Continued-Fraction Bounds for Prime Gaps: Tail Maps, Uniformity, and Leakage-Free Baselines

This preprint develops an elementary continued-fraction (CF) framework for consecutive primes and their gap. We show that the CF view gives exact, lossless bounds for the gap and a deterministic “tail map” that depends only on the gap and the residue class of the smaller prime. This separates two roles cleanly: CF is used to formalize and interpret geometry after a pair is known, while all prediction baselines are designed to avoid any information about the next prime (“leakage-free”). What’s in the paper Exact CF bounds and coding. The first CF coefficient is exactly how many times the gap fits into the smaller prime; the remaining CF “tail” is determined purely by the gap and residue class. Tail catalogue and conditional uniformity. For each fixed even gap, admissible residue classes map bijectively to finite CF tails. Under standard assumptions (Dirichlet equidistribution and the classical Hardy–Littlewood pair heuristic), tails occur with uniform frequency across admissible classes; we confirm this with chi-square tests and Wilson intervals, both globally and within moving prime windows. Leakage-free baselines. We provide a sieve-style gate that filters twin-prime candidates using only the current prime and small-prime congruences, and a principled ranking score proportional to the twin density. Log-axis filter (“H-filter”). On a logarithmic axis we calibrate a bandwidth that matches typical prime separations, so unusually small gaps reinforce energy without ever querying the next prime. Companion work Insights originated from the companion paper Completed Zeta via Operator Blocks (DOI: 10.5281/zenodo.17435807).