Paper I — Geometry First

Discrete-spacetime substrate for the A=1 Discrete Causal Lattice series: a bipartite octahedral causal lattice yielding emergent Lorentz invariance and the Dirac equation as O_h-averaged consequences.

Abstract

We propose that conservation of probability — the single axiom \mathcal{A}=1 — is sufficient to derive the principal results of quantum mechanics without additional postulates. On a bipartite three-dimensional causal lattice \mathbb{T}_\diamond, each particle is represented as a causal session: an independent phase oscillator that propagates amplitude across the lattice subject to |\psi_R|^2 + |\psi_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 we derive:

(i) the Dirac equation in the continuum limit a \to 0, with the gamma matrices identified as geometric encodings of the RGB/CMY bipartite sublattice geometry; rotational invariance is recovered on O_h-averaged observables (the same mechanism by which staggered fermions recover Lorentz invariance in lattice QCD), and the residual lattice birefringence aligned with the optical axis \mathbf{V}_1+\mathbf{V}_2+\mathbf{V}_3 = (1,1,-1) is a falsifiable prediction with closed-form coefficients;

(ii) wave–particle duality and the Born rule as consequences of amplitude redistribution under \mathcal{A}=1;

(iii) quantization of atomic orbitals as Arnold tongue attractors of the two-body phase dynamics, recovering the hydrogen Bohr radius to four significant figures 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/r^2 force law with a falsifiable discrete correction at short range;

(v) photon emission as a structural consequence of \mathcal{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; a first numerical implementation (exp_19c) preserves per-session unitarity at all swept emission rates, and a controlled three-arm operator comparison (exp_20) confirms a pointwise unitary beam splitter as the joint-\mathcal{A}=1-preserving emission operator (joint \mathcal{A}=1 preserved to machine precision 8.9 \times 10^{-16}); orbital settling remains open, traced to a long-horizon two-body baseline instability isolated in exp_12b (both rows PART);

(vi) the structural form of the induced gauge action (Sakharov / Zeldovich), whose leading-order term has the Maxwell form on O_h-averaged observables together with a closed-form gauge-sector birefringence sharing the same optical axis (1,1,-1) as the kinematic-sector prediction (numerical 1/g^2 prefactor and the empirical test of the birefringence remain open); the predicted concordance of kinematic, gauge, gravitational, and atomic-clock channels along this single axis is the framework’s strongest falsifiable prediction (P9).

Energy conservation and momentum conservation are not independent inputs — they are consequences of \mathcal{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 (the system audit table records the status of each claim); the framework’s remaining derivations — black-hole thermodynamics among them — are established analytically under explicitly stated modeling assumptions and flagged as such where they appear.

The framework suggests a structural reinterpretation of the Standard Model’s gauge degrees of freedom as the minimal accounting required to maintain \mathcal{A}=1 on the bipartite lattice geometry: the particle content is reframed as the catalog of probability imbalances the lattice generates and the session types required to resolve them. Whether the full Standard Model gauge group \mathrm{SU}(3) \times \mathrm{SU}(2) \times \mathrm{U}(1) is the automorphism group of (\mathbb{T}_\diamond, \mathcal{A}=1) is a well-posed finite-dimensional Lie-algebra question, deferred to a companion paper.

What this paper establishes

  • Substrate. Physics is built on a bipartite three-dimensional octahedral lattice \mathbb{T}_\diamond whose six basis vectors split into two chirally-opposite sublattices — RGB on even ticks, CMY on odd ticks. The single ontological commitment is the conservation invariant \mathcal{A}=1: each particle is a causal session, a U(1) phase oscillator carrying a two-component amplitude (\psi_R, \psi_L) that is renormalised to |\psi_R|^2 + |\psi_L|^2 = 1 at every tick. The discrete bipartite tick rule — a chirality-alternating hop between the RGB and CMY sublattices — is the only dynamical law. Energy and momentum conservation, the Born rule, and unitary Hilbert-space evolution are derived from this one constraint, not assumed.

  • Lorentz invariance. The continuum limit a \to 0 of the bipartite tick rule is the Dirac equation, with the gamma matrices reading directly off the RGB/CMY sublattice geometry. Rotational — and hence Lorentz — invariance is not imposed: it is recovered on O_h-averaged observables, the directional cross-terms of the octahedral lattice cancelling in the average exactly as staggered fermions recover Lorentz invariance in lattice QCD. The residual anisotropy survives as a closed-form lattice birefringence aligned with the optical axis \mathbf{V}_1+\mathbf{V}_2+\mathbf{V}_3=(1,1,-1) — a falsifiable departure from exact continuum symmetry.

