UOR Euler Reduction

IRI: https://uor.foundation/reduction/
Prefix: reduction:
Space: kernel
Comment: Sequential composition of ψ-maps into a parameterized reduction ψ = ψ_9 ∘ … ∘ ψ_1. Defines stages, phase gates, rollback, and epochs.

This is a kernel-space namespace in the Define stage of the PRISM pipeline. It provides the immutable algebraic substrate — ring structure, schema vocabulary, and operation algebra.

Learn more: Pipeline Overview

Class hierarchy

Imports

https://uor.foundation/op/
https://uor.foundation/state/
https://uor.foundation/partition/
https://uor.foundation/resolver/
https://uor.foundation/morphism/
https://uor.foundation/observable/
https://uor.foundation/predicate/
https://uor.foundation/effect/
https://uor.foundation/schema/
https://uor.foundation/u/

Classes

Name	Subclass Of	Comment
`EulerReduction`	`Thing`	The composite endofunctor ψ = ψ_9 ∘ … ∘ ψ_1, parameterized by Ω = e^{iπ/6}.
`PhaseRotationScheduler`	`Thing`	Schedule Ω⁰, Ω¹, …, Ω⁵ assigning a phase angle to each stage of the reduction.
`TargetConvergenceAngle`	`Thing`	The angle at which the reduction terminates (default: π).
`PhaseGateAttestation`	`Thing`	Validation at each stage boundary checking that the accumulated phase angle matches the expected Ω^k.
`ComplexConjugateRollback`	`Thing`	Recovery operation when a phase gate fails: z → z̄. Involutory: applying twice yields the original value.
`ReductionStep`	`Thing`	A named stage of the reduction. The standard reduction has six stages (Initialization through Convergence).
`ReductionState`	`Thing`	State of reduction execution at a specific point, including the current stage, phase angle, and pinned mask.
`ReductionRule`	`Thing`	Guard-effect pair governing stage transitions in the reduction. The guard must be satisfied before the effect is applied.
`Epoch`	`Thing`	Temporal segment of reduction execution. Each epoch represents one complete pass through the reduction stages.
`EpochBoundary`	`Thing`	Transition between epochs. Carries metadata about the epoch boundary crossing.
`PredicateExpression`	`Thing`	A Boolean expression over the reduction state. The atomic guard unit.
`GuardExpression`	`Thing`	A conjunction of PredicateExpressions that must hold for a transition to fire.
`TransitionEffect`	`Thing`	State changes applied when a transition fires. Contains PropertyBind steps.
`PropertyBind`	`Thing`	A single site pinning: target site + value.
`ReductionAdvance`	`Thing`	Advancement from one ReductionStep to the next.
`ServiceWindow`	`Thing`	A sliding window over recent epochs providing BaseContext.
`ReductionTransaction`	`Thing`	An atomic group of state changes within the reduction.
`PipelineSuccess`	`Thing`	Successful termination (FullGrounding).
`PipelineFailureReason`	`Thing`	Typed failure: DispatchMiss, GroundingFailure, ConvergenceStall, etc.
`PreflightCheck`	`Thing`	A pre-execution validation: feasibility, dispatch coverage, coherence.
`FeasibilityResult`	`Thing`	Result of a preflight check: feasibility witness or infeasibility witness.
`LeaseState`	`Thing`	Lifecycle of a partitioned context lease: Pending → Active → Released/Expired/Suspended.
`ManagedLease`	`Thing`	A context lease with lifecycle tracking.
`LeaseCheckpoint`	`Thing`	Snapshot of lease state at a point in time.
`BackPressureSignal`	`Thing`	Flow control when reduction produces faster than downstream can consume.
`DeferredQuerySet`	`Thing`	Queries postponed to a future epoch.
`SubleaseTransfer`	`Thing`	Transfer of a lease from one computation to another.
`ComparisonPredicate`	`PredicateExpression`	Predicate comparing a state field against a value.
`ConjunctionPredicate`	`PredicateExpression`	Conjunction (AND) of multiple predicates.
`DisjunctionPredicate`	`PredicateExpression`	Disjunction (OR) of multiple predicates.
`NegationPredicate`	`PredicateExpression`	Negation (NOT) of a single predicate.
`MembershipPredicate`	`PredicateExpression`	Predicate testing membership of an element in a set.
`GroundingPredicate`	`PredicateExpression`	Predicate testing whether grounding exceeds a threshold.
`SiteCoveragePredicate`	`PredicateExpression`	Predicate testing whether a site coverage target is met.
`EqualsPredicate`	`PredicateExpression`	Predicate testing equality of two expressions.
`NonNullPredicate`	`PredicateExpression`	Predicate testing that a field is non-null.
`QuerySubtypePredicate`	`PredicateExpression`	Predicate testing whether a query is a subtype of a given type.
`CompileUnit`	`Thing`	The typed input graph submitted to the reduction pipeline. Packages a root Term, target quantum level, verification domains, and thermodynamic budget. Stage 0 accepts exactly one CompileUnit and initializes the reduction state vector from it.

