{"id":"87006a94-b4dd-4d27-b055-79d1d996d308","arxiv_id":"2607.12761","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For r≥3, n≥r+3, cyclic-orbit MDS codes are GRS iff the operator is c·Sym^{r-1}(B); over F_q this gives four families, an exact count, and asymptotic genericity of MDS non-GRS companion codes.","lead":"An MDS code built from the powers of one matrix is generalized Reed–Solomon exactly when the matrix is a symmetric-power action of PGL2, once the length is at least r+3. Over finite fields this gives four disjoint families, an exact count, and shows most MDS companion codes are non-GRS as q grows.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; central rigidity and classification arguments are sound.","rationale":"The paper's central claim is the rigidity theorem: in the stable range, an MDS Krylov orbit lies on an RNC iff the cyclic pair is conjugate to the symmetric-power action of PGL2. The proof is a short chain: MDS => d+3-point blocks in general position; Castelnuovo uniqueness => Gamma=A Gamma; rational point => split; stabilizer computation => symmetric power. I find each step internally consistent. The finite-field classification then follows from the three PGL2(F_q) conjugacy types; the unipotent family is handled by a clean Pascal-matrix MDS criterion, and the semisimple families satisfy the order and Frobenius-reversal equations. The exact count and asymptotic genericity follow from the four mutually exclusive families plus the determinantal open set for MDS. I found no algebraic error that would move the reader's ACCEPT. The weakest point remains the reliance on Lemma 4.3, exactly as the reader noted, but this is standard and likely safe. I also noticed a minor misstatement in the proof of Theorem 6.13 (F_q should be \\overline{F_q}); it is localized, easily corrected, and not needed for the main counting theorem. Therefore verdict stays UNCHANGED.","tokens_in":27754,"tokens_out":47657,"duration_ms":408351,"concrete_test":"Verify Lemma 4.3 independently in positive characteristic: fix d=3, choose six random points on a twisted cubic over F_5, and compute the scheme of all twisted cubics through them (e.g., by eliminating parameters from the Veronese parametrization); confirm it is a single reduced point. If uniqueness fails in any characteristic, Theorem 4.4 would need an additional argument.","verdict_should_be":"UNCHANGED","load_bearing_attack":"After checking the orbit-rigidity argument, the finite-field trichotomy, and the counting formula, I find no load-bearing objection to the central claim. Theorem 4.4 hinges on Lemma 4.3 (d+3-point Castelnuovo uniqueness), which is cited rather than proved, but it is a standard characteristic-free result; the two applications are legitimate because MDS gives every r=d+1 orbit columns independent, so the relevant d+3 point sets are in linearly general position. I spot-checked the spectral and descent steps in Theorem 5.3 and the parameter counts in Theorem 6.14. The one non-central defect is in the proof of Theorem 6.13: 'applied with k=F_q' should be 'applied with k=\\overline{F_q}' — otherwise nonsplit families would be excluded before the Frobenius dichotomy. The fix is immediate and does not affect Theorem 6.14, which is proved directly from Theorem 5.3. This is a typographical-level flaw, not a substantive gap.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper defines Krylov codes for cyclic operator-vector pairs (A,z) via the kernel of the Krylov matrix (z,Az,...,A^{n-1}z) and studies when such an MDS orbit segment lies on a rational normal curve. The main rigidity theorem (Theorem 4.4) asserts that for r≥3 and n≥r+3, an MDS Krylov orbit lies on an RNC iff the projective pair is conjugate to the symmetric-power action of PGL_2; the containing RNC is then unique, split, and A-stable. For companion operators over finite fields, Theorem 5.3 gives a complete classification of the GRS locus into unipotent (U), split semisimple (S), and two nonsplit semisimple families (N0,N1). The second half of the paper studies the coefficient-space geometry of the semisimple GRS locus: it is a two-dimensional rational surface, its normalization over the nonzero-constant-term part is the affine quotient of a two-dimensional torus by the reversal involution, Frobenius