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Prime-Exponent Transition Geometry and Divisor Barriers Between Consecutive Highly Composite Numbers
Number Theory · Submitted 17 Aug 2026
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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The abstract, sentence by sentence
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
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Let $d(n)$ be the divisor function and let $H<H'$ be consecutive highly composite numbers. | background 1.00 | 0.08 | 0.02 | asserted 0.51 |
| s1 | We study directed unit moves between their prime-exponent vectors under the hard ceiling $z\le H'$. | contribution 0.71 | 0.82 | 0.03 | asserted 0.77 |
| s2 | The normalised capacity of such a geodesic is its smallest divisor count divided by $d(H)$, giving a finite fixed-endpoint maximin problem. | method 0.44 | 0.56 | 0.07 | asserted 0.98 |
| s3 | 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 $4/9$. | result 0.99 | 0.92 | 0.12 | asserted 1.00 |
| s4 | 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. | result 0.98 | 0.94 | 0.32 | asserted 1.00 |
| s5 | 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. | method 0.85 | 0.95 | 0.06 | asserted 1.00 |
| s6 | A computer-assisted enumeration through $10^{70}$ produces $889$ records and $888$ transitions. | result 0.80 | 0.94 | 0.08 | asserted 1.00 |
| s7 | 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. | result 1.00 | 0.93 | 0.33 | asserted 1.00 |
| s8 | Independently checked certificates cover all $301$ transitions with $L_-=2$. | method 0.74 | 0.85 | 0.05 | asserted 1.00 |
| s9 | The corresponding universal half-capacity bound and equality classification remain open beyond the proved strata and the verified range. COMMENTS | limitation 0.74 | 0.26 | 0.13 | asserted 0.55 |
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Original 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
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