IndisputableMonolith.Gravity.SevenGaps.Gap2ContinuumMeasureResidualDAG

   1import Mathlib
   2import IndisputableMonolith.Gravity.SevenGaps.PathSumMeasure
   3import IndisputableMonolith.Gravity.SevenGaps.ExactShellGaugePreflight
   4import IndisputableMonolith.Gravity.SevenGaps.ExactShellGaugeUV
   5import IndisputableMonolith.Gravity.SevenGaps.MeasureSubstrateBlocker
   6import IndisputableMonolith.Gravity.SevenGaps.ZqContinuumBlocker
   7import IndisputableMonolith.Gravity.SevenGaps.CapShellBridge
   8import IndisputableMonolith.Gravity.SevenGaps.GaugeHistoryMeasure
   9import IndisputableMonolith.Gravity.SevenGaps.Gap2MeasureStatusBinding
  10import IndisputableMonolith.Gravity.SevenGaps.FullTheoryLedger
  11import IndisputableMonolith.Gravity.SevenGaps.Gap2TailAutFiberParityBlocker
  12import IndisputableMonolith.Gravity.SevenGaps.Gap2CertifiedFin8PhaseClose
  13import IndisputableMonolith.Gravity.SevenGaps.Gap2PostingCocycleCarrier
  14
  15/-!
  16# Wave C typed residual DAG: `gap2_continuum_and_measure`
  17
  18Names the ordered residuals for Pillar-2 measure + continuum-limit
  19recovery.  Pattern mirrors `SRSConvergesEH4D` §4.
  20
  21## Honest status (2026-07-23 Gap2 certified Fin-8 API bank)
  22
  23* Measure half: R1 blocker closed. R2 was recorded closed via
  24  `GaugeHistoryMeasure` (`nuBuild` / posted-history) and the R6 status Bools
  25  were bound to it; **both are retracted 2026-07-26** and the measure half is
  26  open again. See `Gap2MeasureStatusBinding` for the retraction and for
  27  `history_discharge_is_prior_theorem_rewritten`, which reproves the whole
  28  advertised discharge from a theorem that predates the history module.
  29* Continuum half: R3 Cauchy ↔ `OscillatoryTail` banked; R4
  30  `CapShellCompatibility` closed via `CapShellBridge.capShellCompatibility`.
  31* R4 antipodal re-scope (session 4B): sufficiency bridge banked
  32  (`TailAntipodalShift` ⇒ antipodal balance ⇒ `OscillatoryTail`);
  33  `TailAntipodalShift` ⇒ `TailAutFiberEven` banked; finite Aut-bucket
  34  parity probe MEASURED externally (odd buckets at shells n=2,3);
  35  infinite `TailAutFiberParityBlocker` OPEN; matching /
  36  `TailAntipodalShift` route dead for flip; bare R5 phase residual
  37  decoy-killed (requires certified Fin-8 provenance).
  38* Certified Fin-8 phase-close API banked in
  39  `Gap2CertifiedFin8PhaseClose` (structures + bridge to bare R5);
  40  `TypedResidual_certified_fin8_phase_close` remains OPEN (uninhabited).
  41* Posting-cocycle STOP A banked in `Gap2PostingCocycleCarrier`: product
  42  enrichment forgets the Fin-8 posting coordinate
  43  (`TypedResidual_carrier_forgets_posting_phase` CLOSED); non-forgetful
  44  bridge residual remains OPEN.
  45* Continuum R5 substrate-derived phase with `OscillatoryTail` remains OPEN.
  46* Does **not** flip `gap2_continuum_and_measure` until both halves close.
  47-/
  48
  49namespace IndisputableMonolith
  50namespace Gravity
  51namespace SevenGaps
  52namespace Gap2ContinuumMeasureResidualDAG
  53
  54open MeasureSubstrateBlocker
  55open ZqContinuumBlocker
  56open PathSumMeasure
  57open ExactShellGaugePreflight
  58open ExactShellGaugeUV
  59open CapShellBridge
  60open GaugeHistoryMeasure
  61open Gap2MeasureStatusBinding
  62open FullTheoryLedger
  63open Gap2TailAutFiberParityBlocker
  64open Gap2CertifiedFin8PhaseClose
  65open Gap2PostingCocycleCarrier
  66
  67noncomputable section
  68
  69/-! ## §1. Typed residuals -/
  70
  71/-- **R1 (measure).** Normalized gauge counting holds for the counting mass
  72and fails for the quotient-uniform decoy (blocker package). -/
  73def TypedResidual_measure_gaugeCounting_blocker : Prop :=
  74  ∀ (B : ℕ), 2 ≤ B →
  75    GaugeCountingPrinciple
  76        (gaugeOrbitMass : TriangulationClass B → ℝ) ∧
  77      ¬ GaugeCountingPrinciple
  78        (uniformClassMass : TriangulationClass B → ℝ)
  79
  80/-- **R2 (measure), RETRACTED 2026-07-26 and OPEN again.** This residual was
  81described as "gauge counting derived from richer substrate structure
  82(posted-history / dual-entry anchor), not postulated". The proposition itself
  83is provable and stays; the description was wrong. What it says is that a
  84particular presentation of `gaugeOrbitMass` satisfies gauge counting, which
  85follows from `gaugeOrbitMass_satisfies` with the presentation erased
  86(`Gap2MeasureStatusBinding.history_discharge_is_prior_theorem_rewritten`).
