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NEMORA: Neural Equivariant Multipole Operators for Long-Range Atomistic Learning

Jay L. Kaplan, Samuel Varner, Rebecca Willett, Juan J. de Pablo

arXiv:2610.10776v1cs.LGphysics.chem-phphysics.comp-ph

TL;DR

Finite cutoffs limit long-range interactions in atomistic potentials, while existing extensions do not jointly provide learned equivariant propagation, multiscale many-body coupling, and linear scaling. NEMORA learns equivariant multipole transport on an adaptive hierarchy; on non-local benchmarks, it reduces force and energy errors relative to short-range models by over an order of magnitude and up to three orders of magnitude, respectively.

  • Problem

    Finite spatial cutoffs limit atomistic models’ treatment of long-range interactions, and existing extensions do not jointly offer learned physics-informed propagation, cross-degree coupling, higher-body interactions, and O(N) scaling.

    Source: §1 · p. 2

  • Method

    NEMORA learns E(3)-equivariant multipole transport on an adaptive hierarchy, coupling angular degrees and forming higher-body interactions among distant particles.

    Source: §1 · p. 2

  • Results

    On non-local benchmarks, NEMORA reduces force errors by over an order of magnitude and energy errors by up to three orders of magnitude versus short-range models, outperforming or matching other long-range methods.

    Source: §1 · p. 2

  • Takeaways & Limitations

    NEMORA improves four short-range backbones and has linear runtime and memory up to approximately 340,000 atoms on a single GPU.

    Source: §1 · p. 2

    EFA and LOREM cost comparisons are not accuracy-matched because they use different backbones and feature representations from NEMORA.

    Source: §5.6 · p. 14

Abstract

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Equivariant graph neural networks have emerged as foundational architectures for machine-learned interatomic potentials, approaching quantum-chemical accuracy at a fraction of the computational cost. These models describe local atomic environments accurately, but finite spatial cutoffs truncate long-range information flow, and stacking message-passing layers can lead to over-smoothing and over-squashing. Existing long-range extensions either prescribe a fixed analytical propagation kernel, restrict long-range communication to scalars or degree-preserving channels, are only approximately equivariant, or incur super-linear computational cost. Combining learnable long-range equivariant transport with multiscale many-body expressivity and efficient scaling for larger systems remains a central challenge. We introduce Neural Equivariant Multipole Operators (NEMORA), a neural equivariant extension of the Fast Multipole Method (FMM) for learning long-range tensorial representations. NEMORA generalizes the FMM's analytical multipole expansion and translation operators to learned equivariant counterparts on an adaptive spatial hierarchy. Its operators couple angular degrees and form many-body interactions across length scales, retaining the FMM's hierarchical organization and analytical radial factors as physical inductive biases while learning data-dependent long-range couplings. NEMORA evaluates in linear time and memory complexity, allowing it to treat larger systems than other long-range methods reaching hundreds of thousands of atoms, and it augments both symmetry-constrained and unconstrained short-range backbones. On non-local benchmarks, it reduces force and energy errors relative to the short-range backbones by over an order of magnitude and up to three orders of magnitude, respectively, which is better than or competitive with existing long-range extensions in accuracy.

1 Introduction

Finite-cutoff equivariant models leave long-range interactions incompletely represented, while deeper message passing is costly and can over-smooth or over-squash. NEMORA extends the Fast Multipole Method with learned equivariant, many-body long-range transport, improving non-local benchmark accuracy while retaining linear scaling.

Motivation

  • Finite spatial cutoffs limit equivariant models’ treatment of long-range effects, while stacking message-passing layers raises cost and causes over-smoothing and over-squashing.Cutoff graphs also cannot connect disconnected fragments, motivating dedicated long-range channels with tractable scaling.Source: §1 · p. 2
  • Existing long-range methods do not jointly provide learned propagation with a physics-inductive bias, cross-degree angular coupling, distant higher-body interactions, and O(N) cost on an adaptive hierarchy.The FMM offers a hierarchical, linear-complexity foundation with spherical harmonics and analytical radial factors for equivariant transport.Source: §1 · p. 2

