The Inner Rim Spectrum

A withdrawn prediction: why 0.776c is not an emission threshold, and what the few-hundred-keV band actually shows

WarningStatus: withdrawn

This chapter used to predict a gamma-ray shoulder at T_e \approx 300 keV, caused by electrons shedding modons as they reached the substrate’s inner rotation speed. We no longer make that prediction. We keep the chapter as the record of what was claimed, why it failed, and what would have to be true to reopen it.

The Rim in the Right Units

The framework’s inner rim is the circulation speed v_\text{rot,inner} = c\sqrt{2\alpha_{mf}} \approx 0.776\,c. That speed is where the electron’s mass comes from, and this chapter’s withdrawal does not touch that result.

An electron travelling at the same speed has kinetic energy

v = 0.776\,c \;\Rightarrow\; \gamma = \frac{1}{\sqrt{1-0.776^2}} \approx 1.585 \;\Rightarrow\; T_e = (\gamma-1)\,m_e c^2 \approx 0.585 \times 511\;\text{keV} \approx 299\;\text{keV}.

Throughout this chapter, T_e means the kinetic energy of one electron. It is not a plasma temperature or a photon energy.

What Was Claimed

The earlier version made four linked claims:

1. A threshold. At v_e = 0.776\,c, an electron’s boundary could no longer stay coherent, and it would shed a gamma modon. The argument was causal: the leading edge of its dress would move at v_e + 0.776\,c > c.
2. A shoulder. The Compton breath and the thermal tail would smear this threshold into a shoulder across roughly 200–500 keV.
3. A second channel. The shedding would add excess flux and a more isotropic angular component on top of combined electron–ion and electron–electron bremsstrahlung.
4. A cross-domain signature. The same shoulder would appear in terrestrial gamma flashes, solar flares and tokamak runaways.

Why It Was Withdrawn

• Electrons cross this speed routinely without incident. A 300 kV electron microscope drives electrons to 0.7765\,c with a field stronger than a thundercloud’s, and it produces no such emission in vacuum. Whatever happens at 0.776\,c is not triggered by speed, or by being field-driven through that speed.
• Electrons above this speed do not shed energy in flight. If they did, beta spectra would pile up near 300 keV and sharp conversion-electron lines, such as the 624 keV line of ^{137}Cs, would smear. Stored electron beams far above this speed would also lose more energy than synchrotron radiation accounts for.
• The causal argument gave the wrong speed. The sum v_e + 0.776\,c already exceeds c at v_e = 0.224\,c (T_e \approx 13 keV), not at 0.776\,c. With relativistic velocity addition it never exceeds c: the two speeds combine to 0.969\,c. What happens at v_e = 0.776\,c is that the trailing edge would come to rest, which is a different condition and not one we derived anything from.
• It contradicted the framework’s own account of motion. At steady speed a particle’s bow and stern waves cancel and it coasts without drag. A free electron also cannot emit a photon with nothing to take up the recoil. The base case in How a Moving Charge Makes Light is that acceleration sheds, not speed.
• The direct measurement already exists. Nakel & Pankau (1973) sent a 300 keV electron beam through a carbon target and used electron–photon coincidence to measure the absolute cross sections of electron–electron and electron–nucleus bremsstrahlung. They found agreement with theory within experimental errors. Coincidence work at this energy continued on heavier targets (Nakel 1994 reviews it). These experiments have a controlled, single-energy beam and resolve photon angle, which is a cleaner setting than any of the three plasmas below.
• The chapter never settled what it was claiming. It called shedding “the same event” as electron–electron bremsstrahlung, then required an excess on top of that same channel. It cannot be both.

Two Bremsstrahlungs, One Forbidden

The standard physics in this band is worth keeping, since it is what the earlier version confused for support.

Electron–ion bremsstrahlung is the textbook process: an electron deflects off a nucleus and radiates. It has a strong electric-dipole term, because the electron’s acceleration is enormous while the ion barely moves, so the system’s second dipole moment \ddot{\mathbf d} = \sum_i q_i \mathbf a_i does not cancel. Dipole radiation is allowed at every energy, so this is a smooth continuum with no special scale near m_e c^2.

