Query-Relative Walsh Compression: Sharp Punctured Norms and Signed Bent-Orbit Recovery

We study two query-dependent compressions of a Walsh spectrum on \( F_2^m\). The first asks for the sharp comparison between total variation and a punctured Walsh \(L^p\) norm. We derive the exact Walsh–TV duality formula, obtain the universal answer for \(1≤ p≤2\), and give analytic sharp formulas in every exponent \(2≤ p≤∞\) for \(m≤5\). The five-bit proof uses eight intrinsic four-bit spectral shapes, a single support-overlap estimate, an affine obstruction for the exceptional pair, and a congruence-sensitive interpolation valid for the whole interval of exponents. The endpoint \(160\) and the finite spectral catalogues have classical antecedents; the publication claim is the self-contained punctured-norm argument and its continuous-exponent synthesis, not a first enumeration of Boolean spectra. In every even dimension, quadratic bent signs give lower bounds within a uniform relative factor of the Parseval upper bound, while exactness beyond the settled dimensions remains open. The second query asks which element of a known quadratic bent translation/sign orbit generated sampled signed coefficients. Exact recovery is equivalent to affine spanning, and the induced first-order Reed–Muller distance gives sharp sign-error, erasure, and Euclidean-noise thresholds. The arbitrary-pair norm theorem gives an upper bound for characteristic-function dependence differences, but sharpness under the product-of-marginals constraint is not asserted. In six bits, a congruence argument gives the sharp bent profile for \(p≥2.5590297…\), whereas an explicit flipped bent sign improves on it for \(2≤ p<2.1269266…\); the intermediate range remains open.