Thinking in Waves Episode 53 / Part XVIII — Putting Error Bars On It
Were Episodes 51 and 52 hitting the same wall?
Why the Measurable Range
Is Always Logarithmic
Logarithms appeared in ten places across this series — decades, e-folds, \(q^*\), information, proper distance.
★ They share one root — \(f(xy)=f(x)+f(y)\). Its only continuous solution is \(c\ln x\) (confirmed numerically).
★★ And the companion fact — "range" goes as \(\log N\), "precision" as \(N^{-1/2}\). A millionfold more data buys 100× precision but only 3× range.
Episode 51 hit a ceiling in decades, Episode 52 in \(q^*\) — both set by the logarithm of the data.
★ Coincidence, or a common root? That is this episode.
01Every logarithm in the series, lined up
| Where | Quantity | What it is |
|---|---|---|
| Ep. 5 | ratio to the age of the universe | \(\ln(T_{\rm universe}/T_{\rm signal})\) |
| Ep. 20 | the \(c\cdot t=\)const coordinate | phase \(\propto\ln t\) |
| Ep. 21 | information | \(\log_2(\text{number of states})\) |
| Ep. 22 | the wavelet plane | uniform resolution in \(\ln\omega\) |
| Ep. 39 | AdS proper distance | \(|\ln(z_2/z_1)|\) |
| Ep. 45 | Rindler time | \(at=e^{a\eta}\) |
| Ep. 46 | leaving the measurable band | 74.5 e-folds |
| Ep. 47 | the horizon problem | 63.6 e-folds |
| ★ Ep. 51 | decades of power law | \(\log_{10}(f_{\rm hi}/f_{\rm lo})\) |
| ★ Ep. 52 | reachable \(q^*\) | \(\propto\log_2(\text{boxes})\) |
★ Ten of them. Too many for coincidence.
02The only measure invariant under multiplication is \(dx/x\)
Stretch \([a,b]\) to \([\lambda a,\lambda b]\); for which \(w\) does \(\int w(x)dx\) stay the same?
| \(\lambda\) | \(w=1\) | ★ \(w=1/x\) | \(w=1/x^2\) |
|---|---|---|---|
| 1 | 2.00000000 | 1.09861229 | 0.66666667 |
| 10 | 20.00000000 | 1.09861229 | 0.06666667 |
| 100 | 200.00000000 | ★ 1.09861229 | 0.00666667 |
The point of §02
★★★ Only \(1/x\) is unchanged (always \(\ln3=1.098612\)).
★★ That is the invariant measure of the multiplicative group of scale transformations.
★ If the invariant measure is \(dx/x\), the natural coordinate is \(\int dx/x=\ln x\) — hence the logarithm.
★ Episode 39's AdS proper distance \(|\ln(z_2/z_1)|\) and Episode 22's \(\ln\omega\) axis had the same reason.
03With constant relative resolution, the number of distinguishable steps is logarithmic
| \(Q\) | Meaning | 2 dec | 3 dec | 6 dec | 12 dec |
|---|---|---|---|---|---|
| 1 | one per octave | 6.6 | 10.0 | 19.9 | 39.9 |
| 16.8 | ★ a semitone (Ep. 22) | 79.6 | 119.5 | 238.9 | 477.9 |
| 100 | 1 % resolution | 462.8 | 694.2 | 1388.4 | 2776.9 |
The point of §03
★★ The count of distinguishable steps is exactly proportional to the number of decades.
★★★ Which is why "how many decades" is "how many things could be told apart."
★ Episode 51's decades and Episode 52's \(q^*\) both sit on this way of counting.
04★★★ They all come from the same functional equation
Numerically: set \(f(2)=a\), \(f(3)=b\). Whenever \(2^p\approx3^q\), continuity demands \(pa\approx qb\), hence \(b/a=p/q\to\ln3/\ln2\).
| \(p\) | \(q\) | \(2^p/3^q\) | \(p/q\) | \(\ln3/\ln2\) | difference |
|---|---|---|---|---|---|
| 8 | 5 | 1.053498 | 1.60000000 | 1.58496250 | 1.5e−2 |
| 84 | 53 | 0.997914 | 1.58490566 | 1.58496250 | 5.7e−5 |
| 1054 | 665 | 1.000000 | 1.58496241 | 1.58496250 | ★ 9.5e−8 |
The point of §04 — ★★★ the core of this episode
The moment you decide to turn multiplication into addition, the logarithm is forced.
★ Better approximations drive \(p/q\) toward \(\ln3/\ln2\) — there is no other continuous solution.
★★★ Episode 28 sorted "one source of exponentials is addition becoming multiplication." This episode is that read backwards — turn multiplication back into addition and you get a logarithm.
