{"id":"9daac775-bb91-45c6-8a64-0601790328d9","arxiv_id":"2607.26912","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"fRG critical fluctuations on hydrodynamic freeze-out hypersurfaces yield non-monotonic net-proton C4/C2 versus collision energy, absent in the HRG baseline.","lead":"The paper folds critical-point fluctuations from functional renormalization group calculations into realistic hydrodynamic freeze-out surfaces and computes net-proton cumulant ratios at RHIC beam energies. The critical case produces a non-monotonic C4/C2 energy dependence that the non-critical baseline lacks and that tracks STAR data better than a single freeze-out curve.","discovery_kind":"new_application","skeptic_critique":{"model":"grok-4.5","headline":"Eq. (4)–(10) split of net-baryon critical χ_n into (anti)baryons remains the load-bearing untested step for the proton C4/C2 claim.","rationale":"The reader correctly isolated the weakest internal link. The hydro surface vs single Andronic curve and the HRG control on the same surface are genuine methodological advances and cleanly isolate that something in the fRG critical input drives the low-energy C4/C2 structure. Those controls do not, however, validate the net→(anti)baryon→proton map under acceptance. Critical slowing down and cell independence are important physics caveats for the whole static-encoding class, but they are shared with the prior fRG-on-freeze-out-curve work; the new, paper-specific load-bearing step that turns net-baryon fRG output into the STAR-comparable proton ratios is Eqs. (4) and (10). A concrete alternate-partition test would settle whether the non-monotonic proton signal is robust or an artifact of the HRG-ratio split. Until that (or equivalent) check exists, CONDITIONAL with moderate confidence remains the right verdict—no upgrade to ACCEPT, no downgrade to REJECT.","tokens_in":15099,"tokens_out":756,"duration_ms":49613,"concrete_test":"Recompute net-proton C4/C2 (and C3/C2) at 7.7, 14.5, and 19.6 GeV on the same hypersurfaces under two alternate maps: (i) assign the full critical excess χ_n^B−χ_n^{B,HRG} to net baryons only, with HRG used solely for the regular part before isospin/acceptance/SAM; (ii) put all critical excess into baryons (χ^{B+}_n) and none into antibaryons, appropriate at high μ_B. If either map removes the non-monotonic energy dependence or closes the gap to the HRG+hydro baseline within the plot resolution, Eqs. (4)+(10) are load-bearing for the headline claim.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central qualitative claim—that fRG critical input on the hydro particlization surface produces a non-monotonic net-proton C4/C2 absent in the HRG+hydro baseline—still passes through Eqs. (4) and (10). Only net-baryon χ_n^B are available from fRG; (anti)baryon susceptibilities in each cell are defined by rescaling with HRG ratios, baryon–antibaryon correlations are set to zero, and protons are then obtained by binomial isospin filtering (q=1/2) plus acceptance and SAM. The paper states that exactness holds only when all acceptance cuts are removed. At √s_NN=7.7 GeV, where the non-monotonicity appears, the hypersurface spans large μ_B (Fig. 1) and the critical excess in χ_4^B is large; how that excess is partitioned between B+ and B− before proton filtering is therefore not a peripheral detail but the step that converts a net-baryon critical signal into the reported proton cumulant ratios. No alternate partition, no critical-regime validation of the HRG ratio ansatz, and no theory band on this choice are given. If the qualitative non-monotonicity or the gap to the HRG baseline is sensitive to that partition, the strongest claim weakens from a CEP-driven proton observable to a net-baryon statement dressed by an untested map.