SinAE: A Single-Architecture Flow-Matching Autoencoder for Cross-Domain Atomic Systems

# A Single-Architecture Flow-Matching Autoencoder for Cross-Domain Atomic Systems
Source: [https://arxiv.org/html/2607.12380](https://arxiv.org/html/2607.12380)
Yuxuan Ren Department of Computer Science National University of Singapore yuxuan\.ren@nus\.edu\.sg &Fan Yang Tencent AI for Life Science Lab fionafyang@tencent\.com &Jianhua Yao Tencent AI for Life Science Lab jianhuayao@tencent\.com &Yatao Bian Department of Computer Science National University of Singapore ybian@nus\.edu\.sg

###### Abstract

Small molecules, crystals, and proteins all reduce to atoms in 3D space, yet their generative pipelines remain fragmented across domains, each with its own graph, equivariant, or frame\-based architecture\. Cross\-domain training would mitigate per\-domain data scarcity, but direct generation in 3D coordinate space cannot easily handle the heterogeneous structural priors of all three domains, and no prior latent autoencoder is simultaneously lossless and architecturally general across all three\. We introduceSinAE, a single\-architecture flow\-matching autoencoder for molecules, crystals, and proteins, with vanilla Transformer encoder and decoder and no equivariant, graph, or domain\-specific operators\. Rather than requiring the encoder to capture fine\-grained geometry,SinAEshifts the reconstruction burden into an iterative flow\-matching decoder, achieving near\-lossless reconstruction across domains and reducing reconstruction errors by orders of magnitude relative to prior latent baselines\. The same per\-token latent supports a standard Diffusion Transformer prior that reaches strong performance on molecular, crystal, and protein generation benchmarks\. Joint molecule–crystal training strictly improves both domains, providing direct evidence of cross\-domain transfer through a shared atomic latent\. Code is available at[https://anonymous\.4open\.science/r/SinAE\_anonymous\-523D/](https://anonymous.4open.science/r/SinAE_anonymous-523D/)\.

## 1Introduction

3D atomic structure generation underpins advances in drug discovery, materials design, and protein engineering\. Despite differences in scale and domain conventions, the underlying data is the same: atoms placed inℝ3\\mathbb\{R\}^\{3\}with discrete element types, sharing the same physics of bonding and steric exclusion\. Yet generative pipelines have evolved separately along community lines\. Molecular generators build on equivariant message\-passing networks\(Gilmeret al\.,[2017](https://arxiv.org/html/2607.12380#bib.bib26); Schüttet al\.,[2017](https://arxiv.org/html/2607.12380#bib.bib27); Thomaset al\.,[2018](https://arxiv.org/html/2607.12380#bib.bib28)\); crystal generators must additionally handle periodic boundary conditions and lattice\-basis invariance\(Xieet al\.,[2022](https://arxiv.org/html/2607.12380#bib.bib23); Jiaoet al\.,[2023](https://arxiv.org/html/2607.12380#bib.bib22)\); protein generators rely on residue\-level parameterisations such as SE\(3\) frames\(Yimet al\.,[2023](https://arxiv.org/html/2607.12380#bib.bib47); Watsonet al\.,[2023](https://arxiv.org/html/2607.12380#bib.bib46); Boseet al\.,[2024](https://arxiv.org/html/2607.12380#bib.bib56)\)or triangle attention over pair representations\(Jumperet al\.,[2021](https://arxiv.org/html/2607.12380#bib.bib29); Linet al\.,[2023](https://arxiv.org/html/2607.12380#bib.bib38)\)\. Per\-domain 3D datasets are small: QM9\(Ramakrishnanet al\.,[2014](https://arxiv.org/html/2607.12380#bib.bib37)\)contains134134k molecules and MP\-20\(Xieet al\.,[2022](https://arxiv.org/html/2607.12380#bib.bib23)\)contains4545k crystals, both are low\-data regimes by modern standards\. A single architecture that covers all three domains would let each provide training signal for the others and reuse shared geometric regularities \(bond\-length distributions, coordination patterns, steric exclusion etc\)\.

Table 1:Capability comparison ofSinAEwith latent autoencoders baselines\.✓: supported; ✗: not supported\.*Vanilla Transformer*: backbone uses neither equivariant operators nor graph / message\-passing\.*Joint type\+coord\.*: model generates discrete atom\-type or sequence labels jointly with continuous coordinates\.*Sub\-mÅ recon*: reconstruction RMSD below10−310^\{\-3\}Å on the primary benchmark\.Cross\-domain atomic generation has attracted growing interest along two routes\. The*direct\-generation*route avoids an autoencoder and generates structures in coordinate space or through spatial tokenisation\(Moreheadet al\.,[2026](https://arxiv.org/html/2607.12380#bib.bib52); Luet al\.,[2026](https://arxiv.org/html/2607.12380#bib.bib58); Zhanget al\.,[2025](https://arxiv.org/html/2607.12380#bib.bib59)\)\. This is appealing for its simplicity, but places a heavy burden on a single model: the generator must simultaneously learn valid compositions, precise 3D geometry, and the distinct structural priors of each domain \(periodicity and lattice equivalences for crystals, residue\-level backbone geometry for proteins, bond connectivity and valence rules for molecules\)\. In practice, accommodating these heterogeneous priors within one generator has required either per\-modality tokenization and architectural choices\(Luet al\.,[2026](https://arxiv.org/html/2607.12380#bib.bib58)\), staged sequence\-then\-coordinate generation that has not yet been validated for unconditional de novo sampling\(Zhanget al\.,[2025](https://arxiv.org/html/2607.12380#bib.bib59)\), or restricting domain coverage to molecules and crystals\(Moreheadet al\.,[2026](https://arxiv.org/html/2607.12380#bib.bib52)\)\. The*latent\-autoencoder*route offers an alternative: the autoencoder absorbs per\-domain geometric complexity in its encoder and decoder, and the prior operates over smooth latent codes rather than raw 3D structures\(Joshiet al\.,[2025](https://arxiv.org/html/2607.12380#bib.bib6)\)\. Single\-domain work has shown that near\-lossless latent reconstruction is achievable\(Luoet al\.,[2026](https://arxiv.org/html/2607.12380#bib.bib3); Liet al\.,[2026](https://arxiv.org/html/2607.12380#bib.bib44)\), but extending this to a cross\-domain setting introduces two challenges that existing methods have not jointly resolved\.

The first is*reconstruction fidelity*\. Reconstruction error in the autoencoder directly limits generation quality: the prior cannot correct geometric errors introduced by a lossy decoder\. In atomistic systems this limit is especially strict, because sub\-Å coordinate drift can push bond lengths and angles outside physically valid ranges and corresponds to large shifts in potential energy\. The only cross\-domain latent autoencoder to date achieves a94\.6%94\.6\\%structure match rate on QM9 and84\.5%84\.5\\%on MP\-20\(Joshiet al\.,[2025](https://arxiv.org/html/2607.12380#bib.bib6)\), accepting this loss as a cost of domain generality\. The second is*training complexity*\. Existing autoencoders rely on domain\-specific loss combinations: structure tokenisers for proteins combine FAPE, distance, and violation losses with individual weight tuning\(Hayeset al\.,[2025](https://arxiv.org/html/2607.12380#bib.bib30)\); cross\-domain pipelines use separate coordinate and atom\-type losses with per\-domain weights\(Joshiet al\.,[2025](https://arxiv.org/html/2607.12380#bib.bib6)\); molecular autoencoders add bond\-type and bonded\-distance losses with multi\-way weight vectors\(Luoet al\.,[2026](https://arxiv.org/html/2607.12380#bib.bib3)\)\. Each additional domain means re\-balancing the entire loss landscape, making cross\-domain scaling a loss\-engineering problem rather than a data\-scaling problem\.

To the best of our knowledge, no existing cross\-domain autoencoder is simultaneously lossless, architecturally domain\-general, and trainable with a unified objective \(Table[1](https://arxiv.org/html/2607.12380#S1.T1)\)\.

![Refer to caption](https://arxiv.org/html/2607.12380v1/x1.png)Figure 1:SinAEoverview\.A flow\-matching autoencoder for molecules, crystals, and proteins, with a shared per\-token latent \(N\+3N\{\+\}3tokens for periodic systems, with three lattice tokens\)\. A DiT prior over the same latent enables de novo generation\. Bottom: per\-domain reconstruction RMSD\.Motivated by this gap, we introduceSinAE, a flow\-matching autoencoder for molecules, crystals, and proteins whose encoder and decoder are both vanilla Transformers with no equivariant, graph, or domain\-specific operators\. Following DiTo’s\(Chenet al\.,[2025b](https://arxiv.org/html/2607.12380#bib.bib18)\)finding that a single diffusion loss can match or outperform multi\-loss VAE training, we replace the single\-pass decoder with an iterative flow\-matching decoder trained with one unified reconstruction objective across all domains\. The decoder refines structures over multiple ODE steps, resolving atomic placement early and refining bond geometry and sterics later \(Section[5\.2](https://arxiv.org/html/2607.12380#S5.SS2)\), reaching sub\-milli\-Å RMSD without auxiliary losses or per\-domain weight tuning\. Adding a new domain requires adding training data, not adding loss terms\.

Because the decoder handles the geometric complexity of each domain, the resulting latent space is approximately Gaussian and free of the symmetry constraints of the raw structure space\. A standard DiT\(Peebles and Xie,[2023](https://arxiv.org/html/2607.12380#bib.bib51)\)suffices as the generative prior, with no equivariant layers, no triangle attention, and no lattice\-aware modules, yielding a simpler and more efficient pipeline than methods that generate directly in the coordinate space\. At inference time, generated latents are decoded with the same precision as training reconstructions, without post\-hoc relaxation\. Table[1](https://arxiv.org/html/2607.12380#S1.T1)summarises the resulting capability profile\.

In summary,SinAEmakes three contributions:

1. 1\.A single architecture for atomic\-structure tokenisation across three domains\.A vanilla\-Transformer flow\-matching autoencoder handles molecules, crystals, and protein backbones under one architecture, with no equivariant, graph, or domain\-specific operators\.
2. 2\.Near\-lossless reconstruction across all three domains\.Sub\-milli\-Å all\-atom Kabsch\-aligned RMSD on molecules and crystals \(0\.00020\.0002Å on QM9,0\.00170\.0017Å on MP\-20\) and∼0\.02\\sim 0\.02Å backbone CA\-RMSD on proteins, with100%100\\%match rate on molecules and crystals, achieved with a unified flow\-matching reconstruction objective across all domains\.
3. 3\.Cross\-domain training yields positive transfer; latent generation is competitive\.Joint QM9 \+ MP\-20 training strictly improves both single\-domain baselines on reconstruction and generation, giving direct evidence that a shared atomic latent benefits both chemical populations\. The same latents support a vanilla DiT prior that reaches strong crystal generation on hard metrics and competitive molecular and protein generation\.

