面向物理有效生物分子扩散模型的推理时投影

# Inference-Time Projection for Physically Valid Biomolecular Diffusion Models
Source: [https://arxiv.org/html/2610.07037](https://arxiv.org/html/2610.07037)
Qurat\-ul\-ainYee Whye TehAffiliation:Department of Statistics, University of OxfordCharlotte M\. DeaneAffiliation:Department of Statistics, University of OxfordMatteo CagiadaAffiliation:Department of Statistics, University of OxfordAffiliation:Department of Biology, University of Copenhagen

###### Abstract

AlphaFold 3\-style cofolding models predict biomolecular complexes with high structural accuracy, yet a large fraction of their outputs are physically invalid: chains overlap at interfaces, ligand bond lengths and angles are distorted, rings are non\-planar, and stereocentres are inverted\. Current approaches either steer the sampler with physics\-informed potentials, which multiplies sampling cost and memory overhead making inference impossible on large complexes, or finetune the model, costing time and tying the fix to one architecture\. We observe that, unlike structural accuracy, physical validity is fully verifiable at inference time from quantities the sampler already holds\. We therefore treat physical validity as a constrained inference problem and introduce two closed\-form projection operators applied to the diffusion model’s denoised clean\-coordinate estimate,x^0\\hat\{x\}\_\{0\}: an inter\-chain van der Waals projection that pushes apart the most severely clashing atom pairs, and a ligand distance\-geometry projection that restores bond lengths, angles, internal contacts, planarity and chirality\. Both operators are local, sparse and displacement\-capped, require no network evaluations, gradients or importance sampling, and leave the denoiser and its weights untouched, so they can be dropped into any AF3\-style sampler without retraining\. Applied to two independently developed models, Boltz\-2 and OpenFold\-3, across five benchmarks \(CASP15, CASP16, the PoseBusters monomer and complex sets, and the Boltz physical\-validity test set\), our method recovers perfect physical validity while preserving structural\-accuracy and ligand\-placement metrics\. These gains are achieved with negligible runtime and memory overhead, providing a practical, model\-agnostic route to physically valid all\-atom structure prediction\.

## 1Introduction

Biomolecular structure prediction has moved from single\-chain protein modelling toward the prediction of complete molecular assemblies containing proteins, nucleic acids, small molecules, ions, and modified residues\. AlphaFold 3 \(AF3\)\([Abramson et al\., 2024](https://arxiv.org/html/2610.07037#bib.bib10)\)and related cofolding models\([Passaro et al\., 2025](https://arxiv.org/html/2610.07037#bib.bib9);[Chen et al\., 2025b](https://arxiv.org/html/2610.07037#bib.bib11);[Chai Discovery et al\., 2024](https://arxiv.org/html/2610.07037#bib.bib7);[The OpenFold3 Team, 2025](https://arxiv.org/html/2610.07037#bib.bib2)\)have made this possible by framing structure prediction as conditional all\-atom generation: given sequences and ligand identities, multiple sequence alignment \(MSA\), template\-derived information, and token\- and atom\-level features, a learned diffusion model samples the three\-dimensional coordinates of the full complex\. This shift has substantially broadened the scope of structure prediction, enabling a single model class to encode protein–protein interactions, protein–nucleic\-acid interactions, protein–ligand binding, and a wide range of chemically heterogeneous systems involving ions, modified residues, glycans, and covalent modifications\. As a result, these models are rapidly becoming broader engines for structural biology and drug discovery\.

For a predicted complex to be useful in downstream tasks such as docking, affinity prediction, molecular dynamics, and structure\-based optimization, structural accuracy alone is not sufficient; it must also bephysically valid\([Masters et al\., 2025](https://arxiv.org/html/2610.07037#bib.bib20);[Baker, 2019](https://arxiv.org/html/2610.07037#bib.bib14);[Buttenschoen et al\., 2024](https://arxiv.org/html/2610.07037#bib.bib12);[Hollingsworth and Dror, 2018](https://arxiv.org/html/2610.07037#bib.bib19);[Domagalski et al\., 2014](https://arxiv.org/html/2610.07037#bib.bib15)\)\. Following Boltz\([Wohlwend et al\., 2024](https://arxiv.org/html/2610.07037#bib.bib8)\), we define physical validity as the absence of basic all\-atom geometry violations: generated coordinates should satisfy stereochemical constraints, preserve allowable local ligand geometry, avoid steric clashes, and maintain a physically plausible organization of the complex\. Details of violation classes considered in this work are given in Appendix[A](https://arxiv.org/html/2610.07037#A1)\. These failures are not rare in AF3\-style structure predictors: for 26% of CASP15 targets, AF3 produces no physically valid structure in any of five samples \(Fig\.[1](https://arxiv.org/html/2610.07037#S1.F1)\)\. A breakdown of physical validity failures in Boltz\-2\([Passaro et al\., 2025](https://arxiv.org/html/2610.07037#bib.bib9)\)and OpenFold3\([The OpenFold3 Team, 2025](https://arxiv.org/html/2610.07037#bib.bib2)\)is given in Table[3](https://arxiv.org/html/2610.07037#A1.T3)\. Further evaluation of these models, likewise, reports persistent steric and stereochemical limitations in generated complexes\([Ishitani and Moriwaki, 2025](https://arxiv.org/html/2610.07037#bib.bib28);[Masters et al\., 2025](https://arxiv.org/html/2610.07037#bib.bib20)\)\. When these constraints are violated, structures can exhibit strained bond lengths, invalid angles, steric clashes, distorted side\-chain packing, chirality inversions, or overlapping chains and interfaces\. These defects can corrupt binding\-pocket geometry, misplace interaction partners, and make otherwise accurate\-looking predictions unreliable for downstream tasks\.

![Refer to caption](https://arxiv.org/html/2610.07037v1/figures/casp15.png)Figure 1:Physical validity performance of AlphaFold 3 family models on CASP15\.Oracle \(best of 5 samples\) per target \(n=66n=66\)\. Bars show mean across targets; error bars indicate 95% bootstrap confidence intervals\.Compared to previous structure prediction methods which operated on representations that enforce local chemistry by construction\([Anand and Achim, 2022](https://arxiv.org/html/2610.07037#bib.bib23);[Trippe et al\., 2022](https://arxiv.org/html/2610.07037#bib.bib24);[Wu et al\., 2024](https://arxiv.org/html/2610.07037#bib.bib25);[Yim et al\., 2023](https://arxiv.org/html/2610.07037#bib.bib26)\), the current state\-of\-the\-art cofolding models diffuse on raw atom coordinates where physical validity is not imposed as a strict constraint during training\. Aside from a bond loss applied to bonded ligands and glycans, AF3 and related models do not optimize physical\-validity losses in the diffusion objective\. As a result, physically realistic local geometry is encouraged only indirectly through the data distribution, making it possible for globally accurate predictions to remain locally invalid\.

In AF3, physical failures are handled only partially and after sampling: inter\-chain clashes are reflected as a high penalty in the confidence score, down\-ranking problematic samples; however, the structure produced itself is not corrected\. Another strategy is post\-hoc relaxation, where predicted coordinates are energy\-minimized before downstream use; AlphaFold2[Jumper et al\. \(2021\)](https://arxiv.org/html/2610.07037#bib.bib1), for example, used post\-prediction gradient descent in the Amber force field[Hornak et al\. \(2006\)](https://arxiv.org/html/2610.07037#bib.bib21)to enforce peptide\-bond geometry and remove stereochemical violations\. However, relaxation is straightforward only for proteins; other molecule types require additional force fields and accurate ligand parameterisation, making its application to all atom biomolecular structure prediction impractical at large scale\.

To our knowledge, only two methods attempt to directly correct physical invalidity in biomolecular diffusion models\. The first is Boltz\-2x\([Wohlwend et al\., 2024](https://arxiv.org/html/2610.07037#bib.bib8);[Passaro et al\., 2025](https://arxiv.org/html/2610.07037#bib.bib9)\), which improves physical plausibility at inference time using Feynman–Kac steering\([Singhal et al\., 2025](https://arxiv.org/html/2610.07037#bib.bib39)\)\. This approach introduces physics\-informed potentials over seven constraint families, uses importance sampling over multiple candidates, and applies gradient\-based refinement to bias diffusion toward more valid structures\. The second approach is the Gauss–Seidel Projection \(GSP\) method\([Chen et al\., 2025a](https://arxiv.org/html/2610.07037#bib.bib22)\), which trains a differentiable projection layer to enforce physical validity as a hard constraint\. GSP finetunes Boltz with a differentiable projection module built on the same seven physical constraint families used by Boltz steering\. While GSP offers fast two\-step sampling and validity guarantees, it traded accuracy for efficiency and required end\-to\-end finetuning of the base model\.

In this work, we ask:Can biomolecular diffusion models be made physically valid without costly pretraining, finetuning, or compromising speed and structural accuracy?We summarize our contributions as follows:

- •A model\-agnostic projection on the clean\-coordinate estimate:We introduce two closed\-form projection operators applied to the denoised estimatex^0\\hat\{x\}\_\{0\}before each reverse update: an inter\-chain van der Waals projection that removes steric clashes, and a ligand distance\-geometry projection that restores bond lengths, angles, internal contacts, planarity and chirality\. We define physical validity as a verifiable, inference\-time constraint, so projection to a valid set can be applied in any diffusion sampler without retraining\. We demonstrate this on two independently developed AF3\-style models, Boltz\-2 and OpenFold\-3\.
- •Physical Validity for free:Our method lifts physical validity to100%100\\%on five extensive benchmarks, CASP15, CASP16, PoseBusters monomers and complexes, and the Boltz test set, for both base models \(Table[1](https://arxiv.org/html/2610.07037#S3.T1)\) while preserving structural\-accuracy and ligand position metrics\. There is no trade\-off between validity, accuracy and efficiency; every constraint is evaluated in closed form, resulting in negligible wall\-clock overhead and no measurable increase in GPU memory\.

### 1\.1Related Work

#### Constrained and projected diffusion\.

Inference\-time guidance steers pretrained diffusion models without retraining by adding gradients of differentiable losses, rewards, or energies to the reverse process\([Song et al\., 2023](https://arxiv.org/html/2610.07037#bib.bib38);[Chung et al\., 2022b](https://arxiv.org/html/2610.07037#bib.bib34);[Hong, 2024](https://arxiv.org/html/2610.07037#bib.bib37);[Bansal et al\., 2023](https://arxiv.org/html/2610.07037#bib.bib36)\)\. These are soft procedures and are strongly dependent on the guidance weight\. If the weight is too weak, the constraint is not satisfied; if it is too strong, sample quality drops and samples drift off the data manifold\([Huang et al\., 2024](https://arxiv.org/html/2610.07037#bib.bib31);[Chung et al\., 2024](https://arxiv.org/html/2610.07037#bib.bib32)\)\. Posterior samplers such as DPS\([Chung et al\., 2022a](https://arxiv.org/html/2610.07037#bib.bib35)\)make this especially clear, because they perturbxtx\_\{t\}using a loss evaluated on a denoised proxy, and an unreliable proxy or an overly large step pushes the trajectory off\-manifold\. Later work recasts guidance as a sequence of local optimization problems\. The update is restricted to the denoiser\-induced manifold\([Chung et al\., 2022b](https://arxiv.org/html/2610.07037#bib.bib34)\), stopped once the proxy becomes unreliable\([Huang et al\., 2024](https://arxiv.org/html/2610.07037#bib.bib31)\), or bounded to stay near the learned trajectory\([Chung et al\., 2024](https://arxiv.org/html/2610.07037#bib.bib32)\)\. A second line of work replaces soft guidance with explicit feasibility corrections\. Projected Diffusion Models\([Christopher et al\., 2024](https://arxiv.org/html/2610.07037#bib.bib29)\)treat reverse diffusion as constrained optimization and project iterates onto the feasible set\. Predict–Project–Renoise \(PPR\)\([Rochman\-Sharabi and Louppe, 2026](https://arxiv.org/html/2610.07037#bib.bib30)\)projects through the denoiser, i\.e\. onx^0\\hat\{x\}\_\{0\}, and alternates projection with re\-noising\.[Christopher et al\. \(2025\)](https://arxiv.org/html/2610.07037#bib.bib33)apply a similar predict–correct–renoise scheme to protein design, enforcing stereochemical and pocket constraints with an ADMM\-based proximal step\. In each case the projection is itself an iterative optimization inside every reverse step, which adds substantial cost per sample\. Closest to our setting,[Chen et al\. \(2025a\)](https://arxiv.org/html/2610.07037#bib.bib22)enforce physical validity in biomolecular cofolding with a Gauss–Seidel projection over all\-atom coordinates\. However, this approach requires fine\-tuning the diffusion module; applying it only at inference degrades structural accuracy[Chen et al\. \(2025a\)](https://arxiv.org/html/2610.07037#bib.bib22)\.

Our method keeps thex^0\\hat\{x\}\_\{0\}\-space correction of PPR\-style samplers, but replaces the inner solver with a single closed\-form, displacement\-bounded correction\. This correction is applied once per denoising step during the final, low\-noise portion of sampling; the bounded corrections accumulate across these steps and drive samples toward physical validity\. A detailed account of contemporary methods in biomolecular diffusion models is provided in Appendix[B](https://arxiv.org/html/2610.07037#A2)\.

## 2Methods

Algorithm 1ProjectedSampleDiffusion1:

\{f∗\},\{siinputs\},\{sitrunk\},\{zi​jtrunk\}\\\{f^\{\\ast\}\\\},\\\{s\_\{i\}^\{\\mathrm\{inputs\}\}\\\},\\\{s\_\{i\}^\{\\mathrm\{trunk\}\}\\\},\\\{z\_\{ij\}^\{\\mathrm\{trunk\}\}\\\}, noise schedule

\[c0,c1,…,cT\]\[c\_\{0\},c\_\{1\},\\dots,c\_\{T\}\],

ℬproj\\mathcal\{B\}\_\{\\mathrm\{proj\}\},

γ0=0\.8\\gamma\_\{0\}=0\.8,

γmin=1\.0\\gamma\_\{\\min\}=1\.0, noise scale

λ=1\.003\\lambda=1\.003, step scale

η=1\.5\\eta=1\.5
2:

x→l∼c0⋅𝒩⁡\(0→,I3\)\\vec\{x\}\_\{l\}\\sim c\_\{0\}\\cdot\\mathcal\{N\}\(\\vec\{0\},I\_\{3\}\)x→l∈ℝ3\\vec\{x\}\_\{l\}\\in\\mathbb\{R\}^\{3\}

3:for

cτ∈\[c1,…,cT\]c\_\{\\tau\}\\in\[c\_\{1\},\\dots,c\_\{T\}\]do

4:

\{x→l\}←CentreRandomAugmentation​\(\{x→l\}\)\\\{\\vec\{x\}\_\{l\}\\\}\\leftarrow\\textsc\{CentreRandomAugmentation\}\(\\\{\\vec\{x\}\_\{l\}\\\}\)
5:

γ^←γ0​if​cτ\>γmin​else​0\\hat\{\\gamma\}\\leftarrow\\gamma\_\{0\}\\;\\textbf\{if\}\\;c\_\{\\tau\}\>\\gamma\_\{\\min\}\\;\\textbf\{else\}\\;0
6:

t^←cτ−1​\(1\+γ^\);tτ←1−\(τ−1\)/T\\hat\{t\}\\leftarrow c\_\{\\tau\-1\}\(1\+\\hat\{\\gamma\}\);\{\\color\[rgb\]\{1,0,0\}\\quad t\_\{\\tau\}\\leftarrow 1\-\(\\tau\-1\)/T\}
7:

ξ→l←λ​t^2−cτ−12​𝒩​\(0→,I3\)\\vec\{\\xi\}\_\{l\}\\leftarrow\\lambda\\sqrt\{\\hat\{t\}^\{2\}\-c\_\{\\tau\-1\}^\{2\}\}\\,\\mathcal\{N\}\(\\vec\{0\},I\_\{3\}\)ξ→l∈ℝ3\\vec\{\\xi\}\_\{l\}\\in\\mathbb\{R\}^\{3\}

8:

x→lnoisy←x→l\+ξ→l\\vec\{x\}^\{\\mathrm\{noisy\}\}\_\{l\}\\leftarrow\\vec\{x\}\_\{l\}\+\\vec\{\\xi\}\_\{l\}
9:

\{x→ldenoised\}←DiffusionModule​\(\{x→lnoisy\},t^,\{f∗\},\{siinputs\},\{sitrunk\},\{zi​jtrunk\}\)\\\{\\vec\{x\}^\{\\mathrm\{denoised\}\}\_\{l\}\\\}\\leftarrow\\textsc\{DiffusionModule\}\\big\(\\\{\\vec\{x\}^\{\\mathrm\{noisy\}\}\_\{l\}\\\},\\hat\{t\},\\\{f^\{\\ast\}\\\},\\\{s\_\{i\}^\{\\mathrm\{inputs\}\}\\\},\\\{s\_\{i\}^\{\\mathrm\{trunk\}\}\\\},\\\{z\_\{ij\}^\{\\mathrm\{trunk\}\}\\\}\\big\)
10:if

ProjectionActive​\(tτ\)\\textsc\{ProjectionActive\}\(\{t\_\{\\tau\}\}\)then

11:

\{x→~ldenoised\}←Π𝒞⁡\(ℬproj\)​\(\{x→ldenoised\}\)\\\{\\tilde\{\\vec\{x\}\}^\{\\mathrm\{denoised\}\}\_\{l\}\\\}\\leftarrow\\Pi\_\{\\mathcal\{C\}\(\\mathcal\{B\}\_\{\\mathrm\{proj\}\}\)\}\\big\(\\\{\\vec\{x\}^\{\\mathrm\{denoised\}\}\_\{l\}\\\}\\big\)
12:else

13:

\{x→~ldenoised\}←\{x→ldenoised\}\\\{\\tilde\{\\vec\{x\}\}^\{\\mathrm\{denoised\}\}\_\{l\}\\\}\\leftarrow\\\{\\vec\{x\}^\{\\mathrm\{denoised\}\}\_\{l\}\\\}
14:endif

15:

δ→l←\(x→lnoisy−x→~ldenoised\)/t^\\vec\{\\delta\}\_\{l\}\\leftarrow\(\\vec\{x\}^\{\\mathrm\{noisy\}\}\_\{l\}\-\{\\color\[rgb\]\{1,0,0\}\\tilde\{\\vec\{x\}\}^\{\\mathrm\{denoised\}\}\_\{l\}\}\)/\\hat\{t\}
16:

d​t←cτ−t^dt\\leftarrow c\_\{\\tau\}\-\\hat\{t\}
17:

x→l←x→lnoisy\+η​d​t​δ→l\\vec\{x\}\_\{l\}\\leftarrow\\vec\{x\}^\{\\mathrm\{noisy\}\}\_\{l\}\+\\eta\\,dt\\,\\vec\{\\delta\}\_\{l\}
18:endfor

19:return

\{x→l\}\\\{\\vec\{x\}\_\{l\}\\\}

Our starting point is an observation about what makes physical validity different from structural accuracy\. Accuracy can only be scored against an experimental reference that is unavailable at inference time, but validity is defined entirely by quantities an AF3 sampler already has such as van der Waals radii, the molecular graph of a ligand and its RDKit[Landrum et al\. \(2026\)](https://arxiv.org/html/2610.07037#bib.bib6)distance\-geometry bounds, ring planarity and assigned stereochemistry\. Every violation is therefore exactly verifiable on any coordinate estimate at any step\. We treat physical validity as a constrained inference problem and exploit this verifiability with a lightweight projection on the diffusion model’s denoised clean\-coordinate estimate,x^0\\hat\{x\}\_\{0\}\.

Algorithm[1](https://arxiv.org/html/2610.07037#alg1)is a modified version of the AlphaFold 3 sample\-diffusion procedure \(Algorithm 18 in the Supplementary Information of[Abramson et al\. \(2024\)](https://arxiv.org/html/2610.07037#bib.bib10)\); our modifications are shown in red\. Here,ℬproj\\mathcal\{B\}\_\{\\mathrm\{proj\}\}denotes the batch\-level structural metadata used to construct the projection constraints: atom masks, atom\-to\-token mappings, chain and entity indices, bonded connectivity, elements \(used to assign van der Waals radii\), and, for ligands, RDKit\-derived distance\-geometry bounds for bonded, angle\-implied and non\-neighbouring atom pairs, together with masks indicating each constraint subtype\.

