{"id":"9d0ec2a6-d048-43b7-bd95-0c4f90e28dc1","arxiv_id":"2507.01322","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A relevant scalar operator S persists from SO(5) to O(4) deconfined quantum criticality on the fuzzy sphere, indicating the O(4) transition is pseudo-critical rather than a genuine fixed point.","lead":"This paper uses a numerical technique called fuzzy sphere regularization to study a quantum phase transition with O(4) symmetry, a candidate for deconfined quantum criticality. It finds a relevant scalar operator that persists from SO(5) to O(4), suggesting the transition is only pseudo-critical, not a true continuous transition.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The pseudo-criticality claim hinges on whether u5/u=-0.75 is close enough to the would-be O(4) fixed point that Δ_S≈2.845 is the true fixed-point dimension; the paper admits the fixed point is not exactly hit and never quantifies the resulting systematic error.","rationale":"Read in good faith, the paper does what it claims: it tracks low-lying operator flows in a tunable SO(5)→O(4) model on the fuzzy sphere, finds an avoided level crossing, and extracts a low-lying parity-even singlet with Δ≈2.845 at u5/u=-0.75. The central physical conclusion—that the O(4) transition is pseudo-critical rather than a genuine CFT—follows only if this singlet is indeed relevant at the would-be O(4) fixed point. The single most load-bearing condition is therefore the proximity of the u5/u=-0.75 spectra to that fixed point, exactly the reader's weakest assumption. This is not a disagreement with the consensus; it is an internal-correctness risk that the paper itself partially concedes. The unclear operator content of S is real but less load-bearing: even if the precise operator that creates S is not pinned down, the existence of any parity-even scalar with Δ<3 at the near-critical point would already imply pseudo-criticality. The ancestry of S matters for the 'persists from SO(5) to O(4)' narrative and for the overlap analysis, but not for the minimal claim. The fixed-point proximity, by contrast, is required for the number 2.845 to mean anything about the IR. The supplement's O(3)→O(2) benchmark shows that an offset of order 0.05 in the extracted S dimension can occur at a deliberately chosen working point in a well-understood CFT flow; the authors do not translate this calibration into an estimated systematic error for the SO(5)→O(4) case. Without that translation, the error bar ±0.010 reflects only statistical and finite-size spread within their fitting recipe, not the distance to the (nonexistent) fixed point. A self-consistent finite-size scaling fit that includes the leading correction from the relevant singlet itself would settle whether the data are compatible with a stable Δ_S below 3. This is a single, concrete, data-reusing check. If the fit fails or the extrapolated Δ_S moves toward or above 3 when the anisotropy is varied, the CONDITIONAL verdict should be maintained or tightened; if it succeeds, the pseudo-criticality conclusion stands on firmer ground.","tokens_in":19178,"tokens_out":11453,"duration_ms":134970,"concrete_test":"Perform a self-consistent finite-size scaling fit of the lowest singlet gap at u5/u=-0.75 using δE_S(R) = (v/R)(Δ_S + c R^{-(3-Δ_S)} + d R^{-2}), fitting Δ_S, c, d, and v to the No=7–10 data (and No=11–12 if available), with the normalizations Δ(T^μν)=3 and Δ(J^μ)=2 imposed. If the fitted Δ_S stays below 3 and the correction term c R^{-(3-Δ_S)} is subdominant at the largest sizes, the proximity assumption is internally validated. Repeat the same fit at u5/u=-0.5 and -1.0; if the extrapolated Δ_S moves by more than 0.05 across the plateau, the quoted 2.845±0.010 is not representative of the would-be O(4) fixed point.