N_galactic

formal source

 196def galactic_constraint : ℝ := 2 * N_tau_galactic - N_r_galactic
 197
 198/-- The φ-ladder rung for the galactic memory timescale (integer approximation). -/
 199def N_galactic : ℝ := 142
 200
 201/-- The timescale constraint linking a₀ and r₀.
 202    Given τ★ and r₀, the acceleration scale is forced. -/
 203def a0_from_tau_r0 (tau_star r0 : ℝ) : ℝ := 2 * Real.pi * r0 / tau_star ^ 2
 204
 205/-- The timescale constraint linking a₀ and r₀.
 206    Given τ★ and a₀, the length scale is forced. -/
 207def r0_from_tau_a0 (tau_star a0 : ℝ) : ℝ := a0 * tau_star ^ 2 / (2 * Real.pi)
 208
 209theorem tau_constraint_consistency (tau_star r0 a0 : ℝ)
 210    (hτ : tau_star ≠ 0) (ha : a0 = a0_from_tau_r0 tau_star r0) :
 211    r0 = r0_from_tau_a0 tau_star a0 := by
 212  unfold a0_from_tau_r0 r0_from_tau_a0 at *
 213  rw [ha]
 214  have hτ2 : tau_star ^ 2 ≠ 0 := pow_ne_zero 2 hτ
 215  field_simp [hτ2]
 216
 217/-- **THEOREM: φ-Ladder Formula for a₀**
 218
 219    In φ-ladder coordinates, a₀ is determined by the rungs:
 220    a₀ = 2π·c/τ₀ · φ^(N_r - 2N_τ)
 221
 222    **Derivation**:
 223    If τ★ = τ₀·φ^N_τ and r₀ = ℓ₀·φ^N_r (where ℓ₀ = c·τ₀), then:
 224
 225      a₀ = 2π r₀/τ★² = 2π·(ℓ₀·φ^N_r)/(τ₀·φ^N_τ)²
 226         = 2π·(c·τ₀·φ^N_r)/(τ₀²·φ^(2N_τ))
 227         = 2π·c/τ₀ · φ^(N_r - 2N_τ)
 228
 229    With N_τ ≈ 142.4 and N_r ≈ 126.3: