{"id":"0d2ddfa0-0539-4f2c-b926-a74c63d0fb71","arxiv_id":"2608.06367","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For the planar Dacorogna-Marcellini energy, quasiconvexity and rank-one convexity coincide with threshold |γ| ≤ 2/√3, and the same equivalence holds for all SO(2)×SO(2)-invariant quartic polynomials.","lead":"This paper proves that a family of energy functions from the calculus of variations, the Dacorogna-Marcellini energies, is quasiconvex exactly when it is rank-one convex, closing a gap open since 1988. The proof uses a heat-flow monotonicity argument and extends also to all rotation-invariant quartic polynomials on two-by-two matrices.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central proof rests on an unverified algebraic identity in Appendix A; a symbolic re-expansion is needed before the quasiconvexity conclusion is secure.","rationale":"The reader identified the missing interpolation in Lemma 4 and the sketched estimates in Lemma 3 as the fragile premises. I agree that the interpolation is a real gap, but it is fillable by a standard convex-combination step and is not the deepest risk. The proof of Theorem 2 in the appendix covers the full range without interpolation, so the central claim can be rescued even if Lemma 4 is incomplete. The more load-bearing issue is the central algebraic identity in Appendix A: it is the decisive step that turns the exact two-form correction into a pointwise nonnegativity statement, and it is simply asserted as 'collecting squares' without derivation. A single sign error there would destroy the sufficiency proof for all t, and the endpoint check in Lemma 4 does not validate the general t-family. The paper’s own AI-use statement notes that the exact two-form was found autonomously by an agentic system, which makes independent verification of the algebra especially important. My suggested symbolic-expansion test is precise, cheap, and would settle whether the concern lands. Since the proof is otherwise coherent and the gap is methodological rather than demonstrably fatal, the reader’s conditional verdict remains appropriate.","tokens_in":10767,"tokens_out":51493,"duration_ms":489998,"concrete_test":"Expand both sides of the sum-of-squares identity in Appendix A symbolically for arbitrary t, rho, xi1, xi2, xi3, eta1, eta2, eta3 using a computer algebra system (e.g., SymPy) and verify that their difference is identically zero. Independently verify that the matrix M1(rho) in Lemma 4 has principal minors Delta1=(rho-sqrt(3))^2, Delta2=3/4(rho-sqrt(3))^2[(rho-sqrt(3)/9)^2+8/27], Delta3=1/6(rho-sqrt(3))^2(1-rho^2)^2 and is positive semidefinite for all rho in R. If the identity or the minors fail, the correction form is wrong and the proof collapses; if they succeed, the central claim’s algebraic core is sound.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper’s central claim reduces to a pointwise nonnegativity statement: after adding an exact two-form, the second-variation density is a sum of squares. This is the load-bearing step. In Appendix A it appears as the displayed identity after (A.4), asserted without derivation: the right-hand side is claimed to equal the left-hand side for all t, rho, and derivative matrices. If this identity has any sign or coefficient error, the integrated second-variation inequality (2.1) is not established for W_t, the heat-flow criterion in Lemma 3 does not apply, and Theorem 2 — hence Theorem 1 as a special case — fails. The endpoint version in Lemma 4 is partially checked by the explicit matrix M1(rho) and its principal minors, but the general t-identity is not derived or cross-checked; the proof’s correctness for the entire interval |gamma| <= 2/sqrt(3) depends on this unverified algebra. A secondary gap is the promised but omitted interpolation in Lemma 4 for intermediate gamma; it is fillable by a convex-combination argument, but as written the lemma is incomplete. The algebraic identity, however, is the more fundamental risk: it is exactly where a computational error would be invisible without an independent check.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves that the Dacorogna--Marcellini energy f_gamma(A)=|A|^4 - 2 gamma |A|^2 det A is quasiconvex for |gamma| <= 2/sqrt(3), and hence, in view of the known rank-one convexity threshold, quasiconvex if and only if rank-one convex. The proof introduces a heat-flow monotonicity criterion (Lemma 3): if a suitable integrated second-variation inequality holds along perturbations v(x)=Fx+psi, then f_gamma is quasiconvex. The authors then construct an SO(2) x SO(2)-invariant exact two-form correction that makes the second-variation density pointwise nonnegative (Lemma 4), reducing the verification to a one-parameter family of 3x3 matrices whose principal minors are computed. Theorem 2 extends the equivalence to all SO(2) x SO(2)-invariant homogeneous quartic polynomials, which are classified and reduced to a one-parameter boundary family