Important status notice: This version has been superseded by v2.0. Subsequent work shows that the single-domain three-diagonal construction used here produces an experimentally excluded common-mode photon-speed anisotropy. Readers should consult the revised version and 10.5281/zenodo.21464971. ================================================================ABSTRACT================================================================ We propose that conservation of probability — the single axiom A=1— is sufficient to derive the principal results of quantum mechanicswithout additional postulates. On a bipartite three-dimensionalcausal lattice (T³_diamond), each particle is represented as acausal session: an independent phase oscillator that propagatesamplitude across the lattice subject to |ψ_R|² + |ψ_L|² = 1 atevery tick. No Hamiltonian and no measurement postulate areassumed; a Hilbert-space structure with unitary evolution emergesfrom the U(1) oscillator and the per-session amplitude constraintrather than being postulated a priori. From this single constraint we derive: (i) the Dirac equation in the continuum limit a → 0, with thegamma matrices identified as geometric encodings of the RGB/CMYbipartite sublattice geometry; rotational invariance is recoveredon O_h-averaged observables (the same mechanism by which staggeredfermions recover Lorentz invariance in lattice QCD), and theresidual lattice birefringence aligned with the optical axisV₁+V₂+V₃ = (1,1,-1) is a falsifiable prediction with closed-formcoefficients; (ii) wave-particle duality and the Born rule as consequences ofamplitude redistribution under A=1; (iii) quantization of atomic orbitals as Arnold-tongue attractorsof the two-body phase dynamics, recovering the hydrogen Bohr radiusto four significant figures without fitting; (iv) gravity as a clock-density gradient rather than spacetimecurvature — a refraction of causal paths through regions ofhigher session density, reproducing the 1/r² force law with afalsifiable discrete correction at short range; (v) photon emission as a structural consequence of A=1: theCoulomb interaction is a probability source that a single sessioncannot accommodate, mandating creation of a new photon sessionat orbit lock-in; a first numerical implementation (exp_19c)preserves per-session unitarity at all swept emission rates, anda controlled three-arm operator comparison (exp_20) confirms apointwise unitary beam splitter as the joint-A=1-preservingemission operator (joint A=1 preserved to machine precision8.9×10⁻¹⁶); orbital settling remains open, traced to a long-horizon two-body baseline instability isolated in exp_12b(audit table; both rows PART); (vi) the structural form of the induced gauge action (Sakharov /Zeldovich, Appendix B), whose leading-order term has the Maxwellform on O_h-averaged observables together with a closed-formgauge-sector birefringence sharing the same optical axis (1,1,-1)as the kinematic-sector prediction (numerical 1/g² prefactor andthe empirical test of the birefringence remain open); thepredicted concordance of kinematic, gauge, gravitational, andatomic-clock channels along this single axis is the framework'sstrongest falsifiable prediction (P9). Energy conservation and momentum conservation are not independentinputs — they are consequences of A=1 applied to closed sessionsets. The numerically tractable results are verified or partiallyverified by a suite of more than fifteen experiments with no freeparameters beyond the Planck-scale calibration of the latticespacing (audit table records the status of each claim); theframework's remaining derivations — black-hole thermodynamicsamong them — are established analytically under explicitly statedmodelling assumptions and flagged as such where they appear. The framework suggests a structural reinterpretation of theStandard Model's gauge degrees of freedom as the minimalaccounting required to maintain A=1 on the bipartite latticegeometry: the particle content is reframed as the catalogue ofprobability imbalances the lattice generates and the session typesrequired to resolve them. Whether the full Standard Model gaugegroup SU(3) × SU(2) × U(1) is the automorphism group of(T³_diamond, A=1) is a well-posed finite-dimensional Lie-algebraquestion, deferred to a companion paper. ================================================================WHAT'S NEW IN v0.98-RC (since v0.97-RC)================================================================ A release-candidate update building on v0.97-RC's structuralenrichment. Two new experiments completed their first runs, twonew entries were added to the audit table (both PART), and thepaper text was systematically reconciled with the audit-tablestatuses with the help of an automated consistency-check toolintroduced in this release. ----------------------------------------------------------------HEADLINE CHANGES---------------------------------------------------------------- EXP_20 — THREE-ARM EMISSION OPERATOR COMPARISON First experiment to test whether a unitary 2-mode rotation (beamsplitter) is the correct emission operator for the lattice's jointA=1 conservation, compared directly against the as-written drainin exp_19c (arm A) and the phase-rotation drain (arm C). Arm A (control, drain): A_drift = 1.0×10⁰ Arm B (treatment, beam splitter): A_drift = 8.9×10⁻¹⁶ ← machine precision Arm C (alternative, phase rot): A_drift = 1.0×10⁻¹² Joint A=1 conservation experimentally validated to machineprecision under the unitary beam splitter. The as-written drain inexp_19c reproduces the row-5 arithmetic (1 - 2m + 2m²)|ψ_e|² —joint amplitude drift = 1.0 from photon inflation at the firstemission tick. Phase-dependent non-monotonic transfer in arm B: photon amplitudegrew to 0.236 at tick 2019 then receded to 0.005 by tick 4019 —a refinement to the operation-algebra row 6 prediction that afollow-on calculation will need to address. Orbital settling not achieved at the swept rate; see exp_12b. EXP_12B — LONG-DURATION TWO-BODY