Foundation Dalembert Stability Cost Stability Calibrated

Stability of the cost

The d'Alembert functional equation, named for Jean le Rond d'Alembert's work in the 1740s on vibrating strings, asks for functions H where H(t+u) + H(t-u) equals 2·H(t)·H(u). Its smooth solutions are the hyperbolic cosine and cosine families. The equation expresses a deep symmetry: the value at a sum and difference is determined by the product of the values at the parts.

In the Recognition Science framework, the cost function, a measure of the price of a recognition event, is forced by a separate chain of theorems to be J(x) = (x + 1/x)/2 - 1. The d'Alembert equation appears because J is related to the hyperbolic cosine by a logarithmic change of variables. The framework's library, a machine-checked collection of formal theorems, proves a stability result: if a function H is almost a solution to the d'Alembert equation, then H is almost the hyperbolic cosine, and by transfer, almost the cost function.

The central declaration, cost_stability_calibrated, states this precisely. It assumes H is smooth, even, and normalized, and that its second derivative at zero, the curvature, is 1. It also assumes the d'Alembert defect, the amount by which H fails the equation, is uniformly small. Under these conditions, the theorem derives an explicit bound: for every x in a finite interval, the absolute difference between H(log x) - 1 and the cost J(x) is at most a computable error term times J(x). The error term involves a parameter h that can be chosen to balance approximation error against the interval width.

The theorem is a quantitative statement of robustness. It does not merely say that exact solutions are unique; it says that near-solutions are near the unique solution. This is what makes the framework's derivation of J credible: the cost function is not a knife-edge artifact, but a stable attractor. Any function that approximately obeys the forced composition law is forced to be approximately J, with a bound that shrinks as the defect shrinks.

What the theorem does not claim is as important as what it proves. It does not assert that any real physical system obeys the d'Alembert equation, nor that the cost function is the only function with the stated stability property. It establishes a conditional statement: given the hypotheses, the conclusion follows. The hypotheses include the curvature being exactly 1, a calibration choice. The theorem also does not claim that the error bound is optimal, only that it exists and is explicit. It is a theorem about the mathematics of the framework, not a claim about the empirical world.

/-- When a = 1, the cost stability simplifies to |F(x) - J(x)| ≤ δ · J(x). -/
theorem cost_stability_calibrated
    (H : ℝ → ℝ) (T : ℝ) (hyp : StabilityHypotheses H T)
    (h_a1 : hyp.curvature = 1)
    (bounds : StabilityBounds H T)
    (h_stab : StabilityEstimate H T hyp.curvature bounds)
    (h_transfer : CostStabilityTransferHypothesis H T hyp bounds)
    (h : ℝ) (hh_pos : 0 < h) (hh_le : h ≤ T) :
    ∀ x : ℝ, Real.exp (-(T - h)) < x → x < Real.exp (T - h) →
    |H (Real.log x) - 1 - Cost.Jcost x| ≤
      δ_error bounds.ε bounds.B bounds.K h * Cost.Jcost x := by
  intro x hx_lo hx_hi
  have h_main := cost_stability_transfer H T hyp bounds h_stab h_transfer h hh_pos hh_le x hx_lo hx_hi
  simp only [h_a1, Real.sqrt_one, one_mul, div_one] at h_main
  -- Need to show cosh(|log x|) - 1 = J(x) when x > 0
  have hx_pos : 0 < x := by linarith [Real.exp_pos (-(T-h))]
  have hJ : Cost.Jcost x = Real.cosh (Real.log x) - 1 := by
    have h1 := Cost.Jcost_exp_cosh (Real.log x)
    simp only [Real.exp_log hx_pos] at h1
    exact h1
  have h_cosh : Real.cosh (Real.log x) - 1 = Cost.Jcost x := by
    symm
    exact hJ
  simpa [h_cosh] using h_main

/-- When a = 1, the cost stability simplifies to |F(x) - J(x)| ≤ δ · J(x). -/
theorem cost_stability_calibrated
    (H : ℝ → ℝ) (T : ℝ) (hyp : StabilityHypotheses H T)
    (h_a1 : hyp.curvature = 1)
    (bounds : StabilityBounds H T)
    (h_stab : StabilityEstimate H T hyp.curvature bounds)
    (h_transfer : CostStabilityTransferHypothesis H T hyp bounds)
    (h : ℝ) (hh_pos : 0 < h) (hh_le : h ≤ T) :
    ∀ x : ℝ, Real.exp (-(T - h)) < x → x < Real.exp (T - h) →
    |H (Real.log x) - 1 - Cost.Jcost x| ≤
      δ_error bounds.ε bounds.B bounds.K h * Cost.Jcost x := by
  intro x hx_lo hx_hi
  have h_main := cost_stability_transfer H T hyp bounds h_stab h_transfer h hh_pos hh_le x hx_lo hx_hi
  simp only [h_a1, Real.sqrt_one, one_mul, div_one] at h_main
  -- Need to show cosh(|log x|) - 1 = J(x) when x > 0
  have hx_pos : 0 < x := by linarith [Real.exp_pos (-(T-h))]
  have hJ : Cost.Jcost x = Real.cosh (Real.log x) - 1 := by
    have h1 := Cost.Jcost_exp_cosh (Real.log x)
    simp only [Real.exp_log hx_pos] at h1
    exact h1
  have h_cosh : Real.cosh (Real.log x) - 1 = Cost.Jcost x := by
    symm
    exact hJ
  simpa [h_cosh] using h_main

What this page does not claim

The theorem does not claim that any physical system obeys the d'Alembert equation. The theorem does not claim the error bound is optimal. The theorem does not establish the uniqueness of J without the forcing chain.

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/Stability.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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