From Distinction to a Functional Triad: A Framework-Relative Ontology of Coherent Domain-Relative Self-Representation with Oriented Defects, Common Stationary Geometry, and Audited Physics Bridges
This paper develops a framework-relative ontology of determinacy. Its core question is not what the world is made of, but what must minimally be accounted for if a stable deter- minate identity is internally self-representing. The primitive and functional core of Version 3.8 is retained: Being, the possibility of distinction, and domain-relative realizability define the sparse primitive basis; coherent self-representation then requires three non-equivalent func- tional questions—Support, Internal Encoding, and Correspondence. Explicit finite witness pairs establish uniform non-determination of each role from the other two in the unrestricted baseline class, while stronger canonical or equivariant subclasses remain open. Version 4.1.1 is a corrective consolidation of Version 4.1 rather than a new upstream on- tology step. It retains the Version 4.0 bridge revision and the Version 4.1 constructional layer. That layer adds the reverse evaluative direction Physics ⇒ Mathematics ⇒ Ontology and records where downstream structures constrain, but do not retroactively constitute, the ontol- ogy. Several previous bridge claims are weakened or rejected. Symmetry classification, tensor factorization, entanglement of a state, and path-dependent dynamics are kept as four distinct mathematical layers. A conditional expectation is treated as a localization map rather than as actualization; a branch instrument defines possible conditional events but does not select one realized branch; and a projection onto admissible spectral sectors is not called conservative dynamics without an independently derived dilation. The causal and propagation bridge is sharpened in two directions. First, a previous can- didate definition of a null relation by vanishing accumulated interval volume is proved degen- erate: every comparable pair admits a chain of links, so that criterion vanishes throughout the causal order. Link ensembles remain useful near-null approximants, but the null sector is therefore a limiting or measure-relative object rather than a pair-local primitive in the de- clared causal-set model. Second, the long-range free-chain calculation at 𝛼 = 2 is separated into two non-commuting limits. The infinite-range spectral symbol has maximal group veloc- ity 𝜋𝐽𝑎, whereas the envelope of finite-cutoff maxima tends to 2𝐽𝑎 Si(𝜋). These are spectral group-velocity statements; a causal signal cone still requires separate Lieb–Robinson-type hy- potheses. Isotropy and a unique physical Lorentz cone remain open. The defect bridge is updated with the completed 𝐾 = 512 data layer of the frozen ratio campaign. All 903 prescribed values of 𝜃 are present with eight branches each; 897 aggregate points are admissible and six lie in a short rejected interval. Neither √ 2 nor 𝜙 is a local or global minimum. The original preregistered primary ratio test nevertheless remains non- evaluable because its 𝐾 = 64 training gate is universally degenerate. Thus 𝐾 = 512 completes a characterization block; it does not retrospectively repair the primary estimator. Provenance registration and cross-platform verification remain pending. Version 4.0 turned Test 28A into an explicit reconstruction audit, and Version 4.1 retains that audit unchanged. For the standard three one-body extractions on ℋ = (C 3 ) ⊗3 , the combined linear map has exact rank 25 and kernel dimension 704. The hidden space splits as 192 two-body plus 512 fully traceless three-body directions; even the full 𝑆3-invariant class retains an exact 156-dimensional hidden sector. The preregistered continuation identifies an exact energy-selected fixed point, the totally antisymmetric state |𝜀⟩, while holonomy and variance objectives yield set-valued argmin correspondences. The historical non-triviality gate is therefore preserved but shown not to be a universal criterion, and no retroactive 𝑉28.4 verdict is declared. A prospectively frozen two-local QPH family provides an exact computer-assisted existence witness: its cycle-phase differential has rank 704/704 on the marginal kernel for the primary family and two preregistered replications, so marginals plus relational cycle phases are locally in- formationally complete in the fixed regularity stratum. Canonical modular-mismatch selectors are then audited rather than assumed. Identical additive qutrit references and every single pure mismatch under exact 𝑆3 protection obey strict capacity/rank no-go bounds. On the protected 156-dimensional domain a physically generated mixed mismatch has sufficient raw capacity but standard phase-only