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REVIEW 3 major objections 5 minor 75 references

Statistical and non-statistical $\gamma$-decay properties of $^{64}$Zn

T0 review · 3 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read Direct gamma decay from the quasi-continuum to the $^{64}$Zn ground state is suppressed by about a factor of two, indicating non-statistical effects.

desk verdict Solid Oslo-method plus (gamma,n) paper for 64Zn with new NLD, gammaSF, and a genuinely interesting but model-dependent hint of hindered ground-state decay; the hindrance claim needs more work before it is taken as established. read the letter →

arxiv 2608.03943 v2 pith:YE6Q5TJ5 submitted 2026-08-04 nucl-ex nucl-th

A. C. Larsen , M. Guttormsen , T. K. Eriksen , G. M. Tveten , H. Utsunomiya , J. K. Dahl , N. Shimizu , T. Ari-izumi
show 18 more authors
F. L. Bello Garrote L. T. Bell M. M. Bj{o}r{o}en F. W. Furmyr D. Gjestvang A. Görgen V. W. Ingeberg K. C. W. Li E. Lima M. Markova E. F. Matthews A. H. Mj{o}s S. Miyamoto V. Modamio T. Renstr{o}m E. Sahin S. Siem M. Wiedeking
This is my paper · ORCID
classification nucl-exnucl-th PACS 25.40.-h23.20.-g21.10.Ma25.20.-x
keywords nuclearleveldensitygamma-raystrengthfunctionOslomethod64Znhinderedground-statedecaylow-energyenhancementshapegiantdipoleresonance
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

The paper sets out to map the statistical and non-statistical gamma-decay behavior of $^{64}$Zn by extracting its nuclear level density and gamma-ray strength function from $^{64}$Zn($p,p'\gamma$) coincidences with the Oslo method and combining these with photoneutron data. It reports that the quasi-continuum level density follows a constant-temperature form and, as its central claim, that direct gamma transitions to the $0^+$ ground state are hindered by a factor $\kappa \approx 0.5$ relative to statistical expectations. If correct, this means statistical-decay models for $^{64}$Zn and nearby nuclei need a structure-dependent correction for ground-state feeding, and that wave-function or shape effects can survive in the high-level-density regime. The paper also finds a low-energy enhancement in the strength function for $E_\gamma < 4$ MeV and no clearly identifiable scissors mode, extending the empirical systematics for zinc isotopes.

What carries the argument

The load-bearing object is the diagonal decomposition of the primary gamma-ray matrix: D1, D2, and D3 count primary transitions feeding the $0^+$ ground state, the first $2^+$ level at 992 keV, and the second $2^+$ and $0^+$ levels, respectively. Since dipole decay to a $0^+$ final state can only come from $J_i = 1$, while a $2^+$ final state can receive $J_i = 1,2,3$, the ratio of integrated level densities under the corresponding peaks in the Oslo level density carries the spin-selection information. The shape-method formula (Eq. 13) converts those diagonal counts into a strength function, and comparing the same ratios with the spin distribution from the adopted rigid-body spin-cutoff parameterization determines $\kappa$. The constant-temperature model supplies the functional form used to extrapolate the level density to the neutron separation energy for normalization.

What would settle it

Measure the spin distribution of $^{64}$Zn states near 7.4 MeV excitation with an independent method, such as spin-tagged or angular-correlation experiments that can identify $J=1$ levels, and compare the observed fraction with the value near 0.251 predicted by the adopted spin-cutoff model; a measured $J=1$ fraction close to 0.14 would remove the need for $\kappa \approx 0.5$, while a fraction near 0.25 would confirm the hindrance. An independent reaction that populates the quasi-continuum with a known spin population and measures ground-state feeding directly would settle whether the suppression is a property of the decay or of the population.

