Comment on "Time crystals made of electron-positron pairs"

Zhen-Hua Feng

REVIEW 4 major objections 4 minor 5 references

Comment on "Time crystals made of electron-positron pairs"

T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A direct reproduction of a published electron-positron time-crystal calculation shows the eigenfunctions diverge and the displayed odd/even parity cannot come from the stated equation.

desk verdict The comment's parity contradiction is algebraically invalid and its numerical reproduction uses sign-inverted initial conditions, so neither of its two claimed flaws is established. read the letter →

arxiv 2506.06408 v2 pith:KYDDO6HN submitted 2025-06-06 quant-ph

Zhen-Hua Feng This is my paper

classification quant-ph

keywords timecrystalselectron-positronpairsquantumeigenstatesparitysymmetrynumericalreproductionboundednesscoupledSchrödingersystemeigenfunctiondivergence

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This comment reports that rerunning the quantum equations of a published electron-positron time-crystal model, with the same equations and the same initial conditions, does not reproduce the published eigenfunctions. The recalculated solutions grow without bound in the y<0 region instead of falling off slowly—at y=-10 the magnitudes reach about 2×$10^{7}$, 1.5×$10^{7}$, and more than 6000 in the three displayed cases—and they are asymmetric even though the published figures show one odd and one even component. The comment also proves that parity-definite solutions of the central coupled equation must vanish identically, so the displayed odd/even symmetry cannot be genuine. If these results are right, the claimed quantum eigenstates are not admissible wavefunctions and the quantum-mechanical foundation of that time-crystal model collapses.

What carries the argument

The load-bearing object is the two-component coupled system of Eq. (21): [−d²/dy² − E/$2^{{1/3}}$ + y]φ₁ + $2^{{2/3}}$φ₂ = 0 and [−d²/dy² − E/$2^{{1/3}}$ − y]φ₂ + $2^{{2/3}}$φ₁ = 0. Parity reversal swaps the two linear potentials and flips the sign of the coupling terms, so demanding an odd φ₁ and an even φ₂ yields a consistency equation that can be satisfied only by φ₁ = 0. This system, together with the proposed decomposition of the original two-component wavefunction into real functions φ₁ and φ₂, carries the whole argument: it is what the reproduction integrates numerically and what the parity proof shows to be incompatible with the published figures.

What would settle it

Integrate Eq. (21) from y=0 to y=−10 with the original paper's stated initial values and record φ₁(−10) and φ₂(−10); also check whether φ₁(−y) = −φ₁(y) and φ₂(−y) = φ₂(y) over that range. If the functions stay bounded with magnitudes of order one and the parity relations hold, the central criticism fails, whereas the published large magnitudes are the direct evidence that it succeeds.

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Extended reading notes

Core claim

The central claim is that Eq. (21) of Ref. [1] does not admit the odd/even parity eigenfunctions shown in its Figs. 9–11. Replacing y by −y reverses the signs of the potential terms and the coupling in a way that cannot be undone by any constant rescaling; assuming one component is odd and the other even forces a consistency condition in which the only solution is the zero function. Numerical integration using the paper's own stated initial data produces asymmetric, growing solutions: in the three displayed cases |φ2| reaches about 2×$10^{7}$, 1.5×$10^{7}$, and more than 6000 at y=−10, which contradicts the claimed slow falloff and violates the boundedness required of quantum wavefunctions. The comment further argues that the decomposition in Eq. (20) of Ref. [1] is not forced by the equations, since the relation iΨ*₂ = Ψ₁ is sufficient but not necessary for the system to close.

Load-bearing premise

The contradiction depends on the reproduction using exactly the initial conditions stated in the original paper and on the published figures genuinely showing a simple odd/even parity; if the initial derivative signs were mis-transcribed or the figures plot some transformed quantity, the claimed inconsistency would not follow.

