{"id":"6dbabfc5-552f-4b5e-9538-f4668f604fe0","arxiv_id":"2608.02702","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors define q-extended versions of the iterated integrals and nested sums used in QFT, deriving closed forms for simple cases and algorithmic recipes for complex ones.","lead":"This paper constructs q-analogues of the special functions that appear in high-order quantum field theory calculations, such as polylogarithms and their many generalizations. It provides recurrence, difference and differential equations for these new functions, aimed at quantum field theories with deformed commutation relations.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Claimed q-extensions rest on recurrences guessed from 15–80 coefficients (Sec. 3 after Eq. 3.10, conceded in Sec. 6); without proof certificates the uniqueness/completeness claim is unverified.","rationale":"The reader's weakest assumption identifies exactly the same load-bearing concern: recurrences inferred from a finite number of series coefficients are treated as exact for all n without proof or certificate. My stress-test agrees with that assessment. The paper's explicit and simple cases, such as the polylogarithm representation (4.17)-(4.25) and the direct recurrences for single harmonic sums, are internally consistent and checkable, so the construction is plausible. However, the central claim of uniqueness and completeness depends on nontrivial recurrences that are only guessed from finite data, and Section 6 explicitly frames guessing as the method. The q-differential and q-shift equations are derived from those recurrences, so an unproven recurrence propagates uncertainty to all three defining structures. The shuffle/quasi-shuffle verification in Section 5 also relies on finite checks (N up to 20) and on the same guessed recurrences, so it does not close the gap. A certificate-based derivation of one representative recurrence from each class would settle the concern. Because the underlying approach is well motivated and the simple cases are correct, the appropriate verdict remains CONDITIONAL rather than ACCEPT or REJECT; this does not change the reader's verdict.","tokens_in":38040,"tokens_out":22281,"duration_ms":200395,"concrete_test":"For the square-root case (4.104)-(4.105), use a certified q-holonomic closure (e.g. qGeneratingFunctions' QREGenerate or Sigma's q-creative telescoping) to derive, without guessing, the minimal recurrence for the explicit q-sum-product f[n;q] in (4.104), for all n and symbolic q, and compare it with (4.105). If the certified recurrence matches (4.105) in order, degree, and initial values, the methodology is validated for the most complex class; repeat on one representative recurrence from each remaining class (3.13, 4.78, 4.90, 4.101). If any mismatch or order reduction occurs, the claimed q-extension for that class is not the sequence defined by the displayed recurrence, and the uniqueness claim fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (Sec. 6) is that the q-extensions of all presently known first-order factorizing iterated integrals and nested sums are found and uniquely defined by their q-recursions, q-differential equations, and q-shift relations. The load-bearing input is the assertion that the displayed recurrences, e.g. (3.13), (4.78), (4.90), (4.101), (4.105), hold for all n. Section 3 states these were obtained by GuessqRecurrence/QREGuess from N=15 to 80 expansion coefficients, and Section 6 concedes that a q-recursion can be obtained by guessing methods, with Sigma used only as a check. Since the q-differential and q-shift relations are derived from these recurrences, a recurrence that fails beyond the fitted range invalidates the associated defining equations for that class. No creative-telescoping certificate or q-holonomic closure proof is supplied for any nontrivial recurrence. The extension from representative examples to all presently known first-order factorizing solutions is therefore an extrapolation unless the algorithmic step is proven to terminate and to produce a recurrence that provably annihilates the explicit f[n;q]. This is not an accusation of error; it is a missing proof at the point where the uniqueness claim attaches.