theorem
proved
tactic proof
a0_phi_ladder_formula
show as:
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formal statement (Lean)
238theorem a0_phi_ladder_formula (tau0 ell0 c_light N_tau N_r : ℝ)
239 (hτ0 : tau0 > 0) (hlight : ell0 = c_light * tau0) :
240 let tau_star := tau0 * phi ^ N_tau
proof body
Tactic-mode proof.
241 let r0 := ell0 * phi ^ N_r
242 let a0 := a0_from_tau_r0 tau_star r0
243 a0 = 2 * Real.pi * c_light / tau0 * phi ^ (N_r - 2 * N_tau) := by
244 intro tau_star r0 a0
245 -- a0 = 2π r0 / τ★² = 2π (ℓ₀ φ^N_r) / (τ₀ φ^N_τ)²
246 -- = 2π (c τ₀ φ^N_r) / (τ₀² φ^{2N_τ})
247 -- = 2π c / τ₀ · φ^{N_r - 2N_τ}
248 dsimp [a0, r0, tau_star, a0_from_tau_r0]
249 rw [hlight]
250 have hphi : phi > 0 := phi_pos
251 have hτ0_ne : tau0 ≠ 0 := ne_of_gt hτ0
252 have hphi_pow_ne : phi ^ N_tau ≠ 0 := Real.rpow_pos_of_pos hphi N_tau |>.ne'
253 have htau_star_ne : tau0 * phi ^ N_tau ≠ 0 := mul_ne_zero hτ0_ne hphi_pow_ne
254 field_simp [htau_star_ne, hτ0_ne]
255 -- Goal: c_light * phi ^ N_r = c_light * (phi ^ N_tau) ^ 2 * phi ^ (N_r - 2 * N_tau)
256 have h1 : (phi ^ N_tau) ^ 2 = phi ^ (2 * N_tau) := by
257 have hpos : phi ^ N_tau > 0 := Real.rpow_pos_of_pos hphi N_tau
258 rw [sq, ← Real.rpow_add hphi]
259 congr 1
260 ring
261 rw [h1]
262 have h2 : phi ^ (2 * N_tau) * phi ^ (N_r - 2 * N_tau) = phi ^ N_r := by
263 rw [← Real.rpow_add hphi]
264 ring_nf
265 calc c_light * phi ^ N_r
266 = c_light * (phi ^ (2 * N_tau) * phi ^ (N_r - 2 * N_tau)) := by rw [h2]
267 _ = c_light * phi ^ (2 * N_tau) * phi ^ (N_r - 2 * N_tau) := by ring
268
269/-! ## 8. The Fibonacci-Square Conjecture
270
271**Conjecture:** The galactic timescale rung N_τ = 142 has deep structure:
272
273 N_τ = F_12 - 2 = 144 - 2 = 142
274
275where F_12 = 144 = 12² is the **unique non-trivial Fibonacci square**.
276
277This makes rung 144 geometrically special — it's the only φ-ladder rung that is
278both a Fibonacci number AND a perfect square (after the trivial F_0 = F_1 = 1).
279
280**Supporting observations:**
2811. The "-2" correction could arise from 2D disk geometry
2822. The radius rung is offset by 16: N_r = N_τ - 16 = 142 - 16 = 126
2833. 16 = 2^4 = 4² is the second non-trivial perfect square
2844. 16 = 2 × 8 (two 8-tick cycles)
285
286If this conjecture is correct, the model has **ZERO phenomenological parameters**.
287-/
288
289/-- F_12 is the unique non-trivial Fibonacci-square.
290 F_12 = 144 = 12² is both a Fibonacci number and a perfect square. -/