Aspden-Hearld: mathematician's review (PhD mathematician)

Source gap (repeated from Step 1). Physics Unified (Unified-Physics.pdf) is a scan with no text layer. For this review I re-rendered all 212 pages at 150 dpi, OCRed them with Tesseract, and read Chapter 1 (pp.1-12), the vacuum-particle chapter (pp.108-118, eqs 145-168), the G section (pp.114-115, equations read from page images [read: page image p.114, p.115]), the particle chapters (pp.134-136), p.151 and the cosmic-background section (pp.176-177, OCR only). I did NOT read every page, and I did not review the geomagnetism, solar-system-angular-momentum, Planck/Schroedinger/g-factor derivations or the ball-lightning chapters, so my verdict on those parts is limited to saying so. The 2-page plasma essay was read in full. Every experimental paper the essay cites (Kobel 1930; Graneau 1983/85; Roth 1978; Sethian 1978; Aspden 1969/77/85/86) and the Correa patents were NOT available; statements about them rest on the essay's description plus the web checks listed at the end.

Notation. Gaussian-type units with c = 1 are used by Aspden: charges q, Q with masses m, M and velocities v, V; r is the vector from q to Q, r = |r|. A dot is the scalar (dot) product: a.b = |a||b|cos(angle). "Closed circuit" means a loop of steady current.

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Corrections to Step 1

1. Essay date. facts.json dates the essay "printed 2019, papers 1969-1986". The essay itself is signed "30 October 2005" [read: p.2 of the PDF]. It says the 1969 paper "was resolved ... some 37 years ago", which is loose arithmetic for 2005 (36 years). Minor.
2. Equation 163 is now confirmed from the page image [read: page image p.115 of the PDF; printed p.115]. It is G = (e/m_e)^2 (4 pi)^2 (1/108 pi)^6 (1/g)^8, with g = graviton mass in electron masses. Step 1 left the exponents unconfirmed. Recomputed with e/m_e = 5.273e17 esu/g and g = 5063: G = 6.665e-8 cgs; with Aspden's stated "factor 0.0004" (it means multiply by 1.0004) 6.668e-8. With g = 5062.75 (his 1966 value) 6.668e-8, x1.0004 = 6.671e-8. The CODATA 2022 value is 6.67430e-8 cgs (6.67430e-11 SI, relative uncertainty 2.2e-5) [web: https://physics.nist.gov/cgi-bin/cuu/Value?bg]. So the book is 0.05-0.1 percent low, which is about 20-40 times the current uncertainty but within what was known c.1980. [read; own arithmetic]
3. The cosmic-background exponent that Step 1 flagged (10^-18 versus 10^-13). The correct value is m/k = 2.7e-13 K per (erg/g) in cgs units. I recomputed: m = 0.0408 m_e = 3.716e-29 g, k = 1.3806e-16 erg/K, m/k = 2.69e-13. The printed "10^-18" on p.177 is a misprint or OCR error; eq. 231 uses 10^-13 and the numerical result then works out (see claim 6). [read, OCR p.177; own arithmetic]
4. "1843 chosen to minimise electric potential energy" needs sharpening. The book's own lattice-sum calculation (eq. 148, with a 1972 computer check giving r/d = 0.302874) implies a lattice-particle-to-electron volume ratio of about 1844.6 (I recomputed: 9/(4 (0.302874)^2)/2, cubed = 1844.6; the book says 1844.5). That is not an odd integer. The nearest odd integers are 1843 and 1845 and 1845 is the closer one. Aspden picks 1843 by a directional argument (r/d "must be increased", so m_e/m must be "reduced") [read: p.108-111 and p.119]. Choosing 1843 rather than 1845 is the step that decides the answer 137.036 versus 137.011 (computed below). Step 1 did not mention this.
5. The 5063 graviton mass is circular. Step 1 treats 5063 as derived independently and then used to get G. p.151 says the reverse: "This mass gives the value of G exactly ... The 5063 graviton quantum is supported by the derivation of G and the psi particle connection", and it admits the 1843-quantum route to g "puts in doubt the early empirical derivation of the graviton mass" [read: OCR p.151]. So g is fixed by requiring G, then G is "predicted". See claim 4.
6. "Parallel velocities". Step 1 writes that equal masses and "parallel velocities" give an inverse-square attraction. The cancellation of the first two terms of eq. 14 needs v = V as vectors (same direction and same speed), not just parallel lines. For opposite velocities the terms add. Also q = Q and M = m are required. (Details in claim 1.) [read: eq. 14-15, p.11]
7. "Fine-structure constant differs by 0.6 ppm." Step 1 is right (I get 0.615 ppm). But the book itself, in the footnote on p.112, quotes Taylor and Cohen's 1975 value 137.03592 (0.8 ppm uncertainty) and Williams and Olsen's 137.035963(15). So in c.1980 the formula was within the quoted error of one of the two measurements. The CODATA 2022 value is alpha = 7.2973525643(11)e-3, i.e. 1/alpha = 137.035999177, relative uncertainty 1.6e-10 [web: https://physics.nist.gov/cgi-bin/cuu/Value?alph]. The formula is now about 3,800 standard uncertainties away (own arithmetic).
8. Step 1's high-confidence [memory] mainstream values (alpha, proton/electron mass ratio, CMB 2.725 K) are now [web]: m_p/m_e = 1836.152 673 426(32) [web: https://physics.nist.gov/cgi-bin/cuu/Value?mpsme]; CMB T = 2.7255 +- 0.0009 K, FIRAS spectrum deviations at most 0.03 percent of peak [web: search result summarising Fixsen et al., https://arxiv.org/pdf/astro-ph/9705101 and https://iopscience.iop.org/article/10.1086/178173].
9. Step 1's claim 3 and the facts.json statement that the law is "the unique law consistent with Ampere-type closed-circuit data" is Aspden's assertion, not something the text proves (see claim 1a).

