c·t = CONST, THAT CLICKS BONUS ① ／ dug after the main series closed

Turning "cannot be told apart" into "here is what can"

What it means
for mass to vary $\mu=m_p/m_e$ does not move under a conformal transformation — which is why nothing can be told apart.
Digging turned up a degeneracy we had not expected.

What you need: Episode 16's weight table, Episode 30's amplification, Episode 37's anomaly, Episode 48's priorsThe worst direction is 99.6% $m_q/\Lambda$ — a 10.2-bit hole

Across 50 episodes this series kept saying it: "you cannot tell whether space expanded or atoms shrank". That is correct. So what can be told apart? This bonus episode digs into that one point. The answer lies in $\mu=m_p/m_e$ — and digging turned up a degeneracy we had not expected.

01Why $\mu$?

Under a conformal transformation, mass has weight $-1$ (Episode 16) $$\tilde m_p=\Omega^{-1}m_p,\qquad \tilde m_e=\Omega^{-1}m_e\qquad\Longrightarrow\qquad \frac{\tilde m_p}{\tilde m_e}=\frac{m_p}{m_e}$$

$\mu=1836.15267$ has weight $0$ — no pure conformal transformation can move it

That is exactly why "expansion versus shrinking" cannot be told apart. All masses move with the same weight, so the ratio does not change. Which also means — if $\mu$ ever moved, it was not a conformal transformation.

02But $m_p$ and $m_e$ come from different places

$m_e=y_e\,v$ — 100 per cent of Higgs origina Yukawa coupling and a vacuum expectation value, nothing else

$m_p$ — 87 to 93 per cent is $\Lambda_{\rm QCD}$the Higgs part is only the sigma terms, 65–120 MeV out of 938.3

And that $\Lambda_{\rm QCD}$ is born of dimensional transmutation= a product of the trace anomaly (Episode 37)

Conclusion of §02

$\mu$ is the ratio of anomaly-generated mass to Higgs-generated mass.
── Episode 37 said "quantum theory writes a number into the zero column";
$\mu$ is the only ratio with that number in its numerator and the Higgs in its denominator.

03Which makes $\mu$ an amplifier

Dimensional transmutation $$\Lambda_{\rm QCD}=M\exp\!\left(-\frac{2\pi}{b_0\,\alpha_s(M)}\right) \qquad\Longrightarrow\qquad \frac{d\ln\Lambda}{d\ln\alpha_s}=\frac{2\pi}{b_0\alpha_s}$$

specified at $M_Z$, an amplification of about 7; if $\alpha_s$ and $\alpha$ unify, the literature estimates $R\approx30\text{–}50$ (model-dependent)

Episode 30's "amplification factor $K$" reappears — this time on the theory side. Oklo was strong not because the measurement was precise but because it had an amplification of $10^7$; with $\mu$, the amplification is built into the theory.

04How much tighter than $\alpha$?

Measurement	Quantity	Bound	bits
Methanol masers ($z\approx0.89$)	$\Delta\mu/\mu$	$10^{-7}$	$23.3$
Ammonia absorption ($z\approx0.68$)	$\Delta\mu/\mu$	$3.5\times10^{-7}$	$21.4$
H$_2$ quasar absorption	$\Delta\mu/\mu$	$5\times10^{-6}$	$17.6$
Many-multiplet method (Ep. 30)	$\Delta\alpha/\alpha$	$10^{-5}$	$16.6$

At the same redshifts, $\mu$ is 100 times tighter. Converted through the amplification into a bound on the underlying coupling, it is 700 to 3500 times (9.5 to 11.8 bits) stronger. And the reported $\Delta\alpha/\alpha\approx6\times10^{-6}$, assuming the variation is unified, overshoots the $\mu$ bound by 420 to 3000 times.

◇　◇　◇

05The core — digging turned up a degeneracy

All of the above is known. What digging further showed is that "the constants vary" is not a one-dimensional problem. There are three fundamental directions:

Three independent directions $$x_1=\frac{\Delta\alpha}{\alpha},\qquad x_2=\frac{\Delta X_e}{X_e}\ \left(X_e=\frac{m_e}{\Lambda_{\rm QCD}}\right),\qquad x_3=\frac{\Delta X_q}{X_q}\ \left(X_q=\frac{m_q}{\Lambda_{\rm QCD}}\right)$$

Measurement	$\alpha$	$X_e$	$X_q$	$1\sigma$
Mg/Fe multiplets	$1$	$0$	$0$	$10^{-5}$
H$_2$ Lyman–Werner	$0$	$-1$	$0.048$	$5\times10^{-6}$
NH$_3$ and methanol	$0$	$-1$	$0.048$	$10^{-7}$
21 cm versus optical	$2$	$-1$	$-0.039$	$10^{-6}$
OH 18 cm	$3.70$	$-1.85$	$0.002$	$10^{-5}$

