Mathematician review: Koen J. van Vlaenderen (Step 2)

Reviewer: PhD mathematician role. Date: 2026-10-04. Tags: [read] from the documents or the Step 1 deep reads (re-read by me where marked), [check] my own algebra or numerics (SymPy/NumPy, script in the session scratchpad, not a project file), [web] with URL, [memory] with confidence. No source gap: all three papers have a text layer. The 2015 paper pp.16-25 (experiments review, appendices) was not deep-read in Step 1; I read it myself from the extracted text [read: pp.16-23].

Notation, once: Phi = electric scalar potential (V); A = magnetic vector potential (V s/m); E = -grad Phi - dA/dt; B = curl A; eps0, mu0 = vacuum permittivity and permeability, c = 1/sqrt(eps0 mu0); rho, J = charge and current density; "Lorenz condition" = div A + eps0 mu0 dPhi/dt = 0; "gauge" = a change Phi -> Phi + dchi/dt, A -> A - grad chi that leaves E and B unchanged. S = -eps0 mu0 dPhi/dt - div A is the paper's "scalar field" (equal to the Lorenz expression, unit tesla). In the 2015 paper the same quantity splits as B_l = -div A (the "scalar magnetic field") and B_Phi = -dPhi/dt. "N3LM" = Newton's third law of motion.

What I verified (Step 1 is largely right)

• 2003 paper: with the potential wave equations imposed without a gauge condition, eqs (18)-(21), the energy theorem (33) and force theorem (34) hold identically. [check] I re-ran this with arbitrary Phi(x,y,z,t), A(x,y,z,t): residual 0 for (18), (19) and (33). The plane longitudinal wave S = S0 cos(kx - wt), E = -c S0 cos(kx - wt) x-hat, B = 0 satisfies (18)-(19) in vacuum, residual 0. [check]
• 2015 paper: the reciprocal kernel F(x,x') = (mu0/4pi r^3)[(J'.r)J + (J.r)J' - (J'.J) r] is antisymmetric under x <-> x'. The Grassmann kernel (J x B_t) is not. [read: 2015 p.5-6; check by hand]. Appendix A (eq 6.1) is correct: dl x (dl' x r) = dl'(dl.r) - r(dl.dl') and the term (dl.grad(1/r)) integrates to an end-point difference that vanishes only for a closed line [read: p.22].
• 2015 Rutherford "4/3" numbers: [check] for a charged shell of radius R, the momentum from eps0 E x B_t is (2/3) mu0 q^2 v/(4 pi R), the extra eps0 B_l E term is (1/3) of the same unit, the sum is mu0 q^2 v/(4 pi R), which equals eq 4.36. The shell field energy is q^2/(8 pi eps0 R), which is half of the paper's eq 4.35.

