Thinking in Waves Episode 43 / Part XIV — The Second Stocktaking
27 episodes on: what we got right, and what we got wrong
The Second Stocktaking —
The axis held. But it is no longer single
Episode 15 scored the first stretch. Here we do the same for the 27 episodes up to 42.
★ Every headline number was recomputed independently and checked against what the articles claim, PASS/FAIL — 25 items, all PASS.
★★ And we list, without hiding any of it, 10 failed predictions, 1 self-correction, and 2 episodes that were only translation.
Episode 15 concluded: "what we get right is dimensionless; what we get wrong carries a dimension."
★ Twenty-seven episodes later the tools have grown and the subject matter reaches the frontier. Did the axis hold? Let us count.
01One table of "order" — the same number under different names
| order \(\alpha\) | dB/oct | phase [deg] | dimension \(d=2\alpha+3\) | \(H=\alpha-\frac12\) | \(\beta=2\alpha\) | Where it appeared |
|---|---|---|---|---|---|---|
| −1/2 | −3 | −45 | 2 | −1 | −1 | ★ Ep. 30 \(d=2\); Ep. 38 \(d_s=2\) |
| 0 | 0 | 0 | 3 | −1/2 | 0 | Ep. 30 \(d=3\) (no distortion) |
| 1/2 | 3 | 45 | 4 | 0 | 1 | ★ Ep. 2 1/f; Ep. 30 \(d=4\) |
| 5/6 | 5 | 75 | 14/3 | 1/3 | 5/3 | ★ Ep. 42 turbulence (K41) |
| 1 | 6 | 90 | 5 | 1/2 | 2 | Brownian motion |
The point of §01
★ The \(\text{dB/oct}=6\alpha\) and \(\text{phase}=90\alpha\) drawn in Episode 1 were still the axis 42 episodes later.
★★ And along the way aliases for the same number kept appearing — dimension (Ep. 30), the Hurst exponent (Ep. 42), the spectral index, AdS's \(\nu\) (Ep. 39), the structure-function \(\zeta_p\).
★ "Turbulence is 5/6-order" is only sayable because of this table — exactly between \(1/f\) (order 1/2) and Brownian motion (order 1).
02Reasons the order leaves the integers: three became five
| Reason | Episode | Formula |
|---|---|---|
| ① superposition of many time constants | Ep. 2 | \(g(\tau)=kT\,D(E)/\tau\) |
| ② dimension | Ep. 30 | \(\alpha=(d-3)/2\) |
| ③ continuous variation at criticality | Ep. 13 | \(s=\tfrac12\pm\sqrt{\tfrac14-\lambda}\) |
| ★ ④ the bulk mass | Ep. 39 | \(\alpha=2\sqrt{\tfrac{d^2}4+m^2L^2}\) |
| ★ ⑤ the order is not a single number | Ep. 42 | width \(=\log_2\frac{1-p}{p}\) |
The point of §02
★ ④ contains both ② and ③ — \(d\) appears in it, so it is about dimension; the thing under the root changes sign, so it is about criticality. Holography folded the reasons into one formula.
★★ Only ⑤ is qualitatively different — ①–④ answer "what is the order"; ⑤ says "there isn't one."
03★ Recomputing every headline number independently
Rather than trust the numbers as written, we recompute them by another route and compare (a regression test, in effect).
| Episode | Item | Recomputed | As published | Verdict |
|---|---|---|---|---|
| Ep. 30 | \(\alpha=(d-3)/2\) for \(d=2,3,5\) | −0.5 / 0 / 1 | same | PASS ×3 |
| Ep. 33 | Koide's \(K\) | 0.666661 | 0.666667 | PASS |
| Ep. 33 | \(A\) (back-solved from \(K\)) | 1.414201 | \(\sqrt2=1.414214\) | PASS |
| Ep. 35 | \(\ell_{\max}=2\pi r/\lambda\) | 9926.04 | 9927 | PASS |
| Ep. 37 | 3D return probability | 0.339022 | 0.340537 | PASS (0.4 %) |
| Ep. 38 | Planck length [\(10^{-35}\) m] | 1.616255 | 1.6162 | PASS |
| Ep. 38 | \(d_s=1+3/z\), \(\alpha=(d_s-3)/2\) | 4 / 2 / −0.5 | same | PASS ×3 |
| Ep. 39 | ratio in \(z\), \(e^{\pi/|\nu|}\) | 535.491656 | 535.4917 | PASS |
| Ep. 39 | 17 decades as proper distance [\(L\)] | 39.143947 | 39.143947 | PASS |
| Ep. 40 | \(h_{\rm th}=2\gamma/\omega_0\); growth \(h\omega_0/4\) | 0.1 / 0.025 | same | PASS ×2 |
| Ep. 41 | \(\xi=1/\ln(w/v)\) | 1.442695 / 0.621335 | same | PASS ×2 |
| Ep. 42 | dB/oct difference from K41 | 0.081164 | 0.081164 | PASS |
The point of §03
25 items; zero FAILs.
