REVIEW 3 major objections 3 minor 6 references
Reply to "Comments on work of Yang and Nevels 'Direct, analytic solution for the electromagnetic vector potential in any gauge' (arXiv:2507.02104)" by Onoochin (arXiv:2507.08042)
T0 review · 3 major / 3 minor · reviewed 2026-08-05 · deepseek-v4-flash
Pith's one-line read This paper proves, by four independent methods, that the disputed vector-potential formula is an exact solution of the wave equation for arbitrary localized sources.
desk verdict A mathematically sound defense of eq. (10) for finitely switched-on sources, but it sidesteps the infinite-motion case that triggered the dispute. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is the time-integrated difference of scalar potentials, A^(v)_2 = c∇∫(Φ_c − Φ^v)dt. This single expression is what the four methods are all checking. It encodes the two speeds appearing in the velocity gauge: Φ^v propagates at speed v and Φ_c at speed c, and their difference, integrated and then gradient-differentiated, supplies the part of A that conversion between gauges requires. The paper's key maneuvers are the Green's function identities in Method 3 (composition of the v-speed and c-speed propagators, plus derivative identities eqs (21)-(22)) and the gauge function χ = c∫(Φ_c − Φ^v)dt in Method 4. These maneuvers let the authors move from a solution expressed as
What would settle it
Numerically evaluate both sides of the disputed wave equation (∇² − c^{-2}∂²_t)A^(v)_2 = c(−1/v²+1/c²)∂_t∇Φ^(v) for a finite, localized source with a definite turn-on, keeping all boundary terms when the time integral is differentiated; a nonzero residual would identify the omitted term. A sharper version: compute the Coulomb-gauge potential from eq (25) and from the commenter's preferred retarded-integral formula for the moving point charge, and test whether their difference is exactly a pure gradient.
Extended reading notes
Core claim
The central claim is that A^(v)_2(r,t) = c∇∫[Φ_c(r,t) − Φ^(v)(r,t)]dt is an exact solution of (∇² − c^{-2}∂²_t)A^(v)_2 = c(−1/v²+1/c²)∂_t∇Φ^(v) for arbitrary localized time-dependent sources, with no need for regularizations or additional homogeneous solutions. The paper proves this four times. Method 1 applies the wave operator directly to the formula and uses the wave equations for Φ^v and Φ_c. Method 2 differentiates eq (8), rearranges, and solves with a c-retarded Green's function. Method 3 builds a composite two-speed Green's function and reduces it to the same expression. Method 4 obtains the formula as a gauge transformation of the Lorenz-gauge potentials. Since all four routes coinci
Load-bearing premise
The proofs rely on treating the time integral as an antiderivative with zero integration constant and on dropping boundary terms when the wave operator is moved inside the integral; if those boundary terms are nonzero, eq (10) picks up an extra homogeneous solution.
Editorial extensions
If this is right
- For any localized sources, the velocity-gauge vector potential can be written as A_c + c∇∫(Φ_c − Φ^v)dt, reducing the velocity-gauge problem to already-known c-retarded potentials.
- Setting v→∞ gives the Coulomb-gauge formula, so the Coulomb-gauge vector potential is fixed once Φ^C and Φ_c are known.
- Any potentials satisfying eq (30) produce the same E and B, so all potentials presented in the original paper are gauge-invariant.
- The commenter's Coulomb-gauge result, if computed correctly within the same integration conventions, must agree with this formula up to a gradient.
- Because the four methods mutually support eq (10), an objection to it must identify a specific illegal step rather than a disagreement between methods.
Reading between the lines
- The same four-proof strategy should apply to any pair of gauges linked by a gauge function of the form c∫(Φ_a − Φ_b)dt; eq (30) may be read as a completeness relation for inhomogeneous potential solutions.
- Because the proof relies on an antiderivative with zero integration constant and vanishing boundary terms, exploring whether homogeneous terms selected by a finite source turn-on account for the commenter's differing result would settle the dispute.