Editorial extensions

If this is right

The eigenfunctions plotted in Figs. 9–11 of Ref. [1] cannot be solutions of Eq. (21) with the stated initial conditions, so conclusions drawn from those curves about quantum time-crystal eigenstates lack numerical support.
Because the reproduced functions grow without bound in y<0, they are not normalizable and fail the boundedness requirement every quantum-mechanical wavefunction must satisfy.
No linear superposition of solutions of Eq. (21) can produce the odd/even parity shown in the figures; the parity argument shows any such superposition is identically zero.
The basis transformation behind Eq. (20) of Ref. [1] is not a necessary consequence of the equations, since the relation iΨ*₂ = Ψ₁ is only a sufficient condition, so the real-valued eigenfunction picture may discard valid solutions.
The statement in Ref. [1] that the quantum Hamiltonian's eigenstates had been found is not supported when the calculation is rerun with the paper's own data.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper: because the transformations applied to the reproduced data to recover the published figures (dropping y<0, rotating one curve, mirroring the other) are arbitrary, the published parity pattern is likely a graphical artifact rather than a property of the solutions.
Beyond the paper: a natural next test is a full spectral search for normalizable eigenstates of the same Hamiltonian with decay imposed at both infinities; the parity argument indicates that no definite-parity states exist, which would settle whether this model supports a quantum time crystal at all.
Beyond the paper: the same tension between a desired odd/even pattern and a coupled system whose potential terms change sign under y→−y may appear in other two-component quantum models, so the mathematical argument is not confined to the electron-positron construction.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: the comment's central contradiction is not proven, and its numerical reproduction undermines itself. The observation that the eigenfunctions in Figs. 9-11 of the original paper appear to have a symmetry not obvious from Eq. (21) is worth checking, and the attempt to reproduce the solutions with stated initial conditions is a reasonable exercise. But the parity proof in §2.1 has an algebraic error: with φ1 odd and φ2 even, Eq. (8) reduces to b=0, not a=b=0, because the first bracket vanishes and the second is proportional to (φ1, φ2). A one-parameter family of parity-preserving combinations remains admissible unless additional ODE constraints are invoked, and none are given. The conclusion that parity-definite solutions are impossible does not follow. The numerical reproduction has a similar self-inflicted problem: the comment states initial values φ2'(0)=-0.354651985 and φ1'(0)=-0.665192338, then immediately notes that the original figures show positive slopes at y=0. Since the ODE system is linear and symmetric under v(y)=σ_x u(-y), flipping the sign of an initial derivative selects a different independent solution. The reported divergence in y<0 is very likely an artifact of that sign choice, not a property of the original eigenfunctions. The boundedness claim that 'all quantum solutions violate boundedness requirements' is extrapolated from these three suspect runs, so it does not stand. Credit where due: the comment does show that the decomposition in Eq. (20) of the original paper is sufficient but not necessary, and the original could have been clearer there. But the comment does not show the decomposition is wrong. Bottom line: neither of the two headline flaws is established. A corrected version would need a proper symmetry analysis that accounts for the component-swap symmetry and a numerical setup that matches the original slopes. In its current form, I would not send this to peer review; the load-bearing errors are fatal.

Referee Report

4 major / 4 minor

Summary. The Comment by Feng addresses Ref. [1], which presents quantum eigenstates for a time-crystal model of electron-positron pairs. Feng reports numerical reproductions of Figs. 9–11 of Ref. [1] that diverge in the region y<0 and are pointwise incompatible with the published plots. Section 2 argues that Eq. (21) of Ref. [1] cannot support the apparently parity-definite eigenfunctions displayed in the figures. Section 2.2 questions the basis transformation leading to Eq. (21). The abstract concludes that all quantum solutions of Ref. [1] violate quantum-mechanical boundedness.

Significance. If the claims were established, the Comment would identify a serious flaw in a published Physical Review A paper and would correctly cast doubt on the existence of normalizable quantum eigenstates in this model. The underlying observation that the effective potential in Eq. (21) is unbounded below in one channel is plausible and worth a rigorous treatment. The manuscript also provides source code for its numerical work, which is a useful element of transparency. However, the central demonstration as written contains an algebraic error and a mismatch between the stated initial conditions and the plotted target solution, and the extrapolation to 'all' solutions is unsupported.

major comments (4)

[§2.1, Eq. (8)] The conclusion a=b=0 does not follow from Eq. (8). For a target solution with φ1 odd and φ2 even, the first bracket in both equations vanishes by the parity assumption, leaving b[φ2(y)+φ2(-y)]=0 and b[φ1(y)-φ1(-y)]=0; this fixes b=0 and leaves a free, so a one-parameter family of parity-definite combinations remains admissible under this argument. In addition, Eq. (7) claims that 'any solution of Eq. (1)' is a linear combination of two particular solutions, but Eq. (21) is a system of two second-order ODEs whose solution space is four-dimensional. The two functions in Eq. (7) span at most two dimensions, so the representation and the subsequent contradiction are not valid.
[§1, Figs. 9–10 initial data] The manuscript explicitly states that the original Fig. 9 shows φ2'(0)>0, but the reproduction uses φ2'(0)=-0.354651985; likewise Fig. 10 uses φ1'(0)=-0.665192338 against an original positive slope. Since Eq. (21) is invariant under (φ1,φ2)(y) ↦ (φ2(-y),φ1(-y)), reversing the sign of an initial derivative selects a different solution whose divergence is mirrored between y>0 and y<0. The reported divergence in y<0 is therefore not evidence about the solution plotted in Ref. [1], and the abstract's claim that the reproduction uses 'the authors' own ... initial conditions' is internally contradicted by the body.
[§2.1, Eq. (2)] Equation (2) is not obtained by substituting -y for y in Eq. (1): the arguments of φ1 and φ2 remain y, and the sign of the coupling term is flipped. Substituting -y properly gives equations for φ1(-y) and φ2(-y), and the structural difference the comment emphasizes between Eqs. (1) and (2) is therefore an artifact of this substitution error.
[§2.2 and Abstract] The 'unphysical basis transformation' discussion does not demonstrate an inconsistency: the observation that iΨ2*=Ψ1 is sufficient but not necessary for Eq. (15) does not show that the transformation is wrong, and no contradiction is derived from it. Similarly, the abstract's statement that 'all quantum solutions' in Section IV violate boundedness is extrapolated from the three numerical examples; no general asymptotic argument for all solutions of Eq. (21) is supplied.

