{"id":"7b5a1297-ab73-45e9-86c6-c72fe21e3071","arxiv_id":"2608.10201","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A source-dependent matching formalism connects BOEFT to the Peskin OPE and yields two partial magnetic Wilson coefficients for 1S heavy quarkonium.","lead":"The paper connects two effective-theory frameworks used for heavy quark-antiquark bound states and shows they describe the same gluonic response in their common domain. It also computes two new magnetic correction terms for the lowest 1S state, along with the ratio between magnetic and electric pieces.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (42) is internally inconsistent with Eqs. (43)-(44) and the ⟨r²⟩ sum rule in the Coulombic limit, so the claimed reduction to the Bhanot-Peskin electric moments is not established as written.","rationale":"The reader's conditional verdict remains appropriate, but for a more concrete reason than the acknowledged q_H/q_d smallness. I followed the magnetic sector independently: the M1 inverse moment leading to c_B,GG^(1)=(5π/16)α_s²(c_F V_iso^(s))²δ and the covariant-kinetic seagull c_B,dia^(1)=−(π/4)α_s²δ are consistent with the stated conventions and with the partial-coefficient caveat in Eq. (67). The electric sector, however, contains an internal inconsistency among Eqs. (42), (43), (44), and (46) that can be checked by direct momentum-space evaluation. The final quoted c_E,BP=14π/(3N_c²) is independently obtainable, so the paper's concluding numbers may survive correction, but the central derivation as printed does not reduce the stated spectral measure to the Bhanot-Peskin moments. I also note the unresolved Eq. (??) after Eq. (15), which is a smaller but real completeness defect. The paper is honest about the partial nature of the magnetic claims and the unproven hierarchy, and I do not dispute those caveats; the spectral-density inconsistency is a more definite obstacle and should be fixed before acceptance.","tokens_in":15713,"tokens_out":60594,"duration_ms":550017,"concrete_test":"Recompute Eq. (42) from first principles: with ψ_1S(p)=8√π a0^{3/2}/(1+a0²p²)^2, evaluate ρ^(r)(Δ)=∫d³p/(2π)^3 Σ_i|∂_{p_i}ψ_1S(p)|² δ(Δ−(p²/m_Q+ε0)). The result should be 256/(π ε0) a0² y^{3/2}/(1+y)^6, giving ∫dΔ ρ^(r)=3a0². Then compute the N=1 moment ⟨r^i Δ_o^{-1} r^i⟩=7a0²/(4ε0) and compare with Eq. (46) using d_2 from Eq. (45), which gives 7a0²/(1024ε0). If the discrepancy is confirmed, Eqs. (42)-(46) require correction before the claimed reproduction of the Bhanot-Peskin electric moments can stand.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The Coulombic section's single spectral measure is internally inconsistent. Eq. (42) defines ρ^(r)(Δ)=Σ_i⟨1S|r_iδ(Δ−Δ_o)r_i|1S⟩=(28/π)a0²ε0 y^{3/2}/(1+y)^6, y=Δ/ε0−1. Direct evaluation from the stated wavefunction ψ_1S(p)=8√π a0^{3/2}/(1+a0²p²)^2, using Σ_i|∂_{p_i}ψ|² and the free octet Δ_o=p²/m_Q+ε0, gives ρ^(r)=(256/π)(a0²/ε0) y^{3/2}/(1+y)^6. The printed version fails the elementary sum rule ∫dΔ ρ^(r)=⟨r²⟩=3a0² unless ε0²=64/7, and it is dimensionally 1/M rather than 1/M³. Substituting Eq. (42) into the optical-theorem relation Eq. (44) yields σ=(224π/(3N_c²))a0³ε0³ y^{3/2}/(1+y)^5, while Eq. (43) reads σ=(16/(3πN_c²))a0³ε0 y^{3/2}/(1+y)^5; the two agree only for the special value ε0²=1/(14π²). Because Eq. (46) derives the Bhanot-Peskin moments from this same ρ, the central 'one spectral measure' reduction is not demonstrated as written. This is a checkable algebraic defect, not merely an unproven hierarchy.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs a source-dependent QCD-to-BOEFT matching formalism for the gluonic response of compact heavy quarkonium. The central object is the projected resolvent on a gauge-covariant static-source space, which is then reduced to a channel-factorized local OPE under multipole, source-gap, and weak-coupling hierarchies. In the Coulombic limit the paper claims that the Bhanot-Peskin electric moments and dissociation cut arise from a single spectral measure, and it derives new magnetic Wilson-coefficient contributions for a spin-singlet 1S state: c_{B,GG}^{(1)ij}=(5\\pi/16)\\alpha_s^2(c_F V_{iso}^{(s)})^2\\delta^{ij} and c_{B,dia}^{(1)ij}=-(\\pi/4)\\alpha_s^2\\delta^{ij}, with the explicit caveat that these are identifiable