Randell Mills's classical atom and the hydrino: a primer

Project folder: Researchers/Mills-Randal. It holds one document: R. L. Mills, The Grand Unified Theory of Classical Physics (GUT-CP), a 1,876-page PDF whose title page reads "December 10, 2018 Edition, Volume 1 of 3" although the file holds all three volumes and cites a 2022 paper; the file name says "2020 Ed Web" [read: claims.md]. A newer edition (February 2025, 1,888 pages) exists at the same web address [read: sites.md, Leads 1].

Interested party. Mills founded and runs the company that sells the hydrino technology: HydroCatalysis (1991), then BlackLight Power, now Brilliant Light Power (BLP) [read: sites.md, Leads 4]. The book, the company's validation reports and nearly all the experimental papers the book cites come from that side. That is a fact about the sources, not a judgement of their data [read: arbiter.md].

Source gap (repeated from claims.md and arbiter.md, as required). - The deep reads (line-by-line reads with equations checked on page images) first covered about 40 percent of the book's pages. After the quality check (QE, qe.md) the remaining central chapters were deep-read too: Ch.3-4 (free electron, photon), Ch.9 (helium excited states), Ch.11 (the hydrogen molecule, source of the 4.478 eV bond energy), the prose of Ch.15 (pdf pp.801-1082), Ch.24-30 (statistics, superconductivity, quantum Hall effect, Aharonov-Bohm effect, pair production, positronium), Ch.31, the first half of Ch.32 and Ch.34 (gravity, Schwarzschild metric, perihelion, light bending, gravitational waves, the Higgs boson), Ch.35 (the "fifth force") and the Introduction [read: arbiter.md, QE revision 0]. Claim 7 and the general-relativity part of claim 6 now have full verdicts. - Still not deep-read by anyone: Ch.17-21, Ch.23, Ch.33 and pdf pp.1661-1665 (start of Ch.34), the numerical error columns of the Ch.15 tables, the amino-acid and phosphate sections; the Jansson, Booker and Nansteel reports and Mills's 165-page reply to Rathke (read in part) [read: arbiter.md, QE revision 0]. - The book's displayed equations do not survive text extraction; the numbers checked by the deep readers were read from page images, except where the arbiter says a deep reader worked from the printed text alone (for example the free-electron disc profile, Eq. 3.3) [read: claims.md; arbiter.md, QE revision 2]. - Not obtained: the November 1995 edition (the record behind the claim to have predicted cosmic acceleration and the top-quark mass in 1995); the company's full calorimetry data and the 194-page redacted Jansson report (image only, not read); the Hagen-Mills supplementary controls; the reports of the outside laboratories the company lists as confirmations; the IIT Kanpur replication results (not yet published) [read: arbiter.md, Source gap note].

Tags. [read] = from the project documents, deep reads, the Investigator's downloaded files, or review files that quote them; [web] = Internet source with URL; [memory] = recalled from training, with confidence high/medium/low; [computed] = calculated for this primer (run example.py). Verdicts are the arbiter's (arbiter.md); this primer adds none.

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1. The idea in one paragraph

A hundred years ago physicists could not understand why an electron circling a proton does not spiral into it: classical electromagnetism says an accelerating charge radiates and loses energy. Quantum mechanics answered by replacing the circling electron with a wave whose standing patterns do not radiate, and whose lowest pattern (the "ground state", n = 1) is the bottom of the ladder. Randell Mills, by the book's own account a medical doctor (MD, Harvard Medical School) with a chemistry degree from Franklin and Marshall College who also studied electrical engineering at MIT [read: p.1875, also p.439, p.1518; this is the publisher's description, and the publisher is Mills's side], took a different road. He kept classical physics and made the electron itself a thin spherical bubble of charge and current, sized so that, by a known rule of classical radiation theory, it does not radiate. From this bubble he computes spin, the electron's magnetic moment, atomic and molecular energies, and much else, in closed formulas. The same model has room for something quantum mechanics forbids: smaller, more tightly bound bubbles below the ordinary ground state, which he calls "hydrinos". He says a hydrogen atom can drop into them when a nearby atom or ion (a "catalyst") absorbs a resonant chunk of energy, a multiple of 27.2 electron-volts, releasing tens to hundreds of electron-volts per atom, about a hundred times what burning hydrogen gives. That is the basis of his company's energy technology, and he also proposes hydrinos as the universe's dark matter. This primer explains the real physics each step starts from, works through the one place where the theory makes a sharp, precise prediction (the electron's magnetic moment), and sets out what the laboratory evidence does and does not show.

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2. What you need to know first

Circular motion and Coulomb's law. A mass m moving at speed v on a circle of radius r needs an inward force m v^2 / r. Two charges q1, q2 a distance r apart attract or repel with force q1 q2 / (4 pi eps0 r^2), where eps0 = 8.854 x 10^-12 F/m is the permittivity of free space. Setting the two equal is the "force balance" that both Bohr's 1913 model and Mills's model use. Read: Taylor, Classical Mechanics; Griffiths, Introduction to Electrodynamics, Ch. 2.

The virial theorem for an inverse-square force. For a particle bound by a 1/r^2 force in a circular orbit, the kinetic energy T and potential energy V obey V = -2T, so the total energy is E = T + V = -T = V/2. This is why the hydrogen numbers 13.6 eV and 27.2 eV appear together all through Mills's book: 13.6 eV is the kinetic energy (and the binding energy) of the ground state, 27.2 eV is the size of its potential energy. Read: Taylor, Classical Mechanics (central forces chapter).

The electron-volt. 1 eV = 1.602 x 10^-19 J, the energy an electron gains crossing 1 volt. A photon of energy E has wavelength lambda = hc/E = 1239.84 nm / (E in eV), where h is Planck's constant and c the speed of light. So 13.6 eV is 91.2 nm, in the extreme ultraviolet (EUV).

Accelerating charges radiate (the Larmor formula). A point charge q with acceleration a radiates power P = q^2 a^2 / (6 pi eps0 c^3). A classical electron circling at the Bohr radius would radiate its energy and spiral in within about 1.6 x 10^-11 s [computed: a0^3 / (4 r_e^2 c), with r_e the classical electron radius]. This is the puzzle that started quantum theory. Read: Griffiths, Introduction to Electrodynamics, Ch. 11 (Radiation).

Non-radiating sources. A single point charge on a circle must radiate, but an extended distribution of charge and current need not: if the distribution's Fourier transform has no component that travels in step with light (no component at wavenumber k = omega/c, where omega is the angular frequency), it does not radiate. Examples have been in the mainstream literature since Goedecke (1964) and Devaney and Wolf (1973) [memory, high]; H. A. Haus stated a version of the condition in the 1980s [memory, medium]. This is real physics and it is Mills's starting point.

The Fourier transform. It rewrites a function of position and time as a sum of travelling waves, each with a wavenumber k (2 pi / wavelength) and frequency omega. Only the waves with omega = c|k| can carry energy off to infinity as light. You need only this idea, not the mathematics. Read: any mathematical-methods text, e.g. Boas, Mathematical Methods in the Physical Sciences.

The quantum hydrogen atom. The Schroedinger equation for hydrogen has bound solutions only at energies E_n = -13.6 eV / n^2, with n = 1, 2, 3, ... The integer n is not assumed: it appears because the wave function psi must stay finite and fall to zero far away (be "normalisable"). For a "stationary" state the charge density e|psi|^2 does not change in time, so it does not radiate. Light is emitted only during a transition, when two states are mixed. Read: Griffiths, Introduction to Quantum Mechanics, Ch. 4.

