stillness_is_creative

formal source

 394    6. Creation barrier is finite: 0 < J(φ) < 1
 395    7. Fibonacci cascade populates all rungs: φ^n + φ^{n+1} = φ^{n+2}
 396    8. Ledger symmetry gives negative rungs: J(φ^n) = J(φ^{-n}) -/
 397theorem stillness_is_creative {N : ℕ} (hN : 0 < N) :
 398    -- (1) x = 1 is the unique zero-defect state
 399    InitialCondition.total_defect (InitialCondition.unity_config N hN) = 0
 400    -- (2) Uniqueness: zero defect forces all entries to 1
 401    ∧ (∀ c : InitialCondition.Configuration N,
 402        InitialCondition.total_defect c = 0 → ∀ i, c.entries i = 1)
 403    -- (3) T4: uniform ground state cannot support recognition
 404    ∧ ¬ T4_Recognition (InitialCondition.unity_config N hN)
 405    -- (4) T7: uniform cycle is degenerate (period 1 ≠ 8)
 406    ∧ (∀ v : ℝ, ¬ cycle_nondegenerate (fun _ : Fin 8 => v))
 407    -- (5) T6: φ is unique forced ratio (closure → r = φ)
 408    ∧ (∀ r : ℝ, r > 0 → r ^ 2 = r + 1 → r = PhiForcing.φ)
 409    -- (6) Finite barrier: 0 < J(φ) < 1
 410    ∧ (0 < LawOfExistence.J PhiForcing.φ ∧ LawOfExistence.J PhiForcing.φ < 1)
 411    -- (7) Fibonacci cascade: φ^n + φ^{n+1} = φ^{n+2}
 412    ∧ (∀ n : ℤ, PhiForcing.φ ^ n + PhiForcing.φ ^ (n + 1) = PhiForcing.φ ^ (n + 2))
 413    -- (8) Ledger symmetry: J(φ^n) = J(φ^{-n})
 414    ∧ (∀ n : ℤ, Jcost (PhiForcing.φ ^ n) = Jcost (PhiForcing.φ ^ (-n)))
 415    := by
 416  exact ⟨
 417    InitialCondition.unity_defect_zero hN,
 418    fun c hc => (InitialCondition.zero_defect_iff_unity hN c).mp hc,
 419    ground_state_recognition_impossible hN,
 420    uniform_cycle_degenerate,
 421    fun r hr heq => PhiForcing.phi_forced r hr heq,
 422    ⟨phi_cost_pos, phi_perturbation_bounded⟩,
 423    fibonacci_cascade,
 424    ledger_symmetry_negative_rungs
 425  ⟩
 426
 427/-! ## Part XI: Corollaries -/