W_t.","tokens_in":1270,"tokens_out":1450,"duration_ms":492378,"significance":"The result, if correct, resolves a long-standing open problem for this canonical planar family and provides the first complete equivalence between quasiconvexity and rank-one convexity for a nontrivial class of 2x2 rotationally invariant quartics. The endpoint verification via explicit principal minors is a strength, as is the absence of fitted parameters in the proof. The method, combining flow monotonicity with exact differential corrections, appears novel and potentially transferable. However, the manuscript currently leaves two load-bearing algebraic steps insufficiently supported, so the significance is conditional on those being completed.","major_comments":[{"comment":"Lemma 4 is stated for all |gamma| <= 2/sqrt(3), but the proof verifies the corrected pointwise inequality only at the endpoint gamma = 2/sqrt(3). The opening sentence 'It is enough to check this property for gamma = 2/sqrt(3)' is never justified. Since Theorem 1 needs the inequality for the whole interval, this is a load-bearing gap. The gap is repairable: for 0 <= gamma <= 2/sqrt(3) one can write E_gamma = lambda E_{2/sqrt(3)} + (1-lambda) E_0 with lambda = gamma sqrt(3)/2, noting that E_0 >= 0 pointwise because |A|^4 is convex, and negative gamma can be handled through f_{-gamma}(A) = f_gamma(A J) with a reflection J. This argument must be written out.","section":"Lemma 4, proof"},{"comment":"The pointwise nonnegativity of W_t rests entirely on the sum-of-squares identity following (A.4), which is introduced with 'collecting squares gives' and no supporting calculation. This identity is load-bearing for Theorem 2 and is exactly where a sign or coefficient error would be invisible without an independent check. Sampling several monomial coefficients (for example xi1^2, eta2^2, xi1 xi3) indicates that the identity is consistent, but the manuscript should provide the intermediate expansion or a verifiable algebraic derivation, since the claim is not otherwise checkable by the reader.","section":"Appendix A, displayed identity after (A.4)"}],"minor_comments":[{"comment":"Several coefficients appear with missing fraction bars: for instance 8 rho sqrt(3) should read 8 rho / sqrt(3), and 2 rho sqrt(3) should read 2 rho / sqrt(3). As printed, these formulas are inconsistent with the subsequent identity kappa(rho) = (rho - 1/sqrt(3))^2 + 2/3. Please correct the typesetting.","section":"Equations (3.8), (3.10), and kappa(rho)"},{"comment":"The word 'skwe-symmetry' is a typo for 'skew-symmetry'; also, the equality omega_{A tilde}(B tilde_1, B tilde_2) = omega_A(B_1, B_2) uses det U = 1, which should be stated explicitly.","section":"Lemma 4, Step 3"},{"comment":"The symbol F is used both for the fixed matrix in v(x) = F x + psi and for the functional evaluated at tau; this notation clash makes the proof harder to follow. Rename the functional.","section":"Lemma 3, proof"},{"comment":"The reduction from diag(sigma, tau) to diag(1, rho) should explicitly use the degree-two homogeneity of E_gamma in its first argument and state the trivial case sigma = 0.","section":"Lemma 4, Step 3"}],"recommendation":"major_revision","confidential_remarks":"The central algebraic identity in Appendix A is the main risk. I recommend that the editor require the authors to provide a full expansion or a supplementary computer-algebra verification before acceptance. The interpolation gap in Lemma 4 is more readily repairable but must also be closed. If these points are addressed, the paper would be a strong contribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Main take: this is a serious attack on a problem that's been open for decades, and the heat-flow criterion is genuinely new. But the written proof has two gaps and one unverified algebraic identity; I'd want those fixed before calling it settled.\n\nWhat's new and what's good. The result claimed is substantial: quasiconvexity of the Dacorogna-Marcellini energy on the full interval |γ| ≤ 2/√3, matching the rank-one convexity threshold, plus the same equivalence for all SO(2)×SO(2)-invariant quartics. Previous work only reached 1+ε. The proof strategy departs from the usual direct estimates: Lemma 3 reduces quasiconvexity to an integrated second-variation inequality along the heat flow, which is a nice idea. The exact invariant two-form correction is also new, and at the endpoint γ=2/√3 the authors do real work: they give an explicit matrix M1(ρ) with principal minors.\n\nThe soft spots, in order of seriousness. First, Lemma 4 is stated for all |γ| ≤ 2/√3 but only verified at the endpoint. The sentence 'it is enough to check this property for γ=2/√3' appears, but no argument is given. This is likely fixable by a convex-combination or scaling argument, but it's missing.\n\nSecond, the appendix is the load-bearing part for Theorem 2. The sum-of-squares identity after (A.4) is just asserted: 'Differentiating ... and collecting squares, gives ...' with no derivation. That identity is exactly where a sign or coefficient error would be invisible without an independent check. The endpoint computation in Lemma 4 provides some confidence, but the general t-identity is not cross-checked.