Watch

Extended reading notes

Core claim

The central discovery is that the integrated level density under the $0^+$ ground state is far too small for spin-selection rules alone: the Oslo-normalized level density gives $N(0^+_{\mathrm{gs}})/N(2^+_1) \approx 0.13$, whereas the adopted spin distribution with $\sigma_J = 3.53$ at an average excitation energy of 7.41 MeV predicts a feeding ratio of about 0.25 from the $J=1$ initial levels alone. The paper interprets the extra suppression as a hindrance factor $\kappa \approx 0.5$ acting on the gamma decay to the ground state, and shows that applying this factor to the D1 diagonal makes all three shape-method strength functions (D1D2, D1D3, D2D3) consistent with the Oslo-method strength-function slope. It explicitly tests the alternative hypotheses that both $0^+$ levels are hindered or that the $J=1$ level density is overestimated, finds the ground-state-hindrance hypothesis the most consistent, and does not fully rule out the alternatives.

Load-bearing premise

The deduction of $\kappa \approx 0.5$ rests on the assumption that the spin distribution of the decaying quasi-continuum states near 7.4 MeV excitation is correctly given by the adopted spin-cutoff model, specifically that the $J=1$ fraction is about 0.251; if the true $J=1$ population is lower, the same data can be explained without any hindrance of the decay itself.

Editorial extensions

If this is right

  • Statistical-model calculations that feed the $^{64}$Zn ground state through the quasi-continuum will need a spin- or structure-dependent hindrance correction of about 0.5 to reproduce the observed level-density ratios.
  • The shape method becomes a diagnostic for non-statistical decay: with $\kappa = 0.52$, the three diagonal combinations D1D2, D1D3, and D2D3 give consistent gamma-strength-function slopes, so inconsistencies would signal a violation of the assumed statistical decay.
  • The constant-temperature behavior of the level density, with a temperature of about 1.17 MeV for the adopted normalization, gives a baseline for comparing neighboring zinc isotopes and testing level-density systematics.
  • The combined strength function from roughly 2 to 19 MeV, including the low-energy enhancement below 4 MeV and the giant-dipole-resonance region, provides more complete input for astrophysical reaction-rate calculations on $^{64}$Zn.

Reading between the lines

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

  • Beyond the paper, if the hindrance is caused by a shape or wave-function mismatch between the ground state and the quasi-continuum, similar suppression of direct ground-state feeding should appear in other nuclei with coexisting or triaxial shapes, and existing data sets for neighboring isotopes could be re-examined for the same signature.
  • A testable extension would be to repeat the D1/D2 analysis on the heavier zinc isotopes, including $^{70}$Zn where the authors mention an ongoing analysis; a smoothly changing $\kappa$ with neutron number would support a structural origin.
  • Because the alternative that the $J=1$ level density is overestimated remains viable, $\kappa$ should be read as an effective suppression that may mix a true decay hindrance with a population correction until the $J=1$ fraction is measured independently.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper reports a simultaneous extraction of the nuclear level density (NLD) and γ-ray strength function (γSF) of 64Zn using the Oslo method on (p,p′γ) data, combined with photoneutron cross sections from the NewSUBARU facility. The NLD is found to follow a constant-temperature-like behavior, and the γSF exhibits a low-energy enhancement below 4 MeV and a GDR-dominated region above the neutron separation energy. The central new claim is that direct γ decay from the quasi-continuum to the 0+ ground state is strongly hindered, quantified by a hindrance factor κ≈0.5, which the authors suggest may reflect non-statistical effects or shape differences. The analysis is anchored to external data, including neutron-resonance systematics, previous particle-evaporation data, and large-scale shell-model calculations, and the authors test three hypotheses for the observed ground-state feeding deficit in Appendix B.