At each reverse step, the diffusion module maps the noised coordinates \(line[8](https://arxiv.org/html/2610.07037#alg1.l8)\) to a denoised estimate of the clean all\-atom structure \(line[9](https://arxiv.org/html/2610.07037#alg1.l9)\)\. The sampler converts this estimate into a reverse drift \(line[15](https://arxiv.org/html/2610.07037#alg1.l15)\) and takes an Euler step fromx→lnoisy\\vec\{x\}^\{\\mathrm\{noisy\}\}\_\{l\}toward it \(line[17](https://arxiv.org/html/2610.07037#alg1.l17)\)\. We make a small, displacement bound correction tox→ldenoised\\vec\{x\}^\{\\mathrm\{denoised\}\}\_\{l\}\(i\.e\.x^0\\hat\{x\}\_\{0\}\), which changes the direction of the subsequent drift \(line[15](https://arxiv.org/html/2610.07037#alg1.l15)\)\. WhenProjectionActive\(tτ\)=𝟏\[tτ≤projection\_start\_t\]\\textsc\{ProjectionActive\}\(t\_\{\\tau\}\)=\\mathbf\{1\}\[t\_\{\\tau\}\\leq\\texttt\{projection\\\_start\\\_t\}\]holds, wheretτt\_\{\\tau\}is the normalised sampling time of Appendix[C](https://arxiv.org/html/2610.07037#A3), the raw estimate is replaced by its projectionΠ𝒞⁡\(ℬproj\)\\Pi\_\{\\mathcal\{C\}\(\\mathcal\{B\}\_\{\\mathrm\{proj\}\}\)\}\(lines 9–10\); otherwise it passes through unchanged \(line[13](https://arxiv.org/html/2610.07037#alg1.l13)\)\.

Π𝒞⁡\(ℬproj\)\\Pi\_\{\\mathcal\{C\}\(\\mathcal\{B\}\_\{\\mathrm\{proj\}\}\)\}is a bounded local correction operator composed of two projections applied in sequence to the same denoised estimate \(Appendix[C](https://arxiv.org/html/2610.07037#A3)\)\. The van der Waals projection \(Algorithm[2](https://arxiv.org/html/2610.07037#alg2)\) targets inter\-chain steric clashes: it evaluates candidate atom pairs across chains against their van der Waals lower bounds, selects theKKmost deeply penetrating pairs, and applies a symmetric push\-apart to each\. The ligand distance\-geometry \(LDG\) projection \(Algorithm[3](https://arxiv.org/html/2610.07037#alg3)\) corrects the internal geometry of each ligand toward the constraints implied by its molecular graph: bonded and angle pairs are held within RDKit\-derived distance intervals, non\-bonded intra\-ligand pairs are kept above a clash lower bound, aromatic rings and double\-bond groups are flattened toward a fitted plane, and sp3stereocentres whose signed volume disagrees with the assigned handedness receive a Newton step that re\-inverts them\. Both operators accumulate the contributions of all selected violated constraints involving atomllinto a single displacementD→l\\vec\{D\}\_\{l\}and cap only the sum,x→~ldenoised=x→ldenoised\+D→l\\tilde\{\\vec\{x\}\}^\{\\mathrm\{denoised\}\}\_\{l\}=\\vec\{x\}^\{\\mathrm\{denoised\}\}\_\{l\}\+\\vec\{D\}\_\{l\}where‖D→l‖2≤δmax\\\|\\vec\{D\}\_\{l\}\\\|\_\{2\}\\leq\\delta\_\{\\max\}, so an atom involved in many violations is bounded on its net motion rather than on each contribution\. The cap is applied once per call of the van der Waals projection and once per pass of the ligand projection\. Neither operator evaluates or differentiates through the network or modifies its weights; each correction is computed in closed form from the current denoised coordinates andℬproj\\mathcal\{B\}\_\{\\mathrm\{proj\}\}\. In practice, the corrections are small and shrink over the tail: the RMS and maximum per\-atom displacement decrease over the final 40 steps while the mean van der Waals penetration depth falls toward zero, so projection acts as a sparse local repair rather than a global rearrangement of the structure \(Appendix[D\.1](https://arxiv.org/html/2610.07037#A4.SS1)\)\.

#### When and where to project:

The AlphaFold 3 sampler re\-injects noise at every step withcτ\>γmin=1c\_\{\\tau\}\>\\gamma\_\{\\min\}=1Å \(“churn”[Karras et al\. \(2022\)](https://arxiv.org/html/2610.07037#bib.bib40), lines[5](https://arxiv.org/html/2610.07037#alg1.l5)–[8](https://arxiv.org/html/2610.07037#alg1.l8)\)\. Oncecτ≤γminc\_\{\\tau\}\\leq\\gamma\_\{\\min\}, no fresh noise is injected, and the remaining steps are deterministic\. Two observations \(Appendix[D](https://arxiv.org/html/2610.07037#A4)\) shape our design\.

First, the structure is essentially fixed by the time churn switches off \(step 154 out of 200\)\. The denoised estimatex^0\\hat\{x\}\_\{0\}comes within11Å of the final structure just before the churn boundary, then drops to∼\\sim0\.10\.1Å, and its LDDT levels off \(Fig\.[6](https://arxiv.org/html/2610.07037#A4.F6)\)\. In the deterministic tail the denoiser therefore moves atoms by less than0\.10\.1Å, well below the projection capδmax=0\.45\\delta\_\{\\max\}=0\.45Å, so projected corrections are not overwritten and accumulate\. During churn, each step injects noise an order of magnitude larger than the correction, which erases it\. This is visible directly in Fig\.[8](https://arxiv.org/html/2610.07037#A4.F8): projecting only during churn undoes the projection due to high noise and violation count remains similar to no projection baseline\. This is also observed in \(Table[2](https://arxiv.org/html/2610.07037#S3.T2)\) where projection only during churn leaves physical validity at0\.020\.02whereas projection during the tail reduces violations steadily and reaches1\.001\.00\.

Second, steric constraints are meaningful on a clean structure\. Until the tail,xtx\_\{t\}sits at a distance of≈σ\\approx\\sigmafrom the final structure and has near\-zero LDDT, whereasx^0\\hat\{x\}\_\{0\}is a coherent structure from the first step \(Figs\.[5](https://arxiv.org/html/2610.07037#A4.F5),[7](https://arxiv.org/html/2610.07037#A4.F7)\)\. We therefore projectx^0\\hat\{x\}\_\{0\}and activate projection only in the tail \(projection\_start\_t=0\.2\\texttt\{projection\\\_start\\\_t\}=0\.2, i\.e\. from step≈160\\approx 160\)\. Once churn is off,xtx\_\{t\}andx^0\\hat\{x\}\_\{0\}differ by less thanσ≤1\\sigma\\leq 1Å, and projecting either gives the same validity and accuracy \(Table[2](https://arxiv.org/html/2610.07037#S3.T2)\)\.

## 3Results

### 3\.1How projection interacts with the learned prior

The projection is local, sparse, and displacement\-limited: it moves only the atoms involved in the most severe violations; pushing clashing pairs apart, pulling over\-stretched bonds together, flattening distorted rings and re\-inverting flipped stereocentres, while capping the maximum per\-atom movement\. Whether such a scheme can reach physical validity depends on how the denoiser responds to each correction: if the network simply reverts it at the next step, nothing accumulates; if some of it is retained, the effect compounds\. To measure this directly we instrument the Boltz\-2 sampler and track inter\-chain penetration along the trajectory\.

#### Setup:

For a coordinate set𝐱\\mathbf\{x\},P⁡\(𝐱\)=∑\(i,j\)∈𝒫\[ℓi​j−‖𝐱i−𝐱j‖2\]\+P\(\\mathbf\{x\}\)=\\sum\_\{\(i,j\)\\in\\mathcal\{P\}\}\\big\[\\ell\_\{ij\}\-\\\|\\mathbf\{x\}\_\{i\}\-\\mathbf\{x\}\_\{j\}\\\|\_\{2\}\\big\]\_\{\+\}is the total inter\-chain penetration over the candidate pairs of Algorithm[2](https://arxiv.org/html/2610.07037#alg2), withℓi​j=\(1−β\)​\(rivdw\+rjvdw\)\\ell\_\{ij\}=\(1\-\\beta\)\(r\_\{i\}^\{\\mathrm\{vdw\}\}\+r\_\{j\}^\{\\mathrm\{vdw\}\}\)andβ=0\.15\\beta=0\.15\. We selected four CASP16 targets with the highest projection activity and ran five samples each, giving 20 trajectories\. At every stepssof the projection windows∈\[161,200\]s\\in\[161,200\]we logPPon the denoiser’s clean estimate at three points: before projection,𝐱^0raw​\(s\)\\hat\{\\mathbf\{x\}\}\_\{0\}^\{\\mathrm\{raw\}\}\(s\); immediately after projection,𝐱^0proj​\(s\)=Π⁡\(𝐱^0raw​\(s\)\)\\hat\{\\mathbf\{x\}\}\_\{0\}^\{\\mathrm\{proj\}\}\(s\)=\\Pi\\big\(\\hat\{\\mathbf\{x\}\}\_\{0\}^\{\\mathrm\{raw\}\}\(s\)\\big\); and in the model’s next prediction one step later,𝐱^0raw​\(s\+1\)\\hat\{\\mathbf\{x\}\}\_\{0\}^\{\\mathrm\{raw\}\}\(s\{\+\}1\)\. Comparing the first two isolates what the projection removes at stepss, comparing the last two measures how much of the removed overlap the model re\-introduces in its next prediction, which we call the*rebound*: withΔs=P⁡\(𝐱^0raw​\(s\)\)−P⁡\(𝐱^0proj​\(s\)\)\\Delta\_\{s\}=P\(\\hat\{\\mathbf\{x\}\}\_\{0\}^\{\\mathrm\{raw\}\}\(s\)\)\-P\(\\hat\{\\mathbf\{x\}\}\_\{0\}^\{\\mathrm\{proj\}\}\(s\)\),ρs=\[P⁡\(𝐱^0raw​\(s\+1\)\)−P⁡\(𝐱^0proj​\(s\)\)\]/Δs\\rho\_\{s\}=\\big\[P\(\\hat\{\\mathbf\{x\}\}\_\{0\}^\{\\mathrm\{raw\}\}\(s\+1\)\)\-P\(\\hat\{\\mathbf\{x\}\}\_\{0\}^\{\\mathrm\{proj\}\}\(s\)\)\\big\]/\\Delta\_\{s\}and retention=1−ρs=1\-\\rho\_\{s\}\. As a reference we run the identical sampler with the projection disabled \(same seeds, same hyper\-parameters\) and record𝐱^0\\hat\{\\mathbf\{x\}\}\_\{0\}over the same window \(*control*\); any divergence between the control and𝐱^0raw\\hat\{\\mathbf\{x\}\}\_\{0\}^\{\\mathrm\{raw\}\}is therefore attributable to the projection alone\. Figure[2](https://arxiv.org/html/2610.07037#S3.F2)shows the full trajectories and the projection window; confidence intervals below are 95% bootstrap intervals over the 20 trajectories\.

![Refer to caption](https://arxiv.org/html/2610.07037v1/figures/absorption/absorption_row.png)Figure 2:Inter\-chain penetration along the Boltz\-2 reverse trajectoryfor the four most projection\-active CASP16 targets, five samples each\.Top:total penetration over all 200 steps \(log scale\); shading marks the 40\-step projection window\.Bottom:that window on a linear axis, showing𝐱^0\\hat\{\\mathbf\{x\}\}\_\{0\}for the seed\-matched control \(green, dashed\),𝐱^0raw\\hat\{\\mathbf\{x\}\}\_\{0\}^\{\\mathrm\{raw\}\}before projection \(blue\) and𝐱^0proj\\hat\{\\mathbf\{x\}\}\_\{0\}^\{\\mathrm\{proj\}\}after \(red\); means over 5 trajectories, bands 95% bootstrap intervals\. The blue–red gap is what one projection removes; the green–blue gap is what the model has internalized\.
#### Each correction is small and absorbed by the learned prior\.

Averaged over the window, the model retains around∼8%\{\\sim\}8\\%of each correction at each step and re\-expresses the rest from its own learned geometry \(reboundρ=0\.919\\rho=0\.919, 95% CI\[0\.915,0\.925\]\[0\.915,0\.925\]\)\. This is by design: the per\-step correction is deliberately small and displacement\-capped, intended to nudge the model repeatedly rather than override it in one shot\. The projection is iterative and its effect is cumulative: the fraction retained at each step, though small, is reinforced at every subsequent step, and across the 40\-step window this steadily carries the structure to validity\.

#### The trajectory internalizes the corrections\.

Measuring penetration on𝐱^0raw\\hat\{\\mathbf\{x\}\}\_\{0\}^\{\\mathrm\{raw\}\}, i\.e\. on the structure the model proposes on its own before we touch it, the penetration of the model’s unprojected predictions falls by80\.3%80\.3\\%across the window \(95% CI\[77\.5%,83\.0%\]\[77\.5\\%,83\.0\\%\]; per\-target trajectories in Fig\.[2](https://arxiv.org/html/2610.07037#S3.F2), bottom\)\. Over the same steps the seed\-matched control shows no reduction at all: its penetration rises by12\.0%12\.0\\%\(95% CI\[9\.1%,14\.9%\]\[9\.1\\%,14\.9\\%\]\), so this convergence is produced by our method\. The model’s own geometry becomes nearly clash\-free i\.e\. it has internalized the corrections rather than being held apart by force at the final step\. This is also what distinguishes in\-loop projection from a post\-hoc correction applied once to the final coordinates \(Appendix[G\.1](https://arxiv.org/html/2610.07037#A7.SS1)\)\.

### 3\.2Experimental Setup

We evaluate each method along three complementary axes: physical validity, which measures whether the predicted all\-atom coordinates satisfy basic stereochemical and steric constraints; structural accuracy, which measures agreement with the experimental reference structure; and computational efficiency, measured as wall\-clock time and peak GPU memory per sample\. We evaluated five benchmark sets: CASP15, CASP16, the PoseBusters monomer and complex sets, and Test, the physical\-validity dataset released with Boltz\-1[Wohlwend et al\. \(2024\)](https://arxiv.org/html/2610.07037#bib.bib8)\. We apply our method to two independently developed AF3\-style base models, Boltz\-2\([Passaro et al\., 2025](https://arxiv.org/html/2610.07037#bib.bib9)\)and OpenFold\-3[The OpenFold3 Team \(2025\)](https://arxiv.org/html/2610.07037#bib.bib2), and compare each projected model against its unmodified base\. We additionally run Boltz\-2x i\.e\. Boltz\-2 with gradient\-based FK steering\([Singhal et al\., 2025](https://arxiv.org/html/2610.07037#bib.bib39)\)for physical validity, for comparison with an existing inference\-time method\. Boltz\-2x exhausts an 80 GB A100 on nine targets: H1236, R1283v2 and R1285 \(CASP16\); 7b2c, 7cnq, 7suc, 8aql and 8g0v \(PoseBusters complexes\); and 8sro \(Test\)\. All nine complete under the base model and reach full physical validity under our projection\. A direct comparison to GSP[Chen et al\. \(2025a\)](https://arxiv.org/html/2610.07037#bib.bib22)was not possible due to a lack of publicly released model checkpoints\. Further details of the experimental set up are given in Appendix[E](https://arxiv.org/html/2610.07037#A5)\.

### 3\.3Accuracy and Validity

Table 1:Structural quality and physical validity, oracle over 5 samples, with non\-parametric bootstrap standard errors \(10,000 resamples over targets\)\. DockQ\>0\.23is the fraction of scored interfaces with DockQ above 0\.23, and LRMSD<2the fraction of ligands placed within 2 Å\. DockQ requires an interface and LDDT\-PLI/LRMSD a scored ligand\. PoseBusters monomers have no interfaces, hence no DockQ\.MethodLDDT↑\\uparrowDockQ\>0\.23↑\{\}\_\{\>0\.23\}\\uparrowLDDT\-PLI↑\\uparrowLRMSD<2↑\{\}\_\{<2\}\\uparrowPhys\. Valid↑\\uparrowCASP15n6614121266Boltz\-20\.457±\\pm0\.0380\.786±\\pm0\.1090\.564±\\pm0\.0850\.453±\\pm0\.1260\.697±\\pm0\.056Ours \(Boltz\)0\.456±\\pm0\.0380\.786±\\pm0\.1100\.571±\\pm0\.0830\.453±\\pm0\.1271\.000±\\pm0\.000Boltz\-2x0\.457±\\pm0\.0380\.714±\\pm0\.1190\.524±\\pm0\.0920\.425±\\pm0\.1311\.000±\\pm0\.000OpenFold\-30\.440±\\pm0\.0390\.821±\\pm0\.0960\.510±\\pm0\.0750\.408±\\pm0\.1280\.818±\\pm0\.048Ours \(OpenFold\)0\.440±\\pm0\.0390\.821±\\pm0\.0960\.514±\\pm0\.0760\.408±\\pm0\.1291\.000±\\pm0\.000CASP16n52189952Boltz\-20\.453±\\pm0\.0440\.500±\\pm0\.1030\.371±\\pm0\.0620\.363±\\pm0\.0760\.538±\\pm0\.069Ours \(Boltz\)0\.452±\\pm0\.0440\.500±\\pm0\.1030\.372±\\pm0\.0610\.363±\\pm0\.0761\.000±\\pm0\.000Boltz\-2x0\.451±\\pm0\.0440\.519±\\pm0\.0970\.372±\\pm0\.0630\.322±\\pm0\.0811\.000±\\pm0\.000OpenFold\-30\.449±\\pm0\.0440\.583±\\pm0\.0930\.332±\\pm0\.0530\.237±\\pm0\.1040\.692±\\pm0\.064Ours \(OpenFold\)0\.449±\\pm0\.0440\.583±\\pm0\.0930\.332±\\pm0\.0530\.237±\\pm0\.1041\.000±\\pm0\.000PoseBusters Monomern1340134134134Boltz\-20\.954±\\pm0\.003–0\.741±\\pm0\.0240\.687±\\pm0\.0400\.948±\\pm0\.019Ours \(Boltz\)0\.954±\\pm0\.003–0\.741±\\pm0\.0240\.687±\\pm0\.0401\.000±\\pm0\.000Boltz\-2x0\.954±\\pm0\.003–0\.727±\\pm0\.0240\.649±\\pm0\.0411\.000±\\pm0\.000OpenFold\-30\.951±\\pm0\.003–0\.830±\\pm0\.0190\.817±\\pm0\.0330\.858±\\pm0\.030Ours \(OpenFold\)0\.951±\\pm0\.003–0\.831±\\pm0\.0190\.810±\\pm0\.0341\.000±\\pm0\.000PoseBusters Complexn164161164164164Boltz\-20\.908±\\pm0\.0080\.628±\\pm0\.0360\.689±\\pm0\.0220\.621±\\pm0\.0370\.500±\\pm0\.039Ours \(Boltz\)0\.908±\\pm0\.0080\.628±\\pm0\.0360\.689±\\pm0\.0220\.621±\\pm0\.0371\.000±\\pm0\.000Boltz\-2x0\.907±\\pm0\.0080\.630±\\pm0\.0360\.687±\\pm0\.0220\.603±\\pm0\.0371\.000±\\pm0\.000OpenFold\-30\.909±\\pm0\.0080\.602±\\pm0\.0370\.777±\\pm0\.0200\.731±\\pm0\.0340\.585±\\pm0\.038Ours \(OpenFold\)0\.909±\\pm0\.0070\.602±\\pm0\.0380\.778±\\pm0\.0200\.731±\\pm0\.0341\.000±\\pm0\.000Testn541344251251541Boltz\-20\.831±\\pm0\.0060\.714±\\pm0\.0220\.646±\\pm0\.0220\.619±\\pm0\.0280\.608±\\pm0\.021Ours \(Boltz\)0\.831±\\pm0\.0060\.714±\\pm0\.0220\.645±\\pm0\.0220\.619±\\pm0\.0281\.000±\\pm0\.000Boltz\-2x0\.834±\\pm0\.0060\.707±\\pm0\.0220\.649±\\pm0\.0220\.623±\\pm0\.0271\.000±\\pm0\.000OpenFold\-30\.816±\\pm0\.0070\.686±\\pm0\.0230\.599±\\pm0\.0230\.594±\\pm0\.0270\.678±\\pm0\.020Ours \(OpenFold\)0\.816±\\pm0\.0070\.686±\\pm0\.0220\.599±\\pm0\.0230\.594±\\pm0\.0271\.000±\\pm0\.000Our method achieves perfect physical validity without sacrificing structural accuracy\.Table[1](https://arxiv.org/html/2610.07037#S3.T1)shows that our method achieves 100% pass rates on physical validity on every dataset with structural accuracy unchanged\. The two projection operators \(vdW and LDG\) are applied to two independently trained samplers, so the correction is model\-agnostic\. Accuracy differences between Boltz\-2 and Boltz\-2x are within noise at these sample sizes; Appendix[E\.3](https://arxiv.org/html/2610.07037#A5.SS3)shows a failure mode steering can produce: Boltz\-2x collapses the five sampled poses onto a single, physically clean, but incorrect binding mode\.

