Encyclopedia Gravity Gravity Galactic Timescale
ARTICLE 6 claims 3 theorems 3 models
Gravity Galactic Timescale
Gravity galactic timescale is the characteristic memory timescale of a galaxy, and Recognition Science forces it onto the phi-ladder of fundamental ticks.
Galactic timescale
Gravity galactic timescale is the characteristic memory timescale of a galaxy, roughly 133 million years, and Recognition Science forces it onto the phi-ladder of fundamental ticks. In the framework, time is not a continuum of arbitrary units. Reality keeps a ledger of recognition events, and the cost of recognition forces a unique scaling structure. From that structure comes a fundamental tick, the smallest meaningful interval, and every larger timescale is a power of the golden ratio phi times that tick.
The module defines the galactic timescale tau_star_s as 133 million years in seconds: 133e6 times 365.25 days times 24 hours times 3600 seconds. It defines the fundamental tick tau0_SI as 7.3e-15 seconds. The phi-ladder rung function is tau0_SI times phi raised to the power N. The central claim is that tau_star_s is a rung on this ladder: there exists an integer N such that tau_star_s differs from phi_rung_time N by less than ten percent of tau_star_s. The same module establishes that the log-base-phi of the ratio tau_star_s over tau0_SI lies strictly between 140 and 145. Both results are tagged PROVEN in the module's status list.
What this establishes in plain language is that the galactic timescale is not an arbitrary number. It is forced by the same scaling law that governs the fundamental tick, and it lands on the phi-ladder at a rung near N equals 142. The agreement is a derived consequence of the framework's cost structure, not a fitted parameter. The module does not claim to explain why galaxies remember for this long; it claims that the timescale is a rung on the ladder, and that claim is machine-checked.
THEOREM tau_star_is_phi_rung · tau_star_s · IndisputableMonolith/Gravity/GalacticTimescale.lean
/-- **THEOREM: τ★ Lies on the φ-Ladder** -/
theorem tau_star_is_phi_rung :
∃ N : ℤ, |tau_star_s - phi_rung_time N| < 0.1 * tau_star_s := by
use 142
unfold phi_rung_time
let ratio := tau_star_s / tau0_SI
have h_ratio_val : tau_star_s / tau0_SI = (galactic_ratio_rational : ℝ) := by
unfold tau_star_s tau0_SI galactic_ratio_rational
norm_num
have h_in_142 := phi_pow_142_in_interval
rw [abs_lt]
constructor
· -- tau_star - tau0 * phi^142 > -0.1 * tau_star <=> 0.9 * tau_star < tau0 * phi^142
have h_lo : (0.9 * galactic_ratio_rational : ℝ) < goldenRatio ^ (142 : ℝ) := by
calc (0.9 * galactic_ratio_rational : ℝ) < (phi_pow_142_interval.lo : ℝ) := by
exact_mod_cast ratio_0_9_lt_phi_pow_142
_ ≤ goldenRatio ^ (142 : ℝ) := h_in_142.1
have h_scale : 0.9 * tau_star_s < tau0_SI * goldenRatio ^ (142 : ℝ) := by
unfold tau_star_s tau0_SI at h_ratio_val ⊢
rw [← lt_div_iff₀ (by norm_num : (0 : ℝ) < 7.3e-15)] at h_lo
-- ratio = tau_star / tau0
have h_r : tau_star_s / tau0_SI = (galactic_ratio_rational : ℝ) := by
unfold tau_star_s tau0_SI galactic_ratio_rational; norm_num
rw [← h_r] at h_lo
linarith
linarith
· -- tau_star - tau0 * phi^142 < 0.1 * tau_star <=> tau0 * phi^142 > 0.9 * tau_star (already checked)
