The condition being paid is not size and not smoothness but being an integer.
Paper 158 treated Nash: every Riemannian manifold embeds isometrically, with no hypothesis at all. Kodaira’s theorem has the opposite shape, since an embedding into projective space exists exactly when a hypothesis holds, and that hypothesis is not vacuous. NO NEW MATHEMATICAL THEOREM IS CLAIMED: Kodaira’s embedding theorem, the Riemann relations, the fact that a general complex torus is not algebraic, and Nash’s embedding theorem are all standard. The search reported below is bounded: what is shown is that no integral form with entries up to two works, not that no polarisation exists, and the general statement rests on a dimension count that is a standard fact.
First, the convention was checked. Write a complex two-torus with period matrix consisting of the identity beside a two-by-two complex matrix. For an integral alternating form on the lattice, the Riemann relations require one expression to vanish and an associated Hermitian form to be positive definite. With the complex matrix diagonal the torus is a product of elliptic curves and is projective, and for the principal polarisation the first relation is zero while the smallest eigenvalue is 2.2 and 1.8 in two examples. The condition for the principal polarisation is that the matrix be symmetric with positive imaginary part, and indeed the first relation is zero when it is symmetric and 0.583095 when it is not: what the first relation asks for is symmetry. The sign is a matter of choosing the form or its negative and is not part of the condition.
Second, and this is the core. Building every integral alternating form with entries from minus two to two, that is 15624 of them, and counting those satisfying both relations, the abelian surface is found at once at search bound one, with two of them. For randomly chosen period matrices, all three samples give zero, and the best value of the first relation is 0.0139392, 0.00445366 and 0.0273488, never coming near zero. Such a torus is a compact complex manifold that cannot be embedded in any projective space, so Kodaira’s hypothesis genuinely bites.
Third, behind that lies a dimension count. All complex two-tori form a family of complex dimension four, that is real dimension eight, while the abelian surfaces form one of complex dimension three, that is real dimension six, so the codimension is two. Allowing every integral polarisation rather than only the principal one replaces one such subset by a countable union of them, and the union is still of measure zero. Almost every complex torus of dimension two or more is therefore not algebraic. In complex dimension one the count reads two against two, since every one-torus is an elliptic curve, so the obstruction is invisible until complex dimension two. Invisible and absent are different, the same shape as Paper 105 on what appears only in dimension four.
Fourth, the fence is integrality. Rescaling a working polarisation by one, two, three, one half and the square root of two leaves the first relation at zero in every case, with smallest eigenvalues 1.3801961, 0.69009805, 0.46006537, 2.7603922 and 0.97594602, all positive. The only thing that can break is whether the entries are integers. Positivity survives rescaling and integrality does not, so the fence is integrality. A Kahler class can be positive and still fail to lie in the image of the integral second cohomology, and then there is no embedding. This is the same fence as in Paper 146, where what quantises is that the gluing closes up, and Paper 147, where the same second cohomology with different coefficients quantises nothing.
Fifth, Nash pays in dimension. His bound for a compact n-manifold gives 7, 17, 30, 46 and 205 at dimensions one, two, three, four and ten. The two theorems are not a staircase of strength. Nash demands nothing and pays with an ambient space that keeps growing, while Kodaira demands a condition and needs only one kind of ambient space, where moreover the image is cut out by polynomials, so that what being embedded buys is far more. They preserve different things too: Nash preserves the metric and Kodaira the complex structure, the metric only up to the Fubini-Study class. One word, embedding, names two different things, one step beyond the distinction Paper 158 drew between the intrinsic and the extrinsic.
A census. Nash has no hypothesis, what moves it is the dimension, and it preserves the metric. Kodaira requires a positive line bundle, what moves it is integrality, and it preserves the complex structure. The fence is integrality rather than positivity and is broken by rescaling. The counterexample is a general complex torus in complex dimension two or more, of codimension two and measure zero. In complex dimension one everything is algebraic and the obstruction is invisible. The sentence that manifolds can be embedded has no truth value until it says which structure the embedding preserves.
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.
⚠️ Scope ── what is not claimed
a census of established standard facts (Layer 2). What is not shown: any new mathematical theorem, none is claimed; a proof that no polarisation exists, since the search is bounded. What is added: running the search over polarisations exhaustively and reporting the numbers, showing by rescaling that positivity survives and only integrality breaks, and setting Nash’s and Kodaira’s hypotheses, preserved structures and ambient spaces into one table. What remains: Nash’s theorem says it can be done and Kodaira’s says it can be done when this holds. The one with a condition is not the weaker, since what the condition buys is a better ambient space, and the condition being paid is not size and not smoothness but being an integer.
Reference
Yamagishi, Y. (2026). Kodaira’s Fence Is Integrality, Not Positivity [Paper 173]. Zenodo.
DOI: 10.5281/zenodo.22094507
ORCID: 0009-0006-8587-3450