No fair split at infinity?

0:00The question

A matroid captures what independence means, in linear algebra or in a graph. Call a set of edges independent if it contains no cycle, or split a set into blocks and call a subset independent if it takes at most one element from each block. The maximal independent sets are the bases, a set is spanning if it contains a basis, and every matroid has a dual, whose bases are the complements of the original bases. Two matroids on the same set can compete. Take the edges of this graph, with one matroid allowing one edge at each left vertex, and the other one edge at each right vertex. A set independent in both is a matching. For finite matroids, Edmonds’ intersection theorem, background that the paper does not cite, guarantees a split: some common independent set divides into two parts, so that what the first part spans in one matroid, together with what the second spans in the other, is everything. Here a matching of 3 edges does it. A second finite fact concerns packing and covering. A packing is a choice of disjoint spanning sets, one per matroid; a covering is a choice of independent sets, one per matroid, that together use every element. In our example, two copies of the forest matroid of this graph have neither, but the ground set splits into a part with a packing and a part with a covering. Infinite matroids got workable axioms in 2013, from Bruhn, Diestel, Kriesell, Pendavingh and Wollan, who demand that independent sets extend to maximal ones inside every subset. Nash-Williams conjectured the intersection property for finitary matroids, where a set is independent whenever all its finite subsets are. Bowler and Carmesin, in a paper published in 2015, conjectured that every family has a packing/covering partition, and showed this is equivalent to the intersection property for every pair of infinite matroids. Joó proved positive cases on countable sets, and asked about two partitional matroids, sums of uniform ones, on a countable set.

2:07The claim

Result 185 is one OpenAI preprint, 20 pages, dated September 24, 2026. It claims, in ordinary set theory with the axiom of choice: there is a countably infinite set with two matroids on it, each equal to its own dual, such that no independent set of the first together with an independent set of the second covers the whole set. The pair has no packing/covering partition, so the packing/covering conjecture is false, with just two matroids. Through an equivalence proved by Bowler and Carmesin, a corollary says the same pair, both partitional, fails intersection, so the unrestricted intersection conjecture is false, and Joó’s question has a negative answer. A further corollary refutes their separate covering and packing conjectures. Along the way, the paper builds a self-dual uniform matroid on a countable set without extra axioms, which it says answers a question Bowler and Geschke left open; their examples assumed Martin’s axiom for countable partial orders, which follows from the continuum hypothesis. Not claimed: anything about Nash-Williams’ original finitary conjecture, which the paper says it does not refute; its matroids are neither finitary nor cofinitary. Nothing computable either: the construction rests on choice, through an ultrafilter and a transfinite recursion as long as the continuum, so it gives no way to list a basis. The paper uses choice openly; what it stresses is that it needs no further axioms, such as the continuum hypothesis. And for the separate covering and packing conjectures, it does not say which families break them. Finite matroids cannot do this: in our check, all 41 matroids on 6 labelled elements that equal their own duals share a common basis with each other, so every such pair covers.

4:02How the proof goes

The proof first simplifies. For two matroids that equal their own duals, any packing/covering partition can be turned into two independent sets that cover everything, so it is enough to build a self-dual pair that can never be covered. The pair lives on a doubly infinite path. Between consecutive integer vertices runs a bundle, a labelled copy of one countable set D. At every vertex sits a copy of a gadget matroid, Q, on its two bundles, with its side L toward a central bundle and its side R pointing outward. The copies at even vertices form the first matroid, the copies at odd vertices the second, so every bundle is shared by one of each. Q is self-dual and uniform; in particular, every finite set is independent, but the whole set is not. Fix an ultrafilter on D, which calls every subset large or small, with exactly one of a set and its complement large, and an ordinal rank on large sets that can only go down as the sets grow. Q’s bases obey a one-way rule: if a basis takes a large part of its L side, it leaves a large part of its R side untaken, with strictly smaller rank. Now suppose the two matroids covered everything; trim the cover so each element is covered once. Hand each element of a bundle to the endpoint whose matroid covers it. On the central bundle, one of the two vertices gets a large share. Walk outward from it: each vertex receives a large set, and passes on a large set to the next with strictly smaller rank. That would be an infinite strictly decreasing sequence of ordinals, which cannot exist. In our finite experiments with ordinary pieces, the share keeps the same size all the way along; it takes a piece like Q to force a strict drop in rank. Most of the paper builds Q. D is cut into blocks of rapidly growing size, 2 to the 2 to the m, and sets are compared up to a vanishing proportion. Within each class, probes measure a kind of net count at infinity, and a set in a chosen class is a basis when that count, relative to a prototype, lies in a half-open window of length one, so swapping one element for another keeps a basis. An interpolation argument supplies the maximal extensions that the axioms demand, and a transfinite recursion, as long as the continuum, inserts new classes of bases until every interval of subsets is served, keeping the one-way rule. The paper credits the basis criterion and the recursion pattern to Bowler and Geschke.

6:38Where it stands

The preprint is unrefereed and has not been revised. OpenAI’s Lean files state the main theorem and its corollaries, including the self-dual uniform gadget, and prove them in 33 files and about 4,500 lines, with no gaps we could find by search; we did not run the check. The Lean proof derives the needed direction of Bowler and Carmesin’s equivalence itself, and for the separate conjectures it goes further than the paper, producing pairs of matroids that break them. The paper is not among those OSVerify reviewed, and we have seen no expert review; the claimed answer to Bowler and Geschke’s question especially deserves one. If it holds, the packing/covering conjecture for infinite matroids fails already for two matroids on a countable set, and so does the unrestricted intersection conjecture, while Nash-Williams’ finitary conjecture is untouched.