Mathematician review: Znidarsic-Frank (Step 2)

Reviewer role: PhD mathematician. Scope: derivations, notation, units, hidden assumptions. Sources: "The Control of the Natural Forces" (Znidarsic, rev. 2/2011, 15 pp.) and "The Flow of Energy" (Znidarsic and Robertson, Physics Procedia 20 (2011) 457-464). Tags: [read] from the document, [calc] my own computation (NumPy, CODATA 2018/2019 constants), [web] with URL, [memory] with confidence.

No source gap: both papers have a text layer. Equations were read from page images (Step 1 pp.2-8 of each; I re-read Control pp.9-10 as images, Eqs. 24-31, and the printed text of Control pp.9-15 including Eqs. 32-41 and the "New Technologies" section, which Step 1 had covered by digest only).

0. The one-paragraph result (what a reader needs first)

Everything numerical in both papers hangs on one number, Vt = 1.094e6 m/s. I recomputed it: Vt = e^2/(4 eps0 h) = alpha c / 2 = 1,093,845.6 m/s, where alpha = e^2/(4 pi eps0 hbar c) = 1/137.036 is the fine-structure constant [calc]. This is exactly half the speed of the electron in Bohr's ground-state hydrogen orbit (alpha c = 2.188e6 m/s). The papers present it as an observed nuclear sound speed that "produces" Planck's constant h, the Bohr radius a0 and alpha. In fact the tabulated Compton frequency fc = me c^2/h already contains h, so h, a0 and alpha go in with Vt and come out again. The algebra is mostly correct; the logic is circular. One step (Eq. 6, the "nucleon on a spring") is not an identity and is a numerical match to 0.02-0.1% that I examine in section 3.

1. Verification of Step 1 (what I re-did)

[calc] All reproduced: - re = 2.81794e-15 m; Fmax = e^2/(4 pi eps0 re^2) = 29.0535 N; rp = re/2 = 1.40897e-15 m (no h in these). - 2 pi fc rp = 1,093,845.63 m/s, identical to e^2/(4 eps0 h) to 15 digits. (Eq. 1 of Control = Eq. 1 of Flow.) - Fmax rp^2 = 5.7677e-29 J m; e^2/(4 pi eps0) = 2.3071e-28 J m; ratio exactly 4.000. So Control Eq. 3 (E = K-e rp^2 with K-e = Fmax/rx) gives one quarter of the Coulomb energy of Eq. 2. - Bracket of Control Eq. 19: Fmax rp^2/(Vt^2 me) = 5.29177e-11 m = a0 (the factor 4 missing in Eq. 3 cancels the factor 4 in Vt^2 = (alpha c)^2/4). - fe (Flow Eq. 26) = Vt/(4 pi a0) = 1.644921e15 Hz = half the Rydberg frequency (R_inf c/2). rJ (Flow Eq. 27) = 8 pi a0 = 1.32997e-9 m. - Control Eq. 31 bracket: 4 pi Fmax rp^2/Vt = e^2/(4 eps0 Vt) = h exactly (again an identity).

Step 1 findings I confirm without change: the Vt identity; the factor-4 slip in Eq. 3; "n cancels" in Eq. 1; the e = 1.60e-19 (stated) versus exact e (used) in Flow Eq. 13; the e = 3.2e-19 C (stated) versus 1.6e-19 C (used) in Flow Eq. 27 (stated value gives 3.33e-10 m, not 1.33e-9 m); nx/ni ~ 1e-24 not reproducible (the paper's own Eq. 1 gives 2.8e-8); the Robertson product 1e14 x 1e-9 = 1e5 m/s, not 1e6; fe being 100-1000 times the stated LENR frequency; undefined rp, M_-e, nB, omega_2e in Flow.