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Claim 1: The force between two moving point charges is F = (qQ/r^3){(V.r)v - (M/m)(v.r)V - (v.V)r}

Restating the derivation in plain steps [read: pp.3-11]

Two charges q (mass m, velocity v) and Q (mass M, velocity V), constant velocities. Aspden argues: 1. An interaction force F acts along r and, by symmetry and dimensional arguments (a bilinear form in v and V, proportional to 1/r^2), must be F = -K (v.V) r, with K = qQ/r^3 (eq. 9-10). This is the "Neumann potential" form, -i i' (ds.ds')/r in current-element language. 2. F does work F.V on Q, but Q's velocity is constant, so another force Z must exist with F.V + Z.V = 0 (eq. 1). He solves this as Z = K (V.r) v (eq. 11). 3. Z has a turning moment, so a third force P is needed to cancel the torque; he sets it along V, and requires Z = (M/m)P' and Z' = (m/M)P (eq. 2-3), giving P = -K (M/m)(v.r)V (eq. 13). 4. Sum: eq. 14.

a. Is the mathematics internally consistent?

What checks out. - Algebra of step 2: F.V = -K (v.V)(r.V) and Z.V = K (V.r)(v.V), so they cancel. Correct. - Eq. 14 contains the standard magnetic force. The ordinary (Biot-Savart plus Lorentz) magnetic force of charge q on Q, at low speed, in units with c = 1, is (qQ/r^3) V x (v x r) = (qQ/r^3){ v (V.r) - r (V.v) }, using the vector identity a x (b x c) = b (a.c) - c (a.b). These are exactly the first and third terms of eq. 14. So eq. 14 = standard magnetic force + one extra term -(qQ/r^3)(M/m)(v.r)V acting on Q along its own velocity. I verified this identity myself; Aspden does not state it. [read; own derivation] - The extra term vanishes for a closed current loop. If the source charges q move around a closed loop, summing (v.r)/r^3 over the loop with ds = v dt gives the loop integral of ds.r/r^3 = loop integral of d(-1/r) = 0, because it is the integral of an exact differential around a closed path. I checked this. When the carriers are all electrons, M/m = 1, and this is the zero-for-closed-circuits term that Maxwell and Whittaker discussed. So the essay's statement that the extra term "integrates to zero for the closed circuit condition" is correct. [read: essay p.1; own derivation] - Units/dimensions: each term is qQ v V/r^2, a force in Gaussian units with c = 1, and (M/m) is dimensionless. Consistent. (Restoring c: every velocity-squared term carries 1/c^2.) - The reduction in eq. 15: with M = m, q = Q and v = V (vectors), the first two terms cancel and F = -(q^2/r^3)(v.V) r = -(q^2 v^2/r^3) r. Since r points from q to Q, the force on Q points toward q: attraction, magnitude q^2 v^2/r^2, equal to (v/c)^2 times the Coulomb repulsion q^2/r^2. Correct.

What does not check out. - The derivation is not a derivation. Step 2 only fixes the component of Z along V (Z.V = -F.V); the component perpendicular to V is free. "Z = K(V.r)v" (direction along v) is an additional unstated choice. Step 3's rule that P points along V because "reaction ... is that associated with a maximization of energy transferred" is a verbal assumption, not a theorem. The split of the pair condition into Z = (M/m)P' and Z' = (m/M)P (eq. 2-3) is one way to satisfy a single vector equation; it is not shown to be the only way. The text then says "it simply has to be, because we have relied upon the principle of energy conservation ... and little else". The claim of uniqueness is not established. [read: pp.4-11] - Energy conservation is used and then broken. The reason for introducing Z is that F.V must be balanced. But then P is added, and P.V = -K (M/m)(v.r) V^2 is not zero in general. So the final law does not conserve energy for the two charges. The same for momentum: the force on the pair, Z + P + Z' + P', equals K[(V.r)v (1 + m/M) - (v.r)V (1 + M/m)], which is not zero. Aspden accepts this and assigns the balance to the aether (the essay says so explicitly). So "conservation of energy" is the stated starting point and a violation of it (compensated by an unobserved reservoir) is the stated outcome. That is a logic gap, not an arithmetic error. - The a = b step (pp.9-10). Aspden shows the coefficients of two interaction types are equal (a = b) by arguing that a force F_x does no work on Q and "therefore" the reaction on the q charges must do none, so F_x = 0. This assumes the pair forces are equal and opposite (Newton's third law), which is the property the final law abandons. So the early steps assume balanced action-and-reaction while the later steps drop it. [read: pp.9-10, OCR, with eqs 4-5 not re-derived by me] - Frame dependence. The velocities v, V are measured relative to a preferred frame (the aether). In the rest frame of the two charges v = V = 0 and the law gives zero force between them. Gravity as claimed in claims 4 is therefore not invariant under a change of inertial frame. This is intended by Aspden (he rejects relativity's postulates) but it is a consequence he does not confront mathematically; the test is in 1b.

