{"id":"cd8e12a6-0e4a-4c58-8bbf-c776cbdfd728","arxiv_id":"2412.00641","paper_version":3,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Magnetic fields generated at electroweak symmetry breaking have a k^4 energy spectrum, not k^3 as previously claimed, according to non-dynamical Kibble-mechanism simulations.","lead":"A simulation based on random Higgs-field domains shows that magnetic fields generated during electroweak symmetry breaking have a k^4 energy spectrum, correcting the older k^3 estimate. These initial conditions matter because they feed the MHD evolution that determines present-day cosmological magnetic fields, which the paper estimates at 10^-13 G on kpc scales for non-helical fields and 10^-10 G on Mpc scales for maximally helical fields.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central k^4 claim assumes, rather than derives, that the final divergence-free field is fully captured by the Higgs-gradient term with the ∇×A contribution set to zero; if that neglected term contributes comparably at small k, the spectral slope and the present-epoch field estimates are…","rationale":"The reader's weakest_assumption is exactly the load-bearing issue: Eq. (1) contains a ∇×A term that the paper sets to zero in Eq. (6), and the divergence-free field is constructed non-dynamically from plaquette fluxes. The stress-test pass found no additional concern that changes the verdict. The internal numerical evidence is strong: the k^4 scaling is shown across N=256, 512, 1024 in Fig. 3 with convergence, and Appendix B independently derives B_V ∝ k^2, which combined with the window-function calculation forces n ≥ 4, consistent with k^4 and excluding the previously claimed k^3. The authors also provide open code and data, and the remaining limitations (coherence scale, 10% magnetic energy fraction, MHD prefactors) are explicitly acknowledged in Secs. V and VI. Those limitations do make the present-epoch amplitudes (10^-13 G, 10^-10 G) conditional on external assumptions, which is why the original CONDITIONAL verdict is appropriate; but the central spectral result is reproducible and internally consistent, so the verdict should remain unchanged rather than being upgraded or downgraded.","tokens_in":16355,"tokens_out":1737,"duration_ms":15608,"concrete_test":"Recompute the small-k energy spectrum from a full electroweak field-theory simulation (or a controlled model with a dynamical gauge field) using the same lattice resolution and box size as in Fig. 3, and compare EM(k) ∝ k^s with s determined from a run at N=1024 without subtraction of the Higgs-gradient term. If s deviates from 4 by more than ~0.2 in the causal range k < k_H, the central spectral claim is incomplete; if s=4 is reproduced including ∇×A dynamics, the k^4 result is robust within the modeled dynamics and the remaining caveat is only the normalization/coherence assumptions in Eqs. (34)-(35).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central new spectral result is EM(k) ∝ k^4 at small k, obtained by keeping only the Higgs-gradient term in Eq. (1) and by constructing a divergence-free field from plaquette fluxes. The weakest load-bearing premise is that the final, divergence-free electromagnetic field after string decay is fully captured by the non-dynamical plaquette algorithm: in Sec. IV the magnetic field is defined as the curl of the Higgs-derived gauge field A in Eq. (7), with the Maxwellian ∇×A contribution in Eq. (1) set to zero, and the conversion of Z-strings into electromagnetic flux is imposed geometrically rather than derived. The authors explicitly flag this in Sec. VI ('we have ignored the contribution of the A term in (1)'), but the size of that term relative to the Higgs-gradient term at the coherence scale is not estimated. If the gauge-field contribution were comparable anywhere in the causal small-k range, the predicted slope and, more importantly, the present-epoch amplitudes in Eqs. (34)-(35) would not be the full EWSB prediction. In addition, Appendix B only excludes n=3 and proves n≥4; the sharp k^4 result rests on the numerics combined with that inequality. The MHD projection to present-epoch fields then assumes a 10% magnetic energy fraction, horizon-scale coherence, and the recently proposed Hosking-integral decay laws with universal prefactors (Table I), none of which carry quoted error bars; the stated B ~ 10^-13 G and B ~ 10^-10 G are therefore conditional estimates rather than robust predictions. None of this contradicts the internal consistency of the numerical spectral calculation, but the full physical claim of the abstract is conditional on external inputs.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies magnetic fields generated during electroweak symmetry breaking via a non-dynamical, Kibble-mechanism-based lattice algorithm. Using only the Higgs-gradient term of the electromagnetic field (Eq. 6) and a plaquette flux construction (Eqs. 13-16), it obtains a magnetic energy spectrum EM(k) ∝ k^4 at small wavenumbers (Fig. 3, Eq. 24), together with a nearly flat compensated helicity-variance spectrum k^-2 Sp(h) (Fig. 4). Appendix B revises an earlier analytic argument that gave k^3 and shows consistency with n ≥ 4. The paper then combines this initial spectrum with MHD decay laws and several astrophysical assumptions to estimate present-epoch fields: B ~ 10^-13 G on kpc scales for non-helical fields, B ~ 10^-10 G on Mpc scales for maximally helical fields, and interpolating scalings for partial helicity (Eqs. 34-35, 42).","tokens_in":16659,"tokens_out":4186,"duration_ms":40081,"significance":"If the spectral prediction holds, the paper corrects a long-standing expectation of EM(k) ∝ k^3 for electroweak-symmetry-breaking magnetic fields. The k^4 slope matters for the small-scale behavior of primordial fields and for the subsequent MHD evolution that determines present-epoch amplitudes. Strengths of the paper include the convergence of the numerical result over 256^3, 512^3, and 1024^3 meshes, the self-contained random-walk argument in Appendix B, the explicit treatment of the divergence-free condition, the open data and code availability statement, and the clear acknowledgment of several modeling limitations. The present-epoch field strengths, however, are conditional on a chain of assumptions—the dropped ∇×A term, the non-dynamical string-conversion model, horizon-scale initial coherence, and universal MHD prefactors—so the headline estimates should be read as order-of-magnitude scenarios rather than firm predictions.","major_comments":[{"comment":"The spectral prediction is computed from the Higgs-gradient term alone, with the ∇×A term in Eq. (1) set to zero. This is explicitly acknowledged as a limitation in Sec. VI, but no estimate is given for the relative magnitude of the omitted term on the small-k scales that anchor Eq. (24) and the later MHD evolution. If the gauge-field contribution is comparable on these scales, the central k^4 claim and the amplitudes in Eqs. (34)-(35) are not the full electroweak-symmetry-breaking prediction. The authors should either bound the neglected term using the dynamical simulations of Refs. [2-4] or state more narrowly that the result applies only to the Higgs-gradient contribution.","section":"Sec. III (Eq. (6)) and Sec. VI"},{"comment":"The plaquette flux algorithm imposes geometrically the conversion of Z-strings into electromagnetic flux rather than deriving it from the dynamics of string decay. The paper cites Ref. [14] for this conversion, but it does not compare the large-scale spectrum produced by the algorithm with the spectrum obtained from a full dynamical electroweak simulation. Because the small-k slope is the central result, a validation of the algorithm against even one dynamical simulation, or an explicit argument for why the conversion dynamics cannot affect the slope, is needed.","section":"Sec. IV (Eqs. (13)-(16))"},{"comment":"The analytical random-walk argument excludes n = 3 and requires n ≥ 4, but it does not uniquely determine n = 4. The paper should state this explicitly: the sharp k^4 result rests on the convergence of the numerical spectra in Fig. 3 combined with the inequality, not on Appendix B alone. The current wording 'we must have E_M(k) ∝ k^n for n ≥ 4' and the subsequent claim that 'E_M ∝ k^3 does not follow' do not make this dependence on the numerics clear.","section":"Appendix B (Eq. (B9))"},{"comment":"The present-epoch numbers in Eqs. (34)-(35) depend on the assumed 10% magnetic energy fraction, horizon-scale initial coherence, and the universal prefactors collected in Table I, several of which are marked 'somewhat uncertain' in Sec. V.E. No error bars are attached to these inputs, so the quoted 10^-13 G and 10^-10 G values are conditional estimates. The authors should either propagate uncertainties from Table I and the coherence-scale assumption, or present these numbers as illustrative order-of-magnitude scenarios rather than as firm predictions.","section":"Secs. V.B-V.E"}],"minor_comments":[{"comment":"The symbol γ is used both for the temporal growth exponent in Eq. (27) and for a Hopf angle in Eq. (12); the text notes the clash, but renaming one of the two would improve readability.","section":"Sec. V.B (Eq. (27))"},{"comment":"The normalization of the compensated spectra is described only in the captions; adding the normalization integrals explicitly in the main text would help readers reproduce the amplitudes.","section":"Figs. 3 and 4"},{"comment":"The symbol V is used both for the lattice volume in the discrete spectral relations and for the averaging volume in the analytical argument of Appendix B; please distinguish these two uses.","section":"Eqs. (19)-(22) and Eq. (B1)"},{"comment":"The derivation of τ* = τ_EW r_h,EW^{-3/2} from conservation of magnetic helicity omits the intermediate step connecting the helicity at τ* to the initial helicity; one explanatory sentence would make the result easier to follow.","section":"Sec. V.C (Eq. (39))"},{"comment":"Reference [9] is cited both as the source of the earlier k^3 estimate and as a basis for the corrected argument in Appendix B; splitting the citations would make the corrected derivation easier to attribute.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is potentially a valuable correction to a widely quoted spectral exponent, and the numerical part is reproducible and openly documented. The main risk is scope: the title claims spectra from electroweak symmetry breaking, while the calculation excludes one of the two terms in Eq. (1) and models string conversion non-dynamically. I would like to see the authors either quantify the omitted contributions or explicitly narrow the scope of the claim in the abstract and conclusions."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this paper fixes a real error in the literature. The non-dynamical Kibble calculation shows the magnetic energy spectrum from electroweak symmetry breaking goes as k^4 at small k, not the k^3 claimed in Ref. [9]. The numerical convergence across 256^3, 512^3, and 1024^3 runs is convincing, and the corrected analytic argument in Appendix B shows where the old k^3 estimate went wrong. The result is parameter-free and the code and data are public, so the central spectral claim deserves to be taken seriously.\n\nWhat is genuinely new: the plaquette-flux algorithm (Eq. 16) that includes string contributions to enforce div B = 0, the k^4 energy spectrum in Fig. 3, the nearly k^-2 compensated helicity variance spectrum in Fig. 4, and the corrected random-walk argument that rules out k^3 and forces n >= 4. None of this involves hidden fitting or circular reasoning. The paper is honest about what it does and does not do.\n\nTwo soft spots, both acknowledged in the text. First, the sharp n = 4 exponent is not proven analytically; Appendix B only excludes k^3 and establishes n >= 4, so the exact value rests on the numerics. That is a minor weakness because the numerics are clean and converged. Second, the paper sets the curl-of-A term in Eq. (1) to zero and models Z-string conversion geometrically rather than dynamically. The authors flag this in Sec. VI but do not estimate how large that neglected term is at small k. If it contributes comparably, the k^4 slope is not the full electroweak symmetry breaking prediction. I see this as a legitimate limitation, not a fatal flaw: the paper frames itself as a symmetry-based, non-dynamical estimate and is transparent about the conditionality.\n\nThe present-epoch field strengths, B ~ 10^-13 G on kpc scales and B ~ 10^-10 G on Mpc scales, inherit assumptions about the 10% magnetic energy fraction, horizon-scale coherence, and MHD decay prefactors with no quoted error bars. Again, this is stated clearly, but readers should treat those numbers as order-of-magnitude estimates, not robust predictions. The partially helical scaling in Sec. V.C is a useful interpolation between the two extremes.\n\nWho this is for: anyone working on primordial magnetogenesis, initial conditions for MHD evolution, or the Kibble mechanism. It deserves a serious referee because the central result is reproducible and it corrects a published claim. My recommendation: send it to peer review. I would ask the authors to quantify the neglected curl-of-A term if they can and to give uncertainty ranges on the MHD prefactors, but neither should block acceptance.","headline":"Corrects the published k^3 slope for electroweak-symmetry-breaking magnetic fields to k^4 using a clean, parameter-free Kibble algorithm; the cosmological amplitudes are conditional on openly acknowledged inputs.","tokens_in":17295,"tokens_out":2122,"would_cite":true,"duration_ms":22598,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["98.80.Cq","12.15.-y"],"model":"deepseek-v4-flash","headline":"Electroweak-symmetry-breaking magnetic fields scale as k⁴ on large scales, not k³; fed through magnetohydrodynamic decay they predict ~10⁻¹³ G on kpc scales today, or ~10⁻¹⁰ G on Mpc if maximally helical.","keywords":["electroweak symmetry breaking","primordial magnetic fields","Kibble mechanism","magnetic helicity","Hosking integral","magnetohydrodynamic decay","inverse cascade","cosmological magnetogenesis"],"falsifier":"Run a fully dynamical electroweak field-theory simulation with enough dynamic range to resolve the small-wavenumber part of the spectrum after symmetry breaking, keeping both terms of Eq. (1) rather than only the Higgs-gradient term: if the energy spectrum at the largest resolved scales comes out as $k^3$ — or anything other than $k^4$ — the central claim fails. A cheaper check is the paper's own variance diagnostic: the volume-averaged field must fall as $B_V \\propto \\lambda^{-2}$ on large volumes, whereas a $\\lambda^{-4}\\ln\\lambda$ falloff would signal the old $k^3$ spectrum.","tokens_in":16059,"feed_emoji":"🧲","tokens_out":15796,"duration_ms":125525,"temperature":0.7,"pith_summary":"The paper claims that magnetic fields created when the electroweak symmetry broke have an energy spectrum $E_M(k) \\propto k^4$ on large length scales, not the $k^3$ scaling previously assumed, and that the distinction matters: the small-$k$ slope sets how the field's peak migrates and decays through the cosmic plasma. The claim is obtained from Kibble-mechanism arguments rather than expensive dynamical simulations — random Higgs orientations on the vacuum sphere in horizon-size domains guarantee magnetic fields, and a new plaquette-flux algorithm converts the resulting monopole–Z-string network into a divergence-free field whose spectrum can be measured at any resolution. Alongside the $k^4$ energy spectrum sits a nearly flat compensated helicity-variance spectrum $k^{-2}S_p(h)$, giving the field well-defined initial helicity fluctuations with zero mean. Combined with conserved-quantity scaling laws for magnetohydrodynamic turbulence, these spectra yield concrete present-epoch numbers: $B \\sim 10^{-13}$ G with kpc coherence for non-helical fields, $B \\sim 10^{-10}$ G with Mpc coherence for maximally helical fields, and interpolating scalings for partial helicity. A sympathetic reader would care because these are among the few ab initio, parameter-light predictions for the strength and scale of intergalactic magnetic fields, and because the correction changes results that multiple cosmological magnetogenesis estimates have built on.","feed_headline":"Primordial magnetic fields obey a k⁴ spectrum, not k³","feed_subtitle":"Kibble-mechanism simulations set the slope; MHD decay gives 10⁻¹³ G on kpc scales, or 10⁻¹⁰ G on Mpc if helical.","key_machinery":"The argument is carried by three linked mechanisms. First, the Kibble-mechanism vacuum: during electroweak symmetry breaking the Higgs vacuum expectation value takes independent random orientations on the vacuum manifold $S^3$ in domains of roughly horizon size, so $\\nabla\\Phi$ is generically nonzero and the Higgs-gradient term of Eq. (6) produces magnetic field. Second, the plaquette-flux algorithm (Eqs. 13–16): the associated gauge field $A = -i(2\\sin\\theta_w/g\\eta^2)\\Phi^\\dagger\\nabla\\Phi$ is integrated around each lattice plaquette, and because the line integral picks up both the $+2\\pi$ flux of the embedded magnetic monopoles and the $-2\\pi$ flux of the Z-strings (Dirac strings) connecting them, the resulting field is exactly divergence-free — this is what lets a static, non-dynamical calculation stand in for the full electroweak evolution that converts Z-strings into electromagnetic