{"id":"2af20da8-7b55-45a3-836c-5ea51c704855","arxiv_id":"2608.12298","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Rank-one convex non-quasiconvex integrands are explicitly constructed in R^{2x4}, R^{4x4}_sym, and R^{3x3}_sym, via restrictions and transpositions of Grabovsky's example and a modification of Sverak's example.","lead":"The paper constructs explicit energy functions on 2 by 4 and symmetric 3 by 3 matrix spaces that are convex along rank-one lines but still violate quasiconvexity, the key stability condition in the calculus of variations. The constructions settle new low-dimensional cases of Morrey's famous question and provide the first such examples for symmetric matrices.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The counterexample hinges on the computer-algebra identity (3.7); if its sign or factor is wrong, the negative fourth-order variation disappears.","rationale":"The reader's weakest assumption is exactly equation (3.7), the computer-algebra identity. I agree that this is the most load-bearing unverified step. I checked the surrounding algebra: the derivation of (3.1) from the quaternionic inner product is correct; substituting the chosen u,v,s,t into (3.1) and using the definition of Sτ reproduces (3.3) including the absence of ε^3 terms; integration of Sτ vanishes because it is a sum of determinants of Hessian-type blocks; and the embedding/restriction arguments in Corollaries 1.3 and 1.4 are sound because ι and T preserve rank-one directions. The symmetric 3×3 construction in Theorem 1.5 is also internally consistent: (4.1) is correct, f is affine along the four rank-one directions, and the Fourier average is 1/8. The extension from L to all of R^{3×3}_sym is delegated to the standard Šverák extension lemma, which is acceptable though not reproduced. No internal inconsistency or mathematical error was found. The concern about (3.7) is not a demonstrated flaw but a verification gap: the decisive sign of the fourth-order variation is outsourced to a machine computation. Since the notebook is public and the identity is finite and checkable, this does not warrant changing the reader's acceptance; however, an independent recomputation would convert moderate confidence into high confidence.","tokens_in":10036,"tokens_out":23766,"duration_ms":180323,"concrete_test":"Independently recompute (3.7) using a different symbolic algebra system (e.g., SymPy or Sage) that parses the 12-mode ψτ from (3.6), forms Eτ exactly as in (3.4), and evaluates the torus average by summing the finite Fourier products. Confirm the result is exactly 96τ(17τ−2), and then verify at τ=1/12 that the limiting fourth-order variation equals −7/12. As a secondary check, directly evaluate ∫ G^T_im(M+εDφ_{1/12}) dx numerically at a small ε (e.g., 10^{-1} or 10^{-2}) and confirm it is less than G^T_im(M)=4.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The non-quasiconvexity of G^T_im, and hence of F_{2,4} and the Hessian examples, rests on the fourth-order term in the expansion (3.8). The first-order term integrates to zero because Sτ is a null Lagrangian (3.5), and the ε^2 term vanishes upon integration by design. Therefore the sign of ∫ Eτ dx decides the inequality: it must be negative for the chosen τ=1/12. The paper asserts (3.7), ∫ Eτ dx = 96τ(17τ−2), after 'With the help of a computer algebra software' and explicitly states that the calculation requires determining 84 scalar coefficients. The printed text does not reproduce the computation; it delegates to the Mathematica notebook in [1]. This is the single step on which the central construction depends. If the identity were wrong—say, if the coefficient of τ^2 were not 1632 or the constant term were nonzero—the limit in (3.8) would not equal −7/12, and (3.9) could fail. The rest of the proof is either elementary algebra (verified by expanding (3.1) directly) or standard extension arguments from [23], so this computer-assisted identity is the load-bearing assumption.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript constructs explicit rank-one convex, non-quasiconvex integrands in R^{2×4} and in the spaces of symmetric matrices R^{4×4}_sym and R^{3×3}_sym. The construction is based on Grabovsky's example in R^{8×2}; the authors identify a relationship with Šverák's example, then restrict and transpose to lower dimensions. For the R^{2×4} example, a twelve-mode map derived from a scalar potential produces a negative fourth-order variation whose sign is fixed by a computer-algebra identity. A modified Šverák-type construction yields the symmetric 3×3 example.","tokens_in":10258,"tokens_out":14631,"duration_ms":121517,"significance":"If the computations are correct, these are the first explicit counterexamples to 'rank-one convex implies quasiconvex' in R^{2×4} and in symmetric matrix spaces, resolving open cases in Morrey's problem. The examples are fully explicit and the key algebraic identity is