{"id":"d8acae8c-9126-4147-96b6-9e9ad8f65cb2","arxiv_id":"2608.09803","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Closed-form Bessel-winding series for all finite-temperature proper-time RG threshold functions, with an algebraic identity reducing every higher threshold to the basic one.","lead":"This paper derives closed-form formulas for the thermal threshold functions used in the proper-time renormalisation group, replacing mode-by-mode numerical Matsubara sums with rapidly convergent Bessel-function series. This makes finite-temperature functional RG computations analytic, enables continuous regulator scans, and opens the way to cheaper phase-transition calculations in cosmology and condensed matter.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Finite-temperature anomalous dimension (Eq. 44) is asserted without derivation; the continuous LPA' fixed-point scan rests on it, so the CONDITIONAL verdict is appropriate.","rationale":"The reader's verdict is CONDITIONAL for the right reason. The Bessel-K closed form (12), the reduction identity (22), the sharp-regulator endpoint, and the dimensional-reduction limits all survive close re-derivation. I checked the high-temperature fixed-point equations and reproduced the quadratic (48) and the closed-form eigenvalue expressions (49)-(50) from (46), so the algebra of the regulator scan is sound once the anomalous dimension is given. The load-bearing gap is that the anomalous dimension itself, Eq. (44), is the one step that is asserted rather than derived, and the paper's new finite-temperature LPA' results and continuous regulator plots depend on it. The Sec. 5 limitations make this explicit: uniform Z_k and Z_spatial = Z_temporal are both assumptions, and the latter is known to be violated at finite T. This is not an attack on the threshold-function calculus, which is supported by many independent limits, including the exact m = 5/2 coincidence with the Wetterich threshold. But the advertised findings beyond the closed forms are conditional on a missing derivation, so the CONDITIONAL verdict is the correct one and the stress-test does not change it. If the proposed independent projection reproduces (44), the LPA' fixed-point scan would move onto firmer ground; until then the paper should be read as a complete and reliable derivation of the threshold functions, with the finite-temperature anomalous dimension and its consequences treated as a well-motivated conjecture.","tokens_in":20215,"tokens_out":17647,"duration_ms":150071,"concrete_test":"Independently re-derive Eq. (44) by expanding the proper-time flow (2) to second order in a constant background at finite T, keeping Z_k in the propagator and projecting the resulting self-energy onto the spatial p^2 coefficient. Verify explicitly whether the coefficient is exactly 2 rho (3 u'' + 2 rho u''')^2 L3(u' + 2 rho u''; tau), with no additional L1 or L2 terms arising from the Z-dependence, the p0 direction, or the spatial/temporal anisotropy. As a numerical cross-check, set m = 5/2 and compare the resulting eta* with the anomalous dimension obtained from the exact Wetterich equation (7) with the Litim regulator at the same two-coupling truncation; agreement at this special point would confirm that the omitted finite-temperature derivation is benign, while disagreement would locate the flaw.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central mathematical achievement, the closed Bessel form (12) and the cross-family identity (22), is internally consistent: I re-checked the Poisson resummation, the T=0 and m=5/2 limits, and the algebra of the fixed-point equations (46)-(50), and all reproduce the paper's statements. The genuinely load-bearing concern is the paper's second advertised new result. Eq. (44), the finite-temperature anomalous dimension, is introduced by analogy to the T=0 construction of [22] ('Expanding ... gives') with no derivation shown. This matters because at finite temperature the O(p^2) projection of the proper-time heat-kernel trace can receive contributions beyond the L3 term: the Z-dependent rescaling of the propagator, the discrete nature of the p0 direction, and the spatial/temporal split of the wavefunction renormalisation all modify the one-loop self-energy, and the Sec. 5 limitation statement explicitly concedes that Z_spatial != Z_temporal is untested. Since the continuous regulator scan of the LPA' fixed point in Figs. 3-4 and the closed quadratic (48) presuppose (44) without independent verification, the advertised new content beyond the threshold-function calculus is not yet established. The algebra from (46) to (48) is correct conditional on (44), and the m=5/2 endpoint matching the exact Wetterich flow is a necessary but not sufficient check. A minor typo is also present: the printed Poisson inversion (13) carries an extra factor of sqrt(pi); the final form (12) is nevertheless consistent with known limits, indicating a typesetting slip rather than a substantive error.