Chronometric Closure Paper II: Channel-Projection Compatibility and the Algebraic Origin of the Chronometric Source Channels Tovi Zituny — Independent Researcher, May 2026 Overview This paper is the second in the Chronometric Closure Series. It addresses the principal open assumption of Chronometric Closure Paper I: channel-projection compatibility (condition E7). Paper I established that the generalized chronometric maintenance source J_source^general = κ_R D∂_τ log R + κ_D R∂τ log D − Ξ∇ ∇² log χ + O(∂_τ², ∇⁴, ℓ²) arises as the local Markovian limit of an admissible open chronometric environment — but only under the assumption that the environmental response projects onto the three chronometric channels D∂_τ log R, R∂_τ log D, and ∇² log χ. Paper I treated this mapping as a given. Paper II derives it as a conditional algebraic projection theorem. Main Result The paper formulates channel-projection compatibility as an algebraic projection problem over local chronometric diamonds equipped with von Neumann record algebras, Araki relative entropy, modular record transport, and split finite-rank control sectors. Three projection lemmas are proved: Lemma CP1 establishes the record-channel projection: the environmental response onto the record sector is identified with D∂_τ log R through the algebraic record-flow identity ∂_τ log R = −∂_τΣ_R, weighted by the distinguishability factor D. Lemma CP2 establishes the distinguishability-channel projection: the logarithmic variation of the chronometric product structure χ_ℓ = C_ℓ[DR] naturally decomposes into a record channel and a distinguishability channel via ∂_τ(DR) = DR(∂_τ log D + ∂_τ log R). Lemma CP3 establishes the spatial chronometric projection: local, short-range, isotropic split-sector spatial response acting on the logarithmic chronometric variable log χ yields the leading Laplacian term ∇² log χ. Conditional Theorem CP assembles the three lemmas into the full channel-projection compatibility result: under modular-record regularity, product-structure preservation, log-support compatibility, local isotropic spatial response, and admissible projection system conditions, the three E7 channels are recovered as conditional low-energy projections with explicit error terms. This upgrades E7 from an unconstrained matching assumption in Paper I to a conditional algebraic projection theorem. The admissibility conditions in Definition 4.1 are stated independently of the E7 conclusion — a non-tautology note is included to make this explicit. Claim Status The result is a conditional algebraic projection theorem at the derivation level of mathematical physics. It does not prove full algebraic universality, derive all environmental operators from a UV-complete chronometric algebra, or show that every microscopic completion must realize the same projections. An explicit limitation is that the distinguishability factor D is not derived from the record algebra in this paper; it enters as a preserved CIFT input and remains a target for deeper information-geometric or operator-algebraic work. Thirteen failure modes are identified and classified in a dedicated section. Structure The paper contains eleven sections. Sections 1–2 inherit the problem from Paper I and formulate the algebraic projection task. Section 3 establishes the algebraic record setting using local diamonds, von Neumann algebras, Araki relative entropy, and record-log flow. Section 4 develops modular record transport and three projection regimes. Sections 5–7 prove Lemmas CP1, CP2, and CP3. Section 8 assembles Conditional Theorem CP. Section 9 identifies thirteen failure modes and scope limits. Section 10 assesses implications for Paper I, CUP, and the closure programme. Section 11 concludes. A claim-status summary table is provided.
