{"id":"ffc37fb5-9416-4f1c-99f2-c7b469524b93","arxiv_id":"2512.11037","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Curved surfaces pattern energy injection, open a finite-size spectral gap, and create thresholdless defect-bound and Rayleigh-edge oscillations in odd elastic solids.","lead":"Extends the theory of 'odd' elasticity—non-reciprocal active solids—to curved surfaces: curvature gaps the vibration spectrum and concentrates oscillatory motion at topological defects and open edges. Provides a general covariant framework and lattice simulations that could guide experiments on curved biological tissues and active metamaterials.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Equation (2) does not reduce to Eq. (3): direct substitution yields factors of 2 in the odd coupling and shear term, changing the exceptional-point condition from B²=4K_o² to (B−μ)²=16K_o².","rationale":"The reader's weakest assumption focused on the uniqueness of Eq. (1) and the completeness of the Papkovich–Neuber representation. Those are valid concerns, but I find a more direct and more load-bearing problem: the paper's own governing equation (2) is inconsistent with its central reduced equation (3). A straightforward flat-space substitution of the Helmholtz–Hodge ansatz into Eq. (2) produces factor-of-two discrepancies in both the odd coupling and the shear term for ψ. This changes the exceptional-point condition and therefore all quantitative predictions (sphere spectrum, torus spectrum, EP threshold). The numerical threshold reported in Fig. 2(f) matches the paper's B²=4K_o² condition, which suggests Eq. (3) is the intended effective model, but the derivation as written is invalid. This is a concrete, checkable algebraic inconsistency, independent of any debate about curvature-dependent moduli. The reader's uniqueness concern is secondary: even if Eq. (1) were the unique covariant extension, Eq. (3) does not follow from Eq. (2) as stated. The paper should be revised to correct Eq. (2) or to derive Eq. (3) from a consistent equation. The verdict remains CONDITIONAL because the numerics and the independent coarse-graining provide positive evidence for Eq. (3), but the internal inconsistency must be resolved before the theoretical claims can be accepted.","tokens_in":15665,"tokens_out":53733,"duration_ms":704686,"concrete_test":"In flat space, take a plane-wave ansatz (χ,ψ)=(χ0,ψ0)e^{iqx} in Eq. (2). Projecting Eq. (2) onto ∇χ and ε∇ψ gives the 2×2 matrix [[B+μ, −2K_o],[2K_o, 2μ]] q², whose eigenvalues are [B+3μ ± sqrt((B−μ)² − 16K_o²)]/2. Compare this with Eq. (5) and with the numerically observed EP threshold. If the matrix from Eq. (2) is not what the paper uses, the derivation of Eq. (3) from Eq. (2) must be shown explicitly; otherwise the central spectral predictions do not follow from the stated governing equation.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Substituting the Helmholtz–Hodge ansatz u_i = ∇_iχ + ε_i^j∇_jψ into the flat-space limit of Eq. (2) and projecting onto the gradient and co-gradient directions yields the scalar system γ∂_tχ = (B+μ)Δχ − 2K_oΔψ, γ∂_tψ = 2K_oΔχ + 2μΔψ. This differs from Eq. (3) by a factor of 2 in both the odd off-diagonal coupling and the shear term in the ψ equation. The eigenvalues of this 2×2 matrix are not the λ in Eqs. (4)–(5); the exceptional-point condition becomes (B−μ)² = 16K_o² instead of B² = 4K_o². The same factor appears when Eq. (2) is derived from the divergence of the stress σ^ab = C^abcd u_cd with C^odd in Eq. (1); the odd force is K_o(ε^{ik}∂_kΔχ − ∂^iΔψ), not 2K_o(...). Thus Eq. (3) does not follow from the stated governing equation. This is not a curvature or uniqueness issue—it is an internal algebra inconsistency even in flat space. The numerical threshold and the coarse-grained moduli (SM §IV) satisfy B²=4K_o², suggesting Eq. (3) is the intended result, but Eq. (2) as printed is incorrect and must be amended.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper develops a covariant formulation of odd elasticity on curved manifolds. Starting from a two-metric elasticity action, the authors write down the odd Navier–Cauchy equation (Eq. 2) and reduce it via Helmholtz–Hodge potentials to two coupled scalar equations (Eq. 3) in which curvature enters only through the Laplace–Beltrami operator. They solve Eq. (3) on a sphere and a slender torus, obtaining spectra (Eqs. 4–5), an exceptional-point threshold B²=4K_o², and a finite-size gap ∼1/r². They support the theory with simulations of a non-reciprocal honeycomb lattice on torus, sphere, and spherical cap, observing curvature-textured power injection, thresholdless defect-bound modes around disclinations, and Rayleigh edge modes. The coarse-grained moduli of the lattice are used to predict the bulk exceptional threshold, which