## 2Background and Related Work

Molecules\.Early deep\-learning approaches for 3D molecules build on equivariant message\-passing over molecular graphs\(Gilmeret al\.,[2017](https://arxiv.org/html/2607.12380#bib.bib26); Schüttet al\.,[2017](https://arxiv.org/html/2607.12380#bib.bib27); Thomaset al\.,[2018](https://arxiv.org/html/2607.12380#bib.bib28)\)\. Diffusion and flow\-matching generators such as EDM\(Hoogeboomet al\.,[2022](https://arxiv.org/html/2607.12380#bib.bib9)\), EQGAT\-diff\(Leet al\.,[2024](https://arxiv.org/html/2607.12380#bib.bib8)\), SemlaFlow\(Irwinet al\.,[2024](https://arxiv.org/html/2607.12380#bib.bib4)\), and FlowMol\(Dunn and Koes,[2024](https://arxiv.org/html/2607.12380#bib.bib7)\)operate directly in 3D coordinate space with specialised equivariant architectures\. On the latent side, GeoLDM\(Xuet al\.,[2023](https://arxiv.org/html/2607.12380#bib.bib31)\)compresses molecules into SE\(3\)\-equivariant VAE codes, but its encoder–decoder bottleneck leaves0\.250\.25Å RMSD on GEOM\-Drugs, which propagates into bond\-length and angle errors at generation time\. UAE\-3D\(Luoet al\.,[2026](https://arxiv.org/html/2607.12380#bib.bib3)\)addresses this by achieving near\-lossless molecular reconstruction with a Relational Transformer encoder and SO\(3\) augmentation, but is limited to small molecules and does not extend to crystals or proteins\.

Crystals\.Crystal generation must additionally respect periodic boundary conditions and lattice\-basis invariance\(Xieet al\.,[2022](https://arxiv.org/html/2607.12380#bib.bib23); Jiaoet al\.,[2023](https://arxiv.org/html/2607.12380#bib.bib22)\)\. CDVAE\(Xieet al\.,[2022](https://arxiv.org/html/2607.12380#bib.bib23)\)pioneered VAE\-based crystal latents; FlowMM\(Milleret al\.,[2024](https://arxiv.org/html/2607.12380#bib.bib21)\)introduced Riemannian flow matching on the crystal lattice manifold; FlowLLM\(Sriramet al\.,[2024](https://arxiv.org/html/2607.12380#bib.bib41)\)uses a large language model as the base distribution\. ADiT\(Joshiet al\.,[2025](https://arxiv.org/html/2607.12380#bib.bib6)\)unifies molecules and crystals through a shared VAE \+ DiT pipeline, but its single\-pass VAE decoder leaves substantial reconstruction loss on both domains\. Concurrent Zatom\-1\(Moreheadet al\.,[2026](https://arxiv.org/html/2607.12380#bib.bib52)\)takes the opposite approach, eliminating the autoencoder and running flow matching directly inℝ3\\mathbb\{R\}^\{3\}; it does not produce a reusable latent representation and does not address proteins\.SinAEshows that, with a flow\-matching decoder, the latent route retains a smooth latent space while surpassing both lines on hard generation metrics \(Section[4\.4](https://arxiv.org/html/2607.12380#S4.SS4)\)\.

Proteins\.Protein backbone generation is dominated by SE\(3\) frame\-diffusion methods such as FrameFlow\(Yimet al\.,[2023](https://arxiv.org/html/2607.12380#bib.bib47)\)and RFdiffusion\(Watsonet al\.,[2023](https://arxiv.org/html/2607.12380#bib.bib46)\)that operate directly on residue frames; their architectures are tightly coupled to protein geometry and cannot be shared with molecule or crystal encoders\. On the latent side, ProteinAE\(Liet al\.,[2026](https://arxiv.org/html/2607.12380#bib.bib44)\)compresses backbones via a diffusion autoencoder and leads prior latent methods in reconstruction fidelity, but is single\-domain and does not generate residue\-type sequences jointly with coordinates\.SinAEuses the same vanilla\-Transformer architecture as its molecule \+ crystal model to achieve1010–20×20\\timestighter protein reconstruction while additionally supporting sequence–structure co\-design \(Section[4\.5](https://arxiv.org/html/2607.12380#S4.SS5)\)\.

## 3Methodology

### 3\.1Problem Formulation and Overview

Across all domains we represent an atomic structure as𝒢=\(𝐗,𝐀\)\\mathcal\{G\}=\(\\mathbf\{X\},\\mathbf\{A\}\), where𝐗∈ℝN×3\\mathbf\{X\}\\in\\mathbb\{R\}^\{N\\times 3\}are 3D coordinates and𝐀∈\{1,…,K\}N\\mathbf\{A\}\\in\\\{1,\\dots,K\\\}^\{N\}are categorical atom types\. For periodic systems \(crystals\) we additionally include a lattice matrix𝐋∈ℝ3×3\\mathbf\{L\}\\in\\mathbb\{R\}^\{3\\times 3\}; for aperiodic systems \(molecules, proteins\) no lattice is used\.

SinAEis trained for both reconstruction and generation\. Given𝒢\\mathcal\{G\}, the autoencoder produces a latent code𝐳\\mathbf\{z\}from which𝒢\\mathcal\{G\}can be recovered with near\-exact fidelity; a learned priorp​\(𝐳\)p\(\\mathbf\{z\}\)then enables de novo sampling of novel structures\. These capabilities are provided by two components trained in sequence:

1. 1\.Flow\-matching autoencoder\.A Transformer encoder maps𝒢\\mathcal\{G\}to a compact latent𝐳\\mathbf\{z\}, while a non\-equivariant Transformer decoder reconstructs𝒢\\mathcal\{G\}from𝐳\\mathbf\{z\}via conditional flow matching\. The decoder, not the encoder, is responsible for reconstruction fidelity, allowing the encoder to remain simple and domain\-general\.
2. 2\.Latent flow matching prior\.A vanilla Diffusion Transformer \(DiT\) learns the distributionp​\(𝐳\)p\(\\mathbf\{z\}\)over encoder outputs\. At generation time, a new latent is sampled from this prior and passed through the frozen decoder to produce a 3D structure\.

The architecture and procedures are summarized in Figure[2](https://arxiv.org/html/2607.12380#S3.F2)\.

![Refer to caption](https://arxiv.org/html/2607.12380v1/x2.png)Figure 2:SinAEarchitecture and procedures\.Left \(a–d\):\(a\) shared embedding; \(b\) Transformer encoder mapping\(𝐗,𝐀\)\(\\mathbf\{X\},\\mathbf\{A\}\)to a per\-token latent𝐳∈ℝN′×d\\mathbf\{z\}\\in\\mathbb\{R\}^\{N^\{\\prime\}\\times d\}; \(c\) flow decoder with two cross\-attention output heads predicting clean\(𝐱^1,𝐚^1\)\(\\hat\{\\mathbf\{x\}\}\_\{1\},\\hat\{\\mathbf\{a\}\}\_\{1\}\); \(d\) latent DiT prior over𝐳\\mathbf\{z\}\.Right \(e–g\):\(e\) reconstruction training, where the encoder𝐳\\mathbf\{z\}conditions the decoder’s flow\-matching reconstruction of\(𝐗,𝐀\)\(\\mathbf\{X\},\\mathbf\{A\}\); \(f\) DiT prior training on noised encoder latents; \(g\) generation, in which DiT ODE integration produces a sample𝐳\\mathbf\{z\}that conditions the decoder ODE over\(𝐱t,𝐚t\)\(\\mathbf\{x\}\_\{t\},\\mathbf\{a\}\_\{t\}\)\.
### 3\.2Input Featurization

#### Coordinate handling\.

We avoid hand\-crafted geometric preprocessing\. For aperiodic systems \(molecules, proteins\), the model consumes raw Cartesian coordinates𝐱i∈ℝ3\\mathbf\{x\}\_\{i\}\\in\\mathbb\{R\}^\{3\}after centering at the centroid, without PCA alignment, canonical\-frame construction, or invariant descriptors \(pairwise distances, sortedkk\-NN, bond angles\)\. For periodic systems \(crystals\), we append the three lattice vectors𝐋=\[𝐥1,𝐥2,𝐥3\]\\mathbf\{L\}=\[\\mathbf\{l\}\_\{1\},\\mathbf\{l\}\_\{2\},\\mathbf\{l\}\_\{3\}\]as three virtual\-atom tokens at the end of the sequence, so the model processes the full3×33\\times 3lattice matrix in the same way as atomic coordinates\.

#### Equivariance via random augmentation\.

To ensure the non\-equivariant Transformer does not learn orientation\-dependent features, we apply a random SO\(3\) rotation \(and, for aperiodic systems, a random translation\) to every training structure on the fly\. Empirically this suffices to reach the reconstruction RMSD reported in Section[4\.2](https://arxiv.org/html/2607.12380#S4.SS2)without any equivariant layer or invariant featurization\.

#### Token construction\.

Each atomiiis represented by the sum of a learned element embedding \(looked up from a table of sizeKK\) and a linear projection of its coordinate vector𝐱i∈ℝ3\\mathbf\{x\}\_\{i\}\\in\\mathbb\{R\}^\{3\}, both projected to the Transformer model dimensiondmodeld\_\{\\mathrm\{model\}\}\. The three lattice\-vector tokens use a dedicated learned type embedding in place of the element lookup\. For proteins, each residue is represented as a single token at its Cα\\alphaposition, with the residue type as the categorical label in place of the element type\.

