Thinking in Waves Episode 15 ／ Part VII · Interim summing-up

The protagonist was never the constants — it was the order

What physics can be written in wave language,
and what cannot ── plus the questions nobody has closed Folding fourteen episodes onto one page. The opening question (are constants integrals?) came out half right.
But somewhere along the way the protagonist had changed ──
★ what this series was really chasing was not the constants but the order.

Prerequisites: Episodes 1–14Computation: all 13 scripts in kenshou/

It began with one observation: differentiating audio makes it brighter.
We walked from there to the physical constants; now let us count how far we got.

01The protagonist had changed

The opening question was "are all physical constants integrals of something?" Episode 7 scored it ── half right. Yet the series did not stop there; seven more episodes followed. What were they chasing?

Ep.	Nominal subject	What was actually being decided
1・2	calculus, and fractional orders	the order $\alpha$ itself
8	causality and Kramers–Kronig	analyticity of $(i\omega)^\alpha$
9	the uncertainty principle	joint resolution in time and frequency
10	the arrow of time	★ parity of the order
11	why heat does not travel	★ a mismatch: 1 in time, 2 in space
12	why the sky is blue	★ squaring a second derivative
13	imaginary-order differentiation	★ widening the order into a plane
14	the ladder of spectral exponents	★ three ways to leave the integers

The heart of §01

From Part IV onward, this series' protagonist was the order.

Causality, uncertainty, the arrow of time, why heat does not travel, why the sky is blue, turbulence's $-5/3$ ── all of it reduced to statements about order.
★ "Are constants integrals?" was the doorway to "what is an order?"

02What stood

Claim	Verdict	Evidence	Ep.
differentiation = treble, integration = bass	◎	$\pm6.02$ dB/oct, phase $\pm90°$; 9 digits	1
the order can be moved continuously	◎	dB/oct $=6\alpha$, phase $=90\alpha$	2
half-differentiation is real	◎	Warburg impedance $(i\omega)^{-1/2}$	2
constants are integrals (couplings)	◎	$1/\alpha:137.036\to128.940$; matches literature	5
mass = cutoff frequency	◎	Yukawa's 1.41 fm, $E^2=(pc)^2+(mc^2)^2$	6
causality = analyticity	◎	Kramers–Kronig verified to 13 digits	8
uncertainty = bandwidth theorem	◎	exactly $0.5000$ for a Gaussian	9
arrow of time = parity of the order	◎	dissipation $\propto\sin(\alpha\pi/2)$ to $10^{-14}$	10
heat and skin effect share one equation	◎	$\delta=\sqrt{2D/\omega}$; soil 2.24 m, copper 9.23 mm	11
radiation = second derivative; scattering $\propto\omega^4$	◎	blue 4.35× red	12
imaginary order = generator of scale transformations	◎	all-pass to 12 digits; group delay $-\beta/\omega$	13
exponent denominators divide $\|\det M\|$	◎	consistent across all six cascades	14

03What fell

Claim	Verdict	Why	Ep.
mass = integral of the universe's gravity	×	Sciama's 1/2 is an identity; Brans–Dicke excluded at $10^{-5}$	7
light speed = integral of information	×	$c$ is defined; $l_P/t_P=c$ is a definition, not a discovery	7
the constants vary in time	×	bounds only, in every band; $\|\dot\alpha/\alpha\|<10^{-17}$/yr	4
turbulence's $-5/3$ proves fractional order	×	★ Navier–Stokes is integer-order; the fraction is dimensional	14
measuring log-periodicity from the impulse response	×	★ the tail is crushed by $e^{-\beta\pi}$ and loses to a DC artefact	13

§02–03 ── two dividing lines

Line 1 (Episode 7): hits are dimensionless, misses are dimensionful.
$\alpha$ and $\mu=m_p/m_e$ really are integrals and really run. $c$, $\hbar$, $k_B$ turn into unit conventions and differentiate to zero by definition.

★ Line 2 (Episode 14): the order of a law and the exponent of a statistic are different things.
Warburg's $\alpha=-1/2$ is an equation that is genuinely half-order. Turbulence's $-5/3$ comes from an integer-order equation, after statistics.
── The same "fraction", entirely different provenance.

04Limits of the framework itself

Precisely because it worked, it is worth writing down where it breaks.

①

It assumes stationarityFourier is the tool for things that oscillate the same way forever. A one-off jump spreads thinly over all frequencies and escapes (Episode 4)

②

It assumes linearitywaves superpose, but general relativity is non-linear. In strong fields nothing can be written as a "filter"

③

$\omega=0$ is out of reachin principle, because it is the value itself (Episode 3)

④

Small effects are invisible★ we failed at this in Episode 13. The log-periodic tail is crushed by $e^{-\beta\pi}$, $10^{-6}$ at $\beta=3$ ── smaller than the numerical artefact

Putting a number on ② ── how linear is it? $$\frac{GM}{Rc^2}:\quad \text{Earth's surface}\ 7\times10^{-10},\quad \text{solar surface}\ 2\times10^{-6},\quad \text{neutron star}\ 0.2,\quad \text{horizon}\ 0.5$$

On Earth it is linear to 10 digits ── which is why wave language works so well
★ Near a black hole the framework itself is unusable

05The map of orders ── where does physics live?

