import Semantics.RRCLogogramProjection /-! # ManifoldBoundaryAtlas A ManifoldBoundaryAtlas is a deterministic seed-generated set of candidate boundary coordinates for RRC-style projection checks. It generalizes `HexLogogramAtlas`: the seed does not claim a tear, proof, or manifold truth; it generates a finite boundary candidate surface that RRC can identify against receipts and residual evidence. Promotion is byte-law gated. If the boundary seed, law, residual, and receipt do not beat an explicit boundary-coordinate table, the atlas remains a HOLD surface. -/ namespace Semantics.ManifoldBoundaryAtlas /-- Finite laws for generating manifold boundary candidates. -/ inductive BoundaryLaw where | affineCut | genusInversion | massGradient | rrcProjectionEdge deriving Repr, DecidableEq /-- A generated boundary candidate in a finite coordinate domain. -/ structure BoundaryCandidate where index : Nat coordinate : Nat side : Nat confidence : Nat deriving Repr, DecidableEq /-- Seeded boundary-atlas packet. -/ structure BoundaryAtlasPacket where hexSeed : Nat hexDigitWidth : Nat blockBase : Nat law : BoundaryLaw dimensionCount : Nat coordinateDomain : Nat boundaryCount : Nat startIndex : Nat length : Nat stride : Nat genus : Nat massBasin : Nat torsionClass : Nat phaseClass : Nat explicitBoundaryBytes : Nat seedBytes : Nat lawBytes : Nat residualBytes : Nat receiptBytes : Nat rrcShapeOk : Bool residualDeclared : Bool deriving Repr, DecidableEq /-- Hex block base for a fixed-width boundary seed. -/ def expectedHexBase (p : BoundaryAtlasPacket) : Nat := 16 ^ p.hexDigitWidth /-- Structural gate before byte-law promotion. -/ def boundaryAtlasStructurallyValid (p : BoundaryAtlasPacket) : Bool := p.hexDigitWidth > 0 && p.blockBase == expectedHexBase p && p.dimensionCount > 0 && p.coordinateDomain > 0 && p.boundaryCount > 0 && p.length > 0 && p.stride > 0 && p.explicitBoundaryBytes > 0 && p.rrcShapeOk && p.residualDeclared /-- Raw deterministic boundary value before projection into finite domains. -/ def boundaryRawValueAt (p : BoundaryAtlasPacket) (i : Nat) : Nat := let j := p.startIndex + i match p.law with | BoundaryLaw.affineCut => p.hexSeed + (j * p.stride) + p.phaseClass | BoundaryLaw.genusInversion => p.hexSeed + (p.genus * p.dimensionCount) + (j * p.stride) + p.phaseClass | BoundaryLaw.massGradient => p.hexSeed + (p.massBasin * p.stride) + j + p.torsionClass | BoundaryLaw.rrcProjectionEdge => p.hexSeed + (j * p.stride) + p.massBasin + p.torsionClass + p.phaseClass /-- Generate the i-th RRC boundary candidate. -/ def boundaryCandidateAt (p : BoundaryAtlasPacket) (i : Nat) : BoundaryCandidate := let raw := boundaryRawValueAt p i { index := i coordinate := raw % p.coordinateDomain side := raw % p.boundaryCount confidence := (raw / p.boundaryCount) % p.dimensionCount } /-- Replay the finite boundary atlas. -/ def replayBoundaryAtlas (p : BoundaryAtlasPacket) : List BoundaryCandidate := (List.range p.length).map (boundaryCandidateAt p) /-- Explicit table cost for boundary candidates. -/ def explicitBoundaryTableBytes (p : BoundaryAtlasPacket) : Nat := p.length * p.explicitBoundaryBytes /-- Encoded boundary-atlas cost with residual and receipt accounting. -/ def boundaryAtlasEncodedBytes (p : BoundaryAtlasPacket) : Nat := p.seedBytes + p.lawBytes + p.residualBytes + p.receiptBytes /-- The byte law: generated boundary candidates must beat explicit candidates. -/ def boundaryAtlasByteLawHolds (p : BoundaryAtlasPacket) : Bool := boundaryAtlasEncodedBytes p < explicitBoundaryTableBytes p /-- Promotion gate for a deterministic boundary atlas. -/ def boundaryAtlasPromotable (p : BoundaryAtlasPacket) : Bool := boundaryAtlasStructurallyValid p && boundaryAtlasByteLawHolds p /-- Boundary atlases feed RRC identification, not merge admission. A promoted boundary atlas gives candidate boundary evidence; actual tear projection still needs the RRC receipt fields. -/ def rrcBoundaryReceiptFromAtlas (p : BoundaryAtlasPacket) : Semantics.RRCLogogramProjection.LogogramReceipt := { shape := Semantics.RRCLogogramProjection.RRCShape.logogramProjection status := Semantics.RRCLogogramProjection.WitnessStatus.candidate regime := Semantics.RRCLogogramProjection.SemanticRegime.horribleManifoldTearing