Foundation Dalembert Full Unconditional Log Consistency Of Mult Consistency

The bridge to log coordinates

The declaration log_consistency_of_mult_consistency is a formal bridge between two ways of writing the same rule. Suppose a function F on positive numbers obeys a multiplicative consistency law: combining two inputs by multiplication or division, the outputs add up to some fixed function P of the individual outputs. The theorem says that if you switch to logarithmic coordinates, writing G(t) = F(e^t), the same law becomes an additive one: G(t+u) + G(t-u) = P(G(t), G(u)). This is the d'Alembert equation, a classical functional equation studied since Jean le Rond d'Alembert in 1750.

The bridge matters because additive equations are far easier to solve than multiplicative ones. In log coordinates, the unknown function G satisfies a linear-looking relation that forces it to be a hyperbolic cosine, G(t) = cosh(t) - 1, once a few regularity conditions hold. Translating back, F(x) = (x + 1/x)/2 - 1. This is the cost function J in the Recognition Science framework: a measure of how expensive it is for a ledger to recognize a ratio x. The theorem itself does not solve the equation; it only shows that the two formulations are equivalent, so any result proved for the additive version applies to the multiplicative one.

What the declaration does not claim is just as precise. It assumes nothing about the function P beyond its existence; P can be any function of two variables. The theorem does not require P to be symmetric, polynomial, or even continuous. It also does not assume F is smooth, normalized, or symmetric. The bridge holds for any F and P satisfying the single multiplicative consistency law. The stronger conclusions, that F must equal J and P must equal 2uv + 2u + 2v, come from separate theorems that add hypotheses such as smoothness and calibration.

In the framework's library, this theorem is a lemma, not the final result. It feeds into the full unconditional inevitability theorem, which shows that both F and P are forced with no assumption on P. The practical consequence is that the framework's central cost function is not chosen but derived: any cost function satisfying five plain conditions must equal J. The log-consistency bridge is what makes that derivation tractable, by converting a hard multiplicative problem into a solvable additive one.

/-- From F-consistency to G-consistency in log coordinates. -/
theorem log_consistency_of_mult_consistency
    (F : ℝ → ℝ)
    (P : ℝ → ℝ → ℝ)
    (hCons : ∀ x y : ℝ, 0 < x → 0 < y → F (x * y) + F (x / y) = P (F x) (F y)) :
    LogConsistency (G F) P := by
  intro t u
  simp only [G]
  have hexp_t : 0 < Real.exp t := Real.exp_pos t
  have hexp_u : 0 < Real.exp u := Real.exp_pos u
  have h := hCons (Real.exp t) (Real.exp u) hexp_t hexp_u
  rw [← Real.exp_add, ← Real.exp_sub] at h
  exact h

/-- From F-consistency to G-consistency in log coordinates. -/
theorem log_consistency_of_mult_consistency
    (F : ℝ → ℝ)
    (P : ℝ → ℝ → ℝ)
    (hCons : ∀ x y : ℝ, 0 < x → 0 < y → F (x * y) + F (x / y) = P (F x) (F y)) :
    LogConsistency (G F) P := by
  intro t u
  simp only [G]
  have hexp_t : 0 < Real.exp t := Real.exp_pos t
  have hexp_u : 0 < Real.exp u := Real.exp_pos u
  have h := hCons (Real.exp t) (Real.exp u) hexp_t hexp_u
  rw [← Real.exp_add, ← Real.exp_sub] at h
  exact h

/-- From F-consistency to G-consistency in log coordinates. -/
theorem log_consistency_of_mult_consistency
    (F : ℝ → ℝ)
    (P : ℝ → ℝ → ℝ)
    (hCons : ∀ x y : ℝ, 0 < x → 0 < y → F (x * y) + F (x / y) = P (F x) (F y)) :
    LogConsistency (G F) P := by
  intro t u
  simp only [G]
  have hexp_t : 0 < Real.exp t := Real.exp_pos t
  have hexp_u : 0 < Real.exp u := Real.exp_pos u
  have h := hCons (Real.exp t) (Real.exp u) hexp_t hexp_u
  rw [← Real.exp_add, ← Real.exp_sub] at h
  exact h

What this page does not claim

The theorem does not assume P is symmetric, polynomial, or even continuous. The theorem does not prove that F must equal J or that P must equal 2uv + 2u + 2v. The theorem does not require F to be smooth, normalized, or symmetric.

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

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