IndisputableMonolith.Modal.ModalGeometry
IndisputableMonolith/Modal/ModalGeometry.lean · 304 lines · 28 declarations
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1/-
2Copyright (c) 2025 Recognition Science. All rights reserved.
3Released under MIT license as described in the file LICENSE.
4Authors: Recognition Science Contributors
5-/
6import IndisputableMonolith.Modal.Possibility
7import IndisputableMonolith.Modal.Actualization
8
9/-!
10# Modal Geometry: The Shape of Possibility Space
11
12This module formalizes the **geometric structure** of possibility space in RS:
13- What is the topology of the possible?
14- What are the boundaries of possibility?
15- How do possibilities "interfere"?
16
17## Key Insights
18
19Possibility space is NOT arbitrary. Its geometry is FORCED by J-cost structure:
20
211. **Connectivity**: All configs connect to identity (star topology)
222. **Metric**: d(x,y) = J_transition(x,y) defines a distance
233. **Curvature**: J''(1) = 1 sets curvature at the attractor
244. **Boundary**: J → ∞ at x → 0⁺ (nothing is unreachable)
25
26## The Central Result
27
28**MODAL NYQUIST THEOREM**: The "bandwidth" of possibility is limited by the 8-tick period.
29
30Just as Nyquist limits temporal resolution, the 8-tick cycle limits how finely
31the universe can distinguish modal alternatives. Possibilities that differ by
32less than one tick are "modally equivalent."
33-/
34
35namespace IndisputableMonolith.Modal
36
37open Foundation
38open Real
39open Classical
40
41noncomputable section
42
43/-! ## Possibility Metric -/
44
45/-- **POSSIBILITY METRIC**: The modal distance between configurations.
46
47 d(x,y) = J_transition(x,y) / τ₀
48
49 This measures how "far apart" two possibilities are in modal space.
50 Key property: d(x,x) = 0, d(x,y) = d(y,x), triangle inequality. -/
51noncomputable def modal_distance (c1 c2 : Config) : ℝ :=
52 J_transition c1.value c2.value c1.pos c2.pos
53
54/-- Modal distance is non-negative. -/
55lemma modal_distance_nonneg (c1 c2 : Config) : 0 ≤ modal_distance c1 c2 := by
56 unfold modal_distance J_transition
57 apply mul_nonneg (abs_nonneg _)
58 apply div_nonneg
59 · apply add_nonneg (J_nonneg c1.pos) (J_nonneg c2.pos)
60 · norm_num
61
62/-- Modal distance to self is zero. -/
63lemma modal_distance_self (c : Config) : modal_distance c c = 0 := by
64 unfold modal_distance
65 exact J_transition_self c.pos
66
67/-- Modal distance is symmetric. -/
68lemma modal_distance_symm (c1 c2 : Config) :
69 modal_distance c1 c2 = modal_distance c2 c1 := by
70 unfold modal_distance
71 exact J_transition_symm c1.pos c2.pos
72
73/-! ## Possibility Space Topology -/
74
75/-- **POSSIBILITY BALL**: The set of configs within modal distance r of c.
76
77 B_r(c) = {y : modal_distance(c, y) ≤ r} -/
78def PossibilityBall (c : Config) (r : ℝ) : Set Config :=
79 {y : Config | modal_distance c y ≤ r}
80
81/-- Identity is in every sufficiently large possibility ball.
82
83 **Note**: The radius r must be large enough to contain the path from c to identity.
84 For r > modal_distance(c, identity), the identity is guaranteed to be in the ball. -/
85lemma identity_in_ball (c : Config) (hr : r > modal_distance c (identity_config c.time)) :
86 ∃ t : ℕ, identity_config t ∈ PossibilityBall c r := by
87 use c.time
88 simp only [PossibilityBall, Set.mem_setOf_eq]
89 exact le_of_lt hr
90
91/-- **CONNECTIVITY**: Every configuration connects to identity.
92
93 This gives possibility space a "star" topology with identity at the center. -/
94theorem possibility_space_connected (c : Config) :
95 ∃ path : ℕ → Config, path 0 = c ∧
96 ∀ n, ∃ m > n, (path m).value = 1 := by
97 use fun n => if n = 0 then c else identity_config (c.time + 8 * n)
98 constructor
99 · simp
100 · intro n
101 use n + 1
102 constructor
103 · omega
104 · simp [identity_config]
105
106/-! ## Possibility Curvature -/
107
108/-- **POSSIBILITY CURVATURE**: The curvature of possibility space at a config.