Chronometric Closure Paper II: Channel-Projection Compatibility and the Algebraic Origin of the Chronometric Source Channels Tovi Zituny — Independent Researcher, May 2026 Overview This paper is the second in the Chronometric Closure Series. It addresses the principal open assumption of Chronometric Closure Paper I: channel-projection compatibility (condition E7). Paper I established that the generalized chronometric maintenance source J_source^general = κ_R D∂_τ log R + κ_D R∂τ log D − Ξ∇ ∇² log χ + O(∂_τ², ∇⁴, ℓ²) arises as the local Markovian limit of an admissible open chronometric environment — but only under the assumption that the environmental response projects onto the three chronometric channels D∂_τ log R, R∂_τ log D, and ∇² log χ. Paper I treated this mapping as a given. Paper II derives it as a conditional algebraic projection theorem. Main Result The paper formulates channel-projection compatibility as an algebraic projection problem over local chronometric diamonds equipped with von Neumann record algebras, Araki relative entropy, modular record transport, and split finite-rank control sectors. Three projection lemmas are proved: Lemma CP1 establishes the record-channel projection: the environmental response onto the record sector is identified with D∂_τ log R through the algebraic record-flow identity ∂_τ log R = −∂_τΣ_R, weighted by the distinguishability factor D. Lemma CP2 establishes the distinguishability-channel projection: the logarithmic variation of the chronometric product structure χ_ℓ = C_ℓ[DR] naturally decomposes into a record channel and a distinguishability channel via ∂_τ(DR) = DR(∂_τ log D + ∂_τ log R). Lemma CP3 establishes the spatial chronometric projection: local, short-range, isotropic split-sector spatial response acting on the logarithmic chronometric variable log χ yields the leading Laplacian term ∇² log χ. Conditional Theorem CP assembles the three lemmas into the full channel-projection compatibility result: under modular-record regularity, product-structure preservation, log-support compatibility, local isotropic spatial response, and admissible projection system conditions, the three E7 channels are recovered as conditional low-energy projections with explicit error terms. This upgrades E7 from an unconstrained matching assumption in Paper I to a conditional algebraic projection theorem. The admissibility conditions in Definition 4.1 are stated independently of the E7 conclusion — a non-tautology note is included to make this explicit. Claim Status The result is a conditional algebraic projection theorem at the derivation level of mathematical physics. It does not prove full algebraic universality, derive all environmental operators from a UV-complete chronometric algebra, or show that every microscopic completion must realize the same projections. An explicit limitation is that the distinguishability factor D is not derived from the record algebra in this paper; it enters as a preserved CIFT input and remains a target for deeper information-geometric or operator-algebraic work. Thirteen failure modes are identified and classified in a dedicated section. Structure The paper contains eleven sections. Sections 1–2 inherit the problem from Paper I and formulate the algebraic projection task. Section 3 establishes the algebraic record setting using local diamonds, von Neumann algebras, Araki relative entropy, and record-log flow. Section 4 develops modular record transport and three projection regimes. Sections 5–7 prove Lemmas CP1, CP2, and CP3. Section 8 assembles Conditional Theorem CP. Section 9 identifies thirteen failure modes and scope limits. Section 10 assesses implications for Paper I, CUP, and the closure programme. Section 11 concludes. A claim-status summary table is provided.
Chronometric Closure Paper VII addresses the foundational burden of operationally deriving and stabilizing the distinguishability factor D within the CIFT-CMM-CUP programme. The paper constructs the transition-local quantity Dk as a calibrated finite-transition distinguishability functional over admissible finite-rank record sectors, with primary form Dk = Nl dB^2(rho_k, rho_k+1), where dB is the Bures distance and Nl = 1/FQ,ref(l) is the default calibration inherited from CIFT. The central result is a conditional theorem showing that, under finite-rank admissibility, positivity and normalization, operational comparability, non-background parameterization, data-processing monotonicity, coarse-graining stability, and bounded calibration, Dk is non-negative, finite, operationally interpretable, monotone under admissible information-loss channels, compatible with Cl, and suitable for insertion into the Paper VI integrand. A further conditional refinement shows that, where an admissible local record-state assignment and transition localization are available, Dk may be read as the calibrated pullback/restriction of the local distinguishability structure along a record-transition edge. The paper does not claim a universal distinguishability-field theorem for all physical systems.
