A single two-dimensional ordinary differential equation — originally constructed to model co-evolutionary dynamics between a process variable and a geometric complexity parameter — is shown to generate four nested classes of physical behavior without any physics-specific terms in the equations. Gravity-class: The system produces an emergent force law (v ∝ μ^{−α}) through co-descent on a free energy landscape, where the exponent α is fully determined by the coupled 2D transient dynamics. Discretization-class: A pitchfork bifurcation when the effective linear coefficient γ_eff < 0 produces exactly two or three discrete energy basins. Kramers escape rates between basins follow the exponential suppression exp(−2ΔV/σ²) with slope 1.979 (R² = 0.985), matching the mathematical structure of WKB tunneling — the one genuine mathematical overlap between co-descent dynamics and quantum mechanics. Measurement-class: Asymmetric coupled pairs produce one-directional state determination: one subsystem's basin structure constrains the other (r = 0.865, 85.6% entropy reduction) without back-reaction. The measurement rule is a linear volume ratio P = ΔV_B/(ΔV_A + ΔV_B), not the Born rule, and the threshold is a sigmoid crossover, not decoherence. Indeterminacy-class: Periodic forcing at ω_c = 0.51 creates binary committed/circulating domains with layered protection (10× the critical coupling of the measurement mechanism alone). Symmetric coupling with asymmetric stability drives detector destabilization — a back-action principle without quantum measurement axioms. GPT-2 transformer layers 3–9 are shown to access the γ_eff < 0 bifurcation regime, confirming that the discretization mechanism is not merely theoretical. Systematic comparison across four structural mappings (discretization, collapse, threshold, superposition) reveals that co-descent dynamics and quantum mechanics share architectural resemblances but differ mathematically at every mapping point. Co-descent is identified as a sibling physics — same geometric architecture, substrate-specific algebra. All numerical results are drawn from nine pre-registered experiments with explicit pass/fail criteria. The paper includes 11 figures, 14 quantitative tables, and reports both positive and negative results. This paper extends prior work on the co-descent ODE (Rodriguez, 2026l; DOI: 10.5281/zenodo.19528436) and its variational foundation (Rodriguez, 2026m; DOI: 10.5281/zenodo.19529770).
A single two-dimensional ordinary differential equation — originally constructed to model co-evolutionary dynamics between a process variable and a geometric complexity parameter — is shown to generate four nested classes of physical behavior without any physics-specific terms in the equations. Gravity-class: The system produces an emergent force law (v ∝ μ^{−α}) through co-descent on a free energy landscape, where the exponent α is fully determined by the coupled 2D transient dynamics. Discretization-class: A pitchfork bifurcation when the effective linear coefficient γ_eff < 0 produces exactly two or three discrete energy basins. Kramers escape rates between basins follow the exponential suppression exp(−2ΔV/σ²) with slope 1.979 (R² = 0.985), matching the mathematical structure of WKB tunneling — the one genuine mathematical overlap between co-descent dynamics and quantum mechanics. Measurement-class: Asymmetric coupled pairs produce one-directional state determination: one subsystem's basin structure constrains the other (r = 0.865, 85.6% entropy reduction) without back-reaction. The measurement rule is a linear volume ratio P = ΔV_B/(ΔV_A + ΔV_B), not the Born rule, and the threshold is a sigmoid crossover, not decoherence. Indeterminacy-class: Periodic forcing at ω_c = 0.51 creates binary committed/circulating domains with layered protection (10× the critical coupling of the measurement mechanism alone). Symmetric coupling with asymmetric stability drives detector destabilization — a back-action principle without quantum measurement axioms. GPT-2 transformer layers 3–9 are shown to access the γ_eff < 0 bifurcation regime, confirming that the discretization mechanism is not merely theoretical. Systematic comparison across four structural mappings (discretization, collapse, threshold, superposition) reveals that co-descent dynamics and quantum mechanics share architectural resemblances but differ mathematically at every mapping point. Co-descent is identified as a sibling physics — same geometric architecture, substrate-specific algebra. All numerical results are drawn from nine pre-registered experiments with explicit pass/fail criteria. The paper includes 11 figures, 14 quantitative tables, and reports both positive and negative results. This paper extends prior work on the co-descent ODE (Rodriguez, 2026l; DOI: 10.5281/zenodo.19528436) and its variational foundation (Rodriguez, 2026m; DOI: 10.5281/zenodo.19529770).
