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309 lines
9.3 KiB
Text
309 lines
9.3 KiB
Text
import Semantics.PhysicsScalar
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import Semantics.PhysicsEuclidean
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import Semantics.PhysicsLagrangian
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import Semantics.RegimeCore
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namespace Semantics.ExoticSpacetime
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open Semantics.PhysicsScalar
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open Semantics.PhysicsEuclidean
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open Semantics.PhysicsLagrangian
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open Semantics.RegimeCore
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abbrev TemporalOrder := UInt16
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abbrev CurveId := UInt16
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abbrev ConnectorId := UInt16
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abbrev RegionClockId := UInt16
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inductive SignatureClass
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| spacelike
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| timelike
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| nullLike
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| mixed
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deriving Repr, DecidableEq
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inductive TemporalRegime
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| monotonic
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| dilated
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| branched
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| cyclic
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| suspended
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| unresolved
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deriving Repr, DecidableEq
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inductive CausalStatus
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| admissible
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| guarded
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| rejected
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deriving Repr, DecidableEq
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inductive ConnectorKind
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| throatBridge
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| wormholeLike
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| foldBridge
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| sliceBridge
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| delayBridge
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deriving Repr, DecidableEq
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structure TemporalDifferential where
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localStep : Q16_16
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externalStep : Q16_16
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drift : Q16_16
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dilation : Q16_16
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coherence : Q16_16
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deriving Repr, DecidableEq
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structure CausalCone where
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forwardWeight : Q16_16
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backwardWeight : Q16_16
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lateralWeight : Q16_16
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signature : SignatureClass
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deriving Repr, DecidableEq
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structure ExoticRegionProfile where
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regionId : RegionId
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clockId : RegionClockId
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temporalRegime : TemporalRegime
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baseDifferential : TemporalDifferential
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cone : CausalCone
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permitsClosedTraversal : Bool
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permitsDimFold : Bool
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deriving Repr, DecidableEq
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structure TimelikeCurve where
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curveId : CurveId
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label : String
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sourceRegionId : RegionId
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targetRegionId : RegionId
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temporalOrder : TemporalOrder
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differential : TemporalDifferential
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cone : CausalCone
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stable : Bool
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deriving Repr, DecidableEq
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structure WormholeConnector where
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connectorId : ConnectorId
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label : String
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kind : ConnectorKind
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mouthARegionId : RegionId
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mouthBRegionId : RegionId
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entryDifferential : TemporalDifferential
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exitDifferential : TemporalDifferential
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requiresResolvedGate : Bool
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active : Bool
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deriving Repr, DecidableEq
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structure ExoticTransitionRequest (n : Nat) where
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state : PhysicsLagrangian n
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sourceRegionId : RegionId
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targetRegionId : RegionId
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temporalDifferential : TemporalDifferential
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requestedOrder : TemporalOrder
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deriving Repr, DecidableEq
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structure ExoticTransitionResult (n : Nat) where
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status : CausalStatus
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resultingState : PhysicsLagrangian n
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resultingRegime : TemporalRegime
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usedConnector? : Option ConnectorId
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deriving Repr, DecidableEq
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def zeroDifferential : TemporalDifferential :=
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{ localStep := Q16_16.zero
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, externalStep := Q16_16.zero
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, drift := Q16_16.zero
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, dilation := Q16_16.one
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, coherence := Q16_16.one }
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def unitCone : CausalCone :=
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{ forwardWeight := Q16_16.one
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, backwardWeight := Q16_16.zero
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, lateralWeight := Q16_16.zero
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, signature := .timelike }
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def classifySignature (cone : CausalCone) : SignatureClass :=
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if Q16_16.gt cone.forwardWeight cone.backwardWeight && Q16_16.gt cone.forwardWeight cone.lateralWeight then
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.timelike
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else if Q16_16.eq cone.forwardWeight cone.backwardWeight && Q16_16.gt cone.forwardWeight Q16_16.zero then
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.nullLike
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else if Q16_16.gt cone.lateralWeight cone.forwardWeight then
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.spacelike
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else
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.mixed
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def temporalRatio (differential : TemporalDifferential) : Q16_16 :=
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Q16_16.divQ16_16 differential.externalStep (Q16_16.max differential.localStep Q16_16.one)
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def temporalGradient (differential : TemporalDifferential) : Q16_16 :=
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Q16_16.absDiff differential.externalStep differential.localStep
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def classifyTemporalRegime (differential : TemporalDifferential) (cone : CausalCone) : TemporalRegime :=
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if Q16_16.isZero differential.coherence then
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.unresolved
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else if Q16_16.eq cone.backwardWeight Q16_16.zero && Q16_16.ge differential.dilation Q16_16.one then
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.monotonic
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else if Q16_16.gt differential.dilation Q16_16.one then
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.dilated
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else if Q16_16.nonZero cone.backwardWeight then
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.cyclic
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else if Q16_16.nonZero differential.drift then
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.branched
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else
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.suspended
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def permitsTimelikeTraversal (cone : CausalCone) : Bool :=
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match classifySignature cone with
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| .timelike | .nullLike => true
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| .spacelike | .mixed => false
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def differentialCompatible
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(source target : ExoticRegionProfile)
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(request : TemporalDifferential) : Bool :=
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let localOk := Q16_16.ge request.localStep source.baseDifferential.localStep
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let coherenceOk := Q16_16.ge request.coherence (Q16_16.min source.baseDifferential.coherence target.baseDifferential.coherence)
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let dilationOk :=
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Q16_16.betweenInclusive request.dilation Q16_16.half (Q16_16.four)
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localOk && coherenceOk && dilationOk
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def causalStatusFor
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(source target : ExoticRegionProfile)
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(request : TemporalDifferential) : CausalStatus :=
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if !permitsTimelikeTraversal source.cone || !permitsTimelikeTraversal target.cone then
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.rejected
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else if !differentialCompatible source target request then
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.guarded
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else
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.admissible
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def applyTemporalDifferential (state : PhysicsLagrangian n) (differential : TemporalDifferential) : PhysicsLagrangian n :=
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let scaledVelocity := PhysicsEuclidean.scale state.velocity differential.dilation
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let shiftedMomentum := PhysicsEuclidean.scale state.momentum (Q16_16.max differential.coherence Q16_16.half)
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let updatedAction := Q16_16.add state.actionDensity (temporalGradient differential)
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{ state with
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velocity := scaledVelocity
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momentum := shiftedMomentum
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actionDensity := updatedAction }
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def findRegionProfile?
