Conservative Modular Counting at Every Odd Prime

We classify modular counting constraint satisfaction for every fixed finite conservative relational template and every fixed odd prime. At primes 3 and 5, a Mal'tsev polymorphism yields a deterministic polynomial-time algorithm, and its absence yields modular counting completeness. At every prime at least 7, the tractable condition is exact integer strong balance. On the conservative Mal'tsev branch, strong balance is equivalent to the absence of an ordinarily primitive-positive-definable four-ary parity relation on an actual three-element subset with quotient fibres of sizes two and one. We prove this structural equivalence by exact witness selection and a minimal-domain integer matrix argument. An explicit unweighted inverse-path gadget converts the obstruction to Boolean NAND modulo every prime at least 7. For primes 3 and 5, we give the full counting-preserving domain-descent algorithm, including unary-weight elimination and finite-field affine phase evaluation. All criteria are decidable by finite tests. The exact strong-balance algorithm, balance equivalences, conservative algebraic structure, and no-Mal'tsev hardness implications are inherited from established results.