Magnetic moment, spin and the g factor. A loop of current I enclosing area A has magnetic moment mu = I A. The electron behaves as a tiny magnet of moment (g/2) mu_B, where mu_B = e hbar / (2 m_e) = 9.274 x 10^-24 J/T is the Bohr magneton (e the electron charge, hbar = h/2pi, m_e the electron mass) and g is a pure number very close to 2. The Dirac equation gives g = 2 exactly; the small excess, g/2 - 1 = 0.00116, comes from quantum electrodynamics (QED, the quantum theory of electrons and light). Read: Griffiths, Introduction to Quantum Mechanics, Ch. 4 (spin); Eisberg and Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles.

Power series and error propagation. If a quantity is written as a series in a small number x, f = 1 + c1 x + c2 x^2 + c3 x^3 + ..., each term is about x times smaller than the one before; the coefficients c1, c2, ... decide the answer at each level of precision. If an input x has relative uncertainty dx/x, the output moves by (df/dx) dx. Part 5 of this primer uses both ideas. Read: Taylor, An Introduction to Error Analysis.

Resonant (non-radiative) energy transfer. Two atoms or molecules close together can pass energy from one to the other without a photon being emitted, through the near-field coupling of their charge distributions. This is Foerster transfer in chemistry and biology, and it is how lamp phosphors pass energy from a sensitiser ion to an emitting ion. It works best when the energy given up matches an energy the receiver can absorb. Mills uses this real mechanism as his "catalysis" step [read: deep read Ch.5 2d]. Read: Lakowicz, Principles of Fluorescence Spectroscopy (Foerster transfer) [memory, high].

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3. The textbook view

3.1 Hydrogen by numbers

Symbols: m_e = 9.109 x 10^-31 kg (electron mass), e = 1.602 x 10^-19 C, hbar = 1.055 x 10^-34 J s, eps0 as above.

• Bohr radius: a0 = 4 pi eps0 hbar^2 / (m_e e^2) = 0.0529 nm. With the electron-proton reduced mass it is a_H = 0.05295 nm (Mills's symbol) [read: deep read Ch.5 1].
• Energy levels: E_n = -13.6 eV / n^2, n = 1, 2, 3, ... [memory, high].
• At n = 1: kinetic energy T = 13.6 eV, potential energy V = -27.2 eV (the "Hartree", E_h = e^2 / (4 pi eps0 a0) = 27.2114 eV), total E = -13.6 eV.
• A hydrogen-like ion with nuclear charge Z (He+ has Z = 2, Li2+ has Z = 3) has radius a0/Z and binding energy Z^2 x 13.6 eV, because both the charge and the tighter orbit scale with Z.

Bohr got all this in 1913 by balancing forces with one extra rule: the angular momentum m_e v r equals n hbar. Quantum mechanics recovers the same energies from the Schroedinger equation without that rule.

3.2 Why the textbook says there is nothing below n = 1

In quantum mechanics the integer n comes from a mathematical requirement: solutions with non-integer n blow up far from the nucleus, so they are not states [read: deep read Ch.5 7a]. The lowest allowed energy is -13.6 eV. A collision or near-field coupling can move an atom between allowed states, but cannot put the electron in a state that is not in the list. The relativistic Dirac equation gives the same conclusion [memory, high].

There is a fine point, and the review keeps it because it matters to fairness. For a point proton, the zero-angular-momentum equation has a second, singular solution near r = 0 that is still square-integrable, so the mathematics admits a family of alternative boundary conditions, some with an extra very deep state. J. Naudts (2005, Klein-Gordon equation) and A. Khelashvili and T. Nadareishvili (Dirac equation) used this route to get "hydrino-like" deep states inside standard quantum mechanics. N. Dombey (2006) showed these states vanish if the proton has any non-zero size, however small, and Khelashvili and Nadareishvili themselves concluded the extra Dirac solution "must be ignored" [read: mathematician.md, Corrections 5; files/arXiv-Dombey-2006, abstract]. So "there is no state below n = 1" is, strictly, a statement about real protons with finite size, which is what the arbiter's verdict says [read: arbiter.md, Claim 2].

3.3 What radiates and what does not

A static charge does not radiate; nor does a steady current. A stationary quantum state has a static charge density and a steady current, so it does not radiate: that is quantum mechanics' answer to the collapse puzzle of section 2 [memory, high]. In classical physics, special extended distributions of charge and current, oscillating in time, can also be non-radiating (section 2). Mills's electron is one such distribution, and for the simplest case (a uniformly spinning shell with no time-varying charge) the conclusion is correct and standard [read: mathematician.md 1a].

3.4 The electron's magnetic moment

A uniformly charged spherical shell spinning with angular momentum hbar has magnetic moment (2/3) mu_B, not mu_B [read: mathematician.md 1a, re-derived]. The Dirac equation gives a spin-1/2 electron a moment of exactly 1 mu_B (g = 2). Julian Schwinger in 1948 computed the first QED correction:

g/2 = 1 + alpha/(2 pi) + ...

where alpha = e^2 / (4 pi eps0 hbar c) = 1/137.036 is the fine-structure constant (a pure number measuring the strength of electromagnetism). alpha/(2 pi) = 0.0011614. QED continues the series in powers of (alpha/pi):

g/2 = 1 + C1 (alpha/pi) + C2 (alpha/pi)^2 + C3 (alpha/pi)^3 + C4 (alpha/pi)^4 + C5 (alpha/pi)^5 + (small terms from muons, taus, hadrons and the weak force)

with C1 = 1/2, C2 = -0.328479, C3 = +1.181241 [memory, high], C4 = -1.9122, C5 = about 6.7 [memory, medium]. Each coefficient comes from a separate, very long calculation; the fifth-order term required thousands of diagrams [memory, high]. The measured value is g/2 = 1.00115965218059 with an uncertainty of 0.00000000000013 (1.3 x 10^-13) [web: Fan et al. 2023, via physicist.md section 1, https://cfp.physics.northwestern.edu/documents/2023MeasurementElectronMangeticMoment.pdf]. alpha itself is measured independently, by atom-recoil experiments on rubidium and caesium, so the comparison is not circular [read: deep read Ch.1 2.3; web: https://www.cnrs.fr/en/french-team-has-improved-measurement-fundamental-physical-constant].

3.5 Molecules, Bell tests, cosmology and collective effects, briefly

• Molecules. Bond energies come from solving the many-electron Schroedinger equation numerically (coupled-cluster methods reach about 0.04 eV for small molecules; density functional theory a few times that) [memory, high]. For large organic molecules, "group additivity" schemes (Benson) add up fitted values per chemical group and reach similar accuracy [memory, medium].
• Bell tests. John Bell proved that any theory in which each particle carries its own pre-set answers, and in which distant measurements cannot influence each other ("local hidden variables"), obeys a limit on correlations. Experiments violate the limit; since 2015 tests have closed the main loopholes together (Delft: S = 2.42 +/- 0.20, where the local limit is 2) [web: https://arxiv.org/abs/1508.05949].
• Cosmology. The Planck satellite fit gives a Hubble constant H0 = 67.4 +/- 0.5 km/s/Mpc [web: https://arxiv.org/abs/1807.06209]; nearby-supernova methods give about 73 [memory, high]; the cosmic microwave background (CMB) temperature is 2.7255 K [memory, high]; the tensor-to-scalar ratio r < 0.036 [web: https://arxiv.org/abs/2110.10902]. Dark matter is inferred to be non-baryonic (not made of protons and neutrons) from the CMB peak heights and the deuterium made in the Big Bang [memory, high].
• Superconductivity and the quantum Hall effect (claim 7). A superconductor loses all electrical resistance below a critical temperature Tc, which differs widely from metal to metal (aluminium 1.18 K, lead 7.2 K, niobium 9.25 K; copper, silver and gold do not superconduct at ordinary pressure) [memory, high]; the standard (BCS, Bardeen-Cooper-Schrieffer) theory explains it by electrons bound in pairs. In the quantum Hall effect a thin conducting layer in a strong magnetic field shows a Hall resistance fixed at h/(n e^2), with n an integer or certain fractions, mostly with odd denominators [memory, high]. The plateau value h/e^2 = 25,812.807 ohm is known to many digits.

