Abstract
A metric has four axioms — non-negativity, identity of indiscernibles, symmetry, and the triangle inequality. This paper drops each of them in turn and checks in numbers what is left, and then turns the last one around without dropping it. NO NEW MATHEMATICAL THEOREM AND NO NEW LAW IS CLAIMED.
First, identity goes. Putting $d=|x_1-x_2|$ on the plane leaves three distinct points at distance $0$ from one another (Section 2).
Second, symmetry goes. On a one-way ring $d(0,1)=1$ while $d(1,0)=3$, and $d(0,3)=3$ while $d(3,0)=1$ (Section 3).
Third, the triangle inequality goes. With $d=(x-y)^2$ the direct distance $d(0,2t)$ is exactly twice $d(0,t)+d(t,2t)$, which at $t=5$ reads $100$ against $50$ (Section 4).
Fourth, the triangle inequality becomes stronger. In the $2$-adic distance all $19986$ random triples satisfy $d(a,c)\le\max$, and every triangle is isosceles ($100.00\%$) (Section 5).
Fifth, and this is the core. In the Lorentzian interval the triangle inequality points the other way — every detour is shorter, running $10.0\to9.798\to9.165\to8.0\to6.0$. Straight is the longest (Section 6).
Sixth, satisfying all four axioms still does not fix the distance. The chordal and geodesic distances on the sphere are both genuine metrics, and their ratio runs from $1.000004$ to $1.570000$ (Section 7).
Scope (what is not claimed)
NO NEW MATHEMATICAL THEOREM AND NO NEW LAW IS CLAIMED — the definitions of pseudometric, quasimetric, semimetric and ultrametric, the $p$-adic distance, Ostrowski's theorem, and the reversed triangle inequality of Minkowski geometry are all standard. The twin paradox is not "resolved" here; all that is claimed is that it is a statement about a metric axiom, and neither acceleration nor general relativity is treated. No topology is discussed; which distance generates which topology is left aside. No taxonomy of generalised distances is built; the six are a census and not an exhaustive list. No $p$-adic analysis is entered; the $2$-adic distance appears only as a witness for ultrametricity. No ruling is made on which is the real distance; Section 7 shows precisely that the axioms do not settle it. Relation to earlier papers: Paper 201 counted four distinct roots of "complete" and showed that what ties them together is positive definiteness; this paper counts the step before that, the distance itself. Paper 217 showed that demanding only distances be preserved drags linearity along with them; this paper counts first how many things the distance to be preserved can be. Paper 121 treated a metric on a space of metrics; this paper treats the axioms themselves. Paper 233 counted six distinct roots of "differentiable"; this is the same shape for distance. What is added is the side-by-side measurement of each axiom's failure, the isosceles rate in the $2$-adic distance, the reversed triangle inequality tabulated as a function of the detour, and the demonstration that the chordal and geodesic distances diverge by a factor of $\pi/2$.
1. Notation and Position
Definition 1.1 a function $d\colon X\times X\to\mathbb{R}$ is a metric when it satisfies four conditions.
- Non-negativity: $d(x,y)\ge0$.
- Identity of indiscernibles: $d(x,y)=0$ implies $x=y$.
- Symmetry: $d(x,y)=d(y,x)$.
- The triangle inequality: $d(x,z)\le d(x,y)+d(y,z)$.
This paper drops the second, the third and the fourth in turn, and finally changes the direction of the fourth without dropping it.
Position: Paper 201 counted the distinct roots of "complete" and showed that positive definiteness is what ties the four together — this paper is one step before that. What is asked is not which is the real distance but what stops working when each axiom is let go.