Because the projection moves atoms, we verified in Appendix[G](https://arxiv.org/html/2610.07037#A7)that validity is not bought at the expense of structure quality elsewhere\. Three independent analyses agree\. \(i\) Applying projection after sampling instead of during inference either leaves most violations in place \(one pass removes14%14\\%of targeted violations\) or, when iterated to feasibility, significantly lowers LDDT; post\-hoc Phenix\([Liebschner et al\., 2019](https://arxiv.org/html/2610.07037#bib.bib5)\)minimization reaches only0\.8810\.881validity while costing0\.0220\.022LDDT, with the largest losses on the most accurate structures \(§[G\.1](https://arxiv.org/html/2610.07037#A7.SS1)\)\. \(ii\) Covalent geometry is essentially unchanged: bond and angle RMSD from ideal stay within0\.00020\.0002Å and0\.015∘0\.015^\{\\circ\}of Boltz\-2 and angle RMSD remains closer to ideal than the deposited experimental structures \(§[G\.2](https://arxiv.org/html/2610.07037#A7.SS2)\)\. \(iii\) Packing is preserved: after an identical short relaxation, projected and base structures reach the same Lennard\-Jones energy on every target, and the number of contacts in the solvent\-accessible alpha complex is unchanged at the median \(≤0\.01%\\leq 0\.01\\%\), with the largest per\-target change below 2% \(§[G\.3](https://arxiv.org/html/2610.07037#A7.SS3)\)\. The projection therefore removes physical violations rather than redistributing them into other geometric defects\. Unlike fine\-tuned projection\([Chen et al\., 2025a](https://arxiv.org/html/2610.07037#bib.bib22)\), it preserves accuracy without changing any weights; unlike post\-hoc refinement, it corrects the trajectory while the denoiser can still adapt to the correction; and unlike inference\-time steering \(Boltz\-2x\), it adds no measurable runtime or memory \(§[3\.4](https://arxiv.org/html/2610.07037#S3.SS4)\)\.

### 3\.4Efficiency

![Refer to caption](https://arxiv.org/html/2610.07037v1/figures/performance/atom_memory.png)\(a\)Peak GPU memory versus structure size\.
![Refer to caption](https://arxiv.org/html/2610.07037v1/figures/performance/atom_time.png)\(b\)Inference time stratified by atom\-count bin\.

Figure 3:Cost of inference\-time physical correction on CASP16 and PoseBusters Complex\.\(a\)Peak PyTorch CUDA memory per target; thin grey lines join the three methods on the same target\.\(b\)Mean wall\-clock time per target by atom count; error bars are the standard error of the mean,nnthe number of targets per bin\.Our method achieves perfect physical validity at no measurable time or memory overhead\.Our central motivation is that physical validity should be improved without substantially increasing sampling cost\. Figure[3\(a\)](https://arxiv.org/html/2610.07037#S3.F3.sf1)shows that our method adds no memory overhead: the base model and our method coincide over the full size range on both benchmarks as projection adds no network evaluations and no gradients\. Steering is expensive at inference time\. Boltz\-2x exceeds the 80 GB A100 budget on nine targets, including three CASP16 and five PoseBusters Complex targets \(Figure[3\(a\)](https://arxiv.org/html/2610.07037#S3.F3.sf1)\), and Figure[3\(b\)](https://arxiv.org/html/2610.07037#S3.F3.sf2)shows the same picture in wall\-clock time\. Across all size bins, runtime with projection stays within3\.5%3\.5\\%of the base model, and differences of both signs indicate that any remaining gap is below run\-to\-run variation\. Boltz\-2x costs1\.51\.5–2\.1×2\.1\\timesthe base model throughout, and its absolute overhead grows steeply with size: on complexes above10,00010\{,\}000atoms it adds341341s on CASP16 and660660s on PoseBusters Complex\. This is a major limitation as AF3\-style models are often used in high\-throughput settings to screen many ligands, sample multiple seeds, or generate alternative structural hypotheses\. Our method therefore provides a scalable and practical route to physically valid all\-atom predictions while largely preserving the speed and memory profile of the base model\. A detailed account of the time and memory comparison of both base models on all datasets is given in Table[4](https://arxiv.org/html/2610.07037#A6.T4)\.

### 3\.5Ablations

Table 2:Where and when the projection is applied, on 134 CASP15, CASP16 and PoseBusters Monomer and Complex targets on which Boltz\-2 produced at least one physically invalid sample\. Higher is better for all metrics\. Dashed, shaded row is the final configuration; red cells are worse than it under a Holm\-corrected paired test\.Tables[2](https://arxiv.org/html/2610.07037#S3.T2)validates the design choices introduced in §[2](https://arxiv.org/html/2610.07037#S2.SS0.SSS0.Px1)\. Projecting only while churn is active is useless: validity reaches just0\.020\.02\(x^0\\hat\{x\}\_\{0\}\) and0\.040\.04\(xtx\_\{t\}\), because each churn step injects noise an order of magnitude larger than the correction it overwrites, so nothing accumulates\. Once churn ends,xtx\_\{t\}andx^0\\hat\{x\}\_\{0\}lie withinσ≤1\\sigma\\leq 1Å of each other and of the final structure, and the table confirms that in this regime the two are interchangeable; both reach1\.001\.00validity with identical accuracy\. What matters is therefore the window, not which iterate is projected\. Moving the same operator outside the sampler fails in two distinct ways\. A single post\-hoc pass removes only14%14\\%of the targeted violations and reaches0\.220\.22validity, because each pass is displacement\-capped and one is not enough to attain feasibility in one go\. Applying projection after sampling multiple times does attain it \(0\.900\.90\), but there is no longer a denoiser to absorb the corrections: they are imposed on the final coordinates instead of being integrated into the trajectory, and LDDT falls from0\.810\.81to0\.790\.79\. In\-loop projection is what buys both, feasibility and an intact structure, and this is the same absorption mechanism measured directly in §[3\.1](https://arxiv.org/html/2610.07037#S3.SS1)\. Table[8](https://arxiv.org/html/2610.07037#A8.T8)ablates each hyperparameter of the two operators\.

## 4Discussion and Limitations

Our results suggest that physical\-validity failures in AF3\-style cofolding models are not, for the most part, failures of global structural reasoning\. They are small residual violations that remain after the learned denoiser has already placed the molecular components in approximately correct positions\. On the benchmarks we study, a bounded inference\-time projection toward a physically valid constraint set is enough to remove these failures while leaving the structural prior of the base model intact\. Physical correction therefore does not necessarily require retraining, expensive relaxation, or diffusion\-time steering with multiple learned potentials\. More broadly, this points to a practical division of labour for biomolecular diffusion models: learned generative models supply global structural hypotheses, while small constrained inference operators enforce the local physical constraints that are hard to guarantee from data fitting alone\. Such predictions can then support not only global structural hypotheses but also atomistic reasoning about binding, specificity, designability and molecular mechanism, which requires structures that are locally correct as well as globally plausible\.

Limitations:Our operator is a bounded partial projection, not a hard satisfaction onto the feasible set\. Each step moves an atom by at mostδmax\\delta\_\{\\max\}, and only theKKmost deeply penetrating pairs are corrected per step, so validity is reached empirically rather than guaranteed\. Moreover, the validity criteria used to evaluate all methods in this study \(following[Wohlwend et al\. \(2024\)](https://arxiv.org/html/2610.07037#bib.bib8)\) should be read as evidence that the targeted failure modes are removed, not as a general certificate of physical realism\. We address this with independent checks: MolProbity covalent geometry, OpenMM van der Waals energies, and alpha\-complex contact analysis \(§[G](https://arxiv.org/html/2610.07037#A7)\)\. Additionally, validity is measured against the constraint families that the projection enforces, inter\-chain contacts, ligand bond lengths and angles, internal clashes, planarity and chirality, so a criterion outside them, such as torsional strain, protein backbone geometry or hydrogen placement, is neither corrected nor assessed here\.

## References

- Abramsonet al\.\(2024\)J\. Abramson, J\. Adler, J\. Dunger, R\. Evans, T\. Green, A\. Pritzel, O\. Ronneberger, L\. Willmore, A\. J\. Ballard, J\. Bambrick, S\. W\. Bodenstein, D\. A\. Evans, C\. Hung, M\. O’Neill, D\. Reiman, K\. Tunyasuvunakool, Z\. Wu, A\. Žemgulytė, E\. Arvaniti, C\. Beattie, O\. Bertolli, A\. Bridgland, A\. Cherepanov, M\. Congreve, A\. I\. Cowen\-Rivers, A\. Cowie, M\. Figurnov, F\. B\. Fuchs, H\. Gladman, R\. Jain, Y\. A\. Khan, C\. M\. R\. Low, K\. Perlin, A\. Potapenko, P\. Savy, S\. Singh, A\. Stecula, A\. Thillaisundaram, C\. Tong, S\. Yakneen, E\. D\. Zhong, M\. Zielinski, A\. Žídek, V\. Bapst, P\. Kohli, M\. Jaderberg, D\. Hassabis, and J\. M\. JumperAccurate structure prediction of biomolecular interactions with alphafold 3\.Nature630\(8016\),pp\. 493–500\.External Links:ISSN 1476\-4687,[Link](http://dx.doi.org/10.1038/s41586-024-07487-w),[Document](https://dx.doi.org/10.1038/s41586-024-07487-w)Cited by:[§B\.1](https://arxiv.org/html/2610.07037#A2.SS1.p2.1),[§1](https://arxiv.org/html/2610.07037#S1.p1.1),[§2](https://arxiv.org/html/2610.07037#S2.p2.1)\.
- Anand and Achim \(2022\)N\. Anand and T\. AchimProtein structure and sequence generation with equivariant denoising diffusion probabilistic models\.External Links:2205\.15019Cited by:[§B\.1](https://arxiv.org/html/2610.07037#A2.SS1.p1.1),[§1](https://arxiv.org/html/2610.07037#S1.p3.1)\.
- Baeket al\.\(2021\)M\. Baek, F\. DiMaio, I\. Anishchenko, J\. Dauparas, S\. Ovchinnikov, G\. R\. Lee, J\. Wang, Q\. Cong, L\. N\. Kinch, R\. D\. Schaeffer, C\. Millán, H\. Park, C\. Adams, C\. R\. Glassman, A\. DeGiovanni, J\. H\. Pereira, A\. V\. Rodrigues, A\. A\. van Dijk, A\. C\. Ebrecht, D\. J\. Opperman, T\. Sagmeister, C\. Buhlheller, T\. Pavkov\-Keller, M\. K\. Rathinaswamy, U\. Dalwadi, C\. K\. Yip, J\. E\. Burke, K\. C\. Garcia, N\. V\. Grishin, P\. D\. Adams, R\. J\. Read, and D\. BakerAccurate prediction of protein structures and interactions using a three\-track neural network\.Science373\(6557\),pp\. 871–876\(en\)\.Cited by:[§B\.1](https://arxiv.org/html/2610.07037#A2.SS1.p1.1)\.
- Baeket al\.\(2024\)M\. Baek, R\. McHugh, I\. Anishchenko, H\. Jiang, D\. Baker, and F\. DiMaioAccurate prediction of protein\-nucleic acid complexes using RoseTTAFoldNA\.Nat\. Methods21\(1\),pp\. 117–121\(en\)\.Cited by:[§B\.1](https://arxiv.org/html/2610.07037#A2.SS1.p1.1)\.
- Baker \(2019\)D\. BakerWhat has de novo protein design taught us about protein folding and biophysics?\.Protein Science28\(4\),pp\. 678–683\.External Links:ISSN 1469\-896X,[Link](http://dx.doi.org/10.1002/pro.3588),[Document](https://dx.doi.org/10.1002/pro.3588)Cited by:[§1](https://arxiv.org/html/2610.07037#S1.p2.1)\.
- Bansalet al\.\(2023\)A\. Bansal, H\. Chu, A\. Schwarzschild, S\. Sengupta, M\. Goldblum, J\. Geiping, and T\. GoldsteinUniversal guidance for diffusion models\.InProceedings of the IEEE/CVF conference on computer vision and pattern recognition,pp\. 843–852\.Cited by:[§1\.1](https://arxiv.org/html/2610.07037#S1.SS1.SSS0.Px1.p1.1)\.
- Buttenschoenet al\.\(2024\)M\. Buttenschoen, G\. M\. Morris, and C\. M\. DeanePoseBusters: ai\-based docking methods fail to generate physically valid poses or generalise to novel sequences\.Chemical Science15\(9\),pp\. 3130–3139\.External Links:ISSN 2041\-6539,[Link](http://dx.doi.org/10.1039/D3SC04185A),[Document](https://dx.doi.org/10.1039/d3sc04185a)Cited by:[§1](https://arxiv.org/html/2610.07037#S1.p2.1)\.
- Carugo and Pongor \(2001\)O\. Carugo and S\. PongorA normalized root\-mean\-square distance for comparing protein three\-dimensional structures\.Protein Sci\.10\(7\),pp\. 1470–1473\(en\)\.Cited by:[§G\.1](https://arxiv.org/html/2610.07037#A7.SS1.SSS0.Px2.p2.1)\.
- Chai Discoveryet al\.\(2024\)Chai Discovery, J\. Boitreaud, J\. Dent, M\. McPartlon, J\. Meier, V\. Reis, A\. Rogozhnikov, and K\. WuChai\-1: decoding the molecular interactions of life\.bioRxiv\.Cited by:[§B\.1](https://arxiv.org/html/2610.07037#A2.SS1.p2.1),[§1](https://arxiv.org/html/2610.07037#S1.p1.1)\.
- Chenet al\.\(2025a\)S\. Chen, M\. Guo, C\. Wang, A\. H\. Chen, Y\. Zhang, J\. Chai, Y\. Yang, W\. Matusik, and P\. Y\. ChenPhysically valid biomolecular interaction modeling with gauss\-seidel projection\.arXiv\.External Links:[Document](https://dx.doi.org/10.48550/ARXIV.2510.08946),[Link](https://arxiv.org/abs/2510.08946)Cited by:[§1\.1](https://arxiv.org/html/2610.07037#S1.SS1.SSS0.Px1.p1.1),[§1](https://arxiv.org/html/2610.07037#S1.p5.1),[§3\.2](https://arxiv.org/html/2610.07037#S3.SS2.p1.1),[§3\.3](https://arxiv.org/html/2610.07037#S3.SS3.p2.1)\.
- Chenet al\.\(2025b\)X\. Chen, Y\. Zhang, C\. Lu, W\. Ma, J\. Guan, C\. Gong, J\. Yang, H\. Zhang, K\. Zhang, S\. Wu, K\. Zhou, Y\. Yang, Z\. Liu, L\. Wang, B\. Shi, S\. Shi, and W\. XiaoProtenix \- advancing structure prediction through a comprehensive alphafold3 reproduction\.External Links:[Link](http://dx.doi.org/10.1101/2025.01.08.631967),[Document](https://dx.doi.org/10.1101/2025.01.08.631967)Cited by:[§B\.1](https://arxiv.org/html/2610.07037#A2.SS1.p2.1),[§1](https://arxiv.org/html/2610.07037#S1.p1.1)\.
- Christopheret al\.\(2024\)J\. K\. Christopher, S\. Baek, and F\. FiorettoConstrained synthesis with projected diffusion models\.Advances in Neural Information Processing Systems37,pp\. 89307–89333\.Cited by:[§1\.1](https://arxiv.org/html/2610.07037#S1.SS1.SSS0.Px1.p1.1)\.
- Christopheret al\.\(2025\)J\. K\. Christopher, A\. Seamann, J\. Cui, S\. Khare, and F\. FiorettoConstrained diffusion for protein design with hard structural constraints\.arXiv preprint arXiv:2510\.14989\.Cited by:[§1\.1](https://arxiv.org/html/2610.07037#S1.SS1.SSS0.Px1.p1.1)\.
- Chuet al\.\(2024\)A\. E\. Chu, J\. Kim, L\. Cheng, G\. El Nesr, M\. Xu, R\. W\. Shuai, and P\. HuangAn all\-atom protein generative model\.Proc\. Natl\. Acad\. Sci\. U\. S\. A\.121\(27\),pp\. e2311500121\(en\)\.Cited by:[§B\.1](https://arxiv.org/html/2610.07037#A2.SS1.p1.1)\.
- Chunget al\.\(2022a\)H\. Chung, J\. Kim, M\. T\. Mccann, M\. L\. Klasky, and J\. C\. YeDiffusion posterior sampling for general noisy inverse problems\.arXiv preprint arXiv:2209\.14687\.Cited by:[§1\.1](https://arxiv.org/html/2610.07037#S1.SS1.SSS0.Px1.p1.1)\.
- Chunget al\.\(2024\)H\. Chung, J\. Kim, G\. Y\. Park, H\. Nam, and J\. C\. YeCfg\+\+: manifold\-constrained classifier free guidance for diffusion models\.arXiv preprint arXiv:2406\.08070\.Cited by:[§1\.1](https://arxiv.org/html/2610.07037#S1.SS1.SSS0.Px1.p1.1)\.
- Chunget al\.\(2022b\)H\. Chung, B\. Sim, D\. Ryu, and J\. C\. YeImproving diffusion models for inverse problems using manifold constraints\.Advances in Neural Information Processing Systems35,pp\. 25683–25696\.Cited by:[§1\.1](https://arxiv.org/html/2610.07037#S1.SS1.SSS0.Px1.p1.1)\.
- Daviset al\.\(2007\)I\. W\. Davis, A\. Leaver\-Fay, V\. B\. Chen, J\. N\. Block, G\. J\. Kapral, X\. Wang, L\. W\. Murray, W\. B\. Arendall, J\. Snoeyink, J\. S\. Richardson, and D\. C\. RichardsonMolProbity: all\-atom contacts and structure validation for proteins and nucleic acids\.Nucleic Acids Res\.35\(Web Server issue\),pp\. W375–83\(en\)\.Cited by:[§G\.2](https://arxiv.org/html/2610.07037#A7.SS2.p1.1)\.
- Domagalskiet al\.\(2014\)M\. J\. Domagalski, H\. Zheng, M\. D\. Zimmerman, Z\. Dauter, A\. Wlodawer, and W\. MinorThe quality and validation of structures from structural genomics\.Methods Mol\. Biol\.1091,pp\. 297–314\(en\)\.Cited by:[§1](https://arxiv.org/html/2610.07037#S1.p2.1)\.
- Eastmanet al\.\(2023\)P\. Eastman, R\. Galvelis, R\. P\. Peláez, C\. R\. A\. Abreu, S\. E\. Farr, E\. Gallicchio, A\. Gorenko, M\. M\. Henry, F\. Hu, J\. Huang, A\. Krämer, J\. Michel, J\. A\. Mitchell, V\. S\. Pande, J\. P\. Rodrigues, J\. Rodriguez\-Guerra, A\. C\. Simmonett, S\. Singh, J\. Swails, P\. Turner, Y\. Wang, I\. Zhang, J\. D\. Chodera, G\. De Fabritiis, and T\. E\. MarklandOpenMM 8: molecular dynamics simulation with machine learning potentials\.The Journal of Physical Chemistry B128\(1\),pp\. 109–116\.External Links:ISSN 1520\-5207,[Link](http://dx.doi.org/10.1021/acs.jpcb.3c06662),[Document](https://dx.doi.org/10.1021/acs.jpcb.3c06662)Cited by:[§G\.3](https://arxiv.org/html/2610.07037#A7.SS3.SSS0.Px3.p1.1)\.
- Hollingsworth and Dror \(2018\)S\. A\. Hollingsworth and R\. O\. DrorMolecular dynamics simulation for all\.Neuron99\(6\),pp\. 1129–1143\.External Links:ISSN 0896\-6273,[Link](http://dx.doi.org/10.1016/j.neuron.2018.08.011),[Document](https://dx.doi.org/10.1016/j.neuron.2018.08.011)Cited by:[§1](https://arxiv.org/html/2610.07037#S1.p2.1)\.
- Hong \(2024\)S\. HongSmoothed energy guidance: guiding diffusion models with reduced energy curvature of attention\.Advances in Neural Information Processing Systems37,pp\. 66743–66772\.Cited by:[§1\.1](https://arxiv.org/html/2610.07037#S1.SS1.SSS0.Px1.p1.1)\.
- Hornaket al\.\(2006\)V\. Hornak, R\. Abel, A\. Okur, B\. Strockbine, A\. Roitberg, and C\. SimmerlingComparison of multiple amber force fields and development of improved protein backbone parameters\.Proteins: Structure, Function, and Bioinformatics65\(3\),pp\. 712–725\.Cited by:[§1](https://arxiv.org/html/2610.07037#S1.p4.1)\.
- Huanget al\.\(2024\)W\. Huang, Y\. Jiang, T\. Van Wouwe, and K\. LiuConstrained diffusion with trust sampling\.Advances in Neural Information Processing Systems37,pp\. 93849–93873\.Cited by:[§1\.1](https://arxiv.org/html/2610.07037#S1.SS1.SSS0.Px1.p1.1)\.
- Ishitani and Moriwaki \(2025\)R\. Ishitani and Y\. MoriwakiImproving stereochemical limitations in protein\-ligand complex structure prediction\.ACS Omega10\(46\),pp\. 56075–56084\(en\)\.Cited by:[§1](https://arxiv.org/html/2610.07037#S1.p2.1)\.
- Jumperet al\.\(2021\)J\. Jumper, R\. Evans, A\. Pritzel, T\. Green, M\. Figurnov, O\. Ronneberger, K\. Tunyasuvunakool, R\. Bates, A\. Žídek, A\. Potapenko, A\. Bridgland, C\. Meyer, S\. A\. A\. Kohl, A\. J\. Ballard, A\. Cowie, B\. Romera\-Paredes, S\. Nikolov, R\. Jain, J\. Adler, T\. Back, S\. Petersen, D\. Reiman, E\. Clancy, M\. Zielinski, M\. Steinegger, M\. Pacholska, T\. Berghammer, S\. Bodenstein, D\. Silver, O\. Vinyals, A\. W\. Senior, K\. Kavukcuoglu, P\. Kohli, and D\. HassabisHighly accurate protein structure prediction with AlphaFold\.Nature596\(7873\),pp\. 583–589\(en\)\.Cited by:[§B\.1](https://arxiv.org/html/2610.07037#A2.SS1.p1.1),[§1](https://arxiv.org/html/2610.07037#S1.p4.1)\.
- Karraset al\.\(2022\)T\. Karras, M\. Aittala, T\. Aila, and S\. LaineElucidating the design space of diffusion\-based generative models\.arXiv\.External Links:[Document](https://dx.doi.org/10.48550/ARXIV.2206.00364),[Link](https://arxiv.org/abs/2206.00364)Cited by:[§2](https://arxiv.org/html/2610.07037#S2.SS0.SSS0.Px1.p1.1)\.
- Krishnaet al\.\(2024\)R\. Krishna, J\. Wang, W\. Ahern, P\. Sturmfels, P\. Venkatesh, I\. Kalvet, G\. R\. Lee, F\. S\. Morey\-Burrows, I\. Anishchenko, I\. R\. Humphreys, R\. McHugh, D\. Vafeados, X\. Li, G\. A\. Sutherland, A\. Hitchcock, C\. N\. Hunter, A\. Kang, E\. Brackenbrough, A\. K\. Bera, M\. Baek, F\. DiMaio, and D\. BakerGeneralized biomolecular modeling and design with RoseTTAFold All\-Atom\.Science384\(6693\),pp\. eadl2528\(en\)\.Cited by:[§B\.1](https://arxiv.org/html/2610.07037#A2.SS1.p1.1)\.
- Landrumet al\.\(2026\)G\. Landrum, P\. Tosco, R\. Rodriguez, B\. Kelley, D\. Cosgrove, R\. Vianello, sriniker, P\. Gedeck, G\. Jones, D\. Nealschneider, E\. Kawashima, NadineSchneider, tadhurst\-cdd, A\. Dalke, N\. Maeder, M\. Swain, Y\. Pechersky, B\. Cole, K\. Boyd, S\. Turk, A\. Savelev, R\. Walker, A\. Vaucher, M\. Wójcikowski, H\. Faara, I\. Take, V\. F\. Scalfani, S\. Kearnes, K\. Ujihara, and D\. ProbstRdkit/rdkit: 2026\_03\_6 \(q1 2026\) release\.External Links:[Link](https://zenodo.org/doi/10.5281/zenodo.22140358)Cited by:[§2](https://arxiv.org/html/2610.07037#S2.p1.1)\.
- Liebschneret al\.\(2019\)D\. Liebschner, P\. V\. Afonine, M\. L\. Baker, G\. Bunkóczi, V\. B\. Chen, T\. I\. Croll, B\. Hintze, L\. Hung, S\. Jain, A\. J\. McCoy, N\. W\. Moriarty, R\. D\. Oeffner, B\. K\. Poon, M\. G\. Prisant, R\. J\. Read, J\. S\. Richardson, D\. C\. Richardson, M\. D\. Sammito, O\. V\. Sobolev, D\. H\. Stockwell, T\. C\. Terwilliger, A\. G\. Urzhumtsev, L\. L\. Videau, C\. J\. Williams, and P\. D\. AdamsMacromolecular structure determination using x\-rays, neutrons and electrons: recent developments in phenix\.Acta Crystallographica Section D Structural Biology75\(10\),pp\. 861–877\.External Links:ISSN 2059\-7983,[Link](http://dx.doi.org/10.1107/S2059798319011471),[Document](https://dx.doi.org/10.1107/s2059798319011471)Cited by:[§G\.1](https://arxiv.org/html/2610.07037#A7.SS1.SSS0.Px2.p1.1),[§G\.2](https://arxiv.org/html/2610.07037#A7.SS2.SSS0.Px1.p1.1),[§3\.3](https://arxiv.org/html/2610.07037#S3.SS3.p2.1)\.
- Masterset al\.\(2025\)M\. R\. Masters, A\. H\. Mahmoud, and M\. A\. LillInvestigating whether deep learning models for co\-folding learn the physics of protein\-ligand interactions\.Nature Communications16\(1\)\.External Links:ISSN 2041\-1723,[Link](http://dx.doi.org/10.1038/s41467-025-63947-5),[Document](https://dx.doi.org/10.1038/s41467-025-63947-5)Cited by:[§1](https://arxiv.org/html/2610.07037#S1.p2.1)\.
- Passaroet al\.\(2025\)S\. Passaro, G\. Corso, J\. Wohlwend, M\. Reveiz, S\. Thaler, V\. R\. Somnath, N\. Getz, T\. Portnoi, J\. Roy, H\. Stark, D\. Kwabi\-Addo, D\. Beaini, T\. Jaakkola, and R\. BarzilayBoltz\-2: towards accurate and efficient binding affinity prediction\.External Links:[Link](http://dx.doi.org/10.1101/2025.06.14.659707),[Document](https://dx.doi.org/10.1101/2025.06.14.659707)Cited by:[§B\.1](https://arxiv.org/html/2610.07037#A2.SS1.p2.1),[Appendix D](https://arxiv.org/html/2610.07037#A4.SS0.SSS0.Px3.p2.1),[§1](https://arxiv.org/html/2610.07037#S1.p1.1),[§1](https://arxiv.org/html/2610.07037#S1.p2.1),[§1](https://arxiv.org/html/2610.07037#S1.p5.1),[§3\.2](https://arxiv.org/html/2610.07037#S3.SS2.p1.1)\.
- Rochman\-Sharabi and Louppe \(2026\)O\. Rochman\-Sharabi and G\. LouppePredict\-project\-renoise: sampling diffusion models under hard constraints\.arXiv preprint arXiv:2601\.21033\.Cited by:[§1\.1](https://arxiv.org/html/2610.07037#S1.SS1.SSS0.Px1.p1.1)\.
- Singhalet al\.\(2025\)R\. Singhal, Z\. Horvitz, R\. Teehan, M\. Ren, Z\. Yu, K\. McKeown, and R\. RanganathA general framework for inference\-time scaling and steering of diffusion models\.arXiv preprint arXiv:2501\.06848\.Cited by:[§1](https://arxiv.org/html/2610.07037#S1.p5.1),[§3\.2](https://arxiv.org/html/2610.07037#S3.SS2.p1.1)\.
- Songet al\.\(2023\)J\. Song, Q\. Zhang, H\. Yin, M\. Mardani, M\. Liu, J\. Kautz, Y\. Chen, and A\. VahdatLoss\-guided diffusion models for plug\-and\-play controllable generation\.InInternational Conference on Machine Learning,pp\. 32483–32498\.Cited by:[§1\.1](https://arxiv.org/html/2610.07037#S1.SS1.SSS0.Px1.p1.1)\.
- \[36\]The GUDHI ProjectGUDHI user and reference manual\.GUDHI Editorial Board\.External Links:[Link](https://gudhi.inria.fr/)Cited by:[§G\.3](https://arxiv.org/html/2610.07037#A7.SS3.SSS0.Px3.p1.1)\.
- The OpenFold3 Team \(2025\)The OpenFold3 TeamOpenFold3\-preview\.External Links:[Document](https://dx.doi.org/10.5281/zenodo.19001000),[Link](https://github.com/aqlaboratory/openfold-3)Cited by:[§B\.1](https://arxiv.org/html/2610.07037#A2.SS1.p2.1),[§1](https://arxiv.org/html/2610.07037#S1.p1.1),[§1](https://arxiv.org/html/2610.07037#S1.p2.1),[§3\.2](https://arxiv.org/html/2610.07037#S3.SS2.p1.1)\.
- Trippeet al\.\(2022\)B\. L\. Trippe, J\. Yim, D\. Tischer, D\. Baker, T\. Broderick, R\. Barzilay, and T\. JaakkolaDiffusion probabilistic modeling of protein backbones in 3D for the motif\-scaffolding problem\.External Links:2206\.04119Cited by:[§B\.1](https://arxiv.org/html/2610.07037#A2.SS1.p1.1),[§1](https://arxiv.org/html/2610.07037#S1.p3.1)\.
- Wohlwendet al\.\(2024\)J\. Wohlwend, G\. Corso, S\. Passaro, N\. Getz, M\. Reveiz, K\. Leidal, W\. Swiderski, L\. Atkinson, T\. Portnoi, I\. Chinn, J\. Silterra, T\. Jaakkola, and R\. BarzilayBoltz\-1 democratizing biomolecular interaction modeling\.External Links:[Link](http://dx.doi.org/10.1101/2024.11.19.624167),[Document](https://dx.doi.org/10.1101/2024.11.19.624167)Cited by:[Appendix A](https://arxiv.org/html/2610.07037#A1.p1.1),[§B\.1](https://arxiv.org/html/2610.07037#A2.SS1.p2.1),[§1](https://arxiv.org/html/2610.07037#S1.p2.1),[§1](https://arxiv.org/html/2610.07037#S1.p5.1),[§3\.2](https://arxiv.org/html/2610.07037#S3.SS2.p1.1),[§4](https://arxiv.org/html/2610.07037#S4.p2.1)\.
- Wuet al\.\(2024\)K\. E\. Wu, K\. K\. Yang, R\. van den Berg, S\. Alamdari, J\. Y\. Zou, A\. X\. Lu, and A\. P\. AminiProtein structure generation via folding diffusion\.Nat\. Commun\.15\(1\),pp\. 1059\(en\)\.Cited by:[§B\.1](https://arxiv.org/html/2610.07037#A2.SS1.p1.1),[§1](https://arxiv.org/html/2610.07037#S1.p3.1)\.
- Yanget al\.\(2020\)J\. Yang, I\. Anishchenko, H\. Park, Z\. Peng, S\. Ovchinnikov, and D\. BakerImproved protein structure prediction using predicted interresidue orientations\.Proc\. Natl\. Acad\. Sci\. U\. S\. A\.117\(3\),pp\. 1496–1503\(en\)\.Cited by:[§B\.1](https://arxiv.org/html/2610.07037#A2.SS1.p1.1)\.
- Yimet al\.\(2023\)J\. Yim, B\. L\. Trippe, V\. De Bortoli, E\. Mathieu, A\. Doucet, R\. Barzilay, and T\. JaakkolaSE\(3\) diffusion model with application to protein backbone generation\.External Links:2302\.02277Cited by:[§B\.1](https://arxiv.org/html/2610.07037#A2.SS1.p1.1),[§1](https://arxiv.org/html/2610.07037#S1.p3.1)\.

