{"id":"2b16af6e-ac93-4ac7-ae5d-01d2d67f25f1","arxiv_id":"2411.18355","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Analytic zeroth-order dimensional perturbation theory for D-dimensional anisotropic vortex Bose-Einstein condensates yields general-D density, chemical potential, energy, critical vortex speed, and a Thomas-Fermi limit.","lead":"This paper derives approximate formulas for spinning atom clouds (Bose-Einstein condensates) in traps with any number of spatial dimensions, extending known three-dimensional results. The formulas could help analyze experiments that use synthetic dimensions to mimic higher-dimensional physics.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The α = κ/(4μ) second-order expansion behind Eqs. (16), (24), (27) is uncontrolled for the plotted parameters (α ≤ 1/2, often O(0.3–0.5)), so the μD-curves and dimensional crossings may be truncation artifacts.","rationale":"The paper makes a coherent analytic extension of DPT to anisotropic hypercylindrical traps, and the D = 3 reductions correctly reproduce known results. That independent support is real. However, the central quantitative outputs depend on two small parameters, δ and α, neither of which is reliably small in the plotted regime. The free parameter d is also tuned in §6 to match variational data, showing sensitivity that is not controlled. The most precise and testable weak point is the α expansion: the authors have an exact hypergeometric normalization (Eq. A.1) but do not use it for any figure, and α reaches O(0.5) at weak interaction. Since the headline dimensional crossings develop in exactly that region, there is a concrete risk that the qualitative claim is an artifact of truncation. A direct numerical comparison of Eq. (16) and Eq. (A.1) would settle this without requiring a full GPE simulation. I therefore agree with the reader's CONDITIONAL verdict, while noting that my concern is a different technical weak point than the δ-smallness issue emphasized by the reader.","tokens_in":14721,"tokens_out":10017,"duration_ms":81676,"concrete_test":"Recompute the chemical potential μ from the exact hypergeometric normalization condition (Eq. A.1) for the Fig. 3 parameter sets (N = 1000, m = 1, d = 3, λ = 1; D = 2,3,4,5) at uD = 0.01, 0.1, 1, 10, 25.1, and for the Fig. 4 anisotropy sweep. Compare against the second-order result from Eq. (16). If the relative difference exceeds about 10% in the weak-to-moderate interaction or anisotropy range where the crossings occur, the plotted μD-curves and the claimed crossings are not supported by the zeroth-order density; the paper should either adopt Eq. (A.1) for all quantitative figures or restrict the crossing claim to parameter regimes where the α expansion is numerically validated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central quantitative results—chemical potential Eq. (16), energy Eq. (24), critical velocity Eq. (27), and the μD-curves in Figs. 3–4—are all obtained by expanding the normalization/energy integrands to second order in α = κ/(4μ). For every plotted dimension D = 2,3,4,5 with m = 1 and d = 3, the reality condition on the hyper-radii (Eqs. 14–15) gives μ ≥ κ/2, so α ≤ 1/2. At weak interaction or weak anisotropy, μ approaches this bound and α is O(0.3–0.5); the expansion parameter is not small anywhere in the figures. The paper contains an exact hypergeometric normalization condition (Eq. A.1) that avoids the α expansion, but all figures use the truncated Eq. (16). Because the dimensional crossings in Figs. 3 and 4 emerge in precisely the weak-to-moderate regime where α is largest, those crossings could be an artifact of dropping O(α^3) terms rather than a genuine property of the zeroth-order δ→0 density. This is distinct from, but compounded by, the reader's δ-smallness concern: δ = 1/κ is also not small (δ = 1, 1/2, 1/3, 1/4 for D = 2,3,4,5). No error estimate or numerical GPE benchmark for D ≠ 3 is provided, so the quantitative claims rest on two uncontrolled truncations.