Abstract
Heat is measured by air temperature. This paper shows that at the same air temperature the wet-bulb moves by $\mathbf{11.4005}$ degrees with humidity—so $35 ^\circ$C can be more dangerous than $50$. No new theorem or law is claimed.
First, at $35 ^\circ$C and $30$ per cent humidity the wet-bulb is $\mathbf{22.1440} ^\circ$C (Section 2).
Second, at the same temperature and $90$ per cent it is $\mathbf{33.5445}$, a span of $\mathbf{11.4005}$ (Section 2).
Third, this is the core of the paper. $35 ^\circ$C at $90$ per cent gives $33.5445$, above the $31.6032$ of $50 ^\circ$C at $25$ (Section 3).
Fourth, reaching the $35$ limit at an air temperature of $35$ requires $\mathbf{99.2530}$ per cent humidity (Section 3).
Fifth, at saturation the wet-bulb returns to the air temperature, $35.1163$ in the fit, within $0.5$ degrees (Section 2).
Sixth, the separator is whether one quantity measures it, or a pair (Section 4).
Keywords: wet-bulb temperature, humidity, heat stress, evaporative cooling, air temperature.
Scope (what is not claimed)
No new theorem or law is claimed—the wet-bulb temperature, evaporative cooling and physiological limits to heat are all standard. No medical advice is given—diagnosis, response and treatment are not the subject; in a hazardous situation follow professional guidance. No survival threshold is asserted—the $35 ^\circ$C wet-bulb figure is a representative value from the literature and varies with individual, duration and activity. One empirical fit is used—Stull’s single expression; near saturation it departs by about $0.5$ degrees. No weather is forecast—where and how often such conditions occur is not the subject. Radiation and wind are excluded—sunlight, radiant load and wind speed all matter to a person, but only air temperature and humidity are treated. Relation to earlier papers: Paper 459 showed that a round trip multiplies the loss by $1.9$—there too, one quoted number did not stand for the whole. Paper 300 showed that common root and distinct root can be decided—“heat measured by air temperature” and “by the wet-bulb” are distinct roots by that criterion. Paper 403 showed that the order of temperatures is not the order of hotness—there too, one word for temperature hid two quantities. What is added is giving the wet-bulb at four combinations, giving the span at fixed temperature as $11.4005$ degrees, exhibiting a pair where $35 ^\circ$C exceeds $50$, solving for the critical humidity as $99.2530$ per cent by bisection, confirming that the fit returns to the air temperature at saturation, and placing the separator at one quantity versus a pair.
1. Notation and the Question
Setting: The wet-bulb temperature is what a thermometer wrapped in a wet cloth reads. Evaporation takes heat, so it lies below the air temperature and approaches it as the humidity rises. Stull’s empirical expression from air temperature $T$ [$^\circ$C] and relative humidity $H$ [%] is
$$T_{w}=T\arctan\!\bigl(0.151977\sqrt{H+8.313659}\bigr)+\arctan(T+H)-\arctan(H-1.676331)$$ $$\qquad\qquad +\,0.00391838\,H^{1.5}\arctan(0.023101H)-4.686035$$A wet-bulb of $35 ^\circ$C is often taken as the point beyond which a person cannot shed heat.
Observation 1.1 Heat is measured by air temperature. Forecasts, records and warnings all speak in it—one number, easily ordered.
Observation 1.2 But a person cools because sweat evaporates. In damp air it does not—so the same air temperature can offer an escape or none.
Question: By how much do they differ.
2. An Eleven-Degree Span at One Air Temperature
Computation 2.1: Take the wet-bulb at four combinations.
| Air temperature [$^\circ$C] | Humidity [%] | Wet-bulb [$^\circ$C] |
|---|---|---|
| $35$ | $30$ | $\mathbf{22.1440}$ |
| $35$ | $60$ | $28.4883$ |
| $35$ | $90$ | $\mathbf{33.5445}$ |
| $40$ | $50$ | $30.8939$ |
Observation 2.2 At one air temperature the wet-bulb moves $\mathbf{11.4005}$ degrees. From $22.1440$ to $33.5445$—a single figure for air temperature does not carry that span.
Observation 2.3 The wet-bulb never exceeds the air temperature. Evaporation only cools—so air temperature is a ceiling, not the load.
Observation 2.4 At saturation it returns to the air temperature. $35.1163$ at $100$ per cent—within $0.5$ degrees even in the fit, an independent check that the expression is set up correctly.
Observation 2.5 $40 ^\circ$C at $50$ per cent gives $30.8939$, below the $33.5445$ of $35 ^\circ$C at $90$. Five degrees hotter in air—and the order has already reversed.
3. Thirty-Five Can Beat Fifty
Computation 3.1: Trade air temperature against humidity and tabulate the wet-bulb.
| Air [$^\circ$C] | Humidity [%] | Wet-bulb [$^\circ$C] | To the limit [$^\circ$C] |
|---|---|---|---|
| $35$ | $90$ | $\mathbf{33.5445}$ | $1.4555$ |
| $40$ | $60$ | $32.9810$ | $2.0190$ |
| $45$ | $40$ | $32.6437$ | $2.3563$ |
| $50$ | $25$ | $\mathbf{31.6032}$ | $3.3968$ |
Observation 3.2 The two orderings are exactly opposite. Air temperature climbs from $35$ to $50$ while the wet-bulb falls from $33.5445$ to $31.6032$—monotonically, in the other direction.
Observation 3.3 $50 ^\circ$C sits $1.9413$ degrees further from the limit than $35$. Because dry air lets the sweat go—a dry desert at $50$ and a humid $35$ are different phenomena.
Computation 3.4: Solve by bisection for the humidity that reaches the $35$ limit at an air temperature of $35$.
$\mathbf{99.2530}$ per cent (inverting gives a wet-bulb of $35.000000$).
Observation 3.5 Practically saturation is required. At $35 ^\circ$C, even $90$ per cent leaves $1.4555$ degrees—the limit comes from neither quantity alone.
Observation 3.6 None of the four reaches the limit. The nearest, $35 ^\circ$C at $90$ per cent, still has $1.4555$ degrees—but the order of those margins is not the order of air temperature.
4. The Separator
The separator: does one quantity measure it, or a pair.
Observation 4.1 One quantity, and there is an order. $50$ is hotter than $35$—easy to compare and easy to build warnings on.
Observation 4.2 A pair, and there is no order. Neither air temperature alone nor humidity alone suffices—neither component carries an ordering by itself.
Observation 4.3 These are distinct roots by the criterion of Paper 300. Both are called “heat”—one a component of the state of the air, one the room left for evaporation.
Observation 4.4 This stands where Paper 403 stands. There, the order of temperatures was not the order of hotness—here the order of air temperatures is not the order of danger. In both, one word serves two quantities.
Observation 4.5 It overlaps Paper 459 as well. There, the quoted $0.90$ did not stand for the whole path—here the quoted air temperature does not stand for the whole load.
Observation 4.6 So when records are compared, print the humidity. “The highest air temperature on record” is not necessarily the greatest heat—it counts a different quantity.
Observation 4.7 To be honest—the $35$ wet-bulb figure is itself only representative. It moves with individual, activity, duration, wind and sunlight—this paper can say that the ordering reverses, not where the danger lies.
5. Roll Call and Closing
- $35 ^\circ$C at $30$ per cent gives $22.1440$ (Computation 2.1): ample margin.
- At $90$ per cent it gives $33.5445$ (Computation 2.1): same air temperature.
- A span of $11.4005$ (Observation 2.2): not carried by air temperature.
- Saturation returns the air value (Observation 2.4): $35.1163$, within $0.5$.
- $50 ^\circ$C at $25$ gives $31.6032$ (Computation 3.1): the core.
- The orderings are opposite (Observation 3.2): monotone, reversed.
- $50$ is $1.9413$ degrees safer (Observation 3.3): a different phenomenon.
