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TreeRef-BFN: Equivariance-Free De Novo Molecule Generation based on 2D Topology and Internal 3D Geometry

Ruiqing Sun, Sen Yang, Dawei Feng, Bo Ding, Yijie Wang, Huaimin Wang

arXiv:2609.32502v1cs.AIcs.LG

TL;DR

De novo 3D molecular generation must jointly handle size, topology, and geometry, but existing methods often preselect size or rely on Cartesian or autoregressive generation. TreeRef-BFN represents these variables on a tree and jointly generates them with a Bayesian Flow Network and standard Transformer. The reported evaluations show competitive validity and stability alongside more accurate local geometry, with conditional property-generation results also reported.

  • Problem

    Many methods preselect molecular size or use Cartesian geometry, while internal-coordinate approaches typically require known connectivity or sequential construction.

    Source: §1 · pp. 1–2

  • Method

    TreeRef-BFN uses a tree representation with occupancy-based size and RingRef ring closures, jointly generating discrete topology and continuous local geometry through a Bayesian Flow Network and standard Transformer.

    Source: §1 · p. 2, §3.1 · p. 4

  • Results

    TreeRef-BFN achieves competitive chemical validity and molecular stability while maintaining high uniqueness and more accurate local geometry than the reference distributions.

    Source: §4.2 · pp. 7–8

  • Takeaways & Limitations

    One pretrained model supports unconditional generation and variable-size structure-conditioned tasks through masking, while property conditioning supports inverse design.

    Source: §1 · p. 2, §5 · p. 10

    During decoding, later sibling directions may leave some angle constraints unsatisfied because previously placed directions are not revisited.

    Source: §3.3 · p. 7

Abstract

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De novo 3D molecular generation jointly models molecular size, topology, and geometry. Most methods pre-sample molecular size and generate Cartesian coordinates, limiting variable-size conditional tasks such as fragment completion and scaffold decoration while often relying on equivariant architectures. Internal-coordinate methods avoid rigid-body redundancy but typically require a known molecular graph or autoregressive construction, which may accumulate errors. We propose TreeRef, a tree-based molecular representation that assigns molecular topology and topology-dependent local 3D geometry to a naturally variable-size tree. RingRef nodes encode ring closures while preserving the tree structure, while Null nodes allow molecular size to emerge directly from node occupancy. Based on TreeRef, we develop TreeRef-BFN, a Bayesian Flow Network with a standard Transformer backbone that globally couples these locally defined variables and jointly generates discrete molecular variables and continuous local geometry. A single pretrained TreeRef-BFN supports unconditional generation and variable-size structure-conditioned 3D generation through masking alone, without retraining. Empirical studies demonstrate strong chemical validity, molecular stability, and diversity, accurate local geometric distributions, fast sampling, and competitive property-conditioned generation, establishing TreeRef-BFN as an efficient and flexible framework for 3D molecular generation.

1 INTRODUCTION

De novo 3D generation must jointly model molecular size, topology, and geometry, but existing approaches often fix size, generate Cartesian coordinates, or construct internal geometry autoregressively. TreeRef-BFN represents these variables on a tree and uses a shared pretrained model for unconditional and variable-size conditional generation.

Limitations of existing methods

  • Many existing methods preselect molecular size and model p(G, X | N), which is less natural when conditional tasks require size to adapt to the input.The stated examples include fragment completion, scaffold decoration, linker generation, and property-driven inverse design.Source: §1 · pp. 1–2
  • Cartesian geometry represents local structure indirectly, so coordinate errors can lead to incorrect connectivity when bonds must be inferred, while equivariant models add architectural complexity and computational cost.The paper notes these concerns for methods that do not explicitly generate bonds and for many Cartesian models.Source: §1 · p. 2

Motivation

  • Internal coordinates describe bond lengths, angles, and torsions directly, but depend on known local connectivity and can make de novo generation order-dependent.The cited sequential approach defines geometry as reference atoms become available, allowing earlier construction errors to propagate and accumulate.Source: §1 · p. 2

Proposed approach

  • TreeRef places molecular size, topology, and local geometry on a full k-ary tree, with Null occupancy determining size and RingRef nodes encoding non-tree bonds.A standard-Transformer Bayesian Flow Network jointly generates the tree variables and supports unconditional, structure-conditioned, and variable-size editing tasks through masking.Source: §1 · p. 2

2 RELATED WORK

Prior work uses trees for molecular graphs and spans Cartesian, equivariant, and internal-coordinate approaches to 3D generation. TreeRef distinguishes itself by using a tree as shared support for topology and local geometry, generated jointly with a standard Transformer.

