Abstract
No function can do what the delta is asked to do. This paper shows that constructively and confirms in numbers that the delta is a map and not a collection of values. NO NEW MATHEMATICAL THEOREM IS CLAIMED — all of this is standard since Schwartz.
First, a value at one point cannot move an integral. Holding the area at $1$ while shrinking the width drives the height from $10$ to $10^{6}$, and the value at $x=0.01$ drops to $0$ along the way — the pointwise limit is not what is wanted (Section 2).
Second, and this is the core. At width $0.01$ the box, the Gaussian and the sinc take the values $\mathbf{0}$, $\mathbf{5.399}$ and $\mathbf{14.472}$ at $x=0.02$ — nothing like one another. Tested against a test function, all three converge to $\varphi(0)=0.367879441171$ (Section 3).
Third, what must be moved is therefore the space of test functions. Narrow what one tests against and fewer objects get through — a measure can represent the delta but not its derivative, which returns $-\varphi'(0)=-0.331091497054$ (Section 4).
Fourth, the payoff is a derivative that was not there. The difference quotient of the step function runs away, while integrating by parts against a test function gives $0.367879441171$ — agreeing with $\varphi(0)$ to $1.2\times10^{-13}$ (Section 5).
Fifth, the fence is that the product is not defined. Testing the square of the approximating family gives $1.033$, $10.377$, $103.777$, and diverges (Section 6).
Scope (what is not claimed)
NO NEW MATHEMATICAL THEOREM IS CLAIMED — Schwartz's theory of distributions, delta sequences, the distributional derivative, and the fact that distributions cannot in general be multiplied (Schwartz's impossibility theorem, $1954$) are all standard. Nor is Dirac criticised — no framework for distributions existed in $1927$, and the object he used was made rigorous afterwards. What is added is constructing in numbers the claim that no function can do it, setting three approximating families side by side to show that they differ utterly pointwise and agree against a test function, actually computing the distributional derivative of the step by parts, and measuring how fast the square diverges.
1. Notation and position
The test function used throughout is smooth with compact support, $$\varphi(x)=e^{-1/(1-x^2)}(1+0.9\,x)\quad(|x|<1),\qquad \varphi(x)=0\quad(|x|\ge1).$$ It is deliberately not symmetric, so that the mistake of simply evaluating at the origin cannot agree by accident. Here $\varphi(0)=1/e=0.367879441171$.
Position: Paper 155 treated spaces with no points and showed that points cannot be recovered from the open sets. This paper is its pair — here there is nothing at the point, and an object all the same.
2. A value at one point cannot move an integral
Observation 2.1 Two things are demanded of the delta at once — that it vanish away from the origin and that its integral be $1$.
Observation 2.2 A function carrying a value at a single point contributes nothing to an integral, however large the value. Getting integral $1$ takes width, and with width it no longer vanishes off the origin. A function cannot meet both demands.
Verification 2.3: Shrink the width while holding the area at $1$.
| width $w$ | height $1/w$ | integral | value at $x=0.01$ |
|---|---|---|---|
| $0.1$ | $10.0$ | $1.0$ | $10.0$ |
| $0.01$ | $100.0$ | $1.0$ | $\mathbf{0.0}$ |
| $0.001$ | $1000.0$ | $1.0$ | $0.0$ |
| $10^{-6}$ | $10^{6}$ | $1.0$ | $0.0$ |
(QED)
Observation 2.4 The integral stays at $1$ while the values run to infinity at one point and to $0$ everywhere else. What this family converges to pointwise is not the thing being sought.
3. The core: utterly different pointwise, identical against a test
Verification 3.1: Three families of area $1$, all at width $0.01$.
| $x$ | box | Gaussian | sinc |
|---|---|---|---|
| $0$ | $100.0$ | $39.894228$ | $31.830989$ |
| $0.02$ | $\mathbf{0.0}$ | $\mathbf{5.3990967}$ | $\mathbf{14.471918}$ |
| $0.13$ | $0.0$ | $7.9988278\times10^{-36}$ | $1.0287948$ |
Observation 3.2 At $x=0.02$ the box gives $0$, the Gaussian $5.4$ and the sinc $14.5$. As functions they are nothing like one another. And the sinc still stands at $1.03$ at $x=0.13$: it does not even converge pointwise.
Verification 3.3: The same three, tested against $\varphi$.
| width | box | Gaussian | sinc |
|---|---|---|---|
| $0.1$ | $0.367572644909$ | $0.36414456767$ | $0.362742785466$ |
| $0.01$ | $0.367876375486$ | $0.367842647708$ | $0.367880239715$ |
| $0.001$ | $\mathbf{0.367879410515}$ | $\mathbf{0.367879073291}$ | $0.359945061448$ |
The value to compare against is $\varphi(0)=\mathbf{0.367879441171}$. (QED)
Observation 3.4 All three head for the same number, and that number is $\varphi(0)$. The sinc converges slowly and oscillates (hovering near $0.36$) — which is to say that a family failing to converge pointwise can still have the same limit against a test function.
