IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.HardProblemCertificateAudits

   1/-
   2  PrimitiveRecognitionCalculus/HardProblemCertificateAudits.lean
   3
   4  Concrete finite-certificate audit schemas for the first four hard-problem
   5  application stubs.
   6
   7  This file does not solve RH, Navier-Stokes, Yang-Mills, or Hodge. It closes the
   8  Delta-native method gap one level lower: each continuum-facing problem now has
   9  a typed finite certificate inventory and a concrete `ProblemAudit` whose
  10  finite-reduction theorem can be cited.
  11
  12  First pass: identity audits on the finite certificate layer.
  13
  14  Second pass: certified display audits. Each analytic display object carries an
  15  explicit finite certificate. This is stronger than identity bookkeeping and is
  16  the useful bridge shape for later analytic work: a continuum-facing display is
  17  admissible only when its finite certificate is part of the object.
  18
  19  No project-local axioms. No sorry.
  20-/
  21
  22import Mathlib
  23import IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.QuantizedProofMethod
  24
  25namespace IndisputableMonolith
  26namespace Foundation
  27namespace PrimitiveRecognitionCalculus
  28namespace HardProblemCertificateAudits
  29
  30open CompletionConservativity
  31open QuantizedProofMethod
  32
  33/-- Finite certificate inventory for the prime-critical-line audit. -/
  34inductive PrimeCriticalLineCert where
  35  | finitePrimeWindow
  36  | EulerProductBalanceLedger
  37  | zeroObstructionWitness
  38  deriving DecidableEq, Repr
  39
  40/-- Finite certificate inventory for the Navier-Stokes energy-transfer audit. -/
  41inductive NavierStokesEnergyCert where
  42  | finiteCellDecomposition
  43  | energyTransferLedger
  44  | blowupObstructionWitness
  45  deriving DecidableEq, Repr
  46
  47/-- Finite certificate inventory for the Yang-Mills mass-gap audit. -/
  48inductive YangMillsGapCert where
  49  | finitePlaquetteLedger
  50  | excitationGapWitness
  51  | zeroModeObstructionWitness
  52  deriving DecidableEq, Repr
  53
  54/-- Finite certificate inventory for the Hodge algebraic-witness audit. -/
  55inductive HodgeAlgebraicCert where
  56  | finiteCycleLedger
  57  | algebraicWitness
  58  | transcendentalObstructionWitness
  59  deriving DecidableEq, Repr
  60
  61/-- Native audit predicate: every finite certificate is legitimate at the schema
  62layer. Later analytic interfaces can refine this predicate. -/
  63def CertificateLegitimate {C : Type*} (_ : C) : Prop := True
  64
  65/-- Native pathology predicate: every obstruction certificate is admissible at the
  66schema layer. Later analytic interfaces can refine this to the problem-specific
  67bad event. -/
  68def CertificatePathology {C : Type*} (_ : C) : Prop := True
  69
  70/-- Identity audit over a finite certificate type. -/
  71def identityCertificateAudit (C : Type*) : ProblemAudit C C C where
  72  completion := identityCompletion C
  73  legitimate := CertificateLegitimate
  74  pathology := CertificatePathology
  75  legitimate_conservative := identity_conservative C CertificateLegitimate
  76  pathology_conservative := identity_conservative C CertificatePathology
  77
  78def primeCriticalLineAudit : ProblemAudit PrimeCriticalLineCert PrimeCriticalLineCert PrimeCriticalLineCert :=
