{"id":"7a350abb-ce40-406e-84f9-117a1e0eb8f6","arxiv_id":"2502.05030","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":9,"one_line_summary":"For Schrödinger-Poisson eigenstates, the support grows quadratically with excitation index n, the amplitude decay exponent approaches -1, and properly rescaled eigenvelocity profiles collapse to a common shape.","lead":"This paper computes highly excited spherically symmetric Schrödinger-Poisson eigenstates up to n=80 and proposes scaling laws for their size, oscillation amplitude, and velocity profiles, including a collapse onto a universal curve after rescaling. Its value is mainly for the mathematics of the Choquard equation and for dark matter models that use excited states, though the results are empirical fits rather than derivations.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The universal rescaling claim rests on a visual collapse; no quantitative residual is reported, and normalizing by each curve's outermost extremum makes the collapse partially self-fulfilling.","rationale":"The reader's weakest assumption concerns numerical accuracy up to n=80, and that is indeed a necessary condition for every fitted law. I agree with that concern but identify a more direct risk to the paper's headline claim: the universal collapse is asserted from a single figure without a quantitative spread, and the normalizers are fit to the same data they are used to collapse. Any polynomial or power-law fit with enough parameters can make curves pass near a common point at the outermost extremum; the physically meaningful statement requires that the whole rescaled profile converge, and that convergence needs to be measured. The proposed residual test and the alternative normalization test would settle whether the collapse is intrinsic or an artifact of the chosen scaling point. This does not change the reader's conditional verdict—it strengthens the conditionality—so the verdict is unchanged.","tokens_in":12835,"tokens_out":10346,"duration_ms":118718,"concrete_test":"Compute, from independent or the authors' solutions for n=20, 40, 60, 80, the rescaled curves V_n(R)=v_n(r)/vtilde_2n with R=r/rtilde_2n, and report the maximum absolute deviation from a reference curve (e.g., V_80) on R in [0.1, 0.9]; if this residual does not decrease with n and reach a few percent, the claimed universality is not established. Additionally, repeat the rescaling using the second-outermost extremum (rtilde_{2n-2}, vtilde_{2n-2}) instead of the outermost; if the collapse degrades substantially, the universality is an artifact of the chosen normalization point.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim—universal shape of eigenvelocities after rescaling (Eq. 13)—is supported only by the visual collapse in Fig. 13, with no residual metric. Since the normalizing constants rtilde_2n(n) and vtilde_2n(n) are themselves least-squares fits (Eqs. 12a-12b) to the same n<=80 dataset, the collapse is not an independent test: systematic errors or post-hoc choices in the fits can be absorbed into the rescaling. Moreover, the linear mid-range fits (Eq. 10) exclude the first and last two extrema; sigma(n) and hence the expected rescaled shape depend on this arbitrary cut, so the claimed convergence of slopes to zero (Eq. 11) and the resulting universality are not shown to be robust.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper numerically computes spherically symmetric stationary states of the Schrödinger-Poisson equation up to excitation index n=80 and proposes heuristic scaling laws: the effective support rhat_n(n) ~ 131 n^2, the outer node z_n(n) and outer nodal distance d_{n-1}(n) with parabolic fits, the amplitude modulation |f_i| = b(n) rhat_i^{a(n)} with a(n) approaching -1, the mid-range eigenvelocity slope sigma(n) ~ n^{-2.86}, and the outer extremum scaling vtilde_{2n}(rtilde_{2n}) = 0.27 rtilde_{2n}^{-0.5}. The paper's central claim is that all eigenvelocity curves collapse onto a single universal shape after the rescaling R = r/rtilde_{2n}(n), V = v/vtilde_{2n}(n) defined in Eq. (13).","tokens_in":13123,"tokens_out":3677,"duration_ms":41622,"significance":"If the claimed universal rotation-curve shape holds, it would provide a compact empirical characterization of highly excited Schrödinger-Poisson eigenstates and could inform studies of multimodal dark-matter configurations. The paper is honest that the laws are heuristic, and it documents numerical validation through grid refinements. However, the universal collapse is not an independent prediction: the normalizing quantities in Eq. (13) are least-squares fits to the same data being collapsed, and no residual or scatter metric is reported. The absence of error bars on fitted coefficients, post hoc fit-region choices, and the lack of code or data make the central claims plausible but not yet