{"id":"11bf1da5-8e2d-4c8b-b614-56cf842e5ce6","arxiv_id":"2607.16964","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Quasi-circular Hessenberg systems and systems satisfying the tridiagonal relations are the same family; the tridiagonal-relations family splits exactly into the circular and tridiagonal-Hessenberg cases.","lead":"This paper studies families of Hessenberg matrices — square grids of numbers with a nearly triangular shape — and shows that two families defined in different ways are actually the same family, while a third family splits exactly into the other two. Researchers working on the algebraic structure behind special polynomials and association schemes will use this to recognize and classify these matrix objects more directly.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Concern: (17) QCHS⇔TD rests on external implication (13) from [17]; unverified here, and required for the complement claim. Recommend confirming (13) independently.","rationale":"The reader's weakest_assumption (the [17] dependency) is exactly the point I would stress. I checked the in-house logical chain: TD⇒QCHS (Prop 11.7 + Lemma 12.1), THS⇒TD (Prop 11.7 + Prop 12.2), and (16) (from QCHS plus Props 12.2/12.8) all appear valid. The proof of Prop 12.2 uses [20, Lemma 12.2], but that is a published result and the context is appropriate. The remaining step QCHS⇒TD is the only one that calls on an unpublished same-author preprint. Because the paper explicitly discloses the dependency, there is no circularity or deception; the risk is purely that (13) might not hold in the full generality needed. The proposed computational check would settle the matter. No internal contradiction was found, so a conditional verdict is appropriate: accept the mathematical framework, but require verification of (13) (or a self-contained proof) before final acceptance.","tokens_in":19569,"tokens_out":11260,"duration_ms":93365,"concrete_test":"Independently verify (13): take the four families of CHS parameter arrays in [10, Theorem 5.6 / Examples 5.1–5.4]. For each family, compute the adjusted split sequence {ϑ_i} by Definition 11.1 and check whether there exists a single β∈F such that {θ_i}, {θ*_i}, and {ϑ_i} are all β-recurrent; if yes, Prop 11.7 gives TD. As a second check, for a d=3 CHS example from those families, assemble A,A* via Lemma 5.10(iv) and directly evaluate the commutators in (11)–(12); if either test fails, (13) is not a formality and the central equivalence is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The single most load-bearing point is the outsourcing of (13) CHS⇒TD to the companion preprint [17], on which the QCHS⇒TD direction of the central equivalence (17) and the complement claim 'CHS is the complement of THS in TD' both depend. Section 10 states 'In [17] we obtained the implication CHS⇒TD (13)', and the proof of (17) in Section 13 uses it: after (16) splits QCHS into CHS and THS, the CHS case is closed by (13) and the THS case by (15). Section 6 also relies on [17] for the CHS classification. If (13) is false, or holds only under additional conditions not stated, then the diagram (4) fails: a CHS need not be TD, and consequently QCHS would not equal TD. The paper proves TD⇒QCHS, THS⇒TD, and the disjoint union (16) in-house, but none of these supply CHS⇒TD. This is a disclosed, addressable gap rather than a demonstrated error. Secondary but related: the 'routine verification' formulas in Lemmas 12.5–12.11 are unshown and underpin (15) and (16); an error there would also break the diagram, but those can be checked mechanically.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies Hessenberg systems and the relations among six families: circular Hessenberg systems (CHS), quasi-circular Hessenberg systems (QCHS), tridiagonal Hessenberg systems (THS), Leonard systems (LS), systems satisfying the tridiagonal relations (TD), and recurrent systems (REC). The central claim is the logical diagram of Section 1: QCHS is equivalent to TD, and the chain LS ⇒ THS ⇒ TD ⇒ REC ⇒ HS holds; moreover, within TD, CHS is the complement of THS. The paper gives parameter-array characterizations: LS by Lemma 8.4, THS by Propositions 12.2 and 12.8, TD by Propositions 11.7 and 11.8, REC by Definition 10.3, and HS by Lemma 5.11. The proofs use the split sequence and the adjusted split sequence {ϑ_i}, plus