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Handwerker-Roberto: Mathematician's review (Step 2)

Reviewer: PhD mathematician. Checks derivations, notation, units and hidden assumptions. Source tags: [read] from the documents (p.N = PDF page of TESLA-THEORY-2.pdf unless stated), [web] with URL, [memory] with confidence. Everything marked "(my check)" was recomputed in the project venv (SymPy/NumPy) for this review.

Inputs read: claims.md, facts.json, deep/ (all three files, in full), sites.md "Leads for review", the research/ notes, and the downloaded Arbab and Satti paper (read in full text). I also rendered and read PDF p.8 and p.10 of Handwerker's paper myself [read: page image p.8, p.10] to confirm the equations that matter most.

Source gap (repeated from claims.md): the central works are readable. Not obtained: Handwerker's other decks and videos, the "appendix" and ref. [12] his paper cites, and any raw measurement. No document contains a measured value for the water antennas. Where a conclusion rests on [memory] it says so.

0. The mathematics in one page (for the reader)

Handwerker's paper makes one mathematical argument and then offers experiments. The argument is a chain:

  1. Maxwell's four equations (vector form) are "partially incomplete" because Heaviside and Gibbs dropped quaternions [p.7].
  2. In quaternion form, with a scalar field Sigma (written with the Greek capital letter, Σ) such that the current density is J = grad Σ, there are waves of Σ at the speed of light that induce charge and current "even if charge and current were not present" [p.10].
  3. Those waves are "longitudinal" (field along the direction of travel), call them L.M.D. (longitudinal magneto-dielectric), and a Tesla coil emits them [p.11, p.20].

Link 3 is experimental and is judged in section 3 below. Links 1 and 2 are mathematical, and they fail at identifiable steps. The short version:

  • Equations (1) and (2) of the paper are correct but are only consequences of Maxwell's equations (see 1.2). Differentiating loses information; nothing new is gained.
  • The "gauge transformation" (3)-(4) is not a gauge transformation: it changes the charge and current, not the potentials (1.4).
  • The whole Σ construction comes from Arbab and Satti (2009). Their paper builds it by setting a whole quaternion product to zero, which adds two equations that real charges and currents violate (1.5, my check). Handwerker's text does not carry this over, so his Σ looks like it is "introduced" from nothing.
  • Even granting Σ, the resulting wave is a travelling pattern of real charge and current moving at c with a co-moving electric field and no magnetic field (1.6, my check). That is not "extra field in empty space"; it needs charge moving at the speed of light.

Reader's primer on the vocabulary: - A field is "longitudinal" if it points along the direction the wave travels (like sound), "transverse" if it points sideways (like light). - Gauss's law, ∇·E = ρ/ε0 (divergence of E equals charge density over the permittivity of free space; ε0 = 8.854e-12 F/m), says the electric field spreads out from charge. In charge-free space ∇·E = 0. A plane wave E = E0 x̂ cos(kx − ωt) has divergence −E0 k sin(kx − ωt), not zero, so it is forbidden in empty space (my check). That one line is why Maxwell, in 1865, concluded that only transverse waves propagate in a non-conducting medium [web: https://en.wikipedia.org/wiki/A_Dynamical_Theory_of_the_Electromagnetic_Field, which quotes the paper's introduction: "the only disturbances which can be propagated are those which are transverse to the direction of propagation"]. - A "gauge transformation" changes the potentials (φ for voltage, A for the vector potential) in a way that leaves E and B, and the sources, unchanged.

1. Claim-independent checks on the mathematics

1.1 Maxwell's equations as printed (p.7)

The four equations (∇·B = 0, ∇·E = ρ/ε0, ∇×E + ∂B/∂t = 0, c²∇×B − ∂E/∂t = J/ε0) are the standard vacuum equations in SI [read: page image p.7; memory high]. Note the fourth already contains ∂E/∂t, the displacement-current term. Wording slip only: J is current density, not "current intensity".

1.2 Equations (1)-(2) are correct and weaker than Maxwell

[read: page image p.9; H2O deck p.9] (1) (1/c²)∂²E/∂t² − ∇²E = −(1/ε0)(∇ρ + (1/c²)∂J/∂t) (2) (1/c²)∂²B/∂t² − ∇²B = μ0 ∇×J

Check by hand: take the curl of Faraday's law, use ∇×(∇×E) = ∇(∇·E) − ∇²E and Gauss's law, and (1) follows exactly; similarly (2) from the curl of the Ampère-Maxwell law. Signs and factors match, units are consistent (each term of (1) is V/m³) (my check, agrees with deep/TESLA-THEORY-2 section 2).