NEMORA

  • NEMORA replaces scalar FMM charges and analytical operators with learned equivariant tensor charges and E(3)-equivariant operators on an adaptive hierarchy.Learned angular-degree coupling, radial modulation, invariant gates, and symmetric contractions create long-range many-body interactions; the channel updates backbone features and adds a residual energy term.Source: §1 · p. 2

Global reaction-field channel

  • A global reaction-field channel uses a learnable Kelvin transform to turn pooled multipoles into a same-center local expansion and broadcast an equivariant continuum boundary response to atoms.This channel complements hierarchical transport.Source: §1 · p. 2

Accuracy, portability, and scaling

  • Force errors fall by over an order of magnitude and energy errors by up to three orders of magnitude versus short-range models on non-local benchmarks.NEMORA is better than or competitive with other long-range methods, and evaluation remains linear in time and memory up to approximately 340,000 atoms on a single GPU.Source: §1 · p. 2
  • NEMORA improves four short-range backbones, including symmetry-constrained and unconstrained architectures; scalar-only hosts recover part of the native-equivariant gain.The method is also reported to have linear runtime and memory scaling up to approximately 340,000 atoms on a single GPU.Source: §1 · p. 2

2 Background

The background introduces equivariant machine-learned interatomic potentials and the Fast Multipole Method as complementary ingredients: the former represent atomic environments, while the latter organizes long-range interactions hierarchically. Their operators and scaling properties provide context for NEMORA.

2.1 Equivariant Machine-Learned Interatomic Potentials

  • An MLIP maps atomic positions to per-atom energies, with forces given by the negative energy gradient; cutoff graphs define the neighbors used by message-passing layers.Learned message and update functions aggregate neighbor information into node features and invariant energy readouts.Source: §2.1 · p. 3
  • Equivariant graph networks represent features by angular degree and magnetic component, and Clebsch–Gordan tensor products couple feature blocks along allowed paths.Learned weights may depend on invariants, while symmetric tensor contractions build higher-body-order features; correlation order ν gives terms of up to ν + 1 atoms.Source: §2.1 · p. 3

2.2 Fast Multipole Method

  • The FMM reduces direct Coulomb evaluation from O(N^2) to O(N) by grouping particles in a spatial hierarchy and representing distant groups with truncated multipole expansions.Upward stages aggregate source moments; M2L converts well-separated cluster multipoles into local expansions, which downward stages carry to particles.Source: §2.2 · p. 3
  • At fixed expansion order, FMM evaluation takes O(N) time and memory when the hierarchy has linear size, bounded leaf occupancy, and bounded interaction lists.The far-field series converges for separated source and target enclosures.Source: §2.2 · p. 4

3 Related Work

Related work extends long-range modeling through scalar, tensorial, global, and hierarchical approaches, but the cited approaches leave distinct trade-offs in learned angular transport, many-body expressivity, equivariance, or cost. NEMORA is positioned as combining these capabilities through FMM-style hierarchy.

Scalar Long-Range Channels and Charge Equilibration

  • Scalar long-range channels and charge equilibration communicate invariant quantities, while NEMORA transmits scalar and higher-degree tensor charges through a learned kernel at O(N) cost without requiring a Fourier grid or periodicity.Some charge-equilibration formulations use dense solves with O(N^3) time and O(N^2) memory, although alternative formulations improve scaling.Source: §3 · p. 4

Tensorial Transport and Many-Body Representations

  • Many tensorial long-range methods retain prescribed kernels or degree-preserving filters, while global matrix-function approaches provide equivariant many-body representations at O(N^3) time and O(N^2) memory.NEMORA conditions operators on directions to learn cross-degree coupling and uses symmetric contractions for many-body terms at linear cost.Source: §3 · p. 4

Global and Hierarchical Communication

  • Global all-to-all attention costs O(N^2), scalar virtual-node aggregation transmits only invariants, and fixed-quadrature tensorial attention is exactly equivariant only up to its quadrature degree.Other approaches use enlarged finite cutoffs or graph hierarchies for communication across scales.Source: §3 · p. 4
  • NEMORA uses the FMM well-separation criterion for its hierarchy, unlike cited tree architectures, and therefore its analytical limit inherits multipole convergence guarantees.The related approaches include coarsened graph hierarchies, spatial trees, and fragment-based corrections.Source: §3 · p. 5

FMM-Inspired Neural Architectures

  • For atomistic potentials, NEMORA translates multipole features between well-separated regions using equivariant tensor transport, learned angular coupling, modulated analytical radial factors, and many-body contractions.The related FMM-inspired neural methods target PDE operators, point clouds, or differentiable solvers with prescribed kernels.Source: §3 · p. 5

4 NEMORA Architecture

NEMORA makes FMM stages learnable for equivariant tensor features on an adaptive hierarchy. Its design interprets local message passing as a learnable multipole expansion, then extends that correspondence across scales and into energy readout.

