# Verification report

The accepted scope is the explicit finite-stability theorem proved in the manuscript: saturated unit-ball packings in R² and R³ with trivial translational stabilizer, exact grid contact graphs and rational coordinates for rational parameters. The same family has a nontrivial uniformly bounded global flex. The unrestricted original jamming question is not declared solved.

The proof checks nonoverlap and saturation for all integer indices. It constructs a strictly positive planar equilibrium stress, proves that every finitely supported admissible velocity vanishes, and obtains actual finite stability by a normalized-displacement compactness argument. The three-dimensional argument reduces horizontal motion to the planar theorem and vertical motion to monotonicity along finite-support chains. Aperiodicity follows from labeled contact vectors and nonperiodic positive increments. Density and uniform continuity of the global flex are proved separately.

The standard-library Python supplement uses exact fractions, not numerical tolerances. Four parameter values give 263652 all-pair tests, 4356 stress vertices, 1028 scalar indices, a 98-edge summation identity, four uniform-bound checks and three negative controls. Normal and optimized outputs agree. These finite regressions supplement the written universal proof; they are not formal verification or an independent review.

The bibliography credits Connelly's reciprocal-stress and rigid-ordering method, Bezdek--Bezdek--Connelly's finite-stability theory, and the source problem and upstream locally jammed attempt. Novelty is limited to the explicitly displayed construction and combination of properties; absolute priority is not certified.

The manuscript is an AI-assisted, self-audited preprint. It does not claim unrestricted collective jamming, strict jamming, optimal density or a determination of uniform finite-rearrangement stability.