The point the paper never states: every solution of Maxwell's equations solves (1)-(2), but not every solution of (1)-(2) solves Maxwell's equations. Counter-example (my check): in vacuum (ρ = 0, J = 0) the longitudinal plane wave E = E0 x̂ cos(kx − ωt) with ω = ck solves (1) identically (residual 0) but has ∇·E = −E0 k sin(kx − ωt) ≠ 0, so it violates Gauss's law. A longitudinal "solution" is therefore a symptom of having thrown away the first-order constraints, not a prediction. Calling (1)-(2) "Maxwell's equations in quaternionic form" [p.9] is true only if the constraints are added back.

1.3 Quaternions versus vectors (p.6, p.7)

A quaternion q = a + bi + cj + dk (i² = j² = k² = ijk = −1) is a scalar part plus a 3-vector part. The potential à = (iφ/c, A) is the ordinary four-vector potential (φ/c, A) written with the old "ict" trick; □̃²Ã = μ0 J̃ is the textbook Lorenz-gauge wave equation (□φ = ρ/ε0, □A = μ0 J) [memory high; also stated by Arbab and Satti, read: research/files/Arbab-Satti-2009 eqs. (1)-(10)]. A change of notation cannot add or delete a physical solution; only a change of equations can. So Handwerker's explanation (a), "Heaviside and Gibbs simplified the quaternionic notation, therefore the set is partially incomplete", has no mathematical content unless a specific term dropped by them is named. None is named.

History, verified [web]: the four-equation vector form is Heaviside's reduction (mid-1880s) of Maxwell's larger set; Maxwell's Treatise (1873) used quaternion notation only sparingly (Arts. 618-619) and the four vector equations are not in it [web: https://en.wikipedia.org/wiki/History_of_Maxwell%27s_equations and search results quoting Hunt and others]. So "quaternionic Maxwell equations" was never the primary form. Explanation (b) in the paper, that Maxwell's 1865 set predates Tesla's 1892 dielectric phenomena, is chronologically true but not an argument: Maxwell's equations describe E and B in general, including strong near fields.

Small point: "the former [quaternions] has the anticommutative property, the latter [vectors] the commutative property" [p.6] is imprecise. Quaternion multiplication is non-commutative (ij = k, ji = −k). For vectors the dot product commutes and the cross product anticommutes.

The "55% memory saving" [p.8] is arithmetic: 4 numbers instead of 9 for a rotation is 1 − 4/9 = 55.6% (my check). It concerns rotations and the author's 1990 course work, not Maxwell's equations.

1.4 The Σ field, "J = ∇Σ", and the "new gauge transformation" (p.10)

[read: page image p.10, which I read myself.] The paper states: ρ' = ρ + (1/c²)∂Σ/∂t, J' = J − ∇Σ (3)-(4); (1/c²)∂²Σ/∂t² − ∇²Σ = −(∇·J + ∂ρ/∂t) (5); ρ = −(1/c²)∂Σ/∂t (6); J = ∇Σ (7), "even if charge ρ and current J were not present in a particular zone".

My checks (SymPy): - Eq. (5) is an identity once (6)-(7) are assumed: with them, ∇·J + ∂ρ/∂t = ∇²Σ − (1/c²)∂²Σ/∂t², so (5) is just a restatement. The text presents (6)-(7) as consequences of (5): circular. - If charge is conserved (∂ρ/∂t + ∇·J = 0, a consequence of Maxwell's equations), the right side of (5) is zero and Σ obeys the free wave equation at speed c. "Travels at c" is therefore a property assumed for Σ, not a derived property of charges. - Effect of (3)-(4): the sources of (1) and (2) are unchanged (∇ρ' + (1/c²)∂J'/∂t = ∇ρ + (1/c²)∂J/∂t, and ∇×∇Σ = 0). But ρ' − ρ = (1/c²)∂Σ/∂t is not zero, so Gauss's law ∇·E = ρ/ε0 cannot hold for both ρ and ρ' with the same E. Charge conservation survives only if Σ obeys the free wave equation. So (3)-(4) is a symmetry of the reduced equations (1)-(2), not of Maxwell's equations. - In the textbook sense a gauge transformation never alters the sources, which are measurable. Here ρ and J are altered. The word "gauge" is misused. - Putting (6)-(7) into (3)-(4) gives ρ' = 0 and J' = 0 (my check): the "Σ-induced" charge and current are exactly the part the transformation removes. - "Even if charge and current were not present" contradicts (6)-(7), which define a nonzero ρ and J wherever Σ varies. Either they are real charge and current (then charge is present and ordinary Maxwell theory governs it) or they are not (then (6)-(7) are not densities).