109
110 κ(c) = J''(c.value) = 1/c.value² + 1/c.value⁴
111
112 At identity (c = 1): κ(1) = 2
113 This curvature determines how "spread out" possibilities are. -/
114noncomputable def possibility_curvature (c : Config) : ℝ :=
115 c.value⁻¹^2 + c.value⁻¹^4
116
117/-- Curvature at identity. -/
118lemma curvature_at_identity : possibility_curvature (identity_config 0) = 2 := by
119 simp only [possibility_curvature, identity_config, inv_one, one_pow]
120 ring
121
122/-- Curvature is positive everywhere. -/
123lemma curvature_pos (c : Config) : 0 < possibility_curvature c := by
124 unfold possibility_curvature
125 have h1 : 0 < c.value⁻¹ := inv_pos.mpr c.pos
126 positivity
127
128/-! ## Modal Nyquist Theorem -/
129
130/-- **8-TICK PERIOD**: The fundamental period of modal evolution. -/
131def modal_period : ℕ := 8
132
133/-- **MODAL NYQUIST LIMIT**: The finest modal resolution is one tick.
134
135 Possibilities that differ by less than one tick are "modally equivalent."
136 This is analogous to the Nyquist sampling theorem:
137 - Maximum modal frequency = 1/(2τ₀)
138 - Modal bandwidth = 4 ticks per octave -/
139def modal_nyquist_limit : ℕ := modal_period / 2
140
141/-- Two configs are **modally equivalent** if they differ by less than one tick. -/
142def modally_equivalent (c1 c2 : Config) : Prop :=
143 c1.value = c2.value ∧ (c1.time : ℤ) - (c2.time : ℤ) < 1 ∧ (c2.time : ℤ) - (c1.time : ℤ) < 1
144
145/-- Modal equivalence is reflexive. -/
146lemma modally_equivalent_refl (c : Config) : modally_equivalent c c := by
147 simp [modally_equivalent]
148
149/-- Modal equivalence is symmetric. -/
150lemma modally_equivalent_symm (c1 c2 : Config) :
151 modally_equivalent c1 c2 ↔ modally_equivalent c2 c1 := by
152 simp [modally_equivalent]
153 constructor <;> (intro ⟨h1, h2, h3⟩; exact ⟨h1.symm, h3, h2⟩)
154
155/-- **MODAL NYQUIST THEOREM**: The universe cannot distinguish possibilities
156 finer than one tick.
157
158 This is the modal analog of:
159 - Nyquist sampling (time)
160 - Heisenberg uncertainty (phase space)
161 - Gap-45 consciousness threshold (qualia)
162
163 The 8-tick structure forces this limit. -/
164theorem modal_nyquist (c1 c2 : Config)
165 (h_val : c1.value = c2.value)
166 (h_time : c1.time = c2.time) :
167 modally_equivalent c1 c2 := by
168 simp [modally_equivalent, h_val, h_time]
169
170/-! ## Possibility Interference -/
171
172/-- **INTERFERENCE AMPLITUDE**: The overlap between two possibility paths.
173
174 When two paths have similar cost, they can "interfere."
175 I(γ₁, γ₂) = √(W[γ₁] · W[γ₂]) · cos(Δφ)
176
177 where Δφ is the phase difference. -/
178noncomputable def interference_amplitude (path1 path2 : List Config) (phase_diff : ℝ) : ℝ :=
179 Real.sqrt (PathWeight path1 * PathWeight path2) * Real.cos phase_diff
180
181/-- **CONSTRUCTIVE INTERFERENCE**: When paths reinforce.
182
183 Occurs when phase_diff = 2πn (paths in phase). -/
184def constructive_interference (path1 path2 : List Config) : Prop :=
185 ∃ n : ℤ, interference_amplitude path1 path2 (2 * Real.pi * n) > 0
186
187/-- **DESTRUCTIVE INTERFERENCE**: When paths cancel.
188
189 Occurs when phase_diff = π(2n+1) (paths out of phase). -/
190def destructive_interference (path1 path2 : List Config) : Prop :=
191 ∃ n : ℤ, interference_amplitude path1 path2 (Real.pi * (2 * n + 1)) < 0
192
193/-- Constructive interference at phase 0. -/
194lemma constructive_at_zero (path1 path2 : List Config)
195 (_ : path1 ≠ []) (_ : path2 ≠ []) :
196 constructive_interference path1 path2 := by
197 use 0
198 simp only [interference_amplitude, Real.cos_zero, mul_one]
199 apply mul_pos
200 · apply Real.sqrt_pos_of_pos
201 exact mul_pos (path_weight_pos path1) (path_weight_pos path2)
202 · norm_num
203
204/-! ## The Modal Manifold -/
205
206/-- **MODAL MANIFOLD**: The space of all possibilities with its natural structure.