- The limit needs $99.2530$ per cent (Computation 3.4): practically saturation.
- The separator is one or a pair (Observation 4.3): distinct roots by Paper 300.
- Print the humidity with records (Observation 4.6): a different quantity.
Closing: Heat is measured by air temperature—forecasts, records and warnings all speak in it, one number and easily ordered. But a person cools because sweat evaporates, and in damp air it does not: the same air temperature can offer an escape or none. Within one air temperature there is an eleven-degree span—at $35 ^\circ$C the wet-bulb is $\mathbf{22.1440}$ at $30$ per cent humidity, $28.4883$ at $60$ and $\mathbf{33.5445}$ at $90$, a span of $\mathbf{11.4005}$ degrees. The wet-bulb never exceeds the air temperature, since evaporation only cools—so air temperature is a ceiling and not the load. At saturation it returns to $35.1163$, within $0.5$ degrees, which is an independent check on the expression. And the ordering reverses—$35 ^\circ$C at $90$ per cent gives $\mathbf{33.5445}$, $40$ at $60$ gives $32.9810$, $45$ at $40$ gives $32.6437$ and $50$ at $25$ gives $\mathbf{31.6032}$: air temperature climbs from $35$ to $50$ while the wet-bulb falls monotonically. $50 ^\circ$C sits $1.9413$ degrees further from the limit than $35$, because dry air lets the sweat go. A dry desert at $50$ and a humid $35$ are different phenomena. Reaching the $35$ limit at an air temperature of $35$ requires $\mathbf{99.2530}$ per cent humidity, and inverting returns a wet-bulb of $35.000000$. So practically saturation is required: even $90$ per cent leaves $1.4555$ degrees, and the limit comes from neither quantity alone. One thing separates them—does one quantity measure it, or a pair. One quantity, and there is an order: $50$ is hotter than $35$, easy to compare and easy to build warnings on. A pair, and there is none: neither air temperature alone nor humidity alone suffices, because neither component carries an ordering by itself. They are distinct roots by the criterion of Paper 300, one a component of the state of the air and one the room left for evaporation. This stands where Paper 403 stands, where the order of temperatures was not the order of hotness: in both, one word serves two quantities. It overlaps Paper 459 too, where the quoted $0.90$ did not stand for the whole path: here the quoted air temperature does not stand for the whole load. So when records are compared, print the humidity—“the highest air temperature on record” is not necessarily the greatest heat, because it counts a different quantity. To be honest—the $35$ wet-bulb figure is itself only representative, moving with individual, activity, duration, wind and sunlight. This paper can say that the ordering reverses, not where the danger lies.
References
- Stull, R. (2011). Wet-Bulb Temperature from Relative Humidity and Air Temperature. Journal of Applied Meteorology and Climatology.
- Sherwood, S. C., & Huber, M. (2010). An Adaptability Limit to Climate Change Due to Heat Stress. Proceedings of the National Academy of Sciences.
- Vecellio, D. J., Wolf, S. T., Cottle, R. M., & Kenney, W. L. (2022). Evaluating the $35 ^\circ$C Wet-Bulb Temperature Adaptability Threshold for Young, Healthy Subjects. Journal of Applied Physiology.
- Yamagishi, Y. (2026). The Three Hundredth Census: Whether Two Things Share a Root Is Decidable [Paper 300]. Zenodo. 10.5281/zenodo.22171914
- Yamagishi, Y. (2026). The Order of Temperatures Is Not the Order of Hotness [Paper 403]. Zenodo. 10.5281/zenodo.22699405
- Yamagishi, Y. (2026). Ninety Per Cent Twice Leaves Eighty-One [Paper 459]. Zenodo. 10.5281/zenodo.22733103
Reference
Yamagishi, Y. (2026). The Limit of Heat Is Set by Temperature and Humidity Together [Paper 460]. YOI: https://www.yamagishiyuuki.com/yoi/460
DOI: 10.5281/zenodo.22733136
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 460. This paper is a preprint and has not been peer reviewed.