### 3\.3Architecture

#### Encoder\.

The encoderϕ\\phimaps a sequence ofN′N^\{\\prime\}input tokens to a sequence of latent tokens𝐳∈ℝN′×d\\mathbf\{z\}\\in\\mathbb\{R\}^\{N^\{\\prime\}\\times d\}, whereN′=NN^\{\\prime\}=Nfor aperiodic systems andN′=N\+3N^\{\\prime\}=N\{\+\}3for periodic systems \(the three trailing tokens encode the lattice vectors\)\. A per\-token \(rather than pooled\) latent preserves per\-atom granularity: local geometric information scales withNN, so a fixed\-size global bottleneck would force lossy compression for larger systems\. We use a standard pre\-norm Transformer withLencL\_\{\\text\{enc\}\}layers and multi\-head self\-attention; after the final layer, two parallel linear heads produce a Gaussian posterior𝐳i=𝝁i\+𝝈i⊙𝜺i\\mathbf\{z\}\_\{i\}=\\bm\{\\mu\}\_\{i\}\+\\bm\{\\sigma\}\_\{i\}\\odot\\bm\{\\varepsilon\}\_\{i\},𝜺i∼𝒩​\(𝟎,𝐈\)\\bm\{\\varepsilon\}\_\{i\}\\sim\\mathcal\{N\}\(\\mathbf\{0\},\\mathbf\{I\}\)\. We use a small latent channeldd\(see Appendix[F](https://arxiv.org/html/2607.12380#A6)\), keeping the total latent size compact while preserving enough information for near\-exact decoding\.

#### Flow\-matching decoder\.

The decoderψ\\psiis a non\-equivariant Transformer that reconstructs\(𝐗,𝐀\)\(\\mathbf\{X\},\\mathbf\{A\}\)from𝐳\\mathbf\{z\}conditioned on partially noised inputs at timet∈\[0,1\]t\\in\[0,1\]\. We follow DiTo’s\(Chenet al\.,[2025b](https://arxiv.org/html/2607.12380#bib.bib18)\)single\-objective design but replace its score\-matching formulation with a continuous conditional flow\-matching objective for coordinates and a discrete flow\-matching objective for atom types\. The decoder receives noisy\(𝐗t,𝐀t\)\(\\mathbf\{X\}\_\{t\},\\mathbf\{A\}\_\{t\}\)projected through separate linears, summed with a sinusoidal time embedding and a learnable positional embedding:

𝐡i\(0\)=Wx​𝐱t,i\+WA​𝐚t,i\+𝐞time​\(t\)\+𝐞ipos,\\mathbf\{h\}\_\{i\}^\{\(0\)\}=W\_\{x\}\\,\\mathbf\{x\}\_\{t,i\}\+W\_\{A\}\\,\\mathbf\{a\}\_\{t,i\}\+\\mathbf\{e\}^\{\\text\{time\}\}\(t\)\+\\mathbf\{e\}^\{\\text\{pos\}\}\_\{i\},\(1\)whereWx∈ℝd×3W\_\{x\}\\in\\mathbb\{R\}^\{d\\times 3\},WA∈ℝd×KW\_\{A\}\\in\\mathbb\{R\}^\{d\\times K\}\. The joint sequence\[𝐡\(0\);𝐳\]\[\\mathbf\{h\}^\{\(0\)\};\\,\\mathbf\{z\}\]\(concatenated along the sequence dimension, yielding2​N′2N^\{\\prime\}tokens\) is processed byLdecL\_\{\\text\{dec\}\}pre\-norm Transformer blocks\(Vaswaniet al\.,[2017](https://arxiv.org/html/2607.12380#bib.bib34)\)\. Following the two\-stream output design of Tabasco\(Vonessenet al\.,[2026](https://arxiv.org/html/2607.12380#bib.bib1)\), two parallel cross\-attention output heads, one for coordinates and one for atom types, use the final hidden states as queries and the embedded noisy\-input tokens\{𝐜i\}\\\{\\mathbf\{c\}\_\{i\}\\\}as keys/values to produce𝐗^1θ\\hat\{\\mathbf\{X\}\}\_\{1\}^\{\\theta\}and𝐀^1θ\\hat\{\\mathbf\{A\}\}\_\{1\}^\{\\theta\}\(Figure[2](https://arxiv.org/html/2607.12380#S3.F2)\(c\)\)\. No domain\-specific attention, equivariant layer, or self\-conditioning\(Chenet al\.,[2023](https://arxiv.org/html/2607.12380#bib.bib35)\)is used in the backbone; random SO\(3\) augmentation at training time provides rotation robustness \(Appendix[D](https://arxiv.org/html/2607.12380#A4)\)\.

### 3\.4Training Objective

The autoencoder is trained end\-to\-end by minimising a fixed\-weight sum of three terms,

ℒ=ℒcoord\+λatom​ℒatom\+λKL​ℒKL,\\mathcal\{L\}=\\mathcal\{L\}\_\{\\text\{coord\}\}\+\\lambda\_\{\\text\{atom\}\}\\,\\mathcal\{L\}\_\{\\text\{atom\}\}\+\\lambda\_\{\\text\{KL\}\}\\,\\mathcal\{L\}\_\{\\text\{KL\}\},\(2\)whereℒcoord\\mathcal\{L\}\_\{\\text\{coord\}\}is continuous conditional flow matching on coordinates\(Lipmanet al\.,[2023](https://arxiv.org/html/2607.12380#bib.bib33)\),ℒatom\\mathcal\{L\}\_\{\\text\{atom\}\}is discrete flow matching on atom types\(Campbellet al\.,[2024](https://arxiv.org/html/2607.12380#bib.bib25)\), andℒKL\\mathcal\{L\}\_\{\\text\{KL\}\}is a light KL regulariser pushing the encoder’s posterior towards𝒩​\(𝟎,𝐈\)\\mathcal\{N\}\(\\mathbf\{0\},\\mathbf\{I\}\)\. For periodic systems the three lattice\-vector tokens are treated as virtual atoms, so their coordinates are reconstructed through the sameℒcoord\\mathcal\{L\}\_\{\\text\{coord\}\}as ordinary atoms\. Unlike DiTo\(Chenet al\.,[2025b](https://arxiv.org/html/2607.12380#bib.bib18)\), which uses a deterministic encoder, we use a stochastic encoder so that the aggregate posterior approximates the𝒩​\(𝟎,𝐈\)\\mathcal\{N\}\(\\mathbf\{0\},\\mathbf\{I\}\)prior of the DiT \(Section[3\.5](https://arxiv.org/html/2607.12380#S3.SS5)\)\. Exact loss definitions, interpolants, and weight values are in Appendix[E](https://arxiv.org/html/2607.12380#A5)\.

### 3\.5Latent Flow Matching for Generation

Because the KL term in Eq\.[2](https://arxiv.org/html/2607.12380#S3.E2)pushes the encoder’s posterior towards𝒩​\(𝟎,𝐈\)\\mathcal\{N\}\(\\mathbf\{0\},\\mathbf\{I\}\), the aggregate latent distribution is approximately Gaussian, and de novo generation reduces to learningp​\(𝐳\)p\(\\mathbf\{z\}\)and decoding\. We train a vanilla DiT\(Peebles and Xie,[2023](https://arxiv.org/html/2607.12380#bib.bib51)\)vθv\_\{\\theta\}as a flow\-matching velocity field\(Lipmanet al\.,[2023](https://arxiv.org/html/2607.12380#bib.bib33)\)with the linear interpolant𝐳t=\(1−t\)​𝐳0\+t​𝐳1\\mathbf\{z\}\_\{t\}\{=\}\(1\{\-\}t\)\\mathbf\{z\}\_\{0\}\+t\\mathbf\{z\}\_\{1\}and the standard FM objectiveℒFM=𝔼​‖vθ​\(𝐳t,t\)−\(𝐳1−𝐳0\)‖2\\mathcal\{L\}\_\{\\text\{FM\}\}\{=\}\\mathbb\{E\}\\\|v\_\{\\theta\}\(\\mathbf\{z\}\_\{t\},t\)\{\-\}\(\\mathbf\{z\}\_\{1\}\{\-\}\\mathbf\{z\}\_\{0\}\)\\\|^\{2\}, without any equivariant layer or domain\-specific module\. Since𝐳\\mathbf\{z\}is variable\-length \(N′=NN^\{\\prime\}\{=\}NorN\+3N\{\+\}3\), we pad and mask within each mini\-batch during training\. At generation time we sample the sequence length from the training\-set size distribution before running the DiT ODE\. Since the joint molecule–crystal model covers two domains, we condition the DiT prior on a domain label so that the user can specify which domain to generate at sampling time\. During training, each sample carries its ground\-truth domain label; at inference, we apply classifier\-free guidance\(Ho and Salimans,[2021](https://arxiv.org/html/2607.12380#bib.bib36)\)on this label to steer generation toward the target domain\.

A new structure is generated by sampling𝐳0∼𝒩​\(𝟎,𝐈\)\\mathbf\{z\}\_\{0\}\{\\sim\}\\mathcal\{N\}\(\\mathbf\{0\},\\mathbf\{I\}\), integratingvθv\_\{\\theta\}with fixed\-step Euler to obtain𝐳1\\mathbf\{z\}\_\{1\}, and decoding from pure noise\(𝐗0,𝐀0\)\(\\mathbf\{X\}\_\{0\},\\mathbf\{A\}\_\{0\}\)conditioned on𝐳1\\mathbf\{z\}\_\{1\}\. Because the decoder is near\-lossless, generated structures match the geometric precision of reconstructed training structures without post\-hoc relaxation\.

## 4Experiments

We evaluateSinAEon three primary structural domains \(small molecules, crystals, proteins\) and one secondary domain \(metal–organic frameworks; Appendix[N](https://arxiv.org/html/2607.12380#A14)\), focusing on two capabilities: \(1\) reconstruction fidelity of the flow\-matching autoencoder, and \(2\) de novo generation quality of the latent DiT prior\.

### 4\.1Datasets, Training, and Evaluation Protocol

Datasets\.We train and evaluate on standard benchmarks:QM9\(Ramakrishnanet al\.,[2014](https://arxiv.org/html/2607.12380#bib.bib37)\)andGEOM\-Drugs\(Axelrod and Gomez\-Bombarelli,[2022](https://arxiv.org/html/2607.12380#bib.bib13)\)for small molecules,MP\-20\(Xieet al\.,[2022](https://arxiv.org/html/2607.12380#bib.bib23)\)\(from the Materials Project\(Jainet al\.,[2013](https://arxiv.org/html/2607.12380#bib.bib12)\)\) for crystals, andAFDB\-FS\(a filtered single\-chain subset of the AlphaFold Protein Structure Database\(Jumperet al\.,[2021](https://arxiv.org/html/2607.12380#bib.bib29); Linet al\.,[2023](https://arxiv.org/html/2607.12380#bib.bib38)\)\) for proteins, with CASP14/CASP15 held out for protein reconstruction evaluation\. As an additional periodic benchmark we useQMOF\(Rosenet al\.,[2021](https://arxiv.org/html/2607.12380#bib.bib14)\)for MOF reconstruction and generation; the full QMOF results and a head\-to\-head MOF generation comparison with ADiT are reported in Appendix[N](https://arxiv.org/html/2607.12380#A14)\. All splits follow prior work\(Hoogeboomet al\.,[2022](https://arxiv.org/html/2607.12380#bib.bib9); Vignacet al\.,[2023](https://arxiv.org/html/2607.12380#bib.bib11); Irwinet al\.,[2024](https://arxiv.org/html/2607.12380#bib.bib4); Xieet al\.,[2022](https://arxiv.org/html/2607.12380#bib.bib23); Joshiet al\.,[2025](https://arxiv.org/html/2607.12380#bib.bib6); Van Kempenet al\.,[2024](https://arxiv.org/html/2607.12380#bib.bib40)\); full specification in Appendix[C](https://arxiv.org/html/2607.12380#A3)\.