backed by a public Mathematica notebook. The observed relation between Grabovsky's and Šverák's constructions is interesting in its own right and may guide further work on lower-dimensional and symmetric cases.","major_comments":[{"comment":"The proof of non-quasiconvexity of G^T_im, and hence of Corollaries 1.3 and 1.4, depends on the identity ∫ Eτ dx = 96τ(17τ−2). The text states that this is obtained with a computer algebra system and refers to a notebook, but does not reproduce the computation or even the structure of the Fourier-coefficient summation. Since the sign of this average is exactly what makes the variation negative, the central claim is not self-contained. Please either include a human-readable derivation, or move the notebook into the submission as supplementary material and add a short verification protocol, for example by listing the 84 scalar coefficients or by showing a symmetry reduction that makes the calculation transparent.","section":"Section 3, Eq. (3.7)"},{"comment":"The extension from the subspace L to all of R^{3×3}_sym is dismissed with the phrase 'follows exactly as in [23]'. In the symmetric setting the relevant rank-one directions are matrices a⊗a, and the orthogonal projection of a⊗a onto L is not generally a rank-one matrix; for instance, with a=(1,2,3) the projection has full rank. The argument in [23] may still apply, but the authors should confirm that the same choice of α,λ,μ works in this setting, or give a reference to a symmetric-matrix version of the extension lemma. As written, this is a gap in the proof of Theorem 1.5.","section":"Section 4, Theorem 1.5"}],"minor_comments":[{"comment":"The notation for the restricted map alternates between φ1/12 and φ~1/12 without consistency; please add tildes consistently so the reader can follow which domain is meant.","section":"Corollary 1.3 proof"},{"comment":"The heading 'Grabovsky's and Šveráks's examples' contains a typo; it should read 'Šverák's example'.","section":"Section 2 heading"},{"comment":"It would be helpful to remark that the sign of 96τ(17τ−2) changes at τ=2/17 and that the choice τ=1/12 is one admissible value; a short sentence on the admissible range would aid reproducibility.","section":"Section 3, Eq. (3.7)"},{"comment":"The AI disclosure is transparent, but it would fit more naturally in the acknowledgements than in the main text; this is purely a presentational suggestion.","section":"Introduction, AI disclosure"}],"recommendation":"major_revision","confidential_remarks":"I do not doubt the validity of the computer algebra result; the main issue is the self-containedness of the proof. The authors should make the Mathematica notebook an integral part of the submission or provide an appendix with the essential steps of the verification. The second major comment on the symmetric 3×3 extension can likely be resolved by a short clarification, but it is load-bearing for Theorem 1.5 and should not be left to a bare reference to Šverák's paper. The use of AI in the research is not a reason for concern."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, this is the real advance it claims to be: it brings the best known m=2 case down from n~1000 to n=4, and it produces the first examples separating Hessian-quasiconvexity from symmetric rank-one convexity in R^{3x3}_sym and R^{4x4}_sym. Second, the whole thing hangs on one computer-algebra identity, (3.7), that is not reproduced in the text. The stress-test note has exactly the right spot.\n\nWhat is actually new: the constructions are explicit and remarkably simple. The 12-mode scalar-potential map, the transposition trick on Grabovsky's quaternionic example, and the use of a null Lagrangian to kill the epsilon^2 term all work. Lemma 2.1 is a clean exact computation, and the compositions in Corollaries 1.3 and 1.4 are checkable in a few lines. The paper also does something rare: it states openly which step is delegated to software, and puts the notebook online.\n\nThe soft spot is (3.7). The sign and coefficient of E_tau decide whether the variation is negative; the text says 'with the help of a computer algebra software' and points to the notebook. There are 84 scalar coefficients, so this is not something a referee can eyeball. It is a finite, symbolic computation, so it is verifiable in principle, but it is exactly the kind of step that needs independent confirmation before the result is fully trusted. I would not call it fatal—the notebook is public and the rest of the proof is explicit—but it is a genuine caveat, and a referee should be asked to run or re-derive (3.7). The parameter tau is chosen after the sign is known, but any tau in an open interval works, so this is standard counterexample design, not curve-fitting. Theorem 1.5 defers parameter choice to Šverák's paper; minor.\n\nThis paper is for anyone working on Morrey's problem, quasiconvexity, or Hessian quasiconvexity. It shortens a long-standing gap and gives concrete material to test conjectures. If (3.7) checks out, it is a strong advance.