import QFD.Physics.Postulates import Mathlib.Analysis.SpecialFunctions.Pow.Real import Mathlib.Analysis.Calculus.Deriv.Basic import Mathlib.Tactic.Linarith import Mathlib.Topology.Order.Basic import Mathlib.Tactic.Positivity import Mathlib.Tactic.Polyrith noncomputable section namespace QFD.Lepton /-- **Topological Twist Energy (Phase 3).** The energy of a leptonic soliton is determined by three terms: 1. **Vacuum Stiffness**: β * Q^2 (Toroidal circulation) 2. **Topological Charge**: Q (Geometric coupling) 3. **Twist Resonance**: (gamma * N^2) / Q (Poloidal twist) Where: - Q : Winding number (continuous for minimization, but integer-locked at stable roots) - N : Harmonic mode (integer) - beta : Vacuum bulk modulus - gamma : Twist coupling constant -/ def topological_twist_energy (beta : ℝ) (gamma : ℝ) (Q : ℝ) (N : ℤ) : ℝ := beta * Q^2 + Q + (gamma * (N : ℝ)^2) / Q /-- The stability condition: ∂E/∂Q = 0. This determines the 'stable' winding number Q* for a given harmonic mode N. -/ def is_stable_configuration (beta : ℝ) (gamma : ℝ) (Q : ℝ) (N : ℤ) : Prop := HasDerivAt (fun q => topological_twist_energy beta gamma q N) 0 Q /-- The cubic polynomial whose positive root gives the stable Q*. At the critical point ∂E/∂Q = 0, we have: 2βQ + 1 - γN²/Q² = 0 Multiplying by Q² gives: 2βQ³ + Q² - γN² = 0 -/ def stability_polynomial (beta gamma : ℝ) (N : ℤ) (Q : ℝ) : ℝ := 2 * beta * Q^3 + Q^2 - gamma * (N : ℝ)^2 /-- The stability polynomial is continuous -/ theorem stability_polynomial_continuous (beta gamma : ℝ) (N : ℤ) : Continuous (stability_polynomial beta gamma N) := by unfold stability_polynomial continuity /-- P(0) = -γN² < 0 when N ≠ 0 and γ > 0 -/ theorem stability_polynomial_neg_at_zero (beta gamma : ℝ) (h_gamma : gamma > 0) (N : ℤ) (hN : N ≠ 0) : stability_polynomial beta gamma N 0 < 0 := by unfold stability_polynomial simp only [ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true, zero_pow, mul_zero, add_zero] have hN_ne : (N : ℝ) ≠ 0 := Int.cast_ne_zero.mpr hN have hN_sq : (N : ℝ)^2 > 0 := sq_pos_iff.mpr hN_ne linarith [mul_pos h_gamma hN_sq] /-- For large enough Q, P(Q) > 0 (cubic with positive leading coefficient) -/ theorem stability_polynomial_pos_for_large_Q (beta gamma : ℝ) (h_beta : beta > 0) (h_gamma : gamma > 0) (N : ℤ) : ∃ Q_big > 0, stability_polynomial beta gamma N Q_big > 0 := by -- Choose Q_big = γN² + 1 (simpler choice that works) use gamma * (N : ℝ)^2 + 1 have hN2_nonneg : (N : ℝ)^2 ≥ 0 := sq_nonneg _ constructor · -- Q_big > 0 have h1 : gamma * (N : ℝ)^2 ≥ 0 := by positivity linarith · -- P(Q_big) > 0 unfold stability_polynomial set Q := gamma * (N : ℝ)^2 + 1 with hQ_def have hQ_ge_1 : Q ≥ 1 := by simp only [Q]; linarith [mul_nonneg (le_of_lt h_gamma) hN2_nonneg] have hQ_pos : Q > 0 := by linarith -- For Q ≥ 1: Q² ≥ 1, Q³ ≥ 1 -- 2βQ³ + Q² ≥ 2β + 1 > γN² when Q = γN² + 1 -- Actually need: 2βQ³ + Q² - γN² > 0 -- Since Q = γN² + 1, we have Q > γN², so Q² > (γN²)² ≥ 0 -- And Q³ ≥ Q ≥ 1, so 2βQ³ ≥ 2β -- Also Q² ≥ Q = γN² + 1 > γN² -- So 2βQ³ + Q² > 0 + γN² = γN², hence P(Q) > 0 have h_key : Q^2 > gamma * (N : ℝ)^2 := by have : Q^2 ≥ Q := by nlinarith calc Q^2 ≥ Q := this _ = gamma * (N : ℝ)^2 + 1 := rfl _ > gamma * (N : ℝ)^2 := by linarith have h_cubic_pos : 2 * beta * Q^3 ≥ 0 := by positivity linarith /-- **Existence of Stable Configurations (IVT Application).