  87
  88The real R2 obligation is unchanged and open: a proof of
  89`MeasureSubstrateBlocker.GaugeCountingPrinciple` in which some substrate
  90premise is load-bearing. Note that the closure cannot be detected by asking
  91whether the resulting measure moves, since gauge counting has a unique
  92solution; it must be detected in the premises of the proof. -/
  93def TypedResidual_measure_history_presentation : Prop :=
  94  TypedResidual_gap2_gauge_counting_from_history
  95
  96/-- **R3 (continuum).** Cap-free Cauchy criterion equals oscillatory tail. -/
  97def TypedResidual_continuum_cauchy_iff_oscillatoryTail : Prop :=
  98  ∀ (phase : ∀ n : ℕ, ExactPathClass n → ℝ),
  99    CauchySeq (Zcap phase) ↔ OscillatoryTail phase
 100
 101/-- **R4 (continuum).** Cap API equals exact-shell cutoff for some
 102compatible phase family (canonical transport). -/
 103def TypedResidual_continuum_capShellCompatibility : Prop :=
 104  ∃ (P : CapPhaseFamily) (phase : ∀ n : ℕ, ExactPathClass n → ℝ),
 105    CapShellCompatibility P phase
 106
 107/-- **R5 (continuum, OPEN).** A substrate-derived phase has oscillatory
 108tail cancellation (zero phase is a discriminating decoy).
 109
 110Honest ledger close requires the stronger certified Fin-8 residual
 111`Gap2CertifiedFin8PhaseClose.TypedResidual_certified_fin8_phase_close`
 112(API banked; inhabitation OPEN). Bare R5 alone is decoy-killed.
 113Abstract discharge:
 114`Gap2CertifiedFin8PhaseClose.typedResidual_continuum_substrate_oscillatoryTail_of_certified_fin8`
 115(bridges to `BareR5ResidualShape`, definitionally this residual). -/
 116def TypedResidual_continuum_substrate_oscillatoryTail : Prop :=
 117  ∃ phase : ∀ n : ℕ, ExactPathClass n → ℝ,
 118    OscillatoryTail phase ∧ ¬ OscillatoryTail zeroPhase
 119
 120/-! ## §2. Closed residuals -/
 121
 122theorem typedResidual_measure_gaugeCounting_blocker :
 123    TypedResidual_measure_gaugeCounting_blocker := by
 124  intro B hB
 125  exact ⟨(substrate_measure_blocker_certificate B hB).1,
 126    (substrate_measure_blocker_certificate B hB).2.2⟩
 127
 128/-- Presentation result, NOT an R2 closer (retracted 2026-07-26).
 129History-built `nuBuild` satisfies gauge counting and equals `gaugeOrbitMass`
 130via the uniqueness IFF. -/
 131theorem typedResidual_measure_history_presentation :
 132    TypedResidual_measure_history_presentation :=
 133  typedResidual_gap2_gauge_counting_from_history_closed
 134
 135theorem typedResidual_continuum_cauchy_iff_oscillatoryTail :
 136    TypedResidual_continuum_cauchy_iff_oscillatoryTail :=
 137  fun phase => cauchySeq_Zcap_iff_oscillatoryTail phase
 138
 139/-- **R4 closer.** Canonical transport supplies a compatible capped family
 140for every exact-shell phase (including the zero-phase discriminant). -/
 141theorem typedResidual_continuum_capShellCompatibility :
 142    TypedResidual_continuum_capShellCompatibility :=
 143  ⟨capPhaseFamily zeroPhase, zeroPhase, capShellCompatibility zeroPhase⟩
 144
 145theorem decoy_zeroPhase_not_oscillatoryTail :
 146    ¬ OscillatoryTail zeroPhase :=
 147  zeroPhase_not_oscillatoryTail
 148
 149/-! ## §3. Status (gap2 unflipped; continuum R5 still open; measure R6 bound;
 150R4 antipodal parity re-scope banked; certified Fin-8 API banked / residual OPEN) -/
 151
 152structure Gap2ResidualDAGStatus where
 153  measureBlockerClosed : Bool
 154  measureSubstrateDerivedClosed : Bool
 155  measureStatusBoolsBound : Bool
 156  continuumCauchyIffClosed : Bool
 157  capShellCompatibilityClosed : Bool
 158  r4AntipodalSufficiencyClosed : Bool
 159  r4FiniteParityProbeMeasured : Bool
 160  r4InfiniteParityBlockerOpen : Bool
 161  r5BarePhaseDecoyKilled : Bool
 162  r5CertifiedFin8PhaseCloseApiBanked : Bool
 163  r5CertifiedFin8PhaseCloseOpen : Bool
 164  r5PostingCocycleCarrierForgetsPhaseClosed : Bool
 165  r5PostingCocycleBridgeOpen : Bool
 166  substrateOscillatoryTailOpen : Bool
 167  gap2ContinuumAndMeasure : Bool
 168
 169/-- `measureSubstrateDerivedClosed` and `measureStatusBoolsBound` were `true`
 170after Wave C1 R6 and are `false` after the 2026-07-26 retraction; see
 171`Gap2MeasureStatusBinding`. Nothing else in this record changed. -/
 172def gap2ResidualDAGStatus : Gap2ResidualDAGStatus where
 173  measureBlockerClosed := true
 174  measureSubstrateDerivedClosed := false
 175  measureStatusBoolsBound := false
 176  continuumCauchyIffClosed := true
 177  capShellCompatibilityClosed := true
 178  r4AntipodalSufficiencyClosed := true
 179  r4FiniteParityProbeMeasured := true
 180  r4InfiniteParityBlockerOpen := true
 181  r5BarePhaseDecoyKilled := true
 182  r5CertifiedFin8PhaseCloseApiBanked := true
 183  r5CertifiedFin8PhaseCloseOpen := true
 184  r5PostingCocycleCarrierForgetsPhaseClosed := true
 185  r5PostingCocycleBridgeOpen := true
 186  substrateOscillatoryTailOpen := true
 187  gap2ContinuumAndMeasure := false
 188
 189/-- Historical status record: the `gap2ContinuumAndMeasure = false` conjunct
 190below is the module's own status Bool at DAG authoring, not the ledger flag.