Significance. If the hindrance claim holds, it would be an important result: statistical-decay codes and nuclear-reaction models for 64Zn and nearby nuclei would need a spin- or structure-dependent correction for ground-state feeding. The paper also provides a new, independent NLD and γSF measurement for 64Zn, with the γSF covering a wide energy range (≈2–19 MeV), and the comparison with prior (γ,n) data and Ohio particle-evaporation data is valuable. The analysis is generally careful and honest about uncertainties, including an inflation-factor treatment of systematic errors and an explicit discussion of alternative hypotheses. However, the significance of the work is currently tempered by the model dependence of the central hindrance claim, which the authors themselves acknowledge is not uniquely determined by the data.

major comments (3)
  1. [Abstract and Appendix B] The abstract states that transitions to the ground state are 'strongly hindered with a hindrance factor of κ≈0.5', but Appendix B explicitly concludes that hypotheses H2 and H3 cannot be completely ruled out. In particular, H3 with κ=0.45 reproduces the three level-density ratios (0.130, 0.116, 0.884) within approximately 1.6σ, 1.6σ, and 1.3σ of the experimental values (0.130(1), 0.10(1), 0.77(9)). Thus the data do not uniquely determine a γ-decay hindrance; an alternative explanation is a ~45% reduction in the effective J=1 feeding. The language of the abstract and Section IV should be qualified accordingly, or the analysis must be extended to discriminate H1 from H3 (for example, by computing the initial spin distribution with a reaction model rather than assuming it follows the EB parameterization).
  2. [Eq. (13), Section IV] The shape-method analysis assumes that the population probability p_level(E_i,J_i) is independent of spin. This is a strong assumption for the (p,p′) reaction at 16 MeV, where the angular-momentum transfer likely produces a spin-dependent population. If p_level(J) is not flat, the observed deficit in ground-state feeding could be explained without invoking any hindrance of the γ decay itself. The paper does not provide a quantitative test of this assumption. Please estimate the spin dependence of the (p,p′) population (e.g., from a distorted-wave Born approximation or a Hauser-Feshbach model) and show that the extracted ratios are robust to plausible variations in p_level(J), or explicitly state that the hindrance interpretation is conditional on spin-independent feeding.
  3. [Fig. 9 and Section III] The shell-model comparison is offered as support for the EB spin distribution, but the KSHELL calculation is truncated at 200 levels per spin and the paper states that the most common spins (J=2–6) are 'spent' by E_x≈8.8 MeV. The relevant average excitation energy for the hindrance analysis is ⟨E_i⟩=7.41 MeV, and at E_x=7.4–8.0 MeV the cap may already distort the relative spin populations, especially for J=2–6. Since the alternative hypothesis H3 involves a 45% reduction in the J=1 fraction, the shell-model result cannot rule out this possibility unless the number of levels per spin at E_x≈7.4 MeV is reported and the truncation effect on g(J=1) is quantified. Please provide this information or temper the claim that the shell model supports the EB spin distribution.
minor comments (5)
  1. [Abstract] Consider softening 'seem to be strongly hindered' to 'are consistent with a hindrance factor of κ≈0.5' in light of the admitted degeneracy with H3; this would align the abstract with the Appendix B conclusion.
  2. [Section IV and Fig. 12] The background subtraction used to extract the integrated level numbers N(0+_gs), N(2+_1), and N(2+_2+0+_2) is described only as a 'simple, linear background'. The extracted ratios are sensitive to this choice, so a more detailed description (e.g., the fit range and the sensitivity to the background parametrization) would strengthen the presentation.
  3. [Table II] The PDR parameters (E, σ, Γ) carry very large uncertainties, which the text acknowledges; it would improve clarity to explicitly state that the PDR is not well constrained, and that the quoted parameters should be interpreted as a phenomenological representation rather than a firm resonance extraction.
  4. [Section IV, first paragraph] The term 'diagonal' is used without a formal definition; a brief explanation that 'diagonal' refers to a constant value of E_i - E_f (the γ energy) for a fixed final state would help readers.
  5. [Appendix B, Eqs. (B1)–(B9)] The notation g(J) is introduced without an explicit definition in the appendix; it would be helpful to refer back to Eq. (A1) and define g(J) as the spin-distribution function evaluated at ⟨E_i⟩.

Circularity Check

1 steps flagged · score 2.0 of 10

No significant circularity: the κ hindrance claim is underdetermined rather than circular; only a mild CT-normalization self-consistency is present.