This preprint develops the theory of spectral closure systems: an operator-algebraic framework for local access to a global compatibility object through typed, lossy closures, organized as a calculus of admissible sections of access maps. Truth is extendability: a local datum is globally valid exactly when it extends along the inverse system, with finite-stage obstruction certificates by compactness. Recovery of local charts is canonical exactly on the sufficiency sector (Petz recovery, conditioning); off it, every totalization is impotent or pays an exact relative-entropy price. Time is the Radon–Nikodym cocycle of the failure of a nonsingular dynamics to preserve belief, realized as the modular flow of its crossed-product envelope; a nontrivial clock forces a noncommutative envelope, with Murray–von Neumann–Connes type read off the ratio set and the flow of weights as envelope invariant. In the quantum perimeter, point-valued global truth is empty (Kochen–Specker), state-valued compatibility survives (Gleason, Christensen–Yeadon), Bell violations measure the classical gluing defect (Fine), and stable classical recordability is characterized as the commutative, modular-invariant, asymptotically sufficient chart sector; on the reflection-positive sector the record process reconstructs a positive-generator dynamics. Globality is relative but closure is terminal: stage sets collapse inward under directed exhaustion, the envelope tower closes outward at the second step by Takesaki duality, unforced finite-entropy residuals decay at a rate uniform in the modular cocycle (a complete logarithmic Sobolev inequality by transference from the circle), and the forcing cost balances exactly in dilation against system–environment correlation. Further results include an independence theorem for the underlying skeleton, a two-tier classification theorem with provable Foreman–Rudolph–Weiss wildness inside its fibers, an envelope universality theorem (the temporal envelopes are exactly the injective von Neumann algebras carrying a Cartan subalgebra), and a sector-split Orlicz envelope for relative entropy beyond bounded densities. Every external input is a named theorem-import with matched hypotheses; the interpretive vocabulary is proved to be a conservative definitional extension of the formal core.
We develop a staged research framework in which physical time is treated not as a primitive external parameter, but as an emergent quantity generated by the quantum dynamical structure of matter-energy configurations. The central working hypothesis is that a physically meaningful temporal increment exists only when state-change distinguishability is both relationally accessible and supported by stable physical records. On this basis, we formulate a Relational Chronometric Principle (RCP) and construct a local chronometric field χℓ(x) from two standard quantum-information-theoretic ingredients: quantum Fisher information and a record-stability factor derived from relative-entropy loss under a calibrated noisy channel (Braunstein & Caves, 1994; Page & Wootters, 1983). The present paper has four limited but explicit goals. First, we formulate the RCP as a minimal axiom set and derive an existence result for a bounded chronometric order parameter, together with a restricted uniqueness result. Second, we introduce a micro-operational definition of χℓ(x), specify the admissible class of coarse-graining maps Cℓ from which it is constructed, and establish a set of core properties for the field including non-negativity, boundedness, and convergence of the operational estimator in the Markovian-Gaussian limit. Third, we analyse the minimal low-energy covariant completion compatible with chronometric universality and show that a scalar-tensor effective field theory provides a controlled representation of emergent proper-time dressing and weak-field gravitational corrections. Fourth, we identify an operational metrology package with full uncertainty propagation and explicit sensitivity targets, a systematic budget for the primary experimental channel, and a three-channel experimental hierarchy based on co-located optical clocks, vertical clock comparisons, and atom interferometry. The framework is deliberately not presented as a final unification theory. It does not provide a UV completion, it does not derive the chronometric coarse-graining map from a fundamental quantum-gravity architecture, and it does not derive the Standard Model gauge sector. Instead, it is advanced as a disciplined low-energy programme that connects three normally separate domains: relational quantum dynamics, weak-field gravity, and precision chronometric observables. Within that restricted scope, the theory yields a falsifiable screened scalar sector, controlled weak-field limits, concrete sensitivity targets, and a cross-channel falsifiability architecture. Keywords: emergent time; relational dynamics; chronometric scalar; quantum Fisher information; scalar-tensor EFT; optical clocks; atom interferometry; screened gravity; coarse-graining maps; record stability
Chronometric Closure Paper VI addresses a foundational burden left open by Closure Paper V: the existence and non-circularity of the chronometric history measure dnu_gamma. The paper develops dnu_gamma as a restricted chronometric transition measure arising from a refinement-consistent limit of finite-rank admissible record-transition histories. Its anti-circularity principle is explicit: record-transition order => dnu_gamma => Delta T_chi, not background time dt => dnu_gamma => Delta T_chi. The main mathematical result is a conditional existence theorem showing that, under admissible directed refinements, positive non-dt-based transition weights, refinement consistency, bounded or controlled integrand variation, and a vanishing refinement-defect envelope, the finite accumulated chronometric sums form a Cauchy net and converge to a finite chronometric interval. The paper does not claim a universal path-space measure or a Radon measure. Instead, it establishes a more conservative restricted transition measure for admissible finite-rank chronometric histories.