3.6 Gravity, gravitational waves and the Higgs boson, briefly

General relativity (GR) is not assumed in the reader profile, so here is the minimum. GR describes gravity as the geometry of space and time. That geometry is written as a metric: a rule that turns small coordinate steps (dt, dr, ...) into a measured time or distance. Outside a round mass M, the solution of Einstein's equations is the Schwarzschild metric, which has two key factors: clocks run slow by the factor sqrt(1 - 2GM/(c^2 r)) (G is Newton's gravitational constant, r the distance from the centre), and radial distances are stretched by the reciprocal of that same factor. Both factors are needed to get the measured bending of starlight grazing the Sun (1.75 arcseconds, twice what the clock factor alone gives) and the 43 arcseconds per century extra turning of Mercury's orbit [memory, high]. In GR this reciprocal relation follows from the field equations (Birkhoff's theorem); it is not an extra assumption [read: arbiter.md, QE revision 6].

GR also predicts gravitational waves: ripples of the metric, transverse to their direction of travel, moving at the speed of light. They were first detected directly on 14 September 2015 (GW150914) by both LIGO (Laser Interferometer Gravitational-Wave Observatory) detectors, with a peak strain (fractional length change) of 1.0 x 10^-21 and a waveform matching GR's prediction for two merging black holes [web: https://arxiv.org/abs/1602.03837]. In 2017 a neutron-star merger (GW170817) arrived together with a burst of gamma rays: the speed of gravity differs from the speed of light by no more than a few parts in 10^15 [web: https://arxiv.org/abs/1710.05834].

The Higgs boson is the particle the Standard Model of particle physics needs to give mass to the W and Z bosons. ATLAS and CMS, the two large detectors at CERN's Large Hadron Collider (LHC), announced it on 4 July 2012; their combined mass is 125.09 +/- 0.24 GeV (giga-electron-volts) [web: https://arxiv.org/abs/1503.07589, abstract; number memory, high], and its couplings to other particles have since been measured [web: https://arxiv.org/abs/2207.00092].

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4. Where Mills departs

4.1 Side by side

4.2 What holds

These parts survive every reviewer's check and are fair to state first:

• The arithmetic. Across the hundreds of numbers the deep readers recomputed, the printed numbers almost always follow from the printed equations; the company-hired reviewer R. Booker reports the same, chapter by chapter [read: arbiter.md Claim 1, Investigator's findings]. The later deep reads found a few exceptions (H2+ and D2+ vibration energies and Table 11.1 distances not produced by the stated recipe; the positronium level constant 6.795 eV against the formula's own 6.8028 eV; the fifth-force threshold printed with Earth's mass-to-radius ratio) [read: arbiter.md, QE revision 4, 6, 7]. The main problems are in the premises, not the sums.
• Non-radiating extended sources exist in classical electrodynamics (section 2), and a static spinning shell does not radiate.
• The hydrogen numbers come out right because, for one electron, the force balance is Bohr's model and the relativistic formula (Eq. 1.294) is Dirac's: IP = m_e c^2 (1 - sqrt(1 - (Z alpha)^2)). Table 1.5 reproduces 39 ions to every printed digit [read: mathematician.md 1b]. This is agreement with Dirac theory, not independent support.
• Schwinger's term appears exactly as the first correction in Eq. 1.228 [read: deep read Ch.1 2.3].
• The hydride ion. Mills's formula gives the binding energy of H- (one proton, two electrons) as 0.754175 eV against the measured 0.754195 +/- 0.000019 eV (Lykke, Murray and Lineberger 1991), about one standard deviation apart [read: deep read Ch.5 3]. Two cautions: a special reduced mass (m_e m_p / (m_e/sqrt(3/4) + m_p)) carries the agreement (with the ordinary reduced mass the formula is 5 standard deviations off), and ordinary quantum calculations also reproduce H- [read: mathematician.md 1b; memory, medium-high].
• The energy-release bookkeeping (Part G of example.py) is consistent: if hydrogen could go from p = 1 to p = 1 + m, it would release [(1+m)^2 - 1] x 13.6 eV, of which m x 27.2 eV goes to the catalyst; for m = 1, 2, 3 that is 40.8, 108.8 and 204 eV, with EUV continuum edges at 91.2, 22.8 and 10.1 nm [computed; read: deep read Ch.5 2c]. Burning hydrogen gives 1.48 eV per H atom [computed], so H to H(1/4) would be about 140 times more.
• Non-radiative resonant transfer is a real process (section 2), and Mills cites real examples: Franck-Hertz collisions, Sb3+ to Mn2+ transfer in lamp phosphors, excimer-to-hydrogen transfer in discharges [read: deep read Ch.5 2d].

4.3 Where the inferences fail

Each item is a finding of the reviewers or the deep readers, confirmed by the arbiter.

1. hbar is put in, then taken out. Eq. 1.35 sets the shell's speed so that m_e v r = hbar; Eq. 1.37 integrates that speed and "derives" hbar [read: mathematician.md 1a]. The spin-1/2 construction is chosen with its target stated in advance (Mills's own numerical result is 0.495 hbar), and the full Bohr magneton needs an extra postulate (a precessing photon at 60 degrees) and a factor 3/2 inserted by hand [read: physicist.md Claim 1a].
2. The non-radiation proof works only for the static shell. For the excited-state patterns (angular number l >= 1) the vanishing relies on setting the wavelength equal to the radius, which conflicts with the book's other definition 2 pi r_n = lambda_n. With the correct radial factor, the spherical Bessel function j_l(k r_n), the result is not zero: j_1(2 pi) = -0.159 [read: mathematician.md 1a]. The mathematician also showed with symbolic algebra that a rotating charge wave of angular shape l, m on a sphere satisfies the book's own wave equation only if its speed is m omega R / sqrt(l(l+1)), never the omega R the book uses (because l(l+1) = m^2 has no solution with l >= 1) [read: mathematician.md Corrections 3; adopted by arbiter.md Claim 1].
3. The shell's push on itself is removed by postulate. A charged shell has an outward electrostatic pressure on itself, sigma^2 / (2 eps0), with sigma the surface charge density. Including it in Mills's own force balance gives a ground-state radius of 2 a0, not a0 [read: mathematician.md 1a]. The book avoids this by postulating that no point of the surface acts on another.
4. The force balance does not choose the integers for hydrinos. Put r = a_H / Z_eff into Mills's balance hbar^2 / (m_e r^3) = Z_eff e^2 / (4 pi eps0 r^2) and the Z_eff's cancel: it holds for Z_eff = 1.5 or 2.7 as well as 2 or 3 (Part G of example.py shows the residual is zero for every Z_eff) [computed; read: deep read Ch.5 2b]. The integers come from a separate postulate (a "trapped photon" field obeying Laplace's equation, Eq. 5.27), and the 27.2 eV unit from an assumption about counting multipoles (Eqs. 5.44-5.46) [read: deep read Ch.5 2d]. Algebraically a hydrino is a hydrogen-like ion of charge p, with the extra charge attributed to the electron's own field rather than to protons.
5. The catalyst table is not a test by itself. Table 5.2 lists 48 sums of ionisation energies that lie near multiples of 27.2 eV (median miss 0.58 eV). A random energy lands within 0.58 eV of some multiple 4.3 percent of the time, so the table beats chance if each element had one fixed candidate, and is at chance if each had about 16 candidate sums to choose from; the book does not say how many were tried [read: arbiter.md, recomputation table].
6. "The Schroedinger density must radiate" (Ch.40) assumes the density oscillates in time. A stationary state's density does not. The Fourier algebra is fine; the premise is wrong [read: arbiter.md Claim 5].
7. The "classical" prediction for the Aspect Bell test (Eq. 42.23) is the quantum-mechanical formula from Aspect's own paper; the mechanism the book describes in words (pairs sharing a random polarisation, Malus's law at each polariser) gives half the modulation and obeys Bell's limit [read: arbiter.md Claim 5].
8. Cosmology numbers are set by one chosen mass. H0 = 4 pi c^3 / (G m_U), with m_U = 2 x 10^54 kg chosen "for internal consistency", gives 78.3 km/s/Mpc; the CMB temperature needs a different mass (2.56 x 10^54 kg, which would give H0 = 61) [read: arbiter.md Claim 6].
9. The Schwarzschild metric is reached, not derived. The wave equation supplies only that a change in gravity spreads at speed c. The clock factor comes from a light-clock argument using the Newtonian escape velocity sqrt(2GM/r), a known heuristic; the radial factor is then asserted to be its reciprocal, which is exactly the part GR's field equations supply and which carries half the light bending. The result is GR's metric, so the book's perihelion and light-bending tables (Mercury 43.03 arcsec per century; 1.75 arcsec at the Sun's edge) are GR's numbers and cannot tell Mills's theory from GR [read: arbiter.md, QE revision 6].
10. Fixed choices can look like predictions. The Ch.9 helium formulas fit no continuous parameter, but about six discrete choices (a factor 2 for triplets, an orientation sign and others) are made once, and the book does not say whether they were fixed before or after comparison with the NIST tables. "Over 100 states" is 104 table rows but 61 independent levels from 4 formulas, with a mean relative error of 1.0 x 10^-3, not "within experimental error"; for high angular momentum the rows equal hydrogen's -13.6/n^2 eV, which any model with a screened core gets. In its favour, the fixed factors give S-series quantum defects 0.144 and 0.289 against observed about 0.145 and 0.305 [read: arbiter.md, QE revision 2]. The same pattern (a value chosen from a small set with a verbal reason, then compared) appears in the Ch.15 molecular groups and in the superconductor formula, where the dimension f = 1, 2 or 3 changes Tc by a factor of exactly 27 [read: arbiter.md, QE revision 4 and 7].
11. Momentum is not conserved in the "space drive". The book's own example gives a free electron 1.73 x 10^-27 kg m/s of momentum from a 5 GHz photon carrying 1.1 x 10^-32 kg m/s, about 160,000 times more, with "spacetime" taking up the difference. A free electron cannot absorb a single photon and conserve both energy and momentum [read: arbiter.md, QE revision 2; memory, high].

