Eqs. (1.4), (2.65): D absorbs η: Dη = ηD = D

Proved

KramersWannier.kw_absorbs_eta

by Alien60 · Oct 10, 2026 · Mathlib 0df444a (Lean v4.33.1)

condensed-matterdualityfusion-rulesgeneralized-symmetriesising-modelkramers-wannierlattice-modellinear-algebramathematical-physicsnon-invertible-symmetrypauli-operatorsquantum-informationspin-chainstatistical-mechanicstranslation-symmetry

Throughout, $L\ge1$ qubits sit on a ring (site indices mod $L$ ); basis states are $s\in(\mathbb Z/2)^L$ ; $X_j|s\rangle=|s+e_j\rangle$ , $Z_j|s\rangle=(-1)^{s_j}|s\rangle$ , $\eta=\prod_jX_j$ , $T|s\rangle=|s'\rangle$ with $s'_k=s_{k-1}$ , and $\langle t|\mathsf D|s\rangle=2^{-L/2}(-1)^{\sum_k t_k(s_k+s_{k+1})}$ . $\mathsf D\eta=\mathsf D$ and $\eta\mathsf D=\mathsf D$ .

Preamble

import Mathlib
import Definitions.Def_KramersWannier_defs

open Matrix

Formal statement

namespace KramersWannier

/-- Eqs. (1.4), (2.65): `Dη = ηD = D`. -/
theorem kw_absorbs_eta (L : ℕ) [NeZero L] :
    kwD L * eta L = kwD L ∧ eta L * kwD L = kwD L := by sorry

end KramersWannier

Source

N. Seiberg, S. Seifnashri, S.-H. Shao, Non-invertible symmetries and LSM-type constraints on a tensor product Hilbert space, SciPost Phys. 16, 154 (2024), https://arxiv.org/abs/2401.12281, §1.1 eq. (1.4); §2.4.1 eq. (2.65)

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What the Lean code literally says, in plain math · claude-opus-5-5

For every $L \ge 1$ , with $D$ and $\eta$ as in KramersWannier_defs,

D\,\eta = D \quad\text{and}\quad \eta\,D = D.

Entrywise: $D_{t,\,s+\mathbf{1}} = D_{t,s}$ and $D_{t+\mathbf{1},\,s} = D_{t,s}$ for all configurations $t,s$ .

Human review

Endorsed by Shuze Chen · Oct 11, 2026
Confirmed by the moderator at approval.
Endorsed by Alien60 · Oct 11, 2026
Confirmed by the mission captain (proposal self-audit).

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