- The composite two-speed Green's function in Method 3 suggests an explicit closed form for a 'two-speed propagator'; testing it against direct numerical convolution for a finite source would give an independent check.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. This manuscript is a reply to Onoochin's comment, which disputed the Coulomb-gauge vector potential derived by Yang and Nevels. The reply asserts that the velocity-gauge vector-potential component A^(v)_2(r,t) = c∇∫[Φ_c(r,t) − Φ^(v)(r,t)]dt, Eq. (10), is a mathematically correct solution of Eq. (8) for localized charge-current distributions turned on at t0. Four methods are presented: direct substitution into the d'Alembert equation (Method 1), solution via a retarded Green's function (Method 2), a Green's-function composition argument (Method 3), and a gauge transformation from the Lorenz gauge (Method 4). The reply concludes that all well-defined potentials satisfying Eq. (30) are gauge invariant and that the dispute with Onoochin is settled.
Significance. If the finite-turn-on restriction is accepted, Methods 1, 2, and 4 provide a largely self-contained verification of Eq. (10), and the explicit construction of velocity-gauge potentials from the Lorenz gauge via Eq. (27)–(29) is a useful and pedagogically clear result. The manuscript's strength is that the main formula is checked by direct substitution, not merely asserted. However, the paper's broader conclusion—that the specific discrepancy with Onoochin's infinite-motion calculation is resolved—is not established, because Eq. (10) is an indefinite time integral whose value depends on boundary conditions, and the disputed example has no turn-on time. In addition, Method 3 relies on an unpublished preprint for its decisive identity, so the 'four methods' claim is overstated as it stands.
major comments (3)
- [Eq. (10) and Methods 1–2; final paragraph] The central formula Eq. (10) is an indefinite time integral, made unambiguous only by the opening assumption that sources are 'turned on at t0'. For the specific case that triggered Onoochin's comment—a point charge moving at constant speed for all time—there is no finite t0 at which the charge density vanishes. As t→−∞, Φ_c(r,t) and Φ^(v)(r,t) both decay as ~1/|t| but with different speed-dependent coefficients, so their difference is ~const/|t| and the integral diverges logarithmically. Eq. (10) is therefore not defined in the disputed case unless a different lower-limit regularization is imposed. Methods 1 and 2 implicitly select the finite-turn-on/no-incoming regularization: Eq. (14) drops the integration constant when replacing ∫∂²Φ/∂t² dt by ∂Φ/∂t, and Eq. (17) selects the retarded solution. A different regularization changes A^(v)_2 by a time-independent gradient, i.e. by a pure g
- [Method 3, Eqs. (18)–(24)] The decisive simplification in Method 3, Eq. (23), is not proved in this manuscript. The text introduces Eqs. (21)–(22) with 'they can be derived', and the step 'Then from eq. (18), we do the following (see Ref. [4])' invokes the authors' own unpublished arXiv preprint for the central identity. The composition of the c-speed and v-speed retarded Green's functions in Eq. (19), and its reduction to the difference G(r,t|c|r',t') − G(r,t|v|r',t'), is nontrivial. Since the paper promises four analytic proofs, Method 3 as written is not a self-contained proof unless the identity is supplied or a published source is cited. This is a load-bearing gap in the 'four methods' claim.
- [Final paragraph, Eq. (30)] Eq. (30) is stated without proof in this reply, yet it is used to assert that any potentials differing 'in substance' from the authors' potentials would risk violating gauge invariance. This is a stronger claim than the correctness of Eq. (10) and goes beyond what Methods 1–4 establish. If Eq. (30) is intended as a general theorem, it requires a proof or a precise statement of its domain, including boundary conditions and integration constants; otherwise it should be removed or explicitly labelled as a conjecture.
minor comments (3)
- [Eq. (22)] As printed, ∂/∂t' G(r,t|c|v|r',t') = −∂/∂t G(r,t|c|v|r',t); the right-hand side appears to be missing the prime on the time argument and should read −∂/∂t G(r,t|c|v|r',t') (or equivalent). Please check.
- [Ref. [4]] The citation to an unpublished arXiv preprint for a central identity in Method 3 is not ideal. If a published version exists, it should be cited; otherwise the proof should be included in the reply.
- [Eq. (14)] The exchange of the d'Alembertian with the integral and gradient, and the suppression of the integration constant, should be stated explicitly. As written, it is justified only by the t0 turn-on condition, which is exactly the point at issue in the dispute.
