Wheeler-Ken: Mathematician review (Step 2)

Reviewer: PhD mathematician. Subject: Ken L. Wheeler, Uncovering the Missing Secrets of Magnetism (3rd ed., 25 Sept 2014, 236 pp.). Role: check derivations, notation, units and hidden assumptions; answer a-d per claim; give a verdict from the prompt.txt scale. The physics (what mainstream says about magnets, light, relativity) is the physicist's job; I touch it only where a calculation settles it.

Tags: [read] = a source document (book text, page image, deep-read file, research file); [calc] = I computed it in the project venv (SymPy/NumPy; scripts in my scratch directory); [web] = checked on the web, URL given; [memory] = recalled, confidence stated.

0. What I read myself, and how

• All of deep/magnetism1small-6bca80.md (the five deep reads: pp.1-42, 43-53, 112-127, 149-156, 220-236), claims.md, facts.json, sites.md and the research notes on ferrocell, Natural Philosophy wiki, kathodos and ferrocell.us.
• The book text layer for pp.97-111 (bismuth chapter) and pp.207-219 (light and photoelectric chapter), which Step 1 treated as digest-only [read: _text/magnetism1small-6bca80.txt]. I read both ranges in full as text. I did not render page images for them, so equations that exist only inside figures are not seen (the deep reader flagged equations on pp.210 and 212; the text layer carries the prose and the few formulas quoted below).
• The Tufaile 2024 horocycle paper in research/files/ (first pages by OCR only; its body is page images). The Tufaile ferrocell guide and the Dollard PDFs were not re-read; I rely on the Investigator's quotations of them.
• Recomputed the golden-ratio identities, the Poincare metric, the Bi-209 alpha Q-value, the solar light deflection, and (independently of the deep reader) the field of a uniformly magnetised cylinder for the "field formula". My cylinder numbers reproduce the deep reader's table (see claim 4).
• Not read: pp.54-96, 128-148, 157-206 beyond the digests (the RGB LED, ferrocell gallery, gravity, atom and galactic-jet chapters). Anything I say about them is from digests only.

Source gap: none for the text; but no document in the folder reports a measurement. Every quantitative verdict below therefore rests on arithmetic, units and my own calculations, not on Wheeler's data.

1. Corrections to Step 1

1. claims.md "Coverage note" says the bismuth chapter (pp.97-111) is digest-only. Still true of the deep/ folder, but I have now read the full text of pp.97-111 [read]. What it contains: the setup is not a bismuth disk alone. Caption p.99: "using my 'programmed' bismuth disk. Single disk levitation of 3mm N45 cube neodymium with 1" cube N50 5 inches above. On right identical setup using standard bismuth 'non-programmed' disk with same lifter magnet, with resultant much lower levitation." So a 1-inch N50 "lifter" cube is part of the system. "Programming" is described only as "bismuth casting with applied charge and field in casting" (p.99); no field strength, charge, voltage, temperature or procedure is given. The Step 1 question "what was done to program the disk" is answered: nothing quantitative is stated in the book. Claim 6 should be read with that.
2. facts.json / deep read pp.43-53 on CODATA. The deep read cites CODATA 2018, 42.577 478 518(18) MHz/T. The current value is CODATA 2022: 42.577 478 461(18) MHz/T, relative uncertainty 4.3e-10 [web: https://physics.nist.gov/cgi-bin/cuu/Value?gammapbar]. Nothing in the argument changes. Wheeler's replacement 42.4923 is 0.2001% below it. The gap is 0.0852 MHz/T, which is about 4.7 million times the stated uncertainty (the deep read says "10^5 to 10^6 times"; that understates it) [calc]. At 3 T the proton line would sit 256 kHz away [calc].
3. Deep read pp.43-53 item 4(e): "omega = gamma e H counts e twice, dimensionally wrong." This holds only if Wheeler's gamma is the gyromagnetic ratio. On p.48 he defines gamma in his own words as "the scalar proportionality constant between the precessional frequency and the centripetal field force" [read: p.48], so gamma-times-e could be a lumped constant of undefined units. The statement should read: the formula is dimensionally correct only if "gamma e" is one undefined constant; it is wrong if gamma is the NMR gyromagnetic ratio, which is how the book labels it ("Larmor frequency", "compare to the magnetogyric ratio"). Either way, no units are given [read].
4. facts.json mainstream_conflict and the deep reads accept "the p.125 Poincare metric lacks a factor 4". Confirmed, and sharpened: as printed, ds^2 = (dx^2+dy^2)/(1-x^2-y^2)^2 has Gaussian curvature -4 (SymPy: K = -4); with the factor 4 it is -1, and 4 is exactly what the pull-back of the Minkowski metric from t^2 = x^2+y^2+1 gives [calc]. Not a new error, a quantification.
5. sites.md lead on Gardi (2020 Hall-sensor test). The Investigator reports it as evidence against Wheeler's "stronger grade, larger throw" claim. Scale invariance of magnetostatics changes how it should be read (claim 6, section 6.4). The sign of Wheeler's expectation is right; the size of the effect is small; the wiki summary may not be able to resolve it. Not an error in the lead, a caution on weight.
6. Everything else I checked in the deep reads stands: the construction of 42.4923, 68.7539, 21.246, 85, 3.23606 from phi; C1 = phi^-2, C4 = phi^-3, C2 = 1 + phi^-3 = 2/phi in units of C3; "C2 + C3 = phi cubed" false (it is sqrt 5); "360/85 = phi^3" off by 0.018%; "C squared" has units of m^2/s^2 [calc, all re-verified]. One refinement to the deep reader's "85 is to Phi as Phi is to 1 is false": agreed with the pp.149-156 reader's later note (85 degrees = "1", 137.5 = "Phi" gives the exact ratio phi : 1).

