{"id":"f3626deb-f4c3-4d73-801a-ca3fa9d551d9","arxiv_id":"2608.03488","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"For every p>1 there are p-homogeneous rank-one convex integrands on R^{2xm}, and on large square matrix spaces, that are nowhere quasiconvex; for m large enough and in R^{4x2} when p≠2.","lead":"This paper answers a 70-year-old question in the calculus of variations by constructing, for large numbers of columns m, energy functions on 2 by m matrices that are convex along rank-one lines yet fail the much stronger quasiconvexity condition, even at every point. The result settles the homogeneous version of Morrey's problem in these dimensions and gives new test cases for the still-open 2 by 2 case and Iwaniec's conjecture.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Quasiconvexity lower bound (17) depends on an unproved transfer of the unbounded-domain identity (15) to Dacorogna/torus test maps; if the heavy-tail estimate is lost under periodization/truncation, Theorem 1.2's threshold separation collapses.","rationale":"I read the paper and the reader's verdict in good faith. The construction is internally coherent: Lemma 4.1's rank-one identity is exact, the pass from rank-one Rademacher martingales to the UMD transform is sound (Proposition 4.1 uses Prop 2.7(5) with the dimension-free UMD_{C,p}(ℓ_q)), and the envelope argument of Proposition 2.3 correctly converts rank-one convexity at 0 into a global rank-one convex nowhere-quasiconvex integrand once the threshold separation is granted. I checked the polar-coordinate identity (15) and the order-statistic estimate in Lemma 4.2; both are plausible and the heavy-tail computation is legitimate on the unbounded domain. The single load-bearing gap is the sentence 'By standard limiting arguments, this implies (17)' in Section 4.3. The lower bound C_qc must ultimately be witnessed by test maps on a bounded domain or torus, as in Dacorogna's formula, but the only computation supplied is for the Schwartz map on R^{2n}. The transfer is quite likely routine—truncation and periodization of a rapidly decaying function should preserve the quotient—but it is not written, and the n-dependent exponent n^δ enters exclusively through this step. Since Theorem 1.2 collapses if (17) fails, this is the most load-bearing concern. I agree with the reader's weakest_assumption. The explicit p=3,q=6 minorant in Proposition 4.2 and the elementary zig-zag concavity proof are independent support for the rank-one side, but they do not obviate the lower-bound transfer. The claimed Lean formalisation is self-reported and AI-assisted, so it does not close the gap. Therefore the appropriate verdict remains CONDITIONAL, matching the reader's assessment; no adjustment is needed.","tokens_in":27534,"tokens_out":26428,"duration_ms":293779,"concrete_test":"Let χ be a smooth cutoff with χ=1 on B_K(0) and χ=0 outside B_{2K}, K=10. For n and L>K set w_{n,L}(x)= (χ(x/L)u_n(x))^{per}_L(x) and rescale to the unit torus: v_{n,L}(y)=w_{n,L}(Ly). Compute Q_{n,L}=||Q_n(∇v_{n,L})||_{L^p(T^{2n})}/||P_n(∇v_{n,L})||_{L^p(T^{2n})}. Show lim_{L→∞} Q_{n,L} ≥ c n^δ with c>0 independent of n by isolating the event {min_j |z_j| ≤ c' n^{-1/(p+2)} ∩ {|z|≤K}} and proving the boundary/periodization error is O(e^{-cL^2}). If the limit drops below n^δ/2, the passage (15)→(17) fails. A simpler check: re-derive (17) by explicitly approximating u_n with W^{1,∞}_0(B_R) maps and verify the cutoff term (∇χ_L)u_n is negligible relative to χ_L∇u_n in both P and Q L^p-norms.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim Theorem 1.2 requires C_qc(p,q,n) > C_rc(p,q,n) for large n (Section 4.4). C_rc ≤ UMD_{C,p}(ℓ_q) is dimension-independent and well-sourced. The lower bound on C_qc is the load-bearing part: identity (15) computes the quotient for the Schwartz map u_n on all of R^{2n}; Lemma 4.2 extracts n^δ; then 'By standard limiting arguments, this implies (17)' moves to the torus/ball test maps of Lemma 2.2/Prop 2.2. This transfer is not shown. The issue is specific: the numerator in (15) is dominated by the rare event {min_j R_j ≤ c n^{-1/(p+2)}}, a thin neighbourhood of the coordinate hyperplanes. A truncation/periodization must keep that neighbourhood in the region where the cutoff is 1 and must not introduce boundary terms that reduce the numerator by an n-dependent factor or change the denominator's normalization. Dacorogna's formula uses normalized averages on a fixed domain, so the transfer requires rescaling the periodized function to the unit torus and verifying the quotient is asymptotically preserved. Without this, (17) is unproved, so no C lies strictly between the thresholds and no rank-one convex non-quasiconvex integrand is constructed. This is the single point on which the theorem depends.