Dunning-Davies and Norman: Mathematician's review (PhD mathematician)

Scope: for each of the seven central claims in claims.md I answer (a) is the mathematics internally consistent, (b) is it consistent with established experiments, (c) has it been tested, (d) what would have to change in mainstream physics. Then a verdict from the project scale.

Source gap (repeated from Step 1). The 2024 Hadronic Journal paper and the CIA memo are scans with no text layer, and the appendices of "Gravity v.2" were only skimmed. My mathematical conclusions below rest on the text-layer documents I read myself: the 2020 preprint "Deductions from the Quaternion Form of Maxwell's Electromagnetic Equations" (full text, every equation) and "Consequences of the Existence of an Aether" (2018, the scalar-wave and D/Dt sections), plus the front and Carstoiu sections of "Gravity v.2". Where a conclusion rests on [memory] alone I say so.

Source tags: [read] from the documents; [web] with URL; [memory] with confidence; [calc] my own derivation (SymPy used for the gauge check).

0. The mathematics, checked once (used by claims 1 and 2)

Notation (Gaussian-style units as in the paper): phi is the scalar potential, A the vector potential, c the speed of light, E = -grad phi - (1/c) dA/dt, B = curl A. Subscript t means partial time derivative. "lap" is the Laplacian (sum of second spatial derivatives). A gauge transformation changes the potentials but not E or B: A -> A + grad L, phi -> phi - (1/c) L_t, for any function L(t,x,y,z).

The paper [read, eqs. 1-8] defines T = -(1/c) phi_t + div A (eq. 1) and postulates

• (3) curl B = (1/c) E_t + grad T
• (5) div E = (1/c) T_t
• the Lorenz-type condition (7): div A = -(1/c) phi_t, which gives T = 2 div A.

Three checks:

1. T is not gauge-invariant. [calc, SymPy] Under the gauge transformation above, T changes by lap L + (1/c^2) L_tt. E and B do not change. So T is not a property of the electromagnetic field; it is a property of the arbitrary potential choice. (For comparison, the standard Lorenz combination is (1/c) phi_t + div A, which changes by lap L - (1/c^2) L_tt, the wave operator; the paper's T has the opposite relative sign and so does not even give the Lorenz expression.) Jack's own abstract, as listed on arXiv, speaks of a component that "partially constrains gauge freedom" [web: https://arxiv.org/abs/math-ph/0307038], which is consistent with T being a gauge-fixing quantity, not new physics. Setting T = 0 gives the standard Lorenz gauge (up to the sign issue) and ordinary Maxwell. [calc plus web]

2. The modified equations make T obey an elliptic equation, not a wave equation. Take the divergence of (3). The divergence of a curl is zero, so 0 = (1/c)(div E)_t + lap T. Substitute (5): 0 = (1/c^2) T_tt + lap T. The wave equation has a minus sign: lap T - (1/c^2) T_tt = 0. With a plus sign this is Laplace's equation in the variables (ct, x, y, z). Such equations have no travelling-wave solutions and are ill-posed as initial-value problems. Physically: (3) and (5) read like Maxwell with an effective charge density T_t/c and current density grad T, and charge conservation (continuity) then requires T_tt/c^2 + lap T = 0, so the sign convention chosen is inconsistent with conservation. [calc]

3. The c*sqrt(3) wave does not satisfy the paper's own equations. The paper takes A = grad S, T = 2 lap S, and gets lap S - S_tt/(3c^2) = 0 by dropping a "constant of integration" (they integrate grad of a quantity and set the function of time to zero). Granting that, S_tt = 3c^2 lap S. Now insert T = 2 lap S into the elliptic condition of check 2: T_tt/c^2 + lap T = 2 lap( S_tt/c^2 + lap S ) = 2 lap(3 lap S + lap S) = 8 lap lap S. This must vanish, so lap S is itself harmonic (lap lap S = 0). A harmonic lap S then gives T_tt = -c^2 lap T = 0, so T is at most linear in time. There is no oscillating, travelling "csqrt(3)" solution. Separately, condition (7) with the memo's choice phi = -(1/c) S_t demands lap S = S_tt/c^2, which contradicts S_tt = 3c^2 lap S unless both sides vanish. Two different wave equations for the same S (speed c and speed csqrt(3)) cannot both hold. [calc; the second conflict was also found in Step 1 and I confirm it.]