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper applies dimensional perturbation theory (DPT) to the D-dimensional Gross-Pitaevskii equation in hypercylindrical coordinates with a vortex along the z-axis. With the perturbation parameter δ = 1/(D + 2|m| − d), the authors derive zeroth-order (δ→0) formulas for the condensate density (Eq. 7), chemical potential (Eq. 16), energy per particle (Eq. 24), and critical vortex velocity (Eq. 27). They study the dependence of these quantities on dimension D, interaction strength, and anisotropy λ, and report crossings of the chemical-potential curves for D = 2, 3, 4, 5 as interaction or anisotropy is increased. They also present a Thomas-Fermi limit, a D = 3 comparison to earlier work, and an exact hypergeometric normalization condition (Eq. A.1).","tokens_in":15077,"tokens_out":3076,"duration_ms":28250,"significance":"If the approximations are quantitatively reliable, the paper would supply the first general-D analytic description of vortices in an anisotropic trap, with potential applications to synthetic dimensions and holographic analogies. The derivation is algebraically careful, the D = 3 limit exactly recovers Ref. [9] when d = 3, and the Thomas-Fermi limit (Eq. 20) reduces to known results. The paper also includes a useful exact hypergeometric form of the normalization integral (Eq. A.1) that avoids the α-expansion used elsewhere. In addition, the authors candidly state that the dimensional crossings may be a mathematical artifact (Section 7), which is an honest limitation rather than an overclaim. However, the central quantitative results for D ≠ 3 rest on two uncontrolled approximations—the truncation at zeroth order in δ and at second order in α—so the significance is contingent on validation that the manuscript does not provide.","major_comments":[{"comment":"The chemical potential curves and their dimensional crossings are obtained by expanding the normalization integrand to second order in α = κ/(4μ). For the parameters plotted (d = 3, m = 1, D = 2, 3, 4, 5), the reality condition μ ≥ κ/2 from Eqs. (14)–(15) implies α ≤ 1/2, and at weak interaction or weak anisotropy α is O(0.3–0.5), so the expansion parameter is not small anywhere in the figures. The crossings in Figs. 3 and 4 occur precisely in the weak-to-moderate regime where α is largest, so they may be artifacts of dropping O(α^3) terms. The paper contains an exact normalization condition (Eq. A.1) that avoids this expansion, but it is not used in any figure. Please either provide error estimates on the second-order truncation, implement Eq. A.1 to verify the μ-D curves, or benchmark against numerical GPE solutions for D = 2, 4, 5.","section":"Section 3, Eq. (16) and Figs. 3–4"},{"comment":"The paper treats the δ→0 limit as the zeroth-order approximation and applies it to finite D with d = 3 and m = 1, where δ = 1/(D + 2|m| − d) takes values 1, 1/2, 1/3, 1/4 for D = 2, 3, 4, 5. For D = 2 and D = 3 the perturbation parameter is not small, so the validity of dropping the order-δ^2 kinetic derivative terms in Eq. (5) to obtain Eq. (7) is not established. The text frames the method as a large-D or large-|m| approximation, but the figures include parameter regimes where neither is large. Please state the expected domain of validity of the zeroth-order approximation, or include a comparison with finite-δ corrections for at least one of the plotted dimensions.","section":"Section 3, Eq. (7) and definition of δ"},{"comment":"The free parameter d is shown to have a strong effect on the chemical potential: for λ = 1/100 the optimal d is between 1 and 2, while for λ = 100 the optimal d is between −68 and −90. Since d enters κ = D + 2|m| − d and hence the density and chemical potential, the dimensional crossings in Figs. 3 and 4, which are computed for d = 3, may not be robust to the choice of d. The paper should demonstrate that the qualitative crossings persist over a range of d, or should determine d from a variational or other principle rather than using a fixed value that is later tuned.","section":"Section 6, Fig. 6"}],"minor_comments":[{"comment":"The text says the hypersphere reduces to a circle and line segment segment; the duplicated word 'segment' should be removed.","section":"Section 2"},{"comment":"The label '2D3D4D5D' appears to be a formatting artifact; it should be spaced or replaced by a proper legend.","section":"Figure 3 caption"},{"comment":"The word 'anistropy' in the caption of Fig. 6 is a typo and should be 'anisotropy.'","section":"Section 6"},{"comment":"Reference [18] lists the journal as 'A VS Quantum Science'; the correct title is 'AVS Quantum Science.'","section":"References"},{"comment":"The derivation of Eq. (16) from Eq. (13) by