  79  identityCertificateAudit PrimeCriticalLineCert
  80
  81def navierStokesEnergyAudit :
  82    ProblemAudit NavierStokesEnergyCert NavierStokesEnergyCert NavierStokesEnergyCert :=
  83  identityCertificateAudit NavierStokesEnergyCert
  84
  85def yangMillsGapAudit : ProblemAudit YangMillsGapCert YangMillsGapCert YangMillsGapCert :=
  86  identityCertificateAudit YangMillsGapCert
  87
  88def hodgeAlgebraicAudit : ProblemAudit HodgeAlgebraicCert HodgeAlgebraicCert HodgeAlgebraicCert :=
  89  identityCertificateAudit HodgeAlgebraicCert
  90
  91theorem primeCriticalLine_finiteReduction : HasFiniteReduction primeCriticalLineAudit :=
  92  problemAudit_finiteReduction primeCriticalLineAudit
  93
  94theorem navierStokesEnergy_finiteReduction : HasFiniteReduction navierStokesEnergyAudit :=
  95  problemAudit_finiteReduction navierStokesEnergyAudit
  96
  97theorem yangMillsGap_finiteReduction : HasFiniteReduction yangMillsGapAudit :=
  98  problemAudit_finiteReduction yangMillsGapAudit
  99
 100theorem hodgeAlgebraic_finiteReduction : HasFiniteReduction hodgeAlgebraicAudit :=
 101  problemAudit_finiteReduction hodgeAlgebraicAudit
 102
 103/-- **Hard-problem audit headline.** Each application stub now has a concrete
 104finite certificate inventory and a finite-reduction theorem. These are audit
 105schemas, not solutions of the underlying continuum problems. -/
 106theorem hard_problem_certificate_audits_headline :
 107    HasFiniteReduction primeCriticalLineAudit
 108      ∧ HasFiniteReduction navierStokesEnergyAudit
 109      ∧ HasFiniteReduction yangMillsGapAudit
 110      ∧ HasFiniteReduction hodgeAlgebraicAudit :=
 111  ⟨primeCriticalLine_finiteReduction, navierStokesEnergy_finiteReduction,
 112    yangMillsGap_finiteReduction, hodgeAlgebraic_finiteReduction⟩
 113
 114/-! ## Certified display audits -/
 115
 116/-- A continuum-facing display object that carries a finite certificate. `Payload`
 117is the analytic/display-side tag; `cert` is the native finite witness that gives
 118the display authority. -/
 119structure CertifiedDisplay (Cert Payload : Type*) where
 120  cert : Cert
 121  payload : Payload
 122  deriving Repr
 123
 124/-- Completion interface from finite certificates to certified displays. The
 125display map uses a default payload; arbitrary display objects are certified by
 126the certificate they carry. -/
 127def certifiedDisplayCompletion (Cert Payload : Type*) [Inhabited Payload] :
 128    Completion Cert (CertifiedDisplay Cert Payload) Cert where
 129  display := fun c => ⟨c, default⟩
 130  certifies := fun c d => c = d.cert
 131
 132def CertifiedDisplayLegitimate {Cert Payload : Type*} (_ : CertifiedDisplay Cert Payload) : Prop := True
 133def CertifiedDisplayPathology {Cert Payload : Type*} (_ : CertifiedDisplay Cert Payload) : Prop := True
 134
 135theorem certifiedDisplay_conservative
 136    (Cert Payload : Type*) [Inhabited Payload] :
 137    ConservativeFor (certifiedDisplayCompletion Cert Payload) (@CertifiedDisplayLegitimate Cert Payload)
 138      ∧ ConservativeFor (certifiedDisplayCompletion Cert Payload) (@CertifiedDisplayPathology Cert Payload) := by
 139  constructor
 140  · intro d _
 141    exact ⟨d.cert, rfl⟩
 142  · intro d _
 143    exact ⟨d.cert, rfl⟩
 144
 145/-- Display payload tags for the prime-critical-line audit. -/
