{"id":"8df5c1eb-ec44-45b6-b01e-e51a20f8a4c1","arxiv_id":"2506.06036","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Path operators on Macdonald symmetric functions are shown to represent Negut shuffle algebra elements, yielding a unique PDE characterization of a (q,t)-tau function and a new proof of a key result behind the extended delta conjecture.","lead":"The paper introduces a new family of operators on q,t-symmetric functions, built from lattice paths, and shows they match known shuffle algebra elements. These operators give a unique system of PDEs characterizing a (q,t)-deformed tau function that unifies several generating series in geometry, gauge theory, and integrability.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Uniqueness proof's q=t=1 specialization in Proposition 3.9 needs an explicit regularity check, but the gap is likely fixable via the path-operator formula.","rationale":"The reader correctly identifies Proposition 3.9 as the load-bearing step for the uniqueness theorem. The proof's q=t=1 specialization is terse: it assumes without comment that the matrix entries can be evaluated at q=t=1 and that invertibility of the specialized determinant implies invertibility over Q(q,t)[[U,V]]. This is the weakest point in the argument because a pole in any coefficient would break the specialization. However, the path-operator expression from Theorem 3.12 supplies exactly the needed regularity: the valley weights (qt)^h are polynomial, and h⊥_n[MX] has polynomial coefficients in q,t in the power-sum basis, so a_{F,λ} lies in Q[q,t][[U,V]]. Therefore the concern is real as a missing justification but does not threaten the correctness of the central claim; a short clarification would resolve it. I recommend keeping the reader's conditional verdict and asking the authors to add that sentence. Agreement is 'agree' because the same step is the weakest assumption, though I judge it fixable rather than fatal.","tokens_in":29351,"tokens_out":27912,"duration_ms":255946,"concrete_test":"Compute h⊥_n[MX] for n=1,2,3 in the power-sum basis of differential operators and verify that every coefficient is a polynomial in q,t with no factor of M in the denominator. Alternatively, use a computer algebra system to construct a_{F,λ} for all partitions λ with |λ|≤4 via the path-operator formula of Theorem 3.12, specialize to q=t=1, and check that the elementary-basis coefficient matrix is triangular with diagonal entries (Σ_{k≥0} a_k)^{λ_i} whose constant term is 1. If this holds, the q=t=1 specialization is well-defined and the basis argument is sound.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.9's uniqueness claim rests on Proposition 3.9, which asserts that {a_{F,λ}} is a basis of Λ_K. The proof evaluates at q=t=1, shows triangularity of the matrix M_n in the elementary basis with invertible diagonal entries, and concludes that det(M_n) is invertible in K = Q(q,t)[[U,V]]. This inference requires each entry of M_n to have a well-defined q=t=1 limit; otherwise the evaluation is not a ring homomorphism and the triangularity argument is invalid. The paper does not explicitly justify this regularity. The operators A_F^{(ℓ)} are originally defined via commutators that divide by M=(1-q)(1-t), so a priori the coefficients of a_{F,λ} could have poles at q=t=1. The gap is fillable: Theorem 3.12 gives A_F^{(ℓ)} = Σ_{α∈Z^ℓ_{\\ge 0}} a_α Q_α, and each Q_α is a sum of path operators whose valley weights are (qt)^h and whose step operators O(m) are either (-1)^m e_m (for m>0) or h⊥_n[MX] (for m<0). Writing h⊥_n[MX] in the power-sum basis shows that its coefficients are polynomials in q,t (for instance h_1[MX]=(1-q)(1-t)p_1 and h_2[MX] involves (1-q)(1-t) and (1-q^2)(1-t^2), both polynomial), so no denominators in q,t occur. Hence a_{F,λ} has coefficients in Q[q,t][[U,V]] and the specialization is legitimate. The paper should state this explicitly, but the absence of the statement is a presentation issue rather than a substantive flaw.