BASELINE (NO EMISSION) Bare exp_12 setup run for 6000 ticks. r_peak escapes from R₁ togrid edge (~83) by tick ~2000 and never returns. A=1 preserved atmachine precision throughout. Refines but does not invalidate exp_12's 4-sig-fig PASS, whichscores a different (k-scan resonance peak) claim. The orbitalescape seen in all three arms of exp_20 is inherited from thisbaseline, not caused by the emission operator. Exp_12's scoringwindow is now noted explicitly in Table 1's caption. TWO NEW AUDIT-TABLE ENTRIES (BOTH PART, BOTH ADDITIVE) - "Two-body long-horizon stability" — captures the exp_12b finding. - "Emission-operator joint A=1 conservation" — captures the exp_20 arm B confirmation. The existing "Photon emission as A=1 necessity" entry's evidencecolumn was refreshed; the entry remains PART. PAPER-TEXT RECONCILIATION Four sites with pre-exp_20 exp_19c overstatement aligned with theaudit-table: - paper/sections/abstract.tex items (v) and (vi) - paper/sections/introduction.tex (×2 mentions) - paper/sections/hydrogen_spectrum.tex - paper/sections/vacuum_twist_field_equations.tex The row-5 amplitude-drain non-unitarity arithmetic(1 - 2m + 2m²)|ψ_e|² is now explicitly cited in thehydrogen-spectrum write-up of exp_19c. "ONE LAW AT THREE RESOLUTIONS" — NEW INTRODUCTION PARAGRAPH Identifies the clock fluid density ρ_φ(x) = Σᵢ ωᵢ |ψᵢ(x)|² as theω-weighted form of the joint amplitude density A_joint(x) =Σᵢ |ψᵢ(x)|². Three structural resolutions of A=1: per-sessionunitarity (axiom), joint amplitude density (operation-algebratag), local clock-fluid continuity (gives gravity). Energyconservation is the ω-weighted spatial integral. Probabilityconservation, mass-energy conservation, and gravity all share onestructural origin. REVIEWER-DRIVEN FRAMING REFINEMENTS Five external reviewers' feedback was incorporated: - "No Hilbert space is assumed" softened: a Hilbert-space structure is acknowledged to emerge implicitly from the U(1) oscillator and the per-session amplitude constraint rather than being postulated a priori. - "The Standard Model is not a theory — it is an accounting system" replaced with a structural-reinterpretation framing: the Lie-algebra question Aut(T³_diamond, A=1) =? SO(3,1) × SU(3) × SU(2) × U(1) is now explicitly a deferred follow-on. - PASS / PART / STUB legend added at first explicit use in the introduction; Table 1 caption extended with a scoring-window note for exp_12. - New Appendix C: Reproducibility Quickstart. Compact 1-2 page reviewer-targeted summary with a 10-line minimal verification snippet (EXP12B_TICKS=20 run → reference output r_peak ≈ 9.877, A_drift ≈ 4.4×10⁻¹⁶). Complements the comprehensive Appendix A. - Conclusion's "What remains open" subsection extended with three new items: long-horizon two-body stability (exp_12b finding), numerical 1/g² prefactor for the induced gauge action, and the empirical test of the gauge-sector birefringence prediction (P9, separated from the analytical derivation). - Software-engineering vocabulary at the meta-prose layer sanitized to physics-friendly equivalents. Technical sections (CausalSession, tick scheduler, RGB/CMY) intentionally unchanged — their lingo is load-bearing. - Front-matter metadata polish: CC BY 4.0 license statement, active ORCID and email links in the author block, Data Availability statement, Keywords block. ----------------------------------------------------------------QUALITY-OF-LIFE CHANGES---------------------------------------------------------------- - Custom consistency-check tool committed at .claude/agents/dcl-claim-auditor.md, validated against the v0.97-RC abstract during this cycle's reconciliation pass (caught two high-severity overstatements that the manual sweep had missed). - Eight follow-on or speculative notes moved out of the project repository
Important status notice: This version has been superseded by v2.0. Subsequent work shows that the single-domain three-diagonal construction used here produces an experimentally excluded common-mode photon-speed anisotropy. Readers should consult the revised version and 10.5281/zenodo.21464971. ================================================================ABSTRACT================================================================ We propose that conservation of probability — the single axiom A=1— is sufficient to derive the principal results of quantum mechanicswithout additional postulates. On a bipartite three-dimensionalcausal lattice (T³_diamond), each particle is represented as acausal session: an independent phase oscillator that propagatesamplitude across the lattice subject to |ψ_R|² + |ψ_L|² = 1 atevery tick. No Hamiltonian and no measurement postulate areassumed; a Hilbert-space structure with unitary evolution emergesfrom the U(1) oscillator and the per-session amplitude constraintrather than being postulated a priori. From this single constraint we derive: (i) the Dirac equation in the continuum limit a → 0, with thegamma matrices identified as geometric encodings of the RGB/CMYbipartite sublattice geometry; rotational invariance is recoveredon O_h-averaged observables (the same mechanism by which staggeredfermions recover Lorentz invariance in lattice QCD), and theresidual lattice birefringence aligned with the optical axisV₁+V₂+V₃ = (1,1,-1) is a falsifiable prediction with closed-formcoefficients; (ii) wave-particle duality and the Born rule as consequences ofamplitude redistribution under A=1; (iii) quantization of atomic orbitals as Arnold-tongue attractorsof the two-body phase dynamics, recovering the hydrogen Bohr radiusto four significant figures without fitting; (iv) gravity as a clock-density gradient rather than spacetimecurvature — a refraction of causal paths through regions ofhigher session density, reproducing the 1/r² force law with afalsifiable discrete correction at short range; (v) photon emission as a structural consequence of A=1: theCoulomb interaction is a probability source that a single sessioncannot accommodate, mandating creation of a new