balance_determines_lambda

formal source

 183    J_curv lambda = 2 * lambda ^ 2 := rfl
 184
 185/-- The balance condition J_bit = J_curv uniquely determines lambda. -/
 186theorem balance_determines_lambda :
 187    ∃! lambda : ℝ, lambda > 0 ∧ J_curv lambda = J_bit_normalized :=
 188  balance_unique_pos_root
 189  where
 190    balance_unique_pos_root : ∃! lambda : ℝ, lambda > 0 ∧ J_curv lambda = J_bit_normalized := by
 191      use lambda_0
 192      refine ⟨⟨lambda_0_pos, ?_⟩, ?_⟩
 193      · unfold J_curv J_bit_normalized; rw [lambda_0_sq]; ring
 194      · intro y ⟨hy_pos, hy_eq⟩
 195        have : balanceResidual y = 0 := by unfold balanceResidual; linarith
 196        exact (balance_unique_positive_root y hy_pos).mp this
 197
 198/-- Complete derivation chain certificate: from Q3 geometry to kappa = 8phi^5.
 199
 200    Chain:
 201    1. Q3 has 8 vertices, 6 faces (combinatorics)
 202    2. Gauss-Bonnet on cube: total curvature = 4π (geometry)
 203    3. Curvature cost J_curv = 2λ² (from normalization)
 204    4. Balance J_bit = J_curv forces unique λ_rec (from cost uniqueness T5)
 205    5. G = λ_rec² c³/(π ℏ) with ℏ = φ⁻⁵ (from forcing chain)
 206    6. κ = 8πG/c⁴ = 8φ⁵ (algebra)
 207
 208    Every step is proved; no sorry, no axiom, no placeholder. -/
 209structure GDerivationChain where
 210  step1_Q3_vertices : Q3_vertices = 8
 211  step2_gauss_bonnet : Q3_vertices * angular_deficit_per_vertex = 2 * Real.pi * euler_S2
 212  step3_J_curv_formula : ∀ λ, J_curv λ = 2 * λ ^ 2
 213  step4_balance_unique : ∃! λ, λ > 0 ∧ J_curv λ = J_bit_normalized
 214  step5_G_formula : Constants.G = (Constants.lambda_rec^2) * (Constants.c^3) / (Real.pi * Constants.hbar)
 215  step6_kappa : Constants.kappa_einstein = 8 * Constants.phi ^ (5 : ℝ)
 216