recognitionStep_iterates_succ
plain-language theorem explainer
Recognition steps advance the fundamental time quantum by exactly one successor operation on the tick count. Researchers modeling the emergence of temporal structure from recognition processes would cite this to connect ledger dynamics to the natural-number object. The proof extracts the advance property of the step and reduces it to the successor definition via case analysis on the tick constructors.
Claim. For any recognition step $s$, the tick of the output equals the successor of the tick of the input: $τ(output(s)) = succ(τ(input(s)))$, where $τ$ maps snapshots to their tick values and $succ$ is the successor on the tick structure.
background
The Recognition Science framework derives time as the canonical natural-number object forced by recognition iterations. In this module, the tick sequence is shown to be equivalent to the orbit construction from logic-derived arithmetic, forming a Lawvere natural-number object under the successor operation that increments the index by one. Upstream results include the step definition from cellular automata, which applies a local rule to produce the successor tape state, and the tick constant defined as the fundamental time quantum of unity in RS-native units.
proof idea
The proof names the tick advance property from the given step. It performs case analysis on the mk constructors for the input and output ticks, rewrites the advance equality into these cases, simplifies using the successor definition, and concludes exactly with the advance fact.
why it matters
This result provides the recognition advances succ component of the TimeAsOrbitCert definition, which certifies that the temporal sequence is the natural-number object forced by recognition. It completes the bridge from the ledger dynamics of TimeEmergence to the abstract natural-number object on Tick, realizing the module's claim that time is the canonical iteration object of recognition. It relates to the T7 eight-tick octave in the forcing chain but leaves metric time and Berry-phase monotonicity for separate treatment.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.