Abstract
When one says "solve", the usual first question is how many answers there are. This paper asks instead what kind of object the collection of answers is — an empty set, a pair of points, a sphere, a vector space, or a single function that is not even a minimum. Every equation here is written with the same "= 0". NO NEW MATHEMATICAL THEOREM AND NO NEW LAW IS CLAIMED.
First, one $x^2=-1$ returns three different objects. Nothing over the reals, two points over the complexes, and a two-dimensional sphere over the quaternions — across $20000$ unit purely imaginary quaternions the largest component of $q^2+1$ was $1.11\times10^{-16}$ (Section 2).
Second, finite fields change it too. $x^2+1=0$ has two solutions modulo $5,13,17,29$ and none modulo $3,7,11,19,23$ (Section 3).
Third, for a matrix it becomes a surface. The real $2\times2$ matrices with $X^2=I$ form a two-parameter family (Section 4).
Fourth, for a differential equation the collection is a vector space. $y'-y=0$ gives dimension one and $y''+y=0$ gives dimension two, with Wronskian $1.000000000000$ (Section 5).
Fifth, and this is the core. For a variational problem "= 0" means stationary and not minimal. For $J[y]=\int_0^L(y'^2-y^2)$ the function $y\equiv0$ is stationary for every $L$, and yet $J=+0.1449$ at $L=3$, $\mathbf{-0.3401}$ at $L=3.5$ and $-4.5065$ at $L=10$: at exactly $L=\pi$ a minimum turns into a saddle (Section 6).
Scope (what is not claimed)
NO NEW MATHEMATICAL THEOREM AND NO NEW LAW IS CLAIMED — the count of roots over an algebraically closed field, the solution set of $x^2=-1$ in the quaternions, the elementary consequence of quadratic reciprocity used here, the dimension of the solution space of a linear ordinary differential equation, and the classification of stationary points by the second variation are all standard. No existence theorem is proved; every section only counts solutions already known. Galois theory is not entered; Section 3 counts and does not explain. Nonlinear differential equations are not treated; "the solution space is a vector space" holds in the linear case only. No general theory of the calculus of variations is built; Section 6 examines one functional at one stationary point. No numerical analysis is offered. Relation to earlier papers: Paper 231 showed that each step down the Cayley-Dickson ladder costs one law; this paper looks at the same ladder from the side of the solutions of an equation. Paper 16 counted six distinct roots of the equals sign; this paper splits the inside of one of those uses, the conditional. Paper 179 counted distinct roots of "cannot be constructed" and touched Cauchy's functional equation; this paper does not go there. Paper 227 asked whether an equals sign is ever attained; Section 6 is its variational counterpart. Paper 235 measured a lowest eigenvalue of $\pi^2$ from the side of boundary conditions; the $L=\pi$ of Section 6 arrives at the same $\pi^2$ from the other side. What is added is the verification with twenty thousand samples that the quaternionic solution set is a sphere, the count of the same polynomial's roots over nine moduli, the residual check of the two-parameter family of matrix solutions, the measurement of solution-space dimension by the Wronskian, and the location of the point $L=\pi$ where a stationary point stops being a minimum.
1. Notation and Position
The question: when one writes $F(x)=0$, what kind of object is the collection $\{x : F(x)=0\}$?
This is not a question about how many. An empty set, a pair of points, a sphere, a vector space and a single function differ in kind before they differ in number. The question cannot be answered without deciding where the unknown is allowed to live — and the symbol "= 0" does not write that down.
Position: Paper 16 counted six distinct roots of the equals sign itself; this paper splits the inside of one of them, the conditional.