2. How Wheeler uses mathematics (overview)

Four kinds of statement, kept apart:

• Standard mathematics reproduced correctly: golden-ratio identities (phi^2 = phi+1, phi^3 = 2phi+1 = 4.2361, phi^-3 = sqrt5 - 2 = 0.23607); the golden angle 360/phi^2 = 137.5078 deg; hyperboloid projection to the Poincare disk; geodesics as orthogonal arcs; stereographic projection; a complex-conic identity (deep read p.117) [read, calc].
• Arithmetic slips inside that mathematics: listed under claims 2-4.
• Numbers built from phi and then given physical names: "gyromagnetic ratio/precession angle" 42.4923, "maximum precession" 68.7539, "dielectric:magnetic ratio" 3.23606, field radii 0.236/0.382/1.236. No measurement, instrument or error bar is attached to any of them. Wheeler's own method statement is "retroduction": work backwards from the desired conclusion (p.10). Mathematically, a number constructed to equal 180/phi^3 can match any physical quantity only by accident or by choice of units.
• Equals signs used as metaphysics: "dielectric inertial plane = 0 = 1 = infinite inertia" (pp.112, 114, 117). No quantity, unit or map to a measurable is defined, so the statement cannot be true or false as mathematics.

A general hidden assumption throughout: that a dimensionless number must come from the golden ratio because it can be written with phi. The golden ratio's ring of numbers, Z[phi] (a + b phi with whole a, b), is dense on the real line, so any measured ratio can be approximated by an element of it to as many digits as you like, with small a, b only if you are lucky. This is why a single hit (for example C2 = 1.236 R at one height, claim 4) carries almost no evidential weight without a prior prediction of which quantity, at which height.

3. Claim-by-claim review

Claim 1. "Magnetic attraction does not exist; it is dielectric voidance; a magnet is an electrified dielectric object" (pp.3, 8, 26, 54-56, 96, 138)

a. Is the mathematics internally consistent? There is almost no mathematics. The one formula is the "Newton corrected" law (p.30): D.V. = M1 x M2 / Distance^2 [read: page image p.30]. - Units: with M a mass this is kg^2/m^2, not a force; the constant G is dropped. With M a magnetic "pole strength" (A m) it would need mu0/4pi to be a force. No definition of D.V. or M is given, so the formula does not close dimensionally [calc: dimension count]. - Falloff: Wheeler uses 1/d^2 for the force between magnets. The measured and calculated far-field force between two magnets is a dipole-dipole force that falls as d^-4 on axis (F = 3 mu0 m1 m2 / (2 pi d^4)) [memory: high; standard]. For a piece of soft iron attracted by a magnet it falls faster still, since the force is proportional to the gradient of B^2 (roughly d^-7 in the far field of a dipole) [memory: high]. His law is inconsistent with the standard result and he gives no data for it. - Faraday "missing three-fourths" (pp.30-31): doubling distance reduces an inverse-square force to one quarter. Nothing is "missing": the work to move a body from r to 2r is the finite integral of F dr, and the force law is just a function. The text offers no equation for the "translational exchange" it asserts. - Internal tension (kept from the deep read): the text first says acceleration to a magnet "decreases movement" (p.30) and elsewhere that increasing movement increases inertia (pp.116, 119), while p.113 says "an increasingly moving body is losing inertia". Definitions of "inertia" drift between pages. - Units on p.35: "Dielectric field = Phi/(L x T) (Weber per cm-second)" and "Magnetic field = Psi/(L x T) (Coulombs per cm-second)": the names of the fields are swapped relative to the quantities the book assigns (Phi = magnetism in Weber, Psi = dielectricity in Coulomb) [corrected 2026-10-09: the "swapped" point is withdrawn; see arbiter.md QE revision, Verdict changes item 2. Wb/(m s) = V/m (electric field strength) and C/(m s) = A/m (magnetic field strength H), so the names fit the units. The siemens point below stands]. The ratio "dielectric : magnetic" has units of coulomb per weber = siemens (1/ohm) [calc: C/Wb = A s / (V s) = 1/ohm]. It becomes a pure number only after choosing a normalising impedance (for example 376.7 ohm for free space). The "3.23606 : 1" of claim 3 therefore has no meaning until that choice is stated; the book states none.

b. Consistent with established experiment? The mathematical statement that matters: for induced iron the force is F = grad(U) with U = -(chi V / 2 mu0) B^2 (chi the susceptibility, V the volume), derived from energy; it predicts the sign, the distance dependence and the saturation of the pull, and matches force-distance measurements routinely made on magnets [memory: high]. Wheeler's "dielectric voidance" yields no formula to compare. A direct consequence of his stated mechanism is checkable: if forces were 80-90% dielectric (p.76 estimate), immersing magnets in a high-permittivity liquid (water, relative permittivity about 80) should change the force strongly. Magnet forces under water and oil are, to the precision of ordinary tests, those in air [memory: high; common observation, not a controlled study I can cite]. Flag for the physicist.

c. Tested? Not by Wheeler: no force-distance data, no shielding or immersion test. The Investigator found no test by others of "dielectric voidance" as a force law. Wheeler's later work moves further: 2025 Substack says "magnetic attraction" is "identical to the phenomenon called gravity" [web: https://kenwheeler.substack.com/p/the-codex-of-gravity, via sites.md], a change of wording from the 2014 book's "dielectric voidance", so the claim is not stable even across his own writings.

d. If true, what would change? Static magnetostatics (Maxwell's equations with magnetisation, the Maxwell stress tensor, the dipole-dipole force law, all of electrical-machine design) would need a replacement that reproduces the same force numbers; the rewrite would be an equivalent restatement unless it predicts a difference, and the book names none.

Verdict (split): - Weak form (the attractive force is a relabelling: "voidance" instead of force): Not testable as stated, no measurable difference is predicted. - Strong form (the only equation offered, D.V. = M1 M2 / d^2, is a law for magnet forces): Internally inconsistent (no units, G dropped) and in conflict with the measured dipole falloff.

