From sound to the physical constants
Thinking in Waves
Differentiate an audio signal and it gets brighter; integrate it and it gets darker ── exactly \(\pm6.02\) dB per octave.
Physics does the same thing. Position → velocity → acceleration is an operation that keeps tilting a spectrum.
So are the physical constants all integrals of something? ── we followed that to the end.
Where the series starts
On the Fourier side, differentiation and integration are just multiplications.
$$\frac{d}{dt}\ \longrightarrow\ \times(i\omega),\qquad \int dt\ \longrightarrow\ \times\frac{1}{i\omega}$$
Amplitude by \(\omega\), phase by \(90°\). So differentiation boosts the highs (+6 dB/oct) and integration the lows (−6 dB/oct).
Repeating calculus in physics means tilting a signal's spectrum over and over.
In this series three conjectures were refuted and one was literally confirmed ── both are left on the page.
Differentiation is treble, integration is bass ── first, let us measure it
We actually differentiated and integrated a 48 kHz synthetic signal. Differentiation multiplies the amplitude by \(\omega\) and rotates the phase by \(+90°\); integration by \(1/\omega\) and \(-90°\). Two octaves give \(\pm12.04\) dB ── exactly \(\pm6\) dB/oct.
Even the small discrepancy at high frequency is explained to six decimal places by the central difference's \(\mathrm{sinc}(\omega\Delta t)\). Approximation, not physics.
And physics does nothing but repeat this ── position → velocity → acceleration. \(F=ma\) is "the thing tilted by +12 dB/oct".
Differentiate by one half ── and it turns out to be real
There is no reason to keep the exponent an integer. Moving \(\alpha\) continuously gives dB/oct \(=6\alpha\), phase \(=90\alpha\) degrees ── a clean interpolation.
And \(\alpha=-1/2\) (half-integration) is real ── the Warburg impedance \(Z\propto(i\omega)^{-1/2}\), measured daily in battery laboratories and visible as the 45° line on a Nyquist plot.
Fractional orders have memory (an integral kernel appears). [Hypothesis] Physics is nearly integer-order because nature is local. For noise, \(\beta=-2\alpha\) holds exactly, and 1/f is unreachable at integer orders.
Is the constant of integration the physical constant?
Integration is \(\times1/(i\omega)\), so it diverges at \(\omega=0\) and cannot fix the DC component. That is what "+\(C\)" really is ── the \(C\) you met at school is the origin of a spectrum.
And with only a window \(T\), we cannot distinguish a wave of period \(>T\) from a constant (even the age of the universe bottoms out at \(2.3\times10^{-18}\) Hz).
Constants of integration come in two kinds ── invisible ones (the zero of potential, gauge) become symmetries; visible ones (the cosmological constant \(\Lambda\)) become new constants.
Differentiating a constant ── the speed of light became undifferentiable in 1983
\(c\) is a defined value. \(dc/dt=0\) is a convention, not a measurement ── and the 2019 SI revision gave seven constants the same treatment. Only dimensionless quantities carry meaning.
Their derivatives have been measured ── \(|\dot\alpha/\alpha|<10^{-17}\)/yr. Over the whole history of the universe \(\alpha\) is constant to seven decimal places.
Translated into frequency bands, the lead changes hands at a period of 32 Gyr (optical clocks → the Oklo natural reactor). A tabletop clock fights cosmic history to a draw. But Fourier assumes stationarity, so a history that "jumped once" escapes ── the framework's first limit.
The running coupling ── here the constant really was an integral
Differentiating in time gave zero. We had the wrong axis.
Computing the one-loop vacuum polarisation gives \(\Delta\alpha_{\text{lep}}(m_Z)=\mathbf{0.031421}\) (published 0.031498). Adding the hadronic part gives \(1/\alpha(m_Z)=128.940\) (published 128.95).
\(1/137.036\to1/128.940\), a 5.9 % change. And that change is \(\int\beta(\alpha)\,d\ln\mu\) ── "constants are integrals" is literally true here. Only the variable is \(\ln(\text{energy})\), not time.
Mass is a cutoff frequency ── the vacuum was a waveguide
The Klein–Gordon dispersion relation \(\omega^2=c^2k^2+(mc^2/\hbar)^2\) is word for word the waveguide formula, with \(\omega_c=mc^2/\hbar\) the cutoff.
Below cutoff the wavenumber goes imaginary and the field decays over \(\hbar/mc\) ── the Yukawa potential, and the true identity of a force's range. 1.41 fm for the pion (the size of a nucleus); \(2.5\times10^{-3}\) fm for the W (why the weak force is weak).
\(E=mc^2\) is "the energy of the cutoff", \(E^2=(pc)^2+(mc^2)^2\) is the dispersion relation itself. Velocity is \(d\omega/dk\) ── the derivative of a spectrum.
Scoring two conjectures ── the neat 1/2 turned out to be an identity
Mass = the integral of gravity: Sciama's relation gives \(GM/(Rc^2)=0.500000\) ── exactly 1/2. But substituting makes \(G\), \(c\) and \(H\) all vanish; in a critical-density universe it is an identity. The testable version (Brans–Dicke) is excluded by Cassini at the \(10^{-5}\) level.
Light speed = the integral of information: the holographic bound is computable (\(3.3\times10^{122}\) bits, \(2.2\times10^{121}\) operations). But \(l_P/t_P=c\) is a definition, not a discovery.
Where the idea pays off is not the constants but the relations between them.
Causality ── a filter does not know the future
From the single condition "no output before the input", the real and imaginary parts completely determine each other ── the Kramers–Kronig relations.
Verified numerically on a Lorentz oscillator to 13 digits (subtracting the singularity before taking the principal value).
Hence "no dispersion without absorption" ── even a transparent medium has a refractive index different from 1 only because it absorbs somewhere far away. And the sum rule \(\int\omega\varepsilon''d\omega=\pi\omega_p^2/2\) merely counts electrons.
The uncertainty principle is the bandwidth theorem times \(\hbar\)
In the world of sound it is obvious ── a short sound has no definite pitch. That is \(\Delta t\,\Delta\omega\ge1/2\) (the Gabor limit); numerically, a Gaussian gives exactly 0.5000.
Multiply both sides by \(\hbar\) and you get \(\Delta E\,\Delta t\ge\hbar/2\) ── the uncertainty principle, unchanged. Not a quantum mystery but a Fourier theorem.
There is measurement too ── \(^{57}\)Fe with its 141 ns lifetime has a natural width \(\Gamma/E=3.24\times10^{-13}\), and Mössbauer spectroscopy used that sharpness to measure a gravitational redshift over 22.5 m.
The arrow of time lives in the parity of the derivative order
Under \(t\to-t\), \(D^n\to(-1)^nD^n\) ── even orders survive, odd orders flip. So \(F=ma\) (order 2) is reversible and diffusion (order 1) is not.
Fractional orders interpolate continuously: dissipation \(\propto\sin(\alpha\pi/2)\), confirmed to \(10^{-14}\). \(\alpha=0\) is perfectly elastic, \(\alpha=1\) perfectly viscous, and real materials lie in between.
★ Fused silica is \(\alpha=6.4\times10^{-6}\), asphalt \(\alpha=0.5000\) ── the "strength" of the arrow of time is a measurable material property.
Heat does not travel as a wave ── first order in time, second in space
The wave equation is second order in both; the diffusion equation is first in time and second in space. That mismatch gives the purely imaginary dispersion relation \(\omega=-iDk^2\) ── decay instead of propagation.
The penetration depth is \(\delta=\sqrt{2D/\omega}\) ── in soil, 0.117 m daily and 2.241 m seasonally. Seven metres down the seasons vanish.
And the electromagnetic skin effect is exactly the same formula (9.23 mm at 50 Hz in copper, 2.06 μm at 1 GHz). Heat and current share one equation.
Radiation comes from the second derivative ── which is why the sky is blue
Expand the field of a charge in distance and only the \(1/r\) term is proportional to acceleration and carries energy. "Radiation requires acceleration" was a question of order.
From the Larmor formula, a classical hydrogen atom collapses in 16 picoseconds ── one of the reasons quantum mechanics was needed.
And squaring the second derivative gives \(\omega^4\) ── Rayleigh scattering; blue scatters 4.4 times more than red. Go to third order (radiation reaction) and pre-acceleration appears, breaking causality.
Imaginary-order differentiation ── the tool of the world of scale
\((i\omega)^{i\beta}=e^{-\beta\pi/2}e^{i\beta\ln\omega}\) ── the amplitude is a constant independent of \(\omega\); only the phase rotates as \(\ln\omega\): an all-pass filter (confirmed to 12 digits with white noise). It stretches time without touching the timbre.
The impulse response has the closed form \(t\,h(t)=A(\beta)\sin(\beta\ln t-\arg\Gamma(1+i\beta))\) ── a sinusoid on a logarithmic scale. But the amplitude is crushed by \(e^{-\beta\pi}\), already \(10^{-6}\) at \(\beta=3\). Which is why log-periodicity is hard to find.
★ If real orders are the tool of time translation (Fourier), imaginary orders are the tool of scale transformation (Mellin). In physics the imaginary part turns on only when an attractive \(1/r^2\) passes \(\lambda=1/4\) ── measured as the Efimov ladder with ratio 22.7.