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim—that a relevant parity-even scalar S with Δ_S≈2.845 persists to the O(4) DQCP and forces pseudo-criticality—requires that the spectra at u5/u=-0.75 are close enough to the would-be O(4) fixed point for the state-operator correspondence to yield the true IR scaling dimensions. The paper admits this fixed point is not exactly hit: 'Even though the fixed point is not exactly hit since the conformal symmetry is not exact, the scaling dimensions of low-lying O(4) primaries drift very slowly within u5/u ∈ [-1,-0.5]'. The extracted dimensions therefore include corrections from the relevant perturbation S itself, which is the very operator whose dimension is being measured; no controlled expansion is provided for these corrections. The slow drift of Δ_S with system size (2.833 to 2.845 from No=7 to No=10) and with anisotropy is consistent with walking, but it does not demonstrate that the infinite-volume, fixed-point value lies below 3. The supplement's O(3)→O(2) benchmark at rz=-0.5 calibrates a systematic offset of about 0.05 (extracted S dimension 1.5609 versus bootstrap 1.51136), but the distance to the fixed point in that flow need not match the SO(5)→O(4) case, so this does not bound the systematic error in 2.845±0.010.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies the Néel-to-VBS transition in a four-flavor interacting fermion model regularized on the fuzzy sphere, with a parameter u5 that breaks the global symmetry from SO(5) to O(4). Using exact diagonalization and the state-operator correspondence, the authors track the running scaling dimensions of low-lying O(4) primaries as a function of u5/u. They find that SO(5) primaries decompose into O(4) multiplets, that the two singlet operators S and T[0,0] undergo an avoided level crossing with an exchange of operator content, and that a parity-even scalar S remains relevant, with Δ_S ≈ 2.845 ± 0.010 at u5/u = -0.75. On this basis they conclude that the O(4) DQCP is not a genuine conformal fixed point and instead exhibits pseudo-critical behavior analogous to the SO(5) case. The supplemental material supplies branching rules, spectral-flow data, and an O(3)→O(2) benchmark flow.","tokens_in":19428,"tokens_out":10708,"duration_ms":117811,"significance":"If confirmed, the result is significant: it extends the relevant-singlet pseudo-criticality mechanism from the SO(5) DQCP to the O(4) DQCP and offers a microscopic view of how conformal operator content evolves along an RG flow. The paper is methodologically valuable, demonstrating that fuzzy-sphere exact diagonalization can resolve operator decomposition and avoided crossings in a controlled way, and it provides explicit finite-size data and group-theoretic branching rules. The extraction of Δ_S is not circular in the usual sense: the velocity is calibrated with the conserved current and stress tensor, while the scalar dimension is read from the spectrum rather than fitted. The O(2) benchmark in the supplement is a useful check, though it also shows that systematic offsets of order 0.05 must be controlled before the central quantitative claim can be taken at face value.","major_comments":[{"comment":"The central claim that Δ_S ≈ 2.845 ± 0.010 at u5/u = -0.75 lies below 3 requires the spectrum to be close to the would-be O(4) fixed point, but the text concedes that \"the fixed point is not exactly hit since the conformal symmetry is not exact\" and no estimate is given for the systematic correction from the relevant perturbation S itself (the operator whose dimension is being measured) or from irrelevant operators. The O(3)→O(2) benchmark in Supplement §V gives an offset of about 0.05 for the analogous scalar (1.5609 extracted versus 1.51136 from bootstrap), but the distance from the fixed point in that flow need not match the SO(5)→O(4) case, so it does not bound the error in 2.845 ± 0.010. Please provide a controlled estimate of these corrections, for example by including leading correction-to-scaling terms in the finite-size/flow-parameter extrapolation, by extrapolating u5/u toward the would-be fixed point, or by using the O(2) benchmark to assign a conservative systematic error that still keeps Δ_S below 3.","section":"Numerical results, Table I"},{"comment":"The supplement states that \"the precise operator content of the relevant singlet S at the approximate SO(5) and O(4) fixed points remains unclear, potentially due to the corresponding complex fixed points lying a finite distance away from the real axis in the complex plane.