\n\nThird, Lemma 3 depends on the decay estimate (2.2) and continuity estimate (2.4), which are only sketched. They look standard for heat flow and quartic growth, but a referee should ask for the details.\n\nThe citation pattern is clean, and the historical account matches what I know. If the algebra is correct, this is a major result. I'd send it to a serious expert referee, with instructions to verify the appendix identity and fill the interpolation gap. Not a desk reject.","headline":"Serious attack on the Dacorogna-Marcellini gap; the heat-flow criterion is new, but the appendix's sum-of-squares identity and missing interpolation need checking.","tokens_in":11563,"tokens_out":4784,"would_cite":true,"duration_ms":52302,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["49J45","74B20"],"pacs":[],"model":"deepseek-v4-flash","headline":"The Dacorogna–Marcellini energy is quasiconvex exactly when it is rank-one convex, for $|\\gamma|\\le 2/\\sqrt3$.","keywords":["Dacorogna–Marcellini energy","quasiconvexity","rank-one convexity","Morrey's problem","heat-flow method","exact two-form","2 x 2 matrices","homogeneous quartic polynomial"],"falsifier":"Check the principal minors of the matrix $M_1(\\rho)$ given in the paper for all real $\\rho$; any $\\rho$ where a minor is negative would contradict Lemma 4 at $\\gamma=2/\\sqrt3$. Alternatively, run a direct numerical test of inequality (1.1) for $f_{2/\\sqrt3}$ at affine maps with oscillatory perturbations concentrated near matrices with distinct singular values.","tokens_in":10528,"feed_emoji":"📐","tokens_out":7529,"duration_ms":78878,"temperature":0.7,"pith_summary":"The paper proves that the Dacorogna–Marcellini energy $f_\\gamma(A)=|A|^4-2\\gamma|A|^2\\det A$ on $2\\times 2$ matrices is quasiconvex precisely when it is rank-one convex, that is, for $|\\gamma|\\le 2/\\sqrt3$. This closes the interval $1<|\\gamma|\\le 2/\\sqrt3$, where the energy was known to be rank-one convex and not polyconvex but quasiconvexity had been open. The proof uses a heat-flow monotonicity argument and an invariant exact two-form correction rather than a direct estimate of Morrey's condition. The method extends to show that for any homogeneous quartic polynomial on $2\\times2$ matrices invariant under left and right rotations, quasiconvexity and rank-one convexity coincide.","feed_headline":"Planar quartic energy is quasiconvex exactly when rank-one convex","feed_subtitle":"For the Dacorogna–Marcellini energy, quasiconvexity matches rank-one convexity exactly, yielding explicit non-polyconvex examples.","key_machinery":"The load-bearing objects are the heat-flow criterion of Lemma 3 and the exact invariant correction $\\omega_\\gamma=d\\alpha_\\gamma$. Lemma 3 says $f_\\gamma$ is quasiconvex if the integrated second-variation inequality holds for all affine-plus-Schwartz maps; it is proved by evolving the perturbation by the heat semigroup and showing that the relative energy is nonincreasing and decays to zero. The correction is chosen from the four-parameter ansatz $\\alpha=c_1|A|^2\\theta+c_2(\\det A)\\zeta+c_3|A|^2\\zeta+c_4(\\det A)\\theta$, with invariant one-forms $\\theta$ and $\\zeta$; because $\\omega_\\gamma$ is exact, its integral along any gradient vanishes, so it can be added to the second-variation density without changing the integrated quantity. Rotational invariance reduces the pointwise inequality to $A=\\operatorname{diag}(1,\\rho)$, and the endpoint check is completed by the explicit matrix $M_1(\\rho)$, whose principal minors are shown nonnegative.","core_discovery":"The central claim is Theorem 1: for $|\\gamma|\\le2/\\sqrt3$, $f_\\gamma(A)=|A|^4-2\\gamma|A|^2\\det A$ is quasiconvex. Since quasiconvexity always implies rank-one convexity and the converse threshold $|\\gamma|\\le2/\\sqrt3$ was already known, the two notions are equivalent for this family. The proof's strategy is to differentiate the relative energy through the heat flow and establish an integrated second-variation inequality; because the integrand's second variation is not pointwise nonnegative, an exact $\\mathrm{SO}(2)\\times\\mathrm{SO}(2)$-invariant two-form is added whose pullback by any gradient integrates to zero. After reducing to diagonal matrices by singular value decomposition, the endpoint $\\gamma=2/\\sqrt3$ yields an explicit quadratic form whose principal minors are nonnegative, and the paper states that the intermediate values follow by interpolation. Theorem 2 generalizes the same mechanism to every homogeneous quartic polynomial with the unique representation $Q(F)=a|F|^4+b|F|^2\\det F+c(\\det F)^2$.","pith_inferences":["If the interpolation step in Lemma 4 is made fully explicit, the proof would reduce the entire interval $|\\gamma|\\le 2/\\sqrt3$ to a finite check at the endpoint, making the result easier to verify