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.
Figures & tables
Endgame
Positions
Term%
Capt%
Quiet%
v_total
full_v
KQvK
47,136
0.3
6.0
93.6
0
0
KRvK
51,044
0.1
5.6
94.4
0
0
KPvK
42,538
0.0
13.8
86.2
0
0
KQQvK
1,235,796
4.1
8.9
87.0
0
0
KQRvK
2,631,408
1.8
9.3
88.9
0
0
KQBvK
2,733,056
0.9
9.4
89.7
0
0
Table 1: CQD verification on 20 representative 3- and 4-piece chess endgames (detailed per-category breakdown). Term% , Capt% , Quiet% : fraction of positions in each category. v_total : total violations (decomposed). full_v : violations from full retrograde baseline. All 20 endgames match exactly. The remaining 15 four-piece endgames are verified by the chain builder with identical results.
Endgame
Positions
tgen (s)
tverify (s)
Ratio
KQvK
47,136
0.2
0.8
4.0
KRvK
51,044
0.1
0.8
8.0
KQQvK
1,235,796
6.2
7.5
1.2
KQRvK
2,631,408
16.3
14.5
0.9
KQBvK
2,733,056
12.8
15.5
1.2
KQNvK
2,802,307
12.5
15.7
1.3
Table 2: Wall-clock time (seconds) for retrograde generation vs. CQD verification on one CPU core. t_gen : iterative retrograde BFS until fixpoint. t_verify : CQD single-pass (terminal check; sub-model lookup for captures; local consistency for quiet). Ratio : t_verify/t_gen . At 4 pieces both approaches are comparable; at higher piece counts CQD’s single pass is expected to outperform multi-round BFS.
Pieces
Capture %
Quiet %
Est. Speedup
3
8.5
91.1
1.1×
4
19.0
80.6
1.2×
5
25–30
70–75
1.3 – 1.4×
6
35–45
55–65
1.5 – 2×
12
50–70
30–50
2 – 3×
20
60–80
20–40
3 – 5×
Table 3: Capture position fractions by piece count. 3–5 piece values are measured; 6–20 piece values are estimated from material combinations.