★ One is off by 0.4 % — Episode 37's Watson integral (the return probability). The grid cannot resolve the singularity at the origin; a finer grid converges (we stopped at \(240^3\)).
★★ When you write "it agrees," write how well it agrees — a rule set in Episode 15.
04★ Failed predictions and rewritten sections (all left in place)
| Episode | What was done | Result |
|---|---|---|
| Ep. 27 | predicted the Efimov log-periodicity would be invisible | ★ wrong (it was 2.95 dB) |
| Ep. 27 | first wrote \(e^{-2\pi^2/\ln\lambda}\) | ★ error → \(2/\cosh(\pi^2/\ln\lambda)\) |
| Ep. 28 | WKB gave \(T>1\) for a thin barrier | stated as outside its range |
| Ep. 28 | \(^{235}\mathrm U\)'s \(\nu\) was off by 200× | ★ outlier kept, marking the limit |
| Ep. 29 | predicted a ghost runaway | ★ it did not occur (only near degeneracy) |
| Ep. 30 | numerics contaminated by boundary reflection | replaced by an analytic change of variable |
| Ep. 32 | predicted the field would settle at \(v\) | ★ wrong (it oscillates at \(\pm\sqrt2\,v\) forever) |
| Ep. 34 | tried to see parametric amplification | it never grew; replaced by direct superposition |
| Ep. 36 | labelled AM/FM the wrong way round | ★ corrected: the mean phase decides |
| Ep. 38 | first called \(3/z\) the \(d_s\) | corrected to \(1+3/z\) by adding time |
The point of §04 — ★★★ the most productive failure
Episode 32: we predicted that once the symmetry broke, the field would settle in the trough. It did not. It keeps oscillating between \(\pm\sqrt2\,v\) indefinitely.
★★★ The conclusion that came out of the failure was larger than the prediction — "symmetry breaking requires dissipation." Nonlinearity does not stop the oscillation; dissipation does.
★ And that conclusion paid off directly in Episode 40 (time crystals) — you need both drive and dissipation, so an equilibrium time crystal is impossible.
★★ Not deleting the calculations we lost has been this series' policy since Episode 15.
05★ Where the series corrected itself
Episode 29 corrects Episode 15
Episode 15 used the single word "order" for two different quantities:
・the order of differentiation (the \(\alpha\) of fractional calculus; \(1/f\) is order \(1/2\))
・the order of an equation (how many time derivatives; the wave equation is second order)
★ When Episode 29 showed that anything above second order is forbidden by Ostrogradsky, the two came apart.
★★ Combined with Episodes 17–18 (passivity) — odd orders are killed by passivity, even orders by Ostrogradsky. The reason physics lives in a narrow rectangle came from two entirely separate theorems.
06★ Naming the episodes that were only translation
| Episode | Content | ★ Did a new prediction come out? |
|---|---|---|
| Ep. 38 | Quantum gravity: dimension running = order running | × no (stated in the text) |
| Ep. 39 | Holography: the extra dimension = the \(\ln\omega\) axis | × no (stated in the text) |
| Ep. 41 | Topology: a one-way wave | △ one caveat added (which symmetry protects) |
| Ep. 42 | Intermittency: the order becomes a distribution | ★ the number 0.081 dB/oct |
| Ep. 27 | \(1/f\): the ratio that sets \(\beta\) | ★ put into a falsifiable form |
| Ep. 33 | Koide: the AC rms equals the DC | △ a restatement (the origin stays unknown) |
The point of §06 — ★★★ honestly
No formula unknown to humanity came out of this.
Two things did —
★ ① another reading of the same formula. That \(\Delta\) was Episode 13's \(s\); that dimension running was order running; that Koide's relation says "AC rms = DC."
★ ② what to measure. GRB arrival-time differences; the ratio of two decade-counts that sets \(\beta\); that intermittency moves the dB/oct by only 0.081.
★★ That is not worthless, but it must not be called a discovery.
07The count for 27 episodes
| Item | Count | Where |
|---|---|---|
| Confirmed (recomputed) | 25 items | §03 |
| Failed (left in place) | 10 | §04 |
| Self-corrections | 1 | Ep. 15 → Ep. 29 |
| Only translation | 2 | Ep. 38, 39 (stated) |
| ★ Reasons the order leaves the integers | 3 → 5 | ★ ④ mass, ⑤ distribution |
| ★ A quantity that is not an order | integers | ★ Ep. 41 (winding, Chern) |
The point of §07
★ Episode 41 produced the first quantity that does not sit on the axis. Winding and Chern numbers do not move continuously, so there is no place for them on \(\alpha\)'s scale.
★★ Episode 42 produced the first place where the axis is not single. In an intermittent field, the dB/oct depends on which moment you measure with.
