UOR Partitions

IRI
https://uor.foundation/partition/
Prefix
partition:
Space
bridge
Comment
Irreducibility partitions produced by type resolution. A partition divides the ring into four disjoint components: Irreducible, Reducible, Units, and Exterior.

Imports

  • https://uor.foundation/schema/
  • https://uor.foundation/type/

Classes

NameIRISubclass OfDisjoint WithComment
Partitionhttps://uor.foundation/partition/Partitionhttp://www.w3.org/2002/07/owl#ThingA four-component partition of R_n produced by resolving a type declaration. The four components — Irreducible, Reducible, Units, Exterior — are mutually disjoint and exhaustive over the carrier.
Componenthttps://uor.foundation/partition/Componenthttp://www.w3.org/2002/07/owl#Thinghttps://uor.foundation/partition/SiteIndex, https://uor.foundation/partition/FreeRankA single component of a partition: a set of datum values belonging to one of the four categories.
IrreducibleSethttps://uor.foundation/partition/IrreducibleSethttps://uor.foundation/partition/Componenthttps://uor.foundation/partition/ReducibleSet, https://uor.foundation/partition/UnitGroup, https://uor.foundation/partition/ComplementThe set of irreducible elements under the active type: elements whose only factorizations involve units or themselves. Analogous to prime elements in a ring.
ReducibleSethttps://uor.foundation/partition/ReducibleSethttps://uor.foundation/partition/Componenthttps://uor.foundation/partition/IrreducibleSet, https://uor.foundation/partition/UnitGroup, https://uor.foundation/partition/ComplementThe set of reducible non-unit elements: elements that can be expressed as a product of two or more non-unit elements.
UnitGrouphttps://uor.foundation/partition/UnitGrouphttps://uor.foundation/partition/Componenthttps://uor.foundation/partition/IrreducibleSet, https://uor.foundation/partition/ReducibleSet, https://uor.foundation/partition/ComplementThe set of invertible elements (units) in the carrier: elements with a multiplicative inverse. In Z/(2^n)Z, the units are the odd integers.
Complementhttps://uor.foundation/partition/Complementhttps://uor.foundation/partition/Componenthttps://uor.foundation/partition/IrreducibleSet, https://uor.foundation/partition/ReducibleSet, https://uor.foundation/partition/UnitGroupElements of R_n that fall outside the active carrier — i.e., outside the type's domain. These are ring elements that do not participate in the current type resolution.
SiteIndexhttps://uor.foundation/partition/SiteIndexhttp://www.w3.org/2002/07/owl#Thinghttps://uor.foundation/partition/FreeRank, https://uor.foundation/partition/ComponentA single site coordinate in the iterated Z/2Z fibration. Each site represents one binary degree of freedom in the ring's structure. The total number of sites equals the quantum level n.
FreeRankhttps://uor.foundation/partition/FreeRankhttp://www.w3.org/2002/07/owl#Thinghttps://uor.foundation/partition/SiteIndex, https://uor.foundation/partition/ComponentThe site budget for a partition: an accounting of how many sites are pinned (determined by constraints) versus free (still available for further refinement). A closed budget means all sites are pinned and the type is fully resolved.
SiteBindinghttps://uor.foundation/partition/SiteBindinghttp://www.w3.org/2002/07/owl#ThingA record of a single site being pinned by a constraint. Links a specific site coordinate to the constraint that determined its value.
PartitionProducthttps://uor.foundation/partition/PartitionProducthttp://www.w3.org/2002/07/owl#Thinghttps://uor.foundation/partition/PartitionCoproductThe tensor product of two partitions: partition(A × B) = partition(A) ⊗ partition(B). The four-component structure combines component-wise under the product type construction (PT_2a). Carries leftFactor and rightFactor links to the operand partitions.
PartitionCoproducthttps://uor.foundation/partition/PartitionCoproducthttp://www.w3.org/2002/07/owl#Thinghttps://uor.foundation/partition/PartitionProductThe coproduct (disjoint union) of two partitions: partition(A + B) = partition(A) ⊕ partition(B). The four-component structure combines via disjoint union under the sum type construction (PT_2b). Carries leftSummand and rightSummand links to the operand partitions.

