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 The Partition Decomposition exists as a structural separator between kernel and user concerns.

What Is a Site Bundle?

A site bundle consists of three components:

1. Total space E: the full collection of all typed data
2. Base space B: the address space -- the set of all possible content addresses
3. Site F: the type (or set of types) that lives over each address

For every address b in B, there is a site F_b -- a set of valid types at that address. The total space E is the disjoint union of all sites: E = U_{b in B} F_b.

In UOR, the base space is the UniversalAddress space (grounded in the The Ring Substrate Z/(2^8)Z), and sites are type:Site instances -- the possible types attached to a given address. See Content Addressing for how the base space is defined.

The type Namespace

The type namespace is the User-space implementation of site bundle semantics. Key classes:

• type:Site -- the fundamental typed layer over an address
• type:FlatType -- a site with trivial (non-twisted) holonomy
• type:TwistedType -- a site with nontrivial holonomy (monodromy acts non-trivially)
• CompleteType -- a site whose completeness has been certified
• TypeSynthesisGoal and TypeSynthesisResult -- classes for the psi-pipeline inversion: computing what type is needed given a constraint

The partition Namespace

The UOR Partitions namespace implements the decomposition of the base space. An address space can be partitioned into disjoint subsets (sites in the partition sense), and UOR uses this to implement namespace-level modularity. See The Partition Decomposition for full detail.

Key classes:

• Partition -- a decomposition of the address space
• PartitionProduct -- the Cartesian product of two partitions (the full space)
• PartitionCoproduct -- the disjoint union of two partitions (exclusive choice)

The isExhaustive property asserts that a partition covers the entire space -- a formal completeness claim implemented in OWL.

Connection to Holonomy

When a site is transported around a loop in the base space, it may return to a different element of the site -- this transformation is the holonomy. If all holonomies are trivial (the identity), the bundle is flat (type:FlatType). Otherwise, it is twisted (type:TwistedType).

The HolonomyGroup, Monodromy, and MonodromyClass classes in the Bridge space observe these holonomy phenomena. See Observables & Measurement for how measurement works.

Monodromy -- the representation of the fundamental group of the base space in the holonomy group -- is detected by the MonodromyResolver and recorded in MonodromyClass individuals.

Superposed Site States

A site need not be in a definite type state. The SuperposedSiteState and CollapsedSiteState classes implement quantum superposition at the type level.

• A SuperposedSiteState is a linear combination of site types, weighted by amplitudes
• A CollapsedSiteState is the result of measurement -- a definite type

This superposition is not merely formal. The SuperpositionResolver resolves the superposed state, and the MeasurementCertificate certifies that the collapse event followed Born rule probabilities.

Partition Decomposition and the PRISM Pipeline

The partition structure mediates between the Kernel (Define stage) and the User types (Resolve stage) of the PRISM pipeline:

• Kernel defines the address space and The Ring Substrate arithmetic
• The Partition Decomposition (Bridge) decomposes the address space into sites
• Type (User) assigns typed meaning to each site
• Resolution & Queries (Bridge) computes what type is at each address
• Certification (UOR Certificates) certifies that the typing is consistent

The site bundle semantics is what makes UOR more than a list of algebraic facts -- it is a structured framework for typed computation over a content-addressed address space.