One is mending an injury the world left behind, and the other one it inflicted itself.
Hironaka’s resolution of singularities and surgery in Ricci flow look much alike, since each replaces a bad local model by a standard one, finitely often. This paper writes that likeness as a rhyme and puts three separators under it, in numbers. NO STRUCTURAL IDENTITY IS CLAIMED, and refusing to claim one is the purpose of the paper. No new mathematical theorem is claimed either: Hironaka’s theorem, the local calculation of a blow-up, the round-sphere solution of Ricci flow and Perelman’s surgery are all standard. Paper 110 wrote that surgery and renormalisation are a rhyme, and this paper applies that discipline to a second pair, a rhyme being a licence to look and never a ground to conclude.
First, the blow-up was checked by hand. In the chart where the second coordinate is a product, the map round-trips exactly whenever the first coordinate is nonzero, so that the pair one half and minus three passes through one half and minus one and a half and returns unchanged. But at the origin every value of the second coordinate maps to the same point: a whole line collapses to one point, and over that point stands a projective line. A blow-up is therefore an isomorphism outside a set of lower dimension, a birational map, and it has an inverse, the blow-down.
Second, resolution really terminates. Tracking multiplicity sequences with home-made polynomial arithmetic, the node takes one blow-up with sequence two, the cusp takes one with sequence two then one, and the curves given by the square equalling the fifth, seventh and ninth powers take two, three and four, with sequences of twos followed by a one. The multiplicity two comes down while spending two units of exponent at a time. The procedure terminates, and by Hironaka in 1964 it terminates for every variety over a field of characteristic zero. And the first fence already stands here: in characteristic p the theorem is open in dimension four and above, while Ricci flow has no parameter that could even be set to p. The two surgeries do not share a parameter space.
Third, the first separator is invertibility. A blow-up has a blow-down and the information is kept up to birational equivalence, while surgery has no inverse and the information is discarded. Their effects on the invariants run opposite too: blowing up a surface at a point leaves the fundamental group unchanged and adds one to the second Betti number and to the Euler characteristic, so that the projective plane blown up at k points has Euler characteristic three plus k, that is three, four, five, six, seven. Surgery is performed precisely in order to change the fundamental group, splitting the manifold along its prime decomposition. One preserves the object and the other takes it apart, their purposes pointing in opposite directions.
Fourth, and this is the sharpest separator. Under Ricci flow a round n-sphere of radius r obeys the rule that the square of the radius decreases linearly, so that in dimension three starting from radius one it vanishes at time one quarter. Following the curvature scale gives one at time zero, 1.666666667 at a tenth, five at a fifth, 2500 at 0.2499 and 250000 at 0.249999. At time zero there is no singularity anywhere: the blow-up of the curvature was manufactured by the flow. Hironaka removes a defect that was there before anyone arrived, and Perelman removes one his own tool produced. One phrase, resolving a singularity, names two things of opposite provenance.
Fifth, surgery needs a scale. A blow-up has no free parameter, the exceptional divisor being the projectivised tangent cone with nothing to choose. Surgery cuts a neck where the curvature reaches a chosen threshold, and changing that threshold through ten, a hundred, ten thousand and a million moves the time of the cut to 0.225, 0.2475, 0.249975 and 0.24999975, with neck radii 0.316227766, 0.1, 0.01 and 0.001. In the language of Paper 95 this is a convention, and the theorem’s real work is to show that the answer does not depend on it. Hironaka has no such clause to discharge.
So what is the rhyme worth? It is worth something. Both replace a bad local model by a standard one, both do so finitely often before the object becomes usable, and both secure termination by a monotone quantity that drops at each step. That is a real pattern, but a pattern of method and not of subject. A rhyme of method grants a licence to go and look and grants no conclusion, and the three separators above record where that licence runs out. A census: the provenance of the singularity is already there against made by the flow; invertibility is a blow-down exists against none; the fundamental group is unchanged against changed on purpose; the choice of scale is none against needed; the parameter space has a characteristic against has none. What they share is replacing a bad local model by a standard form, doing it finitely often, and having a monotone quantity that drops.
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; any structural identity, none is claimed and refusing it is the point; resolution in characteristic p, which is open. What is added: tracking multiplicity sequences with home-made polynomial arithmetic to count the blow-ups, showing in numbers that the curvature blow-up is absent at time zero, and varying the threshold to show that where one cuts is a convention. What remains: the two theorems look as though they were performing the same act, but one is mending an injury the world left behind and the other one it inflicted itself. Noticing the likeness is a good thing, and not stopping there is a better one.
Reference
Yamagishi, Y. (2026). Whose Singularity Is It? [Paper 174]. Zenodo.
DOI: 10.5281/zenodo.22094572
ORCID: 0009-0006-8587-3450