Claim 2. "A magnet's field is a magneto-dielectric hyperboloid following the Poincare disk model, with a dielectric inertial plane at the equator = Bloch wall; space is a posterior attribute of magnetic divergence" (pp.58-74, 112-127, 186)

a. Internal consistency. - Correct standard content [read, calc]: projection of the upper sheet of t^2 = x^2+y^2+1 from (-1,0,0) onto the unit disk; geodesics as arcs orthogonal to the boundary; stereographic projection; the complex-conic identity (hyperbola and ellipse as real slices of one conic). Its caption's "proving that the dielectric inertial plane = 0 = 1" has no connection to what the figure computes (deep read p.117). - Errors and mismatches: 1. The printed metric lacks the factor 4 (curvature -4, not -1) [calc]. 2. The "magnet" pictures (pp.116, 119, 125) are hyperboloids of one sheet (x^2+y^2-z^2 = 1, a ruled surface), while the Poincare construction needs the two-sheet hyperboloid t^2 - x^2 - y^2 = 1. One-sheet hyperboloids do not project to the Poincare disk (they have negative Gaussian curvature of a different kind: they are not hyperbolic planes). The book treats them as one object [read; deep read item 7; I confirm]. 3. "1/Phi^-3" is used for both 4.236 (phi^3) and 0.236 (phi^-3) (pp.114, 150). "1/Phi^3 = Phi cubed" (p.150 figure) is false by a dropped minus sign. 4. "0 = 1 = infinite" read literally is a contradiction. The deep reader offered a charitable geometric reading (centre at distance 0, boundary at infinite hyperbolic distance); the author puts both at the centre (p.126), so that reading is not his. 5. "only three infinite rays can intersect the centre" (p.114): in the Poincare disk infinitely many geodesics pass through the centre. The three-way split 137.5/137.5/85 degrees is one of infinitely many choices; the book derives no uniqueness. 6. The "dielectric inertial plane" is identified with the Bloch wall. A Bloch wall is a thin transition layer between magnetic domains inside a ferromagnet (width typically tens of nanometres) across which the magnetisation rotates [memory: high]. A magnet's equatorial plane outside the material, where the field is purely radial/perpendicular to the axis, is not a domain wall. The identification is by name only. - "Vortex": outside the magnet B is curl-free (no current there) and divergence-free, so B = -mu0 grad(phi_m) is a potential field, like the flow of an ideal fluid past a doublet. It has no circulation around any loop that does not link a current. The circulation that does exist is the bound surface current K = M x n on the magnet's side wall (Ampere's model), which is a standard vortex-like current sheet [memory: high; Griffiths ch. 6]. If "magnetic vortex" meant that, it would be standard physics; Wheeler's centripetal/centrifugal fluid language ("pressure gradients") is not a quantity in Maxwell's theory and no mapping to B or H is given. Divergence is correct for H at the pole faces (div H = -div M), which may be what "divergence at the poles" refers to, but B has none. - "Space is only a posterior attribute of magnetic divergence" and "Newton's first law is wrong": definitional, by the author's own statement ("definitionally" appears on nearly every page). He also says the root quantity "cannot be measured" (pp.116, 117, 123). A quantity declared unmeasurable cannot yield a formula that is checked.

b. Consistent with experiment? The facts he accepts (field lines close, div B = 0, filings align with the local field) are the standard ones. His own concessions: filings "show the direction of the magnetic flow" (p.26). What he adds (hidden spiral structure) is not quantified. The related ferrocell literature uses hyperbolic geometry differently (see c).

c. Tested? By Wheeler: no. By others, in related terms: Tufaile, Snyder and Tufaile, "Horocycles of Light in a Ferrocell", Condensed Matter 6(3):30 (2021) describe point-source ferrocell images as horocycles (circles tangent to the ideal boundary in hyperbolic geometry) and interpret them "as analogous to a Mobius transformation by inversion of the magnetic field" [web: https://www.mdpi.com/2410-3896/6/3/30, via research/2026-10-08-google-scholar-ferrocell.md]. Mathematically this is a model of the optical pattern in the image plane, produced by light scattering from field-aligned nanoparticle chains; the authors state the pattern lines are perpendicular to field lines, can cross (field lines cannot), and are "magneto-optical effects and not just magnetic effects" [read: Investigator's quotation of the Tufaile guide, ~pp.65-68]. Wheeler is not cited. It is a real, peer-reviewed use of hyperbolic geometry on a ferrocell, so it belongs in the record as related work in different words; it does not show that the field of the magnet is a hyperbolic plane, and it does not use a Poincare disk model of the field. The Tufaile 2024 paper (JOPR) builds on Stokes vectors and Mueller matrices and the Poincare sphere of polarisation (a different Poincare construction) [read: OCR of the paper's pp.3-5]. I did not read the full body. Also relevant and sympathetic: Lori Anne Gardi (2020) Hall-sensor measurement locating a zero reading at what she calls the "dielectric inertial plane" down the side of the magnet [web: https://wiki.naturalphilosophy.org/..., secondary summary]. A sign change of the field component on the equatorial plane of a magnet is expected in standard theory (by symmetry B_z (axial component) changes sign through the return flux; this is the zero-crossing ring I compute under claim 4). It supports that a measurable feature exists in the equatorial region; it does not support the hyperboloid or Poincare structure.

d. If true, what would change? The metric of space (flat, measured with rulers) would not change. For the claim to be a physical one, the field would have to be shown to be a geodesic field of a hyperbolic metric, and the permeability/permittivity of space would then have to vary with position as that metric dictates; nothing like that is measured in magnetostatics, where B of a given magnet is calculated to percent accuracy in flat space. Equivalently, one would have to explain why a calculation without any hyperbolic geometry already reproduces magnet fields.

Verdict (split): - The geometry as geometry: Established (the standard parts, with the slips listed). - The identification magnet field = Poincare disk/hyperboloid, with "=0=1=infinite" and "space is a posterior attribute": Not testable as stated, and Internally inconsistent where read literally (one-sheet versus two-sheet, factor 4, "1/Phi^-3", 0 = 1). - Related work (Tufaile horocycles of ferrocell images): kept; Plausible, untested as a bridge to Wheeler's picture, which the authors themselves do not make.

