{"id":"7c80f51f-9cd9-4cd1-a68c-3da9ed157973","arxiv_id":"2608.01146","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Every normalized maximal Plücker coordinate of the companion remainder orbit equals, up to sign and a power of the constant coefficient, a Schur polynomial, so the universal MDS polynomial is A0 times all Schur polynomials in a rectangular partition set.","lead":"This paper shows that the MDS property of a family of coefficient-vector codes is governed by a product of Schur polynomials in the roots of a single polynomial, giving exact conditions and boundary geometry. It yields finite-field estimates, sparse-multiple thresholds, and shows that non-Generalized Reed-Solomon members are generic in most first-failure strata.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Schur–Plücker core is self-contained and sound; the paper's non-GRS density and counting results are conditional on four unverified companion-preprint theorems.","rationale":"The reader's verdict is well-calibrated: Theorem 3.2 and Theorem 3.3 are proved by a clean universal/specialization argument together with the index-set bijection, and the divisor, degree, flatness, and sparse-threshold results follow from standard algebra. I checked the potentially fragile internal points—the ordinary-degree claim in Lemma 4.1 (it is κ1, not |κ|, in the coefficient variables, as Example 7.3 confirms), the sign and multiplicity bookkeeping in Proposition 3.5, the complete-intersection equations in Proposition 2.5, and the pure-power small cases in Theorem 5.5—and found no internal inconsistency. The only genuine load-bearing assumption is the paper's dependence on the companion preprint [12] for the non-GRS classification and counting. The paper itself flags this in Remarks 4.12 and 6.16, and the reader identified it as the weakest assumption; I agree. Since the central algebraic theorem does not depend on [12], the core verdict should remain intact; however, the headline 'dense open non-GRS locus' and the quantitative non-GRS bounds are conditional on four specific results from [12]. A single brute-force check of (4.20) for small parameters would settle the most quantitative of those dependencies and is worth running before the separation claims are cited as established.","tokens_in":35167,"tokens_out":26138,"duration_ms":214027,"concrete_test":"Run a brute-force check for r=3, n=6 over F_q for q=5,7,8,11: enumerate all monic cubics, construct the six remainder columns Q_0,...,Q_5 by the recurrence (2.4), test MDS by computing all 3×3 minors, and test GRS type by fitting the unique conic through the first five projective points [Q_0],...,[Q_4] and checking whether [Q_5] lies on it. Compare the number of GRS-type MDS points with the right-hand side of (4.20) for each q. If any q mismatches, the exact-GRS count [12, Theorem 6.14] is wrong and Corollaries 4.10 and 6.18 must be revised; if all match, the most quantitative external input is supported.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Theorem 3.3 and its algebraic consequences (divisor flatness, degree formulas, the MDS test) are derived in-paper from the companion recurrence and Schur identities; that portion of the central claim is solid. The load-bearing weak point is the advertised separation claim in the abstract: for r≥3 and N≥r+3, every nonempty first-failure layer has a dense open non-GRS locus. That statement, together with Corollaries 4.10, 5.11, 5.13, 5.14, 6.18, and 6.19, imports four external results from the companion arXiv preprint [12]: rational-normal-curve rigidity [12, Lemma 4.3], the pure-power MDS criterion [12, Lemma 5.2], the classification of GRS-type coefficient points [12, Theorems 5.3 and 6.6], and the exact GRS count [12, Theorem 6.14]. The paper is transparent about this dependence (Remarks 4.12 and 6.16, and the proofs of Propositions 4.11, 5.13, and 5.14), but transparency does not reduce the logical reliance. In particular, Corollary 6.19 uses [12, Theorem 6.6] to assert that every GRS point is either pure-power or semisimple with roots {γ,γt,...,γt^{r−1}}; if that classification fails over a non-perfect or small-characteristic field, the non-GRS density conclusion collapses. Similarly, Corollary 4.10 subtracts exactly [12, Theorem 6.14]'s count (4.20); a counterexample to that count would invalidate the lower bound. Such failures would not affect Theorem 3.3, but they would remove a major advertised contribution of the paper.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies coefficient-vector codes C_g(n)={u g : deg u < n-r} for a monic degree-r polynomial g, and the coefficient-space MDS locus M_{n,r}. It embeds coefficient space into the big cell of Gr(r,n) via the remainder-orbit columns and proves the master identity that every normalized maximal Plücker coordinate pulls back to ((-1)^r a_0)^{i0} S_{\\kappa(U)}(g), where S_\\kappa is a Schur polynomial in the root multiset of g. Hence the universal MDS polynomial is D_{n,r}=A_0 \\prod_{\\kappa\\subseteq (n-r)^{r-1}} S_\\kappa, and C_g(n) is MDS exactly when a_0\\neq 0 and all S_\\kappa(g)\\neq 0. From this description the paper derives flatness of the universal boundary over Z, exact degree formulas, finite-field estimates, dilation and reciprocity symmetries, a length filtration with first-failure