QPH remains incomplete. Exact cumulant analysis explains why order three is the natural next layer: the 𝑆3-hidden sector decomposes as 156 = 36 + 120, and the third connected cumulant is a complete affine coordinate on fibres of fixed proper marginals. A historical preregistered cubic-moment completion is retained only as a delicate T-NUM wit- ness, not as a theorem premise. This is the declared stopping point of the arity/cubic line: selecting one specific cubic observable would require a new independent physical law. Test 28B remains suspended. Version 4.0 added one further macro-level correction, retained in Version 4.1. Causal and conformal structure, transport geometry, stable matter sectors, and quantum correlations are treated as parallel downstream representations of one substrate model, not as a serial matter– space–time chain. A proposed identification of the matter admissibility projector Πadm with spacetime volume is rejected in its literal form: the projector is matter-branch specific, need not be nested under generative updates, and has no canonical finite trace in a type-III con- tinuum. In its place a conditional causal–transport conformal closure theorem is proved. If a causal reconstruction supplies a Lorentzian conformal class and an independent transport limit supplies a positive spatial metric on the same spacelike distribution, the conformal factor is fixed pointwise exactly when the two induced spatial metrics are conformally compatible. A scale-free generalized-eigenvalue defect provides a direct falsifier. The corresponding clock– cone–matter gate is also corrected by conformal weight: frequencies and dimensionful spectral scales have lapse weight one under a pure lapse deformation, whereas spectral ratios, localiza- tion fractions, and dimensionless stability-window coordinates have weight zero. A Tolman– thermal-time comparison is retained only as a relative lapse-profile gate; absolute time units remain open. Version 4.1 introduced a controlled common-substrate realization. Version 4.1.1 corrects its scope: parity alone excludes only an odd or linear mass/source identification with the same signed coordinate; an even law such as 𝑀 ∝ 𝑑 2 is not excluded by parity. On the declared graph Klein–Gordon sector, charge conjugation reverses Noether/gauge-current orientation while leaving mode energy and localization intensity unchanged. A grounded-graph Green- kernel theorem fixes the spatial uniformity of the cross-term sign, while the Coulomb sign itself is conditional on realizing a non-dynamical 𝐴0-type constraint sector; the same kernel in a dynamical scalar sector gives the opposite sign. A separate no-go proves that matter Dirichlet energy alone cannot select a nontrivial interior geometry. A state-selected positive geometry Hessian yields the conditional linear response 𝑦 = 𝐾 −1 𝐸 𝜏 , so weak 𝐶-even stress sources add while 𝐶-odd currents can cancel. The principal new bridge is a conditional stationary common causal–transport–matter con- struction. If one protected center supplies a common base, the transport branch supplies a positive spatial metric ℎ, and several modular/physical clocks agree on one relative lapse 𝑁, then 𝑔CTM = −𝑐 2 *𝑁2𝑑𝑇2 + ℎ has one timelike sign, predicts the same null front for all canonically normalized matter sectors, and fixes its stationary volume element without an additional scalar measure. Different masses and opposite 𝑈(1) representation weights remain lower-order data and do not alter the characteristic cone. The independently reconstructed HSMI conformal class remains a held-out test rather than an input to this construction. A second independent Version 4.1 result supplies a conditional physical law selecting com- plex rank two. A nonzero-Chern line cannot be closed under conjugation at rank one on 𝑆 2 , whereas 𝐿 ⊕ 𝐿¯ is the minimal conjugation-closed carrier; independently, ClC 2 ≃ 𝑀2(C) has minimal irreducible complex module dimension two. The existing odd-Chern defect-link con struction realizes this pair. If its Clifford algebra is physically admissible, the local carrier has 𝑀2(C) qubit kinematics. This is not an assertion that opposite electric charges form a coherent qubit, nor that every particle is a qubit. The resulting Version 4.1.1 is a constrained foundational proposal with a frozen primitive core, stronger positive and negative bridge results, a first common stationary geometry can- didate, an independently motivated minimal rank-two carrier, and a reproducibility-oriented ledger for every step from ontological determinacy toward physics. It also makes the external spin-sign, soft-graviton, equivalence-principle, and mass-budget inputs explicit without treat- ing them as evidence for the ontology.