Claim 3. "The geometry is fixed by phi: proton gyromagnetic ratio and ideal precession angle 42.4923 (degrees and MHz/T = 180/phi^3), maximum precession 68.7539, dielectric:magnetic ratio 3.23606:1, golden angle 137.5077 with an 85-degree sector" (pp.43-51)

a. Internal consistency. Verified [calc]: 180/phi^3 = 42.49224; 180/phi^2 = 68.75388; 360/phi^2 = 137.50776; 360/phi^3 = 84.9845; 90/phi^3 = 21.2461; 2 phi = 3.23607. The deep reader's table is right. The identities are algebra of phi; they are not discoveries about magnets. - Internal inconsistencies (all from the deep read, I re-checked): "42.5 x Phi = 68.7539" is false (42.5 x phi = 68.766; the figure holds for 42.4923 x phi); "42.4932" on p.49 versus "42.4923"; "21.23606" on p.51 versus "21.246"; "137.52788" versus "137.5077"; "360/85 = 4.23606" (360/85 = 4.23529, a 0.018% error caused by rounding 84.9845 to 85); "137.5077 = Phi" and "68.7539 = 1" (their ratio is 2, not phi); "4.23606 degrees" (the ratio 360/85 is dimensionless). The angle 58.1308 on p.44 has no phi expression that I or the deep reader could find; I tried 79.3769 - 21.2461 = 58.1308 and 137.5077 - 79.3769 = 58.1308, which are consistent additions but leave 79.3769 itself unexplained [calc]. - The bold identification: "42.4923" as an angle in degrees and "the same" as a frequency per tesla in MHz/T. An angle of 42.4923 deg in a diagram and a frequency-per-field of 42.4923 MHz/T are different physical dimensions. The coincidence of the digits depends entirely on using degrees (a 360-part circle), megahertz and tesla. In rad s^-1 T^-1 the proton value is 2.675e8; in kHz per gauss it is 4.2577 [memory: high]. No convention-free identity exists. Dimensionless digits can only be compared after a unit choice is declared to be physical. - His 42.4923 is 0.200% below the measured 42.57748; his quoted "published" 42.5781 matches no standard value (42.5775 and the shielded-water proton value 42.5764) [web: NIST page above; the shielded value from the deep read, memory medium]. - The precession cone angle: in the Larmor relation omega = gamma B the frequency is independent of the tilt angle. A fixed "ideal precession angle" is not a constant of nature; it is set by how the moment was tipped. Quantum mechanically, for spin 1/2 in a definite m = +1/2 state, the angle to the field axis is arccos(1/sqrt 3) = 54.74 degrees (the "magic angle" cone) [calc], not 42.5 or 68.75. - Larmor precession belongs to nuclear spins. A permanent magnet's moment is electron-spin and orbital moment; nuclear magnetic moments are about 1/660 of the Bohr magneton; Fe-56 has nuclear spin 0 [memory: high; deep read]. The mechanism "protons at the apex of every atom precess and create the magnet's field" (p.43) is therefore a different claim from the MRI physics he cites and has no stated calculation.

b. Consistent with experiment? No. The proton gamma/2pi is measured to 4.3 parts in 10^10 by NMR and by Penning-trap proton magnetic moment measurements; every NMR spectrometer and MRI scanner would be off resonance by 0.2% if 42.4923 were right. At 3 T that is about 256 kHz, many linewidths [calc, linewidth statement memory: high]. Whether the "incorrectly published" statement could be true as a claim about measurement: it can only be true if all NMR instruments are mis-calibrated in the same way, which contradicts independent calibrations (water-sample NMR, helion, Penning trap) [memory: high]. Wheeler's own caption says precession "is nothing of my discovery" (p.44), so he accepts the textbook relation while altering its constant.

c. Tested? The numbers were never measured by Wheeler; they were in the 2nd edition (1 July 2014) before any of the ferrocell material (research note: kathodos 2nd ed. contents list), consistent with a construction preceding any demonstration. Phi in phyllotaxis (sunflower seeds at the golden angle) is real botany; the extension to magnetic field lines (pp.49-51) is supported only by drawings, a computer-model torus of spheres, and borrowed images (a Belousov-Zhabotinsky scroll wave credited to Winfree, a moire demonstration). Nobody has tested whether ferrocell rings or filings show a 137.5 degree spiral. Later independent work on ferrocell angles exists (Gardi & Snyder 2024, "Finding the Right Angles of a Ferrocell"; blocked for the Investigator) and could bear on this; not read.

d. If true, what would change? The measured proton gyromagnetic ratio, the calibration of every NMR/MRI machine and the proton magnetic moment (and through it the Rydberg-Bohr/QED consistency tests of the proton g-factor) would have to be revised by 0.2% with no instrument showing the discrepancy. In addition, a dimensionless angle would have to be tied to a dimensionful constant by a law, which is impossible without a conversion constant.

Verdict (split): - The arithmetic identities among 42.4923, 68.7539, 137.5077, 85, 3.23606 and phi: Established (as mathematics; with the slips listed). - "42.4923 is the proton's true gyromagnetic ratio; published value incorrect": Refuted by calculation (it differs from the measured constant by about 4.7 million standard uncertainties, and the match of a degree-valued angle to a MHz/T value is a units artefact). - "Ideal precession angle of a magnet = 42.4923 deg", "dielectric:magnetic = 3.23606", "field lines follow 137.5 deg phyllotaxis": Never properly tested (no measurement on record, none dismissed because none was submitted), with the first two Not testable as stated (no units for the ratio, no definition of the angle).

Claim 4. "Magnetic & Dielectric Field Formula" (7-28-2014): C4 = 0.23606, C1 = 0.381966, C3 = 1, C2 = 1.23606 times the magnet's radius, for aspect ratios 1 to phi, cubes and spheres (pp.149-156)

a. Internal consistency. Taking the four rules in units of C3 = 1 [read: p.149]: C4 = phi^-3 = 0.23607; C1 = phi^-2 = 0.38197; C2 = C1 x 2 phi = 1.23607 = 1 + phi^-3 = 2/phi = sqrt 5 - 1 [calc]. These are exact. Two further exact relations the book does not state: C4 + C1 = phi^-1 = 0.618 and C1 + C2 = phi = 1.618 [calc]. Errors on the same page: "C1r + C2r = Phi cubed" is true only in units where C1 = 1 (in C3 units it equals phi); "C2 + C3 = Phi cubed" is false in either unit choice (in C3 units it is sqrt 5 = 2.236) [calc]; p.152 labels C2 "Phi 1.618033" while the rule and the drawn circle give 1.236; "85/360 = Phi cubed" is inverted (it is phi^-3 = 0.2361; 360/84.98 = phi^3); the 108-36-36 triangle is not part of the p.155 pentagon (arcs of 68.75 degrees give corner angles 111.25 and 103.13, not 108) (deep read; I accept it). The "hypotrochoid" is never given an equation or parameters (no rolling radius, pen offset or step angle), so nothing is derived from it. "Incommensurability" is used (p.154) for a drawing that can be read two ways, which is not the mathematical meaning (an irrational ratio such as phi:1). Undefined: "radius" of what (field strength? direction? a ring seen in a ferrocell?), at what height, for which field component, what "mean" means, and "L to H" (magnet aspect ratio inferred).

