c·t = CONST, THAT CLICKS EPISODE 47 ／ Part VI — examining the procedure
\(c\) can be decreed; \(\alpha\) cannot
Metrology drew the line
in the same place
In 2019 the SI fixed seven constants and redefined the units — the kilogram prototype was retired.
An official, international declaration that dimensionful is bookkeeping.
Last time only the characterisations placed in dimensionless quantities could be tested. This time we draw the map — every dimensionless quantity in physics on one page, with the border between what is physics and what is bookkeeping drawn across it. And what emerges is that in 2019 the International System of Units drew that line in exactly the same place.
01In 2019 the SI redrew the line
| Constant | Fixed value (exact) | Unit defined |
|---|---|---|
| \(\Delta\nu_{\rm Cs}\) | 9 192 631 770 Hz | second |
| \(c\) | 299 792 458 m/s | metre |
| \(h\) | \(6.62607015\times10^{-34}\) J·s | kilogram |
| \(e\) | \(1.602176634\times10^{-19}\) C | ampere |
| \(k_B\) | \(1.380649\times10^{-23}\) J/K | kelvin |
| \(N_A\) | \(6.02214076\times10^{23}\) /mol | mole |
| \(K_{\rm cd}\) | 683 lm/W | candela |
Conclusion of §01
The kilogram prototype was retired. Mass is now built from \(h\).
── This is an official, international declaration that dimensionful quantities are bookkeeping.
The line this series drew by hand in Episode 3 is exactly where the world's metrology drew it.
02The core — but \(\alpha\) could not be fixed
because \(\varepsilon_0\) became a measured quantity — \(1/\alpha=137.035999177(21)\) still has to be measured
The main point of this episode
Being dimensionless, it cannot be legislated.
── \(c\) can be decreed; \(\alpha\) cannot.
This is the sharpest form of Episode 3's line.
03How many constants does it take to fix the units?
Conclusion of §03
What every position agrees on: the content of physics is on the dimensionless side.
04The map of dimensionless quantities
| Quantity | Value | Is there an explanation? |
|---|---|---|
| Information processed by the universe, \(N\) (Ep. 24) | \(3.1\times10^{122}\) | the same number as Eps. 40, 41 |
| \(m_p/m_e\) | \(1836.15\) | unexplained |
| \(1/\alpha\) | \(137.036\) | unexplained |
| Spectral index \(n_s\) | \(0.9649\) | inflation claims one |
| \(\Omega_\Lambda\) | \(0.685\) | unexplained |
| \(v/M_P\) | \(2.0\times10^{-17}\) | the hierarchy problem |
| \(\alpha_G=Gm_p^2/\hbar c\) | \(5.9\times10^{-39}\) | unexplained |
| \(\rho_\Lambda/\rho_{\rm Planck}\) (Ep. 32) | \(1.13\times10^{-123}\) | the cosmological constant problem |
From the largest, \(3.1\times10^{122}\), to the smallest, \(1.13\times10^{-123}\), the map spans 815 bits. And all of it sits in the same column (weight 0); a conformal transformation moves none of it.
Figure: the dimensionless quantities of physics on a single logarithmic axis, spanning 815 bits. Move the "window" to see how many fall inside it — unexplained numbers sit at both the large and the small end.
05How many are there, and how many are explained?
| Framework | Count | Breakdown |
|---|---|---|
| Standard Model (excluding neutrino masses) | \(19\) | only \(\mu^2\) (the Higgs \(v\)) is dimensionful — one |
| Neutrino masses and mixings | \(7\) | 3 masses + 3 angles + 1 phase |
| \(\Lambda\)CDM base parameters | \(6\) | all six dimensionless |
| Total | \(32\) | \(\times5.37=\mathbf{171.7}\) bits |
Conclusion of §05
In the Standard Model Lagrangian, exactly one constant is dimensionful: \(\mu^2\). Everything else is dimensionless — and so are all six \(\Lambda\)CDM parameters.
── Modern physics already writes its parameters in the physics column.
And almost none are derived from first principles — 171.7 bits are unexplained.
06Applying the procedure to ourselves
Here is something one is tempted to do. The 171.7 bits just obtained, and Episode 2's "the whole history of the universe = 140.24 log steps" — they look like the same sort of number. Does that mean anything?
calling it "agreement" would need ±5 per cent → it does not land. Surprise: 0 bits
And even if it had been close, the surprise would have been small — a parameter count of 15 to 30 and a price of 5 to 6 bits are both plausible, so the product ranges over 75 to 180 (a width of 105). Landing within ±5 per cent (a width of 14) is worth 2.9 bits, the bottom of the coincidence band.
Conclusion of §06
This is what having a procedure is worth.
An impression that two things "look alike" can be turned into a number and rejected on the spot.
── Episode 36 said the band of coincidences is a selection effect; this is how the ones that fail the selection fall away.
(1) There are several conventions for counting parameters. The Standard Model's 19 is the standard count excluding neutrino masses; including them gives 26 to 28 (depending on whether Majorana phases are counted), and whether to count \(\theta_{\rm QCD}\) changes it too — "32" is the result of one convention and moves between the high twenties and the low thirties.
(2) The 5.37 bits behind "171.7 bits unexplained" is Episode 5's price. That came from a particular dataset size (\(N=1701\)), and there is no universal price for one parameter — the substance is the structure "32 parameters, almost all unexplained", and the significant figures of 171.7 mean nothing.
