{"id":"e8f98d04-9852-4a08-9eef-a45f9f1c022b","arxiv_id":"2411.12096","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"CAMP is a Cartesian-tensor graph neural network for interatomic potentials that matches or beats several leading spherical-harmonic models on standard benchmarks.","lead":"This paper presents CAMP, a machine learning interatomic potential built entirely from Cartesian tensors, avoiding the spherical harmonics used by most leading models. It reports competitive accuracy, stability, and speed on crystals, liquid water, small molecules, and bilayer graphene, offering a simpler alternative for atomistic simulations.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"CAMP's completeness claim is not supported by the construction: the Eq. (7) contraction rule omits invariants such as tr(M2^3) that require pairing lower-rank moments, so the restricted hyper-moment basis is not the full MTP/ACE basis.","rationale":"The reader's conditional verdict focused on benchmark overclaims and the unproven completeness of the restricted basis. The deeper issue is that the contraction rule in Eq. (7) is not a minor efficiency detail: it changes the space of representable invariants. The MTP/ACE completeness theorems cited by the paper cover unrestricted products of moment tensors, whereas CAMP only allows contractions in which every lower-rank tensor is fully contracted against a single highest-rank tensor. This excludes cycle contractions such as tr(M2^3), so the claimed 'complete description of local atomic environments' is false for the linear feature space as defined. The benchmark results are substantial and the code is available, so the concern is not with the empirical measurements but with the theoretical framing and the abstract's absolute language. The conditional verdict should stand, with the requirement that the authors either prove completeness for their restricted contraction rule or soften the 'complete/systematically improvable' wording and explicitly discuss which invariants are omitted.","tokens_in":23456,"tokens_out":21911,"duration_ms":256331,"concrete_test":"Analytical/experimental check: enumerate all scalar contraction monomials of the atomic moment tensors up to vmax=4 and compare with the span of CAMP's admissible hyper moments at the same body order. Specifically, test whether tr(M2^3) is in that span. If it is not, the complete-description claim fails in the linear-basis sense. Then train CAMP and an unrestricted MTP-style model on a synthetic dataset whose target energy depends only on tr(M2^3), using configurations with central-symmetric environments so M1=M3=0 but M2 varies. If CAMP cannot reduce test error to noise while the unrestricted model fits, the restriction has practical consequences for expressiveness.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Central claim: 'complete description of local atomic environments' and 'systematically improvable potentials.' This requires the restricted hyper moment set (Eq. 7 plus the SI selection rules: every lower-rank moment is fully contracted against one highest-rank moment) to span the full MTP/ACE invariant basis. The paper cites [36,37] for completeness, but those proofs apply to unrestricted products of moment tensors. The restriction discards genuine invariants. Example: for vmax=2, the scalar tr(M2^3) = M2_ab M2_bc M2_ca is an independent SO(3) invariant of the atomic moment M2. It cannot be formed under the CAMP rule: any scalar path must satisfy v = vn - (v1+...+v_{n-1}); with three rank-2 factors, vn=2 and v1+v2=4, so no admissible path exists. Equivalently, the rule only produces 'star' contractions centered on the highest-rank tensor, while tr(M2^3) is a cycle contraction. Since Eq. (9) is a linear update (no activation), missing scalar monomials are not obviously manufactured by adding layers; the final MLP only sees whichever invariants are actually generated. Therefore the linear-basis completeness assertion fails as stated, and the systematic improvability claim is unproven for this architecture. The benchmark results remain empirical evidence for specific systems, but they do not establish the advertised theoretical property.