LFM Equation Reference
Two coupled wave equations on a discrete lattice, parameterized by χ₀ = 19.
Framework Overview
LFM models spacetime as a discrete lattice where each point stores two values: a wave amplitude Ψ and a local stiffness parameter χ.
Each point updates using only nearest-neighbor values. Standard physics results (gravity, electromagnetism, bound states) emerge from the coupled dynamics of these two fields.
The framework has been tested against 105 validation cases across 7 physics domains.
Background Parameter: χ₀ = 19
The vacuum value of χ is determined by fitting to the CMB spectral index. From this single parameter, a family of formulas for fundamental constants can be constructed.
Governing Equations
Two fundamental wave equations (GOV-01, GOV-02) plus three simplifications for specific regimes.
GOV-01: Lattice Wave Equation (Fundamental)
The lattice wave equation with position-dependent mass χ(x,t). Waves propagate through a medium where local "stiffness" χ affects behavior. The Dirac equation (spin-½) emerges from its Lorentz symmetry in the continuum limit.
GOV-01 Level 0: Real Scalar Field (Gravity Only)
Simplified form using real-valued E field. Valid when phase is uniform and only gravity is needed.
GOV-02: χ Field Equation (LFM v14)
The geometry itself ripples like a wave. Where there's concentrated energy, χ drops—creating what we experience as a gravitational well. The Mexican hat V(χ)=λ_H(χ²−χ₀²)² makes χ₀=19 a dynamical attractor.
GOV-03: Single-Equation Form (Simplification)
χ responds to time-averaged energy density with memory parameter τ. The memory creates persistent gravitational effects.
GOV-04: Poisson Limit (Quasi-Static)
Static limit of GOV-02. Similar structure to Poisson equation in Newtonian gravity.
LFM-Specific Formulations
LFM-specific formulations and applications. Many build on established physics concepts (properly attributed below). GOV-02 and the χ₀=19 predictions are LFM's contributions.
Chi-Inversion Formula
Invert the velocity-χ relationship to reconstruct χ profiles from rotation curves. LFM's specific formulation for the χ field.
Phase-Charge Encoding
In LFM, charge is encoded as wave phase. This is the LFM implementation of U(1) gauge symmetry, which underlies electromagnetism in quantum field theory.
χ Memory Mechanism
In LFM, the χ field retains memory of past energy distributions. This is LFM's approach to explaining dark matter phenomena without particles.
Mexican Hat Self-Interaction (GOV-02 v13+)
The GOV-02 self-interaction makes χ₀=19 a dynamical attractor (V′′(χ₀)=373). Z₂ symmetry gives a second stable vacuum at −χ₀=−19 — black hole interiors settle there. The old ad-hoc floor term λ(−χ)³Θ(−χ) is permanently retired; the Mexican hat provides superior physics with a geometric derivation.
LFM's RAR Implementation
LFM's derivation of the Radial Acceleration Relation. The RAR itself was discovered observationally by McGaugh et al. (2016). LFM claims to derive it from first principles.
Coulomb Force in LFM
LFM's derivation of Coulomb force from wave interference. Interference forces are standard wave mechanics; LFM shows how 1/r² emerges in its framework.
Stability Analysis
Standard physics analysis applied to LFM. These are well-known theorems (Ostrogradsky, causality bounds, Hamiltonian analysis) confirming mathematical consistency.
No Ghost Modes
LFM's Hamiltonians are positive-definite. Ghost analysis is standard in modified gravity theories.
Ostrogradsky Stability
Ostrogradsky's theorem (1850) states theories with >2 time derivatives are unstable. GOV-01 and GOV-02 have exactly 2nd-order derivatives.
Causality Preservation
Group velocity in LFM never exceeds c. Causality analysis is standard relativistic physics.
Massless Graviton (Theorem)
In vacuum, GOV-02 becomes a massless wave equation. Gravitons have zero mass, consistent with LIGO bound (m < 1.2×10⁻²² eV).
Derived Equations by Category
All of the following emerge from GOV-01 + GOV-02. Click a category to explore the equations.
Published Documentation
The complete LFM framework is documented and archived for reference and citation.
See It In Action
These aren't just equations on paper — they're running in real-time simulations you can explore.