_dy\\|\n\\leq\n\\|D_HQ_dy\\|.\n\\]\nApply these inequalities to \\(y=\\rho_V(a)x\\), then take the supremum over unit\n\\(x\\in\\ran P_d\\).\n\\end{proof}","arepsilon\\|x\\|^2.\n\\]\nThe commutator identity gives\n\\[\n\\|D_H\\rho_V(a)x\\|\n\\leq\n\\|\\rho_E(a)\\|\\sqrt{\\varepsilon}\\|x\\|\n+\n\\beta(a)\\|x\\|.\n\\]\nBecause \\(P_d\\) and \\(Q_d\\) reduce \\(L_H\\), the two energy components are\northogonal and\n\\[\n\\|D_HQ_dy\\|\\leq\\|D_Hy\\|.\n\\]\nOn \\(\\ran Q_d\\), the spectrum of \\(L_H\\) is bounded below by \\(\\mu\\), so\n\\[\n\\sqrt{\\mu}\\|Q_dy\\|\n\\leq\n\\|D_HQ_dy\\|.\n\\]\nApply these inequalities to \\(y=\\rho_V(a)x\\), then take the supremum over unit\n\\(x\\in\\ran P_d\\).\n\\end{proof}","\\[\n\\|D_H\\rho_V(a)x\\|\n\\leq\n\\|\\rho_E(a)\\|\\sqrt{\\varepsilon}\\|x\\|\n+\n\\beta(a)\\|x\\|.\n\\]\nBecause \\(P_d\\) and \\(Q_d\\) reduce \\(L_H\\), the two energy components are\northogonal and\n\\[\n\\|D_HQ_dy\\|\\leq\\|D_Hy\\|.\n\\]\nOn \\(\\ran Q_d\\), the spectrum of \\(L_H\\) is bounded below by \\(\\mu\\), so\n\\[\n\\sqrt{\\mu}\\|Q_dy\\|\n\\leq\n\\|D_HQ_dy\\|.\n\\]\nApply these inequalities to \\(y=\\rho_V(a)x\\), then take the supremum over unit\n\\(x\\in\\ran P_d\\).\n\\end{proof}","\\begin{corollary}[Recovery of exact operation closure]\nIf \\(d=\\dim\\ker L_H\\), \\(\\varepsilon=0\\), and every edge intertwines \\(a\\)\nexactly, then\n\\[\nQ_d\\rho_V(a)P_d=0.\n\\]\nThus Theorem~\\ref{thm:approx-operation-closure} reduces to the exact\noperation-closure result of Part~III.\n\\end{corollary}","\\begin{corollary}[Closure under projector drift]\\label{cor:closure-drift}\nLet \\(\\widetilde P_d\\) be another rank-\\(d\\) orthogonal projector and set\n\\[\n\\delta_P=\\|P_d-\\widetilde P_d\\|.\n\\]\nThen\n\\[\n\\boxed{\n\\|(I-\\widetilde P_d)\\rho_V(a)\\widetilde P_d\\|\n\\leq\n\\|Q_d\\rho_V(a)P_d\\|\n+2\\|\\rho_V(a)\\|\\delta_P.\n}\n\\]\nCombining this estimate with Theorem~\\ref{thm:spectral-stability} controls\noperation leakage after noisy workspace extraction.\n\\end{corollary}","\\begin{proof}\nSubtract \\(Q_d\\rho_V(a)P_d\\) and use\n\\[\nI-\\widetilde P_d-Q_d=P_d-\\widetilde P_d.\n\\]\nThe two resulting perturbation terms are each bounded by\n\\(\\|\\rho_V(a)\\|\\delta_P\\).\n\\end{proof}","Let a path have vertices\n\\[\ni_0\\xrightarrow{e_1}i_1\\xrightarrow{e_2}\\cdots\n\\xrightarrow{e_m}i_m.\n\\]\nWrite\n\\[\nT_r=T_{e_r},\n\\quad\nH_r=H_{e_r},\n\\quad\nV_r=V_{e_r},\n\\]\nand let\n\\[\n\\iota_i^C:C_i\\hookrightarrow X_i,\n\\qquad\n\\iota_i^R:R_i\\hookrightarrow X_i\n\\]\nbe the canonical inclusions. Empty products below are identities.","\\begin{theorem}[Pathwise core--residual Duhamel formula]\\label{thm:residual-duhamel}\nThe deviation of full path transport from pure core transport is exactly\n\\[\n\\boxed{\n\\begin{aligned}\n&T_m\\cdots T_1\\iota_{i_0}^C\n-\n\\iota_{i_m}^C H_m\\cdots H_1\\\\\n&\\qquad=\n\\sum_{r=1}^{m}\nT_m\\cdots T_{r+1}\n\\iota_{i_r}^R V_r\nH_{r-1}\\cdots H_1.\n\\end{aligned}\n}\n\\]\nConsequently,\n\\[\n\\boxed{\n\\begin{aligned}\n\\|Q_{i_m}T_m\\cdots T_1P_{i_0}\\|\n\\leq\n\\sum_{r=1}^{m}\n&\\left(\\prod_{\\ell=r+1}^{m}\\|T_\\ell\\|\\right)\n\\|V_r\\|","&\\qquad=\n\\sum_{r=1}^{m}\nT_m\\cdots T_{r+1}\n\\iota_{i_r}^R V_r\nH_{r-1}\\cdots H_1.\n\\end{aligned}\n}\n\\]\nConsequently,\n\\[\n\\boxed{\n\\begin{aligned}\n\\|Q_{i_m}T_m\\cdots T_1P_{i_0}\\|\n\\leq\n\\sum_{r=1}^{m}\n&\\left(\\prod_{\\ell=r+1}^{m}\\|T_\\ell\\|\\right)\n\\|V_r\\|\\\\\n&\\times\n\\left(\\prod_{\\ell=1}^{r-1}\\|H_\\ell\\|\\right),\n\\end{aligned}\n}\n\\]\nand the same right-hand side bounds\n\\[\n\\|P_{i_m}T_m\\cdots T_1P_{i_0}-H_m\\cdots H_1\\|.\n\\]\n\\end{theorem}","&\\times\n\\left(\\prod_{\\ell=1}^{r-1}\\|H_\\ell\\|\\right),\n\\end{aligned}\n}\n\\]\nand the same right-hand side bounds\n\\[\n\\|P_{i_m}T_m\\cdots T_1P_{i_0}-H_m\\cdots H_1\\|.\n\\]\n\\end{theorem}","\\begin{proof}\nFor every step,\n\\[\nT_r\\iota_{i_{r-1}}^C\n=\n\\iota_{i_r}^CH_r+\n\\iota_{i_r}^RV_r.\n\\]\nTelescope the sequence\n\\[\nT_m\\cdots T_{r+1}\\iota_{i_r}^C H_r\\cdots H_1,\n\\qquad\nr=0,1,\\ldots,m.\n\\]\nThe difference between consecutive terms is exactly the \\(r\\)-th summand in\nthe displayed formula. Projection onto \\(R_{i_m}\\) or \\(C_{i_m}\\), followed\nby submultiplicativity of the operator norm, gives the two bounds.\n\\end{proof}","\\begin{corollary}[Uniform path bound]\\label{cor:uniform-path-bound}\nIf along the path\n\\[\n\\|T_r\\|\\leq\\tau,\n\\qquad\n\\|H_r\\|\\leq h,\n\\qquad\n\\|V_r\\|\\leq\\nu,\n\\]\nthen the leakage and core-transport defects are bounded by\n\\[\n\\boxed{\n\\nu\\sum_{r=1}^{m}\\tau^{m-r}h^{r-1}\n=\n\\begin{cases}\n\\displaystyle\n\\nu\\frac{\\tau^m-h^m}{\\tau-h},&\\tau\\neq h,","\\begin{corollary}[Uniform path bound]\\label{cor:uniform-path-bound}\nIf along the path\n\\[\n\\|T_r\\|\\leq\\tau,\n\\qquad\n\\|H_r\\|\\leq h,\n\\qquad\n\\|V_r\\|\\leq\\nu,\n\\]\nthen the leakage and core-transport defects are bounded by\n\\[\n\\boxed{\n\\nu\\sum_{r=1}^{m}\\tau^{m-r}h^{r-1}\n=\n\\begin{cases}\n\\displaystyle\n\\nu\\frac{\\tau^m-h^m}{\\tau-h},&\\tau\\neq h,\\\\[1.1em]\n\\displaystyle\nm\\nu\\tau^{m-1},&\\tau=h.\n\\end{cases}\n}\n\\]\nIn particular, if \\(\\tau\\leq1\\) and \\(h\\leq1\\), the bound is at most\n\\(m\\nu\\).\n\\end{corollary}","[1.1em]\n\\displaystyle\nm\\nu\\tau^{m-1},&\\tau=h.\n\\end{cases}\n}\n\\]\nIn particular, if \\(\\tau\\leq1\\) and \\(h\\leq1\\), the bound is at most\n\\(m\\nu\\).\n\\end{corollary}","The linear factor \\(m\\) in the merely nonexpansive regime is important. Small\nsingle-edge leakage does not by itself justify arbitrarily long transfer\nchains. Long-horizon stability requires contraction, cancellation, error\ncorrection, or leakage that decreases with path length.","For two consecutive transfers,\n\\[\nP_{i_2}T_2T_1P_{i_0}\n=\nH_2H_1+U_2V_1,\n\\]\n\\[\nQ_{i_2}T_2T_1P_{i_0}\n=\nV_2H_1+K_2V_1.\n\\]\nThus leakage out of the core is first order in the blocks \\(V_r\\), while an\nexcursion that leaves and then returns to the core contains at least one\n\\(V\\)-block and one later \\(U\\)-block. For a fixed path length, if\n\\[\n\\|U_r\\|,\\|V_r\\|=O(\\delta)\n\\]\nand the diagonal blocks remain bounded, the residual-mediated correction to\npure core-to-core transport is \\(O(\\delta^2)\\), whereas terminal residual\nleakage can be \\(O(\\delta)\\). This distinction separate","For two consecutive transfers,\n\\[\nP_{i_2}T_2T_1P_{i_0}\n=\nH_2H_1+U_2V_1,\n\\]\n\\[\nQ_{i_2}T_2T_1P_{i_0}\n=\nV_2H_1+K_2V_1.\n\\]\nThus leakage out of the core is first order in the blocks \\(V_r\\), while an\nexcursion that leaves and then returns to the core contains at least one\n\\(V\\)-block and one later \\(U\\)-block. For a fixed path length, if\n\\[\n\\|U_r\\|,\\|V_r\\|=O(\\delta)\n\\]\nand the diagonal blocks remain bounded, the residual-mediated correction to\npure core-to-core transport is \\(O(\\delta^2)\\), whereas terminal residual\nleakage can be \\(O(\\delta)\\). This distinction separates loss of core\ninvariance from distortion ca","For two consecutive transfers,\n\\[\nP_{i_2}T_2T_1P_{i_0}\n=\nH_2H_1+U_2V_1,\n\\]\n\\[\nQ_{i_2}T_2T_1P_{i_0}\n=\nV_2H_1+K_2V_1.\n\\]\nThus leakage out of the core is first order in the blocks \\(V_r\\), while an\nexcursion that leaves and then returns to the core contains at least one\n\\(V\\)-block and one later \\(U\\)-block. For a fixed path length, if\n\\[\n\\|U_r\\|,\\|V_r\\|=O(\\delta)\n\\]\nand the diagonal blocks remain bounded, the residual-mediated correction to\npure core-to-core transport is \\(O(\\delta^2)\\), whereas terminal residual\nleakage can be \\(O(\\delta)\\). This distinction separates loss of core\ninvariance from distortion caused by temporary excursions through local\nresidual degrees of fre","For two consecutive transfers,\n\\[\nP_{i_2}T_2T_1P_{i_0}\n=\nH_2H_1+U_2V_1,\n\\]\n\\[\nQ_{i_2}T_2T_1P_{i_0}\n=\nV_2H_1+K_2V_1.\n\\]\nThus leakage out of the core is first order in the blocks \\(V_r\\), while an\nexcursion that leaves and then returns to the core contains at least one\n\\(V\\)-block and one later \\(U\\)-block. For a fixed path length, if\n\\[\n\\|U_r\\|,\\|V_r\\|=O(\\delta)\n\\]\nand the diagonal blocks remain bounded, the residual-mediated correction to\npure core-to-core transport is \\(O(\\delta^2)\\), whereas terminal residual\nleakage can be \\(O(\\delta)\\). This distinction separates loss of core\ninvariance from distortion caused by temporary excursions through local\nresidual degrees of freedom.","[\nQ_{i_2}T_2T_1P_{i_0}\n=\nV_2H_1+K_2V_1.\n\\]\nThus leakage out of the core is first order in the blocks \\(V_r\\), while an\nexcursion that leaves and then returns to the core contains at least one\n\\(V\\)-block and one later \\(U\\)-block. For a fixed path length, if\n\\[\n\\|U_r\\|,\\|V_r\\|=O(\\delta)\n\\]\nand the diagonal blocks remain bounded, the residual-mediated correction to\npure core-to-core transport is \\(O(\\delta^2)\\), whereas terminal residual\nleakage can be \\(O(\\delta)\\). This distinction separates loss of core\ninvariance from distortion caused by temporary excursions through local\nresidual degrees of freedom.","t of the core is first order in the blocks \\(V_r\\), while an\nexcursion that leaves and then returns to the core contains at least one\n\\(V\\)-block and one later \\(U\\)-block. For a fixed path length, if\n\\[\n\\|U_r\\|,\\|V_r\\|=O(\\delta)\n\\]\nand the diagonal blocks remain bounded, the residual-mediated correction to\npure core-to-core transport is \\(O(\\delta^2)\\), whereas terminal residual\nleakage can be \\(O(\\delta)\\). This distinction separates loss of core\ninvariance from distortion caused by temporary excursions through local\nresidual degrees of freedom.","For each \\(i\\in V\\), let\n\\[\ng_i\\in GL(C_i),\n\\qquad\ng=(g_i)_{i\\in V}.\n\\]\nDefine the vertex and edge direct sums\n\\[\nG_V=\\bigoplus_{i\\in V}g_i,\n\\qquad\nG_E=\\bigoplus_{e\\in E}g_{t(e)}.\n\\]\nThe transformed transfers and local operation representations are\n\\[\nH_e^g\n=\n g_{t(e)}H_eg_{s(e)}^{-1},\n\\]\n\\[\n\\rho_i^g(a)\n=\n g_i\\rho_i(a)g_i^{-1}.\n\\]\nThe fundamental covariance identity is\n\\[\n\\boxed{\nD_{H^g}G_V=G_ED_H.\n}\n\\]\nIt follows directly by evaluating the \\(e\\)-component:\n\\[\n\\sqrt{w_e}\n\\left(\n g_{t(e)}c_{t(e)}\n-\n g_{t(e)}H_ec_{s(e)}\n\\right).\n\\]","uad\ng=(g_i)_{i\\in V}.\n\\]\nDefine the vertex and edge direct sums\n\\[\nG_V=\\bigoplus_{i\\in V}g_i,\n\\qquad\nG_E=\\bigoplus_{e\\in E}g_{t(e)}.\n\\]\nThe transformed transfers and local operation representations are\n\\[\nH_e^g\n=\n g_{t(e)}H_eg_{s(e)}^{-1},\n\\]\n\\[\n\\rho_i^g(a)\n=\n g_i\\rho_i(a)g_i^{-1}.\n\\]\nThe fundamental covariance identity is\n\\[\n\\boxed{\nD_{H^g}G_V=G_ED_H.\n}\n\\]\nIt follows directly by evaluating the \\(e\\)-component:\n\\[\n\\sqrt{w_e}\n\\left(\n g_{t(e)}c_{t(e)}\n-\n g_{t(e)}H_ec_{s(e)}\n\\right).\n\\]","V}.\n\\]\nDefine the vertex and edge direct sums\n\\[\nG_V=\\bigoplus_{i\\in V}g_i,\n\\qquad\nG_E=\\bigoplus_{e\\in E}g_{t(e)}.\n\\]\nThe transformed transfers and local operation representations are\n\\[\nH_e^g\n=\n g_{t(e)}H_eg_{s(e)}^{-1},\n\\]\n\\[\n\\rho_i^g(a)\n=\n g_i\\rho_i(a)g_i^{-1}.\n\\]\nThe fundamental covariance identity is\n\\[\n\\boxed{\nD_{H^g}G_V=G_ED_H.\n}\n\\]\nIt follows directly by evaluating the \\(e\\)-component:\n\\[\n\\sqrt{w_e}\n\\left(\n g_{t(e)}c_{t(e)}\n-\n g_{t(e)}H_ec_{s(e)}\n\\right).\n\\]","\\begin{theorem}[Exact gauge covariance]\\label{thm:gauge-covariance}\nThe map\n\\[\nG_V:\\Gamma(H)\\longrightarrow\\Gamma(H^g)\n\\]\nis an isomorphism. Therefore,\n\\[\n\\boxed{\n\\Gamma(H^g)=G_V\\Gamma(H),\n\\qquad\n\\dim\\Gamma(H^g)=\\dim\\Gamma(H).\n}\n\\]\nIf the core transports are invertible and \\(o\\) is a root, then\n\\[\nH_\\gamma^g=g_oH_\\gamma g_o^{-1}\n\\]\nfor every loop \\(\\gamma:o\\to o\\). Hence the holonomy-fixed core transforms by\n\\[\nC_*^g=g_oC_*.\n\\]\n\\end{theorem}","}[Exact gauge covariance]\\label{thm:gauge-covariance}\nThe map\n\\[\nG_V:\\Gamma(H)\\longrightarrow\\Gamma(H^g)\n\\]\nis an isomorphism. Therefore,\n\\[\n\\boxed{\n\\Gamma(H^g)=G_V\\Gamma(H),\n\\qquad\n\\dim\\Gamma(H^g)=\\dim\\Gamma(H).\n}\n\\]\nIf the core transports are invertible and \\(o\\) is a root, then\n\\[\nH_\\gamma^g=g_oH_\\gamma g_o^{-1}\n\\]\nfor every loop \\(\\gamma:o\\to o\\). Hence the holonomy-fixed core transforms by\n\\[\nC_*^g=g_oC_*.\n\\]\n\\end{theorem}","\\begin{proof}\nThe covariance identity implies\n\\[\nD_{H^g}(G_Vc)=G_ED_Hc.\n\\]\nSince \\(G_E\\) is invertible,\n\\[\nD_Hc=0\n\\quad\\Longleftrightarrow\\quad\nD_{H^g}(G_Vc)=0.\n\\]\nThe holonomy formula follows by cancellation of adjacent factors\n\\(g_i^{-1}g_i\\) along a path.\n\\end{proof}","\\begin{theorem}[Unitary gauge covariance of the Laplacian]\\label{thm:unitary-gauge}\nIf every \\(g_i\\) is unitary, then \\(G_V\\) and \\(G_E\\) are unitary and\n\\[\n\\boxed{\nL_{H^g}=G_VL_HG_V^{-1}.\n}\n\\]\nConsequently,\n\\[\n\\spec(L_{H^g})=\\spec(L_H),\n\\qquad\nP_I(L_{H^g})=G_VP_I(L_H)G_V^{-1}\n\\]\nfor every spectral interval \\(I\\).\n\\end{theorem}","\\begin{proof}\nFrom\n\\[\nD_{H^g}=G_ED_HG_V^{-1}\n\\]\nand unitarity,\n\\[\nD_{H^g}^*D_{H^g}\n=\nG_VD_H^*G_E^{-1}G_ED_HG_V^{-1}\n=\nG_VL_HG_V^{-1}.\n\\]\nThe spectral statements follow from unitary similarity.\n\\end{proof}","For a general invertible gauge, define\n\\[\nm_V=\\sigma_{\\min}(G_V),\n\\qquad\nM_V=\\|G_V\\|,\n\\]\n\\[\nm_E=\\sigma_{\\min}(G_E),\n\\qquad\nM_E=\\|G_E\\|.\n\\]\nThe covariance identity yields\n\\[\nm_E^2\\mathcal E_H(c)\n\\leq\n\\mathcal E_{H^g}(G_Vc)\n\\leq\nM_E^2\\mathcal E_H(c),\n\\]\nand\n\\[\nm_V^2\\|c\\|^2\n\\leq\n\\|G_Vc\\|^2\n\\leq\nM_V^2\\|c\\|^2.\n\\]\nTherefore the Rayleigh quotients obey\n\\[\n\\boxed{\n\\frac{m_E^2}{M_V^2}\n\\frac{\\mathcal E_H(c)}{\\|c\\|^2}\n\\leq\n\\frac{\\mathcal E_{H^g}(G_Vc)}{\\|G_Vc\\|^2}\n\\leq\n\\frac{M_E^2}{m_V^2}\n\\frac{\\mathcal E_H(c)}{\\|c\\|^2}.\n}\n\\]\nBy the min--max principle,\n\\[\n\\boxed{\n\\frac{m_E^2}{M_V^2}\\lambda_k(L_H)\n\\leq\n\\lambda_k(L_{H^g})\n\\leq\n\\frac{M_E^2}{m_V^2}\\lambda_k(L_H).\n}\n\\]\nExact zero modes are unchanged, but fixed-metric approximate eigenvalues can be\nstrongly distorted by an ill-conditioned coo","For a general invertible gauge, define\n\\[\nm_V=\\sigma_{\\min}(G_V),\n\\qquad\nM_V=\\|G_V\\|,\n\\]\n\\[\nm_E=\\sigma_{\\min}(G_E),\n\\qquad\nM_E=\\|G_E\\|.\n\\]\nThe covariance identity yields\n\\[\nm_E^2\\mathcal E_H(c)\n\\leq\n\\mathcal E_{H^g}(G_Vc)\n\\leq\nM_E^2\\mathcal E_H(c),\n\\]\nand\n\\[\nm_V^2\\|c\\|^2\n\\leq\n\\|G_Vc\\|^2\n\\leq\nM_V^2\\|c\\|^2.\n\\]\nTherefore the Rayleigh quotients obey\n\\[\n\\boxed{\n\\frac{m_E^2}{M_V^2}\n\\frac{\\mathcal E_H(c)}{\\|c\\|^2}\n\\leq\n\\frac{\\mathcal E_{H^g}(G_Vc)}{\\|G_Vc\\|^2}\n\\leq\n\\frac{M_E^2}{m_V^2}\n\\frac{\\mathcal E_H(c)}{\\|c\\|^2}.\n}\n\\]\nBy the min--max principle,\n\\[\n\\boxed{\n\\frac{m_E^2}{M_V^2}\\lambda_k(L_H)\n\\leq\n\\lambda_k(L_{H^g})\n\\leq\n\\frac{M_E^2}{m_V^2}\\lambda_k(L_H).\n}\n\\]\nExact zero modes are unchanged, but fixed-metric approximate eigenvalues can be\nstrongly distorted by an ill-conditioned coordinate transformation.","nce identity yields\n\\[\nm_E^2\\mathcal E_H(c)\n\\leq\n\\mathcal E_{H^g}(G_Vc)\n\\leq\nM_E^2\\mathcal E_H(c),\n\\]\nand\n\\[\nm_V^2\\|c\\|^2\n\\leq\n\\|G_Vc\\|^2\n\\leq\nM_V^2\\|c\\|^2.\n\\]\nTherefore the Rayleigh quotients obey\n\\[\n\\boxed{\n\\frac{m_E^2}{M_V^2}\n\\frac{\\mathcal E_H(c)}{\\|c\\|^2}\n\\leq\n\\frac{\\mathcal E_{H^g}(G_Vc)}{\\|G_Vc\\|^2}\n\\leq\n\\frac{M_E^2}{m_V^2}\n\\frac{\\mathcal E_H(c)}{\\|c\\|^2}.\n}\n\\]\nBy the min--max principle,\n\\[\n\\boxed{\n\\frac{m_E^2}{M_V^2}\\lambda_k(L_H)\n\\leq\n\\lambda_k(L_{H^g})\n\\leq\n\\frac{M_E^2}{m_V^2}\\lambda_k(L_H).\n}\n\\]\nExact zero modes are unchanged, but fixed-metric approximate eigenvalues can be\nstrongly distorted by an ill-conditioned coordinate transformation.","A pure change of coordinates should transform the metric as well as the\nmatrices. On each transformed space, define\n\\[\n\\langle u,v\\rangle_i^g\n=\n\\left\\langle g_i^{-1}u,g_i^{-1}v\\right\\rangle_i.\n\\]\nUse the analogous pullback metric on every edge copy. Then \\(G_V\\) and\n\\(G_E\\) are unitary between the original and transformed metric spaces.\nConsequently, the complete transfer-Laplacian spectrum is exactly preserved.","\\begin{definition}[Exact static equivalence]\\label{def:static-equivalence}\nTwo static realizations are exactly gauge equivalent,\n\\[\n\\mathfrak R\\sim_{\\mathrm{stat}}\\mathfrak R',\n\\]\nwhen there exists an invertible family \\(g_i\\) such that\n\\[\nH_e'=g_{t(e)}H_eg_{s(e)}^{-1},\n\\qquad\n\\rho_i'(a)=g_i\\rho_i(a)g_i^{-1}.\n\\]\nThe exact static abstract workspace is\n\\[\n\\boxed{\n\\Workspace_{\\mathrm{stat}}\n=\n[\\mathfrak R]_{\\sim_{\\mathrm{stat}}}.\n}\n\\]\n\\end{definition}","For approximate theory, the realization also contains the vertex and edge\nmetrics. Two metric realizations are equivalent only when the gauge transports\nthose metrics, or when the gauge is unitary relative to the fixed metrics.\nAccordingly:\n\\[\n\\boxed{\n\\begin{aligned}\n\\text{exact workspace identity}\n&\\text{ is invariant under }GL\\text{-gauge},\\\\\n\\text{fixed-metric spectral geometry}\n&\\text{ is invariant under unitary gauge},\\\\\n\\text{metric-covariant spectral geometry}\n&\\text{ is invariant under transported }GL\\text{-gauge}.\n\\end{aligned}\n}\n\\]","Parts~IV--V now prove:\n\\[\n\\begin{aligned}\n&\\ker L_H=\\Gamma(H),\\\\\n&\\text{low energy plus a gap implies proximity to an exact core},\\\\\n&\\text{a separated workspace cluster is stable under transfer noise},\\\\\n&\\text{approximately intertwined operations nearly preserve that cluster},\\\\\n&\\text{core leakage and residual reentry admit explicit pathwise bounds},\\\\\n&\\text{exact identity is gauge invariant while approximate geometry is metric aware}.\n\\end{aligned}\n\\]\nThese are structural and perturbative results. They still do not prove that a\nworkspace variable is causally used, predictively sufficient, irreducibly\ndistributed, or temporally self-maintaining. Those questions begin in\nPart~VI.","For a declared graph, set of core dimensions, weights, and transfer maps, the\nstatic analysis can be executed in the following order:\n\\begin{enumerate}\n\\item Assemble the sparse block operator \\(D_H\\) from the edge equations\n\\[\n\\sqrt{w_e}\\bigl(c_{t(e)}-H_ec_{s(e)}\\bigr).\n\\]\n\\item Compute a singular-value decomposition\n\\[\nD_H=U\\Sigma V^*,\n\\qquad\n\\lambda_k(L_H)=\\sigma_k(D_H)^2.\n\\]\nUse a declared numerical tolerance to distinguish computational zero modes\nfrom positive modes.\n\\item Select a candidate rank \\(d\\) only at a reproducible spectral boundary\nwith \\(g_d>0\\). A large observed gap is evidence of stability, not evidence of\ncausal relevance.\n\\item Quantify transfer uncertainty by \\(\\delta_H\\) and verify whether\n\\[\n2\\|D_H\\|\\delta_H+\\delta_H^2<\\frac{g_d}{2}.\n\\]\nIf this gate fails, the extracted projector has no certified perturbative\nidentity at the stated error scale.\n\\item For every","For a declared graph, set of core dimensions, weights, and transfer maps, the\nstatic analysis can be executed in the following order:\n\\begin{enumerate}\n\\item Assemble the sparse block operator \\(D_H\\) from the edge equations\n\\[\n\\sqrt{w_e}\\bigl(c_{t(e)}-H_ec_{s(e)}\\bigr).\n\\]\n\\item Compute a singular-value decomposition\n\\[\nD_H=U\\Sigma V^*,\n\\qquad\n\\lambda_k(L_H)=\\sigma_k(D_H)^2.\n\\]\nUse a declared numerical tolerance to distinguish computational zero modes\nfrom positive modes.\n\\item Select a candidate rank \\(d\\) only at a reproducible spectral boundary\nwith \\(g_d>0\\). A large observed gap is evidence of stability, not evidence of\ncausal relevance.\n\\item Quantify transfer uncertainty by \\(\\delta_H\\) and verify whether\n\\[\n2\\|D_H\\|\\delta_H+\\delta_H^2<\\frac{g_d}{2}.\n\\]\nIf this gate fails, the extracted projector has no certified perturbative\nidentity at the stated error scale.\n\\item For every proposed common operation, compute \\(\\beta(a)\\) and the\noperation-leakage","For a declared graph, set of core dimensions, weights, and transfer maps, the\nstatic analysis can be executed in the following order:\n\\begin{enumerate}\n\\item Assemble the sparse block operator \\(D_H\\) from the edge equations\n\\[\n\\sqrt{w_e}\\bigl(c_{t(e)}-H_ec_{s(e)}\\bigr).\n\\]\n\\item Compute a singular-value decomposition\n\\[\nD_H=U\\Sigma V^*,\n\\qquad\n\\lambda_k(L_H)=\\sigma_k(D_H)^2.\n\\]\nUse a declared numerical tolerance to distinguish computational zero modes\nfrom positive modes.\n\\item Select a candidate rank \\(d\\) only at a reproducible spectral boundary\nwith \\(g_d>0\\). A large observed gap is evidence of stability, not evidence of\ncausal relevance.\n\\item Quantify transfer uncertainty by \\(\\delta_H\\) and verify whether\n\\[\n2\\|D_H\\|\\delta_H+\\delta_H^2<\\frac{g_d}{2}.\n\\]\nIf this gate fails, the extracted projector has no certified perturbative\nidentity at the stated error scale.\n\\item For every proposed common operation, compute \\(\\beta(a)\\) and the\noperation-leakage bound in Theorem~\\ref{thm:approx-operation-closure}.\n\\item For every long transfer chain used in an interpr","in the following order:\n\\begin{enumerate}\n\\item Assemble the sparse block operator \\(D_H\\) from the edge equations\n\\[\n\\sqrt{w_e}\\bigl(c_{t(e)}-H_ec_{s(e)}\\bigr).\n\\]\n\\item Compute a singular-value decomposition\n\\[\nD_H=U\\Sigma V^*,\n\\qquad\n\\lambda_k(L_H)=\\sigma_k(D_H)^2.\n\\]\nUse a declared numerical tolerance to distinguish computational zero modes\nfrom positive modes.\n\\item Select a candidate rank \\(d\\) only at a reproducible spectral boundary\nwith \\(g_d>0\\). A large observed gap is evidence of stability, not evidence of\ncausal relevance.\n\\item Quantify transfer uncertainty by \\(\\delta_H\\) and verify whether\n\\[\n2\\|D_H\\|\\delta_H+\\delta_H^2<\\frac{g_d}{2}.\n\\]\nIf this gate fails, the extracted projector has no certified perturbative\nidentity at the stated error scale.\n\\item For every proposed common operation, compute \\(\\beta(a)\\) and the\noperation-leakage bound in Theorem~\\ref{thm:approx-operation-closure}.\n\\item For every long transfer chain used in an interpretation, report the\npathwise leakage certificate from Theorem~\\ref{thm:residual-duhamel} rather\nthan extrapolating from single-edge","mble the sparse block operator \\(D_H\\) from the edge equations\n\\[\n\\sqrt{w_e}\\bigl(c_{t(e)}-H_ec_{s(e)}\\bigr).\n\\]\n\\item Compute a singular-value decomposition\n\\[\nD_H=U\\Sigma V^*,\n\\qquad\n\\lambda_k(L_H)=\\sigma_k(D_H)^2.\n\\]\nUse a declared numerical tolerance to distinguish computational zero modes\nfrom positive modes.\n\\item Select a candidate rank \\(d\\) only at a reproducible spectral boundary\nwith \\(g_d>0\\). A large observed gap is evidence of stability, not evidence of\ncausal relevance.\n\\item Quantify transfer uncertainty by \\(\\delta_H\\) and verify whether\n\\[\n2\\|D_H\\|\\delta_H+\\delta_H^2<\\frac{g_d}{2}.\n\\]\nIf this gate fails, the extracted projector has no certified perturbative\nidentity at the stated error scale.\n\\item For every proposed common operation, compute \\(\\beta(a)\\) and the\noperation-leakage bound in Theorem~\\ref{thm:approx-operation-closure}.\n\\item For every long transfer chain used in an interpretation, report the\npathwise leakage certificate from Theorem~\\ref{thm:residual-duhamel} rather\nthan extrapolating from single-edge defects.\n\\end{enumerate}","ngular-value decomposition\n\\[\nD_H=U\\Sigma V^*,\n\\qquad\n\\lambda_k(L_H)=\\sigma_k(D_H)^2.\n\\]\nUse a declared numerical tolerance to distinguish computational zero modes\nfrom positive modes.\n\\item Select a candidate rank \\(d\\) only at a reproducible spectral boundary\nwith \\(g_d>0\\). A large observed gap is evidence of stability, not evidence of\ncausal relevance.\n\\item Quantify transfer uncertainty by \\(\\delta_H\\) and verify whether\n\\[\n2\\|D_H\\|\\delta_H+\\delta_H^2<\\frac{g_d}{2}.\n\\]\nIf this gate fails, the extracted projector has no certified perturbative\nidentity at the stated error scale.\n\\item For every proposed common operation, compute \\(\\beta(a)\\) and the\noperation-leakage bound in Theorem~\\ref{thm:approx-operation-closure}.\n\\item For every long transfer chain used in an interpretation, report the\npathwise leakage certificate from Theorem~\\ref{thm:residual-duhamel} rather\nthan extrapolating from single-edge defects.\n\\end{enumerate}","H=U\\Sigma V^*,\n\\qquad\n\\lambda_k(L_H)=\\sigma_k(D_H)^2.\n\\]\nUse a declared numerical tolerance to distinguish computational zero modes\nfrom positive modes.\n\\item Select a candidate rank \\(d\\) only at a reproducible spectral boundary\nwith \\(g_d>0\\). A large observed gap is evidence of stability, not evidence of\ncausal relevance.\n\\item Quantify transfer uncertainty by \\(\\delta_H\\) and verify whether\n\\[\n2\\|D_H\\|\\delta_H+\\delta_H^2<\\frac{g_d}{2}.\n\\]\nIf this gate fails, the extracted projector has no certified perturbative\nidentity at the stated error scale.\n\\item For every proposed common operation, compute \\(\\beta(a)\\) and the\noperation-leakage bound in Theorem~\\ref{thm:approx-operation-closure}.\n\\item For every long transfer chain used in an interpretation, report the\npathwise leakage certificate from Theorem~\\ref{thm:residual-duhamel} rather\nthan extrapolating from single-edge defects.\n\\end{enumerate}","e spectral boundary\nwith \\(g_d>0\\). A large observed gap is evidence of stability, not evidence of\ncausal relevance.\n\\item Quantify transfer uncertainty by \\(\\delta_H\\) and verify whether\n\\[\n2\\|D_H\\|\\delta_H+\\delta_H^2<\\frac{g_d}{2}.\n\\]\nIf this gate fails, the extracted projector has no certified perturbative\nidentity at the stated error scale.\n\\item For every proposed common operation, compute \\(\\beta(a)\\) and the\noperation-leakage bound in Theorem~\\ref{thm:approx-operation-closure}.\n\\item For every long transfer chain used in an interpretation, report the\npathwise leakage certificate from Theorem~\\ref{thm:residual-duhamel} rather\nthan extrapolating from single-edge defects.\n\\end{enumerate}","A minimal structural report is therefore\n\\[\n\\boxed{\n\\left(\n d,\\lambda_d,\\lambda_{d+1},g_d,\n \\delta_H,\n \\|P_d-\\widetilde P_d\\|_{\\mathrm{bound}},\n \\beta(a),\n \\Lambda_\\gamma\n\\right),\n}\n\\]\nwhere \\(\\Lambda_\\gamma\\) denotes the right-hand side of the pathwise leakage\nbound. This structural report is intentionally incomplete as a workspace\ncertificate. Parts~VI--IX add the predictive, causal, null, and irreducibility\ngates required to interpret the low-energy subspace operationally.","\\begin{definition}[Dynamic relational realization]\\label{def:dynamic-realization}\nA dynamic relational realization is defined on a probability space\n\\((\\Omega,\\mathcal F,\\mathbb P)\\) and consists of:\n\\begin{itemize}\n\\item node states \\(X_i(t)\\in\\mathsf X_i\\) for \\(i\\in V\\);\n\\item edge messages \\(M_e(t)\\in\\mathsf M_e\\) for \\(e\\in E\\);\n\\item external inputs \\(E_t\\), local actions \\(A_i(t)\\), and disturbances\n \\(\\xi_i(t),\\zeta_e(t)\\);\n\\item time-dependent transfer data \\(H_e(t)\\), local core vectors \\(c_i(t)\\),\n and transfer Laplacians \\(L_t=L_{H(t)}\\); and\n\\item a future target \\(Y_t^+\\in\\mathsf Y\\).\n\\end{itemize}\nAll state, history, and target spaces are assumed standard Borel unless a\nstronger topology is explicitly invoked.\n\\end{definition}","tion}[Dynamic relational realization]\\label{def:dynamic-realization}\nA dynamic relational realization is defined on a probability space\n\\((\\Omega,\\mathcal F,\\mathbb P)\\) and consists of:\n\\begin{itemize}\n\\item node states \\(X_i(t)\\in\\mathsf X_i\\) for \\(i\\in V\\);\n\\item edge messages \\(M_e(t)\\in\\mathsf M_e\\) for \\(e\\in E\\);\n\\item external inputs \\(E_t\\), local actions \\(A_i(t)\\), and disturbances\n \\(\\xi_i(t),\\zeta_e(t)\\);\n\\item time-dependent transfer data \\(H_e(t)\\), local core vectors \\(c_i(t)\\),\n and transfer Laplacians \\(L_t=L_{H(t)}\\); and\n\\item a future target \\(Y_t^+\\in\\mathsf Y\\).\n\\end{itemize}\nAll state, history, and target spaces are assumed standard Borel unless a\nstronger topology is explicitly invoked.\n\\end{definition}","ization}\nA dynamic relational realization is defined on a probability space\n\\((\\Omega,\\mathcal F,\\mathbb P)\\) and consists of:\n\\begin{itemize}\n\\item node states \\(X_i(t)\\in\\mathsf X_i\\) for \\(i\\in V\\);\n\\item edge messages \\(M_e(t)\\in\\mathsf M_e\\) for \\(e\\in E\\);\n\\item external inputs \\(E_t\\), local actions \\(A_i(t)\\), and disturbances\n \\(\\xi_i(t),\\zeta_e(t)\\);\n\\item time-dependent transfer data \\(H_e(t)\\), local core vectors \\(c_i(t)\\),\n and transfer Laplacians \\(L_t=L_{H(t)}\\); and\n\\item a future target \\(Y_t^+\\in\\mathsf Y\\).\n\\end{itemize}\nAll state, history, and target spaces are assumed standard Borel unless a\nstronger topology is explicitly invoked.\n\\end{definition}","ealization is defined on a probability space\n\\((\\Omega,\\mathcal F,\\mathbb P)\\) and consists of:\n\\begin{itemize}\n\\item node states \\(X_i(t)\\in\\mathsf X_i\\) for \\(i\\in V\\);\n\\item edge messages \\(M_e(t)\\in\\mathsf M_e\\) for \\(e\\in E\\);\n\\item external inputs \\(E_t\\), local actions \\(A_i(t)\\), and disturbances\n \\(\\xi_i(t),\\zeta_e(t)\\);\n\\item time-dependent transfer data \\(H_e(t)\\), local core vectors \\(c_i(t)\\),\n and transfer Laplacians \\(L_t=L_{H(t)}\\); and\n\\item a future target \\(Y_t^+\\in\\mathsf Y\\).\n\\end{itemize}\nAll state, history, and target spaces are assumed standard Borel unless a\nstronger topology is explicitly invoked.\n\\end{definition}","y space\n\\((\\Omega,\\mathcal F,\\mathbb P)\\) and consists of:\n\\begin{itemize}\n\\item node states \\(X_i(t)\\in\\mathsf X_i\\) for \\(i\\in V\\);\n\\item edge messages \\(M_e(t)\\in\\mathsf M_e\\) for \\(e\\in E\\);\n\\item external inputs \\(E_t\\), local actions \\(A_i(t)\\), and disturbances\n \\(\\xi_i(t),\\zeta_e(t)\\);\n\\item time-dependent transfer data \\(H_e(t)\\), local core vectors \\(c_i(t)\\),\n and transfer Laplacians \\(L_t=L_{H(t)}\\); and\n\\item a future target \\(Y_t^+\\in\\mathsf Y\\).\n\\end{itemize}\nAll state, history, and target spaces are assumed standard Borel unless a\nstronger topology is explicitly invoked.\n\\end{definition}","d consists of:\n\\begin{itemize}\n\\item node states \\(X_i(t)\\in\\mathsf X_i\\) for \\(i\\in V\\);\n\\item edge messages \\(M_e(t)\\in\\mathsf M_e\\) for \\(e\\in E\\);\n\\item external inputs \\(E_t\\), local actions \\(A_i(t)\\), and disturbances\n \\(\\xi_i(t),\\zeta_e(t)\\);\n\\item time-dependent transfer data \\(H_e(t)\\), local core vectors \\(c_i(t)\\),\n and transfer Laplacians \\(L_t=L_{H(t)}\\); and\n\\item a future target \\(Y_t^+\\in\\mathsf Y\\).\n\\end{itemize}\nAll state, history, and target spaces are assumed standard Borel unless a\nstronger topology is explicitly invoked.\n\\end{definition}","A representative local update is\n\\[\nX_i(t+1)\n=\nF_i\\!\\left(\nX_i(t),\n\\{M_e(t):t(e)=i\\},\nA_i(t),E_t,\\xi_i(t)\n\\right),\n\\]\nwith messages\n\\[\nM_e(t)\n=\nK_e\\!\\left(\nX_{s(e)}(t),\nW_t,\nE_t,\\zeta_e(t)\n\\right).\n\\]\nThis notation permits, but does not require, the current workspace state to\nmodulate future transfers.","For a coalition \\(A\\subseteq V\\), let \\(\\mathbf H_t^A\\) denote the random\nhistory available to that coalition through time \\(t\\). Its generated\ninformation is\n\\[\n\\mathcal F_t^A=\\sigma(\\mathbf H_t^A).\n\\]\nThe full relational history is \\(\\mathbf H_t=\\mathbf H_t^V\\), with\n\\[\n\\mathcal F_t^A\\subseteq\\mathcal F_t^B\n\\qquad\n\\text{whenever }A\\subseteq B.\n\\]\nThe empty-coalition history \\(\\mathbf H_t^\\varnothing\\) is the trivial\nconstant statistic and represents prediction from the background target law\nalone.\nA concrete history may contain node trajectories, incident messages, actions,\nartifacts, transfer-map estimates, and the declared graph history. The exact\ncontents must be reported because coalition sufficiency is relative to","For a coalition \\(A\\subseteq V\\), let \\(\\mathbf H_t^A\\) denote the random\nhistory available to that coalition through time \\(t\\). Its generated\ninformation is\n\\[\n\\mathcal F_t^A=\\sigma(\\mathbf H_t^A).\n\\]\nThe full relational history is \\(\\mathbf H_t=\\mathbf H_t^V\\), with\n\\[\n\\mathcal F_t^A\\subseteq\\mathcal F_t^B\n\\qquad\n\\text{whenever }A\\subseteq B.\n\\]\nThe empty-coalition history \\(\\mathbf H_t^\\varnothing\\) is the trivial\nconstant statistic and represents prediction from the background target law\nalone.\nA concrete history may contain node trajectories, incident messages, actions,\nartifacts, transfer-map estimates, and the declared graph history. The exact\ncontents must be reported because coalition sufficiency is relative to what a\ncoalition is pe","For a coalition \\(A\\subseteq V\\), let \\(\\mathbf H_t^A\\) denote the random\nhistory available to that coalition through time \\(t\\). Its generated\ninformation is\n\\[\n\\mathcal F_t^A=\\sigma(\\mathbf H_t^A).\n\\]\nThe full relational history is \\(\\mathbf H_t=\\mathbf H_t^V\\), with\n\\[\n\\mathcal F_t^A\\subseteq\\mathcal F_t^B\n\\qquad\n\\text{whenever }A\\subseteq B.\n\\]\nThe empty-coalition history \\(\\mathbf H_t^\\varnothing\\) is the trivial\nconstant statistic and represents prediction from the background target law\nalone.\nA concrete history may contain node trajectories, incident messages, actions,\nartifacts, transfer-map estimates, and the declared graph history. The exact\ncontents must be reported because coalition sufficiency is relative to what a\ncoalition is permitted to observe.","The target \\(Y_t^+\\) can be a next output, a finite future trajectory, a\nresearch decision, a proof outcome, or any measurable downstream quantity.\nEvery predictive workspace claim is therefore target-relative:\n\\[\n\\operatorname{WorkspaceFor}(Y_t^+)\n\\centernot\\Longrightarrow\n\\operatorname{WorkspaceFor}(\\widetilde Y_t^+).\n\\]","Let\n\\[\nc(t)=(c_i(t))_{i\\in V}\n\\in\n\\mathcal C_V(t),\n\\qquad\nP_t=\\mathbf 1_{[0,\\varepsilon_t]}(L_t).\n\\]\nThe instantaneous structural workspace coordinate is\n\\[\n\\boxed{\nW_t^{\\mathrm{str}}=P_tc(t)\n\\in\n\\Gamma_{\\varepsilon_t}(H(t)).\n}\n\\]\nWhen \\(P_t\\) varies with time, these coordinates inhabit a moving family of\nsubspaces. Parts~X--XII will supply the transport required to compare them\nacross time. In what follows, all sufficiency and intervention statements\nare made at a fixed decision time \\(t\\).","L_t).\n\\]\nThe instantaneous structural workspace coordinate is\n\\[\n\\boxed{\nW_t^{\\mathrm{str}}=P_tc(t)\n\\in\n\\Gamma_{\\varepsilon_t}(H(t)).\n}\n\\]\nWhen \\(P_t\\) varies with time, these coordinates inhabit a moving family of\nsubspaces. Parts~X--XII will supply the transport required to compare them\nacross time. In what follows, all sufficiency and intervention statements\nare made at a fixed decision time \\(t\\).","\\begin{remark}[Three objects that must not be conflated]\nThe realization state \\(S_t\\), the complete history \\(\\mathbf H_t\\), and the\nstructural coordinate \\(W_t^{\\mathrm{str}}\\) are different objects. The first\nis an instantaneous physical or computational state, the second contains all\nadmissible past information, and the third is a low-transfer-energy statistic\nextracted from the declared core geometry.\n\\end{remark}","The quotient notation\n\\([\\mathbf H_t]_{\\sim_Y}\\) expresses the right intuition but hides\nmeasurability and null-set issues. A canonical construction is obtained from\nregular conditional distributions.","\\begin{definition}[Predictive kernel and canonical state]\\label{def:predictive-kernel}\nLet \\(\\mathsf H_t\\) and \\(\\mathsf Y\\) be standard Borel spaces. A regular\nconditional kernel is a measurable map\n\\[\nK_t:\\mathsf H_t\\times\\mathcal B(\\mathsf Y)\\to[0,1]\n\\]\nsatisfying\n\\[\nK_t(h,A)\n=\n\\mathbb P(Y_t^+\\in A\\mid\\mathbf H_t=h)\n\\]\nfor \\(\\mathbb P_{\\mathbf H_t}\\)-almost every \\(h\\). Let\n\\(\\mathcal P(\\mathsf Y)\\) carry its evaluation sigma-algebra. The canonical\npredictive state is the random probability measure\n\\[\n\\boxed{\nZ_t\n=\nK_t(\\mathbf H_t,\\cdot)\n\\in\n\\mathcal P(\\mathsf Y).\n}\n\\]\nTwo histories are predictively equivalent precisely when they induce the same\nconditional law of \\(Y_t^+\\).\n\\end{definition}","d \\(\\mathsf Y\\) be standard Borel spaces. A regular\nconditional kernel is a measurable map\n\\[\nK_t:\\mathsf H_t\\times\\mathcal B(\\mathsf Y)\\to[0,1]\n\\]\nsatisfying\n\\[\nK_t(h,A)\n=\n\\mathbb P(Y_t^+\\in A\\mid\\mathbf H_t=h)\n\\]\nfor \\(\\mathbb P_{\\mathbf H_t}\\)-almost every \\(h\\). Let\n\\(\\mathcal P(\\mathsf Y)\\) carry its evaluation sigma-algebra. The canonical\npredictive state is the random probability measure\n\\[\n\\boxed{\nZ_t\n=\nK_t(\\mathbf H_t,\\cdot)\n\\in\n\\mathcal P(\\mathsf Y).\n}\n\\]\nTwo histories are predictively equivalent precisely when they induce the same\nconditional law of \\(Y_t^+\\).\n\\end{definition}",")\n=\n\\mathbb P(Y_t^+\\in A\\mid\\mathbf H_t=h)\n\\]\nfor \\(\\mathbb P_{\\mathbf H_t}\\)-almost every \\(h\\). Let\n\\(\\mathcal P(\\mathsf Y)\\) carry its evaluation sigma-algebra. The canonical\npredictive state is the random probability measure\n\\[\n\\boxed{\nZ_t\n=\nK_t(\\mathbf H_t,\\cdot)\n\\in\n\\mathcal P(\\mathsf Y).\n}\n\\]\nTwo histories are predictively equivalent precisely when they induce the same\nconditional law of \\(Y_t^+\\).\n\\end{definition}","\\begin{theorem}[Predictive-state sufficiency and minimality, \\(T_{12}\\)]\n\\label{thm:predictive-state}\nUnder Definition~\\ref{def:predictive-kernel}:\n\\begin{enumerate}\n\\item \\(Z_t\\) is predictively sufficient:\n\\[\nY_t^+\\indep\\mathbf H_t\\mid Z_t.\n\\]\n\\item If \\(S_t=s(\\mathbf H_t)\\) is any other history statistic satisfying\n\\[\nY_t^+\\indep\\mathbf H_t\\mid S_t,\n\\]\nthen there is a measurable map \\(\\phi\\) such that\n\\[\nZ_t=\\phi(S_t)\n\\qquad\\text{almost surely}.\n\\]\n\\item Consequently, any two minimal sufficient predictive states are\nmeasurably isomorphic after removal of null sets.\n\\end{enumerate}\n\\end{theorem}","orem}[Predictive-state sufficiency and minimality, \\(T_{12}\\)]\n\\label{thm:predictive-state}\nUnder Definition~\\ref{def:predictive-kernel}:\n\\begin{enumerate}\n\\item \\(Z_t\\) is predictively sufficient:\n\\[\nY_t^+\\indep\\mathbf H_t\\mid Z_t.\n\\]\n\\item If \\(S_t=s(\\mathbf H_t)\\) is any other history statistic satisfying\n\\[\nY_t^+\\indep\\mathbf H_t\\mid S_t,\n\\]\nthen there is a measurable map \\(\\phi\\) such that\n\\[\nZ_t=\\phi(S_t)\n\\qquad\\text{almost surely}.\n\\]\n\\item Consequently, any two minimal sufficient predictive states are\nmeasurably isomorphic after removal of null sets.\n\\end{enumerate}\n\\end{theorem}","\\begin{proof}\nFor every bounded measurable \\(f:\\mathsf Y\\to\\mathbb R\\), define\n\\[\nZ_t(f)=\\int_{\\mathsf Y}f(y)\\,Z_t(\\mathrm dy).\n\\]\nBy construction,\n\\[\n\\mathbb E[f(Y_t^+)\\mid\\mathbf H_t]=Z_t(f).\n\\]\nSince \\(Z_t\\) is a measurable function of \\(\\mathbf H_t\\), the tower property\ngives\n\\[\n\\mathbb E[f(Y_t^+)\\mid Z_t]\n=\n\\mathbb E[Z_t(f)\\mid Z_t]\n=\nZ_t(f).\n\\]\nHence conditioning additionally on \\(\\mathbf H_t\\) does not change the\nconditional expectation, which is the stated conditional independence.","Now let \\(S_t=s(\\mathbf H_t)\\) be sufficient. Then, for every measurable\n\\(A\\subseteq\\mathsf Y\\),\n\\[\nK_t(\\mathbf H_t,A)\n=\n\\mathbb P(Y_t^+\\in A\\mid\\mathbf H_t)\n=\n\\mathbb P(Y_t^+\\in A\\mid S_t)\n\\qquad\\text{a.s.}\n\\]\nChoose a countable determining class for probability measures on the standard\nBorel space \\(\\mathsf Y\\). The values of \\(K_t(\\mathbf H_t,\\cdot)\\) on this\nclass are measurable functions of \\(S_t\\), and therefore the complete random\nmeasure \\(Z_t\\) is a measurable function of \\(S_t\\). Applying this conclusion\nin both directions to two minimal sufficient states gives the final claim.\n\\end{proof}","w let \\(S_t=s(\\mathbf H_t)\\) be sufficient. Then, for every measurable\n\\(A\\subseteq\\mathsf Y\\),\n\\[\nK_t(\\mathbf H_t,A)\n=\n\\mathbb P(Y_t^+\\in A\\mid\\mathbf H_t)\n=\n\\mathbb P(Y_t^+\\in A\\mid S_t)\n\\qquad\\text{a.s.}\n\\]\nChoose a countable determining class for probability measures on the standard\nBorel space \\(\\mathsf Y\\). The values of \\(K_t(\\mathbf H_t,\\cdot)\\) on this\nclass are measurable functions of \\(S_t\\), and therefore the complete random\nmeasure \\(Z_t\\) is a measurable function of \\(S_t\\). Applying this conclusion\nin both directions to two minimal sufficient states gives the final claim.\n\\end{proof}","or every measurable\n\\(A\\subseteq\\mathsf Y\\),\n\\[\nK_t(\\mathbf H_t,A)\n=\n\\mathbb P(Y_t^+\\in A\\mid\\mathbf H_t)\n=\n\\mathbb P(Y_t^+\\in A\\mid S_t)\n\\qquad\\text{a.s.}\n\\]\nChoose a countable determining class for probability measures on the standard\nBorel space \\(\\mathsf Y\\). The values of \\(K_t(\\mathbf H_t,\\cdot)\\) on this\nclass are measurable functions of \\(S_t\\), and therefore the complete random\nmeasure \\(Z_t\\) is a measurable function of \\(S_t\\). Applying this conclusion\nin both directions to two minimal sufficient states gives the final claim.\n\\end{proof}","hsf Y\\),\n\\[\nK_t(\\mathbf H_t,A)\n=\n\\mathbb P(Y_t^+\\in A\\mid\\mathbf H_t)\n=\n\\mathbb P(Y_t^+\\in A\\mid S_t)\n\\qquad\\text{a.s.}\n\\]\nChoose a countable determining class for probability measures on the standard\nBorel space \\(\\mathsf Y\\). The values of \\(K_t(\\mathbf H_t,\\cdot)\\) on this\nclass are measurable functions of \\(S_t\\), and therefore the complete random\nmeasure \\(Z_t\\) is a measurable function of \\(S_t\\). Applying this conclusion\nin both directions to two minimal sufficient states gives the final claim.\n\\end{proof}","\\begin{remark}[Meaning of minimality]\nThe theorem does not say that \\(Z_t\\) has the fewest numerical coordinates in\na chosen parametrization. It says that every sufficient history statistic\nmust determine \\(Z_t\\) almost surely. This is an information order, invariant\nunder one-to-one reparametrization.\n\\end{remark}","For an arbitrary history statistic \\(S_t=s(\\mathbf H_t)\\), let\n\\[\nK_t^{S}(s,\\cdot)\n=\n\\mathbb P(Y_t^+\\in\\cdot\\mid S_t=s).\n\\]\nWhenever the conditional relative entropy is finite, define the sufficiency\ndefect\n\\[\n\\boxed{\n\\delta_{\\mathrm{suff}}(S_t)\n=\n\\mathbb E\\!\\left[\n\\KL\\!\\left(\nK_t(\\mathbf H_t,\\cdot)\n\\,\\middle\\|\\,\nK_t^S(S_t,\\cdot)\n\\right)\n\\right].\n}\n\\]","\\begin{proposition}[Conditional-information identity]\\label{prop:conditional-information}\nIf \\(S_t\\) is a measurable function of \\(\\mathbf H_t\\), then\n\\[\n\\boxed{\n\\delta_{\\mathrm{suff}}(S_t)\n=\nI(Y_t^+;\\mathbf H_t\\mid S_t).\n}\n\\]\nIn particular, the defect vanishes if and only if \\(S_t\\) is predictively\nsufficient, up to the usual almost-sure qualifications.\n\\end{proposition}","\\begin{proof}\nThe conditional mutual information is the expected relative entropy between\nthe conditional law given \\((\\mathbf H_t,S_t)\\) and the conditional law given\n\\(S_t\\). Because \\(S_t=s(\\mathbf H_t)\\), the first of these is exactly\n\\(K_t(\\mathbf H_t,\\cdot)\\).\n\\end{proof}","Sufficiency has a direct decision-theoretic interpretation. Use the\nconvention\n\\[\n\\TV(P,Q)=\\sup_{A}|P(A)-Q(A)|.\n\\]\nLet \\(\\mathsf A\\) be a finite action set and\n\\(\\ell:\\mathsf Y\\times\\mathsf A\\to[0,1]\\) a bounded loss. Define\n\\[\n\\Risk_\\ell(S_t)\n=\n\\inf_{d}\n\\mathbb E\\!\\left[\n\\ell\\bigl(Y_t^+,d(S_t)\\bigr)\n\\right].\n\\]","\\begin{proposition}[Approximate sufficiency bounds excess Bayes risk]\n\\label{prop:risk-pinsker}\nUnder the preceding assumptions,\n\\[\n\\boxed{\n0\n\\leq\n\\Risk_\\ell(S_t)-\\Risk_\\ell(\\mathbf H_t)\n\\leq\n\\mathbb E\\!\\left[\n\\TV\\!\\left(K_t(\\mathbf H_t),K_t^S(S_t)\\right)\n\\right]\n\\leq\n\\sqrt{\\frac{\\delta_{\\mathrm{suff}}(S_t)}{2}}.\n}\n\\]\n\\end{proposition}","For a finite target alphabet and logarithmic scoring loss\n\\[\n\\ell_{\\log}(y,q)=-\\log q(y),\n\\qquad\nq\\in\\mathcal P(\\mathsf Y),\n\\]\nthe identity is exact:\n\\[\n\\boxed{\n\\Risk_{\\log}(S_t)=H(Y_t^+\\mid S_t),\n\\qquad\n\\Risk_{\\log}(S_t)-\\Risk_{\\log}(\\mathbf H_t)\n=\nI(Y_t^+;\\mathbf H_t\\mid S_t).\n}\n\\]\nNatural logarithms measure the gap in nats; base-two logarithms measure it in\nbits.","Observational prediction may merge histories that behave identically under the\ncurrent policy but respond differently to future interventions. Let\n\\(\\mathcal I\\) be a finite or countable family of interventions applied only\nafter time \\(t\\). For each \\(\\iota\\in\\mathcal I\\), let\n\\[\nK_t^{\\iota}(h,\\cdot)\n=\n\\mathbb P^{\\iota}(Y_t^+\\in\\cdot\\mid\\mathbf H_t=h)\n\\]\nbe a regular post-intervention kernel on the common pre-intervention history\nspace. Define\n\\[\n\\boxed{\nZ_t^{\\mathcal I}\n=\n\\bigl(K_t^{\\iota}(\\mathbf H_t,\\cdot)\\bigr)_{\\iota\\in\\mathcal I}\n\\in\n\\mathcal P(\\mathsf Y)^{\\mathcal I}.\n}\n\\]","ehave identically under the\ncurrent policy but respond differently to future interventions. Let\n\\(\\mathcal I\\) be a finite or countable family of interventions applied only\nafter time \\(t\\). For each \\(\\iota\\in\\mathcal I\\), let\n\\[\nK_t^{\\iota}(h,\\cdot)\n=\n\\mathbb P^{\\iota}(Y_t^+\\in\\cdot\\mid\\mathbf H_t=h)\n\\]\nbe a regular post-intervention kernel on the common pre-intervention history\nspace. Define\n\\[\n\\boxed{\nZ_t^{\\mathcal I}\n=\n\\bigl(K_t^{\\iota}(\\mathbf H_t,\\cdot)\\bigr)_{\\iota\\in\\mathcal I}\n\\in\n\\mathcal P(\\mathsf Y)^{\\mathcal I}.\n}\n\\]","\\begin{corollary}[Minimal simultaneous interventional state]\n\\label{cor:interventional-state}\nThe random element \\(Z_t^{\\mathcal I}\\) is sufficient simultaneously for all\npost-intervention target laws. If a statistic \\(S_t=s(\\mathbf H_t)\\)\ndetermines the conditional target law for every \\(\\iota\\in\\mathcal I\\), then\nthere exists a measurable \\(\\phi_{\\mathcal I}\\) such that\n\\[\nZ_t^{\\mathcal I}=\\phi_{\\mathcal I}(S_t)\n\\qquad\\text{a.s.}\n\\]\n\\end{corollary}","The observational state is the special case\n\\(\\mathcal I=\\{\\operatorname{id}\\}\\). A causal workspace claim should align\nwith \\(Z_t^{\\mathcal I}\\) for a declared intervention family rather than only\nwith the observational state.","A structural workspace coordinate becomes operational only when it retains the\ncanonical predictive or interventional state relevant to the declared task.\nLet\n\\[\nW_t=\\pi_t(\\mathbf H_t)\n\\]\nbe a measurable candidate workspace statistic satisfying the structural support\ncondition\n\\[\nW_t\\in\\Gamma_{\\varepsilon_t}(H(t))\n\\qquad\\text{almost surely}.\n\\]\nThe map \\(\\pi_t\\) may include encoding into a moving low-energy bundle; no\ncanonical operational interpretation follows from the spectral projector\nalone.","\\begin{definition}[Exact structural--operational alignment]\nThe candidate \\(W_t\\) is observationally aligned when there exists a measurable\nmap \\(\\phi_t\\) such that\n\\[\nZ_t=\\phi_t(W_t)\n\\qquad\\text{a.s.}\n\\]\nIt is \\(\\mathcal I\\)-causally aligned when\n\\[\nZ_t^{\\mathcal I}=\\phi_t^{\\mathcal I}(W_t)\n\\qquad\\text{a.s.}\n\\]\n\\end{definition}","\\begin{theorem}[Exact alignment criterion]\\label{thm:alignment-criterion}\nFor any workspace statistic \\(W_t=\\pi_t(\\mathbf H_t)\\), the following are\nequivalent:\n\\begin{enumerate}\n\\item \\(Y_t^+\\indep\\mathbf H_t\\mid W_t\\);\n\\item \\(I(Y_t^+;\\mathbf H_t\\mid W_t)=0\\);\n\\item the canonical predictive state factors through \\(W_t\\):\n\\[\nZ_t=\\phi_t(W_t)\n\\quad\\text{a.s.}\n\\]\nfor some measurable \\(\\phi_t\\).\n\\end{enumerate}\nThe analogous equivalence holds simultaneously over \\(\\mathcal I\\) with\n\\(Z_t^{\\mathcal I}\\).\n\\end{theorem}","iterion}\nFor any workspace statistic \\(W_t=\\pi_t(\\mathbf H_t)\\), the following are\nequivalent:\n\\begin{enumerate}\n\\item \\(Y_t^+\\indep\\mathbf H_t\\mid W_t\\);\n\\item \\(I(Y_t^+;\\mathbf H_t\\mid W_t)=0\\);\n\\item the canonical predictive state factors through \\(W_t\\):\n\\[\nZ_t=\\phi_t(W_t)\n\\quad\\text{a.s.}\n\\]\nfor some measurable \\(\\phi_t\\).\n\\end{enumerate}\nThe analogous equivalence holds simultaneously over \\(\\mathcal I\\) with\n\\(Z_t^{\\mathcal I}\\).\n\\end{theorem}","kspace statistic \\(W_t=\\pi_t(\\mathbf H_t)\\), the following are\nequivalent:\n\\begin{enumerate}\n\\item \\(Y_t^+\\indep\\mathbf H_t\\mid W_t\\);\n\\item \\(I(Y_t^+;\\mathbf H_t\\mid W_t)=0\\);\n\\item the canonical predictive state factors through \\(W_t\\):\n\\[\nZ_t=\\phi_t(W_t)\n\\quad\\text{a.s.}\n\\]\nfor some measurable \\(\\phi_t\\).\n\\end{enumerate}\nThe analogous equivalence holds simultaneously over \\(\\mathcal I\\) with\n\\(Z_t^{\\mathcal I}\\).\n\\end{theorem}","\\begin{proof}\nThe equivalence of the first two statements is the zero-divergence\ncharacterization of conditional mutual information. The first statement and\nTheorem~\\ref{thm:predictive-state} imply the factorization in the third.\nConversely, if \\(Z_t=\\phi_t(W_t)\\), then\n\\[\n\\mathbb P(Y_t^+\\in A\\mid\\mathbf H_t)\n=Z_t(A)\n=\\phi_t(W_t)(A),\n\\]\nwhich depends on the history only through \\(W_t\\), proving sufficiency. The\ninterventional statement follows from\nCorollary~\\ref{cor:interventional-state}.\n\\end{proof}","Define the observational alignment defect by\n\\[\n\\boxed{\n\\delta_{\\mathrm{align}}^{\\mathrm{obs}}(W_t)\n=\nI_{\\mathbb P}(Y_t^+;\\mathbf H_t\\mid W_t).\n}\n\\]\nFor a countable intervention family with declared weights\n\\(q_\\iota>0\\), \\(\\sum_\\iota q_\\iota=1\\), define\n\\[\n\\boxed{\n\\delta_{\\mathrm{align}}^{\\mathcal I}(W_t)\n=\n\\sum_{\\iota\\in\\mathcal I}\nq_\\iota\nI_{\\mathbb P^\\iota}\n(Y_t^+;\\mathbf H_t\\mid W_t).\n}\n\\]\nFor bounded decision losses, Proposition~\\ref{prop:risk-pinsker} applies under\neach intervention and gives the weighted bound\n\\[\n\\sum_{\\iota}q_\\iota\n\\left[\n\\Risk_\\ell^{\\iota}(W_t)\n-\n\\Risk_\\ell^{\\iota}(\\mathbf H_t)\n\\right]\n\\leq\n\\sqrt{\\frac{\n\\delta_{\\mathrm{align}}^{\\mathcal I}(W_t)}{2}}.\n\\]","hbf H_t\\mid W_t).\n}\n\\]\nFor a countable intervention family with declared weights\n\\(q_\\iota>0\\), \\(\\sum_\\iota q_\\iota=1\\), define\n\\[\n\\boxed{\n\\delta_{\\mathrm{align}}^{\\mathcal I}(W_t)\n=\n\\sum_{\\iota\\in\\mathcal I}\nq_\\iota\nI_{\\mathbb P^\\iota}\n(Y_t^+;\\mathbf H_t\\mid W_t).\n}\n\\]\nFor bounded decision losses, Proposition~\\ref{prop:risk-pinsker} applies under\neach intervention and gives the weighted bound\n\\[\n\\sum_{\\iota}q_\\iota\n\\left[\n\\Risk_\\ell^{\\iota}(W_t)\n-\n\\Risk_\\ell^{\\iota}(\\mathbf H_t)\n\\right]\n\\leq\n\\sqrt{\\frac{\n\\delta_{\\mathrm{align}}^{\\mathcal I}(W_t)}{2}}.\n\\]","The static defect\n\\[\n\\delta_{\\mathrm{coh}}(W_t)\n\\]\nand the operational defect\n\\[\n\\delta_{\\mathrm{align}}^{\\mathcal I}(W_t)\n\\]\nanswer different questions. The first measures compatibility under transfer.\nThe second measures loss of task-relevant conditional law.","Thus neither low transfer energy nor high predictive performance alone is\nenough. The combined gate is\n\\[\n\\boxed{\n\\delta_{\\mathrm{coh}}(W_t)\n\\leq\n\\varepsilon_{\\mathrm{coh}},\n\\qquad\n\\delta_{\\mathrm{align}}^{\\mathcal I}(W_t)\n\\leq\n\\varepsilon_{\\mathrm{align}}.\n}\n\\]","\\begin{example}[Coherent but irrelevant archive]\nSuppose every node perfectly copies a common random variable \\(R\\), so the\ncorresponding transfer energy is zero. If \\(R\\indep Y_t^+\\), then\n\\[\n\\delta_{\\mathrm{coh}}(R)=0,\n\\qquad\n\\delta_{\\mathrm{align}}^{\\mathrm{obs}}(R)\n=I(Y_t^+;\\mathbf H_t\\mid R),\n\\]\nwhich can be as large as the entire predictive information in the history.\nPerfect agreement therefore does not imply workspace functionality.\n\\end{example}","\\begin{example}[Sufficient but localized statistic]\nIf one node contains the canonical state \\(Z_t\\) and all transfers from that\nnode are unstable, then \\(W_t=Z_t\\) is sufficient but need not lie in a stable\nlow-energy transfer subspace. This is a local controller or archive, not yet a\nstratified shared workspace.\n\\end{example}","For \\(A\\subseteq V\\), the coalition statistic is \\(\\mathbf H_t^A\\). Given an\naction space \\(\\mathsf A\\) and loss \\(\\ell\\), define its unrestricted Bayes\nrisk by\n\\[\n\\boxed{\n\\Risk_\\ell(A)\n=\n\\inf_d\n\\mathbb E\\!\\left[\n\\ell\\bigl(Y_t^+,d(\\mathbf H_t^A)\\bigr)\n\\right].\n}\n\\]\nThe infimum is over measurable decision rules for which the expectation is\ndefined. The full-system risk is \\(\\Risk_\\ell(V)\\).","\\begin{proposition}[Coalition monotonicity]\\label{prop:coalition-monotonicity}\nIf \\(A\\subseteq B\\subseteq V\\), then\n\\[\n\\Risk_\\ell(B)\\leq\\Risk_\\ell(A).\n\\]\n\\end{proposition}","\\begin{proof}\nEvery decision rule available to coalition \\(A\\) is also available to coalition\n\\(B\\), which can ignore its additional information.\n\\end{proof}","For finite \\(\\mathsf Y\\), logarithmic loss converts risk gaps into conditional\ninformation exactly.","\\begin{theorem}[Coalition log-risk identity]\\label{thm:coalition-log-risk}\nAssume \\(H(Y_t^+)<\\infty\\). If \\(A\\subseteq B\\subseteq V\\), then\n\\[\n\\boxed{\n\\Risk_{\\log}(A)-\\Risk_{\\log}(B)\n=\nI\\!\\left(\nY_t^+;\\mathbf H_t^B\n\\mid\n\\mathbf H_t^A\n\\right).\n}\n\\]\nIn particular,\n\\[\n\\Risk_{\\log}(A)-\\Risk_{\\log}(V)\n=\nI\\!\\left(\nY_t^+;\\mathbf H_t\n\\mid\n\\mathbf H_t^A\n\\right).\n\\]\n\\end{theorem}","\\begin{proof}\nUnder logarithmic loss the optimal report is the conditional target law, so\n\\[\n\\Risk_{\\log}(A)=H(Y_t^+\\mid\\mathbf H_t^A).\n\\]\nBecause \\(\\mathbf H_t^A\\) is a function of \\(\\mathbf H_t^B\\), subtracting the\ntwo conditional entropies gives the stated conditional mutual information.\n\\end{proof}","Define the coalition gap\n\\[\n\\Delta_A^{\\ell}\n=\n\\Risk_\\ell(A)-\\Risk_\\ell(V)\n\\geq0.\n\\]\nThe node-level joint advantage is\n\\[\n\\boxed{\n\\Delta_{\\mathrm{joint}}^{\\ell}\n=\n\\min_{i\\in V}\n\\Delta_{\\{i\\}}^{\\ell}.\n}\n\\]\nUnder logarithmic loss,\n\\[\n\\boxed{\n\\Delta_{\\mathrm{joint}}^{\\log}\n=\n\\min_{i\\in V}\nI\\!\\left(\nY_t^+;\\mathbf H_t\n\\mid\n\\mathbf H_t^{\\{i\\}}\n\\right).\n}\n\\]\nTherefore\n\\[\n\\Delta_{\\mathrm{joint}}^{\\log}>0\n\\]\nmeans that every individual component omits some predictive information\navailable to the complete system.","This quantity should not automatically be identified with a particular notion\nof partial-information synergy. A positive gap can arise from complementary\ninformation distributed across components, from higher-order interaction, or\nfrom both. The theory uses the operationally weaker and more directly\ntestable statement:\n\\[\n\\text{no individual node is sufficient for the declared target.}\n\\]","For \\(0\\leq r\\leq |V|\\), define the order-\\(r\\) coalition gap profile by\n\\[\n\\boxed{\n\\Delta^{(r)}_\\ell\n=\n\\min_{\\substack{A\\subseteq V","For \\(0\\leq r\\leq |V|\\), define the order-\\(r\\) coalition gap profile by\n\\[\n\\boxed{\n\\Delta^{(r)}_\\ell\n=\n\\min_{\\substack{A\\subseteq V\\\\|A|\\leq r}}\n\\left[\n\\Risk_\\ell(A)-\\Risk_\\ell(V)\n\\right].\n}\n\\]\nThe profile is nonincreasing in \\(r\\) and reaches zero by \\(r=|V|\\).\nIt records how rapidly operative capacity becomes reconstructible as larger\ncoalitions are permitted.","|A|\\leq r}}\n\\left[\n\\Risk_\\ell(A)-\\Risk_\\ell(V)\n\\right].\n}\n\\]\nThe profile is nonincreasing in \\(r\\) and reaches zero by \\(r=|V|\\).\nIt records how rapidly operative capacity becomes reconstructible as larger\ncoalitions are permitted.","For tolerance \\(\\varepsilon\\geq0\\), define\n\\[\n\\boxed{\nk_\\varepsilon\n=\n\\min\\left\\{\n|A|:\n\\Risk_\\ell(A)\n\\leq\n\\Risk_\\ell(V)+\\varepsilon\n\\right\\}.\n}\n\\]\nThe set of minimal coalitions is\n\\[\n\\mathcal K_\\varepsilon\n=\n\\left\\{\nA\\subseteq V:\n|A|=k_\\varepsilon,\n\\Risk_\\ell(A)\n\\leq\n\\Risk_\\ell(V)+\\varepsilon\n\\right\\}.\n\\]","\\begin{corollary}[Information form of coalition size]\nUnder the assumptions of Theorem~\\ref{thm:coalition-log-risk},\n\\[\n\\boxed{\nk_\\varepsilon\n=\n\\min\\left\\{\n|A|:\nI(Y_t^+;\\mathbf H_t\\mid\\mathbf H_t^A)\n\\leq\\varepsilon\n\\right\\}.\n}\n\\]\nEquivalently, \\(k_\\varepsilon\\) is the first \\(r\\) for which\n\\(\\Delta_{\\log}^{(r)}\\leq\\varepsilon\\).\n\\end{corollary}","This yields a graded classification:\n\\[\n\\begin{aligned}\nk_0=0\n&\\quad\\Longrightarrow\\quad\n\\text{the background law alone is sufficient},\\\\\nk_0=1\n&\\quad\\Longrightarrow\\quad\n\\text{some individual node is sufficient},\\\\\nk_0\\geq2\n&\\quad\\Longrightarrow\\quad\n\\text{node-level relational irreducibility},\\\\\nk_0=|V|\n&\\quad\\Longrightarrow\\quad\n\\text{every proper coalition is insufficient}.\n\\end{aligned}\n\\]\nThe last condition is much stronger and should not be inferred from the first.","Let \\(d_{\\mathcal P}\\) be a bounded metric generating the measurable structure\non \\(\\mathcal P(\\mathsf Y)\\). Define the reconstruction defect of the canonical\nstate from coalition \\(A\\) by\n\\[\n\\delta_A^Z\n=\n\\inf_{\\psi_A}\n\\mathbb E\\!\\left[\n d_{\\mathcal P}\\!\\left(\n Z_t,\\psi_A(\\mathbf H_t^A)\n \\right)\n\\right].\n\\]","\\begin{proposition}[Sufficiency--reconstruction equivalence]\n\\label{prop:sufficiency-reconstruction}\nFor a coalition \\(A\\subseteq V\\), the following are equivalent:\n\\begin{enumerate}\n\\item \\(\\mathbf H_t^A\\) is sufficient for \\(Y_t^+\\) relative to the full\n history;\n\\item \\(Z_t\\) is measurable with respect to \\(\\mathcal F_t^A\\);\n\\item there exists a measurable \\(\\psi_A\\) such that\n\\[\nZ_t=\\psi_A(\\mathbf H_t^A)\n\\qquad\\text{almost surely}.\n\\]\n\\end{enumerate}\nEach condition implies \\(\\delta_A^Z=0\\). Under finite-target logarithmic loss,\nthey are also equivalent to\n\\[\n\\Risk_{\\log}(A)=\\Risk_{\\log}(V).\n\\]\nIf the reconstruction infimum is known to be attained, then\n\\(\\delta_A^Z=0\\) is itself equivalent to the three conditions.\n\\end{proposition}","-reconstruction}\nFor a coalition \\(A\\subseteq V\\), the following are equivalent:\n\\begin{enumerate}\n\\item \\(\\mathbf H_t^A\\) is sufficient for \\(Y_t^+\\) relative to the full\n history;\n\\item \\(Z_t\\) is measurable with respect to \\(\\mathcal F_t^A\\);\n\\item there exists a measurable \\(\\psi_A\\) such that\n\\[\nZ_t=\\psi_A(\\mathbf H_t^A)\n\\qquad\\text{almost surely}.\n\\]\n\\end{enumerate}\nEach condition implies \\(\\delta_A^Z=0\\). Under finite-target logarithmic loss,\nthey are also equivalent to\n\\[\n\\Risk_{\\log}(A)=\\Risk_{\\log}(V).\n\\]\nIf the reconstruction infimum is known to be attained, then\n\\(\\delta_A^Z=0\\) is itself equivalent to the three conditions.\n\\end{proposition}","gin{enumerate}\n\\item \\(\\mathbf H_t^A\\) is sufficient for \\(Y_t^+\\) relative to the full\n history;\n\\item \\(Z_t\\) is measurable with respect to \\(\\mathcal F_t^A\\);\n\\item there exists a measurable \\(\\psi_A\\) such that\n\\[\nZ_t=\\psi_A(\\mathbf H_t^A)\n\\qquad\\text{almost surely}.\n\\]\n\\end{enumerate}\nEach condition implies \\(\\delta_A^Z=0\\). Under finite-target logarithmic loss,\nthey are also equivalent to\n\\[\n\\Risk_{\\log}(A)=\\Risk_{\\log}(V).\n\\]\nIf the reconstruction infimum is known to be attained, then\n\\(\\delta_A^Z=0\\) is itself equivalent to the three conditions.\n\\end{proposition}","item \\(Z_t\\) is measurable with respect to \\(\\mathcal F_t^A\\);\n\\item there exists a measurable \\(\\psi_A\\) such that\n\\[\nZ_t=\\psi_A(\\mathbf H_t^A)\n\\qquad\\text{almost surely}.\n\\]\n\\end{enumerate}\nEach condition implies \\(\\delta_A^Z=0\\). Under finite-target logarithmic loss,\nthey are also equivalent to\n\\[\n\\Risk_{\\log}(A)=\\Risk_{\\log}(V).\n\\]\nIf the reconstruction infimum is known to be attained, then\n\\(\\delta_A^Z=0\\) is itself equivalent to the three conditions.\n\\end{proposition}","For an arbitrary enriched workspace variable \\(W_t\\), local\nnon-reconstructibility need not imply predictive irreducibility: \\(W_t\\) may\ncontain task-irrelevant material. The equivalence is exact for the canonical\nstate \\(Z_t\\), or for a candidate \\(W_t\\) known to be minimally aligned with\nit.","Let \\(P\\) be the intact law of\n\\((Y_t^+,\\mathbf H_t^1,\\ldots,\\mathbf H_t^n)\\). Define the local-marginal null\nclass\n\\[\n\\mathcal N_{\\mathrm{marg}}(P)\n=\n\\left\\{\nQ:\nQ_{\\mathbf H_t^i}=P_{\\mathbf H_t^i}\n\\text{ for every }i\n\\right\\},\n\\]\nand the stronger local-predictive-matched class\n\\[\n\\boxed{\n\\mathcal N_{\\mathrm{pred}}(P)\n=\n\\left\\{\nQ:\nQ_{(Y_t^+,\\mathbf H_t^i)}\n=\nP_{(Y_t^+,\\mathbf H_t^i)}\n\\text{ for every }i\n\\right\\}.\n}\n\\]\nBecause \\(\\mathbf H_t^i\\) can include the complete local trajectory, the latter\nclass can preserve all single-component temporal and predictive behavior while\naltering cross-component organization.","((Y_t^+,\\mathbf H_t^1,\\ldots,\\mathbf H_t^n)\\). Define the local-marginal null\nclass\n\\[\n\\mathcal N_{\\mathrm{marg}}(P)\n=\n\\left\\{\nQ:\nQ_{\\mathbf H_t^i}=P_{\\mathbf H_t^i}\n\\text{ for every }i\n\\right\\},\n\\]\nand the stronger local-predictive-matched class\n\\[\n\\boxed{\n\\mathcal N_{\\mathrm{pred}}(P)\n=\n\\left\\{\nQ:\nQ_{(Y_t^+,\\mathbf H_t^i)}\n=\nP_{(Y_t^+,\\mathbf H_t^i)}\n\\text{ for every }i\n\\right\\}.\n}\n\\]\nBecause \\(\\mathbf H_t^i\\) can include the complete local trajectory, the latter\nclass can preserve all single-component temporal and predictive behavior while\naltering cross-component organization.","\\begin{proposition}[Local-risk invariance under predictive matching]\n\\label{prop:local-risk-null}\nIf \\(Q\\in\\mathcal N_{\\mathrm{pred}}(P)\\), then for every node \\(i\\) and every\nloss determined by \\((Y_t^+,d(\\mathbf H_t^i))\\),\n\\[\n\\boxed{\n\\Risk_{\\ell,Q}(\\{i\\})\n=\n\\Risk_{\\ell,P}(\\{i\\}).\n}\n\\]\n\\end{proposition}","\\begin{proof}\nEvery such risk depends only on the joint law of\n\\((Y_t^+,\\mathbf H_t^i)\\), which is identical under \\(P\\) and \\(Q\\).\n\\end{proof}","For a predeclared null \\(Q\\), define\n\\[\n\\boxed{\n\\Delta_{\\mathrm{tr}}(Q)\n=\n\\Risk_{\\ell,Q}(V)\n-\n\\Risk_{\\ell,P}(V).\n}\n\\]\nFor a finite validated panel\n\\(\\mathcal Q=\\{Q_1,\\ldots,Q_m\\}\\subseteq\\mathcal N_{\\mathrm{pred}}(P)\\), report\nboth the complete vector\n\\[\n\\boldsymbol\\Delta_{\\mathrm{tr}}(\\mathcal Q)\n=\n\\bigl(\n\\Delta_{\\mathrm{tr}}(Q_1),\\ldots,\n\\Delta_{\\mathrm{tr}}(Q_m)\n\\bigr)\n\\]\nand the robust panel minimum\n\\[\n\\Delta_{\\mathrm{tr}}^{\\min}(\\mathcal Q)\n=\n\\min_{Q\\in\\mathcal Q}\\Delta_{\\mathrm{tr}}(Q).\n\\]\nThe panel must be fixed independently of the observed favorable gap. Otherwise\nnull selection can manufacture apparent relational dependence.","declared null \\(Q\\), define\n\\[\n\\boxed{\n\\Delta_{\\mathrm{tr}}(Q)\n=\n\\Risk_{\\ell,Q}(V)\n-\n\\Risk_{\\ell,P}(V).\n}\n\\]\nFor a finite validated panel\n\\(\\mathcal Q=\\{Q_1,\\ldots,Q_m\\}\\subseteq\\mathcal N_{\\mathrm{pred}}(P)\\), report\nboth the complete vector\n\\[\n\\boldsymbol\\Delta_{\\mathrm{tr}}(\\mathcal Q)\n=\n\\bigl(\n\\Delta_{\\mathrm{tr}}(Q_1),\\ldots,\n\\Delta_{\\mathrm{tr}}(Q_m)\n\\bigr)\n\\]\nand the robust panel minimum\n\\[\n\\Delta_{\\mathrm{tr}}^{\\min}(\\mathcal Q)\n=\n\\min_{Q\\in\\mathcal Q}\\Delta_{\\mathrm{tr}}(Q).\n\\]\nThe panel must be fixed independently of the observed favorable gap. Otherwise\nnull selection can manufacture apparent relational dependence.","\\begin{definition}[Relational witness pair]\\label{def:relational-witness}\nA pair of laws \\((P,Q)\\) is a predictive relational witness pair for the\ndecomposition \\(V\\) and target \\(Y_t^+\\) when\n\\[\nQ\\in\\mathcal N_{\\mathrm{pred}}(P)\n\\]\nbut\n\\[\n\\Risk_{\\ell,Q}(V)\n\\neq\n\\Risk_{\\ell,P}(V).\n\\]\nThe two systems agree on every single-node predictive marginal but differ in\nfull-system operative capacity.\n\\end{definition}","\\begin{example}[XOR witness pair]\\label{ex:xor-witness}\nUnder the intact law \\(P\\), let\n\\[\nX_1,X_2\\overset{\\mathrm{iid}}{\\sim}\\operatorname{Bernoulli}(1/2),\n\\qquad\nY=X_1\\oplus X_2.\n\\]\nThen each \\((X_i,Y)\\) pair is independent and uniform, so\n\\[\nH_P(Y\\mid X_i)=1\\text{ bit},\n\\qquad\nH_P(Y\\mid X_1,X_2)=0.\n\\]\nLet \\(Q\\) make \\(X_1,X_2,Y\\) mutually independent fair bits. Then\n\\[\nQ_{(X_i,Y)}=P_{(X_i,Y)}\n\\qquad(i=1,2),\n\\]\nbut\n\\[\nH_Q(Y\\mid X_1,X_2)=1\\text{ bit}.\n\\]\nTherefore\n\\[\n\\Delta_{\\mathrm{joint}}^{\\log}=1\\text{ bit},\n\\qquad\nk_0=2,\n\\qquad\n\\Delta_{\\mathrm{tr}}(Q)=1\\text{ bit}.\n\\]\nThis proves predictive relational structure relative to the declared variables.\nIt does not, by itself, prove that any workspace variable is causally active.\n\\end{example}","ness pair]\\label{ex:xor-witness}\nUnder the intact law \\(P\\), let\n\\[\nX_1,X_2\\overset{\\mathrm{iid}}{\\sim}\\operatorname{Bernoulli}(1/2),\n\\qquad\nY=X_1\\oplus X_2.\n\\]\nThen each \\((X_i,Y)\\) pair is independent and uniform, so\n\\[\nH_P(Y\\mid X_i)=1\\text{ bit},\n\\qquad\nH_P(Y\\mid X_1,X_2)=0.\n\\]\nLet \\(Q\\) make \\(X_1,X_2,Y\\) mutually independent fair bits. Then\n\\[\nQ_{(X_i,Y)}=P_{(X_i,Y)}\n\\qquad(i=1,2),\n\\]\nbut\n\\[\nH_Q(Y\\mid X_1,X_2)=1\\text{ bit}.\n\\]\nTherefore\n\\[\n\\Delta_{\\mathrm{joint}}^{\\log}=1\\text{ bit},\n\\qquad\nk_0=2,\n\\qquad\n\\Delta_{\\mathrm{tr}}(Q)=1\\text{ bit}.\n\\]\nThis proves predictive relational structure relative to the declared variables.\nIt does not, by itself, prove that any workspace variable is causally active.\n\\end{example}","law \\(P\\), let\n\\[\nX_1,X_2\\overset{\\mathrm{iid}}{\\sim}\\operatorname{Bernoulli}(1/2),\n\\qquad\nY=X_1\\oplus X_2.\n\\]\nThen each \\((X_i,Y)\\) pair is independent and uniform, so\n\\[\nH_P(Y\\mid X_i)=1\\text{ bit},\n\\qquad\nH_P(Y\\mid X_1,X_2)=0.\n\\]\nLet \\(Q\\) make \\(X_1,X_2,Y\\) mutually independent fair bits. Then\n\\[\nQ_{(X_i,Y)}=P_{(X_i,Y)}\n\\qquad(i=1,2),\n\\]\nbut\n\\[\nH_Q(Y\\mid X_1,X_2)=1\\text{ bit}.\n\\]\nTherefore\n\\[\n\\Delta_{\\mathrm{joint}}^{\\log}=1\\text{ bit},\n\\qquad\nk_0=2,\n\\qquad\n\\Delta_{\\mathrm{tr}}(Q)=1\\text{ bit}.\n\\]\nThis proves predictive relational structure relative to the declared variables.\nIt does not, by itself, prove that any workspace variable is causally active.\n\\end{example}","\\begin{example}[Redundant-copy control]\nLet \\(Y\\) be a fair bit and set \\(X_1=X_2=Y\\). The two nodes are perfectly\ncoherent and the full system predicts perfectly, but\n\\[\nH(Y\\mid X_i)=0,\n\\qquad\n\\Delta_{\\mathrm{joint}}^{\\log}=0,\n\\qquad\nk_0=1.\n\\]\nDuplication is therefore not relational irreducibility.\n\\end{example}","A notation such as \\(\\operatorname{do}(W_t=w)\\) is meaningful only if the\ncandidate workspace is an endogenous variable in a structural causal model or\nif an intervention on underlying variables is shown to realize that value.\nAbstractly setting a projection coordinate need not correspond to any possible\nphysical or computational manipulation.","\\begin{definition}[Time-unrolled causal realization]\\label{def:causal-realization}\nA time-unrolled causal realization is a structural causal model\n\\[\n\\mathcal M\n=\n(\\mathcal U,\\mathcal V,F,\\mathbb P_U)\n\\]\nwhose endogenous variables include the local states, messages, actions,\nartifacts, target variables, and a measurable workspace extractor\n\\[\n\\pi_t:\\mathsf S_t\\to\\mathsf W_t,\n\\qquad\nW_t=\\pi_t(S_t).\n\\]\nThe unrolled graph is acyclic even when the rolled process is recurrent.\n\\end{definition}","\\begin{definition}[Feasible implementation intervention]\nLet \\(B_t\\) denote declared background variables that should remain matched.\nA family \\(\\{\\iota_w:w\\in\\mathsf W_t^\\circ\\}\\) is a feasible implementation of\nworkspace values on the experimental domain \\(\\mathsf W_t^\\circ\\) when:\n\\begin{enumerate}\n\\item each \\(\\iota_w\\) acts only on declared manipulable parents, channels, or\n artifacts at or after time \\(t\\);\n\\item the intervened realization satisfies\n\\[\n\\pi_t(S_t^{\\iota_w})=w\n\\qquad\\text{almost surely};\n\\]\n\\item the background law is matched across compared interventions; and\n\\item the intervention does not directly modify \\(Y_t^+\\) except through\n paths included in the declared causal design.\n\\end{enumerate}\n\\end{definition}","ition}[Feasible implementation intervention]\nLet \\(B_t\\) denote declared background variables that should remain matched.\nA family \\(\\{\\iota_w:w\\in\\mathsf W_t^\\circ\\}\\) is a feasible implementation of\nworkspace values on the experimental domain \\(\\mathsf W_t^\\circ\\) when:\n\\begin{enumerate}\n\\item each \\(\\iota_w\\) acts only on declared manipulable parents, channels, or\n artifacts at or after time \\(t\\);\n\\item the intervened realization satisfies\n\\[\n\\pi_t(S_t^{\\iota_w})=w\n\\qquad\\text{almost surely};\n\\]\n\\item the background law is matched across compared interventions; and\n\\item the intervention does not directly modify \\(Y_t^+\\) except through\n paths included in the declared causal design.\n\\end{enumerate}\n\\end{definition}","(B_t\\) denote declared background variables that should remain matched.\nA family \\(\\{\\iota_w:w\\in\\mathsf W_t^\\circ\\}\\) is a feasible implementation of\nworkspace values on the experimental domain \\(\\mathsf W_t^\\circ\\) when:\n\\begin{enumerate}\n\\item each \\(\\iota_w\\) acts only on declared manipulable parents, channels, or\n artifacts at or after time \\(t\\);\n\\item the intervened realization satisfies\n\\[\n\\pi_t(S_t^{\\iota_w})=w\n\\qquad\\text{almost surely};\n\\]\n\\item the background law is matched across compared interventions; and\n\\item the intervention does not directly modify \\(Y_t^+\\) except through\n paths included in the declared causal design.\n\\end{enumerate}\n\\end{definition}","xperimental domain \\(\\mathsf W_t^\\circ\\) when:\n\\begin{enumerate}\n\\item each \\(\\iota_w\\) acts only on declared manipulable parents, channels, or\n artifacts at or after time \\(t\\);\n\\item the intervened realization satisfies\n\\[\n\\pi_t(S_t^{\\iota_w})=w\n\\qquad\\text{almost surely};\n\\]\n\\item the background law is matched across compared interventions; and\n\\item the intervention does not directly modify \\(Y_t^+\\) except through\n paths included in the declared causal design.\n\\end{enumerate}\n\\end{definition}","Let \\(D\\) be a divergence or metric on probability laws. For a matched\nbackground value \\(b\\), define\n\\[\n\\CE_{\\Workspace}(w,w';b)\n=\nD\\!\\left(\n\\mathbb P^{\\iota_w}(Y_t^+\\in\\cdot\\mid B_t=b),\n\\mathbb P^{\\iota_{w'}}(Y_t^+\\in\\cdot\\mid B_t=b)\n\\right).\n\\]\nFor a declared comparison design \\(\\nu\\) over \\((w,w',b)\\), define\n\\[\n\\boxed{\n\\eta_{\\mathrm{causal}}^{\\nu}\n=\n\\mathbb E_{(w,w',b)\\sim\\nu}\n\\left[\n\\CE_{\\Workspace}(w,w';b)\n\\right].\n}\n\\]\nThe maximum\n\\[\n\\eta_{\\mathrm{causal}}^{\\max}\n=\n\\sup_{w,w',b}\\CE_{\\Workspace}(w,w';b)\n\\]\nmay be useful for discovery, but a design-weighted effect is harder to inflate\nby selecting a single exceptional pair after inspection.","ility laws. For a matched\nbackground value \\(b\\), define\n\\[\n\\CE_{\\Workspace}(w,w';b)\n=\nD\\!\\left(\n\\mathbb P^{\\iota_w}(Y_t^+\\in\\cdot\\mid B_t=b),\n\\mathbb P^{\\iota_{w'}}(Y_t^+\\in\\cdot\\mid B_t=b)\n\\right).\n\\]\nFor a declared comparison design \\(\\nu\\) over \\((w,w',b)\\), define\n\\[\n\\boxed{\n\\eta_{\\mathrm{causal}}^{\\nu}\n=\n\\mathbb E_{(w,w',b)\\sim\\nu}\n\\left[\n\\CE_{\\Workspace}(w,w';b)\n\\right].\n}\n\\]\nThe maximum\n\\[\n\\eta_{\\mathrm{causal}}^{\\max}\n=\n\\sup_{w,w',b}\\CE_{\\Workspace}(w,w';b)\n\\]\nmay be useful for discovery, but a design-weighted effect is harder to inflate\nby selecting a single exceptional pair after inspection.","\\begin{proposition}[Coordinate invariance of intervention effects]\n\\label{prop:causal-gauge}\nLet \\(g:\\mathsf W_t\\to\\mathsf W_t'\\) be a measurable bijection and suppose the\nreparametrized intervention family satisfies\n\\[\n\\iota_{g(w)}'=\\iota_w.\n\\]\nThen every divergence-based causal effect is preserved:\n\\[\n\\CE_{\\Workspace'}(g(w),g(w');b)\n=\n\\CE_{\\Workspace}(w,w';b).\n\\]\n\\end{proposition}","A causal effect on the target is not workspace-specific if an equal-cost\nperturbation outside the candidate workspace produces the same effect. Let\n\\(c(\\iota)\\) be a declared intervention cost and let\n\\[\n\\mathcal I_{\\Workspace}(r),\n\\qquad\n\\mathcal I_{\\perp}(r)\n\\]\nbe matched families with cost \\(r\\), the first changing workspace coordinates\nand the second changing complementary coordinates while preserving the\nworkspace value as closely as the implementation permits.","Define\n\\[\n\\eta_{\\Workspace}(r)\n=\n\\mathbb E_{\\iota,\\iota'\\sim\\mathcal I_{\\Workspace}(r)}\nD\\!\\left(\n\\mathbb P^{\\iota}(Y_t^+),\n\\mathbb P^{\\iota'}(Y_t^+)\n\\right),\n\\]\n\\[\n\\eta_{\\perp}(r)\n=\n\\mathbb E_{\\jmath,\\jmath'\\sim\\mathcal I_{\\perp}(r)}\nD\\!\\left(\n\\mathbb P^{\\jmath}(Y_t^+),\n\\mathbb P^{\\jmath'}(Y_t^+)\n\\right),\n\\]\nand\n\\[\n\\boxed{\n\\Delta_{\\mathrm{specific}}(r)\n=\n\\eta_{\\Workspace}(r)-\\eta_{\\perp}(r).\n}\n\\]\nA positive value supports causal concentration in the candidate workspace at\nthe declared perturbation scale.","If the complement is defined by \\(\\Gamma_\\varepsilon(H)^\\perp\\), the result is\nmetric dependent. Under nonunitary gauge changes, either the metric must be\ntransported with the coordinates or interventions must be matched by a\ncoordinate-independent physical cost. Otherwise specificity can change merely\nbecause the representation was rescaled.","Let \\(\\jmath_E\\) randomize selected communication edges while preserving their\nmarginal message laws, rates, and timing budget. The causal transfer gap is\n\\[\n\\boxed{\n\\Delta_{\\mathrm{tr}}^{\\operatorname{do}}(\\jmath_E)\n=\n\\Risk_{\\ell,\\mathbb P^{\\jmath_E}}(V)\n-\n\\Risk_{\\ell,\\mathbb P}(V).\n}\n\\]\nUnlike an observational null law, this quantity is anchored to an implemented\nchannel intervention. A positive effect is evidence that the intact alignment\nof messages, rather than merely their marginal content, contributes to task\nperformance.","A stronger mediation test equalizes the workspace state after two distinct\nchannel interventions. Let \\(\\mu\\) be a common imposed workspace distribution,\nimplemented by a feasible stochastic intervention in the sense of\nDefinition~\\ref{def:causal-realization}. The notation below is shorthand for\nthat implementation. Define the residual mediation defect\n\\[\n\\delta_{\\mathrm{med}}\n=\n\\sup_{\\jmath,\\jmath'}\nD\\!\\left(\n\\mathbb P^{\\jmath,\\operatorname{do}(W_t\\sim\\mu)}(Y_t^+),\n\\mathbb P^{\\jmath',\\operatorname{do}(W_t\\sim\\mu)}(Y_t^+)\n\\right).\n\\]\nA small value means that once the workspace state is equalized, little of the\nchannel intervention's effect remains. This supports, but does not by itself\nidentify without the causal assumptions, the claim that \\(W_t\\) mediates the\nrelevant relational influence.","t\n\\[\n\\delta_{\\mathrm{med}}\n=\n\\sup_{\\jmath,\\jmath'}\nD\\!\\left(\n\\mathbb P^{\\jmath,\\operatorname{do}(W_t\\sim\\mu)}(Y_t^+),\n\\mathbb P^{\\jmath',\\operatorname{do}(W_t\\sim\\mu)}(Y_t^+)\n\\right).\n\\]\nA small value means that once the workspace state is equalized, little of the\nchannel intervention's effect remains. This supports, but does not by itself\nidentify without the causal assumptions, the claim that \\(W_t\\) mediates the\nrelevant relational influence.","Temporal correlation is weaker than causal recurrence. Define the total\nworkspace recurrence effect at lag \\(\\tau\\) by\n\\[\n\\boxed{\n\\eta_{\\mathrm{rec}}(\\tau)\n=\n\\mathbb E_{(w,w',b)\\sim\\nu}\nD\\!\\left(\n\\mathbb P^{\\iota_w}(W_{t+\\tau}\\in\\cdot\\mid B_t=b),\n\\mathbb P^{\\iota_{w'}}(W_{t+\\tau}\\in\\cdot\\mid B_t=b)\n\\right).\n}\n\\]\nA nonzero value shows that manipulating the present workspace changes a later\nworkspace state. It may, however, be explained by direct copying or a retained\nmemory register.","To isolate reentry, consider the unrolled causal graph and remove every direct\nretention edge whose source and target are both workspace variables at\nsuccessive times. Let\n\\[\n\\mathcal M^{\\setminus\\mathrm{dir}}\n\\]\ndenote the resulting path-specific model, preserving paths that leave the\nworkspace through a nonworkspace participant, action, message, or artifact and\nlater return. Define\n\\[\n\\boxed{\n\\eta_{\\mathrm{loop}}(\\tau)\n=\n\\mathbb E_{(w,w',b)\\sim\\nu}\nD\\!\\left(\n\\mathbb P_{\\mathcal M^{\\setminus\\mathrm{dir}}}^{\\iota_w}\n(W_{t+\\tau}\\in\\cdot\\mid B_t=b),\n\\mathbb P_{\\mathcal M^{\\setminus\\mathrm{dir}}}^{\\iota_{w'}}\n(W_{t+\\tau}\\in\\cdot\\mid B_t=b)\n\\right).