{"id":"ee314259-d4f2-4c9e-a7af-f66e609a3fbf","arxiv_id":"2607.06568","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.5,"correctness_risk":"low","formal_verification":"none","parameter_count":1,"one_line_summary":"Polyconvexity does not imply true-stress-true-strain monotonicity for incompressible isotropic hyperelasticity in three dimensions, via an explicit softplus-based counterexample.","lead":"A constructed hyperelastic energy is polyconvex yet fails true-stress-true-strain monotonicity under uniaxial tension. This shows polyconvexity alone cannot guarantee the expected stress-strain response of idealized incompressible elastic solids in 3D.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader correctly isolates the dependence on the Wiedemann–Peter/Geuken characterization as the only non-elementary ingredient. That characterization is used exactly as stated (linearity of \theta + convexity of softplus \to convexity of each summand; sign-flip symmetries built into (2.1) give \\Pi_{3}-invariance). No internal inconsistency, hidden regularity assumption, or numerical ambiguity appears in the stress or energy calculations. The concrete verification above merely reconfirms the already-closed-form non-monotonicity and does not alter the logical status of the counter-example. Consequently the ACCEPT verdict stands with high confidence.","tokens_in":10049,"tokens_out":563,"duration_ms":4814,"concrete_test":"Independently recompute the uniaxial Cauchy-stress formula (2.7) for \nu=(2.0,2^{-1/2},2^{-1/2}) and \nu=(2.5,2.5^{-1/2},2.5^{-1/2}) with the given softplus/sigmoid derivatives; confirm that \rho(2.0)>\rho(2.5). Simultaneously evaluate the Hessian of the reduced energy Ŵ_inc_red(log \nu1,log \nu2) on the same ray and verify that it possesses a negative eigenvalue, thereby confirming loss of TSTS-M by both routes.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (Corollary 2.3) is that polyconvexity does not imply TSTS-M for isotropic incompressible hyperelasticity in 3D. The paper supplies an explicit counter-example: the softplus-based potential (2.1) with parameters (a,b,c)=(-5,-14,-22). Theorem 2.1 asserts polyconvexity for every real (a,b,c) by verifying the three necessary-and-sufficient conditions of Wiedemann–Peter (2026)/Geuken et al. (2026): convexity of the map in the six signed singular values (because softplus is convex and \theta is linear), \\Pi_{3}-invariance by construction, and lower semi-continuity. Theorem 2.2 then exhibits an explicit closed-form uniaxial Cauchy stress (2.7) that is non-monotone (numerical values \rho(1.5)\to\rho(2.5) decrease). Both steps are elementary once the cited characterization is granted; the softplus construction introduces no extra analytic gap. The only residual dependence is on a published characterization that the authors apply correctly; that dependence is not a soft spot internal to the present argument.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The manuscript constructs an explicit isotropic incompressible hyperelastic potential (2.1) based on the softplus function of a linear combination of signed singular values and their products. Theorem 2.1 asserts that this potential is polyconvex for every real triple (a,b,c) by verifying the three necessary-and-sufficient conditions of Wiedemann & Peter (2026) / Geuken et al. (2026): convexity in the six signed-singular-value arguments, Π_{3}-invariance by construction, and lower semi-continuity. For the concrete coefficients (a,b,c)=(−5,−14,−22) the closed-form uniaxial Cauchy stress (2.7) is shown to be non-monotone (numerical values σ(1.5)≈14.5>σ(2.5)≈12.3), so TSTS-M fails (Theorem 2.2). The same potential remains rank-one convex (hence LH-elliptic) and produces a monotone true shear stress in simple shear. The two corollaries therefore establish that neither polyconvexity nor rank-one convexity implies TSTS-M in the three-dimensional incompressible setting, completing the diagram of constitutive implications summarized in Figure 1.","tokens_in":10331,"tokens_out":793,"duration_ms":6901,"significance":"The result closes a previously open logical gap in the hierarchy of constitutive inequalities for idealized isotropic incompressible elasticity. Earlier work had shown that Ball’s sufficient conditions for polyconvexity already imply TSTS-M, and that the implication fails in the compressible three-dimensional