Reachability of maximal zeros by successive splitting

Determine which maximal zeros of the ordered \(\operatorname{Tr}(\phi^3)\) amplitude can be obtained from ordinary hidden zeros by successive exact splitting constructions, and establish whether every maximal zero defined by the fixed \(P/U/I/Q\) alphabet is obtainable in this way; additionally, determine whether a splitting adapted to a new zero can expose further zeros not revealed by the original exact splitting formulas.

Background

The paper develops exact splitting formulas that express ordered Tr⁡(ϕ3)\operatorname{Tr}(\phi^3) amplitudes, nonlinear sigma model amplitudes, and mapped cosmological tree wavefunctions in terms of lower-point factors multiplied by linear numerators associated with hidden-zero conditions. These formulas are then used to construct five all-multiplicity families of new zeros, as well as additional non-blob zeros on small branched trees.

The authors leave unresolved the relationship between the full space of maximal zeros and those generated recursively from ordinary hidden zeros. They also ask whether adapting the splitting construction to a newly discovered zero can produce further zero loci beyond those exposed by the original splitting formulas.

References

An open question is which zeros can be reached from the ordinary ones by successive splitting constructions. Within a fixed alphabet, can every maximal zero defined by that alphabet be obtained in this way? Can a splitting adapted to a new zero reveal further zeros that the original exact splitting formulas do not expose?

— From Hidden Zeros to Exact Splittings and New Zeros  (2610.01775 - Li et al., 1 Oct 2026) in Section Conclusions

We have not yet established an all-multiplicity family of regular maximal non-blob zeros on branched trees. The examples below give new zeros for three small trees. We allow linear relations among their channels that make exact splitting terms vanish separately or cancel in groups. Extending these examples to arbitrary tree size remains an open question.

— From Hidden Zeros to Exact Splittings and New Zeros  (2610.01775 - Li et al., 1 Oct 2026) in Section 5.3, subsection "New non-blob zeros in cosmological wavefunctions"

We leave the origin and more general structure of these zeros for future work.

— From Hidden Zeros to Exact Splittings and New Zeros  (2610.01775 - Li et al., 1 Oct 2026) in Section Conclusions

It remains to determine which of these zeros can be obtained recursively from ordinary zeros and how the finite catalogue extends to families at arbitrary multiplicity.

— From Hidden Zeros to Exact Splittings and New Zeros  (2610.01775 - Li et al., 1 Oct 2026) in Section 5.4, subsection "An alphabet for zero conditions"