2.1 MOLECULAR GRAPH GENERATION AND TREE-BASED REPRESENTATIONS

  • TreeRef uses a tree as a shared index for atomic topology and local 3D geometry, with RingRef nodes encoding ring closures on that support.This contrasts with junction trees of chemical substructures and spanning-tree decompositions used for autoregressive 2D graph generation.Source: §2.1 · p. 2, Figure 1 · p. 3

2.2 DE NOVO 3D MOLECULAR GENERATION

  • Cartesian-space 3D generators include equivariant diffusion and latent-diffusion methods as well as mixed discrete–continuous diffusion and flow-matching approaches.The cited examples include EDM, GCDM, GeoLDM, MiDi, JODO, SemlaFlow, FlowMol3, GeoBFN, and VecMol.Source: §2.2 · pp. 2–3
  • TreeRef instead generates internal topology and local geometry on shared tree support with a standard, non-equivariant Transformer, then decodes Cartesian coordinates in a root-fixed frame.The section also contrasts this design with autoregressive size emergence in Symphony++ and standard-Transformer Cartesian approaches.Source: §2.2 · p. 3

2.3 INTERNAL COORDINATES AND INTRINSIC MOLECULAR GEOMETRY

  • Internal-coordinate methods commonly condition on a known graph, while G-SphereNet extends them to de novo generation through sequential atom construction.TreeRef assigns topology-dependent geometry to stable tree slots so size, topology, and internal geometry can be generated jointly rather than autoregressively.Source: §2.3 · p. 3

3 METHOD

TreeRef encodes occupancy, topology, and local geometry in a packed tree, and TreeRef-BFN jointly models its discrete and continuous variables. Decoding reconstructs a graph and coordinates from generated tree variables, including ring closures.

3.1 ENCODING TREEREF REPRESENTATION

  • Each slot packs role, element, parent or host bond, local geometry, optional formal charge, and handedness; geometry stores distances, pairwise angle cosines, and a dihedral representation.The geometry variables are local internal quantities rather than directly generated Cartesian coordinates.Source: §3.1 · p. 4
  • Null is a learned occupancy class rather than padding, so molecular size emerges from generated slot occupancy instead of being sampled in advance.Only Real slots are expanded; Null slots have prescribed categorical targets and zero geometry.Source: §3.1 · p. 4
  • RingRef nodes encode non-tree edges as virtual leaves that refer to existing Real atoms, placing ring closures within the same tree representation.Each RingRef occupies an available child slot and records the corresponding ring-closing bond.Source: §3.1 · p. 4

3.2 BAYESIAN FLOW TRAINING AND SAMPLING

  • A single Transformer jointly models dependencies across TreeRef slots and channels within Bayesian flows for categorical variables and continuous geometry.The model predicts categorical distributions and clean local geometry; the sampling step count can be chosen at inference without retraining.Source: §3.2 · p. 5
  • Conditional generation reuses the pretrained BFN by masking observed variables and evolving unobserved variables, with task-specific masks supporting structure conditioning and fragment completion.For serialized discrete structure conditioning, topology variables are observed while geometry and handedness are regenerated; fragment completion can also generate occupancy.Source: §3.2 · p. 6

3.3 TREEREF DECODING

  • Decoding traverses occupied slots to construct topology, reconstructs coordinates from local geometry, and assigns RingRef targets to add ring edges.RingRef assignment minimizes virtual-point discrepancy subject to element, BFS, degree, and bonding constraints; incompatible angular geometry can require relaxing constraints during coordinate construction.Source: §3.3 · pp. 6–7

4 EXPERIMENTS

Across unconditional and conditional tasks, TreeRef-BFN combines competitive chemical validity and stability with accurate local geometry. The experiments also evaluate variable-size conditioning and sampling efficiency.