This paper develops a framework-relative ontology of determinacy. Its core question is not what the world is made of, but what must minimally be accounted for if a stable deter- minate identity is internally self-representing. The primitive and functional core of Version 3.8 is retained: Being, the possibility of distinction, and domain-relative realizability define the sparse primitive basis; coherent self-representation then requires three non-equivalent func- tional questions—Support, Internal Encoding, and Correspondence. Explicit finite witness pairs establish uniform non-determination of each role from the other two in the unrestricted baseline class, while stronger canonical or equivariant subclasses remain open. Version 4.1.1 is a corrective consolidation of Version 4.1 rather than a new upstream on- tology step. It retains the Version 4.0 bridge revision and the Version 4.1 constructional layer. That layer adds the reverse evaluative direction Physics ⇒ Mathematics ⇒ Ontology and records where downstream structures constrain, but do not retroactively constitute, the ontol- ogy. Several previous bridge claims are weakened or rejected. Symmetry classification, tensor factorization, entanglement of a state, and path-dependent dynamics are kept as four distinct mathematical layers. A conditional expectation is treated as a localization map rather than as actualization; a branch instrument defines possible conditional events but does not select one realized branch; and a projection onto admissible spectral sectors is not called conservative dynamics without an independently derived dilation. The causal and propagation bridge is sharpened in two directions. First, a previous can- didate definition of a null relation by vanishing accumulated interval volume is proved degen- erate: every comparable pair admits a chain of links, so that criterion vanishes throughout the causal order. Link ensembles remain useful near-null approximants, but the null sector is therefore a limiting or measure-relative object rather than a pair-local primitive in the de- clared causal-set model. Second, the long-range free-chain calculation at 𝛼 = 2 is separated into two non-commuting limits. The infinite-range spectral symbol has maximal group veloc- ity 𝜋𝐽𝑎, whereas the envelope of finite-cutoff maxima tends to 2𝐽𝑎 Si(𝜋). These are spectral group-velocity statements; a causal signal cone still requires separate Lieb–Robinson-type hy- potheses. Isotropy and a unique physical Lorentz cone remain open. The defect bridge is updated with the completed 𝐾 = 512 data layer of the frozen ratio campaign. All 903 prescribed values of 𝜃 are present with eight branches each; 897 aggregate points are admissible and six lie in a short rejected interval. Neither √ 2 nor 𝜙 is a local or global minimum. The original preregistered primary ratio test nevertheless remains non- evaluable because its 𝐾 = 64 training gate is universally degenerate. Thus 𝐾 = 512 completes a characterization block; it does not retrospectively repair the primary estimator. Provenance registration and cross-platform verification remain pending. Version 4.0 turned Test 28A into an explicit reconstruction audit, and Version 4.1 retains that audit unchanged. For the standard three one-body extractions on ℋ = (C 3 ) ⊗3 , the combined linear map has exact rank 25 and kernel dimension 704. The hidden space splits as 192 two-body plus 512 fully traceless three-body directions; even the full 𝑆3-invariant class retains an exact 156-dimensional hidden sector. The preregistered continuation identifies an exact energy-selected fixed point, the totally antisymmetric state |𝜀⟩, while holonomy and variance objectives yield set-valued argmin correspondences. The historical non-triviality gate is therefore preserved but shown not to be a universal criterion, and no retroactive 𝑉28.4 verdict is declared. A prospectively frozen two-local QPH family provides an exact computer-assisted existence witness: its cycle-phase differential has rank 704/704 on the marginal kernel for the primary family and two preregistered replications, so marginals plus relational cycle phases are locally in- formationally complete in the fixed regularity stratum. Canonical modular-mismatch selectors are then audited rather than assumed. Identical additive qutrit references and every single pure mismatch under exact 𝑆3 protection obey strict capacity/rank no-go bounds. On the protected 156-dimensional domain a physically generated mixed mismatch has sufficient raw capacity but standard phase-only QPH remains incomplete. Exact cumulant analysis explains why order three is the natural next