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper addresses Morrey's problem in the class of p-homogeneous integrands on R^{2×m}. The main theorem (Theorem 1.2) asserts that for every p∈(1,∞) and all m≥m_0(p) there is a continuous p-homogeneous rank-one convex integrand F:R^{2×m}→R that is nowhere quasiconvex. The proof uses a threshold-separation scheme: for projections P_n,Q_n onto ℓ_q^n, define F_{n,p,q,C}=C^p∥P_n A∥^p − ∥Q_n A∥^p. The rank-one convexity threshold is bounded above by a UMD martingale-transform constant (independent of n), while quasiconvexity at 0 is bounded below by the ratio for an explicit Schwartz test map u_n, which grows like n^{1/(p+2)−1/q} for q>p+2. Choosing C between the two thresholds and taking the rank-one convex envelope yields the counterexample. Variants give the square case and dimension 4×2.","tokens_in":27837,"tokens_out":24942,"duration_ms":273110,"significance":"If correct, this is a major advance: it settles the homogeneous Morrey problem negatively in all sufficiently large dimensions (and the ordinary Morrey problem in 2×m for large m), a question open since Morrey's original work. The martingale/UMD method is a new and potentially powerful tool in the calculus of variations, and the paper also contains several by-products (non-invariance of quasiconvexity under transposition, smooth counterexamples, non-locality, an explicit p=3 example, and a claimed Lean formalization). The estimates are parameter-free in the sense that no fitted constants are used: C_rc is controlled by UMD constants and C_qc by an explicit test map. However, the proof as written has a false algebraic identity in Lemma 4.1 and an unproved limiting step from (15) to (17); both are repairable but require non-negligible corrections.","major_comments":[{"comment":"The identity Q_nR = lambda(R)P_nR is false as stated. With R=a⊗ξ, the proof's own formulas give alpha_j = a\\bar v_j/2 and beta_j = a v_j/2, so beta_j = (a/\\bar a) \\overline{alpha_j}, not (a/\\bar a)alpha_j. For a=(1,0), x=(1,0,0,1), P=(1/2,-i/2), Q=(1/2,i/2), no single lambda works. This invalidates the proofs of Proposition 4.1 and Proposition 4.2 as written. The repair is straightforward: replace Q_n by its coordinatewise conjugate (an isometry of ell_q^n) or apply the UMD transform bound to \\overline{P_n(M)}. Please correct the lemma and all dependent statements.","section":"Section 4.2, Lemma 4.1"},{"comment":"The passage from the unbounded-domain identity (15) for the Schwartz function u_n to the lower bound (17) for C_qc is not shown. C_qc is defined via Dacorogna's formula on the unit ball/torus, so one must truncate u_n to a bounded domain, make it zero on the boundary or periodic, rescale to the unit cell, and prove that the quotient converges to the unbounded ratio. This is load-bearing because the numerator is concentrated on the thin event {min_j R_j <= c n^{-1/(p+2)}}; the cutoff must be identically 1 on that event and the error must be small compared to n^{delta}. Please provide the full limiting argument; without it (17) is unsupported.","section":"Section 4.3, (15) to (17)"}],"minor_comments":[{"comment":"The stated bounds for q=p+2 and q=∞ are given without proof. If they are not used in the main theorems, label them as conjectures or provide sketches; otherwise a reader cannot verify them.","section":"Section 4.3, Remark 4.2"},{"comment":"The statement 'quasiconvexity at any point implies quasiconvexity at 0' for homogeneous integrands is used to conclude 'nowhere quasiconvex' in Theorem 1.2. A short proof or reference would improve readability; the argument via the finite-valued homogeneous quasiconvex envelope is not spelled out.","section":"Section 1.3 / Proposition 2.3"},{"comment":"After the conjugation fix in Lemma 4.1, the phrase 'thanks to Lemma 4.1 it suffices to prove that U is zig-zag concave' should be updated: the rank-one variation becomes (x+th, y+tεh) with ε = \\bar λ, not ε=λ. This is not merely notational.","section":"Section 4.5, Proposition 4.2"},{"comment":"There are a few typographical issues (e.g., the citation of Lemma 2.2 contains a stray 'and'; the acknowledgements contain a corrupted name). These do not affect the mathematics.