Self-Generated Guarded Counter Loops and Exact Lifetimes of Recurrent Self-Interacting Walkers

This paper studies deterministic self-interacting lattice walkers that write integer-valued memory into a shared register and later make decisions from that accumulated wake. The analysis conditions on an autonomously formed recurrent sector and asks what can be proved about its future continuation. Once a recurrent route is identified, its repeated write geometry generates a fixed integer packet. The decision-relevant wake can then be represented by finitely many unbounded integer counters whose state advances by a fixed translation on each recurrence cycle. The complete deterministic realization policy defines a semilinear policy-output region in this counter space, and the recurrent sector follows an integer ray through that region. Pairwise walker decisions are represented by two related guard families: raw-field margins and cost-normalized integer margins. At each recurrent phase, both evolve affinely with packet count. Their exact roots identify all packet counts at which the policy ordering can change. Evaluating the complete policy only at a finite set of root-adjacent packet counts therefore yields either an exact finite lifetime or an indefinite continuation certificate for the recurrent word. The framework is demonstrated on previously archived P14, P28, and P32 recurrent sectors. The P14 sector admits an indefinite strict-margin continuation certificate. The P28 and P32 sectors have exactly predicted first departures at recorded rows 1420 and 7769, respectively. In the P32 case, an earlier guard crossing changes the internal blocked/rechoice mechanism without changing the realized action, illustrating the distinction between crossing a pairwise decision wall and exiting the full policy-output region.