A triangular window walk on random binary matrices: a conjectured arcsine limit and a candidate mechanism

A deterministic map is defined on n×2 binary matrices with ⌈n/2⌉ ones per column. An ant walks on the triangular lattice of intervals; two comparisons inside the window — whether its mean exceeds that of its first half, and whether the lag-one autocorrelation is positive — select one of four moves. Leaving the region cuts off the smaller triangle and the walk restarts; reaching the diagonal produces a singular row δ. We conjecture: the normalised position X=δ/(n−1) converges to the arcsine law with standard deviation 1/√8; the probability of a singular row tends to about 0.315, convention-dependent; the residual standard deviation of X against density autocorrelation tends to 0.23; and the reversal rate obeys s(m)=arccos ρ(m)/π with ρ(m)≈1−1/m. Exact enumeration (n≤12) and Monte Carlo (n≤8192) support all four; none is proved. A six-link mechanism consistent with the arcsine shape is described, with five models that fail.