Two Premises Reciprocal Cost

Two premises

Imagine keeping the universe's books. A cost assigned to one change must still make sense when two changes are combined, and the scale of that cost must be fixed at the point where nothing changes. In the cost theorem, those are the two premises that matter: the composition law and unit log-curvature, a local statement that fixes the second-order scale at the identity. On paper, only the composition law and the calibration are needed.

The composition law says that the costs of a product and its quotient fit together in one consistent rule. Move to logarithmic coordinates and the rule becomes a d'Alembert equation, a functional equation whose doubling step reads like the Chebyshev relation H(2s) = 2H(s)² - 1. The calibration then opens a small window around the identity where the transformed function is above one. Repeated doubling carries that local information to every scale. The result is exact, rather than an approximation: the cost is J(x) = (x + 1/x)/2 - 1 for every positive input.

What do these premises leave open? They select the cost curve and its scale, but they do not by themselves explain why physical recognition should obey that composition law, or how a ledger of events becomes a physical system. They also do not make every nearby hypothesis unnecessary in every formal statement. The paper's result concerns the assumptions of the uniqueness argument, and the Lean theorem shows the corresponding conclusion under its formal regularity package. The curve is fixed; the physical interpretation and the route from recognition to that law remain separate questions.

/-- **The cost theorem on two premises.** The composition law and unit log
curvature force `F = J` on the positives. Normalization, nonnegativity, and
continuity are all conclusions rather than hypotheses; compare
`law_of_logic_forces_jcost`, which assumes all of them. -/
theorem composition_logCurvature_forces_jcost [AczelSmoothnessPackage]
    (F : ℝ → ℝ)
    (hComp : SatisfiesCompositionLaw F)
    (hκ : HasLogCurvature (H F) 1) :
    ∀ x : ℝ, 0 < x → F x = Cost.Jcost x := by
  have hNorm : IsNormalized F := logCurvature_forces_normalized F hComp hκ
  have hN : F 1 = 0 := hNorm
  have hH0 : H F 0 = 1 := by simp [H, G, hN]
  have hCosh : CoshAddIdentity F := (composition_law_equiv_coshAdd F).mp hComp
  have hdA : ∀ t u, H F (t + u) + H F (t - u) = 2 * H F t * H F u := by
    intro t u
    have hG := hCosh t u
    have hgoal :
        (G F (t + u) + 1) + (G F (t - u) + 1) =
          2 * (G F t + 1) * (G F u + 1) := by
      calc
        (G F (t + u) + 1) + (G F (t - u) + 1)
            = (G F (t + u) + G F (t - u)) + 2 := by ring
        _ = (2 * (G F t * G F u) + 2 * (G F t + G F u)) + 2 := by simpa [hG]
        _ = 2 * (G F t + 1) * (G F u + 1) := by ring
    simpa [H] using hgoal
  have hcont : Continuous (H F) :=
    dAlembert_continuous_of_log_curvature (H F) hH0 hdA hκ
  have hsm : ContDiff ℝ ⊤ (H F) := aczel_dAlembert_smooth (H F) hH0 hcont hdA
  have heven : Function.Even (H F) := dAlembert_even (H F) hH0 hdA
  have hd0 : deriv (H F) 0 = 0 :=
    even_deriv_at_zero (H F) heven
      (hsm.differentiable (by decide : (⊤ : WithTop ℕ∞) ≠ 0) 0)
  have hd2 : deriv (deriv (H F)) 0 = 1 :=
    deriv2_of_logCurvature (H F) hsm hH0 hd0 hκ
  have hcosh : ∀ t, H F t = Real.cosh t :=
    dAlembert_cosh_solution_aczel (H F) hH0 hcont hdA hd2
  intro x hx
  have hGc : G F (Real.log x) = Real.cosh (Real.log x) - 1 := by
    have h := hcosh (Real.log x)
    simp only [H] at h
    linarith
  have ht : Real.exp (Real.log x) = x := Real.exp_log hx
  have hJG : G Cost.Jcost (Real.log x) = Real.cosh (Real.log x) - 1 :=
    Jcost_G_eq_cosh_sub_one (Real.log x)
  calc
    F x = F (Real.exp (Real.log x)) := by rw [ht]
    _ = G F (Real.log x) := rfl
    _ = Real.cosh (Real.log x) - 1 := hGc
    _ = G Cost.Jcost (Real.log x) := by simp only [hJG]
    _ = Cost.Jcost (Real.exp (Real.log x)) := by simp [G]
    _ = Cost.Jcost x := by simp [ht]

What this page does not claim

That the two premises explain the physical origin of the composition law. That every formal regularity assumption has been removed from every Lean statement of the theorem.

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

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