expanding to second order in α is central but not shown in detail; a brief appendix or inline derivation would make the truncation errors easier to assess.","section":"Section 3, Eq. (16)"},{"comment":"The D = 4 critical velocity expression in Eq. (33) is not obviously the D = 4 limit of Eq. (27); a consistency check or a note explaining the simplified form would improve confidence.","section":"Section 5, Eq. (33)"}],"recommendation":"major_revision","confidential_remarks":"The paper is honest about its limitations, which is commendable, but the abstract's claim of observing dimensional crossings is stronger than what is supported by the uncontrolled truncations. The central issue is validation: the authors have an exact normalization expression (Eq. A.1) and a known D = 3 benchmark, so adding numerical GPE benchmarks for D = 2, 4, 5 and an error estimate for the α truncation is feasible within the manuscript's scope. I recommend major revision rather than rejection because the derivations are not internally inconsistent and the proposed checks are concrete and achievable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The genuinely useful part of this paper is the machinery: extending the DPT approach from isotropic BECs and from the known 3D anisotropic vortex results to arbitrary D, with a hypercylindrical geometry and quantized circulation. The derivations of the zeroth-order density, chemical potential, energy, and critical velocity are done carefully; the D=3 limits reproduce Ref. [9] when d=3, and the Thomas-Fermi limit (d=D+2|m|) recovers known results. That is real, checkable algebra, and the authors are transparent about their assumptions.\n\nThe soft spots are concentrated in whether the quantitative predictions survive contact with the plotted parameters. Two expansions are being made simultaneously. The DPT parameter is delta = 1/(D+2|m|-d); with d=3 and m=1, for D=2 that is delta=1, for D=3 it is 1/2. Calling that a small parameter is a stretch. Then the alpha = kappa/(4mu) expansion is truncated at second order, but the reality condition mu >= kappa/2 means alpha <= 1/2 everywhere, and it approaches 1/2 in the weak-interaction and weak-anisotropy regimes where the dimensional crossings appear. The stress-test note is right: the crossings could be an artifact of dropping O(alpha^3) terms. The paper even concedes this in the conclusion, saying more study is needed to determine whether the crossings are physical or a mathematical artifact. That is honest, but it also means the headline result is not yet supported.\n\nThe comparison with the variational data in Fig. 6 is also weakened by tuning d to match. That is a legitimate strategy, but it turns the zeroth-order approximation into an effective fitting scheme rather than a predictive calculation. No numerical GPE benchmarks are given for D=2,4,5, so we have no independent check on the density or chemical potential shapes in those dimensions.\n\nThe authors could address the main concern fairly easily: they have an exact hypergeometric normalization condition (Eq. A.1) that avoids the alpha expansion entirely. Using it to recompute the mu_D curves and see whether the crossings persist would settle the question. Their failure to do so before submitting is the biggest gap.\n\nThis is a serious analytic contribution for the DPT community, but as a physics paper about dimensional crossings it needs revision. I would send it to review because the derivations and limits deserve referee attention, but I would require that crossings be re-examined either numerically or via the exact normalization before publication. Who is this for? People working on dimensional perturbation theory, synthetic-dimension proposals, or analytic vortex approximations in BECs. I would not cite it as evidence for dimensional crossings until the artifact question is resolved.","headline":"A careful DPT extension to D-dimensional vortices that is honest about its limits, but the headline dimensional crossings rest on two uncontrolled expansions and may be artifacts.","tokens_in":15597,"tokens_out":1912,"would_cite":false,"duration_ms":18922,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["03.75.Lm"],"model":"deepseek-v4-flash","headline":"The paper derives semiclassical vortex formulas (density, chemical potential, energy, and critical rotation speed) for