Diophantine representation interfaces

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factorial_residues: Reflection supplies bounded families at one prime, while first-hit distributions and prime cofactor support need separate information.

local_congruences: Interval packing, fixed prime multipliers, moving prime cutoffs and local powerful-number solvability have distinct global limitations.

mobius_split: A pointwise Mertens exponent theta gives squarefree error exponent one over three minus theta through this absolute-value split, leaving the quarter-power target.

nondividing_sums: Density zero of representables implies pairwise-coprime exceptions; an isolated rectangle-summary counterexample does not judge the primary formal proof.

power_sums: Positive sectors, unit coefficients, average collisions and divisor discriminants separate the available power-sum results from the remaining targets.

rational_coefficients: Restricting a floored fifth-power sequence to a denominator progression would give an integer Sidon polynomial, conditional on a rational good coefficient.

square_cubes: A uniform genus-two point bound would bound square cubes even with repeated subset sums, while curve-by-curve finiteness does not.

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These notes retain explicit deductions, source specializations and useful obstructions. Each note states the exact remaining hypothesis or test. All original arguments are author-recorded by Plasma AI; source premises keep their stated reading and acceptance limits.