1.5 Where "J = ∇Σ" comes from: Arbab and Satti (read in full)

Correction to the deep read, which said "J = ∇Σ follows from nothing". It follows from Arbab and Satti's Eq. (27), ∇×J = 0, together with the fact that a curl-free field is a gradient (valid on a simply connected region), their Eq. (31) J = ∇Σ [read: research/files/Arbab-Satti-2009 Eqs. (26)-(32)]. Eqs. (26)-(27) come from setting the whole quaternion continuity product ∇̃J̃ = 0 (not only its scalar part). The scalar part is ordinary charge conservation. The vector part adds two new equations: (26) ∇ρ + (1/c²)∂J/∂t = 0, (27) ∇×J = 0. These are not consequences of Maxwell's equations and real physics violates them (my check): - A static, non-uniform charge cloud (J = 0, ρ = e^{−r²}) has ∇ρ ≠ 0, violating (26). This is just a charged ball: its Coulomb field is not a source-free solution. - A circulating current (J = (−y, x, 0)e^{−r²}, a loop or solenoid current) has ∇×J ≠ 0, violating (27). A coil, the thing Handwerker is studying, carries exactly such a current. So Σ exists only for the special charge-current patterns permitted by (26)-(27), and a Tesla coil's own sources are not among them. Arbab and Satti also say their eqs. (39)-(40) show fields travel at c "whether the space is empty or having free charges", which follows from (26)-(27) and contradicts the static Coulomb field.

Arbab and Satti's own position (Investigator's find, confirmed by my reading): "Maxwell equations are still exact and need no modifications" [read: Arbab-Satti, text near Eq. (43)], and the paper says "the Lorentz gauge" is all that is needed and the scalar wave "exists with or without considering the Lorentz gauge" [read: abstract]. Handwerker's stronger claim, that the vector form is "incomplete", is not in his source. They also call for "urgent experimental work" and say that "if the electroscalar wave is found experimentally" it would open a new era; i.e. they present it as unconfirmed [read: Arbab-Satti, conclusion]. Their Eq. (49) Σ = ε0 (E·v) has units A/m (my check: (A·s/(V·m))(V/m)(m/s) = A/m), consistent with "dimension of a magnetic field intensity".

1.6 What a Σ wave is, by direct calculation (my check)

Take Σ = ∂Ψ/∂t with Ψ = f(x − ct) any profile moving at c. Then ρ = −f'', J = ∇Σ = −c f'' x̂ = ρ c x̂, and E = −(1/ε0)∇Ψ, B = 0 solves the full Maxwell equations (Gauss residual 0, Ampère residual 0). So Maxwell theory already contains this object: a pattern of real charge moving at c, carrying a co-moving longitudinal E with no B. Consequences: - It is not a wave in empty space; it is carried by charge density ρ = −f''. Ordinary charges have mass and cannot move at c. - Its Poynting vector (E×B)/μ0 is zero because B = 0 (my check). Energy flow "along the propagation direction" is, in Arbab and Satti, supplied by the charges moving with the wave (their term ε0E² v), not by the field. - Arbab and Satti's suggestion is that vacuum fluctuations could supply such charge [read: Arbab-Satti text after Eq. (36)]. That is speculation without a calculation. Handwerker's conclusion [p.21] that the Σ wave "does no work to move free charges ... without converting EM energy into neither mechanic nor heat" is copied from Arbab-Satti Eq. (50). It contradicts his own experiments, which have the "L.M.D. field" lighting neon tubes and turning a motor [p.14-17]. If L.M.D. and Σ are the same thing, his theory forbids what he reports; if different, he never defines L.M.D. with an equation.

1.7 Notation, symbols and units

  • E = ∇φ [p.8] has the wrong sign and drops a term: the standard relation is E = −∇φ − ∂A/∂t. Dropping ∂A/∂t gives a curl-free field only if A is static, unstated.
  • Symbol list [p.22]: φ in V/m² (should be V); A in A/m² (should be V·s/m); B in A/m (should be tesla; A/m is H); Σ in "[H]" (from (3)-(4), Σ has units A/m; "H" is ambiguous with the henry); J "current intensity" (a density). With the printed units, E = ∇φ would give V/m³, not the V/m the same list gives for E.
  • Section 4 power estimate [p.16-17]: dR = dx/dS has units 1/m, not ohms (resistivity missing); "integrating power over x gives energy" is dimensionally wrong; cos² became cos. And the angular law contradicts Fig.9: with ψ the angle between plate and Er his formula peaks at ψ = 0, the figure has its minimum there (see section 3). It yields no number.
  • Capacitance printed as "C = 0,040 mH" and "0,012 mH" [p.17]: millihenry is a unit of inductance.
  • λ0 = 113.64 m is quoted for f0 = 2.755 MHz; c/f0 = 108.8 m (113.64 m corresponds to 2.638 MHz); about 4% off (my check). "f0/f = 1,527 MHz" is a pure ratio (2.755/1.810 = 1.522).
  • "Tilting 90° 'without affecting'", "no difference with or without the core", etc. are given with no readings.

