{"id":"21ad00d4-9ac4-4e51-bddc-4ed2bc7bb0db","arxiv_id":"2501.14714","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Explicit vertex-operator super-Hamiltonians are conjectured whose eigenfunctions are the super-Macdonald polynomials of earlier work.","lead":"This paper constructs explicit super-Hamiltonians, operator expressions in bosonic and Grassmann variables, whose proposed eigenfunctions are super-Macdonald polynomials. It gives four closed formulas, two of which are claimed to suffice to determine all these polynomials, and reports computational checks up to super-diagrams of order 25/2.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central eigenfunction claim rests on conjectural formulas: (98)/(100) are only checked to |λ|=25/2, and the H^{-} pair is built from an ad hoc correlator (101), so the finite check does not establish the general statement.","rationale":"I agree with the reader's conditional assessment. The central object is a set of conjectural operator identities; the only stated evidence is a finite check to |λ|=25/2, and the construction of the negative pair explicitly uses an ad hoc correlator. My concern does not identify a contradiction, and I credit the paper for giving explicit formulas, a finite check, and the correct bosonic limit. The appropriate outcome is therefore to keep the reader's CONDITIONAL verdict: accept with the requirement that the conjecture be either proved or verified independently and more extensively.","tokens_in":13578,"tokens_out":9648,"duration_ms":94629,"concrete_test":"Develop an independent implementation of super-Macdonald polynomials from the triangularity/Cauchy definition of Refs. [2,4], without using (98)/(100), and compute residual r_λ = H^{(1),+} M^{q,t}_λ - E^{(1),+}_λ M^{q,t}_λ for all diagrams with |λ| ≤ 25/2 and then up to |λ| = 50/2, including odd diagrams; any nonzero r_λ refutes (98). For the H^{-} pair, repeat with (100) and additionally test sensitivity by replacing correlator (101) with a one-parameter deformation that preserves ⟨s^a⟩=1 and the fermionic algebra; if the eigenfunction property fails under a small deformation, the original check is a fit rather than a structurally forced result. This would not replace a proof, but it would distinguish a numerical accident from a stable pattern.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing gap is the eigenvalue claim for (98) and (100): it is explicitly conjectural and verified only through super-diagrams of size |λ|=25/2. The H^{(3),-} and H^{(4),-} operators are produced with an auxiliary bosonic operator s and a fermionic pair ν,ν†, and the correlator (101) is described by the authors as a 'mere simplification trick.' Because this correlator is chosen ad hoc, the ansatz has enough freedom to reproduce the target eigenvalues on any finite list of diagrams, so the finite check is not independent evidence for the all-orders identity. The unproved Pierri rules (102)–(103) create a second gap, since the commuting family (106)–(107) is derived from them. The paper is internally consistent, and the even-sector reduction to ordinary Macdonald Hamiltonians is a credible check; the issue is the absence of a general proof or of verification that is not fitted to the claimed eigenvalues.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a supersymmetric generalization of the Macdonald finite-difference Hamiltonian. After reviewing the Schur, super-Schur, and Macdonald cases, the authors introduce Grassmann time variables θ_k and define four super-Hamiltonians whose claimed eigenfunctions are super-Macdonald polynomials. The eigenvalue formula (97) has the expected sum-over-boxes form with weights q^{2j}t^{-2i}. Closed vertex-operator expressions for the four Hamiltonians are given in Eqs. (98) and (100); the authors state that these formulas are conjectural and have been checked only up to diagrams of order |λ| = 25/2. The paper also constructs a commuting family of higher Hamiltonians in Eqs. (106)-(107), based on Pieri rules for super-Macdonald polynomials stated in Eqs. (102)-(103). Sections 2-4 provide background and set up the general \"sum-over-boxes plus Pieri rules\" framework.","tokens_in":13844,"tokens_out":4390,"duration_ms":40239,"significance":"If the conjectural formulas are correct, the paper provides an explicit Hamiltonian characterization of super-Macdonald polynomials in a vertex-operator form, with potential applications to super-Yangian and DIM representations. The paper's strengths are its explicit formulas, the consistency checks at order |λ| = 25/2, and the reduction to ordinary Macdonald Hamiltonians on even diagrams. However, the main eigenfunction statement is not proven at all orders, and the verification of the H^{-} operators relies on an auxiliary correlator that appears chosen to reproduce the target eigenvalues; the result is therefore conditional. The paper is internally consistent, but the load-bearing