payloadBound := p.rrcShapeOk contradictionWitness := false tearBoundary := boundaryAtlasPromotable p detachedMass := false residualLane := p.residualDeclared } /-! ## Canonical witnesses -/ /-- Small hand-checkable boundary atlas. -/ def smallBoundaryAtlas : BoundaryAtlasPacket := { hexSeed := 3 hexDigitWidth := 2 blockBase := 256 law := BoundaryLaw.affineCut dimensionCount := 4 coordinateDomain := 16 boundaryCount := 4 startIndex := 0 length := 4 stride := 2 genus := 3 massBasin := 1 torsionClass := 0 phaseClass := 1 explicitBoundaryBytes := 2 seedBytes := 1 lawBytes := 1 residualBytes := 0 receiptBytes := 1 rrcShapeOk := true residualDeclared := true } /-- Compression-sized witness for an inverted-genus boundary surface. -/ def canonicalBoundaryAtlas : BoundaryAtlasPacket := { hexSeed := 0xC0FFEE hexDigitWidth := 6 blockBase := 16777216 law := BoundaryLaw.genusInversion dimensionCount := 16 coordinateDomain := 4096 boundaryCount := 12 startIndex := 0 length := 96 stride := 7 genus := 3 massBasin := 5 torsionClass := 2 phaseClass := 4 explicitBoundaryBytes := 3 seedBytes := 3 lawBytes := 1 residualBytes := 4 receiptBytes := 2 rrcShapeOk := true residualDeclared := true } /-- Too short to beat explicit boundary coordinates. -/ def tinyBoundaryAtlas : BoundaryAtlasPacket := { canonicalBoundaryAtlas with length := 2 } /-- Bad coordinate domain: no finite manifold address space. -/ def badBoundaryDomainAtlas : BoundaryAtlasPacket := { canonicalBoundaryAtlas with coordinateDomain := 0 } /-- Missing residual declaration: cannot promote to RRC identification. -/ def missingResidualBoundaryAtlas : BoundaryAtlasPacket := { canonicalBoundaryAtlas with residualDeclared := false } theorem smallBoundaryReplay : replayBoundaryAtlas smallBoundaryAtlas = [{ index := 0, coordinate := 4, side := 0, confidence := 1 }, { index := 1, coordinate := 6, side := 2, confidence := 1 }, { index := 2, coordinate := 8, side := 0, confidence := 2 }, { index := 3, coordinate := 10, side := 2, confidence := 2 }] := by native_decide theorem canonicalBoundaryAtlasPromotable : boundaryAtlasPromotable canonicalBoundaryAtlas = true := by native_decide theorem tinyBoundaryAtlasNotPromotable : boundaryAtlasPromotable tinyBoundaryAtlas = false := by native_decide theorem badBoundaryDomainAtlasNotPromotable : boundaryAtlasPromotable badBoundaryDomainAtlas = false := by native_decide theorem missingResidualBoundaryAtlasNotPromotable : boundaryAtlasPromotable missingResidualBoundaryAtlas = false := by native_decide theorem promotedBoundaryAtlasStructurallyValid (p : BoundaryAtlasPacket) : boundaryAtlasPromotable p = true -> boundaryAtlasStructurallyValid p = true := by unfold boundaryAtlasPromotable intro h cases hStruct : boundaryAtlasStructurallyValid p · simp [hStruct] at h · simp theorem promotedBoundaryAtlasSatisfiesByteLaw (p : BoundaryAtlasPacket) : boundaryAtlasPromotable p = true -> boundaryAtlasByteLawHolds p = true := by unfold boundaryAtlasPromotable intro h cases hStruct : boundaryAtlasStructurallyValid p · simp [hStruct] at h cases hBytes : boundaryAtlasByteLawHolds p · simp [hStruct, hBytes] at h · simp /-- A promoted atlas contributes tear-boundary evidence to the RRC receipt. -/ theorem promotedBoundaryAtlasSetsRrcTearBoundary (p : BoundaryAtlasPacket) : boundaryAtlasPromotable p = true -> (rrcBoundaryReceiptFromAtlas p).tearBoundary = true := by intro h unfold rrcBoundaryReceiptFromAtlas simp [h] /-- Boundary evidence alone is intentionally insufficient for projection admission: contradiction and detached-mass evidence must still be supplied by later RRC checks. -/ theorem boundaryAtlasAloneDoesNotProject : Semantics.RRCLogogramProjection.projectionAdmissible (rrcBoundaryReceiptFromAtlas canonicalBoundaryAtlas) = false := by native_decide #eval replayBoundaryAtlas smallBoundaryAtlas #eval boundaryAtlasPromotable canonicalBoundaryAtlas #eval boundaryAtlasPromotable tinyBoundaryAtlas #eval boundaryAtlasPromotable badBoundaryDomainAtlas #eval boundaryAtlasPromotable missingResidualBoundaryAtlas #eval Semantics.RRCLogogramProjection.projectionAdmissible (rrcBoundaryReceiptFromAtlas canonicalBoundaryAtlas) end Semantics.ManifoldBoundaryAtlas