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces path operators acting on symmetric functions with two parameters q and t. For any integer sequence β it defines operators R_β as weighted sums over alternating lattice paths, proves a vertex-operator formula for R_β (Theorem 1.5), identifies R_β with the images of Negut shuffle-algebra elements (Corollary 1.7), and derives Schur/elementary/monomial expansions for R_β · 1 (Corollary 2.7). The paper then defines a G-weighted (q,t)-tau function τ_G(z,X,Y), claims that it satisfies a family of PDEs built from path operators, and asserts that these PDEs characterize τ_G uniquely (Theorems 1.9, 3.4, 3.7). Finally, the path-operator formalism is used to give a new proof of the key reformulation [BHM+23a, Theorem 4.4.1] underlying the extended delta conjecture (Theorem 5.1). The proof is organized so that the combinatorial path-operator results of Sections 2 and 4 feed into the tau-function results of Section 3.","tokens_in":29640,"tokens_out":15969,"duration_ms":171939,"significance":"If the main results are correct, the paper gives a clean combinatorial description of a q,t-deformed tau function and of the relevant shuffle-algebra representation, with explicit expansion formulas and a uniqueness statement for the tau function. The connection to the extended delta conjecture provides an independent verification against a substantial external result. The paper is largely self-contained and the main combinatorial proofs are detailed and inductive. However, the tau-function part contains a load-bearing issue in the proof of the key functional equation, as detailed below, so the advertised characterization and uniqueness are not currently established as written.","major_comments":[{"comment":"The proof of Lemma 3.6 drops the squared norms that appear in Eqs. (18) and (19) and in the definition of τ_G. Writing N_λ = ⟨H_λ,H_λ⟩_*, the correct coefficient identity obtained from the displayed inner products is (G(λ)/N_λ) a^{(2)}_{λ,ξ} = (G(ξ)/N_ξ) a^{(1)}_{λ,ξ}. The lemma as stated concludes a^{(2)}_{λ,ξ} = G(ξ)/G(λ) a^{(1)}_{λ,ξ}, which would require N_λ = N_ξ. This already fails for λ = ∅ and ξ = (1), where N_(1)/N_∅ = -(1-q)(1-t) ≠ 1. Thus the proof of Theorem 3.4 is invalid as written, and since Theorem 3.4 is the functional-equation input to Theorem 1.9, this gap is load-bearing. A repair may be possible by working with normalized Macdonald polynomials or by inserting the appropriate norm factors into the adjoint equation, but the current text does not provide such a repair.","section":"Section 3.2, Lemma 3.6"},{"comment":"The proof of Proposition 3.9 evaluates the matrix M_n at q = t = 1 and uses triangularity with invertible diagonal entries to conclude that det(M_n) is invertible over K = Q(q,t)[[U,V]]. This inference requires that each entry of M_n, which a priori is an element of Q(q,t)[[U,V]], has a well-defined q = t = 1 specialization. Since evaluation at q = t = 1 is not a ring homomorphism on all of Q(q,t), the paper must justify that no poles occur. This is very plausibly true: using Theorem 3.12 and the explicit form of the operators O(m), the coefficients of the operators Q_α are polynomials in q and t. But the manuscript does not state or prove this regularity before performing the specialization, so the basis claim in Proposition 3.9 and the uniqueness theorem depending on it are not fully supported.","section":"Section 3.3, Proposition 3.9"}],"minor_comments":[{"comment":"The proof refers to 'Theorem 2.3' when applying Lemma 2.3; the reference should be to the lemma itself.","section":"Section 2.2, proof of Theorem 1.5"},{"comment":"Near the end of the proof, the text says τ(z) is determined by 'the equations Eq. (25)', but Eq. (25) is the path-operator formula of Theorem 3.12; the intended reference is the system of functional equations, Eq. (22).","section":"Section 3.3, proof of Theorem 3.7"},{"comment":"The proofs refer to 