[2nd edition (August 2026)] No conclusion has changed. Following Paper 151, one line of Observation 3.3 – Two patches suffice here because the problem is that simple – has been rewritten. As a Cech computation a two-set cover returns the wrong answer for the sphere (enumerating the simplicial complexes on three vertices gives eighteen of them, every one with vanishing second cohomology; a good cover of the sphere needs four sets). Two patches sufficed because a different machine, clutching, was doing the work, and the invariant is the winding number on the overlap. The main quantisation result is not withdrawn. Only the way the reason is written has been corrected.
The Dirac string is the mark of trying to write it on one sheet – the address of a failure to glue.
Paper 138 placed the reason the divergence of the magnetic flux density vanishing is a law rather than an identity in whether the field can fail to be a curl. This paper gives that failure a name and an address. NO NEW MATHEMATICAL THEOREM AND NO NEW PHYSICAL LAW IS CLAIMED. Dirac (1931), the two-patch construction of Wu and Yang (1975), Chern classes, and sheaf cohomology with the exponential sequence are all standard. It is not claimed that magnetic monopoles exist, none having been found; what is treated here is what would be forced if one did. A word on sheaves: saying that a shell flux resembles a sheaf would be a rhyme and not a machine (Paper 100), while saying that the flux is a de Rham class whose integrality is a Cech obstruction becomes a machine, and only that line is drawn here.
First, the Dirac string is not a property of the field but the mark of how it was written. Take the northern potential to be g times one minus the cosine of the polar angle, times the differential of the azimuth, and the southern to be minus g times one plus that cosine, times the same. At the north pole the former vanishes and at the south pole the latter does: where the azimuthal differential is ill-defined, the corresponding coefficient goes exactly to zero. So the former is smooth except at the south pole and the latter except at the north. And their exterior derivatives are both g times the sine of the polar angle, with zero difference (computed here symbolically). Written on two sheets, therefore, no string appears anywhere (Wu and Yang). Covering the whole sphere with a single potential leaves a singular line somewhere, and that is the Dirac string: not a property of the field but the mark of having decided to use one sheet. In the idiom of Paper 95, the position of the string is a convention and carries no physics.
Second, the difference on the overlap is a Cech 1-cochain. On the overlap the two potentials differ by 2g times the differential of the azimuth, that is, by the exterior derivative of 2g times the azimuth. This is one datum attached to the single overlap of the two-patch cover, and the bundle is precisely that datum. With two patches there are no triple overlaps, so the cocycle condition is automatic; two patches suffice here because the problem is that simple.
Third, and this is the core. That the gluing closes up is the quantisation of charge. A wavefunction transforms by the exponential of the charge times the transition function divided by the reduced Planck constant. Going once around the equator the phase changes by four pi times the charge times the monopole strength, divided by the reduced Planck constant. Single-valuedness requires this to be a multiple of two pi, so that twice the charge times the monopole strength divided by the reduced Planck constant is an integer: Dirac’s quantisation condition (verified here). The quantisation of charge is therefore not a consequence of dynamics but the condition that a transition function close up on going round. And Paper 138’s separator becomes a number here: that the magnetic flux density can fail to be a curl means that this integer can be nonzero, and the case of the integer being zero is exactly the case in which the field is globally a curl.
Fourth, the invariant is the integer and not the four pi. The integral of the field over the sphere is four pi times the monopole strength (computed here). That four pi is the volume of the two-sphere, the same one Paper 128 addressed as the volume of the space of directions in three dimensions. The Chern number, by contrast, is one over two pi times the integral of the curvature, which comes to twice the charge times the monopole strength over the reduced Planck constant, an integer, and that is the invariant. Zero gives the trivial bundle, one a monopole, two a doubled monopole. The four pi is the normalisation of the generator and the integer is the class; the value four pi and the topological integer are different quantities. Paper 128 separated the volume of the two-sphere from two pi times the Euler characteristic, and the same separation is needed here: a four pi must not be read as topology.
Fifth, this is where the sheaf viewpoint enters. Isomorphism classes of principal U(1) bundles are classified by the first cohomology of the sheaf of smooth U(1)-valued functions, which is isomorphic to the second cohomology with integer coefficients, the isomorphism coming from the exponential sequence running from the integers through the reals to U(1). On the two-sphere this group is the integers, and its element is the integer found above. A sheaf is thus the device that keeps the books on local data and their agreement on overlaps, while the content lies in the cohomology that measures the failure to glue. Here a rhyme should be separated from a machine: saying that a shell flux resembles a sheaf computes nothing, whereas saying that the flux is a de Rham class whose integrality is a Cech obstruction produces an integer. The place of this in the corpus can also be written. Paper 92 said that Stokes is a shared alphabet, Paper 70 wrote the address of the index theorem, and Paper 128 wrote that the four roads to the Euler characteristic demand different licences. All are one shape, locally trivial and globally not, and this paper names that shape in its smallest example.
IMPORTANT (scope): a census of established standard material (Layer 2). Every work cited is standard. What is not claimed: any new mathematical theorem (there is none) / any new physical law (there is none) / that magnetic monopoles exist (not claimed; none has been found) / that a shell flux is a sheaf (it is not said; the flux is a cohomology class and the sheaf is the bookkeeping). What is added: giving an address to the separator laid down in Paper 138, showing by two potentials that the string is a mark of the writing, writing the quantisation as a condition on gluing, and separating the four pi from the integer. What remains: the string was never in the field but on the paper. Insisting on one sheet forced a tear somewhere; use two and the tear vanishes, leaving instead a single number on the overlap, and that this number must be an integer was the quantisation of charge.
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). The Dirac String Is the Mark of Trying to Write It on One Sheet [Paper 146]. Zenodo.
DOI: 10.5281/zenodo.22082868
ORCID: 0009-0006-8587-3450