4.4 Where the question is experimental, not theoretical

Showing that Mills's derivation is inconsistent does not settle whether something unusual happens in his reactors. Anomalous measurements exist and the review keeps them: excess heat in a light-water nickel cell at NASA Lewis in 1996, small EUV continua, heat from differential scanning calorimetry (DSC) at the University of Illinois, a structured electron paramagnetic resonance (EPR) spectrum measured at Delft (and broad EPR patterns from other reactor types reported in 2024), and unusually fast hydrogen atoms in certain plasmas. One more experimental question sits outside the hydrino: whether a free electron has gravitational mass, where the book leans on a real 1967 experiment. Their status is in the evidence table (section 6). The arbiter's point is that "the theory is wrong" and "the lab anomalies are not hydrinos" are separate statements, and only the first has been settled [read: arbiter.md Claim 2, ruling 1].

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5. Worked example: does Mills's closed-form g factor survive today's measurements?

This is the one place where the theory makes a sharp numerical prediction that can be compared with a measurement accurate to 13 significant figures, and where a single input (alpha) is known independently. Every number below is printed by example.py [computed].

The formula. Mills's Eq. 1.228 [read: deep read Ch.1 2.1, page images p.138-139]:

g/2 = 1 + alpha/(2 pi) + (2/3) alpha^2 (alpha/(2 pi)) - (4/3) (alpha/(2 pi))^2

The three correction terms come, in his account, from the stored magnetic energy of a "fluxon" (a magnetic flux quantum h/2e captured during a spin flip), the stored electric energy, and a dissipated energy J.E (current density times electric field) [read: deep read Ch.1 2.1]. Every other constant cancels, so alpha is the only input. No number in it is fitted, though several modelling choices fix its factors (a radius scaled by 2 pi; speed c in one term and alpha c in another; only the internal 2/3 of the field energy used; a minus sign for the dissipated term) [read: deep read Ch.1 2.4].

Step 1. The part everyone agrees on. With alpha = 1/137.035999206 (rubidium, 2020): alpha/(2 pi) = 0.001161409732. So 1 + alpha/(2 pi) = 1.001161409732. The measurement is 1.00115965218059. Schwinger's term alone is high by 1.758 x 10^-6. Any theory must supply about -1.758 x 10^-6 from its next terms.

Step 2. The book's comparison. The book uses alpha^-1 = 137.03604 (the 1973 recommended value, cited from the 1987 CRC Handbook) and the 1987 measurement g/2 = 1.001159652188(4) by Van Dyck, Schwinberg and Dehmelt [read: deep read Ch.1 2.2]. Then:

• alpha/(2 pi) = 0.00116140940
• -(4/3)(alpha/(2 pi))^2 = -1.79850 x 10^-6
• (2/3) alpha^2 (alpha/(2 pi)) = +4.1231 x 10^-8
• Sum: g/2 = 1.0011596521215 (the book prints 1.001159652120)
• Minus the 1987 measurement: -6.65 x 10^-11.

The two agree in the first eleven figures (1.0011596521...). This is the book's "eleven-figure agreement" [read: deep read Ch.1 2.3].

Step 3. How sharp was that test? The result depends on alpha. The derivative is d(g/2)/d(alpha) = 0.1587, so a relative error d(alpha)/alpha moves g/2 by about 0.00116 x d(alpha)/alpha. - The 1973 alpha had uncertainty 0.00011 in 137.03604, a relative 8.0 x 10^-7. That moves g/2 by +/- 9.3 x 10^-10. - The book's own computed alpha (Eq. 1.235) carries +/- 0.00082, relative 6.0 x 10^-6, which moves g/2 by +/- 6.9 x 10^-9. So in 1987 any formula landing within about 10^-9 of the truth would have "agreed", whatever its terms. The test could not see a discrepancy of a few times 10^-10.

Step 4. Today's numbers. alpha has since been measured by atom interferometry, independently of g, to a relative 8 x 10^-11: alpha^-1 = 137.035999206(11) (Rb, 2020) [web: CNRS, above] and 137.035999046(27) (Cs, 2018) [memory, high]. That moves g/2 by only +/- 9.3 x 10^-14. Put the Rb value in Eq. 1.228: - g/2 (Mills) = 1.00115965246619 - g/2 (measured, 2023) = 1.00115965218059 +/- 0.00000000000013 - Difference: +2.86 x 10^-10. - Divided by the measurement uncertainty 1.3 x 10^-13: about 2,200. Including the alpha uncertainty: about 1,800 (Rb) or 1,100 (Cs). The rubidium and caesium values of alpha disagree with each other by about 5 standard deviations, but that shifts g/2 by only 1.3 x 10^-12, 200 times smaller than the miss [computed; web: CNRS, above].

Step 5. Why it came so close: term by term. Write each term in powers of alpha/pi = 0.0023228:

Mills's second-order term is too negative by 2.62 x 10^-8; his third-order term is too positive by 2.64 x 10^-8. The two errors nearly cancel, leaving 2.4 x 10^-10, which grows to the 2.9 x 10^-10 of Step 4 once QED's higher orders are counted [computed; QED coefficients memory, high]. QED with the same alpha, adding orders one at a time, gives: - through order 2: 1.5 x 10^-8 too low (example.py prints -1.475 x 10^-8); - through order 3: 5.0 x 10^-11 too high; - through order 5, plus the small muon, tau, hadronic and weak contributions (about 4.5 x 10^-12 [memory, medium]): 3.4 x 10^-13 too low, about 2 standard deviations [computed]. So QED lands about 850 times closer than Eq. 1.228.