'Definition 4.4' and 'Theorem 4.7'/'Theorem 4.13' for objects that are numbered as Definition 4.4, Lemma 4.7, and Proposition 4.13; the cross-references should be corrected for readability.","section":"Section 4.2, Lemma 4.7 and Section 4.3, Proposition 4.13"},{"comment":"The displayed recurrence uses both 'sign(m−α_i−1)' in the statement and 'sign(m−α_i)' in the final formula of the proof; the sign convention and the bounds on r_1,r_2 should be stated consistently.","section":"Section 4.3, Proposition 4.15"},{"comment":"The proof cites 'Theorem 1.7' for the shuffle-algebra identification used to rewrite the sum of R-operators; the intended statement is Corollary 1.7.","section":"Section 5, proof of Theorem 5.1"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a substantial paper and should be refereed. The authors extend their Jack-case path operators to the (q,t)/Macdonald setting, showing they are images of Negut shuffle algebra elements via a clean residue formula (Thm 1.5), producing monomial/elementary/Schur expansions of R_β·1, and using the operators to characterize a (q,t)-tau function by a family of PDEs with uniqueness (Thm 1.9). They also give a new proof of the key technical result behind the extended delta conjecture, cross-checked against BHM+23a.\n\nWhat is genuinely new: the (q,t) path operators, their shuffle algebra identification, the expansions, and the PDE characterization. The proofs are detailed, and Section 4's commutation relations—interpreted as path extensions—are intricate but convincing. The independent proof of the BHM+ result is a significant payoff.\n\nSoft spots, in proportion. The uniqueness proof (Thm 3.7) depends on Prop 3.9, which establishes that the a_{F,λ} form a basis by specializing to q=t=1. The paper does not explicitly justify that the relevant matrix entries have well-defined q=t=1 limits. This is a genuine presentation gap, but not a substantive flaw: via Thm 3.12, each operator is a sum of path operators whose step operators are either (-1)^m e_m or h⊥_n[MX], and h⊥_n[MX] written in power sums has polynomial coefficients in q,t (e.g., h_1[MX]=(1-q)(1-t)p_1), so no denominators occur. The specialization is legitimate; a referee should ask the authors to add a sentence. The other soft spot is Remark 5.2, where the bijection ψ is said to match BHM's correspondence with 'not very hard to see'—that deserves a short proof or precise reference, but it is not load-bearing.\n\nWho it is for: people working on Macdonald polynomials, shuffle algebras, and delta-type conjectures. The main theorems are new and look correct; the one gap is easily fixable. I would send it to peer review, and ask the referee to verify the specialization step and the ψ correspondence.","headline":"A substantial (q,t) extension of path operators with a new PDE characterization of a (q,t)-tau function and a fresh proof of a key delta-conjecture result; the main gap is a terse q=t=1 specialization argument that is fixable.","tokens_in":30228,"tokens_out":10926,"would_cite":true,"duration_ms":94622,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05E05","05E10"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper constructs path operators on $(q,t)$-symmetric functions and proves these operators give a unique PDE characterization of the $(q,t)$-tau function.","keywords":["path operators","symmetric functions","Macdonald polynomials","shuffle algebra","Negut elements","tau functions","extended delta conjecture","functional equations"],"falsifier":"Compute the determinant of the change-of-basis matrix from $\\{(-1)^{|\\lambda|}a_{F,\\lambda}\\}_{\\lambda\\vdash n}$ to the elementary basis for $n\\le 3$ as an explicit element of $\\mathbb{Q}(q,t)[[U,V]]$; if its denominator vanishes at $q=t=1$ for some $F$ (for instance $G_1=G_2=1$), then the specialization step in the proof of Proposition 3.9 would fail, and a direct check of the PDE system at low degree would be needed to decide whether uniqueness