素数指数过渡几何与连续高合数之间的除数障碍
Prime-Exponent Transition Geometry and Divisor Barriers Between Consecutive Highly Composite Numbers
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中文总结 AI 辅助
该研究探讨连续高合数间素数指数向量的移动问题，通过动态规划、计算机辅助枚举等方法，揭示除数障碍相关性质，发现静态边界失效情况，得到测地线容量等关键结果。
中文摘要 AI 辅助
设d(n)为除数函数，令H<H'为连续的高合数，我们研究在硬上限z≤H'下，它们的素数指数向量之间的有向单位移动。此类测地线的归一化容量为其最小除数计数除以d(H)，构成有限的固定端点极大极小问题。精确记录枚举首先在48,886,437,600<64,250,746,560处发现静态替代式d(gcd(H,H'))≥d(H)/2的失效，此处比值为4/9。然而，对于指数盒中的每个状态z，我们证明d(z)d(HH'/z)≥d(H)d(H')，并推导出记录盒间隙：无盒状态在数值上严格介于两个记录之间。我们还给出精确的动态规划递归式，求解层L₋≤1，并将完整的L₋=2问题归约为显式除数选择泛函。通过10⁷⁰的计算机辅助枚举得到889个记录和888个过渡。尽管静态半gcd边界失效119次，但所有计算的测地线容量至少为1/2；等式恰好出现在失去一个指数1支撑素数的124个过渡中。独立验证的证书覆盖所有301个L₋=2的过渡。对应的通用半容量边界及等式分类在已证明层和已验证范围之外仍为开放问题。
英文摘要
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