2. Corrections to Step 1

1. Eq. 6 requires Mn and rn together, not rn alone, and the match depends on which mass. Step 1 says the 1.36 fm value matches "to about 0.1%". I find: with Mn = 1.67e-27 kg (the paper's table) Eq. 6 gives Vt = 1.09483e6 m/s (+0.09%); with the proton mass 1.67262e-27 kg it gives 1.09397e6 (+0.014%); with the neutron mass used in Flow Eq. 5 (1.6749e-27 kg) it gives 1.09322e6 (-0.06%). The "required" rn for exact alpha c/2 is 1.3576 fm, 1.3597 fm or 1.3616 fm respectively [calc]. So the quality of the match is set by the mass choice; all three are within the rounding of "1.36". This is a small correction, but it shows the paper's two versions (Control: nucleon; Flow: neutron) are not the same calculation.
2. Step 1's remark that the factor-4 error "does not show up in the hydrogen result" is right but incomplete. The more serious point is the reason: Eqs. 18-19 only work because the "spring" Fmax/rx is multiplied by the lever arm rp = re/2 instead of the orbit radius rx. I show this in section 4 (Claim 3).
3. Control Eq. 32 was not examined in Step 1. I check it in section 4: it is an identity again, and it differs by a factor of 2 from the standard mean-square displacement.
4. The "Step 1 vs claims.md" statement that the Josephson 1.33 nm "shares" the LENR transition speed needs a sharper form: 1.33 nm is not a measured junction scale in the paper; it is 8 pi a0, computed from atomic constants, with no data on any junction.
5. Step 1 flags the Poole 1e6 m/s statement only as a lead. I add the mathematics: 1e6 m/s is the order of the Fermi velocity of conduction electrons (a few eV kinetic energy) and the Bohr velocity scale; it has no connection to the lattice sound speed (section 4, Claims 1 and 4).

3. Claim by claim

Claim 1: Vt = 1.094e6 m/s, observed in LENR and Podkletnov-type experiments, is the speed of sound in the nucleus

a. Internal consistency. Not consistent. - Vt is defined two ways: (i) as angular frequency x length, Vt = (2 pi fc/n)(n rp), in which n cancels so the "harmonic" does nothing [read: Control p.2, Eq. 1; calc]; (ii) as frequency x length for Podkletnov: 3 MHz x (1/3) m = 1e6 m/s, with plain f and a diameter, no 2 pi [read: Control p.2; calc]. These differ by 2 pi in the definition (and by a factor 2 between radius and diameter). A velocity "observed" in two experiments must be computed one way. - The "observed" quantity is an order of magnitude only. 50 nm x omega with omega in 1e13-1e14 rad/s spans 5e5 to 5e6 m/s [calc, Flow Sec. 2.1]; the four-digit 1.094e6 comes from fc and rp, not from the lab. The 1997 letter gives a third form, f (MHz) x L (inches) = 37, i.e. 9.4e5 m/s [web: Investigator, https://web.archive.org/web/20080105120935/http://padrak.com/ine/NEN_5_9_7.html]; it rests on one 11 MHz, 3.5-inch point that the inventor says he could not repeat. A one-point "rule" fixes a constant to no better than the unknown error on that point (the third-party wiki notes this too [web: https://wiki.naturalphilosophy.org/index.php?title=Frank_J_Znidarsic]). - Eq. 5-6 (Control; Eq. 5 of Flow) treats a nucleon as a mass on a spring whose stiffness is the electron Coulomb force at re divided by 2 rn: K = Fmax/(2 rn). Fmax is the electrostatic force between two electron charges at 2.8e-15 m. Using it as the stiffness of the strong force between nucleons has no derivation; the text says the two "equal" at points where electromagnetic and strong forces balance [read: Control p.3] but gives no equation for the strong-force stiffness. Dimensionally the result is fine (sqrt(F rn/M) is a speed), so this is a units-correct but physically unmotivated recipe. Also Eq. 5 uses cyclic frequency (1/2 pi) sqrt(K/M) x 2rn, while Eqs. 1 and 16 use angular frequency x rp: the 2 pi is kept in one, dropped in the other, with no explanation.