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper computes net-proton cumulant ratios C2/C1, C3/C2 and C4/C2 on (3+1)D hydrodynamic particlization hypersurfaces for Au+Au collisions at nine beam energies √sNN=7.7–200 GeV. Net-baryon susceptibilities that include both regular and CEP-driven critical fluctuations are taken from prior fRG work and assigned to each fluid cell; STAR pT and rapidity acceptance, isospin randomization (q=1/2), and global baryon conservation via the subensemble acceptance method are then applied. Results are compared with an HRG+hydro baseline on the same surfaces and with earlier fRG evaluations along a single Andronic freeze-out curve. The central claim is that critical input produces a non-monotonic collision-energy dependence in C4/C2 that is absent in the non-critical baseline and is closer to STAR BES-II data than the single-curve calculation.","tokens_in":15465,"tokens_out":1241,"duration_ms":20859,"significance":"If the mapping from net-baryon critical susceptibilities to proton cumulants is reliable, the work supplies a concrete, experimentally comparable prediction that critical fluctuations on a realistic multi-cell freeze-out surface generate a non-monotonic C4/C2 signal. Combining calibrated hydrodynamics with first-principles-inspired susceptibilities, acceptance cuts, isospin filtering and SAM conservation is a clear methodological advance over single-point freeze-out evaluations. The HRG baseline on identical surfaces cleanly isolates the critical contribution, and the acceptance-dependence study in the supplement strengthens the qualitative claim. These elements make the paper a useful step toward quantitative CEP searches even if further validation of intermediate assumptions is required.","major_comments":[{"comment":"Eqs. (4) and (10) constitute the load-bearing step that converts fRG net-baryon χ n^B into (anti)baryon and then net-proton cumulants. Critical (anti)baryon susceptibilities are defined by rescaling the fRG net-baryon result with HRG (anti)baryon-to-net ratios, and baryon–antibaryon correlations are set to zero. The manuscript itself notes that exactness holds only when all acceptance cuts are removed. At √sNN=7.7 GeV, where the non-monotonicity appears, the hypersurface spans a wide μ B range (Fig. 1) and the critical excess in χ4^B is large; the partition of that excess between B+ and B− before isospin filtering therefore directly controls the reported proton C4/C2. No alternate partition, no critical-regime validation of the HRG-ratio ansatz, and no theory band on this choice are provided. A sensitivity test (or an explicit statement that the qualitative non-monotonicity survives reas","section":"Critical fluctuations encoded on the freeze-out hypersurface (Eqs. 4, 10)"},{"comment":"Each fluid element is assumed to be independently equilibrated at its local T and μ B (text preceding Eq. 3). Near the CEP the correlation length can become comparable to or larger than typical cell sizes, so the independent-cell grand-canonical assignment may overestimate the critical contribution that survives after acceptance and SAM. The paper does not estimate the correlation length relative to the hypersurface granularity or discuss how finite-size/critical slowing-down effects would modify the mapped cumulants. Even a qualitative argument or a reference to existing estimates would clarify whether this approximation is under control at the lowest energies.","section":"Method paragraph before Eq. (3); Fig. 1"}],"minor_comments":[{"comment":"Results at √sNN=9.2 and 11.5 GeV are omitted because the hydrodynamic background is less well constrained; a short quantitative remark on residual uncertainty at the neighboring 7.7 and 14.5 GeV points would help the reader judge the robustness of the non-monotonic feature.","section":"Results and discussion"},{"comment":"The constant switching density ε fo=0.26 GeV/fm3 is fixed for all energies. A brief check (or citation to prior work) showing that modest variations of ε fo do not erase the non-monotonicity in C4/C2 would strengthen the presentation.","section":"Results and discussion; Fig. 2 caption"},{"comment":"Figure 1 gray region (χ4^B/χ2^B not computed) overlaps part of the low-energy hypersurface; a sentence on how cells falling into that region are treated would remove ambiguity.","section":"Fig. 1 and surrounding text"},{"comment":"Typographical inconsistencies appear in the figure labels (e.g., C2=C1 versus C2/C1) and in the arXiv header date; these should be standardized.","section":"Fig. 2 and supplement figures"}],"recommendation":"major_revision","confidential_remarks":"The central qualitative result is interesting and the hydro+acceptance+SAM machinery is solid, but the untested Eq. (4)/(10) map is the single point on which the proton-level claim rests. I would not recommend acceptance until the authors either supply a sensitivity study or clearly demote the claim to a net-baryon statement dressed by an explicit ansatz. The paper is otherwise within scope for a serious nuclear-theory journal."