b. Consistent with experiment, by my own calculation. I computed the axial field component B_z of a uniformly magnetised cylinder (radius R, top face at z = 0) on a plane at height h, as the field of the equivalent azimuthal surface current, using elliptic-integral loop fields summed along the wall (check: face-centre B_z/Br = 0.447 for L/D = 1, equal to the closed form 0.5 L/sqrt(L^2+R^2) with L = 2R) [calc]. Radius at which B_z changes sign:

These reproduce the deep reader's table (1.14, 1.26, 1.55, 1.97; 1.24, 1.43, 1.83) to rounding [calc]. Conclusions: - The sign-change ring crosses 1.236 R at about h = 0.09 R for L/D = 1 and at about 0.05 R for L/D = phi, and moves outward continuously as height increases. By the intermediate value theorem a continuous function of h that increases from about 1 R to beyond 2 R must pass through 1.236 R at some height. A match at one unspecified height is therefore guaranteed and carries no evidential weight; a fixed ring at 1.236 R at all heights is contradicted by standard magnetostatics. - No feature of B_z at 0.382 R or 0.236 R is expected over the face (B_z varies smoothly there; the field tilts strongly only near the edge, about 0.9 R) [calc, deep reader's table]. - Scale invariance (hidden consequence worth knowing): the field of a uniformly magnetised body depends only on its shape and Br, and in units of its size it is the same for every size. So fixed ratios of field features to magnet size are not absurd in form: they are what standard theory predicts for geometrically similar magnets at the same relative height. The formula's novelty is in the particular numbers, not in the idea of size-proportional features.

c. Tested? Not by Wheeler: no gaussmeter reading, scale, gap or lighting appears on pp.149-156; the evidence is ferrocell photographs (one of a 1/4-inch cube magnet, p.151, whose photographer is not stated on these pages) and drawings the text calls "empirical demonstration" (p.153) although they are drawings. In a ferrocell the number of overlapping circles in a ring-lit image follows the number of LED light sources and the ring radii change with the gap between magnet and cell [memory: medium; the ferrocell.us product is a "ring of 36 RGBW LED's", per the Investigator]; this is not verified here. Closest independent measurement: Gardi's Hall-sensor test (secondary wiki summary). She reports the same maximum surface reading (1.44 V) for N42 magnets from 5 mm to 2 inches and locates zero at the equatorial plane. By scale invariance, geometrically similar magnets of the same grade have the same surface field, so identical readings are what standard theory predicts for similar shapes (her magnets include a 1 x 1 inch cylinder and a 2 x 2 x 1 inch block, which are not geometrically similar, so a sensor limit may also be involved; I cannot tell from the summary). It neither supports nor refutes 0.236/0.382/1.236 R.

d. If true, what would change? If the radii were fixed fractions regardless of height, the magnetostatic solution for a cylinder (Biot-Savart over the bound current) would have to be wrong at the percent level, which hundreds of Hall-probe maps contradict. If the claim is only the single-height coincidence, nothing changes.

Verdict (split): - The arithmetic of the four rules: Established (as phi algebra; several captions wrong). - Strong form (these radii are fixed features of every such magnet's field): Refuted by calculation for the one feature I can compute (the B_z sign-change ring moves with height), and Never properly tested for any ring Wheeler might mean (he defines none). - Weak form (a feature happens to sit near 1.236 R at some height): Plausible, untested, and expected by continuity. - A cheap test exists (Hall-probe scan at several heights, section 6 below).

Claim 5. "Mainstream physics is wrong: no electrons, negative charge, photons or warped space; relativity and QM false; Newton's first law wrong; 'C squared' limits spatial motion but not longitudinal/dielectric fields; light has a Z-axis radial dielectric component; photoelectric effect 'corrected'" (pp.8, 16, 29-34, 113-125, 207-219)