Corrections to Step 1

1. Step 1 (deep read of the 2015 paper, sec 4.6) correctly notes that eq 4.22 holds for any a given continuity. I add its consequence: claims.md claim 3 says the theory "requires ... rejection of the Lorenz condition". The paper's argument for that rejection (eq 4.22-4.26 and p.13) does not select a. The rejection comes only from requiring the force law J B_l be present (b = c from 4.20) plus the premise that mechanical forces alone must obey N3LM. The claims.md sentence should be read that way. [read: p.12-13; check]
2. Step 1 (2003 deep read, sec 8) leaves open whether the "static charge power source" rho S/eps0 is a net source. I settled it from the paper's own equations: see claim 2, section "Energy bookkeeping". It is an energy imbalance of size c^2 rho S (rest charge), not a conversion of existing field energy, because S^2/(2 mu0) is already counted in the field energy of (33).
3. Step 1 (claims.md, priority note and sites.md lead on Bruhn): Bruhn's first review says S is "completely superfluous" because any (E,B) built from potentials with S not zero can also be built from Lorenz-gauge potentials. That is correct for standard Maxwell theory ([check]: if Psi solves (1/c^2) Psi_tt - Lap Psi = -S, then A + grad Psi, Phi - Psi_t have S = 0 and the same E, B). It does not touch the modified theory, because in the modified theory the sources are not the same for the two gauge choices. His second review states (II) correctly: sources rho' = rho + eps0 dS/dt, J' = J - grad S/mu0 are conserved iff S satisfies the homogeneous wave equation; I confirmed this equals eq (24) of the 2003 paper. [read: research/files/TU-Darmstadt-Bruhn-*.html; check] Bruhn himself writes "Maybe that some reasons can be found for the validity of condition (II)". So the two sides agree on the mathematics and differ on interpretation; the author's reply (S is intended as a modification) is on file. I would not call Bruhn's first review a refutation of the modified theory, nor the author's reply a refutation of Bruhn's algebra.
4. Tesla patent dates as printed in the 2003 paper (US 685,953 as "November 5 1905"): [memory, medium] that patent was granted in 1901. Not web-checked (web budget spent on claims below); immaterial to the mathematics.
5. Step 1 reports "1.12c to 1.2c" for Ignatiev: the 2003 paper says 1.2c, the 2015 paper and the primary text say 1.12c [read: 2015 p.18-19]. Both are on the record; the author attributes the 0.12c to misinterpreted data (e.g. fibre phase speed, slower than c) in 2015 and says the "error margin ... is reviewed" in 2003.
6. The folder spelling "Vlaenederen" is a typo for "Vlaenderen" [read].

---

Claim 1. S = -eps0 mu0 dPhi/dt - div A is a physical, measurable field; the Lorenz condition is a special case, not a free choice.

a. Mathematics consistent? Split. - The construction is consistent. Keep the potential wave equations (16)-(17) without S = 0; then Gauss and Ampere become (18)-(19) automatically [check]. The result is exactly standard Maxwell with the sources replaced by rho' = rho + eps0 dS/dt, J' = J - grad S/mu0. - The supporting argument is not sound. The author calls the usual gauge freedom "circular" because the Maxwell equations contain no derivatives of S [read: 2003 p.4]. The gauge invariance of E and B (eqs 10-13) follows from the definitions E = -grad Phi - dA/dt, B = curl A alone, with no field equation used. What requires a gauge choice is solving the potential equations, not the definition of E and B. Choosing a gauge where S = 0 loses no solution of standard Maxwell (Bruhn's point, [check] above). The circularity charge fails. The author's real position is the different statement "add S terms to the equations", which is a new theory, not a deduction. - The 2015 version defines B_l by the Biot-Savart-like integral and "concludes" it is physical because it appears in the postulated reciprocal force law [read: 2015 p.6]. That is a postulate, not a derivation.

b. Consistent with experiment? In static and conserved-source situations S obeys a source-free wave equation (eq 24 [read: 2003 p.5]; 2015 eq 2.10-2.11 [check]) and standard Maxwell is recovered with S = 0. All static and quasi-static tests of Coulomb's and Ampere's laws are therefore unaffected. The theory's content is confined to dynamic S waves and to situations with non-conserved charge. In ordinary quantum mechanics a tunnelling electron obeys an exact continuity equation for probability, so "tunnelling as local violation of charge conservation" [read: 2003 p.5] departs from the Schrodinger equation [memory, high]. Vacuum solutions with S not zero give rho' not zero with rho = 0, i.e. Gauss's law ∇.E = dS/dt in vacuum; no measurement of such a field is on record.

c. Tested? No test of S as a free field by the author. Related: Minotti and Modanese (EPJ C 83:1086 (2023); J. Phys. Commun. 8:055003 (2024)) develop the same S = d_mu A^mu as a dynamical field sourced only by non-conserved current, cite van Vlaenderen and Waser 2001 [read: reference lists in research/files/arXiv-*.pdf], and propose a bench circuit for detection; I found no report of a positive detection [web: https://arxiv.org/abs/2403.10580]. Reed and Hively (Symmetry 12:2110, 2020) report a "preliminary" copper-penetration test of a scalar-longitudinal wave, by the authors' own account, test report paywalled and not read; Hively is the patentee of antennas based on it (interested party) [web: https://www.mdpi.com/2073-8994/12/12/2110].

d. If true: Gauss's law would not hold dynamically in vacuum (∇.E = rho/eps0 + dS/dt), the Lorentz force and energy-momentum conservation would need S terms (see claim 2), and gauge invariance of electromagnetism would be a property of the Lorenz-gauge subset only. Minimal departures exist (the Stueckelberg-type extensions above), so mainstream quantum electrodynamics (where a gauge-fixing field already appears but is unphysical) would need the longitudinal mode to be physical.