flux. Third, conserved-quantity MHD decay scaling: the Hosking integral $I_H$ (conserved for non-helical decay, giving the peak envelope exponent $\\epsilon = 3/2$) and the mean magnetic helicity $I_M$ (conserved for maximally helical decay, $\\epsilon = 0$) fix how the spectral peak moves to larger scales, and the corrected volume-averaging identity $B_V^2 \\propto \\lambda^{-4}$ (Appendix B) converts the numerical $k^4$ slope into an analytic statement.","core_discovery":"Stated on the paper's own terms, the discovery is that the magnetic field produced by electroweak symmetry breaking has a $k^4$ energy spectrum at small wavenumbers, together with a helicity-variance spectrum $S_p(h) \\propto k^2$ and vanishing mean helicity in the absence of CP violation. The field is computed from the Higgs-gradient term $\\mathbf{B} = -i(2\\sin\\theta_w/g\\eta^2)\\,\\nabla\\Phi^\\dagger \\times \\nabla\\Phi$ on a lattice of independently random Higgs orientations (the Kibble mechanism), with the magnetic field obtained by integrating the associated gauge field around plaquettes; this flux algorithm automatically includes the $+2\\pi$ monopole flux and the $-2\\pi$ Z-string flux that cancels it, so the resulting field is exactly divergence-free and represents the state after monopole annihilation and string conversion. The paper further shows that its earlier analytical estimate of $k^3$ was wrong: evaluating the volume-averaged field with the correct window function gives $B_V^2 \\propto \\lambda^{-4}$, which is compatible only with a magnetic energy spectrum rising at least as $k^4$. Evolving the $k^4$ initial conditions through magnetohydrodynamic decay — with the Hosking integral conserved for non-helical fields ($\\epsilon = 3/2$) and mean helicity conserved for maximally helical fields ($\\epsilon = 0$) — yields peak fields $B \\sim 10^{-13}$ G at $k_{\\rm phys} \\sim (1\\,{\\rm kpc})^{-1}$ for non-helical fields and $B \\sim 10^{-10}$ G at $k_{\\rm phys} \\sim (1\\,{\\rm Mpc})^{-1}$ for maximally helical fields, with partially helical cases interpolating as $k_B \\sim (1\\,{\\rm Mpc})^{-1} r_{h,EW}^{-1/3}$ and $B \\sim 10^{-10} r_{h,EW}^{1/3}$ G.","pith_inferences":["The $k^4$ small-$k$ slope is the steepest allowed for a causally generated magnetic field, so this result places electroweak symmetry breaking at the causal limit: a testable consequence is that a larger dynamical simulation should find the spectrum peaking cleanly at the horizon scale, with no extra power at smaller wavenumbers.","The same Kibble-mechanism-plus-plaquette-flux pipeline could be applied to other symmetry-breaking transitions with comparable vacuum manifolds; if the $k^4$ slope is a property of random vacuum orientation rather than electroweak specifics, analogous spectra should appear in any model with a similar vacuum structure.","If a future dynamical calculation keeps the neglected gauge-field term $\\nabla\\times A$ of Eq. (1), the slope could shift or the prefactor could change while the $k^4$ shape survives — a checkable extension is to evaluate Eq. (1) in full on the same random-Higgs ensembles used here.","The paper's correction of its predecessor's volume-averaging argument suggests that other $k^3$ results quoted in the cosmological magnetogenesis literature may need auditing wherever they rest on the same window-function estimate."],"forward_implications":["Estimates of cosmological magnetic fields built on the earlier $k^3$ initial spectrum — including normalized amplitudes taken from field-theory simulations — should be re-derived with $k^4$ initial conditions, which is a steeper rise into the peak and changes how much power sits at the largest scales.","The predicted present-epoch fields ($10^{-13}$ G on kpc scales non-helical; $10^{-10}$ G on Mpc maximally helical; interpolated values for partial helicity when the initial relative helicity exceeds $10^{-9}$) sit within current observational bounds, so the scenario survives existing constraints while remaining potentially detectable.","Because the peak scale and amplitude after decay are independent of the initial spectral slope $s$ (for $s > \\epsilon$), the field-strength forecasts are robust to the exact small-$k$ slope chosen; the headline numbers stand even if the detailed shape of the initial spectrum varies.","The nearly flat $k^{-2}S_p(h)$ helicity-variance spectrum means the