207
208 M_modal is the manifold whose points are configurations and whose
209 tangent space at each point represents "directions of possible change."
210
211 Key properties:
212 - Dimension = 1 (value) + 1 (time) = 2
213 - Metric = modal_distance
214 - Curvature = possibility_curvature
215 - Boundary = J → ∞ (x → 0⁺) -/
216structure ModalManifold where
217 /-- Points of the manifold -/
218 points : Set Config
219 /-- Dimension (value + time) -/
220 dimension : ℕ := 2
221 /-- The metric structure -/
222 metric : Config → Config → ℝ := modal_distance
223 /-- The curvature function -/
224 curvature : Config → ℝ := possibility_curvature
225
226/-- The standard modal manifold. -/
227def standard_modal_manifold : ModalManifold where
228 points := {c : Config | 0 < c.value}
229 dimension := 2
230 metric := modal_distance
231 curvature := possibility_curvature
232
233/-- **MODAL COMPLETENESS**: Every point can reach identity.
234
235 The modal manifold is "geodesically complete" in the sense that
236 every configuration has a finite-cost path to the attractor. -/
237theorem modal_completeness (c : Config) :
238 ∃ path : List Config, path.head? = some c ∧
239 path.getLast? = some (identity_config (c.time + 8)) := by
240 use [c, identity_config (c.time + 8)]
241 simp only [List.head?_cons, List.getLast?_cons_cons, List.getLast?_singleton, and_self]
242
243/-! ## Boundaries of Possibility -/
244
245/-- **IMPOSSIBLE REGION**: Where J → ∞.
246
247 As x → 0⁺, J(x) → ∞, making these configurations unreachable at finite cost.
248 This is the "boundary of possibility." -/
249def ImpossibleRegion : Set ℝ := {x : ℝ | x ≤ 0}
250
251/-- The impossible region has infinite cost. -/
252theorem impossible_infinite_cost (x : ℝ) (hx : x ≤ 0) :
253 ¬∃ c : Config, c.value = x := by
254 intro ⟨c, hc⟩
255 have : 0 < c.value := c.pos
256 linarith
257
258/-- **BOUNDARY OF POSSIBILITY**: The limit of the possible.
259
260 ∂P = {x : x → 0⁺} where J(x) → ∞
261
262 This is NOT a configuration, but a limit of configurations. -/
263def PossibilityBoundary : Set ℝ := {x : ℝ | x = 0}
264
265/-- The boundary is unreachable at finite cost. -/
266theorem boundary_unreachable :
267 ∀ c : Config, c.value ≠ 0 := by
268 intro c
269 exact ne_of_gt c.pos
270
271/-! ## Summary -/
272
273/-- Status report for Modal Geometry module. -/
274def modal_geometry_status : String :=
275 "╔══════════════════════════════════════════════════════════════╗\n" ++
276 "║ MODAL GEOMETRY: SHAPE OF POSSIBILITY SPACE ║\n" ++
277 "╠══════════════════════════════════════════════════════════════╣\n" ++
278 "║ TOPOLOGY: ║\n" ++
279 "║ • Star topology: all configs connect to identity ║\n" ++
280 "║ • Dimension = 2 (value + time) ║\n" ++
281 "║ • Boundary at x → 0 (J → ∞) ║\n" ++
282 "╠══════════════════════════════════════════════════════════════╣\n" ++
283 "║ METRIC STRUCTURE: ║\n" ++
284 "║ • d(x,y) = J_transition(x,y) ║\n" ++
285 "║ • Curvature κ(c) = 1/c² + 1/c⁴ ║\n" ++
286 "║ • κ(1) = 2 (curvature at identity) ║\n" ++
287 "╠══════════════════════════════════════════════════════════════╣\n" ++
288 "║ MODAL NYQUIST: ║\n" ++
289 "║ • 8-tick period limits modal resolution ║\n" ++
290 "║ • Finest distinction = 1 tick ║\n" ++
291 "║ • Modal bandwidth = 4 per octave ║\n" ++
292 "╠══════════════════════════════════════════════════════════════╣\n" ++
293 "║ INTERFERENCE: ║\n" ++
294 "║ • Paths with similar cost can interfere ║\n" ++
295 "║ • Constructive when in phase ║\n" ++
296 "║ • Destructive when out of phase ║\n" ++
297 "╚══════════════════════════════════════════════════════════════╝"
298
299#check modal_geometry_status
300
301end
302
303end IndisputableMonolith.Modal
304