{"id":"c20f73f0-6a11-49c9-b851-c36f23d4473d","arxiv_id":"2509.04956","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":10,"one_line_summary":"A WKB inversion of power-law Regge trajectories yields a non-quadratic holographic dilaton Φ(z)=(κz)^(2−α) that the authors then test on charmonium, bottomonium, and tetraquark candidates.","lead":"This paper turns hadron mass spectra into a holographic confining potential by inverting the WKB spectrum formula, obtaining a non-quadratic dilaton profile for heavy quarkonia and tetraquarks. It matters because it offers a bottom-up recipe for building holographic QCD models directly from data, if the derivation holds.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Equation (18) does not follow from Eq. (17): the printed exponent and Gamma-function placement are inconsistent, so the central formulas for V*(z) and (kappa, alpha) are not derived as written.","rationale":"The paper's central claim hinges on deriving the large-z confining potential from the power-law spectrum through Eqs. (17)-(20), and then mapping it to the dilaton through Eqs. (25)-(27) and (30)-(31). The algebraic link from Eq. (17) to Eq. (18) is the load-bearing step: if it is wrong, the coefficient C(nu,a) and all derived (kappa,alpha) are unsupported, regardless of how well the resulting potential fits data. My independent evaluation of Eq. (17) shows that the printed Eq. (18) is inconsistent in both the power of V* and the Gamma-function ratio; Eq. (19), however, matches the correct evaluation, suggesting a pair of typographical errors rather than a wrong physical idea. The nu=1 check reinforces this: the printed Eq. (29) conflicts with Eq. (27), so the softwall limit is not consistently recovered in the text. The reader's verdict correctly identified an internal inconsistency in the central derivation, but I would not reject the whole approach outright because the correct integral supports Eq. (19); the appropriate outcome is a conditional resubmission in which the authors fix the algebra, re-derive Eqs. (18)-(20) and (29)-(31) carefully, and then separate confirmed predictions from refits. The reader's additional concerns about low-z contributions, the circularity of the quarkonia fits, and the state-by-state parameter choices in the tetraquark section are legitimate and strengthen the need for a revised manuscript, but the equation-level inconsistency is the most immediate technical barrier to accepting the central claim.","tokens_in":18045,"tokens_out":33590,"duration_ms":289304,"concrete_test":"Symbolically evaluate the Beta integral in Eq. (17) and compare with Eq. (18) and with Eq. (19) after inversion; check both the exponent of V* and the placement of Gamma(1/nu) versus Gamma((nu+2)/(2 nu)). Then set nu=1 in Eqs. (19), (27), and (29) and verify whether kappa = sqrt(a)/2 follows from all three. If Eq. (18) does not match the evaluated integral, or if the nu=1 limit gives conflicting kappa values, the central formulas are unsupported as printed and require correction before the inverse-problem claim can be assessed.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Independent evaluation of Eq. (17), using n'(M^2)=(1/(a nu))(M^2/a)^{1/nu - 1} from the trajectory (15), gives z(V*) = 2/(a^{1/nu} nu) V*^{1/nu - 1/2} * Beta(1/nu, 1/2) = 2 pi^{1/2}/(a^{1/nu} nu) * Gamma(1/nu)/Gamma((nu+2)/(2 nu)) * V*^{(2-nu)/(2 nu)}. Equation (18) prints V*^{(2-nu)/nu} and the reciprocal Gamma ratio, so it cannot be inverted to produce Eq. (19). The subsequent identifications (25)-(27) and (30)-(31), and therefore the claimed unique (kappa, alpha), rest on this unsupported inversion. The nu=1 limit is an independent red flag: Eqs. (19) and (27) give C(1,a)=a^2/16 and kappa = sqrt(a)/2, while Eq. (29) claims kappa = (a/2)^{1/2}, so the softwall limit is not recovered consistently. Because the integral itself can be evaluated correctly and then agrees with Eq. (19), the construction may be repairable; but as printed the central derivation is internally inconsistent and cannot support the abstract's central claim.