★ So the moment you start counting, a logarithm appears.
05Re-reading each logarithm in that light
| Where | What multiplies | What you want to add |
|---|---|---|
| Ep. 5 | ratios of timescales | "how many decades apart" |
| Ep. 21 | putting independent systems side by side | ★ information (which must be additive) |
| Ep. 22 | constant-Q frequency steps | the number of steps |
| Ep. 39 | radial scale transformations | proper distance |
| Ep. 45 | acceleration stretching time | proper time |
| ★ Ep. 51 | ratios of frequency | ★ decades of power law |
| ★ Ep. 52 | boxes doubling each level | ★ the reachable \(q^*\) |
The point of §05
★★ Every one has the form "something grows by multiplication; I want to count it by addition."
★ Information (Episode 21) looks different at first, but — "put independent systems side by side and the state count multiplies while the information adds" — it is exactly the same form.
★★ Shannon's reason for choosing \(\log\) and this series' reason for speaking in decades are the same reason.
06★ How optimistic is the textbook criterion?
Episode 52's \(q^*\) has a standard naive criterion — "if the set of \(\alpha\) dominating that \(q\) is non-empty (\(f(\alpha)\ge0\)), the sample should contain it."
| \(q\) | \(f(\alpha(q))\) | Verdict |
|---|---|---|
| 2 | 0.62263432 | fine by both |
| 4 | 0.20743868 | ★ where Episode 52's measurement stopped |
| 10 | 0.00285631 | fine naively, unreachable in practice |
| 25.9 | 0 (here \(f=0\)) | ★ the naive ceiling |
The point of §06
★★★ The naive criterion allows \(q\approx25.9\); the measurement gives \(q^*\approx4\) — more than six times too optimistic.
★★ Why: \(f(\alpha)\ge0\) says "on average one will be present," and the fluctuation of that one is what destroys the slope.
★ "One on average" is not "one every time" — use Episode 52's measurement, not the criterion.
07★★ What is not logarithmic — precision is a power
| Question | Dependence on \(N\) | From |
|---|---|---|
| How wide a range can you see? | ★ \(\log N\) | Ep. 51, 52 |
| How precisely can you measure? | ★ \(N^{-1/2}\) | Ep. 51 |
| How many steps can you distinguish? | \(\log N\) | Ep. 22; §03 here |
| How far can noise be reduced? | \(N^{-1/2}\) | Ep. 51 §01 |
| How deep into the extremes? | \(\log N\) | Ep. 52 |
| More data | range | precision |
|---|---|---|
| \(\times10^2\) | ×1.00 | ×1.0 |
| \(\times10^4\) | ×2.00 | ×10.0 |
| \(\times10^6\) | ★ ×3.00 | ★ ×100.0 |
| \(\times10^8\) | ×4.00 | ×1000.0 |
The point of §07 — ★★★ the second core
A millionfold more data buys 100× precision
but only 3× range.
★★ "Measure finer" rewards effort; "measure wider" barely does.
★ That is what Episodes 51 and 52 ran into — both were range problems, so both hit the logarithm.
★★ Turned around — a "not enough range" problem cannot be solved by measuring more. The system itself has to be changed (Episode 51 §05's "choose a flow with a wide inertial range").
08What wave language can and cannot say
| Item | Verdict | Basis |
|---|---|---|
| Ten logarithms across the series | ◎ organized | too many for coincidence |
| ★ The scale-invariant measure is \(dx/x\) | ◎ numerics | ★ only \(1/x\) survives scaling |
| Constant relative resolution → log steps | ◎ numerics | Episode 22's constant Q |
| ★★ The root is \(f(xy)=f(x)+f(y)\) | ◎ numerics | ★ only \(c\ln x\) is continuous |
| Episode 28 read backwards | ◎ correspondence | addition ↔ multiplication |
| ★ The naive \(q^*\) criterion is optimistic | ◎ numerics | ★ \(f\ge0\) says \(q\approx26\); measured 4 |
| ★★ Range \(\log N\), precision \(N^{-1/2}\) | ◎ numerics | ★ 10⁶× data → 3× range |
| ★ Why physics is scale-invariant | × no | ★ we only worked backwards from results |
Exercises
- §04 says the only continuous solution is \(c\ln x\). What happens without continuity?
Show answer
Infinitely many pathological solutions appear. Setting \(g(u)=f(e^u)\) turns \(f(xy)=f(x)+f(y)\) into Cauchy's additive equation \(g(u+v)=g(u)+g(v)\).
★ Impose continuity (or measurability, or local boundedness — any one suffices) and the solution is \(g(u)=cu\), i.e. \(f=c\ln x\).
