Prime-Exponent Transition Geometry and Divisor Barriers Between Consecutive Highly Composite Numbers

Abstract

Let $d(n)$ be the divisor function and let $H<H'$ be consecutive highly composite numbers. We study directed unit moves between their prime-exponent vectors under the hard ceiling $z\le H'$. The normalised capacity of such a geodesic is its smallest divisor count divided by $d(H)$, giving a finite fixed-endpoint maximin problem. The exact record enumeration first finds a failure of the static surrogate $d(\gcd(H,H'))\ge d(H)/2$ at $48,886,437,600<64,250,746,560$, where the ratio is $4/9$. For every state $z$ in the exponent box, however, we prove $$d(z)d(HH'/z)\ge d(H)d(H')$$ and deduce the record-box gap: no box state lies numerically strictly between the two records. We also give an exact dynamic-programming recursion, solve the strata $L_-\le 1$, and reduce the complete $L_-=2$ problem to an explicit divisor-selection functional. A computer-assisted enumeration through $10^{70}$ produces $889$ records and $888$ transitions. Although the static half-gcd bound fails $119$ times, every computed geodesic capacity is at least $1/2$; equality occurs in exactly the $124$ transitions that lose an exponent-one support prime. Independently checked certificates cover all $301$ transitions with $L_-=2$. The corresponding universal half-capacity bound and equality classification remain open beyond the proved strata and the verified range. COMMENTS

Disclosure

“capacity certificate. For the 19 support-loss cases with threshold 𝑡 > 2, exact repair-floor competition also succeeds; the largest ratio of frontier-assisted repair cost to left-only repair cost is 11/18. Acknowledgements AI disclosure. OpenAI’s ChatGPT was used as a writing and code-assistance tool to reorganise a substantially longer exploratory manuscript, improve exposition, check notation and LaTeX, assist with bibliographic metadata, and help run and review the reproducibility scrip”

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Classification

Rewriting existing author-written text

Multiplier

4

Verified

2026-08-23T05:20:40.803823Z

Structural counts