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_
This framework preserves the empirical success of existing physical theories while proposing an alternative set of foundational assumptions from which their standard formulations emerge as effective limits. Abstract — Key Innovative Points Discrete Updating FabricThe theory postulates a fundamental discrete fabric whose elementary constituents evolve through local update processes, replacing continuous fields and background spacetime as primary entities. Irreversible Local Dynamics as FundamentalUnlike standard approaches based on unitary or reversible evolution, irreversibility is taken as ontological. Physical law emerges from irreversible local updates rather than from imposed symmetries or variational principles. Emergent Time and SpacetimeTime is not assumed as a fundamental parameter but arises from the counting of update events. Spacetime geometry emerges as a coarse-grained description of update activity across the fabric. Quantum Probability from Microstate CountingQuantum probabilities, including the Born rule, emerge from combinatorial counting of compatible microstates of the fabric, rather than being postulated axiomatically. Gravity as Update-Rate ModulationGravitational phenomena are interpreted as variations in local update rates, leading to effective curvature, time dilation, and relativistic behavior at macroscopic scales. Mass as Resistance to UpdateMass and inertia arise from stable topological structures within the fabric that resist updates, providing a dynamical origin of mass without introducing fundamental mass parameters. Fundamental Dissipation and IrreversibilityDissipation is intrinsic to the fabric dynamics and not an emergent statistical effect. The arrow of time follows directly from the irreversible nature of local updates. Extension Beyond Fundamental PhysicsThe same discrete framework accounts for macroscopic dissipative phenomena—such as friction, stick–slip dynamics, superlubricity, growth, and self-organization—linking fundamental physics and material behavior within a single process-based model. iversality Without Fundamental SymmetryLarge-scale physical laws emerge as dynamical attractors of the updating fabric, arising through elimination of unstable configurations rather than enforcement of fundamental symmetries. Emergent Rules Beyond the Born Rule in the Unified Fabric Framework The same discrete-fabric + local-update + coarse-graining structure generates multiple quantum rules without postulation. 1. Schrödinger Dynamics as an Emergent Long-Wavelength Limit Local deterministic updates on a discrete fabric admit spectral modes.In the long-wavelength limit, their evolution converges to a Schrödinger-type equation. No quantum dynamics is postulated.Only update rules and spectral structure are assumed. 2. Superposition from Coarse-Grained Microstate Counting Macroscopic amplitudes arise as sums over compatible microstates.Linearity follows from additive contributions at the amplitude level. Superposition is therefore emergent.It reflects coarse-graining, not a fundamental axiom. 3. Interference as Cross-Terms in Microstate Combinatorics Probabilities are quadratic because amplitudes sum before squaring.Cross-terms correspond exactly to interference effects. Interference is not assumed.It is a combinatorial consequence of microstate superposition. 4. Tensor Product Structure and the Origin of Entanglement Composite systems correspond to composite regions of the fabric.Global microstates factorize only if compatibility constraints separate. Entanglement arises when constraints couple subsystems.It is structural, not mysterious. 5. Decoherence from Phase Inaccessibility Observers access only coarse-grained variables.Rapidly varying microstate phases average out. Off-diagonal terms are dynamically suppressed.Decoherence emerges without postulation. 6. Measurement Collapse as Constraint Reduction Measurement is an interaction that reduces compatible microstates.The effective state updates accordingly. Collapse is not a new law.It is constraint updating under coarse-graining. 7. The Arrow of Time from Asymmetric State Counting Entropy is defined as the logarithm of compatible microstates.Local updates generically move toward larger compatible sets. Irreversibility is structural.It does not rely on subjective ignorance. 8. Coulomb Friction from Discrete Event Statistics Friction equals dissipated update energy per unit distance.The Coulomb law emerges from counting active interfacial events. The friction coefficient is not fundamental.It decomposes into microscopic event parameters. 9. Superlubricity as the Event-Free Limit Superlubricity occurs when locking events are suppressed.Dissipation is rare, not forbidden. This is a limit regime, not an exception.It confirms friction as emergent. 11. Conservation Laws from Update Invariance If the update rule is invariant under local transformations,conserved quantities emerge automatically. Noether’s theorem becomes emergent.Symmetry is a property of the update, not an axiom. 12. Classical Determinism from Phase Averaging At large scales, phase information becomes inaccessible.Only expectation values survive. Classical trajectories emerge statistically.Determinism is an effective limit. 13. Locality from Finite Update Propagation Updates propagate only through neighboring fabric nodes.Information speed is bounded. Causality is structural.Nonlocality is only apparent at coarse scales. 14. Emergent Speed Limit (c) The maximum update rate defines a universal speed.No signal can exceed it. The speed of light is not fundamental.It is a fabric update bound. 15. Metric Geometry from Update Delays Spatial curvature corresponds to spatially varying update rates.Time dilation is an update slowdown. Geometry is dynamical.The metric is emergent. 