Thinking in Waves Episode 55 / Part XVIII — Putting Error Bars On It
Closing Part XVIII — the revised handbook, and an audit of our own numbers
The Revised Handbook, and an Audit of "Conditional" Numbers —
We had been writing too many digits
We fold Episodes 51–54 into the handbook and measure, by sensitivity, how much this series' numbers depend on what.
★ The handbook grows from 7 steps to 10 — ⑧ split the error in two, ⑨ buy range, ⑩ count the assumptions.
★★ And the honest conclusion — Episode 42's 0.0812 and Episode 47's \(-0.1053\) are both good to two significant figures. We had been writing too many digits.
Four episodes since 51 have counted how far one may speak.
★ This one folds the results into the handbook and puts sensitivities on our own numbers.
01The revised handbook (Episode 49 → Episode 55)
| Step | From | Amendment from Episodes 51–55 |
|---|---|---|
| ① plot it log–log | Ep. 1 | |
| ② measure the local order | Ep. 48 | ★ by window fitting, not differencing (Ep. 51) |
| ③ count the flat decades | Ep. 27, 48 | ★ the reportable span is capped at \(D-w\) (Ep. 51) |
| ④ find the bends | Ep. 16 | ★ centre the window in the inertial range (Ep. 51) |
| ⑤ measure the phase | Ep. 1, 8 | is it \(90\alpha\) degrees? |
| ⑥ change the moment | Ep. 42 | ★ measure \(q^*\) first (Ep. 52) |
| ⑦ compare with the system size | Ep. 5, 48 | |
| ★ ⑧ split the error in two | ★ Ep. 51 | ★ statistical \(5.57/\sqrt K\) dB, and systematic |
| ★ ⑨ if range is short, buy it | ★ Ep. 54 | ★ splice / another observable / change the system |
| ★ ⑩ count the assumptions | ★ here | ★ publish a sensitivity with every number |
02★ Sorting this series' numbers into four kinds
| Episode | Number | Kind | Note |
|---|---|---|---|
| Ep. 1 | \(\text{dB/oct}=6\alpha\) | D translation of a known relation | close to a definition |
| Ep. 30 | \(\alpha=(d-3)/2\) | A computed here | free-field Green function |
| Ep. 33 | Koide's \(K=0.666661\) | B literature value | ★ the masses are literature |
| Ep. 38 | \(d_s=1+3/z\) | ★ C conditional on a model | assumes time is unmodified |
| Ep. 39 | \(\alpha=2\nu\) | ★ C conditional on a model | assumes AdS/CFT |
| Ep. 42 | 0.0812 dB/oct from K41 | ★ C conditional on a model | p-model with \(p=0.7\) |
| Ep. 47 | \(n_s\to-0.1053\) dB/oct | B literature value | ★ \(n_s\) is observed |
| Ep. 51 | 5.57 dB per ordinate | A computed here | ★ fewest assumptions of all |
| Ep. 52 | \(q^*\approx4\) (1M boxes) | ★ C conditional on a model | depends on the 0.1 criterion |
The point of §02
Sorting 16 items — A (computed here) 6, B (literature) 3, C (conditional on a model) 5, D (translation) 2.
★★ Five are "conditional on a model" — the class that needs the most care.
★ This episode attaches a sensitivity to those five.
03★★ Sensitivity — how far do the numbers move when the assumptions do?
(a) How much does Episode 42's 0.0812 dB/oct depend on the p-model's \(p\)?
| \(p\) | \(\zeta_6\) | ★ dB/oct difference | vs literature 1.78 |
|---|---|---|---|
| 0.60 | 1.943416 | 0.019483 | +0.163 |
| 0.65 | 1.875672 | 0.044566 | +0.096 |
| 0.70 | 1.785875 | ★ 0.081164 | +0.006 |
| 0.75 | 1.678072 | 0.131091 | −0.102 |
| 0.80 | 1.556393 | 0.197324 | −0.224 |
§03 (a) — ★★★ the core of this episode
★ Matching \(\zeta_6\) to the literature 1.78 gives \(p\approx0.70\), whence 0.0812.
★★ Swing \(p\) over 0.65–0.75 and 0.0812 moves between 0.0446 and 0.1311.
★★★ So 0.0812 is good to two significant figures — it should have been written "of order 0.08 dB/oct." Writing "0.0812" in Episodes 42, 47, 50 and 53 was too many digits.
(b) How much does Episode 47's \(-0.1053\) move with the observational error?
| \(n_s\) | \(3(n_s-1)\) [dB/oct] |
|---|---|
| 0.9607 | −0.117900 |
| 0.9649 | −0.105300 |
| 0.9691 | −0.092700 |
★ \(-0.1053\pm0.0126\) — again two significant figures. "Of order \(-0.11\) dB/oct."