layers, multiplicity-stratum bad-characteristic criteria, density results on the irreducible Frobenius stratum, and the sparse-threshold identity \\sigma_r(g)=r+\\tau(g)-1. It then invokes the companion classification [12] to claim that, for r\\geq 3 and N\\geq r+3, every nonempty first-failure layer has a dense open non-GRS locus.","tokens_in":35397,"tokens_out":11503,"duration_ms":101262,"significance":"The core Theorem 3.3 is a clean, characteristic-free structural result: it identifies all MDS minors of this principal-ideal family with one complete rectangle of Schur polynomials. This is a substantial advance over treating the MDS test as an unrelated collection of minor equations, and it yields explicit and falsifiable consequences: the universal MDS polynomial, flat boundary divisors, exact degree formulas, finite-field lower bounds, and the exact sparse-threshold relation. The paper is also transparent about where it uses the companion preprint [12]; those external results are confined to the GRS-type applications and the quantitative non-GRS statements. If the companion theorems are correct, the paper makes a strong contribution to both coding theory and the algebraic geometry of the MDS locus.","major_comments":[{"comment":"The advertised claim that, for r≥3 and N≥r+3, every nonempty first-failure layer has a dense open non-GRS locus is not self-contained in this manuscript. The proof of Corollary 6.19 invokes [12, Theorem 6.6] to classify GRS points and, through Proposition 4.11, [12, Lemma 4.3]; Corollary 4.10 uses [12, Theorem 6.14]; and Theorem 5.5 uses [12, Lemma 5.2]. Since [12] is a companion preprint whose statements are not reproduced, the final sentence of the abstract and the stated non-GRS density claims are conditional on external results. Please state this conditionality explicitly in the abstract and in each GRS-type corollary, or include the relevant companion statements with proofs in an appendix. This does not affect Theorem 3.3, but it is load-bearing for the paper's last advertised contribution.","section":"Abstract, Corollary 6.19, Eq. (6.39)"},{"comment":"The quantitative non-GRS results subtract or invoke exact counts and classifications imported from [12]. In particular, Corollary 4.10 (Eqs. (4.20)–(4.21)) subtracts #G_{n,r}(F_q)=[12, Theorem 6.14] exactly, and Propositions 5.13 and 5.14 rely on the factorwise classification [12, Theorem 5.3] without proof. If any of those companion theorems fails over fields of small characteristic or over non-finite fields, the corresponding lower bounds and factor criteria in this paper would need revision. The dependence is acknowledged in Remarks 4.12 and 6.16, but acknowledgment does not make the result self-contained; the statements should be labeled as conditional on [12] or the missing companion proofs should be supplied.","section":"Corollary 4.10, Propositions 5.13 and 5.14"}],"minor_comments":[{"comment":"The determinant identity in Proposition 4.11 is stated after 'direct substitution'; I checked it in a generic case and it is correct. A one-line intermediate expansion would improve verifiability, but this is not a correctness issue.","section":"§4, Eq. (4.27)"},{"comment":"The sentence 'the new outer-layer factors S_{(4)},S_{(4,1)},...,S_{(4,4)} have values 8,9,8,4,0' lists the partitions in their natural order, but the text does not say this; adding the order would avoid ambiguity.","section":"Example 7.5, length-seven factors"},{"comment":"The definition of \\sigma_r(g) uses wt(f)\\leq r, so the term 'sparse multiple' is slightly misleading for multiples with exactly r+1 nonzero coefficients; the later text uses the same convention consistently, so this is only a terminological note.","section":"§2.1, Eq. (2.9)"}],"recommendation":"major_revision","confidential_remarks":"The core Schur–Plücker part is sound and would make a strong paper on its own. The decision largely hinges on editorial policy for companion preprints: if [12] is accepted or will be published with the present paper, the GRS-type corollaries become acceptable as citations, and a minor revision with explicit conditionality would suffice. If the journal requires self-contained proofs of advertised claims, the authors should reproduce the needed statements of [12, Lemma 4.3, Theorems 5.3, 6.6, 6.14] or clearly mark those results as external."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my read. The central theorem—every normalized Plücker coordinate of the companion remainder orbit is, up to a power of a_0, a Schur polynomial in the roots—is proved cleanly, characteristic-free, with a specialization argument that does not depend on [12]. The product D_{n,r} = a_0 ∏ S_κ over the rectangle follows, and from it the flat boundary divisor, exact degree, MDS test, sparse-multiple threshold via σ_r = r + τ − 1, and the first-failure filtration are legitimate consequences. I checked the key steps (Theorem 3.2, Lemma 4.1, Theorem 6.6) and they hold. The paper is honest about what is new: not the generalized Vandermonde identity itself but the complete-rectangle identification and its use for all Plücker coordinates.