c·t = CONST, THAT CLICKS BONUS ③ ／ dug after the main series closed
The zero column was not homogeneous, and almost no constants survived
Perhaps there is
not a single constant
A dimensionless quantity is what you can send by radio — but there are four kinds, with different groups.
And most of what we call constants turned out to be running functions.
For 50 episodes the backbone was "dimensionless is physics, dimensionful is bookkeeping". But what is a dimensionless quantity? And are they really constants? Pressing on both questions gave: the "zero column" is not homogeneous, the logarithm is not a discovery, and almost no constants survive.
01A dimensionless quantity is what you can send by radio
Conclusion of §01
A dimensionless quantity is one that can be conveyed without shipping anything.
── That was the content of "physics".
02But the "zero column" is not homogeneous
| Type | Examples | Group | Natural (Haar) measure | Total measure |
|---|---|---|---|---|
| Ratios (scale-like) | \(\alpha\), \(m_p/m_e\), \(\rho_\Lambda/\rho_P\) | multiplicative \(\mathbb{R}_+\) | \(d(\ln x)\) = log-uniform | divergent |
| Angles and phases | \(\theta_{\rm QCD}\), CKM/PMNS | compact, \(U(1)\) etc. | \(d\theta\) = uniform | finite |
| Counts | 3 generations, 3 colours, \(D=4\) | discrete | counting | finite |
| Exponents (log-derivatives) | \(n_s\), \(\nu\), \(\eta\), \(\omega\) | a tangent space | hard to fix | ── |
For 50 episodes these were treated as one column. They are at least four kinds, and the groups differ.
03"Is it really a logarithm?" — the log is not a discovery, it is an isomorphism
log-uniform is "natural" because it is the Haar measure of the multiplicative group — not a discovery, an isomorphism
Conclusion of §03
Bits are not the logarithm of the quantity. They are the logarithm of a probability.
To measure \(-\log_2(\text{probability})\) you need a measure, and the group fixes the measure.
── That is what this series has been doing for 50 episodes.
04The core — which turns Episode 48's criterion into a theorem
The first main point of this episode
Episode 48's "an angle has a reason for its prior, a mass ratio does not" was —
a restatement of compact versus non-compact.
── Not a criterion. A theorem.
05Testing it — within the compact class, do bits predict concern?
| Angle | value [deg] | surprise [bits] | how it is actually treated |
|---|---|---|---|
| PMNS \(\theta_{23}\) | \(49.0\) | \(0.88\) | near maximal mixing (a different kind of question) |
| PMNS \(\theta_{12}\) | \(33.4\) | \(1.43\) | not regarded as a problem |
| CKM \(\theta_{12}\) (Cabibbo) | \(13.04\) | \(2.79\) | not regarded as a problem |
| PMNS \(\theta_{13}\) | \(8.57\) | \(3.39\) | smallish; discussed |
| CKM \(\theta_{23}\) | \(2.38\) | \(5.24\) | part of the flavour hierarchy |
| CKM \(\theta_{13}\) | \(0.201\) | \(8.81\) | the heart of the flavour puzzle |
| \(\theta_{\rm QCD}\) | \(<10^{-10}\) | \(35.87\) | the only "crisis" |
Conclusion of §05
Within the compact class, bits track concern monotonically — 0 to 3 = nobody mentions it; 5 to 9 = the flavour puzzle; 36 = a crisis.
And the one uncontested fine-tuning problem is the one angle, \(\theta_{\rm QCD}\).
── Everything contested (\(v/M_P\), \(\rho_\Lambda\)) is in the non-compact class.
They are not being argued over. The question is not well-posed, so it cannot settle.
Figure: the dimensionless quantities split into two classes. On the left (compact) bits are defined and track concern; on the right (non-compact) the bits themselves move with the prior. Move the prior-range slider — the left does not budge; only the right does.
