IndisputableMonolith.Verification.PhiAlternativesFailCert

   1import Mathlib
   2import IndisputableMonolith.PhiSupport.Alternatives
   3import IndisputableMonolith.RecogSpec.PhiSelectionCore
   4
   5/-!
   6# Phi Alternatives Fail Certificate
   7
   8This module certifies that common mathematical constants (e, π, √2, √3, √5) **fail**
   9the PhiSelection criterion, demonstrating that φ is uniquely determined by the
  10mathematical structure and not an arbitrary choice among "nice" constants.
  11
  12This addresses the "numerology objection" by proving exclusion: even though φ
  13has a simple closed form (1 + √5)/2, the selection criterion x² = x + 1 is not
  14satisfied by other common mathematical constants.
  15
  16Combined with φ uniqueness (PhiPinnedCert), this shows:
  171. φ is the ONLY positive solution to x² = x + 1 (uniqueness)
  182. Common alternatives all fail (exclusion)
  193. Therefore φ is mathematically forced, not chosen by fitting
  20-/
  21
  22namespace IndisputableMonolith
  23namespace Verification
  24namespace PhiAlternativesFail
  25
  26open IndisputableMonolith.PhiSupport.Alternatives
  27
  28/-- Certificate structure for alternative constants failing selection. -/
  29structure PhiAlternativesFailCert where
  30  deriving Repr
  31
  32/-- Verification predicate: all tested common constants fail PhiSelection. -/
  33@[simp] def PhiAlternativesFailCert.verified (_c : PhiAlternativesFailCert) : Prop :=
  34  -- Euler's number e fails
  35  ¬RecogSpec.PhiSelection (Real.exp 1) ∧
  36  -- π fails
  37  ¬RecogSpec.PhiSelection Real.pi ∧
  38  -- √2 fails
  39  ¬RecogSpec.PhiSelection (Real.sqrt 2) ∧
  40  -- √3 fails
  41  ¬RecogSpec.PhiSelection (Real.sqrt 3) ∧
  42  -- √5 fails (notably, despite φ = (1 + √5)/2)
  43  ¬RecogSpec.PhiSelection (Real.sqrt 5)
  44
  45/-- The certificate verifies by referencing the proven theorems. -/
  46@[simp] theorem PhiAlternativesFailCert.verified_any (c : PhiAlternativesFailCert) :
  47    PhiAlternativesFailCert.verified c :=
  48  common_constants_fail_selection
  49
  50end PhiAlternativesFail
  51end Verification
  52end IndisputableMonolith
  53