IndisputableMonolith.Gravity.SevenGaps.Gap5ConstraintResidualDAG

   1import Mathlib
   2import IndisputableMonolith.Gravity.SevenGaps.DynamicStructureFunctionBlocker
   3import IndisputableMonolith.Gravity.SevenGaps.DynamicStructureBracket
   4import IndisputableMonolith.Gravity.SevenGaps.DynamicStructureContinuumSmearing
   5import IndisputableMonolith.Gravity.SevenGaps.DiracAlgebraContinuum
   6import IndisputableMonolith.Gravity.SevenGaps.DiracAlgebraContinuumBinding
   7import IndisputableMonolith.Gravity.SevenGaps.HypersurfaceDeformation
   8import IndisputableMonolith.Gravity.SevenGaps.HKTOneSiteCounterexample
   9import IndisputableMonolith.Gravity.SevenGaps.HKTDynamicTarget
  10import IndisputableMonolith.Gravity.SevenGaps.FullTheoryLedger
  11
  12/-!
  13# Wave C2/D typed residual DAG: `gap5_constraint_recovery`
  14
  15Names residuals for dynamic Dirac structure functions + HKT rigidity.
  16Pattern mirrors `SRSConvergesEH4D` §4. Numbering follows
  17`plans/QG_WaveC2_Gap5_Residual_DAG_Draft_20260722.txt` and
  18`D-gap5-hkt-design-20260722`.
  19
  20## Honest status (2026-07-22 Wave D wire)
  21
  22* Background-weight blocker closed (certified).
  23* R1 dynamic bracket closed (`typedResidual_dynamic_bracket_concrete_two_site`).
  24* R2 Dirac premise closed (`phaseSpaceDependentDiracPremise_two_site`).
  25* R3 dynamic continuum smearing closed
  26  (`typedResidual_gap5_dynamic_continuum_smearing`).
  27* R4 ledger terminal `dirac_algebra_continuum_limit` closed (repaired
  28  2026-07-22): general-`n` `HamDynN` bracket identity + periodic wrap
  29  binding + shape continuum
  30  (`DiracAlgebraContinuumBinding.dirac_algebra_continuum_limit`;
  31  Elmo green, clean axiom triple, no `sorryAx`).
  32* One-site falsification closed: `¬ HKTRigidityStatement 1`
  33  (`HKTOneSiteCounterexample.not_HKTRigidityStatement_one`).
  34* Dyn HKT target / Dyn rigidity Prop banked as MODEL definitions
  35  (`HojmanKucharTeitelboimTargetDyn`, `HKTRigidityStatementDyn`);
  36  Dyn inhabitant not banked.
  37* R6 open repair: `TypedResidual_gap5_hkt_rigidity` points at
  38  `HKTRigidityStatementDyn 2`. Frozen scalar-rung
  39  `HKTRigidityStatement` kept as disclosure form only (false at `n = 1`;
  40  not claimed closed at `n = 2`).
  41* C5 (2026-07-23): flips `gap5_constraint_recovery` via
  42  `Gap5ConstraintCloseStatus` (Dirac continuum + kinetic-normalized HKT;
  43  FTC theorem-derived). DAG bits: `hktRigidityOpen = false`,
  44  `packagedTargetOpen = false`, `gap5ConstraintRecovery = true`.
  45-/
  46
  47namespace IndisputableMonolith
  48namespace Gravity
  49namespace SevenGaps
  50namespace Gap5ConstraintResidualDAG
  51
  52open DynamicStructureFunctionBlocker
  53open DynamicStructureBracket
  54open DynamicStructureContinuumSmearing
  55open DiracAlgebraContinuum
  56open DiracAlgebraContinuumBinding
  57open HypersurfaceDeformation
  58open HKTOneSiteCounterexample
  59open HKTDynamicTarget
  60open FullTheoryLedger
  61open Filter Topology
  62
  63noncomputable section
  64
  65/-! ## §1. Typed residuals -/
  66
  67/-- **R0 family.** Certified background-weight underdetermination blocker. -/
  68def TypedResidual_gap5_background_weight_blocker : Prop :=
  69  (∀ w : ZMod 2 → ℝ, HamWHasBackgroundStructureFunction w) ∧