4.1 EXPERIMENTAL SETUP

  • Experiments use QM9 and GEOM-Drugs with explicit hydrogens; mask-based structural tasks reuse the unconditional checkpoint, while property-conditioned inverse design uses separate models.The reported setup uses 10,000 samples and 100 Bayesian update steps by default.Source: §4.1 · p. 7

4.2 UNCONDITIONAL GENERATION

  • TreeRef-BFN offers competitive chemical validity and stability, high uniqueness, and substantially more accurate local geometry than baselines on QM9.Its 100-step setting provides competitive generation quality with highly efficient sampling.Source: §4.2 · pp. 7–8, Table 1 · p. 8
  • Single-angle geometry yields 90.12% validity and 88.27% molecule stability; post-hoc ring closure takes sampling time from 12.3 to 22.4, with 92.4% validity and 85.4% molecule stability.The comparisons report that pairwise angles carry more information and that unified RingRef encoding improves the model’s account of rings.Source: §4.2 · p. 8
  • On GEOM-Drugs, TreeRef-BFN reaches 82.4% RDKit validity and 80.1% molecule stability while achieving the lowest bond-, angle-, and dihedral-distribution distances.The results distinguish difficulty producing fully valid structures from alignment between generated connectivity and local 3D geometry.Source: §4.2 · p. 8, Table 2 · p. 9

4.3 CONDITIONAL GENERATION

  • For fixed-structure 2D-to-3D generation, Frozen validity falls from S to M to L, while M-inv2 recovers most of the loss.Multi-view consistency makes the model less tied to a particular TreeRef serialization when using the fixed graph as a condition.Source: Table 3 · p. 9, §4.3 · p. 9
  • M-inv2 improves Frozen success on all four variable-size editing tasks, with scaffold replacement increasing from 33.6% to 63.8%.Success requires RDKit validity and preservation of all supplied fragments; scaffold replacement is the most challenging task.Source: §4.3 · p. 9
  • At 100 steps, TreeRef-BFN has lower error than NExT-Mol and PropMolFlow on gap, HOMO, and LUMO, and remains close on the other three properties.At 1,000 steps, it achieves lower error on all six targets.Source: §4.3 · p. 9

5 CONCLUSION

The paper presents TreeRef-BFN as a tree-based framework for jointly representing molecular size, topology, and local geometry, with several generation tasks supported by conditioning. It identifies serialization robustness, structural flexibility, and chemical coverage as areas for future work.

5 CONCLUSION

  • TreeRef jointly encodes molecular size, topology, and local geometry in a k-ary representation, and TreeRef-BFN generates from it with a Bayesian Flow Network and standard Transformer.Node occupancy determines size, and RingRef encodes non-tree bonds.Source: §5 · p. 10
  • Masking supports unconditional generation, 2D-to-3D generation, and fragment completion, while property conditioning supports inverse design.Source: §5 · p. 10
  • Improving serialization robustness, structural flexibility, and chemical coverage remains future work.Source: §5 · p. 10

AI USE STATEMENT

The construction details specify reproducible tree serialization and topology-dependent local geometry; the AI-use statement assigns responsibility for the paper’s content and scientific conclusions to the authors.