Properties

Name	Kind	Functional	Domain	Range	Comment
`phaseParameter`	Datatype	true	`EulerReduction`	`decimal`	The base phase parameter Ω for this reduction (e.g., e^{iπ/6}).
`stageCount`	Datatype	true	`EulerReduction`	`nonNegativeInteger`	The number of stages in this reduction.
`convergenceAngle`	Datatype	true	`EulerReduction`	`decimal`	The cumulative phase angle at which the reduction converges.
`composedOfMaps`	Object	false	`EulerReduction`	`TermExpression`	The ordered list of ψ-maps that compose this reduction.
`rotationSchedule`	Datatype	true	`PhaseRotationScheduler`	`decimal`	String representation of the rotation schedule Ω⁰, Ω¹, …, Ω⁵.
`baseAngle`	Datatype	true	`PhaseRotationScheduler`	`decimal`	The base angle π/6 from which the schedule is derived.
`targetAngle`	Datatype	true	`TargetConvergenceAngle`	`decimal`	The target convergence angle (default: π).
`gateStage`	Object	true	`PhaseGateAttestation`	`ReductionStep`	The reduction stage at which this gate is applied.
`gateExpectedPhase`	Datatype	true	`PhaseGateAttestation`	`decimal`	The expected phase angle Ω^k at this gate.
`gateResult`	Datatype	true	`PhaseGateAttestation`	`boolean`	Whether the phase gate check passed or failed.
`rollbackTarget`	Object	true	`ComplexConjugateRollback`	`ReductionStep`	The reduction stage to which execution rolls back on gate failure.
`stageIndex`	Datatype	true	`ReductionStep`	`nonNegativeInteger`	Zero-based index of this stage in the reduction.
`stageName`	Datatype	true	`ReductionStep`	`string`	Human-readable name of this reduction stage.
`expectedPhase`	Datatype	true	`ReductionStep`	`decimal`	The expected phase angle Ω^k at this stage.
`entryGuard`	Object	true	`ReductionStep`	`StatePredicate`	A typed predicate evaluated on the current ReductionState. Must be satisfied to enter this stage.
`exitGuard`	Object	true	`ReductionStep`	`StatePredicate`	A typed predicate that must be satisfied before the reduction advances past this stage.
`stageEffect`	Object	true	`ReductionStep`	`Effect`	The effect applied by this stage upon successful exit.
`currentStage`	Object	true	`ReductionState`	`ReductionStep`	The reduction stage at which execution is currently positioned.
`phaseAngle`	Datatype	true	`ReductionState`	`decimal`	The accumulated phase angle at the current point.
`pinnedMask`	Datatype	true	`ReductionState`	`nonNegativeInteger`	Bit mask of sites that are pinned (resolved) at this point.
`freeRank`	Datatype	true	`ReductionState`	`nonNegativeInteger`	The number of free (unresolved) sites at this point.
`transitionGuard`	Object	true	`ReductionRule`	`GuardedTransition`	A typed GuardedTransition from predicate/ governing the stage transition.
`transitionEffect`	Object	true	`ReductionRule`	`TermExpression`	The effect applied when this transition fires.
`transitionAdvance`	Datatype	true	`ReductionRule`	`boolean`	Whether this transition advances to the next stage.
`epochIndex`	Datatype	true	`Epoch`	`nonNegativeInteger`	Zero-based index of this epoch in the reduction execution.
`epochDatum`	Object	true	`Epoch`	`TermExpression`	Metadata or summary datum for this epoch.
`epochBoundaryType`	Object	true	`EpochBoundary`	`TermExpression`	The type of epoch boundary crossing (e.g., normal, forced, timeout).
`predicateField`	Datatype	true	`PredicateExpression`	`string`	The state field this predicate tests.
`predicateOperator`	Datatype	true	`PredicateExpression`	`string`	The comparison operator (e.g., '=', '<', '>=').
`predicateValue`	Datatype	true	`PredicateExpression`	`string`	The value against which the field is compared.
`guardPredicates`	Object	false	`GuardExpression`	`PredicateExpression`	The predicate expressions that compose this guard.