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The new piece is straightforward and useful: they take fRG net-baryon susceptibilities (with the CEP) and fold them cell-by-cell onto MUSIC/iS3D particlization hypersurfaces that were already calibrated to bulk hadrons across the BES range, then apply STAR pT/y cuts, isospin randomization, and SAM conservation. The HRG baseline on the exact same surfaces cleanly isolates the critical contribution. That combination did not exist before; prior fRG work sat on a single Andronic (T,μB) point and prior hydro work used only HRG.\n\nWhat they get is sensible. Low-order ratios barely move. C4/C2 develops a non-monotonic energy dependence only when the critical input is switched on, and the hydro version sits closer to the STAR BES-II points than the old freeze-out-curve fRG curve, which overshot the dip. Acceptance dependence is checked in the supplement and the non-monotonicity survives. Citations and the hydro calibration look solid; the CEP location is consistent with the recent functional/lattice consensus.\n\nThe soft spot is real and load-bearing, but it is the one the authors already flag. Only net-baryon χn come from fRG. They rescale to (anti)baryons with HRG ratios (Eq. 4), drop B–antiB correlations (Eq. 10), then binomial-filter to protons. Exactness is guaranteed only with no acceptance cuts. At 7.7 GeV the surface spans a wide μB band where the critical excess is large, so how that excess is partitioned before proton filtering matters. No alternate split or theory band is shown. Equilibrium freeze-out also skips critical slowing-down, and there are no theory error bars. Those are the right caveats; they do not make the qualitative critical-vs-HRG contrast disappear, but they keep the curves from being quantitative CEP evidence yet.\n\nThis is for people already working the RHIC/FAIR fluctuation program who need a more realistic embedding of first-principles susceptibilities. It deserves a serious referee. I would engage with it, cite the hydro+fRG comparison, and push on the partition assumption in follow-up.","headline":"Solid methods paper that puts fRG critical susceptibilities on calibrated hydro freeze-out surfaces and gets a cleaner non-monotonic C4/C2 than the old freeze-out-curve approach; the (anti)baryon split is the real soft spot but does not erase the advance.","tokens_in":16113,"tokens_out":564,"would_cite":true,"duration_ms":9396,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.5","headline":"Critical fluctuations placed on hydrodynamic freeze-out surfaces produce a non-monotonic net-proton kurtosis versus beam energy that the non-critical baseline does not.","keywords":["net-proton fluctuations","QCD critical end point","hydrodynamics","functional renormalization group","cumulant ratios","baryon conservation","heavy-ion collisions"],"falsifier":"High-statistics net-proton C4/C2 at the lowest BES and fixed-target energies that remains monotonic, or the same hydro-plus-fRG pipeline with a freeze-out energy density or equation of state still consistent with bulk hadrons that erases the non-monotonicity, would falsify the claimed critical imprint.","tokens_in":15955,"feed_emoji":"⚛️","tokens_out":926,"duration_ms":36712,"temperature":0.7,"pith_summary":"This paper asks whether a QCD critical end point can still leave a visible mark in net-proton number fluctuations once those fluctuations are embedded in a realistic heavy-ion collision. The authors take baryon-number susceptibilities from the functional renormalization group, including both ordinary and critical contributions, and assign them cell by cell to the particlization hypersurface of calibrated viscous hydrodynamics at nine RHIC energies. Experimental momentum and rapidity cuts, isospin randomization from baryons to protons, and exact global baryon conservation are applied. Low-order cumulant ratios barely change when critical fluctuations are switched on, but the kurtosis ratio C4/C2 develops a non-monotonic energy dependence at low beam energy that is absent in the pure hadron-resonance-gas baseline and is closer