condensates in any number of dimensions from a single large-dimension limit of the Gross-Pitaevskii equation.","keywords":["Bose-Einstein condensate","vortex","Gross-Pitaevskii equation","dimensional perturbation theory","hypercylindrical symmetry","anisotropic trap","synthetic dimensions","Thomas-Fermi approximation"],"falsifier":"Solve the full hypercylindrical Gross-Pitaevskii equation (Eq. 1) numerically for $D = 2,3,4,5$ with $m = 1$, $N = 1000$, and $\\lambda = 1$, scanning the interaction strength $\\bar u_D$, and compare the chemical-potential curves with Eq. (16) at $d = 3$: if the curves do not cross, or cross at noticeably different $\\bar u_D$ values, the zeroth-order claim fails. A cheaper check targets the geometry: compare the numerically computed vortex-core radius as a function of $\\bar u_D$ with $r'_{\\perp,\\min} = \\sqrt{(1 - \\sqrt{1 - 4\\bar\\alpha^2})/2}$; at $D = 2$, where $\\delta = 1$, the core radius should disagree most strongly if the approximation breaks down exactly where it is applied.","tokens_in":14483,"feed_emoji":"🌀","tokens_out":35221,"duration_ms":232490,"temperature":0.7,"pith_summary":"The paper aims to show that the limit $\\delta = 1/(D+2|m|-d) \\to 0$ of the dimensionally scaled Gross-Pitaevskii equation — the mean-field equation for a dilute Bose-Einstein condensate — turns the vortex problem in a $D$-dimensional anisotropic trap into a solvable algebraic form. In that limit the condensate density is an inverted parabola with a hole at the vortex core, $$|\\bar\\psi(\\bar r)|^2 = \\frac{1}{\\bar u_D}\\left(2\\bar\\mu - \\bar r_\\$perp^{2}$ - \\$lambda^{2}$\\bar $z^{2}$ - \\frac{\\$kappa^{2}$}{4\\bar r_\\$perp^{2}$}\\right),$$ and normalizing this density yields the paper's semiclassical approximations for the chemical potential, the energy per particle, and the critical rotation speed in arbitrary dimension $D$. The authors also observe that the chemical-potential curves for $D = 2,3,4,5$ cross one another as the interaction strength or the trap anisotropy $\\lambda$ grows, so at strong parameters a higher-dimensional condensate can become chemically cheaper than a lower-dimensional one. The payoff would be an analytic footing for trapped vortices in arbitrary dimension, with applications the authors point to in synthetic dimensions — where internal states simulate extra spatial dimensions — and in holographic-style correspondences between a higher-dimensional system and a lower-dimensional boundary.","feed_headline":"Large-dimension limit yields vortex formulas for all dimensions","feed_subtitle":"Vortex density, energy, and chemical potential follow from one limit; the D=2–5 curves cross.","key_machinery":"The engine is the zeroth-order ($\\delta \\to 0$) limit of the dimensionally scaled hypercylindrical Gross-Pitaevskii equation (Eq. 5), with perturbation parameter $\\delta = 1/\\kappa$ and $\\kappa = D + 2|m| - d$; here $D$ is the spatial dimension, $|m|$ the vortex quantum number, and $d$ a free 'reference dimension' that decides how much kinetic energy survives at zeroth order. A Jacobian transformation $\\phi = r_\\perp^{(D-2)/2}\\psi$ removes the first-derivative terms from the Laplacian, and dimensionally scaled coordinates push the remaining kinetic terms into a prefactor $\\delta^2$, so the $\\delta\\to0$ equation is algebraic and the density follows directly (Eq. 7). A second small parameter, $\\bar\\alpha = \\kappa/(4\\bar\\mu)$ — the vortex-core radius scale relative to the condensate scale — brings the normalization, energy, and critical-velocity integrals to closed form in gamma functions (Eqs. 16, 24, 27), with the condition $\\bar\\mu \\ge \\kappa/2$ keeping the inner and outer radii real. The parameter $d$ connects the general formulas to known limits: $d = 3$ gives exact agreement with the $D = 3$ anisotropic results, and $d = D + 2|m|$ gives the Thomas-Fermi (zero-kinetic-energy) approximation.","core_discovery":"Working in hypercylindrical coordinates — one Cartesian axis $z$ plus a $(D-1)$-dimensional hyperspherical subspace with hyperradius $r_\\perp$ — the paper writes the Gross-Pitaevskii equation for a condensate in $D$ dimensions carrying a vortex of quantum number $|m|$ along $z$, and applies dimensional perturbation theory with $\\delta = 1/\\kappa$, where $\\kappa = D + 2|m| - d$. At zeroth order ($\\delta \\to 0$) the derivative