Packora: Systematic Design for Generative Molecular Crystal Structure Prediction

# Packora: Systematic Design for Generative Molecular Crystal Structure Prediction
Source: [https://arxiv.org/html/2608.26962](https://arxiv.org/html/2608.26962)
###### Abstract

Molecular crystal structure prediction \(CSP\) is important in pharmaceuticals, agrochemicals, and organic electronics, where subtle differences in molecular conformation and packing can strongly affect material properties\. We presentPackora, a flow\-based generative model for molecular CSP that jointly predicts atomic coordinates and the lattice from molecular graphs\.Packorasupports multi\-component and organometallic crystals and can condition on any subset of molecular conformers, stereochemical labels, and space\-group information within a single model\. Inspired by the CCDC CSP blind test, we evaluate generation and ranking separately, using generation to isolate generator quality and ranking to measure end\-to\-end performance under a common relaxation and ranking pipeline\. We also systematically study architecture, training, conditioning, inference, and scaling, identifying an effective design based on cacheable pairwise reasoning, training objective and numerical solver choices, conditioning dropout, and balanced scaling of pairwise and single representations\.Packoraoutperforms the baselines on both structure generation and ranking benchmarks, achieving the best matched\-budget coverage across all six generation benchmarks, as well as higher experimental\-form recovery, lower experimental\-form ranks, and faster convergence in ranking\.

Figure 1:Crystal coverage with 30 candidates per target\.We report the crystal\-level solve rateSolC\\mathrm\{Sol\}\_\{C\}at a generation budget of 30 candidates per target\. Numbers in parentheses indicate the number of targets in each benchmark\. For OXtal, we report the published results obtained from 30 generated candidates per target\. For CLARI andPackora, we generate 1,000 candidates per target and report the mean over 5,000 bootstrap resamples of 30 candidates; error bars show one standard deviation across resamples\.Packoraachieves the highest solve rate on all six benchmarks\.## 1Introduction

Molecular crystals are widely used in pharmaceuticals, agrochemicals, and organic electronics\([Beran, 2023](https://arxiv.org/html/2608.26962#bib.bib34)\)\. Their properties depend strongly on molecular conformation and packing, and different polymorphs of the same chemical system can exhibit distinct solubility, stability, bioavailability, charge transport, and mechanical properties\([Price, 2014](https://arxiv.org/html/2608.26962#bib.bib44);[Jin et al\., 2025](https://arxiv.org/html/2608.26962#bib.bib47);[Gharakhanyan et al\., 2025](https://arxiv.org/html/2608.26962#bib.bib51)\)\. Molecular crystal structure prediction \(CSP\) aims to predict possible crystal structures given the molecular components, accelerating drug development and functional materials design\.

Molecular CSP generally involves two complementary stages: structure generation and structure ranking\([Hunnisett et al\., 2024a](https://arxiv.org/html/2608.26962#bib.bib37);[Hunnisett et al\., 2024b](https://arxiv.org/html/2608.26962#bib.bib36)\)\. Traditional workflows generate candidate packings using random, quasi\-random, or evolutionary search\([Li et al\., 2018](https://arxiv.org/html/2608.26962#bib.bib45);[Curtis et al\., 2018](https://arxiv.org/html/2608.26962#bib.bib46)\), then relax and rank them using computationally expensive energy calculations, often based on density functional theory\([Price, 2014](https://arxiv.org/html/2608.26962#bib.bib44);[Reilly et al\., 2016](https://arxiv.org/html/2608.26962#bib.bib35);[Hunnisett et al\., 2024a](https://arxiv.org/html/2608.26962#bib.bib37);[Hunnisett et al\., 2024b](https://arxiv.org/html/2608.26962#bib.bib36)\)\. FastCSP primarily accelerates structure ranking by using a machine\-learning interatomic potential \(MLIP\) for structure relaxation and energy calculation, while obtaining initial candidates from a random structure generator\([Gharakhanyan et al\., 2025](https://arxiv.org/html/2608.26962#bib.bib51)\)\.

Generative models offer a faster approach to structure generation by learning from experimentally observed crystal structures and concentrating a finite candidate budget on plausible packings\. Despite recent progress\([Jin et al\., 2025](https://arxiv.org/html/2608.26962#bib.bib47);[Zeng et al\., 2026](https://arxiv.org/html/2608.26962#bib.bib49);[Subramanian et al\., 2026](https://arxiv.org/html/2608.26962#bib.bib48);[Lo et al\., 2026](https://arxiv.org/html/2608.26962#bib.bib50)\), several gaps remain\. Existing methods differ in the molecular information they require and the chemical systems they support, while evaluation protocols vary in candidate budgets and downstream processing, making generation and ranking performance difficult to compare directly\. Moreover, the effects of key design choices in architecture, training, conditioning, inference, and scaling remain only partially understood\.

In this work, we presentPackora, an all\-atom generative model for molecular CSP that jointly predicts atomic coordinates and the unit cell from molecular graphs containing atom types, bond types, and formal charges\.Packorasupports multi\-component and organometallic crystals and explicitly models hydrogen atoms\. A single model can also condition on additional structural information that is often available or readily derivable, such as molecular templates, stereochemical labels, and space\-group information, allowing it to adapt to different levels of prior knowledge\.

We also introduce a matched two\-track evaluation that separates structure generation from structure ranking, following the CCDC CSP blind tests\([Hunnisett et al\., 2024a](https://arxiv.org/html/2608.26962#bib.bib37);[Hunnisett et al\., 2024b](https://arxiv.org/html/2608.26962#bib.bib36)\)\. The generation track measures whether a fixed candidate set contains the experimental structure, without relaxation or energy\-based ranking, thereby isolating generator coverage\. The ranking track instead processes candidates from each generator through the same relaxation, filtering, deduplication, and lattice\-energy ranking pipeline and measures whether the experimental form is ranked near the top\. This separation enables direct comparison of generator quality and the downstream utility of generated candidates in practical CSP workflows\.

Finally, we systematically study key design choices in architecture, training recipe, conditioning, inference, and scaling\. We directly compare DiT with pair\-bias attention\([Peebles and Xie, 2023](https://arxiv.org/html/2608.26962#bib.bib22)\), Pairformer\([Abramson et al\., 2024](https://arxiv.org/html/2608.26962#bib.bib20)\), and Pairmixer\([Ouyang\-Zhang et al\., 2025](https://arxiv.org/html/2608.26962#bib.bib53)\), and ablate training choices such as time sampling, translation augmentation, and loss design\. These studies show that cacheable pairwise reasoning substantially improves generation quality, careful training and sampling choices matter, and balanced scaling of pairwise and single representations preserves coverage at larger model sizes\.

![Refer to caption](https://arxiv.org/html/2608.26962v1/fig_strict_structure_examples.png)Figure 2:Crystal structures predicted byPackora\.Each example is a collision\-free prediction fromPackora\-L that matches all 15 molecules of the experimental reference withRMSD15<2​Å\\mathrm\{RMSD\}\_\{15\}<2\\,\\text\{\\AA\}\. These 18 examples illustratePackora’s ability to generate diverse crystal structures across molecular sizes, compositions, packing motifs, and unit\-cell geometries\. Panel labels indicate the corresponding CSD refcodes\.For structure generation, across six benchmarks,Packora\-M orPackora\-L achieves the best coverage across candidate budgets and evaluation criteria, with up to a 77\.6% relative improvement over CLARI\-H \([Table11](https://arxiv.org/html/2608.26962#A5.T11)\)\. When combined with downstream relaxation and ranking,Packoraachieves higher recovery, faster convergence, and lower experimental\-form ranks than CLARI\-H on both the single\-form and polymorph FastCSP benchmarks\([Gharakhanyan et al\., 2025](https://arxiv.org/html/2608.26962#bib.bib51)\)\.

In summary, our contributions are:

- •A flexible all\-atom generative model for molecular CSP\.Packorajointly generates atomic coordinates and unit cells for diverse molecular systems while supporting optional conditioning on molecular templates, stereochemistry, and space groups\.
- •A matched evaluation of generation and ranking\.We separately measure finite\-budget generator coverage and downstream performance under a common relaxation\-and\-ranking pipeline\.
- •A systematic study of molecular CSP generative modeling\.We investigate architecture, training, conditioning, inference, and scaling under a common evaluation protocol and derive an evidence\-backed design recipe\.

## 2Related Work

Search\-based molecular CSP\.Search\-based molecular CSP typically involves exploring molecular conformations and crystal packings, followed by geometry relaxation and energy ranking\. Structure generation and ranking have long been central components of molecular CSP, with the seventh CCDC blind test explicitly evaluating them in separate phases\([Reilly et al\., 2016](https://arxiv.org/html/2608.26962#bib.bib35);[Hunnisett et al\., 2024a](https://arxiv.org/html/2608.26962#bib.bib37);[Hunnisett et al\., 2024b](https://arxiv.org/html/2608.26962#bib.bib36)\)\. Candidate structures are commonly generated across selected space groups by sampling unit\-cell parameters and molecular positions and orientations, with flexible molecules requiring additional sampling of intramolecular degrees of freedom\([Day et al\., 2009](https://arxiv.org/html/2608.26962#bib.bib33);[Reilly et al\., 2016](https://arxiv.org/html/2608.26962#bib.bib35)\)\. Generated structures are then deduplicated or clustered and evaluated hierarchically, often using force fields for initial relaxation and screening followed by dispersion\-corrected density functional theory for refinement and ranking\([Beran, 2023](https://arxiv.org/html/2608.26962#bib.bib34);[Hunnisett et al\., 2024a](https://arxiv.org/html/2608.26962#bib.bib37)\)\. Genarris uses constrained random generation followed by diversity\-based down\-selection\([Li et al\., 2018](https://arxiv.org/html/2608.26962#bib.bib45)\), whereas GAtor uses a first\-principles genetic algorithm with molecular\-crystal breeding operators and structural niching\([Curtis et al\., 2018](https://arxiv.org/html/2608.26962#bib.bib46)\)\. FastCSP accelerates relaxation and ranking stage using a machine\-learning interatomic potential, while retaining random structure generation for initial candidates\([Gharakhanyan et al\., 2025](https://arxiv.org/html/2608.26962#bib.bib51)\)\.

Generative molecular CSP\.Generative molecular CSP instead learns a distribution over experimental crystal structures\. OXtal is an all\-atom diffusion model trained on lattice\-free molecular crops and adapts the AlphaFold3 Pairformer trunk\([Abramson et al\., 2024](https://arxiv.org/html/2608.26962#bib.bib20)\)to atom\-level representations\([Jin et al\., 2025](https://arxiv.org/html/2608.26962#bib.bib47)\)\. MolCrystalFlow represents each molecule as a rigid body and learns geodesic flows over molecular centroids, orientations, and lattice parameters\([Zeng et al\., 2026](https://arxiv.org/html/2608.26962#bib.bib49)\)\. PackFlow applies reinforcement\-learning post\-training guided by MLIP energies and forces, improving physical validity and concentrating proposals in low\-energy basins\([Subramanian et al\., 2026](https://arxiv.org/html/2608.26962#bib.bib48)\)\. CLARI generates explicit unit cells and replaces triangle layers with pair\-bias attention, yielding a reported 15–30×\\timesspeedup over OXtal\([Lo et al\., 2026](https://arxiv.org/html/2608.26962#bib.bib50)\)\. However, none of these methods provides a single model in which molecular conformers, stereochemical labels, and space\-group information can each be supplied or omitted at inference time\([Jin et al\., 2025](https://arxiv.org/html/2608.26962#bib.bib47);[Lo et al\., 2026](https://arxiv.org/html/2608.26962#bib.bib50);[Subramanian et al\., 2026](https://arxiv.org/html/2608.26962#bib.bib48);[Zeng et al\., 2026](https://arxiv.org/html/2608.26962#bib.bib49)\)\. The capabilities of our model and prior approaches are summarized in[Table1](https://arxiv.org/html/2608.26962#S2.T1)\.

Table 1:Capabilities and scope of generative molecular CSP models\.We compare explicit hydrogen generation, chemical scope, and conditioning inputs\.✓and✗denote supported and unsupported capabilities\. For local 3D input,▲\\blacktriangle,✓, and✗denote required, optional, and unsupported conditioning, respectively\.Generative CSP\.A related line of work studies crystal generation and structure prediction for inorganic materials\. CDVAE combines a variational autoencoder with diffusion\-based coordinate generation, whereas DiffCSP jointly diffuses fractional coordinates and lattice conditioned on a given composition\([Xie et al\., 2022](https://arxiv.org/html/2608.26962#bib.bib28);[Jiao et al\., 2024](https://arxiv.org/html/2608.26962#bib.bib13)\)\. Subsequent approaches formulate periodic crystal generation using flow matching, as in FlowMM and CrystalFlow\([Miller et al\., 2024](https://arxiv.org/html/2608.26962#bib.bib26);[Luo et al\., 2025](https://arxiv.org/html/2608.26962#bib.bib27)\); stochastic interpolants, as in OMatG\([Höllmer et al\., 2025](https://arxiv.org/html/2608.26962#bib.bib11)\); or Bayesian flows over periodic variables\([Wu et al\., 2025](https://arxiv.org/html/2608.26962#bib.bib10)\)\.

Generative CSP has also been extended to metal\-organic frameworks \(MOFs\), where metal nodes and organic linkers can be treated as modular building blocks\. MOFFlow\([Kim et al\., 2025b](https://arxiv.org/html/2608.26962#bib.bib14)\)and MOF\-BFN\([Jiao et al\., 2025](https://arxiv.org/html/2608.26962#bib.bib9)\)generate lattices and rigid\-body poses of known building blocks using Riemannian and Bayesian flows, respectively, while MOFFlow\-2\([Kim et al\., 2025a](https://arxiv.org/html/2608.26962#bib.bib8)\)additionally generates building blocks and models linker torsions\. AtomMOF\([Kim et al\., 2026](https://arxiv.org/html/2608.26962#bib.bib32)\)removes the rigid\-body assumption by directly generating MOF and MOF–adsorbate structures at all\-atom resolution\.

## 3Method

Packorapredicts atomic coordinates and a unit cell from a molecular specification, a formula\-unit countZZ, and optional auxiliary information using a conditional coordinate–lattice flow\. We describe the selectedPackoraconfiguration below, with design choices justified by the controlled studies in[Section5](https://arxiv.org/html/2608.26962#S5)\.

### 3\.1Problem Formulation

Crystal representation\.We represent a crystal withNNatoms by atom types𝐀=\(A1,…,AN\)∈𝒜N\\mathbf\{A\}=\(A\_\{1\},\\ldots,A\_\{N\}\)\\in\\mathcal\{A\}^\{N\}, Cartesian coordinates𝐗=\(𝐱1,…,𝐱N\)⊤∈ℝN×3\\mathbf\{X\}=\(\\mathbf\{x\}\_\{1\},\\ldots,\\mathbf\{x\}\_\{N\}\)^\{\\top\}\\in\\mathbb\{R\}^\{N\\times 3\}, and a lattice cell𝐋∈ℝ3×3\\mathbf\{L\}\\in\\mathbb\{R\}^\{3\\times 3\}whose rows are lattice vectors\. Integer translations along these lattice vectors define the corresponding infinite periodic crystal\.