photon sessionat orbit lock-in; a first numerical implementation (exp_19c)preserves per-session unitarity at all swept emission rates, anda controlled three-arm operator comparison (exp_20) confirms apointwise unitary beam splitter as the joint-A=1-preservingemission operator (joint A=1 preserved to machine precision8.9×10⁻¹⁶); orbital settling remains open, traced to a long-horizon two-body baseline instability isolated in exp_12b(audit table; both rows PART); (vi) the structural form of the induced gauge action (Sakharov /Zeldovich, Appendix B), whose leading-order term has the Maxwellform on O_h-averaged observables together with a closed-formgauge-sector birefringence sharing the same optical axis (1,1,-1)as the kinematic-sector prediction (numerical 1/g² prefactor andthe empirical test of the birefringence remain open); thepredicted concordance of kinematic, gauge, gravitational, andatomic-clock channels along this single axis is the framework'sstrongest falsifiable prediction (P9). Energy conservation and momentum conservation are not independentinputs — they are consequences of A=1 applied to closed sessionsets. The numerically tractable results are verified or partiallyverified by a suite of more than fifteen experiments with no freeparameters beyond the Planck-scale calibration of the latticespacing (audit table records the status of each claim); theframework's remaining derivations — black-hole thermodynamicsamong them — are established analytically under explicitly statedmodelling assumptions and flagged as such where they appear. The framework suggests a structural reinterpretation of theStandard Model's gauge degrees of freedom as the minimalaccounting required to maintain A=1 on the bipartite latticegeometry: the particle content is reframed as the catalogue ofprobability imbalances the lattice generates and the session typesrequired to resolve them. Whether the full Standard Model gaugegroup SU(3) × SU(2) × U(1) is the automorphism group of(T³_diamond, A=1) is a well-posed finite-dimensional Lie-algebraquestion, deferred to a companion paper. ================================================================WHAT'S NEW IN v0.98-RC (since v0.97-RC)================================================================ A release-candidate update building on v0.97-RC's structuralenrichment. Two new experiments completed their first runs, twonew entries were added to the audit table (both PART), and thepaper text was systematically reconciled with the audit-tablestatuses with the help of an automated consistency-check toolintroduced in this release. ----------------------------------------------------------------HEADLINE CHANGES---------------------------------------------------------------- EXP_20 — THREE-ARM EMISSION OPERATOR COMPARISON First experiment to test whether a unitary 2-mode rotation (beamsplitter) is the correct emission operator for the lattice's jointA=1 conservation, compared directly against the as-written drainin exp_19c (arm A) and the phase-rotation drain (arm C). Arm A (control, drain): A_drift = 1.0×10⁰ Arm B (treatment, beam splitter): A_drift = 8.9×10⁻¹⁶ ← machine precision Arm C (alternative, phase rot): A_drift = 1.0×10⁻¹² Joint A=1 conservation experimentally validated to machineprecision under the unitary beam splitter. The as-written drain inexp_19c reproduces the row-5 arithmetic (1 - 2m + 2m²)|ψ_e|² —joint amplitude drift = 1.0 from photon inflation at the firstemission tick. Phase-dependent non-monotonic transfer in arm B: photon amplitudegrew to 0.236 at tick 2019 then receded to 0.005 by tick 4019 —a refinement to the operation-algebra row 6 prediction that afollow-on calculation will need to address. Orbital settling not achieved at the swept rate; see exp_12b. EXP_12B — LONG-DURATION TWO-BODY BASELINE (NO EMISSION) Bare exp_12 setup run for 6000 ticks. r_peak escapes from R₁ togrid edge (~83) by tick ~2000 and never returns. A=1 preserved atmachine precision throughout. Refines but does not invalidate exp_12's 4-sig-fig PASS, whichscores a different (k-scan resonance peak) claim. The orbitalescape seen in all three arms of exp_20 is inherited from thisbaseline, not caused by the emission operator. Exp_12's scoringwindow is now noted explicitly in Table 1's caption. TWO NEW AUDIT-TABLE ENTRIES (BOTH PART, BOTH ADDITIVE) - "Two-body long-horizon stability" — captures the exp_12b finding. - "Emission-operator joint A=1 conservation" — captures the exp_20 arm B confirmation. The existing "Photon emission as A=1 necessity" entry's evidencecolumn was refreshed; the entry remains PART. PAPER-TEXT RECONCILIATION Four sites with pre-exp_20 exp_19c overstatement aligned with theaudit-table: - paper/sections/abstract.tex items (v) and (vi) - paper/sections/introduction.tex (×2 mentions) - paper/sections/hydrogen_spectrum.tex - paper/sections/vacuum_twist_field_equations.tex The row-5 amplitude-drain non-unitarity arithmetic(1 - 2m + 2m²)|ψ_e|² is now explicitly cited in thehydrogen-spectrum write-up of exp_19c. "ONE LAW AT THREE RESOLUTIONS" — NEW INTRODUCTION PARAGRAPH Identifies the clock fluid density ρ_φ(x) = Σᵢ ωᵢ |ψᵢ(x)|² as theω-weighted form of the joint amplitude density A_joint(x) =Σᵢ |ψᵢ(x)|². Three structural resolutions of A=1: per-sessionunitarity (axiom), joint amplitude density (operation-algebratag), local clock-fluid continuity (gives gravity). Energyconservation is the ω-weighted spatial integral. Probabilityconservation, mass-energy conservation, and gravity all share onestructural origin. REVIEWER-DRIVEN FRAMING REFINEMENTS Five external reviewers' feedback was incorporated: - "No Hilbert space is assumed" softened: a Hilbert-space structure is acknowledged to emerge implicitly from the U(1) oscillator and the per-session amplitude constraint rather than being postulated a priori. - "The Standard Model is not a theory — it is an accounting system" replaced with a structural-reinterpretation framing: the Lie-algebra question Aut(T³_diamond, A=1) =? SO(3,1) × SU(3) × SU(2) × U(1) is