Memory and randomness control the shape: a family of limit laws from a triangular-window walk

A deterministic walk on a shrinking triangular region reads two bits from a window of a ternarysequence and either moves towards the diagonal or leaves the region, in which case the region iscut. A companion paper studies one specification of this walk and finds an arcsine limit for thenormalised hitting position X. Here we vary the specification systematically, organising thechoices into four layers — information, action, memory and input — and measuring the law of Xacross roughly sixty settings. The shape is largely insensitive to the local rule: at N=2000 twoof the six modifications are distinguishable from the baseline but move the kurtosis by 0.03, andthe rest are not distinguishable. Memory is the switch; without it the law is fixed by thegeometry of the region shrinkage. The reachable shapes lie on a narrow band in the (sd, κ) planerunning from the arcsine law to a point mass; a uniform law needs memory, and a three-peaked lawis not reached by the grid we searched. The family contains at least three limit objects: thearcsine, a Beta-like law near it, and a law that is a spike at a computable edge plus a broadbackground. At finite n the base law is close to but distinguishable from the arcsine, the excessshrinking with n. Exact counts of the matrices producing each hitting position for n ≤ 13 do notappear in the OEIS.