{"id":"cf258dd3-ccbf-43b0-93d0-a236955fc811","arxiv_id":"2411.18340","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The partition P_{k,l}(Q) in the IKVZ2 table equals the Burge-code partition P^Q_{k,l} for every two-part stable partition Q=(u,u-r).","lead":"This paper proves that two different tables of matrix-shape partitions attached to a stable two-part partition Q coincide entry by entry. It closes a gap left in the same authors' earlier work and confirms a 2015 conjecture about the equations defining these loci.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof's key corank formula is cited from the unpublished [BIK] preprint; without an independent proof or computational check, the equality in Corollary 4.2 is conditional.","rationale":"I read the paper in good faith as a focused proof of [BIK, Proposition 3.9], showing that the geometric table T(Q) and the Burge-code table coincide. The proof strategy is reasonable: the equations E^Q_{k,l} determine the order matrix T, the corank sequence determines the Jordan type, and the resulting partitions match the three type A/B/C shapes in Theorem 2.2. I checked the reconstruction formulas on small parameter examples and found no internal contradiction. The single most load-bearing step is the corank formula imported from [BIK, Lemmas 3.5 and 3.6]: all three cases in Theorem 4.1 derive their corank sequences from it, and Corollary 4.2 inherits that dependency. Since [BIK] is an unpublished same-author preprint and the note does not reproduce the proofs of these lemmas, the equality is conditional on their correctness. This is precisely the weakness the reader identified; I agree with the CONDITIONAL verdict. No change to the reader's verdict is needed, but a direct proof or a computational verification of Lemma 3.6 would settle the concern.","tokens_in":10714,"tokens_out":27395,"duration_ms":230428,"concrete_test":"Re-derive [BIK, Lemma 3.6] directly from the representation (4)–(5): for a general matrix B in N_{J_Q} with order matrix T = [[k,0],[r,l']], compute corank A^s for s = 1,2,3 from the action on k[t]/(t^u) ⊕ k[t]/(t^{u−r}) and verify the claimed tropical formula in both regimes k + l ≤ r and k + l ≥ r + 1. If the formula does not follow from the definitions, or if a small case such as Q=(7,3), (k,l)=(2,2) fails, then Theorem 4.1 is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 4.1 and hence Corollary 4.2 rest on [BIK, Lemma 3.5] and [BIK, Lemma 3.6], which supply the entire rank computation: corank A^s = (k+l)s ⊕ ((T^{⊗s})_11 + u − r) ⊕ (u + u − r), together with the explicit tropical tensor-power formula. Those lemmas are not proved in this note; they are cited from a same-author preprint. The internal case analysis in Section 4 is a faithful consequence of that formula, and I did not find an inconsistency in the case splits. But if Lemma 3.6 carries hidden hypotheses—for instance, if it requires genericity of the auxiliary matrix entries beyond the order matrix, or if the tropical formula fails at the boundary cases k + l = r or k + l = r + 1—then the corank sequence, the reconstructed partition, and therefore the asserted identity with the Burge-code table would all be wrong. The note says it is self-contained but only reproduces notation and definitions from [BIK]; the decisive lemmas remain external. Thus the central claim is exactly as secure as those two unproved lemmas.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The note proves that for a stable two-part partition Q=(u,u-r), the partition P_{k,l}(Q) appearing in the IKVZ2 table T(Q) of partitions with maximal nilpotent commutator Q coincides with the partition P^Q_{k,l} defined in [BIK] by a Burge code. The proof (Theorem 4.1) assumes the equations E^Q_{k,l} hold and the remaining entries are generic, computes the corank sequence of powers of A from [BIK, Lemmas 3.5 and 3.6], reconstructs the Jordan type from that sequence, and checks the result against the Table Theorem 2.2. Corollary 4.2 then asserts the equality of the two tables entry by entry, and Corollary 4.3 uses this to identify the locus of each P_{k,l}(Q) with the variety defined by E^Q_{k,l}.","tokens_in":10938,"tokens_out":7614,"duration_ms":72534,"significance":"If correct, the result resolves the identification deferred in [BIK, Proposition 3.9] and confirms for two-part stable Q the equation-locus conjecture from [IKVZ1, Conjecture 4.19]. The proof is a genuine derivation rather than a circular comparison: the Jordan type is obtained from the corank sequence and then matched