`effectBindings`	Object	false	`TransitionEffect`	`PropertyBind`	The property bind steps applied by this effect.
`bindTarget`	Datatype	true	`PropertyBind`	`string`	The target site identifier for this binding.
`bindValue`	Datatype	true	`PropertyBind`	`string`	The value to pin the target site to.
`advanceFrom`	Object	true	`ReductionAdvance`	`ReductionStep`	The source stage of the advancement.
`advanceTo`	Object	true	`ReductionAdvance`	`ReductionStep`	The target stage of the advancement.
`windowSize`	Datatype	true	`ServiceWindow`	`nonNegativeInteger`	The number of recent epochs in this service window.
`windowOffset`	Datatype	true	`ServiceWindow`	`nonNegativeInteger`	The starting epoch offset of this service window.
`transactionPolicy`	Datatype	true	`ReductionTransaction`	`string`	The execution policy for this transaction (e.g., AllOrNothing, BestEffort).
`transactionOutcome`	Object	true	`ReductionTransaction`	`TermExpression`	The outcome of this transaction (e.g., committed, rolled back).
`failureKind`	Datatype	true	`PipelineFailureReason`	`string`	The kind of pipeline failure (e.g., DispatchMiss, ConvergenceStall).
`preflightKind`	Datatype	true	`PreflightCheck`	`string`	The kind of preflight check (e.g., feasibility, dispatch coverage).
`preflightResult`	Object	true	`PreflightCheck`	`TermExpression`	The result of the preflight check (e.g., pass, fail).
`groundingReached`	Datatype	true	`PipelineSuccess`	`boolean`	Whether full grounding was achieved.
`feasibilityKind`	Datatype	true	`FeasibilityResult`	`string`	The kind of feasibility result (e.g., Feasible, Infeasible).
`feasibilityWitness`	Datatype	true	`FeasibilityResult`	`string`	The witness justifying the feasibility or infeasibility result.
`leasePhase`	Datatype	true	`LeaseState`	`string`	The lifecycle phase of a lease (e.g., Pending, Active, Released).
`managedLeaseId`	Object	true	`ManagedLease`	`TermExpression`	Unique identifier for this managed lease.
`leaseLifecycle`	Object	true	`ManagedLease`	`LeaseState`	The current lifecycle state of this managed lease.
`checkpointEpoch`	Datatype	true	`LeaseCheckpoint`	`nonNegativeInteger`	The epoch index at which this checkpoint was taken.
`checkpointState`	Object	true	`LeaseCheckpoint`	`ReductionState`	The reduction state captured at this checkpoint.
`pressureLevel`	Datatype	true	`BackPressureSignal`	`string`	The current back-pressure level (e.g., Low, Medium, High).
`pressureThreshold`	Datatype	true	`BackPressureSignal`	`decimal`	The threshold at which back-pressure activates.
`deferredCount`	Datatype	true	`DeferredQuerySet`	`nonNegativeInteger`	The number of queries in this deferred set.
`deferralEpoch`	Datatype	true	`DeferredQuerySet`	`nonNegativeInteger`	The epoch in which these queries were deferred.
`sourceLeaseRef`	Object	true	`SubleaseTransfer`	`TermExpression`	The lease being transferred from.
`targetLeaseRef`	Object	true	`SubleaseTransfer`	`TermExpression`	The lease being transferred to.
`transferredBudget`	Datatype	true	`SubleaseTransfer`	`nonNegativeInteger`	The site budget transferred between leases.
`transferCompleted`	Datatype	true	`SubleaseTransfer`	`boolean`	Whether the sublease transfer has been completed.
`comparisonField`	Object	true	`ComparisonPredicate`	`TermExpression`	The state field tested by this comparison predicate.
`comparisonOperator`	Object	true	`ComparisonPredicate`	`TermExpression`	The comparison operator (e.g., '=', '<', '>=').
`comparisonValue`	Object	true	`ComparisonPredicate`	`TermExpression`	The value against which the comparison is made.
`conjuncts`	Object	false	`ConjunctionPredicate`	`TermExpression`	A conjunct predicate in a conjunction.
`disjuncts`	Object	false	`DisjunctionPredicate`	`TermExpression`	A disjunct predicate in a disjunction.
