mass_gap_from_phi

formal source

 338    ∀ n : ℤ, n ≠ 0 → massGap ≤ Jcost (PhiLadder n) := spectral_gap
 339
 340/-- The mass gap is exactly J(φ). -/
 341theorem mass_gap_from_phi : Jcost phi = massGap := Jcost_phi_eq_massGap
 342
 343/-- **Mass gap numerical bounds**: 0.118 < Δ < 0.119. -/
 344theorem mass_gap_bounds : (0.118 : ℝ) < massGap ∧ massGap < (0.119 : ℝ) :=
 345  massGap_numerical_bound
 346
 347/-! ## §12  The Complete Spacetime Emergence Certificate -/
 348
 349/-- **THE SPACETIME EMERGENCE CERTIFICATE**
 350
 351    Verifies the full structure of 4D Lorentzian spacetime is forced
 352    by the J-cost functional and the RS forcing chain T0–T8. -/
 353structure SpacetimeEmergenceCert where
 354  dim_eq_four : spacetime_dim = 4
 355  temporal_one : temporal_dim = 1
 356  spatial_three : spatial_dim = 3
 357  signature_lorentzian :
 358    (Finset.univ.filter (fun i : Fin 4 => η i i < 0)).card = 1 ∧
 359    (Finset.univ.filter (fun i : Fin 4 => 0 < η i i)).card = 3
 360  metric_trace : ∑ i : Fin 4, η i i = 2
 361  metric_det : ∏ i : Fin 4, η i i = -1
 362  cone_timelike : ∀ v : Displacement,
 363    interval v < 0 ↔ spatial_norm_sq v < temporal_sq v
 364  cone_lightlike : ∀ v : Displacement,
 365    interval v = 0 ↔ spatial_norm_sq v = temporal_sq v
 366  mass_gap_positive : 0 < massGap
 367  mass_gap_universal : ∀ n : ℤ, n ≠ 0 → massGap ≤ Jcost (PhiLadder n)
 368  octave_period : eight_tick = 8
 369  sig_unique : temporal_dim = 1 ∧ spatial_dim = 3
 370  arrow : ∀ t : ℕ, t < t + 1
 371