\n}\n\\]\nThen the distinctions are:\n\\[\n\\begin{aligned}\nI(W_t;W_{t+\\tau})>0\n&\\quad\\text{temporal dependence only},","orkspace through a nonworkspace participant, action, message, or artifact and\nlater return. Define\n\\[\n\\boxed{\n\\eta_{\\mathrm{loop}}(\\tau)\n=\n\\mathbb E_{(w,w',b)\\sim\\nu}\nD\\!\\left(\n\\mathbb P_{\\mathcal M^{\\setminus\\mathrm{dir}}}^{\\iota_w}\n(W_{t+\\tau}\\in\\cdot\\mid B_t=b),\n\\mathbb P_{\\mathcal M^{\\setminus\\mathrm{dir}}}^{\\iota_{w'}}\n(W_{t+\\tau}\\in\\cdot\\mid B_t=b)\n\\right).\n}\n\\]\nThen the distinctions are:\n\\[\n\\begin{aligned}\nI(W_t;W_{t+\\tau})>0\n&\\quad\\text{temporal dependence only},\\\\\n\\eta_{\\mathrm{rec}}(\\tau)>0\n&\\quad\\text{causal persistence},\\\\\n\\eta_{\\mathrm{loop}}(\\tau)>0\n&\\quad\\text{causal exit-and-return reentry}.\n\\end{aligned}\n\\]\nPath-specific effects require the usual consistency and no-unmeasured-confounding\nassumptions appropriate to the implementation. In software systems, edge\ninterventions can often be literal. In human collaboration, these assumptions\nare substantially harder to justify.","\\begin{theorem}[Operational relationality, \\(T_{13}\\)]\n\\label{thm:operational-relationality}\nLet \\(\\mathsf Y\\) be finite, let \\(P\\) be the intact law at a fixed decision\ntime, and let \\(W_t=\\pi_t(\\mathbf H_t)\\) be a structurally supported candidate.\nLet\n\\(\\mathcal Q\\subseteq\\mathcal N_{\\mathrm{pred}}(P)\\) be a nonempty,\npredeclared panel of validated relation-breaking nulls. Assume:\n\\begin{enumerate}\n\\item exact structural--operational alignment,\n\\[\nI_P(Y_t^+;\\mathbf H_t\\mid W_t)=0;\n\\]\n\\item a positive node-level information gap,\n\\[\n\\delta_{\\mathrm{node}}\n=\n\\min_{i\\in V}\nI_P(Y_t^+;\\mathbf H_t\\mid\\mathbf H_t^i)\n>0;\n\\]\n\\item a positive robust matched-null gap,\n\\[\n\\delta_{\\mathrm{null}}\n=\n\\min_{Q\\in\\mathcal Q}\n\\left[\nH_Q(Y_t^+\\mid\\mathbf H_t)\n-","\\begin{theorem}[Operational relationality, \\(T_{13}\\)]\n\\label{thm:operational-relationality}\nLet \\(\\mathsf Y\\) be finite, let \\(P\\) be the intact law at a fixed decision\ntime, and let \\(W_t=\\pi_t(\\mathbf H_t)\\) be a structurally supported candidate.\nLet\n\\(\\mathcal Q\\subseteq\\mathcal N_{\\mathrm{pred}}(P)\\) be a nonempty,\npredeclared panel of validated relation-breaking nulls. Assume:\n\\begin{enumerate}\n\\item exact structural--operational alignment,\n\\[\nI_P(Y_t^+;\\mathbf H_t\\mid W_t)=0;\n\\]\n\\item a positive node-level information gap,\n\\[\n\\delta_{\\mathrm{node}}\n=\n\\min_{i\\in V}\nI_P(Y_t^+;\\mathbf H_t\\mid\\mathbf H_t^i)\n>0;\n\\]\n\\item a positive robust matched-null gap,\n\\[\n\\delta_{\\mathrm{null}}\n=\n\\min_{Q\\in\\mathcal Q}\n\\left[\nH_Q(Y_t^+\\mid\\mathbf H_t)\n-\nH_P(Y_t^+\\mid\\mathbf H_t)\n\\right]\n>0;\n\\]\n\\item a feasibl","\\begin{theorem}[Operational relationality, \\(T_{13}\\)]\n\\label{thm:operational-relationality}\nLet \\(\\mathsf Y\\) be finite, let \\(P\\) be the intact law at a fixed decision\ntime, and let \\(W_t=\\pi_t(\\mathbf H_t)\\) be a structurally supported candidate.\nLet\n\\(\\mathcal Q\\subseteq\\mathcal N_{\\mathrm{pred}}(P)\\) be a nonempty,\npredeclared panel of validated relation-breaking nulls. Assume:\n\\begin{enumerate}\n\\item exact structural--operational alignment,\n\\[\nI_P(Y_t^+;\\mathbf H_t\\mid W_t)=0;\n\\]\n\\item a positive node-level information gap,\n\\[\n\\delta_{\\mathrm{node}}\n=\n\\min_{i\\in V}\nI_P(Y_t^+;\\mathbf H_t\\mid\\mathbf H_t^i)\n>0;\n\\]\n\\item a positive robust matched-null gap,\n\\[\n\\delta_{\\mathrm{null}}\n=\n\\min_{Q\\in\\mathcal Q}\n\\left[\nH_Q(Y_t^+\\mid\\mathbf H_t)\n-\nH_P(Y_t^+\\mid\\mathbf H_t)\n\\right]\n>0;\n\\]\n\\item a feasible intervention design","\\begin{theorem}[Operational relationality, \\(T_{13}\\)]\n\\label{thm:operational-relationality}\nLet \\(\\mathsf Y\\) be finite, let \\(P\\) be the intact law at a fixed decision\ntime, and let \\(W_t=\\pi_t(\\mathbf H_t)\\) be a structurally supported candidate.\nLet\n\\(\\mathcal Q\\subseteq\\mathcal N_{\\mathrm{pred}}(P)\\) be a nonempty,\npredeclared panel of validated relation-breaking nulls. Assume:\n\\begin{enumerate}\n\\item exact structural--operational alignment,\n\\[\nI_P(Y_t^+;\\mathbf H_t\\mid W_t)=0;\n\\]\n\\item a positive node-level information gap,\n\\[\n\\delta_{\\mathrm{node}}\n=\n\\min_{i\\in V}\nI_P(Y_t^+;\\mathbf H_t\\mid\\mathbf H_t^i)\n>0;\n\\]\n\\item a positive robust matched-null gap,\n\\[\n\\delta_{\\mathrm{null}}\n=\n\\min_{Q\\in\\mathcal Q}\n\\left[\nH_Q(Y_t^+\\mid\\mathbf H_t)\n-\nH_P(Y_t^+\\mid\\mathbf H_t)\n\\right]\n>0;\n\\]\n\\item a feasible intervention design with\n\\[\n\\eta_{\\mathrm{causal}}^{\\nu}>0;\n\\]\n\\item positive matched causal speci","\\begin{theorem}[Operational relationality, \\(T_{13}\\)]\n\\label{thm:operational-relationality}\nLet \\(\\mathsf Y\\) be finite, let \\(P\\) be the intact law at a fixed decision\ntime, and let \\(W_t=\\pi_t(\\mathbf H_t)\\) be a structurally supported candidate.\nLet\n\\(\\mathcal Q\\subseteq\\mathcal N_{\\mathrm{pred}}(P)\\) be a nonempty,\npredeclared panel of validated relation-breaking nulls. Assume:\n\\begin{enumerate}\n\\item exact structural--operational alignment,\n\\[\nI_P(Y_t^+;\\mathbf H_t\\mid W_t)=0;\n\\]\n\\item a positive node-level information gap,\n\\[\n\\delta_{\\mathrm{node}}\n=\n\\min_{i\\in V}\nI_P(Y_t^+;\\mathbf H_t\\mid\\mathbf H_t^i)\n>0;\n\\]\n\\item a positive robust matched-null gap,\n\\[\n\\delta_{\\mathrm{null}}\n=\n\\min_{Q\\in\\mathcal Q}\n\\left[\nH_Q(Y_t^+\\mid\\mathbf H_t)\n-\nH_P(Y_t^+\\mid\\mathbf H_t)\n\\right]\n>0;\n\\]\n\\item a feasible intervention design with\n\\[\n\\eta_{\\mathrm{causal}}^{\\nu}>0;\n\\]\n\\item positive matched causal specificity at some declared scale \\(r\\),\n\\[\n\\Delta_{\\mathrm{specific}}(r)>0.\n\\]\n\\end{enumerate}\nThen","]\n\\label{thm:operational-relationality}\nLet \\(\\mathsf Y\\) be finite, let \\(P\\) be the intact law at a fixed decision\ntime, and let \\(W_t=\\pi_t(\\mathbf H_t)\\) be a structurally supported candidate.\nLet\n\\(\\mathcal Q\\subseteq\\mathcal N_{\\mathrm{pred}}(P)\\) be a nonempty,\npredeclared panel of validated relation-breaking nulls. Assume:\n\\begin{enumerate}\n\\item exact structural--operational alignment,\n\\[\nI_P(Y_t^+;\\mathbf H_t\\mid W_t)=0;\n\\]\n\\item a positive node-level information gap,\n\\[\n\\delta_{\\mathrm{node}}\n=\n\\min_{i\\in V}\nI_P(Y_t^+;\\mathbf H_t\\mid\\mathbf H_t^i)\n>0;\n\\]\n\\item a positive robust matched-null gap,\n\\[\n\\delta_{\\mathrm{null}}\n=\n\\min_{Q\\in\\mathcal Q}\n\\left[\nH_Q(Y_t^+\\mid\\mathbf H_t)\n-\nH_P(Y_t^+\\mid\\mathbf H_t)\n\\right]\n>0;\n\\]\n\\item a feasible intervention design with\n\\[\n\\eta_{\\mathrm{causal}}^{\\nu}>0;\n\\]\n\\item positive matched causal specificity at some declared scale \\(r\\),\n\\[\n\\Delta_{\\mathrm{specific}}(r)>0.\n\\]\n\\end{enumerate}\nThen, relative to the declared decomposition, target, log loss, null panel, and\nintervention family:\n\\begin{enumerate}\n\\item no statistic of any single-n…","intact law at a fixed decision\ntime, and let \\(W_t=\\pi_t(\\mathbf H_t)\\) be a structurally supported candidate.\nLet\n\\(\\mathcal Q\\subseteq\\mathcal N_{\\mathrm{pred}}(P)\\) be a nonempty,\npredeclared panel of validated relation-breaking nulls. Assume:\n\\begin{enumerate}\n\\item exact structural--operational alignment,\n\\[\nI_P(Y_t^+;\\mathbf H_t\\mid W_t)=0;\n\\]\n\\item a positive node-level information gap,\n\\[\n\\delta_{\\mathrm{node}}\n=\n\\min_{i\\in V}\nI_P(Y_t^+;\\mathbf H_t\\mid\\mathbf H_t^i)\n>0;\n\\]\n\\item a positive robust matched-null gap,\n\\[\n\\delta_{\\mathrm{null}}\n=\n\\min_{Q\\in\\mathcal Q}\n\\left[\nH_Q(Y_t^+\\mid\\mathbf H_t)\n-\nH_P(Y_t^+\\mid\\mathbf H_t)\n\\right]\n>0;\n\\]\n\\item a feasible intervention design with\n\\[\n\\eta_{\\mathrm{causal}}^{\\nu}>0;\n\\]\n\\item positive matched causal specificity at some declared scale \\(r\\),\n\\[\n\\Delta_{\\mathrm{specific}}(r)>0.\n\\]\n\\end{enumerate}\nThen, relative to the declared decomposition, target, log loss, null panel, and\nintervention family:\n\\begin{enumerate}\n\\item no statistic of any single-node history is sufficient for \\(Y_t^+\\), and\n \\(k_0\\geq2\\);\n\\item \\(W_t\\) cannot be re…","bseteq\\mathcal N_{\\mathrm{pred}}(P)\\) be a nonempty,\npredeclared panel of validated relation-breaking nulls. Assume:\n\\begin{enumerate}\n\\item exact structural--operational alignment,\n\\[\nI_P(Y_t^+;\\mathbf H_t\\mid W_t)=0;\n\\]\n\\item a positive node-level information gap,\n\\[\n\\delta_{\\mathrm{node}}\n=\n\\min_{i\\in V}\nI_P(Y_t^+;\\mathbf H_t\\mid\\mathbf H_t^i)\n>0;\n\\]\n\\item a positive robust matched-null gap,\n\\[\n\\delta_{\\mathrm{null}}\n=\n\\min_{Q\\in\\mathcal Q}\n\\left[\nH_Q(Y_t^+\\mid\\mathbf H_t)\n-\nH_P(Y_t^+\\mid\\mathbf H_t)\n\\right]\n>0;\n\\]\n\\item a feasible intervention design with\n\\[\n\\eta_{\\mathrm{causal}}^{\\nu}>0;\n\\]\n\\item positive matched causal specificity at some declared scale \\(r\\),\n\\[\n\\Delta_{\\mathrm{specific}}(r)>0.\n\\]\n\\end{enumerate}\nThen, relative to the declared decomposition, target, log loss, null panel, and\nintervention family:\n\\begin{enumerate}\n\\item no statistic of any single-node history is sufficient for \\(Y_t^+\\), and\n \\(k_0\\geq2\\);\n\\item \\(W_t\\) cannot be reconstructed almost surely from any single-node\n history;\n\\item every null \\(Q\\in\\mathcal Q\\) preserves all single-node predictive\n…","al alignment,\n\\[\nI_P(Y_t^+;\\mathbf H_t\\mid W_t)=0;\n\\]\n\\item a positive node-level information gap,\n\\[\n\\delta_{\\mathrm{node}}\n=\n\\min_{i\\in V}\nI_P(Y_t^+;\\mathbf H_t\\mid\\mathbf H_t^i)\n>0;\n\\]\n\\item a positive robust matched-null gap,\n\\[\n\\delta_{\\mathrm{null}}\n=\n\\min_{Q\\in\\mathcal Q}\n\\left[\nH_Q(Y_t^+\\mid\\mathbf H_t)\n-\nH_P(Y_t^+\\mid\\mathbf H_t)\n\\right]\n>0;\n\\]\n\\item a feasible intervention design with\n\\[\n\\eta_{\\mathrm{causal}}^{\\nu}>0;\n\\]\n\\item positive matched causal specificity at some declared scale \\(r\\),\n\\[\n\\Delta_{\\mathrm{specific}}(r)>0.\n\\]\n\\end{enumerate}\nThen, relative to the declared decomposition, target, log loss, null panel, and\nintervention family:\n\\begin{enumerate}\n\\item no statistic of any single-node history is sufficient for \\(Y_t^+\\), and\n \\(k_0\\geq2\\);\n\\item \\(W_t\\) cannot be reconstructed almost surely from any single-node\n history;\n\\item every null \\(Q\\in\\mathcal Q\\) preserves all single-node predictive\n risks but has strictly worse full-system log risk than the intact law;\n\\item the candidate workspace is causally efficacious and its effect exceeds\n that of the declar…","delta_{\\mathrm{node}}\n=\n\\min_{i\\in V}\nI_P(Y_t^+;\\mathbf H_t\\mid\\mathbf H_t^i)\n>0;\n\\]\n\\item a positive robust matched-null gap,\n\\[\n\\delta_{\\mathrm{null}}\n=\n\\min_{Q\\in\\mathcal Q}\n\\left[\nH_Q(Y_t^+\\mid\\mathbf H_t)\n-\nH_P(Y_t^+\\mid\\mathbf H_t)\n\\right]\n>0;\n\\]\n\\item a feasible intervention design with\n\\[\n\\eta_{\\mathrm{causal}}^{\\nu}>0;\n\\]\n\\item positive matched causal specificity at some declared scale \\(r\\),\n\\[\n\\Delta_{\\mathrm{specific}}(r)>0.\n\\]\n\\end{enumerate}\nThen, relative to the declared decomposition, target, log loss, null panel, and\nintervention family:\n\\begin{enumerate}\n\\item no statistic of any single-node history is sufficient for \\(Y_t^+\\), and\n \\(k_0\\geq2\\);\n\\item \\(W_t\\) cannot be reconstructed almost surely from any single-node\n history;\n\\item every null \\(Q\\in\\mathcal Q\\) preserves all single-node predictive\n risks but has strictly worse full-system log risk than the intact law;\n\\item the candidate workspace is causally efficacious and its effect exceeds\n that of the declared matched off-workspace perturbations.\n\\end{enumerate}\nIf, in addition, \\(\\eta_{\\mathrm{loop}}(\\tau)>0…","\\mathrm{node}}\n=\n\\min_{i\\in V}\nI_P(Y_t^+;\\mathbf H_t\\mid\\mathbf H_t^i)\n>0;\n\\]\n\\item a positive robust matched-null gap,\n\\[\n\\delta_{\\mathrm{null}}\n=\n\\min_{Q\\in\\mathcal Q}\n\\left[\nH_Q(Y_t^+\\mid\\mathbf H_t)\n-\nH_P(Y_t^+\\mid\\mathbf H_t)\n\\right]\n>0;\n\\]\n\\item a feasible intervention design with\n\\[\n\\eta_{\\mathrm{causal}}^{\\nu}>0;\n\\]\n\\item positive matched causal specificity at some declared scale \\(r\\),\n\\[\n\\Delta_{\\mathrm{specific}}(r)>0.\n\\]\n\\end{enumerate}\nThen, relative to the declared decomposition, target, log loss, null panel, and\nintervention family:\n\\begin{enumerate}\n\\item no statistic of any single-node history is sufficient for \\(Y_t^+\\), and\n \\(k_0\\geq2\\);\n\\item \\(W_t\\) cannot be reconstructed almost surely from any single-node\n history;\n\\item every null \\(Q\\in\\mathcal Q\\) preserves all single-node predictive\n risks but has strictly worse full-system log risk than the intact law;\n\\item the candidate workspace is causally efficacious and its effect exceeds\n that of the declared matched off-workspace perturbations.\n\\end{enumerate}\nIf, in addition, \\(\\eta_{\\mathrm{loop}}(\\tau)>0\\), the…",">0;\n\\]\n\\item a feasible intervention design with\n\\[\n\\eta_{\\mathrm{causal}}^{\\nu}>0;\n\\]\n\\item positive matched causal specificity at some declared scale \\(r\\),\n\\[\n\\Delta_{\\mathrm{specific}}(r)>0.\n\\]\n\\end{enumerate}\nThen, relative to the declared decomposition, target, log loss, null panel, and\nintervention family:\n\\begin{enumerate}\n\\item no statistic of any single-node history is sufficient for \\(Y_t^+\\), and\n \\(k_0\\geq2\\);\n\\item \\(W_t\\) cannot be reconstructed almost surely from any single-node\n history;\n\\item every null \\(Q\\in\\mathcal Q\\) preserves all single-node predictive\n risks but has strictly worse full-system log risk than the intact law;\n\\item the candidate workspace is causally efficacious and its effect exceeds\n that of the declared matched off-workspace perturbations.\n\\end{enumerate}\nIf, in addition, \\(\\eta_{\\mathrm{loop}}(\\tau)>0\\), the workspace is\nloop-reentrant at lag \\(\\tau\\).\n\\end{theorem}","feasible intervention design with\n\\[\n\\eta_{\\mathrm{causal}}^{\\nu}>0;\n\\]\n\\item positive matched causal specificity at some declared scale \\(r\\),\n\\[\n\\Delta_{\\mathrm{specific}}(r)>0.\n\\]\n\\end{enumerate}\nThen, relative to the declared decomposition, target, log loss, null panel, and\nintervention family:\n\\begin{enumerate}\n\\item no statistic of any single-node history is sufficient for \\(Y_t^+\\), and\n \\(k_0\\geq2\\);\n\\item \\(W_t\\) cannot be reconstructed almost surely from any single-node\n history;\n\\item every null \\(Q\\in\\mathcal Q\\) preserves all single-node predictive\n risks but has strictly worse full-system log risk than the intact law;\n\\item the candidate workspace is causally efficacious and its effect exceeds\n that of the declared matched off-workspace perturbations.\n\\end{enumerate}\nIf, in addition, \\(\\eta_{\\mathrm{loop}}(\\tau)>0\\), the workspace is\nloop-reentrant at lag \\(\\tau\\).\n\\end{theorem}","n design with\n\\[\n\\eta_{\\mathrm{causal}}^{\\nu}>0;\n\\]\n\\item positive matched causal specificity at some declared scale \\(r\\),\n\\[\n\\Delta_{\\mathrm{specific}}(r)>0.\n\\]\n\\end{enumerate}\nThen, relative to the declared decomposition, target, log loss, null panel, and\nintervention family:\n\\begin{enumerate}\n\\item no statistic of any single-node history is sufficient for \\(Y_t^+\\), and\n \\(k_0\\geq2\\);\n\\item \\(W_t\\) cannot be reconstructed almost surely from any single-node\n history;\n\\item every null \\(Q\\in\\mathcal Q\\) preserves all single-node predictive\n risks but has strictly worse full-system log risk than the intact law;\n\\item the candidate workspace is causally efficacious and its effect exceeds\n that of the declared matched off-workspace perturbations.\n\\end{enumerate}\nIf, in addition, \\(\\eta_{\\mathrm{loop}}(\\tau)>0\\), the workspace is\nloop-reentrant at lag \\(\\tau\\).\n\\end{theorem}","at some declared scale \\(r\\),\n\\[\n\\Delta_{\\mathrm{specific}}(r)>0.\n\\]\n\\end{enumerate}\nThen, relative to the declared decomposition, target, log loss, null panel, and\nintervention family:\n\\begin{enumerate}\n\\item no statistic of any single-node history is sufficient for \\(Y_t^+\\), and\n \\(k_0\\geq2\\);\n\\item \\(W_t\\) cannot be reconstructed almost surely from any single-node\n history;\n\\item every null \\(Q\\in\\mathcal Q\\) preserves all single-node predictive\n risks but has strictly worse full-system log risk than the intact law;\n\\item the candidate workspace is causally efficacious and its effect exceeds\n that of the declared matched off-workspace perturbations.\n\\end{enumerate}\nIf, in addition, \\(\\eta_{\\mathrm{loop}}(\\tau)>0\\), the workspace is\nloop-reentrant at lag \\(\\tau\\).\n\\end{theorem}","\\item \\(W_t\\) cannot be reconstructed almost surely from any single-node\n history;\n\\item every null \\(Q\\in\\mathcal Q\\) preserves all single-node predictive\n risks but has strictly worse full-system log risk than the intact law;\n\\item the candidate workspace is causally efficacious and its effect exceeds\n that of the declared matched off-workspace perturbations.\n\\end{enumerate}\nIf, in addition, \\(\\eta_{\\mathrm{loop}}(\\tau)>0\\), the workspace is\nloop-reentrant at lag \\(\\tau\\).\n\\end{theorem}","history;\n\\item every null \\(Q\\in\\mathcal Q\\) preserves all single-node predictive\n risks but has strictly worse full-system log risk than the intact law;\n\\item the candidate workspace is causally efficacious and its effect exceeds\n that of the declared matched off-workspace perturbations.\n\\end{enumerate}\nIf, in addition, \\(\\eta_{\\mathrm{loop}}(\\tau)>0\\), the workspace is\nloop-reentrant at lag \\(\\tau\\).\n\\end{theorem}","\\begin{proof}\nBy Theorem~\\ref{thm:coalition-log-risk}, for each node \\(i\\),\n\\[\n\\Risk_{\\log,P}(\\{i\\})-\n\\Risk_{\\log,P}(V)\n=\nI_P(Y_t^+;\\mathbf H_t\\mid\\mathbf H_t^i)\n\\geq\n\\delta_{\\mathrm{node}}>0.\n\\]\nHence no single-node history, and therefore no statistic of one, is sufficient;\nthis gives \\(k_0\\geq2\\).","Exact alignment and Theorem~\\ref{thm:alignment-criterion} imply that the\ncanonical predictive state factors through \\(W_t\\). If\n\\(W_t=\\psi_i(\\mathbf H_t^i)\\) almost surely for some node, then the canonical\nstate would also be a function of \\(\\mathbf H_t^i\\), contradicting the positive\nnode gap through Proposition~\\ref{prop:sufficiency-reconstruction}.","For every \\(Q\\in\\mathcal Q\\), Proposition~\\ref{prop:local-risk-null} gives\n\\[\n\\Risk_{\\log,Q}(\\{i\\})\n=\n\\Risk_{\\log,P}(\\{i\\})\n\\qquad\\forall i.\n\\]\nThe assumed null gap gives\n\\[\n\\Risk_{\\log,Q}(V)\n>\n\\Risk_{\\log,P}(V).\n\\]\nThus a description determined only by the collection of single-node predictive\nmarginals cannot reproduce the intact joint capacity: it would assign the same\nlocal data to \\(P\\) and \\(Q\\) while their full-system risks differ.","Finally, the feasible intervention effect establishes causal efficacy relative\nto the declared implementation, and the positive specificity gap shows that\nthe effect exceeds the matched complementary perturbation effect. The final\nreentry statement is exactly the definition of \\(\\eta_{\\mathrm{loop}}\\).\n\\end{proof}","\\begin{remark}[Scope of the theorem]\nThe conclusion is operational and indexed by\n\\[\n(V,Y_t^+,\\ell,\\mathcal Q,\\mathcal I,\\nu,r).\n\\]\nChanging the decomposition, target, loss, nulls, or intervention design may\nchange the verdict. The theorem does not establish metaphysical\nirreducibility, autonomous agency, selfhood, or consciousness.\n\\end{remark}","A reproducible dynamic-causal analysis should proceed in the following order:\n\\begin{enumerate}\n\\item Freeze the node decomposition, accessible coalition histories, target,\n horizon, and proper scoring loss.\n\\item Estimate the structural projector on training data only and encode\n \\(W_t\\) without using held-out targets.\n\\item Cross-fit conditional target laws from the full history, the workspace,\n and every declared coalition. Report alignment and coalition risk gaps.\n\\item Construct the null panel before inspecting null performance. Verify\n local-predictive matching and the intended destruction of transfer\n organization separately.\n\\item Implement feasible workspace and complementary interventions with a\n common background and cost design. Report both causal effect and\n specificity.\n\\item Test recurrence and then cut direct retention paths to test loop-mediated\n reentry.\n\\item Report confidence intervals or finite-sample uncertainty for every gate;\n do not substitute trai","The resulting certification report is\n\\[\n\\boxed{\n\\begin{aligned}\n\\mathcal C_{\\mathrm{dyn}}(W_t)\n=\\bigl(&\n\\delta_{\\mathrm{align}}^{\\mathcal I},\n\\{\\Delta_A^\\ell\\}_{A\\subseteq V},\n k_\\varepsilon,\n\\boldsymbol\\Delta_{\\mathrm{tr}}(\\mathcal Q),\\\\\n&\\eta_{\\mathrm{causal}}^\\nu,\n\\Delta_{\\mathrm{specific}}(r),\n\\eta_{\\mathrm{rec}}(\\tau),\n\\eta_{\\mathrm{loop}}(\\tau)\n\\bigr).\n\\end{aligned}\n}\n\\]\nNo single coordinate certifies a relational workspace.","Let\n\\[\n\\Theta\n=\n\\left(\n\\varepsilon_{\\mathrm{coh}},\n\\varepsilon_{\\mathrm{align}},\n\\varepsilon_k,\n\\underline\\eta_c,\n\\underline\\eta_{\\mathrm{loop}},\n\\delta_{\\mathrm{node}},\n\\delta_{\\mathrm{null}},\n\\delta_{\\mathrm{spec}},\nk,\n\\varepsilon_{\\mathrm{pers}}\n\\right).\n\\]\nLet\n\\[\n\\varsigma\n=\n(\\mathcal D,Y^+,\\ell,\\tau,\\mathcal I,\\mathcal Q,\\Theta)\n\\]\nbe the complete declared scientific specification, where\n\\(\\mathcal Q\\subseteq\\mathcal N_{\\mathrm{pred}}(P)\\) is a validated null panel.\nA dynamic realization contains a certified stratified relational workspace\nwhen there exists a candidate \\(W_t\\) satisfying all gates required for the\nclaimed level under \\(\\varsigma\\). The detailed robustness and identifiability\nrequirements for varying \\(\\varsigma\\) are developed in Parts~XVII and XVIII.","rm{null}},\n\\delta_{\\mathrm{spec}},\nk,\n\\varepsilon_{\\mathrm{pers}}\n\\right).\n\\]\nLet\n\\[\n\\varsigma\n=\n(\\mathcal D,Y^+,\\ell,\\tau,\\mathcal I,\\mathcal Q,\\Theta)\n\\]\nbe the complete declared scientific specification, where\n\\(\\mathcal Q\\subseteq\\mathcal N_{\\mathrm{pred}}(P)\\) is a validated null panel.\nA dynamic realization contains a certified stratified relational workspace\nwhen there exists a candidate \\(W_t\\) satisfying all gates required for the\nclaimed level under \\(\\varsigma\\). The detailed robustness and identifiability\nrequirements for varying \\(\\varsigma\\) are developed in Parts~XVII and XVIII.","\\right).\n\\]\nLet\n\\[\n\\varsigma\n=\n(\\mathcal D,Y^+,\\ell,\\tau,\\mathcal I,\\mathcal Q,\\Theta)\n\\]\nbe the complete declared scientific specification, where\n\\(\\mathcal Q\\subseteq\\mathcal N_{\\mathrm{pred}}(P)\\) is a validated null panel.\nA dynamic realization contains a certified stratified relational workspace\nwhen there exists a candidate \\(W_t\\) satisfying all gates required for the\nclaimed level under \\(\\varsigma\\). The detailed robustness and identifiability\nrequirements for varying \\(\\varsigma\\) are developed in Parts~XVII and XVIII.","a)\n\\]\nbe the complete declared scientific specification, where\n\\(\\mathcal Q\\subseteq\\mathcal N_{\\mathrm{pred}}(P)\\) is a validated null panel.\nA dynamic realization contains a certified stratified relational workspace\nwhen there exists a candidate \\(W_t\\) satisfying all gates required for the\nclaimed level under \\(\\varsigma\\). The detailed robustness and identifiability\nrequirements for varying \\(\\varsigma\\) are developed in Parts~XVII and XVIII.","\\subtopic{Structural coherence}\n\\[\n\\delta_{\\mathrm{coh}}(W_t)\n\\leq\n\\varepsilon_{\\mathrm{coh}}.\n\\]","\\subtopic{Interventional structural--operational alignment}\n\\[\n\\delta_{\\mathrm{align}}^{\\mathcal I}(W_t)\n\\leq\n\\varepsilon_{\\mathrm{align}}.\n\\]","\\subtopic{Causal efficacy}\n\\[\n\\eta_{\\mathrm{causal}}^\\nu\n\\geq\n\\underline\\eta_c.\n\\]","\\subtopic{Loop-mediated reentry, when claimed}\n\\[\n\\eta_{\\mathrm{loop}}(\\tau)\n\\geq\n\\underline\\eta_{\\mathrm{loop}}.\n\\]","\\subtopic{Node-level joint advantage}\n\\[\n\\Delta_{\\mathrm{joint}}^{\\ell}\n\\geq\n\\delta_{\\mathrm{node}}.\n\\]","\\subtopic{Matched-null transfer dependence}\n\\[\n\\Delta_{\\mathrm{tr}}^{\\min}(\\mathcal Q)\n\\geq\n\\delta_{\\mathrm{null}}.\n\\]","\\subtopic{Causal specificity}\n\\[\n\\Delta_{\\mathrm{specific}}(r)\n\\geq\n\\delta_{\\mathrm{spec}}.\n\\]","\\subtopic{Distributedness}\n\\[\nk_{\\varepsilon_k}\n\\geq\nk.\n\\]","\\subtopic{Persistence, when claimed}\n\\[\nd_{\\mathrm{id}}(W_t,W_{t+\\tau})\n\\leq\n\\varepsilon_{\\mathrm{pers}}.\n\\]","Symbolically,\n\\[\n\\boxed{\n\\SRW(W_t;\\varsigma)\n=\n\\bigwedge_{q\\in\\mathcal J(\\varsigma)}\n\\operatorname{Gate}_q(W_t;\\varsigma).\n}\n\\]\nThe gate set is claim-relative. A one-step operational workspace need not pass\npersistence or reentry gates, while a persistent reentrant workspace must.\nNo scalar total score is sufficient: failure of a required gate cannot be\nhidden by high performance on another gate.","\\begin{remark}[Structural, predictive, causal, and temporal claims]\nThe formal hierarchy now distinguishes four logically independent questions:\n\\[\n\\begin{aligned}\n&\\text{Does a stable shared coordinate exist?}\\\\\n&\\text{Does it retain the task-relevant conditional law?}\\\\\n&\\text{Does manipulating it change downstream behavior specifically?}\\\\\n&\\text{Does it persist or reenter through time?}\n\\end{aligned}\n\\]\nThe preceding structural and causal analysis answers the first three at the level of definitions and conditional\ntheorems. The following parts develop the fourth.\n\\end{remark}","At time \\(t\\), let\n\\[\n\\mathcal C_V(t)=\\bigoplus_{i\\in V}C_i(t),\n\\qquad\n\\mathcal C_E(t)=\\bigoplus_{e\\in E}C_{t(e)}(t),\n\\]\nand let\n\\[\nD_t:\\mathcal C_V(t)\\longrightarrow\\mathcal C_E(t),\n\\qquad\nL_t=D_t^*D_t\n\\]\nbe the transfer discrepancy and transfer Laplacian. At time \\(t\\), the\ncombinatorial graph is held fixed while the local representations, transfers,\nand metrics may vary. Changes of graph are treated later by a graph\ncorrespondence.","To compare successive times, choose unitary identification maps\n\\[\nU_t^V:\\mathcal C_V(t)\\longrightarrow\\mathcal C_V(t+1),\n\\qquad\nU_t^E:\\mathcal C_E(t)\\longrightarrow\\mathcal C_E(t+1).\n\\]\nThey are comparison maps, not assumptions that the physical carriers are\nidentical. Define the one-step diagram defect\n\\[\n\\boxed{\n\\Delta_t^D\n=\nD_{t+1}U_t^V-U_t^ED_t,\n\\qquad\n\\delta_t^D=\\|\\Delta_t^D\\|.\n}\n\\]\nExact temporal compatibility is the condition \\(\\Delta_t^D=0\\).","ary identification maps\n\\[\nU_t^V:\\mathcal C_V(t)\\longrightarrow\\mathcal C_V(t+1),\n\\qquad\nU_t^E:\\mathcal C_E(t)\\longrightarrow\\mathcal C_E(t+1).\n\\]\nThey are comparison maps, not assumptions that the physical carriers are\nidentical. Define the one-step diagram defect\n\\[\n\\boxed{\n\\Delta_t^D\n=\nD_{t+1}U_t^V-U_t^ED_t,\n\\qquad\n\\delta_t^D=\\|\\Delta_t^D\\|.\n}\n\\]\nExact temporal compatibility is the condition \\(\\Delta_t^D=0\\).","\\begin{proposition}[Temporal Laplacian drift]\\label{prop:temporal-laplacian-drift}\nThe carried Laplacian satisfies\n\\[\n\\boxed{\n\\left\\|\n(U_t^V)^*L_{t+1}U_t^V-L_t\n\\right\\|\n\\leq\n2\\|D_t\\|\\delta_t^D+(\\delta_t^D)^2.\n}\n\\]\n\\end{proposition}","\\begin{proof}\nSet\n\\[\nB_t=(U_t^E)^*\\Delta_t^D.\n\\]\nBecause the comparison maps are unitary,\n\\[\n(U_t^E)^*D_{t+1}U_t^V=D_t+B_t,\n\\qquad\n\\|B_t\\|=\\delta_t^D.\n\\]\nHence\n\\[\n(U_t^V)^*L_{t+1}U_t^V\n=(D_t+B_t)^*(D_t+B_t).\n\\]\nSubtract \\(D_t^*D_t\\) and apply the triangle inequality.\n\\end{proof}","Let \\(P_t\\) project onto a selected rank-\\(d\\) low-energy workspace cluster\nof \\(L_t\\), and set\n\\[\n\\mathcal W_t=\\ran P_t.\n\\]\nWrite\n\\[\ng_t=\\lambda_{d+1}(L_t)-\\lambda_d(L_t)>0\n\\]\nfor the separating spectral gap and\n\\[\n\\eta_t^L=2\\|D_t\\|\\delta_t^D+(\\delta_t^D)^2.\n\\]","\\begin{corollary}[Temporal workspace tracking]\\label{cor:temporal-workspace-tracking}\nIf\n\\[\n\\eta_t^L<\\frac{g_t}{2},\n\\]\nthen the transported and newly extracted workspace projectors satisfy\n\\[\n\\boxed{\nq_t^{\\mathrm{str}}\n:=\n\\left\\|\nP_{t+1}-U_t^VP_t(U_t^V)^*\n\\right\\|\n\\leq\n\\frac{\\eta_t^L}{g_t-\\eta_t^L}\n\\leq\n\\frac{2\\eta_t^L}{g_t}.\n}\n\\]\n\\end{corollary}","\\begin{proof}\nCarry \\(L_{t+1}\\) back to time \\(t\\) using \\(U_t^V\\), apply\nTheorem~\\ref{thm:spectral-stability}, and conjugate the resulting projector\nbound forward.\n\\end{proof}","The quantity \\(q_t^{\\mathrm{str}}\\) measures structural motion after the\ndeclared coordinate change has been removed. A workspace may therefore move\nsubstantially in raw coordinates while remaining structurally unchanged.","Let the realized vertex state satisfy\n\\[\nx_{t+1}=F_tx_t+\\xi_t,\n\\]\nwhere \\(F_t:\\mathcal C_V(t)\\to\\mathcal C_V(t+1)\\) is the actual update and\n\\(\\xi_t\\) is an unmodeled disturbance. Define the off-workspace error\n\\[\n\\boxed{\ne_t=\\|(I-P_t)x_t\\|.\n}\n\\]\nIntroduce\n\\[\n\\alpha_t\n=\n\\left\\|\n(I-P_{t+1})F_t(I-P_t)\n\\right\\|,\n\\]\nwhich measures persistence of prior off-workspace error, and\n\\[\n\\ell_t\n=\n\\left\\|\n(I-P_{t+1})F_tP_t\n\\right\\|,\n\\]\nwhich measures leakage of a valid workspace state under the update.","\\begin{proposition}[Grounded coherence recurrence]\\label{prop:grounded-coherence-recurrence}\nEvery trajectory satisfies\n\\[\n\\boxed{\ne_{t+1}\\leq\\alpha_te_t+\\sigma_t,}\n\\]\nwhere\n\\[\n\\boxed{\n\\sigma_t\n=\n\\ell_t\\|P_tx_t\\|\n+\n\\|(I-P_{t+1})\\xi_t\\|.\n}\n\\]\n\\end{proposition}","\\begin{proof}\nInsert \\(I=P_t+(I-P_t)\\) before \\(x_t\\), apply \\(I-P_{t+1}\\), and use\nsubmultiplicativity:\n\\[\n\\begin{aligned}\ne_{t+1}\n&\\leq\n\\|(I-P_{t+1})F_t(I-P_t)x_t\\|","\\begin{proof}\nInsert \\(I=P_t+(I-P_t)\\) before \\(x_t\\), apply \\(I-P_{t+1}\\), and use\nsubmultiplicativity:\n\\[\n\\begin{aligned}\ne_{t+1}\n&\\leq\n\\|(I-P_{t+1})F_t(I-P_t)x_t\\|\\\\\n&\\quad+\n\\|(I-P_{t+1})F_tP_tx_t\\|\n+\n\\|(I-P_{t+1})\\xi_t\\|.\n\\end{aligned}\n\\]\nThe displayed recurrence follows from the definitions of \\(\\alpha_t\\) and\n\\(\\ell_t\\).\n\\end{proof}","&\\quad+\n\\|(I-P_{t+1})F_tP_tx_t\\|\n+\n\\|(I-P_{t+1})\\xi_t\\|.\n\\end{aligned}\n\\]\nThe displayed recurrence follows from the definitions of \\(\\alpha_t\\) and\n\\(\\ell_t\\).\n\\end{proof}","\\subtopic{32.3 Persistence theorem \\(T_6\\)}","\\begin{theorem}[Nonautonomous persistence theorem]\\label{thm:persistence}\nLet \\(e_t\\geq0\\) satisfy\n\\[\ne_{t+1}\\leq\\alpha_te_t+\\sigma_t,\n\\qquad\n\\alpha_t,\\sigma_t\\geq0.\n\\]\nThen, for every \\(t\\geq1\\),\n\\[\n\\boxed{\n\\begin{aligned}\ne_t\n\\leq\n&\\left(\\prod_{r=0}^{t-1}\\alpha_r\\right)e_0","\\begin{theorem}[Nonautonomous persistence theorem]\\label{thm:persistence}\nLet \\(e_t\\geq0\\) satisfy\n\\[\ne_{t+1}\\leq\\alpha_te_t+\\sigma_t,\n\\qquad\n\\alpha_t,\\sigma_t\\geq0.\n\\]\nThen, for every \\(t\\geq1\\),\n\\[\n\\boxed{\n\\begin{aligned}\ne_t\n\\leq\n&\\left(\\prod_{r=0}^{t-1}\\alpha_r\\right)e_0\\\\\n&+\n\\sum_{j=0}^{t-1}\n\\left(\\prod_{r=j+1}^{t-1}\\alpha_r\\right)\\sigma_j,\n\\end{aligned}\n}\n\\]\nwhere an empty product equals one. If\n\\[\n\\alpha_t\\leq\\bar\\alpha<1,\n\\qquad\n\\sigma_t\\leq\\bar\\sigma,\n\\]\nthen\n\\[\n\\boxed{\ne_t\n\\leq\n\\bar\\alpha^te_0\n+\n\\frac{\\bar\\sigma(1-\\bar\\alpha^t)}{1-\\bar\\alpha},\n}\n\\]\nand consequently\n\\[\n\\boxed{\n\\limsup_{t\\to\\infty}e_t\n\\leq\n\\frac{\\bar\\sigma}{1-\\bar\\alpha}.\n}\n\\]\n\\end{theorem}","&+\n\\sum_{j=0}^{t-1}\n\\left(\\prod_{r=j+1}^{t-1}\\alpha_r\\right)\\sigma_j,\n\\end{aligned}\n}\n\\]\nwhere an empty product equals one. If\n\\[\n\\alpha_t\\leq\\bar\\alpha<1,\n\\qquad\n\\sigma_t\\leq\\bar\\sigma,\n\\]\nthen\n\\[\n\\boxed{\ne_t\n\\leq\n\\bar\\alpha^te_0\n+\n\\frac{\\bar\\sigma(1-\\bar\\alpha^t)}{1-\\bar\\alpha},\n}\n\\]\nand consequently\n\\[\n\\boxed{\n\\limsup_{t\\to\\infty}e_t\n\\leq\n\\frac{\\bar\\sigma}{1-\\bar\\alpha}.\n}\n\\]\n\\end{theorem}","\\begin{proof}\nRepeated substitution yields\n\\[\n\\begin{aligned}\ne_t\n&\\leq\n\\alpha_{t-1}\\cdots\\alpha_0e_0\n+\n\\alpha_{t-1}\\cdots\\alpha_1\\sigma_0\n+\n\\cdots\n+\n\\sigma_{t-1},\n\\end{aligned}\n\\]\nwhich is the first formula. Under the uniform bounds, dominate each product by\nthe corresponding power of \\(\\bar\\alpha\\) and sum the geometric series.\n\\end{proof}","\\begin{corollary}[Vanishing-disturbance convergence]\\label{cor:vanishing-disturbance}\nIf \\(\\alpha_t\\leq\\bar\\alpha<1\\) and \\(\\sigma_t\\to0\\), then\n\\[\ne_t\\to0.\n\\]\nMore generally, if \\(\\sum_{t\\geq0}\\sigma_t<\\infty\\), then the disturbance\ncontribution in Theorem~\\ref{thm:persistence} vanishes asymptotically.\n\\end{corollary}","\\begin{proof}\nSplit the convolution sum into a finite early part, whose coefficients decay\nto zero, and a late tail on which \\(\\sigma_t\\) is uniformly small. The\nsummable case follows by the same argument.\n\\end{proof}","For stochastic systems, pathwise bounded disturbances may be too strong. Let\n\\(\\{\\mathcal F_t\\}\\) be a filtration containing the history through time\n\\(t\\).","\\begin{proposition}[Mean-square persistence]\\label{prop:mean-square-persistence}\nSuppose\n\\[\n\\mathbb E[e_{t+1}^2\\mid\\mathcal F_t]\n\\leq\nq e_t^2+v_t,\n\\qquad\n0\\leq q<1,\n\\]\nand \\(\\mathbb E[v_t]\\leq\\bar v\\). Then\n\\[\n\\boxed{\n\\mathbb E[e_t^2]\n\\leq\nq^t\\mathbb E[e_0^2]\n+\n\\frac{\\bar v(1-q^t)}{1-q},\n}\n\\]\nso\n\\[\n\\limsup_{t\\to\\infty}\\mathbb E[e_t^2]\n\\leq\n\\frac{\\bar v}{1-q}.\n\\]\n\\end{proposition}","Low off-workspace error is not sufficient for meaningful persistence. The\nzero update \\(F_t=0\\) drives every state exactly into every linear workspace\nwhile destroying all content. Define the internal transport\n\\[\nB_t=P_{t+1}F_tP_t:\\mathcal W_t\\longrightarrow\\mathcal W_{t+1}\n\\]\nand its non-collapse margin\n\\[\n\\boxed{\n\\mu_t^{\\mathrm{keep}}\n=\n\\sigma_{\\min}(B_t|_{\\mathcal W_t}).\n}\n\\]\nA content-preserving persistence claim requires\n\\[\n\\mu_t^{\\mathrm{keep}}>0\n\\]\nfor the relevant interval, or an information-theoretic replacement when the\nworkspace state is stochastic. Structural tracking, state tracking, and\ncontent retention are therefore separate gates.","fficient for meaningful persistence. The\nzero update \\(F_t=0\\) drives every state exactly into every linear workspace\nwhile destroying all content. Define the internal transport\n\\[\nB_t=P_{t+1}F_tP_t:\\mathcal W_t\\longrightarrow\\mathcal W_{t+1}\n\\]\nand its non-collapse margin\n\\[\n\\boxed{\n\\mu_t^{\\mathrm{keep}}\n=\n\\sigma_{\\min}(B_t|_{\\mathcal W_t}).\n}\n\\]\nA content-preserving persistence claim requires\n\\[\n\\mu_t^{\\mathrm{keep}}>0\n\\]\nfor the relevant interval, or an information-theoretic replacement when the\nworkspace state is stochastic. Structural tracking, state tracking, and\ncontent retention are therefore separate gates.","A finite-horizon persistence certificate may be recorded as\n\\[\n\\boxed{\n\\mathcal P_{0:T}\n=\n\\left(\n\\max_{t0\\);\n\\item there exists a linear decoder \\(R_S\\) with \\(R_SA_S=I_{C_*}\\).\n\\end{enumerate}\nWhen these conditions hold, the canonical decoder is\n\\[\n\\boxed{\nR_S=A_S^\\dagger=F_S^{-1}A_S^*.\n}\n\\]\n\\end{proposition}","\\begin{proof}\nIn finite dimensions,\n\\[\n\\ker F_S=\\ker A_S\n\\]\nbecause\n\\[\n\\langle c,F_Sc\\rangle=\\|A_Sc\\|^2.\n\\]\nThis proves the equivalence of the first three statements. An injective linear\nmap admits a left inverse, while a left inverse implies injectivity. The\nformula for the Moore--Penrose left inverse follows from full column rank.\n\\end{proof}","The lower frame bound \\(\\alpha_S\\) measures recoverability. The upper bound\n\\(\\beta_S\\) measures amplification in the forward encoding. Their ratio\n\\[\n\\kappa_S=\\sqrt{\\frac{\\beta_S}{\\alpha_S}}\n\\]\nis the coalition condition number.","\\subtopic{35.1 Stable reconstruction theorem \\(T_7\\)}","\\begin{theorem}[Stable core reconstruction]\\label{thm:stable-reconstruction}\nAssume \\(\\alpha_S>0\\) and observe\n\\[\ny=A_Sc+\\eta.\n\\]\nWith the canonical estimate\n\\[\n\\widehat c=A_S^\\dagger y,\n\\]\none has\n\\[\n\\boxed{\n\\|\\widehat c-c\\|\n\\leq\n\\frac{\\|\\eta\\|}{\\sqrt{\\alpha_S}}.\n}\n\\]\n\\end{theorem}","\\begin{proof}\nSince \\(A_S^\\dagger A_S=I_{C_*}\\),\n\\[\n\\widehat c-c=A_S^\\dagger\\eta.\n\\]\nThe singular values of \\(A_S\\) are the square roots of the eigenvalues of\n\\(F_S\\), so\n\\[\n\\|A_S^\\dagger\\|\n=\n\\frac{1}{\\sigma_{\\min}(A_S)}\n=\n\\frac{1}{\\sqrt{\\alpha_S}}.\n\\]\n\\end{proof}","\\begin{corollary}[Encoding mismatch]\\label{cor:encoding-mismatch}\nIf the true observation is\n\\[\ny=(A_S+\\Delta_S)c+\\eta\n\\]\nbut the nominal decoder \\(A_S^\\dagger\\) is used, then\n\\[\n\\boxed{\n\\|\\widehat c-c\\|\n\\leq\n\\frac{\\|\\Delta_S\\|\\,\\|c\\|+\\|\\eta\\|}{\\sqrt{\\alpha_S}}.\n}\n\\]\n\\end{corollary}","\\begin{proof}\nNow\n\\[\n\\widehat c-c=A_S^\\dagger\\Delta_Sc+A_S^\\dagger\\eta,\n\\]\nand the result follows from Theorem~\\ref{thm:stable-reconstruction}.\n\\end{proof}","Geometric recovery becomes behaviorally relevant when the workspace state\ncontrols a predictive kernel. Let\n\\[\nK_w=\\mathbb P(Y\\in\\cdot\\mid W=w)\n\\]\nand suppose\n\\[\n\\TV(K_w,K_{w'})\\leq L_Kd_W(w,w').\n\\]","\\begin{proposition}[Exact reconstruction implies coalition sufficiency]\n\\label{prop:exact-reconstruction-sufficiency}\nLet \\(W\\) be sufficient for \\(Y\\) relative to the full history \\(\\mathbf H\\):\n\\[\nY\\indep\\mathbf H\\mid W.\n\\]\nIf a coalition reconstructs the workspace exactly,\n\\[\nW=R_S(\\mathbf H^S)\n\\qquad\\text{almost surely},\n\\]\nthen\n\\[\n\\boxed{\nY\\indep\\mathbf H\\mid\\mathbf H^S.\n}\n\\]\nUnder logarithmic loss,\n\\[\n\\boxed{\n\\Risk_{\\log}(S)=\\Risk_{\\log}(V).\n}\n\\]\n\\end{proposition}","position}[Exact reconstruction implies coalition sufficiency]\n\\label{prop:exact-reconstruction-sufficiency}\nLet \\(W\\) be sufficient for \\(Y\\) relative to the full history \\(\\mathbf H\\):\n\\[\nY\\indep\\mathbf H\\mid W.\n\\]\nIf a coalition reconstructs the workspace exactly,\n\\[\nW=R_S(\\mathbf H^S)\n\\qquad\\text{almost surely},\n\\]\nthen\n\\[\n\\boxed{\nY\\indep\\mathbf H\\mid\\mathbf H^S.\n}\n\\]\nUnder logarithmic loss,\n\\[\n\\boxed{\n\\Risk_{\\log}(S)=\\Risk_{\\log}(V).\n}\n\\]\n\\end{proposition}","\\begin{proof}\nSufficiency means that the conditional target law given \\(\\mathbf H\\) is a\nmeasurable function of \\(W\\). Exact reconstruction makes that function\nmeasurable with respect to \\(\\mathbf H^S\\), which is the asserted conditional\nindependence. The log-risk equality follows from\nTheorem~\\ref{thm:coalition-log-risk}.\n\\end{proof}","\\begin{proposition}[Reconstruction-to-risk transfer]\\label{prop:reconstruction-risk-transfer}\nLet \\(W\\) be sufficient for a target \\(Y\\), let losses take values in\n\\([0,1]\\), and let a coalition construct \\(\\widehat W\\) from its local data.\nThen its optimal Bayes risk satisfies\n\\[\n\\boxed{\n0\\leq\n\\Risk(S)-\\Risk(W)\n\\leq\n2L_K\\,\\mathbb E[d_W(W,\\widehat W)].\n}\n\\]\n\\end{proposition}","\\begin{proof}\nFor every pair \\(w,\\widehat w\\), using the Bayes action for\n\\(K_{\\widehat w}\\) under the true law \\(K_w\\) incurs conditional regret at\nmost\n\\[\n2\\TV(K_w,K_{\\widehat w}).\n\\]\nThis follows by inserting and subtracting the risk under\n\\(K_{\\widehat w}\\) for both the chosen action and the Bayes action for\n\\(K_w\\). Average the bound and use the Lipschitz assumption. The coalition's\noptimal rule performs at least as well as this decoder-based rule.\n\\end{proof}","For \\(k\\geq0\\), define the worst surviving lower frame bound\n\\[\n\\boxed{\n\\alpha^{(k)}\n=\n\\min_{\\substack{A\\subseteq V","For \\(k\\geq0\\), define the worst surviving lower frame bound\n\\[\n\\boxed{\n\\alpha^{(k)}\n=\n\\min_{\\substack{A\\subseteq V\\\\|A|\\leq k}}\n\\alpha_{V\\setminus A}.\n}\n\\]\nThe sequence \\(\\alpha^{(k)}\\) is nonincreasing in \\(k\\).","|A|\\leq k}}\n\\alpha_{V\\setminus A}.\n}\n\\]\nThe sequence \\(\\alpha^{(k)}\\) is nonincreasing in \\(k\\).","\\begin{theorem}[Uniform erasure resilience]\\label{thm:uniform-erasure-resilience}\nThe encoding family is exactly reconstructible after every loss of at most\n\\(k\\) nodes if and only if\n\\[\n\\boxed{\\alpha^{(k)}>0.}\n\\]\nIf surviving observations have the form\n\\[\ny_{V\\setminus A}=A_{V\\setminus A}c+\\eta_A,\n\\qquad\n|A|\\leq k,\n\\]\nthen the canonical survivor decoder obeys the uniform estimate\n\\[\n\\boxed{\n\\|\\widehat c_A-c\\|\n\\leq\n\\frac{\\|\\eta_A\\|}{\\sqrt{\\alpha^{(k)}}}.\n}\n\\]\n\\end{theorem}","\\begin{proposition}[Dimensional obstruction]\\label{prop:dimension-obstruction}\nLet \\(d=\\dim C_*\\). If\n\\[\n\\sum_{i\\in S}\\dim \\mathsf Y_ik.\n}\n\\]\n\\end{theorem}","\\begin{proof}\nIf \\(d_{\\mathrm{er}}\\leq k\\), choose \\(c\\neq0\\) whose nonzero accessible\nblocks are contained in a set \\(A\\) with \\(|A|\\leq k\\). The surviving analysis\noperator satisfies \\(A_{V\\setminus A}c=0\\), so it cannot distinguish \\(c\\)\nfrom zero. Conversely, if some erasure set \\(A\\) of size at most \\(k\\) destroys\ninjectivity, there exists \\(c\\neq0\\) with \\(E_ic=0\\) for every \\(i\\notin A\\).\nThus \\(\\operatorname{supp}_E(c)\\subseteq A\\), and\n\\(d_{\\mathrm{er}}\\leq k\\).\n\\end{proof}","{proof}\nIf \\(d_{\\mathrm{er}}\\leq k\\), choose \\(c\\neq0\\) whose nonzero accessible\nblocks are contained in a set \\(A\\) with \\(|A|\\leq k\\). The surviving analysis\noperator satisfies \\(A_{V\\setminus A}c=0\\), so it cannot distinguish \\(c\\)\nfrom zero. Conversely, if some erasure set \\(A\\) of size at most \\(k\\) destroys\ninjectivity, there exists \\(c\\neq0\\) with \\(E_ic=0\\) for every \\(i\\notin A\\).\nThus \\(\\operatorname{supp}_E(c)\\subseteq A\\), and\n\\(d_{\\mathrm{er}}\\leq k\\).\n\\end{proof}","\\(c\\neq0\\) whose nonzero accessible\nblocks are contained in a set \\(A\\) with \\(|A|\\leq k\\). The surviving analysis\noperator satisfies \\(A_{V\\setminus A}c=0\\), so it cannot distinguish \\(c\\)\nfrom zero. Conversely, if some erasure set \\(A\\) of size at most \\(k\\) destroys\ninjectivity, there exists \\(c\\neq0\\) with \\(E_ic=0\\) for every \\(i\\notin A\\).\nThus \\(\\operatorname{supp}_E(c)\\subseteq A\\), and\n\\(d_{\\mathrm{er}}\\leq k\\).\n\\end{proof}","The distance criterion is combinatorial. The frame quantity\n\\(\\alpha^{(k)}\\) is quantitative. A system may tolerate \\(k\\) erasures exactly\nwhile amplifying noise severely because \\(\\alpha^{(k)}\\) is close to zero.","Suppose node \\(r\\) is removed and the surviving coalition is\n\\[\nS=V\\setminus\\{r\\}.\n\\]\nA replacement node \\(r'\\) is to receive the encoding\n\\[\nE_{r'}':C_*\\to\\mathsf Y_{r'}'.\n\\]","\\begin{corollary}[Handoff initialization]\\label{cor:handoff-initialization}\nIf \\(\\alpha_S>0\\) and the survivors observe \\(y=A_Sc+\\eta\\), initialize the\nreplacement by\n\\[\nx_{r'}^{\\mathrm{init}}\n=E_{r'}'A_S^\\dagger y.\n\\]\nThen\n\\[\n\\boxed{\n\\|x_{r'}^{\\mathrm{init}}-E_{r'}'c\\|\n\\leq\n\\frac{\\|E_{r'}'\\|\\,\\|\\eta\\|}{\\sqrt{\\alpha_S}}.\n}\n\\]\n\\end{corollary}","Define the minimum reconstructing coalition size\n\\[\n\\boxed{\nm_{\\mathrm{rec}}\n=\n\\min\\{|S|:\\alpha_S>0\\}\n}\n\\]\nand the maximum exact erasure tolerance\n\\[\n\\boxed{\nk_{\\mathrm{erase}}\n=\n\\max\\{k:\\alpha^{(k)}>0\\}.\n}\n\\]\nThe condition\n\\[\nm_{\\mathrm{rec}}\\geq2\n\\]\nmeans that no individual encoding is sufficient for exact core recovery. The\ncondition\n\\[\nk_{\\mathrm{erase}}\\geq1\n\\]\nmeans that the system survives at least one node loss. These can hold\nsimultaneously because nonlocality of encoding is not the same as fragility.","Let\n\\[\nC_*=\\mathbb R^2\n\\]\nand define scalar node encodings\n\\[\nE_1(c_1,c_2)=c_1,\n\\qquad\nE_2(c_1,c_2)=c_2,\n\\qquad\nE_3(c_1,c_2)=c_1+c_2.\n\\]\nEvery singleton has rank one, so no node reconstructs the two-dimensional\ncore. Every pair does reconstruct. For \\(S=\\{1,2\\}\\),\n\\[\nF_S=I,\n\\qquad\n\\alpha_S=1.\n\\]\nFor \\(S=\\{1,3\\}\\) or \\(S=\\{2,3\\}\\), the frame eigenvalues are\n\\[\n\\frac{3\\pm\\sqrt5}{2}.\n\\]\nTherefore\n\\[\n\\boxed{\nm_{\\mathrm{rec}}=2,\n\\qquad\nk_{\\mathrm{erase}}=1,\n\\qquad\nd_{\\mathrm{er}}=2,\n\\qquad\n\\alpha^{(1)}=\\frac{3-\\sqrt5}{2}.\n}\n\\]\nThis realizes a workspace core that is irreducible to any singleton while\nremaining reconstructible after any one node fails.","The geometric quantity \\(m_{\\mathrm{rec}}\\) and the predictive quantity\n\\(k_0\\) from Part~VII answer different questions. The first asks whether a\ncoalition can reconstruct a declared core coordinate. The second asks whether\na coalition is sufficient for a declared future target. Equality between them\nrequires structural--operational alignment and regularity such as\nProposition~\\ref{prop:reconstruction-risk-transfer}. Neither quantity should\nbe substituted for the other without those assumptions.","Let\n\\[\n\\mathfrak R=(G,\\mathbf C,D,\\rho,\\mathcal I,\\mathcal K)\n\\]\nand\n\\[\n\\mathfrak R'=(G',\\mathbf C',D',\\rho',\\mathcal I',\\mathcal K')\n\\]\nbe two realizations. A correspondence begins with a graph isomorphism\n\\(\\varphi:G\\to G'\\). After relabeling by \\(\\varphi\\), let\n\\[\nJ_i:C_i\\to C_i'\n\\]\nbe unitary local identifications and define\n\\[\nJ_V=\\bigoplus_{i\\in V}J_i,\n\\qquad\nJ_E=\\bigoplus_{e\\in E}J_{t(e)}.\n\\]","\\begin{definition}[Exact structural correspondence]\\label{def:exact-structural-correspondence}\nThe correspondence is structurally exact when\n\\[\n\\boxed{D'J_V=J_ED.}\n\\]\nIt is operation exact for \\(a\\in\\mathcal A\\) when\n\\[\n\\boxed{\\rho_V'(a)J_V=J_V\\rho_V(a).}\n\\]\n\\end{definition}","\\begin{theorem}[Exact replacement isomorphism]\\label{thm:exact-replacement-isomorphism}\nUnder an exact structural correspondence,\n\\[\n\\boxed{\n\\Gamma(D')=J_V\\Gamma(D),\n\\qquad\nL'J_V=J_VL.\n}\n\\]\nConsequently, every spectral workspace subspace is transported exactly:\n\\[\nP_I(L')=J_VP_I(L)J_V^*\n\\]\nfor every spectral interval \\(I\\). If the correspondence is also operation\nexact, then the transported workspace actions are intertwined exactly.\n\\end{theorem}","\\begin{proof}\nFor every \\(c\\),\n\\[\nD'J_Vc=J_EDc.\n\\]\nBecause \\(J_E\\) is invertible, \\(Dc=0\\) if and only if \\(D'J_Vc=0\\), proving\nthe kernel statement. Taking adjoints of the exact covariance identity gives\n\\[\nD'^*J_E=J_VD^*.\n\\]\nTherefore\n\\[\nL'J_V=D'^*D'J_V=D'^*J_ED=J_VD^*D=J_VL.\n\\]\nThe spectral and operation claims follow by functional calculus and the\nintertwining assumptions.\n\\end{proof}","For an approximate correspondence define\n\\[\n\\boxed{\n\\Delta_D=D'J_V-J_ED,\n\\qquad\n\\delta_D=\\|\\Delta_D\\|.\n}\n\\]\nIf the graph weights are unchanged, define edgewise comparison defects\n\\[\n\\delta_e\n=\n\\|H_e'J_{s(e)}-J_{t(e)}H_e\\|.\n\\]\nThe discrepancy operator has one changed source block per edge, so\n\\[\n\\boxed{\n\\delta_D^2\n\\leq\n\\max_{i\\in V}\n\\sum_{e:s(e)=i}w_e\\delta_e^2\n\\leq\n\\sum_{e\\in E}w_e\\delta_e^2.\n}\n\\]\nWhen a single node is replaced, every nonincident edge has \\(\\delta_e=0\\).\nThe replacement defect is therefore locally measurable even though its effect\non the global workspace is controlled by the global spectral gap.","'J_V-J_ED,\n\\qquad\n\\delta_D=\\|\\Delta_D\\|.\n}\n\\]\nIf the graph weights are unchanged, define edgewise comparison defects\n\\[\n\\delta_e\n=\n\\|H_e'J_{s(e)}-J_{t(e)}H_e\\|.\n\\]\nThe discrepancy operator has one changed source block per edge, so\n\\[\n\\boxed{\n\\delta_D^2\n\\leq\n\\max_{i\\in V}\n\\sum_{e:s(e)=i}w_e\\delta_e^2\n\\leq\n\\sum_{e\\in E}w_e\\delta_e^2.\n}\n\\]\nWhen a single node is replaced, every nonincident edge has \\(\\delta_e=0\\).\nThe replacement defect is therefore locally measurable even though its effect\non the global workspace is controlled by the global spectral gap.","The corresponding carried-Laplacian defect obeys\n\\[\n\\boxed{\n\\eta_L\n:=\n\\|J_V^*L'J_V-L\\|\n\\leq\n2\\|D\\|\\delta_D+\\delta_D^2.\n}\n\\]\nThis is Proposition~\\ref{prop:temporal-laplacian-drift} with the time indices\nremoved.","Let \\(P\\) and \\(P'\\) be rank-\\(d\\) workspace projectors selected from \\(L\\)\nand \\(L'\\), and let \\(g>0\\) be the separating source gap.","\\begin{theorem}[Approximate structural replacement]\\label{thm:approx-structural-replacement}\nIf\n\\[\n\\eta_L<\\frac g2,\n\\]\nthen\n\\[\n\\boxed{\nq_{\\mathrm{str}}\n:=\n\\|P'-J_VPJ_V^*\\|\n\\leq\n\\frac{\\eta_L}{g-\\eta_L}\n\\leq\n\\frac{2\\eta_L}{g}.\n}\n\\]\nIf \\(q_{\\mathrm{str}}<1\\), then\n\\[\nB=P'J_VP:\\ran P\\longrightarrow\\ran P'\n\\]\nis bijective and satisfies\n\\[\n\\sigma_{\\min}(B)\n\\geq\n\\sqrt{1-q_{\\mathrm{str}}^2}.\n\\]\nIts polar factor\n\\[\n\\boxed{\n\\beta\n=B(B^*B)^{-1/2}\n}\n\\]\nis a canonical unitary workspace correspondence. Moreover,\n\\[\n\\boxed{\n\\|\\beta-J_VP\\|\n\\leq\nq_{\\mathrm{str}}\n+\n1-\\sqrt{1-q_{\\mathrm{str}}^2}.\n}\n\\]\n\\end{theorem}","eorem}[Approximate structural replacement]\\label{thm:approx-structural-replacement}\nIf\n\\[\n\\eta_L<\\frac g2,\n\\]\nthen\n\\[\n\\boxed{\nq_{\\mathrm{str}}\n:=\n\\|P'-J_VPJ_V^*\\|\n\\leq\n\\frac{\\eta_L}{g-\\eta_L}\n\\leq\n\\frac{2\\eta_L}{g}.\n}\n\\]\nIf \\(q_{\\mathrm{str}}<1\\), then\n\\[\nB=P'J_VP:\\ran P\\longrightarrow\\ran P'\n\\]\nis bijective and satisfies\n\\[\n\\sigma_{\\min}(B)\n\\geq\n\\sqrt{1-q_{\\mathrm{str}}^2}.\n\\]\nIts polar factor\n\\[\n\\boxed{\n\\beta\n=B(B^*B)^{-1/2}\n}\n\\]\nis a canonical unitary workspace correspondence. Moreover,\n\\[\n\\boxed{\n\\|\\beta-J_VP\\|\n\\leq\nq_{\\mathrm{str}}\n+\n1-\\sqrt{1-q_{\\mathrm{str}}^2}.\n}\n\\]\n\\end{theorem}","\\frac g2,\n\\]\nthen\n\\[\n\\boxed{\nq_{\\mathrm{str}}\n:=\n\\|P'-J_VPJ_V^*\\|\n\\leq\n\\frac{\\eta_L}{g-\\eta_L}\n\\leq\n\\frac{2\\eta_L}{g}.\n}\n\\]\nIf \\(q_{\\mathrm{str}}<1\\), then\n\\[\nB=P'J_VP:\\ran P\\longrightarrow\\ran P'\n\\]\nis bijective and satisfies\n\\[\n\\sigma_{\\min}(B)\n\\geq\n\\sqrt{1-q_{\\mathrm{str}}^2}.\n\\]\nIts polar factor\n\\[\n\\boxed{\n\\beta\n=B(B^*B)^{-1/2}\n}\n\\]\nis a canonical unitary workspace correspondence. Moreover,\n\\[\n\\boxed{\n\\|\\beta-J_VP\\|\n\\leq\nq_{\\mathrm{str}}\n+\n1-\\sqrt{1-q_{\\mathrm{str}}^2}.\n}\n\\]\n\\end{theorem}","\\begin{proof}\nThe projector estimate is Theorem~\\ref{thm:spectral-stability} applied after\ncarrying \\(L'\\) back by \\(J_V\\). For unit \\(w\\in\\ran P\\),\n\\[\n\\|(I-P')J_Vw\\|\n\\leq q_{\\mathrm{str}}.\n\\]\nOrthogonality gives\n\\[\n\\|Bw\\|^2\n=1-\\|(I-P')J_Vw\\|^2\n\\geq1-q_{\\mathrm{str}}^2.\n\\]\nThus \\(B\\) is injective, hence bijective because the source and target have\nthe same finite dimension. Its singular values lie in\n\\([\\sqrt{1-q_{\\mathrm{str}}^2},1]\\), so polar normalization moves \\(B\\) by at\nmost \\(1-\\sqrt{1-q_{\\mathrm{str}}^2}\\). Finally,\n\\[\n\\|B-J_VP\\|=\\|(I-P')J_VP\\|\\leq q_{\\mathrm{str}}.\n\\]\n\\end{proof}","{proof}\nThe projector estimate is Theorem~\\ref{thm:spectral-stability} applied after\ncarrying \\(L'\\) back by \\(J_V\\). For unit \\(w\\in\\ran P\\),\n\\[\n\\|(I-P')J_Vw\\|\n\\leq q_{\\mathrm{str}}.\n\\]\nOrthogonality gives\n\\[\n\\|Bw\\|^2\n=1-\\|(I-P')J_Vw\\|^2\n\\geq1-q_{\\mathrm{str}}^2.\n\\]\nThus \\(B\\) is injective, hence bijective because the source and target have\nthe same finite dimension. Its singular values lie in\n\\([\\sqrt{1-q_{\\mathrm{str}}^2},1]\\), so polar normalization moves \\(B\\) by at\nmost \\(1-\\sqrt{1-q_{\\mathrm{str}}^2}\\). Finally,\n\\[\n\\|B-J_VP\\|=\\|(I-P')J_VP\\|\\leq q_{\\mathrm{str}}.\n\\]\n\\end{proof}","ity} applied after\ncarrying \\(L'\\) back by \\(J_V\\). For unit \\(w\\in\\ran P\\),\n\\[\n\\|(I-P')J_Vw\\|\n\\leq q_{\\mathrm{str}}.\n\\]\nOrthogonality gives\n\\[\n\\|Bw\\|^2\n=1-\\|(I-P')J_Vw\\|^2\n\\geq1-q_{\\mathrm{str}}^2.\n\\]\nThus \\(B\\) is injective, hence bijective because the source and target have\nthe same finite dimension. Its singular values lie in\n\\([\\sqrt{1-q_{\\mathrm{str}}^2},1]\\), so polar normalization moves \\(B\\) by at\nmost \\(1-\\sqrt{1-q_{\\mathrm{str}}^2}\\). Finally,\n\\[\n\\|B-J_VP\\|=\\|(I-P')J_VP\\|\\leq q_{\\mathrm{str}}.\n\\]\n\\end{proof}","r\ncarrying \\(L'\\) back by \\(J_V\\). For unit \\(w\\in\\ran P\\),\n\\[\n\\|(I-P')J_Vw\\|\n\\leq q_{\\mathrm{str}}.\n\\]\nOrthogonality gives\n\\[\n\\|Bw\\|^2\n=1-\\|(I-P')J_Vw\\|^2\n\\geq1-q_{\\mathrm{str}}^2.\n\\]\nThus \\(B\\) is injective, hence bijective because the source and target have\nthe same finite dimension. Its singular values lie in\n\\([\\sqrt{1-q_{\\mathrm{str}}^2},1]\\), so polar normalization moves \\(B\\) by at\nmost \\(1-\\sqrt{1-q_{\\mathrm{str}}^2}\\). Finally,\n\\[\n\\|B-J_VP\\|=\\|(I-P')J_VP\\|\\leq q_{\\mathrm{str}}.\n\\]\n\\end{proof}","\\(w\\in\\ran P\\),\n\\[\n\\|(I-P')J_Vw\\|\n\\leq q_{\\mathrm{str}}.\n\\]\nOrthogonality gives\n\\[\n\\|Bw\\|^2\n=1-\\|(I-P')J_Vw\\|^2\n\\geq1-q_{\\mathrm{str}}^2.\n\\]\nThus \\(B\\) is injective, hence bijective because the source and target have\nthe same finite dimension. Its singular values lie in\n\\([\\sqrt{1-q_{\\mathrm{str}}^2},1]\\), so polar normalization moves \\(B\\) by at\nmost \\(1-\\sqrt{1-q_{\\mathrm{str}}^2}\\). Finally,\n\\[\n\\|B-J_VP\\|=\\|(I-P')J_VP\\|\\leq q_{\\mathrm{str}}.\n\\]\n\\end{proof}","Write\n\\[\nA=\\rho_V(a),\n\\qquad\nA'=\\rho_V'(a),\n\\]\nand define\n\\[\n\\delta_\\rho(a)=\\|A'J_V-J_VA\\|,\n\\qquad\n\\ell_a=\\|(I-P)AP\\|.\n\\]\nLet\n\\[\nA_{\\mathcal W}=PAP,\n\\qquad\nA_{\\mathcal W'}'=P'A'P'.\n\\]","\\begin{proposition}[Operation transport bound]\\label{prop:operation-transport-bound}\nFor \\(B=P'J_VP\\),\n\\[\n\\boxed{\n\\|A_{\\mathcal W'}'B-BA_{\\mathcal W}\\|\n\\leq\n\\|A'\\|q_{\\mathrm{str}}\n+\\delta_\\rho(a)\n+\\ell_a.\n}\n\\]\nFor the unitary polar correspondence \\(\\beta\\),\n\\[\n\\boxed{\n\\begin{aligned}\n\\|A_{\\mathcal W'}'\\beta-\\beta A_{\\mathcal W}\\|\n\\leq\n&\\|A'\\|q_{\\mathrm{str}}\n+\\delta_\\rho(a)\n+\\ell_a","\\begin{proposition}[Operation transport bound]\\label{prop:operation-transport-bound}\nFor \\(B=P'J_VP\\),\n\\[\n\\boxed{\n\\|A_{\\mathcal W'}'B-BA_{\\mathcal W}\\|\n\\leq\n\\|A'\\|q_{\\mathrm{str}}\n+\\delta_\\rho(a)\n+\\ell_a.\n}\n\\]\nFor the unitary polar correspondence \\(\\beta\\),\n\\[\n\\boxed{\n\\begin{aligned}\n\\|A_{\\mathcal W'}'\\beta-\\beta A_{\\mathcal W}\\|\n\\leq\n&\\|A'\\|q_{\\mathrm{str}}\n+\\delta_\\rho(a)\n+\\ell_a\\\\\n&+\n\\bigl(\\|A_{\\mathcal W'}'\\|+\\|A_{\\mathcal W}\\|\\bigr)\n\\left(1-\\sqrt{1-q_{\\mathrm{str}}^2}\\right).\n\\end{aligned}\n}\n\\]\n\\end{proposition}","\\begin{proof}\nInsert and subtract \\(P'A'J_VP\\) and \\(P'J_VAP\\):\n\\[\n\\begin{aligned}\nA_{\\mathcal W'}'B-BA_{\\mathcal W}\n={}&P'A'(P'-I)J_VP","\\begin{proof}\nInsert and subtract \\(P'A'J_VP\\) and \\(P'J_VAP\\):\n\\[\n\\begin{aligned}\nA_{\\mathcal W'}'B-BA_{\\mathcal W}\n={}&P'A'(P'-I)J_VP\\\\\n&+P'(A'J_V-J_VA)P\\\\\n&+P'J_V(I-P)AP.\n\\end{aligned}\n\\]\nBound the three terms. Replacing \\(B\\) by \\(\\beta\\) adds\n\\[\nA_{\\mathcal W'}'(\\beta-B)-(\\beta-B)A_{\\mathcal W},\n\\]\nwhich gives the second estimate.\n\\end{proof}","&+P'J_V(I-P)AP.\n\\end{aligned}\n\\]\nBound the three terms. Replacing \\(B\\) by \\(\\beta\\) adds\n\\[\nA_{\\mathcal W'}'(\\beta-B)-(\\beta-B)A_{\\mathcal W},\n\\]\nwhich gives the second estimate.\n\\end{proof}","Structural correspondence does not determine operational identity. Let\n\\[\n\\{\\iota_{w,b}:(w,b)\\in\\mathcal D\\}\n\\quad\\text{and}\\quad\n\\{\\iota'_{\\beta w,\\chi b}:(w,b)\\in\\mathcal D\\}\n\\]\nbe matched feasible implementation-intervention families in the sense of\nPart~VIII. Here \\(b\\) denotes declared background variables and \\(\\chi\\) is\nthe background correspondence. Define\n\\[\nK(w,b)\n=\n\\mathbb P^{\\iota_{w,b}}(Y^+\\in\\cdot),\n\\]\n\\[\nK'(\\beta w,\\chi b)\n=\n\\mathbb P^{\\prime\\,\\iota'_{\\beta w,\\chi b}}(Y^{\\prime+}\\in\\cdot).\n\\]\nFor a chosen divergence \\(D_Y\\), define\n\\[\n\\boxed{\n\\delta_{\\mathrm{causal}}(\\beta)\n=\n\\sup_{(w,b)\\in\\mathcal D}\nD_Y\\left(\nK(w,b),\nK'(\\beta w,\\chi b)\n\\right).\n}\n\\]\nThe observational response is included only when the null intervention belongs\nto the declared family. This definition compares actual implementation\ninterventions rather than formally assigning an arbi","Structural correspondence does not determine operational identity. Let\n\\[\n\\{\\iota_{w,b}:(w,b)\\in\\mathcal D\\}\n\\quad\\text{and}\\quad\n\\{\\iota'_{\\beta w,\\chi b}:(w,b)\\in\\mathcal D\\}\n\\]\nbe matched feasible implementation-intervention families in the sense of\nPart~VIII. Here \\(b\\) denotes declared background variables and \\(\\chi\\) is\nthe background correspondence. Define\n\\[\nK(w,b)\n=\n\\mathbb P^{\\iota_{w,b}}(Y^+\\in\\cdot),\n\\]\n\\[\nK'(\\beta w,\\chi b)\n=\n\\mathbb P^{\\prime\\,\\iota'_{\\beta w,\\chi b}}(Y^{\\prime+}\\in\\cdot).\n\\]\nFor a chosen divergence \\(D_Y\\), define\n\\[\n\\boxed{\n\\delta_{\\mathrm{causal}}(\\beta)\n=\n\\sup_{(w,b)\\in\\mathcal D}\nD_Y\\left(\nK(w,b),\nK'(\\beta w,\\chi b)\n\\right).\n}\n\\]\nThe observational response is included only when the null intervention belongs\nto the declared family. This definition compares actual implementation\ninterventions rather than formally assigning an arbitrary projection\ncoordinate.","nce does not determine operational identity. Let\n\\[\n\\{\\iota_{w,b}:(w,b)\\in\\mathcal D\\}\n\\quad\\text{and}\\quad\n\\{\\iota'_{\\beta w,\\chi b}:(w,b)\\in\\mathcal D\\}\n\\]\nbe matched feasible implementation-intervention families in the sense of\nPart~VIII. Here \\(b\\) denotes declared background variables and \\(\\chi\\) is\nthe background correspondence. Define\n\\[\nK(w,b)\n=\n\\mathbb P^{\\iota_{w,b}}(Y^+\\in\\cdot),\n\\]\n\\[\nK'(\\beta w,\\chi b)\n=\n\\mathbb P^{\\prime\\,\\iota'_{\\beta w,\\chi b}}(Y^{\\prime+}\\in\\cdot).\n\\]\nFor a chosen divergence \\(D_Y\\), define\n\\[\n\\boxed{\n\\delta_{\\mathrm{causal}}(\\beta)\n=\n\\sup_{(w,b)\\in\\mathcal D}\nD_Y\\left(\nK(w,b),\nK'(\\beta w,\\chi b)\n\\right).\n}\n\\]\nThe observational response is included only when the null intervention belongs\nto the declared family. This definition compares actual implementation\ninterventions rather than formally assigning an arbitrary projection\ncoordinate.","{\\beta w,\\chi b}:(w,b)\\in\\mathcal D\\}\n\\]\nbe matched feasible implementation-intervention families in the sense of\nPart~VIII. Here \\(b\\) denotes declared background variables and \\(\\chi\\) is\nthe background correspondence. Define\n\\[\nK(w,b)\n=\n\\mathbb P^{\\iota_{w,b}}(Y^+\\in\\cdot),\n\\]\n\\[\nK'(\\beta w,\\chi b)\n=\n\\mathbb P^{\\prime\\,\\iota'_{\\beta w,\\chi b}}(Y^{\\prime+}\\in\\cdot).\n\\]\nFor a chosen divergence \\(D_Y\\), define\n\\[\n\\boxed{\n\\delta_{\\mathrm{causal}}(\\beta)\n=\n\\sup_{(w,b)\\in\\mathcal D}\nD_Y\\left(\nK(w,b),\nK'(\\beta w,\\chi b)\n\\right).\n}\n\\]\nThe observational response is included only when the null intervention belongs\nto the declared family. This definition compares actual implementation\ninterventions rather than formally assigning an arbitrary projection\ncoordinate.",":(w,b)\\in\\mathcal D\\}\n\\]\nbe matched feasible implementation-intervention families in the sense of\nPart~VIII. Here \\(b\\) denotes declared background variables and \\(\\chi\\) is\nthe background correspondence. Define\n\\[\nK(w,b)\n=\n\\mathbb P^{\\iota_{w,b}}(Y^+\\in\\cdot),\n\\]\n\\[\nK'(\\beta w,\\chi b)\n=\n\\mathbb P^{\\prime\\,\\iota'_{\\beta w,\\chi b}}(Y^{\\prime+}\\in\\cdot).\n\\]\nFor a chosen divergence \\(D_Y\\), define\n\\[\n\\boxed{\n\\delta_{\\mathrm{causal}}(\\beta)\n=\n\\sup_{(w,b)\\in\\mathcal D}\nD_Y\\left(\nK(w,b),\nK'(\\beta w,\\chi b)\n\\right).\n}\n\\]\nThe observational response is included only when the null intervention belongs\nto the declared family. This definition compares actual implementation\ninterventions rather than formally assigning an arbitrary projection\ncoordinate.","A reconstructive handoff may also be imperfect. Let\n\\[\n\\delta_{\\mathrm{rec}}(\\beta)\n=\n\\mathbb E\\bigl[d_{\\mathcal W'}(W'_{\\mathrm{init}},\\beta W)\\bigr]\n\\]\nmeasure replacement initialization error. Corollary~\\ref{cor:handoff-initialization}\nprovides one explicit upper bound on this term.","Take two realizations with identical\n\\[\nG,\n\\quad D,\n\\quad L,\n\\quad P,\n\\quad \\rho.\n\\]\nIn the first, let \\(Y=W\\). In the second, let \\(Y'=1-W\\) for a binary\nworkspace coordinate. Then\n\\[\nq_{\\mathrm{str}}=0,\n\\qquad\n\\delta_\\rho(a)=0\n\\]\nfor every operation, while the observational and interventional outcome\nkernels differ maximally under the identity correspondence. Therefore no\npurely structural theorem can prove causal identity without an additional\nkernel-matching assumption.","For an admissible correspondence \\(\\mathfrak J\\), define the operation defect\non a declared unit ball \\(\\mathcal A_1\\) by\n\\[\n\\delta_{\\mathrm{op}}(\\mathfrak J)\n=\n\\sup_{a\\in\\mathcal A_1}\n\\|A_{\\mathcal W'}'\\beta-\\beta A_{\\mathcal W}\\|.\n\\]\nThe primary certification object is the identity profile\n\\[\n\\boxed{\n\\mathbf d_{\\mathrm{id}}(\\mathfrak J)\n=\n\\left(\nq_{\\mathrm{str}},\n\\delta_{\\mathrm{op}}(\\mathfrak J),\n\\delta_{\\mathrm{causal}}(\\beta),\n\\delta_{\\mathrm{rec}}(\\beta);\n\\mu^{\\mathrm{keep}}\n\\right).\n}\n\\]\nThe first four coordinates must remain below declared tolerances, while the\nnon-collapse margin \\(\\mu^{\\mathrm{keep}}\\) must remain above a declared lower\nbound. This vector, rather than a weighted average, determines whether a\nspecific identity","For an admissible correspondence \\(\\mathfrak J\\), define the operation defect\non a declared unit ball \\(\\mathcal A_1\\) by\n\\[\n\\delta_{\\mathrm{op}}(\\mathfrak J)\n=\n\\sup_{a\\in\\mathcal A_1}\n\\|A_{\\mathcal W'}'\\beta-\\beta A_{\\mathcal W}\\|.\n\\]\nThe primary certification object is the identity profile\n\\[\n\\boxed{\n\\mathbf d_{\\mathrm{id}}(\\mathfrak J)\n=\n\\left(\nq_{\\mathrm{str}},\n\\delta_{\\mathrm{op}}(\\mathfrak J),\n\\delta_{\\mathrm{causal}}(\\beta),\n\\delta_{\\mathrm{rec}}(\\beta);\n\\mu^{\\mathrm{keep}}\n\\right).\n}\n\\]\nThe first four coordinates must remain below declared tolerances, while the\nnon-collapse margin \\(\\mu^{\\mathrm{keep}}\\) must remain above a declared lower\nbound. This vector, rather than a weighted average, determines whether a\nspecific identity claim passes.","W'}'\\beta-\\beta A_{\\mathcal W}\\|.\n\\]\nThe primary certification object is the identity profile\n\\[\n\\boxed{\n\\mathbf d_{\\mathrm{id}}(\\mathfrak J)\n=\n\\left(\nq_{\\mathrm{str}},\n\\delta_{\\mathrm{op}}(\\mathfrak J),\n\\delta_{\\mathrm{causal}}(\\beta),\n\\delta_{\\mathrm{rec}}(\\beta);\n\\mu^{\\mathrm{keep}}\n\\right).\n}\n\\]\nThe first four coordinates must remain below declared tolerances, while the\nnon-collapse margin \\(\\mu^{\\mathrm{keep}}\\) must remain above a declared lower\nbound. This vector, rather than a weighted average, determines whether a\nspecific identity claim passes.","A composite identity cost is\n\\[\n\\boxed{\n\\operatorname{Cost}(\\mathfrak J)\n=\na\\,q_{\\mathrm{str}}\n+b\\,\\delta_{\\mathrm{op}}(\\mathfrak J)\n+c\\,\\delta_{\\mathrm{causal}}(\\beta)\n+d\\,\\delta_{\\mathrm{rec}}(\\beta),\n}\n\\]\nwhere \\(a,b,c,d>0\\) are declared scales. Define\n\\[\n\\boxed{\nd_{\\mathrm{id}}(\\mathfrak R,\\mathfrak R')\n=\n\\inf_{\\mathfrak J}\n\\operatorname{Cost}(\\mathfrak J).\n}\n\\]\nThis quantity is a task-relative optimization surrogate until the\nadmissible correspondence class, intervention matching, divergence, and scales\nare fixed. It does not replace the coordinatewise identity profile or the\nnon-collapse gate.","\\subtopic{41.1 Replacement invariance theorem \\(T_{14}\\) }","\\begin{theorem}[Quantitative replacement invariance]\\label{thm:replacement-invariance}\nLet \\(\\mathfrak J\\) be a unitary graph correspondence with diagram defect\n\\(\\delta_D\\). Define\n\\[\n\\eta_L=2\\|D\\|\\delta_D+\\delta_D^2\n\\]\nand assume\n\\[\n\\eta_L<\\frac g2.\n\\]\nSet\n\\[\nq_*\n=\n\\frac{\\eta_L}{g-\\eta_L}.\n\\]\nThen there exists a unitary workspace correspondence \\(\\beta\\) for which\n\\[\nq_{\\mathrm{str}}\\leq q_*.\n\\]\nMoreover,\n\\[\n\\boxed{\n\\begin{aligned}\n\\delta_{\\mathrm{op}}(\\mathfrak J)\n\\leq\n\\sup_{a\\in\\mathcal A_1}\n\\Bigl[&\n\\|\\rho_V'(a)\\|q_*\n+\\delta_\\rho(a)\n+\\ell_a","m}[Quantitative replacement invariance]\\label{thm:replacement-invariance}\nLet \\(\\mathfrak J\\) be a unitary graph correspondence with diagram defect\n\\(\\delta_D\\). Define\n\\[\n\\eta_L=2\\|D\\|\\delta_D+\\delta_D^2\n\\]\nand assume\n\\[\n\\eta_L<\\frac g2.\n\\]\nSet\n\\[\nq_*\n=\n\\frac{\\eta_L}{g-\\eta_L}.\n\\]\nThen there exists a unitary workspace correspondence \\(\\beta\\) for which\n\\[\nq_{\\mathrm{str}}\\leq q_*.\n\\]\nMoreover,\n\\[\n\\boxed{\n\\begin{aligned}\n\\delta_{\\mathrm{op}}(\\mathfrak J)\n\\leq\n\\sup_{a\\in\\mathcal A_1}\n\\Bigl[&\n\\|\\rho_V'(a)\\|q_*\n+\\delta_\\rho(a)\n+\\ell_a\\\\\n&+\n\\bigl(\\|A_{\\mathcal W'}'\\|+\\|A_{\\mathcal W}\\|\\bigr)\n\\left(1-\\sqrt{1-q_*^2}\\right)\n\\Bigr].\n\\end{aligned}\n}\n\\]\nIf, in addition,\n\\[\n\\delta_{\\mathrm{causal}}(\\beta)\\leq\\delta_c,\n\\qquad\n\\delta_{\\mathrm{rec}}(\\beta)\\leq\\delta_r,\n\\]\nthen\n\\[\n\\boxed{\n\\begin{aligned}\nd_{\\mathrm{id}}(\\mathfrak R,\\mathfrak R')\n\\leq\n&a\\,q_*+b\\,\\delta_{\\mathrm{op}}(\\mathfrak J)","&+\n\\bigl(\\|A_{\\mathcal W'}'\\|+\\|A_{\\mathcal W}\\|\\bigr)\n\\left(1-\\sqrt{1-q_*^2}\\right)\n\\Bigr].\n\\end{aligned}\n}\n\\]\nIf, in addition,\n\\[\n\\delta_{\\mathrm{causal}}(\\beta)\\leq\\delta_c,\n\\qquad\n\\delta_{\\mathrm{rec}}(\\beta)\\leq\\delta_r,\n\\]\nthen\n\\[\n\\boxed{\n\\begin{aligned}\nd_{\\mathrm{id}}(\\mathfrak R,\\mathfrak R')\n\\leq\n&a\\,q_*+b\\,\\delta_{\\mathrm{op}}(\\mathfrak J)","&+\n\\bigl(\\|A_{\\mathcal W'}'\\|+\\|A_{\\mathcal W}\\|\\bigr)\n\\left(1-\\sqrt{1-q_*^2}\\right)\n\\Bigr].\n\\end{aligned}\n}\n\\]\nIf, in addition,\n\\[\n\\delta_{\\mathrm{causal}}(\\beta)\\leq\\delta_c,\n\\qquad\n\\delta_{\\mathrm{rec}}(\\beta)\\leq\\delta_r,\n\\]\nthen\n\\[\n\\boxed{\n\\begin{aligned}\nd_{\\mathrm{id}}(\\mathfrak R,\\mathfrak R')\n\\leq\n&a\\,q_*+b\\,\\delta_{\\mathrm{op}}(\\mathfrak J)\\\\\n&+c\\,\\delta_c+d\\,\\delta_r.\n\\end{aligned}\n}\n\\]\nIn particular, if all local structural, operation, closure, causal, and\nreconstruction defects are \\(O(\\delta_{\\mathrm{rep}})\\), the relevant\noperator norms are uniformly bounded, and \\(g\\geq g_0>0\\), then\n\\[\n\\boxed{\nd_{\\mathrm{id}}(\\mathfrak R,\\mathfrak R')\n\\leq\nC_1\\frac{\\delta_{\\mathrm{rep}}}{g_0}\n+C_2\\delta_{\\mathrm{rep}}\n+O\\!\\left(\\frac{\\delta_{\\mathrm{rep}}^2}{g_0}\\right).\n}\n\\]\nAfter a fixed normalization of the non-spectral terms, this recovers the\nschematic target\n\\[\nd_{\\mathrm{id}}\n\\lesssim\nC\\frac{\\delta_{\\mathrm{rep}}}{g_0}.\n\\]\n\\end{theorem}","&+c\\,\\delta_c+d\\,\\delta_r.\n\\end{aligned}\n}\n\\]\nIn particular, if all local structural, operation, closure, causal, and\nreconstruction defects are \\(O(\\delta_{\\mathrm{rep}})\\), the relevant\noperator norms are uniformly bounded, and \\(g\\geq g_0>0\\), then\n\\[\n\\boxed{\nd_{\\mathrm{id}}(\\mathfrak R,\\mathfrak R')\n\\leq\nC_1\\frac{\\delta_{\\mathrm{rep}}}{g_0}\n+C_2\\delta_{\\mathrm{rep}}\n+O\\!\\left(\\frac{\\delta_{\\mathrm{rep}}^2}{g_0}\\right).\n}\n\\]\nAfter a fixed normalization of the non-spectral terms, this recovers the\nschematic target\n\\[\nd_{\\mathrm{id}}\n\\lesssim\nC\\frac{\\delta_{\\mathrm{rep}}}{g_0}.\n\\]\n\\end{theorem}","truction defects are \\(O(\\delta_{\\mathrm{rep}})\\), the relevant\noperator norms are uniformly bounded, and \\(g\\geq g_0>0\\), then\n\\[\n\\boxed{\nd_{\\mathrm{id}}(\\mathfrak R,\\mathfrak R')\n\\leq\nC_1\\frac{\\delta_{\\mathrm{rep}}}{g_0}\n+C_2\\delta_{\\mathrm{rep}}\n+O\\!\\left(\\frac{\\delta_{\\mathrm{rep}}^2}{g_0}\\right).\n}\n\\]\nAfter a fixed normalization of the non-spectral terms, this recovers the\nschematic target\n\\[\nd_{\\mathrm{id}}\n\\lesssim\nC\\frac{\\delta_{\\mathrm{rep}}}{g_0}.\n\\]\n\\end{theorem}","\\begin{proof}\nThe structural bound and existence of \\(\\beta\\) follow from\nTheorem~\\ref{thm:approx-structural-replacement}. The operation estimate is\nProposition~\\ref{prop:operation-transport-bound} with\n\\(q_{\\mathrm{str}}\\leq q_*\\). Insert these estimates and the assumed causal\nand reconstructive bounds into the definition of\n\\(\\operatorname{Cost}(\\mathfrak J)\\). The asymptotic statement follows from\n\\[\nq_*=O\\!\\left(\\frac{\\delta_D}{g_0}\n+\\frac{\\delta_D^2}{g_0}\\right)\n\\]\nand\n\\[\n1-\\sqrt{1-q_*^2}=O(q_*^2).\n\\]\n\\end{proof}","{proof}\nThe structural bound and existence of \\(\\beta\\) follow from\nTheorem~\\ref{thm:approx-structural-replacement}. The operation estimate is\nProposition~\\ref{prop:operation-transport-bound} with\n\\(q_{\\mathrm{str}}\\leq q_*\\). Insert these estimates and the assumed causal\nand reconstructive bounds into the definition of\n\\(\\operatorname{Cost}(\\mathfrak J)\\). The asymptotic statement follows from\n\\[\nq_*=O\\!\\left(\\frac{\\delta_D}{g_0}\n+\\frac{\\delta_D^2}{g_0}\\right)\n\\]\nand\n\\[\n1-\\sqrt{1-q_*^2}=O(q_*^2).\n\\]\n\\end{proof}","~\\ref{thm:approx-structural-replacement}. The operation estimate is\nProposition~\\ref{prop:operation-transport-bound} with\n\\(q_{\\mathrm{str}}\\leq q_*\\). Insert these estimates and the assumed causal\nand reconstructive bounds into the definition of\n\\(\\operatorname{Cost}(\\mathfrak J)\\). The asymptotic statement follows from\n\\[\nq_*=O\\!\\left(\\frac{\\delta_D}{g_0}\n+\\frac{\\delta_D^2}{g_0}\\right)\n\\]\nand\n\\[\n1-\\sqrt{1-q_*^2}=O(q_*^2).\n\\]\n\\end{proof}","\\begin{corollary}[Preservation of coalition frame bounds]\n\\label{cor:replacement-frame-bound}\nIdentify the old and new abstract cores with \\(\\ran P\\) and \\(\\ran P'\\)\nthrough their chosen core parameterizations. Let \\(S\\) be a declared coalition,\nlet\n\\[\nA_S:\\ran P\\to\\mathsf Y_S,\n\\qquad\nA_S':\\ran P'\\to\\mathsf Y_S'\n\\]\nbe the resulting accessible coalition encoders, and let \\(K_S\\) be a unitary\nidentification of their observation spaces. Suppose\n\\[\n\\|A_S'\\beta-K_SA_S\\|\n\\leq\n\\delta_{S}^{\\mathrm{frame}}\n<\n\\sqrt{\\alpha_S}.\n\\]\nThen the replacement coalition remains reconstructive and its new lower frame\nbound satisfies\n\\[\n\\boxed{\n\\alpha_S'\n\\geq\n\\left(\n\\sqrt{\\alpha_S}-\\delta_{S}^{\\mathrm{frame}}\n\\right)^2.\n}\n\\]\nConsequently, \\(k\\)-failure resilience survives whenever the bound holds\nuniformly for every coalition remaining after at most \\(k\\) er","\\begin{corollary}[Preservation of coalition frame bounds]\n\\label{cor:replacement-frame-bound}\nIdentify the old and new abstract cores with \\(\\ran P\\) and \\(\\ran P'\\)\nthrough their chosen core parameterizations. Let \\(S\\) be a declared coalition,\nlet\n\\[\nA_S:\\ran P\\to\\mathsf Y_S,\n\\qquad\nA_S':\\ran P'\\to\\mathsf Y_S'\n\\]\nbe the resulting accessible coalition encoders, and let \\(K_S\\) be a unitary\nidentification of their observation spaces. Suppose\n\\[\n\\|A_S'\\beta-K_SA_S\\|\n\\leq\n\\delta_{S}^{\\mathrm{frame}}\n<\n\\sqrt{\\alpha_S}.\n\\]\nThen the replacement coalition remains reconstructive and its new lower frame\nbound satisfies\n\\[\n\\boxed{\n\\alpha_S'\n\\geq\n\\left(\n\\sqrt{\\alpha_S}-\\delta_{S}^{\\mathrm{frame}}\n\\right)^2.\n}\n\\]\nConsequently, \\(k\\)-failure resilience survives whenever the bound holds\nuniformly for every coalition remaining after at most \\(k\\) erasures.\n\\end{cor","\\begin{corollary}[Preservation of coalition frame bounds]\n\\label{cor:replacement-frame-bound}\nIdentify the old and new abstract cores with \\(\\ran P\\) and \\(\\ran P'\\)\nthrough their chosen core parameterizations. Let \\(S\\) be a declared coalition,\nlet\n\\[\nA_S:\\ran P\\to\\mathsf Y_S,\n\\qquad\nA_S':\\ran P'\\to\\mathsf Y_S'\n\\]\nbe the resulting accessible coalition encoders, and let \\(K_S\\) be a unitary\nidentification of their observation spaces. Suppose\n\\[\n\\|A_S'\\beta-K_SA_S\\|\n\\leq\n\\delta_{S}^{\\mathrm{frame}}\n<\n\\sqrt{\\alpha_S}.\n\\]\nThen the replacement coalition remains reconstructive and its new lower frame\nbound satisfies\n\\[\n\\boxed{\n\\alpha_S'\n\\geq\n\\left(\n\\sqrt{\\alpha_S}-\\delta_{S}^{\\mathrm{frame}}\n\\right)^2.\n}\n\\]\nConsequently, \\(k\\)-failure resilience survives whenever the bound holds\nuniformly for every coalition remaining after at most \\(k\\) erasures.\n\\end{corollary}","e bounds]\n\\label{cor:replacement-frame-bound}\nIdentify the old and new abstract cores with \\(\\ran P\\) and \\(\\ran P'\\)\nthrough their chosen core parameterizations. Let \\(S\\) be a declared coalition,\nlet\n\\[\nA_S:\\ran P\\to\\mathsf Y_S,\n\\qquad\nA_S':\\ran P'\\to\\mathsf Y_S'\n\\]\nbe the resulting accessible coalition encoders, and let \\(K_S\\) be a unitary\nidentification of their observation spaces. Suppose\n\\[\n\\|A_S'\\beta-K_SA_S\\|\n\\leq\n\\delta_{S}^{\\mathrm{frame}}\n<\n\\sqrt{\\alpha_S}.\n\\]\nThen the replacement coalition remains reconstructive and its new lower frame\nbound satisfies\n\\[\n\\boxed{\n\\alpha_S'\n\\geq\n\\left(\n\\sqrt{\\alpha_S}-\\delta_{S}^{\\mathrm{frame}}\n\\right)^2.\n}\n\\]\nConsequently, \\(k\\)-failure resilience survives whenever the bound holds\nuniformly for every coalition remaining after at most \\(k\\) erasures.\n\\end{corollary}","be a declared coalition,\nlet\n\\[\nA_S:\\ran P\\to\\mathsf Y_S,\n\\qquad\nA_S':\\ran P'\\to\\mathsf Y_S'\n\\]\nbe the resulting accessible coalition encoders, and let \\(K_S\\) be a unitary\nidentification of their observation spaces. Suppose\n\\[\n\\|A_S'\\beta-K_SA_S\\|\n\\leq\n\\delta_{S}^{\\mathrm{frame}}\n<\n\\sqrt{\\alpha_S}.\n\\]\nThen the replacement coalition remains reconstructive and its new lower frame\nbound satisfies\n\\[\n\\boxed{\n\\alpha_S'\n\\geq\n\\left(\n\\sqrt{\\alpha_S}-\\delta_{S}^{\\mathrm{frame}}\n\\right)^2.\n}\n\\]\nConsequently, \\(k\\)-failure resilience survives whenever the bound holds\nuniformly for every coalition remaining after at most \\(k\\) erasures.\n\\end{corollary}","lting accessible coalition encoders, and let \\(K_S\\) be a unitary\nidentification of their observation spaces. Suppose\n\\[\n\\|A_S'\\beta-K_SA_S\\|\n\\leq\n\\delta_{S}^{\\mathrm{frame}}\n<\n\\sqrt{\\alpha_S}.\n\\]\nThen the replacement coalition remains reconstructive and its new lower frame\nbound satisfies\n\\[\n\\boxed{\n\\alpha_S'\n\\geq\n\\left(\n\\sqrt{\\alpha_S}-\\delta_{S}^{\\mathrm{frame}}\n\\right)^2.\n}\n\\]\nConsequently, \\(k\\)-failure resilience survives whenever the bound holds\nuniformly for every coalition remaining after at most \\(k\\) erasures.\n\\end{corollary}","pose\n\\[\n\\|A_S'\\beta-K_SA_S\\|\n\\leq\n\\delta_{S}^{\\mathrm{frame}}\n<\n\\sqrt{\\alpha_S}.\n\\]\nThen the replacement coalition remains reconstructive and its new lower frame\nbound satisfies\n\\[\n\\boxed{\n\\alpha_S'\n\\geq\n\\left(\n\\sqrt{\\alpha_S}-\\delta_{S}^{\\mathrm{frame}}\n\\right)^2.\n}\n\\]\nConsequently, \\(k\\)-failure resilience survives whenever the bound holds\nuniformly for every coalition remaining after at most \\(k\\) erasures.\n\\end{corollary}","\\begin{proof}\nFor \\(w'\\in\\ran P'\\), write \\(w'=\\beta w\\). Since \\(\\beta\\) and \\(K_S\\) are\nunitary,\n\\[\n\\begin{aligned}\n\\|A_S'w'\\|\n&\\geq\n\\|K_SA_Sw\\|\n-\n\\|(A_S'\\beta-K_SA_S)w\\|","\\begin{proof}\nFor \\(w'\\in\\ran P'\\), write \\(w'=\\beta w\\). Since \\(\\beta\\) and \\(K_S\\) are\nunitary,\n\\[\n\\begin{aligned}\n\\|A_S'w'\\|\n&\\geq\n\\|K_SA_Sw\\|\n-\n\\|(A_S'\\beta-K_SA_S)w\\|\\\\\n&\\geq\n\\left(\n\\sqrt{\\alpha_S}-\\delta_{S}^{\\mathrm{frame}}\n\\right)\\|w'\\|.\n\\end{aligned}\n\\]\nSquare the inequality.\n\\end{proof}","\\begin{corollary}[Exact carrier replacement]\\label{cor:exact-carrier-replacement}\nIf\n\\[\n\\delta_D=0,\n\\qquad\n\\delta_\\rho(a)=0,\n\\qquad\n\\ell_a=0,\n\\qquad\n\\delta_c=0,\n\\qquad\n\\delta_r=0,\n\\]\nthen\n\\[\nd_{\\mathrm{id}}(\\mathfrak R,\\mathfrak R')=0.\n\\]\nThus a participant, model, artifact, or notation system may be replaced without\nchanging the abstract workspace when the replacement realizes the same\nstructural, operational, causal, and reconstructive organization.\n\\end{corollary}","For a sequence of realizations\n\\[\n\\mathfrak R_0,\\mathfrak R_1,\\ldots,\\mathfrak R_T,\n\\]\nlet \\(\\mathfrak J_t:\\mathfrak R_t\\to\\mathfrak R_{t+1}\\) be admissible\ncorrespondences. Define the path length\n\\[\n\\operatorname{Len}(\\mathfrak J_{0:T})\n=\n\\sum_{t=0}^{T-1}\\operatorname{Cost}(\\mathfrak J_t)\n\\]\nand the induced directed path quasi-pseudometric\n\\[\n\\boxed{\nd_{\\mathrm{path}}^{\\rightarrow}(\\mathfrak R_0,\\mathfrak R_T)\n=\n\\inf_{\\mathfrak J_{0:T}}\n\\operatorname{Len}(\\mathfrak J_{0:T}).