\n\nRecommendation: send it to peer review. The referee should spend time on the notebook.","headline":"First explicit rank-one convex non-quasiconvex integrand in R^{2x4} and first Hessian counterexamples in symmetric spaces—genuinely new, but one unverified computer-algebra identity carries the whole construction.","tokens_in":10843,"tokens_out":2136,"would_cite":true,"duration_ms":18696,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["49J45","49J10","26B25"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper constructs explicit rank-one convex integrands in $\\mathbb{R}^{2\\times4}$, $\\mathbb{R}^{4\\times4}_{\\mathrm{sym}}$, and $\\mathbb{R}^{3\\times3}_{\\mathrm{sym}}$ that fail quasiconvexity, giving the first Hessian counterexamples on…","keywords":["rank-one convexity","quasiconvexity","calculus of variations","Hessian quasiconvexity","symmetric matrices","Fourier modes","explicit counterexample","scalar potential"],"falsifier":"Take the explicit 12-mode potential $\\psi_{1/12}$ from (3.6), form $E_{1/12}$ from (3.4), and compute its average over $\\mathbb{T}^4$ with an independent computer algebra system or high-precision quadrature; it must equal $-14/3$. Equivalently, evaluate $\\int G_{\\mathrm{im}}^{T}(M+\\varepsilon D\\varphi_{1/12})\\,dx$ for several small $\\varepsilon$ and check that the $\\varepsilon^{4}$ coefficient approaches $-7/12$; any other value would break the strict inequality.","tokens_in":9824,"feed_emoji":"🧮","tokens_out":12618,"duration_ms":102598,"temperature":0.7,"pith_summary":"The paper constructs explicit functions where convexity along rank-one directions holds, yet the stronger quasiconvexity inequality—the one that underlies lower semicontinuity in the calculus of variations—fails. The main new example is an integrand on $\\mathbb{R}^{2\\times4}$, obtained by restricting and transposing a known rank-one convex non-quasiconvex integrand on $\\mathbb{R}^{8\\times2}$; a test map with 12 Fourier modes falsifies quasiconvexity at the origin. Because that map is the gradient of a scalar potential, the same mechanism yields integrands on symmetric $4\\times4$ and $3\\times3$ matrices that are convex along symmetric rank-one directions but not Hessian-quasiconvex. These are the first such explicit examples on symmetric matrices, while the original $2\\times2$ square case remains open.","feed_headline":"2-by-4 matrices now have a rank-one convex, non-quasiconvex integrand","feed_subtitle":"A 12-mode test map also yields Hessian counterexamples on symmetric 4-by-4 and 3-by-3 matrices.","key_machinery":"The base object is the integrand $G(\\xi,\\eta)=\\sqrt{\\|\\xi\\|^{2}\\|\\eta\\|^{2}-|\\langle\\xi,\\eta\\rangle|^{2}}$ on pairs of quaternionic vectors, identified with $\\mathbb{R}^{8\\times2}$; it is homogeneous, rank-one convex, and non-quasiconvex. The paper restricts it to purely imaginary quaternionic entries and transposes it to $\\mathbb{R}^{2\\times6}$. The central new device is a scalar potential $\\psi_{\\tau}$ on $\\mathbb{T}^{4}$ with 12 cosine modes, chosen so that, with $\\varphi_{\\tau}=(\\partial_{z_1}\\psi_{\\tau},-\\partial_{y_1}\\psi_{\\tau})$, the expansion takes the form $G_{\\mathrm{im}}^{T}(M+\\varepsilon D\\varphi_{\\tau})^{2}=(4-\\varepsilon^{2}S_{\\tau})^{2}+\\varepsilon^{4}E_{\\tau}$. The $\\varepsilon^{2}$ coefficient $S_{\\tau}$ is a null Lagrangian—a function whose integral has zero average over the torus—while the $\\varepsilon^{4}$ coefficient $E_{\\tau}$ has torus average $96\\tau(17\\tau-2)$, verified symbolically; at $\\tau=1/12$ this average is negative, so the fourth-order term drives the total integral down. For the symmetric $3\\times3$ case, the machinery changes: four rank-one matrices $\\xi_j\\otimes\\xi_j$ span a subspace where the integrand $-r_1r_2r_3r_4$ is convex along symmetric rank-one lines, and a four-mode trigonometric test map gives the negative variation.","core_discovery":"The paper claims that the gap between rank-one convexity and quasiconvexity occurs in matrix spaces that were previously unresolved. Specifically, $F_{2,4}(A)=G_{\\mathrm{im}}^{T}(M+\\iota(A))$ on $\\mathbb{R}^{2\\times4}$ is rank-one convex but not quasiconvex at $0$: for the explicit 12-mode map $\\tilde{\\varphi}_{1/12}$ the averaged value of $F_{2,4}(\\varepsilon D\\tilde{\\varphi}_{1/12})$ is strictly below $F_{2,4}(0)$ for all sufficiently small nonzero $\\varepsilon$. The same construction, read through a scalar potential, gives $H$ on $\\mathbb{R}^{4\\times4}_{\\mathrm{sym}}$ that is convex along every line $A+t(a\\otimes a)$ but fails Hessian quasiconvexity, and a quartic polynomial on $\\mathbb{R}^{3\\times3}_{\\mathrm{sym}}$ with a four-mode test map does the same in dimension three. Along the way, the paper shows that its starting eight-by-two quaternionic example contains the classical three-mode example as a