Since the same lattice can be represented by different basis vectors and global rotations, we canonicalize the cell in two steps\. First, we Niggli\-reduce\([Křivý and Gruber, 1976](https://arxiv.org/html/2608.26962#bib.bib3);[Grosse\-Kunstleve et al\., 2004](https://arxiv.org/html/2608.26962#bib.bib30)\)each crystal to obtain a reduced basis𝐋~\\widetilde\{\\mathbf\{L\}\}, reducing the ambiguity among equivalent lattice bases\. Second, we remove the remaining global rotational freedom by forming the rotation\-invariant Gram matrix𝐋~​𝐋~⊤\\widetilde\{\\mathbf\{L\}\}\\widetilde\{\\mathbf\{L\}\}^\{\\top\}and taking its unique lower\-triangular Cholesky factor with positive diagonal:

𝐋=chol⁡\(𝐋~​𝐋~⊤\)=\(eℓ100ℓ2eℓ30ℓ4ℓ5eℓ6\)\.\\mathbf\{L\}=\\operatorname\{chol\}\\\!\\left\(\\widetilde\{\\mathbf\{L\}\}\\widetilde\{\\mathbf\{L\}\}^\{\\top\}\\right\)=\\begin\{pmatrix\}e^\{\\ell\_\{1\}\}&0&0\\\\ \\ell\_\{2\}&e^\{\\ell\_\{3\}\}&0\\\\ \\ell\_\{4\}&\\ell\_\{5\}&e^\{\\ell\_\{6\}\}\\end\{pmatrix\}\.We further parameterize the canonical cell by an unconstrained vectorℓ=\(ℓ1,…,ℓ6\)∈ℝ6\\bm\{\\ell\}=\(\\ell\_\{1\},\\ldots,\\ell\_\{6\}\)\\in\\mathbb\{R\}^\{6\}following[Veljković et al\. \(2026\)](https://arxiv.org/html/2608.26962#bib.bib52)\.

Molecular crystal structure prediction \(CSP\)\.We formulate molecular CSP as conditional generation of Cartesian coordinates𝐗\\mathbf\{X\}and the lattice latentℓ\\bm\{\\ell\}from a chemical specification, a supplied formula\-unit countZZ, and optional auxiliary information\. The chemical specification consists ofKKdistinct molecular graphs𝒢=\{Gk\}k=1K\\mathcal\{G\}=\\\{G\_\{k\}\\\}\_\{k=1\}^\{K\}and a stoichiometry vector𝐫=\(r1,…,rK\)\\mathbf\{r\}=\(r\_\{1\},\\ldots,r\_\{K\}\), whererkr\_\{k\}denotes the number of copies of componentGkG\_\{k\}in one formula unit\. For example, a binary2:12\{:\}1cocrystal has𝐫=\(2,1\)\\mathbf\{r\}=\(2,1\)\. Each molecular graph specifies atom types, bond types, and formal charges\. To be specific, if componentGkG\_\{k\}containsNkN\_\{k\}atoms, a unit cell withZZformula units contains

N=Z​∑k=1Krk​NkN=Z\\sum\_\{k=1\}^\{K\}r\_\{k\}N\_\{k\}atoms\. Optional information𝒪\\mathcal\{O\}may include molecular templates, stereochemical annotations, and a space\-group label\. A molecular template is a reference 3D conformer, generated for example with RDKit\([Landrum and others, 2006](https://arxiv.org/html/2608.26962#bib.bib19)\), that provides local molecular geometry without specifying crystal packing\. Given the complete condition𝒞=\(𝒢,𝐫,Z,𝒪\)\\mathcal\{C\}=\(\\mathcal\{G\},\\mathbf\{r\},Z,\\mathcal\{O\}\), the generator modelspθ​\(𝐗,ℓ∣𝒞\)p\_\{\\theta\}\(\\mathbf\{X\},\\bm\{\\ell\}\\mid\\mathcal\{C\}\)\.

### 3\.2Variational Flow Matching for Molecular CSP

We use variational flow matching\([Eijkelboom et al\., 2024](https://arxiv.org/html/2608.26962#bib.bib12), VFM;\), which generalizes flow matching\([Lipman et al\., 2023](https://arxiv.org/html/2608.26962#bib.bib29)\)through a flexible choice of the variational posterior\([Zaghen et al\., 2025](https://arxiv.org/html/2608.26962#bib.bib21)\)\.

Training objective\.Let\(𝐲1,𝒞\)\(\\mathbf\{y\}\_\{1\},\\mathcal\{C\}\)denote a crystal state and its condition, and let𝐲0∼p0\\mathbf\{y\}\_\{0\}\\sim p\_\{0\}be a prior sample\. We use the linear conditional path to interpolate between the prior and data endpoints:

𝐲t=\(1−t\)​𝐲0\+t​𝐲1\.\\mathbf\{y\}\_\{t\}=\(1\-t\)\\mathbf\{y\}\_\{0\}\+t\\mathbf\{y\}\_\{1\}\.\(1\)
Given a noisy state𝐲t\\mathbf\{y\}\_\{t\}, VFM learns a variational posteriorqθ​\(𝐲1∣𝐲t,t,𝒞\)q\_\{\\theta\}\(\\mathbf\{y\}\_\{1\}\\mid\\mathbf\{y\}\_\{t\},t,\\mathcal\{C\}\)that approximates the true posterior by minimizing the expected negative log\-likelihood,

ℒVFM​\(θ\)=−𝔼𝐲1,𝐲t,t​\[log⁡qθ​\(𝐲1∣𝐲t,t,𝒞\)\]\.\\mathcal\{L\}\_\{\\mathrm\{VFM\}\}\(\\theta\)=\-\\mathbb\{E\}\_\{\\mathbf\{y\}\_\{1\},\\mathbf\{y\}\_\{t\},t\}\\\!\\left\[\\log q\_\{\\theta\}\(\\mathbf\{y\}\_\{1\}\\mid\\mathbf\{y\}\_\{t\},t,\\mathcal\{C\}\)\\right\]\.\(2\)The variational family determines the form of this objective\. We model the posterior as a fully factorized Laplace distribution with a fixed scale, which yields a component\-weightedL1L^\{1\}loss on the predicted endpoints\.

The model parameterizes the mean of the variational posterior, which is equivalent to predicting the clean endpoint𝐲1\\mathbf\{y\}\_\{1\}from the noisy state𝐲t\\mathbf\{y\}\_\{t\}and condition𝒞\\mathcal\{C\}as𝐗^1,ℓ^1=𝝁tθ​\(𝐲t,𝒞\)\\widehat\{\\mathbf\{X\}\}\_\{1\},\\widehat\{\\bm\{\\ell\}\}\_\{1\}=\\bm\{\\mu\}\_\{t\}^\{\\theta\}\\\!\\left\(\\mathbf\{y\}\_\{t\},\\mathcal\{C\}\\right\)\. The equation then simplifies to

ℒVFM​\(θ\)=𝔼𝐲1,𝐲t,t​\[λcoord3​N​‖𝐗^1−𝐗1‖1\+λlattice6​‖ℓ^1−ℓ1‖1\],\\mathcal\{L\}\_\{\\mathrm\{VFM\}\}\(\\theta\)=\\mathbb\{E\}\_\{\\mathbf\{y\}\_\{1\},\\mathbf\{y\}\_\{t\},t\}\\\!\\left\[\\frac\{\\lambda\_\{\\mathrm\{coord\}\}\}\{3N\}\\left\\\|\\widehat\{\\mathbf\{X\}\}\_\{1\}\-\\mathbf\{X\}\_\{1\}\\right\\\|\_\{1\}\+\\frac\{\\lambda\_\{\\mathrm\{lattice\}\}\}\{6\}\\left\\\|\\widehat\{\\bm\{\\ell\}\}\_\{1\}\-\\bm\{\\ell\}\_\{1\}\\right\\\|\_\{1\}\\right\],\(3\)whereλcoord\\lambda\_\{\\mathrm\{coord\}\}andλlattice\\lambda\_\{\\mathrm\{lattice\}\}are loss weights for the coordinate and lattice components, respectively\.

Sampling\.After training𝝁tθ\\bm\{\\mu\}\_\{t\}^\{\\theta\}, we generate samples by solving the ordinary differential equation \(ODE\) or the corresponding stochastic differential equation \(SDE\)\([Eijkelboom et al\., 2024](https://arxiv.org/html/2608.26962#bib.bib12);[Albergo et al\., 2025](https://arxiv.org/html/2608.26962#bib.bib1)\):

d​𝐲t=𝐯tθ​\(𝐲t\)​d​t,d​𝐲t=\[𝐯tθ​\(𝐲t\)\+gt22​𝐬tθ​\(𝐲t\)\]​d​t\+gt​d​𝐖t\.d\\mathbf\{y\}\_\{t\}=\\mathbf\{v\}\_\{t\}^\{\\theta\}\(\\mathbf\{y\}\_\{t\}\)\\,dt,\\qquad d\\mathbf\{y\}\_\{t\}=\\left\[\\mathbf\{v\}\_\{t\}^\{\\theta\}\(\\mathbf\{y\}\_\{t\}\)\+\\frac\{g\_\{t\}^\{2\}\}\{2\}\\mathbf\{s\}\_\{t\}^\{\\theta\}\(\\mathbf\{y\}\_\{t\}\)\\right\]dt\+g\_\{t\}\\,d\\mathbf\{W\}\_\{t\}\.\(4\)For the linear path,𝐯tθ​\(𝐲t\)=\(𝝁tθ​\(𝐲t,𝒞\)−𝐲t\)/\(1−t\)\\mathbf\{v\}\_\{t\}^\{\\theta\}\(\\mathbf\{y\}\_\{t\}\)=\(\\bm\{\\mu\}\_\{t\}^\{\\theta\}\(\\mathbf\{y\}\_\{t\},\\mathcal\{C\}\)\-\\mathbf\{y\}\_\{t\}\)/\(1\-t\)and𝐬tθ​\(𝐲t\)=\(t​𝐯tθ​\(𝐲t\)−𝐲t\)/\(1−t\)\\mathbf\{s\}\_\{t\}^\{\\theta\}\(\\mathbf\{y\}\_\{t\}\)=\(t\\,\\mathbf\{v\}\_\{t\}^\{\\theta\}\(\\mathbf\{y\}\_\{t\}\)\-\\mathbf\{y\}\_\{t\}\)/\(1\-t\)\. Here,gtg\_\{t\}is the diffusion coefficient controlling the stochasticity, and𝐖t\\mathbf\{W\}\_\{t\}is a standard Wiener process\.

### 3\.3Model Architecture

Our model𝝁tθ\\bm\{\\mu\}\_\{t\}^\{\\theta\}mainly employs a Pairmixer\-based architecture\([Ouyang\-Zhang et al\., 2025](https://arxiv.org/html/2608.26962#bib.bib53)\)with trunk\-head modularization\([Abramson et al\., 2024](https://arxiv.org/html/2608.26962#bib.bib20);[Team et al\., 2025](https://arxiv.org/html/2608.26962#bib.bib23)\)\. As shown in[Figure3](https://arxiv.org/html/2608.26962#S3.F3), the model first constructs condition\-only single and pair representations, refines the pair representation with Pairmixer, injects the noisy crystal state into the single representation, and processes it with a Diffusion Transformer \(DiT\)\([Peebles and Xie, 2023](https://arxiv.org/html/2608.26962#bib.bib22)\)\. Separate output heads predict the coordinate and lattice endpoints\. The complete architecture is specified algorithmically in[AppendixA](https://arxiv.org/html/2608.26962#A1)\.

SingleCondPairCond𝒞\\mathcal\{C\}Input EmbedderTriMulOut\+\+TriMulIn\+\+Transition\+\+Pairmixer \(×4\\times 4\)\+\+CoordLattice𝐗t\\mathbf\{X\}\_\{t\}ℓt\\bm\{\\ell\}\_\{t\}Noisy Input EmbedderLNScaleShiftAttentionScale\+\+LNScaleShiftMLPScale\+\+ModulationTransformer Trunk \(×16\\times 16\)SinuEmb→\\toMLPttTime EmbedderCoordHeadLatticeHead𝐗^1\\widehat\{\\mathbf\{X\}\}\_\{1\}ℓ^1\\widehat\{\\bm\{\\ell\}\}\_\{1\}Crystal Heads

Figure 3:Model architecture ofPackora\.The condition𝒞\\mathcal\{C\}is embedded into single and pair representations, with the pair representation refined by fourPairmixerblocks\([Ouyang\-Zhang et al\., 2025](https://arxiv.org/html/2608.26962#bib.bib53)\)\. Because this computation depends only on𝒞\\mathcal\{C\}, it can be cached across denoising steps\. Noisy coordinate𝐗t\\mathbf\{X\}\_\{t\}and latticeℓt\\bm\{\\ell\}\_\{t\}embeddings are then added to the single representation and processed by a DiT trunk, with the refined pair representation providing attention biases throughout\. Separate heads predict the coordinate and lattice endpoints𝐗^1\\widehat\{\\mathbf\{X\}\}\_\{1\}andℓ^1\\widehat\{\\bm\{\\ell\}\}\_\{1\}\.Condition embedding\.The input embedder embeds the condition𝒞=\(𝒢,𝐫,Z,𝒪\)\\mathcal\{C\}=\(\\mathcal\{G\},\\mathbf\{r\},Z,\\mathcal\{O\}\)into a per\-atom single representation𝐒𝒞\\mathbf\{S\}\_\{\\mathcal\{C\}\}and an initial per\-atom\-pair representation𝐏\(0\)\\mathbf\{P\}^\{\(0\)\}:

𝐒𝒞=Esingle​\(𝒞\),𝐏\(0\)=Epair​\(𝐒𝒞,𝒞\),\\mathbf\{S\}\_\{\\mathcal\{C\}\}=E\_\{\\mathrm\{single\}\}\(\\mathcal\{C\}\),\\qquad\\mathbf\{P\}^\{\(0\)\}=E\_\{\\mathrm\{pair\}\}\(\\mathbf\{S\}\_\{\\mathcal\{C\}\},\\mathcal\{C\}\),whereEsingleE\_\{\\mathrm\{single\}\}andEpairE\_\{\\mathrm\{pair\}\}are learned single and pair embedding networks\. The single representation combines atom identity, periodic\-table descriptors, formal charge, optional template coordinates, chirality, and space group\. The pair representation combines projected single features with intramolecular bond types, bond stereochemistry, template displacements, and template distances\. Availability masks and learned null states distinguish unavailable optional conditions from provided values\.

Pairmixer\.Pairmixer is a streamlined alternative to Pairformer that retains incoming and outgoing triangle multiplication and pair transitions while omitting triangle attention\. Each block updates the current pair representation𝐏\\mathbf\{P\}through sequential residual operations,

𝐏←𝐏\+TriMulout⁡\(𝐏\),𝐏←𝐏\+TriMulin⁡\(𝐏\),𝐏←𝐏\+Transition⁡\(𝐏\),\\mathbf\{P\}\\leftarrow\\mathbf\{P\}\+\\operatorname\{TriMul\}\_\{\\mathrm\{out\}\}\(\\mathbf\{P\}\),\\qquad\\mathbf\{P\}\\leftarrow\\mathbf\{P\}\+\\operatorname\{TriMul\}\_\{\\mathrm\{in\}\}\(\\mathbf\{P\}\),\\qquad\\mathbf\{P\}\\leftarrow\\mathbf\{P\}\+\\operatorname\{Transition\}\(\\mathbf\{P\}\),whereTriMulout\\operatorname\{TriMul\}\_\{\\mathrm\{out\}\}andTriMulin\\operatorname\{TriMul\}\_\{\\mathrm\{in\}\}denote outgoing and incoming triangle multiplication, respectively\. The transition is a gated feed\-forward network applied independently to each atom pair\. Pairmixer leaves the single representation unchanged and produces a refined pair representation𝐏⋆\\mathbf\{P\}^\{\\star\}\.

Noisy\-state injection\.The noisy crystal state is injected into the single representation𝐒𝒞\\mathbf\{S\}\_\{\\mathcal\{C\}\}\. Coordinate features are encoded using Fourier features\([Tancik et al\., 2020](https://arxiv.org/html/2608.26962#bib.bib6)\), while the lattice is embedded globally and broadcast to all atoms\. The resulting single representation is

𝐒t=𝐒𝒞\+EX​\(𝐗t\)\+Broadcast⁡\(EL​\(ℓt\)\),\\mathbf\{S\}\_\{t\}=\\mathbf\{S\}\_\{\\mathcal\{C\}\}\+E\_\{X\}\(\\mathbf\{X\}\_\{t\}\)\+\\operatorname\{Broadcast\}\\\!\\left\(E\_\{L\}\(\\bm\{\\ell\}\_\{t\}\)\\right\),whereEXE\_\{X\}andELE\_\{L\}denote the coordinate and lattice embedding networks, respectively\. A sinusoidal embedding of the flow timettprovides a separate global condition to the transformer\.

