We study signed, weighted affine $p$-adic residual objectives as native encodings of finite-domain constraints.
For primes that separate the finite alphabet, sufficiently weighted positive unary rows pin each coefficient to its allowed set, while negative rows reward unequal endpoints or clause satisfaction.
A coordinatewise domination theorem places every global minimiser in the finite domain; there the loss is, up to an additive constant, the all-different conflict count or the negative number of satisfied CNF clauses.
Standard Sudoku provides an $81$-coefficient case study without a one-hot lift.
A client-side implementation exposes the generated dataframes, arithmetic, diagnostics, and searches.