N_galactic_approx
Galactic dynamics models in Recognition Science rely on the rung number N_galactic = log_φ(τ★/τ₀) for the memory timescale. This theorem shows 140 < N_galactic < 145, confirming the integer approximation 142 used in acceleration derivations. The proof substitutes the explicit ratio τ★/τ₀, converts the bounds via log properties, and applies pre-established inequalities on powers of phi.
claim$140 < N < 145$ where $N = log_φ(τ★/τ₀)$, τ★ is the galactic memory timescale in seconds, and τ₀ is the fundamental RS tick.
background
The module defines τ★ as the characteristic galactic memory timescale (133 million years in SI units) and τ₀ as the base RS tick 7.3e-15 s. N_galactic is the base-φ logarithm of their ratio: N = log(τ★/τ₀) / log φ. Upstream lemma one_lt_phi supplies 1 < φ so that log φ is positive. Bounds phi_pow_140_lt_ratio and ratio_lt_phi_pow_145 from GalacticBounds supply the concrete power comparisons needed for the log inequalities.
proof idea
The tactic proof first equates the ratio to its rational value by unfolding and norm_num. It obtains 0 < log φ from one_lt_phi. The lower bound rewrites via lt_div_iff₀ and log_rpow, reducing to log(ratio) > log(φ^140) which is discharged by phi_pow_140_lt_ratio. The upper bound applies div_lt_iff₀ and reduces to log(ratio) < log(φ^145) using ratio_lt_phi_pow_145.
why it matters in Recognition Science
This result justifies the integer rung 142 adopted in GravityParameters.N_galactic for deriving the galactic acceleration scale from r₀. It closes the approximation step on the φ-ladder for galactic memory, consistent with the eight-tick octave. The bound is tight enough to support the integer claim without further scaffolding.
scope and limits
- Does not establish that N_galactic equals exactly 142.
- Does not derive τ★ from more fundamental RS principles.
- Does not quantify uncertainty in the galactic timescale input.
- Does not connect to Berry creation threshold or Z_cf.
formal statement (Lean)
30theorem N_galactic_approx : 140 < N_galactic ∧ N_galactic < 145 := by
proof body
Tactic-mode proof.
31 have h_ratio_val : tau_star_s / tau0_SI = (galactic_ratio_rational : ℝ) := by
32 unfold tau_star_s tau0_SI galactic_ratio_rational
33 norm_num
34 have hlog_phi : 0 < Real.log phi := Real.log_pos Constants.one_lt_phi
35
36 constructor
37 · unfold N_galactic
38 rw [h_ratio_val]
39 apply (lt_div_iff₀ hlog_phi).mpr
40 rw [← Real.log_rpow Constants.phi_pos]
41 apply Real.log_lt_log (Real.rpow_pos_of_pos Constants.phi_pos 140)
42 exact phi_pow_140_lt_ratio
43 · unfold N_galactic
44 rw [h_ratio_val]
45 apply (div_lt_iff₀ hlog_phi).mpr
46 rw [← Real.log_rpow Constants.phi_pos]
47 apply Real.log_lt_log
48 · have h_pos : (0 : ℝ) < (galactic_ratio_rational : ℝ) := by norm_num [galactic_ratio_rational]
49 exact h_pos
50 · exact ratio_lt_phi_pow_145
51
52/-- **THEOREM: τ★ Lies on the φ-Ladder** -/