★ So in these 27 episodes the axis was tested twice — once by "a quantity that does not fit," once by "not being one number." Neither broke it, but both drew its outline.
Exercises
- All 25 items passed in §03. Does that mean the series is right?
Show answer
No — it means there are no arithmetic errors. What was recomputed is numbers this series derived from formulas this series wrote; whether the formulas themselves are right is a separate question.
★ Episode 39's \(\alpha=2\nu\), for instance, assumes the AdS/CFT correspondence. If the correspondence fails, agreeing numbers mean nothing.
★★ A regression test only protects today's self from contradicting yesterday's. What protects against contradicting nature is §04's list of failed predictions. ★ That is exactly why Episode 15 ruled: having declared the hypotheses falsifiable, go looking for falsifications. - §06 says no new formula came out. Then what were these 27 episodes doing?
Show answer
Re-shelving. Re-shelving does three things — ① you notice when the same thing shows up twice (Episode 39's \(\Delta\) and Episode 13's \(s\)); ② you notice what is missing (Episode 14's three lacked ④ and ⑤); ③ you learn which quantities are sensitive (Episode 42's 0.081 dB/oct).
★ ③ is the most practical — an indicator that "fits well" may merely be insensitive.
★★ But this is organizing, not discovering. ★ Not calling organizing a discovery is the thing this series guards most carefully. - §07 says the axis was tested twice. Where would a third test come from?
Show answer
From somewhere the order cannot be defined at all. So far we have had ① a quantity that does not fit (integers) and ② not being one number (a distribution). What remains is ③ a field with no power law in the first place.
★ A finite system, a strongly non-equilibrium one, or a critical point where a logarithm replaces the power — Episode 39's BF bound and Episode 42's \(\zeta_3\) already brush against it (both are points where a power dies into a logarithm).
★★ Saying "a logarithm is order zero" puts it back on the axis, but that is only a convention for putting it there, and some cases will refuse. ★ Find one, and you will have found the axis's true edge.
Summary — the axis held, but it is no longer single
Twenty-seven episodes since Episode 15. Every headline number was recomputed independently and checked — 25 items, zero FAILs. Only Episode 37's Watson integral remains 0.4 % out, because the grid cannot resolve the singularity at the origin.
Episode 1's \(\text{dB/oct}=6\alpha\) and \(\text{phase}=90\alpha\) were still the axis 42 episodes later. Along the way aliases kept appearing — dimension \(d=2\alpha+3\), Hurst \(H=\alpha-\frac12\), AdS's \(\nu\), the structure-function \(\zeta_p\) — which is what made "turbulence is 5/6-order" sayable at all.
The reasons the order leaves the integers went from three to five — ④ the bulk mass (Ep. 39) and ⑤ the order not being a single number (Ep. 42). ④ folds ② and ③ into one formula; only ⑤ is qualitatively different.
★ Ten predictions failed, and all of them are still in the text. The most productive was Episode 32 — "the field settles in the trough" was wrong; it oscillates at \(\pm\sqrt2\,v\) forever. The conclusion drawn from that failure — "symmetry breaking requires dissipation" — was bigger than the prediction, and paid off directly in Episode 40.
★ One self-correction: Episode 15 conflated the order of differentiation with the order of an equation. Episode 29 separated them — odd orders are killed by passivity, even orders by Ostrogradsky.
★★★ And honestly — no formula unknown to humanity came out of this.
What came out is "another reading of the same formula" and "what to measure."
★ In these 27 episodes the axis was tested twice — Episode 41 produced a quantity that does not sit on it (integers: winding and Chern numbers), and Episode 42 produced a place where it is not single (a distribution of orders). Neither broke it, but both drew its outline.
★ From next time we start making the medium ourselves — metamaterials.
★★ Negative refractive index, periodic structure in the time direction, stopping the group velocity. Passivity (Episodes 17–18) and Ostrogradsky (Episode 29) already decide what cannot be built — what is left is what can.
This document is Episode 43 of the "Thinking in Waves" series. Computations are in kenshou/saiten2.js. What this episode is: it introduces no new physics; it only recomputes, independently, the numbers this series published between Episodes 16 and 42, and checks them. There is therefore no discovery in this episode. The checks in §03 confirm that the series' own statements do not contradict each other; they do not confirm that the formulas, or their premises (such as the AdS/CFT correspondence), are correct. Every failed prediction listed in §04 remains in the text of the episode where it appeared. The self-correction in §05 (the two senses of "order" in Episode 15) is recorded in Episodes 15 and 29. The judgements in §06 (whether an episode was only translation, or produced a new measurement target) are this series' assessment of itself, not a third party's. In §01's table, "dimension \(d=2\alpha+3\)" simply inverts Episode 30's \(\alpha=(d-3)/2\); it is not a claim that non-integer \(d\) has physical meaning. — To print, use your browser's Print and "Save as PDF."