layer: the 𝑆3-hidden sector decomposes as 156 = 36 + 120, and the third connected cumulant is a complete affine coordinate on fibres of fixed proper marginals. A historical preregistered cubic-moment completion is retained only as a delicate T-NUM wit- ness, not as a theorem premise. This is the declared stopping point of the arity/cubic line: selecting one specific cubic observable would require a new independent physical law. Test 28B remains suspended. Version 4.0 added one further macro-level correction, retained in Version 4.1. Causal and conformal structure, transport geometry, stable matter sectors, and quantum correlations are treated as parallel downstream representations of one substrate model, not as a serial matter– space–time chain. A proposed identification of the matter admissibility projector Πadm with spacetime volume is rejected in its literal form: the projector is matter-branch specific, need not be nested under generative updates, and has no canonical finite trace in a type-III con- tinuum. In its place a conditional causal–transport conformal closure theorem is proved. If a causal reconstruction supplies a Lorentzian conformal class and an independent transport limit supplies a positive spatial metric on the same spacelike distribution, the conformal factor is fixed pointwise exactly when the two induced spatial metrics are conformally compatible. A scale-free generalized-eigenvalue defect provides a direct falsifier. The corresponding clock– cone–matter gate is also corrected by conformal weight: frequencies and dimensionful spectral scales have lapse weight one under a pure lapse deformation, whereas spectral ratios, localiza- tion fractions, and dimensionless stability-window coordinates have weight zero. A Tolman– thermal-time comparison is retained only as a relative lapse-profile gate; absolute time units remain open. Version 4.1 introduced a controlled common-substrate realization. Version 4.1.1 corrects its scope: parity alone excludes only an odd or linear mass/source identification with the same signed coordinate; an even law such as 𝑀 ∝ 𝑑 2 is not excluded by parity. On the declared graph Klein–Gordon sector, charge conjugation reverses Noether/gauge-current orientation while leaving mode energy and localization intensity unchanged. A grounded-graph Green- kernel theorem fixes the spatial uniformity of the cross-term sign, while the Coulomb sign itself is conditional on realizing a non-dynamical 𝐴0-type constraint sector; the same kernel in a dynamical scalar sector gives the opposite sign. A separate no-go proves that matter Dirichlet energy alone cannot select a nontrivial interior geometry. A state-selected positive geometry Hessian yields the conditional linear response 𝑦 = 𝐾 −1 𝐸 𝜏 , so weak 𝐶-even stress sources add while 𝐶-odd currents can cancel. The principal new bridge is a conditional stationary common causal–transport–matter con- struction. If one protected center supplies a common base, the transport branch supplies a positive spatial metric ℎ, and several modular/physical clocks agree on one relative lapse 𝑁, then 𝑔CTM = −𝑐 2 *𝑁2𝑑𝑇2 + ℎ has one timelike sign, predicts the same null front for all canonically normalized matter sectors, and fixes its stationary volume element without an additional scalar measure. Different masses and opposite 𝑈(1) representation weights remain lower-order data and do not alter the characteristic cone. The independently reconstructed HSMI conformal class remains a held-out test rather than an input to this construction. A second independent Version 4.1 result supplies a conditional physical law selecting com- plex rank two. A nonzero-Chern line cannot be closed under conjugation at rank one on 𝑆 2 , whereas 𝐿 ⊕ 𝐿¯ is the minimal conjugation-closed carrier; independently, ClC 2 ≃ 𝑀2(C) has minimal irreducible complex module dimension two. The existing odd-Chern defect-link con struction realizes this pair. If its Clifford algebra is physically admissible, the local carrier has 𝑀2(C) qubit kinematics. This is not an assertion that opposite electric charges form a coherent qubit, nor that every particle is a qubit. The resulting Version 4.1.1 is a constrained foundational proposal with a frozen primitive core, stronger positive and negative bridge results, a first common stationary geometry can- didate, an independently motivated minimal rank-two carrier, and a reproducibility-oriented ledger for every step from ontological determinacy toward physics. It also makes the external spin-sign, soft-graviton, equivalence-principle, and mass-budget inputs explicit without treat- ing them as evidence for the ontology.