\" This is load-bearing for the claim that the same relevant scalar S persists from SO(5) to O(4), because the avoided-crossing analysis in Fig. 3 relies on the overlap with T55, and the supplement also reports that those overlaps remain small throughout the flow. Please identify the O(4) S by an independent criterion, such as its descendant tower at Δ_S + integers, a systematic overlap matrix in the singlet sector, or a computed OPE coefficient, rather than by continuity of an eigenvalue alone.","section":"Supplement §V, last paragraph"},{"comment":"The paper acknowledges that the topological θ-term description of the O(4) DQCP predicts a parity-odd scalar ∼ ε_{αβγδ} φ_α ∂_t φ_β ∂_x φ_γ ∂_y φ_δ, yet the numerical spectrum shows no additional relevant parity-odd scalar. This unresolved discrepancy is a correctness risk for the claim that the operator content of the O(4) DQCP has been fully characterized. Please either construct this operator on the fuzzy sphere and bound its scaling dimension or overlap with the parity-odd scalar candidates, or explain explicitly why it is expected to be gapped or absent from the low-lying spectrum in this regularization.","section":"Summary and discussion, final remark"}],"minor_comments":[{"comment":"The control parameter u5/u is used throughout, but u is never defined; please state whether u denotes u_N, u_K, or some combination, and explain how the optimal u at each system size in Table I is determined from the conditions Δ(T_{μν}) = 3 and Δ(J_μ) = 2.","section":"Model and method, after Eq. (1)"},{"comment":"In the definition of T55(r), the operator n(r) in the traceless combination is not defined; if it is the trace ∑_i n_i(r) n_i(r'), please state this explicitly.","section":"Eq. (2)"},{"comment":"The quoted uncertainty Δ_S ≈ 2.845 ± 0.010 is not defined; it should be stated whether this error comes from finite-size extrapolation, from the drift across u5/u ∈ [-1,-0.5], or from both sources.","section":"Table I"},{"comment":"There are minor typos, including \"[46 ? –48]\" in the introduction and \"Beacuse\" in the last paragraph of Supplement §V, and the supplement's caveat about the small overlaps of S and T[0,0] with the probe operators should be reflected in the main-text discussion of Fig. 3.","section":"Introduction and Supplement §V"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a strong fuzzy-sphere ED study with an honest supplementary discussion of its limitations. The main revision needed is a quantitative control of the fixed-point distance / systematic error in Δ_S, together with a more robust operator identification for S. I do not see citation or disclosure problems; the paper is within scope for cond-mat.str-el."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The genuinely new thing here is the operator-flow tracking: decomposing SO(5) primaries into O(4) multiplets, seeing an avoided level crossing between the singlet S and T[0,0], and finding that a relevant parity-even scalar S persists with Δ≈2.845 at u5/u=-0.75. That last number is the load-bearing result, and it is also the softest part.\n\nThe numerics are careful. The velocity is calibrated using the known conformal values Δ(Tμν)=3 and Δ(Jμ)=2, so S is not fitted; it emerges from the spectrum. The descendant towers in Fig. 4 look consistent with approximate conformal symmetry. The supplement's O(3)→O(2) benchmark is a good methodological check, even though it shows a systematic offset in Δ_S of about 0.05 at that flow (1.5609 vs bootstrap 1.51136).\n\nThe soft spots are real and the paper admits most of them. The fixed point is not exactly hit—the main text says so explicitly. Since S is itself the relevant perturbation, the extracted Δ_S at u5/u=-0.75 includes corrections from the very operator being measured; the slow drift with system size (2.833 to 2.845) and with u5/u does not bound the infinite-volume fixed-point value below 3. The O(3)→O(2) benchmark does not transfer quantitatively, so the 0.05 offset does not quantify the error here. Second, the paper acknowledges finding no parity-odd scalar corresponding to the θ-term operator expected from duality. That unresolved discrepancy weakens the claim that this model actually realizes the O(4) DQCP. Third, the supplement admits the operator content of S remains unclear—overlaps with T55 and n^2 are small, possibly because the complex fixed points lie at finite distance from the real axis. That makes the identification of S as the same relevant singlet less secure.\n\nMinor: the claim of being the \"first example of RG flows of conformal operators\" is overclaimed, since the supplement itself shows an O(3)→O(2) flow, and the branching rules are standard group theory. Not a big deal.