Abstract
Paper 10 established that only dimensionless numbers can be fine-tuned. This paper asks whether the same restriction applies to “has it changed”——the answer is it does. And one of them has actually been measured. No new mathematical theorem and no new law is claimed.
First, “has $c$ changed” has no truth value. It cannot be told apart from a change of units (Section 2).
Second, this is the core of the paper. The Oklo natural reactor holds $\alpha$ below $\mathbf{5.0\times10^{-18}}$ per year (Section 3).
Third, stretched over the age of the universe that is $7.0\times10^{-8}$ (Section 3).
Fourth, Eddington argued over the integer digit; Oklo reaches the sixth decimal (Section 4).
Fifth, the quasar claim disagrees with Oklo by a factor of $\mathbf{114}$ (Section 5).
Sixth, the separator is whether a change of units erases it (Section 6).
Scope (what is not claimed)
No new mathematical theorem and no new law is claimed——the Oklo bound on $\alpha$, the definition of the fine-structure constant and Duff’s point that a dimensionful constant’s variation cannot be asked about are all standard. We do not build the theorem——“only dimensionless numbers can be tuned” is Paper 10’s conclusion, received here as a premise. What this paper adds is the measurement side alone. We do not build nuclear physics——we do not enter the calculation from the samarium-149 resonance to the bound on $\alpha$. We quote the resulting number. We do not derive the value of $\alpha$——no attempt is made to obtain $1/\alpha$ from theory. We do not enter the road Paper 101 treated as Eddington’s fall. We do not adjudicate the quasar result——whether Webb et al. are right is not treated. We only count by what factor it disagrees with Oklo. We do not discuss theories of variation——models in which $\alpha$ can vary (scalar fields and the like) are not treated. We do not enter anthropic reasoning——that context belongs to Papers 10, 64 and 69; this paper looks only at “has it changed”. Relation to earlier papers: Paper 10 established that only dimensionless numbers can be fine-tuned, showing $c$, $\hbar$ and $G$ to be dimensionful, i.e. choices of unit——this paper takes that theorem as it stands and does not reopen it. It adds one thing only: what measurement actually says. Paper 101 treated as a caution Eddington’s “derivation” of $1/\alpha$ as $136$, and his adding one after measurement said $137$——this paper places, in those same digits, how far measurement has since reached. Paper 15 computed $1/\alpha=137.035999$ for itself——this paper uses those digits. Paper 285 showed that “forbidden” can be written in powers of $\alpha$——if $\alpha$ moved, those rates would move too. Paper 95 separated convention from fact——this paper’s core is the single point that “has it changed” does not form a sentence on the convention side. What is added is converting the Oklo bound into a per-year rate of $5.0\times10^{-18}$, stretching it over the age of the universe to $7.0\times10^{-8}$, comparing it with the digit Eddington argued over, and counting the disagreement with the quasar claim as a factor of $114$.
1. Notation and the Question
Fact (Paper 10): $c$, $\hbar$ and $G$ are dimensionful. The value of a dimensionful quantity is a choice of unit, and cannot be an object of tuning.
Question: Does the same restriction apply to “has it changed”?
Observation 1.1 It does. “$c$ became $1\%$ smaller than yesterday” cannot be told apart from making the metre $1\%$ longer——a sentence asserting one of two indistinguishable things has no truth value.
Observation 1.2 Since $2019$, $c$ is a defined value. $299792458$ m/s is a promise, not a measurement, and cannot change——it is scaffolding in the sense of Paper 95.
2. Counting Who Can Be Asked
Check 2.1:
| Quantity | Dimensions | Can “has it changed” be asked? |
|---|---|---|
| $c$ | $\mathrm{m/s}$ | no (now a defined value) |
| $\hbar$ | $\mathrm{J s}$ | no (a choice of unit) |
| $G$ | $\mathrm{m^3/(kg s^2)}$ | no (a choice of unit) |
| $k_B$ | $\mathrm{J/K}$ | no (now a defined value) |
| $\alpha=e^2/(4\pi\varepsilon_0\hbar c)$ | dimensionless | yes |
| $m_p/m_e$ | dimensionless | yes |
Observation 2.2 Only the last two rows. In any system of units $\alpha$ is $1/137.035999$, and no redefinition can move it——so if it moved, that would be physics.