Properties

NameKindFunctionalDomainRangeComment
irreduciblesObjecttruehttps://uor.foundation/partition/Partitionhttps://uor.foundation/partition/IrreducibleSetThe irreducible component of this partition.
reduciblesObjecttruehttps://uor.foundation/partition/Partitionhttps://uor.foundation/partition/ReducibleSetThe reducible component of this partition.
unitsObjecttruehttps://uor.foundation/partition/Partitionhttps://uor.foundation/partition/UnitGroupThe units component of this partition.
exteriorObjecttruehttps://uor.foundation/partition/Partitionhttps://uor.foundation/partition/ComplementThe exterior component of this partition.
memberObjectfalsehttps://uor.foundation/partition/Componenthttps://uor.foundation/schema/DatumA datum value belonging to this partition component.
cardinalityDatatypetruehttps://uor.foundation/partition/Componenthttp://www.w3.org/2001/XMLSchema#nonNegativeIntegerThe number of elements in this partition component. The cardinalities of the four components must sum to 2^n.
densityDatatypetruehttps://uor.foundation/partition/Partitionhttp://www.w3.org/2001/XMLSchema#decimalThe irreducible density of this partition: |Irr| / |A|, where A is the active carrier.
sourceTypeObjecttruehttps://uor.foundation/partition/Partitionhttps://uor.foundation/type/TypeDefinitionThe type declaration that was resolved to produce this partition.
wittLengthDatatypetruehttps://uor.foundation/partition/Partitionhttp://www.w3.org/2001/XMLSchema#positiveIntegerThe Witt level n at which this partition was computed. The ring has 2^n elements at this level.
sitePositionDatatypetruehttps://uor.foundation/partition/SiteIndexhttp://www.w3.org/2001/XMLSchema#nonNegativeIntegerThe zero-based position of this site coordinate within the iterated fibration. Position 0 is the least significant bit; position n-1 is the most significant.
siteStateDatatypetruehttps://uor.foundation/partition/SiteIndexhttp://www.w3.org/2001/XMLSchema#nonNegativeIntegerThe current state of this site coordinate: 'pinned' if determined by a constraint, 'free' if still available for refinement.
siteBudgetObjecttruehttps://uor.foundation/partition/Partitionhttps://uor.foundation/partition/FreeRankThe site budget associated with this partition.
totalSitesDatatypetruehttps://uor.foundation/partition/FreeRankhttp://www.w3.org/2001/XMLSchema#nonNegativeIntegerThe total number of site coordinates in this budget, equal to the quantum level n.
pinnedCountDatatypetruehttps://uor.foundation/partition/FreeRankhttp://www.w3.org/2001/XMLSchema#nonNegativeIntegerThe number of site coordinates currently pinned by constraints.
freeRankDatatypetruehttps://uor.foundation/partition/FreeRankhttp://www.w3.org/2001/XMLSchema#nonNegativeIntegerThe number of site coordinates still free (not yet pinned). Equals totalSites - pinnedCount.
isClosedDatatypetruehttps://uor.foundation/partition/FreeRankhttp://www.w3.org/2001/XMLSchema#booleanWhether all sites in this budget are pinned. A closed budget means the type is fully resolved and the partition is complete.
hasSiteObjectfalsehttps://uor.foundation/partition/FreeRankhttps://uor.foundation/partition/SiteIndexA site coordinate belonging to this budget.
pinnedByObjecttruehttps://uor.foundation/partition/SiteBindinghttps://uor.foundation/type/ConstraintThe constraint that pins this site coordinate.
pinsCoordinateObjecttruehttps://uor.foundation/partition/SiteBindinghttps://uor.foundation/partition/SiteIndexThe site coordinate that this pinning determines.
hasBindingObjectfalsehttps://uor.foundation/partition/FreeRankhttps://uor.foundation/partition/SiteBindingA site pinning record in this budget.
ancillaSiteObjecttruehttps://uor.foundation/partition/SiteIndexhttps://uor.foundation/partition/SiteIndexAn ancilla site coordinate paired with this site for reversible computation (RC_1–RC_4 ancilla model).
reversibleStrategyDatatypetruehttps://uor.foundation/partition/FreeRankhttp://www.w3.org/2001/XMLSchema#booleanTrue when this site budget uses a reversible computation strategy preserving information through ancilla sites.
exteriorCriteriaObjecttruehttps://uor.foundation/partition/Complementhttps://uor.foundation/schema/TermExpressionThe formal membership criterion for this Complement: x ∈ Ext(T) iff x ∉ carrier(T). The Complement is context-dependent on the active type T (FPM_9).
isExhaustiveDatatypetruehttps://uor.foundation/partition/Partitionhttp://www.w3.org/2001/XMLSchema#booleanWhether the four components of this partition are exhaustive over R_n: |Irr| + |Red| + |Unit| + |Ext| = 2^n (FPM_8). Set by the kernel after verification.
leftFactorObjecttruehttps://uor.foundation/partition/PartitionProducthttps://uor.foundation/partition/PartitionThe left operand partition of this tensor product.
rightFactorObjecttruehttps://uor.foundation/partition/PartitionProducthttps://uor.foundation/partition/PartitionThe right operand partition of this tensor product.
leftSummandObjecttruehttps://uor.foundation/partition/PartitionCoproducthttps://uor.foundation/partition/PartitionThe left operand partition of this coproduct.
rightSummandObjecttruehttps://uor.foundation/partition/PartitionCoproducthttps://uor.foundation/partition/PartitionThe right operand partition of this coproduct.