The printed hypotheses

A complex VOA $V$, with vacuum $\Omega$ and conformal vector $\nu$, is of CFT type when

Set

The supplied convention calls $V$ strongly rational when it is of CFT type, self-contragredient, rational, and $\dim V/C_2(V)<\infty$. Unitarity includes a normalized positive invariant Hermitian form and its PCT involution.

For a homogeneous vector $a$ of weight $d_a$, polynomial energy bounds are inequalities

The interval algebra is generated by closed smeared fields through their bounded graph or polar data:

Strong locality is the additional assertion

These analytic notions are those of the CKLW framework CKLW. A formal field commutator is not this inclusion.

The supplied document OpenAI280 claims that every simple unitary strongly rational complex VOA $V$ satisfies:

1. polynomial energy bounds and CKLW strong locality;

2. unitarizability of every simple grading-restricted module and positivity of the canonical fusion forms;

3. strong integration of every unitary grading-restricted finite-length module, including equality of the actual closed fields;

4. complete rationality of $A_V$;

5. a braided unitary tensor equivalence

   $$\mathrm{Rep}(V)\simeq {\mathrm{Rep}}^u(V)\stackrel{{F^V}_{\mathrm{CWX}}}{\to }{\mathrm{Rep}}^f(A_V).$$

   The net-side operations are Connes fusion and DHR braiding.

This is a statement of what the supplied source claims. It is not a theorem proved in this companion.

Complete rationality includes irreducibility, the split property, strong additivity, and finite two-interval index. For $E=I_1\cup I_3$,

The index and modular-sector theory have an established operator-algebraic basis KLM.

Under the same base hypotheses and the source's normalization conventions, the supplied preprint claims a bijection, up to isomorphism over the embedded base, between:

1. normalized CFT-type unitary VOA extensions with the same conformal vector and grading-restricted finite-length underlying $V$-module;

2. normalized irreducible finite-index local conformal-net extensions of $A_V$.

It identifies the corresponding commutative algebra objects and Q-systems, and claims complete rationality and the representation comparison for the extended theory OpenAI280. Gui's extension theorems are the identified external mathematical framework GuiExtensions,GuiQSystems. The general application asserted by the supplied source is not verified here.