c·t = CONST, THAT CLICKS BONUS ② ／ dug after the main series closed

A 408-bit problem becomes 8.7 bits

A hierarchy shrinks
to its own logarithm Dimensional transmutation is a compressor, and its gain is exactly the hierarchy.
Out of which comes a second criterion for whether a fine-tuning is a problem.

What you need: Episode 5's balance, Episode 32's cosmon, Episode 48's priors, bonus ①\(B\to\log_2 B\) — exact, no conventions

In bonus episode ①, the source of the amplification was that \(\Lambda_{\rm QCD}\) is born of dimensional transmutation. Here we measure that exponential map itself, as information. What comes out is a short law — a hierarchy of \(B\) bits shrinks to a description length of \(\log_2 B\) bits. And out of it comes a second criterion for whether a fine-tuning is a real problem.

01Dimensional transmutation is an exponential map

A coupling runs, and a scale is born $$\Lambda=M\exp\!\left(-\frac{2\pi}{b_0\,\alpha(M)}\right) \qquad\Longrightarrow\qquad \ln\frac{M}{\Lambda}=\frac{2\pi}{b_0\alpha(M)}\equiv H$$

differentiate with respect to \(\ln\alpha\): \(d\ln\Lambda/d\ln\alpha=+H\) — the gain is exactly the size of the hierarchy

Conclusion of §01

One bit of \(\alpha\) becomes \(H\) bits of \(\Lambda\).
── The bigger the hierarchy, the bigger the gain.

◇　◇　◇

02The core — so a hierarchy costs its own logarithm

Let the hierarchy be \(B\) bits (\(B=H/\ln2\)) $$\text{to fix }\Lambda\text{ within a factor of two} \ \Longrightarrow\ \delta(\ln\Lambda)=\ln2 \ \Longrightarrow\ \frac{\delta\alpha}{\alpha}=\frac{\ln2}{H}=\frac1B$$ $$\therefore\quad\text{precision needed on }\alpha=\log_2\frac{1}{\delta\alpha/\alpha}=\boxed{\log_2 B}$$

The main point of this episode

A hierarchy of \(B\) bits can be bought for \(\log_2 B\) bits, through the exponential map.
── A compression of \(B/\log_2 B\). This is exact: no approximation, no convention.
It is a statement about description length in the sense of Episode 5's MDL.

03Applied to the actual hierarchies

Hierarchyraw \(B\)\(\log_2 B\)compressionexponential map?
\(\Lambda_{\rm QCD}/M_{\rm Planck}\)\(65.0\)\(6.02\)\(10.8\)yes (QCD running)
\(v/M_{\rm Planck}\) (the hierarchy problem)\(55.5\)\(5.79\)\(9.6\)no
\(\Lambda_{\rm QCD}/v\)\(9.5\)\(3.25\)\(2.9\)yes
\(\rho_\Lambda/\rho_{\rm Planck}\) (the CC problem)\(408.4\)\(8.67\)\(47.1\)no

Even the cosmological constant problem's 408 bits would cost only 8.7 with an exponential map (a compression of 47) — which is exactly why everyone goes looking for exponential mechanisms.

Figure: the size of a hierarchy \(B\) against its price after the exponential map, \(\log_2 B\). Blue is without an exponential (the diagonal — the raw price); red is with one (flattened to \(\log_2 B\)). Move the slider — however large you make it, the red line barely rises.

408
no exponential (price = \(B\)) with one (price = \(\log_2B\)) one parameter's worth (5.37 bits)

04Which gives a second criterion for fine-tuning

Fine-tuningreason for the prior (Ep. 48)exponential map (here)price [bits]how it is actually treated
Strong CP (\(\theta_{\rm QCD}\))yes (an angle)no\(35.9\)a real problem
\(\Lambda_{\rm QCD}/v\)noyes\(9.5\to3.2\)nobody calls it a problem
\(v/M_{\rm Planck}\)nono\(55.5\)contested
\(\rho_\Lambda/\rho_{\rm Planck}\)nono\(408.4\)contested

Conclusion of §04

Two criteria alone reproduce which fine-tunings physicists actually treat as problems.
── We built the test and then went to look: the community had already drawn the same line.
The same shape as Episode 47, where the SI had drawn Episode 3's line.