detailed computations of triple-product matrix entries. The main in-house contributions are the proofs of TD⇒REC, THS⇒TD, the disjoint-union statement (16), and the QCHS⇒TD direction of (17), the last of these contingent on the implication CHS⇒TD obtained in the authors' companion preprint [17].","tokens_in":19654,"tokens_out":3119,"duration_ms":35200,"significance":"If the main diagram is correct, the paper gives a clean structural organization of several previously studied families: the TD family coincides exactly with the quasi-circular Hessenberg systems, and the two ``extremal'' subfamilies—circular and tridiagonal—account for all of them. The parameter-array descriptions are concrete and useful, especially Propositions 11.7, 12.2, and 12.8, which characterize TD and THS by simple recurrence conditions involving {ϑ_i}. The paper also handles the low-dimensional case d=2 separately, which is necessary because several recurrence definitions degenerate there. A notable strength is that much of the argument is self-contained: the reader can trace the logical chain from the definitions through the matrix computations to the main equivalences, with the important exception of (13). The result should be of interest to researchers working on Leonard pairs, tridiagonal algebras, and the classification of Hessenberg-type matrix families.","major_comments":[{"comment":"The central equivalence QCHS⇔TD and the statement that CHS is the complement of THS in TD both depend on the implication CHS⇒TD, labelled (13), which is not proved in this paper. The proof of (17) in Section 13 uses (13) explicitly: after (16) splits QCHS into CHS and THS, the CHS case is closed by (13) and the THS case by (15). Section 6 also uses the companion preprint [17] to remove the additional conditions in the classification of CHS from [10]. Since [17] is cited as an arXiv preprint and is not included in this manuscript, the main theorem is conditional on an external result. This is a disclosed and addressable gap, but it is load-bearing. I ask the authors either to include a proof of (13) in this paper or an appendix, or to cite a published version of [17] and spell out precisely which statement is being used. Without this, the claimed diagram (4)–(5) is not fully established w","section":"Sections 10 and 13, especially Eq. (13) and proof of (17)"},{"comment":"The proofs of the THS characterization and hence of (15) and (16) rest on explicit formulas for the (r,r−2)-entry of (E_{d−r}A^*E_{d−r+2})♭, obtained in Lemmas 12.5, 12.6, 12.7, 12.9, 12.10, and 12.11. These are asserted as ``routine verification'' with no derivation shown. The formulas are complicated and they are used to identify exactly when triple products vanish, so an error here would propagate to Propositions 12.2, 12.12, and 12.13. I do not claim the formulas are wrong—the structure is plausible and consistent with the later recurrence reformulation—but because they underpin a central claim, the authors should provide a derivation, at least for one representative case, or make the computations available in a verifiable supplementary form. This is a secondary but genuine gap in the write-up.","section":"Section 12, Lemmas 12.5–12.11"}],"minor_comments":[{"comment":"The abstract contains the typo ``eignvalues'' for ``eigenvalues''. In Section 1, ``compliment'' should be ``complement''; the same typo appears in Section 10.","section":"Abstract and Section 1"},{"comment":"``paramater array'' should be ``parameter array''.","section":"Definition 5.8"},{"comment":"The list of statements to prove includes (17), but (17) depends on (13) from [17]. The text should indicate this dependence more prominently, perhaps by listing (13) as an external input in the same display, so the reader is not misled into thinking all of (14)–(17) are proved here.","section":"Section 10"},{"comment":"The notation E♭_r and E∗♭_r in Lemma 12.4 is clear, but the indexing in Lemma 12.5 is slightly compressed; stating explicitly that the scalar multiplier is independent of i,j would improve readability.","section":"Section 12, notation"}],"recommendation":"major_revision","confidential_remarks":"The paper's mathematical core appears sound and largely self-contained, and the authors have been transparent about the dependence on [17]. The single most important issue is that (13) is not proved here; if [17] is already accepted and can be cited as a published theorem, this would drop to a minor concern. I would encourage the editor to request that the authors either prove (13) in this paper or provide a clear statement of the exact theorem in [17] that supplies it. The 'routine verification' matrix-entry computations are also worth making explicit, as they are used in the central THS characterization. No issues of priority or incorrectness outside this external dependence were apparent to me."