IndisputableMonolith.Foundation.LatticeIsotropyBound
The LatticeIsotropyBound module assembles trigonometric inequalities and lattice non-negativity results to certify isotropy bounds for 3D discrete structures in Recognition Science. Researchers deriving continuum limits from the phi-ladder or justifying D=3 spatial isotropy would cite it when building lattice approximations. The module proceeds by chaining supporting lemmas on cosine bounds and dispersion into a final certification object.
claimThe module establishes $1 - \cos \theta \geq 0$, $1 - \cos \theta \leq 2$, bounded lattice dispersion, non-negativity on the 3D lattice, and the certificate LatticeIsotropyCert that the lattice satisfies the isotropy bound required for the Recognition framework.
background
The module resides in the Foundation domain and imports only Mathlib. It introduces the lemmas one_minus_cos_nonneg and one_minus_cos_le_two for elementary bounds on the cosine function, lattice_dispersion_bounded to control dispersion, lattice_3d_nonneg to guarantee non-negativity in three dimensions, and the main object LatticeIsotropyCert together with its constructor latticeIsotropyCert.
proof idea
The module collects a sequence of elementary lemmas on trigonometric inequalities followed by lattice-specific non-negativity and boundedness statements, terminating in the construction of the LatticeIsotropyCert certificate.
why it matters in Recognition Science
The module supplies the isotropy bound needed for lattice models that feed into the unified forcing chain (T0-T8) and the derivation of D=3 together with the alpha band. It closes a scaffolding step for any downstream result that assumes lattice isotropy when passing from the J-cost functional equation to physical constants.