{"id":"02439cf3-42e9-47e9-8431-83466dd11a3e","arxiv_id":"2610.41386","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Including spin polarization from thermal vorticity and thermal shear in an AMPT simulation splits the asymmetry fluctuations and the eccentricity-asymmetry correlation ratios away from spinless values, with system-size and energy dependent patterns.","lead":"This preprint uses the AMPT transport model to test whether particle spin polarization, driven by swirling and shearing motion inside the collision fireball, changes the event-by-event fluctuations of the initial geometry. It reports that spin polarization splits the computed asymmetry fluctuations and their correlation with eccentricity away from spinless values, with patterns that depend on system size, collision energy, and temperature.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek/deepseek-v4-flash","headline":"Eq. (5), the factor that creates the entire SP splitting, is not derivable from the paper's Eq. (4): the Pauli term is traceless and the factor's Π=0 limit mismatches the stated normalization, so the reported patterns may be an artifact of the inserted ansatz.","rationale":"The reader's CONDITIONAL verdict is reasonable but stops at the weakest point rather than naming it precisely. The central claim is a model-level prediction, so the absence of experimental data does not by itself invalidate it. The load-bearing element is the spin-measure factor in Eq. (5): it is the only place where polarization enters the energy-momentum tensor, and therefore it generates the splitting that all subsequent claims inherit. The manuscript does not connect Eq. (5) to its own spin-distribution ansatz in Eq. (4). A literal trace of Eq. (4) annihilates the linear Pauli term, and the zero-polarization limit of Eq. (5) disagrees with the stated normalization \\int dS=2. This is an internal consistency problem, not a disagreement with external consensus. The missing numerical recipe for extracting u^μ, ϖ, ξ, and Π from AMPT further prevents any independent check of the figures. The authors honestly note that a detailed numerical analysis is forthcoming and that future observables must still be identified; that is credited, but it does not repair Eq. (5). If the authors can supply a correct derivation of Eq. (5) and recompute the figures, the qualitative pattern might survive; until then the central numerical claim is unverdictable rather than merely conditional on improved statistics.","tokens_in":13172,"tokens_out":14568,"duration_ms":99001,"concrete_test":"Independently compute the spin integral \\int dS [\\delta_{mn}+\\Pi\\cdot\\sigma_{mn}] from Eq. (6) and compare it with Eq. (5); also evaluate both sides at Π=0 under the stated \\int dS=2. If the derivation fails or the zero-polarization limits differ, recompute Figs. 1–4 using the direct trace of Eq. (4), which gives no polarization dependence for the energy-momentum tensor, and using the corrected normalization of Eq. (5). If δÃ/δA or C̃_n/C_n changes sign, magnitude, or system-size ordering, the claimed effect is an artifact of the factor rather than of spin polarization as parametrized in Eq. (4).","verdict_should_be":"UNVERDICTED","load_bearing_attack":"All quantitative claims in Figs. 1–4 depend on the polarization-dependent factor in Eq. (5). As written, Eq. (5) does not follow from the model defined in Eq. (4). The trace over spin indices in Eq. (3) — the double sum reduces to the diagonal sum — applied to \\tilde f_{mn}=f[\\delta_{mn}+\\Pi\\cdot\\sigma_{mn}] gives 2f, because the Pauli matrices are traceless; the linear polarization term contributes nothing. At Π=0, the left side of Eq. (5) is (1/2)\\sum_m \\int dS = \\int dS = 2 under the paper's own statement that \\int dS=2, while the right side is 2 - \\sinh(0)/0 = 1. Thus the printed factor is normalization-inconsistent and is not the trace of Eq. (4). Its small-Π expansion, 1 - (χΠ)^2/6 + ..., determines the sign and magnitude of δÃ/δA and of C̃_n/C_n; no derivation from Eq. (4) and Eq. (6) is supplied. If Eq. (5) is instead a quoted result from ref. [28], the manuscript must show the mapping from the Pauli-matrix distribution to that scalar measure. Without this, the 'significant splitting' is an inserted property of the ansatz, and the vorticity/shear competition arguments in Secs. III–IV explain trends but do not validate the factor. A secondary reproducibility gap compounds this: the extraction of u^μ, ϖ_{ρσ}, ξ_{ρσ}, and Π from AMPT at each proper time is never specified, so the numerical implementation of even the intended Eq. (5) cannot be checked from the text.