Paper 2026/1479

Conjectures and Barriers for RS-MCA

Abstract

As a companion to the proved bounds in "Shortening Bounds for Reed-Solomon MCA", we formulate the complete finite Reed-Solomon mutual correlated agreement (MCA) problem and a pole-aware conjectural positive-density profile envelope. An exact first-match compiler and realized-image moment and incidence inequalities isolate the payments required for a safe certificate without confusing supports, pairs, rays, and affine slopes. We prove a ceiling-normalized moment obstruction, a projective incidence theorem and benchmark dimension diagnostic, and a moving-scale consequence of the previously proved implication from a Sidon payment through the Balog-Szemer\'edi-Gowers theorem and Boolean-cube growth. Exact unsafe edges, repository audits, and numerical margins lead to four direct adjacent conjectures - two MCA and two auxiliary list inequalities - whose safe sides still require primitive-fiber, residual-projection, algebraic-routing, and add-back payments. In a collision-nonbinding, subexponential-budget identity-candidate branch, we conjecture a non-oracular exhaustive atlas whose exact unsafe--safe bracket, through a proved crossing reduction, yields \(\delta^*_{C_n,\mathrm{off,sup}}=1-\rho_n-g^*(\rho_n,\log_2 |{\mathbb B_n}|)+o(g_n^*)\); the general identity lower route instead uses an exact pole-adjusted target, and no matching bracket is claimed for the unrestricted smooth or circle problem.