now explicitly a deferred follow-on. - PASS / PART / STUB legend added at first explicit use in the introduction; Table 1 caption extended with a scoring-window note for exp_12. - New Appendix C: Reproducibility Quickstart. Compact 1-2 page reviewer-targeted summary with a 10-line minimal verification snippet (EXP12B_TICKS=20 run → reference output r_peak ≈ 9.877, A_drift ≈ 4.4×10⁻¹⁶). Complements the comprehensive Appendix A. - Conclusion's "What remains open" subsection extended with three new items: long-horizon two-body stability (exp_12b finding), numerical 1/g² prefactor for the induced gauge action, and the empirical test of the gauge-sector birefringence prediction (P9, separated from the analytical derivation). - Software-engineering vocabulary at the meta-prose layer sanitized to physics-friendly equivalents. Technical sections (CausalSession, tick scheduler, RGB/CMY) intentionally unchanged — their lingo is load-bearing. - Front-matter metadata polish: CC BY 4.0 license statement, active ORCID and email links in the author block, Data Availability statement, Keywords block. ----------------------------------------------------------------QUALITY-OF-LIFE CHANGES---------------------------------------------------------------- - Custom consistency-check tool committed at .claude/agents/dcl-claim-auditor.md, validated against the v0.97-RC abstract during this cycle's reconciliation pass (caught two high-severity overstatements that the manual sweep had missed). - Eight follow-on or speculative notes moved out of the project repository
Important status notice: This version has been superseded by v2.0. Subsequent work shows that the single-domain three-diagonal construction used here produces an experimentally excluded common-mode photon-speed anisotropy. Readers should consult the revised version and 10.5281/zenodo.21464971. We propose that conservation of probability — the single axiom A=1 — is sufficient to derive the principal results of quantum mechanics without additional postulates. On a bipartite three-dimensional causal lattice T³_diamond, each particle is represented as a causal session: an independent phase oscillator that propagates amplitude across the lattice subject to |ψ_R|² + |ψ_L|² = 1 at every tick. No Hamiltonian and no measurement postulate are assumed; a Hilbert-space structure with unitary evolution emerges from the U(1) oscillator and the per-session amplitude constraint rather than being postulated a priori. From this single constraint, the paper derives: (i) the Dirac equation in the continuum limit a→0, with the gamma matrices identified as geometric encodings of the bipartite RGB/CMY sublattice geometry; (ii) wave-particle duality and the Born rule as consequences of amplitude redistribution under A=1, with Gleason's theorem securing the |ψ|² uniqueness from the framework's emergent Hilbert space; (iii) quantization of atomic orbitals as Arnold tongue attractors of the two-body phase dynamics, recovering the hydrogen Bohr radius to four significant figures (k_min = 0.0970 vs k_Bohr = 0.0971) without fitting; (iv) gravity as a clock-density gradient rather than spacetime curvature — a refraction of causal paths through regions of higher session density, reproducing the 1/r² force law with an explicit weak-field metric tensor g_μν as a functional of the clock-density field; (v) photon emission as a structural consequence of A=1: the Coulomb interaction is a probability source that a single session cannot accommodate, mandating creation of a new photon session at orbit lock-in; (vi) the structural form of the induced gauge action (Sakharov / Zeldovich), whose leading-order term has the Maxwell form on O_h-averaged observables, with a closed-form gauge-sector birefringence sharing the same optical axis (1,1,−1) as the kinematic-sector prediction. Energy and momentum conservation are not independent inputs — they are consequences of A=1 applied to closed session sets. The numerically tractable results are verified or partially verified by a suite of more than fifteen experiments with no free parameters beyond the Planck-scale calibration of the lattice spacing; an audit table records the status of every claim. The paper conjectures that the Standard Model gauge group SU(3) × SU(2) × U(1), together with Lorentz SO(3,1), is the automorphism group of the framework's extended per-site amplitude — a precise 18-dimensional Lie-algebra equality. The existing framework is shown to carry SO(3,1) × U(1) (dim 7) on the per-site (ψ_R, ψ_L) amplitude; the remaining factors require structural extensions (per-site C² isospin and C³ colour indices) that are the subject of a companion paper. Six computational scaffolding scripts establish the conjecture's components. This is Paper I of the A=1 Discrete Causal Lattice series. Companion papers in preparation address the Standard Model gauge derivation, the Farey-sequence mass spectrum (and its Veneziano-amplitude limiting case), plasma and gradient-ionization phenomenology, the proton-interior structure, quantum measurement theory, and a Hilbert-6th-style combinatorial foundation for continuum physics. What changed in v1.0 (since v0.98-RC): * Three new audit-table rows: long-horizon escape mechanism (PART), lock-in resonance grid-independence across {65, 81, 97, 113}³ grids (PART), and the Standard Model gauge-group conjecture as a precise 18-dimensional Lie-algebra equality (STUB).* A Gleason-uniqueness subsubsection in the Born-rule discussion, addressing the necessity question for the |ψ|² measure.* An explicit metric-tensor formula in the gravity section, identifying g_μν as a functional of ρ_clock in the weak-field limit.* Refined automorphism conjecture with explicit decomposition into established and structural-extension factors.* New Acknowledgments section. Source code, paper sources, and data files: https://github.com/JackDMenendez/discrete-causal-latticeLicense: Paper text and figures CC BY 4.0; source code MIT.
