To XOR a Stone with Six Birds: Closure Diagnostics for Emergent Bits, Gates, and Booleanity

Ioannis Tsiokos

Abstract: Boolean logic is usually treated as a formal starting point. We ask instead when a substrate actually earns a logic layer. In the Six Birds emergence calculus, P1 operator rewrite, P2 gating/constraints, P3 protocol holonomy, P4 sectors/invariants, P5 packaging, and P6 accounting/audit, we define a logic layer as an audited closure of packaged predicates under induced dynamics. This yields two complementary views. In the feasibility view, propositions are definable feasible sets and connectives are their stable closure operations. In the operator view, bits are packaged equivalence classes and gates are macro-updates induced by the underlying dynamics. We implement quantitative diagnostics for both views: metastability, route mismatch, closure defect, gate error, conditional entropy, unretained input information, and entropy-production audits. Across controlled finite Markov laboratories we obtain four main results. First, parity/XOR-type variables are unusually robust coarse variables: the parity lens beats the median random binary partition across the full tested grid. Second, NOT, AND, and reversible XOR embeddings exhibit distinct closure regimes under noise, staging, and timescale stress. Third, when a reversible CNOT embedding is coarse-grained to an erased XOR output, closure degrades sharply and unretained input information increases by more than eightfold relative to the retained view. Fourth, a minimal unsupervised discovery pipeline recovers stable bits and reconstructs a NOT gate directly from transition structure. The result is a falsifiable criterion for when a system has logic at a layer, and for which logic it has, without treating logic as fundamental.