Training and evaluation\.We train autoencoder–prior pairs with the same vanilla\-Transformer architecture but domain\-appropriate sizing, using Adam/AdamW with constant learning rate10−410^\{\-4\}, on a single NVIDIA H200 for∼48\{\\sim\}48GPU\-hours\. Full hyperparameters, loss weights, flow\-matching schedule, and evaluation protocol are in Appendix[F](https://arxiv.org/html/2607.12380#A6)\.

### 4\.2Reconstruction Fidelity

Figure[3](https://arxiv.org/html/2607.12380#S4.F3)and Tables[2](https://arxiv.org/html/2607.12380#S4.T2)–[3](https://arxiv.org/html/2607.12380#S4.T3)report reconstruction across all four domains\. On every dataset,SinAEreaches sub\-milli\-Å RMSD for molecules and crystals and0\.0100\.010–0\.0140\.014Å CA\-RMSD on proteins \(chains truncated to the training\-time maximum of256256residues; longer chains require retraining at a higher length budget, which we leave to future work\), one to three orders of magnitude tighter than prior latent autoencoders\.

![Refer to caption](https://arxiv.org/html/2607.12380v1/x3.png)Figure 3:Reconstruction RMSD across domains \(log scale\)\.Visual summary of Tables[2](https://arxiv.org/html/2607.12380#S4.T2)–[3](https://arxiv.org/html/2607.12380#S4.T3);SinAEreaches2020–100×100\\timestighter RMSD than the previous best baseline on the crystal and protein domains\.Isolating the flow\-matching decoder\.On molecules,SinAEmatches or outperforms prior latent autoencoders\(Xuet al\.,[2023](https://arxiv.org/html/2607.12380#bib.bib31); Luoet al\.,[2026](https://arxiv.org/html/2607.12380#bib.bib3)\)while reaching100%100\\%structure match rate on both QM9 and GEOM\-Drugs \(Table[2](https://arxiv.org/html/2607.12380#S4.T2)\)\. ADiT\(Joshiet al\.,[2025](https://arxiv.org/html/2607.12380#bib.bib6)\)also uses a vanilla Transformer over Cartesian coordinates with a shared latent across molecules and crystals, but relies on a single\-pass VAE decoder; it reaches94\.6%94\.6\\%match rate on QM9 and84\.5%84\.5\\%on MP\-20, compared to100%100\\%forSinAEon both\. This gap isolates the contribution of the iterative flow\-matching decoder\. Reconstruction RMSD saturates by roughly six decoder steps \(Appendix[G](https://arxiv.org/html/2607.12380#A7)\)\.

Cross\-domain transfer\.Joint QM9 \+ MP\-20 training improves*both*domains relative to single\-domain training: QM9 RMSD drops from0\.00070\.0007to0\.0002\\mathbf\{0\.0002\}Å, and MP\-20 RMSD from0\.00410\.0041to0\.0017\\mathbf\{0\.0017\}Å\. Section[5\.1](https://arxiv.org/html/2607.12380#S5.SS1)provides visual evidence through latent\-space visualisation\.

Table 2:Reconstruction on molecules and crystals\.\(a\)structure match rate and Kabsch\-aligned coordinate RMSD on molecule benchmarks\.\(b\)coordinate RMSD and pymatgen structure match rate on periodic benchmarks \(MP\-20; QMOF as an additional secondary domain\)\. “SinAE\(joint\)” on MP\-20 denotes the model trained jointly on QM9 \+ MP\-20\.\(a\) Molecules\.

\(b\) Periodic systems\.

Table 3:Protein backbone reconstruction RMSD \(↓\\downarrow, Å\)\.Mean over evaluation samples; best per columnbold\.SinAEis evaluated on chains truncated to the training\-time maximum of256256residues; longer chains are not supported by the current autoencoder without retraining at higher length\. Full table with±\\pmstd is in Appendix[K](https://arxiv.org/html/2607.12380#A11)\.
### 4\.3Generative Performance: Molecules

With near\-lossless reconstruction established, we next ask whether the resulting latent space is smooth enough for a standard DiT prior to produce chemically valid structures\. Tables[5](https://arxiv.org/html/2607.12380#S4.T5)and[5](https://arxiv.org/html/2607.12380#S4.T5)report molecular generation on QM9 and GEOM\-Drugs\. The metric we emphasise is PB Valid \(PoseBusters validity\), which simultaneously checks bond lengths, bond angles, and intra\-molecular clashes and is therefore the strictest test of geometric quality\. On QM9, the jointSinAEmodel reaches PB Valid0\.97, above all compared baselines including equivariant methods \(EQGAT\-diff\(Leet al\.,[2024](https://arxiv.org/html/2607.12380#bib.bib8)\)and SemlaFlow\(Irwinet al\.,[2024](https://arxiv.org/html/2607.12380#bib.bib4)\)at0\.940\.94, and FlowMol\(Dunn and Koes,[2024](https://arxiv.org/html/2607.12380#bib.bib7)\)at0\.920\.92\)\. On GEOM\-Drugs the margin widens:SinAEachieves PB Valid0\.93against ADiT\(Joshiet al\.,[2025](https://arxiv.org/html/2607.12380#bib.bib6)\)’s0\.860\.86and SemlaFlow’s0\.880\.88\. That a non\-equivariant architecture matches or exceeds equivariant generators on geometric quality reflects the two\-stage design: the autoencoder’s iterative decoder has already learned to produce geometrically precise structures during reconstruction training, and the DiT prior inherits this precision through the latent interface without needing to rediscover bond\-length or angle constraints\. Validity, uniqueness, diversity, and novelty are competitive across both benchmarks \(Tables[5](https://arxiv.org/html/2607.12380#S4.T5)–[5](https://arxiv.org/html/2607.12380#S4.T5)\); the per\-PoseBusters\-check breakdown is reported in Appendix[M](https://arxiv.org/html/2607.12380#A13)\.

Table 4:Molecule generation on QM9 \(Benchmark I\)\.Each row is a separately trained QM9 model;SinAE\(joint\) is jointly trained on QM9\+MP\-20\.
Table 5:Molecule generation on GEOM\-Drugs \(Benchmark II\)\.SinAE\(GEOM\) is trained on GEOM\-Drugs alone; QM9 and GEOM\-Drugs are never jointly trained\.

![Refer to caption](https://arxiv.org/html/2607.12380v1/x4.png)Figure 4:Generated samples fromSinAE\.Unconditional samples from the latent DiT prior decoded with the corresponding flow\-matching decoder \(left: small molecules; middle: crystals; right: protein backbones\)\. Samples are uncurated; per\-domain visualisations at higher resolution are in Appendix[O](https://arxiv.org/html/2607.12380#A15)\.
### 4\.4Generative Performance: Materials

Crystal generation requires producing not only valid arrangements but thermodynamically metastable structures that survive DFT relaxation\. Table[6](https://arxiv.org/html/2607.12380#S4.T6)reports MP\-20 results using S\.U\.N\. \(Stable, Unique, Novel\) and M\.S\.U\.N\. \(S\.U\.N\. with the stability threshold relaxed toEhull<0\.1E\_\{\\mathrm\{hull\}\}\{<\}0\.1eV/atom\)\. The jointSinAEmodel achievesS\.U\.N\. 5\.4/M\.S\.U\.N\. 30\.2, the highest among all compared methods, surpassing FlowMM\(Milleret al\.,[2024](https://arxiv.org/html/2607.12380#bib.bib21)\)\(2\.82\.8/22\.522\.5\) and FlowLLM\(Sriramet al\.,[2024](https://arxiv.org/html/2607.12380#bib.bib41)\)\(4\.74\.7/26\.326\.3\) despite using a generic Transformer with no lattice invariance or symmetry priors\. The joint model improves over MP\-20\-only on every metric \(S\.U\.N\.4\.6→5\.44\.6\\to 5\.4, M\.S\.U\.N\.24\.31→30\.224\.31\\to 30\.2\), consistent with the reconstruction gains in Section[4\.2](https://arxiv.org/html/2607.12380#S4.SS2)\. Structure / composition validity \(99\.2999\.29/90\.4990\.49\) sits just below jointly\-trained ADiT\(Joshiet al\.,[2025](https://arxiv.org/html/2607.12380#bib.bib6)\)\(99\.7499\.74/92\.1492\.14\), withSinAE’s advantage concentrated in the stability\-filtered metrics that probe thermodynamic plausibility rather than structural or compositional validity alone\.

Table 6:Material generation on MP\-20\.Structure and composition validity rates, stable rate \(Ehull<0E\_\{\\mathrm\{hull\}\}\{<\}0eV/atom\), and the hard S\.U\.N\. / M\.S\.U\.N\. metrics \(all in %, higher is better\)\. “–” denotes numbers not reported by the original authors\. Metastable rate and “Overall” validity are subsumed by M\.S\.U\.N\. and byStructure×Composition\\mathrm\{Structure\}\\times\\mathrm\{Composition\}, respectively; full breakdown is in Appendix[L](https://arxiv.org/html/2607.12380#A12)\.
### 4\.5Generative Performance: Proteins

The protein autoencoder uses the same architecture as the molecule–crystal model, so strong performance here confirms that theSinAEarchitecture generalises across domains rather than being tuned for small\-molecule or crystal chemistry\.