Verdict. Strong form (S is a free, physical, measurable vacuum field): Never properly tested (not dismissed without argument: Bruhn's two reviews and the mainstream gauge-invariance argument address the standard theory; independent mainstream-journal work by Minotti-Modanese has proposed detection but not reported it). Weak form (a self-consistent modification of Maxwell, with conserved-charge sources forcing S to be a source-free wave): Plausible, untested. The argument that standard gauge freedom is circular: Refuted by calculation (identity in definitions).

---

Claim 2. Extended Maxwell equations predict longitudinal vacuum waves (E along travel, no B), a force J S, scalar energy S^2/(2 mu0), and a power term (rho/eps0) S that lets static charge be a power source.

a. Mathematics consistent? Split. - Wave, force and energy theorems: consistent [check, symbolic]. LES wave: from B = 0, vacuum: grad S = eps0 mu0 dE/dt and div E = dS/dt give wave equations at speed c. B = 0 forces curl E = 0, so E is longitudinal. Worked numbers: S0 = 1 microtesla gives |E| = c S0 = 300 V/m, energy flux (c S0^2/mu0) cos^2 = 0.24 W/m^2 peak (check: 3e8 x 1e-12 / 1.2566e-6 = 0.239). The wave speed is c, not the 1.2c cited as evidence (the paper says so by omission; the 2015 paper says it explicitly [read: 2015 p.19]). - The wave is a travelling vacuum charge-current: rho' = eps0 dS/dt, J' = -grad S/mu0 are non-zero while rho = J = 0. That is the content of the theory, not a hidden consequence of ordinary Maxwell (which has div E = 0 in vacuum, so no longitudinal E wave). - Energy bookkeeping (new, [check]). The paper's theorem (33), in power per volume after dividing by mu0, reads: J.E + c^2 rho S = -d u/dt - div P with u = eps0 E^2/2 + B^2/(2 mu0) + S^2/(2 mu0). Its force theorem (34) says matter feels f = rho E + J x B + J S. For charge moving with velocity v (J = rho v) the power delivered to matter is f.v = J.E + rho v^2 S (the magnetic term does no work). Adding matter power and field change: d u/dt + div P + f.v = -(c^2 - v^2) rho S. For a charge at rest the imbalance is -c^2 rho S per unit volume, and S^2/2mu0 is already included in u. Example: rho = 1e-6 C/m^3 in S = 1e-6 T gives c^2 rho S = 8.99e16 x 1e-12 = 9e4 W/m^3 (check). So within the paper's own two theorems, total energy (field plus matter) is not conserved wherever rho S is not zero. The paper presents this term as "applied power ... a remarkable prediction" [read: 2003 p.10]. A reader should see it differently: a source of energy is the same thing as a violation of energy conservation, and the paper gives no mechanism for where the energy comes from. (Whether this is fixable by a different coupling is not shown; Reed and Hively derive the same term from a Lagrangian [web, mdpi], and I did not read how they treat it.) - The "priority" dispute (eqs 37-38): [check] D = eps0/2[(grad Phi)^2 + (dPhi/c dt)^2] and P = -eps0 grad Phi dPhi/dt satisfy dD/dt + div P = (dPhi/dt)(eps0 mu0 Phi_tt - Lap Phi) = 0 using only the homogeneous form of (16). So (37)-(38) are a conservation law derivable from (16) by multiplying by dPhi/dt, with no S required; whether they are the physical flux is an interpretive question. The author's statement "This is not true" is about physical identification, not algebra. Wesley and Monstein's own position is known to me only through the 2003 paper (an interested party in the dispute). Priority itself cannot be settled from mathematics.