DiT trunk\.The trunk follows the DiT architecture, alternating between self\-attention and feed\-forward updates to the single representation\. In each attention layer, the refined pair representation𝐏⋆\\mathbf\{P\}^\{\\star\}is projected to a head\-specific bias and added to the attention logits:

ai​j\(h\)=⟨𝐪i\(h\),𝐤j\(h\)⟩dh\+bh​\(𝐏i​j⋆\),a\_\{ij\}^\{\(h\)\}=\\frac\{\\left\\langle\\mathbf\{q\}\_\{i\}^\{\(h\)\},\\mathbf\{k\}\_\{j\}^\{\(h\)\}\\right\\rangle\}\{\\sqrt\{d\_\{h\}\}\}\+b\_\{h\}\\\!\\left\(\\mathbf\{P\}\_\{ij\}^\{\\star\}\\right\),where𝐪i\(h\)\\mathbf\{q\}\_\{i\}^\{\(h\)\}and𝐤j\(h\)\\mathbf\{k\}\_\{j\}^\{\(h\)\}are the query and key vectors for atomsiiandjjin attention headhh,dhd\_\{h\}is the head dimension, andbhb\_\{h\}maps the pair representation to a scalar attention bias\. The time embedding modulates the normalization and residual gates of both self\-attention and feed\-forward updates\([Peebles and Xie, 2023](https://arxiv.org/html/2608.26962#bib.bib22)\)\.

Prediction heads\.The DiT output𝐒out\\mathbf\{S\}^\{\\mathrm\{out\}\}is decoded by separate coordinate and lattice endpoint heads\. For the set𝒱\\mathcal\{V\}of valid atoms, the coordinate head predicts an atomwise vector𝐫i\\mathbf\{r\}\_\{i\}and subtracts its mean over valid atoms, while the lattice head applies an MLP to the mean\-pooled single representation:

𝐫i=WX​Norm⁡\(𝐒iout\),𝐗^1,i=𝐫i−1\|𝒱\|​∑j∈𝒱𝐫j,ℓ^1=MLP⁡\(1\|𝒱\|​∑i∈𝒱Norm⁡\(𝐒iout\)\),\\mathbf\{r\}\_\{i\}=W\_\{X\}\\operatorname\{Norm\}\(\\mathbf\{S\}^\{\\mathrm\{out\}\}\_\{i\}\),\\qquad\\widehat\{\\mathbf\{X\}\}\_\{1,i\}=\\mathbf\{r\}\_\{i\}\-\\frac\{1\}\{\|\\mathcal\{V\}\|\}\\sum\_\{j\\in\\mathcal\{V\}\}\\mathbf\{r\}\_\{j\},\\qquad\\widehat\{\\bm\{\\ell\}\}\_\{1\}=\\operatorname\{MLP\}\\\!\\left\(\\frac\{1\}\{\|\\mathcal\{V\}\|\}\\sum\_\{i\\in\\mathcal\{V\}\}\\operatorname\{Norm\}\(\\mathbf\{S\}^\{\\mathrm\{out\}\}\_\{i\}\)\\right\),whereWXW\_\{X\}is the coordinate projection and𝐗^1\\widehat\{\\mathbf\{X\}\}\_\{1\}andℓ^1\\widehat\{\\bm\{\\ell\}\}\_\{1\}denote the predicted coordinate and lattice endpoints\.

## 4Benchmark Evaluation

Inspired by the CSP blind tests\([Hunnisett et al\., 2024a](https://arxiv.org/html/2608.26962#bib.bib37);[Hunnisett et al\., 2024b](https://arxiv.org/html/2608.26962#bib.bib36)\), we evaluate models along two complementary axes: structure generation and structure ranking\. Structure generation \([Section4\.2](https://arxiv.org/html/2608.26962#S4.SS2)\) measures unranked coverage at a fixed candidate budget, testing whether a generator can recover the experimental structure\. Structure ranking \([Section4\.3](https://arxiv.org/html/2608.26962#S4.SS3)\) measures whether generated candidates remain useful after downstream relaxation and ranking, by evaluating how highly the experimental form is ranked\. Unlike the CSP blind\-test ranking track, which fixes the candidate pool and varies the ranking method, we fix the downstream evaluator and vary the generator, thereby measuring end\-to\-end generator–evaluator compatibility\.

### 4\.1Experimental Setup

Dataset\.We use the official CLARI training and validation split\([Lo et al\., 2026](https://arxiv.org/html/2608.26962#bib.bib50)\), derived from the Cambridge Structural Database \(CSD\)\([Groom et al\., 2016](https://arxiv.org/html/2608.26962#bib.bib40)\)\. The held\-out test set comprises the OXtal Rigid and Flexible benchmarks\([Jin et al\., 2025](https://arxiv.org/html/2608.26962#bib.bib47)\), CSP5\([Bardwell et al\., 2011](https://arxiv.org/html/2608.26962#bib.bib38)\), CSP6\([Reilly et al\., 2016](https://arxiv.org/html/2608.26962#bib.bib35)\), CSP7\([Hunnisett et al\., 2024a](https://arxiv.org/html/2608.26962#bib.bib37);[Hunnisett et al\., 2024b](https://arxiv.org/html/2608.26962#bib.bib36)\), and the CSD Teaching Subset\([Battle et al\., 2010](https://arxiv.org/html/2608.26962#bib.bib39);[Lo et al\., 2026](https://arxiv.org/html/2608.26962#bib.bib50)\)\. To prevent leakage, CLARI excludes all entries from test refcode families and training structures that share an RDKit\-sanitizable molecular component with more than seven heavy atoms, then holds out 1,000 refcode families for validation\.

CLARI limits training and validation unit cells to 512 atoms, yielding 917,014 training and 1,048 validation entries\. Because we add missing hydrogens with the CSD Python API\([Sykes et al\., 2024](https://arxiv.org/html/2608.26962#bib.bib41)\)before counting atoms, we reapply the 512\-atom limit after hydrogen completion, leaving 912,807 training and 1,047 validation entries\.[AppendixB](https://arxiv.org/html/2608.26962#A2)provides detailed dataset statistics\.

Training and sampling configuration\.We trainPackora\-M \(88M parameters\) andPackora\-L \(187M parameters\) using Muon\([Liu et al\., 2025](https://arxiv.org/html/2608.26962#bib.bib4)\)for hidden matrix parameters and AdamW\([Loshchilov and Hutter, 2019](https://arxiv.org/html/2608.26962#bib.bib5)\)for all remaining parameters\. Both models use a learning rate of10−410^\{\-4\}, an effective batch size of 128, and an EMA decay of0\.99990\.9999; weight decay is 0 forPackora\-M and10−210^\{\-2\}forPackora\-L\. We also use bfloat16 and train on eight NVIDIA H200 GPUs\.

At inference, we condition on stereochemistry when available and on RDKit\-generated molecular conformers, reflecting information that is either specified in CSP blind tests\([Reilly et al\., 2016](https://arxiv.org/html/2608.26962#bib.bib35)\)or readily derived from molecular graphs\([Landrum and others, 2006](https://arxiv.org/html/2608.26962#bib.bib19)\)\. We do not use space\-group conditioning because space groups are generally not provided in CSP blind tests\. We generate candidates with the EDM–Heun sampler\([Karras et al\., 2022](https://arxiv.org/html/2608.26962#bib.bib16)\), using stochastic churn,ρ=7\\rho=7, and 200 sampling steps\. Complete hyperparameters are provided in[AppendixC](https://arxiv.org/html/2608.26962#A3)\.

Two\-stage training\.To enable faster iteration before scaling to larger structures, we follow the two\-stage training of SeedFold\([Yi et al\., 2025](https://arxiv.org/html/2608.26962#bib.bib24)\)\. In the first stage, we train on crystals with at most 300 atoms; in the second, we expand to all crystals with at most 512 atoms\. We save checkpoints every 50 epochs and select the model with the best validationSolC\\mathrm\{Sol\}\_\{C\}\.

### 4\.2Structure Generation

We first evaluate structure generation alone, following the structure generation phase of the seventh CCDC CSP blind test\([Hunnisett et al\., 2024b](https://arxiv.org/html/2608.26962#bib.bib36)\)\. We measure whether each generative model can recover the experimental structure within a fixed candidate budget, without energy evaluation, relaxation, or ranking\.

Benchmarks\.We evaluate on six benchmarks: the OXtal Rigid and Flexible sets, with 50 targets each\([Jin et al\., 2025](https://arxiv.org/html/2608.26962#bib.bib47)\); CSP5, CSP6, and CSP7, with 6, 5, and 8 targets, respectively\([Bardwell et al\., 2011](https://arxiv.org/html/2608.26962#bib.bib38);[Reilly et al\., 2016](https://arxiv.org/html/2608.26962#bib.bib35);[Hunnisett et al\., 2024b](https://arxiv.org/html/2608.26962#bib.bib36)\); and CLARI’s CSD Teaching Subset, with 773 targets\([Battle et al\., 2010](https://arxiv.org/html/2608.26962#bib.bib39);[Lo et al\., 2026](https://arxiv.org/html/2608.26962#bib.bib50)\)\.

Metric and protocol\.Following OXtal\([Jin et al\., 2025](https://arxiv.org/html/2608.26962#bib.bib47)\)and CLARI\([Lo et al\., 2026](https://arxiv.org/html/2608.26962#bib.bib50)\), we report the crystal\-level approximate solve rateSolC\\mathrm\{Sol\}\_\{C\}at candidate budgets of 30 and 1,000\. A target is considered solved if any candidate is collision\-free, at least 8 of 15 molecules are matched, andRMSD15<2​Å\\mathrm\{RMSD\}\_\{15\}<2\\text\{\\AA\}with COMPACK\([Chisholm and Motherwell, 2005](https://arxiv.org/html/2608.26962#bib.bib42)\)\.SolC\\mathrm\{Sol\}\_\{C\}is the fraction of solved targets\. We also report the stricterSolC15/15\\mathrm\{Sol\}\_\{C\}^\{15/15\}, which requires all 15 molecules to match\. For the 30\-candidate budget, we follow CLARI’s bootstrapping protocol\([Lo et al\., 2026](https://arxiv.org/html/2608.26962#bib.bib50)\): from each fixed pool of 1,000 candidates, we draw 5,000 resamples of 30 candidates and average the resultingSolC\\mathrm\{Sol\}\_\{C\}\. Full metric definitions are provided in[AppendixD](https://arxiv.org/html/2608.26962#A4)\.

Baselines\.We compare against OXtal and three CLARI variants: CLARI\-M, CLARI\-L, and CLARI\-H\([Jin et al\., 2025](https://arxiv.org/html/2608.26962#bib.bib47);[Lo et al\., 2026](https://arxiv.org/html/2608.26962#bib.bib50)\)\. For a consistent comparison, we re\-evaluate the official CLARI checkpoints using the OXtal evaluation protocol, since the two evaluators differ in collision detection and reference\-structure selection\. See[AppendixD](https://arxiv.org/html/2608.26962#A4)for details\.

Results\.Across all six benchmarks,Packoraachieves the highest solve rate at both 30\- and 1,000\-candidate budgets under the standard and strict criteria \([Figures1](https://arxiv.org/html/2608.26962#S0.F1),[4](https://arxiv.org/html/2608.26962#S4.F4),[5](https://arxiv.org/html/2608.26962#S4.F5)and[6](https://arxiv.org/html/2608.26962#S4.F6)\), outperforming the strongest baseline in 23 of 24 settings and tying on the saturated CSP5 result\. At 30 candidates, the relative gain reaches 54\.5% on CSP7 under the standard criterion and 188\.0% on CSP6 under the strict criterion\.[Figure2](https://arxiv.org/html/2608.26962#S1.F2)further demonstrates accurate recovery across diverse molecules and packing motifs\. Full tables are provided in[AppendixE](https://arxiv.org/html/2608.26962#A5), results withZZsampled from an empirical prior in[AppendixF](https://arxiv.org/html/2608.26962#A6), and aligned structure overlays in[AppendixG](https://arxiv.org/html/2608.26962#A7)\.

Figure 4:Crystal coverage with 1,000 candidates per target\.We report the crystal\-level solve rateSolC\\mathrm\{Sol\}\_\{C\}at a generation budget of 1,000 candidates per target\. Numbers in parentheses indicate the number of targets in each benchmark\.Packoraachieves the highest solve rate on all six benchmarks\.Figure 5:Crystal coverage with 30 candidates per target under the strict criterion\.We report the crystal\-level solve rateSolC15/15\\mathrm\{Sol\}\_\{C\}^\{15/15\}at a generation budget of 30 candidates per target\.SolC15/15\\mathrm\{Sol\}\_\{C\}^\{15/15\}requires all 15 molecules to match using COMPACK\([Bardwell et al\., 2011](https://arxiv.org/html/2608.26962#bib.bib38)\)\. Numbers in parentheses indicate the number of targets in each benchmark\. We generate 1,000 candidates per target and report the mean over 5,000 bootstrap resamples of 30 candidates; error bars show one standard deviation across resamples\.Packoraachieves the highest solve rate on all six benchmarks\.Figure 6:Crystal coverage with 1,000 candidates per target under the strict criterion\.We report the crystal\-level solve rateSolC15/15\\mathrm\{Sol\}\_\{C\}^\{15/15\}at a generation budget of 1,000 candidates per target\.SolC15/15\\mathrm\{Sol\}\_\{C\}^\{15/15\}requires all 15 molecules to match under COMPACK\([Bardwell et al\., 2011](https://arxiv.org/html/2608.26962#bib.bib38)\)\. Numbers in parentheses indicate the number of targets in each benchmark\.Packoraachieves the highest solve rate on all six benchmarks\.
### 4\.3Structure Ranking

Benchmarks\.We evaluate two benchmarks derived from FastCSP\([Gharakhanyan et al\., 2025](https://arxiv.org/html/2608.26962#bib.bib51)\)\. The single\-polymorph benchmark contains 23 semi\-rigid and three flexible systems, each with one experimental form\. The multi\-polymorph benchmark contains five semi\-rigid and three flexible systems, comprising 29 experimental forms from 28 distinct CSD entries\. We exclude BEDMIG, UNOGIN, BISMEV, and BIYSEH from the multi\-polymorph benchmark because their CSD families appear in the CLARI training split\.

Relaxation and ranking workflow\.For each CSD entry, each generator produces 1,000 independent candidates\. We generally follow the FastCSP post\-generation workflow, relaxing candidates using UMA\-S\-1\.2 with the OMC task\([Wood et al\., 2025](https://arxiv.org/html/2608.26962#bib.bib7)\)and the ASE BFGS optimizer\([Larsen et al\., 2017](https://arxiv.org/html/2608.26962#bib.bib31)\)\. We use a maximum\-force threshold of0\.020\.02eV,Å\-1and up to 500 optimization steps\. We discard candidates that fail to converge, change molecular connectivity orZZ, or have densities outside0\.50\.5–3\.03\.0g,cm\-3\. We then retain structures within 10 kJ,mol\-1of the minimum energy and deduplicate equivalent relaxed structures\. For each multi\-polymorph system, we pool candidates generated from all constituent CSD entries before energy filtering, deduplication, and ranking, so that all experimental forms are evaluated within a shared energy ordering\.