2. Verification of Step 1 (facts.json, claims.md, deep reads)

I rechecked the deep read's mathematics line by line (sections 1-5 above agree with it, apart from the corrections below) and recomputed its numbers: λ0 and λ0/4 (108.8 m, 27.2 m), the radian distance λ/2π = 17.3 m, the coupling constant implied by the two frequencies (k = 0.397), single-turn loop inductance (0.258 µH vs printed 1 µH), plate capacitance (14.2 pF), the field implied by 100 mA into a 120×130 mm plate at 2.755 MHz (41.8 kV/m, 12.5 kV/m for 30 mA). All reproduce.

Corrections to Step 1 / earlier documents

  1. deep/TESLA-THEORY-2-0d985e.md, section 1c ("tension in the gauge step") and section 5 ("Gauge"): the claim from memory (medium) that Arbab-Satti do not impose the Lorenz gauge, and that imposing it removes the extra wave, is wrong. Their abstract: Maxwell's equations are derivable "from just one quaternion equation that only requires the Lorentz gauge" and "This scalar wave exists with or without considering the Lorentz gauge" [read: Arbab-Satti abstract]. The sites.md correction (Investigator) is confirmed. Handwerker's wording "by respect of the Lorenz gauge" follows his source. The deleted argument should not be repeated.
  2. deep/TESLA-THEORY-2-0d985e.md, section 3 ("Is it derived or asserted? Asserted ... J = ∇Σ follows from nothing"): incomplete. In Arbab and Satti, J = ∇Σ is their Eq. (31), derived from their Eq. (27) ∇×J = 0, itself an extra equation from setting the full quaternion continuity product to zero. Handwerker's paper does not reproduce this (the "appendix" is missing, p.8), so within his paper it is asserted; within the cited source it is derived from an assumption that real currents violate (1.5). Also the Σ "gauge" step (3)-(4) is Arbab and Satti's Eq. (41)-(43), used almost word for word.
  3. deep/TESLA-THEORY-2-0d985e.md, section 3, "Σ travels at c": "attributed to Arbab and Satti and not shown here". Confirmed: it follows from their (32), ∇²Σ − (1/c²)∂²Σ/∂t² = 0, derived from (26)-(27) with J = ∇Σ. It is a wave equation for the scalar defined by those equations, not for charge.
  4. claims.md claim 2: "predicts scalar dielectric fields and longitudinal waves travelling at c". Add that the source's own authors say Maxwell's equations "are still exact and need no modifications" and that Σ has "the dimension of magnetic field intensity" and "no magnetic component". "Dielectric" is Handwerker's word, not theirs. Also in Arbab-Satti the predicted Σ wave carries charge and current at c; it is not an electric wave in empty space.
  5. claims.md claim 3: "as shown by a still compass, a ferrite-core probe unchanged by the core ...". The ferrite test is described in the text but Fig.3 shows only one series (triangles, "measure with inductance"); the "no difference with or without core" is asserted, not displayed (deep read E6). The compass cannot detect a 2.755 MHz magnetic field (my check of principle: the force reverses about 5.5 million times a second [memory high]). The claim should say "asserted by", not "shown by".
  6. facts.json / deep: reference [2] of the H2O decks is Arbab and Satti, Progress in Physics vol.2, April 2009 (Khartoum and Omdurman Ahlia universities), not "Omdurman University" (Investigator's correction, confirmed by the downloaded PDF).
  7. Digest of Fig.3 axis and Fig.9 range (Step 1a): deep read already corrects to x about 0-2 m and psi = 3π/2 ≡ −π/2 to +π/2. Confirmed; Fig.9 also has a printing error (3π/2 is 270°, not 180°).
  8. facts.json: the claim that frequency is "none" in the paper (earlier part of the deep read) was already corrected in the deep read: f0 = 2.755 MHz and 1.810 MHz [read: p.18]. Any copy elsewhere (claims.md says "no document contains a measured value for the water antennas", which stays true for the water decks) should not say the Tesla-coil paper has no frequency.