\n\nWho benefits: the DQCP and fuzzy-sphere communities. This deserves a serious referee, not a desk reject—the central claim is important if true, and the paper is honest about its own limitations. The referee should ask for a controlled estimate of the systematic error from the finite distance to the fixed point, a clearer discussion of the missing parity-odd scalar, and ideally code or data.\n\nSend to review, but expect heavy revision.","headline":"A careful fuzzy-sphere operator-flow study from SO(5) to O(4) that plausibly shows a relevant singlet persisting, but the pseudo-criticality conclusion rests on an unquantified systematic error and an unresolved theta-term discrepancy.","tokens_in":20041,"tokens_out":2858,"would_cite":true,"duration_ms":32666,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims the O(4) deconfined quantum critical point is not a genuine conformal fixed point: a relevant scalar operator with $\\Delta_S \\approx 2.845 \\pm 0.010$ survives the symmetry reduction from SO(5) to O(4), making the…","keywords":["deconfined quantum criticality","pseudo-criticality","fuzzy sphere regularization","state-operator correspondence","conformal operator flow","SO(5) to O(4) symmetry breaking","avoided level crossing","relevant scalar operator"],"falsifier":"Compute $\\Delta_S$ at larger system sizes and at parameter values between $u_5/u = -0.75$ and the would-be fixed point: if $\\Delta_S$ extrapolates to 3 or above in the thermodynamic limit, the relevant-singlet claim is falsified and the O(4) transition could be genuinely conformal.","tokens_in":18929,"feed_emoji":"🧲","tokens_out":14236,"duration_ms":244993,"temperature":0.7,"pith_summary":"Deconfined quantum criticality was conjectured to be a genuine (2+1)-dimensional conformal field theory, but accumulating evidence suggests it never quite is. This paper studies the Neel-to-VBS transition in a microscopic model whose global symmetry can be tuned continuously from SO(5) to O(4), using the fuzzy sphere regularization to read off the renormalization group flow of conformal operators. The central claim is that the relevant parity-even scalar operator $S$, first found at the SO(5) point, persists at the O(4) transition with scaling dimension $\\Delta_S \\approx 2.845 \\pm 0.010$ at $u_5/u = -0.75$. Because $\\Delta_S < 3$, the would-be O(4) fixed point is unstable, and the transition is pseudo-critical: the flow lingers near a fixed point that is not actually reached, producing an approximate conformal symmetry and a weak first-order transition in the thermodynamic limit. If correct, this closes a long debate about whether the O(4) easy-plane DQCP is a conformal point, and it demonstrates that operator flows themselves can be observed in a microscopic calculation.","feed_headline":"A relevant scalar keeps the O(4) transition pseudo-critical","feed_subtitle":"Fuzzy-sphere spectra trace the SO(5) to O(4) operator flow, ending in a weak first-order transition, not a conformal point.","key_machinery":"The machinery is the fuzzy sphere regularization combined with the state-operator correspondence: the model is projected onto the lowest Landau level of fermions on a sphere threaded by a monopole field, exact diagonalization yields eigenenergies, and the correspondence $\\delta E_n = \\frac{v}{R}(\\Delta_n - \\Delta_0)$ converts energy gaps into the scaling dimensions $\\Delta_n$ of CFT primaries. The RG flow is traced by varying the single parameter $u_5/u$ from 1 (exact SO(5)) to $-1$ (deep O(4)), with the optimal critical coupling fixed by demanding $\\Delta_{T^{\\mu\\nu}} = 3$ and $\\Delta_{J^\\mu} = 2$ simultaneously. The O(4) quantum numbers are the highest weights $[j,k]$ of the two $\\mathrm{SU}(2)$ subgroups of $\\mathrm{SO}(4)$, and the branching rules of $\\mathrm{so}(5) \\supset \\mathrm{so}(4)$ dictate which O(4) fields each SO(5) primary decomposes into. Finally, the avoided level crossing between $S$ and $T_{[0,0]}$ is diagnosed by the operator-content overlap $F_O = \\langle(S, T_{[0,0]})|T_{55}|I\\rangle / \\lVert T_{55}|I\\rangle\\rVert$, where $T_{55}$ is the $n_5 n_5$ component of the SO(5) rank-2 tensor that carries the identity of $T_{[0,0]}$.","core_discovery":"By tracking the low-lying spectrum of a four-flavor interacting fermion model on the fuzzy sphere as the anisotropy $u_5/u$ is lowered from 1, the paper shows how the SO(5) conformal primaries decompose into O(4) representations $[j,k]$ of $\\mathrm{SU}(2)\\times\\mathrm{SU}(2)$: the SO(5) order parameter splits into the O(4) vector $\\phi_{[1/2,1/2]}$ plus a gapped parity-odd scalar, the rank-2 tensor splits into $T_{[1,1]}$, $T_{[0,0]}$ and a gapped component, and so on. Along this flow, the two lowest parity-even scalars, $S$ and $T_{[0,0]}$, show an avoided level crossing: their scaling dimensions approach, reach a minimal separation near $u_5 \\approx 0.1$, and their operator content, diagnosed by the overlap with the traceless tensor $T_{55}$, is exchanged. At $u_5/u = -0.75$ the paper extracts the O(4) conformal data: $\\Delta_{\\phi_{[1/2,1/2]}} \\approx 0.555 \\pm 0.010$ ($\\eta \\approx 0.11 \\pm 0.02$), $\\Delta_{T_{[1,1]}} \\approx 1.453 \\pm 0.025$ ($\\nu \\approx 0.65 \\pm 0.01$), a relevant $6\\pi$-monopole with $\\Delta \\approx 2.717$, and a relevant parity-even scalar $S$ with $\\Delta_S \\approx 2.845 \\pm 0.010$. The low-lying descendants of each primary sit at integer spacings, indicating an approximate conformal symmetry, but the relevant $S$ means no genuine conformal fixed point is reached; the paper concludes that the O(4) DQCP lives in a pseudo-critical regime, sharing the fate of the SO(5) case.","pith_inferences":["If the relevant singlet $S$ is robust to the symmetry-breaking pattern, pseudo-criticality is likely the generic fate of the Neel-to-VBS transition in any lattice realization, which would explain the persistent disagreements in Monte Carlo studies without invoking a true SO(5) or O(4) fixed point.","The paper's own loose end, the absence of the parity-odd scalar $\\sim \\epsilon_{\\alpha\\beta\\gamma\\delta}\\phi^\\alpha \\partial_t \\phi^\\beta \\partial_x \\phi^\\gamma \\partial_y \\phi^\\delta$ expected from the $\\theta$-term description, suggests that either the topological description of the O(4) DQCP needs revision or the operator sits at a scaling dimension too high for the current spectrum to resolve;","The overlap-based identity tracking used here could be applied to other symmetry-reduction flows, such as Wilson-Fisher $\\mathrm{O}(N) \\to \\mathrm{O}(N-1)$ chains or the Potts and loop-model flows, to decide case by case whether an apparent fixed point is genuine or a walking region; the O(3)-to-O(2) control in the Supplemental Material is a first validation of this diagnostic."],"forward_implications":["The easy-plane Neel-to-VBS transition in O(4)-symmetric models is not a genuine conformal transition: the relevant singlet $S$ forces a weak first-order transition, so measured 'critical' behavior is pseudo-criticality.","The O(4) DQCP inherits the key features of the SO(5) DQCP, namely the same relevant singlet $S$ and the same pseudo-critical interpretation, so the two are governed by the same physics despite the reduced symmetry.","The $6\\pi$-monopole operator is relevant ($\\Delta \\approx 2.717$) and higher monopoles are irrelevant, so on lattices that allow $6\\pi$-monopole events (e.g. honeycomb) the O(4) transition is unstable to that perturbation, while other monopole perturbations are harmless.","Operator decomposition under symmetry reduction follows the branching rules with the parity-odd split components becoming gapped non-conformal fields, and the same flow pattern, including avoided level crossings, appears in the well-understood O(3)-to-O(2) Wilson-Fisher case, indicating the phenomenon is generic to RG flows between fixed points.","The extracted exponents $\\nu \\approx 0.65 \\pm 0.01$ and $\\eta \\approx 0.11 \\pm 0.02$ are what an observer would measure near the O(4) DQCP, and they differ from any genuine O(4)-symmetric CFT."],"supporting_citations":[{"why":"Supplies the SO(5) DQCP precedent: the fuzzy-sphere operator spectrum, the identification of the relevant singlet S, and the pseudo-criticality interpretation that this paper extends to O(4).","marker":"[44]"},{"why":"Introduces the fuzzy sphere regularization method (lowest-Landau-level projection plus state-operator correspondence) used throughout.","marker":"[27]"},{"why":"Sets up the O(4) easy-plane DQCP conjecture and the WZW-term field theory that motivates asking whether an O(4)-symmetric fixed point exists.","marker":"[3]"},{"why":"Provides the duality and symmetry analysis of the DQCP including the theta-term description that predicts a relevant parity-odd scalar at the O(4) transition.","marker":"[17]"},{"why":"Gives prior Monte Carlo