A universal 23-dimensional realization of Steiner triple codes

Abstract

For every finite complete Steiner triple system, we construct bounded full-dimensional strictly convex compact sets in 23-dimensional Euclidean space whose code and interior code are exactly the associated Steiner triple code. Three simultaneous quadratic coefficient relations reduce a sextic construction to 23 coordinates. Uniform face estimates, rational residual bounds, and an all-point ball construction establish the universal result. This contradicts Conjecture 16 in Jeffs, Siegel, Staudinger and Wang, arXiv:2309.14862v1. The optimal universal dimension remains between 4 and 23. Version 1.1 states the balanced-support realization lemma separately and proves that the rational-ball construction has O(n^3) total binary description length for n labels. Its sufficient integer certificates can be verified in polynomial time. This is a verification result, not a polynomial-time discovery algorithm. A new exact-arithmetic consumer checks the lemma's hypotheses, emits rational ball descriptions and includes targeted corruption tests. Its examples are rescaled from the archived R=16384 data, not produced by the main theorem's R=65536 exhaustive-grid procedure. The three supplementary notes are unchanged from version 1.0.0. They concern conditional constructions, labeling obstructions, line and plane traces, contact-model limitations, and shared-support recovery. Their finite examples in dimensions 20 and 22 do not improve the universal bound. Exact checkers, source certificates and reproducibility instructions accompany the proofs. Byungwoong Yoo initiated and directed the research. The manuscripts describe the substantive AI assistance used with mathematical arguments, code, and manuscript preparation. This preprint has not undergone external peer review or formal theorem-prover verification. Original text is licensed under CC BY 4.0 and original code under MIT; third-party material retains its own terms.