IndisputableMonolith.Verification.JcostMinimumCert

   1import Mathlib
   2import IndisputableMonolith.Cost
   3
   4/-!
   5# J-cost Unique Minimum Certificate
   6
   7This audit certificate packages the **unique minimum** property of the RS cost kernel:
   8
   9\[
  10  J(x) = \frac{x + x^{-1}}{2} - 1
  11\]
  12
  13has a unique global minimum at x = 1, where J(1) = 0.
  14
  15## Why this matters for the certificate chain
  16
  17The J-cost function is the fundamental "deviation cost" in Recognition Science:
  18- J(1) = 0: No cost at the identity (perfect recognition)
  19- J(x) > 0 for x ≠ 1: Any deviation from identity incurs positive cost
  20
  21Together with strict convexity (JcostConvexityCert), this establishes that:
  221. The identity is the unique optimal state
  232. All deviations are penalized
  243. There are no other local minima
  25
  26## Proof approach
  27
  28The key identity is J(x) = (x - 1)² / (2x), which shows:
  29- Numerator (x - 1)² ≥ 0, with equality iff x = 1
  30- Denominator 2x > 0 for x > 0
  31- Therefore J(x) ≥ 0 with J(x) = 0 iff x = 1
  32-/
  33
  34namespace IndisputableMonolith
  35namespace Verification
  36namespace JcostMinimum
  37
  38open IndisputableMonolith.Cost
  39
  40structure JcostMinimumCert where
  41  deriving Repr
  42
  43/-- Verification predicate: J has a unique minimum at x = 1.
  44
  45This asserts:
  461. J(1) = 0 (the minimum value)
  472. For all x > 0 with x ≠ 1, we have J(x) > 0 (strict positivity away from minimum)
  48-/
  49@[simp] def JcostMinimumCert.verified (_c : JcostMinimumCert) : Prop :=
  50  (Jcost 1 = 0) ∧
  51  (∀ x : ℝ, 0 < x → x ≠ 1 → 0 < Jcost x)
  52
  53@[simp] theorem JcostMinimumCert.verified_any (c : JcostMinimumCert) :
  54    JcostMinimumCert.verified c := by
  55  constructor
  56  · -- J(1) = 0
  57    exact Jcost_unit0
  58  · -- J(x) > 0 for x > 0, x ≠ 1
  59    intro x hx hx1
  60    exact Jcost_pos_of_ne_one x hx hx1
  61
  62end JcostMinimum
  63end Verification
  64end IndisputableMonolith
  65