`negatedPredicate`	Object	true	`NegationPredicate`	`TermExpression`	The predicate being negated.
`membershipSet`	Object	true	`MembershipPredicate`	`TermExpression`	The set against which membership is tested.
`membershipElement`	Object	true	`MembershipPredicate`	`TermExpression`	The element being tested for set membership.
`groundingThreshold`	Datatype	true	`GroundingPredicate`	`nonNegativeInteger`	The grounding threshold above which the predicate holds.
`coverageTarget`	Datatype	true	`SiteCoveragePredicate`	`nonNegativeInteger`	The site coverage target expression.
`equalityLeft`	Object	true	`EqualsPredicate`	`TermExpression`	The left-hand side of an equality test.
`equalityRight`	Object	true	`EqualsPredicate`	`TermExpression`	The right-hand side of an equality test.
`nonNullField`	Datatype	true	`NonNullPredicate`	`nonNegativeInteger`	The field that must be non-null.
`queryTypeRef`	Object	true	`QuerySubtypePredicate`	`TermExpression`	The query type reference for subtype testing.
`siteState`	Object	true	`ReductionState`	`TermExpression`	The site state descriptor within a reduction state.
`transactionScope`	Object	true	`ReductionTransaction`	`TermExpression`	The scope of sites affected by this transaction.
`transactionStatus`	Object	true	`ReductionTransaction`	`TermExpression`	Current status of this transaction (e.g., pending, committed).
`baseContextRef`	Object	true	`ServiceWindow`	`TermExpression`	Reference to the base context provided by this service window.
`leaseRemainingBudget`	Datatype	true	`LeaseCheckpoint`	`nonNegativeInteger`	The remaining site budget at this checkpoint.
`expiryEpoch`	Datatype	true	`ManagedLease`	`nonNegativeInteger`	The epoch at which this managed lease expires.
`leaseBudget`	Datatype	true	`ManagedLease`	`nonNegativeInteger`	The total site budget allocated to this managed lease.
`sourceStage`	Object	true	`BackPressureSignal`	`TermExpression`	The source stage emitting back-pressure.
`targetStage`	Object	true	`BackPressureSignal`	`TermExpression`	The target stage receiving back-pressure.
`deferralReason`	Object	true	`DeferredQuerySet`	`TermExpression`	The reason for deferring these queries.
`infeasibilityKind`	Object	true	`FeasibilityResult`	`TermExpression`	The kind of infeasibility detected.
`failureStage`	Object	true	`PipelineFailureReason`	`TermExpression`	The reduction stage at which the pipeline failure occurred.
`finalGrounding`	Datatype	true	`PipelineSuccess`	`decimal`	The final grounding level achieved on pipeline success.
`preservedGrounding`	Datatype	true	`EpochBoundary`	`boolean`	Whether grounding was preserved across the epoch boundary.
`rootTerm`	Object	true	`CompileUnit`	`TermExpression`	The root term expression of a CompileUnit — the top-level syntactic node from which all sub-expressions descend.
`unitWittLevel`	Object	true	`CompileUnit`	`WittLevel`	The quantum level at which this CompileUnit operates.
`thermodynamicBudget`	Datatype	true	`CompileUnit`	`decimal`	The Landauer-bounded energy budget for this CompileUnit's resolution, measured in k_B T ln 2 units.
`targetDomains`	Object	false	`CompileUnit`	`VerificationDomain`	The verification domain(s) targeted by this CompileUnit.
`unitAddress`	Object	true	`CompileUnit`	`Element`	Content-addressable identifier computed as the u:Element of the root term’s transitive closure. Computed by stage_initialization, not declared by the submitter. Excludes budget, domains, and quantum level to enable memoization.
`preflightOrder`	Datatype	true	`PreflightCheck`	`nonNegativeInteger`	Zero-based execution order for preflight checks. Lower indices execute first. BudgetSolvencyCheck (order 0) must precede all others.

