unity_has_no_phi_structure

formal source

  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
 100
 101This replaces the old T6_Requirement structure with a THEOREM. -/
 102
 103/-- A configuration is **non-trivial** if at least one entry differs from 1.
 104    Equivalently: the configuration is not the uniform ground state. -/
 105def is_nontrivial {N : ℕ} (c : InitialCondition.Configuration N) : Prop :=
 106  ∃ i : Fin N, c.entries i ≠ 1
 107
 108/-- **T4 Recognition Theorem**: Any configuration that supports recognition
 109    events must be non-trivial.
 110