{"id":"4e5a866c-5989-4003-b6bb-9068d19e8595","arxiv_id":"2607.06738","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":4.5,"correctness_risk":"low","formal_verification":"none","parameter_count":1,"one_line_summary":"Type C∨C DAHA and Koornwinder systems mirror type-A Macdonald structures for Hamiltonians, recursions, evaluations and dualities, but lack a usable Noumi-Shiraishi-style universal series and SL(2,Z)-type twisting automorphisms.","lead":"The paper systematically compares DAHA/DIM-inspired integrable systems of type A (Macdonald/Ruijsenaars) with type C∨C (Koornwinder), cataloguing parallel properties of their non-symmetric and symmetric polynomial eigenfunctions while isolating two structural failures for C∨C. It matters because it maps exactly where the triad of universal series, polynomials and Baker-Akhiezer functions breaks, guiding future constructions of non-A integrable systems.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The manuscript is a careful comparative exposition whose strongest claim is a catalogue of verified parallels plus two openly acknowledged obstructions. The reader’s proposed soft spot is not load-bearing: the higher Hamiltonians are supplied independently by van Diejen’s formulas, so the power-sum construction is used only for the first operator and is not required for the rest of the argument. No internal inconsistency, circularity or unsupported algebraic claim appears. The concrete recursion check above is the natural low-cost verification that the non-symmetric side works as stated; once it passes, the ACCEPT verdict stands without modification.","tokens_in":35716,"tokens_out":480,"duration_ms":6022,"concrete_test":"Verify that the monic non-symmetric polynomials generated by the recursion (92)–(94) for n=2, |α|≤2 coincide with the explicit eigenfunctions of the Cherednik operators C_i given in (84); if they match and satisfy C_i E_α=Λ_α^{(i)} E_α with the eigenvalues (86), the claimed parallel for the non-symmetric sector is confirmed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest_assumption (that the spherical-DAHA power-sum construction may fail for higher Koornwinder-van Diejen Hamiltonians) does not land. Equation (76) is only the first Hamiltonian; the full commuting family is written explicitly in (77)–(78) and is already known to be integrable by the classical van Diejen construction. The paper never claims that higher Hamiltonians are obtained by power sums of the Cherednik operators C_i; it only asserts that the first one arises that way and that the eigenfunctions of the C_i are the non-symmetric Koornwinder polynomials. All other listed parallels (triangular expansions, Knop–Sahi-type recursions (92)–(94), orthogonality measures, evaluation formulas, dualities, weak stability) are either derived or cited to the standard literature and hold independently of any missing automorphisms. The two genuine structural gaps (no factorizing branching rule, no twisting) are already isolated by the authors themselves and do not undermine the catalogue of existing properties.","agreement_with_reader":"disagree"},"referee_report":{"model":"grok-4.5","summary":"The paper systematically compares the integrable systems associated with type-A DAHA/DIM (Cherednik operators, Ruijsenaars–Schneider Hamiltonians, non-symmetric and symmetric Macdonald polynomials, Noumi–Shiraishi series and Baker–Akhiezer functions) with their type-C∨C counterparts (Cherednik operators of type C∨C, Koornwinder–van Diejen Hamiltonians, non-symmetric and symmetric Koornwinder polynomials). It records the parallel structures—triangular expansions, Knop–Sahi-type recursions (92)–(94), orthogonality measures, evaluation formulas, dualities and weak stability—while isolating two genuine structural gaps: the absence of a factorizing branching rule that would produce a Noumi–Shiraishi-type universal series, and the lack of enough DAHA automorphisms to generate twisted systems. Rank-one (Askey–Wilson) specializations and explicit Baker–Akhiezer series for reduced root systems are treated in detail.","tokens_in":35959,"tokens_out":775,"duration_ms":8703,"significance":"The manuscript supplies a clear, self-contained catalogue of the algebraic properties of non-symmetric and symmetric Koornwinder polynomials that mirrors the well-known Macdonald theory. The recursive constructions, evaluation formulas and dualities are written explicitly and match the literature they cite (Noumi, Sahi, Stokman, Chalykh). By isolating the two places where the C∨C story diverges from type A, the paper clarifies the precise limits of the DIM/spherical-DAHA