leading-order slice.","pith_inferences":["The 12-mode potential is a perturbation of a 9-mode profile that lies in the kernel of the fourth-order variation. A natural extension is to search for similar nondegenerate perturbations in $\\mathbb{R}^{2\\times3}$ or $\\mathbb{R}^{2\\times2}$, where lower-order cancellations might still be arranged while a higher-order term controls the sign.","The descent pattern—restrict a high-dimensional quaternionic example to an algebraic subspace, transpose, then find a scalar potential—could be tested on other norm-based integrands; one concrete check is to enumerate Fourier modes satisfying the determinant-zero constraint $\\det(k',k'')=0$ and look for negative fourth-order averages in lower dimensions.","An independent proof of the 84-coefficient torus average could expose why the coefficient $17\\tau-2$ has this particular form and may suggest optimal perturbations for other target spaces."],"forward_implications":["The explicit $\\mathbb{R}^{2\\times4}$ integrand shows the rank-one convex/quasiconvex gap does not require a very large number of columns; range dimension two is not by itself enough to force equivalence.","The scalar-potential construction converts the gradient counterexample into Hessian counterexamples on symmetric $4\\times4$ and $3\\times3$ matrices, so for scalar second-order variational problems convexity along symmetric rank-one directions does not imply Hessian quasiconvexity.","The sign of the fourth-order coefficient $96\\tau(17\\tau-2)$ is the deciding quantity; the choice $\\tau=1/12$ is an explicit nondegenerate direction, and a computer-algebra verification is part of the argument.","The construction does not settle the $2\\times2$ or $2\\times3$ cases; transposition is essential to the low-dimensional examples, and quasiconvexity is not invariant under transposition, so the same route cannot be applied directly to the square case."],"supporting_citations":[{"why":"Supplies the starting homogeneous rank-one convex non-quasiconvex integrand on 8-by-2 matrices whose restriction and transposition are the paper's main source.","marker":"[13]"},{"why":"Supplies the classical three-mode counterexample and the perturbative extension procedure used to build the symmetric 3-by-3 example.","marker":"[23]"},{"why":"Contains the symbolic verification of the load-bearing torus average (3.7), fixing the sign of the fourth-order term.","marker":"[1]"}],"fun_headline_variants":["Rank-one convex but not quasiconvex in 2x4 and symmetric 3x3","Explicit counterexamples close the 2x4 and symmetric 3x3 gaps","12-mode map yields rank-one convex non-quasiconvex in 2x4 and 3x3","Morrey's problem resolved for 2x4 and symmetric 3x3 matrices"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument stands on a long symbolic computation, reported but not reproduced in the text, that the torus average of $E_{\\tau}$ is $96\\tau(17\\tau-2)$; at $\\tau=1/12$ its negative sign is exactly what makes the averaged integrand decrease, so if that identity were wrong the counterexample would fail.","fun_headline_variants_meta":{"raw":{"variants":["Rank-one convex but not quasiconvex in 2x4 and symmetric 3x3","Explicit counterexamples close the 2x4 and symmetric 3x3 gaps","12-mode map yields rank-one convex non-quasiconvex in 2x4 and 3x3","Morrey's problem resolved for 2x4 and symmetric 3x3 matrices"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000236,"raw_usage":{"total_tokens":1514,"prompt_tokens":967,"completion_tokens":547,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":583,"completion_tokens_details":{"reasoning_tokens":449}},"tokens_in":583,"tokens_out":547,"duration_ms":3928,"temperature":1.0,"reasoning_tokens":449,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T00:10:42.687738+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the explicit 12-mode potential $\\psi_{1/12}$ from (3.6), form $E_{1/12}$ from (3.4), and compute its average over $\\mathbb{T}^4$ with an independent computer algebra system or high-precision quadrature; it must equal $-14/3$. Equivalently, evaluate $\\int G_{\\mathrm{im}}^{T}(M+\\varepsilon D\\varphi_{1/12})\\,dx$ for several small $\\varepsilon$ and check that the $\\varepsilon^{4}$ coefficient approaches $-7/12$; any other value would break the strict inequality.","supporting_citations":[{"cited_title":"Grabovsky","cited_arxiv_id":null,"evidence_quote":"Supplies the starting homogeneous rank-one convex non-quasiconvex integrand on 8-by-2 matrices whose restriction and transposition are the paper's main source."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the classical three-mode counterexample and the perturbative extension procedure used to build the symmetric 3-by-3 example."},{"cited_title":"Agazzi, G","cited_arxiv_id":null,"evidence_quote":"Contains the symbolic verification of the load-bearing torus average (3.7), fixing the sign of the fourth-order term."}],"review_version":1}