IndisputableMonolith.Verification.JcostStrictPosCert

   1import Mathlib
   2import IndisputableMonolith.Cost
   3
   4/-!
   5# J-cost Strict Positivity Certificate
   6
   7This audit certificate packages **strict positivity** for the J-cost function:
   8
   9> J(x) > 0 for all x > 0 with x ≠ 1
  10
  11## Why this matters
  12
  131. **Unique minimum characterization**: Combined with Jcost_nonneg (J ≥ 0) and
  14   Jcost_unit0 (J(1) = 0), this shows x = 1 is the **unique** minimizer.
  15
  162. **Variational significance**: The cost function has a strict global minimum,
  17   not just a global minimum — there are no flat regions.
  18
  193. **Physical interpretation**: Any deviation from unit ratio incurs strictly
  20   positive cost. The universe "wants" to be at x = 1.
  21
  22## Proof approach
  23
  24Uses the squared representation J(x) = (x-1)²/(2x) and positivity of squares
  25for non-zero arguments.
  26-/
  27
  28namespace IndisputableMonolith
  29namespace Verification
  30namespace JcostStrictPos
  31
  32open IndisputableMonolith.Cost
  33
  34structure JcostStrictPosCert where
  35  deriving Repr
  36
  37/-- Verification predicate: J-cost is strictly positive away from 1.
  38
  39J(x) > 0 for x > 0 and x ≠ 1, proving the minimum at x = 1 is unique. -/
  40@[simp] def JcostStrictPosCert.verified (_c : JcostStrictPosCert) : Prop :=
  41  ∀ (x : ℝ), 0 < x → x ≠ 1 → 0 < Jcost x
  42
  43@[simp] theorem JcostStrictPosCert.verified_any (c : JcostStrictPosCert) :
  44    JcostStrictPosCert.verified c := by
  45  intro x hx hx1
  46  exact Jcost_pos_of_ne_one x hx hx1
  47
  48end JcostStrictPos
  49end Verification
  50end IndisputableMonolith
  51