(c) How much does Episode 46's "1.5 ms" depend on how the band is defined?
| Band | e-folds | ★ time [ms] |
|---|---|---|
| visible → inverse age of the universe | 74.5 | 1.467 |
| radio → inverse age of the universe | 61.3 | 1.209 |
| visible → 1 mHz | 40.8 | 0.803 |
| visible → 1 Hz | 33.8 | 0.667 |
§03 (c) — ★ the good side of the logarithm
★★ Taking the lower edge at 1 Hz gives 0.67 ms; at the inverse age of the universe, 1.47 ms — a factor of two.
★★★ Because it is a logarithm, it is insensitive to the definition.
★ Episode 53's "range goes as \(\log N\)" was bad news; this is the same property's good side — a logarithm is hard to increase and equally hard to shake.
(d) How much does Episode 51's best error depend on \(n\), the estimates per decade?
| inertial range \(R\) | \(n=30\) | \(n=100\) | \(n=300\) | \(n=1000\) |
|---|---|---|---|---|
| 3 decades | 0.2502 | 0.1793 | 0.1531 | ★ 0.1393 |
| 4 decades | 0.1613 | 0.1155 | 0.0944 | 0.0730 |
| 6 decades | 0.0853 | 0.0587 | 0.0410 | 0.0277 |
★ Thirty times more estimates does not even halve the best error (the systematic remains) — at \(R=3\) decades it plateaus near 0.13 dB/oct.
04★★ A worked case — is Koide's \(K\) within experimental error?
| Quantity | Value |
|---|---|
| \(K\) (central) | 0.666660511 |
| \(2/3\) | 0.666666667 |
| ★ difference | \(-6.155\times10^{-6}\) |
| ★ from \(m_\tau=\pm0.12\) MeV | \(\pm6.775\times10^{-6}\) |
| ★ difference ÷ uncertainty | ★ 0.909 |
The point of §04
★★★ The difference is 0.9 of the uncertainty — with present measurements, \(K\) cannot be told apart from \(2/3\).
★ Episode 33 wrote "\(K=2/3\iff A=\sqrt2\)", but whether it is exactly \(2/3\) should have been decided by this comparison.
★★ And if \(m_\tau\)'s uncertainty shrinks, the verdict can change. Only \(m_\tau\) matters here; \(m_e\) (relative \(3\times10^{-10}\)) and \(m_\mu\) (\(2\times10^{-8}\)) do not.
★ So "is Koide's relation exact?" is decided by the tau mass measurement — a question that can be settled.
05Five levels of confidence
| Level | Character | Example |
|---|---|---|
| ① identity or definition | cannot move | \(\text{dB/oct}=6\alpha\) (Ep. 1) |
| ② computed here | reproducible | ★ 5.57 dB per ordinate (Ep. 51) |
| ③ computed from literature values | depends on the literature | \(n_s\) (Ep. 47), lepton masses (Ep. 33) |
| ★ ④ conditional on a model | ★ depends on an assumption | ★ the p-model's 0.0812 (Ep. 42) |
| ★ ⑤ translation under an assumed duality | ★ the premise itself is assumed | ★ AdS/CFT (Ep. 39) |
The point of §05
★★ This series has marked ①–③ as "◎", ④ as "△ conditional," and ⑤ as "translation" (the policy of Episodes 43 and 50).
★★★ What this episode adds is a sensitivity number for ④ and ⑤. Writing "conditional" is not enough; write which assumption, moved how far, moves the answer how much.
06What wave language can and cannot say
| Item | Verdict | Basis |
|---|---|---|
| Handbook 7 → 10 steps | ◎ organized | ⑧ split the error, ⑨ buy range, ⑩ count assumptions |
| Numbers sorted into four kinds | ◎ organized | ★ five are conditional on a model |
| ★ Sensitivity of Episode 42's 0.0812 | ◎ numerics | ★ 0.04–0.13 for \(p=0.65\)–0.75 |
| Sensitivity of Episode 47's \(-0.1053\) | ◎ numerics | ±0.0126 |
| ★ Sensitivity of Episode 46's 1.5 ms | ◎ numerics | ★ within a factor of two |
| ★ Koide's \(K\) versus experiment | ◎ numerics | ★ difference ÷ uncertainty = 0.909 |
| ★ Validation on real data | × not done | ★ all of Part XVIII used synthetic data |
Exercises
- §03 (a) found 0.0812 good to two figures. Does Episode 42's conclusion change?
Show answer
The conclusion does not change; the way it is written does. Episode 42's claim was "intermittency barely moves the dB/oct, which is why the power spectrum cannot see it" — true at 0.04 and at 0.13 alike.
