Hieronymus-TG: Mathematician review (Step 2)

Reviewer role: PhD mathematician. I check derivations, notation, units and hidden assumptions. I read the extracted patent text and the page-1 drawings myself (rendered image, Figs. 1-4) [read: page image p.1], the Step 1 deep read, and the Investigator's notes. Source tags: [read] document, [web] URL, [memory] with confidence. Interested parties are named as such: the inventor, and Hite (a critic; he has no stated commercial stake), Campbell (an enthusiast who built a device).

Source gap: none for the patent. It is the only document in the folder and is fully readable. The inventor's quantitative material is in his 1988 autobiography, which the Investigator read by keyword only (not in the folder). I use those numbers as the Investigator reports them [web: archive.org/details/the-story-of-eloptic-energy] and I did not re-read that book. The patent itself publishes no measurement.

Plain-language note for the reader: the patent describes a coil, two radio tuning capacitors, a glass prism with two thin metal electrodes, a stack of resistors and an amplifier, ending in a plastic-coated plate that the operator strokes with a fingertip. The operator turns a pointer until the plate "feels draggy", reads an angle, and looks the angle up on a chart to get an atomic weight. So the maths to check is (i) whether a prism can turn a "radiation" into an angle that rises with atomic weight, (ii) whether the circuit can handle that radiation, (iii) whether the worked example and the resistance "quantity" method are consistent.

Corrections to Step 1

1. Cited reference. facts.json says Hall US 2,115,578 was "not checked". The Investigator found it is a thermal-pneumatic "Radiant energy receiver" (Hall/Hayes, 1938) [web: https://patents.google.com/patent/US2115578A/en]. I accept that; it concerns ordinary heat radiation, which is a different thing from the patent's element radiations.
2. The 79.2 example is internally consistent with the chart, but not with calcium. The deep read said only that 30 degrees on the scale gives "roughly 80". From Fig. 3 I read line 45 as a straight line starting at 10 degrees on the vertical axis (at atomic weight 0, not at weight 10 as the deep read says) and reaching about 40 degrees at atomic weight about 118 [read: page image p.1]. That gives slope about 0.25 degrees per atomic mass unit, so 30 degrees maps to A = (30-10)/0.25, about 79. So 79.2 is what the patent's own chart returns for a 30 degree reading. Calcium (standard atomic weight 40.08 [memory: high; textbook value]) would read about 10 + 0.25 x 40 = 20 degrees on the same chart. The patent's own example therefore names the wrong element for its own number. The patent says "If atomic weight 79.2 is calcium", which is conditional, and the deep read flagged that. 79.2 sits close to selenium (78.97) and bromine (79.90) [memory: high]. I take this to be a slip in the drawing's worked example.
3. Hydrogen as the "lowest angle". The text says hydrogen passes at "the sharpest angle". The chart line puts hydrogen (A about 1) at about 10.25 degrees, not near 0. This is not a contradiction (lowest of all elements), but it means the chart has a built-in 10 degree offset the text never explains.
4. Claim 4 versus claim 6 (resistance method). The deep read noted the wording difference. The spec says to raise the resistance to the maximum value that still keeps the drag reaction. Claim 6 says to vary it "until the surface frictional characteristics ... are unaffected". Read literally, "unaffected" is the opposite end point (reaction lost), so the claim and the specification define the quantity reading at different settings. claims.md claim 4 follows the spec; it should say the claim language differs.
5. Prism angles. The deep read flags "angles from 30 to 60 degrees" as ambiguous (apex or face angle). My calculation below shows the ambiguity does not matter to the conclusion.
6. "Magnet insulated wire" is magnet wire (enamel-coated copper wire), not a magnetic material. It is a naming point, not an error in Step 1.
7. All 13 claims are independent (none refers to another). I confirmed this against the claim text in the deep read; no correction needed.
8. Hite's device. Step 1 did not mention that Hite tested a Pathoclast, not the prism analyzer. The Investigator is right to flag this: his negative result does not test the patent's prism and angle claim. I keep it on record (see claim 5).