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
We reconstruct nonrelativistic quantum mechanics from exactly three physical axioms: microscopic ontic definiteness and finite localization, physical reality of the spacetime-vacuum substrate, and persistent causal source–response coupling. The single-run physical state is a finite topological core, its actual causal history, the substrate state, its sector, and the associated historical disturbance. A wave function is not identified with that ontic tuple. It is a ray in a Hilbert representation of admissible preparations and response amplitudes; a density operator is a statistical or reduced representation; and a record is a stable macroscopic event. The logical construction is acyclic. Finite-energy sector spaces and their continuous symmetry representations lead, under explicit representation conditions, to a complex Hilbert direct sum. A closed semibounded quadratic form defines the Hamiltonian and its self-adjoint realization. The standard Schrödinger action, unitary propagation, probability current, spectral observables, uncertainty relations, composite-system tensor products, spin representations, exchange sectors, instruments, and completely positive open dynamics then follow on their declared domains. Gleason-type reasoning fixes the Born form only after additivity, noncontextuality, continuity, and preparation–detector calibration are stated. Those operational conditions are not hidden physical axioms. The distribution of unresolved ontic initial data that realizes quantum equilibrium remains a realization certificate, not a theorem of topology alone. The Heisenberg representation is derived as a controlled transport of observables after Hilbert space and the self-adjoint generator have been constructed. Strongly continuous time translation, two-parameter nonautonomous propagation, common-domain Heisenberg equations, spectral matrix mechanics, regular Weyl relations, open-system adjoints, and instrument effects are connected by one certificate. Schrödinger and Heisenberg predictions are proved equivalent for one-time and finite instrument records, while finite-core, historical, truncation, domain, and detector errors remain separately bounded. Three frozen matrix benchmarks verify low-energy oscillator truncation and its state-weighted CCR defect, second-order unitary convergence for a driven doublet, Schrödinger–Heisenberg equality for a sequential instrument, finite auxiliary-history truncation, and quadratic leakage scaling for a conditional Dirac–Pauli interface. Independent reproduction, including the extended-precision audit, is closed for the frozen packet; physical coefficient calibration remains open. The next interface connects an owner-resolved Dirac–Pauli Hamiltonian to spin records through a Ward-compatible vertex, a causal passive history kernel, an independently audited doublet projection, SPAM-aware forward probabilities, nuisance-profiled identifiability, and a noncompensatory coefficient-to-record error theorem. The DPHC identifiability coordinate is executed on a frozen synthetic response family. A single-scale design is rank deficient, whereas an eighteen-setting frequency–gradient–order–time design has full profiled rank, controlled conditioning, near-nominal finite-sample coverage, and a calibrated held-out statistic. A robust twenty-setting design raises the worst registered information eigenvalue by 20.12%. A ten-parameter nonlinear audit exhibits an exact radiative–detector equivalence curve. An independent dark record breaks that symmetry and restores full local rank. Continuous-set robustness, validated global exclusion, and independent implementation are executed, while physical response identification and empirical calibration remain open. Measurement is treated as a dynamics of system, apparatus, environment, and actual history. Unitary premeasurement and decoherence select robust record sectors but do not by themselves select one record. The three-axiom ontology supplies one actual branch history, while a measurable response map from the complete initial state and instrument to the final record supplies the missing dynamical content. Bell correlations require contextual and nonfactorizable ontic response while parameter independence and operational no-signalling are retained; no controllable superluminal channel is introduced. Historical response produces causal non-Markov Hamiltonians and memory-kernel master equations. Their normalization, positivity, complete positivity, and energy ledger are derived from a unitary enlarged realization whenever such a dilation exists. Finite-core form factors and causal memory yield explicit departures from point-core Markov quantum mechanics. Standard quantum mechanics is recovered as the unresolved-core, sector-fixed, quantum-equilibrium, short-memory limit. Quantum field theory is retained