Arithmetic Refinements, Modular Obstructions, and Congruence Diagnostics for Ones-Free Pairwise-Coprime Partitions

We study arithmetic refinements of partitions whose nonunit parts are pairwise coprime. Removing all parts equal to \(1\) gives the exact core transform\[\overline P(q)=\frac{\overline P^{\circ}(q)}{1-q},\qquad\overline p(n)=\sum_{m\le n}\overline p^{\circ}(m),\]so the full sequence is cumulative whereas the ones-free sequence is pointwise. The bivariate transform separates the free string of ones from the core length. A core with \(k\) parts has size at least the sum of the first \(k\) primes, with equality only for the prime staircase; hence the maximal core length among cores of size at most \(n\) is asymptotic to \(2\sqrt{n/\log n}\). We then examine a proposed core-rank approach to modular congruences. A multivariate locality criterion shows that a prime-base Euler product exists exactly for statistics additive over the selected prime bases. The largest-part core rank is not additive, so its proposed Euler product and the resulting eta identity are false. The associated congruences on \(5n+4\), \(7n+5\), and \(11n+6\) already fail at their initial terms. The additive statistic \(\sigma(\lambda)=|\lambda|-\ell(\lambda)\) supplies an exact root-of-unity refinement in place of the nonlocal rank. The positive theory combines exact computation with structural obstructions. We give a finite-state recurrence, Fourier and cyclotomic criteria for equidistribution, arithmetic-progression dissections, and an affine periodicity theorem for the core transform. The ones-free series is a coefficientwise stable limit of weighted independence polynomials of finite prime-support conflict graphs. Boolean support inversion and a finite Bell decomposition yield exact-support recurrences and a support-excess filtration. The first mixed layer begins with \(q^6+q^{10}+q^{11}\), while the second has the unique leading partition \((15,14)\) of size \(29\). Finally, local prime-power series have the unit circle as a natural boundary in characteristic zero. Their root-of-unity specializations are non-D-finite, whereas reduction in the matching prime characteristic produces an exact Artin–Schreier extension of degree \(p\). Christol’s and Cobham’s theorems give a sharp finite-field dichotomy: the local series is algebraic precisely when the prime base equals the residue characteristic. Canonical Euler exponents and a gcd-periodicity criterion provide rigorous tests for finite eta-quotient representations before any Sturm argument is attempted. Keywords Pairwise-coprime partitions; ones-free partitions; prime-power Euler products; congruences; modular obstructions; prime-support graphs; cyclotomic norms; eta quotients; Artin–Schreier extensions; automatic sequences.