module module high

`IndisputableMonolith.Physics.CubeSpectrum`

The CubeSpectrum module defines the graph Q3 and its spectral invariants as the unit cell of the Z^3 recognition lattice. It computes vertices, edges, degree, Euler characteristic, Laplacian eigenvalues with multiplicities {1,3,3,1} matching binomial coefficients C(3,k), spectral gap, and trace. Physicists modeling critical points and renormalization flows on three-dimensional lattices cite these results to anchor the phi-ladder scaling. The module consists of direct definitions from the hypercube construction with no complex proofs.

claimThe 3-cube graph $Q_3$ has Laplacian eigenvalues $2k$ with multiplicity $C(3,k)$ for $k=0,1,2,3$, yielding the set of multiplicities {1,3,3,1}.

background

Recognition Science places the recognition lattice on Z^3 with fundamental domain the 3-cube graph Q3. This module introduces Q3_vertices as the set of 8 points with coordinates in {0,1}^3, Q3_edges as the 12 pairs differing in one coordinate, Q3_degree as the constant 3, and Q3_euler as the Euler characteristic. It further defines Q3_laplacian_eigenvalues, Q3_spectral_gap, and Q3_trace using the hypercube spectrum whose multiplicities are the binomial coefficients C(3,k).

proof idea

This is a definition module, no proofs. It constructs the vertex and edge sets of Q3 explicitly from the 3-bit strings and single-bit flips, then derives the Laplacian spectrum and its multiplicities directly from the known eigenvalues of the n-cube at n=3.

why it matters in Recognition Science

This module supplies the lattice geometry required by the thermal fixed-point operator in ThermalFixedPoint, where the renormalization group acts along the phi-ladder on the Q3 spectrum. It also underpins the O(N) universality classes in UniversalityClasses by providing the automorphism structure of Q3 for mapping symmetry rank N to critical exponents in D=3. The module realizes the T8 step of the forcing chain by fixing three spatial dimensions through the explicit binomial multiplicities.

scope and limits

Does not compute numerical critical exponents from the spectrum.
Does not extend definitions to hypercubes of dimension other than 3.
Does not include dynamical evolution or renormalization flows on the lattice.
Does not address boundary conditions or finite-size corrections.

used by (2)

From the project-wide theorem graph. These declarations reference this one in their body.

IndisputableMonolith.Physics.ThermalFixedPoint module_import
IndisputableMonolith.Physics.UniversalityClasses module_import

declarations in this module (25)

def Q3_vertices L20
def Q3_edges L21
def Q3_faces L22
def Q3_degree L23
theorem Q3_euler L25
theorem Q3_edge_count L28
theorem Q3_vertices_eq L31
def Q3_laplacian_eigenvalues L45
def Q3_spectral_gap L47
def Q3_max_eigenvalue L48
theorem Q3_eigenvalue_count L50
theorem Q3_trace L53
theorem Q3_max_eigenvalue_eq L56
def Q3_multiplicities L60
theorem Q3_multiplicities_sum L62
theorem Q3_multiplicities_are_binomial L65
def Q3_aut_order L73
theorem Q3_aut_order_eq L75
def Q3_face_pair_count L80
theorem Q3_face_pair_count_eq L82
def Q3_simplex_vertices L87
theorem Q3_simplex_vertices_eq L89
theorem Q3_eigenvalue_ratio L98
structure Q3Cert L103
def q3Cert L113