b. Consistency with experiment. No. The measured/derived sound speed in nuclear matter is c_s^2 = K_inc/(9 m) in the simplest estimate (K_inc the incompressibility, about 230 +/- 40 MeV), giving c_s about 0.16-0.23 c, i.e. 5e7 to 7e7 m/s [web: https://arxiv.org/pdf/2207.01884 and a search summary quoting c1 = 0.19-0.23 for relativistic mean-field matter at saturation density; calc: sqrt(230/(9 x 939)) = 0.165]. That is about 45-60 times Vt. Vt = 1.09e6 m/s is 0.36% of c. Conversely, a nucleon Fermi momentum of 1.36 fm^-1 corresponds to a nucleon Fermi speed hbar kF/m = 0.286 c = 8.6e7 m/s [calc]. Neither nuclear speed is anywhere near Vt. The speed Vt is the electron velocity scale of atoms (half the Bohr speed), as the identity shows.

c. Tested? The inventor reports no test of "Vt is nuclear sound speed" at all. His own related bench work: 1997 letter, one marginal anomaly not repeatable, later runs no anomalous energy [web]; 1999 spinning superconductor with RF to 40 MHz at up to ~1000 rpm, "No anomalous energy was observed" [web: https://newenergytimes.com/v2/archives/fic/N/N199903.PDF; read: research/files/NEN-1999-03.pdf, Investigator]. NASA MSFC (Noever and Koczor 1999, NTRS 19990023209) found gravity changes below 2 parts in 1e8 of g in AC-field tests [web/read: Investigator]. These bear on the Podkletnov anchor, not on the nuclear-sound-speed identification; no independent group has tested that identification.

d. If true: nuclear matter would have to have a sound speed 1/50 of what incompressibility measurements (giant monopole resonances) imply, and the Bohr velocity alpha c would have to be a nuclear property, i.e. atomic and nuclear scales would be tied together by a mechanism with no place in QED.

Verdict: Refuted by calculation (the identification of Vt with nuclear sound speed: Vt = alpha c/2 is an atomic constant combination, and standard nuclear-matter sound speed is ~50 times larger). Sub-claim "a velocity of ~1e6 m/s was observed in cold-fusion and Podkletnov data": Never properly tested as a measurement (order-of-magnitude products and one unrepeated bench point only).

Claim 2: Quantum transitions occur where the speed of light in the electron shell equals the nuclear sound speed Vt ("impedance match")

a. The "impedance match" is never written as an equation connecting the two speeds. Eqs. 21-22 (Control) are the transmission-line formulas Vt ~ 1/sqrt(LC), Z = sqrt(L/C) [memory, high], with no L or C given and no number computed [read: Control p.8]. "Speed of light in the electron shell" is undefined: Eq. 16 (Vt = n omega rp) is the only place it appears, and there the "light speed" is omega x rp with rp = re/2, an angular frequency of a fictitious spring times a 1.4 fm lever arm, and ft x lambda_t in the capacitor argument. In Eq. 9-12 the photon wavelength is taken as Vt/ft, which replaces c by Vt by fiat ("sets the velocity of sound in the nucleus equal to the velocity of light within the electronic structure") [read: Control p.5]. Hidden assumption: the capacitor geometry (area = lambda^2, spacing = lambda in Control, but lambda/2 in Flow Eq. 7; the factor 2 added "because the photon has two degrees of freedom") is chosen to give the needed 4 in 4 eps0. Control and Flow disagree: Control Eq. 9: C = eps0 lambda^2/lambda; Flow Eq. 7: C = eps0 lambda^2/(lambda/2). Both then arrive at C = 2 eps0 lambda by different routes [read: Control p.5, Flow p.5]. The final C is the same; the stated geometry is not. b. Experiment: no test is possible from the given math (no quantity is predicted that differs from standard atomic spectra). Standard QM already reproduces hydrogen spectra with no nuclear sound speed. c. Tested? No. d. If true: atomic spectra would depend on a nuclear mechanical property; isotope effects beyond the standard reduced-mass correction would follow, but none is predicted.

