FPRD hunt packet: the anyon-proliferation presentation

Mode: HUNT
Selected: 6 September 2026
Status: recognized open categorical identity; no result claimed here

Exact target

Let $\mathcal T$ be a unitary modular tensor category and let $\mathcal B\subset\mathcal T$ be a full braided fusion subcategory. The braiding gives embeddings

$$\mathcal B\hookrightarrow\mathcal T, \qquad \overline{\mathcal B}\hookrightarrow Z(\mathcal B).$$

Inside $\mathcal T\boxtimes Z(\mathcal B)$, consider the canonical connected étale algebra

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

Meng Cheng proposes the presentation

$$\mathcal T\simeq \frac{\mathcal T\boxtimes Z(\mathcal B)}{A_{\mathcal B}}.$$

The first mathematical task is to make the quotient convention exact. Under the standard anyon-condensation convention, the sharpened question is:

$$\boxed{ (\mathcal T\boxtimes Z(\mathcal B))^0_{A_{\mathcal B}} \stackrel{?}{\simeq}_{\mathrm{br}}\mathcal T .}$$

Here the left side is the braided category of local modules. If Cheng’s physical quotient denotes a generalized orbifold rather than local modules, the first attack must say so and formulate the correct comparison instead of quietly changing the statement.

Why this is a good FPRD target

The problem is an identity between two presentations of the same proposed topological order. It is organized by one intrinsic object, the diagonal algebra $A_{\mathcal B}$. It does not ask us to enumerate modular categories, fusion rules, condensates, or residual anyons.

The identity is also physically consequential. It is the step that “peels off” the chosen non-Abelian anyons into a symmetry-TFT factor, where dynamical matter can be coupled to them. A proof would justify that presentation for arbitrary braided fusion subcategories; a counterexample would identify exactly what structure the proposed quotient forgets.

This passes the origami calibration: seek the canonical fold and the exact translation it induces. The dangerous alternative would be to test families of low-rank categories and accumulate successful examples. That is explicitly not the attack.

Known frontier

Cheng states the identity and says that a completely general mathematical proof is not yet known. The paper establishes or cites three broad regimes:

1. $\mathcal B$ modular, using Müger factorization;
2. $\mathcal B$ symmetric, via modular-extension technology;
3. $\mathcal B$ Abelian, by direct quotient calculations.

The unbridged regime is therefore a genuinely premodular, non-Abelian subcategory with nontrivial Müger center.

Several established results nearly meet the target but do not state it:

• Davydov–Müger–Nikshych–Ostrik construct and classify diagonal connected étale algebras and control their local-module categories and Witt classes.
• Greenough develops relative tensor products and relative centers, giving a universal language for the quotient.
• Mulevičius proves that a condensation can be inverted by generalized orbifold data. This gives an existence result after extra choices; it does not identify Cheng’s particular $A_{\mathcal B}$-quotient with $\mathcal T$.
• Yadav–Shimizu, posted one day before Cheng’s preprint, identify the local modules of the adjoint algebra in $\overline{\mathcal B}\boxtimes\mathcal B$ with the Müger center and prove a relative-center factorization. Their transparent-subalgebra theorem is a plausible missing bridge in precisely the degenerate case.

Exact controls available before attack

The numerical shadow is forced. Since

$$\operatorname{FPdim}Z(\mathcal B)=\operatorname{FPdim}(\mathcal B)^2, \qquad \operatorname{FPdim}A_{\mathcal B}=\operatorname{FPdim}(\mathcal B),$$

the standard local-module dimension formula gives

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

The two categories also have the same Witt class. Neither equality proves a braided equivalence. They are controls: once a fully faithful braided functor is constructed, equal Frobenius–Perron dimension would finish the proof.

First ATTACK

The attack has one constructive gate and one obstruction gate.

Constructive gate

1. Fix the mathematical meaning of Cheng’s quotient.

2. Express condensation by $A_{\mathcal B}$ as a relative tensor product or relative-center construction.

3. Use the canonical central functor $\mathcal B\to Z(\mathcal B)$ and the inclusion $\mathcal B\to\mathcal T$ to construct a braided functor

   $$\Phi:\mathcal T\longrightarrow (\mathcal T\boxtimes Z(\mathcal B))^0_{A_{\mathcal B}}.$$

   The $Z(\mathcal B)$ component must cancel the monodromy of a $\mathcal T$-object with $\mathcal B$; merely taking a free module is not enough.

4. Prove full faithfulness using the double-centralizer theorem together with Yadav–Shimizu’s identification of the transparent remainder.

5. Invoke the dimension equality only at the final step.

The decisive diagram is the localness square for $\Phi(X)$ against $A_{\mathcal B}$. If it commutes by the two opposite half-braidings, the problem may collapse to a universal property rather than a classification.

Obstruction gate

If no canonical $\Phi$ exists, determine the exact level at which the identity survives:

• braided equivalence;
• Morita equivalence;
• Witt equivalence; or
• equivalence only after generalized-orbifold data.

A negative result must exhibit a structural obstruction carried by the Müger center, such as a nontrivial transparent algebra or mismatch of a canonical central functor. It should not be a computer search over fusion rings.

Stopping rule

Stop this route if the quotient is not canonically local modules and the available theorems imply only Witt equivalence with no distinguished lift. In that case the useful result is the corrected categorical statement and the precise missing datum. Do not replace that datum by an example census.

Sources and credit

• Meng Cheng, Note on proliferation transitions of non-Abelian anyons, arXiv:2608.20604v1, especially equations (II.1)–(II.3) and the explicit statement that a general proof is missing: https://arxiv.org/html/2608.20604v1
• Harshit Yadav and Kenichi Shimizu, Transparent Subalgebras and Local Module Categories, arXiv:2608.19153v1, especially Theorems 4.7 and 4.8: https://arxiv.org/html/2608.19153v1
• Alexei Davydov, Michael Müger, Dmitri Nikshych, and Victor Ostrik, The Witt group of non-degenerate braided fusion categories, arXiv:1009.2117, and the related structural paper arXiv:1109.5558: https://arxiv.org/abs/1009.2117
• Justin Greenough, Relative centers and tensor products of tensor and braided fusion categories, arXiv:1102.3411v4: https://arxiv.org/html/1102.3411v4
• Vincentas Mulevičius, Condensation inversion and Witt equivalence via generalised orbifolds, arXiv:2206.02611: https://arxiv.org/abs/2206.02611
• Liang Kong, Anyon condensation and tensor categories, arXiv:1307.8244: https://arxiv.org/abs/1307.8244

The target and its physical interpretation belong to Cheng. The proposed use of the Yadav–Shimizu relative-center machinery is an FPRD attack direction, not a claimed theorem.

Other finalists considered

• Binary X-rays of permutations: a clean conjecture, but the available reduction lands in integer realization of triangular contingency tables, where saturation fails and the general decision problem is NP-complete. That route invites exactly the residual-space expansion the Lab is avoiding.
• Finite-copy NPT distillability: physically important, but the immediate frontier is dominated by parameter regions and witness optimization.
• Minkowski polytopes of spherical designs: an elegant exact translation, but the recent source did not expose a comparably sharp unresolved bridge with a decisive first attack.

The anyon presentation was selected because it combines a recognized open statement, a canonical algebra, a universal-property route, and a clean failure certificate.