# Capture-Quiet Decomposition: A Verification Theorem for Chess Endgame Tablebases

arXiv: https://arxiv.org/abs/2604.07907 · HTML: https://mlsanity.com/paper/2604.07907/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 1 of its 12 references that are in the map.
- First posted on arXiv 9 Apr 2026.

## Builds on (most cited first, 1 of 1)

- [Set-Based Retrograde Analysis: Precomputing the Solution to 24-card Bridge Double Dummy Deals](https://mlsanity.com/paper/2411.09089/) (2024)

## Abstract (arXiv)

We present the Capture-Quiet Decomposition (CQD), a structural theorem for verifying Win-Draw-Loss (WDL) labelings of chess endgame tablebases. The theorem decomposes every legal position into exactly one of three categories -- terminal, capture, or quiet -- and shows that a WDL labeling is correct if and only if: (1) terminal positions are labeled correctly, (2) capture positions are consistent with verified sub-models of smaller piece count, and (3) quiet positions satisfy retrograde consistency within the same endgame. The key insight is that capture positions anchor the labeling to externally verified sub-models, breaking the circularity that allows trivial fixpoints (such as the all-draw labeling) to satisfy self-consistency alone. We validate CQD exhaustively on all 35 three- and four-piece endgames (42 million positions), all 110 five-piece endgames, and all 372 six-piece endgames -- 517 endgames in total -- with the decomposed verifier producing identical violation counts to a full retrograde baseline in every case.