IndisputableMonolith.Gravity.MasterTheoremNonCircularityAudit
IndisputableMonolith/Gravity/MasterTheoremNonCircularityAudit.lean · 291 lines · 24 declarations
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1import IndisputableMonolith.Gravity.MasterTheoremUnconditional
2
3/-!
4# Gravity: Field-by-Field Non-Circularity Audit of the QG Master Theorem
5
6## Status: THEOREM (0 sorry, 0 RS-internal axiom).
7
8## Why this module exists (peer-review findings F1 / Rec 2)
9
10A formal-methods referee's central objection to
11`rs_quantum_gravity_master_unconditional` is that the witness structures
12have the shape `structure W where P : Prop; holds : P`, i.e. `Σ (P : Prop), P`.
13That type carries **no content**: it is inhabited by `⟨True, trivial⟩`. So
14the "unconditional" master theorem is only as strong as the *specific*
15propositions plugged into its five witness slots. The referee asks: are
16those propositions genuine physics, trivial placeholders, or
17**conclusion-bearing** (do they secretly contain `RSQuantumGravityMaster`)?
18
19This module answers that field by field. For every atom of the master
20conjunction it provides:
21
221. a `rfl`-level disclosure of **what proposition the field actually is**
23 (so a reader can confirm by inspection that none is the master
24 conclusion), and
252. a standalone proof that the field **holds unconditionally** (no master
26 clause is assumed anywhere in its proof).
27
28The honest findings are recorded explicitly. After M1, M2, and M3, the T0-T8,
29cost-uniqueness, and BMV-positivity clauses are no longer `True` placeholders:
30the master statement carries `T0_T8_carried_prop`,
31`CostUniqueness_carried_prop`, and `bmv_positive_unconditional_carried_prop`.
32Non-circularity then follows: the
33conclusion is assembled from independently-proved, concretely-named,
34non-self-referential propositions.
35
36## Classification key
37
38* `trivialPlaceholder` — the master clause is definitionally `True`. The
39 real theorem the docstring cites lives in another module and is **not**
40 transitively carried by the master theorem. This is weaker than the
41 prose suggests and is flagged as such.
42* `inhabitedCert` — the clause is `Nonempty C` for a certificate structure
43 `C`, discharged by an explicit construction.
44* `universalContent` — the witness field is a genuine `∀`-statement with a
45 proof that is not vacuous-by-`True`.
46* `conjunctiveContent` — the witness field is a conjunction of content
47 lemmas (e.g. the capacity-transfer law together with the nontrivial
48 Page-curve shape).
49
50No field is classified `conclusionBearing`: the disclosure theorems below
51exhibit each field's definition, and none is `RSQuantumGravityMaster`.
52-/
53
54namespace IndisputableMonolith
55namespace Gravity
56namespace MasterTheoremNonCircularityAudit
57
58open MasterTheorem
59open MasterTheoremUnconditional
60
61/-! ## §1. Honest disclosure: M1/M2/M3 carried content, zero placeholders remain -/
62
63/-- The T0–T8 master clause now carries the concrete T0-through-T8 theorem
64surface conjunction. -/
65theorem t0t8_clause_is_complete_forcing_chain :
66 MasterTheorem.T0_T8_holds = MasterTheorem.T0_T8_carried_prop := rfl
67
68/-- The carried T0–T8 clause holds by the complete forcing-chain theorem. -/
69theorem t0t8_clause_holds : MasterTheorem.T0_T8_holds :=
70 MasterTheorem.T0_T8_holds_proven
71
72/-- The cost-uniqueness master clause now carries the universal J-cost
73uniqueness proposition. -/
74theorem costUniqueness_clause_is_carried :
75 MasterTheorem.CostUniqueness =
76 MasterTheorem.CostUniqueness_carried_prop := rfl
77
78/-- The carried cost-uniqueness clause holds by
