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.
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.
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.
Finite observability is built in two stages. First, abstractly: a finite observer resolving realvaluedregisters carries a dimensionless ledger of distinctions, capped in total at λmax, withtwo coordinates — the capped total λ+ = λu + λv and the free split λ− = λv − λu. Second,the registers are reformulated as a conjugate pair (u, v) by positing the symplectic pairing,which dresses the capped total as a resolution cell of minimum area K = δu δv; a second,per-pair finiteness Luv ceilings the rate at which distinguishable change is exchanged betweenthe members, and marks the distinguished (null-diagonal) cell on the saturation locus. Theshared ledger makes the reciprocal cell a consequence, not a postulate: split-invariance forcesexponential resolution maps, and δu δv = K is the full-budget locus. From this we derive:noncommutativity at scale K; the imaginary unit as the saturated cell’s complex structure; aconstructed wave with Schr¨odinger-form evolution and a Born rule as Fubini–Study transitionprobability; one unconditional causal theorem (subluminal propagation equivalent to a nonnegativedispersion invariant); a forced first-order (Clifford) form; measurement decomposedinto the cell’s own motions, with one testable consequence — collapse of massive superpositionsabove a derived ceiling (∼ 1012–1014 amu) on the Di´osi–Penrose timescale via a conservative,non-radiating mechanism; the non-fixability of the scale; and composites as differentiated cellbundles. The physical reading — position and momentum, c, Dirac, gauge fields and gravity— is deferred entirely to a companion paper. The claim here is structural: the grammar of onefinite-observability conjugate pair, with no particular physical pair assumed.
This paper develops finite temporal observability in two parts, with a third in a companion[1]. Part I isolates the structure of a single finite-observability conjugate pair (u, v) andpushes it as far as it goes with no specific physical identification assumed. A finite observerresolves real registers whose entire content is a dimensionless ledger of distinctions, capped atλmax and carrying a total λ+ and a split λ−; positing the symplectic pairing dresses these asthe cell area K = δu δv and the slope Luv = δv/δu. A single shared distinguishability ledgerthen makes the reciprocal cell a consequence — split-invariance forces exponential resolutionmaps, the cell boundary is the full-budget locus, and the saturated cell carries the compatiblecomplex structure supplying the bracket’s imaginary unit. From this alone follow noncommutativityat scale K, the complex unit, an implied wave with a Born rule read as a Fubini–Studytransition probability, a causal bound with one unconditional theorem (the causal slope equivalentto non-negative mass-squared), a reversible generator, the forced first-order (Clifford) form,the non-fixability of the scale, and composite objects as differentiated cell bundles. The singleresidue of the wave construction — the Wallstrom single-valuedness condition — is reduced tothe cell’s prequantum integrality, leaving the integral period 2πK as the one named quantuminput rather than a free-standing posit. Part II supplies the physical reading — the conjugatepairexploration. Validated on the regular pairs the wave is Schr¨odinger’s, the bound is therelativistic mass-shell with L = c, and the first-order form is Dirac’s; extended to exotic pairs(horizon angle and area, holonomy and flux, modular flow and entangling area, the cosmologicalconstant and four-volume) a single per-pair criterion — whether the exchange rate Luv vanishes— sorts discrete from continuous spectra, consistent with the careful literature each time; and asurvey of laboratory prospects finds the frontier mapped but no forced, allowed, novel signature.The architectural extensions to gauge fields and gravity — fermionic statistics from stability,the gauge coupling from comparing cells, and the metric with the Lovelock field equations —are developed in the companion [1]. Every claim is graded — theorem, conditional construction,or candidate route. The whole is offered as a constraint grammar of possible universes, not aderivation of ours.
Finite determinacy yields an allocation geometry and a causal cone but not quantummechanics: its axioms admit a classical model, so superposition and Born’s rule cannot followfrom them. This paper asks how far one may nonetheless go, and locates the remaining step.The minimal carrier of a continuous recombination is shown to be C, by classification of theconnected abelian Lie groups admitting the loop action by automorphisms, so the complexunit is not imported. Transport equivariance then forces contributions to combine by additionuniquely, while convex recombination is excluded for failing to close under composition. Thedependence of a record’s probability on the carrier is fixed at the square by an involutivesplitter, its normalisation following from involutivity rather than from any norm, with no otherexponent surviving repeated splitting. Phase-invariance of records and the density ceiling arederived. The chain then closes to an inner-product space, the amplitude being a unit vectorin L2 of the phase space; what is not obtained is the reduction to configuration space, and ofthe two polarisations available one rests on a primitive and the other on a theorem, neitherbeing forced. Several negative results are established unconditionally, among them that nopositive transport of a non-negative content can produce cancellation, and that no phase canbe built from a scalar or from any translation-invariant function of a cell.
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.
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