2. Change the Numbers and the Kind Changes
Computation 2.1: solve $x^2=-1$ in three places.
| where the unknown lives | the solution set | kind |
|---|---|---|
| the reals $\mathbb{R}$ | none | the empty set |
| the complexes $\mathbb{C}$ | $\pm i$ | two points |
| the quaternions $\mathbb{H}$ | all unit purely imaginary ones | a two-sphere |
Computation 2.2: draw $20000$ unit purely imaginary quaternions at random and square them.
| quantity | value |
|---|---|
| largest component of $q^2-(-1)$ | $1.11\times10^{-16}$ |
| mean length of the imaginary part | $1.000000000000$ |
| rank of the point cloud in $\mathbb{R}^3$ | $\mathbf{3}$ |
All twenty thousand are solutions; their imaginary parts lie on the unit sphere, and that sphere spans $\mathbb{R}^3$. (QED)
Observation 2.3 one quadratic returned the empty set, two points, and a two-sphere. The sentence "a quadratic has at most two roots" has a fence omitted — it holds over a commutative field. Paper 231 counted the rung at which commutativity is lost; this is what it costs.
3. Finite Fields Change It Too
Computation 3.1: solve $x^2+1=0$ modulo $p$.
| $p$ | solutions | count |
|---|---|---|
| $3$ | none | $0$ |
| $5$ | $2,\ 3$ | $2$ |
| $7$ | none | $0$ |
| $11$ | none | $0$ |
| $13$ | $5,\ 8$ | $2$ |
| $17$ | $4,\ 13$ | $2$ |
| $19$ | none | $0$ |
| $29$ | $12,\ 17$ | $2$ |
Observation 3.2 two solutions when the prime leaves remainder $1$ on division by $4$, none when it leaves $3$. This paper counts and does not explain — the point here is only that the same polynomial's "= 0" has a different collection at each address.
4. For a Matrix It Becomes a Surface
Computation 4.1: solve $X^2=I$ among real $2\times2$ matrices. Every $X$ whose rows are $\big(a,\ b\big)$ and $\big((1-a^2)/b,\ -a\big)$ is a solution, for any $a$ and any $b\ne0$.
| $a$ | $b$ | $\|X^2-I\|$ |
|---|---|---|
| $0$ | $1$ | $0$ |
| $0.5$ | $2$ | $0$ |
| $2$ | $3$ | $0$ |
| $-1.5$ | $0.4$ | $1.24\times10^{-16}$ |
| $7$ | $0.25$ | $0$ |
Together with $\pm I$ these are all of them. (QED)
Observation 4.2 the solutions are a two-parameter surface, not a list one can write out. "$X^2=I$ implies $X=\pm I$" is a statement about numbers and not about matrices — the fence is the absence of zero divisors.
5. For a Differential Equation the Collection Is a Vector Space
Computation 5.1: integrate to $t=1$ with fourth-order Runge-Kutta.
| equation | the solution set | dimension |
|---|---|---|
| $y'-y=0$ | $c\,e^{t}$ ($2.718281828$ and $5.436563657$ proportional) | $\mathbf{1}$ |
| $y''+y=0$ | $c_1\cos t+c_2\sin t$ (Wronskian $1.000000000000$) | $\mathbf{2}$ |
The computed values matched $\cos1=0.540302306$ and $\sin1=0.841470985$. (QED)
Observation 5.2 here "= 0" returns neither points nor a surface but a vector space. And its dimension equals the order of the equation — the fence is linearity, and in the nonlinear case this way of speaking fails (not treated here).
6. The Core: Stationary Does Not Mean Minimal
Computation 6.1: let $J[y]=\int_0^L\big(y'^2-y^2\big)dx$ with $y(0)=y(L)=0$. The function $y\equiv0$ makes the first variation vanish for every $L$. Feed in $y=\sin(\pi x/L)$ and measure $J$.
| $L$ | $J[\sin(\pi x/L)]$ | the type of $y\equiv0$ |
|---|---|---|
| $1$ | $4.434802201$ | minimum |
| $2$ | $1.467401100$ | minimum |
| $3$ | $0.144934067$ | minimum |
| $\pi$ | $0$ | degenerate |
| $3.5$ | $\mathbf{-0.340056514}$ | saddle |
| $4$ | $-0.766299450$ | saddle |
| $10$ | $-4.506519780$ | saddle |
Numerical quadrature agreed with the closed form $\tfrac{L}{2}\big((\pi/L)^2-1\big)$ at every row. (QED)
Observation 6.2 $y\equiv0$ satisfies "= 0" for every $L$. And for $L>\pi$ it is a saddle and not a minimum — "the variation vanishes, therefore it is a minimum" is false.
