Second-order curvature surrogates — natural gradient, K-FAC, Gauss–Newton, and Adam's diagonal empirical-Fisher proxy — are widely deployed on tasks with cyclic categorical structure: modular arithmetic, calendar / clock heads, periodic phase prediction. We prove that at the maximum-entropy point \ (p_*\) of such a head, all of them share an exact structural blind spot: a third-order coupling carried by the Amari–Chentsov cubic that the quadratic curvature class cannot represent. The proof rests on two finite identities that hold exactly at \ (p_*\) on a categorical model with cyclic ring index \ (R = Z/nZ\): the Fisher information form \ (g_*\) is diagonal in the additive-character basis, hence block-diagonal across the conductor-packet decomposition of the tangent space; and the Amari–Chentsov cubic form \ (T_*\) couples distinct conductor packets exactly under the integer selection rule \ (k + + m 0 n\). The cross-packet cubic coupling is therefore outside the representational capacity of \ (g_*\), and outside the surrogates above plus the neural-tangent-kernel Gram by §2. 3's four-tier lift; the same identity lifts at \ (p_*\) to the head-side input of linear gradient-based edge-attribution scores in mechanistic circuit discovery (EAP-family methods). We verify the certificate computationally for \ (n \6, 8, 12, 18, 30\\) in exact integer arithmetic. From the certificate we derive a retraining-free diagnostic \ (_\) (an L1 spectral-mass ratio on \ (p_*\) -anchored selection-rule triples) and pre-specify a single experiment with a built-in negative control. A synthetic-MLP demo (§6. 5) separates ring task from structureless control in the pre-grokking phase; a Pythia checkpoint sweep on a calendar-months head (§6. 6) identifies the diagnostic's off-trajectory boundary. Three head-only cubic-aware interventions on \ (n = 30\) all return Branch B (§8. 6): on this evidence the certificate's role is measurement, not steering — and a replicator (\ (\) -flow) analysis (Remark 5. 9) shows this null is structural, since the head's intrinsic cross-packet content is flow-invariant and the residual off-centroid signal is therefore upstream (Conjecture 5. 8\ ('\) ). Open follow-ups — \ (\) -sweep, parameter-side K-FAC variant, longer-budget rings, trajectory instrumentation — are listed in §8. 6.
Leonardo Murillo Montero (Mon,) studied this question.