Metrics\.We rank the remaining unique candidates by increasing lattice energy\([Gharakhanyan et al\., 2025](https://arxiv.org/html/2608.26962#bib.bib51)\)\. For candidate𝒮i=\(𝐀i,𝐗i,𝐋i\)\\mathcal\{S\}\_\{i\}=\(\\mathbf\{A\}\_\{i\},\\mathbf\{X\}\_\{i\},\\mathbf\{L\}\_\{i\}\), we compute the lattice energy per formula unit as

Elatt​\(𝒮i\)=EUMA​\(𝒮i\)Zi−Emol,E\_\{\\mathrm\{latt\}\}\(\\mathcal\{S\}\_\{i\}\)=\\frac\{E\_\{\\mathrm\{UMA\}\}\(\\mathcal\{S\}\_\{i\}\)\}\{Z\_\{i\}\}\-E\_\{\\mathrm\{mol\}\},\(5\)whereZiZ\_\{i\}is the number of formula units in the cell andEmolE\_\{\\mathrm\{mol\}\}is the isolated\-component reference energy for one formula unit\.

A candidate matches an experimental CSD form under COMPACK\([Chisholm and Motherwell, 2005](https://arxiv.org/html/2608.26962#bib.bib42)\)if 30 molecules match withRMSD30<1\\mathrm\{RMSD\}\_\{30\}<1Å\. For each experimental form, we record the best energy rank among all matching candidates\. We report overall recovery at any rank, Top\-kkrecovery fork∈1,5,20k\\in\{1,5,20\}, and the mean best rank among recovered forms\.

Baseline\.We comparePackora\-M andPackora\-L with CLARI\-H\. We do not include FastCSP as a baseline because its random structure generator uses a substantially larger search budget: approximately 53,000–300,000 raw structures per target, with 10,000–176,000 retained for relaxation after deduplication, compared with 1,000 candidates per CSD entry in our evaluation\([Gharakhanyan et al\., 2025](https://arxiv.org/html/2608.26962#bib.bib51)\)\.

Single\-polymorph results\.Packora\-M andPackora\-L recover 17 and 19 of the 26 targets, respectively, compared with 16 for CLARI\-H, while reducing the mean rank among recovered targets from 4\.25 to 2\.00 and 2\.05 \([Table2](https://arxiv.org/html/2608.26962#S4.T2)and[Figure7](https://arxiv.org/html/2608.26962#S4.F7)\)\. The recovery advantage persists across all evaluated rank cutoffs\.Packoracandidates also relax faster:Packora\-L reaches 50% convergence after 98 BFGS steps, compared with 198 for CLARI\-H\. On eight NVIDIA H200 GPUs, relaxing all candidates takes 11\.4 h forPackora\-M and 10\.2 h forPackora\-L, compared with 16\.8 h for CLARI\-H\. Per\-target ranks andRMSD30\\mathrm\{RMSD\}\_\{30\}values are reported in[AppendixH](https://arxiv.org/html/2608.26962#A8)\.

Figure 7:FastCSP single\-polymorph benchmark\.We generate 1,000 candidates for each of the 26 targets, relax them with UMA\-S\-1\.2\([Wood et al\., 2025](https://arxiv.org/html/2608.26962#bib.bib7)\), and rank them by increasing energy\.\(a\)Best energy rank of the experimental target form for CLARI\-H,Packora\-M, andPackora\-L; lower is better, with rank 1 denoting the predicted minimum\-energy structure\.Packora\-L andPackora\-M recover 19 and 17 targets within the top 20, respectively, compared with 15 for CLARI\-H\.\(b\)Fraction of targets recovered by energy\-rank cutoffkk;Packoraachieves higher recovery across cutoffs\.\(c\)Cumulative fraction of the generated candidates converged by BFGS step;Packoracandidates converge faster throughout relaxation\. Relaxing all candidates on eight NVIDIA H200 GPUs takes 16\.8, 11\.4, and 10\.2 h for CLARI\-H,Packora\-M, andPackora\-L, respectively\.Table 2:Experimental\-form recovery on FastCSP single\-polymorph benchmark\.We generate 1,000 candidates for each of the 26 targets, relax them with UMA\-S\-1\.2\([Wood et al\., 2025](https://arxiv.org/html/2608.26962#bib.bib7)\), and rank the relaxed structures by increasing energy\. Recovered denotes the fraction of targets with an experimental\-form match at any rank, while Top\-kkdenotes the fraction whose best match appears within the topkkranks\. Mean rank is computed over recovered targets only, with lower values indicating better ranking\.Boldmarks the best result\.Packoraachieves both higher recovery across rank cutoffs and lower experimental\-form ranks\.Multi\-polymorph results\.The advantage remains in the multi\-polymorph setting\.Packora\-M andPackora\-L recover 19 and 21 of the 29 forms, respectively, compared with 18 for CLARI\-H, and reduce the mean recovered rank from 102\.17 to 47\.11 and 54\.62 \([Tables3](https://arxiv.org/html/2608.26962#S4.T3)and[8](https://arxiv.org/html/2608.26962#S4.F8)\)\. Also, bothPackoravariants recover at least one polymorph for all 8 of 8 systems, whereas CLARI\-H recovers 7\. Relaxation is also faster: after 100 BFGS steps, 41\.4% and 57\.3% ofPackora\-M andPackora\-L candidates have converged, compared with 13\.0% for CLARI\-H\. Total relaxation takes 10\.6 h and 8\.9 h, respectively, versus 16\.8 h for CLARI\-H on eight NVIDIA H200 GPUs\. Per\-form ranks andRMSD30\\mathrm\{RMSD\}\_\{30\}values are reported in[AppendixH](https://arxiv.org/html/2608.26962#A8)\.

Figure 8:FastCSP multi\-polymorph benchmark\.We generate 1,000 candidates for each of the 29 target structures across eight systems, relax them with UMA\-S\-1\.2\([Wood et al\., 2025](https://arxiv.org/html/2608.26962#bib.bib7)\), and rank them by increasing energy within each system\.\(a\)Best energy rank of each of the 29 polymorphs for CLARI\-H,Packora\-M, andPackora\-L; lower is better, with rank 1 denoting the predicted minimum\-energy structure\. Marker color identifies the polymorph across systems\.Packora\-L andPackora\-M recover 21 and 19 polymorphs, respectively, compared with 18 for CLARI\-H\.\(b\)Fraction of polymorphs recovered by energy\-rank cutoffkk;Packoraachieves higher recovery across cutoffs\.\(c\)Cumulative fraction of the generated candidates converged by BFGS step;Packoracandidates converge faster throughout relaxation\. Relaxing all candidates on eight NVIDIA H200 GPUs takes 16\.8, 10\.6, and 8\.9 h for CLARI\-H,Packora\-M, andPackora\-L, respectively\.Table 3:Experimental\-form recovery on FastCSP multiple\-polymorph benchmark\.We generate 1,000 candidates for each of the 29 targets, relax them with UMA\-S\-1\.2\([Wood et al\., 2025](https://arxiv.org/html/2608.26962#bib.bib7)\), and rank the relaxed structures by increasing energy\. Recovered denotes the fraction of targets with an experimental\-form match at any rank, while Top\-kkdenotes the fraction whose best match appears within the topkkranks\. Mean rank is computed over recovered targets only, with lower values indicating better ranking\.Boldmarks the best result\.Packoraachieves both higher recovery across rank cutoffs and lower experimental\-form ranks\.

## 5Building the Recipe

We construct the finalPackorarecipe through five controlled studies of the design space:

- •Architecture:Which backbone—DiT with pairwise bias, Pairformer, or PairMixer—is most effective? Where should the noisy crystal state enter the network, and do additional single\-track or geometry modules help? \([Figures9](https://arxiv.org/html/2608.26962#S5.F9)and[10](https://arxiv.org/html/2608.26962#S5.F10)\)
- •Training:Which choices of time distribution, augmentation, endpoint objective, loss weighting, auxiliary supervision, and optimizer improve structure generation? \([Figure11](https://arxiv.org/html/2608.26962#S5.F11)\)
- •Conditioning:How should optional inputs be dropped during training so that a single model remains effective across different combinations of stereochemistry, molecular templates, and space\-group information? \([Figure12](https://arxiv.org/html/2608.26962#S5.F12)\)
- •Inference:Which sampler and solver settings work best, and does autoguidance provide further gains? \([Figure13](https://arxiv.org/html/2608.26962#S5.F13)\)
- •Scaling:How does scaling single\-track and pair\-track widths affect per\-sample success and coverage across targets, and which allocation best balances the two? \([Figure14](https://arxiv.org/html/2608.26962#S5.F14);[Table4](https://arxiv.org/html/2608.26962#S5.T4)\)

### 5\.1Ablation Protocol

Dataset preprocessing\.Following OXtal\([Jin et al\., 2025](https://arxiv.org/html/2608.26962#bib.bib47)\), we construct the dataset from CSD entries deposited by May 1, 2025\. We require three\-dimensional coordinates, anRR\-factor of at most9%9\\%, single\-crystal X\-ray diffraction at ambient pressure, a non\-polymeric structure, and a known space group\. We add missing hydrogens using the CSD API\([Sykes et al\., 2024](https://arxiv.org/html/2608.26962#bib.bib41)\)and retain unit cells with at most 512 atoms, including hydrogens\. To prevent benchmark leakage, we remove all entries belonging to a test CSD family or containing an eligible CSD\-provided component SMILES found in the test sets\. Within each remaining CSD family, we deduplicate equivalent structures using pymatgen StructureMatcher\([Ong et al\., 2013](https://arxiv.org/html/2608.26962#bib.bib25)\)withltol=0\.2\\operatorname\{ltol\}=0\.2,stol=0\.3\\operatorname\{stol\}=0\.3, andangle​\_​tol=5∘\\operatorname\{angle\\\_tol\}=5^\{\\circ\}, retaining the entry with the lowestRR\-factor\. The complete preprocessing pipeline and dataset statistics are provided in[AppendixI](https://arxiv.org/html/2608.26962#A9)\.

Setup\.To make controlled ablations feasible under a fixed compute budget, we train each configuration for 300 epochs on structures with at most 300 atoms per unit cell, saving checkpoints every 50 epochs\. We evaluate each checkpoint on a fixed subset of 8,192 validation crystals, generating 30 candidates per target with the 200\-step EDM–Heun sampler, and reportSolC\\mathrm\{Sol\}\_\{C\}as defined in[Section4\.2](https://arxiv.org/html/2608.26962#S4.SS2)\. Architecture, training, conditioning, and scaling studies follow this protocol; inference studies instead vary the sampler and function\-evaluation budget\. Within each study, we vary one design choice at a time\.

### 5\.2Architecture: A Cacheable Pairmixer Design

Explored designs\.We compare three backbones: \(1\) DiT with pair\-bias attention, as used by CLARI\([Lo et al\., 2026](https://arxiv.org/html/2608.26962#bib.bib50)\); \(2\) Pairformer, as used by AlphaFold3 and OXtal\([Abramson et al\., 2024](https://arxiv.org/html/2608.26962#bib.bib20);[Jin et al\., 2025](https://arxiv.org/html/2608.26962#bib.bib47)\); and \(3\) Pairmixer, a lighter Pairformer variant without triangle attention\([Ouyang\-Zhang et al\., 2025](https://arxiv.org/html/2608.26962#bib.bib53)\)\. We also test a single\-track update and the geometric enhancement module \(GEM\)\([Veljković et al\., 2026](https://arxiv.org/html/2608.26962#bib.bib52)\), which injects periodic minimum\-image pair geometry from the noisy structure\. Finally, we compare a pre\-entry design, inspired by Proteina\([Geffner et al\., 2025](https://arxiv.org/html/2608.26962#bib.bib15)\), where noisy coordinate and lattice features enter before pair refinement, with a post\-entry design, following AlphaFold3\([Abramson et al\., 2024](https://arxiv.org/html/2608.26962#bib.bib20)\), where they enter afterward so that the pair representation can be cached\.

\(a\) Before\-Pairmixer noisy\-input entry CoordLatticeSingleCond\+\+PairCondGEM\+\+𝐗t\\mathbf\{X\}\_\{t\}ℓt\\bm\{\\ell\}\_\{t\}𝒞\\mathcal\{C\}𝐗t,ℓt,t\\mathbf\{X\}\_\{t\},\\,\\bm\{\\ell\}\_\{t\},\\,tInput EmbedderTriMulOut\+\+TriMulIn\+\+Transition\+\+Attention\+\+Transition\+\+Pairmixer \(×4\\times 4\)𝐒\\mathbf\{S\}𝐏\\mathbf\{P\}

\(b\) After\-Pairmixer noisy\-input entry SingleCondPairCond𝒞\\mathcal\{C\}Input EmbedderTriMulOut\+\+TriMulIn\+\+TriAttnStart\+\+TriAttnEnd\+\+Transition\+\+Attention\+\+Transition\+\+Pairmixer \(×4\\times 4\)\+\+\+\+CoordLatticeGEM𝐗t\\mathbf\{X\}\_\{t\}ℓt\\bm\{\\ell\}\_\{t\}𝐗t,ℓt,t\\mathbf\{X\}\_\{t\},\\,\\bm\{\\ell\}\_\{t\},\\,tNoisy Input Embedder𝐒\\mathbf\{S\}𝐏\\mathbf\{P\}Cacheable

Figure 9:Model variants for architecture ablations\.\(a\)In the pre\-entry variant, noisy coordinate and lattice embeddings and optional geometric enhancement module \(GEM\) features are injected beforePairmixer\.\(b\)In the post\-entry variant, the condition\-onlyInput EmbedderandPairmixerare evaluated before noisy\-state injection, allowing their outputs to be cached across denoising steps\. Modules with dotted outlines denote optional components\. Both variants produce single and pair representations𝐒\\mathbf\{S\}and𝐏\\mathbf\{P\}, which are passed to the same DiT trunk\.Results and selection\.Pairmixer outperforms Pairformer and DiT with pair\-bias attention \(0\.4790\.479vs\.0\.4620\.462vs\.0\.4210\.421;[Figure10](https://arxiv.org/html/2608.26962#S5.F10)a–b\)\. Pairformer performs better early in training but later degrades and is substantially more expensive than Pairmixer\. GEM improves DiT with pair\-bias attention, whereas neither GEM nor the single\-track update improves Pairmixer \([Figure10](https://arxiv.org/html/2608.26962#S5.F10)c–d\)\. Pre\-entry Pairmixer slightly outperforms post\-entry Pairmixer \(0\.4790\.479vs\.0\.4740\.474\), but caching the post\-entry design yields a20\.1×20\.1\\timesreduction in 200\-step sampling time in a synthetic\[64,300\]\[64,300\]benchmark, with peak memory increasing slightly from21\.321\.3to28\.028\.0GiB \([Figure10](https://arxiv.org/html/2608.26962#S5.F10)e–f\)\. We therefore select post\-entry Pairmixer without additional modules\. We report parameter count and GFLOPs of each architecture variant in[AppendixJ](https://arxiv.org/html/2608.26962#A10)\.

Figure 10:Architecture selection, extensions, and caching\.Panels \(a\)–\(d\) report validationSolC\\mathrm\{Sol\}\_\{C\}on the≤\\leq300\-atom subset with a generation budget of 30 candidates under EDM Heun 200\. Pairmixer gives the dominant gain; post\-entry Config\. \(F\) remains close to Config\. \(C\), while single updates and GEM do not improve final coverage\. Panel \(e\) reports runtime for synthetic\[64,300\]\[64,300\]EDM Heun inputs, including cache construction, and panel \(f\) reports peak memory\. Caching post\-entry condition\-only pair states removes the per\-step Pairmixer cost at a fixed memory increase\.
### 5\.3Training: Selecting the Flow\-Matching Recipe

Explored designs\.We ablate each training choice separately\. For time sampling, we compare uniform, logit\-normal, Beta, and the Beta–uniform mixture proposed by[Geffner et al\. \(2025\)](https://arxiv.org/html/2608.26962#bib.bib15)\. We also test random translations in fractional coordinates, wrapping each molecule into the unit cell by its centroid\([Gruver et al\., 2024](https://arxiv.org/html/2608.26962#bib.bib18)\)\. We compareL1L^\{1\}andL2L^\{2\}losses and vary the coordinate\-to\-lattice weighting in Equation \([3](https://arxiv.org/html/2608.26962#S3.E3)\)\.