\n}\n\\]\nConcatenation gives the directed triangle inequality. Because replacement\ncosts and intervention divergences need not be symmetric, symmetry is not\nassumed. When reverse correspondences exist, define the symmetrized\npseudometric\n\\[\n\\boxed{\nd_{\\mathrm{path}}^{\\mathrm{sym}}(\\mathfrak R,\\mathfrak R')","For a sequence of realizations\n\\[\n\\mathfrak R_0,\\mathfrak R_1,\\ldots,\\mathfrak R_T,\n\\]\nlet \\(\\mathfrak J_t:\\mathfrak R_t\\to\\mathfrak R_{t+1}\\) be admissible\ncorrespondences. Define the path length\n\\[\n\\operatorname{Len}(\\mathfrak J_{0:T})\n=\n\\sum_{t=0}^{T-1}\\operatorname{Cost}(\\mathfrak J_t)\n\\]\nand the induced directed path quasi-pseudometric\n\\[\n\\boxed{\nd_{\\mathrm{path}}^{\\rightarrow}(\\mathfrak R_0,\\mathfrak R_T)\n=\n\\inf_{\\mathfrak J_{0:T}}\n\\operatorname{Len}(\\mathfrak J_{0:T}).\n}\n\\]\nConcatenation gives the directed triangle inequality. Because replacement\ncosts and intervention divergences need not be symmetric, symmetry is not\nassumed. When reverse correspondences exist, define the symmetrized\npseudometric\n\\[\n\\boxed{\nd_{\\mathrm{path}}^{\\mathrm{sym}}(\\mathfrak R,\\mathfrak R')\n=\n\\max\\left\\{\n d_{\\mathrm{path}}^{\\rightarrow}(\\mathfra","For a sequence of realizations\n\\[\n\\mathfrak R_0,\\mathfrak R_1,\\ldots,\\mathfrak R_T,\n\\]\nlet \\(\\mathfrak J_t:\\mathfrak R_t\\to\\mathfrak R_{t+1}\\) be admissible\ncorrespondences. Define the path length\n\\[\n\\operatorname{Len}(\\mathfrak J_{0:T})\n=\n\\sum_{t=0}^{T-1}\\operatorname{Cost}(\\mathfrak J_t)\n\\]\nand the induced directed path quasi-pseudometric\n\\[\n\\boxed{\nd_{\\mathrm{path}}^{\\rightarrow}(\\mathfrak R_0,\\mathfrak R_T)\n=\n\\inf_{\\mathfrak J_{0:T}}\n\\operatorname{Len}(\\mathfrak J_{0:T}).\n}\n\\]\nConcatenation gives the directed triangle inequality. Because replacement\ncosts and intervention divergences need not be symmetric, symmetry is not\nassumed. When reverse correspondences exist, define the symmetrized\npseudometric\n\\[\n\\boxed{\nd_{\\mathrm{path}}^{\\mathrm{sym}}(\\mathfrak R,\\mathfrak R')\n=\n\\max\\left\\{\n d_{\\mathrm{path}}^{\\rightarrow}(\\mathfrak R,\\mathfrak R'),\n d_{\\mathrm{path}}^{\\rightarrow}(\\mathfrak R',\\mathfrak R)\n\\right\\}.\n}\n\\]\nDistinct physical realizations can have zero symmetrized p","For a sequence of realizations\n\\[\n\\mathfrak R_0,\\mathfrak R_1,\\ldots,\\mathfrak R_T,\n\\]\nlet \\(\\mathfrak J_t:\\mathfrak R_t\\to\\mathfrak R_{t+1}\\) be admissible\ncorrespondences. Define the path length\n\\[\n\\operatorname{Len}(\\mathfrak J_{0:T})\n=\n\\sum_{t=0}^{T-1}\\operatorname{Cost}(\\mathfrak J_t)\n\\]\nand the induced directed path quasi-pseudometric\n\\[\n\\boxed{\nd_{\\mathrm{path}}^{\\rightarrow}(\\mathfrak R_0,\\mathfrak R_T)\n=\n\\inf_{\\mathfrak J_{0:T}}\n\\operatorname{Len}(\\mathfrak J_{0:T}).\n}\n\\]\nConcatenation gives the directed triangle inequality. Because replacement\ncosts and intervention divergences need not be symmetric, symmetry is not\nassumed. When reverse correspondences exist, define the symmetrized\npseudometric\n\\[\n\\boxed{\nd_{\\mathrm{path}}^{\\mathrm{sym}}(\\mathfrak R,\\mathfrak R')\n=\n\\max\\left\\{\n d_{\\mathrm{path}}^{\\rightarrow}(\\mathfrak R,\\mathfrak R'),\n d_{\\mathrm{path}}^{\\rightarrow}(\\mathfrak R',\\mathfrak R)\n\\right\\}.\n}\n\\]\nDistinct physical realizations can have zero symmetrized pseudodistance when\nan exact correspondence exists in both directions. This is intentional:\nidentity is assign…","d path quasi-pseudometric\n\\[\n\\boxed{\nd_{\\mathrm{path}}^{\\rightarrow}(\\mathfrak R_0,\\mathfrak R_T)\n=\n\\inf_{\\mathfrak J_{0:T}}\n\\operatorname{Len}(\\mathfrak J_{0:T}).\n}\n\\]\nConcatenation gives the directed triangle inequality. Because replacement\ncosts and intervention divergences need not be symmetric, symmetry is not\nassumed. When reverse correspondences exist, define the symmetrized\npseudometric\n\\[\n\\boxed{\nd_{\\mathrm{path}}^{\\mathrm{sym}}(\\mathfrak R,\\mathfrak R')\n=\n\\max\\left\\{\n d_{\\mathrm{path}}^{\\rightarrow}(\\mathfrak R,\\mathfrak R'),\n d_{\\mathrm{path}}^{\\rightarrow}(\\mathfrak R',\\mathfrak R)\n\\right\\}.\n}\n\\]\nDistinct physical realizations can have zero symmetrized pseudodistance when\nan exact correspondence exists in both directions. This is intentional:\nidentity is assigned to the invariant relational organization rather than to\nthe carrier tokens.","A complete replacement audit must report at least\n\\[\n\\boxed{\n\\mathcal I_{\\mathrm{rep}}\n=\n\\left(\nq_{\\mathrm{str}},\n\\delta_{\\mathrm{op}},\n\\delta_{\\mathrm{causal}},\n\\delta_{\\mathrm{rec}},\n\\mu^{\\mathrm{keep}}\n\\right).\n}\n\\]\nThe failure modes are different:\n\\[\n\\begin{array}{ll}\nq_{\\mathrm{str}}\\text{ large}\n&\\text{the shared subspace changed},","A complete replacement audit must report at least\n\\[\n\\boxed{\n\\mathcal I_{\\mathrm{rep}}\n=\n\\left(\nq_{\\mathrm{str}},\n\\delta_{\\mathrm{op}},\n\\delta_{\\mathrm{causal}},\n\\delta_{\\mathrm{rec}},\n\\mu^{\\mathrm{keep}}\n\\right).\n}\n\\]\nThe failure modes are different:\n\\[\n\\begin{array}{ll}\nq_{\\mathrm{str}}\\text{ large}\n&\\text{the shared subspace changed},\\\\\n\\delta_{\\mathrm{op}}\\text{ large}\n&\\text{the same operations are no longer realized},\\\\\n\\delta_{\\mathrm{causal}}\\text{ large}\n&\\text{interventions no longer have matching consequences},\\\\\n\\delta_{\\mathrm{rec}}\\text{ large}\n&\\text{the new realization was not initialized from the old core},\\\\\n\\mu^{\\mathrm{keep}}\\text{ small}\n&\\text{content collapsed despite apparent coherence}.\n\\end{array}\n\\]\nThe theorem does not permit one successful coordinate to conceal failure of\nanother.","Let\n\\[\nG=(V,E)\n\\]\nbe a finite directed interaction graph with local core spaces \\(C_i\\), core\ntransfers \\(H_e\\), vertex space \\(\\mathcal C_V(G)\\), discrepancy operator\n\\(D_G\\), and exact workspace\n\\[\n\\mathscr W(G)\n:=\n\\Gamma(H|_G)\n=\n\\ker D_G.\n\\]","\\begin{definition}[Finite workspace cover]\\label{def:workspace-cover}\nA finite family of subgraphs\n\\[\n\\mathcal U=\\{G_\\alpha\\}_{\\alpha\\in A}\n\\]\nis an \\emph{edge-complete workspace cover} of \\(G\\) when\n\\[\nV=\\bigcup_{\\alpha\\in A}V_\\alpha,\n\\qquad\nE=\\bigcup_{\\alpha\\in A}E_\\alpha.\n\\]\nEvery edge in \\(E_\\alpha\\) carries the transfer inherited from \\(G\\). For a\nnonempty finite index set\n\\[\nI=\\{\\alpha_0,\\ldots,\\alpha_p\\}\\subseteq A,\n\\]\ndefine the intersection subgraph\n\\[\nG_I\n=\nG_{\\alpha_0\\cdots\\alpha_p}\n:=\n\\bigcap_{\\alpha\\in I}G_\\alpha\n\\]\nby intersecting both vertex and edge sets. Isolated vertices are retained.\n\\end{definition}","ver}\nA finite family of subgraphs\n\\[\n\\mathcal U=\\{G_\\alpha\\}_{\\alpha\\in A}\n\\]\nis an \\emph{edge-complete workspace cover} of \\(G\\) when\n\\[\nV=\\bigcup_{\\alpha\\in A}V_\\alpha,\n\\qquad\nE=\\bigcup_{\\alpha\\in A}E_\\alpha.\n\\]\nEvery edge in \\(E_\\alpha\\) carries the transfer inherited from \\(G\\). For a\nnonempty finite index set\n\\[\nI=\\{\\alpha_0,\\ldots,\\alpha_p\\}\\subseteq A,\n\\]\ndefine the intersection subgraph\n\\[\nG_I\n=\nG_{\\alpha_0\\cdots\\alpha_p}\n:=\n\\bigcap_{\\alpha\\in I}G_\\alpha\n\\]\nby intersecting both vertex and edge sets. Isolated vertices are retained.\n\\end{definition}","If \\(K\\subseteq L\\subseteq G\\) are subgraphs, coordinate restriction gives\n\\[\n\\rho_K^L:\\mathcal C_V(L)\\longrightarrow\\mathcal C_V(K).\n\\]\nBecause every edge equation on \\(K\\) is also an edge equation on \\(L\\),\nrestriction maps exact sections to exact sections:\n\\[\nr_K^L\n:=\n\\rho_K^L|_{\\mathscr W(L)}:\n\\mathscr W(L)\\longrightarrow\\mathscr W(K).\n\\]\nThey obey\n\\[\nr_L^L=I,\n\\qquad\nr_J^K r_K^L=r_J^L\n\\]\nwhenever \\(J\\subseteq K\\subseteq L\\). Thus \\(K\\mapsto\\mathscr W(K)\\) is a\ncontravariant system of local workspaces.","\\subseteq L\\subseteq G\\) are subgraphs, coordinate restriction gives\n\\[\n\\rho_K^L:\\mathcal C_V(L)\\longrightarrow\\mathcal C_V(K).\n\\]\nBecause every edge equation on \\(K\\) is also an edge equation on \\(L\\),\nrestriction maps exact sections to exact sections:\n\\[\nr_K^L\n:=\n\\rho_K^L|_{\\mathscr W(L)}:\n\\mathscr W(L)\\longrightarrow\\mathscr W(K).\n\\]\nThey obey\n\\[\nr_L^L=I,\n\\qquad\nr_J^K r_K^L=r_J^L\n\\]\nwhenever \\(J\\subseteq K\\subseteq L\\). Thus \\(K\\mapsto\\mathscr W(K)\\) is a\ncontravariant system of local workspaces.","aphs, coordinate restriction gives\n\\[\n\\rho_K^L:\\mathcal C_V(L)\\longrightarrow\\mathcal C_V(K).\n\\]\nBecause every edge equation on \\(K\\) is also an edge equation on \\(L\\),\nrestriction maps exact sections to exact sections:\n\\[\nr_K^L\n:=\n\\rho_K^L|_{\\mathscr W(L)}:\n\\mathscr W(L)\\longrightarrow\\mathscr W(K).\n\\]\nThey obey\n\\[\nr_L^L=I,\n\\qquad\nr_J^K r_K^L=r_J^L\n\\]\nwhenever \\(J\\subseteq K\\subseteq L\\). Thus \\(K\\mapsto\\mathscr W(K)\\) is a\ncontravariant system of local workspaces.","The nerve \\(N(\\mathcal U)\\) has one vertex for each chart \\(G_\\alpha\\) and a\n\\(p\\)-simplex\n\\[\n[\\alpha_0\\cdots\\alpha_p]\n\\]\nwhenever \\(G_{\\alpha_0\\cdots\\alpha_p}\\) is nonempty. The nerve records which\nlocal workspaces can compare states directly. It does not replace the\nunderlying interaction graph: \\(G\\) carries the internal transfer equations,\nwhile \\(N(\\mathcal U)\\) carries overlap-consistency equations between already\nconstructed local objects.","For pairwise overlaps write\n\\[\n\\mathscr W_\\alpha=\\mathscr W(G_\\alpha),\n\\qquad\n\\mathscr W_{\\alpha\\beta}=\\mathscr W(G_{\\alpha\\beta}),\n\\]\nand\n\\[\nr_{\\alpha\\beta}^{\\alpha}:\n\\mathscr W_\\alpha\\to\\mathscr W_{\\alpha\\beta},\n\\qquad\nr_{\\alpha\\beta}^{\\beta}:\n\\mathscr W_\\beta\\to\\mathscr W_{\\alpha\\beta}.\n\\]\nThe interface is therefore a cospan\n\\[\n\\mathscr W_\\alpha\n\\xrightarrow{\\ r_{\\alpha\\beta}^{\\alpha}\\ }\n\\mathscr W_{\\alpha\\beta}\n\\xleftarrow{\\ r_{\\alpha\\beta}^{\\beta}\\ }\n\\mathscr W_\\beta,\n\\]\nnot necessarily a direct transfer map from one complete local workspace to the\nother. Both charts are compared through the content they expose on their\ncommon domain.","For \\(p\\geq0\\), define the finite cochain space\n\\[\n\\check C^p(\\mathcal U;\\mathscr W)\n=\n\\bigoplus_{\\alpha_0<\\cdots<\\alpha_p}\n\\mathscr W(G_{\\alpha_0\\cdots\\alpha_p}),\n\\]\nwhere empty intersections contribute the zero space. The coboundary\n\\[\n\\delta^p:\\check C^p\\longrightarrow\\check C^{p+1}\n\\]\nis\n\\[\n\\boxed{\n(\\delta^p s)_{\\alpha_0\\cdots\\alpha_{p+1}}\n=\n\\sum_{j=0}^{p+1}(-1)^j\nr_{\\alpha_0\\cdots\\alpha_{p+1}}^%\n{\\alpha_0\\cdots\\widehat{\\alpha_j}\\cdots\\alpha_{p+1}}\n%\ns_{\\alpha_0\\cdots\\widehat{\\alpha_j}\\cdots\\alpha_{p+1}}.\n}\n\\]\nThe restriction identities imply\n\\[\n\\delta^{p+1}\\delta^p=0.\n\\]\nFor a local family \\(w=(w_\\alpha)_\\alpha\\),\n\\[\n\\boxed{\n(\\delta^0w)_{\\alpha\\beta}\n=\nr_{\\alpha\\beta}^{\\beta}w_\\beta\n-\nr_{\\alpha\\bet","For \\(p\\geq0\\), define the finite cochain space\n\\[\n\\check C^p(\\mathcal U;\\mathscr W)\n=\n\\bigoplus_{\\alpha_0<\\cdots<\\alpha_p}\n\\mathscr W(G_{\\alpha_0\\cdots\\alpha_p}),\n\\]\nwhere empty intersections contribute the zero space. The coboundary\n\\[\n\\delta^p:\\check C^p\\longrightarrow\\check C^{p+1}\n\\]\nis\n\\[\n\\boxed{\n(\\delta^p s)_{\\alpha_0\\cdots\\alpha_{p+1}}\n=\n\\sum_{j=0}^{p+1}(-1)^j\nr_{\\alpha_0\\cdots\\alpha_{p+1}}^%\n{\\alpha_0\\cdots\\widehat{\\alpha_j}\\cdots\\alpha_{p+1}}\n%\ns_{\\alpha_0\\cdots\\widehat{\\alpha_j}\\cdots\\alpha_{p+1}}.\n}\n\\]\nThe restriction identities imply\n\\[\n\\delta^{p+1}\\delta^p=0.\n\\]\nFor a local family \\(w=(w_\\alpha)_\\alpha\\),\n\\[\n\\boxed{\n(\\delta^0w)_{\\alpha\\beta}\n=\nr_{\\alpha\\beta}^{\\beta}w_\\beta\n-\nr_{\\alpha\\beta}^{\\alpha}w_\\alpha.\n}\n\\]\nThus \\(\\delta^0\\) is the overlap-discrepancy operator at the level of exact\nlocal workspaces.","pha_p}),\n\\]\nwhere empty intersections contribute the zero space. The coboundary\n\\[\n\\delta^p:\\check C^p\\longrightarrow\\check C^{p+1}\n\\]\nis\n\\[\n\\boxed{\n(\\delta^p s)_{\\alpha_0\\cdots\\alpha_{p+1}}\n=\n\\sum_{j=0}^{p+1}(-1)^j\nr_{\\alpha_0\\cdots\\alpha_{p+1}}^%\n{\\alpha_0\\cdots\\widehat{\\alpha_j}\\cdots\\alpha_{p+1}}\n%\ns_{\\alpha_0\\cdots\\widehat{\\alpha_j}\\cdots\\alpha_{p+1}}.\n}\n\\]\nThe restriction identities imply\n\\[\n\\delta^{p+1}\\delta^p=0.\n\\]\nFor a local family \\(w=(w_\\alpha)_\\alpha\\),\n\\[\n\\boxed{\n(\\delta^0w)_{\\alpha\\beta}\n=\nr_{\\alpha\\beta}^{\\beta}w_\\beta\n-\nr_{\\alpha\\beta}^{\\alpha}w_\\alpha.\n}\n\\]\nThus \\(\\delta^0\\) is the overlap-discrepancy operator at the level of exact\nlocal workspaces.","the zero space. The coboundary\n\\[\n\\delta^p:\\check C^p\\longrightarrow\\check C^{p+1}\n\\]\nis\n\\[\n\\boxed{\n(\\delta^p s)_{\\alpha_0\\cdots\\alpha_{p+1}}\n=\n\\sum_{j=0}^{p+1}(-1)^j\nr_{\\alpha_0\\cdots\\alpha_{p+1}}^%\n{\\alpha_0\\cdots\\widehat{\\alpha_j}\\cdots\\alpha_{p+1}}\n%\ns_{\\alpha_0\\cdots\\widehat{\\alpha_j}\\cdots\\alpha_{p+1}}.\n}\n\\]\nThe restriction identities imply\n\\[\n\\delta^{p+1}\\delta^p=0.\n\\]\nFor a local family \\(w=(w_\\alpha)_\\alpha\\),\n\\[\n\\boxed{\n(\\delta^0w)_{\\alpha\\beta}\n=\nr_{\\alpha\\beta}^{\\beta}w_\\beta\n-\nr_{\\alpha\\beta}^{\\alpha}w_\\alpha.\n}\n\\]\nThus \\(\\delta^0\\) is the overlap-discrepancy operator at the level of exact\nlocal workspaces.","ndary\n\\[\n\\delta^p:\\check C^p\\longrightarrow\\check C^{p+1}\n\\]\nis\n\\[\n\\boxed{\n(\\delta^p s)_{\\alpha_0\\cdots\\alpha_{p+1}}\n=\n\\sum_{j=0}^{p+1}(-1)^j\nr_{\\alpha_0\\cdots\\alpha_{p+1}}^%\n{\\alpha_0\\cdots\\widehat{\\alpha_j}\\cdots\\alpha_{p+1}}\n%\ns_{\\alpha_0\\cdots\\widehat{\\alpha_j}\\cdots\\alpha_{p+1}}.\n}\n\\]\nThe restriction identities imply\n\\[\n\\delta^{p+1}\\delta^p=0.\n\\]\nFor a local family \\(w=(w_\\alpha)_\\alpha\\),\n\\[\n\\boxed{\n(\\delta^0w)_{\\alpha\\beta}\n=\nr_{\\alpha\\beta}^{\\beta}w_\\beta\n-\nr_{\\alpha\\beta}^{\\alpha}w_\\alpha.\n}\n\\]\nThus \\(\\delta^0\\) is the overlap-discrepancy operator at the level of exact\nlocal workspaces.","=\n\\sum_{j=0}^{p+1}(-1)^j\nr_{\\alpha_0\\cdots\\alpha_{p+1}}^%\n{\\alpha_0\\cdots\\widehat{\\alpha_j}\\cdots\\alpha_{p+1}}\n%\ns_{\\alpha_0\\cdots\\widehat{\\alpha_j}\\cdots\\alpha_{p+1}}.\n}\n\\]\nThe restriction identities imply\n\\[\n\\delta^{p+1}\\delta^p=0.\n\\]\nFor a local family \\(w=(w_\\alpha)_\\alpha\\),\n\\[\n\\boxed{\n(\\delta^0w)_{\\alpha\\beta}\n=\nr_{\\alpha\\beta}^{\\beta}w_\\beta\n-\nr_{\\alpha\\beta}^{\\alpha}w_\\alpha.\n}\n\\]\nThus \\(\\delta^0\\) is the overlap-discrepancy operator at the level of exact\nlocal workspaces.","A family \\(w=(w_\\alpha)_\\alpha\\) is exactly compatible when\n\\[\n\\delta^0w=0.\n\\]\nIts direct mismatch is\n\\[\n\\boxed{\n\\mathfrak m_{\\mathcal U}(w)=\\delta^0w.\n}\n\\]\nAfter choosing inner products and positive overlap weights, the mismatch\nenergy is\n\\[\n\\boxed{\n\\mathcal E_{\\mathrm{ov}}(w)\n=\n\\|\\delta^0w\\|^2\n=\n\\sum_{\\alpha<\\beta}\n\\nu_{\\alpha\\beta}\n\\left\\|\n r_{\\alpha\\beta}^{\\beta}w_\\beta\n-r_{\\alpha\\beta}^{\\alpha}w_\\alpha\n\\right\\|^2.\n}\n\\]\nThis is the appropriate obstruction when the full local states are already\ngiven. Such a family glues exactly if and only if its mismatch vector itself\nvanishes. No cohomological quotient is needed for that direct question.","A different problem arises when only relative transition data are supplied.\nLet\n\\[\nb=(b_{\\alpha\\beta})\\in\\check C^1(\\mathcal U;\\mathscr W)\n\\]\nbe a prescribed family of overlap offsets. The triple-overlap consistency\ncondition is\n\\[\n\\delta^1b=0.\n\\]\nOne seeks local potentials \\(q=(q_\\alpha)\\in\\check C^0\\) satisfying\n\\[\n\\delta^0q=b.\n\\]\nDefine\n\\[\n\\boxed{\n\\check H^1(\\mathcal U;\\mathscr W)\n=\n\\frac{\\ker\\delta^1}{\\operatorname{im}\\delta^0}.\n}\n\\]\nThen the affine gluing problem is solvable precisely when\n\\[\n\\boxed{[b]=0\\quad\\text{in}\\quad\\check H^1(\\mathcal U;\\mathscr W).}\n\\]\nA nonzero class means that all local transition statements can be mutually\nconsistent on triple overlaps while still failing to arise from any family of\nlocal workspace states. This is a cover-level analogue of nontrivial holonomy.","\\in\\check C^1(\\mathcal U;\\mathscr W)\n\\]\nbe a prescribed family of overlap offsets. The triple-overlap consistency\ncondition is\n\\[\n\\delta^1b=0.\n\\]\nOne seeks local potentials \\(q=(q_\\alpha)\\in\\check C^0\\) satisfying\n\\[\n\\delta^0q=b.\n\\]\nDefine\n\\[\n\\boxed{\n\\check H^1(\\mathcal U;\\mathscr W)\n=\n\\frac{\\ker\\delta^1}{\\operatorname{im}\\delta^0}.\n}\n\\]\nThen the affine gluing problem is solvable precisely when\n\\[\n\\boxed{[b]=0\\quad\\text{in}\\quad\\check H^1(\\mathcal U;\\mathscr W).}\n\\]\nA nonzero class means that all local transition statements can be mutually\nconsistent on triple overlaps while still failing to arise from any family of\nlocal workspace states. This is a cover-level analogue of nontrivial holonomy.","Equip the finite cochain spaces with Hilbert inner products and define the\nfirst \\v{C}ech Laplacian\n\\[\n\\boxed{\n\\Delta_1\n=\n\\delta^0(\\delta^0)^*\n+\n(\\delta^1)^*\\delta^1.\n}\n\\]\nLet\n\\[\n\\mathcal H^1_{\\mathcal U}=\\ker\\Delta_1.\n\\]","\\begin{proposition}[Harmonic gluing obstruction]\\label{prop:harmonic-gluing-obstruction}\nFor every cocycle \\(b\\in\\ker\\delta^1\\), there is an orthogonal decomposition\n\\[\n\\boxed{\nb=\\delta^0q^\\dagger+h,\n\\qquad\nh\\in\\mathcal H^1_{\\mathcal U},\n}\n\\]\nwhere \\(q^\\dagger=(\\delta^0)^\\dagger b\\) is a least-squares potential. Moreover,\n\\[\n\\boxed{\n\\inf_q\\|b-\\delta^0q\\|\n=\n\\|h\\|\n=\n\\|P_{\\mathcal H^1_{\\mathcal U}}b\\|.\n}\n\\]\nHence \\(b\\) is exactly integrable if and only if its harmonic component\nvanishes.\n\\end{proposition}","on}[Harmonic gluing obstruction]\\label{prop:harmonic-gluing-obstruction}\nFor every cocycle \\(b\\in\\ker\\delta^1\\), there is an orthogonal decomposition\n\\[\n\\boxed{\nb=\\delta^0q^\\dagger+h,\n\\qquad\nh\\in\\mathcal H^1_{\\mathcal U},\n}\n\\]\nwhere \\(q^\\dagger=(\\delta^0)^\\dagger b\\) is a least-squares potential. Moreover,\n\\[\n\\boxed{\n\\inf_q\\|b-\\delta^0q\\|\n=\n\\|h\\|\n=\n\\|P_{\\mathcal H^1_{\\mathcal U}}b\\|.\n}\n\\]\nHence \\(b\\) is exactly integrable if and only if its harmonic component\nvanishes.\n\\end{proposition}","\\begin{proof}\nThe finite-dimensional Hodge decomposition gives\n\\[\n\\check C^1\n=\n\\operatorname{im}\\delta^0\n\\oplus\n\\ker\\Delta_1\n\\oplus\n\\operatorname{im}(\\delta^1)^*.\n\\]\nIf \\(b\\in\\ker\\delta^1\\), then \\(b\\) is orthogonal to\n\\(\\operatorname{im}(\\delta^1)^*\\). Its remaining two orthogonal components are\ntherefore an exact part \\(\\delta^0q^\\dagger\\) and a harmonic part \\(h\\). The\northogonal projection theorem gives the least-squares identity.\n\\end{proof}","The scalar\n\\[\n\\omega_{\\mathcal U}(b)\n:=\n\\|P_{\\mathcal H^1_{\\mathcal U}}b\\|\n\\]\nis an irreducible overlap-frustration measure. Its vanishing is\ninvariant under admissible invertible coordinate changes, although its numerical\nnorm depends on the chosen metric. It cannot be removed by adding local\npotentials alone.","Consider three charts arranged in a cycle, with one-dimensional overlap\nspaces and oriented offsets\n\\[\nb_{12},\\qquad b_{23},\\qquad b_{31}.\n\\]\nLocal potentials would have to satisfy\n\\[\nq_2-q_1=b_{12},\n\\qquad\nq_3-q_2=b_{23},\n\\qquad\nq_1-q_3=b_{31}.\n\\]\nAdding the equations gives the necessary and sufficient condition\n\\[\n\\boxed{\nb_{12}+b_{23}+b_{31}=0.}\n\\]\nWith the standard edge norm, the harmonic obstruction has magnitude\n\\[\n\\boxed{\n\\omega_{\\mathcal U}(b)\n=\n\\frac{|b_{12}+b_{23}+b_{31}|}{\\sqrt3}.\n}\n\\]\nEvery pairwise relation may be individually meaningful while the complete\ncycle remains globally inconsistent.","ranged in a cycle, with one-dimensional overlap\nspaces and oriented offsets\n\\[\nb_{12},\\qquad b_{23},\\qquad b_{31}.\n\\]\nLocal potentials would have to satisfy\n\\[\nq_2-q_1=b_{12},\n\\qquad\nq_3-q_2=b_{23},\n\\qquad\nq_1-q_3=b_{31}.\n\\]\nAdding the equations gives the necessary and sufficient condition\n\\[\n\\boxed{\nb_{12}+b_{23}+b_{31}=0.}\n\\]\nWith the standard edge norm, the harmonic obstruction has magnitude\n\\[\n\\boxed{\n\\omega_{\\mathcal U}(b)\n=\n\\frac{|b_{12}+b_{23}+b_{31}|}{\\sqrt3}.\n}\n\\]\nEvery pairwise relation may be individually meaningful while the complete\ncycle remains globally inconsistent.","Let\n\\[\nR_{\\mathcal U}:\\mathscr W(G)\n\\longrightarrow\n\\check C^0(\\mathcal U;\\mathscr W)\n\\]\nbe the restriction map\n\\[\nR_{\\mathcal U}(w)\n=\n(w|_{G_\\alpha})_{\\alpha\\in A}.\n\\]","\\begin{theorem}[Exact workspace gluing]\\label{thm:exact-workspace-gluing}\nFor every finite edge-complete workspace cover \\(\\mathcal U\\), the sequence\n\\[\n\\boxed{\n0\n\\longrightarrow\n\\mathscr W(G)\n\\xrightarrow{\\ R_{\\mathcal U}\\ }\n\\check C^0(\\mathcal U;\\mathscr W)\n\\xrightarrow{\\ \\delta^0\\ }\n\\check C^1(\\mathcal U;\\mathscr W)\n}\n\\]\nis exact. Equivalently,\n\\[\n\\boxed{\n\\mathscr W(G)\n\\cong\n\\ker\\delta^0\n\\cong\n\\operatorname{Eq}\n\\left(\n\\prod_\\alpha\\mathscr W_\\alpha\n\\rightrightarrows\n\\prod_{\\alpha<\\beta}\\mathscr W_{\\alpha\\beta}\n\\right).\n}\n\\]\nEvery pairwise compatible family of exact local workspace states therefore\nglues to a unique exact global workspace state.\n\\end{theorem}","\\begin{proof}\nThe cover contains every vertex, so \\(R_{\\mathcal U}\\) is injective. A global\nsection has identical restrictions on every overlap, hence\n\\(\\delta^0R_{\\mathcal U}=0\\).","Conversely, let \\(w=(w_\\alpha)_\\alpha\\in\\ker\\delta^0\\). For each vertex\n\\(i\\in V\\), choose any chart \\(G_\\alpha\\) containing \\(i\\) and set\n\\[\nc_i=(w_\\alpha)_i.\n\\]\nIf another chart \\(G_\\beta\\) contains \\(i\\), then \\(i\\in G_{\\alpha\\beta}\\)\nand overlap compatibility gives\n\\((w_\\alpha)_i=(w_\\beta)_i\\). Thus \\(c_i\\) is well defined.","For every edge \\(e:i\\to j\\), edge completeness provides a chart\n\\(G_\\alpha\\) containing \\(e\\). Since \\(w_\\alpha\\in\\mathscr W(G_\\alpha)\\),\n\\[\nc_j=(w_\\alpha)_j=H_e(w_\\alpha)_i=H_ec_i.\n\\]\nHence \\(c=(c_i)\\in\\mathscr W(G)\\), and by construction\n\\(R_{\\mathcal U}c=w\\). Uniqueness follows because the charts cover all\nvertices.\n\\end{proof}","\\begin{corollary}[Zeroth descent cohomology]\\label{cor:zeroth-descent}\nFor every finite edge-complete cover,\n\\[\n\\boxed{\n\\mathscr W(G)\n\\cong\n\\check H^0(\\mathcal U;\\mathscr W)\n=\n\\ker\\delta^0.\n}\n\\]\nIn finite dimensions,\n\\[\n\\boxed{\n\\dim\\mathscr W(G)\n=\n\\sum_{\\alpha\\in A}\\dim\\mathscr W_\\alpha\n-\n\\operatorname{rank}(\\delta^0).\n}\n\\]\n\\end{corollary}","The theorem does not require the nerve to be acyclic. Cycles in the cover are\nallowed; their effect is encoded in the rank of \\(\\delta^0\\) and, for\nrelative-only data, in \\(\\check H^1\\). It also shows that the local-workspace\nassignment is a genuine sheaf on the finite subgraph cover used here rather\nthan merely a presheaf.","Let \\(G\\) be the directed triangle\n\\[\n1\\xrightarrow{\\ a\\ }2\n\\xrightarrow{\\ b\\ }3\n\\xrightarrow{\\ c\\ }1\n\\]\nwith one-dimensional vertex cores. Cover it by its three edge charts. Their\nlocal workspaces are all one-dimensional:\n\\[\n\\begin{aligned}\n\\mathscr W_{12}&=\\{(x,ax):x\\in\\mathbb R\\},","Let \\(G\\) be the directed triangle\n\\[\n1\\xrightarrow{\\ a\\ }2\n\\xrightarrow{\\ b\\ }3\n\\xrightarrow{\\ c\\ }1\n\\]\nwith one-dimensional vertex cores. Cover it by its three edge charts. Their\nlocal workspaces are all one-dimensional:\n\\[\n\\begin{aligned}\n\\mathscr W_{12}&=\\{(x,ax):x\\in\\mathbb R\\},\\\\\n\\mathscr W_{23}&=\\{(y,by):y\\in\\mathbb R\\},\\\\\n\\mathscr W_{31}&=\\{(z,cz):z\\in\\mathbb R\\}.\n\\end{aligned}\n\\]\nOverlap agreement requires\n\\[\ny=ax,\n\\qquad\nz=by,\n\\qquad\nx=cz.\n\\]\nTherefore\n\\[\n(1-abc)x=0,\n\\]\nand\n\\[\n\\boxed{\n\\dim\\mathscr W(G)\n=\n\\begin{cases}\n1,&abc=1,","\\mathscr W_{31}&=\\{(z,cz):z\\in\\mathbb R\\}.\n\\end{aligned}\n\\]\nOverlap agreement requires\n\\[\ny=ax,\n\\qquad\nz=by,\n\\qquad\nx=cz.\n\\]\nTherefore\n\\[\n(1-abc)x=0,\n\\]\nand\n\\[\n\\boxed{\n\\dim\\mathscr W(G)\n=\n\\begin{cases}\n1,&abc=1,","\\mathscr W_{31}&=\\{(z,cz):z\\in\\mathbb R\\}.\n\\end{aligned}\n\\]\nOverlap agreement requires\n\\[\ny=ax,\n\\qquad\nz=by,\n\\qquad\nx=cz.\n\\]\nTherefore\n\\[\n(1-abc)x=0,\n\\]\nand\n\\[\n\\boxed{\n\\dim\\mathscr W(G)\n=\n\\begin{cases}\n1,&abc=1,\\\\\n0,&abc\\neq1.\n\\end{cases}\n}\n\\]\nThe meta-level gluing obstruction is exactly the original cycle holonomy.\nStrong local workspaces can therefore coexist with a trivial global core.","First assume each local state already lies in its exact local workspace. Define\n\\[\nB_{\\mathcal U}\n:=\n\\delta^0:\n\\check C^0(\\mathcal U;\\mathscr W)\n\\longrightarrow\n\\check C^1(\\mathcal U;\\mathscr W)\n\\]\nand the meta-Laplacian\n\\[\n\\boxed{\nL_{\\mathcal U}^{\\mathrm{meta}}\n=\nB_{\\mathcal U}^*B_{\\mathcal U}.\n}\n\\]\nBy Theorem~\\ref{thm:exact-workspace-gluing},\n\\[\n\\boxed{\n\\ker L_{\\mathcal U}^{\\mathrm{meta}}\n=\nR_{\\mathcal U}\\mathscr W(G).\n}\n\\]\nThus the gluing problem is itself a workspace problem whose ``nodes'' are\nlocal workspaces and whose interfaces are overlaps.","Let\n\\[\n\\gamma_{\\mathcal U}^{\\mathrm{meta}}\n=\n\\lambda_{\\min}^+\n\\left(L_{\\mathcal U}^{\\mathrm{meta}}\\right)\n\\]\nbe the first positive eigenvalue.","\\begin{proposition}[Approximate overlap gluing]\\label{prop:approx-overlap-gluing}\nFor every \\(w\\in\\check C^0(\\mathcal U;\\mathscr W)\\),\n\\[\n\\boxed{\n\\operatorname{dist}\n\\left(\n w,\n R_{\\mathcal U}\\mathscr W(G)\n\\right)^2\n\\leq\n\\frac{\\|B_{\\mathcal U}w\\|^2}\n{\\gamma_{\\mathcal U}^{\\mathrm{meta}}}.\n}\n\\]\n\\end{proposition}","\\begin{proof}\nDecompose \\(w\\) into its component in \\(\\ker L_{\\mathcal U}^{\\mathrm{meta}}\\)\nand its orthogonal complement. On the complement,\n\\(L_{\\mathcal U}^{\\mathrm{meta}}\\geq\n\\gamma_{\\mathcal U}^{\\mathrm{meta}}I\\). Therefore\n\\[\n\\|B_{\\mathcal U}w\\|^2\n=\n\\langle w,L_{\\mathcal U}^{\\mathrm{meta}}w\\rangle\n\\geq\n\\gamma_{\\mathcal U}^{\\mathrm{meta}}\n\\operatorname{dist}\n\\left(w,\\ker L_{\\mathcal U}^{\\mathrm{meta}}\\right)^2.\n\\]\n\\end{proof}","The gap \\(\\gamma_{\\mathcal U}^{\\mathrm{meta}}\\) measures how strongly the cover\npenalizes disagreement. A small overlap mismatch need not imply a nearby\nglobal workspace when this gap is itself very small.","Local candidates may also violate their internal transfer equations. Define\nthe local-copy space\n\\[\n\\widehat{\\mathcal C}_{\\mathcal U}\n=\n\\bigoplus_{\\alpha\\in A}\\mathcal C_V(G_\\alpha)\n\\]\nand the block internal discrepancy\n\\[\n\\widehat D_{\\mathcal U}\n=\n\\bigoplus_{\\alpha\\in A}D_{G_\\alpha}.\n\\]\nFor \\(u=(u_\\alpha)_\\alpha\\in\\widehat{\\mathcal C}_{\\mathcal U}\\), define the\nraw overlap synchronization operator\n\\[\n(S_{\\mathcal U}u)_{\\alpha\\beta}\n=\n\\rho_{G_{\\alpha\\beta}}^{G_\\beta}u_\\beta\n-\n\\rho_{G_{\\alpha\\beta}}^{G_\\alpha}u_\\alpha.\n\\]\nFor an overlap weight \\(\\lambda_{\\mathrm{ov}}>0\\), set\n\\[\n\\boxed{\nK_{\\mathcal U}\n=\n\\begin{pmatrix}\n\\widehat D_{\\mathcal U}","l candidates may also violate their internal transfer equations. Define\nthe local-copy space\n\\[\n\\widehat{\\mathcal C}_{\\mathcal U}\n=\n\\bigoplus_{\\alpha\\in A}\\mathcal C_V(G_\\alpha)\n\\]\nand the block internal discrepancy\n\\[\n\\widehat D_{\\mathcal U}\n=\n\\bigoplus_{\\alpha\\in A}D_{G_\\alpha}.\n\\]\nFor \\(u=(u_\\alpha)_\\alpha\\in\\widehat{\\mathcal C}_{\\mathcal U}\\), define the\nraw overlap synchronization operator\n\\[\n(S_{\\mathcal U}u)_{\\alpha\\beta}\n=\n\\rho_{G_{\\alpha\\beta}}^{G_\\beta}u_\\beta\n-\n\\rho_{G_{\\alpha\\beta}}^{G_\\alpha}u_\\alpha.\n\\]\nFor an overlap weight \\(\\lambda_{\\mathrm{ov}}>0\\), set\n\\[\n\\boxed{\nK_{\\mathcal U}\n=\n\\begin{pmatrix}\n\\widehat D_{\\mathcal U}","ha\\in A}\\mathcal C_V(G_\\alpha)\n\\]\nand the block internal discrepancy\n\\[\n\\widehat D_{\\mathcal U}\n=\n\\bigoplus_{\\alpha\\in A}D_{G_\\alpha}.\n\\]\nFor \\(u=(u_\\alpha)_\\alpha\\in\\widehat{\\mathcal C}_{\\mathcal U}\\), define the\nraw overlap synchronization operator\n\\[\n(S_{\\mathcal U}u)_{\\alpha\\beta}\n=\n\\rho_{G_{\\alpha\\beta}}^{G_\\beta}u_\\beta\n-\n\\rho_{G_{\\alpha\\beta}}^{G_\\alpha}u_\\alpha.\n\\]\nFor an overlap weight \\(\\lambda_{\\mathrm{ov}}>0\\), set\n\\[\n\\boxed{\nK_{\\mathcal U}\n=\n\\begin{pmatrix}\n\\widehat D_{\\mathcal U}","and the block internal discrepancy\n\\[\n\\widehat D_{\\mathcal U}\n=\n\\bigoplus_{\\alpha\\in A}D_{G_\\alpha}.\n\\]\nFor \\(u=(u_\\alpha)_\\alpha\\in\\widehat{\\mathcal C}_{\\mathcal U}\\), define the\nraw overlap synchronization operator\n\\[\n(S_{\\mathcal U}u)_{\\alpha\\beta}\n=\n\\rho_{G_{\\alpha\\beta}}^{G_\\beta}u_\\beta\n-\n\\rho_{G_{\\alpha\\beta}}^{G_\\alpha}u_\\alpha.\n\\]\nFor an overlap weight \\(\\lambda_{\\mathrm{ov}}>0\\), set\n\\[\n\\boxed{\nK_{\\mathcal U}\n=\n\\begin{pmatrix}\n\\widehat D_{\\mathcal U}\\\\[0.2em]\n\\sqrt{\\lambda_{\\mathrm{ov}}}\\,S_{\\mathcal U}\n\\end{pmatrix},\n\\qquad\nL_{\\mathcal U}^{\\mathrm{tot}}\n=\nK_{\\mathcal U}^*K_{\\mathcal U}.\n}\n\\]\nIts energy is\n\\[\n\\boxed{\n\\begin{aligned}\n\\mathcal E_{\\mathcal U}^{\\mathrm{tot}}(u)\n={}&\n\\sum_\\alpha\\|D_{G_\\alpha}u_\\alpha\\|^2","[0.2em]\n\\sqrt{\\lambda_{\\mathrm{ov}}}\\,S_{\\mathcal U}\n\\end{pmatrix},\n\\qquad\nL_{\\mathcal U}^{\\mathrm{tot}}\n=\nK_{\\mathcal U}^*K_{\\mathcal U}.\n}\n\\]\nIts energy is\n\\[\n\\boxed{\n\\begin{aligned}\n\\mathcal E_{\\mathcal U}^{\\mathrm{tot}}(u)\n={}&\n\\sum_\\alpha\\|D_{G_\\alpha}u_\\alpha\\|^2\\\\\n&+\n\\lambda_{\\mathrm{ov}}\n\\sum_{\\alpha<\\beta}\n\\left\\|\n\\rho^{G_\\beta}_{G_{\\alpha\\beta}}u_\\beta\n-\n\\rho^{G_\\alpha}_{G_{\\alpha\\beta}}u_\\alpha\n\\right\\|^2.\n\\end{aligned}\n}\n\\]","Let\n\\[\nJ_{\\mathcal U}:\\mathcal C_V(G)\\to\\widehat{\\mathcal C}_{\\mathcal U},\n\\qquad\nJ_{\\mathcal U}c=(c|_{G_\\alpha})_\\alpha,\n\\]\nbe the duplication map.","\\begin{theorem}[Total descent kernel]\\label{thm:total-descent-kernel}\nFor every finite edge-complete cover,\n\\[\n\\boxed{\n\\ker K_{\\mathcal U}\n=\nJ_{\\mathcal U}\\mathscr W(G).\n}\n\\]\nConsequently, if\n\\[\n\\gamma_{\\mathcal U}^{\\mathrm{tot}}\n=\n\\lambda_{\\min}^+\n\\left(L_{\\mathcal U}^{\\mathrm{tot}}\\right),\n\\]\nthen\n\\[\n\\boxed{\n\\operatorname{dist}\n\\left(\n u,\n J_{\\mathcal U}\\mathscr W(G)\n\\right)^2\n\\leq\n\\frac{\\mathcal E_{\\mathcal U}^{\\mathrm{tot}}(u)}\n{\\gamma_{\\mathcal U}^{\\mathrm{tot}}}.\n}\n\\]\n\\end{theorem}","\\begin{proof}\nIf \\(u=J_{\\mathcal U}c\\) for a global exact section, every local discrepancy\nand every overlap discrepancy vanishes. Conversely, if \\(K_{\\mathcal U}u=0\\),\nthen every \\(u_\\alpha\\) is an exact local section and the family agrees on all\noverlaps. Theorem~\\ref{thm:exact-workspace-gluing} gives a unique\n\\(c\\in\\mathscr W(G)\\) with \\(u=J_{\\mathcal U}c\\). The distance estimate follows\nfrom the spectral lower bound on the orthogonal complement of the kernel.\n\\end{proof}","This theorem combines two distinct failure modes:\n\\[\n\\text{local incoherence}\n\\qquad\\text{and}\\qquad\n\\text{cross-chart disagreement}.\n\\]\nNeither can be hidden by the other because they occupy separate blocks of\n\\(K_{\\mathcal U}\\).","Let\n\\[\nm_i=|\\{\\alpha:i\\in V_\\alpha\\}|\n\\]\nbe the multiplicity of vertex \\(i\\). Then\n\\[\nJ_{\\mathcal U}^*J_{\\mathcal U}=M_{\\mathcal U},\n\\qquad\n(M_{\\mathcal U}c)_i=m_ic_i.\n\\]\nThe pseudoinverse performs coordinatewise averaging:\n\\[\n\\boxed{\n(J_{\\mathcal U}^\\dagger u)_i\n=\n\\frac{1}{m_i}\n\\sum_{\\alpha:i\\in V_\\alpha}(u_\\alpha)_i.\n}\n\\]\nLet\n\\[\n\\widehat u\n=\nP_{\\ker K_{\\mathcal U}}u\n=\nJ_{\\mathcal U}\\widehat c.\n\\]\nThen\n\\[\n\\widehat c\n=\nJ_{\\mathcal U}^\\dagger\\widehat u\n\\]\nand\n\\[\n\\boxed{\n\\|u-J_{\\mathcal U}\\widehat c\\|\n\\leq\n\\frac{\\|K_{\\mathcal U}u\\|}\n{\\sqrt{\\gamma_{\\mathcal U}^{\\mathrm{tot}}}}.\n}\n\\]\nIf \\(m_{\\min}=\\min_i m_i\\), then\n\\[\n\\boxed{\n\\left\\|\n\\widehat c-J_{\\mathcal U}^\\dagger u\n\\right\\|\n\\leq\n\\frac{\\|K_{\\mathcal U}u\\|}\n{\\sqrt{m_{\\min}\\gamma_","Let\n\\[\nm_i=|\\{\\alpha:i\\in V_\\alpha\\}|\n\\]\nbe the multiplicity of vertex \\(i\\). Then\n\\[\nJ_{\\mathcal U}^*J_{\\mathcal U}=M_{\\mathcal U},\n\\qquad\n(M_{\\mathcal U}c)_i=m_ic_i.\n\\]\nThe pseudoinverse performs coordinatewise averaging:\n\\[\n\\boxed{\n(J_{\\mathcal U}^\\dagger u)_i\n=\n\\frac{1}{m_i}\n\\sum_{\\alpha:i\\in V_\\alpha}(u_\\alpha)_i.\n}\n\\]\nLet\n\\[\n\\widehat u\n=\nP_{\\ker K_{\\mathcal U}}u\n=\nJ_{\\mathcal U}\\widehat c.\n\\]\nThen\n\\[\n\\widehat c\n=\nJ_{\\mathcal U}^\\dagger\\widehat u\n\\]\nand\n\\[\n\\boxed{\n\\|u-J_{\\mathcal U}\\widehat c\\|\n\\leq\n\\frac{\\|K_{\\mathcal U}u\\|}\n{\\sqrt{\\gamma_{\\mathcal U}^{\\mathrm{tot}}}}.\n}\n\\]\nIf \\(m_{\\min}=\\min_i m_i\\), then\n\\[\n\\boxed{\n\\left\\|\n\\widehat c-J_{\\mathcal U}^\\dagger u\n\\right\\|\n\\leq\n\\frac{\\|K_{\\mathcal U}u\\|}\n{\\sqrt{m_{\\min}\\gamma_{\\mathcal U}^{\\mathrm{tot}}}}.\n}\n\\]","Let\n\\[\nm_i=|\\{\\alpha:i\\in V_\\alpha\\}|\n\\]\nbe the multiplicity of vertex \\(i\\). Then\n\\[\nJ_{\\mathcal U}^*J_{\\mathcal U}=M_{\\mathcal U},\n\\qquad\n(M_{\\mathcal U}c)_i=m_ic_i.\n\\]\nThe pseudoinverse performs coordinatewise averaging:\n\\[\n\\boxed{\n(J_{\\mathcal U}^\\dagger u)_i\n=\n\\frac{1}{m_i}\n\\sum_{\\alpha:i\\in V_\\alpha}(u_\\alpha)_i.\n}\n\\]\nLet\n\\[\n\\widehat u\n=\nP_{\\ker K_{\\mathcal U}}u\n=\nJ_{\\mathcal U}\\widehat c.\n\\]\nThen\n\\[\n\\widehat c\n=\nJ_{\\mathcal U}^\\dagger\\widehat u\n\\]\nand\n\\[\n\\boxed{\n\\|u-J_{\\mathcal U}\\widehat c\\|\n\\leq\n\\frac{\\|K_{\\mathcal U}u\\|}\n{\\sqrt{\\gamma_{\\mathcal U}^{\\mathrm{tot}}}}.\n}\n\\]\nIf \\(m_{\\min}=\\min_i m_i\\), then\n\\[\n\\boxed{\n\\left\\|\n\\widehat c-J_{\\mathcal U}^\\dagger u\n\\right\\|\n\\leq\n\\frac{\\|K_{\\mathcal U}u\\|}\n{\\sqrt{m_{\\min}\\gamma_{\\mathcal U}^{\\mathrm{tot}}}}.\n}\n\\]\nThe simple overlap average and the globally coherent least-squares assembly\nare close only whe","Let\n\\[\nm_i=|\\{\\alpha:i\\in V_\\alpha\\}|\n\\]\nbe the multiplicity of vertex \\(i\\). Then\n\\[\nJ_{\\mathcal U}^*J_{\\mathcal U}=M_{\\mathcal U},\n\\qquad\n(M_{\\mathcal U}c)_i=m_ic_i.\n\\]\nThe pseudoinverse performs coordinatewise averaging:\n\\[\n\\boxed{\n(J_{\\mathcal U}^\\dagger u)_i\n=\n\\frac{1}{m_i}\n\\sum_{\\alpha:i\\in V_\\alpha}(u_\\alpha)_i.\n}\n\\]\nLet\n\\[\n\\widehat u\n=\nP_{\\ker K_{\\mathcal U}}u\n=\nJ_{\\mathcal U}\\widehat c.\n\\]\nThen\n\\[\n\\widehat c\n=\nJ_{\\mathcal U}^\\dagger\\widehat u\n\\]\nand\n\\[\n\\boxed{\n\\|u-J_{\\mathcal U}\\widehat c\\|\n\\leq\n\\frac{\\|K_{\\mathcal U}u\\|}\n{\\sqrt{\\gamma_{\\mathcal U}^{\\mathrm{tot}}}}.\n}\n\\]\nIf \\(m_{\\min}=\\min_i m_i\\), then\n\\[\n\\boxed{\n\\left\\|\n\\widehat c-J_{\\mathcal U}^\\dagger u\n\\right\\|\n\\leq\n\\frac{\\|K_{\\mathcal U}u\\|}\n{\\sqrt{m_{\\min}\\gamma_{\\mathcal U}^{\\mathrm{tot}}}}.\n}\n\\]\nThe simple overlap average and the globally coherent least-squares assembly\nare close only when the total descent energy is small relative to the cover\ngap.","i=|\\{\\alpha:i\\in V_\\alpha\\}|\n\\]\nbe the multiplicity of vertex \\(i\\). Then\n\\[\nJ_{\\mathcal U}^*J_{\\mathcal U}=M_{\\mathcal U},\n\\qquad\n(M_{\\mathcal U}c)_i=m_ic_i.\n\\]\nThe pseudoinverse performs coordinatewise averaging:\n\\[\n\\boxed{\n(J_{\\mathcal U}^\\dagger u)_i\n=\n\\frac{1}{m_i}\n\\sum_{\\alpha:i\\in V_\\alpha}(u_\\alpha)_i.\n}\n\\]\nLet\n\\[\n\\widehat u\n=\nP_{\\ker K_{\\mathcal U}}u\n=\nJ_{\\mathcal U}\\widehat c.\n\\]\nThen\n\\[\n\\widehat c\n=\nJ_{\\mathcal U}^\\dagger\\widehat u\n\\]\nand\n\\[\n\\boxed{\n\\|u-J_{\\mathcal U}\\widehat c\\|\n\\leq\n\\frac{\\|K_{\\mathcal U}u\\|}\n{\\sqrt{\\gamma_{\\mathcal U}^{\\mathrm{tot}}}}.\n}\n\\]\nIf \\(m_{\\min}=\\min_i m_i\\), then\n\\[\n\\boxed{\n\\left\\|\n\\widehat c-J_{\\mathcal U}^\\dagger u\n\\right\\|\n\\leq\n\\frac{\\|K_{\\mathcal U}u\\|}\n{\\sqrt{m_{\\min}\\gamma_{\\mathcal U}^{\\mathrm{tot}}}}.\n}\n\\]\nThe simple overlap average and the globally coherent least-squares assembly\nare close only when the total descent energy is small relative to the cover\ngap.","ertex \\(i\\). Then\n\\[\nJ_{\\mathcal U}^*J_{\\mathcal U}=M_{\\mathcal U},\n\\qquad\n(M_{\\mathcal U}c)_i=m_ic_i.\n\\]\nThe pseudoinverse performs coordinatewise averaging:\n\\[\n\\boxed{\n(J_{\\mathcal U}^\\dagger u)_i\n=\n\\frac{1}{m_i}\n\\sum_{\\alpha:i\\in V_\\alpha}(u_\\alpha)_i.\n}\n\\]\nLet\n\\[\n\\widehat u\n=\nP_{\\ker K_{\\mathcal U}}u\n=\nJ_{\\mathcal U}\\widehat c.\n\\]\nThen\n\\[\n\\widehat c\n=\nJ_{\\mathcal U}^\\dagger\\widehat u\n\\]\nand\n\\[\n\\boxed{\n\\|u-J_{\\mathcal U}\\widehat c\\|\n\\leq\n\\frac{\\|K_{\\mathcal U}u\\|}\n{\\sqrt{\\gamma_{\\mathcal U}^{\\mathrm{tot}}}}.\n}\n\\]\nIf \\(m_{\\min}=\\min_i m_i\\), then\n\\[\n\\boxed{\n\\left\\|\n\\widehat c-J_{\\mathcal U}^\\dagger u\n\\right\\|\n\\leq\n\\frac{\\|K_{\\mathcal U}u\\|}\n{\\sqrt{m_{\\min}\\gamma_{\\mathcal U}^{\\mathrm{tot}}}}.\n}\n\\]\nThe simple overlap average and the globally coherent least-squares assembly\nare close only when the total descent energy is small relative to the cover\ngap.","forms coordinatewise averaging:\n\\[\n\\boxed{\n(J_{\\mathcal U}^\\dagger u)_i\n=\n\\frac{1}{m_i}\n\\sum_{\\alpha:i\\in V_\\alpha}(u_\\alpha)_i.\n}\n\\]\nLet\n\\[\n\\widehat u\n=\nP_{\\ker K_{\\mathcal U}}u\n=\nJ_{\\mathcal U}\\widehat c.\n\\]\nThen\n\\[\n\\widehat c\n=\nJ_{\\mathcal U}^\\dagger\\widehat u\n\\]\nand\n\\[\n\\boxed{\n\\|u-J_{\\mathcal U}\\widehat c\\|\n\\leq\n\\frac{\\|K_{\\mathcal U}u\\|}\n{\\sqrt{\\gamma_{\\mathcal U}^{\\mathrm{tot}}}}.\n}\n\\]\nIf \\(m_{\\min}=\\min_i m_i\\), then\n\\[\n\\boxed{\n\\left\\|\n\\widehat c-J_{\\mathcal U}^\\dagger u\n\\right\\|\n\\leq\n\\frac{\\|K_{\\mathcal U}u\\|}\n{\\sqrt{m_{\\min}\\gamma_{\\mathcal U}^{\\mathrm{tot}}}}.\n}\n\\]\nThe simple overlap average and the globally coherent least-squares assembly\nare close only when the total descent energy is small relative to the cover\ngap.",":\n\\[\n\\boxed{\n(J_{\\mathcal U}^\\dagger u)_i\n=\n\\frac{1}{m_i}\n\\sum_{\\alpha:i\\in V_\\alpha}(u_\\alpha)_i.\n}\n\\]\nLet\n\\[\n\\widehat u\n=\nP_{\\ker K_{\\mathcal U}}u\n=\nJ_{\\mathcal U}\\widehat c.\n\\]\nThen\n\\[\n\\widehat c\n=\nJ_{\\mathcal U}^\\dagger\\widehat u\n\\]\nand\n\\[\n\\boxed{\n\\|u-J_{\\mathcal U}\\widehat c\\|\n\\leq\n\\frac{\\|K_{\\mathcal U}u\\|}\n{\\sqrt{\\gamma_{\\mathcal U}^{\\mathrm{tot}}}}.\n}\n\\]\nIf \\(m_{\\min}=\\min_i m_i\\), then\n\\[\n\\boxed{\n\\left\\|\n\\widehat c-J_{\\mathcal U}^\\dagger u\n\\right\\|\n\\leq\n\\frac{\\|K_{\\mathcal U}u\\|}\n{\\sqrt{m_{\\min}\\gamma_{\\mathcal U}^{\\mathrm{tot}}}}.\n}\n\\]\nThe simple overlap average and the globally coherent least-squares assembly\nare close only when the total descent energy is small relative to the cover\ngap.","Define the exact meta-workspace\n\\[\n\\boxed{\n\\mathscr W_{\\mathrm{meta}}(\\mathcal U)\n:=\n\\ker B_{\\mathcal U}\n\\subseteq\n\\bigoplus_\\alpha\\mathscr W_\\alpha.\n}\n\\]\nTheorem~\\ref{thm:exact-workspace-gluing} gives the canonical identification\n\\[\n\\boxed{\n\\mathscr W_{\\mathrm{meta}}(\\mathcal U)\n\\cong\n\\mathscr W(G).\n}\n\\]\nThus exact formation of a workspace from local workspaces does not create an\nadditional mysterious carrier. It gives a second presentation of the same\nglobal compatible-section object.","Suppose local operations\n\\[\nA_\\alpha:\\mathscr W_\\alpha\\to\\mathscr W_\\alpha\n\\]\nand overlap operations\n\\[\nA_{\\alpha\\beta}:\\mathscr W_{\\alpha\\beta}\\to\n\\mathscr W_{\\alpha\\beta}\n\\]\nsatisfy\n\\[\nr_{\\alpha\\beta}^{\\alpha}A_\\alpha\n=\nA_{\\alpha\\beta}r_{\\alpha\\beta}^{\\alpha},\n\\qquad\nr_{\\alpha\\beta}^{\\beta}A_\\beta\n=\nA_{\\alpha\\beta}r_{\\alpha\\beta}^{\\beta}.\n\\]\nThen, with \\(A^0=\\bigoplus_\\alpha A_\\alpha\\) and\n\\(A^1=\\bigoplus_{\\alpha<\\beta}A_{\\alpha\\beta}\\),\n\\[\nB_{\\mathcal U}A^0=A^1B_{\\mathcal U}.\n\\]\nTherefore\n\\[\nA^0\\ker B_{\\mathcal U}\\subseteq\\ker B_{\\mathcal U}.\n\\]\nThe operation-closure theorem from Part~III reappears at the meta level:\noperations on subworkspaces descend globally precisely when they agree on\ninterfaces.","W_\\alpha\\to\\mathscr W_\\alpha\n\\]\nand overlap operations\n\\[\nA_{\\alpha\\beta}:\\mathscr W_{\\alpha\\beta}\\to\n\\mathscr W_{\\alpha\\beta}\n\\]\nsatisfy\n\\[\nr_{\\alpha\\beta}^{\\alpha}A_\\alpha\n=\nA_{\\alpha\\beta}r_{\\alpha\\beta}^{\\alpha},\n\\qquad\nr_{\\alpha\\beta}^{\\beta}A_\\beta\n=\nA_{\\alpha\\beta}r_{\\alpha\\beta}^{\\beta}.\n\\]\nThen, with \\(A^0=\\bigoplus_\\alpha A_\\alpha\\) and\n\\(A^1=\\bigoplus_{\\alpha<\\beta}A_{\\alpha\\beta}\\),\n\\[\nB_{\\mathcal U}A^0=A^1B_{\\mathcal U}.\n\\]\nTherefore\n\\[\nA^0\\ker B_{\\mathcal U}\\subseteq\\ker B_{\\mathcal U}.\n\\]\nThe operation-closure theorem from Part~III reappears at the meta level:\noperations on subworkspaces descend globally precisely when they agree on\ninterfaces.","s\n\\[\nA_{\\alpha\\beta}:\\mathscr W_{\\alpha\\beta}\\to\n\\mathscr W_{\\alpha\\beta}\n\\]\nsatisfy\n\\[\nr_{\\alpha\\beta}^{\\alpha}A_\\alpha\n=\nA_{\\alpha\\beta}r_{\\alpha\\beta}^{\\alpha},\n\\qquad\nr_{\\alpha\\beta}^{\\beta}A_\\beta\n=\nA_{\\alpha\\beta}r_{\\alpha\\beta}^{\\beta}.\n\\]\nThen, with \\(A^0=\\bigoplus_\\alpha A_\\alpha\\) and\n\\(A^1=\\bigoplus_{\\alpha<\\beta}A_{\\alpha\\beta}\\),\n\\[\nB_{\\mathcal U}A^0=A^1B_{\\mathcal U}.\n\\]\nTherefore\n\\[\nA^0\\ker B_{\\mathcal U}\\subseteq\\ker B_{\\mathcal U}.\n\\]\nThe operation-closure theorem from Part~III reappears at the meta level:\noperations on subworkspaces descend globally precisely when they agree on\ninterfaces.","A practical hierarchy rarely transmits each complete local workspace. Let\n\\[\nQ^p:\\check C^p(\\mathcal U;\\mathscr W)\\to\\overline C^p\n\\]\nbe compression maps into smaller local and overlap representations. Suppose\nthere is a compressed coboundary \\(\\overline\\delta^p\\) such that\n\\[\n\\boxed{\nQ^{p+1}\\delta^p\n=\n\\overline\\delta^p Q^p.\n}\n\\]\nThe maps \\(Q^\\bullet\\) then form a cochain map.","\\begin{definition}[Descent-faithful compression]\\label{def:descent-faithful}\nThe compression is \\emph{descent-faithful in degree zero} when the induced map\n\\[\n\\boxed{\nH^0(Q):\n\\ker\\delta^0\n\\longrightarrow\n\\ker\\overline\\delta^0\n}\n\\]\nis an isomorphism.\n\\end{definition}","Injectivity prevents two distinct global workspace states from collapsing to\nthe same compressed meta-state. Surjectivity prevents the compressed hierarchy\nfrom introducing spurious compatible states that have no full-resolution\nlift. Under this condition,\n\\[\n\\boxed{\n\\ker\\overline\\delta^0\n\\cong\n\\mathscr W(G).\n}\n\\]\nThis is the exact criterion for a compressed workspace of workspaces to retain\nthe global core.","Exact or approximate overlap coherence does not by itself establish global\npredictive sufficiency. Let \\(W_t\\) denote a glued structural state and let\n\\[\nZ_t^{\\mathrm{loc}}\n=\n(Z_{\\alpha,t})_{\\alpha\\in A}\n\\]\nbe local predictive summaries recovered from its chart restrictions.","\\begin{proposition}[Local-to-global predictive alignment]\\label{prop:local-global-predictive}\nAssume \\(W_t\\) is measurable with respect to the complete history\n\\(\\mathbf H_t\\), every \\(Z_{\\alpha,t}\\) is a measurable function of\n\\(W_t|_{G_\\alpha}\\), and\n\\[\nI(Y_t^+;\\mathbf H_t\\mid Z_t^{\\mathrm{loc}})\n\\leq\n\\varepsilon.\n\\]\nThen\n\\[\n\\boxed{\nI(Y_t^+;\\mathbf H_t\\mid W_t)\n\\leq\n\\varepsilon.\n}\n\\]\nIn particular, exact sufficiency of the complete local-summary tuple implies\nexact sufficiency of the glued workspace.\n\\end{proposition}","\\begin{proof}\nThe tuple \\(Z_t^{\\mathrm{loc}}\\) is a measurable function of \\(W_t\\), while\n\\(W_t\\) is itself a history statistic. Conditioning on the finer statistic\ncannot increase the residual information that the full history carries about\nthe target. Equivalently, use the conditional-law characterization from\nTheorem~\\ref{thm:predictive-state}.\n\\end{proof}","The isomorphism\n\\[\n\\mathscr W_{\\mathrm{meta}}(\\mathcal U)\\cong\\mathscr W(G)\n\\]\nis a compositional statement, not a relational-irreducibility statement. A\nsingle chart may still expose enough accessible information to reconstruct the\nwhole meta-state. A claim that the hierarchy is genuinely distributed must\ntherefore reuse the coalition gates from Parts~VII and XI. At the chart level,\none may require, for example,\n\\[\n\\alpha_{\\{\\alpha\\}}^{\\mathrm{meta}}=0\n\\quad\\text{for every }\\alpha,\n\\qquad\n\\alpha_A^{\\mathrm{meta}}>0\n\\quad\\text{for some }|A|\\geq2,\n\\]\nor the predictive analogue\n\\[\nk_\\varepsilon^{\\mathrm{meta}}\\geq2,\n\\qquad\n\\Delta_{\\mathrm{joint}}^{\\mathrm{meta}}>0.\n\\]\nCapacity-matched local-memory and chart-ablation controls are still required.\nHierarchical notation c","\\mathscr W_{\\mathrm{meta}}(\\mathcal U)\\cong\\mathscr W(G)\n\\]\nis a compositional statement, not a relational-irreducibility statement. A\nsingle chart may still expose enough accessible information to reconstruct the\nwhole meta-state. A claim that the hierarchy is genuinely distributed must\ntherefore reuse the coalition gates from Parts~VII and XI. At the chart level,\none may require, for example,\n\\[\n\\alpha_{\\{\\alpha\\}}^{\\mathrm{meta}}=0\n\\quad\\text{for every }\\alpha,\n\\qquad\n\\alpha_A^{\\mathrm{meta}}>0\n\\quad\\text{for some }|A|\\geq2,\n\\]\nor the predictive analogue\n\\[\nk_\\varepsilon^{\\mathrm{meta}}\\geq2,\n\\qquad\n\\Delta_{\\mathrm{joint}}^{\\mathrm{meta}}>0.\n\\]\nCapacity-matched local-memory and chart-ablation controls are still required.\nHierarchical notation cannot convert a centralized copy into a relational\nworkspace.","nough accessible information to reconstruct the\nwhole meta-state. A claim that the hierarchy is genuinely distributed must\ntherefore reuse the coalition gates from Parts~VII and XI. At the chart level,\none may require, for example,\n\\[\n\\alpha_{\\{\\alpha\\}}^{\\mathrm{meta}}=0\n\\quad\\text{for every }\\alpha,\n\\qquad\n\\alpha_A^{\\mathrm{meta}}>0\n\\quad\\text{for some }|A|\\geq2,\n\\]\nor the predictive analogue\n\\[\nk_\\varepsilon^{\\mathrm{meta}}\\geq2,\n\\qquad\n\\Delta_{\\mathrm{joint}}^{\\mathrm{meta}}>0.\n\\]\nCapacity-matched local-memory and chart-ablation controls are still required.\nHierarchical notation cannot convert a centralized copy into a relational\nworkspace.","A finite workspace hierarchy is a rooted tree \\(\\mathcal T\\). Each node\n\\(a\\in\\mathcal T\\) carries a subgraph \\(G_a\\). The root \\(r\\) carries\n\\[\nG_r=G.\n\\]\nFor every internal node \\(a\\), its children\n\\[\n\\operatorname{Ch}(a)\n\\]\nform a finite edge-complete cover of \\(G_a\\). Leaves carry primitive local\nworkspace spaces\n\\[\nW_\\ell=\\mathscr W(G_\\ell).\n\\]\nFor an internal node, define recursively\n\\[\n\\boxed{\nW_a\n=\n\\ker\n\\left(\n\\delta_a^0:\n\\bigoplus_{b\\in\\operatorname{Ch}(a)}W_b\n\\longrightarrow\n\\bigoplus_{b1\n&\\text{deep hierarchies can amplify early mismatch}.\n\\end{array}\n\\]\nA deep hierarchy is therefore stable only when its level gaps remain separated\nand its assembly maps do not magnify small overlap errors faster than those\nerrors are corrected.","Let \\(K_r\\) and \\(\\widetilde K_r\\) now denote exact and implemented Markov\nkernels at successive hierarchy levels, and suppose\n\\[\n\\sup_x\n\\operatorname{TV}\n\\left(\n\\widetilde K_r(x,\\cdot),\nK_r(x,\\cdot)\n\\right)\n\\leq\n\\zeta_r.\n\\]\nBecause Markov kernels contract total variation, a telescoping argument gives\n\\[\n\\boxed{\n\\sup_x\n\\operatorname{TV}\n\\left(\n\\widetilde K_0\\cdots\\widetilde K_{R-1}(x,\\cdot),\nK_0\\cdots K_{R-1}(x,\\cdot)\n\\right)\n\\leq\n\\sum_{r=0}^{R-1}\\zeta_r.\n}\n\\]\nThus causal semantics can also be transported through a hierarchy, but only\nunder an explicit kernel-level compatibility test. Structural gluing alone\nprovides no such guarantee.","The exact theorem states\n\\[\n\\boxed{\n\\text{recursive exact gluing}\n=\n\\text{one-shot exact gluing}.\n}\n\\]\nThe approximate theorem instead supplies a measurable flattening defect:\n\\[\n\\boxed{\n\\delta_{\\mathrm{flat}}^{(R)}\n=\n\\left\\|\n\\widetilde F_{R-1}\\cdots\\widetilde F_0(x)\n-\nF_{R-1}\\cdots F_0(x)\n\\right\\|.\n}\n\\]\nA hierarchy is empirically justified when it reduces computation,\ncommunication, or local model complexity while keeping\n\\(\\delta_{\\mathrm{flat}}^{(R)}\\), predictive risk, causal-kernel mismatch, and\nanti-centralization gates within declared tolerances. Hierarchy by itself is\nnot evidence of emergence.","boxed{\n\\text{recursive exact gluing}\n=\n\\text{one-shot exact gluing}.\n}\n\\]\nThe approximate theorem instead supplies a measurable flattening defect:\n\\[\n\\boxed{\n\\delta_{\\mathrm{flat}}^{(R)}\n=\n\\left\\|\n\\widetilde F_{R-1}\\cdots\\widetilde F_0(x)\n-\nF_{R-1}\\cdots F_0(x)\n\\right\\|.\n}\n\\]\nA hierarchy is empirically justified when it reduces computation,\ncommunication, or local model complexity while keeping\n\\(\\delta_{\\mathrm{flat}}^{(R)}\\), predictive risk, causal-kernel mismatch, and\nanti-centralization gates within declared tolerances. Hierarchy by itself is\nnot evidence of emergence.","The exact and approximate constructions can be compressed as\n\\[\n\\boxed{\n\\begin{aligned}\n\\text{local transfer systems}\n&\\xrightarrow{\\ \\ker D_\\alpha\\ }\n\\text{local workspaces}\\\\\n&\\xrightarrow{\\ \\ker\\delta^0\\ }\n\\text{meta-workspace}\\\\\n&\\xrightarrow{\\ \\text{finite recursive descent}\\ }\n\\text{global workspace}.\n\\end{aligned}\n}\n\\]\nAt exact resolution, all finite parenthesizations are canonically equivalent.\nAt approximate resolution, every level contributes a defect scaled by its\nspectral gap and by downstream amplification. Relative-only overlap data can\ncarry a harmonic \\(H^1\\) obstruction even when every local chart is internally\ncoherent. Operational and relational claims still require predictive, causal,\ncoalition, and null-model gates in addition to structural descent.","&\\xrightarrow{\\ \\text{finite recursive descent}\\ }\n\\text{global workspace}.\n\\end{aligned}\n}\n\\]\nAt exact resolution, all finite parenthesizations are canonically equivalent.\nAt approximate resolution, every level contributes a defect scaled by its\nspectral gap and by downstream amplification. Relative-only overlap data can\ncarry a harmonic \\(H^1\\) obstruction even when every local chart is internally\ncoherent. Operational and relational claims still require predictive, causal,\ncoalition, and null-model gates in addition to structural descent.","\\begin{definition}[Formation system]\\label{def:formation-system}\nA \\emph{workspace formation system} is a tuple\n\\[\n\\mathfrak F\n=\n\\bigl(\n\\Sigma,\n\\mathcal D_{\\mathrm{form}},\n\\mathcal L,\n\\Omega,\n\\widehat\\theta,\n\\mathcal D_{\\mathrm{cert}},\n\\mathcal I_{\\mathrm{form}}\n\\bigr),\n\\]\nwhere:\n\\begin{itemize}\n\\item \\(\\Sigma\\) is the externally supplied scaffold;\n\\item \\(\\mathcal D_{\\mathrm{form}}\\) is the formation or training history;\n\\item \\(\\mathcal L\\) is the learning or selection rule;\n\\item \\(\\Omega\\) is its internal random seed or stochastic state;\n\\item \\(\\widehat\\theta=\\mathcal L(\\Sigma,\\mathcal D_{\\mathrm{form}},\\Omega)\\)\n is the learned organization;\n\\item \\(\\mathcal D_{\\mathrm{cert}}\\) is held-out certification data; and\n\\item \\(\\mathcal I_{\\mathrm{form}}\\) is a declared family of interventions on\n scaffold and formation history.\n\\end{itemize}\nThe learned organization is decomposed as\n\\[\n\\widehat\\theta\n=\n\\bigl(\n\\widehat V,\n\\widehat E,\n\\widehat H,\n\\widehat P,","\\begin{definition}[Formation system]\\label{def:formation-system}\nA \\emph{workspace formation system} is a tuple\n\\[\n\\mathfrak F\n=\n\\bigl(\n\\Sigma,\n\\mathcal D_{\\mathrm{form}},\n\\mathcal L,\n\\Omega,\n\\widehat\\theta,\n\\mathcal D_{\\mathrm{cert}},\n\\mathcal I_{\\mathrm{form}}\n\\bigr),\n\\]\nwhere:\n\\begin{itemize}\n\\item \\(\\Sigma\\) is the externally supplied scaffold;\n\\item \\(\\mathcal D_{\\mathrm{form}}\\) is the formation or training history;\n\\item \\(\\mathcal L\\) is the learning or selection rule;\n\\item \\(\\Omega\\) is its internal random seed or stochastic state;\n\\item \\(\\widehat\\theta=\\mathcal L(\\Sigma,\\mathcal D_{\\mathrm{form}},\\Omega)\\)\n is the learned organization;\n\\item \\(\\mathcal D_{\\mathrm{cert}}\\) is held-out certification data; and\n\\item \\(\\mathcal I_{\\mathrm{form}}\\) is a declared family of interventions on\n scaffold and formation history.\n\\end{itemize}\nThe learned organization is decomposed as\n\\[\n\\widehat\\theta\n=\n\\bigl(\n\\widehat V,\n\\widehat E,\n\\widehat H,\n\\widehat P,\n\\widehat{\\mathcal O},\n\\widehat B\n\\bigr),","\\begin{definition}[Formation system]\\label{def:formation-system}\nA \\emph{workspace formation system} is a tuple\n\\[\n\\mathfrak F\n=\n\\bigl(\n\\Sigma,\n\\mathcal D_{\\mathrm{form}},\n\\mathcal L,\n\\Omega,\n\\widehat\\theta,\n\\mathcal D_{\\mathrm{cert}},\n\\mathcal I_{\\mathrm{form}}\n\\bigr),\n\\]\nwhere:\n\\begin{itemize}\n\\item \\(\\Sigma\\) is the externally supplied scaffold;\n\\item \\(\\mathcal D_{\\mathrm{form}}\\) is the formation or training history;\n\\item \\(\\mathcal L\\) is the learning or selection rule;\n\\item \\(\\Omega\\) is its internal random seed or stochastic state;\n\\item \\(\\widehat\\theta=\\mathcal L(\\Sigma,\\mathcal D_{\\mathrm{form}},\\Omega)\\)\n is the learned organization;\n\\item \\(\\mathcal D_{\\mathrm{cert}}\\) is held-out certification data; and\n\\item \\(\\mathcal I_{\\mathrm{form}}\\) is a declared family of interventions on\n scaffold and formation history.\n\\end{itemize}\nThe learned organization is decomposed as\n\\[\n\\widehat\\theta\n=\n\\bigl(\n\\widehat V,\n\\widehat E,\n\\widehat H,\n\\widehat P,\n\\widehat{\\mathcal O},\n\\widehat B\n\\bigr),\n\\]\nwhere \\(\\widehat B\\) is a learned operational boundary.\n\\end{definitio","m]\\label{def:formation-system}\nA \\emph{workspace formation system} is a tuple\n\\[\n\\mathfrak F\n=\n\\bigl(\n\\Sigma,\n\\mathcal D_{\\mathrm{form}},\n\\mathcal L,\n\\Omega,\n\\widehat\\theta,\n\\mathcal D_{\\mathrm{cert}},\n\\mathcal I_{\\mathrm{form}}\n\\bigr),\n\\]\nwhere:\n\\begin{itemize}\n\\item \\(\\Sigma\\) is the externally supplied scaffold;\n\\item \\(\\mathcal D_{\\mathrm{form}}\\) is the formation or training history;\n\\item \\(\\mathcal L\\) is the learning or selection rule;\n\\item \\(\\Omega\\) is its internal random seed or stochastic state;\n\\item \\(\\widehat\\theta=\\mathcal L(\\Sigma,\\mathcal D_{\\mathrm{form}},\\Omega)\\)\n is the learned organization;\n\\item \\(\\mathcal D_{\\mathrm{cert}}\\) is held-out certification data; and\n\\item \\(\\mathcal I_{\\mathrm{form}}\\) is a declared family of interventions on\n scaffold and formation history.\n\\end{itemize}\nThe learned organization is decomposed as\n\\[\n\\widehat\\theta\n=\n\\bigl(\n\\widehat V,\n\\widehat E,\n\\widehat H,\n\\widehat P,\n\\widehat{\\mathcal O},\n\\widehat B\n\\bigr),\n\\]\nwhere \\(\\widehat B\\) is a learned operational boundary.\n\\end{definition}","on system} is a tuple\n\\[\n\\mathfrak F\n=\n\\bigl(\n\\Sigma,\n\\mathcal D_{\\mathrm{form}},\n\\mathcal L,\n\\Omega,\n\\widehat\\theta,\n\\mathcal D_{\\mathrm{cert}},\n\\mathcal I_{\\mathrm{form}}\n\\bigr),\n\\]\nwhere:\n\\begin{itemize}\n\\item \\(\\Sigma\\) is the externally supplied scaffold;\n\\item \\(\\mathcal D_{\\mathrm{form}}\\) is the formation or training history;\n\\item \\(\\mathcal L\\) is the learning or selection rule;\n\\item \\(\\Omega\\) is its internal random seed or stochastic state;\n\\item \\(\\widehat\\theta=\\mathcal L(\\Sigma,\\mathcal D_{\\mathrm{form}},\\Omega)\\)\n is the learned organization;\n\\item \\(\\mathcal D_{\\mathrm{cert}}\\) is held-out certification data; and\n\\item \\(\\mathcal I_{\\mathrm{form}}\\) is a declared family of interventions on\n scaffold and formation history.\n\\end{itemize}\nThe learned organization is decomposed as\n\\[\n\\widehat\\theta\n=\n\\bigl(\n\\widehat V,\n\\widehat E,\n\\widehat H,\n\\widehat P,\n\\widehat{\\mathcal O},\n\\widehat B\n\\bigr),\n\\]\nwhere \\(\\widehat B\\) is a learned operational boundary.\n\\end{definition}","\\mathrm{form}},\n\\mathcal L,\n\\Omega,\n\\widehat\\theta,\n\\mathcal D_{\\mathrm{cert}},\n\\mathcal I_{\\mathrm{form}}\n\\bigr),\n\\]\nwhere:\n\\begin{itemize}\n\\item \\(\\Sigma\\) is the externally supplied scaffold;\n\\item \\(\\mathcal D_{\\mathrm{form}}\\) is the formation or training history;\n\\item \\(\\mathcal L\\) is the learning or selection rule;\n\\item \\(\\Omega\\) is its internal random seed or stochastic state;\n\\item \\(\\widehat\\theta=\\mathcal L(\\Sigma,\\mathcal D_{\\mathrm{form}},\\Omega)\\)\n is the learned organization;\n\\item \\(\\mathcal D_{\\mathrm{cert}}\\) is held-out certification data; and\n\\item \\(\\mathcal I_{\\mathrm{form}}\\) is a declared family of interventions on\n scaffold and formation history.\n\\end{itemize}\nThe learned organization is decomposed as\n\\[\n\\widehat\\theta\n=\n\\bigl(\n\\widehat V,\n\\widehat E,\n\\widehat H,\n\\widehat P,\n\\widehat{\\mathcal O},\n\\widehat B\n\\bigr),\n\\]\nwhere \\(\\widehat B\\) is a learned operational boundary.\n\\end{definition}","bigr),\n\\]\nwhere:\n\\begin{itemize}\n\\item \\(\\Sigma\\) is the externally supplied scaffold;\n\\item \\(\\mathcal D_{\\mathrm{form}}\\) is the formation or training history;\n\\item \\(\\mathcal L\\) is the learning or selection rule;\n\\item \\(\\Omega\\) is its internal random seed or stochastic state;\n\\item \\(\\widehat\\theta=\\mathcal L(\\Sigma,\\mathcal D_{\\mathrm{form}},\\Omega)\\)\n is the learned organization;\n\\item \\(\\mathcal D_{\\mathrm{cert}}\\) is held-out certification data; and\n\\item \\(\\mathcal I_{\\mathrm{form}}\\) is a declared family of interventions on\n scaffold and formation history.\n\\end{itemize}\nThe learned organization is decomposed as\n\\[\n\\widehat\\theta\n=\n\\bigl(\n\\widehat V,\n\\widehat E,\n\\widehat H,\n\\widehat P,\n\\widehat{\\mathcal O},\n\\widehat B\n\\bigr),\n\\]\nwhere \\(\\widehat B\\) is a learned operational boundary.\n\\end{definition}","ied scaffold;\n\\item \\(\\mathcal D_{\\mathrm{form}}\\) is the formation or training history;\n\\item \\(\\mathcal L\\) is the learning or selection rule;\n\\item \\(\\Omega\\) is its internal random seed or stochastic state;\n\\item \\(\\widehat\\theta=\\mathcal L(\\Sigma,\\mathcal D_{\\mathrm{form}},\\Omega)\\)\n is the learned organization;\n\\item \\(\\mathcal D_{\\mathrm{cert}}\\) is held-out certification data; and\n\\item \\(\\mathcal I_{\\mathrm{form}}\\) is a declared family of interventions on\n scaffold and formation history.\n\\end{itemize}\nThe learned organization is decomposed as\n\\[\n\\widehat\\theta\n=\n\\bigl(\n\\widehat V,\n\\widehat E,\n\\widehat H,\n\\widehat P,\n\\widehat{\\mathcal O},\n\\widehat B\n\\bigr),\n\\]\nwhere \\(\\widehat B\\) is a learned operational boundary.\n\\end{definition}","is its internal random seed or stochastic state;\n\\item \\(\\widehat\\theta=\\mathcal L(\\Sigma,\\mathcal D_{\\mathrm{form}},\\Omega)\\)\n is the learned organization;\n\\item \\(\\mathcal D_{\\mathrm{cert}}\\) is held-out certification data; and\n\\item \\(\\mathcal I_{\\mathrm{form}}\\) is a declared family of interventions on\n scaffold and formation history.\n\\end{itemize}\nThe learned organization is decomposed as\n\\[\n\\widehat\\theta\n=\n\\bigl(\n\\widehat V,\n\\widehat E,\n\\widehat H,\n\\widehat P,\n\\widehat{\\mathcal O},\n\\widehat B\n\\bigr),\n\\]\nwhere \\(\\widehat B\\) is a learned operational boundary.\n\\end{definition}","\\)\n is the learned organization;\n\\item \\(\\mathcal D_{\\mathrm{cert}}\\) is held-out certification data; and\n\\item \\(\\mathcal I_{\\mathrm{form}}\\) is a declared family of interventions on\n scaffold and formation history.\n\\end{itemize}\nThe learned organization is decomposed as\n\\[\n\\widehat\\theta\n=\n\\bigl(\n\\widehat V,\n\\widehat E,\n\\widehat H,\n\\widehat P,\n\\widehat{\\mathcal O},\n\\widehat B\n\\bigr),\n\\]\nwhere \\(\\widehat B\\) is a learned operational boundary.\n\\end{definition}","The scaffold is itself stratified:\n\\[\n\\boxed{\n\\Sigma\n=\n\\bigl(\n\\Sigma_V,\n\\Sigma_E,\n\\Sigma_H,\n\\Sigma_P,\n\\Sigma_{\\mathcal O},\n\\Sigma_B,\n\\Sigma_Y\n\\bigr).\n}\n\\]\nThe final coordinate \\(\\Sigma_Y\\) records labels, targets, privileged masks,\nor other supervision that could directly reveal the intended workspace. This\ncoordinate is necessary because an architecture may appear to discover a\nboundary even when the desired membership mask was encoded in its training\nlabels.","For any component \\(q\\) of \\(\\widehat\\theta\\), define its scaffold-only Bayes\nrisk under loss \\(\\ell_q\\) by\n\\[\n\\mathcal R_q^{\\Sigma}\n=\n\\inf_f\n\\mathbb E\n\\left[\n\\ell_q\\bigl(q,f(\\Sigma,\\Omega)\\bigr)\n\\right].\n\\]\nDefine the full formation risk by\n\\[\n\\mathcal R_q^{\\Sigma,\\mathcal H}\n=\n\\inf_g\n\\mathbb E\n\\left[\n\\ell_q\\bigl(q,g(\\Sigma,\\mathcal D_{\\mathrm{form}},\\Omega)\\bigr)\n\\right].\n\\]\nThe history advantage is\n\\[\n\\boxed{\n\\Delta_q^{\\mathrm{hist}}\n=\n\\mathcal R_q^{\\Sigma}\n-\n\\mathcal R_q^{\\Sigma,\\mathcal H}.\n}\n\\]\nA positive history advantage means that formation history contains information\nabout \\(q\\) that is not already recoverable from the declared scaffold and\nrandom seed.","component \\(q\\) of \\(\\widehat\\theta\\), define its scaffold-only Bayes\nrisk under loss \\(\\ell_q\\) by\n\\[\n\\mathcal R_q^{\\Sigma}\n=\n\\inf_f\n\\mathbb E\n\\left[\n\\ell_q\\bigl(q,f(\\Sigma,\\Omega)\\bigr)\n\\right].\n\\]\nDefine the full formation risk by\n\\[\n\\mathcal R_q^{\\Sigma,\\mathcal H}\n=\n\\inf_g\n\\mathbb E\n\\left[\n\\ell_q\\bigl(q,g(\\Sigma,\\mathcal D_{\\mathrm{form}},\\Omega)\\bigr)\n\\right].\n\\]\nThe history advantage is\n\\[\n\\boxed{\n\\Delta_q^{\\mathrm{hist}}\n=\n\\mathcal R_q^{\\Sigma}\n-\n\\mathcal R_q^{\\Sigma,\\mathcal H}.\n}\n\\]\nA positive history advantage means that formation history contains information\nabout \\(q\\) that is not already recoverable from the declared scaffold and\nrandom seed.","r)\n\\right].\n\\]\nDefine the full formation risk by\n\\[\n\\mathcal R_q^{\\Sigma,\\mathcal H}\n=\n\\inf_g\n\\mathbb E\n\\left[\n\\ell_q\\bigl(q,g(\\Sigma,\\mathcal D_{\\mathrm{form}},\\Omega)\\bigr)\n\\right].\n\\]\nThe history advantage is\n\\[\n\\boxed{\n\\Delta_q^{\\mathrm{hist}}\n=\n\\mathcal R_q^{\\Sigma}\n-\n\\mathcal R_q^{\\Sigma,\\mathcal H}.\n}\n\\]\nA positive history advantage means that formation history contains information\nabout \\(q\\) that is not already recoverable from the declared scaffold and\nrandom seed.","\\begin{proposition}[Scaffold-only impossibility]\\label{prop:scaffold-only-impossibility}\nIf\n\\[\nq=f(\\Sigma,\\Omega)\n\\qquad\\text{almost surely},\n\\]\nthen for every formation-history intervention that leaves \\((\\Sigma,\\Omega)\\)\nfixed,\n\\[\n\\mathcal L\\!\\left(q\\mid\n\\operatorname{do}(\\mathcal D_{\\mathrm{form}}=h)\n\\right)\n=\n\\mathcal L\\!\\left(q\\mid\n\\operatorname{do}(\\mathcal D_{\\mathrm{form}}=h')\n\\right),\n\\]\nand \\(\\Delta_q^{\\mathrm{hist}}=0\\).\n\\end{proposition}","{proposition}[Scaffold-only impossibility]\\label{prop:scaffold-only-impossibility}\nIf\n\\[\nq=f(\\Sigma,\\Omega)\n\\qquad\\text{almost surely},\n\\]\nthen for every formation-history intervention that leaves \\((\\Sigma,\\Omega)\\)\nfixed,\n\\[\n\\mathcal L\\!\\left(q\\mid\n\\operatorname{do}(\\mathcal D_{\\mathrm{form}}=h)\n\\right)\n=\n\\mathcal L\\!\\left(q\\mid\n\\operatorname{do}(\\mathcal D_{\\mathrm{form}}=h')\n\\right),\n\\]\nand \\(\\Delta_q^{\\mathrm{hist}}=0\\).\n\\end{proposition}","\\begin{proof}\nThe value of \\(q\\) is measurable with respect to \\((\\Sigma,\\Omega)\\), so an\nintervention on formation history cannot change its conditional law when those\nvariables are held fixed. The scaffold-only predictor \\(f\\) attains the same\nrisk as any predictor that also receives formation history, hence the risk gap\nvanishes.\n\\end{proof}","The same sample must not be used both to select a workspace and to certify its\ncoherence, causal efficacy, or relational irreducibility. The required split is\n\\[\n\\mathcal D\n=\n\\mathcal D_{\\mathrm{form}}\n\\sqcup\n\\mathcal D_{\\mathrm{cal}}\n\\sqcup\n\\mathcal D_{\\mathrm{cert}}.\n\\]\nThe formation sample selects \\(\\widehat\\theta\\). The calibration sample fixes\nthresholds, regularization strengths, and intervention magnitudes. The\ncertification sample evaluates the final gates without further structural\nselection. In a multi-environment study, the stronger requirement is\n\\[\n\\mathcal E_{\\mathrm{form}}\n\\cap\n\\mathcal E_{\\mathrm{cert}}\n=\n\\varnothing,\n\\]\nso that the learned organization is tested under environments not used to\nconstruct it.","cal D\n=\n\\mathcal D_{\\mathrm{form}}\n\\sqcup\n\\mathcal D_{\\mathrm{cal}}\n\\sqcup\n\\mathcal D_{\\mathrm{cert}}.\n\\]\nThe formation sample selects \\(\\widehat\\theta\\). The calibration sample fixes\nthresholds, regularization strengths, and intervention magnitudes. The\ncertification sample evaluates the final gates without further structural\nselection. In a multi-environment study, the stronger requirement is\n\\[\n\\mathcal E_{\\mathrm{form}}\n\\cap\n\\mathcal E_{\\mathrm{cert}}\n=\n\\varnothing,\n\\]\nso that the learned organization is tested under environments not used to\nconstruct it.","\\begin{definition}[Formation leakage]\\label{def:formation-leakage}\nA certification variable \\(Y_{\\mathrm{cert}}\\) leaks into formation when the\nconditional law of \\(\\widehat\\theta\\) changes under an intervention on\n\\(Y_{\\mathrm{cert}}\\) that should be inaccessible during formation:\n\\[\n\\mathcal L\n\\left(\n\\widehat\\theta\n\\mid\n\\operatorname{do}(Y_{\\mathrm{cert}}=y)\n\\right)\n\\neq\n\\mathcal L\n\\left(\n\\widehat\\theta\n\\mid\n\\operatorname{do}(Y_{\\mathrm{cert}}=y')\n\\right).\n\\]\nA valid discovery claim requires a documented information barrier or an audit\nshowing that this effect is absent up to tolerance.\n\\end{definition}","The formation provenance of a workspace and its operational certification are\nseparate coordinates. A fully scaffolded structure may satisfy every\nstructural and causal workspace gate. A highly endogenous structure may fail\nto become coherent or causally useful. Symbolically,\n\\[\n\\operatorname{Endogenous}(W)\n\\centernot\\Longrightarrow\n\\operatorname{SRW}_{\\Theta}(W),\n\\]\n\\[\n\\operatorname{SRW}_{\\Theta}(W)\n\\centernot\\Longrightarrow\n\\operatorname{Endogenous}(W).\n\\]\nThe theory therefore treats endogeneity as a provenance axis rather than a\nhigher level on the workspace hierarchy.","\\begin{definition}[Code-relative history contribution]\\label{def:mdl-endogeneity}\nLet \\(\\mathfrak M_q\\) be a declared coding or predictive model class for a\nlearned component \\(q\\). Let\n\\[\nL_{\\mathfrak M_q}(q\\mid Z)\n\\]\nbe its held-out codelength given side information \\(Z\\). Define\n\\[\n\\boxed{\n e_q^{\\mathrm{MDL}}\n =\n \\left[\n \\frac{\n L_{\\mathfrak M_q}(q\\mid\\Sigma)\n -\n L_{\\mathfrak M_q}(q\\mid\\Sigma,\\mathcal D_{\\mathrm{form}})\n }{\n \\max\\{L_{\\mathfrak M_q}(q\\mid\\Sigma),1\\}\n }\n \\right]_{0}^{1},\n}\n\\]\nwhere \\([x]_0^1=\\min\\{1,\\max\\{0,x\\}\\}\\).\n\\end{definition}","a declared coding or predictive model class for a\nlearned component \\(q\\). Let\n\\[\nL_{\\mathfrak M_q}(q\\mid Z)\n\\]\nbe its held-out codelength given side information \\(Z\\). Define\n\\[\n\\boxed{\n e_q^{\\mathrm{MDL}}\n =\n \\left[\n \\frac{\n L_{\\mathfrak M_q}(q\\mid\\Sigma)\n -\n L_{\\mathfrak M_q}(q\\mid\\Sigma,\\mathcal D_{\\mathrm{form}})\n }{\n \\max\\{L_{\\mathfrak M_q}(q\\mid\\Sigma),1\\}\n }\n \\right]_{0}^{1},\n}\n\\]\nwhere \\([x]_0^1=\\min\\{1,\\max\\{0,x\\}\\}\\).\n\\end{definition}","\\begin{definition}[Causal formation sensitivity]\\label{def:causal-formation-sensitivity}\nLet \\(\\iota_h\\) and \\(\\iota_{h'}\\) be matched feasible interventions that alter\nformation history while preserving the scaffold, randomization law, sample\nsize, and intervention cost. For a divergence \\(D\\), define\n\\[\n\\boxed{\n e_q^{\\mathrm{causal}}\n =\n \\mathbb E_{(h,h')\\sim\\nu_q}\n D\\!\\left(\n \\mathcal L^{\\iota_h}(q),\n \\mathcal L^{\\iota_{h'}}(q)\n \\right).\n}\n\\]\nA common null uses temporally or relationally shuffled histories that preserve\nlocal marginals but destroy the candidate formation signal.\n\\end{definition}","\\begin{proposition}[Reparameterization invariance of causal provenance]\n\\label{prop:causal-provenance-invariance}\nLet \\(\\phi\\) be a measurable bijection on the range of \\(q\\), and let the\ndivergence \\(D\\) be invariant under measurable bijections. Then\n\\[\n e_{\\phi(q)}^{\\mathrm{causal}}\n =\n e_q^{\\mathrm{causal}}.\n\\]\n\\end{proposition}","\\begin{proof}\nEvery intervention-induced law of \\(\\phi(q)\\) is the pushforward of the\ncorresponding law of \\(q\\). Bijection invariance of \\(D\\) preserves each\npairwise divergence, and therefore their design-weighted expectation.\n\\end{proof}","For a family \\(\\mathfrak S\\) of admissible scaffold variants, define the\ncross-scaffold identity defect\n\\[\n\\boxed{\n\\delta_q^{\\mathrm{scaf}}\n=\n\\inf_{\\{J_{\\sigma\\sigma'}\\}}\n\\mathbb E_{\\sigma,\\sigma'\\sim\\pi}\n d_q\n\\left(\nJ_{\\sigma\\sigma'}q_{\\sigma},q_{\\sigma'}\n\\right),\n}\n\\]\nwhere \\(J_{\\sigma\\sigma'}\\) transports the learned object between admissible\ncoordinate systems. A low value indicates that the same abstract\norganization is recovered under multiple non-solution-bearing scaffolds.","The complete formation signature is\n\\[\n\\boxed{\n\\mathbf E_{\\mathrm{form}}(W)\n=\n\\left(\n e_V,e_E,e_H,e_P,e_{\\mathcal O},e_B;\n \\delta^{\\mathrm{scaf}},\n \\delta^{\\mathrm{leak}}\n\\right),\n}\n\\]\nwhere every \\(e_q\\) is reported as the pair\n\\[\n e_q\n=\n\\left(\n e_q^{\\mathrm{MDL}},\n e_q^{\\mathrm{causal}}\n\\right).\n\\]\nNo scalar average is permitted to hide a zero history contribution for one of\nthe structures claimed to have been discovered.","For thresholds \\(\\Theta_{\\mathrm{form}}\\), define\n\\[\n\\operatorname{Form}_{\\Theta_{\\mathrm{form}}}(W)=1\n\\]\nonly when all applicable gates pass:\n\\[\n\\begin{aligned}\n& e_q^{\\mathrm{MDL}}\\geq \\tau_q^{\\mathrm{MDL}},\n&& e_q^{\\mathrm{causal}}\\geq \\tau_q^{\\mathrm{causal}},","For thresholds \\(\\Theta_{\\mathrm{form}}\\), define\n\\[\n\\operatorname{Form}_{\\Theta_{\\mathrm{form}}}(W)=1\n\\]\nonly when all applicable gates pass:\n\\[\n\\begin{aligned}\n& e_q^{\\mathrm{MDL}}\\geq \\tau_q^{\\mathrm{MDL}},\n&& e_q^{\\mathrm{causal}}\\geq \\tau_q^{\\mathrm{causal}},\\\\\n& \\delta_q^{\\mathrm{scaf}}\\leq\\varepsilon_q^{\\mathrm{scaf}},\n&& \\delta^{\\mathrm{leak}}\\leq\\varepsilon_{\\mathrm{leak}},\\\\\n& \\operatorname{SRW}_{\\Theta}(W)=1,\n&& \\operatorname{Pass}(N_{\\mathrm{form}})=1.\n\\end{aligned}\n\\]\nThe final workspace gate is included because endogenous discovery of an\nunstable or causally irrelevant pattern is not endogenous workspace formation.","Let\n\\[\nz_i\\in\\{0,1\\},\n\\qquad\na_e\\in[0,1]\n\\]\nbe node-membership and edge-activation variables satisfying\n\\[\na_e\\leq z_{s(e)},\n\\qquad\na_e\\leq z_{t(e)}.\n\\]\nA weighted learned discrepancy operator is\n\\[\n(D_{a,H}c)_e\n=\n\\sqrt{w_ea_e}\n\\left(\n c_{t(e)}-H_ec_{s(e)}\n\\right).\n\\]\nOne generic formation objective is\n\\[\n\\boxed{\n\\begin{aligned}\n\\widehat\\theta\n\\in\n\\argmin_{z,a,H,P,\\mathcal O,B}\n\\Bigl\n\\{&\n\\widehat{\\mathcal R}_{\\mathrm{task}}\n+\n\\lambda_{\\mathrm{coh}}\\|D_{a,H}c\\|^2\n+\n\\lambda_E\\|a\\|_1\n+\n\\lambda_V\\|z\\|_1","Let\n\\[\nz_i\\in\\{0,1\\},\n\\qquad\na_e\\in[0,1]\n\\]\nbe node-membership and edge-activation variables satisfying\n\\[\na_e\\leq z_{s(e)},\n\\qquad\na_e\\leq z_{t(e)}.\n\\]\nA weighted learned discrepancy operator is\n\\[\n(D_{a,H}c)_e\n=\n\\sqrt{w_ea_e}\n\\left(\n c_{t(e)}-H_ec_{s(e)}\n\\right).\n\\]\nOne generic formation objective is\n\\[\n\\boxed{\n\\begin{aligned}\n\\widehat\\theta\n\\in\n\\argmin_{z,a,H,P,\\mathcal O,B}\n\\Bigl\n\\{&\n\\widehat{\\mathcal R}_{\\mathrm{task}}\n+\n\\lambda_{\\mathrm{coh}}\\|D_{a,H}c\\|^2\n+\n\\lambda_E\\|a\\|_1\n+\n\\lambda_V\\|z\\|_1\\\\\n&+\n\\lambda_{\\mathrm{stab}}\\mathcal J_{\\mathrm{stab}}\n+\n\\lambda_{\\mathrm{leak}}\\mathcal J_{\\mathrm{leak}}\n\\Bigr\\}.\n\\end{aligned}\n}\n\\]\nThe objective is schematic rather than universal. Its purpose is to expose\nwhich quantities were fixed and which were optimized. If \\(z\\) and \\(a\\) are\nfixed by \\(\\Sigma\\), only the transfer maps or local channels can be claimed as\nlearned.","&+\n\\lambda_{\\mathrm{stab}}\\mathcal J_{\\mathrm{stab}}\n+\n\\lambda_{\\mathrm{leak}}\\mathcal J_{\\mathrm{leak}}\n\\Bigr\\}.\n\\end{aligned}\n}\n\\]\nThe objective is schematic rather than universal. Its purpose is to expose\nwhich quantities were fixed and which were optimized. If \\(z\\) and \\(a\\) are\nfixed by \\(\\Sigma\\), only the transfer maps or local channels can be claimed as\nlearned.","A learned transfer map is not absolutely identifiable when local coordinates\nare latent. Under local invertible changes\n\\[\nX_u'=g_uX_u,\n\\qquad\nX_v'=g_vX_v,\n\\]\nthe same abstract transfer is represented by\n\\[\nH_{uv}'=g_vH_{uv}g_u^{-1}.\n\\]\nThe identifiable object is therefore the gauge class\n\\[\n[H]_{\\mathcal G}\n=\n\\left\\{\n(g_{t(e)}H_eg_{s(e)}^{-1})_{e\\in E}\n:\n(g_i)_{i\\in V}\\in\\mathcal G\n\\right\\}.\n\\]\nAny formation claim that compares raw matrices across independently trained\nsystems without gauge alignment is ill posed.","\\subtopic{49.3 Transfer identification theorem \\(T_{15}\\) }","\\begin{theorem}[Population transfer identification]\\label{thm:transfer-identification}\nLet \\(X\\in\\mathbb R^{d_u}\\) and \\(Y\\in\\mathbb R^{d_v}\\) satisfy\n\\[\nY=H_*X+\\xi,\n\\qquad\n\\mathbb E[\\xi X^{\\mathsf T}]=0,\n\\]\nand assume\n\\[\n\\Sigma_{XX}\n=\n\\mathbb E[XX^{\\mathsf T}]\n\\succeq\n\\kappa I\n\\]\nfor some \\(\\kappa>0\\). Then \\(H_*\\) is the unique minimizer of\n\\[\n\\mathcal R(H)\n=\n\\mathbb E\\|Y-HX\\|_2^2,\n\\]\nwith\n\\[\n\\boxed{\nH_*\n=\n\\Sigma_{YX}\\Sigma_{XX}^{-1}.\n}\n\\]\nMoreover,\n\\[\n\\boxed{\n\\mathcal R(H)-\\mathcal R(H_*)\n=\n\\operatorname{tr}\n\\left(\n(H-H_*)\\Sigma_{XX}(H-H_*)^{\\mathsf T}\n\\right)\n\\geq\n\\kappa\\|H-H_*\\|_F^2.\n}\n\\]\n\\end{theorem}","thm:transfer-identification}\nLet \\(X\\in\\mathbb R^{d_u}\\) and \\(Y\\in\\mathbb R^{d_v}\\) satisfy\n\\[\nY=H_*X+\\xi,\n\\qquad\n\\mathbb E[\\xi X^{\\mathsf T}]=0,\n\\]\nand assume\n\\[\n\\Sigma_{XX}\n=\n\\mathbb E[XX^{\\mathsf T}]\n\\succeq\n\\kappa I\n\\]\nfor some \\(\\kappa>0\\). Then \\(H_*\\) is the unique minimizer of\n\\[\n\\mathcal R(H)\n=\n\\mathbb E\\|Y-HX\\|_2^2,\n\\]\nwith\n\\[\n\\boxed{\nH_*\n=\n\\Sigma_{YX}\\Sigma_{XX}^{-1}.\n}\n\\]\nMoreover,\n\\[\n\\boxed{\n\\mathcal R(H)-\\mathcal R(H_*)\n=\n\\operatorname{tr}\n\\left(\n(H-H_*)\\Sigma_{XX}(H-H_*)^{\\mathsf T}\n\\right)\n\\geq\n\\kappa\\|H-H_*\\|_F^2.\n}\n\\]\n\\end{theorem}","\\begin{proof}\nSubstitute \\(Y=H_*X+\\xi\\) into the risk. The cross term vanishes because\n\\(\\mathbb E[\\xi X^{\\mathsf T}]=0\\), giving\n\\[\n\\mathcal R(H)\n=\n\\mathcal R(H_*)\n+\n\\mathbb E\\|(H-H_*)X\\|_2^2.\n\\]\nThe remaining expectation is the displayed trace. Positive definiteness of\n\\(\\Sigma_{XX}\\) gives uniqueness and the lower bound. The normal equations\ngive the covariance formula.\n\\end{proof}","\\begin{corollary}[Deterministic empirical recovery bound]\n\\label{cor:empirical-transfer-bound}\nFor samples \\((X_r,Y_r)_{r=1}^n\\), write\n\\[\n\\widehat\\Sigma_{XX}\n=\n\\frac1n\\sum_{r=1}^nX_rX_r^{\\mathsf T},\n\\qquad\n\\widehat M\n=\n\\frac1n\\sum_{r=1}^n\\xi_rX_r^{\\mathsf T}.\n\\]\nIf \\(\\widehat\\Sigma_{XX}\\succeq\\widehat\\kappa I\\), then ordinary least squares\nsatisfies\n\\[\n\\boxed{\n\\|\\widehat H-H_*\\|_F\n\\leq\n\\frac{\\|\\widehat M\\|_F}{\\widehat\\kappa}.\n}\n\\]\nFor ridge parameter \\(\\tau>0\\),\n\\[\n\\boxed{\n\\|\\widehat H_\\tau-H_*\\|_F\n\\leq\n\\frac{\n\\|\\widehat M\\|_F+\\tau\\|H_*\\|_F\n}{\n\\widehat\\kappa+\\tau\n}.\n}\n\\]\n\\end{corollary}","\\label{cor:empirical-transfer-bound}\nFor samples \\((X_r,Y_r)_{r=1}^n\\), write\n\\[\n\\widehat\\Sigma_{XX}\n=\n\\frac1n\\sum_{r=1}^nX_rX_r^{\\mathsf T},\n\\qquad\n\\widehat M\n=\n\\frac1n\\sum_{r=1}^n\\xi_rX_r^{\\mathsf T}.\n\\]\nIf \\(\\widehat\\Sigma_{XX}\\succeq\\widehat\\kappa I\\), then ordinary least squares\nsatisfies\n\\[\n\\boxed{\n\\|\\widehat H-H_*\\|_F\n\\leq\n\\frac{\\|\\widehat M\\|_F}{\\widehat\\kappa}.\n}\n\\]\nFor ridge parameter \\(\\tau>0\\),\n\\[\n\\boxed{\n\\|\\widehat H_\\tau-H_*\\|_F\n\\leq\n\\frac{\n\\|\\widehat M\\|_F+\\tau\\|H_*\\|_F\n}{\n\\widehat\\kappa+\\tau\n}.\n}\n\\]\n\\end{corollary}","mpirical-transfer-bound}\nFor samples \\((X_r,Y_r)_{r=1}^n\\), write\n\\[\n\\widehat\\Sigma_{XX}\n=\n\\frac1n\\sum_{r=1}^nX_rX_r^{\\mathsf T},\n\\qquad\n\\widehat M\n=\n\\frac1n\\sum_{r=1}^n\\xi_rX_r^{\\mathsf T}.