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek/deepseek-v4-flash","summary":"The paper studies whether spin polarization (SP) induced by thermal vorticity and thermal shear modifies event-by-event fluctuations of the asymmetry observable in heavy-ion collisions. Using the AMPT model for O+O, Cu+Cu, and Au+Au collisions at sqrt(s_NN) = 7-200 GeV, the authors compute the asymmetry A, its event-by-event fluctuation delta-A, and correlation functions C_n between the participant eccentricity and the asymmetry, with and without a spin-dependent phase-space distribution. They report that the SP-corrected quantities split from the spinless ones, that the splitting grows with proper time and follows the system-size hierarchy O > Cu > Au, that the correlation ratios C-tilde_n/C_n stay near unity at high temperature but deviate, sometimes non-monotonically, below T approximately 0.12 GeV, and that incorporating SP is therefore essential for modeling asymmetry fluctuations.","tokens_in":13572,"tokens_out":9390,"duration_ms":64972,"significance":"The underlying question is timely and potentially useful: spin polarization is normally studied through final-state hyperon polarization, and using it as a probe of early-time event-by-event geometry fluctuations would be a new model-level constraint. The qualitative pattern reported here, if correct, is internally coherent: growth with proper time, larger effects in smaller systems, and ratios near unity at high temperature with deviations at low temperature. The authors also state several assumptions explicitly, including the neglect of spin polarization at tau = 0.4 fm, and the paper is framed as a model prediction rather than as a data comparison, which is legitimate. However, the central quantitative claim currently rests on an internally inconsistent spin-measure formula, and the numerical procedure is not documented well enough to be checked. Because the reported splitting is controlled by the disputed factor, the significance cannot be assessed in the present form.","major_comments":[{"comment":"Equation (5) is presented as the evaluation of the spin trace in Eq. (3), but it is not: the Pauli-matrix term is traceless, so the linear-in-Pi contribution vanishes, and at Pi = 0 the left-hand side of Eq. (5) equals (1/2) sum_m integral dS = 2 under the paper's own statement that integral dS = 2, while the right-hand side equals 1. The small-Pi expansion of Eq. (5), 1 - (chi Pi)^2/6, determines the sign and magnitude of every reported splitting in Figs. 1-4, so this is not a normalization detail. The authors should either derive Eq. (5) from Eq. (4) or quote it explicitly as a standalone result of ref. [28] and specify the exact mapping to the trace in Eq. (3); without that, the central claim of a significant splitting is an inserted property of the ansatz.","section":"Sec. II B, Eqs. (3)-(5)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The numerical implementation is not reproducible from the text. The paper does not specify how the fluid four-velocity u^mu, the local temperature T, the tensors varpi_rho sigma and xi_rho sigma, and the polarization vector Pi are extracted from AMPT at each proper time, nor how the energy density epsilon is mapped through T = (pi^2 epsilon / 72)^{1/4}. The statement in Sec. III that a more detailed numerical analysis will be reported in a forthcoming study is an acknowledgment (sentence b66), but it does not replace the need to document the procedure behind the results that are actually shown, since every figure depends on this extraction.","section":"Sec. III, Figs. 1-4","authors_acknowledge":"A more de- tailed numerical analysis will be reported in a forthcoming study.","authors_acknowledge_p":0.65},{"comment":"The claims of a 'significant' splitting (b3) and of splitting that 'grows progressively with time' (b51) are made without any statistical quantification. No event counts, error bars, or significance tests are reported for delta-A, delta-A-tilde, or C-tilde_n/C_n in Figs. 1-4. With event-by-event fluctuations calculated over a broad centrality and