We study arithmetic refinements of partitions whose nonunit parts are pairwise coprime. Removing all parts equal to \ (1\) gives the exact core transform\ P (q) = P^ (q) 1-q, p (n) =₌ ₍ p^ (m), 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 \ (2n/ 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 \ ( () =||- () \) 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 \ (q⁶+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.
Kianming(Jianming) Wang (Tue,) studied this question.