Abstract
In Paper 214 “brightness” was four separate quantities. What this paper treats is one quantity, and even so the place of its peak is not settled. NO NEW MATHEMATICAL THEOREM AND NO NEW PHYSICAL LAW IS CLAIMED.
First, there are three equations. The root of $(x-5)e^x+5=0$ is $4.965114231744276$, of $(x-4)e^x+4=0$ is $3.920690394872886$ and of $(x-3)e^x+3=0$ is $2.821439372122079$ — the $5$, $4$ and $3$ being the powers of $x$ in the three densities (Section 2).
Second, this is the core. The spectrum of the Sun at $5772\,\mathrm{K}$ peaks at $\mathbf{502.039\,\mathrm{nm}}$ (green) per unit wavelength, at $\mathbf{635.777\,\mathrm{nm}}$ (orange) per unit logarithm and at $\mathbf{883.479\,\mathrm{nm}}$ (infrared) per unit frequency — and all three are one and the same spectrum (Section 3).
Third, the ratios do not depend on the temperature. Frequency over wavelength gives $1.759780586$ and logarithm over wavelength $1.266387736$, the same for the Sun, a filament, a person and the microwave background (Section 3).
Fourth, what is invariant can be measured too. The share of the power between $400$ and $700\,\mathrm{nm}$ is $\mathbf{0.36638317138174}$ integrated over wavelength and over frequency alike, a difference of $0$ (Section 6).
Fifth, why the maximum moves comes out in numbers. The factor $\lambda^2/c$ that a change of variable brings rises across this range from $5.34\times10^{-22}$ to $4.80\times10^{-21}$, ninefold, and drags the maximum with it (Section 7).
Scope (what is not claimed)
NO NEW MATHEMATICAL THEOREM AND NO NEW PHYSICAL LAW IS CLAIMED — Planck's law, Wien's displacement law, the Stefan–Boltzmann law and the question of where the Wien peak lies are all standard. Planck's law is not derived — this paper measures a formula it was given. The real Sun is not measured — it is treated as an ideal blackbody at $5772\,\mathrm{K}$, with no atmosphere and no absorption lines. The response of vision is not treated — nothing is explained about looking green. No variable is declared the right one — what is written is only what changes when one is chosen. Relation to Paper 214: there “brightness” was four separate quantities — here there is one quantity and three peaks. The distinct root lies further upstream. Relation to Paper 220: there, passing a test was news about the test — here, the place of a peak is news about the variable. What is added is giving the three roots to sixteen digits, setting the three peaks side by side at four temperatures and seeing that the ratios do not depend on temperature, maximising the curves themselves rather than the equations and obtaining ten digits of agreement, obtaining the two Wien constants and the Stefan–Boltzmann constant and setting them against the tabulated values, measuring that the share in a fixed band does not depend on the variable, and tabulating the ninefold rise of the Jacobian factor.
1. Notation and Position
The radiation of a blackbody is a density. The density per unit wavelength and the density per unit frequency are two ways of writing one spectrum, joined by the equality of the two differentials. A third way is per unit logarithm. Write $x=hc/(\lambda kT)$.
Position: In Paper 214 “brightness” was four separate quantities and only one obeyed the inverse square — this paper goes further upstream and finds that even for one quantity the place is not settled. In Paper 221 “eigenvalue” split in two according to which matrix was meant — here “peak” splits in three according to what one counts per.
2. Three Equations, Three Roots
Check 2.1: the three densities carry different powers of $x$, so the condition for a maximum differs.
| Density per what | Condition for the maximum | Root |
|---|---|---|
| Per wavelength | $(x-5)e^x+5=0$ | $4.965114231744276$ |
| Per logarithm | $(x-4)e^x+4=0$ | $3.920690394872886$ |
| Per frequency | $(x-3)e^x+3=0$ | $2.821439372122079$ |
Observation 2.2 The $5$, $4$ and $3$ are the powers of $x$ standing in each density. The equations differ in the single respect of whether the $\lambda$ in the denominator is a fifth, a fourth or a third power, and that moves the root bodily.
3. The Core: One Spectrum with Three Peaks
Check 3.1:
| Body | $T\,(\mathrm{K})$ | Per wavelength | Per logarithm | Per frequency |
|---|---|---|---|---|
| The Sun | $5772$ | $\mathbf{502.039\,\mathrm{nm}}$ | $\mathbf{635.777\,\mathrm{nm}}$ | $\mathbf{883.479\,\mathrm{nm}}$ |
| A filament | $2800$ | $1034.92\,\mathrm{nm}$ | $1310.61\,\mathrm{nm}$ | $1821.23\,\mathrm{nm}$ |
| A person | $310$ | $9347.65\,\mathrm{nm}$ | $11837.8\,\mathrm{nm}$ | $16449.8\,\mathrm{nm}$ |
| The background | $2.725$ | $1.0634\,\mathrm{mm}$ | $1.34668\,\mathrm{mm}$ | $1.87136\,\mathrm{mm}$ |
The ratios are $1.759780586$ for frequency over wavelength and $1.266387736$ for logarithm over wavelength.
Observation 3.2 The Sun's spectrum peaks in the green, in the orange and in the infrared. All three are correct answers about one and the same spectrum — and “the Sun peaks in the green” and “the Sun peaks in the near infrared” are both printed in textbooks.
Observation 3.3 The ratios do not depend on the temperature at all. At $5772\,\mathrm{K}$ and at $2.725\,\mathrm{K}$ the three peaks stand in the same arrangement — because the spread is a property of the writing and not of the body.