as a successful local, spectral, and asymptotic calculus, but field operators, Fock vectors, and vacuum functionals remain effective representations until a noncircular map from finite carriers and their complete causal histories has been certified. The manuscript proves a sequence of previously open analytic, mathematical, measurement, topological, statistical, and computational obligations on explicitly declared domains. These include non-Markov diffusion, graph-norm stability, distorted-wave and pole variation, finite passive-memory realization, Fisher-information identifiability, detector pushforwards, unique-record basins, Born-representation bounds, measurement independence, topology–Dirac charge conjugation, Stern–Gerlach records, action-scale matching, rank-two spin projection, Finkelstein–Rubinstein topology, Clifford reconstruction, and parent-to-record realization maps. Eighteen groups of numerical obligations are executed on frozen reference models. They cover spectral and scattering calculations, oscillator and two-level dynamics, Schrödinger–Heisenberg equivalence, finite-history truncation, Dirac–Pauli and Foldy–Wouthuysen reduction, gauge covariance, interference, Bell and GHZ correlations, open-system memory, Stern–Gerlach records, identifiability, global searches, topological invariants, path products, semiclassical limits, partial-wave optical closure, the relativistic/QFT interface, and conservative-parent two-level dynamics. Every environment explicitly typed as a computational obligation now has an executed frozen-model certificate; empirical identification and separately typed experimental inputs remain open. The present manuscript differs so substantially from the earlier version in scope, logical architecture, proofs, and executable certificates that it is issued as a new upload rather than as a minor revision. Because the subject is both foundational and technically consequential, readers professionally engaged in quantum foundations, mathematical physics, quantum measurement, or related fields are encouraged to consult both versions: the earlier version preserves the developmental route, whereas the present version gives the substantially expanded and reorganized reconstruction. ## Keywords Axiomatic quantum mechanics; finite particle histories; retarded Green functions; Schrödinger equation; Dirac equation; Born rule; topological spin; quantum measurement; non-Markovian dynamics; classical limit.
Abstract We present Kunato Core Ver.5.0.0, a general framework for ontological admission, typed representation transport, empirical identifiability, effective-structure validation, and provenance-preserving epistemic state transition. The framework addresses a recurrent inferential problem across mathematical physics, statistical inference, inverse problems, effective theories, information theory, complex systems, quantum theory, and scientific realism: mathematical definability, empirical adequacy, model identification, and physical realization are distinct achievements and must not be silently promoted into one another. Kunato formalizes this separation through the constraint Formal(M) AND Emp(M; O) DOES NOT IMPLY Ontic(M). No representation receives automatic ontological privilege. Finite observation provides a first source of non-identification. Distinct formal models can agree exactly on a finite empirical record; in particular, uncountably many models can be constructed that coincide on any prescribed finite set of sample points. This establishes non-uniqueness within the relevant formal model class without promoting mathematical alternatives into claims about distinct physical causal histories. Finite measurement precision is treated through fibers of explicit resolution maps rather than by identifying metric neighborhoods with equivalence classes. The framework further distinguishes finite-dimensional state spaces from finite-state ontologies: finite-dimensional complex Hilbert spaces can contain a continuum of pure-state rays, whereas finite operational distinguishability requires additional assumptions such as a uniform positive separation scale on a compact state space. For stochastic observations, Kunato separates distributional identifiability, asymptotic inference, finite-sample discrimination, and zero-error identification. Positive overlap between competing observational laws prevents zero-error discrimination from the corresponding finite record. Increasing statistical evidence can therefore strengthen discrimination without automatically supplying an ontological realization warrant. Identification within a declared candidate class likewise remains conditional on that class and does not establish that physical reality belongs to it. The mathematical core is a typed property-transport calculus. Given a representation map Phi: X -> Y and a property P: X -> Z, P factors through Phi precisely