ground_state_paradox

formal source

 428
 429/-- **The Ground State Paradox**: x = 1 is the unique equilibrium (J = 0)
 430    AND the unique state that must generate non-trivial structure (T4). -/
 431theorem ground_state_paradox {N : ℕ} (hN : 0 < N) :
 432    (∀ x : ℝ, 0 < x → LawOfExistence.defect x = 0 → x = 1)
 433    ∧ ¬ T4_Recognition (InitialCondition.unity_config N hN) :=
 434  ⟨fun _x hx hd => (LawOfExistence.defect_zero_iff_one hx).mp hd,
 435   ground_state_recognition_impossible hN⟩
 436
 437/-- **The Origin Question Resolved** — every sub-question answered by T0–T8:
 438    - What drives creation? T4 (recognition requires content)
 439    - Why φ? T6 (closure on geometric sequence)
 440    - Why is the barrier crossable? J(φ) < 1 (finite cost)
 441    - Why the full ladder? Fibonacci cascade + ledger symmetry
 442    - Why unavoidable? T7 (8-tick non-degeneracy) -/
 443theorem origin_question_resolved :
 444    (PhiForcing.φ ^ 2 = PhiForcing.φ + 1)
 445    ∧ (∀ r : ℝ, r > 0 → r ^ 2 = r + 1 → r = PhiForcing.φ)
 446    ∧ (0 < LawOfExistence.J PhiForcing.φ ∧ LawOfExistence.J PhiForcing.φ < 1)
 447    ∧ (∀ a b : ℤ,
 448        Jcost (phi_ladder (a + b)) ≤
 449        2 * Jcost (phi_ladder a) + 2 * Jcost (phi_ladder b) +
 450        2 * Jcost (phi_ladder a) * Jcost (phi_ladder b))
 451    ∧ (∀ n : ℤ, n ≠ 0 → 0 < Jcost (phi_ladder n)) := by
 452  exact ⟨
 453    PhiForcing.phi_equation,
 454    fun r hr heq => PhiForcing.phi_forced r hr heq,
 455    ⟨phi_cost_pos, phi_perturbation_bounded⟩,
 456    ladder_cascade_bound,
 457    fun n hn => phi_ladder_positive_cost hn
 458  ⟩
 459
 460/-- **Symmetry Breaking**: The ground state (rung 0, J = 0) is forced
 461    off the trivial rung by T4 + T7. The broken-symmetry states