4 NEMORA Architecture

  • NEMORA makes the P2M, M2M, M2L, L2L, and L2P stages learnable for equivariant tensor features up to a chosen maximum degree on an adaptive hierarchy.Unless stated otherwise, the architecture attaches to a MACE backbone.Source: §4 · p. 5

4.1 Neural Equivariant Multipole Operators

  • The analytical subset replaces scalar FMM charges with equivariant tensor charges while retaining the component-wise Coulomb kernel and coupled harmonic blocks.Source degree sets expressivity, expansion order sets accuracy, and these learned charges are not physical atomic multipoles unless kernel derivatives are used.Source: §4.1 · p. 5

4.2 MLIP Messages and Multipole Expansions

  • NEMORA reads pooled equivariant message-passing features as learnable local multipole expansions, with source features as tensor charges and radial and path weights replacing fixed factors.Setting the specified features, radial weights, path weights, and retained degrees to their analytical counterparts recovers classical moments; symmetric contractions build body order as FMM carries them across scales.Source: §4.2 · pp. 6–7

4.3 Learnable Equivariant Multipole Operators

  • Each analytical operator is replaced by a learnable equivariant operator that mixes channels within degrees and learns weights over compatible Clebsch–Gordan paths.Bounded modulation of analytical radial factors and invariant-based gates adapt the transport, while the analytical operator is recovered under specified settings; many-body contractions and Kelvin transforms add further features and boundary response.Source: §4.3 · p. 7

4.4 Energy Fusion and Readout

  • At selected backbone layers, NEMORA uses tensor charges from backbone features, updates those features with long-range messages, and adds a scalar readout as a residual energy term.Forces are obtained by differentiating the energy, as described in the introduction.Source: §4.4 · pp. 7–8

4.5 Adaptive E(3)-Equivariant Hierarchy

  • The adaptive hierarchy subdivides clusters using geometry-dependent frames and stops when cluster diameter falls below the short-range receptive field or atom count falls below a threshold.Near and far interactions are separated by a geometric criterion; near leaf pairs use a gap-distance envelope for a smooth handoff, and evaluation is linear when the hierarchy has O(N) nodes and bounded interaction lists.Source: §4.5 · p. 8

5 Results

Across molecular and periodic benchmarks, NEMORA improves long-range predictions while generally preserving short-range accuracy. The results also test its portability across backbones and its time and memory scaling to systems of hundreds of thousands of atoms.

5.1 Cumulene

  • NEMORA has the lowest cumulene ID energy error and is second to MACE–MFN on the remaining splits and metrics; MACE–MFN has cubic time and quadratic memory scaling.NEMORA substantially improves energy and force predictions over short-range MACE and scalar long-range models.Source: §5.1 · pp. 8–9, Table 1 · p. 9

5.2 Nonlocal charge transfer

  • Across all three nonlocal charge-transfer datasets, NEMORA ranks first or second on five of six metrics, with the lowest force errors on carbon chain and NaCl and lowest energy error on Au2/MgO.On the NaCl cluster, long-range models generally capture binding-profile effects that short-range models miss.Source: §5.2 · p. 10, Figure 2 · p. 10

5.3 MD22

  • On six of seven MD22 systems, NEMORA lowers energy and force errors relative to MACE; it achieves the lowest energy and force errors among MACE, MACE–LES, and NEMORA on four systems.The two regressions versus MACE are 0.06 meV/Å in force error on DHA and 0.026 meV/atom in energy error on Stachyose.Source: Table 1 · p. 9, §5.3 · p. 11