when P is constant on every fiber of Phi. Equivalently, the kernel equivalence relation induced by Phi is contained in the kernel equivalence relation induced by P. This criterion separates state identification from property identification: a non-injective representation may fail to recover individual source states while still determining every property invariant on its fibers. Representation types, identified by their induced fiber equivalence relations, are organized by refinement within the complete lattice of equivalence relations on X. This provides a common mathematical structure for coarse-graining, quotient representations, projections, information loss, recoverability, and comparison of effective descriptions. Refinement is a partial order, so representations may also be incomparable. Effective structures consequently require property-specific transport warrants. Compression of a unitary parent operator, for example, need not preserve unitarity or normality. Conversely, non-normality alone does not entail a non-real spectrum, irreversibility, non-Markovianity, or absence of characteristic timescales. Observable symmetry, effective geometry, statistical regularity, entropy, information structure, temporal scales, and other representation-dependent properties cannot be transported automatically to a parent ontology. Kunato operationalizes these restrictions through methodological admission protocols. The Metric Admission Protocol regulates inference from effective metric structure to fundamental geometry. The Statistical-Invariant Admission Protocol separates existence of a mean from typicality, concentration, and self-averaging. The Timescale Admission Protocol distinguishes observed decay and hazard structure from claims about fundamental temporal dynamics. The Informational-Semantic Admission Protocol constrains inference from probability, entropy, information geometry, and semantic structure to latent ontology. The Criticality Admission Protocol separates phenomenological scaling from stronger claims of critical structure by requiring appropriately scoped diagnostics such as finite-size behavior, exponent stability, data collapse, robustness analysis, and negative controls. These protocols are methodological admission criteria rather than asserted laws of nature. Ontological admission is governed by differentiated semantic, fit, identification, scope, realization, revocation, and provenance warrants. These warrant types prevent empirical fit, interpretation, candidate-class identification, physical realization, and evidential history from being collapsed into an undifferentiated notion of support. Epistemic judgments are represented as contextual and versioned states, allowing changes in evidence, candidate classes, semantics, scope, and warrants to be reconstructed as explicit state transitions rather than hidden revisions. A central distinction is that UNRESOLVED is not REJECTED. Failure to establish an adequate promotion warrant blocks the requested inference but does not establish falsity or nonexistence. Rejection requires an independent refutation condition. Revocation acts on claims and accepted epistemic states while retaining their provenance; it does not imply any operation on authors or other persons. Self-application closes the framework. Continuum models, discrete models, cellular automata, XOR-based constructions, effective theories, quantum representations, and Kunato itself receive no exemption from the admission rules. Failure to identify a continuum ontology does not identify a discrete ontology, empirical adequacy of a constructive model does not establish its fundamental realization, and formal validity of Kunato does not confer physical ontological status on Kunato. The framework therefore entails neither a privileged fundamental model nor ontological nihilism. Not identified does not imply false, and not admitted does not imply nonexistent. Kunato Core Ver.5.0.0 provides a unified formal and computational vocabulary for ontological admission, finite-data non-identification, finite-record discrimination, inverse problems, coarse-graining, property factorization, representation refinement, effective dynamics, statistical invariants, temporal inference, information-theoretic interpretation, criticality assessment, model-class identification, provenance, epistemic-state revision, and scientific realism. Its objective is not to determine a preferred ontology by methodological fiat, but to make the inferential boundaries between representation, observation, identification, and physical realization explicit and auditable. The framework is summarized by four constraints: no hidden epistemic transition; no hidden ontological promotion; no untyped representational transport; and no self-exemption. Its terminal methodological principle is: no representation receives automatic ontological privilege.