2. Pseudometric: Identity Goes
Computation 2.1: put $d\big((x_1,y_1),(x_2,y_2)\big)=|x_1-x_2|$ on the plane.
| $p$ | $q$ | $d(p,q)$ |
|---|---|---|
| $(1,0)$ | $(1,5)$ | $\mathbf{0}$ |
| $(1,0)$ | $(1,-3)$ | $\mathbf{0}$ |
| $(1,5)$ | $(1,-3)$ | $\mathbf{0}$ |
| $(1,0)$ | $(4,0)$ | $3$ |
| $(1,5)$ | $(4,0)$ | $3$ |
Observation 2.2 three distinct points sit at distance $0$ from one another. The repair is to take a quotient, collapsing the space onto the $x$-axis — and the points after collapsing are not the original points. Identifying functions equal almost everywhere in $L^2$ is the same operation, and the same fence as discarding a null set in Section 4 of Paper 233.
3. Quasimetric: Symmetry Goes
Computation 3.1: take the one-way ring $0\to1\to2\to3\to0$ with unit weights, add an expensive shortcut $0\to3$ of weight $7$, and measure shortest paths.
| $(a,b)$ | $d(a,b)$ | $d(b,a)$ | gap |
|---|---|---|---|
| $(0,1)$ | $1$ | $\mathbf{3}$ | $2$ |
| $(0,2)$ | $2$ | $2$ | $0$ |
| $(0,3)$ | $\mathbf{3}$ | $\mathbf{1}$ | $2$ |
| $(1,3)$ | $2$ | $2$ | $0$ |
Observation 3.2 getting from $0$ to $3$ costs $3$, and getting back costs $1$. The triangle inequality holds in both directions — only symmetry has gone. The repair is to symmetrise, by taking a maximum or an average, and the answers change: $d(0,3)$ becomes neither $3$ nor $1$.
4. Semimetric: the Triangle Inequality Goes
Computation 4.1: put $d(x,y)=(x-y)^2$ on the line. It is non-negative, symmetric, and vanishes exactly when $x=y$.
| $t$ | $d(0,t)+d(t,2t)$ | $d(0,2t)$ | verdict |
|---|---|---|---|
| $0.5$ | $0.5$ | $1.0$ | violated |
| $1$ | $2.0$ | $4.0$ | violated |
| $2$ | $8.0$ | $16.0$ | violated |
| $5$ | $50.0$ | $\mathbf{100.0}$ | violated |
Observation 4.2 the detour always costs half. There is no repair here of the kind a quotient or a symmetrisation provides — taking a square root gives a metric, but that is a different function and not the same distance rescaled. Three axioms can hold and the phrase "shortest path" still lose its meaning.
5. Ultrametric: the Triangle Inequality Becomes Stronger
Computation 5.1: put the $2$-adic distance $|a-b|_2 = 2^{-v_2(a-b)}$ on the integers and draw $19986$ random triples of distinct integers below $4096$.
| property | triples satisfying it |
|---|---|
| the ordinary triangle inequality | $19986/19986$ ($100.00\%$) |
| the strong form $d(a,c)\le\max\big(d(a,b),d(b,c)\big)$ | $\mathbf{19986/19986}$ ($\mathbf{100.00\%}$) |
| the two largest sides equal (isosceles) | $\mathbf{19986/19986}$ ($\mathbf{100.00\%}$) |
Observation 5.2 nothing has been dropped here; an axiom has been strengthened. And the geometry stops looking like geometry — every triangle is isosceles, every point of a ball is its centre, and two balls that meet are nested. Strengthening an axiom is also moving to a different object.
6. The Core: the Triangle Inequality Points the Other Way
Computation 6.1: in $1{+}1$ dimensional Minkowski spacetime join the timelike separated points $A=(0,0)$ and $C=(10,0)$, place $B=(5,x)$ between them, and measure $|AB|+|BC|$, where $|PQ|=\sqrt{\Delta t^2-\Delta x^2}$.
| $B$ | $|AB|$ | $|BC|$ | $|AB|+|BC|$ |
|---|---|---|---|
| $(5,0)$ | $5.0$ | $5.0$ | $\mathbf{10.0}$ (equals $|AC|$) |
| $(5,1)$ | $4.898979486$ | $4.898979486$ | $9.797958971$ |
| $(5,2)$ | $4.582575695$ | $4.582575695$ | $9.16515139$ |
| $(5,3)$ | $4.0$ | $4.0$ | $8.0$ |
| $(5,4)$ | $3.0$ | $3.0$ | $6.0$ |
| $(5,4.9)$ | $0.9949874371$ | $0.9949874371$ | $\mathbf{1.989974874}$ |
The straight path gives $|AC| = 10.0$. (QED)
Observation 6.2 every detour is shorter. The triangle inequality is not violated; its sign is reversed — $|AC|\ge|AB|+|BC|$. The phrase "shortest path" names the longest one.
