Four axioms (N1–N4) define N_ref as the accumulated state-transition count of a physical reference system; the definition is realisation-independent and structurally excludes the Pauli obstruction. The mass-shell constraint yields the Pikovski Hamiltonian Ĥ_phys = Ĥ₀/γ within the expansion regime, and the budget equation (dτ/dt)² + v²/c² = 1 as the exact constraint. A forgetful functor F: Math_t → Phys_N identifies surplus automorphisms of the mathematical temporal category that have no counterpart in the physical category. The computational gauge characterisation (CG1–CG3) shows that the surplus is mathematically necessary (Stone's theorem), informationally determined for KMS states (modular analyticity), and physically vacuous (axiom N1). The signature theorem proves that Minkowski (1,1) is forced by the non-compactness of the boost group, itself a theorem of counting and the relativity principle. The extension to the full (1,3) signature follows from spatial isotropy via Schur's lemma on the irreducible SO(3) representation. A universality proposition establishes that every physical clock satisfying N1–N4 dilates by 1/γ, with the bridging assumption A_bridge naming what the quantum universe contributes to the derivation. These results supply the axiomatic and categorical foundations for the surplus structure programme introduced in Part I; Part III extends the k-operator developed here into the quantum regime.
Four axioms (N1–N4) define N_ref as the accumulated state-transition count of a physical reference system; the definition is realisation-independent and structurally excludes the Pauli obstruction. The mass-shell constraint yields the Pikovski Hamiltonian Ĥ_phys = Ĥ₀/γ within the expansion regime, and the budget equation (dτ/dt)² + v²/c² = 1 as the exact constraint. A forgetful functor F: Math_t → Phys_N identifies surplus automorphisms of the mathematical temporal category that have no counterpart in the physical category. The computational gauge characterisation (CG1–CG3) shows that the surplus is mathematically necessary (Stone's theorem), informationally determined for KMS states (modular analyticity), and physically vacuous (axiom N1). The signature theorem proves that Minkowski (1,1) is forced by the non-compactness of the boost group, itself a theorem of counting and the relativity principle. The extension to the full (1,3) signature follows from spatial isotropy via Schur's lemma on the irreducible SO(3) representation. A universality proposition establishes that every physical clock satisfying N1–N4 dilates by 1/γ, with the bridging assumption A_bridge naming what the quantum universe contributes to the derivation. These results supply the axiomatic and categorical foundations for the surplus structure programme introduced in Part I; Part III extends the k-operator developed here into the quantum regime.
Abstract The temporal parameter t ∈ ℝ carries three properties that no experiment has confirmed as features of physical reality: negative extension, loop-admitting topology, and reversal symmetry. We identify these as surplus structure in the sense of Weatherall, applying Gisin's observation on ℝ-modelling to the temporal parameter specifically. Its operational content is N_ref: the accumulated state-transition count of a reference system, non-negative and monotonically non-decreasing. The caesium-133 hyperfine oscillator is one realisation; the framework is characterised axiomatically (N1–N4, Section 4) and is independent of the choice of clock. We establish two theorems. Theorem 1: restricting the Wheeler–DeWitt scalar-field clock to its operationally grounded domain φ ∈ [0, ∞) halves the minisuperspace solution space, excluding independently contracting universes. Theorem 2: within the Page–Wootters framework, the process-accumulation arrow (a spectral property of the clock operator) and the thermodynamic arrow (a boundary condition on the constraint surface) are formally independent. The Bondi k-calculus then derives the full Lorentz transformation — including the relativity of simultaneity — from three operational inputs: transition-count ratios between inertial observers, the relativity principle, and a finite signal speed c. No prior notion of time, metric, or spacetime is assumed; 1/γ is a theorem rather than an input. Closed timelike curves, parameter reversal in T-symmetry, and the block universe as the default interpretation of the formalism are shown to depend on the surplus structure of ℝ and to lack independent empirical support.
Abstract The temporal parameter t ∈ ℝ carries three surplus properties (negative extension, loop-admitting topology, and reversal symmetry) that no experiment has confirmed. This paper traces this conclusion from the mass-shell relation E² = p²c² + m²c⁴, which functions as a conserved-total resource constraint: total energy is conserved, and any increase in spatial momentum physically necessitates a decrease in the rate of internal evolution. The Pikovski Hamiltonian formalizes this as Ĥ₀/γ, the internal Hamiltonian multiplied by the Lorentz factor. The Page-Wootters framework derives time emergence from entanglement correlations in a globally static state. Smith and Ahmadi (2020) showed that PaW conditioning recovers the Pikovski time dilation; Höhn, Smith and Lock (2021) proved the full three-way equivalence. Combined with the SI definition of a second (9,192,631,770 oscillations of cesium-133, an operational definition grounding time in internal state transitions), these results establish that the physical content of the temporal parameter is a constraint on internal state-transition rates. The temporal parameter t ∈ ℝ serves as a continuous surrogate for a quantity accessed only through physical process that is non-negative and monotonically non-decreasing. A relational observable N_ref replaces t. Following Gisin and Weatherall, three ℝ-domain properties of t are identified as surplus structure. Closed timelike curves fall outside GR's initial value formulation (Choquet-Bruhat-Geroch); parameter reversal in T-symmetry is shown to be surplus via the algebraic CPT theorem (Borchers, Guido-Longo, Mund, Swanson); and the block universe loses its default status. The arrow-of-time problem decomposes into a process-accumulation arrow (a spectral property of the clock operator, formally independent of entropy) and a thermodynamic arrow determined by boundary conditions. The argument extends Einstein's 1905 relativization of simultaneity to the temporal parameter itself.
