IndisputableMonolith.Verification.PhiPositivityCert

   1import Mathlib
   2import IndisputableMonolith.Constants
   3
   4/-!
   5# φ Positivity Certificate
   6
   7This audit certificate packages the **positivity bounds** for the golden ratio:
   8
   91. \(\varphi > 0\) (strict positivity)
  102. \(\varphi > 1\) (supercritical - ratio exceeds unity)
  11
  12## Why this matters for the certificate chain
  13
  14These bounds are foundational for the RS framework:
  15
  161. **Positivity (φ > 0)**: Ensures φ can serve as a ratio/scale factor
  172. **Supercriticality (φ > 1)**: Ensures the φ-ladder has genuine hierarchy
  18   - Powers φⁿ grow without bound as n → ∞
  19   - Powers φ⁻ⁿ decay to zero as n → ∞
  20   - The scale separation between rungs is meaningful
  21
  22Without φ > 1, the "φ-ladder" would collapse (if φ = 1) or invert (if φ < 1).
  23
  24## Proof approach
  25
  26From φ = (1 + √5)/2:
  27- φ > 0: The numerator 1 + √5 > 0 and denominator 2 > 0
  28- φ > 1: Since √5 > 1, we have 1 + √5 > 2, so (1 + √5)/2 > 1
  29-/
  30
  31namespace IndisputableMonolith
  32namespace Verification
  33namespace PhiPositivity
  34
  35open IndisputableMonolith.Constants
  36
  37structure PhiPositivityCert where
  38  deriving Repr
  39
  40/-- Verification predicate: φ is positive and greater than 1.
  41
  42This certifies both the strict positivity and the supercritical property of φ. -/
  43@[simp] def PhiPositivityCert.verified (_c : PhiPositivityCert) : Prop :=
  44  (0 < phi) ∧ (1 < phi)
  45
  46@[simp] theorem PhiPositivityCert.verified_any (c : PhiPositivityCert) :
  47    PhiPositivityCert.verified c := by
  48  constructor
  49  · exact phi_pos
  50  · exact one_lt_phi
  51
  52end PhiPositivity
  53end Verification
  54end IndisputableMonolith
  55