Verdict: Not testable as stated (no measurable prediction distinct from standard theory; the "impedance match" has no mathematical content beyond Vt = alpha c/2).

Claim 3: h, the hydrogen radii n^2 a0, alpha, Compton frequency and de Broglie wavelength follow classically from Vt

a. Consistent algebra, circular logic. Step by step [calc]: - Eqs. 7-15: algebra from C = 2 eps0 Vt/ft to E = [Q^2/(4 eps0 Vt)] ft is correct. The bracket equals h only because Vt = e^2/(4 eps0 h) by Eq. 1's constants (h inside fc). Planck's constant is the input. - Eqs. 16-20: Eq. 18 squared reads (2Vt/n)^2 = e^2/(4 pi eps0 me rx), i.e. Newton's circular orbit with orbital speed alpha c/n, which is Bohr's quantisation v_n = v_1/n put in as "n multiplies omega". Hidden assumption beyond Step 1: K-e(rx) = Fmax/rx is not the stiffness of the Coulomb force at rx. The actual Coulomb force at rx is Fmax (re/rx)^2 and its derivative is -2 e^2/(4 pi eps0 rx^3). Using Fmax/rx and multiplying omega by the lever arm rp = re/2 (not by the orbit radius rx) happens to reproduce v^2 = e^2/(4 pi eps0 me rx) only because Fmax re^2 = e^2/(4 pi eps0) is a constant. The "spring" is a relabelling of 1/rx. - Eq. 23 (alpha = 2 Vt/c): the same Vt, rearranged. The fine-structure argument in words (ratio of fine-structure line lengths equals alpha) is wrong in magnitude: the H-alpha fine-structure fractional spacing is ~2e-5, order alpha^2 = 5.3e-5, not alpha = 7.3e-3 [memory, high; calc alpha^2]. Also the text conflates field-induced splitting (Zeeman) with zero-field fine structure [read: Control p.8; memory, high]. - Eqs. 33-34: fc = Vt/(2 rp) is Eq. 1 inverted; Eq. 34 is the same identity at rx = a0 (omega-e = 7.76e20 rad/s = 2 pi fc). - Eqs. 35-41 (de Broglie): phases equal when 2 pi fc t + pi = 2 pi fc t (1+v/c), so ct = c^2/(2 fc v) = h/(2Mv) with fc = Mc^2/h [calc]. Algebra correct. But (i) h is inserted via "contemporary value of the Compton frequency"; (ii) the first-order Doppler factor (1+v/c) is non-relativistic and is applied to a wave of speed c; (iii) the step from "ct" to the wavelength is by assertion: Eq. 40 gives ct = h/(2Mv) and Eq. 41 sets wavelength d = h/(Mv) = 2ct, though Control p.12 says the luminal displacement ct "was replaced" by the wavelength, which would give h/(2Mv) (a factor 2 short; Flow's Eqs. 22-23 insert the factor 2 explicitly). So Control is internally inconsistent here by a factor 2 unless the unstated doubling is applied. [read: Control p.12 images/text; calc.] This is the standard pedigree of the "beat" derivation; it recovers h/(Mv) only by choosing the doubling that gives it. - Eqs. 24-32 (transition intensity): Eq. 31's bracket is 4 pi Fmax rp^2/Vt = h (identity), so Eq. 32, rt^2 = n h/(8 pi^2 M f), is again Eq. 19 rewritten. It is called "the known formulation for the amplitude of electronic harmonic motion squared". For a harmonic oscillator of energy E = n h f the mean-square displacement is E/(M omega^2) = n h/(4 pi^2 M f) [calc/memory, high], twice the paper's value (and the amplitude squared is four times). The quantum oscillator is (n + 1/2) hbar/(M omega) = (n + 1/2) h/(4 pi^2 M f) [memory, high]. So the claimed identification with the standard result is off by a factor of 2 as printed. Further: the "probability of transition proportional to amplitude squared" is stated in words. - Eq. 19B (extension to Z, muonic atoms): solving gives rx = n^2 a0/Z, the hydrogenic Bohr radius. Valid for one-electron ions in Bohr theory, but no multi-electron atoms are computed; the "fractional frequencies" for real atoms are not defined; the muonic result is the textbook reduced-mass scaling (Fmax rp^2 is mass-independent, so the replacement works) [calc]. Also the section is headed "energy levels" but only radii are derived; E_n requires E = -e^2/(8 pi eps0 rx), not written. - Flow Eq. 17-18: r_or = ne^2 r+h with ne the "number of electrons" is a misidentification: in Bohr theory the integer is the principal quantum number [memory, high]. rp, M_-e, nB are undefined there. - Flow Eq. 25: from Eq. 24 with vi = Vt: Vt^2 = 2 pi hbar fe/me. That sets the electron speed equal to Vt = alpha c/2, half the Bohr speed; then Eq. 25 = Vt = 4 pi a0 fe [calc]. fe is half the Rydberg frequency, so it is a known atomic frequency; the paper's "of the range of the LENR" is 100-1000 times the paper's own LENR frequency.

