The Nuclear Construction and Primordial Abundance: Polytope Closure on a Hexagonal Boundary, and the Reduction of Nucleosynthesis to Counting

Abstract

A companion manuscript reported a correspondence between light-nuclear structure and closure conditions on regular triangular polytopes, stating the results without disclosing the derivation. This manuscript supplies the method and extends it to primordial nucleosynthesis. We give the operation converting a subdivided triad into a tetrahedron, and show it is edge identification rather than any spatial folding — a distinction that blocked the construction entirely until it was corrected. We derive the parity condition determining which subdivisions can close, and show that of the eight convex deltahedra only two have a face count that is a perfect square, giving exactly the two doubly magic nuclei A=4 and A=16. We derive the mass-number gaps at A=5 and A=8 by three independent routes each, and identify both as the first four-dimensional member of their respective polytope families. We give the regularity criterion, correcting an earlier and narrower triangularity criterion. We supply the composition rule — balancing nucleon count across a two-by-two grid of type against spin on both marginals — together with the three routes that failed before it, and report an out-of-sample test that the rule passes without having been aimed at it. The extension to nucleosynthesis follows from the two gaps alone. Because A=5 and A=8 are both unbuildable, there is no path past A=4, so primordial nucleosynthesis terminates at helium — obtained by counting, with no reaction network and no stability comparison. We further show that this framework predicts the observed uniformity of light-element abundances at no cost, the standard account purchasing it with synchronised production; and that the circularity by which the baryon density is conventionally inferred through the very network this framework lacks is not load-bearing, a model-independent census agreeing with the inferred value to within 2%. Finally we reduce the entire primordial abundance question to a single count. With Y_p = 2r/(1+r) and r the neutron-to-proton ratio at closure, r = 1/7 returns Y_p = 1/4 and n_He/n_H = 1/12, both exactly. Section XIII states the candidate reading, one promoted site per complete ring, and its status: an identification, not a derivation. Section XIII.2 corrects a claim made in the earlier version of this manuscript, which held the hydrogen-to-helium ratio to be unobtainable from a counting argument. Every derivation below is counting. No reaction rate, cross-section, or binding energy is computed or required, and Section XIV states precisely which quantities are therefore unavailable. The foundational architecture, geometric intuitions, and motivating philosophy of this framework originated independently with the author prior to and separate from any AI involvement. Subsequent mathematical derivation, connection to established physics literature, computational verification, and error-checking were conducted through extended technical work with AI systems (Anthropic’s Claude and Google’s Gemini). The author directed this process, evaluated and selected among proposed derivations, and takes full responsibility for the accuracy and originality of the final work.