Quantum phenomenology from a classical integer automaton: an emergent Schrödinger sector, exact exchange statistics, CHSH violation without randomness, frame-independent Bell statistics, a surviving isotropy no-go, and an engine-measured Planck-scale prediction, and obstruction theorems for dynamic geometry, and a bit-reversible clock field with a critical depth and a lapse theorem

Abstract

New in v6 — a canonical, bit-reversible clock field (section 4c). The scalar lapse field is made itself bit-reversible with a clock-dependent stiffness by a canonical (ψ, p) transformation with an integer band table (0 register mismatches for matter + field at 64³). A pre-registered 3D run passes every gate for a prescribed static source (stationary Hamiltonian well, energy drift 0.06 %); an adversarial audit showed that the 1/r² and exterior-mass gates passed at 64³ only by a torus cancellation, so those rows stay at pilot level. The field has a compactness limit (fold at d_c ≈ 0.17–0.19; the core freezes above it), a fixed cosmological constant is shown not to be self-consistent under such a local change of state, and a bounded background term restores a bounded frozen core (0.15 % of the box). An exact reversible moving charge is constructed in the integer Maxwell core (Esirkepov deposition, residuals exactly 0, acceleration 1.03 of prediction, measured lattice pinning). A lapse theorem — n^{(m+1)/2} harmonic outside sources — fixes the second post-Newtonian coefficient of the static field, β = (1 − q_hop)/4, which no choice of q_hop brings to 1; the matter kinematics is measured to be geodesic in the lapse metric with γ_matter = 0 (κ ≈ −2), so the second post-Newtonian order conflicts with solar-system bounds under the stated identification (perihelion 0.75 GR, η = −4). Two named conflicts are recorded. Twenty-three self-corrections on record; three were caught the same day. Section 4d adds a first-post-Newtonian redesign: exponential or isotropic-Schwarzschild clocks, constant stiffness, n⁴ kinetics and matter reading n as proper time and n² as its speed limit give β = γ = 1 with deep wells stationary (energy drift 0.001 %); a red-team panel shows a complete 1PN theory still needs a gravito-vector sector (α₁ = −8 without it) and a Komar source. Twenty-four self-corrections on record. What this is. A single bit-exactly reversible, integer-valued lattice automaton in which a broad range of quantum phenomenology arises as measurable, checksum-reproducible behaviour. Evolution is a permutation of integer microstates, not approximate numerical integration — which is why the gates below can be bit-exact rather than statistical. This is an existence result strengthened by a survived no-go and one falsifiable prediction; it is not a claim that our universe is this automaton. Quantum sector. An emergent Schrödinger sector (envelope bridge with effective mass fixed by the curvature of the exact discrete dispersion; two microscopically different substrates collapsing onto the single universal spreading law √(1+τ²)); deterministic double-slit single-object interference (p<10⁻⁴) with trajectories matching the weak-measurement reconstruction of Kocsis et al. 2011; CHSH = 2.79 from position-valued outcomes with a no-signalling control; a measured branch-coherence decay law (R²=0.997); exact exchange statistics, with Pauli exclusion holding as a bit-conserved invariant for all time; a quantum eraser with literal bit-reversible un-measurement; and partial coarse-grained Born relaxation with exact Loschmidt reversal. Where the quantum–classical boundary lies. The two-particle interaction is substrate-derived (cross-/self-phase ratio 2.31 against a parameter-free prediction of 2), so both terms of the two-particle Hamiltonian are emergent — but the reconstructed state stays Schmidt rank 1 where the L² reference reaches K=4.9. The substrate derives the Hamiltonian, not the L² state space; entanglement is carried, not derived. Spin is implemented. Lorentz structure. Velocity anisotropy vanishes as k² in the infrared with first-principles coefficients in 2D and 3D (≈1%), so rotational isotropy emerges as an infrared fixed point. Bell statistics are foliation-order-independent for causally separated measurement operations (|ΔS|<0.005, S>2 in every ordering) while 41% of individual trajectory pairs flip — a preferred foliation that trajectories know about and no measurement reveals, with the residual order-dependence scaling away as A⁻⁰⋅⁹³. Full boost covariance of states and fields remains open. New in v4 — a dynamic-geometry sector, reported with its obstructions. Six pre-registered pilots ask whether matter can make the local-time geometry and be acted on by