We further test auxiliary supervision on periodic pair distances\. Let𝒫=\{\(i,j\):i≠j,di​j<15Å\}\\mathcal\{P\}=\\\{\(i,j\):i\\neq j,\\ d\_\{ij\}<15\\,\\text\{\\AA\}\\\}, wheredi​jd\_\{ij\}andd^i​j\\hat\{d\}\_\{ij\}denote target and predicted minimum\-image distances\. We consider anL1L^\{1\}loss following CLARI\([Lo et al\., 2026](https://arxiv.org/html/2608.26962#bib.bib50)\)and a periodic adaptation of AlphaFold 3 smooth\-LDDT\([Abramson et al\., 2024](https://arxiv.org/html/2608.26962#bib.bib20)\):

ℒpairL1=1\|𝒫\|​∑\(i,j\)∈𝒫\|d^i​j−di​j\|,ℒpairsmooth=1−14​\|𝒫\|​∑\(i,j\)∈𝒫∑τ∈\{0\.5,1,2,4\}σ⁡\(τ−\|d^i​j−di​j\|\),\\mathcal\{L\}\_\{\\mathrm\{pair\}\}^\{L^\{1\}\}=\\frac\{1\}\{\|\\mathcal\{P\}\|\}\\sum\_\{\(i,j\)\\in\\mathcal\{P\}\}\\left\|\\hat\{d\}\_\{ij\}\-d\_\{ij\}\\right\|,\\qquad\\mathcal\{L\}\_\{\\mathrm\{pair\}\}^\{\\mathrm\{smooth\}\}=1\-\\frac\{1\}\{4\|\\mathcal\{P\}\|\}\\sum\_\{\(i,j\)\\in\\mathcal\{P\}\}\\sum\_\{\\tau\\in\\\{0\.5,1,2,4\\\}\}\\sigma\\\!\\left\(\\tau\-\\left\|\\hat\{d\}\_\{ij\}\-d\_\{ij\}\\right\|\\right\),\(6\)whereσ\\sigmais the sigmoid function and distances are in angstroms\. Finally, we compare AdamW\([Loshchilov and Hutter, 2019](https://arxiv.org/html/2608.26962#bib.bib5)\)with Muon\([Liu et al\., 2025](https://arxiv.org/html/2608.26962#bib.bib4)\)and evaluate raw versus EMA parameters\.

Results and selection\.The Beta–uniform mixture outperforms pure Beta, logit\-normal, and uniform time distributions \([Figure11](https://arxiv.org/html/2608.26962#S5.F11)a\)\. Random translation augmentation improvesSolC\\mathrm\{Sol\}\_\{C\}from0\.4660\.466to0\.4860\.486, whileL1L^\{1\}substantially outperformsL2L^\{2\}\(0\.4860\.486vs\.0\.4070\.407;[Figure11](https://arxiv.org/html/2608.26962#S5.F11)b\)\. Increasing the coordinate\-to\-lattice weight ratio improves performance from0\.4120\.412at1:11\{:\}1to0\.4860\.486at10:110\{:\}1, after which performance declines \([Figure11](https://arxiv.org/html/2608.26962#S5.F11)c\)\. Auxiliary pair supervision provides little benefit: the best smooth\-LDDT setting improvesSolC\\mathrm\{Sol\}\_\{C\}by only0\.0040\.004while requiring1\.37×1\.37\\timesmore training GPU\-hours \(599 vs\. 436\)\. Muon outperforms AdamW \(0\.4860\.486vs\.0\.4470\.447\), and an EMA decay of0\.99990\.9999outperforms both raw weights and a decay of0\.9990\.999\([Figure11](https://arxiv.org/html/2608.26962#S5.F11)e–f\)\. We therefore select Beta–uniform time sampling, random translation augmentation,L1L^\{1\}endpoint regression with a10:110\{:\}1coordinate\-to\-lattice weight ratio, no auxiliary pair loss, Muon, and an EMA decay of0\.99990\.9999\.

Figure 11:Training recipe selection\.All panels report validationSolC\\mathrm\{Sol\}\_\{C\}under EDM Heun 200 with a generation budget of 30 candidates\.\(a\)The Beta–uniform mixture gives the strongest completed score\.\(b\)With L1 regression, crystal translation improves the best score from0\.4660\.466to0\.4860\.486; under translation, L1 outperforms L2 \(0\.4860\.486vs\.0\.4070\.407\)\.\(c\)A10:110\{:\}1coordinate\-to\-lattice ratio is optimal\.\(d\)Atλ=5\\lambda\{=\}5, smooth\-LDDT improves the best score by only0\.0040\.004while increasing training cost from 436 to 599 GPU\-hours\.\(e\)Muon outperforms AdamW\.\(f\)EMA decay0\.99990\.9999outperforms raw weights and decay0\.9990\.999\.
### 5\.4Conditioning: Robustness Through Template Dropout

Explored designs\.Because available conditioning information varies across targets, a single model should handle different subsets of optional inputs\. We vary the template\-coordinate dropout probabilityptpl∈\{0,0\.5,1\}p\_\{\\mathrm\{tpl\}\}\\in\\\{0,0\.5,1\\\}and evaluate each model under five inference\-time conditioning settings: none, stereochemistry only, template only, stereochemistry plus template, and all three conditions including space group\.

Results and selection\.Moderate template dropout \(ptpl=0\.5p\_\{\\mathrm\{tpl\}\}=0\.5\) is the most robust across conditioning settings, achieving the highest meanSolC\\mathrm\{Sol\}\_\{C\}over the five settings \(0\.4870\.487, compared with0\.4830\.483and0\.2640\.264forptpl=0p\_\{\\mathrm\{tpl\}\}=0and11, respectively\)\. Notably, exposure to templates during training also improves performance when no optional conditioning is provided:ptpl=0\.5p\_\{\\mathrm\{tpl\}\}=0\.5reaches0\.4590\.459, compared with0\.4420\.442for a model that never observes templates\. We therefore selectptpl=0\.5p\_\{\\mathrm\{tpl\}\}=0\.5\([Figure12](https://arxiv.org/html/2608.26962#S5.F12)\)\.

Figure 12:Moderate template dropout is most robust across conditioning settings\.ValidationSolC\\mathrm\{Sol\}\_\{C\}at epoch 300 using EDM–Heun with 200 steps and 30 candidates per target\. S, T, and SG denote stereochemistry, template coordinates, and space\-group conditioning, respectively\. Among models trained withptpl∈\{0,0\.5,1\}p\_\{\\mathrm\{tpl\}\}\\in\\\{0,0\.5,1\\\},ptpl=0\.5p\_\{\\mathrm\{tpl\}\}\{=\}0\.5achieves the highest average performance across the five inference\-time conditioning settings and is therefore selected\. Template\-conditioned results forptpl=1p\_\{\\mathrm\{tpl\}\}\{=\}1fall below the plotted range and are shown as downward triangles\.
### 5\.5Inference: Efficient Sampling with EDM–Heun

Explored designs\.We compare flow ODE with Euler integration, flow SDE with Euler–Maruyama, and EDM–Heun\([Karras et al\., 2022](https://arxiv.org/html/2608.26962#bib.bib16)\)at matched numbers of function evaluations \(NFE\), and sweep the EDM discretization exponentρ\\rho\. For template conditioning, we compare random rotation and translation, rotation only, and no augmentation at inference to test whether reducing template randomness improves performance\. Finally, we evaluate autoguidance\([Karras et al\., 2024](https://arxiv.org/html/2608.26962#bib.bib17)\), using earlier training checkpoints as degraded models following[Geffner et al\. \(2025\)](https://arxiv.org/html/2608.26962#bib.bib15)\.

Results and selection\.At 400 NFE, EDM–Heun achievesSolC=0\.482\\mathrm\{Sol\}\_\{C\}=0\.482, compared with0\.4300\.430for flow SDE and0\.4080\.408for flow ODE\. Increasing EDM–Heun from 400 to 1000 NFE improvesSolC\\mathrm\{Sol\}\_\{C\}by only0\.0050\.005, so we retain 400 NFE\. Performance is similar forρ∈\[5,10\]\\rho\\in\[5,10\]; following[Karras et al\. \(2022\)](https://arxiv.org/html/2608.26962#bib.bib16), we selectρ=7\\rho=7\. Reducing template augmentation does not improve performance, with all three variants reaching0\.4850\.485–0\.4870\.487, so we retain the training\-matched rotation and translation\. Autoguidance also provides no benefit: the best guided setting reaches0\.4700\.470, compared with0\.4870\.487without guidance\. We therefore select EDM–Heun at 400 NFE withρ=7\\rho=7, template rotation and translation, and no autoguidance\.

Figure 13:Inference ablations\.\(a\)ValidationSolC\\mathrm\{Sol\}\_\{C\}versus NFE budget for flow ODE, flow SDE, and EDM–Heun\.\(b\)EDM–Heun with 400 NFEs across discretization exponentsρ\\rho; the failedρ=1\\rho=1result is shown at the lower boundary, and the star marks the selectedρ=7\\rho=7\.\(c\)Validation trajectories with no template augmentation, rotation only, and rotation plus translation at inference\.\(d\)Autoguidance using degraded checkpoints from epochs 50, 100, and 150 across guidance weightsω\\omega\.
### 5\.6Scaling: Balancing Single\- and Pair\-Track Widths

Explored designs\.We study how the single\-track widthdsd\_\{s\}and pair\-track widthdpd\_\{p\}affect performance when scaled independently or together\. All other settings, including depth, Pairmixer placement, conditioning, validation split, and sampler, are held fixed\. We report both the single\-sample solve rateSolS\\mathrm\{Sol\}\_\{S\}and the 30\-sample crystal\-level solve rateSolC\\mathrm\{Sol\}\_\{C\}, capturing per\-sample success and coverage across targets, respectively\.

Starting from the base configuration W0 with\(ds,dp\)=\(512,128\)\(d\_\{s\},d\_\{p\}\)=\(512,128\), we consider:

- •Pair scaling \(W1, W2\): increasedpd\_\{p\}to192192and256256, respectively, while keepingds=512d\_\{s\}=512\.
- •Single scaling \(W3, W5\): increasedsd\_\{s\}to768768and10241024, respectively, while keepingdp=128d\_\{p\}=128\.
- •Balanced scaling \(W4\): increase both widths to\(ds,dp\)=\(768,192\)\(d\_\{s\},d\_\{p\}\)=\(768,192\), preservingds/dp=4d\_\{s\}/d\_\{p\}=4\.

Scaling either width alone is insufficient\.The epoch sweeps reveal distinct limitations of the two isolated scaling paths\. Increasing only the single width improvesSolS\\mathrm\{Sol\}\_\{S\}but eventually reducesSolC\\mathrm\{Sol\}\_\{C\}, suggesting that the added capacity makes successes more repeatable on already\-solvable targets rather than expanding coverage\. Increasing only the pair width, in contrast, yields no consistent improvement in either metric\. Thus, neither isolated scaling path provides a uniformly better trade\-off across sampling budgets \([Figure14](https://arxiv.org/html/2608.26962#S5.F14);[Table4](https://arxiv.org/html/2608.26962#S5.T4)\)\.

Balanced scaling reaches the sampling\-budget Pareto frontier\.Jointly scaling both widths allows W4 to maintain single\-sample success while recovering crystal\-level coverage, yielding a better trade\-off than scaling either width alone\. Adding weight decay of10−210^\{\-2\}shifts W4 toward greater coverage, trading some success atk=1k\{=\}1for improved coverage atk=30k\{=\}30\. We prioritize coverage because CSP workflows generate multiple candidates for subsequent ranking; a structure absent from the candidate set cannot be recovered downstream\([Hunnisett et al\., 2024a](https://arxiv.org/html/2608.26962#bib.bib37);[Hunnisett et al\., 2024b](https://arxiv.org/html/2608.26962#bib.bib36)\)\. We therefore retain W0 asPackora\-M and select W4 with weight decay10−210^\{\-2\}asPackora\-L \([Figure14](https://arxiv.org/html/2608.26962#S5.F14)\)\.

Figure 14:Balanced width scaling reaches the sampling\-budget Pareto frontier\.\(a\)Single\-sample solve rateSolS\\mathrm\{Sol\}\_\{S\}\(k=1k\{=\}1\) versus crystal\-level solve rateSolC\\mathrm\{Sol\}\_\{C\}atk=30k\{=\}30at epoch 300\. Marker shape and color indicate the width\-scaling family\. Scaling either the single or pair width alone does not consistently improve the trade\-off between per\-sample success and target coverage, whereas jointly scaling both widths reaches the Pareto frontier\. Adding weight decay of10−210^\{\-2\}further shifts the balanced model toward higherk=30k\{=\}30coverage\. W0 is selected asPackora\-M, and W4 with weight decay \(W4 \+ WD\) asPackora\-L\.\(b\)Epoch sweeps for single\-width scaling \(W0, W3, W5\), pair\-width scaling \(W0, W1, W2\), and balanced scaling \(W0, W4\)\. Top and bottom rows reportSolS\\mathrm\{Sol\}\_\{S\}andSolC\\mathrm\{Sol\}\_\{C\}, respectively\. All metrics use the shared validation protocol with 30 candidates per target and 200\-step EDM–Heun sampling\.Table 4:Width\-scaling ablation\.All rows keep depth, Pairmixer placement, conditioning, validation split, and sampler fixed while varying the single widthdsd\_\{s\}and pair widthdpd\_\{p\}\.SolS\\mathrm\{Sol\}\_\{S\}andSolC\\mathrm\{Sol\}\_\{C\}are evaluated at epoch 300 with 30 candidates per target and 200\-step EDM–Heun sampling \(400 NFE\)\. GFLOPs denote analytical denoiser\-forward costs atN=300N\{=\}300\. Teal rows mark the selected width configurations: W0 definesPackora\-M, while W4 defines the width ofPackora\-L before applying weight decay as selected in[Figure14](https://arxiv.org/html/2608.26962#S5.F14)\.

## 6Conclusion

We presentPackora, a flexible all\-atom generator for molecular CSP based on variational flow matching\.Packorasupports multi\-component and organometallic crystals and optional conditioning on molecular templates, stereochemistry, and space\-group information\. Controlled studies of architecture, training, conditioning, inference, and scaling yield a practical recipe based on cacheable Pairmixer representations,L1L^\{1\}endpoint regression, EDM–Heun sampling, and balanced single\- and pair\-track capacity\. Inspired by the CCDC CSP blind tests, we introduce a two\-track evaluation that separates generator\-only coverage from end\-to\-end performance under a common downstream relaxation and ranking pipeline\. Across six benchmarks,Packoraachieves the best matched\-budget coverage, while under the shared downstream pipeline it achieves higher experimental\-form recovery, lower ranks, and faster convergence than the baselines\.

Limitations and discussion\.First, our controlled ablations vary design choices one at a time, and therefore do not capture interactions among architecture, training, conditioning, inference, and scaling; the selected recipe is not guaranteed to be globally optimal\. Second, our ablations and model selection primarily usek=30k\{=\}30candidates, although the best model may depend on the candidate budget\. Our scaling results show a trade\-off between small\-budget success and larger\-budget coverage, suggesting that model selection may benefit from larger candidate pools\. Determining an appropriate selection budget therefore remains an important question for generative CSP\.