contributions to, not the complete value of, the magnetic matching coefficient.","tokens_in":16055,"tokens_out":17439,"duration_ms":165900,"significance":"If the formal construction and the Coulombic reduction are correct, the paper offers a useful bridge between BOEFT channel structure and the classic Peskin OPE, and the two derived magnetic coefficients are concrete, falsifiable numbers in a regime where they can be checked against pNRQCD. The manuscript is careful in several respects: it states its axioms explicitly, does not fit any free parameters to the new numbers, and is transparent about the incompleteness of the magnetic coefficient. The algebraic computations for the M1 spectral density, its inverse moments, and the diamagnetic seagull are internally consistent. The significance is therefore real, provided the normalization defect in the electric spectral-measure relations is fixed; as written, the paper's central \"one spectral measure\" claim is not yet established.","major_comments":[{"comment":"The claimed reduction of the Bhanot-Peskin electric moments and dissociation cut to one spectral measure is not internally consistent. Direct evaluation from the stated wavefunction \\psi_{1S}(p)=8\\sqrt{\\pi}a_0^{3/2}/(1+a_0^2p^2)^2 and \\Delta_o=p^2/m_Q+\\epsilon_0 gives \\rho^{(r)}_{1S}(\\Delta)=\\frac{256}{\\pi}\\frac{a_0^2}{\\epsilon_0}\\frac{y^{3/2}}{(1+y)^6}\\theta(y). This physical density satisfies \\int d\\Delta\\,\\rho=3a_0^2 and, through Eq. (33), yields c_E^{(1),ij}=14\\pi\\delta^{ij}/(3N_c^2), so it is evidently the intended normalization. Yet substituting it into Eq. (44) produces \\sigma=(2048\\pi/(3N_c^2))a_0^3\\epsilon_0 y^{3/2}/(1+y)^5, disagreeing with the coefficient 16/(3\\pi N_c^2) in Eq. (43). Similarly, Eq. (46) is off by a factor 256: at N=1 the left-hand side is 7a_0^2/(4\\epsilon_0), while the right-hand side, with d_2 from Eq. (45), is 7a_0^2/(1024\\epsilon_0). The equivalence of the BOEFT measure, the Bhanot-Peskin moments, and the dissociation cut therefore fails as written, and a missing power-of-two/\\pi normalization factor must be identified and corrected.","section":"Section 3, Eqs. (42)-(47)"},{"comment":"The channel-factorized local OPE reduction rests on the inequalities q_H,q_d\\ll 1 in Eq. (35), but the paper explicitly notes that these are sufficient operator-theoretic criteria and do not by themselves establish smallness in QCD. The power-counting estimates in Eq. (37) are schematic rather than derived from \\delta H_{\\rm light} and \\delta D. Because the abstract and Section 4 claim that the full response reduces to the Peskin construction under the hierarchy Eq. (39), this is a load-bearing point: either an explicit estimate of q_H and q_d in the weak-coupling domain m_Q v\\gg\\Lambda_{\\rm QCD} should be supplied, or the statement should be weakened to an assumption. The Coulombic numbers are computed in the strict free-octet limit and are not affected by this issue, but the general formal claim is.","section":"Section 2.4, Eqs. (35)-(39)"},{"comment":"The formal setup relies on several mathematical assertions that are stated but not proved: self-adjointness of the static transfer Hamiltonian with respect to the inner product (5), the transfer-matrix spectral representation for Gauss-law-constrained states, and the existence of the isometry J in Eq. (8). Since the spectral representation of \\Delta_Q in Eq. (16) and the measure in Eq. (22) are built on these assertions, the authors should either provide proofs or explicitly present these as assumptions of the matching construction rather than as derived facts.","section":"Section 2.1, Eqs. (5)-(16)"}],"minor_comments":[{"comment":"The typesetting of the coefficient in Eq. (42) is ambiguous: what should be 2^8/\\pi appears as \"28/\\pi\", and the factor a_0^2/\\epsilon_0 appears as a_0^2\\epsilon_0. Please clarify, because the dimensional and sum-rule checks depend on the correct reading.","section":"Eq. (42)"},{"comment":"There is an unresolved cross-reference \"Equation(??)