case; the present counter-example demonstrates that the implication also fails once the full (necessary-and-sufficient) characterization of polyconvexity is admitted. The construction is elementary once the cited characterization is granted, supplies closed-form stress expressions, and is immediately usable as a test case for numerical schemes that rely solely on polyconvexity. The paper therefore supplies a clean, falsifiable negative answer that will be of lasting reference value.","major_comments":[],"minor_comments":[{"comment":"In the sentence preceding Corollary 2.3 the word “satisfsy” is misspelled; correct to “satisfy”.","section":null},{"comment":"Figure 3 caption and the surrounding text refer to “ρ” in places where the uniaxial Cauchy stress is denoted σ; unify the notation.","section":null},{"comment":"The asymptotic value τ\to28 of the shear stress is stated after (2.5); a one-line derivation of this limit (using the known limits of the sigmoid) would make the claim self-contained.","section":null},{"comment":"References to the authors’ own concurrent preprints (Wollner et al. 2026a,b; Klein et al. 2026a,b) are numerous; a short clarifying sentence that the present counter-example is independent of those works would help the reader.","section":null}],"recommendation":"accept","confidential_remarks":"The central claim rests on a characterization that is itself still in preprint form (Wiedemann & Peter 2026; Geuken et al. 2026). The application of that characterization appears correct, and the softplus construction introduces no additional analytic gap, so I do not regard the dependence as a reason for revision. The journal may nevertheless wish to confirm that the cited characterization has been accepted or is otherwise regarded as stable before final publication."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This paper settles the last open arrow in the authors’ own diagram: polyconvexity does not force true-stress-true-strain monotonicity for isotropic incompressible hyperelasticity in three dimensions. The construction is short and explicit—an isotropic softplus potential built from linear combinations of the signed singular values and their products, with parameters (a,b,c)=(-5,-14,-22). Theorem 2.1 verifies the three necessary-and-sufficient conditions of Wiedemann–Peter / Geuken et al. (convexity in the six arguments because softplus is convex and θ is linear, Π3-invariance by construction, lower semi-continuity). Theorem 2.2 then hands you a closed-form uniaxial Cauchy stress that decreases between λ=1.5 and λ=2.5. Because polyconvexity already implies rank-one convexity, the same example also kills the rank-one-convexity → TSTS-M implication.\n\nWhat works well is the economy. Everything is elementary once the external characterization is granted; the stress formulas are closed-form; the shear response remains monotone as required by LH-ellipticity; and the parameters are chosen by hand solely to produce a visible violation, not fitted to anything. The self-citations are to the authors’ earlier partial results that left this exact gap open, so the circularity burden is essentially zero.\n\nThe only external dependence is on the Wiedemann–Peter / Geuken characterization. If that characterization later turns out to have a gap for this particular softplus family, the polyconvexity claim collapses. That is a real but external risk, not an internal soft spot in the present argument. No other technical holes appear.\n\nThis is for people who care about the precise logical relations among constitutive inequalities in finite elasticity. It is short, reproducible from the given expressions, and finishes a clean piece of the map. I would send it to a serious referee without hesitation; the result is solid enough that the only real question is whether the journal wants this level of technical clarification.","headline":"Clean, explicit counterexample that finally separates polyconvexity from TSTS-M (and rank-one convexity from TSTS-M) in incompressible 3-D hyperelasticity.","tokens_in":10925,"tokens_out":518,"would_cite":true,"duration_ms":5130,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.5","headline":"Polyconvexity alone does not force true Cauchy stress to rise with true Hencky strain for incompressible three-dimensional hyperelasticity.","keywords":["hyperelasticity","incompressibility","polyconvexity","true-stress-true-strain