Named Individuals

Name	Type	Comment
`Initialization`	`ReductionStep`	Stage 0: initialize state vector to identity.
`stageIndex`: 0 `stageName`: Initialization `expectedPhase`: Ω⁰
`Declare`	`ReductionStep`	Stage 1: dispatch resolver (δ selects).
`stageIndex`: 1 `stageName`: Declare `expectedPhase`: Ω¹
`Factorize`	`ReductionStep`	Stage 2: produce valid ring address (G grounds).
`stageIndex`: 2 `stageName`: Factorize `expectedPhase`: Ω²
`Resolve`	`ReductionStep`	Stage 3: resolve constraints (Π terminates).
`stageIndex`: 3 `stageName`: Resolve `expectedPhase`: Ω³
`Attest`	`ReductionStep`	Stage 4: accumulate without contradiction (α consistent).
`stageIndex`: 4 `stageName`: Attest `expectedPhase`: Ω⁴
`Extract`	`ReductionStep`	Stage 5: extract coherent output (P projects).
`stageIndex`: 5 `stageName`: Extract `expectedPhase`: Ω⁵
`Convergence`	`ReductionStep`	Terminal stage: reduction has reached the convergence angle π.
`stageIndex`: 6 `stageName`: Convergence `expectedPhase`: π
`gate_initialization`	`PhaseGateAttestation`	Phase gate at stage 0 boundary: checks Ω⁰ = 1.
`gateStage`: `stage_initialization` `gateExpectedPhase`: Ω⁰ `gateResult`: true
`gate_declare`	`PhaseGateAttestation`	Phase gate at stage 1 boundary: checks Ω¹.
`gateStage`: `stage_declare` `gateExpectedPhase`: Ω¹ `gateResult`: true
`gate_factorize`	`PhaseGateAttestation`	Phase gate at stage 2 boundary: checks Ω².
`gateStage`: `stage_factorize` `gateExpectedPhase`: Ω² `gateResult`: true
`gate_resolve`	`PhaseGateAttestation`	Phase gate at stage 3 boundary: checks Ω³.
`gateStage`: `stage_resolve` `gateExpectedPhase`: Ω³ `gateResult`: true
`gate_attest`	`PhaseGateAttestation`	Phase gate at stage 4 boundary: checks Ω⁴.
`gateStage`: `stage_attest` `gateExpectedPhase`: Ω⁴ `gateResult`: true
`gate_extract`	`PhaseGateAttestation`	Phase gate at stage 5 boundary: checks Ω⁵.
`gateStage`: `stage_extract` `gateExpectedPhase`: Ω⁵ `gateResult`: true
`euler_reduction_instance`	`EulerReduction`	The canonical Euler reduction instance with Ω = e^{iπ/6} and 6 stages.
`phaseParameter`: e^{iπ/6} `stageCount`: 6 `convergenceAngle`: π
`phase_schedule`	`PhaseRotationScheduler`	The canonical phase rotation schedule for the 6-stage reduction.
`rotationSchedule`: Ω⁰, Ω¹, Ω², Ω³, Ω⁴, Ω⁵ `baseAngle`: π/6
`convergence_target`	`TargetConvergenceAngle`	The default convergence target angle π.
`targetAngle`: π
`conjugate_rollback`	`ComplexConjugateRollback`	The canonical complex conjugate rollback operation: z → z̄.
`rollbackTarget`: `stage_initialization`
`DispatchMiss`	`PipelineFailureReason`	Failure: no resolver matched the dispatch query.
`failureKind`: DispatchMiss
`GroundingFailure`	`PipelineFailureReason`	Failure: grounding to a valid ring address failed.
`failureKind`: GroundingFailure
`ConvergenceStall`	`PipelineFailureReason`	Failure: reduction stalled before reaching convergence angle.
`failureKind`: ConvergenceStall
`ContradictionDetected`	`PipelineFailureReason`	Failure: accumulation detected a logical contradiction.