multiplicity range, the reported differences need associated uncertainties before 'significant' can be supported.","section":"Abstract and Sec. IV","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The construction of the temperature-dependent ratios is underspecified. The text says the results are integrated over proper time from tau = 0.8 to 5 fm, but it does not state how the temperature binning is performed, how the ratio is weighted, or how the threshold T > 0.12 GeV is defined. Because the low-temperature non-monotonic behavior is a central qualitative result, this procedure must be described before the temperature dependence can be evaluated.","section":"Sec. III, Figs. 3-4","authors_acknowledge":null,"authors_acknowledge_p":null}],"minor_comments":[{"comment":"The square-bracket notation [T^{xx}] used in Eqs. (2), (10), and (11)-(12) is never defined; it should be stated explicitly to denote the intended average over the transverse plane or over events.","section":"Sec. II A, Eq. (2)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The sentence 'observables whose sensitivity to initial state features is sensitivity and whose dependence on subsequent hydrodynamic evolution is insensitivity' is garbled and should be rewritten.","section":"Sec. I","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"In the version under review, the axis labels and some captions contain unreadable placeholder tokens such as '/s49/s48/s48'; the figures need to be regenerated with readable axis labels and legends.","section":"Fig. 1-4 captions and labels","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"There are several small typographical errors: 'asymmetric' in Sec. II A should be 'asymmetry'; 'These ratios monotonically' in Sec. III is missing a verb; and 'relative relativistic heavy-ion collisions' in Sec. IV appears to be a typo.","section":"Secs. II and IV","authors_acknowledge":null,"authors_acknowledge_p":null}],"recommendation":"major_revision","confidential_remarks":"For the editor: the main risk is Eq. (5), because the reported pattern is controlled by a factor that is not derivable from the model as stated and has an inconsistent zero-polarization limit. I recommend major revision rather than rejection because the underlying idea is interesting and the qualitative trends could survive a corrected derivation; however, if the corrected spin trace changes the sign or magnitude of the quadratic coefficient in the small-Pi expansion, the conclusions will need to be reconsidered."},"author_rebuttal":null,"desk_editor":{"model":"deepseek/deepseek-v4-flash","letter":"Colleague,\n\nQuick take: this is a preliminary model study with a plausible qualitative idea — feed a spin-polarization correction into the asymmetry built from the energy-momentum tensor and see how the event-by-event fluctuations move. That exact computation, in AMPT, for O-O, Cu-Cu, Au-Au, as a function of energy, proper time, and temperature, does not appear in the spin literature they cite. So there is a genuinely new calculation here, and the qualitative pattern they report is internally coherent: splitting grows with time, is largest for O-O, and the correlation ratios sit near unity at high temperature before deviating at low T. For a first pass, that is worth something.\n\nWhat I cannot sign off on is the numerical support. There are no event counts, no error bars, no significance tests. The abstract's word \"constrains\" overstates what a model-out only paper does. And the stress-test note is right about Eq. (5): the claimed factor 2 − sinh(χΠ)/(χΠ) does not follow from Eq. (4) by tracing over the Pauli matrices. The trace of the linear term vanishes, and at Π = 0 the factor gives 1 while the left side should give 2. The entire splitting is second order in Π and its sign and magnitude are set by that inserted factor. The paper leans on ref. [28] for the spin measure, but never shows the mapping from the Pauli-matrix distribution to that scalar measure. Maybe the factor is a legit quoted result, but the manuscript does not demonstrate it, and without that the quantitative claims float on the ansatz.\n\nAlso soft, and acknowledged by the authors: the numerical recipe. They say a detailed analysis will come later, and it shows — the extraction of u, ϖ, ξ, Π from AMPT at each proper time is never specified. That is a reproducibility gap, not a fatal one for a field where follow-up papers routinely fill in details, but it means the central numbers cannot be checked from this text.