The band invariant of central-divisor exception sets: a closed form, its chamber structure, and the squarefree counting theorem

For a positive integer n, let g(n) be the largest divisor of n not exceeding sqrt(n) (OEIS A033676), and let X_a be the central-divisor exception set: the composite m in [a, a^2] for which g(am) = a, equivalently the m for which am has no divisor in the central window (a, sqrt(am)] (the rows of OEIS A163925). A companion note proves the band law that X_a is contained in {m <= a^2/beta(a)}, where beta(a) = min{ d-(d) d-(e) : d,e | a, de > a } is the band invariant and d-(.) denotes the predecessor in the divisor lattice of a. This note develops the squarefree theory of beta and X_a, writing a = p_1...p_k with p_1 < ... < p_k and P_j = p_1...p_j (with P_0 = 1).

The first result is a closed form for the band invariant: beta(a) equals the minimum over 0 <= j <= k-1 of a P_j / p_{j+1}, equivalently the minimum of the staircase products (prod_{i<=j} p_i^2)(prod_{i>=j+2} p_i). The proof reduces the two-divisor optimization to a one-divisor one, showing that beta(a) equals a divided by the largest ratio of consecutive divisors of a, and that this largest ratio is attained at a prime. The formula unifies the prime, two-prime, and three-prime squarefree formulas of the companion notes and complements the prime-power chain formula beta(p^k) = p^{k-1}.

The second result describes the chamber structure: the minimizing index is governed by an explicit rule on the logarithmic shape of a, with the active chamber advancing from j-1 to j exactly when p_{j+1} > p_j^2.

The third result is a universal survivor family. For every chamber j, if x and y are distinct primes, neither a prime factor of a, with P_j < x < p_{j+1} and a/x < y < a/P_j, then xy lies in X_a.

The fourth result is an unconditional counting theorem: for fixed k and fixed logarithmic shape, |X_a| is of order a^2 / (beta(a) log^2 a), with implied constants depending only on k and the limiting shape. The lower bound comes from the survivor family together with the prime number theorem; the upper bound from the divisor-pair fiber decomposition together with a one-dimensional Selberg upper-bound sieve for rough integers, there being only O_k(1) fibers, each with denominator at least beta(a).

The arguments are elementary, resting on the divisor lattice of a, the prime number theorem, and a standard sieve estimate; no unproved hypotheses are used, in particular no Riemann Hypothesis and no primes-in-short-intervals input. The result turns the companion casework into a closed squarefree theory in which the exception-cloud size is governed by a single divisor-lattice invariant: the reciprocal of the largest consecutive-divisor ratio of a. This is part of a series on the central-divisor exception cloud of g(n) = A033676, with the prime, prime-power, two-prime, and three-prime layers treated in the companion notes.