quantitatively established. With targeted robustness and reproducibility analyses, the result would be a useful contribution; as it stands, the evidence for universality is largely visual.","major_comments":[{"comment":"The claimed universal collapse is partly constructed: the rescaling uses rtilde_{2n}(n) and vtilde_{2n}(n) obtained from the same eigenvelocity curves through the fits (12a)-(12b), so systematic fit errors or post hoc choices in those fits are absorbed into the normalization. The collapse is supported only visually, with no quantitative residual or spread metric. Please report the RMS deviation of the rescaled curves from a common shape as a function of n, and provide an independent test, e.g., fit (12) on n ≤ 50 and examine whether the predicted collapse holds for n = 60,...,80.","section":"§3.5, Eq. (13) and Fig. 13"},{"comment":"The amplitude exponent law a(n) = -1 + 0.24 n^{-0.25} depends on post hoc fit choices: amplitudes are fitted only up to 0.95 r_min, the first and last extremum are excluded, and the onset n ≥ 20 is selected after inspecting the data. No error bars or goodness-of-fit measures are reported for a(n) and b(n), so the asymptotic statement a → -1 is an extrapolation from a fit region that was itself chosen from the data. Please provide uncertainties on the fitted coefficients and a sensitivity analysis with respect to the cutoff 0.95 r_min and the onset index.","section":"§3.4, Eqs. (8)-(9)"},{"comment":"The numerical accuracy of the eigenstates is described only qualitatively as 'confirming reliability up to n = 80'. Because every heuristic law in Eqs. (5)-(12) is a fit to these numerical data, the paper should report concrete convergence diagnostics, including grid sizes, tolerances, eigenvalue errors, and node or extremum position errors as functions of n. Releasing the code or data, even as supplementary material, would substantially strengthen the reproducibility of the empirical claims.","section":"§2.1"}],"minor_comments":[{"comment":"There is a typo in 'demostrated' near the discussion of Tod and Moroz; please correct it.","section":"§1.1"},{"comment":"The definition of v_n(r) omits the 4π factor from angular integration; the omission is stated, but it would be clearer to also note that this is an overall normalization convention that does not affect the scaling or universality claims.","section":"§3.5, Eq. (2)"},{"comment":"The parabolic fits for rhat_n(n) and z_n(n) are inconsistent for small n (e.g., at n=1 the fitted z_1 exceeds the fitted rhat_1, although the outermost node must lie inside the outermost extremum); please state the range of n for which each fit is intended to be valid.","section":"§3.3, Eqs. (5) and (7a)"},{"comment":"The caption refers to 'Red regions' to indicate excluded points, but the figure may not be colorblind-safe; please add hatching or a grayscale-readable marker.","section":"Fig. 7"}],"recommendation":"major_revision","confidential_remarks":"The stress-test concern about the self-fulfilling collapse in Fig. 13 lands: the normalizing constants in Eq. (13) are fits to the same data, so the collapse is partly constructed. The paper is otherwise a reasonable empirical study, and the requested quantitative residual analysis, sensitivity checks, and numerical reproducibility measures are within the scope of a revision. I would not recommend rejection, but I would urge the editor to require either code/data release or detailed convergence tables, since the entire paper rests on numerical data."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Good paper to know about if you work on Schrödinger-Poisson/Choquard states or on dark matter profile models. It does something the literature hasn't done: a systematic numerical sweep of excited states up to n=80, with explicit scaling laws for support, nodal spacings, amplitude exponent, and eigenvelocity slopes. The parabolic support is a nice concrete observation, and the amplitude exponent tending to −1 is consistent with the old sin(r)/r approximation while the nodal spacing pattern is a genuine correction to it. The authors are honest throughout that these are heuristic fits, not theorems; they also say they checked grid refinement and excluded n>80 for accuracy.\n\nThe soft spot is the universality claim. The collapse in Fig. 13 is produced by rescaling every curve by its own fitted outermost extremum (Eq. 13), so perfect collapse is partly built in. There is no residual metric, no scatter plot, no test on a held-out subset. On top of that, the fits themselves have no error bars, and the fit regions involve post hoc choices (amplitudes only up to 0.95 rmin, first and last two extrema excluded, onset n≥20). None of these are fatal for a heuristic study, but they are exactly the details a referee would want pinned down before trusting the asymptotics.