5.4 Portability across backbones and the role of angular features

  • Degree-two NEMORA sources reduce cumulene errors across all four tested backbones, and a geometry-based lift recovers much of that improvement for scalar-only backbones.For the predominantly electrostatic NaCl cluster, scalar NEMORA already captures most of the improvement over short-range models.Source: §5.4 · p. 11, Table 2 · p. 11

5.5 Periodic ions and water

  • On the periodic ionic melt, NEMORA and MACE–LES reduce MACE energy errors by two to three orders of magnitude and force errors by more than an order of magnitude; both extrapolate to larger unseen sizes without losing accuracy.NEMORA is slightly more accurate at nearly every tested size.Source: §5.5 · p. 12, Table 3 · p. 12
  • For periodic water, dipolar NEMORA has the lowest tested energy and force errors; in 2048-molecule dynamics, NEMORA and MACE–LES reproduce long-wavelength polarization correlations that short-range MACE does not.The NEMORA potential energy and temperature remain stable along the trajectory, while all three models give nearly identical O–O radial distributions.Source: §5.5 · p. 12, Figure 3 · p. 13

5.6 Time and memory scaling

  • NEMORA scales linearly in time and memory in periodic and non-periodic systems, reaching about 340,000 atoms on one H200; one evaluation takes about 1.2–1.3 s at 262,144 atoms.The fitted exponents for the long-range component are 0.89–1.00.Source: §5.6 · p. 13, Figure 4 · p. 14
  • RANGE and EFA also scale linearly but are less accurate on the long-range benchmarks; RANGE uses more memory than NEMORA at fixed backbone capacity, and EFA costs vary with its grid and boundary setting.The EFA and LOREM scaling comparisons are not accuracy-matched because they use different backbones and feature representations.Source: §5.6 · pp. 13–14

6 Discussion and Conclusion

NEMORA extends short-range potentials with a learned equivariant multipole hierarchy; the conclusion highlights long-range modeling results and notes limits on equivariance and analytical guarantees.

6 Discussion and Conclusion

  • On cumulene, NEMORA captures torsional profiles and ranks second to MACE–MFN on unseen chain lengths at linear rather than cubic cost.The discussion contrasts this with scalar long-range models, which miss the response to terminal-group rotation.Source: §6 · p. 14
  • On ionic clusters, scalar NEMORA already offers a significant improvement where scalar charges describe much of the interaction.The conclusion also reports that higher-degree atomic dipoles improve water predictions.Source: §6 · p. 14
  • The hierarchy is equivariant only up to eigenframe degeneracy, and learned operators lack the convergence guarantees of analytical FMM.The authors identify broader chemical tests on more complex systems as a next step.Source: §6 · p. 14

Reproducibility, data, and code availability

  • The authors report PyTorch and JAX implementations, distinguish SEMs from molecular-dynamics block estimates, and identify single-run measurements separately.Datasets and code are slated for release upon publication; trained models are planned for a future release.Source: Reproducibility, data, and code… · p. 15

A.1 Kernel and expansion equivariance

  • Under geometry-consistent centers, memberships, interaction lists, and tie handling, relative displacements ensure translation invariance and source sums ensure permutation equivariance.The appendix extends the proof to O(3).Source: §A.1 · p. 28

A.2 Stage operators and phase conventions

  • The irregular addition identity requires |a| < |b|; separated source and target enclosures then ensure convergence across the target region.The appendix identifies regular shifts as M2M and, by transposition, L2L; L2P evaluates the local field.Source: §A.2 · p. 29

B.1 Kernel and expansion equivariance

  • Tensor transport through a scalar kernel is distinct from physical multipole interactions, which require kernel derivatives.The appendix establishes equivariance of coupled expansion blocks while retaining spatial order as a multiplicity label.Source: §B.1 · p. 30

B.2 Stage operators and phase conventions

  • The coupled input and output degrees obey triangle bounds, with spatial orders retained as multiplicity labels; setting charge degree to zero recovers scalar FMM.The appendix describes changing the coupling order through recoupling identities.Source: §B.2 · p. 30

C Learnable NEMORA: Proofs

The proofs establish equivariance for learned operators and the readout pipeline, while clarifying how invariant gates are used.