b. Consistent with established experiments?

• The closed-circuit part (Ampere/Biot-Savart, Lorentz force) is consistent, by construction. The mass-ratio term is invisible in closed circuits, because it integrates to zero. [own derivation]
• The extra term acts on a charge along its own direction of motion with strength proportional to M/m. Where it would be visible: in non-closed or time-varying charge flows (my identity in the "checking the plasma energy" section under claim 2 shows it is proportional to the time derivative of the source's electrostatic potential). Direct tests with isolated ions and electrons (e.g. in accelerators and storage rings, which involve high-precision measurement of ion and electron motion in electromagnetic fields) show no anomalous along-track force; I did not find a source for a specific experiment that was designed to test this term, so I cannot cite one. [memory: low confidence; not verified]
• The mainstream view is that the Lorentz force plus Maxwell's equations describes the force on charges; the extra term would add an unobserved longitudinal force. Aspden's essay itself says no orthodox theory sees "electrodynamic forces existing along the line of the ion current flow path" [read: essay p.2].
• The Graneau wire-rupture experiments are the main claimed support. Web search shows (i) the experiments exist (Graneau and Graneau repeated Nasilowski's exploding-wire experiment at MIT, 5-7 kA, 5-10 ms); (ii) Graneau interprets them as Ampere longitudinal tension, (iii) critics argue the rupture follows from an imbalance of self-induced emf and ohmic potential and from ordinary Lorentz and thermal forces [web: https://www.researchgate.net/publication/292387334 and https://www.sciencedirect.com/science/article/abs/pii/0375960185905912, search summaries only; I did not read these papers]. Note also that Graneau's interpretation is of Ampere's 1820s force law, which has a different structure from Aspden's eq. 14.

c. Tested?

• No experiment is cited that was designed to test eq. 14 or its mass-ratio term as such. The essay offers four experimental items (Kobel 1930, Graneau 1983/85, Roth 1978, Sethian 1978) as "support", but none of the original authors tested Aspden's law; Aspden re-reads their results. Sethian et al. propose, "without particular justification", that the anomaly might be the ion/electron mass ratio [read: essay p.2], which is the nearest thing to independent mention. I did not find a replication or a neutral test. Status: never properly tested (and not "dismissed without testing": it was ignored in the literature according to Aspden, which I cannot confirm from sources).
• The Correa patents are cited as the sole uptake [read: essay p.2]. A patent grant is not an experimental test. I did not obtain the patents or any independent replication of the Correa device. [not checked]

d. If true, what would have to change?

Lorentz force and Maxwell's equations would be incomplete for isolated charges (a longitudinal term proportional to mass ratio); Newton's third law and momentum conservation would fail for the charge pair; a preferred electromagnetic rest frame would exist (violating the Lorentz invariance that is built into quantum electrodynamics, which is tested to extreme precision); the equivalence of inertial frames would fail; and mass would enter electrodynamics directly, which breaks the usual separation of charge and mass.

Verdict, claim 1

• Mathematical derivation: Internally inconsistent in its logic (energy conservation invoked and then violated; Newton's third law assumed in step a = b and then dropped; uniqueness asserted but not shown). The final formula is dimensionally and algebraically fine, and it equals the standard magnetic force plus one extra term.
• The law as a physical claim: Never properly tested (no neutral, controlled test of the mass-ratio term on record in what I could obtain). It is not "dismissed without testing" in a way I can document.

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Claim 2: Heavy ions in a plasma gain far more energy than electrons (explaining ion-hot plasmas, the million-degree corona, and anomalous discharge gains); the sun is not fusion powered

Checking the plasma energy argument myself

Aspden's essay lists four cases (slow/fast ion, electron approaching/receding), and concludes "the net effect ... must therefore be to cause each positive heavy ion on average to have kinetic energy far in excess" [read: essay p.1]. He leaves the statistics to "some enthusiastic mathematician". I did the following.