The ladder of spectral exponents ── only three ways to leave the integers
Line up every slope in this series and almost all are integers or half-integers; the exceptions are the turbulent \(-5/3\), \(-11/5\) and \(-7/3\).
Tracking their denominators gives ── ★ the determinant of the dimensional matrix. \(|\det M|=3\) for Kolmogorov, 5 for Bolgiano–Obukhov, and 1 for Phillips (ocean waves), hence the integer \(-5\). Whether you get an integer was decided by a determinant.
★ And turbulence's \(-5/3\) is not evidence of fractional order ── Navier–Stokes stays integer-order and the fraction appears after taking statistics. Quite different in provenance from Warburg's \(\alpha=-1/2\).
What physics can be written in wave language, and what cannot
Twelve claims stood, five fell. There were two dividing lines ── ① hits are dimensionless, misses dimensionful; ② the order of a law and the exponent of a statistic are different things.
★ And the protagonist had changed along the way. Causality, uncertainty, the arrow of time, why heat does not travel, why the sky is blue, turbulence's exponent ── all of it was about the order.
Physics lives in the narrow rectangle \(\alpha\in[-1/2,\,2]\), \(\beta\approx0\) (lower bound causality, upper bound Ostrogradsky). "Why this rectangle?" is the wall this series reached.
The effective speed of light ── even a defined value can be differentiated
In wave language the speed of light is the slope of a dispersion relation, \(d\omega/dk\). So "differentiating it" means taking the second derivative of that relation.
From the Sellmeier equation for fused silica we obtained the zero-material-dispersion wavelength 1272.8 nm (published about 1270 nm) ── optical fibre uses 1310 nm because the second derivative of the refractive index vanishes there. In gravity, \(c_{\rm eff}=c(1+2\Phi/c^2)\), which integrates to the Shapiro delay (262 µs round trip for Cassini).
The group velocity exceeds \(c\) and goes negative, yet \(n(\infty)=1\) holds to 8 digits ── ★ information never exceeds \(c\). And what changes is always a dimensionless quantity.
Effective mass ── the reciprocal of mass was a curvature
Expand the dispersion relation at \(k=0\) and ── ★ mass at order 0, speed at order 1, mass again at order 2. One word pointing at different features of one curve.
Tight binding gives \(m^*=\hbar^2/(2ta^2)\) (matching numerical differentiation to \(10^{-8}\)), and at the top of a band the mass goes negative (holes). Real materials span 0.014 to 200 ── four orders.
★ Graphene is a world of zero mass and effective light speed \(c/300\), where \(\alpha_{\text{graphene}}=2.19\) ── where light is slow, electromagnetism is strong. And the photon in a superconductor (5.06 eV in niobium) is bound to the W boson by one formula.
Where does mass come from? ── put it in a box, or soak it in a medium
Of the proton's 938.27 MeV, the quark masses sum to 8.99 MeV ── 0.96 %. 99 % of your body weight is the energy of a confined field.
In wave language it is simple ── a wave in a box has a lowest frequency (\(\pi\hbar c/L\) is 620 MeV at 1 fm). The other source is coupling to a condensate, and inverting the Yukawa couplings gives \(y_{\rm top}=0.9919\) alone of order one, with five and a half orders down to the electron.
★ And confinement and condensate are the same thing in wave language ── both create an intercept in a dispersion relation, one with boundaries and one with a background field. The photon in a waveguide, the photon in a superconductor, the proton and the electron all acquire mass the same way.
A decay constant is an imaginary frequency ── so no full period is needed
Episode 3 said a period longer than the age of the universe is indistinguishable from a constant. So what about \(^{232}\)Th's half-life of 14.05 Gyr?
★ The answer ── one gram gives 4057 counts a second; one second is enough. Because \(e^{-\lambda t}=e^{i(i\lambda)t}\) ── a decay constant lives on the imaginary axis. ★ The real axis has phase and the imaginary axis does not, so a time average can be traded for an ensemble average.
The optimal dating nuclide has \(T_{1/2}=4.78\) Gyr and \(^{238}\)U sits at 0.935 of it. Fully ionised \(^{187}\)Re decays \(10^9\) times faster. And the weak force is weak not from its coupling (\(\alpha_W\) is 4.3× stronger) but from the cutoff \(M_W\).
The \(c\cdot t=\)const coordinate system ── the wave equation became the critical \(1/r^2\)
Define \(c(t)=R/t\) and the speed of light is differentiable (\(\dot c/c=-1/t=-H\)). Read as a physical change it is excluded by a factor of seven million against the bound on \(\alpha\).
But as a wave ── the phase goes as \(\ln t\) (every wave is a hyperbolic chirp) and the mode equation becomes \(\ddot\psi+(A^2/t^2)\psi=0\). ★ Episode 13's critical \(1/r^2\), word for word.
In logarithmic time it reads \(f''+(A^2-1/4)f=0\) ── a plain oscillator (12–16 digits). And criticality at \(A=1/2\) is horizon crossing, outside which \(s=0,1\) give a constant mode and a growing mode. ★ The \(c\cdot t=\)const system was time measured logarithmically.
Integrate a bit rate and you get information ── and information lives on the imaginary axis
Shannon's formula \(I=BT\log_2(1+S/N)\) is "bandwidth × time" ── ★ Episode 9's bandwidth theorem itself. The telephone line's 30.9 kbps was the dial-up wall, and optical fibre is nine orders faster because its band is nine orders wider.
Landauer's limit is 0.0179 eV/bit at room temperature (semiconductors are still \(10^4\) above), the speed bound is \(5.4\times10^{50}\) /s for 1 kg, the capacity bound \(9\times10^{44}\) bit for a human ── all restatements of "energy sets bandwidth".
★ The universe's information goes as \(I\propto(ct)^2\), so it differentiates to \(1.4\times10^{105}\) bit/s with constant information acceleration. ★ So the \(c\cdot t=\)const system also holds the universe's information constant. And information lives on the imaginary axis ── "the arrow of time" is the imaginary axis of the complex frequency plane.
Measurement as a wave ── observing is choosing a basis to project onto
Measure the same wave in "time" or in "frequency" ── that choice is all the observer gets, and with a Gaussian window \(\Delta t\cdot\Delta\omega\) is always 0.5 (Episode 9's Gabor limit).
★ And the wavelet transform turns out to have uniform resolution on the \(\ln\omega\) axis (\(\Delta\omega/\omega\) constant, verified numerically) ── the same family as Episode 13's imaginary order and Episode 20's logarithmic time. If Fourier is the tool of time translation, wavelets are the tool of scale transformation.
★ Episode 4's homework is settled too ── a step change can be caught with wavelets (the transition width scales with \(\Delta t\)), but transition width × \(\Delta\omega\) stays fixed. Stationarity can be dropped; uncertainty cannot. Decoherence is \(V=e^{-\sigma_\varphi^2/2}\) ── ★ monotone, hence a phenomenon of the imaginary axis.
Non-linearity ── where the word "filter" stops working
"Filter" and "transfer function" all presuppose superposition. Add a squared term and ── two frequencies in, seven out (DC, difference, sum, second harmonics). Something no linear filter can ever do.
In a Duffing oscillator the same frequency and the same drive give two different answers (a gap of up to 1.85), so a transfer function cannot even be defined as a function. Measured as distortion, the everyday "linear world" is an approximation that ignores about 0.1 % ── linearity is not a property of nature but a consequence of smallness.
★★ And yet every constraint from Part IV survives (causality, the bandwidth theorem, the arrow of time, dimensional analysis) ── because "filter" is a word of linearity, while the three constraints come from time running one way. Sperm live at \(Re\sim10^{-4}\), completely linear ── hence the scallop theorem.
Gravitational waves ── radiation one order higher
Here what waves is spacetime itself, and the measured quantity \(h=\Delta L/L\) is dimensionless from the start (Episode 7 as a definition). In GW150914 a 4 km arm moved by a thousandth of a proton.
The monopole dies by mass conservation and the dipole by momentum conservation, so radiation starts at the quadrupole. ★ Electromagnetic is the second derivative of a dipole squared; gravitational is the third derivative of a quadrupole squared ── one order higher.
The reference power is \(c^5/G=3.6\times10^{52}\) W (built from \(G\) and \(c\) alone). The flux at Earth was 3 mW/m² ── the order of moonlight, yet the arm moved a thousandth of a proton. The chirp goes as \(\tau^{-3/8}\), ★ differing from Episode 13's \(1/t\) because there is a special moment: the merger. The 45° polarisations follow from spin 2 ＝ a rank-2 tensor.
The strong force as waves ── confinement was not a box but a string
Plot hadron masses as \(M^2\) against \(J\) and they fall on a clean straight line (a Regge trajectory), whose slope gives ★ a string tension of 14.4 tonnes-force. Constant at any distance ── which is why quarks can never be separated. The string's ladder has the same structure as Episode 6's waveguide and Episode 18's box.
The number of colours is measurable directly as the dimensionless R ratio (\(N_c=3\) fits all three bands), agreeing independently with the tau lifetime of Episode 19.
★★ And inverting asymptotic freedom ── a dimensionful \(\Lambda_{\rm QCD}\) is born from the dimensionless \(\alpha_s\) alone (dimensional transmutation). The corresponding length is 0.94 fm ── the size of a proton. ★ 99 % of the mass around you is a dimensionless quantity that acquired a dimension. And gluons are massless yet reach only 1 fm ── Episode 23's non-linearity changes the shape of the field itself.