claims require either a proof or a substantially stronger and more independent body of evidence.","major_comments":[{"comment":"The central claim of the paper is that the operators defined by (98) and (100) have super-Macdonald polynomials as eigenfunctions with the eigenvalues (97). The authors explicitly label these formulas as conjectural and report a check only up to |λ| = 25/2, with no all-orders derivation. Since this eigenfunction statement is the paper's main result, the absence of a proof is a load-bearing gap: the finite check does not by itself establish the general statement. I recommend either supplying a proof by a normal-ordering or contraction analysis, or, if the paper is intended as a conjecture, stating this status more prominently and substantially extending the numerical evidence.","section":"Section 5, Eqs. (98), (100)"},{"comment":"The auxiliary correlator ⟨∅|νν†s^b|∅⟩_B is introduced ad hoc and described by the authors as a \"mere simplification trick.\" Because the H^{-} operators in (100) are defined through this correlator, and the correlator is chosen so that the desired eigenvalue terms survive, the finite-order check of (100) is not independent of the target eigenvalues in (97). This weakens the evidential value of the check for the all-orders identity. A concrete way to address this is to derive (101) from a principled algebraic requirement such as a free-field realization, or to verify (100) at orders beyond |λ| = 25/2 with a comparison that does not use the desired eigenvalues as input.","section":"Section 5, Eq. (101)"},{"comment":"The Pieri rules for super-Macdonald polynomials are stated without proof and without a precise citation to a place where they are established. The commuting family (106)-(107) and the recursive operators (104)-(105) rest on these rules; as written, this part of the construction is conditional on an additional unproved assumption. Please provide a proof of (102)-(103) or point to the specific statement in the cited literature where these rules are proven.","section":"Section 5, Eqs. (102)-(103)"}],"minor_comments":[{"comment":"In the fourth eigenvalue expression, the summation variables x and y are undefined; they should be the box coordinates i and j as in the preceding lines of (97).","section":"Section 5, Eq. (97)"},{"comment":"The name \"Pierri\" is a misspelling of \"Pieri\" and should be corrected throughout the manuscript.","section":"Throughout"},{"comment":"The notation for the two families of super-Schur Hamiltonians, with two labels and multiple hats, is difficult to follow; a short glossary or table of the operators would improve readability.","section":"Section 3.2"},{"comment":"The sentence beginning \"Operators as being split in brackets in the r.h.s.\" in the discussion after Eq. (64) is grammatically incomplete and should be rephrased.","section":"Section 3.3"},{"comment":"The typesetting of the second Hamiltonian in (100) contains an anomalous vertical bar adjacent to the vacuum ket; please fix the formatting.","section":"Eq. (100)"}],"recommendation":"major_revision","confidential_remarks":"The paper's main formulas are explicitly conjectural, and the finite-order verification is weakened by the ad hoc choice of the auxiliary correlator (101). If the journal accepts conjectural results with computational evidence, the framing should be adjusted and the evidence strengthened; otherwise the authors should be asked to provide an all-orders proof. The paper is a reasonable candidate after such revision, but in its current form the central claim is not established beyond a fitted finite check."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the quick take: the paper gives an explicit conjectural construction of four super-Hamiltonians whose eigenfunctions are supposed to be the super-Macdonald polynomials. The formulas (98) and (100) are concrete and attractive, and the paper does a good job of situating them in the Schur/Macdonald framework. But the evidence for the central eigenfunction claim is thin: the formulas are checked only up to |λ|=25/2, and that check is not independent because the auxiliary bosonic operator s and fermionic pair ν,ν† with correlator (101) are chosen to reproduce the target eigenvalues. In that sense the construction is reverse-engineered, and the finite verification doesn't establish the general statement.\n\nWhat's genuinely new: the Grassmann-time extension of the Macdonald vertex-operator Hamiltonian, the explicit four-Hamiltonian structure, and the observation that the q,t inversion symmetry is broken for super-Macdonald polynomials. That last point is interesting and, as far as I know, not in the earlier literature. The paper is also honest: it labels the formulas conjectural and describes the correlator as a 'mere simplification trick.'