Unconditional generation\.Table[8](https://arxiv.org/html/2607.12380#S4.T8)reports designability, diversity, DPT, and novelty\. Prior latent\-space protein generators face a trade\-off between designability and diversity: ProteinAE achieves0\.930\.93designability but with204204diversity clusters, while LatentDiff produces diverse but largely non\-designable backbones \(0\.170\.17designability\)\. At guidance temperature0\.250\.25,SinAEreaches0\.96designability and278278diversity, exceeding ProteinAE on both metrics simultaneously; at temperature0\.350\.35it attains the highest diversity \(291\) among all compared methods while maintaining0\.880\.88designability\. Two factors contribute: the near\-lossless decoder ensures that diversity in latent space translates into structural diversity without geometric degradation, and jointly modelling residue types alongside coordinates may steer generation toward more physically plausible backbone geometries compared to coordinate\-only approaches\.

Co\-design generation\.Table[8](https://arxiv.org/html/2607.12380#S4.T8)evaluates sequence–structure co\-design at fixed lengths100100and200200\. At length100100,SinAEreaches scRMSD3\.933\.93Å and scTM0\.830\.83, comparable to MultiFlow\(Campbellet al\.,[2024](https://arxiv.org/html/2607.12380#bib.bib25)\)\(0\.860\.86/4\.734\.73Å\) and ProteinGenerator\(Lisanzaet al\.,[2025](https://arxiv.org/html/2607.12380#bib.bib49)\)\(0\.910\.91/3\.753\.75Å\)\. At length200200, scRMSD is4\.444\.44Å, below both MultiFlow \(4\.984\.98Å\) and ProteinGenerator \(6\.246\.24Å\)\. scTM \(0\.830\.83/0\.780\.78\) is below APM\(Chenet al\.,[2025a](https://arxiv.org/html/2607.12380#bib.bib54)\)\(0\.960\.96/0\.890\.89\), a method with protein\-specific architecture\.

Table 7:Unconditional protein generation\.Designability \(scRMSD≤\\leq2\.0 Å\), diversity, DPT, and novelty\.
Table 8:Co\-design at fixed lengths\(100, 200\) via scTM / scRMSD\.∗: without distillation\. All baseline numbers from ProteinBench\(Yeet al\.,[2024](https://arxiv.org/html/2607.12380#bib.bib53)\)\.

## 5Analysis

### 5\.1Latent Space Structure

Sections[4\.2](https://arxiv.org/html/2607.12380#S4.SS2)and[4\.4](https://arxiv.org/html/2607.12380#S4.SS4)showed that joint QM9\+MP\-20 training improves both domains in reconstruction and generation\. A t\-SNE projection of the joint model’s encoder outputs \(Appendix[H](https://arxiv.org/html/2607.12380#A8)\) shows that molecules and crystals occupy separated, internally coherent regions without any explicit domain label\. Within each domain, KMeans clusters align with chemically meaningful groupings: functional\-group composition for molecules, coordination environment for crystals\. The two domains share a common low\-density boundary region, suggesting that the encoder learns geometric regularities \(bond\-length distributions, coordination patterns\) transferable across domains\. This shared structure is consistent with the reconstruction improvement \(0\.0007→0\.00020\.0007\\to 0\.0002Å on QM9,0\.0041→0\.00170\.0041\\to 0\.0017Å on MP\-20\) and the generation gain \(M\.S\.U\.N\.24\.31→30\.224\.31\\to 30\.2\) when the two populations are pooled through a single latent\. Straight\-line interpolations between encoded structures decode into valid intermediates across all three domains \(Appendix[I](https://arxiv.org/html/2607.12380#A9)\)\.

### 5\.2Decoder Trajectory Analysis

Figure[5](https://arxiv.org/html/2607.12380#S5.F5)examines the decoder’s behaviour*during generation*\(decoding from DiT\-sampled latents, not from encoded ground\-truth structures\)\. We track two quantities along the decoder ODE trajectoryt∈\[0,1\]t\\in\[0,1\]: physical energy \(top row\) and Kabsch\-aligned RMSD to the final state att=1t\{=\}1\(bottom row\)\. For molecules, force\-field energy drops by more than ten orders of magnitude; for crystals, CHGNet energy descends from∼6\{\\sim\}6eV/atom into the sub\-11eV/atom metastable window\. Crucially, the two quantities follow different schedules: RMSD plateaus near zero byt≈0\.8t\\approx 0\.8while energy continues to descend through the final20%20\\%of the trajectory\. Early ODE steps thus handle geometric placement, while late steps perform fine local refinement \(bond\-angle planarity, dihedral relaxation, sterics\) that contributes little to RMSD but substantially to physical energy\. The decoder learns denoising directions aligned with the local energy gradient*even though energy is never supervised*, and this holds for prior\-sampled latents that the encoder has never produced\.

![Refer to caption](https://arxiv.org/html/2607.12380v1/figures/fig_energy_combined.png)

![Refer to caption](https://arxiv.org/html/2607.12380v1/x5.png)

Figure 5:Decoder ODE trajectory during generation\(DiT\-sampled latents, not encoded ground truth\)\.\(a\)Physical energy along the trajectory \(molecule force\-field, log scale; crystal CHGNet, eV/atom\)\.\(b\)Kabsch\-aligned RMSD to the final state att=1t\{=\}1\. RMSD plateaus byt≈0\.8t\\approx 0\.8while energy continues to descend: early steps handle geometric placement, late steps perform fine local refinement\.Energy was never used as a training signal\.

## 6Conclusion

We presentedSinAE, a vanilla\-Transformer flow\-matching autoencoder for molecules, crystals, and proteins\. By shifting reconstruction fidelity into an iterative flow\-matching decoder trained with a unified objective,SinAEreaches sub\-milli\-Å Kabsch\-aligned RMSD on molecules and crystals and∼0\.02\\sim 0\.02Å backbone CA\-RMSD on proteins, while supporting strong latent generation across all three domains\. Joint training on QM9 \+ MP\-20 improves both domains in reconstruction and generation, providing direct evidence that cross\-domain training through a shared atomic latent benefits both chemical populations\.Limitations\.Protein evaluation is at the backbone level; all\-atom protein modelling, property\- / composition\- / length\-conditioned generation, and single\-checkpoint training across all three domains are natural directions for future work\.

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Appendix

## Appendix ABroader Impacts

SinAEis an architectural contribution towards 3D structural representation learning; its immediate downstream use cases are molecular, materials, and protein design in academic and industrial R&D\. Positive societal impacts include accelerating the discovery of catalysts for carbon capture and clean energy, stable metal–organic frameworks for gas storage, and therapeutic lead compounds\. We note, however, that the same generative capabilities could in principle be misused to accelerate the design of toxic small molecules, narcotics, or proteins with harmful biological activity\. The models released alongside this work are trained exclusively on publicly available non\-hazardous datasets \(QM9, GEOM\-Drugs, MP\-20, QMOF, AFDB\-FS\), none of which is enriched for hazardous compounds, and we do not anticipateSinAEconferring a meaningful advantage over existing, widely available generative tools for adversarial design\. We encourage downstream users to combine generative models of this class with structure\-aware toxicity and bioactivity filters before experimental validation, and we are not releasing any fine\-tuned weights targeted at restricted chemical or biological endpoints\.

## Appendix BEvaluation Metrics

This appendix defines every metric reported in the main text and specifies the tool, formula, or threshold used for each\.

### B\.1Reconstruction Metrics

#### Coordinate RMSD \(Å,↓\\downarrow\)\.

For a structure withNNatoms, ground\-truth coordinates𝐗∈ℝN×3\\mathbf\{X\}\\in\\mathbb\{R\}^\{N\\times 3\}, and reconstructed coordinates𝐗^∈ℝN×3\\hat\{\\mathbf\{X\}\}\\in\\mathbb\{R\}^\{N\\times 3\}, we first align𝐗^\\hat\{\\mathbf\{X\}\}to𝐗\\mathbf\{X\}using the Kabsch algorithm \(optimal rotationR∗R^\{\*\}, translationt∗t^\{\*\}via SVD\), then report

RMSD​\(𝐗,𝐗^\)=1N​∑i=1N‖𝐱i−\(R∗​𝐱^i\+t∗\)‖22\.\\mathrm\{RMSD\}\(\\mathbf\{X\},\\hat\{\\mathbf\{X\}\}\)\\;=\\;\\sqrt\{\\tfrac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}\\bigl\\\|\\mathbf\{x\}\_\{i\}\-\(R^\{\*\}\\hat\{\\mathbf\{x\}\}\_\{i\}\+t^\{\*\}\)\\bigr\\\|\_\{2\}^\{2\}\}\.For proteins we apply this to theCαC\_\{\\alpha\}trace \(CA\-RMSD\)\. For periodic systems \(crystals, MOFs\), coordinates are first unwrapped into a single fractional cell and Kabsch alignment is applied in the Cartesian space\.

#### Structure Match Rate — Reconstruction \(%,↑\\uparrow\)\.

For molecule*reconstruction*on QM9, the structure match rate usespymatgen\.analysis\.molecule\_matcher\.MoleculeMatcher\.get\_rmsdwith InChI\-based atom mapping at tolerance0\.010\.01Å\. A reconstructed molecule is counted as a match iff this routine returns a finite RMSD, i\.e\. the predicted 3D conformation is chemically isomorphic to the reference under the OpenBabel bond\-perception convention \(identical atom counts, atomic numbers, and inferred bond multiset\)\.

For MP\-20 crystal reconstruction, we compute the structure match rate usingStructureMatcherfrompymatgen, with thresholdsltol=0\.3l\_\{\\mathrm\{tol\}\}=0\.3,stol=0\.5s\_\{\\mathrm\{tol\}\}=0\.5, andangletol=10∘\\mathrm\{angle\}\_\{\\mathrm\{tol\}\}=10^\{\\circ\}\. A 100% match rate means that every reconstructed crystal is matched to its reference under these tolerances\.

#### Valid \(%,↑\\uparrow\)\.

Fraction of generated molecules accepted by RDKit’s sanitization pipeline \(valid valences, no kekulisation failure, parseable SMILES\)\.

#### Unique \(%,↑\\uparrow\)\.

Fraction of valid molecules whose canonical SMILES is distinct from the other molecules in the sampled batch\. Computed on the sameNNsamples used forValid\.

#### Novelty \(%,↑\\uparrow\)\.

Fraction of valid molecules whose canonical SMILES does not appear in the training set\.

#### Diversity \(↑\\uparrow\)\.

Mean pairwise Tanimoto*dissimilarity*of Morgan fingerprints \(radius 2, 2048 bits\) across the valid generated molecules:

Diversity=1−2N​\(N−1\)​∑i<jTanimoto​\(𝐟i,𝐟j\)\.\\mathrm\{Diversity\}=1\-\\tfrac\{2\}\{N\(N\-1\)\}\\sum\_\{i<j\}\\mathrm\{Tanimoto\}\(\\mathbf\{f\}\_\{i\},\\mathbf\{f\}\_\{j\}\)\.

