IndisputableMonolith.RecogGeom.Locality
The Locality module defines local configuration spaces as configuration spaces equipped with neighborhood structures, establishing axiom RG1 of Recognition Geometry. Neighborhoods permit statements about nearby configurations without metrics or full topologies. Researchers developing recognition maps or foundational theorems cite this module for its minimal locality structure. The module supplies type definitions together with discrete and trivial instances.
claimA local configuration space is a pair $(C, N)$ where $C$ is a configuration space and $N$ is a neighborhood structure on $C$ that assigns to each point a collection of subsets containing it.
background
Recognition Geometry derives spatial notions from the structure of recognition maps rather than taking space as primitive. The upstream Core module supplies the basic configuration space types. This module augments those types with neighborhood structures, allowing local comparisons while remaining agnostic about metrics or continuity.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module supplies the locality structure RG1 required by the fundamental theorems in Foundations, the integration summary in Integration, and the recognition maps in Recognizer. It supplies the minimal neighborhood data needed before recognition maps can compare nearby configurations and before spatial dimensions emerge in the larger Recognition Science framework.
scope and limits
- Does not introduce a metric on the configuration space.
- Does not impose separation or continuity axioms beyond neighborhoods.
- Does not address global connectivity or compactness.
- Does not connect neighborhoods to the J-cost or phi-ladder.