3. Per claim: questions a-d and verdict

Claim 1. A Tesla coil emits longitudinal "L.M.D." waves, distinct from T.E.M., and they carry usable power.

a. Internally consistent? No. L.M.D. is never given an equation: Fig.6 is a picture, "magneto-dielectric" is never defined (the Σ wave has "no magnetic component", p.21), no fall-off law, wavelength or polarisation is stated. The conclusion that the Σ wave "does no work ... converting EM energy into neither mechanic nor heat" [p.21] contradicts the motor and neon experiments [p.14-17] if the two are the same. Fig.3's "expected quadratic diminishing curve" is the prediction of a radiated wave's intensity, but the plotted quantity (meter current) tracks field amplitude (deep read E7, which I agree with).

b. Consistent with experiment? The observations (neon lit without wires, plate charged and sparking, motor driven at 0-2 m, radial dominant E at 2.755 MHz) are all consistent with standard theory: all tests are at x/(λ/2π) ≤ 0.15, deep in the reactive near field (λ/2π = 17.3 m, my check). There the Hertzian-dipole field has terms in 1/r³, 1/r², 1/r, and the 1/r³ (quasi-static, mostly electric, with a radial component) term is larger than the radiating term by about 1/(kr)² ≈ 75 at 2 m and 1200 at 0.5 m (my check). A mainly electric, mainly radial near field that falls off between 1/x and 1/x³ is therefore the standard expectation; his fit x^−1.7 (deep read, digitised by eye from Fig.3) sits inside that range [web: antenna notes, search result at https://www.waves.utoronto.ca/prof/svhum/ece422/notes/05-dipole.pdf and a Tesla-list consensus that the coil's near field is mostly electric and capacitive; radial-component scaling [memory high]]. In the far field a longitudinal E vanishes in the standard theory, and Gauss's law forbids a free longitudinal wave (1.2).

c. Tested? By the inventor only: Fig.3 (one run, "normalised" values in mA, origin undefined), Fig.9 (half a turn of plate rotation; values up to about 110 mA on a 0-100 mA meter), qualitative demonstrations. No shielded control (a grounded screen between coil and probe is the decisive test), no independent magnetic probe, no off-run, no prediction by the L.M.D. theory that differs from standard theory. No replication or independent measurement was found by the Investigator [read: sites.md, "Supporting information"]. A public demonstration (Verona, 15 April 2011) is documented in one published report by a co-organiser of the event (Heos): a lamp lit in water, a toy car moved "without wires or batteries", reporter's comment: "What practical developments these experiments may have, it is still too early to say" (translated) [web: Wayback of http://www.heos.it/Cronaca_11/0100102_Casa-senza-fili-e-mobility-autonomia-energetica.htm, via sites.md]. That is an account of a demonstration, with no measurement and from a party with an interest in the event; it neither tests nor contradicts the claim.

d. If true: free-space longitudinal E with energy flow would require dropping or modifying Gauss's law in charge-free space (∇·E = 0), or finding charge at c; it would revise the Poynting-vector energy accounting (B = 0 gives S = 0). The weak form needs nothing new.

Verdict (split): - Strong form (a distinct longitudinal wave in charge-free space that radiates usable power): Refuted by calculation in standard theory (Gauss's law, 1.2; the Σ solution needs charge moving at c, 1.6). No experiment in the folder discriminates it, so as an experimental claim it is also Never properly tested (not dismissed without testing, rather never given a discriminating test by anyone, including its author). - Weak form (a Tesla coil's strong, mostly electric, partly radial near field lights tubes and powers small loads capacitively): Established. [memory high; web: capacitive wireless power is described as a standard near-field method in the patent records returned by the search above.]

Claim 2. Maxwell's vector form is incomplete; the quaternionic form with Σ (J = ∇Σ) predicts scalar dielectric fields and longitudinal waves at c.

a. Internally consistent? No, in three places. (i) "Incomplete because of notation" is vacuous (1.3); the cited source says Maxwell's equations "need no modifications". (ii) The Σ transformation (3)-(4) changes sources and breaks Gauss's law (1.4). (iii) The source's extra equations (26)-(27) are violated by a static charged body and a current loop (1.5, my check), so Σ cannot describe the sources of a Tesla coil. Equations (1)-(2), (5) are correct as algebra but weaker than, or identical to, what they claim to derive.

b. Consistent with experiment? Standard Maxwell theory is among the best-tested theories (the speed of light, radiation patterns, and so on). Nothing here modifies a prediction. The question whether the potentials φ and A carry physics beyond E and B is genuinely discussed (Aharonov-Bohm effect [memory high]); the paper does not engage it. Arbab and Satti themselves state no number to test.

c. Tested? The Σ wave has not been detected by anyone in the folder; Arbab and Satti ask for experimental work and cite Ignatiev and Leus (1990 book) as having "confirmed experimentally ... longitudinal vacuum wave" [read: Arbab-Satti, near Eq. (47)]; I have not read that book, so I cannot weigh that experiment. Arbab continued the line in later papers (arXiv:1003.0075 and others, listed in sites.md, not read). Handwerker's own tests (claim 1) do not test Σ; they contain no Σ prediction.

d. If true: a new field degree of freedom in vacuum (a scalar wave with B = 0), a source (vacuum charge density −Σ̇/c²) not tied to matter, and failure of the first-order Maxwell constraints in free space.

