IndisputableMonolith.Verification.PhiBoundsCert

   1import Mathlib
   2import IndisputableMonolith.Constants
   3
   4/-!
   5# φ Bounds Certificate
   6
   7This audit certificate packages the **tight interval bounds** for the golden ratio:
   8
   9\[
  10  1 < \varphi < 2
  11\]
  12
  13## Why this matters for the certificate chain
  14
  15These bounds establish that φ lies in the open interval (1, 2):
  16
  171. **Lower bound (φ > 1)**: Already certified in PhiPositivityCert
  182. **Upper bound (φ < 2)**: Ensures φ is not "too large"
  19
  20Together they show:
  21- φ ∈ (1, 2) — the golden ratio is strictly between 1 and 2
  22- This is important for numerical bounds and convergence arguments
  23- The φ-ladder has controlled growth: φⁿ < 2ⁿ
  24
  25## Proof approach
  26
  27From φ = (1 + √5)/2:
  28- Upper bound: √5 < 3 (since 5 < 9), so 1 + √5 < 4, thus (1 + √5)/2 < 2
  29- Lower bound: √5 > 1 (since 5 > 1), so 1 + √5 > 2, thus (1 + √5)/2 > 1
  30
  31The actual value is φ ≈ 1.618...
  32-/
  33
  34namespace IndisputableMonolith
  35namespace Verification
  36namespace PhiBounds
  37
  38open IndisputableMonolith.Constants
  39
  40structure PhiBoundsCert where
  41  deriving Repr
  42
  43/-- Verification predicate: 1 < φ < 2.
  44
  45This certifies the tight interval bounds on the golden ratio. -/
  46@[simp] def PhiBoundsCert.verified (_c : PhiBoundsCert) : Prop :=
  47  (1 < phi) ∧ (phi < 2)
  48
  49@[simp] theorem PhiBoundsCert.verified_any (c : PhiBoundsCert) :
  50    PhiBoundsCert.verified c := by
  51  constructor
  52  · exact one_lt_phi
  53  · exact phi_lt_two
  54
  55end PhiBounds
  56end Verification
  57end IndisputableMonolith
  58