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces CAMP, a graph-neural-network interatomic potential that represents atomic environments entirely in Cartesian space using symmetric tensors ('atomic moments') and builds higher-body-order features ('hyper moments') through restricted tensor-product contractions. The two central claims are (i) the construction provides a complete, systematically improvable description of the local atomic environment, and (ii) CAMP matches or surpasses leading spherical-tensor models in accuracy, speed, and molecular-dynamics stability on LiPS, water, MD17, and a new bilayer-graphene dataset. The empirical evaluation is carried out with five independent training seeds, standard train/validation/test splits, published baselines, and released code and data.","tokens_in":23753,"tokens_out":13844,"duration_ms":139250,"significance":"If the theoretical completeness claim were valid, this would be an important result: a fully Cartesian tensor architecture that is competitive with spherical-tensor models while being simpler and, in the tested regimes, faster. The empirical methodology is a real strength: the paper reports held-out energy and force errors, MD stability statistics over multiple runs, diffusion coefficients consistent with AIMD reference values, and speed measurements on the same GPU, and it releases code and data. The benchmark results do show CAMP to be competitive with NequIP and other leading models on several standard tests. However, the completeness and 'systematically improvable' claims are load-bearing for the paper's stated theoretical contribution, and as written they are not supported by the construction; this overstatement needs to be fixed before the manuscript can be accepted.","major_comments":[{"comment":"The completeness claim is not supported by the construction. The contraction rule in Eq. (7) and in SI Section 2 requires every lower-rank atomic moment to be fully contracted against the single highest-rank moment in the sequence. This generates only 'star' tensor-network contractions, and invariants requiring cycle contractions are excluded. A concrete counterexample is the scalar invariant tr(M2^3) for a rank-2 atomic moment M2: forming it requires three rank-2 factors, and under the rule v = vn - (v1+...+v_{n-1}) with vn=2 and v1+v2=4, the output rank is negative, so the invariant is inadmissible. Because the feature update in Eq. (9) is a linear combination of hyper moments (no nonlinear activation that could compose monomials in a non-polynomial way) and the final MLP in Eq. (10) sees only the generated scalar hyper moments, missing monomials such as tr(M2^3) are not manufactured in later layers. The abstract's claim of a 'complete description of local atomic environments' and the statement in the Hyper moment paragraph that the hyper moments 'can provide a complete description of the local atomic environment by increasing the order of interactions [36, 37]' are therefore unjustified. The completeness proofs in Refs. [36,37] apply to the unrestricted MTP/ACE product basis, not to the restricted star-contraction basis used here. Please either prove completeness of the restricted basis (including the multi-layer linear updates and channel mixing), or remove/downgrade the completeness and systematic-improvability claims to empirical statements about the specific star-contraction feature set, supported by convergence tests in vmax and T.","section":"Model Architecture, Hyper moment, Eq. (7); SI Sections 1-2"},{"comment":"The Discussion describes CAMP as 'systematically improvable' and the hyper moments as allowing a 'quick body order expansion of the potential energy surface' (SI Section 2). As noted above, the restricted contraction rule does not reproduce the full MTP/ACE invariant basis, so 'systematically improvable' in the sense of Refs. [36,37] is not established. Even if the basis were complete, the paper does not state a formal notion of improvement for the GNN architecture (e.g., whether increasing vmax, T, and channel count guarantees a decrease in approximation error on a fixed potential-energy surface). I ask the authors to give a precise characterization of what 'systematically improvable' means for CAMP, or to remove the term from the abstract, the hyper-moment paragraph, and the Discussion.","section":"Discussion"}],"minor_comments":[{"comment":"The sentence 'It surpasses NequIP in energy predictions for three out of seven molecules and ranks second in both energy and force predictions across various cases' is not supported for every molecule in Table 4: for aspirin, CAMP's energy error (8.7 meV) ranks fifth behind NequIP, PaiNN, NewtonNet, and sGDML. Please rephrase to a molecule-by-molecule summary or a clear statement of the median/typical rank.","section":"MD17, Table 4"},{"comment":"In Table S2, the Aspirin 'Energy' row appears to list