StClairJohn: Mathematician review (Step 2)

Reviewer role: PhD mathematician. I check derivations, notation, units and hidden assumptions, and I weigh the Investigator's findings. Tags: [read] = from the documents or the Step 1 deep reads; [calc] = my own computation in the project venv (SymPy, mpmath) in this session; [web] = checked on the web (URL given); [memory] = recalled, with confidence.

Source note. The corpus is five US patent applications by John Quincy St. Clair (Hyperspace Research Institute, San Juan, Puerto Rico), filed 2002-2004. No source gap: the text and the figure-page equations were read by the Step 1 deep readers from page images. Every conclusion rests on the inventor's own filings. The Investigator found no other person who tested anything (sites.md, "Supporting information by others": none). I re-read the deep reads and re-ran their load-bearing calculations. I did not re-render the page images myself. I rely on the deep readers' transcriptions of the equations and checked each against the algebra and the numbers they imply.

0. How the mathematics in these filings is built (read this first)

Almost every filing follows one pattern. (1) A correct textbook fragment is quoted: a unit check, a Faraday tensor, a wedge potential, a coaxial inductance formula. (2) A step is added that is not derived: a sign is assigned, a factor is inserted "to make the units work", a metric is written down by hand. (3) The new step is then described as a result of general relativity. Units balancing is treated as proof. Dimensional consistency is necessary but weak evidence. It cannot show that a spring, a force or a wormhole exists. Where I find an error below, I say whether it sits in step (1), (2) or (3).

1. Claims, questions a-d, verdicts

Claim 1. Oppositely wound coils of different radii give a negative "spacetime spring constant" and negative mass, which opens a wormhole to "hyperspace" (US 2003/0197093)

Plain-language set-up. A spring constant k is the restoring force per unit stretch (newton per metre, N/m). A mass on a spring oscillates at angular frequency ω = √(k/M). The patent builds a quantity k^z = I_θ · (n_θ Area) · K · B_θ from the coil current I_θ (amperes), the lamination cross-section Area (m²), a "curvature" K = 1/r² (r is the coil radius), and the magnetic field B_θ (tesla) [read: deep read §2, p.9].

a. Is the mathematics internally consistent? Split by step.

• Units, FIG. 7. Correct: A · m² · m⁻² · T = kg/s² = N/m [read; calc]. It works only because K is taken as 1/r². Standard curvature of a circle is 1/r (units 1/m), which would give newtons, not N/m. The patent's choice is the thing that makes the units come out [read: deep read §1].
• Hidden physics of the product. B = flux/area, so I · Area · B = I · Φ, where Φ is the magnetic flux. For a coil Φ = L·I (L is inductance), so I·Area·B = L I² = 2U, with U the stored magnetic energy [calc/standard]. So k^z = 2U/r² for each coil. That is positive for either winding direction, because reversing the winding reverses I and also reverses B (B follows I), so I·B does not change sign. The patent says the inner coil is negative "due to the triple product of negative signs of current, field and area" [read: p.9]. Physically at most the orientation convention for the area vector could flip, and the product of the three signs is then not -1 as a physical statement. The minus sign is assigned, not derived, and the energy form shows it is wrong in sign if the same physical definition is used for both coils. This goes beyond the deep read, which said only that the sign is put in by hand.
• FIG. 8 algebra. k ∝ 1/(r+a)² − 1/r² = −a(2r+a)/[(r+a)² r²]. With r=1, a=2 this is −8/9 [calc]. Correct arithmetic. It silently sets I·Area·B equal for both coils and drops it, so the result is a pure number in units of 1/length², not N/m as FIG. 7 says [read].
• The 8/9 = cos²(19.47°) coincidence. With normalised form (r/R)² − 1 and r/R = 1/3, the result is −(1 − sin²θ) = −cos²θ for sin θ = 1/3, so the match to the tetrahedral angle is guaranteed by choosing 1/3. It is not an independent confirmation [calc; agrees with deep read].
• Negative spring constant to negative mass. The patent argues (¶0019, ¶0027): ω = √(K/M); if M < 0 and K > 0, ω is imaginary; so to get a real ω, K must be negative and therefore M negative. This is backwards. With K < 0 and M > 0 the equation M x'' = K x has solutions growing and decaying like e^{±√(|K|/M) t}. That is an unstable, non-oscillating system, which is the ordinary meaning of a negative spring constant. Nothing forces M < 0. The step is a non sequitur. [calc/standard]
• FIG. 9 metric and FIG. 10 "negative mass". I re-derived G_tt for the printed metric (cylindrical coordinates t, r, θ, z; signature −,+,+,+; g_tθ = B(cos θ − sin θ) ≡ B f(θ)). SymPy gives G_tt = −B²f² / (r² + B²f²)², with f² = 1 − sin 2θ; this matches the deep reader exactly [calc]. I add a reduction the deep reader did not give. Since g_tθ depends only on θ, the substitution t' = t − B F(θ), with F' = f, is a legal change of time coordinate. The metric becomes ds² = −dt'² + dr² + (r² + B²f²)dθ² + dz². That is a static, time-independent geometry whose only feature is a curved 2-D spatial surface. G_tt then equals the Gaussian curvature of the 2-D metric dr² + (r² + B²f²)dθ², and I checked that this equals the SymPy G_tt identically [calc]. Consequences: (i) "negative mass" here means "this chosen spatial geometry has negative curvature". It is not computed from any coil, current or field. (ii) The text says the field varies "sinusoidally with time" with a 90° phase shift between the coils, yet nothing in the metric depends on t, so the coil frequency and phase play no role. (iii) At r = 0 the circumference of the "axis" is ∫B|f|dθ, not zero, so r = 0 is not a regular coordinate axis; the spikes are artefacts of evaluating an ansatz where the polar coordinates are already degenerate. (iv) A magnetic field enters Einstein's equations through the stress-energy tensor, with coupling 8πG/c⁴. It cannot be inserted into g in tesla. B has no conversion factor in FIG. 9 [read]. (v) The real electromagnetic energy density B²/(2μ₀) is positive, so the physical G_tt sourced by a coil is positive, not negative [memory, high; standard].
• Wormhole. The only support is a citation to Kip Thorne's paper, not reproduced in the application [read]. The Morris-Thorne result is that a traversable wormhole throat needs exotic matter (violating the averaged null energy condition) [web: https://en.wikipedia.org/wiki/Wormhole]. That is a requirement on a wormhole. It does not say that some negative-G_tt source produces one.
• Numbers. No coil current, turns, field, frequency or mass is ever given in physical units. The drive description (thousands of volts, SuperMalloy tape, <20 MHz) has no connection to any equation [read: deep read §9].