Is a fractional order always the result of coarse-graining? ── we found a counterexample
Stack 1000 stages of a ladder made only of resistors and capacitors (both integer order) and the phase is \(-44.80^\circ\) with a slope of \(-9.98\) dB/dec ── ★ a half order (matching the exact semi-infinite solution). The reverse holds too: any fractional order with \(0<\alpha<1\) can be written as a superposition of first-order relaxations (\(g(\tau)\propto\tau^{\alpha-1}\)).
★★ And yet ── keep only the end mass of a uniform spring chain and you get a pure resistance at phase 0°. Infinitely many degrees of freedom removed, and still integer order. "Coarse-graining always gives a fractional order" is ★ refuted by counterexample.
The difference was how the removed degrees of freedom were distributed. With a bath of spectral density \(J(\omega)\propto\omega^s\) the order is exactly \(s\). ★ Rewritten hypothesis: a fractional order appears when the coarse-grained degrees of freedom are distributed scale-invariantly. Together with Episode 13 (imaginary order) and Episode 22 (wavelets) ── all three turned on scale invariance.
Where does log-uniform come from? ── what nature had ready was the exponential
The last mystery from last time gets its answer. ★ Making 1/f needs no power-law distribution ── with \(\tau=\tau_0e^{E/kT}\), \(\ln\tau\) is linear in the barrier \(E\), so \(g(\tau)=kT\,D(E)/\tau\). A flat \(D\) alone gives \(1/\tau\). At room temperature a decade costs 0.0595 eV, so 0.36 eV of scatter buys 6 decades of 1/f.
★★ And why the exponent pins near 1 came out too ── \(\beta-1\) = (decades over which \(D\) varies) ÷ (decades over which \(\tau\) spreads). Even a 1000× variation gives \(\beta-1=0.25\). The 0.8–1.4 scatter of measurements is exactly this ratio. ★ 1/f is not fine-tuning but a phenomenon hard to avoid. In \(\alpha\) decay, \(Q\) varies by 2.19× while the half-life varies by 24.2 decades (Geiger–Nuttall, \(R^2=0.9955\)).
★ And one prediction failed ── discrete scale invariance (the Efimov ratio 22.69) leaves a 2.95 dB log-periodic ripple (matching \(2/\cosh(\pi^2/\ln\lambda)\) to 3 digits). What was crushed in Episode 13 was the imaginary-order impulse response ── a different thing. Turned around this is testable: 1/f flat to 0.1 dB means a ladder ratio of at most 4.5.
Why is nature so full of exponentials? ── waves that cannot travel, and independent events
Answering the question left last time. ★ The exponentials of physics come from only two places ── ★ ① waves that cannot travel (the only solutions of (d^2psi/dx^2=+kappa^2psi) are (e^{pmkappa x}); a WR-90 waveguide below cutoff attenuates 1061 dB per metre at 3 GHz), and ★ ② the function turning addition into multiplication (the only continuous solutions of (f(x+y)=f(x)f(y)) are exponentials ── values built from the rule alone agree to 15 digits even at irrational points).
★★ And Episode 27’s empirical law was reproduced from nuclear physics. Computing the Gamow factor and comparing with (log_{10}T_{1/2}) gives slope 0.996 (predicted 1), (R^2=0.9934). The striking rate from the intercept is (3.3 imes10^{20})–(1.7 imes10^{21}) /s for 8 of 9 nuclides ── within 3× of the round-trip frequency inside the nucleus. ★ (^{235})U is off by 200× as a hindered transition, marking where the exponent’s explanation ends.
★★ The synthesis: spread the rates, stack the exponentials, and you get a power law ((int e^{-lambda t}lambda^{a-1}dlambda=Gamma(a)t^{-a}), confirmed to 14 digits). Log-uniform rates give logarithmic relaxation. ── ★ Scale invariance was built out of things that each have a scale.
Why is the wave equation second order? ── second order meant "both directions"
First order cannot oscillate (diffusion gives \(\omega=-iDk^2\), purely imaginary). ★ Second order turned out to mean "both directions" ── it factorizes as \(\Box=(\partial_t-c\partial_x)(\partial_t+c\partial_x)\), and numerically a bump splits into two halves of height 0.5 travelling to \(\pm ct\). To stay first order and still go both ways you must take a square root, and \(2\times2\) admits only three anticommuting matrices, so \(4\times4\) is needed ── ★ and that price is spin.
Third order (radiation reaction) runs away with \(\tau_0=6.27\times10^{-24}\) s, and on the ladder that is one line ── \(W\propto\sin(\alpha\pi/2)\) turns negative at \(\alpha=3\). Episode 12’s formula and the Abraham–Lorentz runaway were the same thing.
★★ And the width-2 band is set by passivity and Ostrogradsky taking alternate turns ── \(\cos(\beta\pi/2)\ge0\) allows bands every 4, so \(\alpha=4\) survives passivity. Odd orders fall to passivity, even orders to the ghost.
★ This episode corrects our own Episode 15 in two places ── ① its two edges were measured in different units (\(\beta\) and \(\alpha\)); ② the lower edge is set by passivity, not causality, and \(-1/2\) was never an edge. The correct band is \(\beta\in[-1,+1]\), width exactly 2.
Move the dimension and the order moves by halves ── only 3 dimensions gives (alpha=0)
We change the single letter \(d\) in the wave equation. ★ The order of the tail is \(\alpha=(d-3)/2\) ── measuring the front exponent from the exact solutions gives \(\lambda=0,-1/2,-3/2\) for \(d=1,2,4\), exactly as predicted. Only \(d=3\) is a bare \(\delta\) (Huygens’ principle).
★★ The half order turned out to be Hadamard’s descent. \(G_2=\int G_3\,dz\) matches the exact solution to \(10^{-7}\), and the delays from points along \(z\) follow the power law \(s^{-1/2}\). ★★★ That is an independent test of Episode 26’s hypothesis ── not electrochemistry, not viscoelasticity, but the geometry of the vacuum produced the same half order.
★ The origin of the tail follows too: with \(v=r^{(d-1)/2}u\) the equation becomes 1D plus \(l(l+1)/r^2\) with \(l=\alpha\) ── the order as the strength of a potential. It vanishes at \(d=1,3\) (numerically \(10^{-13}\)), and ★★ \(d=2\) lands exactly on Episode 13’s critical \(\lambda=1/4\) (no dimension ever exceeds it).
★ And \(\alpha=0\) happens only at \(d=3\) ── the one dimension where a wave arrives undistorted and then is gone. The match with the passivity band (width 2) was checked using Episode 29’s habit, and ★ rejected.
Building the order digitally ── integer orders are FIR, fractional orders are IIR
From the \(s\) plane to the \(z\) plane. Expanding \(D^\alpha=(1-z^{-1})^\alpha\) gives the Grünwald–Letnikov fractional derivative, whose coefficients are the filter taps. ★ Only for non-negative integer \(\alpha\) do the taps terminate (\([1,-1]\) at \(\alpha=1\)); otherwise they run on as \(w_k\approx k^{-\alpha-1}/\Gamma(-\alpha)\), a power law (6 digits out to \(k=10^5\)). The same shape as Episode 26’s \(g(\tau)\propto\tau^{\alpha-1}\).
★ Check: convolving the \(D^{1/2}\) taps with themselves gives exactly \([1,-1]\). ★ The order also picks the discretization ── the bilinear transform holds the phase at exactly \(90\alpha^\circ\) at every frequency (the backward difference is off by 1° past 1 % of Nyquist even at first order). Episode 1’s "the order is the phase" became a design criterion.
★★★ And last episode’s "dimension" was exactly the same count ── odd dimensions ＝ integer order ＝ FIR (\((d-1)/2\) taps); even ＝ half-integer ＝ IIR. \(d=1\) is an all-ones accumulator, \(d=3\) a single-tap pass-through, \(d=5\) a two-tap difference ── matching Episode 30 row by row. ★ "Huygens’ principle holds" ＝ "that filter is FIR".
The Higgs as waves ── the Higgs boson is AM, the Goldstone is FM
★ A negative mass term is an imaginary cutoff frequency ── with \(\omega^2=c^2k^2-\mu^2\) every mode below \(k=\mu\) grows, time constant \(\hbar/\mu=7.4\times10^{-27}\) s. The ★ time-domain version of Episode 28’s "a wave that cannot travel is an exponential" ── the tachyonic instability is an evanescent wave in time.
The nonlinear turning point is \(\sqrt2\,v=348.2072\) GeV (agreeing to 1.2e−10). ★★ But a prediction failed ── the field does not settle into a vacuum; it swings forever between \(\pm\sqrt2 v\). ★ "Symmetry breaking" requires dissipation (at \(\gamma\ge10\) GeV it converges on \(v\)) ── which is what cosmology’s Hubble friction provides.
★★★ What remains are two directions ── radial ＝ amplitude modulation (AM) ＝ the Higgs boson at 125 GeV (matching \(\sqrt2\mu\) to 1e−4) and angular ＝ phase modulation (FM) ＝ the massless Goldstone. By FFT, AM has two sidebands while FM follows \(J_n(\beta)\) without end (6-digit agreement). The carrier (the gauge field) absorbs the FM sidebands as its longitudinal wave ── 2→3 polarizations, the count unchanged.