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is promising and the overall strategy is sound, but I found a false algebraic identity in Lemma 4.1 that the accompanying reader's report missed. The author should also provide the missing periodization argument in §4.3. I recommend major revision rather than rejection because both issues are repairable without changing the architecture of the proof."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a serious and genuinely new attack on a long-open problem. The architecture is elegant: separate the rank-one threshold, bounded by UMD martingale-transform constants, from the quasiconvexity threshold, which grows with dimension via heavy-tailed test maps. I checked the core algebra and it is consistent. Lemma 4.1, the Wirtinger computations, and the order-statistic argument all hold up. The explicit p=3, q=6 section is a nice sanity check, and the square and 4x2 variants add real corroboration.\n\nBut the paper has one load-bearing gap, and the stress-test note is right on target. The lower bound (17) on C_qc is inferred from identity (15) for the Schwartz map u_n on all of R^{2n} \"by standard limiting arguments.\" That step is not standard in the way the authors imply. Dacorogna's formula uses functions on a fixed ball or torus with normalized averages. You have to truncate or periodize u_n without losing the rare-event contribution near the coordinate hyperplanes, and the scaling factors do not cancel automatically. If the heavy-tail estimate is lost, no constant C sits between the two thresholds and Theorem 1.2 collapses. I don't see a contradiction, but I do see an essential step that is currently missing.\n\nThe UMD upper bound is well sourced and dimension-independent; the martingale-laminate bridge is sketched but plausible. Less central: the Lean formalization and AI-assistance disclosure are external, so I'm not weighting them, and m_0 is left implicit. These are minor relative to (17).\n\nWho is this for? Specialists in the calculus of variations working on Morrey's problem and quasiconvexity. If the gap gets filled, this is a big result. Right now it is a promising manuscript with a named hole.\n\nRecommendation: send to peer review, with instructions to the referee to focus on the passage from (15) to (17). It deserves referee time, not a desk rejection, but I wouldn't cite it yet.","headline":"A genuinely new approach to Morrey's problem, but the key lower bound (17) has a load-bearing gap in the passage from R^{2n} to the torus/ball; worth refereeing, not yet citable.","tokens_in":28452,"tokens_out":3323,"would_cite":false,"duration_ms":40130,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["49J45","46B09","35A15"],"pacs":[],"model":"deepseek-v4-flash","headline":"For all large m, there exist p-homogeneous rank-one convex integrands on 2×m matrices that are nowhere quasiconvex.","keywords":["rank-one convexity","quasiconvexity","Morrey's problem","homogeneous integrands","calculus of variations","UMD spaces","martingale transforms","laminates"],"falsifier":"Take a torus-periodic or ball-truncated version of u_n and compute the ratio ‖Q_n(∇u)‖_{L^p(ℓ_q^n)} / ‖P_n(∇u)‖_{L^p(ℓ_q^n)} for increasing n with q>p+2. If the ratio does not grow like n^{1/(p+2)−1/q}, the gap underpinning Theorem 1.2 collapses; if it does grow, the proof's limiting step is concretely justified.","tokens_in":27328,"feed_emoji":"📐","tokens_out":11922,"duration_ms":128412,"temperature":0.7,"pith_summary":"Quasiconvexity is the convexity condition behind weak lower semicontinuity and existence of minimisers in the calculus of variations; rank-one convexity is a necessary and much weaker condition. This paper proves that, in the space of 2×m matrices with m large, rank-one convexity does not imply quasiconvexity even among positively p-homogeneous integrands: for every p in (1,∞) and all large m there is a p-homogeneous rank-one convex F that is not quasiconvex at any matrix. The construction separates two thresholds, one for rank-one convexity and one for quasiconvexity, and shows the second exceeds the first by a growing power of