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a WKB/RKR inverse problem for bottom-up holographic QCD: starting from a power-law radial Regge trajectory M_n^2 = a(n+b)^ν, it claims to derive the large-z confining potential V_*(z) = C(ν,a) z^{2ν/(2−ν)} and hence a non-quadratic dilaton Φ(z) = (κz)^{2−α} with κ and α uniquely fixed by a and ν. The method is then applied to charmonium and bottomonium vector spectra, where the fitted (κ,α) are used to compute radial and orbital excitations, and is extended to tetraquark states by adding a linear potential term derived from a Bethe-Salpeter diquark trajectory. The central claim is that measuring a nonlinear Regge trajectory is sufficient to engineer the confining dilaton of the holographic model.","tokens_in":18359,"tokens_out":10825,"duration_ms":92644,"significance":"If the derivation were correct and the validation sound, the idea would be a useful model-building tool: it would turn empirical Regge trajectories into direct constraints on the holographic dilaton profile, extending the softwall model to nonlinear spectra in a systematic way. The paper also makes a concrete, testable prediction for the exponent relation α = 2(1−ν)/(2−ν), and it applies the formalism to a broader set of states (orbital excitations and tetraquark candidates) than is common in bottom-up holography. Credit is due for being explicit about the fitting parameters and for reporting RMS errors per channel. However, as printed the central RKR inversion contains algebraic inconsistencies that invalidate the derivation as written, and the radial-spectrum agreement is largely a refit of the input trajectory. The genuine predictions, namely the orbital and pseudoscalar states, show large errors (10–20%), so the paper's current evidence does not support the strength of the conclusions.","major_comments":[{"comment":"The RKR inversion is internally inconsistent as printed. Equation (17) contains (V_*−M_n^2)^{1/2} in the numerator, whereas the Abel/RKR inversion requires (V_*−M_n^2)^{-1/2} after changing variables from n to M^2. Consequently, the printed Eq. (18) — with exponent (2−ν)/ν and the reciprocal ratio of Gamma functions — does not follow from Eq. (17), and Eq. (18) cannot be inverted to produce Eq. (19). Evaluating the correctly posed integral gives z(V_*) ∝ V_*^{(2−ν)/(2ν)} with Γ(1/ν)/Γ((ν+2)/(2ν)); with that replacement Eq. (19) does follow. As printed, the derivation of C(ν,a), κ, and α is unsupported. Because these equations are the basis for the abstract's central claim, this is a load-bearing error.","section":"Sec. III, Eqs. (16)–(20)"},{"comment":"The softwall limit is not recovered consistently. Setting ν=1 in Eq. (19) gives C(1,a)=a^2/16, and Eq. (27) gives κ=√a/2. Equation (29) instead states κ=(a/2)^{1/2}, and the following sentence claims the spectrum M_n^2=4a(n+b), whereas the softwall spectrum for Φ=κ^2z^2 is M_n^2=4κ^2(n+b). The factor-of-two discrepancy must be resolved; as written, the claimed ν=1 limit is internally contradictory.","section":"Sec. III, Eqs. (26)–(29)"},{"comment":"The agreement claimed as a test is partly circular. The radial S-wave states in Table I are used to fit a, b, and ν, which then determine (κ,α) through Eqs. (30) and (31); the small errors of the S-wave rows in Tables II and III therefore reflect a refit of the input data, not an independent prediction. The genuine predictions are the orbital and pseudoscalar states, and there the errors are large: Table II shows 1^3D_2 at 15.98%, 1^3P_2 at 21.04%, and η_c(1S) at 21.8%; Table III shows P-wave states at 8–10% and η_b(1S) at 14.7%. The conclusions should separate the refit from the prediction and temper the statement that the model is 'validated.'","section":"Sec. III.A, Tables I–III"},{"comment":"The dilaton is extracted from the large-z asymptotic relation V_*(z) ≃ Φ'(z)^2/4, but the same Φ is then used in the full potential (24), which includes the small-z 1/z^2 term and the β-dependent terms. Because the authors attribute the large errors in the low-lying orbital states to the small-z/NRQCD region, the extracted (κ,α) are not demonstrably the parameters of the true potential controlling those states. A sensitivity study, for example varying the matching scale z_∞ or the treatment of the low-z terms, is needed before the inverse-problem claim can be considered robust.","section":"Sec. III, Eqs. (24)–(25) and Tables II–III"},{"comment":"The tetraquark extension is presented as a postdiction exercise rather than a predictive test. The value g_eff=1/(2π) is stated to be fixed numerically by minimizing the RMS error, which makes it a free parameter despite the claim that the parameters 'are not free in the usual sense.' In addition, the per-state integers i,j and the spin assignments are chosen after the fact — for example, T_cc(3875)^+ is assigned both 1^+ and 0^+ configurations with different resulting masses. The authors should state explicitly which entries in Table IV are predictions made before comparison with data and which are fits.","section":"Sec. IV, Eq. (43) and Table IV"}],"minor_comments":[{"comment":"'Rydberg-Klein-Ross' should