\n\nThe soft spots, in order of severity. First, the eigenvalue claim for (98) and (100) is load-bearing and unproved. The finite check is a useful sanity test but, because the ansatz has free parameters that are tuned to the eigenvalues, it doesn't rule out a failure at higher degree. Second, the Pierri rules (102)-(103) for super-Macdonald polynomials are assumed without proof, and the commuting family (106)-(107) is derived from them. Third, the paper does not explain how the new Hamiltonians relate to the existing supersymmetric Ruijsenaars operators of [5,29]; a comparison would help readers judge whether this is a genuinely different construction or a repackaging.\n\nThe paper is worth engaging with. I would send it to a serious referee, with the expectation that the referee will ask for either a proof of the eigenvalue claim, a more extensive check that is not fitted to the answer, or at least a clear statement that the central result is a conjecture whose only evidence is the finite check. For a reader working on super-Macdonald polynomials or Yangian/DIM representations, it's a useful pointer to a plausible structure. But I wouldn't treat the Hamiltonians as established until the verification is independent or a proof appears.","headline":"A neat, clearly written conjectural construction of super-Hamiltonians for super-Macdonald polynomials, but the central eigenfunction claim rests on a finite check that is partly fitted to the target eigenvalues, so it is a plausible conjecture rather than an established result.","tokens_in":14333,"tokens_out":2835,"would_cite":true,"duration_ms":23585,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05E05","17B80","81R12"],"pacs":[],"model":"deepseek-v4-flash","headline":"Four explicit super-Hamiltonians, with conjectural vertex-operator formulas checked up to order 25/2, make super-Macdonald polynomials their eigenfunctions.","keywords":["super-Macdonald polynomials","super-Hamiltonians","Grassmann time variables","super-Young diagrams","sum over boxes","Pierri rules","vertex operators","commuting Hamiltonians"],"falsifier":"Compute the action of the operator in (98) on a super-Macdonald polynomial labeled by a diagram of order $|\\lambda|=27/2$, for instance $\\lambda=[13,1/2]$, and compare with the eigenvalue formula (97); a single mismatch would disprove the conjectural Hamiltonian. A counterexample to the assumed Pierri rules (102)-(103) would likewise invalidate the commuting family (106)-(107).","tokens_in":13389,"feed_emoji":"🧮","tokens_out":9895,"duration_ms":82835,"temperature":0.7,"pith_summary":"Super-Macdonald polynomials are symmetric functions in both ordinary variables $p_k$ and Grassmann variables $\\theta_k$, labeled by half-integer super-Young diagrams. This paper tries to establish that these polynomials are eigenfunctions of a set of four explicitly written super-Hamiltonians, two with positive and two with negative powers of the parameters, whose eigenvalues are sums over the diagram of $q^{2j}t^{-2i}$. The paper writes closed vertex-operator formulas for the Hamiltonians and reports verification up to diagrams of order $|\\lambda|=25/2$; the general statement is left as a conjecture. If correct, the construction gives super-Macdonald polynomials an operator definition analogous to the classical operator characterization of ordinary Macdonald polynomials.","feed_headline":"Four Hamiltonians claim super-Macdonald polynomials as eigenfunctions","feed_subtitle":"Vertex-operator formulas give sum-over-boxes eigenvalues; two operators fix every super-Macdonald polynomial.","key_machinery":"The operative machinery is the 'sum over boxes' principle combined with vertex-operator building blocks. The Hamiltonians are contour integrals of two exponentials: one in the bosonic times $p_k$ and Grassmann times $\\theta_k$, the other in the derivatives $\\partial/\\partial p_k$ and $\\partial/\\partial\\theta_k$. Auxiliary fermionic oscillators $\\psi,\\psi^\\dagger$ for the first pair, and an invertible boson $s$ with a fermionic pair $\\nu,\\nu^\\dagger$ for the second pair, project the vertex operators onto the sector that acts on super-Macdonald polynomials; the paper calls $s$ and $\\nu$ a mere simplification trick. The commuting set of higher Hamiltonians (106)-(107) is then generated from box-adding and box-removing operators whose commutators and anticommutators cancel off-diagonal terms, following the Pierri-rule logic set out in Section 2.","core_discovery":"The central claim is that super-Macdonald polynomials $M^{q,t}_\\lambda$ are common eigenfunctions of the four super-Hamiltonians defined by (97) and given in closed form by (98) and (100). For the two positive Hamiltonians the eigenvalues read\n$$\n1+($q^{2}$-1)(1-$t^{{-2}}$)\\sum_{\\square\\in\\$\\lambda$} $q^{{2j}}$$t^{{-2i}}$,\n$$\nwhere the sum runs over the half-boxes of the super-Young diagram $\\lambda$; the two negative Hamiltonians have the same form with $q\\to q^{-1}$, $t\\to t^{-1}$. The paper states that the four eigenvalues are generically distinct on odd diagrams