  1. fitted input called prediction [Sec. II.C (Normalization of the NLD and gammaSF) and Abstract]
    "We observe that the experimental NLD shows an approximately log-linear dependence in the region E_x≈5.0−8.0 MeV... Hence, we make use of the constant-temperature (CT) model of Ericson... and extrapolate up to the anchor point ρ(S_n). ... We observe that the NLD trend in the quasi-continuum region of 64Zn is best characterized by a constant-temperature-like model."

    The constant-temperature model is used to set the slope/scaling of the Oslo-method NLD through the α transformation, and the same normalized data are then presented as being 'best characterized' by a constant-temperature-like model. The CT characterization is therefore partly imposed by the normalization choice rather than being an independent prediction from the data. This is a mild internal-consistency issue, not a forced result: the log-linear region is empirical, and the extracted temperature agrees with the independent Ohio measurement.

full rationale

The paper's central derivation is largely self-contained and anchored to external data: discrete levels from NNDC, the independent Ohio level-density measurement, neutron-resonance systematics for neighboring Zn isotopes, and large-scale shell-model calculations. The κ≈0.5 ground-state hindrance is inferred from the measured N(0+_gs)/N(2+_1) ratio under the assumed EB spin distribution, not defined into existence. The paper explicitly tests hypotheses H1, H2, and H3 and admits that 'hypotheses H2 and H3 cannot be completely ruled out'; thus the non-statistical-decay claim is underdetermined by the data, but underdetermination is not circularity. The shell-model support for the EB spin distribution is truncated and therefore weaker than claimed, but it is an independent calculation rather than a self-citation or a renamed input. The only mild circular step is the CT-model normalization/characterization: the CT extrapolation to ρ(S_n) helps determine the NLD slope, and the same data are later described as CT-like; however, external Ohio data and the shell model corroborate the CT behavior, so this does not compromise the main results. There is no load-bearing self-citation chain, no imported uniqueness theorem, and no prediction that reduces by construction to a fitted parameter. Score 2 reflects the minor CT self-consistency while recognizing the bulk of the analysis is empirically anchored.

Assumptions & free parameters 8 free parameters · 6 assumptions · 0 invented entities

The central results rest on the Oslo-method factorization, on the chosen spin-distribution model, and on normalization parameters (rho(S_n), <Gamma_gamma0>) that are not measured for 64Zn but estimated from systematics of other Zn isotopes. The hindrance factor is model-dependent and is extracted from the same data as the NLD, with only internal consistency checks. The CT-model extrapolation used to normalize the NLD is also used to claim constant-temperature behavior, a mild self-consistency that is cross-checked against an independent Ohio measurement.