\n\nWho gets value from this paper? Practitioners working on spin polarization in heavy-ion collisions who want a quick sense of whether a VSIP-type correction could move fluctuation observables. As a call for attention to a new observable family, it is useful. As a claim of a constraint on initial-state physics, it is not ready.\n\nMy recommendation: send it to peer review. The idea deserves referee time — a serious referee can force the derivation of Eq. (5), request error bars and event statistics, and push for a reproducible numerical recipe. If the factor survives scrutiny, the result is a legitimate model-level contribution; if it does not, the paper needs major revision before publication.","headline":"A preliminary AMPT-based model study claiming spin polarization visibly shifts asymmetry fluctuations and eccentricity-asymmetry correlation ratios, but the central numerical effects rest on a spin-measure factor whose derivation from the stated distribution function is not shown.","tokens_in":14237,"tokens_out":682,"would_cite":false,"duration_ms":7553,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":[],"pacs":["25.75.-q","24.70.+s","24.60.Ky"],"model":"deepseek/deepseek-v4-flash","headline":"This paper claims that adding spin polarization to the parton distribution function in the AMPT model splits the event-by-event asymmetry fluctuation, with the gap following the system-size hierarchy O > Cu > Au.","keywords":["spin polarization","thermal vorticity","thermal shear","event-by-event fluctuations","initial state","asymmetry","heavy-ion collisions","kinetic theory"],"falsifier":"Recompute $\\delta\\tilde{A}/\\delta A$ and $\\tilde{C}_n/C_n$ with the shear term removed (setting $\\Omega = \\varpi$ only) or with a different spin-measure functional form: if the splitting changes sign or vanishes, the reported effect is an artifact of the parametrization. Alternatively, measure the eccentricity-asymmetry correlation ratios in O+O collisions at $\\sqrt{s_{NN}} \\approx 7.7$ GeV and compare against the predicted size hierarchy.","tokens_in":16329,"feed_emoji":"🌀","tokens_out":3252,"duration_ms":47468,"temperature":0.7,"pith_summary":"This is a model-level prediction, not a measurement. The paper claims that including spin polarization induced by thermal vorticity and thermal shear in the parton distribution function shifts the event-by-event fluctuation of the asymmetry observable, $\\delta A$, to a new value $\\delta \\tilde{A}$, and moves the eccentricity-asymmetry correlation ratios $\\tilde{C}_n/C_n$ (n = 1, 2, 3) away from unity. The gap grows with proper time, is largest for the smallest systems (O > Cu > Au), depends only weakly on collision energy, and stays near unity at temperatures above 0.12 GeV while deviating, often non-monotonically, below that. If correct, the claim would make spin polarization a new model-level constraint on early-time geometry fluctuations, which experimental data have been too sparse to pin down. No experimental data enter the paper.","feed_headline":"Spin polarization splits early heavy-ion fluctuations","feed_subtitle":"Thermal-vorticity spin in the AMPT model moves asymmetry correlation ratios away from unity, sharply below 0.12 GeV.","key_machinery":"The mechanism is the spin-dependent phase-space distribution $\\tilde{f}_{mn} = f[\\delta_{mn} + \\Pi\\cdot\\sigma_{mn}]$ (Eq. 4), paired with the invariant spin measure $(1/2)[2 - \\sinh(\\chi\\Pi)/(\\chi\\Pi)]$ (Eq. 5) and the linear VSIP relation $\\Pi \\approx \\Omega$ with $\\Omega_{\\rho\\sigma} = \\varpi_{\\rho\\sigma} + 2\\hat{\\tau}pE\\,\\xi_{\\rho\\sigma}$ (Eq. 9), where $\\varpi$ is thermal vorticity and $\\xi$ is thermal shear. Because the Pauli-matrix term vanishes under the spin trace, the entire splitting enters only at second order in the polarization vector $\\Pi$, with its sign and magnitude set by the functional form of the spin-measure factor and by the debated shear-coupling term. These feed a spin-corrected energy-momentum tensor $\\tilde{T}^{\\mu\\nu}$ (Eq. 3), from which the SP-dependent asymmetry $\\tilde{A}$ and its fluctuation $\\delta\\tilde{A}$ are computed and compared against the spinless $T^{\\mu\\nu}$, $A$, and $\\delta A$ in the AMPT-generated events.","core_discovery":"The central claim is that spin polarization, specifically the combined thermal-vorticity and thermal-shear contribution (VSIP), produces a splitting of the asymmetry fluctuation $\\delta \\tilde{A}$ relative to the spinless $\\delta A$, and shifts the ratios $\\tilde{C}_n/C_n$ (n = 1, 2, 3) of the correlation functions between the participant eccentricity and the asymmetry. The paper reports that the splitting grows progressively with proper time, obeys the system-size hierarchy O > Cu > Au, shows only weak collision-energy dependence, and exhibits strong temperature dependence: the ratios sit near unity for T > 0.12 GeV and deviate below that, monotonically in lighter systems and non-monotonically in heavier or higher-energy systems. The authors conclude that incorporating spin polarization is essential for modeling fluctuations in asymmetry observables and that it offers a sensitive quantitative probe of event-by-event initial-state fluctuations.","pith_inferences":["If the splitting is real, early-proper-time spin observables could act as a discriminator between competing initial-geometry models, because the reported gap grows with time and follows a clean size hierarchy.","A clean theoretical control test would recompute $\\delta\\tilde{A}/\\delta A$ and $\\tilde{C}_n/C_n$ with the shear term $\\xi$ dropped (setting $\\Omega = \\varpi$ only): since the effect enters only at second order in $\\Pi$, a sign change or vanishing of the splitting would reveal it as an artifact of the parametrization.","The low-temperature non-monotonicity is attributed to Cooper-Frye freeze-out hypersurface contributions; a hydrodynamics-only calculation omitting that term should behave differently, providing an independent check of the mechanism.","Because the sign and magnitude of the splitting are set by the debated shear coupling and the specific spin-measure form, a single experimental measurement of asymmetry-fluctuation ratios in small systems at low energy could select among parametrizations."],"forward_implications":["The SP-corrected asymmetry fluctuation $\\delta\\tilde{A}$ splits from the spinless $\\delta A$, and the gap grows progressively with proper time, so early-time observables carry the largest spin effect.","The splitting follows the system-size hierarchy O > Cu > Au, with O-O collisions suppressing fluctuations while larger systems show the non-monotonic enhancement-then-suppression pattern.","The correlation-function ratios $\\tilde{C}_n/C_n$ (n = 1, 2, 3) exhibit linear energy dependence and a system-size-dependent inflection at later times.","In the high-temperature regime T > 0.12 GeV the ratios stay near unity; below that they deviate, monotonically in lighter or lower-energy systems and non-monotonically in heavier or higher-energy ones.","The authors conclude that incorporating spin polarization is essential for modeling asymmetry-observable fluctuations and that SP can serve as a constraint on pre-equilibrium dynamics and early-time symmetry breaking."],"supporting_citations":[{"why":"The AMPT multi-phase transport model, which supplies all simulated events for the O+O, Cu+Cu, and Au+Au collisions studied throughout.","marker":"[2]"},{"why":"Supplies the parametrized spin-dependent distribution $\\tilde{f}_{mn} = f[\\delta_{mn} + \\Pi\\cdot\\sigma_{mn}]$ used in Eq. (4).","marker":"[16]"},{"why":"Source of the thermal-vorticity and thermal-shear definitions in Eqs. (7) and (8).","marker":"[26]"},{"why":"Source of the invariant spin-measure treatment and the factor $2 - \\sinh(\\chi\\Pi)/(\\chi\\Pi)$ in Eq. (5).","marker":"[28]"},{"why":"Source of the combined relation $\\Omega_{\\rho\\sigma} = \\varpi_{\\rho\\sigma} + 2\\hat{\\tau}pE\\,\\xi_{\\rho\\sigma}$ in Eq. (9) coupling the antisymmetric stress-energy tensor with thermal shear.","marker":"[35]"},{"why":"The proposal of spin correlation functions as a probe of initial-state spin density fluctuations, which motivates using SP to constrain initial-state physics.","marker":"[24]"},{"why":"The system-size-dependent spin-polarization ordering P(O) > P(Cu) > P(Au) that the paper invokes to explain the size hierarchy of the splitting.","marker":"[47]"},{"why":"The local-temperature formula $T = (\\pi^2\\epsilon/72)^{1/4}$ used to convert energy density into temperature in the