Calculations used below

All use the vector-free algebra the reader already has. Constants: Planck h = 6.626e-34 J s, Boltzmann k = 1.381e-23 J/K, speed of light c = 3.00e8 m/s [memory: high].

A. How much violet light does matter at room temperature emit? Matter at temperature T emits thermal radiation described by Planck's law. For a violet photon, wavelength 400 nm, frequency f = c/wavelength = 7.5e14 Hz, energy hf = 3.1 eV. At 300 K (about 80 F, inside the patent's 40-90 F), kT = 0.0259 eV. The emission is suppressed by the factor exp(-hf/kT) = exp(-120) = 8.5e-53. Spectral radiance (energy per unit area, solid angle and frequency) is then about 5e-58 W per square metre per steradian per Hz. Over a 1e14 Hz band that is about 1.6e-43 W per square metre, or roughly 3e-25 photons per square metre per second, which is one photon per square metre every 1e17 years or so. For 300 nm and 200 nm the factors are 4e-70 and 7e-105. So thermal emission cannot be the source of a "violet to ultraviolet" radiation at 40-90 F. The patent says the radiations occur at those temperatures and "without special treatment" [read: p.2]. If the radiation is not thermal it would have to be some other process (luminescence or fluorescence need an excitation source; none is given). [memory: high on Planck's law; the number is my calculation.]

B. Can the circuit tune violet frequencies? The tuned circuit is a coil (2 in. spiral) with broadcast-type variable capacitors [read: p.3]. A resonant circuit has f = 1/(2 pi sqrt(LC)). With a typical broadcast capacitor of up to about 365 pF [memory: medium; typical value, not stated in patent], resonance at 1 MHz needs L about 69 microhenries, a plausible coil. To resonate at 7.5e14 Hz with even 10 pF would need L about 4.5e-21 H, far less than any wire. So the "tuning" in the circuit can only act on radio frequencies (about 1e5 to 1e7 Hz), which is 8 orders of magnitude below violet light.

C. Can a prism refract it? A 2 in. coil has a diameter about 125,000 times a 400 nm wavelength, so it would be a poor antenna for light. Conversely, at radio frequency (1 MHz) the wavelength is 300 m and the prism and electrodes (0.5-2 in. spacing, thickness a few mils) are 4 to 5 orders of magnitude smaller than a wavelength. Geometric refraction (Snell's law, n1 sin(a1) = n2 sin(a2), with a prism) needs structures many wavelengths across. At radio frequency the prism region is a "near-field" capacitive gap: electrode 32, the glass, and electrode 38 form a small variable capacitor whose coupling depends on the electrode's position. No ray is formed.

D. Size of the dispersion if the prism did refract violet to UV. Using the Sellmeier formula for fused silica (Malitson 1965 coefficients [memory: medium]; the refractiveindex.info page confirmed the dataset covers 0.21-6.7 micrometre but would not print the numbers [web: https://refractiveindex.info/?shelf=main&book=SiO2&page=Malitson], so the values are my computation): n = 1.4701 at 400 nm, 1.4878 at 300 nm, 1.5384 at 210 nm. The patent allows "glass, quartz" and prism angles 30-60 degrees, and uses an entry angle of "about 5.5 degrees ... to the face" [read: p.3]. I computed the total deviation for apex angles 30, 45 and 60 degrees and for both readings of "5.5 degrees" (5.5 from the normal, or 5.5 from the face which is 84.5 from the normal):

(45 and 60 degree prisms at 5.5 from the normal fail total internal reflection for some or all wavelengths.) So a real glass or quartz prism spreads the whole violet-to-UV band over about 3 to 5 degrees. The patent's chart needs the elements to spread over roughly 10 to 65 degrees (see E). This is a factor of 10 to 20 beyond what normal dispersion in the stated materials offers. One could say the radiation is at a wavelength where the glass has strong resonance and anomalous dispersion, but the patent says only "violet into ultraviolet".

