Structural Stability Classes in Chess960

[Dimitris Panagakos]

Everyone knows that Chess960 promotes creativity and true freedom of movement in the opening—something that classical chess, with its deep and rigid opening theory, simply cannot offer.

However, one question remains largely unexplored:
How do modern engines evaluate this freedom of movement?

In my previous work, I analyzed all 960 Fischer random starting positions using Stockfish 17 and Leela Chess Zero (Lc0) at depth 20.
This time, I went much deeper by using two independent NNUE engines: Stockfish 17 and Berserk 13.

First, I evaluated all 960 legal Fischer random starting positions using both engines under identical fixed‑depth conditions (depth 20), with no opening books, no tablebases, and fully deterministic evaluation.

Then I isolated the 533 positions that both Stockfish 17 and Berserk 13 evaluated below +0.33 pawns at depth 20.
These ‘double‑safe’ positions represent the subset of the Chess960 landscape where both engines independently agree that the starting position is near equilibrium.

So I pushed these 533 positions to depth 30 using two independent NNUE engines to see what happens.
The results were far more structured — and far more revealing — than I expected.

The 533 positions were classified into three categories:
• 203 fully‑defended (FD) positions (all pawns are covered by a friendly piece),
• 330 under‑covered (UC) positions (at least one pawn is uncovered),
• 67 CORE67 positions (king on the e‑file), all of which belong to FD.

The key point:

When the 533 positions were evaluated at depth 30, the two engines converged dramatically:

68.1% agree within ±10 cp

85.9% agree within ±15 cp

93.2% agree within ±20 cp

This is a level of cross‑engine agreement that simply does not exist at depth 20!

The deeper the search, the more the evaluation landscape reveals a hidden structural order.

Why CORE67 (Kings on the e file) Is Favored by Both Engines?
The most striking discovery is that the CORE67 positions form a deep‑search stability basin.

At depth 20:

Berserk already preferred CORE67,

Stockfish did not show a strong preference,

But at depth 30:

Both engines independently converge toward higher stability and near‑equilibrium evaluations for CORE67.
Why?

“I think this happens because CORE67 positions share three structural advantages:

1. Centralized king geometry
King on the (e) file creates symmetric defensive coordination and reduces early tactical volatility.

2. Smoother castling transitions
Both short and long castling routes are structurally coherent, reducing engine uncertainty.

3. Higher piece‑coordination consistency
The e‑file king aligns better with typical NNUE king‑safety heuristics, leading to more stable evaluations.

In short:

CORE67 positions are not just “fully defended” — they are structurally self‑stabilizing under deep search.

This is why both engines converge toward them as equilibrium points.

UC Positions Remain the Main Source of Instability even at depth 30: nearly all large disagreements occur in UC positions, asymmetry in pawn coverage amplifies tactical branching, NNUE heuristics diverge more strongly in these structures, UC positions are inherently volatile — and deep search reduces but does not eliminate this volatility.

What This Study Shows?
The results suggest that the Chess960 landscape may contain a large, stable equilibrium region consisting of 533 positions.
Positions with the king on the e‑file (CORE67) form a genuine stability basin recognized by both engines.
UC positions remain the primary source of residual tactical volatility.
Cross‑engine convergence becomes extremely strong at depth 30.
Deep search reveals structural order that shallow search hides.

The results of this study suggest that deep‑search equilibrium in Chess960 is not random but structurally organized.

Dimitris Panagakos
Independent Researcher
FIDE school instructor