See changelog below. The full derivations are provided in the attached paper. Every result is reproducible from the stated inputs. This paper derives the complete Standard Model, quantum mechanics, spacetime, and gravity — 85 quantitative results at sub-percent precision spanning 15 orders of magnitude — from a single postulate, four structural principles, and one measurement. The Standard Model has seventeen free parameters. General relativity has no quantum theory. The measurement problem has been open for a century. The cosmological constant is wrong by 120 orders of magnitude. The question of why there is something rather than nothing has no physical answer in any existing framework. Each of these failures shares a common omission: none ask where time comes from. This paper takes that question seriously. A single physical principle — information cannot be destroyed — is self-sealing: its own violation is incoherent. Its formal expression is Postulate I: each foreclosed quantum branch produces exactly one chronon — a massless spin-1 gauge boson of U(1)_T, the unique gauge group forced by the complex ray structure of the Hilbert space and the single scalar collapse charge. We call it the chronon. Time is not the background in which physics happens; it is generated by physics, irreversibly, at every collapse. These requirements yield a single Lagrangian, ℒ_QT, that extends the Standard Model by one massless spin-1 field and one coupling constant. From this — combined with one measurement, Newton’s gravitational constant G — we derive: all of quantum mechanics via the Piron–Soler–Gleason chain (not assumed); the complete Standard Model with all dimensionless parameters determined; the complete resolution of the measurement problem, including the environmental decoherence cascade; and 85 quantitative physical results spanning fifteen orders of magnitude in energy. Three spatial dimensions follow from the channel-matching principle (D = N_c) and the Weinberg angle consistency condition (N_c = 3). The Big Bang was not initiated — it was mandatory. The numerical precision is sharp. The worst result where the theoretical prediction exceeds experimental precision is 0.5%. For the majority of results, the framework’s prediction is already more precise than the measurement. The proton-to-electron mass ratio m_p/m_e = 2N_cπ⁵(1 + α²/√(N_c²−1)) agrees to sub-ppb precision. The proton charge radius matches muonic hydrogen to five significant figures. The Higgs boson mass to 0.002% via the full cascade derivation (0.25% from the explicit formula). The W boson mass to 0.001%. The fine structure constant to 0.026% (1/α = 137 exactly at tree level). The cosmological constant — wrong by 120 orders of magnitude in standard QFT — is derived to 0.03%. The Hubble tension, a 5σ discrepancy unresolved in standard cosmology, follows from the same structure: H₀_local = H₀ × 13/12 = 73.0 km/s/Mpc (0.05%). All four PMNS neutrino mixing parameters, previously unexplained, are derived to sub-0.5%. Sub-percent agreement spans from neutrino mass splittings at ~9 meV to the CMB temperature at 2.725 K — fifteen orders of magnitude from a single dimensionful input. The framework makes ten falsifiable predictions currently under test: an electron electric dipole moment of exactly zero (ACME III is live); a spin-1 gravitational wave breathing mode at ~9% of the + polarisation amplitude (testable with LIGO O4/O5 data); no dark matter particle in any direct-detection experiment; no graviton — gravity is geometric drift in a proper-time gradient sourced by the energy-momentum tensor, not exchange of a spin-2 particle; there is nothing to detect. A confirmed non-zero electron EDM or a detected graviton falsifies the framework immediately. The logical order, from bottom to top: information cannot be destroyed → P-I (one chronon per foreclosed branch) → proposition lattice with axioms P0–P4 → Piron–Soler–Gleason chain → complex Hilbert space → ray phase redundancy → single scalar collapse charge → U(1)_T gauge field with Maxwell dynamics → channel-matching principle gives D = N_c → Weinberg angle selects N_c = 3 → D = 3 → adding G through the Planck cascade gives λ_conf = 0.40092 fm → the complete Standard Model follows. Spacetime, quantum mechanics, and the Standard Model are not independent inputs — they follow as consequences of P-I, the structural principles, and one observational consistency condition. ______________________________________________________________________________________________________ v86 Changelog (April 7–8 2026) — All numerical results unchanged from v85. No formulas modified. Foundational structure and derivation justifications rebuilt following adversarial review by two independent AI systems (Claude, GPT-4o). Foundational hardening: §2.2: Formal axiom framework P0–P4 added. Piron lattice conditions now follow as a theorem given five named structural axioms, not as informal physical arguments. P-I motivates four directly; P0 (admissibility closure) is a structural axiom on the proposition space. §2.5: ℝ exclusion replaced. Old iε propagator argument replaced with operational compositionality / local tomography argument requiring no field-theoretic machinery. §3.1: Gauge derivation rebuilt from complex ray structure forward. U(1)_T now derived from phase redundancy + single scalar collapse charge + rank-1 compact Lie group + label-free emission. Removes prior dependence on Noether's theorem invocation and energy minimization principle. §8.1.1: Channel-matching theorem added. D = N_c derived from information-balance between spatial export channels and internal confinement channels via mismatch functional. Replaces prior mutual-reinforcement argument. §1.5.1: Epistemic taxonomy added classifying all 85 results into three tiers: theorem-level (Piron, Soler, Gleason), physical-argument (gauge structure, mass formulas), and