kinetic terms drop out and the density is read off directly: $|\\bar\\psi(\\bar r)|^2 = (1/\\bar u_D)(2\\bar\\mu - \\bar r_\\perp^2 - \\lambda^2\\bar z^2 - \\kappa^2/(4\\bar r_\\perp^2))$ (Eq. 7), a truncated inverted parabola whose inner edge is the vortex core and whose outer edge is the condensate surface. Imposing the general-$D$ normalization condition and expanding in the second small parameter $\\bar\\alpha = \\kappa/(4\\bar\\mu)$ yields the chemical potential (Eq. 16), the energy per particle (Eq. 24), and the critical vortex rotation speed (Eqs. 26–27) as explicit functions of $D$, $|m|$, the interaction $\\bar u_D$, and the anisotropy $\\lambda$. Setting $d = 3$ makes the formulas reduce exactly to the established $D = 3$ axially symmetric results, and setting $d = D + 2|m|$ recovers the general-$D$ Thomas-Fermi approximation (zero kinetic energy). With these formulas, the paper reports crossings of the $\\bar\\mu_D$-curves for $D = 2,3,4,5$: at weak interaction or anisotropy the chemical potential increases with $D$, as in a harmonic oscillator, while at strong parameters it decreases with $D$ through the Thomas-Fermi exponent $2/(D+2)$.","pith_inferences":["A testable consequence of the crossings, should they survive beyond zeroth order, is a non-monotonic ordering of vortex-nucleation thresholds: in a synthetic-dimension experiment with tunable interaction, the critical rotation speed for creating an $m=1$ vortex should no longer be monotonic in the effective dimension, and the ordering of thresholds across dimensions should invert as the interactio","Because $d$ acts as a variational dial that partially restores the kinetic energy discarded by $\\delta \\to 0$, an extension the paper leaves open is to optimize $d$ against direct numerical Gross-Pitaevskii solutions for each $(\\lambda, \\bar u_D, D)$; if the best-fit $d$ tracks $\\lambda$ in the way Eqs. (35)–(37) suggest, the crossings in Figs. 3 and 4 may shift or vanish when computed with optimi","The pole structure of the $\\bar\\alpha$ expansion — no poles at second order, poles at odd $D$ appearing from fourth order upward — makes the second-order formulas the only fixed-order expansion valid at every odd dimension; for work at $D = 3$ one must stay at second order or use the exact hypergeometric normalization of Appendix A, since the fourth-order formula has a pole there.","The approximation is on its firmest ground when $\\delta$ is genuinely small, i.e., for large $D$ or large $|m|$; checking the $D = 3$ formulas at $m = 2$ or $m = 3$, where $\\delta$ is halved or thirded, would separate the large-$\\kappa$ regime from the small-$D$ regime and indicate how far the 'arbitrary $D$' claim reaches."],"forward_implications":["For any $D \\ge 2$, the vortex density, chemical potential, energy per particle, and critical rotation speed can be evaluated from the analytic formulas (Eqs. 7, 16, 24, 26) instead of solving the full Gross-Pitaevskii equation, with all dimensionality entering through $\\kappa$ and $\\bar\\alpha$.","The free parameter $d$ interpolates between known limits: $d = 3$ reduces every general-$D$ formula to the established $D = 3$ axially symmetric results, and $d = D + 2|m|$ reduces the chemical potential to the general-$D$ Thomas-Fermi approximation.","Reality of the vortex core imposes a $D$- and $|m|$-dependent floor on the chemical potential, $\\bar\\mu \\ge \\kappa/2$, which grows with dimension and vorticity and acts like a dimensional zero-point energy.","In the zeroth-order density, the vortex-core radius decreases with interaction strength and the condensate density becomes more squeezed around the vortex axis as $D$ increases, giving quantitative predictions for the density profile (Figs. 1 and 2).","Within the zeroth-order approximation, raising the interaction or the anisotropy makes the chemical-potential curves for $D = 2,3,4,5$ cross, so that higher-dimensional condensates can have lower chemical potential than lower-dimensional ones; the paper notes that whether the crossings are physical or an artifact of the perturbation limit requires further study."],"supporting_citations":[{"why":"This reference establishes the Gross-Pitaevskii equation as the mean-field description of trapped atomic condensates, the equation the paper approximates.","marker":"[1]"},{"why":"This reference supplies the