b. Experiment? Longitudinal E with B = 0 in vacuum contradicts Gauss's law, which is tested to very high accuracy for static fields, but dynamic vacuum fields are less tightly bounded. The evidence cited is claim 4.

c. Tested? See claim 4. Monstein and Wesley (EPL 59:514 (2002)) reported polarisation consistent with longitudinal; the only replication I know of (Butterworth, Allison, Cavazos, Mullen, J. Sci. Explor. 27(1):13-23 (2013)) found results "resemble MW's theoretical pattern more closely than did their own experiment" yet disputed MW's theory and the ground-plasma alternative, concluding "a yet-unidentified mechanism" [web: https://journalofscientificexploration.org/index.php/jse/article/view/477]. The authors state the prior "longitudinal EM waves do not exist" [same URL]. A published Comment (Bray and Britton, EPL 66(1):153-154 (2004)) and Reply (MW, EPL 66(1):155-156) exist [web: https://epljournal.edpsciences.org/articles/epl/abs/2004/07/epl8084/epl8084.html]; I did not read their contents. K. Rebilas (2008) proposes ground plasma currents [not read; Investigator's lead]. None of these tests the J S force or the static charge power term. No test of (rho/eps0) S is on record; the paper says so [read: 2003 p.10]. No patents by the author were found [web per Investigator].

d. If true: energy conservation would need to be given up or supplemented by a source; Gauss's law dynamic form changed; electromagnetic energy-momentum tensor modified.

Verdict. Longitudinal vacuum wave with B = 0 (S wave at c): Never properly tested (one replication, ambiguous on mechanism; a prior-held "no longitudinal waves" position among the replicators; no neutral test of the S field specifically). Force J S: Never properly tested. Static-charge power source (rho/eps0) S: Internally inconsistent with energy conservation as the paper's own theorems (33) and (34) combine ([check] above); no experiment exists.

---

Claim 3. The Grassmann (Maxwell-Lorentz) force violates N3LM for open-circuit stationary currents; adding J B_l fixes it; this needs three waves (Phi at a >> c, LEM at b = c, TEM at c), rejection of Lorenz, and supports Rutherford's resolution of the 4/3 problem.

a. Mathematics consistent? Mixed; I separate five sub-claims. - (i) Grassmann force between open current elements is not reciprocal: true [check, read]. It equals the known fact that the Biot-Savart law with the Lorentz force gives equal and opposite forces only on closed circuits [memory, high]. - (ii) Open-circuit "magnetostatics" is not static: dJ/dt = 0 with div J not zero means rho changes linearly in time, so E_Phi grows linearly in time [check: 2015 eq 2.10]. Hence field momentum eps0 ∫E x B_t is changing. The standard resolution: force on matter = -d/dt (field momentum) - boundary flux, so mechanical forces do not sum to zero but total momentum does [memory, high; Griffiths treats a version]. The paper's premise (p.9) that open-circuit magnetostatics "is always free of massless electromagnetic momentum" is asserted, not derived, and is contradicted by the paper's own eq 4.34, which contains eps0 E x B_t [read: p.14]. The deep read in Step 1 also found the field momentum in the paper's theory changes at rate ∫B_l J_t d^3x. So the N3LM premise is the weak point. - (iii) Equation 4.22 (the "Lorenz condition is invalid" step) is the continuity equation rewritten and holds for any a and b [check, confirming Step 1]. Eq 4.20 (b = c) comes from requiring the form J B_l, not from independent physics. - (iv) The three waves with speeds a, b, c in (4.1)-(4.3) are postulated displacement terms; each is a hyperbolic equation. A system with different characteristic speeds a >> c in the same field equations is not Lorentz covariant [memory, high: the wave operator with speed a is invariant only under transformations with limiting speed a], and a >> c means signal fronts travel at a. The paper adopts "quasi dynamics" with a -> infinity: the scalar potential becomes instantaneous, the same structure as the Coulomb gauge (the paper admits the resemblance [read: p.14]). In the Coulomb gauge instantaneous Phi is unobservable alone because the observable E also contains -dA/dt; here B_Phi is declared physical, which makes the instantaneous propagation observable in principle. - (v) The 4/3 claim. [check] The extra momentum eps0 B_l E equals (1/3) of the unit and makes p = m v with m c^2 = q^2/(4 pi eps0 R) (eq 4.35), but that is twice the field energy q^2/(8 pi eps0 R) that the paper's own power theorem (4.33) gives. The paper takes the doubled energy from Rutherford's "energy density correction" [read: p.15, ref 20; Rutherford documents not read]. Using the paper's own 4.33 energy and 4.34 momentum, the ratio of momentum-mass to energy-mass is 2, not 1. Numbers (R = 2.818e-15 m): energy mass 4.555e-31 kg, E x B momentum mass 6.073e-31 kg, B_l E momentum mass 3.036e-31 kg, total 9.109e-31 kg [check, matching Step 1]. The agreement therefore rests on an energy factor the paper does not derive.