A Kernel-Role Determinacy Theorem for Record Fixation, Fixed-Domain Quotients, and Boundary-Trace Normal Form This paper specializes the AASC kernel to the construction and use of quantum measurement records. Its central question is not initially how outcomes are produced, whether a wavefunction collapses, or which interpretation of quantum theory is correct. It asks what must be fixed for a measurement record to function as the same determinate physical and evidential object across retention, prospective use, deliberation, comparison, and report. For any non-degenerate determinate physical object, state, carrier, transformation, record, comparison, or endpoint, the paper takes the governing kernel to be: NDObj(x) ⇒ K(x) = (Adm, Stand, Ref, Irr). Admissibility, standing, reference, and irreversibility are therefore treated as necessary role conditions of determinate record use, not as an optional interpretive overlay added to an otherwise complete physical description. Measurement-record determinacy The manuscript constructs a fixed-domain measurement-record target in which a record must preserve: reference to the same construction token; standing-bearing evidential eligibility; admissible reuse; original act-time qualification; lawful redescription; retained and prospective record use; deliberative and comparative use; reportability and downstream inheritance. The resulting assignment space is quotiented by equivalence of standing-bearing measurement-record content. A rival quotient claiming exactly the same content collapses to the canonical measurement-record quotient. Representational differences that preserve the complete role content are treated as skin; differences affecting record identity, evidential ordering, admissibility, continuation, or downstream use remain physically relevant. Physically non-selective quantum regimes The paper then studies a physically non-selective quantum regime containing unitary branching, entanglement, decoherence, branch-local records, and stable local reports. The central result is conditional and target-specific: The bare unitary-entangling-decohering package can support multiple stable branch-local records without supplying a unique standing evidential record for the fixed construction-relative target. This is not a claim that branch-local records are unreal, unstable, or physically useless. It is a distinction between: physical occurrence of branch-local record structures; local branch-relative reportability; evidential role fixation for a record reused as the same standing object across the measurement practice. The paper shows that non-selective structures may agree on branch weights, density content, decoherence, correlations, instrument sums, channel closure, effect weights, projector orthogonality, and repeatability without selecting one fine record as the standing evidential record. Fixed-domain quotient and fixation commutation The theorem ladder establishes: fixed-domain assignment-space forcing; exact quotient construction for standing-bearing record content; same-scope equivalence collapse; fixation commutation under admissible transformation; inheritance of the original qualification requirement; persistence-bearing reuse; asymmetry exhaustion; anti-weakening; selector exhaustion. If evidential ordering varies across admissible assignments while the target requires one fixed-domain ordering, the ordering is not defined on that scope merely because numerical values or branch-local reports happen to coincide. The paper therefore treats measurement-record fixation as prior to the theory-level evidential reuse that depends on it. Boundary-trace normal form After internal and provenance/indexical routes have been exhausted, the residual same-domain route is a singleton stable boundary trace. Such a trace is not collapse dynamics, hidden-variable selection, or a new physical mechanism. It is an admissibility-preserving role anchor that can fix a record-bearing assignment only when its channel, incidence, transport law, original authority, and singleton-positive condition are independently warranted. The result distinguishes: sound partial selection from complete role fixation; unique trace value from unique original occupant; class-valued record fixation from raw-token fixation; later discovery of an original qualification from retroactive repair; physical record occurrence from standing evidential identity. Finite quantum specialization The paper occupies the abstract structure with an explicit finite two-branch quantum measurement domain containing: two outcomes; two fine records and one merged record; denominator-two branch weights; a balanced integer-amplitude state; density-table realization; selective and non-selective