ASPIRE: Saddle-Point Discovery through Set Prediction and Physical Refinement

# ASPIRE: Saddle-Point Discovery through Set Prediction and Physical Refinement
Source: [https://arxiv.org/html/2610.10969](https://arxiv.org/html/2610.10969)
Quanyou ZhangAffiliation:University of Tennessee, KnoxvilleShaoxiang QinAffiliation:University of Tennessee, KnoxvilleAffiliation:McGill UniversityHaixuan XuAffiliation:University of Tennessee, KnoxvilleXiongye XiaoAffiliation:University of Tennessee, Knoxville

###### Abstract

Predicting thermally activated diffusion and defect evolution with event\-driven models requires identifying atomic rearrangement mechanisms and their activation barriers\. Discovering the associated saddle points is a major computational bottleneck: multiple rearrangements may originate from one metastable state, while costly local searches can fail or repeatedly converge to the same saddle\. To address this challenge, we introduce*ASPIRE*\(AtomisticSaddle\-PointInference withRefinement forEvents\), a framework that predicts a set of saddle candidates from a single initial atomic environment and refines them through Dimer searches on the original interatomic potential\. The framework’s equivariant set predictor, Ev\-Quiformer, integrates \(i\) geometry\-conditioned scalar/vector event slots for generating multiple saddle\-point proposals and \(ii\) a decoder that maps each slot to a full atomic displacement field by combining atom, slot, and anchor\-relative vectors with invariant coefficients\. We also contribute two datasets: \(i\)BccFe\-4vacAV\-4000, comprising 4,000 four\-vacancy body\-centered cubic iron configurations and 65,450 reference events grouped by initial state for set supervision and post\-refinement evaluation; and \(ii\)BccFe\-1to4Vac, comprising 5,372 configurations with one to four vacancies each\. Theoretically, we establish conditions for proposal equivariance\. Experimentally, ASPIRE achieves 77\.20% reference\-event coverage on this benchmark, compared with 75\.73% for a conventional Dimer baseline, while requiring approximately half as many Dimer force evaluations\. In a timing evaluation on 50 configurations, ASPIRE reduces wall time per configuration from 478\.8 s to 176\.3 s under the stated hardware settings\. The dataset is available at[https://huggingface\.co/datasets/cheesewawa98/BccFe4VacAV\-4000](https://huggingface.co/datasets/cheesewawa98/BccFe4VacAV-4000)\.

11footnotetext:Corresponding author: xxiao9@utk\.edu## 1Introduction

![Refer to caption](https://arxiv.org/html/2610.10969v1/ASPIRE.png)Figure 1:ASPIRE: Atomistic Saddle\-Point Inference with Refinement for Events\.\(a\) Overall workflow\.Ev\-Quiformer generatesMMsaddle\-point proposals from a single initial configuration, followed by physical refinement using Dimer searches on the original interatomic potential\. Panels \(b–d\) detail its three neural modules\.\(b\) Equivariant encoding\.A six\-layer Equiformer extracts invariant scalar and equivariant vector features for each atom\.\(c\) Geometry\-conditioned event slots\.Scalar self\-attention and repeated slot–atom attention update slots initialized from learned scalar seeds and pooled atom vectors\.\(d\) Multi\-source displacement decoding\.Invariant coefficients combine atom, slot, and anchor\-relative vectors into a full atomic displacement field through a gated term and an ungated residual\. Adding this field to the initial configuration yields a candidate for subsequent physical refinement\.Nuclear fission reactors and future fusion systems require structural materials that retain their mechanical integrity and dimensional stability under irradiation and elevated temperatures\([Du et al\., 2023](https://arxiv.org/html/2610.10969#bib.bib10)\)\. In iron\-based alloys, irradiation\-induced clusters and dislocation loops contribute to hardening\([Shimodaira et al\., 2018](https://arxiv.org/html/2610.10969#bib.bib33)\), while the growth of cavities causes swelling\([Lin et al\., 2021](https://arxiv.org/html/2610.10969#bib.bib24)\)\. Understanding these changes requires resolving the individual and collective atomic rearrangements that drive defect migration, recombination, and interactions\. For example, two interstitial loops in irradiated iron can interact to form a new loop structure, a mechanism revealed by atomistic simulation\([Xu et al\., 2013](https://arxiv.org/html/2610.10969#bib.bib39)\)\. Vacancy diffusion also enables dislocation climb during creep, with migration barriers influenced by the local dislocation environment\([Kabir et al\., 2010](https://arxiv.org/html/2610.10969#bib.bib20)\)\.

Molecular dynamics \(MD\) follows atomic motion directly, but sampling rare thermally activated events requires resolving fast atomic vibrations over long waiting times\. Event\-based methods reach longer time scales by identifying the possible rearrangements and using their rates to select events and advance time\([Voter, 2007](https://arxiv.org/html/2610.10969#bib.bib37)\)\. On\-the\-fly methods, including adaptive KMC\([Henkelman and Jónsson, 2001](https://arxiv.org/html/2610.10969#bib.bib18)\), SEAKMC\([Xu et al\., 2012](https://arxiv.org/html/2610.10969#bib.bib38);[Xu et al\., 2015](https://arxiv.org/html/2610.10969#bib.bib40)\), and k\-ART\([Béland et al\., 2011](https://arxiv.org/html/2610.10969#bib.bib5)\), search for these events as the simulation proceeds\. In event\-based simulations, saddle discovery identifies transition mechanisms and activation barriers used to estimate event rates\. Saddle discovery in this setting has several challenges: \(i\) an initial configuration generally admits multiple possible events whose number and product configurations are unknown in advance; \(ii\) the recovered event catalogue affects event selection and estimated waiting times; \(iii\) and new configurations encountered along a trajectory may require renewed searches, making event discovery computationally costly\([Hayakawa and Xu, 2021](https://arxiv.org/html/2610.10969#bib.bib42);[Liang and Xu, 2023](https://arxiv.org/html/2610.10969#bib.bib43)\)\.

Existing learning\-based saddle \(transition\-state\) discovery methods differ in their required inputs and target applications\. React\-OT predicts transition states from reactant and product configurations\([Duan et al\., 2025](https://arxiv.org/html/2610.10969#bib.bib12)\), and OA\-ReactDiff is primarily evaluated in the same double\-ended setting\([Duan et al\., 2023](https://arxiv.org/html/2610.10969#bib.bib11)\)\. Conditioning on both endpoints does not directly address on\-the\-fly saddle discovery when products are unknown\. ReactionAtlas instead proposes transition states from reactants and uses kinetic modeling to expand reaction networks\([Gugler et al\., 2026](https://arxiv.org/html/2610.10969#bib.bib15)\)\.

Our focus is efficient saddle\-point prediction for multi\-vacancy configurations in body\-centered cubic \(bcc\) iron \(Fe\), a model system for studying defect evolution in iron\-based materials\. We use a fixed embedded\-atom method \(EAM\) potential for both reference\-event generation and candidate refinement\. We aim to identify multiple distinct events from a single initial configuration, without supplied product states, while reducing physical search cost\. A propose\-refine paradigm uses learned models to generate candidates for numerical refinement, aiming to shorten the search while retaining the underlying physical model\. Variants of this paradigm have been used in a range of scientific applications, including electronic\-structure calculations\([Schütt et al\., 2019](https://arxiv.org/html/2610.10969#bib.bib32)\), protein structure prediction\([Jumper et al\., 2021](https://arxiv.org/html/2610.10969#bib.bib19)\), and materials discovery\([Merchant et al\., 2023](https://arxiv.org/html/2610.10969#bib.bib30)\)\.

Our contributions\.We formulate saddle discovery in interacting defect systems as a set\-learning problem, where multiple candidate rearrangements are jointly predicted from a single initial configuration without supplied product states\. We introduce*ASPIRE*\(Atomistic Saddle\-Point Inference with Refinement for Events; Figure[1](https://arxiv.org/html/2610.10969#S1.F1)\), which combines geometric prediction with physical refinement, and establish dedicated datasets and a benchmark for evaluating distinct\-event recovery and physical search cost\. Our contributions are threefold:

\(i\) Problem formulation and datasets\.We make multiple rearrangements from the same initial state a joint learning target, without requiring product configurations\. Our bcc Fe benchmark groups 65,450 reference events by 4,000 four\-vacancy configurations, enabling set supervision and evaluation of distinct event recovery after refinement\.

\(ii\) Framework design and theory\.We propose ASPIRE, combining learned saddle proposals with physical refinement\. Its neural predictor, Ev\-Quiformer, integrates geometry\-conditioned scalar/vector event slots and a three\-source decoder to generate full atomic displacement fields, supervised by sequential matching\. We establish conditions for proposal equivariance and relate geometric error to reference\-event recovery under local\-refinement assumptions\.

\(iii\) Empirical evaluation\.We compare ASPIRE with conventional Dimer and event\-reuse baselines, alongside ablation studies\. On 200 four\-vacancy test configurations, ASPIRE achieves 77\.20% reference\-event coverage versus Dimer’s 75\.73%, with roughly half the recorded search\-force calls under the respective search protocols\.

## 2Related Work

#### Rare\-event simulation, saddle search, and reuse\.

Kinetic Monte Carlo \(KMC\) simulates transitions between metastable states by sampling events and waiting times from their rates \(see Appendix[A\.5](https://arxiv.org/html/2610.10969#A1.SS5)\)\([Voter, 2007](https://arxiv.org/html/2610.10969#bib.bib37)\)\. On\-the\-fly KMC discovers events through saddle searches during simulation, rather than requiring a predefined event catalogue\. SEAKMC localizes these searches to active volumes around defects\([Xu et al\., 2012](https://arxiv.org/html/2610.10969#bib.bib38);[Xu et al\., 2015](https://arxiv.org/html/2610.10969#bib.bib40)\)\. The Dimer method\([Henkelman and Jónsson, 1999](https://arxiv.org/html/2610.10969#bib.bib16)\)finds first\-order saddle points without a supplied product configuration \(see Appendix[A\.4](https://arxiv.org/html/2610.10969#A1.SS4)\)\. The kinetic database \(KDB\) matches local atomic geometries to stored mechanisms and supplies saddle\-search initializations\([Terrell et al\., 2012](https://arxiv.org/html/2610.10969#bib.bib34)\)\. k\-ART combines on\-the\-fly discovery with an event catalogue indexed by local topology for event reuse\([Béland et al\., 2011](https://arxiv.org/html/2610.10969#bib.bib5)\)\.

#### Equivariant atomistic models\.

Tensor Field Networks, SE\(3\)\-Transformers, EGNN, and Equiformer encode geometric transformation rules\([Thomas et al\., 2018](https://arxiv.org/html/2610.10969#bib.bib35);[Fuchs et al\., 2020](https://arxiv.org/html/2610.10969#bib.bib14);[Satorras et al\., 2021](https://arxiv.org/html/2610.10969#bib.bib31);[Liao and Smidt, 2022](https://arxiv.org/html/2610.10969#bib.bib23)\)\. NequIP and MACE learn interatomic energies and forces\([Batzner et al\., 2022](https://arxiv.org/html/2610.10969#bib.bib3);[Batatia et al\., 2022](https://arxiv.org/html/2610.10969#bib.bib2)\)\.

#### Set prediction\.

Set Transformer uses learnable seeds for attention\-based pooling and self\-attention among pooled outputs\([Lee et al\., 2019](https://arxiv.org/html/2610.10969#bib.bib22)\)\. DETR combines learned queries, query self\-attention, cross\-attention, and bipartite matching for unordered object prediction\([Carion et al\., 2020](https://arxiv.org/html/2610.10969#bib.bib7)\)\. Slot Attention iteratively refines slots\([Locatello et al\., 2020](https://arxiv.org/html/2610.10969#bib.bib26)\)\. Geometric query designs include DAB\-DETR’s dynamic anchor boxes\([Liu et al\., 2022](https://arxiv.org/html/2610.10969#bib.bib25)\)and Invariant Slot Attention’s attention\-derived slot\-centric reference frames and relative\-coordinate decoding\([Biza et al\., 2023](https://arxiv.org/html/2610.10969#bib.bib6)\)\. For one\-to\-many supervision, Group DETR performs matching within query groups and MS\-DETR supervises multiple queries in the primary decoder\([Chen et al\., 2023](https://arxiv.org/html/2610.10969#bib.bib8);[Zhao et al\., 2024](https://arxiv.org/html/2610.10969#bib.bib41)\)\. We adapt these set\-prediction designs with equivariant atomistic representations to build Ev\-Quiformer for jointly predicting an unordered set of candidate saddle configurations from a single initial configuration\.

## 3Problem Formulation

Given an initial metastable configuration, our goal is to predict a set of saddle candidates for subsequent physical refinement, without requiring product states\. LetX∈ℝN×3X\\in\\mathbb\{R\}^\{N\\times 3\}be an initial configuration at a*metastable state*, a nondegenerate local minimum of the fixed interatomic potentialV:ℝN×3→ℝV:\\mathbb\{R\}^\{N\\times 3\}\\to\\mathbb\{R\}, with∇V​\(X\)=0\\nabla V\(X\)=0and∇2V​\(X\)≻0\\nabla^\{2\}V\(X\)\\succ 0\. Here∇V\\nabla Vand∇2V\\nabla^\{2\}Vare taken in the fixed free\-coordinate space, with the surrounding environment and constraints already imposed \(Appendix[A\.3](https://arxiv.org/html/2610.10969#A1.SS3)\)\. We consider the*open saddle configuration set prediction problem*, in which the number, identities, and outcomes of accessible events are unknown a priori\.

A predictor outputs an unordered collection of displacement fields, denoted by

Fθ​\(X\)=\{D^m\}m=1M,D^m∈ℝN×3\.F\_\{\\theta\}\(X\)=\\\{\\widehat\{D\}\_\{m\}\\\}\_\{m=1\}^\{M\},\\qquad\\widehat\{D\}\_\{m\}\\in\\mathbb\{R\}^\{N\\times 3\}\.\(1\)The corresponding proposal configurations are\{S^m\}m=1M=\{X\+D^m\}m=1M\\\{\\widehat\{S\}\_\{m\}\\\}\_\{m=1\}^\{M\}=\\\{X\+\\widehat\{D\}\_\{m\}\\\}\_\{m=1\}^\{M\}\. After refinement, the set of retained saddle configurations is defined as

\{Sk\}k=1K=\{ℛX​\(S^m\)\}m=1M∖\{⊥\},K≤M,\\\{S\_\{k\}\\\}\_\{k=1\}^\{K\}=\\\{\\mathcal\{R\}\_\{X\}\(\\widehat\{S\}\_\{m\}\)\\\}\_\{m=1\}^\{M\}\\setminus\\\{\\bot\\\},\\qquad K\\leq M,\(2\)where eachSkS\_\{k\}is a configuration at a*transition state*satisfying: \(i\)SkS\_\{k\}is a first\-order saddle onVV, that is,∇V​\(Sk\)=0\\nabla V\(S\_\{k\}\)=0andλ1​\(∇2V​\(Sk\)\)<0<λ2​\(∇2V​\(Sk\)\)\\lambda\_\{1\}\\\!\\bigl\(\\nabla^\{2\}V\(S\_\{k\}\)\\bigr\)<0<\\lambda\_\{2\}\\\!\\bigl\(\\nabla^\{2\}V\(S\_\{k\}\)\\bigr\), whereλ1≤λ2≤⋯\\lambda\_\{1\}\\leq\\lambda\_\{2\}\\leq\\cdotsare the Hessian eigenvalues and the unit eigenvectorv1​\(Sk\)v\_\{1\}\(S\_\{k\}\)associated withλ1\\lambda\_\{1\}is its unstable mode; \(ii\) relaxing fromSk±ϵ​v1​\(Sk\)S\_\{k\}\\pm\\epsilon v\_\{1\}\(S\_\{k\}\)reachesXXand another distinct product configurationXk\+X\_\{k\}^\{\+\}at a metastable local minimum\. The mapℛX:ℝN×3→ℝN×3∪\{⊥\}\\mathcal\{R\}\_\{X\}:\\mathbb\{R\}^\{N\\times 3\}\\to\\mathbb\{R\}^\{N\\times 3\}\\cup\\\{\\bot\\\}is an idealized search\-and\-validation map: it refines each proposed configuration onVVand returnsSkS\_\{k\}only when the target saddle and connectivity conditions above hold, or⊥\\bototherwise\.

We define the resulting event set as

ℰX=\{ek\}k=1K=\{\(X,Sk,Xk\+,Δ​Ek\)\}k=1K,Δ​Ek=V⁡\(Sk\)−V⁡\(X\)\>0\.\\mathcal\{E\}\_\{X\}=\\\{e\_\{k\}\\\}\_\{k=1\}^\{K\}=\\bigl\\\{\(X,S\_\{k\},X\_\{k\}^\{\+\},\\Delta E\_\{k\}\)\\bigr\\\}\_\{k=1\}^\{K\},\\qquad\\Delta E\_\{k\}=V\(S\_\{k\}\)\-V\(X\)\>0\.\(3\)The resulting event set is determined byXXand the validated saddle configurations\. The objective is to recover as many distinct events as possible at low search cost, without using any product configurationXk\+X\_\{k\}^\{\+\}as input\. Throughout this paper, we use Dimer\-based search and validation as the numerical instantiation of this framework \(see Appendix[A\.4](https://arxiv.org/html/2610.10969#A1.SS4)\)\.

## 4Methods

#### Intuition

Several atomic rearrangements may originate from the same metastable configuration, each associated with a saddle\. We therefore predict a set of approximate saddle geometries and use them to initialize Dimer refinement on the original interatomic potential\. Learning proposes alternative rearrangements, while force\-based refinement seeks nearby saddle points\. Proposition[1](https://arxiv.org/html/2610.10969#Thmproposition1)establishes local convergence of the ideal minimum\-mode iteration from a sufficiently close initialization\([Luo et al\., 2022](https://arxiv.org/html/2610.10969#bib.bib27)\)\.

Based on this intuition, we develop*ASPIRE*\(Figure[1](https://arxiv.org/html/2610.10969#S1.F1)\), integrating set prediction with physical refinement\. We first describe its neural predictor,*Ev\-Quiformer*, through equivariant environment encoding \(Section[4\.1](https://arxiv.org/html/2610.10969#S4.SS1)\), geometry\-conditioned event slots \(Section[4\.2](https://arxiv.org/html/2610.10969#S4.SS2)\), and three\-source displacement decoding \(Section[4\.3](https://arxiv.org/html/2610.10969#S4.SS3)\)\. We then present the set\-matching training objective \(Section[4\.4](https://arxiv.org/html/2610.10969#S4.SS4)\) and the optional confidence readout for selecting proposals for Dimer refinement \(Section[4\.5](https://arxiv.org/html/2610.10969#S4.SS5)\)\. The conditions and proof for proposal equivariance are given in Appendix[J\.2](https://arxiv.org/html/2610.10969#A10.SS2)\.

### 4\.1Equivariant attention environment encoding

The encoder reads the initial configurationXX\. A directed radius graph connects atomsa,ba,bwithin a maximum radius distance, with a limit to the number of neighbors per atom and edge vectorsXa−XbX\_\{a\}\-X\_\{b\}, whereXa∈ℝ3X\_\{a\}\\in\\mathbb\{R\}^\{3\}is theaa\-th row ofXX\. Six Equiformer blocks\([Liao and Smidt, 2022](https://arxiv.org/html/2610.10969#bib.bib23)\)attend over these neighbors and return, for every atom, an invariant scalar featureua∈ℝ128u\_\{a\}\\in\\mathbb\{R\}^\{128\}and an equivariant vector featureVa∈ℝ64×3V\_\{a\}\\in\\mathbb\{R\}^\{64\\times 3\}that rotates with the input\. No product configurationsXk\+X\_\{k\}^\{\+\}are supplied\. Edge features and layer sizes are given in Appendix[C\.1](https://arxiv.org/html/2610.10969#A3.SS1)\.

### 4\.2Geometry\-conditioned event slots

#### Slots\.

A slotmmcarries a scalar fingerprint initialized from a learned seedsm∈ℝdss\_\{m\}\\in\\mathbb\{R\}^\{d\_\{s\}\}and an equivariant vector stateQm∈ℝdv×3Q\_\{m\}\\in\\mathbb\{R\}^\{d\_\{v\}\\times 3\}constructed from the encoded atomic geometry\. Learned scalar seeds distinguish hypotheses, while geometry\-pooled vector states retain directional information without introducing fixed learned spatial axes\.

#### Stages\.

The slot module proceeds through initialization, vector–scalar updates, scalar\-only updates, and final readout, using Equations \([4](https://arxiv.org/html/2610.10969#S4.E4)\)–\([8](https://arxiv.org/html/2610.10969#S4.E8)\) with the stage types summarized in Table[1](https://arxiv.org/html/2610.10969#S4.T1)\. At stagett, lets^m\\hat\{s\}\_\{m\}andu^a\\hat\{u\}\_\{a\}denote the corresponding normalized slot and atomic scalar features, and letH\(t\)H^\{\(t\)\}be the number of attention heads with head widthdhd\_\{h\}\. The exact stage schedule, normalizations, initialization, dimensions, and parameter\-sharing choices are given in Appendix[C\.2](https://arxiv.org/html/2610.10969#A3.SS2)\.

#### Slot self attention\.

Fingerprints optionally self\-attend across slots, in place,

sm\(t\)←\(1−η\(t\)\)​sm\(t\)\+η\(t\)​SA\(t\)​\(s1\(t\),…,sM\(t\)\)m,η\(t\)∈\{0,1\}s\_\{m\}^\{\(t\)\}\\leftarrow\(1\-\\eta^\{\(t\)\}\)s\_\{m\}^\{\(t\)\}\+\\eta^\{\(t\)\}\\,\\operatorname\{SA\}^\{\(t\)\}\\\!\\bigl\(s\_\{1\}^\{\(t\)\},\\dots,s\_\{M\}^\{\(t\)\}\\bigr\)\_\{m\},\\qquad\\eta^\{\(t\)\}\\in\\\{0,1\\\}\(4\)

#### Slot–atom attention\.

At stagett, attention headhhuses scalar similarity and, when enabled, vector alignment to compute the slot–atom attention weights:

Am​a\(t,h\)=softmaxa⁡\[\(Wq\(t,h\)​s^m\)⊤​\(Wk\(t,h\)​u^a\)dh\+γ\(t,h\)​⟨W~Q\(t,h\)​Qm\(t\),W~V\(t,h\)​Va⟩F3​dv\]\.A^\{\(t,h\)\}\_\{ma\}=\\operatorname\{softmax\}\_\{a\}\\\!\\left\[\\frac\{\(W\_\{q\}^\{\(t,h\)\}\\hat\{s\}\_\{m\}\)^\{\\top\}\(W\_\{k\}^\{\(t,h\)\}\\hat\{u\}\_\{a\}\)\}\{\\sqrt\{d\_\{h\}\}\}\+\\gamma^\{\(t,h\)\}\\frac\{\\left\\langle\\widetilde\{W\}\_\{Q\}^\{\(t,h\)\}Q\_\{m\}^\{\(t\)\},\\widetilde\{W\}\_\{V\}^\{\(t,h\)\}V\_\{a\}\\right\\rangle\_\{F\}\}\{\\sqrt\{3d\_\{v\}\}\}\\right\]\.\(5\)
Each head pools a scalar and a vector context; scalar contexts are concatenated over heads,

zm\(t,h\)=∑aA\(t,h\)m​aWz\(t,h\)u^a,zm\(t\)=\[zm\(t,1\);…;zm\(t,H\(t\)\)\]∈ℝds,Um\(t,h\)=∑aAm​a\(t,h\)​WV\(t,h\)​Va∈ℝdv×3\.\\begin\{gathered\}z\_\{m\}^\{\(t,h\)\}=\\sum\_\{a\}A^\{\(t,h\)\}\_\{ma\}\\,W\_\{z\}^\{\(t,h\)\}\\hat\{u\}\_\{a\},\\qquad z\_\{m\}^\{\(t\)\}=\\bigl\[z\_\{m\}^\{\(t,1\)\};\\dots;z\_\{m\}^\{\(t,H^\{\(t\)\}\)\}\\bigr\]\\in\\mathbb\{R\}^\{d\_\{s\}\},\\\\ U\_\{m\}^\{\(t,h\)\}=\\sum\_\{a\}A^\{\(t,h\)\}\_\{ma\}\\,W\_\{V\}^\{\(t,h\)\}V\_\{a\}\\in\\mathbb\{R\}^\{d\_\{v\}\\times 3\}\.\\end\{gathered\}\(6\)
The scalar maps act on invariant features, while vector maps mix channels identically across spatial components\. Their exact parameterization and sharing pattern are specified in Appendix[C\.2](https://arxiv.org/html/2610.10969#A3.SS2)\.

#### Update\.

Using the pooled scalar and vector contexts, we update the slot states according to

sm\(t\+1\)=sm\(t\)\+F\(t\)​\(sm\(t\),zm\(t\),\(‖Um,v\(t,h\)‖2\)h,v\),Qm\(t\+1\)=Qm\(t\)\+∑hWU\(t,h\)​Um\(t,h\)\.s\_\{m\}^\{\(t\+1\)\}=s\_\{m\}^\{\(t\)\}\+F^\{\(t\)\}\\\!\\Bigl\(s\_\{m\}^\{\(t\)\},z\_\{m\}^\{\(t\)\},\\bigl\(\\\|U^\{\(t,h\)\}\_\{m,v\}\\\|\_\{2\}\\bigr\)\_\{h,v\}\\Bigr\),\\qquad Q\_\{m\}^\{\(t\+1\)\}=Q\_\{m\}^\{\(t\)\}\+\\sum\_\{h\}W\_\{U\}^\{\(t,h\)\}U\_\{m\}^\{\(t,h\)\}\.\(7\)
F\(t\)F^\{\(t\)\}is zero, a two\-layer MLP on its three arguments, or, in the multi\-head stages,

F\(t\)\(s,z,⋅\)=Wo\(t\)z\+FF\(t\)\(LN\(t\)\(s\+Wo\(t\)z\)\),F^\{\(t\)\}\(s,z,\\cdot\)=W\_\{o\}^\{\(t\)\}z\+\\operatorname\{FF\}^\{\(t\)\}\\\!\\bigl\(\\operatorname\{LN\}^\{\(t\)\}\(s\+W\_\{o\}^\{\(t\)\}z\)\\bigr\),\(8\)which ignores the vector norms\. The final readout returnsAm​a\(T\)A^\{\(T\)\}\_\{ma\},zm\(T\)z\_\{m\}^\{\(T\)\},sm\(T\)s\_\{m\}^\{\(T\)\}, andQm\(T\)Q\_\{m\}^\{\(T\)\}for the decoder in Section[4\.3](https://arxiv.org/html/2610.10969#S4.SS3)\.

Table 1:Slot stage types\.Settings of Equations \([4](https://arxiv.org/html/2610.10969#S4.E4)\)–\([8](https://arxiv.org/html/2610.10969#S4.E8)\)\. “MLP” is a two\-layer perceptron on\[s,z,‖U‖\]\[s,z,\\\|U\\\|\]; “Wo​zW\_\{o\}z\+FF” is Equation \([8](https://arxiv.org/html/2610.10969#S4.E8)\); “–” means the map is absent;γ\\gammaof the vector–scalar stage is learned from initial value 1\. Detailed stage counts and parameterization are given in Appendix[C\.2](https://arxiv.org/html/2610.10969#A3.SS2)\.

### 4\.3Displacement decoding from three vector sources

Using the final readout outputs, with stage indices omitted, we form a bank of 32 atom\-derived vectors, eight slot\-derived vectors, and one direction from the anchorbm=∑aAm​a​Xab\_\{m\}=\\sum\_\{a\}A\_\{ma\}X\_\{a\}to atomaa:

ℬm​a=\[Watom​Va;Wslot​Qm;unitϵ⁡\(Xa−bm\)\]∈ℝ41×3,unitϵ\(v\)=vmax⁡\(‖v‖2,ϵ\),ϵ=10−8,\\begin\{gathered\}\\mathcal\{B\}\_\{ma\}=\\bigl\[\\,W\_\{\\mathrm\{atom\}\}V\_\{a\};\\ W\_\{\\mathrm\{slot\}\}Q\_\{m\};\\ \\operatorname\{unit\}\_\{\\epsilon\}\\\!\\bigl\(X\_\{a\}\-b\_\{m\}\\bigr\)\\,\\bigr\]\\in\\mathbb\{R\}^\{41\\times 3\},\\\\ \\operatorname\{unit\}\_\{\\epsilon\}\(v\)=\\frac\{v\}\{\\max\(\\\|v\\\|\_\{2\},\\epsilon\)\},\\quad\\epsilon=10^\{\-8\},\\end\{gathered\}\(9\)HereWatomW\_\{\\mathrm\{atom\}\}andWslotW\_\{\\mathrm\{slot\}\}are shared, learned, bias\-free vector\-channel maps\.