C.1 Learned operator equivariance

  • Invariant radial weights and same-degree, same-parity channel mixing preserve equivariance, and covariant interaction-list selection preserves it under summation.Analytical recovery follows from specified choices of gates, mixing, and weights, with required blocks and M2L filters retained through 2p.Source: §C.1 · p. 31

C.2 Invariant gates

  • Invariant gates use geometric invariants, block norms, and same-irrep, same-parity alignments; they are not self-consistent polarization equations..Source: §C.2 · p. 31

C.3 Invariant readout and equivariant forces

  • The equivariant backbone, lift, residuals, contractions, Kelvin maps, periodic correction, and even-scalar readouts yield the stated readout and force properties.The force identity holds wherever the energy is differentiable; the periodic correction H is omitted for nonperiodic systems.Source: §C.3 · p. 32

D Symmetric Contraction: Proofs

The proofs derive symmetric contractions from a separable pair expansion and establish their equivariance.

D.1 Kernel factorization and distributivity

  • Under finite factorization and separable pair expansions, accumulated coefficients let the target field be written using basis functions and local coefficients.For FMM, the basis functions are regular solid-harmonic components about the target center, and the accumulated coefficients are local coefficients.Source: §D.1 · p. 32
  • Distributivity yields an exact identity for the represented factorization, with distinct-source terms reaching body order ν + 1 and repeated indices contributing lower orders.Exact reconstruction requires the product basis and product translation maps; Equation 28 instead projects products into configured equivariant feature blocks.Source: §D.1 · p. 32

D.2 Equivariance

  • Parity-compatible generalized Clebsch–Gordan maps make symmetric contractions equivariant on expansion coefficients before evaluation..Source: §D.2 · p. 32

E Global Kelvin: Proofs

The proofs construct local fields from multipole and local states and show that the construction respects equivariance.

E.1 Local-field construction and summation

  • Kelvin inversion maps irregular harmonics to regular harmonics for a support centered at c_u, with convergence when the original sources lie within radius a < R_u.Each support’s field must be evaluated within its convergence domain; fields from different centers are evaluated at a common target rather than added coefficientwise.Source: §E.1 · pp. 32–33
  • The learned map pools support states and broadcasts them to particles as an additive response, without requiring linear feature-dependent maps or a shared dielectric boundary problem..Source: §E.1 · p. 33

E.2 Equivariance

  • Kelvin inversion commutes with rotations, and the learned map is equivariant when support centers, selection, and pooling transform consistently with geometry and particle reindexing.The tensor-valued field’s component action also commutes with the Kelvin map.Source: §E.2 · p. 33

F Cluster-Local Kelvin: Proofs

At each hierarchy node, the local-field addition combines the regular field from Equation 26 with the downward expansion in a common center, basis, normalization, and validity domain. Subsequent equivariant L2L and L2P maps preserve the addition identity, including with feature-dependent gates.

F.1 Local-field addition

  • At each hierarchy node, the regular field from Equation 26 is added to the downward expansion about the same center.The fields are expressed in a common basis, normalization, and validity domain.Source: §F.1 · p. 33

F.2 Equivariance

  • Equivariant L2L and L2P maps preserve the field-addition identity even when feature-dependent gates make them nonlinear.Source: §F.1 · p. 33, §F.2 · p. 34

F.1 Local-field addition

  • The learned degree-diagonal map is a local-state correction, not necessarily an exact tensor-field inversion or dielectric solution.Source: §F.1 · p. 33

G Periodic Correction: Proof

The supplied sections establish equivariance and conditional linear evaluation costs, describe kernel and long-range model families, and detail learned and analytical correction mechanisms. On the charge-transfer benchmarks, NEMORA has the lowest carbon-chain energy error and the lowest or second-lowest force error on every nonlocal charge-transfer system.

