Definitions and conventions

What the input is, in what gauge results are reported, how they are compared with the exact centers, and what is withheld. Each construction's stages are at the top of its page, under its description, and its checks, tolerances and other reference at the bottom.

Input

Every entry is a multiplicity-free fusion category $\mathcal{C}$ from the AnyonWiki database, given by its rank $r$ , fusion coefficients $N_{ab}^c\in\{0,1\}$ , and $F$ -symbols $F^{abc}_{d;ef}$ in double precision. With --sigfigs N the $F$ -symbols are rounded to $N$ significant figures before anything is computed and the check tolerances follow the resulting noise floor; the dropdown above each table selects the precision shown.

$\mathrm{FR}^{r,m,n}_{i}$: The fusion ring: rank, multiplicity, number of non-self-dual objects, index.
$\mathrm{FC}^{r,m,n}_{i,x,y,z}$: A fusion category on it: $x$ indexes the $F$ -solution, $y$ a braiding on it, $z$ a pivotal structure. Together with the ring this is AnyonWiki's seven-integer code $\mathrm{FC}^{r,m,n}_{i,x,y,z}$ ; AnyonWiki's own data directories name the same object cat_x_y_z. The center depends on $x$ only, so entries differing in $y$ or $z$ have the same center.
URLs: Every category has a page at /center/r_m_n_i/x_y_z, the two bare codes. Identifiers in the tables link to the same object on anyonwiki.org.
$d_a$: Pivotal dimensions, read off $|F^{aa^*a}_{a;00}| = 1/|d_a|$ with signs fixed by $d$ being a character of the fusion ring. For a unitary category these are the Frobenius-Perron dimensions.
Non-unitary entries: 192 of the 989 are Galois conjugates with no unitary gauge (Yang-Lee is the smallest). Some of their $d_a$ are negative. Every categorical formula uses the pivotal $d_a$ , and no construction assumes $F^{-1} = \overline{F}$ .

The three constructions

Three independent computations read the same $F$ -symbols and share no code beyond that. Each page has the same order: the construction explained, its stages, the table of results, then its checks, tolerances and other reference at the bottom, in collapsible sections:

Torus: The Levin-Wen string-net Hamiltonian on a $2\times 2$ honeycomb torus; its ground-state degeneracy is $\operatorname{rank}(\mathcal{Z}(\mathcal{C}))$ . Delivers the rank only.
Tube algebra: The Ocneanu tube algebra, whose center has dimension $\operatorname{rank}(\mathcal{Z}(\mathcal{C}))$ . Delivers rank, Wedderburn blocks and quantum dimensions; carries the pseudocode of its five stages, the eight checks and the full table of numerical tolerances.
Ladder algebra: The weak Hopf algebra $A_{\mathcal{C}}^{\mathcal{C}\boxtimes\mathcal{C}^{\mathrm{rev}}}$ , braided equivalent to $\mathcal{Z}(\mathcal{C})$ . Delivers fusion rules, $F$ - and $R$ -symbols and the modular data.

Gauge

The $F$ - and $R$ -symbols of $\mathcal{Z}(\mathcal{C})$ are reported in the gauge the Clebsch-Gordan maps produce. This is generally not a unitary gauge, even for a unitary $\mathcal{C}$ : the irreducible representations are extracted in a Euclidean-orthonormal basis, not one with $\rho(a^*) = \rho(a)^\dagger$ .

Gauge quantities: Unitarity of the $F$ -blocks; unit modulus of individual $R^{ab}_c$ . Recorded as residuals, never counted as failures.
Gauge invariants: Pentagon, hexagon, monodromies $R^{ab}_cR^{ba}_c$ , twists $\theta_a$ , $S$ -matrix. These are what is checked and compared.
Why $F$ is not rotated to a unitary gauge: The $R$ -matrix is exact in the Clebsch-Gordan gauge and the hexagon holds between the $F$ -symbols and the $R$ -symbols there. Moving the $F$ -symbols alone would break the pair.
Conditioning: The identities the tube checks are gauge-invariant, but the size of the $F$ -entries is not. A Galois conjugate arrives in the unitary gauge pushed through the Galois automorphism, where $\max|F|$ can reach $10^4$ against a gauge-invariant floor of $1/|d_a|$ ; most of the tube's ill-conditioning on those inputs is the gauge, not the category.

Exact centers

The TensorCategories.jl database (Mäurer, arXiv:2406.13438) contains the Drinfeld centers of the AnyonWiki categorifications of rank at most five, computed symbolically over number fields by a categorical construction that shares no method with the algebras here. Their $F$ - and $R$ -symbols are evaluated numerically and compared with the ladder's modular data; the tube contributes its rank. Every verdict on this site links to the matched database entry.

Embedding: The number-field embedding is fixed by $\dim(\mathcal{Z}(\mathcal{C})) = \dim(\mathcal{C})^2$ , not read from the file: the stated embeddings label Galois conjugates inconsistently. A ring whose categorifications fall into several Galois classes (Fib × Fib, Fib × Yang-Lee, Yang-Lee × Yang-Lee on $\mathrm{FR}^{4,1,0}_{5}$ ) has each class tried in turn.
Matching: The database indexes categorifications differently from AnyonWiki, so each computed center is matched against every exact center of the same fusion ring.
Tube: rank agrees: The tube's $\operatorname{rank}(\mathcal{Z}(\mathcal{C}))$ equals the exact rank. All exact centers of a fusion ring have the same rank, so no matching is involved; the badge links to the database entry carrying the same code when there is one.
Ladder: agrees: The ladder's $S$ -matrix and the exact $S$ -matrix coincide entry by entry (to about $10^{-14}$ ) under a relabeling of the simples that preserves the fusion rules, up to complex conjugation (opposite braiding convention) and up to the sign character $\chi$ of the spherical structure, $S\mapsto\chi S\chi$ . Both are convention, not freedom.
Ladder: only rank agrees: The exact center exists and has the same rank, but the ladder stopped after stage 7 for this category (no $R$ -symbols), so there is no $S$ -matrix to compare.
Mismatch: A reported rank or $S$ -matrix matches no exact center of the fusion ring. This would be a bug; it does not currently occur.

Results that are not reported

A result is reported only when every check on it passes. When a check fails, the quantity is not shown, not written to the data files, and not used anywhere else on the site; the failing check is recorded under its name. The cases that occur:

Tube: unreliable: The algebra failed associativity, the unit, or an idempotent check, so no rank is reported. Ten categorifications at full precision; see Unreliable results on the Tube page.
Torus: skipped: Physical basis above 7000 states.
Ladder: stages 1 to 7 only: $\dim(A)$ above the stage-8 size limit; rank, blocks and fusion rules are still computed.
Ladder braiding: not determined: The constructed $\mathcal{R}$ failed quasi-triangularity or the hexagon and the fallback search found more than one modular solution. Has not occurred.