observational-consistency (N_c = 3 from Weinberg angle, cascade from G). §12: New section — Structural Analysis. Result classification (5 epistemic tiers, ~12 independent core results, compression ratio 85:19), axiom necessity (what breaks without each of P0–P4 and D1–D4), and ablation tests (N_c = 2, N_c = 4, drop P4, change coupling exponent, allow D ≠ N_c — every perturbation destroys the majority of results). Framework premise expanded from "one postulate" to "one postulate, four structural principles, and one measurement." Front matter aligned with body throughout. One derivation story, no internal contradictions. Derivation upgrades: Proton-to-electron mass ratio (§9.3): Replaced "2 spin states × N_c colours" with rigorous hyperspherical decomposition. S³ forced by massless emission in (3+1)D; S⁵ forced by K=0 ground state of 3-quark system in 6D relative coordinates. Neutral pion mass (§9.9.1): Factor 5 now derived as 2N_c − 1. Denominator 2π identified as one-loop coefficient of U(1)_T vector gauge coupling. Neutron-proton mass difference (§9.3.2): Replaced "circular loop + straight line" with flux tube arm derivation. π from SU(2) isospin rotation; +1 from chromoelectric flux tube arm to Y-junction. CP phase δ (§9.8.2): Replaced "axial-to-spherical flux ratio" with V−A vertex structure. The ratio 4/π is universal across all generation spaces because any unitary rotation lies on a great circle of S². W boson mass corrections (§9.5.1): Both NLO terms fully derived. Direct vertex α_s/(8π): C_F cancelled by non-relativistic reduction. Exchange α_s/(12π²): two vertices × Dirac-to-colour projection ratio. Charm mass (§9.13.4): Factor 4/π identified as axial-to-vector coupling ratio through V−A weak vertex. Systematic correction — QCD dressing factor: The coefficient 1/(2π) appearing in ~10 results was previously attributed to "hemisphere projection geometry." Replaced throughout: mass in the chronon framework is a U(1)_T vector gauge coupling, not a scalar Yukawa coupling. The factor 1/2 confirmed by two independent routes: (i) Dirac algebra (γ^α γ^μ γ_α = −2γ^μ vs scalar = 4); (ii) C_F cancellation by non-relativistic reduction. Label correction: CKM geometric factor (§9.8.1): "four-dimensional solid angle" corrected to "three-dimensional solid angle S² = 4π." Numbers unchanged. ______________________________________________________________________________________________________ v85 Changelog (April 6 2026) — Electroweak corrections, |V_cb| prediction, consistency audit Four physics improvements and a comprehensive numerical audit, all from first principles with zero free parameters. Top quark mass (0.41% → 0.06%): The electroweak self-energy correction factor changed from (N_c+1) to (N_c−1). The factor (N_c−1) = 2 counts the charged weak boson states (W⁺, W⁻) mediating the t→b→t self-energy loop — the SU(2)_L charge-raising/lowering generators. m_t = 172,803 MeV vs observed 172,700 MeV. CKM θ₂₃ / |V_cb| prediction (1.3% → 0.3%): The framework predicts |V_cb| = 42.1 × 10⁻³, essentially identical to the PDG inclusive determination (42.2 ± 0.5) × 10⁻³ and inconsistent with the exclusive determination (39.8 ± 0.6) × 10⁻³. The inclusive/exclusive |V_cb| tension is one of the open questions in flavour physics; the framework picks a side. Falsifiable prediction. Baryon asymmetry η (0.5% → <0.1%): The sphaleron efficiency κ correction changed from α_s/(2π²) to α/(2π). The sphaleron is an SU(2)_L electroweak process; its one-loop vertex correction uses the QED coupling α/(2π), not the QCD coupling. Neutrino mass splitting Δm²₂₃ (0.3% → 0.08%): QED vacuum polarization correction −α/(2π) added to G₃ alongside the existing QCD correction +α_s/π. Sign pattern (QCD adds, QED subtracts) matches the W boson mass derivation. Numerical audit (22 corrections): Full line-by-line verification of every formula, table entry, agreement claim, and cross-reference. Corrections include: D/H value (2.62→2.58 × 10⁻⁵, 1.3σ→1.8σ — stale from pre-κ η); He-4 agreement (0.5σ→0.7σ); z_
Programme status: This record belongs to an active research programme. It is not a finished or final theory. The framework is currently undergoing a systematic audit, together with corrections and improvements prompted by reader feedback. Several significant issues have been identified, and some claims, derivations and interpretations in the current version may be revised, qualified or withdrawn in future versions. Revised versions will be released when this work is complete. See PROGRAM_STATUS.md. Status as of October 2026. We apply 't Hooft's cellular automaton interpretation of quantum mechanics to the GTE cellular automaton f_MDL on Z_7^5, the 16,807-state visible-sector ring encoding the Standard Model generation structure. The complete orbit decomposition of f_MDL () reveals a single cycle: the vacuum fixed point (0,0,0,0,0) with E=0. The physical Hilbert space is one-dimensional; f_MDL belongs to 't Hooft's information-loss regime. SM generation states are transients with tail lengths (gen_1)=3 > (gen_2)=2 > (gen_3)=1, matching the empirical stability hierarchy (). A direct eigenvalue-to-mass correspondence is falsified on every formulation tested (). The Two-Role Structural Principle () separates roles: the cogwheel construction provides Hilbert space structure, unitarity, and the Born rule (, Lean-certified); the GTE N_eff cascade provides the mass spectrum. The seven winding-equivalence classes of Z_7^5 (, gauge\_arithmetic\_identification) are identified as 't Hooft's information-equivalence gauge classes (): the five SM winding sectors \0,2,3,4,6\ yield U(1)_EM, SU(2)_L, and SU(3)_c; sector W=1 predicts the SU(5) Y-leptoquark; the X-leptoquark (Q=+4/3) cohabits W=4. The AFCA causal graph is update-schedule-independent (Lean-certified); discrete Minkowski causal cones are embedded with coordinate surjection certified. Full Minkowski bijection remains open ().