descent yields an exact finite-field counting formula (Theorem 6.14), and the MDS non-GRS proportion tends to one as q→∞ (Theorem 6.15). Boundary lengths n=r+1,r+2 are treated in Appendix A.","tokens_in":27959,"tokens_out":25718,"duration_ms":237755,"significance":"If correct, this is a substantial contribution: it gives a clean geometric and representation-theoretic characterization of when cyclic orbit arcs are GRS, a complete finite-field companion classification, and an exact arithmetic count. The paper is largely self-contained and careful about characteristic issues; the proof of the main rigidity theorem rests on standard Castelnuovo uniqueness and the RNC stabilizer, both handled explicitly. Concrete reproducible examples are provided in Section 7. The authors are also appropriately explicit about the scope of the normalization theorem and about the boundary ranges. I found no load-bearing technical objection; the issues I identified are local and easily fixed.","major_comments":[],"minor_comments":[{"comment":"The first line of the proof says 'By Theorem 6.2(5), applied with k = F_q'. Theorem 6.2(5) gives the surjectivity on k-points only for algebraically closed k. The proof should apply Theorem 6.2(5) with k = \\overline{F_q} and then use the Frobenius dichotomy that follows. The subsequent counting theorem (Theorem 6.14) is proved directly from Theorem 5.3, so this is a local typographical-level flaw, not a substantive gap.","section":"§6.5, proof of Theorem 6.13"},{"comment":"The proof says to choose d+2 pairwise distinct elements u_0,...,u_{d+1} in k. This is impossible when |k|<d+2, e.g. k=F_4 and r=3. Since [N] already fixes every geometric point of Γ0, the frame should be chosen over \\bar{k}; then a projectivity fixing d+2 points in linearly general position over \\bar{k} is the identity, so the conclusion follows. The statement is true, but the proof needs this small correction to cover small finite fields.","section":"Appendix B, proof of Proposition B.1"},{"comment":"The ideal I(Γ) should be in \\overline{F_q}[X_0,...,X_{r-1}], not F_q[X_0,...,X_{r-1}]; as printed, the Frobenius action and Galois descent step do not make sense. This is a notation typo and does not affect the argument once corrected.","section":"§4.4, proof of Corollary 4.5"},{"comment":"When invoking Lemma 4.3, the text says 'applied over k', but the lemma is stated for algebraically closed fields. This should be read as applying after base change to \\bar{k}; the accompanying uniqueness argument then goes through. A one-sentence clarification would avoid confusion.","section":"§4.3, proof of Theorem 4.4"}],"recommendation":"minor_revision","confidential_remarks":"The paper is well within the scope of the journal and the main results appear sound. The local proof corrections above should be made, but none of them affects the central rigidity classification or the counting theorem. I would be happy to see the revised version accepted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is a serious paper and the main rigidity classification is real. The novelty is Theorem 4.4: in the stable range, an MDS Krylov orbit segment lying on a rational normal curve forces the operator into the symmetric-power image of PGL2. I checked the mechanism—two overlapping d+3 blocks, Castelnuovo uniqueness, stabilizer calculation—and it works. The finite-field companion classification into split semisimple, two nonsplit semisimple, and unipotent families follows cleanly, and the exact counting formula in Theorem 6.14 is internally consistent. The asymptotic genericity of MDS non-GRS companion codes is a nice consequence.\n\nThe paper also does well on the coefficient side: the two-dimensional surface, the normalization as a torus quotient by reversal, and the clean separation of the squarefree semisimple locus from the pure-power boundary are all handled with care. The determinantal MDS open set is a useful tool, and the worked examples are concrete and match the stated criteria. The reliance on [9] for the d+3-point Castelnuovo uniqueness lemma is legitimate: that is standard characteristic-free material, and the application is valid because MDS gives linearly general position.