The Odd-Even Digit Walk: Tie-Broken Symmetry, Zero-Area Returns, and Decimal Renewal Geometry

Abstract

We study the Odd-Even Digit Walk, the integer recurrence obtained by taking steps of magnitude nn and choosing the sign from the difference between the sums of the even-valued and odd-valued decimal digits of the current state, with ties assigned to addition. Although the rule differs from the strict sign mirror of the corresponding Even-Odd Digit Walk only at ties, this convention produces a distinct dynamical system. The paper develops several exact structures of the walk. The decimal digit score admits a folded-coordinate representation and cyclotomic generating-polynomial factorization, with exact equidistribution modulo 1010 at every fixed zero-padded digit length. Index-independent same-state returns are characterized as balanced ±1\pm1 paths of zero signed area and counted by a central Gaussian-binomial coefficient. The decimal branch rule is then analyzed through an exact adjacent boundary and reflected arithmetic-progression corridors, leading to the infinite return-length staircase Lk=25⋅10k−2−5,L_k=25\cdot10^{k-2}-5, whose first four levels are realized by the canonical orbit. Longer return segments are described by a two-stage mechanism in which decimal geometry selects a balanced core and the recurrence then forces a deterministic sequence of square-displacement recovery blocks. The paper also develops a discrepancy formulation b(n)=nSn−∑j<nSj,b(n)=nS_n-\sum_{j