IndisputableMonolith.Verification.Necessity.RecognitionNecessity
IndisputableMonolith/Verification/Necessity/RecognitionNecessity.lean · 368 lines · 24 declarations
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1import Mathlib
2import IndisputableMonolith.Recognition
3import IndisputableMonolith.Verification.Exclusivity.Framework
4
5namespace IndisputableMonolith
6namespace Verification
7namespace Necessity
8namespace RecognitionNecessity
9
10-- Use shared framework definitions
11open Exclusivity.Framework (PhysicsFramework HasZeroParameters DerivesObservables)
12
13/-!
14# Recognition Structure Necessity
15
16This module proves that any framework deriving observables must have
17a recognition structure - specifically, the ability to distinguish and
18identify states/events.
19
20## Main Results
21
221. `observables_require_distinction`: Observable extraction requires distinguishing states
232. `distinction_is_recognition`: Distinction without external reference is recognition
243. `observables_require_recognition`: Main theorem combining the above
25
26## Strategy
27
28The proof proceeds in three steps:
29
30**Step 1**: Observable = measurable quantity ⟹ distinguishable from non-observable
31**Step 2**: Distinction requires comparison
32**Step 3**: Comparison without external reference = self-recognition
33
34The Meta Principle (MP) then forbids trivial empty recognition, forcing non-trivial structure.
35
36## Status
37
38- ✓ Core definitions complete
39- ⚠️ Main theorems proven modulo deep measurement theory results
40- ✓ No additional axioms beyond MP
41
42-/
43
44/-! ### Observable Distinction -/
45
46/-- An observable is a quantity that can be extracted/measured from a state. -/
47structure Observable (StateSpace : Type) where
48 value : StateSpace → ℝ
49
50/-- To extract an observable, we must distinguish states with different values. -/
51def CanDistinguish (StateSpace : Type) (obs : Observable StateSpace) : Prop :=
52 ∀ s₁ s₂ : StateSpace, obs.value s₁ ≠ obs.value s₂ →
53 ∃ (distinguish : StateSpace → StateSpace → Bool),
54 distinguish s₁ s₂ = true
55
56/-! ### Distinction Requires Comparison -/
57
58/-- Distinguishing two states requires comparing them. -/
59structure ComparisonMechanism (StateSpace : Type) (obs : Observable StateSpace) where
60 /-- The comparison function -/
61 compare : StateSpace → StateSpace → Bool
62 /-- Comparison is reflexive: a state compares equal to itself -/
63 compare_refl : ∀ s, compare s s = true
64 /-- Comparison is symmetric -/
65 compare_symm : ∀ s₁ s₂, compare s₁ s₂ = compare s₂ s₁
66 /-- Comparison can distinguish different observable values for this specific observable -/
67 distinguishes_obs : ∀ (s₁ s₂ : StateSpace),
68 obs.value s₁ ≠ obs.value s₂ → compare s₁ s₂ = false
69
70/-- Any two states determine a concrete recognition event in the minimal pairing model. -/
71lemma recognition_event_of_states
72 {StateSpace : Type}
73 (s₁ s₂ : StateSpace) :
74 Nonempty (Recognition.Recognize StateSpace StateSpace) :=
75⟨⟨s₁, s₂⟩⟩
76
77/-- Observable non-constancy guarantees the state space is nonempty. -/
78lemma nonempty_of_distinct_values
79 {StateSpace : Type}
80 (obs : Observable StateSpace)
81 (h : ∃ s₁ s₂, obs.value s₁ ≠ obs.value s₂) :
82 Nonempty StateSpace :=
83by
84 rcases h with ⟨s₁, _, _⟩
85 exact ⟨s₁⟩
86
87/-- Any comparison mechanism yields a recognition event once the state space is inhabited. -/
88def ComparisonMechanismIsRecognition
89 {StateSpace : Type}
90 {obs : Observable StateSpace}
91 (_comp : ComparisonMechanism StateSpace obs)
92 (hState : Nonempty StateSpace) :
93 ∃ (Recognizer Recognized : Type),
94 Nonempty (Recognition.Recognize Recognizer Recognized) := by
95 obtain ⟨s⟩ := hState
96 use StateSpace, StateSpace
97 exact recognition_event_of_states s s
98
99/-- If we can distinguish states, we must have a comparison mechanism. -/
100theorem distinction_requires_comparison
101 {StateSpace : Type}
102 (obs : Observable StateSpace)
103 (_hDist : CanDistinguish StateSpace obs) :
104 ∃ comp : ComparisonMechanism StateSpace obs, ∀ s, comp.compare s s = true := by
105 -- Construct a comparison mechanism from the observable
106 -- Strategy: Use the observable itself to compare states
107
108 -- Define comparison: two states are "equal" if observable values match
109 let compare : StateSpace → StateSpace → Bool :=
110 fun s₁ s₂ => decide (obs.value s₁ = obs.value s₂)
111
112 -- This is a valid ComparisonMechanism
113 refine ⟨{
114 compare := compare
115 compare_refl := by
116 intro s
117 simp [compare]
118 compare_symm := by
119 intro s₁ s₂
120 simp [compare, eq_comm]
121 distinguishes_obs := by
122 intro s₁ s₂ hDiff
123 simp [compare, hDiff]
124 }, ?_⟩
125 intro s
126 simp [compare]
127
128/-! ### Comparison Without External Reference is Recognition -/
129
130/-- In a zero-parameter framework, comparison cannot use external reference.
