A mathematical preprint reports that branching can make two successive derived window operations on a finite graded poset disagree with one operation at the combined scale. In the paper's terminology, a strong flow is a system in which successive windows compose in that way; the mismatch shows that this property fails in the affected structures.
The work is a theorem-driven analysis of finite graded posets and their height-bounded window constructions, not an empirical study. Its derived convolution stalks are modeled by homotopy colimits, which combine data across a window while retaining its homological structure, and are computed with bar complexes.
The obstruction is local
The clearest obstruction appears in a length-two interval with at least two interior elements. There, the single-window degree-zero local output is one copy of the underlying field, while the iterated unit-window output has r copies, with r >= 2. The mismatch means the strong flow fails, including for face posets of finite regular cell complexes of dimension at least 2.
At the unit scale, the paper calculates the first derived contribution at a minimal element as c(v)-1. A minimal element with two distinct covers is therefore not exact, whereas convolution is exact on chains.
The analysis also rules out recovering the transformed degree-zero cohomology, or H0, from degree zero alone. For any window scale a >= 1, the transformed H0 of a finite connected regular cell complex with at least one one-dimensional cell cannot be determined from the isomorphism class of the original H0 alone. The paper states the same result for finite connected posets of length at least 1.
The test for composition
The route to composition runs through the meet assignment. When it is total, meaning defined for every pair required by the construction, it supplies a canonical comparison between the iterated and single derived convolutions. If the associated meet functor satisfies the paper's k-homological finality condition, a condition on the relevant fibres' homology, that comparison is a quasi-isomorphism for every sheaf, meaning the two constructions agree in homology.
The full if-and-only-if statement is not proved in general. The paper leaves the converse open when a meet defect may be inessential. With essential defect, however, it proves the sharper result: for a total meet functor, the comparison is a quasi-isomorphism for every sheaf exactly when the functor is k-homologically final.
Rooted forests show where the positive result is robust. Their meet assignment is total and homotopy final at every apex and for all nonnegative window scales, so the paper classifies these posets as tame and obtains a strong derived flow across that domain. Totality is also proved for augmented face posets of finite simplicial complexes and for lower semimodular lattices, but totality supplies the comparison; finality is what makes it a quasi-isomorphism.
A distance with a ceiling
The paper also studies the induced interleaving pseudometric, an abstract distance between tame posets. Under its stated tameness, contractibility and saturation assumptions, values above the saturation threshold are never finite; distance zero means the objects are isomorphic, and distance at most the threshold is equivalent to having isomorphic derived colimits, the combined homological objects produced from the poset.
What remains unresolved
The results do not classify all finite graded posets. In particular, the general equivalence between successful composition and k-homological finality remains a conjecture where defects may be inessential, and the paper leaves open whether such defects can occur.
Paper data and sources
Original title: Meet obstructions and saturation for the constant window convolution on graded posets
Authors: Shinobu Yokoyama
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text