## Appendix

## Appendix ACharacterizing Physical Violations

![Refer to caption](https://arxiv.org/html/2610.07037v1/figures/violations/inter_chain.png)\(a\)Inter\-chain steric clash\.
![Refer to caption](https://arxiv.org/html/2610.07037v1/figures/violations/bonds.png)\(b\)Ligand distance\-geometry violations\.
![Refer to caption](https://arxiv.org/html/2610.07037v1/figures/violations/ez_chiral.png)\(c\)Ligand stereochemistry violations\.
![Refer to caption](https://arxiv.org/html/2610.07037v1/figures/violations/planar.png)\(d\)Ligand planarity violations\.

Figure 4:Examples of physical validity violations in predicted biomolecular complexes\.\(a\)Inter\-chain steric clash, where two chains overlap at the interface; the zoomed inset highlights atoms occupying incompatible van der Waals volumes\.\(b\)Ligand distance\-geometry violations, showing invalid bond length, distorted bond angle, and internal ligand clash\.\(c\)Ligand stereochemistry violations, including incorrect E/Z double\-bond geometry and inverted chirality\.\(d\)Planarity violations in aromatic rings and planar double\-bond groups\.Figure[4](https://arxiv.org/html/2610.07037#A1.F4)illustrates the main classes of violations considered in this work, set by[Wohlwend et al\. \(2024\)](https://arxiv.org/html/2610.07037#bib.bib8)\. These errors can occur even when the overall protein fold or ligand placement appears plausible, and they can compromise downstream use in docking, affinity prediction, molecular dynamics, or structure\-based design\.

Figure[4](https://arxiv.org/html/2610.07037#A1.F4)a: Inter\-chain Clashes: Here, two protein chains are placed so close at the interface that their molecular surfaces overlap\. The zoomed inset highlights the local steric overlap, corresponding to atoms from different chains occupying incompatible van der Waals \(vdW\) volumes\. vdW interactions describe the short\-range physical constraints between atoms that are not covalently bonded\. Each atom can be approximated as occupying a characteristic vdW radius\. If two non\-bonded atoms are closer than the sum of their vdW radii, their electron clouds overlap, producing a strong repulsive interaction\. In our setting, vdW violations, therefore, correspond to steric overlaps: atoms from the ligand, protein, or different chains occupying incompatible volumes in 3D space\.

Figure[4](https://arxiv.org/html/2610.07037#A1.F4)b: Ligand Distance\-geometry Violations: A bond length violation occurs when two covalently bonded atoms are placed at an unrealistic separation, either too close or too far apart relative to chemically allowed bounds\. A bond angle violation occurs when three connected atoms form an implausible local angle\. The same panel also shows an internal ligand clash, where two non\-bonded atoms within the ligand occupy overlapping van der Waals volumes\.

Figure[4](https://arxiv.org/html/2610.07037#A1.F4)c: Stereochemical Violations: An E/Z stereochemistry violation occurs when the substituents around a double bond are arranged with the wrong cis/trans configuration\. A chiral center violation occurs when the tetrahedral arrangement around a stereocentre is inverted, producing the wrong handedness\. These errors are particularly important because stereochemistry determines the three\-dimensional placement of functional groups in the binding pocket\.

Figure[4](https://arxiv.org/html/2610.07037#A1.F4)d: Planarity Violations: Aromatic five\- and six\-membered rings are expected to remain approximately planar, while double\-bond\-containing groups such as alkenes and amides should resist twisting out of plane\. Deviations from these expected planes indicate strained or chemically implausible ligand conformations\.

Table[3](https://arxiv.org/html/2610.07037#A1.T3)shows the physical validity failure modes of Boltz\-2 and Openfold\-3\.

Table 3:Physical validity failure modes of the base models, Boltz\-2 and OpenFold\-3\. For each target the oracle sample is selected \(a physically valid one if any of the five exists, otherwise the one with fewest violations\); columns give the total number of violations summed over those structures, and chain clashes the total number of clashing chain pairs\. Phys\. valid is the percentage of targets for which any of the five samples is valid \(oracle\)\.⋅\\cdot= 0 \(no violations\)\.
## Appendix BRelated Work

### B\.1Biomolecular Diffusion Models

Diffusion models for protein modeling and design have explored a range of structural representations, including 3D point clouds\([Anand and Achim, 2022](https://arxiv.org/html/2610.07037#bib.bib23);[Trippe et al\., 2022](https://arxiv.org/html/2610.07037#bib.bib24)\), backbone internal coordinates\([Wu et al\., 2024](https://arxiv.org/html/2610.07037#bib.bib25)\), and residue\-level 3D frames\([Yim et al\., 2023](https://arxiv.org/html/2610.07037#bib.bib26)\)\. Earlier protein structure prediction models often operated in torsion\-angle, residue\-frame, or otherwise constrained representations which imposed substantial geometric structure on the space of allowable conformations\([Yang et al\., 2020](https://arxiv.org/html/2610.07037#bib.bib17);[Baek et al\., 2021](https://arxiv.org/html/2610.07037#bib.bib18);[Baek et al\., 2024](https://arxiv.org/html/2610.07037#bib.bib16);[Jumper et al\., 2021](https://arxiv.org/html/2610.07037#bib.bib1)\)\. In such parameterizations, much of the local geometry is specified implicitly through internal coordinates or reconstructed from residue\-level frames, so the model explores a narrower and more chemically structured space\. In contrast, current state\-of\-the\-art cofolding models have expanded from backbone\-level generation to all\-atom modeling and design\([Chu et al\., 2024](https://arxiv.org/html/2610.07037#bib.bib27);[Krishna et al\., 2024](https://arxiv.org/html/2610.07037#bib.bib13)\)\. This removes many of these built\-in geometric constraints: atoms are generated directly in Cartesian space rather than through a representation that enforces bonds, angles, steric exclusion, or stereochemistry by construction\. This increased representational flexibility is essential for modeling heterogeneous complexes with arbitrary ligands and modified residues, but it also makes physical validity a more explicit inference problem\.

The current wave of cofolding models is led by AF3, which introduced a diffusion\-based all\-atom architecture for joint prediction of complexes containing proteins, nucleic acids, ligands, ions, and modified residues, and reported strong gains over specialized baselines across protein–ligand, protein–nucleic acid, and antibody–antigen benchmarks\([Abramson et al\., 2024](https://arxiv.org/html/2610.07037#bib.bib10)\)\. AF3 performs prediction by first constructing a trunk representation of the input complex from sequence, MSA, templates, and molecular features, and then using a diffusion module to iteratively denoise noisy all\-atom coordinates into a final 3D structure\. Open\-source models quickly followed this paradigm\([Wohlwend et al\., 2024](https://arxiv.org/html/2610.07037#bib.bib8);[Passaro et al\., 2025](https://arxiv.org/html/2610.07037#bib.bib9);[Chai Discovery et al\., 2024](https://arxiv.org/html/2610.07037#bib.bib7);[The OpenFold3 Team, 2025](https://arxiv.org/html/2610.07037#bib.bib2);[Chen et al\., 2025b](https://arxiv.org/html/2610.07037#bib.bib11)\)\. Boltz\-1 reproduced AF3\-style biomolecular interaction modeling in an open framework and introduced physics\-inspired steering at inference time\([Wohlwend et al\., 2024](https://arxiv.org/html/2610.07037#bib.bib8)\)\. Boltz\-2 extended this family with improved structure prediction and an additional binding\-affinity prediction module\([Passaro et al\., 2025](https://arxiv.org/html/2610.07037#bib.bib9)\)\. OpenFold\-3 aims to be a faithful reproduction of the AF3 architecture and training recipe\([The OpenFold3 Team, 2025](https://arxiv.org/html/2610.07037#bib.bib2)\), while Protenix explored a few\-step diffusion variant that reduces inference cost while maintaining strong predictive accuracy\([Chen et al\., 2025b](https://arxiv.org/html/2610.07037#bib.bib11)\)\. Together, these models define the current state of the art in all\-atom biomolecular cofolding\.