correspondence beyond type A and provides a useful reference for further work on non-reduced root systems and possible elliptizations.","major_comments":[],"minor_comments":[{"comment":"In §2.1.1 the Hecke relation is written (T_i-1)(T_i+t^{-1})=0 while in §3.1.1 it is (T_i-t^2)(T_i+1)=0; a short remark that the two normalizations differ by a rescaling of the generators would help the reader.","section":null},{"comment":"Equation (76) presents only the first Koornwinder Hamiltonian as a Weyl-symmetric combination of Cherednik operators; a one-sentence clarification that the higher Hamiltonians (77)–(78) are taken from the classical van Diejen construction (and are not claimed to arise by power sums) would remove any possible ambiguity.","section":null},{"comment":"The branching-rule formula (137) cites the very involved coefficients of van Diejen–Emsiz; a pointer to the precise equation number in that reference would make the claim easier to verify.","section":null},{"comment":"A few typographical inconsistencies remain (e.g., “eduction” for “reduction” near (57), occasional missing spaces around “=”). A light copy-edit pass would clean them up.","section":null}],"recommendation":"accept","confidential_remarks":"The manuscript is essentially a careful comparative review with a few new explicit formulae (especially the rank-one BA series). It is solid and useful, but its novelty is modest; the journal should decide whether that level of contribution fits its usual standards. No integrity or citation issues."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This is a careful side-by-side of type-A Macdonald/Ruijsenaars/DAHA material against the type-C∨C Koornwinder/van Diejen/DAHA story. The real addition is the systematic comparison that pins down exactly where the parallel breaks: no factorising branching rule that would give a Noumi-Shiraishi-type universal series, and not enough DAHA automorphisms for twisting. That diagnosis is useful.\n\nWhat they do well is the catalogue itself. Sections 3–4 lay out the Cherednik operators, the recursive construction of non-symmetric Koornwinder polynomials (the C∨C Knop–Sahi analogues), the Weyl-orbit sums that produce the symmetric ones, the orthogonality measures, evaluation formulas, dualities and weak stability. All of it is written with explicit formulas that match the literature they cite (Sahi, Stokman, Chalykh, van Diejen, Noumi). The rank-one Baker-Akhiezer series for the reduced systems (B1, C1, B∨, C∨) are new and clean; they reduce to the A1 case after simple parameter changes, which is satisfying. The stress-test note is right: the worry that higher van Diejen Hamiltonians might not arise from power sums of the Ci is a non-issue. The paper only claims the first Hamiltonian that way (eq. 76) and writes the full commuting family explicitly (77–78); everything else stands independently.\n\nSoft spots are minor and already flagged by the authors. The branching-rule discussion in 5.2 correctly shows why you cannot just continue to complex eigenvalues the way you do for Macdonald, and the generic C∨C1 BA coefficients do not factor. The “towards a triad” section is therefore more of a status report than a construction. Self-citations stay within their earlier DIM/DAHA papers and are not load-bearing. No circularity, no free parameters being fitted.\n\nThis is for people who already work with root-system special functions or quantum integrable systems of Ruijsenaars type. It will save them time when they need the C∨C analogues of the standard Macdonald identities. It is not paradigm-shifting, but it is honest, reproducible pure algebra and deserves a serious referee. I would accept it for peer review and would cite the BA series and the obstruction diagnosis if I were writing on Koornwinder or non-A DAHA.","headline":"Solid comparative catalogue of A vs C∨C DAHA/Koornwinder systems that cleanly isolates two real structural gaps; useful reference, not a breakthrough.","tokens_in":36581,"tokens_out":638,"would_cite":true,"duration_ms":7366,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["33D52","33D80","17B37","81R12"],"pacs":["02.30.Ik","03.65.Fd"],"model":"grok-4.5","headline":"Type C∨C DAHA and Koornwinder systems mirror type A Macdonald theory for eigenfunctions, but lose the Noumi-Shiraishi series and twisting automorphisms.","keywords":["DAHA","DIM algebra","Koornwinder polynomials","Macdonald polynomials","Cherednik operators","van Diejen-Koornwinder Hamiltonians","Baker-Akhiezer functions","type C∨C root systems"],"falsifier":"Explicitly compute the second and third commuting van