#### PB Valid \(%,↑\\uparrow\)\.

PoseBusters\-Validity: a molecule is PB\-valid only if it simultaneously passes all PoseBusters sanity checks: chemistry, intra\-molecular clashes, bond lengths and angles within empirical distributions\. PB Valid is the most physically restrictive metric in Tables[5](https://arxiv.org/html/2607.12380#S4.T5)–[5](https://arxiv.org/html/2607.12380#S4.T5)and is the metric we emphasise for geometric quality\.

### B\.2Crystal Generation Metrics

#### Structure / Composition / Overall Validity \(%,↑\\uparrow\)\.

*Structure validity*counts generated crystals that pass pymatgen’s structure\-level sanity \(no atomic overlap, finite lattice\)\.*Composition validity*counts crystals whose composition is charge\-balanced with standard oxidation states\.*Overall validity*is the product \(i\.e\. passes both\)\. We follow the protocol of CDVAE\.

#### Stable and Metastable Rate \(%,↑\\uparrow\)\.

We follow the two\-stage relaxation protocol of ADiT\(Joshiet al\.,[2025](https://arxiv.org/html/2607.12380#bib.bib6)\): each generated crystal is first pre\-relaxed with the CHGNet universal machine\-learning potential\(Denget al\.,[2023](https://arxiv.org/html/2607.12380#bib.bib19)\), and the pre\-relaxed geometry is then DFT\-relaxed\. Stability is evaluated against the Matbench\-Discovery convex hull using the final*DFT*energy\-above\-hullEhullE\_\{\\mathrm\{hull\}\}:

stable:Ehull<0\.0​eV/atom and​\#​unique elements≥2,\\text\{stable\}:E\_\{\\mathrm\{hull\}\}<0\.0\\text\{ eV/atom and \}\\\#\\text\{unique elements\}\\geq 2,metastable:Ehull<0\.1​eV/atom and​\#​unique elements≥2\.\\text\{metastable\}:E\_\{\\mathrm\{hull\}\}<0\.1\\text\{ eV/atom and \}\\\#\\text\{unique elements\}\\geq 2\.The≥2\\geq 2unique\-element constraint matches ADiT and excludes trivial elemental crystals from stability counting\.

#### S\.U\.N\. and M\.S\.U\.N\. \(%,↑\\uparrow\)\.

*S\.U\.N\.*\(Stable, Unique, Novel\) is the fraction of sampled crystals that are simultaneously \(i\) stable under the above definition, \(ii\) structurally unique within the sample batch, and \(iii\) novel with respect to the MP\-20 training set\. Uniqueness and novelty both usepymatgen\.StructureMatcherwith the thresholds listed in[B\.1](https://arxiv.org/html/2607.12380#A2.SS1)for an all\-to\-all equivalence check\.*M\.S\.U\.N\.*replaces “stable” with “metastable” in condition \(i\)\. Following FlowMM, FlowLLM, and ADiT, these are the*hard*metrics for crystal generation and the ones we emphasise when claiming state of the art\.

### B\.3Protein Generation Metrics

For each generated backbone we run ProteinMPNN inverse folding \(88sequences per backbone\) and fold the resulting sequences with ESMFold; all self\-consistency metrics below are computed over this design\-fold loop, following the protocol of ProteinBench\(Yeet al\.,[2024](https://arxiv.org/html/2607.12380#bib.bib53)\)\.

#### scRMSD \(Å,↓\\downarrow\) and scTM \(↑\\uparrow\)\.

Self\-consistency RMSD is the Kabsch\-aligned CA\-RMSD between the generated backbone and the best ESMFold re\-fold among the88designed sequences; scTM is the corresponding best TM\-score\. These are the two primary*quality*metrics in Table[8](https://arxiv.org/html/2607.12380#S4.T8)\.

#### Designability \(“Des”,↑\\uparrow\)\.

Fraction of generated backbones withscRMSD≤2\.0\\mathrm\{scRMSD\}\\leq 2\.0Å, i\.e\. structures for which at least one designed sequence re\-folds into essentially the same backbone\. This is the standard “designable” criterion\(Watsonet al\.,[2023](https://arxiv.org/html/2607.12380#bib.bib46); Yimet al\.,[2023](https://arxiv.org/html/2607.12380#bib.bib47)\)\.

#### Diversity \(“Div”,↑\\uparrow\)\.

Number of unique structural clusters obtained by hierarchical clustering of the designable backbones using pairwise TM\-score with a0\.50\.5cutoff \(single linkage\)\. A higher number indicates that more distinct folds have been reached\.

#### DPT — Designable Pairwise TM\-score \(↓\\downarrow\)\.

Mean pairwise TM\-score computed*only*over the subset of generated backbones that passed the designability filter \(scRMSD≤2\.0\\mathrm\{scRMSD\}\\leq 2\.0Å\)\. Restricting the computation to designable samples removes low\-quality, degenerate structures that would otherwise inflate pairwise dissimilarity; a lower DPT indicates that the useful outputs of the model explore a structurally wider region\. DPT is complementary to*Div*\(which counts cluster number\) in that it measures the geometric spread of the designable samples\.

#### Novelty \(“Nov”,↑\\uparrow\)\.

1−maxp∈PDB⁡TM​\(x^,p\)1\-\\max\_\{p\\in\\mathrm\{PDB\}\}\\mathrm\{TM\}\(\\hat\{x\},p\), the deviation of each generated backbone from its closest match in the full PDB \(measured by TM\-score\); averaged across designable samples\. Higher means further from any known fold\.

### B\.4Latent\-Space Analysis Metrics

#### Per\-atom classical force\-field energy \(eV/atom,↓\\downarrow\)\.

For molecule trajectories in Figure[5](https://arxiv.org/html/2607.12380#S5.F5)we report the per\-structure energy computed with a classical force field \(UFF/MMFF as implemented in RDKit\) evaluated on the decoder’s intermediate structures\(𝐱t,𝐚t\)\(\\mathbf\{x\}\_\{t\},\\mathbf\{a\}\_\{t\}\)\. For crystal trajectories we report the per\-atom energy from the same CHGNet ML potential used for stability evaluation above\. These are reported only to probe whether the decoder’s ODE path visits progressively lower\-energy configurations; they are not used as training signal at any stage\.

## Appendix CFull Dataset Specification

This appendix provides the per\-domain dataset details abbreviated in Section[4\.1](https://arxiv.org/html/2607.12380#S4.SS1)\.

#### Molecules\.

QM9\(Ramakrishnanet al\.,[2014](https://arxiv.org/html/2607.12380#bib.bib37)\)consists of 130,000 stable small organic molecules containing up to nine heavy atoms \(C, N, O, F\) along with hydrogens\. We follow the splits ofHoogeboomet al\.\([2022](https://arxiv.org/html/2607.12380#bib.bib9)\)\.GEOM\-Drugs\(Axelrod and Gomez\-Bombarelli,[2022](https://arxiv.org/html/2607.12380#bib.bib13)\)is a dataset of 1M high\-quality conformers of drug\-like molecules; we use the splits ofVignacet al\.\([2023](https://arxiv.org/html/2607.12380#bib.bib11)\)and, followingIrwinet al\.\([2024](https://arxiv.org/html/2607.12380#bib.bib4)\), discard molecules with more than 72 heavy atoms from the training set \(accounting for 1% of the data\)\. At test time we sample molecule sizes from the \(unchanged\) test\-set size distribution\.

#### Materials\.

MP\-20\(Xieet al\.,[2022](https://arxiv.org/html/2607.12380#bib.bib23)\)contains 45,231 metastable crystal structures from the Materials Project\(Jainet al\.,[2013](https://arxiv.org/html/2607.12380#bib.bib12)\), each with up to 20 atoms per unit cell and spanning 89 elements; we use the CDVAE splits\(Xieet al\.,[2022](https://arxiv.org/html/2607.12380#bib.bib23)\)\.QMOF\(Rosenet al\.,[2021](https://arxiv.org/html/2607.12380#bib.bib14)\)is a database of 14,000 MOF structures with quantum\-chemical properties; we use the splits ofJoshiet al\.\([2025](https://arxiv.org/html/2607.12380#bib.bib6)\)\.

#### Proteins\.

AFDB\-FS\(Jumperet al\.,[2021](https://arxiv.org/html/2607.12380#bib.bib29); Linet al\.,[2023](https://arxiv.org/html/2607.12380#bib.bib38)\)is a filtered single\-chain subset of the AlphaFold Protein Structure Database, obtained via sequence\-based MMseqs2 and structure\-based Foldseek\(Van Kempenet al\.,[2024](https://arxiv.org/html/2607.12380#bib.bib40)\)clustering\. It contains 588,318 structures with lengths in\[32,256\]\[32,256\]\. For training we apply random global SO\(3\) rotations as data augmentation\. For reconstruction evaluation we use the CASP14 and CASP15 benchmark sets\.

## Appendix DMethod Design Rationale

This appendix collects the design justifications abbreviated in the main\-paper architecture section\.

#### Why a per\-token latent instead of a pooled global code\.

Atomistic structures exhibit local geometric features \(bond environments, coordination shells\) whose information content scales with the number of atomsNN; a fixed\-size global latent would force lossy compression asNNgrows\. Our latent is𝐳∈ℝN′×d\\mathbf\{z\}\\in\\mathbb\{R\}^\{N^\{\\prime\}\\times d\}with compact per\-token dimensiondd\(see Appendix[F](https://arxiv.org/html/2607.12380#A6)\), which keeps the total latent size an order of magnitude smaller than the raw structure while preserving enough information for near\-exact decoding\. Because the decoder handles fine\-grained geometry, the encoder can remain lightweight while producing a latent that is smooth and suitable for downstream generative modelling\.

#### Why a non\-equivariant Transformer\.

Equivariant layers encode physical symmetries as hard constraints, but at substantial architectural\-complexity and scaling cost\. We instead rely on random SO\(3\) rotation augmentation during training \(Section[3\.2](https://arxiv.org/html/2607.12380#S3.SS2)\) to induce rotation robustness, and demonstrate empirically that this, combined with the iterative flow\-matching decoder, is sufficient to achieve near\-lossless reconstruction on molecules and crystals and sub\-0\.030\.03Å RMSD on proteins\.