the dimension. Since the envelope of the constructed integrand remains rank-one convex and p-homogeneous, the failure is inherited everywhere, giving a negative answer to the homogeneous Morrey problem in 2×m for large m, and hence to the original problem there. The same scheme also settles the square case for all large d and the 4×2 case for every p different from 2.","feed_headline":"Rank-one convexity fails to imply quasiconvexity in large 2×m","feed_subtitle":"The failure occurs even among p-homogeneous integrands, settling the homogeneous Morrey problem there.","key_machinery":"The carrying object is the blockwise holomorphic/anti-holomorphic pair (P_n,Q_n): each 2×2 block of a 2×2n matrix is split into a component that respects complex multiplication and a component that conjugates, with values in ℓ_q^n. The argument hinges on the exact rank-one identity Q_n(R)=λ(R)P_n(R) for a unimodular scalar λ(R), which converts the rank-one constraint along martingale increments into a complex unimodular martingale transform. UMD-space estimates—bounds on taking predictable ±1 or unimodular transforms of martingales—then control the rank-one convexity threshold by a constant independent of n. On the quasiconvexity side, the explicit test map u_n(z)=(∏_{j=1}^n ar z_j)e^{-|z|^2","core_discovery":"The central claim is Theorem 1.2: take p∈(1,∞); there is m0(p) such that for all m≥m0(p) there is an integrand f:R^{2×m}→R that is p-homogeneous, rank-one convex, and nowhere quasiconvex. The witness is built from f_{C,p}(A)=C^p‖P_n(A)‖_{ℓ_q^n}^p − ‖Q_n(A)‖_{ℓ_q^n}^p, where P_n and Q_n extract, blockwise on 2×2n matrices, the complex-linear and conjugate-linear parts. A rank-one identity makes Q_n(M) a unimodular predictable transform of P_n(M) along any rank-one martingale, so the rank-one convexity threshold is bounded above by a UMD constant independent of n; an explicit smooth rapidly decaying test map gives the opposite threshold growing like n^{1/(p+2)−1/q} when q>p+2. Choosing q>p+2 a","pith_inferences":["Editorial extension: since the separation mechanism needs q>p+2, the proof cannot be pushed to the Hilbertian 2×2 case; this suggests that any proof of a positive result there must exploit structure absent from the ℓ_q spaces used here.","Editorial extension: the threshold-separation template may be tested on other pairs of linear maps (P,Q); the only needed ingredients are a rank-one phase identity and a family of test maps whose ratio grows with n, so one can search for counterexamples below m0(p) by checking those two ingredients.","Editorial extension: the paper leaves m0(p) qualitative; a numerical experiment at moderate n, comparing the explicit heavy-tail lower bound with the best available UMD constants for ℓ_q^n, could locate the first separating dimension and support or refute the paper's suggestion that m0=4 may be reachable."],"forward_implications":["In R^{2×m} with m large, the homogeneous Morrey problem has a negative answer for every p>1, and hence the original Morrey problem is also settled negatively there.","The same threshold-separation scheme gives, for all large d, a conjugation- and transposition-invariant p-homogeneous rank-one convex integrand on R^{d×d} that is nowhere quasiconvex.","In dimension 4×2, for every p∈(1,∞) with p≠2, the problem fails as well, so non-Hilbertian geometry alone can produce counterexamples in a fixed low dimension.","In 2×m for large m there exist smooth, coercive integrands satisfying a uniform Legendre–Hadamard inequality that are nevertheless not quasiconvex at 0; consequently quasiconvexity is a nonlocal condition in this setting.","In the square case with p=2 and q=4, a fourth-degree homogeneous polynomial gives an explicit rank-one convex but not quasiconvex integrand, and degree three is the lowest degree at which such a phenomenon can occur."],"supporting_citations":[{"why":"Provides the baseline real-valued rank-one convex but not quasiconvex integrand in 3×2, the original model for Morrey-problem counterexamples.","marker":"[70]"},{"why":"Gives the earlier 2-homogeneous rank-one convex example whose quasiconvexity fails away from zero; frames the homogeneous Morrey problem.","marker":"[35]"},{"why":"Establishes the rank-one Rademacher martingale representation of dyadic prelaminates used in Proposition 2.5.","marker":"[19]"},{"why":"Supplies the UMD-space