\" in the text immediately after Eq. (15); the intended equation number should be filled in.","section":"Eq. (15)"},{"comment":"The paper should explicitly distinguish the normalizations of \\rho^{(r)}_{1S} (which has dimension length cubed and reproduces \\langle r^2\\rangle) from \\rho^{(0)}_{M,1S} (which is normalized to unity). The current notation invites confusion between a transition-strength density and a probability density.","section":"Eqs. (42), (56), (57)"},{"comment":"The phrase \"standard optical-theorem normalization\" should spell out the color, polarization, and flux factors used to relate \\sigma to \\rho^{(r)}; as written, Eq. (44) is not consistent with the displayed Eqs. (42) and (43), and the missing factors should be exhibited explicitly.","section":"Section 3, around Eq. (44)"},{"comment":"The statement that Eq. (47) follows from Eq. (46) is not supported by the displayed definitions: with d_2 from Eq. (45), one has c_E^{(1),ij}=\\delta^{ij}d_2/2=7\\pi\\delta^{ij}/(384N_c^2), not the quoted 14\\pi\\delta^{ij}/(3N_c^2). The definition of d_n or the prefactor in Eq. (45) must be corrected.","section":"Section 3, Eq. (47) and surrounding text"}],"recommendation":"major_revision","confidential_remarks":"The paper is part of a series with several self-citations (Refs. [18], [28], and the companion Ref. [10]); that pattern is not itself a concern, but it means the normalization defect in the electric section should not be dismissed as a trivial typo. The M1 and diamagnetic results are independently checkable and appear sound; the revision should focus on restoring the internal consistency of Eqs. (42)-(47) and on making the status of the q_H,q_d smallness assumptions explicit."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the formal architecture is worth a referee's time, but the paper's central Coulombic consistency check is broken as printed. Eq. (42) is supposed to be the spectral measure that reproduces Bhanot–Peskin. It doesn't. Direct evaluation from the 1S momentum-space wavefunction gives (256/π)(a0²/ε0) y^{3/2}/(1+y)^6, not the printed (28/π)(a0²/ε0) y^{3/2}/(1+y)^6. The printed version fails the elementary sum rule ∫ρ dΔ = ⟨r²⟩ = 3a0² by a factor 64/7. The stress-test's dimensional aside is wrong—a0²/ε0 is dimensionally fine—but the coefficient defect is real. And even after replacing 28 by 256, plugging ρ into the optical-theorem relation Eq. (44) still doesn't reproduce the Bhanot–Peskin cross section Eq. (43); the coefficients mismatch by a constant factor. So the 'one spectral measure' story, which is the paper's own summary of the electric limit, is not demonstrated as written. This is fixable algebra, but it is load-bearing.\n\nWhat is genuinely new: the source-dependent projected response on the BOEFT static-source space, the Feshbach separation, and the explicit magnetic Wilson-coefficient contributions, c_B,GG and c_B,dia. Those numbers are the interesting part. The paper is also honest about what remains unproven—q_H and q_d smallness in QCD, and the incompleteness of the magnetic coefficient.\n\nSoft spots beyond the Coulombic algebra: the formal matching relies on several asserted operator-theoretic steps (self-adjointness of the transfer Hamiltonian, Gauss-law spectral representation, the isometry J) that are plausible but not fully demonstrated. There is also a dangling 'Eq. (??)' in Section 2.1 that should have been caught. The author's self-citations are used to supply the matching formalism, which is normal in a research program and not by itself a problem.\n\nWho this is for: people working on heavy-quark EFT and quarkonium–gluon responses. It deserves a serious referee, but only after the Coulombic section is reconstructed. Send it to review with a clear request to fix Eq. (42) and re-check the connection between the spectral measure, the cross section, and the moments.","headline":"The BOEFT-to-Peskin bridge and magnetic Wilson coefficients are new and plausible, but the paper's Coulombic consistency check has a concrete algebraic error that must be fixed before the electric reduction is credible.","tokens_in":16605,"tokens_out":16858,"would_cite":false,"duration_ms":146010,"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":"A single projected gluonic response connects heavy-quarkonium Born-Oppenheimer EFT to Peskin's local OPE and yields two new magnetic Wilson coefficients.","keywords":["heavy quarkonium","Born-Oppenheimer EFT","potential NRQCD","Peskin OPE","gluonic response","chromomagnetic matching","gluo-dissociation","quarkonium