The temporal parameter t ∈ ℝ carries three surplus properties (negative extension, loop-admitting topology, and reversal symmetry) that no experiment has confirmed. This paper traces this conclusion from the mass-shell relation E² = p²c² + m²c⁴, which functions as a conserved-total resource constraint. The Pikovski Hamiltonian formalizes this as Ĥ₀/γ, and the Page-Wootters framework derives time emergence from entanglement correlations in a globally static state. Smith and Ahmadi (2020) showed that PaW conditioning recovers Pikovski time dilation; Höhn, Smith and Lock (2021) proved the full three-way equivalence. Combined with the SI definition of a second (grounding time in internal state transitions), these results establish that the physical content of the temporal parameter is a constraint on internal state-transition rates. Following Gisin and Weatherall, three ℝ-domain properties of t are identified as surplus structure per Weatherall's framework: closed timelike curves, parameter reversal in T-symmetry, and the block universe. The arrow-of-time problem decomposes into a process-accumulation arrow (a spectral property of the clock operator) and a thermodynamic arrow determined by boundary conditions. The argument extends Einstein's 1905 relativization of simultaneity to the temporal parameter itself.
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
AbstractAs the third paper on the non-a-priori construction of time, this paper constructs clocks from mathematical temporal-like structures, defines time in quantum theory axiomatically, and then forgets its temporal interpretation to obtain a formulation of quantum evolution in which physical time is not a primitive variable. Numberless time and temporal numbers from Papers I and II are reintroduced as repetition and recurrence through group actions, and as succession and history through the Peano successor operation, respectively. Physical clocks are defined by equipping their combination with readout and calibration, and an ideal equivalence relation between different types of clocks is examined. A “quantum time axiom” is adopted to define clock time through the energy gap and accumulated relative phase of a two-level quantum clock. The mutual reference between quantum theory and clocks is thereby treated as an axiomatic stopping point grounded in concrete physical realization. A forgetting operation then removes the temporal interpretation while retaining successor operations, group actions, and transformations on Hilbert space as pure operations. A dimensionless unitary group is constructed from a self-adjoint energy operator and the energy scale of a reference clock, and Peano updates, continuous extensions, and invariance under changes of reference clock are examined. The standard Schrödinger equation is recovered through the quantum time axiom, and the dimensionless differential equation is presented as one construction of an atemporal Schrödinger equation. A possible adjunction between free construction and forgetting for time in general is also considered. Finally, comparison with the Page–Wootters mechanism and related approaches distinguishes the nonprimitiveness of a time variable from timelessness in a completely closed system. The paper presents a coherent foundational construction of quantum theory: construct time, realize it as a clock, axiomatize time in quantum theory, and then forget its temporal interpretation. Detailed mathematical derivations and explanations of the constructions presented in this paper are provided in a separate supplementary manuscript. AI Assistance DisclosureGenerative AI was used as an auxiliary tool in drafting, adjusting expressions, and organizing the structure of this paper. The author bears final responsibility for the text, claims, citations, and factual verification. Keywords: non-a-priori construction of time, numberless time, temporal numbers, rhythm time, count time, quantum clocks, quantum time axiom, forgetting the temporal interpretation, atemporal quantum mechanics, group actions, Peano arithmetic, reference-clock invariance
Quantum mechanics is the effective theory of physically real unresolved admissible structures and their resolution. This paper establishes the minimal axiomatic foundation from which this claim follows: physical structure is derived from a single primitive, the local structure of admissible transitions between observable states. Four axioms govern this structure: local projective admissibility (A1), structural non-injectivity (A2), admissibility as non-premature selection (A3), and projection locking (A4). Along the transitions A2 asserts to be generically non-injective, A1 and A2 give immediate-image information loss and structural fluctuations; neither permanent non-recovery nor the arrow of time follows, since non-invertible steps admit cycles. From A3, we derive the proto-state as a physically real unresolved configuration shared between two successive states, and show that admissibility forces the retention of phase coherence. Under an explicit representation-theoretic reading of the admissible fibre, A3 forbids invariant factorisation and hence imposes irreducibility. For a carrier of dimension greater than one generated by a conjugate pair, irreducibility implies non-commutation. It does not, however, make the commutator central or select a finite Heisenberg group: an explicit S3 countermodel satisfies the extracted carrier contract and violates that conclusion. The Heisenberg carrier used by the spectral programme therefore remains a supplied realisation, and must be distinguished from its associated Weil representation. From A4, we derive the discrete character of quantum transitions as a consequence of the interaction between continuous Born–Infeld saturation and the discrete shell structure of the observable space. The foundation is stated with the point at which it stops short of carrier selection.