b. Consistent with established experiments? Nothing here is a number that could be compared. Known: negative inertial mass has never been observed; a pair of opposed coils behaves as ordinary inductors (stored energy positive) [memory, high]. The "bucking" geometry (a bifilar coil) is standard in non-inductive winding; bucking cancels the net dipole field rather than creating anything [memory, high].

c. Tested? Not by anyone as far as the record shows. The application was examined (six examiner citations) and is now "Abandoned" [web: Google Patents, via the Investigator, research/2026-10-04-google-patents.md]. The reasons for abandonment are in the USPTO file wrapper, which has not been fetched. The inventor's own sister application US 2004/0102810 reports a flat bifilar bucking coil driven at ten volts, the energy "felt as it rises through the feet", and a nine-second body rocking [web: Investigator note, FPO]. These are subjective body sensations with no instrument, control or second observer stated. I record them; I cannot analyse them mathematically.

d. If true, what in mainstream physics changes? Negative inertial mass; the sign of the energy of a magnetic field; the energy conditions of general relativity (a macroscopic exotic-matter source made with a laboratory coil); and a second spacetime sector ("hyperspace" with different constants) coupled to ours.

Verdict, Claim 1: Internally inconsistent for the derivation chain (the sign is assigned, negative mass is a non sequitur, the metric is not tied to the coils). The physical claim (a wormhole/negative mass from such a coil) is Never properly tested; the application was examined and abandoned, and there is no experiment on record.

Claim 2. Hyperspace energy with very low speed of light and huge permittivity from a braided gold-wire coaxial cable; radii 8.34 and 7.56 micron follow from tetrahedron geometry; a photograph shows "white mist" (US 2004/0164824)

a. Mathematics consistent? The cable derivation uses four constraints on outer radius a, inner radius b and a "projected" radius c [read: deep read]. I re-ran them with mpmath [calc]:

• Constraint 1: exp(ln(a/b) + 1) = 3, so a/b = 3/e = 1.10364. The patent's a/b gives 2.9999984. Satisfied.
• Constraint 2: [ln c − ln a]² = 10. With the patent's a and c = 0.3596358547 micron the left side is 9.885, not 10 [calc]. Not satisfied.
• Constraint 3: (ln a)² / (2π ln(a/b)) = (ln c)². Left 220.71, right 220.17 [calc]; and it only balances if lengths are in metres. In centimetres it is 81.1 vs 104.7; in microns 7.26 vs 1.05. A logarithm of a quantity with units depends on the unit, so this equation is not a physical statement [calc].
• Constraint 4 (inductance and Planck mass): exp(ln m_P − ln L + 1) = 3 with L = 2×10⁻⁷ ln(a/b) H/m gives 2.99972 [calc]. It depends on a and b only through ln(a/b), which constraint 1 has already fixed. It "holds" to 1×10⁻⁴ because m_P (2.1764×10⁻⁸ kg) happens to be within 1×10⁻⁴ of (3/e)·2×10⁻⁷·ln(3/e) when kg and H/m are the units. That is a numerical coincidence of the SI unit choice. It is one match chosen from a very large menu of constants, relations and units; with that menu a 10⁻⁴ coincidence is not surprising (look-elsewhere effect). It does not constrain the cable.
• Reproducing a and b. Solving constraints 1, 2, 3 together with c free gives a = 8.3412 micron, b = 7.5579 micron, c = 0.3531 micron [calc]. That agrees with the patent's a = 8.34246 and b = 7.55906 to 1.5×10⁻⁴ (about 1 nm). So the radii come from a hidden choice (c solved for, logs of metres) and not from the stated c = 0.3596 micron. The ten printed significant figures are not supported: the inputs (h, G, unit choice) already differ in the 4th-5th digit.
• Physics link: the claim that a high permittivity gives a low speed of light is just c = 1/√(μ ε), true in a dielectric [memory, high]. A refractive index of 33,000 (needed for 8971 m/s) in a vacuum-filled micron-scale coax is not the property of any medium. The coax inductance L = (μ/2π) ln(a/b) is the standard textbook line inductance [read; memory, high].

b. Consistent with experiment? The patent reports one photograph (FIG. 1). Per the deep reader's image reading: a small, dark, low-resolution halftone of bright cloud-like patches, no visible braid or cylinder, no scale bar, magnification, voltage or instrument [read: deep read]. In scanning electron microscopy, bright diffuse halos around non-conducting or contaminated objects are a known charging artefact [memory, high that this is a general SEM phenomenon; it is not a finding about this image]. The only experimental sentence is "Electrical experiments ... show that it is capable of generating substantial amounts of hyperspace energy" with no drive, no measurement and no control [read: ¶0002].

c. Tested? Only by the inventor, in one sentence. No replication, no named lab, no instrument data. The Investigator found related self-reports in sister filings [web: FPO; Investigator note] (2004/0200925 sonoluminescence rig and "soft white misty look"; 2007/0285325 pendulum data). They are described in §3 below.

d. If true: Maxwell's equations in vacuum with fixed c; the constancy of c as a property of vacuum; a second "co-dimension" with its own constants, supplied through a 17-micron wire.

Verdict, Claim 2: Never properly tested (one uninstrumented photograph, one sentence of method). The geometric derivation of the radii is not reproducible as stated (a 10-digit result from constraints that are unit-dependent and mutually inconsistent at the 1% level); that part I classify as internally inconsistent.

Claim 3. The tetrahedron diagram (ln mass vs ln wavelength; base ln(h/c) = −95.91546344) derives the constants of physics, shows 10 non-redundant dimensions and shows that electron and proton are the same particle (US 2004/0164824, 0197093)

Definitions. h = Planck's constant, c = speed of light, λ = Compton wavelength h/(mc) of a particle of mass m, m_P = Planck mass, Λ = reduced Planck length ħ/(m_P c).

a. Mathematics consistent? All numbers I checked, with SI units, agree with the printed values [calc]:

• ln(h/c) = −95.915482 with CODATA 2018 h; the patent's −95.91546344 differs by 1.9×10⁻⁵, implying h = 6.626195×10⁻³⁴ J·s, a value that matches no CODATA edition I recall [memory, medium]. Unimportant for the argument.
• The key identity is a definition. For any particle, m · λ_Compton = (h/c). The statement "electron mass × wavelength = proton mass × wavelength = h/c" is therefore true of every particle with that definition. It is not a discovery, and it says nothing about the electron and proton being one particle [calc; deep read agrees].
• The Planck "box" relation (ln m_P + ln 2πΛ = base) is a rewriting of the definition Λ = ħ/(m_P c) [calc].
• Logs of dimensional quantities depend on units. The whole diagram uses ln of numbers in kg and metres (SI). In g and cm every "geometric" relation changes. The patent never says this. Constructions that use a 19.47° line, a circumscribing circle and mirrored tetrahedra on this plot have no unit-independent meaning.
• The nested radical. The "ten dimensional path" gives √(3a √(3b √(3c … √(1j)))) = 80.1104395, "the Planck scale". The target is −ln Λ (with the patent's Ω = c²/G, i.e. G = 6.6732×10⁻¹¹, I get 80.110449) [calc]. Using the printed constants (a=|ln m_e|=69.17083, b=|ln λ_p|=34.26006, 4π, ln 2π, 0.599547652, √2, √10/10, h = ln(m_P c)−1 = 0.8757, 2π/10, √2) I get 80.3027, 0.24% high, and could not reach 80.1104 under any of the readings I tried (squared or plain edge lengths, either sign, a = electron wavelength instead of mass) [calc]. The constant "e = 0.599547652" has no derivation ("related to fractal dimension and the speed of light factor"). It would have to be 0.5553 to give 80.1104395. The "sum equals 1" relation Σ s_n / (2π ln c) = 1.000000000 gives 1.0036 with the same inputs, and would need e = 0.1553 [calc; deep read]. No single value of e satisfies both relations.
• Sensitivity. In a nested square root, constant number i from the outside enters with exponent 2⁻ⁱ. So a changes the result with weight 1/2, b with 1/4, … the tenth with 1/1024. A match to six or seven digits therefore cannot test the later constants: a 7% error in e moves the total by only 0.24% [calc]. This also means one free constant (e) could be tuned to hit a target, and the text does not exclude that.
• Dimension counting. 24 = 2·(4+4+4) edge-traversals of three squares and 10 = the edges in one path are correct arithmetic. Equating edge counts with spatial dimensions is asserted. The attribution to Edward Witten (24 dimensions, 10 non-redundant) is uncited. What I know: bosonic string theory needs 26 spacetime dimensions (24 transverse to the light cone) [web: https://en.wikipedia.org/wiki/Bosonic_string_theory, 26; the "24 transverse" is 26 − 2 [memory, high]]; superstrings need 10 [memory, high]. A statement "space has 24 dimensions, 10 non-redundant" is not one I can match to any string-theory result.
• The tetrahedral 3:1 identities (area/volume ratio of a tetrahedron to its circumsphere = 3; three-square/one-square area = 3; asin(1/3) = 19.4712°) are each correct [calc]. Linking them to each other and to turbomachinery is asserted.

b. Consistent with experiment? The constants used (electron, proton, Planck) are the measured ones, so the diagram cannot disagree with them. It also cannot predict them, because they are the inputs. The electron and the proton differ in charge sign, spin-statistics bookkeeping aside, and in lepton/baryon number and interactions (the proton is a bound state of quarks [memory, high]). "Same particle entering our universe at two places" makes no measurable prediction that differs from the standard picture; it is not testable as stated.

c. Tested? The inventor treats a copyright deposit "The tetrahedron diagram" (1996) as the record. No one else has tested or discussed it in the record the Investigator found. The deposit copies are held by the Copyright Office and have not been read [web: research/2026-10-04-copyright-office.md]; the nested-radical equation and e = 0.599547652 are the only derivation I could check, and they do not reproduce.

d. If true: the Standard Model's free parameters (the mass ratios) would be derivable from geometry; the proton and electron would be two views of one object, contradicting proton structure measured in deep inelastic scattering [memory, high].

Verdict, Claim 3 (split): - Numerical agreement of the quoted constants with CODATA: Established (they are the standard numbers). - "Derives the constants": Internally inconsistent (the printed radical and sum do not reproduce; units; the free constant) and, where it matches, it matches because the constants were inputs. Not a prediction. - "Electron and proton are one particle": Not testable as stated; the supporting identity m·λ = h/c is a definition that holds for any particle. - "10 non-redundant dimensions": Not testable as stated; arithmetic on edge counts.

Claim 4. Crossed electrostatic fields, a static field plus a radio wave, or field angular momentum in a metric produce net lift or propulsion as spacetime curvature pressure, with no reaction mass (US 2003/0209637, 2003/0230675, 2006/0145019)

Three devices, three separate mathematical failures.

