Mathematical successors within the VOA–net programme

The comparison agenda is complete as a separation and finite certificate. The following mathematical programmes are not solved by it.

1. Beyond strong rationality. A locality theorem for simple unitary VOAs lacking the finite $C_2$ and semisimplicity hypotheses needs a replacement for the relevant generating, energy, and representation arguments. This companion does not settle the remaining CKLW conjectural generality CKLW.

2. Logarithmic and non-semisimple settings. The supplied paper uses ordinary nonlogarithmic intertwiners and diagonalizable conformal energy. A logarithmic theory requires explicitly changed module, analytic, and categorical hypotheses. Nontrivial Jordan blocks cannot be imported unchanged into a positive self-adjoint $L_0$ realization.

3. The converse direction. Henriques–Tener provides an independently stated net-to-VOA construction and representation reconstruction HenriquesTener. Its hypotheses and conclusions are not proved here or treated as a verification of Family 280.

4. Full two-dimensional theories. A chiral circle net is not already a constructed non-chiral two-dimensional theory with a specified left-right coupling. Such a successor must state its new algebraic and analytic data.

5. Boundary conditions, defects, and orbifolds. These have established categorical frameworks, including fusion of defects BartelsDouglasHenriques. A new application must specify the defect, extension, positivity, and locality conditions. The flip-fixed construction reported in the source does not verify every such application.

6. Explicit examples and indices. Gui's published comparison work covers named affine, lattice, parafermion, ADE $W$-algebra, tensor-product, and coset classes under its stated conditions GuiUnbounded. New index calculations require independently justified examples, not merely a declared $S$-matrix.

Each programme is a mathematical extension problem. None becomes a metric, Page curve, or gravitational endpoint merely by being placed after the correspondence record.