module module moderate

`IndisputableMonolith.Physics.CubeSpectrum`

CubeSpectrum supplies the Laplacian spectrum of the 3-cube graph Q3 that serves as the unit cell of the recognition lattice ℤ³. Lattice physicists and critical-phenomena researchers cite it for the binomial multiplicity structure of the eigenvalues. The module consists entirely of definitions and supporting lemmas that enumerate vertices, edges, faces, and spectral invariants without any theorem proofs.

claimThe Laplacian eigenvalues of the 3-cube graph $Q_3$ are $2k$ for $k=0,1,2,3$, each with multiplicity $inom{3}{k}$.

background

The module resides in the Physics domain and imports only Mathlib. It defines the 3-cube Q3 as the Cartesian product of three copies of K2 and records its combinatorial and spectral data. Sibling definitions cover Q3_vertices (8 points), Q3_edges, Q3_faces, Q3_degree, Q3_euler, Q3_edge_count, Q3_laplacian_eigenvalues, Q3_spectral_gap, Q3_max_eigenvalue, Q3_eigenvalue_count, and Q3_trace. The module comment states that the multiplicities are the binomial coefficients C(3,k). Downstream modules treat Q3 as the unit cell of the recognition lattice ℤ³.

proof idea

This is a definition module, no proofs. The argument proceeds by successive definitions that first enumerate the combinatorial structure of Q3 and then extract its Laplacian spectrum together with the associated invariants.

why it matters in Recognition Science

The module furnishes the geometric and spectral foundation required by the two modules that import it. ThermalFixedPoint uses the Q3 unit cell to locate the forced thermal eigenvalue φ on the recognition lattice and to run the renormalization group along the φ-ladder. UniversalityClasses maps subgroups of Aut(Q3) to the O(N) universality classes and their critical exponents in D=3. The construction realizes the T8 step that forces three spatial dimensions.

scope and limits

Does not compute the spectrum of Qn for any n other than 3.
Does not derive the binomial multiplicities from the characteristic polynomial of the adjacency matrix.
Does not relate the eigenvalues to the J-cost function or the forcing chain T0-T8.
Does not contain numerical checks or floating-point approximations of the spectrum.

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