This short note proposes that the conserved substance of physics is action, not energy. Energy conservation is, by Noether's theorem, the exact statement of a uniform flow of time — an assumption that fails, even if only slightly, in a relativistic universe where clocks run at different rates in different places. Read the other way, Planck's relation becomes a conservation law rather than an empirical input: Eτ = h, one quantum of action processed per cycle of a local resonance, with ordinary energy conservation recovered as the special case in which the period τ is held fixed. The note carries exactly one assumption beyond the algebra, stated plainly rather than smuggled in: the vacuum is a medium of such resonances, and a particle is a persistent, self-closing cycle within it. Angle and capacity The invariant has two faces. The law pins the magnitude at h — the capacity — leaving the accumulated phase φ as the only freedom: dS = ℏ dφ. Every familiar conjugate pair is this one statement read along a different axis of advance — energy against time, momentum against position, angular momentum against orientation — three orthogonal accounts of one accumulating phase, each closing one unit of capacity per cycle: Eτ = h, pλ = h, L·2π = h. This split is where classical and quantum mechanics part ways. An open trajectory reads the phase continuously, and its stationary-phase condition is the principle of least action; a closed resonance can only count whole turns, and closure quantises. Quantisation is a property of closure, not a property of the phase — the boundary between classical and quantum is a change of topology, not a change of law. What follows mechanically From the capacity law alone, with no further postulate, three familiar structures are recovered. A resonance at rest turns at the Compton frequency ωC = mc²/ℏ, so that one full turn spends exactly one quantum, EτC = h: mass is nothing but a resonance's cycling rate. Driven through the medium, a resonance meets an increased flux that scales as the Lorentz factor; conserving its fixed action budget under that load forces its geometry to yield — contraction along the direction of motion, dilation of its internal period — recovering special relativity as mechanical equilibration rather than a postulate about coordinates, with no rest frame ever exposed because the same yielding that would reveal one also defines the observer's own rulers and clocks. Requiring the accumulated phase along a worldline to stay coherent, expanded in the weak-field limit, returns the classical Lagrangian directly, −Mc²dτ = (T − V − Mc²)dt: least action is not minimised as a primitive rule, it is what capacity conservation looks like when a resonance crosses a gradient of the medium. And the Heisenberg relation σxσp ≥ ℏ/2 is read as the ordinary bandwidth (Gabor) theorem for a wave and its Fourier transform — a floor on occupancy for a phase-locked resonance, not a ceiling on knowledge. Gravity, without a graviton The most interpretive claim in the note, flagged as such throughout: if the metric's invariant is Eτ = h rather than a fixed energy, the metric carries no absolute clock, and the phase-lock condition forces its temporal and spatial responses to be reciprocal, giving the bipolar exponential line element ds² = −e−2ξc²dt² + e+2ξdx². The conceptual point is what this reframes: the decades-long search for a quantum theory of gravity has sought a graviton by quantising the metric on a fixed background, when gravity is the background. On this reading there is no separate field to quantise — the medium's deformation is already quantised at h per cycle, and gravity is simply the shape of phase accumulation across its gradient. The problem is not solved so much as dissolved: relocated from “how do we quantise the metric” to “the metric was never a separate thing to quantise.” Scope The note is deliberately bounded, and says so explicitly rather than leaving the boundary implicit. It derives special relativity, the principle of least action, and the bandwidth reading of the uncertainty relation from the capacity law alone; it advances the gravitational reading as a stance to be tested against its consequences, not a completed theory. It does not derive the particle spectrum, specific masses, electric charge, colour, or the numerical values of G or α, and makes no cosmological claim. This separation is intentional: a conservation principle should be judged on its own terms. A companion note reports a closed-form numerical identity for the proton mass as a separate, independent empirical fact; the two are meant to be read, and to stand or fall, apart.
