The Lattice Field Medium: A Computational Substrate for Emergent Physics

The Lattice Field Medium

Version: 32.0 (May 2026)

Overview

The Lattice Field Medium (LFM) is a discrete substrate framework in which spacetime emerges from a simple cubic lattice carrying two coupled fields: E in R and chi in R. Complex phase structure Psi in C and color structure Psi_a in C^3 are emergent observational extensions, not axioms.

Core Equations

GOV-01: Causal Core

d2E/dt2 = c2 nabla2 E - chi2 E

GOV-02: chi Medium Equation

d2chi/dt2 = c2 nabla2 chi
- (kappa/chi0) chi (sum_a |Psi_a|2 - E0^2)
- 4 lambda_H chi (chi2 - chi0^2)

Canonical Constants

chi0 = 19
kappa = 1/63
lambda_H = 4/31

Canonical Mixed Stencil (v32.0)

GOV-01 canonical 3D propagation stencil: 27-point (faces + edges + corners).

GOV-02 canonical 3D vacuum/substrate stencil: 19-point (center + 6 faces + 12 edges), with corners excluded in the chi vacuum operator.

GOV-02 19-point form:

nabla2 chi = (1/dx2) [ (1/3) sum_faces + (1/6) sum_edges - 4 chi_center ]

Lattice Field Medium GOV-01 and GOV-02 equations in Python form:

def gov01_step(E_now, E_prev, chi_now, lap_E_27pt, dt, c=1.0):
return (
2.0 * E_now
- E_prev
+ dt2 * (c2 * lap_E_27pt - chi_now**2 * E_now)
)

def gov02_step(chi_now, chi_prev, energy_now, lap_chi_19pt, dt,
c=1.0, kappa=1/63, chi0=19.0, lambda_H=4/31, E0_sq=0.0):
return (
2.0 * chi_now
- chi_prev
+ dt2 * (
c2 * lap_chi_19pt
- (kappa / chi0) * chi_now * (energy_now - E0_sq)
- 4.0 * lambda_H * chi_now * (chi_now2 - chi02)
)
)

v32.0 Update: GOV-02 Derived Observables

This release adds Section XIV: GOV-02 Derived Observables and Appendix E: chi Memory / Relaxation / Screening Observables.

GPU validation:
64^3 lattice
NVIDIA RTX 4060
2026-05-04

Central GOV-02 Result

omega_chi = sqrt(8 lambda_H) chi0 = 19.304

l_chi = c / omega_chi
l_chi = 1 / 19.304
l_chi = 0.0518 lattice units

The chi screening length is sub-lattice. Therefore the quasi-static chi limit is not merely an approximation at astrophysical scales. It is forced by the canonical constants.

Key GOV-02 Observable

M(x,t) = chi0 - chi(x,t)

d2M/dt2 - c2 nabla2 M + omega_chi2 M = kappa (|E|2 - E0^2)

Quasi-Static Memory Limit

M_qs asymp [kappa / (8 lambda_H chi0^2)] (|E|2 - E0^2)

Measured coefficient:
4.26 x 10^-5

Confirmed GOV-02 Effects

• chi ringing
• chi relaxation oscillation
• moving chi wake anisotropy
• rotating-source chi memory
• galactic chi-memory extension
• conservative chi decay
• chi wake persistence
• chi correlation structure

Wake anisotropy ratio: 2.308
chi decay: gamma asymp 0

Non-Detections

The following remained below float32 precision and are reported as numerical non-detections, not falsifications:

• chi-Bremsstrahlung
• refractive delay
• two-source superposition

Interpretation

GOV-01 governs excitations. GOV-02 governs the medium itself. The chi field behaves as substrate stiffness, geometry memory, relaxation medium, screening field, refractive structure, and causal propagation regulator.