agrees with numerics.","tokens_in":16027,"tokens_out":36514,"duration_ms":267246,"significance":"If correct, the framework provides a useful minimal continuum description of active odd solids on curved substrates, with testable predictions for living chiral crystals and metamaterials. The paper's strengths include an independent coarse-graining of the lattice moduli (SM §IV) that predicts the numerically observed threshold without fitting, and direct numerical evidence for geometry-controlled oscillatory mode structure. The main theoretical results, however, rest on Eq. (3), and the derivation leading to it from the stated governing equation contains a load-bearing algebra error; the uniqueness and completeness claims also need qualification.","major_comments":[{"comment":"The governing equation does not reduce to Eq. (3), and it is not the divergence of the stress from Eq. (1). In flat Cartesian coordinates, the passive part of Eq. (2), ∇[(B−μ)e] + Δ(2μu), equals (B−μ)∇e + 2μΔu, whereas the divergence of the isotropic stress σ=(B−μ)eδ+2μu is B∇e + μΔu. Substituting u=∇χ+J∇ψ into Eq. (2) (or SM Eq. (12) using the ε_{ik}u^k convention) yields γ∂_tχ=(B+μ)Δχ − 2K_oΔψ, γ∂_tψ=2K_oΔχ+2μΔψ. The resulting exceptional-point condition is (B−μ)²=16K_o², not B²=4K_o² used in Eqs. (4)–(5). Since the numerical threshold matches B²=4K_o², Eq. (3) appears to be the intended effective theory, but Eq. (2) as printed is not its parent equation. The derivation must be corrected.","section":"Eq. (2) and SM Eq. (12)"},{"comment":"The statement that Eq. (1) is 'the unique covariant extension of the flat-space odd elastic tensor' is asserted without proof. On a curved background, symmetry alone allows curvature-dependent odd couplings, e.g. terms proportional to K_o R ε^{ab} or K_o Ric^{ab}ε^{cd} (with a microscopic length squared), which vanish in flat space and appear at the same derivative order as the Laplace–Beltrami operator in Eq. (3). The sphere spectrum Eq. (5), the gap ∼1/r², and the exceptional-point threshold all rely on excluding such terms. The authors should either provide a systematic derivative/curvature expansion showing that these terms are absent to the relevant order, or restate the assumption as a minimal-coupling choice.","section":"after Eq. (1)"},{"comment":"The footnote claims that every solution of the Navier equation admits the representation u_i=∇_iχ+ε_i^j∇_jψ. This is false on a manifold with nontrivial first cohomology. On the torus, H¹(T²)≠0, so there exist nonzero harmonic vector fields (e.g., constant vector fields) with zero divergence and zero curl that are not gradients plus co-gradients of scalar potentials. These modes are absent from the ansatz used to derive Eq. (4), so the analytical torus spectrum is incomplete. The completeness assertion should be removed or restricted to the non-harmonic sector; the impact on the torus mode comparison should be assessed.","section":"SM footnote 2"}],"minor_comments":[{"comment":"The index notation in Eq. (2) (ε^k_i u_k) differs from that in SM Eq. (12) (ε_{ik} u^k). In flat space these differ by a sign, which affects the factor-of-two algebra discussed above. The notation should be made consistent and unambiguous throughout.","section":"Eq. (2) vs SM Eq. (12)"},{"comment":"The defect-bound and Rayleigh modes are presented as numerical observations; the paper does not derive them from Eq. (3) or another continuum limit. A sentence stating this explicitly would avoid the impression that Eq. (3) explains these classes of modes.","section":"Figs. 3–4 and discussion"},{"comment":"The theoretical lines in Fig. 2(f) are not fully defined in the main text. Please specify the branch labels λ± and the toroidal/poloidal wavenumbers m,n in the caption or in the paragraph discussing Eq. (4).","section":"Fig. 2 caption"}],"recommendation":"major_revision","confidential_remarks":"The algebra error in Eq. (2)/SM Eq. (12) is serious but appears fixable: the numerical threshold and spectra support Eq. (3), so the authors likely intended the equation of motion that follows from the stress divergence. The paper should be resent for review after a corrected derivation, with the uniqueness and Hodge-completeness points addressed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper has a real, load-bearing inconsistency right in the main equations. Direct substitution of the Hodge ansatz u_i = ∇_iχ + ε_i^j∇_jψ into Eq. (2) gives γ∂_tχ = (B+μ)Δχ − 2K_oΔψ and γ∂_tψ = 2K_oΔχ + 2μΔψ, not the clean system in Eq. (3). I checked this by both taking divergence/curl and by writing the RHS as a gradient plus ε-gradient. The factors of 2 are unmistakeable. Consequently the exceptional-point condition from Eq. (2) is (B−μ)² = 16K_o², not B² = 4K_o². The paper's numerics and the coarse-grained moduli support B² = 4K_o², so Eq. (3) is presumably the intended result, but Eq. (2) and the SM derivation that leads to it are wrong as printed. This is not a subtle curvature issue; it is flat-space algebra, and every quantitative prediction depends on it.