Every entry in the $X_q$ column is tiny — the proton mass depends on $m_q$ only through the sigma term, $0.048$. Building the Fisher matrix and diagonalising:

Direction	$\sigma$ along it	Composition
1st	$9.9\times10^{-8}$	almost pure $X_e$ ($0.999$)
2nd	$4.9\times10^{-7}$	almost pure $\alpha$ ($0.999$)
3rd	$1.2\times10^{-4}$	$99.6$ per cent $X_q$

The main point of this episode

A condition number of 1196 — the quark-mass direction alone is 10.2 bits worse.
And varying the sensitivity coefficients by a factor of four (sigma term $0.030$–$0.128$, $g_p$ sensitivity $-0.040$–$-0.150$)
always leaves the worst direction at least 98 per cent $X_q$ — this degeneracy is structural, not a detail of the coefficients.

Figure: how well each direction is constrained. $\alpha$ and $X_e$ are pinned down, while $X_q$ alone is a thousand times looser. Move the sigma-term slider — wherever you put it, the bad direction stays $X_q$.

sigma term $\sigma_{\pi N}/m_p$ 0.048

$\alpha$ and $X_e$ (well seen) $X_q$ (not seen)

06The invisible direction was the one that matters most

If $m_q/\Lambda$ moves	sensitivity	what it affects
The pion mass ($m_\pi^2\propto m_q$)	$0.50$	the range of the nuclear force
The deuteron binding energy	$\approx1$	nucleosynthesis itself
The Hoyle state	large	the production of carbon
The proton–neutron mass difference	large	the stability of the proton
The proton mass itself	$0.048$	the only channel spectroscopy sees

Conclusion of §06

The $0.048$ that spectroscopy sees is the dullest channel of all.
The channels nuclear physics cares about ($0.5$ to $1$) do not appear in spectroscopy at all.
── Of all the ways the constants could drift, the one with the largest consequences is the one we see least.

07But hiding in that hole is expensive

The degeneracy is a simple equation $$\frac{\Delta\mu}{\mu}=0.048\,x_3-x_2=0 \qquad\Longleftrightarrow\qquad x_2=0.048\,x_3$$

if $m_e/\Lambda$ drifts by exactly 4.8 per cent of the drift in $m_q/\Lambda$, spectroscopy sees nothing

$x_3$ to hide	precision required on the ratio	bits of tuning
$10^{-4}$	$2.1\times10^{-2}$	$5.6$
$10^{-3}$	$2.1\times10^{-3}$	$8.9$

The hole is there, but getting through it costs 6 to 9 bits of tuning. In Episode 48's terms — a model that hides in the blind spot is itself fine-tuned. That is the honest conclusion.

08And none of this touches $c\cdot t=$const

◎

$\mu$ has weight 0 — a conformal transformation does not move itso the $\mu$ bounds do not damage this series' subject at all

Not because it is strongbecause it makes no claim (Episodes 1 and 3)

VSL, by contrast, claimed $\Delta\alpha\ne0$so §04 becomes an additional debt on its ledger

Conclusion of §08 — the answer to the question of 50 episodes

"Expanded or shrank" cannot be told apart — they are the same metric read in different coordinates.
So what can be?
All masses moving with the same weight → a conformal transformation = notation (indistinguishable)
Masses moving with different weights → physics mediated by a scalar field (distinguishable)
── and the quantity that measures which is exactly $\mu$. Currently confirmed to 23.3 bits.

The honest line

(1) §05's sensitivity coefficients are literature values with real spread. The sigma term $\sigma_{\pi N}$ sits between 45 and 60 MeV depending on lattice versus phenomenology, the strange sigma term ranges more widely (20–60 MeV), and the $m_q$-sensitivity of $g_p$ is model-dependent — which is why we varied them by a factor of four, but do not trust the individual figures to their significant digits.

(2) The $1\sigma$ bounds in §05 are one representative value each. In practice they move by factors of a few between papers depending on systematics — read the condition number 1196 as "of order a thousand". The structure (that the worst direction is $X_q$) is robust; the multiplier is not.

(3) "$X_q$ is not seen" refers to spectroscopy. The Oklo natural reactor constrains $X_q$ strongly (to around $10^{-9}$) because the $^{149}$Sm resonance is sensitive to the nuclear force, and nucleosynthesis constrains it through the deuteron binding — the blind spot is in the redshift range spectroscopy reaches, not across all epochs. But those two points are widely separated and isolated.

