IndisputableMonolith.Verification.PhiSquaredCert

   1import Mathlib
   2import IndisputableMonolith.Constants
   3
   4/-!
   5# φ² = φ + 1 Certificate
   6
   7This audit certificate packages the **fundamental algebraic identity** of the golden ratio:
   8
   9\[
  10  \varphi^2 = \varphi + 1
  11\]
  12
  13## Why this matters for the certificate chain
  14
  15The identity φ² = φ + 1 is the **defining equation** of the golden ratio. It:
  16
  171. **Uniquely determines φ**: This quadratic has exactly one positive root
  182. **Implies self-similarity**: Dividing by φ gives φ = 1 + 1/φ (already certified)
  193. **Enables recursive structure**: Powers of φ follow the Fibonacci recurrence
  204. **Is algebraically verifiable**: Follows directly from φ = (1 + √5)/2
  21
  22This is the most fundamental property of φ - all other φ identities derive from it.
  23
  24## Proof approach
  25
  26Direct calculation: substituting φ = (1 + √5)/2 into φ² and simplifying using (√5)² = 5.
  27-/
  28
  29namespace IndisputableMonolith
  30namespace Verification
  31namespace PhiSquared
  32
  33open IndisputableMonolith.Constants
  34
  35structure PhiSquaredCert where
  36  deriving Repr
  37
  38/-- Verification predicate: φ satisfies the defining quadratic identity.
  39
  40This certifies φ² = φ + 1, the fundamental algebraic equation of the golden ratio. -/
  41@[simp] def PhiSquaredCert.verified (_c : PhiSquaredCert) : Prop :=
  42  phi ^ 2 = phi + 1
  43
  44@[simp] theorem PhiSquaredCert.verified_any (c : PhiSquaredCert) :
  45    PhiSquaredCert.verified c := by
  46  exact phi_sq_eq
  47
  48end PhiSquared
  49end Verification
  50end IndisputableMonolith
  51