free parameters (8)
  • T_CT (constant-temperature model temperature) = 1.17(2) MeV (EB), 1.20(2) MeV (GC)
    Fitted to the Oslo NLD in the approximate 4-8 MeV region and used to extrapolate the level density to S_n; the reported CT behavior depends on this fit.
  • E_0 (CT model energy shift) = not quoted in the paper
    Second free parameter of the CT fit used for the NLD extrapolation to the S_n anchor.
  • kappa (ground-state hindrance factor) = 0.52(1)
    Determined from the ratio of the integrated ground-state and first-excited-state level-density peaks; used to correct the shape-method D1 counts and central to the claim of hindered ground-state decay.
  • rho(S_n) (level density at neutron separation energy) = 5.41(+0.78,-0.68) x 10^4 MeV^-1 (EB)
    Not measured for 64Zn because 63Zn is unstable; estimated from a hierarchical log-linear fit to s-wave and p-wave resonance spacings of other Zn isotopes, then used as the high-energy anchor for the NLD normalization.
  • <Gamma_gamma0> (average total radiative width) = 708(+298,-210) meV
    Not measured for 64Zn; estimated from a log-linear fit to the outlier-cut (core) average radiative widths of the other Zn isotopes; sets the absolute scale of the gammaSF.
  • T_f (GLO temperature parameter) = 0.0(14) MeV (free fit)
    Free parameter in the generalized Lorentzian fit to the combined gammaSF; strongly influences the E1 tail and therefore the inferred LEE parameters.
  • LEE constant C and slope eta = C = 1.5(5) x 10^-8 MeV^-3, eta = 0.16(15) MeV^-1 (free T_f fit)
    Exponential low-energy enhancement fit to the gammaSF below 4 MeV; large uncertainties reflect the uncertain T_f and the absolute normalization.
  • GDR and PDR resonance parameters (E, sigma, Gamma) = GDR: E=18.2 MeV, sigma=109 mb, Gamma=9.1 MeV; PDR: E=8.1 MeV, sigma=0.88 mb, Gamma=3.0 MeV (free T_f fit)
    Nine-parameter phenomenological fit to the combined gammaSF; descriptive parameters, not predictions, with large uncertainties for the PDR and LEE.
assumptions (6)
  • domain assumption Brink-Axel hypothesis: the gamma-ray transmission coefficient T(E_gamma) is independent of excitation energy, spin, and parity, and the primary decay probability factorizes as P(E_gamma,E_i) proportional to rho(E_i - E_gamma) T(E_gamma), Eq. (7).
    The entire Oslo-method extraction of the NLD and gammaSF in the quasi-continuum relies on this factorization; a violation in 64Zn would distort the extracted level density and hence the inferred hindrance factor.
  • domain assumption Dipole (E1/M1) dominance in quasi-continuum decay, used to relate the transmission coefficient to the dipole gammaSF via Eq. (12) and to restrict the initial spins in the shape method.
    Invoked in Section II B and Section IV; established empirically for many nuclei but not independently proven for 64Zn.
  • domain assumption The spin distribution g(E_x,J) follows Eq. (A1) with the EB rigid-body spin-cutoff parameter, Eq. (A2), and the population probability p_level is independent of spin in the shape method.
    The hindrance factor kappa is defined relative to the expected g(J=1) feeding fraction; hypothesis H3 shows the conclusion is sensitive to this assumption. See Section IV and Appendix B.
  • domain assumption The constant-temperature model of Ericson, Eq. (11), is the correct functional form to extrapolate the NLD from about 8 MeV to S_n, and the hierarchical model, Eq. (A4), describes the trend of rho(S_n) across Zn isotopes.
    Used to set the high-energy anchor for the NLD normalization over a 3-4 MeV extrapolation region; the CT model is fit to the same data whose CT shape it is used to claim.
  • standard math Geant4 simulations of the LCS gamma-beam energy profiles and of the 4pi neutron-detector efficiency are accurate.
    The (gamma,n) cross section is unfolded using simulated beam profiles, Eqs. (3)-(5), and the neutron efficiency is determined by Monte Carlo; errors in these simulations shift the photoneutron cross section and the gammaSF above S_n.
  • domain assumption The iterative unfolding and first-generation methods (Refs. 20-22, Appendix C) correctly isolate the primary gamma-ray spectra, with the region E_i = 6-10 MeV and E_gamma > 2 MeV dominated by statistical decay.
    The extracted NLD and gammaSF, and therefore the hindrance factor, are only meaningful if the primary spectra are correctly obtained and if the decay in the selected region is statistical.

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Pith. "Pith review of Statistical and non-statistical $\gamma$-decay properties of $^{64}$Zn." pith.science (2026). https://pith.science/paper/YE6Q5TJ5

@misc{pith2026260803943,
  author       = {Pith},
  title        = {Pith review of: Statistical and non-statistical $\gamma$-decay properties of $^64$Zn},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YE6Q5TJ5}},
  note         = {Machine review of arXiv:2608.03943}
}
abstract

