T2_atomicity
Atomicity of recognition ticks asserts that each natural-number time index posts to at most one unit inside a recognition structure. Researchers formalizing discrete time in the RRF core cite this result to guarantee ledger consistency and prevent event collisions. The proof is a direct term-level re-export of the upstream uniqueness theorem supplied by the AtomicTick class.
claimLet $M$ be a recognition structure equipped with an atomic tick instance. Then for all natural numbers $t$ and units $u,v$ in the carrier of $M$, if the posted-at predicate holds for $(t,u)$ and for $(t,v)$, then $u=v$.
background
A RecognitionStructure consists of a carrier type U together with a binary recognition relation R on U. The AtomicTick class augments such a structure with a postedAt predicate from natural numbers to U and the axiom that each tick t admits a unique witness u satisfying postedAt t u. This module re-exports the definitions from the base Recognition module to provide a bridge into the RRF framework, where chains, ledgers, and flux calculations are built on top of these recognition steps.
proof idea
The proof is a one-line term wrapper that applies the upstream T2_atomicity theorem from the Chain module. That result performs case analysis on the unique_post witness for the given tick, then uses the uniqueness property to equate both posted units to the same witness.
why it matters in Recognition Science
This declaration re-exports the core atomicity result T2 that enforces unique posting per tick inside recognition chains. It is invoked by the parent T2_atomicity theorems in both the Chain and Recognition modules to maintain ledger consistency. Within the Recognition Science framework the result supports the discrete time structure required for the eight-tick octave and the phi-ladder mass assignments.
scope and limits
- Does not extend the result to continuous-time models.
- Does not constrain the underlying recognition relation R.
- Does not address postings that span multiple distinct ticks.
- Does not bound the cardinality of the carrier type U.
formal statement (Lean)
61theorem T2_atomicity {M : RecognitionStructure} [Recognition.AtomicTick M] :
62 ∀ t u v, Recognition.AtomicTick.postedAt (M:=M) t u → Recognition.AtomicTick.postedAt (M:=M) t v → u = v :=
proof body
Term-mode proof.
63 Recognition.T2_atomicity
64
65/-- The bridge theorem: chainFlux is zero for balanced ledgers. -/