Worked number: take S = f(x - 1.732 c t), a plane wave moving at c*sqrt(3). Then lap S = f'' and S_tt = 3c^2 f'', so the paper's equation is satisfied. But T = 2 f'', T_tt/c^2 + lap T = 3 (2 f'''') + 2 f'''' = 8 f''''. For f = sin(kx - ...) this is 8 k^4 sin(...), not zero. So eq. (3)/(5) are violated by the very wave the paper derives. [calc]

Claim 1. A seventh scalar field T exists

a. Mathematics. Not internally consistent as a field theory. T depends on the gauge (check 1), and the extended equations force T to satisfy a Laplace-type equation (check 2). What is true: if you pick a quaternion-style operator d/dr = (1/c) d/dt + (i d/dx + j d/dy + k d/dz), the scalar part of the product with the potential is a real, computable quantity; but it is the gauge-fixing quantity, which any textbook can set to zero or to any prescribed function. The statement "Maxwell's equations were originally written in quaternions and Heaviside's reduction lost a field" [read] is historically fair as to notation; Heaviside and Gibbs dropped the quaternion scalar part because it carried no independent physics [memory: medium; consistent with Jack's note that it constrains gauge].

b. Experiments. The standard equations reproduce every EM measurement from static fields to optics and radio to extraordinary precision (for example the electron g-factor to parts per trillion, [memory: high]). No experiment requires an extra scalar component. Gauge dependence means T cannot be measured as a field at all.

c. Tested? The authors cite no experiment measuring T. Associated claims (T tied to temperature, charge, "negative time", heat extraction) are asserted without data [read]. No test on record.

d. If true. Maxwell's equations would acquire a new source-like term, charge conservation would need to be re-derived, and gauge invariance of electromagnetism (the foundation of QED) would be broken. The photon would need a longitudinal/scalar polarisation that is observable. Note that in QED the scalar and longitudinal polarisations exist but are unphysical (cancel by the Ward identity) [memory: high].

Verdict. Split. Dynamical form as written (eqs 3, 5 with T from eq. 1): Internally inconsistent (gauge-dependent T, elliptic constraint). Strong form "a new physical, measurable seventh field": Not testable as stated (T is a gauge artefact; no measurable difference is specified). Weak form "quaternion notation can be used to rewrite Maxwell": Established (not contested).

Claim 2. Scalar (longitudinal) waves exist, carry energy, and travel at c*sqrt(3)

a. Mathematics. Three separate sub-claims. - Existence with E = B = 0: with A = grad S and phi = -(1/c) S_t, E = grad(S_t/c) - grad(S_t/c) = 0 and B = curl grad S = 0, for any function S [calc; the memo gives the same, [read: page image, Step 1]]. This is a pure gauge configuration: it is the vacuum written in a different gauge. The wave equation for S arises only if one imposes Lorenz gauge; it is a statement about the gauge function, not a field that propagates anything physical. Worked example: S = sin(kx - w t), E = 0, B = 0 identically. All the energy density (E^2 + B^2)/8pi and Poynting vector (c/4pi) E x B are zero. - Energy carried: the electromagnetic energy-momentum is built from E and B only; with E = B = 0 it is exactly zero. The 2018 paper says the wave implies "movement of energy in the aether in the direction of the disturbance" [read], while in the same paragraph conceding that the Poynting theorem implies no power flow. The memo itself says only that its author could not see where energy would come from and that its existence is "not proven" [read: Step 1]. The authors attribute to the memo more than it says. Refuted by calculation. - Speed c*sqrt(3): inconsistent with the paper's own equations (check 3). Also, the equation lap S - (1/c^2)S_tt = 0 in the memo and the equation lap S - (1/(3c^2))S_tt = 0 in the paper contradict each other.