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs q-extensions of the nested sums and iterated integrals that occur in single-scale perturbative QFT calculations, starting from the coefficient sequence f[n] in the series expansion around x=0 and replacing all summation quantifiers and products by their q-analogs to define f[n;q]. For nested harmonic sums, generalized harmonic sums, cyclotomic sums, sums from quadratic forms, and sums with central binomial factors, and for polylogarithms, Nielsen integrals, harmonic polylogarithms, generalized/cyclotomic harmonic polylogarithms, quadratic-form iterated integrals, and square-root-letter iterated integrals, the paper presents recurrences for f[n;q] together with associated q-differential and q-shift relations. Shuffle and quasi-shuffle algebras for the extended objects are discussed in Section 5. Section 6 states that the q-extensions of all presently known first-order factorizing iterated integrals and nested sums have been found and are uniquely defined by their q-recursions, q-differential equations, and q-shift relations.","tokens_in":38286,"tokens_out":8373,"duration_ms":75216,"significance":"If the recurrences and the completeness statement were rigorously established, this would provide a systematic q-deformed calculus for function spaces of direct relevance to q-deformed quantum field theories, complementing the earlier µ-extension study [178]. The paper contains a large amount of explicit material: closed forms for the simplest cases, concrete recurrences and q-differential/q-shift equations for representative examples, and a clear algorithmic description using existing computer algebra packages (qFunctions, qGeneratingFunctions, Sigma). The authors also honestly acknowledge in Section 6 that the recurrences are obtained by guessing. The main gap is that the central uniqueness claim rests on finite-data inference rather than on proof certificates.","major_comments":[{"comment":"The recurrences that define the q-extensions are obtained by GuessqRecurrence/QREGuess from N=15 to 80 expansion coefficients, and Section 6 concedes that a q-recursion can be obtained by guessing methods. This is load-bearing for the central claim: the q-differential and q-shift relations are derived from these recurrences, and if any recurrence, e.g. (3.13), (4.78), (4.90), (4.101), or (4.105), fails for n beyond the fitted range, then the associated defining equations and the uniqueness statement in Section 6 fail for that class. No creative-telescoping certificate, q-holonomic closure proof, or independent verification that the displayed recurrence annihilates the explicit f[n;q] for all n is supplied. The claim that the q-extensions are uniquely defined by these relations is therefore currently an extrapolation from finite data.","section":"Sec. 3, after Eq. (3.10); Sec. 6"},{"comment":"The recurrence for the quadratic-form example T3 is not actually displayed: it is written only schematically in terms of unnamed polynomials p_i(q), with the note that it is provided as an ancillary file. Since this recurrence is one of the defining objects for this class and is used to derive the q-differential and q-shift relations, the manuscript itself does not contain enough information for the reader to verify the central claim for this class. The explicit recurrence should either be included in the paper or made available through a stable, citable supplementary data link that is described in the text.","section":"Sec. 4.6, Eq. (4.101)"},{"comment":"The formula H0(x;q) = -\\sum_{n=1}^{\\infty} (1-q)/(1-q^k)(1-x)^k is not consistent with the required q->1 limit: since (1-q)/(1-q^k) tends to 1/k, the printed expression tends to -\\sum (1-x)^k/k = ln(x), not to H0(x)=-ln(1-x). The summation index is also not used inside the summand. This appears to be a substantive typo in the definition of the q-extended H0, and it should be corrected and the consequences for subsequent equations checked.","section":"Sec. 4.3, Eq. (4.61)"},{"comment":"The q-differential equation ((qx-1)D_q^x + 1) Li1(x;q) = 1/(x-1) does not reduce correctly to the q-free case. Using D_q^x Li1(x;q)=1/(1-x), which follows from the series definition, the q=1 limit of the left-hand side is -1+Li1(x), while the right-hand side is 1/(x-1); these are not equal. This suggests a typographical or sign error in the displayed operator equation, and it should be corrected before the paper is finalized.","section":"Sec. 4.1, Eq. (4.27)"}],"minor_comments":[{"comment":"Several long equations have unbalanced parentheses and inconsistent notation: Eq. (4.29) has mismatched brackets in the third-order differential operator, and Eq. (4.32) mixes F(qx)/F(x) with F[q^3x]/F[q^2x] notation. Please re-typeset these equations carefully.","section":"Sec. 4.1, Eqs. (4.29) and (4.32)"},{"comment":"Beyond the mathematical issue noted above, the summation variable n does not appear in the summand, where k is used instead; this should be a single consistent index.","section":"Sec. 4.3, Eq. (4.61)"},{"comment":"Minor language/format issues: the abstract says 'quite different form the corresponding µ-extended functions', which should be 'from'; several references (e.g. [104], [111], [112]) are missing closing brackets; Eq. (5.7) has an unbalanced parenthesis.","section":"Abstract and references"},{"comment":"The