{"id":"207cf2c6-2e48-4768-a00a-a6eee3b39f4a","arxiv_id":"2606.22202","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Derives complementary potentials for 1D nonlocal integral formulations via Legendre transform, proves equivalence of strain-driven and stress-driven pairs, and shows a mixed potential is ill-conditioned like the strain-driven version.","lead":"The paper derives complementary potentials for one-dimensional nonlocal strain-driven and stress-driven integral formulations using the Legendre transformation and proves their equivalence. A smart generalist might read it to see how dual energy formulations could aid numerical work in nonlocal material models.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Legendre transform requires unstated convexity/regularity on nonlocal kernels for equivalence to hold","rationale":"The reader's weakest_assumption correctly isolates the missing regularity conditions required for the Legendre step. This is the single load-bearing point because all subsequent claims (equivalence, mixed potential, example confirmation) rest on the transform being valid; without explicit verification the argument is formally incomplete even if it holds in the specific cases considered.","tokens_in":1647,"tokens_out":317,"duration_ms":12591,"concrete_test":"In the section proving equivalence via Legendre transform, extract the precise assumptions on the kernel and potential; if convexity is not invoked, substitute a non-convex quadratic strain energy into the 1D integral formulation and recompute both the potential-based and complementary-potential-based stress-driven solutions to check whether they coincide.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim is that Legendre transformation produces well-defined complementary potentials for the 1D nonlocal integral strain-driven and stress-driven formulations, with equivalence proved and a mixed stress-strain potential derived. This requires the underlying energy functionals (involving integral operators with kernels) to be convex and sufficiently regular so that the transform is bijective and the resulting complementary formulations are equivalent. The paper does not state or verify these conditions on the kernels (e.g., positive-definiteness, coercivity) or potentials; nonlocal integral operators can violate strict convexity or introduce non-local singularities that break invertibility of the transform, rendering the equivalence proof conditional rather than general.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper claims to introduce complementary potentials for the one-dimensional nonlocal integral strain-driven and stress-driven formulations (the strain-driven potential being already known), obtain equivalent formulations via the Legendre transformation with a proof of equivalence, postulate a novel mixed stress-strain potential (noted as ill-conditioned), and provide an example confirming that the stress-driven formulations derived from the potential and complementary potential are equivalent.","tokens_in":1776,"tokens_out":487,"duration_ms":13403,"significance":"If the central derivations hold with the required regularity conditions made explicit and verified, the work would supply previously missing complementary potentials for these nonlocal integral models and demonstrate their equivalence, along with a mixed potential. This could strengthen the variational framework for nonlocal continuum mechanics in one dimension. The example provides concrete verification of the stress-driven case.","major_comments":[{"comment":"The central claim relies on the Legendre transformation yielding well-defined complementary potentials and preserving equivalence between strain-driven and stress-driven formulations. However, no conditions (e.g., convexity, coercivity, or positive-definiteness of the nonlocal kernels or energy functionals) are stated or verified to ensure the transform is bijective and the resulting potentials are equivalent; nonlocal integral operators can violate these in general, making the equivalence conditional rather than general. This is load-bearing for the proof and the mixed-potential construction.","section":"Legendre transformation and equivalence proof (abstract and main derivations)"},{"comment":"The example is said to 'practically confirm' equivalence of the stress-driven formulations, but without quantitative metrics (e.g., error norms between solutions from potential vs. complementary potential, or explicit kernel definitions), it is unclear whether it tests the general case or only a special convex instance.","section":"Example section"}],"minor_comments":[{"comment":"The abstract states that 'the complementary potential has not yet been presented in the literature' for the strain-driven case and similarly for stress-driven; explicit citations to prior work on nonlocal potentials would clarify the novelty.","section":"Abstract"},{"comment":"Notation for the nonlocal integral operators and kernels should be introduced with explicit definitions early in the manuscript to aid readability of the potential expressions.","section":"Introduction and formulation sections"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the constructive comments on our manuscript. The points raised highlight areas where explicit assumptions and quantitative verification can strengthen the presentation. We will revise the manuscript accordingly and address each major comment below.","responses":[{"response":"We agree that the regularity conditions required for the Legendre transform to be bijective should be stated explicitly. In the revised manuscript we will insert a new preliminary subsection (before the main derivations) that lists the standing assumptions: convexity and coercivity of the strain energy functional, positive-definiteness of the nonlocal kernel, and sufficient smoothness to guarantee the existence of the inverse operator. Under these hypotheses the equivalence proof already given in the paper holds; the revision will simply make the hypotheses visible rather than implicit. We do not claim the result is unconditional, and the added statement will clarify the scope.","revision_made":"yes","referee_comment":"[Legendre transformation and equivalence proof (abstract and main derivations)] The central claim relies on the Legendre transformation yielding well-defined complementary potentials and preserving equivalence between strain-driven and stress-driven formulations. However, no conditions (e.g., convexity, coercivity, or positive-definiteness of the nonlocal kernels or energy functionals) are stated or verified to ensure the transform is bijective and the resulting potentials are equivalent; nonlocal integral operators can violate these in general, making the equivalence conditional rather than general. This is load-bearing for the proof and the mixed-potential construction."},{"response":"We accept that the example would be more convincing with quantitative measures. In the revision we will (i) state the explicit kernel (a standard exponential kernel with given length-scale parameter) and (ii) report L2-norm differences between the displacement fields obtained from the potential-based and complementary-potential-based stress-driven formulations. These norms will be shown to be on the order of machine precision, confirming numerical equivalence for the chosen convex instance that satisfies the assumptions listed in the new preliminary subsection.","revision_made":"yes","referee_comment":"[Example section] The example is said to 'practically confirm' equivalence of the stress-driven formulations, but without quantitative metrics (e.g., error norms between solutions from potential vs. complementary potential, or explicit kernel definitions), it is unclear whether it tests the general case or only a special convex instance."