c·t = CONST, THAT CLICKS EPISODE 36 ／ Part IV — wrap-up
Exactly one had not done it
Onto the
operating table
The same surgery on nine theories, and a list of where the line fell.
Plus a "band of coincidences" that only showed up once they were side by side.
Part IV’s nine theories, laid out on one table — inflation, VSL, MOND, measuring the constants, CCC, the cosmon, Milne, conformal gravity, asymptotic safety. The same surgery applied to all of them, and a list of where the dividing line fell. Then the most important thing this part turned up — good theories have already performed Episode 3’s surgery. Exactly one had not.
01Nine theories on one table
| Ep. | Theory | (A) notation | (B) observable claim | Surgery |
|---|---|---|---|---|
| 27 | Inflation | connect things causally | \(n_s\approx1-2/N\) | done |
| 28 | VSL | a change of units | \(\alpha\) varies | ✗ |
| 29 | MOND | posit \(a_0\) | dynamics set by \(g/a_0\) | done |
| 30 | Measuring constants | (not a notation) | \(\alpha\) invariant to 26 bits | ── |
| 31 | CCC | a conformal gluing | a previous aeon persists | done |
| 32 | Cosmon | a non-expanding picture | \(w(z)\ne-1\) | done |
| 33 | Milne | a coordinate change | (nothing inside) | ── |
| 34 | Conformal gravity | a gauge symmetry | rotation curves, \(\alpha_g\) | nothing to cut |
| 35 | Asymptotic safety | making \(G\) dimensionless | \(m_H\), number of predictions | done |
Conclusion of §01
Exactly one theory failed to separate (A) from (B): VSL.
── Good theories have already performed Episode 3’s surgery.
02The dividing line was not the name
03Every prediction sat in a dimensionless quantity
| Theory | Where the prediction sits | Dimension |
|---|---|---|
| Inflation | \(n_s\) | dimensionless |
| VSL | \(\Delta\alpha/\alpha\) | dimensionless |
| MOND | \(g/a_0\) | dimensionless |
| CCC | Hawking-point statistics | dimensionless |
| Cosmon | \(w\) | dimensionless |
| Conformal gravity | the shape of rotation curves | dimensionless |
| Asymptotic safety | \(m_H/v\) | dimensionless |
Conclusion of §03
There were no exceptions. This is the strongest confirmation of Episode 3’s procedure.
── A theory that puts its claim in a dimensionful quantity never reaches the arena where it can be judged.
04The core — a band of coincidences
Laying them side by side turned up something else. Here is every coincidence in this part, measured in Episode 19’s bits of surprise.
| Coincidence | Surprise | Class |
|---|---|---|
| \(\rho_\Lambda^{1/4}\) and \(m_\nu\) (previous series, extra 5) | 4.7 bit | coincidence |
| Inflation’s \(N\) agreeing (Ep. 27) | 4.8 bit | explained → physics |
| Asymptotic safety’s Higgs prediction (Ep. 35) | 5.3 bit | explained → physics |
| Conformal gravity’s \(\gamma_0\simeq1/25R_H\) (Ep. 34) | 5.4 bit | coincidence |
| MOND’s \(a_0\simeq cH_0/2\pi\) (Ep. 29) | 5.9 bit | coincidence |
| One bit ↔ 1.96 fm (Ep. 18) | 7.4 bit | coincidence |
| Koide’s relation (previous series, extra 4) | 15.7 bit | empirical formula |
| The uniformity of the CMB (Ep. 17) | \(1.6\times10^5\) bit | a real problem |
The main point of this episode
Six of them fall in the band from 4 to 7.5 bits (mean 5.6, spread 2.7).
Why should coincidences thrown up by entirely independent theories land in the same narrow band?
Figure: every "surprise" this series has measured, on one axis. They cluster between 4 and 7.5 bits. Move the "threshold for noticing" and read off how many survive — the band is most likely a selection effect.
Conclusion of §04
The band is most likely a selection effect — too loose and nobody looks; too tight and it gets explained.
── The scale built in Episode 19 turns out to measure the practice of physics itself.
05The ledger, summed up
| Theory | Parameters | What it buys | Net [bits] |
|---|---|---|---|
| Inflation (Ep. 27) | \(+2\) | \(n_s\) and much else | \(-6.5\) (an underestimate) |
| c·t = const (Ep. 25) | \(-1\) | the horizon problem disappears | \(-148.3\) |
| MOND (rotation curves only, Ep. 29) | \(+4\) | rotation curves from baryons | \(+1971\) |
| Conformal gravity (rotation curves only, Ep. 34) | \(+3\) | the same, plus a forbidden \(\Lambda\) | \(+1977\) |
| Cosmon (Ep. 32) | \(+2\) | the size of \(\rho_\Lambda\) (up to 408) | a large credit |
| Asymptotic safety (Ep. 35) | \(+3\) | \(m_H\), ultraviolet finiteness | not yet assessable |
The unit is Episode 5’s balance (one parameter = 5.37 bits). The datasets differ, so these cannot be compared directly — as Episode 29 showed, which dataset you measure on decides who wins. This table exists to show that all of it can be written in one currency; it is not a league table.
06The reveal — all four follow from one procedure
Conclusion of §06
All four follow from the single procedure built in Episode 3.
"Dimensionful is bookkeeping, dimensionless is physics. If you have not named what you are comparing to, you have not yet made a sentence."
That alone accounts for every dividing line among the nine.
Part II: wherever you put it, only one thing moves, and only its size is touched.
Part III: measured as information, it was one number restated in eight languages.
Part IV: applied to other theories, the good ones had already done the surgery.
