{"ID":5551821,"CreatedAt":"2026-07-02T01:54:51.863792489Z","UpdatedAt":"2026-07-04T08:17:08.509157724Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2607.00603","arxiv_id":"2607.00603","title":"Measuring Dead Directions: Decomposing and Classifying Singular Structure off Canonical Alignment","abstract":"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.","short_abstract":"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 whateve...","url_abs":"https://arxiv.org/abs/2607.00603","url_pdf":"https://arxiv.org/pdf/2607.00603v1","authors":"[\"Tejas Pradeep Shirodkar\"]","published":"2026-07-01T08:29:36Z","proceeding":"cs.LG","tasks":"[\"cs.LG\"]","methods":"[\"Vision Transformer\",\"Transformer\"]","has_code":false}