Finite Determinacy (FD) was developed from a representational question: whether theuse of real numbers for physical quantities should be read as implying arbitrarily precisephysical determination. Replacing that idealisation by objectively finite determinationleads to bounded shared resolution depth for related records. Close finite-informationand relational antecedents in quantum foundations were identified subsequently and areacknowledged here; they were not the route by which the present framework was developed,and no historical priority is claimed for those principles. This companion paper asks whatdynamical restrictions follow once the resulting finite-record geometry is combined withexplicit finite-rate premises.For self-containment the paper first reproduces the common foundation: bounded jointdepth and split-invariant refinement give exponential resolution and a non-zero two-registerwidth-product floor; a homogeneous, finitely additive area regularity promotes that scalarconstraint to a local area measure; and a primitive tick/readout role asymmetry suppliesorientation, after which the two-member sector is symplectic. The positive symmetric formused for compatible complex structure remains an independent structural input. Dynamicsthen adds a universal pace ceiling rmax, a saturated intra-pair exchange ceiling |dv/du| ≤ L,and an inter-pair causal conversion c∗. In the quantum branch, explicit coherent-recordand record-faithfulness assumptions lead through Weyl–Heisenberg representation theoryto finite-step unitary evolution and the Schrödinger law-form. In the gauge branch, recordgeneratedtransport and holonomy lead, under a minimal local reversible continuum closure,to Yang–Mills curvature-square dynamics. In gravity, locality, reversibility and isotropy givea minimal hyperbolic principal part, while an explicit spatially relational continuum closureleaves two transverse-traceless modes.The proposed contribution is therefore not a new finiteness or relationality principle,nor an exclusive derivation of familiar terminal equations. It is an independently developedfinite-geometric route from the physical interpretation of real-number representation to acommon pre-Hilbert, pre-gauge and pre-metric substrate, followed by sector-specific dynamicalconsistency conditions.
This volume develops the formal mathematical representation of the correlational ontology introduced in Book I. Without adding new ontological assumptions, the framework is translated into a precise formal language based on configuration spaces, equivalence classes, order structures, functionals, and operators. Key structures such as the consistency functional, canonical correlational flow, Hessian operator, and projection to observable subspaces are defined rigorously. The mathematical role of gradients, variational structures, spectral analysis, and stability is clarified, while maintaining the non-ontological status of time, dynamics, and laws. The volume emphasizes representational freedom, gauge equivalence, and the distinction between structural objects and effective physical descriptions. It establishes the formal core from which all subsequent physical realizations and predictions are derived. This book functions as the technical backbone of the framework, independent of any particular physical regime.
This paper provides a purely formal and technical development of the mathematical structures underlying a correlational framework for physics. It introduces no ontological assumptions, no physical interpretation, and no experimental claims. All constructions are independent of primitive notions of space, time, dynamics, causality, or physical law. The paper rigorously defines correlational configuration spaces, admissibility conditions, representational equivalence classes, and quotient structures. A central object is the consistency functional, together with its Hessian operator, treated as a (possibly unbounded) self-adjoint operator on infinite-dimensional spaces. Detailed attention is given to domains, spectral properties, stability, and variational structure. Projection operators onto observable subspaces are introduced as mathematically irreversible restrictions, with explicit treatment of information loss and consistency of reduced descriptions. The canonical correlational flow is defined as a gradient flow of the consistency functional, with proofs of local existence, uniqueness, reparametrization freedom, and dissipative character, all without invoking time or dynamics. The paper also analyzes fixed points, attractors, metastable configurations, and the precise formal relations between flow, Hessian, and projection. Finally, the intrinsic limits of the framework are made explicit, including configurations without physical interpretation, breakdowns of projection, and ill-defined spectral structure. This work is intended as a technical companion to foundational, canonical, and applied papers. Its sole purpose is to eliminate formal ambiguity and provide a mathematically explicit backbone for subsequent physical realization and application.