\n\\]\nIf \\(\\widehat\\Sigma_{XX}\\succeq\\widehat\\kappa I\\), then ordinary least squares\nsatisfies\n\\[\n\\boxed{\n\\|\\widehat H-H_*\\|_F\n\\leq\n\\frac{\\|\\widehat M\\|_F}{\\widehat\\kappa}.\n}\n\\]\nFor ridge parameter \\(\\tau>0\\),\n\\[\n\\boxed{\n\\|\\widehat H_\\tau-H_*\\|_F\n\\leq\n\\frac{\n\\|\\widehat M\\|_F+\\tau\\|H_*\\|_F\n}{\n\\widehat\\kappa+\\tau\n}.\n}\n\\]\n\\end{corollary}","\\begin{proof}\nThe empirical normal equations give\n\\[\n\\widehat H-H_*\n=\n\\widehat M\\widehat\\Sigma_{XX}^{-1}.\n\\]\nFor ridge regression,\n\\[\n\\widehat H_\\tau-H_*\n=\n(\\widehat M-\\tau H_*)\n(\\widehat\\Sigma_{XX}+\\tau I)^{-1}.\n\\]\nTaking Frobenius and operator norms yields both estimates.\n\\end{proof}","\\begin{proposition}[Gauge covariance of least squares]\n\\label{prop:ols-gauge-covariance}\nUnder invertible coordinate transformations\n\\[\nX'=g_uX,\n\\qquad\nY'=g_vY,\n\\]\nthe population least-squares solution transforms as\n\\[\nH_*'=g_vH_*g_u^{-1}.\n\\]\n\\end{proposition}","\\begin{proof}\nThe transformed covariances are\n\\[\n\\Sigma_{Y'X'}=g_v\\Sigma_{YX}g_u^{\\mathsf T},\n\\qquad\n\\Sigma_{X'X'}=g_u\\Sigma_{XX}g_u^{\\mathsf T}.\n\\]\nSubstitution into the covariance formula and inversion of the second factor\ngive the result.\n\\end{proof}","The theorem establishes identification only under excitation and exogeneity.\nWhen \\(\\Sigma_{XX}\\) is singular, the transfer is identified only on the\nexcited source subspace. When \\(\\mathbb E[\\xi X^{\\mathsf T}]\\neq0\\), regression\nrecovers a predictive map rather than necessarily the intended causal\ntransfer.","\\subtopic{49.4 Held-out structure-selection theorem \\(T_{16}\\) }","\\begin{theorem}[Uniform structure-selection recovery]\n\\label{thm:uniform-structure-selection}\nLet \\(\\mathfrak T\\) be a finite candidate family with population score\n\\(J:\\mathfrak T\\to\\mathbb R\\). Let\n\\[\n\\theta_*\n=\n\\argmin_{\\theta\\in\\mathfrak T}J(\\theta)\n\\]\nbe unique, with margin\n\\[\n\\Delta_J\n=\n\\min_{\\theta\\neq\\theta_*}\n\\bigl(J(\\theta)-J(\\theta_*)\\bigr)>0.\n\\]\nIf a held-out estimate \\(\\widehat J\\) satisfies\n\\[\n\\sup_{\\theta\\in\\mathfrak T}\n|\\widehat J(\\theta)-J(\\theta)|\n\\leq\\varepsilon,\n\\]\nthen every empirical minimizer \\(\\widehat\\theta\\) obeys\n\\[\n\\boxed{\nJ(\\widehat\\theta)-J(\\theta_*)\n\\leq2\\varepsilon.\n}\n\\]\nIf \\(2\\varepsilon<\\Delta_J\\), then\n\\[\n\\boxed{\\widehat\\theta=\\theta_*.}\n\\]\n\\end{theorem}","orem}[Uniform structure-selection recovery]\n\\label{thm:uniform-structure-selection}\nLet \\(\\mathfrak T\\) be a finite candidate family with population score\n\\(J:\\mathfrak T\\to\\mathbb R\\). Let\n\\[\n\\theta_*\n=\n\\argmin_{\\theta\\in\\mathfrak T}J(\\theta)\n\\]\nbe unique, with margin\n\\[\n\\Delta_J\n=\n\\min_{\\theta\\neq\\theta_*}\n\\bigl(J(\\theta)-J(\\theta_*)\\bigr)>0.\n\\]\nIf a held-out estimate \\(\\widehat J\\) satisfies\n\\[\n\\sup_{\\theta\\in\\mathfrak T}\n|\\widehat J(\\theta)-J(\\theta)|\n\\leq\\varepsilon,\n\\]\nthen every empirical minimizer \\(\\widehat\\theta\\) obeys\n\\[\n\\boxed{\nJ(\\widehat\\theta)-J(\\theta_*)\n\\leq2\\varepsilon.\n}\n\\]\nIf \\(2\\varepsilon<\\Delta_J\\), then\n\\[\n\\boxed{\\widehat\\theta=\\theta_*.}\n\\]\n\\end{theorem}","te candidate family with population score\n\\(J:\\mathfrak T\\to\\mathbb R\\). Let\n\\[\n\\theta_*\n=\n\\argmin_{\\theta\\in\\mathfrak T}J(\\theta)\n\\]\nbe unique, with margin\n\\[\n\\Delta_J\n=\n\\min_{\\theta\\neq\\theta_*}\n\\bigl(J(\\theta)-J(\\theta_*)\\bigr)>0.\n\\]\nIf a held-out estimate \\(\\widehat J\\) satisfies\n\\[\n\\sup_{\\theta\\in\\mathfrak T}\n|\\widehat J(\\theta)-J(\\theta)|\n\\leq\\varepsilon,\n\\]\nthen every empirical minimizer \\(\\widehat\\theta\\) obeys\n\\[\n\\boxed{\nJ(\\widehat\\theta)-J(\\theta_*)\n\\leq2\\varepsilon.\n}\n\\]\nIf \\(2\\varepsilon<\\Delta_J\\), then\n\\[\n\\boxed{\\widehat\\theta=\\theta_*.}\n\\]\n\\end{theorem}","population score\n\\(J:\\mathfrak T\\to\\mathbb R\\). Let\n\\[\n\\theta_*\n=\n\\argmin_{\\theta\\in\\mathfrak T}J(\\theta)\n\\]\nbe unique, with margin\n\\[\n\\Delta_J\n=\n\\min_{\\theta\\neq\\theta_*}\n\\bigl(J(\\theta)-J(\\theta_*)\\bigr)>0.\n\\]\nIf a held-out estimate \\(\\widehat J\\) satisfies\n\\[\n\\sup_{\\theta\\in\\mathfrak T}\n|\\widehat J(\\theta)-J(\\theta)|\n\\leq\\varepsilon,\n\\]\nthen every empirical minimizer \\(\\widehat\\theta\\) obeys\n\\[\n\\boxed{\nJ(\\widehat\\theta)-J(\\theta_*)\n\\leq2\\varepsilon.\n}\n\\]\nIf \\(2\\varepsilon<\\Delta_J\\), then\n\\[\n\\boxed{\\widehat\\theta=\\theta_*.}\n\\]\n\\end{theorem}","argmin_{\\theta\\in\\mathfrak T}J(\\theta)\n\\]\nbe unique, with margin\n\\[\n\\Delta_J\n=\n\\min_{\\theta\\neq\\theta_*}\n\\bigl(J(\\theta)-J(\\theta_*)\\bigr)>0.\n\\]\nIf a held-out estimate \\(\\widehat J\\) satisfies\n\\[\n\\sup_{\\theta\\in\\mathfrak T}\n|\\widehat J(\\theta)-J(\\theta)|\n\\leq\\varepsilon,\n\\]\nthen every empirical minimizer \\(\\widehat\\theta\\) obeys\n\\[\n\\boxed{\nJ(\\widehat\\theta)-J(\\theta_*)\n\\leq2\\varepsilon.\n}\n\\]\nIf \\(2\\varepsilon<\\Delta_J\\), then\n\\[\n\\boxed{\\widehat\\theta=\\theta_*.}\n\\]\n\\end{theorem}","theta)\n\\]\nbe unique, with margin\n\\[\n\\Delta_J\n=\n\\min_{\\theta\\neq\\theta_*}\n\\bigl(J(\\theta)-J(\\theta_*)\\bigr)>0.\n\\]\nIf a held-out estimate \\(\\widehat J\\) satisfies\n\\[\n\\sup_{\\theta\\in\\mathfrak T}\n|\\widehat J(\\theta)-J(\\theta)|\n\\leq\\varepsilon,\n\\]\nthen every empirical minimizer \\(\\widehat\\theta\\) obeys\n\\[\n\\boxed{\nJ(\\widehat\\theta)-J(\\theta_*)\n\\leq2\\varepsilon.\n}\n\\]\nIf \\(2\\varepsilon<\\Delta_J\\), then\n\\[\n\\boxed{\\widehat\\theta=\\theta_*.}\n\\]\n\\end{theorem}","\\begin{proof}\nBy empirical optimality and uniform approximation,\n\\[\nJ(\\widehat\\theta)\n\\leq\n\\widehat J(\\widehat\\theta)+\\varepsilon\n\\leq\n\\widehat J(\\theta_*)+\\varepsilon\n\\leq\nJ(\\theta_*)+2\\varepsilon.\n\\]\nIf \\(\\widehat\\theta\\neq\\theta_*\\), the margin would force an excess larger than\n\\(2\\varepsilon\\), a contradiction.\n\\end{proof}","The provenance classes are now defined relative to\n\\[\n(\\Sigma,\\mathcal L,\\mathcal I_{\\mathrm{form}},\\mathfrak M).\n\\]\nThey are not absolute labels.","\\item[Adaptively parameterized.]\nThe membership and channel topology are fixed, while transfer parameters,\nlocal operators, or readouts are learned from interaction history. Typically\n\\[\ne_H+e_P+e_{\\mathcal O}>0,\n\\qquad\ne_V\\approx e_E\\approx e_B\\approx0.\n\\]","\\begin{definition}[Boundary problem]\\label{def:boundary-problem}\nA boundary problem is a tuple\n\\[\n\\mathfrak B\n=\n\\left(\nV,\n\\mathfrak S,\n\\tau,\n\\mathcal Y,\n\\mathcal I,\n\\Pi,\n\\mathcal G_B\n\\right),\n\\]\nwhere \\(\\mathfrak S\\subseteq2^V\\) is the admissible candidate family,\n\\(\\tau\\) is the temporal scale, \\(\\mathcal Y\\) is the operational target,\n\\(\\mathcal I\\) is the feasible intervention family, \\(\\Pi\\) is the\nenvironment distribution, and \\(\\mathcal G_B\\) is a symmetry group under\nwhich boundaries are considered equivalent.\n\\end{definition}","abel{def:boundary-problem}\nA boundary problem is a tuple\n\\[\n\\mathfrak B\n=\n\\left(\nV,\n\\mathfrak S,\n\\tau,\n\\mathcal Y,\n\\mathcal I,\n\\Pi,\n\\mathcal G_B\n\\right),\n\\]\nwhere \\(\\mathfrak S\\subseteq2^V\\) is the admissible candidate family,\n\\(\\tau\\) is the temporal scale, \\(\\mathcal Y\\) is the operational target,\n\\(\\mathcal I\\) is the feasible intervention family, \\(\\Pi\\) is the\nenvironment distribution, and \\(\\mathcal G_B\\) is a symmetry group under\nwhich boundaries are considered equivalent.\n\\end{definition}","Unless the application requires otherwise, the candidate family excludes\n\\[\n\\varnothing,\n\\qquad\nV,\n\\qquad\n\\{i\\}\\ \\text{for }i\\in V.\n\\]\nThis anti-triviality rule prevents the empty set, the entire realization, or a\nsingle centralized controller from winning solely through a degenerate score.","For \\(S\\in\\mathfrak S\\), define\n\\[\n\\Ret_\\tau(S)\n=\nI\\!\\left(\nX_S(t+\\tau);X_S(t)\n\\mid X_{\\bar S}(t),E_t\n\\right),\n\\]\n\\[\n\\Dep_\\tau(S)\n=\nI\\!\\left(\nX_S(t+\\tau);X_{\\bar S}(t)\n\\mid X_S(t),E_t\n\\right),\n\\]\n\\[\n\\Out_\\tau(S)\n=\nI\\!\\left(\nX_{\\bar S}(t+\\tau);X_S(t)\n\\mid X_{\\bar S}(t),E_t\n\\right).\n\\]\nThese quantities are observational. They can be nonzero because of common\ncauses, selection, or feedback paths and therefore cannot alone establish a\ncausal boundary.","Let \\(\\iota_{S,x}\\) be a feasible intervention that changes the internal\nworkspace coordinate of \\(S\\) to \\(x\\) while matching background variables and\nintervention cost. Define outward causal efficacy\n\\[\n\\boxed{\n\\COut_\\tau(S)\n=\n\\mathbb E_{x,x'\\sim\\nu_S}\nD\\!\\left(\n\\mathcal L^{\\iota_{S,x}}\n(X_{\\bar S}(t+\\tau)),\n\\mathcal L^{\\iota_{S,x'}}\n(X_{\\bar S}(t+\\tau))\n\\right).\n}\n\\]\nDefine inward causal dependence analogously by intervening on \\(\\bar S\\) and\nmeasuring the future law of \\(X_S\\):\n\\[\n\\boxed{\n\\CIn_\\tau(S)\n=\n\\mathbb E_{u,u'\\sim\\nu_{\\bar S}}\nD\\!\\left(\n\\mathcal L^{\\iota_{\\bar S,u}}\n(X_S(t+\\tau)),\n\\mathcal L^{\\iota_{\\bar S,u'}}\n(X_S(t+\\tau))\n\\right).\n}\n\\]\nNeither term should be minimized to zero in general. A us","Let \\(\\iota_{S,x}\\) be a feasible intervention that changes the internal\nworkspace coordinate of \\(S\\) to \\(x\\) while matching background variables and\nintervention cost. Define outward causal efficacy\n\\[\n\\boxed{\n\\COut_\\tau(S)\n=\n\\mathbb E_{x,x'\\sim\\nu_S}\nD\\!\\left(\n\\mathcal L^{\\iota_{S,x}}\n(X_{\\bar S}(t+\\tau)),\n\\mathcal L^{\\iota_{S,x'}}\n(X_{\\bar S}(t+\\tau))\n\\right).\n}\n\\]\nDefine inward causal dependence analogously by intervening on \\(\\bar S\\) and\nmeasuring the future law of \\(X_S\\):\n\\[\n\\boxed{\n\\CIn_\\tau(S)\n=\n\\mathbb E_{u,u'\\sim\\nu_{\\bar S}}\nD\\!\\left(\n\\mathcal L^{\\iota_{\\bar S,u}}\n(X_S(t+\\tau)),\n\\mathcal L^{\\iota_{\\bar S,u'}}\n(X_S(t+\\tau))\n\\right).\n}\n\\]\nNeither term should be minimized to zero in general. A useful workspace may\nreceive substantial external input. The boundary objective must","Let \\(\\iota_{S,x}\\) be a feasible intervention that changes the internal\nworkspace coordinate of \\(S\\) to \\(x\\) while matching background variables and\nintervention cost. Define outward causal efficacy\n\\[\n\\boxed{\n\\COut_\\tau(S)\n=\n\\mathbb E_{x,x'\\sim\\nu_S}\nD\\!\\left(\n\\mathcal L^{\\iota_{S,x}}\n(X_{\\bar S}(t+\\tau)),\n\\mathcal L^{\\iota_{S,x'}}\n(X_{\\bar S}(t+\\tau))\n\\right).\n}\n\\]\nDefine inward causal dependence analogously by intervening on \\(\\bar S\\) and\nmeasuring the future law of \\(X_S\\):\n\\[\n\\boxed{\n\\CIn_\\tau(S)\n=\n\\mathbb E_{u,u'\\sim\\nu_{\\bar S}}\nD\\!\\left(\n\\mathcal L^{\\iota_{\\bar S,u}}\n(X_S(t+\\tau)),\n\\mathcal L^{\\iota_{\\bar S,u'}}\n(X_S(t+\\tau))\n\\right).\n}\n\\]\nNeither term should be minimized to zero in general. A useful workspace may\nreceive substantial external input. The boundary objective must instead\ne","Let \\(\\iota_{S,x}\\) be a feasible intervention that changes the internal\nworkspace coordinate of \\(S\\) to \\(x\\) while matching background variables and\nintervention cost. Define outward causal efficacy\n\\[\n\\boxed{\n\\COut_\\tau(S)\n=\n\\mathbb E_{x,x'\\sim\\nu_S}\nD\\!\\left(\n\\mathcal L^{\\iota_{S,x}}\n(X_{\\bar S}(t+\\tau)),\n\\mathcal L^{\\iota_{S,x'}}\n(X_{\\bar S}(t+\\tau))\n\\right).\n}\n\\]\nDefine inward causal dependence analogously by intervening on \\(\\bar S\\) and\nmeasuring the future law of \\(X_S\\):\n\\[\n\\boxed{\n\\CIn_\\tau(S)\n=\n\\mathbb E_{u,u'\\sim\\nu_{\\bar S}}\nD\\!\\left(\n\\mathcal L^{\\iota_{\\bar S,u}}\n(X_S(t+\\tau)),\n\\mathcal L^{\\iota_{\\bar S,u'}}\n(X_S(t+\\tau))\n\\right).\n}\n\\]\nNeither term should be minimized to zero in general. A useful workspace may\nreceive substantial external input. The boundary objective must instead\nexpress the intended balance between internal recurrence, external dependence,\nand outward efficacy.","ervention that changes the internal\nworkspace coordinate of \\(S\\) to \\(x\\) while matching background variables and\nintervention cost. Define outward causal efficacy\n\\[\n\\boxed{\n\\COut_\\tau(S)\n=\n\\mathbb E_{x,x'\\sim\\nu_S}\nD\\!\\left(\n\\mathcal L^{\\iota_{S,x}}\n(X_{\\bar S}(t+\\tau)),\n\\mathcal L^{\\iota_{S,x'}}\n(X_{\\bar S}(t+\\tau))\n\\right).\n}\n\\]\nDefine inward causal dependence analogously by intervening on \\(\\bar S\\) and\nmeasuring the future law of \\(X_S\\):\n\\[\n\\boxed{\n\\CIn_\\tau(S)\n=\n\\mathbb E_{u,u'\\sim\\nu_{\\bar S}}\nD\\!\\left(\n\\mathcal L^{\\iota_{\\bar S,u}}\n(X_S(t+\\tau)),\n\\mathcal L^{\\iota_{\\bar S,u'}}\n(X_S(t+\\tau))\n\\right).\n}\n\\]\nNeither term should be minimized to zero in general. A useful workspace may\nreceive substantial external input. The boundary objective must instead\nexpress the intended balance between internal recurrence, external dependence,\nand outward efficacy.","ce coordinate of \\(S\\) to \\(x\\) while matching background variables and\nintervention cost. Define outward causal efficacy\n\\[\n\\boxed{\n\\COut_\\tau(S)\n=\n\\mathbb E_{x,x'\\sim\\nu_S}\nD\\!\\left(\n\\mathcal L^{\\iota_{S,x}}\n(X_{\\bar S}(t+\\tau)),\n\\mathcal L^{\\iota_{S,x'}}\n(X_{\\bar S}(t+\\tau))\n\\right).\n}\n\\]\nDefine inward causal dependence analogously by intervening on \\(\\bar S\\) and\nmeasuring the future law of \\(X_S\\):\n\\[\n\\boxed{\n\\CIn_\\tau(S)\n=\n\\mathbb E_{u,u'\\sim\\nu_{\\bar S}}\nD\\!\\left(\n\\mathcal L^{\\iota_{\\bar S,u}}\n(X_S(t+\\tau)),\n\\mathcal L^{\\iota_{\\bar S,u'}}\n(X_S(t+\\tau))\n\\right).\n}\n\\]\nNeither term should be minimized to zero in general. A useful workspace may\nreceive substantial external input. The boundary objective must instead\nexpress the intended balance between internal recurrence, external dependence,\nand outward efficacy.","nate of \\(S\\) to \\(x\\) while matching background variables and\nintervention cost. Define outward causal efficacy\n\\[\n\\boxed{\n\\COut_\\tau(S)\n=\n\\mathbb E_{x,x'\\sim\\nu_S}\nD\\!\\left(\n\\mathcal L^{\\iota_{S,x}}\n(X_{\\bar S}(t+\\tau)),\n\\mathcal L^{\\iota_{S,x'}}\n(X_{\\bar S}(t+\\tau))\n\\right).\n}\n\\]\nDefine inward causal dependence analogously by intervening on \\(\\bar S\\) and\nmeasuring the future law of \\(X_S\\):\n\\[\n\\boxed{\n\\CIn_\\tau(S)\n=\n\\mathbb E_{u,u'\\sim\\nu_{\\bar S}}\nD\\!\\left(\n\\mathcal L^{\\iota_{\\bar S,u}}\n(X_S(t+\\tau)),\n\\mathcal L^{\\iota_{\\bar S,u'}}\n(X_S(t+\\tau))\n\\right).\n}\n\\]\nNeither term should be minimized to zero in general. A useful workspace may\nreceive substantial external input. The boundary objective must instead\nexpress the intended balance between internal recurrence, external dependence,\nand outward efficacy.","For the induced subsystem on \\(S\\), define\n\\[\n\\mathsf{Coh}(S)\n=\n\\operatorname{tr}\n\\left(\nP_{S,d}L_SP_{S,d}\n\\right),\n\\]\nwhere \\(P_{S,d}\\) projects onto the selected low-energy workspace modes. Let\n\\[\n\\mathsf{Cut}(S)\n=\n\\sum_{e:s(e)\\in S,\\ t(e)\\notin S}w_e\n+\n\\sum_{e:s(e)\\notin S,\\ t(e)\\in S}w_e.\n\\]\nLet \\(k_\\varepsilon(S)\\) be the minimal sufficient coalition size inside the\ncandidate subsystem. A candidate is admissible only if\n\\[\n\\operatorname{SRW}_{\\Theta_S}(W_S)=1,\n\\qquad\nk_\\varepsilon(S)\\geq2,\n\\]\nunless the claim explicitly concerns a centralized workspace.","S\\), define\n\\[\n\\mathsf{Coh}(S)\n=\n\\operatorname{tr}\n\\left(\nP_{S,d}L_SP_{S,d}\n\\right),\n\\]\nwhere \\(P_{S,d}\\) projects onto the selected low-energy workspace modes. Let\n\\[\n\\mathsf{Cut}(S)\n=\n\\sum_{e:s(e)\\in S,\\ t(e)\\notin S}w_e\n+\n\\sum_{e:s(e)\\notin S,\\ t(e)\\in S}w_e.\n\\]\nLet \\(k_\\varepsilon(S)\\) be the minimal sufficient coalition size inside the\ncandidate subsystem. A candidate is admissible only if\n\\[\n\\operatorname{SRW}_{\\Theta_S}(W_S)=1,\n\\qquad\nk_\\varepsilon(S)\\geq2,\n\\]\nunless the claim explicitly concerns a centralized workspace.","A general score is\n\\[\n\\boxed{\n\\begin{aligned}\n\\mathcal B_t(S)\n={}&\n a_R\\Ret_\\tau(S)\n+a_O\\Out_\\tau(S)\n+a_C\\COut_\\tau(S)\\\\\n&-a_D\\Dep_\\tau(S)\n-a_I\\CIn_\\tau(S)\n-a_L\\mathsf{Coh}(S)\\\\\n&-a_K\\mathsf{Cut}(S)\n-a_Z\\mathsf{Size}(S)\n-a_T\\Instability_t(S).\n\\end{aligned}\n}\n\\]\nThe coefficients and all normalizations are part of the boundary problem. No\nsingle choice is universal. The selected boundary is\n\\[\n\\widehat S_t\n\\in\n\\argmax_{S\\in\\mathfrak S_{\\mathrm{pass}}}\n\\widehat{\\mathcal B}_t(S),\n\\]\nwhere \\(\\mathfrak S_{\\mathrm{pass}}\\) contains only candidates that satisfy the\nrequired workspace and anti-centralization gates.","&-a_K\\mathsf{Cut}(S)\n-a_Z\\mathsf{Size}(S)\n-a_T\\Instability_t(S).\n\\end{aligned}\n}\n\\]\nThe coefficients and all normalizations are part of the boundary problem. No\nsingle choice is universal. The selected boundary is\n\\[\n\\widehat S_t\n\\in\n\\argmax_{S\\in\\mathfrak S_{\\mathrm{pass}}}\n\\widehat{\\mathcal B}_t(S),\n\\]\nwhere \\(\\mathfrak S_{\\mathrm{pass}}\\) contains only candidates that satisfy the\nrequired workspace and anti-centralization gates.","If a symmetry \\(g\\in\\mathcal G_B\\) leaves the full observational and\ninterventional law invariant, \\(S\\) and \\(gS\\) cannot be distinguished by the\ndeclared experiment. The identifiable boundary is then the orbit\n\\[\n[S]_{\\mathcal G_B}\n=\n\\{gS:g\\in\\mathcal G_B\\}.\n\\]\nA unique subset should not be claimed when only an equivalence class is\nidentifiable.","\\subtopic{51.2 Stable boundary recovery \\(T_{17}\\) }","\\begin{theorem}[Held-out boundary recovery]\\label{thm:heldout-boundary-recovery}\nLet \\(\\mathfrak S_{\\mathrm{pass}}\\) be finite and let\n\\[\nS_t^*\n=\n\\argmax_{S\\in\\mathfrak S_{\\mathrm{pass}}}\n\\mathcal B_t(S)\n\\]\nbe unique. Define the population margin\n\\[\n\\Delta_t^B\n=\n\\mathcal B_t(S_t^*)\n-\n\\max_{S\\neq S_t^*}\\mathcal B_t(S)>0.\n\\]\nIf\n\\[\n\\sup_{S\\in\\mathfrak S_{\\mathrm{pass}}}\n\\left|\n\\widehat{\\mathcal B}_t(S)-\\mathcal B_t(S)\n\\right|\n\\leq\\varepsilon_t,\n\\]\nthen every empirical maximizer \\(\\widehat S_t\\) satisfies\n\\[\n\\boxed{\n\\mathcal B_t(S_t^*)-\n\\mathcal B_t(\\widehat S_t)\n\\leq2\\varepsilon_t.\n}\n\\]\nIf \\(2\\varepsilon_t<\\Delta_t^B\\), then\n\\[\n\\boxed{\\widehat S_t=S_t^*.}\n\\]\n\\end{theorem}","et \\(\\mathfrak S_{\\mathrm{pass}}\\) be finite and let\n\\[\nS_t^*\n=\n\\argmax_{S\\in\\mathfrak S_{\\mathrm{pass}}}\n\\mathcal B_t(S)\n\\]\nbe unique. Define the population margin\n\\[\n\\Delta_t^B\n=\n\\mathcal B_t(S_t^*)\n-\n\\max_{S\\neq S_t^*}\\mathcal B_t(S)>0.\n\\]\nIf\n\\[\n\\sup_{S\\in\\mathfrak S_{\\mathrm{pass}}}\n\\left|\n\\widehat{\\mathcal B}_t(S)-\\mathcal B_t(S)\n\\right|\n\\leq\\varepsilon_t,\n\\]\nthen every empirical maximizer \\(\\widehat S_t\\) satisfies\n\\[\n\\boxed{\n\\mathcal B_t(S_t^*)-\n\\mathcal B_t(\\widehat S_t)\n\\leq2\\varepsilon_t.\n}\n\\]\nIf \\(2\\varepsilon_t<\\Delta_t^B\\), then\n\\[\n\\boxed{\\widehat S_t=S_t^*.}\n\\]\n\\end{theorem}","ass}}\\) be finite and let\n\\[\nS_t^*\n=\n\\argmax_{S\\in\\mathfrak S_{\\mathrm{pass}}}\n\\mathcal B_t(S)\n\\]\nbe unique. Define the population margin\n\\[\n\\Delta_t^B\n=\n\\mathcal B_t(S_t^*)\n-\n\\max_{S\\neq S_t^*}\\mathcal B_t(S)>0.\n\\]\nIf\n\\[\n\\sup_{S\\in\\mathfrak S_{\\mathrm{pass}}}\n\\left|\n\\widehat{\\mathcal B}_t(S)-\\mathcal B_t(S)\n\\right|\n\\leq\\varepsilon_t,\n\\]\nthen every empirical maximizer \\(\\widehat S_t\\) satisfies\n\\[\n\\boxed{\n\\mathcal B_t(S_t^*)-\n\\mathcal B_t(\\widehat S_t)\n\\leq2\\varepsilon_t.\n}\n\\]\nIf \\(2\\varepsilon_t<\\Delta_t^B\\), then\n\\[\n\\boxed{\\widehat S_t=S_t^*.}\n\\]\n\\end{theorem}","B_t(S)\n\\]\nbe unique. Define the population margin\n\\[\n\\Delta_t^B\n=\n\\mathcal B_t(S_t^*)\n-\n\\max_{S\\neq S_t^*}\\mathcal B_t(S)>0.\n\\]\nIf\n\\[\n\\sup_{S\\in\\mathfrak S_{\\mathrm{pass}}}\n\\left|\n\\widehat{\\mathcal B}_t(S)-\\mathcal B_t(S)\n\\right|\n\\leq\\varepsilon_t,\n\\]\nthen every empirical maximizer \\(\\widehat S_t\\) satisfies\n\\[\n\\boxed{\n\\mathcal B_t(S_t^*)-\n\\mathcal B_t(\\widehat S_t)\n\\leq2\\varepsilon_t.\n}\n\\]\nIf \\(2\\varepsilon_t<\\Delta_t^B\\), then\n\\[\n\\boxed{\\widehat S_t=S_t^*.}\n\\]\n\\end{theorem}","ulation margin\n\\[\n\\Delta_t^B\n=\n\\mathcal B_t(S_t^*)\n-\n\\max_{S\\neq S_t^*}\\mathcal B_t(S)>0.\n\\]\nIf\n\\[\n\\sup_{S\\in\\mathfrak S_{\\mathrm{pass}}}\n\\left|\n\\widehat{\\mathcal B}_t(S)-\\mathcal B_t(S)\n\\right|\n\\leq\\varepsilon_t,\n\\]\nthen every empirical maximizer \\(\\widehat S_t\\) satisfies\n\\[\n\\boxed{\n\\mathcal B_t(S_t^*)-\n\\mathcal B_t(\\widehat S_t)\n\\leq2\\varepsilon_t.\n}\n\\]\nIf \\(2\\varepsilon_t<\\Delta_t^B\\), then\n\\[\n\\boxed{\\widehat S_t=S_t^*.}\n\\]\n\\end{theorem}","\\begin{proof}\nApply Theorem~\\ref{thm:uniform-structure-selection} to\n\\(J(S)=-\\mathcal B_t(S)\\).\n\\end{proof}","\\begin{theorem}[Boundary persistence under score drift]\n\\label{thm:boundary-drift-stability}\nAssume \\(S_t^*\\) is the unique maximizer of \\(\\mathcal B_t\\) with margin\n\\(\\Delta_t^B\\), and suppose\n\\[\n\\sup_{S\\in\\mathfrak S_{\\mathrm{pass}}}\n|\\mathcal B_{t+1}(S)-\\mathcal B_t(S)|\n\\leq\\delta_t.\n\\]\nIf\n\\[\n2\\delta_t<\\Delta_t^B,\n\\]\nthen\n\\[\n\\boxed{S_{t+1}^*=S_t^*.}\n\\]\nMore generally,\n\\[\n\\mathcal B_{t+1}(S_{t+1}^*)\n-\n\\mathcal B_{t+1}(S_t^*)\n\\leq2\\delta_t.\n\\]\n\\end{theorem}","\\begin{proof}\nFor every competitor \\(S\\neq S_t^*\\),\n\\[\n\\begin{aligned}\n\\mathcal B_{t+1}(S_t^*)-\n\\mathcal B_{t+1}(S)\n&\\geq\n\\mathcal B_t(S_t^*)-\n\\mathcal B_t(S)-2\\delta_t","\\begin{proof}\nFor every competitor \\(S\\neq S_t^*\\),\n\\[\n\\begin{aligned}\n\\mathcal B_{t+1}(S_t^*)-\n\\mathcal B_{t+1}(S)\n&\\geq\n\\mathcal B_t(S_t^*)-\n\\mathcal B_t(S)-2\\delta_t\\\\\n&\\geq\n\\Delta_t^B-2\\delta_t>0.\n\\end{aligned}\n\\]\nHence the maximizer is unchanged. The approximate bound follows by comparing\nboth time-indexed scores at their respective maximizers.\n\\end{proof}","If the score depends on a parameter \\(\\vartheta_t\\) and satisfies\n\\[\n\\sup_S\n|\\mathcal B_{\\vartheta}(S)-\\mathcal B_{\\vartheta'}(S)|\n\\leq\nL_Bd(\\vartheta,\\vartheta'),\n\\]\nthen the sufficient stability condition becomes\n\\[\n2L_Bd(\\vartheta_t,\\vartheta_{t+1})\n<\n\\Delta_t^B.\n\\]","\\item[Analyst-discovered boundary.]\nAn external procedure fits \\(\\widehat S\\) after observing the system. The\nsystem need not represent or use that boundary.","\\item[System-instantiated boundary.]\nThe system contains a channel, mask, or routing variable causally equivalent to\n\\(\\widehat S\\), and interventions on that variable alter downstream operation.","A boundary \\(\\widehat S\\) is certified as endogenously formed only if:\n\\begin{enumerate}\n\\item the membership mask was not directly encoded in \\(\\Sigma_B\\) or\n \\(\\Sigma_Y\\);\n\\item \\(e_B^{\\mathrm{MDL}}\\) and \\(e_B^{\\mathrm{causal}}\\) exceed declared\n thresholds;\n\\item held-out recovery satisfies Theorem~\\ref{thm:heldout-boundary-recovery};\n\\item the boundary remains stable under admissible environment drift or its\n changes are predicted by score-margin closure;\n\\item matched history-shuffle and label-leakage controls fail to reproduce\n the result, as do full-system, empty-set, singleton, and centralized\n boundary nulls;\n\\item the induced subsystem satisfies its workspace certificate; and\n\\item a system-instantiatio","A boundary \\(\\widehat S\\) is certified as endogenously formed only if:\n\\begin{enumerate}\n\\item the membership mask was not directly encoded in \\(\\Sigma_B\\) or\n \\(\\Sigma_Y\\);\n\\item \\(e_B^{\\mathrm{MDL}}\\) and \\(e_B^{\\mathrm{causal}}\\) exceed declared\n thresholds;\n\\item held-out recovery satisfies Theorem~\\ref{thm:heldout-boundary-recovery};\n\\item the boundary remains stable under admissible environment drift or its\n changes are predicted by score-margin closure;\n\\item matched history-shuffle and label-leakage controls fail to reproduce\n the result, as do full-system, empty-set, singleton, and centralized\n boundary nulls;\n\\item the induced subsystem satisfies its workspace certificate; and\n\\item a system-instantiation intervention demonstrates that the boundary is\n used by the system rather than only by the analyst.\n\\end{enumerate}","A controlled relational realization extends Definition~\\ref{def:realization}\nby the tuple\n\\[\n\\boxed{\n\\mathfrak A_{\\Workspace}\n=\n\\left(\n\\mathfrak R,\n\\mathcal G,\n\\mathcal A_t,\n\\mathcal Y^+,\n\\{U_g\\}_{g\\in\\mathcal G},\n\\Pi,\n\\mathcal I_A,\n\\mathcal I_G,\n\\mathcal I_D,\n\\mathcal I_R\n\\right).\n}\n\\]\nHere:\n\\begin{itemize}\n\\item \\(\\mathfrak R\\) is the underlying relational realization;\n\\item \\(\\mathcal G\\) is a measurable goal-code space;\n\\item \\(\\mathcal A_t\\) is the feasible action set at time \\(t\\);\n\\item \\(\\mathcal Y^+\\) is the relevant future trajectory space;\n\\item \\(U_g:\\mathcal Y^+\\to\\mathbb R\\) evaluates a trajectory under goal\n code \\(g\\);\n\\item \\(\\Pi\\) is an admissible class of workspace-level policies;\n\\item \\(\\mathcal I_A\\) and \\(\\mathcal I_G\\) are feasible action and goal\n interventions; and\n\\item \\(\\mathcal I_D\\) and \\(\\mathcal I_R\\) are feasible damage and repair\n interventions.\n\\end{itemize}","extends Definition~\\ref{def:realization}\nby the tuple\n\\[\n\\boxed{\n\\mathfrak A_{\\Workspace}\n=\n\\left(\n\\mathfrak R,\n\\mathcal G,\n\\mathcal A_t,\n\\mathcal Y^+,\n\\{U_g\\}_{g\\in\\mathcal G},\n\\Pi,\n\\mathcal I_A,\n\\mathcal I_G,\n\\mathcal I_D,\n\\mathcal I_R\n\\right).\n}\n\\]\nHere:\n\\begin{itemize}\n\\item \\(\\mathfrak R\\) is the underlying relational realization;\n\\item \\(\\mathcal G\\) is a measurable goal-code space;\n\\item \\(\\mathcal A_t\\) is the feasible action set at time \\(t\\);\n\\item \\(\\mathcal Y^+\\) is the relevant future trajectory space;\n\\item \\(U_g:\\mathcal Y^+\\to\\mathbb R\\) evaluates a trajectory under goal\n code \\(g\\);\n\\item \\(\\Pi\\) is an admissible class of workspace-level policies;\n\\item \\(\\mathcal I_A\\) and \\(\\mathcal I_G\\) are feasible action and goal\n interventions; and\n\\item \\(\\mathcal I_D\\) and \\(\\mathcal I_R\\) are feasible damage and repair\n interventions.\n\\end{itemize}","e\n\\[\n\\boxed{\n\\mathfrak A_{\\Workspace}\n=\n\\left(\n\\mathfrak R,\n\\mathcal G,\n\\mathcal A_t,\n\\mathcal Y^+,\n\\{U_g\\}_{g\\in\\mathcal G},\n\\Pi,\n\\mathcal I_A,\n\\mathcal I_G,\n\\mathcal I_D,\n\\mathcal I_R\n\\right).\n}\n\\]\nHere:\n\\begin{itemize}\n\\item \\(\\mathfrak R\\) is the underlying relational realization;\n\\item \\(\\mathcal G\\) is a measurable goal-code space;\n\\item \\(\\mathcal A_t\\) is the feasible action set at time \\(t\\);\n\\item \\(\\mathcal Y^+\\) is the relevant future trajectory space;\n\\item \\(U_g:\\mathcal Y^+\\to\\mathbb R\\) evaluates a trajectory under goal\n code \\(g\\);\n\\item \\(\\Pi\\) is an admissible class of workspace-level policies;\n\\item \\(\\mathcal I_A\\) and \\(\\mathcal I_G\\) are feasible action and goal\n interventions; and\n\\item \\(\\mathcal I_D\\) and \\(\\mathcal I_R\\) are feasible damage and repair\n interventions.\n\\end{itemize}","\\mathfrak A_{\\Workspace}\n=\n\\left(\n\\mathfrak R,\n\\mathcal G,\n\\mathcal A_t,\n\\mathcal Y^+,\n\\{U_g\\}_{g\\in\\mathcal G},\n\\Pi,\n\\mathcal I_A,\n\\mathcal I_G,\n\\mathcal I_D,\n\\mathcal I_R\n\\right).\n}\n\\]\nHere:\n\\begin{itemize}\n\\item \\(\\mathfrak R\\) is the underlying relational realization;\n\\item \\(\\mathcal G\\) is a measurable goal-code space;\n\\item \\(\\mathcal A_t\\) is the feasible action set at time \\(t\\);\n\\item \\(\\mathcal Y^+\\) is the relevant future trajectory space;\n\\item \\(U_g:\\mathcal Y^+\\to\\mathbb R\\) evaluates a trajectory under goal\n code \\(g\\);\n\\item \\(\\Pi\\) is an admissible class of workspace-level policies;\n\\item \\(\\mathcal I_A\\) and \\(\\mathcal I_G\\) are feasible action and goal\n interventions; and\n\\item \\(\\mathcal I_D\\) and \\(\\mathcal I_R\\) are feasible damage and repair\n interventions.\n\\end{itemize}","cal A_t,\n\\mathcal Y^+,\n\\{U_g\\}_{g\\in\\mathcal G},\n\\Pi,\n\\mathcal I_A,\n\\mathcal I_G,\n\\mathcal I_D,\n\\mathcal I_R\n\\right).\n}\n\\]\nHere:\n\\begin{itemize}\n\\item \\(\\mathfrak R\\) is the underlying relational realization;\n\\item \\(\\mathcal G\\) is a measurable goal-code space;\n\\item \\(\\mathcal A_t\\) is the feasible action set at time \\(t\\);\n\\item \\(\\mathcal Y^+\\) is the relevant future trajectory space;\n\\item \\(U_g:\\mathcal Y^+\\to\\mathbb R\\) evaluates a trajectory under goal\n code \\(g\\);\n\\item \\(\\Pi\\) is an admissible class of workspace-level policies;\n\\item \\(\\mathcal I_A\\) and \\(\\mathcal I_G\\) are feasible action and goal\n interventions; and\n\\item \\(\\mathcal I_D\\) and \\(\\mathcal I_R\\) are feasible damage and repair\n interventions.\n\\end{itemize}","cal I_G,\n\\mathcal I_D,\n\\mathcal I_R\n\\right).\n}\n\\]\nHere:\n\\begin{itemize}\n\\item \\(\\mathfrak R\\) is the underlying relational realization;\n\\item \\(\\mathcal G\\) is a measurable goal-code space;\n\\item \\(\\mathcal A_t\\) is the feasible action set at time \\(t\\);\n\\item \\(\\mathcal Y^+\\) is the relevant future trajectory space;\n\\item \\(U_g:\\mathcal Y^+\\to\\mathbb R\\) evaluates a trajectory under goal\n code \\(g\\);\n\\item \\(\\Pi\\) is an admissible class of workspace-level policies;\n\\item \\(\\mathcal I_A\\) and \\(\\mathcal I_G\\) are feasible action and goal\n interventions; and\n\\item \\(\\mathcal I_D\\) and \\(\\mathcal I_R\\) are feasible damage and repair\n interventions.\n\\end{itemize}","hcal I_D,\n\\mathcal I_R\n\\right).\n}\n\\]\nHere:\n\\begin{itemize}\n\\item \\(\\mathfrak R\\) is the underlying relational realization;\n\\item \\(\\mathcal G\\) is a measurable goal-code space;\n\\item \\(\\mathcal A_t\\) is the feasible action set at time \\(t\\);\n\\item \\(\\mathcal Y^+\\) is the relevant future trajectory space;\n\\item \\(U_g:\\mathcal Y^+\\to\\mathbb R\\) evaluates a trajectory under goal\n code \\(g\\);\n\\item \\(\\Pi\\) is an admissible class of workspace-level policies;\n\\item \\(\\mathcal I_A\\) and \\(\\mathcal I_G\\) are feasible action and goal\n interventions; and\n\\item \\(\\mathcal I_D\\) and \\(\\mathcal I_R\\) are feasible damage and repair\n interventions.\n\\end{itemize}","(\\mathfrak R\\) is the underlying relational realization;\n\\item \\(\\mathcal G\\) is a measurable goal-code space;\n\\item \\(\\mathcal A_t\\) is the feasible action set at time \\(t\\);\n\\item \\(\\mathcal Y^+\\) is the relevant future trajectory space;\n\\item \\(U_g:\\mathcal Y^+\\to\\mathbb R\\) evaluates a trajectory under goal\n code \\(g\\);\n\\item \\(\\Pi\\) is an admissible class of workspace-level policies;\n\\item \\(\\mathcal I_A\\) and \\(\\mathcal I_G\\) are feasible action and goal\n interventions; and\n\\item \\(\\mathcal I_D\\) and \\(\\mathcal I_R\\) are feasible damage and repair\n interventions.\n\\end{itemize}","underlying relational realization;\n\\item \\(\\mathcal G\\) is a measurable goal-code space;\n\\item \\(\\mathcal A_t\\) is the feasible action set at time \\(t\\);\n\\item \\(\\mathcal Y^+\\) is the relevant future trajectory space;\n\\item \\(U_g:\\mathcal Y^+\\to\\mathbb R\\) evaluates a trajectory under goal\n code \\(g\\);\n\\item \\(\\Pi\\) is an admissible class of workspace-level policies;\n\\item \\(\\mathcal I_A\\) and \\(\\mathcal I_G\\) are feasible action and goal\n interventions; and\n\\item \\(\\mathcal I_D\\) and \\(\\mathcal I_R\\) are feasible damage and repair\n interventions.\n\\end{itemize}","goal-code space;\n\\item \\(\\mathcal A_t\\) is the feasible action set at time \\(t\\);\n\\item \\(\\mathcal Y^+\\) is the relevant future trajectory space;\n\\item \\(U_g:\\mathcal Y^+\\to\\mathbb R\\) evaluates a trajectory under goal\n code \\(g\\);\n\\item \\(\\Pi\\) is an admissible class of workspace-level policies;\n\\item \\(\\mathcal I_A\\) and \\(\\mathcal I_G\\) are feasible action and goal\n interventions; and\n\\item \\(\\mathcal I_D\\) and \\(\\mathcal I_R\\) are feasible damage and repair\n interventions.\n\\end{itemize}","em \\(\\mathcal A_t\\) is the feasible action set at time \\(t\\);\n\\item \\(\\mathcal Y^+\\) is the relevant future trajectory space;\n\\item \\(U_g:\\mathcal Y^+\\to\\mathbb R\\) evaluates a trajectory under goal\n code \\(g\\);\n\\item \\(\\Pi\\) is an admissible class of workspace-level policies;\n\\item \\(\\mathcal I_A\\) and \\(\\mathcal I_G\\) are feasible action and goal\n interventions; and\n\\item \\(\\mathcal I_D\\) and \\(\\mathcal I_R\\) are feasible damage and repair\n interventions.\n\\end{itemize}","A controlled state contains at least\n\\[\nC_t^{\\mathrm{ctl}}\n=\n\\left(\nW_t,G_t,M_t,B_t,E_t\n\\right),\n\\]\nwhere \\(G_t\\) is the represented goal, \\(M_t\\) is a workspace-integrity\nmonitor, \\(B_t\\) is the current boundary or routing state, and \\(E_t\\) is the\ndeclared external context. An action is generated by\n\\[\nA_t\n\\sim\n\\pi_t\n\\left(\n\\cdot\\mid W_t,G_t,M_t,B_t,E_t\n\\right).\n\\]\nThe future controlled state obeys a time-unrolled structural model\n\\[\nS_{t+1}\n=\nF_t(S_t,A_t,E_{t+1},\\xi_{t+1}),\n\\qquad\nW_{t+1}=\\omega_{t+1}(S_{t+1}).\n\\]\nThe intervention semantics are inherited from Part~VII. A symbol such as\n\\(\\operatorname{do}(A_t=a)\\) is meaningful only when an implementation\nintervention changes an actual action mechanism while the declared background\nvariables are matched.","C_t^{\\mathrm{ctl}}\n=\n\\left(\nW_t,G_t,M_t,B_t,E_t\n\\right),\n\\]\nwhere \\(G_t\\) is the represented goal, \\(M_t\\) is a workspace-integrity\nmonitor, \\(B_t\\) is the current boundary or routing state, and \\(E_t\\) is the\ndeclared external context. An action is generated by\n\\[\nA_t\n\\sim\n\\pi_t\n\\left(\n\\cdot\\mid W_t,G_t,M_t,B_t,E_t\n\\right).\n\\]\nThe future controlled state obeys a time-unrolled structural model\n\\[\nS_{t+1}\n=\nF_t(S_t,A_t,E_{t+1},\\xi_{t+1}),\n\\qquad\nW_{t+1}=\\omega_{t+1}(S_{t+1}).\n\\]\nThe intervention semantics are inherited from Part~VII. A symbol such as\n\\(\\operatorname{do}(A_t=a)\\) is meaningful only when an implementation\nintervention changes an actual action mechanism while the declared background\nvariables are matched.","rrent boundary or routing state, and \\(E_t\\) is the\ndeclared external context. An action is generated by\n\\[\nA_t\n\\sim\n\\pi_t\n\\left(\n\\cdot\\mid W_t,G_t,M_t,B_t,E_t\n\\right).\n\\]\nThe future controlled state obeys a time-unrolled structural model\n\\[\nS_{t+1}\n=\nF_t(S_t,A_t,E_{t+1},\\xi_{t+1}),\n\\qquad\nW_{t+1}=\\omega_{t+1}(S_{t+1}).\n\\]\nThe intervention semantics are inherited from Part~VII. A symbol such as\n\\(\\operatorname{do}(A_t=a)\\) is meaningful only when an implementation\nintervention changes an actual action mechanism while the declared background\nvariables are matched.","A persistent internal variable is not automatically a goal. A goal code must\nindex a stable counterfactual preference over relevant future trajectories.\nFor a fixed state \\(w\\), define the goal-conditioned action value\n\\[\nQ_t^g(a\\mid w)\n=\n\\mathbb E\n\\left[\nU_g(Y_t^+)\n\\mid\n\\operatorname{do}(A_t=a),\nW_t=w,\nG_t=g\n\\right].\n\\]\nWhen action costs are relevant, they are incorporated into \\(U_g\\) or declared\nas a constraint on \\(\\mathcal A_t\\).","ariable is not automatically a goal. A goal code must\nindex a stable counterfactual preference over relevant future trajectories.\nFor a fixed state \\(w\\), define the goal-conditioned action value\n\\[\nQ_t^g(a\\mid w)\n=\n\\mathbb E\n\\left[\nU_g(Y_t^+)\n\\mid\n\\operatorname{do}(A_t=a),\nW_t=w,\nG_t=g\n\\right].\n\\]\nWhen action costs are relevant, they are incorporated into \\(U_g\\) or declared\nas a constraint on \\(\\mathcal A_t\\).","Two goal codes are operationally equivalent,\n\\[\ng\\sim_{\\mathrm{goal}}g',\n\\]\nwhen they induce the same action ordering in every declared state and\nintervention background:\n\\[\nQ_t^g(a\\mid w)\n\\geq\nQ_t^g(a'\\mid w)\n\\quad\\Longleftrightarrow\\quad\nQ_t^{g'}(a\\mid w)\n\\geq\nQ_t^{g'}(a'\\mid w).\n\\]\nThe abstract goal is the equivalence class\n\\[\n[g]_{\\mathrm{goal}}.\n\\]\nThis quotient prevents a mere invertible recoding of a goal channel from being\nreported as a change of objective. For quantitative regret bounds, a\nrepresentative normalization of \\(U_g\\) must nevertheless be fixed.","ate and\nintervention background:\n\\[\nQ_t^g(a\\mid w)\n\\geq\nQ_t^g(a'\\mid w)\n\\quad\\Longleftrightarrow\\quad\nQ_t^{g'}(a\\mid w)\n\\geq\nQ_t^{g'}(a'\\mid w).\n\\]\nThe abstract goal is the equivalence class\n\\[\n[g]_{\\mathrm{goal}}.\n\\]\nThis quotient prevents a mere invertible recoding of a goal channel from being\nreported as a change of objective. For quantitative regret bounds, a\nrepresentative normalization of \\(U_g\\) must nevertheless be fixed.","Let \\(\\mathcal H_t^S\\) be the history available to coalition \\(S\\subseteq V\\).\nFor a loss \\(\\ell_G\\), define goal-reconstruction risk\n\\[\n\\Risk_G(S)\n=\n\\inf_f\n\\mathbb E\n\\left[\n\\ell_G\n\\left(\nG_t,\nf(\\mathcal H_t^S)\n\\right)\n\\right].\n\\]\nFor tolerance \\(\\varepsilon\\), define\n\\[\nk_{G,\\varepsilon}\n=\n\\min\n\\left\\{\n|S|:\n\\Risk_G(S)\n\\leq\n\\Risk_G(V)+\\varepsilon\n\\right\\}.\n\\]\nA goal is locally stored when \\(k_{G,\\varepsilon}=1\\). It is distributed\nrelative to the declared decomposition when \\(k_{G,\\varepsilon}\\geq2\\).","r a loss \\(\\ell_G\\), define goal-reconstruction risk\n\\[\n\\Risk_G(S)\n=\n\\inf_f\n\\mathbb E\n\\left[\n\\ell_G\n\\left(\nG_t,\nf(\\mathcal H_t^S)\n\\right)\n\\right].\n\\]\nFor tolerance \\(\\varepsilon\\), define\n\\[\nk_{G,\\varepsilon}\n=\n\\min\n\\left\\{\n|S|:\n\\Risk_G(S)\n\\leq\n\\Risk_G(V)+\\varepsilon\n\\right\\}.\n\\]\nA goal is locally stored when \\(k_{G,\\varepsilon}=1\\). It is distributed\nrelative to the declared decomposition when \\(k_{G,\\varepsilon}\\geq2\\).","For the complete control state, define\n\\[\n\\Risk_{\\mathrm{ctl}}(S)\n=\n\\inf_f\n\\mathbb E\n\\left[\n\\ell_{\\mathrm{ctl}}\n\\left(\nC_t^{\\mathrm{ctl}},\nf(\\mathcal H_t^S)\n\\right)\n\\right],\n\\]\nand let \\(k_{\\mathrm{ctl},\\varepsilon}\\) be its minimal sufficient coalition\nsize. A strong workspace-level agency claim requires\n\\[\n\\boxed{k_{\\mathrm{ctl},\\varepsilon}\\geq2.}\n\\]\nThis is a location criterion. It does not yet establish goal persistence,\ncounterfactual evaluation, or self-maintenance.","Goal-dependent action mediation is measured by feasible interventions on the\ngoal channel while matching the current workspace and external context. For\na divergence \\(D_A\\), define\n\\[\n\\eta_{G\\to A}\n=\n\\sup_{g,g',w,e}\nD_A\n\\left(\n\\mathbb P(A_t\\in\\cdot\\mid\\iota_g,w,e),\n\\mathbb P(A_t\\in\\cdot\\mid\\iota_{g'},w,e)\n\\right).\n\\]\nWorkspace-dependent action mediation is\n\\[\n\\eta_{W\\to A}\n=\n\\sup_{w,w',g,e}\nD_A\n\\left(\n\\mathbb P(A_t\\in\\cdot\\mid\\iota_w,g,e),\n\\mathbb P(A_t\\in\\cdot\\mid\\iota_{w'},g,e)\n\\right).\n\\]\nAn action must also alter the declared future:\n\\[\n\\eta_{A\\to Y}\n=\n\\sup_{a,a',b}\nD_Y\n\\left(\n\\mathbb P(Y_t^+\\in\\cdot\\mid\\iota_a,b),\n\\mathbb P(Y_t^+\\in\\cdot\\mid\\iota_{a'},b)\n\\right).\n\\]\nA positive \\(\\eta_{G\\to A}\\) without \\(\\eta_{A\\to Y}\\) describes goal-sensitive\noutput selection with no demonstrated control of the relevant environment. A\npositive \\(\\eta_{A\\to Y}\\) withou","Goal-dependent action mediation is measured by feasible interventions on the\ngoal channel while matching the current workspace and external context. For\na divergence \\(D_A\\), define\n\\[\n\\eta_{G\\to A}\n=\n\\sup_{g,g',w,e}\nD_A\n\\left(\n\\mathbb P(A_t\\in\\cdot\\mid\\iota_g,w,e),\n\\mathbb P(A_t\\in\\cdot\\mid\\iota_{g'},w,e)\n\\right).\n\\]\nWorkspace-dependent action mediation is\n\\[\n\\eta_{W\\to A}\n=\n\\sup_{w,w',g,e}\nD_A\n\\left(\n\\mathbb P(A_t\\in\\cdot\\mid\\iota_w,g,e),\n\\mathbb P(A_t\\in\\cdot\\mid\\iota_{w'},g,e)\n\\right).\n\\]\nAn action must also alter the declared future:\n\\[\n\\eta_{A\\to Y}\n=\n\\sup_{a,a',b}\nD_Y\n\\left(\n\\mathbb P(Y_t^+\\in\\cdot\\mid\\iota_a,b),\n\\mathbb P(Y_t^+\\in\\cdot\\mid\\iota_{a'},b)\n\\right).\n\\]\nA positive \\(\\eta_{G\\to A}\\) without \\(\\eta_{A\\to Y}\\) describes goal-sensitive\noutput selection with no demonstrated control of the relevant environment. A\npositive \\(\\eta_{A\\to Y}\\) without a goal or counterfactual-value channel may\ndescribe a reactive controller.",". For\na divergence \\(D_A\\), define\n\\[\n\\eta_{G\\to A}\n=\n\\sup_{g,g',w,e}\nD_A\n\\left(\n\\mathbb P(A_t\\in\\cdot\\mid\\iota_g,w,e),\n\\mathbb P(A_t\\in\\cdot\\mid\\iota_{g'},w,e)\n\\right).\n\\]\nWorkspace-dependent action mediation is\n\\[\n\\eta_{W\\to A}\n=\n\\sup_{w,w',g,e}\nD_A\n\\left(\n\\mathbb P(A_t\\in\\cdot\\mid\\iota_w,g,e),\n\\mathbb P(A_t\\in\\cdot\\mid\\iota_{w'},g,e)\n\\right).\n\\]\nAn action must also alter the declared future:\n\\[\n\\eta_{A\\to Y}\n=\n\\sup_{a,a',b}\nD_Y\n\\left(\n\\mathbb P(Y_t^+\\in\\cdot\\mid\\iota_a,b),\n\\mathbb P(Y_t^+\\in\\cdot\\mid\\iota_{a'},b)\n\\right).\n\\]\nA positive \\(\\eta_{G\\to A}\\) without \\(\\eta_{A\\to Y}\\) describes goal-sensitive\noutput selection with no demonstrated control of the relevant environment. A\npositive \\(\\eta_{A\\to Y}\\) without a goal or counterfactual-value channel may\ndescribe a reactive controller.","ota_{g'},w,e)\n\\right).\n\\]\nWorkspace-dependent action mediation is\n\\[\n\\eta_{W\\to A}\n=\n\\sup_{w,w',g,e}\nD_A\n\\left(\n\\mathbb P(A_t\\in\\cdot\\mid\\iota_w,g,e),\n\\mathbb P(A_t\\in\\cdot\\mid\\iota_{w'},g,e)\n\\right).\n\\]\nAn action must also alter the declared future:\n\\[\n\\eta_{A\\to Y}\n=\n\\sup_{a,a',b}\nD_Y\n\\left(\n\\mathbb P(Y_t^+\\in\\cdot\\mid\\iota_a,b),\n\\mathbb P(Y_t^+\\in\\cdot\\mid\\iota_{a'},b)\n\\right).\n\\]\nA positive \\(\\eta_{G\\to A}\\) without \\(\\eta_{A\\to Y}\\) describes goal-sensitive\noutput selection with no demonstrated control of the relevant environment. A\npositive \\(\\eta_{A\\to Y}\\) without a goal or counterfactual-value channel may\ndescribe a reactive controller.","Workspace-dependent action mediation is\n\\[\n\\eta_{W\\to A}\n=\n\\sup_{w,w',g,e}\nD_A\n\\left(\n\\mathbb P(A_t\\in\\cdot\\mid\\iota_w,g,e),\n\\mathbb P(A_t\\in\\cdot\\mid\\iota_{w'},g,e)\n\\right).\n\\]\nAn action must also alter the declared future:\n\\[\n\\eta_{A\\to Y}\n=\n\\sup_{a,a',b}\nD_Y\n\\left(\n\\mathbb P(Y_t^+\\in\\cdot\\mid\\iota_a,b),\n\\mathbb P(Y_t^+\\in\\cdot\\mid\\iota_{a'},b)\n\\right).\n\\]\nA positive \\(\\eta_{G\\to A}\\) without \\(\\eta_{A\\to Y}\\) describes goal-sensitive\noutput selection with no demonstrated control of the relevant environment. A\npositive \\(\\eta_{A\\to Y}\\) without a goal or counterfactual-value channel may\ndescribe a reactive controller.","hbb P(A_t\\in\\cdot\\mid\\iota_{w'},g,e)\n\\right).\n\\]\nAn action must also alter the declared future:\n\\[\n\\eta_{A\\to Y}\n=\n\\sup_{a,a',b}\nD_Y\n\\left(\n\\mathbb P(Y_t^+\\in\\cdot\\mid\\iota_a,b),\n\\mathbb P(Y_t^+\\in\\cdot\\mid\\iota_{a'},b)\n\\right).\n\\]\nA positive \\(\\eta_{G\\to A}\\) without \\(\\eta_{A\\to Y}\\) describes goal-sensitive\noutput selection with no demonstrated control of the relevant environment. A\npositive \\(\\eta_{A\\to Y}\\) without a goal or counterfactual-value channel may\ndescribe a reactive controller.","The observational condition\n\\[\nI(G_{t+\\tau};G_t\\mid E_{t+1:t+\\tau})>0\n\\]\ncan arise from several non-agentic mechanisms:\n\\begin{enumerate}\n\\item a persistent external command writes the same code at every step;\n\\item a clock, episode label, or task identifier drives both variables;\n\\item a frozen memory cell repeats a symbol that is not used by the policy;\n\\item selection on successful trajectories induces apparent temporal\n dependence; or\n\\item a downstream action is copied back into the nominal goal channel.\n\\end{enumerate}\nA goal-persistence claim therefore requires a cue-controlled intervention that\nchanges \\(G_t\\), removes or matches subsequent command inputs, and checks both\nfuture goal state and action consequences.","ock, episode label, or task identifier drives both variables;\n\\item a frozen memory cell repeats a symbol that is not used by the policy;\n\\item selection on successful trajectories induces apparent temporal\n dependence; or\n\\item a downstream action is copied back into the nominal goal channel.\n\\end{enumerate}\nA goal-persistence claim therefore requires a cue-controlled intervention that\nchanges \\(G_t\\), removes or matches subsequent command inputs, and checks both\nfuture goal state and action consequences.","Let \\(C_t\\) denote an external command or task cue. Fix a neutral or matched\nfuture command trajectory \\(c_{t+1:t+\\tau}^{0}\\). For goal interventions\n\\(\\iota_g\\) and \\(\\iota_{g'}\\), define the cue-controlled retention effect\n\\[\n\\eta_G(\\tau;g,g')\n=\nD_G\n\\left(\n\\mathbb P\n\\left(\nG_{t+\\tau}\\in\\cdot\n\\mid\n\\iota_g,\n\\operatorname{do}(C_{t+1:t+\\tau}=c^0)\n\\right),\n\\mathbb P\n\\left(\nG_{t+\\tau}\\in\\cdot\n\\mid\n\\iota_{g'},\n\\operatorname{do}(C_{t+1:t+\\tau}=c^0)\n\\right)\n\\right).\n\\]\nThe corresponding action-retention effect is\n\\[\n\\eta_{G\\to A}(\\tau;g,g')\n=\nD_A\n\\left(\n\\mathbb P(A_{t+\\tau}\\in\\cdot\\mid\\iota_g,c^0),\n\\mathbb P(A_{t+\\tau}\\in\\cdot\\mid\\iota_{g'},c^0)\n\\right).\n\\]\nThe first quantity tests whether a goal difference","Let \\(C_t\\) denote an external command or task cue. Fix a neutral or matched\nfuture command trajectory \\(c_{t+1:t+\\tau}^{0}\\). For goal interventions\n\\(\\iota_g\\) and \\(\\iota_{g'}\\), define the cue-controlled retention effect\n\\[\n\\eta_G(\\tau;g,g')\n=\nD_G\n\\left(\n\\mathbb P\n\\left(\nG_{t+\\tau}\\in\\cdot\n\\mid\n\\iota_g,\n\\operatorname{do}(C_{t+1:t+\\tau}=c^0)\n\\right),\n\\mathbb P\n\\left(\nG_{t+\\tau}\\in\\cdot\n\\mid\n\\iota_{g'},\n\\operatorname{do}(C_{t+1:t+\\tau}=c^0)\n\\right)\n\\right).\n\\]\nThe corresponding action-retention effect is\n\\[\n\\eta_{G\\to A}(\\tau;g,g')\n=\nD_A\n\\left(\n\\mathbb P(A_{t+\\tau}\\in\\cdot\\mid\\iota_g,c^0),\n\\mathbb P(A_{t+\\tau}\\in\\cdot\\mid\\iota_{g'},c^0)\n\\right).\n\\]\nThe first quantity tests whether a goal difference survives. The second tests\nwhether the surviving difference remains behaviorally operative.","\\tau}^{0}\\). For goal interventions\n\\(\\iota_g\\) and \\(\\iota_{g'}\\), define the cue-controlled retention effect\n\\[\n\\eta_G(\\tau;g,g')\n=\nD_G\n\\left(\n\\mathbb P\n\\left(\nG_{t+\\tau}\\in\\cdot\n\\mid\n\\iota_g,\n\\operatorname{do}(C_{t+1:t+\\tau}=c^0)\n\\right),\n\\mathbb P\n\\left(\nG_{t+\\tau}\\in\\cdot\n\\mid\n\\iota_{g'},\n\\operatorname{do}(C_{t+1:t+\\tau}=c^0)\n\\right)\n\\right).\n\\]\nThe corresponding action-retention effect is\n\\[\n\\eta_{G\\to A}(\\tau;g,g')\n=\nD_A\n\\left(\n\\mathbb P(A_{t+\\tau}\\in\\cdot\\mid\\iota_g,c^0),\n\\mathbb P(A_{t+\\tau}\\in\\cdot\\mid\\iota_{g'},c^0)\n\\right).\n\\]\nThe first quantity tests whether a goal difference survives. The second tests\nwhether the surviving difference remains behaviorally operative.","Cue dependence must be reported separately. For matched initial goal \\(g\\),\ndefine\n\\[\n\\delta_G^{\\mathrm{cue}}(\\tau)\n=\n\\sup_{c,c'}\nD_G\n\\left(\n\\mathbb P(G_{t+\\tau}\\in\\cdot\\mid\\iota_g,\\operatorname{do}(C=c)),\n\\mathbb P(G_{t+\\tau}\\in\\cdot\\mid\\iota_g,\\operatorname{do}(C=c'))\n\\right).\n\\]\nHigh cue dependence is not a failure when the claimed agent is intentionally\ninstruction following. It does mean that persistence cannot be attributed to\nan autonomous internal objective without an additional cue-removal result.","\\subtopic{52.7 \\(T_{19}\\): Cue-controlled goal-retention theorem }","\\begin{theorem}[Cue-controlled goal retention]\n\\label{thm:goal-retention}\nLet \\(G_s\\in\\mathbb R^{d_G}\\) obey\n\\[\nG_{s+1}\n=\nR_sG_s+B_sC_{s+1}+\\zeta_{s+1}\n\\qquad\n(s=t,\\ldots,t+\\tau-1).\n\\]\nConsider two intervention runs that set\n\\[\nG_t=g,\n\\qquad\nG_t'=g',\n\\]\nand then use the same command trajectory and the same disturbance realization.\nLet\n\\[\nR_{t+\\tau:t}\n=\nR_{t+\\tau-1}\\cdots R_t.\n\\]\nThen\n\\[\n\\boxed{\nG_{t+\\tau}-G_{t+\\tau}'\n=\nR_{t+\\tau:t}(g-g').\n}\n\\]\nIf\n\\[\nm_s\n\\leq\n\\sigma_{\\min}(R_s),\n\\qquad\n\\|R_s\\|\\leq M_s,\n\\]\nthen\n\\[\n\\boxed{\n\\left(\n\\prod_{s=t}^{t+\\tau-1}m_s\n\\right)\n\\|g-g'\\|\n\\leq\n\\|G_{t+\\tau}-G_{t+\\tau}'\\|\n\\leq\n\\left(\n\\prod_{s=t}^{t+\\tau-1}M_s\n\\right)\n\\|g-g'\\|.\n}\n\\]\n\\end{theorem}","{thm:goal-retention}\nLet \\(G_s\\in\\mathbb R^{d_G}\\) obey\n\\[\nG_{s+1}\n=\nR_sG_s+B_sC_{s+1}+\\zeta_{s+1}\n\\qquad\n(s=t,\\ldots,t+\\tau-1).\n\\]\nConsider two intervention runs that set\n\\[\nG_t=g,\n\\qquad\nG_t'=g',\n\\]\nand then use the same command trajectory and the same disturbance realization.\nLet\n\\[\nR_{t+\\tau:t}\n=\nR_{t+\\tau-1}\\cdots R_t.\n\\]\nThen\n\\[\n\\boxed{\nG_{t+\\tau}-G_{t+\\tau}'\n=\nR_{t+\\tau:t}(g-g').\n}\n\\]\nIf\n\\[\nm_s\n\\leq\n\\sigma_{\\min}(R_s),\n\\qquad\n\\|R_s\\|\\leq M_s,\n\\]\nthen\n\\[\n\\boxed{\n\\left(\n\\prod_{s=t}^{t+\\tau-1}m_s\n\\right)\n\\|g-g'\\|\n\\leq\n\\|G_{t+\\tau}-G_{t+\\tau}'\\|\n\\leq\n\\left(\n\\prod_{s=t}^{t+\\tau-1}M_s\n\\right)\n\\|g-g'\\|.\n}\n\\]\n\\end{theorem}","1}\n=\nR_sG_s+B_sC_{s+1}+\\zeta_{s+1}\n\\qquad\n(s=t,\\ldots,t+\\tau-1).\n\\]\nConsider two intervention runs that set\n\\[\nG_t=g,\n\\qquad\nG_t'=g',\n\\]\nand then use the same command trajectory and the same disturbance realization.\nLet\n\\[\nR_{t+\\tau:t}\n=\nR_{t+\\tau-1}\\cdots R_t.\n\\]\nThen\n\\[\n\\boxed{\nG_{t+\\tau}-G_{t+\\tau}'\n=\nR_{t+\\tau:t}(g-g').\n}\n\\]\nIf\n\\[\nm_s\n\\leq\n\\sigma_{\\min}(R_s),\n\\qquad\n\\|R_s\\|\\leq M_s,\n\\]\nthen\n\\[\n\\boxed{\n\\left(\n\\prod_{s=t}^{t+\\tau-1}m_s\n\\right)\n\\|g-g'\\|\n\\leq\n\\|G_{t+\\tau}-G_{t+\\tau}'\\|\n\\leq\n\\left(\n\\prod_{s=t}^{t+\\tau-1}M_s\n\\right)\n\\|g-g'\\|.\n}\n\\]\n\\end{theorem}","\\begin{proof}\nSubtract the two update equations. The matched command and disturbance terms\ncancel, giving\n\\[\nG_{s+1}-G_{s+1}'\n=\nR_s(G_s-G_s').\n\\]\nIteration gives the exact product formula. The upper bound follows from\nsubmultiplicativity of the operator norm. The lower bound follows from\n\\[\n\\sigma_{\\min}(AB)\n\\geq\n\\sigma_{\\min}(A)\\sigma_{\\min}(B)\n\\]\nfor square linear maps.\n\\end{proof}","\\begin{corollary}[Behavioral readout of a retained goal]\n\\label{cor:goal-action-readout}\nAssume, in addition, that under the matched intervention background the action\nreadout has conditional mean\n\\[\n\\mathbb E[A_{t+\\tau}\\mid G_{t+\\tau},W_{t+\\tau}]\n=\nK_{t+\\tau}G_{t+\\tau}+b(W_{t+\\tau}),\n\\]\nand that the two runs are matched on \\(W_{t+\\tau}\\). Then\n\\[\n\\mathbb E[A_{t+\\tau}\\mid\\iota_g]\n-\n\\mathbb E[A_{t+\\tau}\\mid\\iota_{g'}]\n=\nK_{t+\\tau}R_{t+\\tau:t}(g-g').\n\\]\nIf \\(\\sigma_{\\min}(K_{t+\\tau})\\geq k_{t+\\tau}>0\\), the norm of this action-mean\ndifference is at least\n\\[\nk_{t+\\tau}\n\\left(\n\\prod_{s=t}^{t+\\tau-1}m_s\n\\right)\n\\|g-g'\\|.\n\\]\n\\end{corollary}","el{cor:goal-action-readout}\nAssume, in addition, that under the matched intervention background the action\nreadout has conditional mean\n\\[\n\\mathbb E[A_{t+\\tau}\\mid G_{t+\\tau},W_{t+\\tau}]\n=\nK_{t+\\tau}G_{t+\\tau}+b(W_{t+\\tau}),\n\\]\nand that the two runs are matched on \\(W_{t+\\tau}\\). Then\n\\[\n\\mathbb E[A_{t+\\tau}\\mid\\iota_g]\n-\n\\mathbb E[A_{t+\\tau}\\mid\\iota_{g'}]\n=\nK_{t+\\tau}R_{t+\\tau:t}(g-g').\n\\]\nIf \\(\\sigma_{\\min}(K_{t+\\tau})\\geq k_{t+\\tau}>0\\), the norm of this action-mean\ndifference is at least\n\\[\nk_{t+\\tau}\n\\left(\n\\prod_{s=t}^{t+\\tau-1}m_s\n\\right)\n\\|g-g'\\|.\n\\]\n\\end{corollary}","action\nreadout has conditional mean\n\\[\n\\mathbb E[A_{t+\\tau}\\mid G_{t+\\tau},W_{t+\\tau}]\n=\nK_{t+\\tau}G_{t+\\tau}+b(W_{t+\\tau}),\n\\]\nand that the two runs are matched on \\(W_{t+\\tau}\\). Then\n\\[\n\\mathbb E[A_{t+\\tau}\\mid\\iota_g]\n-\n\\mathbb E[A_{t+\\tau}\\mid\\iota_{g'}]\n=\nK_{t+\\tau}R_{t+\\tau:t}(g-g').\n\\]\nIf \\(\\sigma_{\\min}(K_{t+\\tau})\\geq k_{t+\\tau}>0\\), the norm of this action-mean\ndifference is at least\n\\[\nk_{t+\\tau}\n\\left(\n\\prod_{s=t}^{t+\\tau-1}m_s\n\\right)\n\\|g-g'\\|.\n\\]\n\\end{corollary}","A minimal goal-persistence report contains:\n\\begin{enumerate}\n\\item the goal code and its declared trajectory-level semantics;\n\\item the feasible goal intervention and its manipulation check;\n\\item the matched or neutralized future cue trajectory;\n\\item \\(\\eta_G(\\tau)\\) across a declared horizon grid;\n\\item \\(\\eta_{G\\to A}(\\tau)\\) over the same grid;\n\\item cue dependence \\(\\delta_G^{\\mathrm{cue}}(\\tau)\\);\n\\item goal-channel ablation and permutation controls;\n\\item coalition reconstruction risks \\(\\Risk_G(S)\\); and\n\\item uncertainty intervals over environments, episodes, and random seeds.\n\\end{enumerate}\nA repeated label with no intervention effect on future action fails this\ncertificate even when its observational autocorrelation is one.","jectory-level semantics;\n\\item the feasible goal intervention and its manipulation check;\n\\item the matched or neutralized future cue trajectory;\n\\item \\(\\eta_G(\\tau)\\) across a declared horizon grid;\n\\item \\(\\eta_{G\\to A}(\\tau)\\) over the same grid;\n\\item cue dependence \\(\\delta_G^{\\mathrm{cue}}(\\tau)\\);\n\\item goal-channel ablation and permutation controls;\n\\item coalition reconstruction risks \\(\\Risk_G(S)\\); and\n\\item uncertainty intervals over environments, episodes, and random seeds.\n\\end{enumerate}\nA repeated label with no intervention effect on future action fails this\ncertificate even when its observational autocorrelation is one.","Let \\(\\mathcal A_t(w,e)\\) denote the actions that can be implemented from\nworkspace state \\(w\\) in environment \\(e\\). The interventional action value is\n\\[\nQ_t^g(a\\mid w,e)\n=\n\\mathbb E^{\\iota_a}\n\\left[\nU_g(Y_t^+)\n\\mid\nW_t=w,\nE_t=e,\nG_t=g\n\\right].\n\\]\nThe optimal action set is\n\\[\n\\mathcal A_t^*(w,g,e)\n=\n\\argmax_{a\\in\\mathcal A_t(w,e)}Q_t^g(a\\mid w,e).\n\\]\nA counterfactual evaluator is an internal state\n\\[\n\\widehat Q_t\n=\n\\left(\n\\widehat Q_t(a)\n\\right)_{a\\in\\mathcal A_t(w,e)}\n\\]\nthat estimates these intervention values. It is not enough for\n\\(\\widehat Q_t\\) to predict the action selected by the system. It must be\nvalidated against externally estimated intervention outcomes.","(\\mathcal A_t(w,e)\\) denote the actions that can be implemented from\nworkspace state \\(w\\) in environment \\(e\\). The interventional action value is\n\\[\nQ_t^g(a\\mid w,e)\n=\n\\mathbb E^{\\iota_a}\n\\left[\nU_g(Y_t^+)\n\\mid\nW_t=w,\nE_t=e,\nG_t=g\n\\right].\n\\]\nThe optimal action set is\n\\[\n\\mathcal A_t^*(w,g,e)\n=\n\\argmax_{a\\in\\mathcal A_t(w,e)}Q_t^g(a\\mid w,e).\n\\]\nA counterfactual evaluator is an internal state\n\\[\n\\widehat Q_t\n=\n\\left(\n\\widehat Q_t(a)\n\\right)_{a\\in\\mathcal A_t(w,e)}\n\\]\nthat estimates these intervention values. It is not enough for\n\\(\\widehat Q_t\\) to predict the action selected by the system. It must be\nvalidated against externally estimated intervention outcomes.","value is\n\\[\nQ_t^g(a\\mid w,e)\n=\n\\mathbb E^{\\iota_a}\n\\left[\nU_g(Y_t^+)\n\\mid\nW_t=w,\nE_t=e,\nG_t=g\n\\right].\n\\]\nThe optimal action set is\n\\[\n\\mathcal A_t^*(w,g,e)\n=\n\\argmax_{a\\in\\mathcal A_t(w,e)}Q_t^g(a\\mid w,e).\n\\]\nA counterfactual evaluator is an internal state\n\\[\n\\widehat Q_t\n=\n\\left(\n\\widehat Q_t(a)\n\\right)_{a\\in\\mathcal A_t(w,e)}\n\\]\nthat estimates these intervention values. It is not enough for\n\\(\\widehat Q_t\\) to predict the action selected by the system. It must be\nvalidated against externally estimated intervention outcomes.","\\item[Counterfactual accuracy.]\n\\[\n\\delta_Q\n=\n\\sup_{a\\in\\mathcal A_t}\n|\\widehat Q_t(a)-Q_t^g(a\\mid w,e)|.\n\\]","\\item[Policy use.]\nA feasible intervention that changes \\(\\widehat Q_t(a)\\) while preserving the\naction affordances changes the action distribution in the direction predicted\nby the decision rule.","\\item[Outcome validity.]\nThe resulting action change alters \\(U_g(Y_t^+)\\) as predicted by the external\ninterventional value estimate.\n\\end{description}","\\subtopic{53.3 \\(T_{20}\\): Approximate counterfactual policy theorem }","\\begin{theorem}[Approximate counterfactual policy selection]\n\\label{thm:counterfactual-policy}\nLet \\(\\mathcal A\\) be a finite feasible action set. Let \\(Q:\\mathcal A\\to\n\\mathbb R\\) be the true interventional value and let \\(\\widehat Q\\) satisfy\n\\[\n\\sup_{a\\in\\mathcal A}|\\widehat Q(a)-Q(a)|\\leq\\varepsilon_Q.\n\\]\nSuppose a policy \\(\\widehat\\pi\\) is \\(\\delta_{\\mathrm{opt}}\\)-optimal for\n\\(\\widehat Q\\):\n\\[\n\\max_{\\pi}\n\\mathbb E_{a\\sim\\pi}[\\widehat Q(a)]\n-\n\\mathbb E_{a\\sim\\widehat\\pi}[\\widehat Q(a)]\n\\leq\n\\delta_{\\mathrm{opt}}.\n\\]\nThen, for every true-value-optimal policy \\(\\pi^*\\),\n\\[\n\\boxed{\n\\mathbb E_{a\\sim\\pi^*}[Q(a)]\n-\n\\mathbb E_{a\\sim\\widehat\\pi}[Q(a)]\n\\leq\n2\\varepsilon_Q+\\delta_{\\mathrm{opt}}.\n}\n\\]\nFor a deterministic greedy action \\(\\widehat a\\in\\argmax_a\\widehat Q(a)\\),\n\\[\nQ(a^*)-Q(\\widehat a)\n\\leq2\\","\\begin{theorem}[Approximate counterfactual policy selection]\n\\label{thm:counterfactual-policy}\nLet \\(\\mathcal A\\) be a finite feasible action set. Let \\(Q:\\mathcal A\\to\n\\mathbb R\\) be the true interventional value and let \\(\\widehat Q\\) satisfy\n\\[\n\\sup_{a\\in\\mathcal A}|\\widehat Q(a)-Q(a)|\\leq\\varepsilon_Q.\n\\]\nSuppose a policy \\(\\widehat\\pi\\) is \\(\\delta_{\\mathrm{opt}}\\)-optimal for\n\\(\\widehat Q\\):\n\\[\n\\max_{\\pi}\n\\mathbb E_{a\\sim\\pi}[\\widehat Q(a)]\n-\n\\mathbb E_{a\\sim\\widehat\\pi}[\\widehat Q(a)]\n\\leq\n\\delta_{\\mathrm{opt}}.\n\\]\nThen, for every true-value-optimal policy \\(\\pi^*\\),\n\\[\n\\boxed{\n\\mathbb E_{a\\sim\\pi^*}[Q(a)]\n-\n\\mathbb E_{a\\sim\\widehat\\pi}[Q(a)]\n\\leq\n2\\varepsilon_Q+\\delta_{\\mathrm{opt}}.\n}\n\\]\nFor a deterministic greedy action \\(\\widehat a\\in\\argmax_a\\widehat Q(a)\\),\n\\[\nQ(a^*)-Q(\\widehat a)\n\\leq2\\varepsilon_Q.\n\\]\nIf the true optimal action \\(a^*\\) is unique and i","\\begin{theorem}[Approximate counterfactual policy selection]\n\\label{thm:counterfactual-policy}\nLet \\(\\mathcal A\\) be a finite feasible action set. Let \\(Q:\\mathcal A\\to\n\\mathbb R\\) be the true interventional value and let \\(\\widehat Q\\) satisfy\n\\[\n\\sup_{a\\in\\mathcal A}|\\widehat Q(a)-Q(a)|\\leq\\varepsilon_Q.\n\\]\nSuppose a policy \\(\\widehat\\pi\\) is \\(\\delta_{\\mathrm{opt}}\\)-optimal for\n\\(\\widehat Q\\):\n\\[\n\\max_{\\pi}\n\\mathbb E_{a\\sim\\pi}[\\widehat Q(a)]\n-\n\\mathbb E_{a\\sim\\widehat\\pi}[\\widehat Q(a)]\n\\leq\n\\delta_{\\mathrm{opt}}.\n\\]\nThen, for every true-value-optimal policy \\(\\pi^*\\),\n\\[\n\\boxed{\n\\mathbb E_{a\\sim\\pi^*}[Q(a)]\n-\n\\mathbb E_{a\\sim\\widehat\\pi}[Q(a)]\n\\leq\n2\\varepsilon_Q+\\delta_{\\mathrm{opt}}.\n}\n\\]\nFor a deterministic greedy action \\(\\widehat a\\in\\argmax_a\\widehat Q(a)\\),\n\\[\nQ(a^*)-Q(\\widehat a)\n\\leq2\\varepsilon_Q.\n\\]\nIf the true optimal action \\(a^*\\) is unique and its value gap\n\\[\n\\Delta_Q\n=\nQ(a^*)-\n\\max_{a\\neq a^*}Q(a)","\\begin{theorem}[Approximate counterfactual policy selection]\n\\label{thm:counterfactual-policy}\nLet \\(\\mathcal A\\) be a finite feasible action set. Let \\(Q:\\mathcal A\\to\n\\mathbb R\\) be the true interventional value and let \\(\\widehat Q\\) satisfy\n\\[\n\\sup_{a\\in\\mathcal A}|\\widehat Q(a)-Q(a)|\\leq\\varepsilon_Q.\n\\]\nSuppose a policy \\(\\widehat\\pi\\) is \\(\\delta_{\\mathrm{opt}}\\)-optimal for\n\\(\\widehat Q\\):\n\\[\n\\max_{\\pi}\n\\mathbb E_{a\\sim\\pi}[\\widehat Q(a)]\n-\n\\mathbb E_{a\\sim\\widehat\\pi}[\\widehat Q(a)]\n\\leq\n\\delta_{\\mathrm{opt}}.\n\\]\nThen, for every true-value-optimal policy \\(\\pi^*\\),\n\\[\n\\boxed{\n\\mathbb E_{a\\sim\\pi^*}[Q(a)]\n-\n\\mathbb E_{a\\sim\\widehat\\pi}[Q(a)]\n\\leq\n2\\varepsilon_Q+\\delta_{\\mathrm{opt}}.\n}\n\\]\nFor a deterministic greedy action \\(\\widehat a\\in\\argmax_a\\widehat Q(a)\\),\n\\[\nQ(a^*)-Q(\\widehat a)\n\\leq2\\varepsilon_Q.\n\\]\nIf the true optimal action \\(a^*\\) is unique and its value gap\n\\[\n\\Delta_Q\n=\nQ(a^*)-\n\\max_{a\\neq a^*}Q(a)\n\\]\nsatisfies \\(\\Delta_Q>2\\varepsilon_Q\\), then every greedy maximizer of\n\\","\\begin{theorem}[Approximate counterfactual policy selection]\n\\label{thm:counterfactual-policy}\nLet \\(\\mathcal A\\) be a finite feasible action set. Let \\(Q:\\mathcal A\\to\n\\mathbb R\\) be the true interventional value and let \\(\\widehat Q\\) satisfy\n\\[\n\\sup_{a\\in\\mathcal A}|\\widehat Q(a)-Q(a)|\\leq\\varepsilon_Q.\n\\]\nSuppose a policy \\(\\widehat\\pi\\) is \\(\\delta_{\\mathrm{opt}}\\)-optimal for\n\\(\\widehat Q\\):\n\\[\n\\max_{\\pi}\n\\mathbb E_{a\\sim\\pi}[\\widehat Q(a)]\n-\n\\mathbb E_{a\\sim\\widehat\\pi}[\\widehat Q(a)]\n\\leq\n\\delta_{\\mathrm{opt}}.\n\\]\nThen, for every true-value-optimal policy \\(\\pi^*\\),\n\\[\n\\boxed{\n\\mathbb E_{a\\sim\\pi^*}[Q(a)]\n-\n\\mathbb E_{a\\sim\\widehat\\pi}[Q(a)]\n\\leq\n2\\varepsilon_Q+\\delta_{\\mathrm{opt}}.\n}\n\\]\nFor a deterministic greedy action \\(\\widehat a\\in\\argmax_a\\widehat Q(a)\\),\n\\[\nQ(a^*)-Q(\\widehat a)\n\\leq2\\varepsilon_Q.\n\\]\nIf the true optimal action \\(a^*\\) is unique and its value gap\n\\[\n\\Delta_Q\n=\nQ(a^*)-\n\\max_{a\\neq a^*}Q(a)\n\\]\nsatisfies \\(\\Delta_Q>2\\varepsilon_Q\\), then every greedy maximizer of\n\\(\\widehat Q\\) equals \\(a^*\\).\n\\en","\\begin{theorem}[Approximate counterfactual policy selection]\n\\label{thm:counterfactual-policy}\nLet \\(\\mathcal A\\) be a finite feasible action set. Let \\(Q:\\mathcal A\\to\n\\mathbb R\\) be the true interventional value and let \\(\\widehat Q\\) satisfy\n\\[\n\\sup_{a\\in\\mathcal A}|\\widehat Q(a)-Q(a)|\\leq\\varepsilon_Q.\n\\]\nSuppose a policy \\(\\widehat\\pi\\) is \\(\\delta_{\\mathrm{opt}}\\)-optimal for\n\\(\\widehat Q\\):\n\\[\n\\max_{\\pi}\n\\mathbb E_{a\\sim\\pi}[\\widehat Q(a)]\n-\n\\mathbb E_{a\\sim\\widehat\\pi}[\\widehat Q(a)]\n\\leq\n\\delta_{\\mathrm{opt}}.\n\\]\nThen, for every true-value-optimal policy \\(\\pi^*\\),\n\\[\n\\boxed{\n\\mathbb E_{a\\sim\\pi^*}[Q(a)]\n-\n\\mathbb E_{a\\sim\\widehat\\pi}[Q(a)]\n\\leq\n2\\varepsilon_Q+\\delta_{\\mathrm{opt}}.\n}\n\\]\nFor a deterministic greedy action \\(\\widehat a\\in\\argmax_a\\widehat Q(a)\\),\n\\[\nQ(a^*)-Q(\\widehat a)\n\\leq2\\varepsilon_Q.\n\\]\nIf the true optimal action \\(a^*\\) is unique and its value gap\n\\[\n\\Delta_Q\n=\nQ(a^*)-\n\\max_{a\\neq a^*}Q(a)\n\\]\nsatisfies \\(\\Delta_Q>2\\varepsilon_Q\\), then every greedy maximizer of\n\\(\\widehat Q\\) equals \\(a^*\\).\n\\end{theorem}","gin{theorem}[Approximate counterfactual policy selection]\n\\label{thm:counterfactual-policy}\nLet \\(\\mathcal A\\) be a finite feasible action set. Let \\(Q:\\mathcal A\\to\n\\mathbb R\\) be the true interventional value and let \\(\\widehat Q\\) satisfy\n\\[\n\\sup_{a\\in\\mathcal A}|\\widehat Q(a)-Q(a)|\\leq\\varepsilon_Q.\n\\]\nSuppose a policy \\(\\widehat\\pi\\) is \\(\\delta_{\\mathrm{opt}}\\)-optimal for\n\\(\\widehat Q\\):\n\\[\n\\max_{\\pi}\n\\mathbb E_{a\\sim\\pi}[\\widehat Q(a)]\n-\n\\mathbb E_{a\\sim\\widehat\\pi}[\\widehat Q(a)]\n\\leq\n\\delta_{\\mathrm{opt}}.\n\\]\nThen, for every true-value-optimal policy \\(\\pi^*\\),\n\\[\n\\boxed{\n\\mathbb E_{a\\sim\\pi^*}[Q(a)]\n-\n\\mathbb E_{a\\sim\\widehat\\pi}[Q(a)]\n\\leq\n2\\varepsilon_Q+\\delta_{\\mathrm{opt}}.\n}\n\\]\nFor a deterministic greedy action \\(\\widehat a\\in\\argmax_a\\widehat Q(a)\\),\n\\[\nQ(a^*)-Q(\\widehat a)\n\\leq2\\varepsilon_Q.\n\\]\nIf the true optimal action \\(a^*\\) is unique and its value gap\n\\[\n\\Delta_Q\n=\nQ(a^*)-\n\\max_{a\\neq a^*}Q(a)\n\\]\nsatisfies \\(\\Delta_Q>2\\varepsilon_Q\\), then every greedy maximizer of\n\\(\\widehat Q\\) equals \\(a^*\\).\n\\end{theorem}","thbb R\\) be the true interventional value and let \\(\\widehat Q\\) satisfy\n\\[\n\\sup_{a\\in\\mathcal A}|\\widehat Q(a)-Q(a)|\\leq\\varepsilon_Q.\n\\]\nSuppose a policy \\(\\widehat\\pi\\) is \\(\\delta_{\\mathrm{opt}}\\)-optimal for\n\\(\\widehat Q\\):\n\\[\n\\max_{\\pi}\n\\mathbb E_{a\\sim\\pi}[\\widehat Q(a)]\n-\n\\mathbb E_{a\\sim\\widehat\\pi}[\\widehat Q(a)]\n\\leq\n\\delta_{\\mathrm{opt}}.\n\\]\nThen, for every true-value-optimal policy \\(\\pi^*\\),\n\\[\n\\boxed{\n\\mathbb E_{a\\sim\\pi^*}[Q(a)]\n-\n\\mathbb E_{a\\sim\\widehat\\pi}[Q(a)]\n\\leq\n2\\varepsilon_Q+\\delta_{\\mathrm{opt}}.\n}\n\\]\nFor a deterministic greedy action \\(\\widehat a\\in\\argmax_a\\widehat Q(a)\\),\n\\[\nQ(a^*)-Q(\\widehat a)\n\\leq2\\varepsilon_Q.\n\\]\nIf the true optimal action \\(a^*\\) is unique and its value gap\n\\[\n\\Delta_Q\n=\nQ(a^*)-\n\\max_{a\\neq a^*}Q(a)\n\\]\nsatisfies \\(\\Delta_Q>2\\varepsilon_Q\\), then every greedy maximizer of\n\\(\\widehat Q\\) equals \\(a^*\\).\n\\end{theorem}","the true interventional value and let \\(\\widehat Q\\) satisfy\n\\[\n\\sup_{a\\in\\mathcal A}|\\widehat Q(a)-Q(a)|\\leq\\varepsilon_Q.\n\\]\nSuppose a policy \\(\\widehat\\pi\\) is \\(\\delta_{\\mathrm{opt}}\\)-optimal for\n\\(\\widehat Q\\):\n\\[\n\\max_{\\pi}\n\\mathbb E_{a\\sim\\pi}[\\widehat Q(a)]\n-\n\\mathbb E_{a\\sim\\widehat\\pi}[\\widehat Q(a)]\n\\leq\n\\delta_{\\mathrm{opt}}.\n\\]\nThen, for every true-value-optimal policy \\(\\pi^*\\),\n\\[\n\\boxed{\n\\mathbb E_{a\\sim\\pi^*}[Q(a)]\n-\n\\mathbb E_{a\\sim\\widehat\\pi}[Q(a)]\n\\leq\n2\\varepsilon_Q+\\delta_{\\mathrm{opt}}.\n}\n\\]\nFor a deterministic greedy action \\(\\widehat a\\in\\argmax_a\\widehat Q(a)\\),\n\\[\nQ(a^*)-Q(\\widehat a)\n\\leq2\\varepsilon_Q.\n\\]\nIf the true optimal action \\(a^*\\) is unique and its value gap\n\\[\n\\Delta_Q\n=\nQ(a^*)-\n\\max_{a\\neq a^*}Q(a)\n\\]\nsatisfies \\(\\Delta_Q>2\\varepsilon_Q\\), then every greedy maximizer of\n\\(\\widehat Q\\) equals \\(a^*\\).\n\\end{theorem}",") is \\(\\delta_{\\mathrm{opt}}\\)-optimal for\n\\(\\widehat Q\\):\n\\[\n\\max_{\\pi}\n\\mathbb E_{a\\sim\\pi}[\\widehat Q(a)]\n-\n\\mathbb E_{a\\sim\\widehat\\pi}[\\widehat Q(a)]\n\\leq\n\\delta_{\\mathrm{opt}}.\n\\]\nThen, for every true-value-optimal policy \\(\\pi^*\\),\n\\[\n\\boxed{\n\\mathbb E_{a\\sim\\pi^*}[Q(a)]\n-\n\\mathbb E_{a\\sim\\widehat\\pi}[Q(a)]\n\\leq\n2\\varepsilon_Q+\\delta_{\\mathrm{opt}}.\n}\n\\]\nFor a deterministic greedy action \\(\\widehat a\\in\\argmax_a\\widehat Q(a)\\),\n\\[\nQ(a^*)-Q(\\widehat a)\n\\leq2\\varepsilon_Q.\n\\]\nIf the true optimal action \\(a^*\\) is unique and its value gap\n\\[\n\\Delta_Q\n=\nQ(a^*)-\n\\max_{a\\neq a^*}Q(a)\n\\]\nsatisfies \\(\\Delta_Q>2\\varepsilon_Q\\), then every greedy maximizer of\n\\(\\widehat Q\\) equals \\(a^*\\).\n\\end{theorem}","r\n\\(\\widehat Q\\):\n\\[\n\\max_{\\pi}\n\\mathbb E_{a\\sim\\pi}[\\widehat Q(a)]\n-\n\\mathbb E_{a\\sim\\widehat\\pi}[\\widehat Q(a)]\n\\leq\n\\delta_{\\mathrm{opt}}.\n\\]\nThen, for every true-value-optimal policy \\(\\pi^*\\),\n\\[\n\\boxed{\n\\mathbb E_{a\\sim\\pi^*}[Q(a)]\n-\n\\mathbb E_{a\\sim\\widehat\\pi}[Q(a)]\n\\leq\n2\\varepsilon_Q+\\delta_{\\mathrm{opt}}.\n}\n\\]\nFor a deterministic greedy action \\(\\widehat a\\in\\argmax_a\\widehat Q(a)\\),\n\\[\nQ(a^*)-Q(\\widehat a)\n\\leq2\\varepsilon_Q.\n\\]\nIf the true optimal action \\(a^*\\) is unique and its value gap\n\\[\n\\Delta_Q\n=\nQ(a^*)-\n\\max_{a\\neq a^*}Q(a)\n\\]\nsatisfies \\(\\Delta_Q>2\\varepsilon_Q\\), then every greedy maximizer of\n\\(\\widehat Q\\) equals \\(a^*\\).\n\\end{theorem}","hbb E_{a\\sim\\widehat\\pi}[\\widehat Q(a)]\n\\leq\n\\delta_{\\mathrm{opt}}.\n\\]\nThen, for every true-value-optimal policy \\(\\pi^*\\),\n\\[\n\\boxed{\n\\mathbb E_{a\\sim\\pi^*}[Q(a)]\n-\n\\mathbb E_{a\\sim\\widehat\\pi}[Q(a)]\n\\leq\n2\\varepsilon_Q+\\delta_{\\mathrm{opt}}.\n}\n\\]\nFor a deterministic greedy action \\(\\widehat a\\in\\argmax_a\\widehat Q(a)\\),\n\\[\nQ(a^*)-Q(\\widehat a)\n\\leq2\\varepsilon_Q.\n\\]\nIf the true optimal action \\(a^*\\) is unique and its value gap\n\\[\n\\Delta_Q\n=\nQ(a^*)-\n\\max_{a\\neq a^*}Q(a)\n\\]\nsatisfies \\(\\Delta_Q>2\\varepsilon_Q\\), then every greedy maximizer of\n\\(\\widehat Q\\) equals \\(a^*\\).\n\\end{theorem}","a)]\n\\leq\n\\delta_{\\mathrm{opt}}.\n\\]\nThen, for every true-value-optimal policy \\(\\pi^*\\),\n\\[\n\\boxed{\n\\mathbb E_{a\\sim\\pi^*}[Q(a)]\n-\n\\mathbb E_{a\\sim\\widehat\\pi}[Q(a)]\n\\leq\n2\\varepsilon_Q+\\delta_{\\mathrm{opt}}.\n}\n\\]\nFor a deterministic greedy action \\(\\widehat a\\in\\argmax_a\\widehat Q(a)\\),\n\\[\nQ(a^*)-Q(\\widehat a)\n\\leq2\\varepsilon_Q.\n\\]\nIf the true optimal action \\(a^*\\) is unique and its value gap\n\\[\n\\Delta_Q\n=\nQ(a^*)-\n\\max_{a\\neq a^*}Q(a)\n\\]\nsatisfies \\(\\Delta_Q>2\\varepsilon_Q\\), then every greedy maximizer of\n\\(\\widehat Q\\) equals \\(a^*\\).\n\\end{theorem}","value-optimal policy \\(\\pi^*\\),\n\\[\n\\boxed{\n\\mathbb E_{a\\sim\\pi^*}[Q(a)]\n-\n\\mathbb E_{a\\sim\\widehat\\pi}[Q(a)]\n\\leq\n2\\varepsilon_Q+\\delta_{\\mathrm{opt}}.\n}\n\\]\nFor a deterministic greedy action \\(\\widehat a\\in\\argmax_a\\widehat Q(a)\\),\n\\[\nQ(a^*)-Q(\\widehat a)\n\\leq2\\varepsilon_Q.\n\\]\nIf the true optimal action \\(a^*\\) is unique and its value gap\n\\[\n\\Delta_Q\n=\nQ(a^*)-\n\\max_{a\\neq a^*}Q(a)\n\\]\nsatisfies \\(\\Delta_Q>2\\varepsilon_Q\\), then every greedy maximizer of\n\\(\\widehat Q\\) equals \\(a^*\\).\n\\end{theorem}","pi^*}[Q(a)]\n-\n\\mathbb E_{a\\sim\\widehat\\pi}[Q(a)]\n\\leq\n2\\varepsilon_Q+\\delta_{\\mathrm{opt}}.\n}\n\\]\nFor a deterministic greedy action \\(\\widehat a\\in\\argmax_a\\widehat Q(a)\\),\n\\[\nQ(a^*)-Q(\\widehat a)\n\\leq2\\varepsilon_Q.\n\\]\nIf the true optimal action \\(a^*\\) is unique and its value gap\n\\[\n\\Delta_Q\n=\nQ(a^*)-\n\\max_{a\\neq a^*}Q(a)\n\\]\nsatisfies \\(\\Delta_Q>2\\varepsilon_Q\\), then every greedy maximizer of\n\\(\\widehat Q\\) equals \\(a^*\\).\n\\end{theorem}","E_{a\\sim\\widehat\\pi}[Q(a)]\n\\leq\n2\\varepsilon_Q+\\delta_{\\mathrm{opt}}.\n}\n\\]\nFor a deterministic greedy action \\(\\widehat a\\in\\argmax_a\\widehat Q(a)\\),\n\\[\nQ(a^*)-Q(\\widehat a)\n\\leq2\\varepsilon_Q.\n\\]\nIf the true optimal action \\(a^*\\) is unique and its value gap\n\\[\n\\Delta_Q\n=\nQ(a^*)-\n\\max_{a\\neq a^*}Q(a)\n\\]\nsatisfies \\(\\Delta_Q>2\\varepsilon_Q\\), then every greedy maximizer of\n\\(\\widehat Q\\) equals \\(a^*\\).\n\\end{theorem}","\\begin{proof}\nFor every policy \\(\\pi\\), uniform approximation gives\n\\[\n\\left|\n\\mathbb E_{a\\sim\\pi}[\\widehat Q(a)]\n-\n\\mathbb E_{a\\sim\\pi}[Q(a)]\n\\right|\n\\leq\\varepsilon_Q.\n\\]\nTherefore\n\\[\n\\begin{aligned}\n\\mathbb E_{\\pi^*}[Q]\n-\n\\mathbb E_{\\widehat\\pi}[Q]\n&\\leq\n\\mathbb E_{\\pi^*}[\\widehat Q]\n-\n\\mathbb E_{\\widehat\\pi}[\\widehat Q]\n+2\\varepsilon_Q","\\begin{proof}\nFor every policy \\(\\pi\\), uniform approximation gives\n\\[\n\\left|\n\\mathbb E_{a\\sim\\pi}[\\widehat Q(a)]\n-\n\\mathbb E_{a\\sim\\pi}[Q(a)]\n\\right|\n\\leq\\varepsilon_Q.\n\\]\nTherefore\n\\[\n\\begin{aligned}\n\\mathbb E_{\\pi^*}[Q]\n-\n\\mathbb E_{\\widehat\\pi}[Q]\n&\\leq\n\\mathbb E_{\\pi^*}[\\widehat Q]\n-\n\\mathbb E_{\\widehat\\pi}[\\widehat Q]\n+2\\varepsilon_Q\\\\\n&\\leq\n\\delta_{\\mathrm{opt}}+2\\varepsilon_Q.\n\\end{aligned}\n\\]\nThe deterministic bound is the special case of point-mass policies. If\n\\(\\widehat a\\neq a^*\\), then\n\\[\n\\widehat Q(a^*)\n\\geq\nQ(a^*)-\\varepsilon_Q\n>\nQ(\\widehat a)+\\varepsilon_Q\n\\geq\n\\widehat Q(\\widehat a),\n\\]\ncontradicting greedy optimality.\n\\end{proof}","&\\leq\n\\delta_{\\mathrm{opt}}+2\\varepsilon_Q.\n\\end{aligned}\n\\]\nThe deterministic bound is the special case of point-mass policies. If\n\\(\\widehat a\\neq a^*\\), then\n\\[\n\\widehat Q(a^*)\n\\geq\nQ(a^*)-\\varepsilon_Q\n>\nQ(\\widehat a)+\\varepsilon_Q\n\\geq\n\\widehat Q(\\widehat a),\n\\]\ncontradicting greedy optimality.\n\\end{proof}","Let \\(V_t\\) denote the internal value channel. A counterfactual-mediation test\ncompares:\n\\[\nG_t\n\\longrightarrow\nV_t\n\\longrightarrow\nA_t\n\\longrightarrow\nY_t^+.\n\\]\nThe relevant quantities include:\n\\[\n\\eta_{G\\to V},\n\\qquad\n\\eta_{V\\to A},\n\\qquad\n\\eta_{A\\to Y},\n\\qquad\n\\delta_{G\\to A\\mid V},\n\\]\nwhere the final term measures the residual goal-to-action effect after the\nvalue channel is blocked or randomized.","The action-replay null preserves the empirical action sequence or action\nmarginal while destroying online dependence on \\(V_t\\). If replayed actions\nperform equally well under held-out perturbations, the evidence supports a\ncached or open-loop policy rather than online counterfactual evaluation. A\nsecond null permutes value labels among feasible actions while preserving the\nmultiset of scores. A genuine value-mediated policy should follow the\npermuted ranking, while an ignored critic should not.","Choose a declared set \\(\\mathcal J_{\\mathrm{maint}}\\) of workspace gates that\nmust be protected. Write each gate as a signed margin\n\\[\n\\phi_{j,t}\n\\geq0\n\\quad\\Longleftrightarrow\\quad\n\\text{gate }j\\text{ passes at time }t.\n\\]\nExamples include\n\\[\n\\begin{aligned}\n\\phi_{\\mathrm{gap},t}\n&=g_{\\varepsilon,t}-\\underline g,","Choose a declared set \\(\\mathcal J_{\\mathrm{maint}}\\) of workspace gates that\nmust be protected. Write each gate as a signed margin\n\\[\n\\phi_{j,t}\n\\geq0\n\\quad\\Longleftrightarrow\\quad\n\\text{gate }j\\text{ passes at time }t.\n\\]\nExamples include\n\\[\n\\begin{aligned}\n\\phi_{\\mathrm{gap},t}\n&=g_{\\varepsilon,t}-\\underline g,\\\\\n\\phi_{\\mathrm{coh},t}\n&=\\overline\\delta_{\\mathrm{coh}}-\\delta_{\\mathrm{coh},t},\\\\\n\\phi_{\\mathrm{rec},t}\n&=\\alpha_t^{(k)}-\\underline\\alpha,\\\\\n\\phi_{\\mathrm{keep},t}\n&=\\mu_t^{\\mathrm{keep}}-\\underline\\mu,\\\\\n\\phi_{\\mathrm{causal},t}\n&=\\eta_{\\mathrm{causal},t}-\\underline\\eta,\\\\\n\\phi_{\\mathrm{boundary},t}\n&=\\Delta_t^B-\\underline\\Delta_B.\n\\end{aligned}\n\\]\nThe aggregate integrity margin is\n\\[\n\\boxed{\nv_t\n=\n\\min_{j\\in\\mathcal J_{\\mathrm{maint}}}\\phi_{j,t}.\n}\n\\]\nThe workspace passes the protected gates when \\(v_t\\geq0\\). To maintain a\nreserve \\(\\rho>0\\), define the maintenance deficit\n\\[\n\\boxed{\ne_t\n=(\\rho-v_t)_+.\n}\n\\]\nDefine the zero-reserve and bounded-deficit viability sets\n\\[\n\\boxed{\n\\mathcal V_\\Theta\n=\n\\left\\{\nS:\\phi_j(S)\\geq0\\ \\text{for every }j\\in\\mathcal J_{\\mathrm{maint}}…","\\phi_{\\mathrm{boundary},t}\n&=\\Delta_t^B-\\underline\\Delta_B.\n\\end{aligned}\n\\]\nThe aggregate integrity margin is\n\\[\n\\boxed{\nv_t\n=\n\\min_{j\\in\\mathcal J_{\\mathrm{maint}}}\\phi_{j,t}.\n}\n\\]\nThe workspace passes the protected gates when \\(v_t\\geq0\\). To maintain a\nreserve \\(\\rho>0\\), define the maintenance deficit\n\\[\n\\boxed{\ne_t\n=(\\rho-v_t)_+.\n}\n\\]\nDefine the zero-reserve and bounded-deficit viability sets\n\\[\n\\boxed{\n\\mathcal V_\\Theta\n=\n\\left\\{\nS:\\phi_j(S)\\geq0\\ \\text{for every }j\\in\\mathcal J_{\\mathrm{maint}}\n\\right\\}\n}\n\\]\n\\[\n\\boxed{\n\\mathcal V_{\\Theta,e_{\\max}}\n=\n\\left\\{\nS:e(S)\\leq e_{\\max}\n\\right\\}\n}\n\\]\nThe first set requires every protected gate to pass. The second permits a\ndeclared finite reserve deficit.","_{\\mathrm{boundary},t}\n&=\\Delta_t^B-\\underline\\Delta_B.\n\\end{aligned}\n\\]\nThe aggregate integrity margin is\n\\[\n\\boxed{\nv_t\n=\n\\min_{j\\in\\mathcal J_{\\mathrm{maint}}}\\phi_{j,t}.\n}\n\\]\nThe workspace passes the protected gates when \\(v_t\\geq0\\). To maintain a\nreserve \\(\\rho>0\\), define the maintenance deficit\n\\[\n\\boxed{\ne_t\n=(\\rho-v_t)_+.\n}\n\\]\nDefine the zero-reserve and bounded-deficit viability sets\n\\[\n\\boxed{\n\\mathcal V_\\Theta\n=\n\\left\\{\nS:\\phi_j(S)\\geq0\\ \\text{for every }j\\in\\mathcal J_{\\mathrm{maint}}\n\\right\\}\n}\n\\]\n\\[\n\\boxed{\n\\mathcal V_{\\Theta,e_{\\max}}\n=\n\\left\\{\nS:e(S)\\leq e_{\\max}\n\\right\\}\n}\n\\]\nThe first set requires every protected gate to pass. The second permits a\ndeclared finite reserve deficit.","t\n=\n\\min_{j\\in\\mathcal J_{\\mathrm{maint}}}\\phi_{j,t}.\n}\n\\]\nThe workspace passes the protected gates when \\(v_t\\geq0\\). To maintain a\nreserve \\(\\rho>0\\), define the maintenance deficit\n\\[\n\\boxed{\ne_t\n=(\\rho-v_t)_+.\n}\n\\]\nDefine the zero-reserve and bounded-deficit viability sets\n\\[\n\\boxed{\n\\mathcal V_\\Theta\n=\n\\left\\{\nS:\\phi_j(S)\\geq0\\ \\text{for every }j\\in\\mathcal J_{\\mathrm{maint}}\n\\right\\}\n}\n\\]\n\\[\n\\boxed{\n\\mathcal V_{\\Theta,e_{\\max}}\n=\n\\left\\{\nS:e(S)\\leq e_{\\max}\n\\right\\}\n}\n\\]\nThe first set requires every protected gate to pass. The second permits a\ndeclared finite reserve deficit.","A damage intervention \\(d\\in\\mathcal I_D\\) changes a mechanism that supports\none or more protected gates. Examples include:\n\\begin{itemize}\n\\item deleting or corrupting a transfer edge;\n\\item removing a participant or artifact;\n\\item perturbing the internal boundary mask;\n\\item scrambling a role-binding or routing channel;\n\\item erasing a handoff representation; or\n\\item degrading a reconstruction map.