only the DeepPot-SE value (5.0) with the other model columns blank; the table seems truncated. Please complete the table or clearly indicate which entries are unavailable.","section":"Table S2"},{"comment":"Chebyshev polynomials Q_beta are conventionally defined on the interval [-1,1], while the argument rij/rcut lies in [0,1]. Please specify the affine mapping used (e.g., 2(r/rcut)-1) or otherwise clarify the convention.","section":"Eq. (2)"},{"comment":"The stability criterion in Eq. (16) is written as an integral over r in [0,infinity), but in practice the RDF is only meaningful over a finite range up to the cutoff or a chosen maximum. Please state the integration range actually used, matching the protocol in Ref. [47].","section":"Methods, Eq. (16)"},{"comment":"The statement that the models 'result in small, parameter-efficient models with fewer than 125k parameters' should be reconciled with the hyperparameters in Table S5 (e.g., u=64, vmax=3, T=3 for LiPS 2500). Please report the exact parameter count per dataset or state the approximate count more precisely.","section":"Model Architecture"},{"comment":"The phrase 'we have thus constructed a new DFT dataset of bilayer graphene' is potentially misleading because the data are derived from Ref. [40]. Please clarify exactly what is new (e.g., additional configurations, different train/test split, or re-generation of the data with the same functional).","section":"Two-dimensional materials"}],"recommendation":"major_revision","confidential_remarks":"The main risk is the completeness overclaim: it is exactly the kind of statement that the MTP/ACE community will test, and the counterexample with tr(M2^3) is elementary. If the authors cannot prove completeness of the restricted star-contraction basis, they should soften the theoretical claims and present CAMP as an empirical architecture; the benchmarking and code release would still make a solid contribution. The paper fits the journal's scope, and I do not see grounds for rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nCAMP is worth taking seriously. The architecture is a genuine new mix of MTP-style moment tensors, product-basis hyper moments, and GNN message passing, and the benchmark work is thorough: five seeds, held-out test sets, code and data available. On LiPS it matches NequIP, on water it is best in class, and the bilayer graphene stacking energetics are a nice addition. The empirical story is real.\n\nThe soft spot is the completeness claim. The paper says hyper moments provide a \"complete description\" and are \"systematically improvable,\" citing MTP/ACE proofs. Those proofs apply to unrestricted products of moment tensors. CAMP's contraction rule (Eq. 7 plus the SI selection rules) only allows star-shaped contractions: every lower-rank moment is fully contracted against the single highest-rank tensor. That omits cycle invariants like tr(M2^3) for vmax=2. So the restricted hyper moment basis is not the full MTP/ACE basis, and the completeness statement, as written, is unsupported. The architecture may still be very expressive in practice, especially with message passing, but the theoretical load-bearing claim needs to be either proven for the restricted set or dropped.\n\nThe abstract also overreaches: \"rival or surpass current leading models\" is contradicted by the paper's own MD17 and rMD17 tables, where CAMP trails NequIP, Allegro, and MACE on forces by 2-3x. The speed advantage rests on cross-paper, cross-GPU comparisons, not a controlled head-to-head. And the bilayer graphene section lacks leading MLIP baselines like NequIP or MACE.\n\nI don't think any of this is fatal. The core contribution—a parameter-efficient, fully Cartesian GNN that is competitive on several benchmarks—stands. The fixes are textual and additive: soften the completeness language, add missing baselines, and report controlled speed comparisons where feasible.