Observation 3.5 So the object is not a collection of values. It is what the values do to a test function, and on that the three agree. The delta is the map $\varphi\mapsto\varphi(0)$ itself.
4. What must be moved is the space of test functions
Observation 4.1 Change what $\varphi$ is allowed to be and the family of objects that can act on it changes with it.
| test functions | what may act | note |
|---|---|---|
| smooth, compact support | distributions | the widest class |
| smooth, rapidly decreasing | tempered distributions | Fourier transform stays inside |
| continuous, compact support | measures | the delta's derivative is excluded |
| polynomials only | very little | too few tests to separate anything |
Verification 4.2: The derivative of the delta separates what a coarser test cannot.
| acting on $\varphi$ | the delta | its derivative |
|---|---|---|
| the number returned | $0.367879441171$ | $\mathbf{-0.331091497054}$ |
The derivative of the delta returns $-\varphi'(0)$. (QED)
Observation 4.3 A measure can represent the delta and cannot represent the object returning $-\varphi'(0)$. One symbol names a different object depending on what it is tested against — that is the separator.
5. The payoff: a derivative that was not there
Verification 5.1: The difference quotient of the step function $H$.
| $h$ | $(H(h)-H(-h))/(2h)$ |
|---|---|
| $0.1$ | $5.0$ |
| $0.01$ | $50.0$ |
| $10^{-6}$ | $500000.0$ |
It runs away: classically there is no derivative at the origin. (QED)
Verification 5.2: Yet computing $-\int H\varphi'$ by parts gives —
| $-\int H\varphi'\,dx$ | $\mathbf{0.367879441171}$ |
|---|---|
| $\varphi(0)$ | $\mathbf{0.367879441171}$ |
| difference | $1.22626\times10^{-13}$ |
(QED)
Observation 5.3 The distributional derivative of the step acts exactly as the delta does. The derivative did not exist and now it does — because the question moved from values to action.
Observation 5.4 This payoff sits next to the limit Paper 177 described — there the Cantor function failed the hypothesis of differentiability everywhere and the fundamental theorem broke. Here a function that is not differentiable is given a derivative. Both turn on which sense of “derivative” is meant.
6. The fence: the product is not defined
Observation 6.1 Distributions form a vector space and can be differentiated any number of times. In general they cannot be multiplied.
Verification 6.2: Test the square of the approximating family.
| width | $\int(\delta_n)^2\varphi$ |
|---|---|
| $0.1$ | $1.032540663$ |
| $0.01$ | $10.37716851$ |
| $0.001$ | $\mathbf{103.7768225}$ |
It grows like the reciprocal of the width and does not converge. (QED)
Observation 6.3 “Delta squared” is not a distribution. The fence is that the product is undefined, which is why this definition buys differentiation and not an algebra (Schwartz's impossibility theorem of $1954$ says that the obstruction is not an accident).
Observation 6.4 This is the same shape of bargain as in Paper 168 — there, closure under limits was bought and absolute convergence paid. Here, differentiation without end is bought and multiplication paid.
7. The census
Observation 7.1
| object | what it is | what it can do | what it cannot |
|---|---|---|---|
| a function | values at points | multiply and differentiate | may fail to be differentiable |
| a measure | a number on each set | the delta is a measure | its derivative is not a measure |
| a distribution | a number on each test function | differentiate without end | no product |
Observation 7.2 These three are not a staircase of strength — a function can do things a distribution cannot, and the reverse. What settles it is what one tests against.
8. Conclusion
- A value at one point cannot move an integral (Ver. 2.3): height $10^{6}$, and $0$ at $x=0.01$.
- Utterly different pointwise (Ver. 3.1): $0$, $5.399$, $14.472$ at $x=0.02$.
- Identical against a test function (Ver. 3.3): all three to $\varphi(0)=0.367879441171$.
- The delta's derivative returns $-\varphi'(0)$ (Ver. 4.2): $-0.331091497054$, beyond any measure.
- The step is given a derivative (Ver. 5.2): difference $1.2\times10^{-13}$.
- The square diverges (Ver. 6.2): $1.03$ to $103.78$.
References
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Reference
Yamagishi, Y. (2026). The Delta Is Not a Function but a Map [Paper 178]. YOI: https://www.yamagishiyuuki.com/yoi/178
DOI: 10.5281/zenodo.22098261
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 178. This paper is a preprint and has not been peer reviewed.