\n\nMy overall read: the individual scaling laws are plausible and useful, and the large-n trends (slopes → 0, exponent → −1) are consistent with the known r−1 approximation. The paper will be a useful reference for the community. What it does not do is prove universality; it conjectures it. For that you need residuals, error bars, and ideally an independent check (for example, fitting on half the n range and predicting the other half).\n\nI'd send it to a serious referee. The numerics are nontrivial and extend prior work; the authors are careful about what they claim. The referee should ask for code/data, uncertainty estimates, and a quantitative collapse metric — but the paper deserves the round trip, not a desk reject.","headline":"Useful heuristic scaling laws for excited Schrödinger-Poisson states, but the universal collapse claim needs real residuals and error bars before it convinces.","tokens_in":13599,"tokens_out":2422,"would_cite":true,"duration_ms":26929,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q41","35Q55","81Q05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Highly excited Schrödinger-Poisson eigenstates obey scaling laws that collapse all their rotation curves onto one universal shape.","keywords":["Schrödinger-Poisson","Choquard equation","excited stationary states","eigenvelocities","rotation curves","scaling laws","universality","numerical eigenstates"],"falsifier":"Compute eigenstates for $n=81$ through $n=200$ with an independent high-accuracy solver and test whether equations (5)-(12) hold and whether the rescaled curves from (13) continue to collapse; if the points drift from the fitted curves or the curves spread apart, the universality claim is refuted.","tokens_in":12649,"feed_emoji":"🌀","tokens_out":9514,"duration_ms":93986,"temperature":0.7,"pith_summary":"Using numerical solutions of the spherically symmetric Schrödinger-Poisson system up to excitation index $n=80$, the paper establishes heuristic scaling laws for excited eigenstates. The effective support of the matter density grows quadratically with $n$, nodal spacings follow a regular pattern that converges after normalization, and the oscillation amplitude decays as a power law whose exponent approaches $-1$. From these eigenfunctions the paper defines eigenvelocities, the tangential speeds of test particles in circular orbits, and shows that their mid-range slope decays to zero as $n^{-2.86}$. The central result is that, after rescaling radius and velocity by the outermost eigenvelocity extremum ($\\tilde r_{2n}(n)=133n^2+245n-185$ and $\\tilde v_{2n}=0.27\\tilde r_{2n}^{-1/2}$), all numerically computed eigenvelocity profiles collapse onto one universal curve. The result matters because it gives the Schrödinger-Poisson model an intrinsic, $n$-independent rotation-curve shape that can be compared with observed galaxy rotation curves.","feed_headline":"Excited Schrödinger-Poisson states share one universal rotation curve","feed_subtitle":"Numerical eigenstates up to n=80 obey scaling laws that give every velocity profile the same shape after rescaling.","key_machinery":"The load-bearing construct is the eigenvelocity $v_n(r)=\\sqrt{(\\int_0^r f_n^2(s)s^2\\,ds)/r}$ and the two-point rescaling built from the outermost extremum of $v_n$. The paper's heuristic laws for the outermost radius, $\\tilde r_{2n}(n)=133n^2+245n-185$, and the outermost velocity, $\\tilde v_{2n}=0.27\\tilde r_{2n}^{-1/2}$, define the dimensionless variables $R=r/\\tilde r_{2n}$ and $V=v/\\tilde v_{2n}$; the claim is that plotting $V$ against $R$ erases the dependence on $n$ and exposes the universal shape. The underlying eigenfunctions come from the Choquard stationary problem, the nonlinear eigenvalue equation obtained after eliminating the Poisson potential, solved numerically on a progressively extended domain.","core_discovery":"The paper's central claim is that highly excited spherically symmetric stationary states of the Schrödinger-Poisson problem are governed by quantitative scaling laws in the excitation index $n$, and that their associated eigenvelocities are universal after a one-parameter rescaling. Specifically, the outermost radius of the eigenvelocity, $\\tilde r_{2n}(n)=133n^2+245n-185$, and the velocity at that point, $\\tilde v_{2n}(\\tilde r_{2n})=0.27\\,\\tilde r_{2n}^{-1/2}$, define the rescaling $R=r/\\tilde r_{2n}$, $V=v/\\tilde v_{2n}$, under which the computed profiles for different $n$ collapse onto a common curve. The same analysis yields parabolic support growth, a regular nodal-distance pattern, power-law amplitude decay with exponent $a(n)=-1+0.24n^{-0.25}$, and mid-range velocity slopes $\\sigma(n)=2.82\\times10^{-5}n^{-2.86}$ that vanish in the large-$n$ limit. These are presented as heuristic laws inferred from numerical data, not as proven theorems.","pith_inferences":["If the collapse is exact beyond