SYMMETRY SURVEY · v0.1 DRAFT · APR 2026
Preprint Lean 4 · Mathlib — symmetry classes — theorems

Verified neural compilation of RoPE and gauge equivalences in transformers.

Transformer weights admit non-trivial symmetries under which the function computed by the network is preserved exactly, or up to a measured error budget. This survey catalogs the ones we have isolated so far — RoPE-commuting rotations, sign/phase gauges, parabolic stabilizers, observable quotients, and probe-induced fake equivalences — each formalised as a Lean theorem, with each fake symmetry exposed by refining the probe family.

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Theorems proven
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Symmetry classes
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Deferred
FIG 1 · Taxonomy of 12 symmetry classes · EDGES = subsumption / composition ■ gauge-free   ■ training-invariant   ■ probe-fake
CATALOG

Symmetry catalog

One card per discovered symmetry. Each links to its detail page with the full invariant, derivation, and Lean theorem list.

WIDGETS

Interactive invariants

Three working demos — pure JS, no frameworks. Each illustrates one symmetry class from the catalog. Move a control and the invariant updates in real time.

RoPE rotational equivalence

Rotation $R_\theta$ applied to both $Q$ and $K$ leaves attention logits unchanged. The affected frequency bands depend on $\theta$.

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Channel dormancy

Zeroing rows of $W$ collapses the downstream image. The invariant block — the submatrix untouched by training — is highlighted.

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Sign-phase gauge

Conjugating a hidden layer by $D = \mathrm{diag}(\pm 1)$ yields a gauge-equivalent network. Output is identical up to floating-point noise.

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LEAN STATUS

Proof status

Per-symmetry counts, auto-populated from _data/lean_status.json · last build:

Symmetry Proven Deferred Axiomatic Stub Total Distribution
ARTIFACTS

Paper & code

Manuscript, source, and formal proofs. Cite as below.

pending PDF Manuscript PDF build pending — latexmk not yet run
pending arXiv Preprint submission pending — awaiting PDF
Repository github.com / d3banjan / symmetry-survey-paper Lean 4 · Mathlib · MIT Citation BibTeX entry below · one-click copy 
@article{basu2026symmetry,
  title   = {Symmetry Survey for Verified Neural Compilation},
  author  = {Basu, Debanjan},
  year    = {2026},
  note    = {Lean 4 companion at github.com/d3banjan/symmetry-survey-paper}
}