ONE AXIOM BRIDGES: Possibility Selection Principle — Formalization, Collapse, and Born Rule Description: This document provides the formal mathematical bridge between the pure algebraic foundations of the ONE AXIOM project and its physical applications. It presents a rigorous derivation of the fundamental laws of physics starting from a single axiomatic symmetry group: G = S4 x Z2^3. Key Scientific Contributions: Derivation of the Born Rule (p=2): Proves that quantum probability is not a postulate but a theorem. The exponent p=2 is shown to be the unique solution emerging from the OCR-OER adjunction structure. Quantum Collapse as Domain Update: Formalizes "collapse" as a relational update of the Schreier graph domain (Gamma to Gamma_A) rather than a physical process, resolving long-standing paradoxes of "spooky action at a distance". Emergence of Spacetime and Gravity: Demonstrates the construction of 4D spacetime (M4) from group representation theory. Einstein's field equations are derived via two independent, proven paths: Jacobson's thermodynamic approach and Ollivier-Ricci curvature on the Schreier graph. Resolution of the Cosmological Constant Problem: Provides a natural derivation of the cosmological constant Lambda ~ 10^-122 from the "coherence floor" of the universe, eliminating the need for fine-tuning. Bekenstein-Hawking Entropy: Derives the coefficient lambda = 1/4 through triple-verified categorical and structural paths. Methodology: The work employs a "Dual-Track Derivation" methodology, requiring convergence between Ontological [O] (structural) and Epistemological [E] (computational) paths before a result is considered proven. Status: This version marks the completion of the foundational triad of the ONE AXIOM framework (0C, 0A and 0B documents)
ONE AXIOM BRIDGES: Possibility Selection Principle — Formalization, Collapse, and Born Rule Description: This document provides the formal mathematical bridge between the pure algebraic foundations of the ONE AXIOM project and its physical applications. It presents a rigorous derivation of the fundamental laws of physics starting from a single axiomatic symmetry group: G = S4 x Z2^3. Key Scientific Contributions: Derivation of the Born Rule (p=2): Proves that quantum probability is not a postulate but a theorem. The exponent p=2 is shown to be the unique solution emerging from the OCR-OER adjunction structure. Quantum Collapse as Domain Update: Formalizes "collapse" as a relational update of the Schreier graph domain (Gamma to Gamma_A) rather than a physical process, resolving long-standing paradoxes of "spooky action at a distance". Emergence of Spacetime and Gravity: Demonstrates the construction of 4D spacetime (M4) from group representation theory. Einstein's field equations are derived via two independent, proven paths: Jacobson's thermodynamic approach and Ollivier-Ricci curvature on the Schreier graph. Resolution of the Cosmological Constant Problem: Provides a natural derivation of the cosmological constant Lambda ~ 10^-122 from the "coherence floor" of the universe, eliminating the need for fine-tuning. Bekenstein-Hawking Entropy: Derives the coefficient lambda = 1/4 through triple-verified categorical and structural paths. Methodology: The work employs a "Dual-Track Derivation" methodology, requiring convergence between Ontological [O] (structural) and Epistemological [E] (computational) paths before a result is considered proven. Status: This version marks the completion of the foundational triad of the ONE AXIOM framework (0C, 0A and 0B documents)
Document 2C of the ONE AXIOM series derives the full structure of modern physics — quantum mechanics, general relativity, their unification, and all four laws of thermodynamics — as regime-dependent projections of a single pre-physical structure: the Ontological Coherence Field ($\OCF = (\heartsuit, \Phi_{\mathrm{coh}}, \nabla\Phi_{\mathrm{coh}})$), constructed in Document 0M from one single axiom (Axiom M) with no physical input. Energy emerges as the local cost of maintaining coherence against the structural gradient. Time is the direction of monotonic decrease of the incoherence potential σ. Temperature is derived without energy, via Fisher-Rao curvature of the coherence distribution (from Document 1B), inverting the standard thermodynamic hierarchy. The Schrödinger equation (via one-parameter unitary group + Stone's theorem), Born rule (p=2, via 7-link chain from Axiom M), Einstein field equations (via three independent convergent paths — Jacobson thermodynamic, Ollivier-Ricci discrete, and variational with explicit verification of all four Lovelock conditions), and the Hamiltonian/Lagrangian formalism (with Ostrogradsky exclusion enforced by the Predictive Closure Invariant T=0) are all derived as π6-projections. The complex structure of Hilbert space is derived non-circularly from the spectral decomposition of