An exact identity. The extra force on ion Q from an electron q is -K (M/m)(v.r)V with K = qQ/r^3 (see claim 1). Note that (v.r)/r^3 q is exactly the time derivative of the Coulomb potential q/r at the ion due only to the electron's motion: d/dt (q/r) at fixed ion position = q (v.r)/r^3 (since r points from electron to ion, d(1/r)/dt = v.r/r^3 for a fixed ion). So summing over all electrons, the extra force on the ion is F_extra = -(M/m) Q V (d phi_e / dt), where phi_e is the electrostatic potential at the ion produced by the moving electrons (my derivation, using d/dt(1/r) = (v.r)/r^3). The power delivered to the ion is F_extra.V = -(M/m) Q V^2 (d phi_e/dt).

Consequence 1: a full encounter delivers zero net energy. For a single straight-line fly-by of an electron past a (nearly) stationary ion, the integral of (v.r)/r^3 dt is the total change of 1/r, which is zero when the electron starts and ends far away. I confirmed numerically: for b = 1e-9 cm-scale impact parameter and v = 1e6, the signed integral is -9e-6 versus an absolute-value integral of 2e9, i.e. cancellation to about 1 part in 1e15 (limited by the finite integration window). Aspden's cases (1) and (2), approach and recession, cancel exactly; cases (3) and (4) differ only in the V-dependence (V^2 factor), so they also cancel unless V changes during the encounter. So a net gain requires a correlation between the ion's speed V and the phase (approaching or receding) of encounters, and Aspden shows no such correlation. The effect, if any, is a second-order effect in V-change and is never estimated. Consequence 2: in a steady plasma the time-average of d phi_e/dt is zero. phi_e is bounded and statistically stationary, so its time derivative has zero mean; only the cross-correlation with V^2 survives. Again, no estimate is given.

Size of the term relative to ordinary Coulomb collisions. The ratio of the extra force to the Coulomb force is of order (M/m) V v/c^2 (V, v thermal speeds of ion and electron; c = speed of light). For protons at 1e6 K I get 2e-2; for iron, which has M/m = 56 x 1836 = 1.0e5, about 0.16 (own arithmetic). So the term is not negligible by order of magnitude, and its sign-averaged effect is the whole question. The essay's inference of a large systematic gain is therefore not a computation but an assertion. The consequence-1 cancellation is a real mathematical objection, not a proof the effect vanishes (second-order terms could remain).

a. Mathematics

The essay contains no equation of its own and no calculation; the energy-partition argument is a verbal plausibility argument that the analysis above shows to be incomplete. The "factor of the order of 160" is 1,000,000/6,000 = 167, an arithmetic ratio of two temperatures, not a prediction from the law. The comparison itself is inconsistent: the essay compares corona ion energy (about 1e6 K) with electron "temperature of 6000 K", but the 1e6 K corona is the electron temperature too (measured electron temperature of the corona is about 1 million K [web: summary of review literature in https://arxiv.org/pdf/1206.6097 and https://pmc.ncbi.nlm.nih.gov/articles/PMC4410557/ via search; I did not read the full texts]). The 6,000 K figure is the photosphere (a surface), not the electrons in the corona. So the "ion/electron energy ratio of 160" in the corona is a mismatch of two different places, not a ratio between ions and electrons.

b. Consistency with established experiments

• Real, documented: some coronal ions (e.g. oxygen in coronal holes, per SOHO/UVCS, via the search summary) are far hotter than electrons, and ion temperatures can exceed electron temperatures in bumpy tori and beam-heated plasmas. These are established observations; the question is the mechanism. Mainstream mechanisms: wave/ion-cyclotron heating, reconnection, collisionless dissipation, and the mass-proportional heating that follows from plasma wave physics (a mass-ratio dependence is expected in several of them; Sethian et al. themselves suggested the mass ratio [read: essay]). [web search summary: https://arxiv.org/pdf/1206.6097]
• Fusion powering the sun is supported by solar neutrino detection (flux and spectrum) [memory: high confidence; not re-checked on the web here, and no verdict hinges on a number from it]. The essay's argument against it ("the evidence ... neutrinos is not a measure of temperature") ignores that neutrino flux measures the nuclear reaction rate directly. The claim "the sun is not powered by fusion" is therefore in conflict with data the essay does not discuss.

c. Tested?

• The extra-term hypothesis: not tested by Aspden or anyone I found (no calculation, no simulation). The cited experiments (Kobel, Roth, Sethian) are real reports of anomalous ion energy but I did not read them. Whether other mechanisms explain them is a separate literature question I could not settle here [not checked]. The Sethian et al. quote ("1,000 times faster than can be predicted by classical processes") is the essay's own quotation [read].
• No replication of the Aspden mechanism.

d. If true

Plasma physics would need a new velocity-dependent, mass-ratio-weighted pair force; fusion experiments would have a built-in ion-heating channel; solar structure and coronal heating theory would change; and the Sun's energy source would need to be the aether.

