Pith Number

pith:MGCGHDBA

pith:2026:MGCGHDBASOMB76VDZFX3E555T6

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Uncovering Symmetry Transfer in Large Language Models via Layer-Peeled Optimization

Hangfeng He, Weijie Su, Zhehang Du

Symmetries in next-token targets transfer exactly to circulant logit matrices and equiangular structures in LLM weights and embeddings.

arxiv:2605.12756 v1 · 2026-05-12 · math.OC · cs.AI · stat.ML

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Claims

C1strongest claim

we prove that when the target tokens exhibit a cyclic-shift symmetry (such as the seven days of the week or the twelve months of the year), the optimal logit matrix is exactly circulant, and the Gram matrices of both the output projections and the context embeddings form circulant geometries as well.

C2weakest assumption

The constrained layer-peeled optimization program serves as a mathematically tractable surrogate for LLMs by treating the output projection matrix and last-layer context embeddings as optimization variables.

C3one line summary

Symmetries in next-token prediction targets induce corresponding geometric symmetries such as circulant matrices and equiangular tight frames in the optimal weights and embeddings of a layer-peeled LLM surrogate model.

References

98 extracted · 98 resolved · 5 Pith anchors

[1] Expositiones Mathematicae , volume = 2021

[2] and Johnson, Charles R 2012

[3] 42; Springer: New York 1977 · doi:10.1007/978-1-4684-9458-7

[4] Vaswani, Ashish and Shazeer, Noam and Parmar, Niki and Uszkoreit, Jakob and Jones, Llion and Gomez, Aidan N. and Kaiser,. Attention is All you Need , booktitle =. 2017 , publisher = 2017

[5] Papyan, Vardan and Han, X. Y. and Donoho, David L. , title =. Proceedings of the National Academy of Sciences , volume =. 2020 , doi = 2020

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Cited by

1 paper in Pith

Symmetry-Compatible Principle for Optimizer Design: Embeddings, LM Heads, SwiGLU MLPs, and MoE Routers

Receipt and verification

First computed	2026-05-18T03:09:48.563146Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

6184638c2093981ffaa3c96fb277bd9fa8d8b3ec3447d267c6bc9a7456bdcca3

Aliases

arxiv: 2605.12756 · arxiv_version: 2605.12756v1 · doi: 10.48550/arxiv.2605.12756 · pith_short_12: MGCGHDBASOMB · pith_short_16: MGCGHDBASOMB76VD · pith_short_8: MGCGHDBA

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/MGCGHDBASOMB76VDZFX3E555T6 \
  | 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: 6184638c2093981ffaa3c96fb277bd9fa8d8b3ec3447d267c6bc9a7456bdcca3

Canonical record JSON

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    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.OC",
    "submitted_at": "2026-05-12T21:10:34Z",
    "title_canon_sha256": "b7d1cfa72d9b06fe1aff385abe82f9f0fc3ff0998ae7e143fa2924e8d3da9dcd"
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