initial_state_is_zero_defect

formal source

  26/-- The universe begins in the unique zero-defect configuration.
  27    This IS the Big Bang initial condition — not a singularity,
  28    but the minimum-cost ledger state. -/
  29theorem initial_state_is_zero_defect (N : ℕ) (hN : 0 < N) :
  30    Foundation.InitialCondition.total_defect
  31      (Foundation.InitialCondition.unity_config N hN) = 0 :=
  32  Foundation.InitialCondition.unity_defect_zero hN
  33
  34/-! ## Dark Energy from Ledger Vacuum -/
  35
  36/-- The RS prediction for the dark energy density parameter.
  37    Ω_Λ = 11/16 − α/π where α = α_lock from RS constants.
  38
  39    The value 11/16 = 0.6875 comes from the fraction of ledger modes
  40    that are in vacuum (unexcited) state in the 8-tick cycle.
  41    The correction −α/π accounts for the small perturbation from
  42    matter-coupled modes. -/
  43noncomputable def omega_lambda : ℝ := 11/16 - alphaLock / Real.pi
  44
  45/-- Ω_Λ is positive (dark energy exists). -/
  46theorem omega_lambda_pos : 0 < omega_lambda := by
  47  unfold omega_lambda
  48  have h_alpha := alphaLock_lt_one
  49  have h_alpha_pos := alphaLock_pos
  50  have h_pi := Real.pi_pos
  51  have h_pi_gt3 := Real.pi_gt_three
  52  have h1 : alphaLock / Real.pi < 11 / 16 := by
  53    rw [div_lt_div_iff₀ Real.pi_pos (by norm_num)]
  54    nlinarith [alphaLock_lt_one, Real.pi_gt_three]
  55  linarith
  56
  57/-- Ω_Λ < 1 (subunitary). -/
  58theorem omega_lambda_lt_one : omega_lambda < 1 := by
  59  unfold omega_lambda