#### Invariant conditioning\.

The decoder is conditioned on the invariant features

ξm​a=\[sm,zm,ua′,χm​a,log⁡\(1\+‖Xa−bm‖22\)\],χm​a=\(13​⟨\(GQ​Qm\)g,\(GV​Va\)g⟩\)g=18,\\begin\{gathered\}\\xi\_\{ma\}=\\bigl\[\\,s\_\{m\},\\ z\_\{m\},\\ u^\{\\prime\}\_\{a\},\\ \\chi\_\{ma\},\\ \\log\\bigl\(1\+\\\|X\_\{a\}\-b\_\{m\}\\\|\_\{2\}^\{2\}\\bigr\)\\bigr\],\\\\ \\chi\_\{ma\}=\\Bigl\(\\tfrac\{1\}\{\\sqrt\{3\}\}\\bigl\\langle\(G\_\{Q\}Q\_\{m\}\)\_\{g\},\\,\(G\_\{V\}V\_\{a\}\)\_\{g\}\\bigr\\rangle\\Bigr\)\_\{g=1\}^\{8\},\\end\{gathered\}\(10\)whereua′=ua\(T\)u^\{\\prime\}\_\{a\}=u\_\{a\}^\{\(T\)\}is the readout’s atomic embedding, andGQ,GVG\_\{Q\},G\_\{V\}are learned vector\-channel maps\.

#### Coefficient networks and displacement\.

Four separately parameterized MLPs, shared across atoms and slots, give

pm​a\\displaystyle p\_\{ma\}=σ⁡\(gact​\(ξm​a\)\),\\displaystyle=\\sigma\\bigl\(g\_\{\\mathrm\{act\}\}\(\\xi\_\{ma\}\)\\bigr\),κm​a\\displaystyle\\kappa\_\{ma\}=softplus⁡\(gmag​\(ξm​a\)\),\\displaystyle=\\operatorname\{softplus\}\\bigl\(g\_\{\\mathrm\{mag\}\}\(\\xi\_\{ma\}\)\\bigr\),\(11\)qm​a\\displaystyle q\_\{ma\}=fcoef​\(ξm​a\),\\displaystyle=f\_\{\\mathrm\{coef\}\}\(\\xi\_\{ma\}\),qm​ares\\displaystyle q^\{\\mathrm\{res\}\}\_\{ma\}=fres​\(ξm​a\),\\displaystyle=f\_\{\\mathrm\{res\}\}\(\\xi\_\{ma\}\),whereσ\\sigmais the sigmoid andqm​a,qm​ares∈ℝ41q\_\{ma\},q^\{\\mathrm\{res\}\}\_\{ma\}\\in\\mathbb\{R\}^\{41\}\. The resulting displacement is

D^m,a=pm​a⏟participation​κm​a⏟scale​unitϵ⁡\(qm​a⊤​ℬm​a\)⏟direction\+γres​\(qm​ares\)⊤​ℬm​a,\\widehat\{D\}\_\{m,a\}=\\underbrace\{p\_\{ma\}\}\_\{\\text\{participation\}\}\\;\\underbrace\{\\kappa\_\{ma\}\}\_\{\\text\{scale\}\}\\;\\underbrace\{\\operatorname\{unit\}\_\{\\epsilon\}\\\!\\bigl\(q\_\{ma\}^\{\\top\}\\mathcal\{B\}\_\{ma\}\\bigr\)\}\_\{\\text\{direction\}\}\\;\+\\;\\gamma\_\{\\mathrm\{res\}\}\\,\(q^\{\\mathrm\{res\}\}\_\{ma\}\)^\{\\top\}\\mathcal\{B\}\_\{ma\},\(12\)This gives atomaa’s displacement for proposalmm, with learned global scaleγres\\gamma\_\{\\mathrm\{res\}\}\. The gate permits localized motion without hard sparsity, while the ungated residual accommodates surrounding relaxations\. For a vacancy event, this allows the proposal to describe both the atoms moving toward vacant sites and the smaller displacements of their neighbors, without selecting the moving atoms in advance\. The periodic coordinate construction is described in Appendix[A\.2](https://arxiv.org/html/2610.10969#A1.SS2); network widths and initialization are given in Appendix[C\.3](https://arxiv.org/html/2610.10969#A3.SS3)\.

### 4\.4Set matching and training objective

Letℰx=\{1,…,K\}\\mathscr\{E\}\_\{x\}=\\\{1,\\ldots,K\\\}index the reference set ofXX, with saddle displacement fieldsDe=Se−XD\_\{e\}=S\_\{e\}\-X\. Slots have no prescribed targets\. Using the scalar SmoothL1 penaltyρβ\\rho\_\{\\beta\}, define the proposal–target cost over allNNatoms by

Cm​e=ℓβ​\(D^m−De\),ℓβ​\(E\)=1N​∑a,dρβ​\(Ea​d\)\.C\_\{me\}=\\ell\_\{\\beta\}\(\\widehat\{D\}\_\{m\}\-D\_\{e\}\),\\qquad\\ell\_\{\\beta\}\(E\)=\\frac\{1\}\{N\}\\sum\_\{a,d\}\\rho\_\{\\beta\}\(E\_\{ad\}\)\.\(13\)Rectangular Hungarian matching gives the minimum\-cost one\-to\-one first\-round assignmentΠ1\\Pi\_\{1\}, assigning each target to a distinct slot whenM≥KM\\geq K\. We remove the matched slots and repeat against the full catalogue for at mostRmatch=4R\_\{\\mathrm\{match\}\}=4rounds in total, stopping when no slots remain\. Each slot is used at most once, whereas a target may supervise multiple slots across rounds\. ForRmatch=1R\_\{\\mathrm\{match\}\}=1, the auxiliary term is omitted\. The auxiliary term is also defined to be zero whenever no pair is matched after the first round, including when the first round exhausts all slots\.

LetΠ\+\\Pi\_\{\+\}denote all pairs matched after the first round\. The objective is

ℒx=λD​\[1\|Π1\|​∑\(m,e\)∈Π1Cm​e\+λ\+\|Π\+\|​∑\(m,e\)∈Π\+Cm​e\],λD=10,λ\+=0\.5\.\\mathcal\{L\}\_\{x\}=\\lambda\_\{D\}\\left\[\\frac\{1\}\{\|\\Pi\_\{1\}\|\}\\sum\_\{\(m,e\)\\in\\Pi\_\{1\}\}C\_\{me\}\+\\frac\{\\lambda\_\{\+\}\}\{\|\\Pi\_\{\+\}\|\}\\sum\_\{\(m,e\)\\in\\Pi\_\{\+\}\}C\_\{me\}\\right\],\\qquad\\lambda\_\{D\}=10,\\quad\\lambda\_\{\+\}=0\.5\.\(14\)

### 4\.5Confidence readout for proposal selection

We hypothesize that features learned through displacement matching also encode catalogue agreement\. We freeze the predictor and train a lightweight MLP*judge*on invariant latent and geometric features \(Appendix[E](https://arxiv.org/html/2610.10969#A5)\)\. It outputsp^m∈\(0,1\)\\widehat\{p\}\_\{m\}\\in\(0,1\)without force evaluations; only proposals withp^m≥τ\\widehat\{p\}\_\{m\}\\geq\\tauundergo Dimer refinement\.

For proposalD^m\\widehat\{D\}\_\{m\}and catalogue fieldDeD\_\{e\}, define the moving\-union root\-mean\-square \(RMS\) error by

Um​e=\{a:max⁡\(‖D^m,a‖2,‖De,a‖2\)≥0\.3​Å\},RMSU⁡\(D^m,De\)=\(1\|Um​e\|​∑a∈Um​e‖D^m,a−De,a‖22\)1/2\.\\begin\{gathered\}U\_\{me\}=\\\{a:\\max\(\\\|\\widehat\{D\}\_\{m,a\}\\\|\_\{2\},\\\|D\_\{e,a\}\\\|\_\{2\}\)\\geq 0\.3\\,\\text\{\\AA\}\\\},\\\\ \\operatorname\{RMS\}\_\{U\}\(\\widehat\{D\}\_\{m\},D\_\{e\}\)=\\left\(\\frac\{1\}\{\|U\_\{me\}\|\}\\sum\_\{a\\in U\_\{me\}\}\\\|\\widehat\{D\}\_\{m,a\}\-D\_\{e,a\}\\\|\_\{2\}^\{2\}\\right\)^\{1/2\}\.\\end\{gathered\}\(15\)Empty unions have infinite error\. For the nearest catalogue fieldDe∗D\_\{e^\{\*\}\}under this RMS, setym=1y\_\{m\}=1iff the RMS is below0\.35​Å0\.35\\,\\text\{\\AA\}and the maximum atomwise error overUm​e∗U\_\{me^\{\*\}\}is below0\.5​Å0\.5\\,\\text\{\\AA\}\. Judge selection provides a tunable trade\-off between reference\-event recovery and physical search cost\. The judge is trained on the training set only\.

## 5Experiments and Results

![Refer to caption](https://arxiv.org/html/2610.10969v1/pes_trajectories.png)Figure 2:Energy background, saddles, and trajectories\.\(a\)Sammon\-projected view of the energy function and initial, saddle, and product configurations\.\(b\)MDS\-coordinate views of ASPIRE and random\-start Dimer searches on one configuration\. ASPIRE proposal jumps effectively predict approximate saddles, followed by light Dimer refinement, whereas random Dimer goes long paths; Each trajectory energy point denotes every five translation steps, and color denotes the energy value above initial at the point\. \(Appendix[I](https://arxiv.org/html/2610.10969#A9)\)\. a\.u\.: arbitrary units\.### 5\.1Datasets

Figure 3:Reference\-event recovery versus search cost\.Coverage \(left\) and RWC \(right\) versus the mean recorded Dimer search force calls per configuration, evaluated on 200 test configurations\. Curves vary the judge threshold for ASPIRE \(up to 96 proposals\) or the number of random\-start Dimer searches \(up to 80\)\.We construct both datasets by removing one to four lattice sites from a 128,000\-atom bcc Fe block and relaxing under the M07 EAM potential \(Appendix[A\.1](https://arxiv.org/html/2610.10969#A1.SS1)\)\. Each example pairs a relaxed active volumeXXof radius11​Å11\\,\\text\{\\AA\}with a reference event setℰXref\\mathcal\{E\}\_\{X\}^\{\\mathrm\{ref\}\}collected through 80 random\-start Dimer searches and SEAKMC validation, with barriersΔ​Ek∈\[0\.1,2\.0\]​eV\\Delta E\_\{k\}\\in\[0\.1,2\.0\]\\,\\mathrm\{eV\}\. These reference sets are finite and not exhaustive, but will be updated with more Dimer searches and made more complete\.BccFe\-4vacAV\-4000contains 4,000 symmetry\-distinct four\-vacancy configurations and 65,450 events \(2≤K≤432\\leq K\\leq 43\), with a 3,700/100/200 split\. It is used for all evaluations in this paper\.BccFe\-1to4Vaccontains 5,372 configurations with one to four vacancies and 83,372 events\. It is built for future KMC trajectory simulation and analysis\. Construction details are in Appendix[B](https://arxiv.org/html/2610.10969#A2)\.

### 5\.2Evaluation protocol

Let𝒳\\mathcal\{X\}be the fixed evaluation set andℰXrec⊆ℰXref\\mathcal\{E\}\_\{X\}^\{\\mathrm\{rec\}\}\\subseteq\\mathcal\{E\}\_\{X\}^\{\\mathrm\{ref\}\}the recovered references atXX\. Harmonic reference rates areke=νe−ΔEe/\(kBT\)∈\(0,∞\)k\_\{e\}=\\nu e^\{\-\\Delta E\_\{e\}/\(k\_\{\\mathrm\{B\}\}T\)\}\\in\(0,\\infty\)with common prefactorν\\nu\. We consider eventskkandk′k^\{\\prime\}from the sameXXmatch if and only ifmaxa⁡‖Dk,a−Dk′,a‖2≤0\.05​Å\\max\_\{a\}\\\|D\_\{k,a\}\-D\_\{k^\{\\prime\},a\}\\\|\_\{2\}\\leq 0\.05\\,\\text\{\\AA\}\\qquadand\|Δ​Ek−Δ​Ek′\|≤0\.02​eV\\qquad\|\\Delta E\_\{k\}\-\\Delta E\_\{k^\{\\prime\}\}\|\\leq 0\.02\\,\\mathrm\{eV\}, whereDk=Sk−XD\_\{k\}=S\_\{k\}\-Xis the saddle displacement,Δ​Ek\\Delta E\_\{k\}the barrier, andaaindexes atoms\. Figure[2](https://arxiv.org/html/2610.10969#S5.F2)\(a\) is a visualization of the initial, saddle, and product configurations on the projected energy surface\. We use coverageCovX=\|ℰXrec\|\|ℰXref\|\\mathrm\{Cov\}\_\{X\}=\\frac\{\|\\mathcal\{E\}\_\{X\}^\{\\mathrm\{rec\}\}\|\}\{\|\\mathcal\{E\}\_\{X\}^\{\\mathrm\{ref\}\}\|\}, rate\-weighted coverageRWCX=∑e∈ℰXrecke∑e∈ℰXrefke\\mathrm\{RWC\}\_\{X\}=\\frac\{\\sum\_\{e\\in\\mathcal\{E\}\_\{X\}^\{\\mathrm\{rec\}\}\}k\_\{e\}\}\{\\sum\_\{e\\in\\mathcal\{E\}\_\{X\}^\{\\mathrm\{ref\}\}\}k\_\{e\}\}, and averaged rate\-weighted coverageRWC¯=1\|𝒳\|​∑X∈𝒳RWCX\\overline\{\\mathrm\{RWC\}\}=\\frac\{1\}\{\|\\mathcal\{X\}\|\}\\sum\_\{X\\in\\mathcal\{X\}\}\\mathrm\{RWC\}\_\{X\}for evaluation\. For a fixed reference catalogue,RWCX→CovX\\mathrm\{RWC\}\_\{X\}\\to\\mathrm\{Cov\}\_\{X\}asT→∞T\\to\\infty; forRWCX\>0\\mathrm\{RWC\}\_\{X\}\>0, restricting to the recovered set gives1/RWCX1/\\mathrm\{RWC\}\_\{X\}times the full\-reference mean waiting time, andRWC¯\\overline\{\\mathrm\{RWC\}\}is the expected fraction recovered by the evaluated method when one event is sampled from the reference by rate at eachXX\. These are kinetic\-focused evaluation proxies, not full KMC trajectory kinetics \(Appendix[K](https://arxiv.org/html/2610.10969#A11)\)\. All reported RWC values useT=800​KT=800\\,\\mathrm\{K\}\.

Throughout the evaluation in this paper, the reference setℰXref\\mathcal\{E\}\_\{X\}^\{\\mathrm\{ref\}\}stays fixed: each output is assigned at most one matching reference, and repeated hits to the same reference are counted once \(Appendix[L](https://arxiv.org/html/2610.10969#A12)\)\. Throughout the experiments, eachXXis a SEAKMC active volume of radius11​Å11\\,\\text\{\\AA\}\([Xu et al\., 2012](https://arxiv.org/html/2610.10969#bib.bib38)\), treated as a global state for evaluation\. Random Dimer searches are capped at 450 translation steps, whereas ASPIRE refinement is capped at 300 \(Appendix[F](https://arxiv.org/html/2610.10969#A6)\)\. This is because random dimer takes more translations to converge, and the main purpose of ASPIRE is to reduce geometric error and translation steps to achieve computational efficiency, as shown in Figure[2](https://arxiv.org/html/2610.10969#S5.F2)\(b\)\.

### 5\.3Reference\-event recovery and force cost

Figure 4:Proposal geometry and refinement outcomes\.Each point represents a proposal and its subsequent Dimer search\. Initial error is measured by moving\-union RMS or Euclidean distance to the nearest reference saddle, and refinement effort by the number of translation steps\. Colors distinguish nearest\-reference recovery \(fastest or slower\), recovery of another reference, valid unmatched outputs, invalid outputs, and searches reaching the cap\. The dashed line marks the 300\-step cap\.Table 2:Ablation of Ev\-Quiformer\.Judge selection is disabled for all variants\. O2M denotes the number of sequential matching rounds; Uniform denotes all\-atom SmoothL1 supervision\. The active\-aware objective is specified in Appendix[D\.2](https://arxiv.org/html/2610.10969#A4.SS2)\.F¯search\\bar\{F\}\_\{\\mathrm\{search\}\}is the mean recorded Dimer search force\-call count per configuration \(Appendix[L\.5](https://arxiv.org/html/2610.10969#A12.SS5)\)\. Evaluated on 50 test configurations selected with topology\-layered sampling covering all five vacancy\-connectivity classes \(Appendix[L\.1](https://arxiv.org/html/2610.10969#A12.SS1)\)\.Figure[3](https://arxiv.org/html/2610.10969#S5.F3)compares reference\-event recovery against physical search cost\. ASPIRE achieves higher reference\-event coverage and recovers a comparable fraction of the reference rate mass at substantially lower search cost compared to 80 random dimer searches\. 80 per active volume is chosen as it is a reasonable parameter to get a relatively comprehensive set in SEAKMC deployment\.

The judge controls this recovery–cost tradeoff by selecting proposals before physical refinement\. The ASPIRE curves rise rapidly and then flatten, indicating the possibility of using the judge to further reduce force cost while keeping much of the event set and reference rate mass recovery\. Together, these results support learned saddle proposals and selective refinement as a means of reducing physical search work while retaining high reference\-event recovery\.

Figure[4](https://arxiv.org/html/2610.10969#S5.F4)shows that Geometric error is strongly associated with refinement behavior: proposals with small moving\-union RMS typically converge rapidly to their nearest ground\-truth saddle, whereas larger errors lead to more variable outcomes\. This suggests that geometric error can serve as an effective offline proxy for model quality, consistent with the strong performance of our union\-RMS\-based judge\.

#### Ablation study\.

Table[2](https://arxiv.org/html/2610.10969#S5.T2)examines how proposal design affects this recovery–cost tradeoff\. Judge selection is disabled for all variants, so the comparison does not depend on a confidence threshold\.

The full Ev\-Quiformer achieves the highest coverage, 76\.13%, with 87\.34% RWC and 75,768 mean search force calls per configuration\. Among those with similar coverage and RWC above 87%, full Ev\-Quiformer reduces the most search force calls\. Together, for the objective of higher coverage and RWC first, and reduced cost second, these results favor the default combination of a larger proposal set, slot interaction, repeated matching, and uniform all\-atom supervision\. Among the tested variants with similar RWC, the full model provides the highest coverage at the lowest force cost\.

### 5\.4Wall time and comparison with event reuse

Figure 5:Recovery and wall time compared with event reuse\.\(a\)Reference\-event coverage and rate\-weighted coverage\. After Random Dimer, methods are ordered by increasing RWC\.\(b\)Measured wall time per configuration and per recovered reference event\. Random Dimer and the reuse baselines use 16 CPUs; ASPIRE uses 16 CPUs and one A100 GPU\. ASPIRE uses judge\-based proposal selection with thresholdτ=10−5\\tau=10^\{\-5\}, refining only proposals withp^m≥τ\\widehat\{p\}\_\{m\}\\geq\\tau\. Evaluated on 50 test configurations with topology\-layered samplingForce\-call counts alone do not capture the lookup cost of event reuse\. Cost\-accounting conventions are discussed in Appendix[L](https://arxiv.org/html/2610.10969#A12)\. We therefore compare measured wall time with*adaptations*of KDB and k\-ART\-style keybank baselines in Figure[5](https://arxiv.org/html/2610.10969#S5.F5)\. We adapt KDB’s local geometric matching and implement KeyBank using WL hashing, exact graph isomorphism, and geometric alignment\. KDB transfers aligned saddle coordinates, whereas KeyBank transfers rotated displacement fields; both initialize Dimer refinement\. KDB\-494 denotes 494 event templates \(Appendix[G](https://arxiv.org/html/2610.10969#A7)\); KeyBank\-32 and KeyBank\-128 use 32 and 128 training configurations \(Appendix[H](https://arxiv.org/html/2610.10969#A8)\), respectively\. Querying the test configurations was already time\-consuming at these sizes\. We therefore report the measured recovery–wall\-time tradeoff at these operating points\.

As shown in Figure[5](https://arxiv.org/html/2610.10969#S5.F5), baselines with comparable RWC are substantially slower than ASPIRE, while faster alternatives achieve much lower RWC under the stated library and hardware settings\. These methods do not contradict each other, so they can be deployed together for complementary benefits\.

## 6Conclusion and Limitation

We introduced ASPIRE, a propose\-refine framework for saddle discovery from a single initial atomic configuration\. Its Ev\-Quiformer predictor uses geometry\-conditioned event slots to generate a set of complete displacement fields for refinement on the original potential\. On the four\-vacancy bcc Fe benchmark, ASPIRE achieves higher reference\-event coverage than the reported 80\-start Random Dimer endpoint, at approximately half the recorded search\-force cost under the respective protocols\. The wall\-time comparison further shows lower measured runtime than event\-reuse baselines with comparable RWC under the stated hardware\. These results support learning approximate saddle geometry as a means of accelerating physical event discovery\.

Limitation\.Our reference sets are finite and non\-exhaustive, and the evaluation is restricted to four\-vacancy bcc Fe under a fixed potential\. Reference\-relative coverage and RWC did not establish the accuracy of a full KMC trajectory\. More complete reference catalogues and trajectory\-level evaluation are needed to assess the resulting long\-time kinetics\.

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## Supplementary Material

Appendices[A](https://arxiv.org/html/2610.10969#A1)–[F](https://arxiv.org/html/2610.10969#A6)describe the physical background, dataset construction, neural prediction and screening, and numerical refinement\. Appendices[G](https://arxiv.org/html/2610.10969#A7)–[I](https://arxiv.org/html/2610.10969#A9)describe the event\-reuse baselines and trajectory visualization\. Appendices[J](https://arxiv.org/html/2610.10969#A10)and[K](https://arxiv.org/html/2610.10969#A11)provide the theoretical analyses, and Appendix[L](https://arxiv.org/html/2610.10969#A12)gives the evaluation protocols and cost accounting\. The complete appendix contents follow on two pages\.

### Appendix Contents

## Supplementary Material

### Appendix Contents \(continued\)

## Appendix APhysical Background and Coordinate Conventions

This section develops the physical and representational background for the learning problem\. We useX∈ℝN×3X\\in\\mathbb\{R\}^\{N\\times 3\}for the initial configuration,VVfor the fixed interatomic potential,SeS\_\{e\}for a reference saddle,De=Se−XD\_\{e\}=S\_\{e\}\-Xfor its displacement field, andXe\+X\_\{e\}^\{\+\}for the connected product minimum\. Parent\-crystal, periodic\-cell, and search\-coordinate notation specifies how these quantities are constructed\.

### A\.1bcc Fe, vacancies, and the fixed interatomic potential

#### Parent lattice\.

In conventional cubic axes, an ideal body\-centered\-cubic \(bcc\) lattice with lattice parametera0a\_\{0\}can be written as

Λbcc=a0​ℤ3∪a0​\(ℤ3\+\(12,12,12\)\)\.\\Lambda\_\{\\mathrm\{bcc\}\}=a\_\{0\}\\mathbb\{Z\}^\{3\}\\;\\cup\\;a\_\{0\}\\\!\\left\(\\mathbb\{Z\}^\{3\}\+\(\\tfrac\{1\}\{2\},\\tfrac\{1\}\{2\},\\tfrac\{1\}\{2\}\)\\right\)\.\(16\)There are two lattice sites per conventional cubic cell\. The elemental bcc structure has space groupI​m​3¯​mIm\\overline\{3\}m\(No\. 229\)\([Mehl et al\., 2017](https://arxiv.org/html/2610.10969#bib.bib1)\)\. An equivalent integer\-coordinate description is

Λbcc=a02​\{z∈ℤ3:z1≡z2≡z3\(mod2\)\}\.\\Lambda\_\{\\mathrm\{bcc\}\}=\\frac\{a\_\{0\}\}\{2\}\\left\\\{z\\in\\mathbb\{Z\}^\{3\}:z\_\{1\}\\equiv z\_\{2\}\\equiv z\_\{3\}\\pmod\{2\}\\right\\\}\.\(17\)Thus the integer coordinates are either all even or all odd, enabling exact comparison of parent vacancy\-site patterns\. Relaxed atomic positions and saddle candidates are represented by continuous Cartesian coordinates\.

#### Vacancies and competing rearrangements\.

A vacancy is an unoccupied site of the parent lattice\. Equation \([16](https://arxiv.org/html/2610.10969#A1.E16)\) gives eight nearest\-neighbor vectors from a lattice site,

a02​\(±1,±1,±1\),dnn=32​a0\.\\frac\{a\_\{0\}\}\{2\}\(\\pm 1,\\pm 1,\\pm 1\),\\qquad d\_\{\\mathrm\{nn\}\}=\\frac\{\\sqrt\{3\}\}\{2\}a\_\{0\}\.\(18\)These directions describe nearest\-neighbor atom–vacancy exchange\. Multiple vacancies alter local coordination and induce relaxation of the surrounding Fe atoms, producing rearrangements with different saddle geometries and barriers\. SEAKMC searches directly in atomic coordinate space to discover the corresponding mechanisms\([Xu et al\., 2012](https://arxiv.org/html/2610.10969#bib.bib38)\)\.

The prediction target is a completeN×3N\\times 3displacement field, which describes both motion toward vacant sites and surrounding atomic relaxation\. This representation accommodates varying numbers of participating atoms and multiple accessible rearrangements for the same vacancy configuration\.

#### M07 and the EAM energy\.

The main\-text experiments use the M07 embedded\-atom\-method \(EAM\) potential for Fe\. The NIST repository identifies the standard M07 file with the Fe potential developed by Marinica in 2007 using a point\-defect\-oriented fitting database, and associates it with the studies of[Malerba et al\. \(2010\)](https://arxiv.org/html/2610.10969#bib.bib28)and[Marinica et al\. \(2012\)](https://arxiv.org/html/2610.10969#bib.bib29)\. For a single\-element EAM model, the energy of a physical configurationYYhas the form

Vfull​\(Y\)=∑iFemb​\(∑j≠ifdens​\(ri​j\)\)\+12​∑i∑j≠iϕpair​\(ri​j\),V\_\{\\mathrm\{full\}\}\(Y\)=\\sum\_\{i\}F\_\{\\mathrm\{emb\}\}\\\!\\left\(\\sum\_\{j\\neq i\}f\_\{\\mathrm\{dens\}\}\(r\_\{ij\}\)\\right\)\+\\frac\{1\}\{2\}\\sum\_\{i\}\\sum\_\{j\\neq i\}\\phi\_\{\\mathrm\{pair\}\}\(r\_\{ij\}\),\(19\)whereri​jr\_\{ij\}is evaluated with the physical boundary convention,FembF\_\{\\mathrm\{emb\}\}is the embedding function,fdensf\_\{\\mathrm\{dens\}\}contributes to the local density, andϕpair\\phi\_\{\\mathrm\{pair\}\}is the pair interaction\([Daw and Baskes, 1984](https://arxiv.org/html/2610.10969#bib.bib9)\)\. HereYYdistinguishes the physical system from the local arrayXXsupplied to the predictor\.

The neural model proposes candidate geometries, while the fixed Fe potential supplies energies and forces for initial relaxation, Dimer refinement, and barrier evaluation\. This division connects learned proposals to a consistent physical energy surface throughout the benchmark\.

### A\.2Periodic images and a local Cartesian representation

#### Periodic images are translated copies\.

A periodic simulation cell represents a crystal together with translated copies of that cell\([Abraham et al\., 2026](https://arxiv.org/html/2610.10969#bib.bib17)\)\. LetHsysH\_\{\\mathrm\{sys\}\}be the nonsingular matrix whose rows are the physical cell vectors, and use row vectors for positions\. Thenrrandr\+n​Hsysr\+nH\_\{\\mathrm\{sys\}\}represent the same indexed atom for an allowed integer lattice shiftnn\. In a partially periodic system, components ofnnalong nonperiodic axes are zero\.

