Cost Functional Equation Ode Regularity Differentiable Of Smooth

The differentiability step

The declaration ode_regularity_differentiable_hypothesis is a formal hypothesis used inside the Recognition Science framework's proof that a certain cost function is unique. In plain terms, it states that a function which satisfies a particular functional equation, and is known to be continuous, must also be differentiable. This is a regularity condition: it upgrades a weak property (continuity) to a stronger one (differentiability), which is a standard kind of step in analysis.

The framework's library proves that this differentiability hypothesis, together with several other regularity hypotheses, is enough to force the unique solution. Specifically, the theorem dAlembert_cosh_solution shows that if a function H satisfies the d'Alembert equation H(t+u) + H(t-u) = 2 H(t) H(u), has H(0) = 1, is continuous, has second derivative 1 at 0, and satisfies the differentiability and other regularity hypotheses, then H(t) = cosh(t) for all t. This is a machine-checked theorem in the framework's library of formal theorems.

The differentiability hypothesis is one of five regularity conditions used in the proof. The others are: continuity, a smoothness condition, a condition that the d'Alembert equation implies an ordinary differential equation, and a bootstrap condition that upgrades differentiability to infinite smoothness. Together, these conditions allow the proof to pass from the functional equation to the differential equation H''(t) = H(t), whose unique solution with the given initial conditions is cosh.

The declaration does not claim that differentiability alone forces the solution. Without the other regularity hypotheses, the d'Alembert equation has many wild, non-differentiable solutions. The differentiability hypothesis is a sufficient condition within a larger package, not a standalone result. It also does not claim that the cost function itself is differentiable; it applies to the auxiliary function H, which is defined from the cost function by a log-coordinate change.

In the framework, this result is a step toward the main theorem that the cost function must be J(x) = (x + 1/x)/2 - 1. The differentiability hypothesis is one of the bridges that lets the framework cross from a purely algebraic functional equation to the analytic world of differential equations, where the solution is uniquely pinned down.

theorem dAlembert_cosh_solution
    (H : ℝ → ℝ)
    (h_one : H 0 = 1)
    (h_cont : Continuous H)
    (h_dAlembert : ∀ t u, H (t+u) + H (t-u) = 2 * H t * H u)
    (h_deriv2_zero : deriv (deriv H) 0 = 1)
    (h_smooth_hyp : dAlembert_continuous_implies_smooth_hypothesis H)
    (h_ode_hyp : dAlembert_to_ODE_hypothesis H)
    (h_cont_hyp : ode_regularity_continuous_hypothesis H)
    (h_diff_hyp : ode_regularity_differentiable_hypothesis H)
    (h_bootstrap_hyp : ode_linear_regularity_bootstrap_hypothesis H) :
    ∀ t, H t = Real.cosh t := by
  have h_ode : ∀ t, deriv (deriv H) t = H t := h_ode_hyp h_one h_cont h_dAlembert h_deriv2_zero
  have h_even : Function.Even H := dAlembert_even H h_one h_dAlembert
  have h_deriv_zero : deriv H 0 = 0 := by
    have h_smooth := h_smooth_hyp h_one h_cont h_dAlembert
    have h_diff : DifferentiableAt ℝ H 0 := h_smooth.differentiable (by decide : (⊤ : WithTop ℕ∞) ≠ 0) |>.differentiableAt
    exact even_deriv_at_zero H h_even h_diff
  exact ode_cosh_uniqueness H h_ode h_one h_deriv_zero h_cont_hyp h_diff_hyp h_bootstrap_hyp

theorem dAlembert_cosh_solution
    (H : ℝ → ℝ)
    (h_one : H 0 = 1)
    (h_cont : Continuous H)
    (h_dAlembert : ∀ t u, H (t+u) + H (t-u) = 2 * H t * H u)
    (h_deriv2_zero : deriv (deriv H) 0 = 1)
    (h_smooth_hyp : dAlembert_continuous_implies_smooth_hypothesis H)
    (h_ode_hyp : dAlembert_to_ODE_hypothesis H)
    (h_cont_hyp : ode_regularity_continuous_hypothesis H)
    (h_diff_hyp : ode_regularity_differentiable_hypothesis H)
    (h_bootstrap_hyp : ode_linear_regularity_bootstrap_hypothesis H) :
    ∀ t, H t = Real.cosh t := by
  have h_ode : ∀ t, deriv (deriv H) t = H t := h_ode_hyp h_one h_cont h_dAlembert h_deriv2_zero
  have h_even : Function.Even H := dAlembert_even H h_one h_dAlembert
  have h_deriv_zero : deriv H 0 = 0 := by
    have h_smooth := h_smooth_hyp h_one h_cont h_dAlembert
    have h_diff : DifferentiableAt ℝ H 0 := h_smooth.differentiable (by decide : (⊤ : WithTop ℕ∞) ≠ 0) |>.differentiableAt
    exact even_deriv_at_zero H h_even h_diff
  exact ode_cosh_uniqueness H h_ode h_one h_deriv_zero h_cont_hyp h_diff_hyp h_bootstrap_hyp

theorem ode_cosh_uniqueness (H : ℝ → ℝ)
    (h_ODE : ∀ t, deriv (deriv H) t = H t)
    (h_H0 : H 0 = 1)
    (h_H'0 : deriv H 0 = 0)
    (h_cont_hyp : ode_regularity_continuous_hypothesis H)
    (h_diff_hyp : ode_regularity_differentiable_hypothesis H)
    (h_bootstrap_hyp : ode_linear_regularity_bootstrap_hypothesis H) :
    ∀ t, H t = Real.cosh t := by
  have h_cont : Continuous H := h_cont_hyp h_ODE
  have h_diff : Differentiable ℝ H := h_diff_hyp h_ODE h_cont
  have h_C2 : ContDiff ℝ 2 H := h_bootstrap_hyp h_ODE h_cont h_diff
  exact ode_cosh_uniqueness_contdiff H h_C2 h_ODE h_H0 h_H'0

What this page does not claim

That differentiability alone forces the unique solution, without the other regularity hypotheses. That the cost function itself is differentiable, as opposed to the auxiliary function H. That the framework has proved the Riemann Hypothesis or the fine-structure constant.

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)

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.

Derived articles

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