still holds.","tokens_in":29114,"feed_emoji":"🧮","tokens_out":12125,"duration_ms":97372,"temperature":0.7,"pith_summary":"The paper constructs a new family of operators on symmetric functions in two parameters $q$ and $t$, one for each alternating lattice path, and proves that a certain $(q,t)$-deformation of the classical hypergeometric tau function is uniquely characterized by a system of partial differential equations built from these operators. The tau function $\\tau_G(z,X,Y)$ is the Macdonald-polynomial series that specializes to several known generating functions in enumerative geometry and gauge theory. The operators also give explicit expansions in Schur, monomial, and elementary bases, and yield a new proof of a key reformulation behind the extended delta conjecture. The upshot is that a purely combinatorial structure---paths whose steps alternate up and down---carries enough information to pin down a two-parameter tau function completely.","feed_headline":"Lattice-path operators uniquely fix the (q,t)-tau function","feed_subtitle":"One family of path operators yields a full PDE characterization of this Macdonald-based series and a new proof for the extended delta…","key_machinery":"The central object is the path operator $R_\\beta=\\sum_{\\gamma\\in R_\\beta}\\mathrm{vw}_\\beta(\\gamma)\\,O_{\\gamma_1}\\cdots O_{\\gamma_{2\\ell}}$, where the sum runs over alternating lattice paths staying weakly above a reference path $\\gamma_\\beta$, each weighted by $(qt)^{\\mathrm{ht}_\\beta(V)}$ over its valleys, and $O(m)$ is the step operator $(-1)^m e_m[X]$ for $m>0$ and $h^\\perp_{-m}[MX]$ for $m<0$. The identity that carries the argument is $R_\\beta=[z_1^{\\beta_1}\\cdots z_\\ell^{\\beta_\\ell}]\\,D(z_1)\\cdots D(z_\\ell)\\prod_{i=1}^{\\ell-1}(1-qt\\, z_{i+1}/z_i)$, where $D(z)=\\sum_n R_{(n)}z^n$ is the standard field; this identifies $R_\\beta$ with the image of a Negut element under the shuffle-algebra representation. For the tau function, the moving part is the family $A_F^{(\\ell)}=\\sum_\\alpha a_\\alpha Q_\\alpha$, where $Q_\\alpha$ is the reparametrization of $R_\\beta$ by distances between decorated particles, along with commutation relations that interpret $\\mathrm{ad}_{D_0/M}$ and $\\mathrm{ad}_{e_1/M}$ as adding particles or extending steps.","core_discovery":"On the paper's own terms, the central claim is Theorem 1.9: for every $\\ell\\ge 1$, the identity $z^\\ell A_{G_1}^{(\\ell)}(X)\\cdot \\tau_G(z,X,Y)=\\bigl(A_{G_2}^{(\\ell)}(Y)\\bigr)^*\\cdot \\tau_G(z,X,Y)$ holds, and $\\tau_G$ is the only series in $\\Lambda^K_X\\otimes\\Lambda^K_Y[[z]]$ with constant term $1$ satisfying all these equations. The operators $A_F^{(\\ell)}$ are sums $\\sum_{\\alpha\\in\\mathbb{Z}_{\\ge0}^\\ell} a_\\alpha Q_\\alpha$, where $Q_\\alpha$ is a path operator written in distance coordinates and $a_\\alpha$ are coefficients of $F=G_1$ or $G_2$. The proof of the equations uses the Pieri rule and the Macdonald operator $D_0$ acting on modified Macdonald polynomials; the proof of uniqueness shows that repeated applications of the $A_F^{(\\ell)}$ to $1$ produce a basis of the symmetric-function ring. The identification of these operators with path operators is established through commutation relations that add particles to decorated lattice paths, and the same mechanism supplies the new proof of the extended-delta-conjecture key result.","pith_inferences":["If the PDE system is as strong as claimed, one could compute $\\tau_G$ coefficient-by-coefficient by solving the linear equations degree by degree, an algorithm that may outperform summing over all partitions for small weights.","The distance-sequence bijection $\\psi$ is the same combinatorial map used in [BHM+23a], which suggests the path-operator proof strategy may extend to other shuffle-theoretic delta-type conjectures.","It remains to be seen whether the operators $A_F^{(\\ell)}$ act triangularly on the modified Macdonald basis; if they do, the uniqueness theorem