(i) Rotating electrostatic thruster, 0209637. Symbols: E_r, E_z are radial and vertical electric fields; ε₀ is the permittivity of free space; Ω is the "linear mass of the universe" c²/G; area is the area normal to r. - FIG. 3 units: T^{zr} = −(ε₀/(Ωc²)) E_zE_r/4π = −1/m². Correct if Ω is in kg/m [read; calc]. Ωc² = c⁴/G = 1.21×10⁴⁴ N (the Planck force), so ε₀E²/(Ωc²) = Gε₀E²/c⁴, the correct order of the GR coupling. With E = 3×10⁶ V/m (air breakdown) this is 5×10⁻⁴⁴ m⁻² [calc]; the inventor himself says that is "too small even to notice" [read: ¶0022]. - Unit-system mix: FIG. 6 is the Gaussian form −E_rE_z/4π. FIG. 3 puts ε₀ (SI) and 4π on the same quantity. In SI the shear stress is ε₀E_zE_r with no 4π. A spurious factor 1/4π ≈ 0.08. The unit check cannot catch a pure number [read; calc]. - FIG. 7 "permutation tensor" ε_{rtz} has three indices, one of them time, in a four-dimensional space whose Levi-Civita symbol has four. The angular momentum flux is x^α T^{βγ} − x^β T^{αγ}; the patent keeps one term. FIG. 8's units show the right side is a momentum (kg m/s) while the left side is labelled d/dt S, a rate [read; calc]. Notation internally inconsistent. - "Time lever arm": using x^t = t makes the force depend on where t = 0 is set. Shifting t → t + t₀ adds a term proportional to t₀. A physical force cannot depend on the choice of time origin. The second term in FIG. 9, (iε₀/4π)E_rE_z t ω e^{iωt}·area, grows without bound with t [calc]. "Imaginary newtons" have no physical meaning; one takes the real part, cos ωt − ωt sin ωt [calc]. - Direction: T^{zr}n_r dA is the z-component (vertical) of force through a surface with radial normal. The device claims a horizontal force [read; standard stress-tensor convention]. - Curvature (Ω, c) never enters FIGs 6-9, so the "force" formula is plain electrostatics times area. - The inventor concedes a continuous radial field gives zero net force; segmentation is his fix [read: ¶0030]. Forces between the charged cylinder segment and ring segment are internal. Net momentum change of a closed system equals the radiated momentum (≤ P/c) plus the change of field momentum, which averages to zero over a cycle [memory, high]. - Size illustration [calc]: ε₀E² = 79.7 N/m² for E = 3×10⁶ V/m, so 1 m² gives 80 N (6.3 N with the spurious 1/4π). These are formula values only; the document gives none.

(ii) Rotor inductance, 0230675. - Vector potential of an infinitely long solenoid A = nIa²/(2ε₀c²r') (FIG. 5): textbook (1/(ε₀c²) = μ₀); applied to short single loops, where it does not strictly hold [read; memory, high]. Units FIG. 6 and FIG. 11 are correct [read]. - Imaginary angular momentum. S = −σ r (e^{iωt} − e^{−iωt}) × … = −2iK sin ωt. The factor r e^{iωt} is a position written as a complex number. The physical lever arm is its modulus r, constant. Subtracting the complex positions of two counter-rotating charges gives i times a difference of y-coordinates, i.e. an imaginary number made from a bookkeeping device. The real parts cancel. A metric must be real, so FIG. 15 (g_tθ = 4iK sin ωt / r) is not a metric [memory, high]. The physical electromagnetic angular momentum L_z = q r A_θ of two opposite charges at the same r in nearly the same A cancels to a small constant [memory, high; Feynman Vol. II §17-4]. - I re-ran the Einstein tensor [calc]: G_zz = 4q² sin²ωt (23 r⁴ + 112 q² sin²ωt) / [ r² (r⁴ + 16 q² sin²ωt)² ] (matches deep read). For real q, G_zz > 0 everywhere: no negative spike, no pole. With q = iK the denominator vanishes at r² = 4K|sin ωt|, giving the "huge spike" of FIG. 16. The spike exists only because the angular momentum was made imaginary. - Units of the metric: g_tθ must be a length; K is in kg m²/s. The conversion G/c³ ≈ 2.5×10⁻³⁶ s/kg [memory, high] is never applied (the patent says only "angular momentum is viewed as meter squared"). Order of magnitude with generous lab numbers (σ ≈ 2.7×10⁻⁵ C/m², rotor area 0.5 m², r = 1 m): S ≈ 2×10⁻⁷ kg m²/s and GS/c³ ≈ 5×10⁻⁴³ m² [calc, deep read]. Even on the patent's own imaginary assumption the pole sits ~10⁻²¹ m from the axis. - Magnetic energy FIG. 17: −L K J² A² "taking a hint". K (1/m²) is inserted because (JA)² has units A²m² and K turns it into A². ½ of ½LI² is dropped. Postulated [read]. - Inductance gradient and lift. FIG. 21: dL/dr = μn²·2πw ∂/∂r[r h(r)]. The differentiation is right. But the patent defines "one turn per height of the groove" (¶0010), so n = 1/h(r), not constant. Then L = μ(1/h²)(2π r h w) = 2πμ r w/h and dL/dr = 2πμ w [1/h + r(h₀−h₁)/(W h²)] > 0 for depth decreasing outward. With the patent's own definition the sign of the gradient reverses [calc; deep read §4(e)]. Also, a derivative with respect to r is a radial generalised force, not a vertical one. The grooves are fixed to the rotor and do not move radially, so no work is done: F = ½I² dL/dx applies to a part of a circuit that is free to move along x [memory, high].