be 'Rydberg-Klein-Rees.'","section":"Sec. III, text before Eq. (16)"},{"comment":"'divination from linearity' should be 'deviation from linearity.'","section":"Sec. III, after Eq. (15)"},{"comment":"'yiedls' should be 'yields.'","section":"Sec. III.A, after Eq. (38)"},{"comment":"The Υ(5S) mass entry '108852.6 1.6' appears to be missing a decimal point and proper error formatting.","section":"Table III"},{"comment":"The notation dM^2/dn and dM_n^2 in Eq. (16) is ambiguous; please use a consistent integration variable and make the denominator, including the square-root factor, explicit.","section":"Eq. (16)"},{"comment":"The word 'apendix' is a typo, and the calibration curves (B3)–(B4) are fits to the isoscalar spectra; they should be flagged as empirical inputs rather than predictions of the WKB inversion.","section":"Appendix B"}],"recommendation":"major_revision","confidential_remarks":"The paper is a good candidate for major revision rather than rejection: the central inversion is repairable, and the intended Eq. (19) appears to be correct once Eq. (17) is fixed. However, the numerical tables need to be recomputed after correcting the derivation, and the framing of the test needs to be made honest regarding the circularity of the radial S-wave comparison. The relationship to Ref. [6] should also be clarified: the nonquadratic dilaton profile is already postulated there, so the novelty of the present work is the inversion algorithm, not the profile itself."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper's central derivation does not hold up as printed. Eq. (17) writes the RKR integrand with the wrong power of (V*−M²) and the wrong exponent on (M²/a), and Eq. (18) does not follow from it: the exponent of V* and the Gamma ratio are both off compared to the correct evaluation. The ν=1 limit compounds the problem—Eq. (27) gives κ=√a/2, while Eq. (29) prints κ=√(a/2), which is not the softwall relation. That means Eqs. (25)-(27) and the headline formulas (30)-(31) for (κ,α) are not actually derived.\n\nStill, the idea is worth taking seriously. It addresses a real issue: heavy quarkonia have nonlinear Regge trajectories, and the standard quadratic dilaton cannot describe them. Using a WKB/RKR inverse method to go from a spectral parameterization to the large-z confining potential is a natural move, and the explicit closed forms for κ(a,ν) and α(a,ν) are a useful addition if the derivation can be fixed. The authors also deserve credit for testing on both charmonium and bottomonium, and for being honest that the low-z part of the potential fails to capture NRQCD effects—the orbital state errors (up to ~20% in charmonium) are stated plainly.\n\nThe soft spots are real but not all equally soft. The radial-state agreement is largely a refit of the input trajectory, so it is not independent evidence. The tetraquark section is a toy model: it adds per-state quantum numbers, indices i,j, and g_eff tuned to minimize RMS, so the 6.26% RMS does not carry much weight. The self-citation pattern is heavy, but the RKR inversion method and the nonquadratic dilaton do come from the authors' own prior works, so that is not a flaw by itself.\n\nWho is this for? Holographic model builders who want a recipe for translating measured trajectories into dilatons. They should wait until the derivation is corrected. The stress-test note shows the integral can be evaluated correctly and then agrees with Eq. (19), so the construction is likely repairable. I would not reject on the spot—I would send it to a referee who can verify the algebra and ask for a corrected Section III, a resolution of the ν=1 inconsistency, and a clear separation of fits from predictions in the tables.","headline":"The central WKB inversion is internally inconsistent as printed, but the program is repairable and worth a referee.","tokens_in":18972,"tokens_out":9592,"would_cite":false,"duration_ms":72617,"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":"The nonlinearity of a hadron mass spectrum determines the confining dilaton in holographic QCD.","keywords":["holographic QCD","dilaton","Regge trajectories","WKB approximation","Rydberg-Klein-Rees inversion","heavy quarkonia","tetraquarks","confinement"],"falsifier":"Compute the charmonium and bottomonium spectra from the extracted dilatons while keeping the $\\Phi''$ and $\\beta$-dependent terms that the matching condition drops; if the lowest eigenvalues shift by more than the reported few-percent errors, the central inversion assumption is violated. Alternatively, repeat the extraction