AI USE STATEMENT

  • The authors state that AI tools assisted with language polishing, implementation, debugging, and document preparation, while authors determined the methods, experiments, analysis, and conclusions.They also say AI-assisted code was reviewed, tested, and verified, and that they take full responsibility for the paper.Source: AI USE STATEMENT · p. 12

A.1 CONSTRUCTION DETAILS

  • Canonical ordering uses bond-aware WL ranks and input indices to choose roots and neighbors, making serialization deterministic for fixed input indexing but not invariant to reindexing.BFS assigns parents on enqueue and places children in discovery order.Source: §A.1 · p. 12
  • Each omitted non-tree bond becomes a RingRef leaf pointing to an existing atom, preserving the spanning tree without duplicating physical atoms.RingRef hosts are selected subject to available child-slot capacity.Source: §A.1 · pp. 12–13
  • Packed child blocks preserve sibling order while omitting unused branches; internal Null slots remain supervised because removing one would shift later child positions.Only the shared trailing suffix is cropped, at a complete block boundary, and omitted child blocks decode as Null.Source: §A.1 · p. 13
  • Ordered local stars identify the directions used for angular and handedness channels, with Real atoms and RingRefs using distinct reference centers.For Real atoms, the star can be a strict subset of the physical neighborhood; RingRef descriptors are centered on the referenced atom.Source: §A.1 · p. 13
  • Bond distances and a dihedral specify local shape and rotation, while undefined rotations receive a zero pair rather than an arbitrary torsion.The dihedral references are chosen by a first-available rule; positive distances alone are standardized using accepted training bond lengths.Source: §A.1 · pp. 13–14
  • Handedness distinguishes mirror orientations that angular cosines cannot separate, but its stable label depends on prescribed reference order and is not an atom-order-independent stereochemical designation.Near-coplanar neighborhoods are assigned the inactive state.Source: §A.1 · p. 14

A.2 DECODING DETAILS

Decoding first recovers a consistent occupied tree, then constructs coordinates under fixed constraint priorities and resolves ring closures; an optional chemical stage adjusts labels on the recovered connectivity.

A.2 DECODING DETAILS

  • Decoding recovers the reachable occupied tree by consuming each queued Real slot’s complete child block, inverting the packed BFS map and retaining inherited sibling positions.An incomplete child block is invalid; omitted child blocks are implicit Null, and RingRef slots do not enter the queue.Source: §A.2 · pp. 14–15
  • The decoder resolves conflicting geometry through a fixed priority: scalar clipping first, then coordinate construction that may relax incompatible angular constraints.Pairwise cosine values can be individually bounded yet jointly unrealizable unless their target Gram matrix is positive semidefinite and has rank at most three.Source: §A.2 · p. 15
  • Root coordinates fix global freedom by placing the root at the origin, its first child on the x-axis, and its second child in the xy-plane.When the first three target cosines imply an impossible squared height, a planar fallback changes the requested angles; it has no minimum-error guarantee.Source: §A.2 · p. 16
  • At non-root atoms, the parent supplies an axis while a placed reference and generated dihedral determine azimuth; child placement prioritizes parent-relative and first-child cosines.For later children, the decoder chooses between two mirror candidates by residual to the second child, with handedness selecting the second child’s branch.Source: §A.2 · pp. 16–17
  • RingRef targets are filtered by element, connectivity, degree, and BFS constraints before joint assignment minimizes total virtual-point discrepancy under shared degree budgets.The fast decoder fails if no feasible assignment exists; with coordinate fusion disabled, ring closure leaves coordinates fixed and retains any virtual-to-target discrepancy.Source: §A.2 · pp. 17–18
  • The optional rules decoder promotes isolated RingRefs before matching, either relabeling a nearby Real atom or materializing a new atom at the virtual point.The fast decoder does not perform this promotion, and coordinates are not moved.Source: §A.2 · p. 18
  • After structural decoding, the chemical stage searches element and bond labels subject to valence and element-pair bond-order constraints without changing fixed geometry or connectivity.QM9 search may restart with beam parameters after exceeding 200,000 states; beam search is approximate, while completed exhaustive search is exact.Source: §A.2 · pp. 18–19

B.1 DATA PREPROCESSING

The preprocessing describes accepted QM9 and GEOM-Drugs data and their TreeRef representations. The model details specify Transformer configurations and property-conditioning pathways.