(iii) Triangular spacecraft, 0145019. - Laplace wedge potential. Steps 1-9 are correct textbook electrostatics (Jackson §2.11) [memory, high]. For β = 300° the exponent is π/β = 3/5 and σ ∝ ρ^{−2/5} [calc]. This is an integrable edge singularity: charge in a strip ρ₀ wide goes as ρ₀^{3/5} → 0, so there is no finite "huge line charge", and in air the field is capped near 3 MV/m by breakdown [memory, high]. Printed line potential −(λ/2πε₀) ln((x−x₀)²+(y−y₀)²) is twice too large; the correct form is −(λ/2πε₀) ln r = −(λ/4πε₀) ln(r²) [calc]. - Differential-forms force. ∗d(E ∧ (B_w + E_w ∧ dt)) ε/c: counting degrees, E is a 1-form, the bracket is a 2-form, the wedge is a 3-form, d gives a 4-form, ∗ gives a 0-form: a scalar. A force density needs three components [calc; deep read]. Not derived and mixes SI and c=1. - Stress tensor. With the printed Faraday tensor (static E_x plus wave E_z cos(x−t), no magnetic field) I computed ∂_ν T^{μν} [calc]: t: −E_z² sin(2t−2x)/(8π); x: +E_z² sin(2t−2x)/(8π); z: −E_xE_z sin(t−x)/(4π). Nonzero. A wave with E_z needs a companion magnetic field B_y (Faraday's law). Including it in the consistent sign, all four components of the divergence are exactly zero [calc]. So the printed tensor is not a Maxwell solution, and the "forces" it yields come from the omitted B. The oscillating cross stress −E_xE_z cos(x−t)/(4π) averages to zero over a cycle. What remains is the wave's own radiation pressure (recoil P/c: 3.3 micronewtons for 1 kW [calc]) independent of the corner charges. Maxwell's equations are linear in vacuum, so a static field and a wave add without a mean mutual force [memory, high]. - The inventor's own text gives the wave field as perpendicular to the static field, so E·E_W = 0 (stated in the deep read).

b. Consistent with established experiments? Conservation of momentum, with field momentum included, is among the best-tested laws; closed electrodynamic systems have not shown reactionless net thrust [memory, high]. A force of order 0.03 N/m² (ε₀ × 3 MV/m × 1 kV/m [calc]) at zero mean is an oscillation, not a thrust. Torsion-pendulum tests of "asymmetric capacitor" thrust (Biefeld-Brown "lifters") attribute the effect to ion wind in air; I recall NASA/others finding no thrust in vacuum [memory, medium; not checked on the web, not essential to the math verdict].

c. Tested? No device, voltage, mass or thrust is given in any of the three applications [read]. No test on record. Related filings by others (Pais, US Navy, "inertial mass reduction" etc.) are discussed in §3.

d. If true: conservation of momentum in electrodynamics (field plus matter) would have to fail, and the Maxwell stress-energy tensor would have to produce a net force on its source.

Verdict, Claim 4: Refuted by calculation for the three mechanisms as written (0145019: the vanishing divergence and zero time average; 0209637: closed-system momentum conservation plus the vertical-versus-horizontal and time-origin errors; 0230675: imaginary angular momentum is a complex-number artefact and the energy gradient has the wrong sign under the patent's own n). Honest scope: this refutes the derivations given. It does not refute every conceivable use of fields for propulsion (radiation recoil exists, though far too small to matter), and no experiment was ever run on these designs, so "tested" would be wrong.

Claim 5. The force is amplified enormously by hyperspace permittivity (stated "about a trillion"; the inventor's own two speeds give a billion) or by an inductance gradient on grooved rotors acting with negative curvature (US 2003/0209637, 2003/0230675)

a. Mathematics consistent? - The "trillion". Hyperspace c = 8971 m/s; our c = 299,792,458 m/s; permittivity ∝ 1/c², so the factor is (299792458/8971)² = 1.117×10⁹ [calc]. The stated 10¹² does not follow from the document's own inputs. The "enclosed reference calculation" is not in the published application [read], and the 8971 m/s figure is said to come from the tetrahedron diagram, which I could not tie to any printed edge, so its derivation is unsupported. (e^R with R = 14.838 gives 3.6×10⁻⁷, not a speed.) - Whether larger permittivity ε increases force depends on what is held fixed. At fixed voltage V across a gap d, E = V/d does not depend on ε, so stress εE² grows with ε. At fixed charge Q, E = Q/(εA) falls as 1/ε and the stress Q²/(εA²) falls [calc/standard]. The patent states neither condition. A uniform increase in ε of a medium that permeates the device would also lower the speed of light for everything in it, including the fields producing the force [memory, high]. - The magnets exist in the claims only to bring "hyperspace permittivity" (claim 2, claim 8 of 0209637). Their role is the same as the negative spring constant of Claim 1: a pre-requisite stated, not shown. - The groove "negative inductance gradient × negative curvature × J²A²" sign bookkeeping rests on the imaginary-angular-momentum curvature (Claim 4 ii) and on a gradient whose sign reverses under the patent's own n = 1/h (above) [calc]. Therefore the "positive lift" has two broken signs. - Size illustration [calc]: if the first term of FIG. 9 gave 6.3 N per m² at 3 MV/m, a 10⁹ factor would give ~7×10⁹ N, comparable to the thrust of a launch vehicle from a hardware item with no stated power supply. Energy would also have to be supplied: with no stated power or time scale, the amplification claim is not even an energy accounting.