(4) §03's amplification $R\approx30\text{–}50$ assumes grand unification. In non-unified models $R$ is entirely different and can be below 1 — so §04's "the $\Delta\alpha$ claim is killed" holds only under that assumption; read in reverse, it says "if $\Delta\alpha$ is real, the variation is not unified".

(5) There is no new physics in this document. The $\mu$ amplification, the sigma terms and the sensitivity matrix are all in the literature (Calmet & Fritzsch 2002, Langacker–Segre–Strassler 2002, and Flambaum's body of work) — this series' contribution is the restatement in §08, turning "cannot be told apart" into "here is what can be".

Exercises

Does a conformal transformation move $\mu=m_p/m_e$?
Show the answer
No. Mass has weight $-1$, so $m_p$ and $m_e$ move with the same weight and the ratio has weight $0$. That is exactly why "expansion versus shrinking" cannot be told apart — and conversely, if $\mu$ moved, it was not a conformal transformation.
How do $m_p$ and $m_e$ differ in origin?
Show the answer
$m_e=y_e v$ is 100 per cent Higgs. $m_p$ is 87–93 per cent $\Lambda_{\rm QCD}$ (the Higgs part is only the 65–120 MeV of sigma terms) — and $\Lambda_{\rm QCD}$ is a product of the trace anomaly (Episode 37). So $\mu$ is the only ratio straddling anomaly and Higgs.
What is the worst direction of the Fisher matrix, and how bad is it?
Show the answer
$99.6$ per cent $X_q=m_q/\Lambda$, with $\sigma=1.2\times10^{-4}$ — 1196 times (10.2 bits) worse than the best direction. Varying the coefficients by a factor of four leaves it $X_q$, so the degeneracy is structural.
Does that invisible direction matter physically?
Show the answer
It matters most. $m_q/\Lambda$ drives the pion mass (0.5), the deuteron binding energy ($\approx1$), the Hoyle state and the proton–neutron mass difference. Yet spectroscopy sees only the sigma term's 0.048 — the direction with the largest consequences is the one we see least.
(Harder) "Expanded or shrank" cannot be measured — so what can?
Show the answer
Whether the mass variation is universal. All masses moving alike means a conformal transformation = notation (indistinguishable); masses moving differently means physics mediated by a scalar (distinguishable) — and $\mu$ is the quantity that measures which, currently confirmed to 23.3 bits.

Summary: turning "cannot be told apart" into "here is what can"

$\mu=m_p/m_e$ has weight $0$ — no pure conformal transformation moves it. That is why "did space expand or did atoms shrink" cannot be told apart.

But $m_e$ is 100 per cent of Higgs origin while $m_p$ is 87–93 per cent $\Lambda_{\rm QCD}$ — a product of the trace anomaly (Episode 37). $\mu$ is the only ratio straddling the two origins. That makes it an amplifier (about 7 at $M_Z$, 30–50 if unified), and the observations are 100 times tighter than for $\alpha$ at the same redshifts.

Digging further turned up a degeneracy. Variation of the constants is a three-dimensional problem in $(\alpha,\ X_e,\ X_q)$, and diagonalising the Fisher matrix gives a third direction that is 99.6 per cent $X_q$, with $\sigma=1.2\times10^{-4}$ — 1196 times (10.2 bits) worse than the best. Vary the coefficients by four and the worst direction stays $X_q$: the degeneracy is structural.

And that invisible direction is the one that matters most — the pion mass, the deuteron binding, the Hoyle state. The sigma term's $0.048$ that spectroscopy sees is the dullest channel. But hiding in the hole requires tuning $x_2=0.048\,x_3$ to 6–9 bits, so a model that hides there is itself fine-tuned.

Finally, the answer to the question of 50 episodes. "Expanded or shrank" cannot be told apart — same metric, different coordinates. But whether the mass variation is universal can be. All masses alike means notation; masses differently means physics. The quantity that measures which is $\mu$, and it is currently confirmed to 23.3 bits.

On to bonus episode ② Here the source of the amplification was that $\Lambda_{\rm QCD}$ is born of dimensional transmutation. Next we measure that exponential map itself, as information — "a hierarchy shrinks to its own logarithm". The cosmological constant problem's 408 bits become 8.7 through the exponential map, and out of that comes a second criterion for whether a fine-tuning is a real problem.

