{"id":"e0944041-d6a6-43dd-b5a5-012dcc1d3e33","arxiv_id":"2607.29521","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"An Eulerian Poisson-bracket formulation for solids generates nonlinear terms absent in the Lagrangian frame and reproduces the odd elastic modulus of chiral active solids.","lead":"This paper develops a Poisson-bracket formalism for elastic solids written directly in the Eulerian (deformed-space) frame, although the stored energy is naturally Lagrangian. It shows that this route generates nonlinear terms that must be kept in large-deformation regimes, and uses it to reproduce the odd elastic modulus of chiral active solids.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (27)'s adiabatic elimination of θ via ∂H/∂θ=0 is the load-bearing step for K_o=τ/4; if θ's relaxation is not fast, the claimed stress-strain tensor becomes frequency-dependent, so the central quantitative claim needs a validity range.","rationale":"The paper's central methodological claim is that the Eulerian PB formalism, applied directly to a Hamiltonian expressed in Eulerian variables, reproduces the real-space Cauchy stress and odd elastic modulus K_o=τ/4 that the authors previously obtained via the Lagrangian formalism plus stress transformation. This is a consistency claim, and the algebraic derivation in Secs. 2.2–2.3 and Appendix C appears internally consistent: the Eulerian PBs generate the expected streaming terms, the nonlinear potential transformation changes coefficients as stated, and the cancellation of the antisymmetric ε_ij term in the stress follows from combining κ_c ε_ij ϕ with the −(λ̃+μ̃)ε_ijδ_kl ∇_l u_k term. The final elasticity tensor therefore has the claimed form. The reader's weakest assumption correctly identifies the adiabatic elimination of θ as the decisive physical step. Equation (27) is obtained by setting ℓ≈0 and δH/δθ=0 in Eq. (26), which assumes a clean separation between the fast rotational relaxation and the slow displacement dynamics. The paper states that this is typical for internal degrees of freedom and cites related work, but it does not derive or bound this timescale for the specific chiral active solid model. If the assumption fails, the stress-strain response is not purely elastic with K_o=τ/4; instead, the effective modulus becomes frequency-dependent, and the headline quantitative claim no longer holds as stated. This does not invalidate the formal consistency of the two frameworks, but it makes the physical prediction conditional on an unquantified assumption. The paper also inherits a self-consistency limitation: the input potential Eq. (18) from Ref. [21] already contains the geometric nonlinearities whose coefficients ultimately control K_o, so the derivation is a check of the transformation, not an independent prediction of odd elasticity. This is acknowledged by the reader and is acceptable given the paper's stated goal. A minor typo in Eq. (26) (missing tildes on λ and μ compared to Eq. (27)) should be corrected. Together, these considerations support a CONDITIONAL verdict: the methodological core is sound, but the physical claim requires a validity range for the fast-relaxation assumption, and the typos should be fixed. The reader's verdict already reflects this, so no change is needed.","tokens_in":19148,"tokens_out":10676,"duration_ms":92588,"concrete_test":"Retain θ as a dynamical field by adding a linear damping term −γ δH/δθ (or Γ ℓ) to Eq. (26), solve the coupled linearized equations for u and θ at frequency ω, and compute the effective stress-strain modulus K_o(ω). Verify whether K_o(ω→0) = τ/4 and whether the correction at the elastic wave frequency is small. If the correction is not small, the quasistatic elimination used to obtain Eq. (28) is invalid in the regime where odd elasticity is claimed. A complementary analytical test is to independently re-derive Eq. (28) without setting δH/δθ=0 and check that the zero-frequency limit matches the adiabatic result; if it does not, the derivation has a hidden assumption.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing step is the replacement of the full angular-momentum balance (Eq. 26) by the algebraic relation Eq. (27). This is an adiabatic elimination of the internal rotation θ (or ϕ) that assumes θ relaxes to local mechanical equilibrium δH/δθ=0 (equivalently ℓ≈0) on timescales faster than the displacement field. The odd elastic modulus K_o=τ/4 enters the stress tensor only after substituting this algebraic solution into Eq. (25). If the relaxation of θ is not sufficiently fast at the frequencies of interest, the response is not a static stress-strain tensor; instead, the effective modulus is K_o(ω) with a frequency dependence determined by the θ dynamics, and the central 'stress-strain response' claim must be qualified. The paper asserts this fast relaxation with citations (Refs. [1,20,63–65]) but does not derive the timescale separation for the chiral active solid model, nor specify the regime (e.g., damping strength vs. stiffness) in which the elimination is valid. This is not a flaw in the PB algebra, which appears internally consistent, but it is the point at which the specific prediction K_o=τ/4 could break. The self-consistency check (potential Eq. (18) already encodes the geometric nonlinearities) does not rescue the quantitative claim if the adiabatic assumption is violated.