Quantitative tiling stability from quadratic discrepancy in Hamming spaces
Quadratic ball discrepancy defines an energy on codes in finite Hamming spaces. At perfect-code parameters, its exact minimizers are the perfect codes. We fix the alphabet size, length, and code cardinality and compare all codes with these parameters. We prove tiling-defect stability: excess discrepancy above the perfect-code benchmark controls the squared deviation of the distinguished ball-covering multiplicity from one. For one-error parameters satisfying sphere-packing and Lloyd integrality, the lower coefficient is $\kappa_{n,q}/q^2\geq1$. The uniform floor one is sharp, while the certified parameter-dependent coefficient can be much larger. An explicit parameter-dependent upper estimate is also available, and the two certified coefficients can be far apart. For any two-error parameter pair with $n\geq5$ satisfying sphere-packing divisibility and having two distinct integral Lloyd roots in the Hamming weight range, we obtain an explicit positive coefficient without assuming that a perfect code exists. For alphabets of size at least four, this conditional coefficient has a closed form and fixed-alphabet asymptotics. Direct certificates for the repetition and Golay families, combined with perfect-code classification, give tiling-defect stability for every nontrivial perfect code. Here stability concerns the ball-covering multiplicity profile, not symmetric-difference proximity to a particular perfect code. The defect is also a normalized chi-square smoothing error under uniform ball noise, so excess discrepancy controls holes, overlaps, defective ambient points, total variation, and R\'enyi divergence from uniformity of the ball-noise output. Competing codes need not be linear or satisfy a distance or error-correction constraint.