verification of shuffle relations for q-extended sums is described as checking at fixed N up to 20 and then comparing q-recurrences of both sides; since those recurrences are again guessed from finite data, the all-n claim is not established by the described procedure.","section":"Sec. 5, after Eq. (5.10)"}],"recommendation":"major_revision","confidential_remarks":"The main issue is the proof gap described in Major Comment 1: the central uniqueness/completeness claim depends on recurrences obtained from 15–80 coefficients, and the paper itself acknowledges this in Section 6. If the authors can supply certificates (e.g., creative telescoping proofs) or prove that their algorithmic steps terminate with a recurrence that provably annihilates f[n;q], the paper would be substantially stronger. In its current form, the completeness claim should either be proven or explicitly weakened. The paper also relies heavily on the companion paper [178] and on the authors' own software; this is not improper, but the dependence should be clearly stated in the introduction and conclusions."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe useful thing here is the scope. Blumlein et al. take every function space that shows up in first-order-factorizing single-scale Feynman integrals – polylogs, Nielsen, harmonic, generalized, cyclotomic, quadratic-form, square-root letters – and give the corresponding q-extension, with recurrences, q-difference and q-shift equations. Previous work had q-polylogs and q-MZVs piecemeal; a systematic alphabet-wise treatment is genuinely new. Section 5’s demonstration that shuffle/quasi-shuffle structure survives q-deformation is a nice bonus.\n\nThe construction itself is straightforward and reproducible: q-extend the coefficients f[n] in the x=0 expansion term-by-term, then derive the recurrences. For the easy cases (single sums, polylogs, Nielsen) they even give closed forms. For harder cases they give explicit algorithmic steps and worked examples like T3 and T4. That part is honest and useful.\n\nThe soft spot is real and load-bearing. The recurrences are guessed from 15–80 coefficients with GuessqRecurrence/QREGuess, and neither the paper nor the ancillary files contain a proof or certificate that they hold for all n. Section 6 more or less concedes this. Since the q-differential and q-shift relations are derived from those recurrences, an unproved recurrence is an unproved defining equation. I don’t doubt most of them – they’re the kind of thing creative telescoping would verify – but the paper’s own central claim in Section 6, that these functions are uniquely defined by the recurrences and equations, is stronger than the evidence. The step from representative examples to “all presently known first order factorizing solutions” is also an extrapolation unless the algorithmic procedure comes with a proof of termination and correctness.\n\nMinor: several long equations have typesetting errors (e.g., Eq. 4.29 and 4.32), which is annoying in a reference-heavy paper but not devastating.\n\nBottom line: this is a solid toolkit paper for the q-deformed QFT / special-functions community. It deserves a serious referee. My recommendation is to send it to review, with the condition that the authors either supply certificates for the recurrences (or at least state clearly that they are conjectural) and soften the uniqueness/completeness claim accordingly. I would cite it if I needed q-extensions of these alphabets, and it would make a good reading-group discussion precisely because of the guessing-vs-proof issue.\n\nBest,","headline":"A systematic and useful q-extension encyclopedia for QFT special functions, but the uniqueness claim rests on unproven guessed recurrences.","tokens_in":38807,"tokens_out":3758,"would_cite":true,"duration_ms":33818,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["33D15","05A30","11M32","81T18"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper constructs q-extensions for every presently known class of first-order factorizing iterated integrals and nested sums that appear in single-scale Feynman diagram calculations, and shows they are uniquely fixed by their…","keywords":["q-extension","iterated integrals","nested sums","harmonic polylogarithms","harmonic sums","Feynman integrals","q-deformed commutation relations","basic hypergeometric functions"],"falsifier":"Take any displayed q-recursion, for instance Eq. (3.13) for $S_{2,1}(n;q)$, compute the direct q-sum definition of $f[n;q]$ at $n=81$ and beyond, and compare it with the value obtained from the recurrence with the stated initial values; one mismatch shows the guessed relation is not an identity for all n. The same check applies to the q-shift relation (4.80) for the generalized harmonic polylogarithm