a. Internal consistency. - "C squared" as a speed limit (pp.115, 116, 118): c^2 = 8.988e16 m^2/s^2, units of energy per mass, not speed [calc]. On p.212 he writes correctly that mu x epsilon = 1/c^2, and elsewhere says light travels at "luminal velocity". So the speed-limit sentence is either a unit error or a loose name for c. The attribution "Maxwell's discovery, not Einstein's" mixes two results (Maxwell: wave speed 1/sqrt(mu0 eps0); relativity: limit for bodies) [deep read; memory high]. - Newton's first law (p.113): his objections are about "rest" and "straight line" being absolute. In mechanics both are frame-relative and the law is the definition of an inertial frame. A body curving relative to the galaxy does not contradict it. Objection 3 renames "force" as "field voidance" and makes no different prediction [deep read; I agree]. - Photoelectric effect (pp.216-219), the one place where an equation is at stake. The page keeps Einstein's relation: "Replace E with energy of the dielectric, h Planck and f frequency remain the same" (p.217). So the "correction" keeps K_max = h f - W (K_max maximum kinetic energy of the emitted charge, W the work function) and changes the story only. It then says shorter wavelengths have "more dielectric pulses" per length AND "higher dielectric capacitance", so "the power increases on two levels exponentially as wavelengths decrease" (p.219). If pulses per length scale as 1/lambda and the per-pulse energy also scales as 1/lambda, the energy scales as 1/lambda^2, not linearly with frequency, which contradicts the retained E = h f. "Exponentially" is also wrong (it would be a power law). Experiment: the stopping potential grows linearly with frequency with slope h/e = 4.136e-15 V s, first measured by Millikan (1916) [memory: high]. Quick worked number: sodium (W = 2.28 eV) lit at 400 nm (photon energy hc/lambda = 3.10 eV) gives K_max = 0.82 eV; at 600 nm (2.07 eV) there is no emission at any intensity [calc from memory constants]. A model must reproduce the threshold; "dielectric pulses" gives none. - The "Z-axis radial dielectric component" of light (pp.207-219, 228; Plate 3): the geometry is the coaxial cable. In a coaxial cable the TEM mode has E radial, B azimuthal, propagation along z. "Radial" and "Z-axis" are therefore different directions; the text uses both words for the same "dielectric" axis ("Z-axis radial dielectric"). If Z is the propagation direction, a field component along it in vacuum is longitudinal. Gauss's law and Faraday-Ampere force it to vanish for a plane wave: for E(z,t), div E = 0 gives dEz/dz = 0 and the z-component of Ampere's law gives eps0 mu0 dEz/dt = (curl B)_z = 0, so Ez is a constant, not a wave [calc: derivation]. The energy-flow direction is already carried by the Poynting vector S = E x B / mu0, which points along z: that may be the "third component" in an honest reading, and it is in standard theory. Real beams of finite width do have a small longitudinal field, of relative size about lambda / (2 pi w) for beam waist w, required by div E = 0 [memory: high]; this is where twisted-light (orbital angular momentum) beams, which Wheeler cites as dependent on his component (p.228), have a z-component, and no "dielectric carrier" is needed to derive it. The clippings he reproduces make no reference to his model [read: deep read pp.228-230]. - Light deflection (p.209): "Light does not bend around supermassive objects... rather the accumulative field of both gravity and dielectricity ... are centripetally attractive." No formula is given. Any theory must reproduce the observed solar deflection of 1.75 arcseconds, which general relativity gives as 4GM/(c^2 R) = 8.5e-6 rad = 1.75 arcsec for the Sun (twice the Newtonian 0.875 arcsec) [calc]; he provides no way to compute it. - Nuclear energy from "empty space" (p.121: 99.9999999999% of "nothing" accounts for 100% of the energy "NOT the conversion of any matter"). The energy released by fission is measured at about 200 MeV per U-235 fission; that equals 0.091% of the rest-mass energy of the nucleus (200 / (235.04 x 931.494)) [calc], and equals the mass defect of the fragments times c^2 within measurement error [memory: high]. - Refraction (p.212): "light through a medium is the square root of the inverse of the change of capacitance ... if there is 0 capacitance there is infinite speed." The form v = 1/sqrt(mu eps) is right and v does fall with permittivity, which is proportional to capacitance for fixed geometry. But permittivity cannot go below eps0 (vacuum), so "0 capacitance" is outside the physical range and gives no infinite speed [memory: high].

b. Consistent with experiment? The framework would have to reproduce, with no new constant, time dilation of muons, GPS clock corrections, light deflection, the photoelectric threshold, electron diffraction, the electron g-factor to 12 digits and so on. The book gives none of the calculations. Mathematically, a theory that retains E = h f and c but renames the carriers has the same predictions as the standard theory and so cannot be "more right"; where it changes a quantity (E growing as 1/lambda^2), it fails the photoelectric data.

c. Tested? The standard framework has been tested many times; Wheeler's alternatives have no stated predictions to test, apart from the items above, so there are no tests of his version. Dismissal without testing is not the case here: there is nothing testable stated. "Longitudinal fields exempt from the speed limit" (p.115) is testable in principle (a superluminal signal) but no experiment or derivation is offered; Dollard-type "longitudinal wave" claims are a separate subject for the physicist and contrarian.

d. If true, what would change? All of relativity, quantum electrodynamics and the Standard Model would have to be reproduced by a different structure; nothing in the book approaches this.

Verdict (split): - "C squared" as a speed limit: Internally inconsistent (units). - Newton's first law "wrong": Not testable as stated (definitional). - Photoelectric effect "corrected": Internally inconsistent (keeps E = h f, then claims quadratic-or-exponential growth). - Z-axis dielectric component of light, as a longitudinal vacuum field: Refuted by calculation (Maxwell's equations for a plane wave); in the reading "energy-flow axis" it is Established (Poynting vector) but not new. - "Empty space" is the source of nuclear energy: Refuted by calculation (mass-defect budget). - No electrons, photons, relativity, QM: Tested, not supported (numerous controlled experiments; no alternative calculation given).

Claim 6. "The model is confirmed by demonstrations and promises technology: ferrocell/ferrofluid patterns, RGB LED discovery, 'programmed' bismuth disk, objects resting at the precession angle, vortex videos, vortex technology" (pp.22-25, 97-111, 133-142, 162-176, 226-228)