`failureKind`: ContradictionDetected
`CoherenceViolation`	`PipelineFailureReason`	Failure: coherence constraint violated during extraction.
`failureKind`: CoherenceViolation
`FullGroundingSuccess`	`PipelineSuccess`	Successful termination: all sites saturated.
`groundingReached`: true
`FeasibilityCheck`	`PreflightCheck`	Preflight: checks that the reduction can reach convergence.
`preflightKind`: Feasibility `preflightOrder`: 1
`DispatchCoverageCheck`	`PreflightCheck`	Preflight: checks that every dispatch query has a resolver.
`preflightKind`: DispatchCoverage `preflightOrder`: 2
`PackageCoherenceCheck`	`PreflightCheck`	Preflight: checks package-level coherence constraints.
`preflightKind`: PackageCoherence `preflightOrder`: 3
`default_service_window`	`ServiceWindow`	The default service window: 3 epochs, zero offset.
`windowSize`: 3 `windowOffset`: 0
`advance_init_to_declare`	`ReductionAdvance`	Advancement from Initialization to Declare.
`advanceFrom`: `stage_initialization` `advanceTo`: `stage_declare`
`atomic_transaction`	`ReductionTransaction`	An all-or-nothing atomic reduction transaction.
`transactionPolicy`: AllOrNothing
`empty_guard`	`GuardExpression`	A trivially satisfied guard with no predicates.
`identity_effect`	`TransitionEffect`	The identity effect: no state changes.
`true_predicate`	`PredicateExpression`	A predicate that always evaluates to true.
`predicateField`: * `predicateOperator`: true `predicateValue`: *
`noop_bind`	`PropertyBind`	A no-op property binding that changes nothing.
`bindTarget`: none `bindValue`: unchanged
`Pending`	`LeaseState`	Lease is pending activation.
`leasePhase`: Pending
`Active`	`LeaseState`	Lease is active and resources are allocated.
`leasePhase`: Active
`Released`	`LeaseState`	Lease has been explicitly released.
`leasePhase`: Released
`Expired`	`LeaseState`	Lease has expired due to timeout.
`leasePhase`: Expired
`Suspended`	`LeaseState`	Lease is temporarily suspended.
`leasePhase`: Suspended
`FeasibilityWitness`	`FeasibilityResult`	Preflight result: reduction is feasible.
`feasibilityKind`: Feasible
`InfeasibilityWitness`	`FeasibilityResult`	Preflight result: reduction is infeasible.
`feasibilityKind`: Infeasible `feasibilityWitness`: obstruction detected
`PreflightTiming`	`PreflightCheck`	Preflight: timing feasibility check.
`preflightKind`: Timing `preflightOrder`: 4
`RuntimeTiming`	`PreflightCheck`	Preflight: runtime timing bounds check.
`preflightKind`: RuntimeTiming `preflightOrder`: 5
`BudgetSolvencyCheck`	`PreflightCheck`	Preflight: verifies thermodynamicBudget ≥ bitsWidth(unitWittLevel) × ln 2. Rejects the CompileUnit if the budget is absent or insufficient. Must execute before all other preflights (preflightOrder 0). Cost is O(1) per CS_4.
`preflightKind`: BudgetSolvency `preflightOrder`: 0
`BackPressureDefault`	`BackPressureSignal`	Default back-pressure signal with medium threshold.
`pressureLevel`: Medium `pressureThreshold`: 0.75