b. Experiments. Special relativity's limit on signal speed has been tested in countless ways; no confirmed superluminal signal exists [memory: high]. The related Aharonov-Bohm effect shows potentials matter quantum mechanically but only through gauge-invariant loop integrals [memory: high]; pure-gauge potentials have no effect.

c. Tested? The cited tests are Sweet and Bearden's device claims, quoted from an abstract [read]. I could not find or read the device data; no independent replication is cited [read]. Not tested in any way that bears on the c*sqrt(3) claim.

d. If true. A faster-than-light signal would break causality in special relativity (a signal faster than c in one frame travels backwards in time in another). Energy in a zero-field wave would require a new energy-momentum tensor.

Verdict. Split. "Scalar potential wave with E = B = 0 exists": true but trivial, it is Established as a mathematical gauge artefact with zero physical content. "Carries energy": Refuted by calculation (energy density and Poynting vector are exactly zero). "Travels at csqrt(3)": Internally inconsistent*.

Claim 3. A physical aether exists; replace d/dt by D/Dt

a. Mathematics. The substitution D/Dt = d/dt + u.grad (u being the aether velocity) [read, quoted from Thornhill] is a legitimate move for a fluid, but the resulting Maxwell-type equations are not Lorentz covariant, and the equations Dunning-Davies and Norman present are not shown to be consistent with Faraday induction or charge conservation when u varies with position (the curl of the convective term picks up extra terms that are not discussed). They then say that "away from mass" the medium is stationary so D/Dt returns to d/dt, and "this permits near instantaneous longitudinal wave propagation" [read]. That does not follow: if D/Dt reduces to d/dt then the equation is the ordinary wave equation with speed c, which is what the 2018 paper writes down. No equation in the documents derives an instantaneous speed. The conclusion is asserted. The aether "particle mass 0.497e-39 kg" quoted from Thornhill [read: page image, Step 1] has no derivation I could check. Not internally consistent as a derivation.

b. Experiments. Michelson-Morley (1887) and later improved versions found no aether wind; the Kennedy-Thorndike and Ives-Stilwell tests, and the invariance of c measured by modern optical-cavity experiments, are consistent with Lorentz invariance [memory: high]. A gas-like aether would also give transverse and longitudinal sound-type waves; no longitudinal EM mode in free space has been detected [memory: high].

c. Tested? The aether hypothesis was tested thoroughly in 1887 to the present; the specific "gas-like aether with D/Dt" model (Thornhill 1984) was published in Speculations in Science and Technology [read: cited in the paper; I did not verify beyond that]. No test of its distinctive predictions on record.

d. If true. Special relativity would be valid only in the aether frame, so GPS timing, particle accelerator kinematics and the Lorentz-invariance bounds would need a conspiracy of cancellations.

Verdict. Aether as a physical medium defining a preferred frame: Tested, not supported (Michelson-Morley and successors [memory: high]; a reasonable position in the 19th century, so also instructive as Historically superseded). D/Dt "near-instantaneous waves" step: Internally inconsistent (the stated reduction gives speed c, not infinity). The claim that relativity is "wrong or only valid in vacuum": Tested, not supported.

