Whatever can be measured from inside has a home – but the address is chosen by our own demand.
Paper 91 wrote that curvature is intrinsic, measurable from inside a surface alone. Nash (1956) says the converse: a metric defined intrinsically can always be realised isometrically inside some Euclidean space. This paper sets the two side by side and shows that they are dual and yet distinct roots. NO NEW MATHEMATICAL THEOREM IS CLAIMED. The Theorema Egregium, Whitney’s embedding theorem, the Nash-Kuiper C1 theorem and Nash’s Ck theorem are all standard.
First, the Theorema Egregium was checked. For a sphere, a torus and a cylinder, the curvature computed from the first fundamental form alone, by Brioschi’s formula, was compared with the one computed using the second fundamental form; in all three cases the difference was exactly zero (computed symbolically here). For the sphere it is one over the radius squared, for the torus the cosine of v divided by r times R plus r cosine v, and for the cylinder zero. A curvature obtained without ever using the surrounding space agrees with the one obtained by using it – that is the Theorema Egregium, and it is the content of Paper 91.
Second, bending moves the second fundamental form and not the first. Placing a plane beside a cylinder, the first fundamental form is the same for both. The second forms, however, differ. And the curvature is zero in both cases. Rolling a sheet of paper does not change distances drawn on the paper – an isometry that is not a congruence. So the second form is not a function of the metric: one metric corresponds to many second forms. This single point sets up everything that follows.
Third, a home always exists, but the dimension required moves with the smoothness demanded. For an n-dimensional manifold, a smooth embedding ignoring the metric (Whitney) needs 2n, a C1 isometric one (Nash-Kuiper) needs n plus one, and a Ck isometric one with k at least three (Nash 1956, compact) needs n times three n plus eleven, over two. At n equal to two these are four, three and seventeen. The three numbers answer three different questions, and what must be moved is the smoothness demanded. That C1 needs three while Ck needs seventeen is not a matter of scale but of question – the single word embeddable does not fix a dimension until the smoothness is stated.
Fourth, and this is the core. The flat torus has a C1 home and no C2 one. For a torus whose radii are in the ratio two to one, the curvature changes sign around the tube: one third on the outside, one fifth at sixty degrees, zero at ninety, minus one third at a hundred and twenty, and minus one on the inside. Integrated over the whole surface it comes to exactly zero, agreeing with Gauss-Bonnet. Here the obstruction bites. A compact C2 surface in three-dimensional space must, at the point farthest from the origin, have curvature at least one over that distance squared, since the surface lies inside the sphere of that radius. Checking four shapes of torus, this held in every case. So a flat torus, with zero curvature everywhere, admits no C2 isometric embedding there.
And yet the Nash-Kuiper theorem gives that same flat torus a C1 isometric embedding in three-dimensional space (Borrelli and co-workers constructed one explicitly, and drew it, in 2012). The two do not contradict: they answer different questions. The separator is smoothness, with a line drawn between C1 and C2 across which the answer to has a home changes. And since the notion of curvature itself requires C2, in the C1 world the obstruction cannot even be stated – the obstruction does not vanish so much as the language for writing it does.
The census. Curvature is intrinsic and follows from the first form alone (Paper 91). The second form is extrinsic, needs the ambient space, and moves under bending. The existence of an embedding is extrinsic and always granted (Nash 1956). The dimension required moves with the smoothness demanded. The flat torus has a C1 home and no C2 one. Intrinsic and extrinsic are therefore dual – what is defined inside always has a realisation outside – and also distinct roots, since where that realisation lives is not determined by the metric. What determines it is the smoothness we demanded. This is a shape the corpus has met before: a statement that something exists and a statement of where it is carry different strengths (the same type as Paper 113’s existence can be argued, the value must be measured).
IMPORTANT (scope): a census of established standard material (Layer 2). What is not claimed: any new mathematical theorem (there is none). What is added: placing intrinsic and extrinsic side by side in the corpus’s language of distinct roots, checking the Theorema Egregium independently on three surfaces, writing the address of what moves the required dimension as the smoothness demanded, and exhibiting the C1 versus C2 divide for the flat torus as a numerical lower bound on the curvature of a compact surface. What remains: whatever can be measured from inside has an address outside. But that address is not settled by the information inside. What settles where a thing can live is our own demand about how smoothly it should live there – and loosening the demand by one degree lets in what could not live there before.
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). Whatever Can Be Measured from Inside Has a Home Somewhere [Paper 158]. Zenodo.
DOI: 10.5281/zenodo.22092143
ORCID: 0009-0006-8587-3450