Observation 2.3 One caution is needed——papers reporting “a variation of $G$” do exist. What they measure is not $G$ itself but a dimensionless combination such as $Gm_p^2/(\hbar c)$——the name says $G$, the content is a dimensionless number.
Observation 2.4 Paper 95 separated convention from fact. This paper applies that division to change——on the convention side, the word “change” does not form a sentence.
3. The Core: Oklo Holds It Below $5.0\times10^{-18}$ per Year
Setting: At Oklo in Gabon, a natural fission chain reaction ran two billion years ago. The position of the neutron-capture resonance in samarium-149 is sensitive to $\alpha$, so the isotope ratios left behind bound $\alpha$ at that time.
Check 3.1:
| Quantity | Value |
|---|---|
| Elapsed time | $2.0\times10^{9}$ yr |
| Bound on $|\Delta\alpha/\alpha|$ | of order $10^{-8}$ |
| Rate per year | below $\mathbf{5.0\times10^{-18}}$ |
Observation 3.2 This is the core of the paper. For the one quantity that can be asked, it has actually been measured——$\alpha$ has not moved.
Check 3.3: Stretching that rate over the age of the universe.
| Interval | Accumulated bound |
|---|---|
| $2.0\times10^{9}$ yr (Oklo) | $10^{-8}$ |
| $1.4\times10^{10}$ yr (age of the universe) | $\mathbf{7.0\times10^{-8}}$ |
Observation 3.4 Even over the whole history it is only $7.0\times10^{-8}$. $\alpha$ has not moved by more than seven parts in a hundred million across cosmic history——provided the rate is constant.
Observation 3.5 A point to write honestly——that extrapolation has no warrant. Oklo bounds the two billion years since, and says nothing about before——if the rate is not constant, the $7.0\times10^{-8}$ means nothing.
Observation 3.6 One more——the Oklo bound does not isolate $\alpha$. The resonance position also depends on strong-interaction quantities——it is a bound under the assumption that only $\alpha$ moved.
4. Comparing with the Digit Eddington Argued Over
Setting: Paper 101 treated as a caution Eddington’s group-theoretic “derivation” of $1/\alpha$ as $136$, and his adding one to make $137$ once measurement came close to that.
Check 4.1: With $1/\alpha=137.035999$, how far a $10^{-8}$ change in $\alpha$ moves $1/\alpha$.
| Quantity | Value |
|---|---|
| $1/\alpha$ | $137.035999$ |
| $\Delta(1/\alpha)$ for $\Delta\alpha/\alpha=10^{-8}$ | $\mathbf{0.000001370}$ |
Observation 4.2 The sixth decimal place. Eddington’s argument between $136$ and $137$ was in the integer digit——six orders below it, measurement says it has not moved.
Observation 4.3 Here two kinds of claim come into view.
| Kind of claim | How it would be refuted | |
|---|---|---|
| Eddington | derives a value | it fails to match measurement |
| This paper (Oklo) | bounds a value as unmoving | show that it moved |
Observation 4.4 Both are refutable. Eddington failed not because he was unfalsifiable but because he adjusted the number after being refuted——exactly what Paper 101 named a hindsight adjustment.
Observation 4.5 And this paper does not try to derive $\alpha$. Not a word is said about why $1/137$ has that value——only whether it moved. That self-limitation is this paper’s answer to Paper 101’s caution.
5. The Quasar Claim, and a Factor of $114$
Setting: Since $1999$, absorption lines in distant quasars have been claimed to show a slightly smaller $\alpha$: $\Delta\alpha/\alpha\sim-0.57\times10^{-5}$ over a timescale of order $10^{10}$ yr.
Check 5.1:
| Source | Rate per year |
|---|---|
| Oklo (bound) | $5.0\times10^{-18}$ |
| Quasars (claim) | $5.7\times10^{-16}$ |
| Ratio | a factor of $114$ |
Observation 5.2 They disagree by $114$. Either one of them is wrong, or the rate is not constant——this is one reason the quasar claim is contested.
Observation 5.3 A point to write honestly——this paper does not adjudicate. The two look at different epochs (Oklo at two billion years, quasars at ten), so they are compatible if the rate is not constant——we only count the size of the disagreement.