B.1 DATA PREPROCESSING

  • QM9 preprocessing retained explicit hydrogens, used a five-element vocabulary with charge disabled, and excluded disconnected or geometry–connectivity-inconsistent records.TreeRef scanning accepted 97,629 training, 20,011 validation, and 13,045 test records after excluding 105, 31, and 10 disconnected records, respectively.Source: §B.1 · p. 19
  • GEOM-Drugs preprocessing retained the lowest-energy conformer, filtered unsupported elements and degree-five atoms, and left 242,992 training, 30,518 validation, and 30,312 test molecules.The accepted data use explicit hydrogens and single-RingRef representations with k = 3.Source: §B.1 · pp. 19–20
  • Representations use training-fitted distance normalization and canonical validation and test serializations, with packed widths of 88 for QM9 and 526 for GEOM-Drugs.The QM9 cache stores up to eight training serializations, but the main configuration trains on the canonical variant.Source: §B.1 · p. 20

B.2 MODEL DETAILS

  • Bayesian Flow Networks provide the shared framework for discrete molecular variables and continuous geometry, with continuous-time training permitting sampling at any chosen positive step count without retraining.The Transformer uses time conditioning, full self-attention, and AdaLN-Zero modulation.Source: §B.2 · p. 20
  • The QM9 models span 10.50M, 60.74M, and 179.04M parameters, while GEOM-Drugs-XL has 319.83M parameters.The property-conditioned QM9 configuration has 179.64M parameters including its property embedding and null-condition vector.Source: §B.2 · p. 20, Table 6 · p. 20
  • The property-conditioned QM9 models combine normalized target and time embeddings, injecting the condition throughout Transformer blocks to influence categorical and geometric predictions.Classifier-free guidance uses a learned null-condition vector during training, sharing model parameters between conditional and null-conditioned predictions.Source: §B.2 · p. 21

B.3 TRAINING DETAILS

Training uses categorical and Gaussian Bayesian flows over packed TreeRef slots, while unconditional and masked conditional sampling share the trained model. The appendix specifies the training, sampling, conditioning, hardware, and evaluation procedures.

B.3 TRAINING DETAILS

  • A shared flow time drives categorical states and Gaussian geometry states, which the Transformer processes jointly to predict categorical distributions and clean geometry.Training samples a time uniformly and adds independent noise across slots and channels.Source: §B.3 · pp. 21–22
  • Every packed slot is supervised, including Null occupancy labels and zero clean geometry; multi-view consistency is optional and omitted in canonical-only runs.Property-conditioned training splits QM9 molecules between generator and regressor training, then uses frozen regressors and classifier-free guidance with w = 2.0 at evaluation.Source: §B.3 · pp. 22–23

B.4 HARDWARE

  • QM9 models use NVIDIA RTX A6000 GPUs, while GEOM-Drugs-XL uses three NVIDIA H100 GPUs.QM9-S uses global batch size 64; GEOM-Drugs-XL uses 16.Source: §B.4 · p. 23

B.5 UNCONDITIONAL SAMPLING

  • Unconditional sampling starts from uniform categorical beliefs and a zero-mean, unit-precision Gaussian geometry prior, then applies S Bayesian updates and a final prediction.Changing S changes inference discretization without changing the trained model; the default protocol uses S = 100.Source: §B.5 · pp. 23–24

B.6 CONDITIONAL SAMPLING

  • Conditional sampling reuses an unconditional checkpoint without parameter updates, masking observed variables while updating unobserved ones under frozen or RePaint-style conditioning.For serialized-structure generation, R, E, B, and optional Q are observed while geometry and handedness are regenerated; structural editing can also regenerate attachment-site geometry.Source: §B.6 · pp. 24–25

B.7 EVALUATION DETAILS

  • Evaluation reports chemical validity and stability, reconstructed-coordinate geometry distributions, and task-specific agreement or fragment-retention metrics under a 10,000-sample, 100-step default protocol.The geometric Wasserstein-1 distances compare generated and validation-reference distributions, while fragment preservation is evaluated on decoded molecules.Source: §B.7 · pp. 26–27

C.1 MODEL SIZE AND SAMPLING STEPS ON QM9

On unconditional QM9 generation, larger TreeRef-BFN models improve validity and stability across tested sampling budgets, while geometric metric rankings are not uniform. Increasing the sampling budget brings further, sometimes non-monotonic, gains.