polarizability"],"falsifier":"Compute, in lattice QCD or in a weakly coupled model, the norms $q_H$ and $q_d$ of Eq. (35) for a compact 1S state, together with the source-accessible gap $\\Delta_{\\rm src}$; if either norm is not parametrically small while $\\Delta_{\\rm src}$ is large, or if the subthreshold inverse moments of the projected response fail to satisfy the Bhanot-Peskin moment relation (46), then the channel-factorized local OPE is not the leading description. Equivalently, a direct NRQCD/pNRQCD computation of the magnetic polarizability for the spin-singlet 1S state that disagrees with the partial ratio $\\frac{3}{14}v_C^2$ at tree level would falsify the new matching coefficients.","tokens_in":15461,"feed_emoji":"⚛️","tokens_out":14515,"duration_ms":130697,"temperature":0.7,"pith_summary":"The paper aims to show that the full gluonic response of a compact heavy-quarkonium state, computed on the gauge-covariant static-source Hilbert space of Born-Oppenheimer EFT, is the same object that Peskin's short-distance expansion describes: under explicit hierarchies (multipole locality, a source-accessible spectral gap, small propagation and source-creation corrections, and weak coupling), the response collapses to a channel-factorized local operator product expansion, with the same electric moments and dissociation cut as in the established Bhanot-Peskin construction. The new quantitative content is magnetic: for a spin-singlet 1S Coulombic state, the isotropic sequential M1 inverse moment gives $c_{B,GG}^{(1)ij} = (5\\pi/16)\\alpha_s^2 (c_F V_{\\rm iso}^{(s)})^2 \\delta^{ij}$, and the leading covariant-kinetic diamagnetic seagull gives $c_{B,\\rm dia}^{(1)ij} = -(\\pi/4)\\alpha_s^2 \\delta^{ij}$, both stated as identifiable contributions to, not the complete value of, the magnetic matching coefficient. A sympathetic reader would care because this supplies a bridge between two widely used effective-field-theory descriptions of quarkonium and produces concrete, testable Wilson coefficients at the same velocity order as the electric polarizability.","feed_headline":"Same spectrum yields Peskin moments and new M1 coefficients","feed_subtitle":"A single projected resolvent connects Born-Oppenheimer EFT to the standard local OPE and fixes two magnetic terms.","key_machinery":"The load-bearing object is the source-dependent projected response $\\alpha^{ab}_{BO,ij}(\\omega)$ of Eq. (16), defined as the fixed-singlet contraction of the resolvent $(\\Delta_Q - z_\\pm)^{-1}$ on the auxiliary gauge-covariant static-source Hilbert space $K_{\\rm adj}$, with $\\Delta_Q$ the physical pullback of $Q_\\phi(H_0 - E_\\phi)Q_\\phi$ and $|d_i^a\\rangle$ the short-distance E1 transition vector. The structure that makes the argument run is the Feshbach-Schur separation: retained BO channels stay in a dynamical resolvent $\\mathcal{S}_R(z)^{-1}$, while gapped sectors are integrated out and expanded in inverse powers of the source-accessible gap $\\Delta_{\\rm src}$, producing local Wilson coefficients from inverse moments of a positive semidefinite spectral measure $\\mathrm{d}\\mu^{ab}_{ij}(\\Delta) = \\delta^{ab}\\mathrm{d}\\mu_{ij}(\\Delta)$. The same measure's boundary value on the cut yields the dissociation kernel, so locality (inverse moments) and absorption (cut) are two readings of one object. The sufficient criteria for channel-factorized locality are the propagation-vertex bounds $q_H \\ll 1$, $q_d \\ll 1$ of Eq. (35), with the weak-coupling hierarchy of Eq. (38) making the reference propagator and vertices calculable in pNRQCD.","core_discovery":"The central claim is that the projected, source-dependent response of Eq. (16), built from the auxiliary adjoint static-source space and the singlet map of BOEFT, is the correct matching object: before any derivative expansion or partonic projection, it separates the BO channels kept dynamical from gapped sectors encoded as local inverse moments, and under the hierarchy of Eq. (39) it reduces exactly to Peskin's local OPE. In the Coulombic, leading-E1, free-octet, forward-on-shell-gluon limit, the paper shows the known Bhanot-Peskin electric coefficients $c_{E,BP}^{(N)ij} = \\delta^{ij} d_{2N}/2$ and the dissociation kernel both follow from the boundary value and inverse moments of one spectral measure $\\mathrm{d}\\mu_{ij}(\\Delta)$. As a magnetic extension, for a spin-singlet 1S state the subthreshold inverse moment of the known isotropic M1 spectral density yields $c_{B,GG}^{(1)ij} = \\frac{5\\pi}{16}\\alpha_s^2(c_F V_{\\rm iso}^{(s)})^2\\delta^{ij}$, and the covariant-kinetic seagull, whose coefficient is fixed by Poincar\\'e invariance, yields $c_{B,\\rm dia}^{(1)ij} = -\\frac{\\pi}{4}\\alpha_s^2\\delta^{ij}$. These are explicitly identified as contributions to, not the complete value of, the magnetic matching coefficient; the physical $\\alpha_B/\\alpha_E$ ratio requires the full decomposition of Eq. (67), including Hamiltonian insertions, higher one-field vertices, and irreducible two-field matching terms.","pith_inferences":["The sufficient criteria of Eqs. (35)-(36) could be tested directly in lattice or model calculations: measure the source-accessible gap and the norms of the light-channel perturbation and source correction; if $q_H$ or $q_d$ is not small, the mismatch would appear as nonlocal or channel-dependent corrections that no local Wilson coefficient can absorb.","Because the diamagnetic seagull carries no independent Wilson coefficient, a precision determination of the full magnetic coefficient, say from lattice NRQCD or from quarkonium in a magnetic field, could isolate the Hamiltonian-insertion and irreducible two-field contributions that the paper leaves uncomputed.","The same projected-resolvent construction should extend to spin-triplet, hybrid, and open-flavor channels; in noncentral or coupled BO channels the $GT$ interference between isotropic and traceless magnetic vertices need not vanish and would enter at relative $O(\\alpha_s)$, affecting medium-modification predictions.","One testable extension is to compute the next inverse moment ($N=2$) of the same M1 spectral measure and compare its evolution under the enlarged renormalization group with the twist-two sector, which would probe whether the correlated trace components matter numerically."],"forward_implications":["Under the hierarchy of Eq. (39), the full BOEFT gluonic response reduces to a channel-factorized local OPE, so the standard Bhanot-Peskin electric moments and the gluo-dissociation cut are recovered from a single spectral measure rather than from separate constructions.","The new magnetic coefficient $c_{B,GG}^{(1)ij} = \\frac{5\\pi}{16}\\alpha_s^2(c_F V_{\\rm iso}^{(s)})^2\\delta^{ij}$ for a spin-singlet 1S state comes with definite polarization averages: $1/3$ for a vector state and $1/2$ for the full hyperfine multiplet.","The covariant-kinetic diamagnetic seagull $c_{B,\\rm dia}^{(1)ij} = -\\frac{\\pi}{4}\\alpha_s^2\\delta^{ij}$ is spin-independent and its coefficient is fixed by Poincar\\'e invariance, so the combined partial ratio at strict tree level is $(c_{B,GG} + c_{B,\\rm dia})/c_{E,BP} = \\frac{3}{14} v_C^2$, showing that two M1 insertions are suppressed by $v^2$ relative to two E1 insertions.","The physical magnetic polarizability $\\alpha_B/\\alpha_E$ is the ratio of the full matched coefficients in Eq. (67), not merely the two computed pieces; the remaining Hamiltonian, higher-one-field, and irreducible two-field terms contribute at the same nominal order in the Coulombic counting.","At the bare matching scale the electric tower decomposes into a twist-two gluon operator plus correlated metric-trace components that renormalize separately, so a common bare coefficient does not imply common renormalization-group evolution."],"supporting_citations":[{"why":"Defines the short-distance OPE for compact quarkonium and the electric moments and dissociation cross section that the paper's reduction must reproduce.","marker":"[1, 2]"},{"why":"Gives the pNRQCD octet-response formula, normalization conventions, and the 1/3 isotropic factor used to fix signs and factors in Eq. (2) and the local Lagrangian.","marker":"[6]"},{"why":"Supplies the gauge-covariant BOEFT static-source space and adjoint-fiber transformation rules on which the projected response is built.","marker":"[9]"},{"why":"Provides the leading-twist gluon operator basis, moment function f(x), and d_n normalization used to connect inverse moments to Bhanot-Peskin coefficients.","marker":"[13]"},{"why":"Derives the Bhanot-Peskin limit in pNRQCD, including Coulombic