IndisputableMonolith.Verification.JcostConvexityCert

   1import Mathlib
   2import IndisputableMonolith.Cost.Convexity
   3
   4/-!
   5# J-cost Strict Convexity Certificate
   6
   7This audit certificate packages the **strict convexity** of the RS cost kernel:
   8
   9\[
  10  J(x) = \frac{x + x^{-1}}{2} - 1
  11\]
  12
  13is strictly convex on \((0, \infty)\).
  14
  15## Why this matters for the certificate chain
  16
  17Strict convexity of the cost function is central to uniqueness arguments in Recognition
  18Science. If J were merely convex (not strictly), there could be multiple minimizers or
  19flat regions. Strict convexity ensures:
  20
  211. **Unique minimum**: J has exactly one critical point (at x = 1, where J(1) = 0)
  222. **Optimization uniqueness**: Any cost-minimization problem using J has a unique solution
  233. **T5 foundation**: The uniqueness theorem (T5) relies on strict convexity to establish
  24   that the RS cost kernel is the unique symmetric, normalized, strictly convex function
  25   agreeing with the averaging kernel
  26
  27## Proof approach
  28
  29The strict convexity is proven via the second derivative test:
  30- J''(x) = x^{-3} > 0 for all x > 0
  31- A function with positive second derivative on a convex domain is strictly convex
  32-/
  33
  34namespace IndisputableMonolith
  35namespace Verification
  36namespace JcostConvexity
  37
  38open Set
  39
  40structure JcostConvexityCert where
  41  deriving Repr
  42
  43/-- Verification predicate: J-cost is strictly convex on (0, ∞). -/
  44@[simp] def JcostConvexityCert.verified (_c : JcostConvexityCert) : Prop :=
  45  StrictConvexOn ℝ (Ioi (0 : ℝ)) IndisputableMonolith.Cost.Jcost
  46
  47@[simp] theorem JcostConvexityCert.verified_any (c : JcostConvexityCert) :
  48    JcostConvexityCert.verified c := by
  49  exact IndisputableMonolith.Cost.Jcost_strictConvexOn_pos
  50
  51end JcostConvexity
  52end Verification
  53end IndisputableMonolith
  54