Foundation Dalembert Full Unconditional Consistency Forces Rcl Form Is Theorem

The forced composition law

The d'Alembert functional equation, H(t+u) + H(t-u) = 2H(t)H(u), has a long history. Jean le Rond d'Alembert studied it in 1750 while working on the physics of vibrating strings. The equation asks for functions where the value at a sum plus the value at a difference equals twice the product of the values. Its continuous solutions are the hyperbolic cosine, cosh, and the ordinary cosine, cos.

Recognition Science builds on a similar but more general consistency condition. The framework models recognition as a ledger, a discrete record of events, where each event carries a cost, a number measuring the price of recognizing that event. The cost function F must satisfy F(xy) + F(x/y) = P(F(x), F(y)) for some function P. This says the cost of a product and a quotient must combine in a way that depends only on the individual costs, not on the original inputs. The framework's library, a machine-checked collection of formal theorems, proves that this consistency condition, together with symmetry, normalization, and smoothness, forces P to have the specific form P(u,v) = 2uv + 2u + 2v.

This result, named consistency_forces_RCL_form_is_theorem, is a theorem in the library. It shows that the composition law for costs is not a free choice. The theorem requires P to be a polynomial of degree at most two, symmetric, and to satisfy P(1,1) = 6. Under those conditions, the form of P is uniquely determined. The theorem does not require F to be the specific cost function J; it works for any F meeting the stated conditions.

The consequence is that the way costs combine is as forced as the costs themselves. The framework proves a separate theorem that the cost function must be J(x) = (x + 1/x)/2 - 1. Together, these results mean the entire recognition structure, both the individual costs and the rule for combining them, follows from a few plain assumptions. The composition law is not an additional postulate; it is a consequence.

The theorem's scope has limits. It applies to the range of F, meaning it pins down P only for values that F actually produces. It does not describe what P does for arbitrary inputs outside that range. The theorem also assumes P is a polynomial and symmetric. Without those assumptions, the conclusion does not follow from the library's proofs.

/-- Bridging theorem: polynomial inevitability implies RCL form once the canonical
normalization `P 1 1 = 6` (equivalently `c = 2`) is fixed. -/
theorem consistency_forces_RCL_form_is_theorem
    (F : ℝ → ℝ) (P : ℝ → ℝ → ℝ)
    (_hSymm : ∀ x : ℝ, 0 < x → F x = F x⁻¹)
    (hUnit : F 1 = 0)
    (_hSmooth : ContDiff ℝ 2 F)
    (hCons : ∀ x y : ℝ, 0 < x → 0 < y → F (x * y) + F (x / y) = P (F x) (F y))
    (hPoly : ∃ (a b c d e f : ℝ), ∀ u v, P u v = a + b*u + c*v + d*u*v + e*u^2 + f*v^2)
    (hSymP : ∀ u v, P u v = P v u)
    (hNonTriv : ∃ x : ℝ, 0 < x ∧ F x ≠ 0)
    (hCont : ContinuousOn F (Set.Ioi 0))
    (hP11 : P 1 1 = 6) :
    ∀ x y : ℝ, 0 < x → 0 < y →
      P (F x) (F y) = 2 * F x * F y + 2 * F x + 2 * F y := by
  obtain ⟨c, hc, _⟩ :=
    Inevitability.bilinear_family_forced F P hUnit hCons hPoly hSymP hNonTriv hCont
  have hc_two : c = 2 := by
    have h11_formula : P 1 1 = 2 * 1 + 2 * 1 + c * 1 * 1 := by
      simpa using hc 1 1
    linarith [hP11, h11_formula]
  intro x y hx hy
  calc
    P (F x) (F y) = 2 * F x + 2 * F y + c * F x * F y := by simpa using hc (F x) (F y)
    _ = 2 * F x * F y + 2 * F x + 2 * F y := by rw [hc_two]; ring

/-- Bridging theorem: polynomial inevitability implies RCL form once the canonical
normalization `P 1 1 = 6` (equivalently `c = 2`) is fixed. -/
theorem consistency_forces_RCL_form_is_theorem
    (F : ℝ → ℝ) (P : ℝ → ℝ → ℝ)
    (_hSymm : ∀ x : ℝ, 0 < x → F x = F x⁻¹)
    (hUnit : F 1 = 0)
    (_hSmooth : ContDiff ℝ 2 F)
    (hCons : ∀ x y : ℝ, 0 < x → 0 < y → F (x * y) + F (x / y) = P (F x) (F y))
    (hPoly : ∃ (a b c d e f : ℝ), ∀ u v, P u v = a + b*u + c*v + d*u*v + e*u^2 + f*v^2)
    (hSymP : ∀ u v, P u v = P v u)
    (hNonTriv : ∃ x : ℝ, 0 < x ∧ F x ≠ 0)
    (hCont : ContinuousOn F (Set.Ioi 0))
    (hP11 : P 1 1 = 6) :
    ∀ x y : ℝ, 0 < x → 0 < y →
      P (F x) (F y) = 2 * F x * F y + 2 * F x + 2 * F y := by
  obtain ⟨c, hc, _⟩ :=
    Inevitability.bilinear_family_forced F P hUnit hCons hPoly hSymP hNonTriv hCont
  have hc_two : c = 2 := by
    have h11_formula : P 1 1 = 2 * 1 + 2 * 1 + c * 1 * 1 := by
      simpa using hc 1 1
    linarith [hP11, h11_formula]
  intro x y hx hy
  calc
    P (F x) (F y) = 2 * F x + 2 * F y + c * F x * F y := by simpa using hc (F x) (F y)
    _ = 2 * F x * F y + 2 * F x + 2 * F y := by rw [hc_two]; ring

What this page does not claim

The theorem does not prove that P has the RCL form for all possible inputs, only on the range of F. The theorem does not derive the cost function J itself; that is a separate result. The theorem does not apply without the polynomial and symmetry assumptions on P.

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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