The Dimensionless Constants

Contents

Why only dimensionless numbers carry content
The systematic construction
Universality: the numbers a stranger could check
Relations to mathematical constants
Summary

The Free Parameters of Physics tabulates the free parameters of the Standard Model and of $\Lambda$CDM with their measured values. This appendix asks a different question about the same numbers: which of them mean anything independently of us. The answer is sharp — only the dimensionless ones — and it has a formal statement, a complete construction, and a consequence that reaches beyond physics into what could in principle be compared with a civilization that shares none of our history.

Why only dimensionless numbers carry content

The speed of light in vacuum is exactly $299792458\,\mathrm{m}/\mathrm{s}$. That statement contains no physics whatever. Since 1983 the metre has been defined as the distance light travels in $1/299792458$ of a second (Measurement, SI Units, and the Theory of Errors), so the number records a decision of the Conférence Générale des Poids et Mesures, not a property of light. Had the committee chosen differently, $c$ would read differently and not one experimental outcome would change.

This is not a peculiarity of the SI. Any dimensional constant carries the same defect in weaker form: its numerical value is a statement about the relation between the quantity and an arbitrarily chosen yardstick.

Remark D.1 (Convention and content).

The distinction is the one drawn in Section 1.3.2 between convention and content, and it is why that section insists the conventions be declared. A theory's empirical content is what survives every allowed change of convention. For units, the allowed changes are rescalings of the base quantities, and what survives them is exactly the dimensionless combinations.

The dimension map

Make the claim precise. Let $q_{1},\ldots,q_{n}$ be physical quantities and let there be $k$ base dimensions — in mechanics $\mathsf{M},\mathsf{L},\mathsf{T}$, so $k=3$; in the SI, seven. Each $q_{a}$ has a dimension

\begin{equation}\tag{D.1} \left[q_{a}\right] =\mathsf{X}_{1}^{\,d_{1a}}\mathsf{X}_{2}^{\,d_{2a}} \cdots\mathsf{X}_{k}^{\,d_{ka}}\ec \end{equation}

with rational exponents $d_{ia}$. A monomial in the quantities,

\begin{equation}\tag{D.2} \Pi=q_{1}^{\,p_{1}}q_{2}^{\,p_{2}}\cdots q_{n}^{\,p_{n}}\ec \qquad p\in\Q^{n}\ec \end{equation}

then has dimension $\mathsf{X}_{i}^{\,\sum_{a}d_{ia}p_{a}}$. Writing $D$ for the $k\times n$ matrix of exponents $d_{ia}$, the monomial is dimensionless precisely when

\begin{equation}\tag{D.3} Dp=0\ec \end{equation}

so the dimensionless combinations are the kernel of a linear map. Taking logarithms has turned a multiplicative question into linear algebra, and Figure D.1 is the whole content of the subject in one picture.

[figure: dimension-kernel.pdf]

Dimensional analysis as linear algebra. Monomials in the $n$ quantities are represented by their exponent vectors in $\Q^{n}$; the dimension map $D$ records the exponents of the $k$ base dimensions. The dimensionless combinations are exactly $\ker D$. A change of units acts on $\Q^{n}$ in a way that leaves the kernel pointwise fixed, which is the precise sense in which only dimensionless numbers are convention-free.

Theorem D.2 (Buckingham's $\Pi$ theorem).

Let $n$ quantities involve $k$ base dimensions, and let $r$ be the rank of the dimension matrix $D$. Then the dimensionless monomials form a $\Q$-vector space of dimension $n-r$, and any physical law relating the $n$ quantities can be written as a relation among $n-r$ independent dimensionless groups [Buckingham:1914]. Rests on Equation (D.3).

Remark D.3 (Two footnotes on the theorem's name and its statement).

The attribution is conventional rather than historical: the general statement was published by Vaschy in 1892 and, apparently independently, by Federman and Riabouchinsky in 1911, and Buckingham himself later ceded priority to Riabouchinsky. Second, the theorem is often quoted with $r$ replaced by the number of base dimensions. That is the imprecise version, and it is not Buckingham's: he defines the subtracted quantity as the size of a maximal independent set, which is the rank. Where the dimensional matrix is rank-deficient the two differ, and the careless form undercounts the groups. Bridgman's monograph [Bridgman:1922] is the standard early treatment.

