Dead Directions: Geometric Singular Learning

Singular learning theory characterises the complexity of a deep network through the geometry of its loss singularities. The local learning coefficient (LLC), the standard estimator of Watanabe's real log canonical threshold (RLCT, $λ$), reads this geometry as an integrated Bayesian scalar through SGLD, which needs per-task calibration and $10^4$-$10^6$ forward-backward passes per checkpoint. We introduce Dead-Direction Signatures (DDS), a family of cheap closed-form spectral readings of singular structure: each reads a network's activation matrix or per-sample-gradient Fisher-Gram at a chosen layer, replacing the SGLD posterior chain with spectral linear algebra. The readings rest on a dead-direction framework that predicts a structural correlation between activation- and Fisher-side spectra at any singular minimum, and a rank-multiplicative volume identity that single-eigenvalue monitors cannot produce: the active-volume $\log\det^{+}(G)$ slope counts the dead directions, tracking the rank-deficit $r$ across $r \in \{1,2,3,4\}$ (slope ratios $2.0, 3.1, 4.0$ at $r{=}2,3,4$ against the predicted $2,3,4$), where the smallest eigenvalue is rank-blind. On reduced-rank regression with closed-form $λ$, calibrated LLC recovers $λ$ at $99\%$ mean and the DDS observables rank-track it at the framework-predicted sign; on a non-linear modular-addition transformer DDS separates $d_{\mathrm{model}}$ across eighteen orders of magnitude where calibrated LLC at the protocol budget is rank-flat. Complementary to LLC's integrated posterior reading, DDS gives a directional, layer-local handle on a network's dead directions, read in closed form from its activation and gradient spectra.

We give a descent-free, alignment-free measurement of singular structure on trained networks. At a single frozen checkpoint the read recovers the order $k$ of each dead direction from the directional-Fisher rate, the master invariant from which the per-direction learning coefficient $1/(2k)$ follows exactly, in whatever basis the optimizer left. The same read classifies each direction, separating a genuine singularity, whose order the architecture fixes, from a flat gauge symmetry; the directional-Fisher magnitude settles the cases the order cannot. A pluggable detector supplies the directions for transformer, convolutional, and normalisation layers. The read recovers the architecture-predicted order across constructed cells and trained networks, including a fine-tuned vision transformer whose dead structure is the LayerNorm-kernel gauge and a from-scratch one whose compressed MLP forms a node-death at its activation order. Where the singular structure enumerates, the per-direction orders assemble, through the typed intersection of the loci, into the global coefficient $(λ, m)$ matching the closed form. The method removes the canonical-alignment and descent preconditions of the underlying rate result, turning order-recovery into a deterministic, architecture-general reading. We then map its reach into the Watanabe triple: the order determines the universal singular fluctuation $ν(k)$, though a trained network's realized $ν$ falls below it as the live structure absorbs the dead direction's data fluctuation, and the multiplicity recovers from the dominant structure under a single-locus assumption.

A deep network's loss is invariant to continuous symmetries of its parameters: the logit shift, the ReLU rescaling, the LayerNorm scale, the per-head attention rotation. Adam's per-coordinate preconditioner drifts along each symmetry orbit, which pulls the trajectory off the symmetry quotient where the optimization lives and blurs the singular-learning rate the quotient makes readable. We build DDC, a Dead-Direction Conditioner that lifts a base optimizer into a $G$-equivariant one: it conditions the optimizer's state in the orbit decomposition of a $G$-invariant metric, so the trajectory stays a preconditioned gradient flow on the quotient $\barΘ= Θ/G$. The construction carries four architectural gauges (cross-entropy shift, ReLU and SwiGLU rescaling, LayerNorm and RMSNorm scale, and a per-head $O(d_{\rm head})$ attention rotation matched to RoPE), proves exactly equivariant on an Adam base, and composes with a Muon base through a gauge-equivariant orthogonaliser. Respecting the symmetry changes both the minimum the optimizer reaches and what it leaves measurable there. On a language model trained past the point of fit, DDCAdam resists the over-training collapse AdamW falls into, holding a validation-train loss gap of 0.67 against 5.88, and reads the dead-direction rate in 32 of 65 layer-by-observable cells where AdamW reads it in 7. A vision transformer trained from scratch reaches lower validation loss (1.71 against 2.12) while compressing spare feed-forward capacity a matched AdamW leaves intact. On a Muon base, where the rotation gauge composes exactly, DDCMuon groks ten of eleven seeds at depth 24 that a plain Muon never reaches. Built into the optimizer, a network's gauge symmetry sharpens the minimum it finds and turns that minimum's geometry into something the trajectory can measure.