E. The patent's angle versus atomic weight law, and the inventor's later numbers. From Fig. 3, line 45: angle = 10 + 0.25 x A (degrees, A = atomic weight) [read: page image p.1; my reading of a small drawing, about 10 percent uncertain on the slope]. The autobiography reports a 1600-division quadrant (so one division is 90/1600 = 0.056 degrees) with beryllium near 186 and bismuth near 1097 [web: archive.org autobiography pp.237-238, via the Investigator]. That converts to 10.5 and 61.7 degrees. For Be (A about 9) and Bi (A about 209), the slope is (1097-186)/(209-9) = 4.56 divisions per atomic mass unit = 0.256 degrees per unit. This matches the patent's chart slope (0.25) very closely. So the inventor's two data points, given 40 years later with a different apparatus, are consistent with a straight line in A. That is worth stating plainly, because it is the one place where his numbers agree with each other. Two cautions: the two data points are two numbers read from a book summary, not a data table, and a straight line through two points fits anything. Also a linear law in A is not what dispersion produces: refractive index near a resonance goes as 1/(f0^2 - f^2), not as mass.

F. Resolution test of the isobar claim. The autobiography says a "nuclear weight factor" (protons x 1.00758 + neutrons x 1.00894) separates isobars like Ti-50 and Cr-50 (50.41708 versus 50.41436) [web: autobiography pp.266-267, via the Investigator]. The difference is 0.00272 mass units. At 0.256 degrees per unit that is 0.0007 degrees. His stated reproducibility target is half a graduation, or 0.028 degrees [web: autobiography p.233]. The claimed separation is about 40 times smaller than the instrument's own stated resolution. Also, the quantity in the formula is the mass of free protons and neutrons; real nuclei differ from that sum by the binding energy (about 8.7 MeV per nucleon for Fe-region nuclei, about 0.9 percent of the mass) [memory: high]. A "nuclear weight factor" built from free-nucleon masses is therefore not the actual mass of Ti-50 or Cr-50 and cannot separate them in the way described (the real mass difference of those isobars is about 0.002 to 0.003 u as well, same order, but the factor omits the binding terms that produce it). The reported "smooth curve" of factor against dial setting is not in the folder.

G. The quantity (resistance) method. Unit 40 is a decade resistance box (1 megohm in 100 kilohm steps, then 100 kilohm in 10 kilohm steps, and so on) [read: p.3-4]. The operator raises the resistance "until the maximum amount of resistance has been introduced into the circuit without interrupting the reactions manifested at detector" [read: p.5]. Then "a chart must be prepared" relating switch settings to quantity [read: p.5]. Issues: (i) the chart and the units of "quantity" are never given; (ii) the end point depends on the operator's tactile threshold, a judgement; (iii) the series resistance forms a divider with whatever impedance follows, so the reading depends on the amplifier gain, the tube's age and transformer 58, and the 1972 notes record that the calibration drifted over decades (normal tissue about 15-20 on a 40 scale, later nearer 100) [web: Hite PDF appendix pp.33-34]. A measure with an unspecified, drifting, operator-set threshold is not a calibrated function of quantity.

Claim-by-claim review

Claim 1. Every element emits violet-to-UV "radiations" with both electrical and optical character at 40-90 F

