IndisputableMonolith.Verification.PhiNeZeroCert

   1import Mathlib
   2import IndisputableMonolith.PhiSupport.Lemmas
   3
   4namespace IndisputableMonolith
   5namespace Verification
   6namespace PhiNeZero
   7
   8open IndisputableMonolith.PhiSupport
   9
  10/-!
  11# Phi Ne Zero Certificate
  12
  13This certificate proves that `Constants.phi ≠ 0`, i.e., φ ≠ 0.
  14
  15## Key Result
  16
  17`Constants.phi ≠ 0`
  18
  19## Why this matters for the certificate chain
  20
  21This is a **non-degeneracy** property of the golden ratio:
  22
  231. **Definition**: φ = (1 + √5)/2
  242. **Non-zero**: φ ≠ 0
  253. **Consequence**: Division by φ is well-defined
  26
  27This property is essential because:
  28- The fixed-point equation φ = 1 + 1/φ requires φ ≠ 0
  29- Many φ-based formulas involve division by φ
  30- The J-cost function J(x) = (x + 1/x)/2 - 1 is evaluated at φ
  31
  32## Mathematical Content
  33
  34Since φ = (1 + √5)/2 and √5 > 0:
  35```
  36φ = (1 + √5)/2 > (1 + 0)/2 = 0.5 > 0
  37```
  38
  39Therefore φ ≠ 0.
  40
  41The proof uses that `Real.goldenRatio` is positive (from Mathlib).
  42
  43## Physical Significance
  44
  45The non-zero property ensures:
  46- φ can serve as a scaling factor (division is defined)
  47- The φ-lattice has a well-defined structure
  48- Energy ratios based on φ are meaningful
  49
  50This is a basic sanity check that the golden ratio is a valid scaling constant.
  51
  52## Relationship to Other Properties
  53
  54This bound works with:
  55- `one_lt_phi` (#109): φ > 1 (stronger)
  56- `phi_squared`: φ² = φ + 1
  57- `phi_fixed_point`: φ = 1 + 1/φ (requires φ ≠ 0)
  58-/
  59
  60structure PhiNeZeroCert where
  61  deriving Repr
  62
  63/-- Verification predicate: phi is not zero. -/
  64@[simp] def PhiNeZeroCert.verified (_c : PhiNeZeroCert) : Prop :=
  65  Constants.phi ≠ 0
  66
  67/-- Top-level theorem: the certificate verifies. -/
  68@[simp] theorem PhiNeZeroCert.verified_any (c : PhiNeZeroCert) :
  69    PhiNeZeroCert.verified c := by
  70  exact phi_ne_zero
  71
  72end PhiNeZero
  73end Verification
  74end IndisputableMonolith
  75