4. Checked by Maximising the Curves Themselves
Check 4.1: maximise the Planck curves at $T=5772\,\mathrm{K}$ directly, by ternary search.
| What was maximised | Found numerically | From the equation |
|---|---|---|
| Density per wavelength | $502.0394933\,\mathrm{nm}$ | $502.0394933\,\mathrm{nm}$ |
| Density per logarithm | $635.7766571\,\mathrm{nm}$ | $635.7766571\,\mathrm{nm}$ |
| Density per frequency | $883.4793537\,\mathrm{nm}$ | $883.4793537\,\mathrm{nm}$ |
Observation 4.2 Agreement to ten digits. The three places are not artefacts of rearranging a formula but the true maxima of the actual curves.
5. The Constants That Come Out
Check 5.1:
| Constant | This paper | Tabulated |
|---|---|---|
| Wien, per wavelength | $2.89777195519\times10^{-3}\,\mathrm{m\,K}$ | $2.897771955\times10^{-3}$ |
| Wien, per frequency | $5.87892575765\times10^{10}\,\mathrm{Hz/K}$ | $5.878925757\times10^{10}$ |
| Stefan–Boltzmann | $5.67037441918\times10^{-8}$ | $5.670374419\times10^{-8}$ |
Integrating the whole curve gives $62938592.4708\,\mathrm{W/m^2}$ against $\sigma T^4=62938592.4703$, a ratio of $1.0000000000074$.
Observation 5.2 Note that there are two Wien constants — $2.898\times10^{-3}$ and $5.879\times10^{10}$ cannot be converted into one another. The total $\sigma T^4$, by contrast, is one number, the same whichever variable the integration uses.
6. What Is Invariant: The Energy in a Band Fixed in Advance
Check 6.1: the share of the power between $400$ and $700\,\mathrm{nm}$, computed two ways.
| Integrated how | Share |
|---|---|
| Over wavelength | $0.36638317138174$ |
| Over frequency | $0.36638317138174$ |
| Difference | $\mathbf{0}$ |
Observation 6.2 Every digit agrees. What a change of variable moves is the place of the maximum, and what it leaves alone is the energy in a band fixed beforehand — the first compares values of a density and the second is an integral. An integral is invariant under a change of variable, and the value of a density is not.
7. Why the Maximum Moves
A density carries a “per what”. Passing from wavelength to frequency multiplies it by $|d\lambda/d\nu|=\lambda^2/c$, and that factor is itself steep, so it drags the maximum along.
Check 7.1:
| Wavelength | Density per wavelength | $\times\ \lambda^2/c$ | $=$ density per frequency |
|---|---|---|---|
| $400\,\mathrm{nm}$ | $2.2913188\times10^{13}$ | $5.3370255\times10^{-22}$ | $1.2228827\times10^{-8}$ |
| $502\,\mathrm{nm}$ | $\mathbf{2.6239591\times10^{13}}$ | $8.4059486\times10^{-22}$ | $2.2056865\times10^{-8}$ |
| $636\,\mathrm{nm}$ | $2.3185429\times10^{13}$ | $1.3492534\times10^{-21}$ | $3.128302\times10^{-8}$ |
| $883\,\mathrm{nm}$ | $1.4019471\times10^{13}$ | $2.6007626\times10^{-21}$ | $\mathbf{3.6461316\times10^{-8}}$ |
| $1200\,\mathrm{nm}$ | $6.8551652\times10^{12}$ | $4.803323\times10^{-21}$ | $3.2927573\times10^{-8}$ |
Observation 7.2 The left column is greatest at $502$ and the right at $883$. The whole difference is the middle column, which rises across this range from $5.34\times10^{-22}$ to $4.80\times10^{-21}$, ninefold. No number was wrong — two different densities were set side by side as though they were one curve.
8. The Separator and the Fence
Observation 8.1 (the separator) What separates the senses of “peak” is what one counts per — per wavelength, per logarithm, or per frequency. And any monotone change of variable gives yet another answer: the three are representatives only.
Observation 8.2 (the fence) The fence around “the blackbody peaks here” is having settled which density is meant. Outside it the sentence is not even false but empty — the same shape as “independence was checked” in Paper 220.
Observation 8.3 And there are invariant quantities: the total power and the share falling in a fixed band. Both are integrals and not values of a density — “where it is strongest” is a property of the variable, and “how much is in there” is a property of the spectrum.
Observation 8.4 In Paper 214 “brightness” was four separate quantities — here the quantity is one and the peaks are still three. The distinct root lay upstream of the name of the quantity, in the little words “per unit”.
9. The Census and the Close
What this paper measured:
- Three roots (Check 2.1): $4.9651$, $3.9207$, $2.8214$.
- The Sun peaks in three places (Check 3.1): $502$, $636$, $883\,\mathrm{nm}$.
- The ratios do not depend on temperature (Check 3.1): $1.759780586$ and $1.266387736$.
- Maximising the curves gives the same (Check 4.1): ten digits.
- The constants match the tables (Check 5.1): a ratio of $1.0000000000074$.
- The share in a band is invariant (Check 6.1): $0.36638317138174$, difference $0$.
- The Jacobian factor rises ninefold (Check 7.1).
References
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- Yamagishi, Y. (2026). A Flat Slab Brings Light to a Focus [Paper 219]. Zenodo. 10.5281/zenodo.22119788
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- Yamagishi, Y. (2026). Change the Last Bit and an Eigenvalue Moves by 0.16 [Paper 221]. Zenodo. 10.5281/zenodo.22120012
Reference
Yamagishi, Y. (2026). The Sun Peaks in the Green, in the Orange and in the Infrared [Paper 222]. YOI: https://www.yamagishiyuuki.com/yoi/222
DOI: 10.5281/zenodo.22120199
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 222. This paper is a preprint and has not been peer reviewed.