/- Copyright (c) 2025 Quantum Field Dynamics. All rights reserved. Released under Apache 2.0 license. Authors: Tracy # Vortex Stability: β-ξ Degeneracy Resolution This module formalizes the mathematical breakthrough that resolved the β-parameter offset between V22 (β ≈ 3.15) and Golden Loop (β ≈ 3.043, derived from α). ## Key Results **Theorem 1 (V22 Degeneracy)**: Single-parameter models (ξ=0) are degenerate - any radius R can fit the data by adjusting β. This is the GIGO case. **Theorem 2 (Degeneracy Broken)**: Two-parameter models (β, ξ) uniquely determine the particle scale R for fixed mass. This is the stability condition. **Lemma (Beta Offset)**: The empirical β_fit from V22 relates to true (β, ξ) via: β_fit = β_true + ξ·(C_grad)/(C_comp·R²) This explains the 3% offset as the "Geometric Signature of Gradient Energy." ## Physical Interpretation The gradient term ξ|∇ρ|² contributes ~64% of total energy for Hill vortex. V22 model (ξ=0) compensated by inflating β from ~3.043 to 3.15. Including ξ breaks the (β, R) degeneracy and validates β ≈ 3.043 from α. ## References - Source: complete_energy_functional/D_FLOW_ELECTRON_FINAL_SYNTHESIS.md - MCMC Results: Stage 3b (β = 3.0627 ± 0.1491, ξ = 0.97 ± 0.55) - Missing gradient: GRADIENT_ENERGY_BREAKTHROUGH_SUMMARY.md -/ import Mathlib.Analysis.Calculus.Deriv.Basic import Mathlib.Analysis.Calculus.MeanValue import Mathlib.Analysis.SpecialFunctions.Pow.Real import Mathlib.Data.Real.Basic import Mathlib.Tactic import QFD.Vacuum.VacuumParameters open Real Set Filter Topology noncomputable section namespace QFD.Lepton /-! ## Hill Vortex Geometry Constants -/ structure HillGeometry where C_comp : ℝ C_grad : ℝ h_comp_pos : C_comp > 0 h_grad_pos : C_grad > 0 def standardHillGeometry : HillGeometry := { C_comp := 1 C_grad := 9/5 h_comp_pos := by norm_num h_grad_pos := by norm_num } /-! ## Energy Functional Definitions -/ def totalEnergy (g : HillGeometry) (β ξ R : ℝ) : ℝ := β * g.C_comp * R^3 + ξ * g.C_grad * R def compressionEnergy (g : HillGeometry) (β R : ℝ) : ℝ := β * g.C_comp * R^3 def gradientEnergy (g : HillGeometry) (ξ R : ℝ) : ℝ := ξ * g.C_grad * R /-! ## Theorem 1: V22 Degeneracy -/ theorem v22_is_degenerate (g : HillGeometry) (mass : ℝ) (h_mass : mass > 0) : ∀ R₁ R₂ : ℝ, R₁ > 0 → R₂ > 0 → ∃ β₁ β₂ : ℝ, totalEnergy g β₁ 0 R₁ = mass ∧ totalEnergy g β₂ 0 R₂ = mass := by intro R₁ R₂ hR₁ hR₂ use mass / (g.C_comp * R₁^3), mass / (g.C_comp * R₂^3) constructor · unfold totalEnergy simp have h_ne : g.C_comp * R₁^3 ≠ 0 := mul_ne_zero (ne_of_gt g.h_comp_pos) (pow_ne_zero 3 (ne_of_gt hR₁)) field_simp [h_ne] exact div_self (ne_of_gt g.h_comp_pos) · unfold totalEnergy simp have h_ne : g.C_comp * R₂^3 ≠ 0 := mul_ne_zero (ne_of_gt g.h_comp_pos) (pow_ne_zero 3 (ne_of_gt hR₂)) field_simp [h_ne] exact div_self (ne_of_gt g.h_comp_pos) theorem v22_beta_R_perfectly_correlated (g : HillGeometry) (mass : ℝ) (h_mass : mass > 0) : ∀ β₁ β₂ R₁ R₂ : ℝ, β₁ > 0 → β₂ > 0 → R₁ > 0 → R₂ > 0 → totalEnergy g β₁ 0 R₁ = mass → totalEnergy g β₂ 0 R₂ = mass → β₁ * R₁^3 = β₂ * R₂^3 := by intro β₁ β₂ R₁ R₂ hβ₁ hβ₂ hR₁ hR₂ h_E₁ h_E₂ unfold totalEnergy at h_E₁ h_E₂ simp at h_E₁ h_E₂ have h_C_comp_ne : g.C_comp ≠ 0 := ne_of_gt g.h_comp_pos calc β₁ * R₁^3 = (β₁ * g.C_comp * R₁^3) / g.C_comp := by field_simp _ = mass / g.C_comp := by rw [h_E₁] _ = (β₂ * g.C_comp * R₂^3) / g.C_comp := by rw [← h_E₂] _ = β₂ * R₂^3 := by field_simp /-! ## Theorem 2: Degeneracy Broken -/ theorem energy_derivative_positive (g : HillGeometry) (β ξ R : ℝ) (hβ : β > 0) (hξ : ξ > 0) (hR : R > 0) : 3 * β * g.C_comp * R^2 + ξ * g.C_grad > 0 := by apply add_pos · apply mul_pos · apply mul_pos · apply mul_pos · norm_num · exact hβ · exact g.h_comp_pos · apply sq_pos_of_pos hR · apply mul_pos hξ g.h_grad_pos theorem degeneracy_broken (g : HillGeometry) (β ξ mass : ℝ) (hβ : β > 0) (hξ : ξ > 0) (hm : mass > 0) : ∃! R : ℝ, R > 0 ∧ totalEnergy g β ξ R = mass := by let a := β * g.C_comp let b := ξ * g.C_grad have ha : a > 0 := mul_pos hβ g.h_comp_pos have hb : b > 0 := mul_pos hξ g.h_grad_pos let f : ℝ → ℝ := fun R => a * R ^ 3 + b * R -- Prove f is continuous have hf_cont : Continuous f := by apply Continuous.add · exact Continuous.mul continuous_const (continuous_pow 3) · exact Continuous.mul continuous_const continuous_id -- Prove f is strictly monotonic on (0, ∞) have hf_strict_mono : StrictMonoOn f (Ioi 0) := by intro R1 hR1 R2 hR2 h_lt unfold_let f apply add_lt_add · apply mul_lt_mul_of_pos_left _ ha exact pow_lt_pow_left h_lt (le_of_lt hR1) (by norm_num : 0 < 3) · exact mul_lt_mul_of_pos_left h_lt hb -- Prove f(ε) < mass for small ε have h_small : ∃ ε > 0, f ε < mass := by use min 1 (mass / (2 * (a + b))) constructor · apply lt_min one_pos (div_pos hm (mul_pos two_pos (add_pos ha hb))) · calc f (min 1 (mass / (2 * (a + b)))) = a * (min 1 (mass / (2 * (a + b))))^3 + b * (min 1 (mass / (2 * (a + b)))) := rfl _ ≤ a * 1 + b * (mass / (2 * (a + b))) := by apply add_le_add · apply mul_le_mul_of_nonneg_left _ (le_of_lt ha) apply pow_le_one · apply le_of_lt (lt_min one_pos _) · exact min_le_left _ _ · apply mul_le_mul_of_nonneg_left (min_le_right _ _) (le_of_lt hb) _ < mass := by calc a * 1 + b * (mass / (2 * (a + b))) = a + b * mass / (2 * (a + b)) := by ring _ ≤ (a + b) * mass / (2 * (a + b)) := by apply div_le_div_of_nonneg_right _ (mul_pos two_pos (add_pos ha hb)) linarith _ = mass / 2 := by field_simp; ring _ < mass := half_lt_self hm -- Prove f(R_big) > mass for large R_big have h_large : ∃ R_big > 0, f R_big > mass := by use max 1 ((2 * mass / a) ^ (1/3 : ℝ)) constructor · apply lt_max_of_lt_left one_pos · calc f (max 1 ((2 * mass / a) ^ (1/3 : ℝ))) ≥ a * ((2 * mass / a) ^ (1/3 : ℝ))^3 := by apply le_add_of_nonneg_right apply mul_nonneg (le_of_lt hb) apply le_of_lt (lt_max_of_lt_left _) exact Real.rpow_pos_of_pos (div_pos (mul_pos two_pos hm) ha) _ _ = a * (2 * mass / a) := by rw [← Real.rpow_natCast, ← Real.rpow_mul (le_of_lt (div_pos (mul_pos two_pos hm) ha))] norm_num _ = 2 * mass := by field_simp; ring _ > mass := by linarith -- Apply IVT obtain ⟨ε, hε_pos, hε_mass⟩ := h_small obtain ⟨R_big, hR_big, hR_big_mass⟩ := h_large have hε_R : ε < R_big := by by_contra h_contra push_neg at h_contra have : f ε ≥ f R_big := by by_cases h_eq : ε = R_big · rw [h_eq] · have : ε > R_big := lt_of_le_of_ne h_contra (Ne.symm h_eq) exact le_of_lt (hf_strict_mono hR_big hε_pos this) linarith have : ContinuousOn f (Icc ε R_big) := hf_cont.continuousOn have h_ivt : ∃ r ∈ Icc ε R_big, f r = mass := by apply intermediate_value_Icc (le_of_lt hε_R) this · exact le_of_lt hε_mass · exact le_of_lt hR_big_mass obtain ⟨r, hr_Icc, hr_mass⟩ := h_ivt have hr_pos : r > 0 := lt_of_lt_of_le hε_pos hr_Icc.1 use r constructor · exact ⟨hr_pos, hr_mass⟩ · intro r' ⟨hr'_pos, hr'_mass⟩ by_cases h : r ≤ r' · by_cases h_eq : r = r' · exact h_eq · have : r < r' := lt_of_le_of_ne h h_eq have : f r < f r' := hf_strict_mono hr_pos hr'_pos this linarith · push_neg at h have : f r' < f r := hf_strict_mono hr'_pos hr_pos h linarith lemma beta_offset_relation (g : HillGeometry) (β_true ξ_true R_true : ℝ) (hR : R_true > 0) : let β_fit := β_true + (ξ_true * g.C_grad) / (g.C_comp * R_true^2) totalEnergy g β_fit 0 R_true = totalEnergy g β_true ξ_true R_true := by intro β_fit unfold totalEnergy simp rw [show β_fit = β_true + (ξ_true * g.C_grad) / (g.C_comp * R_true^2) from rfl] rw [add_mul] have h_ne : g.C_comp * R_true^2 ≠ 0 := mul_ne_zero (ne_of_gt g.h_comp_pos) (pow_ne_zero 2 (ne_of_gt hR)) have h_C_ne : g.C_comp ≠ 0 := ne_of_gt g.h_comp_pos congr 1 field_simp [h_ne, h_C_ne] theorem degeneracy_resolution_complete (g : HillGeometry) : (∀ mass : ℝ, mass > 0 → ∃ β₁ β₂ R₁ R₂ : ℝ, β₁ ≠ β₂ ∧ R₁ ≠ R₂ ∧ totalEnergy g β₁ 0 R₁ = mass ∧ totalEnergy g β₂ 0 R₂ = mass) ∧ (∀ β ξ mass : ℝ, β > 0 → ξ > 0 → mass > 0 → ∃! R : ℝ, R > 0 ∧ totalEnergy g β ξ R = mass) := by constructor · intro mass h_mass use mass / g.C_comp, mass / (8 * g.C_comp), 1, 2 constructor · intro h_eq have h_C_ne : g.C_comp ≠ 0 := ne_of_gt g.h_comp_pos have h_m_ne : mass ≠ 0 := ne_of_gt h_mass field_simp [h_C_ne, h_m_ne] at h_eq linarith constructor · norm_num constructor · unfold totalEnergy; simp field_simp [ne_of_gt g.h_comp_pos] · unfold totalEnergy; simp field_simp [ne_of_gt g.h_comp_pos] norm_num · intro β ξ mass hβ hξ hm exact degeneracy_broken g β ξ mass hβ hξ hm end QFD.Lepton