A two-set cover returns the wrong answer for the circle – this paper applies a brake to Paper 146, without withdrawing its conclusion.
Paper 146 wrote the magnetic monopole on two patches and derived, as the condition that the gluing close up, that twice the product of the charge and the monopole strength divided by the reduced Planck constant is an integer. That integer is right. What was not right is the account of why two patches sufficed. NO NEW MATHEMATICAL THEOREM IS CLAIMED. Leray’s theorem, the clutching construction and the cocycle condition of descent are all standard. It is not claimed that magnetic monopoles exist.
First, if the cover is not good, gluing returns the wrong answer. Cover the circle by two arcs. Each arc is contractible, but their intersection falls into two components. The nerve of that cover has two vertices and one edge, that is, it is a segment, and its first cohomology is zero – whereas the first cohomology of the circle is the real line. So a two-set cover returns zero where the answer is the real line (computed here from the ranks of boundary matrices). Cover it by three arcs instead, so that each intersection is connected, and the nerve becomes the boundary of a triangle, which has the shape of a circle, and the first cohomology comes out as the real line, which is correct. Glue the local data and you recover the global is therefore not unconditional: what Leray’s theorem demands is that every finite intersection be empty or contractible.
Second, three sets can never see the sphere. Enumerating the simplicial complexes on three vertices, that is, the downward-closed families of faces, gives eighteen of them, and every one has second cohomology zero (computed here). No cover by three sets, therefore, has a nerve that can see the sphere. This is not the statement that no clever triple has yet been found but one settled by a finite check. Four sets do see it: the four faces of a tetrahedron projected radially form a good cover whose nerve is the boundary of a tetrahedron, with second cohomology the real line. In general a good cover of the n-sphere needs n plus two sets (checked here for n from one to four).
Third, and this is the core. Paper 146’s two patches sat on a cover for which Cech cohomology returns the wrong answer. The nerve of two sets is a segment and returns zero for the second cohomology of the sphere. Indeed the overlap of two enlarged hemispheres is an annulus and is not contractible, so the cover is not good. What was working, then? Clutching. A bundle built from two contractible patches is classified by the homotopy class of the transition function on the overlap; the overlap is an annulus, homotopy equivalent to a circle, so the set of classes is the set of homotopy classes of maps from the circle to the one-dimensional unitary group, which is the integers. Recounting numerically the winding number of Paper 146’s transition function, for each integer value of the exponent from minus two to three the winding number came out exactly at that value (computed here). The invariant is thus the winding number, an element of the fundamental group of the overlap’s homotopy type, and not a Cech computation on that cover.
So what survives and what is corrected? Paper 146’s conclusion is right and is not withdrawn. What is corrected is the single line saying that two patches suffice because the problem is that simple. They suffice because the clutching machine needs only the overlap – and that same cover is precisely one on which Cech would lie. This has the same shape as Paper 149’s finding that what decided the answer was not the determinant but the largest invariant factor: while the number comes out right, a misattributed reason cannot be caught.
Fourth, the cocycle condition bites only from three sets onward. With two patches there are no triple overlaps, so the condition is merely that the product of the two transition functions be one, which holds by definition. Nothing can break and nothing can be tested. With three the condition has content: covering the circle by three arcs with integer transition data written additively, a triple summing to zero glues and a triple summing to three does not (checked here). The obstruction to descent therefore appears only from the third set onward, and with two it can neither be exhibited nor ruled out.
The census comes out as follows. Cech with a good cover demands that finite intersections be contractible, and Leray’s theorem guarantees the answer – correct. Cech with an arbitrary cover demands nothing and is not correct. Clutching with two patches demands that both patches be contractible, and the answer lives in the fundamental group of the overlap – correct. Descent with three or more demands the cocycle condition on triples and is correct if it holds. From local to global is therefore not one operation: which machine is in use decides the demand, the home of the answer, and the reason it is right.
IMPORTANT (scope): a census of established standard material, together with a correction to the corpus itself (Layer 2). What is not claimed: any new mathematical theorem (there is none) / the withdrawal of Paper 146’s conclusion (it is not withdrawn) / that magnetic monopoles exist (not claimed). What is added: exhibiting in numbers, in the smallest example, that a bad cover returns a wrong answer; settling by a finite enumeration that three sets cannot see the sphere; and identifying which machine made Paper 146’s two patches work. What remains: two patches sufficed not because the problem was simple but because the machine in use needed only the overlap. And that same cover, handed to a different machine, lies. Change the tool and the answer changes on the same picture, which is why one must write down what was used.
On the making of this work: The ideas and content of this work stem from the author’s own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author’s. Feedback and corrections are sincerely appreciated.
Reference
Yamagishi, Y. (2026). A Two-Set Cover Returns the Wrong Answer for the Circle [Paper 151]. Zenodo.
DOI: 10.5281/zenodo.22084096
ORCID: 0009-0006-8587-3450