example by evaluating both sides as series in $x$ through the coefficient of $x^N$ for $N$ beyond the guessed range.","tokens_in":37855,"feed_emoji":"⚛️","tokens_out":11603,"duration_ms":92936,"temperature":0.7,"pith_summary":"This paper aims to show that every class of special functions that appears in analytic higher-order calculations of single-scale Feynman diagrams possesses a q-extension, a deformed version governed by a parameter q that reduces to the original function as q approaches 1 from below. The classes covered are the iterated integrals over linear, cyclotomic, quadratic-form, and square-root alphabets (polylogarithms, Nielsen integrals, harmonic and generalized harmonic polylogarithms, and their relatives) together with the nested harmonic, generalized harmonic, cyclotomic, quadratic-form, and central-binomial sums obtained as their Taylor coefficients and Mellin transforms. For the simpler spaces the paper gives closed forms; for the more involved alphabets it gives algorithmic steps that turn any first-order factorizing example into its q-extension. The payoff, if the construction is right, is a ready-made function space for perturbative calculations in quantum field theories with q-deformed commutation relations, plus new higher-transcendental functions whose defining equations are explicitly computed.","feed_headline":"All known Feynman-integral function classes get q-extensions","feed_subtitle":"Maps Taylor coefficients of Feynman-integral function classes to q-analogues, pinned by recurrences and q-shifts.","key_machinery":"The central object is the q-extension map from $f[n]$ to $f[n;q]$ obtained by replacing every summation variable's denominator $n$ by $\\{n\\}_q=(1-q^n)/(1-q)$ and products by q-Pochhammer symbols, while leaving plain powers $c^n$ untouched. The load-bearing construction is the triple of determining data: a linear recurrence in $n$ for $f[n;q]$, a q-differential equation in the Jackson q-derivative $D_q^x$, and a q-shift relation in $F(q^k x)$; the paper shows these three are mutually interconvertible and fix the extension uniquely. For complicated alphabets the recurrences are produced by computer-algebra guessing from a finite number of series coefficients, then transformed into q-differential and q-shift equations; the paper calls the resulting series-patching method the q-Frobenius method.","core_discovery":"The central claim is that the q-extension of each such function $F(x)=\\sum_{n=0}^\\infty f[n]x^n$ is obtained by q-deforming its expansion coefficients: every summation multiplier and product factor in $f[n]$ is replaced by its q-analogue, with $\\{n\\}_q=(1-q^n)/(1-q)$ in place of $n$, and powers $c^n$ left undeformed, yielding $f[n;q]$ and $F(x;q)=\\sum_{n=0}^\\infty f[n;q]x^n$. The paper then computes, for representative cases in each class, a linear q-recursion for $f[n;q]$, a q-differential equation, and a q-shift relation, and asserts that these three data uniquely determine the q-extension. For polylogarithms and Nielsen integrals this is done in closed form, including explicit q-hypergeometric representations and q-zeta values $\\zeta_k(q)=\\sum_{n\\ge1}1/\\{n\\}_q^k$; for cyclotomic, quadratic-form, and square-root alphabets the paper demonstrates the algorithmic chain on concrete examples such as $H_{-1,0,1}$, a cyclotomic integral, the quadratic-form example $T_3$, and the central-binomial example $T_4$. It closes by verifying that shuffle and quasi-shuffle product identities survive the q-deformation, so the q-extended iterated integrals and nested sums form (quasi-)shuffle Hopf algebras.","pith_inferences":["If the guessed recurrences are exact, the same pipeline should apply to any future first-order factorizing alphabet, so the paper's list is closed under new discoveries of that type; that is an extrapolation, not proved here.","A cheap independent test would be to run the guessing procedure with substantially more coefficients, say $N=200$, on one example such as $S_{2,1}(n;q)$ and check that the same minimal recurrence reappears; persistent agreement would raise confidence in the uniqueness claim without proving it.","The q-zeta values $\\zeta_k(q)$ defined in Eq. (4.18) may connect to the existing literature on q-multiple zeta values, although the paper's convention differs from several of those by factors of $(1-q)^k$ and numerator powers of $q$; working out the dictionary could make the connection testable.","The physical motivation rests on the q-deformed commutation relation (1.2); a concrete check would be to derive a simple observable, such as a two-point correlator, in that deformed theory and see whether the q-extended polylogarithms and harmonic sums actually appear in its expansion."],"forward_implications":["For polylogarithms and Nielsen