\n\\end{itemize}","A repair intervention \\(r\\in\\mathcal I_R\\) can add, remove, retrain, rewrite,\nor reroute an admissible component. Let \\(r_0\\) denote matched no-repair. The\npost-repair integrity value at horizon \\(\\tau\\) is\n\\[\nV_t^{\\mathrm{rep}}(r;d)\n=\n\\mathbb E\n\\left[\nv_{t+\\tau}\n\\mid\n\\operatorname{do}(D_t=d),\n\\operatorname{do}(R_t=r),\n\\mathbf H_t\n\\right].\n\\]\nThe repair gain is\n\\[\n\\Delta_t^{\\mathrm{rep}}(r;d)\n=\nV_t^{\\mathrm{rep}}(r;d)\n-\nV_t^{\\mathrm{rep}}(r_0;d).\n\\]\nRepair costs are handled either by restricting the feasible repair family or\nby reporting a separate cost-effect curve. They should not be hidden inside\nan undocumented scalar reward.","Let \\(M_t\\) be an internal integrity monitor and \\(R_t\\) the selected repair\naction. A self-maintenance claim requires the causal chain\n\\[\nD_t\n\\longrightarrow\nM_t\n\\longrightarrow\nR_t\n\\longrightarrow\nv_{t+\\tau}.\n\\]\nThe required tests are:\n\\begin{description}\n\\item[Detection.]\nMatched damage interventions change \\(M_t\\) in a gate-specific direction.","\\item[Detection.]\nMatched damage interventions change \\(M_t\\) in a gate-specific direction.","\\item[Repair mediation.]\nIntervening on \\(M_t\\) changes the repair distribution, and blocking \\(M_t\\)\nreduces the damage-to-repair effect.","\\item[Restoration.]\nThe selected repair improves \\(v_{t+\\tau}\\) relative to matched no-repair.","Define the detection effect \\(\\eta_{D\\to M}\\), mediation effect\n\\(\\eta_{M\\to R}\\), restoration effect \\(\\eta_{R\\to v}\\), and repair\nspecificity\n\\[\n\\Delta_{\\mathrm{repair}}^{\\mathrm{spec}}\n=\n\\eta_{R\\to v}\n-\n\\eta_{R_\\perp\\to v},\n\\]\nwhere \\(R_\\perp\\) is a matched action outside the proposed repair channel.","\\subtopic{53.8 \\(T_{21}\\): Repair-selection margin theorem }","\\begin{theorem}[Repair selection under value-model error]\n\\label{thm:repair-selection}\nFix a damage intervention \\(d\\) and a finite feasible repair set\n\\(\\mathcal R\\). Let\n\\[\nV(r)=V_t^{\\mathrm{rep}}(r;d)\n\\]\nbe the true expected post-repair integrity margin and let \\(\\widehat V\\) obey\n\\[\n\\sup_{r\\in\\mathcal R}|\\widehat V(r)-V(r)|\n\\leq\n\\varepsilon_R.\n\\]\nLet \\(\\widehat r\\in\\argmax_r\\widehat V(r)\\) and\n\\(r^*\\in\\argmax_r V(r)\\). Then\n\\[\n\\boxed{\nV(r^*)-V(\\widehat r)\n\\leq\n2\\varepsilon_R.\n}\n\\]\nIf\n\\[\nV(r^*)\n\\geq\n\\rho+2\\varepsilon_R,\n\\]\nthen the selected repair restores the expected reserve:\n\\[\n\\boxed{V(\\widehat r)\\geq\\rho.}\n\\]\nIf \\(r^*\\) is unique and its repair-value gap exceeds\n\\(2\\varepsilon_R\\), then \\(\\widehat r=r^*\\).\n\\end{theorem}","gin{theorem}[Repair selection under value-model error]\n\\label{thm:repair-selection}\nFix a damage intervention \\(d\\) and a finite feasible repair set\n\\(\\mathcal R\\). Let\n\\[\nV(r)=V_t^{\\mathrm{rep}}(r;d)\n\\]\nbe the true expected post-repair integrity margin and let \\(\\widehat V\\) obey\n\\[\n\\sup_{r\\in\\mathcal R}|\\widehat V(r)-V(r)|\n\\leq\n\\varepsilon_R.\n\\]\nLet \\(\\widehat r\\in\\argmax_r\\widehat V(r)\\) and\n\\(r^*\\in\\argmax_r V(r)\\). Then\n\\[\n\\boxed{\nV(r^*)-V(\\widehat r)\n\\leq\n2\\varepsilon_R.\n}\n\\]\nIf\n\\[\nV(r^*)\n\\geq\n\\rho+2\\varepsilon_R,\n\\]\nthen the selected repair restores the expected reserve:\n\\[\n\\boxed{V(\\widehat r)\\geq\\rho.}\n\\]\nIf \\(r^*\\) is unique and its repair-value gap exceeds\n\\(2\\varepsilon_R\\), then \\(\\widehat r=r^*\\).\n\\end{theorem}","nder value-model error]\n\\label{thm:repair-selection}\nFix a damage intervention \\(d\\) and a finite feasible repair set\n\\(\\mathcal R\\). Let\n\\[\nV(r)=V_t^{\\mathrm{rep}}(r;d)\n\\]\nbe the true expected post-repair integrity margin and let \\(\\widehat V\\) obey\n\\[\n\\sup_{r\\in\\mathcal R}|\\widehat V(r)-V(r)|\n\\leq\n\\varepsilon_R.\n\\]\nLet \\(\\widehat r\\in\\argmax_r\\widehat V(r)\\) and\n\\(r^*\\in\\argmax_r V(r)\\). Then\n\\[\n\\boxed{\nV(r^*)-V(\\widehat r)\n\\leq\n2\\varepsilon_R.\n}\n\\]\nIf\n\\[\nV(r^*)\n\\geq\n\\rho+2\\varepsilon_R,\n\\]\nthen the selected repair restores the expected reserve:\n\\[\n\\boxed{V(\\widehat r)\\geq\\rho.}\n\\]\nIf \\(r^*\\) is unique and its repair-value gap exceeds\n\\(2\\varepsilon_R\\), then \\(\\widehat r=r^*\\).\n\\end{theorem}","mage intervention \\(d\\) and a finite feasible repair set\n\\(\\mathcal R\\). Let\n\\[\nV(r)=V_t^{\\mathrm{rep}}(r;d)\n\\]\nbe the true expected post-repair integrity margin and let \\(\\widehat V\\) obey\n\\[\n\\sup_{r\\in\\mathcal R}|\\widehat V(r)-V(r)|\n\\leq\n\\varepsilon_R.\n\\]\nLet \\(\\widehat r\\in\\argmax_r\\widehat V(r)\\) and\n\\(r^*\\in\\argmax_r V(r)\\). Then\n\\[\n\\boxed{\nV(r^*)-V(\\widehat r)\n\\leq\n2\\varepsilon_R.\n}\n\\]\nIf\n\\[\nV(r^*)\n\\geq\n\\rho+2\\varepsilon_R,\n\\]\nthen the selected repair restores the expected reserve:\n\\[\n\\boxed{V(\\widehat r)\\geq\\rho.}\n\\]\nIf \\(r^*\\) is unique and its repair-value gap exceeds\n\\(2\\varepsilon_R\\), then \\(\\widehat r=r^*\\).\n\\end{theorem}","rm{rep}}(r;d)\n\\]\nbe the true expected post-repair integrity margin and let \\(\\widehat V\\) obey\n\\[\n\\sup_{r\\in\\mathcal R}|\\widehat V(r)-V(r)|\n\\leq\n\\varepsilon_R.\n\\]\nLet \\(\\widehat r\\in\\argmax_r\\widehat V(r)\\) and\n\\(r^*\\in\\argmax_r V(r)\\). Then\n\\[\n\\boxed{\nV(r^*)-V(\\widehat r)\n\\leq\n2\\varepsilon_R.\n}\n\\]\nIf\n\\[\nV(r^*)\n\\geq\n\\rho+2\\varepsilon_R,\n\\]\nthen the selected repair restores the expected reserve:\n\\[\n\\boxed{V(\\widehat r)\\geq\\rho.}\n\\]\nIf \\(r^*\\) is unique and its repair-value gap exceeds\n\\(2\\varepsilon_R\\), then \\(\\widehat r=r^*\\).\n\\end{theorem}","st-repair integrity margin and let \\(\\widehat V\\) obey\n\\[\n\\sup_{r\\in\\mathcal R}|\\widehat V(r)-V(r)|\n\\leq\n\\varepsilon_R.\n\\]\nLet \\(\\widehat r\\in\\argmax_r\\widehat V(r)\\) and\n\\(r^*\\in\\argmax_r V(r)\\). Then\n\\[\n\\boxed{\nV(r^*)-V(\\widehat r)\n\\leq\n2\\varepsilon_R.\n}\n\\]\nIf\n\\[\nV(r^*)\n\\geq\n\\rho+2\\varepsilon_R,\n\\]\nthen the selected repair restores the expected reserve:\n\\[\n\\boxed{V(\\widehat r)\\geq\\rho.}\n\\]\nIf \\(r^*\\) is unique and its repair-value gap exceeds\n\\(2\\varepsilon_R\\), then \\(\\widehat r=r^*\\).\n\\end{theorem}","bey\n\\[\n\\sup_{r\\in\\mathcal R}|\\widehat V(r)-V(r)|\n\\leq\n\\varepsilon_R.\n\\]\nLet \\(\\widehat r\\in\\argmax_r\\widehat V(r)\\) and\n\\(r^*\\in\\argmax_r V(r)\\). Then\n\\[\n\\boxed{\nV(r^*)-V(\\widehat r)\n\\leq\n2\\varepsilon_R.\n}\n\\]\nIf\n\\[\nV(r^*)\n\\geq\n\\rho+2\\varepsilon_R,\n\\]\nthen the selected repair restores the expected reserve:\n\\[\n\\boxed{V(\\widehat r)\\geq\\rho.}\n\\]\nIf \\(r^*\\) is unique and its repair-value gap exceeds\n\\(2\\varepsilon_R\\), then \\(\\widehat r=r^*\\).\n\\end{theorem}","R}|\\widehat V(r)-V(r)|\n\\leq\n\\varepsilon_R.\n\\]\nLet \\(\\widehat r\\in\\argmax_r\\widehat V(r)\\) and\n\\(r^*\\in\\argmax_r V(r)\\). Then\n\\[\n\\boxed{\nV(r^*)-V(\\widehat r)\n\\leq\n2\\varepsilon_R.\n}\n\\]\nIf\n\\[\nV(r^*)\n\\geq\n\\rho+2\\varepsilon_R,\n\\]\nthen the selected repair restores the expected reserve:\n\\[\n\\boxed{V(\\widehat r)\\geq\\rho.}\n\\]\nIf \\(r^*\\) is unique and its repair-value gap exceeds\n\\(2\\varepsilon_R\\), then \\(\\widehat r=r^*\\).\n\\end{theorem}","\\begin{proof}\nGreedy optimality and uniform approximation give\n\\[\n\\begin{aligned}\nV(r^*)\n&\\leq\n\\widehat V(r^*)+\\varepsilon_R\\\\\n&\\leq\n\\widehat V(\\widehat r)+\\varepsilon_R\\\\\n&\\leq\nV(\\widehat r)+2\\varepsilon_R.\n\\end{aligned}\n\\]\nThe reserve and exact-selection statements follow immediately.\n\\end{proof}","Restoration of performance is not automatically restoration of identity. Let\n\\(W_{t^-}\\) be the last certified pre-damage workspace and let\n\\(W_{t+\\tau}^{r}\\) be the post-repair workspace. Define the repair identity\ndefect\n\\[\n\\delta_{\\mathrm{rep-id}}(r)\n=\n d_{\\mathrm{id}}\n\\left(\nW_{t^-},W_{t+\\tau}^{r}\n\\right).\n\\]\nA repair action is identity preserving at tolerance\n\\(\\varepsilon_{\\mathrm{id}}\\) when\n\\[\n\\Delta_t^{\\mathrm{rep}}(r;d)>0,\n\\qquad\n\\delta_{\\mathrm{rep-id}}(r)\n\\leq\n\\varepsilon_{\\mathrm{id}}.\n\\]\nWhen integrity is restored but the identity defect exceeds tolerance, the\nresult is better described as a successor or replacement workspace. This\nterminological distinction prevents any high-performing reconstruction from\nbeing counted as persistence of the same relational orga","Restoration of performance is not automatically restoration of identity. Let\n\\(W_{t^-}\\) be the last certified pre-damage workspace and let\n\\(W_{t+\\tau}^{r}\\) be the post-repair workspace. Define the repair identity\ndefect\n\\[\n\\delta_{\\mathrm{rep-id}}(r)\n=\n d_{\\mathrm{id}}\n\\left(\nW_{t^-},W_{t+\\tau}^{r}\n\\right).\n\\]\nA repair action is identity preserving at tolerance\n\\(\\varepsilon_{\\mathrm{id}}\\) when\n\\[\n\\Delta_t^{\\mathrm{rep}}(r;d)>0,\n\\qquad\n\\delta_{\\mathrm{rep-id}}(r)\n\\leq\n\\varepsilon_{\\mathrm{id}}.\n\\]\nWhen integrity is restored but the identity defect exceeds tolerance, the\nresult is better described as a successor or replacement workspace. This\nterminological distinction prevents any high-performing reconstruction from\nbeing counted as persistence of the same relational organization.","oration of performance is not automatically restoration of identity. Let\n\\(W_{t^-}\\) be the last certified pre-damage workspace and let\n\\(W_{t+\\tau}^{r}\\) be the post-repair workspace. Define the repair identity\ndefect\n\\[\n\\delta_{\\mathrm{rep-id}}(r)\n=\n d_{\\mathrm{id}}\n\\left(\nW_{t^-},W_{t+\\tau}^{r}\n\\right).\n\\]\nA repair action is identity preserving at tolerance\n\\(\\varepsilon_{\\mathrm{id}}\\) when\n\\[\n\\Delta_t^{\\mathrm{rep}}(r;d)>0,\n\\qquad\n\\delta_{\\mathrm{rep-id}}(r)\n\\leq\n\\varepsilon_{\\mathrm{id}}.\n\\]\nWhen integrity is restored but the identity defect exceeds tolerance, the\nresult is better described as a successor or replacement workspace. This\nterminological distinction prevents any high-performing reconstruction from\nbeing counted as persistence of the same relational organization.","\\subtopic{53.10 \\(T_{22}\\): Closed-loop self-maintenance theorem }","\\begin{theorem}[Causal repair stabilizes maintenance deficit]\n\\label{thm:self-maintenance}\nLet \\(e_t\\geq0\\) be the maintenance deficit. Suppose the matched no-repair\ntrajectory satisfies\n\\[\ne_{t+1}^{(0)}\n\\leq\n\\alpha e_t+d_t,\n\\]\nwhere \\(d_t\\geq0\\) is exogenous damage. Suppose the implemented repair policy\nhas the causal one-step effect\n\\[\ne_{t+1}^{(0)}-e_{t+1}\n\\geq\n\\kappa e_t-\\epsilon_t.\n\\]\nAssume\n\\[\n0\\leq\\beta:=\\alpha-\\kappa<1,\n\\qquad\n0\\leq d_t\\leq\\overline d,\n\\qquad\n0\\leq\\epsilon_t\\leq\\overline\\epsilon.\n\\]\nThen\n\\[\n\\boxed{\ne_t\n\\leq\n\\beta^te_0\n+\n\\frac{(\\overline d+\\overline\\epsilon)(1-\\beta^t)}{1-\\beta},\n}\n\\]\nand therefore\n\\[\n\\boxed{\n\\limsup_{t\\to\\infty}e_t\n\\leq\n\\frac{\\overline d+\\overline\\epsilon}{1-\\beta}.\n}\n\\]\nIf the admissible deficit is \\(e_{\\max}\\) and\n\\[\ne_0\\leq e_{\\max},\n\\qquad\n\\overline","\\begin{theorem}[Causal repair stabilizes maintenance deficit]\n\\label{thm:self-maintenance}\nLet \\(e_t\\geq0\\) be the maintenance deficit. Suppose the matched no-repair\ntrajectory satisfies\n\\[\ne_{t+1}^{(0)}\n\\leq\n\\alpha e_t+d_t,\n\\]\nwhere \\(d_t\\geq0\\) is exogenous damage. Suppose the implemented repair policy\nhas the causal one-step effect\n\\[\ne_{t+1}^{(0)}-e_{t+1}\n\\geq\n\\kappa e_t-\\epsilon_t.\n\\]\nAssume\n\\[\n0\\leq\\beta:=\\alpha-\\kappa<1,\n\\qquad\n0\\leq d_t\\leq\\overline d,\n\\qquad\n0\\leq\\epsilon_t\\leq\\overline\\epsilon.\n\\]\nThen\n\\[\n\\boxed{\ne_t\n\\leq\n\\beta^te_0\n+\n\\frac{(\\overline d+\\overline\\epsilon)(1-\\beta^t)}{1-\\beta},\n}\n\\]\nand therefore\n\\[\n\\boxed{\n\\limsup_{t\\to\\infty}e_t\n\\leq\n\\frac{\\overline d+\\overline\\epsilon}{1-\\beta}.\n}\n\\]\nIf the admissible deficit is \\(e_{\\max}\\) and\n\\[\ne_0\\leq e_{\\max},\n\\qquad\n\\overline d+\\overline\\epsilon\n\\leq\n(1-\\beta)e_{\\max},\n\\]\nthen the interval \\([0,e_{\\max}]\\) is forward invariant.\n\\end{theorem}","egin{theorem}[Causal repair stabilizes maintenance deficit]\n\\label{thm:self-maintenance}\nLet \\(e_t\\geq0\\) be the maintenance deficit. Suppose the matched no-repair\ntrajectory satisfies\n\\[\ne_{t+1}^{(0)}\n\\leq\n\\alpha e_t+d_t,\n\\]\nwhere \\(d_t\\geq0\\) is exogenous damage. Suppose the implemented repair policy\nhas the causal one-step effect\n\\[\ne_{t+1}^{(0)}-e_{t+1}\n\\geq\n\\kappa e_t-\\epsilon_t.\n\\]\nAssume\n\\[\n0\\leq\\beta:=\\alpha-\\kappa<1,\n\\qquad\n0\\leq d_t\\leq\\overline d,\n\\qquad\n0\\leq\\epsilon_t\\leq\\overline\\epsilon.\n\\]\nThen\n\\[\n\\boxed{\ne_t\n\\leq\n\\beta^te_0\n+\n\\frac{(\\overline d+\\overline\\epsilon)(1-\\beta^t)}{1-\\beta},\n}\n\\]\nand therefore\n\\[\n\\boxed{\n\\limsup_{t\\to\\infty}e_t\n\\leq\n\\frac{\\overline d+\\overline\\epsilon}{1-\\beta}.\n}\n\\]\nIf the admissible deficit is \\(e_{\\max}\\) and\n\\[\ne_0\\leq e_{\\max},\n\\qquad\n\\overline d+\\overline\\epsilon\n\\leq\n(1-\\beta)e_{\\max},\n\\]\nthen the interval \\([0,e_{\\max}]\\) is forward invariant.\n\\end{theorem}","enance deficit. Suppose the matched no-repair\ntrajectory satisfies\n\\[\ne_{t+1}^{(0)}\n\\leq\n\\alpha e_t+d_t,\n\\]\nwhere \\(d_t\\geq0\\) is exogenous damage. Suppose the implemented repair policy\nhas the causal one-step effect\n\\[\ne_{t+1}^{(0)}-e_{t+1}\n\\geq\n\\kappa e_t-\\epsilon_t.\n\\]\nAssume\n\\[\n0\\leq\\beta:=\\alpha-\\kappa<1,\n\\qquad\n0\\leq d_t\\leq\\overline d,\n\\qquad\n0\\leq\\epsilon_t\\leq\\overline\\epsilon.\n\\]\nThen\n\\[\n\\boxed{\ne_t\n\\leq\n\\beta^te_0\n+\n\\frac{(\\overline d+\\overline\\epsilon)(1-\\beta^t)}{1-\\beta},\n}\n\\]\nand therefore\n\\[\n\\boxed{\n\\limsup_{t\\to\\infty}e_t\n\\leq\n\\frac{\\overline d+\\overline\\epsilon}{1-\\beta}.\n}\n\\]\nIf the admissible deficit is \\(e_{\\max}\\) and\n\\[\ne_0\\leq e_{\\max},\n\\qquad\n\\overline d+\\overline\\epsilon\n\\leq\n(1-\\beta)e_{\\max},\n\\]\nthen the interval \\([0,e_{\\max}]\\) is forward invariant.\n\\end{theorem}","e \\(d_t\\geq0\\) is exogenous damage. Suppose the implemented repair policy\nhas the causal one-step effect\n\\[\ne_{t+1}^{(0)}-e_{t+1}\n\\geq\n\\kappa e_t-\\epsilon_t.\n\\]\nAssume\n\\[\n0\\leq\\beta:=\\alpha-\\kappa<1,\n\\qquad\n0\\leq d_t\\leq\\overline d,\n\\qquad\n0\\leq\\epsilon_t\\leq\\overline\\epsilon.\n\\]\nThen\n\\[\n\\boxed{\ne_t\n\\leq\n\\beta^te_0\n+\n\\frac{(\\overline d+\\overline\\epsilon)(1-\\beta^t)}{1-\\beta},\n}\n\\]\nand therefore\n\\[\n\\boxed{\n\\limsup_{t\\to\\infty}e_t\n\\leq\n\\frac{\\overline d+\\overline\\epsilon}{1-\\beta}.\n}\n\\]\nIf the admissible deficit is \\(e_{\\max}\\) and\n\\[\ne_0\\leq e_{\\max},\n\\qquad\n\\overline d+\\overline\\epsilon\n\\leq\n(1-\\beta)e_{\\max},\n\\]\nthen the interval \\([0,e_{\\max}]\\) is forward invariant.\n\\end{theorem}","-e_{t+1}\n\\geq\n\\kappa e_t-\\epsilon_t.\n\\]\nAssume\n\\[\n0\\leq\\beta:=\\alpha-\\kappa<1,\n\\qquad\n0\\leq d_t\\leq\\overline d,\n\\qquad\n0\\leq\\epsilon_t\\leq\\overline\\epsilon.\n\\]\nThen\n\\[\n\\boxed{\ne_t\n\\leq\n\\beta^te_0\n+\n\\frac{(\\overline d+\\overline\\epsilon)(1-\\beta^t)}{1-\\beta},\n}\n\\]\nand therefore\n\\[\n\\boxed{\n\\limsup_{t\\to\\infty}e_t\n\\leq\n\\frac{\\overline d+\\overline\\epsilon}{1-\\beta}.\n}\n\\]\nIf the admissible deficit is \\(e_{\\max}\\) and\n\\[\ne_0\\leq e_{\\max},\n\\qquad\n\\overline d+\\overline\\epsilon\n\\leq\n(1-\\beta)e_{\\max},\n\\]\nthen the interval \\([0,e_{\\max}]\\) is forward invariant.\n\\end{theorem}","ppa e_t-\\epsilon_t.\n\\]\nAssume\n\\[\n0\\leq\\beta:=\\alpha-\\kappa<1,\n\\qquad\n0\\leq d_t\\leq\\overline d,\n\\qquad\n0\\leq\\epsilon_t\\leq\\overline\\epsilon.\n\\]\nThen\n\\[\n\\boxed{\ne_t\n\\leq\n\\beta^te_0\n+\n\\frac{(\\overline d+\\overline\\epsilon)(1-\\beta^t)}{1-\\beta},\n}\n\\]\nand therefore\n\\[\n\\boxed{\n\\limsup_{t\\to\\infty}e_t\n\\leq\n\\frac{\\overline d+\\overline\\epsilon}{1-\\beta}.\n}\n\\]\nIf the admissible deficit is \\(e_{\\max}\\) and\n\\[\ne_0\\leq e_{\\max},\n\\qquad\n\\overline d+\\overline\\epsilon\n\\leq\n(1-\\beta)e_{\\max},\n\\]\nthen the interval \\([0,e_{\\max}]\\) is forward invariant.\n\\end{theorem}","\\leq\\epsilon_t\\leq\\overline\\epsilon.\n\\]\nThen\n\\[\n\\boxed{\ne_t\n\\leq\n\\beta^te_0\n+\n\\frac{(\\overline d+\\overline\\epsilon)(1-\\beta^t)}{1-\\beta},\n}\n\\]\nand therefore\n\\[\n\\boxed{\n\\limsup_{t\\to\\infty}e_t\n\\leq\n\\frac{\\overline d+\\overline\\epsilon}{1-\\beta}.\n}\n\\]\nIf the admissible deficit is \\(e_{\\max}\\) and\n\\[\ne_0\\leq e_{\\max},\n\\qquad\n\\overline d+\\overline\\epsilon\n\\leq\n(1-\\beta)e_{\\max},\n\\]\nthen the interval \\([0,e_{\\max}]\\) is forward invariant.\n\\end{theorem}","\\begin{proof}\nCombine the no-repair bound with the causal repair effect:\n\\[\n\\begin{aligned}\ne_{t+1}\n&\\leq\n e_{t+1}^{(0)}-\\kappa e_t+\\epsilon_t\\\\\n&\\leq\n(\\alpha-\\kappa)e_t+d_t+\\epsilon_t\\\\\n&\\leq\n\\beta e_t+\\overline d+\\overline\\epsilon.\n\\end{aligned}\n\\]\nIterating this scalar recurrence gives the stated geometric bound. If\n\\(e_t\\leq e_{\\max}\\), the forward-invariance condition gives\n\\[\ne_{t+1}\n\\leq\n\\beta e_{\\max}\n+(1-\\beta)e_{\\max}\n=e_{\\max}.\n\\]\n\\end{proof}","&\\leq\n\\beta e_t+\\overline d+\\overline\\epsilon.\n\\end{aligned}\n\\]\nIterating this scalar recurrence gives the stated geometric bound. If\n\\(e_t\\leq e_{\\max}\\), the forward-invariance condition gives\n\\[\ne_{t+1}\n\\leq\n\\beta e_{\\max}\n+(1-\\beta)e_{\\max}\n=e_{\\max}.\n\\]\n\\end{proof}","Let \\(b_t\\in[0,1]\\) be a boundary-integrity score and let the reserve be\n\\(\\rho\\). Define\n\\[\ne_t=(\\rho-b_t)_+.\n\\]\nSuppose uncorrected damage obeys\n\\[\ne_{t+1}^{(0)}\n\\leq\n1.08e_t+d_t,\n\\]\nso the open-loop boundary is locally unstable. A repair policy with measured\ncausal effect\n\\[\ne_{t+1}^{(0)}-e_{t+1}\n\\geq\n0.25e_t-\\epsilon_t\n\\]\ngives\n\\[\n\\beta=1.08-0.25=0.83.\n\\]\nIf\n\\[\n\\overline d=0.01,\n\\qquad\n\\overline\\epsilon=0.005,\n\\]\nthen\n\\[\n\\limsup e_t\n\\leq\n\\frac{0.015}{1-0.83}\n\\approx0.0883.\n\\]\nThus an open-loop unstable boundary can be maintained within a finite deficit\nband by a sufficiently strong repair channel. The example establishes only a\ncontrol calculation. A complete self-maintenance claim must also show that an","Let \\(b_t\\in[0,1]\\) be a boundary-integrity score and let the reserve be\n\\(\\rho\\). Define\n\\[\ne_t=(\\rho-b_t)_+.\n\\]\nSuppose uncorrected damage obeys\n\\[\ne_{t+1}^{(0)}\n\\leq\n1.08e_t+d_t,\n\\]\nso the open-loop boundary is locally unstable. A repair policy with measured\ncausal effect\n\\[\ne_{t+1}^{(0)}-e_{t+1}\n\\geq\n0.25e_t-\\epsilon_t\n\\]\ngives\n\\[\n\\beta=1.08-0.25=0.83.\n\\]\nIf\n\\[\n\\overline d=0.01,\n\\qquad\n\\overline\\epsilon=0.005,\n\\]\nthen\n\\[\n\\limsup e_t\n\\leq\n\\frac{0.015}{1-0.83}\n\\approx0.0883.\n\\]\nThus an open-loop unstable boundary can be maintained within a finite deficit\nband by a sufficiently strong repair channel. The example establishes only a\ncontrol calculation. A complete self-maintenance claim must also show that an\ninternal monitor detects damage, mediates the repair choice,","Let \\(b_t\\in[0,1]\\) be a boundary-integrity score and let the reserve be\n\\(\\rho\\). Define\n\\[\ne_t=(\\rho-b_t)_+.\n\\]\nSuppose uncorrected damage obeys\n\\[\ne_{t+1}^{(0)}\n\\leq\n1.08e_t+d_t,\n\\]\nso the open-loop boundary is locally unstable. A repair policy with measured\ncausal effect\n\\[\ne_{t+1}^{(0)}-e_{t+1}\n\\geq\n0.25e_t-\\epsilon_t\n\\]\ngives\n\\[\n\\beta=1.08-0.25=0.83.\n\\]\nIf\n\\[\n\\overline d=0.01,\n\\qquad\n\\overline\\epsilon=0.005,\n\\]\nthen\n\\[\n\\limsup e_t\n\\leq\n\\frac{0.015}{1-0.83}\n\\approx0.0883.\n\\]\nThus an open-loop unstable boundary can be maintained within a finite deficit\nband by a sufficiently strong repair channel. The example establishes only a\ncontrol calculation. A complete self-maintenance claim must also show that an\ninternal monitor detects damage, mediates the repair choice, and preserves\nworkspace identity","Let \\(b_t\\in[0,1]\\) be a boundary-integrity score and let the reserve be\n\\(\\rho\\). Define\n\\[\ne_t=(\\rho-b_t)_+.\n\\]\nSuppose uncorrected damage obeys\n\\[\ne_{t+1}^{(0)}\n\\leq\n1.08e_t+d_t,\n\\]\nso the open-loop boundary is locally unstable. A repair policy with measured\ncausal effect\n\\[\ne_{t+1}^{(0)}-e_{t+1}\n\\geq\n0.25e_t-\\epsilon_t\n\\]\ngives\n\\[\n\\beta=1.08-0.25=0.83.\n\\]\nIf\n\\[\n\\overline d=0.01,\n\\qquad\n\\overline\\epsilon=0.005,\n\\]\nthen\n\\[\n\\limsup e_t\n\\leq\n\\frac{0.015}{1-0.83}\n\\approx0.0883.\n\\]\nThus an open-loop unstable boundary can be maintained within a finite deficit\nband by a sufficiently strong repair channel. The example establishes only a\ncontrol calculation. A complete self-maintenance claim must also show that an\ninternal monitor detects damage, mediates the repair choice, and preserves\nworkspace identity.","ndary-integrity score and let the reserve be\n\\(\\rho\\). Define\n\\[\ne_t=(\\rho-b_t)_+.\n\\]\nSuppose uncorrected damage obeys\n\\[\ne_{t+1}^{(0)}\n\\leq\n1.08e_t+d_t,\n\\]\nso the open-loop boundary is locally unstable. A repair policy with measured\ncausal effect\n\\[\ne_{t+1}^{(0)}-e_{t+1}\n\\geq\n0.25e_t-\\epsilon_t\n\\]\ngives\n\\[\n\\beta=1.08-0.25=0.83.\n\\]\nIf\n\\[\n\\overline d=0.01,\n\\qquad\n\\overline\\epsilon=0.005,\n\\]\nthen\n\\[\n\\limsup e_t\n\\leq\n\\frac{0.015}{1-0.83}\n\\approx0.0883.\n\\]\nThus an open-loop unstable boundary can be maintained within a finite deficit\nband by a sufficiently strong repair channel. The example establishes only a\ncontrol calculation. A complete self-maintenance claim must also show that an\ninternal monitor detects damage, mediates the repair choice, and preserves\nworkspace identity.","Let\n\\[\n\\Theta_A\n=\n\\left(\n\\tau_G,\n\\tau_{G\\to A},\n\\varepsilon_Q,\n\\tau_{V\\to A},\n\\tau_{A\\to Y},\n\\tau_{D\\to M},\n\\tau_{M\\to R},\n\\tau_{R\\to v},\n\\tau_{\\mathrm{spec}},\n\\varepsilon_{\\mathrm{ctl}},\n e_{\\max}\n\\right).\n\\]\nA \\(\\Theta_A\\)-certified workspace agent is a controlled realization for\nwhich:\n\\begin{enumerate}\n\\item \\(\\SRW(W_t;\\varsigma)=1\\);\n\\item cue-controlled goal retention exceeds \\(\\tau_G\\);\n\\item goal-to-action mediation exceeds \\(\\tau_{G\\to A}\\);\n\\item counterfactual-value error is at most \\(\\varepsilon_Q\\);\n\\item value-to-action use exceeds \\(\\tau_{V\\to A}\\);\n\\item action-to-outcome control exceeds \\(\\tau_{A\\to Y}\\);\n\\item damage detection, monitor-to-repair mediation, and restoration exceed\n their thresholds;\n\\item repair specificity exceeds \\(\\tau_{\\mathrm{spec}}\\);\n\\item \\(k_{\\mathrm{ctl},\\varepsilon_{\\mathrm{ctl}}}\\geq2\\); and\n\\item repeated maintenance deficit remains at most \\(e_{\\ma","Let\n\\[\n\\Theta_A\n=\n\\left(\n\\tau_G,\n\\tau_{G\\to A},\n\\varepsilon_Q,\n\\tau_{V\\to A},\n\\tau_{A\\to Y},\n\\tau_{D\\to M},\n\\tau_{M\\to R},\n\\tau_{R\\to v},\n\\tau_{\\mathrm{spec}},\n\\varepsilon_{\\mathrm{ctl}},\n e_{\\max}\n\\right).\n\\]\nA \\(\\Theta_A\\)-certified workspace agent is a controlled realization for\nwhich:\n\\begin{enumerate}\n\\item \\(\\SRW(W_t;\\varsigma)=1\\);\n\\item cue-controlled goal retention exceeds \\(\\tau_G\\);\n\\item goal-to-action mediation exceeds \\(\\tau_{G\\to A}\\);\n\\item counterfactual-value error is at most \\(\\varepsilon_Q\\);\n\\item value-to-action use exceeds \\(\\tau_{V\\to A}\\);\n\\item action-to-outcome control exceeds \\(\\tau_{A\\to Y}\\);\n\\item damage detection, monitor-to-repair mediation, and restoration exceed\n their thresholds;\n\\item repair specificity exceeds \\(\\tau_{\\mathrm{spec}}\\);\n\\item \\(k_{\\mathrm{ctl},\\varepsilon_{\\mathrm{ctl}}}\\geq2\\); and\n\\item repeated maintenance deficit remains at most \\(e_{\\max}\\).\n\\end{enum","Let\n\\[\n\\Theta_A\n=\n\\left(\n\\tau_G,\n\\tau_{G\\to A},\n\\varepsilon_Q,\n\\tau_{V\\to A},\n\\tau_{A\\to Y},\n\\tau_{D\\to M},\n\\tau_{M\\to R},\n\\tau_{R\\to v},\n\\tau_{\\mathrm{spec}},\n\\varepsilon_{\\mathrm{ctl}},\n e_{\\max}\n\\right).\n\\]\nA \\(\\Theta_A\\)-certified workspace agent is a controlled realization for\nwhich:\n\\begin{enumerate}\n\\item \\(\\SRW(W_t;\\varsigma)=1\\);\n\\item cue-controlled goal retention exceeds \\(\\tau_G\\);\n\\item goal-to-action mediation exceeds \\(\\tau_{G\\to A}\\);\n\\item counterfactual-value error is at most \\(\\varepsilon_Q\\);\n\\item value-to-action use exceeds \\(\\tau_{V\\to A}\\);\n\\item action-to-outcome control exceeds \\(\\tau_{A\\to Y}\\);\n\\item damage detection, monitor-to-repair mediation, and restoration exceed\n their thresholds;\n\\item repair specificity exceeds \\(\\tau_{\\mathrm{spec}}\\);\n\\item \\(k_{\\mathrm{ctl},\\varepsilon_{\\mathrm{ctl}}}\\geq2\\); and\n\\item repeated maintenance deficit remains at most \\(e_{\\max}\\).\n\\end{enumerate}","{G\\to A},\n\\varepsilon_Q,\n\\tau_{V\\to A},\n\\tau_{A\\to Y},\n\\tau_{D\\to M},\n\\tau_{M\\to R},\n\\tau_{R\\to v},\n\\tau_{\\mathrm{spec}},\n\\varepsilon_{\\mathrm{ctl}},\n e_{\\max}\n\\right).\n\\]\nA \\(\\Theta_A\\)-certified workspace agent is a controlled realization for\nwhich:\n\\begin{enumerate}\n\\item \\(\\SRW(W_t;\\varsigma)=1\\);\n\\item cue-controlled goal retention exceeds \\(\\tau_G\\);\n\\item goal-to-action mediation exceeds \\(\\tau_{G\\to A}\\);\n\\item counterfactual-value error is at most \\(\\varepsilon_Q\\);\n\\item value-to-action use exceeds \\(\\tau_{V\\to A}\\);\n\\item action-to-outcome control exceeds \\(\\tau_{A\\to Y}\\);\n\\item damage detection, monitor-to-repair mediation, and restoration exceed\n their thresholds;\n\\item repair specificity exceeds \\(\\tau_{\\mathrm{spec}}\\);\n\\item \\(k_{\\mathrm{ctl},\\varepsilon_{\\mathrm{ctl}}}\\geq2\\); and\n\\item repeated maintenance deficit remains at most \\(e_{\\max}\\).\n\\end{enumerate}","to M},\n\\tau_{M\\to R},\n\\tau_{R\\to v},\n\\tau_{\\mathrm{spec}},\n\\varepsilon_{\\mathrm{ctl}},\n e_{\\max}\n\\right).\n\\]\nA \\(\\Theta_A\\)-certified workspace agent is a controlled realization for\nwhich:\n\\begin{enumerate}\n\\item \\(\\SRW(W_t;\\varsigma)=1\\);\n\\item cue-controlled goal retention exceeds \\(\\tau_G\\);\n\\item goal-to-action mediation exceeds \\(\\tau_{G\\to A}\\);\n\\item counterfactual-value error is at most \\(\\varepsilon_Q\\);\n\\item value-to-action use exceeds \\(\\tau_{V\\to A}\\);\n\\item action-to-outcome control exceeds \\(\\tau_{A\\to Y}\\);\n\\item damage detection, monitor-to-repair mediation, and restoration exceed\n their thresholds;\n\\item repair specificity exceeds \\(\\tau_{\\mathrm{spec}}\\);\n\\item \\(k_{\\mathrm{ctl},\\varepsilon_{\\mathrm{ctl}}}\\geq2\\); and\n\\item repeated maintenance deficit remains at most \\(e_{\\max}\\).\n\\end{enumerate}","ec}},\n\\varepsilon_{\\mathrm{ctl}},\n e_{\\max}\n\\right).\n\\]\nA \\(\\Theta_A\\)-certified workspace agent is a controlled realization for\nwhich:\n\\begin{enumerate}\n\\item \\(\\SRW(W_t;\\varsigma)=1\\);\n\\item cue-controlled goal retention exceeds \\(\\tau_G\\);\n\\item goal-to-action mediation exceeds \\(\\tau_{G\\to A}\\);\n\\item counterfactual-value error is at most \\(\\varepsilon_Q\\);\n\\item value-to-action use exceeds \\(\\tau_{V\\to A}\\);\n\\item action-to-outcome control exceeds \\(\\tau_{A\\to Y}\\);\n\\item damage detection, monitor-to-repair mediation, and restoration exceed\n their thresholds;\n\\item repair specificity exceeds \\(\\tau_{\\mathrm{spec}}\\);\n\\item \\(k_{\\mathrm{ctl},\\varepsilon_{\\mathrm{ctl}}}\\geq2\\); and\n\\item repeated maintenance deficit remains at most \\(e_{\\max}\\).\n\\end{enumerate}","(\\Theta_A\\)-certified workspace agent is a controlled realization for\nwhich:\n\\begin{enumerate}\n\\item \\(\\SRW(W_t;\\varsigma)=1\\);\n\\item cue-controlled goal retention exceeds \\(\\tau_G\\);\n\\item goal-to-action mediation exceeds \\(\\tau_{G\\to A}\\);\n\\item counterfactual-value error is at most \\(\\varepsilon_Q\\);\n\\item value-to-action use exceeds \\(\\tau_{V\\to A}\\);\n\\item action-to-outcome control exceeds \\(\\tau_{A\\to Y}\\);\n\\item damage detection, monitor-to-repair mediation, and restoration exceed\n their thresholds;\n\\item repair specificity exceeds \\(\\tau_{\\mathrm{spec}}\\);\n\\item \\(k_{\\mathrm{ctl},\\varepsilon_{\\mathrm{ctl}}}\\geq2\\); and\n\\item repeated maintenance deficit remains at most \\(e_{\\max}\\).\n\\end{enumerate}","d goal retention exceeds \\(\\tau_G\\);\n\\item goal-to-action mediation exceeds \\(\\tau_{G\\to A}\\);\n\\item counterfactual-value error is at most \\(\\varepsilon_Q\\);\n\\item value-to-action use exceeds \\(\\tau_{V\\to A}\\);\n\\item action-to-outcome control exceeds \\(\\tau_{A\\to Y}\\);\n\\item damage detection, monitor-to-repair mediation, and restoration exceed\n their thresholds;\n\\item repair specificity exceeds \\(\\tau_{\\mathrm{spec}}\\);\n\\item \\(k_{\\mathrm{ctl},\\varepsilon_{\\mathrm{ctl}}}\\geq2\\); and\n\\item repeated maintenance deficit remains at most \\(e_{\\max}\\).\n\\end{enumerate}","G\\);\n\\item goal-to-action mediation exceeds \\(\\tau_{G\\to A}\\);\n\\item counterfactual-value error is at most \\(\\varepsilon_Q\\);\n\\item value-to-action use exceeds \\(\\tau_{V\\to A}\\);\n\\item action-to-outcome control exceeds \\(\\tau_{A\\to Y}\\);\n\\item damage detection, monitor-to-repair mediation, and restoration exceed\n their thresholds;\n\\item repair specificity exceeds \\(\\tau_{\\mathrm{spec}}\\);\n\\item \\(k_{\\mathrm{ctl},\\varepsilon_{\\mathrm{ctl}}}\\geq2\\); and\n\\item repeated maintenance deficit remains at most \\(e_{\\max}\\).\n\\end{enumerate}","\\varepsilon_Q\\);\n\\item value-to-action use exceeds \\(\\tau_{V\\to A}\\);\n\\item action-to-outcome control exceeds \\(\\tau_{A\\to Y}\\);\n\\item damage detection, monitor-to-repair mediation, and restoration exceed\n their thresholds;\n\\item repair specificity exceeds \\(\\tau_{\\mathrm{spec}}\\);\n\\item \\(k_{\\mathrm{ctl},\\varepsilon_{\\mathrm{ctl}}}\\geq2\\); and\n\\item repeated maintenance deficit remains at most \\(e_{\\max}\\).\n\\end{enumerate}","\\subtopic{53.14 \\(T_{23}\\): Workspace-agency synthesis theorem }","r declared gates]\n\\label{thm:agency-synthesis}\nAssume a controlled realization satisfies:\n\\begin{enumerate}\n\\item the operational relationality assumptions of\n Theorem~\\ref{thm:operational-relationality};\n\\item cue-controlled goal retention and action readout as in\n Theorem~\\ref{thm:goal-retention} and\n Corollary~\\ref{cor:goal-action-readout};\n\\item a counterfactual evaluator with uniform error \\(\\varepsilon_Q\\) and\n optimization error \\(\\delta_{\\mathrm{opt}}\\);\n\\item feasible interventions establishing positive value-to-action and\n action-to-outcome effects;\n\\item the causal self-maintenance assumptions of\n Theorem~\\ref{thm:self-maintenance}; and\n\\item \\(k_{\\mathrm{ctl},\\varepsilon}\\geq2\\), together with a matched local\n policy-null panel that preserves every single-node predictive marginal\n but loses full-system control value.\n\\end{enumerate}\nThen, relative to the declared specification:\n\\begin{enumerate}\n\\item the expected one-step decision regret is at most\n \\(2\\varepsilon_Q+\\delta_{\\mathrm{opt}}\\);\n\\item the long-run maintenance deficit is bounded by\n \\[\n \\frac{\\overline d+\\over…","sis}\nAssume a controlled realization satisfies:\n\\begin{enumerate}\n\\item the operational relationality assumptions of\n Theorem~\\ref{thm:operational-relationality};\n\\item cue-controlled goal retention and action readout as in\n Theorem~\\ref{thm:goal-retention} and\n Corollary~\\ref{cor:goal-action-readout};\n\\item a counterfactual evaluator with uniform error \\(\\varepsilon_Q\\) and\n optimization error \\(\\delta_{\\mathrm{opt}}\\);\n\\item feasible interventions establishing positive value-to-action and\n action-to-outcome effects;\n\\item the causal self-maintenance assumptions of\n Theorem~\\ref{thm:self-maintenance}; and\n\\item \\(k_{\\mathrm{ctl},\\varepsilon}\\geq2\\), together with a matched local\n policy-null panel that preserves every single-node predictive marginal\n but loses full-system control value.\n\\end{enumerate}\nThen, relative to the declared specification:\n\\begin{enumerate}\n\\item the expected one-step decision regret is at most\n \\(2\\varepsilon_Q+\\delta_{\\mathrm{opt}}\\);\n\\item the long-run maintenance deficit is bounded by\n \\[\n \\frac{\\overline d+\\overline\\epsilon}{1-\\beta};\n \\]\n\\item no singl…","n\n Theorem~\\ref{thm:goal-retention} and\n Corollary~\\ref{cor:goal-action-readout};\n\\item a counterfactual evaluator with uniform error \\(\\varepsilon_Q\\) and\n optimization error \\(\\delta_{\\mathrm{opt}}\\);\n\\item feasible interventions establishing positive value-to-action and\n action-to-outcome effects;\n\\item the causal self-maintenance assumptions of\n Theorem~\\ref{thm:self-maintenance}; and\n\\item \\(k_{\\mathrm{ctl},\\varepsilon}\\geq2\\), together with a matched local\n policy-null panel that preserves every single-node predictive marginal\n but loses full-system control value.\n\\end{enumerate}\nThen, relative to the declared specification:\n\\begin{enumerate}\n\\item the expected one-step decision regret is at most\n \\(2\\varepsilon_Q+\\delta_{\\mathrm{opt}}\\);\n\\item the long-run maintenance deficit is bounded by\n \\[\n \\frac{\\overline d+\\overline\\epsilon}{1-\\beta};\n \\]\n\\item no single-node statistic is sufficient for the complete control state;\n\\item matched local policy nulls cannot reproduce the intact control value;\n\\item goal-dependent action selection and workspace repair are causally\n mediated b…","sal self-maintenance assumptions of\n Theorem~\\ref{thm:self-maintenance}; and\n\\item \\(k_{\\mathrm{ctl},\\varepsilon}\\geq2\\), together with a matched local\n policy-null panel that preserves every single-node predictive marginal\n but loses full-system control value.\n\\end{enumerate}\nThen, relative to the declared specification:\n\\begin{enumerate}\n\\item the expected one-step decision regret is at most\n \\(2\\varepsilon_Q+\\delta_{\\mathrm{opt}}\\);\n\\item the long-run maintenance deficit is bounded by\n \\[\n \\frac{\\overline d+\\overline\\epsilon}{1-\\beta};\n \\]\n\\item no single-node statistic is sufficient for the complete control state;\n\\item matched local policy nulls cannot reproduce the intact control value;\n\\item goal-dependent action selection and workspace repair are causally\n mediated by implemented internal channels.\n\\end{enumerate}\nHence the realization satisfies the formal predicate\n\\[\n\\operatorname{Agent}_{\\Theta_A}(\\Workspace)=1.\n\\]\n\\end{theorem}","ctl},\\varepsilon}\\geq2\\), together with a matched local\n policy-null panel that preserves every single-node predictive marginal\n but loses full-system control value.\n\\end{enumerate}\nThen, relative to the declared specification:\n\\begin{enumerate}\n\\item the expected one-step decision regret is at most\n \\(2\\varepsilon_Q+\\delta_{\\mathrm{opt}}\\);\n\\item the long-run maintenance deficit is bounded by\n \\[\n \\frac{\\overline d+\\overline\\epsilon}{1-\\beta};\n \\]\n\\item no single-node statistic is sufficient for the complete control state;\n\\item matched local policy nulls cannot reproduce the intact control value;\n\\item goal-dependent action selection and workspace repair are causally\n mediated by implemented internal channels.\n\\end{enumerate}\nHence the realization satisfies the formal predicate\n\\[\n\\operatorname{Agent}_{\\Theta_A}(\\Workspace)=1.\n\\]\n\\end{theorem}","by\n \\[\n \\frac{\\overline d+\\overline\\epsilon}{1-\\beta};\n \\]\n\\item no single-node statistic is sufficient for the complete control state;\n\\item matched local policy nulls cannot reproduce the intact control value;\n\\item goal-dependent action selection and workspace repair are causally\n mediated by implemented internal channels.\n\\end{enumerate}\nHence the realization satisfies the formal predicate\n\\[\n\\operatorname{Agent}_{\\Theta_A}(\\Workspace)=1.\n\\]\n\\end{theorem}","The following implications are not licensed:\n\\[\n\\operatorname{Agent}_{\\Theta_A}(\\Workspace)\n\\centernot\\Longrightarrow\n\\operatorname{PhenomenallyConscious}(\\Workspace),\n\\]\n\\[\n\\operatorname{SelfMaintaining}(\\Workspace)\n\\centernot\\Longrightarrow\n\\operatorname{SelfAware}(\\Workspace),\n\\]\n\\[\n\\operatorname{GoalPersistent}(\\Workspace)\n\\centernot\\Longrightarrow\n\\operatorname{MoralPatient}(\\Workspace).\n\\]\nA thermostat-like controller can maintain a variable without representing\nitself. A sophisticated planner can evaluate counterfactual actions without\nphenomenal experience. Conversely, failure to pass this engineering\ncertificate would not prove absence of consciousnes","The following implications are not licensed:\n\\[\n\\operatorname{Agent}_{\\Theta_A}(\\Workspace)\n\\centernot\\Longrightarrow\n\\operatorname{PhenomenallyConscious}(\\Workspace),\n\\]\n\\[\n\\operatorname{SelfMaintaining}(\\Workspace)\n\\centernot\\Longrightarrow\n\\operatorname{SelfAware}(\\Workspace),\n\\]\n\\[\n\\operatorname{GoalPersistent}(\\Workspace)\n\\centernot\\Longrightarrow\n\\operatorname{MoralPatient}(\\Workspace).\n\\]\nA thermostat-like controller can maintain a variable without representing\nitself. A sophisticated planner can evaluate counterfactual actions without\nphenomenal experience. Conversely, failure to pass this engineering\ncertificate would not prove absence of consciousness. Agency and\nconsciousness remain distinct axes of inquiry.","Let \\(\\Omega\\) be a finite set of primitive carriers at a declared audit\nresolution. A primitive carrier may be a person, a model instance, a process,\na memory store, an instrument, or another independently tracked mechanism.\nThe word \\emph{primitive} is relative to the audit. It does not assert a final\nmetaphysical atomization of the system.","\\begin{definition}[System decomposition]\\label{def:system-decomposition}\nA decomposition of a carrier presentation is a tuple\n\\[\n\\mathcal D\n=\n\\bigl(\n\\Pi,\\mathbf O,G_{\\Pi},\\mathcal I_{\\Pi}\n\\bigr),\n\\]\nwhere:\n\\begin{itemize}\n\\item \\(\\Pi=\\{B_\\alpha\\}_{\\alpha\\in A}\\) is a partition of \\(\\Omega\\);\n\\item \\(O_\\alpha\\) is the declared local-access map for block \\(B_\\alpha\\),\n producing the local state or history available at that component;\n\\item \\(G_{\\Pi}\\) records the declared communication, transfer, and causal\n interfaces among the blocks; and\n\\item \\(\\mathcal I_{\\Pi}\\) is the family of feasible interventions at that\n resolution.\n\\end{itemize}\nThe accessible information of component \\(\\alpha\\) is the sigma-field\n\\[\n\\mathscr F_\\alpha^{\\mathcal D}\n=\n\\sigma\\!\\left(O_\\alpha(\\mathbf H_t)\\right).\n\\]\nEvery shared bus, common memory, supervisor, or artifact that can alter more\nthan one local stat","\\begin{definition}[System decomposition]\\label{def:system-decomposition}\nA decomposition of a carrier presentation is a tuple\n\\[\n\\mathcal D\n=\n\\bigl(\n\\Pi,\\mathbf O,G_{\\Pi},\\mathcal I_{\\Pi}\n\\bigr),\n\\]\nwhere:\n\\begin{itemize}\n\\item \\(\\Pi=\\{B_\\alpha\\}_{\\alpha\\in A}\\) is a partition of \\(\\Omega\\);\n\\item \\(O_\\alpha\\) is the declared local-access map for block \\(B_\\alpha\\),\n producing the local state or history available at that component;\n\\item \\(G_{\\Pi}\\) records the declared communication, transfer, and causal\n interfaces among the blocks; and\n\\item \\(\\mathcal I_{\\Pi}\\) is the family of feasible interventions at that\n resolution.\n\\end{itemize}\nThe accessible information of component \\(\\alpha\\) is the sigma-field\n\\[\n\\mathscr F_\\alpha^{\\mathcal D}\n=\n\\sigma\\!\\left(O_\\alpha(\\mathbf H_t)\\right).\n\\]\nEvery shared bus, common memory, supervisor, or artifact that can alter more\nthan one local state must appear either as its own carrier or as an explicit\nchanne","\\begin{definition}[System decomposition]\\label{def:system-decomposition}\nA decomposition of a carrier presentation is a tuple\n\\[\n\\mathcal D\n=\n\\bigl(\n\\Pi,\\mathbf O,G_{\\Pi},\\mathcal I_{\\Pi}\n\\bigr),\n\\]\nwhere:\n\\begin{itemize}\n\\item \\(\\Pi=\\{B_\\alpha\\}_{\\alpha\\in A}\\) is a partition of \\(\\Omega\\);\n\\item \\(O_\\alpha\\) is the declared local-access map for block \\(B_\\alpha\\),\n producing the local state or history available at that component;\n\\item \\(G_{\\Pi}\\) records the declared communication, transfer, and causal\n interfaces among the blocks; and\n\\item \\(\\mathcal I_{\\Pi}\\) is the family of feasible interventions at that\n resolution.\n\\end{itemize}\nThe accessible information of component \\(\\alpha\\) is the sigma-field\n\\[\n\\mathscr F_\\alpha^{\\mathcal D}\n=\n\\sigma\\!\\left(O_\\alpha(\\mathbf H_t)\\right).\n\\]\nEvery shared bus, common memory, supervisor, or artifact that can alter more\nthan one local state must appear either as its own carrier or as an explicit\nchannel in \\(G_{\\Pi}\\).\n\\end{defini","\\begin{definition}[System decomposition]\\label{def:system-decomposition}\nA decomposition of a carrier presentation is a tuple\n\\[\n\\mathcal D\n=\n\\bigl(\n\\Pi,\\mathbf O,G_{\\Pi},\\mathcal I_{\\Pi}\n\\bigr),\n\\]\nwhere:\n\\begin{itemize}\n\\item \\(\\Pi=\\{B_\\alpha\\}_{\\alpha\\in A}\\) is a partition of \\(\\Omega\\);\n\\item \\(O_\\alpha\\) is the declared local-access map for block \\(B_\\alpha\\),\n producing the local state or history available at that component;\n\\item \\(G_{\\Pi}\\) records the declared communication, transfer, and causal\n interfaces among the blocks; and\n\\item \\(\\mathcal I_{\\Pi}\\) is the family of feasible interventions at that\n resolution.\n\\end{itemize}\nThe accessible information of component \\(\\alpha\\) is the sigma-field\n\\[\n\\mathscr F_\\alpha^{\\mathcal D}\n=\n\\sigma\\!\\left(O_\\alpha(\\mathbf H_t)\\right).\n\\]\nEvery shared bus, common memory, supervisor, or artifact that can alter more\nthan one local state must appear either as its own carrier or as an explicit\nchannel in \\(G_{\\Pi}\\).\n\\end{definition}","el{def:system-decomposition}\nA decomposition of a carrier presentation is a tuple\n\\[\n\\mathcal D\n=\n\\bigl(\n\\Pi,\\mathbf O,G_{\\Pi},\\mathcal I_{\\Pi}\n\\bigr),\n\\]\nwhere:\n\\begin{itemize}\n\\item \\(\\Pi=\\{B_\\alpha\\}_{\\alpha\\in A}\\) is a partition of \\(\\Omega\\);\n\\item \\(O_\\alpha\\) is the declared local-access map for block \\(B_\\alpha\\),\n producing the local state or history available at that component;\n\\item \\(G_{\\Pi}\\) records the declared communication, transfer, and causal\n interfaces among the blocks; and\n\\item \\(\\mathcal I_{\\Pi}\\) is the family of feasible interventions at that\n resolution.\n\\end{itemize}\nThe accessible information of component \\(\\alpha\\) is the sigma-field\n\\[\n\\mathscr F_\\alpha^{\\mathcal D}\n=\n\\sigma\\!\\left(O_\\alpha(\\mathbf H_t)\\right).\n\\]\nEvery shared bus, common memory, supervisor, or artifact that can alter more\nthan one local state must appear either as its own carrier or as an explicit\nchannel in \\(G_{\\Pi}\\).\n\\end{definition}","\\mathbf O,G_{\\Pi},\\mathcal I_{\\Pi}\n\\bigr),\n\\]\nwhere:\n\\begin{itemize}\n\\item \\(\\Pi=\\{B_\\alpha\\}_{\\alpha\\in A}\\) is a partition of \\(\\Omega\\);\n\\item \\(O_\\alpha\\) is the declared local-access map for block \\(B_\\alpha\\),\n producing the local state or history available at that component;\n\\item \\(G_{\\Pi}\\) records the declared communication, transfer, and causal\n interfaces among the blocks; and\n\\item \\(\\mathcal I_{\\Pi}\\) is the family of feasible interventions at that\n resolution.\n\\end{itemize}\nThe accessible information of component \\(\\alpha\\) is the sigma-field\n\\[\n\\mathscr F_\\alpha^{\\mathcal D}\n=\n\\sigma\\!\\left(O_\\alpha(\\mathbf H_t)\\right).\n\\]\nEvery shared bus, common memory, supervisor, or artifact that can alter more\nthan one local state must appear either as its own carrier or as an explicit\nchannel in \\(G_{\\Pi}\\).\n\\end{definition}","\\);\n\\item \\(O_\\alpha\\) is the declared local-access map for block \\(B_\\alpha\\),\n producing the local state or history available at that component;\n\\item \\(G_{\\Pi}\\) records the declared communication, transfer, and causal\n interfaces among the blocks; and\n\\item \\(\\mathcal I_{\\Pi}\\) is the family of feasible interventions at that\n resolution.\n\\end{itemize}\nThe accessible information of component \\(\\alpha\\) is the sigma-field\n\\[\n\\mathscr F_\\alpha^{\\mathcal D}\n=\n\\sigma\\!\\left(O_\\alpha(\\mathbf H_t)\\right).\n\\]\nEvery shared bus, common memory, supervisor, or artifact that can alter more\nthan one local state must appear either as its own carrier or as an explicit\nchannel in \\(G_{\\Pi}\\).\n\\end{definition}","declared local-access map for block \\(B_\\alpha\\),\n producing the local state or history available at that component;\n\\item \\(G_{\\Pi}\\) records the declared communication, transfer, and causal\n interfaces among the blocks; and\n\\item \\(\\mathcal I_{\\Pi}\\) is the family of feasible interventions at that\n resolution.\n\\end{itemize}\nThe accessible information of component \\(\\alpha\\) is the sigma-field\n\\[\n\\mathscr F_\\alpha^{\\mathcal D}\n=\n\\sigma\\!\\left(O_\\alpha(\\mathbf H_t)\\right).\n\\]\nEvery shared bus, common memory, supervisor, or artifact that can alter more\nthan one local state must appear either as its own carrier or as an explicit\nchannel in \\(G_{\\Pi}\\).\n\\end{definition}","i}\\) is the family of feasible interventions at that\n resolution.\n\\end{itemize}\nThe accessible information of component \\(\\alpha\\) is the sigma-field\n\\[\n\\mathscr F_\\alpha^{\\mathcal D}\n=\n\\sigma\\!\\left(O_\\alpha(\\mathbf H_t)\\right).\n\\]\nEvery shared bus, common memory, supervisor, or artifact that can alter more\nthan one local state must appear either as its own carrier or as an explicit\nchannel in \\(G_{\\Pi}\\).\n\\end{definition}","The same physical realization can support several decompositions. One may\nseparate a model from its executor, merge them into one computational unit, or\ntreat a persistent artifact as its own component. These alternatives are not\nmere renamings when they change local access or feasible interventions. Every certificate should therefore display the full specification\n\\[\n\\SRW(W_t;\\varsigma),\n\\qquad\n\\varsigma=(\\mathcal D,Y^+,\\ell,\\tau,\\mathcal I,\\mathcal N,\\Theta).\n\\]\nThis part varies the decomposition coordinate while holding the remaining\ncoordinates fixed. Part~XVIII treats the full specification family.","\\begin{definition}[Refinement and coarsening]\\label{def:decomposition-refinement}\nLet \\(\\mathcal D_f\\) and \\(\\mathcal D_c\\) be decompositions of the same carrier\npresentation. The fine decomposition refines the coarse decomposition,\nwritten\n\\[\n\\mathcal D_f\\preceq\\mathcal D_c,\n\\]\nwhen there is a surjection\n\\[\nq:V_f\\longrightarrow V_c\n\\]\nsuch that each coarse component is the union of the fine components in one\nfiber of \\(q\\), and its accessible information is generated only from those\nfine components. The aggregation is \\emph{information faithful} when\n\\[\n\\mathscr F_B^{\\mathcal D_c}\n=\n\\bigvee_{i\\in q^{-1}(B)}\n\\mathscr F_i^{\\mathcal D_f}\n\\qquad(B\\in V_c).\n\\]\nIt is \\emph{structurally faithful} when it also retains every internal\ncompatibility condition and every boundary transfer condition ne","\\begin{definition}[Refinement and coarsening]\\label{def:decomposition-refinement}\nLet \\(\\mathcal D_f\\) and \\(\\mathcal D_c\\) be decompositions of the same carrier\npresentation. The fine decomposition refines the coarse decomposition,\nwritten\n\\[\n\\mathcal D_f\\preceq\\mathcal D_c,\n\\]\nwhen there is a surjection\n\\[\nq:V_f\\longrightarrow V_c\n\\]\nsuch that each coarse component is the union of the fine components in one\nfiber of \\(q\\), and its accessible information is generated only from those\nfine components. The aggregation is \\emph{information faithful} when\n\\[\n\\mathscr F_B^{\\mathcal D_c}\n=\n\\bigvee_{i\\in q^{-1}(B)}\n\\mathscr F_i^{\\mathcal D_f}\n\\qquad(B\\in V_c).\n\\]\nIt is \\emph{structurally faithful} when it also retains every internal\ncompatibility condition and every boundary transfer condition needed to\nrecover the o","\\begin{definition}[Refinement and coarsening]\\label{def:decomposition-refinement}\nLet \\(\\mathcal D_f\\) and \\(\\mathcal D_c\\) be decompositions of the same carrier\npresentation. The fine decomposition refines the coarse decomposition,\nwritten\n\\[\n\\mathcal D_f\\preceq\\mathcal D_c,\n\\]\nwhen there is a surjection\n\\[\nq:V_f\\longrightarrow V_c\n\\]\nsuch that each coarse component is the union of the fine components in one\nfiber of \\(q\\), and its accessible information is generated only from those\nfine components. The aggregation is \\emph{information faithful} when\n\\[\n\\mathscr F_B^{\\mathcal D_c}\n=\n\\bigvee_{i\\in q^{-1}(B)}\n\\mathscr F_i^{\\mathcal D_f}\n\\qquad(B\\in V_c).\n\\]\nIt is \\emph{structurally faithful} when it also retains every internal\ncompatibility condition and every boundary transfer condition needed to\nrecover the original global-section problem.\n\\end{definition}","ement and coarsening]\\label{def:decomposition-refinement}\nLet \\(\\mathcal D_f\\) and \\(\\mathcal D_c\\) be decompositions of the same carrier\npresentation. The fine decomposition refines the coarse decomposition,\nwritten\n\\[\n\\mathcal D_f\\preceq\\mathcal D_c,\n\\]\nwhen there is a surjection\n\\[\nq:V_f\\longrightarrow V_c\n\\]\nsuch that each coarse component is the union of the fine components in one\nfiber of \\(q\\), and its accessible information is generated only from those\nfine components. The aggregation is \\emph{information faithful} when\n\\[\n\\mathscr F_B^{\\mathcal D_c}\n=\n\\bigvee_{i\\in q^{-1}(B)}\n\\mathscr F_i^{\\mathcal D_f}\n\\qquad(B\\in V_c).\n\\]\nIt is \\emph{structurally faithful} when it also retains every internal\ncompatibility condition and every boundary transfer condition needed to\nrecover the original global-section problem.\n\\end{definition}","r\npresentation. The fine decomposition refines the coarse decomposition,\nwritten\n\\[\n\\mathcal D_f\\preceq\\mathcal D_c,\n\\]\nwhen there is a surjection\n\\[\nq:V_f\\longrightarrow V_c\n\\]\nsuch that each coarse component is the union of the fine components in one\nfiber of \\(q\\), and its accessible information is generated only from those\nfine components. The aggregation is \\emph{information faithful} when\n\\[\n\\mathscr F_B^{\\mathcal D_c}\n=\n\\bigvee_{i\\in q^{-1}(B)}\n\\mathscr F_i^{\\mathcal D_f}\n\\qquad(B\\in V_c).\n\\]\nIt is \\emph{structurally faithful} when it also retains every internal\ncompatibility condition and every boundary transfer condition needed to\nrecover the original global-section problem.\n\\end{definition}","Let the vertex set of a core-transfer diagram be partitioned as\n\\[\nV=V_1\\sqcup\\cdots\\sqcup V_m.\n\\]\nWrite the transfer discrepancy as\n\\[\nD_H\n=\n\\begin{pmatrix}\nD_{\\mathrm{int}}","Let the vertex set of a core-transfer diagram be partitioned as\n\\[\nV=V_1\\sqcup\\cdots\\sqcup V_m.\n\\]\nWrite the transfer discrepancy as\n\\[\nD_H\n=\n\\begin{pmatrix}\nD_{\\mathrm{int}}\\\\\nD_{\\partial}\n\\end{pmatrix},\n\\qquad\nD_{\\mathrm{int}}\n=\n\\bigoplus_{\\alpha=1}^{m}D_\\alpha,\n\\]\nwhere \\(D_\\alpha\\) contains the edge constraints internal to \\(V_\\alpha\\),\nand \\(D_{\\partial}\\) contains the constraints crossing between blocks. Define\nthe internally coherent macro-state spaces\n\\[\nW_\\alpha=\\ker D_\\alpha,\n\\qquad\nW_{\\Pi}=\\bigoplus_{\\alpha=1}^{m}W_\\alpha.\n\\]\nLet\n\\[\nJ_\\Pi:W_\\Pi\\hookrightarrow\\mathcal C_V\n\\]\nbe the canonical isometric inclusion, and define the quotient discrepancy\n\\[\nD_\\Pi=D_{\\partial}J_\\Pi.\n\\]\nThe quotient does not replace a group of nodes by an arbitrary summary. Its\nmacro-state is the complete internal section space of that group, and its\nboundary operator retains all cross-block compatibility equations.","D_{\\partial}\n\\end{pmatrix},\n\\qquad\nD_{\\mathrm{int}}\n=\n\\bigoplus_{\\alpha=1}^{m}D_\\alpha,\n\\]\nwhere \\(D_\\alpha\\) contains the edge constraints internal to \\(V_\\alpha\\),\nand \\(D_{\\partial}\\) contains the constraints crossing between blocks. Define\nthe internally coherent macro-state spaces\n\\[\nW_\\alpha=\\ker D_\\alpha,\n\\qquad\nW_{\\Pi}=\\bigoplus_{\\alpha=1}^{m}W_\\alpha.\n\\]\nLet\n\\[\nJ_\\Pi:W_\\Pi\\hookrightarrow\\mathcal C_V\n\\]\nbe the canonical isometric inclusion, and define the quotient discrepancy\n\\[\nD_\\Pi=D_{\\partial}J_\\Pi.\n\\]\nThe quotient does not replace a group of nodes by an arbitrary summary. Its\nmacro-state is the complete internal section space of that group, and its\nboundary operator retains all cross-block compatibility equations.","where \\(D_\\alpha\\) contains the edge constraints internal to \\(V_\\alpha\\),\nand \\(D_{\\partial}\\) contains the constraints crossing between blocks. Define\nthe internally coherent macro-state spaces\n\\[\nW_\\alpha=\\ker D_\\alpha,\n\\qquad\nW_{\\Pi}=\\bigoplus_{\\alpha=1}^{m}W_\\alpha.\n\\]\nLet\n\\[\nJ_\\Pi:W_\\Pi\\hookrightarrow\\mathcal C_V\n\\]\nbe the canonical isometric inclusion, and define the quotient discrepancy\n\\[\nD_\\Pi=D_{\\partial}J_\\Pi.\n\\]\nThe quotient does not replace a group of nodes by an arbitrary summary. Its\nmacro-state is the complete internal section space of that group, and its\nboundary operator retains all cross-block compatibility equations.","\\begin{theorem}[Faithful coarse-graining]\\label{thm:faithful-coarse-graining}\nFor every finite partition of the vertex set,\n\\[\n\\boxed{\n\\ker D_H\n=\nJ_\\Pi\\ker D_\\Pi.\n}\n\\]\nEquivalently,\n\\[\n\\boxed{\n\\Gamma(H)\n\\cong\n\\ker D_\\Pi\n=\n\\ker(D_\\Pi^*D_\\Pi).\n}\n\\]\nIn particular,\n\\[\n\\dim\\Gamma(H)\n=\n\\dim W_\\Pi-\\operatorname{rank}(D_\\Pi).\n\\]\nThus faithful coarse-graining preserves the exact structural workspace even\nthough it changes the number and interpretation of the displayed nodes.\n\\end{theorem}","\\begin{proof}\nA vector \\(c\\in\\mathcal C_V\\) lies in \\(\\ker D_H\\) exactly when\n\\[\nD_{\\mathrm{int}}c=0\n\\qquad\\text{and}\\qquad\nD_{\\partial}c=0.\n\\]\nThe first condition is equivalent to \\(c=J_\\Pi w\\) for a unique\n\\(w\\in W_\\Pi\\). Substitution into the second condition gives\n\\[\nD_\\Pi w\n=\nD_{\\partial}J_\\Pi w\n=0.\n\\]\nHence \\(c\\in\\ker D_H\\) if and only if\n\\(c=J_\\Pi w\\) with \\(w\\in\\ker D_\\Pi\\). The Laplacian identity follows from\n\\(\\ker(D_\\Pi^*D_\\Pi)=\\ker D_\\Pi\\), and the dimension formula is rank-nullity.\n\\end{proof}","(c\\in\\mathcal C_V\\) lies in \\(\\ker D_H\\) exactly when\n\\[\nD_{\\mathrm{int}}c=0\n\\qquad\\text{and}\\qquad\nD_{\\partial}c=0.\n\\]\nThe first condition is equivalent to \\(c=J_\\Pi w\\) for a unique\n\\(w\\in W_\\Pi\\). Substitution into the second condition gives\n\\[\nD_\\Pi w\n=\nD_{\\partial}J_\\Pi w\n=0.\n\\]\nHence \\(c\\in\\ker D_H\\) if and only if\n\\(c=J_\\Pi w\\) with \\(w\\in\\ker D_\\Pi\\). The Laplacian identity follows from\n\\(\\ker(D_\\Pi^*D_\\Pi)=\\ker D_\\Pi\\), and the dimension formula is rank-nullity.\n\\end{proof}","\\begin{corollary}[Two-scale coherence bound]\\label{cor:two-scale-decomposition}\nLet \\(P_\\Pi\\) be the orthogonal projector onto \\(J_\\Pi W_\\Pi\\). Let\n\\(\\gamma_{\\mathrm{int}}>0\\) be the smallest positive eigenvalue of\n\\(D_{\\mathrm{int}}^*D_{\\mathrm{int}}\\), and let\n\\(\\gamma_\\Pi>0\\) be the smallest positive eigenvalue of\n\\(D_\\Pi^*D_\\Pi\\). Then every \\(c\\in\\mathcal C_V\\) satisfies\n\\[\n\\boxed{\n\\operatorname{dist}(c,\\Gamma(H))\n\\leq\n\\frac{\\|D_{\\mathrm{int}}c\\|}{\\sqrt{\\gamma_{\\mathrm{int}}}}\n+\n\\frac{\n\\|D_\\Pi J_\\Pi^*P_\\Pi c\\|\n}{\\sqrt{\\gamma_\\Pi}}.\n}\n\\]\nThe first term measures incoherence within macro-components. The second\nmeasures incompatibility among their internally coherent states.\n\\end{corollary}","\\begin{proof}\nBy the triangle inequality,\n\\[\n\\operatorname{dist}(c,\\Gamma(H))\n\\leq\n\\|c-P_\\Pi c\\|\n+\n\\operatorname{dist}(P_\\Pi c,\\Gamma(H)).\n\\]\nThe workspace Poincar\\'e estimate applied to \\(D_{\\mathrm{int}}\\) bounds the\nfirst term. Write \\(P_\\Pi c=J_\\Pi w\\). Theorem~\\ref{thm:faithful-coarse-graining}\nand the corresponding estimate for \\(D_\\Pi\\) bound the second term by\n\\(\\|D_\\Pi w\\|/\\sqrt{\\gamma_\\Pi}\\). Since \\(J_\\Pi\\) is isometric,\n\\(w=J_\\Pi^*P_\\Pi c\\).\n\\end{proof}","By the triangle inequality,\n\\[\n\\operatorname{dist}(c,\\Gamma(H))\n\\leq\n\\|c-P_\\Pi c\\|\n+\n\\operatorname{dist}(P_\\Pi c,\\Gamma(H)).\n\\]\nThe workspace Poincar\\'e estimate applied to \\(D_{\\mathrm{int}}\\) bounds the\nfirst term. Write \\(P_\\Pi c=J_\\Pi w\\). Theorem~\\ref{thm:faithful-coarse-graining}\nand the corresponding estimate for \\(D_\\Pi\\) bound the second term by\n\\(\\|D_\\Pi w\\|/\\sqrt{\\gamma_\\Pi}\\). Since \\(J_\\Pi\\) is isometric,\n\\(w=J_\\Pi^*P_\\Pi c\\).\n\\end{proof}","ty,\n\\[\n\\operatorname{dist}(c,\\Gamma(H))\n\\leq\n\\|c-P_\\Pi c\\|\n+\n\\operatorname{dist}(P_\\Pi c,\\Gamma(H)).\n\\]\nThe workspace Poincar\\'e estimate applied to \\(D_{\\mathrm{int}}\\) bounds the\nfirst term. Write \\(P_\\Pi c=J_\\Pi w\\). Theorem~\\ref{thm:faithful-coarse-graining}\nand the corresponding estimate for \\(D_\\Pi\\) bound the second term by\n\\(\\|D_\\Pi w\\|/\\sqrt{\\gamma_\\Pi}\\). Since \\(J_\\Pi\\) is isometric,\n\\(w=J_\\Pi^*P_\\Pi c\\).\n\\end{proof}","The structural core can be preserved by faithful coarse-graining, while\ncoalition counts generally are not. Let\n\\[\n\\mathscr F_S^{\\mathcal D}\n=\n\\bigvee_{i\\in S}\\mathscr F_i^{\\mathcal D}\n\\]\nbe the information available to coalition \\(S\\) under decomposition\n\\(\\mathcal D\\). For a declared loss \\(\\ell\\), write\n\\[\n\\Risk_{\\mathcal D}(S)\n=\n\\inf_f\n\\mathbb E\\!\\left[\n\\ell\\bigl(Y^+,f(\\mathscr F_S^{\\mathcal D})\\bigr)\n\\right].\n\\]","\\begin{theorem}[Bayes-risk transport under faithful aggregation]\n\\label{thm:risk-aggregation}\nSuppose \\(\\mathcal D_f\\preceq\\mathcal D_c\\) through\n\\(q:V_f\\to V_c\\), and suppose the aggregation is information faithful. Then\nfor every coarse coalition \\(S\\subseteq V_c\\),\n\\[\n\\boxed{\n\\Risk_{\\mathcal D_c}(S)\n=\n\\Risk_{\\mathcal D_f}\\bigl(q^{-1}(S)\\bigr).\n}\n\\]\nConsequently,\n\\[\nk_\\varepsilon(\\mathcal D_c)\n\\leq\nk_\\varepsilon(\\mathcal D_f),\n\\]\nwhere coalition size is counted in displayed nodes.\n\\end{theorem}","\\begin{proof}\nInformation faithfulness gives\n\\[\n\\mathscr F_S^{\\mathcal D_c}\n=\n\\bigvee_{B\\in S}\n\\bigvee_{i\\in q^{-1}(B)}\n\\mathscr F_i^{\\mathcal D_f}\n=\n\\mathscr F_{q^{-1}(S)}^{\\mathcal D_f}.\n\\]\nThe two Bayes risks optimize over the same measurable decision rules and are\ntherefore equal. If a fine coalition of size \\(k\\) is sufficient within\n\\(\\varepsilon\\), the set of coarse blocks meeting that coalition has at most\n\\(k\\) elements and contains at least the same information. This proves the\ncoalition-size inequality.\n\\end{proof}","ation faithfulness gives\n\\[\n\\mathscr F_S^{\\mathcal D_c}\n=\n\\bigvee_{B\\in S}\n\\bigvee_{i\\in q^{-1}(B)}\n\\mathscr F_i^{\\mathcal D_f}\n=\n\\mathscr F_{q^{-1}(S)}^{\\mathcal D_f}.\n\\]\nThe two Bayes risks optimize over the same measurable decision rules and are\ntherefore equal. If a fine coalition of size \\(k\\) is sufficient within\n\\(\\varepsilon\\), the set of coarse blocks meeting that coalition has at most\n\\(k\\) elements and contains at least the same information. This proves the\ncoalition-size inequality.\n\\end{proof}","A more stable quantity measures primitive carrier cost. Let\n\\(w_\\omega>0\\) be a declared cost for each \\(\\omega\\in\\Omega\\), and assign a\nblock the cost\n\\[\nc(B)=\\sum_{\\omega\\in B}w_\\omega.\n\\]\nDefine the sufficient support cost\n\\[\n\\boxed{\ns_\\varepsilon(\\mathcal D)\n=\n\\min_{S\\subseteq V_{\\mathcal D}}\n\\left\\{\n\\sum_{B\\in S}c(B):\n\\Risk_{\\mathcal D}(S)\n\\leq\n\\Risk_{\\mathcal D}(V_{\\mathcal D})+\\varepsilon\n\\right\\}.\n}\n\\]\nIf \\(\\mathcal D_f\\preceq\\mathcal D_c\\) is information faithful, then\n\\[\n\\boxed{\ns_\\varepsilon(\\mathcal D_f)\n\\leq\ns_\\varepsilon(\\mathcal D_c).}\n\\]\nThe fine decomposition permits every union of coarse blocks and may also\npermit cheaper sub-block coalitions. Equality holds whenever an optimal fine\ncoalition is a union of complete coarse fibers. This weighted","A more stable quantity measures primitive carrier cost. Let\n\\(w_\\omega>0\\) be a declared cost for each \\(\\omega\\in\\Omega\\), and assign a\nblock the cost\n\\[\nc(B)=\\sum_{\\omega\\in B}w_\\omega.\n\\]\nDefine the sufficient support cost\n\\[\n\\boxed{\ns_\\varepsilon(\\mathcal D)\n=\n\\min_{S\\subseteq V_{\\mathcal D}}\n\\left\\{\n\\sum_{B\\in S}c(B):\n\\Risk_{\\mathcal D}(S)\n\\leq\n\\Risk_{\\mathcal D}(V_{\\mathcal D})+\\varepsilon\n\\right\\}.\n}\n\\]\nIf \\(\\mathcal D_f\\preceq\\mathcal D_c\\) is information faithful, then\n\\[\n\\boxed{\ns_\\varepsilon(\\mathcal D_f)\n\\leq\ns_\\varepsilon(\\mathcal D_c).}\n\\]\nThe fine decomposition permits every union of coarse blocks and may also\npermit cheaper sub-block coalitions. Equality holds whenever an optimal fine\ncoalition is a union of complete coarse fibers. This weighted support measure\ndoes not remove the need to choo","A more stable quantity measures primitive carrier cost. Let\n\\(w_\\omega>0\\) be a declared cost for each \\(\\omega\\in\\Omega\\), and assign a\nblock the cost\n\\[\nc(B)=\\sum_{\\omega\\in B}w_\\omega.\n\\]\nDefine the sufficient support cost\n\\[\n\\boxed{\ns_\\varepsilon(\\mathcal D)\n=\n\\min_{S\\subseteq V_{\\mathcal D}}\n\\left\\{\n\\sum_{B\\in S}c(B):\n\\Risk_{\\mathcal D}(S)\n\\leq\n\\Risk_{\\mathcal D}(V_{\\mathcal D})+\\varepsilon\n\\right\\}.\n}\n\\]\nIf \\(\\mathcal D_f\\preceq\\mathcal D_c\\) is information faithful, then\n\\[\n\\boxed{\ns_\\varepsilon(\\mathcal D_f)\n\\leq\ns_\\varepsilon(\\mathcal D_c).}\n\\]\nThe fine decomposition permits every union of coarse blocks and may also\npermit cheaper sub-block coalitions. Equality holds whenever an optimal fine\ncoalition is a union of complete coarse fibers. This weighted support measure\ndoes not remove the need to choose primitive carriers, but it prevents a\nlarge macro-component from","A more stable quantity measures primitive carrier cost. Let\n\\(w_\\omega>0\\) be a declared cost for each \\(\\omega\\in\\Omega\\), and assign a\nblock the cost\n\\[\nc(B)=\\sum_{\\omega\\in B}w_\\omega.\n\\]\nDefine the sufficient support cost\n\\[\n\\boxed{\ns_\\varepsilon(\\mathcal D)\n=\n\\min_{S\\subseteq V_{\\mathcal D}}\n\\left\\{\n\\sum_{B\\in S}c(B):\n\\Risk_{\\mathcal D}(S)\n\\leq\n\\Risk_{\\mathcal D}(V_{\\mathcal D})+\\varepsilon\n\\right\\}.\n}\n\\]\nIf \\(\\mathcal D_f\\preceq\\mathcal D_c\\) is information faithful, then\n\\[\n\\boxed{\ns_\\varepsilon(\\mathcal D_f)\n\\leq\ns_\\varepsilon(\\mathcal D_c).}\n\\]\nThe fine decomposition permits every union of coarse blocks and may also\npermit cheaper sub-block coalitions. Equality holds whenever an optimal fine\ncoalition is a union of complete coarse fibers. This weighted support measure\ndoes not remove the need to choose primitive carriers, but it prevents a\nlarge macro-component from being counted as a single inexpensive unit merely\nbecause it was assigned one node label.","ity measures primitive carrier cost. Let\n\\(w_\\omega>0\\) be a declared cost for each \\(\\omega\\in\\Omega\\), and assign a\nblock the cost\n\\[\nc(B)=\\sum_{\\omega\\in B}w_\\omega.\n\\]\nDefine the sufficient support cost\n\\[\n\\boxed{\ns_\\varepsilon(\\mathcal D)\n=\n\\min_{S\\subseteq V_{\\mathcal D}}\n\\left\\{\n\\sum_{B\\in S}c(B):\n\\Risk_{\\mathcal D}(S)\n\\leq\n\\Risk_{\\mathcal D}(V_{\\mathcal D})+\\varepsilon\n\\right\\}.\n}\n\\]\nIf \\(\\mathcal D_f\\preceq\\mathcal D_c\\) is information faithful, then\n\\[\n\\boxed{\ns_\\varepsilon(\\mathcal D_f)\n\\leq\ns_\\varepsilon(\\mathcal D_c).}\n\\]\nThe fine decomposition permits every union of coarse blocks and may also\npermit cheaper sub-block coalitions. Equality holds whenever an optimal fine\ncoalition is a union of complete coarse fibers. This weighted support measure\ndoes not remove the need to choose primitive carriers, but it prevents a\nlarge macro-component from being counted as a single inexpensive unit merely\nbecause it was assigned one node label.","red cost for each \\(\\omega\\in\\Omega\\), and assign a\nblock the cost\n\\[\nc(B)=\\sum_{\\omega\\in B}w_\\omega.\n\\]\nDefine the sufficient support cost\n\\[\n\\boxed{\ns_\\varepsilon(\\mathcal D)\n=\n\\min_{S\\subseteq V_{\\mathcal D}}\n\\left\\{\n\\sum_{B\\in S}c(B):\n\\Risk_{\\mathcal D}(S)\n\\leq\n\\Risk_{\\mathcal D}(V_{\\mathcal D})+\\varepsilon\n\\right\\}.\n}\n\\]\nIf \\(\\mathcal D_f\\preceq\\mathcal D_c\\) is information faithful, then\n\\[\n\\boxed{\ns_\\varepsilon(\\mathcal D_f)\n\\leq\ns_\\varepsilon(\\mathcal D_c).}\n\\]\nThe fine decomposition permits every union of coarse blocks and may also\npermit cheaper sub-block coalitions. Equality holds whenever an optimal fine\ncoalition is a union of complete coarse fibers. This weighted support measure\ndoes not remove the need to choose primitive carriers, but it prevents a\nlarge macro-component from being counted as a single inexpensive unit merely\nbecause it was assigned one node label.","\\begin{proposition}[Coarsening and secret-sharing trivializations]\n\\label{prop:decomposition-trivialization}\nThe following statements hold.\n\\begin{enumerate}\n\\item Any finite realization can be coarsened to a single displayed component.\n Relative to that decomposition, the full-system state is local and\n \\(k_0\\leq1\\).\n\\item Let \\(Z\\) be uniform on a finite abelian group \\(\\mathsf G\\). Let\n \\(K\\) be an independent uniform group element and define\n \\[\n X_1=K,\n \\qquad\n X_2=Z-K.\n \\]\n Then\n \\[\n I(Z;X_1)=I(Z;X_2)=0,\n \\qquad\n H(Z\\mid X_1,X_2)=0.\n \\]\n Under logarithmic loss,\n \\[\n \\Risk(\\{1\\})=\\Risk(\\{2\\})=\\log|\\mathsf G|,\n \\qquad\n \\Risk(\\{1,2\\})=0,\n \\]\n so the declared two-node representation has \\(k_0=2\\) and joint advantage\n \\(\\log|\\mathsf G|\\), even when a single centralized encoder generated both\n shares.\n\\end{enumerate}\n\\end{proposition}","vializations]\n\\label{prop:decomposition-trivialization}\nThe following statements hold.\n\\begin{enumerate}\n\\item Any finite realization can be coarsened to a single displayed component.\n Relative to that decomposition, the full-system state is local and\n \\(k_0\\leq1\\).\n\\item Let \\(Z\\) be uniform on a finite abelian group \\(\\mathsf G\\). Let\n \\(K\\) be an independent uniform group element and define\n \\[\n X_1=K,\n \\qquad\n X_2=Z-K.\n \\]\n Then\n \\[\n I(Z;X_1)=I(Z;X_2)=0,\n \\qquad\n H(Z\\mid X_1,X_2)=0.\n \\]\n Under logarithmic loss,\n \\[\n \\Risk(\\{1\\})=\\Risk(\\{2\\})=\\log|\\mathsf G|,\n \\qquad\n \\Risk(\\{1,2\\})=0,\n \\]\n so the declared two-node representation has \\(k_0=2\\) and joint advantage\n \\(\\log|\\mathsf G|\\), even when a single centralized encoder generated both\n shares.\n\\end{enumerate}\n\\end{proposition}","trivialization}\nThe following statements hold.\n\\begin{enumerate}\n\\item Any finite realization can be coarsened to a single displayed component.\n Relative to that decomposition, the full-system state is local and\n \\(k_0\\leq1\\).\n\\item Let \\(Z\\) be uniform on a finite abelian group \\(\\mathsf G\\). Let\n \\(K\\) be an independent uniform group element and define\n \\[\n X_1=K,\n \\qquad\n X_2=Z-K.\n \\]\n Then\n \\[\n I(Z;X_1)=I(Z;X_2)=0,\n \\qquad\n H(Z\\mid X_1,X_2)=0.\n \\]\n Under logarithmic loss,\n \\[\n \\Risk(\\{1\\})=\\Risk(\\{2\\})=\\log|\\mathsf G|,\n \\qquad\n \\Risk(\\{1,2\\})=0,\n \\]\n so the declared two-node representation has \\(k_0=2\\) and joint advantage\n \\(\\log|\\mathsf G|\\), even when a single centralized encoder generated both\n shares.\n\\end{enumerate}\n\\end{proposition}","alization can be coarsened to a single displayed component.\n Relative to that decomposition, the full-system state is local and\n \\(k_0\\leq1\\).\n\\item Let \\(Z\\) be uniform on a finite abelian group \\(\\mathsf G\\). Let\n \\(K\\) be an independent uniform group element and define\n \\[\n X_1=K,\n \\qquad\n X_2=Z-K.\n \\]\n Then\n \\[\n I(Z;X_1)=I(Z;X_2)=0,\n \\qquad\n H(Z\\mid X_1,X_2)=0.\n \\]\n Under logarithmic loss,\n \\[\n \\Risk(\\{1\\})=\\Risk(\\{2\\})=\\log|\\mathsf G|,\n \\qquad\n \\Risk(\\{1,2\\})=0,\n \\]\n so the declared two-node representation has \\(k_0=2\\) and joint advantage\n \\(\\log|\\mathsf G|\\), even when a single centralized encoder generated both\n shares.\n\\end{enumerate}\n\\end{proposition}","and\n \\(k_0\\leq1\\).\n\\item Let \\(Z\\) be uniform on a finite abelian group \\(\\mathsf G\\). Let\n \\(K\\) be an independent uniform group element and define\n \\[\n X_1=K,\n \\qquad\n X_2=Z-K.\n \\]\n Then\n \\[\n I(Z;X_1)=I(Z;X_2)=0,\n \\qquad\n H(Z\\mid X_1,X_2)=0.\n \\]\n Under logarithmic loss,\n \\[\n \\Risk(\\{1\\})=\\Risk(\\{2\\})=\\log|\\mathsf G|,\n \\qquad\n \\Risk(\\{1,2\\})=0,\n \\]\n so the declared two-node representation has \\(k_0=2\\) and joint advantage\n \\(\\log|\\mathsf G|\\), even when a single centralized encoder generated both\n shares.\n\\end{enumerate}\n\\end{proposition}","(Z\\) be uniform on a finite abelian group \\(\\mathsf G\\). Let\n \\(K\\) be an independent uniform group element and define\n \\[\n X_1=K,\n \\qquad\n X_2=Z-K.\n \\]\n Then\n \\[\n I(Z;X_1)=I(Z;X_2)=0,\n \\qquad\n H(Z\\mid X_1,X_2)=0.\n \\]\n Under logarithmic loss,\n \\[\n \\Risk(\\{1\\})=\\Risk(\\{2\\})=\\log|\\mathsf G|,\n \\qquad\n \\Risk(\\{1,2\\})=0,\n \\]\n so the declared two-node representation has \\(k_0=2\\) and joint advantage\n \\(\\log|\\mathsf G|\\), even when a single centralized encoder generated both\n shares.\n\\end{enumerate}\n\\end{proposition}","\\begin{proof}\nThe first statement follows by taking the indiscrete partition\n\\(\\Pi=\\{\\Omega\\}\\). For the second, conditioning on any value of \\(Z\\) leaves\n\\(K\\), and hence \\(X_1\\), uniform. Translation invariance of the uniform law\nalso leaves \\(X_2=Z-K\\) uniform, so each share is independent of \\(Z\\). The\npair reconstructs \\(Z=X_1+X_2\\), which gives the entropy and Bayes-risk\nidentities.\n\\end{proof}","\\begin{definition}[Admissible decomposition family]\n\\label{def:admissible-decomposition-family}\nAn admissible family \\(\\mathfrak D\\) is a predeclared collection of\ndecompositions of the same carrier presentation satisfying the following\naudit conditions.\n\\begin{enumerate}\n\\item \\textbf{Carrier continuity.} Each component corresponds to a process,\n artifact, or mechanism that can be tracked through the relevant time scale.\n\\item \\textbf{Access locality.} The local state contains only information\n available through the declared component and its recorded incoming channels.\n\\item \\textbf{Channel completeness.} Shared memories, buses, supervisors,\n caches, and synchronized external cues are represented explicitly.\n\\item \\textbf{Intervention separability.} At least some component mechanisms\n or channels can be altered witho","Let \\(m_q(W;\\mathcal D)\\) be the signed margin of gate \\(q\\), oriented so that\n\\[\nm_q(W;\\mathcal D)>0\n\\]\nmeans that the gate passes after its uncertainty correction. Define the\ndecomposition envelope\n\\[\n\\underline m_q\n=\n\\inf_{\\mathcal D\\in\\mathfrak D}m_q(W;\\mathcal D),\n\\qquad\n\\overline m_q\n=\n\\sup_{\\mathcal D\\in\\mathfrak D}m_q(W;\\mathcal D),\n\\]\nand its sensitivity width\n\\[\n\\Delta_q^{\\mathrm{dec}}\n=\n\\overline m_q-\\underline m_q.\n\\]","\\begin{definition}[Decomposition-robust certificate]\n\\label{def:decomposition-robust-certificate}\nA workspace passes the \\(\\mathfrak D\\)-robust certificate at level \\(L\\) when\n\\[\n\\boxed{\n\\operatorname{Certify}_{L,\\mathfrak D}^{\\mathrm{rob}}(W)\n=\n\\bigwedge_{q\\in\\mathcal J(L)}\n\\left\\{\n\\inf_{\\mathcal D\\in\\mathfrak D}\n m_q(W;\\mathcal D)>0\n\\right\\}.\n}\n\\]\nA pass for one selected decomposition is a \\emph{decomposition-specific}\nclaim. A pass throughout \\(\\mathfrak D\\) is a \\emph{family-robust} claim. A\nclaim is \\emph{decomposition fragile} when at least one required gate changes\nsign inside the admissible family.\n\\end{definition}","\\label{def:decomposition-robust-certificate}\nA workspace passes the \\(\\mathfrak D\\)-robust certificate at level \\(L\\) when\n\\[\n\\boxed{\n\\operatorname{Certify}_{L,\\mathfrak D}^{\\mathrm{rob}}(W)\n=\n\\bigwedge_{q\\in\\mathcal J(L)}\n\\left\\{\n\\inf_{\\mathcal D\\in\\mathfrak D}\n m_q(W;\\mathcal D)>0\n\\right\\}.\n}\n\\]\nA pass for one selected decomposition is a \\emph{decomposition-specific}\nclaim. A pass throughout \\(\\mathfrak D\\) is a \\emph{family-robust} claim. A\nclaim is \\emph{decomposition fragile} when at least one required gate changes\nsign inside the admissible family.\n\\end{definition}","Universal invariance over all partitions is neither possible nor desirable,\nas Proposition~\\ref{prop:decomposition-trivialization} shows. The family\n\\(\\mathfrak D\\) must be broad enough to challenge the favored description and\nnarrow enough to exclude partitions with no operational meaning.","Fix a reference decomposition \\(\\mathcal D_0\\). For each\n\\(\\mathcal D\\in\\mathfrak D\\), let\n\\[\n\\Phi_{\\mathcal D}:W^{\\mathcal D_0}\\longrightarrow W^{\\mathcal D}\n\\]\nbe a declared workspace correspondence and let\n\\(\\delta_{\\mathcal D}\\) measure its structural and causal identity defect.\nAssume gate \\(q\\) has a transport bound\n\\[\n\\left|\n m_q(\\Phi_{\\mathcal D}W;\\mathcal D)\n -m_q(W;\\mathcal D_0)\n\\right|\n\\leq\nL_q\\delta_{\\mathcal D}+b_q(\\mathcal D),\n\\]\nwhere \\(L_q\\) is a sensitivity constant and \\(b_q\\) records model mismatch not\ncaptured by the identity distance.","\\(\\mathcal D\\in\\mathfrak D\\), let\n\\[\n\\Phi_{\\mathcal D}:W^{\\mathcal D_0}\\longrightarrow W^{\\mathcal D}\n\\]\nbe a declared workspace correspondence and let\n\\(\\delta_{\\mathcal D}\\) measure its structural and causal identity defect.\nAssume gate \\(q\\) has a transport bound\n\\[\n\\left|\n m_q(\\Phi_{\\mathcal D}W;\\mathcal D)\n -m_q(W;\\mathcal D_0)\n\\right|\n\\leq\nL_q\\delta_{\\mathcal D}+b_q(\\mathcal D),\n\\]\nwhere \\(L_q\\) is a sensitivity constant and \\(b_q\\) records model mismatch not\ncaptured by the identity distance.","\\Phi_{\\mathcal D}:W^{\\mathcal D_0}\\longrightarrow W^{\\mathcal D}\n\\]\nbe a declared workspace correspondence and let\n\\(\\delta_{\\mathcal D}\\) measure its structural and causal identity defect.\nAssume gate \\(q\\) has a transport bound\n\\[\n\\left|\n m_q(\\Phi_{\\mathcal D}W;\\mathcal D)\n -m_q(W;\\mathcal D_0)\n\\right|\n\\leq\nL_q\\delta_{\\mathcal D}+b_q(\\mathcal D),\n\\]\nwhere \\(L_q\\) is a sensitivity constant and \\(b_q\\) records model mismatch not\ncaptured by the identity distance.","\\begin{theorem}[Decomposition-transport robustness]\n\\label{thm:decomposition-robustness}\nIf, for every required gate \\(q\\),\n\\[\n\\boxed{\n m_q(W;\\mathcal D_0)\n>\n\\sup_{\\mathcal D\\in\\mathfrak D}\n\\left(\nL_q\\delta_{\\mathcal D}+b_q(\\mathcal D)\n\\right),\n}\n\\]\nthen \\(W\\) passes the \\(\\mathfrak D\\)-robust certificate. Conversely, if for\nsome required gate\n\\[\n m_q(W;\\mathcal D_0)\n+\n\\sup_{\\mathcal D\\in\\mathfrak D}\n\\left(\nL_q\\delta_{\\mathcal D}+b_q(\\mathcal D)\n\\right)\n<0,\n\\]\nthen that gate fails throughout the family.\n\\end{theorem}","\\begin{proof}\nFor every \\(\\mathcal D\\in\\mathfrak D\\), the transport bound gives\n\\[\n m_q(\\Phi_{\\mathcal D}W;\\mathcal D)\n\\geq\nm_q(W;\\mathcal D_0)\n-\nL_q\\delta_{\\mathcal D}\n-\nb_q(\\mathcal D).\n\\]\nThe first hypothesis makes the right-hand side positive uniformly in\n\\(\\mathcal D\\), which proves robust passage of every required gate. The second\nstatement follows from the corresponding upper bound.\n\\end{proof}","When \\(b_q\\equiv0\\), a convenient certificate radius is\n\\[\n\\boxed{\nr_{\\mathrm{dec}}(W)\n=\n\\min_{q:L_q>0}\n\\frac{m_q(W;\\mathcal D_0)}{L_q}.