Diejen-Koornwinder operators for n=2 or n=3 and check whether they equal the corresponding power sums of the Cherednik operators restricted to Weyl-symmetric functions; any mismatch would break the claimed parallel.","tokens_in":36625,"feed_emoji":"⚛️","tokens_out":856,"duration_ms":7277,"temperature":0.7,"pith_summary":"The paper shows that the integrable systems built from type C∨C double affine Hecke algebras and their spherical projections stand in the same relation to Koornwinder polynomials that type A DAHA and DIM algebra stand to Macdonald polynomials. Non-symmetric Koornwinder polynomials are the common eigenfunctions of the Cherednik operators; their Weyl-group averages are the ordinary (symmetric) Koornwinder polynomials that diagonalize the van Diejen-Koornwinder Hamiltonians. Almost every structural property that makes the Macdonald theory useful—triangular expansions, recursive construction, orthogonality measures, evaluation formulas, dualities, and a weak form of stability—has a direct counterpart. The two places where the parallel breaks are decisive: there is no factorizing branching rule that would produce a Noumi-Shiraishi-type universal power series, and the DAHA of type C∨C lacks the automorphisms that generate twisted (\"integer-ray\") systems. The result therefore both enlarges the catalogue of explicitly solvable many-body models and isolates precisely which algebraic features of type A are responsible for the richest part of the Macdonald triad.","feed_headline":"Koornwinder systems mirror Macdonald, lose the universal series","feed_subtitle":"Type C∨C DAHA gives the same eigenfunction toolkit, but no Noumi-Shiraishi series and no twisting.","key_machinery":"The spherical projection that realises the first Koornwinder Hamiltonian as the Weyl-symmetric combination of Cherednik operators Ci + Ci−1 (eq. 76), together with the recursive action of the affine intertwiners B and Ti that generate all monic non-symmetric Koornwinder polynomials from the constant function.","core_discovery":"Non-symmetric and symmetric Koornwinder polynomials are the eigenfunctions of the type-C∨C Cherednik and Koornwinder-van Diejen Hamiltonians respectively, and they possess direct counterparts to the Macdonald triangular expansions, Knop-Sahi recursions, orthogonality measures, evaluation formulas, dualities and weak stability; the only essential failures are the absence of a factorizing branching rule that would yield a Noumi-Shiraishi-type universal series and the absence of enough DAHA automorphisms to produce twisted systems.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Koornwinder polynomials mirror Macdonald without Noumi-Shiraishi series","Type C∨C DAHA yields Koornwinder eigenfunctions like Macdonald ones","C∨C systems share Macdonald toolkit but lack twisting and universal series","Non-symmetric Koornwinder polynomials from type-C∨C Cherednik Hamiltonians","Type A and C∨C DAHA parallel except branching rule and automorphisms"],"cache_read_input_tokens":32896,"weakest_assumption_plain":"That the same power-sum construction that turns type-A Cherednik operators into the full tower of Ruijsenaars Hamiltonians continues to produce all higher van Diejen-Koornwinder Hamiltonians from the type-C∨C Cherednik operators, even though the richer automorphism group that generates twisting is missing.","fun_headline_variants_meta":{"raw":{"variants":["Koornwinder polynomials mirror Macdonald without Noumi-Shiraishi series","Type C∨C DAHA yields Koornwinder eigenfunctions like Macdonald ones","C∨C systems share Macdonald toolkit but lack twisting and universal series","Non-symmetric Koornwinder polynomials from type-C∨C Cherednik Hamiltonians","Type A and C∨C DAHA parallel except branching rule and automorphisms"]},"model":"grok-4.5","effort":"low","cost_usd":0.006334,"raw_usage":{"total_tokens":1697,"prompt_tokens":866,"num_sources_used":0,"completion_tokens":88,"cost_in_usd_ticks":63340000,"prompt_tokens_details":{"text_tokens":866,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":743,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":866,"tokens_out":88,"duration_ms":9417,"temperature":1.0,"reasoning_tokens":743,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-10T22:23:52.633257+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Explicitly compute the second and third commuting van Diejen-Koornwinder operators for n=2 or n=3 and check whether they equal the corresponding power sums of the Cherednik operators restricted to Weyl-symmetric functions; any mismatch would break the claimed parallel.","supporting_citations":[],"review_version":1}