Claim 4. Gravity is a superluminal longitudinal wave coupling to EM; antigravity and over-unity achieved

a. Mathematics. - Whittaker (1903) [read, quoted at length]: the static 1/r potential can be written as an integral of spherical waves: integral from 0 to infinity of (2/pi)(dmu/mu) sin(mu(Vt - r))/r = 1/r once Vt - r > 0 [read]. This is correct: the integral of (2/pi) sin(y)/y dy from 0 to infinity is 1. But it is a Fourier decomposition of a static field. It does not specify V, and Whittaker himself says V "need not be the same as that of light" and "may be enormously greater" [read]. The decomposition is mathematics, not evidence of a speed. The authors then use it to support "superluminal gravity" without a derivation of V. - The Carstoiu/Brillouin equations quoted in "Gravity v.2" [read]: curl F = -Omega_t, curl Omega = (1/c^2) F_t - (G/c^2) J_g, div F = -G rho_g, div Omega = 0. Units check [calc]: F in m/s^2, Omega in 1/s, G/c^2 in m/kg, J_g in kg/(m^2 s); every term in the curl Omega equation has units 1/(m s). So they balance (up to a missing factor 4 pi in div F = -G rho_g, a convention). These equations carry waves at c, and Brillouin says so: "both types of wave propagate with the same speed" and calls them transverse [read]. So the appendix of the same document contradicts the main text's superluminal longitudinal gravity. - Van Flandern's argument that a delay "destroys angular momentum conservation" [read]: this is the aberration argument. Carlip (2000) showed that in general relativity, velocity-dependent terms cancel the leading aberration, so finite c_g = c is consistent with stable orbits [web: https://arxiv.org/abs/gr-qc/9909087]. - Sweet-Bearden numbers: the abstract quotes "open loop power gains of 5 x 10^4 to 1.5 x 10^6" and in the same breath "signal wave input of 175 microwatts and an output of 1 kilowatt" [read]. 1000 W / 175e-6 W = 5.7e6, above the stated maximum gain by a factor of about 4. The quoted numbers disagree with each other. The word "open loop" with a battery input means the gain compares a small control signal with a battery-fed output, not a free-energy gain, a point the authors do not discuss. [calc]

b. Experiments. GW170817 and its gamma-ray burst GRB 170817A bound the difference between the speed of gravity and light to between -3e-15 and +7e-16 of c [web: https://arxiv.org/abs/1710.05834]. Gravitational lensing is achromatic to measurement precision in GR [memory: high; the "Einstein rings are blue" quote concerns source colour, not diffraction, [memory: medium, not web-checked]]. Orbital decay of the Hulse-Taylor pulsar matches GR's radiation at c to better than 1% [memory: high]. Laithwaite's gyroscope "weight loss" has been explained by ordinary precession mechanics; I did not check a source in this review [memory: medium].

c. Tested? Over-unity and antigravity devices: no independent controlled replication is cited, and none is known to me; the Sweet-Bearden material is a conference abstract [read]. Schauberger's turbines: anecdotal [read]. Brown's electrogravitic effects: later analyses (ion wind) are in the literature [memory: medium; not web-checked]. The proposed experiments "demonstrating over-unity" in the paper [read] had not been done in the documents.

d. If true. General relativity's causal structure and energy conservation (Noether's theorem for time translation) would need to be abandoned, and the GW170817 observation would need to be a coincidence.

Verdict. Split. - "Gravity propagates much faster than c": Tested, not supported (GW170817 [web]). In addition the project's own Carstoiu appendix gives speed c: Internally inconsistent across the documents. - "Lensing is refraction, no dark matter": Tested, not supported for the chromatic-refraction claim (lensing achromatic) [memory: high]; dark matter evidence is wider than lensing, [memory: high]. - "Over-unity gains achieved": Refuted by calculation if the gain is net energy (conservation of energy); the numbers quoted are themselves inconsistent (5.7e6 versus the stated maximum 1.5e6) [calc]. As claimed device results: Never properly tested (no neutral replication on record; not dismissed without testing on the evidence available to me, but also never presented with data in these documents). - "Antigravity (90% weight loss)": Never properly tested.

