Exact Walsh–Hadamard Spectral Analysis of ML–KEM Compression Maps

The standardised Module-Lattice-Based Key-Encapsulation Mechanism uses coefficient compression, yet an abstract residue map has no unique Boolean cube spectrum until a binary representation, domain extension, and input measure are fixed. We analyse these maps under an explicit twelve-bit reduction-based lift. Exact interval character sums yield a general high-modulus theorem for quarter-threshold indicators and show that message decoding has a uniquely dominant high-bit parity. The argument also clarifies the connection among Walsh coefficients, affine approximation, Hamming distance, nonlinearity, agreement probability, and sign correlation. A vectorial extension exhaustively certifies every nonzero scalar component of the standardised compression widths by deterministic integer Walsh–Hadamard transforms. A representation comparison then separates full-cube coefficients from centred, distribution-dependent correlations and delineates how canonical, centred, Montgomery, Barrett, shared, or compiler-generated intermediates require distinct models. Reproduction scripts regenerate the complete certificates and the spectral-gap visualisation without sampling, random choices, physical traces, floating-point decisions in the core certificates, or network access. The results identify mathematically distinguished affine predictors for later implementation-specific validation; they do not establish measured leakage, attack success, or implementation resistance.