Circularity Check
Derivation is self-contained: eq. (10) is verified by direct substitution and independently derived by retarded Green's function and gauge transformation; no circular reduction.
full rationale
The reply's central claim is that A2^(v)=c∇∫(Φc−Φ^(v))dt solves eq. (8). Method 1 is a direct substitution: applying the d'Alembertian to the candidate and using eq. (5) reduces the expression to c(−1/v^2+1/c^2)∂_t∇Φ^(v), which is the RHS of eq. (8). This is a verification, not a result imported from the answer. Method 2 solves eq. (8) independently by time-differentiating, obtaining ∂_t A2^(v)+c∇Φ^(v)=c∇Φc by the retarded Green's function, and integrating to (10). Method 4 constructs (10) from the Lorenz-gauge potentials via the explicit gauge function χ=c∫(Φc−Φ^(v))dt, so (10) is produced rather than assumed. Method 3 cites the authors' earlier preprint for a simplification, but the key identity is displayed in eq. (23); hence the citation is not load-bearing for the central result. The stated turn-on at t0 makes the time integrals well-defined; the disputed infinite-motion case raises a boundary-condition/regularization issue, not an equivalence of input and output. Footnote [6] explicitly restricts eq. (30) to 'exclusively inhomogeneous solutions'; this is a caveat limiting the universality claim, but it does not make the derivation circular. No step reduces the conclusion to its own premise, so there is no significant circularity.
Assumptions & free parameters
free parameters (1)
- v (velocity-gauge parameter) =
arbitrary real value chosen by user (gauge parameter)
assumptions (4)
- standard math The wave operator (∇^2 - c^{-2}∂_t^2) commutes with ∇ and with the time integral, and the indefinite integral is defined with zero lower-limit constant.
- domain assumption Sources ρ and J are localized and turned on at finite t0, so retarded potentials vanish for sufficiently early times and the antiderivative ∫(Φ_c - Φ^(v))dt is well-defined.
- domain assumption All potentials considered are exclusively inhomogeneous solutions of Maxwell's equations for potentials, i.e., no free-field (homogeneous) terms are added.
- ad hoc to paper The identity in eq (23), involving the composition of c-speed and v-speed retarded Green's functions, is valid.
Cite this review
Pith. "Pith review of Reply to "Comments on work of Yang and Nevels 'Direct, analytic solution for the electromagnetic vector potential in any gauge' (arXiv:2507.02104)" by Onoochin (arXiv:2507.08042)." pith.science (2026). https://pith.science/paper/J2JZQXMG
@misc{pith2026250902845,
author = {Pith},
title = {Pith review of: Reply to "Comments on work of Yang and Nevels 'Direct, analytic solution for the electromagnetic vector potential in any gauge' (arXiv:2507.02104)" by Onoochin (arXiv:2507.08042)},
year = {2026},
howpublished = {\url{https://pith.science/paper/J2JZQXMG}},
note = {Machine review of arXiv:2509.02845}
}
read the original abstract
Because of a difference between his and our results for the Coulomb-gauge vector potential, Onoochin (arXiv:2507.08042) suspected that we might have used some (so far unidentified) mathematically illegal operations in our paper (arXiv:2507.02104). In this note, we present four methods to prove the validity of our results in dispute. Onoochin's method for calculating his Coulomb-gauge vector potential is among our four methods used in the proofs.
Reference graph
Works this paper leans on
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[6]
We state that all our potentials are exclusively inhomogeneous solutions of Maxwell’s equations for potentials. 7
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[4]
Velocity, temporal and generalized Kirchhoff gauges
K.-H. Yang and R. D. Nevels, “Velocity, temporal and generalized Kirchhoff gauges,” https://doi.org/10.48550/arXiv.2508.20248
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[1]
V. Onoochin, “Comment on work of Yang and Nevels ‘Direct, analytic solution for the electromagnetic vector potential in any gauge’ (arXiv:2507.02104)” https://doi. org/10.48550/arXiv.2507.08042
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[2]
Direct, analytic solution for the electromagnetic vector potential in any gauge
K.-H. Yang and R. D. Nevels, “Direct, analytic solution for the electromagnetic vector potential in any gauge,” https://doi.org/10.48550/arXiv.2507.02104
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[3]
The physics of gauge transformations,
K.-H. Yang, “The physics of gauge transformations,” Am. J. Phys. 73, 742-751 (2005)
work page 2005
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[5]
From Lorenz to Coulomb and other explicit gauge transformations,
J. D. Jackson, “From Lorenz to Coulomb and other explicit gauge transformations,” Am. J. Phys. 70, 917-928 (2002)
work page 2002
Reviewed August 5, 2026 · model on record in the stance chip above.
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