\n\nThe soft spot is exactly where the stress-test put it. The advertised abstract claim—for r ≥ 3, N ≥ r+3, every nonempty first-failure layer has a dense open non-GRS locus—is imported from four theorems in the companion preprint [12]: the rigidity lemma, the pure-power criterion, the GRS classification, and the exact GRS count. The present paper is transparent about this, and credits it; but a reader who wants the full theorem cannot verify it from this manuscript alone. If any of [12]'s results fail over non-perfect or small-characteristic fields, the lower-bound corollaries (4.10, 4.21, 6.18, 6.19) would need revision. The core Schur system would survive. That separation between core and advertised applications is important and the paper states it, but the abstract does not.\n\nOne more minor point: Proposition 4.11's obstruction uses the rigidity lemma [12, Lemma 4.3] as input; the remainder-coordinate computation is new but the logical dependency is there. The paper's claim that the low-degree obstruction is self-contained once the rigidity lemma is taken as input is honest, but it makes Theorem 4.13 split non-GRS bound conditional too.\n\nOverall, this is a serious paper. It deserves peer review. The referee should have [12] (or a version of it) in hand. I'd recommend sending to a journal with the expectation that the companion preprint is cited and ideally available as a posted companion. For my own work, I'd cite the Schur–Plücker core. Reading group: could be worth a session on the factorization, though the conditional parts make it harder.","headline":"The Schur–Plücker factorization is real and proven in-paper; the non-GRS density claims are real only modulo the companion preprint [12], which is the right thing to flag in review.","tokens_in":36046,"tokens_out":3103,"would_cite":true,"duration_ms":27150,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["94B05","05E05","11T71","14M15"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that every maximal minor of the coefficient-vector remainder matrix for a monic degree-$r$ polynomial $g$ factors, up to a power of $a_0$, as a Schur polynomial evaluated at the roots of $g$; consequently the MDS test…","keywords":["MDS codes","principal-ideal codes","Schur polynomials","Plücker coordinates","Grassmannians","confluent Vandermonde determinants","sparse multiples","length filtration"],"falsifier":"For a concrete counterexample, take $r=3$ and $n=6$ over several small finite fields, compute all $20$ maximal minors of $H_g^{\\mathrm{rem}}=(Q_0\\ \\cdots\\ Q_5)$, and compare their vanishing pattern with the ten-factor Schur product displayed in Example 7.3; any specialization where the product and the minors disagree would refute the master theorem. The identity is characteristic-free, so the same check can be run over any prime field and any extension.","tokens_in":2301,"feed_emoji":"🧮","tokens_out":2783,"duration_ms":107028,"temperature":0.7,"pith_summary":"The paper studies coefficient-vector codes $C_g(n)$ formed by all multiples of a monic degree-$r$ polynomial $g$ with degree below $n$, and asks exactly when such a code is MDS, meaning any $r$ positions determine the whole codeword. Using the remainder orbit of $g$ inside the Grassmannian $\\operatorname{Gr}(r,n)$, it proves that every maximal minor (normalized Plücker coordinate) is, up to a power of the constant coefficient $a_0$, a Schur polynomial $S_\\kappa$ evaluated at the roots of $g$. Hence the universal MDS polynomial is the single product $D_{n,r}=a_0\\prod_{\\kappa\\subseteq (n-r)^{r-1}}S_\\kappa$: the code is MDS if and only if $a_0$ and every $S_\\kappa(g)$ are nonzero. This turns the MDS locus into the complement of a flat boundary divisor with explicit degree, yields finite-field counts, and identifies the first length where MDS fails with the first sparse multiple of $g$. It also shows that, on nonempty first-failure layers, MDS members outside the generalized Reed–Solomon family are dense.","feed_headline":"One Schur product decides which coefficient codes are MDS","feed_subtitle":"The code is MDS exactly when an explicit Schur-type product is nonzero, with flat boundary and density estimates.","key_machinery":"The machinery is the remainder orbit embedded in the standard big cell of $\\operatorname{Gr}(r,n)$: the map $\\Phi_{n,r}(g)=\\operatorname{rowspan}H_g^{\\mathrm{rem}}$ is a closed immersion whose image is the graph of the companion recurrence, cut out by a regular sequence of $r(n-r-1)$ equations. The master identity reduces each Plücker coordinate $p_I(\\Phi_{n,r}(g))$ to $\\det(T_g)^{i_0}\\delta_{U_I}(g)$, where $T_g$ is the companion multiplication matrix and $\\det(T_g)=(-1)^r a_0$. The bijection $U\\mapsto\\kappa(U)$ from reduced index sets to partitions in $(n-r)^{r-1}$, together with the bialternant formula, realizes every reduced minor $\\delta_U(g)$ as the Schur polynomial $s_{\\kappa(U)}$ evaluated at the root multiset of $g$; a confluent Hasse-derivative Vandermonde identity extends this to repeated roots without factorial factors. The Schur coordinates are then packaged into the single rectangular product $D_{n,r}=a_0\\prod_{\\kappa}S_\\kappa$.","core_discovery":"The central discovery is an exact Schur–Plücker factorization for the coefficient-space MDS locus. For the remainder parity-check matrix $H_g^{\\mathrm{rem}}=(Q_0\\ \\cdots\\ Q_{n-1})$, where $Q_i$ is the coefficient vector of $x^i\\bmod g$, the maximal minor indexed by $I=\\{i_0<\\cdots