module module high

`IndisputableMonolith.Physics.CubeSpectrum`

The CubeSpectrum module defines the combinatorial structure and Laplacian spectrum of the three-cube Q3, the unit cell on the ℤ³ recognition lattice, with eigenvalue multiplicities given by binomial coefficients C(3,k). Lattice physicists cite it when building renormalization-group flows or O(N) universality classes in Recognition Science. The module consists entirely of definitions establishing vertices, edges, faces, degree, Euler number, eigenvalues, spectral gap, and trace.

claimThe Laplacian spectrum of the 3-cube graph $Q_3$ has eigenvalues with multiplicities $1,3,3,1$, equal to the binomial coefficients $C(3,k)$ for $k=0,1,2,3$.

background

The module supplies the geometric unit cell for the ℤ³ lattice in Recognition Science. It introduces Q3_vertices, Q3_edges, Q3_faces, Q3_degree, Q3_euler, Q3_edge_count, Q3_vertices_eq, Q3_laplacian_eigenvalues, Q3_spectral_gap, Q3_max_eigenvalue, Q3_eigenvalue_count, and Q3_trace. These definitions encode the hypercube combinatorics whose multiplicities match binomial coefficients, as stated in the module comment. The setting is the D=3 spatial dimensions forced by T8, with Q3 serving as the fixed unit cell for renormalization along the φ-ladder.

proof idea

This is a definition module, no proofs.

why it matters in Recognition Science

CubeSpectrum feeds the ThermalFixedPoint module, where the renormalization group operates on the recognition lattice with unit cell Q3 and the thermal perturbation satisfies the Fibonacci recurrence forced by φ. It also feeds UniversalityClasses, which maps symmetry rank N to critical exponents via the automorphism structure of Q3. The module realizes the D=3 geometry required by the unified forcing chain (T8) and supplies the spectral data used in both downstream physics modules.

scope and limits

Does not derive the eigenvalue multiplicities from the adjacency matrix.
Does not extend the spectrum to higher-dimensional hypercubes.
Does not include dynamical or renormalization flow equations.
Does not reference the J-cost function or recognition composition law.

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