Proof.

Derives Theorem D.2. That the dimensionless monomials are $\ker D$ is Equation (D.3), and the rank–nullity theorem (Linear Algebra and Representation Theory) gives $\dim\ker D=n-r$ at once. Choose a basis $\Pi_{1},\ldots,\Pi_{n-r}$ of the kernel.

For the second statement, let $f(q_{1},\ldots,q_{n})=0$ be a law. A change of units multiplies each $q_{a}$ by $\prod_{i}\lambda_{i}^{d_{ia}}$ for positive scale factors $\lambda_{i}$. If the law is to hold in every system of units — which is what it means for it to be a law about nature rather than about a bureau of standards — then $f$ must be invariant under the whole $k$-parameter group of such rescalings. The orbits of that group are the level sets of the map $q\longmapsto(\Pi_{1},\ldots,\Pi_{n-r})$, since two configurations have the same $\Pi$'s exactly when a rescaling carries one to the other. A function constant on orbits factors through the orbit space, so $f$ is a function of the $\Pi$'s alone.

∎

Example D.4 (The theorem doing work).

A pendulum's period $T$ can depend on its length $\ell$, its mass $m$ and $g$. Here $n=4$ and the dimension matrix in $(\mathsf{M},\mathsf{L},\mathsf{T})$ has rank $3$, so there is $4-3=1$ independent group. It is $\Pi=T\sqrt{g/\ell}$, whence $T=C\sqrt{\ell/g}$ with $C$ a pure number — and the mass cannot appear. Two lines of linear algebra have produced the form of the answer, the independence of the period from the mass, and the impossibility of any other scaling, without solving an equation of motion (Oscillations and Mechanical Waves supplies $C=2\pi$ for small amplitude).

How many dimensionful constants are fundamental?

If dimensional constants are conventions, why keep any? The question was put sharply in a three-way exchange between Duff, Okun and Veneziano, who argued respectively for zero, three and two fundamental dimensionful constants [Duff:2002]. The disagreement is not empirical: all three agree on every measurable prediction. It is about what one calls fundamental, and it resolves once the roles are separated.

One caveat on that source: Veneziano's count of two is explicitly conditional on superstring theory, which this treatise excludes from scope by editorial rule 1. His position is reported here as a position held in the literature, not as a result. And the paper cannot be cited in support of any particular count — its three authors never converged, which is precisely its value: it establishes that the count is not fixed by physics.

Constants such as $c$, $\hbar$ and $G$ do two jobs. As conversion factors they relate quantities we had historically thought distinct — $c$ converts time to length, $\hbar$ converts frequency to energy, $G$ converts mass to length — and in that role they are pure convention, and can be set to $1$, which is what natural units do. As markers of a regime they say where new physics appears: relativity when speeds approach $c$, quantum mechanics when actions approach $\hbar$, and strong gravity near the Planck scale built from all three. That second role is physical, but it is a statement about ratios — $v/c$, $S/\hbar$ — and ratios are dimensionless. Nothing is lost by the reduction.

The systematic construction

Theorem D.2 says how many independent dimensionless numbers exist; this section constructs them for the quantities physics actually has.

Planck units and the mass ratios

From $\hbar$, $c$ and $G$ alone — three quantities, three dimensions, rank three — the kernel is trivial: there is no dimensionless combination of the three. That is precisely why they can be used to build a unique system of units,

\begin{equation}\tag{D.4} m_{\mathrm{P}}=\sqrt{\frac{\hbar c}{G}}\ec\qquad \ell_{\mathrm{P}}=\sqrt{\frac{\hbar G}{c^{3}}}\ec\qquad t_{\mathrm{P}}=\sqrt{\frac{\hbar G}{c^{5}}}\ec \end{equation}

and it is why the Planck scale is the natural yardstick against which every other scale becomes a pure number. Adjoin any fourth quantity and a dimensionless group appears immediately: adjoining a mass $m$ gives the single group $m/m_{\mathrm{P}}$.