For a displacement row vectorvv, the minimum\-image operation is

MICHsys⁡\(v\)=v−n∗​Hsys,n∗∈arg​minn∈ℤper3⁡‖v−n​Hsys‖2\.\\operatorname\{MIC\}\_\{H\_\{\\mathrm\{sys\}\}\}\(v\)=v\-n^\{\*\}H\_\{\\mathrm\{sys\}\},\\qquad n^\{\*\}\\in\\operatorname\*\{arg\\,min\}\_\{n\\in\\mathbb\{Z\}^\{3\}\_\{\\mathrm\{per\}\}\}\\\|v\-nH\_\{\\mathrm\{sys\}\}\\\|\_\{2\}\.\(20\)It chooses a shortest representative among the allowed translations\. For an orthogonal, fully periodic cell, this becomes

MICHsys⁡\(v\)=\(v​Hsys−1−round⁡\(v​Hsys−1\)\)​Hsys,\\operatorname\{MIC\}\_\{H\_\{\\mathrm\{sys\}\}\}\(v\)=\\bigl\(vH\_\{\\mathrm\{sys\}\}^\{\-1\}\-\\operatorname\{round\}\(vH\_\{\\mathrm\{sys\}\}^\{\-1\}\)\\bigr\)H\_\{\\mathrm\{sys\}\},\(21\)apart from image ties\. The row\-vector convention gives the inverseHsys−1H\_\{\\mathrm\{sys\}\}^\{\-1\}\. For a general skew cell, Equation \([20](https://arxiv.org/html/2610.10969#A1.E20)\) defines the shortest\-image search\. At exact ties, preprocessing uses a fixed image\-selection convention compatible with the stated symmetry conditions\.

#### From stored positions toXX\.

A local representation places atoms near a defect in a common image even when their stored coordinates straddle the periodic box\. Letccbe a local reference point, also written as a row vector, and leti⁡\(a\)i\(a\)identify local atomaain the physical configurationYY\. A suitable local representation is

Xa=MICHsys⁡\(Yi⁡\(a\)−c\)\.X\_\{a\}=\\operatorname\{MIC\}\_\{H\_\{\\mathrm\{sys\}\}\}\\bigl\(Y\_\{i\(a\)\}\-c\\bigr\)\.\(22\)Alternatively, an already unwrapped local configuration can be supplied directly\. In either case,XXis the resulting Cartesian array with fixed atom order; cell information andccspecify its construction\.

Pairwise Cartesian differences agree with minimum\-image displacements under the following sufficient geometric condition\. Let

ℓmin=minn∈ℤper3∖\{0\}⁡‖n​Hsys‖2,\\ell\_\{\\min\}=\\min\_\{n\\in\\mathbb\{Z\}^\{3\}\_\{\\mathrm\{per\}\}\\setminus\\\{0\\\}\}\\\|nH\_\{\\mathrm\{sys\}\}\\\|\_\{2\},and suppose all represented positions satisfy‖Xa‖2≤rAV\\\|X\_\{a\}\\\|\_\{2\}\\leq r\_\{\\mathrm\{AV\}\}\. If

4​rAV<ℓmin,4r\_\{\\mathrm\{AV\}\}<\\ell\_\{\\min\},\(23\)then all pair differences in the local array are their unique shortest periodic representatives\. Indeed,‖Xa−Xb‖2≤2​rAV\\\|X\_\{a\}\-X\_\{b\}\\\|\_\{2\}\\leq 2r\_\{\\mathrm\{AV\}\}, whereas any nonzero lattice translationllgives‖Xa−Xb−l‖2≥ℓmin−2​rAV\>2​rAV\\\|X\_\{a\}\-X\_\{b\}\-l\\\|\_\{2\}\\geq\\ell\_\{\\min\}\-2r\_\{\\mathrm\{AV\}\}\>2r\_\{\\mathrm\{AV\}\}\. This criterion makes the relation between the local Cartesian chart and minimum\-image geometry explicit in terms of the actual cell dimensions\.

#### Operations used by the main\-text model\.

After the local representation is fixed, the encoder usesXa−XbX\_\{a\}\-X\_\{b\}in its6​Å6\\,\\text\{\\AA\}radius graph\. The decoder uses

bm=∑aAm​a​Xa,Xa−bm,∑aAm​a=1\.b\_\{m\}=\\sum\_\{a\}A\_\{ma\}X\_\{a\},\\qquad X\_\{a\}\-b\_\{m\},\\qquad\\sum\_\{a\}A\_\{ma\}=1\.\(24\)The anchor is the attention\-weighted mean in the supplied local Cartesian representation\. Preparing a consistent local image before this averaging preserves the intended neighborhood geometry; for example, raw positions0\.49​L0\.49Land−0\.49​L\-0\.49Lstraddle a periodic boundary and have nearest\-image separation0\.02​L0\.02L\.

#### Displacement targets and atom correspondence\.

The initial and saddle arrays must refer to the same atoms in compatible representations before forming

De=Se−X,S^m=X\+D^m\.D\_\{e\}=S\_\{e\}\-X,\\qquad\\widehat\{S\}\_\{m\}=X\+\\widehat\{D\}\_\{m\}\.\(25\)Image tracking along a search gives a consistent physical displacement\. For a localized event, compatible shortest displacements can be selected relative to the initial atoms; trajectories winding around the cell use path unwrapping\.

With consistent atom correspondence and image representation, assignment and regression use the full Cartesian residualD^m−De\\widehat\{D\}\_\{m\}\-D\_\{e\}directly\. The same atom ordering is retained inXX,SeS\_\{e\}, and their displacement field\.

#### Three different meanings of “image\.”

Periodic images are translated representatives of the same atom\. The reflections and inversion used in cubic\-symmetry deduplication are spatial transformations of an entire vacancy pattern\. The two images of a Dimer are nearby trial configurations in configuration space\. Each operation therefore has a defined role in data preparation or numerical search\.

### A\.3Active volumes, surrounding atoms, and free coordinates

SEAKMC localizes event searches to active volumes surrounding defects, reducing the coordinates involved in a search while retaining an atomistic description of the rearrangement\([Xu et al\., 2012](https://arxiv.org/html/2610.10969#bib.bib38);[Xu et al\., 2015](https://arxiv.org/html/2610.10969#bib.bib40);[Liang and Xu, 2023](https://arxiv.org/html/2610.10969#bib.bib43)\)\. In this paper,XXrepresents an active volume of radiusrAV=11​År\_\{\\mathrm\{AV\}\}=11\\,\\text\{\\AA\}\. ItsNNlocal atoms are selected from the 128,000\-atom parent block before vacancy removal\. The active\-volume radius controls the search region, the6​Å6\\,\\text\{\\AA\}graph cutoff controls neural neighborhoods, and the potential’s interaction cutoff controls physical interactions\.

The active\-volume model incorporates its prescribed surrounding environment\. With fixed surroundings, a local trial arrayZZcan be embedded into the parent configuration by a mapιX\\iota\_\{X\}\. The local potential is

V⁡\(Z\)=Vfull​\(ιX​Z\)−CX,V\(Z\)=V\_\{\\mathrm\{full\}\}\(\\iota\_\{X\}Z\)\-C\_\{X\},\(26\)whereCXC\_\{X\}is any configuration\-independent energy offset\. Interactions across the active\-volume boundary are included according to the physical model\. The offset cancels fromV⁡\(Se\)−V⁡\(X\)V\(S\_\{e\}\)\-V\(X\)\. This physical locality model specifies the forces and barriers used for refinement\([Béland et al\., 2015](https://arxiv.org/html/2610.10969#bib.bib4)\)\.

When some represented atoms are fixed during refinement, letℱX\\mathcal\{F\}\_\{X\}index the free atoms\. An admissible search displacement then hasDa=0D\_\{a\}=0fora∉ℱXa\\notin\\mathcal\{F\}\_\{X\}, and the search acts on vectorized free coordinates\. Any additional linear constraints are imposed before taking the gradient and Hessian of the restricted potential\. This is the coordinate interpretation used by the local\-convergence proposition\.

The free\-coordinate representation specifies the nondegeneracy conditions\. A translation\-invariant whole\-system potential has uniform translation zero modes; fixing the environment or removing these modes supplies the space in which a positive\-definite minimum Hessian or nondegenerate index\-one saddle is defined\. Motion of the free atoms relative to a fixed shell remains part of the local physical problem\.

### A\.4Dimer saddle searches and learned initialization

#### What Dimer searches for\.

The Dimer method seeks an index\-one saddle using first derivatives of the potential, without requiring a product minimum as input\([Henkelman and Jónsson, 1999](https://arxiv.org/html/2610.10969#bib.bib16)\)\. Work in the same vectorized free coordinates as above, and denote a trial point byyyand its force byf​\(y\)=−∇V​\(y\)f\(y\)=\-\\nabla V\(y\)\. Dimer combines ascent along an unstable direction with relaxation in transverse directions to locate an index\-one saddle\.

#### Two configurations estimate directional curvature\.

A symmetric description of a Dimer consists of a centeryy, a unit orientationnn, and a half\-separationhdim\>0h\_\{\\mathrm\{dim\}\}\>0:

y\+=y\+hdim​n,y−=y−hdim​n,‖n‖2=1\.y^\{\+\}=y\+h\_\{\\mathrm\{dim\}\}n,\\qquad y^\{\-\}=y\-h\_\{\\mathrm\{dim\}\}n,\\qquad\\\|n\\\|\_\{2\}=1\.\(27\)The orientation assigns a displacement direction to every free coordinate\. A finite difference of the forces estimates a Hessian–vector product,

Bhdim​\(y,n\)=−f⁡\(y\+\)−f⁡\(y−\)2​hdim⟶∇2V​\(y\)​nas​hdim→0,B\_\{h\_\{\\mathrm\{dim\}\}\}\(y,n\)=\-\\frac\{f\(y^\{\+\}\)\-f\(y^\{\-\}\)\}\{2h\_\{\\mathrm\{dim\}\}\}\\longrightarrow\\nabla^\{2\}V\(y\)n\\quad\\text\{as \}h\_\{\\mathrm\{dim\}\}\\to 0,\(28\)with estimated curvatureκhdim=n⊤​Bhdim​\(y,n\)\\kappa\_\{h\_\{\\mathrm\{dim\}\}\}=n^\{\\top\}B\_\{h\_\{\\mathrm\{dim\}\}\}\(y,n\)\. This explains how curvature information can be obtained without assembling a full Hessian\. The symmetric formula describes the method’s principle; implementations can use force reuse or one\-sided estimates to reduce the number of force evaluations\([Kästner and Sherwood, 2008](https://arxiv.org/html/2610.10969#bib.bib21)\)\.

#### Rotation and translation are different updates\.

At fixedyy, the orientation is adjusted toward a low\-curvature mode\. In the small\-separation limit, minimizingn⊤​∇2V​\(y\)​nn^\{\\top\}\\nabla^\{2\}V\(y\)nover the unit sphere gives a tangential gradient proportional to

\(I−n​n⊤\)​∇2V​\(y\)​n,\(I\-nn^\{\\top\}\)\\nabla^\{2\}V\(y\)n,\(29\)which can be estimated using Equation \([28](https://arxiv.org/html/2610.10969#A1.E28)\)\. A translation update then moves the center\. In a negative\-curvature region, the minimum\-mode\-following effective force is

feff\(y,n\)=f\(y\)−2n\(n⊤f\(y\)\)=−\(I−2nn⊤\)∇V\(y\)\.f\_\{\\mathrm\{eff\}\}\(y,n\)=f\(y\)\-2n\\bigl\(n^\{\\top\}f\(y\)\\bigr\)=\-\(I\-2nn^\{\\top\}\)\\nabla V\(y\)\.\(30\)The force component parallel tonnis reversed while perpendicular components are retained\. Ifnnis the unstable mode near an index\-one saddle, this reverses the downhill motion along that mode and leaves relaxation in stable directions\. Rotation and translation are repeated, with numerical step control and convergence tests\. Additional controls handle positive\-curvature escape from the initial basin\([Chill et al\., 2014](https://arxiv.org/html/2610.10969#bib.bib13)\)\.

#### Random and learned initialization\.

The conventional path initializes the search direction randomly, using the axis sampler in Appendix[F](https://arxiv.org/html/2610.10969#A6)\. ASPIRE suppliesS^m=X\+D^m\\widehat\{S\}\_\{m\}=X\+\\widehat\{D\}\_\{m\}as the proposed center configuration\. In free coordinates, this gives

y0=vec⁡\(S^m\|ℱX\)\.y\_\{0\}=\\operatorname\{vec\}\\\!\\left\(\\widehat\{S\}\_\{m\}\|\_\{\\mathcal\{F\}\_\{X\}\}\\right\)\.\(31\)The Dimer also requires an orientation\. For displacement\-derived initialization, its general form is

n0=𝖳X​\(D^m\)‖𝖳X​\(D^m\)‖2,‖𝖳X​\(D^m\)‖2\>0,n\_\{0\}=\\frac\{\\mathsf\{T\}\_\{X\}\(\\widehat\{D\}\_\{m\}\)\}\{\\\|\\mathsf\{T\}\_\{X\}\(\\widehat\{D\}\_\{m\}\)\\\|\_\{2\}\},\\qquad\\\|\\mathsf\{T\}\_\{X\}\(\\widehat\{D\}\_\{m\}\)\\\|\_\{2\}\>0,\(32\)where𝖳X\\mathsf\{T\}\_\{X\}collects the orientation preprocessing and restriction to search coordinates\. In the supplied Cartesian path, the bridge removes the mean displacement for the initial axis, retains the full displacement for the center preload, and can reset the axis during loading \(Appendix[F](https://arxiv.org/html/2610.10969#A6)\)\. The center preload specifies a candidate geometry, while orientation updates estimate the local unstable direction\.

Proposition[1](https://arxiv.org/html/2610.10969#Thmproposition1)analyzes the exact minimum\-mode iteration with its specified step size\. The resulting local convergence region provides a mathematical rationale for geometric initialization\. Appendix[F](https://arxiv.org/html/2610.10969#A6)gives the finite\-difference orientation updates and search controls used in numerical refinement\.

#### Termination, saddle validation, and connectivity\.

The target in Section[3](https://arxiv.org/html/2610.10969#S3)combines stationarity, the index\-one condition, and downhill connectivity fromSe±ϵ​v1​\(Se\)S\_\{e\}\\pm\\epsilon v\_\{1\}\(S\_\{e\}\)toXXand a distinct minimumXe\+X\_\{e\}^\{\+\}\. Together these conditions define the transition tuple and barrier

e=\(X,Se,Xe\+,Δ​Ee\),Δ​Ee=V⁡\(Se\)−V⁡\(X\)\>0\.e=\(X,S\_\{e\},X\_\{e\}^\{\+\},\\Delta E\_\{e\}\),\\qquad\\Delta E\_\{e\}=V\(S\_\{e\}\)\-V\(X\)\>0\.\(33\)The main\-text idealized mapℛX\\mathcal\{R\}\_\{X\}encodes these target conditions\. The native Dimer/SEAKMC implementation uses force/step stopping, two\-sided relaxation, final flags, and cap checks as its operational acceptance protocol \(Appendix[F](https://arxiv.org/html/2610.10969#A6)\)\.

#### Search work in this benchmark\.

The main\-text evaluation caps ASPIRE refinement at 300 translation steps and random\-start Dimer at 450\. The predictor hasM=96M=96slots; with judge selection, fewer thanMMproposals may be submitted\. The random baseline uses up to 80 searches per configuration\. The slot count specifies the proposal pool, the translation cap controls each search, and force\-call counts measure its computational work, including orientation updates\.

The Dimer search\-force ledger counts calls routed through its force routine for recorded searches, including failed and duplicate searches\. It excludes model inference and force evaluations attributed to separate setup and downhill relaxation\. Wall time is reported separately\. Dimer separation, rotation limits, force tolerances, and step\-size controls are run\-specific settings described in the numerical refinement protocol\.

### A\.5Transition rates and kinetic Monte Carlo

#### From a saddle to an event rate\.

Transition\-state theory associates a rate with escape through a transition state\. Under its harmonic approximation, an event rate has Arrhenius form

ke=νe​exp⁡\(−Δ​EekB​T\)\.k\_\{e\}=\\nu\_\{e\}\\exp\\\!\\left\(\-\\frac\{\\Delta E\_\{e\}\}\{k\_\{\\mathrm\{B\}\}T\}\\right\)\.\(34\)The prefactorνe\\nu\_\{e\}accounts for the local vibrational contribution\. Forddunconstrained vibrational degrees of freedom after zero modes have been removed, the classical harmonic expression is

νe=12​π​∏i=1dωi​\(X\)∏i=2dωi​\(Se\),\\nu\_\{e\}=\\frac\{1\}\{2\\pi\}\\frac\{\\prod\_\{i=1\}^\{d\}\\omega\_\{i\}\(X\)\}\{\\prod\_\{i=2\}^\{d\}\\omega\_\{i\}\(S\_\{e\}\)\},\(35\)whereωi\\omega\_\{i\}are angular frequencies from the mass\-weighted Hessian and the unstable saddle mode is omitted from the denominator\([Vineyard, 1957](https://arxiv.org/html/2610.10969#bib.bib36)\)\. The benchmark uses the common\-prefactor conventionke=νexp\[−ΔEe/\(kBT\)\]k\_\{e\}=\\nu\\exp\[\-\\Delta E\_\{e\}/\(k\_\{\\mathrm\{B\}\}T\)\]for reference weights\. The common factor cancels from RWC and relative event probabilities; absolute waiting times retain its scale\.

#### State selection and the clock\.

The jump\-process description assumes that relaxation within a metastable basin is fast relative to escape, so that event rates can be treated as constant while the system remains in that state\. KMC advances the system through discrete transitions using their event rates\([Voter, 2007](https://arxiv.org/html/2610.10969#bib.bib37)\)\. For a nonempty event setℰX\\mathcal\{E\}\_\{X\}with total rate

ΛX=∑e∈ℰXke,\\Lambda\_\{X\}=\\sum\_\{e\\in\\mathcal\{E\}\_\{X\}\}k\_\{e\},\(36\)a residence\-time step draws independentu1,u2∼Uniform⁡\(0,1\)u\_\{1\},u\_\{2\}\\sim\\operatorname\{Uniform\}\(0,1\)and uses

e∗=min\{j:∑e=1jke≥u1ΛX\},X←Xe∗\+,Δ​t=−log⁡u2ΛX\.\\begin\{gathered\}e^\{\*\}=\\min\\\!\\left\\\{j:\\sum\_\{e=1\}^\{j\}k\_\{e\}\\geq u\_\{1\}\\Lambda\_\{X\}\\right\\\},\\qquad X\\leftarrow X\_\{e^\{\*\}\}^\{\+\},\\\\ \\Delta t=\-\\frac\{\\log u\_\{2\}\}\{\\Lambda\_\{X\}\}\.\\end\{gathered\}\(37\)Event indices can be ordered arbitrarily for the cumulative sum\. The event probability iske/ΛXk\_\{e\}/\\Lambda\_\{X\}, and the physical waiting time is exponential with mean1/ΛX1/\\Lambda\_\{X\}\. Event selection and clock advancement use independent random draws\([Voter, 2007](https://arxiv.org/html/2610.10969#bib.bib37)\)\.

#### On\-the\-fly discovery and SEAKMC\.

When the accessible products are unknown, an on\-the\-fly method first constructs an event catalogue at the current state\. SEAKMC combines defect identification, local active\-volume searches, transition processing, and KMC event selection\([Xu et al\., 2012](https://arxiv.org/html/2610.10969#bib.bib38);[Xu et al\., 2015](https://arxiv.org/html/2610.10969#bib.bib40)\)\. After a selected transition, the affected local environments and event catalogues are updated\. Repeated event discovery therefore motivates efficient proposal generation across the configurations encountered during simulation\.

ASPIRE targets the proposal\-and\-refinement stage of this process\. Its experiments evaluate distinct reference\-event recovery and search efficiency from fixed initial configurations\. These measurements assess the saddle\-discovery component that supplies mechanisms for downstream event\-based modeling\.

#### Why distinct events and rate mass both matter\.

Each distinct physical exit channel contributes to the total escape rate\. Repeated searches reaching one event count as one recovered identity, while symmetry\-related exits with equal rates can contribute separately\. Combining channels into a common catalogue entry therefore retains their summed rates or multiplicities \(Appendix[B\.7](https://arxiv.org/html/2610.10969#A2.SS7)\)\.

For a fixed reference set and its recovered subset,

ΛXref=∑e∈ℰXrefke,ΛXrec=∑e∈ℰXrecke,RWCX=ΛXrecΛXref\.\\Lambda\_\{X\}^\{\\mathrm\{ref\}\}=\\sum\_\{e\\in\\mathcal\{E\}\_\{X\}^\{\\mathrm\{ref\}\}\}k\_\{e\},\\qquad\\Lambda\_\{X\}^\{\\mathrm\{rec\}\}=\\sum\_\{e\\in\\mathcal\{E\}\_\{X\}^\{\\mathrm\{rec\}\}\}k\_\{e\},\\qquad\\mathrm\{RWC\}\_\{X\}=\\frac\{\\Lambda\_\{X\}^\{\\mathrm\{rec\}\}\}\{\\Lambda\_\{X\}^\{\\mathrm\{ref\}\}\}\.\(38\)Keeping the original reference rates, a one\-step model restricted to the recovered subset has mean waiting time larger by1/RWCX1/\\mathrm\{RWC\}\_\{X\}whenRWCX\>0\\mathrm\{RWC\}\_\{X\}\>0\. Also, a rate\-weighted draw from the reference set belongs to the recovered subset with probabilityRWCX\\mathrm\{RWC\}\_\{X\}\. These are the single\-state interpretations used in the main text\. An empty recovered set has no finite escape time under that restricted model\.

The benchmark measures distinct reference\-event recovery and search work across held\-out states\. Coverage summarizes recovered event identities, while RWC quantifies their reference escape\-rate mass\. These metrics provide complementary measures of the quality and efficiency of saddle discovery\.

### A\.6Interface to the main\-text learning problem

The physical environment, fixed potential, and consistent Cartesian representation define the inputXXand targetsDe=Se−XD\_\{e\}=S\_\{e\}\-X\. The predictor mapsXXto\{D^m\}m=1M\\\{\\widehat\{D\}\_\{m\}\\\}\_\{m=1\}^\{M\}without product endpoints\. The encoder’sXa−XbX\_\{a\}\-X\_\{b\}, decoder’sXa−bmX\_\{a\}\-b\_\{m\}, full SmoothL1 residual, and reference matcher use this shared representation, connecting the physical task to geometric learning and refinement\.

## Appendix BDataset Construction and bcc\-Symmetry Deduplication

### B\.1Physical construction and recorded examples

Both collections start from a periodic40×40×4040\\times 40\\times 40conventional\-cell bcc Fe block containing 128,000 atoms before vacancy removal\. One to four ideal lattice sites are removed, and the parent is relaxed through SEAKMC under the M07 EAM potential\. The lattice parameter is approximatelya0=2\.85528​Åa\_\{0\}=2\.85528\\,\\text\{\\AA\}and the cubic box length is approximately114\.2112​Å114\.2112\\,\\text\{\\AA\}; the stored cell specifies the exact geometry\. Forces and energies are evaluated with LAMMPS usingeam/fsandM07\_eam\.fs\. Positions and energies use Å and eV, respectively; the physical background is given in Appendix[A\.1](https://arxiv.org/html/2610.10969#A1.SS1)\.

The learning input is the relaxed local active\-volume \(AV\) arrayXXof radius11​Å11\\,\\text\{\\AA\}\. Each example pairsXXwith a variable\-size, unordered reference event set collected through 80 random\-start Dimer searches and SEAKMC validation, retaining barriers in\[0\.1,2\.0\]​eV\[0\.1,2\.0\]\\,\\mathrm\{eV\}\. Table[3](https://arxiv.org/html/2610.10969#A2.T3)gives the dataset sizes and their roles in this paper\. All reported evaluations useBccFe\-4vacAV\-4000;BccFe\-1to4Vacis constructed for future KMC trajectory simulation and analysis\.

Table 3:Dataset records\.Each state is one retained local AV\. The two collections use their respective AV\-selection protocols and have complementary roles in evaluation and future trajectory studies\.#### Reference\-collection settings\.

The common generation settings are summarized in Table[4](https://arxiv.org/html/2610.10969#A2.T4)\. The active, buffer, and fixed settings describe the physical AV construction; they are distinct from the encoder’s graph cutoff\. WithSearchBuffer=False, the search coordinates comprise the active region\.

Table 4:Reference\-catalogue generation settings\.Dataset collection uses the following protocol; evaluation settings are specified in Appendix[F](https://arxiv.org/html/2610.10969#A6)\.Reference collection uses a 900\-translation limit\. The main\-text evaluation uses caps of 300 for ASPIRE refinement and 450 for conventional Dimer \(Appendix[F](https://arxiv.org/html/2610.10969#A6)\); search records belong to their respective collection or evaluation stage\.

### B\.2Four\-vacancy benchmark construction

#### Vacancy topology enumeration and parent sampling\.

ForBccFe\-4vacAV\-4000, vacancies occupy ideal bcc sites represented as\(a0/2\)​\(i,j,k\)\(a\_\{0\}/2\)\(i,j,k\), with the three integers either all even or all odd\. Candidate four\-site patterns have maximum pairwise vacancy distance strictly below11​Å11\\,\\text\{\\AA\}\. Identifying patterns under periodic lattice translations and all 48 operations of the cubic point groupOhO\_\{h\}gives 21,297 symmetry\-distinct candidate topologies\. The symmetry operations and a mathematical canonical key are described below\.

Parent sampling selects 4,000 configurations using seed20260818\. All candidates in the point\-defect\-reduction \(PDR\) fragmentation classes44,3\+13\+1, and2\+22\+2are retained; the remaining classes are sampled across 24 diameter bins\. The selected class counts are given in Table[5](https://arxiv.org/html/2610.10969#A2.T5)\. The seed and stratification specify parent selection for the 4,000\-configuration corpus\.

Table 5:Four\-vacancy parent sampling\.Counts are for the recorded PDR fragmentation signatures used in dataset construction\.PDR denotes point\-defect reduction in the defect\-detection procedure\. The table records the parent\-sampling categories; selection of the fixed 50\-state evaluation subset is described separately in Appendix[L\.1](https://arxiv.org/html/2610.10969#A12.SS1)\.

#### One\-AV Wigner–Seitz construction\.

For each relaxed parent, Wigner–Seitz \(WS\) detection against the full perfect reference first determines the possible PDR groups\. A target group is selected that reproduces itself when the other vacancy markers are hidden\. The physical parent retains all four vacancies\. The case\-specific WS reference is formed by removing the non\-target vacancy sites from the perfect reference, so only the target\-group vacancies appear as WS differences\. This yields exactly one selected AV per parent, with all four physical vacancies inside its active sphere\. The WS matching radius is1\.03785​Å1\.03785\\,\\text\{\\AA\}, obtained fromDCut4Def×cutdefectmax=0\.3×3\.4595​Å\\texttt\{DCut4Def\}\\times\\texttt\{cutdefectmax\}=0\.3\\times 3\.4595\\,\\text\{\\AA\}, andNPredef=0\.

Selecting one AV preserves its reference labels during collection\. The reduced WS reference defines defect detection while retaining the physical parent and potential\-energy surface\. The resulting local\-environment corpus contains 65,450 connected events, with 2–43 per AV and a mean of 16\.36\.

### B\.3One\-to\-four\-vacancy collection

#### Candidate coverage\.

The initial candidate bank contains 8,192 ideal vacancy arrangements: 1, 95, 2,048, and 6,048 with one, two, three, and four vacancies, respectively\. Candidates are filtered under translations and all 48OhO\_\{h\}operations\. Coverage includes compact clusters, fragmented motifs, spatially separated arrangements, and directly constructed vacancy\-hop endpoints\. Hop\-related descendants retain their source family assignment\. This designed candidate bank provides broad structural coverage for the subsequent AV construction and release filtering\.

#### Native CN construction and independent WS audit\.

Production AVs use native coordination\-number \(CN\) defect detection,NPredef=1, native PDR, and the shared11/1/11​Å11/1/11\\,\\text\{\\AA\}settings\. WS is run only on a deep copy of the relaxed full parent against the full perfect reference to audit vacancy content\. An eligible native AV must contain all intended parent vacancies in its active region, have the expected vacancy count, and have no WS interstitial\. Eligibility is evaluated at the native center against the full perfect reference; parents without an eligible native AV are recorded as ineligible\.

