Abstract
Put a function on a surface and it fixes how many minima, saddles and maxima there are. This paper shows, by integer counting with no tolerance anywhere, that all three move a great deal while the alternating sum never moves at all. NO NEW MATHEMATICAL THEOREM AND NO NEW LAW IS CLAIMED.
First, the parts move. Running $100$ random functions on a $24$ by $24$ triangulated torus, $c_0$ spreads over $72$ to $94$, $c_1$ over $145$ to $183$, $c_2$ over $72$ to $91$, and the total over $290$ to $366$ (Section 2).
Second, the alternating sum does not move. Across all $100$, the set of values taken by $c_0-c_1+c_2$ has exactly one element, $0$ (Section 2).
Third, the same procedure returns a different fixed number on a sphere. Across three levels of refinement the total ranges from $16$ to $396$, and the alternating sum is always $2$ (Section 3).
Fourth, this is the core. The floor exists and chance never reaches it. The floor on the torus is $4$ — four thousand random functions never got below $\mathbf{282}$, while one built on purpose reached $4$ at the first attempt (Section 4).
Fifth, every excess cancels. Setting the excess over the floor side by side, $e_0-e_1+e_2=0$ in every trial — one extra minimum or maximum is always paid for with one extra saddle (Section 5).
Scope (what is not claimed)
NO NEW MATHEMATICAL THEOREM AND NO NEW LAW IS CLAIMED — the Morse inequalities, Banchoff's discrete index and the Euler characteristic are all standard. The Morse inequalities are not proved — this paper only counts what they permit and what they forbid. Random values are noise as functions — values drawn independently at the vertices of a grid have no correlation between neighbours, so a large number of critical points is only to be expected, and what is surprising is not the number but that the alternating sum does not move even in that noise. Only two surfaces are treated — a torus and a sphere; no general statement about surfaces is made. Relation to Paper 128: 128 treated Poincaré–Hopf and wrote that the sum of the indices is the Euler characteristic — this paper counts not the sum but how free the individual terms are. Relation to Paper 151: 151 verified a procedure for computing Betti numbers from the ranks of boundary matrices — this paper only uses those numbers as a floor and does not enter the computation. What is added is setting the spread of the parts and the fixity of the sum inside the same trials, confirming across three refinements that the same procedure returns a different fixed number on a different surface, measuring over four thousand trials that chance does not reach the floor, and checking trial by trial that the excess cancels.
1. Notation and Position
Assign distinct values to the vertices of a triangulated surface. Among the link of a vertex $v$, the ring of vertices around it, call those with smaller values lower. If there is no lower vertex, $v$ is a minimum; if all are lower, a maximum; and if the lower ones fall into $k$ separate runs around the ring, count $k-1$ saddles there. This counting uses integers only and carries no tolerance whatever. Write $c_0$, $c_1$, $c_2$ for the totals of minima, saddles and maxima.
Position: Paper 128 treated Poincaré–Hopf and wrote that the sum of the indices of a vector field is the Euler characteristic. What this paper counts is not the sum but how free the individual terms making it up are.
2. The Torus: the Parts Move, the Alternating Sum Does Not
Check 2.1: on a $24$ by $24$ triangulated torus with $576$ vertices, place $100$ random assignments of values.
| Trial | $c_0$ | $c_1$ | $c_2$ | $c_0-c_1+c_2$ |
|---|---|---|---|---|
| $1$ | $80$ | $161$ | $81$ | $0$ |
| $2$ | $85$ | $171$ | $86$ | $0$ |
| $3$ | $81$ | $168$ | $87$ | $0$ |
| $4$ | $87$ | $172$ | $85$ | $0$ |
| $5$ | $88$ | $163$ | $75$ | $0$ |
| $6$ | $76$ | $155$ | $79$ | $0$ |
Check 2.2: gather all $100$.
| Quantity measured | Value |
|---|---|
| Range of $c_0$ | $72$ to $94$ |
| Range of $c_1$ | $145$ to $183$ |
| Range of $c_2$ | $72$ to $91$ |
| Range of the total | $290$ to $366$ |
| Values taken by $c_0-c_1+c_2$ | $\mathbf{0}$ only |
Observation 2.3 Each of the three swings by more than $20$, and the total by $76$. The one number made by adding and subtracting them was the same all $100$ times. Not “close to the same” — the counting is in integers, so it is exactly the same.
