Algebra Cost Algebra Canonical Recognition Cost System Cost One

The canonical cost system

The Recognition Composition Law is the rule that ties together the costs of separate recognition events. If J(x) is the cost of recognizing a ratio x, then the law states that the cost of recognizing both a product and a quotient equals a fixed combination of the individual costs: J(xy) + J(x/y) = 2·J(x)·J(y) + 2·J(x) + 2·J(y). This is the one primitive from which the framework's later results flow.

Within this law, the canonical cost system is the specific choice of cost function that satisfies it. The declaration canonicalRecognitionCostSystem_cost_one establishes the simplest fact about that system: the cost of recognizing the ratio 1, the identity, is exactly zero. In plain terms, recognizing that something is equal to itself costs nothing. This is a theorem, proved in the framework's machine-checked library of formal theorems, and it follows directly from the definition of the cost function J(x) = ½(x + 1/x) − 1.

The zero-cost-at-identity fact is not an isolated curiosity. It is one of the five plain conditions that the framework proves force the unique form of the cost function. The other conditions are reciprocal symmetry (J(x) = J(1/x)), the composition law itself, a calibration condition, and continuity. Together they single out J from all possible cost functions. The declaration is the anchor for the zero-cost condition within the larger proof.

What the declaration does not claim is broader. It does not assert that all recognition events have zero cost; only the identity ratio does. It does not establish the uniqueness of J, which is a separate theorem requiring all five conditions. And it says nothing about what the cost function means physically, only that the identity ratio is cost-free under the canonical choice.

/-- **Normalization**: The multiplicative identity has zero cost. -/
theorem J_at_one : J 1 = 0 := Jcost_unit0

/-- **Cost-composition**: The binary operation on costs induced by the RCL.
    Given two "cost levels" a = J(x) and b = J(y), the composed cost is:
    a ★ b = 2ab + 2a + 2b = 2(a+1)(b+1) − 2

    This captures how costs combine under multiplication of ratios. -/
noncomputable def costCompose (a b : ℝ) : ℝ := 2 * a * b + 2 * a + 2 * b

/-- **THEOREM: The canonical cost algebra is unique.**
    Any CostAlgebraData with the same axioms + calibration J''(1)=1
    must have cost = J. (This is T5 in the forcing chain.) -/
theorem cost_algebra_unique (C : CostAlgebraData)
    (hCalib : deriv (deriv (fun t => C.cost (Real.exp t))) 0 = 1)
    (hCont : ContinuousOn C.cost (Set.Ioi 0))
    (hSmooth : dAlembert_continuous_implies_smooth_hypothesis (IndisputableMonolith.Cost.FunctionalEquation.H C.cost))
    (hODE : dAlembert_to_ODE_hypothesis (IndisputableMonolith.Cost.FunctionalEquation.H C.cost))
    (hContReg : ode_regularity_continuous_hypothesis (IndisputableMonolith.Cost.FunctionalEquation.H C.cost))
    (hDiffReg : ode_regularity_differentiable_hypothesis (IndisputableMonolith.Cost.FunctionalEquation.H C.cost))
    (hBoot : ode_linear_regularity_bootstrap_hypothesis (IndisputableMonolith.Cost.FunctionalEquation.H C.cost)) :
    ∀ x : ℝ, 0 < x → C.cost x = J x := by
  have hRecip : IsReciprocalCost C.cost := by
    intro x hx
    simpa using C.symmetric x hx
  have hNorm : IsNormalized C.cost := by
    simpa [IsNormalized] using C.normalized
  have hComp : SatisfiesCompositionLaw C.cost := by
    intro x y hx hy
    exact C.rcl x y hx hy
  have hCal : IsCalibrated C.cost := by
    simpa [IsCalibrated, G] using hCalib
  intro x hx
  simpa [J] using
    (law_of_logic_forces_jcost_with_regularization C.cost hRecip hNorm hComp hCal hCont
      hSmooth hODE hContReg hDiffReg hBoot x hx)

What this page does not claim

All recognition events have zero cost; only the identity ratio does. The declaration proves the uniqueness of J; that requires all five conditions together. The cost function has a physical meaning beyond being the canonical algebraic choice.

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/Algebra/CostAlgebra.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.

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