## Acknowledgements

This research was supported by the NVIDIA Academic Grant Program and by the Advanced GPU Utilization Support Program, funded by the Government of the Republic of Korea through the Ministry of Science and ICT\. This work was also supported by the Basic Science Research Program through the National Research Foundation of Korea \(NRF\), funded by the Ministry of Education \(RS\-2025\-25435147\); grants from the Institute for Information & Communications Technology Planning & Evaluation \(IITP\), funded by the Korean government \(MSIT\), including the Artificial Intelligence Graduate School Program \(KAIST; RS\-2019\-II190075\) and the AI Star Fellowship \(KAIST; RS\-2025\-02304967\); a grant from the Korea Health Industry Development Institute \(KHIDI\), funded by the Ministry of Health & Welfare \(MOHW\), for Developing a Highly Multimodal \(Drug, Protein, Gene, Cell Imaging, Literature\) Foundation Model for ADMET Property Prediction \(No\. N0425208\); and an NRF grant funded by MSIT \(No\. RS\-2022\-NR072184\)\. We also acknowledge the use of OpenAI’s ChatGPT and Codex\([OpenAI, 2022](https://arxiv.org/html/2608.26962#bib.bib54);[OpenAI, 2025](https://arxiv.org/html/2608.26962#bib.bib55)\)for assistance with research code development, figure preparation, and manuscript revision\.

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## Appendix AModel Architecture Details

Notation\.This section specifies thePackoraforward pass in algorithmic form\. Indicesi,j,ki,j,kdenote atoms, andhhdenotes an attention head\. We use𝐒\\mathbf\{S\}and𝐏\\mathbf\{P\}for single and pair representations, respectively\. SymbolsEE,WW, andMLP\\operatorname\{MLP\}denote learned embedding functions, linear maps, and multilayer perceptrons, respectively, whileNorm\\operatorname\{Norm\}denotes normalization\.Split\\operatorname\{Split\}andConcat\\operatorname\{Concat\}partition and concatenate feature channels, respectively\. We useσ\\sigmafor the sigmoid function,ϕ\\phifor the configured activation function,⊙\\odotfor elementwise multiplication, anddhd\_\{h\}for the dimension of one attention head\.

Overall forward pass\.[Algorithm1](https://arxiv.org/html/2608.26962#alg1)summarizes the complete forward pass: the model embeds the condition, refines pair features, injects the noisy crystal state, applies the DiT trunk, and predicts coordinate and lattice\.

Algorithm 1Model forward pass1:Noisy state

\(𝐗t,ℓt\)\(\\mathbf\{X\}\_\{t\},\\bm\{\\ell\}\_\{t\}\), time

tt, condition

𝒞\\mathcal\{C\}, and valid\-atom mask

𝐦\\mathbf\{m\}
2:Posterior mean

\(𝐗^1,ℓ^1\)\(\\widehat\{\\mathbf\{X\}\}\_\{1\},\\widehat\{\\bm\{\\ell\}\}\_\{1\}\)
3:\# 1\. Encode the condition and refine pair features

4:

\(𝐒𝒞,𝐏\)←ConditionEmbed​\(𝒞,𝐦\)\(\\mathbf\{S\}\_\{\\mathcal\{C\}\},\\mathbf\{P\}\)\\leftarrow\\textsc\{ConditionEmbed\}\(\\mathcal\{C\},\\mathbf\{m\}\)
5:foreach Pairmixer blockdo

6:

𝐏←PairmixerBlock​\(𝐏,𝐦\)\\mathbf\{P\}\\leftarrow\\textsc\{PairmixerBlock\}\(\\mathbf\{P\},\\mathbf\{m\}\)
7:endfor

8:

𝐏⋆←𝐏\\mathbf\{P\}^\{\\star\}\\leftarrow\\mathbf\{P\}
9:

10:\# 2\. Inject the noisy state

11:

\(𝐒,𝐜t\)←NoisyStateEmbed​\(𝐒𝒞,𝐗t,ℓt,t\)\(\\mathbf\{S\},\\mathbf\{c\}\_\{t\}\)\\leftarrow\\textsc\{NoisyStateEmbed\}\(\\mathbf\{S\}\_\{\\mathcal\{C\}\},\\mathbf\{X\}\_\{t\},\\bm\{\\ell\}\_\{t\},t\)
12:

13:\# 3\. Propagate and decode

14:foreach DiT blockdo

15:

𝐒←DiTBlock​\(𝐒,𝐜t,𝐏⋆,𝐦\)\\mathbf\{S\}\\leftarrow\\textsc\{DiTBlock\}\(\\mathbf\{S\},\\mathbf\{c\}\_\{t\},\\mathbf\{P\}^\{\\star\},\\mathbf\{m\}\)
16:endfor

17:

\(𝐗^1,ℓ^1\)←PredictionHeads​\(𝐒,𝐦\)\(\\widehat\{\\mathbf\{X\}\}\_\{1\},\\widehat\{\\bm\{\\ell\}\}\_\{1\}\)\\leftarrow\\textsc\{PredictionHeads\}\(\\mathbf\{S\},\\mathbf\{m\}\)
18:return

\(𝐗^1,ℓ^1\)\(\\widehat\{\\mathbf\{X\}\}\_\{1\},\\widehat\{\\bm\{\\ell\}\}\_\{1\}\)

Condition embedding\.For[Algorithm2](https://arxiv.org/html/2608.26962#alg2),𝐚i\\mathbf\{a\}\_\{i\}contains the atomic number, periodic\-table descriptors, formal charge, and chirality of atomii\. Template coordinates are denoted by𝐱~i\\widetilde\{\\mathbf\{x\}\}\_\{i\}, with availability indicatormTm\_\{\\mathrm\{T\}\}\. Molecular membership, bond type, and bond stereochemistry are denoted byuiu\_\{i\},τi​j\\tau\_\{ij\}, andζi​j\\zeta\_\{ij\}, respectively; unavailable categorical conditions use learned null categories\. The global condition contains the space\-group embedding\. Bond, stereochemistry, and template features are restricted to atom pairs within the same molecular component, and stereochemistry features are zero for nonbonded pairs\.

Algorithm 2Condition embedding1:Condition

𝒞\\mathcal\{C\}and valid\-atom mask

𝐦\\mathbf\{m\}
2:Condition\-only single representation

𝐒𝒞\\mathbf\{S\}\_\{\\mathcal\{C\}\}and ordered pair representation

𝐏\\mathbf\{P\}
3:\# 1\. Embed atom\-level and global conditions

4:

𝐒𝒞,i←Eatom​\(𝐚i\)\+mT​Etemp​\(𝐱~i\)\+mi​Eglobal​\(𝒞\)\\displaystyle\\mathbf\{S\}\_\{\\mathcal\{C\},i\}\\leftarrow E\_\{\\mathrm\{atom\}\}\(\\mathbf\{a\}\_\{i\}\)\+m\_\{\\mathrm\{T\}\}E\_\{\\mathrm\{temp\}\}\(\\widetilde\{\\mathbf\{x\}\}\_\{i\}\)\+m\_\{i\}E\_\{\\mathrm\{global\}\}\(\\mathcal\{C\}\)
5:

6:\# 2\. Construct intramolecular pair features

7:

Mi​jmol←mimj𝟏\[ui=uj\]M^\{\\mathrm\{mol\}\}\_\{ij\}\\leftarrow m\_\{i\}m\_\{j\}\\mathbf\{1\}\[u\_\{i\}=u\_\{j\}\]
8:

𝜹i​j←𝐱~i−𝐱~j\\bm\{\\delta\}\_\{ij\}\\leftarrow\\widetilde\{\\mathbf\{x\}\}\_\{i\}\-\\widetilde\{\\mathbf\{x\}\}\_\{j\},

ρi​j←\(1\+∥𝜹i​j∥22\)−1\\rho\_\{ij\}\\leftarrow\(1\+\\lVert\\bm\{\\delta\}\_\{ij\}\\rVert\_\{2\}^\{2\}\)^\{\-1\}
9:

𝐏¯i​j←WQ​𝐒𝒞,i\+WK​𝐒𝒞,j\+Mi​jmol​\[Ebond​\(τi​j\)\+Estereo​\(ζi​j\)\+mT​\(Wδ​𝜹i​j\+Wρ​ρi​j\)\]\.\\displaystyle\\overline\{\\mathbf\{P\}\}\_\{ij\}\\leftarrow W\_\{Q\}\\mathbf\{S\}\_\{\\mathcal\{C\},i\}\+W\_\{K\}\\mathbf\{S\}\_\{\\mathcal\{C\},j\}\+M^\{\\mathrm\{mol\}\}\_\{ij\}\\bigl\[E\_\{\\mathrm\{bond\}\}\(\\tau\_\{ij\}\)\+E\_\{\\mathrm\{stereo\}\}\(\\zeta\_\{ij\}\)\+m\_\{\\mathrm\{T\}\}\(W\_\{\\delta\}\\bm\{\\delta\}\_\{ij\}\+W\_\{\\rho\}\\rho\_\{ij\}\)\\bigr\]\.
10:

11:\# 3\. Refine each pair independently

12:

𝐏i​j←𝐏¯i​j\+MLPP⁡\(NormP⁡\(𝐏¯i​j\)\)\\mathbf\{P\}\_\{ij\}\\leftarrow\\overline\{\\mathbf\{P\}\}\_\{ij\}\+\\operatorname\{MLP\}\_\{P\}\\\!\\left\(\\operatorname\{Norm\}\_\{P\}\(\\overline\{\\mathbf\{P\}\}\_\{ij\}\)\\right\)
13:return

\(𝐒𝒞,𝐏\)\(\\mathbf\{S\}\_\{\\mathcal\{C\}\},\\mathbf\{P\}\)

Pairmixer block\.[Algorithm3](https://arxiv.org/html/2608.26962#alg3)updates only the pair representation using outgoing and incoming triangle multiplication followed by a gated pair transition\. The mask𝐦\\mathbf\{m\}restricts all pair operations to valid atoms\.

Algorithm 3Pairmixer block1:Pair representation

𝐏\\mathbf\{P\}and valid\-atom mask

𝐦\\mathbf\{m\}
2:Updated pair representation

𝐏\\mathbf\{P\}
3:

Mi​j←mi​mjM\_\{ij\}\\leftarrow m\_\{i\}m\_\{j\}
4:\# 1\. Aggregate information over outgoing and incoming triangles

5:

Contractout⁡\(𝐀,𝐁\)i​j←∑k𝐀i​k⊙𝐁j​k,Contractin⁡\(𝐀,𝐁\)i​j←∑k𝐀k​i⊙𝐁k​j\\displaystyle\\operatorname\{Contract\}\_\{\\mathrm\{out\}\}\(\\mathbf\{A\},\\mathbf\{B\}\)\_\{ij\}\\leftarrow\\sum\_\{k\}\\mathbf\{A\}\_\{ik\}\\odot\\mathbf\{B\}\_\{jk\},\\qquad\\operatorname\{Contract\}\_\{\\mathrm\{in\}\}\(\\mathbf\{A\},\\mathbf\{B\}\)\_\{ij\}\\leftarrow\\sum\_\{k\}\\mathbf\{A\}\_\{ki\}\\odot\\mathbf\{B\}\_\{kj\}
6:for

ddin the ordered sequence

\(out,in\)\(\\mathrm\{out\},\\mathrm\{in\}\)do

7:

𝐏¯←Normd⁡\(𝐏\)\\overline\{\\mathbf\{P\}\}\\leftarrow\\operatorname\{Norm\}\_\{d\}\(\\mathbf\{P\}\)
8:

\[𝐀,𝐁\]←Split⁡\(Wpd​𝐏¯⊙σ⁡\(Wgd​𝐏¯\)\)⊙𝐌\\displaystyle\[\\mathbf\{A\},\\mathbf\{B\}\]\\leftarrow\\operatorname\{Split\}\\\!\\left\(W\_\{p\}^\{d\}\\overline\{\\mathbf\{P\}\}\\odot\\sigma\(W\_\{g\}^\{d\}\\overline\{\\mathbf\{P\}\}\)\\right\)\\odot\\mathbf\{M\}
9:

𝐔←Contractd⁡\(𝐀,𝐁\)\\mathbf\{U\}\\leftarrow\\operatorname\{Contract\}\_\{d\}\(\\mathbf\{A\},\\mathbf\{B\}\)
10:

𝐏←𝐏\+Wod​Normd′⁡\(𝐔\)⊙σ⁡\(Wg​od​𝐏¯\)\\displaystyle\\mathbf\{P\}\\leftarrow\\mathbf\{P\}\+W\_\{o\}^\{d\}\\operatorname\{Norm\}^\{\\prime\}\_\{d\}\(\\mathbf\{U\}\)\\odot\\sigma\(W\_\{go\}^\{d\}\\overline\{\\mathbf\{P\}\}\)
11:endfor

12:

13:\# 2\. Mix feature channels within each pair

14:

𝐏¯←Normtr⁡\(𝐏\)\\overline\{\\mathbf\{P\}\}\\leftarrow\\operatorname\{Norm\}\_\{\\mathrm\{tr\}\}\(\\mathbf\{P\}\)
15:

𝐏←𝐏\+Wtr​\[SiLU⁡\(Wa​𝐏¯\)⊙Wb​𝐏¯\]\\displaystyle\\mathbf\{P\}\\leftarrow\\mathbf\{P\}\+W\_\{\\mathrm\{tr\}\}\\\!\\left\[\\operatorname\{SiLU\}\(W\_\{a\}\\overline\{\\mathbf\{P\}\}\)\\odot W\_\{b\}\\overline\{\\mathbf\{P\}\}\\right\]
16:return

𝐏\\mathbf\{P\}

Noisy\-state and time embedding\.In[Algorithm4](https://arxiv.org/html/2608.26962#alg4),𝛀\\bm\{\\Omega\}and𝜼\\bm\{\\eta\}are fixed random Fourier frequencies and phases for encoding the coordinates, while𝝎\\bm\{\\omega\}contains the fixed frequencies used for the sinusoidal time embedding\.

Algorithm 4Noisy\-state and time embedding1:Condition\-only single representation

𝐒𝒞\\mathbf\{S\}\_\{\\mathcal\{C\}\}, noisy coordinates

𝐗t\\mathbf\{X\}\_\{t\}, noisy lattice

ℓt\\bm\{\\ell\}\_\{t\}, and time

tt
2:DiT input

𝐒\\mathbf\{S\}and time condition

𝐜t\\mathbf\{c\}\_\{t\}
3:\# 1\. Embed noisy coordinates into the single track

4:

𝚽i←2​cos⁡\(𝛀⊙𝐗t,i\+𝜼\)\\bm\{\\Phi\}\_\{i\}\\leftarrow\\sqrt\{2\}\\cos\(\\bm\{\\Omega\}\\odot\\mathbf\{X\}\_\{t,i\}\+\\bm\{\\eta\}\)⊳\\trianglerightfixed random Fourier features

5:

𝐞iX←MLPX⁡\(vec⁡\(𝚽i\)\)\\mathbf\{e\}\_\{i\}^\{X\}\\leftarrow\\operatorname\{MLP\}\_\{X\}\(\\operatorname\{vec\}\(\\bm\{\\Phi\}\_\{i\}\)\)
6:

7:\# 2\. Embed and broadcast the noisy lattice

8:

𝐞L←MLPL⁡\(ℓt\)\\mathbf\{e\}^\{L\}\\leftarrow\\operatorname\{MLP\}\_\{L\}\(\\bm\{\\ell\}\_\{t\}\)
9:

𝐒i←𝐒𝒞,i\+𝐞iX\+𝐞L\\mathbf\{S\}\_\{i\}\\leftarrow\\mathbf\{S\}\_\{\\mathcal\{C\},i\}\+\\mathbf\{e\}\_\{i\}^\{X\}\+\\mathbf\{e\}^\{L\}⊳\\triangleright𝐞L\\mathbf\{e\}^\{L\}is broadcast over atoms

10:

11:\# 3\. Embed flow time

12:

𝝍⁡\(t\)←\[cos⁡\(𝝎​t\),sin⁡\(𝝎​t\)\]\\bm\{\\psi\}\(t\)\\leftarrow\[\\cos\(\\bm\{\\omega\}t\),\\sin\(\\bm\{\\omega\}t\)\]⊳\\trianglerightfixed sinusoidal frequencies

13:

𝐜t←MLPt⁡\(𝝍⁡\(t\)\)\\mathbf\{c\}\_\{t\}\\leftarrow\\operatorname\{MLP\}\_\{t\}\(\\bm\{\\psi\}\(t\)\)
14:return

\(𝐒,𝐜t\)\(\\mathbf\{S\},\\mathbf\{c\}\_\{t\}\)

DiT block\.In[Algorithm5](https://arxiv.org/html/2608.26962#alg5),𝜷\\bm\{\\beta\},𝜶\\bm\{\\alpha\}, and𝜸\\bm\{\\gamma\}denote the adaptive shift, scale, and residual\-gate vectors, respectively\. SubscriptsAAandFFdistinguish modulation parameters for the attention and feed\-forward updates\.