the fitted range, the master curve could be tabulated once and used as a template for multimodal dark-matter halos built from many excited states, avoiding repeated numerical solution of the Choquard equation.","The model's norm-scaling invariance suggests the universal shape may survive renormalization of the total mass, so the same rescaled curve could apply across very different physical mass scales; the paper does not itself make this claim.","The fitted correction $0.24n^{-0.25}$ means plain $r^{-1}$ behavior is only approached slowly; at $n=80$ the exponent is still about $-0.92$, so finite-$n$ rotation curves should retain noticeable residual slope and oscillations even after rescaling."],"forward_implications":["In the large-$n$ limit the eigenfunction amplitudes become $f_n(r)\\sim r^{-1}$, which plugged into the eigenvelocity definition gives an approximately flat mid-range curve; flat rotation-curve plateaux are thus a natural asymptotic feature of the model.","The mid-range slope decays as $\\sigma(n)=2.82\\times10^{-5}n^{-2.86}$, so the flattening is quantitative: higher excited states produce flatter plateaux at a predictable rate.","The universal rescaling (13) reduces every computed eigenvelocity profile to one common shape, giving an $n$-independent curve intrinsic to the Schrödinger-Poisson model.","The parabolic laws for support, outermost node, and outermost eigenvelocity radius connect the excited-state structure to Bohr/Kepler-type scaling, offering a concrete target for analytic derivations."],"supporting_citations":[{"why":"Establishes existence and uniqueness of the minimizing solution of the Choquard equation, grounding the ground state of the Schrödinger-Poisson problem.","marker":"[17]"},{"why":"Proves the existence of the infinite discrete family of spherically symmetric stationary states with excitation index $n$ that the paper analyzes.","marker":"[18]"},{"why":"First connected Schrödinger-Poisson eigenvelocities to observed galactic rotation curves; the paper extends this connection with scaling laws.","marker":"[19]"},{"why":"Supplies the numerical implementation, with two nested iterative procedures, used to compute all eigenstates up to $n=80$.","marker":"[20]"},{"why":"Proposes the $\\sin(r)/r$ density approximation whose irregular nodal spacing and amplitude decay the paper's laws correct.","marker":"[40]"}],"fun_headline_variants":["Schrödinger-Poisson eigenstates reveal universal rotation curve","Excited states collapse onto one scaling curve after rescaling","Universal velocity profile emerges from Schrödinger-Poisson scaling","Scaling laws make Schrödinger-Poisson rotation curves universal","Highly excited Schrödinger-Poisson states obey universal scaling"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"All fitted laws and the claimed universal collapse rest on the numerical solver accurately resolving eigenstates up to $n=80$; no convergence proof or error metrics are reported, and the large-$n$ statements extrapolate from that range.","fun_headline_variants_meta":{"raw":{"variants":["Schrödinger-Poisson eigenstates reveal universal rotation curve","Excited states collapse onto one scaling curve after rescaling","Universal velocity profile emerges from Schrödinger-Poisson scaling","Scaling laws make Schrödinger-Poisson rotation curves universal","Highly excited Schrödinger-Poisson states obey universal scaling"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000198,"raw_usage":{"total_tokens":1367,"prompt_tokens":940,"completion_tokens":427,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":556,"completion_tokens_details":{"reasoning_tokens":344}},"tokens_in":556,"tokens_out":427,"duration_ms":4255,"temperature":1.0,"reasoning_tokens":344,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T20:31:46.871385+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute eigenstates for $n=81$ through $n=200$ with an independent high-accuracy solver and test whether equations (5)-(12) hold and whether the rescaled curves from (13) continue to collapse; if the points drift from the fitted curves or the curves spread apart, the universality claim is refuted.","supporting_citations":[{"cited_title":"Sin, Late-time phase transition and the galactic halo as a Bose liquid, Phys","cited_arxiv_id":null,"evidence_quote":"First connected Schrödinger-Poisson eigenvelocities to observed galactic rotation curves; the paper extends this connection with scaling laws."},{"cited_title":"Bernstein, E","cited_arxiv_id":null,"evidence_quote":"Supplies the numerical implementation, with two nested iterative procedures, used to compute all eigenstates up to $n=80$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proposes the $\\sin(r)/r$ density approximation whose irregular nodal spacing and amplitude decay the paper's laws correct."}],"review_version":1}