Concepts

Deep-dive explanations of the core mathematical and architectural concepts in the UOR Foundation. Each page connects the formal ontology definitions to the intuitions behind the PRISM pipeline.

Suggested Reading Order

New to UOR? Follow this path through the core concepts, from the algebraic foundation to the full certification pipeline:

  1. 1The Ring Substratekernel
  2. 2Witt Levelskernel
  3. 3Content Addressingkernel
  4. 4The Partition Decompositionbridge
  5. 5Site Bundle Semanticsbridge
  6. 6Resolution & Queriesbridge
  7. 7Observables & Measurementbridge
  8. 8Proofs, Derivations & Tracesbridge
  9. 9Homological Analysisbridge
  10. 10Certification and Verificationbridge
  11. 11Morphisms and Transformationsuser
  12. 12State, Sessions, and Accumulationuser

All Concepts

Certification and Verification

The final stage of the PRISM pipeline is **Certify**: every resolution result must be attested with a machine-verifiable certificate before it leaves the pipeline. The cert namespace encodes this attestation layer.

Content Addressing

Content addressing is the foundational principle of UOR: an object is identified by *what it is*, not *where it is*. The u namespace formalizes this with the Element class and the ContentAddressed interface.

Homological Analysis

The homology and cohomology namespaces add an algebraic topology layer to UOR. When constraints interact in complex ways, topological invariants diagnose whether Resolution will converge or stall.

Morphisms and Transformations

The morphism namespace defines the maps between UOR objects. Where the kernel namespaces declare *what* objects are, and bridge namespaces compute *how* to resolve them, morphisms specify *how objects relate to each other* through structure-preserving transformations.

Observables & Measurement

The observable namespace defines what can be measured during resolution. Observables are bridge-space objects that quantify geometric, topological, and algebraic properties of ring elements and their types.

The Partition Decomposition

The partition namespace decomposes the address space into disjoint subsets. Every ring element is classified as irreducible, reducible, a unit, or exterior. This four-way decomposition is the structural backbone of the Resolve stage in the [PRISM](../pipeline/) pipeline.

Proofs, Derivations & Traces

The proof, derivation, and trace namespaces implement the certification pathway of the [PRISM](../pipeline/) pipeline. Every algebraic identity must be *proved*, every proof must be *derived* from axioms, and every derivation must be *traced* for reproducibility.

Resolution & Queries

Resolution is the core operation of the [PRISM](../pipeline/) pipeline's Resolve stage. A Query specifies what to resolve; a Resolver computes the answer by factorizing the input under the dihedral group D_{2^n}.

The Ring Substrate

Every UOR computation operates over a ring — specifically the modular integer ring Z/(2^n)Z, where n is determined by the Witt level. This document explains the ring structure, its physical motivation, and how it grounds the entire ontology.

Site Bundle Semantics

UOR organizes typed data using the mathematical structure of a site bundle. Understanding site bundles explains why types in UOR behave the way they do, and why the Partition exists as a structural separator between kernel and user concerns.

State, Sessions, and Accumulation

The state namespace models the mutable side of the UOR framework. While the kernel is immutable and the bridge is purely computed, state captures what happens when a resolver accumulates bindings across a sequence of queries.

Witt Levels

Witt levels W8--W32 are the four scaling tiers of the UOR Ring substrate. Every computation, identity, and proof in UOR is valid at one or more Witt levels. Understanding Witt levels is essential for reading the algebraic identities and their associated proofs.

Found 12 concept pages.