Exact target

A canonical diagonal condensation

Let $\mathcal T$ be a unitary modular tensor category and let$\mathcal B\subset\mathcal T$be a full braided fusion subcategory. In$\mathcal T\boxtimes Z(\mathcal B)$there is a canonical connected étale algebra

$$A_{\mathcal B}=\bigoplus_{x\in\operatorname{Irr}(\mathcal B)} x\boxtimes i_-(x^*).$$

$i_-(z)=(z,c_{-,z}^{-1})$is the negative central embedding into the Drinfeld center.

Interpreting gauging as passage to local modules sharpens Cheng's proposed presentation to the identity treated in the manuscript:

$$(\mathcal T\boxtimes Z(\mathcal B))^0_{A_{\mathcal B}} \simeq_{\mathrm{br}}\mathcal T.$$

The quotient convention is itself part of the problem. If the physical operation requires generalized-orbifold data rather than local modules, the correct outcome is a repaired statement—not a proof of a silently altered one.

Background to the problem

The missing case is genuinely premodular

The identity is known when $\mathcal B$is modular, symmetric, or Abelian. The regime targeted by the candidate proof is non-Abelian with nontrivial Müger center: exactly where neither factorization nor ordinary group gauging removes the transparent sector.

A timely structural tool is available. Yadav and Shimizu identify local modules of the adjoint algebra with the Müger center and prove a relative-center factorization. This work appeared one day before Cheng's preprint and may supply the missing bridge. Greenough's relative centers provide the universal language; condensation inversion shows that an inverse exists after generalized-orbifold choices, but not that this particular condensation is canonically the identity.

What is already forced

Dimension and Witt class are controls, not proofs

The standard dimension formula gives

$$\operatorname{FPdim} (\mathcal T\boxtimes Z(\mathcal B))^0_{A_{\mathcal B}} =\operatorname{FPdim}(\mathcal T).$$

The two sides also have the same Witt class. These are necessary shadows of the desired equivalence. They become decisive only after constructing a fully faithful braided functor.

First attack

Build the functor, then use dimension once

1. Fix the quotient as local modules or prove that this is not Cheng's operation.
2. Construct a canonical braided functor from$\mathcal T$ to the local-module category. Its center component must cancel monodromy with$\mathcal B$.
3. Prove full faithfulness through relative-center universality and the transparent-subalgebra theorem.
4. Use equal Frobenius–Perron dimension to conclude equivalence.

If the functor cannot be defined canonically, the obstruction gate asks whether the statement survives only as Morita equivalence, Witt equivalence, or a generalized orbifold. No enumeration of fusion categories is part of the plan.

Selection calibration

One intrinsic object, one exact translation

The target is controlled by the diagonal algebra$A_{\mathcal B}$ and a universal property. This distinguishes it from targets whose apparent structure dissolves into a growing census of presentations and residual configurations. Binary X-rays, finite-copy NPT distillability, and spherical-design polytopes were considered; none offered this combination of a recognized open statement, canonical object, and decisive structural attack.

Research record

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