c·t = CONST, THAT CLICKS EPISODE 25 ／ Ranking theories by description length, and where it breaks
Compression ratio turned out to be a stake, not a quality
Are physical laws a
compression algorithm?
\(a\propto t\) is 66 bits; the Einstein equations are 512.
So is shorter better? Compute it seriously and it breaks down along the way.
Physical laws are machines for rewriting the world more shortly. Kepler's three laws fold tables of planetary positions into equations; \(\Lambda\)CDM produces six million multipoles from six numbers. So does ranking theories by compression ratio order them by quality? Today we compute that seriously — and watch it break down along the way.
01First, measure the length of the law itself
Write each equation in LaTeX and count at \(\log_2 95=6.57\) bits per character (95 printable ASCII characters).
| Law | Characters | Bits |
|---|---|---|
| c·t=const (expansion law) \(a\propto t\) | 10 | 66 |
| Newtonian gravity | 13 | 85 |
| Schrödinger equation | 31 | 204 |
| Dirac equation | 33 | 217 |
| Friedmann equation | 63 | 414 |
| Maxwell's equations (tensor form) | 66 | 434 |
| Einstein field equations | 78 | 512 |
| Standard Model Lagrangian (expanded) | ≈ 5000 | ≈ 33000 |
The shortest is \(a\propto t\) at 66 bits — the entire expansion history of the universe in one fortieth of a tweet.
02Next, count the numbers explained
spherical harmonic modes from \(l=2\) to \(2500\)
$$\sum_{l=2}^{2500}(2l+1)=6{,}254{,}997\quad(\text{TT only})\qquad\times3=1.88\times10^7\quad(\text{TT+TE+EE})$$\(\Lambda\)CDM has six parameters, so
$$\text{compression ratio}=\frac{6{,}254{,}997}{6}=1.0\times10^6$$| Theory | Parameters | Numbers explained | Compression ratio |
|---|---|---|---|
| Rydberg formula | 1 (\(R_\infty\)) | 4,950 lines for \(n\le100\) | 4,950× |
| \(\Lambda\)CDM | 6 | \(6.3\times10^6\) modes | \(1.0\times10^6\)× |
| Standard Model | 19 | every cross section | (depends how you count) |
| General relativity | 0 | every spacetime | formally infinite |
So far it is a pleasing story: fewer parameters, better theory. And then —
03Where it breaks — the same calculation reverses the verdict
Total it up properly with MDL (minimum description length). The total is a sum of three things.
reusing Episode 5's values: \(\tfrac12\log_2N=5.37\) bit per parameter at \(N=1701\), and a residual of 153.6 bit for \(\Delta\chi^2=213\)
The problem is that \(L(\text{law})\) changes depending on whether \(a\propto t\) is counted as a constraint added to the Friedmann equation or as something replacing it wholesale.
| Reading A: an added constraint | L(law) | L(param) | L(residual) | Total |
|---|---|---|---|---|
| \(\Lambda\)CDM | 414 | 32.2 | 0.0 | 446 |
| c·t=const | 414+66 | 26.8 | 153.6 | 660 |
| Reading B: a replacement | L(law) | L(param) | L(residual) | Total |
|---|---|---|---|---|
| \(\Lambda\)CDM | 414 | 32.2 | 0.0 | 446 |
| c·t=const | 66 | 26.8 | 153.6 | 246 |
The thing this episode most wants to say
Same model, same data, and the verdict flips from losing by 214 bits to winning by 200.
All that moved was 414 bits of \(L(\text{law})\) — and that is set by which language you write the equation in.
This is Kolmogorov complexity's invariance theorem itself: description length is defined only up to a constant depending on the description language. Textbooks treat that constant as "large but finite, and irrelevant once you have enough data". But for the purpose of comparing theories, it turned out large enough to flip the result.
Figure: MDL's three pillars. Moving \(L(\text{law})\) with the slider swaps the totals, while the comparison using only \(L(\text{param})\) and \(L(\text{residual})\) (the narrow bars at right) does not move.
04So which part of the compression ratio is trustworthy?
Episode 3's verdict depends only on these two terms. Throw away the vague business of equation length entirely and the conclusion is unchanged — not a re-litigation of the verdict, but a check on how narrow a foundation it rests on.
05The reveal — compression ratio is not "quality" but "stake"
| Parameters \(k\) | Room to manoeuvre |
|---|---|
| \(k=0\) | it is impossible in principle to fit to anything |
| \(k=1\) | can move in one direction |
| \(k=6\) (\(\Lambda\)CDM) | six directions |
| \(k=25\) (\(\Lambda\)CDM + Standard Model) | twenty-five directions |
Conclusion of §05
High compression ratio ⟺ few parameters ⟺ nowhere to hide ⟺ easily falsified.
MDL's compression ratio and Popper's falsifiability were two sides of the same axis.
So \(a\propto t\) being "shortest, zero parameters" is not good news but news of a large bet. The stake was big, so losing cost 148 bits. That is what "the common weakness of theories that compress too well" means — there is no room to absorb.
06The universe's own compression ratio
information actually in use (Episodes 6 and 23)
$$k=S/\ln2=4.47\times10^{104}\ \text{bit}$$parameters used by physical law (\(\Lambda\)CDM + Standard Model)
$$25\ \text{parameters}\approx134\ \text{bit}$$It looks as though 134 bits of law explain a \(4.5\times10^{104}\)-bit world, but laws do not compress initial conditions. What holds the breakdown of those \(10^{104}\) bits is not the law but the history. Episode 20's "the universe is a 140-move program" was exactly about that history.
① "One character = 6.57 bits" is an arbitrary way of counting. That is in fact the subject of this episode. Choose a notation other than LaTeX and every equation becomes shorter or longer — so §01's table is not a claim about ranking but material to be demolished in §03.
