Logical Consistency as a Functional Equation on Continuous Positive-Ratio Comparisons

The d'Alembert Inevitability Theorem and the Canonical Reciprocal Cost Uniqueness Theorem establish that a continuous nonconstant cost on positive ratios with polynomial-degree-two composition is forced into the Recognition Composition Law family F(xy) + F(x/y) = 2F(x) + 2F(y) + cF(x)F(y), with calibrated representative J(x) = ½(x + x⁻¹) − 1 on the bilinear branch. The present paper does not reprove that rigidity. Its contribution is to show that the rigidity has a logical reading: the hypotheses the d'Alembert theorem uses are, under one named interpretation of the comparison operator, the operator-level encoding of the four classical Aristotelian conditions. We give the encoding (identity as zero self-cost, non-contradiction as symmetric single-valued comparison, totality as definiteness on the open positive quadrant, composition consistency as route-independence of the combiner), and prove a canonicality theorem (Theorem 16): under the magnitude-of-mismatch interpretation, this encoding is the unique reading of the four Aristotelian conditions that preserves their propositional content. We prove a structural theorem (Theorem 42): any reality structure with definite states, single-valued comparison, decidability, and compositional inference satisfies the four conditions on its comparison operator. The two further conditions a truth-evaluable reality plausibly satisfies, scale-free comparison of magnitudes and no-hidden-state finite composition, force ratios and counted-once bi-affine algebra respectively. The translation theorem (Theorem 22) reduces the encoded conditions to the hypotheses of the d'Alembert theorem; the rigidity then follows. A direct theorem (Theorem 28) gives a short self-contained proof of the Recognition Composition Law in the strict counted-once bi-affine subcase, independent of the d'Alembert route. Combining the canonicality theorem, the structural theorem, and the rigidity, we obtain (Corollary 50) the operative-domain identification: on the operative domain (continuous positive-ratio reality structures supporting counted-once finite logical comparison), the recognition composition law, the law of logic, and the structural form of any reality on this domain are the same mathematical object, with calibrated representative J on the bilinear branch. The polynomial restriction is sharp. Proposition 30 exhibits a continuous operative comparison (ln(x/y))⁴ whose continuous combining rule 2a + 2b + 12√(ab) is closed under iteration on [0, ∞) but is not polynomial of degree at most two; the Recognition Composition Law fails for this example. Proposition 53 shows that real-analyticity of the combiner at the origin also does not force polynomial degree at most two. Counted-once finite logical comparison is therefore not a removable regularity hypothesis. It is the structural condition that distinguishes the Recognition Composition Law family from broader continuous-combiner alternatives. The operative domain contains every pairwise physical measurement on positive quantities, every continuous likelihood ratio between probability distributions on a one-parameter family, and every continuous scale-invariant distance or divergence on a multiplicative state space. We do not claim that every divergence in information geometry is in this class; the Kullback-Leibler divergence and the Bregman family in their general distribution-level form are not scale-invariant in the operator-level sense used here. The discrete propositional case is open. We do not claim a derivation of arithmetic, modal logic, or higher-order extensions. The contribution is the encoding, its canonicality, and the structural reading of the existing rigidity.