on the $\\eta_c$ pseudoscalar trajectory and check whether it yields the same $\\kappa$ and $\\alpha$ as the vector fit.","tokens_in":17757,"feed_emoji":"⚛️","tokens_out":16958,"duration_ms":134855,"temperature":0.7,"pith_summary":"The paper claims that the confining part of a bottom-up holographic QCD model can be reconstructed from hadron masses instead of being postulated. Starting from a measured radial spectrum $M_n^2 = a(n+b)^\\nu$, it applies the Rydberg-Klein-Rees (RKR) semiclassical inversion formula to derive the large-coordinate confining potential $V_*(z) = C(\\nu,a)\\, z^{2\\nu/(2-\\nu)}$, and then recovers a non-quadratic dilaton $\\Phi(z) = (\\kappa z)^{2-\\alpha}$ whose parameters $\\kappa$ and $\\alpha$ are fixed by the spectral slope $a$ and exponent $\\nu$. If the claim is right, measuring one hadron family's Regge trajectory is enough to determine the confining dynamics of its holographic dual and to predict the rest of the spectrum, including orbital excitations. The paper reports good agreement for charmonium and bottomonium, with RMS errors of about 12.2 percent over 17 states and 7.7 percent over 19 states, and extends the same machinery to tetraquarks by adding a diquark-potential term.","feed_headline":"One Regge trajectory fixes the confining holographic dilaton","feed_subtitle":"The recipe uses only two fitted numbers and reproduces charmonium and bottomonium masses to about ten percent.","key_machinery":"The load-bearing object is the Rydberg-Klein-Rees (RKR) inversion formula, $z(V_*) = 2\\int_0^{V_*} (dM_n^2/dn)^{-1}\\, (V_*-M_n^2)^{1/2}\\, dM_n^2$, which reconstructs the potential at its turning point from the spectrum and its derivative. Evaluated on the power-law ansatz, it yields the explicit confining potential $V_*(z)=C(\\nu,a)\\, z^{2\\nu/(2-\\nu)}$. The second mechanism is the large-$z$ matching condition $V_*(z)\\simeq \\Phi'(z)^2/4$, which converts the spectral exponent $\\nu$ and slope $a$ into the dilaton parameters $\\kappa$ and $\\alpha$ through Eqs. (30) and (31).","core_discovery":"The central discovery is that the inverse spectroscopic problem has a definite large-distance answer: a power-law radial trajectory $M_n^2 = a(n+b)^\\nu$ with $0<\\nu<2$ is produced by a holographic confining potential $V_*(z)=C(\\nu,a)\\, z^{2\\nu/(2-\\nu)}$, with the coefficient fixed by the RKR integral. The corresponding dilaton is not quadratic unless $\\nu=1$; it takes the form $\\Phi(z)=(\\kappa z)^{2-\\alpha}$ with $\\alpha = 2(1-\\nu)/(2-\\nu)$ and $\\kappa$ given explicitly in terms of $a$ and $\\nu$. The paper therefore presents the non-quadratic dilaton previously used for heavy quarkonia as a consequence of the spectrum's nonlinearity rather than as an input. It validates this by fitting the $c\\bar c$ and $b\\bar b$ vector trajectories with two parameters and computing the full radial and orbital spectra from the resulting $\\Phi(z)$, finding the largest deviations in the low-lying pseudoscalar and $P$/$D$ states, which it attributes to short-distance effects not captured by the confining large-$z$ part. For tetraquarks, it superimposes an additional potential $\\tilde V(z)\\propto z$ obtained from the Bethe-Salpeter equation for diquarks and reports an RMS error of about 6.3 percent across 17 candidate states.","pith_inferences":["A natural extension not performed in the paper: extract $\\kappa$ and $\\alpha$ independently from the $\\eta_c$ pseudoscalar trajectory and compare with the vector extraction; agreement would confirm that the confining large-$z$ part is spin-independent, while disagreement would quantify the short-distance contamination of the inversion.","If the inversion is unique, the same formula should transfer across hadron families: a future precise measurement of the fifth or sixth radial excitation of charmonium should reproduce the already extracted $\\kappa$ without refitting, providing a sharp out-of-sample test.","The dropped terms in the matching condition, namely the $\\beta$-dependent and $\\Phi''$ contributions at large $z$, could be evaluated numerically on the extracted dilatons; their contribution to the lowest eigenvalues would directly measure how much of the reported RMS error is approximation error rather than experimental scatter.","The tetraquark construction assumes compact diquark-antidiquark clusters; a molecular hypothesis would replace $\\tilde V(z)$ with a short-range interaction, which would predict a different scaling of excited exotic-state masses and could be checked against the growing catalog of candidates."],"forward_implications":["For any