Step 6. Run it backwards. What alpha would Eq. 1.228 need to reproduce today's g? Solving gives alpha^-1 = 137.036033, against the measured 137.035999206(11): a difference of 3.4 x 10^-5, about 3,100 times the uncertainty of the rubidium measurement [computed; read: deep read Ch.1 2.3 gives 137.0360330].

Step 7. A sign check. The dissipated-energy term enters with a minus sign "because it is dissipative". With a plus sign the formula gives 1.0011632, wrong in the 6th decimal [computed; read: deep read Ch.1 2.4]. The sign is a modelling choice the text justifies in words.

What the example shows. Mills's formula contains Schwinger's real first-order result exactly, and its next two terms happen to cancel each other's errors, so with 1970s constants it matched the measurement as well as the constants allowed. With an alpha measured independently of g and 10,000 times more precisely, it misses by about 2,000 standard deviations. The 2018-dated edition still compares against 1973/1987 constants and does not report the modern comparison [read: deep read Ch.1 2.3]. The arbiter's verdict on this item is Tested, not supported [read: arbiter.md Claim 1].

A second calculation you can do (section 4). Part G of example.py solves Mills's hydrino force balance symbolically and shows it is satisfied by any effective charge, then prints the energy ladder (40.8, 108.8, 204 eV) and compares it with burning hydrogen (1.48 eV per atom).

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6. Evidence table

Verdicts are the arbiter's, using the review's scale. "Replicated?" means by a party with no stake.

Sources for the table: arbiter.md (all verdicts and numbers), physicist.md, mathematician.md, contrarian.md, deep reads [read]; Bell [web: https://arxiv.org/abs/1508.05949]; H0 [web: https://arxiv.org/abs/1807.06209]; r [web: https://arxiv.org/abs/2110.10902]; top mass [web: https://pdglive.lbl.gov/view/Q007TP]; EarthTech [web: https://earthtech.org/cold-fusion/mills/]; Phelps-Clementson [web: https://link.springer.com/article/10.1140/epjd/e2012-30114-5]; GW150914 [web: https://arxiv.org/abs/1602.03837]; GW170817 [web: https://arxiv.org/abs/1710.05834]; Higgs [web: https://arxiv.org/abs/1503.07589; https://arxiv.org/abs/2207.00092]; quasar time dilation [web: Crossref, DOI 10.1038/s41550-023-02029-2, title and date only]; Witteborn-Fairbank [web: Crossref, DOI 10.1103/physrevlett.19.1049, title only; the 0.09 m_e figure is the book's, p.1680, p.1686]; Loureiro-Amorim [web: Crossref, DOI 10.1140/epjd/e2011-20028-1, title only]; KEK paper [web, via the deep reader: https://link.aps.org/doi/10.1103/PhysRevLett.78.424]; rows 1f-1k, 2f-2h, 4a-4c, 5e-5f, 6f-6m and 7a-7f [read: arbiter.md, QE revision 1-7, and the deep reads it cites]. Still [memory] in these rows: "many events" for LIGO-Virgo, Schiff-Barnhill physics (medium), supernova ejecta speeds (high) and the SN 1997ex dilation (medium), superconductor Tc values (high), positronium lifetimes (high).

The experimental record in more detail (kept whatever the verdict)

• Hagen and Mills 2022 (International Journal of Hydrogen Energy 47, 23751). W. R. Hagen, an EPR specialist at TU Delft with no financial support from BLP (per the paper), measured Ga(O)OH fibres made by BLP from SunCell products. Strengths: an empty-tube baseline; 11 samples from 11 reactor runs; features that keep a fixed field spacing while the centre keeps a fixed g as the microwave frequency changes; no nitrogen or paramagnetic transition-metal ions found by surface mass spectrometry. Limits, in the paper's own words: the fit is "semi-quantitative with an average error of 0.004 +/- 0.550 gauss over the 11 lines"; the sub-line separation "is not a constant"; their number and intensities "are presently not understood"; "A broad signal underlies" the spectrum, "origin and nature unknown"; a conventional fit with two radicals "approximately reproduces" it, which the authors judge chemically "highly unlikely" [read: physicist.md Claim 2c; contrarian.md Corrections 1-3]. The book's figure "average error 0.097 G" is a different, unstated statistic [read: mathematician.md Corrections 1]. A duplicate at Bruker (Billerica, Massachusetts) is reported by Mills's co-authored paper citing Mills's own preprint; no Bruker-authored record was found [read: sites.md, Corrections].
• NASA Lewis Technical Memorandum 107167 (Niedra, Myers, Fralick, Baldwin, 1996). Up to 11 W excess on 60 W input, gains 1.06-1.68, in a cell lent by HydroCatalysis. The authors say recombination of dissolved hydrogen and oxygen is an ordinary explanation they could not exclude, and call further study "warranted". They also weigh it: explaining all the excess by recombination "requires a recombination efficiency of quite a high magnitude" (above 0.55, rising to about 1 at 5 A), which the data "marginally admit". They quote the cell's maker as reporting "50 W steady excess heat for a continuous period exceeding hundreds of days" (the company's claim, quoted by NASA), and name fusion of the "shrunken" atoms as one hypothesis [read: research/files/NASA-TM-Niedra-1996, p.1, p.4, via qe.md and arbiter.md, QE revision 3]. So the authors' own reading is ambiguous with a slight lean toward an anomaly, not a plain "could not exclude".
• EarthTech International (Austin, 1997). 164-hour run of the same cell family with a palladium recombiner: "no detectable sign of excess heat"; a 10 percent excess would have shown [web: https://earthtech.org/cold-fusion/mills/].
• Thermacore, US Air Force SBIR (Small Business Innovation Research) Phase I (1993-94). 25 W anomalous heat in a pressurised nickel-tube cell; Lehigh University surface analysis (ESCA, electron spectroscopy for chemical analysis) reported the predicted 55 eV peak; Mills and Good consulted; no Phase II located [read: research/files/LENR-CANR-Gernert-nascent-hydrogen.pdf; sites.md Leads 2].
• NASA Institute for Advanced Concepts Phase I (Rowan University, 2002). Took the plasma anomalies as worth propulsion study "regardless of the theoretical explanation" [read: research/files/NIAC-752].
• Company-hosted validation reports. Nansteel 2021 (one test, 169-210 kW, local gain about 4.2); Booker 2020 (194-218 kW); Tse 2020 (Rutgers, hired observer); Glumac 2014 DSC at the University of Illinois (101-148 J/g exotherms "far more exothermic than the predicted thermochemistry"); Crouse DSC; a 2016 summary claiming "1 MW+"; the Jansson report (194 pages, redacted, image only). These are the interested party's selection; the authors are named scientists; none was read in full by a reviewer [read: arbiter.md, Investigator's findings].
• Hagen and Mills 2024 (ChemRxiv, 2026 version). H2(1/4) "produced in multiple reactors of different type" gives broad EPR patterns over 0-8000 G, measured at L-, S-, X- and Q-band microwave frequencies (roughly 1 to 35 GHz [memory, high]), with "anti-Curie behaviour" (the signal does not grow as 1/T on cooling, as an ordinary paramagnet's does) "and a magnetic phase transition around 107 K" [read: research/2026-10-09-brilliantlightpower.md, abstract p.1]. Hagen and Mills also have a paper proposing an EPR link to the cosmic neutrino background (not read) [read: sites.md, Leads; arbiter.md, Investigator's findings]. The samples again come from the company's reactors; the full paper has not been weighed by a reviewer.
• Fast hydrogen atoms (broadened Balmer lines). The book cites seven papers (2005-2010, all with Mills as an author; co-authors Scharer, Phillips, Chen, Mayo and Nansteel may have had outside affiliations [memory, medium-low]) reporting hydrogen atoms in certain plasmas moving far faster than the gas temperature allows, which it reads as energy from hydrino formation [read: arbiter.md, QE revision 3]. Phillips (2008/2009, a Mills co-author) argued it has implications for quantum theory; Loureiro and Amorim, "Anomalous Doppler broadening caused by exothermic reactions: application to hydrogen Balmer lines" (Eur. Phys. J. D, 2011), propose an ordinary chemical mechanism [web: Crossref, DOI 10.1140/epjd/e2011-20028-1, title only; sites.md had 2010]. Neither side has been read in full by this review; it is a distinct experimental anomaly class, kept in the record.
• IIT Kanpur (Pala and Sharma, Electrochemical Society 2026 abstract). The first academic group to announce a replication programme; no results yet; its abstract treats hydrinos as already supported [web: https://iopscience.iop.org/article/10.1149/MA2026-01281445mtgabs].
• Theory critique and reply. Rathke (New Journal of Physics 7, 127, 2005) found "severe inconsistencies in the deterministic model". Mills's 165-page reply (2008) correctly points out a sign slip in Rathke's Eq. (9), but also concedes that the delta-function shell "is not a radial solution of Eq. 4; rather, it defines a constant charge density on a spherical shell"; the mathematician finds the main objections survive the sign correction [read: mathematician.md Corrections 4].
• Patent offices. The US Patent and Trademark Office (USPTO) allowed application 09/009,294 after its examiner reviewed experimental results and samples, then withdrew it in February 2000 after "an outside inquiry"; the Group Director's concern was "the electron going to a lower orbital in a fashion that I knew was contrary to the known laws of physics and chemistry"; the court found it "undisputed that no one involved in the withdrawal had reviewed the '294 patent application before issuance of the Notice" (that is, before the notice of allowance, not "ever"); Wikipedia, citing Rimmer (2011), reports that the Group Director, Esther Kepplinger, said the application appeared to involve cold fusion and perpetual motion (the institutional party's characterization, paraphrased; read only via research/pages/wiki-blp.txt; Rimmer and Bob Park's 2002 account of the "outside inquiry" not read); the Federal Circuit (2002) upheld the Director's authority, not the physics [read: research/2026-10-09-findlaw.md; web: https://caselaw.findlaw.com/court/us-federal-circuit/1362587.html]. In the UK the Intellectual Property Office refused two applications in 2008 as contrary to accepted physical law; on appeal, Floyd J allowed the appeal and remitted the cases with a new test (whether there is "a reasonable prospect that on a full investigation with the benefit of expert evidence" the theory will turn out valid) [web: https://caselaw.nationalarchives.gov.uk/ewhc/pat/2008/2763]; a search summary says the Hearing Officer then refused again (O/170/09; text not opened) [read: arbiter.md, web check 6].
• Mills's patent record. About 115 patent families, including medical patents from before the hydrino work, the 1989 "Energy emission process", "antigravitational force" (1989, 1994) and "Fifth-force apparatus and method for propulsion" (2007), plus a 2025 "Reactionless propulsion" paper. These are the inventor's own statements of what his principles are for; a patent is not a test [read: research/2026-10-09-google-patents.md, via arbiter.md, QE revision 6 and 8].
• What the inventor said in the press. Scientific American (2014) reports Mills saying hydrinos had not yet been isolated [read: research/2026-10-09-wikipedia.md, a pointer only; the article itself not read]; the book says "Hydrinos have been isolated in the laboratory" (p.83). The Harvard Crimson (2000) reports "over $25 million" from investors, a board including a former Nuclear Regulatory Commission official (Shelby Brewer), Vice Admiral Kalleres and Aris Melissaratos (Westinghouse), Mills's claim of Navy research trials (no document in the project folder), critics Howard Georgi and Steven Chu, and law-firm letters to four physicists [read: research/2026-10-09-press-crimson-crossref.md, via arbiter.md, QE revision 8]. Rowan University: the Jansson 1997 thesis was done under a HydroCatalysis service contract (the earliest academic calorimetry for the company) [read: research/2026-10-09-rowan-digital-works.md]. New Jersey Economic Development Authority grant of about $1.1M (2013) [read: research/2026-10-09-wikipedia.md].