evidence for a weakly first-order O(4) transition and approximate O(4) symmetry, the numerical baseline this work's pseudo-criticality conclusion aligns with.","marker":"[8]"},{"why":"Independent conformal bootstrap verification of the relevant singlet S, supporting the pseudo-criticality picture at the SO(5) level.","marker":"[49]"},{"why":"Formalizes the walking and pseudo-criticality scenario (a relevant operator near a complex fixed point) used to interpret a relevant S as implying a weak first-order transition.","marker":"[46–48]"},{"why":"Theory of deconfined pseudo-criticality controlling the RG phase diagram, used to place the O(4) result in the same pseudo-critical regime as the SO(5) case.","marker":"[57]"},{"why":"Supplies the state-operator correspondence that converts eigenenergies on the sphere into scaling dimensions.","marker":"[28]"}],"fun_headline_variants":["O(4) quantum criticality is pseudo-critical, not conformal","Fuzzy sphere reveals SO(5) to O(4) operator switch","Relevant scalar blocks true quantum critical point","Operator flow ends in pseudo-criticality, not fixed point","Avoided crossing exposes pseudo-critical O(4) transition"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the measured energy gaps at $u_5/u = -0.75$ really do encode the scaling dimensions of the would-be O(4) critical theory, meaning the system is close enough to the fixed point that the mapping from spectrum to operator dimensions is trustworthy, even though the fixed point is never exactly reached and conformal symmetry is only approximate.","fun_headline_variants_meta":{"raw":{"variants":["O(4) quantum criticality is pseudo-critical, not conformal","Fuzzy sphere reveals SO(5) to O(4) operator switch","Relevant scalar blocks true quantum critical point","Operator flow ends in pseudo-criticality, not fixed point","Avoided crossing exposes pseudo-critical O(4) transition"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000308,"raw_usage":{"total_tokens":1906,"prompt_tokens":1234,"completion_tokens":672,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":850,"completion_tokens_details":{"reasoning_tokens":587}},"tokens_in":850,"tokens_out":672,"duration_ms":7089,"temperature":1.0,"reasoning_tokens":587,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T20:54:18.118392+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute $\\Delta_S$ at larger system sizes and at parameter values between $u_5/u = -0.75$ and the would-be fixed point: if $\\Delta_S$ extrapolates to 3 or above in the thermodynamic limit, the relevant-singlet claim is falsified and the O(4) transition could be genuinely conformal.","supporting_citations":[{"cited_title":"avoided crossing","cited_arxiv_id":null,"evidence_quote":"Supplies the SO(5) DQCP precedent: the fuzzy-sphere operator spectrum, the identification of the relevant singlet S, and the pseudo-criticality interpretation that this paper extends to O(4)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the fuzzy sphere regularization method (lowest-Landau-level projection plus state-operator correspondence) used throughout."},{"cited_title":"Read and S","cited_arxiv_id":null,"evidence_quote":"Sets up the O(4) easy-plane DQCP conjecture and the WZW-term field theory that motivates asking whether an O(4)-symmetric fixed point exists."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the duality and symmetry analysis of the DQCP including the theta-term description that predicts a relevant parity-odd scalar at the O(4) transition."},{"cited_title":"Serna and A","cited_arxiv_id":null,"evidence_quote":"Gives prior Monte Carlo evidence for a weakly first-order O(4) transition and approximate O(4) symmetry, the numerical baseline this work's pseudo-criticality conclusion aligns with."},{"cited_title":"Gorbenko, S","cited_arxiv_id":null,"evidence_quote":"Independent conformal bootstrap verification of the relevant singlet S, supporting the pseudo-criticality picture at the SO(5) level."},{"cited_title":"Tanaka and X","cited_arxiv_id":null,"evidence_quote":"Theory of deconfined pseudo-criticality controlling the RG phase diagram, used to place the O(4) result in the same pseudo-critical regime as the SO(5) case."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the state-operator correspondence that converts eigenenergies on the sphere into scaling dimensions."}],"review_version":1}