and that two of the Hamiltonians suffice to determine all super-Macdonald polynomials. The closed formulas are labeled conjectural and have been checked up to $|\\lambda|=25/2$.","pith_inferences":["If the conjectural formulas are verified for all orders, the auxiliary $s,\\nu$ correlator is most naturally a remnant of a supersymmetric many-body integrable system, a connection the paper leaves open.","A proof of the closed formulas without assuming the eigenvalues would resolve their conjectural status; the natural route is an independent evaluation of the vertex-operator action on arbitrary super-Macdonald polynomials.","The broken $q\\leftrightarrow q^{-1}$, $t\\leftrightarrow t^{-1}$ symmetry suggests that super-generalizations of the relevant infinite-dimensional algebras carry two independent commuting families of Cartan-like generators, one for each sign, a representational prediction not stated in the paper."],"forward_implications":["Super-Macdonald polynomials become computable as common eigenfunctions of two explicitly written super-Hamiltonians, without relying on Cauchy formulas or triangularity conditions.","The four eigenvalues are independent on generic odd super-diagrams, so the two positive Hamiltonians separate all super-Macdonald polynomials.","The construction predicts a full tower of commuting higher super-Hamiltonians, with one family explicitly built and the second family left to future work.","The broken $q,t$ inversion symmetry of super-Macdonald polynomials implies the super-case carries additional commuting operators, which the paper links to a difference in the commuting structures of two infinite-dimensional algebras."],"supporting_citations":[{"why":"Gives the original Macdonald polynomial theory that forms the bosonic limiting case.","marker":"[1]"},{"why":"Defines super-Macdonald polynomials and earlier constructions that identify the target eigenfunctions.","marker":"[2–5]"},{"why":"Introduces super-Young diagrams with half-integer parts, the combinatorial labels used for super-Macdonald polynomials.","marker":"[4,6,9]"},{"why":"Provides a prior super-Hamiltonian approach to super-Macdonald polynomials that this construction extends and compares with.","marker":"[5,29]"},{"why":"Supplies the ordinary Macdonald Hamiltonian and its sum-over-boxes eigenvalue formula (96), which the super-case lifts.","marker":"[17]"},{"why":"Provides the single-hook expansions used in Section 4.2 to rewrite the Macdonald Hamiltonian.","marker":"[35]"}],"fun_headline_variants":["Four super-Hamiltonians yield super-Macdonald eigenfunctions","Grassmann variables extend Macdonald theory to super case","Super-Macdonald polynomials are eigenfunctions of four operators","New super-Hamiltonians diagonalize super-Macdonald polynomials","Super-Macdonald polynomials solved by quartet of Hamiltonians"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The closed vertex-operator formulas (98) and (100) are conjectural: they have been checked only up to $|\\lambda|=25/2$, and the check uses a correlator chosen to reproduce the target eigenvalues, so the general case is not independently confirmed.","fun_headline_variants_meta":{"raw":{"variants":["Four super-Hamiltonians yield super-Macdonald eigenfunctions","Grassmann variables extend Macdonald theory to super case","Super-Macdonald polynomials are eigenfunctions of four operators","New super-Hamiltonians diagonalize super-Macdonald polynomials","Super-Macdonald polynomials solved by quartet of Hamiltonians"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000199,"raw_usage":{"total_tokens":1297,"prompt_tokens":799,"completion_tokens":498,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":415,"completion_tokens_details":{"reasoning_tokens":416}},"tokens_in":415,"tokens_out":498,"duration_ms":7277,"temperature":1.0,"reasoning_tokens":416,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T14:52:15.373722+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the action of the operator in (98) on a super-Macdonald polynomial labeled by a diagram of order $|\\lambda|=27/2$, for instance $\\lambda=[13,1/2]$, and compare with the eigenvalue formula (97); a single mismatch would disprove the conjectural Hamiltonian. A counterexample to the assumed Pierri rules (102)-(103) would likewise invalidate the commuting family (106)-(107).","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the original Macdonald polynomial theory that forms the bosonic limiting case."},{"cited_title":"On generalized Macdonald polynomials","cited_arxiv_id":"1907.05410","evidence_quote":"Supplies the ordinary Macdonald Hamiltonian and its sum-over-boxes eigenvalue formula (96), which the super-case lifts."},{"cited_title":"Hook variables: cut-and-join operators and $\\tau$-functions","cited_arxiv_id":"1912.00635","evidence_quote":"Provides the single-hook expansions used in Section 4.2 to rewrite the Macdonald Hamiltonian."}],"review_version":1}