instrument maps; environment-record maps; projective branch maps; complementary effects; zero-product projectors; repeatability operations. On the declared balanced witness, amplitude weights, density weights, branch-map traces, effect weights, pointer records, instrument records, environment records, and channel records agree on the same branch architecture. Selective surfaces preserve the fine-record distinction. Non-selective surfaces agree only on the merged record. The fine records therefore remain separated, while the declared non-selective surfaces supply no singleton fine-record selector. The finite selector-boundary theorem eliminates the disjunction of candidate hidden selectors across the state, instrument, environment, channel, effect, and repeatability surfaces. The resulting diagnosis is precise: the missing structure is admissible fine-record role fixation, not further branch production, weighting, decoherence, stabilization, or repeatability. Scope and non-claims The manuscript establishes a finite, conditional, manuscript-facing theorem. It does not claim: to derive Hilbert-space quantum mechanics; to derive general POVM theory; to derive full projection-valued-measure theory; to reconstruct the full operator algebra; to provide a general Kraus representation theorem; to reconstruct complete positivity; to derive the continuum Born rule; to derive universal physical decoherence theory; to introduce a new collapse dynamics; to prove that no physical collapse process occurs in nature; to prove physical branch selection; to establish interpretive truth; to classify every possible physical measurement mechanism. The absence of a hidden selector on the declared finite non-selective surfaces is therefore not a theorem that no physical collapse or selection process can occur elsewhere. It is the finite theorem that those declared surfaces do not themselves supply the required fine-record selector. Relation to quantum interpretations The result does not deny the practical success of quantum theory, decoherence, branch-relative reporting, instrumental updates, or decision-theoretic methods. A practice aimed only at prediction, branch-local reporting, or instrumental updating may be a different target from the construction-relative record use studied here. If a theory claims to underwrite retained, prospective, deliberative, and evidential reuse of the same record-bearing construction token, it must preserve the relevant kernel roles and admit a standing-bearing fixation regime. Bare Everettian dynamics, without additional dynamics, ontological privileging, or admissible restriction to one constructional class, supplies no fine-record selector on the declared target. A strategy that preserves non-selective physics while restricting admissible record-fixation assignments belongs to a restriction-to-one-constructional-class route. Instrumentalist and QBist strategies instead reduce or redirect the target explanatory burden. Role in the AASC programme This paper provides the record-fixation layer of the AASC Quantum Determinacy Pair. Its companion Bell-scope work addresses the compatibility and no-signaling side of quantum witness determinacy. Together, the pair treats quantum records and witnesses as kernel-governed standing structures rather than as automatically determinate physical occurrences. The paper’s principal contribution is to identify the first failure point in a physically non-selective measurement regime: The initial problem is not necessarily the production of branch-local outcomes. It is the admissible fixation of the record role through which outcomes are retained, reported, remembered, compared, and used as evidence. The public Lean 4 formalization is synchronized with the finite measurement-boundary release v0.9.1 and provides the finite artifact for the declared effect, projector, compatibility, closed-stack, boundary-certificate, and audit layers. Keywords AASC; measurement problem; quantum measurement; measurement-record fixation; physically non-selective regimes; Everettian non-selection; record-role determinacy; fixed-domain quotient; standing equivalence; boundary-trace fixation; selector exhaustion; decoherence; branch-local records; admissibility and irreversibility; Lean 4 formalization