b. Experiments? None cited. Nothing measured.

c. Tested? No.

d. If true: the speed of light and the permittivity of vacuum would be locally changeable by a factor of 10⁹ without any change in the field-free spectrum of atoms, which is highly constrained [memory, high: tests of constancy of c and the fine-structure constant].

Verdict, Claim 5: Internally inconsistent (the amplification factor contradicts its own inputs by 10³, the sign chain fails, the fixed-charge versus fixed-voltage condition is missing). The underlying physical claim (hyperspace permittivity) is Never properly tested.

2. Corrections to Step 1

I verified Step 1 (facts.json, claims.md, deep reads). It is accurate. Items I confirmed independently: G_tt for the FIG. 9 metric; G_zz for the FIG. 15 metric; the 1.12×10⁹ ratio; the failure to reproduce 80.1104395 (I get 80.3027); the sum to unity (1.0036); radii a, b from constraints 1-3 with free c; the divergence of T for the printed Faraday tensor in 0145019; the factor 2 in the line-charge potential. Corrections and additions:

1. claims.md, claim 2 says the radii "follow from a tetrahedron geometry". More exactly: they are reproducible to 1.5×10⁻⁴ only by a solution (constraints 1-3 with c free) that the text does not state and that contradicts the printed c = 0.3596 micron.
2. claims.md, claim 5 repeats "about a trillion times". It reports the inventor's number, correctly flagged; keep it, but the verdict above should carry the 10⁹ vs 10¹² mismatch.
3. Deep read of 0197093 §8: "the sign is put in by hand ... three sign flips". Add: I·Area·B is I·Φ = L I² (twice the stored magnetic energy), which is positive for either winding direction (I and B reverse together). So the assigned minus sign cannot come from reversing the winding. This strengthens, not reverses, the deep read.
4. Deep read of 0197093 §7: "At r = 0 and θ = π/4, 5π/4, det g = 0 and G_tt diverges". Exactly at f = 0 and r = 0 the expression is 0/0; the spike minimum is G_tt = −1/(4r²) where B|f| = r, near but not exactly at π/4. The qualitative conclusion stands. Beyond that reading, the metric reduces to a static geometry by t' = t − BF(θ), so the "negative mass" is the curvature of a chosen spatial surface (my addition, §1 Claim 1).
5. Deep read of 0164824: the observation that the radical's target equals −ln Λ holds with the patent's own Ω (G = 6.6732×10⁻¹¹, which Ω = 1.346812891×10²⁷ kg/m implies: c²/Ω = 6.67320×10⁻¹¹ [calc]) to 1×10⁻⁵ (80.110449 vs 80.1104395). With CODATA G = 6.67430×10⁻¹¹ [memory, high] Ω would be 1.34659×10²⁷. Not a Step 1 error, a refinement.
6. Add the sensitivity point (weight 2⁻ⁱ) and the look-elsewhere point for constraint 4; neither is in Step 1.
7. Nothing in Step 1 needed to be withdrawn. The note in the 0145019 deep read that the question's formula f = (1/2c²)ε₀(E·E_W)E_W is absent from the document stands; I did not find it either.

3. The Investigator's findings, weighed mathematically

These stay in the record whatever the verdict. All are [web] from the Investigator's notes unless marked.

