A Canonical Collision–Skip Bijection in a Rounded Hyperbolic Code Sequence

This article proves a new explicit canonical collision–skip bijection for the rounded reciprocal code sequence \[ R_N(m)=RNTE\!(2N^2/m), N≤ m≤ 2N, \] where RNTE denotes round to nearest, ties to even. The height \[ h_j=3N-j-R_N(j) \] forms a nonnegative unimodal Motzkin excursion, whose up-steps and down-steps yield a canonical noncrossing matching between two-code skips and collisions. If a skip at \(s\) omits the output code \(c=R_N(s)-1\), then \[ R_N(c+1)=s, R_N(c)∈\{s,s+1\}, \] and the paired collision has left index \(r=c\) or \(r=c-1\). Consequently, the omitted code reappears as one of the two consecutive inputs in the paired double fiber. The explicit reciprocal construction and the Motzkin matching are proved to coincide. Exact boundary cases reduce to a Pell equation and always give unit transitions. The release contains complete English and Hungarian articles, LaTeX sources, figures, exact-arithmetic generators, verifier programs, tables, numerical data, licensing information, and reproducibility documentation. The full computational baseline covers 11,014 distinct values of \(N\) and 86,911,458 transitions. This work develops the counting result established in Publication 20, Collision Counting in a Rounded Hyperbolic Code Sequence, DOI 10.5281/zenodo.21357521. Publication 21 is a new standalone publication, not a new version of Publication 20. In the targeted literature search conducted for this work, we did not locate an earlier formulation of this specific code-label-preserving empty–double fiber correspondence. This records the scope of the search rather than an exhaustive global priority claim.---