The same H^2, and with different coefficients nothing is quantised – a census of the obstructions from local to global.
The corpus has met the shape locally trivial and globally not many times: Stokes in Paper 92, the index theorem in Paper 70, Poincare-Hopf in Paper 128, and Dirac in Paper 146. This paper calls the roll on those obstructions and writes their addresses and types. NO NEW MATHEMATICAL THEOREM AND NO NEW PHYSICAL LAW IS CLAIMED. De Rham’s theorem, the agreement of Cech and de Rham cohomology, the exponential sequence, and Dirac’s quantisation condition are all standard. Nor is it claimed that magnetic monopoles exist (as in Paper 146).
First, the obstruction is a group, and its dimension is not any of the corpus’s numbers. For the circle, the sphere, the torus and punctured three-space alike, the top cohomology is one-dimensional. The corpus’s four pi is therefore not that dimension: it is the value obtained by integrating a generator, that is, a period, and not the size of a group. That the obstruction is one-dimensional and that the number is four pi are two different statements about the same second cohomology (Paper 17), one giving the size of a group and the other the value of a normalisation. To demand an address is to demand that these two not be confused.
Second, being one-dimensional is exactly why the area form and the Euler form are proportional. On a sphere of radius R the area form integrates to four pi R squared and the Gaussian curvature times the area element integrates to four pi, a ratio of R squared (computed here symbolically). The former depends on the metric and the latter does not. Yet the second cohomology is one-dimensional, so these two 2-forms are elements of one group and constant multiples of each other. Belonging to the same family of cohomology classes therefore says nothing about which of them is forced. Paper 128 separated the volume of the two-sphere from two pi times the Euler characteristic; here that separation is visible as a factor R squared inside a single group. Living in the same group is not being the same thing.
Third, two pi and four pi appear as periods of generators. The generator of the first cohomology of the circle has period two pi, and that of the second cohomology of the unit sphere has period four pi (computed here). In this language the two numbers line up as periods of generators one degree apart. That does not make them share a root. Paper 142’s two pi is the smoothness condition that removes a conical singularity, and Paper 128’s four pi is the volume of the space of directions. Being shelved together and coming from the same place are different things (Paper 100): the shelf is a language and the provenance is a root.
Fourth, and this is the core. The same second cohomology with different coefficients quantises nothing. With real coefficients the group is the real line, a continuum, and permits any flux. With integer coefficients it is the integers, a lattice, and permits only integer classes. De Rham cohomology, having real coefficients, therefore quantises nothing at all. Integrality comes from the integer class, equivalently from requiring a U(1) transition function to be single-valued (Paper 146). So the sentence it is quantised because there is a global obstruction is not right: that an obstruction exists and that its value comes in steps are different statements, and the separator between them is neither the space nor the dimension but the coefficients.
Fifth, so electric charge is not quantised by Gauss’s law. Gauss’s law expresses the flux through the charge but places no restriction whatever on the charge itself, and the flux runs over a real continuum. Then why is charge quantised? Not by this argument. If even one monopole exists, the condition that twice the charge times the monopole strength divided by the reduced Planck constant be an integer constrains the product, so fixing the monopole strength quantises every charge (Dirac). But no monopole has been found. In Paper 146 that integer was read as the quantisation of the monopole strength; here it is read as the quantisation of the charge. It is one and the same integer, and the roles swap according to which quantity is taken as known – the same shape as Paper 16’s account of the equals sign, in which one relation becomes a different statement depending on where one stands.
The census comes out as follows. For Stokes (Paper 92) the obstruction’s address is the trivial group and there is no obstruction: closed implies exact. For Gauss’s law the address is the second de Rham cohomology of punctured three-space with real coefficients, the type is real, and the value moves continuously with the charge. For Dirac (Paper 146) the address is the second cohomology of the two-sphere with integer coefficients, the type is integer, and the value moves in steps. For Poincare-Hopf (Paper 128) the address is the Euler class, the type is integer, and the value is the Euler characteristic. One phrase, locally trivial but globally not, names four different obstructions, and what separates the second from the third is the coefficients. A sheaf is the device that keeps the books on local data and their agreement on overlaps, and a common bookkeeping does not make the entries the same. What Grothendieck’s viewpoint supplies is a common language, not a common root – and what the corpus has been guarding is the refusal to read an agreement of language as an agreement of root.
IMPORTANT (scope): a census of established standard material (Layer 2). What is not claimed: any new mathematical theorem (there is none) / any new physical law (there is none) / that magnetic monopoles exist (not claimed). What is added: setting four obstructions side by side with their addresses and types, making explicit that four pi is a period and not a dimension, exhibiting as a factor R squared that a metric-dependent and a topological form share one one-dimensional group, and writing that the separator for quantisation is the coefficients. What remains: being shelved together is not coming from the same place. Sheaves and cohomology give a common name to a shape the corpus has met again and again, but a common name with different coefficients yields different conclusions – moving continuously, or moving in steps. To write an address is to write not only the group but the coefficients.
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 Same $H^2$, and with Different Coefficients Nothing Is Quantised [Paper 147]. Zenodo.
DOI: 10.5281/zenodo.22083144
ORCID: 0009-0006-8587-3450