UOR Partitions
This is a bridge-space namespace in the Resolve stage of the PRISM pipeline. It provides the resolution infrastructure — queries, partitions, observables, proofs, derivations, and traces that transform inputs into certified results.
Learn more: Pipeline Overview · The Partition Decomposition
Imports
https://uor.foundation/schema/https://uor.foundation/type/
Classes
| Name | Subclass Of | Disjoint With | Comment |
|---|---|---|---|
Partition | Thing | A 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. | |
Component | Thing | SiteIndex, FreeRank | A single component of a partition: a set of datum values belonging to one of the four categories. |
IrreducibleSet | Component | ReducibleSet, UnitGroup, Complement | The set of irreducible elements under the active type: elements whose only factorizations involve units or themselves. Analogous to prime elements in a ring. |
ReducibleSet | Component | IrreducibleSet, UnitGroup, Complement | The set of reducible non-unit elements: elements that can be expressed as a product of two or more non-unit elements. |
UnitGroup | Component | IrreducibleSet, ReducibleSet, Complement | The set of invertible elements (units) in the carrier: elements with a multiplicative inverse. In Z/(2^n)Z, the units are the odd integers. |
Complement | Component | IrreducibleSet, ReducibleSet, UnitGroup | Elements 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. |
SiteIndex | Thing | FreeRank, Component | A 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. |
FreeRank | Thing | SiteIndex, Component | The 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. |
SiteBinding | Thing | A record of a single site being pinned by a constraint. Links a specific site coordinate to the constraint that determined its value. | |
PartitionProduct | Thing | PartitionCoproduct | The 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. |
PartitionCoproduct | Thing | PartitionProduct | The 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
| Name | Kind | Functional | Domain | Range | Comment |
|---|---|---|---|---|---|
irreducibles | Object | true | Partition | IrreducibleSet | The irreducible component of this partition. |
reducibles | Object | true | Partition | ReducibleSet | The reducible component of this partition. |
units | Object | true | Partition | UnitGroup | The units component of this partition. |
exterior | Object | true | Partition | Complement | The exterior component of this partition. |
member | Object | false | Component | Datum | A datum value belonging to this partition component. |
cardinality | Datatype | true | Component | nonNegativeInteger | The number of elements in this partition component. The cardinalities of the four components must sum to 2^n. |
density | Datatype | true | Partition | decimal | The irreducible density of this partition: |Irr| / |A|, where A is the active carrier. |
sourceType | Object | true | Partition | TypeDefinition | The type declaration that was resolved to produce this partition. |
wittLength | Datatype | true | Partition | positiveInteger | The Witt level n at which this partition was computed. The ring has 2^n elements at this level. |
sitePosition | Datatype | true | SiteIndex | nonNegativeInteger | The 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. |
siteState | Datatype | true | SiteIndex | nonNegativeInteger | The current state of this site coordinate: 'pinned' if determined by a constraint, 'free' if still available for refinement. |
siteBudget | Object | true | Partition | FreeRank | The site budget associated with this partition. |
totalSites | Datatype | true | FreeRank | nonNegativeInteger | The total number of site coordinates in this budget, equal to the quantum level n. |
pinnedCount | Datatype | true | FreeRank | nonNegativeInteger | The number of site coordinates currently pinned by constraints. |
freeRank | Datatype | true | FreeRank | nonNegativeInteger | The number of site coordinates still free (not yet pinned). Equals totalSites - pinnedCount. |
isClosed | Datatype | true | FreeRank | boolean | Whether all sites in this budget are pinned. A closed budget means the type is fully resolved and the partition is complete. |
hasSite | Object | false | FreeRank | SiteIndex | A site coordinate belonging to this budget. |
pinnedBy | Object | true | SiteBinding | Constraint | The constraint that pins this site coordinate. |
pinsCoordinate | Object | true | SiteBinding | SiteIndex | The site coordinate that this pinning determines. |
hasBinding | Object | false | FreeRank | SiteBinding | A site pinning record in this budget. |
ancillaSite | Object | true | SiteIndex | SiteIndex | An ancilla site coordinate paired with this site for reversible computation (RC_1–RC_4 ancilla model). |
reversibleStrategy | Datatype | true | FreeRank | boolean | True when this site budget uses a reversible computation strategy preserving information through ancilla sites. |
exteriorCriteria | Object | true | Complement | TermExpression | The 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). |
isExhaustive | Datatype | true | Partition | boolean | Whether 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. |
leftFactor | Object | true | PartitionProduct | Partition | The left operand partition of this tensor product. |
rightFactor | Object | true | PartitionProduct | Partition | The right operand partition of this tensor product. |
leftSummand | Object | true | PartitionCoproduct | Partition | The left operand partition of this coproduct. |
rightSummand | Object | true | PartitionCoproduct | Partition | The right operand partition of this coproduct. |