it. Three results are derived and measured: (i) an exactly conserved geometric ledger forces the static response to be contact-only — the lattice counterpart of Gauss’s law on a compact space — so no two-body force is possible in that class (hypotheses stated explicitly; this is a no-go for a class, not for discrete gravity in general); (ii) a source entering the first-order equation yields a flat plateau in 1D (depth 46.3 against the parameter-free 45.25, constant in time, exactly zero outside the causal cone), 1/t dilution in 2D and nothing in 3D, so transport cannot build a static sourced field in any dimension; (iii) four independently probed force channels are blocked, the only directional one carrying the winding (electromagnetic) sign. What the ledger does permit. Relaxing conservation of the free field while keeping the total exact restores long range without losing bit-exactness (far/source 0.98–1.06 against ≈0 for contact), and a carry-based binding rule makes mass = number of bound tokens a 1% measurement rather than a framing (rate drift 1.3% while the object’s volume changed 3.05×). The substance carries a second bitwise-exact Gauss law (div e − ΔQ = 0 over 2×10⁴ ticks, flux = enclosed charge exactly) whose charge is the conserved energy. Letting the clocks read it closes the matter→geometry→matter loop kinematically: a body slows its own clock by 30%, matching the clock-rate law to 1.3%, with a causal onset. In 2D the clock field refracts a mobile probe, and a binary, untunable band-edge prediction tabulated before the run is confirmed at both clock ratios (transmission 0.008–0.046 in the predicted reflecting region against 0.58–0.93 just below it); the quantitative refraction law is closed as unproven after two attempts, per its own pre-registration. None of this is general relativity: the geometry is a scalar lapse, no attraction is measured, and the obstruction theorem forbids one in the conserving class. Falsifiable prediction, factor-corrected in v4. The quadratically Planck-suppressed photon dispersion coefficient is measured on the engine rather than derived from a formula: ξiso(3D) = 0.024299 ± 1.1×10⁻⁴ against the exact rational 187/7680 (0.20%), with the subluminal sign resolved from zero by 221 numerical standard errors of the fit. Three coefficients are now stated separately: phase (ξ), group velocity (3ξ) and the Lang convention, δγ,2 = −2ξ/EPl² = −3.27×10⁻⁴⁰ GeV⁻². Our earlier headline quoted the phase coefficient under the Lang symbol; that factor of 2 is the nineteenth self-correction on record and was found by an external automated review, credited as such. The prediction sits ~30× below the older model-dependent Lang et al. (2017) bound, but is in tension (~3.3×) with the conditional Pierre Auger (2022) limit for a source scenario with a subdominant proton component — so it is now decidable by UHECR composition measurements, with a threshold signature at E*≈2 EeV. Its fate depends on the proton fraction at the highest energies, the source model, and the assumption of standard interaction-vertex kinematics, which this substrate does not derive. Methodology as a co-equal contribution. Criteria are committed to version control before each run; the commit hash is the timestamp. Nineteen headline results have been retracted or corrected when the project’s own tests exposed artefacts — four in this version. The archive ships the pre-registration designs themselves (specs/), including the correction blocks for changes that failed: a coupling measured to pump energy and reported as “measured, not validated”; a gate left failed on a statistic carrying 0.4% of the signal power; a predicted field inversion that did not occur; and three measuring instruments retired after being shown ill-posed. v4 also discloses and fixes a reproducibility defect in the v3 archive: argparse defaults differed from the published configuration and one report JSON was omitted, so exact command lines are now given verbatim in the README. Prior art. Individual phenomena have known precedents — Philippidis–Dewdney–Hiley 1979 (double-slit Bohmian trajectories); Valentini–Westman 2005 (Born relaxation); Visscher 1991 (the update scheme), with the integer/floor-reversible lifting already noted by Fredkin 1999 and Martin-Delgado 2004; Dürr et al. 1999 (a preferred foliation hidden from equilibrium statistics) — and are framed accordingly. The contribution is their tested, validated realization on one exactly reproducible integer substrate, several new exact constructions, the obstruction results above, and the pre-registered, falsification-first methodology. Produced in an AI-assisted workflow (Anthropic Claude) under the author’s direction.