• (a) Mathematically consistent? Not defined well enough to check. "Both electrical and optical characteristics" is not tied to any quantity (frequency, wavelength, amplitude); the only frequency statement is the zone "from the violet ray portion ... up into the ultra-violet" [read: p.2]. The same "radiation" is then picked up by a coil, tuned by broadcast capacitors, conducted by wire, and refracted by a prism. Those operations need frequencies differing by 8 orders of magnitude (calculation B). The patent hedges that the energy is "probably electrons" [read: p.3], which gives a third identity for the same entity. No equation relates the emission to the element.
• (b) Consistent with experiment? Thermal emission calculation A shows violet-UV emission at room temperature is suppressed by exp(-120) or more; this is the standard result behind why a cool object emits no visible light. Emission of element-specific UV needs excitation (flame, arc, discharge), as in standard emission spectroscopy. Element-specific radiation exists at other frequencies (X-ray fluorescence, radioactivity), and the inventor later attributes extra range to radioactive isotopes. These are known and unrelated to the claim.
• (c) Tested? The patent gives no data. Campbell built a device from the patent alone and reported "positive results" in Astounding June 1956, with operator-dependent success (best for young girls, worst for technically trained men) and the device working as well unplugged; it failed with a defective tube [web: archive.org Astounding Jun and Aug 1956, via the Investigator]. Hite's engineering tests were on a Pathoclast, not on this analyzer, and he found a conventional explanation for the "stick" [web: leehite.org PDF]. The Huntsville "duplication" claim rests on the autobiography's preface and Tompkins and Bird, the report not found [web: via Investigator]. No controlled, blinded test of the element-identification claim is on record in anything I have seen.
• (d) If true? Matter at ambient temperature would emit element-specific UV with no excitation, violating the Planck/Kirchhoff picture for thermal emission and energy conservation unless a source is named. A continuous source of UV photons would also have to come from somewhere (an energy-budget issue for the physicist).
• Verdict: Never properly tested (as a testable claim there is no controlled, neutral test on record; it was not mathematically "tested and failed" either) and, in the quantitative sense above, Refuted by calculation for the thermal-origin reading (Planck suppression exp(-120)). The patent does not say the origin is thermal, so I give a split verdict: thermal emission, refuted by calculation; non-thermal "element radiation", never properly tested.

Claim 2. The radiations travel on wires like current, respond to capacitance, inductance and resistance, and also refract, focus and diffract like light

• (a) Consistent? No, in the sense that wires and optics are different regimes. A wire conducts electric current as electromagnetic fields guided by the metal at radio frequencies; light at 7.5e14 Hz is not guided by a bare metal wire (metals reflect and absorb it within about 10 nm of surface) [memory: high]. A lens or prism works only if its size is much larger than the wavelength; the same entity cannot be handled as a "lumped" circuit element (size much smaller than wavelength) and as a geometric ray (size much larger) unless the wavelength is somewhere impossible for both. The tuned circuit is a lumped RF circuit (calculation B); the prism is only a prism for light (C, D). I find these two statements incompatible as written, unless the entity is a new type of wave for which the patent gives no dispersion relation. The inventor's later rules (Bakelite insulates, Plexiglas transmits, black opaque Plexiglas prism refracts like glass, light "shorts" the energy [web: Hite appendix]) are not consistent with any electromagnetic wave: an opaque material that refracts is not a light-like refractor.
• (b) Consistent with experiment? Refraction and diffraction of light, and current in wires, are among the best-tested results in physics, but the claim that one entity does both is not an established result. No measurement of the angle at which a prism turns the entity is given. If we take the Fig. 3 numbers as measurements (E), they deviate from the normal-dispersion range by a factor of 10-20 (D).
• (c) Tested? Campbell's reported success with a prism "drawn in ink" (his symbolic machine) and the unplugged operation suggest the prism is not what carries the result [web: Astounding and the autobiography, via the Investigator]; Hieronymus's own 1972 notes say "diagnosis works just as well" with power disconnected [web: Hite appendix pp.33-34]. These are reported by an enthusiast and by the inventor's circle. They are evidence that the observed response does not depend on the stated circuit, so the circuit equations are not what the device's response follows. No controlled comparison (real prism versus no prism versus a drawn prism, blinded) exists in the record I have seen.
• (d) If true? Electromagnetism would need a new carrier with dual wire and ray behaviour and an element-specific frequency spectrum, which would require modifying Maxwell's equations or adding a field. This also bears on the dispersion of ordinary glass.
• Verdict: Internally inconsistent as stated (lumped-circuit and ray behaviour at one frequency; claim to both). The experimental side is Never properly tested (no neutral test of prism versus no prism on record).

