The Conductor Blind Spot of the Quadratic Curvature Class on Ring-Structured Categorical Heads

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 $R = \mathbb{Z}/n\mathbb{Z}$: the Fisher information form $g_{p_*}$ 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 $T_{p_*}$ couples distinct conductor packets exactly under the integer selection rule $k + \ell + m \equiv 0 \pmod n$. The cross-packet cubic coupling is therefore outside the representational capacity of $g_{p_*}$, 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 $n \in \{6, 8, 12, 18, 30\}$ in exact integer arithmetic. From the certificate we derive a retraining-free diagnostic $\rho_\times$ (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*. Open follow-ups — $\alpha$-sweep, parameter-side K-FAC variant, longer-budget rings, trajectory instrumentation — are listed in §8.6.