\n\nSetting that aside, the conceptual core is genuinely new and useful. A covariant two-metric formulation of odd elasticity is a natural but necessary step, and the curvature-induced gap, the thresholdless defect-bound modes, and the Rayleigh edge modes are all coherent and backed by lattice simulations on torus, sphere, and cap. The EP threshold is a real prediction: the moduli B, μ, K_o are coarse-grained from the lattice, not fitted to the spectrum, and the agreement with the numerical transition is meaningful evidence in favor of the intended theory.\n\nThe other soft spots are secondary. The claim that Eq. (1) is 'the unique covariant extension' is asserted, not proven, and curvature-dependent odd moduli at the same derivative order are silently excluded. The Hodge decomposition u_i = ∇_iχ + ε_i^j∇_jψ is not complete on the torus (harmonic vector fields are missed), so the analytic torus spectrum may drop modes. No code, data, or error bars are given for the numerics, which makes the agreement harder to judge, though the EP threshold is still a strong point.\n\nWho should read this: people working on active solids on curved substrates, active metamaterials, and tissue mechanics. If the algebra is repaired, the framework will likely be a reference for the subfield. As printed, the internal contradiction between Eqs. (2) and (3) means the paper is not yet suitable for publication. I would send it to a serious referee—the physics is promising and the fix is probably straightforward—but the referee should be instructed to verify the derivation from the stress tensor to the scalar system carefully.","headline":"The paper's core covariant equation is internally inconsistent: Eq. (2) reduces to a scalar system with factor-2 errors, changing the exceptional-point condition; the physics of curved odd elasticity still looks worth refereeing after the algebra is fixed.","tokens_in":769,"tokens_out":753,"would_cite":false,"duration_ms":68873,"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":"This paper extends odd elasticity—chiral, non-reciprocal elastic response—to curved manifolds, claiming that curvature enters only through the Laplace–Beltrami operator and thereby gaps, textures, and localizes the oscillatory spectrum of a","keywords":["odd elasticity","active solids","curved manifolds","Laplace-Beltrami operator","non-reciprocal elasticity","exceptional points","Rayleigh waves","topological defects"],"falsifier":"Measure the low-frequency eigenvalues of a spherical odd-elastic lattice at several radii: Eq. (5) predicts λ_l = (B+2μ ± √(B²−4K_o²)) l(l+1)/(2r²), a pure l(l+1)/r² scaling. Any deviation in the l-dependence or r-dependence, or any mode that depends on the local Ricci scalar beyond the Laplace–Beltrami term, would falsify the claim that curvature enters only through the curved Laplacian.","tokens_in":1463,"feed_emoji":"🌀","tokens_out":1816,"duration_ms":63973,"temperature":0.7,"pith_summary":"This paper argues that odd elasticity—an elastic response in which forces are not reciprocal, so a solid can inject and absorb energy internally—extends naturally to curved manifolds. The central claim is that, once written covariantly, curvature enters the dynamics only through the Laplace–Beltrami operator: the coupled equations for the two scalar potentials have the same non-reciprocal matrix as flat space, and all geometry is contained in the curved Laplacian. On a sphere this yields the exact long-wavelength spectrum with a finite-size gap that closes in the flat limit. On lattices, the same theory captures curvature-textured power injection, thresholdless oscillations bound to topological disclinations, and Rayleigh edge modes at open boundaries. The paper thus provides a geometric baseline for active solids in curved environments, shifting the question from whether a solid oscillates to where and in what form it oscillates.","feed_headline":"Curvature gaps and localizes odd-elastic oscillations","feed_subtitle":"A covariant theory predicts finite-size gaps, defect-bound modes, and Rayleigh edge waves in active curved solids.","key_machinery":"The central object is the covariant odd elasticity tensor C_odd^{abcd} = (K_o/2)( ḡ^{ac} ε̄^{bd} + ḡ^{ad} ε̄^{bc} + ḡ^{bc} ε̄^{ad} + ḡ^{bd} ε̄^{ac}), where ε̄ is the Levi-Civita tensor density and K_o is the odd—non-reciprocal, chiral—modulus. Inserted into the two-metric elasticity action it yields the odd Navier–Cauchy equation. The reduction that carries the argument is the Helmholtz–Hodge (Papkovich–Neuber) decomposition u_i = ∇_i χ + ε̄_i^j ∇_j ψ, which turns