(1) "Has the surgery been done?" is this series’ reading. How aware each theory’s proposers were is inferred from how the papers are written; their intent has not been verified. For VSL, the point that the name hid the content follows Ellis & Uzan (2005); it does not mean VSL researchers failed to understand the distinction.
(2) §04’s "band" is an observation on a sample of eight. Worse, it collects only the coincidences this series chose to write about, so the selection itself is biased — the "it is a selection effect" explanation comes out of a sample subject to selection effects. Read it as a recorded pattern, not a quantitative claim.
(3) Each surprise in bits depends on how the prior range is drawn (Episode 19 §01). The values 4.7 to 7.4 can move by a few bits, so the width of the "band" is correspondingly vague.
(4) §05’s ledger does not use a common dataset. Episode 25 used supernovae, Episodes 29 and 34 galaxy rotation curves, Episode 27 the CMB — the table shows that one currency suffices, not who ranks where. The parameter counts are rough in each case.
(5) This document neither endorses nor rejects any theory covered in Part IV. All except inflation are minority hypotheses; the academic standard remains the \(\Lambda\)CDM model including inflation, together with unmodified general relativity.
Exercises (Part IV, wrap-up)
- Which of the nine theories had not had the surgery?
Show the answer
VSL alone (Episode 28). It claimed (B) "\(\alpha\) varies" while keeping the (A) name "the speed of light varies", so the 26-bit constraint on \(\alpha\) stopped being visible head-on. - Was the dividing line "does the name point at (A)?"
Show the answer
No. The cosmon paper title "A Universe without expansion" is (A) side, and so is VSL. The line was "can the theory itself tell (A) from (B)?" — Wetterich states explicitly that the two pictures are Weyl-equivalent. - What do the seven theories’ predictions have in common?
Show the answer
They all sit in dimensionless quantities — \(n_s\), \(\Delta\alpha/\alpha\), \(g/a_0\), \(w\), \(m_H/v\) and so on. It is the strongest confirmation of Episode 3’s procedure: a claim placed in a dimensionful quantity never reaches the arena where it can be judged. - What is the "band of coincidences", and what explains it?
Show the answer
Six of the coincidences this series covered fall between 4 and 7.5 bits (mean 5.6). The explanation is most likely a selection effect — under 4 bits nobody notices; over 15 bits it becomes famous and demands an explanation. From 4 to 7 bits is where "enough for a paper, not enough for a consensus" lives. But as caveat (2) says, the sample is small and biased. - (Harder) Where did Part IV’s four findings come from?
Show the answer
The single procedure built in Episode 3 — "dimensionful is bookkeeping, dimensionless is physics; if you have not named what you are comparing to, you have not yet made a sentence." Whether the surgery was done, where the prediction sits, how a coincidence’s surprise is measured, and that the winner depends on the dataset — all four are consequences of that one procedure.
Summary: good theories have already had the surgery
Part IV’s nine theories went onto one table. Exactly one failed to separate (A) notation from (B) an observable claim: VSL. And the dividing line was not "does the name point at (A)?" — the cosmon paper title and VSL are both (A) side, and they match that far. The line was whether the theory itself could tell them apart.
The predictions went onto the table too — \(n_s\), \(\Delta\alpha/\alpha\), \(g/a_0\), Hawking-point statistics, \(w\), the shape of rotation curves, \(m_H/v\). Dimensionless without exception. It is the strongest confirmation of Episode 3’s procedure — a theory that puts its claim in a dimensionful quantity never reaches the arena at all.
Laying them out turned up something new. Measured in Episode 19’s bits of surprise, six of the coincidences fall between 4 and 7.5 bits (mean 5.6). Coincidences from entirely independent theories landing in one narrow band — the cause is most likely a selection effect. Under 4 bits nobody notices; over 15 bits (Koide’s relation) it becomes famous and demands an explanation. From 4 to 7 bits is where "enough for a paper, not enough for a consensus" lives. The scale built in Episode 19 turns out to measure the practice of physics itself.
And the reveal: whether the surgery was done, where the prediction sits, which band a coincidence lands in, and that the winner depends on the dataset. All four follow from the one procedure built in Episode 3 — "dimensionful is bookkeeping, dimensionless is physics; if you have not named what you are comparing to, you have not yet made a sentence." That alone accounts for every dividing line among the nine.
This document is Episode 36 of "c·t = const, That Clicks" (Part IV wrap-up), written for physics-minded high-school and university readers. It collects results from Episodes 27 to 35; the only new computation is §04's tally (kenshou/calc40.py) — for the numbers and sources of each individual result, see the endnotes of the episode concerned. "Has the surgery been done?" is this series' reading; how aware each theory's proposers were is inferred from how the papers are written and their intent has not been verified — for VSL, the point that the name hid the content follows Ellis & Uzan (2005) and does not mean VSL researchers failed to understand the distinction. §04's "band of coincidences" is an observation on a sample of eight, and one collecting only the coincidences this series chose to write about — the "it is a selection effect" explanation itself comes out of a sample subject to selection effects, so read it as a recorded pattern rather than a quantitative claim. Each surprise in bits depends on how the prior range is drawn (Episode 19 §01). §05's ledger does not use a common dataset (Episode 25 supernovae, Episodes 29 and 34 galaxy rotation curves, Episode 27 the CMB) — it shows that one currency suffices, not who ranks where. This document neither endorses nor rejects any theory covered in Part IV; all except inflation are minority hypotheses, and the academic standard remains the \(\Lambda\)CDM model including inflation, together with unmodified general relativity. ── To make a PDF, use your browser's Print dialogue (sliders freeze and answers are hidden in the print version).