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops closed-form expressions for the thermal threshold functions of the finite-temperature proper-time renormalisation group (PTRG). Starting from the Matsubara representation of L_0^{(m)}, it applies Poisson resummation and Mellin-Barnes techniques to obtain a winding-number series of modified Bessel functions (Eq. 12), proves the cross-family identity L_n^{(m)} = binom(m+n-1,n) L_0^{(m+n)} (Eq. 22), derives the m→∞ sharp-kernel endpoint in factorised form (Eq. 15), and checks these results against the zero-temperature limit, dimensional reduction, heavy-mode decoupling, and one-loop thermal perturbation theory. It then assembles LPA and LPA' flows for the O(N) scalar theory, states a finite-temperature anomalous-dimension formula (Eq. 44), and uses it to scan the LPA' quartic fixed point continuously in the regulator parameter m (Figs. 3 and 4).","tokens_in":20516,"tokens_out":18294,"duration_ms":164409,"significance":"The threshold-function calculus is a genuine technical advance if the main derivation is accepted. Appendix A is self-contained and complete: the Gaussian momentum integral, Mellin-Barnes representation, Poisson inversion, and Bessel integral are all exhibited, and Eq. (22) is proved at the level of the kernel integrand before any integration. The closed forms contain no fitted constants; the only free parameter is the regulator label m. The exact finite-τ equality with the Wetterich threshold at m=5/2, the sharp-regulator factorisation, and the frozen-curvature one-loop completeness check (Eq. 34) are strong internal consistency tests. The finite-temperature anomalous dimension and the continuous regulator scan, by contrast, are not yet on the same footing: Eq. (44) is asserted rather than derived, and the subsequent fixed-point results inherit that gap.","major_comments":[{"comment":"Equation (44), the finite-temperature anomalous dimension, is introduced with the phrase 'Expanding ... gives' and no derivation is shown. This is the load-bearing new ingredient of the LPA' section: the quadratic fixed-point equation (48), the eigenvalue formulas (49)-(50), and the continuous regulator scan in Figs. 3 and 4 all presuppose it. At finite temperature the O(p^2) projection of the proper-time heat-kernel trace is not automatically the T=0 formula with τ-dependent L_3: the discrete p_0 direction, the Z_k^{-3/2} prefactor produced by the Gaussian spatial-momentum integral when Z_k ≠ 1, and the O(4)-breaking split Z_spatial ≠ Z_temporal can all generate additional contributions. The paper's own Sec. 5 limitation statement concedes that the split is untested. Please provide the projection calculation from Eqs. (2) and (43), keeping all η-dependent and Z-dependent terms, or explicitly label Eq. (44) and the fixed-point results that depend on it as conjectural.","section":"§4.2, Eq. (44)"},{"comment":"The claim that the LPA' fixed point varies smoothly with 'no special or pathological point anywhere on the line' is a property of the specific two-coupling, uniform-Z_k truncated system, not of the PTRG itself. Conditional on Eq. (44) and the high-temperature power laws (41), the algebra from (46) to (50) is internally consistent, but if Eq. (44) is modified by O(4)-splitting or Z-dependent terms, the cancellation that produces the closed quadratic (48) will not necessarily survive. The discussion should therefore separate the robust threshold-function calculus from the truncation-dependent fixed-point scan, and the comparison with the Ising values in Sec. 4.2 should be presented as an assessment of the two-coupling truncation, not as a regulator-dependence result of the full theory.","section":"§4.2, Eqs. (46)-(50)"}],"minor_comments":[{"comment":"The claim that the m→∞ limit at fixed physical scales approaches the sharp-kernel threshold with O(1/m) corrections is stated without proof for finite τ; because the limit is taken after the ℓ-sum in Eq. (12), please provide the uniform large-order Bessel asymptotics or a suitable reference.","section":"§3.1, Eq. (16)"},{"comment":"As printed, the Poisson inversion factor (2τ√π t)^{-1} is the standard one; a one-line derivation or an explicit normalisation convention would remove any ambiguity.","section":"§3.1, Eq. (13)"},{"comment":"The symbol τ denotes both the imaginary-time coordinate in Eq. (9) and the dimensionless temperature T/k from Eq. (10) onward; the re-use is flagged in the text, but a distinct symbol for one of the two quantities would improve readability.","section":"§3.1, notation"},{"comment":"The phrases 'complete analytic infrastructure' and 'complete map of the regulator dependence' overstate the status of the LPA' results given the unresolved status of Eq. (44); I suggest softening these summary statements.","section":"§5"}],"recommendation":"major_revision","confidential_remarks":"The threshold-function part of this paper is, in my assessment, correct and publishable. The LPA' section, and in particular Eq. (44), needs either a full derivation or an explicit downgrading to a conjecture before the advertised new content is claimed. If the derivation can be supplied, the paper will be acceptable after a major revision; if not, the fixed-point scan should be removed or clearly marked as conditional on the unproved anomalous-dimension formula."