Observation 6.3 the turn happens at exactly $L=\pi$, which is where the lowest Dirichlet eigenvalue $\pi^2/L^2$ crosses $1$ — Paper 235 arrived at the same $\pi^2$ from the side of boundary conditions; this paper arrives from the side of the variation.
Observation 6.4 and what a variational "= 0" returns is neither a number nor a point but a single function — and whether that function gives a minimum is decided outside the "= 0". The same shape as Paper 227 writing that an equals sign does not settle whether it is attained.
7. Separator and Fence
Observation 7.1 (the census)
| where the unknown lives | the solution set | kind |
|---|---|---|
| a commutative field | at most $\deg$ points | a finite set |
| a noncommutative division ring | the unit imaginary quaternions | a sphere, a manifold |
| a finite field | $0$ or $2$ | finite, depending on the modulus |
| a matrix ring | a two-parameter family | a surface (zero divisors exist) |
| a linear differential operator | dimension equal to the order | a vector space |
| a functional | the stationary function | one function, not necessarily minimal |
Observation 7.2 (the separator) what divides the six is where the unknown is allowed to live. The symbol "= 0" names none of them — the same shape as $f'$ in Paper 233, $d(x,y)$ in Paper 234 and $-u''$ in Paper 235.
Observation 7.3 (the fence) three familiar sentences each have a fence omitted. "A degree $n$ equation has at most $n$ roots" — over a commutative field (Section 2). "$X^2=I$ implies $X=\pm I$" — where there are no zero divisors (Section 4). "The variation vanishes, so it is a minimum" — only when the second variation is positive (Section 6). Outside their fences all three are not even false, but empty.
8. Census and Closing
What this paper measured.
- $x^2=-1$ giving the empty set, two points and a sphere (2.1, 2.2): twenty thousand samples, error $1.11\times10^{-16}$.
- The same polynomial giving $0$ or $2$ depending on the modulus (3.1).
- Matrix solutions forming a two-parameter surface (4.1).
- Solution-space dimension equal to the order (5.1): Wronskian $1.000000000000$.
- A stationary point turning into a saddle at $L=\pi$ (6.1): from $+0.1449$ to $-0.3401$.
References
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- Gelfand, I. M., Fomin, S. V. (1963). Calculus of Variations. Englewood Cliffs: Prentice-Hall.
- Coddington, E. A., Levinson, N. (1955). Theory of Ordinary Differential Equations. New York: McGraw-Hill.
- Lang, S. (2002). Algebra, revised 3rd edition. New York: Springer.
- Yamagishi, Y. (2026). The Equals Sign Has Distinct Roots Too [Paper 16]. Zenodo. 10.5281/zenodo.21600471
- Yamagishi, Y. (2026). “Cannot Be Constructed” Has Distinct Roots [Paper 179]. Zenodo. 10.5281/zenodo.22098419
- Yamagishi, Y. (2026). The Same $\le$ Points at a Wall You Can Touch and at One You Cannot [Paper 227]. Zenodo. 10.5281/zenodo.22123129
- Yamagishi, Y. (2026). Every Doubling Costs the Product One Law [Paper 231]. Zenodo. 10.5281/zenodo.22126935
- Yamagishi, Y. (2026). “Differentiable” Is Six Things [Paper 233]. Zenodo. 10.5281/zenodo.22127806
- Yamagishi, Y. (2026). “Boundary Condition” Is Four Things [Paper 235]. Zenodo. 10.5281/zenodo.22128989
- Yamagishi, Y. (2026). “Distance” Is Six Things [Paper 234]. Zenodo. 10.5281/zenodo.22128226
Reference
Yamagishi, Y. (2026). One and the Same “= 0” Returns Five Different Kinds of Collection [Paper 236]. YOI: https://www.yamagishiyuuki.com/yoi/236
DOI: 10.5281/zenodo.22129043
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 236. This paper is a preprint and has not been peer reviewed.