would follow from a more algebraic argument not requiring the $q=t=1$ specialization.","One could test whether the functional equations continue to characterize $\\tau_G$ when $G$ is a rational weight with zeros or poles, a case the current formal setup excludes but which appears naturally in some geometric specializations."],"forward_implications":["If Theorem 1.9 holds, the $(q,t)$-tau function is fully pinned down by the path-operator PDE system together with its constant term, so any series satisfying those equations is automatically $\\tau_G$.","The explicit Schur, monomial, and elementary expansions in Theorem 1.8 give closed formulas for $R_\\beta\\cdot 1$ in the classical symmetric-function bases.","Because $R_\\beta$ equals the shuffle-algebra image of Negut elements, the paper supplies an explicit combinatorial formula for that representation.","Theorem 5.1 provides a new proof of the key reformulation [BHM+23a, Theorem 4.4.1] underlying the extended delta conjecture, independent of intermediate elliptic Hall algebra results.","For special choices of $G$, the same PDE characterization covers the character-variety generating function, the Whittaker-vector specialization of the deformed Virasoro algebra, and the $(q,t)$-Toda tau function."],"supporting_citations":[{"why":"Supplies the field $D(z)$, its commutation relation, and the operator $D_0$ with modified Macdonald eigenfunctions used throughout.","marker":"[GHT99]"},{"why":"Provides the modified Macdonald polynomials, the Pieri rule, and the star scalar product defining $\\tau_G$.","marker":"[Mac95]"},{"why":"Introduces the shuffle-algebra elements whose images under the representation are shown to be the path operators.","marker":"[Neg14]"},{"why":"States the key reformulation of the extended delta conjecture that the paper reproves via path operators.","marker":"[BHM+23a]"},{"why":"Defines the $G$-weighted $b$-Hurwitz tau functions that the present $(q,t)$-tau function deforms.","marker":"[CD22]"},{"why":"Constructs analogous operators in the Macdonald setting that motivate the extension.","marker":"[NS15]"}],"fun_headline_variants":["Path operators uniquely fix (q,t)-tau via PDEs","Lattice path operators prove extended delta","Path operators characterize (q,t)-tau uniquely","PDEs from path operators determine (q,t)-tau"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof of uniqueness assumes that the rational functions in $q$ and $t$ expressing the products $A_F^{(\\lambda_1)}\\cdots A_F^{(\\lambda_\\ell)}\\cdot 1$ in the elementary basis can be specialized to $q=t=1$ without poles; if that specialization is not legitimate, the triangularity argument only proves a limit statement and the basis claim over $\\mathbb{Q}(q,t)[[U,V]]$ is left open.","fun_headline_variants_meta":{"raw":{"variants":["Path operators uniquely fix (q,t)-tau via PDEs","Lattice path operators prove extended delta","Path operators characterize (q,t)-tau uniquely","PDEs from path operators determine (q,t)-tau"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000897,"raw_usage":{"total_tokens":3891,"prompt_tokens":1000,"completion_tokens":2891,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":616,"completion_tokens_details":{"reasoning_tokens":2828}},"tokens_in":616,"tokens_out":2891,"duration_ms":20517,"temperature":1.0,"reasoning_tokens":2828,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T06:02:13.977153+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the determinant of the change-of-basis matrix from $\\{(-1)^{|\\lambda|}a_{F,\\lambda}\\}_{\\lambda\\vdash n}$ to the elementary basis for $n\\le 3$ as an explicit element of $\\mathbb{Q}(q,t)[[U,V]]$; if its denominator vanishes at $q=t=1$ for some $F$ (for instance $G_1=G_2=1$), then the specialization step in the proof of Proposition 3.9 would fail, and a direct check of the PDE system at low degree would be needed to decide whether uniqueness still holds.","supporting_citations":[],"review_version":1}