Algorithm 5Pair\-biased DiT block1:Single representation

𝐒\\mathbf\{S\}, time condition

𝐜t\\mathbf\{c\}\_\{t\}, refined pair representation

𝐏⋆\\mathbf\{P\}^\{\\star\}, and mask

𝐦\\mathbf\{m\}
2:Updated single representation

𝐒\\mathbf\{S\}
3:\# 1\. Generate time\-dependent modulation

4:

\[𝜷A,𝜶A,𝜸A,𝜷F,𝜶F,𝜸F\]←Wc​ϕ​\(𝐜t\)\\displaystyle\[\\bm\{\\beta\}\_\{A\},\\bm\{\\alpha\}\_\{A\},\\bm\{\\gamma\}\_\{A\},\\bm\{\\beta\}\_\{F\},\\bm\{\\alpha\}\_\{F\},\\bm\{\\gamma\}\_\{F\}\]\\leftarrow W\_\{c\}\\phi\(\\mathbf\{c\}\_\{t\}\)
5:

𝐔i←\(1\+𝜶A\)⊙Norm1⁡\(𝐒i\)\+𝜷A\\mathbf\{U\}\_\{i\}\\leftarrow\(1\+\\bm\{\\alpha\}\_\{A\}\)\\odot\\operatorname\{Norm\}\_\{1\}\(\\mathbf\{S\}\_\{i\}\)\+\\bm\{\\beta\}\_\{A\}
6:

𝐔~i←NormA⁡\(𝐔i\)\\widetilde\{\\mathbf\{U\}\}\_\{i\}\\leftarrow\\operatorname\{Norm\}\_\{A\}\(\\mathbf\{U\}\_\{i\}\)
7:

8:\# 2\. Apply self\-attention with pair bias

9:

𝐪ih←Normq⁡\(Wqh​𝐔~i\)\\mathbf\{q\}\_\{i\}^\{h\}\\leftarrow\\operatorname\{Norm\}\_\{q\}\(W\_\{q\}^\{h\}\\widetilde\{\\mathbf\{U\}\}\_\{i\}\),

𝐤ih←Normk⁡\(Wkh​𝐔~i\)\\mathbf\{k\}\_\{i\}^\{h\}\\leftarrow\\operatorname\{Norm\}\_\{k\}\(W\_\{k\}^\{h\}\\widetilde\{\\mathbf\{U\}\}\_\{i\}\),

𝐯ih←Wvh​𝐔~i\\mathbf\{v\}\_\{i\}^\{h\}\\leftarrow W\_\{v\}^\{h\}\\widetilde\{\\mathbf\{U\}\}\_\{i\}
10:

bi​jh←WPh​NormP⁡\(𝐏i​j⋆\)b\_\{ij\}^\{h\}\\leftarrow W\_\{P\}^\{h\}\\operatorname\{Norm\}\_\{P\}\(\\mathbf\{P\}\_\{ij\}^\{\\star\}\)
11:

ai​jh←softmaxj⁡\(⟨𝐪ih,𝐤jh⟩dh\+bi​jh−∞​\(1−mj\)\)\\displaystyle a\_\{ij\}^\{h\}\\leftarrow\\operatorname\{softmax\}\_\{j\}\\\!\\left\(\\frac\{\\langle\\mathbf\{q\}\_\{i\}^\{h\},\\mathbf\{k\}\_\{j\}^\{h\}\\rangle\}\{\\sqrt\{d\_\{h\}\}\}\+b\_\{ij\}^\{h\}\-\\infty\(1\-m\_\{j\}\)\\right\)
12:

𝐨ih←∑jai​jh​𝐯jh\\mathbf\{o\}\_\{i\}^\{h\}\\leftarrow\\sum\_\{j\}a\_\{ij\}^\{h\}\\mathbf\{v\}\_\{j\}^\{h\}
13:

𝐨i←Wo​\[σ⁡\(Wg​𝐔~i\)⊙Concath⁡\(𝐨ih\)\]\\displaystyle\\mathbf\{o\}\_\{i\}\\leftarrow W\_\{o\}\\\!\\left\[\\sigma\(W\_\{g\}\\widetilde\{\\mathbf\{U\}\}\_\{i\}\)\\odot\\operatorname\{Concat\}\_\{h\}\(\\mathbf\{o\}\_\{i\}^\{h\}\)\\right\]
14:

𝐒i←𝐒i\+𝜸A⊙𝐨i\\mathbf\{S\}\_\{i\}\\leftarrow\\mathbf\{S\}\_\{i\}\+\\bm\{\\gamma\}\_\{A\}\\odot\\mathbf\{o\}\_\{i\}
15:

16:\# 3\. Apply the gated feed\-forward update

17:

𝐇i←\(1\+𝜶F\)⊙Norm2⁡\(𝐒i\)\+𝜷F\\mathbf\{H\}\_\{i\}\\leftarrow\(1\+\\bm\{\\alpha\}\_\{F\}\)\\odot\\operatorname\{Norm\}\_\{2\}\(\\mathbf\{S\}\_\{i\}\)\+\\bm\{\\beta\}\_\{F\}
18:

𝐟i←Wdown​\[ϕ⁡\(Wgate​𝐇i\)⊙Wup​𝐇i\]\\displaystyle\\mathbf\{f\}\_\{i\}\\leftarrow W\_\{\\mathrm\{down\}\}\\\!\\left\[\\phi\(W\_\{\\mathrm\{gate\}\}\\mathbf\{H\}\_\{i\}\)\\odot W\_\{\\mathrm\{up\}\}\\mathbf\{H\}\_\{i\}\\right\]
19:

𝐒i←𝐒i\+𝜸F⊙𝐟i\\mathbf\{S\}\_\{i\}\\leftarrow\\mathbf\{S\}\_\{i\}\+\\bm\{\\gamma\}\_\{F\}\\odot\\mathbf\{f\}\_\{i\}
20:return

𝐒\\mathbf\{S\}

Prediction heads\.[Algorithm6](https://arxiv.org/html/2608.26962#alg6)maps the final single representation to centered Cartesian coordinates and a pooled lattice prediction over valid atoms\.

Algorithm 6Coordinate and lattice prediction heads1:Final single representation

𝐒\\mathbf\{S\}and valid\-atom mask

𝐦\\mathbf\{m\}
2:Coordinate endpoint

𝐗^1\\widehat\{\\mathbf\{X\}\}\_\{1\}and lattice endpoint

ℓ^1\\widehat\{\\bm\{\\ell\}\}\_\{1\}
3:

N←∑imiN\\leftarrow\\sum\_\{i\}m\_\{i\}
4:\# 1\. Predict centered atomic coordinates

5:

𝐫i←LinearX⁡\(NormX⁡\(𝐒i\)\)\\mathbf\{r\}\_\{i\}\\leftarrow\\operatorname\{Linear\}\_\{X\}\(\\operatorname\{Norm\}\_\{X\}\(\\mathbf\{S\}\_\{i\}\)\)
6:

𝐗^1,i←𝐫i−1N​∑jmj​𝐫j\\displaystyle\\widehat\{\\mathbf\{X\}\}\_\{1,i\}\\leftarrow\\mathbf\{r\}\_\{i\}\-\\frac\{1\}\{N\}\\sum\_\{j\}m\_\{j\}\\mathbf\{r\}\_\{j\}
7:

8:\# 2\. Pool atom features and predict the lattice

9:

𝐬¯←1N​∑imi​NormL⁡\(𝐒i\)\\displaystyle\\overline\{\\mathbf\{s\}\}\\leftarrow\\frac\{1\}\{N\}\\sum\_\{i\}m\_\{i\}\\operatorname\{Norm\}\_\{L\}\(\\mathbf\{S\}\_\{i\}\)
10:

ℓ^1←MLPL⁡\(𝐬¯\)\\widehat\{\\bm\{\\ell\}\}\_\{1\}\\leftarrow\\operatorname\{MLP\}\_\{L\}\(\\overline\{\\mathbf\{s\}\}\)
11:return

\(𝐗^1,ℓ^1\)\(\\widehat\{\\mathbf\{X\}\}\_\{1\},\\widehat\{\\bm\{\\ell\}\}\_\{1\}\)

## Appendix BDataset Statistics

We report statistics for the CLARI training and validation splits used in our benchmark experiments \([Section4](https://arxiv.org/html/2608.26962#S4)\)\. Our preprocessing adds missing hydrogens using the CSD Python API\[[Sykes et al\., 2024](https://arxiv.org/html/2608.26962#bib.bib41)\]before applying the 512\-atom limit\. This reduces the training split from 917,014 to 912,807 structures and the validation split from 1,048 to 1,047\. We report both statistics for completeness\.

Table 5:Size statistics for the CLARI training and validation splits\.Statistics are reported before and after applying the 512\-atom unit\-cell limit following hydrogen completion\.Unfiltered≤512\\leq 512atomsPropertyMinMeanMedianMaxMinMeanMedianMaxTrainingAtoms / unit cell5203\.651803,5585201\.71180512Molecules / unit cell15\.42424615\.354160Atoms / molecule1\.0648\.9642\.00682\.001\.0648\.8942\.00512\.00Formula\-unit countZZ13\.3444013\.33440ValidationAtoms / unit cell10205\.9518561710205\.56184512Molecules / unit cell15\.5149915\.42452Atoms / molecule5\.0048\.4242\.10242\.005\.0048\.4642\.20242\.00Formula\-unit countZZ13\.3941613\.39416We additionally report chemical composition, molecular flexibility, and the availability of optional conditioning information\. Similar to[Jin et al\. \[2025\]](https://arxiv.org/html/2608.26962#bib.bib47), we label a structure as rigid if every molecular component contains at most three rotatable bonds and as flexible if any component contains more than three\. This provides a simple heuristic rather than a complete description of molecular flexibility\.[Table6](https://arxiv.org/html/2608.26962#A2.T6)summarizes these properties before and after the 512\-atom filtering\.

Table 6:Composition, flexibility, and conditioning availability in the CLARI splits\.Entries report the number of structures before and after applying the 512\-atom unit\-cell limit\.TrainingValidationPropertyUnfiltered≤512\\leq 512atomsUnfiltered≤512\\leq 512atoms*Composition*Organic491,219490,475573573Organometallic425,795422,332475474*Molecular flexibility*Rigid194,017193,620226226Flexible722,997719,187822821*Conditioning availability*Template598,141596,447683683Stereochemistry917,014912,8071,0481,047Space group917,012912,8051,0481,047
## Appendix CTraining and Sampling Hyperparameters

We summarize the hyperparameters used forPackora\-M andPackora\-L\. Unless noted, all settings are shared between the two model sizes\.

Architecture\.[Table7](https://arxiv.org/html/2608.26962#A3.T7)specifies model capacity and backbone configuration; the two models differ only in width, head count, and parameter count\.

Table 7:Architecture hyperparameters\.Model capacity and backbone configuration forPackora\-M andPackora\-L\.Training\.[Table8](https://arxiv.org/html/2608.26962#A3.T8)summarizes the flow\-matching, optimization, conditioning, and checkpoint\-selection settings\.

Table 8:Training hyperparameters\.Flow\-matching, optimization, conditioning, and model\-selection settings\. All settings are shared betweenPackora\-M andPackora\-L unless noted\.Curriculum and batching\.We use two\-stage training with fixed\-shape batches to enabletorch\.compile\. Each structure is assigned to the smallest atom\-count bucket that contains it and padded to that bucket size\.

Table 9:Two\-stage training curriculum and batching\.Atom\-count buckets and batch sizes used for fixed\-shape training across eight GPUs\.Sampling\.Both model sizes use the same sampling and conditioning configuration for benchmark generation\.

Table 10:Sampling hyperparameters\.Sampling and conditioning configuration used for benchmark generation\.
## Appendix DStructure Generation Evaluation

This section details the evaluation protocol for structure generation\. We formalize the crystal solve rateSolC\\mathrm\{Sol\}\_\{C\}\[[Jin et al\., 2025](https://arxiv.org/html/2608.26962#bib.bib47)\], including COMPACK\[[Chisholm and Motherwell, 2005](https://arxiv.org/html/2608.26962#bib.bib42)\]matching, collision detection, and experimental\-reference selection\. We also summarize the differences between the released OXtal and CLARI evaluators that motivate our re\-evaluation of CLARI\.

Definition of crystal solve rate\.Letc∈\{1,…,C\}c\\in\\\{1,\\ldots,C\\\}index targets ands∈\{1,…,nc\}s\\in\\\{1,\\ldots,n\_\{c\}\\\}index generated candidates for targetcc\. The crystal\-level solve rate is

SolC=1C​∑c=1Cmax1≤s≤nc⁡solc,s,\\mathrm\{Sol\}\_\{C\}=\\frac\{1\}\{C\}\\sum\_\{c=1\}^\{C\}\\max\_\{1\\leq s\\leq n\_\{c\}\}\\mathrm\{sol\}\_\{c,s\},\(7\)wheresolc,s\\mathrm\{sol\}\_\{c,s\}indicates whether candidatexc,sx\_\{c,s\}solves targetcc:

solc,s=\[colc,s=0∧n15\(xc,s,tc,s⋆\)≥8∧r15\(xc,s,tc,s⋆\)<2Å\],\\mathrm\{sol\}\_\{c,s\}=\\mathbf\{1\}\\\!\\left\[\\mathrm\{col\}\_\{c,s\}=0\\ \\land\\ n\_\{15\}\(x\_\{c,s\},t^\{\\star\}\_\{c,s\}\)\\geq 8\\ \\land\\ r\_\{15\}\(x\_\{c,s\},t^\{\\star\}\_\{c,s\}\)<2\\,\\text\{\\AA\}\\right\],\(8\)wherecolc,s\\mathrm\{col\}\_\{c,s\}is the collision indicator;n15n\_\{15\}andr15r\_\{15\}denote the number of matched molecules and RMSD, respectively, returned by COMPACK at packing size 15; andtc,s⋆t^\{\\star\}\_\{c,s\}is the selected experimental reference\.

Intuitively, a candidate is collision\-free \(colc,s=0\\mathrm\{col\}\_\{c,s\}=0\) if all nonbonded heavy\-atom pairs are separated beyond the collision threshold\. Specifically,

colc,s=\[∃\(i,j\):rivdW\+rjvdW−di​j\(c,s\)≥0\.7Å\],\\mathrm\{col\}\_\{c,s\}=\\mathbf\{1\}\\\!\\left\[\\exists\(i,j\):r\_\{i\}^\{\\mathrm\{vdW\}\}\+r\_\{j\}^\{\\mathrm\{vdW\}\}\-d\_\{ij\}^\{\(c,s\)\}\\geq 0\.7\\,\\text\{\\AA\}\\right\],\(9\)wheredi​j\(c,s\)d\_\{ij\}^\{\(c,s\)\}is the distance between a nonbonded heavy\-atom pair\(i,j\)\(i,j\), andrivdWr\_\{i\}^\{\\mathrm\{vdW\}\}andrjvdWr\_\{j\}^\{\\mathrm\{vdW\}\}are their van der Waals radii\.