\n}\n\\]\nEvery admissible family satisfying\n\\[\n\\sup_{\\mathcal D\\in\\mathfrak D}\\delta_{\\mathcal D}\n<\nr_{\\mathrm{dec}}(W)\n\\]\npreserves the sign of all required gate margins. In applications, the\ntransport terms can be bounded using earlier results:\n\\begin{center}\n\\small\n\\begin{tabularx}{\\textwidth}{@{}>{\\bfseries\\raggedright\\arraybackslash}p{0.22\\textwidth}>{\\raggedright\\arraybackslash}X@{}}\n\\toprule\nGate family & Relevant transport control","The report should state:\n\\[\n\\begin{aligned}\n&\\text{reference decomposition and audit resolution},\\\\\n&\\text{admissible family and reasons for excluded partitions},\\\\\n&\\text{hidden-resource and common-controller checks},\\\\\n&\\text{faithful-aggregation maps and identity defects},\\\\\n&\\text{gatewise minima, maxima, and sensitivity widths},\\\\\n&\\text{strongest decomposition-specific and family-robust conclusions}.\n\\end{aligned}\n\\]","\\begin{definition}[Workspace specification]\\label{def:workspace-specification}\nA workspace specification is a tuple\n\\[\n\\boxed{\n\\varsigma\n=\n\\bigl(\n\\mathcal D,\nY^+,\n\\ell,\n\\tau,\n\\mathcal I,\n\\mathcal N,\n\\Theta\n\\bigr),\n}\n\\]\nwhere:\n\\begin{itemize}\n\\item \\(\\mathcal D\\) is the decomposition and local-access model;\n\\item \\(Y^+\\) is the declared future variable or trajectory;\n\\item \\(\\ell\\) is the normalized loss or utility convention;\n\\item \\(\\tau\\) is the temporal horizon or horizon family;\n\\item \\(\\mathcal I\\) is the feasible intervention family;\n\\item \\(\\mathcal N\\) is the required matched-null family; and\n\\item \\(\\Theta\\) is the gate schema, direction, threshold, and indifference\n region used for certification.\n\\end{itemize}\nThe complete certificate is written\n\\[\n\\operatorname{Cert}(W;\\varsigma)\n=\n\\bigwedge_{q\\in\\mathcal J(\\varsigma)}\n\\mathbf 1\\{m_q(W;\\varsigma)>0\\},\n\\]\nwhere \\(m_q\\) is the signed margin of","\\begin{definition}[Workspace specification]\\label{def:workspace-specification}\nA workspace specification is a tuple\n\\[\n\\boxed{\n\\varsigma\n=\n\\bigl(\n\\mathcal D,\nY^+,\n\\ell,\n\\tau,\n\\mathcal I,\n\\mathcal N,\n\\Theta\n\\bigr),\n}\n\\]\nwhere:\n\\begin{itemize}\n\\item \\(\\mathcal D\\) is the decomposition and local-access model;\n\\item \\(Y^+\\) is the declared future variable or trajectory;\n\\item \\(\\ell\\) is the normalized loss or utility convention;\n\\item \\(\\tau\\) is the temporal horizon or horizon family;\n\\item \\(\\mathcal I\\) is the feasible intervention family;\n\\item \\(\\mathcal N\\) is the required matched-null family; and\n\\item \\(\\Theta\\) is the gate schema, direction, threshold, and indifference\n region used for certification.\n\\end{itemize}\nThe complete certificate is written\n\\[\n\\operatorname{Cert}(W;\\varsigma)\n=\n\\bigwedge_{q\\in\\mathcal J(\\varsigma)}\n\\mathbf 1\\{m_q(W;\\varsigma)>0\\},\n\\]\nwhere \\(m_q\\) is the signed margin of gate \\(q\\). Positive margin means that\nthat","\\begin{definition}[Workspace specification]\\label{def:workspace-specification}\nA workspace specification is a tuple\n\\[\n\\boxed{\n\\varsigma\n=\n\\bigl(\n\\mathcal D,\nY^+,\n\\ell,\n\\tau,\n\\mathcal I,\n\\mathcal N,\n\\Theta\n\\bigr),\n}\n\\]\nwhere:\n\\begin{itemize}\n\\item \\(\\mathcal D\\) is the decomposition and local-access model;\n\\item \\(Y^+\\) is the declared future variable or trajectory;\n\\item \\(\\ell\\) is the normalized loss or utility convention;\n\\item \\(\\tau\\) is the temporal horizon or horizon family;\n\\item \\(\\mathcal I\\) is the feasible intervention family;\n\\item \\(\\mathcal N\\) is the required matched-null family; and\n\\item \\(\\Theta\\) is the gate schema, direction, threshold, and indifference\n region used for certification.\n\\end{itemize}\nThe complete certificate is written\n\\[\n\\operatorname{Cert}(W;\\varsigma)\n=\n\\bigwedge_{q\\in\\mathcal J(\\varsigma)}\n\\mathbf 1\\{m_q(W;\\varsigma)>0\\},\n\\]\nwhere \\(m_q\\) is the signed margin of gate \\(q\\). Positive margin means that\nthat gate passes in the direction declared by \\(\\Theta\\).\n\\end{d","\\begin{definition}[Workspace specification]\\label{def:workspace-specification}\nA workspace specification is a tuple\n\\[\n\\boxed{\n\\varsigma\n=\n\\bigl(\n\\mathcal D,\nY^+,\n\\ell,\n\\tau,\n\\mathcal I,\n\\mathcal N,\n\\Theta\n\\bigr),\n}\n\\]\nwhere:\n\\begin{itemize}\n\\item \\(\\mathcal D\\) is the decomposition and local-access model;\n\\item \\(Y^+\\) is the declared future variable or trajectory;\n\\item \\(\\ell\\) is the normalized loss or utility convention;\n\\item \\(\\tau\\) is the temporal horizon or horizon family;\n\\item \\(\\mathcal I\\) is the feasible intervention family;\n\\item \\(\\mathcal N\\) is the required matched-null family; and\n\\item \\(\\Theta\\) is the gate schema, direction, threshold, and indifference\n region used for certification.\n\\end{itemize}\nThe complete certificate is written\n\\[\n\\operatorname{Cert}(W;\\varsigma)\n=\n\\bigwedge_{q\\in\\mathcal J(\\varsigma)}\n\\mathbf 1\\{m_q(W;\\varsigma)>0\\},\n\\]\nwhere \\(m_q\\) is the signed margin of gate \\(q\\). Positive margin means that\nthat gate passes in the direction declared by \\(\\Theta\\).\n\\end{definition}","tion]\\label{def:workspace-specification}\nA workspace specification is a tuple\n\\[\n\\boxed{\n\\varsigma\n=\n\\bigl(\n\\mathcal D,\nY^+,\n\\ell,\n\\tau,\n\\mathcal I,\n\\mathcal N,\n\\Theta\n\\bigr),\n}\n\\]\nwhere:\n\\begin{itemize}\n\\item \\(\\mathcal D\\) is the decomposition and local-access model;\n\\item \\(Y^+\\) is the declared future variable or trajectory;\n\\item \\(\\ell\\) is the normalized loss or utility convention;\n\\item \\(\\tau\\) is the temporal horizon or horizon family;\n\\item \\(\\mathcal I\\) is the feasible intervention family;\n\\item \\(\\mathcal N\\) is the required matched-null family; and\n\\item \\(\\Theta\\) is the gate schema, direction, threshold, and indifference\n region used for certification.\n\\end{itemize}\nThe complete certificate is written\n\\[\n\\operatorname{Cert}(W;\\varsigma)\n=\n\\bigwedge_{q\\in\\mathcal J(\\varsigma)}\n\\mathbf 1\\{m_q(W;\\varsigma)>0\\},\n\\]\nwhere \\(m_q\\) is the signed margin of gate \\(q\\). Positive margin means that\nthat gate passes in the direction declared by \\(\\Theta\\).\n\\end{definition}","fication is a tuple\n\\[\n\\boxed{\n\\varsigma\n=\n\\bigl(\n\\mathcal D,\nY^+,\n\\ell,\n\\tau,\n\\mathcal I,\n\\mathcal N,\n\\Theta\n\\bigr),\n}\n\\]\nwhere:\n\\begin{itemize}\n\\item \\(\\mathcal D\\) is the decomposition and local-access model;\n\\item \\(Y^+\\) is the declared future variable or trajectory;\n\\item \\(\\ell\\) is the normalized loss or utility convention;\n\\item \\(\\tau\\) is the temporal horizon or horizon family;\n\\item \\(\\mathcal I\\) is the feasible intervention family;\n\\item \\(\\mathcal N\\) is the required matched-null family; and\n\\item \\(\\Theta\\) is the gate schema, direction, threshold, and indifference\n region used for certification.\n\\end{itemize}\nThe complete certificate is written\n\\[\n\\operatorname{Cert}(W;\\varsigma)\n=\n\\bigwedge_{q\\in\\mathcal J(\\varsigma)}\n\\mathbf 1\\{m_q(W;\\varsigma)>0\\},\n\\]\nwhere \\(m_q\\) is the signed margin of gate \\(q\\). Positive margin means that\nthat gate passes in the direction declared by \\(\\Theta\\).\n\\end{definition}","D,\nY^+,\n\\ell,\n\\tau,\n\\mathcal I,\n\\mathcal N,\n\\Theta\n\\bigr),\n}\n\\]\nwhere:\n\\begin{itemize}\n\\item \\(\\mathcal D\\) is the decomposition and local-access model;\n\\item \\(Y^+\\) is the declared future variable or trajectory;\n\\item \\(\\ell\\) is the normalized loss or utility convention;\n\\item \\(\\tau\\) is the temporal horizon or horizon family;\n\\item \\(\\mathcal I\\) is the feasible intervention family;\n\\item \\(\\mathcal N\\) is the required matched-null family; and\n\\item \\(\\Theta\\) is the gate schema, direction, threshold, and indifference\n region used for certification.\n\\end{itemize}\nThe complete certificate is written\n\\[\n\\operatorname{Cert}(W;\\varsigma)\n=\n\\bigwedge_{q\\in\\mathcal J(\\varsigma)}\n\\mathbf 1\\{m_q(W;\\varsigma)>0\\},\n\\]\nwhere \\(m_q\\) is the signed margin of gate \\(q\\). Positive margin means that\nthat gate passes in the direction declared by \\(\\Theta\\).\n\\end{definition}","\\]\nwhere:\n\\begin{itemize}\n\\item \\(\\mathcal D\\) is the decomposition and local-access model;\n\\item \\(Y^+\\) is the declared future variable or trajectory;\n\\item \\(\\ell\\) is the normalized loss or utility convention;\n\\item \\(\\tau\\) is the temporal horizon or horizon family;\n\\item \\(\\mathcal I\\) is the feasible intervention family;\n\\item \\(\\mathcal N\\) is the required matched-null family; and\n\\item \\(\\Theta\\) is the gate schema, direction, threshold, and indifference\n region used for certification.\n\\end{itemize}\nThe complete certificate is written\n\\[\n\\operatorname{Cert}(W;\\varsigma)\n=\n\\bigwedge_{q\\in\\mathcal J(\\varsigma)}\n\\mathbf 1\\{m_q(W;\\varsigma)>0\\},\n\\]\nwhere \\(m_q\\) is the signed margin of gate \\(q\\). Positive margin means that\nthat gate passes in the direction declared by \\(\\Theta\\).\n\\end{definition}","\\item \\(\\ell\\) is the normalized loss or utility convention;\n\\item \\(\\tau\\) is the temporal horizon or horizon family;\n\\item \\(\\mathcal I\\) is the feasible intervention family;\n\\item \\(\\mathcal N\\) is the required matched-null family; and\n\\item \\(\\Theta\\) is the gate schema, direction, threshold, and indifference\n region used for certification.\n\\end{itemize}\nThe complete certificate is written\n\\[\n\\operatorname{Cert}(W;\\varsigma)\n=\n\\bigwedge_{q\\in\\mathcal J(\\varsigma)}\n\\mathbf 1\\{m_q(W;\\varsigma)>0\\},\n\\]\nwhere \\(m_q\\) is the signed margin of gate \\(q\\). Positive margin means that\nthat gate passes in the direction declared by \\(\\Theta\\).\n\\end{definition}",";\n\\item \\(\\mathcal I\\) is the feasible intervention family;\n\\item \\(\\mathcal N\\) is the required matched-null family; and\n\\item \\(\\Theta\\) is the gate schema, direction, threshold, and indifference\n region used for certification.\n\\end{itemize}\nThe complete certificate is written\n\\[\n\\operatorname{Cert}(W;\\varsigma)\n=\n\\bigwedge_{q\\in\\mathcal J(\\varsigma)}\n\\mathbf 1\\{m_q(W;\\varsigma)>0\\},\n\\]\nwhere \\(m_q\\) is the signed margin of gate \\(q\\). Positive margin means that\nthat gate passes in the direction declared by \\(\\Theta\\).\n\\end{definition}","e intervention family;\n\\item \\(\\mathcal N\\) is the required matched-null family; and\n\\item \\(\\Theta\\) is the gate schema, direction, threshold, and indifference\n region used for certification.\n\\end{itemize}\nThe complete certificate is written\n\\[\n\\operatorname{Cert}(W;\\varsigma)\n=\n\\bigwedge_{q\\in\\mathcal J(\\varsigma)}\n\\mathbf 1\\{m_q(W;\\varsigma)>0\\},\n\\]\nwhere \\(m_q\\) is the signed margin of gate \\(q\\). Positive margin means that\nthat gate passes in the direction declared by \\(\\Theta\\).\n\\end{definition}","and\n\\item \\(\\Theta\\) is the gate schema, direction, threshold, and indifference\n region used for certification.\n\\end{itemize}\nThe complete certificate is written\n\\[\n\\operatorname{Cert}(W;\\varsigma)\n=\n\\bigwedge_{q\\in\\mathcal J(\\varsigma)}\n\\mathbf 1\\{m_q(W;\\varsigma)>0\\},\n\\]\nwhere \\(m_q\\) is the signed margin of gate \\(q\\). Positive margin means that\nthat gate passes in the direction declared by \\(\\Theta\\).\n\\end{definition}","The notation absorbs lower-bounded and upper-bounded gates into one sign\nconvention. If a causal effect must exceed \\(c_0\\), then\n\\[\nm_q=\\widehat\\eta-c_0.\n\\]\nIf a defect must remain below \\(\\varepsilon_0\\), then\n\\[\nm_q=\\varepsilon_0-\\widehat\\delta.\n\\]\nThe complete certificate therefore passes exactly when every required signed\nmargin is positive.","This yields the more precise notation\n\\[\n\\boxed{\n\\SRW(W;\\varsigma)=1.\n}\n\\]\nThe earlier expression \\(\\SRW_\\Theta(W;\\mathcal D)\\) is the special case in\nwhich the remaining coordinates of \\(\\varsigma\\) are held fixed.","\\begin{definition}[Specification equivalence]\\label{def:spec-equivalence}\nTwo specifications \\(\\varsigma\\) and \\(\\varsigma'\\) are equivalent, written\n\\(\\varsigma\\simeq_{\\mathrm{spec}}\\varsigma'\\), when there are bijective\ntransport maps that:\n\\begin{enumerate}\n\\item preserve the sigma-field generated by the future target;\n\\item preserve the ordering and normalization of decisions under the loss;\n\\item map feasible interventions to physically equivalent interventions;\n\\item map every required null to a null preserving the same quantities; and\n\\item carry every gate threshold and margin into its counterpart.\n\\end{enumerate}\n\\end{definition}","\\begin{proposition}[Certificate invariance under specification equivalence]\n\\label{prop:spec-equivalence-invariance}\nIf \\(\\varsigma\\simeq_{\\mathrm{spec}}\\varsigma'\\) and \\(\\Phi\\) is the induced\nworkspace correspondence, then\n\\[\n\\operatorname{Cert}(W;\\varsigma)\n=\n\\operatorname{Cert}(\\Phi W;\\varsigma').\n\\]\n\\end{proposition}","\\begin{definition}[Admissible specification family]\n\\label{def:admissible-spec-family}\nAn admissible specification family \\(\\mathfrak S\\) is a predeclared set of\nspecifications that:\n\\begin{enumerate}\n\\item refer to the same scientific claim type and carrier presentation;\n\\item use targets that are equivalent operationalizations or explicitly\n declared nearby stress targets;\n\\item normalize losses so that margins are comparable;\n\\item use horizons justified by the process being studied;\n\\item include every feasible intervention and null needed to distinguish the\n principal alternative explanations;\n\\item use thresholds fixed without access to certification outcomes; and\n\\item exclude only alternatives for reasons stated before evaluation.\n\\end{enumerate}\n\\end{definition}","A robustness family must use a common semantic gate index\n\\(\\mathcal J_*\\). A specification may mark a gate as not required only when the\nclaim level itself does not include that property. It may not remove a failed\ngate after observing the result. Let\n\\[\n\\mathcal J(\\varsigma)\\subseteq\\mathcal J_*\n\\]\nbe fixed before certification. For a family intended to support one fixed\nclaim level, the cleanest design uses\n\\[\n\\mathcal J(\\varsigma)=\\mathcal J_*\n\\qquad\n(\\varsigma\\in\\mathfrak S).\n\\]\nThis prevents apparent robustness from being created by silently changing the\nmeaning of passage across the family.","ily must use a common semantic gate index\n\\(\\mathcal J_*\\). A specification may mark a gate as not required only when the\nclaim level itself does not include that property. It may not remove a failed\ngate after observing the result. Let\n\\[\n\\mathcal J(\\varsigma)\\subseteq\\mathcal J_*\n\\]\nbe fixed before certification. For a family intended to support one fixed\nclaim level, the cleanest design uses\n\\[\n\\mathcal J(\\varsigma)=\\mathcal J_*\n\\qquad\n(\\varsigma\\in\\mathfrak S).\n\\]\nThis prevents apparent robustness from being created by silently changing the\nmeaning of passage across the family.","For each gate \\(q\\in\\mathcal J_*\\), define the specification envelope\n\\[\n\\boxed{\n\\underline m_q^{\\mathfrak S}(W)\n=\n\\inf_{\\varsigma\\in\\mathfrak S}m_q(W;\\varsigma),\n\\qquad\n\\overline m_q^{\\mathfrak S}(W)\n=\n\\sup_{\\varsigma\\in\\mathfrak S}m_q(W;\\varsigma).\n}\n\\]\nIts sensitivity width is\n\\[\n\\Delta_q^{\\mathrm{spec}}(W)\n=\n\\overline m_q^{\\mathfrak S}(W)\n-\n\\underline m_q^{\\mathfrak S}(W).\n\\]\nThe robust certificate margin is\n\\[\n\\boxed{\nr_{\\mathfrak S}(W)\n=\n\\inf_{\\varsigma\\in\\mathfrak S}\n\\min_{q\\in\\mathcal J_*}m_q(W;\\varsigma).\n}\n\\]\nThen\n\\[\n\\operatorname{Cert}_{\\mathfrak S}(W)=1\n\\quad\\Longleftrightarrow\\quad\nr_{\\mathfrak S}(W)>0.\n\\]","in\\mathcal J_*\\), define the specification envelope\n\\[\n\\boxed{\n\\underline m_q^{\\mathfrak S}(W)\n=\n\\inf_{\\varsigma\\in\\mathfrak S}m_q(W;\\varsigma),\n\\qquad\n\\overline m_q^{\\mathfrak S}(W)\n=\n\\sup_{\\varsigma\\in\\mathfrak S}m_q(W;\\varsigma).\n}\n\\]\nIts sensitivity width is\n\\[\n\\Delta_q^{\\mathrm{spec}}(W)\n=\n\\overline m_q^{\\mathfrak S}(W)\n-\n\\underline m_q^{\\mathfrak S}(W).\n\\]\nThe robust certificate margin is\n\\[\n\\boxed{\nr_{\\mathfrak S}(W)\n=\n\\inf_{\\varsigma\\in\\mathfrak S}\n\\min_{q\\in\\mathcal J_*}m_q(W;\\varsigma).\n}\n\\]\nThen\n\\[\n\\operatorname{Cert}_{\\mathfrak S}(W)=1\n\\quad\\Longleftrightarrow\\quad\nr_{\\mathfrak S}(W)>0.\n\\]","n\\mathfrak S}m_q(W;\\varsigma),\n\\qquad\n\\overline m_q^{\\mathfrak S}(W)\n=\n\\sup_{\\varsigma\\in\\mathfrak S}m_q(W;\\varsigma).\n}\n\\]\nIts sensitivity width is\n\\[\n\\Delta_q^{\\mathrm{spec}}(W)\n=\n\\overline m_q^{\\mathfrak S}(W)\n-\n\\underline m_q^{\\mathfrak S}(W).\n\\]\nThe robust certificate margin is\n\\[\n\\boxed{\nr_{\\mathfrak S}(W)\n=\n\\inf_{\\varsigma\\in\\mathfrak S}\n\\min_{q\\in\\mathcal J_*}m_q(W;\\varsigma).\n}\n\\]\nThen\n\\[\n\\operatorname{Cert}_{\\mathfrak S}(W)=1\n\\quad\\Longleftrightarrow\\quad\nr_{\\mathfrak S}(W)>0.\n\\]","\\item[Specification-specific] when it is asserted only for one declared\n \\(\\varsigma_0\\).\n\\item[Specification-robust] when every required gate has positive lower\n envelope across \\(\\mathfrak S\\).\n\\item[Specification-fragile] when it passes for at least one admissible\n specification and fails for at least one other.\n\\end{description}\n\\end{definition}","\\item[Specification-robust] when every required gate has positive lower\n envelope across \\(\\mathfrak S\\).\n\\item[Specification-fragile] when it passes for at least one admissible\n specification and fails for at least one other.\n\\end{description}\n\\end{definition}","A high fraction of passing specifications is informative but does not replace\nthe lower envelope when the claim is universal over \\(\\mathfrak S\\). A family\nin which ninety-nine analyses pass and one scientifically decisive hidden-\ncontroller null fails does not support the complete relational certificate.\nThe noncompensatory logic remains in force at the specification level.","Let \\(d_{\\mathfrak S}\\) be a declared extended pseudometric on the quotient of\n\\(\\mathfrak S\\) by specification equivalence. A convenient decomposition is\n\\[\n\\begin{aligned}\nd_{\\mathfrak S}(\\varsigma,\\varsigma')\n={}&\n w_Dd_D(\\mathcal D,\\mathcal D')\n+w_Yd_Y(Y^+,Y^{+'})\n+w_\\ell d_\\ell(\\ell,\\ell')","Let \\(d_{\\mathfrak S}\\) be a declared extended pseudometric on the quotient of\n\\(\\mathfrak S\\) by specification equivalence. A convenient decomposition is\n\\[\n\\begin{aligned}\nd_{\\mathfrak S}(\\varsigma,\\varsigma')\n={}&\n w_Dd_D(\\mathcal D,\\mathcal D')\n+w_Yd_Y(Y^+,Y^{+'})\n+w_\\ell d_\\ell(\\ell,\\ell')\\\\\n&+w_\\tau d_\\tau(\\tau,\\tau')\n+w_Id_I(\\mathcal I,\\mathcal I')\n+w_Nd_N(\\mathcal N,\\mathcal N')\n+w_\\Theta d_\\Theta(\\Theta,\\Theta').\n\\end{aligned}\n\\]\nNo universal metric is imposed. The target term may use a divergence between\nconditional laws under a declared coupling. The intervention and null terms\nmay use Hausdorff distances between implemented kernels. Threshold terms must\nbe expressed on normalized signed-margin scales. The metric is part of the\naudit and should be varied in sensitivity analysis when more than one\nscientifically defensible choice exists.","For a reference specification \\(\\varsigma_0\\), define the distance to\ncertificate failure by\n\\[\n\\boxed{\nr_{\\mathrm{spec}}(W;\\varsigma_0)\n=\n\\inf\\left\\{\n d_{\\mathfrak S}(\\varsigma,\\varsigma_0):\n \\min_q m_q(W;\\varsigma)\\leq0\n\\right\\}.\n}\n\\]\nIf no failing specification exists in the declared universe, the radius is\ninfinite relative to that universe. A radius near zero means that a small\nscientifically defensible change can reverse the conclusion.","Changing the decomposition can alter the coordinate realization of the\nworkspace, while changing the target or horizon can alter the relevant\npredictive state. Let\n\\[\n\\Phi_{\\varsigma}:W_{\\varsigma_0}\\longrightarrow W_{\\varsigma}\n\\]\nbe the declared correspondence when such transport is needed. For each gate,\nassume a bound\n\\[\n\\left|\n m_q(\\Phi_{\\varsigma}W;\\varsigma)\n -m_q(W;\\varsigma_0)\n\\right|\n\\leq\nL_qd_{\\mathfrak S}(\\varsigma,\\varsigma_0)\n+b_q(\\varsigma),\n\\]\nwhere \\(b_q\\) records mismatch not represented by the metric. The mismatch may\ninclude a change of model class, imperfect target coupling, or an intervention\nwhose implementation cannot be transported exactly.","orkspace, while changing the target or horizon can alter the relevant\npredictive state. Let\n\\[\n\\Phi_{\\varsigma}:W_{\\varsigma_0}\\longrightarrow W_{\\varsigma}\n\\]\nbe the declared correspondence when such transport is needed. For each gate,\nassume a bound\n\\[\n\\left|\n m_q(\\Phi_{\\varsigma}W;\\varsigma)\n -m_q(W;\\varsigma_0)\n\\right|\n\\leq\nL_qd_{\\mathfrak S}(\\varsigma,\\varsigma_0)\n+b_q(\\varsigma),\n\\]\nwhere \\(b_q\\) records mismatch not represented by the metric. The mismatch may\ninclude a change of model class, imperfect target coupling, or an intervention\nwhose implementation cannot be transported exactly.","\\subtopic{62.3 \\(T_{28}\\): Specification-transport robustness theorem}","\\begin{theorem}[Specification-transport robustness]\n\\label{thm:specification-transport}\nLet \\(\\mathfrak S\\) be an admissible specification family with reference\n\\(\\varsigma_0\\). Suppose every required gate satisfies the transport bound\nabove. If\n\\[\n\\boxed{\n m_q(W;\\varsigma_0)\n>\n\\sup_{\\varsigma\\in\\mathfrak S}\n\\left[\nL_qd_{\\mathfrak S}(\\varsigma,\\varsigma_0)\n+b_q(\\varsigma)\n\\right]\n}\n\\]\nfor every \\(q\\in\\mathcal J_*\\), then\n\\[\n\\operatorname{Cert}(\\Phi_{\\varsigma}W;\\varsigma)=1\n\\qquad\n\\text{for every }\\varsigma\\in\\mathfrak S.\n\\]\nIf \\(b_q\\equiv0\\), then\n\\[\n\\boxed{\nr_{\\mathrm{spec}}(W;\\varsigma_0)\n\\geq\n\\min_{q:L_q>0}\n\\frac{m_q(W;\\varsigma_0)}{L_q}.\n}\n\\]\n\\end{theorem}","on-transport robustness]\n\\label{thm:specification-transport}\nLet \\(\\mathfrak S\\) be an admissible specification family with reference\n\\(\\varsigma_0\\). Suppose every required gate satisfies the transport bound\nabove. If\n\\[\n\\boxed{\n m_q(W;\\varsigma_0)\n>\n\\sup_{\\varsigma\\in\\mathfrak S}\n\\left[\nL_qd_{\\mathfrak S}(\\varsigma,\\varsigma_0)\n+b_q(\\varsigma)\n\\right]\n}\n\\]\nfor every \\(q\\in\\mathcal J_*\\), then\n\\[\n\\operatorname{Cert}(\\Phi_{\\varsigma}W;\\varsigma)=1\n\\qquad\n\\text{for every }\\varsigma\\in\\mathfrak S.\n\\]\nIf \\(b_q\\equiv0\\), then\n\\[\n\\boxed{\nr_{\\mathrm{spec}}(W;\\varsigma_0)\n\\geq\n\\min_{q:L_q>0}\n\\frac{m_q(W;\\varsigma_0)}{L_q}.\n}\n\\]\n\\end{theorem}","on family with reference\n\\(\\varsigma_0\\). Suppose every required gate satisfies the transport bound\nabove. If\n\\[\n\\boxed{\n m_q(W;\\varsigma_0)\n>\n\\sup_{\\varsigma\\in\\mathfrak S}\n\\left[\nL_qd_{\\mathfrak S}(\\varsigma,\\varsigma_0)\n+b_q(\\varsigma)\n\\right]\n}\n\\]\nfor every \\(q\\in\\mathcal J_*\\), then\n\\[\n\\operatorname{Cert}(\\Phi_{\\varsigma}W;\\varsigma)=1\n\\qquad\n\\text{for every }\\varsigma\\in\\mathfrak S.\n\\]\nIf \\(b_q\\equiv0\\), then\n\\[\n\\boxed{\nr_{\\mathrm{spec}}(W;\\varsigma_0)\n\\geq\n\\min_{q:L_q>0}\n\\frac{m_q(W;\\varsigma_0)}{L_q}.\n}\n\\]\n\\end{theorem}","nce\n\\(\\varsigma_0\\). Suppose every required gate satisfies the transport bound\nabove. If\n\\[\n\\boxed{\n m_q(W;\\varsigma_0)\n>\n\\sup_{\\varsigma\\in\\mathfrak S}\n\\left[\nL_qd_{\\mathfrak S}(\\varsigma,\\varsigma_0)\n+b_q(\\varsigma)\n\\right]\n}\n\\]\nfor every \\(q\\in\\mathcal J_*\\), then\n\\[\n\\operatorname{Cert}(\\Phi_{\\varsigma}W;\\varsigma)=1\n\\qquad\n\\text{for every }\\varsigma\\in\\mathfrak S.\n\\]\nIf \\(b_q\\equiv0\\), then\n\\[\n\\boxed{\nr_{\\mathrm{spec}}(W;\\varsigma_0)\n\\geq\n\\min_{q:L_q>0}\n\\frac{m_q(W;\\varsigma_0)}{L_q}.\n}\n\\]\n\\end{theorem}","\\begin{proof}\nFor each \\(\\varsigma\\in\\mathfrak S\\),\n\\[\nm_q(\\Phi_{\\varsigma}W;\\varsigma)\n\\geq\nm_q(W;\\varsigma_0)\n-L_qd_{\\mathfrak S}(\\varsigma,\\varsigma_0)\n-b_q(\\varsigma).\n\\]\nThe hypothesis makes the right-hand side positive uniformly over the family.\nEvery required gate therefore passes under every admissible specification.\nWhen \\(b_q=0\\), no gate can cross zero at a distance smaller than its reference\nmargin divided by its Lipschitz constant. Taking the minimum over gates gives\nthe radius bound.\n\\end{proof}","The theorem is deliberately gatewise. A large structural margin cannot absorb\na fragile causal margin. It also clarifies the relation to Part~XVII. The\ndecomposition-transport theorem is recovered by varying only the\n\\(\\mathcal D\\) coordinate while holding the remaining coordinates fixed.","In practice, the reference margin is reduced by three distinct uncertainty\nbudgets:\n\\[\n\\boxed{\nB_q^{\\mathrm{total}}\n=\nB_q^{\\mathrm{model}}\n+B_q^{\\mathrm{spec}}\n+B_q^{\\mathrm{sample}}.\n}\n\\]\nThe first covers variation across observationally compatible realizations. The\nsecond covers variation across admissible specifications. The third covers\nfinite-sample estimation. The sufficient condition\n\\[\n\\widehat m_q(W;\\varsigma_0)\n>\nB_q^{\\mathrm{total}}\n\\qquad(q\\in\\mathcal J_*)\n\\]\nwill reappear below in identified-set and confidence-envelope form. Keeping the\nbudgets separate prevents an apparently narrow confidence interval from being\nmistaken for strong identification or specification robustness.","\\begin{definition}[Experimental design and induced law]\n\\label{def:workspace-experiment-design}\nAn experimental design is a tuple\n\\[\n\\mathcal E\n=\n(\\mathcal O,\\mathcal I,\\mathcal T),\n\\]\nwhere \\(\\mathcal O\\) is the observation map, \\(\\mathcal I\\) is the family of\nimplemented interventions, and \\(\\mathcal T\\) is the set of measured temporal\nwindows. A realization \\(\\mathfrak R\\) induces the family of laws\n\\[\nP_{\\mathfrak R}^{\\mathcal E}\n=\n\\left\n\\{\n\\mathcal L_{\\mathfrak R}\n\\bigl(\\mathcal O(\\mathbf H)\\mid\\operatorname{do}(\\iota)\\bigr):\n\\iota\\in\\mathcal I,\n\\ t\\in\\mathcal T\n\\right\\}.\n\\]\n\\end{definition}","efinition}[Experimental design and induced law]\n\\label{def:workspace-experiment-design}\nAn experimental design is a tuple\n\\[\n\\mathcal E\n=\n(\\mathcal O,\\mathcal I,\\mathcal T),\n\\]\nwhere \\(\\mathcal O\\) is the observation map, \\(\\mathcal I\\) is the family of\nimplemented interventions, and \\(\\mathcal T\\) is the set of measured temporal\nwindows. A realization \\(\\mathfrak R\\) induces the family of laws\n\\[\nP_{\\mathfrak R}^{\\mathcal E}\n=\n\\left\n\\{\n\\mathcal L_{\\mathfrak R}\n\\bigl(\\mathcal O(\\mathbf H)\\mid\\operatorname{do}(\\iota)\\bigr):\n\\iota\\in\\mathcal I,\n\\ t\\in\\mathcal T\n\\right\\}.\n\\]\n\\end{definition}","\\begin{definition}[Experimental equivalence]\nTwo realizations are experimentally equivalent under \\(\\mathcal E\\), written\n\\[\n\\mathfrak R\\equiv_{\\mathcal E}\\mathfrak R',\n\\]\nwhen\n\\[\nP_{\\mathfrak R}^{\\mathcal E}\n=\nP_{\\mathfrak R'}^{\\mathcal E}.\n\\]\nA property \\(\\psi(\\mathfrak R)\\) is identifiable on a model class\n\\(\\mathfrak M\\) under \\(\\mathcal E\\) when it is constant on every\n\\(\\equiv_{\\mathcal E}\\)-equivalence class contained in \\(\\mathfrak M\\).\n\\end{definition}","uivalence]\nTwo realizations are experimentally equivalent under \\(\\mathcal E\\), written\n\\[\n\\mathfrak R\\equiv_{\\mathcal E}\\mathfrak R',\n\\]\nwhen\n\\[\nP_{\\mathfrak R}^{\\mathcal E}\n=\nP_{\\mathfrak R'}^{\\mathcal E}.\n\\]\nA property \\(\\psi(\\mathfrak R)\\) is identifiable on a model class\n\\(\\mathfrak M\\) under \\(\\mathcal E\\) when it is constant on every\n\\(\\equiv_{\\mathcal E}\\)-equivalence class contained in \\(\\mathfrak M\\).\n\\end{definition}","\\subtopic{63.1 \\(T_{29}\\): Non-identifiability impossibility theorem}","\\begin{theorem}[No uniformly correct decision under experimental equivalence]\n\\label{thm:workspace-nonidentifiability}\nLet \\(\\mathfrak R,\\mathfrak R'\\in\\mathfrak M\\) satisfy\n\\[\n\\mathfrak R\\equiv_{\\mathcal E}\\mathfrak R'\n\\]\nbut\n\\[\n\\operatorname{Cert}(W_{\\mathfrak R};\\varsigma)\n\\neq\n\\operatorname{Cert}(W_{\\mathfrak R'};\\varsigma).\n\\]\nThen the workspace certificate is not identifiable on \\(\\mathfrak M\\) under\n\\(\\mathcal E\\). More strongly, for every possibly randomized decision rule\n\\(\\widehat C\\) based only on data generated by \\(\\mathcal E\\),\n\\[\n\\boxed{\n\\mathbb P_{\\mathfrak R}\n\\left(\n\\widehat C\\neq C_{\\mathfrak R}\n\\right)\n+\n\\mathbb P_{\\mathfrak R'}\n\\left(\n\\widehat C\\neq C_{\\mathfrak R'}\n\\right)\n=1,\n}\n\\]\nwhere \\(C_{\\mathfrak R}=\\operatorname{Cert}(W_{\\mathfrak R};\\varsigma)\\).\nConsequently,\n\\[\n\\inf_{\\widehat C}\n\\max\\left\\{\n\\mathbb P_{\\mathfrak R}(\\wideh","\\begin{theorem}[No uniformly correct decision under experimental equivalence]\n\\label{thm:workspace-nonidentifiability}\nLet \\(\\mathfrak R,\\mathfrak R'\\in\\mathfrak M\\) satisfy\n\\[\n\\mathfrak R\\equiv_{\\mathcal E}\\mathfrak R'\n\\]\nbut\n\\[\n\\operatorname{Cert}(W_{\\mathfrak R};\\varsigma)\n\\neq\n\\operatorname{Cert}(W_{\\mathfrak R'};\\varsigma).\n\\]\nThen the workspace certificate is not identifiable on \\(\\mathfrak M\\) under\n\\(\\mathcal E\\). More strongly, for every possibly randomized decision rule\n\\(\\widehat C\\) based only on data generated by \\(\\mathcal E\\),\n\\[\n\\boxed{\n\\mathbb P_{\\mathfrak R}\n\\left(\n\\widehat C\\neq C_{\\mathfrak R}\n\\right)\n+\n\\mathbb P_{\\mathfrak R'}\n\\left(\n\\widehat C\\neq C_{\\mathfrak R'}\n\\right)\n=1,\n}\n\\]\nwhere \\(C_{\\mathfrak R}=\\operatorname{Cert}(W_{\\mathfrak R};\\varsigma)\\).\nConsequently,\n\\[\n\\inf_{\\widehat C}\n\\max\\left\\{\n\\mathbb P_{\\mathfrak R}(\\widehat C\\neq C_{\\mathfrak R}),\n\\mathbb P_{\\mathfrak R'}(\\wide","\\begin{theorem}[No uniformly correct decision under experimental equivalence]\n\\label{thm:workspace-nonidentifiability}\nLet \\(\\mathfrak R,\\mathfrak R'\\in\\mathfrak M\\) satisfy\n\\[\n\\mathfrak R\\equiv_{\\mathcal E}\\mathfrak R'\n\\]\nbut\n\\[\n\\operatorname{Cert}(W_{\\mathfrak R};\\varsigma)\n\\neq\n\\operatorname{Cert}(W_{\\mathfrak R'};\\varsigma).\n\\]\nThen the workspace certificate is not identifiable on \\(\\mathfrak M\\) under\n\\(\\mathcal E\\). More strongly, for every possibly randomized decision rule\n\\(\\widehat C\\) based only on data generated by \\(\\mathcal E\\),\n\\[\n\\boxed{\n\\mathbb P_{\\mathfrak R}\n\\left(\n\\widehat C\\neq C_{\\mathfrak R}\n\\right)\n+\n\\mathbb P_{\\mathfrak R'}\n\\left(\n\\widehat C\\neq C_{\\mathfrak R'}\n\\right)\n=1,\n}\n\\]\nwhere \\(C_{\\mathfrak R}=\\operatorname{Cert}(W_{\\mathfrak R};\\varsigma)\\).\nConsequently,\n\\[\n\\inf_{\\widehat C}\n\\max\\left\\{\n\\mathbb P_{\\mathfrak R}(\\widehat C\\neq C_{\\mathfrak R}),\n\\mathbb P_{\\mathfrak R'}(\\widehat C\\neq C_{\\mathfrak R'})\n\\right\\}\n\\geq\\frac12.\n\\]\n\\end{theorem}","em}[No uniformly correct decision under experimental equivalence]\n\\label{thm:workspace-nonidentifiability}\nLet \\(\\mathfrak R,\\mathfrak R'\\in\\mathfrak M\\) satisfy\n\\[\n\\mathfrak R\\equiv_{\\mathcal E}\\mathfrak R'\n\\]\nbut\n\\[\n\\operatorname{Cert}(W_{\\mathfrak R};\\varsigma)\n\\neq\n\\operatorname{Cert}(W_{\\mathfrak R'};\\varsigma).\n\\]\nThen the workspace certificate is not identifiable on \\(\\mathfrak M\\) under\n\\(\\mathcal E\\). More strongly, for every possibly randomized decision rule\n\\(\\widehat C\\) based only on data generated by \\(\\mathcal E\\),\n\\[\n\\boxed{\n\\mathbb P_{\\mathfrak R}\n\\left(\n\\widehat C\\neq C_{\\mathfrak R}\n\\right)\n+\n\\mathbb P_{\\mathfrak R'}\n\\left(\n\\widehat C\\neq C_{\\mathfrak R'}\n\\right)\n=1,\n}\n\\]\nwhere \\(C_{\\mathfrak R}=\\operatorname{Cert}(W_{\\mathfrak R};\\varsigma)\\).\nConsequently,\n\\[\n\\inf_{\\widehat C}\n\\max\\left\\{\n\\mathbb P_{\\mathfrak R}(\\widehat C\\neq C_{\\mathfrak R}),\n\\mathbb P_{\\mathfrak R'}(\\widehat C\\neq C_{\\mathfrak R'})\n\\right\\}\n\\geq\\frac12.\n\\]\n\\end{theorem}","m:workspace-nonidentifiability}\nLet \\(\\mathfrak R,\\mathfrak R'\\in\\mathfrak M\\) satisfy\n\\[\n\\mathfrak R\\equiv_{\\mathcal E}\\mathfrak R'\n\\]\nbut\n\\[\n\\operatorname{Cert}(W_{\\mathfrak R};\\varsigma)\n\\neq\n\\operatorname{Cert}(W_{\\mathfrak R'};\\varsigma).\n\\]\nThen the workspace certificate is not identifiable on \\(\\mathfrak M\\) under\n\\(\\mathcal E\\). More strongly, for every possibly randomized decision rule\n\\(\\widehat C\\) based only on data generated by \\(\\mathcal E\\),\n\\[\n\\boxed{\n\\mathbb P_{\\mathfrak R}\n\\left(\n\\widehat C\\neq C_{\\mathfrak R}\n\\right)\n+\n\\mathbb P_{\\mathfrak R'}\n\\left(\n\\widehat C\\neq C_{\\mathfrak R'}\n\\right)\n=1,\n}\n\\]\nwhere \\(C_{\\mathfrak R}=\\operatorname{Cert}(W_{\\mathfrak R};\\varsigma)\\).\nConsequently,\n\\[\n\\inf_{\\widehat C}\n\\max\\left\\{\n\\mathbb P_{\\mathfrak R}(\\widehat C\\neq C_{\\mathfrak R}),\n\\mathbb P_{\\mathfrak R'}(\\widehat C\\neq C_{\\mathfrak R'})\n\\right\\}\n\\geq\\frac12.\n\\]\n\\end{theorem}","R,\\mathfrak R'\\in\\mathfrak M\\) satisfy\n\\[\n\\mathfrak R\\equiv_{\\mathcal E}\\mathfrak R'\n\\]\nbut\n\\[\n\\operatorname{Cert}(W_{\\mathfrak R};\\varsigma)\n\\neq\n\\operatorname{Cert}(W_{\\mathfrak R'};\\varsigma).\n\\]\nThen the workspace certificate is not identifiable on \\(\\mathfrak M\\) under\n\\(\\mathcal E\\). More strongly, for every possibly randomized decision rule\n\\(\\widehat C\\) based only on data generated by \\(\\mathcal E\\),\n\\[\n\\boxed{\n\\mathbb P_{\\mathfrak R}\n\\left(\n\\widehat C\\neq C_{\\mathfrak R}\n\\right)\n+\n\\mathbb P_{\\mathfrak R'}\n\\left(\n\\widehat C\\neq C_{\\mathfrak R'}\n\\right)\n=1,\n}\n\\]\nwhere \\(C_{\\mathfrak R}=\\operatorname{Cert}(W_{\\mathfrak R};\\varsigma)\\).\nConsequently,\n\\[\n\\inf_{\\widehat C}\n\\max\\left\\{\n\\mathbb P_{\\mathfrak R}(\\widehat C\\neq C_{\\mathfrak R}),\n\\mathbb P_{\\mathfrak R'}(\\widehat C\\neq C_{\\mathfrak R'})\n\\right\\}\n\\geq\\frac12.\n\\]\n\\end{theorem}","in\\mathfrak M\\) satisfy\n\\[\n\\mathfrak R\\equiv_{\\mathcal E}\\mathfrak R'\n\\]\nbut\n\\[\n\\operatorname{Cert}(W_{\\mathfrak R};\\varsigma)\n\\neq\n\\operatorname{Cert}(W_{\\mathfrak R'};\\varsigma).\n\\]\nThen the workspace certificate is not identifiable on \\(\\mathfrak M\\) under\n\\(\\mathcal E\\). More strongly, for every possibly randomized decision rule\n\\(\\widehat C\\) based only on data generated by \\(\\mathcal E\\),\n\\[\n\\boxed{\n\\mathbb P_{\\mathfrak R}\n\\left(\n\\widehat C\\neq C_{\\mathfrak R}\n\\right)\n+\n\\mathbb P_{\\mathfrak R'}\n\\left(\n\\widehat C\\neq C_{\\mathfrak R'}\n\\right)\n=1,\n}\n\\]\nwhere \\(C_{\\mathfrak R}=\\operatorname{Cert}(W_{\\mathfrak R};\\varsigma)\\).\nConsequently,\n\\[\n\\inf_{\\widehat C}\n\\max\\left\\{\n\\mathbb P_{\\mathfrak R}(\\widehat C\\neq C_{\\mathfrak R}),\n\\mathbb P_{\\mathfrak R'}(\\widehat C\\neq C_{\\mathfrak R'})\n\\right\\}\n\\geq\\frac12.\n\\]\n\\end{theorem}","varsigma).\n\\]\nThen the workspace certificate is not identifiable on \\(\\mathfrak M\\) under\n\\(\\mathcal E\\). More strongly, for every possibly randomized decision rule\n\\(\\widehat C\\) based only on data generated by \\(\\mathcal E\\),\n\\[\n\\boxed{\n\\mathbb P_{\\mathfrak R}\n\\left(\n\\widehat C\\neq C_{\\mathfrak R}\n\\right)\n+\n\\mathbb P_{\\mathfrak R'}\n\\left(\n\\widehat C\\neq C_{\\mathfrak R'}\n\\right)\n=1,\n}\n\\]\nwhere \\(C_{\\mathfrak R}=\\operatorname{Cert}(W_{\\mathfrak R};\\varsigma)\\).\nConsequently,\n\\[\n\\inf_{\\widehat C}\n\\max\\left\\{\n\\mathbb P_{\\mathfrak R}(\\widehat C\\neq C_{\\mathfrak R}),\n\\mathbb P_{\\mathfrak R'}(\\widehat C\\neq C_{\\mathfrak R'})\n\\right\\}\n\\geq\\frac12.\n\\]\n\\end{theorem}",") under\n\\(\\mathcal E\\). More strongly, for every possibly randomized decision rule\n\\(\\widehat C\\) based only on data generated by \\(\\mathcal E\\),\n\\[\n\\boxed{\n\\mathbb P_{\\mathfrak R}\n\\left(\n\\widehat C\\neq C_{\\mathfrak R}\n\\right)\n+\n\\mathbb P_{\\mathfrak R'}\n\\left(\n\\widehat C\\neq C_{\\mathfrak R'}\n\\right)\n=1,\n}\n\\]\nwhere \\(C_{\\mathfrak R}=\\operatorname{Cert}(W_{\\mathfrak R};\\varsigma)\\).\nConsequently,\n\\[\n\\inf_{\\widehat C}\n\\max\\left\\{\n\\mathbb P_{\\mathfrak R}(\\widehat C\\neq C_{\\mathfrak R}),\n\\mathbb P_{\\mathfrak R'}(\\widehat C\\neq C_{\\mathfrak R'})\n\\right\\}\n\\geq\\frac12.\n\\]\n\\end{theorem}","\\begin{proof}\nThe data law is identical under the two realizations, so\n\\(p=\\mathbb P(\\widehat C=1)\\) is the same in both. If\n\\(C_{\\mathfrak R}=0\\) and \\(C_{\\mathfrak R'}=1\\), the two error probabilities\nare \\(p\\) and \\(1-p\\). Their sum is one, and at least one is at least one half.\nThe opposite labeling is identical.\n\\end{proof}","Consider two realizations that emit the same local streams \\(X_1,X_2\\) and the\nsame outcome\n\\[\nY=X_1\\oplus X_2.\n\\]\nIn the distributed realization, the two streams are generated and retained by\nseparately intervenable carriers, and no carrier contains the pair. In the\ncentralized realization, a hidden controller contains \\((X_1,X_2)\\) and emits\nthe two displayed streams. Under passive observation of\n\\((X_1,X_2,Y)\\), the induced laws are identical. Even interventions that alter\nonly the displayed messages may fail to distinguish them if the hidden\ncontroller simply re-emits the manipulated values.","ider two realizations that emit the same local streams \\(X_1,X_2\\) and the\nsame outcome\n\\[\nY=X_1\\oplus X_2.\n\\]\nIn the distributed realization, the two streams are generated and retained by\nseparately intervenable carriers, and no carrier contains the pair. In the\ncentralized realization, a hidden controller contains \\((X_1,X_2)\\) and emits\nthe two displayed streams. Under passive observation of\n\\((X_1,X_2,Y)\\), the induced laws are identical. Even interventions that alter\nonly the displayed messages may fail to distinguish them if the hidden\ncontroller simply re-emits the manipulated values.","Carrier-relative relationality becomes identifiable only when the experiment\ncan test the claimed causal separation. Examples include independently\ninterrupting the candidate carriers, exposing the hidden memory, randomizing a\nshared controller, or measuring whether one physical mechanism reconstructs\nthe complete state before the displayed split. The decomposition controls\n\\(N_{25}\\)--\\(N_{27}\\) are therefore identifiability interventions, not merely\noptional robustness checks.","The full certificate may be unidentified even when some coordinates are\nidentified. Structural dimension might be recoverable from observed transfer\nconstraints while causal specificity remains ambiguous. Conversely, a causal\ncontrast may be identifiable even when the latent workspace coordinates are\nknown only up to gauge. Each margin should therefore be assigned one of four\nstatuses:\n\\[\n\\text{point identified},\n\\quad\n\\text{sign identified},\n\\quad\n\\text{partially identified},\n\\quad\n\\text{unidentified}.\n\\]\nA complete pass requires the sign of every required gate to be identified as\npositive, not merely that one compatible model produces a positive estimate.","\\begin{definition}[Identified realization set]\n\\label{def:identified-realization-set}\nGiven an experimental law \\(P^{\\mathcal E}\\) and model class \\(\\mathfrak M\\),\ndefine\n\\[\n\\boxed{\n\\mathfrak R_{\\mathrm{ID}}\n(P^{\\mathcal E};\\mathfrak M)\n=\n\\left\\{\n\\mathfrak R\\in\\mathfrak M:\nP_{\\mathfrak R}^{\\mathcal E}=P^{\\mathcal E}\n\\right\\}.\n}\n\\]\nFor a gate \\(q\\) and specification \\(\\varsigma\\), its identified margin set is\n\\[\n\\mathcal M_q^{\\mathrm{ID}}\n(P^{\\mathcal E};\\varsigma)\n=\n\\left\\{\n m_q(W_{\\mathfrak R};\\varsigma):\n\\mathfrak R\\in\\mathfrak R_{\\mathrm{ID}}\n\\right\\}.\n\\]\n\\end{definition}","ion}[Identified realization set]\n\\label{def:identified-realization-set}\nGiven an experimental law \\(P^{\\mathcal E}\\) and model class \\(\\mathfrak M\\),\ndefine\n\\[\n\\boxed{\n\\mathfrak R_{\\mathrm{ID}}\n(P^{\\mathcal E};\\mathfrak M)\n=\n\\left\\{\n\\mathfrak R\\in\\mathfrak M:\nP_{\\mathfrak R}^{\\mathcal E}=P^{\\mathcal E}\n\\right\\}.\n}\n\\]\nFor a gate \\(q\\) and specification \\(\\varsigma\\), its identified margin set is\n\\[\n\\mathcal M_q^{\\mathrm{ID}}\n(P^{\\mathcal E};\\varsigma)\n=\n\\left\\{\n m_q(W_{\\mathfrak R};\\varsigma):\n\\mathfrak R\\in\\mathfrak R_{\\mathrm{ID}}\n\\right\\}.\n\\]\n\\end{definition}","The sharp lower and upper identified margins are\n\\[\n\\underline m_q^{\\mathrm{ID}}(\\varsigma)\n=\n\\inf\\mathcal M_q^{\\mathrm{ID}}(\\varsigma),\n\\qquad\n\\overline m_q^{\\mathrm{ID}}(\\varsigma)\n=\n\\sup\\mathcal M_q^{\\mathrm{ID}}(\\varsigma).\n\\]\nA gate is sign-identified as passing if its lower bound is positive. It is\nsign-identified as failing if its upper bound is nonpositive. If the interval\ncrosses zero, the available design does not determine that gate's outcome.","Define\n\\[\n\\boxed{\n\\mathcal C_{\\mathrm{ID}}\n(P^{\\mathcal E};\\varsigma)\n=\n\\left\\{\n\\operatorname{Cert}(W_{\\mathfrak R};\\varsigma):\n\\mathfrak R\\in\\mathfrak R_{\\mathrm{ID}}\n\\right\\}\n\\subseteq\\{0,1\\}.\n}\n\\]\nThe certificate is point identified exactly when this set is a singleton. The\nmarginal gate intervals provide simple sufficient conditions, but they need\nnot be logically necessary because gate margins can be dependent across the\nidentified set.","For robustness across specifications, define the realization-level robust\nmargin\n\\[\nr_{\\mathfrak S}(\\mathfrak R)\n=\n\\inf_{\\varsigma\\in\\mathfrak S}\n\\min_{q\\in\\mathcal J_*}\n m_q(W_{\\mathfrak R};\\varsigma)\n\\]\nand its identified range\n\\[\n\\underline r_{\\mathrm{ID},\\mathfrak S}\n=\n\\inf_{\\mathfrak R\\in\\mathfrak R_{\\mathrm{ID}}}\n r_{\\mathfrak S}(\\mathfrak R),\n\\qquad\n\\overline r_{\\mathrm{ID},\\mathfrak S}\n=\n\\sup_{\\mathfrak R\\in\\mathfrak R_{\\mathrm{ID}}}\n r_{\\mathfrak S}(\\mathfrak R).\n\\]","\\subtopic{64.2 \\(T_{30}\\): Joint specification-identification theorem}","\\begin{theorem}[Identification of a specification-robust certificate]\n\\label{thm:joint-spec-identification}\nLet \\(\\mathfrak R_{\\mathrm{ID}}\\) be the identified realization set and\n\\(\\mathfrak S\\) an admissible specification family.\n\\begin{enumerate}\n\\item If\n\\[\n\\boxed{\n\\underline r_{\\mathrm{ID},\\mathfrak S}>0,\n}\n\\]\nthen every realization compatible with the experimental law passes every\nrequired gate under every admissible specification. The robust certificate is\nidentified as passing.\n\\item If\n\\[\n\\boxed{\n\\overline r_{\\mathrm{ID},\\mathfrak S}\\leq0,\n}\n\\]\nthen no compatible realization passes the complete certificate throughout the\nfamily. The robust certificate is identified as failing.\n\\item If there exist compatible realizations\n\\(\\mathfrak R_+,\\mathfrak R_-\\) with\n\\[\nr_{\\mathfrak S}(\\mathfrak R_+)>0,\n\\qquad\nr_{\\mathfrak S}(\\m","\\begin{theorem}[Identification of a specification-robust certificate]\n\\label{thm:joint-spec-identification}\nLet \\(\\mathfrak R_{\\mathrm{ID}}\\) be the identified realization set and\n\\(\\mathfrak S\\) an admissible specification family.\n\\begin{enumerate}\n\\item If\n\\[\n\\boxed{\n\\underline r_{\\mathrm{ID},\\mathfrak S}>0,\n}\n\\]\nthen every realization compatible with the experimental law passes every\nrequired gate under every admissible specification. The robust certificate is\nidentified as passing.\n\\item If\n\\[\n\\boxed{\n\\overline r_{\\mathrm{ID},\\mathfrak S}\\leq0,\n}\n\\]\nthen no compatible realization passes the complete certificate throughout the\nfamily. The robust certificate is identified as failing.\n\\item If there exist compatible realizations\n\\(\\mathfrak R_+,\\mathfrak R_-\\) with\n\\[\nr_{\\mathfrak S}(\\mathfrak R_+)>0,\n\\qquad\nr_{\\mathfrak S}(\\mathfrak R_-)\\leq0,\n\\]\nthen the robust certificate is n","\\begin{theorem}[Identification of a specification-robust certificate]\n\\label{thm:joint-spec-identification}\nLet \\(\\mathfrak R_{\\mathrm{ID}}\\) be the identified realization set and\n\\(\\mathfrak S\\) an admissible specification family.\n\\begin{enumerate}\n\\item If\n\\[\n\\boxed{\n\\underline r_{\\mathrm{ID},\\mathfrak S}>0,\n}\n\\]\nthen every realization compatible with the experimental law passes every\nrequired gate under every admissible specification. The robust certificate is\nidentified as passing.\n\\item If\n\\[\n\\boxed{\n\\overline r_{\\mathrm{ID},\\mathfrak S}\\leq0,\n}\n\\]\nthen no compatible realization passes the complete certificate throughout the\nfamily. The robust certificate is identified as failing.\n\\item If there exist compatible realizations\n\\(\\mathfrak R_+,\\mathfrak R_-\\) with\n\\[\nr_{\\mathfrak S}(\\mathfrak R_+)>0,\n\\qquad\nr_{\\mathfrak S}(\\mathfrak R_-)\\leq0,\n\\]\nthen the robust certificate is not identified under the declared experiment.\n\\end{enumerate}\n\\end{theorem}","tion}\nLet \\(\\mathfrak R_{\\mathrm{ID}}\\) be the identified realization set and\n\\(\\mathfrak S\\) an admissible specification family.\n\\begin{enumerate}\n\\item If\n\\[\n\\boxed{\n\\underline r_{\\mathrm{ID},\\mathfrak S}>0,\n}\n\\]\nthen every realization compatible with the experimental law passes every\nrequired gate under every admissible specification. The robust certificate is\nidentified as passing.\n\\item If\n\\[\n\\boxed{\n\\overline r_{\\mathrm{ID},\\mathfrak S}\\leq0,\n}\n\\]\nthen no compatible realization passes the complete certificate throughout the\nfamily. The robust certificate is identified as failing.\n\\item If there exist compatible realizations\n\\(\\mathfrak R_+,\\mathfrak R_-\\) with\n\\[\nr_{\\mathfrak S}(\\mathfrak R_+)>0,\n\\qquad\nr_{\\mathfrak S}(\\mathfrak R_-)\\leq0,\n\\]\nthen the robust certificate is not identified under the declared experiment.\n\\end{enumerate}\n\\end{theorem}","mpatible with the experimental law passes every\nrequired gate under every admissible specification. The robust certificate is\nidentified as passing.\n\\item If\n\\[\n\\boxed{\n\\overline r_{\\mathrm{ID},\\mathfrak S}\\leq0,\n}\n\\]\nthen no compatible realization passes the complete certificate throughout the\nfamily. The robust certificate is identified as failing.\n\\item If there exist compatible realizations\n\\(\\mathfrak R_+,\\mathfrak R_-\\) with\n\\[\nr_{\\mathfrak S}(\\mathfrak R_+)>0,\n\\qquad\nr_{\\mathfrak S}(\\mathfrak R_-)\\leq0,\n\\]\nthen the robust certificate is not identified under the declared experiment.\n\\end{enumerate}\n\\end{theorem}","sses every\nrequired gate under every admissible specification. The robust certificate is\nidentified as passing.\n\\item If\n\\[\n\\boxed{\n\\overline r_{\\mathrm{ID},\\mathfrak S}\\leq0,\n}\n\\]\nthen no compatible realization passes the complete certificate throughout the\nfamily. The robust certificate is identified as failing.\n\\item If there exist compatible realizations\n\\(\\mathfrak R_+,\\mathfrak R_-\\) with\n\\[\nr_{\\mathfrak S}(\\mathfrak R_+)>0,\n\\qquad\nr_{\\mathfrak S}(\\mathfrak R_-)\\leq0,\n\\]\nthen the robust certificate is not identified under the declared experiment.\n\\end{enumerate}\n\\end{theorem}","In finite samples, the identified set is itself uncertain. Let\n\\[\n\\widehat{\\mathfrak R}_{1-\\delta}\n\\]\nbe a confidence region satisfying\n\\[\n\\inf_{\\mathfrak R_0\\in\\mathfrak M}\n\\mathbb P_{\\mathfrak R_0}\n\\left(\n\\mathfrak R_0\\in\n\\widehat{\\mathfrak R}_{1-\\delta}\n\\right)\n\\geq1-\\delta.\n\\]\nA conservative robust lower margin is\n\\[\n\\widehat r_{1-\\delta}\n=\n\\inf_{\\mathfrak R\\in\\widehat{\\mathfrak R}_{1-\\delta}}\n\\inf_{\\varsigma\\in\\mathfrak S}\n\\min_q m_q(W_{\\mathfrak R};\\varsigma).\n\\]\nIf \\(\\widehat r_{1-\\delta}>0\\), a false specification-robust certificate has\nprobability at most \\(\\delta\\), provided that the model class and\nspecification family were fixed before the confidence region was computed.\nThis confidence-set approach is useful when gate estimators are strongly\ncoupled or when the latent realiza","In finite samples, the identified set is itself uncertain. Let\n\\[\n\\widehat{\\mathfrak R}_{1-\\delta}\n\\]\nbe a confidence region satisfying\n\\[\n\\inf_{\\mathfrak R_0\\in\\mathfrak M}\n\\mathbb P_{\\mathfrak R_0}\n\\left(\n\\mathfrak R_0\\in\n\\widehat{\\mathfrak R}_{1-\\delta}\n\\right)\n\\geq1-\\delta.\n\\]\nA conservative robust lower margin is\n\\[\n\\widehat r_{1-\\delta}\n=\n\\inf_{\\mathfrak R\\in\\widehat{\\mathfrak R}_{1-\\delta}}\n\\inf_{\\varsigma\\in\\mathfrak S}\n\\min_q m_q(W_{\\mathfrak R};\\varsigma).\n\\]\nIf \\(\\widehat r_{1-\\delta}>0\\), a false specification-robust certificate has\nprobability at most \\(\\delta\\), provided that the model class and\nspecification family were fixed before the confidence region was computed.\nThis confidence-set approach is useful when gate estimators are strongly\ncoupled or when the latent realization is the primary inferential object.","hfrak R}_{1-\\delta}\n\\]\nbe a confidence region satisfying\n\\[\n\\inf_{\\mathfrak R_0\\in\\mathfrak M}\n\\mathbb P_{\\mathfrak R_0}\n\\left(\n\\mathfrak R_0\\in\n\\widehat{\\mathfrak R}_{1-\\delta}\n\\right)\n\\geq1-\\delta.\n\\]\nA conservative robust lower margin is\n\\[\n\\widehat r_{1-\\delta}\n=\n\\inf_{\\mathfrak R\\in\\widehat{\\mathfrak R}_{1-\\delta}}\n\\inf_{\\varsigma\\in\\mathfrak S}\n\\min_q m_q(W_{\\mathfrak R};\\varsigma).\n\\]\nIf \\(\\widehat r_{1-\\delta}>0\\), a false specification-robust certificate has\nprobability at most \\(\\delta\\), provided that the model class and\nspecification family were fixed before the confidence region was computed.\nThis confidence-set approach is useful when gate estimators are strongly\ncoupled or when the latent realization is the primary inferential object.","\\mathbb P_{\\mathfrak R_0}\n\\left(\n\\mathfrak R_0\\in\n\\widehat{\\mathfrak R}_{1-\\delta}\n\\right)\n\\geq1-\\delta.\n\\]\nA conservative robust lower margin is\n\\[\n\\widehat r_{1-\\delta}\n=\n\\inf_{\\mathfrak R\\in\\widehat{\\mathfrak R}_{1-\\delta}}\n\\inf_{\\varsigma\\in\\mathfrak S}\n\\min_q m_q(W_{\\mathfrak R};\\varsigma).\n\\]\nIf \\(\\widehat r_{1-\\delta}>0\\), a false specification-robust certificate has\nprobability at most \\(\\delta\\), provided that the model class and\nspecification family were fixed before the confidence region was computed.\nThis confidence-set approach is useful when gate estimators are strongly\ncoupled or when the latent realization is the primary inferential object.","Let \\(\\widehat m_q(\\varsigma)\\) estimate\n\\(m_q(W;\\varsigma)\\). Suppose there are nonnegative error radii\n\\(\\varepsilon_q(n,\\delta)\\) such that\n\\[\n\\mathbb P\\left(\n\\sup_{\\varsigma\\in\\mathfrak S}\n\\left|\n\\widehat m_q(\\varsigma)-m_q(\\varsigma)\n\\right|\n\\leq\n\\varepsilon_q(n,\\delta)\n\\quad\\text{for all }q\\in\\mathcal J_*\n\\right)\n\\geq1-\\delta.\n\\]\nDefine the empirical lower confidence envelope\n\\[\n\\boxed{\n\\widehat{\\underline m}_{q,1-\\delta}^{\\mathfrak S}\n=\n\\inf_{\\varsigma\\in\\mathfrak S}\n\\widehat m_q(\\varsigma)\n-\n\\varepsilon_q(n,\\delta).\n}\n\\]","\\subtopic{65.1 \\(T_{31}\\): Uniform specification-certificate theorem}","\\begin{theorem}[Uniform finite-sample certification across specifications]\n\\label{thm:uniform-spec-certification}\nUnder the uniform error event above, declare a specification-robust certificate\nonly when\n\\[\n\\widehat{\\underline m}_{q,1-\\delta}^{\\mathfrak S}>0\n\\qquad\n\\text{for every }q\\in\\mathcal J_*.\n\\]\nThen\n\\[\n\\boxed{\n\\sup_{P:\\,\\operatorname{Cert}_{\\mathfrak S}(W)=0}\n\\mathbb P_P\n\\left(\n\\text{the robust certificate is declared}\n\\right)\n\\leq\\delta.\n}\n\\]\n\\end{theorem}","\\begin{proof}\nOn the uniform error event,\n\\[\nm_q(\\varsigma)\n\\geq\n\\widehat m_q(\\varsigma)-\\varepsilon_q\n\\]\nfor every gate and every specification. If all empirical lower envelopes are\npositive, every population margin is positive throughout the family. A false\ndeclaration can therefore occur only when the uniform error event fails, which\nhas probability at most \\(\\delta\\).\n\\end{proof}","Suppose \\(\\mathfrak S\\) and \\(\\mathcal J_*\\) are finite, and each margin is\nestimated as\n\\[\n\\widehat m_q(\\varsigma)\n=\n\\frac1n\\sum_{i=1}^nZ_{i,q,\\varsigma}-\\theta_{q,\\varsigma},\n\\qquad\n0\\leq Z_{i,q,\\varsigma}\\leq1,\n\\]\nfrom independent certification units. Hoeffding's inequality and a union bound\ngive\n\\[\n\\mathbb P\\left(\n\\sup_{q,\\varsigma}\n\\left|\n\\widehat m_q(\\varsigma)-m_q(\\varsigma)\n\\right|\n>\n\\sqrt{\n\\frac{\n\\log\\bigl(2|\\mathcal J_*||\\mathfrak S|/\\delta\\bigr)\n}{2n}}\n\\right)\n\\leq\\delta.\n\\]\nThus the choice\n\\[\n\\boxed{\n\\varepsilon_n(\\delta)\n=\n\\sqrt{\n\\frac{\n\\log\\bigl(2|\\mathcal J_*||\\mathfrak S|/\\delta\\bigr)\n}{2n}\n}\n}\n\\]\nyields an explicit conservative certificate. Dependent episodes require a\nbound appropriate to the t","Suppose \\(\\mathfrak S\\) and \\(\\mathcal J_*\\) are finite, and each margin is\nestimated as\n\\[\n\\widehat m_q(\\varsigma)\n=\n\\frac1n\\sum_{i=1}^nZ_{i,q,\\varsigma}-\\theta_{q,\\varsigma},\n\\qquad\n0\\leq Z_{i,q,\\varsigma}\\leq1,\n\\]\nfrom independent certification units. Hoeffding's inequality and a union bound\ngive\n\\[\n\\mathbb P\\left(\n\\sup_{q,\\varsigma}\n\\left|\n\\widehat m_q(\\varsigma)-m_q(\\varsigma)\n\\right|\n>\n\\sqrt{\n\\frac{\n\\log\\bigl(2|\\mathcal J_*||\\mathfrak S|/\\delta\\bigr)\n}{2n}}\n\\right)\n\\leq\\delta.\n\\]\nThus the choice\n\\[\n\\boxed{\n\\varepsilon_n(\\delta)\n=\n\\sqrt{\n\\frac{\n\\log\\bigl(2|\\mathcal J_*||\\mathfrak S|/\\delta\\bigr)\n}{2n}\n}\n}\n\\]\nyields an explicit conservative certificate. Dependent episodes require a\nbound appropriate to the true independent unit","Suppose \\(\\mathfrak S\\) and \\(\\mathcal J_*\\) are finite, and each margin is\nestimated as\n\\[\n\\widehat m_q(\\varsigma)\n=\n\\frac1n\\sum_{i=1}^nZ_{i,q,\\varsigma}-\\theta_{q,\\varsigma},\n\\qquad\n0\\leq Z_{i,q,\\varsigma}\\leq1,\n\\]\nfrom independent certification units. Hoeffding's inequality and a union bound\ngive\n\\[\n\\mathbb P\\left(\n\\sup_{q,\\varsigma}\n\\left|\n\\widehat m_q(\\varsigma)-m_q(\\varsigma)\n\\right|\n>\n\\sqrt{\n\\frac{\n\\log\\bigl(2|\\mathcal J_*||\\mathfrak S|/\\delta\\bigr)\n}{2n}}\n\\right)\n\\leq\\delta.\n\\]\nThus the choice\n\\[\n\\boxed{\n\\varepsilon_n(\\delta)\n=\n\\sqrt{\n\\frac{\n\\log\\bigl(2|\\mathcal J_*||\\mathfrak S|/\\delta\\bigr)\n}{2n}\n}\n}\n\\]\nyields an explicit conservative certificate. Dependent episodes require a\nbound appropriate to the true independent unit rather than token-level or\nedge-level pseudo-replication.","sigma)\n=\n\\frac1n\\sum_{i=1}^nZ_{i,q,\\varsigma}-\\theta_{q,\\varsigma},\n\\qquad\n0\\leq Z_{i,q,\\varsigma}\\leq1,\n\\]\nfrom independent certification units. Hoeffding's inequality and a union bound\ngive\n\\[\n\\mathbb P\\left(\n\\sup_{q,\\varsigma}\n\\left|\n\\widehat m_q(\\varsigma)-m_q(\\varsigma)\n\\right|\n>\n\\sqrt{\n\\frac{\n\\log\\bigl(2|\\mathcal J_*||\\mathfrak S|/\\delta\\bigr)\n}{2n}}\n\\right)\n\\leq\\delta.\n\\]\nThus the choice\n\\[\n\\boxed{\n\\varepsilon_n(\\delta)\n=\n\\sqrt{\n\\frac{\n\\log\\bigl(2|\\mathcal J_*||\\mathfrak S|/\\delta\\bigr)\n}{2n}\n}\n}\n\\]\nyields an explicit conservative certificate. Dependent episodes require a\nbound appropriate to the true independent unit rather than token-level or\nedge-level pseudo-replication.","Let \\(\\mathfrak S_\\eta\\) be an \\(\\eta\\)-net of\n\\((\\mathfrak S,d_{\\mathfrak S})\\) with covering number\n\\(N(\\eta,\\mathfrak S,d_{\\mathfrak S})\\). If both the population and empirical\nmargin functions are \\(L_q\\)-Lipschitz, then\n\\[\n\\sup_{\\varsigma\\in\\mathfrak S}\n|\\widehat m_q(\\varsigma)-m_q(\\varsigma)|\n\\leq\n\\max_{\\varsigma_j\\in\\mathfrak S_\\eta}\n|\\widehat m_q(\\varsigma_j)-m_q(\\varsigma_j)|\n+2L_q\\eta.\n\\]\nFor bounded empirical means, a valid radius is therefore\n\\[\n\\boxed{\n\\varepsilon_{q,n,\\eta}(\\delta)\n=\n\\sqrt{\n\\frac{\n\\log\\bigl(2|\\mathcal J_*|\nN(\\eta,\\mathfrak S,d_{\\mathfrak S})/\\delta\\bigr)\n}{2n}\n}\n+2L_q\\eta.\n}\n\\]\nThe tuning parameter \\(\\eta\\) balances net cardinality against interpolation\nerror. More refined empirical","Let \\(\\mathfrak S_\\eta\\) be an \\(\\eta\\)-net of\n\\((\\mathfrak S,d_{\\mathfrak S})\\) with covering number\n\\(N(\\eta,\\mathfrak S,d_{\\mathfrak S})\\). If both the population and empirical\nmargin functions are \\(L_q\\)-Lipschitz, then\n\\[\n\\sup_{\\varsigma\\in\\mathfrak S}\n|\\widehat m_q(\\varsigma)-m_q(\\varsigma)|\n\\leq\n\\max_{\\varsigma_j\\in\\mathfrak S_\\eta}\n|\\widehat m_q(\\varsigma_j)-m_q(\\varsigma_j)|\n+2L_q\\eta.\n\\]\nFor bounded empirical means, a valid radius is therefore\n\\[\n\\boxed{\n\\varepsilon_{q,n,\\eta}(\\delta)\n=\n\\sqrt{\n\\frac{\n\\log\\bigl(2|\\mathcal J_*|\nN(\\eta,\\mathfrak S,d_{\\mathfrak S})/\\delta\\bigr)\n}{2n}\n}\n+2L_q\\eta.\n}\n\\]\nThe tuning parameter \\(\\eta\\) balances net cardinality against interpolation\nerror. More refined empirical-process bounds","Let \\(\\mathfrak S_\\eta\\) be an \\(\\eta\\)-net of\n\\((\\mathfrak S,d_{\\mathfrak S})\\) with covering number\n\\(N(\\eta,\\mathfrak S,d_{\\mathfrak S})\\). If both the population and empirical\nmargin functions are \\(L_q\\)-Lipschitz, then\n\\[\n\\sup_{\\varsigma\\in\\mathfrak S}\n|\\widehat m_q(\\varsigma)-m_q(\\varsigma)|\n\\leq\n\\max_{\\varsigma_j\\in\\mathfrak S_\\eta}\n|\\widehat m_q(\\varsigma_j)-m_q(\\varsigma_j)|\n+2L_q\\eta.\n\\]\nFor bounded empirical means, a valid radius is therefore\n\\[\n\\boxed{\n\\varepsilon_{q,n,\\eta}(\\delta)\n=\n\\sqrt{\n\\frac{\n\\log\\bigl(2|\\mathcal J_*|\nN(\\eta,\\mathfrak S,d_{\\mathfrak S})/\\delta\\bigr)\n}{2n}\n}\n+2L_q\\eta.\n}\n\\]\nThe tuning parameter \\(\\eta\\) balances net cardinality against interpolation\nerror. More refined empirical-process bounds may replace this elementary\ncovering arg","Let \\(\\mathfrak S_\\eta\\) be an \\(\\eta\\)-net of\n\\((\\mathfrak S,d_{\\mathfrak S})\\) with covering number\n\\(N(\\eta,\\mathfrak S,d_{\\mathfrak S})\\). If both the population and empirical\nmargin functions are \\(L_q\\)-Lipschitz, then\n\\[\n\\sup_{\\varsigma\\in\\mathfrak S}\n|\\widehat m_q(\\varsigma)-m_q(\\varsigma)|\n\\leq\n\\max_{\\varsigma_j\\in\\mathfrak S_\\eta}\n|\\widehat m_q(\\varsigma_j)-m_q(\\varsigma_j)|\n+2L_q\\eta.\n\\]\nFor bounded empirical means, a valid radius is therefore\n\\[\n\\boxed{\n\\varepsilon_{q,n,\\eta}(\\delta)\n=\n\\sqrt{\n\\frac{\n\\log\\bigl(2|\\mathcal J_*|\nN(\\eta,\\mathfrak S,d_{\\mathfrak S})/\\delta\\bigr)\n}{2n}\n}\n+2L_q\\eta.\n}\n\\]\nThe tuning parameter \\(\\eta\\) balances net cardinality against interpolation\nerror. More refined empirical-process bounds may replace this elementary\ncovering argument without changing the certificate logic.",",d_{\\mathfrak S})\\) with covering number\n\\(N(\\eta,\\mathfrak S,d_{\\mathfrak S})\\). If both the population and empirical\nmargin functions are \\(L_q\\)-Lipschitz, then\n\\[\n\\sup_{\\varsigma\\in\\mathfrak S}\n|\\widehat m_q(\\varsigma)-m_q(\\varsigma)|\n\\leq\n\\max_{\\varsigma_j\\in\\mathfrak S_\\eta}\n|\\widehat m_q(\\varsigma_j)-m_q(\\varsigma_j)|\n+2L_q\\eta.\n\\]\nFor bounded empirical means, a valid radius is therefore\n\\[\n\\boxed{\n\\varepsilon_{q,n,\\eta}(\\delta)\n=\n\\sqrt{\n\\frac{\n\\log\\bigl(2|\\mathcal J_*|\nN(\\eta,\\mathfrak S,d_{\\mathfrak S})/\\delta\\bigr)\n}{2n}\n}\n+2L_q\\eta.\n}\n\\]\nThe tuning parameter \\(\\eta\\) balances net cardinality against interpolation\nerror. More refined empirical-process bounds may replace this elementary\ncovering argument without changing the certificate logic.","igma\\in\\mathfrak S}\n|\\widehat m_q(\\varsigma)-m_q(\\varsigma)|\n\\leq\n\\max_{\\varsigma_j\\in\\mathfrak S_\\eta}\n|\\widehat m_q(\\varsigma_j)-m_q(\\varsigma_j)|\n+2L_q\\eta.\n\\]\nFor bounded empirical means, a valid radius is therefore\n\\[\n\\boxed{\n\\varepsilon_{q,n,\\eta}(\\delta)\n=\n\\sqrt{\n\\frac{\n\\log\\bigl(2|\\mathcal J_*|\nN(\\eta,\\mathfrak S,d_{\\mathfrak S})/\\delta\\bigr)\n}{2n}\n}\n+2L_q\\eta.\n}\n\\]\nThe tuning parameter \\(\\eta\\) balances net cardinality against interpolation\nerror. More refined empirical-process bounds may replace this elementary\ncovering argument without changing the certificate logic.","A robust procedure can be valid yet uninformative when the family is too large\nor the data are insufficient. Each gate should therefore report an\nindifference region \\([-\\kappa_q,\\kappa_q]\\). A conclusion is:\n\\begin{description}\n\\item[Robust pass] when the lower confidence envelope exceeds\n \\(\\kappa_q\\) for every gate.\n\\item[Robust fail] when a required gate's upper confidence envelope is below\n \\(-\\kappa_q\\) throughout the relevant family.\n\\item[Specification conflict] when well-estimated admissible specifications\n lie on opposite sides of the indifference region.\n\\item[Statistically unresolved] when sampling uncertainty intersects the\n indifference region.\n\\end{description}\nThis vocabulary separates genuine sensitivity from low power.","\\item[Robust pass] when the lower confidence envelope exceeds\n \\(\\kappa_q\\) for every gate.\n\\item[Robust fail] when a required gate's upper confidence envelope is below\n \\(-\\kappa_q\\) throughout the relevant family.\n\\item[Specification conflict] when well-estimated admissible specifications\n lie on opposite sides of the indifference region.\n\\item[Statistically unresolved] when sampling uncertainty intersects the\n indifference region.\n\\end{description}\nThis vocabulary separates genuine sensitivity from low power.","\\item[Robust fail] when a required gate's upper confidence envelope is below\n \\(-\\kappa_q\\) throughout the relevant family.\n\\item[Specification conflict] when well-estimated admissible specifications\n lie on opposite sides of the indifference region.\n\\item[Statistically unresolved] when sampling uncertainty intersects the\n indifference region.\n\\end{description}\nThis vocabulary separates genuine sensitivity from low power.","A pointwise valid test does not remain valid after the analyst searches for the\nmost favorable target, horizon, null panel, or threshold. Consider\n\\(K\\) independent null \\(p\\)-values and select the smallest. Then\n\\[\n\\mathbb P\\left(\\min_{1\\leq k\\leq K}p_k\\leq\\alpha\\right)\n=\n1-(1-\\alpha)^K,\n\\]\nwhich exceeds \\(\\alpha\\) whenever \\(K>1\\). The same logic applies when a\nworkspace certificate is reported only for the specification that produced the\nlargest gate margins.","\\begin{definition}[Specification leakage]\nSpecification leakage occurs when certification outcomes influence any of the\nfollowing without a fresh inferential correction:\n\\[\nY^+,\n\\ell,\n\\tau,\n\\mathcal I,\n\\mathcal N,\n\\Theta,\n\\mathfrak S,\n\\mathcal J_*.\n\\]\nThe leakage status is separate from formation leakage. A workspace extractor\nmay be frozen while the analyst still searches over targets or thresholds on\nthe certification split.\n\\end{definition}","\\subtopic{66.2 \\(T_{32}\\): Sample-split selection theorem}","\\begin{theorem}[Validity after independent specification selection]\n\\label{thm:sample-split-spec-selection}\nLet \\(\\widehat\\varsigma\\) be selected using\n\\(\\mathcal D_{\\mathrm{sel}}\\), and let\n\\(\\mathcal D_{\\mathrm{cert}}\\) be independent of\n\\(\\mathcal D_{\\mathrm{sel}}\\). For every fixed\n\\(\\varsigma\\in\\mathfrak S\\), let\n\\(\\phi_\\varsigma(\\mathcal D_{\\mathrm{cert}})\\in\\{0,1\\}\\) be a certificate\nrule satisfying\n\\[\n\\sup_{P:\\operatorname{Cert}(W;\\varsigma)=0}\n\\mathbb P_P\\bigl(\\phi_\\varsigma=1\\bigr)\n\\leq\\alpha.\n\\]\nThen the selected-specification procedure satisfies\n\\[\n\\boxed{\n\\sup_P\n\\mathbb P_P\\left(\n\\phi_{\\widehat\\varsigma}=1,\n\\operatorname{Cert}(W;\\widehat\\varsigma)=0\n\\right)\n\\leq\\alpha.\n}\n\\]\n\\end{theorem}","em}[Validity after independent specification selection]\n\\label{thm:sample-split-spec-selection}\nLet \\(\\widehat\\varsigma\\) be selected using\n\\(\\mathcal D_{\\mathrm{sel}}\\), and let\n\\(\\mathcal D_{\\mathrm{cert}}\\) be independent of\n\\(\\mathcal D_{\\mathrm{sel}}\\). For every fixed\n\\(\\varsigma\\in\\mathfrak S\\), let\n\\(\\phi_\\varsigma(\\mathcal D_{\\mathrm{cert}})\\in\\{0,1\\}\\) be a certificate\nrule satisfying\n\\[\n\\sup_{P:\\operatorname{Cert}(W;\\varsigma)=0}\n\\mathbb P_P\\bigl(\\phi_\\varsigma=1\\bigr)\n\\leq\\alpha.\n\\]\nThen the selected-specification procedure satisfies\n\\[\n\\boxed{\n\\sup_P\n\\mathbb P_P\\left(\n\\phi_{\\widehat\\varsigma}=1,\n\\operatorname{Cert}(W;\\widehat\\varsigma)=0\n\\right)\n\\leq\\alpha.\n}\n\\]\n\\end{theorem}","\\mathrm{sel}}\\), and let\n\\(\\mathcal D_{\\mathrm{cert}}\\) be independent of\n\\(\\mathcal D_{\\mathrm{sel}}\\). For every fixed\n\\(\\varsigma\\in\\mathfrak S\\), let\n\\(\\phi_\\varsigma(\\mathcal D_{\\mathrm{cert}})\\in\\{0,1\\}\\) be a certificate\nrule satisfying\n\\[\n\\sup_{P:\\operatorname{Cert}(W;\\varsigma)=0}\n\\mathbb P_P\\bigl(\\phi_\\varsigma=1\\bigr)\n\\leq\\alpha.\n\\]\nThen the selected-specification procedure satisfies\n\\[\n\\boxed{\n\\sup_P\n\\mathbb P_P\\left(\n\\phi_{\\widehat\\varsigma}=1,\n\\operatorname{Cert}(W;\\widehat\\varsigma)=0\n\\right)\n\\leq\\alpha.\n}\n\\]\n\\end{theorem}","\\begin{proof}\nCondition on \\(\\mathcal D_{\\mathrm{sel}}\\). The selected specification is then\nfixed. Independence leaves the certification data distributed exactly as in\nthe fixed-specification test, whose false-pass probability is at most\n\\(\\alpha\\). Averaging the conditional bound over the selection data preserves\nthe same bound.\n\\end{proof}","A practical design may use:\n\\[\n\\mathcal D_{\\mathrm{form}}\n\\sqcup\n\\mathcal D_{\\mathrm{spec}}\n\\sqcup\n\\mathcal D_{\\mathrm{cert}}\n\\sqcup\n\\mathcal D_{\\mathrm{stress}}.\n\\]\nFormation learns the workspace. The specification split chooses targets,\nhorizons, nulls, and thresholds from a predeclared menu. Certification makes\nthe pass decision once. Stress testing explores broader shifts without\nretroactively changing the certified claim. Cross-fitting can recover sample\nefficiency, but the fold-level selection and evaluation paths must remain\nseparated and the final uncertainty calculation must account for recombination.","Threshold tuning can be as consequential as target selection. If the observed\ncausal effect is \\(0.12\\), choosing a minimum effect threshold of \\(0.10\\)\nafter seeing the estimate is not certification. Likewise, retaining only the\nnulls that the intact system defeats converts a falsification panel into a\nsuccess filter. Thresholds and null families must be frozen or evaluated under\na uniform rule that includes every considered alternative.","\\begin{enumerate}\n\\item State the reference specification \\(\\varsigma_0\\) in full.\n\\item Distinguish equivalent recodings, robustness alternatives, and genuinely\n different scope claims.\n\\item Declare the admissible family \\(\\mathfrak S\\) and justify every major\n exclusion before certification.\n\\item Record the specification pseudometric or other transport bounds used to\n compare family members.\n\\item Compute every gate's lower envelope, upper envelope, sensitivity width,\n and reference margin.\n\\item State the experiment family \\(\\mathcal E\\) and the model class\n \\(\\mathfrak M\\) relative to which identifiability is claimed.\n\\item Report identified sets or bounds whenever point identification fails.\n\\item Apply a uniform finite-sample rule over every gate and sp","\\begin{enumerate}\n\\item State the reference specification \\(\\varsigma_0\\) in full.\n\\item Distinguish equivalent recodings, robustness alternatives, and genuinely\n different scope claims.\n\\item Declare the admissible family \\(\\mathfrak S\\) and justify every major\n exclusion before certification.\n\\item Record the specification pseudometric or other transport bounds used to\n compare family members.\n\\item Compute every gate's lower envelope, upper envelope, sensitivity width,\n and reference margin.\n\\item State the experiment family \\(\\mathcal E\\) and the model class\n \\(\\mathfrak M\\) relative to which identifiability is claimed.\n\\item Report identified sets or bounds whenever point identification fails.\n\\item Apply a uniform finite-sample rule over every gate and specification used\n in the conclusion.\n\\item Preserve the complete specification-selection history and the data split\n on which each choice was made.\n\\item Report the","ternatives, and genuinely\n different scope claims.\n\\item Declare the admissible family \\(\\mathfrak S\\) and justify every major\n exclusion before certification.\n\\item Record the specification pseudometric or other transport bounds used to\n compare family members.\n\\item Compute every gate's lower envelope, upper envelope, sensitivity width,\n and reference margin.\n\\item State the experiment family \\(\\mathcal E\\) and the model class\n \\(\\mathfrak M\\) relative to which identifiability is claimed.\n\\item Report identified sets or bounds whenever point identification fails.\n\\item Apply a uniform finite-sample rule over every gate and specification used\n in the conclusion.\n\\item Preserve the complete specification-selection history and the data split\n on which each choice was made.\n\\item Report the strongest pointwise, family-robust, sign-identified, and\n statistically resolved conclusions separately.\n\\end{enumerate}","scope claims.\n\\item Declare the admissible family \\(\\mathfrak S\\) and justify every major\n exclusion before certification.\n\\item Record the specification pseudometric or other transport bounds used to\n compare family members.\n\\item Compute every gate's lower envelope, upper envelope, sensitivity width,\n and reference margin.\n\\item State the experiment family \\(\\mathcal E\\) and the model class\n \\(\\mathfrak M\\) relative to which identifiability is claimed.\n\\item Report identified sets or bounds whenever point identification fails.\n\\item Apply a uniform finite-sample rule over every gate and specification used\n in the conclusion.\n\\item Preserve the complete specification-selection history and the data split\n on which each choice was made.\n\\item Report the strongest pointwise, family-robust, sign-identified, and\n statistically resolved conclusions separately.\n\\end{enumerate}","The empirical null and audit panel is extended by:\n\\[\n\\begin{aligned}\nN_{29}&=\\text{post hoc target-selection audit},\\\\\nN_{30}&=\\text{horizon cherry-picking audit},\\\\\nN_{31}&=\\text{loss and utility substitution audit},\\\\\nN_{32}&=\\text{restricted intervention-family audit},\\\\\nN_{33}&=\\text{threshold and gate-set tuning audit},\\\\\nN_{34}&=\\text{frozen specification sweep across }\\mathfrak S.\n\\end{aligned}\n\\]\nThe first five reconstruct or bound the analysis search that preceded the\nreported claim. The final control reruns the complete frozen certificate across\nthe admissible family. A claim that survives \\(N_{34}\\) but fails the search\naudit remains vulnerable to selection bias. A claim that passes the search\naudit but changes sign across \\(\\mathfrak S\\) is validly estimated yet\nspecification-fragile.","N_{34}&=\\text{frozen specification sweep across }\\mathfrak S.\n\\end{aligned}\n\\]\nThe first five reconstruct or bound the analysis search that preceded the\nreported claim. The final control reruns the complete frozen certificate across\nthe admissible family. A claim that survives \\(N_{34}\\) but fails the search\naudit remains vulnerable to selection bias. A claim that passes the search\naudit but changes sign across \\(\\mathfrak S\\) is validly estimated yet\nspecification-fragile.","zen specification sweep across }\\mathfrak S.\n\\end{aligned}\n\\]\nThe first five reconstruct or bound the analysis search that preceded the\nreported claim. The final control reruns the complete frozen certificate across\nthe admissible family. A claim that survives \\(N_{34}\\) but fails the search\naudit remains vulnerable to selection bias. A claim that passes the search\naudit but changes sign across \\(\\mathfrak S\\) is validly estimated yet\nspecification-fragile.","The machine-readable record should include:\n\\[\n\\boxed{\n\\begin{aligned}\n\\mathsf{SpecificationRecord}\n=\\{&\n\\mathsf{reference},\n\\mathsf{family},\n\\mathsf{equivalence\\_classes},\n\\mathsf{metric},\\\\\n&\\mathsf{targets},\n\\mathsf{losses},\n\\mathsf{horizons},\n\\mathsf{interventions},\\\\\n&\\mathsf{null\\_panels},\n\\mathsf{thresholds},\n\\mathsf{selection\\_history},\\\\\n&\\mathsf{identified\\_sets},\n\\mathsf{uniform\\_bounds},\n\\mathsf{gate\\_envelopes}\n\\}.\n\\end{aligned}\n}\n\\]\nEvery omitted family member requires a reason. Every changed specification\nrequires a data-split label and time stamp. Every gate requires pointwise and\nfamily-wide status.","The terminal architecture is now explicit. A strong claim must survive three\nseparate questions:\n\\[\n\\boxed{\n\\begin{aligned}\n&\\text{Would it survive another defensible specification?}\\\\\n&\\text{Could a contrary realization generate the same experiments?}\\\\\n&\\text{Does the finite sample support every required margin uniformly?}\n\\end{aligned}\n}\n\\]\nOnce these questions are answered, further extensions concern new mathematical\nsettings or new empirical domains rather than a missing foundational layer.","\\subtopic{\\(T_1\\): Limit--holonomy theorem}\n\\[\n\\Gamma(H)\\cong C_o^{\\operatorname{Hol}_o}.\n\\]","\\subtopic{\\(T_2\\): Operator closure theorem}\n\\[\nH_e\\rho_i(a)=\\rho_j(a)H_e\n\\Longrightarrow\n\\rho_o(a)C_*\\subseteq C_*.\n\\]","\\subtopic{\\(T_{2a}\\): Workspace commutant theorem }\n\\[\n\\mathcal O_{\\Workspace}\n\\cong\n\\operatorname{Comm}(\\Hol_o).\n\\]","\\subtopic{\\(T_3\\): Transfer-Laplacian kernel theorem}\n\\[\n\\ker L_H=\\Gamma(H).\n\\]","\\subtopic{\\(T_4\\): Spectral stability theorem}\nIf \\(\\eta=\\|\\widetilde L_H-L_H\\|0\\), observations\n\\[\ny=A_Sc+\\eta\n\\]\nand canonical decoder \\(A_S^\\dagger\\) satisfy\n\\[\n\\boxed{\n\\|A_S^\\dagger y-c\\|\n\\leq\n\\frac{\\|\\eta\\|}{\\sqrt{\\alpha_S}}.\n}\n\\]\nThe theorem is extended to encoding mismatch, exact support-based erasure\ndistance, uniform node erasures, reconstructive handoff, exact coalition\nsufficiency, and operational risk under a Lipschitz predictive kernel.","\\subtopic{\\(T_8\\): Exact gluing theorem}","For every finite edge-complete workspace cover,\n\\[\n\\boxed{\n0\n\\longrightarrow\n\\mathscr W(G)\n\\xrightarrow{\\ R_{\\mathcal U}\\ }\n\\check C^0(\\mathcal U;\\mathscr W)\n\\xrightarrow{\\ \\delta^0\\ }\n\\check C^1(\\mathcal U;\\mathscr W)\n}\n\\]\nis exact. Hence\n\\[\n\\boxed{\n\\mathscr W(G)\n\\cong\n\\ker\\delta^0\n\\cong\n\\operatorname{Eq}\n\\left(\n\\prod_\\alpha\\mathscr W_\\alpha\n\\rightrightarrows\n\\prod_{\\alpha<\\beta}\\mathscr W_{\\alpha\\beta}\n\\right).\n}\n\\]\nThe same analysis also yields a cover-Laplacian distance bound, a total descent kernel\ntheorem, and a harmonic \\(\\check H^1\\) obstruction for relative-only data.","\\mathcal U;\\mathscr W)\n\\xrightarrow{\\ \\delta^0\\ }\n\\check C^1(\\mathcal U;\\mathscr W)\n}\n\\]\nis exact. Hence\n\\[\n\\boxed{\n\\mathscr W(G)\n\\cong\n\\ker\\delta^0\n\\cong\n\\operatorname{Eq}\n\\left(\n\\prod_\\alpha\\mathscr W_\\alpha\n\\rightrightarrows\n\\prod_{\\alpha<\\beta}\\mathscr W_{\\alpha\\beta}\n\\right).\n}\n\\]\nThe same analysis also yields a cover-Laplacian distance bound, a total descent kernel\ntheorem, and a harmonic \\(\\check H^1\\) obstruction for relative-only data.","\\subtopic{\\(T_9\\): Hierarchical collapse theorem}","For every finite rooted hierarchy whose children edge-cover their parent,\n\\[\n\\boxed{\nW_a\\cong\\mathscr W(G_a)\n\\quad\\text{for every hierarchy node }a,\n}\n\\]\nand in particular\n\\[\n\\boxed{W_{\\mathrm{root}}\\cong\\mathscr W(G).}\n\\]\nIf level \\(r\\) contributes defect \\(\\delta_r\\) and downstream collapse maps\nhave Lipschitz constants \\(L_s\\), then\n\\[\n\\boxed{\n\\delta_{\\mathrm{flat}}^{(R)}\n\\leq\n\\sum_{r=0}^{R-1}\n\\left(\n\\prod_{s=r+1}^{R-1}L_s\n\\right)\\delta_r.\n}\n\\]","\\subtopic{\\(T_{25}\\): Faithful coarse-graining theorem}","For a partition of the vertex set, let \\(W_\\Pi\\) be the direct sum of the\ninternally coherent section spaces and let \\(D_\\Pi\\) retain all boundary\ncompatibility equations. Then\n\\[\n\\boxed{\n\\Gamma(H)\n=\nJ_\\Pi\\ker D_\\Pi.\n}\n\\]\nExact structural workspace content is therefore invariant under faithful\naggregation. Approximate error separates into an internal-coherence term and\na boundary-coherence term controlled by their respective spectral gaps.","\\subtopic{\\(T_{26}\\): Bayes-risk transport theorem}","For an information-faithful coarsening \\(q:V_f\\to V_c\\),\n\\[\n\\boxed{\n\\Risk_{\\mathcal D_c}(S)\n=\n\\Risk_{\\mathcal D_f}(q^{-1}(S))\n\\qquad(S\\subseteq V_c).