Encyclopedia Cost Cost Cont Diff Reduction D Alembert Second Deriv At Zero Of Cont Diff

ARTICLE 4 claims 4 theorems

Cost Cont Diff Reduction D Alembert Second Deriv At Zero Of Cont Diff

A single equation from 1747, the d'Alembert functional equation, ties the curvature of a cost function at zero to its curvature everywhere, and a machine-checked library proves the link for all twice-differentiable solutions.

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The second derivative bridge

The d'Alembert functional equation, d'Alembert equation (a relation that forces a function's value at a sum and difference to equal a product of its values at the inputs), appeared in 1747 in Jean le Rond d'Alembert's work on vibrating strings. For a function H, it reads H(t+u) + H(t−u) = 2·H(t)·H(u). The classical continuous solutions are the hyperbolic cosine and the ordinary cosine; the equation's structure appears across harmonic analysis and probability theory.

The theorem dAlembert_second_deriv_at_zero_of_contDiff in the framework's machine-checked library of formal theorems proves a precise bridge: for any twice continuously differentiable function H satisfying the d'Alembert equation, the second derivative at any point t equals the function's value at t times the second derivative at zero. In symbols, 2·H''(t) = 2·H(t)·H''(0). This is a recognition (the framework's term for an event the universe records as a discrete cost) regularity result: it shows that the curvature of such a function is entirely determined by its curvature at a single point, the origin.

The theorem does not claim that every d'Alembert solution is twice differentiable, nor that the second derivative at zero is any particular number. It only relates the second derivative at zero to the second derivative elsewhere, assuming the function is already smooth enough to have those derivatives. The companion theorem dAlembert_to_ODE_of_contDiff adds the calibration condition H''(0) = 1 and then derives the full differential equation H'' = H, whose solution is the hyperbolic cosine. That step is a separate theorem, not part of the second-derivative bridge itself.

In Recognition Science, this bridge matters because it removes a regularity assumption from the main forcing theorem. The framework models cost as a function J(x) = (x + 1/x)/2 − 1, and the central result proves this form follows from normalization, composition, and calibration alone. The second-derivative theorem shows that on the twice-differentiable surface, reciprocity (the symmetry J(x) = J(1/x)) is derived rather than assumed, so the canonical cost follows from fewer premises. The bridge is one link in that chain: it converts a local property at zero into a global statement about the whole function.

