IndisputableMonolith.Verification.PhiIrrationalityCert

   1import Mathlib
   2import IndisputableMonolith.Constants
   3
   4/-!
   5# φ-Irrationality Certificate
   6
   7This audit certificate packages the mathematical fact that the golden ratio φ
   8(the fundamental RS scale constant) is **irrational**.
   9
  10## Why this matters for the certificate chain
  11
  12The Recognition Science framework pins all dimensionless predictions to φ = (1 + √5)/2.
  13If φ were rational, the entire framework would be a disguised rational approximation
  14that could be rewritten with integer parameters.
  15
  16The irrationality proof uses Mathlib's `Real.goldenRatio_irrational`, which derives
  17from the irrationality of √5 (5 is prime, hence not a perfect square).
  18
  19## What this certificate does NOT do
  20
  21It does not assert φ is *transcendental* (which is an open question for the golden ratio).
  22It only asserts irrationality, which is sufficient to ensure φ encodes irreducible
  23algebraic structure rather than a rational "magic number."
  24-/
  25
  26namespace IndisputableMonolith
  27namespace Verification
  28namespace PhiIrrationality
  29
  30structure PhiIrrationalityCert where
  31  deriving Repr
  32
  33/-- Verification predicate: φ is irrational.
  34
  35This uses Mathlib's definition of `Irrational`: a real `x` is irrational iff
  36there is no rational `q` with `x = q`. -/
  37@[simp] def PhiIrrationalityCert.verified (_c : PhiIrrationalityCert) : Prop :=
  38  Irrational IndisputableMonolith.Constants.phi
  39
  40@[simp] theorem PhiIrrationalityCert.verified_any (c : PhiIrrationalityCert) :
  41    PhiIrrationalityCert.verified c := by
  42  -- Delegate to the proven theorem in Constants.lean
  43  exact IndisputableMonolith.Constants.phi_irrational
  44
  45end PhiIrrationality
  46end Verification
  47end IndisputableMonolith
  48