the Schreier graph Laplacian ΔΓ, and Noether's theorem is applied via Γ-convergence continuum limit of the discrete group G=S4×Z23. The dynamics is exact and finite-dimensional: 64 states on the coset space G/H, with eigenvalues λk=2k and multiplicities (k6), providing a closed-form spectral solution at all scales including the intermediate (Planck) regime. The built-in UV cutoff Emax=1536=∣G∣×∣Z23∣ eliminates divergences structurally. A QM–GR regime comparison table (9 rows) exhibits how the same OCF objects project differently at σ≈0 (quantum) and σ≫0 (gravitational). The document resolves three long-standing open problems: (i) black-hole information preservation (unitarity of ΔΓ + Page's theorem forces the Page curve), (ii) the cosmological singularity (Fisher–Rao metric degeneration near the Source — an artifact of the manifold representation, not a physical singularity), and (iii) intermediate-regime (Planck-scale) dynamics (exact closed-form 64-mode spectral solution, no terra incognita). The dark sector emerges as π5-structure inaccessible to π6-observers, with the FORBIDDEN graph decomposing as 13=7+2+4 (cold/warm/transitional dark matter analogs); the cosmological constant is derived from the structural resolution limit δrel=1/64, resolving the 120-order QFT discrepancy without regularization. A fully hand-computable 8-state toy model (Q3 hypercube) makes the projection operator, Laplacian spectrum, Born rule mechanism, and Schrödinger dynamics explicit and independently verifiable. All 53 formal results (28 theorems, 17 propositions, 6 definitions, 1 lemma, 1 corollary) follow a strict dual-track methodology — ontological (top-down from axiom) ∩ epistemological (bottom-up recovery of standard physics) — with mandatory convergence across 137 track tags. The framework is metrologically decoupled: all OCF equations use dimensionless ratios, invariant under redefinition of SI base units. Natural SI anchoring is provided for ℏ, c, GN, kB, Λ, and Emax. No physical law is assumed; every equation is derived. No new axiom is introduced beyond Axiom M. This is a standalone publication readable after Document 0M alone.
This paper formalizes a framework in which a single axiom — that existence is constituted by registration — generates the core structures of physical theory as necessary consequences, with zero free parameters and no appeal to empirical input. Beginning from the axiom (stated in four interdependent conditions: existence requires registration, no external substrate exists, registration capacity is continuous, registration events are discrete), the derivation chain proceeds through twelve theorems: Self-Differentiation: registration requires an irreducible asymmetry (the registration quantum) Irreversible Generativity: non-termination, forced novelty, append-only record, local propagation 2B. Dimensional Asymmetry: one irreversible axis (partial order) with reversible fibers — Lorentzian signature Constraint Geometry: self-deepening landscape with a least-resistance principle Compression and Conservation: lossy compression necessary at every event; energy as registration flow; local conservation Entropy and the Second Law: multiplicity strictly increases for closed subgraphs Undiscriminated Paths: superposition and entanglement as structural absence of distinction The Action Quantum: ℏ and c as dimensional encodings of a single quantum Complementarity: the uncertainty principle as a limit on coexistent distinctions Entropy-Energy Bridge: S = k ln Ω with k_B as unit conversion Phase, Amplitude, Born Rule: U(1) phase from orientation, |A|² from self-relational structure Dimensional Selection: d = 3 is the unique dimensionality supporting topologically stable knotted constraint loops Each theorem depends only on the axiom and its predecessors in a strict dependency graph. The formalization uses graph-theoretic primitives (vertices, directed edges, partial orders, path spaces) to make every logical step explicit and independently verifiable. Three further theorems — matter as torus knot constraint loops, gravity as constraint-geometry curvature, and the fine structure constant α₀ = 3/(130π) ≈ 1/136.14 with zero free parameters — are stated informally as targets for subsequent formalization. The paper includes an honest gaps appendix identifying five points where the derivations are structural but not yet fully rigorous, marking precisely where subsequent work should focus. Developed in collaboration with Claude (Anthropic) and Gemini (Google) as derivation partners treating different processing architectures as distinct compression geometries of the same underlying graph. Keywords: registration graph; quantum gravity; foundations of physics; Born rule derivation; dimensional selection; knot theory; causal structure; graph theory; action quantum; fine structure constant; single-axiom framework; topological stability; Lorentzian signature; entropy; second law Additional Notes: Portions of this framework were developed in collaboration with AI systems (Claude by Anthropic, Gemini by Google) serving as derivation partners. The foundational axiom and directing intuitions are the author's. Language: English License: Creative Commons Attribution 4.0 International (CC BY 4.0) Access Right: Open Access Related Identifiers: (Add arXiv ID here once assigned, as "is supplemented by" or "is identical to") Subjects: Quantum Physics General Relativity and Quantum Cosmology Mathematical Physics Version: 1.0