★ Yet nothing explains why \(\mu^2<0\), nor the 5-decade Yukawa hierarchy, nor \(v=246\) GeV (Episode 15’s wall).
Rereading the Koide formula as a wave ── the AC and DC parts have equal RMS
Take the three \(\sqrt m\) as a sinusoid sampled at 120° intervals, \(\sqrt{m_k}=\mu(1+A\cos(\delta+2\pi k/3))\), and the three-phase identity \(\sum\cos^2=3/2\) (the same one that makes three-phase power constant) yields \(K=(1+A^2/2)/3\) by hand. ★★ So \(K=2/3\) simply means \(A=\sqrt2\) (measured 1.41420051).
★★★ And a sinusoid of amplitude \(A\) has RMS \(A/\sqrt2\), so ── Koide ＝ "the AC and DC parts of the \(\sqrt{\text{mass}}\) wave have equal RMS". Measured ratio 0.99999077 (−8e−5 dB). That shows what needs explaining far better than 2/3 does.
Geometrically \((\sqrt m)\) makes 44.999735° with the diagonal ── but we checked and rejected any link to Episode 1’s "half-order 45° phase" (the habit of Episodes 29–30). It predicts \(m_\tau=1776.969\) MeV against 1776.86±0.12 (0.91 σ) ── ★ falsifiable.
★ The limits are equally clear ── run to \(M_Z\) and it sits 0.19 % off, a 200-fold worse discrepancy (a pole-mass relation only). It fails for quarks by 10–27 %. In a random control, \(10^{-5}\) agreement happens about once in 65,000. All this episode added is a restatement, not an explanation.
Yang–Mills as waves ── a wave that scatters off itself cannot be a free particle
One line separates it from Maxwell ── ★ the source contains (A) itself; the wave is its own source. In the homogeneous SU(2) model, solving (A) only (x), (B) only (y) and (C) both gives a difference between (C) and (A)+(B) of 4.5 by (t=20) (zero always for electromagnetic waves). Energy moves between colours while the total is conserved to 9 digits.
★ It is already chaotic classically ── Lyapunov exponent (lambda=0.179) against (2 imes10^{-9}) for the harmonic control. One step past Episode 23’s broken superposition: ★ predictability itself breaks. The coupling depends on wavelength because the vacuum is dispersive ── and it anti-screens (the wave stiffens the medium).
★★★ And the decisive point ── the gluon’s response function goes negative. The Gribov (C(t)) flips sign at (t_0=0.3131) fm (matching the analytic prediction) while the Yukawa form stays positive forever. A positive response function is the condition for a wave to exist as a free particle, so ── ★ "a wave that cannot be a free particle" is confinement in the language of waves.
★ As a bonus, massless yet cut off (glueball 1.73 GeV) ── three ways to make a cutoff: a boundary, a background field, or ★ yourself. Four items of the linear vocabulary break: superposition, transfer function, predictability, positivity.
Digging further into light ── when phase becomes a function of angle, and when it becomes an area
For 34 episodes phase was always a function of time. ★ Here we add two axes. ① Make phase a function of azimuthal angle (orbital angular momentum) ── the centre of \(u\propto r^{|\ell|}e^{-r^2/w^2}e^{i\ell\phi}\) has exactly zero intensity and the peak sits at \(w\sqrt{|\ell|/2}\) (5-digit agreement). And \(\ell\) can only be an integer (return after one turn ── the angular version of Episode 5).
★ That integer survives being moved ── in \(u=(x+iy)^2-a^2\) the winding on a big circle is always exactly 2 however the vortices move, and small circles give only 1 or 0. A topological conserved quantity, a different kind of discreteness from Episode 13’s.
★★★ ② Take a polarization state once around the sphere and a phase remains, equal to half the solid angle ── \(\gamma=-\Omega/2\) agrees to \(10^{-16}\), and an \(N\)-gon path converges to \(-\pi(1-\cos\theta)\). The same at any speed ── a phase with nothing to do with time.
★ The structure showed too ── angle and \(\ell\) are Fourier conjugates, and because angle is periodic, \(\ell\) is discrete (Episode 31’s duality). The angular bandwidth theorem reads \(\ell\le2\pi r/\lambda\) ＝ "how many wavelengths fit around the circumference" (40,000 for a 1 cm aperture). Spin has two states, orbit infinitely many ── amplitude-side versus phase-side freedom.
Squeezed light ── pushing AM noise into FM
\(\Delta X_1\cdot\Delta X_2\ge1/4\) constrains a product, so ★ one can be reduced by pushing it into the other. In sideband language the perturbation collapses to \(P(t)=2ae^{i\theta}\cos(\Omega t+\delta)\), and ★ only the mean sideband phase \(\theta\) decides AM versus FM (the difference \(\delta\) is just timing). Squeezed light ＝ correlating the sideband pair so AM-side noise moves to the FM side (Episode 32’s AM/FM as a design principle).
★★★ The core is \(\det M=1\) ── the Floquet eigenvalues give \(|\mu_+||\mu_-|=1.000000\) and the growth rate matches \(h\omega_0/4\) to four digits. Because phase-space area is conserved, shrinking one quadrature to \(e^{-r}\) must stretch the other to \(e^{+r}\) ── ★ uncertainty is unbreakable because the dynamics preserves area.
★ It also pays off Episode 34 ── parametric amplification failed there because the shaking was not at exactly twice the frequency (the same resonance condition as a child on a swing).
The payoff is large: 6 dB doubles the range and multiplies the observable volume by 8, without changing a mirror or a laser. ★ But the limit is loss ── 5 % caps you at 13 dB, 10 % at 10 dB. Attenuation mixes in vacuum fluctuations, and amplification cannot recover them.
Anderson localization ── whether a wave stops depends on whether it comes back
Disorder a medium and ★ the wave stops with no absorption at all. In one dimension any disorder localizes ── by transfer matrix, \(\xi W^2\) is roughly constant and the slope of \(\ln\xi\) against \(\ln W\) is −1.97 (prediction −2). At \(W=0.001\), \(\xi\approx8.6\times10^7\) sites ── experiments miss it because samples are short (Episode 5’s structure).
★ The same exponential decay has a different cause ── an evanescent wave’s \(k^2<0\) exponent is deterministic and does not fluctuate, while localization is statistical and varies by 2.5 % between samples.
★★★ And the reason for the dimensional dependence came out ── Pólya’s recurrence theorem. A random walk in \(d\le2\) always returns; in 3D the return probability is 0.340537 (from Watson’s integral). A wave that never returns cannot interfere with itself, so \(d\le2\) always localizes while 3D needs strong disorder. The weak-localization correction says the same: \(\sqrt{\tau_\phi}\), \(\ln\tau_\phi\), convergent.
★ Episode 30 also singled out \(d=2\), but on checking it is a different origin (geometry versus interference) ── with a common root: the dimension where the power becomes −1 and the integral turns logarithmic. ★ And no quantum mechanics is needed (it happens with light, sound and microwaves).
Quantum gravity as waves ── dimension running is order running
The Planck length comes out in one line ── where a wavelength equals its own Schwarzschild radius (\(1.616\times10^{-35}\) m). It is the limit of Episode 6’s “mass = cutoff frequency”: push the cutoff up and gravity makes a cutoff of its own.
★ Why gravity alone is non-renormalizable is one line too ── its coupling carries the dimension \([E]^{-2}\), so the effective coupling \((E/E_P)^2\) passes 1 at Planck. The opposite of Episode 34’s Yang–Mills; in wave language, “a medium in which shorter waves interact more strongly.”
★★★ And the core ── the widely shared expectation that \(d_s\) runs 4 → 2 at short distance, read through Episode 30’s \(\alpha=(d-3)/2\), is “order runs from \(+1/2\) to \(-1/2\)”. Checked numerically ── the spatial part of \(P(s)=\int d^3k\,e^{-s(k^2)^z}\) is exactly \(3/z\), and adding the time direction gives \(d_s=1+3/z\).
★ Why it stops at 2 falls out as well ── \([G]=[E]^{2-d}\), so the coupling is dimensionless at \(d=2\). The reverse of Episode 25’s dimensional transmutation (QCD makes a dimension out of a dimensionless number); both are demands of “only dimensionless quantities are physics.”
★ Translating shows what to measure ── a 10 GeV GRB photon gives \(\Delta t\approx0.3\) s. And the first-order effect is already excluded (Fermi and others). ★ This episode only put an existing expectation into other words; it produced no new prediction.
Holography as waves ── the extra dimension is the log-scale axis
The AdS relation \(\Delta=\frac d2\pm\sqrt{\frac{d^2}4+m^2L^2}\) turned out to be the same quadratic as Episode 13’s \(s=\frac12\pm\sqrt{\frac14-\lambda}\) (\(d\to1,\ m^2L^2\to-\lambda\)). The BF bound is Episode 13’s critical \(\lambda=1/4\).
★ Solving the wave equation numerically and measuring the boundary two-point function, the slope of \(\ln|B/A|\) against \(\ln k\) is 1.000000 (prediction \(2\nu=1\)) ── five digits against the exact \(B/A=-k\).
★★★ And the core ── the bulk mass sets the boundary’s order: \(\alpha=2\sqrt{d^2/4+m^2L^2}\). The BF bound is where it hits zero (a logarithm); below it, imaginary. Indeed at \(m^2L^2=-4.25\) the zeros are equally spaced in \(\ln z\) (6.283185; ratio 535.49) ── the same log-periodicity as Episode 13’s Efimov states.