\n\nI would send this to peer review, with the expectation that the authors address the completeness gap and the baselines. It is a serious piece of work, just not as cleanly framed as it should be.","headline":"A solid, well-tested Cartesian MLIP whose empirical performance is real, but whose completeness claim is unsupported and needs revision.","tokens_in":24309,"tokens_out":2624,"would_cite":true,"duration_ms":25028,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper introduces CAMP, a graph-neural-network interatomic potential built entirely from Cartesian tensors, and claims it matches or beats leading spherical-tensor models on accuracy, stability, and speed across four benchmark classes.","keywords":["machine learning interatomic potentials","Cartesian tensors","atomic moment tensors","graph neural networks","equivariant representations","molecular dynamics stability","bilayer graphene","moment tensor potentials"],"falsifier":"Train CAMP and an unrestricted Cartesian moment-tensor model, allowing arbitrary contractions among lower-rank moments, with the same $v_{\\max}$, channels, and data on a system with strong many-body effects, and compare force errors. If the unrestricted model is substantially more accurate, or if CAMP's error stops decreasing as $v_{\\max}$ goes from 3 to 4 on a broad benchmark, the restricted contraction rule has dropped information the paper's completeness argument requires.","tokens_in":23223,"feed_emoji":"⚛️","tokens_out":7185,"duration_ms":71229,"temperature":0.7,"pith_summary":"The paper introduces CAMP, a machine-learning interatomic potential built entirely from Cartesian tensors, and argues that such a representation can match or beat current spherical-tensor models on accuracy, stability, and speed. The authors construct atomic moment tensors from neighboring atoms and form hyper moments by tensor products, embedding them in a graph neural network with message passing. Across four benchmark families—an inorganic lithium phosphorus sulfide crystal, bulk water, seven small organic molecules, and bilayer graphene—CAMP achieves errors equal to or below the leading comparison models in most tests, while running roughly 1.4 to 3.9 times faster than the main spherical-tensor competitors. If these results hold, the standard angular-momentum machinery of equivariant potentials is not required for state-of-the-art accuracy.","feed_headline":"Match top accuracy without spherical harmonics","feed_subtitle":"CAMP, built only from Cartesian tensors, ties or beats spherical-tensor models on four benchmark classes.","key_machinery":"The load-bearing objects are the atomic moment tensor $M^i_{uv,p}=\\sum_{j\\in N_i} R_{uvv_1v_2} h^j_{uv_1}\\odot_c D^{ij}_{v_2}$ and the hyper moment $H_{uv,p}=M_{uv_1}\\odot_{c_1} M_{uv_2}\\odot_{c_2}\\cdots\\odot_{c_{n-1}} M_{uv_n}$. Here $D_v=\\hat r\\otimes\\cdots\\otimes\\hat r$ is a symmetric polyadic tensor built from the unit bond vector, $h$ is the atom feature, and $\\odot_c$ contracts $c$ index pairs. A selection rule requires $c=v_1$ in atomic moments and, in hyper moments, that all lower-rank atomic moments are fully contracted against the highest-rank one; this routes information downward to scalars and cuts the number of tensor products. Chebyshev radial basis functions supply the distance dependence, and repeated message-passing layers refine the features. The construction cost scales as $O(3^{v_{\\max}})$ per moment, versus $O(L^6)$ for Clebsch–Gordan products in spherical-tensor models, which is why CAMP is faster at the small ranks used here.","core_discovery":"CAMP's central claim is that a complete, systematically improvable description of a local atomic environment can be built in Cartesian space, using symmetric moment tensors rather than spherical harmonics. Atomic moments are formed by summing over neighbors the radial-weighted contraction of the neighbor's feature tensor with a polyadic tensor of the unit bond vector; hyper moments then multiply these atomic moments together, with contraction rules that force information from higher-rank tensors down to scalars. The resulting features are updated through message-passing layers and mapped to atomic energies. Benchmarks show CAMP matches the accuracy of a leading spherical-tensor potential on the LiPS crystal, produces the lowest energy and force errors on the water test set among all compared models, is competitive on MD17, and reduces bilayer-graphene force errors to 6.3 meV/Å, accurately resolving AB versus AA stacking energetics that empirical carbon potentials miss.","pith_inferences":["If the restricted contraction subspace is indeed sufficient, CAMP suggests that earlier failures of Cartesian approaches came from limiting tensors to rank two, not from Cartesian representations themselves: raising $v_{\\max}$ to three appears to be the step that makes them competitive.","A natural test is to train CAMP on systems with strong high-body-order or anisotropic many-body effects, such as liquid water at high temperature, and compare against an unrestricted moment-tensor basis at the same $v_{\\max}$; a large gap would localize where the contraction rules lose