16. Gravity from Update Resistance Mass corresponds to structures resisting updates.Nearby updates slow down in response. Gravitational attraction emerges.No force field is postulated. 17. Equivalence Principle as a Structural Identity All stable update-resisting structures slow updates identically.Their response is universal. Inertial and gravitational mass coincide.Equivalence is automatic. 18. Quantization from Discrete Spectrum A finite update lattice admits discrete eigenmodes.Energy levels are quantized. Quantization is not imposed.Continuum is an approximation. 19. Noise Spectra (1/f) from Avalanche Dynamics Update cascades produce scale-free event statistics.Long-range correlations appear. 1/f noise is generic.It signals discrete underlying dynamics. 20. Criticality without Tuning The fabric naturally operates near instability thresholds.Small perturbations trigger cascades. Self-organized criticality emerges.No parameter tuning is required. 21. Thermodynamic Laws as Counting Theorems Temperature measures update activity density.Heat is disordered update flow. Thermodynamics is not independent.It is bookkeeping of updates. 22. Growth and Morphogenesis from Update Bias Directional update biases produce structured growth.Forms emerge without templates. Morphogenesis is dynamical.Shape follows update flow. 23. Fine-structure value and other unknow meaning The fine-structure value (~137) corresponds to the effective degeneracyof coherently contributing update configurations in an elementary coupling process.Proton–electron mass ratio the number of internal update cycles required to stabilize a topological vortex (proton), and a simple oscillatory update mode (electron). Mass measures cumulative resistance to updates. Lepton mass hierarchy Different stability thresholds of the same update pattern. Each mass corresponds to a different coherence regime.→ Families are metastable realizations of one structure. Reduced Planck constant Energy × time associated with one elementary update event. → ħ is the discrete unit of updating. Speed of light Maximum update propagation speed across the fabric. → c is the causal limit of the fabric. Bekenstein–Hawking entropy Number of blocked or saturated update configurations at the horizon. → An event horizon is an informational saturation boundary. Universal critical exponents They arise from the topology of update avalanches,independent of microscopic details. → Criticality is a natural fabric regime. Explanations Rarely Given in Existing Theories 1. Why probabilities are quadratic and not cubic or linear Most theories postulate P=∣ψ∣2P=|\psi|^2P=∣ψ∣2.Here it follows from coherent counting plus phase cancellation. The square is not a choice.It is the minimal stable coarse-graining. 2. Why quantum phases are physically unobservable but indispensable Phases affect interference but not outcomes directly.They label micro-configurations, not macrostates. Phase is structural memory.Measurement erases access to it. 3. Why constants are dimensionless and few Only ratios of update counts survive coarse-graining.Dimensional parameters wash out. Dimensionless constants are structural.They encode geometry, not dynamics. 4. Why “free parameters” cluster around simple numbers Many constants are order-one ratios or small inverses.They reflect combinatorial suppression. Fine tuning is apparent.It is state counting in disguise. 5. Why quantum randomness is stable and reproducible Randomness is not noise.It is deterministic micro-dynamics seen through coarse-graining. Statistics are invariant.Details are inaccessible. 6. Why identical particles are truly identical Microstates differ, macrostates do not.Labels are erased by update symmetry. Indistinguishability is enforced.It is not a postulate. 7. Why spin is quantized and half-integer values exist Spin counts topological winding in update cycles.Only discrete windings are stable. Half-integer spin reflects double-cover topology.No mystery remains. 8. Why gauge free
Preprint and supporting numerical material for the TGP emergent-quantum-mechanics paper, a companion to the TGP core paper (Zenodo DOI: 10.5281/zenodo.19670324). Starting from the substrate-field framework fixed by the core paper, we show that quantum mechanics is not fundamental but emergent: measurement, the Born rule, superposition, Bell-type entanglement, spin 1/2, spin-statistics, and decoherence all follow from the self-referential dynamics of the substrate field Φ. Key structural results: (i) ℏ = π·χ·A_tail derived from the scaling symmetry of the soliton ODE, with A(α)=A_0/√α fitted exponent −0.49999 to R²=0.9999999955; (ii) ℏ(Φ)=ℏ_0√(Φ_0/Φ) a position-dependent Planck constant, testable shift Δℏ/ℏ≈−3.5×10⁻¹⁰ near Earth; (iii) Δx·Δp ≥ ℏ from three independent derivations (oscillation period + Nyquist, Fisher information + Cramér-Rao, energy minimisation); (iv) Born rule from measurement asymmetry — detector back-reaction <ΔA_det²> ~ A_part^{2.028} with R²=0.99995, CV=2.3%; (v) CHSH = 2√2 exactly (Tsirelson bound) from a substrate phasor on the topological singlet π_3(S³)=ℤ, with E(a,b)=−cos(a−b) to 3.3×10⁻¹⁶, no-signaling verified; (vi) spin 1/2 from hedgehog-vielbein winding B=1 universal across g_0∈[0.3,0.95], analytic B=2∫_0^1 sin²(πu)du=1; (vii) Fermi-Dirac vs Bose-Einstein statistics from exchange phase (−1)^B, anyons in 2D from π_1(config)=ℤ; (viii) decoherence from three converging channels (ℏ(Φ) damping, non-linear mode mixing, environmental back-reaction) plus quantum Darwinism as geometric tail imprint on the substrate. The repository contains the paper source, compiled PDF, and the seven numerical folders that are directly cited: qm_foundations/, qm_measurement/, qm_superposition/, qm_entanglement/, qm_spin/, qm_statistics/, qm_decoherence/.