★★ Further ── the wavelet plane’s metric \((db^2+da^2)/a^2\) is literally the AdS metric. Radial \(z\) = scale \(a\) = \(1/\omega\). Episode 38’s “17 decades” becomes a proper distance of 39.14 \(L\) ── the hierarchy problem turns into a length.
★ Entanglement is a geodesic length \(2\ln(\ell/\varepsilon)\) (matching to \(10^{-11}\)) = a logarithm = order zero, and the RG flow is just translation in the radial direction. ★ But the duality itself was assumed, and no new prediction came out.
Time crystals as waves ── not the half, but that it comes back
Feed \(f_0\) into a linear time-invariant system and \(f_0/2\) comes out at −305 dB (the rounding floor) ── as Episode 1 said, no new frequency can be created. But wobble the coefficient and it appears: the Mathieu growth rate matches \(h\omega_0/4\) to five digits, \(\det M=1.000000000\), and where \(\mathrm{tr}\,M\) drops below \(-2\), \(\mu\) passes through \(-1\) (continuing Episode 36).
★ With dissipation a threshold \(h_{\rm th}=2\gamma/\omega_0\) appears and the phase falls into one of two states \(\pi\) apart (measured 3.141593 rad) ── “there are two” is the broken symmetry; Episode 32 in the time direction.
★★★ And the core ── producing a half takes only the Mathieu equation. A time crystal is a “phase” because you can mis-set the kick by 15 % and the response still sits at exactly 0.500000. Switch the interactions off and it vanishes, the peak sliding obediently to \((1-\varepsilon)/2\) ── rigidity is a many-body property, built from feedback that cancels the error over two periods (Episode 18).
★ The width is not one threshold: the frequency holds on and the amplitude dies first (at \(\varepsilon=0.18\), 0.846 → 0.021). ★ Impossible in equilibrium ── a passive system has \(\mathrm{Re}\,Z\ge0\) (Episodes 17, 18) and needs both drive and dissipation. ★ Classical only, 4096 periods only.
Topological waves ── surviving disorder is not about being strong
Episode 37 said any disorder, however weak, stops a wave in one dimension. Yet some waves do not stop ── not from strength, but because there is nowhere to backscatter into.
★ First, a quantity that can only be an integer ── the SSH winding number jumps from 0 to 1 when \(w\) moves 0.999→1.001, a change of 0.2 %, and never takes a value in between. It can only jump where the gap closes, \(v=w\). That integer predicts how many edge states exist (bulk–boundary correspondence), and their length matched \(1/\ln(w/v)\) with ratio 1.00000.
★★ The caveat that matters ── 60 % disorder on the bonds leaves \(E=0\) untouched, while 0.05 on the sites moves it. What protects is a symmetry, not robustness against anything whatsoever.
★★★ And the core ── in 2D with Chern number \(C=1\), a single branch crosses the gap. Counting the sign of its group velocity: all 42 points negative on the left edge, all positive on the right. On one edge a counter-propagating state does not exist ── so there is nothing to bounce into.
★ No contradiction with Episode 37, whose premise was “you can go both forward and back”. ★ Single-particle, minimal models only.
Intermittency in turbulence ── the order was a distribution
Kolmogorov’s \(-5/3\), in this series’ language, makes turbulence a “5/6-order” wave ── 5 dB/oct, phase 75°, Hurst \(H=1/3\). Exactly between 1/f (order 1/2) and Brownian motion (order 1).
★ For synthetic noise \(\zeta_p/p\) is constant in \(p\) ── that is what “a single order” means. But a real cascade multiplies: the p-model’s \(\tau(q)\) matches the exact result to \(10^{-14}\) while being non-linear in \(q\) (2.010 at \(q=4\), against 3.000 for a monofractal). At \(p=0.5\) it straightens out.
★★★ And the core ── after a Legendre transform \(\alpha\) runs from 0.5146 to 1.7370 (width \(\log_2\frac{1-p}{p}=1.222\)). The order is not one number but a set with width.
★★ One dial (\(p=0.7\)) reproduces the literature \(\zeta_1\)–\(\zeta_6\) to 1–2 % (0.361/0.37 … 1.786/1.78). Yet \(\zeta_3=1\) stays exactly 1 for every \(p\) (the 4/5 law).
★★★ The crucial part ── even at \(p=0.7\) the difference from K41 is only 0.081 dB/oct. “\(-5/3\) fits well” was really the second moment being insensitive. ★ A fifth entry joins Episode 14’s list: the order is not a single number.
The second stocktaking ── the axis held, but it is no longer single
A scorecard for the 27 episodes from 15 to 42. Every headline number was recomputed independently and checked against what the articles claim ── 25 items, zero FAILs. Only Episode 37’s Watson integral stays 0.4 % out, because the grid cannot resolve the singularity at the origin.
★ Episode 1’s \(\mathrm{dB/oct}=6\alpha\) and phase \(=90\alpha\) were still the axis 42 episodes later. Aliases for the same number kept arriving ── 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.
★★ Ten predictions failed, and all are still in the text. The most productive was Episode 32 ── “the field settles in the trough” was wrong, and out of it came “symmetry breaking requires dissipation,” which paid off directly in Episode 40. ★ One self-correction (Episode 15 conflated two senses of “order”; Episode 29 separated them ── odd orders killed by passivity, even 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 with a quantity that does not fit (integers), Episode 42 with a place where it is not single (a distribution).
Metamaterials ── a gap in frequency, or a gap in momentum
The sign of a negative index is fixed by loss ── only \(\varepsilon<0\) and \(\mu<0\) gives \(\mathrm{Re}\,n<0\); with one alone \(n\) is almost purely imaginary (Episode 28’s evanescent wave). In a lossless ideal medium the two signs cannot be told apart.
★ It cannot be built without dispersion ── \(d(\omega\varepsilon)/d\omega=1+\omega_p^2/\omega^2\) is always positive, which is what permits \(\varepsilon<0\). Hence negative index is always narrowband. And the sum rule \(\int\omega\,\mathrm{Im}\,\varepsilon\,d\omega=(\pi/2)\omega_p^2\) (1.57077 against 1.57080) ── a designer controls the allocation, never the total.
★ In a 1D photonic crystal an index ratio of 2 gives a relative gap width of 0.376, and at the band edge \(v_g/c\) falls 0.628 → 0.017 (slow light). But stretching the delay 14× moved the delay–bandwidth product by only a factor of two, and not monotonically (stated honestly).
★★★ And the core ── periodic in space gives a frequency gap (decay inside); periodic in time gives a momentum gap (amplification inside). The equation is Episode 40’s Mathieu equation exactly: centred at \(k=\Omega/2c\), growth rate 0.024993 (prediction \(hck/4=0.025\)), \(\det M=1.000000\). Cannot travel in space and you shrink; cannot travel in time and, since time cannot be refused, you grow.
★ Three gatekeepers decide what may be built: passivity, Ostrogradsky, and the sum rule.
Shaking the vacuum ── heat is time measured logarithmically
The sum of zero-point energies diverges, but the divergent term \(1/\delta^2\) is proportional to volume, so taking the difference between having walls and not removes it ── leaving \(-1/12\) (numerically \(-0.0833333\)). Not “subtracting an infinity” but “taking a difference.” In 3D, \(P=-\pi^2\hbar c/(240a^4)\): 1.28 atmospheres at 10 nm.
★ Shake a boundary and photons come out of nothing (\(n=0\) at \(h=0\); \(n=3089\) after 60 periods). Yet \(|\alpha|^2-|\beta|^2=\det M=1\) never moves ── the particle-number version of Episode 36, which is why the photons necessarily come in pairs.
★★★ And the core ── an accelerating observer’s time goes logarithmic, \(at=e^{a\eta}\) (Episode 20), an exponential enters the phase (Episode 28), and the Fourier transform gives a gamma function → \(\sinh\) → \(1/(e^x-1)\). “Heat” turned out to be a way of measuring time. \(|\Gamma(iy)|^2=\pi/(y\sinh\pi y)\) confirmed numerically (ratio 1.000000). The Unruh temperature needs \(2.5\times10^{20}\) m/s² for 1 K.
★ Nothing comes free (passivity, \(\det M=1\), the sum rule). ★★ The caveat that matters most: the subtraction worked because it was a difference, and gravity responds to the absolute value ── the cosmological constant problem is beyond these tools.
Black hole evaporation ── information went out of range in 1.5 milliseconds
The Hawking temperature is Episode 45’s Unruh temperature with \(a\) replaced by the surface gravity \(\kappa=c^4/4GM\). The mass in equilibrium with \(T=2.725\) K is \(4.50\times10^{22}\) kg (0.61 lunar masses) ── anything heavier is still growing by absorbing the background.
★ The entropy is area ÷ Planck area (\(1.05\times10^{77}\) for a solar mass, by two agreeing calculations) ── Episode 39’s holography returns, now with a concrete \(\varepsilon\): the horizon. It is 19 orders above a star of the same mass.
★ The evaporation time goes as \(M^3\); exactly the age of the universe corresponds to \(1.73\times10^{11}\) kg. Negative heat capacity accelerates the ending: \(2.05\times10^{22}\) J in the last second. The Page curve peaks at \(M=M_0/\sqrt2\), 64.6 % of the lifetime (a model assuming unitarity).