② The compression ratio (numbers explained ÷ parameters) also depends on how the numerator is counted. \(\Lambda\)CDM's \(6.3\times10^6\) is the number of TT \(a_{lm}\) modes; the genuinely independent information is smaller (Planck's binned \(C_l\) are 1701 points) and larger if TT+TE+EE and other observations are added. The Standard Model row was left blank for the same reason — there was no honest way to count it.
③ \(\Delta\chi^2=213\) is the value used in Episode 5 and depends on the observational set compared. More robust than the number is the structure: the residual term exceeds the parameter term by two orders.
④ Using \(\Delta\chi^2/(2\ln2)\) for MDL's residual term is a Gaussian approximation. Strictly one should take the likelihood ratio itself; for non-Gaussian likelihoods the coefficient changes.
⑤ "General relativity has zero parameters" excludes \(\Lambda\). Count \(\Lambda\) as part of the theory and it is one; \(G\) and \(c\) are unit conversions and are not counted (Episode 2's convention).
⑥ Episode 3's verdict is not moved. §04 is not a re-judgement but a confirmation that the verdict does not depend at all on the vague term \(L(\text{law})\).
Exercises (solvable with this episode's formulas alone)
- Count \(\Lambda\)CDM's compression ratio using TT+TE+EE.
Show the answer
\(3\times6{,}254{,}997/6=3.1\times10^6\)×. But the numerator is a mode count rather than independent information, so this is the generous way of counting. - In one line, why do readings A and B give opposite verdicts?
Show the answer
Because \(L(\text{law})\) depends on the description language. A and B differ by the Friedmann equation's 414 bits, which more than cancels the 148-bit difference. The "language-dependent constant" of Kolmogorov's invariance theorem bites here. - By how much must \(\Delta\chi^2\) fall for an added parameter to pay for itself (\(N=1701\))?
Show the answer
The price is \(\tfrac12\log_2 1701=5.37\) bit, so it pays if \(\Delta\chi^2/(2\ln2)>5.37\), i.e. \(\Delta\chi^2>7.44\) — a far stricter criterion than the naive "\(\Delta\chi^2>1\)". - Is "a theory with a higher compression ratio is better" correct?
Show the answer
No. High compression ⟺ few parameters ⟺ nowhere to hide ⟺ easily falsified. Compression ratio measures not quality but the size of the bet. \(a\propto t\) is shortest with zero parameters, which is exactly why it missed by so much. - (Harder) How can 134 bits of law explain a \(4.5\times10^{104}\)-bit world?
Show the answer
It cannot. Laws compress the rules of time evolution but not the initial conditions. What holds the breakdown of \(10^{104}\) bits is the history — Episode 20's "140-move program" — and the law is merely its rule of execution.
Summary — compression ratio is a stake, not a quality
Counting the length of laws in LaTeX characters, the shortest is \(a\propto t\) at 66 bits, the Einstein equations at 512, the Standard Model at about 33,000. Divided by the numbers explained, \(\Lambda\)CDM compresses \(10^6\)-fold and general relativity formally infinitely.
Totalled properly with MDL, it broke down. Counting \(a\propto t\) as "a constraint added to the Friedmann equation" or as "a replacement" flips the same model on the same data from losing by 214 bits to winning by 200. All that moved was \(L(\text{law})\)'s 414 bits, a language-dependent quantity. The "language-dependent constant" of Kolmogorov's invariance theorem turned out, for the purpose of comparing theories, large enough to decide the outcome.
What is trustworthy is the other two: the parameter count (reparametrisation invariant, 5.37 bit each) and the residual (fixed by the data, 153.6 bits). Compared with those alone, \(-148\) bits, an odds ratio of \(4.3\times10^{44}\). Episode 3's verdict stands on that narrow foundation alone.
And the reveal — high compression ⟺ few parameters ⟺ nowhere to hide ⟺ easily falsified. MDL's compression ratio and Popper's falsifiability are two sides of one axis. That \(a\propto t\) is shortest with zero parameters is not good news but news of a large bet. The weakness of theories that compress too well is having no room to absorb — though that concerns judging; the value of shortness for rewriting remains untouched.
This document is Episode 25 of "c·t = const, That Clicks", written for physics-minded high-school and university readers. The minimum description length principle (Rissanen), the invariance theorem of Kolmogorov complexity, and BIC's \(\tfrac12\log_2N\) are all standard. The convention "one character = \(\log_2 95\) bits" is arbitrary and is itself the subject of this episode — §01's table is material for §03 to demolish, not a claim about ranking. The A/B comparison (losing by 214 bits versus winning by 200) and the comparison with \(L(\text{law})\) discarded (\(-148.3\) bit) are computed here (kenshou/calc29.py). The numerator of a compression ratio (the count of numbers explained) depends strongly on how it is counted: \(\Lambda\)CDM's \(6.3\times10^6\) is the number of TT \(a_{lm}\) modes (the independent information is closer to Planck's 1701 binned \(C_l\)). The Standard Model row was left blank because there was no honest way to count it. \(\Delta\chi^2=213\) is the value used in Episode 5 and depends on the observational set compared; the residual term \(\Delta\chi^2/(2\ln2)\) is a Gaussian approximation. "General relativity has zero parameters" is the counting that excludes \(\Lambda\). Linear expansion (\(c\cdot t=\)const) is a minority model under examination whose judgement was handled in Episode 3 — this document does not re-examine it but confirms that the judgement does not depend on \(L(\text{law})\). The academic standard remains the \(\Lambda\)CDM model including inflation. ── To make a PDF, use your browser's Print dialogue (sliders freeze and answers are hidden in the print version).