This four-part series proposes a mechanism-level, computable program for cross-scale unification based on an Elementary Energy-Quantum Sea (EP-sea): a collision-driven statistical background whose recursive phase transitions generate stable phase-membrane defects and their effective dynamics. Rather than treating geometry or symmetry groups as primitive, the series emphasizes a free-energy/response ledger and event-statistical closure that connect gravity, electromagnetism, nuclear interior phenomenology, and quantum measurement within a single effective-theory dictionary. Falsifiability pack (minimal must-hit constraints):(1) Double-slit visibility scaling under phase diffusion: η(t) ≈ exp[−γ_dec t] with leading-order γ_dec ∼ α_env Θ / ħ_eff.(2) Born statistics as hazard-rate counting: λ(x,t) = Γ(x)|ψ(x,t)|^2 and consistency with absorbing-potential norm loss.(3) Aharonov–Bohm phase calibration: Δφ_AB = (q/ħ_eff)∮A·dl, with q/ħ_eff fixed per tested species in laboratory regimes.(4) Spin-1/2 + Pauli coupling with tree-level g = 2 from SU(2) rotor (tethered-defect) + minimal coupling.(5) (Optional interface) Barrier crossover: activation exp(−ΔW/Θ) vs tunneling exp(−S_E/ħ_eff) with τ_* ~ ω_b^−1 at a quadratic barrier top. Key mechanisms / claims (overview):1) Gravity as a bulk–boundary free-energy response: mass is associated with a defect free-energy gap ΔF, inducing an effective potential Φ_eff through background stress/response gradients.2) Casimir and gravity as two limits of the same free-energy ledger: boundary-constrained fluctuations vs bulk gradients.3) Light speed vs sound speed: two mode identities. Sound is longitudinal (stiffness/inertia), while light is a transverse Maxwell-sector response with c^2 = (ε0 μ0)^−1 in vacuum-like regimes (c ≠ c_s), avoiding a macroscopic preferred frame.4) Operational charge and potentials: charge as boundary flux asymmetry; Aμ as effective response variables; AB phase as a topological consequence of minimal coupling.5) Quantum probability and measurement: |ψ|^2 is interpreted as event-density; measurement is modeled by thresholded absorption/nucleation with hazard Γ(x)|ψ|^2; Born statistics emerges as a frequency limit of Poisson/SDE event dynamics.6) “Collapse” is replaced by (i) absorbing norm loss and (ii) dynamical decoherence via phase diffusion.7) Spin-1/2 from tethered-defect topology (belt-trick): 2π rotation induces a sign change while 4π restores the full configuration (SU(2) double cover as a geometric constraint).8) Tree-level g = 2: once SU(2) rotor structure is in place, minimal coupling forces the Pauli term; deviations (g−2) are assigned to higher-order backreaction/noise-spectrum effects.9) Nuclear interior as a phase-transition landscape: a core–shell (χ1, χ2) closure yields two-scale kernels (e.g., double-Yukawa structure), saturation, and R ∝ A^(1/3), with decays treated as rare events.10) Unifying tunneling and activation: WKB and Kramers/Freidlin limits are related at the exponent level via intrinsic barrier timescales.11) Polarization and Malus’s law: polarization is a geometric projection of two transverse degrees of freedom; Malus’s law follows from deterministic projection plus intensity ∝ amplitude^2. Series structure:Paper I: Gravity and bulk–boundary free-energy closure (including Casimir interface).Paper II: Electromagnetism as transverse response; constitutive closure; why c differs from c_s.Paper III: Nuclear interior (tier-2) closure; two-scale kernels; saturation and rare-event decays.Paper IV: Quantum interface: ψ/|ψ|^2 closure, hazard-rate measurement, AB phase, spin/Pauli sector, decoherence scaling. Status and scope:This release is intended as a coherent, test-driven framework (effective-theory level) with explicit interfaces for numerical templates and data confrontation. Precision QED/GR-level matching is treated as follow-up work guided by the falsifiability pack above.