Order $\alpha+i\beta$	Physics	Character	Ep.
$\alpha=-1$	integration (position, potential)	$-6$ dB/oct	1
$\alpha=-1/2$	diffusion boundary (Warburg)	★ lower bound	2, 11
$\alpha=0$	identity (elastic, conservative)	zero dissipation	10
$0<\alpha<1$	real materials (silica $6.4\times10^{-6}$, asphalt $0.5$)	dissipation $\propto\sin(\alpha\pi/2)$	10
$\alpha=1$	viscosity, diffusion, conduction	fully irreversible	10, 11
$\alpha=2$	Newton's law, the wave equation	reversible	1, 10
$\alpha=3$	radiation reaction (Abraham–Lorentz)	★ past the ceiling; pre-acceleration	2, 12
$\beta\neq0$	critical $1/r^2$, Efimov states	★ log-periodic; ratio 22.7	13

§05 ── a narrow corner

Physics lives in the narrow rectangle $\alpha\in[-1/2,\,2]$, $\beta\approx0$.

Lower bound $-1/2$: below it no causal response can be built (the diffusion boundary).
Upper bound 2: above it the Ostrogradsky instability sets in (and indeed order-3 radiation reaction produces pre-acceleration).
Imaginary part 0: $\beta\neq0$ requires a special balance such as $1/r^2$.

★ "Why does physics live in this rectangle?" ── that is the question this series arrived at.

06Doors still open

Question	State	What would close it
Why is $1/f$ noise universal?	unsolved	a reason for the $1/\tau$ distribution. Our "half-integral" story is a restatement
Is a fractional order always a coarse-graining?	hypothesis	one counterexample refutes it ── falsifiable
Where does $\alpha\approx1/137$ come from?	unsolved	a framework deriving a dimensionless number as an integral. Nobody has one
Non-zero $\omega$ components of the constants	bounds only	a few more digits of sensitivity; the Th-229 nuclear clock is a candidate
The $10^{-11}\sim10^{-17}$ Hz band	thin	periods of $10^4\sim10^{10}$ yr; ice cores and varves are unused records
Why are the bounds on the order what they are?	★ new	can $-1/2$ and $2$ come from a single principle?
Is $\beta\neq0$ only for $1/r^2$?	★ new	is there another mechanism for discrete scale invariance? (Episode 13)

§06 ── two new doors

Parts IV–VI added two questions, both about the order.

① Where the bounds $-1/2$ and $2$ come from. At present they are explained separately (the diffusion boundary; Ostrogradsky). If one principle gave both, that would be close to an answer to "why is physics shaped this way".

② What turns the imaginary part on. The only known case is a critical $1/r^2$. Are there others, or is $1/r^2$ essentially unique? ── if Episode 13's hypothesis is right (that $1/r^2$ is the only potential with the same $r$-dependence as the kinetic energy), then the latter.

※ We have solved neither. Putting them into the form of questions is the result.

07Folding it onto one page

The whole series $$\frac{d}{dt}\ \to\ \times(i\omega),\qquad \int dt\ \to\ \times\frac{1}{i\omega},\qquad D^{\alpha+i\beta}\ \to\ \times\,\omega^\alpha e^{-\beta\pi/2}e^{i(\alpha\pi/2+\beta\ln\omega)}$$

Concept in physics	In wave language
position → velocity → acceleration	tilt the spectrum by $+6$ dB/oct each time
the constant of integration $+C$	the $\omega=0$ component (undeterminable)
physical constants	components with period longer than the observation window
mass	cutoff frequency $mc^2/\hbar$
range of a force	evanescent decay length below cutoff
coupling constants	non-DC components on the $\ln E$ axis
causality	analyticity in the upper half-plane (real and imaginary parts determine each other)
the uncertainty principle	the bandwidth theorem times $\hbar$
the arrow of time	parity of the order; dissipation is $\sin(\alpha\pi/2)$
heat conduction / skin effect	1 in time, 2 in space → $\omega=-iDk^2$
radiation, Rayleigh scattering	second derivative squared → $\omega^4$
scale invariance	the imaginary part of the order; phase $\beta\ln\omega$
turbulence's $-5/3$	the dimensional matrix has determinant 3

The final answer

"Are all physical constants integrals of something?" ──
Half right, and the right half sits at the centre of particle physics.

The right half: couplings are literally the integral of a beta function. $1/\alpha$ runs from 137.036 to 128.940.
The wrong half: dimensionful constants cannot be integrated into. They turn into unit conventions.

★ And the real answer lay outside the question ──
what this series found was not "what constants are" but "the axis of order".