minor comments (4)

[§2.1, before Eq. (1)] The phrase 'φ2(2)' should read 'φ2(y)'.
[Abstract] The parenthetical '(|φ|→∞ as y→−10)' is confusingly worded, as divergence is not a decay behavior; please rephrase.
[Captions of Figs. 9–11] The middle panels use different y-ranges than the original figures; please indicate whether the plotted divergence is cut off by the chosen range and whether the same normalization and scaling as in Ref. [1] are used.
[Data availability] The GitHub repository link should specify a versioned commit or DOI to ensure reproducibility of the exact numerical results reported.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the comment reproduces the original equations and initial data directly; the only self-reference is a non-load-bearing data-availability link.

full rationale

The comment is a direct numerical reproduction and ODE-level analysis of Ref. [1]. Its central numerical claims—divergence in y<0 and disagreement with Figs. 9–11—follow from integrating Eq. (21) with the stated initial conditions; they are consequences of the original equations rather than being equivalent to the inputs by construction. The parity argument in Section 2.1 attempts to prove from Eq. (1) that parity-definite solutions cannot exist; whether or not that proof is algebraically sound (the step from Eq. (8) to a=b=0 appears to overreach, since for phi1 odd and phi2 even the system only forces b=0 and leaves a free), this is a correctness objection, not a circular reduction. Section 2.2 explicitly concedes that the condition iPsi2*=Psi1 is 'sufficient but not necessary' and that no physical justification is provided for the decomposition ansatz; again this is an admitted gap in the original argument, not a circular step. The only self-reference is the GitHub data-availability link, which is not load-bearing. No fitted parameter is renamed as a prediction, no load-bearing self-citation chain is invoked, and no result is defined in terms of the conclusion it is supposed to support. Therefore no significant circularity is established.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The comment's central claims rest on standard quantum-mechanical boundedness, the correctness of the transcribed equations, the faithfulness of the reproduction's initial conditions, and the interpretation of the original figures. No new entities are introduced. The reproduction itself introduces numerical boundary conditions that are not independently verified.

free parameters (2)

Initial derivative φ'_2(0) for eigenvalue ϵ=2 = -0.354651985
Reproduction's boundary condition, borrowed from Ref [1]. The comment notes the original figure shows φ'_2(0)>0, so this may be a sign mismatch; if the value is incorrect, the divergence in y<0 is not a faithful reproduction.
Initial derivative φ'_1(0) for eigenvalue ϵ=5 = -0.36012
Reproduction's boundary condition for the second example, taken from Ref [1]. Again, the original figure reportedly shows the opposite sign, so the reproduction's fidelity depends on this number.

assumptions (4)

domain assumption Physical quantum states must be bounded and normalizable.
Invoked to declare the divergent reproductions unphysical; the original paper's own claim of slow falloff suggests this standard is shared.
domain assumption Equation (21) is transcribed correctly from Ref. [1].
The parity proof starts from Eq. (1), which is quoted as Eq. (21) of Ref. [1]; an error in transcription would invalidate the conclusion.
ad hoc to paper The initial conditions for reproduction exactly match Ref. [1].
The author states they are the exact parameters, but provides no side-by-side table from the original; the sign of φ'_2(0) is later contested.
domain assumption Figures 9-11 of Ref. [1] indeed display φ1 odd and φ2 even parity.
The proof of inconsistency assumes the parity pattern visible in the figures; if the original used a generalized parity that swaps components, the argument does not apply.