05The cosmon against this ruler

Episode 32's cosmon (an exponential potential explaining the cosmological constant) $$\text{theoretical floor}=\log_2(408.4)=\mathbf{8.67\ \text{bits}} \qquad \text{what it actually paid}=2\times5.37=\mathbf{10.73\ \text{bits}}$$

an overhead of only \(2.06\) bits (24 per cent)

Conclusion of §05

The cosmon sits only 2 bits above the information-theoretic floor for any exponential mechanism.
Episode 32 said "it pays 10.7 and buys up to 408" —
what we can now see is that the price of 10.7 was itself nearly minimal.
※ This does not mean the cosmon is right. It is a statement about a floor: no exponential mechanism can be cheaper.

06Converting the floor into a parameter count

Problem\(B\) [bits]floor \(\log_2B\)= how many parametersthe actual mechanism
Strong CP\(35.9\)\(5.17\)\(0.96\)the axion = 1
Hierarchy\(55.5\)\(5.79\)\(1.08\)transmutation = 1
Cosmological constant\(408.4\)\(8.67\)\(1.62\)the cosmon = 2
✓
Strong CP: floor 0.96 → one is enough → the axion uses oneit matches
✓
The CC: floor 1.61 → two are needed → the cosmon uses twoit matches
◎
Since it is \(\log_2 B\), the floor saturates at one or two parameters however large the hierarchy408 bits, 55 bits and 36 bits all need one or two — with an exponential map, the size of the hierarchy barely affects the price

Without an exponential map, you pay \(B\) in full — and \(408\) bits is 76 parameters' worth at Episode 5's price. Which is why it is not solved.

The honest line — what is strong here and what is weak

(1) Strong (exact): \(d\ln\Lambda/d\ln\alpha=H\), and \(B\to\log_2B\). The first follows directly from one-loop dimensional transmutation. The second is a statement about description length — "the precision on \(\alpha\) needed to fix a \(B\)-bit hierarchy to within one bit is \(\log_2B\) bits" — with no approximation and no convention. Everything up to §03 is strong in this sense.

(2) Medium: §04's table. Whether an exponential map exists is objective, but the "how it is actually treated" column summarises the mood of the literature — four cases only, and the selection is mine.

(3) Weak — the weakest point here: §05 and §06 divide \(\log_2B\) by Episode 5's 5.37 bits per parameter. That 5.37 came from a particular dataset size (\(N=1701\)) and is the price of a parameter; there is no guarantee it is the same currency as bits of description precision (the same weakness as Episode 48, caveat 2). Reject that conversion and §05 and §06 do not stand.

(4) §06's agreement (0.96 → axion 1, 1.61 → cosmon 2) rests on two cases. As in Episode 36, caveat 2 — it is a recorded pattern, not a result. A third and fourth case that missed would end it.

(5) "No exponential map" means "none found so far". No one has shown that an exponential mechanism for the cosmological constant or the hierarchy is impossible — indeed, the compression of 47 in §03 is precisely the reason people keep looking.