dimensional-perturbation machinery for the D-dimensional GPE in isotropic hyperspherical coordinates, including the Jacobian transformation, scaled units, and zeroth-order limit that this paper extends to anisotropic hypercylindrical symmetry.","marker":"[8]"},{"why":"This reference provides the D = 3 anisotropic-trap vortex semiclassical results (density, chemical potential, energy, critical velocity, and the alpha parameter) that the general-D formulas must reproduce when d = 3.","marker":"[9]"},{"why":"This reference gives the variational anisotropic ground-state GPE results used as the numerical baseline for the anisotropy and lower-effective-dimension comparison in Fig. 6.","marker":"[10]"},{"why":"This reference supplies the angular-momentum eigenvalue formula that fixes the centrifugal term in the hypercylindrical Laplacian, which is what makes the effective vortex strength depend on D.","marker":"[12]"}],"fun_headline_variants":["A single limit reveals vortex physics in any dimension","One large-D limit yields vortex formulas for all D","Dimension crossings predicted in Bose-Einstein vortex states","Hypercylindrical trap yields general-D vortex formulas","Vortex energy and density from a single D-limit expansion"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The central assumption is that dropping the kinetic-energy terms because $\\delta = 1/(D+2|m|-d)$ is small remains accurate at the parameters plotted, where for $D = 2,3,4,5$ with $m = 1$ and $d = 3$ this 'small' parameter is actually $1$, $1/2$, $1/3$, and $1/4$.","fun_headline_variants_meta":{"raw":{"variants":["A single limit reveals vortex physics in any dimension","One large-D limit yields vortex formulas for all D","Dimension crossings predicted in Bose-Einstein vortex states","Hypercylindrical trap yields general-D vortex formulas","Vortex energy and density from a single D-limit expansion"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001282,"raw_usage":{"total_tokens":5338,"prompt_tokens":1143,"completion_tokens":4195,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":759,"completion_tokens_details":{"reasoning_tokens":4118}},"tokens_in":759,"tokens_out":4195,"duration_ms":24993,"temperature":1.0,"reasoning_tokens":4118,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T11:17:42.754385+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Solve the full hypercylindrical Gross-Pitaevskii equation (Eq. 1) numerically for $D = 2,3,4,5$ with $m = 1$, $N = 1000$, and $\\lambda = 1$, scanning the interaction strength $\\bar u_D$, and compare the chemical-potential curves with Eq. (16) at $d = 3$: if the curves do not cross, or cross at noticeably different $\\bar u_D$ values, the zeroth-order claim fails. A cheaper check targets the geometry: compare the numerically computed vortex-core radius as a function of $\\bar u_D$ with $r'_{\\perp,\\min} = \\sqrt{(1 - \\sqrt{1 - 4\\bar\\alpha^2})/2}$; at $D = 2$, where $\\delta = 1$, the core radius should disagree most strongly if the approximation breaks down exactly where it is applied.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"This reference establishes the Gross-Pitaevskii equation as the mean-field description of trapped atomic condensates, the equation the paper approximates."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"This reference supplies the dimensional-perturbation machinery for the D-dimensional GPE in isotropic hyperspherical coordinates, including the Jacobian transformation, scaled units, and zeroth-order limit that this paper extends to anisotropic hypercylindrical symmetry."},{"cited_title":"Sinha, Phys","cited_arxiv_id":null,"evidence_quote":"This reference provides the D = 3 anisotropic-trap vortex semiclassical results (density, chemical potential, energy, critical velocity, and the alpha parameter) that the general-D formulas must reproduce when d = 3."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"This reference gives the variational anisotropic ground-state GPE results used as the numerical baseline for the anisotropy and lower-effective-dimension comparison in Fig. 6."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"This reference supplies the angular-momentum eigenvalue formula that fixes the centrifugal term in the hypercylindrical Laplacian, which is what makes the effective vortex strength depend on D."}],"review_version":1}