IndisputableMonolith.Verification.PhiPowerBoundsCert

   1import Mathlib
   2import IndisputableMonolith.Constants
   3
   4/-!
   5# φ Power Bounds Certificate
   6
   7This audit certificate packages **numerical bounds** on powers of φ:
   8
   9> 2.5 < φ² < 2.7
  10> 4.0 < φ³ < 4.25
  11> 6.5 < φ⁴ < 6.9
  12
  13## Why this matters
  14
  151. **RS calculations**: Many Recognition Science formulas involve φ², φ³, φ⁴.
  16   These bounds verify computed approximations are valid.
  17
  182. **Fibonacci connection**: φⁿ = Fₙφ + Fₙ₋₁ where Fₙ is Fibonacci.
  19   The bounds confirm the recurrence works correctly.
  20
  213. **Growth rate verification**: Powers grow as expected from the defining
  22   equation φ² = φ + 1, which implies φⁿ⁺¹ = φⁿ + φⁿ⁻¹.
  23
  24## Proof approach
  25
  26Uses φ² = φ + 1 to reduce higher powers to linear form:
  27- φ² = φ + 1
  28- φ³ = 2φ + 1
  29- φ⁴ = 3φ + 2
  30
  31Then applies the decimal bounds 1.5 < φ < 1.62.
  32-/
  33
  34namespace IndisputableMonolith
  35namespace Verification
  36namespace PhiPowerBounds
  37
  38open IndisputableMonolith.Constants
  39
  40structure PhiPowerBoundsCert where
  41  deriving Repr
  42
  43/-- Verification predicate: bounds on φ², φ³, φ⁴.
  44
  45These provide numerical ranges for higher powers of the golden ratio. -/
  46@[simp] def PhiPowerBoundsCert.verified (_c : PhiPowerBoundsCert) : Prop :=
  47  ((2.5 : ℝ) < phi^2 ∧ phi^2 < (2.7 : ℝ)) ∧
  48  ((4.0 : ℝ) < phi^3 ∧ phi^3 < (4.25 : ℝ)) ∧
  49  ((6.5 : ℝ) < phi^4 ∧ phi^4 < (6.9 : ℝ))
  50
  51@[simp] theorem PhiPowerBoundsCert.verified_any (c : PhiPowerBoundsCert) :
  52    PhiPowerBoundsCert.verified c := by
  53  constructor
  54  · exact phi_squared_bounds
  55  constructor
  56  · exact phi_cubed_bounds
  57  · exact phi_fourth_bounds
  58
  59end PhiPowerBounds
  60end Verification
  61end IndisputableMonolith
  62