Canonical Arithmetic Defects for Polynomial Congruence Roots: Higher Tuples, Frobenius Filters, and Resultant-Sharp Finite States

Abstract

We develop a higher-order extension of canonical bounded defects for periodic root-counting systems. If a period contains $r$ marked residue classes and $A(X)$ counts marked integers up to $X$, then the dyadic $k$-tuple functional $\binom{A(2X)}{k} - 2^k \binom{A(X)}{k}$ has a unique anchored polynomial counterterm removing all complete-period dependence. The resulting local defect is $$\binom{A(2X)-2r\lfloor X/q\rfloor}{k}-2^k\binom{A(X)-r\lfloor X/q\rfloor}{k},$$ is uniformly bounded, has a finite universal alphabet, and vanishes identically for $k > 2r$. More generally, for a counting polynomial $P$ of degree $m \ge 2$, the normalization $2^m$ is the unique scalar for which the canonical counterterm drops in quotient degree from $m$ to $m-1$. For polynomial congruence roots, the local root number at an unramified prime is the fixed-point count of Frobenius on the roots. This turns the tuple order into a Frobenius fixed-point filter and gives a group-theoretic threshold isolating the completely split class. In the cubic $S_3$ case the triple defect has the abstract alphabet $\{-7, -4, 0, 1, 2, 4, 10, 12\}$, but a single completely split prime has one of only three six-state signatures. We prove a general diagonal $k = r$ chamber theorem and, for a fixed irreducible polynomial, show that outside a finite set of primes determined by explicit affine resultants, the chamber signature is automatically sharp, uniformly for all prime powers. We also give exact higher-$\gcd$ valuation carriers and an explicit multiplicative quotient realizing the canonical triple defect. Finally, for depressed $S_3$ cubics we compute the chamber volumes $1/4, 1/4, 1/2$; these become prime densities conditional on the ordered-root equidistribution conjectures of Kitaoka. No higher-degree prime-root equidistribution theorem is assumed or claimed.