eight_tick_implies_RG4
Recognition Science's eight-tick cycle on ledger states forces every R̂-neighborhood to carry only finitely many distinguishable measurement outcomes, which directly satisfies Recognition Geometry's finite-resolution axiom RG4 on the induced local configuration space and recognizer. Researchers modeling discrete spacetime or emergent geometry from recognition dynamics would cite this bridge result to justify importing RS discreteness into geometric axioms. The proof is a one-line term wrapper that instantiates the local neighborhood via rs.RHat
claimLet $L$ be a ledger space satisfying the RS configuration axioms and $E$ an event space. Given a locality structure induced by the recognition operator $R̂$ on $L$, a measurement map $m:L→E$, and the hypothesis that every $R̂$-neighborhood has finitely many distinguishable measurement outcomes under the eight-tick cycle, the induced local configuration space and recognizer satisfy the finite-resolution property of Recognition Geometry: for every local configuration $c$, the set of measurement outcomes in the neighborhood of $c$ is finite.
background
The module RecogGeom.RSBridge constructs the structural link between Recognition Science ledger dynamics and Recognition Geometry axioms. RS supplies ledger states as an infinite-dimensional configuration space (RSConfigSpace), the recognition operator $R̂$ as the generator of local neighborhoods (RSLocalityFromRHat), and measurement maps as recognizers (RSMeasurement). The eight-tick hypothesis (EightTickFiniteResolution) asserts that for any ledger state $ℓ$, the image of its $R̂$-neighborhood under $m$ is finite, encoding the fundamental discreteness of recognition events. Upstream, HasFiniteResolution from FiniteResolution is the target RG4 predicate, while the tick definition from Constants supplies the native time quantum whose octave period (T7) supplies the eight-tick structure.
proof idea
The term-mode proof proceeds by introducing an arbitrary local configuration $ℓ$, selecting the $R̂$-neighborhood $rs.RHat ℓ$ as the witnessing set, and constructing a finite-resolution witness via the pair $⟨1, rfl, trivial⟩$ together with the direct application of the hypothesis $h8.finite_local_events ℓ$. No additional lemmas are invoked; the construction simply transfers the finiteness already recorded in EightTickFiniteResolution to the HasFiniteResolution predicate on the transported local space and recognizer.
why it matters in Recognition Science
This theorem supplies the RG4 component of the master bridge statement RS_instantiates_RG, which is invoked by the integration summary and the rsbridge_status diagnostic. It closes the structural gap between the eight-tick octave (T7) of Recognition Science and the finite-resolution axiom required for Recognition Geometry to derive three-dimensional space as a recognition quotient. The result is cited whenever one needs to justify that RS ledger discreteness is compatible with the geometric axioms without introducing extra hypotheses.
scope and limits
- Does not derive the numerical value of the resolution bound or the explicit form of the eight-tick period.
- Does not prove the existence or uniqueness of the eight-tick cycle itself.
- Does not address global resolution or non-local measurement outcomes.
- Does not instantiate concrete ledger data structures from other RS modules.
Lean usage
theorem RS_instantiates_RG {L E : Type*} [RSConfigSpace L] (rs : RSLocalityFromRHat L) (m : RSMeasurement L E) (h8 : EightTickFiniteResolution L E rs m) : HasFiniteResolution (toLocalConfigSpace rs) (toRecognizer m) := eight_tick_implies_RG4 rs m h8formal statement (Lean)
149theorem eight_tick_implies_RG4 {L E : Type*} [RSConfigSpace L]
150 (rs : RSLocalityFromRHat L) (m : RSMeasurement L E)
151 (h8 : EightTickFiniteResolution L E rs m) :
152 HasFiniteResolution (toLocalConfigSpace rs) (toRecognizer m) := by
proof body
Term-mode proof.
153 intro ℓ
154 use rs.RHat ℓ
155 constructor
156 · exact ⟨1, rfl, trivial⟩
157 · exact h8.finite_local_events ℓ
158
159/-! ## Master Bridge Statement -/
160
161/-- **Master theorem (RG4)**: RS's 8-tick finite-resolution hypothesis yields Recognition Geometry's
162finite-resolution axiom (RG4) for the induced locality and recognizer. -/