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(profiles : List ExoticRegionProfile)
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(regionId : RegionId) : Option ExoticRegionProfile :=
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profiles.find? (fun profile => profile.regionId = regionId)
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def findConnector?
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(connectors : List WormholeConnector)
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(sourceRegionId targetRegionId : RegionId) : Option WormholeConnector :=
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connectors.find? (fun connector =>
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connector.active &&
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((connector.mouthARegionId = sourceRegionId && connector.mouthBRegionId = targetRegionId) ||
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(connector.mouthBRegionId = sourceRegionId && connector.mouthARegionId = targetRegionId)))
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def connectorMatchesRequest
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(connector : WormholeConnector)
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(sourceRegionId targetRegionId : RegionId) : Bool :=
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connector.active &&
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((connector.mouthARegionId = sourceRegionId && connector.mouthBRegionId = targetRegionId) ||
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(connector.mouthBRegionId = sourceRegionId && connector.mouthARegionId = targetRegionId))
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def traverseExoticTransition
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(profiles : List ExoticRegionProfile)
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(connectors : List WormholeConnector)
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(request : ExoticTransitionRequest n) : ExoticTransitionResult n :=
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match findRegionProfile? profiles request.sourceRegionId, findRegionProfile? profiles request.targetRegionId with
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| some source, some target =>
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let status := causalStatusFor source target request.temporalDifferential
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let connector? := findConnector? connectors request.sourceRegionId request.targetRegionId
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let gatedStatus :=
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match connector? with
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| some connector =>
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if connector.requiresResolvedGate && status != .admissible then .guarded else status
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| none => status
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let resultingState :=
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match gatedStatus with
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| .admissible => applyTemporalDifferential request.state request.temporalDifferential
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| .guarded | .rejected => request.state
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let resultingRegime := classifyTemporalRegime request.temporalDifferential target.cone
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{ status := gatedStatus
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, resultingState := resultingState
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, resultingRegime := resultingRegime
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, usedConnector? := connector?.map (fun connector => connector.connectorId) }
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| _, _ =>
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{ status := .rejected
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, resultingState := request.state
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, resultingRegime := .unresolved
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, usedConnector? := none }
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def flatlandRegionProfile (regionId : RegionId) : ExoticRegionProfile :=
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{ regionId := regionId
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, clockId := 1
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, temporalRegime := .monotonic
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, baseDifferential :=
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{ localStep := Q16_16.one
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, externalStep := Q16_16.one
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, drift := Q16_16.zero
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, dilation := Q16_16.one
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, coherence := Q16_16.one }
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, cone :=
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{ forwardWeight := Q16_16.one
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, backwardWeight := Q16_16.zero
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, lateralWeight := Q16_16.half
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, signature := .timelike }
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, permitsClosedTraversal := false
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, permitsDimFold := true }
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def wormholeRegionProfile (regionId : RegionId) : ExoticRegionProfile :=
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{ regionId := regionId
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, clockId := 2
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, temporalRegime := .dilated
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, baseDifferential :=
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{ localStep := Q16_16.one
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, externalStep := Q16_16.two
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, drift := Q16_16.half
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, dilation := Q16_16.two
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, coherence := Q16_16.three }
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, cone :=
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{ forwardWeight := Q16_16.three
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, backwardWeight := Q16_16.quarter
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, lateralWeight := Q16_16.one
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, signature := .mixed }
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, permitsClosedTraversal := true
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, permitsDimFold := true }
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def defaultWormholeConnector (sourceRegionId targetRegionId : RegionId) : WormholeConnector :=
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{ connectorId := 1
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, label := "defaultWormholeConnector"
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, kind := .wormholeLike
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, mouthARegionId := sourceRegionId
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, mouthBRegionId := targetRegionId
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, entryDifferential :=
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{ localStep := Q16_16.one
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, externalStep := Q16_16.two
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, drift := Q16_16.half
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, dilation := Q16_16.two
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, coherence := Q16_16.two }
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, exitDifferential :=
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{ localStep := Q16_16.one
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, externalStep := Q16_16.one
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, drift := Q16_16.zero
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, dilation := Q16_16.one
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, coherence := Q16_16.one }
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, requiresResolvedGate := true
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, active := true }
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end Semantics.ExoticSpacetime
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