The primary AV maximizes the minimum vacancy\-to\-active\-boundary margin, with randomized ties within0\.25​Å0\.25\\,\\text\{\\AA\}bands\. Approximately one ninth of topology families may contribute a second distinct native AV, with at most two searched AVs per parent\. This policy builds a structurally diverse collection with explicit parent and family provenance\.

#### Full\-geometry deduplication and event collection\.

Before Dimer search, complete active, buffer, and fixed coordinates are compared under translations and all 48 cubic operations using species\-preserving bijective matching at0\.001​Å0\.001\\,\\text\{\\AA\}tolerance\. A shared registry prevents concurrent searches of congruent geometries\. Subset selection preserves the native coordinates, atom regions, AV indices, and center provenance\. Native per\-AV search and postprocessing use 80 starts, and labels are archived before cross\-AV deletion\. The collection wrapper stops before KMC event selection and propagation\.

The released collection contains 5,372 deduplicated AVs from 4,812 parent IDs and 83,372 events, distributed as in Table[6](https://arxiv.org/html/2610.10969#A2.T6)\. The number of AV examples is therefore different from the number of distinct parent IDs\.

Table 6:One\-to\-four\-vacancy release\.Counts summarize the final AV collection after construction and filtering\.

### B\.4Reference\-event records and corpus distinction

Each example groups the retained reference events by their shared initial AV, providing direct supervision for the one\-to\-many prediction task\. Core labels include initial, saddle, and final positions; barriers; energy biases; connectivity; and event/search identifiers\. Vacancy topology, AV selection, and parent/family metadata record provenance\. Compatible atom ordering and coordinate representations are retained when formingDe=Se−XD\_\{e\}=S\_\{e\}\-X, as described in Appendix[A\.2](https://arxiv.org/html/2610.10969#A1.SS2)\. Vacancy patterns record construction provenance\. Reference rates use the common\-prefactor convention, consistent withCalPrefactor=Falseduring collection\.

The fullBccFe\-1to4Vacrepresentation retains complete AV geometry and region masks\. Its compact representation stores active\-atom arrays and keeps the static buffer/fixed coordinates once incontext\_pos0; original ordering and metadata are retained undersource\_\_\*\. The providedcompact\_io\.load\_full\_avreader reconstructs the full representation, preserving the inputXXand displacement fields used by the learning objective\.

The 80\-start collection protocol and barrier window define the observed reference catalogue for each state\. Deduplication retains distinct events, and evaluation reports recovery relative to these catalogues\.

The corpora provide complementary local\-environment collections\. The four\-vacancy benchmark has one WS\-selected AV per parent; the one\-to\-four\-vacancy collection uses native\-CN AVs with an independent WS audit and can retain a second AV\. Each corpus is deduplicated using its construction protocol\.

### B\.5The 48 cubic operations used for configuration deduplication

Configuration deduplication uses the 48 operations of the full cubic point groupOh=m​3¯​mO\_\{h\}=m\\overline\{3\}m, comprising 24 proper rotations and 24 improper operations\. In conventional cubic coordinates, it has the explicit matrix representation

𝒢bcc=\{G=diag\(ς1,ς2,ς3\)Cπ:ςi∈\{−1,\+1\},π∈𝔖3\},\|𝒢bcc\|=233\!=48,\\mathcal\{G\}\_\{\\mathrm\{bcc\}\}=\\left\\\{G=\\operatorname\{diag\}\(\\varsigma\_\{1\},\\varsigma\_\{2\},\\varsigma\_\{3\}\)C\_\{\\pi\}:\\varsigma\_\{i\}\\in\\\{\-1,\+1\\\},\\ \\pi\\in\\mathfrak\{S\}\_\{3\}\\right\\\},\\qquad\|\\mathcal\{G\}\_\{\\mathrm\{bcc\}\}\|=2^\{3\}\\,3\!=48,\(39\)whereCπC\_\{\\pi\}is a3×33\\times 3coordinate\-permutation matrix\. This is different from theN×NN\\times Natom\-permutation matrixPPused in the main\-text equivariance statement\.

The six coordinate orders are

\(x,y,z\),\(x,z,y\),\(y,x,z\),\(y,z,x\),\(z,x,y\),\(z,y,x\)\.\\begin\{gathered\}\(x,y,z\),\\ \(x,z,y\),\\ \(y,x,z\),\\\\ \(y,z,x\),\\ \(z,x,y\),\\ \(z,y,x\)\.\\end\{gathered\}\(40\)For each order, all eight independent sign choices are included\. Every matrix in Equation \([39](https://arxiv.org/html/2610.10969#A2.E39)\) is orthogonal\. Its determinant is\(ς1​ς2​ς3\)​sgn⁡\(π\)\(\\varsigma\_\{1\}\\varsigma\_\{2\}\\varsigma\_\{3\}\)\\operatorname\{sgn\}\(\\pi\), so 24 matrices have determinant\+1\+1and 24 have determinant−1\-1\. The latter include reflections and inversion combined with rotations; for example,diag⁡\(−1,1,1\)\\operatorname\{diag\}\(\-1,1,1\)reflects one coordinate and−I\-Iis inversion\.

These matrices preserve the bcc lattice: a coordinate permutation and sign changes preserve the all\-even or all\-odd condition in Equation \([17](https://arxiv.org/html/2610.10969#A1.E17)\)\. The point group is finite, while lattice translations belong to the space\-group description and are handled separately\. A symmetric vacancy pattern can have fewer than 48 distinct transformed representatives because multiple group operations may yield the same unordered set\.

### B\.6Equivalence of vacancy patterns and an explicit canonical key

Letnvn\_\{\\mathrm\{v\}\}be the number of initial vacancies and let𝒱=\{v1,…,vnv\}⊂Λbcc\\mathcal\{V\}=\\\{v\_\{1\},\\ldots,v\_\{n\_\{\\mathrm\{v\}\}\}\\\}\\subset\\Lambda\_\{\\mathrm\{bcc\}\}be their unordered parent\-lattice positions, represented as row vectors\. The vacancy\-pattern reduction uses one common cubic operation and a lattice translation\. In a consistent local representation, the equivalence criterion is

𝒱′∼𝒱⟺∃G∈𝒢bcc,t∈Λbcc:𝒱′=\{vG⊤\+t:v∈𝒱\}\.\\mathcal\{V\}^\{\\prime\}\\sim\\mathcal\{V\}\\quad\\Longleftrightarrow\\quad\\exists G\\in\\mathcal\{G\}\_\{\\mathrm\{bcc\}\},\\ t\\in\\Lambda\_\{\\mathrm\{bcc\}\}:\\mathcal\{V\}^\{\\prime\}=\\\{vG^\{\\top\}\+t:v\\in\\mathcal\{V\}\\\}\.\(41\)A common transform acts on the whole pattern and preserves its geometry\. Translation removal defines the comparison key; the stored AV retains its native coordinate representation\.

For completeness, one exact canonicalization of Equation \([41](https://arxiv.org/html/2610.10969#A2.E41)\) is as follows\. Writezi=2​vi/a0∈ℤ3z\_\{i\}=2v\_\{i\}/a\_\{0\}\\in\\mathbb\{Z\}^\{3\}\. For each of the 48 matrices and each choice of vacancyjjas a translation anchor, form the unordered relative set

𝒵G,j=\{\(zi−zj\)G⊤:i=1,…,nv\}\.\\mathcal\{Z\}\_\{G,j\}=\\\{\(z\_\{i\}\-z\_\{j\}\)G^\{\\top\}:i=1,\\ldots,n\_\{\\mathrm\{v\}\}\\\}\.\(42\)Sort its rows lexicographically, concatenate the sorted triples, and take the lexicographically smallest resulting integer tuple:

keybcc⁡\(𝒱\)=minG∈𝒢bcc1≤j≤nv⁡lex⁡\(sortrow⁡\(𝒵G,j\)\)\.\\operatorname\{key\}\_\{\\mathrm\{bcc\}\}\(\\mathcal\{V\}\)=\\min\_\{\\begin\{subarray\}\{c\}G\\in\\mathcal\{G\}\_\{\\mathrm\{bcc\}\}\\\\ 1\\leq j\\leq n\_\{\\mathrm\{v\}\}\\end\{subarray\}\}\\operatorname\{lex\}\\\!\\left\(\\operatorname\{sort\}\_\{\\mathrm\{row\}\}\(\\mathcal\{Z\}\_\{G,j\}\)\\right\)\.\(43\)Sorting handles the absence of vacancy labels; enumeration of all anchors removes dependence on which vacancy was chosen as the origin\. For four vacancies, this construction inspects at most48×4=19248\\times 4=192candidate relative sets, accounting for all four choices of translation anchor\.

To see why the key works, a lattice translation cancels in everyzi−zjz\_\{i\}\-z\_\{j\}, a vacancy reordering only permutes candidate anchors and sorted rows, and applying a cubic operation permutes the 48 group elements by composition\. Hence equivalent patterns have equal keys\. Conversely, equality of two such sorted relative sets supplies a common cubic transformation and translation between their parent vacancy positions\. This construction therefore realizes the stated equivalence without a floating\-point tolerance on ideal lattice coordinates\.

Equation \([43](https://arxiv.org/html/2610.10969#A2.E43)\) gives an explicit canonicalization of the stated equivalence relation\. A fixed\-center implementation can enumerate 48 centered transforms using its chosen origin convention\. The canonical construction provides a mathematical reference for these implementations\.

#### Local representation and relaxed coordinates\.

Before canonicalization, the parent vacancy sites are expressed in the consistent local Cartesian representation described in Appendix[A\.2](https://arxiv.org/html/2610.10969#A1.SS2)\.

The integer key acts on parent vacancy sites\. The one\-to\-four\-vacancy collection additionally compares relaxed full\-AV geometries at0\.001​Å0\.001\\,\\text\{\\AA\}tolerance \(Appendix[B\.3](https://arxiv.org/html/2610.10969#A2.SS3)\)\. Together, canonical vacancy patterns and full\-geometry checks distinguish configuration identity at their respective stages\.

#### Transforming a complete event record\.

When changing the representative of a configuration, the same spatial operation and atom correspondence must be applied to its entire record\. For atom permutationPP, cubic operationGG, and row translationtt,

X′=PXG⊤\+𝟏t,S′e=PSeG⊤\+𝟏t,\(Xe\+\)′=PXe\+G⊤\+𝟏t,D′e=PDeG⊤\.\\begin\{gathered\}X^\{\\prime\}=PXG^\{\\top\}\+\\mathbf\{1\}t,\\qquad S^\{\\prime\}\_\{e\}=PS\_\{e\}G^\{\\top\}\+\\mathbf\{1\}t,\\\\ \(X\_\{e\}^\{\+\}\)^\{\\prime\}=PX\_\{e\}^\{\+\}G^\{\\top\}\+\\mathbf\{1\}t,\\qquad D^\{\\prime\}\_\{e\}=PD\_\{e\}G^\{\\top\}\.\\end\{gathered\}\(44\)All atom\-indexed masks and any physical boundary data must transform consistently\. Because the EAM expression depends on interatomic distances, these transformations preserve its energies and barriers when the complete physical environment is transformed consistently, as follows from Equation \([19](https://arxiv.org/html/2610.10969#A1.E19)\)\. The neural transformation law is established separately in Appendix[J\.2](https://arxiv.org/html/2610.10969#A10.SS2)\.

### B\.7Configuration deduplication, event matching, and data splitting

#### Configuration deduplication\.

The 48\-operation rule groups cubic\-symmetry\-equivalent initial vacancy environments into configuration classes\. Each retained state then has its own reference set of saddle mechanisms and connected products\.

#### Event identity within a fixed state\.

The main\-text reference matcher compares events from the same representedXXusing their full atomwise displacements and barriers:

maxa⁡‖De,a−De′,a‖2≤0\.05​Å,\|Δ​Ee−Δ​Ee′\|≤0\.02​eV\.\\max\_\{a\}\\\|D\_\{e,a\}\-D\_\{e^\{\\prime\},a\}\\\|\_\{2\}\\leq 0\.05\\,\\text\{\\AA\},\\qquad\|\\Delta E\_\{e\}\-\\Delta E\_\{e^\{\\prime\}\}\|\\leq 0\.02\\,\\mathrm\{eV\}\.\(45\)This is the event\-matching criterion in Section[5](https://arxiv.org/html/2610.10969#S5), with no additional minimization over cubic operations\. It is distinct from both initial\-configuration deduplication and the moving\-union RMS used for judge labels\. During recovery evaluation the reference catalogue stays fixed, an output receives at most one matching reference identity, and repeated hits to that identity count once\. Fixed reference identities define the recovery denominator, providing a consistent basis for comparing outputs across methods\.

Retaining spatial exit channels preserves their contribution to escape rates\. For an isolated vacancy, nearest\-neighbor exchanges in different directions can be related by cubic symmetry and have equal barriers\. They move the vacancy to different positions and contribute separately to the total escape rate\. Event\-level evaluation retains these channels and their multiplicities\.

#### Grouping for the four\-vacancy train/monitor/test split\.

All targets sharing an initial state belong to the same split\. The reportedBccFe\-4vacAV\-4000split contains 3,700 training, 100 monitoring, and 200 held\-out states\. The 200\-state recovery evaluation uses that test split; the fixed 50\-state ablation and wall\-time subset is described in Appendix[L\.1](https://arxiv.org/html/2610.10969#A12.SS1)\. Both evaluation populations are drawn from the four\-vacancy test bank\.

State\-level grouping keeps the reference events of each configuration together\. Parent\-corpus construction and assignment to training, monitoring, and test partitions are separate data\-preparation stages\.

For a split intended to separate symmetry\-equivalence classes, all symmetry\-equivalent copies must remain in one group\. If𝒦train\\mathcal\{K\}\_\{\\mathrm\{train\}\},𝒦monitor\\mathcal\{K\}\_\{\\mathrm\{monitor\}\}, and𝒦test\\mathcal\{K\}\_\{\\mathrm\{test\}\}are the configuration\-key sets, the integrity condition is

𝒦train∩𝒦monitor=𝒦train∩𝒦test=𝒦monitor∩𝒦test=∅\.\\mathcal\{K\}\_\{\\mathrm\{train\}\}\\cap\\mathcal\{K\}\_\{\\mathrm\{monitor\}\}=\\mathcal\{K\}\_\{\\mathrm\{train\}\}\\cap\\mathcal\{K\}\_\{\\mathrm\{test\}\}=\\mathcal\{K\}\_\{\\mathrm\{monitor\}\}\\cap\\mathcal\{K\}\_\{\\mathrm\{test\}\}=\\varnothing\.\(46\)This criterion defines symmetry\-level split integrity and can be checked directly from the released configuration\-key sets\.

#### One\-to\-four\-vacancy release assignments\.

The separateBccFe\-1to4Vacrelease manifest supplies 4,171/543/658 train/validation/test AV assignments\. The generation record reports that no parent ID or family ID is shared across these partitions\. These release assignments support future studies using the broader vacancy collection; the reported experiments use the four\-vacancy split\. The single one\-vacancy AV supplies an elementary reference case within the release\.

## Appendix CArchitecture and Training Details

### C\.1Encoder

The encoder uses the represented configurationXXdefined in the main text, after the background coordinate preparation in Appendix[A\.2](https://arxiv.org/html/2610.10969#A1.SS2)\. A directed edgeb→ab\\to ais present for a nonzero separation within6​Å6\\,\\text\{\\AA\}, with at most 128 neighbors per atom\. Its features are

rb​a=Xa−Xb,db​a=∥rb​a∥2,r^b​a=rb​a/db​a,eb​a=\(ϕ⁡\(db​a\),⨁ℓ=02Y\(ℓ\)​\(r^b​a\)\),\\begin\{gathered\}r\_\{ba\}=X\_\{a\}\-X\_\{b\},\\qquad d\_\{ba\}=\\\|r\_\{ba\}\\\|\_\{2\},\\qquad\\widehat\{r\}\_\{ba\}=r\_\{ba\}/d\_\{ba\},\\\\ e\_\{ba\}=\\left\(\\phi\(d\_\{ba\}\),\\ \\bigoplus\_\{\\ell=0\}^\{2\}Y^\{\(\\ell\)\}\(\\widehat\{r\}\_\{ba\}\)\\right\),\\end\{gathered\}\(47\)whereϕ\\phiis a 128\-dimensional Gaussian radial expansion and the spherical harmonics extend through degree two\. Equation \([47](https://arxiv.org/html/2610.10969#A3.E47)\) uses the prepared Cartesian edge vector directly for graph inclusion, distances, and angular features\.

Species embeddings are augmented by an equivariant edge\-degree embedding\. Every atomic representation contains

ha=ha\(0\)⊕ha\(1\)⊕ha\(2\),ha\(ℓ\)∈ℝCℓ×\(2​ℓ\+1\),\(C0,C1,C2\)=\(128,64,32\)\.h\_\{a\}=h\_\{a\}^\{\(0\)\}\\oplus h\_\{a\}^\{\(1\)\}\\oplus h\_\{a\}^\{\(2\)\},\\qquad h\_\{a\}^\{\(\\ell\)\}\\in\\mathbb\{R\}^\{C\_\{\\ell\}\\times\(2\\ell\+1\)\},\\qquad\(C\_\{0\},C\_\{1\},C\_\{2\}\)=\(128,64,32\)\.\(48\)The last axis transforms under the degree\-ℓ\\ellrepresentation ofR∈SO⁡\(3\)R\\in\\mathrm\{SO\}\(3\), giving128\+3​\(64\)\+5​\(32\)=480128\+3\(64\)\+5\(32\)=480components per atom\. All benchmark atoms are Fe\.

Suppressing the layer index, each of the six Equiformer blocks constructs an edge representation using equivariant source and destination projections and a radially conditioned depthwise tensor product\([Liao and Smidt, 2022](https://arxiv.org/html/2610.10969#bib.bib23)\):

tb​a=TPψ⁡\(ϕ⁡\(db​a\)\)⁡\(Wsrc​hb\+Wdst​ha,Y⁡\(r^b​a\)\)\.t\_\{ba\}=\\operatorname\{TP\}\_\{\\psi\(\\phi\(d\_\{ba\}\)\)\}\\left\(W\_\{\\mathrm\{src\}\}h\_\{b\}\+W\_\{\\mathrm\{dst\}\}h\_\{a\},\\ Y\(\\widehat\{r\}\_\{ba\}\)\\right\)\.\(49\)Hereψ\\psipredicts invariant tensor\-product weights\. For each of four heads, scalar channels provide an invariant logit, while equivariant projection, gating, and a further separable tensor product provide the valuevb​a\(h\)v\_\{ba\}^\{\(h\)\}:

sb​a\(h\)=wh⊤act\(Ph\(0\)tb​a\),αb​a\(h\)=softmaxb∈𝒩⁡\(a\)sb​a\(h\),ma\(h\)=∑b∈𝒩⁡\(a\)αb​a\(h\)​vb​a\(h\)\.\\begin\{gathered\}s\_\{ba\}^\{\(h\)\}=w\_\{h\}^\{\\top\}\\operatorname\{act\}\(P\_\{h\}^\{\(0\)\}t\_\{ba\}\),\\qquad\\alpha\_\{ba\}^\{\(h\)\}=\\operatorname\{softmax\}\_\{b\\in\\mathcal\{N\}\(a\)\}s\_\{ba\}^\{\(h\)\},\\\\ m\_\{a\}^\{\(h\)\}=\\sum\_\{b\\in\\mathcal\{N\}\(a\)\}\\alpha\_\{ba\}^\{\(h\)\}v\_\{ba\}^\{\(h\)\}\.\\end\{gathered\}\(50\)The heads are concatenated and equivariantly projected\. Normalization, residual connections, and an equivariant feed\-forward update complete the block\. Attention\-weight dropout is0\.20\.2during training; Equation \([50](https://arxiv.org/html/2610.10969#A3.E50)\) gives the pre\-dropout weights\. The output features used in the main text areua∈ℝ128u\_\{a\}\\in\\mathbb\{R\}^\{128\}andVa∈ℝ64×3V\_\{a\}\\in\\mathbb\{R\}^\{64\\times 3\}\.

The equivariance result in Appendix[J\.2](https://arxiv.org/html/2610.10969#A10.SS2)assumes atom\-reindexing\-compatible graph construction and neighbor truncation, including a compatible convention when the neighbor cap splits a distance tie\.

### C\.2Slot refinement and cross\-attention

The main model hasM=96M=96,ds=256d\_\{s\}=256, anddv=16d\_\{v\}=16\. We use the main\-text stage index throughout: initialization ist=0t=0, the two vector–scalar stages aret=1,2t=1,2, the three scalar\-only stages aret=3,4,5t=3,4,5, and the final readout isT=6T=6\. The state entering initialization is the learned scalar seedsm\(0\)=sms\_\{m\}^\{\(0\)\}=s\_\{m\}and the zero vector stateQm\(0\)=0Q\_\{m\}^\{\(0\)\}=0\. Each slot acquires directional information through attention\-weighted pooling of the encoded atomic vectors\.

#### Initialization\.

The optional slot self\-attention in Equation \([4](https://arxiv.org/html/2610.10969#S4.E4)\) acts in place on the scalar seeds\. It is enabled for the full\-model setting and disabled for the cross\-attention\-only ablation in Table[2](https://arxiv.org/html/2610.10969#S5.T2)\. With these scalar states, the initialization attention and vector readout are

Am​a\(0,1\)=softmaxa⁡\[\(Wq​sm\(0\)\)⊤​\(Wk​ua\(0\)\)ds\],Um\(0,1\)=∑aA\(0,1\)m​aWVVa,sm\(1\)=sm\(0\),Qm\(1\)=Um\(0,1\)\.\\begin\{gathered\}A^\{\(0,1\)\}\_\{ma\}=\\operatorname\{softmax\}\_\{a\}\\\!\\left\[\\frac\{\(W\_\{q\}s\_\{m\}^\{\(0\)\}\)^\{\\top\}\(W\_\{k\}u\_\{a\}^\{\(0\)\}\)\}\{\\sqrt\{d\_\{s\}\}\}\\right\],\\\\ U\_\{m\}^\{\(0,1\)\}=\\sum\_\{a\}A^\{\(0,1\)\}\_\{ma\}W\_\{V\}V\_\{a\},\\qquad s\_\{m\}^\{\(1\)\}=s\_\{m\}^\{\(0\)\},\\qquad Q\_\{m\}^\{\(1\)\}=U\_\{m\}^\{\(0,1\)\}\.\\end\{gathered\}\(51\)Hereua\(0\)u\_\{a\}^\{\(0\)\}is an MLP embedding128→256→256128\\to 256\\to 256ofuau\_\{a\}\. The zero initial vector state andγ\(0,1\)=0\\gamma^\{\(0,1\)\}=0reduce the main\-text template to this scalar\-attention bootstrap\.

#### Two vector–scalar updates\.

Fort=1,2t=1,2, writeua\(t\)=ua\(0\)u\_\{a\}^\{\(t\)\}=u\_\{a\}^\{\(0\)\}\. The shared scalar maps areWq,Wk,WzW\_\{q\},W\_\{k\},W\_\{z\}, and the shared atomic vector maps areWK,WVW\_\{K\},W\_\{V\}\. The slot\-vector maps, scalar update MLPs, vector update maps, and compatibility coefficients are stage specific\. Omitting only the single head index, the main\-text attention becomes

Am​a\(t\)=softmaxa⁡\[\(Wq​sm\(t\)\)⊤​\(Wk​ua\(t\)\)ds\+γ\(t\)3​dv​⟨W~Q\(t\)​Qm\(t\),WK​Va⟩F\]\.A\_\{ma\}^\{\(t\)\}=\\operatorname\{softmax\}\_\{a\}\\\!\\left\[\\frac\{\(W\_\{q\}s\_\{m\}^\{\(t\)\}\)^\{\\top\}\(W\_\{k\}u\_\{a\}^\{\(t\)\}\)\}\{\\sqrt\{d\_\{s\}\}\}\+\\frac\{\\gamma^\{\(t\)\}\}\{\\sqrt\{3d\_\{v\}\}\}\\left\\langle\\widetilde\{W\}\_\{Q\}^\{\(t\)\}Q\_\{m\}^\{\(t\)\},W\_\{K\}V\_\{a\}\\right\\rangle\_\{F\}\\right\]\.\(52\)The mapsWq,Wk,Wz,WK,WVW\_\{q\},W\_\{k\},W\_\{z\},W\_\{K\},W\_\{V\}are shared with initialization where applicable\. Initial logits use the scalar term with zero vector\-compatibility coefficient, while each update\-stageγ\(t\)\\gamma^\{\(t\)\}starts at one\. The contexts and updates are

zm\(t\)\\displaystyle z\_\{m\}^\{\(t\)\}=∑aAm​a\(t\)​Wz​ua\(t\),\\displaystyle=\\sum\_\{a\}A\_\{ma\}^\{\(t\)\}W\_\{z\}u\_\{a\}^\{\(t\)\},Um\(t\)\\displaystyle U\_\{m\}^\{\(t\)\}=∑aAm​a\(t\)​WV​Va,\\displaystyle=\\sum\_\{a\}A\_\{ma\}^\{\(t\)\}W\_\{V\}V\_\{a\},\(53\)sm\(t\+1\)\\displaystyle s\_\{m\}^\{\(t\+1\)\}=sm\(t\)\+F\(t\)​\(sm\(t\),zm\(t\),\(‖Um,v\(t\)‖2\)v=1dv\),\\displaystyle=s\_\{m\}^\{\(t\)\}\+F^\{\(t\)\}\\\!\\left\(s\_\{m\}^\{\(t\)\},z\_\{m\}^\{\(t\)\},\(\\\|U\_\{m,v\}^\{\(t\)\}\\\|\_\{2\}\)\_\{v=1\}^\{d\_\{v\}\}\\right\),Qm\(t\+1\)\\displaystyle Q\_\{m\}^\{\(t\+1\)\}=Qm\(t\)\+WU\(t\)​Um\(t\)\.\\displaystyle=Q\_\{m\}^\{\(t\)\}\+W\_\{U\}^\{\(t\)\}U\_\{m\}^\{\(t\)\}\.The scalar update is a two\-layer MLP\. The invariant vector norms communicate geometric information to scalar fingerprints; vector\-channel mixing preserves orientation\. The scalar query/key/value maps in these stages and all vector\-channel maps are bias free\.

#### Scalar\-only updates and readout\.

The three scalar\-only stages use an independent embeddingua′∈ℝ256u^\{\\prime\}\_\{a\}\\in\\mathbb\{R\}^\{256\}from another128→256→256128\\to 256\\to 256MLP, with eight heads, pre\-normalization, and the residual feed\-forward update of Equation \([8](https://arxiv.org/html/2610.10969#S4.E8)\)\. They leaveQm\(t\)Q\_\{m\}^\{\(t\)\}unchanged\. In the attention notation of the main text,s^m=N\(t\)​\(sm\(t\)\)\\hat\{s\}\_\{m\}=\\mathrm\{N\}^\{\(t\)\}\(s\_\{m\}^\{\(t\)\}\)andu^a=N\(t\)​\(ua\(t\)\)\\hat\{u\}\_\{a\}=\\mathrm\{N\}^\{\(t\)\}\(u\_\{a\}^\{\(t\)\}\), with the appropriate separately parameterized normalizations;N\(t\)\\mathrm\{N\}^\{\(t\)\}is identity in the unnormalized stages and layer normalization in the scalar\-only stages\.

The final single\-head readout atT=6T=6has its ownWq,Wk,Wz,W~Q,W~VW\_\{q\},W\_\{k\},W\_\{z\},\\widetilde\{W\}\_\{Q\},\\widetilde\{W\}\_\{V\}\. It recomputes attention withγ\(T\)=1\\gamma^\{\(T\)\}=1and returnsAm​a\(T\),zm\(T\),sm\(T\),Qm\(T\)A\_\{ma\}^\{\(T\)\},z\_\{m\}^\{\(T\)\},s\_\{m\}^\{\(T\)\},Q\_\{m\}^\{\(T\)\}for decoding, retaining the final slot states\. Stage\-specific vector maps mix channels identically across the three spatial components and use zero spatial bias\.