\n}\n\\]\nHence displayed coalition size can decrease under coarsening. Weighted\nprimitive support cost is a more stable localization measure and cannot\nimprove merely because several carriers were relabeled as one macro-node.","\\subtopic{\\(T_{27}\\): Decomposition-transport robustness theorem}","If every gate margin at a reference decomposition exceeds its worst-case\ntransport and model-mismatch defect across \\(\\mathfrak D\\), then the complete\ncertificate passes throughout the admissible family:\n\\[\n\\boxed{\n m_q(W;\\mathcal D_0)\n>\n\\sup_{\\mathcal D\\in\\mathfrak D}\n\\bigl(L_q\\delta_{\\mathcal D}+b_q(\\mathcal D)\\bigr)\n\\quad\\forall q\n\\Longrightarrow\n\\operatorname{Certify}_{L,\\mathfrak D}^{\\mathrm{rob}}(W)=1.\n}\n\\]","\\subtopic{\\(T_{10}\\): Approximate universality theorem }","For the first \\(d\\) transfer-Laplacian modes, with\n\\(\\lambda_d\\leq\\varepsilon<\\lambda_{d+1}=\\mu\\),\n\\[\n\\boxed{\n\\|(I-P_d)\\rho_V(a)P_d\\|\n\\leq\n\\frac{\\|\\rho_E(a)\\|\\sqrt{\\varepsilon}+\\beta(a)}{\\sqrt{\\mu}}.\n}\n\\]\nThus low transfer energy, a separated complement, and small intertwining defect\nimply approximate closure under the common operation algebra.","\\subtopic{\\(T_{11}\\): Core--residual accumulation theorem }","For a path \\(\\gamma=e_m\\cdots e_1\\),\n\\[\n\\boxed{\n\\begin{aligned}\n\\|Q_{t(\\gamma)}T_\\gamma P_{s(\\gamma)}\\|\n\\leq\n\\sum_{r=1}^{m}\n&\\left(\n\\prod_{\\ell=r+1}^{m}\\|T_{e_\\ell}\\|\n\\right)\n\\|V_{e_r}\\|","For a path \\(\\gamma=e_m\\cdots e_1\\),\n\\[\n\\boxed{\n\\begin{aligned}\n\\|Q_{t(\\gamma)}T_\\gamma P_{s(\\gamma)}\\|\n\\leq\n\\sum_{r=1}^{m}\n&\\left(\n\\prod_{\\ell=r+1}^{m}\\|T_{e_\\ell}\\|\n\\right)\n\\|V_{e_r}\\|\\\\\n&\\times\n\\left(\n\\prod_{\\ell=1}^{r-1}\\|H_{e_\\ell}\\|\n\\right).\n\\end{aligned}\n}\n\\]\nThe same bound controls the defect between full core-to-core path transport and\nthe pure core product. An exact Duhamel\nidentity yields both estimates.","\\subtopic{\\(T_{12}\\): Predictive-state minimality and alignment theorem }","On standard Borel spaces, the canonical conditional-law state\n\\[\nZ_t\n=\n\\mathbb P(Y_t^+\\in\\cdot\\mid\\mathbf H_t)\n\\]\nis sufficient and minimal:\n\\[\nY_t^+\\indep\\mathbf H_t\\mid Z_t,\n\\]\nand every sufficient statistic \\(S_t=s(\\mathbf H_t)\\) satisfies\n\\[\nZ_t=\\phi(S_t)\n\\qquad\\text{a.s.}\n\\]\nMoreover, a structural candidate \\(W_t\\) is exactly operationally aligned if\nand only if\n\\[\nZ_t=\\phi_W(W_t)\n\\qquad\\text{a.s.}\n\\]\nThe countable intervention-indexed version holds for\n\\(Z_t^{\\mathcal I}\\).","Z_t\n=\n\\mathbb P(Y_t^+\\in\\cdot\\mid\\mathbf H_t)\n\\]\nis sufficient and minimal:\n\\[\nY_t^+\\indep\\mathbf H_t\\mid Z_t,\n\\]\nand every sufficient statistic \\(S_t=s(\\mathbf H_t)\\) satisfies\n\\[\nZ_t=\\phi(S_t)\n\\qquad\\text{a.s.}\n\\]\nMoreover, a structural candidate \\(W_t\\) is exactly operationally aligned if\nand only if\n\\[\nZ_t=\\phi_W(W_t)\n\\qquad\\text{a.s.}\n\\]\nThe countable intervention-indexed version holds for\n\\(Z_t^{\\mathcal I}\\).","\\subtopic{\\(T_{13}\\): Operational relationality theorem }","For a finite target, exact alignment, a positive node-level conditional\ninformation gap, a positive robust local-predictive-matched null gap, a feasible\nworkspace intervention effect, and positive matched specificity jointly imply\nthat:\n\\[\n\\begin{aligned}\n&\\text{no single-node statistic is sufficient},\\\\\n&k_0\\geq2,\\\\\n&\\text{no single node reconstructs the aligned workspace},\\\\\n&\\text{matched local nulls preserve all node risks but lose joint capacity},\\\\\n&\\text{and the workspace coordinate is causally efficacious at the tested scale.}\n\\end{aligned}\n\\]\nIf \\(\\eta_{\\mathrm{loop}}(\\tau)>0\\), the conclusion additionally includes\nloop-mediated reentry.","&\\text{and the workspace coordinate is causally efficacious at the tested scale.}\n\\end{aligned}\n\\]\nIf \\(\\eta_{\\mathrm{loop}}(\\tau)>0\\), the conclusion additionally includes\nloop-mediated reentry.","\\subtopic{\\(T_{14}\\): Replacement invariance theorem }","For a unitary realization correspondence with discrepancy defect\n\\[\n\\delta_D=\\|D'J_V-J_ED\\|,\n\\]\nset\n\\[\n\\eta_L=2\\|D\\|\\delta_D+\\delta_D^2.\n\\]\nIf \\(\\eta_L0\\), and held-out uniform score error at most \\(\\varepsilon\\),\n\\[\n\\boxed{\nJ(\\widehat\\theta)-J(\\theta_*)\\leq2\\varepsilon.\n}\n\\]\nIf \\(2\\varepsilon<\\Delta_J\\), then \\(\\widehat\\theta=\\theta_*\\). The theorem\napplies to node membership, edge selection, gauge-normalized transfer models,\nand finite boundary families.","\\subtopic{\\(T_{17}\\): Stable boundary recovery theorem }","For a finite admissible boundary family with unique maximizer \\(S_t^*\\), margin\n\\(\\Delta_t^B\\), and uniform held-out score error \\(\\varepsilon_t\\),\n\\[\n\\mathcal B_t(S_t^*)-\n\\mathcal B_t(\\widehat S_t)\n\\leq2\\varepsilon_t,\n\\]\nand exact recovery holds when \\(2\\varepsilon_t<\\Delta_t^B\\). If the population\nscore drifts uniformly by at most \\(\\delta_t\\), then\n\\[\n2\\delta_t<\\Delta_t^B\n\\Longrightarrow\nS_{t+1}^*=S_t^*.\n\\]","\\subtopic{\\(T_{18}\\): Model-specific formation transition theorem }","For\n\\[\nm_{t+1}=\\tanh(\\lambda m_t),\n\\]\nthere is a unique zero fixed point for \\(0\\leq\\lambda\\leq1\\). For\n\\(\\lambda>1\\), the zero state is unstable and exactly two stable nonzero fixed\npoints appear. Their amplitude satisfies\n\\[\n\\boxed{\nm_\\lambda\n=\n\\sqrt{\\frac{3(\\lambda-1)}{\\lambda^3}}\n+\nO\\bigl((\\lambda-1)^{3/2}\\bigr).\n}\n\\]\nThis is an exact structural benchmark, not a universal law for complete\nrelational workspaces.","\\subtopic{\\(T_{19}\\): Cue-controlled goal-retention theorem }","For matched future command and disturbance trajectories under\n\\[\nG_{s+1}=R_sG_s+B_sC_{s+1}+\\zeta_{s+1},\n\\]\ntwo goal interventions satisfy\n\\[\n\\boxed{\nG_{t+\\tau}-G_{t+\\tau}'\n=\nR_{t+\\tau:t}(g-g').\n}\n\\]\nIf\n\\[\nm_s\\leq\\sigma_{\\min}(R_s),\n\\qquad\n\\|R_s\\|\\leq M_s,\n\\]\nthen\n\\[\n\\boxed{\n\\left(\\prod_s m_s\\right)\\|g-g'\\|\n\\leq\n\\|G_{t+\\tau}-G_{t+\\tau}'\\|\n\\leq\n\\left(\\prod_s M_s\\right)\\|g-g'\\|.\n}\n\\]\nA linear action readout gives a corresponding lower bound on the induced\naction-mean separation. The result isolates internal retention from repeated\nexternal goal rewriting.","\\subtopic{\\(T_{20}\\): Approximate counterfactual-policy theorem }","If the interventional action-value estimator satisfies\n\\[\n\\sup_a|\\widehat Q_t(a)-Q_t(a)|\\leq\\varepsilon_Q\n\\]\nand the implemented policy has optimization error at most\n\\(\\delta_{\\mathrm{opt}}\\) relative to \\(\\widehat Q_t\\), then\n\\[\n\\boxed{\n\\operatorname{Reg}_t(\\widehat\\pi_t)\n\\leq\n2\\varepsilon_Q+\\delta_{\\mathrm{opt}}.\n}\n\\]\nFor deterministic greedy action, the bound is \\(2\\varepsilon_Q\\). A unique\ntrue optimum with value gap greater than \\(2\\varepsilon_Q\\) is recovered\nexactly. The guarantee is relative to the declared utility, feasible action\nsupport, causal identification, and horizon.","tor satisfies\n\\[\n\\sup_a|\\widehat Q_t(a)-Q_t(a)|\\leq\\varepsilon_Q\n\\]\nand the implemented policy has optimization error at most\n\\(\\delta_{\\mathrm{opt}}\\) relative to \\(\\widehat Q_t\\), then\n\\[\n\\boxed{\n\\operatorname{Reg}_t(\\widehat\\pi_t)\n\\leq\n2\\varepsilon_Q+\\delta_{\\mathrm{opt}}.\n}\n\\]\nFor deterministic greedy action, the bound is \\(2\\varepsilon_Q\\). A unique\ntrue optimum with value gap greater than \\(2\\varepsilon_Q\\) is recovered\nexactly. The guarantee is relative to the declared utility, feasible action\nsupport, causal identification, and horizon.","\\subtopic{\\(T_{21}\\): Repair-selection margin theorem }","For a finite repair set and a repair-value estimator with uniform error\n\\(\\varepsilon_R\\), the selected repair \\(\\widehat r\\) satisfies\n\\[\n\\boxed{\nV(r^*)-V(\\widehat r)\n\\leq\n2\\varepsilon_R.\n}\n\\]\nIf the optimal repair reaches reserve \\(\\rho+2\\varepsilon_R\\), then the selected\nrepair reaches reserve \\(\\rho\\). A unique optimal repair whose value gap\nexceeds \\(2\\varepsilon_R\\) is selected exactly.","\\subtopic{\\(T_{22}\\): Closed-loop self-maintenance theorem }","If matched no-repair deficit obeys\n\\[\ne_{t+1}^{(0)}\\leq\\alpha e_t+d_t\n\\]\nand the implemented repair has measured causal effect\n\\[\ne_{t+1}^{(0)}-e_{t+1}\\geq\\kappa e_t-\\epsilon_t,\n\\]\nset \\(\\beta=\\alpha-\\kappa\\). Under\n\\[\n0\\leq\\beta<1,\n\\qquad\n d_t\\leq\\overline d,\n\\qquad\n \\epsilon_t\\leq\\overline\\epsilon,\n\\]\n\\[\n\\boxed{\ne_t\n\\leq\n\\beta^te_0\n+\n\\frac{(\\overline d+\\overline\\epsilon)(1-\\beta^t)}{1-\\beta}.\n}\n\\]\nHence\n\\[\n\\limsup_{t\\to\\infty}e_t\n\\leq\n\\frac{\\overline d+\\overline\\epsilon}{1-\\beta}.\n\\]\nA corresponding margin condition makes the declared bounded-deficit viability\nset forward invariant.","no-repair deficit obeys\n\\[\ne_{t+1}^{(0)}\\leq\\alpha e_t+d_t\n\\]\nand the implemented repair has measured causal effect\n\\[\ne_{t+1}^{(0)}-e_{t+1}\\geq\\kappa e_t-\\epsilon_t,\n\\]\nset \\(\\beta=\\alpha-\\kappa\\). Under\n\\[\n0\\leq\\beta<1,\n\\qquad\n d_t\\leq\\overline d,\n\\qquad\n \\epsilon_t\\leq\\overline\\epsilon,\n\\]\n\\[\n\\boxed{\ne_t\n\\leq\n\\beta^te_0\n+\n\\frac{(\\overline d+\\overline\\epsilon)(1-\\beta^t)}{1-\\beta}.\n}\n\\]\nHence\n\\[\n\\limsup_{t\\to\\infty}e_t\n\\leq\n\\frac{\\overline d+\\overline\\epsilon}{1-\\beta}.\n\\]\nA corresponding margin condition makes the declared bounded-deficit viability\nset forward invariant.","\\subtopic{\\(T_{23}\\): Workspace-agency synthesis theorem }","Operational relationality, cue-controlled and behaviorally operative goals,\ncalibrated counterfactual values, feasible value-to-action and action-to-outcome\ninterventions, causal closed-loop maintenance, distributed control-state\nsufficiency, and matched local policy nulls jointly imply:\n\\[\n\\begin{aligned}\n&\\operatorname{Reg}_t\\leq2\\varepsilon_Q+\\delta_{\\mathrm{opt}},\\\\\n&\\limsup e_t\\leq\n \\frac{\\overline d+\\overline\\epsilon}{1-\\beta},\\\\\n&k_{\\mathrm{ctl},\\varepsilon}\\geq2,\n\\end{aligned}\n\\]\ntogether with causal mediation of goal-dependent action and repair through the\nimplemented internal channels. This is a sufficient, domain-relative agency\ncertificate. It does not imply consciousness, personhood, or unrestricted\nautonomy.","\\subtopic{\\(T_{24}\\): Gatewise certification-validity theorem }","Suppose a certificate requires every gate \\(G_1,\\ldots,G_m\\) to pass, and the\npredeclared test for gate \\(j\\) has false-pass probability at most \\(\\alpha_j\\)\nwhenever that required property is absent. Under the global null that at least\none required property is absent,\n\\[\n\\boxed{\n\\mathbb P(\\text{complete certificate passes})\n\\leq\n\\max_j\\alpha_j.\n}\n\\]\nNo independence assumption among gate tests is required. The guarantee fails\nunder adaptive gate deletion, selective reporting, or reuse of the\ncertification split.","\\subtopic{\\(T_{25}\\): Faithful coarse-graining theorem}\nFor an exact partition into internally coherent macro-components,\n\\[\n\\boxed{\n\\ker D_H=J_\\Pi\\ker D_\\Pi.\n}\n\\]\nThus faithful aggregation preserves the exact shared workspace and its\ndimension while changing the displayed node resolution.","\\subtopic{\\(T_{26}\\): Bayes-risk transport under aggregation}\nFor an information-faithful coarsening map \\(q\\),\n\\[\n\\boxed{\n\\Risk_{\\mathcal D_c}(S)\n=\n\\Risk_{\\mathcal D_f}\\bigl(q^{-1}(S)\\bigr).\n}\n\\]\nPredictive content is preserved, although the visible coalition count may\ncontract. Primitive-carrier support cost is therefore preferred when claims\nmust be compared across resolutions.","\\subtopic{\\(T_{27}\\): Decomposition-transport robustness theorem}\nIf every required gate has reference margin larger than its worst admissible\ndecomposition-transport defect, then the complete certificate passes\nthroughout \\(\\mathfrak D\\). When the mismatch term vanishes, the certificate\nradius is bounded below by the smallest gate margin divided by its transport\nsensitivity.","\\subtopic{\\(T_{28}\\): Specification-transport robustness theorem}\nFor a reference specification \\(\\varsigma_0\\), if\n\\[\nm_q(W;\\varsigma_0)\n>\n\\sup_{\\varsigma\\in\\mathfrak S}\n\\left[\nL_qd_{\\mathfrak S}(\\varsigma,\\varsigma_0)+b_q(\\varsigma)\n\\right]\n\\]\nfor every gate, then the transported workspace passes every admissible\nspecification. If \\(b_q=0\\),\n\\[\nr_{\\mathrm{spec}}(W;\\varsigma_0)\n\\geq\n\\min_{q:L_q>0}\\frac{m_q(W;\\varsigma_0)}{L_q}.\n\\]","\\subtopic{\\(T_{29}\\): Experimental non-identifiability theorem}\nIf two realizations induce the same law under every declared observation and\nintervention but have different certificate values, no decision rule based on\nthat experiment can be uniformly correct. For every such rule, the two error\nprobabilities sum to one, so the minimax error is at least one half.","\\subtopic{\\(T_{30}\\): Joint specification-identification theorem}\nLet \\(r_{\\mathfrak S}(\\mathfrak R)\\) be the worst gate margin across the\nspecification family. If its infimum over the identified realization set is\npositive, the robust certificate is identified as passing. If its supremum is\nnonpositive, the certificate is identified as failing. Compatible realizations\non opposite sides of zero establish non-identifiability.","\\subtopic{\\(T_{31}\\): Uniform finite-sample specification theorem}\nIf all gate estimators are uniformly within their error radii across\n\\(\\mathfrak S\\), then requiring every empirical lower envelope to be positive\ncontrols the probability of a false robust certificate by the failure\nprobability of the uniform event.","\\subtopic{\\(T_{32}\\): Independent selection and certification theorem}\nA specification chosen on data independent of the certification split may be\nevaluated by its fixed-specification test without inflating the false-pass\nbound. Selection and certification on the same outcomes require simultaneous\nor post-selection correction.","The symbolic outline proposed a coupling threshold \\(\\lambda_c\\). A general\ntheory must first state what is allowed to vary with \\(\\lambda\\). A threshold\nmay concern structural coherence, causal efficacy, relational irreducibility,\nor the conjunction of all certification gates. These need not occur at the\nsame parameter value.","\\begin{proposition}[Threshold values are parameterization dependent]\n\\label{prop:threshold-reparam}\nSuppose a model indexed by \\(\\lambda\\) passes a declared formation certificate\nexactly for \\(\\lambda>\\lambda_c\\). For every strictly increasing bijection\n\\(f\\), the reparameterized family indexed by \\(\\mu=f(\\lambda)\\) has threshold\n\\[\n\\mu_c=f(\\lambda_c).\n\\]\nConsequently, the numerical value of a coupling threshold is not invariant\nwithout a physically or operationally fixed parameterization.\n\\end{proposition}","\\begin{proof}\nThe reparameterized realization at \\(\\mu\\) is the original realization at\n\\(f^{-1}(\\mu)\\). It passes precisely when\n\\(f^{-1}(\\mu)>\\lambda_c\\), equivalently when\n\\(\\mu>f(\\lambda_c)\\).\n\\end{proof}","There can therefore be no model-independent numerical value of \\(\\lambda_c\\).\nThe invariant object is the change in the certified organization and its local\ncritical behavior, not the label placed on the control parameter.","For a one-parameter family \\(\\mathfrak R_\\lambda\\), define\n\\[\n\\begin{aligned}\n\\lambda_{\\mathrm{str}}\n&=\n\\inf\\{\\lambda:\n d_{\\varepsilon}(\\lambda)>0,\n g_{\\varepsilon}(\\lambda)>0\\},","For a one-parameter family \\(\\mathfrak R_\\lambda\\), define\n\\[\n\\begin{aligned}\n\\lambda_{\\mathrm{str}}\n&=\n\\inf\\{\\lambda:\n d_{\\varepsilon}(\\lambda)>0,\n g_{\\varepsilon}(\\lambda)>0\\},\\\\\n\\lambda_{\\mathrm{op}}\n&=\n\\inf\\{\\lambda:\n \\delta_{\\mathrm{align}}(\\lambda)\n \\leq\\varepsilon_{\\mathrm{op}}\\},\\\\\n\\lambda_{\\mathrm{causal}}\n&=\n\\inf\\{\\lambda:\n \\eta_{\\mathrm{causal}}(\\lambda)\n \\geq\\tau_{\\mathrm{causal}}\\},\\\\\n\\lambda_{\\mathrm{rel}}\n&=\n\\inf\\{\\lambda:\n k_\\varepsilon(\\lambda)\\geq2,\n \\Delta_{\\mathrm{tr}}(\\lambda)\n \\geq\\tau_{\\mathrm{tr}}\\},\\\\\n\\lambda_{\\mathrm{form}}\n&=\n\\inf\\{\\lambda:\n \\operatorname{Form}_{\\Theta_{\\mathrm{form}}}\n (W_\\lambda)=1\\}.\n\\end{aligned}\n\\]\nIn general,\n\\[\n\\lambda_{\\mathrm{str}},\n\\lambda_{\\mathrm{op}},\n\\lambda_{\\mathrm{causal}},\n\\lambda_{\\mathrm{rel}},\n\\lambda_{\\mathrm{form}}\n\\]\ncan be distinct. A synchronized representation can appear before it becomes\ncausally useful, and causal coordination can appear while remaining reducible\nto one centralized component.","\\lambda_{\\mathrm{form}}\n&=\n\\inf\\{\\lambda:\n \\operatorname{Form}_{\\Theta_{\\mathrm{form}}}\n (W_\\lambda)=1\\}.\n\\end{aligned}\n\\]\nIn general,\n\\[\n\\lambda_{\\mathrm{str}},\n\\lambda_{\\mathrm{op}},\n\\lambda_{\\mathrm{causal}},\n\\lambda_{\\mathrm{rel}},\n\\lambda_{\\mathrm{form}}\n\\]\ncan be distinct. A synchronized representation can appear before it becomes\ncausally useful, and causal coordination can appear while remaining reducible\nto one centralized component.","\\subtopic{71.3 A solvable nonlinear formation model \\(T_{18}\\) }","A minimal exact model is the scalar recurrent coherence equation\n\\[\n\\boxed{\nm_{t+1}=\\tanh(\\lambda m_t),\n\\qquad\n\\lambda\\geq0.\n}\n\\]\nThe order parameter \\(m_t\\) represents a signed coherent mode. This model is\nnot a complete relational workspace. It isolates one mechanism by which a\nnonzero shared mode can appear continuously from a symmetric zero state.","\\begin{theorem}[Pitchfork formation transition]\\label{thm:pitchfork-formation}\nFor the map\n\\[\nF_\\lambda(m)=\\tanh(\\lambda m),\n\\qquad\nm\\in[-1,1],\n\\]\nthe following hold.\n\\begin{enumerate}\n\\item If \\(0\\leq\\lambda\\leq1\\), the unique fixed point is \\(m=0\\).\n It is globally attracting for \\(\\lambda<1\\) and nonexpanding at\n \\(\\lambda=1\\).\n\\item If \\(\\lambda>1\\), there are exactly three fixed points\n \\[\n -m_\\lambda,\n \\quad0,\n \\quad m_\\lambda,\n \\qquad\n m_\\lambda\\in(0,1).\n \\]\n The zero fixed point is unstable and the two nonzero fixed points are\n locally asymptotically stable.\n\\item As \\(\\lambda\\downarrow1\\),\n \\[\n \\boxed{\n m_\\lambda\n =\n \\sqrt{\\frac{3(\\lambda-1)}{\\lambda^3}}\n +\n O\\bigl((\\lambda-1)^{3/2}\\bigr).\n }\n \\]\n\\end{enumerate}\n\\end{theorem}","\\item If \\(0\\leq\\lambda\\leq1\\), the unique fixed point is \\(m=0\\).\n It is globally attracting for \\(\\lambda<1\\) and nonexpanding at\n \\(\\lambda=1\\).\n\\item If \\(\\lambda>1\\), there are exactly three fixed points\n \\[\n -m_\\lambda,\n \\quad0,\n \\quad m_\\lambda,\n \\qquad\n m_\\lambda\\in(0,1).\n \\]\n The zero fixed point is unstable and the two nonzero fixed points are\n locally asymptotically stable.\n\\item As \\(\\lambda\\downarrow1\\),\n \\[\n \\boxed{\n m_\\lambda\n =\n \\sqrt{\\frac{3(\\lambda-1)}{\\lambda^3}}\n +\n O\\bigl((\\lambda-1)^{3/2}\\bigr).\n }\n \\]\n\\end{enumerate}\n\\end{theorem}","leq1\\), the unique fixed point is \\(m=0\\).\n It is globally attracting for \\(\\lambda<1\\) and nonexpanding at\n \\(\\lambda=1\\).\n\\item If \\(\\lambda>1\\), there are exactly three fixed points\n \\[\n -m_\\lambda,\n \\quad0,\n \\quad m_\\lambda,\n \\qquad\n m_\\lambda\\in(0,1).\n \\]\n The zero fixed point is unstable and the two nonzero fixed points are\n locally asymptotically stable.\n\\item As \\(\\lambda\\downarrow1\\),\n \\[\n \\boxed{\n m_\\lambda\n =\n \\sqrt{\\frac{3(\\lambda-1)}{\\lambda^3}}\n +\n O\\bigl((\\lambda-1)^{3/2}\\bigr).\n }\n \\]\n\\end{enumerate}\n\\end{theorem}","\\begin{proof}\nA fixed point satisfies\n\\[\n\\operatorname{artanh}(m)=\\lambda m.\n\\]\nFor \\(m>0\\), the function\n\\[\nh(m)=\\frac{\\operatorname{artanh}(m)}{m}\n\\]\nis strictly increasing from \\(1\\) to \\(+\\infty\\). Hence there is no positive\nsolution when \\(\\lambda\\leq1\\), and exactly one positive solution when\n\\(\\lambda>1\\). Odd symmetry gives the negative solution.","The derivative is\n\\[\nF_\\lambda'(m)\n=\n\\lambda\\operatorname{sech}^2(\\lambda m).\n\\]\nAt zero this equals \\(\\lambda\\), proving stability for \\(\\lambda<1\\) and\ninstability for \\(\\lambda>1\\). At a nonzero fixed point,\n\\[\nF_\\lambda'(m_\\lambda)\n=\n\\lambda(1-m_\\lambda^2)\n=\n\\frac{(1-m_\\lambda^2)\\operatorname{artanh}(m_\\lambda)}{m_\\lambda}\n<1,\n\\]\nbecause\n\\[\n\\operatorname{artanh}(m)\n<\n\\frac{m}{1-m^2}\n\\qquad(01\\). At a nonzero fixed point,\n\\[\nF_\\lambda'(m_\\lambda)\n=\n\\lambda(1-m_\\lambda^2)\n=\n\\frac{(1-m_\\lambda^2)\\operatorname{artanh}(m_\\lambda)}{m_\\lambda}\n<1,\n\\]\nbecause\n\\[\n\\operatorname{artanh}(m)\n<\n\\frac{m}{1-m^2}\n\\qquad(0\\lambda_c\n\\Longrightarrow\n\\operatorname{Form}_{\\Theta_{\\mathrm{form}}}(W_\\lambda)=1\n\\]\nwith a stable nonzero structural mode, positive causal efficacy, and\nrelational irreducibility. The theorem above establishes this only for one\nstructural order parameter in a solvable nonlinear model. The conjunction of\nall workspace gates remains model dependent.","For appropriate recurrent systems,\n\\[\nW^{(r+1)}\n=\n\\Gamma_{\\varepsilon_r}(W^{(r)})\n\\]\nmay converge to a stable workspace type:\n\\[\nd_{\\mathrm{id}}\\!\\left(\nW^{(r+1)},W^{(r)}\n\\right)\n\\to0.\n\\]","Let\n\\[\nB_t^{\\mathrm{sys}}\n\\]\nbe an internal membership or routing state used by the system, and let\n\\[\nS_t^*\n=\n\\argmax_{S\\in\\mathfrak S_{\\mathrm{pass}}}\n\\mathcal B_t(S)\n\\]\nbe the population boundary defined by the external certification protocol.\nThe strong endogenous boundary conjecture asserts that there are adaptive\nsystems and nontrivial environments for which:\n\\[\n\\boxed{\n\\begin{aligned}\n&d_B(B_t^{\\mathrm{sys}},S_t^*)\\to0,","Let\n\\[\nB_t^{\\mathrm{sys}}\n\\]\nbe an internal membership or routing state used by the system, and let\n\\[\nS_t^*\n=\n\\argmax_{S\\in\\mathfrak S_{\\mathrm{pass}}}\n\\mathcal B_t(S)\n\\]\nbe the population boundary defined by the external certification protocol.\nThe strong endogenous boundary conjecture asserts that there are adaptive\nsystems and nontrivial environments for which:\n\\[\n\\boxed{\n\\begin{aligned}\n&d_B(B_t^{\\mathrm{sys}},S_t^*)\\to0,\\\\\n&\\operatorname{Form}_{\\Theta_{\\mathrm{form}}}(W_{B_t^{\\mathrm{sys}}})=1,\\\\\n&\\operatorname{CE}\n\\left(\nB_t^{\\mathrm{sys}}\\to\n\\operatorname{Routing}_{t+1}\n\\right)>0,\\\\\n&\\operatorname{CE}\n\\left(\n\\operatorname{BoundaryDamage}_t\\to\n\\operatorname{RepairAction}_{t+1}\n\\right)>0,\n\\end{aligned}\n}\n\\]\nwhile the scaffold contains no direct encoding of \\(S_t^*\\).","&\\operatorname{CE}\n\\left(\n\\operatorname{BoundaryDamage}_t\\to\n\\operatorname{RepairAction}_{t+1}\n\\right)>0,\n\\end{aligned}\n}\n\\]\nwhile the scaffold contains no direct encoding of \\(S_t^*\\).","Before the certification split is opened, record:\n\\begin{enumerate}\n\\item the full reference specification\n \\(\\varsigma_0=(\\mathcal D,Y^+,\\ell,\\tau,\\mathcal I,\\mathcal N,\\Theta)\\);\n\\item the admissible decomposition family \\(\\mathfrak D\\) and specification\n family \\(\\mathfrak S\\);\n\\item the target trajectory \\(Y^+\\), loss, horizon, and environment family;\n\\item the candidate structural workspace extractor;\n\\item the goal family \\(\\mathcal G\\), utility normalization, and feasible\n action set;\n\\item the intervention families \\(\\mathcal I_W,\\mathcal I_G,\\mathcal I_A,\n \\mathcal I_D,\\mathcal I_R\\);\n\\item the null panel and the local distributions each null must preserve;\n\\item the required certificate level and every numerical threshold;\n\\item the formation, calibration, certification, and stress-test splits;\n\\item the uncertainty procedure and resampling unit; and\n\\ite","Before the certification split is opened, record:\n\\begin{enumerate}\n\\item the full reference specification\n \\(\\varsigma_0=(\\mathcal D,Y^+,\\ell,\\tau,\\mathcal I,\\mathcal N,\\Theta)\\);\n\\item the admissible decomposition family \\(\\mathfrak D\\) and specification\n family \\(\\mathfrak S\\);\n\\item the target trajectory \\(Y^+\\), loss, horizon, and environment family;\n\\item the candidate structural workspace extractor;\n\\item the goal family \\(\\mathcal G\\), utility normalization, and feasible\n action set;\n\\item the intervention families \\(\\mathcal I_W,\\mathcal I_G,\\mathcal I_A,\n \\mathcal I_D,\\mathcal I_R\\);\n\\item the null panel and the local distributions each null must preserve;\n\\item the required certificate level and every numerical threshold;\n\\item the formation, calibration, certification, and stress-test splits;\n\\item the uncertainty procedure and resampling unit; and\n\\item the stopping and model-selection rules;\n\\item the experim","Before the certification split is opened, record:\n\\begin{enumerate}\n\\item the full reference specification\n \\(\\varsigma_0=(\\mathcal D,Y^+,\\ell,\\tau,\\mathcal I,\\mathcal N,\\Theta)\\);\n\\item the admissible decomposition family \\(\\mathfrak D\\) and specification\n family \\(\\mathfrak S\\);\n\\item the target trajectory \\(Y^+\\), loss, horizon, and environment family;\n\\item the candidate structural workspace extractor;\n\\item the goal family \\(\\mathcal G\\), utility normalization, and feasible\n action set;\n\\item the intervention families \\(\\mathcal I_W,\\mathcal I_G,\\mathcal I_A,\n \\mathcal I_D,\\mathcal I_R\\);\n\\item the null panel and the local distributions each null must preserve;\n\\item the required certificate level and every numerical threshold;\n\\item the formation, calibration, certification, and stress-test splits;\n\\item the uncertainty procedure and resampling unit; and\n\\item the stopping and model-selection rules;\n\\item the experiment family and model class used for identi","Before the certification split is opened, record:\n\\begin{enumerate}\n\\item the full reference specification\n \\(\\varsigma_0=(\\mathcal D,Y^+,\\ell,\\tau,\\mathcal I,\\mathcal N,\\Theta)\\);\n\\item the admissible decomposition family \\(\\mathfrak D\\) and specification\n family \\(\\mathfrak S\\);\n\\item the target trajectory \\(Y^+\\), loss, horizon, and environment family;\n\\item the candidate structural workspace extractor;\n\\item the goal family \\(\\mathcal G\\), utility normalization, and feasible\n action set;\n\\item the intervention families \\(\\mathcal I_W,\\mathcal I_G,\\mathcal I_A,\n \\mathcal I_D,\\mathcal I_R\\);\n\\item the null panel and the local distributions each null must preserve;\n\\item the required certificate level and every numerical threshold;\n\\item the formation, calibration, certification, and stress-test splits;\n\\item the uncertainty procedure and resampling unit; and\n\\item the stopping and model-selection rules;\n\\item the experiment family and model class used for identifiability; and\n\\item the complete spe","opened, record:\n\\begin{enumerate}\n\\item the full reference specification\n \\(\\varsigma_0=(\\mathcal D,Y^+,\\ell,\\tau,\\mathcal I,\\mathcal N,\\Theta)\\);\n\\item the admissible decomposition family \\(\\mathfrak D\\) and specification\n family \\(\\mathfrak S\\);\n\\item the target trajectory \\(Y^+\\), loss, horizon, and environment family;\n\\item the candidate structural workspace extractor;\n\\item the goal family \\(\\mathcal G\\), utility normalization, and feasible\n action set;\n\\item the intervention families \\(\\mathcal I_W,\\mathcal I_G,\\mathcal I_A,\n \\mathcal I_D,\\mathcal I_R\\);\n\\item the null panel and the local distributions each null must preserve;\n\\item the required certificate level and every numerical threshold;\n\\item the formation, calibration, certification, and stress-test splits;\n\\item the uncertainty procedure and resampling unit; and\n\\item the stopping and model-selection rules;\n\\item the experiment family and model class used for identifiability; and\n\\item the complete specification-selection history and multiplicity rule.\n\\end{enumerate}\nChanging any of these after observing certification…","D,Y^+,\\ell,\\tau,\\mathcal I,\\mathcal N,\\Theta)\\);\n\\item the admissible decomposition family \\(\\mathfrak D\\) and specification\n family \\(\\mathfrak S\\);\n\\item the target trajectory \\(Y^+\\), loss, horizon, and environment family;\n\\item the candidate structural workspace extractor;\n\\item the goal family \\(\\mathcal G\\), utility normalization, and feasible\n action set;\n\\item the intervention families \\(\\mathcal I_W,\\mathcal I_G,\\mathcal I_A,\n \\mathcal I_D,\\mathcal I_R\\);\n\\item the null panel and the local distributions each null must preserve;\n\\item the required certificate level and every numerical threshold;\n\\item the formation, calibration, certification, and stress-test splits;\n\\item the uncertainty procedure and resampling unit; and\n\\item the stopping and model-selection rules;\n\\item the experiment family and model class used for identifiability; and\n\\item the complete specification-selection history and multiplicity rule.\n\\end{enumerate}\nChanging any of these after observing certification outcomes creates a new\nanalysis that must be reported as exploratory or evaluated on a fresh spli…","Use at least four logically distinct partitions:\n\\[\n\\boxed{\n\\mathcal D\n=\n\\mathcal D_{\\mathrm{form}}\n\\sqcup\n\\mathcal D_{\\mathrm{cal}}\n\\sqcup\n\\mathcal D_{\\mathrm{cert}}\n\\sqcup\n\\mathcal D_{\\mathrm{stress}}.\n}\n\\]\nTheir roles are:\n\\begin{description}\n\\item[Formation.] Learn membership, transfer maps, workspace channels,\n boundaries, policies, and repair models.\n\\item[Calibration.] Select thresholds, intervention amplitudes, and\n uncertainty procedures without making the final pass decision.\n\\item[Certification.] Estimate the declared gates once under the frozen\n analysis plan.\n\\item[Stress test.] Evaluate distribution shift, repeated damage, replacement,\n and long-horizon persistence.\n\\end{description}\nWhen independent partitions are impossible, the report must state the precise\ncross-fitting or nested-resampling procedure and preserve formation leakage\ncontrols.","Report the complete low-energy spectrum relevant to the selected threshold:\n\\[\n\\spec(L_H),\n\\qquad\n d_\\varepsilon,\n\\qquad\n g_\\varepsilon,\n\\qquad\n \\delta_{\\mathrm{coh}}.\n\\]\nAt minimum include:\n\\begin{itemize}\n\\item the construction and normalization of \\(D_H\\);\n\\item edge weights and estimated transfer uncertainties;\n\\item the selected eigenspace dimension and threshold rule;\n\\item the spectral gap with uncertainty or perturbation bound;\n\\item stability under bootstrap, seed, and representation replacement; and\n\\item the exact or approximate kernel residual.\n\\end{itemize}","Report\n\\[\n\\eta_{\\mathrm{int}},\n\\qquad\n\\delta_e^{\\mathrm{tr}},\n\\qquad\n\\Omega_\\gamma,\n\\qquad\n\\Lambda_\\gamma,\n\\]\ntogether with:\n\\begin{itemize}\n\\item the paths and cycle basis used;\n\\item path-length dependence of leakage;\n\\item the gauge-alignment procedure;\n\\item held-out transfer-prediction error;\n\\item operation-intertwining defects; and\n\\item the approximate closure bound from Theorem~\\(T_{10}\\).\n\\end{itemize}","Report\n\\[\n\\delta_{\\mathrm{align}}^{\\mathcal I},\n\\qquad\n\\{\\Delta_A^\\ell\\}_{A\\subseteq V},\n\\qquad\n k_\\varepsilon,\n\\qquad\n\\delta_A^Z.\n\\]\nWhen the number of coalitions is too large for exhaustive evaluation, state the\nsampling design, include every singleton and every leave-one-out coalition,\nand report the most favorable coalition found for the reducibility null. The\nclaim must not be based only on average coalition performance.","Report\n\\[\n\\eta_{\\mathrm{causal}}^\\nu,\n\\qquad\n\\Delta_{\\mathrm{specific}}(r),\n\\qquad\n\\delta_{\\mathrm{med}},\n\\qquad\n\\eta_{\\mathrm{rec}}(\\tau),\n\\qquad\n\\eta_{\\mathrm{loop}}(\\tau).\n\\]\nEvery causal estimate must be paired with:\n\\begin{enumerate}\n\\item a concrete mechanism-level implementation;\n\\item a manipulation check;\n\\item matched background variables;\n\\item declared direct paths that were blocked or preserved;\n\\item intervention magnitude and cost;\n\\item an off-workspace matched intervention; and\n\\item the downstream outcome window.\n\\end{enumerate}","Report\n\\[\nd_{\\mathrm{id}}(W_t,W_{t+\\tau}),\n\\qquad\n\\Rec_k,\n\\qquad\n e_t,\n\\qquad\n\\alpha^{(k)},\n\\qquad\n\\mu_t^{\\mathrm{keep}}.\n\\]\nThe report must separate:\n\\begin{itemize}\n\\item subspace tracking from state tracking;\n\\item coherence from non-collapse;\n\\item reconstruction from mere prediction;\n\\item repair from successor-workspace creation; and\n\\item structural replacement from causal and reconstructive replacement.\n\\end{itemize}","Report\n\\[\n\\mathbf E_{\\mathrm{form}},\n\\qquad\n\\delta^{\\mathrm{scaf}},\n\\qquad\n\\delta^{\\mathrm{leak}},\n\\qquad\n\\Delta_q^{\\mathrm{hist}}.\n\\]\nInclude the external scaffold, information available to the learner, history\nshuffles, cross-scaffold retraining, and every route by which certification\nlabels or target boundaries could have entered formation.","Report\n\\[\n\\|\\widehat H-H_*\\|,\n\\qquad\n\\Delta_J,\n\\qquad\n\\Delta_t^B,\n\\qquad\n\\COut_\\tau(\\widehat S),\n\\qquad\n\\CIn_\\tau(\\widehat S).\n\\]\nBoundary reports must include the admissible candidate family, full-system and\nempty-set controls, singleton and centralized controls, score components,\nmargin to the runner-up, and temporal stability under held-out environment\ndrift.","Report the agency coordinates\n\\[\n\\eta_G(\\tau),\n\\qquad\n\\delta_G^{\\mathrm{cue}}(\\tau),\n\\qquad\n\\eta_{G\\to A},\n\\qquad\n\\delta_Q,\n\\qquad\n\\eta_{V\\to A},\n\\qquad\n\\eta_{A\\to Y},\n\\qquad\n k_{\\mathrm{ctl},\\varepsilon}.\n\\]\nThe required experiments include:\n\\begin{enumerate}\n\\item cue-controlled goal interventions;\n\\item goal-channel ablation and permutation;\n\\item counterfactual-value calibration against action interventions;\n\\item value-label permutation among feasible actions;\n\\item critic or value-channel blocking;\n\\item action replay under held-out perturbations;\n\\item policy regret or value-gap estimation; and\n\\item coalition reconstruction of the complete control state.\n\\end{enumerate}","For every damage family, report\n\\[\n\\eta_{D\\to M},\n\\qquad\n\\eta_{M\\to R},\n\\qquad\n\\eta_{R\\to v},\n\\qquad\n\\Delta_{\\mathrm{repair}}^{\\mathrm{spec}},\n\\qquad\n\\delta_{\\mathrm{rep-id}},\n\\qquad\n\\sup_t e_t.\n\\]\nThe repeated-disturbance experiment must contain:\n\\begin{enumerate}\n\\item a no-damage baseline;\n\\item damage with repair disabled;\n\\item damage with the intact repair loop;\n\\item monitor ablation;\n\\item cost-matched random repair;\n\\item off-workspace repair;\n\\item repeated and compound damage; and\n\\item a held-out disturbance distribution.\n\\end{enumerate}","The core null panel is:\n\\[\n\\begin{aligned}\nN_1&=\\text{shuffled transfer maps with validated local matching},\\\\\nN_2&=\\text{shuffled cross-node temporal correspondence},\\\\\nN_3&=\\text{matched-capacity local memory},\\\\\nN_4&=\\text{matched-size irrelevant artifact},\\\\\nN_5&=\\text{participant ablation},\\\\\nN_6&=\\text{artifact ablation},\\\\\nN_7&=\\text{workspace implementation intervention},\\\\\nN_8&=\\text{matched off-workspace intervention},\\\\\nN_9&=\\text{representation replacement},\\\\\nN_{10}&=\\text{context reset and reconstruction},\\\\\nN_{11}&=\\text{matched formation-history shuffle},\\\\\nN_{12}&=\\text{cross-scaffold re-encoding and retraining},\\\\\nN_{13}&=\\text{boundary-label and certification leakage audit},\\\\\nN_{14}&=\\text{empty, full, singleton, and centralized boundary nulls},\\\\\nN_{15}&=\\text{held-out transfer and membership re-estimation},\\\\\nN_{16}&=\\text{boundary damage, use, and repair intervention},\\\\\nN_{17}&=\\text{goal-code permutation with cues and actions matched},\\\\\nN_{18}&=\\text{persistent external-cue rewrite with internal retention cut},\\\\\nN_{19}&=\\text{counterfactual-value label perm…","A successful claim requires\n\\[\n\\Pass(N_1,\\ldots,N_{28})\n\\]\nonly for the subset relevant to its stated level. The subset must be frozen\nbefore certification. Failed applicable nulls, invalid nulls, and unmeasured\nnulls are retained separately in the final report. They must not be merged\ninto a generic ``not significant'' category.","Use a vector rather than one aggregate score:\n\\[\n\\boxed{\n\\begin{aligned}\n\\mathcal C_{\\mathrm{full}}(W)\n=\\bigl(&\n d_\\varepsilon,\n g_\\varepsilon,\n \\delta_{\\mathrm{coh}},\n \\delta_{\\mathrm{align}}^{\\mathcal I},\n \\Delta_{\\mathrm{joint}}^\\ell,\n k_\\varepsilon,\\\\\n&\\boldsymbol\\Delta_{\\mathrm{tr}}(\\mathcal Q),\n \\eta_{\\mathrm{causal}}^\\nu,\n \\Delta_{\\mathrm{specific}}(r),\n \\eta_{\\mathrm{loop}}(\\tau),\\\\\n&d_{\\mathrm{id}},\n \\Rec_k,\n \\underline{\\mathbf m}_{\\mathfrak D},\n \\boldsymbol\\Delta_{\\mathrm{dec}},\n \\mathbf E_{\\mathrm{form}},\n \\delta^{\\mathrm{scaf}},\n \\delta^{\\mathrm{leak}},\\\\\n&\\Delta_J,\n \\Delta_t^B,\n \\COut_\\tau(\\widehat S),\n \\CIn_\\tau(\\widehat S),\\\\\n&\\eta_G(\\tau),\n \\delta_G^{\\mathrm{cue}}(\\tau),\n \\eta_{G\\to A},\n \\delta_Q,\n \\eta_{V\\to A},\n \\eta_{A\\to Y},\\\\\n&\\eta_{D\\to M},\n \\eta_{M\\to R},\n \\eta_{R\\to v},\n \\Delta_{\\mathrm{repair}}^{\\mathrm{spec}},\n \\delta_{\\mathrm{rep-id}},\n \\sup_t e_t\n\\bigr).\n\\end{aligned}\n}\n\\]\nA claim passes only when every required coordinate lies in its predeclared\nadmissible region. No weighted average may compensate for a failed required\ngate.","Target and horizon & \\(Y^+\\), loss, utility, prediction horizon, control\nhorizon, and maintenance horizon.","Workspace extractor & Frozen construction of \\(D_H\\), \\(L_H\\), spectral\nthreshold, and state readout.","Null panel & Applicable \\(N_j\\), preservation targets, and fit budgets.","\\midrule\n\\endhead\nNull ID & One of \\(N_1,\\ldots,N_{34}\\) or a declared extension.","\\midrule\nGoal semantics & Held-out \\(Q_t^g\\) ordering & Distinct goals reverse or alter\nat least one declared action ordering.","Goal retention & \\(\\eta_G(\\tau)\\) & Goal intervention with future cues matched\nor removed.","Goal use & \\(\\eta_{G\\to A}\\) & Goal intervention versus goal-label replay.","Counterfactual accuracy & \\(\\delta_Q\\) & Internal value versus external action\ninterventions.","Value use & \\(\\eta_{V\\to A}\\) & Value-label permutation and value-channel\nablation.","Outcome control & \\(\\eta_{A\\to Y}\\) & Feasible action intervention.","Damage detection & \\(\\eta_{D\\to M}\\) & Damage versus matched no-damage.","Repair mediation & \\(\\eta_{M\\to R}\\) & Monitor intervention and monitor\nablation.","Restoration & \\(\\eta_{R\\to v}\\) & Repair versus matched no-repair.","Repair specificity & \\(\\Delta_{\\mathrm{repair}}^{\\mathrm{spec}}\\) & Intended\nrepair versus cost-matched generic action.","Identity preservation & \\(\\delta_{\\mathrm{rep-id}}\\) & Pre-damage workspace\nversus post-repair workspace.","Long-run maintenance & \\(\\sup_t e_t\\) & Repeated held-out disturbances.","Relational location & \\(k_{\\mathrm{ctl},\\varepsilon}\\) & Full system versus\nall single nodes and matched local nulls.","Pass decisions should use one-sided uncertainty bounds. For a lower-bounded\ngate \\(\\theta\\geq\\tau\\), pass only when the lower confidence bound exceeds\n\\(\\tau\\). For an upper-bounded defect \\(\\delta\\leq\\varepsilon\\), pass only when\nthe upper confidence bound is below \\(\\varepsilon\\).","When a conclusion ranges over \\(\\mathfrak S\\), the uncertainty rule must be\nuniform over every gate and specification included in the claim. Report the\npointwise interval, the family-wide lower envelope, the covering or\nmultiplicity correction, and the identification status. A narrow pointwise\ninterval does not establish a robust result when model or specification\nuncertainty remains wide.","\\subtopic{76.8 \\(T_{24}\\): Gatewise certification validity }","\\begin{theorem}[Intersection-gate false-certification control]\n\\label{thm:gatewise-validity}\nLet a certificate require all gates \\(G_1,\\ldots,G_m\\) to pass. For each gate\n\\(j\\), let \\(H_{0j}\\) denote the null that the required property is absent, and\nassume the predeclared test satisfies\n\\[\n\\sup_{P\\in H_{0j}}\n\\mathbb P_P(G_j\\text{ passes})\n\\leq\n\\alpha_j.\n\\]\nThe global null is\n\\[\nH_0\n=\n\\bigcup_{j=1}^m H_{0j},\n\\]\nmeaning that at least one required property is absent. If the certificate is\ndeclared only when every gate passes, then\n\\[\n\\boxed{\n\\sup_{P\\in H_0}\n\\mathbb P_P(\\text{certificate passes})\n\\leq\n\\max_{1\\leq j\\leq m}\\alpha_j.\n}\n\\]\nNo independence assumption among gate tests is required.\n\\end{theorem}","dots,G_m\\) to pass. For each gate\n\\(j\\), let \\(H_{0j}\\) denote the null that the required property is absent, and\nassume the predeclared test satisfies\n\\[\n\\sup_{P\\in H_{0j}}\n\\mathbb P_P(G_j\\text{ passes})\n\\leq\n\\alpha_j.\n\\]\nThe global null is\n\\[\nH_0\n=\n\\bigcup_{j=1}^m H_{0j},\n\\]\nmeaning that at least one required property is absent. If the certificate is\ndeclared only when every gate passes, then\n\\[\n\\boxed{\n\\sup_{P\\in H_0}\n\\mathbb P_P(\\text{certificate passes})\n\\leq\n\\max_{1\\leq j\\leq m}\\alpha_j.\n}\n\\]\nNo independence assumption among gate tests is required.\n\\end{theorem}","\\begin{proof}\nFor any \\(P\\in H_0\\), choose an index \\(j_P\\) such that \\(P\\in H_{0j_P}\\).\nThe event that every gate passes is contained in the event that gate \\(j_P\\)\npasses. Hence\n\\[\n\\mathbb P_P(\\text{certificate passes})\n\\leq\n\\mathbb P_P(G_{j_P}\\text{ passes})\n\\leq\n\\alpha_{j_P}\n\\leq\n\\max_j\\alpha_j.\n\\]\nTaking the supremum over \\(P\\in H_0\\) proves the claim.\n\\end{proof}","The machine-readable record should contain at least the following top-level\nobjects:\n\\[\n\\boxed{\n\\begin{aligned}\n\\mathsf{ResultRecord}\n=\\{&\n\\mathsf{claim},\n\\mathsf{specification},\n\\mathsf{specification\\_family},\n\\mathsf{decomposition},\n\\mathsf{data\\_splits},\n\\mathsf{formation},\\\\\n&\\mathsf{workspace\\_extractor},\n\\mathsf{goals},\n\\mathsf{actions},\n\\mathsf{interventions},\\\\\n&\\mathsf{nulls},\n\\mathsf{gates},\n\\mathsf{identified\\_sets},\n\\mathsf{specification\\_envelopes},\n\\mathsf{uncertainty},\n\\mathsf{failures},\n\\mathsf{artifacts}\n\\}.\n\\end{aligned}\n}\n\\]\nEach gate object records its symbol, estimator, direction, threshold, point\nestimate, interval, split, seed set, status, and artifact reference. Each\nintervention and null object records its manipulation check and validity\nstatus. The schema must preserve failed and invalid entries rather than only\nthe passing subset.","Let \\(\\mathcal J(L)\\) be the required gate set for claim level \\(L\\). Then\n\\[\n\\boxed{\n\\operatorname{Certify}_L(W;\\varsigma)\n=\n\\bigwedge_{j\\in\\mathcal J(L,\\varsigma)}\n\\operatorname{Pass}_j(W;\\varsigma).\n}\n\\]\nFor a workspace-agent claim, the required set includes every structural,\noperational, relational, persistent, goal, counterfactual-control, and\nself-maintenance gate declared in Parts~IX and XVI. A specification-robust\nclaim additionally requires the uniform rule of Part~XVIII across\n\\(\\mathfrak S\\). Failure of","Let \\(\\mathcal J(L)\\) be the required gate set for claim level \\(L\\). Then\n\\[\n\\boxed{\n\\operatorname{Certify}_L(W;\\varsigma)\n=\n\\bigwedge_{j\\in\\mathcal J(L,\\varsigma)}\n\\operatorname{Pass}_j(W;\\varsigma).\n}\n\\]\nFor a workspace-agent claim, the required set includes every structural,\noperational, relational, persistent, goal, counterfactual-control, and\nself-maintenance gate declared in Parts~IX and XVI. A specification-robust\nclaim additionally requires the uniform rule of Part~XVIII across\n\\(\\mathfrak S\\). Failure of an earlier level\nprevents certification of a l","Let \\(\\mathcal J(L)\\) be the required gate set for claim level \\(L\\). Then\n\\[\n\\boxed{\n\\operatorname{Certify}_L(W;\\varsigma)\n=\n\\bigwedge_{j\\in\\mathcal J(L,\\varsigma)}\n\\operatorname{Pass}_j(W;\\varsigma).\n}\n\\]\nFor a workspace-agent claim, the required set includes every structural,\noperational, relational, persistent, goal, counterfactual-control, and\nself-maintenance gate declared in Parts~IX and XVI. A specification-robust\nclaim additionally requires the uniform rule of Part~XVIII across\n\\(\\mathfrak S\\). Failure of an earlier level\nprevents certification of a later level, even when later-level behavior appears\nstrong on a task benchmark.","rname{Certify}_L(W;\\varsigma)\n=\n\\bigwedge_{j\\in\\mathcal J(L,\\varsigma)}\n\\operatorname{Pass}_j(W;\\varsigma).\n}\n\\]\nFor a workspace-agent claim, the required set includes every structural,\noperational, relational, persistent, goal, counterfactual-control, and\nself-maintenance gate declared in Parts~IX and XVI. A specification-robust\nclaim additionally requires the uniform rule of Part~XVIII across\n\\(\\mathfrak S\\). Failure of an earlier level\nprevents certification of a later level, even when later-level behavior appears\nstrong on a task benchmark.","\\begin{definition}[Case-study realization]\\label{def:case-study-realization}\nA case-study realization is a tuple\n\\[\n\\mathfrak K\n=\n\\bigl(\n\\mathfrak R,\n\\mathfrak B,\n\\mathfrak C,\n\\mathfrak P,\n\\mathcal E,\n\\mathfrak D,\n\\mathfrak S,\n\\mathcal I,\n\\mathcal Q,\n\\mathcal G\n\\bigr),\n\\]\nwhere:\n\\begin{itemize}\n\\item \\(\\mathfrak R\\) is the declared relational realization;\n\\item \\(\\mathfrak B\\) is the boundary problem and component decomposition;\n\\item \\(\\mathfrak C\\) is the structural core--transfer model;\n\\item \\(\\mathfrak P\\) is the downstream predictive or control problem;\n\\item \\(\\mathcal E\\) is the evidence or generated-data family;\n\\item \\(\\mathfrak D\\) is the admissible decomposition family;\n\\item \\(\\mathfrak S\\) is the admissible family of complete scientific\n specifications, including targets, losses, horizons, interventions, nulls,\n and thresholds;\n\\item \\(\\mathcal I\\) is the feasible intervention family;\n\\item \\(\\mathcal Q\\) is the matched null family; and\n\\item","\\begin{definition}[Case-study realization]\\label{def:case-study-realization}\nA case-study realization is a tuple\n\\[\n\\mathfrak K\n=\n\\bigl(\n\\mathfrak R,\n\\mathfrak B,\n\\mathfrak C,\n\\mathfrak P,\n\\mathcal E,\n\\mathfrak D,\n\\mathfrak S,\n\\mathcal I,\n\\mathcal Q,\n\\mathcal G\n\\bigr),\n\\]\nwhere:\n\\begin{itemize}\n\\item \\(\\mathfrak R\\) is the declared relational realization;\n\\item \\(\\mathfrak B\\) is the boundary problem and component decomposition;\n\\item \\(\\mathfrak C\\) is the structural core--transfer model;\n\\item \\(\\mathfrak P\\) is the downstream predictive or control problem;\n\\item \\(\\mathcal E\\) is the evidence or generated-data family;\n\\item \\(\\mathfrak D\\) is the admissible decomposition family;\n\\item \\(\\mathfrak S\\) is the admissible family of complete scientific\n specifications, including targets, losses, horizons, interventions, nulls,\n and thresholds;\n\\item \\(\\mathcal I\\) is the feasible intervention family;\n\\item \\(\\mathcal Q\\) is the matched null family; and\n\\item \\(\\mathcal G\\) is the set of gates being cla","\\begin{definition}[Case-study realization]\\label{def:case-study-realization}\nA case-study realization is a tuple\n\\[\n\\mathfrak K\n=\n\\bigl(\n\\mathfrak R,\n\\mathfrak B,\n\\mathfrak C,\n\\mathfrak P,\n\\mathcal E,\n\\mathfrak D,\n\\mathfrak S,\n\\mathcal I,\n\\mathcal Q,\n\\mathcal G\n\\bigr),\n\\]\nwhere:\n\\begin{itemize}\n\\item \\(\\mathfrak R\\) is the declared relational realization;\n\\item \\(\\mathfrak B\\) is the boundary problem and component decomposition;\n\\item \\(\\mathfrak C\\) is the structural core--transfer model;\n\\item \\(\\mathfrak P\\) is the downstream predictive or control problem;\n\\item \\(\\mathcal E\\) is the evidence or generated-data family;\n\\item \\(\\mathfrak D\\) is the admissible decomposition family;\n\\item \\(\\mathfrak S\\) is the admissible family of complete scientific\n specifications, including targets, losses, horizons, interventions, nulls,\n and thresholds;\n\\item \\(\\mathcal I\\) is the feasible intervention family;\n\\item \\(\\mathcal Q\\) is the matched null family; and\n\\item \\(\\mathcal G\\) is the set of gates being claimed.\n\\end{itemize}\nA conclusion is always relative to this tu","]\\label{def:case-study-realization}\nA case-study realization is a tuple\n\\[\n\\mathfrak K\n=\n\\bigl(\n\\mathfrak R,\n\\mathfrak B,\n\\mathfrak C,\n\\mathfrak P,\n\\mathcal E,\n\\mathfrak D,\n\\mathfrak S,\n\\mathcal I,\n\\mathcal Q,\n\\mathcal G\n\\bigr),\n\\]\nwhere:\n\\begin{itemize}\n\\item \\(\\mathfrak R\\) is the declared relational realization;\n\\item \\(\\mathfrak B\\) is the boundary problem and component decomposition;\n\\item \\(\\mathfrak C\\) is the structural core--transfer model;\n\\item \\(\\mathfrak P\\) is the downstream predictive or control problem;\n\\item \\(\\mathcal E\\) is the evidence or generated-data family;\n\\item \\(\\mathfrak D\\) is the admissible decomposition family;\n\\item \\(\\mathfrak S\\) is the admissible family of complete scientific\n specifications, including targets, losses, horizons, interventions, nulls,\n and thresholds;\n\\item \\(\\mathcal I\\) is the feasible intervention family;\n\\item \\(\\mathcal Q\\) is the matched null family; and\n\\item \\(\\mathcal G\\) is the set of gates being claimed.\n\\end{itemize}\nA conclusion is always relative to this tuple. Changing the downstream target,\ntemporal horizon, intervent…","s a tuple\n\\[\n\\mathfrak K\n=\n\\bigl(\n\\mathfrak R,\n\\mathfrak B,\n\\mathfrak C,\n\\mathfrak P,\n\\mathcal E,\n\\mathfrak D,\n\\mathfrak S,\n\\mathcal I,\n\\mathcal Q,\n\\mathcal G\n\\bigr),\n\\]\nwhere:\n\\begin{itemize}\n\\item \\(\\mathfrak R\\) is the declared relational realization;\n\\item \\(\\mathfrak B\\) is the boundary problem and component decomposition;\n\\item \\(\\mathfrak C\\) is the structural core--transfer model;\n\\item \\(\\mathfrak P\\) is the downstream predictive or control problem;\n\\item \\(\\mathcal E\\) is the evidence or generated-data family;\n\\item \\(\\mathfrak D\\) is the admissible decomposition family;\n\\item \\(\\mathfrak S\\) is the admissible family of complete scientific\n specifications, including targets, losses, horizons, interventions, nulls,\n and thresholds;\n\\item \\(\\mathcal I\\) is the feasible intervention family;\n\\item \\(\\mathcal Q\\) is the matched null family; and\n\\item \\(\\mathcal G\\) is the set of gates being claimed.\n\\end{itemize}\nA conclusion is always relative to this tuple. Changing the downstream target,\ntemporal horizon, intervention support, component decomposition, null family,\nor threshol…",",\n\\mathfrak P,\n\\mathcal E,\n\\mathfrak D,\n\\mathfrak S,\n\\mathcal I,\n\\mathcal Q,\n\\mathcal G\n\\bigr),\n\\]\nwhere:\n\\begin{itemize}\n\\item \\(\\mathfrak R\\) is the declared relational realization;\n\\item \\(\\mathfrak B\\) is the boundary problem and component decomposition;\n\\item \\(\\mathfrak C\\) is the structural core--transfer model;\n\\item \\(\\mathfrak P\\) is the downstream predictive or control problem;\n\\item \\(\\mathcal E\\) is the evidence or generated-data family;\n\\item \\(\\mathfrak D\\) is the admissible decomposition family;\n\\item \\(\\mathfrak S\\) is the admissible family of complete scientific\n specifications, including targets, losses, horizons, interventions, nulls,\n and thresholds;\n\\item \\(\\mathcal I\\) is the feasible intervention family;\n\\item \\(\\mathcal Q\\) is the matched null family; and\n\\item \\(\\mathcal G\\) is the set of gates being claimed.\n\\end{itemize}\nA conclusion is always relative to this tuple. Changing the downstream target,\ntemporal horizon, intervention support, component decomposition, null family,\nor threshold rule creates a different specification of the case. A stronger\ncase s…",",\n\\mathcal Q,\n\\mathcal G\n\\bigr),\n\\]\nwhere:\n\\begin{itemize}\n\\item \\(\\mathfrak R\\) is the declared relational realization;\n\\item \\(\\mathfrak B\\) is the boundary problem and component decomposition;\n\\item \\(\\mathfrak C\\) is the structural core--transfer model;\n\\item \\(\\mathfrak P\\) is the downstream predictive or control problem;\n\\item \\(\\mathcal E\\) is the evidence or generated-data family;\n\\item \\(\\mathfrak D\\) is the admissible decomposition family;\n\\item \\(\\mathfrak S\\) is the admissible family of complete scientific\n specifications, including targets, losses, horizons, interventions, nulls,\n and thresholds;\n\\item \\(\\mathcal I\\) is the feasible intervention family;\n\\item \\(\\mathcal Q\\) is the matched null family; and\n\\item \\(\\mathcal G\\) is the set of gates being claimed.\n\\end{itemize}\nA conclusion is always relative to this tuple. Changing the downstream target,\ntemporal horizon, intervention support, component decomposition, null family,\nor threshold rule creates a different specification of the case. A stronger\ncase study reports whether the conclusion survives the predeclared fa…","em \\(\\mathfrak R\\) is the declared relational realization;\n\\item \\(\\mathfrak B\\) is the boundary problem and component decomposition;\n\\item \\(\\mathfrak C\\) is the structural core--transfer model;\n\\item \\(\\mathfrak P\\) is the downstream predictive or control problem;\n\\item \\(\\mathcal E\\) is the evidence or generated-data family;\n\\item \\(\\mathfrak D\\) is the admissible decomposition family;\n\\item \\(\\mathfrak S\\) is the admissible family of complete scientific\n specifications, including targets, losses, horizons, interventions, nulls,\n and thresholds;\n\\item \\(\\mathcal I\\) is the feasible intervention family;\n\\item \\(\\mathcal Q\\) is the matched null family; and\n\\item \\(\\mathcal G\\) is the set of gates being claimed.\n\\end{itemize}\nA conclusion is always relative to this tuple. Changing the downstream target,\ntemporal horizon, intervention support, component decomposition, null family,\nor threshold rule creates a different specification of the case. A stronger\ncase study reports whether the conclusion survives the predeclared family\n\\(\\mathfrak S\\) and whether the relevant gates are identi…","structural core--transfer model;\n\\item \\(\\mathfrak P\\) is the downstream predictive or control problem;\n\\item \\(\\mathcal E\\) is the evidence or generated-data family;\n\\item \\(\\mathfrak D\\) is the admissible decomposition family;\n\\item \\(\\mathfrak S\\) is the admissible family of complete scientific\n specifications, including targets, losses, horizons, interventions, nulls,\n and thresholds;\n\\item \\(\\mathcal I\\) is the feasible intervention family;\n\\item \\(\\mathcal Q\\) is the matched null family; and\n\\item \\(\\mathcal G\\) is the set of gates being claimed.\n\\end{itemize}\nA conclusion is always relative to this tuple. Changing the downstream target,\ntemporal horizon, intervention support, component decomposition, null family,\nor threshold rule creates a different specification of the case. A stronger\ncase study reports whether the conclusion survives the predeclared family\n\\(\\mathfrak S\\) and whether the relevant gates are identified by the available\nexperiment family.\n\\end{definition}","the downstream predictive or control problem;\n\\item \\(\\mathcal E\\) is the evidence or generated-data family;\n\\item \\(\\mathfrak D\\) is the admissible decomposition family;\n\\item \\(\\mathfrak S\\) is the admissible family of complete scientific\n specifications, including targets, losses, horizons, interventions, nulls,\n and thresholds;\n\\item \\(\\mathcal I\\) is the feasible intervention family;\n\\item \\(\\mathcal Q\\) is the matched null family; and\n\\item \\(\\mathcal G\\) is the set of gates being claimed.\n\\end{itemize}\nA conclusion is always relative to this tuple. Changing the downstream target,\ntemporal horizon, intervention support, component decomposition, null family,\nor threshold rule creates a different specification of the case. A stronger\ncase study reports whether the conclusion survives the predeclared family\n\\(\\mathfrak S\\) and whether the relevant gates are identified by the available\nexperiment family.\n\\end{definition}","(\\mathcal E\\) is the evidence or generated-data family;\n\\item \\(\\mathfrak D\\) is the admissible decomposition family;\n\\item \\(\\mathfrak S\\) is the admissible family of complete scientific\n specifications, including targets, losses, horizons, interventions, nulls,\n and thresholds;\n\\item \\(\\mathcal I\\) is the feasible intervention family;\n\\item \\(\\mathcal Q\\) is the matched null family; and\n\\item \\(\\mathcal G\\) is the set of gates being claimed.\n\\end{itemize}\nA conclusion is always relative to this tuple. Changing the downstream target,\ntemporal horizon, intervention support, component decomposition, null family,\nor threshold rule creates a different specification of the case. A stronger\ncase study reports whether the conclusion survives the predeclared family\n\\(\\mathfrak S\\) and whether the relevant gates are identified by the available\nexperiment family.\n\\end{definition}","hed null family; and\n\\item \\(\\mathcal G\\) is the set of gates being claimed.\n\\end{itemize}\nA conclusion is always relative to this tuple. Changing the downstream target,\ntemporal horizon, intervention support, component decomposition, null family,\nor threshold rule creates a different specification of the case. A stronger\ncase study reports whether the conclusion survives the predeclared family\n\\(\\mathfrak S\\) and whether the relevant gates are identified by the available\nexperiment family.\n\\end{definition}","For each case we report the vector\n\\[\n\\boxed{\n\\mathsf{Card}(\\mathfrak K)\n=\n\\left(\n\\mathsf S,\n\\mathsf O,\n\\mathsf R,\n\\mathsf P,\n\\mathsf F,\n\\mathsf A,\n\\mathsf D,\n\\mathsf X,\n\\mathsf C\n\\right),\n}\n\\]\nwhere:\n\\[\n\\begin{array}{ll}\n\\mathsf S & = \\text{structural core and spectral coherence},","For each case we report the vector\n\\[\n\\boxed{\n\\mathsf{Card}(\\mathfrak K)\n=\n\\left(\n\\mathsf S,\n\\mathsf O,\n\\mathsf R,\n\\mathsf P,\n\\mathsf F,\n\\mathsf A,\n\\mathsf D,\n\\mathsf X,\n\\mathsf C\n\\right),\n}\n\\]\nwhere:\n\\[\n\\begin{array}{ll}\n\\mathsf S & = \\text{structural core and spectral coherence},\\\\\n\\mathsf O & = \\text{operational alignment and causal efficacy},\\\\\n\\mathsf R & = \\text{relational irreducibility and matched-null gap},\\\\\n\\mathsf P & = \\text{persistence, reconstruction, and identity},\\\\\n\\mathsf F & = \\text{formation provenance and boundary status},\\\\\n\\mathsf A & = \\text{goal-conditioned agency and self-maintenance},\\\\\n\\mathsf D & = \\text{decomposition robustness and interface audit},\\\\\n\\mathsf X & = \\text{specification robustness, identification, and uniform uncertainty},\\\\\n\\mathsf C & = \\text{consciousness or personhood claim}.\n\\end{array}\n\\]\nThe decomposition coordinate distinguishes a result established only for the\ndeclared interface from one stable across an admissible family. The\nspecification coordinate records whether the same conclusion survives\nscientifically defensible alterna…","m one stable across an admissible family. The\nspecification coordinate records whether the same conclusion survives\nscientifically defensible alternatives, whether its sign is identified by the\navailable experiments, and whether its lower margin remains positive after\nuniform finite-sample uncertainty. The final coordinate is included to make\nthe non-implication explicit. In all examples below,\n\\[\n\\mathsf C=\\text{not established}.\n\\]","Let \\(L(\\mathfrak K)\\in\\{0,1,\\ldots,6\\}\\) denote the highest level from\nPart~XXIV whose complete required gate set is established. The level is\nconjunctive and pointwise in the declared scientific specification unless the\n\\(\\mathsf D\\) and \\(\\mathsf X\\) coordinates also pass:\n\\[\nL(\\mathfrak K)\\geq \\ell\n\\quad\\Longleftrightarrow\\quad\n\\bigwedge_{j\\in\\mathcal J(\\ell)}\\Pass_j(\\mathfrak K).\n\\]\nA case may have strong higher-level-looking behavior while remaining at a lower\ncertified level because an earlier gate is absent. For example, successful\nrepair behavior does not establish self-maintenance when no matched no-repair\ncounterfactual is available, and coherent shared documents do not establish a\nrelational workspace when one partici","Let \\(L(\\mathfrak K)\\in\\{0,1,\\ldots,6\\}\\) denote the highest level from\nPart~XXIV whose complete required gate set is established. The level is\nconjunctive and pointwise in the declared scientific specification unless the\n\\(\\mathsf D\\) and \\(\\mathsf X\\) coordinates also pass:\n\\[\nL(\\mathfrak K)\\geq \\ell\n\\quad\\Longleftrightarrow\\quad\n\\bigwedge_{j\\in\\mathcal J(\\ell)}\\Pass_j(\\mathfrak K).\n\\]\nA case may have strong higher-level-looking behavior while remaining at a lower\ncertified level because an earlier gate is absent. For example, successful\nrepair behavior does not establish self-maintenance when no matched no-repair\ncounterfactual is available, and coherent shared documents do not establish a\nrelational workspace when one participant contains the complete operative\nstate.","The positive examples use a recurring distributed code. Let\n\\[\nc=(c_1,c_2,c_3)^{\\mathsf T}\\in\\mathbb R^3\n\\]\nand define three local encodings\n\\[\nE_1c=(c_1,c_2),\n\\qquad\nE_2c=(c_2,c_3),\n\\qquad\nE_3c=(c_1,c_3).\n\\]\nThen\n\\[\n\\sum_{i=1}^3E_i^*E_i=2I_3.\n\\]\nNo individual encoding is injective, while every two-node coalition is\ninjective. The pairwise frame operators are\n\\[\n\\begin{aligned}\nE_{\\{1,2\\}}^*E_{\\{1,2\\}}&=\\operatorname{diag}(1,2,1),","The positive examples use a recurring distributed code. Let\n\\[\nc=(c_1,c_2,c_3)^{\\mathsf T}\\in\\mathbb R^3\n\\]\nand define three local encodings\n\\[\nE_1c=(c_1,c_2),\n\\qquad\nE_2c=(c_2,c_3),\n\\qquad\nE_3c=(c_1,c_3).\n\\]\nThen\n\\[\n\\sum_{i=1}^3E_i^*E_i=2I_3.\n\\]\nNo individual encoding is injective, while every two-node coalition is\ninjective. The pairwise frame operators are\n\\[\n\\begin{aligned}\nE_{\\{1,2\\}}^*E_{\\{1,2\\}}&=\\operatorname{diag}(1,2,1),\\\\\nE_{\\{1,3\\}}^*E_{\\{1,3\\}}&=\\operatorname{diag}(2,1,1),\\\\\nE_{\\{2,3\\}}^*E_{\\{2,3\\}}&=\\operatorname{diag}(1,1,2).\n\\end{aligned}\n\\]\nHence\n\\[\n\\alpha_{\\{i,j\\}}=1\n\\qquad(i\\neq j),\n\\]\nand one-node erasure recovery is exact. Every nonzero core vector is visible\nat at least two nodes, so\n\\[\nd_{\\mathrm{er}}=2.\n\\]\nThis code is distributed but redundant: no node is sufficient, yet any one\nnode may be lost without destroying the core.","E_{\\{2,3\\}}^*E_{\\{2,3\\}}&=\\operatorname{diag}(1,1,2).\n\\end{aligned}\n\\]\nHence\n\\[\n\\alpha_{\\{i,j\\}}=1\n\\qquad(i\\neq j),\n\\]\nand one-node erasure recovery is exact. Every nonzero core vector is visible\nat at least two nodes, so\n\\[\nd_{\\mathrm{er}}=2.\n\\]\nThis code is distributed but redundant: no node is sufficient, yet any one\nnode may be lost without destroying the core.","Let \\(C_1,C_2,C_3\\) be independent uniform bits and let\n\\[\nY=C_1\\oplus C_2\\oplus C_3.\n\\]\nEach node sees two of the three bits. Therefore\n\\[\nH(Y\\mid E_iC)=1\\text{ bit}\n\\qquad\\text{for every }i,\n\\]\nwhile\n\\[\nH(Y\\mid E_1C,E_2C,E_3C)=0.\n\\]\nUnder logarithmic loss,\n\\[\n\\Risk_{\\log}(\\{i\\})=1,\n\\qquad\n\\Risk_{\\log}(V)=0,\n\\qquad\n\\Delta_{\\mathrm{joint}}^{\\log}=1\\text{ bit},\n\\qquad\nk_0=2.\n\\]\nMoreover, \\(Y\\) is independent of every individual local state. The null law\nthat keeps all three core bits independent and makes \\(Y\\) an independent\nuniform bit preserves every pair \\((Y,E_iC)\\) while destroying the joint parity\nrelation. It is therefore a predictive-marginal-matched relational null with\n\\[\n\\Delta_{\\mathrm{tr}}=1\\tex","Let \\(C_1,C_2,C_3\\) be independent uniform bits and let\n\\[\nY=C_1\\oplus C_2\\oplus C_3.\n\\]\nEach node sees two of the three bits. Therefore\n\\[\nH(Y\\mid E_iC)=1\\text{ bit}\n\\qquad\\text{for every }i,\n\\]\nwhile\n\\[\nH(Y\\mid E_1C,E_2C,E_3C)=0.\n\\]\nUnder logarithmic loss,\n\\[\n\\Risk_{\\log}(\\{i\\})=1,\n\\qquad\n\\Risk_{\\log}(V)=0,\n\\qquad\n\\Delta_{\\mathrm{joint}}^{\\log}=1\\text{ bit},\n\\qquad\nk_0=2.\n\\]\nMoreover, \\(Y\\) is independent of every individual local state. The null law\nthat keeps all three core bits independent and makes \\(Y\\) an independent\nuniform bit preserves every pair \\((Y,E_iC)\\) while destroying the joint parity\nrelation. It is therefore a predictive-marginal-matched relational null with\n\\[\n\\Delta_{\\mathrm{tr}}=1\\text{ bit}.\n\\]\nThis test will be reused with different semantic interpre","Let \\(C_1,C_2,C_3\\) be independent uniform bits and let\n\\[\nY=C_1\\oplus C_2\\oplus C_3.\n\\]\nEach node sees two of the three bits. Therefore\n\\[\nH(Y\\mid E_iC)=1\\text{ bit}\n\\qquad\\text{for every }i,\n\\]\nwhile\n\\[\nH(Y\\mid E_1C,E_2C,E_3C)=0.\n\\]\nUnder logarithmic loss,\n\\[\n\\Risk_{\\log}(\\{i\\})=1,\n\\qquad\n\\Risk_{\\log}(V)=0,\n\\qquad\n\\Delta_{\\mathrm{joint}}^{\\log}=1\\text{ bit},\n\\qquad\nk_0=2.\n\\]\nMoreover, \\(Y\\) is independent of every individual local state. The null law\nthat keeps all three core bits independent and makes \\(Y\\) an independent\nuniform bit preserves every pair \\((Y,E_iC)\\) while destroying the joint parity\nrelation. It is therefore a predictive-marginal-matched relational null with\n\\[\n\\Delta_{\\mathrm{tr}}=1\\text{ bit}.\n\\]\nThis test will be reused with different semantic interpretations.","he three bits. Therefore\n\\[\nH(Y\\mid E_iC)=1\\text{ bit}\n\\qquad\\text{for every }i,\n\\]\nwhile\n\\[\nH(Y\\mid E_1C,E_2C,E_3C)=0.\n\\]\nUnder logarithmic loss,\n\\[\n\\Risk_{\\log}(\\{i\\})=1,\n\\qquad\n\\Risk_{\\log}(V)=0,\n\\qquad\n\\Delta_{\\mathrm{joint}}^{\\log}=1\\text{ bit},\n\\qquad\nk_0=2.\n\\]\nMoreover, \\(Y\\) is independent of every individual local state. The null law\nthat keeps all three core bits independent and makes \\(Y\\) an independent\nuniform bit preserves every pair \\((Y,E_iC)\\) while destroying the joint parity\nrelation. It is therefore a predictive-marginal-matched relational null with\n\\[\n\\Delta_{\\mathrm{tr}}=1\\text{ bit}.\n\\]\nThis test will be reused with different semantic interpretations.","\\qquad\\text{for every }i,\n\\]\nwhile\n\\[\nH(Y\\mid E_1C,E_2C,E_3C)=0.\n\\]\nUnder logarithmic loss,\n\\[\n\\Risk_{\\log}(\\{i\\})=1,\n\\qquad\n\\Risk_{\\log}(V)=0,\n\\qquad\n\\Delta_{\\mathrm{joint}}^{\\log}=1\\text{ bit},\n\\qquad\nk_0=2.\n\\]\nMoreover, \\(Y\\) is independent of every individual local state. The null law\nthat keeps all three core bits independent and makes \\(Y\\) an independent\nuniform bit preserves every pair \\((Y,E_iC)\\) while destroying the joint parity\nrelation. It is therefore a predictive-marginal-matched relational null with\n\\[\n\\Delta_{\\mathrm{tr}}=1\\text{ bit}.\n\\]\nThis test will be reused with different semantic interpretations.","d\n\\Risk_{\\log}(V)=0,\n\\qquad\n\\Delta_{\\mathrm{joint}}^{\\log}=1\\text{ bit},\n\\qquad\nk_0=2.\n\\]\nMoreover, \\(Y\\) is independent of every individual local state. The null law\nthat keeps all three core bits independent and makes \\(Y\\) an independent\nuniform bit preserves every pair \\((Y,E_iC)\\) while destroying the joint parity\nrelation. It is therefore a predictive-marginal-matched relational null with\n\\[\n\\Delta_{\\mathrm{tr}}=1\\text{ bit}.\n\\]\nThis test will be reused with different semantic interpretations.","Consider two researchers and one persistent notebook:\n\\[\nV_{\\mathrm{HH}}=\\{P,C,N\\},\n\\]\nwhere:\n\\begin{itemize}\n\\item \\(P\\) is a proposer who tracks the theorem statement and open proof\n obligations;\n\\item \\(C\\) is a critic who tracks proof obligations and falsification status;\n\\item \\(N\\) is a notebook that tracks the theorem statement and falsification\n status.\n\\end{itemize}\nThe abstract core is\n\\[\nc=(q,p,f)^{\\mathsf T}\\in\\mathbb R^3,\n\\]\nwith\n\\[\n\\begin{aligned}\nq&=\\text{statement coordinate},\n&\np&=\\text{proof-obligation coordinate},","onsider two researchers and one persistent notebook:\n\\[\nV_{\\mathrm{HH}}=\\{P,C,N\\},\n\\]\nwhere:\n\\begin{itemize}\n\\item \\(P\\) is a proposer who tracks the theorem statement and open proof\n obligations;\n\\item \\(C\\) is a critic who tracks proof obligations and falsification status;\n\\item \\(N\\) is a notebook that tracks the theorem statement and falsification\n status.\n\\end{itemize}\nThe abstract core is\n\\[\nc=(q,p,f)^{\\mathsf T}\\in\\mathbb R^3,\n\\]\nwith\n\\[\n\\begin{aligned}\nq&=\\text{statement coordinate},\n&\np&=\\text{proof-obligation coordinate},","ook:\n\\[\nV_{\\mathrm{HH}}=\\{P,C,N\\},\n\\]\nwhere:\n\\begin{itemize}\n\\item \\(P\\) is a proposer who tracks the theorem statement and open proof\n obligations;\n\\item \\(C\\) is a critic who tracks proof obligations and falsification status;\n\\item \\(N\\) is a notebook that tracks the theorem statement and falsification\n status.\n\\end{itemize}\nThe abstract core is\n\\[\nc=(q,p,f)^{\\mathsf T}\\in\\mathbb R^3,\n\\]\nwith\n\\[\n\\begin{aligned}\nq&=\\text{statement coordinate},\n&\np&=\\text{proof-obligation coordinate},\\\\\nf&=\\text{falsification-status coordinate}.\n\\end{aligned}\n\\]\nThe accessible encodings are\n\\[\nE_P=\n\\begin{pmatrix}\n1&0&0","f&=\\text{falsification-status coordinate}.\n\\end{aligned}\n\\]\nThe accessible encodings are\n\\[\nE_P=\n\\begin{pmatrix}\n1&0&0\\\\\n0&1&0\n\\end{pmatrix},\n\\quad\nE_C=\n\\begin{pmatrix}\n0&1&0\\\\\n0&0&1\n\\end{pmatrix},\n\\quad\nE_N=\n\\begin{pmatrix}\n1&0&0\\\\\n0&0&1\n\\end{pmatrix}.\n\\]\nThe semantics are stylized. The point is not that real researchers contain\nonly two numbers. The point is to exhibit, without metaphor, a system in which\nno participant or artifact contains the entire operative state while the\ncoupled system does.","Order the local coordinates as\n\\[\nx=(q_P,p_P,p_C,f_C,q_N,f_N)^{\\mathsf T}.\n\\]\nCompatibility requires\n\\[\np_P=p_C,\n\\qquad\nf_C=f_N,\n\\qquad\nq_N=q_P.\n\\]\nThe discrepancy operator is\n\\[\nD_{\\mathrm{HH}}\n=\n\\begin{pmatrix}\n0&1&-1&0&0&0","Order the local coordinates as\n\\[\nx=(q_P,p_P,p_C,f_C,q_N,f_N)^{\\mathsf T}.\n\\]\nCompatibility requires\n\\[\np_P=p_C,\n\\qquad\nf_C=f_N,\n\\qquad\nq_N=q_P.\n\\]\nThe discrepancy operator is\n\\[\nD_{\\mathrm{HH}}\n=\n\\begin{pmatrix}\n0&1&-1&0&0&0\\\\\n0&0&0&1&0&-1\\\\\n-1&0&0&0&1&0\n\\end{pmatrix}.\n\\]\nTherefore\n\\[\n\\ker D_{\\mathrm{HH}}\n=\n\\left\\{\n(q,p,p,f,q,f)^{\\mathsf T}:q,p,f\\in\\mathbb R\n\\right\\}\n\\cong\\mathbb R^3.\n\\]\nThe rows are orthogonal with squared norm two, so\n\\[\nD_{\\mathrm{HH}}D_{\\mathrm{HH}}^*=2I_3.\n\\]\nConsequently the transfer Laplacian has spectrum\n\\[\n\\boxed{\n\\spec(L_{\\mathrm{HH}})\n=\n\\{0,0,0,2,2,2\\},\n}\n\\]\nand the exact workspace gap is\n\\[\n\\gamma_{\\mathrm{HH}}=2.\n\\]\nThe low-energy subspace is therefore stable under any Laplacian perturbation\nwhose operator norm is strictly below one, with the quantitative projector\nbound from Theorem~\\ref{thm:spectral-stability}.","-1&0&0&0&1&0\n\\end{pmatrix}.\n\\]\nTherefore\n\\[\n\\ker D_{\\mathrm{HH}}\n=\n\\left\\{\n(q,p,p,f,q,f)^{\\mathsf T}:q,p,f\\in\\mathbb R\n\\right\\}\n\\cong\\mathbb R^3.\n\\]\nThe rows are orthogonal with squared norm two, so\n\\[\nD_{\\mathrm{HH}}D_{\\mathrm{HH}}^*=2I_3.\n\\]\nConsequently the transfer Laplacian has spectrum\n\\[\n\\boxed{\n\\spec(L_{\\mathrm{HH}})\n=\n\\{0,0,0,2,2,2\\},\n}\n\\]\nand the exact workspace gap is\n\\[\n\\gamma_{\\mathrm{HH}}=2.\n\\]\nThe low-energy subspace is therefore stable under any Laplacian perturbation\nwhose operator norm is strictly below one, with the quantitative projector\nbound from Theorem~\\ref{thm:spectral-stability}.","The full frame operator is\n\\[\nF_{PCN}=E_P^*E_P+E_C^*E_C+E_N^*E_N=2I_3.\n\\]\nFor every two-node coalition \\(S\\),\n\\[\n\\lambda_{\\min}(F_S)=1.\n\\]\nThus every two-node coalition reconstructs the complete core and, for noisy\nobservations \\(y_S=A_Sc+\\eta_S\\),\n\\[\n\\|\\widehat c-c\\|\n\\leq\n\\|\\eta_S\\|.\n\\]\nNo individual node reconstructs \\(c\\), because\n\\[\n\\operatorname{rank}(E_P)\n=\n\\operatorname{rank}(E_C)\n=\n\\operatorname{rank}(E_N)\n=2<3.\n\\]\nThe realization is therefore simultaneously:\n\\[\n\\text{individually incomplete}\n\\quad\\text{and}\\quad\n\\text{robust to any one-node loss}.\n\\]","For every two-node coalition \\(S\\),\n\\[\n\\lambda_{\\min}(F_S)=1.\n\\]\nThus every two-node coalition reconstructs the complete core and, for noisy\nobservations \\(y_S=A_Sc+\\eta_S\\),\n\\[\n\\|\\widehat c-c\\|\n\\leq\n\\|\\eta_S\\|.\n\\]\nNo individual node reconstructs \\(c\\), because\n\\[\n\\operatorname{rank}(E_P)\n=\n\\operatorname{rank}(E_C)\n=\n\\operatorname{rank}(E_N)\n=2<3.\n\\]\nThe realization is therefore simultaneously:\n\\[\n\\text{individually incomplete}\n\\quad\\text{and}\\quad\n\\text{robust to any one-node loss}.\n\\]","Let \\(q,p,f\\) now be independent uniform bits and define the downstream\nacceptance decision\n\\[\nY=q\\oplus p\\oplus f.\n\\]\nThe interpretation is abstract: acceptance depends on the statement, the state\nof the proof obligations, and the current falsification record. No individual\nnode has sufficient information:\n\\[\nH(Y\\mid X_P)\n=\nH(Y\\mid X_C)\n=\nH(Y\\mid X_N)\n=1.\n\\]\nEvery two-node coalition has all three bits and therefore zero conditional\nentropy. Hence\n\\[\n\\boxed{\n\\Delta_{\\mathrm{joint}}^{\\log}=1\\text{ bit},\n\\qquad\nk_0=2.\n}\n\\]\nLet \\(Q_{\\mathrm{HH}}\\) preserve the independent uniform local bit pairs but\nreplace \\(Y\\) by an independent uniform bit. Because parity is independent of\nevery two-bit local view, \\(Q","Let \\(q,p,f\\) now be independent uniform bits and define the downstream\nacceptance decision\n\\[\nY=q\\oplus p\\oplus f.\n\\]\nThe interpretation is abstract: acceptance depends on the statement, the state\nof the proof obligations, and the current falsification record. No individual\nnode has sufficient information:\n\\[\nH(Y\\mid X_P)\n=\nH(Y\\mid X_C)\n=\nH(Y\\mid X_N)\n=1.\n\\]\nEvery two-node coalition has all three bits and therefore zero conditional\nentropy. Hence\n\\[\n\\boxed{\n\\Delta_{\\mathrm{joint}}^{\\log}=1\\text{ bit},\n\\qquad\nk_0=2.\n}\n\\]\nLet \\(Q_{\\mathrm{HH}}\\) preserve the independent uniform local bit pairs but\nreplace \\(Y\\) by an independent uniform bit. Because parity is independent of\nevery two-bit local view, \\(Q_{\\mathrm{HH}}\\) preserves each\n\\((Y,X_i)\\)-marginal. Nevertheless,\n\\[\n\\Risk_{\\log,Q_{\\mathrm{HH}}}(V)\n-","Let \\(q,p,f\\) now be independent uniform bits and define the downstream\nacceptance decision\n\\[\nY=q\\oplus p\\oplus f.\n\\]\nThe interpretation is abstract: acceptance depends on the statement, the state\nof the proof obligations, and the current falsification record. No individual\nnode has sufficient information:\n\\[\nH(Y\\mid X_P)\n=\nH(Y\\mid X_C)\n=\nH(Y\\mid X_N)\n=1.\n\\]\nEvery two-node coalition has all three bits and therefore zero conditional\nentropy. Hence\n\\[\n\\boxed{\n\\Delta_{\\mathrm{joint}}^{\\log}=1\\text{ bit},\n\\qquad\nk_0=2.\n}\n\\]\nLet \\(Q_{\\mathrm{HH}}\\) preserve the independent uniform local bit pairs but\nreplace \\(Y\\) by an independent uniform bit. Because parity is independent of\nevery two-bit local view, \\(Q_{\\mathrm{HH}}\\) preserves each\n\\((Y,X_i)\\)-marginal. Nevertheless,\n\\[\n\\Risk_{\\log,Q_{\\mathrm{HH}}}(V)\n-\n\\Risk_{\\log,P_{\\mathrm{HH}}}(V)\n=1\\text{ bit}.\n\\]\nThis is an exact relational witness: the predictive capacity lies in the joint\norganization rather than in any local predictive marginal.","ne the downstream\nacceptance decision\n\\[\nY=q\\oplus p\\oplus f.\n\\]\nThe interpretation is abstract: acceptance depends on the statement, the state\nof the proof obligations, and the current falsification record. No individual\nnode has sufficient information:\n\\[\nH(Y\\mid X_P)\n=\nH(Y\\mid X_C)\n=\nH(Y\\mid X_N)\n=1.\n\\]\nEvery two-node coalition has all three bits and therefore zero conditional\nentropy. Hence\n\\[\n\\boxed{\n\\Delta_{\\mathrm{joint}}^{\\log}=1\\text{ bit},\n\\qquad\nk_0=2.\n}\n\\]\nLet \\(Q_{\\mathrm{HH}}\\) preserve the independent uniform local bit pairs but\nreplace \\(Y\\) by an independent uniform bit. Because parity is independent of\nevery two-bit local view, \\(Q_{\\mathrm{HH}}\\) preserves each\n\\((Y,X_i)\\)-marginal. Nevertheless,\n\\[\n\\Risk_{\\log,Q_{\\mathrm{HH}}}(V)\n-\n\\Risk_{\\log,P_{\\mathrm{HH}}}(V)\n=1\\text{ bit}.\n\\]\nThis is an exact relational witness: the predictive capacity lies in the joint\norganization rather than in any local predictive marginal.","tion is abstract: acceptance depends on the statement, the state\nof the proof obligations, and the current falsification record. No individual\nnode has sufficient information:\n\\[\nH(Y\\mid X_P)\n=\nH(Y\\mid X_C)\n=\nH(Y\\mid X_N)\n=1.