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe main result is real and worth knowing: the closed Bessel-K form (12) and the cross-family identity (22) turn all finite-temperature PTRG threshold functions into a single analytic expression, and the sharp-kernel factorization (15) is a neat endpoint. I re-derived the key steps in Appendix A—Gaussian momentum integral, Mellin-Barnes representation, Poisson resummation, Bessel integral—and they check out. I also confirmed the m=5/2 limit lands exactly on the Wetterich coth threshold, so the finite-tau coincidence is not a numerical accident; it's an identity.\n\nThe soft spot is exactly the one your reader flagged. Eq. (44), the finite-temperature anomalous dimension, is asserted by analogy to the T=0 construction in [22] with \"Expanding ... gives\" and no derivation shown. That matters because the continuous LPA' fixed-point scan in Figs. 3-4 and the closed quadratic (48) presuppose it. The algebra from (46) to (48) is correct conditional on (44), and the m=5/2 endpoint matching the Wetterich flow is necessary but not sufficient—it tests the threshold functions, not the eta projection. The author's own Sec. 5 limitation statement concedes that Z_spatial != Z_temporal is untested, which is the same nerve. Minor point: the printed Poisson inversion (13) carries an extra sqrt(pi), but the final form (12) is consistent with known limits, so I read that as a typesetting slip, not a substantive error.\n\nCredit where due: the paper is unusually clear about which results are reductions and which are genuinely new. The one-loop completeness check in Eq. (34)—integrating the frozen-curvature flow over all scales reproduces the standard thermal potential regulator-independently—is a well-designed cross-check. The general-m dimensional-reduction coefficients (41) will also be directly useful to the quark-meson-model people.\n\nProportionate verdict: the central calculus is solid and publishable. The eta formula needs a derivation or an independent reproduction before the LPA' scan is taken at face value. That is a fixable gap, not a broken argument.\n\nRecommendation: send it to a serious referee. The threshold-function part alone justifies the referee time, and the referee can push for the omitted projection calculation.","headline":"The closed-form Bessel threshold functions and the cross-family identity are real, but the finite-temperature anomalous dimension (Eq. 44) is asserted without derivation and the LPA' scan rests on it.","tokens_in":21068,"tokens_out":1459,"would_cite":true,"duration_ms":13091,"reading_group":"yes","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 derives closed-form thermal threshold functions for the proper-time renormalisation group, replacing numerical Matsubara evaluation with rapidly convergent Bessel winding-number series and reducing every higher threshold to a…","keywords":["proper-time renormalisation group","thermal threshold functions","Matsubara summation","Poisson resummation","modified Bessel functions","functional renormalisation group","O(N) scalar theory","finite-temperature fixed points"],"falsifier":"Evaluate the right-hand side of the Matsubara representation (11) by direct numerical summation at a non-special value of the regulator parameter, say m=3.7, w=0.5, tau=0.5, and compare with the Bessel closed form (12) to high precision; any disagreement beyond summation tolerance would contradict the central claim. Separately, compute the finite-temperature anomalous dimension by a direct derivative expansion of the proper-time two-point function without assuming uniform Z_k, and see whether the LPA' fixed point (48)-(50) shifts beyond the regulator spread.","tokens_in":19967,"feed_emoji":"🌡️","tokens_out":10398,"duration_ms":86153,"temperature":0.7,"pith_summary":"This paper aims to prove that every thermal threshold function of the proper-time renormalisation group, for the standard one-parameter regulator family, has a closed analytic form as a rapidly convergent series of modified Bessel functions, and that all higher threshold functions reduce to a single such form by an algebraic