Absence of evidence

About 35 years of company existence, "over $150 million" raised by the company's own 2025 statement (Form D filings show $18.3M sold by 2018), validation reports and a 100 kW prototype claimed since 2020, but no product sale found, no independent demonstration with an outsider's own instruments, and the reports of the "13 laboratories" the company lists (Atomic Energy of Canada Limited (AECL) at Chalk River, Westinghouse, MIT Lincoln Laboratory, the Idaho National Engineering Laboratory (INEL), Hokkaido, Moscow Power Engineering Institute, Brookhaven, Lehigh, Harvard CfA) not found [read: arbiter.md, Absence-of-evidence analysis].

The book's own count, and the record beside it. The Introduction says the hydrino states have been "confirmed by the data presented in over 100 published journal articles and over 50 independent test reports and articles", citing only the company's website (ref 22) [read: p.37]. That is the inventor's and company's characterization. Beside it: of the evidence references 23-42, Mills is an author of 19 of 21; the one single-author validation (Bykanov) is a file on the company's server; no laboratory or test-report author is named in pp.1-86, and no negative result is mentioned; the independent datum with a declared absence of company funding is the Hagen EPR measurement on a company-supplied sample [read: arbiter.md, QE revision 3 and 8]. Neither statement is the neutral fact without the other.

The arbiter sets out five explanations: 1. A real effect that is hard to engineer. For: repeat investors over decades; outside measurements (NASA, Glumac, Hagen) the measurers did not explain. Against: no sale or independent run in 35 years; the NASA authors named recombination; EarthTech negative; the theory's numbers are still moving (p limit 137 to 126; EPR statistic 0.097 G to +/-0.55 G). 2. Trade secrecy. For: heavily redacted reports. Against: the company publishes the book and invites academia "to repeat and extend"; a sample test under a non-disclosure agreement would not reveal the process. 3. Renaming and absorption. Explains the trail of company names, not the silence. 4. Artifacts (recombination, electrode ions, calorimetry error, ordinary paramagnetic centres). For: NASA's caveat, EarthTech, Phelps-Clementson, the conventional EPR fit, interested-party calorimetry. Against: anomalies across several methods and labs, and no artifact yet shown for each. 5. Institutional refusal to test. For: the USPTO withdrawal, decided by people who had not reviewed the application before the notice of allowance; Rathke's own remark that discussion had been "restricted to the sweeping statement that the hydrino model is in contradiction to quantum mechanics". Against: NASA, NIAC and several universities did measure. The arbiter's weighing: 1 and 4 remain open, 2 is plausible for details, 3 and 5 explain parts of the trail; the silence lowers the odds of 1 but is not proof of 4. The deciding record is one neutral laboratory running the claimed signatures on a sample it made itself, with a hydrogen-free control [read: arbiter.md].

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7. What a learner should take away

What is solid. - The classical collapse problem is real, and so is the answer that certain extended charge-current distributions do not radiate. Mills's static spinning shell is a legitimate non-radiating object. - For one electron, force balance plus hbar reproduces Bohr's energies, and Mills's relativistic formula is Dirac's. Getting right numbers from a model does not show the model is right when it is algebraically the same as a known theory. - Schwinger's alpha/(2 pi) is in the formula exactly; the virial-theorem bookkeeping of 13.6 eV and 27.2 eV is correct; resonant non-radiative energy transfer is real. - The book's arithmetic reproduces almost everywhere it was checked (exceptions found in the deep reads: some Ch.11 table entries not computed in the text, the positronium level constant, the fifth-force threshold). - Some numbers agree with experiment because they are standard physics: the perihelion and light-bending tables are GR's numbers; the speed of gravity equals c, as GW170817 confirmed; the statistical-mechanics results of Ch.24 are the textbook ones; the von Klitzing constant h/e^2 is correct. - The helium formulas, with no fitted continuous parameter, give S-series quantum defects close to observed (0.144 and 0.289 against about 0.145 and 0.305): a genuine, if retrospective, point in favour.