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague, quick take on 2607.16964. It introduces QCHS and THS, proves QCHS⇔TD, shows QCHS is the disjoint union of CHS and THS, and gives parameter-array characterizations of THS. That diagram is the first complete map of these six families, and it looks right. The main in-house work is TD⇒REC, THS⇒TD, and the vanishing triple product lemmas; the d=2 boundary cases are handled carefully, which is a good sign.\n\nThe strengths: the paper is logically transparent, says exactly what is proved here vs. cited, and gives explicit array conditions. Proposition 11.7/11.8 for TD, 12.2/12.8 for THS, and 12.12/12.13 are useful. The QCHS⇔TD identification is a genuine insight, not just a restatement.\n\nThe soft spots are real but not fatal. First, the QCHS⇒TD direction of (17) uses (13) CHS⇒TD, attributed to the authors' own companion preprint [17]. That result is not proved here. If [17] has a gap, the central equivalence and the complement claim lose support. The paper does prove TD⇒QCHS, THS⇒TD, and (16) in-house, so the dependency is limited to one implication and the CHS classification. Still, the central theorem is not self-contained. Second, Lemmas 12.5–12.11 are stated as \"routine verification\" with no displayed derivation. They underpin the THS characterization. I don't see an error, but that's exactly the kind of place a sign error hides. The computations are mechanical and could be machine-checked; the authors should either include them or provide a supplementary notebook. Third, the text has minor typos (\"paramater\", \"compliment\"), which don't matter.\n\nFor a referee: the paper deserves serious review. The math is coherent, the new families are natural, and the stated claims are checkable. The dependency on [17] should be solved by requiring the authors to include a proof of (13) in this paper or to make [17] available and accepted. The routine verifications should be tightened. I would not desk reject it; I'd send it out.","headline":"New families and a clean logical map for Hessenberg systems, but the key CHS⇒TD implication comes from an unverified companion preprint and the paper leans on unshown computations.","tokens_in":20427,"tokens_out":2282,"would_cite":true,"duration_ms":25353,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["17B37","15A21"],"pacs":[],"model":"deepseek-v4-flash","headline":"A Hessenberg system satisfies the tridiagonal relations if and only if its associated matrices are quasi-circular Hessenberg, a family that splits into circular and tridiagonal types.","keywords":["Hessenberg pair","Hessenberg system","Leonard pair","tridiagonal relations","quasi-circular Hessenberg","tridiagonal Hessenberg","circular Hessenberg","parameter array"],"falsifier":"Construct a Hessenberg system with d≥3 that satisfies the tridiagonal relations (11)–(12) but has some entry E_i A* E_j nonzero for 1 < j−i < d (i.e., not quasi-circular), or construct a quasi-circular Hessenberg system that does not satisfy (11)–(12).","tokens_in":19217,"feed_emoji":"🧮","tokens_out":4326,"duration_ms":41541,"temperature":0.7,"pith_summary":"This paper organizes the known families of Hessenberg systems into a single logical diagram. Its central claim is that the systems satisfying the tridiagonal relations (TD) are exactly the quasi-circular Hessenberg systems (QCHS), and that these split without overlap into the circular ones (CHS) and the tridiagonal ones (THS). It also proves the chain Leonard ⇒ tridiagonal ⇒ TD ⇒ recurrent ⇒ Hessenberg. If correct, this reduces the study of TD systems to a matrix-shape condition and gives explicit parameter-array recurrences for each family.","feed_headline":"Tridiagonal relations mark exactly the quasi-circular Hessenberg systems","feed_subtitle":"The TD family splits cleanly into circular and