IndisputableMonolith.Gravity.GravityParameters
IndisputableMonolith/Gravity/GravityParameters.lean · 341 lines · 35 declarations
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1import Mathlib
2import IndisputableMonolith.Constants
3
4/-!
5# Gravity Parameters Derived from φ
6
7This module derives the phenomenological galactic gravity parameters from
8Recognition Science first principles. Each parameter is either:
9
101. **DERIVED**: Mathematically proven from φ
112. **HAS RS BASIS**: Formula matches observations, physical motivation exists
123. **PHENOMENOLOGICAL**: No known RS connection
13
14## The Seven Parameters
15
16| Parameter | Status | RS Formula | Match |
17|-----------|--------|------------|-------|
18| α | DERIVED | 1 - 1/φ | 1.8% |
19| C_ξ | HAS RS BASIS | 2φ⁻⁴ | 2% |
20| p | HAS RS BASIS | 1 - αLock/4 | 0.2% |
21| A | HAS RS BASIS | 1 + αLock/2 | 3% |
22| Υ★ | DERIVED | φ | — (convention) |
23| a₀, r₀ | LINKED | via τ★ = √(2πr₀/a₀) | — |
24
25## Axiom Status
26
27| Axiom | Nature | Status |
28|-------|--------|--------|
29| a0_phi_ladder_formula | RS prediction | THEOREM ✓ (PROVEN) |
30
31-/
32
33namespace IndisputableMonolith
34namespace Gravity
35namespace GravityParameters
36
37open Real
38open Constants
39
40noncomputable section
41
42/-! ## 1. α (Dynamical-Time Exponent) — DERIVED
43
44The dynamical-time exponent α is exactly 1 - 1/φ.
45This is proven in `ParameterizationBridge.lean`. -/
46
47/-- The RS-derived dynamical-time exponent.
48 α_gravity = 2 · alphaLock = 1 - 1/φ ≈ 0.382 -/
49def alpha_gravity : ℝ := 1 - 1 / phi
50
51theorem alpha_gravity_eq_two_alphaLock : alpha_gravity = 2 * alphaLock := by
52 unfold alpha_gravity alphaLock
53 ring
54
55/-- Paper fitted value: 0.389 ± 0.015
56 RS prediction: 1 - 1/φ ≈ 0.382
57 Match: 1.8% -/
58theorem alpha_gravity_pos : 0 < alpha_gravity := by
59 unfold alpha_gravity
60 have h := one_lt_phi -- 1 < φ
61 have hphi_pos := phi_pos
62 have : 1 / phi < 1 := by
63 rw [div_lt_one hphi_pos]
64 exact h
65 linarith
66
67/-! ## 2. Υ★ (Mass-to-Light Ratio) — DERIVED
68
69The stellar mass-to-light ratio is φ.
70This is proven in `Astrophysics/MassToLight.lean`. -/
71
72/-- The RS-derived stellar mass-to-light ratio.
73 Υ★ = φ ≈ 1.618 -/
74def upsilon_star : ℝ := phi
75
76theorem upsilon_star_eq_phi : upsilon_star = phi := rfl
77
78theorem upsilon_star_bounds : 1.5 < upsilon_star ∧ upsilon_star < 1.62 := by
79 unfold upsilon_star
80 exact ⟨phi_gt_onePointFive, phi_lt_onePointSixTwo⟩
81
82/-- Upsilon-star bounds imply positivity of the stellar mass-to-light ratio. -/
83theorem upsilon_star_bounds_implies_pos (h : 1.5 < upsilon_star ∧ upsilon_star < 1.62) :
84 0 < upsilon_star := by
85 linarith [h.1]
86
87/-! ## 3. C_ξ (Morphology Coupling) — HAS RS BASIS
88
89The morphology coupling coefficient is 2φ⁻⁴.
90
91**Physical motivation:** The factor 2 is fundamental (from 2³ = 8 tick structure).
92The φ⁻⁴ is one power above E_coh = φ⁻⁵. -/
93
94/-- The RS-based morphology coupling coefficient.
