Exact Clocks and Reachability in the SBA State-11 Neutral Dynamics

We study a finite-support neutral counter system associated with the Specular Bit Architecture (SBA). The term State 11 denotes the elementary ordered pair (positive, negative) $=(1,1),$ which has zero signed value but retains one neutral unit. Its three firing rules replace two units at a source by units at prescribed Fibonacci-related destinations, including two exceptional lower-boundary rules. The system is the duplicate-only subsystem of the Zeckendorf game, realized on equal positive and negative counters.

We classify all linear clocks with unit decrement, identify the minimal nonnegative integer and real clocks, and derive exact odometers for saturated intervals with a single added unit. The clocks yield a sharp lifetime bound and are related explicitly to existing Zeckendorf-game length weights. As a self-contained specialization of established abelian and chip-firing theory, we prove termination, uniqueness of the firing-count vector and stable endpoint, and schedule independence of capped execution. These properties describe the finite reachability lattice, its occurrence antimatroid, and its hypercube metric.

Triangular integer elimination and capped bulk evaluation decide endpoint reachability in $O(L^{4})$ bit operations after input processing, where L is the bit length of the conserved load; infeasible nonnegative count candidates admit $O(L^{2})$ -bit algebraic barriers. We also prove earliest completion for one-firing-per-source snapshot rounds and compute the stable endpoint, odometer, and lifetime by uncapped bulk evaluation. All statements concern the specified half-line boundary rules and finite-support inputs; scalar firing counts are kept distinct from compact input complexity and physical time.

Index Terms:

Fibonacci splitting, State 11, abelian stabilization, chip-firing, odometer, exact clock, reachability, antimatroid