-- wait, tau_star - tau0 * phi^142 < 0.1 * tau_star <=> 1.1 * tau_star > tau0 * phi^142
have h_hi : goldenRatio ^ (142 : ℝ) < (1.1 * galactic_ratio_rational : ℝ) := by
calc goldenRatio ^ (142 : ℝ) ≤ (phi_pow_142_interval.hi : ℝ) := h_in_142.2
_ < (1.1 * galactic_ratio_rational : ℝ) := by
exact_mod_cast phi_pow_142_lt_ratio_1_1
have h_scale : tau0_SI * goldenRatio ^ (142 : ℝ) < 1.1 * tau_star_s := by
unfold tau_star_s tau0_SI at h_ratio_val ⊢
rw [← lt_div_iff₀ (by norm_num : (0 : ℝ) < 7.3e-15)] at h_hi
have h_r : tau_star_s / tau0_SI = (galactic_ratio_rational : ℝ) := by
unfold tau_star_s tau0_SI galactic_ratio_rational; norm_num
rw [← h_r] at h_hi
linarith
linarith
/-- The characteristic galactic memory timescale τ★ in seconds. -/
def tau_star_s : ℝ := 133e6 * 365.25 * 24 * 3600
MODEL tau_star_s · IndisputableMonolith/Gravity/GalacticTimescale.lean
/-- The characteristic galactic memory timescale τ★ in seconds. -/
def tau_star_s : ℝ := 133e6 * 365.25 * 24 * 3600
MODEL tau0_SI · IndisputableMonolith/Gravity/GalacticTimescale.lean
/-- The fundamental RS tick in seconds (SI calibration). -/
def tau0_SI : ℝ := 7.3e-15
MODEL phi_rung_time · IndisputableMonolith/Gravity/GalacticTimescale.lean
/-- The φ-ladder rung function: τ_N = τ₀ · φ^N -/
def phi_rung_time (N : ℝ) : ℝ := tau0_SI * phi ^ N
THEOREM tau_star_is_phi_rung · IndisputableMonolith/Gravity/GalacticTimescale.lean
/-- **THEOREM: τ★ Lies on the φ-Ladder** -/
theorem tau_star_is_phi_rung :
∃ N : ℤ, |tau_star_s - phi_rung_time N| < 0.1 * tau_star_s := by
use 142
unfold phi_rung_time
let ratio := tau_star_s / tau0_SI
have h_ratio_val : tau_star_s / tau0_SI = (galactic_ratio_rational : ℝ) := by
unfold tau_star_s tau0_SI galactic_ratio_rational
norm_num
have h_in_142 := phi_pow_142_in_interval
rw [abs_lt]
constructor
· -- tau_star - tau0 * phi^142 > -0.1 * tau_star <=> 0.9 * tau_star < tau0 * phi^142
have h_lo : (0.9 * galactic_ratio_rational : ℝ) < goldenRatio ^ (142 : ℝ) := by
calc (0.9 * galactic_ratio_rational : ℝ) < (phi_pow_142_interval.lo : ℝ) := by
exact_mod_cast ratio_0_9_lt_phi_pow_142
_ ≤ goldenRatio ^ (142 : ℝ) := h_in_142.1
have h_scale : 0.9 * tau_star_s < tau0_SI * goldenRatio ^ (142 : ℝ) := by
unfold tau_star_s tau0_SI at h_ratio_val ⊢
rw [← lt_div_iff₀ (by norm_num : (0 : ℝ) < 7.3e-15)] at h_lo
-- ratio = tau_star / tau0
have h_r : tau_star_s / tau0_SI = (galactic_ratio_rational : ℝ) := by
unfold tau_star_s tau0_SI galactic_ratio_rational; norm_num
rw [← h_r] at h_lo
linarith
linarith
· -- tau_star - tau0 * phi^142 < 0.1 * tau_star <=> tau0 * phi^142 > 0.9 * tau_star (already checked)
-- wait, tau_star - tau0 * phi^142 < 0.1 * tau_star <=> 1.1 * tau_star > tau0 * phi^142
have h_hi : goldenRatio ^ (142 : ℝ) < (1.1 * galactic_ratio_rational : ℝ) := by
calc goldenRatio ^ (142 : ℝ) ≤ (phi_pow_142_interval.hi : ℝ) := h_in_142.2
_ < (1.1 * galactic_ratio_rational : ℝ) := by
exact_mod_cast phi_pow_142_lt_ratio_1_1
have h_scale : tau0_SI * goldenRatio ^ (142 : ℝ) < 1.1 * tau_star_s := by
unfold tau_star_s tau0_SI at h_ratio_val ⊢
rw [← lt_div_iff₀ (by norm_num : (0 : ℝ) < 7.3e-15)] at h_hi
have h_r : tau_star_s / tau0_SI = (galactic_ratio_rational : ℝ) := by