Methodological role

First step in the methodological arc: geometry first → geometry forces physics → geometry axiomatizes physics. Paper I is the substrate-priority paper — it fixes the lattice, the conservation axiom, and the tick rule, then shows that the principal results of quantum mechanics, special relativity, and Newtonian gravity follow as consequences rather than inputs. Everything later in the series builds on this substrate: Paper II asks whether the Standard Model gauge group is the lattice’s automorphism group; Paper III turns the clock-density picture of gravity into a falsifiable astrophysical prediction. Paper I is the foundation those papers stand on, and the audit table here is the authority all downstream claims are reconciled against.

Audit-row impact

See the Claim map for a one-page, reviewer-facing table of every claim below — proven vs. conjectured, with audit tags — across Papers I and II.

Rows that reach PASS in this paper, grouped by sector (each backed by a no-free-parameter experiment or an analytic derivation):

  • Kinematics. Speed-of-light limit c=1 (exp_00); inertial persistence (exp_01); 1/r^2 discrete path-count correction (exp_06); the Dirac equation and spin-\tfrac{1}{2} (derived analytically); the equivalence principle m_g = m_i (exp_01, exp_02).
  • Quantum phenomena. Interference (exp_03); decoherence (exp_04); measurement / irreversibility (exp_05); the Born rule (exp_03–exp_05); mass quantization as Arnold-tongue locking and photon dispersion (exp_09).
  • Gravity. Gravitational attraction (exp_02); gravitational time dilation (exp_02, exp_07); clock-density conservation / continuity equation (exp_07).
  • Electromagnetism. EM-vs-gravity curl-vs-div force geometry and photon emission at orbit lock-in (exp_08).
  • Hydrogen and atomic physics. Hydrogen E_n \sim -1/n^2 (exp_10); spontaneous quantization at n=1 (exp_11); two-body hydrogen to four significant figures, k_\text{min}=0.0970 vs k_\text{Bohr}=0.0971 (exp_12); bound state requires an active proton (exp_11 n{=}2, exp_12); three-body helium-like system (exp_13); two-electron helium (exp_14).

Rows that are PART (mechanism established, full numerical or long-horizon confirmation open): black-hole event horizon + Hawking temperature and Bekenstein–Hawking entropy (analytic, under stated modeling assumptions); photon emission as an \mathcal{A}=1 necessity and the emission-operator joint-\mathcal{A}=1 beam splitter (exp_19c, exp_20); the induced gauge action in Sakharov form; two-body long-horizon stability and its escape mechanism (exp_12b, exp_12c); lock-in grid-independence (exp_12d_tight); the proton-mass lower bound (exp_16).

Rows that are STUB (predicted with closed-form structure, not yet tested): kinematic lattice birefringence; the Zitterbewegung-source dual hypothesis; gauge-sector birefringence; the Standard-Model-gauge-group automorphism conjecture; and the multi-channel concordance prediction (P9) along the (1,1,-1) optical axis.

What changed in v1.01 vs v1.0

v1.01 is a metadata and front-matter release (deposited 2026-05-19; v1.0 deposited 2026-05-07). It brings the title-page author block into alignment with the project domain geometryinducedphysics.org and reorders the front matter so the abstract follows immediately after the title. The derivations, experiments, audit table, and results are unchanged from v1.0; the v1.01 numbering signals the metadata scale, and parallels Paper II’s own v1.01 metadata release.

Open questions deferred downstream

  • Standard Model gauge group as an automorphism → Paper II. Whether \mathrm{Aut}(\mathbb{T}_\diamond, \mathcal{A}=1) equals SU(3) \times SU(2) \times U(1) (times the Lorentz factor) is posed here as a well-defined finite-dimensional Lie-algebra question and answered in Geometry Forces Physics.
  • Quantum Roche limit / tidal ionization → Paper III. The clock-density picture of gravity opens a third (gradient-detuning) ionization channel and an atomic ISCO; developed in Tidal Ionization and the Quantum Roche Limit (Paper III, in preparation).
  • Gauge-coupling prefactor 1/g^2 → planned discrete-probability paper. The induced gauge action’s Maxwell form is established but its numerical prefactor is open; closure is the centerpiece of the planned discrete-probability companion.
  • Long-horizon two-body settling. Orbital lock-in survives only on a finite-volume grid; the bare two-body baseline escapes (exp_12b, exp_12c). Resolving genuine long-horizon settling — distinct from the short-run resonance match — remains open.

Citation and version

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