finite_logical_has_counted_once
plain-language theorem explainer
Finite logical comparison on positive ratios includes counted-once composition as one of its defining algebraic properties. Researchers deriving the Recognition Composition Law from logical constraints cite this step when assembling the finite case of the forcing chain. The proof is a direct field projection from the FiniteLogicalComparison structure.
Claim. Let $C$ be a comparison operator on positive reals. If $C$ is a finite logical comparison (satisfying identity, non-contradiction, excluded middle, scale invariance, nontriviality, and counted-once algebra), then $C$ satisfies counted-once composition: there exists a combiner $P$ such that the derived cost obeys the addition rule for products and quotients under the counted-once and symmetric conditions.
background
FiniteLogicalComparison is a structure on a ComparisonOperator $C$ that packages the Aristotelian laws (identity, non-contradiction, excluded middle) with scale invariance, nontriviality, and the counted-once algebra condition. CountedOnceComposition asserts existence of a symmetric counted-once combiner $P$ such that derivedCost $C$ satisfies derivedCost $C$ $(x y)$ + derivedCost $C$ $(x / y)$ equals the appropriate expression in the combiner for positive $x, y$. The module packages the paper's sharpened theorem that finite logical comparison on positive ratios forces the RCL family, with counterexamples establishing necessity of the finite pairwise polynomial condition.
proof idea
The proof is a one-line wrapper that projects the counted_once_algebra field from the hypothesis $h$ : FiniteLogicalComparison $C$.
why it matters
This supplies the counted-once composition property required by the downstream theorem finite_logical_comparison_forces_rcl, which establishes that finite logical comparison forces the RCL family. It fills the algebraic content step in the sharpened theorem that finite logical comparison forces the Recognition Composition Law, a key landmark toward deriving physics from the single functional equation. It touches the open question of why the finite condition is necessary, as demonstrated by the continuous and analytic counterexamples in the same module.
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