This paper formalizes a framework in which a single axiom — that existence is constituted by registration — generates the core structures of physical theory as necessary consequences, with zero free parameters and no appeal to empirical input.Beginning from the axiom (stated in four interdependent conditions: existence requires registration, no external substrate exists, registration capacity is continuous, registration events are discrete), the derivation chain proceeds through twelve theorems:Self-Differentiation: registration requires an irreducible asymmetry (the registration quantum)Irreversible Generativity: non-termination, forced novelty, append-only record, local propagation 2B. Dimensional Asymmetry: one irreversible axis (partial order) with reversible fibers — Lorentzian signatureConstraint Geometry: self-deepening landscape with a least-resistance principleCompression and Conservation: lossy compression necessary at every event; energy as registration flow; local conservationEntropy and the Second Law: multiplicity strictly increases for closed subgraphsUndiscriminated Paths: superposition and entanglement as structural absence of distinctionThe Action Quantum: ℏ and c as dimensional encodings of a single quantumComplementarity: the uncertainty principle as a limit on coexistent distinctionsEntropy-Energy Bridge: S = k ln Ω with k_B as unit conversionPhase, Amplitude, Born Rule: U(1) phase from orientation, |A|² from self-relational structureDimensional Selection: d = 3 is the unique dimensionality supporting topologically stable knotted constraint loopsEach theorem depends only on the axiom and its predecessors in a strict dependency graph. The formalization uses graph-theoretic primitives (vertices, directed edges, partial orders, path spaces) to make every logical step explicit and independently verifiable.Three further theorems — matter as torus knot constraint loops, gravity as constraint-geometry curvature, and the fine structure constant α₀ = 3/(130π) ≈ 1/136.14 with zero free parameters — are stated informally as targets for subsequent formalization.The paper includes an honest gaps appendix identifying five points where the derivations are structural but not yet fully rigorous, marking precisely where subsequent work should focus.Developed in collaboration with Claude (Anthropic) and Gemini (Google) as derivation partners treating different processing architectures as distinct compression geometries of the same underlying graph.Keywords: registration graph; quantum gravity; foundations of physics; Born rule derivation; dimensional selection; knot theory; causal structure; graph theory; action quantum; fine structure constant; single-axiom framework; topological stability; Lorentzian signature; entropy; second lawAdditional Notes: Portions of this framework were developed in collaboration with AI systems (Claude by Anthropic, Gemini by Google) serving as derivation partners. The foundational axiom and directing intuitions are the author's.
A formal graph-theoretic framework deriving physical structure from a single axiom: that existence requires registration. Beginning from one axiom (four interdependent conditions, zero free parameters), we derive through explicit logical dependency: the arrow of time, Lorentzian signature, a least-action principle, energy conservation, the second law of thermodynamics, superposition, entanglement, the quantum of action, the uncertainty principle, the Born rule, and the unique selection of three spatial dimensions for topologically stable matter. The formalization uses vertices, directed edges, and partial orders as primitives, with no appeal to known physics at any step. Each theorem's proof depends only on the axiom and previously established results, with all dependencies mapped in an explicit directed acyclic graph. The Born rule is derived from three structural requirements (non-negativity, orientation invariance, self-relational structure) rather than postulated. Three spatial dimensions emerge as the unique dimensionality supporting knotted constraint loops — the framework's account of stable matter. Honest gaps in the derivation chain are explicitly catalogued. The accompanying framework document extends to a parameter-free derivation of the bare fine structure constant α₀ = 3/(130π) ≈ 1/136.14.