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper develops a systematic Eulerian Poisson-bracket (PB) formalism for elastic systems whose Hamiltonians are originally written in Lagrangian variables. It derives the transformation between Lagrangian and Eulerian field dynamics (Eq. 14), identifies two sources of additional nonlinearities in the Eulerian description (the Jacobian volume factor in the Hamiltonian and the spatial-gradient terms in Eulerian PBs), and verifies the relation between Cauchy and first Piola–Kirchhoff stresses. As an illustration, the formalism is applied to a 2D chiral active solid with internal particle rotations; the resulting stress and elasticity tensors (Eqs. 28–29) recover the odd elastic modulus K_o=τ/4 of the authors' previous Lagrangian treatment (Ref. [21]).","tokens_in":19555,"tokens_out":9707,"duration_ms":85676,"significance":"If the formal claims hold, Eq. (14) and the PB list in Appendix B provide a useful and clean bridge between Lagrangian and Eulerian descriptions of solids with internal rotation. The derivation of Eq. (14) and its use to recover the standard stress transformation are elegant, and the presentation of the Eulerian PBs with 3D generalization is valuable. However, the illustrative recovery of K_o=τ/4 is a consistency check against Ref. [21] rather than an independent prediction: the input potential Eq. (18) already contains the geometric nonlinearities that produce the odd modulus, and the result depends on an adiabatic elimination of the internal rotation that is asserted rather than derived. These points do not undermine the formal methodology, but they temper the strength of the 'recovering' claim.","major_comments":[{"comment":"The coefficient of the φ(∇·u) term is printed as λ+μ, but the potential V in Eq. (20) contains −(λ̃+μ̃)φ²(∇·u), whose derivative with respect to φ is −2(λ̃+μ̃)φ(∇·u). Equation (27) then uses λ̃+μ̃, so Eq. (26) as printed is not the equation that yields Eq. (27). Since Eq. (27) is substituted into Eq. (25) to obtain the central stress tensor Eq. (28), please correct the coefficient and check the signs in the intermediate algebra.","section":"§2.3, Eq. (26)"},{"comment":"The central quantitative result K_o=τ/4 is obtained by replacing the full angular-momentum equation by the algebraic relation Eq. (27) (∂H/∂θ=0). This is an adiabatic elimination of the internal rotation θ. The paper asserts that θ relaxes fast and cites Refs. [1,20,63–65], but no timescale separation is derived for the present model, and the reactive PB equations (25)–(26) contain no dissipative relaxation. If the rotational degree of freedom is not fast at the frequencies of interest, the effective modulus is frequency-dependent, K_o(ω), rather than the static τ/4. Please state the validity regime (e.g., damping vs κ_c) or derive the elimination from a dissipative model.","section":"§2.3, Eqs. (26)–(28)"}],"minor_comments":[{"comment":"The wording 'recovering' and 'demonstrate its ability to capture emergent nonlinear elastic behavior' overstates the example. The potential Eq. (18) already contains the geometric nonlinearities that produce K_o, so the computation is a consistency check of the Eulerian route against Ref. [21], not an independent prediction. Please qualify the language accordingly.","section":"Abstract and §2.3"},{"comment":"Please show the intermediate steps in transforming V° to V; the origin of the −κ_c ε_li δ_jk term is not transparent from the one-line description following Eq. (20). A fuller derivation would aid reproducibility.","section":"§2.2, Eq. (20)"},{"comment":"The notation λ, μ versus λ̃, μ̃ is confusing. The tilde coefficients are imported from Ref. [21] but are not defined in this paper; please define them explicitly at the point of introduction.","section":"§2.1, Eq. (18)"}],"recommendation":"major_revision","confidential_remarks":"The formal transformation result and the PB list are solid. The odd-elasticity example is drawn from the authors' own Ref. [21]; the new contribution is the Eulerian route rather than the physics of odd elasticity. The main risk is the adiabatic-elimination gap: if the authors cannot justify the fast-relaxation assumption, the static K_o=τ/4 claim becomes fragile. The coefficient inconsistency in Eq. (26) is fixable but needs attention."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The thing to know about this paper is that the methodological core is solid and genuinely new: a systematic Eulerian Poisson-bracket route to elastic problems whose potentials are naturally written in Lagrangian form, plus a direct transformation between the resulting field dynamics (Eq. 14) that recovers the usual stress transformation without a separate step. Appendix B's PB list is careful and the derivation of Eq. (14) is clean. The authors do not oversell the physics: they frame the odd-modulus result as a demonstration of consistency with their earlier Lagrangian calculation, not as a new effect.