IndisputableMonolith.Verification.JcostSymmetryCert

   1import Mathlib
   2import IndisputableMonolith.Cost
   3
   4/-!
   5# J-cost Symmetry Certificate
   6
   7This audit certificate packages the **inversion symmetry** of the RS cost kernel:
   8
   9\[
  10  J(x) = J(x^{-1}) \quad \text{for all } x > 0
  11\]
  12
  13## Why this matters for the certificate chain
  14
  15The symmetry J(x) = J(1/x) is a fundamental structural property that:
  16
  171. **Reflects reciprocal invariance**: The cost of being "too big" equals the cost of being "too small"
  182. **Ensures balance**: Ratios and their inverses are treated equally
  193. **Enables log-coordinate treatment**: J(exp(t)) = J(exp(-t)), so Jlog is even
  204. **Underpins functional equation**: The cosh-add identity relies on this symmetry
  21
  22This is a proven theorem from the explicit formula J(x) = (x + x⁻¹)/2 - 1.
  23
  24## Proof approach
  25
  26Direct calculation: exchanging x ↔ x⁻¹ leaves (x + x⁻¹)/2 - 1 unchanged because
  27addition is commutative.
  28-/
  29
  30namespace IndisputableMonolith
  31namespace Verification
  32namespace JcostSymmetry
  33
  34open IndisputableMonolith.Cost
  35
  36structure JcostSymmetryCert where
  37  deriving Repr
  38
  39/-- Verification predicate: J is symmetric under inversion.
  40
  41For all x > 0, we have J(x) = J(1/x). -/
  42@[simp] def JcostSymmetryCert.verified (_c : JcostSymmetryCert) : Prop :=
  43  ∀ x : ℝ, 0 < x → Jcost x = Jcost x⁻¹
  44
  45@[simp] theorem JcostSymmetryCert.verified_any (c : JcostSymmetryCert) :
  46    JcostSymmetryCert.verified c := by
  47  intro x hx
  48  exact Jcost_symm hx
  49
  50end JcostSymmetry
  51end Verification
  52end IndisputableMonolith
  53