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

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

Problem Statement
The paper investigates the geometric and arithmetic structure of the transition between two consecutive highly composite numbers (HCNs), denoted $H < H'$. An HCN is a strict record holder of the divisor function $d(n)$, meaning $d(m) < d(H)$ for all $m < H$. The transition is modeled as a directed path in the prime-exponent lattice, where the state space is constrained by an arithmetic "ceiling" $z \le H'$.

The central object of study is the maximin capacity of a geodesic (a shortest path in the exponent lattice) between $H$ and $H'$. Specifically, the paper seeks to determine the maximum possible value of the minimum divisor count encountered along such a path, normalized by the initial divisor count $d(H)$. This is formulated as a fixed-endpoint maximin problem:
$$\beta_{\text{geo}}(H, H') = \frac{1}{d(H)} \max_{\gamma} \min_{z \in \gamma} d(z)$$
where the maximum is taken over all $H'$-admissible geodesics $\gamma$.

Methodology
The approach combines exact arithmetic enumeration, dynamic programming, and structural number theory:

1. State Space and Recursion: The transition is represented as a path through a finite "exponent box" defined by the coordinate-wise minimum and maximum of the prime exponents of $H$ and $H'$. The paper establishes an exact dynamic programming recursion (Theorem 2.3) to compute the optimal bottleneck capacity for any fixed pair $(H, H')$ and ceiling.
2. Complementary Reflection: A key structural tool is the involution $z \mapsto z^\sharp = HH'/z$. The paper proves that for any state $z$ in the exponent box, the product of divisor counts satisfies $d(z)d(z^\sharp) \ge d(H)d(H')$.
3. Static vs. Dynamic Analysis: The author contrasts the "static" overlap, measured by $d(\gcd(H, H'))$, with the "dynamic" capacity of an optimally scheduled path. They demonstrate that the static gcd route is not necessarily optimal because it forces a "delete-then-insert" sequence, whereas interleaving deletions and insertions can maintain higher divisor counts.
4. Stratification by Deletion Distance: The problem is analyzed based on $L^-$, the number of prime layers that must be deleted (where the exponent decreases). The paper provides exact solutions for $L^- \le 1$ and reduces the $L^- = 2$ case to a finite divisor-selection functional.
5. Computer-Assisted Verification: An exhaustive enumeration of HCNs up to $10^{70}$ was performed. This involved generating 889 records and 888 consecutive transitions. The exact maximin capacity was computed for all transitions using integer arithmetic, while heuristic greedy schedulers were verified against these optima using arbitrary-precision interval arithmetic.

Key Contributions and Results

• Failure of the Static Half-Bound: The paper disproves the conjecture that the static gcd overlap always retains at least half the divisor count, i.e., $d(\gcd(H, H')) \ge d(H)/2$. The first counterexample occurs at $H = 48,886,437,600$, where the ratio is $4/9$. In the range up to $10^{70}$, this static bound fails 119 times.
• The Record-Box Gap: Using the reflection principle, the author proves that no integer state $z$ exists strictly between $H$ and $H'$ that is coordinate-wise between their exponent vectors. Consequently, any admissible geodesic must descend below $H$ before returning to $H'$.
• Dynamic Half-Capacity: Despite the failure of the static bound, the computed dynamic capacity $\beta_{\text{geo}}(H, H')$ is never less than $1/2$ in the verified range.
• Equality Classification: The paper identifies that $\beta_{\text{geo}}(H, H') = 1/2$ occurs exactly when a prime with exponent 1 in $H$ is removed (i.e., the support size decreases). In these cases, the loss of the support prime forces a factor of $1/2$ that cannot be avoided.
• Exact Reductions:
  • For $L^- = 1$, the capacity is exactly $\frac{a}{a+1}$ where $a$ is the exponent of the deleted prime.
  • For $L^- = 2$, the problem reduces to a divisor-selection functional $R_s(C; g)$, isolating the "repair" (insertions) that must be accumulated before the second deletion.
• Certificates and Scheduling: The paper provides several arithmetic certificates (crossing, source-profile packing, and compensated matching) that guarantee the $1/2$ lower bound without enumerating all paths. It also proposes a record-specific greedy scheduler (Rule G1-G2) that matches the dynamic programming optimum for all 888 transitions in the computed range.

Significance and Claims
The paper claims to establish a tractable state space and a set of proved mechanisms for analyzing transitions between HCNs. Its primary contribution is the distinction between static arithmetic overlap and dynamic scheduling geometry. The author demonstrates that while the "standard" gcd path can dip significantly below half the divisor count, an optimally interleaved path can always recover to at least half.

The work remains modest regarding universal proofs:

• The Universal Half-Capacity Conjecture ($\beta_{\text{geo}} \ge 1/2$ for all consecutive HCNs) is stated as open, though it holds for all proved strata ($L^- \le 1$) and the entire verified range ($10^{70}$).
• The Equality Classification (equality iff a support prime is lost) is also stated as a conjecture outside the proved cases, though it holds in the verified data.
• The paper does not claim to solve the general maximin problem for arbitrary integers or to prove the conjecture universally, but rather provides the "exact two-deletion reduction" and "finite certificates" that narrow the remaining open problem to specific arithmetic control of divisor-selection functionals.

The significance lies in the rigorous formulation of the transition as a constrained scheduling problem on the prime-exponent lattice, the discovery of the "record-box gap," and the provision of a reproducible computational framework that challenges the sufficiency of static gcd analysis in divisor function theory.