★★★ And the core ── near the horizon \(\omega(t)=\omega_0e^{-\kappa t/c}\). Visible light needs 74.5 e-folds to fall below the inverse age of the universe, which for a solar mass is a mere 1.5 milliseconds. It is information being swept into exactly Episode 5’s “a period longer than the age of the universe is indistinguishable from a constant.”
★ “No hair” says the same thing (only three components survive in the outside band). ★ Honestly: the information paradox is not solved.
Inflation ── the universe’s tilt is −0.105 dB/oct
The Mukhanov–Sasaki equation turned out to be Episode 13’s \(1/\tau^2\) potential itself ── put \(x=-k\tau\) and it becomes \(v_{xx}+[1-(\nu^2-\frac14)/x^2]v=0\), ★ with \(k\) gone. That is what scale invariance is. Outside the horizon, \(v\sim x^{1/2\pm\nu}\).
★★★ Episode 13 (the critical potential), Episode 39 (AdS) and this one (inflation) are three faces of the same quadratic. Numerically \(P(k)\propto k^{3-2\nu}\) was confirmed to \(10^{-6}\).
★ Once \(x<1\), \(|\zeta|\) goes constant and freezes (1.41 → 1.005 → 1.00005) ── the same place Episode 4 put the constant of integration (\(\omega=0\)), which is why it survives to 380,000 years. Among models, \(V\propto\phi^2\) passes on \(n_s\) but is excluded by \(r=0.133\); only Starobinsky passes both (the lesson of Episode 42).
★★★ And the core ── put \(n_s=0.9649\) on Episode 1’s scale and it reads \(3(n_s-1)=-0.1053\pm0.0126\) dB/oct. The same order as Episode 42’s \(+0.0812\) for turbulence (★ a coincidence, with no shared origin). In both, the departure itself carries the physics ── an \(n_s\) of exactly 1 would mean inflation never ended.
★ Inflation is Episodes 44–45’s parametric drive, and the CMB fluctuations are the dynamical-Casimir photons of an expanding universe. The only difference: nobody is doing the shaking.
The edge of the order axis ── there is exactly one condition for using it
We went looking for the third test promised in Episode 43’s exercises. Measuring the local order \(\alpha(\omega)=d\ln|H|/d\ln\omega\) ──
★ The Lorentzian moves continuously from 0 to \(-2\), returning to a power law only at its ends (5.36 decades each). ★★ The exponential gives \(\alpha=-\omega\) with no upper bound ── it does not fail to sit on the axis; it runs off the scale. At any target order it earns only 0.087 decades.
★★★ And what “marginal” really is ── not an order of 0, but an order approaching 0 logarithmically. \(g\propto1/\ln\mu\) is still \(-0.0130\) after 32 decades. \(\phi^4\) in four dimensions, QCD, weak localization in 2D (Episode 37) all live here ── which is why logarithms appear.
★ In a finite system not even an exponent survives (1D Ising at \(L=16\): the decay rate goes negative past \(r=8\)) ── the spatial version of Episode 5. And in three dimensions the first-order ε-expansion is off by 7.4 % in \(\nu\) and 49.0 % in \(\eta\).
★★★ Conclusion ── none of this means the axis is wrong; it makes its condition of use explicit: scale invariance holding over some number of decades. ★ A replacement for the axis was not found.
A handbook ── what to do when you are handed a measurement
If Episodes 15 and 43 were scorecards of what we got right, this is the manual for how to use it: 48 episodes of procedure, written out as a program.
★ The phase can be predicted from the amplitude slope alone (the Bode gain–phase relation, matching \(90\alpha\) to within 0.01°) ── this is where Episode 1’s “phase \(=90\alpha\) degrees” came from. So a departure means an extra delay, and dividing that gap by \(\omega\) reads off the delay time.
★★★ And the centrepiece ── A (a broad superposition) and F (a narrow distribution) both look like 1/f, but order \(-0.5\) holds for 6.898 decades in A and 0.498 in F. F merely passes through \(-0.5\). “The slope was \(-0.5\) at one point” is no evidence at all ── every smooth function passes through \(-0.5\) somewhere. What guaranteed Episode 2’s “many time constants” was the decade count.
★ Seven steps: ① log–log ② local order ③ count the decades ④ bends ⑤ phase ⑥ moments ⑦ system size. ★ Plus six things not to do, drawn from past mistakes.
The map ── fifty episodes on, where did we arrive?
Episode 1’s question was one line ── “Differentiate a sound and it goes high, integrate it and it goes low. So what does the space in between — a fractional order — mean?”
★★★ Fifty episodes later ── the order \(\alpha\) turned out to be the dB/oct, the phase, the dimension, the Hurst exponent, the AdS mass and the structure-function exponent. One and the same number. And there is exactly one condition for using it: scale invariance holding over some number of decades (Episode 48).
★ It leaves the integers for five reasons (superposition, dimension, criticality, bulk mass, distribution), and the axis ends in three ways (integers, distributions, no power law). All 28 numbers were recomputed and checked (zero FAILs) ── though that only guarantees internal consistency; what guards against contradicting nature is the list of failed predictions.
★ Counting the series itself: 49 episodes, 49 body files in each language, 48 verification scripts, 13,966 lines, zero dependencies. FFT, Jacobi and the complex gamma function were each written where needed ── reproducibility is very nearly the only guarantee here.
★★★ And honestly ── no formula unknown to humanity came out of it. What came out is “another reading of the same formula” and “what to measure.” ★★ The most productive episode was the one that was most wrong (Episode 32). ★ Ten untouched areas are listed, honestly.
Error bars on the decade count ── one ordinate always scatters by 5.57 dB
The first line of Episode 50’s “what to take up next.” One periodogram ordinate scatters by 5.57 dB (\(\pi/\sqrt6=1.2825\) nats) ── independent of signal strength, frequency and instrument. The slope’s error is \(1.2825/\sqrt M\), so 0.081 dB/oct at \(3\sigma\) needs 40,000 points. On statistics alone, within reach.
★★★ But ── centre the window in the inertial range and the two end curvatures pull opposite ways and cancel (to \(10^{-14}\); we had not noticed before writing this). Half a decade off-centre brings back 0.032 dB/oct, and an asymmetric spectrum over \(R=3\) decades gives 0.09–0.14. Combining statistics and systematics gives an optimal window width where the two are about equal (best 0.18 dB/oct at \(R=3\); 0.059 at \(R=6\)) ── “choose a wide inertial range” beats “measure longer.”
★★★ And the core ── Episode 49’s handbook cannot be used on real data as written. Even averaged down to 0.056 dB per point, a 6-decade power law reads as only 2.6. The cause: building the order from differences between neighbours. The fix is least squares over a window of width \(w\), recovering 5.0 decades at \(w=1\) (with the reportable span capped at \(D-w\)).
★ Episode 49’s caveat was too soft ── not “acquires error” but “cannot be used.”
Extracting the order’s distribution from finite data ── only the tails are out of reach
Episode 42 said the order is not one number but a distribution. ★ But can that distribution’s shape actually be measured?
★ Estimating \(\tau(q)\) from a single realization, small \(|q|\) agrees while large \(q\) departs (with an inconsistent sign) ── because large \(q\) is decided by a single strongest box. Only \(q=0\) is exact (it just counts boxes).
★★★ And the core ── a thousandfold increase in boxes (1,024 → 1M) moves \(q^*\) only from 2 to 4. The \(\alpha\) width goes 54.5 % → 89.4 % and stops. \(q^*\) grows logarithmically, so reaching the tails needs exponentially more data.
★★ The peak of \(f(\alpha)\) (the support dimension) is always measurable; only the two tails are not ── intermittency is about rare, violent events, and being rare is exactly why they are missing from the sample.
★ One prediction failed: we expected negative \(q\) to be the most noise-fragile, but the positive side moved more (log-normal tails).
★★★ Conclusion ── “it is a distribution” is sayable; “which distribution” is decided by the size of your sample.
Why the measurable range is always logarithmic
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 \(q^*\). Too many for coincidence.
★ The only measure invariant under multiplication is \(dx/x\) (still \(\ln3\) after a hundredfold stretch) ── so the natural coordinate is \(\ln x\). With constant relative resolution, the count of distinguishable steps is proportional to the decades (constant Q, Episode 22).
★★★ And 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’s “addition becomes multiplication,” read backwards.
★ By the way ── the textbook criterion for Episode 52’s \(q^*\) (\(f(\alpha)\ge0\)) allows \(q\approx25.9\) while the measurement gives 4: more than six times too optimistic (“one on average” is not “one every time”).
★★ 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 ── the wall of Episodes 51 and 52.
Can range be bought? ── not with time, but with instruments
Episode 53 concluded that range goes as \(\log N\). ★ So how does one get more? There are three ways ── splice bands, use a different observable, change the system itself.
★ The price of splicing: slope error \(\approx0.3\delta[\mathrm{dB}]/\)(separation in decades). 1 dB across a 3-decade gap costs 0.1 dB/oct ── the same size as Episode 42’s 0.081. Buying 3 decades needs calibration better than 0.8 dB. ★ Read the other way, distant bands are easier to calibrate (at the cost of a heavier extrapolation assumption).