 146inductive PrimeDisplayPayload where
 147  | zetaDisplay
 148  | criticalStripDisplay
 149  | offLineZeroDisplay
 150  deriving DecidableEq, Repr
 151
 152instance : Inhabited PrimeDisplayPayload := ⟨PrimeDisplayPayload.zetaDisplay⟩
 153
 154/-- Display payload tags for the Navier-Stokes audit. -/
 155inductive NavierStokesDisplayPayload where
 156  | smoothFlowDisplay
 157  | energyCascadeDisplay
 158  | blowupDisplay
 159  deriving DecidableEq, Repr
 160
 161instance : Inhabited NavierStokesDisplayPayload := ⟨NavierStokesDisplayPayload.smoothFlowDisplay⟩
 162
 163/-- Display payload tags for the Yang-Mills mass-gap audit. -/
 164inductive YangMillsDisplayPayload where
 165  | connectionDisplay
 166  | excitationDisplay
 167  | zeroModeDisplay
 168  deriving DecidableEq, Repr
 169
 170instance : Inhabited YangMillsDisplayPayload := ⟨YangMillsDisplayPayload.connectionDisplay⟩
 171
 172/-- Display payload tags for the Hodge audit. -/
 173inductive HodgeDisplayPayload where
 174  | cohomologyClassDisplay
 175  | algebraicCycleDisplay
 176  | transcendentalClassDisplay
 177  deriving DecidableEq, Repr
 178
 179instance : Inhabited HodgeDisplayPayload := ⟨HodgeDisplayPayload.cohomologyClassDisplay⟩
 180
 181def certifiedDisplayAudit (Cert Payload : Type*) [Inhabited Payload] :
 182    ProblemAudit Cert (CertifiedDisplay Cert Payload) Cert where
 183  completion := certifiedDisplayCompletion Cert Payload
 184  legitimate := CertifiedDisplayLegitimate
 185  pathology := CertifiedDisplayPathology
 186  legitimate_conservative := (certifiedDisplay_conservative Cert Payload).1
 187  pathology_conservative := (certifiedDisplay_conservative Cert Payload).2
 188
 189def primeCertifiedDisplayAudit :
 190    ProblemAudit PrimeCriticalLineCert (CertifiedDisplay PrimeCriticalLineCert PrimeDisplayPayload)
 191      PrimeCriticalLineCert :=
 192  certifiedDisplayAudit PrimeCriticalLineCert PrimeDisplayPayload
 193
 194def navierStokesCertifiedDisplayAudit :
 195    ProblemAudit NavierStokesEnergyCert (CertifiedDisplay NavierStokesEnergyCert NavierStokesDisplayPayload)
 196      NavierStokesEnergyCert :=
 197  certifiedDisplayAudit NavierStokesEnergyCert NavierStokesDisplayPayload
 198
 199def yangMillsCertifiedDisplayAudit :
 200    ProblemAudit YangMillsGapCert (CertifiedDisplay YangMillsGapCert YangMillsDisplayPayload)
 201      YangMillsGapCert :=
 202  certifiedDisplayAudit YangMillsGapCert YangMillsDisplayPayload
 203
 204def hodgeCertifiedDisplayAudit :
 205    ProblemAudit HodgeAlgebraicCert (CertifiedDisplay HodgeAlgebraicCert HodgeDisplayPayload)
 206      HodgeAlgebraicCert :=
 207  certifiedDisplayAudit HodgeAlgebraicCert HodgeDisplayPayload
 208
 209/-- **Certified display audit headline.** The hard-problem stubs now have a
 210non-identity display interface: every admissible display object carries an
 211explicit finite certificate, and the finite-reduction theorem applies to each
 212display audit. This is still not a solution of the four problems; it is the
 213correct Delta bridge shape for later analytic interfaces. -/
 214theorem certified_display_audits_headline :
 215    HasFiniteReduction primeCertifiedDisplayAudit
 216      ∧ HasFiniteReduction navierStokesCertifiedDisplayAudit
 217      ∧ HasFiniteReduction yangMillsCertifiedDisplayAudit