Exercises

  1. What is the "gain" of dimensional transmutation?
    Show the answer
    \(H=\ln(M/\Lambda)\) — the hierarchy itself. Differentiating \(\Lambda=M\exp(-2\pi/b_0\alpha)\) with respect to \(\ln\alpha\) gives \(d\ln\Lambda/d\ln\alpha=2\pi/(b_0\alpha)=H\). One bit of \(\alpha\) becomes \(H\) bits of \(\Lambda\).
  2. What does a \(B\)-bit hierarchy cost through the exponential map?
    Show the answer
    \(\log_2 B\) bits. Fixing \(\Lambda\) within a factor of two needs \(\delta\alpha/\alpha=\ln2/H=1/B\), and specifying that precision takes \(\log_2B\) bits. A compression of \(B/\log_2B\), and this is exact.
  3. What do the CC problem's 408 bits become with an exponential map?
    Show the answer
    \(\log_2(408.4)=8.67\) bits — a compression of 47. Which is exactly why everyone goes looking for exponential mechanisms.
  4. What are the two criteria for whether a fine-tuning is a problem?
    Show the answer
    (i) Is there a reason for the prior? (Episode 48 — an angle has one, a mass ratio does not.) (ii) Is an exponential map available? (here.) Strong CP has (i) and lacks (ii), so it is real; \(\Lambda_{\rm QCD}/v\) has (ii), so it is not a problem — and that reproduces how they are actually treated.
  5. (Harder) What is the weakest part of this episode?
    Show the answer
    Dividing \(\log_2B\) by 5.37 bits per parameter (§05, §06). That 5.37 came from \(N=1701\) and is the price of a parameter — there is no guarantee it is the same currency as bits of description precision. Everything up to §03 is exact; beyond that it leans on the conversion.

Summary: a hierarchy shrinks to its own logarithm

Differentiate dimensional transmutation \(\Lambda=M\exp(-2\pi/b_0\alpha)\) with respect to \(\ln\alpha\) and you get \(d\ln\Lambda/d\ln\alpha=H\) — the gain is exactly the size of the hierarchy. One bit of \(\alpha\) becomes \(H\) bits of \(\Lambda\).

So a hierarchy of \(B\) bits shrinks to a description length of \(\log_2B\) bits (a compression of \(B/\log_2B\), exactly). \(\Lambda_{\rm QCD}/M_P\)'s 65 bits become 6.0, \(v/M_P\)'s 55.5 become 5.8, and the cosmological constant problem's 408 bits become 8.7 — a compression of 47. Which is exactly why everyone goes looking for exponential mechanisms.

From this comes a second criterion for fine-tuning — alongside Episode 48's "is there a reason for the prior?", now "is an exponential map available?". Strong CP has the first and lacks the second, so it is a real problem; \(\Lambda_{\rm QCD}/v\) has the second, so nobody calls it one; \(v/M_P\) and \(\rho_\Lambda\) have neither, and both are contested — exactly the distribution of what physicists actually treat as problems.

Then Episode 32's cosmon. The floor for an exponential mechanism is \(\log_2(408.4)=8.67\) bits, and the cosmon paid \(10.73\) — an overhead of only 2.06 bits. Episode 32 said "it pays 10.7 and buys up to 408"; what we can now see is that the 10.7 was itself nearly minimal.

And finally, how \(\log_2B\) behaves. However large the hierarchy, the floor saturates at one or two parameters — 408 bits, 55 bits and 36 bits all need one or two. With an exponential map, the size of the hierarchy barely affects the price. Without one you pay \(B\) in full — and 408 bits is 76 parameters' worth. Which is why it is not solved.

This document is bonus episode ② 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/calc57.py. Dimensional transmutation, running couplings, the hierarchy problem and the cosmological constant problem are all standard material. Sections 01 to 03 are strong: \(d\ln\Lambda/d\ln\alpha=H\) follows directly from one-loop transmutation, and \(B\to\log_2B\) is a statement about description length — "the precision on \(\alpha\) needed to fix a \(B\)-bit hierarchy to within one bit is \(\log_2B\) bits" — with no approximation and no convention. §04's table is medium: whether an exponential map exists is objective, but the "how it is actually treated" column summarises the mood of the literature, with four cases and a selection that is mine. §05 and §06 are the weakest part: dividing \(\log_2B\) by Episode 5's 5.37 bits per parameter assumes that a parameter's price (from \(N=1701\)) is the same currency as bits of description precision, and there is no guarantee that it is (the same weakness as Episode 48, caveat 2). §06's agreement rests on two cases and is a recorded pattern, not a result. "No exponential map" means "none found so far" — nobody has shown such a mechanism to be impossible, and the compression of 47 is precisely why people keep looking. ── 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, enlarge the hierarchy and watch the red line refuse to rise. "Show the answer" opens each solution.