b. Experiment? The longitudinal Ampere force: the paper cites Ampere's hairpin and Peoglos, Moyssides, Graneau, Saumont, Johansson, Marinov [read: 2015 p.17; UJPA 2016 per Investigator, not looked up]. For the hairpin, standard field theory explains the longitudinal motion by the diverging current in the mercury at the ends, as Hillas interpreted [web: summarised in search results for Kirk McDonald's hairpin note http://kirkmcd.princeton.edu/examples/hairpin.pdf, which I did not open]; so the experiment does not by itself discriminate. No experiment in the record distinguishes J B_l from the field-momentum explanation.

c. Tested? Longitudinal Ampere force: multiple historical experiments by others, ambiguous in interpretation; no controlled test of J B_l on a geometry where the two accounts differ is on record that I could find. The Aharonov-Bohm "force" reading (2015 p.17): the author argues Caprez, Barwick and Batelaan's null result was resolution-limited; that is his assumption ("we assume that the resolution ... was insufficient") [read: p.17]. The Phi-wave sub-claim (a >> c): de Sangro et al., arXiv:1211.2913, found results "compatible with an electric field rigidly carried by the beam itself" [web: https://arxiv.org/abs/1211.2913]; the Investigator's note that a published counter-reading exists is [memory, low], not verified here; in standard theory a uniformly moving charge's field is the velocity field with no delay, so the observation does not require a >> c [memory, high].

d. If true: the Lorentz force law and the electromagnetic stress-energy would need J B_l; Lorentz covariance would break (a >> c); the speed of light would no longer be the limiting signal speed; the canonical resolutions of the 4/3 problem (Poincare stresses, covariant energy-momentum) would be replaced.

Verdict. Non-reciprocity of Grassmann force for open pieces of circuit: Established (mathematically). That it signals a violation of momentum conservation requiring a new force: Refuted by calculation (the missing momentum is the changing eps0 E x B field momentum, and the paper's own eq 4.34 contains it). Need for J B_l as a distinct force: Never properly tested. Phi wave at a >> c with LEM at c: Internally inconsistent as a selection (eq 4.22 holds for any a) and not Lorentz covariant; speed test Never properly tested. Rutherford 4/3 support: Refuted by calculation inside the paper (the doubled energy is not supplied by eq 4.33).

---

Claim 4. Experiments (Tesla, Ignatiev and Leus, Monstein and Wesley, Tzontchev et al., exploding wires, Ampere hairpin, Aharonov-Bohm) are qualitatively consistent with scalar/longitudinal effects.

a. Mathematics consistent? The comparison is qualitative: the 2003 paper gives no quantitative theory-to-measurement comparison [read: deep read sec 11]. The theory predicts speed c; Ignatiev measured 1.12c (phase shift against a 0.5 km fibre reference at 2.5 km wavelength [read: research/files/wayback-raccoon-Art-p16.pdf per Investigator]); the 2015 paper dismisses the 0.12c as data misinterpretation and keeps the experiment as support for the existence of the wave [read: 2015 p.19]. That is selective use: the same measurement is evidence when it shows a longitudinal wave and discounted where it conflicts with the prediction. (A 0.5 km baseline against a 2.5 km wavelength is a fifth of a wavelength, near-field to intermediate region; the paper's claim "definitely a far field" is an assumption [read: p.19].)