#### Why output heads instead of stream\-specific mid\-backbone cross\-attention\.

Following Tabasco\(Vonessenet al\.,[2026](https://arxiv.org/html/2607.12380#bib.bib1)\), we localise the coordinate / atom\-type specialisation in two parallel cross\-attention output heads rather than interleaving stream\-specific cross\-attention inside the backbone\. This keeps every Transformer block in the backbone fully shared across its constituent structural categories, and places all cross\-attention compute at the output, where it directly conditions the two prediction heads on the embedded noisy inputs\.

#### Why we do not use self\-conditioning\.

We found a single\-pass output head with cross\-attention from the noisy input to be sufficient; self\-conditioning\(Chenet al\.,[2023](https://arxiv.org/html/2607.12380#bib.bib35)\)does not provide measurable gains on our atomistic reconstruction benchmarks in preliminary tests and increases per\-step cost\.

## Appendix EFull Training Objective

This appendix gives the exact interpolants, loss formulae, and weight values abbreviated in Section[3\.4](https://arxiv.org/html/2607.12380#S3.SS4)\. All three terms are summed in Eq\.[2](https://arxiv.org/html/2607.12380#S3.E2)\.

#### Coordinate flow matching\(Lipmanet al\.,[2023](https://arxiv.org/html/2607.12380#bib.bib33)\)\.

With the interpolant𝐗t=t​𝐗1\+\(1−t\)​ϵ\\mathbf\{X\}\_\{t\}=t\\,\\mathbf\{X\}\_\{1\}\+\(1\-t\)\\,\\bm\{\\epsilon\},ϵ∼𝒩​\(𝟎,𝐈\)\\bm\{\\epsilon\}\\sim\\mathcal\{N\}\(\\mathbf\{0\},\\mathbf\{I\}\),t∼𝒰​\(0,1\)t\\sim\\mathcal\{U\}\(0,1\), the coordinate head is trained withx1x\_\{1\}\-prediction:

ℒcoord=𝔼ϵ,t​\[1N​‖𝐗^1θ​\(𝐗t,𝐀t,t,𝐳\)−𝐗1‖22\]\.\\mathcal\{L\}\_\{\\text\{coord\}\}=\\mathbb\{E\}\_\{\\bm\{\\epsilon\},\\,t\}\\\!\\left\[\\tfrac\{1\}\{N\}\\bigl\\\|\\hat\{\\mathbf\{X\}\}\_\{1\}^\{\\theta\}\(\\mathbf\{X\}\_\{t\},\\mathbf\{A\}\_\{t\},t,\\mathbf\{z\}\)\-\\mathbf\{X\}\_\{1\}\\bigr\\\|\_\{2\}^\{2\}\\right\]\.\(3\)

#### Atom\-type discrete flow matching\(Campbellet al\.,[2024](https://arxiv.org/html/2607.12380#bib.bib25)\)\.

Ground\-truth atom types are corrupted by interpolating with a uniform categorical prior,𝐀t∼Cat​\(t​δ​\(𝐀1\)\+\(1−t\)​1K​𝟏\)\\mathbf\{A\}\_\{t\}\\sim\\mathrm\{Cat\}\(t\\,\\delta\(\\mathbf\{A\}\_\{1\}\)\+\(1\-t\)\\,\\tfrac\{1\}\{K\}\\mathbf\{1\}\), and the head is trained with endpoint cross\-entropy:

ℒatom=𝔼t​\[−1N​∑i=1N𝐀1,i⊤​log⁡𝐀^1,iθ​\(𝐗t,𝐀t,t,𝐳\)\]\.\\mathcal\{L\}\_\{\\text\{atom\}\}=\\mathbb\{E\}\_\{t\}\\\!\\left\[\-\\tfrac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}\\mathbf\{A\}\_\{1,i\}^\{\\top\}\\log\\hat\{\\mathbf\{A\}\}\_\{1,i\}^\{\\theta\}\(\\mathbf\{X\}\_\{t\},\\mathbf\{A\}\_\{t\},t,\\mathbf\{z\}\)\\right\]\.\(4\)

#### KL regularisation\.

Unlike DiTo\(Chenet al\.,[2025b](https://arxiv.org/html/2607.12380#bib.bib18)\), we pair a stochastic encoder with the decoder and add a light KL term pushingqϕ​\(𝐳\|𝒢\)q\_\{\\phi\}\(\\mathbf\{z\}\|\\mathcal\{G\}\)towards𝒩​\(𝟎,𝐈\)\\mathcal\{N\}\(\\mathbf\{0\},\\mathbf\{I\}\):

ℒKL=DKL​\(qϕ​\(𝐳\|𝒢\)∥𝒩​\(𝟎,𝐈\)\)=12​∑j\(μj2\+σj2−log⁡σj2−1\)\.\\mathcal\{L\}\_\{\\text\{KL\}\}=D\_\{\\mathrm\{KL\}\}\\bigl\(q\_\{\\phi\}\(\\mathbf\{z\}\|\\mathcal\{G\}\)\\,\\\|\\,\\mathcal\{N\}\(\\mathbf\{0\},\\mathbf\{I\}\)\\bigr\)=\\tfrac\{1\}\{2\}\\sum\_\{j\}\\bigl\(\\mu\_\{j\}^\{2\}\+\\sigma\_\{j\}^\{2\}\-\\log\\sigma\_\{j\}^\{2\}\-1\\bigr\)\.\(5\)Aligning the aggregate posterior with a Gaussian simplifies the prior\-fitting problem for the latent flow\-matching model \(Section[3\.5](https://arxiv.org/html/2607.12380#S3.SS5)\)\.

Weight values \(λatom=0\.1\\lambda\_\{\\text\{atom\}\}\{=\}0\.1,λKL=10−6\\lambda\_\{\\text\{KL\}\}\{=\}10^\{\-6\}\) are listed alongside the rest of the training configuration in Appendix[F](https://arxiv.org/html/2607.12380#A6); we find the model insensitive to these weights across an order of magnitude each\.

## Appendix FFull Implementation Details

This appendix gives the complete architectural, training, and evaluation specification for theSinAEmodels summarised in Section[4\.1](https://arxiv.org/html/2607.12380#S4.SS1)\.

### F\.1Architecture

The autoencoder uses a pre\-norm vanilla Transformer with hidden dimensiondmodel=64d\_\{\\text\{model\}\}\{=\}64and88attention heads throughout\. No equivariant layer, message\-passing operator, graph attention, or pretrained discriminator is used anywhere in the architecture\.

- \-Joint molecule–crystal autoencoder\.Encoder depthLenc=4L\_\{\\text\{enc\}\}\{=\}4, decoder depthLdec=16L\_\{\\text\{dec\}\}\{=\}16, per\-token latent dimensiond=8d\{=\}8\(so𝐳∈ℝN′×8\\mathbf\{z\}\\in\\mathbb\{R\}^\{N^\{\\prime\}\\times 8\}\)\.
- \-Joint molecule–crystal latent DiT\.Ldit=24L\_\{\\text\{dit\}\}\{=\}24layers,dmodel=1024d\_\{\\text\{model\}\}\{=\}1024,1616attention heads\.
- \-Protein autoencoder\.Encoder depthLenc=4L\_\{\\text\{enc\}\}\{=\}4, decoder depthLdec=16L\_\{\\text\{dec\}\}\{=\}16, per\-token latent dimensiond=16d\{=\}16\.
- \-Protein latent DiT prior\.Ldit=15L\_\{\\text\{dit\}\}\{=\}15layers,dmodel=768d\_\{\\text\{model\}\}\{=\}768,1212attention heads\.

### F\.2Training

All stages use Adam \(autoencoder\) and AdamW \(latent prior\) with a constant learning rate of1×10−41\{\\times\}10^\{\-4\}, weight decay0, no warm\-up, and no scheduler\. The joint molecule–crystal autoencoder is trained with batch size20482048\(each batch drawn5050/5050from QM9 and MP\-20\); the protein autoencoder uses batch size256256\.

The total reconstruction loss follows Eq\.[2](https://arxiv.org/html/2607.12380#S3.E2)with the fixed weights

λcoord=1\.0,λatom=0\.1,λKL=1×10−6\.\\lambda\_\{\\text\{coord\}\}\{=\}1\.0,\\quad\\lambda\_\{\\text\{atom\}\}\{=\}0\.1,\\quad\\lambda\_\{\\text\{KL\}\}\{=\}1\{\\times\}10^\{\-6\}\.The flow\-matching schedule uses a lower time cutofftmin=10−2t\_\{\\min\}\{=\}10^\{\-2\}; no additionalσmin\\sigma\_\{\\min\}is imposed\.

The joint molecule–crystal model was trained on a single NVIDIA H200 GPU for approximately4848GPU\-hours\.

### F\.3Evaluation Protocol

#### Reconstruction RMSD\.

All reconstruction RMSDs are computed after Kabsch SVD alignment between the decoded and reference coordinates\. Protein CA\-RMSD uses the same Kabsch pipeline on theCαC\_\{\\alpha\}trace\.

#### Crystal Structure\-Match Rate\.

Both crystal generation and crystal reconstruction structure\-match rates use the samepymatgen\.StructureMatcherpipeline with the standard thresholdsltol=0\.3\\mathrm\{ltol\}\{=\}0\.3,stol=0\.5\\mathrm\{stol\}\{=\}0\.5,angle​\_​tol=10∘\\mathrm\{angle\\\_tol\}\{=\}10^\{\\circ\}\(Xieet al\.,[2022](https://arxiv.org/html/2607.12380#bib.bib23)\); we additionally report Kabsch\-aligned RMSD for reconstruction as a continuous fidelity metric\.

#### Generation NFE\.

At generation time we use NFE=100\{=\}100Euler steps for both the DiT latent prior and the flow\-matching decoder of the joint molecule–crystal model\. Protein latent sampling uses a smaller latent step sizedtlatent=2\.5×10−3\\mathrm\{dt\}\_\{\\text\{latent\}\}\{=\}2\.5\{\\times\}10^\{\-3\}\(NFE=400\{=\}400\), because the longer protein latents empirically benefit from more steps\. Classifier\-free guidance \(used only for domain\-selection in the joint molecule–crystal model\) doubles per\-step compute but does not change NFE\.

