finite_resolution_cell_finite_events
plain-language theorem explainer
If a recognizer has finite local resolution at configuration c, then the image of its resolution cell under the recognizer map is finite. Researchers formalizing discrete geometries from Recognition Science axioms cite this result to derive local discreteness. The proof is a direct term extraction that unpacks the existential witness from the HasFiniteLocalResolution hypothesis and reasserts the finiteness of the image.
Claim. If recognizer $r$ satisfies finite local resolution at configuration $c$, then there exists a neighborhood $U$ of $c$ such that $r(R(U))$ is finite.
background
The module formalizes axiom RG4 of Recognition Geometry: every recognizer has finite local resolution, meaning that in any bounded neighborhood only finitely many configurations are distinguished. This supplies the bridge from the 8-tick cycle to the appearance of discreteness at fundamental scales. HasFiniteLocalResolution is the predicate asserting that for a local configuration space L and recognizer r at point c there exists U in the neighborhood filter L.N c such that the image r.R '' U is finite.
proof idea
The term proof obtains the triple witness U, membership in L.N c, and finiteness of the image directly from the hypothesis h. It then supplies the same U to the existential goal and verifies the universal quantification over c' in U by exhibiting the membership pair and the already-known finiteness fact.
why it matters
The declaration closes the local half of RG4 inside the Recognition Geometry development and thereby supports the claim that the 8-tick octave forces finite resolution everywhere. It sits upstream of any later results on discrete local recognition geometry and supplies the concrete mechanism by which the universe appears discrete at voxel scales.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.