b/c. Experiments and tests. Monstein-Wesley: one replication, ambiguous (claim 2, c). Tzontchev, Chubykalo, Rivera-Juarez: "speed approximately c", apparatus not described [read: 2003 p.9]; not looked up. Exploding wires (Nasilowski, Graneau): no numbers given in the paper [read: 2003 p.9-10]. The 2015 paper states that induction law 3.2 "should be proven by experiments" and "no experiments have been done yet" [read: p.7, p.17]. The Aharonov-Bohm effect has a standard quantum treatment with no force (phase from the integral of A.dl), measured in many experiments [memory, high]; the author's force reading is a reinterpretation with no calculated delay figure, and he says the null time-delay result was probably under-resolved [read: p.17].

d. If true: as in claims 2-3.

Verdict. Never properly tested as support for the scalar theory (the cited experiments are by others, are not quantitative against this theory, and each has a standard or unresolved alternative). Not dismissed without testing for Monstein-Wesley (one replication, mechanism open); the other items are cited, not tested against the theory.

---

Claim 5. Phi or LEM radiation underlies anomalous devices (Podkletnov, Moray, Correa, Chernetsky) via a "Phi photoelectric effect" and macroscopic tunnelling as local violation of charge conservation.

a. Mathematics consistent? There is no mathematics for the effect: the 2015 paper says the Phi photoelectric effect "has not been researched and it has not been verified nor falsified" [read: p.20]; no equation, cross-section, or power estimate. The statement "we have no reason to doubt Dr. Moray's device" rests on one affidavit by Harvey Fletcher (as cited) [read: p.20]; the Investigator found a search-engine removal notice on a Moray/Fletcher query and has not identified what was removed. I do not draw anything from it. The assumed link "Podkletnov signal >= 64c = Phi wave with a >> c" [read: p.19] rests on one reported speed measurement (reference [18], not read) and no other source I can cite.

b. Experiment? I have no source in hand for independent replication of any of these devices and do not state any. Moray's receiver and Podkletnov's gravity impulse are reported by their authors and supporters (interested parties); Correa and Chernetsky are cited from the 2015 paper only [read: p.21].

c. Tested? Not by neutral controlled tests that I could find. Whether any attempted replication exists was not established here.

d. If true: local charge non-conservation (against the continuity equation of quantum mechanics), superluminal signalling (a >> c), and a free energy source (see energy bookkeeping above).

Verdict. Not testable as stated for the Phi photoelectric effect (no quantitative model). Device claims: Never properly tested (no neutral test on record that I found; absence of one is itself unexplained: the Moray and Podkletnov records are worth the Contrarian's attention). The energy source version inherits the claim 2 inconsistency.

---

Claim 6. 1999 paper: psi is a real charge density with velocity; Maxwell's equations follow from charge/mass conservation; points are Coulomb or gravity sources only where psi satisfies self-potential equations; the two mass models are joined by rho_mass c^2 = rho Phi; charge has fractal dimension 3/2.

a. Mathematics consistent? The algebra checks (Step 1 deep read; I re-read its derivation and agree with its conclusions, not re-running every step). Key points: - "Derivation" of Maxwell's equations: the sources are defined (eq 2.11-2.12) as the wave operator on the potentials, so Gauss and Ampere follow for any potentials in the Lorenz gauge. The only content is that the Lorenz condition equals the mass continuity equation if A is proportional to rho_mass u, Phi to rho_mass u^2 and |u| is constant [read: deep read 4.5]. That is a restatement of textbook electrodynamics run backwards, not a derivation from conservation. - Self-potential equations (3.11)-(3.14) reduce to the single equation (1/u^2) Phi_tt - Lap Phi = mu Phi, a Klein-Gordon-type equation with the wrong-sign ("tachyonic") mass term; the paper does not note it [read: deep read 4.7]. - Eq 3.4 (mc^2 = q Phi) is asserted by analogy; eq 3.16/3.29 follows only because eqs 3.13 and 3.26 are adopted as definitions; the Step 1 reader found no region where both hold at once, consistent with the paper's "only at some points" wording [read: deep read 4.9]. I did not independently reproduce this. - Units: charge = sqrt(kg m^3) is consistent as a unit system (cgs-like). The step from the exponent 3/2 of metres to "fractal dimension 3/2" is a reading of a unit convention as geometry; no argument is given [read: p.6]. - Typos: eq 2.15 (curl for divergence), 2.22 (scalar crossed with vector). Minor.