## Appendix CMethods

### C\.1Van der Waals Projection

Algorithm 2VDWProjection1:Denoised coordinates

X^∈ℝB×Natom×3\\hat\{X\}\\in\\mathbb\{R\}^\{B\\times N\_\{\\mathrm\{atom\}\}\\times 3\}; batch features

ℬproj\\mathcal\{B\}\_\{\\mathrm\{proj\}\}; normalised sampling time

tτ∈\(0,1\]t\_\{\\tau\}\\in\(0,1\]
2:Buffer

β\\beta, damping

α\\alpha, pair budget

KK, displacement cap

δmax\\delta\_\{\\max\}, numerical floor

ε\\varepsilon
3:Projected coordinates

X′X^\{\\prime\}
4:if

tτ\>projection\_start\_tt\_\{\\tau\}\>\\texttt\{projection\\\_start\\\_t\}then

5:return

X^\\hat\{X\}⊳\\trianglerightprojection inactive outside the window

6:endif

7:

\(𝒫,ℓ\)←BuildVdwArgs​\(ℬproj\)\(\\mathcal\{P\},\\ell\)\\leftarrow\\textsc\{BuildVdwArgs\}\(\\mathcal\{B\}\_\{\\mathrm\{proj\}\}\)⊳\\trianglerightℓi​j=\(1−β\)​\(rivdw\+rjvdw\)\\ell\_\{ij\}=\(1\-\\beta\)\\,\(r^\{\\mathrm\{vdw\}\}\_\{i\}\+r^\{\\mathrm\{vdw\}\}\_\{j\}\)

8:

X′←clone⁡\(X^\)X^\{\\prime\}\\leftarrow\\mathrm\{clone\}\(\\hat\{X\}\)
9:for

b=1,…,Bb=1,\\dots,Bdo

10:

C←Xb′C\\leftarrow X^\{\\prime\}\_\{b\}
11:

di​j←max⁡\(‖Ci−Cj‖2,ε\)d\_\{ij\}\\leftarrow\\max\(\\\|C\_\{i\}\-C\_\{j\}\\\|\_\{2\},\\ \\varepsilon\)⊳\\trianglerightfor all\(i,j\)∈𝒫\(i,j\)\\in\\mathcal\{P\}

12:

pi​j←max⁡\(ℓi​j−di​j,0\)p\_\{ij\}\\leftarrow\\max\(\\ell\_\{ij\}\-d\_\{ij\},\\ 0\)⊳\\trianglerightpenetration depth

13:

𝒱←\{\(i,j\)∈𝒫:pi​j\>0\}\\mathcal\{V\}\\leftarrow\\\{\(i,j\)\\in\\mathcal\{P\}:p\_\{ij\}\>0\\\}
14:if

𝒱=∅\\mathcal\{V\}=\\emptysetthen

15:continue⊳\\trianglerightno violation in this batch element

16:endif

17:

𝒯←TopKp​\(𝒱,min⁡\(K,\|𝒱\|\)\)\\mathcal\{T\}\\leftarrow\\textsc\{TopK\}\_\{p\}\\big\(\\mathcal\{V\},\\ \\min\(K,\|\\mathcal\{V\}\|\)\\big\)⊳\\trianglerightKKmost severe violations

18:

D←0D\\leftarrow 0⊳\\trianglerightper\-atom displacement accumulator

19:for all

\(i,j\)∈𝒯\(i,j\)\\in\\mathcal\{T\}do

20:

ui​j←12​α​pi​j​\(Ci−Cj\)/di​ju\_\{ij\}\\leftarrow\\tfrac\{1\}\{2\}\\,\\alpha\\,p\_\{ij\}\\,\(C\_\{i\}\-C\_\{j\}\)/d\_\{ij\}⊳\\trianglerightsymmetric push\-apart

21:

Di←Di\+ui​jD\_\{i\}\\leftarrow D\_\{i\}\+u\_\{ij\};

Dj←Dj−ui​jD\_\{j\}\\leftarrow D\_\{j\}\-u\_\{ij\}
22:endfor

23:for

i=1,…,Natomi=1,\\dots,N\_\{\\mathrm\{atom\}\}do

24:

Di←Di⋅min⁡\(1,δmax/max⁡\(‖Di‖2,ε\)\)D\_\{i\}\\leftarrow D\_\{i\}\\cdot\\min\\\!\\big\(1,\\ \\delta\_\{\\max\}/\\max\(\\\|D\_\{i\}\\\|\_\{2\},\\ \\varepsilon\)\\big\)⊳\\trianglerightper\-atom displacement cap

25:endfor

26:

Xb′←C\+DX^\{\\prime\}\_\{b\}\\leftarrow C\+D
27:endfor

28:return

X′X^\{\\prime\}

Algorithm[2](https://arxiv.org/html/2610.07037#alg2)is a van der Waals projection layer between the denoising prediction and the drift computation\. Each atom carries an effective excluded\-volume radius, and two non\-bonded atoms are not expected to approach closer than the sum of their radii; distances below this threshold indicate overlapping excluded volumes and strong steric repulsion, which we treat as clash violations\.

Letτ=1,…,T\\tau=1,\\dots,Tindex the denoising steps \(T=200T=200in AF3\) and lettτ=1−\(τ−1\)/Tt\_\{\\tau\}=1\-\(\\tau\-1\)/Tdenote the normalised sampling time, so thatt1=1t\_\{1\}=1at the first reverse step andtT=1/Tt\_\{T\}=1/Tat the last\. Given denoised all\-atom coordinatesX^∈ℝB×Natom×3\\hat\{X\}\\in\\mathbb\{R\}^\{B\\times N\_\{\\mathrm\{atom\}\}\\times 3\}and batch\-level structural metadataℬproj\\mathcal\{B\}\_\{\\mathrm\{proj\}\}containing the allowed VDW lower\-bound distances, the projection is active over the late part of the reverse trajectory, whenevertτ≤t\_\{\\tau\}\\leqprojection\_start\_t; outside that windowX^\\hat\{X\}is returned unchanged\. WithT=200T=200andprojection\_start\_t=0\.2\\texttt\{projection\\\_start\\\_t\}=0\.2this is the final4040steps\.

We construct a set of candidate atom pairs𝒫vdw=\{\(i,j\):i<j\}\\mathcal\{P\}\_\{\\mathrm\{vdw\}\}=\\\{\(i,j\):i<j\\\}and corresponding lower\-bound distancesℓi​j\\ell\_\{ij\}fromℬproj\\mathcal\{B\}\_\{\\mathrm\{proj\}\}\. Candidate pairs are restricted to valid, non\-padding atoms whose chains are not joined by covalent connectivity; single\-atom ion chains are also excluded\. For each remaining pair the lower bound is defined from the vdW radii of the two atoms, relaxed by a toleranceβ\\beta,

ℓi​j=\(1−β\)​\(rivdw\+rjvdw\)\.\\ell\_\{ij\}=\(1\-\\beta\)\\left\(r\_\{i\}^\{\\mathrm\{vdw\}\}\+r\_\{j\}^\{\\mathrm\{vdw\}\}\\right\)\.The buffer admits the mild contacts that occur in experimentally resolved structures, so that only clear overlaps are treated as violations: at the value we use,β=0\.15\\beta=0\.15, two atoms must approach within85%85\\%of their summed van der Waals radii before the pair is flagged\. The projection therefore targets inter\-chain steric overlap\.

For each batch element we compute all pairwise distances over the candidate set,

di​j=max⁡\(‖Xi′−Xj′‖2,ε\),d\_\{ij\}=\\max\\\!\\left\(\\\|X^\{\\prime\}\_\{i\}\-X^\{\\prime\}\_\{j\}\\\|\_\{2\},\\ \\varepsilon\\right\),whereε\\varepsilonis a small positive constant guarding division by zero, and convert these into penetration depths,

pi​j=max⁡\(ℓi​j−di​j,0\)\.p\_\{ij\}=\\max\(\\ell\_\{ij\}\-d\_\{ij\},\\ 0\)\.Pairs withpi​j\>0p\_\{ij\}\>0are considered violated\. If no candidate pair violates its lower bound, the batch element is left unchanged\. To keep the projection minimal and stable we do not update all violated pairs\. Instead we select the top\-KKpairs by penetration depth, whereK=200K=200\. Each selected pair\(i,j\)\(i,j\)contributes the damped, symmetric push\-apart update

ui​j=12​α​pi​j​r^i​j,r^i​j=Xi′−Xj′di​j,u\_\{ij\}=\\tfrac\{1\}\{2\}\\,\\alpha\\,p\_\{ij\}\\,\\hat\{r\}\_\{ij\},\\qquad\\hat\{r\}\_\{ij\}=\\frac\{X^\{\\prime\}\_\{i\}\-X^\{\\prime\}\_\{j\}\}\{d\_\{ij\}\},which is added to atomiiand subtracted from atomjj\. The factor12\\tfrac\{1\}\{2\}splits the correction evenly between the two atoms, and the damping factorα\\alphasets how much of the remaining overlap a single step removes: the pair separates byα​pi​j\\alpha\\,p\_\{ij\}\. For the Boltz\-2 base model we useα=0\.25\\alpha=0\.25, which removes a quarter of the penetration per step and leaves the remainder to the denoiser; for OpenFold\-3 we useα=1\\alpha=1\. The under\-relaxation in the Boltz\-2 setting is deliberate: the projection is applied at every step of a4040\-step window, so a small repeated nudge is sufficient to reach feasibility while keeping any individual correction far smaller than the structural changes the denoiser itself makes\.

The selected pair updates are accumulated into a per\-atom displacement tensorDDbefore being applied\. This ensures that if an atom participates in several violated pairs, its total displacement is capped atδmax\\delta\_\{\\max\}after summing all pairwise corrections rather than each correction being clipped independently:

Di←Di​min⁡\(1,δmaxmax⁡\(‖Di‖2,ε\)\)\.D\_\{i\}\\leftarrow D\_\{i\}\\min\\\!\\left\(1,\\ \\frac\{\\delta\_\{\\max\}\}\{\\max\(\\\|D\_\{i\}\\\|\_\{2\},\\ \\varepsilon\)\}\\right\)\.The clipped displacement is added to the coordinates, returningX′=X^\+DX^\{\\prime\}=\\hat\{X\}\+D\.

Our design makes the projection intentionally local, sparse, and minimal: local because each update acts only on the atoms involved in a violated pair; sparse because only the largest vdW overlaps are selected; and minimal because each correction is damped byα\\alphaand the accumulated displacement of each atom is explicitly capped\. The goal is therefore not to solve a full molecular optimization problem or enforce every chemical validity constraint, but to remove the most severe inter\-chain steric conflicts while staying as close as possible to the denoised structure predicted by the model\.

### C\.2Ligand Distance\-Geometry Projection

LDGProjection\(Algorithm[3](https://arxiv.org/html/2610.07037#alg3)\) corrects the internal geometry of every ligand in the denoised estimateX^\\hat\{X\}toward the distance\-geometry, planarity and, optionally, chirality constraints implied by its molecular graph\. It is active only inside the late\-step windowtτ≤projection\_start\_tt\_\{\\tau\}\\leq\\texttt\{projection\\\_start\\\_t\}, moves atoms by bounded, per\-atom\-accumulated displacements, requires no additional network evaluations, and leaves the denoiser and its weights untouched\. Constraints are built once per trajectory byBuildLdgArgs; the operator returns the corrected coordinates together with a chirality latchLLthat the sampler carries from one step to the next\.

Algorithm 3LDGProjection1:Denoised coordinates

X^∈ℝB×Natom×3\\hat\{X\}\\in\\mathbb\{R\}^\{B\\times N\_\{\\mathrm\{atom\}\}\\times 3\}; batch features

ℬproj\\mathcal\{B\}\_\{\\mathrm\{proj\}\}; normalised sampling time

tτ∈\(0,1\]t\_\{\\tau\}\\in\(0,1\]
2:Buffers

βb,βa,βc,βp\\beta\_\{\\mathrm\{b\}\},\\beta\_\{\\mathrm\{a\}\},\\beta\_\{\\mathrm\{c\}\},\\beta\_\{\\mathrm\{p\}\}, number of sweeps

NswN\_\{\\mathrm\{sw\}\}, displacement cap

δldg\\delta\_\{\\mathrm\{ldg\}\}; numerical floor

ε\\varepsilon
3:Chirality latch

L∈\{0,1\}B×\|𝒬\|L\\in\\\{0,1\\\}^\{B\\times\|\\mathcal\{Q\}\|\}, zero at the start of each trajectory

4:Projected coordinates

X′X^\{\\prime\}; updated latch

LL
5:if

tτ\>projection\_start\_tt\_\{\\tau\}\>\\texttt\{projection\\\_start\\\_t\}then

6:return

\(X^,L\)\(\\hat\{X\},L\)⊳\\trianglerightprojection inactive outside the window

7:endif

8:

\(𝒫BA,𝒫CL,ℓ,h,𝒢,𝒬\)←BuildLdgArgs​\(ℬproj\)\(\\mathcal\{P\}\_\{\\mathrm\{BA\}\},\\mathcal\{P\}\_\{\\mathrm\{CL\}\},\\ell,h,\\mathcal\{G\},\\mathcal\{Q\}\)\\leftarrow\\textsc\{BuildLdgArgs\}\(\\mathcal\{B\}\_\{\\mathrm\{proj\}\}\)⊳\\trianglerightcached

9:

X′←clone⁡\(X^\)X^\{\\prime\}\\leftarrow\\mathrm\{clone\}\(\\hat\{X\}\)
10:for

b=1,…,Bb=1,\\dots,Bdo

11:

C←Xb′C\\leftarrow X^\{\\prime\}\_\{b\}
12:

\(C,Lb\)←ChiralityStep​\(C,Lb\)\(C,L\_\{b\}\)\\leftarrow\\textsc\{ChiralityStep\}\(C,L\_\{b\}\)⊳\\trianglerightAlg\.[6](https://arxiv.org/html/2610.07037#alg6)

13:for

s=1,…,Nsws=1,\\dots,N\_\{\\mathrm\{sw\}\}do

14:

C←IntervalSweep​\(C,𝒫BA\)C\\leftarrow\\textsc\{IntervalSweep\}\(C,\\mathcal\{P\}\_\{\\mathrm\{BA\}\}\)⊳\\trianglerightAlg\.[4](https://arxiv.org/html/2610.07037#alg4): bonds, angles

15:

C←IntervalSweep​\(C,𝒫CL\)C\\leftarrow\\textsc\{IntervalSweep\}\(C,\\mathcal\{P\}\_\{\\mathrm\{CL\}\}\)⊳\\trianglerightAlg\.[4](https://arxiv.org/html/2610.07037#alg4): internal clashes

16:

C←IntervalSweep​\(C,𝒫BA\)C\\leftarrow\\textsc\{IntervalSweep\}\(C,\\mathcal\{P\}\_\{\\mathrm\{BA\}\}\)⊳\\trianglerightAlg\.[4](https://arxiv.org/html/2610.07037#alg4): re\-seat bonds, angles

17:

C←PlanarSweep​\(C,𝒢\)C\\leftarrow\\textsc\{PlanarSweep\}\(C,\\mathcal\{G\}\)⊳\\trianglerightAlg\.[5](https://arxiv.org/html/2610.07037#alg5): planarity

18:endfor

19:

Xb′←CX^\{\\prime\}\_\{b\}\\leftarrow C
20:endfor

21:return

\(X′,L\)\(X^\{\\prime\},L\)

Algorithm 4IntervalSweep1:Coordinates

C∈ℝNatom×3C\\in\\mathbb\{R\}^\{N\_\{\\mathrm\{atom\}\}\\times 3\}; pair set

ℛ\\mathcal\{R\}with bounds

\[ℓi​j,hi​j\]\[\\ell\_\{ij\},h\_\{ij\}\]
2:Updated coordinates

3:

D←0D\\leftarrow 0⊳\\trianglerightper\-atom displacement accumulator

4:for all

\(i,j\)∈ℛ\(i,j\)\\in\\mathcal\{R\}do⊳\\trianglerightevery pair; no top\-KK

5:

di​j←max⁡\(‖Ci−Cj‖2,ε\)d\_\{ij\}\\leftarrow\\max\(\\\|C\_\{i\}\-C\_\{j\}\\\|\_\{2\},\\ \\varepsilon\)
6:

Δi​j←\{ℓi​j−di​jdi​j<ℓi​jhi​j−di​jdi​j\>hi​j0otherwise\\Delta\_\{ij\}\\leftarrow\\begin\{cases\}\\ell\_\{ij\}\-d\_\{ij\}&d\_\{ij\}<\\ell\_\{ij\}\\\\ h\_\{ij\}\-d\_\{ij\}&d\_\{ij\}\>h\_\{ij\}\\\\ 0&\\text\{otherwise\}\\end\{cases\}⊳\\trianglerightsigned violation

7:

ui​j←12​Δi​j​\(Ci−Cj\)/di​ju\_\{ij\}\\leftarrow\\tfrac\{1\}\{2\}\\,\\Delta\_\{ij\}\\,\(C\_\{i\}\-C\_\{j\}\)/d\_\{ij\}⊳\\trianglerightpush apart or pull together

8:

Di←Di\+ui​jD\_\{i\}\\leftarrow D\_\{i\}\+u\_\{ij\};

Dj←Dj−ui​jD\_\{j\}\\leftarrow D\_\{j\}\-u\_\{ij\}
9:endfor

10:for

i=1,…,Natomi=1,\\dots,N\_\{\\mathrm\{atom\}\}do

11:

Di←Di⋅min⁡\(1,δldg/max⁡\(‖Di‖2,ε\)\)D\_\{i\}\\leftarrow D\_\{i\}\\cdot\\min\\\!\\big\(1,\\ \\delta\_\{\\mathrm\{ldg\}\}/\\max\(\\\|D\_\{i\}\\\|\_\{2\},\\ \\varepsilon\)\\big\)⊳\\trianglerightper\-atom displacement cap

12:endfor

13:return

C\+DC\+D

Algorithm 5PlanarSweep1:Coordinates

C∈ℝNatom×3C\\in\\mathbb\{R\}^\{N\_\{\\mathrm\{atom\}\}\\times 3\}; planar groups

𝒢\\mathcal\{G\}; planarity buffer

βp\\beta\_\{\\mathrm\{p\}\}
2:Updated coordinates

3:

D←0D\\leftarrow 0
4:for all

g∈𝒢g\\in\\mathcal\{G\}do

5:

μg←1\|g\|​∑k∈gCk\\mu\_\{g\}\\leftarrow\\frac\{1\}\{\|g\|\}\\sum\_\{k\\in g\}C\_\{k\}
6:

Σg←∑k∈g\(Ck−μg\)​\(Ck−μg\)⊤\\Sigma\_\{g\}\\leftarrow\\sum\_\{k\\in g\}\(C\_\{k\}\-\\mu\_\{g\}\)\(C\_\{k\}\-\\mu\_\{g\}\)^\{\\top\}
7:

ng←n\_\{g\}\\leftarrowsmallest\-eigenvalue eigenvector of

Σg\\Sigma\_\{g\}⊳\\trianglerightplane normal

8:for all

k∈gk\\in gdo

9:

ζk←\(Ck−μg\)⊤​ng\\zeta\_\{k\}\\leftarrow\(C\_\{k\}\-\\mu\_\{g\}\)^\{\\top\}n\_\{g\}⊳\\trianglerightsigned out\-of\-plane offset

10:

Dk←Dk−sign⁡\(ζk\)​max⁡\(\|ζk\|−βp,0\)​ngD\_\{k\}\\leftarrow D\_\{k\}\-\\mathrm\{sign\}\(\\zeta\_\{k\}\)\\,\\max\(\|\\zeta\_\{k\}\|\-\\beta\_\{\\mathrm\{p\}\},\\ 0\)\\;n\_\{g\}
11:endfor

12:endfor

13:for

i=1,…,Natomi=1,\\dots,N\_\{\\mathrm\{atom\}\}do

14:

Di←Di⋅min⁡\(1,δldg/max⁡\(‖Di‖2,ε\)\)D\_\{i\}\\leftarrow D\_\{i\}\\cdot\\min\\\!\\big\(1,\\ \\delta\_\{\\mathrm\{ldg\}\}/\\max\(\\\|D\_\{i\}\\\|\_\{2\},\\ \\varepsilon\)\\big\)⊳\\trianglerightper\-atom displacement cap

15:endfor

16:return

C\+DC\+D

Algorithm 6ChiralityStep1:Coordinates

C∈ℝNatom×3C\\in\\mathbb\{R\}^\{N\_\{\\mathrm\{atom\}\}\\times 3\}; latch

L∈\{0,1\}\|𝒬\|L\\in\\\{0,1\\\}^\{\|\\mathcal\{Q\}\|\}; quadruples

𝒬\\mathcal\{Q\}with handedness

χq\\chi\_\{q\}, margin

vqv\_\{q\}and signed volume

VqV\_\{q\}\( equation[1](https://arxiv.org/html/2610.07037#A3.E1)\)

2:Updated coordinates and latch

3:

Vq←Vq​\(C\)V\_\{q\}\\leftarrow V\_\{q\}\(C\)for all

q∈𝒬q\\in\\mathcal\{Q\}
4:

𝒜←\{q∈𝒬:Lq=1∨χq​Vq<0\}\\mathcal\{A\}\\leftarrow\\\{q\\in\\mathcal\{Q\}:L\_\{q\}=1\\ \\lor\\ \\chi\_\{q\}V\_\{q\}<0\\\}⊳\\trianglerightinverted, or still latched

5:if

𝒜=∅\\mathcal\{A\}=\\emptysetthen

6:return

\(C,L\)\(C,L\)⊳\\trianglerightcorrect centres are never edited

7:endif

8:

D←0D\\leftarrow 0
9:for all

q∈𝒜q\\in\\mathcal\{A\}with

χq​Vq<vq\\chi\_\{q\}V\_\{q\}<v\_\{q\}do

10:

λq←\(χq​vq−Vq\)/max⁡\(‖∇Vq‖F2,ε\)\\lambda\_\{q\}\\leftarrow\(\\chi\_\{q\}v\_\{q\}\-V\_\{q\}\)\\,/\\,\\max\(\\\|\\nabla V\_\{q\}\\\|\_\{F\}^\{2\},\\ \\varepsilon\)⊳\\trianglerightNewton step

11:for

k=0,…,3k=0,\\dots,3do

12:

Dqk←Dqk\+λq​∇qkVqD\_\{q\_\{k\}\}\\leftarrow D\_\{q\_\{k\}\}\+\\lambda\_\{q\}\\,\\nabla\_\{q\_\{k\}\}V\_\{q\}
13:endfor

14:endfor

15:for

i=1,…,Natomi=1,\\dots,N\_\{\\mathrm\{atom\}\}do

16:

Di←Di⋅min⁡\(1,δldg/max⁡\(‖Di‖2,ε\)\)D\_\{i\}\\leftarrow D\_\{i\}\\cdot\\min\\\!\\big\(1,\\ \\delta\_\{\\mathrm\{ldg\}\}/\\max\(\\\|D\_\{i\}\\\|\_\{2\},\\ \\varepsilon\)\\big\)⊳\\trianglerightper\-atom displacement cap

17:endfor

18:

C←C\+DC\\leftarrow C\+D
19:

Lq←𝟏\[q∈𝒜∧χqVq\(C\)<vq\]∀q∈𝒬L\_\{q\}\\leftarrow\\mathbf\{1\}\\big\[q\\in\\mathcal\{A\}\\land\\chi\_\{q\}V\_\{q\}\(C\)<v\_\{q\}\\big\]\\ \\ \\forall q\\in\\mathcal\{Q\}
20:return

\(C,L\)\(C,L\)

#### Constraint construction\.

BuildLdgArgs\(Algorithm[3](https://arxiv.org/html/2610.07037#alg3)\) builds the constraints once per trajectory from the ligand molecular graphs inℬproj\\mathcal\{B\}\_\{\\mathrm\{proj\}\}\. For every ligand with at least two heavy atoms, the candidate set𝒫\\mathcal\{P\}contains all heavy\-atom pairs\(i,j\)\(i,j\),i<ji<j, with RDKit distance\-geometry bounds\[ℓi​jrd,hi​jrd\]\[\\ell^\{\\mathrm\{rd\}\}\_\{ij\},h^\{\\mathrm\{rd\}\}\_\{ij\}\]\. Bonded \(1–2\) pairs𝒫bond\\mathcal\{P\}\_\{\\mathrm\{bond\}\}and angle \(1–3\) pairs𝒫ang\\mathcal\{P\}\_\{\\mathrm\{ang\}\}form𝒫BA=𝒫bond∪𝒫ang\\mathcal\{P\}\_\{\\mathrm\{BA\}\}=\\mathcal\{P\}\_\{\\mathrm\{bond\}\}\\cup\\mathcal\{P\}\_\{\\mathrm\{ang\}\}, and all remaining pairs, those that are neither bonded nor separated by one atom, form the clash set𝒫CL\\mathcal\{P\}\_\{\\mathrm\{CL\}\}\. The bounds are widened by the buffers,

\[ℓi​j,hi​j\]=\[\(1−β\)​ℓi​jrd,\(1\+β\)​hi​jrd\],β=\{βb1–2 pairs onlyβa1–3 pairs onlymin⁡\(βb,βa\)both\[\\ell\_\{ij\},h\_\{ij\}\]=\\big\[\(1\-\\beta\)\\,\\ell^\{\\mathrm\{rd\}\}\_\{ij\},\\ \(1\+\\beta\)\\,h^\{\\mathrm\{rd\}\}\_\{ij\}\\big\],\\quad\\beta=\\begin\{cases\}\\beta\_\{\\mathrm\{b\}\}&\\text\{1\-\-2 pairs only\}\\\\ \\beta\_\{\\mathrm\{a\}\}&\\text\{1\-\-3 pairs only\}\\\\ \\min\(\\beta\_\{\\mathrm\{b\}\},\\beta\_\{\\mathrm\{a\}\}\)&\\text\{both\}\\end\{cases\}for\(i,j\)∈𝒫BA\(i,j\)\\in\\mathcal\{P\}\_\{\\mathrm\{BA\}\}, while pairs in𝒫CL\\mathcal\{P\}\_\{\\mathrm\{CL\}\}receive only a lower bound,\[ℓi​j,hi​j\]=\[\(1−βc\)ℓi​jrd,\+∞\)\[\\ell\_\{ij\},h\_\{ij\}\]=\[\(1\-\\beta\_\{\\mathrm\{c\}\}\)\\,\\ell^\{\\mathrm\{rd\}\}\_\{ij\},\\ \+\\infty\)\. The planar groups𝒢\\mathcal\{G\}are the aromatic five\- and six\-membered rings and the planarC=C\\mathrm\{C\{=\}C\}groups \(the double bond with its four substituents\)\.𝒬\\mathcal\{Q\}holds the assigned sp3stereocentres with at least three heavy neighbours, each as a quadrupleq=\(q0,…,q3\)q=\(q\_\{0\},\\dots,q\_\{3\}\)of the centre and its three heavy neighbours, whose handedness is the sign of the signed volume

Vq​\(C\)=\(Cq1−Cq0\)⋅\[\(Cq2−Cq0\)×\(Cq3−Cq0\)\]\.V\_\{q\}\(C\)=\(C\_\{q\_\{1\}\}\-C\_\{q\_\{0\}\}\)\\cdot\\big\[\(C\_\{q\_\{2\}\}\-C\_\{q\_\{0\}\}\)\\times\(C\_\{q\_\{3\}\}\-C\_\{q\_\{0\}\}\)\\big\]\.\(1\)The target handednessχq=sign​Vq​\(Xref\)\\chi\_\{q\}=\\mathrm\{sign\}\\,V\_\{q\}\(X^\{\\mathrm\{ref\}\}\)and the marginvq=κ​\|Vq​\(Xref\)\|v\_\{q\}=\\kappa\\,\|V\_\{q\}\(X^\{\\mathrm\{ref\}\}\)\|are read from the RDKit\-embedded reference conformerXrefX^\{\\mathrm\{ref\}\}\.

#### Interval constraints\.

Every heavy\-atom pair carries an admissible distance interval: bonded and angle pairs are held to a two\-sided interval around their RDKit bounds, all other pairs to a lower bound only, each widened by a class\-specific tolerance so that only clear violations are corrected\.IntervalSweep\(Algorithm[4](https://arxiv.org/html/2610.07037#alg4)\) visits every pair in a set, computes the signed violationΔi​j\\Delta\_\{ij\}, positive when the pair is too close, negative when too far, and applies the symmetric updateui​j=12​Δi​j​r^i​ju\_\{ij\}=\\tfrac\{1\}\{2\}\\Delta\_\{ij\}\\,\\hat\{r\}\_\{ij\}to atomiiand−ui​j\-u\_\{ij\}to atomjj, pushing the pair apart or pulling it together as required\. All updates in a pass are accumulated into a per\-atom tensorDD, and each atom’s total displacement is capped atδldg\\delta\_\{\\mathrm\{ldg\}\}before it is applied, so an atom in many violated pairs is bounded on its net motion rather than on each contribution\.

#### Sweep order\.

Each of theNswN\_\{\\mathrm\{sw\}\}sweeps runs the interval pass on𝒫BA\\mathcal\{P\}\_\{\\mathrm\{BA\}\}, then on𝒫CL\\mathcal\{P\}\_\{\\mathrm\{CL\}\}, then on𝒫BA\\mathcal\{P\}\_\{\\mathrm\{BA\}\}again, and finallyPlanarSweep\. Resolving an internal clash can stretch the bonds adjacent to it, so bonds and angles are re\-seated after the clash pass; planarity comes last because it moves whole ring systems that the pairwise passes would otherwise disturb\.PlanarSweep\(Algorithm[5](https://arxiv.org/html/2610.07037#alg5)\) fits a plane to each aromatic ring orC=C\\mathrm\{C\{=\}C\}group by taking the smallest\-eigenvalue eigenvector of the group’s coordinate covariance as its normalngn\_\{g\}, and moves each member atom toward the plane by its out\-of\-plane offset in excess ofβp\\beta\_\{\\mathrm\{p\}\}\. The sign ofngn\_\{g\}is arbitrary, but the update depends only onsign⁡\(ζk\)​ng\\mathrm\{sign\}\(\\zeta\_\{k\}\)\\,n\_\{g\}, which is not\. Because the cap is applied within each pass, an atom may move by up toδldg\\delta\_\{\\mathrm\{ldg\}\}per pass over the course of one diffusion step\.

#### Chirality\.

ChiralityStep\(Algorithm[6](https://arxiv.org/html/2610.07037#alg6)\) runs once per call before the sweeps\. Each stereocentre in𝒬\\mathcal\{Q\}is represented by the signed volumeVqV\_\{q\}of Eq\. equation[1](https://arxiv.org/html/2610.07037#A3.E1), whose sign gives its handedness\. A centre whose sign disagrees with the targetχq\\chi\_\{q\}receives a damped Newton step onVqV\_\{q\}towardχq​vq\\chi\_\{q\}v\_\{q\}, distributed over its four atoms along∇Vq\\nabla V\_\{q\}\. BecauseVqV\_\{q\}passes through zero as a centre inverts, a correction that stopped at the first sign change would leave a nearly planar centre that the denoiser could flip back; the latchLqL\_\{q\}therefore keeps a centre active untilχq​Vq\\chi\_\{q\}V\_\{q\}exceeds the marginvqv\_\{q\}\. The latch is initialised to zero once per trajectory and threaded through every call, so a centre latched at one step remains active at the next\. Correct, unlatched centres are never edited\.

#### Ours \(Boltz\)\.

Unless stated otherwise the vdW projection usesprojection\_start\_t=0\.2\\texttt\{projection\\\_start\\\_t\}=0\.2,α=0\.25\\alpha=0\.25,δmax=0\.45\\delta\_\{\\max\}=0\.45Å,K=200K=200,β=0\.15\\beta=0\.15,ε=10−8\\varepsilon=10^\{\-8\}, and one sweep per diffusion step\. Unless stated otherwise the LDG projection usesδldg=0\.2\\delta\_\{\\mathrm\{ldg\}\}=0\.2Å,βb=0\.24\\beta\_\{\\mathrm\{b\}\}=0\.24,βa=0\.24\\beta\_\{\\mathrm\{a\}\}=0\.24,βc=0\.19\\beta\_\{\\mathrm\{c\}\}=0\.19,βp=0\.19\\beta\_\{\\mathrm\{p\}\}=0\.19Å,Nsw=3N\_\{\\mathrm\{sw\}\}=3,

#### Ours \(OpenFold\)\.

Unless stated otherwise the vdW projection usesα=1\.0\\alpha=1\.0,δmax=0\.3\\delta\_\{\\max\}=0\.3Å,K=200K=200,β=0\.15\\beta=0\.15,ε=10−8\\varepsilon=10^\{\-8\}, and one sweep per diffusion step\. Unless stated otherwise the LDG projection usesδldg=0\.25\\delta\_\{\\mathrm\{ldg\}\}=0\.25Å,βb=0\.24\\beta\_\{\\mathrm\{b\}\}=0\.24,βa=0\.10\\beta\_\{\\mathrm\{a\}\}=0\.10,βc=0\.19\\beta\_\{\\mathrm\{c\}\}=0\.19,βp=0\.20\\beta\_\{\\mathrm\{p\}\}=0\.20Å,κ=0\.1\\kappa=0\.1,Nsw=3N\_\{\\mathrm\{sw\}\}=3and the sameprojection\_start\_tas Algorithm[2](https://arxiv.org/html/2610.07037#alg2)\.

## Appendix DSampler dynamics and the projection window

To understand where in the reverse process a correction can take effect, we record the iteratextx\_\{t\}and the denoised estimatex^0\\hat\{x\}\_\{0\}at every step of an unprojected Boltz\-2 run \(200 steps\) on four CASP16 targets \(M1212, M1296, T1234, T1249v2\) and compare each against the final structure\. Churn is active up to step 154 and off thereafter \(shaded region in all figures\)\.

#### The structure is fixed when churn ends\.

The iteratextx\_\{t\}is dominated by noise for most of sampling: its RMSD to the final structure tracks the noise levelσ\\sigmaalmost exactly until the churn boundary \(Fig\.[5](https://arxiv.org/html/2610.07037#A4.F5)\)\. The denoised estimate behaves differently\. It starts 10–40 Å from the final structure, approaches it steadily, and then drops by an order of magnitude, from∼\\sim11Å to∼\\sim0\.10\.1Å, at the step where churn switches off \(Fig\.[6](https://arxiv.org/html/2610.07037#A4.F6), left\)\. The all\-atom LDDT ofx^0\\hat\{x\}\_\{0\}levels off within a few steps of the boundary \(Fig\.[6](https://arxiv.org/html/2610.07037#A4.F6), right\)\. The remaining∼\\sim45 steps refine the structure by less than0\.10\.1Å\.

#### Why projectx^0\\hat\{x\}\_\{0\}\.

Van der Waals and ligand\-geometry constraints are defined on clean structures\. The denoised estimate is a clean structure from the first step: its backbone LDDT is at or near its final value from step 0, and only the all\-atom LDDT improves over sampling \(Fig\.[7](https://arxiv.org/html/2610.07037#A4.F7)\)\. The iteratextx\_\{t\}, by contrast, has near\-zero all\-atom LDDT until∼\\simstep 160, and its backbone LDDT only rises between steps 100 and 160\. Projectingx^0\\hat\{x\}\_\{0\}therefore applies the constraints to the model’s current best estimate of the structure rather than to noise, and the correction reaches the trajectory through the drift \(line[15](https://arxiv.org/html/2610.07037#alg1.l15)\)\. Once churn is off,xtx\_\{t\}andx^0\\hat\{x\}\_\{0\}are withinσ≤1\\sigma\\leq 1Å of each other, and in this regime projecting either one gives the same validity and accuracy \(Table[2](https://arxiv.org/html/2610.07037#S3.T2)\)\.

#### Why project only after churn\.

Figure[8](https://arxiv.org/html/2610.07037#A4.F8)tracks the number of violated van der Waals pairs inxtx\_\{t\}over the last 60 steps\. Projecting only during churn has no visible effect: at no point does the count fall below the unprojected run, before or after the churn boundary\. Each correction is at mostδmax=0\.45\\delta\_\{\\max\}=0\.45Å per atom, whereas each churn step injects noise with per\-coordinate standard deviation of at least1\.51\.5Å, so the correction is overwritten before it can accumulate\. Withprojection\_start\_t=0\.2\\texttt\{projection\\\_start\\\_t\}=0\.2, the count separates from the unprojected run at∼\\simstep 160 and falls steadily until the end of sampling\. In this tail no fresh noise is injected, and the denoiser itself changesx^0\\hat\{x\}\_\{0\}by less than0\.10\.1Å RMSD over all remaining steps \(Fig\.[6](https://arxiv.org/html/2610.07037#A4.F6)\)\. Successive bounded corrections are therefore retained and add up\. The benchmark results agree: churn\-only projection leaves physical validity at0\.020\.02\(x^0\\hat\{x\}\_\{0\}\) and0\.040\.04\(xtx\_\{t\}\), while tail projection reaches1\.001\.00with unchanged accuracy \(Table[2](https://arxiv.org/html/2610.07037#S3.T2)\)\.

The counts in Figure[8](https://arxiv.org/html/2610.07037#A4.F8)do not reach zero violations because the diagnostic is stricter than the evaluation: it is the projection’s own criterion\. A pair counts as violated if its two atoms lie in different chains \(excluding covalently linked chains and single\-atom ions\) and are closer than\(1−β\)​\(ri\+rj\)\(1\-\\beta\)\(r\_\{i\}\+r\_\{j\}\)withβ=0\.15\\beta=0\.15, i\.e\.0\.85​\(ri\+rj\)0\.85\\,\(r\_\{i\}\+r\_\{j\}\), with no further tolerance\. Physical validity in our benchmarks follows the Boltz definition\([Passaro et al\., 2025](https://arxiv.org/html/2610.07037#bib.bib9)\), whose steric\-clash test flags a pair of chains as clashing if any two of their atoms are closer than0\.75​\(ri\+rj\)0\.75\\,\(r\_\{i\}\+r\_\{j\}\), using the same van der Waals radii\. Almost all pairs remaining in the figure are mild contacts between these two thresholds, inside the spacing the projection aims for but outside what the Boltz criterion treats as a clash\. At the last step of thex^0\\hat\{x\}\_\{0\}run withprojection\_start\_t=0\.2=0\.2, 1,401 of the 1,402 remaining pairs \(four targets, five samples each\) lie between0\.750\.75and0\.85​\(ri\+rj\)0\.85\\,\(r\_\{i\}\+r\_\{j\}\), and 19 of the 20 structures are clash\-free\. Without projection, 1,037 of 2,309 pairs fall below0\.75​\(ri\+rj\)0\.75\\,\(r\_\{i\}\+r\_\{j\}\)and no structure is clash\-free\. The figure is meant to show when corrections are retained, not to measure validity\.

![Refer to caption](https://arxiv.org/html/2610.07037v1/figures/dynamics/fig_rmsd_to_final.png)Figure 5:RMSD to the final structure of the denoised estimatex^0\\hat\{x\}\_\{0\}and the iteratextx\_\{t\}at each sampler step \(log scale\), with the noise levelσ\\sigmafor reference\.xtx\_\{t\}tracksσ\\sigmauntil churn switches off;x^0\\hat\{x\}\_\{0\}drops by an order of magnitude at the churn boundary\. Shaded: steps without churn\.![Refer to caption](https://arxiv.org/html/2610.07037v1/figures/dynamics/fig_summary.png)Figure 6:Summary across the four targets\. Left: RMSD ofx^0\\hat\{x\}\_\{0\}to the final structure; all targets cross11Å \(dashed\) just before the churn boundary and fall to∼\\sim0\.10\.1Å immediately after\. Right: LDDT ofx^0\\hat\{x\}\_\{0\}, which levels off within a few steps of the boundary\.![Refer to caption](https://arxiv.org/html/2610.07037v1/figures/dynamics/fig_convergence.png)Figure 7:All\-atom and backbone LDDT ofx^0\\hat\{x\}\_\{0\}\(solid\) andxtx\_\{t\}\(dashed\) at each sampler step\. The backbone ofx^0\\hat\{x\}\_\{0\}is close to its final accuracy from the first step;xtx\_\{t\}only becomes a meaningful structure after churn switches off\.![Refer to caption](https://arxiv.org/html/2610.07037v1/figures/dynamics/fig_violations.png)Figure 8:Number of violated van der Waals pairs inxtx\_\{t\}over steps 140–199\. Projecting in the tail \(projection\_start\_t=0\.2\\texttt\{projection\\\_start\\\_t\}=0\.2, green\) reduces violations steadily to the end of sampling\.
### D\.1Projection Dynamics

![Refer to caption](https://arxiv.org/html/2610.07037v1/figures/dynamics/rms_proj.png)\(a\)RMS projection step\.
![Refer to caption](https://arxiv.org/html/2610.07037v1/figures/dynamics/displacement.png)\(b\)Maximum per\-atom projection displacement\.
![Refer to caption](https://arxiv.org/html/2610.07037v1/figures/dynamics/penetration.png)\(c\)Mean VdW penetration depth\.

Figure 9:Each line corresponds to a different system in CASP15, with colors and line styles as labeled\.We posit that minimal, local coordinate updates can remove a large fraction of physical\-validity violations while remaining compute\- and memory\-efficient\. Figure[9](https://arxiv.org/html/2610.07037#A4.F9)illustrates the vdW projection dynamics on CASP15 targets\. Projection is activated only in the late refinement regime; in this experiment,projection\_start\_t=0\.2=0\.2, corresponding to the final4040steps of the standard200200\-step AF3\-style sampler\. We therefore plot only steps160160–200200, since all projection quantities are zero before activation\. The observed dynamics show that the correction remains small and displacement\-limited while VDW penetration decreases over the final denoising steps\.