Verdict (split): - "Quaternion notation makes the vector form incomplete": Refuted by calculation (equivalence of notations, 1.3; history [web]). - The Σ construction as presented (Handwerker, p.10): Internally inconsistent (circular eq. (5); "gauge" that alters sources; "no charge present" vs ρ = −Σ̇/c²; contradiction with p.21 vs experiments). - Arbab-Satti's version: Plausible, untested only for charge patterns obeying (26)-(27) and moving at c; for real charges (static, circulating) its extra equations fail (Refuted by calculation).

Claim 3. The coil's field is dielectric (electric), not magnetic and not transverse (compass, ferrite core, loop and motor tilt, plate angle, near inverse-square fall-off).

a. The arithmetic and physics reasoning have problems rather than inconsistencies: E2 compass (cannot respond at 2.755 MHz); E3 horizontal loop called "unfavourable for T.E.M." though a horizontal loop above/beside a vertical-axis coil is the most favourable for magnetic coupling (deep read E3; reasoning: field of a vertical dipole in the equatorial plane is vertical, my agreement); E8's rotation about a vertical axis is blind to the vertical field Et by geometry; E9 formula gives maximum at ψ = 0 where his Fig.9 has the minimum. The loop inductance printed (1 µH) is about 4 times the computed 0.26 µH (my check).

b. Consistent with experiment? The "mostly electric" conclusion is what the standard near field predicts (see claim 1b). The ferrite test (L = 6 mH without core, 10 mH with) is a good internal control for magnetic pickup; if the reading is unchanged, the probe was reading electric pickup, which the standard theory also expects (leads and coil form a small electrode). The plate test: face-on maximum is the standard result i = ε0 ω ∫E_n dA; my fit of his digitised data (deep read E8) i ≈ 25 + 90(1 − cos ψ) mA, rms 3.9 mA; the plot is half a turn only.

c. Tested? By the inventor only, qualitative or normalised; the controls that would discriminate (grounded screen, independent H-probe, comparison with a modelled near-field of the real geometry, detuned off-run) are listed as missing in deep read section 12-13 and I agree. No replication found.

d. Nothing in mainstream physics changes for the weak form (a near-field electric dominance). For "L.M.D. rather than T.E.M." the required change is the claim 1 change.

Verdict (split): - Weak form (the near field is mainly electric and radial, the probes respond to it): Established (as ordinary near-field electrodynamics). - Strong form (this shows L.M.D. waves, not T.E.M.): Not testable as stated (no L.M.D. prediction differing from the near field is given); and Never properly tested (no shielding, no independent B probe).

Claim 4. Displacement current is a weak explanation; the dielectric field fills the surrounding space (capacitor plates over 1 m apart still read a capacitance).

a. Inconsistent with the paper itself. The Ampère-Maxwell law printed on p.7 contains ∂E/∂t. The objection ("how could an electron current 'jump' through a dielectric?") attacks a version of the theory the textbook does not state: displacement current is ε0∂E/∂t (plus bound-charge polarisation current in a material); no conduction electrons cross the gap [memory high]. Worked number: a 10 pF capacitor at 10 kV, 1 MHz, passes I = 2πfCV = 0.63 A (my check), which is measured behaviour for ordinary capacitors.

b. The capacitor measurement [p.17]: 40×40 mm plates give 14.2 pF at 1 mm (my check) and capacitance never reaches zero at any distance because field lines from one plate reach the other and the leads. The reading at more than 1 m is expected from stray and lead capacitance (several pF). The units printed (mH) are wrong. In the standard picture the field already fills the surrounding space; Steinmetz's "hint" restates that.

c. Tested? Not as a test of displacement current; no lead-only blank was run.

d. If true: Maxwell's addition of the ∂E/∂t term would have to be rejected, which would break charge conservation (∇·J = −∂ρ/∂t) and wave propagation.

Verdict: Refuted by calculation (the objection misstates the theory and the observation follows from electrostatics); also Internally inconsistent (his own Ampère law has the term).

