A flow with conditions
An arXiv preprint finds that a window-based operation on finite graded posets can follow a clean composition rule under the right order conditions. The rule says that applying derived convolution at one scale and then another should be equivalent to applying it once at the combined scale. The analysis supplies a sufficient condition for that equivalence, but also shows a form of branching that breaks it. The work is theoretical: its objects are finite posets, sheaves and bounded derived categories, not sampled people or clinical data.
How the convolution is calculated
At a nonnegative scale, ordinary convolution first restricts a sheaf along one projection of the window and then extends it along the other, using a construction called a left Kan extension. The derived construction evaluates each stalk through a homotopy colimit over a finite window-comma poset, calculated by a finite bar complex. Its degree-zero homology is the ordinary convolution. Higher degrees are part of the exactness test because the criterion asks whether higher relative homology vanishes.
The key comparison is made through a meet assignment. When that assignment is total, it gives a map from the iterated derived convolution to the single convolution at the combined scale. If it is also k-homologically final, the map is a quasi-isomorphism for every sheaf, meaning the two constructions agree in the derived sense. The supplied analysis treats this as a sufficient condition; it does not establish the full converse in general.
Branching breaks the rule
The clearest failure appears in a branching interval of length two with at least two interior elements. At the bottom stalk, the single convolution at scale two has degree-zero homology equal to one copy of the coefficient field. Two successive unit convolutions instead have a stalk with one copy for each interior element. Their dimensions are therefore different, so the strong flow law fails. The analysis says the same obstruction reaches face posets of finite regular cell complexes of dimension at least two.
At a minimal apex, the paper gives a sharper unit-window test. If the meet assignment for the two unit windows is total, the comparison works for every sheaf exactly when each interval of length two above that apex contains one, and only one, interior element. This isolates the short branching pattern that decides the comparison in that setting.
Where the construction works
Rooted forests form the clearest positive domain in the analysis. For every apex and every pair of nonnegative window scales, their meet assignments are total and homotopy final. The resulting comparisons are quasi-isomorphisms for every sheaf, and the posets are tame, so the associated interleaving distance can be defined there.
On a tame poset, that distance is an extended pseudometric: finite values occur in nonnegative integer steps, while infinity is also allowed. It places an object no more than the window scale away from its convolution, and applying the same convolution to two objects cannot increase their distance. These are structural bounds from a mathematical construction, not estimates from observed data.
At the saturation threshold, the distance has a further interpretation. Under the stated saturation assumptions, no finite value is available above that threshold. A distance of zero occurs exactly when the two derived objects are isomorphic, while a distance no greater than the threshold occurs exactly when their derived colimits over the poset are equivalent. This ties the metric's finite-scale endpoint to a global colimit.
A result with an open edge
Because the document is a preprint rather than a report of an experiment, its contribution is a set of mathematical criteria and counterexamples. It is arXiv version 2, dated 31 August 2026. The central unresolved issue is how broadly the finality criterion can be reversed: the supplied analysis says the converse is not established in full generality.
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