Derivation Audit and Canonical Equation Registry for the Lattice Field Medium Framework

This paper presents a systematic derivation audit and canonical equation registry for the Lattice Field Medium (LFM) framework. Starting from a single governing wave equation,

∂2E∂t2=c2∇2E−χ(x,t)2E,\frac{\partial^2 E}{\partial t^2} = c^2 \nabla^2 E - \chi(x,t)^2 E,∂t2∂2E=c2∇2E−χ(x,t)2E,

we examine every equation appearing across a paper LFM corpus and classify each according to its epistemic status: DERIVED, LIMIT, PHENOM, or EXTERNAL.

Under a frozen Tier-1 assumption set, permitting only algebraic manipulation, calculus, Fourier methods, boundary conditions, and explicitly named limits, we provide complete step-by-step derivations for all equations that follow from the governing equation. Where derivations fail, we identify and document the precise mathematical obstructions, rather than appealing to phenomenology or analogy.

The audit reveals that:

• 22 equations follow exactly from the governing equation (DERIVED),

• 4 equations emerge only in explicitly stated limiting regimes (LIMIT),

• 8 equations remain phenomenological fits to data (PHENOM),

• 2 equations require external physics not contained in the framework (EXTERNAL).

Key results that emerge as effective descriptions include the dispersion relation, Hamiltonian structure, energy conservation, electromagnetic analogues via Helmholtz decomposition, bound-state quantization under confinement, ray dynamics in χ-gradients, and cosmological scaling from χ-relaxation. The paper also documents unresolved issues, including exponent mismatches in velocity mappings and the phenomenological status of several commonly used χ-profiles.

This work does not propose new physics. Its purpose is to establish a transparent epistemic foundation for the LFM program by clearly separating what is derived from what is assumed. The resulting registry provides a reproducible reference against which all future LFM developments can be audited and classified.