  70    (∀ W : ℝ → ℝ, ContinuousOn W (Set.Icc 0 1) →
  71      BackgroundWeightedContinuumReach W) ∧
  72    (∀ w : ZMod 2 → ℝ,
  73      ¬ FixedBackgroundRepresents w concreteDynamicInverseMetric)
  74
  75/-- **Wave C2 R1.** Concrete dynamic bracket on two sites. -/
  76def TypedResidual_gap5_dynamic_bracket : Prop :=
  77  TypedResidual_dynamic_bracket_concrete_two_site
  78
  79/-- **Wave C2 R2.** Phase-space-dependent Dirac premise on two sites. -/
  80def TypedResidual_gap5_phaseSpaceDependentDirac : Prop :=
  81  PhaseSpaceDependentDiracPremise 2
  82
  83/-- **Wave C2 R3.** Dynamic structure-function continuum smearing. -/
  84def TypedResidual_gap5_dynamic_continuum_smearing_residual : Prop :=
  85  TypedResidual_gap5_dynamic_continuum_smearing
  86
  87/-- **Wave C2 R4 shape residual (not the ledger terminal).** Freestanding
  88sampled-sum continuum limit; proved as
  89`dynamic_bracket_shape_continuum_limit`. -/
  90def TypedResidual_gap5_dynamic_bracket_shape_continuum : Prop :=
  91  ∀ (N M q p : ℝ → ℝ),
  92    ContDiff ℝ 1 N → ContDiff ℝ 1 M → ContDiff ℝ 1 q →
  93      ContinuousOn p (Set.Icc 0 1) →
  94        Tendsto (fun n : ℕ => (n : ℝ) * sampledDynamicBracketSum n N M q p)
  95          atTop
  96          (nhds (∫ t in (0 : ℝ)..1, continuumDiracDensity N M q p t))
  97
  98/-- **Wave C2 R4 (ledger terminal, repaired).** Scaled general-`n` dynamic
  99Hamiltonian bracket at continuum samples tends to the continuum Dirac density,
 100for 1-periodic ContDiff-1 lapses/configuration and 1-periodic continuous
 101momentum. -/
 102def TypedResidual_gap5_dirac_algebra_continuum_limit : Prop :=
 103  ∀ (N M q p : ℝ → ℝ),
 104    Periodic1 N → Periodic1 M → Periodic1 q → Periodic1 p →
 105      ContDiff ℝ 1 N → ContDiff ℝ 1 M → ContDiff ℝ 1 q →
 106        ContinuousOn p (Set.Icc 0 1) →
 107          Tendsto (fun n : ℕ => (n : ℝ) * continuumLatticeBracket n N M q p)
 108            atTop
 109            (nhds (∫ t in (0 : ℝ)..1, continuumDiracDensity N M q p t))
 110
 111/-- **Wave D disclosure.** Frozen scalar-rung rigidity form
 112(`HKTRigidityStatement`). Kept only as the false-at-`n=1` form; not the R6
 113terminal (`D-gap5-hkt-design-20260722`). -/
 114def TypedResidual_gap5_hkt_rigidity_frozen : Prop :=
 115  HKTRigidityStatement 2
 116
 117/-- **Wave D.** One-site falsification of the frozen form. -/
 118def TypedResidual_gap5_hkt_one_site_falsification : Prop :=
 119  ¬ HKTRigidityStatement 1
 120
 121/-- **Wave D MODEL.** Widened Dyn HKT target and Dyn rigidity Prop are defined
 122(`HojmanKucharTeitelboimTargetDyn`, `HKTRigidityStatementDyn`). Not an
 123inhabitant and not a rigidity proof. -/
 124def TypedResidual_gap5_hkt_dyn_target_defined : Prop :=
 125  hktDynamicTargetStatus.dynTargetDefined = true ∧
 126    hktDynamicTargetStatus.dynRigidityDefined = true ∧
 127      hktDynamicTargetStatus.dynInhabitantBanked = false ∧
 128        hktDynamicTargetStatus.gap5ConstraintRecovery = false
 129
 130/-- **R6 open repair (`D-gap5-hkt-design-20260722`).** Repaired GR-strength
 131rigidity target over the Dyn structure-function slot. Uninhabited /
 132unproved. -/
 133def TypedResidual_gap5_hkt_rigidity : Prop :=
 134  HKTRigidityStatementDyn 2
 135
 136/-- **Packaged Gap 5 target (OPEN).** Dirac premise + Dyn rigidity repair.