the vector problem into two coupled scalar equations controlled by the Laplace–Beltrami operator. The spectrum then follows from the eigenvalues of the Laplace–Beltrami operator on each surface—spherical harmonics on a sphere, Four","core_discovery":"The paper establishes a covariant low-energy theory of odd elasticity on curved manifolds. Its starting point is the claim that the flat-space odd elasticity tensor has a unique covariant extension, built from the reference metric and the Levi-Civita tensor density. Using the Papkovich–Neuber decomposition of the displacement into two scalar potentials, the odd Navier–Cauchy equation reduces to two coupled scalar equations whose only curved ingredient is the Laplace–Beltrami operator. Applied to a sphere this gives the long-wavelength spectrum λ = (B+2μ ± √(B²−4K_o²)) l(l+1)/(2r²), predicting a finite-size gap that closes as the radius grows. Numerical simulations on a torus, sphere, and sph","pith_inferences":["The uniqueness of the covariant odd tensor is asserted, not proven; curvature-dependent odd moduli proportional to K_o R̄ or Ricci contractions are symmetry-allowed and would enter at the same derivative order, so the theory is best read as a leading-order geometric truncation.","On the torus, the two-scalar Helmholtz–Hodge representation misses harmonic vector fields because H¹≠0, so the analytic spectrum may omit global modes that the numerical lattice still contains.","The predicted 1/r² gap and defect-bound oscillations suggest a design route: curvature alone can turn a homogeneous active solid into spatially localized micro-oscillators for active metamaterial resonators or mechanical logic.","The same covariant construction can be extended to torque densities, Cosserat or micropolar structure, and out-of-plane displacements, where curvature would likely couple to new active moduli—a direction the paper explicitly leaves open."],"forward_implications":["On a sphere, curvature opens a finite-size frequency gap ∼1/r²; smaller active shells oscillate at higher frequencies, and the gap closes in the flat-space limit.","On surfaces with nonzero Euler characteristic, topologically enforced disclinations host thresholdless oscillatory defect modes for any nonzero odd modulus, before the bulk exceptional transition.","Open boundaries support Rayleigh surface waves that are thresholdless and sit above the bulk modes for any odd modulus, so edges destabilize before the bulk.","In curved odd solids, eigenmodes develop spatial texture in power injection and dissipation: some plaquettes locally consume energy even when the net power is positive, an experimentally accessible signature of curvature.","Coarse-graining the non-reciprocal honeycomb lattice predicts the bulk exceptional transition at k_o*/k = √3(1+6k′/k)/18, matching the simulated threshold."],"fun_headline_variants":["Curved odd elasticity: spectrum gaps and defect modes","Curvature patterns activity in odd-elastic solids","Odd elasticity on curved manifolds: gaps and edge states","Curvature-induced gaps and localized modes in odd elasticity","Finite-size gaps from curvature in active odd-elastic solids"],"cache_read_input_tokens":17792,"weakest_assumption_plain":"The entire quantitative spectrum rests on the assertion, stated but not proven, that Eq. (1) is the unique covariant extension of flat odd elasticity, so curvature enters only through the Laplace–Beltrami operator and no curvature-coupled odd moduli appear; the same reduction also assumes two scalar potentials span every displacement mode on a torus.","fun_headline_variants_meta":{"raw":{"variants":["Curved odd elasticity: spectrum gaps and defect modes","Curvature patterns activity in odd-elastic solids","Odd elasticity on curved manifolds: gaps and edge states","Curvature-induced gaps and localized modes in odd elasticity","Finite-size gaps from curvature in active odd-elastic solids"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000344,"raw_usage":{"total_tokens":1679,"prompt_tokens":648,"completion_tokens":1031,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":392,"completion_tokens_details":{"reasoning_tokens":951}},"tokens_in":392,"tokens_out":1031,"duration_ms":9266,"temperature":1.0,"reasoning_tokens":951,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T06:30:42.387702+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure the low-frequency eigenvalues of a spherical odd-elastic lattice at several radii: Eq. (5) predicts λ_l = (B+2μ ± √(B²−4K_o²)) l(l+1)/(2r²), a pure l(l+1)/r² scaling. Any deviation in the l-dependence or r-dependence, or any mode that depends on the local Ricci scalar beyond the Laplace–Beltrami term, would falsify the claim that curvature enters only through the curved Laplacian.","supporting_citations":[],"review_version":1}