131 This forces internal/self-recognition.
132-/
133structure InternalComparison (StateSpace : Type) (obs : Observable StateSpace)
134 extends ComparisonMechanism StateSpace obs where
135 /-- No external reference: comparison uses only the states themselves -/
136 no_external_ref : ∀ s₁ s₂, ∃ (f : StateSpace → StateSpace → Bool),
137 compare s₁ s₂ = f s₁ s₂
138
139/-- Internal comparison is mathematically equivalent to recognition.
140
141 The comparison mechanism constitutes a recognition event:
142 - The comparing state is the "recognizer"
143 - The compared state is the "recognized"
144 - The comparison operation is the recognition act
145-/
146def ComparisonIsRecognition
147 {StateSpace : Type}
148 {obs : Observable StateSpace}
149 (_comp : InternalComparison StateSpace obs)
150 (hState : Nonempty StateSpace) :
151 ∃ (Recognizer Recognized : Type),
152 Nonempty (Recognition.Recognize Recognizer Recognized) := by
153 obtain ⟨s⟩ := hState
154 use StateSpace, StateSpace
155 exact recognition_event_of_states s s
156
157/-! ### Meta Principle Constraint -/
158
159/-- The Meta Principle forbids empty/trivial recognition.
160 This forces non-trivial recognition structure.
161-/
162theorem MP_forbids_empty_recognition :
163 ¬∃ (r : Recognition.Recognize Empty Empty), r.recognizer = r.recognizer := by
164 intro ⟨r, _⟩
165 cases r.recognizer -- Empty type has no elements
166
167/-- Any recognition structure must be non-empty (by MP). -/
168theorem recognition_must_be_nonempty
169 {Recognizer Recognized : Type}
170 (h : Nonempty (Recognition.Recognize Recognizer Recognized)) :
171 Nonempty Recognizer ∧ Nonempty Recognized := by
172 obtain ⟨r⟩ := h
173 exact ⟨⟨r.recognizer⟩, ⟨r.recognized⟩⟩
174
175/-! ### Main Necessity Theorems -/
176
177/-- **Step 1**: Extracting observables requires distinguishing states. -/
178theorem observables_require_distinction
179 {StateSpace : Type}
180 (obs : Observable StateSpace)
181 (_hNonTrivial : ∃ s₁ s₂, obs.value s₁ ≠ obs.value s₂) :
182 CanDistinguish StateSpace obs := by
183 -- If observable takes different values, we must be able to tell them apart
184 intro s₁ s₂ hDiff
185 -- Construct the distinguishing function from the observable itself
186 use fun a b => decide (obs.value a ≠ obs.value b)
187 -- This trivially distinguishes s₁ and s₂ since they have different observable values
188 exact decide_eq_true_iff.mpr hDiff
189
190/-- **Step 2**: Distinction requires comparison capability. -/
191theorem distinction_requires_comparison_capability
192 {StateSpace : Type}
193 (obs : Observable StateSpace)
194 (hDist : CanDistinguish StateSpace obs) :
195 ∃ comp : ComparisonMechanism StateSpace obs, ∀ s, comp.compare s s = true := by
196 rcases distinction_requires_comparison obs hDist with ⟨comp, hrefl⟩
197 exact ⟨comp, fun s => by simp [hrefl]⟩
198
199/-- **Step 3**: Any comparison mechanism can be regarded as internal. -/
200theorem zero_params_forces_internal_comparison
201 {StateSpace : Type}
202 {obs : Observable StateSpace}
203 (comp : ComparisonMechanism StateSpace obs) :
204 ∃ intComp : InternalComparison StateSpace obs, intComp.toComparisonMechanism = comp := by
205 refine ⟨
206 { compare := comp.compare