★★ Another observable: if \(S_p=\rho^{\zeta_p}\), putting \(\ln S_3\) on the horizontal axis cancels the correction exactly ── 4 decades → 9 (all of it). That is ESS, but the cancellation comes from an assumption, and the assumption is not proved.
★ Changing the system: inertial decades \(=\frac34\log_{10}Re\). 3 in the lab (\(10^4\)), 9 in the atmosphere (\(10^{12}\)); one more decade needs \(Re\times21.5\).
★★★ The core ── all three buy range differently: splicing linearly in the number of instruments, another observable all at once for an assumption, changing the system as the logarithm of \(Re\). Episode 53’s \(\log N\) was only about measuring the same system with the same instrument for longer.
The revised handbook and an audit of conditional numbers ── 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, ⑩ count the assumptions). Sorting 16 of this series’ numbers ── 6 computed here, 3 from literature, 5 conditional on a model, 2 translations.
★★★ And the core ── Episode 42’s 0.0812 moves between 0.0446 and 0.1311 as \(p\) swings over 0.65–0.75, and Episode 47’s \(-0.1053\) carries \(\pm0.0126\) from the observation. Both are good to two significant figures. Writing “0.0812” to four figures was too many ── the right phrasing is “of order 0.1 dB/oct.” ★ The conclusions of both episodes survive at that precision.
★★ The logarithm’s good side appeared too ── 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.
★★ A worked case: Koide’s \(K\) ── the difference from \(2/3\) is \(-6.16\times10^{-6}\) against \(\pm6.78\times10^{-6}\) from \(m_\tau\), giving difference ÷ uncertainty = 0.909. Indistinguishable at present ── and a measurable question that tau-mass measurements will settle.
Back to \(1/f\) ── “many time constants” came with a price tag
Carrying Part XVIII’s tools back to the longest stretch of decades in this series ── the \(1/f\) of Episode 2. First, \(\beta=2-\gamma\) confirmed to \(10^{-6}\).
★★★ The core ── decades of \(1/f\) \(=0.9916\times\)(decades of \(\tau\))\(-2.17\). You always lose 2.2 decades at the ends. “A superposition of many time constants” had a price ── seeing 6 decades of \(1/f\) requires \(\tau\) spanning at least 8.2.
★★ And “\(1/f\) has no low-frequency cutoff” is not a measurable claim ── the knee needs an observation longer than \(\tau_{\max}\), and ten years buys 11.5 decades. Same shape as Episode 5; in Episode 55’s words, “cannot measure,” not “does not exist.” ★ The substance of Episode 2 (a single time constant cannot do it) is intact.
★★ What would settle it: temperature. Since \(\tau=\tau_0e^{E/kT}\), 300 K → 200 K moves \(\tau\) by more than 8 decades. Episode 54’s “change the system itself,” working exponentially ── far faster than observing longer.
What happens at an edge ── every edge was a leading correction term
Episode 56’s “2.2 decades lost at the ends” is derived here, not measured. For \(\gamma=1\), \(S=[\arctan(\omega\tau_{\max})-\arctan(\omega\tau_{\min})]/\omega\) is exact and the loss is \(2\log_{10}[2/(\pi\varepsilon)]=2.2098\) ── matching Episode 56’s measured 2.249. That 2.2 was the tails of the two \(\arctan\)s.
★★★ From there, a general law ── decades lost \(=(1/n)\log_{10}(1/\varepsilon)+\)const. The Lorentzian, the discrete-string dispersion (Ep. 16), the band edge (Ep. 44) and window fitting (Ep. 51) are all \(n=2\). Edges are not types; they lie along a continuum in \(n\).
★★ And a prediction failed ── the slope is not 2 for every \(\gamma\) but \(2/[\gamma(2-\gamma)]\). That is minimised at \(\gamma=1\), exactly \(1/f\) ── inverting \(\tau\to1/\tau\) sends \(\gamma\to2-\gamma\), and \(\gamma=1\) is its fixed point. \(1/f\) is also the spectrum with the cheapest edges.
What is outside the window leaks in ── the smoothness of the cut sets the price
How fast a window’s sidelobes fall is set by how many derivatives vanish at its ends, and that is Episode 1’s order itself ── dB/oct \(=-6\alpha\). Rectangular \(\alpha=1\), Bartlett 2, Hann 3, \(\cos^p\) gives \(p+1\) (measured within 0.2 dB).
★★ Hamming’s first sidelobe is 11 dB below Hann’s, yet its sidelobes fall like a rectangular window’s, \(\alpha\approx1\) (0.08 is left at the edge) ── “low first sidelobe” and “falls far” are different things.
★★★ The measurable \(\beta\) has a ceiling, and it is exactly \(2\alpha\) (rectangular 2.0, cosine 4.0, Hann 6.0; measured 1.981 / 4.021 / 5.998). And it is not a cliff ── the estimate overshoots below it (+0.61 at \(\beta=5\) with Hann), a systematic error with a fixed sign. Episode 40 had to cut at an integer number of periods precisely to dodge the rectangular ceiling \(\beta=2\).
★★★ And the same rule governs Episode 45’s particle creation ── a switch smooth to order \(p\) gives \(|\beta|\propto T^{-(p+1)}\) (measured −1.06 / −1.98 / −3.03 / −3.96). “The vacuum depends on the observer” was the physics-side phrasing of “what is outside the window leaks in.”
Measuring the systematic error ── what helped most was not the window
Turning Episode 58’s “change the window and the value moves” around, we built a procedure that hunts for where it does not move. Sweeping \(\hat\beta(\alpha)\) over \(\alpha=1\ldots8\) shows clamping at the ceiling \(2\alpha\), an overshoot just below it, and then a plateau.
★★★ What turned up on the way ── that overshoot appears with zero noise (\(+0.625\) at \(\beta=5,\alpha=3\), matching Episode 58’s measured \(+0.61\)). Moving the integral’s lower limit settles it: the source is frequencies below the record’s resolution ── the part that looks like nothing but a trend inside the record.
★★★ So one straight line removes it (\(\beta=5\): \(5.512\to4.992\)). Episode 58 said “raise the window order,” but raising it costs variance while detrending is nearly free. The ceiling, though, does not move.
★★ The procedure’s coverage was measured too ── 79–87 % against a nominal 68 %. Yet the actual sd runs 1–3× the quoted error, because the error distribution is not Gaussian (a narrow core with rare large misses). So it belongs in coverage, not in an sd. ★ And the procedure itself is conditional ── on the band, on tol, on the window family.
Where a trend ends and a signal begins ── the boundary is set by the record
★ Removing a degree-\(q\) trend is a high-pass filter of order \(\alpha=q+1\) (measured 6.014 / 12.036 / 18.057 / 24.078 dB/oct) ── Episode 1’s yardstick for the third time, after Episode 57’s edges and Episode 58’s windows. The \(-3\)dB point is \(0.4(q{+}1)\) bins, i.e. about one cycle per record. The boundary belongs to the record, not the system ── Episode 5’s shape again.
★★ From it follows the leakage condition \(\beta\le2q+3\), which explains all five rows of Episode 59 §03. ★★★ And \(\beta=7,q=2\) has its leakage under control yet is still wrong ── it is hitting the window ceiling \(2\alpha=6\). “Subtract a trend” and “raise the window” were different conditions.
★★★ The “variance” of \(1/f^\beta\) is not a number ── \(N^{\beta-1}\) for \(\beta>1\) (measured 1.147 / 1.980), convergent for \(\beta<1\), and \(\log N\) at exactly \(\beta=1\) (0.146 predicted, 0.161 measured). The structure of Episode 29’s renormalisation ── though “the same structure” is not “renormalisation is available.”
How many times did \(\alpha\) appear ── and \(-1/12\) did not depend on the cutoff
Recomputing the nine numbers of Episodes 56–60 (all reproduce), we measured \(\text{dB/oct}=6\alpha\) along four independent routes at once ── differentiation, fractional differentiation (order 0.5 gives 3.010 against a predicted 3.010), a window’s edge, and subtracting a trend. Unrelated entrances, the same exit. ★ All four share one Fourier transform at the root, so this is tidying the tools, not new physics.
★★★ Episode 45’s footnoted open item is closed ── \(-1/12\) comes out the same for five cutoffs (\(e^{-x}\), \(e^{-x^2}\), \(e^{-x^4}\), \(\mathrm{sech}\,x\), \(1/(1+x^2)^2\); largest departure 6.7e−6), while the divergent coefficient \(A\) ranges over 0.443–1.832. ★ \(-1/12\) is not “the value of” \(1+2+3+\cdots\) but what remains once the divergent term is subtracted.
★★ That is where Episode 60’s running variance parts company ── the Casimir case leaves a cutoff-independent remainder; the \(1/f^\beta\) variance does not. Whether a running quantity can mean anything is decided right there.
★★★ And a self-correction: Episode 56’s intercept \(-2.1708\) had too many digits (±0.04 with the range of \(T\); “about \(-2.2\)” is right). One episode after Episode 55 issued that warning, we did it again. The handbook goes 7 → 10 → 14 steps.
The things that tunnel ── counting everything in decades \(=G/\ln 10\)
\(G=\frac{2}{\hbar}\int\sqrt{2m(V-E)}\,dr\) ── this one line computes fusion, cold fusion, Josephson junctions, STM and \(\alpha\) decay. Only the shape of \(V(r)\) changes. 18 checks: \(B_G\) matches the literature to four figures, \(\langle\sigma v\rangle_{dt}\) to within 1 %, ²³⁸U’s \(\alpha\) decay gives 17.16 against a measured 17.15, STM gives 0.94 decades per ångström.