Claim 5. Quantum and nuclear phenomena have a causal aether basis

a. Mathematics. The 2018 paper offers words, not equations, for tunnelling, the Coulomb barrier and the strong force [read]. No potential, barrier transmission formula or numerical prediction (for example the tunnel-diode I-V curve, or the Gamow factor) is derived from scalar waves. Consequently it cannot be checked, and it is not shown to agree with the standard result T ~ exp(-2 kappa L) for tunnelling, which does match diode and alpha-decay data.

A calculation that does test an assertion: the neutron as a proton-electron bound state. The mass difference m_n - m_p - m_e = 939.565 - 938.272 - 0.511 = 0.782 MeV [calc from standard masses, [memory: high]]. This is positive, so a p-e state would have mass larger than the constituents: it would be anti-bound (needs input energy), not bound. The authors call 0.782 MeV a "positive binding energy" [read], a wording problem: standard binding energy is negative mass defect. Further, confining an electron to a nucleon-size region of radius r = 0.84 fm requires, by the uncertainty relation, a momentum of about hbar c / r = 197.3 MeV fm / 0.84 fm = about 235 MeV/c, hundreds of times the 0.78 MeV available [calc, using standard quantum mechanics]. Santilli's hadronic mechanics avoids this by changing quantum mechanics itself, so this is a refutation within standard quantum mechanics, not within the authors' framework.

The "Enigma that is Light" manuscript [read] argues that Bell's inequality mixes "abstract" and "concrete" objects. I find no mathematical derivation in the text that identifies a flaw in the CHSH algebra, which is a short, elementary proof from locality and realism. The proof stands independently of probabilistic interpretation.

b. Experiments. Tunnelling matches the Schrodinger equation (tunnel diodes, STM, alpha decay). The neutrino was detected directly (Cowan-Reines 1956) and neutrino oscillations are measured [memory: high]. The quark model is tested by deep inelastic scattering [memory: high]. The loophole-free Bell test (Hensen et al.) found S = 2.42 ± 0.20 against the local limit 2, p = 0.039 [web: https://arxiv.org/abs/1508.05949]; two other 2015 tests agree [memory: high]. Note the authors accept non-local superluminal effects, so a Bell violation is compatible with their stance; but their claim that the inequalities "do not rule out local-realistic explanations" is contradicted.

c. Tested? Santilli's hadronic claims are tested only by his own group and journal (Hadronic Journal, Hadronic Press) [memory: medium, not web-checked this round]. Mind-matter double-slit claims (Radin, Jahn): no accepted independent replication [memory: medium]. Unbuilt as quantitative predictions.

d. If true. Quantum mechanics's probabilistic structure, the quark and neutrino sectors of the Standard Model, and spin-statistics would all need replacing.

Verdict. Split. Tunnelling and strong-force "scalar-wave" accounts: Not testable as stated (no equations or numbers). Neutron as proton-electron bound state in standard quantum mechanics: Refuted by calculation (momentum scale 235 MeV/c versus 0.78 MeV). Local-hidden-variable reading of Bell: Tested, not supported (loophole-free Bell tests [web]). Mind-matter effects: Never properly tested here (no data in the documents).

Claim 6. Hadronic mechanics yields practical clean energy

a. Mathematics. The documents give claims, not derivations: resonant photons stimulating electron capture to neutralise radioactive waste [read: page images]. No cross-section, rate or energy balance is given in the pages I read. The proposed mechanism needs the neutron to be a p-e compound (see claim 5) and cannot be evaluated here. Hadronic mechanics itself is a non-unitary, non-linear generalisation of quantum mechanics; its internal consistency is outside what these documents show, and mathematically it is disputed (non-unitary evolution changes the meaning of expectation values) [memory: medium; not web-checked].

b. Experiments. Mainstream nuclear physics finds no accelerated decay of long-lived waste by photon resonance at accessible intensities. Reported claims of "PlasmaArcFlow reactors in industrial operation" [read: page image] are commercial statements with no independent data cited.

c. Tested? Independent testing: none cited, none known to me. I could not verify the commercial claims on the web this round (source budget used on the theoretical claims), so this paragraph rests on [read] plus [memory: medium].

d. If true. Nuclear waste could be burned without reactors; electron capture rates would be switchable by photon fields, contrary to the established weak-interaction rate calculations.