Verdict, claim 2

• Ion-hot plasmas exist (the observation): Established (as an observation, with mainstream explanations).
• That Aspden's term explains them: Never properly tested; and the proposed argument is incomplete (my exact result shows that a complete encounter gives zero net energy at first order). Not "Refuted by calculation", because second-order and correlated effects were not ruled out by me.
• "The sun is not fusion powered": Tested, not supported in the sense that the neutrino measurements (which I cite from memory, high confidence) are direct evidence for fusion; the essay offers no counter-calculation. [memory: high confidence; not web-verified]
• The "ratio of 160": Internally inconsistent (compares ion energy at 1e6 K with an electron temperature taken from the photosphere).

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Claim 3: A structured vacuum lattice (aether) sets a preferred frame, supplies the balancing momentum and energy, is compatible with the classic aether experiments, and underlies the Correa "power from space" device

a. Mathematics

• The book's lattice chapters (pp.108-118) are a sequence of definitions: lattice particles of charge e in a uniform opposite-charge background, with constants (r/d, m/m_e, the 3 x 3 x 3 "photon unit") fixed by energy-minimisation conditions. I checked the arithmetic that connects them: r/d = (3/4)(8/1843)^(1/6) = 0.30292; m/m_e = 4 (r/d)^2 / 9 = 0.04078 (book: 0.0408); the claimed hc/(2 pi e^2) = 144 pi (r/d). These are algebraically consistent with one another. [read: eqs 147-157, p.108-112; own arithmetic]
• No transformation analysis (Lorentz covariance or Galilean covariance, with the preferred frame) is done by me; I read that the book asserts "special-relativity formulae" are reproduced. I did not verify that derivation [not checked: chapters on vacuum structure and the Michelson-Morley, Sagnac, Kennedy-Thorndike, Ives-Stilwell discussions].
• The aether's balancing role is an unquantified bookkeeping device: the energy and momentum "supplied" by the aether are whatever is needed, with no equation of state or conservation law for the aether itself. A reservoir with unlimited, unspecified capacity cannot be falsified by an energy-balance argument.

b. Consistency with experiment

• Michelson-Morley and the other cited experiments have been interpreted since 1905 within special relativity; Aspden says an aether can be made compatible with them. A preferred frame compatible with all null results exists in principle (Lorentz aether theory is observationally equivalent to special relativity) [memory: high confidence]. The preferred-frame claim therefore has no measurable difference unless the aether's special properties are tested. Aspden's specific laws (claims 1 and 4) do imply frame-dependent forces, which would be testable.
• The only decisive test in the folder is the Correa device, which I have no data for. [not checked]

c. Tested?

Not by any source I had. The Correa patents and Aspden's 1996 report "Power from Space: The Correa Invention" were not obtained, and I make no statement about whether independent tests of the device were run. I found no source and will not claim one. [not checked]

d. If true

Special relativity's postulates would become derived consequences of a medium; energy conservation would hold only if the aether is included; excess-energy devices would be possible in principle (against the second law only if closed cycles were claimed, which I did not examine).

Verdict, claim 3

• Preferred-frame aether as an interpretation compatible with the null aether experiments: Not testable as stated (observationally equivalent to relativity unless extra properties are specified).
• Aether as a usable energy source / Correa device: Never properly tested (no neutral, controlled test in the materials; patents and report not obtained). I cannot say from these sources whether it was dismissed without testing.

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Claim 4: Gravitation is an electrodynamic effect mediated by 2.587 GeV gravitons (5063 m_e), from which G is calculated

Checking the numbers [read: page image p.114-115, p.151; own arithmetic]

• Eq. 163: G = (e/m_e)^2 (4 pi)^2 (1/108 pi)^6 (1/g)^8. Dimensional check: in cgs, (e/m_e)^2 has units esu^2/g^2 = (g cm^3 s^-2)/g^2 = cm^3 g^-1 s^-2, which is the unit of G. The remaining factors are dimensionless, so eq. 163 is dimensionally consistent. This is a reassuring check that is easy to pass: the ratio e^2/m_e^2 is about 2.8e35 cgs, 43 orders of magnitude above G, and any gravitational "theory" built this way has to bridge a 1e-43 gap by dimensionless factors.
• The bridge here is (1/g)^8 times (1/108 pi)^6 times (4 pi)^2: with g = 5063, g^8 = 4.3e29 and (108 pi)^6 = 1.5e15 (own arithmetic). The eighth power makes G extremely sensitive to g: a 0.01 percent change in g changes G by 0.08 percent. So "agreement" of 0.1 percent with measured G corresponds to knowing g to about 0.013 percent.
• Where does g come from? Two routes: (i) a 1966 argument via the lattice (5062.75) and a table of hypothetical decays through the 1843-quantum plus muon pairs and hadrons (pp.112-114); (ii) p.151: "This mass gives the value of G exactly ... supported by the derivation of G and the psi particle connection", with a statement that the 1843-based route is now in doubt. So g is effectively fixed by requiring G on the second route, which makes the "prediction" of G circular. On the first route, 5062.75 is quoted but I did not re-derive it; the table [read: p.113] is a sum-of-quanta counting argument with a free "exclusion rule" ("number of muon pairs could not exceed the 1843-quantum pairs by more than one") that was introduced to select the entries.
• The result: G(g = 5063) x 1.0004 = 6.668e-8; CODATA 6.6743e-8; 0.1 percent low (about 40 standard uncertainties of today's value) [web: https://physics.nist.gov/cgi-bin/cuu/Value?bg].

