Paper I — Geometry First (v1.0)

First stable release of the substrate-priority paper of the A=1 Discrete Causal Lattice series, deposited to Zenodo on 2026-05-07.

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Paper I of the A=1 Discrete Causal Lattice series — emergent Lorentz invariance, the Dirac equation, and the Born rule from a single conservation law on a discrete-spacetime causal lattice.
Author

Jack D. Menendez

Published

May 7, 2026

Paper I (Geometry First) is on Zenodo with DOI 10.5281/zenodo.20078529. It is the substrate-priority paper of the A=1 Discrete Causal Lattice series: it fixes the lattice, the conservation axiom, and the discrete tick rule, then shows that the principal results of quantum mechanics, special relativity, and Newtonian gravity follow as consequences rather than inputs.

From the single axiom \mathcal{A}=1 — conservation of probability — the paper recovers the Dirac equation in the continuum limit (with the gamma matrices reading directly off the bipartite RGB/CMY sublattice geometry), the Born rule and wave–particle duality, quantization of atomic orbitals as Arnold-tongue attractors that reproduce the hydrogen Bohr radius to four significant figures with no fitting, and gravity as a clock-density gradient that reproduces the 1/r^2 force law. Lorentz invariance is not imposed: rotational symmetry is recovered on O_h-averaged observables, the directional cross-terms of the octahedral lattice cancelling exactly as staggered fermions recover it in lattice QCD. Energy and momentum conservation are outputs of \mathcal{A}=1 on closed session sets, not separate postulates.

The numerically tractable claims are backed by a suite of more than fifteen experiments with no free parameters beyond the Planck-scale lattice spacing; the audit table on the landing page records the PASS / PART / STUB status of each. The framework’s sharpest falsifiable prediction is a residual lattice birefringence aligned with the optical axis (1,1,-1) — a closed-form departure from exact continuum symmetry.

Paper I is the foundation the rest of the series stands on: Paper II takes up its central open conjecture (is the Standard Model gauge group the lattice’s automorphism group?), and the planned Paper III turns the clock-density picture of gravity into a falsifiable astrophysical prediction.

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