This short note proposes that the conserved substance of physics is action, not energy. Energy conservation is, by Noether's theorem, the exact statement of a uniform flow of time — an assumption that fails, even if only slightly, in a relativistic universe where clocks run at different rates in different places. Read the other way, Planck's relation becomes a conservation law rather than an empirical input: Eτ = h, one quantum of action processed per cycle of a local resonance, with ordinary energy conservation recovered as the special case in which the period τ is held fixed. The note carries exactly one assumption beyond the algebra, stated plainly rather than smuggled in: the vacuum is a medium of such resonances, and a particle is a persistent, self-closing cycle within it. Angle and capacity The invariant has two faces. The law pins the magnitude at h — the capacity — leaving the accumulated phase φ as the only freedom: dS = ℏ dφ. Every familiar conjugate pair is this one statement read along a different axis of advance — energy against time, momentum against position, angular momentum against orientation — three orthogonal accounts of one accumulating phase, each closing one unit of capacity per cycle: Eτ = h, pλ = h, L·2π = h. This split is where classical and quantum mechanics part ways. An open trajectory reads the phase continuously, and its stationary-phase condition is the principle of least action; a closed resonance can only count whole turns, and closure quantises. Quantisation is a property of closure, not a property of the phase — the boundary between classical and quantum is a change of topology, not a change of law. What follows mechanically From the capacity law alone, with no further postulate, three familiar structures are recovered. A resonance at rest turns at the Compton frequency ωC = mc²/ℏ, so that one full turn spends exactly one quantum, EτC = h: mass is nothing but a resonance's cycling rate. Driven through the medium, a resonance meets an increased flux that scales as the Lorentz factor; conserving its fixed action budget under that load forces its geometry to yield — contraction along the direction of motion, dilation of its internal period — recovering special relativity as mechanical equilibration rather than a postulate about coordinates, with no rest frame ever exposed because the same yielding that would reveal one also defines the observer's own rulers and clocks. Requiring the accumulated phase along a worldline to stay coherent, expanded in the weak-field limit, returns the classical Lagrangian directly, −Mc²dτ = (T − V − Mc²)dt: least action is not minimised as a primitive rule, it is what capacity conservation looks like when a resonance crosses a gradient of the medium. And the Heisenberg relation σxσp ≥ ℏ/2 is read as the ordinary bandwidth (Gabor) theorem for a wave and its Fourier transform — a floor on occupancy for a phase-locked resonance, not a ceiling on knowledge. Gravity, without a graviton The most interpretive claim in the note, flagged as such throughout: if the metric's invariant is Eτ = h rather than a fixed energy, the metric carries no absolute clock, and the phase-lock condition forces its temporal and spatial responses to be reciprocal, giving the bipolar exponential line element ds² = −e−2ξc²dt² + e+2ξdx². The conceptual point is what this reframes: the decades-long search for a quantum theory of gravity has sought a graviton by quantising the metric on a fixed background, when gravity is the background. On this reading there is no separate field to quantise — the medium's deformation is already quantised at h per cycle, and gravity is simply the shape of phase accumulation across its gradient. The problem is not solved so much as dissolved: relocated from “how do we quantise the metric” to “the metric was never a separate thing to quantise.” Scope The note is deliberately bounded, and says so explicitly rather than leaving the boundary implicit. It derives special relativity, the principle of least action, and the bandwidth reading of the uncertainty relation from the capacity law alone; it advances the gravitational reading as a stance to be tested against its consequences, not a completed theory. It does not derive the particle spectrum, specific masses, electric charge, colour, or the numerical values of G or α, and makes no cosmological claim. This separation is intentional: a conservation principle should be judged on its own terms. A companion note reports a closed-form numerical identity for the proton mass as a separate, independent empirical fact; the two are meant to be read, and to stand or fall, apart.
This work continues the framework developed in Ref. [1], in which time is treated relationally as the +1 evolution parameter associated with the dimensional configuration space of a physical phenomenon, and quantum-mechanical structure is proposed to emerge from the information-preserving dimensional recording of a higher-dimensional wave field. Here we extend that construction beyond the single-system description and investigate whether additional central elements of quantum mechanics can arise without being introduced as independent quantum postulates. We first clarify the recording process in terms of physically distinguishable equivalence classes of parent configurations and show how changes of the parent field can produce effective measurement-induced state updating in the reduced description. For independently preparable systems, preservation of local linear superposition leads to bilinear composition and, subject to completeness of the accessible composite state space, to Hilbert-space tensor products. Joint interactions then generate nonfactorizable reduced states, providing a structural origin for entanglement before any probabilistic interpretation is introduced. We subsequently derive the Born probability rule from the geometry of orthogonal reduced alternatives together with unitary invariance, noncontextual consistency, continuity, normalization, and additive coarse-graining. The resulting probability structure is compared with the independent robustness-based reconstruction of De Raedt, Katsnelson, and Michielsen [2]. Using the derived composite structure and probability rule, Bell–CHSH correlations are obtained from generic interaction-generated pure entangled states without assuming a singlet state or Pauli spin operators, while operational no-signalling is preserved. Finally, we explore a possible geometric extension toward internal gauge symmetries: aligned three-dimensional, tilted three-dimensional, and fully four-dimensional parent-field deformations lead respectively to one, two, and three component complex internal representations, suggesting U(1), SU(2), and SU(3) type symmetry structures. Unlike approaches that take the quantum formalism as their starting point, or reconstruction programs based primarily on operational or informational axioms, the present framework seeks a concrete physical mechanism from which these structures emerge together. The fact that a single modification in the dimensional role of time, combined with information-preserving reduction, consistently generates multiple otherwise independent features of quantum theory makes this possibility physically significant even before genuinely new empirical predictions are established.