IndisputableMonolith.Verification.PhiSelfSimilarityCert

   1import Mathlib
   2import IndisputableMonolith.PhiSupport.Lemmas
   3
   4/-!
   5# φ Self-Similarity Certificate
   6
   7This audit certificate packages the **self-similarity** (fixed-point) identity of φ:
   8
   9\[
  10  \varphi = 1 + \frac{1}{\varphi}
  11\]
  12
  13## Why this matters for the certificate chain
  14
  15The self-similarity identity is the defining algebraic property that makes φ unique
  16among positive reals. It encodes:
  17
  181. **Scale invariance**: φ contains a copy of itself at the next scale down (1/φ)
  192. **Recursive structure**: The whole equals 1 plus the reciprocal of the whole
  203. **Uniqueness**: This fixed-point equation (equivalent to φ² = φ + 1) has exactly
  21   one positive solution
  22
  23This identity is central to the Recognition Science framework's claim that φ emerges
  24from self-similarity constraints rather than being chosen by fitting.
  25
  26## Proof approach
  27
  28From φ² = φ + 1 (Mathlib's `Real.goldenRatio_sq`), divide both sides by φ:
  29- φ²/φ = (φ + 1)/φ
  30- φ = φ/φ + 1/φ
  31- φ = 1 + 1/φ
  32-/
  33
  34namespace IndisputableMonolith
  35namespace Verification
  36namespace PhiSelfSimilarity
  37
  38open IndisputableMonolith.PhiSupport
  39
  40structure PhiSelfSimilarityCert where
  41  deriving Repr
  42
  43/-- Verification predicate: φ satisfies the self-similarity fixed-point equation.
  44
  45This certifies φ = 1 + 1/φ, the defining recursive structure of the golden ratio. -/
  46@[simp] def PhiSelfSimilarityCert.verified (_c : PhiSelfSimilarityCert) : Prop :=
  47  IndisputableMonolith.Constants.phi = 1 + 1 / IndisputableMonolith.Constants.phi
  48
  49@[simp] theorem PhiSelfSimilarityCert.verified_any (c : PhiSelfSimilarityCert) :
  50    PhiSelfSimilarityCert.verified c := by
  51  exact phi_fixed_point
  52
  53end PhiSelfSimilarity
  54end Verification
  55end IndisputableMonolith
  56