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
| Name | IRI | Subclass Of | Disjoint With | Comment |
|---|---|---|---|---|
| Partition | https://uor.foundation/partition/Partition | http://www.w3.org/2002/07/owl#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 | https://uor.foundation/partition/Component | http://www.w3.org/2002/07/owl#Thing | https://uor.foundation/partition/SiteIndex, https://uor.foundation/partition/FreeRank | A single component of a partition: a set of datum values belonging to one of the four categories. |
| IrreducibleSet | https://uor.foundation/partition/IrreducibleSet | https://uor.foundation/partition/Component | https://uor.foundation/partition/ReducibleSet, https://uor.foundation/partition/UnitGroup, https://uor.foundation/partition/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 | https://uor.foundation/partition/ReducibleSet | https://uor.foundation/partition/Component | https://uor.foundation/partition/IrreducibleSet, https://uor.foundation/partition/UnitGroup, https://uor.foundation/partition/Complement | The set of reducible non-unit elements: elements that can be expressed as a product of two or more non-unit elements. |
| UnitGroup | https://uor.foundation/partition/UnitGroup | https://uor.foundation/partition/Component | https://uor.foundation/partition/IrreducibleSet, https://uor.foundation/partition/ReducibleSet, https://uor.foundation/partition/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 | https://uor.foundation/partition/Complement | https://uor.foundation/partition/Component | https://uor.foundation/partition/IrreducibleSet, https://uor.foundation/partition/ReducibleSet, https://uor.foundation/partition/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 | https://uor.foundation/partition/SiteIndex | http://www.w3.org/2002/07/owl#Thing | https://uor.foundation/partition/FreeRank, https://uor.foundation/partition/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 | https://uor.foundation/partition/FreeRank | http://www.w3.org/2002/07/owl#Thing | https://uor.foundation/partition/SiteIndex, https://uor.foundation/partition/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 | https://uor.foundation/partition/SiteBinding | http://www.w3.org/2002/07/owl#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 | https://uor.foundation/partition/PartitionProduct | http://www.w3.org/2002/07/owl#Thing | https://uor.foundation/partition/PartitionCoproduct, https://uor.foundation/partition/CartesianPartitionProduct | 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 | https://uor.foundation/partition/PartitionCoproduct | http://www.w3.org/2002/07/owl#Thing | https://uor.foundation/partition/PartitionProduct, https://uor.foundation/partition/CartesianPartitionProduct | 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. |
| CartesianPartitionProduct | https://uor.foundation/partition/CartesianPartitionProduct | http://www.w3.org/2002/07/owl#Thing | https://uor.foundation/partition/PartitionProduct, https://uor.foundation/partition/PartitionCoproduct | The Cartesian product of partitions. Classifies the nerve topology of A ⊠ B as a simplicial product (χ multiplicative per CPT_3, Betti by Künneth per CPT_4) rather than a site-disjoint union (χ additive — PartitionProduct). Site budget is |S_A| + |S_B| per CPT_1 — the bit width of the product state space. Partition-ness is asserted via leftCartesianFactor / rightCartesianFactor (both ranged at Partition), matching the sibling pattern for PartitionProduct and PartitionCoproduct. Satisfies CPT_1–CPT_6 per this amendment. |
| TagSite | https://uor.foundation/partition/TagSite | https://uor.foundation/partition/SiteIndex | The distinguishing site in a PartitionCoproduct whose value (0 or 1) selects the variant. Logically, the tag is not a data site of either operand (ST_6) and carries exactly the ln 2 entropy quantum (ST_2). Its physical placement in a flat constraint layout follows the foundation layout convention: layoutTagSite = max(SITE_COUNT(A), SITE_COUNT(B)), so the tag does not collide with any inherited bookkeeping sites when operands are themselves coproducts. | |
| FreeRankObservable | https://uor.foundation/partition/FreeRankObservable | https://uor.foundation/observable/Observable | Observes the free-rank of the partition associated with a Datum's site context, recording the count of unbound sites at the moment of observation. Used as the bound observable for the siteConstraintKind BoundConstraint. | |
| PartitionComponent | https://uor.foundation/partition/PartitionComponent | http://www.w3.org/2002/07/owl#Thing | Closed enumeration of partition component kinds: Irreducible (non-factorizable), Reducible (factorizable into non-trivial parts), Units (invertible), Exterior (outside the factorization domain). Codegen treats this as an enum class with exactly 4 individuals. |
Properties
| Name | Kind | Functional | Domain | Range | Comment |
|---|---|---|---|---|---|
| irreducibles | Object | true | https://uor.foundation/partition/Partition | https://uor.foundation/partition/IrreducibleSet | The irreducible component of this partition. |
| reducibles | Object | true | https://uor.foundation/partition/Partition | https://uor.foundation/partition/ReducibleSet | The reducible component of this partition. |
| units | Object | true | https://uor.foundation/partition/Partition | https://uor.foundation/partition/UnitGroup | The units component of this partition. |
| exterior | Object | true | https://uor.foundation/partition/Partition | https://uor.foundation/partition/Complement | The exterior component of this partition. |
| member | Object | false | https://uor.foundation/partition/Component | https://uor.foundation/schema/Datum | A datum value belonging to this partition component. |
| cardinality | Datatype | true | https://uor.foundation/partition/Component | http://www.w3.org/2001/XMLSchema#nonNegativeInteger | The number of elements in this partition component. The cardinalities of the four components must sum to 2^n. |
