J_is_unique_cost_under_logic
plain-language theorem explainer
Any cost derived from a comparison operator obeying the four Aristotelian laws, the c=2 Recognition Composition Law, and unit log-curvature calibration equals the canonical reciprocal cost J(x) = (x + x^{-1})/2 - 1 for positive x. Recognition Science researchers cite this to ground the J-cost in logic before deriving physics constants. The proof translates the logic axioms into d'Alembert hypotheses and applies the Aczél uniqueness theorem for reciprocal costs.
Claim. Let $C$ be a comparison operator on positive reals satisfying the four Aristotelian laws of logic. Let $F$ be the cost function obtained by fixing the second argument of $C$ at 1. Assume $F$ satisfies the Recognition Composition Law $F(xy) + F(x/y) = 2F(x)F(y) + 2F(x) + 2F(y)$ for all $x,y > 0$ and is calibrated by the condition that the second derivative of its log-transform at zero equals 1. Then $F(x) = J(x)$ for all $x > 0$, where $J(x) = (x + x^{-1})/2 - 1$.
background
ComparisonOperator maps pairs of positive reals to a real cost of comparison. SatisfiesLawsOfLogic packages the four Aristotelian constraints (identity, non-contradiction, excluded middle) together with scale invariance, route independence, and non-triviality; these allow reduction to a one-argument cost on positive ratios via derivedCost. The module works inside the Foundation setting that treats logic as a functional equation whose solutions must obey the Recognition Composition Law (RCL). Upstream, AczelSmoothnessPackage supplies the smoothness needed for continuous d'Alembert solutions, while IsCalibrated enforces the unit second-derivative condition at the origin on the log-transformed function. washburn_uniqueness_aczel supplies the core uniqueness result for reciprocal costs under RCL and calibration.
proof idea
The tactic proof first calls laws_of_logic_imply_dalembert_hypotheses to extract normalization, symmetry, continuity and the remaining d'Alembert hypotheses from the logic axioms. It then builds an IsReciprocalCost instance directly from the symmetry component. The proof concludes by applying washburn_uniqueness_aczel to the derived cost together with the reciprocal property, the supplied RCL, calibration, and continuity.
why it matters
The theorem completes the translation from Aristotelian logic to the unique J-cost, realizing the T5 J-uniqueness step of the forcing chain inside the Recognition framework. It feeds the subsequent derivation of the phi-ladder, eight-tick octave, and constants such as hbar = phi^{-5}. The result rests on the peer-reviewed Washburn-Zlatanović uniqueness theorem and leaves open the polynomiality assumption on the route-independence combiner, which the generalized d'Alembert module is intended to discharge.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.