Claim 3. The prism angle rises with atomic weight (hydrogen smallest, isotopes included); scale plus chart identifies the element

• (a) Consistent? The chart is an empirical, per-instrument lookup (the patent says the maker builds chart 46 by running a small quantity of each element [read: p.4]), so the claim is a calibration, not a derivation. The patent offers no formula. Points to check: (i) the straight line 45 plus a boundary curve 46 is never explained; (ii) a function angle = 10 + 0.25 A is monotonic, so it is invertible in principle; (iii) the worked example is arithmetically consistent with the chart but names the wrong element (Corrections 2); (iv) the mapping from dispersion to mass is not derived (D, E); (v) composite signature of mixtures, "composite frequency", is undefined (a sum? a weighted mean? a set of lines?), so the claim that mixtures can be identified has no content in the maths. For a mixture of elements with different A, one angle cannot represent more than one element at once, and no deconvolution rule is given.
• (b) Consistent with experiment? Real element identification uses spectral lines, which are not ordered by atomic weight. Wavelengths of the strongest line for the alkali metals, for example, go 671 nm (Li), 589 (Na), 767 (K) [memory: high], not monotonic in A; X-ray characteristic lines do scale with atomic number (Moseley's law, f proportional to (Z-1)^2) [memory: high], which is the nearest real ordering of element radiation. It is a quadratic in Z, not linear in A, and requires X-ray excitation, not room temperature. A linear law with a single line is not what any known element radiation does.
• (c) Tested? The inventor's autobiography reports a test plan (inter-operator and inter-instrument agreement within half a graduation, change of prism angle) and results on Analyzer No. 508 (peak counts matching isotope counts for Ti, V, Cr, Mn, Fe, Ni) [web: autobiography pp.233, 266-267, via the Investigator]. Whether the inter-operator test was done, who did it, and whether it was blind is not stated in the passages the Investigator read. I give his two numerical data points (Be 186 and Bi 1097 divisions) real weight for what they are: a record by the inventor, internally consistent with the patent's chart slope to within about 4 percent (E). That consistency is evidence the inventor's readings were repeatable over decades, or that the chart was used to generate them; I cannot tell which. Campbell reports operator-dependent success. Hite's test is on a different instrument.
• (d) If true? A new force/radiation whose refractive dispersion tracks nuclear mass would amount to a new interaction between nuclei and a medium at room temperature, not found in nuclear or atomic physics. Also, mass number is not a property that atoms or nuclei radiate; two isotopes with the same chemistry but different A would have to emit differently at the 0.25 degrees per mass unit scale.
• Verdict: Never properly tested (a testable calibration claim; inventor reports data but no blind, neutral test is on record), with the stated physical mechanism Refuted by calculation (dispersion in the stated materials spans a few degrees against the 55 or more degrees the chart needs; D, E). The isobar-separation extension in the later book is Refuted by calculation against the inventor's own resolution (F).

Claim 4. Quantity is measured by raising a series resistance to its maximum that still keeps the detector reaction

• (a) Consistent? The decade resistance box is described clearly [read: p.3-4]. The method has no stated model: quantity Q is never related to R_max by an equation, and the "chart that must be prepared" is not given. Claim 6 and the spec use different end points (Corrections 4). A monotonic dependence of R_max on Q is plausible for any signal that falls with source strength, but the threshold of tactile "drag" is a function of the operator, the amplifier gain and the series impedance, so R_max is not a function of Q alone.
• (b) Consistent with experiment? Series attenuation to a threshold is an ordinary technique, so the form is not weird. What is missing is any calibration against a known mass or concentration, and the calibration drifts by a factor of 5 or more according to the 1972 notes [web: Hite appendix].
• (c) Tested? No data in the patent. Only the 1972 notes (inventor's statement as recorded) mention drift. No replication.
• (d) If true? No change to mainstream physics for the circuit part. The part that depends on the "radiation" carries the cost of claims 1 to 3.
• Verdict: Not testable as stated (no defined units, chart or end-point rule; spec and claim 6 differ). It becomes Never properly tested once a chart and units are supplied.