to STAR measurements than earlier calculations that used a single chemical freeze-out curve. A sympathetic reader cares because this is a concrete bridge between first-principles critical physics and the actual observables used to hunt for the critical end point.","feed_headline":"Critical point leaves non-monotonic kurtosis after hydro freeze-out","feed_subtitle":"fRG fluctuations on realistic hypersurfaces, not a single freeze-out curve, track STAR net-proton C4/C2","key_machinery":"Cell-by-cell encoding of fRG susceptibilities on the hydro particlization hypersurface: each fluid element contributes a local volume times χ_n(T,µ_B), followed by binomial acceptance, isospin filtering to protons, hypersurface summation, and canonical correction via the subensemble acceptance method.","core_discovery":"When fRG net-baryon susceptibilities that include critical end-point fluctuations are evaluated on the hydrodynamic freeze-out hypersurface, then filtered by STAR acceptance, isospin randomization and subensemble baryon conservation, the net-proton ratio C4/C2 shows a non-monotonic dependence on collision energy at low √s_NN; the identical pipeline without critical fluctuations does not, and the critical result is more comparable to STAR BES-II data than fRG evaluated on a simple freeze-out curve.","pith_inferences":["Once fixed-target points are added, residual data tension can jointly constrain both the critical-end-point location and the freeze-out energy density.","Because the critical excess grows with cumulant order, sixth-order ratios computed the same way would be a sharper experimental discriminant if measured.","The cell-by-cell hydro encoding is a reusable template for inserting any other first-principles susceptibility set into the same dynamical background."],"forward_implications":["A critical end point can still imprint non-monotonic C4/C2 after realistic hydrodynamics, acceptance cuts and baryon conservation.","Evaluating critical susceptibilities on a single freeze-out curve overstates the non-monotonic signal relative to a full hypersurface.","Low-order ratios such as C2/C1 remain dominated by non-critical physics, acceptance and conservation.","The same pipeline extended below 7.7 GeV is a direct next test of the critical contribution."],"fun_headline_variants":["Critical fRG modes leave non-monotonic net-proton C4/C2 on hydro freeze-out","Hydro particlization preserves CEP-driven non-monotonic kurtosis in net-protons","C4/C2 non-monotonic only with critical fluctuations on realistic hydro surface","fRG critical susceptibilities on hydro hypersurface track STAR net-proton C4/C2","Baseline without CEP stays monotonic; critical hydro C4/C2 does not"],"cache_read_input_tokens":128,"weakest_assumption_plain":"Critical fluctuations for baryons and antibaryons separately are obtained by rescaling the theoretically known net-baryon susceptibilities with ordinary hadron-gas ratios, and baryon–antibaryon correlations are neglected.","fun_headline_variants_meta":{"raw":{"variants":["Critical fRG modes leave non-monotonic net-proton C4/C2 on hydro freeze-out","Hydro particlization preserves CEP-driven non-monotonic kurtosis in net-protons","C4/C2 non-monotonic only with critical fluctuations on realistic hydro surface","fRG critical susceptibilities on hydro hypersurface track STAR net-proton C4/C2","Baseline without CEP stays monotonic; critical hydro C4/C2 does not"]},"model":"grok-4.5","effort":"low","cost_usd":0.004456,"raw_usage":{"total_tokens":1319,"prompt_tokens":810,"num_sources_used":0,"completion_tokens":98,"cost_in_usd_ticks":44564000,"prompt_tokens_details":{"text_tokens":810,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":411,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":810,"tokens_out":98,"duration_ms":7346,"temperature":1.0,"reasoning_tokens":411,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-30T17:14:42.507679+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"High-statistics net-proton C4/C2 at the lowest BES and fixed-target energies that remains monotonic, or the same hydro-plus-fRG pipeline with a freeze-out energy density or equation of state still consistent with bulk hadrons that erases the non-monotonicity, would falsify the claimed critical imprint.","supporting_citations":[],"review_version":1}