What is wrong, and why the mistake is instructive. - Precision tests need precise inputs. The g-factor "eleven-figure agreement" was real arithmetic, but the alpha of 1973 could not distinguish a formula 3 x 10^-10 off from a correct one. Always ask how much the inputs' uncertainty moves the prediction (Part C of example.py). When alpha improved 10,000-fold, the formula failed by about 2,000 standard deviations. - Errors can cancel. Two wrong coefficients (-1/3 instead of -0.3285; pi^2/3 instead of 1.181) nearly cancel. A good total does not certify each term. - An equation that holds for every value does not select any value. The hydrino force balance is satisfied by any Z_eff; the integers come from added postulates. When a derivation "produces" integers, find the line where they enter. - Circular inputs. hbar in Eq. 1.35 returns as hbar in Eq. 1.37; the muon g formula takes the measured muon g as input. - Stationary means static. The argument that quantum orbitals "must radiate" assumes a time-varying density that a stationary state does not have. - Look-elsewhere. A dense ladder of possible energies (p up to 137, several m) will match some observed line; a match counts only if the line list was fixed in advance. - A prediction kept in print after the decisive observation. The book says gravity has "no transverse time-dependent gravity wave effect in the far field", citing the then-current absence of a detection (LIGO 2009, Shawhan 2010) as support (p.1642, p.1678). LIGO detected gravitational waves in September 2015 and in 2017 saw one arrive with light. The project copy is dated December 2018, and p.1642 cites 2017 papers, so the page was edited after the detection and the sentence was left in [read: arbiter.md, QE revision 6; web: https://arxiv.org/abs/1602.03837]. The same pattern appears with the Higgs boson: "will not be observed", then, after 2012, a reinterpretation of the observed particle. The lesson is general: "not yet observed" is weak support, and a theory earns credit by naming the observation that would refute it and then reporting the result when it comes. (Whether the 2025 edition changed the passage has not yet been checked.) - Reproducing the metric is not deriving it. Getting the Schwarzschild metric and GR's perihelion and light-bending numbers shows agreement with GR, not an independent theory, when the decisive step (the reciprocal radial factor) is asserted rather than derived. - Cite what the experiment measured. The KEK paper cited as confirming the free-electron disc measures the running of the electromagnetic coupling; the Witteborn-Fairbank experiment set an upper limit, not a zero. Read the cited paper before counting it as confirmation.

What is genuinely open. - Whether the laboratory anomalies (the Delft EPR spectrum, the DSC exotherms, the NASA 1996 excess heat, the EUV continua) have ordinary causes. No neutral lab has produced its own sample, run a hydrogen-free control, and measured the predicted hydrino signatures (EPR, Raman lines near 1950 cm^-1 with 16 times the H2 rotational spacing, nuclear magnetic resonance (NMR) shifts of the hydride). - The muon-to-electron mass ratio matching to 1.6 x 10^-8 after correction factors justified in words: striking, retrospective, and untested prospectively (would the same rules predict the tau ratio blind?). - What the November 1995 edition actually said; what the "outside inquiry" at the USPTO was; what the 13 listed laboratories found; what IIT Kanpur will report. - Whether Mills's molecular scheme, with all its rules frozen, can predict molecules it was not built on (a pre-registered blind test). The same test applies to the helium formulas on lithium-like and beryllium-like ions, and to the Tc formula on a blind list of materials with the dimension rule stated first. - Whether the fast-hydrogen Balmer broadening has an ordinary cause (Loureiro-Amorim) or not; settled by a neutral laboratory measuring broadening with and without the catalyst. - Whether a free electron falls like a bound one: an electron free-fall test with the shielding field removed or independently measured. The book's claim rests on a real, disputed 1967 experiment, and its patents propose propulsion from it. - Hydrino-catalysed fusion: whether any of the company's runs show neutrons, tritium or helium-3 above background.

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8. Check yourself

1. Mills's force balance for a hydrino is hbar^2 / (m_e r^3) = Z_eff e^2 / (4 pi eps0 r^2). Show that r = a_H / Z_eff solves it for any Z_eff. What does this tell you about where the integer p comes from?
2. If hydrogen went from n = 1 to n = 1/3 with a catalyst taking m x 27.2 eV, how much energy is released in total, how much goes to the catalyst, and at what wavelength is the continuum edge?
3. In 1987 alpha^-1 was 137.03604 +/- 0.00011. Estimate how much that uncertainty moves g/2, using g/2 - 1 approximately alpha/(2 pi). Could the 1987 comparison have detected a discrepancy of 3 x 10^-10?
4. Why does a hydrogen atom in a stationary quantum state not radiate, even though Mills's Ch.40 shows the Fourier transform of a smooth radial density is non-zero on the light cone?
5. The book's Table 5.2 lists ionisation-energy sums close to multiples of 27.2 eV, with a median miss of 0.58 eV. What extra piece of information would you need before counting the table as evidence for resonant catalysis?

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9. Further reading

Mainstream. - D. J. Griffiths, Introduction to Electrodynamics: Ch. 2 (electrostatics), Ch. 5 (magnetostatics), Ch. 11 (radiation, Larmor formula). - D. J. Griffiths, Introduction to Quantum Mechanics: Ch. 4 (hydrogen atom, angular momentum, spin). - R. Eisberg and R. Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles (Bohr model, spin, magnetic moments). - J. Schwinger, "On quantum-electrodynamics and the magnetic moment of the electron", Phys. Rev. 73, 416 (1948) [memory, high]. - X. Fan, T. G. Myers, B. A. D. Sukra, G. Gabrielse, "Measurement of the electron magnetic moment", Phys. Rev. Lett. 130, 071801 (2023) [web: via physicist.md]. - L. Morel et al., "Determination of the fine-structure constant with an accuracy of 81 parts per trillion", Nature 588, 61 (2020) [memory, high; result web-checked via CNRS link above]. - A. J. Devaney and E. Wolf, "Radiating and nonradiating classical current distributions and the fields they generate", Phys. Rev. D 8, 1044 (1973) [memory, medium]; G. H. Goedecke, Phys. Rev. 135, B281 (1964) [memory, medium]. - N. Dombey, "The hydrino and other unlikely states", arXiv:physics/0608095 (2006) [read: research/files]. - A. Rathke, "A critical analysis of the hydrino model", New J. Phys. 7, 127 (2005) [read: research/files]. - J. Naudts, "On the hydrino state of the relativistic hydrogen atom", arXiv:physics/0507193 (2005) [read: research/files]. - B. Hensen et al., "Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres", Nature 526, 682 (2015) [web: https://arxiv.org/abs/1508.05949]. - A. V. Phelps and J. Clementson, Eur. Phys. J. D 66, 120 (2012), on the EUV continua [web: https://link.springer.com/article/10.1140/epjd/e2012-30114-5]. - J. R. Taylor, An Introduction to Error Analysis (propagation of uncertainties). - B. F. Schutz, A First Course in General Relativity (metrics, the Schwarzschild solution, gravitational waves, at an undergraduate level) [memory, high]. - B. P. Abbott et al. (LIGO and Virgo), "Observation of gravitational waves from a binary black hole merger", Phys. Rev. Lett. 116, 061102 (2016) [web: https://arxiv.org/abs/1602.03837; journal reference memory, high]. - B. P. Abbott et al., "Gravitational waves and gamma-rays from a binary neutron star merger: GW170817 and GRB 170817A", Astrophys. J. Lett. 848, L13 (2017) [web: https://arxiv.org/abs/1710.05834; journal reference memory, high]. - ATLAS and CMS, "Combined measurement of the Higgs boson mass in pp collisions at sqrt(s) = 7 and 8 TeV", Phys. Rev. Lett. 114, 191803 (2015) [web: https://arxiv.org/abs/1503.07589; journal reference memory, high]. - F. C. Witteborn and W. M. Fairbank, "Experimental comparison of the gravitational force on freely falling electrons and metallic electrons", Phys. Rev. Lett. 19, 1049 (1967) [web: Crossref, DOI 10.1103/physrevlett.19.1049]. - L. L. Loureiro and J. Amorim, "Anomalous Doppler broadening caused by exothermic reactions: application to hydrogen Balmer lines", Eur. Phys. J. D (2011) [web: Crossref, DOI 10.1140/epjd/e2011-20028-1]. - G. F. Lewis and B. J. Brewer, "Detection of the cosmological time dilation of high-redshift quasars", Nature Astronomy (2023) [web: Crossref, DOI 10.1038/s41550-023-02029-2; author initials memory, medium].