Claim 5. Detection is a coated plate whose "drag" under the fingers changes when radiation reaches it; mechanism "is not known"

• (a) Consistent? The patent is open about it: "What actually happens at detector 12 ... is not known" [read: p.4]. This is an honest admission but it leaves the detector without a model, so no circuit equation can be checked. The claims (4, 5, 6, 7, 9, 12, 13) rely on a "conductive solid having a smooth planar surface" varying "surface frictional characteristics", which is a statement of an observation, not a mechanism.
• (b) Consistent with experiment? Known physics offers a conventional candidate. Electro-adhesion and electrostatic or ultrasonic "electro-vibration" tactile effects exist: a voltage between the finger and a coated plate changes the friction felt [memory: medium; the effect is known as electrovibration, used in touch-screen haptics, which is why the lighting-control patents in the Investigator's citing list cite the touch idea]. Hite's oscilloscope measurements of an RF signal pulsed at 60 Hz on the plate of his Pathoclast, an amplifier that oscillated at certain dial settings, and settings shifting with the operator's posture, fit that explanation [web: leehite.org PDF pp.6-14, Hite is an independent critic]. His claim that Hieronymus admitted the patent was worded so a working device could not be built has no source in the text I have seen; I treat it as his assertion. Against the conventional explanation: Campbell and the inventor's 1972 notes both say it works with power off [web, via Investigator], which a purely electrical mechanism cannot do unless there is a source of stray voltage (body static, mains pickup). Operator dependence (Campbell: best for young girls) fits a human-perception effect. I cannot decide this from the maths.
• (c) Tested? Hite: one engineering study of a related instrument (negative for substance correlation). Campbell: positive but non-blind, operator-dependent. No double-blind trial of the prism analyzer.
• (d) If true? If it were a real new energy, instruments with a non-electrical detector would show it, contradicting the amplifier model the patent offers. If it is electrovibration, mainstream physics is unchanged.
• Verdict: Never properly tested for the prism analyzer (no neutral controlled test; the closest test, Hite's, is on a related instrument and was not dismissal without testing, it was negative on that device). Tested, not supported applies only to the Pathoclast and not to this patent's device; I flag that limit.

Claim 6. Wired plants in a dark cellar stayed green while an unwired control lacked chlorophyll, showing element radiations are conducted through wires

• (a) Consistent? The experiment is stated qualitatively with no numbers [read: p.3]. Logic: the conclusion "element radiations from the light were conducted through the wires" needs the elements to be fed to the plants (the premise says elements "have been transmitted to the plants through metallic conductors") but the observed outcome is greenness. Mass balance: a plant's dry mass comes from matter (CO2, water, minerals); a radiation carrying no mass cannot supply atoms. If instead it supplies "light energy", the quantity must be compared: any mV-level voltage or microwatt power collected on a plate of area 0.1 m^2 in sunshine (1000 W per square metre incident) as an electrical collector at, say, 1 percent efficiency would be 1 W, delivered as ordinary electric current, not a flow of elements. The patent gives no electrical measurement, so this cannot be computed from the document. The control is a single box, with no replication, species, duration or blinding (deep read).
• (b) Consistent with experiment? In angiosperms, dark-grown seedlings are etiolated and mostly lack chlorophyll, and greening is light-triggered [web: https://en.wikipedia.org/wiki/Etiolation]. So the control behaving as pale is expected. The treated boxes being green implies some light reached them or some other difference. Possible conventional causes, which I cannot exclude from the patent: light leaks in the cellar (very low light suffices to start greening in many species [memory: medium]); the boxes differing in more than the wire; seed type (the Tompkins and Bird account adds aluminium-lined boxes, a water-pipe ground and "seed grain" [web: Secret Life of Plants p.334, via Investigator]). A conventional electrical hypothesis (weak electric field or ground potential differences affecting germination) is also not excluded. The "darkened basement" and "kept in a dark room" are the inventor's descriptions.
• (c) Tested? The patent text, the later reprint ("Growing Plants in the Dark", JBR 1988) and the Tompkins and Bird retelling are all the inventor's own account; the 1988 reprint adds no data and the inventor dates it to 1931 [web: Investigator]. Simon Martin's 1988 cress experiment (lengths, colour, light versus dark, no wires) is not a replication [web: JBR vol. 44 no. 3, via Investigator]. I found no replication with wires.
• (d) If true? Plants would need to receive light-triggered signals by a wire without light, i.e. an unknown conductive channel for photo-induced greening, plus a mechanism by which the chlorophyll pathway responds to it. This would not change basic physics directly but would require new biology and a new energy carrier.
• Verdict: Never properly tested (a reported single experiment with no data and no replication; testable by anyone with seeds, boxes and wires, and not dismissed after a controlled test). Worth stating: it is the cheapest claim here to test.

