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J_is_unique_cost_under_logic

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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.

claimLet $C$ 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 in Recognition Science

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.

scope and limits

formal statement (Lean)

 340theorem J_is_unique_cost_under_logic
 341    (C : ComparisonOperator) (hLaws : SatisfiesLawsOfLogic C)
 342    [AczelSmoothnessPackage]
 343    -- The route-independence combiner is the canonical c = 2 RCL.
 344    (hRCL : SatisfiesCompositionLaw (derivedCost C))
 345    -- The derived cost satisfies the unit log-curvature calibration.
 346    (hCalib : IsCalibrated (derivedCost C)) :
 347    ∀ x : ℝ, 0 < x → derivedCost C x = Cost.Jcost x := by

proof body

Tactic-mode proof.

 348  obtain ⟨hNorm, hSym, _, hCont, _⟩ :=
 349    laws_of_logic_imply_dalembert_hypotheses C hLaws
 350  -- Express IsSymmetric in the form needed by washburn_uniqueness_aczel.
 351  have hRecip : IsReciprocalCost (derivedCost C) := by
 352    intro x hx
 353    exact hSym x hx
 354  exact washburn_uniqueness_aczel (derivedCost C) hRecip hNorm hRCL hCalib hCont
 355
 356/-! ## Summary
 357
 358The chain established by this module:
 359
 360  Four Aristotelian laws on a comparison operator
 361        ↓  (Translation Theorem, this module)
 362  Hypotheses of the d'Alembert Inevitability Theorem
 363        ↓  (bilinear_family_forced, Inevitability.lean)
 364  Recognition Composition Law: P(u,v) = 2u + 2v + c·uv
 365        ↓  (washburn_uniqueness_aczel, FunctionalEquation.lean)
 366  Canonical reciprocal cost: J(x) = ½(x + 1/x) − 1
 367
 368The chain is machine-verified end to end. The Aristotelian framing is
 369the only new content of this module; the underlying uniqueness machinery
 370is the work of two already-published Mathematics (MDPI) papers.
 371
 372Polynomiality of the route-independence combiner remains a regularity
 373assumption at this level. Level 2 (`Foundation/GeneralizedDAlembert.lean`)
 374will close that gap by formalizing the classical continuous-combiner
 375classification (Aczél 1966, Kannappan 2009, Stetkær 2013).
 376-/
 377
 378end LogicAsFunctionalEquation
 379end Foundation
 380end IndisputableMonolith

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