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

For signed differences of measures on \(\mathbb F_2^m\), we determine sharp comparisons between total variation and punctured (zero-frequency-deleted) Walsh \(L^p\) spectra. For every \(m\) and \(1\le p\le2\), the optimal constant is \(\Lambda_{m,p}=2^{m/p-1}\). For all dimensions \(m\le5\) we also give analytical sharp formulas throughout \(2\le p\le\infty\); in particular the five-bit formula is \(\Lambda_{5,p}=\frac{32^{1/p-1}}2(16\cdot4^q+12\cdot8^q)^{1/q}\), where \(q=p/(p-1)\). The result is a continuous-exponent punctured-norm theorem, not a claim that the classical five-bit endpoint value 160 or the underlying Boolean spectral catalogues were first discovered here. For observations of a known quadratic bent-function translation/sign orbit, exact identification is equivalent to affine spanning by sampled frequencies. The first-order Reed–Muller code determines optimal worst-case sign-error, erasure and Euclidean-noise thresholds. We also identify a sharp six-bit bent regime for \(p\ge2.5590297\ldots\) and an explicit counterexample to bent optimality for \(2\le p<2.1269266\ldots\); the intermediate regime remains unresolved. Sharpness for arbitrary signed differences is not automatically claimed for the more restricted class of characteristic-function dependence defects.