a. Mathematics

Eq. 163 is consistent as written (dimensions OK, the exponents follow from eq. 162 with x/d = (1/g)(1/108 pi), x/a = 1/g [read: p.115]). The inputs (the factor 108 pi, the exponent 8, the integer 5063) are chosen by the construction described above; the derivation of eq. 54, the starting point (G in terms of lattice quantities), is in the first chapters and I did not check it. The electrodynamic law in claim 1 gives gravity only as (v/c)^2 times the Coulomb repulsion between charges that move at equal velocity v; for individual charges at ordinary speeds that is absurdly small: to match gravity between two electrons (e^2/(G m_e^2) = 4.2e42) you would need (v/c)^2 = 2.4e-43. So the actual gravitational constant must come entirely from the vacuum-lattice ansatz (eqs 161-163), not from the electrodynamic law alone. The book's chain "electrodynamics gives gravity" is therefore carried by the lattice assumptions. [own arithmetic]

b. Consistency with experiment

• Gravity is observed to be independent of the velocity of the attracting bodies relative to any preferred frame at the level of the Eotvos and lunar laser-ranging tests, and the strict equivalence of inertial and gravitational mass has been tested to about 1e-13 [memory: medium confidence; not web-verified; I do not rely on a number]. A gravity built from (v/c)^2 terms in a fixed frame conflicts with this in kind.
• No 2.587 GeV "graviton" has been found; the mainstream expectation for a quantum of gravity is a massless spin-2 particle [memory: high confidence]. The book itself reports that a 2.60 GeV resonance seen in 1977 was not confirmed by independent searches [read, per Step 1, p.? summary].

c. Tested?

The graviton mass prediction was compared to a 1976-77 Fermilab announcement that the book itself says failed independent confirmation. Beyond that, nothing. [read: Step 1 summary, not re-checked by me page by page]

d. If true

General relativity would be an approximation to a lattice electrodynamics with a preferred frame; the gravitational constant would be a derived number; a heavy graviton would exist in the GeV range (this would change particle physics substantially and would be seen in accelerator data at GeV energies).

Verdict, claim 4

• Dimensional consistency of eq. 163: holds. Predictive content: Internally inconsistent in its logic (g is tuned to G; the first "derivation" of g is a counting table with an ad hoc rule; the electrodynamic law cannot supply the gravitational strength by itself, per the (v/c)^2 estimate above).
• A 2.587 GeV graviton: Tested, not supported per the book's own admission about the 1977 resonance (not confirmed in independent searches), and general particle searches since; I rely on the book's statement [read] and have no independent source. I will say plainly: this conclusion rests on [memory] and the book alone, not on a source I checked.
• G from first principles: Refuted by calculation? No. It is not refuted; it is not predictive.

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Claim 5: The fine-structure constant (137.035915), proton/electron mass ratio (1836.152), electron g-factor, Planck's law and the Schroedinger equation follow from vacuum-lattice geometry, with no free parameters other than integers such as 1843

Checking the numbers [read: pp.108-119, 123; own arithmetic]

