Two Causally Necessary Attention Heads, Two Different Geometries: OV-Circuit Rank Divergence in a Chess-Playing Transformer

We ask whether two components that are each causally necessary for the same behavior must also share similar internal structure. In a chess-playing transformer, two attention heads, L1H1 and L2H7, are each independently necessary for detecting a forced mate in one move: zero-ablating either head collapses mate-in-1 accuracy from a 51.0\% baseline to 2.5\% and 18.0\% respectively. Despite this shared causal role, the two heads organize their OV maps with sharply different spectral structure. We decompose each head's OV circuit by singular value and find that L1H1 concentrates 84.7\% of its squared Frobenius energy in a single singular direction, while L2H7 concentrates only 10.7\%, indicating sharply different OV spectral concentration. Comparing all four heads we examined places L1H1 and L2H7 at opposite ends of the rank spectrum, not near each other, even though only these two are behaviorally load-bearing. We read this as a caution for representational-geometry analysis of trained networks: causal equivalence at the level of behavior does not imply geometric equivalence at the level of mechanism, and a component's rank structure has to be measured, not inferred from its causal importance.