95 C_ξ = 2 · φ⁻⁴ ≈ 0.292 -/
96def C_xi : ℝ := 2 * phi ^ (-(4 : ℝ))
97
98/-- C_ξ is positive.
99 Paper fitted value: 0.298 ± 0.015
100 RS prediction: 2/φ⁴ ≈ 0.292
101 Match: 2% -/
102theorem C_xi_pos : 0 < C_xi := by
103 unfold C_xi
104 have hphi_pos := phi_pos
105 apply mul_pos (by norm_num : (0:ℝ) < 2)
106 exact Real.rpow_pos_of_pos hphi_pos _
107
108/-! ## 4. p (Spatial Profile Steepness) — HAS RS BASIS
109
110The steepness exponent is 1 - αLock/4.
111
112**Physical motivation:** The transition steepness differs from 1 by a quarter
113of the fine-structure exponent, linking spatial structure to α. -/
114
115/-- The RS-based spatial profile steepness.
116 p = 1 - αLock/4 = 1 - (1 - 1/φ)/8 ≈ 0.952 -/
117def p_steepness : ℝ := 1 - alphaLock / 4
118
119theorem p_steepness_eq : p_steepness = 1 - (1 - 1/phi) / 8 := by
120 unfold p_steepness alphaLock
121 ring
122
123/-- p is between 0 and 1.
124 Paper fitted value: 0.95 ± 0.02
125 RS prediction: 1 - αLock/4 ≈ 0.952
126 Match: 0.2% -/
127theorem p_steepness_pos : 0 < p_steepness ∧ p_steepness < 1 := by
128 unfold p_steepness
129 have ha := alphaLock_pos
130 have hb := alphaLock_lt_one
131 constructor <;> linarith
132
133/-! ## 5. A (Spatial Profile Amplitude) — HAS RS BASIS
134
135The amplitude is 1 + αLock/2.
136
137**Physical motivation:** The outer enhancement is 1 plus half the fine-structure
138exponent, linking spatial structure amplitude to α. -/
139
140/-- The RS-based spatial profile amplitude.
141 A = 1 + αLock/2 = 1 + (1 - 1/φ)/4 ≈ 1.096 -/
142def A_amplitude : ℝ := 1 + alphaLock / 2
143
144theorem A_amplitude_eq : A_amplitude = 1 + (1 - 1/phi) / 4 := by
145 unfold A_amplitude alphaLock
146 ring
147
148/-- A is between 1 and 2.
149 Paper fitted value: 1.06 ± 0.04
150 RS prediction: 1 + αLock/2 ≈ 1.096
151 Match: 3% (within 1σ) -/
152theorem A_amplitude_bounds : 1 < A_amplitude ∧ A_amplitude < 2 := by
153 unfold A_amplitude
154 have ha := alphaLock_pos
155 have hb := alphaLock_lt_one
156 constructor <;> linarith
157
158/-! ## 6-7. a₀ and r₀ — LINKED via τ★
159
160The characteristic acceleration a₀ and radius r₀ are linked through
161the memory timescale τ★:
162
163 τ★ = √(2π r₀/a₀)
164
165This constraint reduces (a₀, r₀) from 2 independent parameters to 1.
166
167## Key Result: a₀ is Determined by the φ-Ladder
168
169If τ★ = τ₀·φ^N_τ and r₀ = ℓ₀·φ^N_r (where ℓ₀ = c·τ₀), then:
170
171 a₀ = 2π·c/τ₀ · φ^(N_r - 2N_τ)
172
173With the paper values (N_τ ≈ 142.4, N_r ≈ 126.3):
174 a₀ = 2π·c/τ₀ · φ^(-158.5) ≈ 1.96 × 10⁻¹⁰ m/s²
175
176This matches the paper's fitted a₀ = 1.95 × 10⁻¹⁰ m/s² to 0.5%!
177
178**Conclusion:** a₀ is NOT a free parameter. Only one φ-ladder rung
179(either N_τ or N_r) remains as the single phenomenological input.
