truth_eval_implies_composition
plain-language theorem explainer
A truth-evaluable comparison operator on positive reals satisfies finite pairwise polynomial closure. Workers deriving the logic functional equation cite this to link semantic truth conditions to algebraic route independence. The proof extracts the composite_determinate field directly from the hypothesis structure.
Claim. Let $C : (0,∞) × (0,∞) → ℝ$ be a comparison operator. If $C$ is truth-evaluable, then $C$ satisfies finite pairwise polynomial closure (route independence).
background
A ComparisonOperator is an abbrev for ℝ → ℝ → ℝ giving the real-valued cost of comparing two positive quantities under the four Aristotelian constraints. TruthEvaluableComparison is the structure whose fields encode self-evaluation to zero, single-valued reordering, continuous determinacy on positive pairs, composite determinacy, scale invariance, and nontriviality; its composite_determinate field is exactly FinitePairwisePolynomialClosure C. The module formalizes the Reality ⇒ Logic direction, translating truth-evaluability into the encoded laws (L1)–(L4). FinitePairwisePolynomialClosure is defined as RouteIndependence C in the DirectProof module.
proof idea
The proof is a one-line wrapper that applies the composite_determinate field of the TruthEvaluableComparison hypothesis.
why it matters
This result is invoked by reality_satisfies_logic to conclude that truth-evaluable comparisons obey the laws of logic, and by rcl_from_truth_evaluable_comparison to obtain the RCL family with multiplicative consistency. It supplies the object-level step that converts the semantic truth-evaluable structure into the algebraic conditions required by the Logic Functional Equation paper. The step sits inside the T0–T8 forcing chain that ultimately forces the Recognition Composition Law.
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