initial conditions carry a well-defined Hosking integral $I_H$, so non-helical decay is governed by a conserved quantity and the late-time peak evolution has no free parameters in its scaling form.","If the initial relative helicity exceeds $10^{-9}$ the field becomes maximally helical before the present epoch and the coherence scale grows to Mpc, switching the decay law mid-course; below that threshold the field decays as if non-helical."],"supporting_citations":[{"why":"Establishes that the Higgs field acquiring its vacuum expectation value generates a primordial magnetic field and supplies the electromagnetic field definition in Eq. (1) that the paper's Eq. (6) is built from.","marker":"[1]"},{"why":"The earlier review whose volume-averaging estimate of a $k^3$ spectrum is corrected in Appendix B; also supplies normalization and observational-bounds context.","marker":"[9]"},{"why":"Introduces the Kibble mechanism of random vacuum orientations in causally disconnected domains, the premise for the lattice of independent Higgs directions.","marker":"[10]"},{"why":"Establishes the Kibble-mechanism network of electroweak magnetic monopoles joined by Z-strings that the numerical algorithm reproduces.","marker":"[13]"},{"why":"Shows annihilating electroweak dumbbells convert Z-string flux into electromagnetic flux, the physical process the plaquette-flux algorithm mimics.","marker":"[14]"},{"why":"Defines the Hosking integral $I_H$ and establishes it as the conserved quantity governing non-helical MHD decay, the basis of the $\\epsilon = 3/2$ scaling.","marker":"[20]"},{"why":"Supplies the classes of hydromagnetic turbulent decay and the scaling exponents that connect the initial $k^s$ spectrum to late-time growth.","marker":"[24]"},{"why":"Provides the inverse-cascade scaling for initial spectra between Saffman and Batchelor, fixing the exponent $\\gamma = 10/9$ used in Eq. (27) and the universal prefactor comparisons.","marker":"[25]"},{"why":"Gives the decay laws for hydromagnetic turbulence from the electroweak phase transition, the source of the non-helical $\\epsilon = 3/2$ envelope and peak tracking behind Eqs. (30)–(34).","marker":"[26]"}],"fun_headline_variants":["Electroweak fields get k⁴ spectrum, not k³","Magnetic fields from symmetry breaking: k⁴ slope, 10⁻¹³ G on kpc","Kibble fields yield k⁴ spectrum; MHD decay gives 10⁻¹³ G","Higgs-gradient fields show k⁴, helical case reaches 10⁻¹⁰ G"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The prediction rests on the premise that the surviving magnetic field comes entirely from spatial variations of the Higgs field's direction — the electromagnetic gauge-field term $\\nabla\\times A$ is set to zero, and the conversion of Z-strings into electromagnetic flux is handled by a static flux rule rather than the actual electroweak dynamics — so if the omitted pieces generate a comparable large-scale field, the $k^4$ slope and the $10^{-13}$ G and $10^{-10}$ G numbers are not the full electroweak prediction.","fun_headline_variants_meta":{"raw":{"variants":["Electroweak fields get k⁴ spectrum, not k³","Magnetic fields from symmetry breaking: k⁴ slope, 10⁻¹³ G on kpc","Kibble fields yield k⁴ spectrum; MHD decay gives 10⁻¹³ G","Higgs-gradient fields show k⁴, helical case reaches 10⁻¹⁰ G"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000251,"raw_usage":{"total_tokens":1653,"prompt_tokens":1135,"completion_tokens":518,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":751,"completion_tokens_details":{"reasoning_tokens":422}},"tokens_in":751,"tokens_out":518,"duration_ms":5539,"temperature":1.0,"reasoning_tokens":422,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T05:09:43.455605+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run a fully dynamical electroweak field-theory simulation with enough dynamic range to resolve the small-wavenumber part of the spectrum after symmetry breaking, keeping both terms of Eq. (1) rather than only the Higgs-gradient term: if the energy spectrum at the largest resolved scales comes out as $k^3$ — or anything other than $k^4$ — the central claim fails. A cheaper check is the paper's own variance diagnostic: the volume-averaged field must fall as $B_V \\propto \\lambda^{-2}$ on large volumes, whereas a $\\lambda^{-4}\\ln\\lambda$ falloff would signal the old $k^3$ spectrum.","supporting_citations":[],"review_version":1}