 218      ∧ HasFiniteReduction hodgeCertifiedDisplayAudit :=
 219  ⟨problemAudit_finiteReduction primeCertifiedDisplayAudit,
 220    problemAudit_finiteReduction navierStokesCertifiedDisplayAudit,
 221    problemAudit_finiteReduction yangMillsCertifiedDisplayAudit,
 222    problemAudit_finiteReduction hodgeCertifiedDisplayAudit⟩
 223
 224/-! ## Domain-specific analytic display interfaces -/
 225
 226/-- Domain-specific analytic display record for the prime-critical-line bridge. -/
 227structure PrimeAnalyticDisplay where
 228  cert : PrimeCriticalLineCert
 229  primeWindowRadius : ℕ
 230  balanceDepth : ℕ
 231  displayKind : PrimeDisplayPayload
 232  deriving Repr
 233
 234/-- Domain-specific analytic display record for the Navier-Stokes bridge. -/
 235structure NavierStokesAnalyticDisplay where
 236  cert : NavierStokesEnergyCert
 237  cellResolution : ℕ
 238  energyDepth : ℕ
 239  displayKind : NavierStokesDisplayPayload
 240  deriving Repr
 241
 242/-- Domain-specific analytic display record for the Yang-Mills bridge. -/
 243structure YangMillsAnalyticDisplay where
 244  cert : YangMillsGapCert
 245  plaquetteResolution : ℕ
 246  excitationLevel : ℕ
 247  displayKind : YangMillsDisplayPayload
 248  deriving Repr
 249
 250/-- Domain-specific analytic display record for the Hodge bridge. -/
 251structure HodgeAnalyticDisplay where
 252  cert : HodgeAlgebraicCert
 253  complexDimension : ℕ
 254  cohomologicalDegree : ℕ
 255  displayKind : HodgeDisplayPayload
 256  deriving Repr
 257
 258def primeAnalyticCompletion : Completion PrimeCriticalLineCert PrimeAnalyticDisplay PrimeCriticalLineCert where
 259  display := fun c => ⟨c, 0, 0, PrimeDisplayPayload.zetaDisplay⟩
 260  certifies := fun c d => c = d.cert
 261
 262def navierStokesAnalyticCompletion :
 263    Completion NavierStokesEnergyCert NavierStokesAnalyticDisplay NavierStokesEnergyCert where
 264  display := fun c => ⟨c, 0, 0, NavierStokesDisplayPayload.smoothFlowDisplay⟩
 265  certifies := fun c d => c = d.cert
 266
 267def yangMillsAnalyticCompletion : Completion YangMillsGapCert YangMillsAnalyticDisplay YangMillsGapCert where
 268  display := fun c => ⟨c, 0, 0, YangMillsDisplayPayload.connectionDisplay⟩
 269  certifies := fun c d => c = d.cert
 270
 271def hodgeAnalyticCompletion : Completion HodgeAlgebraicCert HodgeAnalyticDisplay HodgeAlgebraicCert where
 272  display := fun c => ⟨c, 0, 0, HodgeDisplayPayload.cohomologyClassDisplay⟩
 273  certifies := fun c d => c = d.cert
 274
 275def PrimeAnalyticLegitimate (_ : PrimeAnalyticDisplay) : Prop := True
 276def PrimeAnalyticPathology (_ : PrimeAnalyticDisplay) : Prop := True
 277def NavierStokesAnalyticLegitimate (_ : NavierStokesAnalyticDisplay) : Prop := True
 278def NavierStokesAnalyticPathology (_ : NavierStokesAnalyticDisplay) : Prop := True
 279def YangMillsAnalyticLegitimate (_ : YangMillsAnalyticDisplay) : Prop := True
 280def YangMillsAnalyticPathology (_ : YangMillsAnalyticDisplay) : Prop := True
 281def HodgeAnalyticLegitimate (_ : HodgeAnalyticDisplay) : Prop := True
 282def HodgeAnalyticPathology (_ : HodgeAnalyticDisplay) : Prop := True
 283
 284theorem primeAnalytic_conservative :
 285    ConservativeFor primeAnalyticCompletion PrimeAnalyticLegitimate
 286      ∧ ConservativeFor primeAnalyticCompletion PrimeAnalyticPathology := by
 287  constructor <;> intro d _ <;> exact ⟨d.cert, rfl⟩
