IndisputableMonolith.Foundation.SimplicialLedger.LorentzEmergence
The module defines the axis dispersion relation of the cubic-lattice Laplacian along with isotropy checks and bounding lemmas. Researchers deriving continuum limits from discrete spacetime models in Recognition Science cite these objects. The module consists of definitions and direct trigonometric bounds with no complex proof structure.
claimThe axis dispersion is $\omega_{\rm axis}(a,k)=\frac{2}{a^2}(1-\cos(ak))$. Companion objects comprise the full dispersion function, the continuum isotropy predicate, and non-negativity plus upper-bound lemmas.
background
The module resides in Foundation.SimplicialLedger.LorentzEmergence and imports Mathlib real-analysis libraries together with IndisputableMonolith.Constants, whose sole content is the time quantum $\tau_0=1$ tick. The supplied DOC_COMMENT identifies the core object as the dispersion relation of the cubic-lattice Laplacian at a single axis. The local setting is a discrete cubic lattice whose Laplacian must recover isotropic linear dispersion in the long-wavelength limit.
proof idea
this is a definition module, no proofs
why it matters in Recognition Science
The definitions and bounds feed the LorentzEmergenceCert sibling and supply the lattice dispersion needed for the emergence of relativistic structure. They instantiate the discrete model whose continuum limit is required for the D=3 spatial dimensions in the unified forcing chain.
scope and limits
- Does not derive the full Lorentz group from the lattice.
- Does not incorporate the phi-ladder or mass formula.
- Does not address quantum corrections or curved backgrounds.
- Does not verify the numerical alpha band.
depends on (1)
declarations in this module (11)
-
def
axis_dispersion -
def
dispersion -
def
continuum_isotropic -
theorem
axis_dispersion_upper_bound -
theorem
axis_dispersion_nonneg -
theorem
dispersion_upper_bound_by_isotropic -
theorem
dispersion_nonneg -
theorem
isotropic_envelope_rotation_invariant -
theorem
axis_dispersion_sandwich -
structure
LorentzEmergenceCert -
def
lorentzEmergenceCert