EightTickFiniteResolution
Recognition Science's eight-tick cycle enforces finite resolution on local measurements by requiring that every R̂-neighborhood yields only finitely many distinguishable outcomes under the measurement map. Researchers deriving discrete geometry from ledger states would cite this definition when establishing the RG4 axiom from the fundamental time quantum. The structure is introduced directly with a single field asserting finiteness of the image set under the measurement map.
claimLet $L$ be a ledger space satisfying the configuration space axioms and $E$ an event space. Given a locality structure induced by the recognition operator $R̂$ on $L$ and a measurement map $m:L→E$, the finite-resolution condition holds if for every ledger state $ℓ$ the set of measurement outcomes on the $R̂$-neighborhood of $ℓ$ is finite.
background
In the Recognition Geometry bridge module, Recognition Science supplies ledger states as the configuration space via RSConfigSpace, which requires the space to be nonempty and equipped with decidable equality. The RSLocalityFromRHat structure defines neighborhoods through the R̂ operator, with self-inclusion, refinement, and transitivity ensuring that reachability behaves like a neighborhood system. RSMeasurement supplies the map from ledger states to observable events together with a nontriviality condition that at least two states are distinguishable.
proof idea
This declaration is a pure structural definition containing no proof body. It directly encodes the finite cardinality condition on the image of the measurement map restricted to each R̂-neighborhood.
why it matters in Recognition Science
It supplies the hypothesis needed for the theorem eight_tick_implies_RG4, which establishes that RS satisfies the HasFiniteResolution axiom of Recognition Geometry. This closes the link from the eight-tick octave (T7) to the finite-resolution property (RG4) in the Recognition Science framework. The master result RS_instantiates_RG then uses this to show full instantiation of the geometric axioms from the 8-tick cycle.
scope and limits
- Does not derive finiteness from the tick definition alone.
- Does not bound the cardinality of the finite outcome set.
- Does not address global resolution across the entire ledger.
- Does not specify the concrete mechanism by which the 8-tick period enforces local finiteness.
formal statement (Lean)
143structure EightTickFiniteResolution (L E : Type*) [RSConfigSpace L]
144 (rs : RSLocalityFromRHat L) (m : RSMeasurement L E) : Prop where
145 /-- Every R̂ neighborhood has finitely many distinguishable outcomes -/
146 finite_local_events : ∀ ℓ, (m.measure '' rs.RHat ℓ).Finite
147
148/-- 8-tick finite resolution implies RG4 -/