origin_question_resolved

formal source

 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
 462    (rungs n ≠ 0, J > 0) are connected by the d'Alembert cascade. -/
 463theorem symmetry_breaking_mechanism :
 464    Jcost (PhiForcing.φ ^ (0 : ℤ)) = 0
 465    ∧ (∀ n : ℤ, n ≠ 0 → 0 < Jcost (PhiForcing.φ ^ n))
 466    ∧ (∀ a b : ℤ,
 467        Jcost (PhiForcing.φ ^ a * PhiForcing.φ ^ b) ≤
 468        2 * Jcost (PhiForcing.φ ^ a) + 2 * Jcost (PhiForcing.φ ^ b) +
 469        2 * Jcost (PhiForcing.φ ^ a) * Jcost (PhiForcing.φ ^ b)) := by
 470  exact ⟨by simp [Jcost_unit0],
 471    fun n hn => phi_ladder_positive_cost hn,
 472    fun a b => Jcost_submult (zpow_pos PhiForcing.phi_pos a) (zpow_pos PhiForcing.phi_pos b)⟩
 473