Verdict. Never properly tested (no neutral, controlled test on record that I could find; whether it was dismissed without testing I cannot establish from these documents). Conclusion rests partly on [memory] and the unread appendices.

Claim 7. Suppression: government and military already use this science

a. Mathematics. Not a mathematical claim. The mathematical content the authors say is suppressed is, by the checks above, not usable physics, so there is nothing to conceal.

b-c. Evidence cited. The CIA memo [read: page image] says scalar waves' "existence is not proven" and the source of energy is unclear. It is a 1987 analyst's note from a reading-room document released 2001. A memo that discusses a topic is not evidence that it works, and this one says the opposite of working. The other cited evidence (Bearden books, Winterhaven 1952 and Electrogravitics 1954 quotes) was not read in full.

d. If true. Secret programmes would need to rely on the mathematics in claims 1-2, which is inconsistent as shown.

Verdict. Not testable as stated (no checkable evidence; the cited primary document contradicts the claim). A sociological question outside mathematical review.

Corrections to Step 1

1. facts.json says the sign of T "differs from the standard Lorenz-gauge combination, which is why T does not vanish." Confirmed, but incomplete: the larger issue is that T is gauge-dependent (changes by lap L + L_tt/c^2) and that eqs (3) and (5) force T_tt/c^2 + lap T = 0 (elliptic). The Step 1 "needs adjudication" is resolved here: the c*sqrt(3) wave is inconsistent with the paper's own equations (Section 0, check 3).
2. facts.json (Gravity v.2) lists the Carstoiu/Brillouin appendix as supporting EM-gravity coupling; it does not mention that the appendix states the waves are transverse and travel at the same speed as light, which contradicts the main text's superluminal longitudinal gravity.
3. facts.json and claims.md reproduce the Sweet-Bearden "gains of 5e4 to 1.5e6" without noting that the same abstract gives 175 microwatts in and 1 kilowatt out, a gain of 5.7e6, so the quoted figures are mutually inconsistent.
4. facts.json states the GW170817 bound as "about 1e-15 [memory: medium]". Now [web]: -3e-15 to +7e-16 of c, https://arxiv.org/abs/1710.05834.
5. The Step 1 text calls 0.782 MeV a "binding energy" (following the paper). It is the mass excess of the neutron over proton plus electron, positive, so the p-e state would be anti-bound; calling it a binding energy is a misuse of the term.
6. facts.json for the 2020 preprint says the paper's eq. 7 is the Lorenz-type condition; confirmed. One note: the text layer of eq. 7 and later equations lost the time derivatives (garbled); I checked the page content against the derivation and the algebra is consistent only with phi_t in eq. 7.
7. claims.md claim 3 says special/general relativity "wrong or only valid in vacuum"; the 2020 preprint does not itself argue this, the "Enigma" manuscript and Gravity v.2 do. Minor attribution fix.

Web checks done (5 calls)

• https://arxiv.org/abs/math-ph/0307038 (Jack, abstract; confirmed author, title, gauge-constraint remark)
• https://arxiv.org/abs/1710.05834 (GW170817 speed bound)
• https://arxiv.org/abs/gr-qc/9909087 (Carlip, aberration; rebuts Van Flandern)
• https://arxiv.org/abs/1508.05949 (loophole-free Bell test)
• https://en.wikipedia.org/wiki/GW170817 (context only) Not web-checked, [memory] only: Michelson-Morley and successors (high), Cowan-Reines neutrino (high), Hulse-Taylor (high), Laithwaite and Brown explanations (medium), Hadronic Press status (medium), mind-matter replication status (medium).

Summary table