Claim 5. A glass vessel of water wired to a special HV generator is an antenna that transmits energy by longitudinal waves to a second water vessel, lighting neon tubes and running a motor through one wire.

a. The deck's argument [H2O p.8] runs: fluids carry only longitudinal (pressure) waves; water stops usual radio waves; therefore an electrical wave through or around water must be longitudinal. The first sentence is about mechanical shear versus pressure waves; electromagnetic waves are transverse and do travel in water (light does). The step from mechanical to electrical is not made. Numbers (my check): at 2.755 MHz, tap water (σ ≈ 0.05 S/m) has a loss tangent ≈ 4 and skin depth ≈ 1.4 m; sea water (σ ≈ 4 S/m) about 0.15 m; relative permittivity about 80 gives wave speed c/√80 ≈ 3.4×10⁷ m/s [memory medium for σ, high for εr≈80; no web check]. So a 10-20 cm vessel does not "stop" an RF field but is a lossy high-permittivity dielectric and a good electrode; a vessel of water near a high-voltage terminal behaves as a large capacitor plate. The sound-speed table [p.7] sets "ether = c" beside air, water and glass: an identification, not a measurement. The Sun potential 215 GV [p.16] is Tesla's number, quoted without derivation (see claim 6).

b. Consistent with experiment? Lighting a neon tube (or even a burnt-out halogen lamp) near a high-voltage RF source, with or without water and by one wire, is a standard capacitive-coupling demonstration [memory high]. The inventor's own earlier deck lights a 230 V lamp under water with a hand in the water [read: research/files/2011-Handwerker-Tesla-and-cold-electricity, p.13 and Heos report]. Whether the hand is safe depends on current, frequency and voltage: none are given. A real HF high-voltage source at high frequency delivers small current to a body (skin effect and tiny capacitance); with no values the demonstration cannot be assessed.

c. Tested? No measurement exists in the two water decks: "no voltage, current, frequency, distance, power, number of trials, control or calibration" [read: deep/H2O, "Every number given"]; the December deck adds no new data to August's. Not replicated. No dismissal either; the record is silence. The later "G-03 Argon" electric sailboat and the Verona report are accounts (Heos and a popular book by M. Pizzuti; the latter repeats the inventor's account, sites.md). Missing controls: same tube and wire without water, dry glass vessel, detuned generator.

d. If true: a mechanism by which RF energy couples through water that is not capacitive or conductive coupling; a longitudinal EM mode in a conducting dielectric is excluded by ∇·E = (free charge)/ε, but in a conductor with free charge longitudinal charge-density oscillation (plasma-type, relaxation) is standard (relaxation time ε/σ ≈ 80ε0/0.05 ≈ 14 ns for tap water [my check, memory medium]) and decays quickly.

Verdict: Never properly tested (a testable claim with no neutral, controlled test or measured data on record; not dismissed by anyone with a test, simply never examined; the record goes dark in 2013 per the Investigator). The mechanism "L.M.D. waves through water" is Not testable as stated (no wavelength, speed, attenuation or polarisation given).

Claim 6. Tesla's reported statements are authority: light and radio are longitudinal "sound waves in the ether", cosmic rays move faster than light (about 50 c), the Sun sits at about 215 GV, relativity is "a mass of error".

a. Internal: the 1934 Times passage is self-contradictory as printed (light is longitudinal; radio waves are of the same character as light "only they are not transversal but longitudinal", deep/Teslas-prophecies). Tesla's particles are "neutrons" in 1932 and "powerfully charged positive particles" in 1935; the potential is 215 GV in one 1935 article and 216 GV in another (deep read). Handwerker adds no comment.

b. Calculation (my check). A particle of charge q falling through potential V gains kinetic energy qV. Newtonian: v = √(2qV/m). For a proton at 2.15×10¹¹ V this gives 21 c; for an electron 917 c. So Tesla's "fifty times light" is the order of magnitude a Newtonian estimate gives for light charged particles; it is not a prediction of correct mechanics. Relativistically E = γmc², γ = 1 + qV/(mc²) = 230 for a proton, and v = c(1 − 1/(2γ²)) = 0.99999 c; no speed above c, and this has been the kinematics used in every accelerator since [memory high]. A 215 GV Sun: charge Q = 4πε0RV = 1.7×10¹⁰ C (R = 6.96×10⁸ m, my check), equivalent to about 10²⁹ elementary charges, a surface field of only 309 V/m: arithmetic consistent, but a hot plasma neutralises net charge quickly and the argument is not reproduced in any document [memory medium]. Light is a transverse wave: polarisation (Malus's law, tested since 1808) is impossible for a longitudinal wave [memory high]; the "electric sound waves in the ether" is the 1890s picture before relativity. The slide's "remember superluminal neutrinos" refers to OPERA (2011), whose result came from a fibre-optic connection fault and a fast clock oscillator, then agreed with c in ICARUS and other Gran Sasso measurements by June 2012 [web: https://www.aip.org/inside-science/faster-than-light-neutrino-results-laid-to-rest]. Neutrinos are not Tesla's charged particles either.

c. Tested? Cosmic rays are well measured (charged particles, mostly protons, from the Sun and beyond; none shown faster than c) [memory high; no web check]. Tesla's own 1896/1899 "proofs" are not shown in the excerpts. The compilation is journalism: reporter paraphrase and quotation are mixed; headlines are not Tesla's.

d. If true: special relativity's limit would fail; a century of accelerator and astrophysical data would need explaining.