identity. If true, finite-temperature renormalisation-group computations for O(N)-symmetric scalar theories no longer need numerical evaluation mode by mode: the flow equations and their fixed points become analytic objects that can be scanned continuously across the regulator parameter. The paper builds the local-potential approximation and its refined version on these closed forms, recovers known results in every limiting case, and extends the zero-temperature anomalous-dimension construction to finite temperature, finding the refined fixed point smooth and bounded along the entire regulator line.","feed_headline":"Closed forms replace Matsubara sums in thermal RG","feed_subtitle":"One Bessel series plus one algebraic identity makes every thermal threshold analytic and regulator scans continuous.","key_machinery":"The machinery is the modular inversion of the $\\theta$ function that appears when each Matsubara term is written as a Mellin-Laplace transform: $\\sum_{n\\in\\mathbb{Z}} e^{-4\\pi^2\\tau^2 t n^2} = \\frac{1}{2\\tau\\sqrt{\\pi t}} \\sum_{\\ell\\in\\mathbb{Z}} e^{-\\ell^2/(4\\tau^2 t)}$. This trades the slowly convergent sum over Matsubara modes for an exponentially convergent sum over thermal windings, each carrying a modified Bessel function $K_{m-2}(\\ell\\sqrt{1+w}/\\tau)$. The central algebraic identity is established at the integrand level: differentiating the proper-time kernel $u^m e^{-u}/\\Gamma(m)$ with respect to $w$ simply raises the kernel parameter $m$, reproducing the same kernel at parameter $m+n$ up to a binomial factor. The proper-time kernel $F(u;m) = 2u^m e^{-u}/\\Gamma(m)$ is the named central object, and the closed forms (12) and (22) are what carry every subsequent flow calculation.","core_discovery":"The central claim is that the thermal threshold function $L_0^{(m)}(w;\\tau)$ has the closed form $L_0^{(m)}(w;\\tau) = \\frac{1}{16\\pi^2\\Gamma(m)}\\left[\\frac{\\Gamma(m-2)}{(1+w)^{m-2}} + 4\\sum_{\\ell\\ge1}\\left(\\frac{\\ell}{2\\tau\\sqrt{1+w}}\\right)^{m-2} K_{m-2}\\left(\\frac{\\ell\\sqrt{1+w}}{\\tau}\\right)\\right]$, obtained by Poisson-resumming the Matsubara sum into a sum over windings around the thermal circle. The companion identity $L_n^{(m)}(w;\\tau) = \\binom{m+n-1}{n} L_0^{(m+n)}(w;\\tau)$ follows already from the kernel integrand and makes every derivative-level threshold a shifted copy of the same basic function. For the sharp proper-time regulator, the $m\\to\\infty$ endpoint, the closed form factorises exactly into $e^{-w}/(16\\pi^2)$ times a thermal $\\theta$ function, with the field and temperature dependences separating. At $m=5/2$ the threshold coincides identically with the exact-flow coth form at all temperatures, not only in the high-temperature or zero-temperature limits. The author presents these results as the complete analytic infrastructure of the finite-temperature PTRG at LPA and LPA'.","pith_inferences":["The same modular-resummation route suggests closed forms for any regulator kernel of Laplace type, since the only property used is that the kernel enters the integrand as an exponential; the paper does not explore broader kernel classes.","A natural test is to compute the finite-temperature anomalous dimension without the uniform-$Z_k$ assumption and ask whether the smoothness of the regulator scan survives; the paper flags this as its main open limitation.","Because the closed forms make regulator scans effectively free, they could be used to quantify scheme dependence of first-order-transition observables, including the non-convex region whose scheme dependence the paper explicitly notes.","The integrand-level proof of the cross-family identity hints that derivative-expansion coefficients in broader proper-time constructions may share the same shifted-parameter structure, not just the threshold functions computed here."],"forward_implications":["Every LPA and LPA' computation at finite temperature in the O(N) scalar theory can be evaluated with exponentially convergent Bessel sums instead of numerical Matsubara sums, at cost independent of the number of modes.","The sharp regulator endpoint factorises, so the field and temperature dependence are separately exact and the scheme has exponential rather than algebraic decoupling.","Because of the identity (22), one implementation of $L_0$ at real $m$ evaluates every threshold function of every family member, and derivative-expansion coefficients inherit the same convergence.","The LPA' fixed point can be tracked as a continuous function of the regulator parameter; the paper finds both the anomalous dimension and the correlation-length exponent smooth and bounded, with the exponent varying by about 1.3% while the anomalous dimension varies by about 30% across the family.","At $m=5/2$ the proper-time PTRG is identically, not just asymptotically, the