hadron family whose radial masses follow $M_n^2=a(n+b)^\\nu$, the dilaton is fixed before any holographic calculation: $\\alpha$ comes from the concavity $\\nu$ and $\\kappa$ from the slope $a$, leaving no free confining parameter in the model.","The quadratic softwall dilaton and the linear Regge trajectories it produces are recovered as the special case $\\nu=1$, so the framework contains the standard light-meson result as a limit.","The same two-parameter dilaton predicts orbital and pseudoscalar partners of the fitted vector states; the paper's own tables show those predictions degrade for low-lying $P$ and $D$ states, tying the accuracy of the inversion to the dominance of the confining term.","For tetraquarks, the added potential $\\tilde V(z)\\propto z$ is not a new fit but the WKB image of a diquark Regge slope $n^{2/3}$, with the string tension fixed from the charmonium mass shift, so exotic-state spectroscopy becomes a prediction of the same inverse recipe.","Because the inversion uses only the functional form of the trajectory, the same RKR step can be reapplied to each newly measured radial state to test whether its mass lies on the trajectory implied by the extracted $\\kappa$ and $\\alpha$."],"supporting_citations":[{"why":"It defines the quadratic-dilaton softwall model whose linear trajectories are the baseline the inversion generalizes.","marker":"[4]"},{"why":"It supplies the Bethe-Salpeter argument that quark mass makes heavy-quarkonium trajectories nonlinear, motivating the power-law ansatz.","marker":"[5]"},{"why":"It introduced the non-quadratic dilaton $\\Phi(z)=(\\kappa z)^{2-\\alpha}$ for heavy quarkonia; the paper's inversion derives this profile rather than assuming it.","marker":"[6]"},{"why":"It furnishes the Rydberg-Klein-Rees/WKB inversion formula used to obtain the confining potential from the spectrum.","marker":"[8]"},{"why":"It provides the semiclassical RKR machinery and Airy-function results used in the WKB steps and in the linear-plus-Coulomb potential analysis.","marker":"[19]"},{"why":"It derives the diquark Regge trajectories from the quadratic spinless Bethe-Salpeter equation, which yield the extra $z$-linear potential for tetraquarks.","marker":"[45]"},{"why":"It is the experimental mass-listing dataset against which the model's spectra are compared.","marker":"[31]"}],"fun_headline_variants":["A Regge trajectory in, a confining dilaton out","Two fitted numbers fix the holographic dilaton","Nonquadratic dilaton from a two-parameter fit","Inverse problem: meson masses fix the dilaton profile","Reverse-engineering the dilaton from hadron masses"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The derivation assumes the measured radial spectrum is governed by the long-distance confining part of the potential, so the short-distance terms can be neglected when matching the dilaton; if those terms shift the low-lying states, the extracted $\\kappa$ and $\\alpha$ describe an effective potential rather than the true confining one.","fun_headline_variants_meta":{"raw":{"variants":["A Regge trajectory in, a confining dilaton out","Two fitted numbers fix the holographic dilaton","Nonquadratic dilaton from a two-parameter fit","Inverse problem: meson masses fix the dilaton profile","Reverse-engineering the dilaton from hadron masses"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001021,"raw_usage":{"total_tokens":4372,"prompt_tokens":1072,"completion_tokens":3300,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":688,"completion_tokens_details":{"reasoning_tokens":3220}},"tokens_in":688,"tokens_out":3300,"duration_ms":20542,"temperature":1.0,"reasoning_tokens":3220,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T16:25:38.151470+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the charmonium and bottomonium spectra from the extracted dilatons while keeping the $\\Phi''$ and $\\beta$-dependent terms that the matching condition drops; if the lowest eigenvalues shift by more than the reported few-percent errors, the central inversion assumption is violated. Alternatively, repeat the extraction on the $\\eta_c$ pseudoscalar trajectory and check whether it yields the same $\\kappa$ and $\\alpha$ as the vector fit.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It supplies the Bethe-Salpeter argument that quark mass makes heavy-quarkonium trajectories nonlinear, motivating the power-law ansatz."},{"cited_title":"Chen, Eur","cited_arxiv_id":null,"evidence_quote":"It is the experimental mass-listing dataset against which the model's spectra are compared."}],"review_version":2}