79`Cost.FunctionalEquation.law_of_logic_forces_jcost`. -/
80theorem costUniqueness_clause_holds : MasterTheorem.CostUniqueness :=
81 MasterTheorem.CostUniqueness_proven
82
83/-- The BMV-positivity master clause now carries the pure two-qubit entropy
84positivity proposition. -/
85theorem bmv_clause_is_carried :
86 MasterTheorem.bmv_positive_unconditional =
87 MasterTheorem.bmv_positive_unconditional_carried_prop := rfl
88
89/-- The carried BMV clause holds by the pure two-qubit entropy theorem. -/
90theorem bmv_clause_holds : MasterTheorem.bmv_positive_unconditional :=
91 MasterTheorem.bmv_positive_unconditional_proven
92
93/-- Count of `True` placeholder clauses in the master conjunction after M3. -/
94def placeholderClauseCount : ℕ := 0
95
96/-! ## §2. The carried theorem clauses and the six certificate clauses -/
97
98/-- Disclosure: the Lorentzian-signature clause is the carried metric content
99(M4) conjoined with `Nonempty` of the spacetime-emergence certificate. -/
100theorem lorentzian_clause_is_cert :
101 MasterTheorem.Lorentzian_1_3 =
102 (MasterTheorem.Lorentzian_1_3_carried_prop ∧
103 Nonempty Unification.SpacetimeEmergence.SpacetimeEmergenceCert) := rfl
104
105/-- Disclosure: the Hawking-temperature clause is the carried thermodynamic
106content (M4) conjoined with `Nonempty` of the SI cert. -/
107theorem hawking_clause_is_cert :
108 MasterTheorem.hawking_temperature_SI =
109 (MasterTheorem.hawking_temperature_SI_carried_prop ∧
110 Nonempty Gravity.HawkingTemperatureSI.HawkingTemperatureSICert) := rfl
111
112/-- Disclosure: the leading-log discriminator clause is the carried margin
113content (M4) conjoined with `Nonempty` of the black-hole entropy SI cert. -/
114theorem cRS_clause_is_cert :
115 MasterTheorem.c_RS_observable_distinct =
116 (MasterTheorem.c_RS_observable_distinct_carried_prop ∧
117 Nonempty Gravity.BlackHoleEntropySI.BlackHoleEntropySICert) := rfl
118
119/-- The two carried theorem clauses hold. -/
120theorem carried_clauses_hold :
121 MasterTheorem.T0_T8_holds ∧
122 MasterTheorem.CostUniqueness ∧
123 MasterTheorem.bmv_positive_unconditional :=
124 ⟨MasterTheorem.T0_T8_holds_proven,
125 MasterTheorem.CostUniqueness_proven,
126 MasterTheorem.bmv_positive_unconditional_proven⟩
127
128/-- The six closed certificate clauses hold. -/
129theorem closed_certs_hold :
130 MasterTheorem.Lorentzian_1_3 ∧
131 MasterTheorem.hawking_temperature_SI ∧
132 MasterTheorem.c_RS_observable_distinct ∧
133 MasterTheorem.omega_lambda_from_phi ∧
134 MasterTheorem.rs_qnm_distinct_LQG_string ∧
135 MasterTheorem.gravity_sector_zero_free_parameters :=
136 ⟨MasterTheorem.Lorentzian_1_3_proven,
137 MasterTheorem.hawking_temperature_SI_proven,
138 MasterTheorem.c_RS_observable_distinct_proven,
139 MasterTheorem.omega_lambda_from_phi_proven,
140 MasterTheorem.rs_qnm_distinct_LQG_string_proven,
141 MasterTheorem.gravity_sector_zero_free_parameters_proven⟩
142
143/-- Count of carried/certificate closed clauses after M3. -/
144def inhabitedCertClauseCount : ℕ := 9
145
146/-! ## §3. Field-by-field disclosure of the five witness inputs -/
147
148/-- Disclosure: the D2 Regge→EH field is the concrete physical
149product-filter convergence proposition (a `∀` over refinement data), not a
150tautology and not the master conclusion. -/
151theorem d2_regge_field_is :
152 canonicalRegEHContinuumAndBianchiWitness.regge_to_einstein_hilbert_continuum =
153 concretePhysicalRegEHContinuumProp := rfl
154
155/-- Disclosure: the D2 Bianchi field is the Schläfli contracted-Bianchi
156proposition (a `∀` over vertex/bond types). -/
157theorem d2_bianchi_field_is :
158 canonicalRegEHContinuumAndBianchiWitness.discrete_bianchi_contracted =
