Homological Analysis

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

Simplicial Complexes

The Simplex class represents a single simplex -- a generalized triangle. A collection of simplices that is closed under taking faces forms a SimplicialComplex.

In UOR, the Cech nerve is the key simplicial complex. Each constraint in the resolution process is a vertex. A set of constraints that can be simultaneously satisfied forms a simplex. The resulting complex encodes the combinatorial structure of constraint compatibility.

Chain Homology and Betti Numbers

Chain homology measures the "holes" in a simplicial complex:

• Betti number beta_0 counts connected components. If beta_0 = 1, the constraint space is connected -- all constraints can reach each other.
• Betti number beta_1 counts independent loops. Nonzero beta_1 means the constraints contain cyclic dependencies that may cause resolution to stall.
• Higher Betti numbers detect higher-dimensional voids in the constraint structure, signaling more complex obstructions.

When all Betti numbers are trivial (zero except beta_0 = 1), the constraint space is contractible -- resolution converges without obstruction. This is the ideal case.

Sheaf Cohomology

The Sheaf class models data that varies smoothly over the constraint space. Sheaf cohomology detects when local constraint satisfaction fails to globalize.

A GluingObstruction in the first cohomology group H^1 pinpoints exactly where local solutions cannot be assembled into a global resolution. This is the algebraic formalization of "works locally, fails globally."

Spectral Sequences

The SpectralSequencePage class models the pages of a spectral sequence -- a tool that refines the homological analysis level by level. At each Witt Levels scale, the spectral sequence provides increasingly precise information about obstructions.

The spectral sequence connects the homological analysis to the Witt level tower: obstructions visible at W8 may dissolve at W16, or new obstructions may appear at higher levels.

Euler Characteristics

The nerveEulerCharacteristic property records the Euler characteristic of the Cech nerve -- a single integer that summarizes the topological complexity. A positive Euler characteristic suggests contractibility (good for convergence); a negative one signals complexity.

Connection to the PRISM Pipeline

Homological analysis operates within the Resolve stage. When Resolution & Queries stalls, the Cech nerve's topology explains why. The Site Bundle Semantics bundle structure provides the geometric substrate that homology and cohomology analyze. The Observables & Measurement namespace measures the topological invariants (holonomy groups, monodromy classes) that feed into this analysis.