a. Internal consistency / calculability. 6.1 Bismuth levitation (pp.97-111, text read in full [read]). Setup: a 3 mm N45 cube floats "5 inches above" a 1-inch N50 cube lifter; a "programmed" bismuth disk gives a higher float than a standard disk (p.99). My estimate [calc], treating both as dipoles with m = Br V / mu0 and assuming the typical remanences Br = 1.43 T (N50), 1.35 T (N45), magnetic parameters from memory medium confidence: - m(lifter) = 18.6 A m^2; m(3 mm cube) = 0.029 A m^2; weight of the cube (density 7500 kg/m^3) = 1.99 mN. - Axial repulsion of two opposed dipoles: F = 3 mu0 m1 m2 / (2 pi d^4) = 1.25 mN at d = 127 mm (5 in); F equals the cube's weight at d = 113 mm = 4.45 in (dipole approximation; finite size shifts this by a few percent). So the observed height of about 5 inches is what two magnets alone give; bismuth is not needed to explain the height. - Bismuth's role by calculation: with volume susceptibility chi = -1.66e-4 [web: table of volume susceptibilities, via search result https://arxiv.org/pdf/1506.09199 listing bismuth at -16.6e-5 SI] the repulsion of a semi-infinite bismuth slab on the 3 mm cube is F = (3/h)(|chi|/2 mu0)(mu0 m /4 pi)^2 (pi / 2 h^3) = 0.16 mN at a 2 mm gap, 0.004 mN at 5 mm, 0.0003 mN at 10 mm, and drops to 1.6e-5 mN at 20 mm. It equals the cube's weight only below about 1 mm [calc; dipole approximation, crude at that gap]. A bismuth disk sitting in the field of the lifter carries an induced moment of order |chi| B V / mu0 = 1e-3 A m^2 against the lifter's 18.6 A m^2, a fraction 6e-5 [calc, assuming 0.4 T at the disk]; so it can change the force on the floating cube at 127 mm by about 6e-5 of itself, i.e. the height by about 1e-5 of 127 mm. - Conclusion (with the caveat that the book does not say where the disk sits or its size): bismuth cannot lift the cube; what bismuth does, as for every diamagnetic levitation, is provide the lateral and tilt stability that Earnshaw's theorem denies to fixed magnets (the energy -(chi/2mu0) B^2 has minima for chi < 0) [memory: high]. A "programmed" disk can therefore be expected to differ from a "standard" one only in stability, not in the height set by the lifter, unless some other variable differed between the two photographs (disk position, magnet separation, orientation, temperature). That variable is exactly what the photograph pair cannot control. 6.2 Bismuth nuclear remarks (pp.98-103). (i) "Bismuth also exhibits overunity since its 209 nucleus has a higher energy state than the sum of its components, thallium 205 and helium-4... bismuth breaks the rule of conservation" (p.103). The mass difference m(Bi-209) - m(Tl-205) - m(He-4) times c^2 = 3.137 MeV [calc; atomic masses from memory, medium; the Orsay line is 3.14 MeV]. That positive Q-value is exactly the energy the alpha particle carries away; energy is conserved, and a positive Q is why alpha decay is possible. The same page cites the 3.14 MeV line. (ii) The half-life measured at Orsay was (1.9 +- 0.2) x 10^19 years from 128 alpha events [read: p.103 quoting the excerpt]; confirmed at Gran Sasso (2.01 +- 0.08) x 10^19 years [web: https://physicsworld.com/a/bismuth-breaks-half-life-record-for-alpha-decay/ and https://wikipedia.com/wiki/Bismuth-209, via search]. The book says the half-life is "false" because contamination by Bi-210/Po-209 produced the alphas (p.102). Contamination would produce lines at the contaminants' alpha energies (Po-210 about 5.3 MeV; Po-209 about 4.9 MeV, per my recall, medium), not at the 3.14 MeV the same page quotes, which matches the Bi-209 to Tl-205 Q-value [calc]. (iii) "Golden ratio balance between protons and neutrons": 126 neutrons / 83 protons = 1.518, not 1.618 (6% off) [calc]. 126 is a magic number of nuclear shell theory [memory: high]. 6.3 Ferrofluid/ferrocell/RGB LED: no measurement, quantity, exposure, geometry or repeat count (deep reads; claims.md). "Cone volume gives magnetic strength and velocity" (p.23) is a formula-free assertion; the standard normal-field instability gives a spike spacing set by the capillary length sqrt(sigma / (rho g)) in the Rosensweig regime [memory: high], which includes surface tension and gravity, the two things Wheeler says do not matter (p.23). RGB LED: pp.133-176 digest only; I cannot assess the mathematics. The Tufaile guide, which describes a ferrocell with a ring of RGB LEDs, explains pattern lines as perpendicular to field lines (a geometric relation: scattering by chains of nanoparticles aligned with the field), so the same image has a published account in which Wheeler's "dielectric inertial plane" does not appear [read: Investigator's quote]. 6.4 Third-party measurement (Gardi 2020, secondary summary). Wheeler "implied in his videos that a stronger magnet of the same size would have a greater throw"; Gardi concludes grade "makes no difference to either the surface reading or the distance at which a given reading is found" [web: https://wiki.naturalphilosophy.org/..., via research/2026-10-08-cnps-wiki.md]. Scale invariance gives the expected size of the effect: B(x) = Br f(x/L) for fixed shape, with f decreasing, so a higher remanence means a larger throw to any fixed threshold. The surface reading should rise with Br: N52 (Br about 1.45 T) over N42 (about 1.32 T) is +10% [memory: medium on the Br values]. In the dipole far field B ~ Br (L/x)^3, so the distance at fixed threshold goes as Br^(1/3), +3%. Wheeler's expectation has the right sign and is small; whether Gardi's gauge could resolve 10% at the surface and 3% in distance is unknown, and that "the same maximum 1.44 V" appeared for different sizes might also indicate output saturation of the sensor. So her result is a measured, controlled-looking test whose resolution I cannot judge; weigh it as evidence against a large effect only. 6.5 Vortex videos (Tinmanpower, hydrogen-bubble vortex over a magnet, p.226-227): captions give no current, voltage, field or sense of rotation. A bubble swirl over a magnet pole in electrolysis is a standard magnetohydrodynamic effect: the Lorentz force J x B on the current in the electrolyte drives rotation, with torque proportional to the product of the radial current and the axial field [memory: high]. Rotation direction reverses with current or field polarity. Nothing in the caption separates this from Wheeler's reading. 6.6 Dated discoveries (ice figure 5-1-2014, p.234; "Magnetic Egg" video): no experimental description; unknown.

b. Consistent with experiment? The bismuth calculation is consistent with a floating height set by the two magnets; the existence of a difference between the two disks is not computed by Wheeler and, if real, must come from something other than bismuth's diamagnetic force. The ferrocell images are real observations of a real device and nobody disputes the patterns, only their interpretation.

c. Tested? Wheeler: demonstrations by photograph, no controls. Others: Gardi (above); Ufopolitics induction test on a forum (Energetic Forum, Dec 2015, pages 2-27 unread by the Investigator); Tinmanpower tests described only by Wheeler's 2026 account; Kris Lang's RGB LED confirmation appears only as a photograph caption (p.136). None is a neutral replication with data. The claim "bismuth programmed" has no replication on record.

d. If true, what would change? A casting method that raises levitation height while the force law stays the same would be a metallurgy/susceptibility finding, not a change in physics; if the magnetic force on a magnet at 127 mm changes by a factor needed for "much lower" levitation (tens of percent), the disk would need a susceptibility about 10^3 to 10^4 times larger than any known bismuth value [calc: from the 6e-5 fraction above]. That would be a discovery in condensed matter, and could be tested by measuring the disk's susceptibility on a balance (Faraday method) or with a SQUID.