Observation 6.3 this is the twin paradox. The twin who stays ages by $10$ and the twin who turns around ages by $6$ — there is no paradox anywhere, only a triangle inequality pointing the other way. That is all this paper claims; neither acceleration nor general relativity is treated.
Observation 6.4 the triangle inequality is not the only casualty — non-negativity and identity of indiscernibles fail as well (distinct points at lightlike separation have interval $0$). So the Lorentzian interval is not a generalised distance but a different object — this boundary is what Paper 201 meant by saying that positive definiteness ties the four together.
7. All Four Axioms, and Still Not One Distance
Computation 7.1: on the unit sphere compare the chord through the surrounding space with the great circle along the surface. Both satisfy all four axioms.
| central angle | chordal | geodesic | ratio |
|---|---|---|---|
| $0.01$ | $0.00999995833339$ | $0.01$ | $1.000004167$ |
| $0.1$ | $0.0999583385414$ | $0.1$ | $1.000416788$ |
| $1$ | $0.958851077208$ | $1$ | $1.042914821$ |
| $2$ | $1.68294196962$ | $2$ | $1.188395106$ |
| $3$ | $1.99498997321$ | $3$ | $1.503766956$ |
| $3.14$ | $1.99999936586$ | $3.14$ | $\mathbf{1.570000498}$ |
Observation 7.2 the ratio opens to $\pi/2$. Both are genuine metrics and the axioms choose neither — "which one is the distance" is not a theorem but a declaration of what one wants to measure. An aircraft uses the geodesic and a tunnel uses the chord.
8. Separator and Fence
Observation 8.1 (the census)
| kind | axiom released or altered | what stops working |
|---|---|---|
| metric | none | — |
| pseudometric | identity of indiscernibles | telling points apart |
| quasimetric | symmetry | calling a round trip one quantity |
| semimetric | the triangle inequality | the phrase "shortest path" |
| ultrametric | strengthened | triangles having general shapes |
| Lorentzian interval | reversed | the shortest being shortest |
Observation 8.2 (the separator) what divides the six is how each of the four axioms is treated. The symbol $d(x,y)$ writes none of it — the same shape as $f'$ in Paper 233.
Observation 8.3 (the fence) some failures can be repaired and some cannot. A pseudometric is fixed by a quotient. A quasimetric is fixed by symmetrising, and the answers change. A semimetric is not fixed at all: the square root is a different function. And an ultrametric and the Lorentzian interval are not things to be fixed — they are different objects, not broken distances.
Observation 8.4 what Section 7 shows is that satisfying all four axioms still does not fix the distance. The fence is not inside the axioms — "which distance" can only be declared before measuring.
9. Census and Closing
What this paper measured.
- Three distinct points at distance $0$ from one another (2.1).
- $d(0,3)=3$ and $d(3,0)=1$ (3.1).
- The detour always costing half (4.1): $50$ against $100$ at $t=5$.
- The strong triangle inequality at $19986/19986$, isosceles at $19986/19986$ (5.1).
- Every detour shorter (6.1): from $10.0$ down to $1.99$.
- The chordal-to-geodesic ratio opening to $1.570000$ (7.1).
References
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Reference
Yamagishi, Y. (2026). “Distance” Is Six Things, and One of Them Satisfies the Triangle Inequality Backwards [Paper 234]. YOI: https://www.yamagishiyuuki.com/yoi/234
DOI: 10.5281/zenodo.22128226
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 234. This paper is a preprint and has not been peer reviewed.