RMS projection stepis the Root Mean Square \(RMS\) of the final capped per\-atom displacement,DiD\_\{i\}, produced by Algorithm[2](https://arxiv.org/html/2610.07037#alg2)\. The plotted quantity is

RMSproj=1Natom​∑i=1Natom‖𝐃i‖22\.\\mathrm\{RMS\}\_\{\\mathrm\{proj\}\}=\\sqrt\{\\frac\{1\}\{N\_\{\\mathrm\{atom\}\}\}\\sum\_\{i=1\}^\{N\_\{\\mathrm\{atom\}\}\}\\\|\\mathbf\{D\}\_\{i\}\\\|\_\{2\}^\{2\}\}\.This value decreases throughout the trajectory, indicating that the projection requires progressively smaller corrections as the structure becomes more physically consistent\.

Maximum per\-atom displacementshows the largest single\-atom movement given by

dmax=maxi⁡‖𝐃i‖2\.d\_\{\\max\}=\\max\_\{i\}\\\|\\mathbf\{D\}\_\{i\}\\\|\_\{2\}\.
We cap the maximum per\-atom displacement by the parameter*projection\_max\_atom\_disp*in Algorithm[2](https://arxiv.org/html/2610.07037#alg2)\. Although some atoms reach this cap in the early steps, the maximum displacement rapidly decreases toward the end of sampling, indicating that the correction becomes increasingly minimal as violations disappear\.

VDW penetration depthshows the mean van der Waals penetration depth over violated pairs

pi​j=max⁡\(ℓi​j−‖𝐱i−𝐱j‖2,0\),p\_\{ij\}=\\max\\\!\\left\(\\ell\_\{ij\}\-\\\|\\mathbf\{x\}\_\{i\}\-\\mathbf\{x\}\_\{j\}\\\|\_\{2\},\\,0\\right\),whereℓi​j\\ell\_\{ij\}is the VDW lower\-bound distance\. This quantity is nonzero only when two atoms are too close, i\.e\. when they clash\. Therefore, lower penetration directly means weaker or fewer steric overlaps, andpi​j=0p\_\{ij\}=0means that the pair satisfies the van der Waals separation constraint\. Across CASP15 targets, the mean penetration consistently decreases toward zero, showing that steric overlaps are progressively reduced as sampling approaches the terminal structure\.

Interpreted together with Figures[9\(a\)](https://arxiv.org/html/2610.07037#A4.F9.sf1)–[9\(b\)](https://arxiv.org/html/2610.07037#A4.F9.sf2), this reduction happens with small corrections: the RMS projection step is small, the per\-atom displacement is capped atδmax\\delta\_\{\\max\}, and both shrink as the clashes are resolved\. This fits the dynamics described in Appendix[D](https://arxiv.org/html/2610.07037#A4)\. By the time projection is activated, the denoiser has settled the fold:x^0\\hat\{x\}\_\{0\}is within∼\\sim0\.10\.1Å of the final structure, and its LDDT has stopped changing\. What remains are local steric clashes that the model does not remove on its own during the remaining steps\. Projection targets exactly these clashes, so it acts as a sparse local repair rather than a global energy minimization\. The resulting corrections are cheap and stable, and they remove the remaining clashes without rearranging the structure the model has already settled on\.

## Appendix EExperimental Setup

### E\.1Evaluation Criteria

We evaluated all methods on the same target sets, using identical MSA settings, random seed \(42\), and the same post\-processing and metric scripts\. For each target, we ran inference for 5 samples, with the default sampling settings for Boltz\-2 and OpenFold\-3, unless otherwise stated\. Our projection method uses the same weights and sampling schedule, with only the additional inference\-time projection enabled\.

We measured structural accuracy with OpenStructure using the same evaluation pipeline as Boltz\-1\. We computed physical validity using the Boltz physical\-validity checks, which include ligand bond lengths, bond angles, internal ligand clashes, chirality, E/Z stereochemistry, planarity, and inter\-chain steric clashes\. A prediction was counted as physically valid only if it passes all considered violation checks\. We used the same validity definition as Boltz, but corrected a bug in the planarity check; as a result, Boltz validity values differ \(in the positive direction\) from those reported in the original Boltz paper\.

We measured runtime as wall\-clock inference time per target, including diffusion sampling and excluding pre\-processing steps like MSA generation\. We evaluated GPU memory with PyTorch CUDA peak\-memory counters\. All runtime and memory comparisons were run on A100 80GB GPUs under the same software environment\. For each method and dataset, we reported mean metrics across targets; uncertainty is computed by bootstrap resampling over targets\.

We have used the following metrics to evaluate our method:

- •LDDT\(↑\\uparrow\): Local Distance Difference Test, measuring local structural accuracy by comparing predicted and reference inter\-atomic distances\. Higher values indicate better local geometry\.
- •DockQ\(↑\\uparrow\): Interface quality score for protein–protein or biomolecular docking interfaces, combining interface RMSD, ligand RMSD, and native contact recovery\. Higher values indicate more accurate interface placement\.
- •LDDT\-PLI\(↑\\uparrow\): Protein–ligand interaction score, measuring the accuracy of predicted protein–ligand contacts and binding\-site geometry\. Higher values indicate better recovery of ligand interactions\.
- •LRMSD<2<2\(↑\\uparrow\): Fraction of protein–ligand predictions with ligand RMSD below2​Å2\\,\\text\{\\AA\}, typically after pocket or receptor alignment\. Higher values indicate a larger fraction of near\-native ligand poses\.
- •Valid\(↑\\uparrow\): Fraction of predictions passing the physical validity checks, including steric clash, ligand distance\-geometry, stereochemistry, and planarity criteria\. Higher values indicate fewer physical violations\.

Per\-metric target counts are given in thennrow of Table[1](https://arxiv.org/html/2610.07037#S3.T1)\. DockQ requires a scored interface and LDDT\-PLI/LRMSD<2a scored ligand, so those columns rest on subsets\.

### E\.2Benchmark targets

Every method was run on the same query set per dataset: CASP15 \(66\), CASP16 \(55\), PoseBusters Monomer \(134\), PoseBusters Complex \(169\), Test \(542\), 966 targets in total\. CASP targets are listed by their CASP target ID \(subunit\-level queries keep their suffix, e\.g\. s1\), all other targets by PDB ID\. Targets inredare excluded from Table[1](https://arxiv.org/html/2610.07037#S3.T1)because Boltz\-2x runs out of memory on them on an 80 GB A100 \(CASP16: H1236, R1283v2, R1285; PoseBusters Complex: 7B2C, 7CNQ, 7SUC, 8AQL, 8G0V; Test: 8SRO\)\. Boltz\-2 and Ours \(Boltz\) complete all of them, and all comparisons in the table use the targets every Boltz method completed\.

#### CASP15 \(66 targets\)\.

H1106, H1129, H1151, H1157, H1185, R1107, R1108, R1116, R1117, R1117v2, R1128, R1136, R1149, R1156, T1104, T1106s1, T1106s2, T1112, T1113, T1114s1, T1114s3, T1118v1, T1119, T1120, T1122, T1124, T1125, T1129s2, T1132, T1134s1, T1137s1, T1137s2, T1137s3, T1137s4, T1137s5, T1137s6, T1137s7, T1137s8, T1137s9, T1145, T1147, T1151s2, T1152, T1154, T1155, T1157s1, T1157s2, T1158, T1158v1, T1158v2, T1158v3, T1158v4, T1160, T1161, T1173, T1174, T1176, T1178, T1179, T1183, T1185s1, T1185s2, T1185s4, T1187, T1188, T1194

#### CASP16 \(55 targets \(52 of 55 in Table[1](https://arxiv.org/html/2610.07037#S3.T1)\)\)\.

H1202, H1204, H1213, H1222, H1223, H1225, H1232, H1233,H1236, M1209, M1211, M1212, M1296, R1203, R1205, R1209, R1212, R1242, R1260, R1261, R1261v1, R1262, R1262v1, R1263, R1263v1, R1264, R1264v1, R1283v1,R1283v2,R1285, R1286, R1296, T1201, T1206, T1210, T1212, T1214, T1220s1, T1227s1, T1234, T1235, T1249v1, T1249v2, T1266, T1269, T1272s2, T1272s3, T1272s4, T1272s5, T1272s6, T1272s7, T1272s8, T1272s9, T1280, T1299

#### PoseBusters Monomer \(134 targets\)\.

5SAK, 5SB2, 5SD5, 5SIS, 6M2B, 6M73, 6T88, 6TW7, 6WTN, 6XBO, 6XCT, 6XG5, 6XM9, 6YMS, 6YQW, 6YRV, 6Z0R, 6Z14, 6Z1C, 6ZAE, 6ZCY, 6ZPB, 7A9E, 7AKL, 7BCP, 7BJJ, 7BTT, 7CL8, 7CTM, 7D5C, 7DKT, 7ELT, 7EPV, 7ES1, 7F51, 7F5D, 7F8T, 7JG0, 7JMV, 7JXX, 7L5F, 7LCU, 7LOE, 7LT0, 7LZD, 7MGT, 7MMH, 7MWU, 7MY1, 7N03, 7N4W, 7N6F, 7N7H, 7NF3, 7NGW, 7NP6, 7NPL, 7NR8, 7NU0, 7O1T, 7OEO, 7OFF, 7OMX, 7OSO, 7OZC, 7P4C, 7P5T, 7PGX, 7PK0, 7PL1, 7PRM, 7Q27, 7Q2B, 7Q5I, 7QF4, 7QGP, 7QPP, 7R59, 7R7R, 7RC3, 7RH3, 7RKW, 7ROU, 7RWS, 7SDD, 7TOM, 7TUO, 7U3J, 7UQ3, 7USH, 7UYB, 7V14, 7V3N, 7V3S, 7V43, 7VBU, 7VC5, 7W05, 7W06, 7WCF, 7WDT, 7WJB, 7WPW, 7X9K, 7XBV, 7XFA, 7XI7, 7XQZ, 7YZU, 7Z1Q, 7ZF0, 7ZHP, 7ZOC, 7ZU2, 7ZXV, 8A1H, 8A2D, 8AAU, 8AEM, 8AP0, 8BOM, 8BTI, 8CNH, 8D39, 8DHG, 8DP2, 8EX2, 8EXL, 8EYE, 8FLV, 8G6P, 8GFD, 8HFN, 8HO0

#### PoseBusters Complex \(169 targets \(164 of 169 in Table[1](https://arxiv.org/html/2610.07037#S3.T1)\)\)\.

6TW5, 6VTA, 6XHT, 6YJA, 6YQV, 6YR2, 6YSP, 6YT6, 6YYO, 6Z2C, 6Z4N, 6ZC3, 6ZK5, 7A1P, 7A9H, 7AFX, 7AN5,7B2C, 7B94, 7BKA, 7BMI, 7BNH, 7C0U, 7C3U, 7C8Q, 7CD9, 7CIJ,7CNQ, 7CNS, 7CUO, 7DQL, 7DUA, 7E4L, 7EBG, 7ECR, 7ED2, 7FB7, 7FHA, 7FT9, 7JHQ, 7JY3, 7K0V, 7KB1, 7KC5, 7KM8, 7KQU, 7KRU, 7KZ9, 7L00, 7L03, 7L7C, 7LEV, 7LJN, 7LMO, 7LOU, 7M3H, 7M6K, 7MFP, 7MGY, 7MOI, 7MSR, 7MWN, 7MYU, 7N4N, 7N7B, 7NF0, 7NFB, 7NLV, 7NSW, 7NUT, 7NXO, 7O0N, 7ODY, 7OFK, 7OLI, 7OPG, 7OZ9, 7P1F, 7P1M, 7P2I, 7PIH, 7PJQ, 7POM, 7PRI, 7PT3, 7PUV, 7Q25, 7QE4, 7QFM, 7QHG, 7QHL, 7QTA, 7R3D, 7R6J, 7R9N, 7RNI, 7ROR, 7RSV, 7RZL, 7SCW, 7SFO, 7SIU,7SUC, 7SZA, 7T0D, 7T1D, 7T3E, 7TB0, 7TBU, 7TE8, 7TH4, 7THI, 7TM6, 7TS6, 7TSF, 7TXK, 7TYP, 7U0U, 7UAS, 7UAW, 7UJ4, 7UJ5, 7UJF, 7ULC, 7UMW, 7UTW, 7UXS, 7UY4, 7VB8, 7VKZ, 7VQ9, 7VWF, 7VYJ, 7WKL, 7WL4, 7WQQ, 7WUX, 7WUY, 7WY1, 7X5N, 7XG5, 7XJN, 7XPO, 7XRL, 7Z2O, 7Z7F, 7ZCC, 7ZDY, 7ZL5, 7ZTL, 7ZZW, 8AIE,8AQL, 8AUH, 8AY3, 8B8H, 8C3N, 8C5M, 8CSD, 8D19, 8D5D, 8DKO, 8DSC, 8EAB, 8F8E, 8FAV, 8FO5,8G0V, 8SLG

#### Test \(542 targets \(541 of 542 in Table[1](https://arxiv.org/html/2610.07037#S3.T1)\)\)\.

7FCT, 7QA1, 7QNL, 7QPK, 7QPL, 7QQD, 7QRR, 7QSJ, 7R1N, 7R5I, 7S2J, 7SC4, 7SFY, 7SPQ, 7T4W, 7TL6, 7TPU, 7TZL, 7U08, 7U71, 7UBA, 7UDL, 7UG2, 7UI4, 7UI8, 7UK8, 7UNH, 7UOI, 7UQU, 7UR2, 7URP, 7UWV, 7UYX, 7V34, 7VS6, 7VXT, 7W91, 7WJU, 7WKP, 7WMY, 7WNH, 7WOI, 7WRN, 7WUY, 7WWC, 7X2X, 7X6G, 7XBJ, 7XCA, 7XCC, 7XF1, 7XF2, 7XFS, 7XFU, 7XGS, 7XGT, 7XGV, 7XH4, 7XHZ, 7XL7, 7XMJ, 7XMW, 7XNJ, 7XP0, 7XPC, 7XRX, 7XS3, 7XSG, 7XVI, 7XWY, 7XYO, 7XZ3, 7Y01, 7Y16, 7Y1S, 7Y3Z, 7Y43, 7Y4H, 7Y7O, 7Y8H, 7Y8S, 7Y95, 7Y9A, 7Y9G, 7YDQ, 7YDW, 7YFX, 7YH1, 7YH2, 7YH5, 7YIA, 7YK3, 7YKG, 7YKM, 7YL4, 7YLQ, 7YLZ, 7YN2, 7YNX, 7YOJ, 7YPD, 7YPQ, 7YPR, 7YR9, 7YSI, 7YTU, 7YUJ, 7YUK, 7YUN, 7YYG, 7Z3I, 7Z64, 7Z8E, 7ZAO, 7ZCA, 7ZCB, 7ZDS, 7ZFR, 7ZIE, 7ZLQ, 7ZPN, 7ZRZ, 8A0A, 8A16, 8A39, 8A50, 8A82, 8A90, 8A9N, 8AA9, 8AAS, 8ABV, 8ACA, 8ACC, 8ACR, 8ADC, 8ADD, 8AEP, 8AF9, 8AGG, 8AGR, 8AHD, 8AHZ, 8AMU, 8AN0, 8ANT, 8ANU, 8ARD, 8ARE, 8AU6, 8AXF, 8AXJ, 8AXT, 8AY2, 8AYR, 8AYV, 8B2E, 8B2H, 8B2S, 8B3E, 8B3W, 8B3Z, 8B45, 8B4C, 8B4D, 8B4E, 8B5W, 8B6A, 8B6P, 8B6Q, 8B6U, 8BAT, 8BBR, 8BBZ, 8BCK, 8BDQ, 8BF8, 8BFY, 8BH8, 8BH9, 8BID, 8BJ3, 8BL6, 8BNB, 8BRD, 8BRP, 8BRY, 8BUX, 8BVE, 8BVI, 8BVK, 8BW6, 8BY5, 8BZ2, 8BZ4, 8C26, 8C27, 8C4D, 8C9V, 8CB2, 8CBV, 8CEG, 8CH4, 8CHW, 8CHX, 8CID, 8CIH, 8CIL, 8CJG, 8CK0, 8CLU, 8CLZ, 8CMQ, 8CPK, 8CPN, 8CPQ, 8CQM, 8CQO, 8CR7, 8D0P, 8D40, 8D49, 8D4A, 8D4B, 8DAJ, 8DEO, 8DHJ, 8DQ6, 8DSP, 8DTQ, 8DVQ, 8DWJ, 8DX0, 8DZJ, 8E2B, 8E55, 8E5D, 8E5E, 8E7F, 8EB5, 8EBB, 8EC3, 8EEL, 8EFZ, 8EHN, 8EHT, 8EIL, 8EIN, 8ENR, 8EOV, 8ER5, 8ERC, 8ERM, 8ESL, 8EWG, 8EX9, 8EXA, 8EZW, 8F5D, 8F5G, 8F70, 8F72, 8F7N, 8F9Y, 8FBJ, 8FBN, 8FIT, 8FJD, 8FJE, 8FKL, 8FNR, 8FTV, 8FUQ, 8FW7, 8FX9, 8FXF, 8G0N, 8G2V, 8G32, 8G49, 8G53, 8G5S, 8G8E, 8GA6, 8GAQ, 8GDW, 8GH8, 8GI1, 8GKH, 8GKV, 8GLD, 8GMX, 8GMZ, 8GOY, 8GPP, 8GPS, 8GSR, 8GSX, 8GSY, 8GTY, 8GXR, 8GY0, 8GY4, 8GYG, 8GYR, 8H0L, 8H18, 8H1J, 8H3Z, 8H55, 8H5S, 8H7A, 8H8P, 8HAW, 8HBB, 8HBR, 8HDU, 8HDV, 8HE2, 8HF2, 8HFC, 8HFP, 8HGU, 8HHL, 8HHM, 8HI8, 8HIG, 8HIO, 8HIU, 8HL8, 8HLG, 8HM4, 8HML, 8HN0, 8HNO, 8HP8, 8HR5, 8HTX, 8HUZ, 8HVC, 8I3F, 8I3J, 8I3X, 8IA3, 8IBW, 8IBX, 8IBY, 8IBZ, 8IF8, 8IFK, 8IHA, 8IIB, 8IJT, 8IKS, 8IL8, 8ILL, 8IMD, 8IMS, 8IN6, 8IN8, 8IOI, 8IOM, 8IOO, 8IP1, 8IQ8, 8IQC, 8IR2, 8IRQ, 8ISZ, 8ITN, 8ITO, 8IUB, 8IVI, 8IVZ, 8IW5, 8IWP, 8J0Q, 8J12, 8J1J, 8J3R, 8J48, 8J69, 8J8H, 8J9G, 8JAH, 8JDG, 8JFR, 8JFU, 8JHR, 8JHU, 8JI2, 8JMR, 8JNM, 8JNX, 8JP0, 8JQE, 8JU7, 8JW3, 8JWS, 8JZX, 8K1F, 8K1I, 8K1R, 8K34, 8K3D, 8K3F, 8K4L, 8K5K, 8K5U, 8K76, 8K7X, 8K7Y, 8K86, 8K8A, 8KGZ, 8OEI, 8OJN, 8OK3, 8OKH, 8OLJ, 8OML, 8OQJ, 8OU1, 8OW8, 8OWF, 8OWI, 8OXU, 8OYY, 8P0C, 8P5Q, 8P6J, 8P6M, 8PAG, 8PAQ, 8PE3, 8PHB, 8PM4, 8PMU, 8POF, 8POG, 8PR6, 8PSN, 8PSO, 8PSQ, 8PSS, 8PSU, 8PT2, 8PT6, 8PT7, 8PTH, 8PTJ, 8PX4, 8PXC, 8PZH, 8Q1K, 8Q2A, 8Q3Z, 8Q40, 8Q41, 8Q42, 8Q43, 8Q44, 8Q66, 8Q70, 8QCW, 8QDO, 8QFL, 8QK8, 8QLD, 8R79, 8R7D, 8R8A, 8RB4, 8RH3, 8RXO, 8S0U, 8S9W, 8SDZ, 8SIU, 8SM6, 8SM9, 8SME, 8SMQ, 8SOT, 8SQU,8SRO, 8SSC, 8SSU, 8SUT, 8SVA, 8SW0, 8SXS, 8SXU, 8SY3, 8T0P, 8T1M, 8T3J, 8T4C, 8T4R, 8T65, 8T66, 8T8L, 8TA2, 8TFS, 8TGH, 8TJG, 8TKA, 8TN8, 8TNM, 8TP8, 8TPK, 8TTO, 8TVL, 8U0M, 8U1K, 8UA5, 8UK7, 8UPI, 8UQ9, 8URN, 8UZ8, 8V14, 8V1K, 8VRM, 8VXA, 8VXC, 8W7P, 8W8S, 8W9O, 8WBR, 8WCT, 8WIL, 8WO7, 8WU2, 8X51, 8X6V, 8X7X, 8XAT, 8XBI, 8XX9, 8Y4U, 8YA7

### E\.3Qualitative Results

![Refer to caption](https://arxiv.org/html/2610.07037v1/figures/qualitative/7n7h_panels_all.png)\(a\)7n7h
![Refer to caption](https://arxiv.org/html/2610.07037v1/figures/qualitative/7p5t_panels_all.png)\(b\)7p5t

Figure 10:All five sampled ligand poses per method, superposed on the crystal ligand \(green\) after alignment on the receptor\. Left: Boltz\-2 \(blue\); middle: our projection \(magenta\); right: Boltz\-2x \(orange\)\. Panel titles give the oracle \(best\-of\-5\) ligand RMSD and the standard deviationσ\\sigmaand range \(spread\) of the five per\-sample ligand RMSDs\. Our projection reproduces the base\-model ensemble almost exactly and keeps the same oracle pose, whereas steering collapses the ensemble onto a single tight but incorrect binding mode\.Fig\.[10](https://arxiv.org/html/2610.07037#A5.F10)illustrates the accuracy–validity trade\-off of our approach vs\. Boltz\-2x on two PoseBusters monomer targets: 7n7h and 7p5t\. On both, our projection leaves the base\-model ensemble essentially intact: the oracle RMSD is unchanged \(0\.310\.31Å on 7n7h,1\.371\.37Å on 7p5t\) while achieving physical validity\. Boltz\-2x behaves differently\. Its ensembles are often tighter \(σ=0\.05\\sigma=0\.05and0\.210\.21Å versus1\.581\.58and4\.894\.89Å\), which is the expected consequence of importance\-sampling over candidates, and the resulting consensus pose is sometimes wrong: the oracle RMSD degrades to3\.713\.71Å on 7n7h and to11\.1611\.16Å on 7p5t, where every steered sample places the ligand tail outside the crystallographic channel\. This suggests that Boltz\-2x steering may sometimes achieve physical validity at the expense of prediction quality, reweighting the sampler decisively towards a physically clean but biologically incorrect binding mode\.