The inventor's own key works. - R. L. Mills, The Grand Unified Theory of Classical Physics, December 2018 edition (the project copy); February 2025 edition at brilliantlightpower.com [read: sites.md]. - W. R. Hagen and R. L. Mills, "Electron paramagnetic resonance proof for the existence of molecular hydrino", Int. J. Hydrogen Energy 47 (2022) 23751 [read: research/files]. - R. L. Mills, 2008 reply to Rathke (165 pages) [read in part: research/files/Millsian-Mills-response-to-Rathke-2008.pdf]. - Mills and Hagen, 2025 FeOOH hydrino paper (p <= 126) [read: research/files/BLP-FeOOH_Hydrino_Paper.pdf].

Independent tests of the earlier cell. - J. M. Niedra et al., NASA TM-107167 (1996) [read: research/files]. - EarthTech International, Mills replication report (1997) [web: https://earthtech.org/cold-fusion/mills/].

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Corrections to earlier documents

1. The "about 7 x 10^-9" band for the 1987 g-factor test (physicist.md Claim 1b; mathematician.md 1b; contrarian.md Claim 1b; arbiter.md recomputation table, row "What the 1987 comparison could resolve"). The arbiter could not reproduce 7 x 10^-9 from alpha's uncertainty and did not repeat it. The source is found: the deep reader propagated the uncertainty Mills prints on his own computed alpha in Eq. 1.235 (137.03603 +/- 0.00082, relative 6.0 x 10^-6), which gives +/- 6.9 x 10^-9 [read: deep read Ch.1 2.3, finding 2; computed]. The uncertainty of the alpha the book actually uses for Eq. 1.229 (CODATA 1973, the Committee on Data for Science and Technology recommended value, 137.03604 +/- 0.00011, relative 8.0 x 10^-7) gives +/- 9.3 x 10^-10 [computed]. The physicist's wording "alpha was known to 1e-8 then" is not right for that value (it was known to 8 x 10^-7). Both bands are larger than the 2.9 x 10^-10 discrepancy, so the arbiter's conclusion (the old test could not resolve it) stands. Suggested fix: wherever "about 7e-9" is quoted, say "+/- 9.3e-10 with the 1973 alpha the book uses (+/- 6.9e-9 with the uncertainty the book prints for its own Eq. 1.235)".
2. Standard deviations of the modern g miss. Reviewers wrote "about 2,000"; the arbiter "about 2,200" (measurement uncertainty only). Including alpha's own uncertainty gives about 1,800 (Rb) or 1,100 (Cs) [computed]. All are correct for their stated denominators; this primer gives all three.
3. UK patent appeal. As the arbiter found, physicist.md's "appeal dismissed" is wrong: Floyd J allowed the appeal and remitted [web: https://caselaw.nationalarchives.gov.uk/ewhc/pat/2008/2763]. This primer uses the corrected wording.

Corrections to this primer's own earlier version (QE revision, 2026-10-09; see qe.md and arbiter.md "QE revision"):

4. Section 5, step 5. The earlier text said QED "through order 2" is "1.6 x 10^-8 too low". example.py (Part E) prints -1.475 x 10^-8, so it is 1.5 x 10^-8. Fixed.
5. Section 1, Mills's training. It was tagged [memory, medium]. The book itself states it (p.1875, also p.439 and p.1518); it is now tagged [read] and marked as the publisher's description.
6. Section 6, rows 6f and 7. Row 6f said the general-relativity part was "Not read / Not assessed", and row 7 gave a provisional, digest-only verdict. Both chapters have now been deep-read; the rows are replaced by the arbiter's QE-revision verdicts (rows 6f-6m, 7a-7f). The earlier "factor 25" for the superconductor dimension choice (arbiter's first pass) is exactly 27.
7. USPTO quote. The earlier text quoted "no one involved in the withdrawal had reviewed the application before", which can be misread as "ever". The court's full words are "...had reviewed the '294 patent application before issuance of the Notice" [read: research/2026-10-09-findlaw.md]. Fixed.
8. NASA TM-107167. The earlier text gave only "recombination could not be excluded", which understates the authors' own weighing (recombination would need "quite a high" efficiency, which the data "marginally admit"). Fixed in the experimental record and row 2e.
9. Omissions restored. The earlier version left out pertinent material on both sides: the book's statements that gravitational waves do not exist and that the Higgs "will not be observed" (both against the book), the free-electron gravity and fifth-force claims with the Witteborn-Fairbank experiment, hydrino-catalysed fusion, the fast-hydrogen evidence class, the Hagen-Mills 2024 broad-line EPR, the Maris point on helium mobility peaks, the book's "over 50 independent test reports" statement, the 2014 Scientific American remark, and the patent record. All are now in sections 4, 6 and 7.
10. Loureiro-Amorim date. sites.md and arbiter.md's first pass give 2010; Crossref lists publication on 28 June 2011 (Eur. Phys. J. D) [web, via arbiter.md, QE revision 0.1]. This primer uses 2011; an earlier online date has not been checked.

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Answers

1. Substitute r = a_H / Z_eff with a_H = 4 pi eps0 hbar^2 / (m_e e^2). Left side: hbar^2 Z_eff^3 / (m_e a_H^3). Right side: Z_eff e^2 Z_eff^2 / (4 pi eps0 a_H^2) = Z_eff^3 e^2 / (4 pi eps0 a_H^2). Both carry Z_eff^3, which cancels, leaving hbar^2 / (m_e a_H) = e^2 / (4 pi eps0), the definition of a_H. So the balance holds for every Z_eff, integer or not. The integer p must come from somewhere else: in the book, a separate "trapped photon" postulate (Eq. 5.27).
2. n = 1/3 means p = 3, so m = 2. Released: (3^2 - 1^2) x 13.6 = 108.8 eV. To the catalyst: 2 x 27.2 = 54.4 eV. Left for emission: 54.4 eV, edge at 1239.84 / 54.4 = 22.8 nm.
3. Relative error 0.00011 / 137.036 = 8.0 x 10^-7. g/2 moves by about (alpha/(2 pi)) x 8.0 x 10^-7 = 0.00116 x 8.0 x 10^-7 = 9.3 x 10^-10. A 3 x 10^-10 discrepancy is a third of that band: the 1987 comparison could not detect it.
4. For a stationary state psi_n(r) e^(-i E_n t / hbar), the density |psi|^2 has no time dependence (the phase factor cancels), and the probability current is steady. A static charge and a steady current do not radiate. Mills's Fourier argument multiplies the spatial transform by a time factor at frequency omega_n, assuming the density oscillates; that premise is what fails. Radiation in quantum mechanics comes only from a mixture of two states, during a transition.
5. How many candidate sums were considered for each element (how many electrons t were allowed), and what tolerance counts as "resonant". Any number lies within 13.6 eV of a multiple of 27.2 eV; a random one is within 0.58 eV 4.3 percent of the time. With about 16 candidates per element, a hit that close is a coin flip, so without the count the table cannot support or refute the resonance claim.