The one non-identity: Eq. 6 / Flow Eq. 5. I computed the closed form for the needed rn: rn = (pi^2/2) alpha^3 (M/me) (hbar/(me c)) = 1.3597e-15 m for the proton mass [calc]. It is a product of pure numbers (pi^2/2, alpha^3, the mass ratio 1836) and the electron reduced Compton wavelength, and it lands within 0.02% of the tabulated 1.36 fm. A match of this type has to be judged by how much freedom the author had. The free choices: which mass (three options differ by 0.2%), whether to use K = F/rn or F/2rn, the displacement rn or 2rn, cyclic or angular frequency (factor 2 pi), and which electron-scale force to substitute. Nuclear length scales span roughly 0.8-1.4 fm, so a value matching to 0.1% by chance has probability of order 1% for one trial, and larger once the choices above are counted [calc/my estimate; the count of choices is my judgement]. The match is real arithmetic. It is not evidence unless rn = 1.36 fm comes independently from nuclear data. The Control table calls rn "the nuclear Fermi momentum spacing" and cites Modarres (1987), which exists and concerns nucleon momentum distributions n(k) in nuclear matter, with the gap at the Fermi surface 0.96 [web: https://iopscience.iop.org/article/10.1209/0295-5075/3/10/005]. A momentum-distribution paper gives momenta (fm^-1), not a nucleon spacing in fm. That 1.36 matches the usual kF of symmetric nuclear matter in fm^-1 is from memory [memory, medium-high: kF ~ 1.33-1.36 fm^-1]; I did not open the paper's text, so I cannot confirm that it prints 1.36 or uses it as a length. This remains an open check, flagged but not asserted as a units error.

b. Experiment: h, a0 and alpha are measured; the paper's numbers agree with them because they are built from them. Nothing here is a test. c. Tested: no independent test is possible beyond agreement of constants. d. If true: Planck's constant would be derived from classical electrodynamics plus a nuclear mechanical constant; the success of QED to parts in 1e10 for alpha, g-2, etc. would be a consequence of a classical spring model. No consequences are calculated.

Verdict, split: the claimed derivation of h, a0, alpha, Compton frequency from Vt: Internally inconsistent in the sense of circular/self-referential (outputs are inputs), with arithmetic errors (Eq. 3 factor 4, Eq. 41 vs Eq. 40 factor 2, Eq. 32 vs standard factor 2, Flow Eq. 17 integer misidentified). The Eq. 6 numerical match (nucleon spring): Never properly tested (a coincidence of undetermined significance; needs independent sourcing of rn).