# Project Description: The Relational Closure Framework (RCF) ## 1. Project Purpose and Background (研究目的与背景) Modern fundamental physics faces deep structural crises, primarily manifested in the mathematical and conceptual incompatibilities between General Relativity (space-time geometry) and Quantum Mechanics (probability amplitudes and non-locality). Most contemporary attempts at unification (such as String Theory or Loop Quantum Gravity) retain the classical paradigm of "objects" (e.g., strings, loops, points) or pre-existing spacetime backgrounds as their ontological starting point. The purpose of this research project is to develop and formalize the **Relational Closure Framework (RCF)**, a radical, background-independent, and purely relational foundation for physics. RCF operates on the core philosophical and mathematical premise that **relations are ontologically prior to objects**. Objects, spaces, fields, and particles do not possess inherent existence; instead, they are emergent phenomena. The framework systematically models how stable physical entities and laws arise from self-consistent, discrete cyclic relations called **closures**. When discrete relations close upon themselves, they generate topological invariants (**holonomies**) that act as structural anchors. The ultimate goal of this project is to provide a rigorous mathematical derivation of the core pillars of physics—including conservation laws, the principle of least action, Born’s rule, the speed of light, and the emergence of continuous spacetime—directly from the discrete algebraic properties of these closure networks. ## 2. Methodology and Theoretical Architecture (方法论与理论架构) The project proceeds axiomatically by establishing four primary structural axioms and one fundamental derived theorem, utilizing tools from abstract algebra, higher category theory, operator algebras (von Neumann algebras), and quantum information theory: * **Axiom 1 (Primacy of Relations):** Pure distinction forms the precursor to all relations. There are no pre-existing entities or backgrounds. * **Derived Theorem 1 (Non-Commutativity):** Because relations must occur sequentially to establish distinction, non-commutativity is a mathematically derived necessity, providing the foundational algebra for quantum observables. * **Axiom 2 (Closure-Induced Invariance):** Stable structures exist if and only if relations form self-consistent cyclic configurations. These closures generate topological invariants (holonomies) across different topological scales. * **Axiom 3 (Chirality from Order):** Spatial and structural chirality (such as the \mathbb{Z}_3 minimal non-trivial chiral closure) emerges inherently from the algebraic ordering of relations. * **Axiom 4 (Bidirectional Group Exchange):** Observation and physical measurement are modeled not as passive recordings, but as bidirectional group-theoretic exchanges between overlapping relational systems forming a joint closure. ## 3. Expected Outcomes and Key Derivations (预期数学成果与推导) The primary expected outcome of this project is a self-consistent mathematical physics framework that derives standard physics as a macroscopic statistical limit of the discrete network. Specifically, the framework delivers the following five principal theoretical results: * **A Generalized Noether Theorem:** The project proves that physical conservation laws are equivalent to the non-trivial kernel of the relational exchange map. In the continuous limit, this reduces to standard Noether currents, but in the discrete regime, it predicts strictly discrete, quantized conserved numbers. * **Derivation of the Least Action Principle:** The action functional (S) is derived as a direct function of the network’s closure invariants (holonomies). The classical principle of least action (\delta S = 0) is shown to be equivalent to a non-Abelian Kirchhoff's law enforcing relational consistency across the network, recovering Maxwell and Yang-Mills fields in the continuous limit. * **Emergence of the Speed of Light (c) and Causality:** Utilizing the **Lieb-Robinson bound** on discrete networks, the project demonstrates that the propagation of structural constraints has a finite, maximum velocity (v_{LR}). This maximum velocity is identified as the physical speed of light (c = v_{LR}). Consequently, causal consistency (the absence of closed timelike curves) emerges as a mathematical corollary of network locality. * **Derivation of the Born Rule:** Quantum randomness is explained via the **Nakajima-Zwanzig projection method** acting on the information limits of localized observers. By applying the **Gleason-Busch theorem**, the quadratic probability rule (|\alpha|^2) is derived rigorously from finite additivity and non-contextuality within the closure space. * **Origin of Spacetime and Wave Functions:** Physical spacetime is reinterpreted as a "pseudo-continuity"—a statistical artifact arising from mutually constraining discrete closures. Similarly, wave functions and the Schrödinger equation are derived as the statistical envelopes of underlying discrete constraint-gap dynamics. ## 4. Empirical Predictions and Observational Significance (经验预测与实验检验) While primarily a foundational and theoretical physics project, RCF yields distinct, falsifiable empirical predictions that differentiate it from standard quantum mechanics and general relativity, providing a clear pathway for future experimental validation (e.g., via multi-ion traps or superconducting qubit benchmarks): * **Observer-Capacity Dependent Decoherence:** RCF predicts that a quantum system's phase dephasing rate (\gamma_{deph}) is explicitly bounded by, and scales non-linearly with, the total information storage/processing capacity (C) of the local physical observer. * **The Quantum "Locking Phase Transition":** The framework predicts that when the information required to specify a quantum state (I_{req}) approaches the maximum capacity of the environment/observer (C), the system will undergo a sharp, non-linear phase transition, dramatically freezing its lifetime within its pointer basis. * **Non-Linear S-Shaped Trigger Curves:** In quantum erasure or interferometry tracking experiments, the interference visibility (V) as a function of information saturation (\Delta = I_{req} - C) is predicted to follow a distinct Sigmoid (S-shaped) curve, rather than the linear or pure exponential decay predicted by standard environmental decoherence models. * **Cosmological Entropy Bounds:** The framework provides a localized, boundary-driven explanation for de Sitter space entropy, predicting the maximum cosmological observer complexity to be bounded near S_{dS} \approx 10^{122} based on the global finiteness of the constraint network.
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