gluo-dissociation and the finite-N_c octet final-state correction delimiting the strict limit.","marker":"[15]"},{"why":"Introduces the normalized source-dependent residue and pole definition adopted here, fixing how the response is extracted from two-time correlators.","marker":"[18]"},{"why":"Calculates the M1 singlet-octet gluo-dissociation mechanism and Coulombic spectral density used for the isotropic M1 inverse moment.","marker":"[19]"},{"why":"Provides the spin-resolved pNRQCD M1 vertex tensors and the tensor-basis map used in Eqs. (50)-(51).","marker":"[20]"},{"why":"Fixes the covariant-kinetic coefficient to unity via Poincar\\'e invariance, so the diamagnetic seagull carries no independent Wilson coefficient.","marker":"[30]"},{"why":"Supplies the conserved-pseudomomentum construction used to derive the diamagnetic seagull in a constant Cartan background.","marker":"[31]"}],"fun_headline_variants":["Same spectrum yields Peskin moments and M1 contributions","One projected response links BOEFT to Peskin OPE","Magnetic matching coefficients from BOEFT-Peskin connection","New magnetic terms via single spectral measure in BOEFT"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the corrections to how gluonic channels propagate between the two source insertions, and to how the source creates those channels, are genuinely small in QCD; the paper proves sufficient bounds for these corrections but does not itself establish that smallness from the EFT hierarchy.","fun_headline_variants_meta":{"raw":{"variants":["Same spectrum yields Peskin moments and M1 contributions","One projected response links BOEFT to Peskin OPE","Magnetic matching coefficients from BOEFT-Peskin connection","New magnetic terms via single spectral measure in BOEFT"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000422,"raw_usage":{"total_tokens":2241,"prompt_tokens":1089,"completion_tokens":1152,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":705,"completion_tokens_details":{"reasoning_tokens":1087}},"tokens_in":705,"tokens_out":1152,"duration_ms":10109,"temperature":1.0,"reasoning_tokens":1087,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T04:10:55.391636+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, in lattice QCD or in a weakly coupled model, the norms $q_H$ and $q_d$ of Eq. (35) for a compact 1S state, together with the source-accessible gap $\\Delta_{\\rm src}$; if either norm is not parametrically small while $\\Delta_{\\rm src}$ is large, or if the subthreshold inverse moments of the projected response fail to satisfy the Bhanot-Peskin moment relation (46), then the channel-factorized local OPE is not the leading description. Equivalently, a direct NRQCD/pNRQCD computation of the magnetic polarizability for the spin-singlet 1S state that disagrees with the partial ratio $\\frac{3}{14}v_C^2$ at tree level would falsify the new matching coefficients.","supporting_citations":[{"cited_title":"Brambilla, G","cited_arxiv_id":null,"evidence_quote":"Gives the pNRQCD octet-response formula, normalization conventions, and the 1/3 isotropic factor used to fix signs and factors in Eq. (2) and the local Lagrangian."},{"cited_title":"Complex chromoelectric polarizability of a heavy-quarkonium resonance: pole definition and channel-complete pNRQCD matching","cited_arxiv_id":"2608.00611","evidence_quote":"Introduces the normalized source-dependent residue and pole definition adopted here, fixing how the response is extracted from two-time correlators."},{"cited_title":"Chen and M","cited_arxiv_id":null,"evidence_quote":"Calculates the M1 singlet-octet gluo-dissociation mechanism and Coulombic spectral density used for the isotropic M1 inverse moment."},{"cited_title":"Yang and X","cited_arxiv_id":null,"evidence_quote":"Provides the spin-resolved pNRQCD M1 vertex tensors and the tensor-basis map used in Eqs. (50)-(51)."},{"cited_title":"Brambilla, D","cited_arxiv_id":null,"evidence_quote":"Fixes the covariant-kinetic coefficient to unity via Poincar\\'e invariance, so the diamagnetic seagull carries no independent Wilson coefficient."},{"cited_title":"Alford and M","cited_arxiv_id":null,"evidence_quote":"Supplies the conserved-pseudomomentum construction used to derive the diamagnetic seagull in a constant Cartan background."}],"review_version":1}