b. Experiment? None cited, none numerical [read: whole paper]. Illustrative: for an electron, Phi = mc^2/e = 5.11e5 V and q Phi equals the classical-radius electrostatic energy at 2.82e-15 m [check]; this is standard electrostatic-mass reasoning (Petkov, Collins as cited), not a test.

c. Tested? No.

d. If true: a real-valued psi as charge density replaces the Born interpretation (|psi|^2 as probability), which gives interference and entanglement phenomena a different footing; no consequence is worked out in the paper.

Verdict. "Maxwell's equations derived from conservation": Refuted by calculation as a derivation (they follow from the definitions; a restatement of the Lorenz-gauge route). Psi as real charge density, self-potential source criterion, mass-model unification, fractal dimension 3/2: Not testable as stated (no measurable prediction, one unit-convention reading).

---

Investigator findings kept in the record (with how they bear on the mathematics)

• Own work: 2001 Hadronic Journal paper with A. Waser (received 13 Nov 2001; source of S theorems), 2004 Rinton chapter (uses a = b, b > c; the 2015 paper rejects that choice [read: p.13]), 2015 viXra and 2016 UJPA version ("consistent only if a >> c"; Whittaker's reciprocal force law), the 2001 preprint line "A single point charge cannot be a source for the scalar field, because the scalar field that is derived from the Liénard-Wiechert potentials is zero" [read: research notes]. That line agrees with my check that conserved sources leave S as a source-free wave (so ordinary charges are not sources of S), which undercuts any scheme where a static charge acts as a power source unless charge conservation fails.
• Support: Butterworth et al. 2013 (claim 2); Ignatiev and Leus primary text (claim 4); de Sangro et al. (claim 3); Hively and Loebl 2019 / Reed and Hively 2020 (independent derivation of the same S, J.C force, rho C power term; own-account preliminary test; Hively holds US 9,306,527); Minotti and Modanese (mainstream-journal S theory, cite the 2001 paper; S sourced only by non-conserved current).
• Against: Bruhn's two reviews (TU Darmstadt), Bray and Britton Comment, Rebilas 2008 (titles only).
• Related: Arbab, Simulik, Tomilin and Panchelyuga (RUDN Metaphysics 2025; the 2015 paper cites Tomilin's Generalized Electrodynamics as the source of the induction-law pattern [read: p.7]). I did not read those.
• Historical record: Ethergy B.V. (no output found), IBR/Hadronic Press venue, no arXiv submissions after 2003, 2023 Prague lecture on Moray-type converters. Silence of the Ethergy record is an open anomaly (competing explanations: small consultancy, trade secrecy, no device built); no mathematical bearing.

What would settle the main disagreements (mathematical view)

1. Open-circuit force: compute total mechanical force plus d/dt field momentum for two charging capacitors joined by wires, with and without J B_l, including the plate charges. If standard theory closes the momentum budget (it should), the J B_l term is not needed; a measured residual in a geometry designed for it would be needed to favour the paper.
2. S as a vacuum field: build the Minotti-Modanese bench circuit or an equivalent; a repeated, shielded detection of a B = 0, E longitudinal signal would be the decisive test.
3. Energy: ask the author or a supporter for the Lagrangian behind (33)-(34), or an experiment measuring heat released by a static charged body in an S field. Without it, rho S/eps0 stands as energy non-conservation.

Summary of verdicts