06The next question — are they constants?
| Category | count | constant with respect to what | verdict |
|---|---|---|---|
| Running couplings (3 gauge, 9 Yukawa, \(\lambda\), and the mixing angles run weakly) | \(\approx24\) | they change with scale | not constants |
| The six \(\Lambda\)CDM parameters | \(6\) | a description of this universe's state | not constants of law |
| \(\theta_{\rm QCD}\) | \(1\) | an RG-invariant angle | a genuine constant |
Most of what we call constants are running functions. As Episode 37 showed, \(\alpha\) moves 7 per cent between \(0\) and \(M_Z\) — when we say "\(\alpha=1/137\)" that number includes the convention of having chosen \(M_Z\). And the six \(\Lambda\)CDM parameters are different again: they are initial conditions of this universe, not laws. We do not call the Earth's orbital radius a constant of nature, for the same reason.
07So what is genuinely invariant?
| Quantity | independent of |
|---|---|
| Critical exponent \(\nu=0.6300\) | scale, scheme, microscopic content (Ep. 44) |
| Anomalous dimension \(\eta=0.0363\) | the same (Ep. 14) |
| Correction exponent \(\omega=0.8303\) | the same |
| Anomaly coefficients \(a\), \(c\) | fixed by the field content alone (Ep. 37) |
| \((D-1)(D-2)=6\) | fixed by the dimension alone (Ep. 38) |
Conclusion of §07
But — these are not constants of nature. They are theorems.
\(\nu=0.6300\) is a mathematical fact about the 3D Ising fixed point, not a measured input.
── What is genuinely invariant turned out not to be a constant but a theorem.
08One remains — \(\theta_{\rm QCD}\). And then
The second main point of this episode
"It cannot be that there are many constants — perhaps there is not a single one" is largely right.
More precisely — among independent inputs, almost nothing is scale-independent. What looks like a constant is an output (a theorem), a fact about this universe, or the value of a running function.
09And a correction to bonus ②
The numbers (\(408\to8.67\) bits) do not change. What changes is the subject of the sentence. ── Episode 3's practice exactly: if you have not named what you are comparing to (here, the prior), you have not yet made a sentence. Bonus ② was loose about that, so it is marked rather than deleted.
(1) Physical mass ratios really are RG-invariant. \(m_p/m_e\) is a ratio of pole masses and does not run — it is a genuine dimensionless constant. So "zero constants" is an overstatement, and §08 has to be restricted to "among independent inputs". The reply that \(m_p/m_e\) is an output (in principle computable from the Lagrangian) is a weak reply — in practice it is not computed, so for now it must be treated like an input.
(2) "Running" is itself half a convention. Choosing \(\mu\) is a human act, and RG-invariant combinations can always be formed — but they tend to come out dimensionful, like \(\Lambda_{\rm QCD}\), and dimensionful is bookkeeping (Episode 3). That round trip is this series' whole subject.
(3) Calling the six \(\Lambda\)CDM parameters "state" may be too strong. \(n_s\) is an initial condition but also a prediction of inflationary models — the border between law and state cannot be drawn sharply and depends on disciplinary custom.
(4) §05's monotonicity is an observation on seven cases. The "how it is treated" column summarises the mood of the literature and carries my bias (the same caveat as Episode 36). PMNS \(\theta_{23}\) is discussed for being near maximal, not for being small — it is not measured on the same ruler.
(5) §04's Haar-measure argument is standard mathematics, but reading it as "therefore the fine-tuning question is well-posed or ill-posed" is this series' move — statistics has finer machinery (invariant priors, Jeffreys priors). All that is claimed here is the single point that compact means normalisable and non-compact does not.
Exercises
- What is a dimensionless quantity, operationally?
Show the answer
Something you can send by radio — conveyed without shipping an object. "One metre" cannot be sent; "\(\alpha=1/137.036\)" can. The same reason the SI could decree \(c\) but not \(\alpha\) (Episode 47). - Why is the logarithm natural for ratios but not for angles?
Show the answer
Ratios form the multiplicative group \(\mathbb{R}_+\), whose Haar measure is \(d(\ln x)\) — the log is not a discovery but the isomorphism to the additive group. Angles form a compact group whose Haar measure is \(d\theta\) — taking a log has no meaning. - What was Episode 48's "is there a reason for the prior?" a restatement of?
Show the answer
Compact versus non-compact. Compact → finite Haar measure → normalisable → the prior is uniquely fixed → the question is well-posed. Non-compact → not normalisable → ill-posed. It was a theorem, not a criterion. - Why is \(\theta_{\rm QCD}\) the only uncontested fine-tuning problem?