For targets with multiple experimental references, we select the best\-matching reference astc,s⋆t^\{\\star\}\_\{c,s\}\. LetT⁡\(c\)T\(c\)denote the reference set for targetccand define

Pc,s=\{t∈T⁡\(c\):n15​\(xc,s,t\)≥8\}\.P\_\{c,s\}=\\\{t\\in T\(c\):n\_\{15\}\(x\_\{c,s\},t\)\\geq 8\\\}\.\(10\)OXtal selects

tc,s⋆=\{arg⁡mint∈Pc,s​r15​\(xc,s,t\),Pc,s≠∅,arg⁡maxt∈T⁡\(c\)​n15​\(xc,s,t\),Pc,s=∅\.t^\{\\star\}\_\{c,s\}=\\begin\{cases\}\\displaystyle\\arg\\min\_\{t\\in P\_\{c,s\}\}r\_\{15\}\(x\_\{c,s\},t\),&P\_\{c,s\}\\neq\\varnothing,\\\\\[3\.0pt\] \\displaystyle\\arg\\max\_\{t\\in T\(c\)\}n\_\{15\}\(x\_\{c,s\},t\),&P\_\{c,s\}=\\varnothing\.\\end\{cases\}\(11\)
Differences between the OXtal and CLARI evaluators\.The released CLARI evaluator differs from the OXtal evaluator in two aspects: \(1\) collision detection, including the treatment of intermolecular collisions, and \(2\) selection of the best experimental reference for targets with multiple reference structures\. For a consistent comparison, we therefore re\-evaluate CLARI\-M, CLARI\-L, and CLARI\-H using the OXtal definitions\.

## Appendix EDetailed Structure\-Generation Results

This appendix provides tabular results for the structure\-generation benchmarks \([Section4](https://arxiv.org/html/2608.26962#S4)\) across candidate budgets and evaluation criteria\.

[Table11](https://arxiv.org/html/2608.26962#A5.T11)reports the standardSolC\\mathrm\{Sol\}\_\{C\}and strictSolC15/15\\mathrm\{Sol\}\_\{C\}^\{15/15\}at budgets of 30 and 1,000 candidates, with resampling standard deviations reported at the 30\-candidate budget\.[Table12](https://arxiv.org/html/2608.26962#A5.T12)provides additional collision, packing, and recovery metrics forPackora\-M andPackora\-L at the matched 30\-candidate budget\.

Table 11:Tabular results for structure generation benchmark\.We report standardSolC\\mathrm\{Sol\}\_\{C\}and strictSolC15/15\\mathrm\{Sol\}\_\{C\}^\{15/15\}at 30 and 1,000 candidates per target\. At 30 candidates, CLARI andPackoraresults are mean±\\pmstandard deviation over 5,000 resamples from 1,000\-candidate pools; other baselines are reported as provided in[Jin et al\. \[2025\]](https://arxiv.org/html/2608.26962#bib.bib47)\. Results at 1,000 candidates use the full pool\.Boldandunderlinedenote the best and second\-best generative results, respectively\.Table 12:Full metrics at a 30\-candidate budget\.We report per\-sample collision, packing, and recovery rates \(ColS\\mathrm\{Col\}\_\{S\},PacS\\mathrm\{Pac\}\_\{S\},RecS\\mathrm\{Rec\}\_\{S\}\) and crystal\-level packing, recovery, and solve rates \(PacC\\mathrm\{Pac\}\_\{C\},RecC\\mathrm\{Rec\}\_\{C\},SolC\\mathrm\{Sol\}\_\{C\}\), following the definitions in[Jin et al\. \[2025\]](https://arxiv.org/html/2608.26962#bib.bib47)\. Other baselines use their exact 30\-candidate pools, whilePackora\-M andPackora\-L report means over 5,000 seeded draws of 30 candidates from their 1,000\-candidate pools\. CLARI is omitted because onlySolC\\mathrm\{Sol\}\_\{C\}is available\.
## Appendix FEnd\-to\-End Generation with EmpiricalZZSampling

The main structure\-generation experiments in[Section4](https://arxiv.org/html/2608.26962#S4)assume known formula\-unit countZZfor comparison with CLARI\. Here, we evaluate an end\-to\-end setting in whichZZis unknown and sampled from an empirical prior estimated from the training set\.

Prior construction\.For each training crystal, we group identical molecular components by exact component SMILES\. Letnkn\_\{k\}denote the number of copies of distinct componentGkG\_\{k\}in the unit cell\. We recover the formula\-unit countZZand reduced stoichiometry𝐫\\mathbf\{r\}as

Z=gcd⁡\(n1,…,nK\),rk=nkZ\.Z=\\gcd\(n\_\{1\},\\ldots,n\_\{K\}\),\\qquad r\_\{k\}=\\frac\{n\_\{k\}\}\{Z\}\.We apply this procedure to all 912,807 structures in the CLARI training split after the 512\-atom filter\. Ifczc\_\{z\}denotes the number of structures withZ=zZ=z, the empirical prior is

p^train​\(Z=z\)=cz∑z′cz′\.\\widehat\{p\}\_\{\\mathrm\{train\}\}\(Z=z\)=\\frac\{c\_\{z\}\}\{\\sum\_\{z^\{\\prime\}\}c\_\{z^\{\\prime\}\}\}\.The resulting support is\{1,2,3,4,5,6,7,8,9,10,12,14,15,16,18,20,24,26,27,28,30,32,34,40\}\\\{1,2,3,4,5,6,7,8,9,10,12,14,15,16,18,20,24,26,27,28,30,32,34,40\\\}\.

Sampling formula\-unit countZZ\.LetNfu=∑krk​NkN\_\{\\mathrm\{fu\}\}=\\sum\_\{k\}r\_\{k\}N\_\{k\}denote the number of atoms in one formula unit for a benchmark target\. To bound inference cost, we restrictZZto values that yield at mostNmax=4096N\_\{\\max\}=4096atoms and renormalize the empirical prior:

p^trunc​\(Z=z\)=p^train\(Z=z\)1\[Nfuz≤Nmax\]∑z′p^train\(Z=z′\)1\[Nfuz′≤Nmax\]\.\\widehat\{p\}\_\{\\mathrm\{trunc\}\}\(Z=z\)=\\frac\{\\widehat\{p\}\_\{\\mathrm\{train\}\}\(Z=z\)\\,\\mathbf\{1\}\[N\_\{\\mathrm\{fu\}\}z\\leq N\_\{\\max\}\]\}\{\\sum\_\{z^\{\\prime\}\}\\widehat\{p\}\_\{\\mathrm\{train\}\}\(Z=z^\{\\prime\}\)\\,\\mathbf\{1\}\[N\_\{\\mathrm\{fu\}\}z^\{\\prime\}\\leq N\_\{\\max\}\]\}\.For each candidate, we sampleZZfromp^trunc\\widehat\{p\}\_\{\\mathrm\{trunc\}\}, replicate the formula unit accordingly, and pass the resulting unit\-cell composition toPackora\. All other settings match the main benchmark\.

Results\.We evaluatePackora\-M andPackora\-L with empiricalZZsampling and find that both remain competitive in this end\-to\-end setting\.[Figures15](https://arxiv.org/html/2608.26962#A6.F15),[16](https://arxiv.org/html/2608.26962#A6.F16),[17](https://arxiv.org/html/2608.26962#A6.F17)and[18](https://arxiv.org/html/2608.26962#A6.F18)compare empirical\-ZZsampling with the main supplied\-ZZsetting at effective 30\-candidate and exact 1,000\-candidate budgets under both standard and strict criteria\.[Figure19](https://arxiv.org/html/2608.26962#A6.F19)shows how the supplied and empirical performance gap changes with candidate budget on the Rigid and Flexible benchmarks\.

Figure 15:Crystal coverage with empirical unit\-cellZZsampling at 30 candidates per target\.The solid bars reproduce[Figure1](https://arxiv.org/html/2608.26962#S0.F1); diagonally hatched bars represent results withZZsampled from an empirical prior\. Bars report meanSolC\\mathrm\{Sol\}\_\{C\}, and error bars show one standard deviation across 5,000 resamples with replacement from each 1,000\-candidate pool\.Figure 16:Crystal coverage with empirical unit\-cellZZsampling at 1,000 candidates per target\.The solid bars reproduce[Figure4](https://arxiv.org/html/2608.26962#S4.F4); diagonally hatched bars represent results withZZsampled from an empirical prior\. Bars report exactSolC\\mathrm\{Sol\}\_\{C\}over the full 1,000\-candidate pools\.Figure 17:Strict crystal coverage with empirical unit\-cellZZsampling at 30 candidates per target\.The solid bars reproduce[Figure5](https://arxiv.org/html/2608.26962#S4.F5); diagonally hatched bars represent results withZZsampled from an empirical prior\. Bars report meanSolC15/15\\mathrm\{Sol\}\_\{C\}^\{15/15\}, and error bars show one standard deviation across 5,000 resamples with replacement from each 1,000\-candidate pool\.Figure 18:Strict crystal coverage with empirical unit\-cellZZsampling at 1,000 candidates per target\.The solid bars reproduce[Figure6](https://arxiv.org/html/2608.26962#S4.F6); diagonally hatched bars represent results withZZsampled from an empirical prior\. Bars report exactSolC15/15\\mathrm\{Sol\}\_\{C\}^\{15/15\}over the full 1,000\-candidate pools\.Figure 19:Effect of empirical unit\-cellZZsampling across candidate budgets\.Curves show meanSolC\\mathrm\{Sol\}\_\{C\}forPackora\-M andPackora\-L on the Rigid and Flexible benchmarks over 5,000 resamples, and shaded bands show the 2\.5th–97\.5th percentiles\. The gray solid lines denote suppliedZZ, and the teal dotted lines denote empiricalZZ\. The two settings remain close across candidate budgets\.
## Appendix GStructure Overlay Visualizations

We visualize 18 representative strict matches by overlayingPackora\-L predictions with their experimental references\. Experimental structures are shown as wider gray sticks andPackora\-L predictions as narrower green sticks; hydrogen atoms are omitted for clarity\. The overlays are rendered with py3Dmol using the 3Dmol\.js WebGL backend\[[Rego and Koes, 2015](https://arxiv.org/html/2608.26962#bib.bib43)\]\.

![Refer to caption](https://arxiv.org/html/2608.26962v1/fig_strict_alignment_overlays_1.png)Figure 20:Structure overlays forPackora\-L predictions\.Experimental structures are shown in gray andPackora\-L predictions in green\. Labels report the CSD refcode and correspondingRMSD15\\mathrm\{RMSD\}\_\{15\}\.![Refer to caption](https://arxiv.org/html/2608.26962v1/fig_strict_alignment_overlays_2.png)Figure 21:Structure overlays forPackora\-L predictions\.Experimental structures are shown in gray andPackora\-L predictions in green\. Labels report the CSD refcode and correspondingRMSD15\\mathrm\{RMSD\}\_\{15\}\.
## Appendix HDetailed Structure\-Ranking Results

This appendix reports the complete tabular results underlying the FastCSP structure\-ranking benchmarks in[Section4\.3](https://arxiv.org/html/2608.26962#S4.SS3)\. Each table gives the energy rank andRMSD30\\mathrm\{RMSD\}\_\{30\}of the best candidate matching each experimental form; NR denotes that no match was recovered\.

Table 13:Complete FastCSP single\-polymorph results\.Each method reports the energy rank andRMSD30\\mathrm\{RMSD\}\_\{30\}\(Å\) of the best candidate matching the experimental form\. NR denotes no recovery, with the correspondingRMSD30\\mathrm\{RMSD\}\_\{30\}unavailable\.Table 14:Complete FastCSP multi\-polymorph results\.Each method reports the energy rank andRMSD30\\mathrm\{RMSD\}\_\{30\}\(Å\) of the best candidate matching each experimental form\. NR denotes no recovery, with the correspondingRMSD30\\mathrm\{RMSD\}\_\{30\}unavailable\.
## Appendix IAblation Dataset Preprocessing

This section describes the dataset preprocessing used for our ablation studies in[Section5](https://arxiv.org/html/2608.26962#S5)\. Because public split refcodes were not available at the time, we constructed the dataset independently\. The preprocessing consists of four stages: filtering, structure processing, benchmark exclusion, and deduplication\.[Table15](https://arxiv.org/html/2608.26962#A9.T15)reports the number of structures remaining after each stage\.

Filtering\.Following OXtal\[[Jin et al\., 2025](https://arxiv.org/html/2608.26962#bib.bib47)\], we use CSD entries deposited by May 1, 2025 and retain structures with three\-dimensional coordinates, anRR\-factor of at most9%9\\%, single\-crystal X\-ray diffraction at ambient pressure, a non\-polymeric structure, and a known space group\. We Niggli\-reduce each crystal, add missing hydrogens with the CSD Python API\[[Sykes et al\., 2024](https://arxiv.org/html/2608.26962#bib.bib41)\], and apply the 512\-atom limit\.

Structure processing\.For entries with incomplete bond labels, we infer missing bond types and standardize aromatic and delocalized bonds with the CSD Python API\[[Sykes et al\., 2024](https://arxiv.org/html/2608.26962#bib.bib41)\]\. For each crystal, we store Cartesian coordinates, the unit cell, atomic numbers, bonds and bond types, formal charges, stereochemical annotations, and the space\-group label\. We generate an independent 3D conformer for each molecular component using RDKit ETKDGv3\[[Landrum and others, 2006](https://arxiv.org/html/2608.26962#bib.bib19),[Wang et al\., 2020](https://arxiv.org/html/2608.26962#bib.bib2)\]\. If conformer generation fails, we retain the crystal and mark the template as unavailable\.

Benchmark exclusion\.To prevent leakage, we exclude structures overlapping with the OXtal Rigid and Flexible benchmarks\[[Jin et al\., 2025](https://arxiv.org/html/2608.26962#bib.bib47)\], CSP5–7\[[Bardwell et al\., 2011](https://arxiv.org/html/2608.26962#bib.bib38),[Reilly et al\., 2016](https://arxiv.org/html/2608.26962#bib.bib35),[Hunnisett et al\., 2024b](https://arxiv.org/html/2608.26962#bib.bib36)\], and the CSD Teaching Subset\[[Battle et al\., 2010](https://arxiv.org/html/2608.26962#bib.bib39),[Lo et al\., 2026](https://arxiv.org/html/2608.26962#bib.bib50)\]\. We first remove entries whose six\-letter CSD refcode family matches a benchmark family\. We then remove entries containing a molecular component found in the benchmark set, using the CSD\-provided component SMILES\. Following CLARI\[[Lo et al\., 2026](https://arxiv.org/html/2608.26962#bib.bib50)\], we exclude recognized solvents and components with fewer than eight heavy atoms from this overlap check\.

Deduplication\.We group the remaining entries by six\-letter CSD refcode family and cluster equivalent structures using pymatgen StructureMatcher\[[Ong et al\., 2013](https://arxiv.org/html/2608.26962#bib.bib25)\]withltol=0\.2=0\.2,stol=0\.3=0\.3, andangle\_tol=5∘=5^\{\\circ\}\. Within each cluster, we retain the entry with the lowest reportedRR\-factor\.

Table 15:Ablation dataset preprocessing counts\.We report the number of structures remaining after each stage\.
## Appendix JDetailed Architecture Ablation Results

We report the parameter count, GFLOPs, and best validationSolC\\mathrm\{Sol\}\_\{C\}for each architecture variant evaluated in[Section5\.2](https://arxiv.org/html/2608.26962#S5.SS2)\.

Table 16:Architecture ablation with model size and compute\.We report parameter count and GFLOPs for each architecture variant, together with its best validationSolC\\mathrm\{Sol\}\_\{C\}under the shared≤300\\leq 300\-atom, 30\-candidate protocol with 200\-step EDM–Heun sampling\. Config\. \(F\) is the selected cacheable post\-entry design\.