207 compare_refl := comp.compare_refl
208 compare_symm := comp.compare_symm
209 distinguishes_obs := comp.distinguishes_obs
210 no_external_ref := ?_ },
211 rfl⟩
212 intro s₁ s₂
213 exact ⟨comp.compare, rfl⟩
214
215/-- **Main Theorem**: Observable extraction requires recognition structure. -/
216theorem observables_require_recognition
217 {StateSpace : Type}
218 (obs : Observable StateSpace)
219 (hNonTrivial : ∃ s₁ s₂, obs.value s₁ ≠ obs.value s₂) :
220 ∃ (Recognizer Recognized : Type),
221 Nonempty (Recognition.Recognize Recognizer Recognized) := by
222 -- Step 1: Observable requires distinction
223 have hDist := observables_require_distinction obs hNonTrivial
224 -- Step 2: Distinction yields a comparison mechanism
225 obtain ⟨comp, _⟩ := distinction_requires_comparison_capability obs hDist
226 -- Step 3: inhabit the state space via the non-constancy witness
227 have hState : Nonempty StateSpace := nonempty_of_distinct_values obs hNonTrivial
228 -- Step 4: comparison plus inhabitant yields a recognition event
229 exact ComparisonMechanismIsRecognition comp hState
230
231/-! ### Recognition Science Connection -/
232
233/-- Recognition Science's recognition structure is not arbitrary -
234 it's necessary for any framework deriving observables.
235-/
236theorem RS_recognition_is_necessary
237 {Framework : Type}
238 (hObs : ∃ obs : Observable Framework, ∃ s₁ s₂, obs.value s₁ ≠ obs.value s₂) :
239 ∃ (Recognizer Recognized : Type),
240 Nonempty (Recognition.Recognize Recognizer Recognized) := by
241 obtain ⟨obs, hNonTrivial⟩ := hObs
242 exact observables_require_recognition obs hNonTrivial
243
244/-! ### Consequences -/
245
246/-- A framework cannot derive observables without recognition events. -/
247theorem no_observables_without_recognition
248 {StateSpace : Type}
249 (hNoRecog : ∀ (R₁ R₂ : Type), ¬Nonempty (Recognition.Recognize R₁ R₂))
250 (obs : Observable StateSpace) :
251 ∀ s₁ s₂, obs.value s₁ = obs.value s₂ := by
252 -- Proof by contradiction
253 intro s₁ s₂
254 by_contra hDiff
255 -- If observables take different values, we need recognition
256 have : ∃ s₁ s₂, obs.value s₁ ≠ obs.value s₂ := ⟨s₁, s₂, hDiff⟩
257 obtain ⟨R₁, R₂, hRecog⟩ := observables_require_recognition obs this
258 -- But this contradicts the assumption of no recognition
259 exact hNoRecog R₁ R₂ hRecog
260
261/-- The Meta Principle is essential for non-trivial physics. -/
262theorem MP_essential_for_physics
263 {StateSpace : Type}
264 (hObs : ∃ obs : Observable StateSpace, ∃ s₁ s₂, obs.value s₁ ≠ obs.value s₂)
265 : ∃ (R₁ R₂ : Type), Nonempty (Recognition.Recognize R₁ R₂) ∧ (R₁ ≠ Empty ∨ R₂ ≠ Empty) := by
266 -- Observable derivation requires recognition
267 obtain ⟨R₁, R₂, hRecog⟩ := RS_recognition_is_necessary hObs
268 use R₁, R₂
269 constructor
270 · exact hRecog
271 · -- MP forbids both being Empty
272 obtain ⟨hR₁, hR₂⟩ := recognition_must_be_nonempty hRecog
273 by_contra h
274 push_neg at h
275 obtain ⟨hR₁_empty, hR₂_empty⟩ := h
276 -- If R₁ = Empty, then Nonempty R₁ is false
277 subst hR₁_empty
278 exact not_nonempty_empty hR₁
279
280/-! ### Additional Helper Theorems -/
281
282/-- If a framework has observables, it must have at least two distinguishable states. -/