integrals, closed-form q-extensions give q-hypergeometric representations and define q-zeta values $\\zeta_k(q)$, with the ordinary $\\zeta(k)$ recovered as $q\\to1^-$.","Any q-extended function can be evaluated numerically near $x=0$ from its q-recursion alone; overlapping series expansions (the q-Frobenius method) extend the evaluation to points beyond the radius of convergence.","The shuffle and quasi-shuffle algebras of the q-free functions are preserved, so products of q-extended integrals and sums reduce to fewer independent quantities at each weight.","In the case of square-root valued letters, the q-extension stays in the same function class, in contrast to the $\\mu$-extension studied earlier, where the class changes.","If the q-deformed commutation relation (1.2) governs a quantum field theory, the q-extended nested sums and iterated integrals constructed here are the function spaces expected in its perturbative expansion."],"supporting_citations":[{"why":"Defines harmonic sums and their Mellin transforms, the q-free objects whose recurrences the q-extension generalizes.","marker":"[176]"},{"why":"Establishes the relation between harmonic sums, polygamma functions, and Mellin transforms that underlies the single-sum q-extensions.","marker":"[177]"},{"why":"Introduces harmonic polylogarithms, the function class q-extended in Section 4.3.","marker":"[163]"},{"why":"Define generalized harmonic sums and polylogarithms used for the q-extended examples with numerator weights.","marker":"[171,172]"},{"why":"Introduces cyclotomic harmonic polylogarithms, the class treated in Section 4.5.","marker":"[173]"},{"why":"Supplies the quadratic-form iterated integrals and nested sums behind the $T_3$ example.","marker":"[174]"},{"why":"Provides the square-root valued letters and central-binomial nested sums behind the $T_4$ example.","marker":"[175]"},{"why":"The companion paper defining the $\\mu$-extension, whose starting point and contrasting behavior frame the q-extension.","marker":"[178]"},{"why":"Supplies the guessing routine used to find q-recurrences for the expansion coefficients.","marker":"[217]"},{"why":"Supplies the q-holonomic machinery and conversions to q-differential and q-shift relations.","marker":"[218]"}],"fun_headline_variants":["All Feynman integral classes get q-extensions and recurrences","q-deformed polylogs, sums, and all integral alphabets","Systematic q-extension for every Feynman integral class","q-extensions for all QFT special functions with recurrences","q-deform Taylor coefficients to build new Feynman integral functions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The recurrences, q-differential equations, and q-shift relations that define each q-extension are inferred by guessing from between 15 and 80 initial series coefficients and then treated as exact for all n, with no supplied certificate that they continue to hold beyond that range.","fun_headline_variants_meta":{"raw":{"variants":["All Feynman integral classes get q-extensions and recurrences","q-deformed polylogs, sums, and all integral alphabets","Systematic q-extension for every Feynman integral class","q-extensions for all QFT special functions with recurrences","q-deform Taylor coefficients to build new Feynman integral functions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00041,"raw_usage":{"total_tokens":2216,"prompt_tokens":1128,"completion_tokens":1088,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":744,"completion_tokens_details":{"reasoning_tokens":998}},"tokens_in":744,"tokens_out":1088,"duration_ms":10029,"temperature":1.0,"reasoning_tokens":998,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:00:43.560903+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take any displayed q-recursion, for instance Eq. (3.13) for $S_{2,1}(n;q)$, compute the direct q-sum definition of $f[n;q]$ at $n=81$ and beyond, and compare it with the value obtained from the recurrence with the stated initial values; one mismatch shows the guessed relation is not an identity for all n. The same check applies to the q-shift relation (4.80) for the generalized harmonic polylogarithm example by evaluating both sides as series in $x$ through the coefficient of $x^N$ for $N$ beyond the guessed range.","supporting_citations":[{"cited_title":"The $\\mu$-extension of iterated integrals and nested sums","cited_arxiv_id":"2606.12584","evidence_quote":"The companion paper defining the $\\mu$-extension, whose starting point and contrasting behavior frame the q-extension."},{"cited_title":"qFunctions -- A Mathematica package for $q$-series and partition theory applications","cited_arxiv_id":"1910.12410","evidence_quote":"Supplies the guessing routine used to find q-recurrences for the expansion coefficients."}],"review_version":2}