180-/
181
182/-- The φ-ladder rung for the galactic memory timescale.
183 τ★ = τ₀ · φ^N_galactic where N_galactic ≈ 142.4.
184
185 142 ≈ 144 - 2 = F_12 - 2, suggesting possible Fibonacci structure. -/
186def N_tau_galactic : ℝ := 142.4
187
188/-- The φ-ladder rung for the characteristic galactic radius.
189 r₀ = ℓ₀ · φ^N_r_galactic where N_r_galactic ≈ 126.3.
190
191 This is constrained by the τ★ relation: N_r = 2·N_τ - 158.5 -/
192def N_r_galactic : ℝ := 126.3
193
194/-- The "galactic constraint number": 2N_τ - N_r ≈ 158.5
195 This determines the acceleration scale exponent. -/
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:
230 N_r - 2N_τ ≈ -158.5
231 a₀ ≈ 2π·(3×10⁸)/(7.3×10⁻¹⁵)·φ^(-158.5)
232 ≈ 1.96×10⁻¹⁰ m/s²
233
234 This matches the paper's a₀ = 1.95×10⁻¹⁰ m/s² to 0.5%.
235
236 **RS Significance**: This is a TESTABLE PREDICTION. The MOND acceleration
237 scale is not a free parameter but is fixed by the φ-ladder structure. -/
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
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. -/
291def F_12 : ℕ := 144
292
293theorem F_12_is_fibonacci_12 : F_12 = Nat.fib 12 := by native_decide
294
295theorem F_12_is_perfect_square : F_12 = 12 ^ 2 := by native_decide
296
297/-- The conjectured galactic timescale rung.
298 N_τ = F_12 - 2 = 144 - 2 = 142 -/
299def N_tau_conjecture : ℕ := F_12 - 2
300
301theorem N_tau_conjecture_eq_142 : N_tau_conjecture = 142 := by native_decide
302
303/-- The 16-rung offset is 2^4 = 4² (second non-trivial perfect square). -/
304def rung_offset : ℕ := 16
305
306theorem rung_offset_is_power_of_2 : rung_offset = 2 ^ 4 := by native_decide
307
308theorem rung_offset_is_perfect_square : rung_offset = 4 ^ 2 := by native_decide
309
310theorem rung_offset_is_two_8tick_cycles : rung_offset = 2 * 8 := by native_decide
311
312/-- The conjectured galactic radius rung.
313 N_r = N_τ - 16 = 142 - 16 = 126 -/
314def N_r_conjecture : ℕ := N_tau_conjecture - rung_offset
315
316theorem N_r_conjecture_eq_126 : N_r_conjecture = 126 := by native_decide
317
318/-- If the conjecture is correct, N_r = N_τ - 16 exactly. -/
319theorem rung_relationship : N_r_conjecture = N_tau_conjecture - rung_offset := rfl
320
321/-! ## Summary
322
323| Parameter | RS Formula | Status |
324|-----------|------------|--------|
325| α = 1 - 1/φ ≈ 0.382 | **DERIVED** (algebraic) |
326| Υ★ = φ ≈ 1.618 | **DERIVED** (MassToLight.lean) |
327| C_ξ = 2φ⁻⁴ ≈ 0.292 | **HAS RS BASIS** (numerical match 2%) |
328| p = 1 - αLock/4 ≈ 0.952 | **HAS RS BASIS** (numerical match 0.2%) |
329| A = 1 + αLock/2 ≈ 1.096 | **HAS RS BASIS** (numerical match 3%) |
330| a₀, r₀ | **LINKED** via τ★ constraint |
331| N_τ = F_12 - 2 | **CONJECTURED** (Fibonacci-square) |
332
333If the Fibonacci-square conjecture is correct, this model has ZERO phenomenological
334parameters — all seven are derived from φ plus the Fibonacci-square selection. -/
335
336end
337
338end GravityParameters
339end Gravity
340end IndisputableMonolith
341