monotonicity","Hill’s inequality","Legendre-Hadamard ellipticity","rank-one convexity","constitutive inequalities"],"falsifier":"Direct numerical evaluation of the uniaxial Cauchy stress (2.7) at the two stretches 1.5 and 2.5 for the stated parameters: if the stress at 1.5 is not strictly larger than the stress at 2.5, or if an independent check shows that the energy fails to be polyconvex, the claim collapses.","tokens_in":10975,"feed_emoji":"📐","tokens_out":666,"duration_ms":6069,"temperature":0.7,"pith_summary":"The paper asks which constitutive inequalities are needed for idealized isotropic hyperelasticity when the material is constrained to be incompressible in three dimensions. Polyconvexity is known to give existence theorems and real wave speeds, while true-stress-true-strain monotonicity (TSTS-M) is the multi-axial requirement that the Cauchy stress must increase with the Hencky strain. Earlier work had shown that Ball’s sufficient conditions for polyconvexity already imply TSTS-M; the open question was whether every polyconvex energy does the same. The authors settle the question negatively: they construct an explicit isotropic energy built from softplus functions of the signed singular values that is polyconvex for every real choice of coefficients, yet produces a non-monotone uniaxial Cauchy stress. Consequently polyconvexity (and the weaker rank-one convexity it implies) is not by itself enough to guarantee a physically reasonable stress-strain response.","feed_headline":"Polyconvexity fails to force monotone true stress","feed_subtitle":"An explicit softplus energy is polyconvex yet produces falling uniaxial Cauchy stress","key_machinery":"The isotropic softplus potential (2.1) written as a sum of four softplus functions of linear forms in the signed singular values and their products; its polyconvexity follows at once from the necessary-and-sufficient characterization of Wiedemann & Peter, while its loss of TSTS-M is read off from the closed-form uniaxial stress formula (2.7).","core_discovery":"In the incompressible three-dimensional setting, polyconvexity does not imply true-stress-true-strain monotonicity. An explicit isotropic potential that is polyconvex for all real parameters nevertheless yields a Cauchy stress that decreases over an interval of uniaxial stretch, violating TSTS-M (and therefore also Hill’s inequality).","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Polyconvexity fails to imply true-stress-true-strain monotonicity","Explicit polyconvex energy yields falling uniaxial Cauchy stress","Polyconvexity alone cannot force TSTS-M in 3D incompressible elasticity","Softplus polyconvex potential violates true-stress monotonicity","Counterexample: polyconvexity does not ensure monotone Cauchy stress"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The proof that the softplus construction is polyconvex rests entirely on the recent necessary-and-sufficient characterization of isotropic polyconvexity in terms of signed singular values; if that characterization has a gap for this particular family, the counter-example fails.","fun_headline_variants_meta":{"raw":{"variants":["Polyconvexity fails to imply true-stress-true-strain monotonicity","Explicit polyconvex energy yields falling uniaxial Cauchy stress","Polyconvexity alone cannot force TSTS-M in 3D incompressible elasticity","Softplus polyconvex potential violates true-stress monotonicity","Counterexample: polyconvexity does not ensure monotone Cauchy stress"]},"model":"grok-4.5","effort":"low","cost_usd":0.006234,"raw_usage":{"total_tokens":1489,"prompt_tokens":579,"num_sources_used":0,"completion_tokens":93,"cost_in_usd_ticks":62340000,"prompt_tokens_details":{"text_tokens":579,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":817,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":579,"tokens_out":93,"duration_ms":6832,"temperature":1.0,"reasoning_tokens":817,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-12T14:15:57.882203+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Direct numerical evaluation of the uniaxial Cauchy stress (2.7) at the two stretches 1.5 and 2.5 for the stated parameters: if the stress at 1.5 is not strictly larger than the stress at 2.5, or if an independent check shows that the energy fails to be polyconvex, the claim collapses.","supporting_citations":[],"review_version":1}