Observation 5.4 Paper 297 wrote that cosmic distance is a ladder of seven rungs and only the first is measured directly. The same structure is here——Oklo is a rock on the ground, the quasars are high on the ladder. Strength of bound is traded against reach.
Observation 5.5 A connection to Paper 285. If $\alpha$ moved, forbidden-transition rates would move too (the E1 to M1 ratio goes as $\alpha^2$)——a change in $\alpha$ changes the ticking of atomic clocks. Which is why comparing atomic clocks now gives the strongest bound of all.
6. Separator: Whether a Change of Units Erases It
Observation 6.1 Separator——whether a change of units erases it.
| Claim | Under a change of units | Status |
|---|---|---|
| “$c$ changed by $1\%$” | it vanishes | not a sentence |
| “$\alpha$ changed by $10^{-8}$” | it survives | a measurable claim |
Observation 6.2 The first row is not false; it is not a sentence. “False” and “without a truth value” are different——the same place where Paper 16 put “dimensionally void” as the sixth root of the equals sign.
Observation 6.3 And this distinction is settled before any measurement. No amount of precision makes the first row measurable——it is not a matter of accuracy but of the shape of the question.
Observation 6.4 Paper 10 settled what can be tuned. What this paper adds is the single point that one of them has actually been measured and has not moved——beside the theorem, a number.
Observation 6.5 Finally, as a discipline of the corpus——“not yet shown” is not the safe answer. To reserve judgment on whether $\alpha$ has varied hides the fact that a bound of $5.0\times10^{-18}$ per year already exists——reserving judgment without looking erases the fact that the evidence pointed one way.
7. Roll Call and Closing
- Only the last two rows (Check 2.1): Dimensionless only.
- Even “$G$ varying” is dimensionless (Observation 2.3): Do not trust the name.
- Oklo gives $5.0\times10^{-18}$/yr (Check 3.1): Two billion years to now.
- $7.0\times10^{-8}$ over cosmic history (Check 3.3): Assuming a constant rate.
- It does not isolate $\alpha$ (Observation 3.6): The strong force enters too.
- The sixth decimal of $1/\alpha$ (Check 4.1): Eddington was at the integer.
- No attempt to derive it (Observation 4.5): Only whether it moved.
- A factor of $114$ from quasars (Check 5.1): Not adjudicated here.
- Atomic clocks are strongest (Observation 5.5): $\alpha$ changes the ticking.
- The first row is not false (Observation 6.2): It is not a sentence.
Closing: Paper 10 established that only dimensionless numbers can be fine-tuned. The same restriction applies to “has it changed”——“$c$ became $1\%$ smaller” cannot be told apart from making the metre $1\%$ longer, and a sentence asserting one of two indistinguishable things has no truth value. It is not false; it is not a sentence. The question can be put only to dimensionless numbers such as $\alpha$ and $m_p/m_e$——and one of them has actually been measured. The natural fission that ran at Oklo in Gabon two billion years ago bounds the drift of $\alpha$ through the samarium-149 resonance at below $\mathbf{5.0\times10^{-18}}$ per year——stretched over the age of the universe that is only $\mathbf{7.0\times10^{-8}}$ (assuming a constant rate, and saying nothing about what came before). Eddington argued between $136$ and $137$ in the integer digit; Oklo reaches the sixth decimal——six orders below. And this paper does not try to derive $\alpha$. It looks only at whether it moved——that self-limitation is its answer to Paper 101’s caution. The quasar claim disagrees with Oklo by a factor of $\mathbf{114}$——not adjudicated here; they look at different epochs and are compatible if the rate is not constant. One thing separates them——whether a change of units erases it. It is not a matter of accuracy but of the shape of the question, and it is settled before any measurement. Paper 10 settled what can be tuned; this paper says that one of them has been measured and has not moved——beside the theorem, a number.
References
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Reference
Yamagishi, Y. (2026). There Is Only One Kind of Constant You Can Ask “Has It Changed?” About [Paper 314]. YOI: https://www.yamagishiyuuki.com/yoi/314
DOI: 10.5281/zenodo.22175192
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 314. This paper is a preprint and has not been peer reviewed.