phi_cost_pos

formal source

  66theorem phi_cost_eq : LawOfExistence.J PhiForcing.φ = PhiForcing.φ - 3 / 2 :=
  67  PhiForcing.J_phi
  68
  69theorem phi_cost_pos : 0 < LawOfExistence.J PhiForcing.φ := by
  70  rw [phi_cost_eq]; linarith [PhiForcing.phi_gt_onePointSix]
  71
  72theorem phi_perturbation_bounded : LawOfExistence.J PhiForcing.φ < 1 := by
  73  rw [phi_cost_eq]; linarith [PhiForcing.phi_lt_two]
  74
  75/-! ## Part III: φ-Structure in Configurations -/
  76
  77def has_phi_structure {N : ℕ} (c : InitialCondition.Configuration N) : Prop :=
  78  ∃ i j : Fin N, ∃ n : ℤ, n ≠ 0 ∧ c.entries i / c.entries j = PhiForcing.φ ^ n
  79
  80theorem unity_has_no_phi_structure {N : ℕ} (hN : 0 < N) :
  81    ¬ has_phi_structure (InitialCondition.unity_config N hN) := by
  82  intro ⟨i, j, n, hn, hratio⟩
  83  simp only [InitialCondition.unity_config] at hratio
  84  have h_div : (1 : ℝ) / 1 = PhiForcing.φ ^ n := hratio
  85  simp at h_div
  86  exact phi_zpow_ne_one hn h_div.symm
  87
  88/-! ## Part IV: DERIVED — Recognition Forces Non-Trivial Content (Gap A)
  89
  90T4 (Recognition Forcing): Recognition requires distinguishing states.
  91A configuration where all entries are identical contains zero information
  92and cannot support any recognition event. This is the RS formalization
  93of "nothing cannot recognize itself" (MP/T1) applied to configurations:
  94a uniform ledger is informationally equivalent to nothing.
  95
  96The derivation:
  97  T4 (recognition requires substrate) + T1 (nothing has infinite cost)
  98  → at least one entry must differ from the ground state
  99  → the configuration is non-trivial