information.","The completeness argument, inherited from atomic-cluster-expansion theory, applies to the unrestricted moment basis; extending that guarantee to CAMP's restricted basis would require a dedicated proof, which the paper does not supply.","A practical consequence is that future spherical-tensor results should be benchmarked against a Cartesian model at rank 3, since the default assumption that spherical representations are needed for top accuracy no longer holds."],"forward_implications":["A Cartesian-only architecture can serve as a practical alternative to spherical-tensor potentials for molecular dynamics, with accuracy that is at least competitive on standard benchmarks.","CAMP's moment tensors can be extended to arbitrary ranks and body orders, so users can trade cost against accuracy by raising $v_{\\max}$ instead of adding angular-momentum channels.","Because only four hyperparameters are tuned per dataset, CAMP should transfer to new materials with minimal adjustment.","The same symmetric Cartesian features can be repurposed for tensorial outputs such as NMR chemical-shielding tensors with small changes to the output layer."],"supporting_citations":[{"why":"Supplies the moment-tensor construction and the systematic-improvement idea that CAMP's atomic moments are based on.","marker":"[36]"},{"why":"Establishes the completeness of the moment-tensor/atomic-cluster-expansion basis, the theoretical ground for CAMP's completeness claim.","marker":"[37]"},{"why":"Defines the atomic cluster expansion whose B-basis plays the role CAMP's hyper moments play.","marker":"[35]"},{"why":"Provides the LiPS dataset and the leading spherical-tensor baseline whose errors CAMP compares against.","marker":"[17]"},{"why":"Supplies the bulk water dataset and its reference energies and forces, plus the BPNN baseline.","marker":"[39]"},{"why":"Provides the MD17 small-molecule trajectories on which CAMP is benchmarked.","marker":"[8]"},{"why":"Defines the MD stability protocol, stability criteria, timesteps, and the comparison results for many models used in the MD tests.","marker":"[47]"},{"why":"Provides the Cartesian-cluster-expansion model comparison values for water and is the closest Cartesian-space alternative to CAMP.","marker":"[38]"},{"why":"Supplies the bilayer graphene dataset, the hybrid-neural-network baseline, and the carbon-potential comparisons CAMP improves on.","marker":"[40]"}],"fun_headline_variants":["No spherical harmonics needed for top MLIP accuracy","Cartesian moments match leading spherical-tensor models","CAMP: Cartesian-only potential ties state-of-the-art MLIPs","Eliminate spherical harmonics with Cartesian tensor MLIPs","Cartesian tensors rival spherical harmonics in MLIPs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The paper assumes the restricted set of contractions it allows still describes the local environment completely enough for any target accuracy; that completeness is inherited from earlier moment-tensor theory for the unrestricted basis, not proved for CAMP's truncated rules.","fun_headline_variants_meta":{"raw":{"variants":["No spherical harmonics needed for top MLIP accuracy","Cartesian moments match leading spherical-tensor models","CAMP: Cartesian-only potential ties state-of-the-art MLIPs","Eliminate spherical harmonics with Cartesian tensor MLIPs","Cartesian tensors rival spherical harmonics in MLIPs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001052,"raw_usage":{"total_tokens":4382,"prompt_tokens":872,"completion_tokens":3510,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":488,"completion_tokens_details":{"reasoning_tokens":3440}},"tokens_in":488,"tokens_out":3510,"duration_ms":26538,"temperature":1.0,"reasoning_tokens":3440,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T17:54:28.650097+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Train CAMP and an unrestricted Cartesian moment-tensor model, allowing arbitrary contractions among lower-rank moments, with the same $v_{\\max}$, channels, and data on a system with strong many-body effects, and compare force errors. If the unrestricted model is substantially more accurate, or if CAMP's error stops decreasing as $v_{\\max}$ goes from 3 to 4 on a broad benchmark, the restricted contraction rule has dropped information the paper's completeness argument requires.","supporting_citations":[{"cited_title":"Batzner, A","cited_arxiv_id":null,"evidence_quote":"Provides the MD17 small-molecule trajectories on which CAMP is benchmarked."}],"review_version":1}