}],"tokens_in":1285,"tokens_out":496,"duration_ms":17100,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main thing here is that the authors derive the complementary potentials for the strain-driven and stress-driven one-dimensional nonlocal integral formulations by applying the Legendre transformation. They show the resulting formulations are equivalent and back that up with an example that confirms the stress-driven versions match. They also define a mixed stress-strain potential from the same results, though they note it inherits the ill-conditioning of the pure strain-driven case.\n\nWhat is actually new is the explicit construction of those complementary potentials, which the abstract says had not appeared before for these nonlocal setups. The example provides a concrete check that the two routes to the stress-driven formulation agree, which is useful for seeing the math work in practice.\n\nThe soft spot is that the Legendre transform requires the underlying functionals to be convex and the nonlocal kernels to satisfy regularity conditions that make the transform bijective. The provided abstract and stress-test note give no indication that these conditions are stated or verified for the kernels. Nonlocal integral operators can fail strict convexity or introduce singularities that break the needed invertibility, so the equivalence claim may hold only under assumptions that are not laid out. If the full text does not close this gap, the result stays conditional.\n\nThis is for researchers working on variational principles in nonlocal continuum mechanics, especially those focused on one-dimensional integral formulations. A reader already familiar with strain-driven and stress-driven models would get value from the derivations and the example.\n\nI would send it to peer review. The core extension is narrow but fills a stated gap, and the example adds a practical check even if the regularity conditions need more attention in revision.","headline":"The paper derives complementary potentials for 1D nonlocal integral models via Legendre transform, proves equivalence, and checks it with an example, but leaves the needed convexity and kernel regularity conditions unstated.","tokens_in":2251,"tokens_out":405,"would_cite":false,"duration_ms":26646,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Legendre transformation produces equivalent potentials and complementary potentials for one-dimensional nonlocal strain-driven and stress-driven formulations.","keywords":["nonlocal integral formulations","potentials","complementary potentials","Legendre transformation","strain-driven","stress-driven","one-dimensional"],"falsifier":"A boundary-value problem in which the total energy computed from the potential differs from the energy computed from the complementary potential after the Legendre transformation is applied.","tokens_in":2546,"feed_emoji":"🔄","tokens_out":633,"duration_ms":21823,"temperature":0.7,"pith_summary":"The paper shows how to construct complementary potentials for both the pure strain-driven and pure stress-driven nonlocal integral models in one dimension, using the Legendre transformation to establish their equivalence. This supplies the missing variational structure for the stress-driven case and confirms that the two formulations are interchangeable at the level of potentials. The same transformation also permits definition of a mixed stress-strain potential, although the authors demonstrate that this mixed version inherits the same ill-conditioning already known for the pure strain-driven version. An explicit example is solved to verify that the stress-driven model derived from the potential matches the one derived from the complementary potential.","feed_headline":"Legendre transform equates nonlocal strain and stress potentials","feed_subtitle":"It proves equivalence between strain-driven and stress-driven one-dimensional models and shows a mixed potential remains equally ill-conditi","key_machinery":"Legendre transformation applied to the nonlocal integral operators, which converts the known strain-driven potential into a complementary potential for the stress-driven formulation while preserving equivalence.","core_discovery":"The equivalent formulations are obtained by resorting to the Legendre transformation, and their equivalence is proved. It is also shown that these results can be used to postulate a novel potential, i.e. a kind of mixed stress-strain potential, which is, however, as ill-conditioned as the pure strain-driven formulation. Finally, an example is given that practically confirms that the stress-driven formulations resulting from the potential and the complementary potential are equivalent.","pith_inferences":["The equivalence may allow modelers to switch between strain-driven and stress-driven descriptions without changing the underlying energy functional.","Extension of the same Legendre construction to two or three dimensions would supply complementary potentials for nonlocal continuum models in higher dimensions.","The persistent ill-conditioning of the mixed potential indicates that any practical implementation would still require additional regularization or stabilization techniques."],"forward_implications":["Strain-driven and stress-driven nonlocal formulations become variationally equivalent once their potentials and complementary potentials are related by the Legendre transformation.","A mixed stress-strain potential can be defined directly from the transformed quantities.","The mixed potential remains ill-conditioned to the same degree as the pure strain-driven potential.","Numerical solutions of stress-driven problems obtained from either the potential or the complementary potential coincide."],"fun_headline_variants":["Nonlocal potentials equated by Legendre transform","Equivalence shown for strain and stress nonlocal models","Mixed stress-strain potential as ill-conditioned as strain-driven","Example confirms stress-driven nonlocal formulations equivalent"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The nonlocal integral operators and associated potentials admit a Legendre transformation that yields well-defined complementary potentials and preserves equivalence.","fun_headline_variants_meta":{"raw":{"variants":["Nonlocal potentials equated by Legendre transform","Equivalence shown for strain and stress nonlocal models","Mixed stress-strain potential as ill-conditioned as strain-driven","Example confirms stress-driven nonlocal formulations equivalent"]},"model":"grok-4.3","cost_usd":0.004386,"raw_usage":{"total_tokens":2153,"prompt_tokens":582,"num_sources_used":0,"completion_tokens":54,"cost_in_usd_ticks":43862000,"prompt_tokens_details":{"text_tokens":582,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1517,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":582,"tokens_out":54,"duration_ms":10916,"temperature":1.0,"reasoning_tokens":1517,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-26T11:09:41.624248+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A boundary-value problem in which the total energy computed from the potential differs from the energy computed from the complementary potential after the Legendre transformation is applied.","supporting_citations":[],"review_version":1}