Exercises

Name this series' biggest hit and biggest miss.
Show answer
Hit: mass = cutoff frequency (Episode 6). The Yukawa potential, $E=mc^2$ and the relativistic energy relation all came out of one dispersion relation.
Miss: mass = the integral of the universe's gravity (Episode 7). A beautiful 1/2 appeared, but it was an identity, and the testable version is excluded at $10^{-5}$.
What is striking is that the same "mass" succeeded completely one way and failed completely the other. The difference: seen as a wave, or seen as an integral.
★ And the biggest surprise was Episode 13 ── making the order imaginary moved us from time translation to scale transformation, an entirely different symmetry.
There are two kinds of "fractional order". Distinguish them.
Show answer
① The law itself is fractional-order: the Warburg impedance $(i\omega)^{-1/2}$, the springpot of viscoelasticity. A fractional operator literally appears in the equation.
② The exponent of a statistic is fractional: turbulence's $-5/3$. The governing equation (Navier–Stokes) stays integer-order; the fraction comes from dividing dimensions.
★ How to tell: ① has memory (an integral kernel), ② does not. ① is one system's response; ② is the statistical average over many degrees of freedom.
Episode 14 is where we noticed we had been conflating them.
Where should this view be taken next?
Show answer
Two directions.
A: toward dimensionless quantities (Episode 8's answer). Of the Standard Model's twenty-odd dimensionless parameters, not one has been derived as "the integral of" anything. Couplings run, but their initial values remain inputs.
B: toward the boundaries of the order (this episode's answer). Why $\alpha\in[-1/2,\,2]$? The lower bound is causality, the upper stability ── but both look as though they come from time running one way only.
★ B has the narrower reach but a sharper question, which makes it easier to start on ── that is the impression after fifteen episodes.

Summary　The framework stood, and the protagonist changed

It began with "differentiating audio makes it brighter". Measured: $\pm6.02$ dB/oct, phase $\pm90°$, nine digits ── the foundation was exact.

Making the order continuous, $\alpha=-1/2$ turned out to be real, and at $\omega=0$ the $1/\omega$ diverges ── the true identity of the constant of integration, where the physical constants live. In time they are flat, but change the axis to $\ln E$ and $1/\alpha$ runs from 137.036 to 128.940 ── "constants are integrals" is literally true there.

Seen as a wave, mass is a cutoff frequency. Seen as an integral of gravity, the beautiful 1/2 is an identity and the testable form is excluded. Hits are dimensionless, misses are dimensionful.

And from Part IV the protagonist changed ── causality is analyticity, uncertainty is the bandwidth theorem, the arrow of time is the parity of the order, heat does not travel because the orders are mismatched, the sky is blue because a second derivative was squared, and turbulence's $-5/3$ is a determinant. Then in Episode 13 the order widened into the imaginary direction and became the generator of scale transformations.

★ Physics lives in the narrow rectangle $\alpha\in[-1/2,\,2]$, $\beta\approx0$.
"Why this rectangle?" ── that is the wall this series reached.

A pause here (Episodes 1–15) Doors closed: mass = integral of gravity (refuted), light speed = integral of information (a matter of definition), time variation of the constants (bounds only), turbulence = evidence of fractional order (a different animal).
Doors still open: the origin of $1/f$, fractional order versus coarse-graining, the value of $\alpha$, the $10^{4}\sim10^{10}$ yr observational gap, why the bounds on the order are what they are, and whether $\beta\neq0$ needs $1/r^2$.
── And Part VIII begins. We have said repeatedly that $c$ is a defined value and cannot be differentiated; but an effective speed of light can be. Do the same for mass and ── ★ the Taylor expansion of the dispersion relation gives mass, speed and mass again at orders 0, 1 and 2.

This document is Episode 15 of the series "Thinking in Waves". The computations are the 13 scripts in kenshou/ (nami.js, bunsuu.js, teisu.js, hashiru.js, shadan.js, shitsuryo.js, inga.js, fukakutei.js, yajirushi.js, netsu.js, housha.js, kyosu.js, hashigo.js), all plain Node with no dependencies. On this episode: there is no new computation; it summarises Episodes 1–14. We folded the series here once, but reader questions have led to Part VIII and beyond (effective light speed, effective mass, the origin of mass, decay constants, the $c\cdot t=$const coordinate system, information). The $GM/(Rc^2)$ values in §04 are rough figures from representative parameters. The "open doors" in §06 are our own framing and do not imply that researchers in those fields see them that way ── in particular "filling the $\alpha$ gap with ice cores and varves", "deriving both bounds on the order from one principle" and "is $\beta\neq0$ only $1/r^2$?" are all our own speculations. That the origin of $1/f$ noise is unsolved is widely accepted, but our "half-integral" story is a restatement rather than an explanation. The bounds $-1/2$ and $2$ are boundaries within the scope we treated, not statements about the mathematical domain of fractional calculus. "Dimensionful is bookkeeping, dimensionless is physics" is the framing established in the sister series "Understanding c·t = constant", rediscovered here from the wave side. ── To print, use your browser's Print → Save as PDF.