IndisputableMonolith.Verification.PhiDecimalBoundsCert

   1import Mathlib
   2import IndisputableMonolith.Constants
   3
   4/-!
   5# φ Decimal Bounds Certificate
   6
   7This audit certificate packages **tight decimal bounds** on φ:
   8
   9> 1.5 < φ < 1.62
  10
  11## Why this matters
  12
  131. **Numerical precision**: The interval (1.5, 1.62) provides two-decimal
  14   accuracy for φ ≈ 1.618. Useful for sanity-checking computed values.
  15
  162. **Verification of approximations**: When RS uses φ ≈ 1.618 in calculations,
  17   these bounds prove the approximation is valid.
  18
  193. **Ruling out degenerate cases**: φ is well-separated from both 1 and 2,
  20   confirming it's in the "interesting" range of the cost function.
  21
  22## Proof approach
  23
  24- Lower bound: √5 > 2 (since 4 < 5), so (1 + √5)/2 > 1.5
  25- Upper bound: √5 < 2.24 (since 5 < 5.0176), so (1 + √5)/2 < 1.62
  26-/
  27
  28namespace IndisputableMonolith
  29namespace Verification
  30namespace PhiDecimalBounds
  31
  32open IndisputableMonolith.Constants
  33
  34structure PhiDecimalBoundsCert where
  35  deriving Repr
  36
  37/-- Verification predicate: φ has tight decimal bounds.
  38
  391.5 < φ < 1.62, giving two-decimal precision around φ ≈ 1.618. -/
  40@[simp] def PhiDecimalBoundsCert.verified (_c : PhiDecimalBoundsCert) : Prop :=
  41  ((1.5 : ℝ) < phi) ∧ (phi < (1.62 : ℝ))
  42
  43@[simp] theorem PhiDecimalBoundsCert.verified_any (c : PhiDecimalBoundsCert) :
  44    PhiDecimalBoundsCert.verified c := by
  45  constructor
  46  · exact phi_gt_onePointFive
  47  · exact phi_lt_onePointSixTwo
  48
  49end PhiDecimalBounds
  50end Verification
  51end IndisputableMonolith
  52