Show the answer
Because it is the only one in the compact class. For \(v/M_P\) and \(\rho_\Lambda\) the bit count itself moves with the prior — they are not being argued over; the question is not well-posed, so it cannot settle. - (Harder) How far is "perhaps there is not a single constant" correct?
Show the answer
Largely correct. Of the 32, about 24 are running functions, 6 are this universe's state, and what is genuinely invariant (\(\nu\), \(\eta\), \(\omega\)) is a theorem, not a constant. The remaining \(\theta_{\rm QCD}\) is precisely what the axion would make dynamical — if PQ is right, none remain. But per honest line (1), \(m_p/m_e\) really is RG-invariant, so the claim must be restricted to independent inputs.
Summary: the zero column was not homogeneous, and almost no constants survived
A dimensionless quantity is what you can send by radio — conveyed without shipping anything. That was the content of "physics".
But the zero column is not homogeneous. There are ratios, angles, counts and exponents, and the groups differ. Ratios form the multiplicative group \(\mathbb{R}_+\); angles form a compact group. And the logarithm is not a discovery but the isomorphism from the multiplicative group to the additive one — log-uniform is natural because it is \(\mathbb{R}_+\)'s Haar measure, while taking the log of an angle means nothing. Bits were never the log of the quantity; they were the log of a probability.
That turns Episode 48's criterion into a theorem: compact → finite Haar measure → normalisable → the prior is uniquely fixed (well-posed); non-compact → not normalisable → not fixed (ill-posed). Testing it, within the compact class bits track concern monotonically (PMNS \(\theta_{23}\) 0.88 → CKM \(\theta_{13}\) 8.81 → \(\theta_{\rm QCD}\) 35.87), and the one uncontested fine-tuning problem is the one angle. Everything contested is non-compact — not argued over, but unable to settle.
And "are they constants?" Re-sorted, about 24 are running functions, 6 are this universe's state, and one remains: \(\theta_{\rm QCD}\). What is genuinely invariant (\(\nu\), \(\eta\), \(\omega\), \(a\), \(c\)) is a theorem, not a constant — as \(\pi\) is a theorem about circles.
And finally — that \(\theta_{\rm QCD}\) is exactly what the axion would turn into a field. If Peccei–Quinn is right, not a single constant remains. The intuition that "it cannot be that there are many constants" was largely right.
Bonus ②: a hierarchy shrinks to its own logarithm (the arithmetic stands; the reading is corrected here).
Bonus ③: the zero column is not homogeneous, Episode 48's criterion was a theorem, and almost no constants survive.
── All three stand on the single procedure of Episode 3. And in the third, that procedure deleted one of the criteria this series had built. The tool is still working.
This document is bonus episode ③ of "c·t = const, That Clicks", written after the main 50 episodes closed, for physics-minded high-school and university readers. The numbers are computed in kenshou/calc58.py and calc59.py. Haar measure, the running of couplings under the renormalisation group, the universality of critical exponents and the Peccei–Quinn mechanism are all standard material. The strongest objection is honest line (1): \(m_p/m_e\) is a ratio of pole masses and really is RG-invariant — a genuine dimensionless constant — so "zero constants" is an overstatement and the claim must be restricted to independent inputs (and the reply that it is an "output" is weak, since in practice it is not computed). "Running" is itself half a convention: RG-invariant combinations can always be formed, but they tend to come out dimensionful. Calling the six \(\Lambda\)CDM parameters "state" may be too strong — \(n_s\) is also a prediction of inflationary models, and the law/state border cannot be drawn sharply. §05's monotonicity is an observation on seven cases whose "how it is treated" column carries my bias, and PMNS \(\theta_{23}\) is discussed for being near maximal rather than small, so it is not on the same ruler. §04's Haar-measure argument is standard mathematics, but reading it as "therefore well-posed or ill-posed" is this series' move — statistics has finer machinery (invariant and Jeffreys priors); the only claim made here is that compact means normalisable and non-compact does not. ── To make a PDF, use your browser's Print dialogue (sliders freeze and answers are hidden in the print version).