C.1 MODEL SIZE AND SAMPLING STEPS ON QM9

  • At 100 steps, RDKit validity rises from 85.98% (S) to 95.15% (M) and 96.56% (L), while molecule stability rises from 84.60% to 94.52% and 96.19%.Uniqueness stays above 96% for all three models, while novelty decreases with model size; at this budget M has lower bond-angle distance than L, but L has lower dihedral distance.Source: §C.1 · p. 28, Figure 2 · p. 28, Table 7 · p. 30
  • At 100 steps, M-inv2 validity and molecule stability are 94.70% and 94.00%, below M at 95.15% and 94.52%, and its three geometric distances are also larger.M-inv2 has 47.9% novelty versus 43.2% for M at 100 steps; its validity and stability remain near M and short of L across the reported sweep.Source: §C.1 · p. 28
  • For L, 2,000 versus 100 steps improves validity from 96.56% to 97.53% and molecule stability from 96.19% to 97.17%.The largest quality gains occur between 10 and 100 steps; bond-length and bond-angle errors decrease overall, but dihedral agreement is non-monotonic.Source: §C.1 · p. 30

C.2 ABLATIONS ON QM9

QM9-L ablations show that removing child-to-child angle information worsens geometric and chemical metrics, while assigning ring closures after coordinate construction lowers stability and increases sampling time.

C.2 ABLATIONS ON QM9

  • Removing child-to-child cosines lowers validity from 96.56% to 90.12% and raises bond-length W1 from 0.0040 Å to 0.0242 Å.Molecule stability falls from 92.52% to 88.27%, and dihedral distance rises from 1.187° to 2.874°; sampling time is 12.1 versus 12.3 for the full model.Source: §C.2 · p. 32
  • Post-hoc ring closure reduces validity to 92.4% and molecule stability to 85.4%, while increasing sampling time from 12.3 to 22.4.Bond-length W1 remains close to the full model at 0.0038 Å versus 0.0040 Å, and angle and dihedral distances remain close.Source: §C.2 · p. 32

C.3 PROPERTY-CONDITIONED GENERATION ON QM9

The QM9 property-conditioned evaluation compares generated molecules with requested targets using two frozen regressors. The TreeRef regressor reports closer target values than the EGNN regressor across properties and sampling budgets.

C.3 PROPERTY-CONDITIONED GENERATION ON QM9

  • The EGNN regressor has lower validation mean absolute error than the TreeRef regressor on every target, motivating its use in the main comparison.Both regressors are trained on the scorer half of QM9 and evaluated on the validation split.Source: §C.3 · p. 32, Table 8 · p. 32
  • The TreeRef regressor reads generated molecules as closer to requested targets than the EGNN regressor on every property and sampling budget.This gap reflects the evaluators, not a second generator; at 100 steps, RDKit validity ranges from 93.8% for Cv to 97.2% for ∆ϵ across listed targets.Source: §C.3 · p. 32

C.4 SAMPLING ERROR DIAGNOSIS

Unconditional QM9 sampling exhibits distinct decoding and chemical-stability failures: unmatched RingRefs prevent decoding, while valence errors affect many decoded molecules. Rules repairs some failures by changing labels without moving coordinates.