Every particle mass therefore contributes one dimensionless number. Two families of ratio are worth separating, because they are physically very different:

Ratios to the Planck mass, such as

\begin{equation}\tag{D.5} \frac{m_{e}}{m_{\mathrm{P}}}\approx4.19\times 10^{-23}\ec\qquad \frac{m_{p}}{m_{\mathrm{P}}}\approx7.69\times 10^{-20}\ec \end{equation}

which measure how feebly gravity acts on elementary particles. Their extreme smallness is the hierarchy problem (Gravitational Coupling, Planck Units, and the Hierarchy Problem). Equivalently one writes the gravitational coupling

\begin{equation}\tag{D.6} \alpha_{G}=\frac{Gm_{e}^{2}}{\hbar c} =\left(\frac{m_{e}}{m_{\mathrm{P}}}\right)^{2} \approx1.75\times 10^{-45}\ec \end{equation}

the exact gravitational analogue of Equation (D.8) below. The ratio $\alpha/\alpha_{G}\approx4\times 10^{42}$ is the number Dirac found suspicious enough to build a cosmology on [Dirac:1938]; that cosmology, which required $G$ to vary with time, is excluded by the bounds of Is it actually constant?.
Ratios among the particles themselves, which involve no gravity at all:

\begin{equation}\tag{D.7} \frac{m_{p}}{m_{e}}\approx1836.15\ec\qquad \frac{m_{\mu}}{m_{e}}\approx206.768\ec\qquad \frac{m_{\tau}}{m_{\mu}}\approx16.817\ep \end{equation}

These are the numbers that fix chemistry, the size of atoms relative to nuclei, and the pattern of the three generations. In the Standard Model they are not independent inputs but consequences of the Yukawa couplings, themselves dimensionless (Electroweak and Higgs).

The couplings

Definition D.5 (Fine-structure constant).

\begin{equation}\tag{D.8} \alpha:=\frac{e^{2}}{4\pi\varepsilon_{0}\hbar c}\ep \end{equation}

Every dimensional quantity in Equation (D.8) cancels: $\alpha$ is a pure number, the same in SI, Gaussian or natural units, and the same for any observer anywhere. It measures the strength of the electromagnetic interaction — more precisely, it is the square of the electric charge in units where $\hbar=c=1$, and it is the expansion parameter of the perturbation series of quantum electrodynamics (Quantum Electrodynamics and Renormalization). Its smallness is why that series works.

The recommended value is

\begin{equation}\tag{D.9} \alpha^{-1}=137.035999177(21)\ec \end{equation}

a relative uncertainty of $1.6\times 10^{-10}$ [Mohr:2025].

Remark D.6 (The last digits are less settled than they look).

Equation (D.9) is not a straight statistical combination. The two most precise determinations — caesium recoil at Berkeley, $\alpha^{-1}=137.035999046(27)$ [Parker:2018], and rubidium recoil at the Kastler–Brossel laboratory, $\alpha^{-1}=137.035999206(11)$ [Morel:2020] — disagree with each other by more than $5\sigma$, and the disagreement is unresolved. The recommended uncertainty in Equation (D.9) has been inflated by an expansion factor to cover it. Quoting ten significant figures without that caveat would misrepresent the state of the measurement, which is precisely the kind of thing the availability grading of Epistemology and the Scientific Method exists to prevent.

The other interactions contribute their own couplings, $\alpha_{s}$ for the strong force and the electroweak pair $g,g'$; unlike $\alpha$ they are not small at all accessible energies, which is why perturbation theory fails for the strong interaction at low energy (Quantum Chromodynamics).

The full catalogue

Collecting: the dimensionless numbers of established physics are the gauge couplings, the Yukawa couplings (equivalently the mass ratios), the CKM and PMNS mixing angles and phases, the strong CP angle $\theta_{\mathrm{QCD}}$, the Higgs self-coupling and the ratio of the Higgs vacuum expectation value to the Planck mass, together with the cosmological parameters — the density fractions $\Omega_{i}$, the baryon-to-photon ratio $\eta$, the scalar spectral index $n_{s}$. Their measured values occupy The Free Parameters of Physics; the count is about twenty-six for the Standard Model plus a handful for cosmology.

Remark D.7 (What a ``theory of everything'' would have to do).