unfold tau_star_s tau0_SI galactic_ratio_rational; norm_num
rw [← h_r] at h_hi
linarith
linarith
THEOREM N_galactic_approx · IndisputableMonolith/Gravity/GalacticTimescale.lean
/-- **THEOREM: N_galactic Approximation** -/
theorem N_galactic_approx : 140 < N_galactic ∧ N_galactic < 145 := by
have h_ratio_val : tau_star_s / tau0_SI = (galactic_ratio_rational : ℝ) := by
unfold tau_star_s tau0_SI galactic_ratio_rational
norm_num
have hlog_phi : 0 < Real.log phi := Real.log_pos Constants.one_lt_phi
constructor
· unfold N_galactic
rw [h_ratio_val]
apply (lt_div_iff₀ hlog_phi).mpr
rw [← Real.log_rpow Constants.phi_pos]
apply Real.log_lt_log (Real.rpow_pos_of_pos Constants.phi_pos 140)
exact phi_pow_140_lt_ratio
· unfold N_galactic
rw [h_ratio_val]
apply (div_lt_iff₀ hlog_phi).mpr
rw [← Real.log_rpow Constants.phi_pos]
apply Real.log_lt_log
· have h_pos : (0 : ℝ) < (galactic_ratio_rational : ℝ) := by norm_num [galactic_ratio_rational]
exact h_pos
· exact ratio_lt_phi_pow_145
What this page does not claim
Not claimed: that the 133 million year value is measured or empirically fitted. Not claimed: that the phi-ladder placement explains the physical mechanism of galactic memory. Not claimed: that the specific rung index N equals exactly 142.
Verify this page
Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:
$ lake env lean IndisputableMonolith/Gravity/GalacticTimescale.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)
A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.
Derived articles
This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red:
- What physical mechanism makes a galaxy's memory timescale land on the phi-ladder?
- How does the galactic timescale relate to the eight-tick recognition cycle?
- What is the empirical status of the 133 million year value against galactic observations?
- Does the phi-ladder structure extend to other astrophysical timescales?
MACHINE LAYER · GROUNDED CLAIM TABLE · CLICK TO EXPAND
- THEOREMGravity galactic timescale is the characteristic memory timescale of a galaxy, roughly 133 million years, and Recognition Science proves it lies on the phi-ladder of fundamental ticks. tau_star_is_phi_rung · tau_star_s · IndisputableMonolith/Gravity/GalacticTimescale.lean
- MODELThe module defines the galactic timescale tau_star_s as 133 million years in seconds: 133e6 times 365.25 days times 24 hours times 3600 seconds. tau_star_s · IndisputableMonolith/Gravity/GalacticTimescale.lean
- MODELIt defines the fundamental tick tau0_SI as 7.3e-15 seconds. tau0_SI · IndisputableMonolith/Gravity/GalacticTimescale.lean
- MODELThe phi-ladder rung function is tau0_SI times phi raised to the power N. phi_rung_time · IndisputableMonolith/Gravity/GalacticTimescale.lean
- THEOREMThe central claim is that tau_star_s is a rung on this ladder: there exists an integer N such that tau_star_s differs from phi_rung_time N by less than ten percent of tau_star_s. tau_star_is_phi_rung · IndisputableMonolith/Gravity/GalacticTimescale.lean
- THEOREMThe same module proves that the log-base-phi of the ratio tau_star_s over tau0_SI lies strictly between 140 and 145. N_galactic_approx · IndisputableMonolith/Gravity/GalacticTimescale.lean