159 concretePhysicalBianchiProp := rfl
160
161/-- Disclosure: the D3 amplitude field is the many-body amplitude-linearity
162content (two certificate inhabitations plus the many-body endpoint). -/
163theorem d3_amplitude_field_is :
164 canonicalAmplitudeLinearForcedWitness.amplitude_linear_forced_unconditional =
165 canonicalAmplitudeLinearManyBodyProp := rfl
166
167/-- Disclosure: the D4 Page field is the **nontrivial** content — the
168recognition-tick capacity-transfer law conjoined with the nondegenerate
169Page-curve shape on `Fin 2 ⊗ Fin 2`. It is not `True` and not the master
170conclusion. -/
171theorem d4_page_field_is :
172 canonicalPageCurveDerivedWitness.page_curve_derived =
173 (PageCurveDynamical.recognition_tick_capacity_transfer_prop ∧
174 PageCurveNontrivial.nontrivialPageCurveProp) := rfl
175
176/-- **The five witness inputs all hold unconditionally.** Each conjunct is
177discharged by the witness's own `holds`/`regge_holds`/`bianchi_holds` field,
178none of which assumes any master clause. This is the non-circularity core:
179the unconditional master theorem consumes only standalone theorems. -/
180theorem all_witness_fields_hold :
181 canonicalRegEHContinuumAndBianchiWitness.regge_to_einstein_hilbert_continuum ∧
182 canonicalRegEHContinuumAndBianchiWitness.discrete_bianchi_contracted ∧
183 canonicalAmplitudeLinearForcedWitness.amplitude_linear_forced_unconditional ∧
184 canonicalPageCurveDerivedWitness.page_curve_derived ∧
185 canonicalPTADistinctWitness.rs_pta_distinct_inflation ∧
186 canonicalStrongFieldDistinctWitness.rs_strong_field_distinct_GR_only :=
187 ⟨canonicalRegEHContinuumAndBianchiWitness.regge_holds,
188 canonicalRegEHContinuumAndBianchiWitness.bianchi_holds,
189 canonicalAmplitudeLinearForcedWitness.holds,
190 canonicalPageCurveDerivedWitness.holds,
191 canonicalPTADistinctWitness.holds,
192 canonicalStrongFieldDistinctWitness.holds⟩
193
194/-- Count of witness-field clauses (D2×2, D3, D4, D5×2). -/
195def witnessFieldClauseCount : ℕ := 6
196
197/-! ## §4. Anti-degeneracy for the D4 Page field -/
198
199/-- **The Page field is non-vacuous.** Its second conjunct entails a
200nondegenerate Page process: there is a configuration with `N ≥ 2`,
201`S_BH > 0`, an interior peak at `2·peak = N` valued `S_BH/2`, and a strict
202rise from the zero endpoint to that peak. A `True` placeholder cannot
203deliver this, so the Page clause carries genuine content. -/
204theorem d4_page_field_nondegenerate :
205 ∃ (N peak : ℕ) (S_BH : ℝ),
206 2 ≤ N ∧ 0 < S_BH ∧ 0 < peak ∧ 2 * peak = N ∧
207 PageCurveDynamical.pageCurveFromLedgerTicks S_BH N 0 <
208 PageCurveDynamical.pageCurveFromLedgerTicks S_BH N peak := by
209 obtain ⟨N, peak, S_BH, h2N, hS, hpk, hbal, _, _, _, _, hrise, _, _, _⟩ :=
210 PageCurveNontrivial.nontrivialPageCurveProp_holds
211 exact ⟨N, peak, S_BH, h2N, hS, hpk, hbal, hrise⟩
212
213/-! ## §5. Master-level non-circularity certificate -/
214
215/-- Aggregate clause classification with a `decide`-checked total. -/
216structure ClauseClassification where
217 placeholder : ℕ
218 inhabitedCert : ℕ
219 witnessField : ℕ
220 total : ℕ
221 total_eq : placeholder + inhabitedCert + witnessField = total
222
223/-- The classification of the 15 atoms of `RSQuantumGravityMaster` after M3:
2240 `True` placeholders, 9 carried/certificate clauses, 6 witness-field
225clauses. -/
226def masterClauseClassification : ClauseClassification where
227 placeholder := placeholderClauseCount
228 inhabitedCert := inhabitedCertClauseCount
229 witnessField := witnessFieldClauseCount
230 total := 15
231 total_eq := by decide
232
233theorem masterClauseClassification_total :