The catalogue is the honest statement of what physics does not explain. Feynman's verdict on $\alpha$ — “a magic number that comes to us with no understanding by man” [Feynman:1985] — applies to every entry. A theory that derived even one of them from something more basic would be a decisive advance, and none has. This is worth stating plainly because it is often obscured: the Standard Model is not incomplete in the sense of failing a measurement, but in the sense that roughly two dozen numbers must be handed to it from outside. Programmes that claim to derive them are assessed on that claim and no other, and Quantum Gravity: The Honest Status records that none currently succeeds.

Universality: the numbers a stranger could check

Why $\alpha$ and not a metre

Suppose we wished to establish, with a civilization that shares nothing of our history, that we had found the same physics. Almost everything we might send fails. A length in metres is meaningless without the metre. A temperature in kelvin, a mass in kilograms, an energy in joules: all encode the Earth, the water molecule, a platinum cylinder, a French committee. Worse, much of what we would think to describe is local: the luminosity of our star, the surface gravity of our planet, the composition of our atmosphere, the strength of our magnetic field. A recipient orbiting a different star has different values for every one of them, and no disagreement about those would indicate any disagreement about physics.

What survives is $\ker D$. The dimensionless constants are the same for them as for us — not by convention but because they are what remains when all conventions are quotiented away. If their measurement of $\alpha$ disagreed with ours, that would be a discovery of the first order; if their measurement of the local gravitational acceleration disagreed with ours, that would be a weather report.

The message that needs no dictionary

$\alpha$ can be communicated with no shared units, no shared numerals and no shared language, because it is a ratio of counts, and counting is not a convention. Figure D.2 draws it: one stroke, a divider, one hundred and thirty-seven strokes.

[figure: alpha-tally.pdf]

The fine-structure constant drawn without conventions. Above: the same count of $137$ grouped in fives, in eights, and irregularly — the grouping is notation and varies freely, the total does not. Below: the invariant itself, one stroke against $137$. A recipient needs no unit, no base, no numeral system and no language, only the ability to count. The approximation is discussed in Remark D.8.

This is not an idle conceit. The Pioneer plaque used exactly this strategy, taking as its unit the hyperfine transition of neutral hydrogen — the most common object in the universe, with a frequency any radio astronomer anywhere must know — and expressing every other quantity as a multiple of it [Sagan:1972]. The design principle is the one above: build the message out of $\ker D$, never out of a chosen yardstick.

A primer built from nothing

A single ratio is not a message. To make $\alpha$ mean anything to a recipient one must first establish what a count is, what a ratio is, and what the two things are whose coupling is being reported — and each of those must be established using only what has already been established. Figures D.3 and D.4 carry that construction out.

The governing constraint is strict and is what makes the exercise instructive: no glyph may be used before a panel has defined it by exhibiting instances. Nothing is translated, because there is no shared language to translate into; everything is shown. That forces an order. Counting must come before equality, since equality is defined as sameness of count. Equality must come before the marks for TRUE and FALSE, which are introduced by attaching them to one true and one false statement — an ostensive definition, and the only kind available. Only then can anything be asserted, so only then can the naturals, sets, division and the reals follow.

[figure: xeno-board-1.pdf]

An ostensive primer, first board: mathematics. Counting is established by tally, then the equality sign as sameness of count, then the marks for TRUE and FALSE by exhibiting a true and a false statement. Those three panels make assertion possible; the rest follow. The unit square in panel 7 is drawn to scale, its side equal to the $0$-to-$1$ spacing of the line, so the transferred diagonal really does land at $\sqrt{2}$ — a point no sharing of counts reaches, which is what forces $\R$ beyond $\N/\N$.

[figure: xeno-board-2.pdf]

Second board: physics and language. The electron is introduced as a thing of which every instance is identical, the photon as a thing with no mass moving always at the greatest speed, and $\alpha$ as the strength of the single joining by which the two act on each other — which is what $\alpha$ is, before it is a number. Panel 11 then gives its size as a count. Panels 12 and 13 attach a written language to referents already fixed on the boards: an alphabet in isolation is uninterpretable, so the words are grounded in things the recipient has just been shown.