Verdict (split): - "Ferrocell/ferrofluid patterns confirm the model": Never properly tested (patterns are real; no quantitative comparison; alternative published explanation exists). - "Programmed bismuth lifts a magnet higher": Never properly tested (no replication, no controls); the stated mechanism, a change in diamagnetic lift, is Refuted by calculation for a height set by the lifter at 5 inches (the diamagnetic force at that distance is about 10^-4 of the weight). - "Bismuth exhibits over-unity / breaks conservation": Refuted by calculation (3.14 MeV is the alpha-decay Q-value). - "Weaker/stronger magnet grade changes throw": Tested, not supported only at the resolution of one secondary-source test; by calculation the expected change is a few percent. - "Vortex technology" (100+ TB platter, communications): Not testable as stated (no device, no data).

4. Investigator's findings I weigh and keep (pertinent, whatever the verdict)

1. 2nd edition (1 July 2014, 110 pp.) [web via Wayback: https://web.archive.org/web/20140731145835/http://www.kathodos.com/magnetism.pdf]. The phi numbers (42.4923, 137.5077) and "Bloch wall" already appear, before the ferrocell/bismuth/field-formula material; the text layer lacks bismuth, RGB, ferrocell, 0.381966. For claims 3, 4, 6: the numbers precede the demonstrations. Open point: the 3rd edition dates the RGB LED discovery 18 June 2014, earlier than the 2nd edition's date; the images of the 2nd edition were not checked.
2. Substack 2024-2026: "magnetic attraction" = gravity (2025); "the plane of inertia is the dielectric 45 degrees from magnetic flux and 90 degrees from gyromagnetic precession". Both are statements without measurement. The first changes claim 1; the second assigns three angles (45, 90) not in the 2014 book's phi set: the 42.4923 versus 45 relation is not explained.
3. Tufaile et al. horocycles (claim 2): kept, see above. A mathematical fact worth giving the reader: a horocycle in the upper half-plane model of hyperbolic geometry is a circle tangent to the boundary line (or a horizontal line) [memory: high], and a Mobius transformation z to (az+b)/(cz+d) maps circles to circles; the ferrocell "inversion" is of that kind. It is a model for how light patterns map to field geometry. No Poincare disk with phi appears.
4. Gardi/Hall sensor (claims 2, 4, 6): a zero reading at the equatorial plane is expected (it is the symmetry plane); the same maximum for different magnets is expected from scale invariance for similar shapes; grade effect small. Primary video unwatched.
5. Dollard, Adams (counterspace), Krafft (vortex atoms): sources of vocabulary. The mathematically relevant point: Adams's counterspace is projective geometry's dual (points and planes interchanged) [memory: medium], a legitimate mathematical duality; Wheeler maps it to dielectric versus magnetic without stating the duality's equations. Dollard's "conjugate" fields correspond to the standard E/D and H/B pairs (Maxwell's treatment of the two inductions); for Dollard's oil-cube transfer (p.18) a dielectric moved between plates retains polarisation if it is an electret or retains charge by conduction; not enough detail to assess.
6. Absence of evidence (4th edition not found; no patent; Kris Lang only in the book). I do not draw a conclusion; competing explanations are in sites.md. The mathematician's one contribution: 'the formula and the 42.4923 claim are each testable cheaply, so a sceptic and a supporter can settle them without the 4th edition'.
7. Ferrocell USA (Vanderelli) stake: he sells the Ferrocell and co-authors the Tufaile papers; his description of the effect makes no mention of Wheeler. Wheeler thanks him for "more empirical proof"; Wheeler stands to sell the book. Both parties have stakes.

5. Summary table

6. What would settle it (cheap tests the reader can do)

1. Claim 3: nothing to measure, a unit statement: write 42.4923 deg and 42.4923 MHz/T in rad and in rad s^-1 T^-1 and see that no common physical constant relates them; compare with NMR of water in a known field.
2. Claim 4: Hall-probe scan of B_z across the face of a disc magnet with L/D between 1 and 1.618 at heights 0.05, 0.1, 0.25, 0.5 R. If the zero-crossing ring stays at 1.236 R at all heights, Wheeler's strong form is supported; if it moves as in the table above (1.15 to 1.98 R for L/D = 1), the formula describes one height only. Look for any feature at 0.236 R and 0.382 R. For the ferrocell, photograph the same magnet at several gaps and with different LED counts and measure ring radii against a scale in the frame.
3. Claim 6 bismuth: measure the lift of the 3 mm cube with and without the disk at the same separation using a balance; or measure the disk's volume susceptibility (Faraday balance). The calculation predicts a difference of order 1e-4 in force at 5 inches.
4. Claim 1: weigh the pull on a steel ball from a magnet in air, oil and water at the same distance.
5. Claim 5: Millikan-type measurement of stopping voltage against frequency (slope h/e).

Corrections to earlier documents

(For the orchestrator to apply.) 1. claims.md "Coverage note" and facts.json how_read: state that pp.97-111 and pp.207-219 were read in full from the text layer by the mathematician in Step 2 (page images not viewed), and that the bismuth setup includes a 1-inch N50 lifter magnet (p.99). Digest-only remains true of the deep/ folder. 2. deep/magnetism1small-6bca80.md section pp.43-53, items 4(b) and 8: CODATA 2018 value 42.577 478 518(18) should read CODATA 2022, 42.577 478 461(18) MHz/T [web: https://physics.nist.gov/cgi-bin/cuu/Value?gammapbar]; "10^5 to 10^6 times the published uncertainty" should read about 4.7 x 10^6 [calc]. 3. Same section, item 4(e): "counts e twice" is conditional on gamma being the gyromagnetic ratio; Wheeler defines gamma on p.48 as an undefined proportionality constant. The dimensional error stands only on the first reading. 4. sites.md "Leads for review", Gardi item: add the scale-invariance caution (6.4): identical surface readings across sizes are expected for geometrically similar magnets; the grade effect predicted is about 10% at the surface and 3% in throw; the lead is a secondary wiki summary of a video. 5. Nothing in facts.json is withdrawn.