\n\nWhere the paper is soft is exactly where the stress-test note points. The quasistatic replacement of the angular momentum balance by the algebraic condition ∂H/∂θ=0 (Eq. 27) is asserted with citations, not derived for this model. Since K_o=τ/4 enters only after that adiabatic elimination, the central quantitative claim is conditional on a timescale separation that the paper does not characterize. If θ does not relax fast, the response becomes frequency-dependent and the claimed stress-strain tensor is not the full story. That is not a flaw in the PB algebra, but it is a real gap in the physical claim, and it should be stated as such.\n\nThere is a second, related soft spot. The input potential (Eq. 18) is imported from the same group's Ref. [21], including the geometric nonlinearities whose coefficients control the recovered modulus. So the illustrative 'recovery' is partly a self-consistency check. That is fine for a methodological paper, but it means the paper does not independently validate odd elasticity. Finally, there is a minor coefficient mismatch between Eqs. (26) and (27): Eq. (26) has λ+μ in the φ(∇·u) term, while the potential in Eq. (20) gives λ̃+μ̃, which is what Eq. (27) uses. The odd modulus does not depend on this, so it is cosmetic, but it should be fixed.\n\nOverall, the formal method deserves a serious referee. The soft spots are present but not fatal; they are about scope and attribution, not about the internal consistency of the derivation. The paper is for people working on Poisson-bracket formulations of elasticity, active solids, or odd elasticity, and it will be a reasonable reference for the Eulerian route even if the illustrative example is not a new prediction.\n\nI would send it to peer review, and would cite the formal transformation if I worked in this area.","headline":"The formal Eulerian-to-Lagrangian dynamics transformation (Eq. 14) is a genuine methodological step forward; the odd-elasticity recovery is a consistency check that inherits an asserted adiabatic assumption rather than providing a new physical prediction.","tokens_in":19986,"tokens_out":1827,"would_cite":true,"duration_ms":18403,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Applying the Poisson-bracket formalism in real space directly reproduces the Cauchy stress and odd modulus K_o = τ/4 of chiral active solids, bypassing the Lagrangian stress transform.","keywords":["Poisson-bracket formalism","Eulerian elasticity","odd elasticity","chiral active solids","micropolar (Cosserat) elasticity","Cauchy stress","geometric nonlinearities","coarse-grained dynamics"],"falsifier":"Measure the shear-stress response of a chiral active solid in the regime |∇u| ≪ θ ≪ 1 as a function of driving frequency, or solve the full coupled dynamics of displacement and internal rotation numerically without imposing ∂H/∂θ = 0: if the effective odd modulus deviates from τ/4 or shows strong frequency dependence where the paper predicts a quasistatic constant, the central reduction is falsified.","tokens_in":19005,"feed_emoji":"🌀","tokens_out":6148,"duration_ms":56279,"temperature":0.7,"pith_summary":"The paper sets out to establish that the Poisson-bracket (PB) formalism, a standard route for deriving coarse-grained field dynamics, can be applied directly to elastic solids described in Eulerian (real-space) variables even though their elastic potentials are naturally written in Lagrangian (reference-space) variables. It shows that the Eulerian formulation is not a passive coordinate change: transforming the Hamiltonian and evaluating Eulerian brackets produces extra nonlinear terms, from coarse-grained volume changes and from particle flow between neighbouring volumes, that the Lagrangian truncation omits. The illustrative case is a two-dimensional chiral active solid with rotating particles, where active torques create geometric nonlinearities that yield an odd elastic modulus coupling the two shear modes non-reciprocally. Recovering this modulus directly from Eulerian brackets, rather than by transforming the Lagrangian stress, demonstrates the consistency of the two frameworks and provides a route for materials whose measured responses live in real space. The paper thus argues that the Eulerian PB formalism is a valid and useful tool for nonlinear elastodynamics of driven solids.","feed_headline":"Real-space math recovers the odd shear response of chiral solids","feed_subtitle":"A direct real-space derivation gives the non-reciprocal shear modulus K_o=τ/4, matching the reference-frame route without the detour.","key_machinery":"The load-bearing object is the Eulerian Poisson bracket, computed with canonical pairs defined in the real/deformed space, so that derivatives of delta functions generate the convective terms that the Lagrangian bracket misses; the most important examples are {g_i^c(R), u_j(R′)} = (δ_ij − ∇′_i u_j) δ(R−R′) and {g_i^c(R), θ(R′)} = −∇′_i θ(R′) δ(R−R′). The second load-bearing identity is Eq. (14), dΨ°/dt = J[∂Ψ/∂t + ∇_j(v_j^c Ψ)], which converts Eulerian