locality_status
This definition supplies a status string that records completion of the RG1 locality structure in recognition geometry. It confirms the LocalConfigSpace type, its neighborhood axioms, and supporting lemmas are in place for local recognition. A researcher auditing the Recognition Science monolith would cite it to confirm the minimal neighborhood conditions before proceeding to geometry or forcing-chain steps. The definition is a direct string concatenation with no computational or logical content.
claimThe locality status is the string reporting that the neighborhood structure on configuration spaces satisfies the RG1 axioms: LocalConfigSpace is defined, neighborhoods obey membership, non-emptiness, intersection closure, and refinement, with discrete and trivial examples supplied.
background
The module RecogGeom.Locality implements axiom RG1 of recognition geometry: recognition occurs inside finite neighborhoods of configurations rather than globally. A LocalConfigSpace on a type C extends ConfigSpace C by a neighborhood map N : C → Set (Set C) together with the four axioms mem_of_mem_N, N_nonempty, intersection_closed, and refinement. These axioms let one speak of “nearby” configurations without choosing a metric or topology. The module imports the Neighborhood structure from cellular automata and evaluation maps from the linear-logic and scalar-field bridges; those upstream objects supply the concrete carriers on which the abstract locality axioms are later instantiated.
proof idea
The declaration is a plain definition that builds the status string by successive string concatenation of fixed messages. No tactics or lemmas are applied; the body simply lists the completed components and terminates with the completion banner.
why it matters in Recognition Science
The definition serves as the explicit marker that the RG1 locality axiom has been discharged inside the Recognition Science framework. It supplies the neighborhood infrastructure required by the forcing chain (T0–T8) and by the Recognition Composition Law before any spatial dimension or coupling-constant derivations can proceed. Because used_by_count is zero, the marker currently functions only as an internal checkpoint rather than a lemma feeding a parent theorem.
scope and limits
- Does not prove any topological or metric properties beyond the four listed axioms.
- Does not instantiate the structure on any concrete physical configuration space.
- Does not derive dimension D = 3 or any forcing-chain step.
- Does not supply a falsifier or counter-model for the locality axioms.
formal statement (Lean)
130def locality_status : String :=
proof body
Definition body.
131 "✓ LocalConfigSpace defined (RG1)\n" ++
132 "✓ Neighborhood axioms: mem_of_mem_N, intersection_closed, refinement\n" ++
133 "✓ Basic lemmas: has_neighborhood, self_mem_neighborhood, common_refinement\n" ++
134 "✓ Neighborhood filter construction\n" ++
135 "✓ Discrete and trivial examples\n" ++
136 "\n" ++
137 "LOCALITY STRUCTURE (RG1) COMPLETE"
138
139#eval locality_status
140
141end RecogGeom
142end IndisputableMonolith