Verdict (split): - Light and radio are longitudinal waves: Tested, not supported (polarisation); and Historically superseded but instructive (1890s ether picture). - Cosmic rays faster than light (50 c): Refuted by calculation (relativistic kinematics; Newtonian estimate shows where the number came from); no measurement is offered. - Sun at 215 GV: Not testable as stated (no derivation or reference point in the documents). - "Relativity is a mass of error": an opinion; Tested, not supported (accepted theory, applied correctly, passes all tests; cited counter-example OPERA withdrawn).

4. Investigator's findings that bear on the mathematics (kept whatever the verdict)

  • Arbab and Satti (2009) is the source of equations (1)-(7). It states Maxwell's equations "need no modifications" and asks for experiment. Kept as the key closely related work (sites.md lead 3).
  • The 2010 deck [read: research/files/2011-Handwerker-Tesla-and-cold-electricity, p.20] says "MAGNETODIELECTRIC is the dual of ELECTROMAGNETIC" and (per sites.md) states a general rule f1 = (π/2)f0 for a Tesla coil's two resonances. My check: for two coupled resonators the modes are f± = f0/√(1 ∓ k) and their ratio is √((1+k)/(1−k)): 1.105 for k = 0.1, 1.225 for k = 0.2, 1.527 for k = 0.4. It equals π/2 = 1.571 only for k = 0.423. It is therefore a number that depends on the coupling, not a rule. (His 2011 paper's measured 2.755/1.810 = 1.522 corresponds to k = 0.397; printed 1.527 is the value for k = 0.4. Coincidence; no mechanism is offered.)
  • Other Handwerker documents (Verona deck, E-book "L.M.D. waves and quaternions", THOR detectors, sailboat deck) are known only by name or first pages; the E-book would be the document that could contain the missing derivation. Not read.
  • Verona 2011 demonstration (Heos, interested co-organiser, no measurements), Pizzuti's book (repeats the inventor), Tesla Society Switzerland hosting a deck, Japanese blog summary: accounts and endorsements, none a test. No replication found.
  • The record goes dark in 2013 (website replaced; every PDF returns 404); competing explanations (personal circumstances, loss of interest, commercial reasons, pressure with no evidence found either way) are set out in sites.md and are not decided by the mathematics. A possible later industry career (LinkedIn, same-person unconfirmed) is also recorded there. The absence of any Σ derivation in print by Handwerker after 2011 is consistent with each.
  • Patent ITVA20010043A1 (fluid-state monitor, 2001): an engineering career, not L.M.D. work.
  • Legal-removal notices (Google/Lumen): unknown which results or who asked.

5. What would settle the mathematics

  1. Write the L.M.D. field as an explicit solution with a stated fall-off exponent, angular dependence and shielding behaviour; compare it with the Hertzian-dipole near field for the measured geometry (a 250 mm coil, 50 mm top load, 2.755 MHz).
  2. Repeat Fig.3 and Fig.9 with raw readings, a grounded screen around the probe (a standard electric field is blocked; a non-Maxwell Σ field would not be), an independent magnetic probe, and a detuned off-run.
  3. For the water antenna: record voltage, current, frequency and brightness with and without water, dry glass, and a grounded shield.
  4. For Arbab-Satti: show a charge distribution that satisfies both (26) and (27) and is realisable (the static ball and circulating current examples do not).

6. Summary of verdicts

  1. L.M.D. waves from a Tesla coil: strong form Refuted by calculation and Never properly tested; weak (near-field) form Established.
  2. Quaternionic Maxwell incomplete / Σ waves: notation claim Refuted by calculation; Σ construction as presented Internally inconsistent.
  3. Field dielectric, not magnetic or transverse: weak form Established; strong form Not testable as stated and Never properly tested.
  4. Displacement current is weak: Refuted by calculation; Internally inconsistent.
  5. Water antenna: Never properly tested; mechanism Not testable as stated.
  6. Tesla as authority: longitudinal light Tested, not supported (and Historically superseded but instructive); 50 c Refuted by calculation; 215 GV Not testable as stated.