dynamics into Lagrangian dynamics, carries the nonlinear streaming and volume-change contributions, and yields the Cauchy-to-first-Piola–Kirchhoff stress relation. In the chiral-solid example, these brackets, combined with the adiabatic elimin","core_discovery":"The paper's central claim is that the Eulerian PB formalism, applied to the Eulerian Hamiltonian obtained from the Lagrangian one through the Jacobian factor J, generates the full real-space stress tensor and the odd elastic modulus K_o = τ/4 in a chiral active solid, matching the result of a previous Lagrangian derivation. The key step is the fast-relaxation (adiabatic) elimination of the internal rotation θ, which reduces the rotational dynamics to the algebraic relation ϕ ≈ (τ/2κ_c)(1 + ((λ̃+μ̃)/κ_c) ∇·u). Substituting this into the Eulerian momentum balance (Eq. 25) produces the Cauchy stress with an active prestress and an elasticity tensor that contains the odd modulus proportional to","pith_inferences":["If the fast-variable assumption on θ is relaxed, the quasistatic modulus τ/4 should become a frequency-dependent odd response; measuring the shear modulus as a function of driving frequency in a chiral active solid would provide a clean test the paper does not itself perform.","Eq. (14) is likely a general tool beyond elasticity: any coarse-grained theory with a dynamic reference configuration, such as plastic flow, growth, or active remodeling, could use this transformation to connect Eulerian observations to Lagrangian material behavior.","The derivation suggests that odd elastic response does not require a microscopic model of contact forces, only a Hamiltonian with an active torque potential and geometric nonlinearities, so the same Eulerian PB route could predict odd moduli in other torque-driven field theories such as chiral liquid crystal elastomers."],"forward_implications":["The real-space (Cauchy) stress becomes directly accessible from the Eulerian PB formalism, so experimental or computational studies that measure stress in deformed coordinates can compare with theory without reference-frame transformations.","In any elastic system with large internal rotations, the extra Eulerian nonlinearities—neglected in linear elasticity—must be kept; ignoring them hides the odd response and misstates the stress-strain relation.","The recovered odd modulus K_o = τ/4 implies non-reciprocal shear coupling: a simple shear strain induces pure shear stress, and a pure shear strain induces negative simple shear stress, with consequences such as tilting under uniaxial compression and growing wave modes as established in the broader odd-elasticity literature.","The transformation Eq. (14) gives a general dictionary between Eulerian and Lagrangian dynamics, allowing the two formulations to be cross-checked even in systems such as viscoelastic, elastoplastic, or active materials where a fixed reference frame is not always available.","For three-dimensional micropolar solids, the axis-angle canonical-momentum construction in Appendix A extends the same Eulerian PB scheme, so the approach is not limited to two-dimensional chiral solids."],"fun_headline_variants":["Eulerian route to odd elasticity in chiral solids","Real-space derivation nails chiral solid's odd shear","Direct Eulerian formalism yields odd modulus of chiral active solids","Chiral solids' odd shear from real-space Poisson brackets","No detour: Eulerian math captures odd elastic response"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The argument assumes that the internal rotation θ relaxes so quickly that it can be set to local mechanical equilibrium, ∂H/∂θ = 0, turning it into an algebraic function of the displacement gradient; if θ does not relax fast, the quasistatic odd modulus K_o = τ/4 becomes a frequency-dependent quantity and the central stress-strain claim no longer holds as stated.","fun_headline_variants_meta":{"raw":{"variants":["Eulerian route to odd elasticity in chiral solids","Real-space derivation nails chiral solid's odd shear","Direct Eulerian formalism yields odd modulus of chiral active solids","Chiral solids' odd shear from real-space Poisson brackets","No detour: Eulerian math captures odd elastic response"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000479,"raw_usage":{"total_tokens":2262,"prompt_tokens":852,"completion_tokens":1410,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":596,"completion_tokens_details":{"reasoning_tokens":1332}},"tokens_in":596,"tokens_out":1410,"duration_ms":9249,"temperature":1.0,"reasoning_tokens":1332,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T05:19:45.382881+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure the shear-stress response of a chiral active solid in the regime |∇u| ≪ θ ≪ 1 as a function of driving frequency, or solve the full coupled dynamics of displacement and internal rotation numerically without imposing ∂H/∂θ = 0: if the effective odd modulus deviates from τ/4 or shows strong frequency dependence where the paper predicts a quasistatic constant, the central reduction is falsified.","supporting_citations":[],"review_version":1}