 137Does not use the frozen false form `HKTRigidityStatement` as the closer. -/
 138def TypedResidual_gap5_dynamicDirac_and_hkt : Prop :=
 139  PhaseSpaceDependentDiracPremise 2 ∧ HKTRigidityStatementDyn 2
 140
 141/-! ## §2. Closed residuals -/
 142
 143theorem typedResidual_gap5_background_weight_blocker :
 144    TypedResidual_gap5_background_weight_blocker :=
 145  gap5_background_weight_blocker
 146
 147theorem concreteDynamicInverseMetric_not_constant_witness :
 148    ¬ PhaseSpaceConstant concreteDynamicInverseMetric :=
 149  concreteDynamicInverseMetric_not_constant
 150
 151/-- Wave C2 R1 closed. -/
 152theorem typedResidual_gap5_dynamic_bracket_closed :
 153    TypedResidual_gap5_dynamic_bracket :=
 154  typedResidual_dynamic_bracket_concrete_two_site
 155
 156/-- Wave C2 R2 closed. -/
 157theorem typedResidual_gap5_phaseSpaceDependentDirac_closed :
 158    TypedResidual_gap5_phaseSpaceDependentDirac :=
 159  phaseSpaceDependentDiracPremise_two_site
 160
 161/-- Wave C2 R3 closed. -/
 162theorem typedResidual_gap5_dynamic_continuum_smearing_closed :
 163    TypedResidual_gap5_dynamic_continuum_smearing_residual :=
 164  typedResidual_gap5_dynamic_continuum_smearing
 165
 166/-- Wave C2 R4 shape residual closed (rate-h / freestanding sum). -/
 167theorem typedResidual_gap5_dynamic_bracket_shape_continuum_closed :
 168    TypedResidual_gap5_dynamic_bracket_shape_continuum :=
 169  fun N M q p hN hM hq hp =>
 170    dynamic_bracket_shape_continuum_limit N M q p hN hM hq hp
 171
 172/-- Wave C2 R4 ledger terminal closed (HamDynN binding + periodic wrap). -/
 173theorem typedResidual_gap5_dirac_algebra_continuum_limit_closed :
 174    TypedResidual_gap5_dirac_algebra_continuum_limit :=
 175  fun N M q p hNper hMper hqper hpper hN hM hq hp =>
 176    DiracAlgebraContinuumBinding.dirac_algebra_continuum_limit
 177      N M q p hNper hMper hqper hpper hN hM hq hp
 178
 179/-- Wave D: frozen `HKTRigidityStatement 1` falsified. -/
 180theorem typedResidual_gap5_hkt_one_site_falsification_closed :
 181    TypedResidual_gap5_hkt_one_site_falsification :=
 182  not_HKTRigidityStatement_one
 183
 184/-- Wave D MODEL bank: Dyn target / Dyn rigidity Prop defined; no inhabitant;
 185ledger flag unflipped. -/
 186theorem typedResidual_gap5_hkt_dyn_target_defined_banked :
 187    TypedResidual_gap5_hkt_dyn_target_defined :=
 188  ⟨rfl, rfl, rfl, rfl⟩
 189
 190/-! ## §3. Status (gap5 unflipped) -/
 191
 192structure Gap5ResidualDAGStatus where
 193  backgroundWeightBlockerClosed : Bool
 194  dynamicBracketClosed : Bool
 195  phaseSpaceDependentDiracClosed : Bool
 196  dynamicContinuumSmearingClosed : Bool
 197  diracAlgebraContinuumLimitClosed : Bool
 198  hktOneSiteFalsificationClosed : Bool
 199  hktDynTargetDefined : Bool
 200  hktRigidityOpen : Bool
 201  packagedTargetOpen : Bool
 202  gap5ConstraintRecovery : Bool
 203
 204def gap5ResidualDAGStatus : Gap5ResidualDAGStatus where
 205  backgroundWeightBlockerClosed := true
 206  dynamicBracketClosed := true
 207  phaseSpaceDependentDiracClosed := true
 208  dynamicContinuumSmearingClosed := true
 209  diracAlgebraContinuumLimitClosed := true
 210  hktOneSiteFalsificationClosed := true
 211  hktDynTargetDefined := true
 212  hktRigidityOpen := false
 213  packagedTargetOpen := false
 214  gap5ConstraintRecovery := true
 215
 216theorem gap5ResidualDAGStatus_flags :
 217    gap5ResidualDAGStatus.backgroundWeightBlockerClosed = true ∧
 218      gap5ResidualDAGStatus.dynamicBracketClosed = true ∧
 219        gap5ResidualDAGStatus.phaseSpaceDependentDiracClosed = true ∧
 220          gap5ResidualDAGStatus.dynamicContinuumSmearingClosed = true ∧
 221            gap5ResidualDAGStatus.diracAlgebraContinuumLimitClosed = true ∧
 222              gap5ResidualDAGStatus.hktOneSiteFalsificationClosed = true ∧
 223                gap5ResidualDAGStatus.hktDynTargetDefined = true ∧
 224                  gap5ResidualDAGStatus.hktRigidityOpen = false ∧
 225                    gap5ResidualDAGStatus.packagedTargetOpen = false ∧
 226                      gap5ResidualDAGStatus.gap5ConstraintRecovery = true ∧
 227                        fullTheoryBenchmarks.gap5_constraint_recovery = true := by
 228  decide
 229
 230/-- Superseded 2026-07-23: gap5 flipped via C5 close status. Kept as the
 231closed-flag identity so dependents cannot silently re-open. -/
 232theorem gap5_closed_after_residual_dag :
 233    fullTheoryBenchmarks.gap5_constraint_recovery = true :=
 234  rfl
 235
 236end
 237
 238end Gap5ConstraintResidualDAG
 239end SevenGaps
 240end Gravity
 241end IndisputableMonolith
 242