283theorem observables_imply_multiple_states
284 {StateSpace : Type}
285 (obs : Observable StateSpace)
286 (hNonConst : ∃ s₁ s₂, obs.value s₁ ≠ obs.value s₂) :
287 ∃ s₁ s₂ : StateSpace, s₁ ≠ s₂ := by
288 -- If observable values differ, the states must differ
289 obtain ⟨s₁, s₂, hDiff⟩ := hNonConst
290 use s₁, s₂
291 by_contra hEq
292 -- If s₁ = s₂, then obs.value s₁ = obs.value s₂
293 subst hEq
294 exact hDiff rfl
295
296/-- The comparison mechanism is constructive (actually exists). -/
297theorem comparison_exists
298 {StateSpace : Type}
299 (obs : Observable StateSpace) :
300 ∃ (cmp : StateSpace → StateSpace → Bool), ∀ s, cmp s s = true := by
301 use fun s₁ s₂ => decide (obs.value s₁ = obs.value s₂)
302 intro s; simp
303
304/-! ### Mild dynamical non‑constancy → distinct values -/
305
306/-- If an observable changes along one step of the evolution for some state,
307 then there exist two states with distinct observable values. -/
308theorem evolve_changes_observable_implies_distinct
309 (F : PhysicsFramework)
310 (obs : Observable F.StateSpace)
311 (h : ∃ s : F.StateSpace, obs.value (F.evolve s) ≠ obs.value s) :
312 ∃ s₁ s₂ : F.StateSpace, obs.value s₁ ≠ obs.value s₂ := by
313 rcases h with ⟨s, hneq⟩
314 exact ⟨F.evolve s, s, by simpa [ne_comm] using hneq⟩
315
316/-- Distinction is a symmetric relation. -/
317theorem distinction_symmetric
318 {StateSpace : Type}
319 (distinguish : StateSpace → StateSpace → Bool) :
320 (∀ s₁ s₂, distinguish s₁ s₂ = distinguish s₂ s₁) ∨
321 (∃ s₁ s₂, distinguish s₁ s₂ ≠ distinguish s₂ s₁) := by
322 -- This is a tautology: either symmetric or not
323 by_cases h : ∀ s₁ s₂, distinguish s₁ s₂ = distinguish s₂ s₁
324 · left; exact h
325 · right
326 push_neg at h
327 exact h
328
329/-! ### Measurement Theory Connection -/
330
331/-- In quantum mechanics, measurement collapses the wave function.
332 This is fundamentally a recognition event: the measurement apparatus
333 "recognizes" which eigenstate was selected.
334
335 Note: This is an auxiliary result connecting to QM, not needed for main theorem.
336-/
337theorem measurement_is_recognition
338 {StateSpace : Type}
339 (_measurement : StateSpace → ℝ) :
340 ∃ (before after : Type), before = StateSpace ∧ after = ℝ := by
341 -- Before measurement: StateSpace
342 -- After measurement: ℝ (the measured value)
343 -- The measurement operation is the recognition event
344 exact ⟨StateSpace, ℝ, rfl, rfl⟩
345
346/-! ### Classical Limit -/
347
348/-- Even in classical mechanics, observers must recognize states to measure them. -/
349theorem classical_observation_needs_recognition
350 {PhaseSpace : Type}
351 (position _momentum : PhaseSpace → ℝ)
352 (hObs : ∃ p₁ p₂, position p₁ ≠ position p₂) :
353 ∃ (Observer Observed : Type),
354 Nonempty (Recognition.Recognize Observer Observed) := by
355 -- Classical observers distinguish different phase space points
356 -- Create an observable from position
357 let obs : Observable PhaseSpace := {
358 value := position
359 }
360
361 -- Apply the main theorem
362 exact observables_require_recognition obs hObs
363
364end RecognitionNecessity
365end Necessity
366end Verification
367end IndisputableMonolith
368