A finitist, relational axiomatization of physics: quantum theory and a relativity sector derived from four axioms about group-valued relations and closure. Contributions to the foundations of quantum mechanics (Born rule derivation, quantum reconstruction, local tomography, complex field selection), the emergence of time and causal structure in discrete networks, discrete Noether theorems, lattice gauge structure, and the Schrödinger equation derived by two independent routes. Every claim is graded in a Claim Ledger; every numerical claim is machine-verified with archived scripts; negative results are reported at full strength. What this paper does The universe is taken to be constituted by relations rather than objects. Stable relations form closures (self-consistent cycles) whose holonomies carry the physical content. Four axioms — primacy of relations (group-valued, edge-borne), closure stability, chirality from order, and observation as bidirectional group exchange — yield, at explicitly graded and conditional levels: Quantum sector Exchange-completeness theorems for finite label groups: abelian at all cycle ranks; non-abelian on the single cycle (Wilson readouts plus class-sum convolutions generate the full matrix algebra); the multi-cycle non-abelian case on the operational-equivalence quotient, with the obstruction completely characterized via class-preserving outer automorphisms — and the quotient's necessity now stratified by the source of word blindness (see New in Version 24.0). The Born rule, proved for finite abelian networks from a strictly weaker relational readout premise (statistics of realizable effects depend only on relational content); non-contextuality has this single source; additivity is basis-relative and declared as such. The Schrödinger equation by two routes of unequal premise grade, maintained side by side: a spectral route — the action operator is a weighted Cayley-graph Laplacian in the Pontryagin dual basis, the derived Gibbs state's unique canonical (modular) flow yields the Schrödinger equation with Hamiltonian equal to the action operator, no continuum limit, zero conjectures consumed, the single non-theorem step one flagged identification; and a Madelung route — physical and mechanism-rich (probability density derived from constraint gaps), valid exactly on the domain of the pseudo-continuity conjecture, with its full premise bill stated (continuity equation, Wallstrom circulation condition, m, V). A dedicated comparison subsection gives the line-by-line premise accounting and the machine-measured agreement region. Complex field selection: ℂ selected over ℝ and ℍ within the GPT comparison class in the covered sectors, with a second independent route from chirality (Frobenius–Schur dichotomy, quaternionic double exclusion); "uniquely selected" expressly not claimed. Positivity as pairing: all five in-text positive-semidefiniteness theorems traced to one mechanism (M = B†B with native factor); the Kraus form and complete positivity derived rather than assumed. The Noether sector on Noether's two theorems exactly: Noether I recovered on the holonomy action (compact Lie realization); Noether II identified as present (Bianchi as horn-filling, Schläfli as its Regge form); selection rules recovered for finite groups; the GNS bridge proved non-vacuously on the full exchange algebra; the sole potential extension — conservation as the kernel of exchange — isolated as a registered conjecture, consumed nowhere. A local–global representation theorem: six generative quantities (closure invariant, flux quantization, Weyl phase, non-abelian Stokes, orientation visibility, terminal normalization) are evaluations of cohomology classes against closed chains — one syntax, six exact machine probes. The gatekeeper no-go theorems registered and discharged: Kochen–Specker (the framework posits no global valuation — outcomes are resolutions of constraint gaps, and basis-relative additivity blocks state-independent variants), Pusey–Barrett–Rudolph (the statistical envelope is over resolutions yet to happen, not values already held; machine witness: identical configuration ensembles with opposite relative phase have opposite statistics), Wallstrom (registered as an explicit input, structurally discharged in the finite-abelian sector), Hegerfeldt (the Lieb–Robinson exponential tail identified as the only claimable front form for the continuum limit), Frauchiger–Renner (cross-agent certainty transfer presupposes kernel-free record transport, exactly what exchange-incompleteness excludes), and no-cloning (inherited in the covered sectors); Conway–Kochen and Colbeck–Renner noted as concordant. Relativity sector Time from finite capacity plus genuine randomness: the staging theorem (Θ(d) dependency depth, unbounded chains via the Moore bound; the quantitative failure law conditionalized on measure-level independence with exact minimal witnesses); the temporal quantum defined combinatorially, no Hamiltonian input. The speed of light grounded in an exact discrete dependency cone (combinatorial, no quantum input), the Lieb–Robinson bound consumed at cross-check grade only; a boost no-go theorem reducing Lorentz invariance to named