** For any non-zero harmonic mode N, there exists a positive winding number Q* satisfying the stability condition. This is the geometric origin of discrete lepton masses - each harmonic mode N selects a unique Q*. -/ theorem stable_configuration_exists (beta gamma : ℝ) (h_beta : beta > 0) (h_gamma : gamma > 0) (N : ℤ) (hN : N ≠ 0) : ∃ Q > 0, stability_polynomial beta gamma N Q = 0 := by -- Get P(0) < 0 have h_P0 : stability_polynomial beta gamma N 0 < 0 := stability_polynomial_neg_at_zero beta gamma h_gamma N hN -- Get Q_big with P(Q_big) > 0 obtain ⟨Q_big, hQ_big_pos, h_P_big⟩ := stability_polynomial_pos_for_large_Q beta gamma h_beta h_gamma N -- Apply IVT: continuous function from negative to positive value crosses zero have h_cont := stability_polynomial_continuous beta gamma N -- IVT on [0, Q_big]: P(0) < 0 < P(Q_big), so ∃ x ∈ (0, Q_big), P(x) = 0 have h_ivt := intermediate_value_Icc (le_of_lt hQ_big_pos) h_cont.continuousOn have h_mem : (0 : ℝ) ∈ Set.Icc (stability_polynomial beta gamma N 0) (stability_polynomial beta gamma N Q_big) := ⟨le_of_lt h_P0, le_of_lt h_P_big⟩ obtain ⟨Q_root, hQ_in_interval, hQ_zero⟩ := h_ivt h_mem -- Q_root > 0 because P(0) < 0 and P(Q_root) = 0 use Q_root constructor · -- Q_root > 0 rcases hQ_in_interval with ⟨hQ_ge, hQ_le⟩ by_contra h_not_pos push_neg at h_not_pos have hQ_root_zero : Q_root = 0 := le_antisymm h_not_pos hQ_ge rw [hQ_root_zero] at hQ_zero linarith · exact hQ_zero /-- **Theorem G.6: The Lepton Mass Hierarchy.** If the vacuum parameters (beta, gamma) are fixed, stable configurations exist for different harmonic modes, with energy increasing with |N|. Assignment (Phase 3 Validation): - Electron: N = 1 => E_min ≈ 0.511 MeV - Muon: N = 19 => E_min ≈ 105.66 MeV -/ theorem lepton_mass_hierarchy_existence (beta gamma : ℝ) (h_beta : beta > 0) (h_gamma : gamma > 0) : ∃ (Q_e Q_mu : ℝ), Q_e > 0 ∧ Q_mu > 0 ∧ stability_polynomial beta gamma 1 Q_e = 0 ∧ stability_polynomial beta gamma 19 Q_mu = 0 ∧ Q_e < Q_mu := by -- Get stable configurations for N=1 and N=19 obtain ⟨Q_e, hQe_pos, hQe_stable⟩ := stable_configuration_exists beta gamma h_beta h_gamma 1 (by norm_num) obtain ⟨Q_mu, hQmu_pos, hQmu_stable⟩ := stable_configuration_exists beta gamma h_beta h_gamma 19 (by norm_num) use Q_e, Q_mu refine ⟨hQe_pos, hQmu_pos, hQe_stable, hQmu_stable, ?_⟩ -- Show Q_e < Q_mu using stability equations -- From P(Q) = 0: Q²(2βQ + 1) = γN² -- Since γ·19² > γ·1² and Q²(2βQ+1) is increasing for Q > 0, -- we must have Q_mu > Q_e have h_eq_e : Q_e^2 * (2 * beta * Q_e + 1) = gamma * 1^2 := by have := hQe_stable simp only [stability_polynomial] at this -- this : 2 * beta * Q_e^3 + Q_e^2 - gamma * 1^2 = 0 -- Goal: Q_e^2 * (2 * beta * Q_e + 1) = gamma * 1^2 -- Note: Q_e^2 * (2 * beta * Q_e + 1) = 2*beta*Q_e^3 + Q_e^2 ring_nf ring_nf at this linarith have h_eq_mu : Q_mu^2 * (2 * beta * Q_mu + 1) = gamma * 19^2 := by have := hQmu_stable simp