toolkit: dyadic martingale reductions, decoupling, and transference arguments that control the rank-one convexity threshold.","marker":"[42]"},{"why":"Provides the complex unimodular martingale transform bound used to make the rank-one convexity threshold independent of n.","marker":"[73]"},{"why":"Supplies the UMD estimate for Schatten spaces used in the square-case rank-one convexity bound.","marker":"[67]"},{"why":"Gives the sharp lower bound for powers of the Beurling–Ahlfors operator used in the 4×2 construction.","marker":"[28]"},{"why":"Provides the scalar zig-zag function used in the explicit p=3, q=6 rank-one convexity proof.","marker":"[15]"}],"fun_headline_variants":["Nowhere quasiconvex rank-one convex integrands for large 2×m","Homogeneous rank-one convex fails to imply quasiconvex for large 2×m","Morrey's problem solved: rank-one convex not quasiconvex for large 2×m","Rank-one convex, nowhere quasiconvex integrands for large 2×m"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"Everything rests on the unshown step that the large-ratio calculation made on the whole plane, with the smooth test map u_n, survives passage to the torus or unit ball in the quasiconvexity formula; the paper invokes 'standard limiting arguments' without proving that the heavy-tail growth is preserved by periodization or truncation.","fun_headline_variants_meta":{"raw":{"variants":["Nowhere quasiconvex rank-one convex integrands for large 2×m","Homogeneous rank-one convex fails to imply quasiconvex for large 2×m","Morrey's problem solved: rank-one convex not quasiconvex for large 2×m","Rank-one convex, nowhere quasiconvex integrands for large 2×m"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001167,"raw_usage":{"total_tokens":4603,"prompt_tokens":618,"completion_tokens":3985,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":362,"completion_tokens_details":{"reasoning_tokens":3894}},"tokens_in":362,"tokens_out":3985,"duration_ms":32531,"temperature":1.0,"reasoning_tokens":3894,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T18:07:44.696750+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a torus-periodic or ball-truncated version of u_n and compute the ratio ‖Q_n(∇u)‖_{L^p(ℓ_q^n)} / ‖P_n(∇u)‖_{L^p(ℓ_q^n)} for increasing n with q>p+2. If the ratio does not grow like n^{1/(p+2)−1/q}, the gap underpinning Theorem 1.2 collapses; if it does grow, the proof's limiting step is concretely justified.","supporting_citations":[{"cited_title":"Rank-one convexity does not imply quasiconvexity,","cited_arxiv_id":null,"evidence_quote":"Provides the baseline real-valued rank-one convex but not quasiconvex integrand in 3×2, the original model for Morrey-problem counterexamples."},{"cited_title":"From Microstructure-Independent Formulas for Composite Materials to Rank-One Convex, Non- quasiconvex Functions,","cited_arxiv_id":null,"evidence_quote":"Gives the earlier 2-homogeneous rank-one convex example whose quasiconvexity fails away from zero; frames the homogeneous Morrey problem."},{"cited_title":"Martingales, laminates and minimal Korn inequalities,","cited_arxiv_id":null,"evidence_quote":"Establishes the rank-one Rademacher martingale representation of dyadic prelaminates used in Proposition 2.5."},{"cited_title":"Hytönen, J","cited_arxiv_id":null,"evidence_quote":"Supplies the UMD-space toolkit: dyadic martingale reductions, decoupling, and transference arguments that control the rank-one convexity threshold."},{"cited_title":"Even fourier multipliers and martingale transforms in infinite dimensions,","cited_arxiv_id":null,"evidence_quote":"Provides the complex unimodular martingale transform bound used to make the rank-one convexity threshold independent of n."},{"cited_title":"Non-commutative martingale transforms,","cited_arxiv_id":null,"evidence_quote":"Supplies the UMD estimate for Schatten spaces used in the square-case rank-one convexity bound."},{"cited_title":"Some remarks on theLp estimates for powers of the Ahlfors–Beurling operator,","cited_arxiv_id":null,"evidence_quote":"Gives the sharp lower bound for powers of the Beurling–Ahlfors operator used in the 4×2 construction."},{"cited_title":"Sharp inequalities for martingales and stochastic integrals,","cited_arxiv_id":null,"evidence_quote":"Provides the scalar zig-zag function used in the explicit p=3, q=6 rank-one convexity proof."}],"review_version":1}