physical_interpretation_finite_resolution
plain-language theorem explainer
A locally discrete recognition geometry ensures that for any configuration c there exists a neighborhood U such that the recognizer maps U to a finite set of states. Physicists deriving discrete spacetime from recognition principles cite this to ground finite resolution in the 8-tick atomicity. The proof is a one-line wrapper that unfolds the definition of local discreteness.
Claim. Let $L$ be a local configuration space and $r$ a recognizer. If the geometry is locally discrete, then for every configuration $c$ there exists a neighborhood $U$ of $c$ such that the image of $U$ under $r$ is finite.
background
Recognition geometry imposes finite local resolution (RG4): in any bounded neighborhood only finitely many configurations are distinguishable. The predicate of local discreteness is defined to mean that the recognizer has finite resolution on the local configuration space, so every neighborhood contains only finitely many distinguishable configurations. The module states that the 8-tick cycle supplies this finite resolution and that the property is a general constraint any recognition-based geometry must satisfy.
proof idea
The proof is a one-line wrapper that applies the definition of local discreteness: introduce the configuration $c$ and invoke the hypothesis directly.
why it matters
This result supplies the physical interpretation of RG4, linking finite resolution to the 8-tick atomicity (T7) in which the number of distinguishable states is bounded by ledger updates. It feeds the downstream status definition that records the module's coverage of finite-resolution theorems. The declaration closes the bridge from abstract recognition geometry to the discrete appearance of fundamental scales.
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