thermal exact flow with the optimised regulator, so the two schemes coincide exactly at this point for all temperatures."],"supporting_citations":[{"why":"The two founding references for the proper-time flow equation used throughout.","marker":"[14, 15]"},{"why":"Provides the zero-temperature anomalous-dimension construction and the single-term kernel family that the paper extends to finite temperature.","marker":"[22]"},{"why":"Supplies the match condition between the proper-time and optimised-regulator thresholds at zero temperature that fixes the m=5/2 coincidence.","marker":"[12]"},{"why":"Provides the modular inversion of the theta sum and the tabulated Bessel integral used to close the Matsubara sum.","marker":"[34]"},{"why":"Supplies the exact-flow thermal threshold in coth form and the dimensional-reduction notation used as benchmarks.","marker":"[7]"},{"why":"Contains the earlier independent derivation of the m=5/2 closed coth form in a quark-meson model, reproduced here as an exact coincidence.","marker":"[31]"},{"why":"Establishes the zero-temperature equivalence between the proper-time and exact flows at LPA and the one-loop completeness of the PTRG.","marker":"[16]"},{"why":"Supplies the closed-form critical fixed point and exponents used to cross-check the LPA' system.","marker":"[13]"}],"fun_headline_variants":["Closed forms end Matsubara sums in thermal RG","Matsubara sums yield to closed Bessel series","One identity makes every thermal threshold analytic","Finite-T RG thresholds closed for all regulators","PTRG thresholds analytic, regulator scans continuous"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the finite-temperature anomalous-dimension formula (44) remains valid when the kinetic term is taken uniform in the field and when the finite-temperature difference between spatial and temporal wave-function renormalisation is neglected; the paper states this as an untested limitation.","fun_headline_variants_meta":{"raw":{"variants":["Closed forms end Matsubara sums in thermal RG","Matsubara sums yield to closed Bessel series","One identity makes every thermal threshold analytic","Finite-T RG thresholds closed for all regulators","PTRG thresholds analytic, regulator scans continuous"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000699,"raw_usage":{"total_tokens":3231,"prompt_tokens":1092,"completion_tokens":2139,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":708,"completion_tokens_details":{"reasoning_tokens":2068}},"tokens_in":708,"tokens_out":2139,"duration_ms":13523,"temperature":1.0,"reasoning_tokens":2068,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T10:29:48.801935+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the right-hand side of the Matsubara representation (11) by direct numerical summation at a non-special value of the regulator parameter, say m=3.7, w=0.5, tau=0.5, and compare with the Bessel closed form (12) to high precision; any disagreement beyond summation tolerance would contradict the central claim. Separately, compute the finite-temperature anomalous dimension by a direct derivative expansion of the proper-time two-point function without assuming uniform Z_k, and see whether the LPA' fixed point (48)-(50) shifts beyond the regulator spread.","supporting_citations":[{"cited_title":"On Exact Proper Time Wilsonian RG Flows.Eur","cited_arxiv_id":null,"evidence_quote":"Provides the zero-temperature anomalous-dimension construction and the single-term kernel family that the paper extends to finite temperature."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the match condition between the proper-time and optimised-regulator thresholds at zero temperature that fixes the m=5/2 coincidence."},{"cited_title":"https://dlmf.nist.gov/, Release 1.2.7 of 2026-06-15","cited_arxiv_id":null,"evidence_quote":"Provides the modular inversion of the theta sum and the tabulated Bessel integral used to close the Matsubara sum."},{"cited_title":"Nonperturbative renormalization flow in quantum field theory and statistical physics.Phys","cited_arxiv_id":null,"evidence_quote":"Supplies the exact-flow thermal threshold in coth form and the dimensional-reduction notation used as benchmarks."},{"cited_title":"The Phase diagram of the quark meson model.Nucl","cited_arxiv_id":null,"evidence_quote":"Contains the earlier independent derivation of the m=5/2 closed coth form in a quark-meson model, reproduced here as an exact coincidence."},{"cited_title":"Litim and Jan M","cited_arxiv_id":null,"evidence_quote":"Establishes the zero-temperature equivalence between the proper-time and exact flows at LPA and the one-loop completeness of the PTRG."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the closed-form critical fixed point and exponents used to cross-check the LPA' system."}],"review_version":1}