• Inventor's own experiments in sister applications (bears on Claims 1-2):
• US 2004/0200925: a person spun on a palletizing ring "99 times per day for a few months", right arm said to stay raised "without making any exertion" [Investigator: FPO]. No count of subjects, no measurement, no control.
• US 2004/0102810: bifilar coil at ten volts, felt through the feet, nine-second body rocking. Subjective.
• US 2007/0285325: pendulum beside an 8.5-inch clay dome with a slot antenna. ¾-inch alternating swings every 20 kHz from 20 to 120 kHz; a six-inch swing at 2.78 MHz, compared with e^14.838168 Hz = 2.7806 MHz from the diagram. I computed e^14.838168 = 2,780,574 Hz and e^14.83817383 = 2,780,590 Hz [calc]. So the figure matches the quoted 2.7806 MHz to 5 digits, but that is only the arithmetic. The input R = (√(4/3) − 1)·ln(h/c) is a dimensionless number built from logs of SI numbers; reading it as ln of a frequency in hertz is a unit choice (exp(R) = 2780.6 if the unit were kHz, which would mean 2780.6 kHz = the same number only by relabelling; the diagram gives no reason to choose Hz over another unit, and the same R read in another unit would name a different frequency). The size of the match depends on how wide the pendulum response is and how many frequencies were scanned; neither is reported. A response that alternates every 20 kHz over a 100 kHz range suggests structure (cable, antenna or mechanical resonances) at many frequencies. Without the width and a blind control I cannot say the 2.7806 MHz coincidence is more than chance. This is a statement about the missing information, not a finding that it is chance.
• Ω = c²/G = 1.346812891×10²⁷ kg/m [Investigator: FPO]. Useful: it fixes the factor ε₀/(Ωc²) in 0209637 (checked above) and equals c²/G for G = 6.6732×10⁻¹¹ [calc].
• Beaconing sonoluminescent bubble and Horsburgh thesis (Claim 1/2 background): the inventor derives f_drag = 2.83 Hz from ω_drag = (4/3)(G/c²)(m/R)ω modified by the fine-structure constant, and calls it "close to the 3 Hz observed". The unmodified formula is the standard Thirring frame-dragging for a rotating shell [memory, high]. Magnitude: G·m/(c²R) is about 7×10⁻²⁸ for 1 kg at 1 m, and smaller for lighter objects [calc]. A laboratory object therefore drags frames at ~10⁻²² rad/s for ω ~ 100 rad/s. Reaching ~3 Hz would need an extra factor near 10²². The fine-structure constant (1/137) does not supply that. I have not read the filing text (only the Investigator's note), so I cannot say which m, R and ω the inventor inserted. The Horsburgh thesis itself has not been read; what it reports about a ~3 Hz precession should be checked directly.
• Copyright registrations 1996-2008: "The tetrahedron diagram" (VAu000372532, 1996-07-11) and later items. They show the diagram is older than the patents and was kept up for twelve years. The deposits are unread and are the inventor's fullest statement of Claim 3.
• Full filing family: 16+ applications; all abandoned, US only. The mathematical issues above recur in the sister filings by his own reuse of the same moves (Ω, tetrahedron constants, bucking coils); I have not read them, so I do not extend the verdicts to them.
• Related work, Jefimenko / gravitoelectromagnetism: the "co-gravitational K field" with ∇×K = −(4πG/c²)J + (1/c²)∂g/∂t uses Jefimenko's name and symbol [web: https://en.wikipedia.org/wiki/Oleg_D._Jefimenko, via Investigator]. Gravitoelectromagnetism is the weak-field limit of general relativity and is legitimate mathematics [memory, high]. It supports the idea that moving mass-energy has a "gravitomagnetic" effect; at laboratory field strengths the effect is of order G/c⁴ × energy density (the 10⁻⁴⁴ m⁻² computed above). It does not rescue the claims by itself.
• Absence of evidence, historical record. The record goes quiet after 2008. The Investigator set out competing explanations: (a) a private one-person effort that stopped; (b) the inventor moved to Madison, Wisconsin and continued privately; (c) death or incapacity. Evidence for (a) is self-filing, no assignee and a forum remark; for (b) a LinkedIn profile and Minnesota clay in a 2006 filing; for (c) nothing about the inventor himself (an anonymous Reddit snippet is about the father). The records that would decide are the USPTO file wrappers, the Puerto Rico registry and obituaries; none reached. As a mathematician I note one thing the file wrappers could answer: whether an examiner raised the sign, units or momentum issues above, and whether the inventor replied.
• Commentary (IPWatchdog 2013, unmakeme 2021, forums linking the triangular craft to "TR-3B" lore): characterisations by commentators, not tests.

4. Verdict summary

What would change my mind (one calculation or experiment each): for Claim 1, a physical derivation of the g_tθ term from the coil's stress-energy tensor with the 8πG/c⁴ factor; for Claim 2, a published scale-calibrated micrograph of the cable with drive conditions and a powered/unpowered control; for Claim 3, the Copyright Office deposits showing the derivation with units fixed and a prediction of a not-yet-used constant; for Claims 4-5, a thrust measurement on any of the three designs in vacuum with a momentum-conservation check, and in the 0230675 device a field-solver check of the sign of dL/dr using the patent's n = 1/h.

Web checks done: Wikipedia "Wormhole" (exotic matter for Morris-Thorne wormholes), Wikipedia "Bosonic string theory" (26 dimensions). Not checked on the web: CODATA edition values for h (medium), Biefeld-Brown vacuum tests (medium), SEM charging artefact (high); none of the verdicts depends on them.

Scratch computations: /tmp/stc_math/ (a.py to f.py). Output: review/StClairJohn/mathematician.md