G.1 Equivariance

  • The degree-diagonal correction commutes with D(ℓ)(R), establishing equivariance but not exact periodic electrostatics.Source: §G.1 · p. 34

H Linear-Time Complexity

  • Evaluation runtime and memory are O(N) under bounded hierarchy and interaction-list sizes, fixed angular degrees and channel widths, and fixed K and image shell; hierarchy construction is excluded.A separation test alone does not guarantee bounded interaction lists.Source: H Linear-Time Complexity · p. 34

I Analytical Kernel Families; J Long-Range Models

  • Analytical free-space kernel families require their own addition coefficients and, where applicable, radial branches or tensor structure; Table 5 compares long-range methods across six architectural axes.Only the Coulomb-based model is benchmarked here; the other families are not claims of implemented NEMORA variants.Source: I Analytical Kernel Families · p. 35, Table 4 · p. 35, Table 5 · p. 36

K.1 Symmetric contractions

  • Symmetric contractions add degree-ν products to pooled gated M2L fields, providing cross-cluster features for non-additive interactions; disabling contractions leaves the gated M2L operator.The stated linear cost is at fixed degrees, channels, and contraction order.Source: §K.1 · p. 37

K.2 Kelvin transform; K.3 Periodic boundary correction

  • The Kelvin response pools and broadcasts a learned continuum-inspired field, while the periodic branch applies a finite-shell, degree-diagonal root correction that does not enforce charge neutrality.The degree-diagonal response need not be an exact tensor Kelvin inversion; the periodic correction is a modeled root response, not an exact tinfoil lattice sum.Source: §K.2 · p. 37, §K.3 · p. 37

K.4 Analytical NEMORA; L.4 Backbone portability

  • The analytical NEMORA variant uses scalar latent charges and fixed FMM coefficients without learned transport or contractions; Table 9 compares short-range, scalar-source, and degree-two models across backbones.Tensor-source extensions add latent dipoles or traceless quadrupoles but retain analytical translations and fixed kernel derivatives.Source: §K.4 · p. 38, §L.4 · p. 41, Table 9 · p. 41

L.2 Charge-equilibration and electrostatic references

  • Published nonlocal charge-transfer rankings are not controlled comparisons because their splits may not match ours; reported ranks compare means, not statistical significance.Energy errors are normalized per atom before aggregation, while force RMSE pools Cartesian components within each run.Source: §L.2 · p. 39, §L.1 · p. 39
  • NEMORA has the lowest carbon-chain energy error and the lowest or second-lowest force error on every nonlocal charge-transfer system.Charge equilibration gives lower energy errors on the monopolar Na8Cl+8/Na9Cl+8 and Au2/MgO benchmarks, including with two-interaction CELLI–MACE.Source: §L.2 · p. 39, Table 6 · p. 40

L.5 Component ablations

Component ablations identify hierarchical transport and angular feature degree as key contributors to accuracy, while ionic, water, and numerical tests probe size transfer and solver behavior. The later scaling protocol fits long-range cost separately from matched short-range cost.

L.5 Component ablations

  • Removing the hierarchy returns carbon-chain, NaCl, and Au2/MgO force errors close to short-range MACE; the Kelvin branch alone helps, but the hierarchy is needed for the lowest errors.The ablation controls are retrained and compared with their corresponding FULL controls.Source: §L.5 · pp. 41, 43
  • Carbon-chain force error rises from 2.07 to 17.00 meV/Å when NEMORA is restricted to scalar features and operators; scalar NEMORA still reaches 3.50 meV/Å on NaCl versus 43.89 meV/Å for short-range MACE.Cumulene is especially sensitive to feature degree, with degree-zero and degree-one variants approaching the short-range error.Source: §L.5 · p. 43
  • FMMCLASSIC lowers Au2/MgO force error to 4.86 meV/Å but worsens cumulene test errors, consistent with simpler transport sufficing for some predominantly electrostatic systems while cumulene benefits from more flexible operators.Removing the standalone Kelvin branch has little effect on two small clusters but raises Au2/MgO force error from 8.50 to 10.28 meV/Å; removing per-box Kelvin contributions raises NaCl error from 1.89 to 2.40 meV/Å.Source: §L.5 · p. 43

L.6 Ionic systems

  • For non-periodic ionic clusters, NEMORA has 0.2–1.3% lower force RMSE at every size; scalar-source NEMORA matches LES closely and extrapolates to larger systems under both boundary conditions.The benchmark uses fixed ±0.6e charges, with larger test sizes excluded from training.Source: §L.6 · pp. 43–44