## Appendix GFidelity–Efficiency Trade\-off \(Decoder NFE\)

Because reconstruction is iterative, inference cost scales linearly with the number of function evaluations \(NFE\) in the decoder\. Figure[6](https://arxiv.org/html/2607.12380#A7.F6)plots reconstruction RMSD against NFE on QM9 and MP\-20\. On both datasets, RMSD drops sharply between NFE=1\{=\}1and NFE=4\{=\}4and saturates by NFE≈6\{\\approx\}6: the QM9 decoder reaches its asymptotic RMSD of∼0\.0003\\sim 0\.0003Å after only66steps, and MP\-20 saturates at∼0\.0016\\sim 0\.0016Å over the same range\. Even at NFE=2\{=\}2,SinAEalready outperforms every VAE baseline in Table[2](https://arxiv.org/html/2607.12380#S4.T2), confirming that the fidelity advantage is not from expensive sampling\. In practice we use NFE=10\{=\}10as a comfortable operating point\.

In wall\-clock terms, at NFE=10\{=\}10on a single NVIDIA H200 the decoder takes0\.130\.13s per batch of44QM9\-scale molecules \(∼36\\sim 36atoms total\), i\.e\.∼32\\sim 32ms per molecule, or∼3\.6\\sim 3\.6ms per atom\. Generating1010k molecules takes∼5\\sim 5minutes including overhead\.

![Refer to caption](https://arxiv.org/html/2607.12380v1/figures/reconstruction_convergence_full.png)Figure 6:Reconstruction fidelity vs decoder NFE\.RMSD on QM9 \(blue\) and MP\-20 \(green\) as a function of the number of decoder function evaluations\. Both curves drop sharply between NFE=1\{=\}1and NFE=4\{=\}4and saturate by NFE≈6\{\\approx\}6\.
## Appendix Ht\-SNE of the Joint Latent Space

Section[5\.1](https://arxiv.org/html/2607.12380#S5.SS1)interprets the cross\-domain transfer through a t\-SNE projection of the joint molecule \+ crystal model’s encoder outputs\. Figure[7](https://arxiv.org/html/2607.12380#A8.F7)visualises this projection: QM9 molecules \(circles\) and MP\-20 crystals \(squares\) occupy separated, internally coherent regions of the latent space without any explicit domain label\. Colours indicate within\-domain KMeans clusters; molecular clusters separate by functional\-group composition while crystal clusters align with coordination environment\. The two domains share a low\-density boundary region, consistent with shared low\-level geometric regularities \(bond\-length distributions, coordination patterns\) being captured by a common encoder\.

![Refer to caption](https://arxiv.org/html/2607.12380v1/figures/latent_joint_tsne.png)Figure 7:t\-SNE ofSinAElatents\.QM9 \(circles\) and MP\-20 \(squares\) separate without explicit domain conditioning; colours are within\-domain KMeans clusters\.
## Appendix ILatent\-Space Interpolation Across Domains

Section[5\.1](https://arxiv.org/html/2607.12380#S5.SS1)reports that straight\-line interpolations between pairs ofSinAElatents decode into chemically and geometrically valid intermediates across all three domains; we visualise representative trajectories in Figure[8](https://arxiv.org/html/2607.12380#A9.F8)\. Molecular interpolations preserve ring connectivity while smoothly varying functional groups; crystal interpolations transition between unit\-cell geometries while keeping atoms well\-coordinated; protein interpolations evolve continuously from highly helical backbones toward more compact mixed\-fold structures\. This smoothness is consistent with the strong generative results in Section[4](https://arxiv.org/html/2607.12380#S4)\.

![Refer to caption](https://arxiv.org/html/2607.12380v1/figures/fig_interp_molecules.png)

![Refer to caption](https://arxiv.org/html/2607.12380v1/figures/fig_interp_crystals.png)

![Refer to caption](https://arxiv.org/html/2607.12380v1/figures/fig_interp_proteins.png)

Figure 8:Latent\-space interpolation across domains\.Structures decoded along straight\-line interpolations between pairs ofSinAElatents: molecules \(top\), crystals \(middle\), proteins \(bottom\)\.
## Appendix JLength Generalisation on Proteins

Figure[9](https://arxiv.org/html/2607.12380#A10.F9)examines whether theSinAEprotein model, trained with chains truncated to a maximum of256256residues, generalises to longer chains and to generation lengths it was never explicitly conditioned on\. Figure[9](https://arxiv.org/html/2607.12380#A10.F9)\(a\) plots per\-sample reconstruction CA\-RMSD against protein length on AFDB\-FS; the binned mean \(right panel\) stays below0\.0150\.015Å across the full6060–260260residue range, increasing only mildly from∼0\.006\\sim 0\.006Å at 60 residues to∼0\.014\\sim 0\.014Å at 260 residues\. This sub\-linear scaling indicates that the flow\-matching decoder does not require length\-conditioned training to remain accurate on longer chains\. Figure[9](https://arxiv.org/html/2607.12380#A10.F9)\(b\) reports the analogous picture on the generative side: the distribution of self\-consistency scRMSD as a function of generated length\. The mean scRMSD remains in a narrow band between 60 and 128 residues without any monotonic upward trend; quantitative per\-length numbers are reported in Table[8](https://arxiv.org/html/2607.12380#S4.T8)\.

![Refer to caption](https://arxiv.org/html/2607.12380v1/figures/fig_length.png)Figure 9:Length generalisation on proteins\.\(a\)Per\-sample reconstruction CA\-RMSD vs protein length on AFDB\-FS; the binned mean \(right\) stays below0\.0150\.015Å across the full 60–260 residue range\.\(b\)Distribution of generation scRMSD as a function of generated sequence length; the mean \(orange\) is flat across 60–128 residues\.
## Appendix KProtein Reconstruction: Full Table with±\\pmstd

Table[9](https://arxiv.org/html/2607.12380#A11.T9)reproduces the protein\-backbone reconstruction RMSD numbers from Table[3](https://arxiv.org/html/2607.12380#S4.T3)with the full mean±\\,\\pm\\,std\. uncertainties from ProteinBench\(Yeet al\.,[2024](https://arxiv.org/html/2607.12380#bib.bib53)\)\.

Table 9:Protein backbone reconstruction RMSD — full table\.Mean±\\,\\pm\\,std\. in Å \(↓\\downarrow\)\. Best per columnbold\.SinAEis evaluated on chains truncated to the training\-time maximum of256256residues\. All baseline numbers are taken from ProteinBench\(Yeet al\.,[2024](https://arxiv.org/html/2607.12380#bib.bib53)\)\.
## Appendix LMaterial Generation: Full MP\-20 Breakdown

Table[10](https://arxiv.org/html/2607.12380#A12.T10)reports the complete MP\-20 generation breakdown, including the Overall validity column \(Overall=Structure×Composition\\mathrm\{Overall\}=\\mathrm\{Structure\}\\times\\mathrm\{Composition\}, which is derivable from the main\-text table\) and the Metastable rate \(Ehull<0\.1E\_\{\\mathrm\{hull\}\}\{<\}0\.1eV/atom\)\. Both columns are omitted from the main paper because Overall is redundant and Metastable rate is the input to the M\.S\.U\.N\. metric that we already report\.

Table 10:Material generation on MP\-20 — full breakdown\.All columns in %, higher is better\. “–” denotes numbers not reported by the original authors\.
## Appendix MMolecule Generation: Per\-PoseBusters\-Check Breakdown

Table[11](https://arxiv.org/html/2607.12380#A13.T11)reports the per\-check pass rates for each of the seven PoseBusters sanity checks\(Buttenschoenet al\.,[2024](https://arxiv.org/html/2607.12380#bib.bib57)\)forSinAEon QM9 \(joint model\) and GEOM\-Drugs \(1010k samples each\), together with the aggregate*PB Valid*rate \(passes all seven checks\) reported in Tables[5](https://arxiv.org/html/2607.12380#S4.T5)–[5](https://arxiv.org/html/2607.12380#S4.T5)\. The breakdown shows that on QM9 the residual is dominated by the*internal energy*check \(96\.8%96\.8\\%\), and on GEOM\-Drugs by*internal steric clash*\(96\.7%96\.7\\%\) and*bond angles*\(97\.8%97\.8\\%\); other checks are at or near100%100\\%\.

Table 11:Per\-PoseBusters\-check pass rates \(%,↑\\uparrow\) forSinAEon QM9 and GEOM\-Drugs\.Last row is the aggregate*PoseBusters Valid*rate \(passes all seven checks\), corresponding to the PB Valid column in Tables[5](https://arxiv.org/html/2607.12380#S4.T5)–[5](https://arxiv.org/html/2607.12380#S4.T5)\.
## Appendix NMOF Generation: Full MOFChecker Sanity Checks

Table[12](https://arxiv.org/html/2607.12380#A14.T12)reports the full set of MOFChecker sanity checks for 1,000 unconditionally sampled MOF structures fromSinAE\(trained on QMOF\) and the jointly\-trained ADiT baseline\. The main text reports MOF reconstruction in Table[2](https://arxiv.org/html/2607.12380#S4.T2); this appendix provides the generation\-side comparison\.SinAEoutperforms ADiT by\+6\.1\+6\.1points on the aggregate validity rate \(16\.28%16\.28\\%vs10\.2%10\.2\\%\), with consistent improvements on nearly every individual sanity check\.

Table 12:MOF generation benchmark \(full MOFChecker breakdown\)\.Sanity checks from MOFChecker for 1,000 sampled structures are reported forSinAEand Joint ADiT\. Higher values are better for metrics marked with↑\\uparrow, lower values are better for those marked with↓\\downarrow\.
## Appendix OGenerated Sample Visualisations

Figures[10](https://arxiv.org/html/2607.12380#A15.F10)–[12](https://arxiv.org/html/2607.12380#A15.F12)show unconditional samples drawn from the latent DiT prior ofSinAEacross all evaluated domains \(QM9 small molecules, GEOM\-Drugs drug\-like molecules, MP\-20 crystals, and AFDB\-FS proteins\), decoded with the corresponding flow\-matching decoder\. Samples are uncurated unless otherwise stated\.

![Refer to caption](https://arxiv.org/html/2607.12380v1/x6.png)Figure 10:Generated samples on QM9 and GEOM\-Drugs\.Top: unconditional samples from the joint molecule \+ crystalSinAEprior \(QM9\)\. Bottom: unconditional samples from the GEOM\-DrugsSinAEprior\.![Refer to caption](https://arxiv.org/html/2607.12380v1/x7.png)Figure 11:Generated samples on MP\-20\.Unconditional crystal samples from the joint molecule \+ crystalSinAEprior, decoded with their lattice tokens to inorganic crystal structures\.![Refer to caption](https://arxiv.org/html/2607.12380v1/x8.png)Figure 12:Generated samples on AFDB\-FS\.Unconditional protein backbones from the protein\-onlySinAEprior; structures shown as cartoon traces of the Cα\\alphachain\.