Certified finite packings of Platonic solids: a generalized soft-to-rigid search and its computational development

We describe an open computational framework for placing congruent regular polyhedra in the smallest homothetic regular-polyhedral container found by numerical search. A cube-packing procedure introduced by Yohei Nakajima is generalized by replacing a convex piece P, with a centered inscribed ball of radius rho, by S_r=(1-r/rho)P + B(0,r). This family interpolates from the inscribed ball to the full piece, remains inside the piece, and grows monotonically as the rounding radius decreases. Search combines overlap-penalized heavy-ball dynamics, analytic rigid-motion gradients, numerical legalization, constrained local tightening, and archive-based perturbations. A separate exact checker proves strict containment and pairwise separation of the stored configurations using arithmetic in Q(sqrt(2),sqrt(5)); numerical optimization is not used as evidence of validity. The experiment covers five same-solid problems and four dual-pair problems for every integer population from 2 to 40, producing 351 certified upper bounds. Approximately 121,109 autonomous attempts followed 1,912 initial raw runs. We document the development sequence, geometric derivations, shaking criterion, solver comparisons, unsuccessful approaches, and reproducibility limits. The principal result is a general search-and-certification method with an inspectable data archive. Neither global optimality nor a physical theory of crystallization follows from the computations. Tools used: Claude (Anthropic) assisted computational development under the author's direction; Codex assisted source inspection, reproducibility checks, quantitative analysis, and manuscript drafting and exposition. Computational tools are described in the methodology. Alejandro Zarzuelo Urdiales is the human author responsible for the work; no AI system is listed as an author. Scientific source snapshot: e15b496ae7306287e0e70bb5b14be79f5dfdde84. The data cover 351 stored configurations in nine shape pairs and every count from 2 to 40. Exact feasibility certificates establish stored upper bounds, not general global optimality. The source repository provides inspectable construction files and certificates.