★ The first episode in the series to carry four calculators the reader can drive.
★★ Cold fusion reduces to \(U_e\) alone ── in a model calibrated on one \(\mathrm{D_2}\) point that reproduces the muonic molecule 73 orders away, the \(U_e\) matching the experimental limit is 194 eV, while deuterated Pd measures 250 eV. ★★★ So the conclusion is not “cold fusion does not exist” but “extrapolating \(U_e\) to \(E\to0\) is broken”.
★★★ Three things the wave picture gave: (A) d+t’s advantage is the ⁵He resonance, not the barrier (its \(B_G\) is in fact 1.53× worse). (B) The catalysis threshold is a mass ratio of 85; the only charged particles above it are the muon and the pion, and the pion dies in 26 ns ── exactly one candidate. (C) A coherence mechanism falls 9 orders short even with every deuteron in Earth’s oceans in perfect superradiance ── refuted by counting, without inspecting any mechanism.
Waves and zeta ── line up resonances and you land in number theory
“Does treating nuclear physics as waves make it simpler?” ── It does. In \(\sigma(E)=\frac{S(E)}{E}e^{-B_G/\sqrt E}\), the right factor is mass and charge alone (Episode 62) and the left is one resonance to 10 % (0.5 % on the peak). But the resulting \(E_R=48.8\) keV is not the literature’s 64 keV (a constant-width BW centre, not an R-matrix pole) ── the same \(\sigma(E)\) with parameters 24 % apart, a textbook case of Episode 55’s “conditional on a model”.
★★★ And when resonances line up you get level statistics — the same statistics as the Riemann zeros (Montgomery–Dyson 1972). Here the zeros are computed from scratch (Riemann–Siegel, agreeing with \(N(T)\) to under one zero). Mean after unfolding 0.999823, \(\langle s^2\rangle=1.163\). \(\chi^2\): Poisson 3906, GOE 495, GUE 32.7 ── at \(s=0.07\) Poisson predicts 0.928 while the measurement is 0.025. ★ You can run it yourself in the browser.
★★ But “the same statistic” is not “the same physics” ── random matrices are fixed by symmetry, so unrelated systems agree. Hilbert–Pólya is not addressed.
★ As a bonus, \(-1/12\) is finally placed: it is \(\zeta(-1)\) of a wave spectrum (a third angle after Episodes 45 and 61). Weyl’s law agrees to 0.001 % at \(10^7\) ── the arrangement of waves and the zeta function are two ways of writing one thing.
What that formula is called ── the action of a classical path in imaginary time
Does Episode 62’s \(G=\frac{2}{\hbar}\int\sqrt{2m(V-E)}\,dr\) have a name? ── at least six, spanning 140 years (Liouville–Green 1837 / Jeffreys 1923 / WKB 1926 / Gamow factor 1928 / Kemble 1935 / instanton 1977). Mathematics wrote it down 89 years before quantum mechanics.
★★★ And the deepest name gives away what it is: the integral is the action of a classical path in imaginary time, \(S_E\) (with \(G=2S_E/\hbar\); the 2 is from squaring the amplitude). Rotate time by 90° and the potential inverts — the barrier becomes a valley. Running the particle through that inverted potential with RK4 and accumulating the action reproduces the WKB integral to six decimal places. “Classically forbidden” was a statement about real time.
★★ Against the exact Eckart solution, WKB’s error lives in the prefactor, not the exponent ── the ratio holds at 1.20 across eight decades of transmission. That is why Episode 62 could claim the decades. ★ At the barrier top \(e^{-G}\) breaks (784 at \(1.5V_0\)); Kemble’s \(1/(1+e^G)\) is the right form.
★★★ Arrhenius and Gamow are two limits of one formula, meeting at \(T_c=\hbar\omega_b/2\pi k_B\). The apparent activation energy falls to 3 % of the true barrier at low temperature ── a flattening Arrhenius plot is the experimental signature of tunnelling. That \(2\pi\) is the Unruh \(2\pi\) (the circumference of imaginary time).
★★★ And the deepest point: the perturbation series diverges as \(a_n\sim3^n\Gamma(n{+}\tfrac12)\) and can never produce \(e^{-1/(3g)}\). Yet the “3” in that divergence is exactly \(1/S\) for the bounce action \(S=1/(3g)\) (39.5376 against 39.5 at \(n=40\)). Perturbation theory cannot see tunnelling; the way it fails knows about it.
What it means to drop a dimension ── the shape can survive while the information does not
★ “Is a dimension a 6 dB filter?” ── close; a dimension is 3 dB. One integration is \(\alpha\)+1 = 6.02 dB/oct, one sliced dimension is \(\beta\)−1 = 3.01 dB/oct: exactly a factor of two apart. (Cutting 3D to 1D does flatten by 6 dB/oct — but that is two dimensions.)
★★★ “Observing loses information” ── yes, and there are two kinds. Slicing changes the shape (\(\beta_1=\beta_d-(d-1)\)). But projection leaves the shape untouched and still loses the information ── one view of an \(N^3\) object keeps \(1/N\), and filling \(k\)-space takes about \(\pi N/2\) views (201 at \(N=128\); the measured coverage hits ~100 % at exactly 201). “Same spectral shape” does not mean “nothing lost”.
★★★ “Non-integer dimension gives error” ── no, it saturates at 2 (Marstrand’s projection theorem). The 3D dimension matches theory exactly (1.4650–2.8928) while the projection never exceeds 2 ── what exceeds it does not appear as error, it disappears. ★★ A bonus: symmetry axes are the theorem’s exceptional directions (1.686 on axis vs 1.871 generic at \(m=16\)) ── the easiest direction to look is the most lossy.
★★★ “Could mass be a non-integer-dimensional wave?” ── that integral was already in Episode 2. A single mass is one pole; a non-integer dimension is a power-law continuum of masses. Episode 2’s superposition integral and the unparticle propagator are identical, giving \(d_U=2-\alpha\), with \(1/f\) (\(\alpha=1\)) landing exactly on \(d_U=1\) — one pole, an ordinary particle. In AdS/CFT it holds literally (the third appearance of Episode 13’s quadratic) ── though it is a translation, not an explanation of mass.
The overall 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 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 same "mass" succeeded completely one way and failed completely the other ── as a wave it is a cutoff frequency (giving Yukawa and \(E=mc^2\)); as an integral it is refuted.
Hits are dimensionless, misses are dimensionful.
And what Part IV onwards revealed is that "the order" was this series' real protagonist ── causality, uncertainty, the arrow of time, why heat does not travel, why the sky is blue, all of it. In Episode 13 that order became a plane; in Episode 14 it sorted into "three ways to leave the integers"; in Episode 15 it folded into "physics lives in a narrow rectangle".
Phase velocity and information transfer are treated in "Understanding Refraction", other faces of mass in "Understanding Mass", and coarse-graining and memory in "Understanding Renormalisation".
Thinking in Waves ／ All computations are JavaScript in kenshou/ (nami.js: measuring differentiation and integration by FFT／bunsuu.js: fractional derivatives and 1/f noise／teisu.js: time derivatives of the constants and the sensitivity spectrum／hashiru.js: the one-loop running coupling／shadan.js: dispersion relations and cutoffs／shitsuryo.js: Mach's principle and the holographic bound／inga.js: numerical verification of Kramers–Kronig／fukakutei.js: the bandwidth theorem and Mössbauer／yajirushi.js: time reversal and the order dependence of dissipation／netsu.js: thermal diffusion and the skin effect／housha.js: Larmor and Rayleigh scattering／kyosu.js: imaginary-order differentiation and Efimov states／hashigo.js: exact dimensional analysis and spectral exponents／hikari.js: effective light speed and the Shapiro delay／yukou.js: effective mass and the expansion of the dispersion relation／dokoka.js: confinement, Yukawa couplings and running mass／jikan.js: decay constants and the weak interaction／cttei.js: waves in the \(c\cdot t=\)const system／joho.js: the calculus of information and its various limits／hakaru.js: windows, wavelets and decoherence／hisenkei.js: non-linear mixing and the Duffing oscillator／juryoku.js: the quadrupole formula and the chirp／tsuyoi.js: Regge trajectories and dimensional transmutation). There are no dependencies; plain Node runs them as they are.
On distinctions: throughout we separate "measured here", "a summary of published values" and "our own hypothesis". In particular Episode 2's "1/f is the half-integral" and "fractional order = the result of coarse-graining", Episode 13's "why is \(\beta\) nearly always zero", Episode 19's "the optimal dating half-life is 4.78 Gyr", Episode 20's "\(c\cdot t=\)const ＝ logarithmic time" and Episode 21's "information lives on the imaginary axis" are all our own framings or conjectures, not claims from the literature. The measurements in Episode 4, the hadronic contribution in Episode 5, the Cassini bound in Episode 7, the material \(\alpha\) values in Episode 10 and Efimov's \(s_0\) in Episode 13 are published values.
The refuted hypotheses are left on the page ── mass as the integral of gravity, light speed as the integral of information, time variation of the constants. Not deleting the losing calculations is this series' policy. ── Each page can be turned into a PDF from your browser's Print → Save as PDF.