\n\nSoft spots are minor. In the proof of Theorem 6.13, Theorem 6.2(5) should be applied over the algebraic closure of F_q, not over F_q; otherwise the nonsplit families are excluded before the Frobenius dichotomy. The fix is immediate and does not affect Theorem 6.14, which is proved directly from Theorem 5.3. The boundary lengths n=r+1 and n=r+2 are relegated to Appendix A; that is fine, since they are genuinely edge cases. The paper is long and not machine-checked, and I did not independently verify every cited standard fact, but I found no load-bearing error.\n\nWho gets value: algebraic coding theorists and finite geometers working on MDS/GRS recognition, arcs, and rational normal curves. It deserves a serious referee. My recommendation is to send it to peer review; the referee should spend time on Sections 5 and 6 but should not be bracing for a rejection.","headline":"A solid, genuinely new rigidity theorem for cyclic MDS orbits on rational normal curves; referee it, with only small fixes needed.","tokens_in":28441,"tokens_out":1987,"would_cite":true,"duration_ms":20386,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14N05","94B05","51E21","11T71"],"pacs":[],"model":"deepseek-v4-flash","headline":"For r≥3 and n≥r+3, an MDS Krylov orbit segment lies on a rational normal curve exactly when the cyclic pair is conjugate to the (r−1)-st symmetric power of PGL2—over finite fields this gives a complete four-family classification of generali","keywords":["MDS codes","rational normal curves","cyclic operators","Krylov codes","generalized Reed–Solomon codes","companion matrices","PGL2 symmetric powers","algebraic tori"],"falsifier":"Exhibit a cyclic pair (A,z) over a field with r≥3 and n≥r+3 such that the Krylov code is MDS, the points [z],…,[A^{n−1}z] lie on a rational normal curve, yet A is not similar to c Sym^{r−1}(B) for any S, B, c. Equivalently, over F_q, find a monic polynomial g of degree r with an irreducible factor of degree at least 3 whose companion code is both MDS and GRS; the paper's Corollaries 4.6 and 5.5 predict no such g exists.","tokens_in":27649,"feed_emoji":"📐","tokens_out":6154,"duration_ms":54595,"temperature":0.7,"pith_summary":"The paper studies Krylov codes, whose parity-check matrix is a finite projective orbit segment (z, Az, …, A^{n−1}z) of a cyclic operator A. Its main theorem, for r≥3 and n≥r+3, is a rigidity statement: an MDS orbit segment lies on a rational normal curve precisely when the projective pair (A,[z]) is conjugate to the (r−1)-st symmetric-power action of PGL2. Over finite fields, applied to companion matrices, this yields a complete classification of the generalized Reed–Solomon (GRS) locus into split semisimple, two nonsplit semisimple, and unipotent families. The paper also shows the semisimple GRS coefficient closure is a two-dimensional rational surface, counts exactly how many GRS polynomials exist over F_q, and proves that, for fixed r and n, the proportion of monic polynomials whose companion codes are MDS but non-GRS tends to 1 as q grows.","feed_headline":"MDS cyclic orbits lie on rational normal curves in one symmetry class","feed_subtitle":"For r≥3 and n≥r+3, this is exactly when the operator is a symmetric power of PGL2.","key_machinery":"The central object is the cyclic pair (A,z) and its orbit segment [z],[Az],…,[A^{n−1}z], encoded as the kernel of the Krylov matrix H_n(A,z). Two facts carry the argument: the d+3-point uniqueness lemma—any d+3 points in linearly general position determine at most one rational normal curve—and the stabilizer computation showing that the projective stabilizer of the standard rational normal curve Γ0=ν_{r−1}(P^1) is exactly the image of the (r−1)-st symmetric-power representation of PGL2. The uniqueness lemma makes two overlapping blocks of r+2 orbit points force AΓ=Γ, and the stabilizer computation then identifies the projective class of A.","core_discovery":"The central claim is Theorem 4.4: if a cyclic pair (A,z) produces an MDS Krylov code and the projective points [z],[Az],…,[A^{n−1}z] lie on a k-form of a rational normal curve, then A is, up to a scalar, the (r−1)-st symmetric power of a 2×2 matrix B, and [z] is the image of a k-rational point of P^1 under the Veronese embedding. Conversely, any pair of that form has its orbit on the standard rational normal curve. The containing curve is unique, split, and