\n\\]\nEvery two-node coalition has all three bits and therefore zero conditional\nentropy. Hence\n\\[\n\\boxed{\n\\Delta_{\\mathrm{joint}}^{\\log}=1\\text{ bit},\n\\qquad\nk_0=2.\n}\n\\]\nLet \\(Q_{\\mathrm{HH}}\\) preserve the independent uniform local bit pairs but\nreplace \\(Y\\) by an independent uniform bit. Because parity is independent of\nevery two-bit local view, \\(Q_{\\mathrm{HH}}\\) preserves each\n\\((Y,X_i)\\)-marginal. Nevertheless,\n\\[\n\\Risk_{\\log,Q_{\\mathrm{HH}}}(V)\n-\n\\Risk_{\\log,P_{\\mathrm{HH}}}(V)\n=1\\text{ bit}.\n\\]\nThis is an exact relational witness: the predictive capacity lies in the joint\norganization rather than in any local predictive marginal.","igations, and the current falsification record. No individual\nnode has sufficient information:\n\\[\nH(Y\\mid X_P)\n=\nH(Y\\mid X_C)\n=\nH(Y\\mid X_N)\n=1.\n\\]\nEvery two-node coalition has all three bits and therefore zero conditional\nentropy. Hence\n\\[\n\\boxed{\n\\Delta_{\\mathrm{joint}}^{\\log}=1\\text{ bit},\n\\qquad\nk_0=2.\n}\n\\]\nLet \\(Q_{\\mathrm{HH}}\\) preserve the independent uniform local bit pairs but\nreplace \\(Y\\) by an independent uniform bit. Because parity is independent of\nevery two-bit local view, \\(Q_{\\mathrm{HH}}\\) preserves each\n\\((Y,X_i)\\)-marginal. Nevertheless,\n\\[\n\\Risk_{\\log,Q_{\\mathrm{HH}}}(V)\n-\n\\Risk_{\\log,P_{\\mathrm{HH}}}(V)\n=1\\text{ bit}.\n\\]\nThis is an exact relational witness: the predictive capacity lies in the joint\norganization rather than in any local predictive marginal.","[\nH(Y\\mid X_P)\n=\nH(Y\\mid X_C)\n=\nH(Y\\mid X_N)\n=1.\n\\]\nEvery two-node coalition has all three bits and therefore zero conditional\nentropy. Hence\n\\[\n\\boxed{\n\\Delta_{\\mathrm{joint}}^{\\log}=1\\text{ bit},\n\\qquad\nk_0=2.\n}\n\\]\nLet \\(Q_{\\mathrm{HH}}\\) preserve the independent uniform local bit pairs but\nreplace \\(Y\\) by an independent uniform bit. Because parity is independent of\nevery two-bit local view, \\(Q_{\\mathrm{HH}}\\) preserves each\n\\((Y,X_i)\\)-marginal. Nevertheless,\n\\[\n\\Risk_{\\log,Q_{\\mathrm{HH}}}(V)\n-\n\\Risk_{\\log,P_{\\mathrm{HH}}}(V)\n=1\\text{ bit}.\n\\]\nThis is an exact relational witness: the predictive capacity lies in the joint\norganization rather than in any local predictive marginal.","\\mid X_N)\n=1.\n\\]\nEvery two-node coalition has all three bits and therefore zero conditional\nentropy. Hence\n\\[\n\\boxed{\n\\Delta_{\\mathrm{joint}}^{\\log}=1\\text{ bit},\n\\qquad\nk_0=2.\n}\n\\]\nLet \\(Q_{\\mathrm{HH}}\\) preserve the independent uniform local bit pairs but\nreplace \\(Y\\) by an independent uniform bit. Because parity is independent of\nevery two-bit local view, \\(Q_{\\mathrm{HH}}\\) preserves each\n\\((Y,X_i)\\)-marginal. Nevertheless,\n\\[\n\\Risk_{\\log,Q_{\\mathrm{HH}}}(V)\n-\n\\Risk_{\\log,P_{\\mathrm{HH}}}(V)\n=1\\text{ bit}.\n\\]\nThis is an exact relational witness: the predictive capacity lies in the joint\norganization rather than in any local predictive marginal.","ition has all three bits and therefore zero conditional\nentropy. Hence\n\\[\n\\boxed{\n\\Delta_{\\mathrm{joint}}^{\\log}=1\\text{ bit},\n\\qquad\nk_0=2.\n}\n\\]\nLet \\(Q_{\\mathrm{HH}}\\) preserve the independent uniform local bit pairs but\nreplace \\(Y\\) by an independent uniform bit. Because parity is independent of\nevery two-bit local view, \\(Q_{\\mathrm{HH}}\\) preserves each\n\\((Y,X_i)\\)-marginal. Nevertheless,\n\\[\n\\Risk_{\\log,Q_{\\mathrm{HH}}}(V)\n-\n\\Risk_{\\log,P_{\\mathrm{HH}}}(V)\n=1\\text{ bit}.\n\\]\nThis is an exact relational witness: the predictive capacity lies in the joint\norganization rather than in any local predictive marginal.","A feasible statement intervention must alter both local realizations of the\nstatement coordinate:\n\\[\n\\iota_{q\\leftarrow a}:\n(q_P,q_N)\\leftarrow(a,a),\n\\]\nwhile holding \\((p,f)\\) fixed. For configurations with fixed \\(p\\) and \\(f\\),\nchanging \\(a\\) flips the downstream decision with total-variation effect one:\n\\[\n\\TV\n\\left(\nP(Y\\mid\\iota_{q\\leftarrow0}),\nP(Y\\mid\\iota_{q\\leftarrow1})\n\\right)\n=1.\n\\]\nA matched intervention on a residual notebook-format coordinate leaves \\(Y\\)\nunchanged. Thus the core intervention has positive causal specificity in this\nmodel.","f the\nstatement coordinate:\n\\[\n\\iota_{q\\leftarrow a}:\n(q_P,q_N)\\leftarrow(a,a),\n\\]\nwhile holding \\((p,f)\\) fixed. For configurations with fixed \\(p\\) and \\(f\\),\nchanging \\(a\\) flips the downstream decision with total-variation effect one:\n\\[\n\\TV\n\\left(\nP(Y\\mid\\iota_{q\\leftarrow0}),\nP(Y\\mid\\iota_{q\\leftarrow1})\n\\right)\n=1.\n\\]\nA matched intervention on a residual notebook-format coordinate leaves \\(Y\\)\nunchanged. Thus the core intervention has positive causal specificity in this\nmodel.","Suppose the proposer becomes unavailable. The remaining coalition \\(\\{C,N\\}\\)\nreconstructs \\((q,p,f)\\) exactly. A replacement proposer \\(P'\\) can therefore\nbe initialized by\n\\[\nE_{P'}c=(q,p).\n\\]\nIf \\(P'\\) uses a different local coordinate system\n\\[\nJ_{P'}\\in GL(2),\n\\]\nthen the incoming and outgoing overlap maps are conjugated accordingly. The\nglobal section and all downstream kernels remain unchanged. This is exact\nreplacement invariance, not literal carrier identity.","If the notebook alone survives, the core cannot be reconstructed because the\nproof-obligation coordinate is missing. The model therefore predicts a sharp\nhandoff boundary:\n\\[\n\\text{one surviving node is insufficient},\n\\qquad\n\\text{any two surviving nodes are sufficient}.\n\\]","Inside the declared finite model,\n\\[\n\\boxed{\n\\begin{aligned}\n\\mathsf S&=\\checkmark_{\\mathrm{analytic}},\n&\\mathsf O&=\\checkmark_{\\mathrm{analytic}},\\\\\n\\mathsf R&=\\checkmark_{\\mathrm{analytic}},\n&\\mathsf P&=\\checkmark_{\\mathrm{analytic}},\\\\\n\\mathsf F&=\\text{scaffolded},\n&\\mathsf A&=\\text{not tested},\\\\\n\\mathsf D&=\\text{fixed decomposition},\n&\\mathsf C&=\\text{not established}.\n\\end{aligned}\n}\n\\]\nThe highest supported level is Level~4 when the handoff dynamics and identity\ncorrespondence are included. The example does not contain an operational goal\nstate, counterfactual policy, damage monitor, or repair loop, so it does not\nsupport Levels~5 or 6.","Consider four heterogeneous components:\n\\[\nV_{\\mathrm{HAI}}=\\{H,M,X,A\\},\n\\]\nwhere:\n\\begin{itemize}\n\\item \\(H\\) is a human researcher who maintains goals and semantic direction;\n\\item \\(M\\) is a language-model process that constructs local formalizations;\n\\item \\(X\\) is an execution and verification process that compiles, tests, or\n falsifies candidate artifacts; and\n\\item \\(A\\) is a persistent artifact store containing exact source, results,\n and versioned handoff state.\n\\end{itemize}\nLet the abstract research state be\n\\[\nc=(g,d,p,v)^{\\mathsf T}\\in\\mathbb R^4,\n\\]\nwhere:\n\\[\n\\begin{array}{ll}\ng=\\text{goal and acceptance criterion},\n&d=\\text{current derivation or theory state},","rcher who maintains goals and semantic direction;\n\\item \\(M\\) is a language-model process that constructs local formalizations;\n\\item \\(X\\) is an execution and verification process that compiles, tests, or\n falsifies candidate artifacts; and\n\\item \\(A\\) is a persistent artifact store containing exact source, results,\n and versioned handoff state.\n\\end{itemize}\nLet the abstract research state be\n\\[\nc=(g,d,p,v)^{\\mathsf T}\\in\\mathbb R^4,\n\\]\nwhere:\n\\[\n\\begin{array}{ll}\ng=\\text{goal and acceptance criterion},\n&d=\\text{current derivation or theory state},","tem \\(M\\) is a language-model process that constructs local formalizations;\n\\item \\(X\\) is an execution and verification process that compiles, tests, or\n falsifies candidate artifacts; and\n\\item \\(A\\) is a persistent artifact store containing exact source, results,\n and versioned handoff state.\n\\end{itemize}\nLet the abstract research state be\n\\[\nc=(g,d,p,v)^{\\mathsf T}\\in\\mathbb R^4,\n\\]\nwhere:\n\\[\n\\begin{array}{ll}\ng=\\text{goal and acceptance criterion},\n&d=\\text{current derivation or theory state},\\\\\np=\\text{open proof or implementation obligations},\n&v=\\text{verification and regression status}.\n\\end{array}\n\\]\nDefine local encodings\n\\[\nE_Hc=(g,d),\n\\qquad\nE_Mc=(g,p),\n\\qquad\nE_Xc=(p,v),\n\\qquad\nE_Ac=(d,v).\n\\]\nEach coordinate is represented at exactly two nodes, and the incidence pattern\nforms a four-cycle.","p=\\text{open proof or implementation obligations},\n&v=\\text{verification and regression status}.\n\\end{array}\n\\]\nDefine local encodings\n\\[\nE_Hc=(g,d),\n\\qquad\nE_Mc=(g,p),\n\\qquad\nE_Xc=(p,v),\n\\qquad\nE_Ac=(d,v).\n\\]\nEach coordinate is represented at exactly two nodes, and the incidence pattern\nforms a four-cycle.","Order the local coordinates as\n\\[\nx=(g_H,d_H,g_M,p_M,p_X,v_X,d_A,v_A)^{\\mathsf T}.\n\\]\nThe overlap discrepancies are\n\\[\ng_H-g_M,\n\\qquad\np_M-p_X,\n\\qquad\nv_X-v_A,\n\\qquad\nd_A-d_H.\n\\]\nThus\n\\[\nD_{\\mathrm{HAI}}\n=\n\\begin{pmatrix}\n1&0&-1&0&0&0&0&0","Order the local coordinates as\n\\[\nx=(g_H,d_H,g_M,p_M,p_X,v_X,d_A,v_A)^{\\mathsf T}.\n\\]\nThe overlap discrepancies are\n\\[\ng_H-g_M,\n\\qquad\np_M-p_X,\n\\qquad\nv_X-v_A,\n\\qquad\nd_A-d_H.\n\\]\nThus\n\\[\nD_{\\mathrm{HAI}}\n=\n\\begin{pmatrix}\n1&0&-1&0&0&0&0&0\\\\\n0&0&0&1&-1&0&0&0\\\\\n0&0&0&0&0&1&0&-1\\\\\n0&-1&0&0&0&0&1&0\n\\end{pmatrix}.\n\\]\nThe rows are orthogonal with squared norm two. Therefore\n\\[\n\\ker D_{\\mathrm{HAI}}\n\\cong\\mathbb R^4,\n\\]\n\\[\n\\boxed{\n\\spec(L_{\\mathrm{HAI}})\n=\n\\{0,0,0,0,2,2,2,2\\},\n\\qquad\n\\gamma_{\\mathrm{HAI}}=2.\n}\n\\]\nThe full accessible frame operator is\n\\[\nF_{HMXA}=2I_4.\n\\]\nNo single node is injective. Every three-node coalition reconstructs the full\nstate, and the two opposite pairs\n\\[\n\\{H,X\\},\n\\qquad\n\\{M,A\\}\n\\]\nare already sufficient. Adjacent pairs observe only three of the four\ncoordinates. Hence\n\\[\nk_{\\mathrm{rec}}=2,\n\\qquad\nd_{\\mathrm{er}}=2,\n\\]\nwhile the identity of the sufficient two-node coalition depends on the graph\ngeometry.","0&-1&0&0&0&0&1&0\n\\end{pmatrix}.\n\\]\nThe rows are orthogonal with squared norm two. Therefore\n\\[\n\\ker D_{\\mathrm{HAI}}\n\\cong\\mathbb R^4,\n\\]\n\\[\n\\boxed{\n\\spec(L_{\\mathrm{HAI}})\n=\n\\{0,0,0,0,2,2,2,2\\},\n\\qquad\n\\gamma_{\\mathrm{HAI}}=2.\n}\n\\]\nThe full accessible frame operator is\n\\[\nF_{HMXA}=2I_4.\n\\]\nNo single node is injective. Every three-node coalition reconstructs the full\nstate, and the two opposite pairs\n\\[\n\\{H,X\\},\n\\qquad\n\\{M,A\\}\n\\]\nare already sufficient. Adjacent pairs observe only three of the four\ncoordinates. Hence\n\\[\nk_{\\mathrm{rec}}=2,\n\\qquad\nd_{\\mathrm{er}}=2,\n\\]\nwhile the identity of the sufficient two-node coalition depends on the graph\ngeometry.","&0&0&0&1&0\n\\end{pmatrix}.\n\\]\nThe rows are orthogonal with squared norm two. Therefore\n\\[\n\\ker D_{\\mathrm{HAI}}\n\\cong\\mathbb R^4,\n\\]\n\\[\n\\boxed{\n\\spec(L_{\\mathrm{HAI}})\n=\n\\{0,0,0,0,2,2,2,2\\},\n\\qquad\n\\gamma_{\\mathrm{HAI}}=2.\n}\n\\]\nThe full accessible frame operator is\n\\[\nF_{HMXA}=2I_4.\n\\]\nNo single node is injective. Every three-node coalition reconstructs the full\nstate, and the two opposite pairs\n\\[\n\\{H,X\\},\n\\qquad\n\\{M,A\\}\n\\]\nare already sufficient. Adjacent pairs observe only three of the four\ncoordinates. Hence\n\\[\nk_{\\mathrm{rec}}=2,\n\\qquad\nd_{\\mathrm{er}}=2,\n\\]\nwhile the identity of the sufficient two-node coalition depends on the graph\ngeometry.","thbb R^4,\n\\]\n\\[\n\\boxed{\n\\spec(L_{\\mathrm{HAI}})\n=\n\\{0,0,0,0,2,2,2,2\\},\n\\qquad\n\\gamma_{\\mathrm{HAI}}=2.\n}\n\\]\nThe full accessible frame operator is\n\\[\nF_{HMXA}=2I_4.\n\\]\nNo single node is injective. Every three-node coalition reconstructs the full\nstate, and the two opposite pairs\n\\[\n\\{H,X\\},\n\\qquad\n\\{M,A\\}\n\\]\nare already sufficient. Adjacent pairs observe only three of the four\ncoordinates. Hence\n\\[\nk_{\\mathrm{rec}}=2,\n\\qquad\nd_{\\mathrm{er}}=2,\n\\]\nwhile the identity of the sufficient two-node coalition depends on the graph\ngeometry.","Let \\(g,d,p,v\\) be independent uniform bits and define\n\\[\nY=g\\oplus d\\oplus p\\oplus v.\n\\]\nEvery single node sees only two bits, so\n\\[\nH(Y\\mid X_i)=1\n\\qquad\\text{for all }i.\n\\]\nThe opposite pairs \\(\\{H,X\\}\\) and \\(\\{M,A\\}\\) determine all four bits and\ntherefore determine \\(Y\\). Every adjacent pair remains one bit short. It\nfollows that\n\\[\n\\boxed{\n\\Delta_{\\mathrm{joint}}^{\\log}=1\\text{ bit},\n\\qquad\nk_0=2.\n}\n\\]\nAs in the preceding case, parity is independent of every individual local\nview. Replacing \\(Y\\) by an independent uniform bit produces a matched local\npredictive null and a full-system transfer gap of one bit.","ode sees only two bits, so\n\\[\nH(Y\\mid X_i)=1\n\\qquad\\text{for all }i.\n\\]\nThe opposite pairs \\(\\{H,X\\}\\) and \\(\\{M,A\\}\\) determine all four bits and\ntherefore determine \\(Y\\). Every adjacent pair remains one bit short. It\nfollows that\n\\[\n\\boxed{\n\\Delta_{\\mathrm{joint}}^{\\log}=1\\text{ bit},\n\\qquad\nk_0=2.\n}\n\\]\nAs in the preceding case, parity is independent of every individual local\nview. Replacing \\(Y\\) by an independent uniform bit produces a matched local\npredictive null and a full-system transfer gap of one bit.","The idealized loop can be read as the recurrent sequence\n\\[\n\\boxed{\nH\\xrightarrow{g}M\n\\xrightarrow{p}X\n\\xrightarrow{v}A\n\\xrightarrow{d}H.\n}\n\\]\nA more explicit cycle is:\n\\begin{enumerate}\n\\item the human supplies a goal and acceptance criterion;\n\\item the model converts that goal into a formal derivation and explicit\n obligations;\n\\item the executor compiles, tests, or searches for counterexamples;\n\\item the artifact store preserves the exact source and verification record;\n\\item the human evaluates the returned derivation and evidence, revising the\n next goal.\n\\end{enumerate}\nThe operative state is reentrant because products of one cycle become inputs to\nthe next. The artifact is not merely a record when its contents alter the next\nformalization, test, or acceptance decision.","Exact structural core \\(\\Gamma(H)\\) & Unmeasured & No frozen local coordinate\nsystem or learned transfer family has yet been estimated from held-out data.","Transfer-Laplacian gap & Unmeasured & The idealized value \\(2\\) belongs to the\ntoy four-cycle, not to the actual collaboration.","The correct present statement is therefore:\n\\[\n\\boxed{\n\\text{candidate persistent collaborative workspace}\n\\quad\\text{rather than}\\quad\n\\text{certified relational workspace agent}.\n}\n\\]","ext-stage task family;\n\\item the downstream target, such as accepted theorem patch, successful clean\n compile, and absence of a declared regression;\n\\item the local information available to each component;\n\\item the workspace extractor and transfer maps;\n\\item all intervention and null families; and\n\\item the required gate thresholds.\n\\end{itemize}\nThe following conditions can then be compared:\n\\[\n\\begin{array}{ll}\n\\mathsf H & =\\text{human-side externalized state only},\\\\\n\\mathsf M & =\\text{fresh model instance with model-side state only},\\\\\n\\mathsf A & =\\text{artifact package without live human or model state},\\\\\n\\mathsf X & =\\text{executor and test outputs without the research context},\\\\\n\\mathsf Full & =\\text{the intact recurrent loop}.\n\\end{array}\n\\]\nThe design must avoid pretending that a person's unobserved internal state has\nbeen fully measured. The local conditions certify only the declared accessible\nhistories.","Let \\(B_t\\) be the source-and-results bundle at the end of stage \\(t\\). A blind\nhandoff trial replaces the active model instance and optionally the human\nanalyst, then asks the replacement coalition to reconstruct:\n\\[\n\\left(\n\\text{current definitions},\n\\text{proved results},\n\\text{open obligations},\n\\text{next admissible research step}\n\\right).\n\\]\nThe reconstruction error is measured against a frozen adjudication set. A\nreplacement invariance test additionally compares:\n\\begin{itemize}\n\\item the transported structural workspace;\n\\item operation behavior on held-out transformations;\n\\item response to matched interventions; and\n\\item preservation of failed and unresolved claims.\n\\end{itemize}\nEquivalen","Let \\(B_t\\) be the source-and-results bundle at the end of stage \\(t\\). A blind\nhandoff trial replaces the active model instance and optionally the human\nanalyst, then asks the replacement coalition to reconstruct:\n\\[\n\\left(\n\\text{current definitions},\n\\text{proved results},\n\\text{open obligations},\n\\text{next admissible research step}\n\\right).\n\\]\nThe reconstruction error is measured against a frozen adjudication set. A\nreplacement invariance test additionally compares:\n\\begin{itemize}\n\\item the transported structural workspace;\n\\item operation behavior on held-out transformations;\n\\item response to matched interventions; and\n\\item preservation of failed and unresolved claims.\n\\end{itemize}\nEquivalent benchmark prose alone is insufficient. A replacement that","Let \\(B_t\\) be the source-and-results bundle at the end of stage \\(t\\). A blind\nhandoff trial replaces the active model instance and optionally the human\nanalyst, then asks the replacement coalition to reconstruct:\n\\[\n\\left(\n\\text{current definitions},\n\\text{proved results},\n\\text{open obligations},\n\\text{next admissible research step}\n\\right).\n\\]\nThe reconstruction error is measured against a frozen adjudication set. A\nreplacement invariance test additionally compares:\n\\begin{itemize}\n\\item the transported structural workspace;\n\\item operation behavior on held-out transformations;\n\\item response to matched interventions; and\n\\item preservation of failed and unresolved claims.\n\\end{itemize}\nEquivalent benchmark prose alone is insufficient. A replacement that silently\nchanges which interventions alter later reasoning has a nonzero causal identity\ndefect even when it produces a similarly polished manuscript.","For the idealized four-cycle model,\n\\[\n\\mathsf S,\\mathsf O,\\mathsf R,\\mathsf P\n\\]\ncan all be made analytic by specifying the downstream kernel and temporal\nupdates. For the motivating collaboration itself, the conservative card is\n\\[\n\\boxed{\n\\begin{aligned}\n\\mathsf S&=\\text{observational candidate},\n&\\mathsf O&=\\text{unmeasured},","For the idealized four-cycle model,\n\\[\n\\mathsf S,\\mathsf O,\\mathsf R,\\mathsf P\n\\]\ncan all be made analytic by specifying the downstream kernel and temporal\nupdates. For the motivating collaboration itself, the conservative card is\n\\[\n\\boxed{\n\\begin{aligned}\n\\mathsf S&=\\text{observational candidate},\n&\\mathsf O&=\\text{unmeasured},\\\\\n\\mathsf R&=\\text{unmeasured},\n&\\mathsf P&=\\text{observational candidate},\\\\\n\\mathsf F&=\\text{mixed and unmeasured},\n&\\mathsf A&=\\text{not established},\\\\\n\\mathsf D&=\\text{unmeasured},\n&\\mathsf C&=\\text{not established}.\n\\end{aligned}\n}\n\\]\nThis audit is intentionally stricter than ordinary conversational language. It\npreserves the possibility that the collaboration instantiates a real relational\nworkspace while refusing to treat that possibility as already certified.","Consider three agents:\n\\[\nV_{\\mathrm{MA}}=\\{P,M,R\\},\n\\]\ncalled the planner, monitor, and recorder. The global workspace-control state\nis\n\\[\nc_t=(g_t,b_t,h_t)\\in\\mathbb F_2^3,\n\\]\nwhere:\n\\[\n\\begin{array}{ll}\ng_t=\\text{goal bit},\n&b_t=\\text{environment or belief bit},","Consider three agents:\n\\[\nV_{\\mathrm{MA}}=\\{P,M,R\\},\n\\]\ncalled the planner, monitor, and recorder. The global workspace-control state\nis\n\\[\nc_t=(g_t,b_t,h_t)\\in\\mathbb F_2^3,\n\\]\nwhere:\n\\[\n\\begin{array}{ll}\ng_t=\\text{goal bit},\n&b_t=\\text{environment or belief bit},\\\\\nh_t=\\text{workspace-health bit},\n&h_t=1\\text{ means intact}.\n\\end{array}\n\\]\nThe local accessible states are\n\\[\nX_P(t)=(g_t,b_t),\n\\qquad\nX_M(t)=(b_t,h_t),\n\\qquad\nX_R(t)=(g_t,h_t).\n\\]\nThis is the three-node distributed code from Topic~77. Consequently:\n\\[\n\\dim\\Gamma=3,\n\\qquad\n\\spec(L)=\\{0,0,0,2,2,2\\},\n\\qquad\nk_{\\mathrm{rec}}=2,\n\\qquad\nd_{\\mathrm{er}}=2.\n\\]\nThe organization is fully scaffolded. No endogenous-formation claim is made.","h_t=\\text{workspace-health bit},\n&h_t=1\\text{ means intact}.\n\\end{array}\n\\]\nThe local accessible states are\n\\[\nX_P(t)=(g_t,b_t),\n\\qquad\nX_M(t)=(b_t,h_t),\n\\qquad\nX_R(t)=(g_t,h_t).\n\\]\nThis is the three-node distributed code from Topic~77. Consequently:\n\\[\n\\dim\\Gamma=3,\n\\qquad\n\\spec(L)=\\{0,0,0,2,2,2\\},\n\\qquad\nk_{\\mathrm{rec}}=2,\n\\qquad\nd_{\\mathrm{er}}=2.\n\\]\nThe organization is fully scaffolded. No endogenous-formation claim is made.","In the absence of an explicit goal intervention, let\n\\[\ng_{t+1}=g_t.\n\\]\nUnder matched cue-cut trials,\n\\[\n\\operatorname{do}(g_t=0)\n\\quad\\text{and}\\quad\n\\operatorname{do}(g_t=1)\n\\]\nremain distinguishable at every later time:\n\\[\n|g_{t+\\tau}-g_{t+\\tau}'|=1.\n\\]\nThus the cue-controlled retention factor is exactly one:\n\\[\n\\eta_G(\\tau)=1.\n\\]\nThe goal is not reloaded by an external channel after the initial intervention.","After a possible damage event, let the pre-repair health be\n\\[\nh_t^-\\in\\{0,1\\}.\n\\]\nThe system chooses a joint action\n\\[\nu_t=(a_t,r_t)\\in\\{0,1\\}^2,\n\\]\nwhere \\(a_t\\) is a task action and \\(r_t\\) is a repair action. Define the\noptimal task action\n\\[\na_t^*=g_t\\oplus b_t\\oplus h_t^-\n\\]\nand optimal repair action\n\\[\nr_t^*=1-h_t^-.\n\\]\nNo individual node determines the full pair \\((a_t^*,r_t^*)\\):\n\\begin{itemize}\n\\item the planner lacks \\(h_t^-\\);\n\\item the monitor lacks \\(g_t\\); and\n\\item the recorder lacks \\(b_t\\).\n\\end{itemize}\nEvery two-node coalition determines all three workspace bits and therefore the\ncomplete action. Under independent uniform bits,\n\\[\nH(a_t^*\\mid X_i(t))=1\n\\qquad\\text{for every }i,\n\\]\nso the task-action component has a one-bit joint advantage and\n\\[\nk_{\\","After a possible damage event, let the pre-repair health be\n\\[\nh_t^-\\in\\{0,1\\}.\n\\]\nThe system chooses a joint action\n\\[\nu_t=(a_t,r_t)\\in\\{0,1\\}^2,\n\\]\nwhere \\(a_t\\) is a task action and \\(r_t\\) is a repair action. Define the\noptimal task action\n\\[\na_t^*=g_t\\oplus b_t\\oplus h_t^-\n\\]\nand optimal repair action\n\\[\nr_t^*=1-h_t^-.\n\\]\nNo individual node determines the full pair \\((a_t^*,r_t^*)\\):\n\\begin{itemize}\n\\item the planner lacks \\(h_t^-\\);\n\\item the monitor lacks \\(g_t\\); and\n\\item the recorder lacks \\(b_t\\).\n\\end{itemize}\nEvery two-node coalition determines all three workspace bits and therefore the\ncomplete action. Under independent uniform bits,\n\\[\nH(a_t^*\\mid X_i(t))=1\n\\qquad\\text{for every }i,\n\\]\nso the task-action component has a one-bit joint advantage and\n\\[\nk_{\\mathrm{ctl},0}=2.\n\\]","the pre-repair health be\n\\[\nh_t^-\\in\\{0,1\\}.\n\\]\nThe system chooses a joint action\n\\[\nu_t=(a_t,r_t)\\in\\{0,1\\}^2,\n\\]\nwhere \\(a_t\\) is a task action and \\(r_t\\) is a repair action. Define the\noptimal task action\n\\[\na_t^*=g_t\\oplus b_t\\oplus h_t^-\n\\]\nand optimal repair action\n\\[\nr_t^*=1-h_t^-.\n\\]\nNo individual node determines the full pair \\((a_t^*,r_t^*)\\):\n\\begin{itemize}\n\\item the planner lacks \\(h_t^-\\);\n\\item the monitor lacks \\(g_t\\); and\n\\item the recorder lacks \\(b_t\\).\n\\end{itemize}\nEvery two-node coalition determines all three workspace bits and therefore the\ncomplete action. Under independent uniform bits,\n\\[\nH(a_t^*\\mid X_i(t))=1\n\\qquad\\text{for every }i,\n\\]\nso the task-action component has a one-bit joint advantage and\n\\[\nk_{\\mathrm{ctl},0}=2.\n\\]","in\\{0,1\\}.\n\\]\nThe system chooses a joint action\n\\[\nu_t=(a_t,r_t)\\in\\{0,1\\}^2,\n\\]\nwhere \\(a_t\\) is a task action and \\(r_t\\) is a repair action. Define the\noptimal task action\n\\[\na_t^*=g_t\\oplus b_t\\oplus h_t^-\n\\]\nand optimal repair action\n\\[\nr_t^*=1-h_t^-.\n\\]\nNo individual node determines the full pair \\((a_t^*,r_t^*)\\):\n\\begin{itemize}\n\\item the planner lacks \\(h_t^-\\);\n\\item the monitor lacks \\(g_t\\); and\n\\item the recorder lacks \\(b_t\\).\n\\end{itemize}\nEvery two-node coalition determines all three workspace bits and therefore the\ncomplete action. Under independent uniform bits,\n\\[\nH(a_t^*\\mid X_i(t))=1\n\\qquad\\text{for every }i,\n\\]\nso the task-action component has a one-bit joint advantage and\n\\[\nk_{\\mathrm{ctl},0}=2.\n\\]","nt action\n\\[\nu_t=(a_t,r_t)\\in\\{0,1\\}^2,\n\\]\nwhere \\(a_t\\) is a task action and \\(r_t\\) is a repair action. Define the\noptimal task action\n\\[\na_t^*=g_t\\oplus b_t\\oplus h_t^-\n\\]\nand optimal repair action\n\\[\nr_t^*=1-h_t^-.\n\\]\nNo individual node determines the full pair \\((a_t^*,r_t^*)\\):\n\\begin{itemize}\n\\item the planner lacks \\(h_t^-\\);\n\\item the monitor lacks \\(g_t\\); and\n\\item the recorder lacks \\(b_t\\).\n\\end{itemize}\nEvery two-node coalition determines all three workspace bits and therefore the\ncomplete action. Under independent uniform bits,\n\\[\nH(a_t^*\\mid X_i(t))=1\n\\qquad\\text{for every }i,\n\\]\nso the task-action component has a one-bit joint advantage and\n\\[\nk_{\\mathrm{ctl},0}=2.\n\\]","plus b_t\\oplus h_t^-\n\\]\nand optimal repair action\n\\[\nr_t^*=1-h_t^-.\n\\]\nNo individual node determines the full pair \\((a_t^*,r_t^*)\\):\n\\begin{itemize}\n\\item the planner lacks \\(h_t^-\\);\n\\item the monitor lacks \\(g_t\\); and\n\\item the recorder lacks \\(b_t\\).\n\\end{itemize}\nEvery two-node coalition determines all three workspace bits and therefore the\ncomplete action. Under independent uniform bits,\n\\[\nH(a_t^*\\mid X_i(t))=1\n\\qquad\\text{for every }i,\n\\]\nso the task-action component has a one-bit joint advantage and\n\\[\nk_{\\mathrm{ctl},0}=2.\n\\]","(a_t^*,r_t^*)\\):\n\\begin{itemize}\n\\item the planner lacks \\(h_t^-\\);\n\\item the monitor lacks \\(g_t\\); and\n\\item the recorder lacks \\(b_t\\).\n\\end{itemize}\nEvery two-node coalition determines all three workspace bits and therefore the\ncomplete action. Under independent uniform bits,\n\\[\nH(a_t^*\\mid X_i(t))=1\n\\qquad\\text{for every }i,\n\\]\nso the task-action component has a one-bit joint advantage and\n\\[\nk_{\\mathrm{ctl},0}=2.\n\\]","Let the one-step utility be\n\\[\nU_g(a,r;b,h^-)\n=\n\\mathbf 1\\{a=g\\oplus b\\oplus h^-\\}\n+2\\mathbf 1\\{h^-\\lor r=1\\}\n-\\frac12r.\n\\]\nFor fixed \\((g,b,h^-)\\), the unique maximizer is\n\\[\n\\boxed{\n(a^*,r^*)\n=\n(g\\oplus b\\oplus h^-,1-h^-).\n}\n\\]\nIndeed:\n\\begin{itemize}\n\\item a wrong task action loses one unit;\n\\item omitting repair when \\(h^-=0\\) loses \\(3/2\\) units; and\n\\item performing unnecessary repair when \\(h^-=1\\) loses \\(1/2\\) unit.\n\\end{itemize}\nThe smallest optimal-action gap is therefore\n\\[\n\\Delta_Q=\\frac12.\n\\]\nBy Theorem~\\ref{thm:counterfactual-policy}, any uniformly accurate value model with\n\\[\n\\varepsilon_Q<\\frac14\n\\]\nrecovers the exact greedy joint action. More generally, its regret is bounded\nby\n\\[\n\\operatorname{Reg}\\leq2\\varepsilon_Q+\\delta_{\\mathrm{opt}}.\n\\]","et the one-step utility be\n\\[\nU_g(a,r;b,h^-)\n=\n\\mathbf 1\\{a=g\\oplus b\\oplus h^-\\}\n+2\\mathbf 1\\{h^-\\lor r=1\\}\n-\\frac12r.\n\\]\nFor fixed \\((g,b,h^-)\\), the unique maximizer is\n\\[\n\\boxed{\n(a^*,r^*)\n=\n(g\\oplus b\\oplus h^-,1-h^-).\n}\n\\]\nIndeed:\n\\begin{itemize}\n\\item a wrong task action loses one unit;\n\\item omitting repair when \\(h^-=0\\) loses \\(3/2\\) units; and\n\\item performing unnecessary repair when \\(h^-=1\\) loses \\(1/2\\) unit.\n\\end{itemize}\nThe smallest optimal-action gap is therefore\n\\[\n\\Delta_Q=\\frac12.\n\\]\nBy Theorem~\\ref{thm:counterfactual-policy}, any uniformly accurate value model with\n\\[\n\\varepsilon_Q<\\frac14\n\\]\nrecovers the exact greedy joint action. More generally, its regret is bounded\nby\n\\[\n\\operatorname{Reg}\\leq2\\varepsilon_Q+\\delta_{\\mathrm{opt}}.\n\\]","utility be\n\\[\nU_g(a,r;b,h^-)\n=\n\\mathbf 1\\{a=g\\oplus b\\oplus h^-\\}\n+2\\mathbf 1\\{h^-\\lor r=1\\}\n-\\frac12r.\n\\]\nFor fixed \\((g,b,h^-)\\), the unique maximizer is\n\\[\n\\boxed{\n(a^*,r^*)\n=\n(g\\oplus b\\oplus h^-,1-h^-).\n}\n\\]\nIndeed:\n\\begin{itemize}\n\\item a wrong task action loses one unit;\n\\item omitting repair when \\(h^-=0\\) loses \\(3/2\\) units; and\n\\item performing unnecessary repair when \\(h^-=1\\) loses \\(1/2\\) unit.\n\\end{itemize}\nThe smallest optimal-action gap is therefore\n\\[\n\\Delta_Q=\\frac12.\n\\]\nBy Theorem~\\ref{thm:counterfactual-policy}, any uniformly accurate value model with\n\\[\n\\varepsilon_Q<\\frac14\n\\]\nrecovers the exact greedy joint action. More generally, its regret is bounded\nby\n\\[\n\\operatorname{Reg}\\leq2\\varepsilon_Q+\\delta_{\\mathrm{opt}}.\n\\]","bf 1\\{h^-\\lor r=1\\}\n-\\frac12r.\n\\]\nFor fixed \\((g,b,h^-)\\), the unique maximizer is\n\\[\n\\boxed{\n(a^*,r^*)\n=\n(g\\oplus b\\oplus h^-,1-h^-).\n}\n\\]\nIndeed:\n\\begin{itemize}\n\\item a wrong task action loses one unit;\n\\item omitting repair when \\(h^-=0\\) loses \\(3/2\\) units; and\n\\item performing unnecessary repair when \\(h^-=1\\) loses \\(1/2\\) unit.\n\\end{itemize}\nThe smallest optimal-action gap is therefore\n\\[\n\\Delta_Q=\\frac12.\n\\]\nBy Theorem~\\ref{thm:counterfactual-policy}, any uniformly accurate value model with\n\\[\n\\varepsilon_Q<\\frac14\n\\]\nrecovers the exact greedy joint action. More generally, its regret is bounded\nby\n\\[\n\\operatorname{Reg}\\leq2\\varepsilon_Q+\\delta_{\\mathrm{opt}}.\n\\]","\\oplus b\\oplus h^-,1-h^-).\n}\n\\]\nIndeed:\n\\begin{itemize}\n\\item a wrong task action loses one unit;\n\\item omitting repair when \\(h^-=0\\) loses \\(3/2\\) units; and\n\\item performing unnecessary repair when \\(h^-=1\\) loses \\(1/2\\) unit.\n\\end{itemize}\nThe smallest optimal-action gap is therefore\n\\[\n\\Delta_Q=\\frac12.\n\\]\nBy Theorem~\\ref{thm:counterfactual-policy}, any uniformly accurate value model with\n\\[\n\\varepsilon_Q<\\frac14\n\\]\nrecovers the exact greedy joint action. More generally, its regret is bounded\nby\n\\[\n\\operatorname{Reg}\\leq2\\varepsilon_Q+\\delta_{\\mathrm{opt}}.\n\\]","ing repair when \\(h^-=0\\) loses \\(3/2\\) units; and\n\\item performing unnecessary repair when \\(h^-=1\\) loses \\(1/2\\) unit.\n\\end{itemize}\nThe smallest optimal-action gap is therefore\n\\[\n\\Delta_Q=\\frac12.\n\\]\nBy Theorem~\\ref{thm:counterfactual-policy}, any uniformly accurate value model with\n\\[\n\\varepsilon_Q<\\frac14\n\\]\nrecovers the exact greedy joint action. More generally, its regret is bounded\nby\n\\[\n\\operatorname{Reg}\\leq2\\varepsilon_Q+\\delta_{\\mathrm{opt}}.\n\\]","The intervention family contains:\n\\[\n\\operatorname{do}(g_t=a),\n\\qquad\n\\operatorname{do}(b_t=b),\n\\qquad\n\\operatorname{do}(h_t^-=h),\n\\]\nimplemented consistently across every local chart that contains the manipulated\ncoordinate. Holding the other two bits fixed, each core-bit intervention\nchanges the prescribed task action. A matched intervention on an independent\nresidual bit leaves the task and repair actions unchanged. Hence\n\\[\n\\Delta_{\\mathrm{specific}}>0.\n\\]\nA damage intervention sets\n\\[\nh_t^-=0.\n\\]\nThe matched no-repair counterfactual holds \\(r_t=0\\), while the intact policy\nsets \\(r_t=1\\).","s:\n\\[\n\\operatorname{do}(g_t=a),\n\\qquad\n\\operatorname{do}(b_t=b),\n\\qquad\n\\operatorname{do}(h_t^-=h),\n\\]\nimplemented consistently across every local chart that contains the manipulated\ncoordinate. Holding the other two bits fixed, each core-bit intervention\nchanges the prescribed task action. A matched intervention on an independent\nresidual bit leaves the task and repair actions unchanged. Hence\n\\[\n\\Delta_{\\mathrm{specific}}>0.\n\\]\nA damage intervention sets\n\\[\nh_t^-=0.\n\\]\nThe matched no-repair counterfactual holds \\(r_t=0\\), while the intact policy\nsets \\(r_t=1\\).",",\n\\qquad\n\\operatorname{do}(h_t^-=h),\n\\]\nimplemented consistently across every local chart that contains the manipulated\ncoordinate. Holding the other two bits fixed, each core-bit intervention\nchanges the prescribed task action. A matched intervention on an independent\nresidual bit leaves the task and repair actions unchanged. Hence\n\\[\n\\Delta_{\\mathrm{specific}}>0.\n\\]\nA damage intervention sets\n\\[\nh_t^-=0.\n\\]\nThe matched no-repair counterfactual holds \\(r_t=0\\), while the intact policy\nsets \\(r_t=1\\).","tently across every local chart that contains the manipulated\ncoordinate. Holding the other two bits fixed, each core-bit intervention\nchanges the prescribed task action. A matched intervention on an independent\nresidual bit leaves the task and repair actions unchanged. Hence\n\\[\n\\Delta_{\\mathrm{specific}}>0.\n\\]\nA damage intervention sets\n\\[\nh_t^-=0.\n\\]\nThe matched no-repair counterfactual holds \\(r_t=0\\), while the intact policy\nsets \\(r_t=1\\).","tains the manipulated\ncoordinate. Holding the other two bits fixed, each core-bit intervention\nchanges the prescribed task action. A matched intervention on an independent\nresidual bit leaves the task and repair actions unchanged. Hence\n\\[\n\\Delta_{\\mathrm{specific}}>0.\n\\]\nA damage intervention sets\n\\[\nh_t^-=0.\n\\]\nThe matched no-repair counterfactual holds \\(r_t=0\\), while the intact policy\nsets \\(r_t=1\\).","Post-action health evolves according to\n\\[\nh_{t+1}=h_t^-\\lor r_t.\n\\]\nDefine the maintenance deficit\n\\[\ne_t=1-h_t.\n\\]\nUnder the intact policy,\n\\[\nh_{t+1}=1\n\\qquad\\text{and}\\qquad\ne_{t+1}=0\n\\]\nfor both damage states. Under the matched no-repair intervention and\n\\(h_t^-=0\\),\n\\[\ne_{t+1}^{(0)}=1.\n\\]\nThus the measured causal repair gain is\n\\[\n\\boxed{\n\\Delta_{\\mathrm{maint}}=1.\n}\n\\]\nThe closed-loop recurrence has\n\\[\n\\beta=0,\n\\qquad\n\\overline d=0,\n\\qquad\n\\overline\\epsilon=0,\n\\]\nso Theorem~\\ref{thm:self-maintenance} gives\n\\[\n\\sup_t e_t=0\n\\]\nafter each repair action. Monitor-blind random repair fails the specificity\ncontrol whenever it either omits necessary repair or performs unnecessary\nrepair with positive probability.","health evolves according to\n\\[\nh_{t+1}=h_t^-\\lor r_t.\n\\]\nDefine the maintenance deficit\n\\[\ne_t=1-h_t.\n\\]\nUnder the intact policy,\n\\[\nh_{t+1}=1\n\\qquad\\text{and}\\qquad\ne_{t+1}=0\n\\]\nfor both damage states. Under the matched no-repair intervention and\n\\(h_t^-=0\\),\n\\[\ne_{t+1}^{(0)}=1.\n\\]\nThus the measured causal repair gain is\n\\[\n\\boxed{\n\\Delta_{\\mathrm{maint}}=1.\n}\n\\]\nThe closed-loop recurrence has\n\\[\n\\beta=0,\n\\qquad\n\\overline d=0,\n\\qquad\n\\overline\\epsilon=0,\n\\]\nso Theorem~\\ref{thm:self-maintenance} gives\n\\[\n\\sup_t e_t=0\n\\]\nafter each repair action. Monitor-blind random repair fails the specificity\ncontrol whenever it either omits necessary repair or performs unnecessary\nrepair with positive probability.","r the intact policy,\n\\[\nh_{t+1}=1\n\\qquad\\text{and}\\qquad\ne_{t+1}=0\n\\]\nfor both damage states. Under the matched no-repair intervention and\n\\(h_t^-=0\\),\n\\[\ne_{t+1}^{(0)}=1.\n\\]\nThus the measured causal repair gain is\n\\[\n\\boxed{\n\\Delta_{\\mathrm{maint}}=1.\n}\n\\]\nThe closed-loop recurrence has\n\\[\n\\beta=0,\n\\qquad\n\\overline d=0,\n\\qquad\n\\overline\\epsilon=0,\n\\]\nso Theorem~\\ref{thm:self-maintenance} gives\n\\[\n\\sup_t e_t=0\n\\]\nafter each repair action. Monitor-blind random repair fails the specificity\ncontrol whenever it either omits necessary repair or performs unnecessary\nrepair with positive probability.","The repaired health bit becomes part of the next workspace state:\n\\[\nh_t^-\n\\longrightarrow\nr_t\n\\longrightarrow\nh_{t+1}\n\\longrightarrow\nX_M(t+1),X_R(t+1).\n\\]\nCutting the direct storage of \\(h_t^-\\) while preserving the action-mediated\npath still leaves a return path through repair. Cutting the repair-to-health\nedge destroys that return. The system therefore contains explicit\nloop-mediated reentry rather than mere temporal autocorrelation.","nal action values and zero policy regret;\n\\item goal and workspace interventions that alter action;\n\\item damage detection and repair selection;\n\\item positive matched repair gain and bounded maintenance deficit; and\n\\item causal reentry through the action-mediated repair loop.\n\\end{enumerate}\nTherefore, relative to the declared bits, actions, utility, intervention family,\nand one-step horizon,\n\\[\n\\boxed{\n\\operatorname{Agent}_{\\Theta_A}(\\Workspace)=1.\n}\n\\]\nThe case reaches formal Level~6 by construction. It remains:\n\\[\n\\text{fully scaffolded},\n\\qquad\n\\text{domain restricted},\n\\qquad\n\\text{non-evidence for consciousness}.\n\\]","nd repair selection;\n\\item positive matched repair gain and bounded maintenance deficit; and\n\\item causal reentry through the action-mediated repair loop.\n\\end{enumerate}\nTherefore, relative to the declared bits, actions, utility, intervention family,\nand one-step horizon,\n\\[\n\\boxed{\n\\operatorname{Agent}_{\\Theta_A}(\\Workspace)=1.\n}\n\\]\nThe case reaches formal Level~6 by construction. It remains:\n\\[\n\\text{fully scaffolded},\n\\qquad\n\\text{domain restricted},\n\\qquad\n\\text{non-evidence for consciousness}.\n\\]","\\[\n\\boxed{\n\\begin{aligned}\n\\mathsf S&=\\checkmark_{\\mathrm{analytic}},\n&\\mathsf O&=\\checkmark_{\\mathrm{analytic}},\\\\\n\\mathsf R&=\\checkmark_{\\mathrm{analytic}},\n&\\mathsf P&=\\checkmark_{\\mathrm{analytic}},\\\\\n\\mathsf F&=\\text{fully scaffolded},\n&\\mathsf A&=\\checkmark_{\\mathrm{analytic}},\\\\\n\\mathsf D&=\\text{fixed decomposition},\n&\\mathsf C&=\\text{not established}.\n\\end{aligned}\n}\n\\]\nThe example demonstrates logical compatibility among relationality, redundancy,\nagency, and self-maintenance. It does not show that those properties arise\nspontaneously, scale to open environments, or imply subjective experience.","Let every node store a complete copy of the same core:\n\\[\nE_i=I_d\n\\qquad\\text{for every }i.\n\\]\nThen coherence is perfect and reconstruction is robust, but\n\\[\nk_0=1,\n\\qquad\n\\min_i\\delta_i^{\\mathrm{local}}=0.\n\\]\nIf the archive is never used in downstream control, then also\n\\[\n\\eta_{\\mathrm{causal}}=0.\n\\]\nThis is a shared representation or archive, not a relationally irreducible\nworkspace.","Suppose node \\(o\\) contains the complete predictive state \\(Z_t\\), while every\nother node receives only messages generated by \\(o\\). The full system can have\nexcellent performance, rich communication, and recurrent activity. Yet\n\\[\nY_t^+\\indep\\mathbf H_t\\mid X_o(t),\n\\qquad\nk_0=1,\n\\qquad\n\\Delta_{\\mathrm{joint}}=0.\n\\]\nRemoving the decorative nodes may change latency or style without changing the\noperative capacity. Multi-agent appearance therefore does not imply a\nrelational workspace.","Let three local scalar charts have relative overlap demands\n\\[\nb_{12}=b_{23}=b_{31}=1.\n\\]\nThe cycle obstruction is\n\\[\n\\omega_{\\mathcal U}(b)\n=\n\\frac{|1+1+1|}{\\sqrt3}\n=\\sqrt3>0.\n\\]\nEach local chart is internally coherent, but no global scalar assignment\nsatisfies all three overlap equations. The system contains compatible-looking\nsubworkspaces without a global workspace.","Let\n\\[\nG_{t+1}=C_{t+1},\n\\]\nwhere \\(C_{t+1}\\) is an external instruction repeated at every step. Goal and\naction can be strongly correlated, but after cutting the cue channel,\n\\[\nI(G_{t+\\tau};G_t\\mid\\operatorname{cut}(C))=0.\n\\]\nThe system tracks an external command. It has not established internal goal\nretention.","Let two realizations have identical benchmark output distributions,\n\\[\nP(Y_{\\mathrm{bench}})=P'(Y_{\\mathrm{bench}}),\n\\]\nbut different intervention kernels:\n\\[\nK(w,b)\\neq K'(\\beta w,b)\n\\]\nfor some matched workspace intervention. Then task behavior is preserved while\n\\[\nd_{\\mathrm{causal}}(W,W')>0.\n\\]\nThe later system may be a successor or replacement rather than the same\nworkspace under the identity criterion of Part~XII.","A recorded repair sequence may restore average task performance after a fixed\ndamage schedule. If the same sequence is replayed when the damage timing or\ntype changes, it need not respond appropriately. When\n\\[\nR_t\\indep M_t\\mid\\text{clock and replay index},\n\\]\nmonitor-to-repair mediation is absent. Successful replay is robustness under a\nscript, not self-maintenance.","The near misses establish six independent exclusions:\n\\[\n\\boxed{\n\\begin{aligned}\n\\text{workspace}\n\\neq{}&\\text{replicated storage},\\\\\n\\neq{}&\\text{centralized control with decorative communication},\\\\\n\\neq{}&\\text{locally coherent but obstructed charts},\\\\\n\\neq{}&\\text{externally reloaded goals},\\\\\n\\neq{}&\\text{task-equivalent identity replacement},\\\\\n\\neq{}&\\text{open-loop repair replay}.\n\\end{aligned}\n}\n\\]\nA complete certificate must distinguish the intended realization from all\nnear misses relevant to its claimed level.","Spectral gap & \\(2\\) & Not measured & \\(2\\)","Minimal sufficient coalition & \\(k_0=2\\) & Not measured & \\(k_0=2\\)","Matched-null transfer gap & \\(1\\) bit & Not measured & \\(1\\) bit for task parity","The first benchmark is the exact parity family from Topics~85--80. For\n\\(m\\geq3\\), distribute \\(m\\) independent bits among nodes so that no individual\nnode sees all bits, but specified coalitions do. The target is\n\\[\nY=\\bigoplus_{j=1}^m C_j.\n\\]\nThe benchmark measures:\n\\begin{itemize}\n\\item the learned transfer-Laplacian spectrum;\n\\item coalition conditional entropy and log risk;\n\\item recovery of \\(k_0\\);\n\\item robustness to edge and node ablations;\n\\item matched local-predictive null gaps; and\n\\item causal response to consistent versus inconsistent coordinate\n interventions.\n\\end{itemize}\nThe ground truth is known exactly, so structural and relational estimators can\nbe calibrated before they are applied to open-ended research tasks.","e first benchmark is the exact parity family from Topics~85--80. For\n\\(m\\geq3\\), distribute \\(m\\) independent bits among nodes so that no individual\nnode sees all bits, but specified coalitions do. The target is\n\\[\nY=\\bigoplus_{j=1}^m C_j.\n\\]\nThe benchmark measures:\n\\begin{itemize}\n\\item the learned transfer-Laplacian spectrum;\n\\item coalition conditional entropy and log risk;\n\\item recovery of \\(k_0\\);\n\\item robustness to edge and node ablations;\n\\item matched local-predictive null gaps; and\n\\item causal response to consistent versus inconsistent coordinate\n interventions.\n\\end{itemize}\nThe ground truth is known exactly, so structural and relational estimators can\nbe calibrated before they are applied to open-ended research tasks.","A theorem-patch task supplies:\n\\[\n\\left(\n\\text{definitions},\n\\text{partial proof},\n\\text{counterexample tests},\n\\text{style constraints}\n\\right)\n\\]\nacross distinct components. No component receives all four. The coupled\nsystem must produce a patch that:\n\\begin{enumerate}\n\\item is formally consistent with the definitions;\n\\item closes the declared proof obligation or marks it unresolved;\n\\item passes the frozen counterexample suite;\n\\item compiles from a clean environment; and\n\\item survives a blind replacement-handoff trial.\n\\end{enumerate}\nMatched nulls permute proof fragments, substitute irrelevant artifacts of equal\nsize, or replay test outputs from a different theorem. This family targets the\nstructural, relational, persistence, and identity layers without requiring an\nagency claim.","A certified workspace is first formed and measured. A damage intervention then\nremoves or corrupts one of:\n\\[\n\\text{a transfer channel},\n\\quad\n\\text{a participant},\n\\quad\n\\text{an artifact},\n\\quad\n\\text{a goal encoding},\n\\quad\n\\text{a verification path}.\n\\]\nThe system receives a repair-action family with matched costs. The benchmark\nmeasures:\n\\[\n\\eta_{D\\to M},\n\\qquad\n\\eta_{M\\to R},\n\\qquad\n\\eta_{R\\to v},\n\\qquad\n\\Delta_{\\mathrm{repair}}^{\\mathrm{spec}},\n\\qquad\n\\mathbf d_{\\mathrm{id}}.\n\\]\nOpen-loop replay, random repair, off-workspace repair, and identity-breaking\ntask recovery are mandatory controls.","Train the same formation rule under several scaffolds\n\\[\n\\Sigma^{(1)},\\ldots,\\Sigma^{(K)}\n\\]\nthat differ in node labels, coordinate systems, and communication topology but\nsupport the same abstract task. After gauge alignment, measure whether the\nlearned organizations transport to a common equivalence class:\n\\[\n[\\widehat H^{(1)}]_{\\mathcal G}\n\\cong\\cdots\\cong\n[\\widehat H^{(K)}]_{\\mathcal G}.\n\\]\nA structure that appears only under one answer-encoding scaffold has weak\nendogeneity evidence. A structure that recurs across alternative scaffolds,\npasses held-out task and causal gates, and cannot be compressed into the\nscaffold specification has stronger evidence.","When a case combines exact mathematics with observed data, report separate\ncolumns:\n\\[\n\\boxed{\n\\text{proved in model}\n\\quad\\big|\\quad\n\\text{estimated in system}\n\\quad\\big|\\quad\n\\text{not established}.\n}\n\\]\nFor example, the four-cycle Laplacian has an analytic gap of two, but an actual\nhuman--AI collaboration has that gap only if its measured discrepancy operator\nhas the corresponding spectrum. Importing the toy value into the empirical\ncase is invalid.","The audit uses five labels:\n\\[\n\\begin{array}{ll}\n\\mathsf{STD} & = \\text{established concept, theorem, or technique},\\\\\n\\mathsf{ADP} & = \\text{established machinery adapted to relational workspaces},\\\\\n\\mathsf{SYN} & = \\text{nontrivial synthesis of established components},\\\\\n\\mathsf{CAN} & = \\text{candidate original contribution requiring priority review},\\\\\n\\mathsf{EMP} & = \\text{empirical claim not yet established in an uncontrolled system}.\n\\end{array}\n\\]\nThese labels describe the status of a claim, not its importance. A standard\nresult may be indispensable. A synthesis may be scientifically useful without\ncontaining a new theorem. A candidate contribution may later be narrowed or\nwithdrawn when a closer precedent is found.","Failure to find the phrase ``Stratified Relational Workspace Theory'' in prior\nliterature is weak evidence. Priority is determined by mathematical and\noperational equivalence, not by terminology. Let \\(C\\) be a claim in the\npaper and \\(S\\) a prior source. Write\n\\[\nS\\preccurlyeq C\n\\]\nwhen the main content of \\(C\\) follows from \\(S\\) after admissible relabeling,\nrestriction, or direct specialization. A substantive novelty residual exists\nonly when the claim contains content not absorbed by the closure of known\nsources:\n\\[\n\\operatorname{Res}(C)\n=\nC\\setminus\n\\operatorname{Cl}\n\\left(\n\\bigcup_{S\\in\\mathcal L}S\n\\right).\n\\]\nThis is a conceptual audit device rather than a set-theoretic definition of\nideas. Its purpose is to force an explicit account of what remains after the\nstandard machinery is removed.","mined by mathematical and\noperational equivalence, not by terminology. Let \\(C\\) be a claim in the\npaper and \\(S\\) a prior source. Write\n\\[\nS\\preccurlyeq C\n\\]\nwhen the main content of \\(C\\) follows from \\(S\\) after admissible relabeling,\nrestriction, or direct specialization. A substantive novelty residual exists\nonly when the claim contains content not absorbed by the closure of known\nsources:\n\\[\n\\operatorname{Res}(C)\n=\nC\\setminus\n\\operatorname{Cl}\n\\left(\n\\bigcup_{S\\in\\mathcal L}S\n\\right).\n\\]\nThis is a conceptual audit device rather than a set-theoretic definition of\nideas. Its purpose is to force an explicit account of what remains after the\nstandard machinery is removed.","The analysis does not make one global novelty claim. It uses a priority\nprofile\n\\[\n\\boxed{\n\\mathbf p(C)\n=\n\\left(\n p_{\\mathrm{concept}},\n p_{\\mathrm{object}},\n p_{\\mathrm{theorem}},\n p_{\\mathrm{certificate}},\n p_{\\mathrm{experiment}}\n\\right).\n}\n\\]\nFor example, the philosophical concept of distributed cognition may be\nstandard while a specific mathematical object is adapted, a theorem is a\nspecialization of known spectral theory, the certificate is a synthesis, and\nthe empirical claim is unresolved. This prevents originality in one\ncoordinate from being used to imply originality in every coordinate.","Accordingly,\n\\[\n\\operatorname{ExtendedCognition}\n\\centernot\\Longleftrightarrow\n\\operatorname{SRW}.\n\\]\nAn externally scaffolded process may be cognitively extended without being\nrelationally irreducible. Conversely, an institution or multi-agent system may\ninstantiate a certified relational workspace without being an extension of any\nsingle person's cognition.","The distinction is:\n\\[\n\\begin{aligned}\n\\text{transactive memory}\n&\\approx\n\\text{distributed indexing and retrieval of expertise},\\\\\n\\text{collective intelligence}\n&\\approx\n\\text{system-level task capacity},\\\\\n\\text{relational workspace}\n&\\approx\n\\text{an invariant state plus transfer, causal, and identity structure}.\n\\end{aligned}\n\\]\nA transactive-memory system may still be centralized by one coordinator. A\nhigh-performing group may still be explainable by the best individual member.\nThe certificate therefore requires matched coalition risks and nulls\nrather than inferring relationality from group success.","These literatures address questions that the theory deliberately keeps\nseparate:\n\\[\n\\begin{array}{ll}\n\\text{epistemic coordination} & \\text{what members know about one another},\\\\\n\\text{shared intention} & \\text{how intentions and plans are jointly organized},\\\\\n\\text{group agency} & \\text{when collective attitudes and actions form an agent},\\\\\n\\text{SRW structure} & \\text{how a distributed operative state exists and persists}.\n\\end{array}\n\\]\nThe workspace-agent predicate in Part~XVI is domain-relative and causal. It is\nnot a general account of corporate personhood, moral responsibility, or\ncollective intentionality.","The relation is therefore analogical rather than identificatory:\n\\[\n\\boxed{\n\\operatorname{SRW}(W)\n\\centernot\\Longrightarrow\n\\operatorname{GlobalNeuronalWorkspace}(W)\n\\centernot\\Longrightarrow\n\\operatorname{Conscious}(W).\n}\n\\]\nThe term is retained because it denotes a domain in which jointly\navailable operations can be carried out. Every consciousness claim remains on\nan explicitly separate axis.","Blackboard architectures coordinate specialized modules through a shared data\nstructure \\cite{HayesRoth1985}. They are an important engineering precedent\nfor artifact-mediated collaboration. They also provide a canonical near\nmiss. When the blackboard itself contains the complete operative state and\nall modules merely read or write it, the system can fail the relational\nirreducibility gate:\n\\[\nk_0=1.\n\\]\nA blackboard implementation can instantiate a relational workspace, but only\nwhen no single board or controller is sufficient and when the transfer and\ncoalition tests show genuinely distributed operative capacity.","Network consensus studies how local interactions drive distributed states\ntoward agreement \\cite{OlfatiSaberEtAl2007}. Agreement is not the same as a\nworkspace. Consensus often removes heterogeneity by converging all nodes to a\ncommon value. Stratified relational workspaces preserve heterogeneous local\nrepresentations and may require complementary disagreement:\n\\[\nX_i\\not\\cong X_j,\n\\qquad\nE_i c\\neq E_j c,\n\\qquad\n(c_i)_i\\in\\Gamma(H).\n\\]\nThe global state is compatible, not necessarily copied. This distinction is\ncentral to the anti-replication criterion.","The exact compatible-section object\n\\[\n\\Gamma(H)=\\varprojlim C_i\n\\]\nis a standard categorical limit construction \\cite{MacLane1998}. For\ninvertible transfers on a connected graph, identifying global sections with\nthe fixed subspace of holonomy is standard local-system and bundle mathematics\n\\cite{Hatcher2002,Bredon1997}. Accordingly, Theorem~\\(T_1\\) is not claimed as\na new abstract theorem. Its role is foundational: it translates the workspace\nquestion into an exact invariant-subspace question and exposes cycle\nobstructions.","Cellular sheaves encode heterogeneous local data and restriction maps, while\nglobal sections express consistency across a cell complex. Their Laplacians\nhave kernels equal to spaces of global sections and support spectral analysis\nof approximate consistency \\cite{HansenGhrist2019,Robinson2017}. The transfer\nLaplacian\n\\[\nL_H=D_H^*D_H\n\\]\nis therefore best understood as a workspace-specific sheaf or connection\nLaplacian. The kernel theorem \\(\\ker L_H=\\Gamma(H)\\) is standard once the\ncoboundary operator is constructed.","data and restriction maps, while\nglobal sections express consistency across a cell complex. Their Laplacians\nhave kernels equal to spaces of global sections and support spectral analysis\nof approximate consistency \\cite{HansenGhrist2019,Robinson2017}. The transfer\nLaplacian\n\\[\nL_H=D_H^*D_H\n\\]\nis therefore best understood as a workspace-specific sheaf or connection\nLaplacian. The kernel theorem \\(\\ker L_H=\\Gamma(H)\\) is standard once the\ncoboundary operator is constructed.","The appropriate priority statement is:\n\\[\n\\begin{aligned}\n\\text{sheaf/Laplacian mechanism} &:\\ \\mathsf{STD},\\\\\n\\text{workspace interpretation and gate integration} &:\\ \\mathsf{ADP}/\\mathsf{SYN}.\n\\end{aligned}\n\\]\nThe transfer Laplacian itself is not presented as an\nunprecedented mathematical object.","Connection Laplacians compare vector-valued data transported across a graph\nand naturally transform under local changes of coordinates\n\\cite{SingerWu2012}. The gauge rule\n\\[\nH_e^g=g_{t(e)}H_eg_{s(e)}^{-1}\n\\]\nand the conjugacy of holonomy are therefore standard. The theory uses\nthis structure to define realization identity and to prevent raw coordinate\nmatrices from being mistaken for invariant workspace content.","The exact descent theorems \\(T_8\\) and \\(T_9\\) should therefore be presented as\nspecialized consequences of established descent. The candidate contribution\nis the combination:\n\\[\n\\text{descent}\n+\n\\text{predictive sufficiency}\n+\n\\text{causal equivalence}\n+\n\\text{coalition anti-centralization}.\n\\]\nStandard gluing alone does not certify an operative workspace of workspaces.","The projector-stability results in Part~IV belong to the classical perturbation\ntheory of invariant subspaces \\cite{DavisKahan1970,Kato1995}. Bounds of the\nform\n\\[\n\\|P-\\widetilde P\\|\n\\lesssim\n\\frac{\\|L-\\widetilde L\\|}{\\text{spectral gap}}\n\\]\nare not new. Their workspace-specific role is to turn uncertainty in learned\ntransfers into uncertainty about identity, operation closure, and replacement.\nThe novelty, if any, lies in the downstream certificate built on the bound, not\nin the perturbation inequality.","The conditional-law state\n\\[\nZ_t=\\mathbb P(Y_t^+\\in\\cdot\\mid\\mathbf H_t)\n\\]\nis a canonical sufficient statistic. Computational mechanics develops causal\nstates as equivalence classes of histories with identical future conditional\nlaws and proves their minimal predictive character\n\\cite{ShaliziCrutchfield2001}. Theorem~\\(T_{12}\\) is therefore an established\nminimal-sufficiency result specialized to the declared future target and\nintervention family.","The framework adds a decomposition-relative question:\n\\[\n\\text{what is the smallest coalition whose accessible history is sufficient?}\n\\]\nThe statistic \\(k_\\varepsilon\\) is an adaptation of sufficiency to distributed\ncarrier structure. Its interpretation depends on the declared nodes and\naccessible encodings.","Partial-information decomposition distinguishes redundant, unique, and\nsynergistic information in multivariate systems, building on standard\ninformation-theoretic quantities \\cite{CoverThomas2006,WilliamsBeer2010}. The\nXOR examples in Part~XXII are canonical synergy constructions and are not new.\nThey are used because they provide exact ground truth for relational\nirreducibility:\n\\[\nI(Y;X_i)=0\n\\quad\\text{for every }i,\n\\qquad\nI(Y;X_1,\\ldots,X_m)>0.\n\\]\nThe matched-predictive-null test is more restrictive than merely\nreporting synergy. It asks whether local predictive marginals can be held\nfixed while the relational organization and full-system capacity change.\nThat joint null design is a candidate synthesis, not a claim that XOR or\nsynergy is new.","Interventions, counterfactuals, mediation, and causal graphs are established\nmachinery \\cite{Pearl2009}. The theory adopts the interventionist\nrequirement that a workspace variable can be manipulated only through a\nfeasible implementation intervention. This avoids treating\n\\(\\operatorname{do}(W=w)\\) as meaningful when no physical or computational\noperation realizes \\(w\\).","The causal gate is an adaptation:\n\\[\n\\text{SCM intervention theory}\n\\longrightarrow\n\\text{implemented workspace and repair interventions}.\n\\]\nThe potentially distinctive contribution is that causal efficacy is tested\njointly with coherence, sufficiency, coalition irreducibility, matched nulls,\nand identity preservation.","Stable reconstruction from distributed encodings is standard frame theory.\nLower frame bounds control noise amplification, and frame robustness under\nerasures is well studied \\cite{CasazzaKovacevic2003,Christensen2016}. Theorem\n\\(T_7\\) and the node-erasure distance therefore instantiate known recovery\nlogic.","The adaptation is to distinguish:\n\\[\n\\overline E_i\n=\n\\text{complete mathematical chart},\n\\qquad\nE_i=O_i\\overline E_i\n=\n\\text{locally accessible encoding}.\n\\]\nThis distinction is essential to the anti-centralization gate, but the frame\ninequalities themselves are standard.","Bounds for recurrences such as\n\\[\ne_{t+1}\\leq\\alpha_t e_t+\\sigma_t\n\\]\nare elementary stability results. The persistence theorem is not novel as an\ninequality. Its conceptual role is to force three independent checks:\n\\begin{enumerate}\n\\item the workspace subspace remains trackable;\n\\item the state remains close to it; and\n\\item the retained workspace content does not collapse to zero.\n\\end{enumerate}\nThe non-collapse gate is a modeling contribution because it rules out a common\nbut misleading form of ``stability.''","Population least squares, excitation conditions, ridge bias, and empirical\nmodel selection are standard system-identification and statistical-learning\nideas \\cite{Ljung1999}. Theorems \\(T_{15}\\)--\\(T_{17}\\) are finite-dimensional\nspecializations. Their workspace-specific content is the demand that transfer\nmaps be compared modulo gauge and that formation, calibration, and\ncertification data remain separated.","The value-error-to-regret bound in Theorem~\\(T_{20}\\) is elementary decision\ntheory and aligns with standard reinforcement-learning analysis\n\\cite{SuttonBarto2018}. Its role is evidential. An internal value channel\ncounts only when interventions on that channel alter action selection in the\ndirection predicted by counterfactual value and outperform replay or\ncritic-ablation controls.","Multi-agent research contains extensive models of planning, communication,\nconsensus, epistemic coordination, and decentralized control. The present\nreview did not identify one established framework that jointly uses all of the\nfollowing as necessary gates:\n\\[\n\\begin{aligned}\n&\\text{heterogeneous core transport and residuals},\\\\\n&\\text{exact or spectral global-section structure},\\\\\n&\\text{predictive sufficiency and feasible causal interventions},\\\\\n&\\text{coalition irreducibility under matched predictive nulls},\\\\\n&\\text{persistent identity under replacement},\\\\\n&\\text{formation provenance and leakage controls},\\\\\n&\\text{goal-conditioned repair of the certificate itself}.\n\\end{aligned}\n\\]\nThis negative search result is provisional. It supports a synthesis claim, not\nan absolute claim of firstness.","After Mycelium and the shared-workspace systems above, the novelty residual\nshould not be stated as ``human and AI form an active shared scientific\nworkspace.'' After CCP, it should not be stated as ``groups solve tasks no\nmember can represent.'' Both the operational architecture and a rigorous\nrepresentational planning model are already present. The narrower candidate residual is:\n\\[\n\\boxed{\n\\begin{aligned}\n&\\text{persistent core--transfer workspace structure}\\\\\n&+\\text{causal and coalition anti-triviality}\\\\\n&+\\text{identity under replacement}\\\\\n&+\\text{formation provenance}\\\\\n&+\\text{certificate-directed self-maintenance}\\\\\n&+\\text{robustness across admissible decompositions}.\n\\end{aligned}\n}\n\\]\nThis is a stronger and more defensible formulation.","\\midrule\n\\endhead\n\\(T_1\\) & Limit--holonomy equivalence & Categorical limits, local systems, holonomy fixed spaces & \\(\\mathsf{STD}/\\mathsf{ADP}\\). Standard theorem in workspace notation.","\\(T_2,T_{2a}\\) & Operation closure and commutant & Intertwiners and representation commutants & \\(\\mathsf{STD}/\\mathsf{ADP}\\).","\\(T_3\\) & Transfer-Laplacian kernel & Sheaf and connection Laplacians & \\(\\mathsf{STD}/\\mathsf{ADP}\\).","\\(T_4,T_5\\) & Spectral stability and gauge covariance & Davis--Kahan, Kato, connection Laplacians & \\(\\mathsf{STD}/\\mathsf{ADP}\\).","\\(T_{10},T_{11}\\) & Approximate operation closure and path leakage & Spectral projection estimates and telescoping block products & Mostly \\(\\mathsf{ADP}\\). Exact constants may be manuscript-specific, but the method is standard.","\\(T_{12}\\) & Minimal predictive state and alignment & Sufficient statistics and computational mechanics & Minimality is \\(\\mathsf{STD}\\). Alignment with a transfer workspace is \\(\\mathsf{ADP}/\\mathsf{SYN}\\).","\\(T_{13}\\) & Operational relationality & Conditional information, coalition risk, SCM interventions & Individual identities are standard. Their conjunctive matched-null certificate is \\(\\mathsf{SYN}/\\mathsf{CAN}\\).","\\(T_6\\) & Moving-subspace persistence & Stability recurrences and perturbation theory & \\(\\mathsf{STD}/\\mathsf{ADP}\\). The explicit non-collapse gate is a modeling synthesis.","\\(T_7\\) & Stable coalition reconstruction & Frame theory and erasure recovery & \\(\\mathsf{STD}/\\mathsf{ADP}\\).","\\(T_{14}\\) & Replacement invariance profile & Perturbation, process equivalence, causal comparison & Bound ingredients are standard. The coordinatewise structural--causal--reconstructive identity profile is \\(\\mathsf{SYN}/\\mathsf{CAN}\\).","\\(T_8,T_9\\) & Gluing and hierarchical collapse & Sheaf descent and \\v{C}ech theory & Exact results are \\(\\mathsf{STD}/\\mathsf{ADP}\\). Descent plus anti-centralization and causal alignment is \\(\\mathsf{SYN}\\).","\\(T_{15}\\)--\\(T_{17}\\) & Transfer learning, selection, boundary recovery & System identification and finite-margin selection & \\(\\mathsf{STD}/\\mathsf{ADP}\\). Gauge and leakage audit is integrative.","\\(T_{18}\\) & Formation transition example & Standard pitchfork bifurcation & \\(\\mathsf{STD}\\). Included as a solvable benchmark, not a priority claim.","\\(T_{19}\\)--\\(T_{22}\\) & Goal retention, regret, repair, maintenance & Linear systems, decision theory, viability, feedback control & Mathematical bounds are \\(\\mathsf{STD}/\\mathsf{ADP}\\). Applying them to certificate repair is \\(\\mathsf{SYN}\\).","\\(T_{23}\\) & Workspace-agency synthesis & Agency theory, causal control, viability & The conjunctive domain-relative certificate is \\(\\mathsf{SYN}/\\mathsf{CAN}\\).","\\(T_{24}\\) & Gatewise false-certification bound & Intersection testing and elementary probability & \\(\\mathsf{STD}\\). The gate architecture is the relevant synthesis.","\\(T_{25}\\)--\\(T_{27}\\) & Faithful coarse-graining, risk transport, and decomposition robustness & Kernel factorization, sigma-field aggregation, and margin stability & Exact identities are \\(\\mathsf{STD}/\\mathsf{ADP}\\). Their integration with anti-centralization and certificate envelopes is \\(\\mathsf{SYN}/\\mathsf{CAN}\\).","\\(T_{28}\\)--\\(T_{32}\\) & Specification transport, non-identifiability, partial identification, uniform certification, and independent selection & Sensitivity analysis, experimental equivalence, identified sets, uniform concentration, and post-selection inference & The statistical principles are \\(\\mathsf{STD}/\\mathsf{ADP}\\). Their gatewise integration with the full relational certificate is \\(\\mathsf{SYN}/\\mathsf{CAN}\\).","Part XXII & Positive cases and near misses & Distributed parity, frames, causal tests & Toy calculations are standard. The common positive/near-miss benchmark architecture is \\(\\mathsf{SYN}/\\mathsf{CAN}\\).","The following should always be described as inherited machinery:\n\\[\n\\begin{aligned}\n&\\text{inverse limits, global sections, holonomy invariants, commutants},\\\\\n&\\text{sheaf and connection Laplacians, \\v{C}ech gluing, spectral gaps},\\\\\n&\\text{Davis--Kahan and Kato perturbation, frame reconstruction, erasures},\\\\\n&\\text{minimal sufficient states, conditional information, SCM interventions},\\\\\n&\\text{least squares, ridge recovery, finite-margin model selection},\\\\\n&\\text{sensitivity analysis, partial identification, uniform concentration, post-selection inference},\\\\\n&\\text{pitchfork bifurcation, feedback stability, viability, policy regret}.\n\\end{aligned}\n\\]\nRenaming these objects does not create novelty.","The strongest current candidate is not a single equation. It is the following\nconjunctive object:\n\\[\n\\boxed{\n\\mathfrak C_{\\mathrm{SRW}}\n=\n\\left(\n\\begin{array}{c}\n\\text{heterogeneous core--transfer diagram and residuals},\\\\\n\\text{exact or spectral shared-section state},\\\\\n\\text{operation and predictive alignment},\\\\\n\\text{feasible causal interventions},\\\\\n\\text{coalition irreducibility under matched local-predictive nulls},\\\\\n\\text{non-collapse, reconstruction, and replacement identity},\\\\\n\\text{descent across subworkspaces without hidden centralization},\\\\\n\\text{formation provenance and certification leakage control},\\\\\n\\text{goal-conditioned action and certificate-directed repair},\\\\\n\\text{robustness across admissible decompositions and scientific specifications},\\\\\n\\text{identified-set and uniform finite-sample certification}\n\\end{array}\n\\right).\n}\n\\]\nThe reviewed sources contain many of these coordinates separately and some in\npairs. This audit did not locate a prior source that defines their conjunction\nas one falsifiable workspace certificate.","Each coordinate excludes a distinct near miss:\n\\[\n\\begin{array}{ll}\n\\text{coherence} & \\text{rejects incompatible local states},\\\\\n\\text{predictive alignment} & \\text{rejects inert structure},\\\\\n\\text{causal efficacy} & \\text{rejects correlational epiphenomena},\\\\\n\\text{coalition irreducibility} & \\text{rejects centralized sufficiency},\\\\\n\\text{matched nulls} & \\text{reject local-marginal explanations},\\\\\n\\text{non-collapse} & \\text{rejects convergence to a trivial zero state},\\\\\n\\text{replacement identity} & \\text{rejects task-equivalent but causally different successors},\\\\\n\\text{provenance} & \\text{rejects scaffold-decoded organization},\\\\\n\\text{repair specificity} & \\text{rejects passive robustness and open-loop replay},\\\\\n\\text{specification robustness} & \\text{rejects favorable analytical tuning},\\\\\n\\text{identifiability} & \\text{rejects conclusions unsupported by the experiment family}.\n\\end{array}\n\\]\nBecause the excluded systems differ, the certificate is not equivalent to any\none inherited component.","\\subtopic{Level 0: Archive}\n\\[\n\\delta_{\\mathrm{coh}}\\text{ may be low},\n\\qquad\n\\eta_{\\mathrm{causal}}=0.\n\\]","\\subtopic{Level 1: Shared representation}\n\\[\nd_\\varepsilon>0,\n\\qquad\n\\delta_{\\mathrm{coh}}\\leq\\varepsilon.\n\\]","\\subtopic{Level 2: Coordinated operational workspace}\n\\[\n\\delta_{\\mathrm{align}}^{\\mathcal I}\\leq\\varepsilon,\n\\qquad\n\\eta_{\\mathrm{causal}}^\\nu>0.\n\\]","\\subtopic{Level 3: Relational workspace}\n\\[\nk_\\varepsilon\\geq2,\n\\qquad\n\\Delta_{\\mathrm{joint}}^\\ell>0,\n\\qquad\n\\Delta_{\\mathrm{tr}}^{\\min}(\\mathcal Q)>0.\n\\]","\\subtopic{Level 4: Persistent relational workspace}\n\\[\nd_{\\mathrm{id}}(W_t,W_{t+\\tau})\n\\leq\\varepsilon.\n\\]","\\subtopic{Level 5: Self-maintaining workspace}\n\\[\n\\eta_{D\\to M}>0,\n\\qquad\n\\eta_{M\\to R}>0,\n\\qquad\n\\eta_{R\\to v}>0,\n\\qquad\n\\Delta_{\\mathrm{repair}}^{\\mathrm{spec}}>0,\n\\qquad\n\\sup_t e_t\\leq e_{\\max}.\n\\]","\\subtopic{Level 6: Workspace agent}\n\\[\n\\begin{aligned}\n&\\eta_{G\\to A}>0,\n\\qquad\n\\eta_{W\\to A}>0,\n\\qquad\n\\delta_Q\\leq\\varepsilon_Q,\\\\\n&\\operatorname{Reg}_t(\\pi)\n \\leq2\\varepsilon_Q+\\delta_{\\mathrm{opt}},\n\\qquad\nk_{\\mathrm{ctl},\\varepsilon}\\geq2,\n\\qquad\n\\operatorname{SelfMaintenance}.\n\\end{aligned}\n\\]","\\subtopic{Orthogonal formation-provenance axis}\n\\[\n\\operatorname{Scaffolded}\n\\longrightarrow\n\\operatorname{AdaptivelyParameterised}\n\\longrightarrow\n\\operatorname{StructurallyAdaptive}\n\\longrightarrow\n\\operatorname{EndogenouslyConstituted}.\n\\]\nThis axis is independent of Levels~0--6. A scaffolded system can instantiate a\nhigh-level relational workspace, while an endogenously learned organization\ncan fail every operational workspace gate.","Every level is first defined relative to a decomposition \\(\\mathcal D\\). A\nfamily-robust version carries the additional requirement\n\\[\n\\inf_{\\mathcal D\\in\\mathfrak D}\n m_q(W;\\mathcal D)>0\n\\qquad\n\\text{for every gate required at that level}.\n\\]\nThis is a robustness qualifier rather than a higher agency level. A system can\nbe highly agentive under one fixed interface while remaining decomposition\nfragile.","Every level is also relative to a complete scientific specification\n\\(\\varsigma\\). A specification-robust version requires\n\\[\n\\inf_{\\varsigma\\in\\mathfrak S}\n m_q(W;\\varsigma)>0\n\\qquad\n\\text{for every gate required at that level}.\n\\]\nAn identified version additionally requires the sign of each required lower\nenvelope to be constant across the realization set compatible with the\navailable experiment family. An empirical claim must finally subtract the\nuniform finite-sample error radius before declaring the lower envelope\npositive. These are epistemic qualifiers, not additional agency levels.","\\subtopic{Separate, unresolved axis}\n\\[\n\\operatorname{PhenomenalConsciousness}.\n\\]","The complete static-to-agentive construction is\n\\[\n\\boxed{\n\\begin{aligned}\n&\n(\\Omega,\\mathfrak D,\\mathfrak S,\\Sigma,\\mathcal D_{\\mathrm{form}},\\mathcal L)\n\\\\[2mm]\n&\\Downarrow\n\\qquad\n(\\mathcal D,\\widehat V,\\widehat E,\\widehat H,\\widehat P,\\widehat B)\n\\\\[2mm]\n&\\Downarrow\n\\qquad\n(G,\\{X_i\\},\\{T_e\\})\n\\\\[2mm]\n&\\Downarrow\n\\qquad\nX_i=C_i\\oplus R_i\n\\\\[2mm]\n&\n(\\{C_i\\},\\{H_e\\},\\mathcal A,\\{\\rho_i\\})\n\\\\[2mm]\n&\\Downarrow\n\\qquad\n\\Gamma(H)=\\varprojlim C_i\n\\\\[2mm]\n&\\Downarrow\n\\qquad\nL_H=D_H^*D_H\n\\\\[2mm]\n&\\Downarrow\n\\qquad\n\\Gamma_\\varepsilon(H)\n=\n\\ran\\,\\mathbf 1_{[0,\\varepsilon]}(L_H)\n\\\\[2mm]\n&\\Downarrow\n\\qquad\nW_t^{\\mathrm{str}}\n\\\\[2mm]\n&\\xrightarrow{\\text{alignment}}\nZ_t=\\mathbb P(Y_t^+\\in\\cdot\\mid\\mathbf H_t)\n\\\\[2mm]\n&\\Downarrow\n\\qquad\n\\SRW_\\Theta(W_t;\\mathcal D)\n\\\\[2mm]\n&\\Downarrow\n\\qquad\n(W_t,G_t,\\widehat Q_t,M_t,B_t)\n\\\\[2mm]\n&\\xrightarrow{\\text{goal and value use}}\n(A_t,R_t)\n\\\\[2mm]\n&\\xrightarrow{\\text{causal repair}}\ne_t\\leq e_{\\max}\n\\\\[2mm]\n&\\Downarrow\n\\qquad\n\\Workspace\n=\n[\\mathfrak R]_{\\text{gauge}+\\text{causal equivalence}}.\n\\end{aligned}\n}\n\\]\nCompactly:\n\\[\n\\boxed{\n\\begin{aligned}\n\\text{sca…","[2mm]\n&\\Downarrow\n\\qquad\n\\Workspace\n=\n[\\mathfrak R]_{\\text{gauge}+\\text{causal equivalence}}.\n\\end{aligned}\n}\n\\]\nCompactly:\n\\[\n\\boxed{\n\\begin{aligned}\n\\text{scaffold and history}\n&\\to\n\\text{learned organization and boundary}\n\\\\\n&\\to\n\\text{local core--transfer realization}\n\\\\\n&\\to\n\\text{compatible sections and global descent}\n\\\\\n&\\to\n\\text{causal operation and persistent identity}\n\\\\\n&\\to\n\\text{cue-controlled goal retention and counterfactual action}\n\\\\\n&\\to\n\\text{monitor-mediated repair and bounded maintenance deficit}\n\\\\\n&\\to\n\\text{decomposition- and specification-robust identified certification}.\n\\end{aligned}\n}\n\\]\nThe final two arrows are optional extensions rather than consequences of the\nearlier ones. A relational workspace can exist without being an agent, and an\nagentive controller can fail to maintain the workspace that realizes it.","The resulting position is neither that decomposition is arbitrary nor that one\npartition is uniquely privileged. Decompositions are constrained by access,\ncausal interfaces, intervention separability, temporal continuity, and explicit\nshared resources. A certificate established only at one interface is a valid\nbut limited claim. A stronger claim reports the full envelope\n\\[\n\\left(\n\\widetilde m_q,\n\\underline m_q,\n\\overline m_q,\n\\Delta_q^{\\mathrm{dec}}\n\\right)\n\\]\nfor each required gate, where \\(\\tilde m_q\\) is the reference margin and the\nremaining terms summarize its behavior across \\(\\mathfrak D\\).","ly privileged. Decompositions are constrained by access,\ncausal interfaces, intervention separability, temporal continuity, and explicit\nshared resources. A certificate established only at one interface is a valid\nbut limited claim. A stronger claim reports the full envelope\n\\[\n\\left(\n\\widetilde m_q,\n\\underline m_q,\n\\overline m_q,\n\\Delta_q^{\\mathrm{dec}}\n\\right)\n\\]\nfor each required gate, where \\(\\tilde m_q\\) is the reference margin and the\nremaining terms summarize its behavior across \\(\\mathfrak D\\).","rability, temporal continuity, and explicit\nshared resources. A certificate established only at one interface is a valid\nbut limited claim. A stronger claim reports the full envelope\n\\[\n\\left(\n\\widetilde m_q,\n\\underline m_q,\n\\overline m_q,\n\\Delta_q^{\\mathrm{dec}}\n\\right)\n\\]\nfor each required gate, where \\(\\tilde m_q\\) is the reference margin and the\nremaining terms summarize its behavior across \\(\\mathfrak D\\).","re distinguishes truth under a declared specification\nfrom robustness across an admissible family. It also distinguishes robustness\ninside one fitted realization from identification across all realizations\ncompatible with the available experiments. A positive point estimate at one\nfavorable analysis is the weakest admissible conclusion. A stronger claim\nrequires a positive lower envelope across \\(\\mathfrak S\\), an identified sign\nacross the compatible realization set, and a finite-sample confidence bound\nthat remains positive after uniform error control.","The framework is realist about relational organization but does not posit a free-standing substance\nbetween the components. A workspace exists only through a realization, and every change in the\nworkspace must be implemented by a change in participants, artifacts, channels, or their states. The\nquotient\n\\[\n\\Workspace=[\\mathfrak R]_{\\simeq}\n\\]\nis therefore a mathematical account of identity across realizations, not the name of an additional\nphysical object.","\\item \\textbf{Admissible-specification burden.} Part~XVIII makes the target, loss, horizon,\nintervention family, null panel, and thresholds explicit. The admissible family \\(\\mathfrak S\\) still\nrequires scientific judgment. A family chosen after seeing the results can conceal fragility even when\nevery reported analysis is technically correct.","The central identity principle is therefore\n\\[\n\\boxed{\n\\text{workspace identity}\n=\n\\text{persistent invariant causal organization under admissible transport}.\n}\n\\]\nThis principle is compatible with changing carriers, representations, and participants, but only when\nstructural, operational, causal, reconstructive, and non-collapse defects remain within declared tolerances.\nIt provides a formal language for studying distributed collaboration without turning that collaboration\ninto a mysterious third substance and without inferring consciousness from organization alone."]}