only [stability_polynomial] at this ring_nf ring_nf at this linarith have h_gamma_ineq : gamma * (1 : ℝ)^2 < gamma * (19 : ℝ)^2 := by have : (1 : ℝ)^2 < (19 : ℝ)^2 := by norm_num exact mul_lt_mul_of_pos_left this h_gamma -- Q²(2βQ+1) is strictly increasing for Q > 0 -- So Q_e < Q_mu by_contra h_not_lt push_neg at h_not_lt -- If Q_e ≥ Q_mu, then Q_e²(2βQ_e+1) ≥ Q_mu²(2βQ_mu+1) have h_Qe2 : Q_e^2 ≥ Q_mu^2 := by nlinarith have h_coeff : 2 * beta * Q_e + 1 ≥ 2 * beta * Q_mu + 1 := by nlinarith have h_pos1 : 2 * beta * Q_mu + 1 > 0 := by nlinarith have h_pos2 : Q_e^2 > 0 := sq_pos_of_pos hQe_pos have h_prod : Q_e^2 * (2 * beta * Q_e + 1) ≥ Q_mu^2 * (2 * beta * Q_mu + 1) := by have h_pos3 : Q_mu^2 > 0 := sq_pos_of_pos hQmu_pos nlinarith rw [h_eq_e, h_eq_mu] at h_prod linarith /-- At stability P(Q) = 0, the energy simplifies to E = 3βQ² + 2Q. This is because γN²/Q = Q(2βQ + 1) at the critical point. -/ theorem energy_at_stability (beta gamma : ℝ) (Q : ℝ) (N : ℤ) (hQ_pos : Q > 0) (h_stable : stability_polynomial beta gamma N Q = 0) : topological_twist_energy beta gamma Q N = 3 * beta * Q^2 + 2 * Q := by unfold topological_twist_energy stability_polynomial at * -- From 2βQ³ + Q² - γN² = 0, we get γN² = Q²(2βQ + 1) -- So γN²/Q = Q(2βQ + 1) = 2βQ² + Q have h_relation : gamma * (N : ℝ)^2 / Q = 2 * beta * Q^2 + Q := by have hQ_ne : Q ≠ 0 := ne_of_gt hQ_pos field_simp at h_stable ⊢ linarith rw [h_relation] ring /-- **Theorem G.7: Energy Hierarchy from Winding Hierarchy.** If Q_1 < Q_2 (both satisfying stability), then E_1 < E_2. Combined with the Q ordering from stability_polynomial, this gives the lepton mass hierarchy. -/ theorem energy_increases_with_Q (beta gamma : ℝ) (h_beta : beta > 0) (Q1 Q2 : ℝ) (h_Q1_pos : Q1 > 0) (h_Q2_pos : Q2 > 0) (N1 N2 : ℤ) (h_stable1 : stability_polynomial beta gamma N1 Q1 = 0) (h_stable2 : stability_polynomial beta gamma N2 Q2 = 0) (h_Q_lt : Q1 < Q2) : topological_twist_energy beta gamma Q1 N1 < topological_twist_energy beta gamma Q2 N2 := by rw [energy_at_stability beta gamma Q1 N1 h_Q1_pos h_stable1] rw [energy_at_stability beta gamma Q2 N2 h_Q2_pos h_stable2] -- 3βQ1² + 2Q1 < 3βQ2² + 2Q2 when Q1 < Q2 and β > 0 have h_sq : Q1^2 < Q2^2 := sq_lt_sq' (by linarith) h_Q_lt have h1 : 3 * beta * Q1^2 < 3 * beta * Q2^2 := by nlinarith linarith /-- **Corollary: The Complete Lepton Mass Hierarchy.** Combining the Q ordering from harmonic modes with energy monotonicity gives E(N=1) < E(N=19), i.e., electron mass < muon mass. -/ theorem lepton_energy_hierarchy (beta gamma : ℝ) (h_beta : beta > 0) (h_gamma : gamma > 0) : ∃ (Q_e Q_mu : ℝ), Q_e > 0 ∧ Q_mu > 0 ∧ stability_polynomial beta gamma 1 Q_e = 0 ∧ stability_polynomial beta gamma 19 Q_mu = 0 ∧ topological_twist_energy beta gamma Q_e 1 < topological_twist_energy beta gamma Q_mu 19 := by obtain ⟨Q_e, Q_mu, hQe_pos, hQmu_pos, hQe_stable, hQmu_stable, hQ_lt⟩ := lepton_mass_hierarchy_existence beta gamma h_beta h_gamma use Q_e, Q_mu refine ⟨hQe_pos, hQmu_pos, hQe_stable, hQmu_stable, ?_⟩ exact energy_increases_with_Q beta gamma h_beta Q_e Q_mu hQe_pos hQmu_pos 1 19 hQe_stable hQmu_stable hQ_lt end QFD.Lepton