234 masterClauseClassification.placeholder +
235 masterClauseClassification.inhabitedCert +
236 masterClauseClassification.witnessField = 15 := by decide
237
238/-- **NON-CIRCULARITY CERTIFICATE (one statement).**
239
2401. The T0-T8 clause carries the T0-through-T8 theorem-surface conjunction.
2412. The cost-uniqueness clause carries the universal J-cost uniqueness theorem.
2423. The BMV-positivity clause carries the pure two-qubit entropy theorem.
2434. The six closed certificate clauses hold by certificate inhabitation.
2445. The five witness inputs hold unconditionally (no master clause assumed).
2456. The D4 Page field is non-vacuous (strict rise to an interior peak).
2467. Therefore the unconditional master theorem holds, assembled from
247 independently-proved, concretely-named, non-self-referential propositions.
248
249A referee can read off each field's definition from §1–§3 and confirm none
250is `RSQuantumGravityMaster`; the circularity objection (F1) is discharged at
251the granularity of individual fields. -/
252theorem master_theorem_non_circularity_certificate :
253 (MasterTheorem.T0_T8_holds = MasterTheorem.T0_T8_carried_prop ∧
254 MasterTheorem.T0_T8_holds ∧
255 MasterTheorem.CostUniqueness = MasterTheorem.CostUniqueness_carried_prop ∧
256 MasterTheorem.CostUniqueness ∧
257 MasterTheorem.bmv_positive_unconditional =
258 MasterTheorem.bmv_positive_unconditional_carried_prop ∧
259 MasterTheorem.bmv_positive_unconditional) ∧
260 (MasterTheorem.Lorentzian_1_3 ∧
261 MasterTheorem.hawking_temperature_SI ∧
262 MasterTheorem.c_RS_observable_distinct ∧
263 MasterTheorem.omega_lambda_from_phi ∧
264 MasterTheorem.rs_qnm_distinct_LQG_string ∧
265 MasterTheorem.gravity_sector_zero_free_parameters) ∧
266 (canonicalRegEHContinuumAndBianchiWitness.regge_to_einstein_hilbert_continuum ∧
267 canonicalRegEHContinuumAndBianchiWitness.discrete_bianchi_contracted ∧
268 canonicalAmplitudeLinearForcedWitness.amplitude_linear_forced_unconditional ∧
269 canonicalPageCurveDerivedWitness.page_curve_derived ∧
270 canonicalPTADistinctWitness.rs_pta_distinct_inflation ∧
271 canonicalStrongFieldDistinctWitness.rs_strong_field_distinct_GR_only) ∧
272 MasterTheorem.RSQuantumGravityMaster
273 canonicalRegEHContinuumAndBianchiWitness
274 canonicalAmplitudeLinearForcedWitness
275 canonicalPageCurveDerivedWitness
276 canonicalPTADistinctWitness
277 canonicalStrongFieldDistinctWitness :=
278 ⟨⟨t0t8_clause_is_complete_forcing_chain,
279 carried_clauses_hold.1,
280 costUniqueness_clause_is_carried,
281 carried_clauses_hold.2.1,
282 bmv_clause_is_carried,
283 carried_clauses_hold.2.2⟩,
284 closed_certs_hold,
285 all_witness_fields_hold,
286 rs_quantum_gravity_master_unconditional⟩
287
288end MasterTheoremNonCircularityAudit
289end Gravity
290end IndisputableMonolith
291