eps^2 = 0 is a definition of a ring and (dW)^2 = dt is a fact about a path – a census of the word point.
One of Grothendieck’s moves was to reread a point not as an element inside an object but as a morphism from a test object. This paper calls the roll on that word and sets it against distinctions the corpus already holds. NO NEW MATHEMATICAL THEOREM AND NO NEW PHYSICAL LAW IS CLAIMED. That the epsilon-points of the dual numbers are tangent vectors, the functor of points, nilpotents and multiplicity, and Ito’s lemma with quadratic variation are all standard. The juxtaposition of eps^2 = 0 with (dW)^2 = dt is likewise a familiar observation to anyone who knows both. What this paper adds is only writing which of the two is a convention and which a fact, naming the separator and showing it in numbers, and giving addresses to five distinct roots of one word.
First, how many points there are has no answer until one names the field. Factoring x squared plus one over a finite field, the number of closed points is one or two according to the prime: when the prime is one more than a multiple of four it splits into two, and when it is three more it stays one, that single point having a residue field of degree two. At the prime two a single point is doubled. Yet the total degree is two for every prime up to twenty-nine (computed here). The count depends on the field and the total degree does not. A closed point whose residue field is a quadratic extension is one point over the prime field and two geometric points – one word carrying two numbers, the same shape as the equals sign of Paper 16.
Second, the same zero set can carry different rings. The zero set of the ideal generated by x and that of the ideal generated by x squared are both the origin alone, yet the dimensions of the quotient rings are one and two (computed here). Multiplicity is therefore the length of a ring, not a number of points. Written in its smallest Bezout case, the ideal of the intersection of a parabola with a line is, by the Groebner basis computed here, generated by x squared and y, so the quotient has length two. One point, counted twice: discard the nilpotents and this two has nowhere to live. The same as a set is not the same object.
Third, a fat point is a tangent vector, and it sees singularity. Counting morphisms from the ring obtained by adjoining one element whose square is zero, a smooth point leaves one linear condition and a solution space of dimension one, which is the tangent line. But at the node of a nodal cubic and at the cusp of a cuspidal curve the condition is empty and the dimension is two (computed here): the curve is one-dimensional and the tangent space is two-dimensional. The fat point sees the singularity. And this is Grothendieck’s move itself – a tangent vector is a morphism from the ring with a square-zero element, so once point has become morphism from a test object, changing the test object changes the answer. Change the field and one gets the first section; take the dual numbers and one gets the third.
Fourth, and this is the core. eps^2 = 0 and (dW)^2 = dt have different roots. In the ring of dual numbers, squares and cubes close at first order and no second-order term survives; the epsilon-part was checked against the derivative for seven functions. For Brownian motion, by contrast, the sum of squared increments converges to the time T however fine the partition, and does not vanish; the first-order sum diverges and the cubic sum dies, so only the exponent two is finite. For a smooth path the same quadratic sum dies in inverse proportion to the number of steps. And moving the roughness moves the number: generating fractional Brownian motion exactly from its covariance by Cholesky and taking the Hurst exponent from zero point two to zero point eight, the quadratic sum at two thousand and forty-eight steps runs from ninety-five point seven down to zero point zero zero nine two, in good agreement with the theoretical n to the power one minus twice H. It is finite and nonzero only at Hurst exponent one half. So eps^2 = 0 is a definition of a ring and cannot be moved, while (dW)^2 = dt is a fact about a path and moves when the roughness moves. The two share the shape of stopping at second order and are not the same statement: one discards the second order, the other keeps it, demoted to first.
Fifth, the Ito term is exactly the term the dual numbers discard. Written by the ordinary chain rule, the differential of W squared is twice W dW, whereas the truth is twice W dW plus dt. Comparing the left-endpoint sum with the exact value, the gap at ten million steps is zero point nine nine nine three eight four, converging to T (computed here). And that gap is nothing other than the sum of squared increments measured in the fourth section. The Ito correction is therefore exactly the term the dual numbers throw away, and what decides whether it may be thrown away is not the definition of a ring but the roughness of the path. Paper 132 wrote that Ito’s two is not a dimension but the reciprocal of the roughness; here that two is set beside the algebraic two and the pair is separated by whether it moves.
The census comes out as follows. A point of the topological space is a prime ideal; a closed point is a maximal ideal and separates residue fields; the generic point is the zero ideal and corresponds to the function field; a geometric point is a morphism from an algebraically closed field and separates counts; a fat point is a morphism from the ring with a square-zero element and separates singularity. One word names five different things, and what separates them is not the space but which test object one maps in from. In Paper 147 the separator was the coefficients; here it is the test object. Both have the same shape: that a thing can be written in a common language does not make it share a 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). What is added: writing which of eps^2 = 0 and (dW)^2 = dt is a convention and which a fact, naming the separator (the roughness of the path) and showing it in numbers, and giving addresses to five distinct roots of the word point. What remains: there were two things that looked like stopping at second order. One is how a ring was defined and cannot be moved. The other is how a path is, and moves when the roughness moves. A convention is not safe because it cannot be moved; that it does not move is the evidence that it is a convention.
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). $\varepsilon^2=0$ Is a Definition of a Ring and $(dW)^2=dt$ Is a Fact About a Path [Paper 148]. Zenodo.
DOI: 10.5281/zenodo.22083237
ORCID: 0009-0006-8587-3450