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

Abstract

Let $d(n)$ be the divisor function and let $H<H^{\prime }$ be consecutive highly composite numbers. We study directed unit moves between their prime-exponent vectors under the hard ceiling $z\le H^{\prime }$. 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 $d(\gcd (H,H^{\prime }))\ge d(H)/2$ 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 $$d(z)d(H{H}^{\prime }/z)\ge d(H)d(H^{\prime })$$ 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

29 pages, 2 figures, 4 tables. Accompanying reproducibility archive, version 1.0.0: https://doi.org/10.5281/zenodo.21983543