Finite Determinacy was developed from a representational question: whether the use ofreal numbers for physical quantities should be read as implying arbitrarily precise physicaldetermination. Replacing that idealisation by objectively finite determination leads to abounded shared resolution depth for related records. Close finite-information and relationalantecedents in quantum foundations were identified subsequently and are acknowledged here;no historical priority is claimed for those principles. The distinctive step investigated is theirfinite-geometric implementation. Split-invariant refinement forces exponential resolutionand a non-zero two-register width product. An explicit homogeneous, finitely additive arearegularity promotes that scalar cell constraint to a local area measure, while a primitivetick/readout role asymmetry supplies orientation; only then is the two-member sectorsymplectic. Three downstream developments are studied. Oriented inter-copy recordsgenerate compatible complex and unitary internal structure and gauge-connection kinematics.Finite record contexts, with an explicit noncontextual complete-context invariance assumptionand finite additivity, yield a quadratic primitive weight and the finite Born form. Persistentrelational frustration gives a graph-Laplacian equilibrium with a 1/r exterior response inan isotropic three-dimensional continuum. The proposed contribution is therefore not anew finiteness principle but an independently developed finite-geometric route from therepresentation of physical quantities to a common pre-Hilbert, pre-gauge and pre-metricsubstrate for these three sectors.
Finite Determinacy was developed from a representational question: whether the use ofreal numbers for physical quantities should be read as implying arbitrarily precise physicaldetermination. Replacing that idealisation by objectively finite determination leads to abounded shared resolution depth for related records. Close finite-information and relationalantecedents in quantum foundations were identified subsequently and are acknowledged here;no historical priority is claimed for those principles. The distinctive step investigated is theirfinite-geometric implementation. Split-invariant refinement forces exponential resolutionand a non-zero two-register width product. An explicit homogeneous, finitely additive arearegularity promotes that scalar cell constraint to a local area measure, while a primitivetick/readout role asymmetry supplies orientation; only then is the two-member sectorsymplectic. Three downstream developments are studied. Oriented inter-copy recordsgenerate compatible complex and unitary internal structure and gauge-connection kinematics.Finite record contexts, with an explicit noncontextual complete-context invariance assumptionand finite additivity, yield a quadratic primitive weight and the finite Born form. Persistentrelational frustration gives a graph-Laplacian equilibrium with a 1/r exterior response inan isotropic three-dimensional continuum. The proposed contribution is therefore not anew finiteness principle but an independently developed finite-geometric route from therepresentation of physical quantities to a common pre-Hilbert, pre-gauge and pre-metricsubstrate for these three sectors.
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 study asks which structures can be derived from a single finite relational system to which no prior physical meaning, coordinates, or time are assigned. A selector-free relational lift and its minimal self-sustaining orbit generate an intrinsic separator order and grading, providing a directed order axis without an external clock. Boolean-diamond incidence defines transverse locality, and five-K4 incidence produces an exact S3 minimal persistent complex whose canonical continuation yields a four-dimensional associated realization. An explicit finite witness gives a nonzero commutator between longitudinal incidence and transverse locality, ruling out a simple product of fixed space with independent time. A complementary stable-quotient readout contains a refinement branch that closes as a one-dimensional metric-graph continuum. Combined with the S2 fiber of the standard S4 representation, this gives a regular spatial sector locally of the form (0,1) x S2(R). The S2 Laplace spectrum provides a branch-independent nonzero rest gap for every l >= 1; under the stated internal second-order wave representation this gap is read as a rest-mass parameter. Gravity is represented by an exact finite source-to-field map using the Moore-Penrose Green operator, and the regular three-dimensional branch yields the local 1/r^2 acceleration exponent in the weak-static regime. The same relational development also supplies three distinct internal Hilbert representations. Conditional on the full orthogonal-projector event structure and standard noncontextual probability consistency, Gleason's theorem fixes the Born form in the spatial representation. The incidence representation is noncommutative and yields the Robertson uncertainty relation, while procyclic reaction gives exact discrete unitary dynamics by Haar-measure invariance. No canonical identification of these Hilbert spaces, unique continuous Hamiltonian, measurement-update rule, spin-1/2 structure, or Bose/Fermi statistics is claimed. Complete derivations and branch audits are provided in a Supplementary Formula Atlas.
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
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 manuscript presents a synthesis of the quantum branch of the Finite Relational Closure Framework (FRCF). The central claim is that finite-dimensional Hilbert-space quantum kinematics can be constructed from finite measurement primitives rather than assumed as an independent foundation. In standard presentations of quantum mechanics, a Hilbert space is usually introduced first, with measurement represented afterward by operators on that space. FRCF reverses this explanatory order. It begins with finite contexts, admissible relational assignments, finite-resolution equivalence, and measurement partitions. From these structures, the finite-resolution admissible classes Σε(C)=Σ(C)/∼C,ε are obtained. The associated complex contribution space HC,ε=CΣε(C), equipped with its finite inner product, is a finite-dimensional Hilbert space. The manuscript shows how finite measurement maps induce orthogonal decompositions, projection operators, Born-type probabilities from squared norms, and measurement update as conditioning on recorded finite outcomes. It also distinguishes this finite Hilbert-space kinematics from the separate generator problem governing contribution transport between contexts. The construction does not derive all of quantum mechanics. It does not determine a Hamiltonian, the Schrödinger equation, quantum field theory, or a unique physical generator. Its contribution is kinematic and foundational: it identifies Hilbert-space structure as an effective representation of finite admissible measurement distinctions within FRCF.
A physical quantity is taken to have finitely many determinate digits rather than a realvalue partially resolved, the continuum being an idealisation the world does not honour.This paper develops the consequences by rank. An object is a family of related quantitiessharing one bounded joint depth — relatedness only is assumed, not conjugacy — and theexponential resolution law and a conversion cocycle follow, while parity separates closedfamilies from constrained ones. Reciprocal resolution, the antisymmetric form and a compatiblecomplex structure are each derived rather than posited, so what is usually takenas the canonical structure of a pair is here an output. A counting argument shows that norelation between pairs exists below rank two, so the single-pair theory is complete ratherthan deficient. The antisymmetric form is then assembled from a relation, which suppliescoupling, and from roles, which supply orientation and are primitive. A second and strictlystronger premise, a pointwise ceiling on record density, yields exact content-space results:an entropy floor, a characterisation of saturation as uniform product support, and a sharpvariance bound from confinement alone. Criticality is characterised as a crossover betweencompeting ceilings, first within a pair and then, at rank two and above, between them. Aclosing section collects results establishing what the construction provably cannot do, includingthat no scale-invariant datum can fix a scale-covariant normalisation. The axiomsare shown to admit a classical model, so quantum structure is not claimed to follow fromthem.