### C\.3Decoder details

Omitting the final stage index as in the main text, the anchor is

bm=∑aAm​a​Xa,∑aAm​a=1\.b\_\{m\}=\\sum\_\{a\}A\_\{ma\}X\_\{a\},\\qquad\\sum\_\{a\}A\_\{ma\}=1\.\(54\)The offsetXa−bmX\_\{a\}\-b\_\{m\}encodes each atom’s position relative to the attention\-weighted anchor\. Computing this mean in the prepared local Cartesian environment gives the decoder a geometry\-conditioned spatial reference\.

The atom\-derived generators are the 32 rows ofWatom​VaW\_\{\\mathrm\{atom\}\}V\_\{a\}, and the slot\-derived generators are the eight rows ofWslot​QmW\_\{\\mathrm\{slot\}\}Q\_\{m\}\. Higher\-order encoder channels contribute through equivariant message passing\. Together with the anchor\-relative direction, they form the 41\-row vector bankℬm​a\\mathcal\{B\}\_\{ma\}in Equation \([9](https://arxiv.org/html/2610.10969#S4.E9)\), combining three sources of geometric information for displacement synthesis\.

The decoder uses the invariant conditioning

ξm​a=\[sm,zm,ua′,χm​a,log⁡\(1\+‖Xa−bm‖22\)\],χm​a=\(13​⟨\(GQ​Qm\)g,\(GV​Va\)g⟩\)g=18\.\\begin\{gathered\}\\xi\_\{ma\}=\\left\[s\_\{m\},z\_\{m\},u^\{\\prime\}\_\{a\},\\chi\_\{ma\},\\log\(1\+\\\|X\_\{a\}\-b\_\{m\}\\\|\_\{2\}^\{2\}\)\\right\],\\\\ \\chi\_\{ma\}=\\left\(\\frac\{1\}\{\\sqrt\{3\}\}\\left\\langle\(G\_\{Q\}Q\_\{m\}\)\_\{g\},\(G\_\{V\}V\_\{a\}\)\_\{g\}\\right\\rangle\\right\)\_\{g=1\}^\{8\}\.\\end\{gathered\}\(55\)The coordinates are numerically expressed in the fixed input length unit\. Withsm,zm,ua′∈ℝ256s\_\{m\},z\_\{m\},u^\{\\prime\}\_\{a\}\\in\\mathbb\{R\}^\{256\},ξm​a∈ℝ777\\xi\_\{ma\}\\in\\mathbb\{R\}^\{777\}\. The mapsGQG\_\{Q\}andGVG\_\{V\}have channel widths16→816\\to 8and64→864\\to 8, respectively\.

The four MLPs follow Equation \([11](https://arxiv.org/html/2610.10969#S4.E11)\)\. Gate and scale networksgact,gmagg\_\{\\mathrm\{act\}\},g\_\{\\mathrm\{mag\}\}have widths777→256→1777\\to 256\\to 1; coefficient networksfcoef,fresf\_\{\\mathrm\{coef\}\},f\_\{\\mathrm\{res\}\}have widths777→256→256→41777\\to 256\\to 256\\to 41\. Hidden activations are SiLU, and the outputs arepm​ap\_\{ma\},κm​a\\kappa\_\{ma\},qm​aq\_\{ma\}, andqm​aresq\_\{ma\}^\{\\mathrm\{res\}\}\.

The output layers ofgact,gmag,fresg\_\{\\mathrm\{act\}\},g\_\{\\mathrm\{mag\}\},f\_\{\\mathrm\{res\}\}start with zero weights and biases−2,−3\.9,0\-2,\-3\.9,0, respectively\. The learned residual scaleγres=σ⁡\(ϑ\)\\gamma\_\{\\mathrm\{res\}\}=\\sigma\(\\vartheta\)starts at0\.050\.05\. Initial participation isσ⁡\(−2\)≈0\.12\\sigma\(\-2\)\\approx 0\.12, the magnitude parameter issoftplus⁡\(−3\.9\)≈0\.02​Å\\operatorname\{softplus\}\(\-3\.9\)\\approx 0\.02\\,\\text\{\\AA\}, and the residual output is zero\. The realized displacement magnitude also depends on the directional combination\.

The residual lies outside the activity gate in Equation \([12](https://arxiv.org/html/2610.10969#S4.E12)\), allowing small\-participation atoms to express surrounding relaxation\. Radial normalization usesv/max⁡\(‖v‖2,ϵ\)v/\\max\(\\\|v\\\|\_\{2\},\\epsilon\)withϵ=10−8\\epsilon=10^\{\-8\}, giving normmin⁡\(1,‖v‖2/ϵ\)\\min\(1,\\\|v\\\|\_\{2\}/\\epsilon\)and zero output atv=0v=0\. This stabilized geometric operation preserves the rotation law used in the equivariance proof\.

## Appendix DDisplacement Supervision and Active\-Aware Matching

### D\.1Displacement fields and the uniform objective

For each initial configurationX∈ℝN×3X\\in\\mathbb\{R\}^\{N\\times 3\}, the predictor producesMMdisplacement fieldsD^m\\widehat\{D\}\_\{m\}, supervised byKKreference fieldsDe=Se−XD\_\{e\}=S\_\{e\}\-X\. All fields share atom correspondence and Cartesian units ofÅ\. Training uses the full residualD^m−De\\widehat\{D\}\_\{m\}\-D\_\{e\}directly; inference takes the initial configuration as input\.

The scalar SmoothL1 penalty and its all\-atom reduction are

ρβ​\(t\)=\{t2/\(2​β\),\|t\|<β,\|t\|−β/2,\|t\|≥β,ℓβ​\(E\)=1N​∑a=1N∑d=13ρβ​\(Ea​d\)\.\\rho\_\{\\beta\}\(t\)=\\begin\{cases\}t^\{2\}/\(2\\beta\),&\|t\|<\\beta,\\\\ \|t\|\-\\beta/2,&\|t\|\\geq\\beta,\\end\{cases\}\\qquad\\ell\_\{\\beta\}\(E\)=\\frac\{1\}\{N\}\\sum\_\{a=1\}^\{N\}\\sum\_\{d=1\}^\{3\}\\rho\_\{\\beta\}\(E\_\{ad\}\)\.\(56\)The objective sums the three coordinate penalties and averages over allNNatoms, including those with zero reference displacement\. The main Uniform model usesCm​e=ℓβ​\(D^m−De\)C\_\{me\}=\\ell\_\{\\beta\}\(\\widehat\{D\}\_\{m\}\-D\_\{e\}\)for both assignment and matched regression, withβ=0\.03​Å\\beta=0\.03\\,\\text\{\\AA\}\.

### D\.2The active\-aware ablation

The active\-aware ablation changes both the proposal–reference assignment cost and matched displacement loss\. For each reference event, define

ge,a=∥De,a∥2,am​a=∥D^m,a∥2,ae∗=arg​maxage,a,𝒜e=\{\{a:ge,a≥δact\},if this set is nonempty,\{ae∗\},otherwise,δact=0\.05Å\.\\begin\{gathered\}g\_\{e,a\}=\\\|D\_\{e,a\}\\\|\_\{2\},\\qquad a\_\{ma\}=\\\|\\widehat\{D\}\_\{m,a\}\\\|\_\{2\},\\qquad a\_\{e\}^\{\*\}=\\operatorname\*\{arg\\,max\}\_\{a\}g\_\{e,a\},\\\\ \\mathcal\{A\}\_\{e\}=\\begin\{cases\}\\\{a:g\_\{e,a\}\\geq\\delta\_\{\\mathrm\{act\}\}\\\},&\\text\{if this set is nonempty\},\\\\ \\\{a\_\{e\}^\{\*\}\\\},&\\text\{otherwise\},\\end\{cases\}\\qquad\\delta\_\{\\mathrm\{act\}\}=0\.05\\,\\text\{\\AA\}\.\\end\{gathered\}\(57\)The active set is defined from each event’s reference displacement\. The fallback ensures a nonempty set, with tied maxima resolved by the first maximum in stored atom order\.

Let the coordinatewise penalty at one atom be

Em​e,a=∑d=13ρβ​\(D^m,a​d−De,a​d\)\.E\_\{me,a\}=\\sum\_\{d=1\}^\{3\}\\rho\_\{\\beta\}\(\\widehat\{D\}\_\{m,ad\}\-D\_\{e,ad\}\)\.The five enabled components are

ℓall,m​e\\displaystyle\\ell\_\{\\mathrm\{all\},me\}=1N​∑aEm​e,a,\\displaystyle=\\frac\{1\}\{N\}\\sum\_\{a\}E\_\{me,a\},ℓA,m​e\\displaystyle\\ell\_\{A,me\}=1\|𝒜e\|​∑a∈𝒜eEm​e,a,\\displaystyle=\\frac\{1\}\{\|\\mathcal\{A\}\_\{e\}\|\}\\sum\_\{a\\in\\mathcal\{A\}\_\{e\}\}E\_\{me,a\},\(58\)ℓmax,m​e\\displaystyle\\ell\_\{\\max,me\}=Em​e,ae∗,\\displaystyle=E\_\{me,a\_\{e\}^\{\*\}\},ℓmag,m​e\\displaystyle\\ell\_\{\\mathrm\{mag\},me\}=1\|𝒜e\|​∑a∈𝒜eρβ​\(am​a−ge,a\),\\displaystyle=\\frac\{1\}\{\|\\mathcal\{A\}\_\{e\}\|\}\\sum\_\{a\\in\\mathcal\{A\}\_\{e\}\}\\rho\_\{\\beta\}\(a\_\{ma\}\-g\_\{e,a\}\),ℓcos,m​e\\displaystyle\\ell\_\{\\cos,me\}=1\|𝒜e\|​∑a∈𝒜eωm,a​\[1−cosϵcos⁡\(D^m,a,De,a\)\]\.\\displaystyle=\\frac\{1\}\{\|\\mathcal\{A\}\_\{e\}\|\}\\sum\_\{a\\in\\mathcal\{A\}\_\{e\}\}\\omega\_\{m,a\}\\bigl\[1\-\\cos\_\{\\epsilon\_\{\\cos\}\}\(\\widehat\{D\}\_\{m,a\},D\_\{e,a\}\)\\bigr\]\.Here the cosine and its detached amplitude gate are

cosϵcos⁡\(u,v\)=⟨u,v⟩max⁡\(‖u‖2,ϵcos\)​max⁡\(‖v‖2,ϵcos\),ϵcos=10−4,ωm,a=stopgrad⁡\[clip⁡\(am​amax⁡\(δact,10−6\),0,1\)\]\.\\begin\{split\}\\cos\_\{\\epsilon\_\{\\cos\}\}\(u,v\)&=\\frac\{\\langle u,v\\rangle\}\{\\max\(\\\|u\\\|\_\{2\},\\epsilon\_\{\\cos\}\)\\max\(\\\|v\\\|\_\{2\},\\epsilon\_\{\\cos\}\)\},\\qquad\\epsilon\_\{\\cos\}=10^\{\-4\},\\\\ \\omega\_\{m,a\}&=\\operatorname\{stopgrad\}\\\!\\left\[\\operatorname\{clip\}\\\!\\left\(\\frac\{a\_\{ma\}\}\{\\max\(\\delta\_\{\\mathrm\{act\}\},10^\{\-6\}\)\},0,1\\right\)\\right\]\.\\end\{split\}\(59\)The stabilizers use the supplied length convention\. The detached gate reduces directional supervision near a zero prediction\. Each active mean is normalized by its event’s active\-atom count, and the maximum\-atom term evaluates the atom with the largest reference displacement\.

The enabled pair objective is

Cm​eactive=ℓactive,m​e=0\.25​ℓall,m​e\+ℓA,m​e\+ℓmax,m​e\+0\.5​ℓmag,m​e\+0\.1​ℓcos,m​e\.C^\{\\mathrm\{active\}\}\_\{me\}=\\ell\_\{\\mathrm\{active\},me\}=0\.25\\ell\_\{\\mathrm\{all\},me\}\+\\ell\_\{A,me\}\+\\ell\_\{\\max,me\}\+0\.5\\ell\_\{\\mathrm\{mag\},me\}\+0\.1\\ell\_\{\\cos,me\}\.\(60\)The all\-atom term remains present at reduced weight\. The active variant retainsβ=0\.03​Å\\beta=0\.03\\,\\text\{\\AA\}and the same sequential matching and configuration\-level aggregation as Uniform\. The0\.05​Å0\.05\\,\\text\{\\AA\}training activity cutoff is distinct from the judge’s0\.3​Å0\.3\\,\\text\{\\AA\}moving\-atom cutoff and its score thresholdτ\\tau\.

### D\.3Sequential assignment and loss aggregation

HereK=\|ℰXref\|K=\|\\mathcal\{E\}\_\{X\}^\{\\mathrm\{ref\}\}\|counts reference targets, following the training notation of Section[4\.4](https://arxiv.org/html/2610.10969#S4.SS4)\. Let𝒪r\\mathcal\{O\}\_\{r\}denote the slots available at roundrr, starting with𝒪1=\{1,…,M\}\\mathcal\{O\}\_\{1\}=\\\{1,\\ldots,M\\\}\. For a nonempty reference set, define the rectangular one\-to\-one assignment family

𝔐\(𝒪r,\[K\]\)=\{Π⊆𝒪r×\[K\]:\|Π\|=min⁡\(\|𝒪r\|,K\),degΠ\(m\)≤1,degΠ\(e\)≤1\}\.\\begin\{split\}\\mathfrak\{M\}\(\\mathcal\{O\}\_\{r\},\[K\]\)=\\\{\\Pi\\subseteq\\mathcal\{O\}\_\{r\}\\times\[K\]:\\;&\|\\Pi\|=\\min\(\|\\mathcal\{O\}\_\{r\}\|,K\),\\\\ &\\deg\_\{\\Pi\}\(m\)\\leq 1,\\quad\\deg\_\{\\Pi\}\(e\)\\leq 1\\\}\.\\end\{split\}\(61\)At each round, Hungarian matching solves

Πr∈arg​minΠ∈𝔐⁡\(𝒪r,\[K\]\)∑\(m,e\)∈ΠCm​e,𝒪r\+1=𝒪r∖\{m:\(m,e\)∈Πrfor somee\}\.\\Pi\_\{r\}\\in\\operatorname\*\{arg\\,min\}\_\{\\Pi\\in\\mathfrak\{M\}\(\\mathcal\{O\}\_\{r\},\[K\]\)\}\\sum\_\{\(m,e\)\\in\\Pi\}C\_\{me\},\\qquad\\mathcal\{O\}\_\{r\+1\}=\\mathcal\{O\}\_\{r\}\\setminus\\\{m:\(m,e\)\\in\\Pi\_\{r\}\\text\{ for some \}e\\\}\.\(62\)UseCm​e=ℓall,m​eC\_\{me\}=\\ell\_\{\\mathrm\{all\},me\}for Uniform andCm​e=Cm​eactiveC\_\{me\}=C^\{\\mathrm\{active\}\}\_\{me\}for the active\-aware ablation\. The detached cost matrix determines the discrete assignment; regression then propagates gradients through the predicted fields of the selected pairs\.

The procedure performs at mostRmatch=4R\_\{\\mathrm\{match\}\}=4rounds, stopping when no slots remain\. The full reference set is available again at every round\. A slot is used at most once, whereas each target can receive one slot per round\. With finite pair costs andK\>0K\>0, the total number of matched pairs is

∑r\|Πr\|=min⁡\(M,Rmatch​K\)\.\\sum\_\{r\}\|\\Pi\_\{r\}\|=\\min\(M,R\_\{\\mathrm\{match\}\}K\)\.\(63\)The number of supervised slots adapts to the reference\-set size: for example,K=2K=2gives eight matched slots whenM=96M=96\. WhenM≥KM\\geq K, the first round covers every target, and subsequent rounds assign the remaining slots against the full reference set\. Each round solves its own residual assignment problem\.

#### Primary, duplicate, and batch normalization\.

WriteΠ\+=⋃r≥2Πr\\Pi\_\{\+\}=\\bigcup\_\{r\\geq 2\}\\Pi\_\{r\}\. Using the chosen pair objectiveCm​eC\_\{me\}, define

L1=1\|Π1\|​∑\(m,e\)∈Π1Cm​e,L\+=\{\|Π\+\|−1​∑\(m,e\)∈Π\+Cm​e,\|Π\+\|\>0,0,\|Π\+\|=0\.L\_\{1\}=\\frac\{1\}\{\|\\Pi\_\{1\}\|\}\\sum\_\{\(m,e\)\\in\\Pi\_\{1\}\}C\_\{me\},\\qquad L\_\{\+\}=\\begin\{cases\}\|\\Pi\_\{\+\}\|^\{\-1\}\\displaystyle\\sum\_\{\(m,e\)\\in\\Pi\_\{\+\}\}C\_\{me\},&\|\\Pi\_\{\+\}\|\>0,\\\\ 0,&\|\\Pi\_\{\+\}\|=0\.\\end\{cases\}\(64\)The configuration loss uses the main\-text notationℒx\\mathcal\{L\}\_\{x\}and aggregation

ℒx=λD​\(L1\+λ\+​L\+\),λD=10,λ\+=0\.5\.\\mathcal\{L\}\_\{x\}=\\lambda\_\{D\}\(L\_\{1\}\+\\lambda\_\{\+\}L\_\{\+\}\),\\qquad\\lambda\_\{D\}=10,\\quad\\lambda\_\{\+\}=0\.5\.\(65\)The primary and duplicate terms are averaged separately before applying the0\.50\.5duplicate weight\. All later\-round pairs contribute to the common duplicate mean\. With one matching round or an empty duplicate set,L\+=0L\_\{\+\}=0; an empty reference set gives zero loss\. Batch reduction weights each configuration equally\.

Repeated supervision gives a reference event multiple assigned proposals across rounds\. Assigned pairs provide displacement gradients through the shared predictor, while distinct event recovery is measured after Dimer refinement and reference matching\.

### D\.4Predictor optimization and checkpoint scope

The predictor training recipe uses AdamW with learning rate8×10−58\\times 10^\{\-5\}, weight decay10−510^\{\-5\}, batch size two, gradient clipping at five, and a 120\-epoch cosine schedule\. The data split comprises 3,700 training, 100 monitoring, and 200 held\-out configurations\. Monitoring\-set first\-round displacement loss selects the predictor checkpoint\. Downstream recovery and search\-force metrics evaluate its physical\-search performance\. Appendix[E](https://arxiv.org/html/2610.10969#A5)specifies the frozen Full\-model checkpoint and monitoring procedure for the judge\.

## Appendix EInitial\-State\-Only Neural RMS Judge

### E\.1Frozen predictor and inference inputs

The neural judge scores proposals using only the initial configuration, its frozen encoder features, and the complete predicted displacement pool\. For the Full\-model pairing described here, the predictor checkpoint is epoch 103, loaded with strict parameter matching and kept in evaluation mode\. Its scalar featuresua∈ℝ128u\_\{a\}\\in\\mathbb\{R\}^\{128\}, vector channelsVa∈ℝ64×3V\_\{a\}\\in\\mathbb\{R\}^\{64\\times 3\}, and proposalsD^m\\widehat\{D\}\_\{m\}are detached from judge optimization\. This frozen\-feature readout scores the proposal pool without force evaluations\. Reference displacements supply offline training and monitoring labels; inference uses the initial state and its predicted proposals\.

### E\.2Atom features, pooling, and network widths

Letam​a=‖D^m,a‖2a\_\{ma\}=\\\|\\widehat\{D\}\_\{m,a\}\\\|\_\{2\}\. To avoid confusing the judge feature with the decoder coefficientsqm​aq\_\{ma\}, denote it byζm​a\\zeta\_\{ma\}:

ζm​a=\[ua,\(⟨D^m,a,Va,v⟩\)v=164,am​a,log⁡\(1\+am​a\)\]∈ℝ194\.\\zeta\_\{ma\}=\\left\[u\_\{a\},\\;\\bigl\(\\langle\\widehat\{D\}\_\{m,a\},V\_\{a,v\}\\rangle\\bigr\)\_\{v=1\}^\{64\},\\;a\_\{ma\},\\;\\log\(1\+a\_\{ma\}\)\\right\]\\in\\mathbb\{R\}^\{194\}\.\(66\)Numerical amplitudes in logarithms are expressed in the fixedÅconvention\. A shared atom network produceshm​aJ=ϕJ​\(ζm​a\)∈ℝ128h\_\{ma\}^\{J\}=\\phi\_\{J\}\(\\zeta\_\{ma\}\)\\in\\mathbb\{R\}^\{128\}, where

ϕJ:194→Linear192→SiLU,Dropout⁡\(0\.2\)192→Linear128→SiLU128\.\\phi\_\{J\}:194\\xrightarrow\{\\mathrm\{Linear\}\}192\\xrightarrow\{\\mathrm\{SiLU\},\\;\\mathrm\{Dropout\}\(0\.2\)\}192\\xrightarrow\{\\mathrm\{Linear\}\}128\\xrightarrow\{\\mathrm\{SiLU\}\}128\.A learned affine mapwJ⊤​h\+bJw\_\{J\}^\{\\top\}h\+b\_\{J\}gives the attention score\. The complete readout is

αm​aJ=softmaxa⁡\[wJ⊤​hm​aJ\+bJ\+3​log⁡\(1\+am​a\)\],vmJ=\[∑aαm​aJ​hm​aJ,maxa⁡hm​aJ,gm\]∈ℝ266,tmJ=ψJ\(vmJ\),p^m=σ\(tmJ\),\\begin\{gathered\}\\alpha^\{J\}\_\{ma\}=\\operatorname\{softmax\}\_\{a\}\\bigl\[w\_\{J\}^\{\\top\}h^\{J\}\_\{ma\}\+b\_\{J\}\+3\\log\(1\+a\_\{ma\}\)\\bigr\],\\\\ v\_\{m\}^\{J\}=\\left\[\\sum\_\{a\}\\alpha^\{J\}\_\{ma\}h^\{J\}\_\{ma\},\\;\\max\_\{a\}h^\{J\}\_\{ma\},\\;g\_\{m\}\\right\]\\in\\mathbb\{R\}^\{266\},\\\\ t\_\{m\}^\{J\}=\\psi\_\{J\}\(v\_\{m\}^\{J\}\),\\qquad\\widehat\{p\}\_\{m\}=\\sigma\(t\_\{m\}^\{J\}\),\\end\{gathered\}\(67\)where the maximum is channelwise andgm∈ℝ10g\_\{m\}\\in\\mathbb\{R\}^\{10\}is defined below\. The output network is

ψJ:266→Linear128→SiLU,Dropout⁡\(0\.2\)128→Linear1\.\\psi\_\{J\}:266\\xrightarrow\{\\mathrm\{Linear\}\}128\\xrightarrow\{\\mathrm\{SiLU\},\\;\\mathrm\{Dropout\}\(0\.2\)\}128\\xrightarrow\{\\mathrm\{Linear\}\}1\.Both dropout layers are disabled for scoring\. The head’s sigmoid output provides the score used for threshold selection\.

### E\.3The ten geometric descriptors, in implementation order

All descriptors are computed*before*filtering, from the full proposal pool of the configuration\. Define

ammax=maxa⁡am​a,jm∗=arg​maxa⁡am​a,𝒰m=\{a:am​a≥0\.3​Å\}\.a\_\{m\}^\{\\max\}=\\max\_\{a\}a\_\{ma\},\\qquad j\_\{m\}^\{\*\}=\\operatorname\*\{arg\\,max\}\_\{a\}a\_\{ma\},\\qquad\\mathcal\{U\}\_\{m\}=\\\{a:a\_\{ma\}\\geq 0\.3\\,\\text\{\\AA\}\\\}\.The descriptor vector is

gm=\(fmmax,fband,fnjump,fnhalo,fnclust,fsecond,fsum,fred,frank,fcoher\)m\.g\_\{m\}=\(f\_\{\\mathrm\{mmax\}\},f\_\{\\mathrm\{band\}\},f\_\{\\mathrm\{njump\}\},f\_\{\\mathrm\{nhalo\}\},f\_\{\\mathrm\{nclust\}\},f\_\{\\mathrm\{second\}\},f\_\{\\mathrm\{sum\}\},f\_\{\\mathrm\{red\}\},f\_\{\\mathrm\{rank\}\},f\_\{\\mathrm\{coher\}\}\)\_\{m\}\.The following definitions specify the raw features supplied to the head; no feature standardization is inserted\.

#### Amplitude and atom counts \(features 1–4 and 7\)\.

fmmax,m\\displaystyle f\_\{\\mathrm\{mmax\},m\}=ammax,\\displaystyle=a\_\{m\}^\{\\max\},\(68\)fband,m\\displaystyle f\_\{\\mathrm\{band\},m\}=max⁡\{0,0\.8​Å−ammax,ammax−1\.6​Å\},\\displaystyle=\\max\\\{0,\\;0\.8\\,\\text\{\\AA\}\-a\_\{m\}^\{\\max\},\\;a\_\{m\}^\{\\max\}\-1\.6\\,\\text\{\\AA\}\\\},fnjump,m\\displaystyle f\_\{\\mathrm\{njump\},m\}=\|\{a:am​a≥0\.5​Å\}\|,\\displaystyle=\\bigl\|\\\{a:a\_\{ma\}\\geq 0\.5\\,\\text\{\\AA\}\\\}\\bigr\|,fnhalo,m\\displaystyle f\_\{\\mathrm\{nhalo\},m\}=\|\{a:0\.05​Å≤am​a<0\.5​Å\}\|,\\displaystyle=\\bigl\|\\\{a:0\.05\\,\\text\{\\AA\}\\leq a\_\{ma\}<0\.5\\,\\text\{\\AA\}\\\}\\bigr\|,fsum,m\\displaystyle f\_\{\\mathrm\{sum\},m\}=∑aam​a\.\\displaystyle=\\sum\_\{a\}a\_\{ma\}\.The count features retain absolute atom counts, andfsum,mf\_\{\\mathrm\{sum\},m\}sums their displacement amplitudes\.

#### Moving\-atom clusters \(features 5–6\)\.

Construct an undirected graph on𝒰m\\mathcal\{U\}\_\{m\}, connecting two moving atoms when their*initial*positions are separated by at most3​Å3\\,\\text\{\\AA\}\. Let𝒞m,1,…,𝒞m,cm\\mathcal\{C\}\_\{m,1\},\\ldots,\\mathcal\{C\}\_\{m,c\_\{m\}\}be its connected components and letPm,j=maxa∈𝒞m,j⁡am​aP\_\{m,j\}=\\max\_\{a\\in\\mathcal\{C\}\_\{m,j\}\}a\_\{ma\}be their peak amplitudes\. Thenfnclust,m=cmf\_\{\\mathrm\{nclust\},m\}=c\_\{m\}andfsecond,mf\_\{\\mathrm\{second\},m\}is the second\-largest component peak\. An empty moving set gives\(0,0\)\(0,0\), and fewer than two components givesfsecond,m=0f\_\{\\mathrm\{second\},m\}=0\. The epoch\-103 judge computes distances as‖Xa−Xb‖2\\\|X\_\{a\}\-X\_\{b\}\\\|\_\{2\}in the prepared nonperiodic local\-coordinate representation of the active volume\.

#### Pool\-relative redundancy and rank \(features 8–9\)\.

Letℋm=\{n≠m:anmax\>ammax\}\\mathcal\{H\}\_\{m\}=\\\{n\\neq m:a\_\{n\}^\{\\max\}\>a\_\{m\}^\{\\max\}\\\}\. The redundancy distance is

fred,m=\{minn∈ℋm⁡\(1N​∑a‖D^m,a−D^n,a‖22\)1/2,ℋm≠∅,5​Å,ℋm=∅\.f\_\{\\mathrm\{red\},m\}=\\begin\{cases\}\\displaystyle\\min\_\{n\\in\\mathcal\{H\}\_\{m\}\}\\left\(\\frac\{1\}\{N\}\\sum\_\{a\}\\\|\\widehat\{D\}\_\{m,a\}\-\\widehat\{D\}\_\{n,a\}\\\|\_\{2\}^\{2\}\\right\)^\{1/2\},&\\mathcal\{H\}\_\{m\}\\neq\\varnothing,\\\\ 5\\,\\text\{\\AA\},&\\mathcal\{H\}\_\{m\}=\\varnothing\.\\end\{cases\}\(69\)This feature measures all\-atom Cartesian RMS separation within the proposal pool\. The strict\>\>test selects higher\-amplitude proposals\. Rank is the zero\-based position in the descending sort ofammaxa\_\{m\}^\{\\max\}, ranging from00toM−1M\-1, with ties following the array\-sort ordering\.