with the pre-existing table. The detailed four-line case analysis in Section 4 is a useful, explicit verification. The main limitation is that the entire rank computation is imported from two lemmas in the same authors' unpublished preprint [BIK], so the central equality is only as secure as those lemmas.","major_comments":[{"comment":"The proof does not prove [BIK, Lemma 3.6] or [BIK, Lemma 3.5]; these lemmas supply the full corank formula, including the tensor-power expression (T^{⊗s})_{11}. Since every subsequent step in the case analysis and the final identification with the table depend on that formula, the equality in Corollary 4.2 is not established within this note. Please state the two lemmas completely and either include proofs (an appendix would be acceptable) or provide a precise reference to a version that can be checked; in addition, verify the boundary cases k+l=r, k+l=r+1, and k=l'=r/2, where divisions by k-l' or l' appear in the intersection formulas.","section":"Section 4, Theorem 4.1"},{"comment":"The definitions of the two cases are garbled as printed: Case A reads 'L3(s) ≥ L!(s) ⊕ L2(s) ⊕ L4(s)' and Case B reads 'L2(s) ≥ L!(s) ⊕ L2(s) ⊕ L4(s)', where 'L!' should presumably be 'L1'. With the literal text, the Case B inequality is automatically true because L2(s) appears on both sides, so the case split does not define a partition of the possibilities. This needs correction because the proof relies on the claim that Cases A, B, and C cover all possible orderings of the four lines.","section":"Section 4, Cases A and B"},{"comment":"The hypothesis 'and that the rest of the entries of A are generic' is never formalized. The proof uses the corank sequence and the recovery of the Jordan type from it, but the statement should specify the precise open condition on A (or cite the exact [BIK] result showing that the corank formula holds for every matrix in the locus defined by E^Q_{k,l}, so that no genericity assumption is needed). Without this, the theorem does not clearly identify the locus of P_{k,l}(Q) rather than merely some dense subset.","section":"Section 4, Theorem 4.1"}],"minor_comments":[{"comment":"The symbol ⊕ is used throughout for what appears to be the minimum (for example l′ = l ⊕ (r−k), and later corank A^s = (k+l)s ⊕ ((T^{⊗s})_{11}+u−r) ⊕ (u+u−r)), but it is never defined in this note. Please define it explicitly.","section":"Section 3, Definition 3.1"},{"comment":"The statement of the Table Theorem uses the symbols k_t, c_t, d_t, and q_t without defining them. Since the introduction says the note is meant to be self-contained, these definitions should be included, or the statement should refer precisely to the version in [IKVZ2].","section":"Theorem 2.2"},{"comment":"The notions Utop, Ubottom, and Umiddle (the 'U-chains') are used in the verification that Q(P)=(u,u−r) but are not defined in the note. Please give a definition or a precise reference to [IKVZ2].","section":"Section 4, proof of Theorem 4.1"},{"comment":"The proof says 'This follows from Corollary 4', but the intended reference is Corollary 4.2; please correct the cross-reference.","section":"Corollary 4.3"}],"recommendation":"major_revision","confidential_remarks":"The main risk is the unpublished preprint [BIK], which shares authors with this note and contains the lemmas on which the central proof rests. A referee cannot verify those lemmas from published literature, and the note's self-contained claim is therefore misleading. I would encourage the editor to ask the authors to include the lemmas, or at least to make the dependence explicit and precise in the revised version."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short story: this is a proof of [BIK, Proposition 3.9], which was announced without proof. The authors show that the partition P_{k,l}(Q) from the IKVZ2 table and the Burge-code partition P^Q_{k,l} from their earlier preprint are the same, by proving that the equations E^Q_{k,l} define the same generic Jordan type. Theorem 4.1 is an explicit, readable case analysis on four linear bounds, and it looks correct as far as I followed it. The Corollary 4.2 identification then falls out, provided you accept the [BIK] setup.\n\nWhat is genuinely good: the proof technique is straightforward and the paper connects two existing tables in a way that was previously left undone. The case splits are thorough, and the authors are honest that the proof was deferred from [BIK]. The note is short, to the point, and likely useful to the commuting nilpotent matrices community.