Mathematical successor questions within the source programme

The comparison stages are complete as conditional local results and separations. The following mathematical extensions are not proved.

1. Local-to-global implementation. Construct the required generators or automorphisms, prove their integration, identify the appropriate conformal covering group, and establish covariance on the whole bounded net or its specified extension. The source explicitly leaves this separate.

2. Infinite scaling support or accumulation. Dropping finite scaling support or local finiteness requires new convergence and domain control. Finite field-label expansions do not justify an unrestricted infinite expansion.

3. Vacuum uniqueness. A conclusion about standard cone algebras does not establish a unique Poincaré- and scale-invariant vector. Any uniqueness theorem requires its own hypotheses.

4. Gauge-theory enlargement. An application beyond the chosen gauge-invariant local algebra must identify physical fields, currents, constraints, domains, and observable-net realizations. Gauge-dependent formal potentials cannot replace physical membership.

5. {Comparison with dilaton and $a$-theorem arguments.} Such a comparison should explicitly identify the state, scattering or analyticity assumptions, anomaly conventions, and improvement freedom. The papers KomargodskiSchwimmer,LPR,DKST,DFKLP are external approaches, not automatic premises of the reported modular proof.

6. Comparison with Family 280. Chiral circle nets and four-dimensional bounded nets have different geometric and representation-theoretic interfaces. A comparison may be useful, but neither source statement derives the other merely by sharing the word “conformal”.

No item is completed by naming it. None becomes a Page curve, metric derivation, or gravitational endpoint without additional identified constructions.