#### Directional coherence \(feature 10\)\.

When\|𝒰m\|\>1\|\\mathcal\{U\}\_\{m\}\|\>1andammax\>0a\_\{m\}^\{\\max\}\>0,

fcoher,m=1\|𝒰m\|​∑a∈𝒰m⟨D^m,amax⁡\(am​a,10−8\),D^m,jm∗ammax⟩;f\_\{\\mathrm\{coher\},m\}=\\frac\{1\}\{\|\\mathcal\{U\}\_\{m\}\|\}\\sum\_\{a\\in\\mathcal\{U\}\_\{m\}\}\\left\\langle\\frac\{\\widehat\{D\}\_\{m,a\}\}\{\\max\(a\_\{ma\},10^\{\-8\}\)\},\\;\\frac\{\\widehat\{D\}\_\{m,j\_\{m\}^\{\*\}\}\}\{a\_\{m\}^\{\\max\}\}\\right\\rangle;\(70\)otherwise it equals one\. The mean includes the maximum\-displacement atom itself\.jm∗j\_\{m\}^\{\*\}is the first maximum in the stored atom order\.

#### Geometric invariance and tie conventions\.

Norms, dot products, local distances, and atom pooling preserve rotations under the predictor’s transformation laws\. Exact atom\-permutation invariance also requires compatible handling of tied maximum amplitudes in coherence; proposal\-rank ties follow pool order\. The redundancy and rank features provide context from the complete proposal pool, so scores reflect both individual geometry and the alternatives proposed for the same state\.

### E\.4Offline labels and judge training

For each proposal–reference pair, use the main\-text moving union

Um​e=\{a:max⁡\(‖D^m,a‖2,‖De,a‖2\)≥0\.3​Å\}\.U\_\{me\}=\\\{a:\\max\(\\\|\\widehat\{D\}\_\{m,a\}\\\|\_\{2\},\\\|D\_\{e,a\}\\\|\_\{2\}\)\\geq 0\.3\\,\\text\{\\AA\}\\\}\.For nonempty unions, compute

rm​eU=\(1\|Um​e\|​∑a∈Um​e‖D^m,a−De,a‖22\)1/2,dm​eU=maxa∈Um​e⁡‖D^m,a−De,a‖2\.r^\{U\}\_\{me\}=\\left\(\\frac\{1\}\{\|U\_\{me\}\|\}\\sum\_\{a\\in U\_\{me\}\}\\\|\\widehat\{D\}\_\{m,a\}\-D\_\{e,a\}\\\|\_\{2\}^\{2\}\\right\)^\{1/2\},\\qquad d^\{U\}\_\{me\}=\\max\_\{a\\in U\_\{me\}\}\\\|\\widehat\{D\}\_\{m,a\}\-D\_\{e,a\}\\\|\_\{2\}\.\(71\)Empty unions are ineligible, equivalently having infinite RMS\. First select the referenceem∗e\_\{m\}^\{\*\}of smallestrm​eUr^\{U\}\_\{me\}, keeping the first reference in catalogue order at a tie\. Then set

ym=𝟏\{rm​em∗U<0\.35Åanddm​em∗U<0\.5Å\}\.y\_\{m\}=\\mathbf\{1\}\\\{r^\{U\}\_\{me\_\{m\}^\{\*\}\}<0\.35\\,\\text\{\\AA\}\\text\{ and \}d^\{U\}\_\{me\_\{m\}^\{\*\}\}<0\.5\\,\\text\{\\AA\}\\\}\.\(72\)If no eligible reference exists,ym=0y\_\{m\}=0\. Both tests are strict and apply to the RMS\-nearest reference selected above\. These geometric\-accuracy labels train proposal screening; the final event evaluation uses the all\-atom displacement and barrier criteria\.

#### Head training and checkpoint selection\.

Gradient updates use training configurations only, as stated in Section[4\.5](https://arxiv.org/html/2610.10969#S4.SS5)\. The stage\-1 head\-training recipe uses the first 200 configurations oftrain\_strictfor those updates and the first 50 monitoring configurations for head selection; no held\-out test configurations enter either role\. One optimizer step processes all slots of one configuration, and the binary cross\-entropy is averaged over those slots\.

Letπ\+\\pi\_\{\+\}be the positive\-label fraction obtained by concatenating all training labels\. The positive\-class weight is

w\+=1−π\+max⁡\(π\+,0\.001\),0<π\+<1\.w\_\{\+\}=\\frac\{1\-\\pi\_\{\+\}\}\{\\max\(\\pi\_\{\+\},0\.001\)\},\\qquad 0<\\pi\_\{\+\}<1\.\(73\)For one configuration, the logit loss is

ℒJ=−1M∑m\[w\+ymlogσ\(tmJ\)\+\(1−ym\)log\(1−σ\(tmJ\)\)\]\.\\mathcal\{L\}\_\{J\}=\-\\frac\{1\}\{M\}\\sum\_\{m\}\\left\[w\_\{\+\}y\_\{m\}\\log\\sigma\(t\_\{m\}^\{J\}\)\+\(1\-y\_\{m\}\)\\log\(1\-\\sigma\(t\_\{m\}^\{J\}\)\)\\right\]\.The implementation uses binary cross\-entropy with logits for numerical stability\. The head uses AdamW with learning rate8×10−48\\times 10^\{\-4\}and weight decay10−410^\{\-4\}\. Torch and Python random seeds are reset to zero, and training configurations are shuffled each epoch\. Training lasts at most 40 epochs\. After each epoch, monitoring ROC AUC is computed from the concatenated proposal scores and labels\. Strictly improved AUC saves the head; six consecutive non\-improving epochs stop training\. The best saved head is restored with strict parameter matching and placed in evaluation mode\.

### E\.5Deployment and proposal identifiers

Compute the ten descriptors on the complete proposal pool, evaluate the head, and refine exactly the indices

IX​\(τ\)=\{m:p^m≥τ\}\.I\_\{X\}\(\\tau\)=\\\{m:\\widehat\{p\}\_\{m\}\\geq\\tau\\\}\.\(74\)Selection retains the original slot/search identifiers and uses descriptors computed on the full pool\. The default score thresholdτ=10−5\\tau=10^\{\-5\}is used in the 50\-configuration wall\-time comparison \(Figure[5](https://arxiv.org/html/2610.10969#S5.F5)\)\. The 200\-configuration recovery curves vary this threshold, while the ablations in Table[2](https://arxiv.org/html/2610.10969#S5.T2)disable selection\. The sigmoid readout is used as a geometric proposal\-selection score, with physical recovery assessed after refinement\.

## Appendix FNative Dimer Refinement and Validation

### F\.1Enabled path and coordinate convention

The physical search uses the native SEAKMC Dimer implementation on the fixed interatomic potential\. The geometric background of Dimer is given in Appendix[A\.4](https://arxiv.org/html/2610.10969#A1.SS4); here we specify the supplied numerical path\. Searches use Cartesian active\-atom coordinates with the native fixed/buffer surroundings\. The optionsSNC,ApplyMass,SearchBuffer,LocalRefine, andEarlyStopare disabled, whileLocalRelaxis enabled\. The main Dimer search refines every submitted proposal;LocalRefine=Falsespecifies that its additional local\-refinement branch is disabled\.

The paper writes atomic fields asN×3N\\times 3arrays\. The native search stores their transpose as3×N3\\times Narrays\. Below,yyandnndenote the vectorized active\-coordinate center and orientation, respectively, andf​\(y\)=−∇V​\(y\)f\(y\)=\-\\nabla V\(y\)denotes the physical force in that same coordinate space\. The potential includes fixed surroundings, whilennspans the active coordinates\. Native magnitude operations usemagnative\\operatorname\{mag\}\_\{\\mathrm\{native\}\}, corresponding to themat\_maghelper and its stopping convention\.

### F\.2Random axes, proposal loading, and orientation reset

For a random\-start search, the native initial\-axis sampler draws an array with independent entries from\[−0\.5,0\.5\)\[\-0\.5,0\.5\)\. The wrapper sets

seed⁡\(c,i\)=\(2026091400\+10000​c\+i\)mod232,\\operatorname\{seed\}\(c,i\)=\(2026091400\+10000c\+i\)\\bmod 2^\{32\},\(75\)before the native draw, whereccis the configuration ordinal andiiis the search identifier\. Angle checking against previous axes follows the native settings, completing the initial\-axis sampling protocol\.

For a learned or transferred fieldD∈ℝN×3D\\in\\mathbb\{R\}^\{N\\times 3\}, the bridge transposes the full field for preload and forms a separate translation\-removed direction\. In paper notation,

D¯=1N​∑aDa,n~​\(D\)=vec⁡\(D−𝟏​D¯\)‖D−𝟏​D¯‖F\.\\bar\{D\}=\\frac\{1\}\{N\}\\sum\_\{a\}D\_\{a\},\\qquad\\widetilde\{n\}\(D\)=\\frac\{\\operatorname\{vec\}\(D\-\\mathbf\{1\}\\bar\{D\}\)\}\{\\\|D\-\\mathbf\{1\}\\bar\{D\}\\\|\_\{F\}\}\.\(76\)The helper accepts finite fields whose translation\-removed norm exceeds10−1210^\{\-12\}\. Mean removal defines the axis; the complete fieldDDpreloads the candidate centerX\+DX\+D\.

Native loading pads or restricts the field to the configured active coordinates and applies any type/fixed\-coordinate scaling\. Write its actual center field asDexecD^\{\\mathrm\{exec\}\}, and letnpren\_\{\\mathrm\{pre\}\}be the native axis immediately before loading\. The bridge suppliesn~​\(D\)\\widetilde\{n\}\(D\); any intervening native axis handling remains governed by the job settings\. Let𝖲\\mathsf\{S\}denote the native type\-scaling operator on the axis, or the identity when no such scaling is enabled\. The loading/reset rule is

n0=\{unit⁡\(𝖲⁡\[rreset​npre\+vec⁡\(Dexec\)\]\),ResetVN04Preload=True,unit⁡\(𝖲​npre\),otherwise\.n\_\{0\}=\\begin\{cases\}\\operatorname\{unit\}\\\!\\left\(\\mathsf\{S\}\\bigl\[r\_\{\\mathrm\{reset\}\}n\_\{\\mathrm\{pre\}\}\+\\operatorname\{vec\}\(D^\{\\mathrm\{exec\}\}\)\\bigr\]\\right\),&\\texttt\{ResetVN04Preload=True\},\\\\ \\operatorname\{unit\}\(\\mathsf\{S\}n\_\{\\mathrm\{pre\}\}\),&\\text\{otherwise\}\.\\end\{cases\}\(77\)The configured coefficient isrreset=0\.2r\_\{\\mathrm\{reset\}\}=0\.2in the implementation’s numericalÅconvention\. In the unscaled case with no intervening axis change, this reduces to normalizing0\.2​n~​\(D\)\+vec⁡\(D\)0\.2\\widetilde\{n\}\(D\)\+\\operatorname\{vec\}\(D\)\. This initializes the orientation from the proposal\. Subsequent centering, rescaling, and angle controls follow the resolved job settings\.

On an admissible, already represented displacement fieldDD, the complete initialization in the conditional analysis is written

ℐX​\(D\)=\(X\+D,n0​\(D\)\)\.\\mathcal\{I\}\_\{X\}\(D\)=\\bigl\(X\+D,\\;n\_\{0\}\(D\)\\bigr\)\.\(78\)Heren0​\(D\)n\_\{0\}\(D\)includes the configured orientation processing\. The admissible domain𝒟X\\mathcal\{D\}\_\{X\}enforces coordinate constraints and a finite, nonzero processed orientation\. The recovery analysis applies to the executed field and this configured initialization through its loading condition and local\-success assumption\.

### F\.3Curvature estimation, rotation, and force projection

The native implementation reuses the center force to estimate endpoint curvature\. At centeryy, letf0=f⁡\(y\)f\_\{0\}=f\(y\)andf1=f⁡\(y\+hdim​n\)f\_\{1\}=f\(y\+h\_\{\\mathrm\{dim\}\}n\)\. WithLocalRefine=False, it computes

f2=2​f0−f1,c=⟨f2−f1,n⟩2​hdim,frot=\(f1−f2\)−n⁡⟨f1−f2,n⟩2​hdim\.f\_\{2\}=2f\_\{0\}\-f\_\{1\},\\qquad c=\\frac\{\\langle f\_\{2\}\-f\_\{1\},n\\rangle\}\{2h\_\{\\mathrm\{dim\}\}\},\\qquad f\_\{\\mathrm\{rot\}\}=\\frac\{\(f\_\{1\}\-f\_\{2\}\)\-n\\langle f\_\{1\}\-f\_\{2\},n\\rangle\}\{2h\_\{\\mathrm\{dim\}\}\}\.\(79\)The reused\-force estimatef2f\_\{2\}supplies the opposite\-endpoint approximation\. The curvatureccand tangential force drive the native orientation update, with separationhdim=0\.005​Åh\_\{\\mathrm\{dim\}\}=0\.005\\,\\text\{\\AA\}\.

#### Native rotation and force projection\.

The rotation update maintains a conjugate direction, with a restart based on the current and previous rotation forces\. Its coefficient has the form

γ⁡\(f,fold\)=\{⟨f,f−fold⟩⟨fold,fold⟩,\|⟨f,fold⟩\|≤12​⟨fold,fold⟩,⟨fold,fold⟩≠0,0,otherwise\.\\gamma\(f,f\_\{\\mathrm\{old\}\}\)=\\begin\{cases\}\\displaystyle\\frac\{\\langle f,f\-f\_\{\\mathrm\{old\}\}\\rangle\}\{\\langle f\_\{\\mathrm\{old\}\},f\_\{\\mathrm\{old\}\}\\rangle\},&\\begin\{gathered\}\|\\langle f,f\_\{\\mathrm\{old\}\}\\rangle\|\\leq\\tfrac\{1\}\{2\}\\langle f\_\{\\mathrm\{old\}\},f\_\{\\mathrm\{old\}\}\\rangle,\\\\\[\-2\.0pt\] \\langle f\_\{\\mathrm\{old\}\},f\_\{\\mathrm\{old\}\}\\rangle\\neq 0,\\end\{gathered\}\\\\ 0,&\\text\{otherwise\}\.\\end\{cases\}\(80\)For rotation, the accumulated direction is normalized to a search vectoruu\. A forward angular sample updates the orientation and search vector by

\(n,u\)⟼\(n​cos⁡θ\+u​sin⁡θ,u​cos⁡θ−n​sin⁡θ\),θ=π/4\.\(n,u\)\\longmapsto\\bigl\(n\\cos\\theta\+u\\sin\\theta,\\;u\\cos\\theta\-n\\sin\\theta\\bigr\),\\qquad\\theta=\\pi/4\.Writing the directional rotation\-force samples asr1,r2r\_\{1\},r\_\{2\}, the return stage applies the incremental angle

θinc=\{−12​arctan⁡\(r1/r2\),r2≠0,−12​π,r2=0\+12π1\{r2\>0\}−14π\.\\theta\_\{\\mathrm\{inc\}\}=\\begin\{cases\}\-\\tfrac\{1\}\{2\}\\arctan\(r\_\{1\}/r\_\{2\}\),&r\_\{2\}\\neq 0,\\\\ \-\\tfrac\{1\}\{2\}\\pi,&r\_\{2\}=0\\end\{cases\}\+\\tfrac\{1\}\{2\}\\pi\\,\\mathbf\{1\}\\\{r\_\{2\}\>0\\\}\-\\tfrac\{1\}\{4\}\\pi\.\(81\)Native normalization and enabled axis controls follow this update\. Rotation\-force tests and the rotation counter useFMin4Rot,FThres4Rot, andNMax4Rot\.

For translation, writef∥=n⁡⟨f0,n⟩f\_\{\\parallel\}=n\\langle f\_\{0\},n\\rangle\. The effective force is curvature dependent:

feff=\{f0−2​f∥,c<0,−f∥,c≥0​and the horizon mode is off,f0−f∥,c≥0​and the horizon mode is on\.f\_\{\\mathrm\{eff\}\}=\\begin\{cases\}f\_\{0\}\-2f\_\{\\parallel\},&c<0,\\\\ \-f\_\{\\parallel\},&c\\geq 0\\text\{ and the horizon mode is off\},\\\\ f\_\{0\}\-f\_\{\\parallel\},&c\\geq 0\\text\{ and the horizon mode is on\}\.\\end\{cases\}\(82\)The horizon switch follows the native curvature, energy, and initial\-well logic, selecting the appropriate positive\-curvature branch\.

### F\.4Two\-stage translation and fine\-step controls

The translation direction is also updated by a restarted conjugate\-force rule using Equation \([80](https://arxiv.org/html/2610.10969#A6.E80)\) and is normalized touu\. Lettft\_\{f\}andtmt\_\{m\}be the current trial and maximal translation settings\. The forward stage moves the current center bytf​ut\_\{f\}u\. With effective forces before and after that trial, define

a1=⟨fbefore,u⟩,a2=⟨ftrial,u⟩,ct=a1−a2tf\.a\_\{1\}=\\langle f\_\{\\mathrm\{before\}\},u\\rangle,\\qquad a\_\{2\}=\\langle f\_\{\\mathrm\{trial\}\},u\\rangle,\\qquad c\_\{t\}=\\frac\{a\_\{1\}\-a\_\{2\}\}\{t\_\{f\}\}\.The second stage adds a further displacementtc​ut\_\{c\}u\. Its native rule is

t∗=\(a1\+a2\)/2ct\+10−12when​ct≥0,tc=\{tm,ct<0,tm−tf,ct≥0,\|t∗\|\>tm,t∗≥0,tf−tm,ct≥0,\|t∗\|\>tm,t∗<0,t∗−tf/2,ct≥0,\|t∗\|≤tm\.\\begin\{gathered\}t\_\{\*\}=\\frac\{\(a\_\{1\}\+a\_\{2\}\)/2\}\{c\_\{t\}\+10^\{\-12\}\}\\quad\\text\{when \}c\_\{t\}\\geq 0,\\\\ t\_\{c\}=\\begin\{cases\}t\_\{m\},&c\_\{t\}<0,\\\\ t\_\{m\}\-t\_\{f\},&c\_\{t\}\\geq 0,\\ \|t\_\{\*\}\|\>t\_\{m\},\\ t\_\{\*\}\\geq 0,\\\\ t\_\{f\}\-t\_\{m\},&c\_\{t\}\\geq 0,\\ \|t\_\{\*\}\|\>t\_\{m\},\\ t\_\{\*\}<0,\\\\ t\_\{\*\}\-t\_\{f\}/2,&c\_\{t\}\\geq 0,\\ \|t\_\{\*\}\|\\leq t\_\{m\}\.\\end\{cases\}\\end\{gathered\}\(83\)The combined center displacement is\(tf\+tc\)​u\(t\_\{f\}\+t\_\{c\}\)u, withtct\_\{c\}determined by the trial forces and the configured maximal\-step parameter in this two\-stage update\.

A preloaded search uses the nativeFineStepsettings\. The supplied ordinary trial/maximal defaults are0\.015/0\.05​Å0\.015/0\.05\\,\\text\{\\AA\}, while the fine\-step defaults are0\.003/0\.01​Å0\.003/0\.01\\,\\text\{\\AA\}\. When the decay style is not fixed, the source updates them in the form

tf​\(j\)=tf\(0\)​d​\(j\)\+tmin,tm​\(j\)=tm\(0\)​d​\(j\)\+tmin​rmin,t\_\{f\}\(j\)=t\_\{f\}^\{\(0\)\}d\(j\)\+t\_\{\\min\},\\qquad t\_\{m\}\(j\)=t\_\{m\}^\{\(0\)\}d\(j\)\+t\_\{\\min\}r\_\{\\min\},with the corresponding ordinary or fine\-step parameter family\. The decay factor is eitherd⁡\(j\)=d0⌊j/J⌋d\(j\)=d\_\{0\}^\{\\lfloor j/J\\rfloor\}ord0j/Jd\_\{0\}^\{j/J\}under the configured style\. The resolved job settings select the ordinary or fine\-step family and its decay parameters for each search\.

Table 7:Native numerical controls and defaults\.Defaults identify the implementation’s parameter families\. Each experiment uses its resolved settings and the reported translation cap\.
### F\.5Search loop, stopping, and cap handling

The loop callsdimer\_forcebefore the first preload to establish the initial energy\. At the first iteration, an available preload is then loaded, the runner is prepared again, and anotherdimer\_forcecall evaluates the preloaded configuration\. Subsequent iterations alternate the native rotation and translation states\. Rotation completion advances the nativeNTSITRcounter, which supplies the recorded translation count\.

In the enabled branch, the force\-convergence flagFO\_CONuses

magnative⁡\(FC\)<FConv,\\operatorname\{mag\}\_\{\\mathrm\{native\}\}\(\\texttt\{FC\}\)<\\texttt\{FConv\},after the appropriate projection/update, andST\_CONrequiresNTSITR \> IgnoreTransSteps\. The search converges when both flags hold\. The enabled branch uses these force and step flags for convergence; energy convergence is recorded separately\. Native relaxation and connectivity checks complete the acceptance protocol\.

The cap check is applied whenNTSITR \>= NMax4Trans; it sets the maximum\-translation flag and terminates the search\. Cap\-reaching outputs are invalidated in native finalization\. The downstream scorer also requires a translation count strictly below the configured cap\. Optional native displacement, barrier, and angle checks remain governed by the job configuration\.

### F\.6Native two\-sided relaxation and connectivity

WithLocalRelaxenabled and the additional refinement/early\-stop branches disabled, the two relaxation seeds are constructed from the last search segment\. In row\-coordinate notation, letYcY\_\{c\}denoteXCR,Y0Y\_\{0\}denoteX0, andrrelr\_\{\\mathrm\{rel\}\}denoteDRATIO\. The seeds are

Z\+\(0\)=Y0\+rrel​\(Yc−Y0\),Z−\(0\)=Y0−rrel​\(Yc−Y0\)\.Z^\{\(0\)\}\_\{\+\}=Y\_\{0\}\+r\_\{\\mathrm\{rel\}\}\(Y\_\{c\}\-Y\_\{0\}\),\\qquad Z^\{\(0\)\}\_\{\-\}=Y\_\{0\}\-r\_\{\\mathrm\{rel\}\}\(Y\_\{c\}\-Y\_\{0\}\)\.\(84\)Each seed is relaxed by the nativeSPSRELAXrunner with the configured fixed surroundings\. Relaxed active coordinates are shifted back relative to the initial configuration by the native periodic reboxing convention\. Thus the final search segment provides both seeds for the native two\-sided connectivity test\.

The implementation first identifies the farther and nearer relaxed branches using their overall native displacement magnitude fromXX\. For those selected branches, it then computes

dfar=maxa,d⁡\|\(Zfar−X\)a​d\|,dnear=maxa,d⁡\|\(Znear−X\)a​d\|\.d\_\{\\mathrm\{far\}\}=\\max\_\{a,d\}\|\(Z\_\{\\mathrm\{far\}\}\-X\)\_\{ad\}\|,\\qquad d\_\{\\mathrm\{near\}\}=\\max\_\{a,d\}\|\(Z\_\{\\mathrm\{near\}\}\-X\)\_\{ad\}\|\.Ifdfar<Tol4Connectd\_\{\\mathrm\{far\}\}<\\texttt\{Tol4Connect\}, it marks the event invalid and disconnected\. Otherwise the event is connected whendnear<Tol4Connectd\_\{\\mathrm\{near\}\}<\\texttt\{Tol4Connect\}; when this latter test fails, the relaxation routine can return a valid but disconnected result\. Final native barrier and displacement cuts can additionally invalidate an output\. Connectivity uses a maximum coordinate\-component tolerance; reference matching uses the all\-atom Euclidean displacement gate\.

### F\.7Returned saddle geometry and the numerical map

The wrapper returns the final active saddle coordinates fromXCR\.T, the recorded barrierBARR, native valid and connected flags, the cap flag, translation count, and search\-force count\. The saddle field for scoring is

Sout=saddle\_active\_coords,Dout=Sout−X\.S\_\{\\mathrm\{out\}\}=\\texttt\{saddle\\\_active\\\_coords\},\\qquad D\_\{\\mathrm\{out\}\}=S\_\{\\mathrm\{out\}\}\-X\.\(85\)Scoring uses this saved saddle geometry and its associated barrier\. NativeXDISPrecords the displacement of a selected downhill branch seed during finalization\.

LetℛXnum\\mathcal\{R\}\_\{X\}^\{\\mathrm\{num\}\}denote this configured initialization, search, and native acceptance pipeline\. It accepts a configurationX\+DX\+Dand returns the numerical saddle output when the native flags and cap checks pass, or⊥\\bototherwise\. For completed search indicesII,

AI=\{ℛXnum\(X\+Dmexec\):m∈I,ℛXnum\(X\+Dmexec\)≠⊥\}\.A\_\{I\}=\\\{\\mathcal\{R\}\_\{X\}^\{\\mathrm\{num\}\}\(X\+D\_\{m\}^\{\\mathrm\{exec\}\}\):m\\in I,\\ \\mathcal\{R\}\_\{X\}^\{\\mathrm\{num\}\}\(X\+D\_\{m\}^\{\\mathrm\{exec\}\}\)\\neq\\bot\\\}\.\(86\)The main\-text ideal map specifies the saddle and connectivity target\. The numerical map implements the configured search and native acceptance checks\. Its connection to reference recovery is formalized by the local\-success assumption for this map and the final reference matcher\.

### F\.8Search\-force accounting and run configuration

Each entry toDimer\.dimer\_forceincrements the search counter once per routine invocation\. The ledger covers preloading and search work, including failed, duplicate, and capped searches with completed records\. LAMMPS initialization and nativeSPSRELAXforce calls lie outside this routine\.

Reproduction uses the native helpers together with each run’s resolved step\-size and decay settings, axis controls, fixed\-coordinate masks, displacement cuts, and timer boundaries\. The specifications above document the numerical path and default parameter families\.

## Appendix GKDB Event\-Reuse Adaptation

### G\.1Stored mechanisms and the scope of the baseline

The KDB comparison adapts the kinetic\-database approach\([Terrell et al\., 2012](https://arxiv.org/html/2610.10969#bib.bib34)\)\. Insertion and querying retain the upstream geometric kernel at commita03dd2f, including periodic clumping, graph mapping, feasibility tests, and overlap checks\.

The template library is built exclusively fromtrain\_strict\. A stored mechanism contains its training reactantXlibX^\{\\mathrm\{lib\}\}, saddleSlibS^\{\\mathrm\{lib\}\}, productX\+,libX^\{\+,\\mathrm\{lib\}\}, mobile\-atom indices, periodic clump orders, barriers, and source/event identifiers\. The in\-memory adapter retains every successfully inserted template and records failed insertions separately\. Candidate deduplication is performed after querying as specified below\.

KDB\-494 denotes 494 stored event templates\. Every template is queried in both directions: one directed entry uses the training reactant as its minimum, and the other uses the training product\. Both share the stored saddle and swap their minimum/product roles and clump order\. Thus the 494\-template library provides 988 directed query entries\. Reverse barriers equal the forward barrier minus the stored product energy bias\. Lookup and ranking use the target initial state and the training\-library records\.

### G\.2Mobile atoms and the retained local template

LetMICH\\operatorname\{MIC\}\_\{H\}denote the periodic displacement convention for the stored cellHH, with row\-vector coordinates as in the background\. For each training atom, define

ma=max\{‖MICH⁡\(Salib−Xalib\)‖2,‖MICH⁡\(Salib−Xa\+,lib\)‖2,∥MICH\(Xa\+,lib−Xalib\)∥2\}\.\\begin\{split\}m\_\{a\}=\\max\\\{&\\\|\\operatorname\{MIC\}\_\{H\}\(S\_\{a\}^\{\\mathrm\{lib\}\}\-X\_\{a\}^\{\\mathrm\{lib\}\}\)\\\|\_\{2\},\\;\\\|\\operatorname\{MIC\}\_\{H\}\(S\_\{a\}^\{\\mathrm\{lib\}\}\-X\_\{a\}^\{\+,\\mathrm\{lib\}\}\)\\\|\_\{2\},\\\\ &\\\|\\operatorname\{MIC\}\_\{H\}\(X\_\{a\}^\{\+,\\mathrm\{lib\}\}\-X\_\{a\}^\{\\mathrm\{lib\}\}\)\\\|\_\{2\}\\\}\.\\end\{split\}\(87\)The mobile set consists of atoms withma\>0\.7​Åm\_\{a\}\>0\.7\\,\\text\{\\AA\}\. If it is empty, insertion retains the atom with largest reactant\-to\-product displacement\. The local template retains the mobile atoms and their neighbors in any of the reactant, saddle, or product states\. The neighbor threshold uses the upstream elemental radii:

ra​bneigh=\(raelem\+rbelem\)​\(1\+nf\),nf=0\.2\.r\_\{ab\}^\{\\mathrm\{neigh\}\}=\(r\_\{a\}^\{\\mathrm\{elem\}\}\+r\_\{b\}^\{\\mathrm\{elem\}\}\)\(1\+n\_\{f\}\),\\qquad n\_\{f\}=0\.2\.\(88\)The upstream routine applies its periodic unwrapping/clumping procedure and size checks using these elemental radii\.