• alpha^-1. 108 pi (8/1843)^(1/6) = 137.035915 (I recompute 137.03591485). Confirmed. For nearby odd integers: N = 1841 gives 137.0607; 1843 gives 137.0359; 1845 gives 137.0111; 1847 gives 136.9864. The spacing of the candidates is about 0.025 in alpha^-1 (180 parts per million), so the odd-integer rule gives a 1-in-6000 lattice of possible answers, and the observed value falls within 0.6 ppm of one of them. If one picks the odd integer to fit, the chance that the closest of the candidates lands within about 0.6 ppm (of either sign) of the target is about 0.7 percent. That is low but not astronomically so; and it ignores that the form of the formula (108 pi, the exponent 1/6, the number 8) was itself chosen. Look-elsewhere effect: not quantified.
• The decisive step is the integer. From Aspden's own lattice sum the volume ratio is 1844.6 (not odd). Choosing 1843 instead of the nearest odd integer 1845 (which gives alpha^-1 = 137.011) is where the agreement with experiment is produced. The justification is a direction argument ("r/d must be increased") and requires the sign of an electric energy; I could not verify that argument from the text. Also the rigorous lattice-sum value r/d = 0.302874 differs from the 1843-formula value 0.302916 by 1.4e-4 (relative), which is 230 times the 0.6 ppm "agreement", meaning the book's own physics only fixes r/d to about 1e-4, and the 0.6 ppm match comes from the integer step. [own arithmetic]
• Current data. alpha^-1 = 137.035999177 (CODATA 2022), uncertainty 2.2e-8 (1.6e-10 relative); the formula's 137.035915 is 8.4e-5 low, i.e. 0.615 ppm, about 3,800 standard uncertainties [web: https://physics.nist.gov/cgi-bin/cuu/Value?alph; own arithmetic]. In c.1980 the footnote values were 137.03592 (0.8 ppm) and 137.035963(15) [read: p.112 footnote]; so the match was then good to within the larger uncertainty, and has since been excluded by the measurement.
• m_p/m_e. The book derives 1836.15232 (eq. 168) against 1836.15152(70) then [read: p.116]. CODATA 2022: 1836.152 673 426(32) [web: https://physics.nist.gov/cgi-bin/cuu/Value?mpsme]. The derived value is off by 3.5e-4 absolute (1.9e-7 relative) which is about 1e4 times today's relative uncertainty (1.7e-11), and also 0.0008 away from the then-measured value, i.e. 1.1 of the quoted "(70)" uncertainty units: 0.00080/0.00070. Acceptable then, excluded now. [own arithmetic]
• Pion and muon. The book gets pion 273.1262 (versus 273.1266 +- 0.0039 [read: p.136]) and muon 206.7688 (versus 206.76859(29)). The pion calculation uses the parameter n = 0.1488809 obtained by solving the equation (178 = 179), where g = 5063 enters, and the muon calculation then adds alpha and subtracts the electron mass "as required by the D reaction" (eq. 186-187). These are multi-step constructions with an energy-bookkeeping model of unobserved intermediate states; I did not re-derive them. The muon figure matches 1 part in 1e5 (206.7688 versus 206.7686), with several adjustable choices upstream (k = 3/2, the Q = E_0 assignment, and the added alpha). The number of such choices is not small, so the claim "no free parameters other than integers" is not accurate: k = 3/2, the exclusion rule, g, the "-1 electron", and "+alpha" are all selected. [read: pp.115-117, 134-136; partially checked]
• g-factor, Planck's law, Schroedinger equation. Not checked by me (Chapter on wave mechanics); Step 1 summarises them as derived from lattice rotation. I cannot give a mathematical verdict on them.

a. Mathematics

Arithmetic is correct where I checked it. The logic: the derivations take a model with several choices, fix the free quantities by energy-minimisation conditions, and require an integer that is picked by a directional argument. The fine-structure constant is the best-presented case; its sensitivity analysis above shows it is a one-integer fit with a 0.7 percent accident probability, not a parameter-free prediction. "Derives Planck's law" is a stronger claim than the text supports (the lattice rotation gives a relation with ħ, but the constant h is then expressed through e and c and the geometric number 144 pi (r/d), which is simply eq. 152 restated).

b. Consistent with experiment?

Not at today's precision (alpha off by 0.6 ppm, 3,800 sigma; m_p/m_e off by 1.9e-7). The book predates these measurements; the agreement of c.1980 does not carry over. Note that the vacuum-polarisation and QED calculations of the electron g-factor agree with experiment to better than 1 part in 1e10 [memory: high confidence; no web check; no verdict hinges on the exact number], so a lattice g-factor would have to equal this.

c. Tested?

Compared with measured values only in the sense of the book's own comparison, not by independent predictions made before the data. alpha and m_p/m_e were already known when the formulas were assembled; the muon and pion values too. Aspden also published a Fermilab-resonance connection which failed [read: book]. No replication by others of the derivations that I know of [memory: low confidence].

d. If true

The constants would be geometric numbers of a vacuum lattice, quantum mechanics would be derived from classical lattice mechanics, and QED's renormalised calculations would be an effective description.

Verdict, claim 5

• Numerical agreement c.1980: Historically superseded but instructive (a good fit to the then-quoted values, now excluded by 3,800 sigma for alpha).
• "Derivation of alpha": Not a derivation as stated (one fitted odd integer, chosen by a non-rigorous argument): Internally inconsistent in the claim of no free parameters; for today's value, Tested, not supported.
• g-factor, Planck, Schroedinger derivations: not reviewed by me (source gap; I decline to give a verdict).