This document is bonus episode ① 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/calc55.py and calc56.py. There is no new physics here — the $\mu$ amplification, the sigma terms and the sensitivity matrix are all in the literature (Calmet & Fritzsch 2002, Langacker–Segre–Strassler 2002, and Flambaum's body of work); this series' contribution is the restatement in §08, turning "cannot be told apart" into "here is what can". §05's sensitivity coefficients are literature values with real spread (sigma term 45–60 MeV, strange sigma term 20–60 MeV, and the $g_p$ sensitivity model-dependent) — which is why they were varied by a factor of four, but the individual figures should not be trusted to their significant digits. The $1\sigma$ bounds are one representative value each and move by factors of a few between papers, so read the condition number 1196 as "of order a thousand". "$X_q$ is not seen" refers to spectroscopy — Oklo (the $^{149}$Sm resonance) and nucleosynthesis (the deuteron binding) do constrain it strongly, but those two points are widely separated and isolated. §03's $R\approx30\text{–}50$ assumes grand unification; in non-unified models it is entirely different, so §04's conclusion is a conditional statement. ── To make a PDF, use your browser's Print dialogue (sliders freeze and answers are hidden in the print version).

Print / PDF: ⌘+P (Ctrl+P on Windows). On screen, move the sigma term and watch the bad direction refuse to change. "Show the answer" opens each solution.

Measurement	Quantity	Bound	bits
Methanol masers (\(z\approx0.89\))	\(\Delta\mu/\mu\)	\(10^{-7}\)	\(23.3\)
Ammonia absorption (\(z\approx0.68\))	\(\Delta\mu/\mu\)	\(3.5\times10^{-7}\)	\(21.4\)
H\(_2\) quasar absorption	\(\Delta\mu/\mu\)	\(5\times10^{-6}\)	\(17.6\)
Many-multiplet method (Ep. 30)	\(\Delta\alpha/\alpha\)	\(10^{-5}\)	\(16.6\)

Measurement	\(\alpha\)	\(X_e\)	\(X_q\)	\(1\sigma\)
Mg/Fe multiplets	\(1\)	\(0\)	\(0\)	\(10^{-5}\)
H\(_2\) Lyman–Werner	\(0\)	\(-1\)	\(0.048\)	\(5\times10^{-6}\)
NH\(_3\) and methanol	\(0\)	\(-1\)	\(0.048\)	\(10^{-7}\)
21 cm versus optical	\(2\)	\(-1\)	\(-0.039\)	\(10^{-6}\)
OH 18 cm	\(3.70\)	\(-1.85\)	\(0.002\)	\(10^{-5}\)

Direction	\(\sigma\) along it	Composition
1st	\(9.9\times10^{-8}\)	almost pure \(X_e\) (\(0.999\))
2nd	\(4.9\times10^{-7}\)	almost pure \(\alpha\) (\(0.999\))
3rd	\(1.2\times10^{-4}\)	\(99.6\) per cent \(X_q\)

\(x_3\) to hide	precision required on the ratio	bits of tuning
\(10^{-4}\)	\(2.1\times10^{-2}\)	\(5.6\)
\(10^{-3}\)	\(2.1\times10^{-3}\)	\(8.9\)

What it means
for mass to vary \(\mu=m_p/m_e\) does not move under a conformal transformation — which is why nothing can be told apart.
Digging turned up a degeneracy we had not expected.

01Why \(\mu\)?

02But \(m_p\) and \(m_e\) come from different places

03Which makes \(\mu\) an amplifier

04How much tighter than \(\alpha\)?

05The core — digging turned up a degeneracy

06The invisible direction was the one that matters most

07But hiding in that hole is expensive

08And none of this touches \(c\cdot t=\)const

Summary: turning "cannot be told apart" into "here is what can"

If \(m_q/\Lambda\) moves	sensitivity	what it affects
The pion mass (\(m_\pi^2\propto m_q\))	\(0.50\)	the range of the nuclear force
The deuteron binding energy	\(\approx1\)	nucleosynthesis itself
The Hoyle state	large	the production of carbon
The proton–neutron mass difference	large	the stability of the proton
The proton mass itself	\(0.048\)	the only channel spectroscopy sees

What it meansfor mass to vary \(\mu=m_p/m_e\) does not move under a conformal transformation — which is why nothing can be told apart.Digging turned up a degeneracy we had not expected.

01Why \(\mu\)?

02But \(m_p\) and \(m_e\) come from different places

03Which makes \(\mu\) an amplifier

04How much tighter than \(\alpha\)?

05The core — digging turned up a degeneracy

06The invisible direction was the one that matters most

07But hiding in that hole is expensive

08And none of this touches \(c\cdot t=\)const

Summary: turning "cannot be told apart" into "here is what can"

What it means
for mass to vary \(\mu=m_p/m_e\) does not move under a conformal transformation — which is why nothing can be told apart.
Digging turned up a degeneracy we had not expected.