Two features of the second board deserve comment. First, $\alpha$ is introduced as a coupling before it is a number: panel 10 shows the vertex at which an electron emits or absorbs a photon, and identifies $\alpha$ as how strongly that joining happens. This is the right order, because Equation (D.8) is not a definition a stranger could parse — it presupposes $e$, $\varepsilon_{0}$, $\hbar$ and $c$, none of which is available — whereas the vertex is a picture. Second, the alphabet is introduced last and is anchored: panel 13 pairs each written word with the thing already drawn, because a list of 26 shapes carries no information whatever until at least one of them is tied to something the recipient can independently identify. That is the same requirement Frege's sense and reference imposes (Section 3.7.1), arriving here as an engineering constraint rather than a philosophical one.

What the boards cannot do is worth stating as plainly as what they can. They convey $\alpha\approx1/137$ and not $\alpha^{-1}=137.035999177(21)$; they do not specify the energy scale at which the coupling was measured, for the reason given in Remark D.8; and they assume the recipient shares our arithmetic, which is the one assumption the construction cannot itself justify. They are a demonstration of what a convention-free message would have to look like, not a protocol ready to transmit.

Remark D.8 (Three honest qualifications).

The picture is a good deal less tidy than the slogan, and the qualifications matter more than the slogan does.

First, $137$ is not the value. The measured quantity is $\alpha^{-1}=137.035999$ to the digits shown, and no tally can draw a non-integer. What Figure D.2 transmits is a rational approximation good to about four parts in $10^{5}$ — impressive for a row of scratches, useless for physics. Eddington's insistence that the value must be exactly an integer, first $136$ and then $137$ when measurement moved, is the cautionary tale here [Eddington:1946]: the integer was the artefact, and the digits after the decimal point were the physics.

Second, $\alpha$ runs. The number $1/137$ is the low-energy limit, measured at zero momentum transfer. At the mass of the $Z$ boson the same coupling is near $1/128$ (The Renormalization Group). A civilization measuring at a different scale and reporting the result without saying so would appear to disagree with us while agreeing perfectly. Any serious message would have to specify the scale — which can itself be done dimensionlessly, as a ratio to a particle mass.

Third, agreement is expected, not guaranteed. That $\alpha$ takes the same value everywhere is an empirical claim, and Is it actually constant? reports how well it is established. It is exactly the kind of claim that would be most interesting to find false.

Is it actually constant?

The universality argument is worth only as much as the evidence that $\alpha$ does not vary. Three independent probes bound it.

Laboratory clocks give the cleanest bound, with no astrophysical modelling at all. Comparing an aluminium-ion with a mercury-ion optical clock over a year — transitions whose frequencies depend on $\alpha$ with different powers — gives

\begin{equation}\tag{D.10} \frac{\dot{\alpha}}{\alpha} =-1.6(23)\times 10^{-17}\,/\mathrm{yr}\ec \end{equation}

consistent with zero [Rosenband:2008] [Uzan:2011].
The Oklo natural reactor. Isotope ratios left by a natural fission reactor that operated in Gabon about $2\times 10^{9}$ years ago constrain the position of a samarium neutron resonance, and through it $\alpha$ at that epoch [Damour:1996]:

\begin{equation}\tag{D.11} -0.9\times 10^{-7}<\frac{\Delta\alpha}{\alpha}<1.2\times 10^{-7} \end{equation}

at two standard deviations.
Quasar absorption spectra probe $\alpha$ at large redshift and in distant directions — the only probe that tests spatial universality. Here the evidence is contested, and must be reported as such. The many-multiplet analyses of Keck and VLT data are mutually inconsistent: Keck/HIRES gives $\Delta\alpha/\alpha=-0.57(11)\times 10^{-5}$, a nominal five-sigma detection of variation, while VLT/UVES on the same method gives $-0.06(6)\times 10^{-5}$, consistent with none [Uzan:2011]. The proposed reconciliation — that $\alpha$ varies spatially, the two telescopes viewing different hemispheres — was reported as a $4.2\sigma$ dipole [Webb:2011]. That claim has since been substantially undercut, and from inside the original collaboration: long-range wavelength-scale distortions of order $\pm200\,\mathrm{m}/\mathrm{s}$ per $100\,\mathrm{nm}$ are ubiquitous in both instruments' archives, and reproduce important features of the reported signal [Whitmore:2015].