THEOREM dAlembert_second_deriv_at_zero_of_contDiff · IndisputableMonolith/Cost/ContDiffReduction.lean
dAlembert_second_deriv_at_zero_of_contDiff · IndisputableMonolith/Cost/ContDiffReduction.lean:80 
/-- Differentiate the first-derivative identity at `u = 0` to relate `H''(t)` to `H''(0)`. -/
theorem dAlembert_second_deriv_at_zero_of_contDiff
    (Hf : ℝ → ℝ)
    (h_dAlembert : ∀ t u, Hf (t + u) + Hf (t - u) = 2 * Hf t * Hf u)
    (h_diff : ContDiff ℝ 2 Hf) :
    ∀ t, 2 * deriv (deriv Hf) t = 2 * Hf t * deriv (deriv Hf) 0 := by
  intro t
  have h_first :
      (fun u => deriv Hf (t + u) - deriv Hf (t - u)) =
        ((fun _ : ℝ => 2 * Hf t) * deriv Hf) := by
    funext u
    simpa [Pi.mul_apply, mul_assoc] using
      dAlembert_first_deriv_of_contDiff Hf h_dAlembert h_diff t u
  have h_plus :
      HasDerivAt (fun u => deriv Hf (t + u)) (deriv (deriv Hf) t) 0 := by
    have h_inner : HasDerivAt (fun u => t + u) 1 0 := by
      simpa using (hasDerivAt_const 0 t).add (hasDerivAt_id 0)
    simpa using (hasDerivAt_deriv_of_contDiffTwo h_diff (t + 0)).comp 0 h_inner
  have h_minus_raw :
      HasDerivAt (fun u => deriv Hf (t - u)) (-deriv (deriv Hf) t) 0 := by
    have h_inner : HasDerivAt (fun u => t - u) (-1) 0 := by
      simpa using (hasDerivAt_const 0 t).sub (hasDerivAt_id 0)
    simpa using (hasDerivAt_deriv_of_contDiffTwo h_diff (t - 0)).comp 0 h_inner
  have h_left_raw :
      HasDerivAt (fun u => deriv Hf (t + u) - deriv Hf (t - u))
        (deriv (deriv Hf) t + deriv (deriv Hf) t) 0 := by
    simpa using h_plus.sub h_minus_raw
  have h_const : HasDerivAt (fun _ : ℝ => 2 * Hf t) 0 0 :=
    hasDerivAt_const 0 (2 * Hf t)
  have h_right :
      HasDerivAt (((fun _ : ℝ => 2 * Hf t) * deriv Hf))
        (2 * (Hf t * deriv (deriv Hf) 0)) 0 := by
    simpa [mul_assoc] using h_const.mul (hasDerivAt_deriv_of_contDiffTwo h_diff 0)
  have h_deriv_eq := congrArg (fun f : ℝ → ℝ => deriv f 0) h_first
  change deriv (fun u => deriv Hf (t + u) - deriv Hf (t - u)) 0 =
      deriv (((fun _ : ℝ => 2 * Hf t) * deriv Hf)) 0 at h_deriv_eq
  rw [h_left_raw.deriv, h_right.deriv] at h_deriv_eq
  linarith
THEOREM dAlembert_second_deriv_at_zero_of_contDiff · IndisputableMonolith/Cost/ContDiffReduction.lean
dAlembert_second_deriv_at_zero_of_contDiff · IndisputableMonolith/Cost/ContDiffReduction.lean:80 
/-- Differentiate the first-derivative identity at `u = 0` to relate `H''(t)` to `H''(0)`. -/
theorem dAlembert_second_deriv_at_zero_of_contDiff
    (Hf : ℝ → ℝ)
    (h_dAlembert : ∀ t u, Hf (t + u) + Hf (t - u) = 2 * Hf t * Hf u)
    (h_diff : ContDiff ℝ 2 Hf) :
    ∀ t, 2 * deriv (deriv Hf) t = 2 * Hf t * deriv (deriv Hf) 0 := by
  intro t
  have h_first :
      (fun u => deriv Hf (t + u) - deriv Hf (t - u)) =
        ((fun _ : ℝ => 2 * Hf t) * deriv Hf) := by
    funext u
    simpa [Pi.mul_apply, mul_assoc] using
      dAlembert_first_deriv_of_contDiff Hf h_dAlembert h_diff t u
  have h_plus :
      HasDerivAt (fun u => deriv Hf (t + u)) (deriv (deriv Hf) t) 0 := by
    have h_inner : HasDerivAt (fun u => t + u) 1 0 := by
      simpa using (hasDerivAt_const 0 t).add (hasDerivAt_id 0)
    simpa using (hasDerivAt_deriv_of_contDiffTwo h_diff (t + 0)).comp 0 h_inner
  have h_minus_raw :
      HasDerivAt (fun u => deriv Hf (t - u)) (-deriv (deriv Hf) t) 0 := by
    have h_inner : HasDerivAt (fun u => t - u) (-1) 0 := by
      simpa using (hasDerivAt_const 0 t).sub (hasDerivAt_id 0)
    simpa using (hasDerivAt_deriv_of_contDiffTwo h_diff (t - 0)).comp 0 h_inner
  have h_left_raw :
      HasDerivAt (fun u => deriv Hf (t + u) - deriv Hf (t - u))
        (deriv (deriv Hf) t + deriv (deriv Hf) t) 0 := by
    simpa using h_plus.sub h_minus_raw
  have h_const : HasDerivAt (fun _ : ℝ => 2 * Hf t) 0 0 :=
    hasDerivAt_const 0 (2 * Hf t)
  have h_right :
      HasDerivAt (((fun _ : ℝ => 2 * Hf t) * deriv Hf))
        (2 * (Hf t * deriv (deriv Hf) 0)) 0 := by
    simpa [mul_assoc] using h_const.mul (hasDerivAt_deriv_of_contDiffTwo h_diff 0)
  have h_deriv_eq := congrArg (fun f : ℝ → ℝ => deriv f 0) h_first
  change deriv (fun u => deriv Hf (t + u) - deriv Hf (t - u)) 0 =
      deriv (((fun _ : ℝ => 2 * Hf t) * deriv Hf)) 0 at h_deriv_eq
  rw [h_left_raw.deriv, h_right.deriv] at h_deriv_eq
  linarith
THEOREM dAlembert_to_ODE_of_contDiff · IndisputableMonolith/Cost/ContDiffReduction.lean
dAlembert_to_ODE_of_contDiff · IndisputableMonolith/Cost/ContDiffReduction.lean:119 
/-- A `C²` d'Alembert solution with calibrated second derivative satisfies `H'' = H`. -/
theorem dAlembert_to_ODE_of_contDiff
    (Hf : ℝ → ℝ)
    (h_dAlembert : ∀ t u, Hf (t + u) + Hf (t - u) = 2 * Hf t * Hf u)
    (h_diff : ContDiff ℝ 2 Hf)
    (h_deriv2_zero : deriv (deriv Hf) 0 = 1) :
    ∀ t, deriv (deriv Hf) t = Hf t := by
  intro t
  have h_rel := dAlembert_second_deriv_at_zero_of_contDiff Hf h_dAlembert h_diff t
  rw [h_deriv2_zero] at h_rel
  linarith
THEOREM composition_law_forces_reciprocity · IndisputableMonolith/Cost/ContDiffReduction.lean
composition_law_forces_reciprocity · IndisputableMonolith/Cost/ContDiffReduction.lean:138 
/-- A normalized composition-law cost is automatically reciprocal. -/
theorem composition_law_forces_reciprocity
    (F : ℝ → ℝ)
    (hNorm : IsNormalized F)
    (hComp : SatisfiesCompositionLaw F) :
    IsReciprocalCost F := by
  intro x hx
  let Hf : ℝ → ℝ := H F
  have h_H0 : Hf 0 = 1 := by
    dsimp [Hf]
    simpa [H, G, IsNormalized] using hNorm
  have hCoshAdd : CoshAddIdentity F := (composition_law_equiv_coshAdd F).mp hComp
  have h_direct : DirectCoshAdd (G F) := CoshAddIdentity_implies_DirectCoshAdd F hCoshAdd
  have h_dAlembert : ∀ t u, Hf (t + u) + Hf (t - u) = 2 * Hf t * Hf u := by
    intro t u
    have hG := h_direct t u
    have h_goal :
        (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 [Hf, H] using h_goal
  have h_even : Function.Even Hf := dAlembert_even Hf h_H0 h_dAlembert
  have h_even_at_log := h_even (Real.log x)
  have h_eq_plus :
      F x + 1 = F x⁻¹ + 1 := by
    simpa [Hf, H, G, Real.exp_log hx, Real.exp_neg] using h_even_at_log.symm
  linarith

What this page does not claim

The theorem does not prove that all d'Alembert solutions are twice differentiable. The theorem does not assign a particular value to the second derivative at zero. The theorem does not by itself derive the canonical cost function; that requires the companion ODE theorem and the composition-law reciprocity result.

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

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