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.
Papers I and II derived gravitational geometry and second-order dynamics from finite accessibility without invoking measurement, observation protocols, or collapse as primitive notions. This paper extends the operational program to the measurement sector, deriving the structure of record formation and probabilistic weights from the same finite accessibility and compositional constraints. Finite accessibility restricts admissible operations on a subsystem to a proper operator subalgebra. Because bounded certification excludes operators requiring nonlocal coordination or arbitrarily fine phase resolution, the accessible algebra is strictly smaller than the full operator algebra. When a system becomes correlated with orthogonal environment sectors, off-diagonal operators between those sectors fall outside the accessible algebra. The reduced state is therefore indistinguishable, under all admissible operations, from its diagonal ensemble. Definiteness emerges as an algebraic consequence of sector restriction, not as a collapse postulate. The weights of these sectors are fixed uniquely by compositional consistency. Phase invariance, multiplicativity under independent refinement, and stability under record coarse-graining constrain any admissible assignment of sector weights to satisfy a functional equation whose only continuous, non-contextual solution is quadratic in amplitude. The Born rule therefore arises as the unique invariant measure compatible with finite accessibility. Irreversibility follows from action scaling: reversal requires operators outside the admissible tensor algebra and incurs action cost proportional to the number of correlated subsystems. Since admissible operations are bounded by the action gap, macroscopic record reversal is excluded. Measurement introduces no new physical primitives beyond finite accessibility, locality, and compositional stability. Record formation, classical definiteness, and Born probabilities are structural consequences of algebraic restriction under finite accessibility. **This paper is part of the Finite Accessibility series, which reconstructs gravitational geometry, quantum dynamics, gauge structure, cosmological perturbation theory, and the dark sector from a single operational primitive: a finite certification ceiling ω* governing the maximum rate at which physical relations can be maintained across extended domains.** **Introductory papers** - Geometric Limits to Coherent Timekeeping in Extended Reference Frames — https://doi.org/10.5281/zenodo.18824556 - Accessibility Selection Law (ASL) — https://doi.org/10.5281/zenodo.18824649 - Phase Boundary Gravity — https://doi.org/10.5281/zenodo.18824707 **Pillars (numbered I–IV)** - Paper I: Deriving Gravitational Geometry and Schrödinger Dynamics from Finite Accessibility — https://doi.org/10.5281/zenodo.18601600 - Paper II: Deriving Inertia, Motion, and Wave Propagation from Finite Accessibility — https://doi.org/10.5281/zenodo.18625174 - Paper III: Accessible State Spaces, Record Stability, and the Emergence of Quantum Statistics — https://doi.org/10.5281/zenodo.18625192 - Paper IV: Frame Dissolution, the Certification Ceiling, and Observable Deformations — https://doi.org/10.5281/zenodo.18654458 **Extension papers** - Dark Matter from Finite Accessibility — https://doi.org/10.5281/zenodo.18654433 - Dark Energy from Finite Accessibility — https://doi.org/10.5281/zenodo.18654523 - No-Go Theorems and the Admissibility Corridor — https://doi.org/10.5281/zenodo.18654493 - Gauge Structure from Finite Accessibility — https://doi.org/10.5281/zenodo.18928152 - Perturbation Theory from Finite Accessibility — https://doi.org/10.5281/zenodo.18928213 **Experimental paper** - Experimental Probes of Finite Accessibility: GW Dispersion and Vacuum Path Independence — https://doi.org/10.5281/zenodo.18931126 **Keywords:** Born rule derivation, measurement problem, quantum statistics, finite accessibility, operator subalgebra, record formation, definiteness emergence, wave function collapse, compositional consistency, accessible state spaces, action gap, irreversibility, sector restriction, quantum foundations, algebraic quantum mechanics, decoherence