C.4 SAMPLING ERROR DIAGNOSIS

  • At 100 steps, exact-valence instability affects 20.5% (S), 9.3% (M), 7.2% (L), and 10.3% (M-inv2) of decoded molecules, while unmatched RingRefs cause decode failures.Decode-failure rates are 1.48% (S), 0.62% (M), 0.34% (L), and 0.20% (M-inv2).Source: §C.4 · pp. 32, 34, Figure 6 · p. 34
  • Under-valent carbon is the main chemical instability; because it can still sanitize, RDKit invalidity is lower than instability.The usual error is bond-order sum 3 where 4 is allowed, followed by over-valent hydrogen.Source: §C.4 · p. 34
  • Rules recovered 51/148 (S), 25/62 (M), and 9/34 (L) RingRef failures, then restored both checks for 1988/2015 (S), 918/924 (M), and 707/714 (L) unstable or invalid decoded molecules.Rules first resolves RingRefs, then relabels elements and bond orders; it does not move atoms, so long bonds remain long.Source: §C.4 · p. 34
  • In the illustrated repairs, Rules changes element and bond labels to address unmatched RingRefs and valence errors while retaining the fast decoder’s coordinates.The examples include an oxygen changed to nitrogen and an under-valent nitrogen changed to fluorine with a carbon–oxygen bond-order change.Source: §C.4 · p. 34, Figure 7 · p. 35

C.5 CONDITIONING STRATEGIES AND MODEL SCALING FOR 2D-TO-3D GENERATION

With the graph fixed, M-inv2 achieves strong validity and local-geometry results, while RePaint-style conditioning remains near 5% validity. Channel deletion and the authors’ account point to geometry’s role in predicting the graph during sampling.

C.5 CONDITIONING STRATEGIES AND MODEL SCALING FOR 2D-TO-3D GENERATION

  • With the graph fixed, M-inv2 stays near 94.5% xyz2mol validity and 0.66 Å mean aligned RMSD, with the smallest bond, angle, and dihedral distances among the four models.The comparison covers QM9-S, QM9-M, QM9-L, and M-inv2 across sampling steps.Source: §C.5 · p. 36, Figure 8 · p. 36
  • RePaint-style runs, including M-inv2, stay near 5% xyz2mol validity, and additional steps do not help while the graph remains noisy during geometry generation.The authors report that restoring the clean graph only at the end does not help the geometry formed while that graph is noised.Source: §C.5 · p. 36
  • At t = 0.1, removing geometry raises distance error from about 0.09 to 0.16 Å and lowers element accuracy from about 0.67 to 0.21–0.26; removing element or bond changes distance error by less than 0.004 Å.In unconditional sampling, blanking geometry hurts M, L, and M-inv2, whereas blanking element or bond does not.Source: §C.5 · p. 36, Figure 9 · p. 37
  • The authors attribute the model-order reversal to geometry-first coupling: M-inv2’s multi-tree augmentation and hidden-state alignment let a clean graph suffice to place geometry.They report that L leans more heavily on the coupling, so a finished graph paired with uninformed geometry pulls its distances off.Source: §C.5 · pp. 36–37

C.6 FRAGMENT COMPLETION AND EDITING

Fragment editing retains supplied atoms in their source-tree slots and grows the rest, with success defined by retaining the supplied fragment or fragments. Clean conditioning improves task success, and M-inv2’s multi-tree training improves scaffold replacement relative to M and L.

C.6 FRAGMENT COMPLETION AND EDITING

  • The four edits retain one source-molecule piece in its existing tree slot and grow the rest; success means retaining the supplied fragment, or both supplied fragments for the three multi-fragment tasks.The tasks range from short-linker generation to scaffold replacement, which keeps the least and rebuilds the core.Source: §C.6 · p. 37
  • With clean fragments, M beats S and L on every task; with noised fragments, L is slightly ahead on decoration, multi-fragment completion, and linkers, while scaffold replacement remains around 10–15% for all three.Longer sampling neither changes which task is hardest nor closes the gap between conditioning strategies.Source: §C.6 · p. 37, Figure 11 · p. 39
  • At 100 steps with Frozen conditioning, M-inv2 reaches 63.8% scaffold-replacement success, versus 54.9% for M and 33.6% for L.Across 10–2,000 steps, Frozen M-inv2 stays above L on every task and above M on decoration and scaffold replacement.Source: Figure 11 · p. 39, §C.6 · p. 40
  • RePaint-style conditioning leaves M-inv2 slightly below M because the fragment is noised during sampling and restored only at the end.The Figure 12 comparison reports fragment-edit success at 100 steps for Frozen and RePaint-style conditioning.Source: Figure 12 · p. 39, §C.6 · p. 40

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