The laboratory and geophysical bounds are robust and consistent with no variation; the astrophysical detection is contested and is recorded here as contested rather than omitted, because a claim of that kind is exactly what the universality argument stands or falls on. Reporting only the comfortable evidence would be a failure of the editorial rule, and the contested item is in any case the more interesting one: it is the one that could still turn out to matter.

The universality of Why $\alpha$ and not a metre is therefore an experimentally supported statement about this universe, not an assumption — and it is falsifiable, which by Remark 1.2 is the most that can be asked of it.

Relations to mathematical constants

If the dimensionless constants are pure numbers, it is natural to ask whether they are recognizable numbers — whether $\alpha$ or the mass ratios are built from $\pi$, from $e$, from small integers. The history of that question is mostly a history of error, and it is worth setting out carefully, because the failure mode is seductive and the criterion for distinguishing sense from nonsense is sharp.

When a mathematical constant legitimately appears

Mathematical constants appear throughout physics with impeccable credentials. The $2\pi$ in the pendulum period of Example D.4 comes from integrating the equation of motion; the $\pi$ in the Gaussian integral of Probability and Statistics comes from the integral; the $4\pi$ in Equation (D.8) comes from the surface area of a sphere in Gauss's law. In every case the constant enters as the output of a derivation, and one can point to the step that produced it.

Definition D.9 (The criterion).

A proposed relation between a measured dimensionless constant and a mathematical one is physics if and only if it is derived — if there is an argument from stated premises whose conclusion is the relation, and which would have predicted the value before it was measured. A relation obtained by searching for a formula that reproduces a known value is numerology, however many digits it matches.

The asymmetry is the same one that separates a prediction from a fit (Section 1.4), and the reason for the strictness is counting: the supply of short expressions built from $\pi$, $e$ and small integers is enormous, so agreement to a few digits with some such expression is expected by chance and carries no information.

Eddington, and the failure mode

Eddington argued that $\alpha^{-1}$ must be exactly the integer $136$, on a group-theoretic counting argument; when measurements moved, he produced a revised derivation giving exactly $137$ [Eddington:1946]. The modern value is $137.035999$, and neither derivation survives. The instructive part is not that he was wrong but how: the argument was adjusted to track the data while retaining the appearance of derivation, which is the signature Definition D.9 is designed to catch.

The Koide relation

The most serious modern case is Koide's [Koide:1982] [Koide:1983a]. Form, from the three charged-lepton masses, the ratio

\begin{equation}\tag{D.12} Q:=\frac{m_{e}+m_{\mu}+m_{\tau}} {\left(\sqrt{m_{e}}+\sqrt{m_{\mu}}+\sqrt{m_{\tau}}\right)^{2}}\ep \end{equation}

$Q$ is dimensionless by construction, and it is bounded: writing $v_{i}=\sqrt{m_{i}}$, Equation (D.12) is $\abs{\vect{v}}^{2}/\left(\vect{v}\cdot\vect{u}\right)^{2}$ with $\vect{u}=(1,1,1)/\sqrt{3}$ up to normalization, so by Cauchy–Schwarz

\begin{equation}\tag{D.13} \frac{1}{3}\leq Q\leq1\ec \end{equation}

the lower bound attained when the three masses are equal and the upper when one dominates. The measured value sits almost exactly at the midpoint of that range: with the current pole masses,

\begin{equation}\tag{D.14} Q_{\text{obs}}=0.6666645(51)\ec\qquad \frac{2}{3}=0.6666667\ec \end{equation}

a deviation of $0.43\sigma$, or agreement at the level of a few parts in $10^{6}$.

What should be made of it? Three observations, in the spirit of Definition D.9.