### G\.3Graph matching and geometric screening

For the target initial stateXX, the upstream kernel builds the reactant graph and searches subgraph mappings for the directed library entries\. Each surviving mapping is subjected to the upstream periodic ordered clump construction, geometric alignment, and feasibility/overlap checks\.

The acceleration layer caches target distances and applies the same necessary pair\-distance condition earlier during mapping\. For a newly mapped atom and each already mapped atom, at least one of the upstream periodic\-image distances must satisfy

\|dtarget\(j\)−dtemplate\|<2​dc,dc=0\.3​Å\.\\bigl\|d\_\{\\mathrm\{target\}\}^\{\(j\)\}\-d\_\{\\mathrm\{template\}\}\\bigr\|<2d\_\{c\},\\qquad d\_\{c\}=0\.3\\,\\text\{\\AA\}\.\(89\)The image indexjjranges over the upstreamatomAtomPermDistancecandidates\. Caching and JIT execution accelerate this pair\-distance check while preserving its image candidates; surviving mappings pass the fullfeasibility\(\)path\.

#### Orthogonal alignment and the geometric gate\.

After correspondence and periodic clumping have been established, letAAandBBbe the centered template\-minimum and mapped target\-coordinate matrices\. Compute

A⊤​B=U​Σ​W⊤,𝖮=U​W⊤\.A^\{\\top\}B=U\\Sigma W^\{\\top\},\\qquad\\mathsf\{O\}=UW^\{\\top\}\.\(90\)The orthogonal map𝖮\\mathsf\{O\}acts on row vectors from the right and permits both rotations and reflections when aligning a stored template to the query configuration\.

The alignment is computed by an SVD least\-squares fit and then screened by the maximum atomwise residual,

dfit=maxa⁡‖\(A​𝖮−B\)a‖2≤dc\.d\_\{\\mathrm\{fit\}\}=\\max\_\{a\}\\\|\(A\\mathsf\{O\}\-B\)\_\{a\}\\\|\_\{2\}\\leq d\_\{c\}\.\(91\)The SVD supplies the least\-squares alignment, anddfitd\_\{\\mathrm\{fit\}\}supplies the maximum\-residual query score and acceptance gate\.

For a nearly coplanar template, the kernel also tries the normal\-flipped alignment\. The coplanarity test is

σmin​\(A⊤​A\)ntpl<dc2,\\frac\{\\sigma\_\{\\min\}\(A^\{\\top\}A\)\}\{n\_\{\\mathrm\{tpl\}\}\}<d\_\{c\}^\{2\},wherentpln\_\{\\mathrm\{tpl\}\}is the retained template size\. The alternative is obtained by changing the sign of the final row ofW⊤W^\{\\top\}before forming𝖮\\mathsf\{O\}\. Alignments passing the distance gate subsequently undergo the upstream product/mobile overlap checks\.

### G\.4Aligned state transfer and displacement extraction

Letμtpl\\mu\_\{\\mathrm\{tpl\}\}andμX\\mu\_\{X\}be the fitted template\-minimum and mapped\-target centroids, and letι⁡\(a\)\\iota\(a\)be the target atom mapped to template atomaa\. For a surviving unconstrained mapping, the aligned saddle coordinates are placed into a copy of the target initial state:

S~ι⁡\(a\)=\(Salib−μtpl\)𝖮\+μX,S~b=Xbfor unfilled atomsb\.\\widetilde\{S\}\_\{\\iota\(a\)\}=\(S\_\{a\}^\{\\mathrm\{lib\}\}\-\\mu\_\{\\mathrm\{tpl\}\}\)\\mathsf\{O\}\+\\mu\_\{X\},\\qquad\\widetilde\{S\}\_\{b\}=X\_\{b\}\\quad\\text\{for unfilled atoms \}b\.\(92\)The training product is transferred in the same way\. Native constraints, periodic reboxing, and overlap checks remain part of the upstream path\. The wrapper extracts the saddle and product displacement fields relative toXX\. In the supplied cell representation, its extraction is

u=S~−X,Dcand=u−rint⁡\(u​H−1\)​H,u=\\widetilde\{S\}\-X,\\qquad D^\{\\mathrm\{cand\}\}=u\-\\operatorname\{rint\}\(uH^\{\-1\}\)H,\(93\)and analogously for the product\. This fractional\-rounding extraction uses the supplied cell representation\.

Before the final reboxing, the mapped saddle displacement decomposes as

S~ι⁡\(a\)−Xι⁡\(a\)=\(Salib−Xalib\)​𝖮\+\[\(Xalib−μtpl\)​𝖮\+μX−Xι⁡\(a\)\]\.\\begin\{split\}\\widetilde\{S\}\_\{\\iota\(a\)\}\-X\_\{\\iota\(a\)\}=\{\}&\(S\_\{a\}^\{\\mathrm\{lib\}\}\-X\_\{a\}^\{\\mathrm\{lib\}\}\)\\mathsf\{O\}\\\\ &\+\\bigl\[\(X\_\{a\}^\{\\mathrm\{lib\}\}\-\\mu\_\{\\mathrm\{tpl\}\}\)\\mathsf\{O\}\+\\mu\_\{X\}\-X\_\{\\iota\(a\)\}\\bigr\]\.\\end\{split\}\(94\)The bracket is the initial\-state fitting residual\. KDB coordinate transfer therefore combines the transformed stored displacement with this alignment correction\. KeyBank uses the direct displacement transfer described below\.

### G\.5Global candidate deduplication and Dimer submission

Entry\-internal deduplication is disabled in the query path\. After all directed entries have been queried, candidates are sorted by increasing geometric fit score and greedily deduplicated\. LetDcandD^\{\\mathrm\{cand\}\}andD\+,candD^\{\+,\\mathrm\{cand\}\}be one candidate’s saddle and product displacements\. It is discarded when an already retained candidatejjsatisfies both

maxa⁡‖Dacand−Dj,acand‖2≤0\.3​Å,maxa⁡‖Da\+,cand−Dj,a\+,cand‖2≤0\.3​Å\.\\max\_\{a\}\\\|D^\{\\mathrm\{cand\}\}\_\{a\}\-D^\{\\mathrm\{cand\}\}\_\{j,a\}\\\|\_\{2\}\\leq 0\.3\\,\\text\{\\AA\},\\qquad\\max\_\{a\}\\\|D^\{\+,\\mathrm\{cand\}\}\_\{a\}\-D^\{\+,\\mathrm\{cand\}\}\_\{j,a\}\\\|\_\{2\}\\leq 0\.3\\,\\text\{\\AA\}\.\(95\)These fields use the target atom correspondence\. Score\-ordered greedy deduplication compares each candidate against retained candidates, using the transferred saddle and product predictions from the training library\. Source, event, and direction identifiers are preserved\.

Retained saddle displacements initialize native Dimer searches with a 300\-translation cap\. Physical acceptance and final recovery use the common protocol, including the all\-atom0\.05​Å0\.05\\,\\text\{\\AA\}displacement and0\.02​eV0\.02\\,\\mathrm\{eV\}barrier gates\. Library size, directed entries, returned proposals, and recovered references summarize successive pipeline stages\.

## Appendix HRooted\-Topology KeyBank Adaptation

### H\.1Rooted graphs and directed library entries

KeyBank adapts the rooted\-topology catalogue idea of k\-ART\([Béland et al\., 2011](https://arxiv.org/html/2610.10969#bib.bib5)\)to retrieve proposals for the common native Dimer pipeline\. KeyBank\-32 and KeyBank\-128 use libraries built from 32 and 128train\_strictconfigurations, respectively\.

For a root atomrrin a configuration, retain

𝒩r=\{a:∥Xa−Xr∥2≤5Å\},pa=Xa−Xr\(a∈𝒩r\)\.\\mathcal\{N\}\_\{r\}=\\\{a:\\\|X\_\{a\}\-X\_\{r\}\\\|\_\{2\}\\leq 5\\,\\text\{\\AA\}\\\},\\qquad p\_\{a\}=X\_\{a\}\-X\_\{r\}\\quad\(a\\in\\mathcal\{N\}\_\{r\}\)\.\(96\)The local graph has an atom\-species/root label\(za,𝟏\{a=r\}\)\(z\_\{a\},\\mathbf\{1\}\\\{a=r\\\}\)on each vertex and an undirected edge when

‖pa−pb‖2<3\.145​Å,a≠b\.\\\|p\_\{a\}\-p\_\{b\}\\\|\_\{2\}<3\.145\\,\\text\{\\AA\},\\qquad a\\neq b\.\(97\)The neighborhood radius is inclusive and the bond cutoff is strict\. A four\-iteration Weisfeiler–Lehman hash with node labels identifies lookup buckets; exact graph mappings and geometric checks then resolve candidates\.

KeyBank operates directly on local active\-volume coordinates, with the stated assumption that the represented neighborhoods remain within a consistent cell image\.

#### Forward and reverse library entries\.

For every training event, construct a forward entry with initial stateXlibX^\{\\mathrm\{lib\}\}and a reverse entry with initial stateX\+,libX^\{\+,\\mathrm\{lib\}\}\. In either direction, letXdirX^\{\\mathrm\{dir\}\}be the chosen initial state,X\+,dirX^\{\+,\\mathrm\{dir\}\}the opposite minimum, andSlibS^\{\\mathrm\{lib\}\}the shared saddle\. The full directed fields are

Dlib=Slib−Xdir,D\+,lib=X\+,dir−Xdir\.D^\{\\mathrm\{lib\}\}=S^\{\\mathrm\{lib\}\}\-X^\{\\mathrm\{dir\}\},\\qquad D^\{\+,\\mathrm\{lib\}\}=X^\{\+,\\mathrm\{dir\}\}\-X^\{\\mathrm\{dir\}\}\.Choose the training root as the atom with largest‖Dalib‖2\\\|D\_\{a\}^\{\\mathrm\{lib\}\}\\\|\_\{2\}, using the stored\-order first maximum at a tie\. The rooted graph is constructed from the directed initial stateXdirX^\{\\mathrm\{dir\}\}\. An entry is retained only when the saddle\-displacement tail outside its5​Å5\\,\\text\{\\AA\}neighborhood satisfies

maxa∉𝒩r⁡‖Dalib‖2<0\.3​Å,\\max\_\{a\\notin\\mathcal\{N\}\_\{r\}\}\\\|D\_\{a\}^\{\\mathrm\{lib\}\}\\\|\_\{2\}<0\.3\\,\\text\{\\AA\},\(98\)with an empty tail assigned value zero\. Entries with equality at the threshold are rejected\. The stored record contains local initial coordinates, the colored graph, saddle and product displacements within the neighborhood, source/event identifiers, direction, and tail maximum\. The entry filter applies the stated saddle\-displacement tail criterion\.

### H\.2Exact graph mappings and necessary distance checks

At query time,*every*target atom is considered as a root\. Target root selection uses no reference moving atoms, saddle, or product\. For entries in the corresponding WL bucket, exact colored graph isomorphism is tested using the VF2 matcher\. The root label must match the root label and chemical species must match\.

The geometric matcher additionally prunes a candidate correspondence\(i,j\)\(i,j\)against every already mapped pair\(a,b\)\(a,b\)unless

\|‖pi−pa‖2−‖qj−qb‖2\|≤0\.600001​Å\.\\bigl\|\\\|p\_\{i\}\-p\_\{a\}\\\|\_\{2\}\-\\\|q\_\{j\}\-q\_\{b\}\\\|\_\{2\}\\bigr\|\\leq 0\.600001\\,\\text\{\\AA\}\.\(99\)The0\.6​Å0\.6\\,\\text\{\\AA\}scale is twice the0\.3​Å0\.3\\,\\text\{\\AA\}atomwise fit tolerance, with the implementation’s small numerical allowance\. This pair\-distance filter precedes the final fit gate\. Candidate vertices are ordered by root status, root\-relative distance, and index, followed by exact colored\-isomorphism and geometric tests\.

### H\.3Orthogonal fit and direct displacement transfer

For an exact graph mapping, reorder the target local coordinates to the template order\. Subtract each local point set’s own centroid, obtainingAAandBB, and form

A⊤​B=U​Σ​W⊤,𝖮=U​W⊤\.A^\{\\top\}B=U\\Sigma W^\{\\top\},\\qquad\\mathsf\{O\}=UW^\{\\top\}\.The orthogonal fit permits both rotations and reflections\. Its acceptance criterion is

maxa⁡‖\(A​𝖮−B\)a‖2≤0\.3​Å\.\\max\_\{a\}\\\|\(A\\mathsf\{O\}\-B\)\_\{a\}\\\|\_\{2\}\\leq 0\.3\\,\\text\{\\AA\}\.\(100\)The maximum residual on the least\-squares fit serves as both acceptance criterion and candidate score\.

KeyBank transfers stored displacement vectors by the fitted orthogonal map\. With target indicesι⁡\(a\)\\iota\(a\),

Dι⁡\(a\)cand\\displaystyle D^\{\\mathrm\{cand\}\}\_\{\\iota\(a\)\}=Dalib​𝖮,\\displaystyle=D\_\{a\}^\{\\mathrm\{lib\}\}\\mathsf\{O\},Dι⁡\(a\)\+,cand\\displaystyle D^\{\+,\\mathrm\{cand\}\}\_\{\\iota\(a\)\}=Da\+,lib​𝖮,\\displaystyle=D\_\{a\}^\{\+,\\mathrm\{lib\}\}\\mathsf\{O\},\(101\)Dbcand\\displaystyle D^\{\\mathrm\{cand\}\}\_\{b\}=0,\\displaystyle=0,Db\+,cand\\displaystyle D^\{\+,\\mathrm\{cand\}\}\_\{b\}=0for all unfilled target atomsb\.\\displaystyle=0\\quad\\text\{for all unfilled target atoms \}b\.These fields directly transfer the library motion to matched target atoms, with zero displacement assigned to the remaining atoms\.

### H\.4Mapping limit, sorting, and candidate deduplication

The query has a global mapping\-enumeration protection limit of 100,000 per target configuration, accumulated across roots and library entries\. The counter increments for each yielded graph mapping before the alignment gate\. When it exceeds the limit, enumeration stops andtruncated=Truerecords that retrieval ended at the mapping limit\.

Candidates are sorted lexicographically by geometric fit score, source, event, reverse\-direction flag, and query root\. They are then greedily deduplicated against already retained candidates using both the saddle and product conditions in Equation \([95](https://arxiv.org/html/2610.10969#A7.E95)\), with0\.3​Å0\.3\\,\\text\{\\AA\}for each\. The transferred library product provides the product field used in these candidate\-to\-candidate comparisons\.

Retained fields initialize native Dimer searches capped at 300 translations\. Native acceptance and common reference matching determine recovery using the all\-atom0\.05​Å0\.05\\,\\text\{\\AA\}displacement and0\.02​eV0\.02\\,\\mathrm\{eV\}barrier criteria\. Retrieval uses the graph, fit, tail, and deduplication settings above\.

## Appendix IShared Multiscale Embedding of Search Trajectories

#### Complementary views of the energy landscape and search paths\.

Figure[2](https://arxiv.org/html/2610.10969#S5.F2)\(a\) presents the Sammon\-projected energy background with initial, saddle, and product configurations\. Panel \(b\) compares ASPIRE and conventional Dimer trajectories in one shared MDS coordinate system\. The multiscale construction for panel \(b\) is detailed below\.

### I\.1Illustrative sample and one shared coordinate system

Figure[2](https://arxiv.org/html/2610.10969#S5.F2)\(b\) embeds 33 saved successful searches: 17 ASPIRE and 16 Random trajectories, totaling 4,994 frames\. This selected sample visualizes search geometry; the quantitative recovery and cost experiments use their separately specified evaluation populations\.

Both methods, all retained frames, and the geometric landmarks share one embedding with fixed atom identities and ordering\. For a trajectory with stored initial configurationX\(ℓ\)X^\{\(\\ell\)\}, a single translation

tℓ=1N​∑a\(Xa−Xa\(ℓ\)\)t\_\{\\ell\}=\\frac\{1\}\{N\}\\sum\_\{a\}\(X\_\{a\}\-X\_\{a\}^\{\(\\ell\)\}\)aligns that initial configuration to the shared native stateXX\. Agreement is checked after the shift, which is then applied unchanged to every full\-atom frame of that trajectory\.

### I\.2Configuration features, landmarks, and radial preprocessing

For an aligned configurationZZ, define

𝐟⁡\(Z\)=vec⁡\(Z−X\)N\.\\mathbf\{f\}\(Z\)=\\frac\{\\operatorname\{vec\}\(Z\-X\)\}\{\\sqrt\{N\}\}\.\(102\)Consequently,

‖𝐟⁡\(Zi\)−𝐟⁡\(Zj\)‖2=\(1N​∑a‖Zi,a−Zj,a‖22\)1/2\.\\\|\\mathbf\{f\}\(Z\_\{i\}\)\-\\mathbf\{f\}\(Z\_\{j\}\)\\\|\_\{2\}=\\left\(\\frac\{1\}\{N\}\\sum\_\{a\}\\\|Z\_\{i,a\}\-Z\_\{j,a\}\\\|\_\{2\}^\{2\}\\right\)^\{1/2\}\.\(103\)This metric is the Cartesian RMS over all corresponding atoms\. Coordinates and energies retain their recorded atom correspondence\.

The reference list contains the native initial state, followed by 41 separately supplied landmarks: another initial\-state record, 20 saddles, and 20 products\. In the code’s zero\-based reference list,refs\[2:22\]selects the 20 saddle landmarks\. Exactly equal feature rows across landmarks and trajectories share one two\-dimensional variable, with first\-occurrence ordering preserved\. Associated energy records are retained separately for coloring\.

#### Radial compression before dimensionality reduction\.

Let the reference radius be the median saddle distance from the initial configuration in the full feature space,

rref=mediane=120​‖𝐟⁡\(Selandmark\)‖2,rmap=0\.4​rref\.r\_\{\\mathrm\{ref\}\}=\\operatorname\{median\}\_\{e=1\}^\{20\}\\\|\\mathbf\{f\}\(S\_\{e\}^\{\\mathrm\{landmark\}\}\)\\\|\_\{2\},\\qquad r\_\{\\mathrm\{map\}\}=0\.4r\_\{\\mathrm\{ref\}\}\.\(104\)The displayed selected map first applies the radial transformation

𝐆⁡\(𝐟\)=\{rmap​asinh⁡\(‖𝐟‖2rmap\)​𝐟‖𝐟‖2,‖𝐟‖2\>0,0,𝐟=0\.\\mathbf\{G\}\(\\mathbf\{f\}\)=\\begin\{cases\}\\displaystyle r\_\{\\mathrm\{map\}\}\\operatorname\{asinh\}\\\!\\left\(\\frac\{\\\|\\mathbf\{f\}\\\|\_\{2\}\}\{r\_\{\\mathrm\{map\}\}\}\\right\)\\frac\{\\mathbf\{f\}\}\{\\\|\\mathbf\{f\}\\\|\_\{2\}\},&\\\|\\mathbf\{f\}\\\|\_\{2\}\>0,\\\\ 0,&\\mathbf\{f\}=0\.\\end\{cases\}\(105\)The common radial transformation acts in the full3​N3N\-dimensional feature space before the shared two\-dimensional fit\. Fitting targets are distances between the transformed features𝐆i\\mathbf\{G\}\_\{i\}\. The display thus provides a compressed geometric view of the trajectories, with axes expressed in arbitrary units and physical search performance reported by the recovery and force\-call metrics\.

### I\.3Six classes of distance constraints

Let the deduplicated feature nodes be indexed byii\. The edge collection is constructed once from the original features𝐟i\\mathbf\{f\}\_\{i\}with random seed 20260924, and is kept fixed when changing the fitted metric\. It contains the following six classes\.

#### Global pairs\.

Generate 60,000 pairs by sampling a trajectory uniformly and then a saved frame uniformly within that trajectory, independently for the two endpoints\. Self\-pairs are removed\. Uniform trajectory selection balances the contribution of the 33 selected successful searches at this stage\.

#### Frame–reference pairs\.

Connect every saved frame of each trajectory to all reference\-list entries\. Each trajectory forms a separate normalized block with initial importance1/331/33\.

#### Adjacent\-step pairs\.

For each trajectory, connect all consecutive saved frames\. The input check requires consecutive recorded translation\-step indices\. Each trajectory’s adjacent\-step block has initial importance1/331/33\.

#### Temporal\-window pairs\.

For each trajectory and each lag in\{2,4,8,16,32\}\\\{2,4,8,16,32\\\}shorter than the trajectory, connect frames at that lag\. Each available block has initial importance1/\(33×5\)1/\(33\\times 5\)\.

#### Reference–reference pairs\.

Include all unordered pairs of reference\-list entries, removing pairs whose node identifiers are identical\. The block has initial importance one\.

#### Original\-space nearest neighbors\.

Connect each deduplicated feature node to five nearest neighbors in the original Euclidean feature space\. This couples nearby configurations across trajectories and methods, with initial block importance one\. Neighbor selection uses the original Euclidean feature distances\.

Each block removes self\-pairs and retains its own weights\. Contributions from multiple blocks together define the weighted geometric constraints\.

### I\.4Block normalization and weighted stress

The optimizer uses the internal numerical scalesnum=0\.05s\_\{\\mathrm\{num\}\}=0\.05\. Define original and compressed target distances in those internal coordinates by

di​j0=‖𝐟i−𝐟j‖2snum,di​j=‖𝐆i−𝐆j‖2snum\.d^\{0\}\_\{ij\}=\\frac\{\\\|\\mathbf\{f\}\_\{i\}\-\\mathbf\{f\}\_\{j\}\\\|\_\{2\}\}\{s\_\{\\mathrm\{num\}\}\},\\qquad d\_\{ij\}=\\frac\{\\\|\\mathbf\{G\}\_\{i\}\-\\mathbf\{G\}\_\{j\}\\\|\_\{2\}\}\{s\_\{\\mathrm\{num\}\}\}\.\(106\)For a blockBB, the base weighting is

bB​\(d\)=\{1/max⁡\(d,2×10−4\),B​is frame–reference or nearest\-neighbor,1,otherwise\.b\_\{B\}\(d\)=\\begin\{cases\}1/\\max\(d,2\\times 10^\{\-4\}\),&B\\text\{ is frame\-\-reference or nearest\-neighbor\},\\\\ 1,&\\text\{otherwise\}\.\\end\{cases\}\(107\)The2×10−42\\times 10^\{\-4\}stabilizing floor is expressed in the optimizer’s internally scaled distance units\.

LetαB\\alpha\_\{B\}be the initial block importance above, withαB=1\\alpha\_\{B\}=1for the global block\. Original\-space weights are

wi​j,B0=αB​bB​\(di​j0\)max⁡\(∑\(u,v\)∈BbB​\(du​v0\)​\(du​v0\)2,10−15\)\.w^\{0\}\_\{ij,B\}=\\frac\{\\alpha\_\{B\}b\_\{B\}\(d^\{0\}\_\{ij\}\)\}\{\\max\\\!\\left\(\\sum\_\{\(u,v\)\\in B\}b\_\{B\}\(d^\{0\}\_\{uv\}\)\(d^\{0\}\_\{uv\}\)^\{2\},10^\{\-15\}\\right\)\}\.\(108\)When substituting the compressed metric, the implementation preservesαBeff=∑\(i,j\)∈Bwi​j,B0​\(di​j0\)2\\alpha\_\{B\}^\{\\mathrm\{eff\}\}=\\sum\_\{\(i,j\)\\in B\}w^\{0\}\_\{ij,B\}\(d^\{0\}\_\{ij\}\)^\{2\}, which equals the intendedαB\\alpha\_\{B\}when the normalization floor is inactive, and recomputes

wi​j,B=cB​αBeff​bB​\(di​j\)max⁡\(∑\(u,v\)∈BbB​\(du​v\)​du​v2,10−15\),cB=\{12,adjacent step,6,temporal window,1,all other blocks\.w\_\{ij,B\}=c\_\{B\}\\frac\{\\alpha\_\{B\}^\{\\mathrm\{eff\}\}b\_\{B\}\(d\_\{ij\}\)\}\{\\max\\\!\\left\(\\sum\_\{\(u,v\)\\in B\}b\_\{B\}\(d\_\{uv\}\)d\_\{uv\}^\{2\},10^\{\-15\}\\right\)\},\\qquad c\_\{B\}=\\begin\{cases\}12,&\\text\{adjacent step\},\\\\ 6,&\\text\{temporal window\},\\\\ 1,&\\text\{all other blocks\}\.\\end\{cases\}\(109\)These block weights balance trajectory, temporal, local, and global geometric constraints in the shared fit\.

The two\-dimensional variablesYi∈ℝ2Y\_\{i\}\\in\\mathbb\{R\}^\{2\}minimize the weighted metric stress

𝒮⁡\(Y\)=∑B∑\(i,j\)∈Bwi​j,B​\(‖Yi−Yj‖2−di​j\)2\.\\mathcal\{S\}\(Y\)=\\sum\_\{B\}\\sum\_\{\(i,j\)\\in B\}w\_\{ij,B\}\\bigl\(\\\|Y\_\{i\}\-Y\_\{j\}\\\|\_\{2\}\-d\_\{ij\}\\bigr\)^\{2\}\.\(110\)The implementation accumulates the two opposite endpoint contributions of each edge using the stabilized expression

2​wi​j,B​r~i​j−di​jr~i​j​\(Yi−Yj\),r~i​j=max⁡\(‖Yi−Yj‖2,10−12\)\.2w\_\{ij,B\}\\frac\{\\widetilde\{r\}\_\{ij\}\-d\_\{ij\}\}\{\\widetilde\{r\}\_\{ij\}\}\(Y\_\{i\}\-Y\_\{j\}\),\\qquad\\widetilde\{r\}\_\{ij\}=\\max\(\\\|Y\_\{i\}\-Y\_\{j\}\\\|\_\{2\},10^\{\-12\}\)\.Numerical stress evaluation also usesr~i​j\\widetilde\{r\}\_\{ij\}\. Above the10−1210^\{\-12\}floor, the expression is the stress gradient; the floor stabilizes updates for coincident or nearly coincident nodes\. Fitting uses geometric constraints from the selected successful trajectories and landmark coordinates\. Energies provide the separate color overlay described below\.

### I\.5Initialization, optimization, and reproducibility

The selected map starts from a saved shared raw\-metric MDS embedding\. That baseline fits the uncompressed distances using the same edge construction, with three stages of adjacent\-step multipliers2,8,242,8,24and temporal\-window multipliers equal to half those values\. Each baseline stage allows up to 900 L\-BFGS\-B iterations\. Its saved initial coordinates come from the common Sammon map in this workflow\.

For the displayed map, the saved two\-dimensional baseline is radially compressed withrmapr\_\{\\mathrm\{map\}\}and divided bysnums\_\{\\mathrm\{num\}\}for initialization\. The compressed\-space distances and weights in Equations \([106](https://arxiv.org/html/2610.10969#A9.E106)\)–\([109](https://arxiv.org/html/2610.10969#A9.E109)\) are then optimized in two consecutive L\-BFGS\-B stages, each allowing 800 iterations\. The optimization settings areftol=2×10−10=2\\times 10^\{\-10\},gtol=10−6=10^\{\-6\},maxcor=12=12, andmaxls=30=30\. The initial\-state node is recentered to the origin after each stage\.

Both selected\-map stages ran to their 800\-iteration limits\. Reproduction uses the saved baseline and initial coordinates together with the frames, landmarks, sampled edges, and hyperparameters\. Final stress and distance distortion are evaluated from the saved optimization outputs\.