(3) §02's "\(\alpha\) cannot be fixed" is a statement about the design of the SI. Fixing \(e\), \(\hbar\) and \(c\) makes \(\varepsilon_0\) (and \(\mu_0\)) measured quantities whose uncertainty is set by that of \(\alpha\) — "a dimensionless quantity cannot be legislated" is this series' phrasing; more precisely, "defining units does not determine dimensionless quantities".
(4) §03's "how many fundamental constants" remains a matter of disagreement. The 2002 trialogue in which Duff, Okun and Veneziano argued for 0, 3 and 2 respectively is well known, and it is not settled — this document takes only the uncontested part.
(5) §04's "unexplained" means "not derived from first principles". Many of these quantities have partial understanding, or relations within a model (\(m_p/m_e\), for instance, should in principle be computable from QCD and the electroweak theory) — it does not mean nothing is known about them.
Exercises
- What did the 2019 SI fix in order to define the units?
Show the answer
The exact values of seven constants — \(\Delta\nu_{\rm Cs}\), \(c\), \(h\), \(e\), \(k_B\), \(N_A\), \(K_{\rm cd}\). The kilogram prototype was retired and mass is now built from \(h\) — an official, international declaration that dimensionful quantities are bookkeeping. - \(e\), \(\hbar\) and \(c\) are all exact, so why is \(\alpha\) not determined?
Show the answer
Because \(\varepsilon_0\) became a measured quantity. The uncertainty in \(\alpha=e^2/4\pi\varepsilon_0\hbar c\) is carried entirely by \(\varepsilon_0\) — defining units does not determine dimensionless quantities. \(c\) can be decreed, \(\alpha\) cannot: the sharpest form of Episode 3's line. - How many dimensionful constants does the Standard Model Lagrangian have?
Show the answer
One: \(\mu^2\) (the Higgs mass term, which sets \(v\)). Every other coupling and Yukawa is dimensionless — modern physics already writes its parameters in the physics column. All six \(\Lambda\)CDM base parameters are dimensionless too. - Do 171.7 bits and 140.24 log steps agree?
Show the answer
No — a 22 per cent relative difference. Calling it agreement would need ±5 per cent, so it does not land and the surprise is 0 bits. Even if it had, the surprise would be 2.9 bits, the bottom of the band — with a procedure, an impression can be rejected on the spot. - (Harder) How wide is the map of dimensionless quantities?
Show the answer
From \(3.1\times10^{122}\) to \(1.13\times10^{-123}\): 815 bits. And all of it is in the same column (weight 0), unmovable by any conformal transformation — which is exactly why this column can serve as the arena for judging.
Summary: the world's metrology drew the line in the same place
Since 20 May 2019, the SI has defined its units by fixing the exact values of seven constants — \(\Delta\nu_{\rm Cs}\), \(c\), \(h\), \(e\), \(k_B\), \(N_A\), \(K_{\rm cd}\). The kilogram prototype was retired; mass is built from \(h\). This is an official, international declaration that dimensionful quantities are bookkeeping — the line this series drew by hand in Episode 3, drawn in the same place by the world's metrology.
But \(\alpha\) could not be fixed. Fixing \(e\), \(\hbar\) and \(c\) makes \(\varepsilon_0\) a measured quantity, and \(1/\alpha=137.035999177(21)\) still has to be measured — being dimensionless, it cannot be legislated. \(c\) can be decreed and \(\alpha\) cannot: the sharpest form of Episode 3's line.
The map of dimensionless quantities runs from \(3.1\times10^{122}\) (Episode 24's \(N\)) to \(1.13\times10^{-123}\) (the cosmological constant), 815 bits wide, all in the same column. Counting the parameters: 19 for the Standard Model, 7 for neutrinos, 6 for \(\Lambda\)CDM — 32, of which exactly one, \(\mu^2\), is dimensionful, and almost none are derived from first principles: 171.7 bits unexplained.
Finally we applied the procedure to ourselves. The 171.7 bits and Episode 2's 140.24 log steps look like the same sort of number, but the relative difference is 22 per cent: it does not land, and the surprise is 0 bits. Even if it had, 2.9 bits — the bottom of the band. An impression that two things look alike can be turned into a number and rejected on the spot. That is what having a procedure is worth.
This document is Episode 47 of "c·t = const, That Clicks" (the second of Part VI), written for physics-minded high-school and university readers. The 2019 SI redefinition, the parameter counts of the Standard Model and \(\Lambda\)CDM, and the list of dimensionless quantities are all standard, and nothing here is a new claim — the numbers are computed in kenshou/calc51.py. §05's parameter count follows one of several conventions — the Standard Model's 19 excludes neutrino masses; including them gives 26 to 28 (depending on Majorana phases), and counting \(\theta_{\rm QCD}\) changes it too. The 5.37 bits behind "171.7 bits" is Episode 5's price (from a dataset of \(N=1701\)), and there is no universal price for a parameter — the substance is the structure, not the significant figures. §02's "\(\alpha\) cannot be fixed" is a statement about the SI's design; more precisely, "defining units does not determine dimensionless quantities" (fixing \(e,\hbar,c\) makes \(\varepsilon_0\) a measured quantity whose uncertainty is that of \(\alpha\)). §03's "how many fundamental constants" remains disputed (the 2002 trialogue in which Duff, Okun and Veneziano argued for 0, 3 and 2 is well known), and this document takes only the uncontested part. §04's "unexplained" means "not derived from first principles" — many of these have partial understanding or relations within a model, and it does not mean nothing is known about them. ── To make a PDF, use your browser's Print dialogue (sliders freeze and answers are hidden in the print version).