The agreement is genuinely striking, and much better than the bounds Equation (D.13) alone would make likely. It has the predictive character Definition D.9 demands: the relation was written down when the $\tau$ mass was poorly known, and it survived the subsequent measurements — indeed the agreement improved when the world average moved after the Belle II determination. That is the reason it is discussed here rather than dismissed.
It has no accepted derivation. Lepton masses in the Standard Model are free Yukawa parameters with no reason to satisfy any relation, and no mechanism in or beyond it produces Equation (D.14).
The precision is a fact about pole masses, and only about pole masses — which is the crux, and is usually stated carelessly. Masses run with scale (The Renormalization Group), so “the mass of the electron” is a function of where it is defined. Two results must be held together. Li and Ma showed the running Koide ratio is very nearly scale-independent, drifting by less than one part in $10^{5}$ from $1\,\mathrm{GeV}$ to $2\times10^{16}$\,$\mathrm{GeV}$ [Li:2006] — but the value it is frozen at is about $0.19\,\mathrm{\%}$ above $2/3$, not $2/3$. Sumino showed the same thing from the other side: the QED self-energy term $-\tfrac{3\alpha}{4\pi}m_{i}\log m_{i}^{2}$, coming from the infrared region of the loop, rotates the vector $(\sqrt{m_{e}},\sqrt{m_{\mu}},\sqrt{m_{\tau}})$, so that if the running masses satisfied the relation at a high scale the pole masses would violate it by roughly $0.1\,\mathrm{\%}$ — about $120$ times the experimental error [Sumino:2009].

The two are not in conflict, and it is worth being explicit because they are routinely quoted against each other: the running ratio is stable but sits off $2/3$, while the pole ratio sits on $2/3$. Nothing carries the running value onto $2/3$. And that is precisely what is suspicious — pole masses are infrared-sensitive, low-energy quantities that no known symmetry singles out, whereas a fundamental relation would be expected to hold at a high scale where a symmetry is unbroken.
The extension to quarks does not work, and fails asymmetrically: $Q\approx0.669$ for the heavy triplet and $Q\approx0.57$ for the light one, and it must be applied to running masses there — the opposite prescription from the charged leptons. A relation needing a different mass definition in each sector to look good is a relation whose status is unresolved.

The honest verdict is that the Koide relation is an unexplained numerical regularity of the right shape to be meaningful, which in four decades has been neither derived nor explained away. Koide himself allows that it “may be an accidental coincidence”. This treatise records it as exactly that, and Definition D.9 says what would have to be supplied to settle it: a derivation, from stated premises, of why the relation should hold for pole masses.

Remark D.10 (A cautionary detail about which paper says what).

The relation $Q=2/3$ is that of Koide's earlier “model I” [Koide:1982] [Koide:1983a]. The often-cited 1983 Brief Report [Koide:1983b] proposes a modified relation with an extra parameter $\delta=0.040026$, giving $Q=(2+\delta^{2})/3$ and predicting $m_{\tau}=1786.45\,\mathrm{MeV}$ — which the measured $1776.93(9)\,\mathrm{MeV}$ excludes overwhelmingly. Citing that paper for $Q=2/3$, as is common, attributes the surviving relation to the one Koide paper whose headline prediction is dead.

Fine-tuning and the anthropic move

Several of the constants appear to lie in narrow windows outside which complex structure would not form. Weinberg's bound on the cosmological constant is the sharpest instance: he argued, before the measurement, that $\Lambda$ could not greatly exceed a value set by the requirement that galaxies form, and the measured value (Lambda-CDM Cosmological Parameters) is of that order [Weinberg:1987].

Whether such reasoning explains anything is disputed, and this treatise does not adjudicate it. What can be said within its editorial rules is narrow and worth saying: an anthropic argument requires an ensemble of universes with varying constants for the selection to act on, and there is at present no observational evidence that such an ensemble exists. It therefore falls under the same rule as the programmes named in Quantum Gravity: The Honest Status — it may be named, and it is named here, but it is not evidence-based physics and is not presented as such. The measured values in The Free Parameters of Physics stand as measurements whatever their eventual explanation.

Summary

The content of physics that is independent of every human choice is exactly $\ker D$: the dimensionless combinations. There are about thirty of them in established physics; none is derived; and they are the same throughout the observable universe to the precision of Is it actually constant?. They are therefore simultaneously the deepest unexplained facts we possess and the only quantitative statements we could compare, without any shared convention, against a measurement made by anyone else anywhere.

Figure D.2 is the smallest complete expression of that idea: one stroke, then one hundred and thirty-seven. It is not the value of $\alpha$, and it does not say at what scale it was measured. But it is the part of our physics that needs no dictionary — and the part that a stranger could confirm or refute.