| density | Datatype | true | https://uor.foundation/partition/Partition | http://www.w3.org/2001/XMLSchema#decimal | The irreducible density of this partition: |Irr| / |A|, where A is the active carrier. |
| sourceType | Object | true | https://uor.foundation/partition/Partition | https://uor.foundation/type/TypeDefinition | The type declaration that was resolved to produce this partition. |
| wittLength | Datatype | true | https://uor.foundation/partition/Partition | http://www.w3.org/2001/XMLSchema#positiveInteger | The Witt level n at which this partition was computed. The ring has 2^n elements at this level. |
| sitePosition | Datatype | true | https://uor.foundation/partition/SiteIndex | http://www.w3.org/2001/XMLSchema#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 | https://uor.foundation/partition/SiteIndex | http://www.w3.org/2001/XMLSchema#nonNegativeInteger | The current state of this site coordinate: 'pinned' if determined by a constraint, 'free' if still available for refinement. |
| siteBudget | Object | true | https://uor.foundation/partition/Partition | https://uor.foundation/partition/FreeRank | The site budget associated with this partition. |
| totalSites | Datatype | true | https://uor.foundation/partition/FreeRank | http://www.w3.org/2001/XMLSchema#nonNegativeInteger | The total number of site coordinates in this budget, equal to the quantum level n. |
| pinnedCount | Datatype | true | https://uor.foundation/partition/FreeRank | http://www.w3.org/2001/XMLSchema#nonNegativeInteger | The number of site coordinates currently pinned by constraints. |
| freeRank | Datatype | true | https://uor.foundation/partition/FreeRank | http://www.w3.org/2001/XMLSchema#nonNegativeInteger | The number of site coordinates still free (not yet pinned). Equals totalSites - pinnedCount. |
| isClosed | Datatype | true | https://uor.foundation/partition/FreeRank | http://www.w3.org/2001/XMLSchema#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 | https://uor.foundation/partition/FreeRank | https://uor.foundation/partition/SiteIndex | A site coordinate belonging to this budget. |
| pinnedBy | Object | true | https://uor.foundation/partition/SiteBinding | https://uor.foundation/type/Constraint | The constraint that pins this site coordinate. |
| pinsCoordinate | Object | true | https://uor.foundation/partition/SiteBinding | https://uor.foundation/partition/SiteIndex | The site coordinate that this pinning determines. |
| hasBinding | Object | false | https://uor.foundation/partition/FreeRank | https://uor.foundation/partition/SiteBinding | A site pinning record in this budget. |
| ancillaSite | Object | true | https://uor.foundation/partition/SiteIndex | https://uor.foundation/partition/SiteIndex | An ancilla site coordinate paired with this site for reversible computation (RC_1–RC_4 ancilla model). |
| reversibleStrategy | Datatype | true | https://uor.foundation/partition/FreeRank | http://www.w3.org/2001/XMLSchema#boolean | True when this site budget uses a reversible computation strategy preserving information through ancilla sites. |
| exteriorCriteria | Object | true | https://uor.foundation/partition/Complement | https://uor.foundation/schema/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 | https://uor.foundation/partition/Partition | http://www.w3.org/2001/XMLSchema#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 | https://uor.foundation/partition/PartitionProduct | https://uor.foundation/partition/Partition | The left operand partition of this tensor product. |
| rightFactor | Object | true | https://uor.foundation/partition/PartitionProduct | https://uor.foundation/partition/Partition | The right operand partition of this tensor product. |
| leftSummand | Object | true | https://uor.foundation/partition/PartitionCoproduct | https://uor.foundation/partition/Partition | The left operand partition of this coproduct. |
| rightSummand | Object | true | https://uor.foundation/partition/PartitionCoproduct | https://uor.foundation/partition/Partition | The right operand partition of this coproduct. |
| leftCartesianFactor | Object | true | https://uor.foundation/partition/CartesianPartitionProduct | https://uor.foundation/partition/Partition | The left operand partition of this Cartesian partition product. |
| rightCartesianFactor | Object | true | https://uor.foundation/partition/CartesianPartitionProduct | https://uor.foundation/partition/Partition | The right operand partition of this Cartesian partition product. |
| tagSiteOf | Object | true | https://uor.foundation/partition/Partition | https://uor.foundation/partition/TagSite | The tag site distinguishing the variants of a PartitionCoproduct. Logically distinct from every data site of either operand (ST_6) and carries the ln 2 entropy quantum of ST_2. |
| tagValue | Datatype | true | https://uor.foundation/partition/TagSite | http://www.w3.org/2001/XMLSchema#boolean | The boolean value (false = 0, true = 1) assigned to a tag site. false selects the left variant of the PartitionCoproduct; true selects the right variant. |
| productCategoryLevel | Datatype | true | https://uor.foundation/partition/Partition | http://www.w3.org/2001/XMLSchema#string | The categorical level at which this construction is a product / coproduct. Values: 'partition_classification' (PartitionProduct, PartitionCoproduct), or 'nerve_topology' (CartesianPartitionProduct). Prevents misreading the product vs coproduct distinction across levels. |
Named Individuals
| Name | Type | Properties | Comment |
|---|---|---|---|
| Irreducible | https://uor.foundation/partition/PartitionComponent | — | The irreducible component: elements that admit no non-trivial factorization within the ring. |
| Reducible | https://uor.foundation/partition/PartitionComponent | — | The reducible component: elements that factor into non-trivial parts. |
| Units | https://uor.foundation/partition/PartitionComponent | — | The unit component: invertible elements of the ring. |
| Exterior | https://uor.foundation/partition/PartitionComponent | — | The exterior component: elements outside the factorization domain (e.g., zero or ring-boundary values). |