Claim 4: Josephson-junction scale (~1.33 nm) and LENR domain share the ~1e6 m/s transition speed, so LENR releases are discrete bursts

a. Not consistent. rJ = 2 (4 pi hbar/e)^2 (eps0/me) = 8 pi a0 = 1.3300e-9 m with e = 1.602e-19 C [calc]. With the stated Cooper-pair charge e = 3.2e-19 C the result is 3.33e-10 m, not the quoted 1.33 nm [calc, confirms Step 1]. The Robertson product (omega_e rJ = 1e14 x 1e-9) is 1e5 m/s, not 1e6. Eq. 27 equates "omega_2e" (an angular frequency) with fe/ne (an ordinary frequency) without 2 pi [read: Flow p.7; calc]. The two sides of the "similarity" (LENR: 50 nm x 1e13-1e14 rad/s; junction: ~1 nm x 1e14 rad/s) both produce 1e5-1e7 m/s, so "similar to within a factor of 10-100" is all that is shown. The jump to "so LENR energy is released in discrete bursts of electron-pair energy" is a postulate ("Here it is postulated...") [read: Flow p.3]. Why 1e6 m/s keeps appearing: any atomic-scale length (0.5-50 nm) times an atomic-scale angular frequency is ~alpha c. Conduction-electron Fermi speeds in metals are ~1e6 m/s (e.g. 5 eV gives sqrt(2 x 5 eV/me) = 1.3e6 m/s [calc]), and the Bohr speed is 2.2e6 m/s. A "speed" near 1e6 m/s is therefore the generic electron speed in matter, not a distinctive constant; this is also likely what Poole's textbook remark that the Investigator flagged refers to (Fermi velocity; the sentence is cited by Znidarsic, not read by me [memory, low]). Typical Josephson barrier thickness is 1-3 nm [memory, medium] set by fabrication, so "1.33 nm is in range" does not distinguish 8 pi a0 from other values. b. Not contradicted or confirmed by experiment; the cited LENR experiments (Kitamura 2009; Mosier-Boss 2007) are cited without any data used in the equations [read: Flow]. Lawson 1957 is cited for a claim about heavy-element transmutation; Lawson's paper concerns thermonuclear energy balance [memory, high]. c. Tested? No. Peer review was by the co-author's own institute (a fact; the effect on quality is for the reader to weigh). The junction-mass-change mechanism of Robertson (NTRS 19990046249) is a separate, earlier, NASA-era theory [Investigator, read]. d. If true: LENR product distributions would have to follow discrete Josephson-like energies; no spectrum is predicted.

Verdict: Internally inconsistent (as stated: charge and frequency inputs do not give the quoted 1.33 nm and 1e6 m/s) for the numerical coincidence; the physical link (burst model) Not testable as stated (no quantitative prediction).

Claim 5: at transition speeds >= ~1e6 m/s new phenomena occur and "free" vacuum/dark energy can be pulled from ~1e-9 m interactions

a. No equation exists. The text is hedged ("would seem to imply", "appear to place", "may lead to") [read: Flow pp.3-4, 7-8]. No energy balance, power, or mechanism: a "minimum velocity" threshold is stated, but threshold of what quantity is undefined, and Vt = alpha c/2 is a combination of constants, not a variable that can be exceeded. b. The principle of energy conservation (Noether, time-translation invariance) and no measured vacuum-energy extraction; Casimir force measurements show vacuum energy differences between configurations but no net extraction cycle [memory, high]. c. Never tested; no experiment reported. d. If true: energy conservation as formulated for closed systems would have to fail, or the vacuum would be a source requiring a specific coupling not given.

Verdict: the quantitative content is Not testable as stated; the strong reading (net free energy from vacuum) is outside the mathematics supplied; I assign no stronger verdict because no calculation is given to refute.

