A Finite-Field, Zero-Less Toy Model of Correlated Residues: A Classical Shared-Seed Correlate in the Correlation Polytope

Classical correlations satisfy the CHSH bound (|S| ≤ 2), quantum correlations reach S = 2√2, and Popescu–Rohrlich (PR) boxes achieve S = 4. We give an explicit, arithmetic-honest example of the classical corner. Within the Zero-Less Framework (ZLFF), the physical alphabet is F_q = {L, 2L, ..., (p-1)L} with 0 not in F_q and p an odd prime. A correlated pair is defined by co-generation: a public seed x in F_q and a random label a in F_q with a ≠ x are drawn, and the partner is set to b = x ⊖ a, where ⊖ denotes subtraction mod p on ℓ-labels. The condition a ≠ x guarantees b ≠ 0, so the alphabet remains zero-less and the map a → b is a bijection on the allowed set. The correlation is pre-encoded in the shared preparation x. The model reproduces anti-correlation in the preparation basis only, not in arbitrary rotated matched bases, unlike the singlet state. Being local-realistic with λ = x, it satisfies |S| ≤ 2 by the standard CHSH factorization, and is therefore distinguishable from quantum mechanics by any CHSH test with randomly chosen settings. Both this model and the PR-box are no-signaling; they differ in locality, which S quantifies. The construction is offered as a pedagogical note and is not a replacement for quantum entanglement.