L.7 Periodic water

  • Adding atomic dipoles further improves energy and force errors over scalar-source NEMORA in the single-seed water experiment, at additional computational cost; no quadrupolar accuracy comparison is reported.Scalar analytical NEMORA and LES evaluate the same long-range energy functional by different numerical methods, with periodic runs differing in neutrality projection.Source: §L.7 · pp. 44–45

L.8 Numerical convergence

  • A factor of 4.46 reduction in force error follows tightening θ from 0.5 to 0.35 for periodic ions, while direct interaction rows increase from 312,989 to 915,502.At θ = 0.4, every pair is routed to P2P, recovering direct summation to roundoff; the accuracy tests do not establish linear runtime for the complete analytical backend because the reference smoothing correction enumerates all within-graph pairs.Source: §L.8 · pp. 45–46

L.9 Water NVE energy conservation; L.10 Elongated-water molecular dynamics and dipole correlations

  • In water, the NVE energy-fluctuation log–log slope is 2.06, consistent with ∆t^2 scaling; as k→0, gL(k) approaches a dielectric-set finite value, unlike the spurious rise without long-range electrostatics.At 0.25 fs, the NVE fluctuation approaches a fitted floor of about 2×10^-3 meV/atom, consistent with single-precision round-off.Source: §L.9 · p. 46, Figure 5 · p. 46, §L.10 · p. 47

L.11 Scaling protocol and complete measurements

  • Long-range scaling exponents are fitted after subtracting matched short-range cost, but absolute timing and memory statistics are not directly comparable across frameworks.The timing and memory protocol excludes dataset preparation and compilation.Source: §L.11 · p. 47

M.6 Periodic Ionic Benchmark

The appendix specifies ionic and periodic-water datasets and the model protocols used for comparisons. It also documents evaluation controls and the interface used to supply equivariant or lifted tensor features to NEMORA.

M.6 Periodic Ionic Benchmark

  • The ionic benchmark uses equimolar systems with fixed ±0.6e charges and Lennard-Jones interactions; because charges are assigned after geometry sampling, the corpus does not develop equilibrium ionic screening correlations.Training sizes are N = 256, 512, and 1024, with N = 2048 and 4096 reserved for test size extrapolation.Source: §M.6 · p. 51

M.7 Periodic Water; N.7.7 Periodic water

  • Periodic water configurations contain 128 or 256 molecules labeled with revPBE-D3, while the shared model uses one MACE interaction, a 6 Å cutoff, and long-range evaluation in float64.The periodic water systems contain 384 or 768 atoms; the analytical settings use p = 3, θ = 0.35, and image-shell radius w = 2.Source: §M.7 · p. 51, §N.7.7 · p. 57

N Model Settings; N.1 MACE; N.2 MACE–LES; N.5 MACE–RANGE

  • MACE-based comparisons share a short-range architecture where specified, while LOREM, EFA, and MACE–RANGE retain their own training implementations.For LOREM and EFA, settings prioritize comparable receptive fields, angular resolution, maximum irrep degree, and balanced parameter counts.Source: N Model Settings · p. 52, §N.1 · p. 52, §N.2 · p. 53

N.7.5 MD22

  • MD22 comparisons use shared splits, backbone, batch size, and schedule for MACE, MACE–LES, and NEMORA; EFA test predictions repeat identically across 21 runs, though CPU/GPU parity was not checked for every test checkpoint.The MD22 models were evaluated on held-out test splits using validation-selected checkpoints, except MACE–RANGE, which uses its final 400-epoch checkpoint.Source: §N.7.5 · p. 56

N.8 Backbone interfaces

  • For scalar-only hosts, a geometry-based lift combines scalar features with relative-position angular information; it preserves equivariance for invariant scalar inputs but does not make the entire PET backbone exactly equivariant.Native equivariant hosts instead provide their typed scalar and tensor features directly.Source: §N.8 · p. 57

N.11 Component ablation settings

Table 27 defines the component-ablation arms and identifies joint changes.

N.11 Component ablation settings

  • Table 27 defines the component-ablation arms.The table distinguishes the changes made in each arm.Source: Table 27 · p. 59
  • All component-ablation arms are retrained.The definitions apply to retrained arms.Source: Table 27 · p. 59
  • Joint changes are identified explicitly in the ablation definitions.The table marks arms that change multiple components together.Source: Table 27 · p. 59
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