 191The ledger roll-up flipped to true on 2026-07-31 (flag 9, `Gap2GaugeTransport`,
 192Jon's criterion ruling), and the live-ledger conjunct was dropped from this
 193theorem that day. -/
 194theorem gap2ResidualDAGStatus_flags :
 195    gap2ResidualDAGStatus.measureBlockerClosed = true ∧
 196      gap2ResidualDAGStatus.measureSubstrateDerivedClosed = false ∧
 197        gap2ResidualDAGStatus.measureStatusBoolsBound = false ∧
 198          gap2ResidualDAGStatus.continuumCauchyIffClosed = true ∧
 199            gap2ResidualDAGStatus.capShellCompatibilityClosed = true ∧
 200              gap2ResidualDAGStatus.r4AntipodalSufficiencyClosed = true ∧
 201                gap2ResidualDAGStatus.r4FiniteParityProbeMeasured = true ∧
 202                  gap2ResidualDAGStatus.r4InfiniteParityBlockerOpen = true ∧
 203                    gap2ResidualDAGStatus.r5BarePhaseDecoyKilled = true ∧
 204                      gap2ResidualDAGStatus.r5CertifiedFin8PhaseCloseApiBanked =
 205                        true ∧
 206                        gap2ResidualDAGStatus.r5CertifiedFin8PhaseCloseOpen =
 207                          true ∧
 208                          gap2ResidualDAGStatus.r5PostingCocycleCarrierForgetsPhaseClosed =
 209                            true ∧
 210                            gap2ResidualDAGStatus.r5PostingCocycleBridgeOpen =
 211                              true ∧
 212                              gap2ResidualDAGStatus.substrateOscillatoryTailOpen =
 213                                true ∧
 214                                gap2ResidualDAGStatus.gap2ContinuumAndMeasure =
 215                                  false ∧
 216                                    pathSumMeasureStatus.substrate_measure_derived =
 217                                      false ∧
 218                                      gaugePreflightStatus.counting_principle_derived_from_ledger =
 219                                        false ∧
 220                                        gap2TailAutFiberParityBlockerStatus.bareR5DecoyCertificateBanked =
 221                                          true ∧
 222                                          gap2CertifiedFin8PhaseCloseStatus.certifiedApiBanked =
 223                                            true ∧
 224                                            gap2CertifiedFin8PhaseCloseStatus.certifiedCloseInhabited =
 225                                              false ∧
 226                                              gap2PostingCocycleCarrierStatus.carrierForgetsPhaseResidualClosed =
 227                                                true ∧
 228                                                gap2PostingCocycleCarrierStatus.certifiedCloseInhabited =
 229                                                  false := by
 230  decide
 231
 232/-- R6 retraction (2026-07-26): both measure-side status Bools are false. The
 233`gap2_measure_derived = false` conjunct was dropped on 2026-07-30 (flag 8
 234flip, `Gap2MeasureDerivation`) and the roll-up conjunct on 2026-07-31 (flag 9
 235flip, `Gap2GaugeTransport`, Jon's criterion ruling). -/
 236theorem typedResidual_measure_status_retracted :
 237    pathSumMeasureStatus.substrate_measure_derived = false ∧
 238      gaugePreflightStatus.counting_principle_derived_from_ledger = false :=
 239  ⟨gap2_measure_status_retracted.1, gap2_measure_status_retracted.2.1⟩
 240
 241theorem gap2_rollup_closed_after_residual_dag :
 242    fullTheoryBenchmarks.gap2_continuum_and_measure = true :=
 243  rfl
 244
 245end
 246
 247end Gap2ContinuumMeasureResidualDAG
 248end SevenGaps
 249end Gravity
 250end IndisputableMonolith
 251