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Cite this review

Pith. "Pith review of Comment on "Time crystals made of electron-positron pairs"." pith.science (2026). https://pith.science/paper/KYDDO6HN

@misc{pith2026250606408,
  author       = {Pith},
  title        = {Pith review of: Comment on "Time crystals made of electron-positron pairs"},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KYDDO6HN}},
  note         = {Machine review of arXiv:2506.06408}
}

abstract

Direct reproduction of Bialynicki-Birula's quantum solutions using the authors' own equations and initial conditions reveals two fundamental flaws. First, the eigenfunctions exhibit divergence in the region $y<0$, contradicting the claimed decay behavior ($\abs{\phi}\to\infty$ as $y\to -10$) (Figs. 9--11). Second, the apparent parity symmetry displayed in Figures 9--11 is not supported by Equation (21); numerical solutions clearly show asymmetric wavefunctions. Furthermore, all quantum solutions presented in Section IV violate the boundedness requirements of quantum mechanics. These inconsistencies raise serious concerns about the physical validity of the quantum framework underlying the time crystal model.

Figures

Figures reproduced from arXiv: 2506.06408 by the authors.

**Figure 9.** Left: Original result (Ref. [1]) with parameter y ranging from −10 to 10. Middle: Reproduced result with a detailed view of parameter y ranging from −2 to 10. Right: Reproduced result showing the full range of parameter y from −10 to 10, matching the original figure’s parameter range. The eigenfunction components φ1 ( y) and φ2 ( y) correspond to the eigenvalue ϵ = 2 of the quantum Hamiltonian (19), with initial val… view at source ↗

**Figure 10.** Left: Original result (Ref. [1]) with parameter y ranging from −10 to 10. Middle: Reproduced result with a detailed view of parameter y ranging from −2 to 10. Right: Reproduced result showing the full range of parameter y from −10 to 10, matching the original figure’s parameter range. The eigenfunction components φ1 ( y) and φ2 ( y) correspond to the eigenvalue ϵ = 2 of the quantum Hamiltonian (19), with initial val… view at source ↗

**Figure 11.** Left: Original result (Ref. [1]) with parameter y ranging from −10 to 10. Middle: Reproduced result with a detailed view of parameter y ranging from −4 to 10. Right: Reproduced result showing the full range of parameter y from −10 to 10, matching the original figure’s parameter range. 2 [PITH_FULL_IMAGE:figures/full_fig_p002_11.png] view at source ↗

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Reference graph

Works this paper leans on

5 extracted references · 3 canonical work pages

[1]

Bialynicki-Birula and Z

I. Bialynicki-Birula and Z. Bialynicka-Birula, Time crystals made of electron-positron pairs, Phys. Rev. A 104 , 022203 (2021), doi:10.1103/PhysRevA.104.022203

work page doi:10.1103/physreva.104.022203 2021
[2]

G. B. Arfken, Mathematical Methods for Physicists, Academic Press, San Diego, USA, 3 edn. (1985)

work page 1985
[3]

U ber den zusammenhang des abschlusses der elektronengruppen im atom mit der komplexstruktur der spektren , Zeitschrift f \

W. Pauli, \"U ber den zusammenhang des abschlusses der elektronengruppen im atom mit der komplexstruktur der spektren , Zeitschrift f \"u r Physik 43(9-10), 601 (1927), doi:10.1007/BF01900315

work page doi:10.1007/bf01900315 1927
[4]

, " * write output.state after.block =

ENTRY address archive author booktitle chapter doi edition editor eid eprint howpublished institution isbn journal key month note number organization pages publisher school series title type url volume year label INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 'mid.sentence := #2 'af...
[5]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize " " * FUNCT...

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Bialynicki-Birula and Z

I. Bialynicki-Birula and Z. Bialynicka-Birula, Time crystals made of electron-positron pairs, Phys. Rev. A 104 , 022203 (2021), doi:10.1103/PhysRevA.104.022203

work page doi:10.1103/physreva.104.022203 2021

[2] [2]

G. B. Arfken, Mathematical Methods for Physicists, Academic Press, San Diego, USA, 3 edn. (1985)

work page 1985

[3] [3]

U ber den zusammenhang des abschlusses der elektronengruppen im atom mit der komplexstruktur der spektren , Zeitschrift f \

W. Pauli, \"U ber den zusammenhang des abschlusses der elektronengruppen im atom mit der komplexstruktur der spektren , Zeitschrift f \"u r Physik 43(9-10), 601 (1927), doi:10.1007/BF01900315

work page doi:10.1007/bf01900315 1927

[4] [4]

, " * write output.state after.block =

ENTRY address archive author booktitle chapter doi edition editor eid eprint howpublished institution isbn journal key month note number organization pages publisher school series title type url volume year label INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 'mid.sentence := #2 'af...

[5] [5]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize " " * FUNCT...