Pith Number

pith:FBM372NA

pith:2026:FBM372NADXDRALWOQEOMAXRNEF

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Dynamics of the Transformer Residual Stream: Coupling Spectral Geometry to Network Topology

Grigori Guitchounts, Jesseba Fernando

Training installs a monotonic spectral gradient in LLMs from non-normal early layers to near-symmetric late layers, creating a low-rank bottleneck for perturbations.

arxiv:2605.14258 v1 · 2026-05-14 · cs.LG · cs.AI

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Claims

C1strongest claim

training installs a monotonic spectral gradient through depth -- from non-normal, rotation-dominated early layers to near-symmetric late layers -- together with a cumulative low-rank bottleneck that funnels perturbations into a small fraction of the residual stream's effective dimensions. ... the topological positioning of graph communities predicts whether the Jacobian amplifies or suppresses them, with the sign of the coupling determined by the local operator type, a relationship absent at initialization.

C2weakest assumption

That the local linearization given by the Jacobian at each layer remains a faithful description of perturbation propagation even though the actual layer update is nonlinear, and that the chosen graph-community detection procedure yields communities whose functional role is independent of the Jacobian analysis itself.

C3one line summary

Training installs a depth-dependent spectral gradient and low-rank bottleneck in LLM residual streams whose amplification or suppression of graph communities is predicted by local operator type.

References

36 extracted · 36 resolved · 9 Pith anchors

[1] Dubey, Abhimanyu and Jauhri, Abhinav and Pandey, Abhinav and Kadian, Abhishek and Al-Dahle, Ahmad and Letman, Aiesha and Mathur, Akhil and Schelten, Alan and Yang, Amy and Fan, Angela and others , jou 2024

[2] arXiv preprint arXiv:2512.13961 , year = · arXiv:2512.13961

[3] International Conference on Learning Representations (ICLR) , year =

[4] and Waltman, Ludo and van Eck, Nees Jan , journal = 2019

[5] Advances in Neural Information Processing Systems (NeurIPS) , volume =

Receipt and verification

First computed	2026-05-17T23:39:10.508262Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

2859bfe9a01dc7102ece811cc05e2d21771e876b088515806634f0e2f559c134

Aliases

arxiv: 2605.14258 · arxiv_version: 2605.14258v1 · doi: 10.48550/arxiv.2605.14258 · pith_short_12: FBM372NADXDR · pith_short_16: FBM372NADXDRALWO · pith_short_8: FBM372NA

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/FBM372NADXDRALWOQEOMAXRNEF \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 2859bfe9a01dc7102ece811cc05e2d21771e876b088515806634f0e2f559c134

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "3c82a4cd4a13228faa8e3f8f0452df549492de1680669473df93427cd848dcde",
    "cross_cats_sorted": [
      "cs.AI"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "cs.LG",
    "submitted_at": "2026-05-14T01:57:47Z",
    "title_canon_sha256": "33c4e39637b4370251316a412bb40e8ff8a91afb48e987994e2aaed1d6ea0104"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.14258",
    "kind": "arxiv",
    "version": 1
  }
}