★ What changes is having written "0.081" to three figures — it should read "of order 0.1 dB/oct."
★★ And the comparison with Episode 47 ("both the universe and turbulence are of order 0.1 dB/oct") is more accurate in this phrasing — since neither is known better than one figure.
★ Saying "the same order" was right; setting "0.0812 against 0.1053" side by side overstated the precision. - §03 (c) says the logarithm makes the answer insensitive to definitions. Does that contradict Episode 53?
Show answer
No — the same property, seen from both sides. A logarithm barely moves when its input moves a lot.
★ In Episode 53 that was bad news (\(10^6\) times the data buys 3× the range).
★★ Here it is good news (changing the band definition by \(10^{18}\) changes the answer by 2×).
★★★ "Hard to increase" and "hard to shake" are the same thing. So quantities spoken of in e-folds can be discussed without pinning down the definition — which is why "60 to 75" sufficed in Episodes 46 and 47. - §04 found Koide's \(K\) indistinguishable from \(2/3\). So is the relation right?
Show answer
It cannot be called right, and it cannot be called wrong — that is the present state. Difference ÷ uncertainty = 0.909 means only that the discrepancy is within chance.
★ What matters is that this is a measurable question — reduce \(m_\tau\)'s uncertainty tenfold and the ratio becomes 9, settling it.
★★ Episode 15 ruled that having declared the hypotheses falsifiable, we go looking for falsifications. Koide's relation is exactly of that shape, and tau-mass measurements will decide it automatically.
★ Conversely, the reading this series offered in Episode 33 — "\(K=2/3\iff\) the AC rms equals the DC" — is independent of whether the relation is exact: it is a translation, not a claim.
Summary — we had been writing too many digits
Episodes 51–54 are folded into the handbook, taking it from 7 steps to 10 — ⑧ split the error into statistical and systematic, ⑨ buy range if it is short, ⑩ count the assumptions.
Sorting 16 of this series' numbers — 6 computed here, 3 from literature, 5 conditional on a model, 2 translations. The five conditional ones now carry sensitivities.
★★★ The core —
Episode 42's 0.0812 moves between 0.0446 and 0.1311 as \(p\) swings over 0.65–0.75.
Episode 47's \(-0.1053\) carries \(\pm0.0126\) from the observation.
Both are good to two significant figures.
★ This series has been writing "0.0812" and "\(-0.1053\)" to four figures — too many. The right phrasing is "of order 0.1 dB/oct." ★ But the conclusions of Episodes 42 and 47 (intermittency is hard to see; universe and turbulence are the same order) survive at this precision.
★ The logarithm's good side also appeared — Episode 46's "1.5 ms" changes by only a factor of two when the band's lower edge moves by \(10^{18}\). Episode 53's "hard to increase" and this "hard to shake" are one property seen twice.
★★ As a worked case, Koide's \(K\) — the difference from \(2/3\) is \(-6.16\times10^{-6}\) against an uncertainty of \(\pm6.78\times10^{-6}\) from \(m_\tau\). Difference ÷ uncertainty = 0.909: indistinguishable at present. ★ What matters is that this is a measurable question — tau-mass measurements will settle it.
★ Honestly: all of Part XVIII (Episodes 51–55) used synthetic data and was never once applied to a real measurement.
★ From next time we take them back to physics — Part XIX.
★★ We return first to the place with the longest span — Episode 2's \(1/f\). How many decades does it really hold, and where does it end? ★ Episode 27's "ratio that sets \(\beta\)" gets re-measured with the tools of Episodes 51–55.
This document is Episode 55 of the "Thinking in Waves" series. Computations are in kenshou/kaitei.js. On the numbers here: the sensitivity calculations (the p-model's \(p\), \(n_s\)'s observational error, the band definition, the estimates per decade) and the comparison of Koide's \(K\) with experimental uncertainty were all computed here. The lepton masses and their uncertainties, \(n_s=0.9649\pm0.0042\), and turbulence's \(\zeta_6\approx1.78\) are literature values. The propagation in §04 accounts only for \(m_\tau\)'s uncertainty; those of \(m_e\) and \(m_\mu\) are below \(10^{-8}\) relative and do not contribute. The value used, \(m_\tau=1776.86\pm0.12\) MeV, is a representative literature figure; a different adopted value changes "difference ÷ uncertainty," and this article's conclusion (indistinguishable at present) holds only within that. The sensitivity in §03 (a) rests on one particular way of matching (fitting the p-model's \(\zeta_6\) to the literature); other models (log-normal, She–Lévêque) give different sensitivities. The classification in §02 is this series' assessment of itself, not a third party's. All of Part XVIII (Episodes 51–55) used synthetic data and was not applied to any real measurement. — To print, use your browser's Print and "Save as PDF."