Correction residues in the plus subspace

Establish that, for every finite homogeneous strong generating set S of a vertex operator algebra V, every level n, every word length N≥2, every tuple of generators and positive vacuum-descendant indices, every m≥k≥0, every z∈V, and every 0≤t≤N−2, the correction residue defined by the termwise one-sided truncated-iterate expansion lies in the global plus subspace O_n^∘(N,S,V)^+.

Background

The paper introduces one-sided iterated expansions of fields associated with vacuum words built from a finite homogeneous strong generating set S. These expansions consist of a final product of truncated fields together with correction terms involving singular iterates and explicit diagonal kernels.

The conjecture asserts a termwise strengthening: each correction residue, rather than merely their total sum, belongs to the more restrictive plus subspace O_n∘(N,S,V)+. Proving this would support finite-length generation and the proposed reductions of higher-level Zhu circle relations.

References

For words in S we state the following termwise refinement as a conjecture.

— On generators and relations for $O_n^\circ(V)$ in higher-level Zhu algebras, Part I  (2610.19501 - Werf, 14 Oct 2026) in Conjecture 4.8, Section 4.2, “Finite-length circle subspaces and conjectures”

The proposed word-length bound is N=n+1, under the auxiliary-degree condition eq:auxiliary-degree-drop. The precise statement is Conjecture~\ref{conj:finite-length-zhu-relation}.

eq:auxiliary-degree-drop:

∑μ=1ℓρd(t(ρ,μ))≤d(a)+d(b)−1.\sum_{\mu=1}^{\ell_\rho}d(t^{(\rho,\mu)})\le d(a)+d(b)-1.

— On generators and relations for $O_n^\circ(V)$ in higher-level Zhu algebras, Part I  (2610.19501 - Werf, 14 Oct 2026) in Conjecture 4.10, Section 4.2, “Finite-length circle subspaces and conjectures”