### I\.6Common display alignment, markers, and energy colors

After optimization, a single common orthogonal Procrustes alignment of the landmark coordinates places the map in the orientation of the saved baseline\. No fitted scale factor is applied\. The display coordinates are

yidisplay=snumrref​\(Yi−Yinitial\)​𝖮2​D,y\_\{i\}^\{\\mathrm\{display\}\}=\\frac\{s\_\{\\mathrm\{num\}\}\}\{r\_\{\\mathrm\{ref\}\}\}\(Y\_\{i\}\-Y\_\{\\mathrm\{initial\}\}\)\\mathsf\{O\}\_\{2D\},\(111\)so the axes are labeled in arbitrary units \(a\.u\.\)\. Both methods share this transformation and scale; panel view limits can provide a closer view of the ASPIRE region alongside the wider Random trajectories\.

Lines retain all saved frames\. Markers highlight translation\-step indices divisible by five, together with each trajectory’s first and last frames\.

## Appendix JTheoretical Analysis

### J\.1Local convergence motivation

###### Proposition 1\(Local convergence,[Luo et al\., 2022](https://arxiv.org/html/2610.10969#bib.bib27), Theorem 4\.2\)\.

In fixed, vectorized free coordinates, letssbe a nondegenerate first\-order saddle ofVV\. SupposeV∈C2V\\in C^\{2\}and, throughoutB⁡\(s,R0\)B\(s,R\_\{0\}\), its Hessian isL3L\_\{3\}\-Lipschitz, has exactly one negative eigenvalue, and satisfiesμ≤\|λi\|≤L\\mu\\leq\|\\lambda\_\{i\}\|\\leq Lfor allii, where0<μ<L0<\\mu<L\. We takeL3\>0L\_\{3\}\>0\. Consider the ideal minimum\-mode iteration

yj\+1=yj−2L\+μ\(I−2v1\(yj\)v1\(yj\)⊤\)∇V\(yj\),y\_\{j\+1\}=y\_\{j\}\-\\frac\{2\}\{L\+\\mu\}\\bigl\(I\-2v\_\{1\}\(y\_\{j\}\)v\_\{1\}\(y\_\{j\}\)^\{\\top\}\\bigr\)\\nabla V\(y\_\{j\}\),\(112\)wherev1​\(yj\)v\_\{1\}\(y\_\{j\}\)is an exact unit eigenvector associated with the lowest Hessian eigenvalue\. Writingrj=‖yj−s‖2r\_\{j\}=\\\|y\_\{j\}\-s\\\|\_\{2\}, ifr0<min⁡\{R0,ρ\}r\_\{0\}<\\min\\\{R\_\{0\},\\rho\\\},ρ=2​μ/L3\\rho=2\\mu/L\_\{3\}, then the iterates remain in the neighborhood and converge geometrically toss, withrj≤\(L\+μL\+3​μ\)j​ρ​r0ρ−r0r\_\{j\}\\leq\\left\(\\frac\{L\+\\mu\}\{L\+3\\mu\}\\right\)^\{j\}\\frac\{\\rho r\_\{0\}\}\{\\rho\-r\_\{0\}\}\.

### J\.2Proof of Geometric Consistency

The positions and vector\-channel rows use the main\-text row\-vector convention\. The input transformation is

X′=P​X​R⊤,R∈SO⁡\(3\),X^\{\\prime\}=PXR^\{\\top\},\\qquad R\\in\\mathrm\{SO\}\(3\),\(113\)wherePPreindexes atoms and all atom\-indexed auxiliary features consistently\. We establish rotation and atom\-permutation equivariance for deterministic inference on the prepared Cartesian representation\.

###### Lemma 1\(Attention and weighted readouts\)\.

LetA=softmaxrow⁡\(L\)A=\\operatorname\{softmax\}\_\{\\mathrm\{row\}\}\(L\)\. IfL′=L​P⊤L^\{\\prime\}=LP^\{\\top\}, scalar atomic features satisfyZ′=P​ZZ^\{\\prime\}=PZ, and each vector channel satisfiesV′=P​V​R⊤V^\{\\prime\}=PVR^\{\\top\}, then

A′=A​P⊤,A′​Z′=A​Z,A′​V′=A​V​R⊤\.A^\{\\prime\}=AP^\{\\top\},\\qquad A^\{\\prime\}Z^\{\\prime\}=AZ,\\qquad A^\{\\prime\}V^\{\\prime\}=AVR^\{\\top\}\.\(114\)The vector identity also applies toX′=P​X​R⊤X^\{\\prime\}=PXR^\{\\top\}\.

###### Proof\.

Row\-wise softmax commutes with a permutation of its entries\. ThusA′=A​P⊤A^\{\\prime\}=AP^\{\\top\}\. The two readout identities follow by multiplication andP⊤​P=IP^\{\\top\}P=I\. A vector\-channel projection acts on channels, so it commutes with right multiplication byR⊤R^\{\\top\}\. ∎

###### Lemma 2\(Slot transformation law\)\.

Suppose encoder scalars are invariant under rotations, encoder vector channels rotate covariantly, and both are reindexed byPP\. The slot stages and final readout satisfy, for each fixed slotmm,

sm′=sm,zm′=zm,Qm′=Qm​R⊤,A′=A​P⊤\.s^\{\\prime\}\_\{m\}=s\_\{m\},\\qquad z^\{\\prime\}\_\{m\}=z\_\{m\},\\qquad Q^\{\\prime\}\_\{m\}=Q\_\{m\}R^\{\\top\},\\qquad A^\{\\prime\}=AP^\{\\top\}\.\(115\)

###### Proof\.

The learned scalar seeds and their scalar self\-attention are unchanged by input rotations and atom permutations\. The zero initial vector state is covariant\. At initialization, scalar query–key logits only reindex across atoms, so Lemma[1](https://arxiv.org/html/2610.10969#Thmlemma1)gives the required vector law forQm\(1\)Q\_\{m\}^\{\(1\)\}\.

Assume the transformation laws hold at a vector–scalar stage\. The scalar part of Equation \([5](https://arxiv.org/html/2610.10969#S4.E5)\) is invariant, and the vector part is invariant because

⟨U​R⊤,V​R⊤⟩F=⟨U,V⟩F\.\\langle UR^\{\\top\},VR^\{\\top\}\\rangle\_\{F\}=\\langle U,V\\rangle\_\{F\}\.Consequently, the attention weights reindex with the atoms\. The pooled scalar context is invariant, and the vector context rotates byR⊤R^\{\\top\}\. Vector norms are invariant, so the MLP update preserves the scalar law; bias\-free channelwise vector maps preserve the vector law\. This proves the induction step for both vector–scalar stages\.

The subsequent scalar\-only stages use invariant fingerprints and atomic scalars\. Their headwise dot products, normalizations, attention\-weighted sums, and feed\-forward updates therefore preserve scalar invariance and leave the vector state unchanged\. The final readout uses invariant scalar products and vector inner products again\. Applying Lemma[1](https://arxiv.org/html/2610.10969#Thmlemma1)to its contexts yields Equation \([115](https://arxiv.org/html/2610.10969#A10.E115)\)\. ∎

###### Proposition 2\(Proposal equivariance\)\.

Assume deterministic inference, rotation\- and permutation\-compatible graph construction and neighbor selection, an equivariant encoder, the stated slot updates, invariant scalar decoder conditioning, bias\-free channelwise vector maps, and the radial normalization in Equation \([9](https://arxiv.org/html/2610.10969#S4.E9)\)\. Then every fixed slot satisfies

D^m​\(P​X​R⊤\)=P​D^m​\(X\)​R⊤,S^m​\(P​X​R⊤\)=P​S^m​\(X\)​R⊤\.\\widehat\{D\}\_\{m\}\(PXR^\{\\top\}\)=P\\widehat\{D\}\_\{m\}\(X\)R^\{\\top\},\\qquad\\widehat\{S\}\_\{m\}\(PXR^\{\\top\}\)=P\\widehat\{S\}\_\{m\}\(X\)R^\{\\top\}\.\(116\)

###### Proof\.

Corresponding Cartesian edges rotate asrb​a′=rb​a​R⊤r^\{\\prime\}\_\{ba\}=r\_\{ba\}R^\{\\top\}after atom reindexing\. Their lengths and radial features are unchanged, and their spherical harmonics transform in the appropriate rotation representations\. Compatible neighbor selection preserves the same reindexed graph\. The equivariant encoder therefore supplies the scalar and vector laws required by Lemma[2](https://arxiv.org/html/2610.10969#Thmlemma2)\.

The final attention satisfiesA′=A​P⊤A^\{\\prime\}=AP^\{\\top\}\. Hence

bm′=∑aAm​a′​Xa′=bm​R⊤\.b^\{\\prime\}\_\{m\}=\\sum\_\{a\}A^\{\\prime\}\_\{ma\}X^\{\\prime\}\_\{a\}=b\_\{m\}R^\{\\top\}\.The offsetsXa−bmX\_\{a\}\-b\_\{m\}rotate byR⊤R^\{\\top\}under the same atom reindexing\. The 32 atom\-derived vectors and eight slot\-derived vectors also rotate, since their learned maps only mix vector channels\. The stabilized direction satisfies

unitϵ⁡\(v​R⊤\)=v​R⊤max⁡\(‖v‖2,ϵ\)=unitϵ⁡\(v\)​R⊤,\\operatorname\{unit\}\_\{\\epsilon\}\(vR^\{\\top\}\)=\\frac\{vR^\{\\top\}\}\{\\max\(\\\|v\\\|\_\{2\},\\epsilon\)\}=\\operatorname\{unit\}\_\{\\epsilon\}\(v\)R^\{\\top\},including atv=0v=0\. Thus every row ofℬm​a\\mathcal\{B\}\_\{ma\}is covariant\.

All scalar entries ofξm​a\\xi\_\{ma\}are invariant: they are invariant scalar representations, inner products, or squared norms\. The shared coefficient networks therefore produce the samepm​a,κm​a,qm​a,qm​aresp\_\{ma\},\\kappa\_\{ma\},q\_\{ma\},q^\{\\mathrm\{res\}\}\_\{ma\}at corresponding atom–slot pairs\. Both the gated directional term and the ungated residual in Equation \([12](https://arxiv.org/html/2610.10969#S4.E12)\) rotate covariantly\. This proves the displacement law\. AddingX′=P​X​R⊤X^\{\\prime\}=PXR^\{\\top\}gives the configuration law\. ∎

#### Representation and implementation conditions\.

The deterministic inference setting uses atom\-reindexing\-compatible graph truncation, disabled dropout, and a common Cartesian chart\. The arithmetic anchor is evaluated in that chart as described in Appendix[A\.2](https://arxiv.org/html/2610.10969#A1.SS2), with common origin changes handled during input preparation\. These conditions establish the neural proposal transformation law\.

## Appendix KSingle\-Step Reference Interpretation of RWC

### K\.1Reference rates and restricted one\-step models

Fix a stateXXand a finite, nonempty reference setℰXref\\mathcal\{E\}\_\{X\}^\{\\mathrm\{ref\}\}with positive rateskek\_\{e\}\. For a nonempty subsetG⊆ℰXrefG\\subseteq\\mathcal\{E\}\_\{X\}^\{\\mathrm\{ref\}\}, write

ΛG=∑e∈Gke,pG\(e\)=keΛG\(e∈G\),ΔtG∼Exp\(ΛG\)\.\\Lambda\_\{G\}=\\sum\_\{e\\in G\}k\_\{e\},\\qquad p\_\{G\}\(e\)=\\frac\{k\_\{e\}\}\{\\Lambda\_\{G\}\}\\quad\(e\\in G\),\\qquad\\Delta t\_\{G\}\\sim\\operatorname\{Exp\}\(\\Lambda\_\{G\}\)\.\(117\)These are the single\-step event distribution and waiting time described in Appendix[A\.5](https://arxiv.org/html/2610.10969#A1.SS5)\([Voter, 2007](https://arxiv.org/html/2610.10969#bib.bib37)\), providing a kinetic interpretation of reference recovery at a fixed state\.

###### Proposition 3\(Single\-step interpretation of RWC\)\.

Let the recovered reference subsetℰXrec\\mathcal\{E\}\_\{X\}^\{\\mathrm\{rec\}\}be nonempty and retain the same rates as the full reference set\. Extend its event distribution by zero outside that subset\. Then

TV⁡\(pref,prec\)=1−RWCX,𝔼⁡\[Δ​trec\]𝔼⁡\[Δ​tref\]=1RWCX,\\operatorname\{TV\}\(p\_\{\\mathrm\{ref\}\},p\_\{\\mathrm\{rec\}\}\)=1\-\\mathrm\{RWC\}\_\{X\},\\qquad\\frac\{\\mathbb\{E\}\[\\Delta t\_\{\\mathrm\{rec\}\}\]\}\{\\mathbb\{E\}\[\\Delta t\_\{\\mathrm\{ref\}\}\]\}=\\frac\{1\}\{\\mathrm\{RWC\}\_\{X\}\},\(118\)wherepref=pℰXrefp\_\{\\mathrm\{ref\}\}=p\_\{\\mathcal\{E\}\_\{X\}^\{\\mathrm\{ref\}\}\}andprec=pℰXrecp\_\{\\mathrm\{rec\}\}=p\_\{\\mathcal\{E\}\_\{X\}^\{\\mathrm\{rec\}\}\}\.

###### Proof\.

Putc=Λrec/Λref=RWCX∈\(0,1\]c=\\Lambda\_\{\\mathrm\{rec\}\}/\\Lambda\_\{\\mathrm\{ref\}\}=\\mathrm\{RWC\}\_\{X\}\\in\(0,1\]\. On the recovered subset,prec​\(e\)=pref​\(e\)/c≥pref​\(e\)p\_\{\\mathrm\{rec\}\}\(e\)=p\_\{\\mathrm\{ref\}\}\(e\)/c\\geq p\_\{\\mathrm\{ref\}\}\(e\)\. Consequently,

TV⁡\(pref,prec\)\\displaystyle\\operatorname\{TV\}\(p\_\{\\mathrm\{ref\}\},p\_\{\\mathrm\{rec\}\}\)=12​\[∑e∈ℰXrec\(prec​\(e\)−pref​\(e\)\)\+∑e∉ℰXrecpref​\(e\)\]\\displaystyle=\\frac\{1\}\{2\}\\left\[\\sum\_\{e\\in\\mathcal\{E\}\_\{X\}^\{\\mathrm\{rec\}\}\}\(p\_\{\\mathrm\{rec\}\}\(e\)\-p\_\{\\mathrm\{ref\}\}\(e\)\)\+\\sum\_\{e\\notin\\mathcal\{E\}\_\{X\}^\{\\mathrm\{rec\}\}\}p\_\{\\mathrm\{ref\}\}\(e\)\\right\]=12​\[\(1−c\)\+\(1−c\)\]=1−c\.\\displaystyle=\\frac\{1\}\{2\}\[\(1\-c\)\+\(1\-c\)\]=1\-c\.The exponential mean is1/ΛG1/\\Lambda\_\{G\}, so the ratio of waiting\-time means isΛref/Λrec=1/c\\Lambda\_\{\\mathrm\{ref\}\}/\\Lambda\_\{\\mathrm\{rec\}\}=1/c\. ∎

### K\.2Average recovery probability and the high\-temperature limit

Sample one reference eventEXE\_\{X\}at each fixed evaluation state using its normalized reference rates\. Then

Pr⁡\(EX∈ℰXrec\)=RWCX\.\\Pr\(E\_\{X\}\\in\\mathcal\{E\}\_\{X\}^\{\\mathrm\{rec\}\}\)=\\mathrm\{RWC\}\_\{X\}\.For the fixed evaluation bank𝒳\\mathcal\{X\}, linearity of expectation gives

𝔼\[1\|𝒳\|∑X∈𝒳𝟏\{EX∈ℰXrec\}\]=1\|𝒳\|∑X∈𝒳RWCX=RWC¯\.\\mathbb\{E\}\\\!\\left\[\\frac\{1\}\{\|\\mathcal\{X\}\|\}\\sum\_\{X\\in\\mathcal\{X\}\}\\mathbf\{1\}\\\{E\_\{X\}\\in\\mathcal\{E\}\_\{X\}^\{\\mathrm\{rec\}\}\\\}\\right\]=\\frac\{1\}\{\|\\mathcal\{X\}\|\}\\sum\_\{X\\in\\mathcal\{X\}\}\\mathrm\{RWC\}\_\{X\}=\\overline\{\\mathrm\{RWC\}\}\.\(119\)Linearity of expectation makes this identity valid for dependent samples as well\. Evaluation configurations receive uniform weight, consistently with the main\-text metric\.

Under the main\-text common prefactor,ke=νexp\[−ΔEe/\(kBT\)\]k\_\{e\}=\\nu\\exp\[\-\\Delta E\_\{e\}/\(k\_\{\\mathrm\{B\}\}T\)\]\. For a fixed finite reference set and a fixed recovered subset, every exponential approaches one asT→∞T\\to\\infty\. Therefore

limT→∞RWCX=\|ℰXrec\|\|ℰXref\|=CovX\.\\lim\_\{T\\to\\infty\}\\mathrm\{RWC\}\_\{X\}=\\frac\{\|\\mathcal\{E\}\_\{X\}^\{\\mathrm\{rec\}\}\|\}\{\|\\mathcal\{E\}\_\{X\}^\{\\mathrm\{ref\}\}\|\}=\\mathrm\{Cov\}\_\{X\}\.This formal limit describes how the weights of a fixed catalogue become uniform under the common\-prefactor convention\. With event\-specific prefactors, the limiting normalized weights are proportional to those prefactors\.

### K\.3Metric interpretation and conventions

The proposition uses fixed reference rates\. Numerical barriers enter the matching gate, while matched events retain their reference rates in the sums\. For an empty recovered set, RWC is zero and the restricted exit distribution and finite mean waiting time are undefined\. Aggregating event identities into product\-basin labels can decrease total variation, yielding a corresponding comparison at the basin level\.

RWC measures the fraction of reference escape\-rate mass recovered at each state\. A common scaling of all rates preserves this fraction and rescales the clock, separating relative event recovery from absolute waiting times\. Averaging RWC over the fixed state bank summarizes this reference\-relative recovery across configurations\.

## Appendix LPhysical Evaluation, Event Matching, and Cost Accounting

### L\.1Fixed 50\-configuration evaluation set and topology coverage

The ablation and wall\-time experiments use the same fixed set𝒳50\\mathcal\{X\}\_\{50\}of 50 held\-out configurations\. It augments an existing 30\-configuration evaluation set with 20 additional cases selected to broaden topology coverage\. The set is drawn from the held\-out configuration bank; the judge uses a separately defined 50\-configuration monitoring subset\.

The five vacancy\-connectivity classes are the component\-size signatures recorded asfragmentation\_signature:44,3\+13\+1,2\+22\+2,2\+1\+12\+1\+1, and1\+1\+1\+11\+1\+1\+1\. For example,3\+13\+1denotes one connected component of three vacancies and one isolated vacancy, whereas1\+1\+1\+11\+1\+1\+1denotes four separate vacancy components\. These signatures summarize connectivity in the dataset’s vacancy graph\. Each class can contain multiple geometric arrangements and fine\-grained topologies\. Encoder, judge, and KeyBank atom\-neighbor graphs are defined in their respective method sections\.

Table 8:Vacancy\-connectivity coverage of the fixed evaluation subset\.The existing 30 cases are augmented by 20 cases to obtain the 50\-case set used for both ablation and wall\-time comparisons\. Counts report the configuration selection in each connectivity class\.After excluding the existing 30 configurations from the held\-out bank, the augmentation included all remaining cases in classes44and2\+22\+2\. It prioritized the available rare target\-PDR label combinations\(1,3\)\(1,3\),\(0,3\)\(0,3\), and\(0,1,2,3\)\(0,1,2,3\), then filled the remaining positions deterministically by configuration ID within the relevant connectivity classes until 20 additional cases had been selected\. The target\-PDR combinations refer to the storedtarget\_pdr\_groupmetadata; they complement the vacancy component\-size signatures in offline subset construction\. At query time, the predictor, judge, and event\-reuse methods use their specified initial\-state inputs\.

The 20\-case augmentation uses topology, target\-label coverage, and configuration ID as its selection criteria\. Together with the existing 30 cases, it forms a fixed evaluation set covering all five vacancy\-connectivity classes\. Every compared method uses this same designed subset\.

### L\.2Numerical acceptance and reference\-event matching

The evaluation state is the relaxed11​Å11\\,\\text\{\\AA\}SEAKMC active volumeXXin the offline protocol of Section[5](https://arxiv.org/html/2610.10969#S5)\. All methods use the same fixed local reference catalogueℰXref\\mathcal\{E\}\_\{X\}^\{\\mathrm\{ref\}\}, so coverage measures event recovery for that active\-volume configuration\.

For a completed search, an output is eligible for reference matching when the native valid and connected flags are true, the maximum\-translation flag is false, and the recorded translation count is strictly below its configured cap\. These operational acceptance criteria follow Appendix[F](https://arxiv.org/html/2610.10969#A6); the reported pipeline uses them without an additional independent Hessian\-index check\.

#### Saddle geometry, recorded barriers, and reference identities\.

The output saddle coordinates are the savedsaddle\_active\_coords, obtained from nativeXCR\.T\. SetDout=Sout−XD\_\{\\mathrm\{out\}\}=S\_\{\\mathrm\{out\}\}\-Xand use the associated recorded barrierΔ​Eout\\Delta E\_\{\\mathrm\{out\}\}\. NativeXDISPrecords the branch\-dependent finalization displacement\. For referenceeefrom the sameXX, the eligible set is

ℳX\(Sout\)=\{e:max1≤a≤N⁡‖Dout,a−De,a‖2≤0\.05​Å,\|ΔEout−ΔEe\|≤0\.02eV\}\.\\begin\{split\}\\mathcal\{M\}\_\{X\}\(S\_\{\\mathrm\{out\}\}\)=\\bigl\\\{e:\\;&\\max\_\{1\\leq a\\leq N\}\\\|D\_\{\\mathrm\{out\},a\}\-D\_\{e,a\}\\\|\_\{2\}\\leq 0\.05\\,\\text\{\\AA\},\\\\ &\|\\Delta E\_\{\\mathrm\{out\}\}\-\\Delta E\_\{e\}\|\\leq 0\.02\\,\\mathrm\{eV\}\\bigr\\\}\.\\end\{split\}\(120\)This main\-text criterion uses both non\-strict inequalities and compares all corresponding atoms in the fixed Cartesian representation\. Matching uses the recorded saddle displacement and barrier directly\.

The matcherJX​\(Sout\)J\_\{X\}\(S\_\{\\mathrm\{out\}\}\)assigns at most one reference identity inℳX​\(Sout\)\\mathcal\{M\}\_\{X\}\(S\_\{\\mathrm\{out\}\}\), or⊥\\botif the set is empty\. Distinct recovery is the set of assigned reference identities: repeated hits to one reference count once\. The reference catalogue remains fixed and accepted unmatched outputs receive no reference\-coverage credit\. Accepted unmatched outputs are candidates for further mechanism analysis and expansion of the reference catalogue\.

### L\.3Distinct thresholds serve distinct operations

Training activity, judge labels, retrieval, numerical connectivity, and reference matching are separate definitions\. Their use is summarized below to make clear which operations depend on reference data\.

Table 9:Thresholds in the implementation\.“Reference” denotes the target configuration’s reference catalogue\. Event\-reuse methods retrieve stored mechanisms from their training libraries\.KDB mobile cutoffs and KeyBank graph/tail criteria are specified in their respective sections\. Dataset deduplication applies the 48 bcc operations to initial configurations, while event recovery uses the fixed\-XXmatching predicate above\.

### L\.4Reference metrics and experiment\-specific settings

Coverage and RWC use the recovered\-reference subset and the reference rates, exactly as in Section[5](https://arxiv.org/html/2610.10969#S5):

CovX=\|ℰXrec\|\|ℰXref\|,RWCX=∑e∈ℰXrecke∑e∈ℰXrefke,RWC¯=1\|𝒳\|∑X∈𝒳RWCX,ke=νexp\(−Δ​EekB​T\)\.\\begin\{gathered\}\\mathrm\{Cov\}\_\{X\}=\\frac\{\|\\mathcal\{E\}\_\{X\}^\{\\mathrm\{rec\}\}\|\}\{\|\\mathcal\{E\}\_\{X\}^\{\\mathrm\{ref\}\}\|\},\\qquad\\mathrm\{RWC\}\_\{X\}=\\frac\{\\sum\_\{e\\in\\mathcal\{E\}\_\{X\}^\{\\mathrm\{rec\}\}\}k\_\{e\}\}\{\\sum\_\{e\\in\\mathcal\{E\}\_\{X\}^\{\\mathrm\{ref\}\}\}k\_\{e\}\},\\\\ \\overline\{\\mathrm\{RWC\}\}=\\frac\{1\}\{\|\\mathcal\{X\}\|\}\\sum\_\{X\\in\\mathcal\{X\}\}\\mathrm\{RWC\}\_\{X\},\\qquad k\_\{e\}=\\nu\\exp\\\!\\left\(\-\\frac\{\\Delta E\_\{e\}\}\{k\_\{\\mathrm\{B\}\}T\}\\right\)\.\\end\{gathered\}\(121\)All reported RWC values useT=800​KT=800\\,\\mathrm\{K\}and the common prefactorν\\nu, as stated in the main text\. The average weights evaluation configurations uniformly\. Matched outputs retain their assigned reference rates, and the common prefactor cancels from RWC \(Appendix[A\.5](https://arxiv.org/html/2610.10969#A1.SS5)\)\. The reference catalogue, temperature, and prefactor convention are fixed across judge\-threshold and random\-start comparisons\.

Table 10:Settings of the main\-text evaluation populations\.The 50\-state ablation and wall\-time studies use the same held\-out subset, but different judge settings\. Proposal count, number of submitted searches, and translation cap are distinct quantities\.
### L\.5Recorded search\-force count

For the fixed evaluation bank𝒳\\mathcal\{X\}, letIXI\_\{X\}index recorded completed searches andFX,mF\_\{X,m\}count entries toDimer\.dimer\_forcein searchmm\. The mean search\-force cost is

F¯search=1\|𝒳\|​∑X∈𝒳∑m∈IXFX,m\.\\bar\{F\}\_\{\\mathrm\{search\}\}=\\frac\{1\}\{\|\\mathcal\{X\}\|\}\\sum\_\{X\\in\\mathcal\{X\}\}\\sum\_\{m\\in I\_\{X\}\}F\_\{X,m\}\.\(122\)The ledger includes recorded force calls from failed, capped, duplicate, and accepted\-unmatched searches\. Each entry counts one invocation of the search\-force routine for the configuration\.

The search\-force ledger excludes LAMMPS initialization, nativeSPSRELAXforce work outside the routine, neural inference, and catalogue lookup\.

ASPIRE and reuse searches use a 300\-translation cap; the Random baseline uses up to 80 searches with a 450\-translation cap\. Preloaded searches use the native fine\-step parameter family\. The reported comparisons therefore evaluate these specified operating protocols\. Each translation may involve multiple force calls for orientation updates\.

### L\.6Wall time, retrieval size, and experiment records

Wall time is reported separately from search\-force counts\. In the main\-text wall\-time comparison, Random and event\-reuse baselines use 16 CPUs, while ASPIRE uses 16 CPUs and one A100 GPU\. This comparison uses the shared 50\-state subset, with ASPIRE judge thresholdτ=10−5\\tau=10^\{\-5\}as stated in Figure[5](https://arxiv.org/html/2610.10969#S5.F5)\. The timing results complement search\-force counts with recorded elapsed time under these hardware settings\.

KDB\-494 contains 494 stored templates with 988 directed query entries; KeyBank\-32/128 designate libraries from 32/128 training configurations\. Retained proposals undergo native Dimer refinement\. KeyBank’s truncation flag identifies retrievals stopped at the mapping limit\. Detailed baseline construction appears in Appendices[G](https://arxiv.org/html/2610.10969#A7)and[H](https://arxiv.org/html/2610.10969#A8)\.

The experiment populations comprise the 200\-configuration recovery test, the shared 50\-configuration ablation and wall\-time subset of Appendix[L\.1](https://arxiv.org/html/2610.10969#A12.SS1), the judge’s 200 training/50 monitoring configurations, and the 33 selected successful MDS trajectories\. Each result is interpreted with its state selection, checkpoints, thresholds, library, Dimer settings, scorer, and force/timing protocol\. The MDS sample provides a complementary illustration of search geometry\.