We present a study on the $\gamma$-decay properties of $^{64}$Zn using the Oslo method on $^{64}$Zn($p,p^\prime \gamma$) data combined with $^{64}$Zn$(\gamma,n)$ cross-section measurements at the NewSUBARU facility. With the Oslo method, we have measured the $\gamma$-ray strength function ($\gamma$SF) and the nuclear level density (NLD) below the neutron threshold. We observe that the NLD trend in the quasi-continuum region of $^{64}$Zn is best characterized by a constant-temperature-like model. %with temperature parameter $T_{\rm CT}=1.21(5)$ MeV. Surprisingly, we find that $\gamma$-ray transitions from the quasi-continuum decaying directly to the $0^+$ ground state seem to be strongly hindered with a hindrance factor of $\kappa \approx 0.5$, which could be an indication of non-statistical effects in the ground-state decay due to, \textit{e.g.}, differences in nuclear shapes. For $\gamma$ energies above the neutron separation energy, the NewSUBARU ($\gamma, n$) data set probes a significant part of the giant dipole resonance. Furthermore, we find that the Oslo-method $\gamma$SF shows a rather smooth behavior, with a clear low-energy enhancement (LEE) for $E_{\gamma} < 4$ MeV.

Figures

Figures reproduced from arXiv: 2608.03943 by the authors.

Figure 1
Figure 1. As seen from Eq. (3), to obtain an estimate for the true [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (Color online) Unfolded data from this work (azure filled cir [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. (Color online) Proton- [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (18 more)
Figure 4
Figure 4. Figure 4: FIG. 4. (Color online) Distribution of residuals after multiplying the [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. (Color online) Probability density distributions of primary [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. (Color online) Level density (filled data points) of [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 10
Figure 10. Figure 10: FIG. 10. (Color online) The Oslo level density (filled data points) of [PITH_FULL_IMAGE:figures/full_fig_p008_10.png]
Figure 9
Figure 9. Figure 9: FIG. 9. (Color online) Calculated spin distribution of [PITH_FULL_IMAGE:figures/full_fig_p008_9.png]
Figure 11
Figure 11. Figure 11: FIG. 11. (Color online) (a) The primary [PITH_FULL_IMAGE:figures/full_fig_p009_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. (Color online) The level density of [PITH_FULL_IMAGE:figures/full_fig_p010_12.png]
Figure 14
Figure 14. Figure 14: FIG. 14. (Color online) Comparison with [PITH_FULL_IMAGE:figures/full_fig_p011_14.png]
Figure 15
Figure 15. Figure 15: FIG. 15. (Color online) Same as Fig- 14, but with the [PITH_FULL_IMAGE:figures/full_fig_p011_15.png]
Figure 13
Figure 13. Figure 13: The resulting parameters are listed in Table II. [PITH_FULL_IMAGE:figures/full_fig_p012_13.png]
Figure 16
Figure 16. Figure 16: FIG. 16. (Color online) Estimate of the level density [PITH_FULL_IMAGE:figures/full_fig_p013_16.png]
Figure 17
Figure 17. Figure 17: FIG. 17. (Color online) Estimate of the level density [PITH_FULL_IMAGE:figures/full_fig_p013_17.png]
Figure 18
Figure 18. Figure 18: To obtain an absolute normalization of the γSF, we have made use of available s-wave resonance data from Mughabghab’s 2018 Atlas [12]. The Γγ0 values for the in￾dividual resonances of the compound nuclei 65,67,68,69Zn are binned and shown as histograms in [PITH_FULL_…
Figure 19
Figure 19. Figure 19: FIG. 19. (Color online) Histograms of [PITH_FULL_IMAGE:figures/full_fig_p015_19.png]
Figure 20
Figure 20. Figure 20: FIG. 20. (Color online) Estimate of [PITH_FULL_IMAGE:figures/full_fig_p016_20.png]
Figure 21
Figure 21. Figure 21: FIG. 21. (Color online) Normalized [PITH_FULL_IMAGE:figures/full_fig_p016_21.png]
Figure 23
Figure 23. Figure 23: FIG. 23. The raw (a), unfolded (b) and primary (c) [PITH_FULL_IMAGE:figures/full_fig_p018_23.png]
Figure 24
Figure 24. Figure 24: FIG. 24. The raw (a), unfolded (b) and primary (c) [PITH_FULL_IMAGE:figures/full_fig_p018_24.png]

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