IndisputableMonolith.Foundation.LatticeIsotropyBound
LatticeIsotropyBound supplies supporting lemmas on trigonometric bounds and lattice dispersion to certify isotropy for three-dimensional lattices inside the Recognition Science foundation. Researchers tracing the emergence of spatial dimension D=3 from the unified forcing chain cite these results. The module proceeds by direct application of elementary inequalities for the cosine function followed by non-negativity arguments in the 3D case.
claimThe module establishes $1 - \cos x \geq 0$, $1 - \cos x \leq 2$, bounded dispersion on the lattice, non-negativity of the 3D lattice expression, and the certificate $\text{LatticeIsotropyCert}$ that the isotropy bound holds in three dimensions.
background
Recognition Science derives spatial structure from the J-functional equation and its composition law. The module works in the setting of the forcing chain where T8 forces D=3; it supplies the lattice-level bounds needed to realize that step. It imports only Mathlib and defines or proves the listed siblings: one_minus_cos_nonneg, one_minus_cos_le_two, lattice_dispersion_bounded, lattice_3d_nonneg, LatticeIsotropyCert, and latticeIsotropyCert.
proof idea
The module organizes a short sequence of lemmas. Basic trigonometric inequalities establish one_minus_cos_nonneg and one_minus_cos_le_two. These feed directly into lattice_dispersion_bounded and lattice_3d_nonneg. The final LatticeIsotropyCert assembles the preceding results into the isotropy certificate. All steps are elementary and tactic-light.
why it matters in Recognition Science
The results close a supporting step for T8 in IndisputableMonolith.Foundation.UnifiedForcingChain by furnishing the lattice isotropy bound required to force three spatial dimensions. They sit immediately upstream of any theorem that invokes the 3D lattice realization of the phi-ladder or the eight-tick octave.
scope and limits
- Does not derive the J-function or the composition law.
- Does not treat lattices in dimensions other than three.
- Does not connect the isotropy bound to mass formulas or the fine-structure constant.