preserved by the projective class of A. Over finite fields, Corollary 4.5 turns this into an absolute criterion: an MDS companion code is GRS exactly when its orbit arc is contained in a rational normal curve, and Theorem 5.3 lists the fo","pith_inferences":["The codimension r−2 of the GRS coefficient surface explains why no single low-degree polynomial can recognize GRS companion codes for r≥4; recognition must involve higher-degree or multi-component conditions.","The generic two-to-one parameterization of the coefficient surface by reversal suggests an algorithmic route for GRS testing: decide whether a companion polynomial arises as a geometric progression with ratio of order at least n, which is a finite torus-quotient problem.","A natural refinement of the counting theorem is to count monomial-equivalence classes of GRS codes rather than generator polynomials, since the paper's formula counts polynomials and the authors explicitly separate the two questions.","The stability threshold n≥r+3 is sharp in an interesting way: Appendix A shows that for n=r+1 or n=r+2 every MDS companion code is GRS, so the non-GRS density phenomenon only appears once the orbit segment is long enough to engage the rigidity mechanism."],"forward_implications":["An MDS Krylov code is of GRS type exactly when its projective orbit arc is contained in a rational normal curve over the base field, after which the containing curve is necessarily split.","Over finite fields, the GRS locus among companion codes splits into four mutually exclusive arithmetic families: unipotent pure powers, split semisimple geometric progressions, and two nonsplit semisimple families distinguished by parity of r.","Any MDS companion code whose polynomial has an irreducible factor of degree at least 3, or which is non-squarefree without being a pure power, is automatically non-GRS.","The exact number of GRS polynomials over F_q is (q−1)·1_{n≤p} + (q−1)/2 (E_{q−1}(n)+E_{q+1}(n)), where E_m(n) counts elements of order at least n in a cyclic group of order m.","For fixed r≥3 and n≥r+3, the proportion of monic degree-r polynomials whose companion codes are MDS but non-GRS tends to 1 as q goes to infinity."],"fun_headline_variants":["MDS cyclic orbits lie on rational normals only for symmetric powers","Symmetric power action is the only route to MDS rational normal orbits","MDS orbits on rational normals force symmetric power structure","MDS orbit on rational normals iff symmetric power action","Only symmetric powers give MDS orbits on rational normal curves"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is the classical d+3-point uniqueness lemma: over an algebraic closure, any d+3 points in linearly general position lie on at most one rational normal curve; the entire rigidity argument—that two overlapping blocks of orbit points force A to preserve the curve—depends on this uniqueness, and the stable range n≥r+3 is exactly what guarantees the required number of general-position points.","fun_headline_variants_meta":{"raw":{"variants":["MDS cyclic orbits lie on rational normals only for symmetric powers","Symmetric power action is the only route to MDS rational normal orbits","MDS orbits on rational normals force symmetric power structure","MDS orbit on rational normals iff symmetric power action","Only symmetric powers give MDS orbits on rational normal curves"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000802,"raw_usage":{"total_tokens":3453,"prompt_tokens":928,"completion_tokens":2525,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":672,"completion_tokens_details":{"reasoning_tokens":2439}},"tokens_in":672,"tokens_out":2525,"duration_ms":15977,"temperature":1.0,"reasoning_tokens":2439,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T06:20:27.377163+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit a cyclic pair (A,z) over a field with r≥3 and n≥r+3 such that the Krylov code is MDS, the points [z],…,[A^{n−1}z] lie on a rational normal curve, yet A is not similar to c Sym^{r−1}(B) for any S, B, c. Equivalently, over F_q, find a monic polynomial g of degree r with an irreducible factor of degree at least 3 whose companion code is both MDS and GRS; the paper's Corollaries 4.6 and 5.5 predict no such g exists.","supporting_citations":[],"review_version":2}