computationally.","The same two-form correction is likely adaptable to higher-degree rotationally invariant homogeneous integrands, as Theorem 2 already suggests; the size of the ansatz would grow with the degree.","Assuming the proof stands, the theorem decides previously conflicting numerical reports on the Dacorogna–Marcellini family in favor of quasiconvexity: any numerical violation found in this range would have to be an artifact rather than a genuine counterexample.","The reliance on a global polynomial primitive for the correction connects the method to the theory of null Lagrangians; integrands admitting such corrections may be exactly those for which the heat-flow criterion succeeds."],"forward_implications":["For every $\\gamma$ with $1<|\\gamma|\\le 2/\\sqrt3$, the paper supplies explicit quasiconvex integrands on $\\mathbb{R}^{2\\times2}$ that are not polyconvex.","For the whole Dacorogna–Marcellini family, the rank-one-convexity threshold is the quasiconvexity threshold, so the two classical conditions cannot be separated within this family.","The equivalence carried by the heat-flow argument extends to all homogeneous quartic polynomials on $2\\times2$ matrices invariant under left and right rotations; for those, quasiconvexity is exactly rank-one convexity.","The proof pattern—heat-flow monotonicity plus an exact differential correction—provides a template for other variational integrands whose second-variation density is indefinite."],"supporting_citations":[{"why":"Proves quasiconvexity for $|\\gamma|\\le 1+\\varepsilon$, the partial result this paper extends to the full rank-one-convex range.","marker":"[1]"},{"why":"Establishes the convexity and rank-one-convexity thresholds for the family and supplies the early numerical evidence at the endpoint.","marker":"[6]"},{"why":"Introduces the Dacorogna–Marcellini family as a testing ground for the planar gap between rank-one convexity and quasiconvexity.","marker":"[9]"},{"why":"Supplies the flow-interchange viewpoint that motivates the heat-flow monotonicity criterion used in Lemma 3.","marker":"[18]"},{"why":"Defines polyconvexity and provides the framework for the statement that the interval consists of quasiconvex integrands that are not polyconvex.","marker":"[4]"}],"fun_headline_variants":["Quasiconvexity equals rank-one convexity for Dacorogna–Marcellini","Heat flow yields quasiconvexity criterion for quartic energy","D–M energy: quasiconvex iff |gamma| ≤ 2/√3","Rank-one convexity suffices for quasiconvexity here","Quartic energy threshold: quasiconvexity = rank-one convexity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the heat-flow sufficiency lemma: on the strength of sketched decay and continuity estimates, quasiconvexity is reduced to the integrated second-variation inequality, and the pointwise correction that proves that inequality for all $|\\gamma|\\le 2/\\sqrt3$ is checked in detail only at the endpoint, with the interior range handled by an asserted interpolation.","fun_headline_variants_meta":{"raw":{"variants":["Quasiconvexity equals rank-one convexity for Dacorogna–Marcellini","Heat flow yields quasiconvexity criterion for quartic energy","D–M energy: quasiconvex iff |gamma| ≤ 2/√3","Rank-one convexity suffices for quasiconvexity here","Quartic energy threshold: quasiconvexity = rank-one convexity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000894,"raw_usage":{"total_tokens":3830,"prompt_tokens":901,"completion_tokens":2929,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":517,"completion_tokens_details":{"reasoning_tokens":2828}},"tokens_in":517,"tokens_out":2929,"duration_ms":24526,"temperature":1.0,"reasoning_tokens":2828,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T04:14:12.529375+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check the principal minors of the matrix $M_1(\\rho)$ given in the paper for all real $\\rho$; any $\\rho$ where a minor is negative would contradict Lemma 4 at $\\gamma=2/\\sqrt3$. Alternatively, run a direct numerical test of inequality (1.1) for $f_{2/\\sqrt3}$ at affine maps with oscillatory perturbations concentrated near matrices with distinct singular values.","supporting_citations":[{"cited_title":"Alibert and B","cited_arxiv_id":null,"evidence_quote":"Proves quasiconvexity for $|\\gamma|\\le 1+\\varepsilon$, the partial result this paper extends to the full rank-one-convex range."},{"cited_title":"Dacorogna, J","cited_arxiv_id":null,"evidence_quote":"Establishes the convexity and rank-one-convexity thresholds for the family and supplies the early numerical evidence at the endpoint."},{"cited_title":"Dacorogna and P","cited_arxiv_id":null,"evidence_quote":"Introduces the Dacorogna–Marcellini family as a testing ground for the planar gap between rank-one convexity and quasiconvexity."},{"cited_title":"Matthes, R","cited_arxiv_id":null,"evidence_quote":"Supplies the flow-interchange viewpoint that motivates the heat-flow monotonicity criterion used in Lemma 3."}],"review_version":1}