Overall mathematical assessment

1. The one quantitative law in the document (angle rises with atomic weight, chart in Fig. 3) is a calibration, not a derivation, and it is nearly linear (about 0.25 degrees per mass unit). The inventor's later numbers (Be and Bi) are consistent with it within about 4 percent (E). That is real internal consistency, and it stays in the record beside the points against it.
2. The stated physics (prism dispersion of violet and UV) cannot produce the claimed span of angle, by a factor of 10 to 20 (D), and the circuit cannot tune violet frequencies (B). The prism-and-wire hybrid is incompatible as a single wave picture (C).
3. The patent's worked example names calcium for a reading that the chart maps to atomic weight 79.2 (Corrections 2). The example is a conditional ("If atomic weight 79.2 is calcium"), so it is a slip, not a proven fraud, but a careful applicant could have checked it.
4. The strongest alternative reading of the observations (operator dependence, works unplugged, works with a drawn prism, stick at amplifier-oscillation settings) is that the response is produced in the operator and at the plate, with the prism and the chart not playing the role claimed. That explanation is partly supported (Hite, on a different instrument) and partly untested.
5. What would settle it: a blinded protocol in which an operator reads a chart angle for 10 sealed vials of different pure elements, with a dummy prism, a real prism and power off each presented in random order; 30 or more trials per condition would distinguish chance from reproducible readings. Any reader with the circuit from Fig. 1 can try a simplified version.

Investigator's findings I carry forward (kept whatever my verdict)

• Patent family GB 663,978 and CA 486,008; the GB abstract confirms the drawing numerals (amplifier 70, detector 72, transformer 74) that the US text layer garbles [web: https://patents.google.com/patent/GB663978A/en].
• Autobiography numbers (Be about 186, Bi about 1097 of 1600; 40+ element specimens, about 35 run; Analyzer No. 508 isotope peaks; the nuclear-weight factor) [web: archive.org/details/the-story-of-eloptic-energy]. These are the inventor's statements (interested party); I could not see the data tables.
• Campbell 1956: independent build from the patent, positive but operator-dependent results, works unplugged, fails with a bad tube [web: archive.org Astounding].
• Huntsville engineers' duplication, Apollo 8 monitoring and the 300-copy report (single circle, report not found); submarine test (one source, no data). I have no mathematical content to add; I flag them as unverified.
• Hite's Pathoclast study and 1972 "Random Notes" (including power-off operation and the resistance-to-ground intensity method) [web: leehite.org PDF].
• Radiation Laboratories, Inc. ending about 1950 (his stated reasons: banking rules and AMA/FDA pressure), the assignments, Advanced Sciences R and D Corp (Florida 1967-1996), biography. These do not affect the mathematics; they belong to the historical record for the later steps. The absence of a published data set after the patent is an anomaly worth analysing (commercial failure, regulatory pressure, secrecy among customers); I cannot resolve it from the maths.
• Date conflicts the Investigator noted (GB 1951 vs 1952; corporation 1967 vs 1975) are not mine to resolve.

Verdict summary