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Claim 6: The 2.7 K background is a local lattice temperature set by gravitational potential (2.6 K near Earth), not a Big Bang relic; geomagnetism, solar-system formation and field reversals arise from rotating vacuum domains

Checking eq. 230-231 [read: OCR pp.176-177; own arithmetic]

• Eq. 230: T = T_0 + H^2/(8 pi k N) - phi m/k, with k Boltzmann's constant, N = 1/d^3 = 3.87e30 per cc (d = 6.37e-11 cm), H the magnetic field, phi the local gravitational potential (negative), m the lattice-particle mass = 0.0408 m_e = 3.716e-29 g.
• m/k = 3.716e-29 / 1.38065e-16 = 2.69e-13 [own arithmetic; the printed text has an exponent misprint, see Correction 3].
• phi near Earth, taken by Aspden as G(M_sun/AU + M_earth/R_earth) = 6.67e-8 (1.989e33/1.496e13 + 5.977e27/6.378e8) = 9.49e12 cgs. I reproduce 9.49e12.
• T = 0 + 9.49e12 x 2.69e-13 = 2.56 K. The book's 2.6 K is reproduced. The arithmetic is right.
• Hidden assumption: the zero of potential. In Newtonian gravity potential is defined only up to an additive constant (conventionally zero at infinity). The formula assigns a physical temperature to the potential's value, so the answer depends on which masses are included and where "zero" is. Aspden includes only the Sun and Earth. The potential of the Milky Way at the Sun's position is roughly (v_escape)^2/2 with v_escape about 550 km/s: 1.5e15 cgs, 160 times larger than the Sun-Earth sum (own arithmetic; the 550 km/s figure is [memory: medium confidence; not web-checked]). With that term the formula would give about 400 K. That is, the "2.6 K" agreement results from omitting the largest terms in the potential. Including the Local Group or the Universe, the potential grows further.
• Hidden assumption: "T_0 = 0 near Earth". The text says "if we look at the vacuous medium near the Earth we must expect T_0 to be zero". This is an assumption, not derived.
• Also, the thermal motion model uses two degrees of freedom per lattice particle ("energy related to kT, not 3kT/2") chosen earlier; the number m/k relies on m, which was fixed by the alpha fit (the integer 1843). So a failure in claim 5 propagates here.

b. Consistency with experiment

• The CMB is a near-perfect blackbody at T = 2.7255 +- 0.0009 K, with deviations from a blackbody below about 0.03 percent of the peak [web: https://arxiv.org/pdf/astro-ph/9705101; https://iopscience.iop.org/article/10.1086/178173]. A thermal spectrum with a single temperature that has the same value in every direction needs a mechanism; the book offers a temperature that depends on the local gravitational potential, which differs by large factors from place to place (and from satellite orbit to orbit), but the same 2.7255 K is measured by FIRAS at the Sun-Earth L2 point (COBE was in low Earth orbit) and by Planck at L2 [memory: medium confidence on L2]. I have not verified these specifics; the main point is that the observed spectrum is isotropic to about 1 part in 1e5 after the dipole is removed [memory: high confidence].
• The mainstream relic-radiation interpretation also predicts T(z) = T_0 (1 + z) at high redshift, which is measured through molecular excitation and the Sunyaev-Zeldovich effect [memory: medium confidence; not web-verified here]. Aspden's local-potential formula has no such scaling. I did not obtain a source for these measurements and give no number.

c. Tested?

The CMB temperature has been measured extensively; Aspden's formula is "tested" only by comparison with that single number. Its prediction for other places (other potentials) is not discussed and appears not to have been compared with anything. Geomagnetism, field reversals, solar-system formation: not examined by me.

d. If true

The Big Bang interpretation of the CMB would be replaced by a local vacuum thermal effect; the cosmological redshift-temperature relation would fail; the vacuum would have a temperature set by gravitational potential.

Verdict, claim 6

• 2.7 K as local lattice temperature from gravitational potential: Refuted by calculation (in its stated form): with the same eq. 230 and a complete potential (adding the Galaxy's, 1.5e15 cgs) the formula gives about 400 K, not 2.7 K, and a potential defined only up to a constant cannot fix a temperature. The 2.6 K comes from omitting the dominant term. (The calculation is shown above; the galactic potential is from [memory: medium confidence], so this verdict is firm only if the 1.5e15 figure is accepted; the argument that the value depends on an arbitrary zero stands without it.)
• Geomagnetism, field reversals, solar-system formation from rotating vacuum domains: Not reviewed by me (source gap); I decline to give a verdict.

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Summary table

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Web verification log (budget about 6)

1. alpha: https://physics.nist.gov/cgi-bin/cuu/Value?alph (CODATA 2022 alpha = 7.2973525643(11)e-3). Success.
2. m_p/m_e: https://physics.nist.gov/cgi-bin/cuu/Value?mpsme. Success.
3. G: https://physics.nist.gov/cgi-bin/cuu/Value?bg. Success.
4. Graneau exploding-wire controversy: search results only (summaries; no paper read). Partial.
5. COBE/FIRAS CMB temperature and blackbody deviation: search results (summaries). Partial.
6. Solar corona electron temperature and ion/electron differences: search results (summaries). Partial. Not web-checked (kept as [memory], tagged where they matter): neutrino evidence for solar fusion (high confidence), galactic escape speed 550 km/s (medium), equivalence-principle precision (medium), T(z) scaling of the CMB (medium), QED g-factor precision (high). None of the verdicts that depend on these says more than "rests on [memory]" where noted.

Scratch computations (SymPy/NumPy) were run in the project venv; no files were written to the source folder.