★★ Drop it, and using a basis of \(\mathbb{R}\) over \(\mathbb{Q}\) one constructs everywhere-discontinuous solutions (the axiom of choice is needed). Their graphs are dense in the plane.
★ In physics continuity is taken for granted, so effectively "only the logarithm" — but when writing "of course," one should write what was assumed. - How does §07's "range \(\log N\), precision \(N^{-1/2}\)" translate into experiment design?
Show answer
First, distinguish "not enough range" from "not enough precision."
★ If precision is short, more measuring time helps as \(N^{-1/2}\) — 4× the time for 2× better. Effort is repaid.
★★ If range is short, more time helps only as \(\log N\) — 1000× the time for 1.3×. Nearly wasted.
★★★ Then you change not the apparatus but the subject — Episode 51's "choose a flow with a wide inertial range," where 6 decades reaches 0.059 dB/oct. As experiment design, that decision comes first. - §02 says "the invariant measure is \(dx/x\), hence the logarithm." Is that an explanation?
Show answer
Only half of one. Granted the premise "we want quantities invariant under scale transformations," it does explain why a logarithm appears.
★ But it does not explain why physics is scale-invariant in the first place — this article only worked backwards from results.
★★ The same shape as Episode 48's conclusion — we learned that the axis works where scale invariance holds, but not where the scale invariance comes from.
★ At a critical point there is an explanation (the correlation length diverges and the characteristic length disappears). But "why are so many phenomena near a critical point?" is another question again.
Summary — two faces of the same wall
Logarithms appeared in ten places across the series — Episode 5's ratio to the age of the universe, Episode 21's information, Episode 22's wavelet plane, Episode 39's proper distance, Episode 45's Rindler time, Episodes 46–47's e-folds, and Episodes 51–52's decades and \(q^*\).
★ The only measure invariant under multiplication is \(dx/x\) (confirmed numerically: still \(\ln3\) after a hundredfold stretch). With \(dx/x\) invariant, the natural coordinate is \(\ln x\).
★ And with constant relative resolution, the count of distinguishable steps is proportional to the decades (constant Q, Episode 22) — so "how many decades" is "how many things could be told apart."
★★★ The core — one root: \(f(xy)=f(x)+f(y)\).
Its only continuous solution is \(c\ln x\) (\(p/q\to\ln3/\ln2\) to \(10^{-8}\)).
The moment you decide to turn multiplication into addition, the logarithm is forced.
★ Episode 28 sorted "one source of exponentials is addition becoming multiplication." This is that read backwards. Information (Episode 21) has the same form — side by side, states multiply while information adds.
★ A by-product — for Episode 52's \(q^*\), the textbook criterion (\(f(\alpha)\ge0\)) allows \(q\approx25.9\) while the measurement gives 4: more than six times too optimistic. "One on average" is not "one every time."
★★ And the second core —
"range" goes as \(\log N\), "precision" as \(N^{-1/2}\).
A millionfold more data buys 100× precision but only 3× range.
★ "Measure finer" rewards effort; "measure wider" barely does — that is what Episodes 51 and 52 ran into. ★ Honestly: why physics is scale-invariant is not explained here; we only worked backwards from results.
★ Next time we look for the way around it — can range be bought at all?
★★ Splicing several measurements, using a different observable, changing the system itself — what experimentalists actually do, counted in this series' language. ★ Episode 51's "choose the inertial range" was one of them.
This document is Episode 53 of the "Thinking in Waves" series. Computations are in kenshou/taisuu2.js. On the numbers here: the numerical check of the invariant measure, the constant-Q step counts, the convergence of \(p/q\) to \(\ln3/\ln2\), the comparison with the naive \(q^*\) criterion, and the \(N\)-dependence of range and precision were all computed here. That the Haar measure of the scaling group is \(dx/x\), that the continuous solutions of Cauchy's equation \(f(xy)=f(x)+f(y)\) are exactly \(c\ln x\), that Shannon information is logarithmic because it must be additive, and the \(f(\alpha)\ge0\) sampling criterion for multifractals are all well-known results and none are discoveries of this article — what this article did was line up the ten logarithms of this series, show they reduce to the same functional equation, and put numbers on the contrast "range is \(\log N\), precision is \(N^{-1/2}\)." §04's "only \(c\ln x\)" assumes continuity (or measurability, or local boundedness); without it, everywhere-discontinuous solutions exist (Exercise 1). The comparison in §06 depends on Episode 52's measurement conditions (\(|\Delta\tau|\le0.1\), median of five seeds); a looser criterion extends \(q^*\). The \(N^{-1/2}\) of §07 assumes independent estimates; correlations slow the improvement. Why scale invariance arises in physics at all is beyond this article. — To print, use your browser's Print and "Save as PDF."