Cost Functional Equation Aczel Law Of Logic Forces Jcost Aczel

The uniqueness theorem

The cost of a recognition event is a number that measures how much work it takes to tell two things apart. In the Recognition Science framework, the ledger, a discrete record of such events, assigns each pair a cost. The framework's central question is whether the form of that cost is forced by a few natural conditions, or whether many different cost functions could serve equally well.

The answer, proved in the framework's machine-checked library of formal theorems, is that only one cost function survives. If a cost function F is reciprocal, meaning F(1/x) = F(x); normalized so that F(1) = 0; obeys a composition law that ties the cost of a pair to the costs of its parts; is calibrated so that the cost of a pair equals the cost of its reciprocal pair; and is continuous for positive arguments, then F(x) must equal J(x) = (x + 1/x)/2 - 1 for every positive x. The theorem's name, law_of_logic_forces_jcost_aczel, records that it uses a classical closure argument due to János Aczél to finish the uniqueness proof.

The proof is not a sketch. It is a formal derivation checked by the Lean kernel, with no gaps and no extra axioms beyond the three standard ones of the ambient type theory. The declaration law_of_logic_forces_jcost_aczel is a thin compatibility wrapper: it states the same result as the core theorem law_of_logic_forces_jcost, but packages it so that callers need not supply regularity hypotheses themselves. The Aczél-based version relies on a global axiom internally, which is why the framework's main development prefers the core module.

What the theorem does not claim is just as important. It does not say that the five conditions are the only possible starting points, nor that the J-cost function is the only cost function that could ever be imagined. It says that within this exact set of assumptions, the solution is unique. The result is a mathematical fact about a functional equation, not a statement about which physical systems actually use this cost. That empirical question is separate and remains open.

The consequence is practical. Once the uniqueness is established, the framework can build on J without worrying that a different cost function might also satisfy the same laws. The golden ratio, the eight-tick cycle, and the three spatial dimensions that the framework derives later all rest on this single pinned-down function. The theorem is the load-bearing wall; everything else in the framework's forcing chain stands on it.

/-- **Law of Logic cost theorem, Aczél closure**: The J-cost function is the unique
    reciprocal cost satisfying the RCL, normalization, calibration, and continuity.

    This version uses the global Aczél axiom internally and requires NO regularity
    hypothesis parameters from the caller. -/
theorem law_of_logic_forces_jcost_aczel (F : ℝ → ℝ)
    (hRecip : IsReciprocalCost F)
    (hNorm : IsNormalized F)
    (hComp : SatisfiesCompositionLaw F)
    (hCalib : IsCalibrated F)
    (hCont : ContinuousOn F (Set.Ioi 0)) :
    ∀ x : ℝ, 0 < x → F x = Cost.Jcost x := by
  exact law_of_logic_forces_jcost F hRecip hNorm hComp hCalib hCont

/-- **Law of Logic cost theorem, Aczél closure**: The J-cost function is the unique
    reciprocal cost satisfying the RCL, normalization, calibration, and continuity.

    This version uses the global Aczél axiom internally and requires NO regularity
    hypothesis parameters from the caller. -/
theorem law_of_logic_forces_jcost_aczel (F : ℝ → ℝ)
    (hRecip : IsReciprocalCost F)
    (hNorm : IsNormalized F)
    (hComp : SatisfiesCompositionLaw F)
    (hCalib : IsCalibrated F)
    (hCont : ContinuousOn F (Set.Ioi 0)) :
    ∀ x : ℝ, 0 < x → F x = Cost.Jcost x := by
  exact law_of_logic_forces_jcost F hRecip hNorm hComp hCalib hCont

/-- **Law of Logic cost theorem, Aczél closure**: The J-cost function is the unique
    reciprocal cost satisfying the RCL, normalization, calibration, and continuity.

    This version uses the global Aczél axiom internally and requires NO regularity
    hypothesis parameters from the caller. -/
theorem law_of_logic_forces_jcost_aczel (F : ℝ → ℝ)
    (hRecip : IsReciprocalCost F)
    (hNorm : IsNormalized F)
    (hComp : SatisfiesCompositionLaw F)
    (hCalib : IsCalibrated F)
    (hCont : ContinuousOn F (Set.Ioi 0)) :
    ∀ x : ℝ, 0 < x → F x = Cost.Jcost x := by
  exact law_of_logic_forces_jcost F hRecip hNorm hComp hCalib hCont

What this page does not claim

The theorem does not claim that the five conditions are the only possible axioms for a cost function. The theorem does not claim that any physical system actually uses the J-cost function. The theorem does not claim that the Aczél-based version is the framework's preferred proof route.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Cost/FunctionalEquationAczel.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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