## Appendix FTime and Memory

Table 4:Per\-target inference cost on five datasets\. Runtime is in seconds and peak GPU memory in GiB, both over five samples: median, interquartile range \(IQR, p25–p75\) and mean or maximum over targets\. Rel\. is the median over targets of the per\-target runtime ratio to the method’s base model \(Boltz\-2 for Boltz\-2x and Ours \(Boltz\); OpenFold\-3 for Ours \(OpenFold\)\)\. Runtimes are comparable within a model family, not across families\. Only targets completed by every method are included\.Runtime \(s\)Peak GPU memory \(GiB\)MethodnnMedianIQRMeanRel\.MedianIQRMaxCASP15Boltz\-26660\.322\.8–148193–5\.72\.8–10\.231\.8Ours \(Boltz\)6658\.023\.6–1461931\.005\.72\.8–10\.231\.8Boltz\-2x6612065\.1–2663021\.776\.03\.4–10\.759\.1OpenFold\-36628\.324\.0–43\.955\.5–7\.72\.8–16\.625\.1Ours \(OpenFold\)6628\.223\.9–47\.056\.01\.017\.72\.8–16\.625\.1CASP16Boltz\-25265\.426\.4–167201–5\.32\.6–10\.633\.6Ours \(Boltz\)5267\.427\.8–1652011\.005\.32\.6–10\.633\.6Boltz\-2x5212054\.4–2853181\.786\.42\.9–11\.453\.0OpenFold\-35232\.124\.8–49\.258\.1–9\.53\.5–12\.530\.7Ours \(OpenFold\)5231\.424\.1–51\.457\.60\.999\.53\.5–12\.530\.7PoseBusters MonomerBoltz\-213449\.437\.9–70\.967\.5–7\.45\.5–9\.414\.8Ours \(Boltz\)13450\.439\.1–72\.668\.21\.017\.45\.5–9\.414\.8Boltz\-2x13410679\.9–1471322\.057\.45\.5–9\.414\.8OpenFold\-313426\.124\.4–30\.230\.2–6\.44\.5–9\.423\.8Ours \(OpenFold\)13427\.124\.8–31\.030\.81\.026\.44\.5–9\.423\.8PoseBusters ComplexBoltz\-216417091\.5–345295–14\.89\.3–22\.144\.3Ours \(Boltz\)16417191\.6–3492961\.0114\.89\.3–22\.144\.3Boltz\-2x164294177–5744931\.7516\.59\.7–23\.878\.2OpenFold\-316449\.134\.6–85\.972\.4–13\.29\.9–18\.233\.1Ours \(OpenFold\)16447\.533\.4–84\.871\.01\.0013\.29\.9–18\.233\.1TestBoltz\-254165\.141\.7–191178–6\.43\.9–9\.836\.8Ours \(Boltz\)54167\.341\.6–1931771\.006\.43\.9–9\.836\.8Boltz\-2x54114472\.6–3412961\.777\.24\.2–12\.275\.3OpenFold\-354129\.823\.5–52\.948\.4–9\.24\.1–14\.724\.3Ours \(OpenFold\)54130\.824\.9–54\.549\.81\.039\.24\.1–14\.724\.3Table[4](https://arxiv.org/html/2610.07037#A6.T4)reports the inference cost of every method on NVIDIA A100\-SXM4\-80GB\. The projection adds almost nothing:Ours \(Boltz\)runs at 1\.001–1\.007×\\timesits time per target \(median ratio per dataset\), and its peak GPU memory is unchanged \(largest median per\-target difference 0\.00 GiB\), because it operates on coordinates the sampler already holds and adds no network evaluations\.Ours \(OpenFold\)behaves the same way relative to OpenFold\-3 \(0\.99–1\.03×\\timesits time, memory unchanged\)\. Inference\-time steering is far more expensive: Boltz\-2x takes roughly 1\.7–2\.1×\\timesthe time of Boltz\-2, needs up to 78 GiB where Boltz\-2 needs at most 44 GiB on the same targets, and runs out of memory on 9 targets that Boltz\-2 and Ours \(Boltz\) both complete\. The absolute runtimes of the two model families are not directly comparable, since the OpenFold timer covers only the predict step\.

## Appendix GStructural effects of the projection

The projection moves atoms, so a natural concern is that it resolves steric violations at the expense of other aspects of structural quality\. We test this in three ways: we compare in\-loop projection with post\-hoc correction \(§[G\.1](https://arxiv.org/html/2610.07037#A7.SS1)\), and we check covalent geometry \(§[G\.2](https://arxiv.org/html/2610.07037#A7.SS2)\), and atomic packing and non\-bonded energies \(§[G\.3](https://arxiv.org/html/2610.07037#A7.SS3)\)\. All three point to the same conclusion: the projection removes steric violations while leaving accuracy, covalent geometry, and local packing statistically indistinguishable from the base model\.

### G\.1In\-loop projection versus post\-hoc correction

We apply the projection inside the sampler rather than to the final structure\. To test whether this matters, we compare it with two post\-hoc alternatives applied to the output of the unmodified model\.

#### Post\-hoc projection\.

We apply the same operatorΠ𝒞⁡\(ℬproj\)\\Pi\_\{\\mathcal\{C\}\(\\mathcal\{B\}\_\{\\mathrm\{proj\}\}\)\}, with identical hyperparameters, to final Boltz\-2 structures, either once or iterated until no targeted violation remains \(40 passes\)\. A single pass removes only14%14\\%of the targeted violations and raises physical validity to0\.220\.22, compared with1\.001\.00for in\-loop projection \(Table[2](https://arxiv.org/html/2610.07037#S3.T2)\)\. This is expected, because each pass moves an atom by at mostδmax\\delta\_\{\\max\}\. Iterating to feasibility recovers validity \(0\.900\.90\) but significantly lowers LDDT \(0\.790\.79versus0\.810\.81, Holm\-corrected paired test\)\. Once sampling is over, the denoiser can no longer absorb the corrections, so they are imposed on the structure instead of being integrated into it, hence, affecting the structural quality\.

#### Post\-hoc geometry minimization\.

We also relax each predicted structure withphenix\.geometry\_minimization\([Liebschner et al\., 2019](https://arxiv.org/html/2610.07037#bib.bib5)\), which minimizes against a standard stereochemical restraint library covering bonds, angles, dihedrals, planarity, chirality and non\-bonded clash terms\. Phenix could not generate restraints for ligand\-bearing targets, so this comparison covers the 42 of 52 CASP16 targets without ligands\. Minimization improves validity from0\.5950\.595to0\.8810\.881but lowers accuracy on every per\-atom metric \(except DockQ\) \(Table[5](https://arxiv.org/html/2610.07037#A7.T5)\)\. Our projection instead reaches1\.0001\.000validity \(\+0\.405\+0\.405\[\+0\.262\+0\.262,\+0\.548\+0\.548\],p=1\.2×10−4p=1\.2\\times 10^\{\-4\}\) while remaining statistically indistinguishable from the base model on every structural metric \(all n\.s\.,\|Δ\|≤0\.003\|\\Delta\|\\leq 0\.003for LDDT, backbone LDDT and TM\-score\)\. Head\-to\-head, projection significantly outperforms minimization on LDDT \(\+0\.022\+0\.022,p=7\.4×10−5p=7\.4\\times 10^\{\-5\}\) and backbone LDDT \(\+0\.027\+0\.027,p=1\.6×10−6p=1\.6\\times 10^\{\-6\}\)\. The accuracy loss from minimization is also largest where the prediction is best\. Per\-target LDDT change correlates at−0\.57\-0\.57with baseline accuracy: it averages\+0\.001\+0\.001for models below0\.40\.4LDDT,−0\.020\-0\.020for0\.40\.4–0\.70\.7, and−0\.052\-0\.052above0\.70\.7\. Post\-hoc minimization therefore trades accuracy for validity, and it does so most on the models that are already accurate\. In\-loop projection avoids this trade\-off and can be applied on all types of biomolecules\.

Protocol:Each structure is minimized for three macro\-cycles with the built\-in monomer library\. We record the RMSD between the input and minimized structure and verified that minimization did not move atoms by more than a size\-dependent RMSD threshold\. The threshold \(RMSDthresh\\mathrm\{RMSD\_\{thresh\}\}\) scales with the size of the query protein\([Carugo and Pongor, 2001](https://arxiv.org/html/2610.07037#bib.bib41)\), defined as:

RMSDthresh=\(1\+ln⁡N100\)​RMSD100\\displaystyle\\mathrm\{RMSD\_\{thresh\}\}=\\left\(1\+\\ln\\sqrt\{\\frac\{N\}\{100\}\}\\right\)\\mathrm\{RMSD\}\_\{100\}\(2\)
whereNNis the size of the protein and RMSD100is a selected reference threshold for a protein of length 100 residues, selected here to be 1\.0Å\. We found that none of the structures exceeded the corresponding threshold\. All comparisons use the oracle model \(best of five\) and paired Wilcoxon signed\-rank tests with Holm correction\.

Table 5:In\-loop projection versus post\-hoc Phenix minimization on the 42 CASP16 targets for which Phenix can build restraints \(ligand\-free\)\. Oracle \(best\-of\-five\) model per target\. Arrows indicate the direction of improvement; DockQ\>0\.23is computed over the 17 multi\-chain targets\.

### G\.2Covalent geometry

The van der Waals projection moves atoms without reference to their bonds, so it could in principle stretch bonds or distort angles\. We assess covalent geometry of the protein chains with MolProbity[Davis et al\. \(2007\)](https://arxiv.org/html/2610.07037#bib.bib42)\(Fig\.[11](https://arxiv.org/html/2610.07037#A7.F11), Table[6](https://arxiv.org/html/2610.07037#A7.T6)\)\. Bonded geometry is essentially unchanged by the projection\. The median bond RMSD from ideal is0\.00620\.0062Å with projection and0\.00600\.0060Å for Boltz\-2, against0\.00500\.0050Å for the experimental structures, i\.e\.∼0\.4%\{\\sim\}0\.4\\%of a1\.521\.52Å C–C bond\. The median angle RMSD is0\.955∘0\.955^\{\\circ\}versus0\.940∘0\.940^\{\\circ\}\(experimental1\.051∘1\.051^\{\\circ\}\), or∼0\.85%\{\\sim\}0\.85\\%of a tetrahedral angle\. The worst\-case angle and dihedral deviations are slightly lower with projection \(median7\.32∘7\.32^\{\\circ\}vs\.8\.13∘8\.13^\{\\circ\}and78\.4∘78\.4^\{\\circ\}vs\.80\.8∘80\.8^\{\\circ\}\)\. The one metric that moves upward is the worst\-case bond deviation \(median0\.1140\.114vs\.0\.0700\.070Å\); its upper tail is lower with projection \(p950\.5680\.568vs\.0\.6250\.625Å\)\. Both predicted models are closer to ideal than the deposited structures on angle RMSD \(0\.955∘0\.955^\{\\circ\}and0\.940∘0\.940^\{\\circ\}versus1\.051∘1\.051^\{\\circ\}\), while their bond RMSD is marginally higher \(0\.0062Å and 0\.0060Å versus 0\.0050Å\)\. The angle result likely reflects the bias of cofolding models toward the idealized geometry learned during training\. The overall MolProbity score is essentially unchanged \(median 1\.780 with projection versus 1\.744 for Boltz\-2, against 1\.550 for the experimental structures\)\.

#### Protocol:

We runphenix\.molprobity\([Liebschner et al\. \(2019\)](https://arxiv.org/html/2610.07037#bib.bib5), Phenix 2\.1\-6048\) on the CASP15 and CASP16 protein chains\. Structures are first prepared withphenix\.pdbtools, removing waters, ligands and ions and keeping only the primary alternate conformation \(selectionnot water and \(altloc " " or altloc A\) and protein\)\. For large complexes, wherepdbtoolswrites mmCIF, the mmCIF file is passed to MolProbity directly\. Hydrogens are added withreduce, which optimizes the hydrogen\-bond network and flips Asn/Gln/His side chains where needed, so that all\-atom contacts are computed on fully protonated models\. Bond and angle deviations are evaluated against the standard monomer \(geostd\) library\. Each target contributes the mean over its five predicted models; the experimental structure contributes once\. Because ligands are removed, ligand\-internal geometry is assessed only by the validity criterion\.

Figure 11:MolProbity covalent\-geometry metrics on CASP15 and CASP16 protein chains\.Kernel density estimates across targets for the experimental structures \(blue\), Boltz\-2 \(orange\) and our projection \(green\): MolProbity score, bond and angle RMSD, and maximum bond, angle and dihedral deviation\. Dotted lines mark the medians\. The angle maximum deviation is shown on a logarithmic axis\.Table 6:Covalent geometry and overall quality of CASP protein chains \(n=46n=46\)\. Each target contributes its mean over the five predicted models; the table reports the distribution across targets as median \[Q1, Q3\]\. Lower is better throughout \(↓\\downarrow\)\.

### G\.3Packing and non\-bonded energetics

Finally, we check that the projection does not disturb the local environment of each residue\. We use two independent measures on the CASP16 systems: Lennard\-Jones energies under a molecular\-mechanics force field, and contact counts in the alpha complex\.

#### Lennard\-Jones energy\.

Before relaxation, the per\-residue Lennard\-Jones energies of the two methods are similar on most targets, with a lower median under projection \(224\.5224\.5vs\.249\.5249\.5kJ/mol per residue; Table[7\(a\)](https://arxiv.org/html/2610.07037#A7.T7.st1)\)\. The distribution is heavy\-tailed\. On a few targets a single severe overlap dominates the Lennard\-Jones term, and these are exactly the targets where the projection helps most: on T1272s5, for example, the energy drops from∼6\.4×106\{\\sim\}6\.4\\times 10^\{6\}to∼1\.1×104\{\\sim\}1\.1\\times 10^\{4\}kJ/mol per residue\. This is why the means differ by more than two orders of magnitude\. After an identical short relaxation, the two methods reach the same energy \(median145\.1145\.1vs\.145\.5145\.5kJ/mol per residue\), agreeing to within a fraction of a kJ/mol on every target\. The projection therefore removes the severe contacts without moving the structure away from the same local energy minimum that the base model reaches\.

#### Alpha\-complex packing\.

The number of non\-covalent neighbours per residue in the solvent\-accessible alpha complex is essentially unchanged \(Table[7\(b\)](https://arxiv.org/html/2610.07037#A7.T7.st2)\): the median per\-target change is\+0\.003\+0\.003neighbours per residue \(≤0\.01%\\leq 0\.01\\%\), and per\-target differences are distributed symmetrically around zero\. The largest changes occur on RNA and large multimeric targets, and even these stay below2%2\\%and come with no degradation in covalent geometry or energy\. The projection removes clashes without changing the contact topology of the structure\.

#### Protocol\.

Lennard\-Jones energies are computed with OpenMM\([Eastman et al\., 2023](https://arxiv.org/html/2610.07037#bib.bib4)\)and the AMBER ff14SB force field\. We evaluate each raw prediction \(pre\-relaxation\) and the same structure after an identical minimization of 100 steps with ff14SB and GBn2 implicit solvent \(post\-relaxation\)\. The relaxation removes the artefactual energy of a few severe initial contacts and evaluates the structure in a nearby low\-energy state\. The alpha complex is built with GUDHI\([The GUDHI Project,](https://arxiv.org/html/2610.07037#bib.bib3)\), which uses the CGAL alpha\-shape implementation, on atoms with van der Waals radii inflated by a1\.41\.4Å solvent probe, atα=0\\alpha=0\. Two atoms are neighbours when their inflated spheres overlap without occlusion; covalently bonded pairs are excluded\.

Table 7:Packing on CASP16 systems for Boltz\-2 and our projection\.↓\\downarrow: lower is better\.\(a\)Lennard\-Jones energy per residue \(kJ/mol\) before and after an identical short relaxation \(↓\\downarrow\)\.
\(b\)Non\-covalent neighbour counts in the solvent\-accessible alpha complex \(α=0\\alpha=0\), reported as median \[Q1, Q3\] \(max\)\. Neither direction is better: the test is whether the projection leaves the counts unchanged relative to the base model\.

## Appendix HHyperparameter Sensitivity

The sweeps in Tables[2](https://arxiv.org/html/2610.07037#S3.T2)and[8](https://arxiv.org/html/2610.07037#A8.T8)use a pooled set of134134targets drawn from CASP15, CASP16, PoseBusters Monomer and PoseBusters Complex: every target on which the unmodified Boltz\-2 sampler produced at least one physically invalid sample\. One setting is varied at a time; all others stay at the final configuration\.δmax\\delta\_\{\\max\}\(projection\_max\_atom\_disp, Å\) caps the net per\-atom displacement;α\\alpha\(projection\_alpha\) scales the per\-pair correction;β\\beta\(vdw\_projection\_buffer\) sets the vdW lower bound\(1−β\)​\(ri\+rj\)\(1\-\\beta\)\(r\_\{i\}\+r\_\{j\}\);KK\(projection\_max\_pairs\) caps the violating pairs corrected per step;tstartt\_\{\\mathrm\{start\}\}\(projection\_start\_t\) opens a projection window that runs to the end of sampling\.

Because the set is enriched for hard, clash\-heavy structures, the accuracy columns are lower than the benchmark averages and are best read as relative comparisons between rows, not as absolute accuracy\. Full benchmark\-level comparison is available in main text Table[1](https://arxiv.org/html/2610.07037#S3.T1)\.

Table 8:Sensitivity of the projection to its hyperparameters on the sensitivity set \(CASP15, CASP16, PoseBusters Monomer and Complex\)\. Values are mean±\\pmnon\-parametric bootstrap standard error over targets \(10,000 resamples\), oracle over 5 samples, computed as in Table[1](https://arxiv.org/html/2610.07037#S3.T1)\. Higher is better for all metrics\. Dashed, shaded rows are the final configuration used in the main experiments\. Red cells are worse than the final configuration under a paired test \(Wilcoxon; exact McNemar for physical validity\) surviving Holm correction over the table\.