Electron–electron bremsstrahlung is the other process: two electrons scatter and radiate. Here the dipole term vanishes identically. For two identical charges the dipole moment is \mathbf d = -e(\mathbf r_1 + \mathbf r_2), so

\ddot{\mathbf d} = -e(\mathbf a_1 + \mathbf a_2) = 0,

because the two electrons have equal mass and feel equal-and-opposite mutual forces. The leading radiation is therefore quadrupole (and magnetic-dipole), down from the allowed dipole rate by a factor of order (v/c)^2. As v \to c the penalty lifts, and the channel becomes comparable to electron–ion bremsstrahlung near T_e \sim m_e c^2 (Haug 1975; Gould 1980; Stepney & Guilbert 1983).

The earlier version treated this as the same event as the rim. It isn’t. The factor (v/c)^2 rises smoothly and has no special value at 0.776\,c, and T_e = m_e c^2 corresponds to 0.866\,c. Standard QED already explains why this band is interesting, with no substrate involved.

What the Data Shows

None of the three plasma domains shows evidence for a substrate feature.

Domain	What is measured near the rim	Reading
Solar flares	Upward spectral breaks at varying energies	Accounted for by standard physics, with variable breaks that do not fit a fixed threshold
Tokamak runaways	Limited coverage of the 300–500 keV band	No search has been made, and a thin-target beam is a better test
Terrestrial gamma flashes	Hard continuum reshaped by Compton scattering in the sub-MeV band	No native feature in this band

Solar flares. Kontar et al. (2007) found that including electron–electron bremsstrahlung naturally accounts for the upward break near 400 keV in the RHESSI spectrum of the 17 January 2005 flare. Kong, Li & Chen (2013) catalogue flares whose continuum hardens at breaks of 0.3–1.3 MeV, and find that the largest hardenings are too large to explain by electron–electron bremsstrahlung alone. But that break energy varies from flare to flare, which points to a variable electron spectrum rather than a fixed speed.

Tokamaks. The earlier version called this the cleanest test, citing routine coverage of the band. That overstated it. The DIII-D Gamma Ray Imager starts at 0.5 MeV (Cooper et al. 2016), some older systems leave a detector gap at 300–500 keV, and the SPARC and ITER hard-X-ray monitors have not yet measured a plasma. Runaway populations are also broad rather than single-energy.

Terrestrial gamma flashes. The characteristic break is at \sim\!7 MeV, the endpoint of the avalanche bremsstrahlung spectrum, with a hard continuum below it (Dwyer et al. 2012). The sub-MeV band of a satellite-detected flash is reshaped by Compton scattering in the atmosphere and spacecraft (Fitzpatrick et al. 2014).

The Rim Across Three Plasmas

The earlier version argued that a shoulder at the same energy in all three plasmas would be hard to explain except through a shared substrate scale. That argument was wrong. QED already provides a shared scale, since m_e c^2 is the same everywhere, so a common feature near that energy would not point to the substrate.

What Would Reopen It

A revived version would have to do all of the following:

• Stay silent for electrons in vacuum, at any speed, including when they are driven by a strong field.
• Fit inside the errors of thin-target measurements at 300 keV, which means emission tied to collisions could only be a small correction.
• Name what an atmospheric cascade, a flare and a tokamak share that a beam on a foil lacks, and derive why that matters.
• Compute the size and spectrum of the excess, and say how a threshold in electron energy maps onto a photon spectrum.

We do not currently have any of these.

Honest Accounting

First, the failure was visible from the start. The objection comes from ordinary electron microscopes, and we should have checked the prediction against them before calling it a clean test.

Second, the limit from beam data is not yet quantified. We have relied on the published conclusion of Nakel & Pankau, not on a re-analysis of their error bars, so we cannot state the largest excess those measurements allow.

Third, there was never a first-principles lineshape. The shoulder’s width and skew rested on a breathing waveform that has not been derived (WIP-12).

Place in the Framework

The inner rim now does one job in the paper: it is the rotation speed that sets the electron’s mass. It has no independent spectral read. The other chapters that relied on this prediction — Lightning, Thermal Dynamics, Follow the Energy — have been updated to match.