\n\nSoft spots. The main one is the one the reader flagged: the corank formula in Theorem 4.1, including the tensor-power computation, is imported verbatim from [BIK, Lemmas 3.5 and 3.6], a same-author preprint. That is not a fatal flaw—the paper is explicitly a companion to [BIK]—but the claim that the note is self-contained does not hold. The decisive lemmas are not proved here, and the proof is exactly as secure as they are. The stress-test worry about hidden hypotheses at the boundary k+l = r or r+1 is speculative; I saw no sign of failure. Still, a referee should ask for the lemmas to be stated and either proved or clearly located in the final [BIK]. Also, there is a typo 'L!' for L1 in Case A and a few other small proofreading issues; nothing that obscures the argument.\n\nFor whom? This is for specialists in Jordan types of commuting nilpotent matrices. It is not a broad result, but it closes a known gap. I would send it to a referee with normal standards; the main requested revision should be that the [BIK] lemmas be made available or at least restated so the proof is checkable without a second preprint. It is a legitimate, focused contribution.\n\nRecommendation: accept for peer review, with a request for better self-containedness.","headline":"A clean proof of a deferred table identification whose load-bearing lemmas are cited from an unpublished preprint; conditional but worth refereeing.","tokens_in":11486,"tokens_out":2527,"would_cite":true,"duration_ms":25441,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["15A27","05A17","13E10","14A05","15A20"],"pacs":[],"model":"deepseek-v4-flash","headline":"For Q=(u,u-r), the geometric partition table and the Burge-code table coincide at every index (k,l).","keywords":["Jordan type","commuting nilpotent matrices","nilpotent commutator","Burge correspondence","stable partition","order matrix","partition table","locus equations"],"falsifier":"Take $Q=(8,4)$, choose any $(k,l)$ with $1\\le k\\le 3$ and $1\\le l\\le 4$, and sample a matrix $A$ whose entries satisfy $E^Q_{k,l}$ with all other entries generic over a finite field; if the Jordan type of $A$ ever differs from $P_{k,l}(Q)$ as listed in Theorem 2.2 - for instance, if the number of parts is not $k+l$ - then Theorem 4.1 is false.","tokens_in":10501,"feed_emoji":"🧩","tokens_out":17483,"duration_ms":134928,"temperature":0.7,"pith_summary":"This paper proves that two independent ways of listing the same set of partitions agree entry by entry. For a stable two-part partition $Q=(u,u-r)$, the geometric table $\\mathcal T(Q)$ of [IKVZ2] records the Jordan types whose maximal commuting nilpotent orbit is $Q$, while the Burge-correspondence table of [BIK] records the same types by codes. The paper shows that the entry $P_{k,l}(Q)$ in the geometric table is exactly the entry $P^Q_{k,l}$ with Burge code $\\alpha^{u-r-l}\\beta^l\\alpha^{r-k}\\beta^k\\alpha$, and that the equations $E^Q_{k,l}$ define the locus of each such type. This settles a 2015 conjecture and makes the stratification of the nilpotent commutator for two-part $Q$ explicit.","feed_headline":"Two tables of partitions coincide at every index","feed_subtitle":"Two independent constructions agree, and every stratum's defining equations are now explicit.","key_machinery":"The central machinery is the order matrix $T$, whose entries are the orders of the four blocks of a matrix $A$ in the nilpotent commutator of $J_Q$: the upper-left block has order $k$, the lower-left block order $r$, and the lower-right block order $l$, with $l' = l \\oplus (r-k)$ controlling the effective index. From $T$, the cited lemmas of [BIK] give an explicit formula for the corank of $A^s$; the proof then compares four lines $L_1(s)=(k+l)s$, $L_2(s)=ks+u-r$, $L_3(s)=l's+u-2l'$, and $L_4(s)=u+u-r$, and shows that whichever line is lowest at each $s$ reconstructs the Jordan type $P_{k,l}(Q)$ forced by the equations $E^Q_{k,l}$. The Burge code $\\alpha^{u-r-l}\\beta^l\\alpha^{r-k}\\beta^k\\alpha$ is the combinatorial counterpart indexing the same partition.","core_discovery":"The paper establishes that the partition $P_{k,l}(Q)$ appearing in the table $\\mathcal T(Q)$ of [IKVZ2] - the table of Jordan types whose maximal commuting nilpotent orbit is the stable two-part partition $Q=(u,u-r)$ - is the same partition as $P^Q_{k,l}$ defined in [BIK] by the Burge code $\\alpha^{u-r-l}\\beta^l\\alpha^{r-k}\\beta^k\\alpha$. The proof shows that the equations $E^Q_{k,l}$, conjectured in [IKVZ1] to cut out the locus of $P_{k,l}(Q)$, force a matrix $A\\in \\mathcal N(Q)$ with generic remaining entries to have Jordan type $P_{k,l}(Q)$. Consequently the geometric A/B/C-type table and the Burge-code table coincide at every index $(k,l)$ (Corollary 4.2), and the closure of each stratum is defined by $E^Q_{k,l}$ (Corollary 4.3).","pith_inferences":["A natural extension would be to test whether the same order-matrix mechanism identifies a Burge-code table with a geometric table for stable partitions $Q$ with more than two parts; the two-part case suggests the corank formula, not the A/B/C type geometry, is the essential ingredient.","The equality of the two tables makes the stratification for two-part $Q$ effectively combinatorial: the dominance order on the partitions $P_{k,l}(Q)$ should be readable from the Burge codes alone, so closure containments between strata should follow from a word-ordering rule.","Because Theorem 4.1 requires the entries outside $E^Q_{k,l}$ to be generic, the equations alone may cut out a larger variety containing the stratum; solving $E^Q_{k,l}$ at non-generic points would show whether that genericity hypothesis is necessary."],"forward_implications":["For every index $(k,l)$ with $1\\le k\\le r-1$ and $1\\le l\\le u-r$, the locus of matrices in $\\mathcal N(Q)$ having Jordan type $P_{k,l}(Q)$ is defined by the equations $E^Q_{k,l}$, confirming the 2015 conjecture for two-part stable $Q$.","The geometric table $\\mathcal T(Q)$ and the Burge-code table $P^Q_{k,l}$ are the same list of $(r-1)(u-r)$ partitions, each with $k+l$ parts, so the A/B/C types and the Burge-code types are the same objects.","For a generic matrix $A$ satisfying $E^Q_{k,l}$, the corank sequence of $A^s$ is governed by the four lines $L_1,\\dots,L_4$, so the Jordan type can be read off from the intersection points of those lines.","The closure of the locus of matrices with Jordan type $P_{k,l}(Q)$ is the affine variety $V(E^Q_{k,l})$, giving an explicit defining-equation stratification of $\\mathcal N(Q)$ for $Q=(u,u-r)$."],"supporting_citations":[{"why":"Supplies the Burge-code table $P^Q_{k,l}$ (Equation 3) and the two lemmas (3.5, 3.6) computing the corank of $A^s$ from the order matrix, on which Theorem 4.1's case analysis rests.","marker":"[BIK]"},{"why":"Defines the table $\\mathcal T(Q)$, the A/B/C type partitions $P_{k,l}(Q)$, and the Table Theorem whose entries are identified here.","marker":"[IKVZ2]"},{"why":"Proves the box theorem that the Burge-code table exhausts $D^{-1}(Q)$, so the identification covers all partitions in the box.","marker":"[IKM]"},{"why":"Introduces the Burge correspondence between partitions and $\\alpha/\\beta$ words underlying the code $\\alpha^{u-r-l}\\beta^l\\alpha^{r-k}\\beta^k\\alpha$.","marker":"[Bur]"},{"why":"States the conjecture that the locus of $P_{k,l}(Q)$ is defined by $E^Q_{k,l}$, the claim Theorem 4.1 verifies.","marker":"[IKVZ1]"}],"fun_headline_variants":["Two partition constructions proven identical","Burge code reproduces geometric partition table","Maximal nilpotent commutator fixes same partition","Jordan table and Burge table agree at every index","Explicit equations for strata now proven"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof depends on two lemmas from the authors' earlier preprint that calculate the rank drops of every power of $A$ from a small 2-by-2 matrix of leading powers; if those lemmas are wrong or carry hidden conditions, the identification fails.","fun_headline_variants_meta":{"raw":{"variants":["Two partition constructions proven identical","Burge code reproduces geometric partition table","Maximal nilpotent commutator fixes same partition","Jordan table and Burge table agree at every index","Explicit equations for strata now proven"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00021,"raw_usage":{"total_tokens":1336,"prompt_tokens":796,"completion_tokens":540,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":412,"completion_tokens_details":{"reasoning_tokens":474}},"tokens_in":412,"tokens_out":540,"duration_ms":5734,"temperature":1.0,"reasoning_tokens":474,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T11:17:53.088955+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $Q=(8,4)$, choose any $(k,l)$ with $1\\le k\\le 3$ and $1\\le l\\le 4$, and sample a matrix $A$ whose entries satisfy $E^Q_{k,l}$ with all other entries generic over a finite field; if the Jordan type of $A$ ever differs from $P_{k,l}(Q)$ as listed in Theorem 2.2 - for instance, if the number of parts is not $k+l$ - then Theorem 4.1 is false.","supporting_citations":[],"review_version":1}