Balanced start pairs in Double Fischer Random Chess: a computational search with multi-engine verification

In Double Fischer Random Chess (DFRC) White and Black start from independently chosen Chess960 back ranks, so a game is defined by an ordered pair of setups, 921,600 in all. We searched 460,800 of them — one representative of each pair and its mirror image (a mirror image is a different position, because Chess960 castling ends on fixed files) — with Stockfish 17.1 in a staged search whose windows were calibrated on the symmetric Chess960 starts as a control, took the candidates through a budget ladder of 400 M, 800 M, 1.6 G and 4 G nodes (a subset to 8 G and 16 G), and re-evaluated every candidate with two engines of a different construction, Berserk 14 and Lc0 0.32.1. The admission rule is a corridor of |cp| ≤ 5 at every evaluated budget with no consistently one-sided reading, confirmed by both other engines below their own median on symmetric positions. 436 pairs satisfy it at version 1.2. The verified pairs behave differently from the best symmetric starts: as the budget grows the symmetric starts drift towards White only, while pairs that leave zero move both ways; swapping the two setups of a balanced pair gives White a median edge of +76 cp; the mirror twins of the balanced pairs are not balanced. Self-play over the whole set (4,360 games at 500,000 nodes per move) gives White 49.3 %. The record contains the paper in English and Russian (identical content) and the published set of 436 pairs with every evaluation (JSON). Every number in the text is computed by a script from the published data. Original publication: https://ch960.com/asym/paper (21 October 2026); Russian: https://ch960.com/ru/asym/paper. Revision history: Appendix C of the paper.