 288
 289theorem navierStokesAnalytic_conservative :
 290    ConservativeFor navierStokesAnalyticCompletion NavierStokesAnalyticLegitimate
 291      ∧ ConservativeFor navierStokesAnalyticCompletion NavierStokesAnalyticPathology := by
 292  constructor <;> intro d _ <;> exact ⟨d.cert, rfl⟩
 293
 294theorem yangMillsAnalytic_conservative :
 295    ConservativeFor yangMillsAnalyticCompletion YangMillsAnalyticLegitimate
 296      ∧ ConservativeFor yangMillsAnalyticCompletion YangMillsAnalyticPathology := by
 297  constructor <;> intro d _ <;> exact ⟨d.cert, rfl⟩
 298
 299theorem hodgeAnalytic_conservative :
 300    ConservativeFor hodgeAnalyticCompletion HodgeAnalyticLegitimate
 301      ∧ ConservativeFor hodgeAnalyticCompletion HodgeAnalyticPathology := by
 302  constructor <;> intro d _ <;> exact ⟨d.cert, rfl⟩
 303
 304def primeAnalyticAudit : ProblemAudit PrimeCriticalLineCert PrimeAnalyticDisplay PrimeCriticalLineCert where
 305  completion := primeAnalyticCompletion
 306  legitimate := PrimeAnalyticLegitimate
 307  pathology := PrimeAnalyticPathology
 308  legitimate_conservative := primeAnalytic_conservative.1
 309  pathology_conservative := primeAnalytic_conservative.2
 310
 311def navierStokesAnalyticAudit :
 312    ProblemAudit NavierStokesEnergyCert NavierStokesAnalyticDisplay NavierStokesEnergyCert where
 313  completion := navierStokesAnalyticCompletion
 314  legitimate := NavierStokesAnalyticLegitimate
 315  pathology := NavierStokesAnalyticPathology
 316  legitimate_conservative := navierStokesAnalytic_conservative.1
 317  pathology_conservative := navierStokesAnalytic_conservative.2
 318
 319def yangMillsAnalyticAudit : ProblemAudit YangMillsGapCert YangMillsAnalyticDisplay YangMillsGapCert where
 320  completion := yangMillsAnalyticCompletion
 321  legitimate := YangMillsAnalyticLegitimate
 322  pathology := YangMillsAnalyticPathology
 323  legitimate_conservative := yangMillsAnalytic_conservative.1
 324  pathology_conservative := yangMillsAnalytic_conservative.2
 325
 326def hodgeAnalyticAudit : ProblemAudit HodgeAlgebraicCert HodgeAnalyticDisplay HodgeAlgebraicCert where
 327  completion := hodgeAnalyticCompletion
 328  legitimate := HodgeAnalyticLegitimate
 329  pathology := HodgeAnalyticPathology
 330  legitimate_conservative := hodgeAnalytic_conservative.1
 331  pathology_conservative := hodgeAnalytic_conservative.2
 332
 333/-- **Domain-specific analytic audit headline.** The four hard-problem stubs now
 334have named analytic display records with certificate fields and domain-specific
 335parameters. The reductions still do not solve the problems; they give the exact
 336display interface that future analytic proofs must refine. -/
 337theorem domain_specific_analytic_audits_headline :
 338    HasFiniteReduction primeAnalyticAudit
 339      ∧ HasFiniteReduction navierStokesAnalyticAudit
 340      ∧ HasFiniteReduction yangMillsAnalyticAudit
 341      ∧ HasFiniteReduction hodgeAnalyticAudit :=
 342  ⟨problemAudit_finiteReduction primeAnalyticAudit,
 343    problemAudit_finiteReduction navierStokesAnalyticAudit,
 344    problemAudit_finiteReduction yangMillsAnalyticAudit,
 345    problemAudit_finiteReduction hodgeAnalyticAudit⟩
 346
 347end HardProblemCertificateAudits
 348end PrimitiveRecognitionCalculus
 349end Foundation
 350end IndisputableMonolith
 351