IndisputableMonolith.Verification.EulerLagrangeCert

   1import Mathlib
   2import IndisputableMonolith.Cost
   3
   4/-!
   5# Euler-Lagrange Stationarity Certificate
   6
   7This audit certificate packages the **Euler-Lagrange conditions** for the log-domain cost:
   8
   91. \(\frac{d}{dt} J_{\log}(t)\big|_{t=0} = 0\) (stationarity at origin)
  102. \(J_{\log}(0) \leq J_{\log}(t)\) for all \(t\) (global minimum)
  11
  12## Why this matters for the certificate chain
  13
  14The Euler-Lagrange conditions are the variational calculus foundation:
  15
  161. **Stationarity**: Jlog'(0) = 0 means t = 0 is a critical point
  172. **Global minimum**: Jlog(0) ≤ Jlog(t) for all t
  183. **Uniqueness connection**: Combined with strict convexity, this proves
  19   the origin is THE unique global minimizer
  20
  21These conditions are essential for:
  22- Variational principles in physics (action minimization)
  23- Cost-minimization interpretation of Recognition Science
  24- Bridge to T5 uniqueness theorem
  25
  26## Proof approach
  27
  28- Stationarity: Jlog' = sinh, and sinh(0) = 0
  29- Global minimum: Jlog ≥ 0 everywhere, and Jlog(0) = 0
  30-/
  31
  32namespace IndisputableMonolith
  33namespace Verification
  34namespace EulerLagrange
  35
  36open IndisputableMonolith.Cost
  37
  38structure EulerLagrangeCert where
  39  deriving Repr
  40
  41/-- Verification predicate: Euler-Lagrange conditions hold at t = 0.
  42
  43This certifies:
  441. Jlog'(0) = 0 (stationarity)
  452. Jlog(0) ≤ Jlog(t) for all t (global minimum) -/
  46@[simp] def EulerLagrangeCert.verified (_c : EulerLagrangeCert) : Prop :=
  47  (deriv Jlog 0 = 0) ∧ (∀ t : ℝ, Jlog 0 ≤ Jlog t)
  48
  49@[simp] theorem EulerLagrangeCert.verified_any (c : EulerLagrangeCert) :
  50    EulerLagrangeCert.verified c := by
  51  constructor
  52  · exact EL_stationary_at_zero
  53  · exact EL_global_min
  54
  55end EulerLagrange
  56end Verification
  57end IndisputableMonolith
  58
  59