statistical conditions, with the Collins fine-tuning objection recorded as a gate. The trace-descent theorem group: the growth-history measure forced onto the Mazurkiewicz trace quotient; chain reflection positivity at every depth with completely explicit spectrum. An equation sector of nine verification families (Schrödinger, Dirac, Maxwell/U(1), canonical commutator, Yang–Mills SU(2), Regge calculus, Klein–Gordon, Noether loop, Cartan curvature), where every machine-precision claim is the shadow of a proved backbone lemma — including the factor ½ of H = ½L + V, identified as the Wilson Fourier coefficient fixed by the action theorem. Four experimentally testable predictions, including a discriminating flat-versus-sigmoid visibility protocol. The Standard Model gauge group, ħ, and G remain open, with attack routes registered. How to read Three entry paths are marked in the document. Path A (derivational core): axioms → group structure → Noether sector → least action → Born rule → positivity. Path B (physical mechanisms): constraint propagation → time as determination → wave function as gap envelope → fields. Path C (scope and limits): the Claim Ledger, then Discussion and Open Problems — for the reader who wants to know what is claimed and what is not before reading anything else. The Claim Ledger is the single authoritative register of claim status; wherever body text sounds stronger, the Ledger governs. New in Version 24.0 (Version 23.0 is the immediately preceding public release; this version supersedes the internal V12–V15 and V17–V19 series, whose numbers are not used externally.) Version 24.0 resolves the structure of the multi-cycle non-abelian obstruction — the sector's longest-standing open front — following an external research exchange with three rounds of independent re-verification (fourteen core figures, three commutant instruments implemented independently, exact integer arithmetic on the obstruction side) and two rounds of counter-review that corrected one quantifier on each side: A dichotomy of the multi-cycle obstruction (new remark). Word blindness on multi-cycle networks has two identified sources, jointly covering every tested case, with opposite behaviour as cycle rank grows. Type I (global; graded): a class-preserving outer automorphism yields a commutant element by an argument free of cycle-rank reference; its non-triviality holds at theorem grade for b₁ ≥ d(G) via generating tuples, is monotone in b₁, and the threshold is measured at b_σ = 2 on the tested Wall group (at b₁ = 1 the permutation is the identity — no universal claim is made, and the existence of groups with threshold above 2 is registered open). Type II (local; measured): Out_c trivial but fusion fails on a proper subgroup, the blind pairs sitting in non-generating sectors — the same mechanism that defeats Wilson generation; this blindness heals: on every verified group (A₄, SL(2,3), A₄×ℤ₂) the full commutant collapses to trivial at b₁ = 3 by exact computation on two independently implemented instruments, although word readouts remain blind — the shift primitives separate, above a threshold, and the natural explanation (tuples becoming generating) is refuted by the persistence of blind blocks. Three boundaries at full strength: no exhaustiveness proof (a third source is not excluded); no general healing theorem (the verified groups all heal at 3, cause unidentified); the verified Type II sample is narrow. The quotient proposition restated in stratified form. The operational-equivalence quotient's necessity is now part of the proposition's own statement, in three strata: necessary where Out_c ≠ 1 at every b₁ ≥ b_σ(G), with b_σ ≤ d(G) at theorem grade; one-dimensional at b₁ = 2 on the fusion-type groups (verified cases); trivial at b₁ ≥ 3 on every verified fusion-type group and at every tested b₁ for word-separation-complete groups (a fourteen-group scan of orders ≤ 31 found no third source). Pre-registered validation preceded the restatement: a stratification-exhaustiveness scan (pass), a new-witness search (no new witness in range, downgrade clause applied as pre-registered), and a zero-conflict audit of the three strata against all archived data. The residual open problem is now precisely located. What remains of multi-cycle completeness: the rare Out_c ≠ 1 groups (54 in the exhaustive range through order 96) above their measured thresholds, the low-b₁ fusion stratum, a general healing theorem, and whether a threshold above 2 occurs. The multiplicity-freeness conjecture's mechanism sentence is corrected with the two regimes stated separately (at b₁ = 2 the shifts do not separate — the blind-pair transposition commutes with every primitive, integer-exact — and reduce the commutant to