analysis.","marker":"[34]"}],"fun_headline_variants":["Spin polarization splits asymmetry fluctuations event by event","Thermal vorticity spin splits heavy-ion asymmetry ratios","Spin polarization splits fluctuation ratios below 0.12 GeV","Spin-induced splitting marks initial-state fluctuations in heavy ions"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The entire splitting is manufactured by the assumed spin parametrization, including the linear VSIP relation with its debated shear-coupling term and the specific spin-measure factor, and no experimental data enter anywhere to anchor or verify it.","fun_headline_variants_meta":{"raw":{"variants":["Spin polarization splits asymmetry fluctuations event by event","Thermal vorticity spin splits heavy-ion asymmetry ratios","Spin polarization splits fluctuation ratios below 0.12 GeV","Spin-induced splitting marks initial-state fluctuations in heavy ions"]},"model":"deepseek/deepseek-v4-flash","effort":"low","cost_usd":0.000127,"raw_usage":{"cost":0.000126854,"is_byok":false,"cost_details":{"upstream_inference_cost":0.000126854,"upstream_inference_prompt_cost":2.73e-05,"upstream_inference_completions_cost":9.9554e-05},"total_tokens":1507,"prompt_tokens":960,"completion_tokens":547,"prompt_tokens_details":{"audio_tokens":0,"video_tokens":0,"cached_tokens":825,"cache_write_tokens":0},"completion_tokens_details":{"audio_tokens":0,"image_tokens":0,"reasoning_tokens":496}},"tokens_in":960,"tokens_out":547,"cost_source":"openrouter_usage","duration_ms":5601,"image_pages":[],"temperature":1.0,"billed_attempts":1,"reasoning_tokens":496,"transport_provider":"openrouter","cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-11-02T15:21:33.039266+00:00","model_set":{"reader":"deepseek/deepseek-v4-flash"},"falsifier":"Recompute $\\delta\\tilde{A}/\\delta A$ and $\\tilde{C}_n/C_n$ with the shear term removed (setting $\\Omega = \\varpi$ only) or with a different spin-measure functional form: if the splitting changes sign or vanishes, the reported effect is an artifact of the parametrization. Alternatively, measure the eccentricity-asymmetry correlation ratios in O+O collisions at $\\sqrt{s_{NN}} \\approx 7.7$ GeV and compare against the predicted size hierarchy.","supporting_citations":[{"cited_title":"A Multi-Phase Transport Model for Relativistic Heavy Ion Collisions","cited_arxiv_id":"nucl-th/0411110","evidence_quote":"The AMPT multi-phase transport model, which supplies all simulated events for the O+O, Cu+Cu, and Au+Au collisions studied throughout."},{"cited_title":"Generating spin polarization from vorticity through nonlocal collisions","cited_arxiv_id":"2005.01506","evidence_quote":"Supplies the parametrized spin-dependent distribution $\\tilde{f}_{mn} = f[\\delta_{mn} + \\Pi\\cdot\\sigma_{mn}]$ used in Eq. (4)."},{"cited_title":"Relativistic distribution function for particles with spin at local thermodynamical equilibrium","cited_arxiv_id":"1303.3431","evidence_quote":"Source of the thermal-vorticity and thermal-shear definitions in Eqs. (7) and (8)."},{"cited_title":"Relativistic hydrodynamics for spin-polarized fluids","cited_arxiv_id":"1811.04409","evidence_quote":"Source of the invariant spin-measure treatment and the factor $2 - \\sinh(\\chi\\Pi)/(\\chi\\Pi)$ in Eq. (5)."},{"cited_title":"Spin-thermal shear coupling in a relativistic fluid","cited_arxiv_id":"2103.10917","evidence_quote":"Source of the combined relation $\\Omega_{\\rho\\sigma} = \\varpi_{\\rho\\sigma} + 2\\hat{\\tau}pE\\,\\xi_{\\rho\\sigma}$ in Eq. (9) coupling the antisymmetric stress-energy tensor with thermal shear."},{"cited_title":"Initial-state-driven spin correlations in high-energy nuclear collisions","cited_arxiv_id":"2502.13102","evidence_quote":"The proposal of spin correlation functions as a probe of initial-state spin density fluctuations, which motivates using SP to constrain initial-state physics."},{"cited_title":"$\\Lambda$ spin polarization in event-by-event relativistic heavy-ion collisions","cited_arxiv_id":"2203.15718","evidence_quote":"The system-size-dependent spin-polarization ordering P(O) > P(Cu) > P(Au) that the paper invokes to explain the size hierarchy of the splitting."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The local-temperature formula $T = (\\pi^2\\epsilon/72)^{1/4}$ used to convert energy density into temperature in the analysis."}],"review_version":1}