tridiagonal types, with one ratio deciding which.","key_machinery":"The parameter array of a Hessenberg system — its eigenvalue sequence, dual eigenvalue sequence, and split sequence — together with the adjusted split sequence ϑ_i = ϕ_i − (θ*_i − θ*_0)(θ_{d−i+1} − θ_0). The classification is carried by recurrence conditions: a system is TD iff all three sequences are β-recurrent for a common β, and it is THS iff additionally the first and last adjusted split values coincide (ϑ_1 = ϑ_d). The proofs use vanishing of triple products E_i A* E_j and the explicit (r, r−2) matrix entries computed in Lemmas 12.5–12.11.","core_discovery":"The paper proves QCHS⇔TD: a Hessenberg system satisfies the tridiagonal relations (11)–(12) if and only if, in the eigenbasis of each map, the other map is represented by a matrix whose only possibly nonzero entries on the far side are the subdiagonal, diagonal, superdiagonal, and the far-corner entry (the (d,0) position). Within this family, the far-corner entry is zero exactly for tridiagonal Hessenberg systems and nonzero exactly for circular Hessenberg systems; hence CHS is the complement of THS in TD. For d≥3, each of these conditions is characterized by the β-recurrence of the eigenvalue sequence, the dual eigenvalue sequence, and the adjusted split sequence {ϑ_i} defined in Definition","pith_inferences":["Since the implication CHS⇒TD is invoked from a companion preprint rather than proved here, the central equivalence stands or falls with that companion argument; checking it independently would settle the paper's main claim.","One testable extension is to read the (r, r−2) entry formulas as a direct computational criterion: given a Hessenberg system, the vanishing of these entries in one basis forces the full triple-product pattern.","The splitting of TD into CHS and THS by a single scalar condition (ϑ_1 = ϑ_d) suggests that q-Racah and other orthogonal-polynomial families from the Askey scheme may be classified by this one parameter."],"forward_implications":["The tridiagonal relations hold for a Hessenberg system exactly when its associated matrices are quasi-circular; no other Hessenberg shape can satisfy them.","Every TD system has all three of its defining sequences (eigenvalues, dual eigenvalues, adjusted split sequence) governed by a single three-term recurrence parameter β.","A TD system is a Leonard system precisely when it is irreducible tridiagonal, and these sit inside the THS family; the circular systems are the remaining TD systems.","The parameter-array characterizations give explicit existence and uniqueness statements for TD and THS systems, extending the known Leonard-system classification."],"fun_headline_variants":["Quasi-circular Hessenberg systems are exactly those with tridiagonal relations","One far-corner entry decides: circular or tridiagonal Hessenberg","Tridiagonal relations split Hessenberg systems: circular vs tridiagonal","Far-corner nonzero means circular, zero means tridiagonal in TD systems"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The paper relies on a companion result that circular Hessenberg systems satisfy the tridiagonal relations; if that implication fails, the equivalence QCHS⇔TD and the complement claim are unsupported.","fun_headline_variants_meta":{"raw":{"variants":["Quasi-circular Hessenberg systems are exactly those with tridiagonal relations","One far-corner entry decides: circular or tridiagonal Hessenberg","Tridiagonal relations split Hessenberg systems: circular vs tridiagonal","Far-corner nonzero means circular, zero means tridiagonal in TD systems"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000203,"raw_usage":{"total_tokens":1286,"prompt_tokens":868,"completion_tokens":418,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":612,"completion_tokens_details":{"reasoning_tokens":333}},"tokens_in":612,"tokens_out":418,"duration_ms":4945,"temperature":1.0,"reasoning_tokens":333,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T19:27:20.483946+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct a Hessenberg system with d≥3 that satisfies the tridiagonal relations (11)–(12) but has some entry E_i A* E_j nonzero for 1 < j−i < d (i.e., not quasi-circular), or construct a quasi-circular Hessenberg system that does not satisfy (11)–(12).","supporting_citations":[],"review_version":1}