Claim 6: forcing a macroscopic body into a single transitional quantum state via Bose-condensate vibration at ~1.094 MHz-m induces gravitational and long-range nuclear forces (propulsion, waste reduction, energy)

a. Control p.13 ("New Technologies") has three sentences, no equation [read: Control p.13 text]. The "theorem" quoted on p.11: "constants of the motion tend toward the electromagnetic in a Bose condensate that is stimulated at a dimensional frequency of 1.094 megahertz-meters" [read]. The unit "MHz-m" equals 1e6 m/s: 1.094 MHz-m = 1.094e6 m/s [calc], consistent with the 1997 rule 37 MHz-inch = 9.4e5 m/s. A mechanical disk driven so that f x L = Vt would be a standing-wave condition with wave speed Vt, but the speed of sound in YBCO ceramics is of order a few km/s [memory, medium], about 300 times smaller; and the electromagnetic wavelength at 3 MHz is 100 m. So f x L = 1e6 m/s is neither the acoustic nor the electromagnetic resonance condition of a 0.3 m disk. The product has no stated wave meaning. Control p.11 also claims rp is where "the strength of a proton's electrical field equals the strength of its strong nuclear field", but gives no formula; rp is numerically re/2 [calc]. b. Experiments: the inventor's own tests (1997, 1999 above) saw no anomalous energy [web]. NASA MSFC's AC-field tests found <2 parts in 1e8 of g [Investigator, read]. I take no position here on Podkletnov's original reports beyond noting that their strength is disputed and that the papers use them as input without quantities. c. Tested? The RF-plus-rotation configuration: partly by the inventor (null); the NASA tests were different configurations; the inventor's account is that funding ended before MSFC ran the RF method (his characterization, one source: his 2012 book, rexresearch1) [web/read: Investigator]. No neutral, controlled test of the "1.094 MHz-m" prediction on record. The Podkletnov-type effect has had few replication attempts; Hathaway, Cleveland and Bao reported a null [web: third-party wiki, Investigator; not checked by me]. d. If true: new long-range gravitational and nuclear interactions would be induced in bulk matter by vibration; this contradicts the short range of the nuclear force and the weakness of gravity as formulated in GR; no coupling constant is given.

Verdict, split: the testable core (vibrating a superconductor at f x L = 1.094 MHz-m produces anomalous force or energy): Never properly tested by a neutral party; own-bench tests Tested, not supported (null results by the inventor in 1997-1999). The propulsion, waste-reduction and energy applications: Not testable as stated.

4. Pertinent material from the Investigator (retained)

• Origin of Vt differs between accounts: 1997 letter (own 3.5-inch disk, 37 MHz-inch, patent application filed 19 Oct 1997, anomaly at 11 MHz not repeatable) versus the 2012/2019 book (Podkletnov 3 MHz x 1/3 m, Poole's remark, then 20 THz x 50 nm in Pd). Mathematically, a constant defined from one unrepeated point (or from order-of-magnitude products) cannot be tested to 4 digits; the 4-digit value comes from fc and rp.
• Third-party wiki: independently states Vt = alpha c/2 and credits the author for reporting null results. Wiki is run by a dissident-science society; its Hathaway-Cleveland-Bao citation was not checked by me.
• Robertson (NTRS 1999) and Noever-Koczor 1999: documented NASA MSFC context. They are not replications of Znidarsic's RF idea.
• Patent: none found in Google Patents; the 1997 application would be unpublished if abandoned [memory, high]. The record goes quiet here; competing explanations (abandonment, rejection, never pursued) are listed by the Investigator; I add no mathematical evidence either way.
• NEN 1999 null experiment: spinning disk, 100 kHz-40 MHz RF, up to ~1000 rpm, "No anomalous energy was observed". Editor's comment that other work used higher rpm (unpublished report: no effects below 10,000 rpm) means this null did not cover that regime.
• Biography and the other Znidarsic obituaries (not the inventor): irrelevant to the mathematics.

5. Summary of the mathematical state

What would settle the open items: (1) the text of Modarres (1987) to see whether 1.36 appears and in what units; (2) a neutral replication of a rotating, RF-driven YBCO disk at f x L = 1.094 MHz-m with a gravimeter at the 1e-8 g level; (3) any equation giving the energy extracted in Claim 5.

Web checks done (4 of ~6): Modarres 1987 exists (IOPscience URL above); nuclear-matter sound speed (arXiv 2207.01884 and search summary). Not checked: Poole p.4 text, Hathaway-Cleveland-Bao, speed of sound in YBCO (left as [memory, medium]).