3. The Sphere: the Same Procedure Returns a Different Fixed Number
Check 3.1: on a subdivided icosahedron, repeat at three levels of refinement, $40$ trials each.
| Subdivision | Vertices | $c_0-c_1+c_2$ | Range of the total |
|---|---|---|---|
| $1$ | $42$ | $\mathbf{2}$ only | $16$ to $30$ |
| $2$ | $162$ | $\mathbf{2}$ only | $74$ to $104$ |
| $3$ | $642$ | $\mathbf{2}$ only | $342$ to $396$ |
Observation 3.2 Multiplying the vertices by fifteen and the total by twenty leaves the alternating sum at $2$. It is $0$ for the torus and $2$ for the sphere — the procedure is the same and only the number returned depends on the surface.
Observation 3.3 This paper does not derive what the $0$ and the $2$ are — that they are the Euler characteristic is a standard fact, and what is checked here is the behaviour of not moving.
4. The Core: the Floor Exists and Chance Never Reaches It
The Morse inequalities say that $c_i$ cannot fall below the Betti number $b_i$. On the torus $b_0,b_1,b_2=1,2,1$, so at least $4$ critical points are required.
Check 4.1:
| How the function was built | $(c_0,c_1,c_2)$ | Total |
|---|---|---|
| Random values, the least of $4000$ trials | $(73,141,68)$ | $\mathbf{282}$ |
| One cosine in each direction | $(1,2,1)$ | $\mathbf{4}$ |
| The floor (the Betti numbers) | $(1,2,1)$ | $4$ |
Observation 4.2 The floor is $4$, and a function built on purpose lands on it at the first attempt. Random functions, over four thousand trials, came down only to $282$ — seventy times the floor.
Observation 4.3 This must not be read as “randomness makes a poor function”. Independent values at the points of a grid are noise, and with no correlation between neighbours it is only to be expected that roughly half the vertices are critical. What is surprising is not the number — it is that even inside that noise the alternating sum never moved once. Two hundred and eighty-two meaningless comings and goings cancelled exactly.
5. Every Excess Cancels
Check 5.1: on a $16$ by $16$ torus, set out the excess over the floor, $e_i=c_i-b_i$.
| $e_0$ | $e_1$ | $e_2$ | Total excess | $e_0-e_1+e_2$ |
|---|---|---|---|---|
| $30$ | $67$ | $37$ | $134$ | $\mathbf{0}$ |
| $37$ | $74$ | $37$ | $148$ | $\mathbf{0}$ |
| $32$ | $68$ | $36$ | $136$ | $\mathbf{0}$ |
| $33$ | $71$ | $38$ | $142$ | $\mathbf{0}$ |
| $33$ | $67$ | $34$ | $134$ | $\mathbf{0}$ |
| $37$ | $77$ | $40$ | $154$ | $\mathbf{0}$ |
Observation 5.2 One extra minimum or maximum is paid for with exactly one extra saddle. An extra critical point is never born alone — it always brings a partner that cancels it. There is freedom to leave the floor, and the leaving has a direction.
6. The Separator and the Fence
Observation 6.1 (the separator) What separates what moves from what does not is whether it belongs to the function or to the surface. The three counts are fixed by the function that was placed — change the function and they move by tens. The alternating sum is fixed by the surface — whatever function is placed, the same number comes back. Two kinds of quantity are mixed together inside the same trial.
Observation 6.2 (the fence) The fence around “count the critical points and you learn the shape” is taking the alternating sum. Inside the fence the answer is settled by the surface alone; outside it — that is, looking at $c_0$ alone, or $c_1$ alone — the answer is swallowed by the habits of the function. The swing from $290$ to $366$ is the size of that swallowing.
Observation 6.3 The floor, by contrast, lives outside the fence — $c_i\ge b_i$ is a statement about each $c_i$ separately, and it bites before any sum is taken. So the two constraints are different things: one puts a floor under each term, and the other fixes a combination.
7. The Census and the Close
What this paper checked:
- The parts move a great deal (Check 2.2): the total runs from $290$ to $366$.
- The alternating sum never moves (Check 2.2): $0$ in all $100$, exactly.
- A different surface, a different fixed number (Check 3.1): always $2$ on the sphere, even at fifteen times the vertices.
- The floor is $4$ and the least by chance is $282$ (Check 4.1): first attempt on purpose, seventy times the floor by chance.
- What is surprising is not the number but that the sum holds inside noise (Observation 4.3).
- Every excess cancels (Check 5.1): $e_0-e_1+e_2=0$ in every trial.
- The fence is taking the alternating sum (Observation 6.2).
References
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- Yamagishi, Y. (2026). The Two Hundredth Census [Paper 200]. Zenodo. 10.5281/zenodo.22108829
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Reference
Yamagishi, Y. (2026). The Counts of Critical Points Move a Great Deal, and Their Alternating Sum Never Moves at All [Paper 203]. YOI: https://www.yamagishiyuuki.com/yoi/203
DOI: 10.5281/zenodo.22109443
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 203. This paper is a preprint and has not been peer reviewed.